The Quasi-Geostrophic Omega Equation (without friction and diabatic terms) We will now develop the Trenberth (1978)* modification to the QG Omega equation.

p

VfVp

fp

f ggg

202

2

02

The Quasi-Geostrophic Omega Equation (without friction and diabatic terms)

We will now develop the Trenberth (1978)* modification to the QG Omega equation

THE TRENBERTH (1978) INTERPRETATION

*Trenberth, K.E., 1978: On the Interpretation of the Diagnostic Quasi-Geostrophic Omega Equation. Mon. Wea. Rev., 106, 131–137

Trenberth (1978) argued that carrying out all of the derivatives on the RHS on the Equation could simplify the forcing function for .

PROBLEM: TERM 1 and 2 on the RHS are often large and opposite leading to ambiguity about the sign and magnitude of when analyzing weather maps

p

VfVp

fp

f ggg

202

2

02

py

vpx

uy

fv

x

fu

pf

p

fgg

gg

gg

222

02

202

QG OMEGA EQUATION:

EXPAND THE ADVECTION TERMS:

yfug

1xf

vg

1 2

0

1

fg

USE THE EXPRESSIONS FOR THE GEOSTROPHIC WIND AND GEOSTROPHIC VORTICITY:

To Get:

pyxpxyf

ffyxf

ffxyfp

fp

f

22

22

0

2

002

202 11111

pyxpxyf

ffyxf

ffxyfp

fp

f

22

22

0

2

002

202 11111

x

B

y

A

y

B

x

ABAJ ,

EXPAND ALL THE DERIVATIVESTHEN USE THE JACOBIAN OPERATOR TO SIMPLIFY NOTATION

RESULT:

212

202 1

FFfp

f

pyy

Jpxx

Jp

Jp

JF22

221 ,,2,,

p

Jffp

Jp

JF 2

02

2 ,,,

pyy

Jpxx

Jp

Jp

JF22

221 ,,2,,

p

Jffp

Jp

JF 2

02

2 ,,,

Same term:opposite sign

DeformationTerms in Sutcliff eqn

Trenberth: Ignore deformation terms (removes frontogenetic effects)

212

202 1

FFfp

f

Opposite term:opposite sign = same term

02

21 ,,211

ffp

Jp

Jf

FFf

02

21 ,,21

ffp

Jp

Jf

FFf

Approximate last term = 2 last term

02

21 ,,21

ffp

Jp

Jf

FFf

Expand Jacobian terms:

fp

Vf

p

fg

g

02

202 2

This result says that large scale vertical motions can be diagnosed by Examining the advection of absolute vorticity by the thermal wind

fVVVVf gg 000

RECALL SUTCLIFF’S EQUATION

SAME INTERPRETATION!!

The Geostrophic paradox

Confluent geostrophic flow will tighten

temperature gradient, leading to an increase in

shear via the thermal wind relationship……..

……but advection of geostrophic momentum

by geostrophic wind decreases the vertical shear in the column

so…geostrophic flow destroysgeostrophic balance!

00

aggg ufvVt

0

pV

t g

0

gg vVt

0

pV

t g

000

y

vv

x

vu

t

v

pfvV

tpf g

gg

gg

gg

y momentum equation (QG)

Thermodynamic energy equation (QG)

For the moment, let’s ignore the ageostrophy (no uag and no )

The geostrophic paradox: a mathematical interpretation

Take vertical derivative of first equation

Let’s look at this equation

000

y

vv

x

vu

t

v

pfvV

tpf g

gg

gg

gg

Expand the derivative:

0000

y

v

p

v

x

v

p

uf

p

vfV

tvV

tpf ggggg

ggg

Substitute using the thermal wind relationship:

pxp

vf g

2

0

000

px

V

p

vfV

tvV

tpf gg

ggg

to get:

pyp

uf g

2

0

Remember equation in blue box

00

aggg ufvVt

0

pV

t g

0

gg vVt

0

pV

t g

y momentum equation (QG)

Thermodynamic energy equation (QG)

For the moment, let’s ignore the ageostrophy (no uag and no )

The geostrophic paradox: a mathematical interpretation

Take x derivative of second equation

Now let’s look at this equation

0

py

vpx

uptxp

Vtx ggg

0

py

vpx

uptxp

Vtx ggg

Expand the derivative and use vector notation:

