Advanced SynopticM. D. Eastin Dynamic Meteorology: A Review

Advanced Synoptic M. D. Eastin

Dynamic Meteorology:A Review


Total Derivatives

• The rate of change of something following a fluid element is called the Lagrangian rate of change

• Example: How temperature changes following an air parcel as is moves around

Partial Derivatives

• The rate of change of something at a fixed point is called the Eulerian rate of change

• Example: The temperature change at a surface weather station

Euler’s Relation

• Shows how a total derivative can be decomposed into a local rate of change and advection terms

Total Vs. Partial Derivatives

dt

d

Dt

D

t

x

y

p

z

Tw

y

Tv

x

Tu

t

T

Dt

DT


Scalar: Has only a magnitude (e.g. temperature)

Vector: Has a magnitude and direction (e.g. wind)Usually represented in bold font (V) or as ( )

Unit Vectors: Represented by the letters i, j, k Magnitude is 1.0 Point in the x, y, and z (or p) directions

Total Wind Vector: Defined as V = ui + vj + wk, where u, v, w are the scalar components of the zonal, meridional, and vertical wind

Vector Addition/Subtraction: Simply add the scalars of each component together

Vector Multiplication:

Dot Product: Defined as the product of the magnitude of the vectorsResults in a scalar

The dot product of any unit vector with another = 0

Vectors

V

V1+V2 = (u1+u2)i + (v1+v2)j + (w1+w2)k

i•i = j•j = k•k = 1

V1•V2 = u1u2+v1v2+w1w2


Vector Multiplication:

Cross Product: Results in a third vector that points perpendicular to the first twoFollows the “Right Hand Rule”Often used in meteorology when rotation is involved (e.g. vorticity)

Differential “Del” Operator:

Definition:

Del multiplied by a scalar (“gradient” of the scalar):

Dot product of Del with Total Wind Vector (“divergence”):

Vectors

V1 x V2 = i(v1w2 – v2w1)+j(u2w1-u1w2)+k(u1v2-u2v1)

zyx

kji

z

a

y

a

x

aa

kji

z

w

y

v

x

u

V


Differential “Del” Operator:

Cross product of Del with Total Wind Vector (“vorticity”):

Note: The third term is rotation in the horizontal plane about the vertical axis This is commonly referred to “relative vorticity” (ζ) We can arrive at this by taking the dot product with the k unit vector

Dot product of Del with itself (“Laplacian” operator)

If we apply the Laplacian to a scalar:

Vectors

y

u

x

v

x

w

z

u

z

v

y

wkjiV

y

u

x

v

Vk

2

2

2

2

2

2

22

z

a

y

a

x

aaa


Euler’s Relation Revisited:

If we dot multiply the gradient of a scalar (e.g. Temperature) with the total wind vector we get the advection of temperature by the wind:

Recall, the total derivative of temperature can be written as (in scalar form)

Or as (in vector form) upon substituting from above:

Vectors

z

Tw

y

Tv

x

TuT

V

z

Tw

y

Tv

x

Tu

t

T

Dt

DT

Tt

T

Dt

DT

V


The equations of motion describe the forces that act on an air parcel in a three-dimensional rotating system → describe the conservation of momentum

Fundamental Forces:

Pressure Gradient Force (PGF) → Air parcels always accelerate down the pressure gradient from regions of high to low pressure

Gravitational Force (G) → Air parcels always accelerate (downward) toward the Earth’s center of mass (since the Earth’s mass is much greater than an air parcel’s mass)

Frictional Force (F) → Air parcels always decelerate due to frictional drag forces both within the atmosphere and at the boundaries

Apparent Forces (due to a rotating reference frame):

Centrifugal Force (CE) → Air parcels always accelerate outward away from theiraxis of rotation

Coriolis Force (CF) → Air parcels always accelerate 90° to the right of their current direction (in the Northern Hemisphere)

Equations of Motion


The equation of motion for 3D flow can be written symbolically as:

Normally, this equation is decomposed into three equations:

What are each of these terms?

Equations of Motion

FCFGPGFCEDt

D

V

rzFugz

p

a

vu

Dt

Dw

cos2122

rxFwvx

p

a

uw

a

uv

Dt

Du

cos2sin2

1tan

ryFuy

p

a

vw

a

u

Dt

Dv

sin2

1tan2


The equations of motion for 3D flow:

where: Total Derivative of WindPressure Gradient ForceGravitational ForceFrictional ForceCurvature TermsCoriolis Force

Are all of these terms significant? Can we simplify the equations?

