The analysis of the behavior features, wave characteristics, and spatial-wave structure of axial-symmetric and azimuthal solitary oscillations at the boundary of a cold semi-infinite plasma is carried out using a closed system of equations describing the interdependence of the curvature of the surface and the properties of the nonlinear surface charge of the plasma. The specificity of the nature of motion, spatial distribution, and interconnection of the parameters of wave formations of each type is described. Numerical calculations have been performed, and an approximate analytical solution has been found.

Waves on the surface of the plasma have been known for a long time1 and are interesting, among other things, because these perturbations are the first to interact with any external influence on the plasma. Therefore, the parameters of the latter undergo a more or less strong distortion, which must be taken into account if the goal is to exert a strictly defined influence on the deep regions of the plasma. However, to do this, it is necessary to know the properties of these arising perturbations quite well, including the vast variety of types of nonlinear waves.2,3 Among them, the so-called nonlinear waves of surface charge4 are distinguished by their peculiarities, which easily interact with external electromagnetic beams5,6 and are capable of generating a number of interesting effects.7,8 All the physical properties of these waves are described in sufficient detail and clearly by the formulas obtained within the framework of the potential theory,9 which greatly facilitates the analysis of various processes of external influence even in difficult conditions such as the appearance of curvatures on the surface of the plasma boundary.10 In a simple model of a semi-infinite cold plasma with a sharp boundary, a feature of the nonlinear surface charge (NSC) is the absence of charge perturbation in the plasma and the environment, as well as the specific nature of the motion of electrons, which are driven by a potential of the velocity.4–8,10 Therefore, the main manifestations of NSC are associated with the surface of the plasma boundary. Since practically all external influences are associated with the emergence of the motion of this boundary, a reasonable interest arises in solving the general problem of how the acceleration of electrons along the normal to the boundary in some limited area or point affects the velocity along the surface of the plasma, and what equilibrium configurations of the resulting caverns are capable of propagating, changing, and exerting a reverse effect. The common feature of these effects, associated with the change in the state of electrons on the surface of the plasma, makes it relevant to study the problem of excitation of NSC in various forms by the forced movement of electrons at the boundary of the plasma in a certain local region of its surface.

In this paper, we consider the self-consistent problem of the occurrence of interdependent curvatures of the surface of electrons and NSC perturbations caused by the acceleration of electrons at the boundary at a separate point of its surface along the normal to it. In a cylindrical coordinate system with an axis along the normal to the boundary, axially symmetric and azimuth-dependent NSC perturbations, as well as the shapes of caverns on the plasma surface, were analyzed. The occurrence of such nonlinear waves can have a significant impact on the processes of interaction of the external force with the plasma surface. Therefore, they are an important object of study. Moreover, the number of different types of such perturbations can be easily increased if, for example, electromagnetic effects are taken into account, as is done in Refs. 11–13 for nonlinear surface waves. Under certain conditions, such effects play a decisive role. Alternatively, it is possible to generalize the model under consideration by adding an inhomogeneous transition layer at the boundary, which, as is known,14,15 leads to the appearance of resonant absorption and additional dispersion of surface oscillations in a linear approximation. Of course, not all kinds of surface perturbations interact in the same way with any type of external influence. However, in practice, for any chosen specific method of influencing the properties of the plasma, it is always possible to find such a form of nonlinear surface excitation that will either significantly increase the effect of the influence or significantly reduce its effectiveness. However, the selection and provision of the appropriate conditions for the implementation of the appropriate process almost always helps to achieve the desired goal.

The NSC waves themselves can also be interesting in terms of their direct use. For example, the occurrence of such a disturbance is accompanied by the creation of strong electrostatic fields that weakly attenuate at a distance from the plasma boundary. This, in turn, can generate local beams of fast particles. The possible application of NSC waves in plasma work is associated not only with the direct use of their already discovered and studied properties but also with the consideration of their capabilities in new promising conditions. For example, the presented theory could be generalized in the future to use phase-mixing,16,17 quantum effects,18 and perspective numerical schemes.19 The described approach can be also useful for the design and production of new functional materials and processes.20 Thus, it can be considered that there are convincing grounds for a consistent and thorough study of nonlinear wave phenomena on the surface of the plasma.

