Flow patterns governed by fluid equation in earth sciences-numerical study from view points of dynamical systems-

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Flow pattern formation is studied in the framework of fluid equations employed in earth sciences, especially of the 2D Navier-Stokes equations on a rotating sphere. When the flow region is a whole sphere, westward circumpolar jets are observed to emerge from turbulent initial conditions. A scaling law is proposed for the strength and the width of these polar jets, which shows that the whole energy of the flow field is absorbed in these jets in the limit of large rotation rate. The zonal jet formation is also studied in the case of shallow water equation, where the jets emerge in the equatorial region rather than the polar regions. When the flow region is a rotating hemisphere, the boundary of which coincides with meridional lines, the westward intensified flow emerges under the forcing of zonal winds. The lstability of the westward intensified flow on the rotating hemisphere is studied, and the critical Hopf mode is found to have the largest amplitude near the point where the boundary flow leaves the west boundary. Flow pattern and its stability are studied also in a rotating polar cap, a circular bounded region with the center at the north pole and with inlet and outlet on the boundary. While in the linear solution the inlet flow goes through all the flow region, there emerges an isolated vortex in the central region near the north pole, and this nonlinear flow becomes unstable as the inlet flow rate is increased. In some chaotic dynamical system with small degrees of freedom, averaged dynamical quantities over unstable periodic orbits are studied especially when the period of the periodic orbits is large. Numerical results show that statistical properties of the orbital averages along the unstable periodic orbits agree with those along chaotic orbits in, for example, 1D map, but do not in 2D map or in some systems of ordinary differential equations.