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Vadim Urpin*

- AF Ioffe Institute of Physics and Technology, Russia

**Received:** January 25, 2018; **Published:** February 08, 2018

**Corresponding author:** Vadim Urpin, AF Ioffe Institute of Physics and Technology, Russia

**DOI:** 10.26717/BJSTR.2018.02.000744

We consider diffusion caused by a combined influence of the electric current and Hall effect, and argue that such diffusion can form in homogeneities of a chemical composition in plasma. Such current-driven diffusion can be accompanied by prop-agation of a particular type of waves in which the impurity number density oscillates alone. These compositional waves exist if the magnetic pressure in plasma is greater than the gas pressure.

**Keywords: **Plasma magnetic fields; Plasma waves; Plasma chemical spots

Often laboratory and astrophysical plasmas are multicomponent, and diffusion plays an important role in many phenomena in such plasmas. For instance, diffusion can be responsible for the formation of chemical inhomogeneities which influence emission, heat transport, conductivity, etc see, e.g., [1-3]. In fusion experiments, the source of trace elements is usually the chamber walls, and diffusion determines the penetration depth of these elements and their distribution in plasma see, e.g., [4-6]. Even a small admixture of heavy ions increases drastically radiative losses of plasma and changes its thermal properties. In astrophysical conditions, diffusion leads to the formation of element spots detected on the surface of many stars see, e.g., [7-9].

Diffusion in chatged gases or fluids can differ qualitatively from that in media con-sisting of neutral paticles because of the presence of electrons and electric currents. A mean motion of electrons caused by electric currents provides an additional internal force those results in diffusion of trace elements. One more important contribution of electrons in diffusion is relevant to the Hall effect. The magnetic field can magnetize the charged particles that lead to anisotropic transport. In the case of electron transport, such anisotropy is characterized by the Hall parameter, *x _{e} = ω_{Be}τ_{e}* where

In this paper, we consider the diffusion process that can lead to formation of chemical inhomogeneities in plasma. This process is caused by a combined influence of electric currents and the Hall effect. Using a simple model, we show that the interaction of the electric current with trace elements leads to their diffusion in the direction perpendicular to both the electric current and magnetic field. This type of diffusion can alter the distribution of chemical elements in plasma and contribute to formation of chemical spots even if the magnetic field is relatively weak and does not magnetize electrons (x_{e} «: 1). We also argue that the current-driven diffusion in combination with the Hall effect can be the reason of the particular type of modes in which the number density of a trace element oscillates alone. The considered diffusive process is rather general and can be important in any medium consisting of charges particles.

Consider plasma with the magnetic field parallel to the axis z, *B = ^{B} ~ ^{}*e

We suppose that *j _{ϕ}* → 0 at large s and, hence,

The dot denotes the partial time derivative. Here, *m _{α}* and

If there are no mean hydrodynamic velocity and only diffusive velocities of trace elements are non-vanishing, the partial momentum equation for particles α reads

The friction forces *R _{i}* for trace particles i can be represented as

Where is the force acting on the electron gas [15]. In this case, is determined by Scattering of electrons on protons but scattering on ions i gives a small contribution. Therefore we can use for the expression for [15] for hydrogen plasma. In our model, this expression reads

Where is the deference between the mean velocities of electrons and protons *α _{⊥} , α_{∧} , β_{⊥}^{uT}* and

The force consists of two parts as well, and which are proportional to the relative velocity of ions i and protons and to the temperature gradient, respectively. The thermoforce is vanishing in our model. The Friction force is given by

See [16] Where, is the characteristic timescale of ion-proton scattering; we assume that Coulomb logarithms are the same for all types of scattering. The momentum equation for the species i (see Eq. (3)) contains components of the electric field, E_{s} and E_{ϕ}. These components can be determined from the momentum equations (3) for electrons and protons. Taking into account the condition of hydrostatic equilibrium and quasi-neutrality (*n _{e}* ≈

