The generalized self-similarity and scaling invariance in fluid motions

摘要

ATMOSPHERIC and oceanic flows stride across very wide scale. For example, the scale of large vortexes is of the order of 10~6 m, while the scale of small vortexes is of only a few millimeters in the atmosphere, The scales of turbulent flows in the inertial range stride across three orders of magnitude in atmospheric surface layer. The monthly mean surface air temperature data in the Northern Hemisphere from 1851 to 1984 include the four different time scales (moon, year, 10~1 years and 10~2 years). These phenomena in the geophysical fluid motions are all of non-characteristic scale. The nonlinear interaction between different scale motions results in very complex flow pattern such as turbulence and climate. Some physical quantities vary even with the changing scales. For instance, the number of turbulent vortexes increases with decreasing scale. The climate also changes with time scales , just as the length of coastlines increases with decreasing scale. In this note the scaling invariance of equations of geophysical fluid dynamics is demonstrated in theory and the invariance in different physical processes is deduced. The self-similar relations between the different scale solutions are also given.