Nonlinear Bénard convection with rotation

Abstract

The equations for nonlinear Bénard convection with rotation for a layer of fluid, thickness d, are derived using the Glansdorff & Prigogine (1964) evolutionary criterion as used by Roberts (1966) in his paper on non-rotational Bénard convection. The parameters of the problem in this case are the Rayleigh number R = αgΔθd/vK, the Taylor number T = 4d4Ω23/v2 and the Prandtl number Pr = v/K, where α is the coefficient of volume expansion, g the acceleration due to gravity, Δθ the temperature difference between the horizontal surfaces, v the kinematic viscosity, K the thermal diffusivity and Ω3 the rotation rate about the vertical direction. The asymptotic solution for two-dimensional cells (rolls) is investigated for large Rayleigh numbers and large Taylor numbers. For rolls the convection equations are found t o be independent of the Prandtl number. However, the solutions depend upon the Prandtl number for another reason. The rotational problem differs from the non-rotational one in that the Rayleigh number and the horizontal wavenumber a of the convection are now functions of the Taylor number. These are taken to be R ∼ ρTα′ and α ∼ ATβ, where α′ and β are positive numbers. Thermal layers develop as R becomes large with ρ or T becoming large. The order in which ρ and T are allowed to increase is important since the horizontal wavenumber a also increases with T and the convection equations can be reduced in this case. A liquid of large Prandtl number such as water has v [Gt ] K. Since R ∼ O (1/vK) and T ∼ O(1/v2), ρ will be greater than T for a given (large) Δθ and Ω3. Similarly, for a liquid of small Prandtl number such as mercury v [Lt ] K, and T is greater than ρ for a given Δθ and Ω3. For rigid-rigid horizontal boundaries with ρ large and then T large the ρ thermal layer has the same structure as for the non-rotating problem. As T → ∞ three types of thermal layers are possible: a linear Ekman layer, a nonlinear Ekman layer and a Blasius-type thermal layer. When the horizontal boundaries are both free the ρ thermal layer is again of the same structure as for non-rotating BBnard convection. As T → ∞ a nonlinear Ekman layer and a Blasius-type thermal layer are possible.When T is large and then ρ made large the differential equations governing the convection are reduced from eighth order to sixth order owing to a becoming large as T → ∞. There are Ekman layers as T → ∞, when the horizontal boundaries are both rigid. The ρ thermal layers now have a different structure from the non-rotating problem for both rigid-rigid and free-free horizontal boundaries. The equation for small amplitude convection near to the marginal case is derived and the solution for free-free horizontal boundaries is obtained.