Chapter 6: The equations of fluid motion - Forsiden - · PDF fileGEF 1100 – Klimasystemet Chapter 6: The equations of fluid motion 1 GEF1100–Autumn 2016 13.09.2016 Prof. Dr. Kirstin

GEF 1100 ndash Klimasystemet

Chapter 6 The equations of fluid motion

1

GEF1100ndashAutumn 2016 13092016

Prof Dr Kirstin Kruumlger (MetOs UiO) Email kkruegergeouiono

Research interests

bull Stratosphere-ocean interactions bull Dynamics of the stratosphere-troposphere bull Stratospheric ozone and ozone depleting substances bull Influence of volcanic eruptions on climate earth system

Who am I

Prof Dr Kirstin Kruumlger

kkruegergeouiono

climate modelling

observations

ship expeditions

Volcanic excursion

1 Motivation

2 Basics 21 Lagrange - Euler 22 Mathematical add ons Vorticity

3 Equation of motion for a nonrotating fluid 31 Forces 32 Equations of motion 33 Hydrostatic balance

4 Conversation of mass

5 Thermodynamic equation

6 Equation of motion for a rotating fluid 61 Forces 62 Equations of motion

7 Take home messages

Lect

ure

Ou

tlin

e ndash

Ch

6 Ch 6 - The equations of fluid motions

6

Addition not in book

Motivation 1 Motivation

Marshall and Plumb (2008)

7

- Derivation of the general equation of motion - Different horizontal wind forms - General circulation of atmosphere ocean

Equations of fluid motions

bull Hydrodynamics focuses on moving liquids and gases

bull Hydrodynamics and thermodynamics are fundamental for meteorology and oceanography

bull Equations of fluid motions for the atmosphere and ocean are constituted by macroscopic conservation laws for substances impulses and internal energy (hydrodynamics and thermodynamics)

The characteristics of very small volume elements are looked at (eg overall mass average speed)

Derivation of hydrodynamics is based on the conservation of momentum mass and energy

8 hydro (greek) water

2 Basics

State variables for atmosphere and ocean

u Wind u = (u v w) in ms T Temperature in degC or K p Pressure in Nm2 = 1 Pa (1 hPa = 102 Pa = 1 mbar) Specifically for the atmosphere q Specific humidity in kg water vapourkg moist air Specifically for the ocean S Salinity in kg saltkg seawater (often the salt content is expressed in practical salinity unit psu = kg salt1000 kg seawater) 9

2 Basics

Differentiation following the motion - Euler ndash Lagrange derivatives

10

2 Basics

Example wind blows over hill cloud forms at the

ridge of the lee wave steady state assumption eq cloud does not change in time

Eulerian derivative (after Euler 1707-1783)

120597119862

120597119905 fixed point in space= 0

Lagrangian derivative (after Lagrange 1736-1813)

120597119862

120597119905 fixed particlene 0

C = C(xyzt) eg cloud amount 120597

120597119905 partial derivatives other variables

are kept fixed during the differentiation

Leonhard Euler (CH)

Joseph-Louis Lagrange (I)

Differentiation following the motion - Euler and Lagrange perspectives

T(Xt) T(Xt+1)

χ=Xt+1 Lagrange

Parcel of air

trajectory

Euler

T(x1t2) u(x1t2) T(x1t1) u(x1t1)

Fixed box

in space

Euler perspective The control volume is anchored stationary within the coordinate system eg a cube through which the air or the seawater flows through

Lagrange perspective The control volume is an air or seawater parcel that always consists of the same particles and which moves along with the wind or the current

11

2 Basics

Think about examples in your everyday life for Eulerrsquos and Lagrangersquos differentiation Discuss in groups of 2 and give one example

Euler and Lagrangian differentiation

12

2 Basics

Differentiation following the motion mathematically

13

2 Basics

For arbitrary small variations

120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905

Dividing by δt and in limit of small variations

120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513

Advection term of a quantity 119958 ∙ 120629 lthellipgt ldquo

δ Greek small delta infinitesimal small size

Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics

Lagrangian (or material substantial or total) derivative (after Lagrange 1736-1813)

