ALow-order Model of Two-dimensional Fluid Dynamics on the Surface of a Sphere

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A low-order model of two-dimensional fluid dynamics on the surface of a sphere

Without any other approximations apart from the spectral method which is employed, the energy spectra corresponding to two kinds of “negative temperatures” are simulated with a symmetric trapezium truncation. The simulated results with either of the two negative temperatures are reasonably consistent with those from the statistical theory of turbulence. The more usual case for two positive temperatures evolves differently from the theoretical prediction. The viscosity influence on the ergodicity is discussed. It is shown that two-dimensional (2D) ideal flows on the sphere have a less pronounced tendency to be ergodic than those on planar geometry due to the curvature of the spherical surface that weakens the interaction between different parts of the flow, enabling these parts to behave in more relative isolation. The expressions for the standard deviations from a canonical ensemble for the two different options of coefficients are shown to be proportional to √N (N is the total number of independent modes in the system), independent of the initial conditions of the system.