^{1}, Shin-ichi Takehiro

^{1}and Michio Yamada

^{1}

### Abstract

Stability and bifurcation of Boussinesq thermal convection in a moderately rotating spherical shell are investigated by obtaining finite-amplitude solutions with the Newton method instead of the numerical time integration. The ratio of the inner and outer radii of the shell and the Prandtl number are fixed to 0.4 and 1, respectively, while the Taylor number is varied from 52^{2} to 500^{2} and the Rayleigh number is from about 1500 to 10 000. In this range of the Taylor number, the stable finite-amplitude solutions, which have four-fold symmetry in the longitudinal (azimuthal) direction, bifurcate supercritically at the critical points and become unstable when the Rayleigh number is increased up to about 1.2 to 2 times the critical values. When the Taylor number is larger than 340^{2}, propagating direction of the solutions changes from prograde to retrograde continuously as the Rayleigh number is increased. The associated transition of the convection structure is also continuous.

Numerical calculations were performed with the computer systems of the Institute for Information Management and Communication (IIMC) of Kyoto University. For the calculation of the critical modes and the nonlinear finite-amplitude solutions, we used the library for spectral transform ISPACK (http://www.gfd-dennou.org/library/ispack/) and its Fortran90 wrapper library SPMODEL library (http://www.gfd-dennou.org/library/spmodel/). The subroutines of the Fujitsu SSL II were used in the calculation of eigenvalues to examine the stability of solutions. The products of the Dennou Ruby project (http://www.gfd-dennou.org/library/ruby/) were used to draw the figures. This work was supported by a Grant in-Aid for the Global COE Program “Fostering top leaders in mathematics–broadening the core and exploring new ground” from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan.

I. INTRODUCTION

II. MODEL AND NUMERICAL METHOD

III. RESULTS

A. Marginal stability solutions of the conductive state

B. Nonlinear solutions and a bifurcation diagram

IV. CONCLUSION AND DISCUSSION

### Key Topics

- Convection
- 33.0
- Zonal flows
- 25.0
- Rotating flows
- 19.0
- Thermal convection
- 13.0
- Bifurcations
- 12.0

## Figures

A schematic picture of the configuration of the problem.

A schematic picture of the configuration of the problem.

(Color online) The phase velocity of the critical modes [*v* _{ p }]_{ c } in slowly and moderately rotating cases (0 ≤ τ ≤ 1000). Positive values mean prograde propagation. *m* _{ c } means the critical longitudinal (azimuthal) wavenumber. The inset is the enlarged drawing of the transition region of [*v* _{ p }]_{ c }.

(Color online) The phase velocity of the critical modes [*v* _{ p }]_{ c } in slowly and moderately rotating cases (0 ≤ τ ≤ 1000). Positive values mean prograde propagation. *m* _{ c } means the critical longitudinal (azimuthal) wavenumber. The inset is the enlarged drawing of the transition region of [*v* _{ p }]_{ c }.

(Color online) Distributions of the radial velocity *u* _{ r } in the equatorial plane (θ = 90°) and the axial components of vorticity ω_{ z } = ** k ** · (▿ ×

**) in the meridional plane (Φ = 22.5°) for the critical modes (**

*u**R*=

*R*

_{ c }). The rotation rate τ increases from (a) to (f). Convection patterns of (a)–(d) propagate in the retrograde direction, while those of (e) and (f) propagate in the prograde direction. The detailed parameters for each panel are listed in Table I.

(Color online) Distributions of the radial velocity *u* _{ r } in the equatorial plane (θ = 90°) and the axial components of vorticity ω_{ z } = ** k ** · (▿ ×

**) in the meridional plane (Φ = 22.5°) for the critical modes (**

*u**R*=

*R*

_{ c }). The rotation rate τ increases from (a) to (f). Convection patterns of (a)–(d) propagate in the retrograde direction, while those of (e) and (f) propagate in the prograde direction. The detailed parameters for each panel are listed in Table I.

(Color online) A bifurcation diagram of the stable finite-amplitude solutions which have four-fold symmetry in the longitudinal (azimuthal) direction (TW4s). The propagating direction of the solution is shown by a blue circle (retrograde) and a red triangle (prograde). The lower solid curve shows the marginal stability of the stationary (conductive) solution, where the blue curve (τ < 340) shows that the propagating direction is retrograde, and the red curve (τ ≥ 340) prograde. All circles and triangles mean that the nonlinear solutions are stable. TW4s become unstable above the upper black solid line. The propagating velocity *v* _{ p } vanishes on the dashed line. The blue crosses mean that the nonlinear solutions propagating in the retrograde direction are unstable.

(Color online) A bifurcation diagram of the stable finite-amplitude solutions which have four-fold symmetry in the longitudinal (azimuthal) direction (TW4s). The propagating direction of the solution is shown by a blue circle (retrograde) and a red triangle (prograde). The lower solid curve shows the marginal stability of the stationary (conductive) solution, where the blue curve (τ < 340) shows that the propagating direction is retrograde, and the red curve (τ ≥ 340) prograde. All circles and triangles mean that the nonlinear solutions are stable. TW4s become unstable above the upper black solid line. The propagating velocity *v* _{ p } vanishes on the dashed line. The blue crosses mean that the nonlinear solutions propagating in the retrograde direction are unstable.