119863

119863119905 fixed particle equiv 120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)

ldquoTime rate of change of some characteristic of a particular element of fluid which in general is changing its positionrdquo


120597

120597119905 fixed point

ldquoTime rate of change of some characteristic at a fixed point in space but with constantly changing fluid element because the fluid is movingrdquo

119958 = (uvw) velocity vector

120513 equiv120597

120597119909

120597

120597119910120597

120597119911 gradient operator

called ldquonablardquo

Forward trajectories started 850 hPa - top between 2604-29041986 - bottom between 3004-05051986

Example nuclear reactor accident Chernobyl on 26041986 0130 am Trajectories (=flight path) are used for the prediction of air movement

15

2 Basics

Kraus (2004)

Differentiation following the motion - Lagrangian perspective

16

Where is the missing Malaysian Airline MH370 (8 March 2014)

GEOMAR press release from 1 September 2015

22 Basics - Mathematical add ons

- Nomenclature

- Vector operations

- Nabla operator

17

22 Basics - Mathematical add ons September 13 2016

Nomenclature summary

bull δ Greek small delta infinitesimal small size

eg δV= small air volume

bull partial derivative of a quantity with respect to

a coordinate eg x

bull total differential derivative of a quantity

with respect to all dependent coordinates (xzy)

bull Lagrangian (or total) derivative

18

x

dtd

22 Mathematical add ons

Vector operations - Multiplication with a scalar a -

z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva

bull With a scalar multiplication the vector stays a

vector

bull Each element of a vector is multiplied individually with a

scalar a

bull The vector extends (or shortens) itself by the factor a

bull Convention With the multiplication scalar ndash vector we

donrsquot use a point (like with scalar ndash scalar)


19

wind

vector

Vector operations - Scalar product -

ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv

bull The scalar product of two vectors is a scalar

bull It is multiplied component-wise then added

bull Convention The scalar product is marked by a multiplication sign (middot)

bull It is at a maximum with parallel vectors and disappears when the

vectors are perpendicular to each other

f

v

cosfvfv


20

Vector operations ndash vector product

bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b

bull The vector product of two vectors is again a vector

bull Convention The vector product or cross product is marked by an ldquoxrdquo

bull If is turned to on the fastest route possible the vector (from the

vector product) points in the direction in which a right-hand screw

would move (right hand rule)

bull With parallel vectors it disappears and it reaches its maximum when the

two vectors are at right angles to each other

21


Graphically

- Pooling of the spatial gradients towards the spatial coordinate axes rarr vector

- Gradient points towards the largest increase of the quantity

- Amount is the magnitude of the derivative pointing towards the largest increase

Nabla operator ndash spatial gradient

with

Operator -Nabla

z

y

x

i

22

bull Nabla is a (vector) operator that works to

the right

bull It only results in a value if it stands left of

an arithmetic expression

bull If it stands to the right of an arithmetic

expression it keeps its operator function

(and ldquowaitsrdquo for application)


TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x

Vector product ldquoxrdquo

vector

Scalar product ldquordquo

scalar

Product with a

Scalar vector

Nabla operator 22 Mathematical add ons

23

Unit vectors

i x direction

j y direction

k z direction

Nabla operator ndash algorithms

z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x


Note Nabla is a (vector) operator ie the sequence must not be changed here

24

Divergence Convergence

0 Hu 0 Hu

Characteristics of horizontal wind

25


Rotation and vorticity (comes from the word ldquovortexrdquo)

26

Rotation of horizontal flow

Horizontal

wind Rotation

L



Rotation of a vector field - Vector product of the Nabla operator with a vector -

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x

Definition vorticity

ζ Greek bdquoZetaldquo

The vorticity is a scalar

The rotation of a flow field is constituted by

The vorticity ζ is defined as a vertical component of the velocity fieldsrsquos rotation

u u rot

y

u

x

v

-uk

22 Basics - Vorticity

Relative vorticity ζ

Rotation relative to the earth

Absolute vorticity η

Sum of relative vorticity and planetary vorticity

Rotation in an absolute coordinate system oriented towards a fixed star

Planetary vorticity f

Contribution of earth rotation f = 2 Ω sinφ f Coriolis parameter φ latitude Ω angular velocity Ω = 2π24h is the angular velocity of the earth rotation

yx

uvuk

f

Application examples eg vorticity advection in synoptic strength of a vortex (eg stratospheric polar vortex) stratosphere vorticity correlates with ozone

