Nonlinear dispersion relation
A nonlinear dispersion relation (NDR) is a dispersion relation that assigns the correct phase velocity
As an example of how diverse and intricate the underlying description can be, we deal with plane electrostatic wave structures
Such structures are ubiquitous, for example in the magnetosphere of the Earth, in fusion reactors, or in a laboratory setting.
Correct means that this must be done according to the governing equations, in this case the Vlasov-Poisson system, and the conditions prevailing in the plasma during the wave formation process.
This means that special attention must be paid to the particle trapping processes acting on the resonant electrons and ions, which requires phase space analyses.
Since the latter is stochastic, transient and rather filamentary in nature, the entire dynamic trapping process eludes mathematical treatment, so that it can be adequately taken into account “only” in the asymptotic, quiet regime of wave generation, when the structure is close to equilibrium.
[3] In the Schamel method the Vlasov equations for the species involved are first solved and only in the second step Poisson's equation to ensure self-consistency.
The Schamel method is generally considered the preferred method because it is best suited to describing the immense diversity of electrostatic structures, including their phase velocities.
These structures are also known under Bernstein–Greene–Kruskal modes or phase space electron and ion holes, or double layers, respectively.
With the S method, the distribution functions for electrons
, which solve the corresponding time-independent Vlasov equation,
, respectively, are described as functions of the constants of motion.
Thereby the unperturbed plasma conditions are adequately taken into account.
are the single particle energies of electrons and ions, respectively ;
are the signs of the velocity of untrapped electrons and ions, respectively.
Of particular interest is the range in phase space where particles are trapped in the potential wave trough.
This area is (partially) filled via the stochastic particle dynamics during the previous creation process and is expressed and parameterized by so-called trapping scenarios.
are the corresponding densities obtained by velocity integration of
the points stand for the different trapping parameters
they fell victim to the necessary constraint that the gradient of the potential
and leads to the term in question, the nonlinear dispersion relation (NDR).
of the structure to be determined in terms of the other parameters and to eliminate
These expressions are valid for a current-carrying, thermal background plasma described by Maxwellians (with a drift
between electrons and ions) and for the presence of the perturbative trapping scenarios
A well known example is the Thumb-Teardrop dispersion relation, which is valid for single harmonic waves and is given as a simplified version of the NDR above with zero trapping parameters and a vanishing drift.
It has been thoroughly discussed in [4] but mistakenly[5] as a linear dispersion relation.
for a more general type of structures, the periodic cnoidal electron holes,[6] is presented in Fig 1. for the case
) showing the effect of the electron trapping parameter
Finally, it should be mentioned that an NDR has the nice property that it remains valid even if the potential
has an undisclosed form, i.e. can no longer be described by mathematically known functions.
