Theory of a Turbulent Stationary State of a Plasma

Abstract

We first establish a self-consistent scheme to determine the fluctuation spectrum for a class of turbulent plasmas in which a conventional linear theory predicts an exponential growth of the density-fluctuation excitations, or instability. This theoretical approach is based on the fundamental postulate that a proper inclusion of the correlation effects or the existence of the fluctuations in the stationary state should be able to remove the instability and lead to a description of the most stable state for the turbulent plasma. A dielectric response function $\epsilon (\mathbf{k},\omega )$ for such a turbulent plasma is calculated within the validity of the hydrodynamic description. In this calculation, there are involved various polarization processes associated with the interaction between the external test charge (or the induced fluctuations) and the turbulent fluctuations existing in the plasma; the nature of those processes is clarified with the aid of diagrammatic considerations. The fluctuations of the internal electric field give rise to an additional mechanism for particle diffusion; the effective diffusion coefficient in the turbulent plasma is obtained by investigating the behavior of $\epsilon (\mathbf{k},\omega )$ in the limit of long wavelengths and low frequencies. The effects of turbulence upon the properties of the ion acoustic wave are studied. Following the self-consistent scheme, an integral equation is derived for the fluctuation spectrum associated with the acoustic mode; it is then solved for values of the electron drift velocity ${\mathbf{V}}_d$ above and below the critical one, $V_c$. We thus find that the results indeed support our original postulate, and the dielectric response function remains stable for the entire range of the drift velocity. The over-all structure of the fluctuation spectrum is investigated. In terms of the small plasma parameter, $g\equiv \frac{1}{n_0\lambda D^3}$, the energy $\epsilon \left(\mathbf{k}\right)$ in the acoustic mode with wave vector k is of the order of $g$ (i.e., around the thermal level) in the stable region; as the plasma enters the transition region, $\hat{k}$. ${\mathbf{V}}_d\simeq V_c$, the order of $\epsilon \left(\mathbf{k}\right)$ goes up to $g^{\frac{1}{2}}$; in the turbulent region, $\epsilon \left(\mathbf{k}\right)$ contains a part of the order of $g^0$. It is also shown that a certain domain of the turbulence spectrum can be explained with the aid of a dimensional argument. The results of the calculations are compared with a fluctuation spectrum measured by a microwave scattering experiment.