Equations of Motion

rzFugz

p

a

vu

Dt

Dw

cos2122

rxFwvx

p

a

uw

a

uv

Dt

Du

cos2sin2

1tan

ryFuy

p

a

vw

a

u

Dt

Dv

sin2

1tan2


Scale Analysis:

• Method by which to determine which terms in the equations can be neglected: [Neglect terms much smaller than other terms (by several orders of magnitude)]

• Use typical values for parameters in the mid-latitudes on the synoptic scale

Horizontal velocity (U) ≈ 10 m s-1 (u,v)Vertical velocity (W) ≈ 10-2 m s-1 (w)Horizontal Length (L) ≈ 106 m (dx,dy)Vertical Height (H) ≈ 104 m (dz)Angular Velocity (Ω) ≈ 10-4 s-1 (Ω)Time Scale (T) ≈ 105 s (dt)Frictional Acceleration (Fr) ≈ 10-3 m s-2 (Frx, Fry, Frz)Gravitational Acceleration (G) ≈ 10 m s-2 (g)Horizontal Pressure Gradient (∆p) ≈ 103 Pa (dp/dx, dp/dy)Vertical Pressure Gradient (Po) ≈ 105 Pa (dp/dz)Air Density (ρ) ≈ 1 m3 kg-1 (ρ)Coriolis Effect (C) ≈ 1 (2sinφ, 2cosφ)

Using these values, you will find that numerous terms can be neglected…..

Equations of Motion


The “simplified” equations of motion for synoptic-scale 3D flow:

where: f = 2ΩsinΦ and Φ is the latitude

This set of equations is often called the “primitive equations” for large-scale motion

Note: The total derivatives have been decomposed into their local and advective terms

The vertical equation of motion reduces to the hydrostatic approximation – vertical velocity can NOT be predicted using the vertical equation of motion – other approaches must be used

Equations of Motion

gz

p

1

0

fvx

p

z

uw

y

uv

x

uu

t

u

Dt

Du

1

fuy

p

z

vw

y

vv

x

vu

t

v

Dt

Dv

1


The continuity equation describes the conservation of mass in a 3D system

• Mass can be neither created or destroyed• Must account for mass in synoptic-scale numerical prediction

Mass Divergence Form:

Velocity Divergence Form:

Scale Analysis results in:

Mass Continuity Equation

z

w

y

v

x

u

t

1 Interpretation: Net mass change

is equal to the 3-D convergenceof mass into the column

z

w

y

v

x

u

Dt

D

1 Form commonly used by

numerical models to predictdensity changes with time

z

w

y

v

x

u

0Form commonly used byobservational studies toidentify regions of vertical motion


If we isolate the vertical velocity term on one side:

OR

Thus, changes in the vertical velocity can be induced from the horizontal convergence/divergence fields

Example:

Mass Continuity Equation

y

v

x

u

z

w hVz

w

Convergence near the surface(low pressure) leads to upwardmotion that increases with height

LDivergence near the surface(e.g. high pressure) leads to downward motion increasingwith height


The thermodynamic equation describes the conservation of energy in a 3D system

Begin with the First Law of Thermodynamics:

After some algebra….

Decomposed into local and advective components:

What are each of these terms?

Thermodynamic Equation

Dt

DQ

cDt

Dp

cDt

DT

pp

11

DpDTcDQ p 1

Dt

DQ

cDt

Dp

cz

Tw

y

Tv

x

Tu

t

T

pp

11

Dt

DQ

cDt

Dp

cz

Tw

y

Tv

x

Tu

t

T

pp

11

Diabatic temperature change from condensation, evaporation, and radiation

Adiabatic temperature change due to expansion and contraction

Advection of temperature

Local change in temperature


Isobaric Coordinates

Advantages of Isobaric Coordinates:

• Simplifies the primitive equations• Remove density (or mass) variations that are difficult to measure• Upper air maps are plotted on isobaric surfaces

Characteristics of Isobaric Coordinates:

• The atmosphere is assumed to be in hydrostatic balance• Vertical coordinate is pressure → [x,y,p,t]• Vertical velocity (ω)

ω > 0 for sinking motionω < 0 for rising motion

• Euler’s relation in isobaric coordinates

What are the primitive equations in isobaric coordinates?

