Bifurcation and Stability of Low-Order Steady Flows in Horizontally and Vertically Forced Convection

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  • 1 Martin Marietta Aerospace, Denver Division, Denver, CO 80201
  • | 2 Department of Meteorology, The Pennsylvania Stage University, University Park 16802
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Abstract

The response of a convecting fluid to externally imposed horizontal and vertical temperature gradients is fundamentally different from that obtained when only vertical forcing is present. Using a three-component spectral model of two-dimensional shallow convection, we display the different forms and stability hierarchies of the stationary solutions as functions of both the vertical and the horizontal temperature differences. The structures of the possible steady states for zero and nonzero horizontal forcing are markedly different; the case of vertical heating only is singular and hence unobservable. Thus we demonstrate that the, generic low-order convective model that most accurately reproduces some observations of Bénard convection, for example. must contain both horizontal and vertical heating parameters.

The steady behavior of the three-component model is described by three cusps in the thermal parameter plane for all values of domain aspect ratio and Prandtl number. Branching to periodic solutions occurs for all values of domain aspect ratio and Prandtl number. However, the qualitative nature of the sets of Hopf bifurcation points depends on the values of these parameters; three distinct types of Hopf bifurcation curves are identified. For some values of the horizontal and vertical heating, both thermally direct and indirect steady flows are stable and hence observable. The location of these cusps and Hopf bifurcation curves determines the regions of multiple equilibria and their stability. When the steady states lose stability via Hopf bifurcation, temporally periodic solutions exist nearby, and when all steady states are unstable, time-dependent flows must exist. This model also exhibits steady-state hysteresis and a mechanism whereby catastrophe, or sudden large change in the magnitude of the stationary solution, can occur.

Abstract

The response of a convecting fluid to externally imposed horizontal and vertical temperature gradients is fundamentally different from that obtained when only vertical forcing is present. Using a three-component spectral model of two-dimensional shallow convection, we display the different forms and stability hierarchies of the stationary solutions as functions of both the vertical and the horizontal temperature differences. The structures of the possible steady states for zero and nonzero horizontal forcing are markedly different; the case of vertical heating only is singular and hence unobservable. Thus we demonstrate that the, generic low-order convective model that most accurately reproduces some observations of Bénard convection, for example. must contain both horizontal and vertical heating parameters.

The steady behavior of the three-component model is described by three cusps in the thermal parameter plane for all values of domain aspect ratio and Prandtl number. Branching to periodic solutions occurs for all values of domain aspect ratio and Prandtl number. However, the qualitative nature of the sets of Hopf bifurcation points depends on the values of these parameters; three distinct types of Hopf bifurcation curves are identified. For some values of the horizontal and vertical heating, both thermally direct and indirect steady flows are stable and hence observable. The location of these cusps and Hopf bifurcation curves determines the regions of multiple equilibria and their stability. When the steady states lose stability via Hopf bifurcation, temporally periodic solutions exist nearby, and when all steady states are unstable, time-dependent flows must exist. This model also exhibits steady-state hysteresis and a mechanism whereby catastrophe, or sudden large change in the magnitude of the stationary solution, can occur.

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