A distinctive feature of the electrostatic fields of NSC is the absence of perturbations of the electron density n inside the plasma, which occupies the region of x > x0(t,y,z) in Cartesian coordinates, as well as in its environment with a dielectric constant εd. Such perturbations for this kind of deviation from the equilibrium state exist only exactly on the surface of the boundary described by the function x = x0(t,y,z), where y and z are the remaining spatial coordinates, and t is the time. The calculation of the space-time distribution of the electrostatic potential φe(t,r) in the entire space is carried out within the framework of the potential theory4,9 and leads to the following well-known result:5–8 
φ e ( r , t ) = 1 2 π d y d z | x x 0 ( y , z , t ) | χ ( y , z , t ) Φ 0 ( y , z , t ) [ ( y y ) 2 + ( z z ) 2 + ( x x 0 ( y , z , t ) ) 2 ] 3 / 2 .
(1)
Here r = {x,y,z} is the radius vector, χ ( t , y , z ) = 1 + H y , z 2 , H y , z = y , z x 0 , and Φ(t,y,z) = φe(t,x=x0,y,z). The “+” sign in (1) is chosen when the radius vector points to a point inside the plasma (x > x0(t,y,z)). Otherwise, the “−” sign is taken.
Equation (1) is necessary to obtain the values of φe(x → x0 ± 0, y, z, t), which are used in the boundary condition that can be received after integration of the Poison equation
Δ φ e = e ( N e n 0 ) / ε 0
(2)
near the boundary over the layer of the thickness δ → 0.4–8 The main peculiarity of considered nonlinear perturbations consists of the localization of the electron density Ne on the plasma boundary only4–8 (n0 is the equilibrium number density that coincides with the ions one, e is the elementary charge, and ε0 is the vacuum permittivity). It means that the potential φe satisfies the Laplace equation as inside the plasma as well in the surrounding dielectric with a permittivity εd since the right side of the Poison equation in the plasma is equal zero. After such integration in (2), this boundary condition is described by the following relation:4–8,
x φ e | x = x 0 + 0 ε d x φ e | x = x 0 0 = e n S / ε 0 .
(3)
The surface charge characterized by the density nS appears at the plasma boundary because of electrostatic perturbations4–8 and defines entirely existence of the potential φe as
n S ( z , y , t ) = lim δ 0 x 0 δ x 0 + δ d x N e ( x , y , z , t ) .
(4)
Another boundary condition consists of requiring the continuity of the potential φe at the boundary x = x0(y,z,t).
To satisfy Eq. (3), it is necessary to find the derivatives x φ e | x = x 0 ± 0, for which one should write the expansion of the function φe(t,x,y,z) from (1) near x = x0(y,z,t) ± 04–8 
φ e ( x x 0  ±  0 , y , z , t ) = Φ 0 ( y , z , t ) ± ( x x 0 ) [ a 1 | | Φ 0 + H y 2 y Φ 0 + H z 2 z Φ 0 χ ( y , z , t ) ] ( x x 0 ) 2 2 Δ | | Φ 0 a 1 [ H y | | y 2 Φ 0 + H z | | x 2 Φ 0 ] χ ( y , z , t ) + , Δ | | = | | | | , | | = { 0 , y , z } , a 1 = ln ( 2 + 1 ) / ( 2 π ) 0.2 , | | = y + z .
(5)
It gives the possibility to determine nS described by (4) from (3) after inserting into (5) as
n S ( z , t ) = a 1 ( 1 + ε d ) ε 0 | | Φ 0 / e .
(6)
The motion of electrons is carried out with a speed v, which is governed by a change in the potential Ψ(t,r), i.e., v = ∇Ψ(t,r). At the same time, the following equation was obtained for the value of this potential at the boundary Ψ0 = Ψ(t,x=x0,z,y)5,6,8 from the equation of the motion of electrons:5–8,
t Ψ 0 + a 2 { ( y Ψ 0 ) 2 + ( z Ψ 0 ) 2 } + a 1 2 y Ψ 0 z Ψ 0 = e Φ 0 ( y , z , t ) / m , Ψ 0 = Ψ 0 ( x = x 0 + 0 , y , z , t ) , a 2 = ( 1 + a 1 2 ) / 2.
(7)
Equation (7) was derived with the help of preposition that inside plasma x > x0(y,z,t) + 0 the quasi neutrality takes place Ne = n0 that provides the validity of the relation ΔΨ = 0, in accordance with the equation of the continuity:5–8,
t N e + ( N e v ) = 0.
(8)
It means that the solution (1) and the expansion (5) are applicable also for Ψ(r,t) in the region x > x0(y,z,t) + 0. An integration of Eq. (8) similar to one made in (3) thus results in5–8 
t n S + n 0 u 1 + y ( n S u 2 ) + z ( n S u 3 ) = 0 ,
(9)
where u 1 , 2 , 3 = v x , y , z | x = x 0 + 0 .
Equation (8) can be rewritten in the dimensionless form
τ | | 2 F 0 + | | ( | | F 0 | | F ) + | | F = 0 ,
(10)
where the following dimensionless notations are used: ζ = z ωS/c, η = y ωS/c, c=const,
F 0 = τ F + a 2 { ( ζ F ) 2 + ( η F ) 2 } + a 1 2 ζ F η F + a 1 | | F { H ζ 2 ζ F + H η 2 η F } , | | = { 0 , η , ζ } , a 1 = ln ( 2 + 1 ) / ( 2 π ) 0.2 , a 2 = ( 1 + a 1 2 ) / 2 , | | = ζ + η ,

