Nonlinear dispersion relation in Vlasov-Poisson plasmas

{{Short description|Relation assigning the phase velocity}}

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{{Notability|date=May 2024}}

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A nonlinear dispersion relation (NDR) is a dispersion relation that assigns the correct phase velocity $v\_0$ to a nonlinear wave structure.

As an example of how diverse and intricate the underlying description can be, we deal with plane electrostatic wave structures $\backslash phi(x-v\_0t)$ which propagate with $v\_0$ in a collisionless plasma. 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.

This is where the pseudo-potential method in the version of Schamel, also known as Schamel method, comes into play, which is an alternative to the method described by Bernstein, Greene and Kruskal.

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 $f\_e(x,v)=F\_e(\backslash epsilon\_e,\backslash sigma\_e)$ and ions $f\_i(x,u)=F\_i(\backslash epsilon\_i,\backslash sigma\_i)$, which solve the corresponding time-independent Vlasov equation,

$(v\backslash partial\_x\; +\backslash phi\text{'}(x)\backslash partial\_v)f\_e(x,v)=0$ and $(u\backslash partial\_x\; -\backslash theta\backslash phi\text{'}(x)\backslash partial\_u)f\_i(x,u)=0$, respectively,

are described as functions of the constants of motion. Thereby the unperturbed plasma conditions are adequately taken into account. Here $\backslash epsilon\_e=\backslash frac\{v^2\}\{2\}-\backslash phi$ and

$\backslash epsilon\_i=\backslash frac\{u^2\}\{2\}-\backslash theta(\backslash psi-\backslash phi)$ are the single particle energies of electrons and ions, respectively ; $\backslash sigma\_e=\backslash frac\{v\}$

v

and $\backslash sigma\_i=\backslash frac\{u\}$

u

are the signs of the velocity of untrapped electrons and ions, respectively. Normalized quantities have been used, $\backslash theta=\backslash frac\{T\_e\}\{T\_i\}$ and it is assumed that $0\; \backslash le\backslash phi\; \backslash le\backslash psi$, where $\backslash psi$ is the amplitude of the structure.

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.

In the second step, Poisson's equation, $\backslash phi\_\{xx\}\; =\; n\_e\; -\; n\_i=:\; -\; \backslash mathcal\; \{V\}\; \text{'}(\backslash phi;...)$, where $n\_e,n\_i$ are the corresponding densities obtained by velocity integration of $F\_e(\backslash epsilon\_e,\backslash sigma\_e)$ and $F\_i(\backslash epsilon\_i,\backslash sigma\_i)$, is integrated whereby the pseudo-potential $\backslash mathcal\{V\}(\backslash phi;...)$ is introduced. The result is $\backslash frac\{\backslash phi\_x^2\}\{2\}\; +\; \backslash mathcal\{V(\backslash phi;...)\}\; =\; 0$, which represents the pseudo-energy. In

$\backslash mathcal\{V\}(\backslash phi;...)$ the points stand for the different trapping parameters $B\_s,\; D\_\{1s\},\; D\_\{2s\},\; C\_s,\; s=e,i$. Integration of the pseudo-energy results in $x(\backslash phi)=\; \backslash int\_\backslash phi^\backslash psi\; \backslash frac\{d\backslash xi\}\{\backslash sqrt\{-2\backslash mathcal\{V\}(\backslash xi;...)\}\}$, which yields by inversion the desired $\backslash phi(x)$. In these expressions the canonical form of $\backslash mathcal\{V\}(\backslash phi;...)$ is used already.

There are, however, two further trapping parameters

$\backslash Gamma\_s,s=e,i$, which are missing in the canonical pseudo-potential. In its extended previous version $\backslash mathcal\{V\}\_0(\backslash phi;...,\backslash Gamma\_e,\backslash Gamma\_i,v\_0)$ they fell victim to the necessary constraint that the gradient of the potential $\backslash phi(x)$ vanishes at its maximum. This requirement is

$\backslash mathcal\{V\}\_0(\backslash psi;...,\backslash Gamma\_e,\backslash Gamma\_i,v\_0)=0$

and leads to the term in question, the nonlinear dispersion relation (NDR). It allows the phase speed $v\_0$ of the structure to be determined in terms of the other parameters and to eliminate $v\_0$ from $\backslash mathcal\{V\}\_0(\backslash phi;...,\backslash Gamma\_e,\backslash Gamma\_i,v\_0)$ to obtain the canonical $\backslash mathcal\{V\}(\backslash phi;...)$. Due to the central role, the two expressions $\backslash mathcal\; \{V\}(\backslash phi;...)$ and $\backslash mathcal\; \{V\}\_0(\backslash phi;...,\backslash Gamma\_e,\backslash Gamma\_i\; ,v\_0)$ are playing, the Schamel method is sometimes also called Schamel's pseudo-potential method (SPP method). For more information, see references 1 and 2.

A typical example for the canonical pseudo-potential $\backslash mathcal\; \{V\}(\backslash phi;...)$ and the NDR is given by

$-\backslash mathcal\{V\}(\backslash phi;B\_e,\; B\_i)/\backslash psi^2=\backslash frac\{k\_0^2\}\{2\}\backslash varphi(1-\backslash varphi)$

+B_e\frac{\varphi^2}{2}(1-\sqrt\varphi) + B_i\frac{\theta^{3/2}}{2}(1 - (1-\varphi)^{5/2} - \frac{1}{2}\varphi(5-3\varphi))

and by

$k\_0^2\; -\; \backslash frac\{1\}\{2\}Z\_r\text{'}(\backslash frac$

v_D-v_0

{\sqrt2}) - \frac{\theta}{2}Z_r'(\sqrt{\frac{\theta}{2\delta}}v_0)=B_e + \frac{3}{2}\theta^{3/2}B_i -\Gamma_e -\Gamma_i

where $\backslash varphi:=\backslash phi/\backslash psi$ and $\backslash delta:=m\_e/m\_i$, respectively.

These expressions are valid for a current-carrying, thermal background plasma described by Maxwellians (with a drift $v\_D$ between electrons and ions) and for the presence of the perturbative trapping scenarios $(\backslash Gamma\_s,\; B\_s)$, s=e,i.

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 reads

$k\_0^2\; -\; \backslash frac\{1\}\{2\}Z\_r\text{'}(\backslash frac\{v\_0\}\{\backslash sqrt2\})\; -\; \backslash frac\{\backslash theta\}\{2\}Z\_r\text{'}(\backslash sqrt\{\backslash frac\{\backslash theta\}\{2\backslash delta\}\}v\_0)=0$.

It has been thoroughly discussed in but mistakenly as a linear dispersion relation.

File:NonlinearDispersionRelation.pdf

A plot of $\backslash omega\_0:=k\_0\; v\_0$ for a more general type of structures, the periodic cnoidal electron holes, is presented in Fig 1. for the case $B\_e=B,B\_i=0$ and immobile ions ($\backslash theta=0$) showing the effect of the electron trapping parameter $B$ for positive and negative values.

Finally, it should be mentioned that an NDR has the nice property that it remains valid even if the potential

$\backslash phi(x)$ has an undisclosed form, i.e. can no longer be described by mathematically known functions.