Substituting Eqs.(6), (8), and (9) into Eq.(3) for the trace particles i, we arrive to the expression for a diffusion velocity V_{i},

*V _{nI}* Are the velocities of ordinary diffusion and

Eqs. (10)-(12) describe the drift of ions i under a combined influence of ∇n_{i} and . If magnetic field is weak and x « 1, then Eq.(12) yields ^{i}

Where *c ^{2}_{A} = B^{2}*1(

In a strong field, diffusion in the radial direction is suppressed because all sorts of particles are magnetized. For instance, the Coefficients D (Eq. (14)) and the corresponding diffusion velocity *V _{ni}* which characterize the standard diffusion in the s-direction are ≈

It is generally believed that standard diffusion smoothes chemical inhomogeneities on a diffusion timescale ~ *L ^{2} / D* where L is the length scale of a non-uniformity. This is not the case, however, for diffusion given by Eq. (12). In this case, chemical inhomogeneities can exist during a much longer time than ~

Where *B _{4} =*

If the distribution of impurities is non-axisymmetric then such diffusion in the azimuthal direction should lead to slow variations in the abundance peculiarities. Note that in the case of a strong field (q »1) all sorts of trace particles rotate around the axis with the same period that depends only on the number density and electric current. The condition of hydrostatic equilibrium in our model is given by

Where p and ρ are the pressure and density, respectively. Since the background plasma is hydrogen, *p ≈ 2nk _{B}T* where

Where *β= 8π _{o}/B*

The term on the r.h.s. describes the effect of electric currents on the distribution of trace elements. We consider first the case of a weak magnetic field with x ≪ 1. Then, one has from Eq. (17)

Substituting Eq. (19) into Eq.(18) and integrating, we obtain

And *n _{i0}* is the value of

Where, γ _{i0} = *n*_{i0} /n_{0} Local abundances turn out to be flexible to the field strength and, particularly, these concerns the ions with large charge numbers. If other mechanisms of diffusion are negligible and the distribution of elements is basically current- driven, then the exponent *(μ-*1) can reach large negative values for elements with large *Z _{i}* and, hence, produce strong abundance anormalies. For instance,

A distribution of the impurities can be substantially different if the magnetic field is strong and q ≫ 1. Using the same procedure as in the case of a weak field, we obtain

Therefore, all trace elements with Zi > 1 are overabundant in the regions with the magnetic field weaker than B0. On the contrary, these elements are under abundant in the regions with a stronger magnetic field.

In our simplified model of plasma cylinder with the velocity given by Eq.(10), the continuity equation for trace ions i reads

Consider the behaviour of small disturbances of the number density of trace ions by making use of a linear analysis of Eq. (25). In the basic (unperturbed) state, plasma is assumed to be in diffusive equilibrium and, hence, the unperturbed impurity number density satisfies Eq. (19). Since the number density of impurity i is small, its influence on parameters of the basic state is negligible. For the sake of simplicity, we consider disturbances that do not depend on z. Denoting disturbances of the impurity number density by *δn _{i}* and linearizing Eq.(25), we obtain the equation governing the evolution of such small disturbances,

For the purpose of illustration, we consider only disturbances with wavelengths shorter than the length scale of unperturbed quantities. In this case, we can use the so called local approximation for a consideration of linear waves and assume that small disturbances are *∞ exp* *(-iks - iMϕ)* where k is the wave vector *(ks»* 1) and M is the azimuthal wave number. Since the basic state does not depend on time, *δn _{i}* can be represented as

This dispersion equation describes spiral waves in which only the number density of impurities oscillates and, therefore, such waves can be called "compositional". The quantity *ω _{R}* characterizes decay of these waves with the characteristic timescale ~ (

