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.
I present estimates to justify previously proposed by me heuristic Dipole Dynamical Model (DDM) of Ball Lightning (BL). The movement and energy supplying to the dipole BL are due to the atmospheric electric field. Crucial for the detailed analysis of
BL is using the new relation of balance of the force of atmospheric electric field (per unit mass of electron cloud) and dipole forces electrons-ions within BL dipole (per unit mass of BL) as the first necessary condition for the existance of BL as an integer. This model is unique because, unlike existing static models, fundamental condition for the existence of Ball Lightning is its forward motion. The virial theorem limiting BL power does not apply to BL which is not closed system like the Sun or Galaxy systems and is strongly dependent part of the infinitely extended in time and space large system. Stability of BL is due to two free parameters with the fundamental role of thermodynamic non-equilibrium, ionization, recombination and translational movement with energy loss by radiation and also excess volumetric positive charge. Stability of BL is not related to the presence of any external shells. Polarization degree of BL plasma is characterized by polarizability factor {gamma}. An example is presented of calculating the stability of an option of BL. There is also a possible connection of stability BL with statistical distributions of the atmospheric electric field in time and space. Destruction of BL can also occur due to arising kinematical instability at its accelerating (or decelerating) movement. Maximal energy density in BL DDM does not exceed the value Espec<(10(8) - 10(9)) J/m(3) decreasing with the growing BL radius. Resulting indefinitely long BL lifetime is also discussed. BL has no outer shell and no any inner rigid or elastic microstructure elements.
The nature of ball lightning (BL) is pure electric and can be described by simple equations following to elementary considerations of equality of translational acceleration and velocity of the ions and electrons, a spherical-like dipole BL as a whole
and balance of the energy influx of atmospheric electricity and radiation losses. From these equations follows a linear relationship between the size of BL and the tension of the atmospheric field E. A typical size of the fireball (FB) r ~ 5 cm corresponds to the calculated electron temperature T(e) ~ 8000K at a pressure p = 1 at with a horizontal component of the electric field E a few kV/cm. I estimate the energy of BL and characterize the conditions of its possible experimental generation. The estimation is given of the surface tension of BL. The possibility of the hot and the most realistic thermodynamic non-equilibrium cold BL is discussed. Here we presented preliminary evaluations preceding the more detailed work in Arxiv.org [11].
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 unifor
m 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.
After approximate replacing of Maxwellian distribution exponent with the rational polynomial fraction we have obtained precise analytical expression for and calculated the principal value of logarithmically divergent integral in the electron wave dis
persion equation. At the same time our calculations have shown the presence of strong collisionless damping of the electromagnetic low-velocity (electron) wave in plasmas with Maxwellian-like electron velocity distribution function at some small, of the order of several per cents, differences from Maxwellian distribution in the main region of large electron densities, however due to the differences in the distribution tail, where electron density itself is negligibly small.
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.
