25pXB: Symposium on structure formation and dynamics in turbulence:
March 25, 2005, at the 60th annual meeting of the Physical Society of Japan
2D Turbulence and Structure in
Planetary Atmospheres
Shigeo YODEN, Keiichi ISHIOKA,
and Jitsuko HASEGAWA
Department of Geophysics, Kyoto University
1.
2.
3.
4.
5.
6.
Introduction
Model and Numerical Procedure
Jet Formation
Asymmetry of Eastward and Westward Jet Profiles
Mechanism: Wave-Mean Flow Interaction
Concluding Remarks
1. Introduction
A series of approximations of the motions of
planetary atmospheres
• incompressible, homogeneous fluid (like oceans)
Navier-Stokes Eqs. of 3D fluid on a rotating sphere
• shallowness (L~10000km, H~1km for oceans)
Shallow Water Eqs. of 2D flow
• slow and nearly nondivergent flow (Charney, 1948)
Quasi-Geostrophic Potential Vorticity Eq.
• purely nondivergent flow
2D Vorticity Eq. on a rotating sphere
A briew on 2D turbulence
• Decaying turbulence governed by
2D vorticity equation for
incompressible, homogeneous fluid:
¶z
+ J ( y , z ) = - n 2D 2z ,
¶t
where y (x , y , t ) : stream function,
z = Ñ 2 y : relative vorticity,
n 2 : hyperviscosity coefficient
• McWilliams(1984):
Emergence of coherent vortices from
a random flow field
Evolution of
vorticity field
McWilliams
(1984)
• 2D decaying turbulence on a beta-plane
¶z
¶y
+ J (y , z ) + b
= - n 2D 2z ,
¶t
¶x
where the Coriolis parameter f = f 0 + b y
• Rhines(1975):
Charney-Hasegawa-Mima
equation
y
zonal band structure in -field
β= 0
β= 52
¶y
¶
(z - y ) + C R
+ J ( y , z ) = 0,
¶t
¶x
where z = Ñ 2 y
Charney, J.G., 1948: On the scale of atmospheric
motions. Geofys Publik. XVII, 1-27.
Hasegawa, A. & K. Mima, 1978: Pseudo-threedimensional turbulence in magnetized nonuniform
plasma. Phys. Fluids, 21, 87-92.
• Williams(1978):
Forced 2D turbulence on a rotating sphere
→ zonal band structure similar to Jupiter
snapshot of forcing
Zonal mean
http://nssdc.gsfc.nasa.gov/image/ zonal flow
planetary/jupiter/jupiter_gany.jpg
• 2D turbulence on a rotating sphere with
high resolution models:
Yoden & Yamada(1993), Cho & Polvani(1996a,b),
Nozawa & Yoden(1997a,b), Yoden et al.(1999), . . .
• Yoden & Yamada(1993):
emergence of
a westward
circumpolar vortex
+
zonal jet structure
in low-latitudes
• Ishioka et al. (1999, Nagare Multimedia )
– Pattern Formation from 2D Decaying
Turbulence on a Rotating Sphere
– http://www.nagare.or.jp/mm/99/ishioka/
Ω=0
Ω=50
Ω=400
In this study, we perform a numerical
experiment on 2D decaying turbulence
on a beta-plane, to survey the nature of
zonal jet formation from a random
initial field.
• When, where and how do the jets form?
• Are there any asymmetric difference
between the eastward and westward jets?
2. Model and Numerical Procedure
• 2D vorticity equation for incompressible,
homogeneous fluid on a doubly periodic
beta-plane:
¶y
¶z
+ J (y , z ) + b
= - n 2 D 2z ,
¶t
¶x
z = Ñ 2y
(0 £ x £ 2p , 0 £ y £ 2p )
• Initial condition:
1D kinetic energy spectrum for a scalar
wavenumber k :
Ak g / 2
E (k , t = 0) =
g ,
(k + k 0 )
with g = 1000 and E = 0.5.
• a spectral model: ISPACK (Ishioka, 1999)
y (x , y , t ) =
å å
k
Y lk (t ) e i (kx + ly )
l
- a low resolution model
k max = l max = 128
k 0 = 26
n 2 = 1 ´ 10- 8
(512 ´ 512 grids)
- a high resolution model
k max = l max = 1024
k 0 = 228
n 2 = 1 ´ 10- 10
(4096 ´ 4096 grids)
• time integrations:
- 4th-order Runge-Kutta method
t max = 20 (low),
12 (high)
Vt = 0.004 (low), 0.0004 (high)
3. Jet Formation
• Parameter-sweep in a low resolution model
• characteristic (Rhines) wavenumber
kb = b / U
example : b = 200 ; k b : 14 < k 0 = 26
• Time evolution of relative vorticity ζ
and zonal mean zonal flow
(Animation) Rossby wave motions
• Time evolution of potential vorticity
q = ζ+ f0 + βy and dq/dy
(Animation)
• Time evolutions of zonal mean zonal flow,
total energy, total enstrophy, and
enstrophy dissipation rate
4. Asymmetry of Eastward and Westward
Jet Profiles
• ensembles with a high resolution model
• jet: local max. or min.