τ = ωS t, F = ωS Ψ0/c2, and ωS = ωp (1 + εd)−1/2 where ωp = (n0 e2/mε0)1/2 and m is the electron mass.

For the boundary curvature included in (10) and described by the function x0(t,y,z), the following relation is established with the velocity potential at the boundary:7,21
τ Λ + | | F | | Λ + a 1 | | F = 0 .
(11)

Here Λ = x0ωS/c.

Equations (10) and (11) form a closed system for two variables Λ and F. Its solution makes it possible to determine the electrostatic potential Φ(z,t) using Eq. (7). Formulas (1), (10), and (11) make it possible to determine the spatiotemporal structure of all the characteristics of the process under consideration for the boundary and initial conditions given at the surface x=x0(t,y,z).5–8,21

As a rule, the external influence on the plasma is carried out in a limited area on the surface of its boundary. Therefore, it is more convenient to write Eqs. (10) and (11) in cylindrical coordinates {ρ,θ} on a plane centered at the point ρ = 0, where the boundary conditions of the problem should be formulated. To simplify the process of solving without losing the analysis of the main features of the problem, we can consider two types of perturbations that characterize nonlinear phenomena on the surface of the plasma. These are axially symmetric wave structures and azimuth-oriented excitations of electron density on the surface of the plasma. On the example of these two limiting cases of the spatial structure of the NSC, it is possible to study all the main features of the phenomenon under discussion in the most simplified form.

