Nonlinear Plasma Theory

Comments on “Nonlinear Oscillations in a Collisionless Plasma”

In order to realize which approximations that are made in the descriptions of plasmas that we generally use, it is instructive to start from the most general description which includes all individual particles and their correlations in the six dimensional phase space (r,v). I the absence of particle sources or sinks we must have a continuity equation for the delta function density N:

$$ N(X,t) = \sum\limits_{{i = 1}}^N {(X - {{X}_i}} (t))\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,X = ({\mathbf{r}},{\mathbf{v}}) $$



$$ \frac{\partial }{{\partial t}}N + \sum\limits_i {\frac{\partial }{{\partial {{r}_i}}}} \left( {N\frac{{\partial {{r}_i}}}{{\partial t}}} \right) + \sum\limits_i {\frac{\partial }{{\partial {{\hbox{v}}_i}}}} \left( {N\frac{{\partial {{\text{v}}_i}}}{{\partial t}}} \right) = 0, $$

Different Ways of Describing Plasma Dynamics

A Korteweg-de Vries (KdV) equation including the effect of Landau damping is derived to study the propagation of weakly nonlinear and weakly dispersive ion acoustic waves in a collisionless unmagnetized plasma consisting of warm adiabatic ions and two different species of electrons at different temperatures. The hotter energetic electron species follows the nonthermal velocity distribution of Cairns et al. [Geophys. Res. Lett. 22, 2709 (1995)] whereas the cooler electron species obeys the Boltzmann distribution. It is found that the coefficient of the nonlinear term of this KdV like evolution equation vanishes along different family of curves in different parameter planes. In this context, a modified KdV (MKdV) equation including the effect of Landau damping effectively describes the nonlinear behaviour of ion acoustic waves. It has also been observed that the coefficients of the nonlinear terms of the KdV and MKdV like evolution equations including the effect of Landau damping, are simultaneously equal to zero along a family of curves in the parameter plane. In this situation, we have derived a further modified KdV (FMKdV) equation including the effect of Landau damping to describe the nonlinear behaviour of ion acoustic waves. In fact, different modified KdV like evolution equations including the effect of Landau damping have been derived to describe the nonlinear behaviour of ion acoustic waves in different region of parameter space. The method of Ott & Sudan [Phys. Fluids 12, 2388 (1969)] has been applied to obtain the solitary wave solution of the evolution equation having the nonlinear term $(\phi^{(1)})^{r}\frac{\partial \phi^{(1)}}{\partial \xi}$, where $\phi^{(1)}$ is the first order perturbed electrostatic potential and $r =1,2,3$. We have found that the amplitude of the solitary wave solution decreases with time for all $r =1,2,3$.

Effect of Landau damping on ion acoustic solitary waves in a multi-species collisionless unmagnetized plasma consisting of nonthermal and isothermal electrons.

The nonlinear saturation of parametric decay instability in a plasma is investigated analytically under a wide range of conditions. Nonlinear coupling of four Langmuir waves is studied as a new saturation mechanism, which is treated together with three-wave cascade coupling and ion nonlinear Landau damping. It is found that when the ele ctron and ion temperatures are almost equal, nonlinear four-wave coupling is the dominant saturation mechanism, which produces a single-peaked Langmuir wave spectrum. When the electron temperature far exceeds the ion temperature, cascade mode coupling is more important and the resulting spectrum exhibits many well-resolved peaks. All these features are in agreement with laboratory experiments.

Nonlinear Plasma Theory

Citations

Comments on “Nonlinear Oscillations in a Collisionless Plasma”

Different Ways of Describing Plasma Dynamics

Effect of Landau damping on ion acoustic solitary waves in a multi-species collisionless unmagnetized plasma consisting of nonthermal and isothermal electrons.

Steady state spectrum and wave energy of parametric decay instability in plasma

Nonlinear dynamical-information magnetosphere models for space weather forecasting

Related Papers (5)

Methods in nonlinear plasma theory

Large-amplitude Alfvén waves in the interplanetary medium, 2

Reviews of Plasma Physics

Principles of Plasma Physics

Toward a theory of interstellar turbulence. 2. Strong Alfvenic turbulence