BALANCED FLOW: EXAMPLES (PHH lecture 3) Potential Vorticity in the real atmosphere Need to introduce a new measure of the buoyancy Potential temperature θ In a compressible fluid, the relevant measure of buoyancy is the density that a particle would have if it were moved adiabatically to a reference pressure. For a gas this density is simply a function of the entropy, or alternatively, of the potential temperature θ = T( p /p ) κ the temperature that a parcel would have if it were moved adiabatically to a reference pressure. The vertical gradient of θ is a measure of the stratification. It is convenient to use θ as a vertical co-ordinate. In adiabatic flow (relevant for time scales of a few days) parcels do not cross θ surfaces the vertical velocity is zero. (κ is the ratio of specific heats) [For the ocean, density is a function of salinity as well as pressure and temperature; use potential density surfaces but see McDougall, 1984 ] Rossby Ertel potential vorticity In frictionless flow: PHH 3/1 P = ζ (a). χ ρ is conserved following the fluid motion ζ ( a) = u + 2Ω relative vorticity Ω = rotation of Earth χ is any quantity such that (1) Dχ = 0 (2) χ.( ρ p) = 0 Dt e.g. for dry atmosphere: ρ = ρ( p,θ), take χ = f (θ): P is then conserved in frictionless and adiabatic motion. For Boussinesq fluid: take χ = ρ or χ = ρ θ (potential density this is conventional large-scale oceanographic approach). PHH 3/2
Atmospheric potential vorticity: Cut-off cyclone 300K isentrope 20 25 September 1982 40 90 N 120 W 0 W (from Hoskins et al. 1985) Atmospheric potential vorticity: Blocking anticyclone 330K isentrope PHH 3 / 3 30 September 7 October 1982 30 80 N (from Hoskins et al. 1985) 60 W 60 E PHH 3 / 4
Simple axisymmetric circulations (from Hoskins et al. QJRMS 1985, calc. by Thorpe) CYCLONE (+ve potential vorticity anomaly) ANTICYCLONE ( ve potential vorticity anomaly) These potential vorticity anomalies might be thought of as arising from advection along isentropic surfaces (not across isentropic surfaces) Oceanic potential vorticity PHH 3 / 5 PV on σ θ = 25 (Talley, 1988) Tritium on σ θ = 23.9 (Fine et al., 1981) σ θ = 26.02 PV on σ θ = 26 PHH 3 / 6
Simple forced quasi-geostrophic flows Changes in PV due to given forcing e.g. symmetric buoyancy (or thermal) forcing mechanical forcing no variation in along stream direction, hence no PV advection Take x-independent for simplicity: assume given force ( y, z) exerted in x-direction and buoyancy forcing Q( y, z) q.-g. p.v. equation becomes 2 ψ t y 2 + 2 f 0 ψ z N 2 z = y + f 0 Q ρ 0 z (+ boundary conditions) Response: PHH 3 / 7 Take Q = 0 (i.e. no buoyancy forcing). Resulting ψ will generally be function of y, z, i.e. ψ z 0 Thus the response appears in the buoyancy field as well as the velocity field Correspondingly if = 0 (buoyancy forcing only) response appears in velocity field as well as buoyancy field If and Q are localised then the response tends to spread away from the forcing region and to be isotropic in co-ordinates scaled by Prandtl s ratio f 0 / N PHH 3 / 8
The ageostrophic (secondary) circulation Consider the ageostrophic circulation (0,v a, w a ) associated with such forcing. Define the stream function χ a for the ageostrophic circulation by v a = χ a z w a = χ a y It follows that 2 2 χ a y 2 + f 0 2 χ a N 2 z 2 = f 0 N 2 z + 1 Q ρ 0 N 2 y (so χ a too satisfies an elliptic equation, forced by a combination of and Q) Note that some combinations of and Q will lead only to a change in the PV. and others will lead only to an ageostrophic circulation Response to mechanical forcing only PHH 3 / 9 PHH 3 / 10
Response to thermal buoyancy forcing only PHH 3 / 11 Effect of the ageostrophic circulation: Mechanical forcing: ageostrophic circulation causes the response to be narrower in the horizontal and taller in the vertical than the forcing (particularly when the forcing is shallow in f / N scaled sense) Thermal forcing: ageostrophic circulation causes the response to be broader in the horizontal and shallower in the vertical than the forcing (particularly when the forcing is deep in f / N scaled sense) N.B. This model predicts unlimited increase in u and ρ if and Q are sustained in practice there will be dissipative effects that limit the response be functions of u and ρ. and Q will In large-scale atmospheric flow necessary to take account of decrease in density with height (non-boussinesq effects) PHH 3 / 12
Numerical model simulation of a sudden stratospheric warming fv a ut meridional mass circulation = region where force is exerted From Dunkerton et al. (1981) PHH 3 / 13 TEMPERATURE OZONE CHANGES ACCOMPANYING A NH PLANETARY WAVE EVENT (1 14 JANUARY 1980) dt/dt dχ O3 /dt CORRELATIONS WITH VERTICAL HEAT FLUX AT 60 N 50 mb RANDEL, 1993 (short time scale response) d dt (column O 3) PHH 3 / 14
