AOSS 401, Fall 2006 Lecture 10 September 28 , 2007

AOSS 401, Fall 2006Lecture 10

September 28, 2007

Richard B. Rood (Room 2525, SRB)[email protected]

734-647-3530Derek Posselt (Room 2517D, SRB)

[email protected]

mailto:[email protected]

mailto:[email protected]

Class News

• Homework 2 returned today

• Homework 3 due today (questions?)

• Homework 4 posted Monday

• Exam 1 October 10—covers chapters 1-3 in Holton

Weather

• NCAR Research Applications Program– http://www.rap.ucar.edu/weather/

• National Weather Service– http://www.nws.noaa.gov/dtx/

http://www.rap.ucar.edu/weather/



http://www.nws.noaa.gov/dtx/



Correction…

• I made a mistake in my last set of lectures (September 19th)

• Geostrophic wind is only non-divergent if pressure is the vertical coordinate…

• Corrected lecture 6 posted to ctools by Monday.

Today:Material from Chapter 3

• Natural coordinates

• Balanced flow

Another Coordinate System?

• We want to simplify the equations of motion• For horizontal motions on many scales, the

atmosphere is in balance– Mass (p, Φ) fields in balance with wind (u)– It is easy to observe the pressure or geopotential

height, much more difficult to observe the wind

• Balance provides a way to infer the wind from the observed (p, Φ)

• Wind is useful for prediction (remember the advection homework and in-class problems?)

The horizontal momentum equation

p

pp

pp

fDt

D

fuydt

d

fxdt

du

uku

v

v

Assume no viscosity and no vertical wind

Geostrophic balance

High Pressure

Low Pressure

Flow initiated by pressure gradient

Flow turned by Coriolis force

Geostrophic & observed wind 300 mb

Describe previous figure. What do we see?

• At upper levels (where friction is negligible) the observed wind is parallel to geopotential height contours.

• (On a constant pressure surface)

• Wind is faster when height contours are close together.

• Wind is slower when height contours are farther apart.

Geopotential (Φ) in upper troposphere

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north


eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy


eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

δΦ = Φ0 – (Φ0+2ΔΦ)


eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

yy

2


p

pp

pp

fDt

D

fuydt

d

fxdt

du

uku

v

v

Assume no viscosity

Geostrophic approximation

g

p

p

fuy

fx

gv


eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

yfug

2


eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

ΔΦ > 0

south

north

Δy

yfug


• Think about the observed wind– Flow is parallel to geopotential height lines– There is curvature in the flow

Geostrophic & observed wind 300 hPa


• Think about the observed (upper level) wind– Flow is parallel to geopotential height lines– There is curvature in the flow

• Geostrophic balance describes flow parallel to geopotential height lines

• Geostrophic balance does not account for curvature

• How to best describe balanced flow with curvature?

Another Coordinate System?

• We want to simplify the equations of motion• For horizontal motions on many scales, the

atmosphere is in balance– Mass (p, Φ) fields in balance with wind (u)– It is easy to observe the pressure or geopotential

height, much more difficult to observe the wind

• Balance provides a way to infer the wind from the observed (p, Φ)

• Need to describe balance between pressure gradient, coriolis, and curvature

“Natural” Coordinate System

• Follow the flow

• From hydrodynamics—assumes no local changes – No local change in geopotential height– No local change in wind speed or direction

• Assume– Horizontal flow only (no vertical component)– No friction

Return to Geopotential (Φ) in upper troposphere

eastwest

Φ0+3ΔΦ

Φ0

ΔΦ > 0

south

northDefine one component of the horizontal wind as

tangent to the direction of the wind. t

t t

t


eastwest

Φ0+3ΔΦ

Φ0

ΔΦ > 0

south

north

t t

t

Define the other component of the horizontal wind

as normal to the direction of the wind. n

nn n

• Regardless of position (i,j)– t always points in the direction of flow– n always points perpendicular to the direction of

the flow toward the left

• Remember the “right hand rule” for vectors? Take k x t to get n

• Assume– Pressure as a vertical coordinate– Flow parallel to contours of geopotential height


• Advantage: We can look at a height (on a pressure surface) and pressure (on a height surface) and estimate the wind.– It is difficult to directly measure winds– We estimate winds from pressure (or

hydrostatically equivalent height), a thermodynamic variable.

– Natural coordinates are useful for diagnostics and interpretation.



• For diagnostics and interpretation of flows, we need an equation…


eastwest

ΔΦ > 0

south

north

t t

t

Geostrophic assumption. Do you notice that those n vectors point towards something out in the distance?

nn n

HIGH

Low


eastwestsouth

north

HIGH t t

tnn nLow

Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle.

Time to look at themathematics

First simplification: the velocity

• Always positive

• Always points in the positive t direction

V

tV

V

VDefine velocity as:

Definition of magnitude:

One direction: no (u,v)

Goal: Quantify Acceleration

Dt

DV

Dt

DV

Dt

DDt

VD

Dt

D

tt

V

tV

)(

acceleration is:

Change in speed

Change in Direction

(Chain Rule)

How to get as a function of V, R ?Dt

Dt

Remember our circle geometry…

this is not rotation of the Earth!It is an element of curvature in the flow.

Δφ

R=radius of curvature t

t

Δtt+Δt

Δs

Δs=RΔφ


this is not rotation of the Earth!It is an element of curvature in the flow.

