reinisch_asd_85.5151 vorticity in terms of horizontal circulation

Reinisch_ASD_85.515 1

Vorticity in terms of horizontal circulation

0 0

. If then

d d × d (Stokes' theorem)

d × dA

dlim lim 4.8A A

w d dx dy

dA A

C

A A

C

A

U V k l i j

V l U l U A

V l U k

V l


4.2.1 Vorticity in Natural Coordinates

0

It is convenient to express the vertical component of the

relative vorticity in natural cooridinates. Consider the

circulation around the tiny loop shown in Fig. 4.5. Then

lim . Now:C

C V s d sn s

,

the two segments along the n direction cancel.

and

where is the radius of curvature of the streamline.

and

ss

s

s s

VV n

n

VC V d s n

nn s

d s n R s d sR

R

V V VC n s

R n R

4.9V

n

Rs

V

n


Shear and curvature vorticity

The vorticity has two terms, shear vorticity ,

and curvature vorticity . See examples in Fig. 4.6 .s

V

nV

R


4.3 Potential Vorticity

1

Assume adiabatic flow on a surface of

. The density as

function of p is given by the Poisson equation:

1

constant potential

temperature (isentropic surface)

p p

p

R c R c

s

s

R c

pp pTp R p

pR

1.

Therefore 0 and from 4.3

0 (Kelvin's circulation theorem), or from 4.3

. But for horizontal relative circulation:

4.11

p v p pR c c c R cs s

a

a

p p p pR

dp

DC

DtC C f A const

CC A f A const

A


Ertel’s Potential Vorticity EquationAssume an air parcel extending from to + moves along

an isentropic surface =const. The mass of the parcel is

is conserved following

the motion, therefore A

pM V z A A

g

Mg Mg

p p

for constant . Substitute into 4.11 :

4.12

P is Ertel's potential vorticity in isentropic coordinates.

When using 4.11 , we assumed that the parcel moves

approximatel

const gp

P f g constp

y horizontally. The -sign makes P positive in NH.


Applications of Kelvin’s TheoremThe large-scale motion of the atmosphere is, to first order, adiabatic.

Conservation of the potential vorticity is a very useful constraint

discribing air movements. For example, flow over a large mountain

barrier when is large. p

0 0

0 0 0 0 0

0 0

But first consider for simple zonal flow at x , andp

, 0. Then . Since is constant, the value

downstream , . ., . For the flow to curve northward

downstream,

const y

x y f f

f f i e f f

must be < 0 (since 2 sin ) to assure conservation of .

This is the case for easterly flow (see Fig. 4.8 for NH).

Southward curvature requires >0. Again this is the case for easterly flow.

Westerly

f

flow must be purely zonal!


Westerly and easterly flows are discusssed in Figures 4.9 and 4.10.


Flow over mountain barrier

Now consider the case where changes following the motion. In this

case the potential vorticity P is conserved. To study adiabatic flow over a

mountain barrier we consider a homogeneous incompress

p

ible fluid.

For such fluid 4.5 becomes

1, since 0.

With .

From

4.13

dpC f A const dp

C A f A const

M constM A z A z

z

f z const


East/West Flow Asymmetry

0

0

Consider a vertical air column between the potential surfaces (ground)

and . Westerly adiabatic flow over a ridge will produce a decrease

in z Fig. 4.9 . To satisfy 4.13 , must decrease. Upstr

eam, before the

barrier, z streaches, so must increase from its value further upstream.

If the initial circulation upstream is = 0 (steady westerly flow), the flow

turns northward (cyclonic flow) b

efore reaching the barrier creating a ridge,

then southward over the barrier forming a trough, then northward on the

east side of the barrier (Fig. 9).

Very different behavior for eastward flow, see Fig. 4.10. Discuss.


East/West Flow Asymmetry


East/West Flow Over Large Mountain Summary

• Steady westerly flow results in cyclonic flow immediately east of the barrier, followed by ridges and troughs downstream

• Steady easterly flow creates one trough at the location of the barrier, followed by straight easterly flow downstream


4.4 Vorticity Equation

DIn general 0, i.e., the flow is adiabatic. How does the absolute

DtD

vorticity change when moving with the flow? ?Dt

To find out we start with the approximate horizontal momentum equations,

2.

f

Vnot

24 and 2.24 :

1 1, or

14.14

14.15

Du p u pfv fv

Dt x t x

u u u u pu v w fv

t x y z x

v v v v pu v w fu

t x y z y

U u


2 2 2

2

2

2

Differentiate 4.14 and 4.15 :

1

1

1 11

y x

u u u u pu v w fv

t x y z x

v v v v pu v w fu

t x y z y

uu u u u u v u w u v fy

u v w f vt x y y z y y x y y y z y y

p p

x y y x

v

2 2 2

2

2

2

1 12

v v v v v v v w v u fx u v w f ut x x y x z x x x y x z x x

p p

x y x y


Total derivative of absolute vorticity

2

2 1 and considering that = :

14.16

Considering that 0, and 0 gives the

v u

x y

u v w v w u dfu v w f v

t x y z x y x z y z dy

p p

x y y x

f Df f fu f vx Dt t y

U

2

vorticity eq

:

1

uation

D u v w v w u p pf f

Dt x y x z y z x y y x

divergence tilt solenoid


4.4.3 Scale Analysis of the Vorticity Equation

2

1D u v w v w u p pf f

Dt x y x z y z x y y x

reinisch_asd_85.5151 vorticity in terms of horizontal circulation

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