Elementary properties of non-Linear Rossby-Haurwitz planetary waves revisited in terms of the underlying spherical symmetry

We revisit Rossby-Haurwitz planetary wave modes of a two-dimensional fluid along the surface of a rotating planet, as elements of irreducible representations of the so(3) Lie algebra. Key questions addressed are, firstly, why it is that the non-linear self-interaction of any Rossby-Haurwitz wave mode is zero, and secondly,
why the phase velocity of these wave modes is insensitive to their orientation with respect to the axis of rotation of the planet, while at the same time the very rotation of the planet is a precondition for the existence of the waves. As we show, answers to both questions can be rooted in Lie group and representation theory.
In our study the Rossby-Haurwitz modes emerge in a coordinate-free, as well as in a Ricci tensor rank-free manner. We find them with respect to a continuum of spherical coordinate systems, that are arbitrarily oriented with respect to the planet. Furthermore, we show that, in the same sense in which the Lie derivative of Ricci tensor fields is rank-free, the wave equation for Rossby-Haurwitz modes is rank-free. We find that, for each irreducible representation of so(3), there is a corresponding sufficient condition for existence of Rossby-Haurwitz modes as solutions that are separable with respect to space and time. This condition comes in the form of a system of equations of motion
for the coordinate systems. Coordinate systems that move along with Rossby-Haurwitz modes emerge as special cases of these. In these coordinate systems the waves appear as stationary spatial fields, so that the motion of the coordinate system coincides with the wave phase propagation. The general solution of the existence condition is a
continuum of moving spherical coordinate systems that precess about the axes of the Rossby-Haurwitz modes. Within a single irreducible representation of so(3), the waves are dispersionless.