Stress exerted at horizontal boundaries Represent by vertically varying stress τ which varies in the vertical through a thin layer close to the boundary The associated force disrupts geostrophic balance in the boundary layer z Interior Boundary Layer τ f u = p ρ 0 1 ρ 0 τ z = f u g + f u E Integrate over the boundary layer: ρ 0 f interior 0 u E dz = τ(0) 1 42 43 Ekman transport 123 Net force acting on the flow due to the boundary In boundary layer at a rigid surface, we might expect τ(0) to be the opposite sign to u g u E dz PHH 3 / 15 f > 0, i.e. u g τ(0) cyclonic relative vorticity in interior anticyclonic relative vorticity in interior convergence in boundary layer divergence in boundary layer There is a resulting vertical velocity at top of lower boundary layer (Ekman pumping) w E = interior 0 h.u E dz = h.{ekman transport} PHH 3 / 16
Oceanic surface forcing Consider stress τ applied at upper surface τ u E Response is Ekman flow in surface layer Positive stress curl Divergence in surface layer Upward Ekman pumping velocity Negative stress curl Convergence in surface layer Downward Ekman pumping velocity Ekman pumping velocity in terms of surface stress w E = 1 fρ 0 ( τ).k PHH 3 / 17 Surface forces in quasi-geostrophic equations Surface friction or applied stress appears in the q.-g. boundary conditions through Ekman pumping velocity The effect of boundary forces (e.g. friction, wind stress) is communicated to the interior through the secondary (ageostrophic) circulation [No immediate change in q.-g. p.v. in interior, but convergence / divergence in boundary layer effectively produces surface temperature gradients] The convergence/divergence associated with the secondary circulation compresses or stretches the vorticity in the interior, as in the tea-cup case discussed by MEM. But the response is substantially modified by stable stratification. PHH 3 / 18
Simple example: stratified spin-down Assume τ( 0) = α u g Then: Consider x -independent problem: interior: lower boundary: assume with interior: lower boundary: l 0 N f 0 and w E = h Ekman transport 2 ψ t y + f 2 0 2 ψ 2 N 2 z 2 = 0 ( ) = ˆ ψ y, z, t ψ = N 2 t z f 0 ψ ( z,t)sin l 0 y ψ ˆ ( z, t) = ψ ˆ 0 at t = 0 w E = N2 f 0 2 α 2 ψ ρ 0 y 2 2 ˆ ψ z l 2 2 0 ψ ˆ = l 02 ˆ ψ 0 ψ ˆ = ψ ˆ 0 + A( t)e Nl 0 z f 0 da dt = N 2 2 f 0 ˆ ψ ( z,t) = ˆ { } = ˆ ψ = N 2 α t z ρ 0 f l 2 2 0 ψ ˆ 0 α 2 l 0 [ ˆ ψ 0 + A], hence A = ˆ ψ 0 e Nlαt f 0ρ 0 1 ρ 0 [ ] ψ 0 1+ e Nlαt f 0ρ ( 0 1)e Nl 0z f 0 α ζ g f ρ 0 0 ( ) PHH 3/19 Note: 1) Effect of boundary penetrates only into interior f 0 N 1 l 0 2) Velocity of lower boundary reduced to rest in steady state. Ageostrophic circulation gives rise to density gradients compatible with vertical shear. f 0 N 1 l 0 t = 0 t PHH 3/20
Wind-driven ocean circulation Switch on surface forcing interior flow spins up How can it reach a steady state? Planetary vorticity gradient β in interior, initial forced flow changes PV distribution and hence flow evolves (through Rossby wave propagation etc.) Steady Sverdrup balance 1 ρ 0 ( τ).k = βv vertically integrated rate at which fluid is spun up by wind stress vertically integrated rate at which fluid is spun down by supplying PV from a different latitude Shortcomings: how does circulation close? (western boundary currents) what is vertical structure? PHH 3 / 21 PV view of ocean circulation Steady flow in interior u. q = 0 (no direct forcing) q contours act as barriers to the flow, if they intersect lateral boundaries Subsurface wind-driven circulation if (1) q homogenised by eddies, or (2) q contours intersect the ocean surface rather than lateral boundary (ventilation) Evidence from PV observations shows that both (1) and (2) seem to play a role in determining the circulation in the interior. See Rhines (1986), Pedlosky (1996) for full discussion PHH 3 /22
From Keffer, 1985 pressure on σ θ = 26.15 PV σ θ 26.05 26.25 (calculated under large-scale oceanographic approximation) PV σ θ 26.25 26.75 PV σ θ 27.0 27.3 σ θ = (ρ θ 1000) (kg m 3 ) PHH 3 / 23 Layer 1 Layer 2 q.-g. p.v. in numerical simulation of Antarctic Circumpolar Current (Marshall et al., 1993) PHH 3 / 24
Summary PHH lectures 1-3 In rotating, stratified flows there are two important classes of motion: I: 'Fast' inertio-gravity waves, 'sloshing modes ( 2) II: 'Slow' involving potential vorticity advection ( 1) (e.g. Rossby waves, barotropic and baroclinic instabilities see following lectures) Motion on time scales of a day or more in extratropical atmosphere and oceans, eg weather systems (cyclones and anticyclones), ocean eddies, but not tides, convective clouds, mountain waves, is 'slow'. Slow motion described by prognostic equation for one field (usually interior PV + boundary potential temperature). All other fields are 'slaved'to this (and may be calculated from it, eg by PV inversion). QG system is a valuable example of above, but is not accurate enough for practical use (eg weather forecasting). Seeking improved sets of 'slow'equations is ongoing research problem Inversion of PV field in shallow-water model (McIntyre & Norton 1990) [flow parameters well outside regime where QG theory accurate] u, h. u PHH 3 / 25 simulated by full model PV fields (two views) u, h. u inverted from PV field PHH 3 / 26