(Color online) Distributions of the radial velocity *u* _{ r } in the equatorial plane (θ = 90°) and the axial components of vorticity ω_{ z } = ** k ** · (▿ ×

**) in the meridional plane (Φ = 22.5°) for the nonlinear stable solutions of TW4s at slightly supercritical points (**

*u**R*≃ 1.01

*R*

_{ c }). The rotation rate τ increases from (a) to (f). Convection patterns of (a)–(d) propagate in the retrograde direction, while those of (e) and (f) propagate in the prograde direction. The detailed parameters for each panel are listed in Table I.

(Color online) Distributions of the radial velocity *u* _{ r } in the equatorial plane (θ = 90°) and the axial components of vorticity ω_{ z } = ** k ** · (▿ ×

**) in the meridional plane (Φ = 22.5°) for the nonlinear stable solutions of TW4s at slightly supercritical points (**

*u**R*≃ 1.01

*R*

_{ c }). The rotation rate τ increases from (a) to (f). Convection patterns of (a)–(d) propagate in the retrograde direction, while those of (e) and (f) propagate in the prograde direction. The detailed parameters for each panel are listed in Table I.

(Color online) The phase velocity *v* _{ p } of the stable solutions of TW4s at τ = 52, 70, 100, 200, 300, 400, and 500. The phase velocity of the critical modes is shown as a dashed line.

(Color online) The phase velocity *v* _{ p } of the stable solutions of TW4s at τ = 52, 70, 100, 200, 300, 400, and 500. The phase velocity of the critical modes is shown as a dashed line.

(Color online) Distributions of the axial component of vorticity ω_{ z } in the meridional plane (Φ = 22.5°) of the nonlinear stable solutions of TW4 at τ = 400 for several values of the Rayleigh number. The Rayleigh number is increased from (i) to (iv). TW4s of (i) and (ii) propagate in the prograde direction, while TW4s of (iii) and (iv) propagate in the retrograde direction. The detailed parameters for each panel are listed in the Table II.

(Color online) Distributions of the axial component of vorticity ω_{ z } in the meridional plane (Φ = 22.5°) of the nonlinear stable solutions of TW4 at τ = 400 for several values of the Rayleigh number. The Rayleigh number is increased from (i) to (iv). TW4s of (i) and (ii) propagate in the prograde direction, while TW4s of (iii) and (iv) propagate in the retrograde direction. The detailed parameters for each panel are listed in the Table II.

(Color online) Distributions of the radial velocity *u* _{ r } in the equatorial plane (θ = 90°) and the axial components of vorticity ω_{ z } = ** k ** · (▿ ×

**) in the meridional plane (Φ = 22.5°) for the stable large amplitude solutions of TW4s. The Rayleigh number of each panel is increased to the value slightly below the marginal stability of TW4 from that of each panel in Figure 5. From (I) to (VI) the rotation rate τ is increased. All the convection patterns propagate in the retrograde direction. The detailed parameters for each panel are listed in Table III.**

*u*(Color online) Distributions of the radial velocity *u* _{ r } in the equatorial plane (θ = 90°) and the axial components of vorticity ω_{ z } = ** k ** · (▿ ×

**) in the meridional plane (Φ = 22.5°) for the stable large amplitude solutions of TW4s. The Rayleigh number of each panel is increased to the value slightly below the marginal stability of TW4 from that of each panel in Figure 5. From (I) to (VI) the rotation rate τ is increased. All the convection patterns propagate in the retrograde direction. The detailed parameters for each panel are listed in Table III.**

*u*(Color online) Distributions of the mean meridional fields of the stable large amplitude solutions of TW4s. Left six panels show the distributions of mean zonal flows 〈*u* _{φ}〉. Right six panels illustrate mean meridional circulations 〈*u* _{ r }〉 and 〈*u* _{θ}〉 (vectors) with the mean temperature disturbance 〈Θ〉 (tone). From (I) to (VI) the rotation rate τ is increased. The Rayleigh number is the value slightly below the marginal stability. The detailed parameters for each panel are listed in Table III.

(Color online) Distributions of the mean meridional fields of the stable large amplitude solutions of TW4s. Left six panels show the distributions of mean zonal flows 〈*u* _{φ}〉. Right six panels illustrate mean meridional circulations 〈*u* _{ r }〉 and 〈*u* _{θ}〉 (vectors) with the mean temperature disturbance 〈Θ〉 (tone). From (I) to (VI) the rotation rate τ is increased. The Rayleigh number is the value slightly below the marginal stability. The detailed parameters for each panel are listed in Table III.

(Color online) Schematic diagrams of the propagation mechanism of vortices. (Figure 8 in Takehiro (Ref. 13), revised.)

(Color online) Schematic diagrams of the propagation mechanism of vortices. (Figure 8 in Takehiro (Ref. 13), revised.)

