Open AccessA simple model of high Prandtl and high Rayleigh number convection bounded by thin low‐viscosity layers
Author(s) -
Busse F. H.,
Richards M. A.,
Lenardic A.
Publication year - 2006
Publication title -
geophysical journal international
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.302
H-Index - 168
eISSN - 1365-246X
pISSN - 0956-540X
DOI - 10.1111/j.1365-246x.2005.02836.x
Subject(s) - prandtl number , rayleigh number , nusselt number , convection , viscosity , thermodynamics , mechanics , mantle convection , natural convection , convection cell , physics , materials science , combined forced and natural convection , lithosphere , geology , reynolds number , turbulence , paleontology , tectonics
SUMMARY 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 ∼ R 1/3 (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∼ℓ 6 δ −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 ( R τ 2/3 ) 1/3 at ℓ= (τδ 3 /9) 1/6 π/2 . This result suggests that the aspect ratio of convection increases with viscosity contrast.