Nonlinear theory of the E × B electron drift instability in high-β plasmas with T i ∼ T e

A nonlinear theory of the E × B electron drift instability for Te∼Ti is developed on the basis of the wave kinetic equation for weakly turbulent plasmas in a uniform magnetic field. It is found that nonlinear electron cyclotron resonances stabilize short wavelength—but destabilize long wavelength modes; the total wave energy, however, is stabilized by this mechanism. In a two-dimensional model (the waves propagate in a narrow fan encompassing the plane orthogonal to B), wave-wave scattering processes can transfer energy from the unstable small k region of the spectrum toward larger k's where nonlinear electron damping dominates. Either balancing the relevant stabilizing and destabilizing processes for small k's, or considering over-all energy balance, predicts that the theoretical turbulence level at k = kD (the inverse of the Debye length) is a few hundred times thermal. The estimated effective electron-ion collision frequency is found to be much too small to explain the anomalous heating observed in collisionless orthogonal shocks suggesting that another instability, not confined to propagate in a narrow fan, is required to explain the anomalous resistance.