Inertial Waves in a Rotating Spherical Shell with Homogeneous Boundary Conditions

Abstract

We find the analytical form of inertial waves in an incompressible, rotating fluid constrained by concentric inner and outer spherical surfaces with homogeneous boundary conditions on the normal components of velocity and vorticity. These fields are represented by Galerkin expansions whose basis consists of toroidal and poloidal vector functions, i.e., products and curls of products of spherical Bessel functions and vector spherical harmonics. These vector basis functions also satisfy the Helmholtz equation and this has the benefit of providing each basis function with a well-defined wavenumber. Eigenmodes and associated eigenfrequencies are determined for both the ideal and dissipative cases. These eigenmodes are formed from linear combinations of the Galerkin expansion basis functions. The system is truncated to numerically study inertial wave structure, varying the number of eigenmodes. The largest system considered in detail is a 25 eigenmode system and a graphical depiction is presented of the five lowest dissipation eigenmodes, all of which are non-oscillatory. These results may be useful in understanding data produced by numerical simulations of fluid and magnetofluid turbulence in a spherical shell that use a Galerkin, toroidal–poloidal basis as well as qualitative features of liquids confined by a spherical shell.

Transition to Equilibrium and Coherent Structure in Ideal MHD Turbulence

“…Our approach to resolving this mystery is to study three-dimensional (3-D) homogeneous, incompressible MHD turbulence, i.e., to examine a turbulent magnetofluid confined to a 3-D periodic box (a three-torus). This allows for the use of Fourier expansions to represent turbulent velocity and magnetic fields in order to study the nonlinear dynamics of a magnetofluid without the complicating factor of boundary conditions; in fact, introducing boundaries with no-slip conditions also requires introducing compressibility [11] and further complicates the problem. Fourier methods transform the problem from the partial differential equations of MHD (presented in Section 2.1) in x-space (physical space) to a dynamical system of nonlinear, coupled ordinary differential equations in k-space (Fourier space), which describe the evolution of the Fourier modes associated with velocity and magnetic field; this is discussed in Section 2.3.…”