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

在地球科学中使用的流体方程的框架下，特别是在旋转球体上的二维N-S方程的框架内，研究了流型的形成。当流动区域是一个完整的球体时，观察到从湍流初始条件中出现向西的绕极喷流。给出了这些极地射流的强度和宽度的标度律，表明在大转速范围内，流场的能量全部被这些极地射流吸收。在浅水方程的情况下，也研究了纬向急流的形成，其中急流出现在赤道区域而不是极地区域。当流动区域为旋转半球，其边界与子午线重合时，在纬向风的强迫作用下，出现向西加强的气流。研究了旋转半球上西向强化流的不稳定性，发现临界Hopf模在边界流离开西边界点附近最大。文中还研究了旋转极帽、圆周有界区域、中心在北极、进出口在边界上的流态及其稳定性。而在线性解中，进口流动贯穿整个流动区域，在靠近北极的中心区域出现孤立涡，并且随着进口流量的增加，这种非线性流动变得不稳定。在一些小自由度的混沌动力系统中，研究了不稳定周期轨道上的平均动力量，特别是当周期轨道的周期较大时。数值结果表明，不稳定周期轨道上轨道平均的统计特性与一维映射中混沌轨道上的统计特性一致，而在二维映射或某些常微分方程组中则不同。

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.