Flow Pattern Formation in Turbulence on a Rotating Sphere.

在旋转的球体上，由于流域的紧致性和差动旋转的影响，流体运动经常表现出平面运动中未曾见过的花纹形成现象，并且与地球物理流动的基本性质密切相关。在本项目中，我们主要研究了湍流初始条件引起的流型形成，以及流体力学和数值分析中的相关问题。我们首先研究了旋转球体上圆形区域内二维流体的运动，发现在赤道具有刚性边界的南半球，自发地形成了向东的绕极流，而在整个球体的情况下，出现了向西的绕极急流。Rossby波的动量传递在这一过程中起着重要的作用，我们在弱非线性理论的框架下给出了喷流形成的理论描述。以强迫风强度为分叉参数，研究了西向强化气流的不稳定性。结果表明，随着强风的增加，涡旋的不稳定存在Hopf型不稳定性和超临界分叉，并讨论了这种不稳定性与涡旋构型之间的关系。水的相变是影响地球大气环流的重要因素，本文从热带活跃雨区的角度研究了湿对流产生的流型及其与全球环流的关系。数值方法在这些研究中非常重要，我们编制了几种用于数值模拟的计算机程序，并提出了一些新的数值方法，用于球面上的流体方程的保角投影到平面圆盘上的数值计算，以及有界域上的流体方程的带有Jacobi多项式的谱方法。研究了与湍流动力学有关的流体方程的数学性质。

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.