Transitions in Shallow Convection: An Explanation for Lateral Cell Expansion

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  • 1 Department of Meteorology, The Pennsylvania State University, Park, PA 16802
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Abstract

A generalized seven-coefficient model of two-dimensional Rayleigh-Bénard convection is presented. The model simulates successfully one means by which lateral cell expansion can occur as the value of the imposed vertical temperature difference is changed. Such changes in the horizontal wavelengths of the convective rolls are accomplished by the nonlinear transfer of energy from cells to other cells with smaller wavenumbers. The crucial effect is one represented by the advective term v˙∇v in the equation of motion, and as a consequence an interacting triad of harmonics must be included in the spectral model. Thus, the generalized model has basically the same form as that used by Saltzman or Shirer and Dutton, but in the generalized model the triad of interacting wavenumbers is varied as the vertical heating rate is varied. Actual values of the horizontal wavenumbers are determined by assuming that the first unstable wave will have the largest growth rate, or equivalently that the bifurcation point will have the smallest value. Thus, only the energetically active components are retained; in this way, transitional behavior within two-dimensional convective flow can be simulated properly, and can be interpreted physically as representing the cell expansion process, via successive secondary branching.

When the solutions are compared with those obtained by Clever and Busse from a large three-dimensional spectral model, it is found that for small values of the Prandtl number P (∼0.1), a two-dimensional cell broadening mechanism is likely to operate, but for larger values of P (∼0.7), a three-dimensional mechanism is expected. Consequently, these results suggest that this generalized seven-component model can be used to simulate successfully some transitions in a system having more degrees of freedom, because the seven components apparently form the basic unit by which steady two-dimensional flow develops. Moreover, the modeling philosophy presented here can provide the basis for development of simple atmospheric convection models.

Abstract

A generalized seven-coefficient model of two-dimensional Rayleigh-Bénard convection is presented. The model simulates successfully one means by which lateral cell expansion can occur as the value of the imposed vertical temperature difference is changed. Such changes in the horizontal wavelengths of the convective rolls are accomplished by the nonlinear transfer of energy from cells to other cells with smaller wavenumbers. The crucial effect is one represented by the advective term v˙∇v in the equation of motion, and as a consequence an interacting triad of harmonics must be included in the spectral model. Thus, the generalized model has basically the same form as that used by Saltzman or Shirer and Dutton, but in the generalized model the triad of interacting wavenumbers is varied as the vertical heating rate is varied. Actual values of the horizontal wavenumbers are determined by assuming that the first unstable wave will have the largest growth rate, or equivalently that the bifurcation point will have the smallest value. Thus, only the energetically active components are retained; in this way, transitional behavior within two-dimensional convective flow can be simulated properly, and can be interpreted physically as representing the cell expansion process, via successive secondary branching.

When the solutions are compared with those obtained by Clever and Busse from a large three-dimensional spectral model, it is found that for small values of the Prandtl number P (∼0.1), a two-dimensional cell broadening mechanism is likely to operate, but for larger values of P (∼0.7), a three-dimensional mechanism is expected. Consequently, these results suggest that this generalized seven-component model can be used to simulate successfully some transitions in a system having more degrees of freedom, because the seven components apparently form the basic unit by which steady two-dimensional flow develops. Moreover, the modeling philosophy presented here can provide the basis for development of simple atmospheric convection models.

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