A New Type of Nonlinear Plasma Equation I. Formulation

Abstract

In part I we derive a model nonlinear equation which describes the collective behavior of trapped particles or the nonlinear transport caused by electrostic waves both in a microplasma and in a macroplasma. The equation is found to be approximately of the form, n_t+ann_x+bn_xxx²=0. In a Vlasov plasma n stands for the velocity distribution function modified by narrow-band waves and x for the velocity, whereas in a magnetized macroplasma n stands for the density distribution deformed by waves and x for a coordinate variable perpendicular to the magnetic field. In the course of deriving the nonlinear equation a general formula of saturation amplitudes for collisionless drift instabilities is derived, which is the same as previously derived for collisional drift instabilities. The saturation amplitudes of both collisionless and collisional instabilities are found to be essentially given by (κ₀l⁄2π), κ₀ being the initial logarithmic distribution gradient and l representing the unstable range in the direction of the density gradient.