Spontaneous gravity wave radiation fromunsteady rotational flows

in a rotating shallow water system

Norihiko SUGIMOTO1), H. Kobayashi1), Y. Shimomura1), & K. Ishioka2)

(1)Dept. Phys., Keio Univ., Japan (2)Dept. Geophys., Kyoto Univ., Japan

AGU Chapman Conference on Atmospheric Gravity Waves and Their Effects on General Circulation and Climate

Honolulu, Hawaii, 28 February – 4 March 2011Jet/Imbalance Sources

Presiding: Riwal Plougonven, Friday, 4 March, 9:40–10:00

This work was supported by a Grant-in-Aid for the Young Scientists (B) (19740290) from the Ministry of Education Culture, Sports, Science and Technology in Japan

Outline1. Motivation

• Atmospheric Gravity Wave (GW) and their roles• Spontaneous Gravity Wave Radiation (Sp-GWR)• Concept of balance and its limitation

2. Analytical study• Setup & Estimation of the far field• Effect of the Earth’s rotation

3. Numerical simulation• Computational method• Preliminary result

4. Summary

PANSY project (2005)

winter summer

GW in the atmospheric science

GW radiation, propagation, and dissipation

1. Motivation

Observational studyYoshiki and Sato(00): polar night jetKitamura and Hirota(89): sub tropical jetPfister et al.(93): hurricane

Experimental study Williams et al. (05): 2-layer fluid in rotating annuls

Numerical study (GCM, Meso-scale model)O’Sullivan and Dunkerton(95), Zhang(04),Plougonven and Snyder(05): sub tropical jetSato et al.(99): polar night jetSnyder et al.(07), Viudez(07), Wang et al.(09): dipole

Wang & Zhang(10)

GW radiation from rotational flow(Spontaneous GW radiation: Sp-GWR)

Sato(00)

Plougonven et al.(05)

Williams et al.(05)

Viudes(07)

5

•The most simplified system in which rotational flows and GW exist. •Same as 2D compressible gas fluid if the earth’s rotation is negligible. •GW are analogous to Sound Waves.

( )

,

,

+ 0.

u u duu v fv gt x dy xv v dvu v fu gt x dy y

d u vu v Ht x dy x y

η

η

η η η η

∂ ∂ ∂+ + − = −

∂ ∂ ∂∂ ∂ ∂

+ + + = −∂ ∂ ∂

⎛ ⎞∂ ∂ ∂ ∂+ + + + =⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠

η

Numerical study (simplified model ＝f -plane Shallow Water, f-SW)

Ford (1994): vorticity stripeSugimoto et al. (2005, 2007b, 2008etc):

unsteady jet with relaxation forcing McIntyre (2009): outside the scope of SH dynamics

Lighthill theory

Sp-GWR in simplified model

Ford (1994)

6

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂+ − Δ = + + −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂ ∂ ∂⎛ ⎞+ + − + − +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

144424443 14444444244444443

144444424444443

GW pr opagat i on Vor t ex sour ce（ １ ）

Vor t ex source（ ２ ）

2 2 22 2

0 02 2

2 2 22 2

2

( ) ( )2

(2 ) ( )

h hu gf gH fhuv h Ht t x t t

huv hvfhv fhu fhuvx y t y t

⎛ ⎞∂−⎜ ⎟∂⎝ ⎠14444444244444443

Vor t ex sour ce（ ３ ）

20( )

2g h Ht

Gravity wave source (Lighthill-Ford eq.)

Vortex sound theoryLighthill (1952),Powell(1964), Howe(1975) etc.

PV dh/dt & Source Geopotential height

T=5.0

Analogy with vortex sound (Lighthill theory)

Lorenz(80), Leith(80), Warn(97), Ford et al.(00): Balanced flow and slow manifoldGent & McWilliams(83), Spall & McWilliams(92), Sugimoto et al. (07a): Several balanced regimes

Concept of balance

Sp-GWR violate balance … but Earth’s rotation enhance balance!

Sugimoto et al. (2007b)

This Study

We investigate Sp-GWR from vortex pair in f-ShW

Derive analytical estimation of Sp-GWR in the far-fieldCheck the results using numerical simulationInvestigate the effect of the Earth’s rotation on Sp-GWRDiscuss parameter regimes to hold balance

θj =67.5

Vorticity Divergence Geopotential

Sp-GWR ：freq. of the source＞Coriolis freq.

