Draft:Schamel method

The Schamel method (S method) is a method for finding electrostatic equilibrium solutions $$\phi(x-v_0t)$$ for the Vlasov-Poisson system. It is an alternative to the method described by Bernstein, Greene and Kruskal, as it first solves the Vlasov equations for the species involved and only in a second step Poisson's equation to ensure self-consistency. The S method is generally considered the preferred method because it is best suited to describing the immense variety of electrostatic structures, including their phase velocities. These structures, also known under Bernstein–Greene–Kruskal modes or phase space electron and ion holes, or double layers, respectively, are ubiquitously found in collisionless plasmas such as in the Earth's magnetosphere, in fusion machines, or in the laboratory.

With the S method, the distribution functions for electrons $$f_e(x,v)=F_e(\epsilon_e,\sigma_e)$$ and ions $$f_i(x,u)=F_i(\epsilon_i,\sigma_i)$$, which solve the corresponding time-independent Vlasov equation,

$$(v\partial_x +\phi'(x)\partial_v)f_e(x,v)=0$$ and $$(u\partial_x -\theta\phi'(x)\partial_u)f_i(x,u)=0$$, respectively,

are described as functions of the constants of motion. Thereby the unperturbed plasma conditions are adequately taken into account. Here $$\epsilon_e=\frac{v^2}{2}-\phi$$ and $$\epsilon_i=\frac{u^2}{2}-\theta(\psi-\phi)$$ are the single particle energies of electrons and ions, respectively ; $$\sigma_e=\frac{v}{|v|}$$ and  $$\sigma_i=\frac{u}{|u|}$$ are the signs of the velocity of untrapped electrons and ions, respectively. Normalized quantities have been used, $$\theta=\frac{T_e}{T_i}$$ and it is assumed that $$0 \le\phi \le\psi$$, where $$\psi$$ 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, $$\phi_{xx} = n_e - n_i=: - \mathcal {V} '(\phi;...) $$, where $$n_e,n_i$$ are the corresponding densities obtained by velocity integration of $$F_e(\epsilon_e,\sigma_e)$$ and $$F_i(\epsilon_i,\sigma_i)$$, is integrated whereby  the pseudo-potential $$\mathcal{V}(\phi;...)$$ is introduced. The result is $$\frac{\phi_x^2}{2} + \mathcal{V(\phi;...)} = 0$$, which represents the pseudo-energy. In $$\mathcal{V}(\phi;...)$$ 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 $$x(\phi)= \int_\phi^\psi \frac{d\xi}{\sqrt{-2\mathcal{V}(\xi;...)}} $$, which yields by inversion the desired $$\phi(x)$$. In these expressions the canonical form of $$\mathcal{V}(\phi;...)$$ is used already.

There are, however, two further trapping parameters $$\Gamma_s,s=e,i$$, which are missing in the canonical pseudo-potential. In its extended previous version $$\mathcal{V}_0(\phi;...,\Gamma_e,\Gamma_i,v_0)$$ they fell victim to the necessary constraint that the gradient of the potential $$\phi(x)$$ vanishes at its maximum. This requirement is

$$\mathcal{V}_0(\psi;...,\Gamma_e,\Gamma_i,v_0)=0$$

and leads to what is called 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 $$\mathcal{V}_0(\phi;...,\Gamma_e,\Gamma_i,v_0)$$ to obtain the canonical $$\mathcal{V}(\phi;...)$$. Due to the central role, the two expressions $$\mathcal {V}(\phi;...)$$ and $$\mathcal {V}_0(\phi;...,\Gamma_e,\Gamma_i ,v_0)$$ are playing, the S method is sometimes also called Schamel's pseudo-potential method (SPP method). For more information, see references 1. and 3

A typical example for the canonical pseudo-potential and the NDR is given by $$-\mathcal{V}(\phi;B_e, B_i)/\psi^2=\frac{k_0^2}{2}\varphi(1-\varphi) +B_e\frac{\varphi^2}{2}(1-\sqrt\varphi) + B_i\frac{\theta^{3/2}}{2}(1 - (1-\varphi)^{5/2} - \frac{1}{2}\varphi(5-3\varphi))$$

and by

$$k_0^2 - \frac{1}{2}Z_r'(\frac{|v_D-v_0|}{\sqrt2}) - \frac{\theta}{2}Z_r'(\sqrt{\frac{\theta}{2\delta}}v_0)=B_e + \frac{3}{2}\theta^{3/2}B_i -\Gamma_e -\Gamma_i$$

where $$\varphi:=\phi/\psi$$ and $$\delta:=m_e/m_i $$, 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 $$(\Gamma_s, B_s)$$, s=e,i.

