PV Thinking
CH 06
» 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 understandthe dynamics.
» In certain circumstances the approach can be generalized.
What is PV thinking?
» Davies and Emanuel (1991)
Y “ … a proper integration of the equations of motion is not synonymous with a conceptual grasp of the
phenomena being predicted”
» Answer
Y “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?
Vorticity – Streamfunction method
Assumptions: homogeneous, two-dimensional, inviscid, non-rotating flow.
D 0
Dt ζ =
v u
x y
∂ ∂
ζ = −
∂ ∂
D DT t
≡ ∂ + ⋅∇
∂ u
u v
x y 0
∂ +∂ =
∂ ∂
vorticity material
derivative
continuity u v
y x
∂ψ ∂ψ
= − =
∂ ∂
∇ ψ = ζ2
1
2 conservation
invertibility
Rotating flow on a β-plane
Assumptions: homogeneous, two-dimensional, inviscid, rotating flow on a β-plane.
Dq 0 Dt =
q= ζ +f
u v
x y 0
∂ +∂ =
∂ ∂
absolute vorticity
continuity u v
y x
∂ψ ∂ψ
= − =
∂ ∂
∇ ψ = ζ2
1
2 conservation
invertibility
Tropical cyclone thought experiment
f ζ f
ζ
. .
10oN 40oN
cyclonic asymmetry
anticyclonic asymmetry
«
Tropical cyclone thought experiment
. .
10oN 40oN
Asymmetries induce a poleward flow across
the vortex centre
The symmetric vortex rotates the asymmetry
Tropical cyclone thought experiment
.
.
10oN 40oN
The symmetric vortex rotates the asymmetry
The asymmetric flow across the symmetric vortex advects the the vortex northwestwards
Relative vorticity Streamfunction
Relative vorticity Streamfunction
Relative vorticity Streamfunction
Relative vorticity Absolute vorticity
Divergent flow, variable depth
Assumptions: homogeneous, divergent, inviscid, variable depth h(x,y,t).
Dq 0 Dt = q f
h
=ζ +
Dh h Dt = − ∇ ⋅u absolute
vorticity continuity
1 Conservation
No invertibility!
h(x,y,t)
Except for QG-flow
Quasi-geostrophic motion
Assumptions: stratified (Boussinesq), rotating, three- dimensional, adiabatic, inviscid, flow.
Dq 0 Dt =
2
q f 2
z
= ζ + + ε∂ ψ
∂ g
D DT t
≡ ∂ + ⋅∇
∂ u
g 0
∇⋅u = potential
vorticity
material derivative
continuity g
o
u 1 p
= f ∧ ∇ = ∧ ∇ψ
ρ k k
1
2 Conservation
Invertibility
2 2
h f ∂ ψ2 q
∇ ψ + + ε =
∂
f2
ε =
General adiabatic motion of a rotating stratified fluid
Assumptions: compressible, stratified, rotating, three- dimensional, adiabatic, inviscid, flow.
DP 0 Dt = P=1( + ⋅∇θ)
ρ ω f D
DT t
≡ ∂ + ⋅∇
∂ u ( ) 0
t
∂ρ+ ∇⋅ ρ =
∂ u
Ertel potential
vorticity
material derivative continuity
Conservation 1
No invertibility! unless …flow balanced
Isentropic coordinates
Assumptions: compressible, stratified, rotating, three- dimensional, adiabatic, inviscid, flow.
DP 0 Dt =
P g( f ) 1( f )
p z
∂θ ∂θ
= − ζ + = ζ +
∂ ρ ∂
D u v
DT t x y
∂ ∂ ∂
≡ + +
∂ ∂ ∂
Ertel potential vorticity material derivative Conservation
» 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 θ, and
iii. Solve the inversion principle globally, with proper attention to boundary conditions.
Formulation of an invertibility principle
Diabatic and frictional effects
PV is no longer materially conserved:
a
DP 1 1
Dt 1
= ⋅∇θ + ⋅∇θ
ρ ρ
= − ∇ ⋅ ρ
K Y
ζ ζa= absolute vorticity
= absolute vorticity K = ∇ ∧F
= −θ + ∇θ ∧a
Y ζ K
V S a
d PdV ( ) ˆdS
dt
∫
ρ =∫
θ + θζ K ⋅n Vnˆ
d
∫
ρPdV=0 Frictionless, adiabaticθ
Physical interpretation
a
1 ⋅∇θ ρζ
1 ⋅∇θ ρ
= ∇ ∧ K
K F
Diabatic term
Friction term
ζa ζa
«
«
heat cool
. .
∇ ∧F
F
» Consider the case of an axisymmetric vortex with tangential velocity distribution v(r).
» Assume that a linear frictional force F = -μv(r)acts at the ground z = 0.
» Then K = -μζk, where ζis the vertical component of relative vorticity.
» Then PV is destroyed at the rate Example
z 0
1 1
z =
⎛∂θ⎞
⋅∇θ = − μζ⎜ ⎟
ρK ρ ⎝∂ ⎠
