Fast Thermoviscous Convection

This paper studies the asymptotic structure of convection in an infinite Prandtl number fluid with strongly temperature-dependent viscosity, in the limit where the dimensionless activation energy 1/ε is large, and the Rayleigh number R, defined (essentially) with the basal viscosity and the prescribed temperature drop, is also large. We find that the Nusselt number N is given by N~CεR^1/5, where C depends on the aspect ratio a. The relative error in this result is O(R^-1/10ε^-1/4, ε^1/2, R^-2/5ε^-2, R^-2/20ε^-1/24), so that we cannot hope to find accurate confirmation of this result at moderate Rayleigh numbers, though it should serve as a useful indicator of the relative importance of R and ε. For the above result to be valid, we require R ≳ 1/ε⁵ ≫1. More important, however, is the asymptotic structure of the flow: there is a cold (hence rigid) lid with sloping base, beneath which a rapid, essentially isoviscous, convection takes place. This convection is driven by plumes at the sides, which generate vorticity due to thermal buoyancy, as in the constant viscosity case (Roberts, 1979). However, the slope of the lid base is sufficient to cause a large shear stress to be generated in the thermal boundary layer which joins the lid to the isoviscous region underneath (though a large velocity is not generated); consequently, the layer does not "see" the shear stress exerted by the interior flow (at leading order), and therefore the thermal boundary layer structure is totally self-determined: it even has a similarity structure (as a consequence). This fact makes it easy to analyse the problem, since the boundary layer uncouples from the rest of the flow. In addition, we find an alternative scaling (in which the lid base is "almost" flat), but it seems that the resulting boundary layer equations have no solution, though this is certainly open to debate: the results quoted above are not for this case. When a free slip boundary condition is applied at the top surface, one finds that there exists a thin "skin" at the top of the lid which is a stress boundary layer. The shear stress changes rapidly to zero, and there exists a huge longitudinal stress (compressive/tensile) in this skin. For earthlike parameters, this stress far exceeds the fracture strength of silicate rocks.