Spontaneous gravity wave radiation from unsteady rotational flows in a rotating shallow water system

Transcription

1 AGU Chapman Conference on Atmospheric Gravity Waves and Their Effects on General Circulation and Climate Honolulu, Hawaii, 8 February 4 March 011 Jet/Imbalance Sources Presiding: Riwal Plougonven, Friday, 4 March, 9:40 10:00 Spontaneous gravity wave radiation from unsteady rotational flows in a rotating shallow water system Norihiko SUGIMOTO 1), H. Kobayashi 1), Y. Shimomura 1), & K. Ishioka ) (1)Dept. Phys., Keio Univ., Japan ()Dept. Geophys., Kyoto Univ., Japan This work was supported by a Grant-in-Aid for the Young Scientists (B) ( ) from the Ministry of Education Culture, Sports, Science and Technology in Japan

2 Outline 1. Motivation Atmospheric Gravity Wave (GW) and their roles Spontaneous Gravity Wave Radiation (Sp-GWR) Concept of balance and its limitation. Analytical study Setup & Estimation of the far field Effect of the Earth s rotation 3. Numerical simulation Computational method Preliminary result 4. Summary

3 1. Motivation GW in the atmospheric science winter GW radiation, propagation, and dissipation summer PANSY project (005)

4 GW radiation from rotational flow (Spontaneous GW radiation: Sp-GWR) Observational study Yoshiki and Sato(00): polar night jet Kitamura and Hirota(89): sub tropical jet Pfister et al.(93): hurricane Experimental study Williams et al. (05): -layer fluid in rotating annuls Numerical study (GCM, Meso-scale model) O Sullivan and Dunkerton(95), Zhang(04), Plougonven and Snyder(05): sub tropical jet Sato et al.(99): polar night jet Snyder et al.(07), Viudez(07), Wang et al.(09): dipole Viudes(07) Williams et al.(05) Sato(00) Plougonven et al.(05) Wang & Zhang(10)

5 Sp-GWR in simplified model Numerical study (simplified model =f -plane Shallow Water, f-sw) Ford (1994): vorticity stripe Sugimoto et al. (005, 007b, 008etc): unsteady jet with relaxation forcing McIntyre (009): outside the scope of SH dynamics η u u du η + u + v fv= g, t x dy x v v dv η + u + v + fu = g, t x dy y η η dη u v + u + v + ( H + η ) + = 0. t x dy x y Ford (1994) The most simplified system in which rotational flows and GW exist. Same as D compressible gas fluid if the earth s rotation is negligible. GW are analogous to Sound Waves. Lighthill theory 5

6 Analogy with vortex sound (Lighthill theory) Gravity wave source (Lighthill-Ford eq.) h ( hu ) g + f gh0δ = + fhuv+ ( h H0) t t x t t GWpropagat i on Vor t ex sour ce( 1) ( huv) ( hv ) g + + fhv fhu + + fhuv ( h H0) xy t y t t Vor t ex sour ce( ) Vor t ex sour ce( 3) Concept of balance Lorenz(80), Leith(80), Warn(97), Ford et al.(00): Balanced flow and slow manifold Gent & McWilliams(83), Spall & McWilliams(9), Sugimoto et al. (07a): Several balanced regimes Sp-GWR violate balance but Earth s rotation enhance balance! Vortex sound theory Lighthill (195), Powell(1964), Howe(1975) etc. PV dh/dt & Source Geopotential height T=5.0 Sugimoto et al. (007b) 6

7 Vorticity Divergence Geopotential θj =67.5 Sp-GWR :freq. of the source>coriolis freq. This Study Sugimoto & Ishii (011, to be submitted) We investigate Sp-GWR from vortex pair in f-shw Derive analytical estimation of Sp-GWR in the far-field Check the results using numerical simulation Investigate the effect of the Earth s rotation on Sp-GWR Discuss parameter regimes to hold balance