ζ Zeta η Eta f Coriolis parameter

Vorticity

28


Curvature vorticity (VKs)

Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)

Note In general the horizontal wind field contains both the curvature as well as the shear forces (varying strengths of winds on the verticals)

x

y

z

ζ lt 0

ζ gt 0

Surface


NH stratospheric polar vortex

Potential vorticity unit (PVU) at ~20 km altitude

Stratospheric

polar vortex

Edge of vortex

Filament

- High vorticity (PV) in the centre of the polar vortex (low pressure centre) - Greatest PV gradient at the edge of the polar vortex (rarr wind maximum) 30

L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31


I Laws of motion for a fluid parcel in x-y-z- directions (applying Newtonrsquos 1 and 2 law)

II Conservation of mass

III Law of thermodynamics (including motion)

rarr5 equations for the evolution of the fluid (5 unknowns

(u p T)) 32

3 Equations of motion - nonrotating fluid


Momentum conservation The conservation of momentum is represented by

Newtonrsquos first law (ldquoLex primardquo)

(momentum = mass middot velocity)

momentum = const at absence of forces

Momentum sentence (ldquoLex secundardquo)

temporal change in momentum = Force

(mass middot acceleration = Force)

33

Fu

dt

dM t time M mass u velocity vector F Force vector




fluid parcel - cube

34



Forces on a fluid parcel

35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)

120588 density (120575119909 120575119910 120575119911) fluid parcel with infinitesimal dimensions 120575119872mass of the parcel 120575119872 = 120588 120575119909 120575119910 120575119911 119958 parcel velocity F net force

119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36

The gravitational force g 120575119872 is directed downward

Fgravity= minusg 120588119963 120575119909 120575119910 120575119911 (Eq 6-3)

Fgravity Gravitational force M mass g gravity acceleration g asymp const 120588 density 119963 unit vector in upward direction





Pressure gradient force

37

x-component of the pressure force is

F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)

Apply Taylor expansion (A21) at midpoint of parcel and neglect small terms (see book)

Fx =minus120597119901

120597119909120575119909 120575119910 120575119911

Apply for all sides (y and z)

119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39


- Friction force operates on the earthrsquos surface

- the greater the surface roughness the higher the friction force

- the greater the wind velocity the higher the friction force

- friction force takes effect in vertical distances of ~ 100 m up to 1000 m (rarr atmospheric boundary layer Chapter 7)

Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)

ℱ frictional force per unit mass (see book chapters 742 and 101) 120588 density



40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion

bull All three forces together in Eq 6-2 gives

120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899

bull Re-arranging leads to equation of motion for fluid parcels

119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42


120588 density u velocity vector F Force vector ℱ frictional force per unit mass p pressure g gravity acceleration 119963 unit vector in z-direction

Equations of motion ndash component form

(Eq 6-6) in Cartesian coordinates

bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43


119906 zonal velocity 119907meridional velocity w vertical velocity 120588 density ℱ frictional force per unit mass p pressure g gravity acceleration

Hydrostatic balance

If friction ℱ119911 and vertical acceleration 119863119908

119863119905 are negligible we

derive from the vertical eq of motion (6-7c) the hydrostatic balance (Ch3 Eq3-3)

120597119901

120597119911 = - 120588g (Eq 6-8)

Note This approximation holds for large-scale atmospheric and oceanic circulation with weak vertical motions

44


Balance between vertical pressure gradient and gravitational force ldquoPressure decreases with height in proportion to the weight of the overlying atmosphererdquo

