Abstract
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 rankfree, 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.
Highlights
Key QuestionsA striking property of the non-linear potential vorticity equation, for dynamics of a shallow fluid layer on a rotating planet, is that, as if it were a linear wave equation, it supports linear scaling of its normal modes, the planetary Rossby-Haurwitz [9, 20] waves
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. 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
A specific goal of this paper is to show that the existence of non-trivial sets of linearly independent, non-self-interacting fundamental wave modes, in the case at hand is implied by the near spherical symmetry of the surface of the planet, given only the algebraic structure of the non-linear interaction term and the fact that the elementary operators from which the non-linear interaction operator is composed all commute with generators of the rotation group
Summary
A striking property of the non-linear potential vorticity equation, for dynamics of a shallow fluid layer on a rotating planet, is that, as if it were a linear wave equation, it supports linear scaling of its normal modes, the planetary Rossby-Haurwitz [9, 20] waves. A second remarkable feature of the Rossby-Haurwitz modes, especially in view of the fact that their very existence has the rotation of the planet about an axis as a premise, lies in the fact that their orientation with respect to the rotation axis of the planet can be any [14], to the extent that this orientation even is of no influence on their phase velocity These features are both well-known and straightforward to demonstrate by mere substitution and calculation, e.g. based on calculation rules for spherical harmonics [14, 22, 25]. Why, on the one hand, the wave modes apparently are a consequence of the rotation of the planet, which defines an axis and by itself breaks the spherical symmetry, while on the other hand the wave modes apparently are not sensitive at all to the direction of this axis of rotation
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