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Register a new userSolving non-linear equations of longitudinal and transverse electron waves in collisionless Maxwellian plasma
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V. N. Soshnikov
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We have considered an expansion of solutions of the non-linear equations for both longitudinal and transverse waves in collisionless Maxwellian plasma in series of non-damping overtones of the field E(x,t) and electron velocity distribution function f=f(0) +f(1) where f(0) is background Maxwellian electron distribution function and f(1) is perturbation. The electrical field and perturbation f(1) are presented as a series of non-damping harmonics with increasing frequencies of the order n and the same propagation speed. It is shown presence of recurrent relations for arising overtones. Convergence of the series is provided by a power law parameter series convergence. There are proposed also successive procedures of cutting off the distribution function f(1) to the condition of positivity f near the singularity points where kinetic equation becomes inapplicable. In this case, at poles absence the solution reduces to non-damping Vlasov waves (oscillations). In the case of transverse waves, dispersion equation has two roots, corresponding to the branches of fast electromagnetic and slow electron waves. There is noted a possibility of experimental testing appearing exotic results with detecting frequencies and amplitudes of n-order overtones.
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In this paper we have criticized the so-called Landau damping theory. We have analyzed solutions of the standard dispersion equations for longitudinal (electric) and transversal (electromagnetic and electron) waves in half-infinite slab of the uniform collisionless plasmas with non-Maxwellian and Maxwellian-like electron energy distribution functions. One considered the most typical cases of both the delta-function type distribution function (the plasma stream with monochromatic electrons) and distribution functions, different from Maxwellian ones as with a surplus as well as with a shortage in the Maxwellian distribution function tail. It is shown that there are present for the considered cases both collisionless damping and also non-damping electron waves even in the case of non-Maxwellian distribution function.
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.
It is shown in linear approximation that in the case of one-dimensional problem of transverse electron waves in a half-infinite slab of homogeneous Maxwellian collisionless plasma with the given boundary field frequency two wave branches of solution of the dispersion equation are simultaneously realizing. These are the branch of fast forward waves determined mainly by Maxwell equations of electromagnetic field, as well as the branch of forward and backward slow waves determined in the whole by kinetic properties of electrons in the collective electrical field. The physical nature of wave movements is revealed. A relation is found between electric field amplitudes of fast and slow waves. Multiform dividing the coupled slow waves into standing and traveling parts leads to a necessity of additional requirements to a selection of the type of a device analyzing these waves and its response interpretation.
We determine the growth rate of linear instabilities resulting from long-wavelength transverse perturbations applied to periodic nonlinear wave solutions to the Schamel-Korteweg-de Vries-Zakharov-Kuznetsov (SKdVZK) equation which governs weakly nonlinear waves in a strongly magnetized cold-ion plasma whose electron distribution is given by two Maxwellians at slightly different temperatures. To obtain the growth rate it is necessary to evaluate non-trivial integrals whose number is kept to minimum by using recursion relations. It is shown that a key instance of one such relation cannot be used for classes of solution whose minimum value is zero, and an additional integral must be evaluated explicitly instead. The SKdVZK equation contains two nonlinear terms whose ratio $b$ increases as the electron distribution becomes increasingly flat-topped. As $b$ and hence the deviation from electron isothermality increases, it is found that for cnoidal wave solutions that travel faster than long-wavelength linear waves, there is a more pronounced variation of the growth rate with the angle $theta$ at which the perturbation is applied. Solutions whose minimum value is zero and travel slower than long-wavelength linear waves are found, at first order, to be stable to perpendicular perturbations and have a relatively narrow range of $theta$ for which the first-order growth rate is not zero.
To better understanding the principal features of collisionless damping/growing plasma waves we have implemented a demonstrative calculation for the simplest cases of electron waves in two-stream plasmas with the delta-function type electron velocity distribution function of each of the streams with velocities v(1) and v(2). The traditional dispersion equation is reduced to an algebraic 4th order equation, for which numerical solutions are presented for a variant of equal stream densities. In the case of uniform half-infinite slab one finds two dominant type solutions: non-damping forward waves and forward complex conjugated exponentially both damping and growing waves. Beside it in this case there is no necessity of calculation any logarithmically divergent indefinite integrals. The possibility of wave amplifying might be useful in practical applications.
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