Sugimoto & Ishii (2011, to be submitted)

Far field of vortex sound derived by use of Green’s functionAnalytical estimation of vortex sound for 3D case (Howe, 92)

δ δ τ τ⎛ ⎞∂

− ∇ = − − = <⎜ ⎟∂⎝ ⎠x y where for

22

2 20

1 ( ) ( ), 0G t G tc t

τ δ τπ

−− = − −

−x y

x yx y 0

1( , , ) ( )4

G t tc

∂⎛ ⎞∂− ∇ =⎜ ⎟∂ ∂ ∂⎝ ⎠

222

2 20

1 ij

i j

TP

c t x xFor quadruple source

τπ

∞

−∞

− − −∂=

∂ ∂ −∫y x y

x yx y

20 3( , / )1( , )

4ij

i j

T t cp t d

x x

Green’s function (3D)

Far field

2. Analytical study

π πρ θ⎡ ⎤⎛ ⎞

= − − Ω − +⎢ ⎥⎜ ⎟⎝ ⎠⎣ ⎦

Ω→ ∞

x

at

32 2

00

0

( , ) 4 cos 2 2 ,4

l rp t U M tr c

rc

Mitchell, Lele and Moin (JFM95)：Sound generation from co-rotating vortex pair (theoretical study & numerical simulation)

ρθ θ θπ

⎛ ⎞⎛ ⎞ ⎛ ⎞− Γ Ω Ω= + Ω + + Ω⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠

40 0 0 0

2 0 2 03 4 20 0 0

2 2( , , ) J sin(2 2 ) Y cos(2 2 ) ,64

r rp r t t tR c c c

Far field of co-rotating vortex pair for 3D case

Historical integral appears for 2D case (more complicated)

Green’s function (Nuclear force for quantum physics)

This study：Sp-GWR from co-rotating vortex pair in f-ShW

μ τ δ δ τ μ⎛ ⎞∂∇ − − = − − −⎜ ⎟∂⎝ ⎠

x y x y =2

2 22 2

1 ( , , , ) ( ) ( ), fG t tc t c

μ ττ θ τ

π τ

− − −− = − − −

− − −

x yx y x y

x y

22 2

22 2

cos ( )( , , ) ( ( ) )

2 ( )

c tcG t c tc t

4 22 2 2 20

0 02

2( , ) N 4 sin(2 2 )-J 4 cos(2 2 ) .l Hh t r rf t f tt c c c

θ θΓΩ ⎡ ⎤∂ ⎛ ⎞ ⎛ ⎞= Ω − − Ω Ω − − Ω⎜ ⎟ ⎜ ⎟⎢ ⎥∂ ⎝ ⎠ ⎝ ⎠⎣ ⎦

x

Correspond to vortex sound (Mitchell et al., 95) for f=0Sp-GWR ：freq. of the source＞Coriolis freq.

Far field (include both historical integral and Earth’s rotation)

Examples of far field (2Ω=0.18)

Numerical simulation (in the double periodic boundary)f =0 (Ro=inf , Fr=0.6)

Weak Sp-GWR by finite f

f =0.15 (Ro=4, Fr=0.6)f =0.03 (Ro=20, Fr=0.3)

Basic equation：f -ShW in polar coordinate (r, θ)ψ ψ ψ χ

θχ ψ ψ ψ

θ

θ

∂∇ ∂ ∇ ∂ ∇= − − − ∇

∂ ∂ ∂∂∇ ∂ ∇ ∂ ∇

= − + ∇ − ∇ + Φ∂ ∂ ∂

∂Φ ∂ Φ ∂ Φ= − −

∂ ∂ ∂

2 2 22

2 2 22 2

( ) ( ) ,

( ) ( ) ( ),

( ) ( ).

rv u ft r r r

ru v f Et r r r

rv ut r r r

3. Numerical simulationSetup

Initial state：Gaussian vortex (in balance with surface elevation)

Experimental parameter：

θ σ θ π σ= − − − − − − +r r r r 22 2 21 1 2 1exp ( , ) /2 exp ( , ) /2q A r A r

= =× Φ

1

,i iU URo Frf r

θ

r

Vorticity u Geopotential height

Computational scheme：ISPACK-0.81Domain：unbounded domainDiscretization：spherical harmonicsResolution：M=341(512×1024grids)Time integration：4th-order Runge-KuttaInertial freq.：f =0.01Π-0.5Π, Average Depth：Φ=0.5, 1, 2, 4Pseudo hyper viscosity：3rd order Laplacian

hyper viscosity in near field, sponge layer in far field

φ π⎛ ⎞= +⎜ ⎟⎝ ⎠

2 tan2 4

r R

Projection spherical coordinate (λ, φ) of radius R to polar coordinate (r, θ) in 2D plane by the relation

φφ θ λ

φ

∂ − ∂ ∂ ∂= =

∂ ∂ ∂ ∂

−∇ = ∇

then

Laplacian 2

2 22

1 sin 1, ,2

(1 sin )4 s

r R r

R

ν −= 810

λφ

Computational method：(Ishioka, 2008)（λ, φ）

（r, θ）

Spectral method in unbounded domain

Sp-GWR from co-rotating vortex pair & dissipation in the far field

Vorticity Divergence Geopotential height

Preliminary result (Ro=20, Fr=0.6): Flow field

1 40

Similarity of wavelength, frequency, and phase speed Comparison with analytical & numerical result (Ro=20, Fr=0.6)

