SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 1, Page 38-64, January 2021.
In this article, we aim to classify the dynamic transitions and bifurcations for a family of axisymmetric geophysical fluid problems of a generic fourth-second order structure. A transition theorem is established by reducing the governing partial differential equations to a complex-valued ordinary differential equation, derived by employing approximate invariant manifolds. We develop an algorithm for the numerical determination of the transition/bifurcation types. Finally we apply the transition theorem and algorithm to examine the baroclinic instability in a two-layer quasi-geostrophic system in an annular channel and with different bathymetry profiles. Our numerical results show that with concave bathymetry the transition (bifurcation) is always continuous (supercritical Hopf bifurcation), whereas for convex bathymetry a jump transition (subcritical Hopf bifurcation) may occur in the basic azimuthal currents that rotate in the same direction.

中文翻译：

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一类轴对称地球物理流体流的动态转变和分叉

SIAM应用动力系统杂志，第20卷，第1期，第38-64页，2021年1月。
在本文中，我们旨在对通用的四阶二阶结构的一系列轴对称地球物理流体问题的动态过渡和分支进行分类。通过将控制性偏微分方程式简化为复数值常微分方程式（通过采用近似不变流形导出）来建立过渡定理。我们开发了一种用于确定过渡/分支类型的数值的算法。最后，我们应用转换定理和算法来检查环形通道中具有不同测深剖面的两层拟地转系统中的斜压不稳定性。我们的数值结果表明，采用凹式测深法，过渡（分叉）始终是连续的（超临界霍夫夫分叉），