Dt

Dp

pyv

xu

tDt

D


Isobaric Coordinates

Primitive Equations (for large-scale flow) in Isobaric Coordinates:

See Holton Chapter 3 for a complete description of the transformations

We will be working with (starting from) these equations most of the semester!!!

fvx

zg

p

u

y

uv

x

uu

t

u

Dt

Du

fuy

zg

p

v

y

vv

x

vu

t

v

Dt

Dv

p

RT

p

zg

0

py

v

x

u

Dt

DQ

cpc

RT

p

T

y

Tv

x

Tu

t

T

Dt

DT

pp

1

Zonal Momentum

Meridional Momentum

Hydrostatic Approximation

Mass Continuity

Thermodynamic

RTp Equation of State


Hypsometric Equation

What it means: The thickness between any two pressure levels is proportional to the mean temperature within that layer

Warmer layer → Greater thickness Pressure decrease slowly with height

Colder layer → Less thickness Pressure decreases rapidly with height

Derivation: Integrate the Hydrostatic Approximation between two pressure levels

2

112 ln

p

p

g

TRzz


Hypsometric Equation

Application: Can infer the mean vertical structure of the atmosphere:

• Location/structure of pressure systems• Location/structure of jet streams• Precipitation type (rain/snow line)

From Lackmann (2011)

500-mb Heights – 0600 UTC 22 Jan 2004 1000-500-mb Thickness – 0600 UTC 22 Jan 2004


Geostrophic Balance

Recall the horizontal momentum equations:

• Scale analysis for large-scale (synoptic) motions above the surface reveals that the total derivatives are one order of magnitude less than the PGF and CF.

• Neglect the total derivatives and do some algebra….

• The PGF exactly balances the CF

• There are no accelerations acting on the parcel (once balance is achieved)

fvx

zg

Dt

Du

fuy

zg

Dt

Dv

x

z

f

gvg

y

z

f

gug


Geostrophic Balance

CoriolisForce

Geostrophic Wind

PressureGradient

Force


What is Means: The vertical shear of the geostrophic wind over a layeris directly proportional to the horizontal temperature (or thickness) gradient through the layer

Derivation: Differentiate the geostrophic balance equations with respect to pressure and apply the hydrostatic approximation

Characteristics:

• Relates the temperature field to the wind field

• Describes how much the geostrophic wind will change with height (pressure) for a given horizontal temperature gradient

• The thermal wind is the vector difference between the two geostrophic winds above and below the pressure level where the horizontal temperature gradient resides

• The thermal wind always blows parallel to the mean isotherms (or lines of constant thickness) within a layer with cold air to the left and warm air to the right

Thermal Wind

p

g

x

T

fp

R

p

v

p

g

y

T

fp

R

p

u


Thermal Wind: Application

The thermal wind can be used to diagnose the mean horizontal temperature advection within a layer of the atmosphere

V850V850

V500

V500Vtherm

Vtherm

Warm Air Advection (WAA)(within a layer)

Warm

Warm

Cold

Cold

Geostrophic winds turn clockwise (or “veer”) with height through the layer

Cold Air Advection (CAA)(within a layer)

Geostrophic winds turn counterclockwise (or “back”) with height through the layer



850 mb

500 mb

International Falls, MN

• Winds turn counterclockwise (“back”) with height between 850 and 500 mb

• We should expect CAA within the layer

Note that CAA appears to be occurring at both 850 and 500 mb

Buffalo, NY

• Winds turn clockwise (“veer”) with height between 850 and 500 mb

• We should expect WAA within the layer



Minneapolis / Saint Paul (MSP)

We can infer WAA and CAA with a single sounding from the vertical profile of wind direction

Winds are veering with height → WAA

Winds are backing with height → CAA


Surface Pressure Tendency

What it means: The net divergence (convergence) of mass out of (in to) a column of air will lead to a decrease (increase) in surface pressure

Derivation: Integrate the Continuity Equation (in isobaric coordinates) through the entire depth of the atmosphere and apply boundary conditions

Characteristics:

• Provide qualitative information concerning the movement (approach) of pressure systems

• Difficult to apply as a forecasting technique since small errors in wind (i.e. divergence) field can lead to large pressure tendencies

• Also, divergence at one level is usually offset by convergence at another level

Note: Temperature changes in the column do not have a direct effect on the surface pressure – they change the height of the pressure levels, not the net mass

dpy

v

x

u

t

p sp

s

0


Circulation and VorticityCirculation: The tendency for a group of air parcels to rotate

