Singularity and energy spectrum in a two-dimensional incompressible inviscid flow

A solution is found for a two-dimensional Euler equation which describes the development of singularity and the energy spectrum of an incompressible inviscid flow. The mean square of the nth curl of velocity in the flow is computed by a Pade approximation. It is shown that velocity increases exponentially in time where n is greater than 2, while energy and entropy remain unchanged. The energy spectrum reaches an equilibrium state which is expressed in terms of a power law k exp(-4.6) in the inviscid inertial range. For small wavenumbers the equilibrium state is approached more rapidly than for large wavenumbers. Graphic illustrations of the solution are provided.