Nonlinear Waves of a Surface Charge at the Boundary of a Semi-Infinite Cold Plasma in a Constant Magnetic Field

Abstract

In this paper, an equation describing nonlinear wave phenomena on the surface of magnetically active plasma in the approximation of the complete homogeneity of processes along the direction of the constant magnetic field is obtained. One of its solutions, in the form of a pulse having the shape of rapidly decaying oscillations with a changing period, is found to essentially depend on the magnitude of the magnetic field and shown to be approximately described by a specially selected analytical function. A detailed analytical analysis of the properties of another solitary wave formation existing under conditions of resonant coincidence of its carrier frequency with the corresponding value of its eigen surface oscillations in the considered cold semi-infinite plasma, in which a constant magnetic field is directed along its boundary, is also carried out. The conditions for the excitation of wave disturbances are determined, and analytical expressions that adequately describe the space–time structure of nonlinear waves are proposed.

1. Introduction

The consideration of the behavior of a nonlinear surface charge (NSC) in a constant magnetic field significantly brings the formulation of the problem closer to real experimental conditions, in which, in most cases, magnetic confinement of plasma takes place. In addition to describing the modifications of effects revealed in the absence of a magnetic field [1,2,3,4], such a formulation of the problem makes it possible to study phenomena whose existence is actually impossible without a magnetic field. The previous theoretical studies on NSCs were based on the methods of the theory of potential [5], making it possible to solve a wide range of boundary value problems in quite a simple way. Therefore, the solution of quite complex problem of the self-consistent interaction of the nonlinear wave motion of density perturbations and the resulting curvature of the surface could even be presented in an approximate analytical form. This theory greatly facilitates the identification and analysis of the features of the revealed regularities. The study of nonlinear surface waves by other methods [6,7,8,9,10,11] often brings to the analysis of the numerical solutions obtained, since it is quite rare to obtain simple enough analytical formulas as the final result. The exceptional opportunities that arise from the use of potential theory methods make it possible to study a wide range of various nonlinear effects created by the interaction of the plasma surface with different types of external influence, for example, with electromagnetic radiation beams. This paper presents the analysis of nonlinear surface phenomena using a simple enough example of a two-dimensional problem, where, along the direction of the magnetic field collinear boundary, all processes are spatially homogeneous, and the potential theory is used in a simplified form using a complex representation of all the functions used. It is shown that the entire system of initial equations with boundary conditions can be reduced to a single differential equation for the electron velocity potential. The exact solution of this equation, obtained in the form of an impulse of oscillations, the amplitude of which decreases with growing distance from its center, turns out to be approximated by quite a simple analytical function by selecting numerical coefficients depending on the magnetic field. The settings for the existence of solitary pulses under conditions when their carrier frequency is close to the eigen frequency of surface oscillations of a cold semi-infinite plasma in a constant magnetic field is also determined. The generation of such signals, which do not change their shape when propagating over a long distance, can be used both for diagnostic purposes and to exert a certain effect on selected areas of the plasma surface.

2. Theory and Method

A number of nonlinear effects can exist in plasma simultaneously under the same conditions. By influencing the existing parameters, those effects create a complex picture, which can be understood, for example, by studying the features of the manifestation of these effects separately. In particular, the properties of nonlinear processes on the plasma surface are most conveniently identified within the framework of a simplified model of a semi-bounded cold plasma with a plane boundary, which can self-consistently curve under the influence of nonlinear perturbations. If the space x-axis is directed along the normal to the unperturbed boundary, then the y-axis can be linked to the direction of the strength vector of the constant magnetic field B₀, which holds the plasma. For the electrostatic potential φ_e(r,t), depending on coordinates r (radius vector) and time t, the density of single-charged ions N_i, the density of electrons n_e(r,t) having an electric charge e, and the velocity of electrons v(r,t), the following system of hydrodynamic equations describing the behavior of electrons [1,2,3,4] in time and space can be written:

$$\Delta {\varphi }_e=e(n_e-N_i)/{\epsilon }_0,$$

(1)

$${\displaystyle \frac{\partial n_{\mathrm{e}}}{\partial t}}+\nabla \cdot \left(n_{\mathrm{e}}v\right)=0,$$

(2)

$${\displaystyle \frac{\partial v}{\partial t}}+(v\cdot \nabla )v={\displaystyle \frac{e}{m}}\nabla {\varphi }_e+v\times B.$$

(3)

Here, m is the electron mass, ε₀ is the vacuum permittivity, and B is the total strength of a magnetic field in a chosen location. For the considered approximation, B = B₀.

