Effective attraction between oscillating electrons in a plasmoid via acoustic waves exchange

We consider the effective interaction between electrons due to the exchange of virtual acoustic waves in a low temperature plasma. Electrons are supposed to participate in rapid radial oscillations forming a spherically symmetric plasma structure. We show that under certain conditions this effective interaction can result in the attraction between oscillating electrons and can be important for the dynamics of a plasmoid. Some possible applications of the obtained results to the theory of natural long-lived plasma structures are also discussed.


Introduction
The interaction between a test charged particle, embedded in a warm plasma, and the collective response of a plasma results in the concept of a dressed particle.
Accounting for the collective plasma interaction, for a particle at rest one gets the screening of the vacuum Coulomb interaction leading to the Debye-Hückel potential (Landau & Lifshitz, 1980). Nambu & Akama (1985) showed that a test particle moving in plasma is a source of ion acoustic waves forming in its wake. If the speed of a test particle is close to the speed of the ion acoustic waves propagation, the resulting wake potential prevails the Debye-Hückel interaction leading to the effective attraction of charged particles of the same polarity (Nambu & Akama, 1985). This process may be responsible for the formation of complex structures in dusty plasmas (Shukla & Mamun, 2002;Tsytovich, 2007).
There is, however, another possibility for charged particles of equal polarities to experience an effective attraction in a plasma. It happens when charged particles interact via an acoustic wave appearing in the neutral component of plasma (Vlasov & Yakovlev, 1978). This process is analogous to the phonon exchange be-tween two electrons in metal (Madelung, 1978). For the implementation of this mechanism the temperature of plasma should be at least less than the ionization potential of an atom -otherwise neutral atoms can exist only in a small fraction.
The results of Vlasov & Yakovlev (1978) may have various applications. In particular, they can be used for the explanation of the stability of atmospheric plasma structures (Stenhoff, 1999).
In the present work, using the formalism of Vlasov & Yakovlev (1978), we will study the effective attraction between charged particles participating in spherically symmetric oscillations of a low temperature plasma. Recently radial plasma pulsations were considered by Dvornikov & Dvornikov (2007); Dvornikov (2010Dvornikov ( , 2011, in frames of both quantum and classical approaches, as a theoretical model of natural plasmoids (Stenhoff, 1999).

effects.
This work is organized as follows. In Sec. 2 we develop the general theory of the effective interaction between two charged particles via the exchange of an acoustic wave and apply it to electrons performing spherically symmetric oscillations in plasma. Then, in Sec. 3, we evaluate the parameters of the effective potential and analyze the conditions when it results in the attraction between oscillating electrons. In Sec. 4 we consider the application of the described effective interaction to the dynamics of natural long-lived plasmoids. Finally, in Sec. 5, we summarize our results.
2 Effective interaction in a spherically symmetric plasma structure In this section we formulate the general dynamics of a spherically symmetric oscillation of a low temperature plasma. The electron temperature should be less than the ionization potential of an atom. In this case both electron, ion, and neutral components can coexist in a plasma. For hydrogen plasma this critical temperature is ∼ 10 5 K. Then we consider the effective interaction between two oscillating electrons due to the exchange of a virtual acoustic wave and establish conditions when this interaction corresponds to a repulsion and an attraction between particles.
Suppose that one has excited a plasma oscillation in a low temperature plasma. In this situation both electrons and ions will oscillate with respect to neutral atoms. Note that electrons will oscillate with higher frequency than that of ions since their mobility is much bigger. If the density of neutral atoms is sufficiently high, charged particles will collide with neutral atoms generating perturbations of their density or even waves in the neutral component of plasma. Typically this process will result in the acoustic waves emission and finally in the energy dissipation in the system. However, if an acoustic wave is emitted coherently to be absorbed by another charged particle, one can expect the appearance of an effective interaction. The analogous phenomenon is well known in solid state physics (Madelung, 1978). It was suggested by Vlasov & Yakovlev (1978) that such an effective interaction may well happen in a low temperature plasma.
In this work we will mainly discuss spherically symmetric plasma oscillations. The possibility of the existence of such a plasma structure is pre-dicted in both classical (Škorić & ter Haar, 1980;Laedke & Spatschen, 1984;Dvornikov, 2011) and quantum (Dvornikov & Dvornikov, 2007;Haas & Shukla, 2009) cases. Stable classical plasmoids can exist due to the various plasma nonlinearities, which arrest the collapse of oscillations (Goldman, 1984), whereas the stability of quantum plasmoids is provided by the additional quantum pressure. In our study we do not specify physical processes underlying the stability of the plasma structure in question.
