Critical stability of almost adiabatic convection in a rapidly rotating thick spherical shell
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TICAL, NONLINEAR, AND SOFT MATTER PHYSICS
Critical Stability of Almost Adiabatic Convection in a Rapidly Rotating Thick Spherical Shell1 S. V. Starchenkoa and M. S. Kotelnikovab a
Institute of Terrestrial Magnetism, the Ionosphere, and Radiowave Propagation, Russian Academy of Sciences, Troitsk, Moscow, 142190 Russia bLavrentyev Institute of Hydrodynamics, pr. Akademika Lavrent’eva 15, Novosibirsk, 630090 Russia email: [email protected] Received June 28, 2012
Abstract—In this work, the convection equations in the almost adiabatic approximation is studied for which the choice of physical parameters is primarily based on possible applications to the hydrodynamics of the deep interiors of the Earth and planets and moons of the terrestrial group. The initial system of partial differ ential equations (PDEs) was simplified to a single secondorder ordinary differential equation for the pressure or vertical velocity component to investigate the linear stability of convection. The critical frequencies, mod ified Rayleigh numbers, and distributions of convection are obtained at various possible Prandtl numbers and in different thick fluid shells. An analytical WKBtype solution was obtained for the case when the inner radius of the shell is much smaller than the outer radius and convective sources are concentrated along the inner boundary. DOI: 10.1134/S1063776113020179 1
1. INTRODUCTION
This work is a continuation of [1]. The theory of planetary magnetic field generation is the main appli cation of studies of convection in a rapidly rotating spherical shell. The conducting fluid in deep planetary interiors is in a state very close almost adiabatic; e.g., for the main physical parameters, the deviations from the adiabatic state in the Earth’s outer core are about or less than 10–3%. Therefore, the choice of the almost adiabatic approximation is quite natural for the con struction of magnetic field generation models. All considered objects, planets and moons of the terres trial type are in a state of rapid rotation, which means that fluid motion in its deep convective interiors is characterized by a large Reynolds number, or, equiva lently, by a small Ekman number (~10–15 for the Earth’s outer core). The presence of this small param eter allows us to use asymptotic analysis to study the convection equations. In earlier works on this problem, this ordinary dif ferential equation was obtained from the initial system of equations by taking curl and double curl from the equation of motion and substituting a toroidal–poloi dal representation of nondivergent velocity [1–5]. In the next section, we use the initial linear system of equations of almost adiabatic planetary convection to obtain, for the first time, a simplified system of ordi nary differential equations for the pressure, z compo nent of velocity, and entropy variations. With this orig 1 The article was translated by the authors.
inal approach, there is no risk of surplus solutions, unlike the method which resulted in an order of equa tions with a cu
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