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.
Highlights
Homogeneous Boundary Conditions.The desire to understand fluid motion in the Earth’s outer core, and the geodynamo, provides the physical motivation for our study
Understanding magnetohydrodynamic inertial modes through Galerkin T–P expansions is a further goal and the present work is viewed as a necessary step in that direction)
The choice of T–P basis functions that satisfy the Helmholtz equation in a spherical shell leads to the second boundary condition: the normal component of vorticity vanishes at the boundaries
Summary
We use Galerkin toroidal–poloidal (T–P) expansions to focus on velocity and vorticity, where ‘Galerkin’ means each term satisfies designated boundary conditions (e.g., see [4]). The choice of T–P basis functions that satisfy the Helmholtz equation in a spherical shell leads to the second boundary condition: the normal component of vorticity vanishes at the boundaries. We do not use the no-slip or stress-free or inviscid boundary conditions that were compared by [11], but something else, which we will call homogeneous boundary conditions, i.e., the normal components of velocity and vorticity vanish at the boundaries. At this point, let us set the context for our mathematical model and T–P vector basis functions (or equivalent C–K functions) that we use, followed by our results
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