A characteristic feature of an NSC is the absence of perturbations in the plasma occupying a half-space x > 0, when the densities of electrons and ions are equal to each other (n_e(x > 0,t) = N_i ≡ const) and homogeneous in space. Perturbations in electron density, velocity, and potential exist only at the plasma boundary, and in this case, for the sake of simplicity of calculations, one can assume without prejudice to generality that all these characteristics do not depend on the y coordinate. This means that inside the plasma (x > 0) and in the region of the medium with the dielectric constant ε_d surrounding the plasma (x < 0), the Poisson equation (1) transforms into the Laplace equation

$$\Delta {\varphi }_e=0.$$

(4)

To describe the behavior of an NSC at the boundary (x = 0), one can use a parameter n_S:

$$n_{\mathrm{S}}(z,t)=\underset{\delta \to 0}{\lim }{\displaystyle \underset{-\delta }{\overset{+\delta }{\int }}\mathrm{d}x{n}_{\mathrm{e}}(x,z,t}),$$

(5)

named in Refs. [1,2,3,4], the surface charge density.

Meantime, from Equation (2) and from the peculiarities of the behavior of the densities n_e and N_i, it follows that the velocity potential Ψ(z,t), introduced to describe the NSC [1,2,3,4] by the formula v = ∇Ψ and determined by Equation (3), must also satisfy the Laplace equation, similar to the electrostatic potential. It is possible to use the solution in the framework of a complex representation as a function φ_e(z ± ix,t) [12], where the sign “+” is used for the solution inside the plasma (x > 0), and the sign “-” is used to describe the behavior of the potential in the environment surrounding the plasma (x < 0). As a result, the initial system of equations, together with the boundary conditions, can be reduced to a single equation for the velocity potential Ψ₀(t,z) at the plasma boundary and similarly for the electrostatic potential Φ₀(t,z):

$${\Phi }_0(t,z)=\varphi (t,z,x=0),{\Psi }_0(t,z)=\Psi (t,z,x=0).$$

(6)

The equation for describing the behavior of the density of the surface charge (5) can be obtained from the equation of continuity (2) by integrating Equation (2) [1] along a narrow transition layer near the plasma boundary:

$${\displaystyle \frac{\partial n_{\mathrm{S}}}{\partial t}}+n_0{u}_1+{\displaystyle \frac{\partial \left(n_{\mathrm{S}}{u}_2\right)}{\partial z}}=0,$$

(7)

where $u_{1,2}={\left.v_{x,z}\right|}_{\mathrm{x}=+0}.$

Similarly, following Refs. [1,2,3,4], the boundary condition for the normal component of the electrostatic field can be obtained from the Poisson equation:

$${\left.{\displaystyle \frac{{\partial }_x{\varphi }_{\mathrm{e}}}{\partial x}}\right|}_{x=x_0+0}-{\epsilon }_{\mathrm{d}}{\left.{\displaystyle \frac{{\partial }_x{\varphi }_{\mathrm{e}}}{\partial x}}\right|}_{x=x_0-0}=-e{n}_{\mathrm{S}}/{\epsilon }_0.$$

(8)

The second boundary condition is the continuity of the potential at the boundary.

In a vector form, Equation (3) represents two equations for the velocity components v_x and v_z. In a narrow neighborhood near the plasma boundary, these equations are satisfied by the following expression for the electrostatic potential:

$${\displaystyle \frac{e}{m}}{\varphi }_{\mathrm{e}}(x\to +0,z,t)=-{\displaystyle \frac{\partial \Psi }{\partial t}}-i{B}_0\Psi .$$

(9)

The relation (9) provides the connection between the potentials Ψ and Φ. After the substitution of the considered solution in the complex form φ_e(z ± ix,t) into Equation (8), one obtains the following value for the density of the surface charge:

$$n_{\mathrm{S}}(z,t)=i(1+{\epsilon }_{\mathrm{d}}){\displaystyle \frac{{\epsilon }_0}{e}}{\displaystyle \frac{{\partial }_{\mathrm{z}}{\Phi }_0}{\partial z}}.$$

(10)

Using definitions (8) and (9), Equation (7) reduces to a description of the behavior of the potential Ψ₀ in time and space, and in a dimensionless form, reads:

$${\displaystyle \frac{{\partial }^{2}F}{\partial {\tau }^2}}-iB{\displaystyle \frac{\partial F}{\partial \tau }}+{\displaystyle \frac{{\partial }^{2}F}{\partial \tau \partial \eta }}\cdot {\displaystyle \frac{\partial F}{\partial \eta }}+iB{\left({\displaystyle \frac{\partial F}{\partial \eta }}\right)}^2+F=0.$$

(11)

The following designations are used: η = z ω_S/c, τ = ω_S t, F = ω_S Ψ₀/c², ω_S = ω_p (1 + ε_d)^−1/2, ω_p = (N_i e²/mε₀)^1/2, B = B₀/ω_S, and c denotes the speed of light.

