A PV Dynamics for Rotating Shallow Water on the Sphere

Transcription

1 A PV Dynamics for Rotating Shallow Water on the Sphere search for a balance dynamics on the full sphere potential vorticity inversion & advection ray theory for short scale waves David J Muraki & Andrea Blazenko, Simon Fraser University Chris Snyder, NCAR

2 Midlatitude Balanced Dynamics Rotating Shallow Water (rsw), β-scaled winds, u & height field, H = 1 + ɛ η longitude-latitude local coordinates, (λ, φ) (λ 0, φ 0 ) ɛ 2 D u ɛ Dη + (1 + β(φ φ 0 ))(ˆr 0 u) ɛ η + (1 + ɛ η) ( u) = 0 dimensionless parameters: Rossby number & Coriolis-β gh0 ɛ = (2Ω sin φ 0 ) r 1 ; β = cot φ 0 Midlatitude Quasigeostrophy (QG) balanced dynamics: slow Rossby waves & NO fast gravity waves restrict to short deformation scales: λ λ 0 = ɛx ; φ φ 0 = ɛy geostrophy: ˆr 0 u ɛ η non-divergent winds limit as ɛ 0, geostrophic degeneracy advection & inversion of potential vorticity (PV) geometric obstacle: local Rossby number singular at Equator, ɛ 1

3 rsw on the Full Sphere Spherical Coordinates longitude-latitude global coordinates, (λ, φ) ( ɛ 2 Du + vˆλ D ˆφ ) 1 v sin φ = ɛ cos φ η λ ( ɛ 2 Dv + u ˆφ Dˆλ ) + u sin φ = ɛ η φ ɛ Dη + {1 + ɛ η} u λ + (v cos φ) φ cos φ = 0 Lamb parameter ɛ = 4Ω2 r 2 gh 0 Balanced Dynamics! 1/2 1 PV Inversion on a Hemisphere, McIntyre/Norton 1999 landmark for PV as prognostic variable non-divergent winds (at leading-order) dynamical oddity: mirror symmetry across Equator on midlatitude scales, limit as ɛ 0 is now inconsistent! 2

4 A Case for Global Balanced Dynamics Grose & Hoskins (1979) rsw flow past mountain at 30 N steady, super-rotation zonal wind steady waves by Rayleigh-damped perturbations perturbation vorticity after 17 days (with damping) & ɛ 0.33 waves propagate smoothly across Equator nearly non-divergent flow 3

5 A Case for Global Balanced Dynamics Grose & Hoskins (1979) rsw flow past mountain at 30 N steady, super-rotation zonal wind steady waves by Rayleigh-damped perturbations perturbation height after 17 days (with damping) & ɛ 0.33 essentially zero height disturbance at Equator non-divergent balance relations, with streamfunction ψ u = ɛ ψ φ ; v = ɛ 1 cos φ ψ λ ; η = ψ sin φ 4

6 rsw on the Sphere Redone ɛ 2 Du ɛ 2 Dv v sin φ ɛ + u sin φ ɛ «1 η sin φ cos φ sin φ λ «η sin φ sin φ φ ɛ Dη + {1 + ɛ η} u λ + (v cos φ) φ cos φ = 0 non-divergent balance relations, with streamfunction ψ u = ɛ ψ φ ; v = ɛ 1 cos φ ψ λ ; η = ψ sin φ 5

7 rsw on the Sphere Redone ɛ 2 Du ɛ 2 Dv v sin φ ɛ 1 cos φ ψ λ sin φ + u sin φ ɛ ψ φ sin φ ɛ ψ cos φ ɛ Dη + {1 + ɛ η} u λ + (v cos φ) φ cos φ = 0 Geostrophic Degeneracy Restored on midlatitude scales, limit as ɛ 0 is consistently degenerate non-divergent balance relations β-effect displaced: meridional advection of planetary PV Potential Vorticity total PV is advected quantity: DQ/ = 0 Q = sin φ + ɛ q = sin φ + ɛ2 ˆr ( u) 1 + ɛη 6

