PV Thinking. What is PV thinking?
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1 PV Thinking CH 06 What is PV thinking? Two main approaches to solving fluid flow problems:. We can integrate the momentum, continuity and thermodynamic equations (the primitive equations) directly.. In certain cases we can use the streamfunction formulation Often () is more insightful than (). The streamfunction approach can provide a neat conceptual framework in which to understand the dynamics. In certain circumstances the approach can be generalized.
2 How can PV thinking help the forecaster? Davies and Emanuel (99) 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. Vorticity Streamfunction method Assumptions: homogeneous, two-dimensional, inviscid, non-rotating flow. conservation Dζ = 0 Dt v u ζ= x y material derivative D DT + u t u v + = 0 ψ ψ continuity u = v = x y y x invertibility ψ= ζ
3 Rotating flow on a β-plane Assumptions: homogeneous, two-dimensional, inviscid, rotating flow on a β-plane. conservation absolute q Dq Dt = 0 =ζ+ f u v + = 0 ψ ψ continuity u = v = x y y x invertibility ψ= ζ Tropical cyclone thought experiment 40 o N cyclonic asymmetry f.. ζ f ζ 0 o N anticyclonic asymmetry 3
4 Tropical cyclone thought experiment 40 o N.. Asymmetries induce a poleward flow across the vortex centre The symmetric vortex rotates the asymmetry 0 o N Tropical cyclone thought experiment 40 o N. The symmetric vortex rotates the asymmetry 0 o N. The asymmetric flow across the symmetric vortex advects the the vortex northwestwards 4
5 Relative Streamfunction Relative Streamfunction 5
6 Relative Streamfunction Relative Absolute 6
7 Divergent flow, variable depth Assumptions: homogeneous, divergent, inviscid, variable depth h(x,y,t). h(x,y,t) Conservation absolute continuity Dq 0 Dt = ζ + f q = h Dh h Dt = u No invertibility! Except for QG-flow Quasi-geostrophic motion Assumptions: stratified (Boussinesq), rotating, threedimensional, adiabatic, inviscid, flow. Conservation Dq Dt = 0 potential continuity Invertibility ψ material D q =ζ+ f +ε + u g z derivative DT t ug = 0 ψ h f u ψ+ +ε = q z g = p= ψ ρf k k o f ε= N 7
8 General adiabatic motion of a rotating stratified fluid Assumptions: compressible, stratified, rotating, threedimensional, adiabatic, inviscid, flow. Conservation Ertel potential DP 0 Dt = P = ( ω + f) θ ρ material derivative D DT + u t continuity ρ + ( ρ u) = 0 t No invertibility! unless flow balanced Isentropic coordinates Assumptions: compressible, stratified, rotating, threedimensional, adiabatic, inviscid, flow. Conservation Ertel potential material derivative DP 0 Dt = θ θ P= g( ζ+ f) = ( ζ+ f) p ρ z D + u + v DT t x y 8
9 Formulation of an invertibility principle 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. Diabatic and frictional effects PV is no longer materially conserved: DP = ζa θ + K θ Dt ρ ρ = Y ρ V ˆn d dt Frictionless, adiabatic ζ a = absolute θ = absolute K= F Y= θ ζ + θ K ρ = θ ˆ a +θ V ζ K n S a PdV ( ) ds d dt V ρ PdV = 0 9
10 Physical interpretation Diabatic term ζ ρ a θ ζ a heat ζ a cool Friction term K θ ρ K = F F.. F Example 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 ζ is the vertical component of relative. Then PV is destroyed at the rate θ K θ= μζ ρ ρ z z = 0 0
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