PV Thinking

(1)

PV Thinking

(2)

 Two main approaches to solving fluid flow problems:

1. We can integrate the momentum, continuity and thermodynamic equations (the primitive equations) directly.

2. In certain cases we can use the vorticity – streamfunction formulation

 Often (2) is more insightful than (1).

 The vorticity – streamfunction approach can provide a neat conceptual framework in which to understand the dynamics.

 In certain circumstances the approach can be generalized.

What is PV thinking?

(3)

 Davies and Emanuel (1991)

 “ … a proper integration of the equations of motion is not synonymous with a conceptual grasp of the

phenomena being predicted”

 Answer

 “PV thinking” can provide the forecaster with a sound conceptual framework in which to interpret numerical analyses and prognoses.

How can PV thinking help the forecaster?

(4)

Vorticity – Streamfunction method

Assumptions: homogeneous, two-dimensional, inviscid, non-rotating flow.

D 0

Dt

 

v u

x y

 

  

 

D

DT t

   



u

u v

x y 0

   

 

vorticity material

derivative

continuity

u v

y x

 

  

 

   

2

1

2 conservation

invertibility

(5)

Rotating flow on a -plane

Assumptions: homogeneous, two-dimensional, inviscid, rotating flow on a -plane.

Dq 0 Dt 

q    f

u v

x y 0

   

 

absolute vorticity

continuity

u v

y x

 

  

 

1 conservation

(6)

Tropical cyclone thought experiment

f 

  f 



. .

10^oN 40^oN

cyclonic asymmetry

anticyclonic asymmetry





(7)

Tropical cyclone thought experiment

. .

10 N 40^oN

Asymmetries induce a poleward flow across

the vortex centre

The symmetric vortex rotates the asymmetry

(8)

Tropical cyclone thought experiment

.

10^oN 40^oN

The symmetric vortex rotates the asymmetry

The asymmetric flow across the symmetric vortex advects the the vortex northwestwards

(9)

Relative vorticity Streamfunction

(10)

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Relative vorticity Absolute vorticity

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Divergent flow, variable depth

Assumptions: homogeneous, divergent, inviscid, variable depth h(x,y,t).

Dq 0 Dt  q f

h

  

Dh     h

u absolute

vorticity

1 Conservation

h(x,y,t)

(14)

Quasi-geostrophic motion

Assumptions: stratified (Boussinesq), rotating, three- dimensional, adiabatic, inviscid, flow.

Dq 0 Dt 

2

q f

2

z

      



^g

D

DT t

   



u

g

0  

u



potential

vorticity

material derivative

continuity g

o

u 1 p

 f     



k k

1

2 Conservation

Invertibility

2

f  

2

q

     

_{ } ^f²

(15)

General adiabatic motion of a rotating stratified fluid

Assumptions: compressible, stratified, rotating, three- dimensional, adiabatic, inviscid, flow.

DP 0 Dt  P  1 (   )

 

f

D

DT t

   



u

( ) 0

t

     



u

Ertel potential

vorticity

material derivative

continuity

1 Conservation

(16)

Isentropic coordinates

Assumptions: compressible, stratified, rotating, three- dimensional, adiabatic, inviscid, flow.

DP 0 Dt 

P g( f ) 1 ( f )

p z

 

      

  

D u v

DT t x y

  

  

  

Ertel potential

vorticity

material derivative Conservation

(17)

 To formulate an invertibility principle one must:

i. Specify some kind of balance condition, the simplest, but least accurate option being ordinary quasi-geostrophic balance,

ii. Specify some sort of reference state, expressing the mass distribution of , and

iii. Solve the inversion principle globally, with proper attention to boundary conditions.

Formulation of an invertibility principle

(18)

Diabatic and frictional effects

PV is no longer materially conserved:

a

DP 1 1

Dt

1    

 

   



K Y



 ^^a = absolute vorticity

 = absolute vorticity .

K =   F

    

a

Y 



K

V S a

d PdV ( ) dS ˆ

dt       

^

^

^{K n}



V

n

ˆ

d 

(19)

Physical interpretation

a

1 

 



1 



  

K

K F

Diabatic term

Friction term

_a _a



 



heat cool

. .

 

F

(20)

 Consider the case of an axisymmetric vortex with tangential velocity distribution v(r).

 Assume that a linear frictional force F = - v(r) acts at the ground z = 0.

 Then K = -k, where  is the vertical component of relative vorticity.