地球对流层中分层流体的Rayleigh-Bénard对流运动
Rayleigh-Bénard convective motion of stratified fluids in the Earth's troposphere
AI总结：
本研究基于Boussinesq近似推导了地球对流层无界与有界区域的Rayleigh-Bénard对流波色散关系，发现两种情况的不稳定性条件显著不同，并分析了临界Rayleigh数和热波不稳定性增长率。
摘要：
最近，Kaladze和Misra [Phys. Scr. 99 (2024) 085013] 证明了对流层分层流体流动可能因负温度梯度效应和由热膨胀引起的温度相关密度不均匀性而变得不稳定。他们还预测，由密度不均匀性引起的Brunt-Väisälä频率的改变可导致对流层无界层中出现Rayleigh-Bénard对流不稳定性。本工作的目的是通过同时考虑无界和有界对流层，更详细地重新审视Rayleigh-Bénard对流不稳定性。从一组具有温度梯度的不可压缩中性流体方程出发，并使用Boussinesq近似，我们推导了无界和有界对流层区域中Rayleigh-Bénard对流波的广义色散关系，并通过解析和数值方法对一些特定情况进行了分析。我们表明，这两种情况下的不稳定性条件存在显著差异。此外，我们获得并分析了这两种情况下的Rayleigh数的临界值以及热波不稳定性增长率的表达式。在有界区域的情况下，我们还推导了必要的边界条件，并指出垂直波数是量子化的，且相应的特征值问题是适定的。
英文摘要：
Recently, Kaladze and Misra [Phys. Scr. 99 (2024) 085013] showed that the tropospheric stratified fluid flows may be unstable by the effects of the negative temperature gradient and the temperature-dependent density inhomogeneity arising from the thermal expansion. They also predicted that the modification in the Brunt-V{ä}is{ä}l{ä} frequency by the density inhomogeneity can lead to Rayleigh-B{é}nard convective instability in the tropospheric unbounded layers. The purpose of the present work is to revisit the Rayleigh-B{é}nard convective instability in more detail by considering both unbounded and bounded tropospheric layers. Starting from a set of fluid equations for incompressible neutral fluids with temperature gradients and using the Boussinesq approximation, we derive the general dispersion relations for Rayleigh-B{é}nard convective waves in unbounded and bounded tropospheric domains and analyze them with some particular cases both analytically and numerically. We show that the conditions for instability in these two cases significantly differ. Furthermore, we obtain and analyze the critical values of the Raleigh numbers and the expressions for the instability growth rates of thermal waves in the two cases. In the case of the bounded region, we also derive the necessary boundary conditions and note that the vertical wave number is quantified, and the corresponding eigenvalue problem is well-set.
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