119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911

Conserved quantity A quantity of which the derivative equals zero

Conservation of mass for liquids is also called

continuity equation

Conservation of mass ndash continuity equation

45

4 Conservation of mass

Conservation of mass

46




47

Conservation of mass -1

Mass continuity requires

120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911

= (net mass flux into the volume)

bull mass flux in x-direction per unit time into the volume

120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911

bull mass flux in x-direction per unit time out of the volume

120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48


bull net mass flux in x-direction into the volume is then (employing Taylor expansion A22)

minus120597

120597119909120588119906 120575119909 120575119910 120575119911

bull net mass flux in y- and z- direction accordingly (see book)

bull Net mass flux into the volume

minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911 bull substituting into mass continuity

120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49


bull leads to equation of continuity

120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)

which has the form of conservation law

120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0

bull Using DDt (Eq 6-1) and 120571 ∙ 120588119958 =120588120571 ∙ 119958 + 119958 ∙ 120571120588 (see A22)

we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513

Approximations for continuity equation

incompressible - compressible flows

bull Incompressible flow (eg ocean)

120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)

bull Compressible flow (eg air ρ varies) expressed in pressure coordinate p applying hydrostatic assumption

120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50


Thermodynamic equation

bull Temperature evolution can be derived from first law of thermodynamics (Ch4 4-2)

119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)

bull If the heating rate is zero (119863119876

119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51


The temperature of a parcel will decrease increase in ascent descent with decreasingincreasing pressure

rarr Introduction of potential temperature 120579 (Eq 4-17)

120579 = 119879 1199010

119901

119876 heat 119863119876

119863119905 diabatic heating rate per unit mass

119888119901 specific heat at constant pressure T temperature

120588 density p pressure p0 = 1000 hPa κ=Rcp R gas constant for dry air

Thermodynamic equation - potential temperature

bull Eq 6-13 for potential temperature 120579 (see book for details)

119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)

Note For adiabatic motions 120575119876 = 0 120579 is conserved

52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54

Summary Summary 3-5)

rarr5 equations for the evolution of the fluid (5 unknowns)

Restrictions - application to average motion is often incorrect ie turbulence - fixed coordinate system

bull Eulerian ndash Lagrangian derivations perspectives

bull Equations of motion on a non-rotating fluid Pressure gradient force gravitational force and friction force act

119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ

bull 5 equations for the evolution of the fluid with 5 unknowns (uvwpT) equations of motion (3) conservation of mass (1) thermodynamic equation (1)

Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56


Forces on rotating sphere

Fictitious forces

- occur in revolvingaccelerating system

(eg car in curve)

- occur on the earthrsquos surface

(Fixed system on earthrsquos surface forms an accelerated system because a circular motion is performed once a day Movement of earth around the sun compared to earthrsquos rotation is insignificant)

rarr Coriolis force

rarr Centrifugal force

57

5 Equations of motion - rotating fluid

Coordinate systems in meteorology

bull Cartesian coordinates (xyz)

bull Spherical polar coordinates (λφz)

bull Height coordinates

geometrical altitude (z)

pressure (p) and

potential temperature (θ)


a radius Ω Greek omega rotation vector spherical polar coordinates φ 119897119886119905119894119905119906119889119890 λ 119897119900119899119892119894119905119906119889119890 119911 119903119886119889119894119886119897 119889119894119904119905119886119899119888119890



59

Inertial - rotating frames

119955 position vector Ω Greek omega rotation vector

60


Transformation into rotating coordinates

Consider figure 69

bull 119958119894119899 = 119958119903119900119905 + Ω times 119955 (Eq 6-24) We set A =r The transformation rule for DDt acting on a vector is given by

bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)

Then we set A rarr 119958119894119899 in Eq 6-26 using 6-24 we derive

119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905

119863119905 119903119900119905+ 2Ω times 119958119903119900119905 + Ω times Ω times 119955 (Eq 6-27)

119863119955

119863119905 119903119900119905= 119958119903119900119905

119955 position vector A any vector Ω rotation vector

Rotating equations of motion

Substituting 119863119958119894119899

119863119905 119894119899 from Eq 6-27 into Eq 6-6 (inertial frame

equation of motion) we derive in rotating frame dropping subscript ldquorotldquo 119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1

120588120571119901 + g119963 = minus 2Ω times 119958 minus Ω times Ω times 119955 + ℱ (Eq 6-28)

