Stationary Spectrum of Pseudo-Three-Dimensional Electrostatic Turbulence in Magnetized Plasmas

We present a stationary k -spectrum of pseudo-three-dimensional electrostatic plasma turbulence in a uniform, loss-less, magnetized plasma based on the model equation of Hasegawa and Mima; \(∂/∂ t(φ-{∇}²φ)+[{∇}φ̂{·}{∇}]{∇}²φ{=}0\). The k -spectrum for a potential φ k is given by <|φ k | 2 >=(1+ k 2 ) -1 (α+β k 2 ) -1 , where α and β are constants. The existence of the two constants, α and β, in the spectrum is the consequence of the two rugged constants of motion. The spectrum is obtained using three different methods, namely, the method of Gibbs distribution, of Hopf's equation and of Wiener Hermite expansion. Because of the similarity of the equation, the obtained spectrum should also apply to the hydrodynamic turbulence of geostrophic vortices.