Surface zonal flows
induced by thermal convection
in a rapidly rotating thin spherical shell

Youhei SASAKI∗1 , Shin-ichi Takehiro1 ,Ken-suke Nakajima2 , Yoshi-Yuki Hayashi3∗

contact: uwabami@gfd-dennou.org

1 Kyoto Univ. , 2 Kyushu Univ.,3 Kobe Univ.

Outline

Background

Formulation and experiments setup

Results

Summary

Outline

Background

Formulation and experiments setup

Results

SummarySurface flows of gas giant planets• Surface flows of Jupiter andSaturn are characterized by thebroad prograde jets around theequator and the narrow alternatingjets in mid- and high-latitudes.
• It is not yet clear whether thosesurface jets are produced byconvective motions in the “deep”
region, or are the result of fluid
motions in the “shallow” weatherlayer.
(Vasavada and Showman, 2005)

“Deep” models and “Shallow” models• “Shallow” models◦ 2D turbulence on a rotating sphere◦ Primitive model• Result: Narrow alternating jets in mid-and high-latitudes.
• Problem: the equatorial jets are notnecessarily prograde• “Deep” models◦ Convection in rotating sphericalshells• Result: Produce equatorial progradeflows easily• Problem: difficult to generatealternating jets in mid- andhigh-latitudes

“Thin” spherical shell model• Heimpel and Aurnou(2007)(hereafter, HA2007)◦ “Thin” spherical shell modelwith large Rayleigh number,small Ekman number.
◦ Prograde jets and alternating
jets in mid- and high-latitudes
can produce simultaneously◦ However, eight-fold symmetry in
the longitudinal direction isassumed.

Purpose• HA2007: eight-fold symmetry in the longitudinal direction is assumed.
◦ The artificial limitation of the computational domain may influence on thestructure of the global flow field.
• Zonal flows may not develop efficiently due to the insufficient upward cascade of

two-dimensional turbulence• Stability of mean zonal flows may change with the domain size in the longitudinal

direction.
•• In the present study:◦ Numerical simulations in the whole thin spherical shell domain.
◦ Coarse spatial resolution and slow rotation rate are used due to the limit ofcomputational resources.

Outline

Background

Formulation and experiments setup

Results

SummaryModel setup• Boussinesq fluid in a rotating spherical shell.
◦ scaling: the shell thickness, viscous diffusion time, temperature difference.
∇·u = 0,{ ∂u}RaE2 rE+ (u · ∇)u − ∇2 u + 2k × u + ∇p =T,∂tPr ro∂T
1+ (u · ∇)T = ∇2 T.
∂tPr• Parameters:ν◦ Plandtl number: Pr =κ
αg ∆T D3
◦ Rayleigh number: Ra = oκνν◦ Ekman number: E =ΩD2r◦ radius ratio: η = iroExperimental setup• Boundary condition: Isothermal, Impermeable and Stress free.
• Input parameters:parameters
Prandtl number: PrRadius ratio: ηEkman number: E
Modified Rayleigh number: Ra∗present study

HA20070.1
0.75
10−40.05, 0.1

0.10.85, 0.9
10−60.05◦ the definition of modified Rayleigh number: Ra∗ =• the ratio of Coriolis term and buoyancy term

RaE2 αg∆T D=Pr
Ω2 D

• Output parameters:◦ (local) Reynolds number, Re, is equivalent to the non-dimensional velocity inthe chosen scaling.

◦ (local) Rossby number: Ro = EReNumerical methods• Traditional spectral method.
◦ Toroidal and Poloidal potentials of velocity are introduced.
◦ The total wave number of spherical harmonics is truncated at 170, and theChebychev polynomials are calculated up to the 48th degree.
• The numbers of grid points:512, 256, and 48 in the longitudinal, latitudinal, andradial directions, respectively.

• In order to save computational resources, we use hyperdiffusion with thesame functional form as the previous studies{ν=ν0 ,for l ≤ l0 ,
ν[1 + ε(l − l0 )2 ], for l > l0 .

◦ we choose l0 = 85, ε = 10−2
• The time integration is performed using the Crank-Nicolson scheme forthe diffusion terms and the second-order Adams-Bashforth scheme forthe other terms.

Outline

Background

Formulation and experiments setup

Results

SummaryResults: Ra∗ = 0.05
Zonal velocity (Ra* = 0.05)
Zonal velocity (Ra*=0.05)

Axial Vorticity

00-400

-200

0200

400

Zonal velocity (Ra*=0.05)(1)80• Broad prograde equatorial jet

60• Alternating zonal jets do not
emerge in mid- andhigh-latitudes.

20Latitude

40

0-20-40-60
-80
-500

-400-300

-200

-100

0100

Reynolds number

200300400500(1)

Results: Ra∗ = 0.1
Zonal velocity (Ra* = 0.1)
Zonal velocity (Ra*=0.1)

Axial Vorticity

00-1500-750

0750

1500

Zonal velocity (Ra*=0.1)(1)• Broad prograde equatorial jet

8060• Outer surface: alternating
zonal jets in high latitudes ?

40
20

Latitude• Mean zonal flow: alternating
zonal jets emerge in mid- andhigh-latitudes

0-20-40-60-80-15-10-50

Reynolds number

51015(×100 1)

Results: output parameters• Input parametersparameters
Prandtl number: Pr
Radius ratio: ηEkman number: E
Modified Rayleigh number: Ra∗present studyHA2007

0.1
0.75
10−40.05, 0.1

0.10.85, 0.9
10−60.05• Output parametersparameters
local Reynolds numnber: Re
local Rossby number: Ro◦ small Re, i.e. weak jet◦ Ro, i.e. slow rotation ratepresent study

HA2007

3.59 × 102 , 1.13 × 103
3.59 × 10−2 , 1.13 × 10−1

5 × 104
1.2 × 10−2 , 2.5 × 10−2Outline

Background

Formulation and experiments setup

Results

SummarySummary and discussion.
Summary.
• the profile of mean zonal flow:◦ Broad prograde equatorial jet◦ Alternating zonal jets emerge in mid- and high-latitudes• Outer surface: alternating zonal jets in high latitudes ?
◦ thick shell? , small Ra∗ ?, large E ?
. ◦ hyperdiffusivity?
.
In the future...
.
• More ’thin’, ’fast rotating’ spherical shell convection◦ η = 0.75 η = 0.8, 0.85, 0.9◦ E = 10−4 → E = 3 × 10−5
• Investigation of generation mechanism. ◦ Comparison with the Rhines scale