Omitting the motion of ions we can represent the plasma characteristics, like the electron density n e , the electron velocity v e , and the electric field E, in the following way: where n 0 is the background electron density and the index "1" stays for the oscillating parts of the electron density n 1 , the electron velocity v 1 , and the electric field E 1 . As we study spherically symmetric plasma oscillation all the functions in Eq.
(1) depend only on the radial coordinate r as well as the vectors v 1 and E 1 have only the radial component. Note that the frequency of the plasma oscillation ω in Eq. (1) is typically less than the Langmuir frequency for electrons ω p = 4πe 2 n 0 /m, where e > 0 is the proton charge and m is the mass of the electron. The discrepancy between ω and ω p depends on the physical processes proving the plasma structure stability. For example, for a classical plasmoid, which is treated byŠkorić & ter Haar (1980); Laedke & Spatschen (1984); Dvornikov (2011) with help of the perturbation theory, it can be about 10%. However, if a system possesses a high degree of nonlinearity, the difference between ω and ω p can be significant. We will suppose that ω = ξω p , with ξ < 1.
The quantities n 1 , v 1 , and E 1 are related by the plasma hydrodynamic equations derived byŠkorić & ter Haar (1980), Therefore we can describe the plasma oscillation in terms of only one function, e.g., E 1 = |E 1 |. The form of this function depends on the peculiar type of a plasmoid and it is rather difficult to find it analytically. Thus one should rely only on the numerical simulations of the plasmoid dynamics. Nevertheless we can use the following ansatz proposed by Anderson (1983), which describes quite accurately the dynamics of a plasmoid (Bang et al., 2002). Now a plasmoid is described in terms of the amplitude A and the width σ. In Fig. 1 we show the schematic behaviour of the perturbations of the electric field and the normalized number density of electrons f n = (3 − r 2 /σ 2 ) exp(−r 2 /2σ 2 ). As we mentioned above, rapidly oscillating electrons, described by Eqs. (1) and (3), can interact with neutral atoms generating oscillations in the neutral component of plasma. Under certain conditions these acoustic waves can be absorbed by other electrons producing an effective interaction between charged particles. The process of the exchange of a virtual acoustic wave between two electrons is schematically shown in Fig. 2.
At the second order of the perturbation theory the matrix element of the electrons interaction due to the exchange of the "quantum" of an  Figure 2: The schematic illustration of the exchange between two oscillating electrons by a virtual acoustic wave. Initially electrons are in the states ψ k and ψ k ′ whereas their final states are ψ k−q and ψ k ′ +q . The definition of the "quantum" of an acoustic wave, ρ 1 , is given in Eq. (6).
acoustic wave has the form (Madelung, 1978), where ω q is the frequency of a virtual acoustic wave, E(k) are the energy levels of an electron participating in rapid oscillations, and M k,k−q is the matrix element of the electron's interaction with the "quantum" of an acoustic wave taken in the first order of the perturbation theory. Note that an electron state in Eq. (4) is specified with help of the single quantum number k rather than of a vector k since here we consider a radial motion of charged particles.
To determine the type of the effective interaction (4) one should examine the sign of the denominator in Eq. (4). The situation when |E(k)−E(k−q)| > ω q corresponds to the repulsion between electrons and |E(k) − E(k − q)| < ω q to the effective attraction. In the next section we will evaluate the energy change ∆E = |E(k) − E(k − q)| in a collision with a neutral atom as well as the typical frequency of an acoustic wave ω q .

Parameters of the effective potential
In this section we evaluate the parameters of the effective interaction (4) and show that under certain conditions it can be attractive. Then we estimate the magnitude of the effective interaction and compare it with the typical kinetic energy of oscillating electrons.
To evaluate the energy change ∆E of an electron while it emits an acoustic wave we should notice that this acoustic wave is generated when an electron collides with neutral atoms. Thus knowing the total power of acoustic waves emitted and the mean number of collisions with neutral atoms per unit time we can evaluate the energy change in a collision.
