Weak Interactions of Equatorial Waves in a One-Layer Model. Part I: General Properties

Abstract

Dispersive equatorial waves are labeled by the zonal slowness s, the meridional quantum number n and the vertical separation constant c. The slowness (reciprocal of phase speed) is a variable more useful than the wavenumber to relate the interactions among equatorial waves. For instance, frequency is a simpler function of slowness than it is of wavenumber, and the four classes of equatorial waves are separated in s-space; viz., Rossby (R): sc ≤ ?2n ? 1, mixed Rossby-gravity (M): sc < 1, gravity (G): ?1 < sc < 1, and Kelvin (K): sc = 1. Moreover, total energy and pseudo-momentum conservation require for the component with intermediate slowness of each triad to gain (loose) energy from (to) the other two. (If the triad is resonant, the wave with intermediate s must also have maximum absolute frequency.) Nonlinear effects are parameterized by a single variable, the interaction coefficient ? for each resonant triad (RT). The interaction and resonance conditions are reduced to finding the zeros of a polynomial of, at most, sixth degree is s; allowing for classification of all possible resonant triads: There are three types of RT for n > 0: RRR, GGR, and GGG; resonant triads with M (n = 0) and/or K (n = ?1) components have the properties of one of these three classes, depending on the frequency of the wave(s) with n < 1 (namely, the M and K may be taken as an R for ?2 ≤ ?c/2 or as a G otherwise). Non-local resonant triads in frequency space include: the packets of Rossby or inertia?gravity waves interacting with a long Rossby mode; short Rossby or inertia?gravity waves with different meridional quantum numbers interacting with a long Rossby or Kelvin mode (geostrophic flow); and the scattering of a short westward propagating inertia?gravity wave into a short eastward propagating inertia?gravity, mixed Rossby?gravity or Kelvin wave, by a short Rossby (or a mixed Rossby?gravity) wave with twice the wavenumber. Unlike the problems of quasi-geostrophic flow at midlatitude and internal gravity waves in a vertical plane, there are resonant triads of equatorial waves with the same speed, which have a finite interaction coefficient.