8 . Analytical study Analytical estimation of vortex sound for 3D case (Howe, 9) Far field of vortex sound derived by use of Green s function Green s function (3D) 1 c0 t For quadruple source Far field G = δ( x y ) δ( t τ), where G = 0 for t < τ 1 x y G( xy,, t τ) = δ( t τ ) 4π x y c0 1 T ij P = c0 t xi x 1 Tij ( y, t τ x y / c ) p( x, t) = 4π d x x x y i j j 0 3 y

9 Far field of co-rotating vortex pair for 3D case 3 πl π r p( x, t) = 4 ρ0u M cos θ Ω t +, r c0 4 Ωr at c 0 Historical integral appears for D case (more complicated) 4 ρ 0Γ0 Ω0r Ω0r pr (, θ, t) = J θ + Ω + θ + Ω 3 4 π sin( 0t) Y cos( 0t), 64 Rc 0 c0 c0 Mitchell, Lele and Moin (JFM95): Sound generation from co-rotating vortex pair (theoretical study & numerical simulation)

10 This study:sp-gwr from co-rotating vortex pair in f-shw Green s function (Nuclear force for quantum physics) 1 f μ G( x, t, y, τ) = δ( x y ) δ( t τ), μ= c t c c cos μ c ( t τ) x y G( xy,, t τ) = θ( c( t τ) x y) π c ( t τ ) x y Far field (include both historical integral and Earth s rotation) 4 h( x, t) ΓΩ lh 0 r r = N 0 4Ω f sin(θ Ωt)-J0 4Ω f cos(θ Ωt). t c c c Correspond to vortex sound (Mitchell et al., 95) for f=0 Sp-GWR :freq. of the source>coriolis freq.

11 Examples of far field (Ω=0.18) Weak Sp-GWR by finite f f =0 (Ro=inf, Fr=0.6) f =0.03 (Ro=0, Fr=0.3) f =0.15 (Ro=4, Fr=0.6) Numerical simulation (in the double periodic boundary)

12 3. Numerical simulation Setup Basic equation:f -ShW in polar coordinate (r, θ) ψ ( rv ψ) ( u ψ) = f χ, t r r r θ χ ( ru ψ) ( v ψ) = + f ψ ( E +Φ), t r r r θ Φ ( rvφ) ( uφ) =. t r r r θ θ r Vorticity u Geopotential height Initial state: Gaussian vortex (in balance with surface elevation) { r r1 1 θ σ } { r r 1 θ π σ } q = Aexp ( r, ) / Aexp ( r, + ) / Experimental parameter: Ui Ui Ro =, Fr = f r Φ 1

13 Computational method:(ishioka, 008) Spectral method in unbounded domain Projection spherical coordinate (λ, φ) of radius R to polar coordinate (r, θ) in D plane by the relation φ π r = Rtan sinφ 1 =, =, then r R φ r θ λ (1 sin φ) Laplacian = 4R s Computational scheme:ispack-0.81 Domain:unbounded domain λ Discretization:spherical harmonics Resolution:M=341(51 104grids) Time integration:4th-order Runge-Kutta Inertial freq.:f =0.01Π-0.5Π, Average Depth:Φ=0.5, 1,, 4 Pseudo hyper viscosity:3 rd 8 order Laplacian ν = 10 hyper viscosity in near field, sponge layer in far field φ (λ, φ) (r, θ)

14 Preliminary result (Ro=0, Fr=0.6): Flow field Sp-GWR from co-rotating vortex pair & dissipation in the far field Vorticity Divergence Geopotential height 1 40

15 Comparison with analytical & numerical result (Ro=0, Fr=0.6) Similarity of wavelength, frequency, and phase speed

16 t GW flux (Ro=0, Fr=0.6) A e + F = dq A = Φ ( + ) + Φ + Φ + Φ e u v uu u/ q, dr F = Φ +Φ Φ r u uv v, dq Fθ = uφ u + uφ + ( Φ + u ) u Φ + Φ u / q. dr Conservation of GW flux at r=40

17 Parameter sweep exp: Anti-cyclone (Ro=0, Fr=0.6) Vorticity Divergence Geopotential height Cyclone (C) & Anti-cyclone (AC) asymmetry

18 GW flux in Ro-Fr parameter space Dependence on Ro for Fr=0.6 C & AC Dependence on Fr for Ro=, 0, inf Anti-cyclone > Cyclone Local maximum at medium Ro for AC Power law of Fr and its breakdown? ζ + f is small for AC : Effective Rossby number?

