Physical interpretation of unstable modes of a linear shear flow in shallow water on an equatorial beta-plane

Abstract

Unstable modes of a linear shear flow in shallow water on an equatorial $\beta$-plane are obtained over a wide range of values of a non-dimensional parameter and are interpreted in terms of resonance between neutral waves. The non-dimensional parameter in the system is $E \,{\equiv}\, \gamma^{4} / (gH\beta^{2})$, where $\gamma$, $g$, $H$ and $\beta$ are the meridional shear of basic zonal flow, gravitational constant, equivalent depth and the north–south gradient of the Coriolis parameter, respectively. The value of $E$ is varied within the range $-2.50 \,{\le}\,\log E \,{\le}\,7.50$.The problem is solved numerically in a channel of width $5\gamma/\beta$. The structures of the most unstable modes, and the combinations of resonating neutral waves that cause the instability, change according to the value of $E$ as follows. For $\log E \,{<}\, 2.00$, the most unstable modes have zonally non-symmetric structures; the most unstable modes for $\log E \,{<}\, 1.00$ are caused by resonance between equatorial Kelvin modes and continuous modes, and those for $1.00 \,{\le}\,\log E \,{<}\, 2.00$ are caused by resonance between equatorial Kelvin modes and westward mixed Rossby–gravity modes. The most unstable modes for $\log E \,{\ge}\, 2.00$ have symmetric structures and are identical with inertially unstable modes. Examinations of dispersion curves suggest that non-symmetric unstable modes for $1.00 \,{\le}\,\log E \,{<}\, 2.00$ and inertially unstable modes for $\log E \,{\ge}\, 2.00$ are the same kind of instability.