Disappearance of surface banded structure
produced by thermal convection
in a rapidly rotating thin spherical shell
Shin-ichi Takehiro
Research Institute for Mathematical Sciences, Kyoto University

May 14, 2015

In collaboration withYouhei Sasaki (Kyoto Univ.), Keiichi Ishioka (Kyoto Univ.),
Kensuke Nakajima (Kyushu Univ), Yoshi-Yuki Hayashi (Kobe Univ.)
Japanese-French model studies of planetary atmospheres @ CPS, Kobe Univ.
(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 2015

1 / 20Outline

1Background2

Numerical experiments

3

References(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 2015

2 / 20Outline

1Background2

Numerical experiments

3References(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 2015

3 / 20Surface flows of gas giant planets

Surface flows of Jupiter andSaturn are characterized by thebroad prograde jets around theequator and the narrowalternating jets in mid- andhigh-latitudes.
It is not yet clear whether thosesurface jets are produced byconvective motions in the“deep” region, or are the resultof fluid motions in the “shallow”weather layer.

(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 2015

4 / 20“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

Recent improvements
Strong Newtonian cooling (shallowwater)
MHD drag (primitive model)“Deep” models
Convection in rotating spherical shells
Result: Produce equatorial prograde
flows easily
Problem: diﬃcult to generatealternating jets in mid- andhigh-latitudes(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 2015

5 / 20“Thin” spherical shell model

Heimpel and Aurnou (2007)(hereafter, HA2007)“Thin” spherical shellmodel with large Rayleighnumber, small Ekman
number.
Prograde jets andalternating jets in midand high-latitudes can
produce simultaneously
However, eight-foldsymmetry in thelongitudinal direction isassumed.

(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 2015

6 / 20Purpose
HA2007 explains that the banded structure in high latitudes is causedby forced 2D turbulence on the β plane (Rhiens eﬀect)
Forced 2D barotropic turbulence on a rotating sphere :banded structure disappears after long time integration(Obuse et al. 2010).
Question : does the banded structure of HA2007 disappear after longtime integration or not?

In the present study:
Let us perform long time integration of thermal convection in arotating thin spherical shell.
Calculate in the whole domain.

(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 2015

7 / 20Forced 2D barotropic turbulence on a rotating sphere
Obuse et al. (2010)
Merger and disappearance of jets ⇒ banded structure disappears

Time development of latitudinal distribution of angular momentum (left) and final zonal windprofile (right) (Obuse et al. 2010)(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 20158 / 20Outline

1

Background2

Numerical experiments

3References(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 2015

9 / 20Model setup
Boussinesq fluid in a rotating spherical shell.
Scaling: the shell thickness, rotation period, temperature diﬀerence.
∇·u = 0,∂u
RaE2 r+ (u · ∇)u − E∇2 u + 2k × u + ∇p =T,∂tPr ro∂TE 2+ (u · ∇)T =∇ T.
∂tPrParameters:νκ
αgo ∆T D3
Rayleigh number: Ra =κνν
Ekman number: E =ΩD2ri
radius ratio: η =roPrandtl number: Pr =(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 201510 / 20

Experimental setup

Boundary condition: Isothermal, Impermeable and Stress free.
Input parameters:Prandtl number: Pr
Radius ratio: ηEkman number: E
Modified Rayleigh number: Ra∗

0.10.85
3 × 10−60.05

the definition of modified Rayleigh number: Ra∗ =RaE2αg∆T= 2PrΩ D

Ratio of Coriolis term and buoyancy term

(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 201511 / 20

Numerical methods
Traditional spectral method.
Toroidal and Poloidal potentials of velocity are introduced.
The total wave number of spherical harmonics is truncated at 341, andthe Chebychev polynomials are calculated up to the 48th degree.
The numbers of grid points:1024, 512, and 65 in the longitudinal,latitudinal, and radial directions, respectively.

In order to save computational resources, we use hyperdiﬀusion withthe same functional form as the previous studies{
ν0 ,for l ≤ l0 ,ν=2
ν[1 + ε(l − l0 ) ], for l > l0 .
we choose l0 = 21, 42, 85, 170, ε = 10−2
(l0 is gradually increased)

The time integration is performed using the Crank-Nicolson scheme
for the diﬀusion terms and the second-order Adams-Bashforth scheme
for the other terms.
(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 201512 / 20

Convective activity

Radial velocity at t = 64030 (about 10000 rotation).

(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 201513 / 20

Vorticity

Axial vorticity at t = 64030 (zonal mean component removed)(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 201514 / 20

Mean zonal flows
Mean zonal flow at t = 47630 (about 7500 rotation).
Banded structure appears in mid- and high-latitudes.

Meridional plane(RIMS, Kyoto Univ.)

Outer surface

Thermal convection in rotating thin shells

May 14, 201515 / 20

Development of zonal flows
Further time integration.
Number of jets decreases

t = 47630(RIMS, Kyoto Univ.)t = 64030

Thermal convection in rotating thin shells

t = 80430
May 14, 201516 / 20

Summary
Disappearance of alternating jets in high latitudes
Prograde acceleration in high latitudes
Retrograde acceleration around the tangential cylinder

What is acceleration mechanism? One possible story is...
Thermal convection occurs in high latitudes
It excites 2-dim. axial vortices
The vortices propagate as Rossby waves outward and are absorbed near
the tangent cylinder
Associated with the propagation of Rossby waves, negative angularmomentum is transferred outward

Deep models may by diﬃcult to explain the banded structure ofJupiter and Saturn?
Acknowledgement
The numerical computation of thermal convection in a rotating thin spherical shell wascarried out on the Earth Simulator (ES2) at the Japan Agency for Marine Earth Scienceand Technology.
(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 201517 / 20

Outline

1

Background2

Numerical experiments

3References(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 201518 / 20

References

Heimpel, M., Aurnou, J., 2007 : Turbulent convection in rapidly
rotating spherical shells: A model for equatorial and high latitude jets
on Jupiter and Saturn. Icarus, 187, 540–557.
Obuse, K, Takehiro, S., Yamada, M., 2010 : Long-time asymptotic
states of forced two-dimensional barotropic incompressible flows on arotating sphere. Phys. Fluids, 22, 056601.
Sukoriansky, S., Galperin, B., Dikovskaya, N., 2002 : Universal
spectrum of two-dimensional turbulence on a rotating sphere andsome basic features of atmospheric circulation on giant planets. Phys.
Rev. Lett., 89, 124501-1–4.

(RIMS, Kyoto Univ.)

Thermal convection in rotating thin shells

May 14, 201519 / 20