Failure of conventional wave theories
Probably the most important achievement of the S method is the demonstration of the failure of linear wave theories in connection with stationary solutions. The reason for this is that coherence and trapping are interdependent and remain relevant even in the limit range of vanishingly small amplitudes.

Unlimited variety of potential structures $$\phi(x)$$
The theory based on the S method predicts an unlimited variety of potential structures of the solitary and cnoidal types, not just $$sech^4(x) $$ or $$e^{-x^2} $$ solitary structures or single harmonic waves extending the few discrete modes known from linear wave theories appreciably. Most of them appear in an undisclosed form, that is, $$x(\phi)$$ cannot be represented by mathematically known functions and therefore neither can its inverse $$\phi(x)$$.

Function space of solutions
The function space of Vlasov-Poisson (VP) solutions is completely detached from the corresponding linear Vlasov-Poisson (lVP) space. There is no overlap and neither the linear Landau theory nor the Van Kampen theory can bridge this gap.

Trapping scenarios - resonance broadening
The chaotic behavior of the particles near resonance when captured by the wave is addressed through so-called trapping scenarios. Various expressions are used with corresponding parameters that stand for the different trapping channels. Two of them ($$\Gamma_e$$, $$\Gamma_i$$) have no effect on the structure because they are missing in the canonical form of $$\mathcal{V}(\phi)$$. However, they have an impact on the NDR as they allow $$v_0$$ to be shifted. This results in “resonance broadening” as predicted but not approved in the 1960s on the basis of linear wave theory.

Nonlinear dispersion relation - constraints
As said, $$\mathcal{V}_0(\psi;...,\Gamma_e,\Gamma_i,v_0)=0$$, represents a closed connection between the various parameters and is therefore suitable for a unique determination the phase speed $$v_0$$ in dependence on the other parameters.

The conditions for the existence of these structures can moreover only be accessed and derived from the NDR, such as $$\theta > 3.5$$ for long-wavelength ion-acoustic structures or $$\theta < 3.5$$ for solitary ion holes , It should be noted also that in the special case of a single wave the NDR takes the form of a "thumb-teardrop" dispersion relation, which is therefore strictly nonlinear.

Slow electron and ion acoustic modes
In a two-component, Maxwellian background plasma there are two acoustic modes, the slow electron acoustic $$v_0=1.307$$ and the slow ion acoustic $$u_0=1.307 \sqrt{\theta/\delta}$$ mode, where $$\delta=m_e/m_i$$ is the mass ratio. They are strictly nonlinear, complement the linear wave spectrum and are the basis for the propagation of solitary electron and ion holes.

Privileged solitary electron hole and Schamel equation
The showpiece of the S method is the privileged solitary electron hole because it has the smoothest of all trapped particle distributions. It therefore has the best chance of surviving collisions in a weakly dissipative plasma environment, , ,. It reads

$$-\mathcal{V(\phi)}= B_e \frac{\phi^2}{2}(1-\sqrt{\frac{\phi}{\psi}}) ; \phi(x)= \psi sech^4(\frac{\sqrt B_e}{4} x) ; v_0= 1.307( 1-B_e+ \Gamma_e)$$

Note that these expressions only contain the two parameters $$B_e,\Gamma_e$$, which are assumed to be small. It is a solution to the Schamel equation derived in by the reduced scaling method using appropriate stretched coordinates.

Solitary holes of different polarity
By the appropriate selection of trapping scenarios the existence of solitary electron holes of positive $$\phi(x)>0$$ and negative $$\phi(x)<0$$ polarity was confirmed,. This meant that corresponding observations in the magnetosphere could be explained theoretically for the first time.

Marginal stability of single waves
Contrary to some statements in the literature, there is currently no reliable linear stability theory for these structures, especially solitary structures. An exception, however, are harmonic single wave structures, for which marginal stability is predicted regardless of the drift in a current-carrying plasma. This statement contradicts Landau's theory for corresponding linear single waves.

Negative wave energy holes - anomalous transport
The energy of a structured plasma associated with these nonlinear waves can typically be negative. These negative energy holes therefore act as attractors in this dynamical system, providing an abundance of secondary wave excitations that contribute to anomalous resistivity and heating in intermittent turbulence and anomalous transport. , ,

Extensions
Based on the S method, various extensions have been presented in the past. In a generalization to relativistic electrons and the inclusion of magnetic field effects were presented. The structures, observed ubiquitously in circular particle accelerators and storage rings, could be explained in by use of the S method considering the focusing property of wall impedances. Quantum corrections of holes, investigated by, revealed the tendency to bring this extreme non-equilibrium state closer to thermodynamic equilibrium, an effect that is due to the tunneling of particles through the potential barrier. Finally reference dealt with the nonlinear shielding of planar test charges and pointed out the important role of particle trapping.