61


119958 velocity vector t time p pressure g gravity acceleration119963 unit vector in zminusdirection Ω rotation vector ℱ frictional force per unit mass



62

Coriolis acceleration

119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31

NH (viewed from above the Northpole) Ω gt 0 rotation anticlockwise SH (viewed from above the Southpole) Ω lt 0 rotation clockwise Absence of other forces

120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63

Assuming that 1) Ωz is negligible as Ωu ltlt g 2) wltltuh leads to

minus2120512 times 119958 ≃(minus2Ω sinφ119907 2Ω sinφ 1199060) (Eq 6-41)

= 119891119963 times 119958 Coriolis parameter 119943

119891 = 2Ω sin120593 (Eq 6-42)

Note that Ω sinφ is the vertical component of the Earths rotation rate only component that matters

Coriolis force on the sphere ndash Coriolis parameter

Ω= 727 x 10-5 s-1 a= 637 x 106 m

Coriolis force In direction of movement

in

NH

deflection of air

particles to the right

SH

deflection of air

particles to the left

takes place

Directional movement

64

Schematic picture for Coriolis force


0

0

cyclonic motion in

anti-clockwise direction low pressure

anticyclonic motion in

clockwise direction gt high pressure

Rotation Cyclonic ndash Anticyclonic

65

We call a rotating movement ldquocyclonicrdquo if it follows the same rotational

direction as the earth

As f gt 0 in the NH the rotary movement is

ζ vorticity

f Coriolis parameter

What is the rotary motion like in the SH

In the SH because of f lt 0 it is exactly the opposite

Centripetal force FCp Centrifugal force FC

Angular velocity Ω

Body (mass M) Centre

FCp = - FC

Centrifugal force ndash Centripetal force

r

66

FC = minusM Ω x (Ω x r)

bull Directed radially outward

bull If no other forces are present the particle would accelerate outwards bull Fictitious force balanced by Centripetal force


Substituting 119863119958119894119899


equation of motion ) we derive in rotating frame dropping subscript ldquorotldquo 119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67



Centrifugal acceleration

Friction acceleration

Pressure gradient accel

Gravi- tational accel


Centrifugal acceleration ndash modified gravitational potential

Combine the gradient of Centrifugal potential minusΩ times Ω times 119955 = 120571Ω2r2

2 and

Gravitational potential g119963 = 120571(gz) in Eq 6-28

119863119854

119863119905 +

1

120588120571119901 + 120571120567 = minus 2Ω times 119958 + ℱ Eq 6minus29

where 120567 = g119911 minusΩ2r2

2 Eq 6minus30

120567(Greek ldquoPhildquo) is a modified (by centrifugal accelerations) gravitational potential ldquomeasuredrdquo in the rotating frame

68




Gravitational +Centrifugal accelerations


Marshall and Plumb (2008) 69


hellipon the sphere ldquoShallow atmosphererdquo approx (a+z ≃ a)

r=(a+z)cos120593 ≃ a cos120593 hence Eq 6-30 becomes

120567 = g119911 minusΩ21198862cos2120593

2

Definition of geopotential surfaces

119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m

rarrAt the equator 119911lowast 119894119904 Ω21198862

2g asymp 11 km

higher than at the Pole (119911lowast=z)

Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)

bull Fluid in a thin spherical shell on a rotating sphere applying hydrostatic balance for vertical component and neglecting ℱ119911

compared with gravity

119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ

bull Equations of motion on a rotating fluid Pressure gradient force

modified gravitational potential (gravitational and centrifugal force)

Coriolis force and friction force act 119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ

Take home message I+II

71

Research interests

bull Stratosphere-ocean interactions bull Dynamics of the stratosphere-troposphere bull Stratospheric ozone and ozone depleting substances bull Influence of volcanic eruptions on climate earth system

Who am I

Prof Dr Kirstin Kruumlger

kkruegergeouiono

climate modelling

observations

ship expeditions

Volcanic excursion

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


6




7









2 Basics



2 Basics


10

2 Basics




120597119862



120597119862





Leonhard Euler (CH)



T(Xt) T(Xt+1)

