Q-G vorticity equation Q-G thermodynamic equation We now have two equations in two unknowns, and We will solve these to find an equation for , the.

p

ffVftf

g

0

2

0

2

0

11

PV

Ptg

Q-G vorticity equation

Q-G thermodynamic equation

We now have two equations in two unknowns, and

We will solve these to find an equation for , the vertical motion in pressure

coordinates, and for , the change of geopotential height with time.t

Wave-cyclones from the Q-G perspective

p

ffVftf

g

0

2

0

2

0

11

pV

ptg

1

2

Derive the Q-G height tendency equation (equation for height changes of a pressure surface)

1. Assume is constant2. Change order of differentiation on left side of (2)

3. Multiply (2) by and (1) by f0

4. Differentiate (2) with respect to p4. Add to resultant equation to (1)

0f

p

Vp

ff

fVf

tp

fgg

202

002

2202 1

Both the QG omega equation and the QG height tendency equation can be derivedIncluding the friction term and the diabatic heating term. We will not do this here,But I will simply show the result.

p

Vff

Vp

f

p

fgg

22

0

02

2202 11

dt

dQ

Cp

RK

p

f

Pg

120

dt

dQ

cp

R

p

fKf

Pg

120

0

QG OMEGA EQUATION

QG HEIGHT TENDENCY EQUATION

P

Vp

ff

fVf

tp

fgg

202

002

2202 1

dt

dQ

cP

R

p

fKf

Pg

120

0

QG HEIGHT TENDENCY EQUATION

p

Vp

ff

fVf

tP

fgg

202

002

2202 1

First term is proportional tot

First term represents the rate at which geopotential height decreases with time

First term represents the rate at which geopotential height decreases with time

trough propagating500 mb maps -- 24 hours between panels

trough deepening

First term represents the rate at which geopotential height decreases with time

trough propagating500 mb maps -- 24 hours between panels

trough deepening

0t

0t

0t

0t

dt

dQ

cp

R

p

fKf

Pg

120

0

QG HEIGHT TENDENCY EQUATION

p

Vp

ff

fVf

tp

fgg

202

002

2202 1

The first term on the right side of the equation describes the propagation of the height field.

Propagation of the height field depends on :

the advection of the relative vorticity (spin imparted by shear and curvature)

and the advection of the planetary vorticity (spin imparted by the earth rotation)

20

1

f

f

A “battle” between advection of planetary vorticity and relative vorticity occurs in ridge-trough systems.

High Latitudes

Low Latitudes

20

1

f

f

Height rises will occurdue to the advectionof relative vorticity

Height falls will occurdue to the advectionof planetary vorticity

Height falls will occurdue to the advectionof relative vorticity

Height rises will occurdue to the advectionof planetary vorticity

Which process will win the battle?

Short waves in flow

Due to strong shear and sharp curvature changeRelative vorticity changes substantially from ridge to trough

f doesn’t change much – little deviation in latitude

Advection of absolute vorticity is dominated by advection of relative vorticity:

The troughs and ridges will propagate rapidly eastward

Long waves in flow

f changes significantly – large deviation in latitude

weak shear and wide curvature:relative vorticity changes small

from ridge to trough

As a result, advection of absolute vorticity is nearly equal to zero

Long waves are stationary, or propagate very slowly eastward (g > f) or westward (g < f)

Note the speed of propagation of the waves on these maps: the shorter thewavelength becomes, the faster the wave propagates

00 Hours +12 Hours

+24 Hours +36 Hours

dt

dQ

cp

R

p

fKf

Pg

120

0

QG HEIGHT TENDENCY EQUATION

P

Vp

ff

fVf

tp

fgg

202

002

2202 1

We will examine these two terms together:

The vertical derivative of thickness advection(Differential thickness advection)

The vertical derivative of diabatic heating(Differential diabatic heating)

Za

Zb

Z

A

B

Za

Zb

Z

A

B

Za

Zb

ZA

B

Differential thickness (mean layer temperature) advectionor Diabatic Heating

Suppose we have a layer of air bounded by height Za and Zb with level Z in the

middle.

Warm advection ordiabatic heating in A

causes layer to expand

Cold advection ordiabatic heating in B

causes layer to contract

Height Z falls to loweraltitude

Cold advection or diabatic cooling in A

causes layer to contract

Warm advection or diabatic heating in B

causes layer to expand

Height Z rises to higheraltitude

In the real atmosphere

warm and cold advection and diabatic heating and coolingdecrease with height

and are always strongest in the lower atmosphere

850 mb 2 Mar 03 00 UTC 700 mb 2 Mar 03 00 UTC 500 mb 2 Mar 03 00 UTC

Red circles: Strong warm advection pattern at 850, weaker at 700, very weak at 500 mbBlue circles: Strong cold advection pattern at 850, weaker at 700, very weak at 500 mb

dt

dQ

cp

R

p

fKf

Pg

120

0

QG HEIGHT TENDENCY EQUATION

p

Vp

ff

fVf

tp

fgg

202

002

2202 1

Summary

Cold advection in the lower atmosphere will produce height falls amplifying trough aloft

Warm advection in the lower atmosphere will produce height risesamplifying ridge aloft

dt

dQ

CP

R

P

fKf

Pg

120

0

QG HEIGHT TENDENCY EQUATION

p

Vp

ff

fVf

tp

fgg

202

002

2202 1

Surface anticyclone (term outlined > 0)

Divergence, descending air and height falls

Surface cyclone (term outlined < 0)

Convergence, ascending air and height rises