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 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.

How can PV thinking help the forecaster? 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.

Vorticity – Streamfunction method Assumptions: homogeneous, two-dimensional, inviscid, non-rotating flow. 1 conservation material derivative vorticity continuity invertibility 2

Rotating flow on a b-plane Assumptions: homogeneous, two-dimensional, inviscid, rotating flow on a b-plane. 1 conservation absolute vorticity continuity invertibility 2

Tropical cyclone thought experiment cyclonic asymmetry . . f  z f  z    anticyclonic asymmetry 10oN

Tropical cyclone thought experiment Asymmetries induce a poleward flow across the vortex centre . . The symmetric vortex rotates the asymmetry 10oN

Tropical cyclone thought experiment The symmetric vortex rotates the asymmetry . . The asymmetric flow across the symmetric vortex advects the the vortex northwestwards 10oN

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). h(x,y,t) 1 Conservation absolute vorticity continuity No invertibility! Except for QG-flow

Quasi-geostrophic motion Assumptions: stratified (Boussinesq), rotating, three-dimensional, adiabatic, inviscid, flow. 1 Conservation potential vorticity material derivative continuity 2 Invertibility

General adiabatic motion of a rotating stratified fluid Assumptions: compressible, stratified, rotating, three-dimensional, adiabatic, inviscid, flow. 1 Conservation Ertel potential vorticity material derivative continuity No invertibility! unless …flow balanced

Isentropic coordinates Assumptions: compressible, stratified, rotating, three-dimensional, adiabatic, inviscid, flow. Conservation Ertel potential vorticity material derivative

Formulation of an invertibility principle To formulate an invertibility principle one must: Specify some kind of balance condition, the simplest, but least accurate option being ordinary quasi-geostrophic balance, Specify some sort of reference state, expressing the mass distribution of q, and Solve the inversion principle globally, with proper attention to boundary conditions.

Diabatic and frictional effects PV is no longer materially conserved: za = absolute vorticity . q = absolute vorticity K =   F Suppose each string of particles along lines of latitude (y = constant) is given a sinusoidal meridional (north-south) displacement (x), as shown. The corresponding perturbation in potential vorticity for each string, q(x), is as shown also. From this, and using the string analogy (keeping in mind that a positive force is analogous to a negative q) we deduce the streamfunction distribution and finally, using the second component of (6.3), v = x, we infer the meridional velocity component to have the variation shown. Comparison of the meridional displacement with the meridional velocity shows: (I) that the disturbance propagates westwards, and (ii) since the longitudes of displacement maxima are places of zero velocity, the wave amplitude does not vary with time; in particular, the wave is stable. V Frictionless, adiabatic

Physical interpretation Diabatic term za za    heat  cool Friction term . .

Example Consider the case of an axisymmetric vortex with tangential velocity distribution v(r). Assume that a linear frictional force F = - mv(r) acts at the ground z = 0. Then K = -mzk, where z is the vertical component of relative vorticity. Then PV is destroyed at the rate

The End