Analytic solutions for equatorial, Kelvin, Rossby, and Yanai beams

Abstract

Wind-driven equatorial Kelvin, Rossby, and Yanai waves are known to propagate vertically, as well as zonally, and packets of them can form “beams” that descend into the deep ocean along ray paths consistent with wave-group theory. Here, we obtain analytic solutions to a simplified ocean model that provide a more complete description of beam properties and dynamics than in previous studies.

The model is a linear, continuously stratified (LCS) system, in which the bottom is ignored and the background Vaisala frequency $N_b$ is constant. Solutions are forced by an oscillatory wind stress, ${\tau }^{\alpha }={\tau }_o^{\alpha }X\left(x\right)exp\left(-i\sigma t\right)$, where: $\alpha$ is $x$ or $y$; $X\left(x\right)$ is confined to the region $-L<x<L$ and increases and decreases monotonically; and ${\tau }^{\alpha }$ enters the ocean as a body force with the profile $Z\left(z\right)=\left(2/\pi \right)h/\left(z^2+h^2\right)$. Under these restrictions, solutions can be represented as cosine transforms in $z$ that can be readily inverted.

Beam solutions for all three wave types have similar mathematical forms, and hence share many properties. Among other things, the solutions show how the structure and amplitude of beams depend on the above model parameters. Potential impacts of processes neglected in the solutions are noted.