We describe the dynamics of neutral gas using a set of the inhomogeneous hydrodynamic equations proposed by Ingard (1966), where ρ a is the mass density of neutral gas, u is its velocity, p is the neutral gas pressure and F is the rate of the momentum transfer per unit volume from an external source, which is the electron subsystem in our case, to the neutral gas. In Eq. (5) we suppose that the generation of acoustic waves happens at the constant entropy. We should also supply Eq. (5) with the equation of state of the neutral gas, p = p(ρ a ). Taking into account the oscillatory character of the electron number density (1) we get that the external force F should also have the similar behaviour, F = F 1 e −iωt + F * 1 e iωt . Linearizing Eq. (5) we obtain that the pressure, the density, and the velocity of the neutral gas also oscillate on the same frequency ω: where p 0 and ρ 0 are the equilibrium values. As in Eq. (1) the index "1" stays here for the perturbed quantities and higher harmonics are omitted. Using Eq. (6) we can represent the linearized Eq. (5) for the neutral gas pressure as the inhomogeneous Helmholtz equation, where k = ω/c a is the wave vector of the acoustic wave and c a = (∂p/∂ρ a ) S is the sound velocity in the neutral gas, with the derivative being taken at the constant entropy. Note that acoustic waves will be emitted with the same frequency as the plasma oscillation, ω q = ω. Now we should specify the type of the interaction between electrons and neutral atoms. According to the definition of the external force it can be represented as F 1 = −n a ∇U , where n a = ρ 0 /M a is the background number density of neutral atoms and M a is the mass of an atom. We also suggest that only oscillating electrons contribute to the effective potential U (r) of the interaction of an acoustic wave with electrons, where K(r) is the potential of the interaction between an electron and a neutral atom. Let us choose the charge-dipole potential first proposed by Buchingham (1937), which is a good model of the interaction between electrons and neutral atoms in a low energy plasma, where r 0 is the cut-off radius, which is of the order of the atomic size, and α ∼ r 3 0 is the electric dipole polarizability of an atom. The corrections to Eq. (9) due to the collective plasma effects were studied by Redmer et al. (1987). It was found that for a hydrogen plasma the deviations from Eq. (9) are negligible at the low plasma temperature ∼ 10 3 K used below in this section.
Taking into account the general solution of Eq. (7), we can represent the total power, radiated in the form of the acoustic waves, in the following form: To derive Eq. (11) we suppose that the function U (r) rapidly decreases at large distances, cf. Eq. (3). Now let us evaluate the number of collisions with neutral atoms per unit time at rapid oscillations of electrons. For this purpose it is more convenient to use the Lagrange variables: (r, t) → (ρ, τ ). Suppose that an oscillating electron is at the distance ρ from the center of the system and its law of motion has the form, r = ρ + A 0 (ρ) sin ωτ , where A 0 is the amplitude of oscillations. Taking into account the continuity equation in Lagrange variables for a spherically symmetric system found by Dvornikov (2011), n 0 ρ 2 = n e r 2 (∂r/∂ρ), and supposing that the amplitude A 0 is not so large, i.e. n 1 (r) ≈ n 1 (ρ), we obtain the number of collisions per unit time aṡ where σ s is the cross section of the electron scattering off a neutral atom. To derive Eq. (12) we suppose that A 0 (0) = 0, i.e. oscillations vanish at the center of the system. Now we can evaluate the energy change in a collision with a neutral atom as ∆E ∼ Ė /Ṅ . As it was mentioned above, the situation when |∆E| is less than the typical energy of the "quantum" of an acoustic wave, ω q , corresponds to the effective attraction between electrons. Besides this condition, for the effective interaction to become important for the dynamics of the system, its magnitude should be comparable with the typical kinetic energy of oscillating electrons. That is why we have to evaluate the matrix element of an acoustic wave generation by an oscillating electron M k,k−q in Eq. (4).
This matrix element can be expressed in the following form: where V (r) is the energy of interaction of an electron and the acoustic field. The wave function of an electron ψ k (r) in Eq. (13) can be taken in the form of a normalized spherical wave, ψ k (r) = k π sin(kr) r , since we study a localized oscillation. Generally speaking, in a realistic situation the wave function of an electron can differ from that in Eq. (3). However, to get a rough estimate we can choose it in such a form. Analogously to Eq. (8) we suggest that an electron scatters off the density perturbation caused by an acoustic wave, i.e. at the first order of the perturbation theory the effective potential V (r) has the following form proposed by Vlasov & Yakovlev (1978): where the energy of interaction between an electron and a neutral atom K(r) is given in Eq. (9).
