Evolution of energy and enstrophy containing scales in decaying, two-dimensional turbulence with friction

Abstract

Decaying, two-dimensional (2D), isotropic, incompressible, turbulence with finite Newtonian (lateral) friction and linear bottom drag is examined analytically. The classical result for inviscid, 2D turbulence of an upscale transfer of energy is shown to be robust to the presence of certain types of friction. It is shown that the energy-weighted mean wave number decreases with time, regardless of the Newtonian viscosity (i.e., for all Reynolds number) or linear bottom drag coefficients. For a given flow state, the time rate of change of the energy-weighted mean wave number is shown to be independent of the bottom friction but to increase with the Newtonian viscosity and the rate of spreading of energy in wave number space. There is an implicit dependence upon bottom friction since the second time derivative of the energy-weighted mean wave number involves a term linear in bottom drag coefficient. The forward transfer of enstrophy, on the other hand, is counter-balanced by the enstrophy decay due to finite Newtonian friction. The enstrophy-weighted mean wavelength decreases for sufficiently large Reynolds number, but increases when the Reynolds number becomes small enough. For a given flow state, the first time derivative of the enstrophy-weighted mean wavelength is independent of bottom friction. However, it implicitly affects the evolution through its influence upon the rate of spreading.