Numerical study of the onset of thermosolutal convection in rotating spherical shells

Abstract

The influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell is studied for Ekman numbers E = 10−3 and E = 10−6, Prandtl numbers σ = 0.1, 0.001, Lewis numbers τ = 0.01, 0.1, 0.8, and radius ratio η = 0.35. The Boussinesq approximation of the governing equations is derived by taking the denser component of the mixture for the equation of the concentration. Differential and internal heating, an external compositional gradient, and the Soret and Dufour effects are included in the model. By neglecting these two last effects, and by considering only differential heating, it is found that the critical thermal Rayleigh number Rec depends strongly on the direction of the compositional gradient. The results are compared with those obtained previously for pure fluids of the same σ. The influence of the mixture becomes significant when the compositional Rayleigh number Rc is at least of the same order of magnitude as the known Rec computed without mixture. For positive and sufficiently large compositional gradients, Rec decreases and changes sign, indicating that the compositional convection becomes the main source of instability. Then the critical wave number mc decreases, and the drifting waves slow down drastically giving rise to an almost stationary pattern of convection. Negative gradients delay the onset of convection and determine a substantial increase of mc and ωc for Rc sufficiently high. Potential laws are obtained numerically from the dependence of Rec and of the critical frequency ωc on Rc, for the moderate and small Ekman numbers explored.Postprint (published version

The onset of thermo-compositional convection in rotating spherical shells

“…We employ the Boussinesq approximation in that variations in density are assumed important only when affecting the gravitational force −ργr, with γ a constant. In order to isolate the effects induced by differences in thermal and chemical diffusivities we follow (Busse 2002, Net et al 2012) and disregard any differences in source-sink distribution and in boundary conditions for the temperature T and the concentration C. Static profiles T (r) and C(r) with radial gradients ∂ r T = −β T r and ∂ r C = −β C r then exist assuming the temperature and concentration are fixed at the boundaries and have uniformly distributed sources with constant densities β T and β C , respectively.…”

“…When the instability is driven by the small-diffusivity component it is called fingering; when the instability driven by the large-diffusivity component it is known as diffusive convection; see (Radko 2013). The linear onset in a two-component fluid layer constrained between two horizontal boundaries was studied by (Stern 1960, Nield 1967, Veronis 1968, Baines and Gill 1969 who naturally focussed on the fingering and diffusive regimes. The effect of rotation with respect to a vertical axis was considered by Pearlstein (1981) who found that a non-rotating layer can be destabilized by rotation, and that a rotating layer can be destabilized by the addition of a bottom-heavy solute gradient.…”

“…It was also observed that while known asymptotic expressions for the critical Rayleigh number and frequency derived by Busse (2002) describe the onset of convection over an extended range of non-asymptotic parameter values but do not capture the full complexity of the critical curves. Net et al (2012) studied numerically the influence of an externally enforced compositional gradient on the onset of convection of a mixture of two components in a rotating fluid spherical shell. Both positive and negative compositional gradients were considered in the latter study and it was found that the influence of the mixture is significant in both cases, in particular the critical values of the thermal Rayleigh number, of the frequency and of the wave number depends strongly on the direction of the compositional gradient.…”

Radial collocation methods for the onset of convection in rotating spheres

“…The authors have developed several time integration and stability analysis codes, using this approach, for the convection in spherical shells of pure and binary fluids with very good results [33][34][35][36][37][38]. The main reasons for using collocation methods were simplicity and that changing boundary conditions was easy.…”

Computation of azimuthal waves and their stability in thermal convection in rotating spherical shells with application to the study of a double-Hopf bifurcation

“…From a mathematical point of view the difficulties of the spherical geometry in the rotating case make the attempts of analytical treatment extremely challenging, requiring very careful sequences of approximations to preserve the relevant features. The tools used to study the problem have been those of hydrodynamic stability theory, to obtain the critical value of the parameters, and the structure of the solutions at the onset of convection by using analytical, semianalytical [1][2][3][4][5][6], or numerical methods [7][8][9][10], and time evolution codes to obtain the nonlinear solutions at low [11][12][13] or high supercritical Rayleigh numbers [14][15][16][17], to mention just a few of all the available references.…”