The transition from the Cartesian coordinate system to the polar coordinate system on the plane {x,y} for Eqs. (10) and (11) can be carried out following the calculations performed in Refs. 5–8. At the same time, in dimensionless form, these equations retain their previous form, but some terms change their form. For example, η = ρ ωS/c, ζ → θ, and ∂/∂ζ → ρ−1 ∂/∂θ. For axially symmetric perturbations, the system of equations (2) and (3) takes the following form:
τ η 2 F 0 + 1 η η ( η η F 0 η F ) + η F = 0 , F 0 = τ F + a 2 ( η F ) 2 + a 1 ( η F ) 2 ( η Λ ) 2 , τ Λ + η F η Λ + a 1 η F = 0 .
(12)
In a linear approximation, the equation describes the natural surface oscillations of a semi-bounded plasma with a frequency of ωS. It gives some reason to consider nonlinear perturbations in which frequencies ω close to this characteristic value are present in the spectrum and their harmonics. As a result, such a solution can be sought in the form of the following expansion4,10 in the vast region kη ≫ 1, which is of maximal interest
F ( τ , η ) = n = 1 e xp { i n ( ω τ k η ) } { w Rn ( ξ ) + i w In ( ξ ) } + c . c . , ξ = η Ω τ .
(13)
In this case, Eq. (12) takes the following form for the amplitude wR1 of the first harmonic:
Ω 2 d 2 w R 1 ( ξ ) d ξ 2 + B 4 w R 1 3 ( ξ ) + B 5 w R 1 5 ( ξ ) + δ ω    w R 1 ( ξ ) = 0 , δ ω = 1 ω 2 , B 4 = 2 ω k 2 [ B 1 ( a 2 + 1 ) ω k 2 ] , B 5 = 2 k 2 { a 1 k 3 [ a 1 2 k ω B 1 B 2 ] + 9 ω B 1 B 3 6 k 2 B 3 } , B 1 = ω k 2 2 a 2 + 1 4 ω 2 1 , B 2 = a 1 k 3 2 ω 2 { 1 + 2 ω k 2 B 1 } , B 3 = 2 k 2 9 ω 2 1 { a 2 k 2 ω B 1 ( 6 a 2 + 1 ) } .
(14)
Its relation to the other harmonics of decomposition (13) in the accepted approximation is as follows:
w R 2 ( ξ ) = i B 1 w R 1 2 ( ξ ) , w R 3 ( ξ ) = B 3 w R 1 3 ( ξ ) .
(15)
It should be noted that the magnitude of the harmonic decreases noticeably with the increase in its number. However, in order to correctly calculate all the nonlinear terms included in (12), it is necessary to take into account the first three harmonics. Taking into account other harmonics produces additional small terms of order of the relation of amplitudes in comparison with unity.
It should be noted that simultaneously with the function F(τ,η), the expansion of type (13) was also applied to the value Λ(τ,η), the harmonics of which are related according to (12) by the following relations with the variable wR1(ξ):
Λ R 1 ( ξ ) = a 1 k ω w R 1 ( ξ ) , Λ R 2 ( ξ ) = i B 2 w R 1 2 ( ξ ) .
(16)
FIG. 1.

The comparative dependence of the velocity potential amplitude of electrons and the approximating function (dashed line) on the combined space-time coordinate at the flat half-space plasma boundary for the following task parameter values: k = 3, δω = −0.0001, Ω = 1, wR1(0) = 0.5, dξwR1(0) = −2.

FIG. 1.

The comparative dependence of the velocity potential amplitude of electrons and the approximating function (dashed line) on the combined space-time coordinate at the flat half-space plasma boundary for the following task parameter values: k = 3, δω = −0.0001, Ω = 1, wR1(0) = 0.5, dξwR1(0) = −2.

Close modal
The solution of Eq. (14), which determines the value of the function wR1(ξ), as well as all other harmonics (15) and (16), can be represented both as the result of an exact numerical calculation and with the help of approximate analytical formulas. Figure 1 shows both of these variants with the analytic description proposed as a function f(ξ), the expression for which can be written using the following formula (cf. Refs. 4 and 10):
f ( ξ ) = A | ξ | + κ , A , κ = const .
(17)
Under resonance conditions |δω| → 0, the function f(ξ) coincides as closely as possible with the exact solution of Eq. (14) for those values of the coefficients included in (17) that satisfy the following relation
A = 15 2 4 1 k 2 , κ = A 2 f 2 ( ξ = 0 ) .
(18)
Formula (18) gives an adequate, though approximate, idea of the behavior of the amplitude of the electron velocity potential as the wavelength of surface perturbations changes under resonance conditions ω ≈ ωS.

Equation (14) does not accurately describe the behavior in space of surface perturbations in a narrow region η → 0 due to the use of the approximation made (kη ≫ 1), but the main part of the perturbed surface transforms its structure closely to the exact solutions of Eq. (12). By virtue of the same assumption, it is impossible also to describe the change in the amplitude of the wave during its propagation by means of this approximate Eq. (14) obtained from (12) in the zero order of the small parameter (kη)−1, but the other characteristics of motion resulting from it remain close to the solution of the exact Eq. (12). The latter conclusion can be easily proved by a perturbation theory with this small parameter (kη)−1 in the next approximation step. Figure 2 shows this structure for two different time points.

FIG. 2.