μ±1.645σ
Number: 36
Number: 42
Max.: 0.6509
Min.: -1.0289
Westward Jets
Westward Jet
Eastward Jets
Eastward Jet
• composites of strong jets
5. Mechanism: Wave-Mean Flow Interaction
• Rossby wave motion is dominant
in the period of jet formation
¶ (2)
• conservation law of pseudo-momentum:
u - v 'z ' = 0
¶ é (2)
ù» 0
u
+
A
ú
û
¶ t êë
¶t
¶ z '2
+ bˆ v ' z ' = 0
¶t 2
deceleration of mean zonal wind ~
2
¶ é (2)
z
'
ù= 0, A =
change of Rossby-wave
activity
u
+
A
ê
ú
¶t ë
û
• behavior of Rossby waves in zonal jets
determines the acceleration of jets
2bˆ
• Meridional propagation of Rossby-wave packet
in sinusoidal zonal jets
– under WKB approximation,
conservation of frequency and zonal wavenumber
change of meridional wavenumber and group velocity
U 0.5 sin y, 6400, k 64, l0 64
ˆk 22
ˆkˆ
ˆ
ˆ
2
2
l
kl k
k
k 2c 2 k l
gy
U yU
ˆ
ˆU
U U crt
crt
(k 2 l 2 ) 2
k crt
2
U ytrn
Westward Jet
basic
zonal flow
k
k2
large dissipation of Rossby waves
and
large l
small cgy
large acceleration of westward jet
meridional
wavenumber
meridional
group velocity
• Formation mechanism of
asymmetric zonal jets
– initial random field ~
superposition of Rossby wave packets
– propagation of Rossby wave packets
in meridionally varying mean zonal flow
– large meridional wavenumber and
slow propagation in westward jets
– selective damping in westward jets produces
sharper and stronger westward jets
• Confirmation with a linear model
– linearized vorticity equation:
¶z '
¶y '
= - J (Y, z ')- J (y ', Z )- b
- n 2 D 2z '
¶t
¶x
with time-constant sinusoidal mean zonal flow
– evaluation of the acceleration of mean zonal
flow by the wave effect
monochromatic wave
basic flow
acceleration
random field
basic flow
acceleration
6. Concluding Remarks
• Zonal jets form at an early stage just after
the maximum of enstrophy dissipation,
and persist quite robustly.
• Asymmetry of the eastward and westward
jet profiles exists:
Westward jets are sharper and stronger.
• A weakly nonlinear interaction theory
between Rossby waves and mean zonal flow
explains the asymmetry of the jet profiles.
Wave-Mean Flow Interaction
wave activity equation :
ur
¶A
+ Ñ ×F = 0,
¶t
where wave activity A º z '2/ (2bµ), bµ = b - U y y ,
ur
and its flux F = (U A + (v '2 + u '2 ) / 2, - u ' v ')
zonal mean zonal flow equation :
ur
¶ (2)
u - Ñ ×F = 0,
¶t
where
denotes zonal mean
conservation law of pseudo - momentum :
¶
{u (2) + A } = 0.
¶t
▸ロスビー波の擬運動量保存則
u U u 'u ( 2) ,
v'v ( 2) ,
v
Z ' ( 2)
基本場と擾乱
→ 擾乱の振幅展開
•運動量保存則
( 2)
u v' ' 0
t
•エンストロフィー保存則
'2 ˆ
v' ' 0
t 2
•擬運動量保存則
( 2)
u A 0,
t
Wave activityの流出
⇒
西風加速
Wave activityの流入
⇒
東風加速
'2
A
2 ˆ
▸ロスビー波パケットの運動
•擬運動量保存則の確認
southward
northward
kˆ
分散関係は U k 2 2 U k ˆ で表され、
k l
kˆ
波数kと U yinit k 2 2 を保存して運動。
k l
ˆ
波数lは分散関係から l
k2
U / k
2 ˆ k l
2
また、ｙ方向群速度は cgy
(k l )
2
2 2
で決まる。
である。
▸粘性の比較（高階粘性・通常の粘性）
ここではx方向波数が変化しないから、y方向の1次元で書ける。
•非線形実験に即した状況
linear-1d-nocrt
→緩やかな西風加速と
鋭い東風加速
•普通の粘性を与えた場合
程度は小さいものの、
東風に鋭い加速
• Vallis and Maltrud(1993): Dumbbell of
anisotropic wave-turbulence boundary
(k, l ) = k b (cos3 / 2 q, cos1/ 2 q sin q ),
where Rhines wavenumber k b = b / U ,
and q = tan- 1(l / k )
• qualitatively OK
• anisotropy even
for large k > k b
β = 100 ; kβ = 10
• Time evolution of 2-D Energy Spectrum
κ0 = 26
• 1-D Energy Spectrum energy weighted k kβ
• arguments with dimension
L = 5 ´ 106 m ( ~ radius of the earth)
U = 10 m/ s
dimensional value of beta
b * = (U / L2 )b = 4 ´ 10-
11
(ms)-
1
for b = 100
• If we take 8L as a new length scale and
pick up only 1/8 in both directions x, y,
then the results in the subdomain are
regarded as another experiment:
b * = (U / L2 )b / 64
b * = 6 ´ 10- 13 (ms)-
1
for b = 100
• Zonal mean zonal flow for different β*
• Lateral scale of jets
k0 :
energy weighted
wavenumber for
k = 0 component
κ0
k β*
k0
• Dagnosis of zonal-jet fluctuations
(low-resolution model, β = 200)
U(y)
U(y,t)
＾β(y) δ[A]/δt
δU/δt
[div F]
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