Dynamics of Ideal Fluid on a Sphere

This chapter deals with an ideal and unforced fluid on a sphere. The motion of is described by the vorticity equation which takes into account such important dynamic processes as the nonlinear interaction, dispersion, and sphere rotation. The four groups of solutions to this equation are known up to now, namely the classical infinitely differentiable solutions (Legendre-polynomial flows and Rossby–Haurwitz waves) and weak solutions (modons and Wu–Verkley waves). The solutions satisfy infinite number of integral conservation laws. Due to its relative simplicity, the model of an ideal fluid is a very convenient object for the application of mathematical methods in the study of nonlinear fluid dynamics (nonlinear interaction of waves and turbulence, construction of exact solutions and their stability, etc.). The chapter is mainly devoted to the study of exact wave solutions of the vorticity equation and the dynamics of perturbations of such solutions. In Sect. 4.1, the main conservation laws of inviscid fluid dynamics are considered, and Fjörtoft theorem that estimates the transfer of energy to small scales is given. The main properties of the triad interaction coefficients K βαγ describing the nonlinear interaction of three spherical harmonics are presented in Sect. 4.2, and in Sect. 4.3, a recurrence relation for calculating these coefficients is derived for the particular case when β = (0, j) and j is a natural number. This recurrence formula will be used in Sect. 7.4 to analyze the structure of stability matrix and the stable manifold of the basic flow in the form of a single Legendre polynomial. A theorem by Szeptycki on the existence and uniqueness of a weak solution is given in Sect. 4.4. The well-known class of infinitely differentiable wave solutions by Thompson which include all the Rossby–Haurwitz waves is described in Sect. 4.5. One more class of BVE solutions representing a set of modons on a sphere is considered in Sect. 4.6. In contrast to smooth Rossby–Haurwitz waves, the stream function of modons has continuous derivatives on a sphere only up to the second order. In Sect. 4.7, the distance between different BVE solutions is estimated in the norms related with the energy and/or enstrophy. In Chaps. 5 and 6 , these estimates are used to prove the Liapunov instability of non-zonal Rossby–Haurwitz waves and dipole modons. Section 4.8 contains auxiliary information related to the Euler angles.