Damping of transversal plasma-electron oscillations and waves in low-collision electron-ion plasmas
Previously developed method for finding asymptotic solutions of Vlasov equations using two-dimensional (in coordinate x and time t) Laplace transform is here applied to consider transversal oscillations and waves in low-collision quasi-neutral (n_i \simeq n_e) Maxwellian electron-ion plasmas. We obtain two branches of electron waves: the ubiquitous one of high-frequency and high-velocity oscillations and the unusual low-velocity one. Taking into account Coulomb collisions in the limit m_e << m_i, \bar{v_i} << \bar{v_e}, and T_e m_e << T_i m_i results in expressions for transversal plasma-electron oscillation/wave decrements with a damping of the low-velocity electron branch \sim n_i^{1/3}/\bar{v}_e^{4/3}, where n_i is the ion density and \bar{v}_e is the mean electron velocity. It ought to rehabilitate Vlasov principal value prescription for relevant integrals, but to supplement it with representation of an asymptotical solution as a sum of exponents (not a single one). "Non-damping" kinematical waves in low-collision plasma transform in the damping ones at reasonably chosen iteration process.
Non-linear equations for electron waves in Maxwellian low-collision ion-electron plasmas
The before described general principles and methodology of calculating electron wave propagation in homogeneous isotropic half-infinity slab of Maxwellian plasma with indefinite but in principal value sense taken integrals in characteristic equations, and the use of 2D Laplace transform method are applied to an evaluation of collision damping decrements of plane electron longitudinal and transverse waves. Damping decrement tends to infinity when the wave frequency tends to electron Langmuir frequency from above values. We considered recurrent relations for amplitudes of the overtones which form in their sum the all solution of the plasma wave non-linear equations including collision damping and quadratic (non-linear) terms. Collisionless damping at frequencies more the Langmuir one is possible only in non-Maxwellian plasmas.
General dispersion equation for oscillations and waves in non-collisional Maxwellian plasmas
We propose a new and effective method to find plasma oscillatory and wave modes. It implies searching a pair of poles of two-dimensional (in coordinate $x$ and time $t$) Laplace transform of self-consistent plasma electric field $E(x,t) \to E_{p_1p_2}$, where $p_1 \equiv -i \omega$, $p_2 \equiv i k$ are Laplace transform parameters, that is determining a pair of zeros of the following equation $$\frac1{E_{p_1p_2}} = 0 .$$ This kind of conditional equation for searching double poles of $E_{p_1p_2}$ we call ``general dispersion equation'', so far as it is used to find the pair values ($\omega^{(n)}, k^{(n)}$), $n=1, 2, ...$ . It differs basically from the classic dispersion equation $\epsilon_l(\omega,k) = 0$ (and is not its generalization), where $\epsilon_l$ is longitudinal dielectric susceptibility, its analytical formula being derived according to Landau analytical continuation. In distinction to $\epsilon_l$, which is completely plasma characteristic, the function $E_{p_1p_2}$ is defined by initial and boundary conditions and allows one to find all the variety of asymptotical plasma modes for each concrete plasma problem. In this paper we demonstrate some possibilities of applying this method to the simplest cases of collisionless ion-electron plasma and to electron plasma with collisions described by a collision-relaxation term $-\nu f^{(1)}$.