Δφ


t

Δtt+Δt

Δs

Δs=RΔφ

n

n

tt

t

R

s


If Δs is very small, Δt is parallel to n.So, Δt points in the direction of n

Δφ


t

Δtt+Δt

Δs

Δs=RΔφ

n

n

tt

t

R

s

nt

ntt

nt

tt

t

t

R

V

Dt

D

VDt

DsDt

Ds

RDt

Ds

Ds

D

Dt

D

RDs

D

s

t

R

s

RVDt

D

st

lim

), of (function ?

0,

Remember, we want anexpression for

From circle geometrywe have:

Rearrange and take the limit

Use the chain rule

Remember the definitionof velocity

Goal: Quantify Acceleration

ntV

nt

tt

V

tV

R

V

Dt

DV

Dt

D

R

V

Dt

DDt

DV

Dt

DV

Dt

DDt

VD

Dt

D

2

)(

acceleration defined as:

(Chain Rule)

We just derived:

So the total acceleration is

Acceleration in Natural Coordinates

ntV

R

V

Dt

DV

Dt

D 2

Along-flowspeed change

?


222

2

1

RRRR

V

RDt

DRV

RsDt

DsV

R

V

Dt

DV

Dt

D

nn

ntVThe total

acceleration is

Definition ofwind speed

Circle geometry

Plug in for Δs

Centrifugal force

angular velocity

angle of rotation


CentrifugalAcceleration

ntV

R

V

Dt

DV

Dt

D 2

Along-flowspeed change

We have seen that Coriolis force is normal to the velocity.

nVk fVf - force coriolis

Pressure gradient (by definition)

nsp nt


nsfV

R

V

Dt

DV

fDt

Dp

ntnnt

uku

2

onssubstitutiour all make we

The horizontal momentum equation(in natural coordinates)

nfV

R

V

sDt

DV

nsfV

R

V

Dt

DV

2

2

formcomponent in and

ntnnt

Along-flowdirection (t)

Across-flowdirection (n)

• Simplification?

• Which coordinate system is easier to interpret?

fuyDt

D

fxDt

Du

pp

pp

v

v

nfV

R

V

sDt

DV

2

• We are only looking at flow parallel to geopotential height contours

00

nfV

R

V

2

• Simplification?

• Which coordinate system is easier to interpret?

• We are only looking at flow parallel to geopotential height contours

fuyDt

D

fxDt

Du

pp

pp

v

v

nfV

R

V

2

Curved flow (Centrifugal Force)

Coriolis Pressure Gradient

One Diagnostic Equation

Uses of Natural Coordinates

• Geostrophic balance– Definition: coriolis and pressure gradient in

exact balance.– Parallel to contours straight line R is

infinite

nfV

R

V

2

0

Geostrophic balance in natural coordinates

nfV

Which actually tells us the geostrophic wind can only be equal to the real wind if the height contours are straight.

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

south

northn

fVg

Δn

Therefore

• If the contours are curved then the real wind is not geostrophic.

How does curvature affect the wind?(cyclonic flow/low pressure)

nfV

R

V

2

R

t

n

Δn

Φ0

Φ0+ΔΦ

Φ0-ΔΦ

HIGH

Low

1. Mathematical Perspective

ag

gag

agg

fVR

V

nfVfV

R

V

nVVf

R

V

nfV

R

V

2

2

2

2

Equation of motion

Split Coriolis intogeostrophic and

ageostrophic parts

Use definition of geostrophic wind


g

agg

ag

ag

VV

VVV

V

fVR

V

0

0 0

2Total centrifugal force balances ageostrophic

part of coriolis

Total wind is sum of its parts

Real wind speed is slower than geostrophic

for cyclonic flow!

Look at sign of terms (R > 0)

2. Physical Perspective

V

PGF

COR

Geostrophic balance Add curvature (centrifugal force)

V

PGF

COR

CEN

Pressure gradient force is the same in each case. With curvature less coriolis force is needed to balance the pressure gradient.

gfV fV>



Observed:95 knots

Geostrophic:140 knots

How does curvature affect the wind?(anticyclonic flow/high pressure)

nfV

R

V

2

R

t

n

Δn

Φ0

Φ0+ΔΦ

Φ0-ΔΦ

HIGH

Low


g

agg

ag

ag

VV

VVV

V

fVR

V

0

0 0

2

Total wind is sum of its parts

Real wind speed is faster than geostrophic

for anticyclonic flow!

Look at sign of terms (R > 0)

Total centrifugal force balances ageostrophic

part of coriolis

2. Physical Perspective

V

PGF

COR

Geostrophic balance Add curvature

V

PGF

COR

CEN

Pressure gradient force is the same in each case. With curvature more coriolis force is needed to balance the pressure gradient.

gfV fV<



Observed:30 knots

Geostrophic:25 knots

What did we just do?

• Found a way to describe balances between pressure gradient, coriolis, and curvature

• We assumed friction was unimportant and only looked at flow at a particular level

• We assumed flow was on pressure surfaces• We saw that the simplified system can be used

to describe real flows in the atmosphere• Can we describe other flow patterns? (Different

scales? Different regions of the Earth?)

Next Time:Finish balanced flows

• Cyclostrophic flow (tornados, water spouts, dust devils)

• Gradient wind (general description of curved flow anywhere on the globe—if friction is not important…)

Documents

AOSS 401, Fall 2006 Lecture 10 September 28 , 2007