Stability and bifurcations of two-dimensional zonal jet flows on a rotating sphere

In planetary atmospheres in Jupiter or Saturn, for example, strong zonal jets have been observed. The existence of the zonal jet flow has been considered as one of the robust properties of planetary atmospheres. The two-dimensional incompressible Navier-Stokes flow on a rotating sphere is considered to be one of the simplest and most fundamental models of the atmospheric motions taking into account the effect of the planetary rotation. The Reynolds number of the planetary atmospheres is so large that properties of the Navier-Stokes turbulence on a rotating sphere should be relevant to some aspect of the dynamics of the atmosphere. However, even in this simplest model, it is far from straightforward to obtain global properties of fully nonlinear solutions. In this thesis we discuss the Navier-Stokes flows on a rotating sphere, with an attention focused on the stability problem, the bifurcation structure of the zonal jet flows and chaotic solutions at high Reynolds numbers. First we show the inviscid stability of the zonal jet flows on a rotating sphere. The semi-circle theorem obtained by Howard (1961) on a non-rotating planer domain is extended to the rotating sphere. We also study the linear stability of the zonal jet flows (l-jet flow) the streamfunction of which is expressed by a single spherical harmonics function Y 0 l . This linear stability problem was first studied by Baines (1976), and his numerical result has been considered as a standard result for the zonal jet flows. We show that the critical rotation rates obtained by Baines include numerical errors caused by an emergence of singularities (critical layers), and we give accurate numerical results for the critical rotation rate by using a power-series expansion and a shooting methods taking into account the singular points. Next, we study the viscous stability problem and the bifurcation diagram of the zonal jet flows, by introducing a forcing term balancing with the viscous dissipation terms. This setting is similar to the Kolmogorov problem, in which the stability and the bifurcation diagram of two-dimensional Navier-Stokes flows on a flat torus is considered. We prove rigorously that the 2-jet zonal flow is globally and asymptotically stable for an arbitrary Reynolds number and rotation rate. Then we study the linear stability of l-jet zonal flow for l ≥ 3 and find an interesting phenomenon that the inviscid limit of the critical stability point does not coincide with the inviscid critical stability point. We also show that this is not a contradiction because the inviscid limit of the growth rates of the viscous unstable modes coincides with that of the inviscid unstable mode. In the numerical simulation by Obuse et al. (2010), the asymptotic states of forced two-dimensional turbulence are only the 2or 3-jet zonal flows. A discussion is given on their results and our result on stability of laminar jets. We study the bifurcation structure arising from the 3-jet zonal flow. In non-rotating case, at the critical Reynolds number, a steady traveling wave solution arises from the 3-jet zonal flow through the Hopf bifurcation. As the Reynolds number increases, several traveling solutions arise only through the pitchfork bifurcations and at high Reynolds numbers the steady bifurcating solutions become Hopf unstable. For the steady bifurcating solutions in