Chapter 7: Circulation and Vorticity Circulation

€

C = u • d s ∫

Integration is performed in a counterclockwise direction

C is positive for counterclockwise flow!!! Kelvin’s Circulation Theorem The rate of change of circulation can be expressed as:

€

DCDt

=D u Dt∫ • d s

= −1ρ

∫ ∂p∂s

ds +∂Φ∂s

ds∫ + friction∫

What is the magnitude of each term in this equation? For a barotropic fluid (density is a function only of pressure):

€

−1

ρ p( )∫ ∂p∂sds = −

1ρ p( )

dp∫

= −1

ρ p2( )−

1ρ p1( )

&

' (

)

* +

= 0 around a closed circuit

Geopotential term:

€

∂Φ∂s∫ ds = dΦ∫

= Φ2 −Φ1[ ]= 0 around a closed circuit

In a frictionless (inviscid) flow the friction term is also zero. In a barotropic, inviscid fluid the circulation is constant!!!

€

DCDt

= 0

Bjerknes’ Circulation theorem Changes in circulation can arise due to friction or baroclinicity. The sea breeze circulation

How does the temperature over the land and over the ocean vary during the course of a day? What impact does this have on the thickness of an atmospheric column? How does the atmosphere respond to this horizontal variation in thickness? Example: Calculating the circulation associated with a sea breeze

Relative circulation Circulation in the atmosphere arises due to our rotating frame of reference.

€

Cabsolute = Cearth + Crelative where,

€

Cearth = u earth • d s ∫

≈ RΩ×2πRsinφ

Cearth ≈ 2πΩR2 sinφ

If Cabsolute is conserved what does this imply about changes in Crelative for meridional flow? Vorticity

€

ζ =∂C∂A

For solid body rotation:

€

ζ =δCδA

=2πδrVπδr 2

=2Vδr

= 2ω

The earth undergoes solid body rotation with an angular velocity of

€

ω =Ωsinφ , so:

€

ζearth = 2Ωsinφ = f

What is the sign of ζearth in the Northern and Southern hemispheres?

The circulation around ABCD can be calculated as:

€

C = u • d s = udx + vdy( )∫∫

€

δC = CAB + CBC + CCD + CDA

= uδx + v+∂v∂xδx

$

% &

'

( ) δy− u +

∂u∂yδy

$

% &

'

( ) δx − vδy

=∂v∂xδxδy− ∂u

∂yδyδx

=∂v∂x

−∂u∂y

$

% &

'

( ) δA

Then the relative vorticity, ζ, is given by:

€

ζ =δCδA

€

ζ =∂v∂x

−∂u∂y

What is the sign of ζ for clockwise and counterclockwise flow? What does this imply about the sign of ζ for flow around low and high pressure centers? Example: Calculation of relative vorticity from a weather map

Conservation of Potential Vorticity What conditions were required for constant circulation according to Kelvin’s circulation theorem? On a constant potential temperature, θ, surface:

€

ρ =pRT

=pRθ

p0p

$

% &

'

( )

Rd cp

= pcv cp p0Rd cp

Rθ

$

% & &

'

( ) )

This is analogous to a barotropic fluid. Therefore, on a constant potential temperature surface the pressure gradient term is zero and Kelvin’s circulation theorem is satisfied.

€

DCabsolute

Dt θ

= 0

This implies that:

€

Cabsolute = ζθ + f( )dAA∫∫

ζθ + f = limδA→0

Cabsolute

δA

€

δA ζθ + f( ) = constant Consider an air parcel that is confined between two potential temperature surfaces,

€

θ and

€

θ + δθ , separated by pressure interval

€

δp. The motion of this air parcel will be adiabatic. The mass of the parcel is given by:

€

δM = ρδzδA = − δp g( )δA and must be conserved following the motion.

This gives:

€

δA = −δMgδp

≡ −δMgδp

×δθδθ

δA = −δMgδθ

δθδp'

( )

*

+ ,

€

δA = −constant × gδθδp

Combining this result with

€

δA ζθ + f( ) = constant gives an expression for Rossby-Ertel potential vorticity, P:

€

constant ×−gδθδp

&

' (

)

* + ζθ + f( ) = constant

∴P = −gδθδp

ζθ + f( ) = constant

What does the

€

δθ δp term represent? Potential vorticity depends on the depth of the fluid and the absolute vorticity.

Example: Conservation of potential vorticity and flow over the Rocky Mountains

Air column depth

Increase

€

δp ↑ Decrease

€

δp ↓ Increase

€

δp ↑ Decrease (return to

original value) Change in

€

δθδp

Decrease

Increase

Decrease

Increase (return to

original value)

Change in

€

ζ + f

Increase

Decrease

Increase

Decrease Sign of ζ Positive Negative Positive Negative Resulting

motion northward southward lee side trough southward

Change in

€

f

Increase

Decrease

Increase

Decrease

For westerly flow across a mountain range a lee wave will form downstream of the mountain. Stretching a column of the atmosphere results in generation of cyclonic vorticity. Shrinking of a column of the atmosphere results in generation of anticylonic vorticity.

The Vorticity Equation Using Navier-Stokes equations scaled for mid-latitude weather systems we can derive an equation for the time rate of change of vorticity.

€

Dh u h

Dt= −

1ρ∂pd

∂x i + ∂pd

∂y j

%

& '

(

) * − f k × u h

⇒Dh uDt

= −1ρ∂pd

∂x+ fv (1) Dhv

Dt= −

1ρ∂pd

∂y− fu (2)

∂∂y

(1)⇒ Dh

Dt∂u∂y%

& '

(

) * = −

1ρ∂ 2 pd

∂x∂y+ f ∂v

∂y+ v∂f

∂y(1 - )

∂∂x

(2)⇒ Dh

Dt∂v∂x%

& '

(

) * = −

1ρ∂ 2 pd

∂x∂y− f ∂u

∂x− u∂f

∂x(2 - )

(2 - ) − (1 - ) ⇒ Dh

Dt∂v∂x

−∂u∂y

%

& '

(

) * = − f ∂u

∂x+∂v∂y

%

& '

(

) * − u∂f

∂x− v∂f

∂y

∴DhζDt

+ u∂f∂x

+ v∂f∂y

= − f ∂u∂x

+∂v∂y

%

& '

(

) *

€

∂ζ∂t

+ u∂ζ∂x

+ v∂ζ∂y

+ u∂f∂x

+ v∂f∂y

= − f ∂u∂x

+∂v∂y

%

& '

(

) *

What causes the relative vorticity, at a fixed location, to change in time? How does vorticity change for a non-divergent flow? For quasi-geostrophic flow:

€

Dg

Dtζg + f( ) = − f ∂ua

∂x+∂va∂y

%

& '

(

) *

€

Dg

Dtζg + f( ) = f ∂w

∂z

What does this equation tell us about changes in relative vorticity in a quasi-geostrophic framework?