Note that in Eq. (15) we take into account only the oscillating part of the neutral gas density ρ 1 , cf. Eq. (6). Accounting for Eqs. (9) and (13)-(15) we obtain the matrix element M k,k−q as dr rU (r) sin kr. (16) Here we take into account that both ρ 1 and ρ * 1 contribute to the matrix element.
It is convenient to compare the potential of the effective interaction between oscillating electrons (4) with the typical value of the kinetic energy of an oscillating electron. Using Eq. (3) one gets the kinetic energy, where we take into account the relation between the main harmonic amplitudes of the electron gas presented in Eq.
(2). Note that in Eq. (17) we take into account only the part of the kinetic energy due to the oscillatory motion rather than the total energy which also includes the thermal contribution.
To analyze the behaviour of the effective interaction first we mention that one can explicitly calculate the common integral in Eqs. (11) and (16), where we use Eqs.
(2), (3), and (8). Then we rewrite Eq. (4) as V kk ′ q / E k = W/(R−1), where we introduce the new functions, Here f R (x) = x 16 e −2x 2 and f W (x) = x 11 e −x 2 . To derive Eq. (19) we use Eq. (18) and the following plasma parameters: α ∼ 10 −24 cm 3 , r 0 ∼ 10 −8 cm, σ s ∼ 10 −15 cm 2 , M a ∼ 10 −24 g, and c a ∼ 10 5 cm/s. The effective interaction turns out to be attractive when R < 1. The function W is the "magnitude" of the interaction expressed in terms of the mean kinetic energy of oscillating electrons, cf. Eq. (17). In Fig. 3 we show the behavior of the functions f R,W versus the dimensionless variable kσ = ωσ/c a . As we can see these functions have the maxima due to the enhanced emissivity of acoustic waves with the wave length of the order of the size of a plasmoid, 2π/k ∼ σ. Using Eq. (19) and Fig. 3 we get the lower bound on the width of a plasmoid at which the effective attraction between electrons takes place, σ > 0.3 × 10 −7 cm × ξ −1 . To obtain this constraint we account for the maximum of the function (f R ) max ≈ 22.
In our analysis we take into account only the interaction of oscillating electrons with neutral atoms. It is however known that oscillations of plasma are possible if the total number of collisions of background electrons per unit time ν is less than the oscillations frequency. Note that one should account for collisions only inside the plasmoid. Thus we get for this quantity, ν = v T σ s n a V eff n 0 , where v T = T /m is the thermal electron velocity, V eff = (4π/3)R 3 eff = 17σ 3 is the effective plasmoid volume, and R eff = 1.6σ is the effective plasmoid radius calculated as R 2 eff = E 2 r 2 d 3 r/ E 2 d 3 r, cf. Eq. (3). Taking T = 10 3 K and using Eq. (19) at the maximal value of the function W , corresponding to the strongest effective attraction, we get that plasma oscillations are possible (i.e. the condition ν < ω is satisfied) if σ < 1.4×10 −6 cm×ξ 2/7 . Combining this result with the previously ob-tained lower bound for σ, one gets that the width of a plasma structure should be in the range, 0.3 × 10 −7 cm × ξ −1 < σ < 1.4 × 10 −6 cm × ξ 2/7 . Therefore, for the effective attraction between oscillating electrons to take place, the frequency of oscillations should lie in the following interval: 0.05ω p < ω < ω p . Note that for the chosen parameters the Debye length for electrons λ D = v T /ω p is of the order of R eff .

Applications
In Sec. 3 we showed that the exchange of a virtual acoustic wave between two electrons participating in spherically symmetric oscillations may result in their effective attraction. The typical size of the plasma structure when the effective attraction becomes important, i.e. when it is comparable with the mean kinetic energy of oscillating particles, is quite small ∼ 10 −6 cm. Although in the present work we do not analyze the physical processes which underlie the plasmoid existence, such a small size is the indication to the quantum nature of this plasma structure. For example, quantum plasmoids of the similar size were described by Dvornikov & Dvornikov (2007) on the basis of the solution of the nonlinear Schrödinger equation.
If the effective attraction between oscillating electrons is sufficiently strong, we may suggest that these particles can form a bound state. Previously Nambu & Akama (1985); Nambu et al. (1995) suggested that bound states of charged particles in plasma can be formed due to the exchange of ion acoustic waves. Another mechanism based on the magnetic interaction was proposed by Meierovich (1984). Note that in our case the presence of a strong nonlinear effect is required to significantly diminish the fre-quency of oscillations, ω < ω p . We suggest that the phenomenon of the bound states formation of electrons in plasma can be implemented inside a stable atmospheric plasmoid (Stenhoff, 1999), called a ball lightning (BL). These kind of plasma structures appears during a thunderstorm and have the lifetime up to several minutes. Despite a great variety of models of BL summarized by Bychkov et al. (2010, pp. 270-296), these objects are likely to be plasma based phenomena.
