A simple model of high Prandtl and high Rayleigh number convection bounded by thin low-viscosity layers

Motivated by recent numerical results on convection in the Earth's mantle in the presence of a low-viscosity zone, an analytical model is derived for 2-D steady convection with symmetric low-viscosity layers at the upper and lower boundaries. The asymptotic limits of high Rayleigh and high Prandtl number are assumed. The low- viscosity layers carry with minimal dissipation most of the horizontal component of the convection flow and thereby increase the efficiency of the convective heat transport. A Nusselt number (Nu)–Rayleigh number (R) relationship of the form Nu∼R1/3 (ℓ/δ)2/3 is obtained for an aspect ratio 2ℓ of the convection cell of order unity or less. Here δ denotes the fraction of the layer depth occupied by each of the low-viscosity channels. For large ℓ and τ∼ℓ6δ−3, where τ denotes the ratio between the viscosity of the interior and that of the thin low-viscosity layers, Nu reaches a maximum value of the order (Rτ2/3)1/3 at ℓ = (τδ3/9)1/6π/2. This result suggests that the aspect ratio of convection increases with viscosity contrast.