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Stability and bifurcation diagram of Boussinesq thermal convection in a moderately rotating spherical shell
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10.1063/1.3602917
/content/aip/journal/pof2/23/7/10.1063/1.3602917
http://aip.metastore.ingenta.com/content/aip/journal/pof2/23/7/10.1063/1.3602917

Figures

Image of FIG. 1.
FIG. 1.

A schematic picture of the configuration of the problem.

Image of FIG. 2.
FIG. 2.

(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 .

Image of FIG. 3.
FIG. 3.

(Color online) Distributions of the radial velocity u r in the equatorial plane (θ = 90°) and the axial components of vorticity ω z  =  k · (▿ ×  u ) in the meridional plane (Φ = 22.5°) for the critical modes (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.

Image of FIG. 4.
FIG. 4.

(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.

Image of FIG. 5.
FIG. 5.

(Color online) Distributions of the radial velocity u r in the equatorial plane (θ = 90°) and the axial components of vorticity ω z  =  k · (▿ ×  u ) in the meridional plane (Φ = 22.5°) for the nonlinear stable solutions of TW4s at slightly supercritical points (R ≃ 1.01R 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.

Image of FIG. 6.
FIG. 6.

(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.

Image of FIG. 7.
FIG. 7.

(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.

Image of FIG. 8.
FIG. 8.

(Color online) Distributions of the radial velocity u r in the equatorial plane (θ = 90°) and the axial components of vorticity ω z  =  k · (▿ ×  u ) 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.

Image of FIG. 9.
FIG. 9.

(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.

Image of FIG. 10.
FIG. 10.

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

Image of FIG. 11.
FIG. 11.

(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).

Image of FIG. 12.
FIG. 12.

(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.

Image of FIG. 13.
FIG. 13.

(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.

Image of FIG. 14.
FIG. 14.

(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

Generic image for table
Table I.

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.

Generic image for table
Table II.

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.

Generic image for table
Table III.

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 14v 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).

Generic image for table
Table IV.

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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/content/aip/journal/pof2/23/7/10.1063/1.3602917
2011-07-01
2014-04-21
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Stability and bifurcation diagram of Boussinesq thermal convection in a moderately rotating spherical shell
http://aip.metastore.ingenta.com/content/aip/journal/pof2/23/7/10.1063/1.3602917
10.1063/1.3602917
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