[Paper Review] Damping of electromagnetic waves in low-collision electron-ion plasmas

В. Н. Сошников|ArXiv.org|May 13, 2002

Ionosphere and magnetosphere dynamics3 citations

TL;DR

This paper presents a new iteration method for calculating collisional damping of electromagnetic waves in low-collision electron-ion plasmas using a two-dimensional Laplace transform approach. Despite differences in the dispersion equation formulation, the resulting wave damping decrements for fast and slow modes exactly match prior results when higher-order terms in $v_x^2/c^2$ are neglected, validating the method's consistency and physical accuracy.

ABSTRACT

Using previously developed method of two-dimensional Laplace transform we obtain the characteristic equations k(ω) for electromagnetic waves in low-collision fully ionized plasma of a plane geometry. We apply here a new, different from the one used in our previous paper, iteration procedure of taking into account the Coulomb collisions. The waves are collisionally damping in the same extent as electromagnetic waves. Despite the different from previous paper form of the dispersion (poles) equation, the obtained decrements for fast and slow wave modes coincide with results obtained in our earlier paper, if one neglects the terms of higher orders in v^2/c^2, (v and c are electron and light velocities). We point out how one can determine mutually dependent boundary conditions allowing to eliminate simultaneously both the backward and kinematical waves for transversal as well as for longitudinal oscillations.

Motivation & Objective

• To develop a more physically intuitive iteration procedure for incorporating Coulomb collisions into the kinetic description of electromagnetic waves in low-collision plasmas.
• To resolve inconsistencies in boundary conditions that lead to unphysical backward and kinematical waves in the solution of the Vlasov-Maxwell system.
• To derive characteristic equations for wave propagation and damping that correctly account for collisional effects while preserving the structure of Landau's original approach.
• To demonstrate that the new method yields the same damping decrements as previous approaches, validating its consistency under the same approximations.
• To establish a self-consistent framework for determining the electron distribution function at the plasma boundary that eliminates non-physical wave components.

Proposed method

• Applies a two-dimensional Laplace transform to the coupled Vlasov-Maxwell equations to derive asymptotic solutions for electromagnetic waves in a plane-geometry plasma.
• Introduces a novel iteration scheme where the collision term is approximated as $ Q_{p_1p_2}^{(1)} = f_{p_1p_2}^{(1)} \cdot \left[ Q_{p_1p_2}^o / f_{p_1p_2}^o \right] $, replacing the previous method with improved physical clarity.
• Uses the principal value prescription for divergent integrals arising from the Laplace-transformed Vlasov equation, consistent with Landau's method.
• Imposes boundary conditions that eliminate both backward-propagating and kinematical waves by enforcing zero amplitude for these components in the distribution function.
• Derives a characteristic equation for double poles $ p_1, p_2 $ of the Laplace images of the electric field and electron distribution function, leading to a dispersion relation.
• Performs approximations by replacing $ v_x^2 $ with $ v_{0x}^2 \simeq kT_e/m_e $, and evaluates integrals using mean values to simplify the dispersion equation.

Experimental results

Research questions

• RQ1 How can Coulomb collisions be consistently incorporated into the kinetic description of electromagnetic wave propagation in low-collision plasmas using a Laplace transform method?
• RQ2 What boundary conditions on the electron distribution function are required to eliminate both unphysical backward waves and kinematical waves in the solution?
• RQ3 Does the new iteration procedure for the collision term yield the same damping decrements as previous methods, despite differences in the form of the dispersion equation?
• RQ4 How does the collisional damping of electromagnetic waves depend on frequency, especially near the plasma frequency $ \omega_L $?
• RQ5 To what extent do relativistic corrections and higher-order terms in $ v_x^2/c^2 $ affect the damping rates, and when do they become significant?

Key findings

• The new iteration method produces damping decrements for fast and slow electromagnetic wave modes that exactly match those from the prior method when higher-order terms in $ v_x^2/c^2 $ are neglected.
• The damping decrement for the fast mode is $ \delta^{(1)} = \pm \frac{2\pi e^4 n_i L \omega_L^2}{3\sqrt{3} v_{0x} m_e kT_e c \omega^2 \sqrt{1 - \omega_L^2/\omega^2}} $, showing a strong enhancement near $ \omega \to \omega_L^+ $.
• The damping decrement for the slow mode is $ \delta^{(2)} = \pm \left( \frac{\pi e^4 n_i L \omega^2}{3\sqrt{3} v_{0x}^4 m_e kT_e} \right)^{1/3} $, indicating a power-law dependence on frequency.
• The characteristic equation derived in this work differs in form from the one in [2], yet yields identical damping results under the same approximations, confirming methodological consistency.
• The method successfully eliminates both backward and kinematical waves by enforcing single-valued dependence of the electron distribution function on $ \vec{v}, x, t $ through boundary condition constraints.
• The analysis reveals a sharp increase in collisional absorption near $ \omega \approx \omega_L $, contrasting with the collisionless evanescent reflection that dominates at $ \omega < \omega_L $.