GW flux (Ro=20, Fr=0.6)( )

( )θ

⎧ ⎛ ⎞′ ′ ′ ′ ′ ′= Φ + + Φ + Φ + Φ⎜ ⎟⎪ ⎝ ⎠⎪⎪ ′ ′ ′ ′= Φ + Φ Φ⎨⎪ ⎛ ⎞⎪ ′ ′ ′ ′ ′= Φ + Φ + Φ + Φ + Φ⎜ ⎟⎪ ⎝ ⎠⎩

2 2 2 2 2

2 2 2 2 2 2

1 1 /2 ,2 2

,

/2 .

e

r

dqA u v uu u qdr

F u u v vdqF u u u u u u qdr

∂+ ∇ =

∂F 0.eAt

Conservation of GW flux at r=40

Vorticity Divergence

Parameter sweep exp: Anti-cyclone (Ro=20, Fr=0.6)

Cyclone (C) & Anti-cyclone (AC) asymmetry

Geopotential height

Local maximum at medium Ro for AC

GW flux in Ro-Fr parameter space

Dependence on Ro for Fr=0.6 C & AC Dependence on Fr for Ro=2, 20, inf

Power law of Fr and its breakdown?

Anti-cyclone > Cyclone

| ζ + f | is small for AC : Effective Rossby number?

Spontaneous gravity wave radiation from co-rotating vortex pair in f -plane shallow water system

Analytical estimation for the far field Difficulty in 2D case (historical integral of the source)Weak radiation by the Earth’s rotation

Numerical simulation in the unbounded domainDissipation in the far filedCyclone and anti-cyclone asymmetry for GW fluxLocal maximum of GW flux at medium Ro for AC

Future workEstimate 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.

4. Summary

Balanced flowBalance 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, (2007), p353-377.

Spontaneous GW radiation Fr dependencies (f-plane SH)

Froude Number Dependence of Gravity Wave Radiation From UnsteadyRotational Flow in f-plane Shallow Water System, Norihiko Sugimoto, Keiichi Ishioka, and Shigeo Yoden, Theoretical and Applied Mechanics Japan, 54, (2005), p299-304.

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. 11-12, (2007), p731-754.

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, (2008), p235-249.

Latitudinal change of the earth rotation (Spherical SH)The effect of the Earth Rotation of Spontaneous Gravity Wave Radiation in Shallow WaterSystem on a Rotating Sphere, Norihiko Sugimoto and Katsuya Ishii, Journal of the Meteorological Society Japan, to be submitted.

References

Future work

Parameterizations of spontaneous GW radiation

Frequency spectra of GW sourc

2 layer shallow water model (collaborated with K. Ishii)(Spontaneous internal GW radiation from baroclinic unstable jet)

22

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂+ − Δ = + + −⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂ ∂ ∂⎛ ⎞+ + − + − +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠

144424443 14444444244444443

144444424444443

GW pr opagat i on Vor t ex sour ce（ １ ）

Vor t ex source（ ２ ）

2 2 22 2

0 02 2

2 2 22 2

2

( ) ( )2

(2 ) ( )

h hu gf gH fhuv h Ht t x t t

huv hvfhv fhu fhuvx y t y t

⎛ ⎞∂−⎜ ⎟∂⎝ ⎠14444444244444443

Vor t ex sour ce（ ３ ）

20( )

2g h Ht

Gravity wave source (Lighthill-Ford eq.)

Vortex sound theoryLighthill (1952),Powell(1964), Howe(1975) etc.

GW amplitude for far field

( )∂′ ′ ′ ′ ′= + − − − −

∂ ∫ 0

0 00

, 1 [ ( ( ), ) ( ( ), ),2

( , ) :

t

t

h y tdt G y c t t t G y c t t t

t c

G y t Zonally averaged source

PV dh/dt & Source Geopotential height

T=5.0

Sugimoto et al. (2007b)

( )ω ω −1

2 3 2 2 20

e

B U fF Fr

gCorresponds to the power law of Mach number if f=0

Power law of GW flux

Analogy with vortex sound (Lighthill theory)

Ro dependence Fr=0.1 Fr=0.7

Ro Ro

Fr dependence

Local maximum at Ro=10

Time mean of gravity wave flux averaged at Y=40

Breakdown of the power law of Fr

Ro=10 Ro=100

Fr Fr

Sugimoto et al. (2008)

Parameter dependence of gravity wave flux （Fr, Ro）

Frequency spectra of GW source(Ro dependence for Fr=0.1)

200000 date series between T=5.0～15.0

Increase of the source related to Coriolis term for small Ro

Cut-off ( f /2π)Ro=5 Ro=1

Ro=100 Ro=10

( )∂ ∂= − +

∂ ∂

2 2

22 2hv g hT fhuvt t

= 0URofB

Increase of inertial frequency cut-off for small Ro

Sugimoto et al. (2008)

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