The Mechanics of Vacillation

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  • 1 Massachusetts Institute of Technology
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Abstract

The equations governing a symmetrically heated rotating viscous fluid are reduced to a system of fourteen ordinary differential equations, by a succession of approximations. The equations contain two external parameters-an imposed thermal Rossby number and a Taylor number.

Solutions where the blow is purely zonal, and solutions with superposed “steady” waves which progress without changing their shape, are obtained analytically. Additional solutions exhibiting vacillation, where the waves change shape in a regular periodic manner in addition to their progression, and solutions exhibiting irregular nonperiodic flow, are obtained by numerical integration.

For a given imposed thermal Rossby number, the flow becomes more complicated as the Taylor number increases. Exceptions occur at very high Taylor numbers, where the equations become unrealistic because of truncation.

For values of the external parameters where steady-wave solutions are found, solutions with purely zonal flow also exist, but are unstable, Where vacillating solutions are found, steady-wave solutions also exist, but are unstable. A transition between unsymmetric and symmetric vacillation is not associated with the instability of either form of vacillation. It is hypothesized that where irregular nonperiodic solutions are found, vacillating solutions also exist but are unstable.

Abstract

The equations governing a symmetrically heated rotating viscous fluid are reduced to a system of fourteen ordinary differential equations, by a succession of approximations. The equations contain two external parameters-an imposed thermal Rossby number and a Taylor number.

Solutions where the blow is purely zonal, and solutions with superposed “steady” waves which progress without changing their shape, are obtained analytically. Additional solutions exhibiting vacillation, where the waves change shape in a regular periodic manner in addition to their progression, and solutions exhibiting irregular nonperiodic flow, are obtained by numerical integration.

For a given imposed thermal Rossby number, the flow becomes more complicated as the Taylor number increases. Exceptions occur at very high Taylor numbers, where the equations become unrealistic because of truncation.

For values of the external parameters where steady-wave solutions are found, solutions with purely zonal flow also exist, but are unstable, Where vacillating solutions are found, steady-wave solutions also exist, but are unstable. A transition between unsymmetric and symmetric vacillation is not associated with the instability of either form of vacillation. It is hypothesized that where irregular nonperiodic solutions are found, vacillating solutions also exist but are unstable.

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