正压涡度方程的Lie对称性和精确解
Lie symmetries and exact solutions of the barotropic vorticity equation
摘要：
李群方法用于研究正压涡度方程的对称性和精确解的各种问题。确定了正压涡度方程在$f$ -和$β$ -平面上以及在旋转和静止参考系中的球面上的Lie对称性。给出了将旋转参考系简化为静止系的对称背景。这两个方程的李不变性代数的一维和二维不等价子代数进行了详尽的分类，然后用于计算不变解的涡度方程。这提供了大量的精确解，其中包括Rossby和Rossby-Haurwitz波作为特殊情况。我们还讨论了$β$ -平面方程的部分不变性的可能性，从而进一步扩展了它的精确解族。这是在一个更系统和完整的方式比以前在文献中。
英文摘要：
Lie group methods are used for the study of various issues related to symmetries and exact solutions of the barotropic vorticity equation. The Lie symmetries of the barotropic vorticity equations on the $f$- and $β$-planes, as well as on the sphere in rotating and rest reference frames, are determined. A symmetry background for reducing the rotating reference frame to the rest frame is presented. The one- and two-dimensional inequivalent subalgebras of the Lie invariance algebras of both equations are exhaustively classified and then used to compute invariant solutions of the vorticity equations. This provides large classes of exact solutions, which include both Rossby and Rossby--Haurwitz waves as special cases. We also discuss the possibility of partial invariance for the $β$-plane equation, thereby further extending the family of its exact solutions. This is done in a more systematic and complete way than previously available in literature.
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