(Color online) The propagating velocity *v* _{ p } (black) of the stable TW4 solutions and the maximum (red) and the minimum (blue) values of their mean zonal flow in the meridional plane (τ = 400).

(Color online) The propagating velocity *v* _{ p } (black) of the stable TW4 solutions and the maximum (red) and the minimum (blue) values of their mean zonal flow in the meridional plane (τ = 400).

(Color online) Distributions of mean zonal flow of the stable TW4s at τ = 400. The isolines of the axial components of vorticity in the meridional plane (Φ = 22.5°) are also shown. The detailed parameters for each panel are the same as in Figure 7, and are listed in Table II.

(Color online) Distributions of mean zonal flow of the stable TW4s at τ = 400. The isolines of the axial components of vorticity in the meridional plane (Φ = 22.5°) are also shown. The detailed parameters for each panel are the same as in Figure 7, and are listed in Table II.

(Color online) Distributions of mean zonal flow of the stable TW4s at τ = 340. The isolines of the axial components of vorticity in the meridional plane (Φ = 22.5°) are also shown. The detailed parameters for each panel are listed in Table IV.

(Color online) Distributions of mean zonal flow of the stable TW4s at τ = 340. The isolines of the axial components of vorticity in the meridional plane (Φ = 22.5°) are also shown. The detailed parameters for each panel are listed in Table IV.

(Color online) Distributions of mean zonal flow of the large amplitude stable TW4s at each τ. The isolines of the axial components of vorticity ω_{ z } in the meridional plane (Φ = 22.5°) are also shown. The detailed parameters for each panel are the same as in Figure 8 and are listed in Table III.

(Color online) Distributions of mean zonal flow of the large amplitude stable TW4s at each τ. The isolines of the axial components of vorticity ω_{ z } in the meridional plane (Φ = 22.5°) are also shown. The detailed parameters for each panel are the same as in Figure 8 and are listed in Table III.

## Tables

The parameters of critical modes shown in the panels of Figure 3 and those of typical finite-amplitude solutions near the critical points shown in the panels of Figure 5. Cases (a) to (f) correspond to those in Figure 3 and in Figure 5. *R* _{ c } and [*v* _{ p }]_{ c } are the critical Rayleigh number and the critical propagating velocity for each τ, respectively. *v* _{ p } is the propagating velocity of finite-amplitude stable solution of TW4 for corresponding *R* and τ. Both critical modes and TW4s of (a)–(d) propagate in the retrograde direction, while those of (e) and (f) propagate in the prograde direction.

The parameters of critical modes shown in the panels of Figure 3 and those of typical finite-amplitude solutions near the critical points shown in the panels of Figure 5. Cases (a) to (f) correspond to those in Figure 3 and in Figure 5. *R* _{ c } and [*v* _{ p }]_{ c } are the critical Rayleigh number and the critical propagating velocity for each τ, respectively. *v* _{ p } is the propagating velocity of finite-amplitude stable solution of TW4 for corresponding *R* and τ. Both critical modes and TW4s of (a)–(d) propagate in the retrograde direction, while those of (e) and (f) propagate in the prograde direction.

The Rayleigh number and the propagating velocity of the stable solutions of TW4s at τ = 400. Cases (i)–(iv) correspond to those in Figures 7 and 12. TW4s of (i) and (ii) propagate in the prograde direction, while TW4s of (iii) and (iv) propagate in the retrograde direction.

The Rayleigh number and the propagating velocity of the stable solutions of TW4s at τ = 400. Cases (i)–(iv) correspond to those in Figures 7 and 12. TW4s of (i) and (ii) propagate in the prograde direction, while TW4s of (iii) and (iv) propagate in the retrograde direction.

The parameters of typical large amplitude solutions shown in the panels of Figures 8, 9, and 14. Cases (I) to (VI) correspond to those in Figures 8, 9, and 14*v* _{ p } is the propagating velocity of TW4 solution for corresponding *R* and τ. The last column shows the Rayleigh number where the TW4 solution becomes unstable for each τ. All the TW4s propagate in the retrograde direction. The asterisk appearing in case (VI) means the result using the truncation wavenumbers (*N,L*) = (16,21).

The parameters of typical large amplitude solutions shown in the panels of Figures 8, 9, and 14. Cases (I) to (VI) correspond to those in Figures 8, 9, and 14*v* _{ p } is the propagating velocity of TW4 solution for corresponding *R* and τ. The last column shows the Rayleigh number where the TW4 solution becomes unstable for each τ. All the TW4s propagate in the retrograde direction. The asterisk appearing in case (VI) means the result using the truncation wavenumbers (*N,L*) = (16,21).

The Rayleigh number and the propagating velocity of the stable solutions of TW4s at τ = 340. Cases (i)–(iv) correspond to those in Figure 13. TW4 of (i) propagates in the prograde direction, while TW4s of (ii), (iii), and (iv) propagate in the retrograde direction.

The Rayleigh number and the propagating velocity of the stable solutions of TW4s at τ = 340. Cases (i)–(iv) correspond to those in Figure 13. TW4 of (i) propagates in the prograde direction, while TW4s of (ii), (iii), and (iv) propagate in the retrograde direction.

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