Flow Pattern Formation in Turbulence on a Rotating Sphere.

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Fluid motion on a rotating sphere often shows pattern formation phenomena never observed in a planer motion because of the influence of the compactness of the flow domain and the differential rotation, and is strongly related to fundamental properties of geophysical flows. Mainly in this project, we studied flow pattern formation arising from turbulent initial conditions, and related problems in fluid dynamics and numerical analysis. We first studied 2D fluid motion in a circular domain on a rotating sphere, and found that in the case of southern hemisphere with the rigid boundary at the equator, eastward circumpolar flow is spontaneously formed, in contrast with the whole sphere case where westward circumpolar jets emerge. The momentum transfer by the Rossby wave is important in this process, and we gave a theoretical description for the jet formation in the frame of weakly nonlinear theory. Instability of westward intensified flows were also studied with the strength of the forcing wind being a bifurcation parameter. A Hopf-type instatility followed by a supercritical bifurcation, is found as the forcing wind increases, and a relation between the instability and configuration of gyres is discussed. Phase change of water is an important factor of general circulation on the earth, and we study flow patterns generated by moist convection and its relation to global circulation from a view point of active rainfall area in tropical zone. Numerical methods are quite important in these studies, and we prepared several types of computer codes for numerical simulation, and also proposed some new numerical methods for a fluid equation on a sphere using a conformal projection to a planer disk, and also for fluid equation on a bounded domain by using a, spectral method with Jacobi polynomials. Mathematical properties of fluid equation related to turbulence dynamics are also studied.