Spatial structure of surface perturbations for two selected moments of time: (а) τ = 0 and (b) τ = 7 resulting from the use of the following task parameters: k = 3, δω = 0.0001, Ω = 1, wR1(0) = 0.6, dξwR1(0) = −5.8.

FIG. 2.

Spatial structure of surface perturbations for two selected moments of time: (а) τ = 0 and (b) τ = 7 resulting from the use of the following task parameters: k = 3, δω = 0.0001, Ω = 1, wR1(0) = 0.6, dξwR1(0) = −5.8.

Close modal

If, at the initial moment of time τ = 0, the excitation structure shown in Fig. 2(a) is created on the surface of the plasma, then in the course of time it moves and, for example, at some selected moment τ = 7 takes the form shown in Fig. 2(b). The nature and peculiarities of the motion of the surface disturbance are determined by the self-similarity of its behavior existing within the framework of the accepted approximations. However, the distortion of the exact picture of the development of events that inevitably arises is not so great as to decisively affect the correct perception of the true behavior of the nonlinear surface perturbation. It is necessary also to underline that the physics solution having a finite values everywhere and presented in Fig. 2 was found by correct choosing the initial conditions for used constants of the problem.

In the other limit of the spatial behavior of surface perturbations, their dependence on spatial coordinates is mainly due to noticeable changes in azimuthal angle variations and to a weak influence of the radius value on their magnitude when displaced along this polar coordinate. In the vicinity of the R value of the radius, where the magnitude of the disturbance under consideration reaches its maximum value, in Eq. (10), it is possible to neglect the derivatives of the ρ coordinate (or η in dimensionless form in the transformation to the polar coordinate system), and the value of R itself can be considered a free parameter. Equation (10) can then be rewritten as
τ F 0 + q F 0 q F + F = 0 , q = ω R θ c , F 0 = τ F + a 2 ( q F ) 2 + a 2 ( q F ) 2 ( q Λ ) 2 .
(19)
Its solution, which depends on a single argument in the form of a combination of the independent initial variables ξ = q − τ like the case of axially symmetric oscillations, is presented here both as the result of an exact numerical calculation and with the help of an approximate analytical formula. The corresponding calculations are quite similar to those given above and are based on the expansion (6) with the calculation of the harmonic values and the derivation of the equation for the first harmonic, which for Ω  = 1 is the same in shape as (14), but has a different independent variable. The latter circumstance imposes a certain additional restriction on the type of solution obtained since if the angular variable θ is replaced by the value θ + 2πp (p = 1, 2, 3, …), the function satisfying (19) must remain unchanged. This means that the analytic solution of Eq. (19) must be a periodic function with a period of 2π. Such a requirement, taking into account the definition of the argument ξ, leads to the conclusion that the azimuth oscillations under consideration exist only for a discrete series of values of radius R given by the formula
R = c ω S p , p = 1 , 2 , 3 , .
(20)
The result of the exact solution of Eq. (19) existing for δω > 0 is shown in Fig. 3 together with a curve of some analytic function (dotted line) that is not a solution (19) but closely repeats the behavior of the desired function that represents this exact solution. The type of such a function is defined by the formula
f ( ξ ) = A | sin ( κ ξ ) | + δ ω , A , κ = const .
(21)
Together with the selected value of the radius (20), the parameters (21) of the function f(ξ) form a set of quantities that characterize the wave process in question to a certain extent. These parameters are related to each other by the following empiric ratios:
A = 9 δ ω 4 k , κ = 2 δ ω .
(22)
As can be seen from (22), the value of the coefficients included in formula (21) depends on the value of the frequency detuning, which plays the role of a small parameter (δω ≪ 1) under resonance conditions ω  ≈ ωS.
FIG. 3.

The behavior of the velocity potential amplitude of electrons in comparison with the approximating function (dashed line) depending on the combined space-time coordinate at the flat half-space plasma boundary for the following values of the problem parameters: k = 3, δω = 0.1, Ω = 1, wR1(0) = 0.65, dξwR1(0) = −2.

FIG. 3.

The behavior of the velocity potential amplitude of electrons in comparison with the approximating function (dashed line) depending on the combined space-time coordinate at the flat half-space plasma boundary for the following values of the problem parameters: k = 3, δω = 0.1, Ω = 1, wR1(0) = 0.65, dξwR1(0) = −2.