χ=Xt+1 Lagrange

Parcel of air

trajectory

Euler

T(x1t2) u(x1t2) T(x1t1) u(x1t1)

Fixed box

in space



11

2 Basics



12

2 Basics


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


6




7









2 Basics



2 Basics


10

2 Basics




120597119862



120597119862





Leonhard Euler (CH)



T(Xt) T(Xt+1)

χ=Xt+1 Lagrange

Parcel of air

trajectory

Euler

T(x1t2) u(x1t2) T(x1t1) u(x1t1)

Fixed box

in space



11

2 Basics



12

2 Basics


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



7









2 Basics



2 Basics


10

2 Basics




120597119862



120597119862





Leonhard Euler (CH)



T(Xt) T(Xt+1)

χ=Xt+1 Lagrange

Parcel of air

trajectory

Euler

T(x1t2) u(x1t2) T(x1t1) u(x1t1)

Fixed box

in space



11

2 Basics



12

2 Basics


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71








2 Basics



2 Basics


10

2 Basics




120597119862



120597119862





Leonhard Euler (CH)



T(Xt) T(Xt+1)

χ=Xt+1 Lagrange

Parcel of air

trajectory

Euler

T(x1t2) u(x1t2) T(x1t1) u(x1t1)

Fixed box

in space



11

2 Basics



12

2 Basics


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



2 Basics


10

2 Basics




120597119862



120597119862





Leonhard Euler (CH)



T(Xt) T(Xt+1)

χ=Xt+1 Lagrange

Parcel of air

trajectory

Euler

T(x1t2) u(x1t2) T(x1t1) u(x1t1)

Fixed box

in space



11

2 Basics



12

2 Basics


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


10

2 Basics




120597119862



120597119862





Leonhard Euler (CH)



T(Xt) T(Xt+1)

χ=Xt+1 Lagrange

Parcel of air

trajectory

Euler

T(x1t2) u(x1t2) T(x1t1) u(x1t1)

Fixed box

in space



11

2 Basics



12

2 Basics


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


T(Xt) T(Xt+1)

χ=Xt+1 Lagrange

Parcel of air

trajectory

Euler

T(x1t2) u(x1t2) T(x1t1) u(x1t1)

Fixed box

in space



11

2 Basics



12

2 Basics


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



12

2 Basics


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


13

2 Basics


120575119862 =120597119862

120597119905120575119905 +

120597119862

120597119909120575119909 +

120597119862

120597119910120575119910 +

120597119862

120597119911120575119911

120575119862 119891119894119909119890119889 119901119886119903119905119894119888119897119890 =

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 120575119905


120597119862

120597119905 119891119894119909119890119889 119901119886119903119905119894119888119897119890=

120597119862

120597119905+ 119906

120597119862

120597119909+ 119907

120597119862

120597119910 + w

120597119862

120597119911 = 119863119862

119863119905

119863

119863119905

119863

119863119905=

120597

120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911 equiv

120597

120597119905 + 119958 ∙ 120513



Insert 120575119909 = u120575119905 120575119910 = v120575119905 120575119911 = w120575119905


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


14

2 Basics


119863


120597119905+ 119906

120597

120597119909+ 119907

120597

120597119910 + w

120597

120597119911

equiv 120597

120597119905 + 119958 ∙ 120513 (6-1)



120597




120513 equiv120597

120597119909

120597

120597119910120597





15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



15

2 Basics

Kraus (2004)