8 PV Dynamics on the Sphere (spv) Inversion, Streamfunction & Advection PV-streamfunction relation with topography, b(λ, φ): ɛ 2 2 ψ (sin 2 φ) ψ = q b sin φ balance relations for non-divergent winds: u = ɛ ψ φ ; v = ɛ disturbance PV advection: j q ɛ t + u q cos φ λ + v q ff + v cos φ = 0 φ 1 cos φ ψ λ ; η = ψ sin φ Mountain Waves, Disturbance PV (t = 4) 7

9 PV Dynamics on the Sphere (spv) Inversion, Streamfunction & Advection PV-streamfunction relation with jet; ū(φ), ɛ η(φ) = O(1) & topography; b(λ, φ): ɛ 2 (1 + ɛ η) 2 ψ (sin 2 φ) ψ = (1 + ɛ η) 2 q b sin φ balance relations for non-divergent winds: u = ɛ ψ φ ; v = ɛ disturbance PV advection: j q ɛ t + ū + u q cos φ λ + v q ff φ 1 cos φ ψ λ ; η = ψ sin φ «sin φ + v 1 + ɛ η φ = 0 Mountain Waves, Disturbance PV (t = 4) 8

10 spv on the Sphere: Computations Mountain Waves, Disturbance Height (t = 8) Mountain Waves, Disturbance PV (t = 8) 9

11 spv on the Sphere: Comparison with rsw Disturbance Height: spv (colour) & rsw (contour) Disturbance PV: ɛ thanks to Joe Klemp, NCAR 10

12 spv is not a Global Asymptotic Theory? ɛ 2 Du ɛ 2 Dv v sin φ ɛ 1 cos φ ψ λ sin φ + u sin φ ɛ ψ φ sin φ ɛ ψ cos φ ɛ Dη + {1 + ɛ η} u λ + (v cos φ) φ cos φ = 0 Midlatitude Synoptic-Scale Truncation leading-order balance assumptions on deformation scales / λ, / φ = O(1/ɛ) at midlatitudes sin φ 0 there is NO expectation of asymptotic validity at the Equator spv is not globally accurate, but is well-posed PV inversion & velocity-streamfunction relationship are non-singular Q: So, Why are Equatorial Crossings Faithfully Represented? 11

13 spv Rossby Waves 0 rossby wave frequencies (8 meridional modes: n) frequency: ω mn zonal wavenumber: m Rotating Waves, ψ(λ ct, φ) ODE eigenfunction [Verkley (2007), Schubert (2008)] exact nonlinear spv solutions (ɛ 0.337) ɛ 2 2 ψ (sin 2 φ) ψ + (1/c) ψ = 0 ; ψ(±π/2) = 0 compare to modes for rsw on sphere (Margules, Hough, Longuet-Higgins,... ) only longest planetary waves have O(1) wavespeed errors 12

14 spv Rossby Waves rossby wave contours plane wave decomposition: phasefronts Rotating Waves, ψ(λ ct, φ) ODE eigenfunction [Verkley (2007), Schubert (2008)] exact nonlinear spv solutions (ɛ 0.337) ɛ 2 2 ψ (sin 2 φ) ψ + (1/c) ψ = 0 ; ψ(±π/2) = 0 compare to modes for rsw on sphere (Margules, Hough, Longuet-Higgins,... ) only longest planetary waves have O(1) wavespeed errors 13

15 Wavepacket Limit of Short-Scale Waves: ɛ = 1 2 ɛ gh & 1 4 ɛ gh 14

16 Ray Theory Geometrical Optics Limit as ɛ 0 (flow) = (steady, zonal ū(φ)) + (wave amplitude) e is(λ,φ,t)/ɛ phase S(λ, φ, t) satisfies ɛ-independent Hamilton-Jacobi PDE «S 2! λ S t + ū cos φ S λ midlatitude replacements... S t ω ; cos 2 φ + S2 φ + sin2 φ 1 + ɛ η S λ sin φ 1 + ɛ η «φ (1 + ɛ η) = 0 S λ cos φ k ; S φ l ; sin φ f ; cos φ β ; ɛ η 0... give midlatitude Rossby wave dispersion (Hoskins & Karoly, 1981) ω = ū k βk k 2 + l 2 + f 2 Q: So, Why are Equatorial Crossings Faithfully Represented? 15

