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Stability and bounding of flow quantities in time-modulated convection


The daily cycle of heating and cooling from the rising and setting of the sun creates the potential for periodically-forced convection in the world's atmosphere and bodies of water. In this dissertation, we model a body of water experiencing daily heating from the sun and examine the onset of convection using linear and nonlinear stability analysis. We also find the first exact bound on a flow quantity in time-modulated convection. The setup considered consists of fluid between two surfaces that are infinite in the horizontal directions, with one of the surfaces heated in a time-varying manner. The most important parameters are the Rayleigh number, $Ra$, which captures the strength of the heating, and the nondimensional frequency of the time modulation, $\omega$.

Radiative heating from the sun is modeled as a source term with exponential decay in the vertical. When confined to a thin layer near the surface, the effects of radiative heating may be approximated using an imposed temperature or heat flux boundary condition, and this approach forms the main focus of the dissertation. Each method of heating has time dependence to capture the sun's daily cycle. For stability analysis, a sinusoidal time dependence is used, and for bounding, results are found for a generic profile.

The stability results show that while the critical Rayleigh numbers for both linear and nonlinear stability increase with increasing modulation frequency, the linear and nonlinear stability thresholds show very different behavior at high modulation frequencies. As expected, the nonlinear stability threshold is always below the linear stability threshold. For the linear stability threshold, the ratio $Ra \omega^{-3/2}$ or $Ra \omega^{-2}$ approaches a constant, with the power of the frequency depending on the type of heating. This is found to match the result from considering a semi-infinite domain. For the nonlinear stability threshold, the growth of the Rayleigh number with the frequency is approximately linear, and no semi-infinite domain result is found. Beyond stability, we find an exact bound on the temperature dissipation that grows like $\sqrt{Ra}$ and $\sqrt{\omega}$ in the large $Ra$ and $\omega$ limits, respectively.

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