16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

16




- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


- Nomenclature

- Vector operations

- Nabla operator

17






a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71





a coordinate eg x




18

x

dtd



z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


z

y

x

z

y

x

ii

av

av

av

v

v

v

aavavavva


vector


scalar a





19

wind

vector


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


ii

i

iiiizzyyxx

z

y

x

z

y

x

vfvfvffvfvfv

f

f

f

v

v

v

vffv






f

v

cosfvfv


20


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


bcacbabac

baba

baba

baba

babakbabajbabai

bbb

aaa

kji

abbac

xyyx

zxxz

yzzy

xyyxzxxzyzzy

zyx

zyx

sin

)()()(

a

b








21


Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

Graphically





with

Operator -Nabla

z

y

x

i

22


the right







TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

TTgradT

z

y

x

zT

yT

xT

z

w

y

v

x

u

w

v

u

udivu

z

y

x

y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x


vector


scalar

Product with a

Scalar vector


23

Unit vectors

i x direction

j y direction

k z direction


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


z

y

x

z

y

x

zT

yT

xT

T

T

T

TTT

zw

yv

xu

w

v

u

u

z

w

y

v

x

u

w

v

u

udivu

z

y

x

z

y

x



24


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


0 Hu 0 Hu


25



26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


26


Horizontal

wind Rotation

L




y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


y

u

x

vx

w

z

u

z

v

y

w

wvu

kji

w

v

u

urotuzyx

z

y

x






u u rot

y

u

x

v

-uk









yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71








yx

uvuk

f



Vorticity

28



Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


Shear vorticity

H

L

Tropopause

STJ

Examples

STJ Sub-

tropical Jet

Vorticity (NH)


x

y

z

ζ lt 0

ζ gt 0

Surface




Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



Stratospheric

polar vortex

Edge of vortex

Filament


L

x Oslo


1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


31






(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71





(u p T)) 32










33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71








33

Fu

dt





fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



fluid parcel - cube

34




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71




35

120588 120575119909 120575119910 120575119911 119863119958

119863119905 = F (Eq 6-2)


119863119958

119863119905 =

120597119958

120597119905 + u

120597119958

120597119909 +v

120597119958

120597119910+w

120597119958

120597119911

= 120597119958

120597119905 + (119958 ∙ 120571) 119958

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

Gravity

36









37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71




37


F(A) = p ∙ 119860 = 119901 (x minus 120575119909

2 y z) 120575119910 120575119911

F(B) = p ∙ 119861 = minus119901 (x + 120575119909

2 y z) 120575119910 120575119911

Fx = F(A) + F(B)


Fx =minus120597119901

120597119909120575119909 120575119910 120575119911


119901 pressure

pressure force

(inward)

pressure

force

(inward)




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71




38

Fpressure= 119865119909 119865119910 119865119911

=minus120597119901

120597119909120597119901

120597119910120597119901

120597119911120575119909 120575119910 120575119911

=minus 120571119901 120575119909 120575119910 120575119911 (Eq 6-4)

119901 pressure

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

Friction force

39






Ffriction = 120588 ℱ 120575119909 120575119910 120575119911 (Eq 6-5)




40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



40



1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


41


Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

Equations of motion


120588 120575119909 120575119910 120575119911 119863119854

119863119905 = F119901119903119890119904119904119906119903119890 + F119892119903119886119907119894119905119910 + F119891119903119894119888119905119894119900119899


119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ (Equation 6-6)

42





bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



bull120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

bull120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

bull120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

43



Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

Hydrostatic balance




120597119901

120597119911 = - 120588g (Eq 6-8)


44



119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863119908

119863119905=

120597119908

120597119905+ 119906

120597119908

120597119909+ 119907

120597119908

120597119910 + w

120597119908

120597119911



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



continuity equation


45



46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


46




47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


47



120597

120597119905 (120588 120575119909 120575119910 120575119911) =

120597120588

120597119905 120575119909 120575119910 120575119911



120588119906 119909 minus1

2 120575119909 119910 119911 120575119910 120575119911


120588119906 119909 +1

2 120575119909 119910 119911 120575119910 120575119911



48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


48



minus120597

120597119909120588119906 120575119909 120575119910 120575119911




120597

120597119905 (120588 120575119909 120575119910 120575119911) = minus 120571 ∙ 120588119958 120575119909 120575119910 120575119911



49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


49



120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (Eq 6-9)


120597 119862119900119899119888119890119899119905119903119886119905119894119900119899

120597119905+ 120571 ∙ 119891119897119906119909 = 0


we can rewrite

119863120588

119863119905+ 120588120571 ∙ 119958 = 0 (Eq 6-10)


119863

119863119905 119871119886119892119903119886119899119892119894119886119899 119889119890119903119894119907119886119905119894119907119890

