Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio

Abstract

a b s t r a c tIn this paper we analyse convective solutions of a two dimensional fluid layer in which viscosity depends exponentially on temperature. This problem takes in features of mantle convection, since large viscosity variations are to be expected in the Earth's interior. These solutions are compared with solutions obtained at constant viscosity. Special attention is paid to the influence of the aspect ratio in the solutions presented. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, systematic method for gaining insight into the possible stationary solutions satisfied by the basic equations. One feature presented by the fluid with non constant viscosity is the presence of pitchfork and saddle-node subcritical bifurcations and the presence of convective solutions below the linear critical threshold. The analysis also provides limits of existence of stationary solutions and draws the boundaries for time dependent convection.

Influence of Horizontal Temperature Gradients on Convective Instabilities with Geophysical Interest

“…(36; 46; 49) and numerical results in Ref. (48). It is remarkable that the horizontal temperature gradient favours a threedimensional structure after the bifurcation, while the pattern continues being axisymmetric after the bifurcation in the only vertical gradient case.…”

“…The evaluation rules are detailed in Ref. (48). Therefore, the eigenvalue problem in its discrete form is,…”

“…The benchmarking of the method can be done with results in Ref. (48). The critical wave number for Γ = 2.936, η = 0.0862 and δ = 0isk c = 0, so for these values of the parameters the results reported in Ref.…”

RB (Reduced basis) for RB (Rayleigh–Bénard)

“…We consider infinite Pr number as in [17,18], then the left hand side term in Eq. (2) can be made equal to zero.…”

“…In [17] the different solutions and successive bifurcations when the temperature gradients increase are obtained based on a branch continuation technique. In the study of these bifurcation problems the model of partial differential equations must be solved for lots of values of the bifurcation parameter and a linear stability analysis has to be performed for each solution in order to know its linear stability properties and the succession of bifurcations.…”