Weakly nonlinear analysis of the onset of convection in rotating spherical shells

Calum S. Skene, Steven M. Tobias
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Abstract

A weakly nonlinear study is numerically conducted to determine the behaviour near the onset of convection in rotating spherical shells. The mathematical and numerical procedure is described in generality, with the results presented for an Earth-like radius ratio. Through the weakly nonlinear analysis a Stuart--Landau equation is obtained for the amplitude of the convective instability, valid in the vicinity of its onset. Using this amplitude equation we derive a reduced order model for the saturation of the instability via nonlinear effects and can completely describe the resultant limit cycle without the need to solve initial value problems. In particular the weakly nonlinear analysis is able to determine whether convection onsets as a supercritical or subcritical Hopf bifurcation through solving only linear 2D problems, specifically one eigenvalue and two linear boundary value problems. Using this, we efficiently determine that convection can onset subcritically in a spherical shell for a range of Prandtl numbers if the shell is heated internally, confirming previous predictions. Furthermore, by examining the weakly nonlinear coefficients we show that it is the strong zonal flow created through Reynolds and thermal stresses that determines whether convection is supercritical or subcritical.
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旋转球壳中开始对流的弱非线性分析
对旋转球壳中开始对流的行为进行了弱非线性数值研究。对数学和数值过程进行了一般性描述,并给出了类地球半径比的结果。通过弱非线性分析,得到了对流不稳定性振幅的斯图尔特-兰道方程,该方程在对流开始附近有效。利用这个振幅方程,我们推导出了不稳定性非线性效应饱和的低阶模型,并可以完全描述由此产生的极限循环,而无需求解初值问题。特别是,弱非线性分析只需求解线性二维问题,特别是一个特征值和两个线性边界值问题,就能确定对流是作为超临界还是次临界霍普夫分岔开始。利用这一方法,我们有效地确定了如果球壳内部加热,对流可以在一定范围的普朗特尔数下在球壳内亚临界起始,这证实了之前的预测。此外,通过对弱非线性系数的研究,我们发现是通过雷诺和热应力产生的强烈带状流决定了对流是超临界还是亚临界。
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