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Title: Scaling and excitation of combined convection in a rapidly rotating plane layer

Abstract

The optimum (to my mind) scaling of the combined thermal and compositional convection in a rapidly rotating plane layer is proposed.This scaling follows from self-consistent estimates of typical physical quantities. Similarity coefficients are introduced for the ratio convection dissipation/convection generation (s) and the ratio thermal convection/compositional convection (r). The third new and most important coefficient δ is the ratio of the characteristic size normal to the axis of rotation to the layer thickness. The faster the rotation, the lower δ. In the case of the liquid Earth core, δ ~ 10{sup –3} substitutes for the generally accepted Ekman number (E ~ 10{sup –15}) and s ~ 10{sup –6} substitutes for the inverse Rayleigh number 1/Ra ~ 10{sup –30}. It is found that, at turbulent transport coefficients, number s and the Prandtl number are on the order of unity for any objects and δ is independent of transport coefficients. As a result of expansion in powers of δ, an initially 3D system of six variables is simplified to an almost 2D system of four variables without δ. The problem of convection excitation in the main volume is algebraically solved and this problem for critical values is analytically solved. Dispersion relations andmore » general expressions for critical wavenumbers, numbers s (which determine Rayleigh numbers), other critical parameters, and asymptotic solutions are derived. Numerical estimates are made for the liquid cores in the planets that resemble the Earth. Further possible applications of the results obtained are proposed for the interior of planets, moons, their oceans, stars, and experimental objects.« less

Authors:
 [1]
  1. Russian Academy of Sciences, Pushkov Institute of Terrestrial Magnesium, Ionosphere and Radio Wave Propagation (Russian Federation)
Publication Date:
OSTI Identifier:
22617059
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Experimental and Theoretical Physics; Journal Volume: 124; Journal Issue: 2; Other Information: Copyright (c) 2017 Pleiades Publishing, Inc.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ASYMPTOTIC SOLUTIONS; CONVECTION; DISPERSION RELATIONS; DISPERSIONS; EXCITATION; EXPANSION; LAYERS; LIQUIDS; PLANETS; PRANDTL NUMBER; RAYLEIGH NUMBER; ROTATION; STARS; THICKNESS

Citation Formats

Starchenko, S. V., E-mail: sstarchenko@mail.ru. Scaling and excitation of combined convection in a rapidly rotating plane layer. United States: N. p., 2017. Web. doi:10.1134/S1063776117020091.
Starchenko, S. V., E-mail: sstarchenko@mail.ru. Scaling and excitation of combined convection in a rapidly rotating plane layer. United States. doi:10.1134/S1063776117020091.
Starchenko, S. V., E-mail: sstarchenko@mail.ru. Wed . "Scaling and excitation of combined convection in a rapidly rotating plane layer". United States. doi:10.1134/S1063776117020091.
@article{osti_22617059,
title = {Scaling and excitation of combined convection in a rapidly rotating plane layer},
author = {Starchenko, S. V., E-mail: sstarchenko@mail.ru},
abstractNote = {The optimum (to my mind) scaling of the combined thermal and compositional convection in a rapidly rotating plane layer is proposed.This scaling follows from self-consistent estimates of typical physical quantities. Similarity coefficients are introduced for the ratio convection dissipation/convection generation (s) and the ratio thermal convection/compositional convection (r). The third new and most important coefficient δ is the ratio of the characteristic size normal to the axis of rotation to the layer thickness. The faster the rotation, the lower δ. In the case of the liquid Earth core, δ ~ 10{sup –3} substitutes for the generally accepted Ekman number (E ~ 10{sup –15}) and s ~ 10{sup –6} substitutes for the inverse Rayleigh number 1/Ra ~ 10{sup –30}. It is found that, at turbulent transport coefficients, number s and the Prandtl number are on the order of unity for any objects and δ is independent of transport coefficients. As a result of expansion in powers of δ, an initially 3D system of six variables is simplified to an almost 2D system of four variables without δ. The problem of convection excitation in the main volume is algebraically solved and this problem for critical values is analytically solved. Dispersion relations and general expressions for critical wavenumbers, numbers s (which determine Rayleigh numbers), other critical parameters, and asymptotic solutions are derived. Numerical estimates are made for the liquid cores in the planets that resemble the Earth. Further possible applications of the results obtained are proposed for the interior of planets, moons, their oceans, stars, and experimental objects.},
doi = {10.1134/S1063776117020091},
journal = {Journal of Experimental and Theoretical Physics},
number = 2,
volume = 124,
place = {United States},
year = {Wed Feb 15 00:00:00 EST 2017},
month = {Wed Feb 15 00:00:00 EST 2017}
}
  • Fully nonlinear three-dimensional convection in a rotating layer is studied for large Taylor numbers. In this regime, the leading order nonlinearity arises from the distortion of the horizontally averaged temperature profile. As a result, steady rolls, squares, hexagons, triangles, and a pattern called patchwork quilt all have identical Nusselt numbers. A similar degeneracy is present in overstable convection with six patterns having identical time-averaged Nusselt numbers and oscillation frequencies. These results are obtained via an asymptotic expansion in the Taylor number that determines, for each Rayleigh number, the time-averaged Nusselt number and oscillation frequency from the solution of a nonlinearmore » eigenvalue problem for the vertical temperature profile. A number of other patterns are determined by a weakly nonlinear analysis that cannot be extended into the fully nonlinear regime by the present methods, but these patterns are necessarily unstable. {copyright} {ital 1999 American Institute of Physics.}« less
  • 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 differential equations (PDEs) was simplified to a single second-order ordinary differential equation for the pressure or vertical velocity component to investigate the linear stability of convection. The critical frequencies, modified Rayleigh numbers, and distributions of convection are obtained at various possible Prandtl numbers and in different thick fluid shells. An analyticalmore » WKB-type 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.« less
  • A Comment on the Letter by Uwe R. Fischer, Phys. Rev. Lett. 93, 160403 (2004). The author of the Letter offers a Reply.
  • A Comment on the Letter by V.thinspM. Canuto and M.thinspS. Dubovikov, Phys.thinspthinspRev.thinspthinspLett.thinspthinsp{bold 80}, 281 (1998). The authors of the Letter offer a Reply. {copyright} {ital 1999} {ital The American Physical Society}