17 Ray Theory A: Both spv & rsw have the same global ray theory! spv dynamics & steady group rays with phasefronts 16

18 20-24 April 2009 NCEP Operational Analysis 250mB wind vectors (5-day average) thanks to Mel Shapiro, NCAR Rossby wave flow in Equatorial Pacific 17

19 20-24 April 2009 NCEP Operational Analysis 250mB meridional wind anomaly (5-day average) Rossby wave flow in Equatorial Pacific 18

20 20-24 April 2009 NCEP Operational Analysis 250mB geopotential height anomaly (5-day average) Rossby wave flow in Equatorial Pacific 19

21 2-Layer spv Dynamics + an Unstable Jet [A. Blazenko, MSc] PV-Streamfunction: 2 coupled elliptic equations, q 1, q 2 ψ 1, ψ 2 (k = 1, 2) " ɛ 2 γ 1 2 " ɛ 2 γ 2 2 «# 1 sin 2 φ ψ 1 1 δ + «# 1 sin 2 φ ψ 2 1 δ + «1 sin 2 φ ψ 2 1 δ = γ1 2 q 1 «δ sin 2 φ ψ 1 1 δ = γ2 2 q 2 fractional depths ««α 1 γ 1 = + ɛ ( η 1 + α 1 η 2 ) ; γ 2 = + ɛ η 1 + α 2 spv Balance Relations: ψ k u k, v k, η k u = ɛ ψ k φ ; v = ɛ ψ k cos φ λ ; η = ψ k sin φ. Disturbance PV Advection: time evolution DQ k = ɛ j qk t + ūk + u k cos φ ff q k λ + v q k φ «sin φ + v k γ k φ = 0 20

22 2-Layer rsw Baroclinic Instability Linearized rsw, Warn (1976) identify unstable eigenvalues; expect spv results to compare closely for small ɛ Jet Profile mean jet in upper layer ū 1 = 1 8 sin2 φ cos 3 φ max jet speed 22 m/s at 40 no flow in lower layer ū 2 = 0 Parameter values ɛ = p 2/ α H 1 /H 2 = 1 δ ρ 1 /ρ 2 = 0.9 windspeed (m/s) SP 60 S 30 S EQ 30 N 60 N NP latitude 21

23 2-Layer rsw Baroclinic Instability Linearized rsw, Warn (1976) identify unstable eigenvalues; expect spv results to compare closely for small ɛ Jet Profile disturbance height in upper layer ɛ η 1 = sin6 φ 1 4 sin4 φ + 11 «420 ɛ η 2 = δ 1 δ ɛ η 1 mean depth: 1 + (ɛ η 1 ɛ η 2 ) Parameter values atmosphere depth (m) COLD WARM COLD ɛ = p 2/ α H 1 /H 2 = 1 δ ρ 1 /ρ 2 = SP 60 S 30 S EQ 30 N 60 N NP latitude 22

24 Unstable Growth Rates: (ɛ 0.4) Maximum Growth Rates (day 1 ) m spv rsw * most unstable mode: wavenumber m = 6 short wave cutoff [Charney (1947), Eady (1949)] growth rate m 23

25 Most Unstable Mode (m = 6) Upper Layer PV (q1) streamfunction (ψ1) Lower Layer PV (q2) streamfunction (ψ2) 24

26 Nonlinear Evolution: m = 6 & t = 2.58 days Total PV & Streamfunction/Height 25

27 Nonlinear Evolution: m = 6 & t = 8.77 days Total PV & Streamfunction/Height 26

28 Nonlinear Evolution: m = 6 & t = days Total PV & Streamfunction/Height 27

29 In Closing PV Dynamics for the rsw Sphere local asymptotic validity in midlatitudes accurate for global Rossby waves at scales smaller than planetary recreates Grose & Hoskins (1979) equatorial wave crossing shares identical (linear) ray theory with rsw rsw ray theory is degenerate theory, but globally valid spv includes nonlinear PV advection 28