Where and *c _{s}* is the sound speed,

Both conditions (28) and (29) require the magnetic field such strong that the magnetic pressure is substantially greater than the gas pressure. The frequency of com-positional waves is higher in the region where the magnetic field has a stronger gradient or, in other words, where the density of electric currents is greater. Note that different impurities oscillate with different frequences. Consider first the radial waves with M = 0. Substituting M = o into Eq. (27), we obtain the dispersion equation for such waves in the form

This dispersion equation describes waves in which only the number density of trace particles oscillates and oscillations of *n _{i}* occur only in the radial direction. The order of magnitude estimate of

Where *l _{i}* =

In non-axisymmetric waves, trace ions rotate around the cylindrical axis with the frequency *ω _{ϕ}* and decay slowly on the diffusion timescale ~

Since these estimates are justified only in the case of a weak magnetic field *(x ≪ * 1), the period of non-axisymmetric waves is shorter for waves with *M > A _{i}x(ks*). The ratio of diffusion timescale and period of non-axisymmetric waves is

And can be large. Therefore, azimuthal waves can be oscillatory as well.

Strong magnetic field (q »1) in a strong magnetic field, the order of magnitude estimates of the characteristic frequences are

Like the case of a weak field, the frequency of compositional waves is higher in the region where the density of electric currents is greater. Oscillations of different trace ions occur with different frequences in radial waves but azinuthal oscillations have the same frequency for different impurities. The frequency of azimuthal waves is higher than that of radial waves if

If the magnetic field is such strong that q *>>* 1 then the azimuthal waves oscillate with a higher frequency than the radial ones even for not very large M. The condition that radial waves exist in a strong magnetic field, *|ω _{S}| ≫ ω_{R} * |, is given by

Like the case of a weak magnetic field, compositional waves occur in plasma only if the magnetic pressure is greater than the gas one. The analgous condition for azinuthal waves, *ω _{ϕ} » ω_{R}* , reads

Note that this condition can be satisfied even if the magnetic pressure is smaller than the gas one but q and M are large.

We have considered diffusion of heavy ions under the influence of electric currents. Generally, the diffusion velocity in this case can be comparable to or even greater than that caused by other diffusion mechanisms. The current-driven diffusion can form chemical inhomogeneities even if the magnetic field is relatively weak whereas other diffusion mechanisms require a substantially stronger magnetic field. The current-driven diffusion is relevant to the Hall effect and, therefore, it leads to a drift of ions in the direction perpendicular to both the magnetic field and electric current. As a result, a distribution of chemical elements in plasma depends essentially on the geometry of the magnetic fields and electric current. Chemical inhomogeneities can manifest themselves, for example, by emission in spectral lines and a nonuniform plasma temperature. Usually, diffusion processes play an important role in plasma if hydrodynamic motions are very slow. In some cases, however, chemical spots can be formed even in flows with a relatively large velocity but with some particular topology (for example, a rotating flow).

Our study reveals that a particular type of waves may exist in multicomponent plasma in the presence of electric currents. These waves are slowly decaying and characterized by oscillations of the impurity number density alone. They exist only if the magnetic field is such strong that the magnetic pressure is greater than the gas pressure. Generally, the frequency of such waves turns out to be different for different impurities. This frequency is rather low and is determined mainly by a diffusion timescale. If M = 0, it can be estimated as where *λ =* 2π/ *k* is the wavelength of waves. In astrophysical conditions, such waves can manifest themselves in the atmospheres of magnetic stars where the magnetic field is of the order of 10^{4} G and the number density and temperature are 10^{14} cm^{-3} and 10^{4} K, respectively. If the length scale, L, and the wavelength, λ, are of the same order of magnitude (for in-stance, cm), then the period of compositional waves is yrs. This is much shorter than the stellar lifetime (see, e.g., [17, 18]) and generation of such waves in the atmospheres should lead to spectral variability with the corresponding timescale.

Compositional waves can occur in laboratory plasmas as well but their frequency is essentially higher. If *B~* 10^{5}G, *n~* 10^{15}*cm ^{}*

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