Linear Stability of Viscous Zonal Jet Flows on a Rotating Sphere

Abstract

Linear stability of viscous zonal jet flows on a rotating sphere is investigated. This problem setting is similar to the Kolmogorov problem on the point that the basic solutions are expressed by a single eigenfunction of Laplacian of each manifold, a sphere and a torus. In the non-rotating case, as the number of jets increases the critical Reynolds number increases monotonically with the critical azimuthal wavenumber being \(m_{\text{c}}=2\). In the rotating cases, the zonal flows are stable for large rotation rates while the effect of small rotation does not always stabilize the zonal flows. We find that the critical rotation rate in the inviscid limit does not coincide with the inviscid one. This seeming contradiction between the inviscid limit stability and the inviscid stability is resolved by an observation that, as the Reynolds number increases, the growth rates of viscous unstable mode converge to zero in the regions where the inviscid zonal flow is stable while the viscous zonal flow is unstable.