A Uniform PV Framework for Balanced Dynamics vertical structure of the troposphere surface quasigeostrophic models Dave Muraki Simon Fraser University Greg Hakim University of Washington Chris Snyder NCAR Boulder http://grads.iges.org 1
Dynamical Structures of the Troposphere Two-Dimensional Idealizations 2D barotropic vorticity: rotating dynamics shallow water: balanced and gravity wave dynamics & vertical displacement surface QG: balanced dynamics & uniform PV with vertical structure surface pressure & thickness vis-à-vis 500 hpa vorticity & geopotential http://grads.iges.org 2
PV Structure of the Lower Atmosphere Well-Mixed Troposphere PV gradients are relatively weak in troposphere dominant PV influence from tropopause displacement PV on 330K & 310K surfaces, mean PV gradient morgan & nielsen-gammon (1999) http:/www.atmos.washington.edu/ hakim 3
Vertical Structure of the Lower Atmosphere Upper-Level Disturbances disturbance amplitudes peaked at tropopause level, decrease in troposphere vertical structure of geopotential (zonal fourier amplitudes) at 40 N tropopause map of potential temperature Pressure REAN Geopotential EOF 10/95--3/96 (40N) 30 50 70 100 150 200 250 300 400 500 600 700 850 925 1000 0 0.2 0.4 0.6 0.8 Amplitude k = 0: 99.9979% k = 1: 93.0974% k = 5: 96.7242% k = 10: 95.4953% k = 20: 87.0621% http:/www.atmos.washington.edu/ hakim tropopause potential temperature as key dynamical variable for simple tropospheric model 4
Vertical Structure of the Lower Atmosphere II Upper-Level Disturbances disturbance amplitudes peaked at tropopause level, decrease in troposphere vertical structure of winds (zonal fourier amplitudes) at 40 N Pressure 30 50 70 100 150 200 250 300 400 REAN U EOF 10/95--3/96 (40N) 500 600 700 850 925 1000 0 0.2 0.4 0.6 Amplitude k = 0: 99.0392% k = 1: 91.07% k = 5: 92.769% k = 10: 88.9822% k = 20: 83.4804% Pressure 30 50 70 100 150 200 250 300 400 REAN V EOF 10/95--3/96 (40N) 500 600 700 850 925 1000 0 0.2 0.4 0.6 Amplitude k = 0: 82.7959% k = 1: 89.3682% k = 5: 95.5604% k = 10: 95.1176% k = 20: 87.9461% http:/www.atmos.washington.edu/ hakim 5
Zero PV Quasigeostrophic Dynamics Well-Mixed Troposphere q 0 dynamically active, rigid tropopause (z = H) & passive, topographic lower surface (z = 0) H D Dt t θ = 0 ; Φ θ z = t 2 Φ = 0 0 Φ z = -b (x, y ) Surface QG for a Finite Depth Troposphere (HsQG) 3D PV inversion from tropopause & topographic BCs: 2 Φ = 0 ; Φ z (z = H) = θ t (x, y, t) ; Φ z (z = 0) = b(x, y) 2D advection of tropopause potential temperature, θ t : Dθ t Dt = θt t + ut θ t x + vt θ t y = 0 ; ut = Φ y (z = H) ; v t = Φ x (z = H) sqg interface as model for tropopause: Rivest, et.al. (1992); Juckes (1994) 6
PV Inversion & Baroclinic/Barotropic Dynamics Zero PV Fourier Inversion fourier transform of streamfunction (m = k 2 + l 2 ) { } { } cosh mz cosh m(z H) ˆΦ(k.l; z, t) = ˆθt (k, l; t) + ˆb(k, l) m sinh mh m sinh mh vertical decay away from boundaries small scales are more localized to tropopause/surface (mh 1 ) larger scales extend deeper into troposphere (mh 1) Small & Large-Scale Dynamics fourier transform of tropopause-level streamfunction (w/o topography) ˆ Φ t (x, y; H, t) = { } coth mh ˆθt (k, l; t) m ˆ θ t m mh 1 ˆ θ t /H m 2 mh 1 small horizontal scales (relative to depth) invert as sqg (mh 1) larger horizontal scales large invert as barotropic vorticity (mh 1) on the large scales, potential temperature gradient ( θ t /H) evolves as barotropic vorticity ζ 7
Topographic Flows Sloping Bottom Topography (b = σy) illustration of dynamic similarity between horizontal PV gradient & sloping bottom streamfunction: Eady shear with simple travelling fourier mode (m = k 2 + l 2 ) { } cosh mz Φ(x, y, z, t) = σyz + A cos k(x ct) cos ly m sinh mh topographic Rossby wave dispersion relation, analogous to Rhines (1970) { } coth mh c = σh 1 mh Flow over Bottom Topography tropopause streamfunction for gaussian topography, b = e αr2 Φ(x, y, H, t) = U y 0 J 0 (mr) sinh mh e m2 /4α 2α dm 8
Two-Surface Edge Wave Finite Rossby Number Corrections nonlinear edge wave solution with simple Eady shear, correct to O(R) square wave k = l = 1, vertical mode number m = k 2 + l 2 = 2.5 beyond short-wave stability criterion: m > m c 2.399 upper-level cyclone asymmetry for R = 0.1 nonlinear wavespeed same as neutral linear edge waves Upper boundary potential temperature Lower boundary potential temperature 1.5 1 1.5 2 1.5 1 1 0 1 0.5 0.5 0.5 1 0 0 0 0.5 2 0.5 0.5 1 1 3 1 1.5 1.5 1.5 1 0.5 0 0.5 1 1.5 4 1.5 1.5 1 0.5 0 0.5 1 1.5 2 9
Free-Surface Dynamics Uniform PV Inversion (in progress, R Tulloch) moving free-surface at z = Rh(x, y; t) total surface potential temperature, θ s (x, y; t) = h(x, y; t) + θ(x, y, Rh(x, t; t), t) surface BCs: kinematic conditions with continuity of potential temperature and pressure Fourier solution of the 3D streamfunction (m = k 2 + l 2 ) + { } Φ(x, y, z; t) = ˆθ s 1 (k, l; t) m + σ 1 e i(kx+ly) dk dl surface value of potential temperature is σ freely decaying vortex organizations sqg +1 random IVs fsqg +1 10
Baroclinic Instability 2sQG +1 : Downstream Development tropopause baroclinic wave 11
Uniform PV Thinking Dynamics of Uniform PV Layers significant part of tropospheric dynamics are strongly influenced by tropopause & ground simple formulation for understanding rotating, stratified flows dominated by surfaces/boundaries surface dynamics embeds large & small-scale limits: large-scale barotropic vorticity dynamics small-scale surface-trapped dynamics moving interface formulations: free-surface dynamics, as a continuously-stratified shallow-water analog tropopause dynamics Computational Efficiency of sqg Fourier Inversion resolution of vertical structure is exact for given horizontal discretization only 2D FFTs required to evolve 3D tropospheric flow finite Rossby number corrections also computed with 2D efficiencies 12
QG+ Reformulation Exact Reformulation of PE three-potential representation: Φ, F, G v = Φ x G z u = Φ y + F z θ = Φ z + G x F y R w = F x + G y potential inversions 2 Φ = q R { [ ] } θ ( u H ) { ( ) ( ) } 2 Dθ Dv F = R + Dt x Dt z { ( ) ( ) } 2 Dθ Du G = R Dt Dt surface boundary conditions R w s = (F x + G y ) s ; θ t = (Φ z + G x F y ) s advection dynamics (interior & surface) Dq Dt = 0 ; Dθ t Dt + w s = 0 y z 13