If an area of atmosphere is of interest, you compute the circulation

Vorticity: The tendency for the wind shear at a given point to induce rotation If a point in the atmosphere is of interest, you compute the vorticity

Planetary Vorticity: Vorticity associated with the Earth’s rotation

Relative Vorticity: Vorticity associated with 3D shear in the wind field

Only the vertical component of vorticity (the k component) is of interest for large-scale (synoptic) meteorology

Absolute Vorticity: The sum of relative and planetary vorticity

sin2f

y

u

x

v

xp

u

p

v

ykjiV

y

u

x

v

f


Circulation and VorticityCirculation: The tendency for a group of air parcels to rotate

If an area of atmosphere is of interest, you compute the circulation

Vorticity: The tendency for the wind shear at a given point to induce rotation If a point in the atmosphere is of interest, you compute the vorticity


Absolute Vorticity (η = ζ + f)500-mb Heights


Circulation and VorticityVorticity Types:


Absolute Vorticity (η = ζ + f)

fsin2f

y

u

x

v

y

u

x

v

Relative Vorticity (ζ)



Positive Vorticity: Associated with cyclonic (counterclockwise) circulations in the Northern Hemisphere

Negative Vorticity: Associated with anticyclonic (clockwise) circulations in the Northern Hemisphere



Shear Vorticity: Associated with gradients along local straight-line wind maxima

Curvature Vorticity: Associated with the turning of flow along a stream line

Shear Vorticity Curvature Vorticity

_+

+



Vorticity EquationDescribes the factors that alter the magnitude of the absolute vorticity with time

Derivation: Start with the horizontal momentum equations (in isobaric coordinates)

Take of the meridional equation and subtract of the zonal equation

After use of the product rule, some simplifications, and cancellations:

fvx

zg

p

u

y

uv

x

uu

t

u

fuy

zg

p

v

y

vv

x

vu

t

v

Zonal Momentum

Meridional Momentum

x

y

p

v

xp

u

yy

v

x

uf

y

fv

pyv

xu

t


Vorticity EquationWhat do the terms represent?

Local rate of change of relative vorticity ~10-10

Horizontal advection of relative vorticity ~10-10

Vertical advection of relative vorticity ~10-11

Meridional advection of planetary vorticity ~10-10

Divergence Term ~10-9

Tilting Terms ~10-11

p

v

xp

u

yy

v

x

uf

y

fv

pyv

xu

t

What are the significant terms? → Scale analysis and neglect of “small” terms yields:

y

v

x

uf

y

fv

yv

xu

t


Vorticity EquationPhysical Explanation of Significant Terms:

Horizontal Advection of Relative Vorticity

• The local relative vorticity will increase (decrease) if positive (negative) relative vorticity is advected toward the location → Positive Vorticity Advection (PVA) and

→ Negative Vorticity Advection (NVA)

• PVA often leads to a decrease in surface pressure (intensification of surface lows)

Meridional Advection of Planetary Vorticity

• The local relative vorticity will decrease (increase) if the local flow is southerly (northerly) due to the advection of planetary vorticity (minimum at Equator; maximum at poles)

Divergence Term

• The local relative vorticity will increase (decrease) if local convergence (divergence) exists

y

v

x

uf

y

fv

yv

xu

t


Vorticity EquationPhysical Explanation: Horizontal Advection of Relative Vorticity

y

v

x

uf

y

fv

yv

xu

t


Relative Vorticity (ζ) Relative Vorticity Advection


Quasi-Geostrophic Theory

Most meteorological forecasts:

• Focus on Temperature, Winds, and Precipitation (amount and type)**

• Are largely a function of the evolving synoptic-scale weather patterns

Quasi-Geostrophic Theory:

• Makes further simplifying assumptions about the large-scale dynamics

• Diagnostic methods to estimate: Changes in large-scale surface pressureChanges in large-scale temperature

(thickness)Regions of large-scale vertical motion

• Despite the simplicity, it provides accurate estimates of large-scale changes

• Will provide the basic analysis framework for remainder of the semester

Next Time……


Summary

Important Dynamic Meteorology (METR 3250) Concepts:

• Total / Partial Derivatives and Vector Notation• Equation of Motion (Components and Simplified Terms)• Mass Continuity Equation• Thermodynamic Equation• Isobaric Coordinates and Equations• Hypsometric Equation• Geostrophic Balance• Thermal Wind • Surface Pressure Tendency• Circulation and Vorticity• Vorticity Equation

Documents

Advanced SynopticM. D. Eastin Dynamic Meteorology: A Review