19 4. Summary Spontaneous gravity wave radiation from co-rotating vortex pair in f -plane shallow water system Analytical estimation for the far field Difficulty in D case (historical integral of the source) Weak radiation by the Earth s rotation Numerical simulation in the unbounded domain Dissipation in the far filed Cyclone and anti-cyclone asymmetry for GW flux Local maximum of GW flux at medium Ro for AC Future work Estimate gravity wave flux in a wide parameter space. Derive gravity wave source and integrate the source analytically. Discuss how much balance maintained by the Earth s rotation.

20 References Balanced flow Balance regimes for the stability of a jet in an f-plane shallow water system, Norihiko Sugimoto, Keiichi Ishioka, and Shigeo Yoden, Fluid Dynamics Research, Vol. 39, No. 5, (007), p Spontaneous GW radiation Fr dependencies (f-plane SH) Froude Number Dependence of Gravity Wave Radiation From Unsteady Rotational Flow in f-plane Shallow Water System, Norihiko Sugimoto, Keiichi Ishioka, and Shigeo Yoden, Theoretical and Applied Mechanics Japan, 54, (005), p GW source (f-plane SH) Gravity wave radiation from unsteady rotational flow in an f-plane shallow water system, Norihiko Sugimoto, Keiichi Ishioka, and Shigeo Yoden, Fluid Dynamics Research, Vol. 39, No. 11-1, (007), p Parameter sweep experiments (f-plane SH) Parameter Sweep Experiments on Spontaneous Gravity Wave Radiation From Unsteady Rotational Flow in an F-plane Shallow Water System, Norihiko Sugimoto, Keiichi Ishioka, and Katsuya Ishii, Journal of the Atmospheric Sciences, Vol. 65, No. 1, (008), p Latitudinal change of the earth rotation (Spherical SH) The effect of the Earth Rotation of Spontaneous Gravity Wave Radiation in Shallow Water System on a Rotating Sphere, Norihiko Sugimoto and Katsuya Ishii, Journal of the Meteorological Society Japan, to be submitted.

21 Future work layer shallow water model (collaborated with K. Ishii) (Spontaneous internal GW radiation from baroclinic unstable jet) Frequency spectra of GW sourc Parameterizations of spontaneous GW radiation

22 Gravity wave source (Lighthill-Ford eq.) h ( hu ) g + f gh0δ = + fhuv+ ( h H0) t t x t t GWpropagat i on Vor t ex sour ce( 1) ( huv) ( hv ) g + + fhv fhu + + fhuv ( h H0) xy t y t t Vor t ex sour ce( ) Vor t ex sour ce( 3) GW amplitude for far field ( ) h y, t 1 t = + dtgy [ ( c0( t t), t) Gy ( c0( t t), t), t c t 0 0 G( y, t): Zonally averaged source F e Analogy with vortex sound (Lighthill theory) Power law of GW flux ωbu ( ω f ) g Fr Vortex sound theory Lighthill (195), Powell(1964), Howe(1975) etc. PV dh/dt & Source Geopotential height T=5.0 Sugimoto et al. (007b) Corresponds to the power law of Mach number if f=0

23 Parameter dependence of gravity wave flux (Fr, Ro) Fr dependence Ro=10 Ro=100 Time mean of gravity wave flux averaged at Y=40 Breakdown of the power law of Fr Fr Fr Ro dependence Fr=0.1 Fr=0.7 Sugimoto et al. (008) Local maximum at Ro=10 Ro Ro

24 Frequency spectra of GW source (Ro dependence for Fr=0.1) date series between T=5.0~15.0 T ( ) hv g h = fhuv + t t Ro=100 Ro=10 Ro U 0 = fb Increase of the source related to Coriolis term for small Ro Ro=5 Cut-off ( f /π) Ro=1 Sugimoto et al. (008) Increase of inertial frequency cut-off for small Ro

25 Time evolution: energy, enstrophy, umax, Fr, and qmax