To describe the long lifetime of a natural plasmoid, Dijkhuis (1980) suggested that BL is a spherical vortex composed of a dense superconducting plasma. Recently Zelikin (2008) revisited the idea that the plasma superconductivity may be implemented in BL. Zelikin (2008) suggested that a natural plasmoid can have the positively charged kernel and the superconducting electron envelop. Nevertheless those models were based on the phenomenological assumption of the plasma superconductivity without pointing out a physical mechanism which underlies it.
As we mentioned in Sec. 2 the incoherent emission of acoustic waves results in the energy dissipation in the plasmoid. On the contrary, the coherent exchange by an acoustic wave leads to the attractive interaction between electrons. As we showed in Sec. 3, this attraction can result in the formation of bound states, analogous to Cooper pairs in metals (Madelung, 1978). Thus one may expect that the electron component of plasma will experience the phase transition followed by the formation of the condensate of electrons. The collective oscillations of the electrons condensate can be responsible for reducing the resistance of plasma and thus can provide the observed lifetime of a natural plasma structure. Although this hypothesis requires the additional analysis which would carefully account for all thermal effects, the estimates given in Sec. 3 show that the described process is quite possible.
For the existence of plasma oscillations the number of collisions of oscillating particles per unit time should be much less than the oscillations frequency, ν ≪ ω. Otherwise oscillations will decay. In Sec. 3 we checked that the weaker inequality, ν < ω, is satisfied. However, if a plasma is in the reduced resistance state and supposing that rather strong nonlinearity is present, i.e. a significant fraction of electrons participates in oscillations, we may expect that the condition ν ≪ ω is satisfied as well. In Sec. 3 we also obtained that for the chosen parameters the Debye length is of the order of the plasmoid size. If R eff ≪ λ D , oscillating electrons sometimes can pass through the plasma structure that results in the energy dissipation (Goldman, 1984). Although we do not violate the condition R eff λ D , the excessive energy dissipation can be avoided in the reduced resistance plasma state.
In the present work we considered a plasmoid with a very small core ∼ 10 −6 cm, whereas according to observations the visible diameter of a natural plasma structure is of the order of several centimeters (Bychkov et al., 2010). Nevertheless, if one analyzes the available photographs of this phenomenon published by Burt (2007); Stenhoff (1999, pp. 129-161), one can conclude that, while moving in the atmosphere, a plasmoid leaves a trace which is much smaller than its visible size. It can be possible if a small and hot kernel exists inside a plasmoid. This core ionizes the air and produces the observed trace. The visible dimensions of a plasmoid are likely to be attributed to auxiliary effects.
One more indication that the actual size of a natural plasma structure is much smaller than the visible one, is in the fact that sometimes it can pass through tiny holes and cracks, with the structure of the plasmoid being unchanged (Bychkov et al., 2010, pp. 203-246). In many cases the materials which a plasmoid passes through are not damaged either. The most natural explanation of this unusual behavior is the suggestion of the small scale internal structure of the object. Although it is not directly related to our description of natural plasmoids, we should mention that recently Muldrew (2010) discussed the model of BL in which the visible size of the object is bigger than its core having the radius ∼ 0.1 − 2 cm.
Another evidence of the small-sized core of long-lived plasma structures can be obtained from the laboratory experiments (Bychkov et al., 2010, pp. 263-265). Moreover recently Klimov & Kutlaliev (2010) reported that plasmoids with a nano-scale kernel were generated in the studies of high frequency discharges in dusty plasmas. The experiments with silicon discharges carried out by Abrahamson (2002); Lazarouk et al. (2006); Dikhtyar & Jerby (2006); Paiva et al. (2007);Mitchell et al. (2008), where glowing structures resembling natural plasmoids were generated, should be also mentioned. Although the interpretation of the results of those experiments involves another plasmoid model, the generated objects have nano-sized cores as well.