From Equation (11), it follows that, in a linear approximation, there are oscillations [13] with a frequency

$$\Omega =-{\displaystyle \frac{B}{2}}+\sqrt{{\displaystyle \frac{B^2}{4}}+1}.$$

(12)

In the geometry of the problem under consideration, such oscillations have been extensively studied [13,14,15] not only in magnetically active plasma but also in other anisotropic media, and a description of non-potential electromagnetic branches of the surface wave spectrum has been presented.

3. Results

Transferring from the function F(τ,η) to another complex parameter using the formula

$$F(\tau ,\eta )=iy(\tau ,\eta )$$

(13)

and introducing a new variable ξ = η − iτ, Equation (11) transforms to

$${\displaystyle \frac{d^{2}y(\xi )}{d{\xi }^2}}+B{\displaystyle \frac{dy(\xi )}{d\xi }}-{\displaystyle \frac{d^{2}y(\xi )}{d{\xi }^2}}\cdot {\displaystyle \frac{dy(\xi )}{d\xi }}+B{\left({\displaystyle \frac{dy(\xi )}{d\xi }}\right)}^2+y=0.$$

(14)

The numerical solution of Equation (13) is presented in Figure 1 for different values of the magnetic field strength value B and some initial values of the functions y(ξ) и dy(ξ)/dξ. It can be seen that both the amplitude of damped oscillations and their period significantly depend on the magnitude of the magnetic field.

In Figure 1, the dotted line shows the auxiliary function f(ξ), which has the following analytical representation:

$$f(\xi )=A{\displaystyle \frac{\exp (-a\xi )\cos (q{\xi }^{\alpha })}{1+b\mathrm{sh}(q|\xi {|}^2)}}.$$

(15)

The constant coefficients A, a, q, b, and α included in Equation (15) can be selected in such a way that, for the selected parameters of the function y(ξ), the function f(ξ) gets the values close to the values of y(ξ), providing an approximate analytical representation of the wanted characteristic y(ξ). It should be mentioned that the function (15) is not an exact solution of Equation (14), but represents quite a close description of the function y(ξ), according to Figure 1, in the selected range of the parameter ξ values. Approximating an exact solution using an analytical function offers important advantages over numerical exact calculations, not only because of the visual representation of the wave characteristics, but also because of the ability to perform the necessary calculations and make various estimates with it, such as determining the initial and boundary conditions to efficiently excite the nonlinear structures.

Since only the value Re[F(τ,η)] has a physical meaning, the analytical approximation of the function u(τ,η) = Re[if (τ,η)] using the definition (15) is of particular interest

$$\mathrm{Re}[F(\xi )]\approx \mathrm{Re}[if(\xi )]=A{\displaystyle \frac{\exp (-a\xi )}{1+b\mathrm{sh}(q[{\eta }^2+{\tau }^2])}}\Lambda (\tau ,\eta ),$$

(16)

where

$$\begin{array}{l}\Lambda (\tau ,\eta )=\cos \beta \mathrm{ch}{\beta }_1\sin (a\tau )+\sin \beta \mathrm{sh}{\beta }_1\cos (a\tau ),q_0=q{({\eta }^2+{\tau }^2)}^{\alpha /2},\mathrm{with} \beta =q_0\cos (\alpha \mathsf{\chi }),\\ {\beta }_1=q_0\sin (\alpha \mathsf{\chi }),\mathrm{where} \mathsf{\chi }=\mathrm{Arccos}{\displaystyle \frac{\eta }{\sqrt{{\eta }^2+{\tau }^2}}}.\end{array}$$

A graphical representation of the function (16) is shown in Figure 2 in some selected range of values of the variables τ and η and parameters corresponding to Figure 1d. It can be seen that the signal weakens when moving away from the source of disturbances. However, additional research is required to determine all the features of the actual behavior of the perturbations in the velocity potential in real space.