02

px

V

pxV

tpV

txg

gg

Now recall first saved equation:

000

px

V

p

vfV

tvV

tpf gg

ggg

Let’s take these two blue boxed equations and compare them…..

px

V

pxV

tg

g

2

px

V

p

vfV

tgg

g

0

pxp

vf g

2

0

Following the geostrophic wind themagnitude of the temperature gradientand the vertical shear have opposite

Tendencies

TIGHTENING THE TEMPERATURE GRADIENT WILL REDUCE THE SHEAR!

thermal windbalance

The Geostrophic paradox:

RESOLUTION

A separate “ageostrophiccirculation” must exist thatrestores geostrophic balance thatsimultaneously:

1) Decreases the magnitude of the horizontal temperature gradient

2) Increases the vertical shear

The Q-Vector interpretation of the Q-G Omega Equation(Hoskins et al. 1978)

px

V

pxV

tg

g

2

px

V

p

vfV

tgg

g

0

From consideration of the geostrophic wind, we derived

these equations:

Let’s denote the term on the RHS:

px

VQ g

1

00

aggg ufvVt

0

pV

t g

y momentum equation (QG)

Thermodynamic energy equation (QG)

If we start with our original equations, below,

and perform the same operations as before, but with the ageostrophic terms included….

We arrive at:

02010

p

ufQ

p

vfV

tagg

g

01

2

xQ

pxV

t g

01

2

Qpx

Vt g

010

Qp

vfV

tg

g

With ageostrophic terms

Only geostrophic terms

Note that the additional terms represent the ageostrophic circulation that works to reestablish geostrophic balance as air accelerates in unbalanced flow.

02010

p

ufQ

p

vfV

tagg

g

01

2

xQ

pxV

t g

With ageostrophic terms

Let’s multiply the bottom equation by -1 and add it to the top equation, recalling

that pyp

uf g

2

0

p

uf

xQ ag

2012

Let’s do the same operations with the x equation of motion and the thermodynamicequation. If we do, we find that:

p

vf

yQ ag

2022

Let’s do the same operations with the x equation of motion and the thermodynamicequation.

p

vf

yQ ag

2022

02020

p

vfQ

p

ufV

tagg

g

02

2

yQ

pxV

t g

py

VQ g

2Where:

and:

p

vf

yQ ag

2022

p

uf

xQ ag

2012

A

B

Takey

B

x

A

y

v

x

u

pf

yxy

Q

x

Q agag202

2

2

2212

2

2

02

2

2

2212

pf

yxy

Q

x

Q

Substitute continuity equation

And use vector notation to get:

y

Q

x

Q

pf 21

2

2

02 2

p

Vff

Vp

fp

fgg

22

002

2202 1

COMPARE THIS EQUATION WITH THE TRADITIONAL QG EQUATION!

Qp

f

22

2

02

We can write the Q-vector form of the QG equation as:

Where the components of the Q vector are

jpy

Vi

px

VQ gg ˆ,ˆ

jpy

Vi

px

VQ gg ˆ,ˆ

Using the hydrostatic relationship, we can write Q more simply as:

jTy

ViT

x

V

p

RQ gg ˆ,ˆ

jy

T

y

v

x

T

y

ui

y

T

x

v

x

T

x

u

p

RQ gggg ˆˆ

or in scalar notation as

02 Q

First note that if the Q vector is convergent

0 Q

Qp

f

22

2

02

02

2

02

p

f

0

Therefore air is rising when the Q vector is convergent

jy

T

y

v

x

T

y

ui

y

T

x

v

x

T

x

u

p

RQ gggg ˆˆ

Let’s go back to our jet entrance region

Note that there is no in this particular jety

T

jy

ui

y

v

x

T

p

Rj

x

T

y

ui

x

T

x

u

p

RQ gggg ˆˆˆ,ˆ

y

Vk

x

T

p

RQ g

ˆ

Q c

onve

rgen

ce

Q d

iver

genc

e

The Q vectors capture the sense of the ageostrophic circulation and allow us to see where the rising motion is occurring

y

Vk

x

T

p

RQ g

ˆ

Q c

onve

rgen

ce

Q d

iver

genc

e

The Q vectors capture the sense of the ageostrophic circulation and allow us to see where the rising motion is occurring

