SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 208-231, January 2021.
The thermal convection of rotating fluids in spherical geometry is a classical problem with application to many geophysical and astrophysical problems. The study of the transition to periodic solutions from the steady conduction state of a rotating and self-gravitating fluid sphere, heated uniformly from the inside, is discussed here. The continuation of double Hopf points is used to determine the region of the parameter space in which the first bifurcation is to solutions independent of the longitude (axisymmetric solutions). It is limited by three segments of curves separated by two triple Hopf points. This type of so-called torsional solutions was recently found, and it is shown here that they are the preferred solutions at the onset of convection for a wide range of fluids of Prandtl numbers, ${Pr}$, extending from ${Pr}=0$ to ${Pr}\approx 0.9$, which includes, for instance, liquid metals and gases. Although the corresponding interval of Ekman numbers, ${E}$, narrows when ${Pr}\rightarrow 0$, it is shown that there is always a small gap of parameters, relevant to geophysics and astrophysics, where the torsional solutions are preferred. The limits of the double Hopf curves when ${Pr\rightarrow 0}$ follow linear laws of the form ${E}=c(m_1,m_2){Pr}$, $c(m_1,m_2)$ being constant depending on the two azimuthal wavenumbers, $m_1$ and $m_2$, of the eigenfunctions that define the double Hopf problem.

中文翻译：

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旋转流体球体热对流中双Hopf点的连续性

SIAM应用动力系统杂志，第20卷，第1期，第208-231页，2021年1月。
球形几何体中旋转流体的热对流是一个经典问题，适用于许多地球物理和天体物理问题。本文讨论了从内部均匀加热的旋转自重流体球的稳态传导状态到周期解的过渡过程。双Hopf点的连续性用于确定参数空间的区域，在该区域中，第一分叉是独立于经度的解（轴对称解）。它受到由两个三重Hopf点分隔的三段曲线的限制。最近发现了这种所谓的扭转解决方案，并显示出它们是对流开始时对$ {Pr} $（从$ {Pr}扩展）的各种Prandtl流体的首选解决方案。 = 0 $到$ {Pr} \约0。9 $，其中包括液态金属和气体。尽管当$ {Pr} \ rightarrow 0 $时，对应的Ekman数间隔$ {E} $变窄，但表明与地球物理学和天体物理学有关的参数总是存在很小的间隙，在这种情况下，扭转解是首选的。当$ {Pr \ rightarrow 0} $遵循形式为$ {E} = c（m_1，m_2）{Pr} $，$ c（m_1，m_2）$的线性定律时，双重Hopf曲线的极限定义双重Hopf问题的本征函数的两个方位波数$ m_1 $和$ m_2 $。