The marginal Raleigh number is plotted against the aspect ratio for different values of at the onset of hydrothermal convection.
Sketch of the domain decomposition in spherical-shell geometry.
Kinetic energies of nonlinear convection (a) at the aspect ratio for five different Rayleigh numbers and (b) at the aspect ratio for six different Rayleigh numbers are shown as a function of time.
Isosurfaces of the temperature of nonlinear convection at : (a) for the Rayleigh number and (b) for . Red contours indicate positive temperature and blue contours correspond to negative temperature .
Isosurfaces of the temperature of nonlinear convection at and obtained from four different initial conditions: (a) with , , (b) with , , (c) with , , (d) with , .
Isosurfaces of of nonlinear convection for at , where , obtained with different initial conditions: (a) , (b) , (c) , (d) , (e) , (f) , (g) , and (h) .
Typical values of the marginal Rayleigh number as a function of for various values of . The most unstable mode is denoted by the bold number.
Dominant spherical harmonics in weakly nonlinear stationary convection solutions at with obtained by starting simulation from different initial conditions.
The accuracy of the numerical solutions at different spatial resolution, where , , and are the grid numbers in the azimuthal, latitudinal, and radial directions and denotes the time step in numerical simulations.
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