Spectral considerations in two-dimensional incompressible flow

The spectrum of two-dimensional incompressible motion is studied with particular attention paid to non-linear interactions. Conditions for the absence of direct interactions between high and low wave numbers are developed. The non-linear tendency of an initial spectrum is computed to illustrate these points. The importance of the angular distribution of the spectrum is illustrated in a spectral stability problem for an isotropic stream function spectrum. Finally, a special inertial equilibrium spectrum is derived exhibiting the properties of an energy barrier in wave number space.