The Nonhydrostatic Balanced-Geostrophic Equations

Keith Julien
University of Colorado, Boulder

Convection under the influence rotation has been the subject of a great deal of theoretical and experimental research. This problem is relevant to convectively driven fluid flows in the Earth's ocean and interior and also in the Sun and
other stars, where the influence of rotation is generally important. In general numerical simulations of rotationally constrained flows are unable to reach realistic parameter values, e.g., Reynolds Re and Richardson Ri numbers. In particular, low values of Rossby number Ro, defining the extent of rotational constraint, compound the already prohibitive temporal and
spatial restrictions present for high-Re simulations by engendering high frequency inertial waves and the development of thin (Ekman) boundary layers.

Recent work in the development of reduced partial differential equations (pde's) that filter fast waves and relax the need to resolve boundary layers has been extended to construct a hierarchy of balanced equations that span the stably- and unstably-stratified limits. By varying the aspect ratio for spatial anisotropy characterizing horizontal and vertical scales, rapidly rotating convection and stably-stratified quasi-geostrophic motions can be described within the same framework.

In this talk, the asymptotic pde's relevant for rotating convection are explored. Special classes of fully nonlinear exact solutions are identified and discussed. Direct numerical solutions that correctly capture the regular
vortex columnar and irregular geostrophic turbulence regime of recent laboratory experiments are also presented and discussed.

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