Close modal

The curves presented in Fig. 3 show a good coincidence of the results of the exact numerical solution of Eq. (19) and its approximate analytical description (21), which does not fully satisfy this equation. Nevertheless, it gives a fairly good understanding of the characteristic kind of perturbation in question.

The spatial structure of azimuthal perturbations is determined, first of all, by the exact solution of Eq. (19), which describes their shape along the azimuthal coordinate, and by a certain smooth dependence on the radial component, with respect to which, within the framework of the accepted assumptions, there is only a requirement that its first derivative in radius be small relative to the magnitude itself. Many different functions can satisfy this condition. Of these, a hyperbolic dependence of the type sechν(κη) (ν, κ ≪ 1) that fully satisfies the above requirements was selected to represent the potential perturbation moving at a fixed radius value along the azimuth coordinate, the spatial structure of which is depicted in Fig. 4 for the two selected moments of time. It can be seen that when it moves along the ring, the disturbance retains its form for an indefinite amount of time and can be destroyed only by some external influence in the approximation made. Taking into account dissipation and studying the stability of a nonlinear wave require additional independent research.

FIG. 4.

Spatial structure of surface perturbations for two selected moments of time: (a) τ = 0 and (b) τ = 7 obtained as a result of using the following parameters of the problem: k = 3, δω = 0.1, Ω = 1, wR1(0) = 0.6, dξwR1(0) = −2.

FIG. 4.

Spatial structure of surface perturbations for two selected moments of time: (a) τ = 0 and (b) τ = 7 obtained as a result of using the following parameters of the problem: k = 3, δω = 0.1, Ω = 1, wR1(0) = 0.6, dξwR1(0) = −2.

Close modal

Thus, potential perturbations in the density and velocity of electrons can arise and propagate across the surface of the plasma in the form of nonlinear waves of various shapes. The limiting cases of nonlinear excitation considered in this paper—from radial axially symmetric to azimuthal with a fixed radius of structures—form the boundaries of the range of forms within which various types of nonlinear perturbations can arise. The peculiarities of their special movement may also be interesting for the use of an applied nature.

Some of the nonlinear phenomena of the surface charge of electrons presented in this paper are obtained within the framework of a simple model using a number of rather rough approximations and assumptions and being, on first look, very far away from real conditions of processes under consideration. Nevertheless, such model studies are important parts of many research. The fact is that in real life there are many different processes that overlap and create an extremely complex picture of their bizarre combination. In order to distinguish from this set of events and study the features of a particular phenomenon, a model approximation is often used. The construction of a specific model makes it possible to immediately discard many different processes that have nothing to do with the object of study, but are capable of “masking” the peculiarities of the manifestation of those state changes that are the goal of the study. If we add else the assumptions and simplifications adopted in the course of the work, it becomes clear that the picture of the phenomenon created in the model approximation and the existing reality cannot fully correspond to each other. However, the most important feature of their similarity is the presence of the main peculiarities of the process under consideration, which are identical in their appearances in both cases. In particular, the two types of nonlinear surface charge waves discussed in this paper have such characteristic features of behavior and spatial structure that cannot cause difficulties in registration by modern diagnostic tools. Therefore, the features of excitation, behavior, as well as the relationships between parameters revealed within the framework of the model are only important guidelines for experimental verification or an attempt to use in practice, since they are not able to provide absolutely accurate information within the framework of the model's simplified perception of reality.

The analysis of the properties of nonlinear manifestations of surface charge such as axially symmetric and azimuthal solitary waves on the surface of plasma can be generalized in the future to study similar effects in liquids and gases. For example, there is every reason to expect from the study of the properties of azimuthal oscillations on the surface of a liquid in the theory of “shallow water” a number of important explanations of such features of the emergence and behavior of solitary waves on the surface of the liquid, the nature and essence of which are not yet properly understood.

The publication was carried out within the State Assignment on Fundamental Research to the Kurnakov Institute of General and Inorganic Chemistry of Russian Academy of Sciences.

The authors have no conflicts to disclose.

O. M. Gradov: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal); Writing – review & editing (equal).

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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