Enumeration, orthogonality and completeness of the incompressible Coriolis modes in a tri-axial ellipsoid

Inertial waves often occur in geophysics and astrophysics since fluids dominated by rotation are common. A simple model to study inertial waves consists of a uniform incompressible fluid filling a rigid tri-axial ellipsoid, which rotates about an arbitrary axis fixed in an inertial frame. The waves are due to the Coriolis force, which can be treated mathematically as a skew-symmetric bounded linear operator C acting on smooth inviscid flows in the ellipsoid. It is shown that the space of incompressible polynomial flows in the ellipsoid of degree N or less is invariant under C. The symmetry of -iC thus implies the Coriolis operator C is non-defective with an orthogonal set of eigenmodes - Coriolis modes - in the finite-dimensional space of inviscid polynomial flows in the ellipsoid. The modes with non-zero eigenvalues are the inertial modes; the zero-eigenvalue modes are geostrophic. This shows the Coriolis modes are polynomials, enables their enumeration and leads to proof of their completeness by using the Weierstrass polynomial approximation theorem. The modes are tilted if the rotation axis is not aligned with a principal axis of the ellipsoid. A basic tool is that the solution of the polynomial Poisson-Neumann problem, i.e. Poisson's equation with Neumann boundary condition and polynomial data, in an ellipsoid is a polynomial. The tilted Coriolis modes of degree one are explicitly constructed and shown to be the only modes with non-zero angular momentum in the boundary frame. All tilted geostrophic modes are also explicitly constructed.