Another solution to the complex Equation (11) can be sought, as it was performed in Refs. [8,12], in the following form:

$$\begin{gathered} F(\tau ,\eta )={\displaystyle \sum _{\mathrm{n}=1}^2{w}_{\mathrm{n}}(\zeta )\exp (-in\omega \tau +ink\eta )}, \\ {w}_{\mathrm{n}}(\xi )=w_{\mathrm{nR}}(\zeta )+i{w}_{\mathrm{nI}}(\zeta ), \\ \zeta =\tau -\eta ,n=1,2. \end{gathered}$$

(17)

Here, the functions w_R,I,n are real functions of one variable ζ, and k, ω >> |∂lnw_R/∂ζ|. The latter conditions can be met under conditions of resonance ω >> |ω − Ω|, when the frequency of the disturbance ω under consideration is close to the natural frequency Ω.

By substituting Equation (17) into Equation (11) and performing a harmonic analysis as in Refs. [1,3,12], one obtains the following equations for amplitudes can be obtained for w_Rn,In:

$$\begin{array}{l}{\displaystyle \frac{d^2{w}_{\mathrm{R}1}}{d{\tau }^2}}+\omega {\displaystyle \frac{d{w}_{\mathrm{I}1}}{d\tau }}+\delta \omega w_{\mathrm{R}1}+2k^2(2B-3\omega )(w_{\mathrm{R}2}{w}_{\mathrm{I}1}-w_{\mathrm{I}2}{w}_{\mathrm{R}1})=0,\end{array}$$

(18)

$${\displaystyle \frac{d^2{w}_{\mathrm{I}1}}{d{\tau }^2}}-\omega {\displaystyle \frac{d{w}_{\mathrm{R}1}}{d\tau }}+\delta \omega w_{\mathrm{I}1}-k^2(2\omega -B)(w_{\mathrm{R}2}{w}_{\mathrm{R}1}+w_{\mathrm{I}2}{w}_{\mathrm{I}1})=0, \delta \omega =1-{\omega }^2-\omega {\rm B},$$

(19)

$$w_{\mathrm{R}2}={\displaystyle \frac{k^2(4\omega -B)}{1-4{\omega }^2+2\omega B}}{w}_{\mathrm{R}1}{w}_{\mathrm{I}1},w_{\mathrm{I}2}={\displaystyle \frac{k^2(2\omega -B)}{1-4{\omega }^2+2\omega B}}(w_{\mathrm{I}1}^2-w_{\mathrm{R}1}^2).$$

(20)

As it follows from the system (16)–(20), for w_R1 ≡ 0, the sytem reduces to a single equation:

$${\displaystyle \frac{d{w}_{\mathrm{I}1}}{d\tau }}+\delta \omega w_{\mathrm{I}1}+k^4{\displaystyle \frac{{(2\omega -B)}^2}{4{\omega }^2-1-2\omega B}}{w}_{\mathrm{I}1}^3=0.$$

(21)

The solution of Equation (21) is a known one [1,3,6] and for |δω| << 1, it can be presented in the following form:

$$w_{\mathrm{R}1}(\zeta )=A\sec \mathrm{h}(\kappa \zeta ),$$

(22)

where

$$A=\kappa (2\omega -B)\sqrt{{\displaystyle \frac{2}{4{\omega }^2-1-2\omega B}}},\kappa =\sqrt{-\delta \omega }$$

The conditions for the existence of the solution (22) are not only the requirement of the smallness of the magnitude of the parameter |δω| compared to the value of the natural frequency Ω but also the negative range of its values (δω < 0) for realization (22).

4. Discussion

In magnetically active plasma, there can be solitary nonlinear structures of surface charge in the form of damped oscillations depending on the magnitude of the magnetic field. The presence of such a field in plasma, for example, in a linear approximation, leads to the appearance of new spectra of eigen oscillations, and among nonlinear waves, to new forms and wave structures. One of these structures has been presented in this paper as one of the types of NSCs. In addition, when the resonance conditions of the carrier frequency of nonlinear perturbations coincide with the corresponding value of eigen surface oscillations in the magnetic field, it is possible to generate NSC pulses propagating along the plasma boundary. Such waves can be straightforwardly excited on the surface of the plasma by resonance and then used for diagnostics, as well as for other research [16] or technological [17] purposes. The method of approximation of the exact solution with the help of the specially selected analytical function used in the present paper makes it possible to visualize its most significant features and estimate the range of acceptable values of initial and boundary conditions that are optimal from the point of view of the goals of excitation of a nonlinear structure. Nonlinear perturbations in NSCs are localized on the surface of the plasma and therefore interact with external influences earlier than other internal volumes, distorting, to a greater or lesser extent, the expected effect of those perturbations on the plasma. This circumstance may be used for practical purposes, or, at least, taken into account when planning any special interaction with plasma.