Degenerate Modes of Symmetric Instability

Abstract

A basic ow, which is symmetric in one of the horizontal direction (say, y direction), is unstable for y-symmetric disturbances when the basic potential vorticity is negative (so called symmetric instability). The temporal dependence of the growing modes is exponential, that is, e^σt. In the case of vanishing growth rate, that is, σ → 0, there emerge degenerate modes as shown by Xu (2007). Under the thermal wind-balanced initial condition, Xu (2007) showed that the temporal dependence of the degenerate modes is linear, that is, they grow as a linear function of time. In this note, we show that in general the degener ate modes grow quadratically in time, and that in particular they are reduced to the result of Xu (2007), (i.e., linear growth in time) under the thermal-wind-balanced initial conditions.