Langmuir turbulence and modulational instability

In this paper we discuss the turbulent state of an unmagnetized, completed ionized hydrogen plasma for the case when the electron temperature is much higher than the ion temperature so that the main modes to be considered are the Langmuir and ion-sound modes, while the transverse, electromagnetic mode can for most purposes be neglected. We give in the introduction a brief discussion of the quasi-linear theory of weak Langmuir turbulence as developed by Tsytovich and coworkers. We discuss why the peak of the resulting spectrum occurs at a finite wavenumber; the reason is that the removal of energy through electron collisions is faster than either thermalization of the ion-component or the transfer to yet lower wavenumbers through scattering by ions. In section 2 we sketch Zakharov's derivation of the basic equations which in the linear form describe the three modes mentioned earlier and which describe the nonlinear modulational instability (M.I.). In section 3 we give the linear analysis of the equations derived in section 2 and show that the decay instability of finite-amplitude Langmuir waves is fully covered by these equations. We also discuss the general dispersion relations for perturbations of such finite-amplitude waves and show that they lead to both the M.I. and the decay instability. We evaluate the growth rates for various kinds of perturbations. It is shown that all our results are valid only provided the energy density of the Langmuir modes W is less than the kinetic energy density nT_e of the electrons. In section 4 we discuss under what circumtances in the linear approximation the M.I. dominates over other dissipative mechanisms and we show that it may well be the dominant mechanism for the case of strong turbulence, when $\text{W/nT}{}_{\mathrm{e}}\text{≡}\text{W}\text{?}\text{> = m}{}_{\mathrm{e}}\text{/m}{}_{\mathrm{i}}$. Section 5 is devoted to a discussion of the one-dimensional variant of our basic equations. We give a brief discussion of various soliton-bearing equations such as the Kortweg-de Vries equation (KdV), the sine-Gordon equation (SG), and the non-linear Schrödinger equation (NLS).We discuss the possible connection between the number of polynomial conserved densities (p.c.d.) which lead to integrals of motion and the number of solitons in the strict sense of the term which can occur in a single exact solution of the equation. We also discuss the evolution of a system of Langmuir solitons in terms of their interactions with one another and with ion-sound. In section 6 we first of all show that in three-dimensions soliton-like structures are unstable against adiabatic collapse. We then discuss the dynamics of such collapsing “cavitons” and show that the theory of collapsing cavitons predicts that the turbulent energy should increase as the $\text{2}\text{3}$ power of the pumping rate. We discuss the fact that it may be difficult to avoid overcrowding of cavitons which would violate their independence. We finally briefly discuss the few relevant experiments and mention some ideas for future research.