119863

119863119905=

120597

120597119905 + 119958 ∙ 120513




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71




120571 ∙ 119958 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119911= 0 (Eq6-11)


120571119901 ∙ 119958119901 =120597119906

120597119909+

120597119907

120597119910+

120597119908

120597119901= 0 (Eq6-12)

50




119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (Eq 6-13)


119863119905 = 0 Ch 431) then

119863119879

119863119905 =

1

120588119888119901

119863119901

119863119905

51




120579 = 119879 1199010

119901

119876 heat 119863119876






119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



119863120579

119863119905=

119901

1199010

minusκ 119863119876

119863119905

119888119901

(Eq 6-14)


52


119876 heat 119863119876



53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

53

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

Ia) 120597119906

120597119905 + 119906

120597119906

120597119909 + 119907

120597119906

120597119910 + w

120597119906

120597119911 +

1

120588

120597119901

120597119909 = ℱx (6-7a)

Ib) 120597119907

120597119905 + 119906

120597119907

120597119909 + 119907

120597119907

120597119910 + w

120597119907

120597119911 +

1

120588

120597119901

120597119910 = ℱ119910

(6minus7b)

Ic) 120597119908

120597119905 + 119906

120597119908

120597119909 + 119907

120597119908

120597119910 + w

120597119908

120597119911 +

1

120588

120597119901

120597119911 + g = ℱz (6-7c)

II) 120597120588

120597119905+ 120571 ∙ 120588119958 = 0 (6-9 or 6-116-12)

III) 119863119876

119863119905= 119888119901

119863119879

119863119905 minus

1

120588 119863119901

119863119905 (6-13 or 6-14)

54






119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ


Take home message I

55

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

1 Motivation







Lect

ure

Ou

tlin

e ndash

Ch


56



Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


Fictitious forces


(eg car in curve)



rarr Coriolis force


57







pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71






pressure (p) and






59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



59



60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

60



Consider figure 69


bull119863119912

119863119905 119894119899=

119863119912

119863119905 119903119900119905+ Ω times A (Eq 6-26)


119863119958119894119899

119863119905 119894119899=

119863

119863119905 119903119900119905+ Ω times 119958119903119900119905 + Ω times 119955

=119863119958

119903119900119905


119863119955

119863119905 119903119900119905= 119958119903119900119905



Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


61





62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



62


119863119854

119863119905 = minus 2120512 times 119958 Eq 6-31


120512 =0 Ωy

Ωz

=0

Ω 119888119900119904φΩ sinφ

Angular velocity



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



63




119891 = 2Ω sin120593 (Eq 6-42)



Ω= 727 x 10-5 s-1 a= 637 x 106 m


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


in

NH

deflection of air


SH

deflection of air


takes place


64



0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

0

0

cyclonic motion in





65




ζ vorticity





Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


Angular velocity Ω


FCp = - FC


r

66





Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71


Substituting 119863119958119894119899



119863119905 +

1

120588120571119901 + g119963 = ℱ (Eq 6-6)

119863119854

119863119905 +

1


67










2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71



2 and


119863119854

119863119905 +

1



2 Eq 6minus30


68










120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71





120567 = g119911 minusΩ21198862cos2120593

2


119911lowast = 119911 +Ω21198862cos2120593

2g (Eq 6-40)

Ω= 727 x 10-5 s-1 a = 637 x 106 m


2g asymp 11 km


Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

Equations of motion

119863119854

119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ (Eq 6minus43)



119863u

119863119905 +

1

120588

120597119901

120597119909minus 119891119907 = ℱ119909 (Eq 6minus44)

119863119907

119863119905 +

1

120588

120597119901

120597119910+ 119891119906 = ℱ119910

1

120588

120597119901

120597119911+ g = 0

70





119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71




119863119854

119863119905 +

1

120588120571119901 + g119963 = ℱ




119863119905 +

1

120588120571119901 + 120571120567 +f 119963 times 119958 = ℱ


71

Documents

Chapter 6: The equations of fluid motion - Forsiden - · PDF fileGEF 1100 – Klimasystemet Chapter 6: The equations of fluid motion 1 GEF1100–Autumn 2016 13.09.2016 Prof. Dr. Kirstin