The reports of the energy content of natural plasma structures are rather different: along the objects with relatively small internal energy, the plasmoids possessing huge energy were observed (Bychkov et al., 2010, pp. 203-246). The analysis of the present work is applicable for a low energy plasma structure which does not seem to have an internal energy source. That is why in Sec. 3, while making numerical estimates, we took the temperature of electrons ∼ 10 3 K. We remind that the maximal electron temperature is ∼ 10 5 K. Note that the existence of low temperature plasmoids is not excluded by observations (Bychkov et al., 2010, pp. 203-246). It is worth mentioning that, if the plasma of such a low energy BL is in the superconducting state, one can avoid the energy losses, plasma recombination, and provide the plasma structure stability.
We showed that the model of a nano-sized plasmoid is able to describe some of the observed properties of BL. Nevertheless we may also suggest that in a realistic natural plasma structure there could be multiple tiny kernels where intense electron oscillations happen. Note that the analogous model of a composite BL was discussed by Nikitin (2006). The separate oscillatory centers can be held together by the attractive quantum exchange forces (Kulakov & Rumyantsev, 1991), which are relevant in our situation since the predicted size of a single core is tiny, ∼ 10 −6 cm. However the detailed description of the coagulation process using the results of Kulakov & Rumyantsev (1991) requires additional special analysis.
The analysis of the observed characteristics of natural plasmoids summarized by Bychkov et al. (2010, pp. 203-246) shows that they are likely to be the phenomena of different origin. Therefore a unique model which would explain all the observations does not seem to exist. For instance, the plasmoid model described in the present work does not explain the electromagnetic action of BL, like the generation of strong electric currents and the emission of radio-waves. Nevertheless in frames our approach we could naturally explain some of the BL properties which are hard to account for in alternative models of natural plasmoids.
At the end of this section we may say a few words how an atmospheric plasmoid based on spherically symmetric oscillations of electrons appears in natural conditions. As we revealed in Sec. 3, for the existence of a plasmoid the frequency of plasma oscillations should be comparable with ω p which is in the GHz region. It is very difficult to generate such a high frequency during a thunderstorm since, e.g., a linear lightning is a quite low frequency phenomenon (Rakov & Uman, 2006). Thus the possible BL generation scenario looks as follows. Suppose that during a linear lightning stroke a natural capacitor with a small capacity ∼ 10 pF is charged up to a very high voltage. Then, if this capacitor is discharged on a thin point, a GHz electromagnetic oscillation can be created, provided the discharge channel inductance is ∼ 0.1 µH. However many other factors, like shape of the point, air humidity etc, should be properly combined for the successful BL generation.

Conclusion
In this work we have discussed the effective interaction between electrons in a low temperature plasma due to the exchange of virtual acoustic waves. The electron temperature of such a plasma should be low enough to allow the existence of both electrons, ions, and neutral atoms. In Sec. 2 we have derived the expression for the potential of this effective interaction (4) and considered a particular case of the effective interaction between electrons participating in spherically symmetric oscillations. We supposed that charged particles in plasma perform nonlinear oscillations and form a stable plasma structure. However in our analysis we just used the com-monly adopted ansatz for the electric field distribution (3) without going into details what kind of physical processes provides the plasmoid stability.
Then, in Sec. 3, we have evaluated the parameters of the effective interaction. It has been established that under certain conditions the interaction is attractive and its strength can be comparable with the mean kinetic energy of oscillating electrons. For this situation to happen, the typical size of a plasmoid should be quite small, ∼ 10 −6 cm. It is the indication that one should use the concept of quantum plasmas to describe the stability of the plasma structure.
Note that analogous effective interaction will also appear between ions and neutral atoms as well as between ions and electrons since Eq. (4) is of the universal type. However, in the former case the frequency of the ion oscillations is significantly less than ω p and the interaction will be always repulsive. In the latter case we should recall that electrons and ions interact mainly electromagnetically and the small contribution due to the collisions will be negligible.
Finally, in Sec. 4, we have discussed the possible application of the obtained results to the description of stable natural plasma structures (Stenhoff, 1999;Bychkov et al., 2010). According to the model of Dvornikov & Dvornikov (2007);Dvornikov (2010Dvornikov ( , 2011 these objects can be implemented as spherically symmetric plasma oscillations. Note that analogous idea was also discussed by Shmatov (2003). To explain the long lifetime (up to several minutes) of a natural plasmoid we put forward a hypothesis that plasma could be in a reduced resistance state. The mechanism underlying the existence of such a state could be the exchange of virtual acoustic waves between oscillating electrons, described in Secs. 2 and 3. We have also considered how the characteristics of a plasma structure, predicted in frames of our model, conform to the observed properties of natural plasmoids.