Q vectors diagnose a thermally direct circulation

Adiabatic cooling of rising warm air Adiabatic warming of sinking cold air

Counteracts the tendency of the geostrophictemperature advection in confluent flow

Under influence of Coriolis force, horizontalbranches tend to increase shear

Counteracts the tendency of the geostrophicMomentum advection in the confluent flow

Resolution of the Geostrophic Paradox

A natural coordinate version of the Q vector(Sanders and Hoskins 1990)

jy

T

y

v

x

T

y

ui

y

T

x

v

x

T

x

u

p

RQ gggg ˆˆ

Consider a zonally oriented confluent

entrance region of a jet where 0x

T

jy

vi

x

v

y

T

p

RQ gg ˆˆ

jx

ui

x

v

y

T

p

RQ gg ˆˆ

Use non-divergence of geostrophic wind

or

x

Vk

y

T

p

RQ g

ˆ

A natural coordinate version of the Q vector(Sanders and Hoskins 1990)

jy

T

y

v

x

T

y

ui

y

T

x

v

x

T

x

u

p

RQ gggg ˆˆ

Consider a meridionally oriented confluent

entrance region of a jet where 0y

T

jy

ui

x

u

x

T

p

RQ gg ˆˆ

jy

ui

y

v

x

T

p

RQ gg ˆˆ

Use non-divergence of geostrophic wind

or

y

Vk

x

T

p

RQ g

ˆ

A natural coordinate version of the Q vector(Sanders and Hoskins 1990)

Using these two expressions, let’s adoptA natural coordinate expression for Q

y

Vk

x

T

p

RQ g

ˆ

x

Vk

y

T

p

RQ g

ˆ

s

Vk

n

T

p

RQ g

ˆ

Adopt a coordinate system where is directed along the isotherms is directed normal to the isotherms

ns ˆ,ˆ sn

Q vector oriented perpendicular to the vector change in the geostrophic windalong the isotherms. Magnitude proportional to temperature gradient and inversely proportional to pressure.

Simple Application #1 Train of cyclones and anticyclones

At center of highs and lows:

Black arrows:

Gray arrows =

Bold arrows =

0gV

gV

s

Vg

s

Vk

n

T

p

RQ g

ˆ

sinkingmotion

risingmotion

Note also that because of divergence/convergence, train of cyclones and anticyclones propagates east along direction of thermal wind

Simple Application #2 Pure deformation flow with a temperature gradient

Along axis of dilitation

Black arrows:

Gray arrows =

Bold arrows =

gV

gV

s

Vg

s

Vk

n

T

p

RQ g

ˆ

sinkingmotion

risingmotion

Increases toward east

Simple Application #3 Homogeneous warm advection

No variation in

Black arrows:

Gray arrows =

Bold arrows =

gV

gV

0s

Vg

0ˆ

s

Vk

n

T

p

RQ g

along an isotherm

No heterogeneity in the warm advection field = No rising motion!

Note that the Q vector form of the QG -equation contains the deformation terms

(unlike the Sutcliff and Trenberth forms)

And combines the vorticity and thermal advection terms into a single diagnostic

(unlike the traditional QG -equation)

Sutcliff/Trenberth approximation Deformation term contribution to

The along and across-isentrope components of the Q vector

pv CC

p

p

fp

R/

00

f

p

Begin with the hydrostatic equation in potential temperature form

where:

And the definition of the Q vector:

jpy

Vi

px

VQ gg ˆ,ˆ

(which is constant on an isobaric surface)

Substituting:

jy

Vi

x

VfQ gg ˆ,ˆ

This expression is equivalent to:

gdt

dfQ

gdt

dfQ

The Q-vector describes the rate of change of the potential temperature alongThe direction of the geostrophic flow

Let’s consider separately the components of Q along and across the isentropes

isentropesalongQs

isentropesacrossQn

isentropesalongQs

isentropesacrossQn

nQ

Is parallel to and can only affect changes in the magnitude of

sQ

Is perpendicular to and can only affect changes in the direction of

nQQ

Q nn ˆ

sQkkQ

Q ss ˆˆˆ

sQnQQ sn ˆˆ

sQnQQ sn ˆˆ

Qp

f

22

2

02

Returning to QG equation

Components of vertical motion can be distributed in couplets across (transverse to)the thermal wind (mean isotherms) and along (shearwise) the thermal wind.

We will see later that the transverse component of Q is related to the dynamics offrontal zones.