1 INTRODUCTION. COMPLEX STRUCTURE OF MAGNETOPAUSES IN SPACE PLASMA

The knowledge of the structure of the boundary between two regions of collisionless plasma with different parameters and different magnetic fields is important for many problems of the physics of space and laboratory plasmas, particularly concerning so-called magnetopauses where collisions between particles are rare [1–12]. In this work, we give an exact analytical description of such a planar layered boundary for a very general case of the multicomponent plasma with arbitrary parameters, arbitrary energy distributions, and arbitrary magnetic fields on both sides of the boundary. Such a one-dimensional description of magnetopauses was previously limited to a narrow class of distribution functions of particles, primarily Maxwellian, and did not allow the simulation of co-mplex profiles of the current of different populations of particles and their total magnetic field; cf., e.g., [12–18].

The proposed description can be applied to some problems such as the detailed qualitative analysis of a wide range of phenomena in regions of magnetopauses formed by the stellar (solar) wind. They occur at the contact of the wind with magnetospheres of (exo)planets with high coronal arcs and with the surrounding interstellar plasma, in particular, magnetized one, as well as at the contact between neighboring regions of the wind with different parameters of the plasma and different magnetic fields [1–10, 12, 17, 18]. Obviously, the flow of the stellar wind on a magnetoplasma obstacle is not generally one-dimensional and often results in the formation of a shock wave and developed turbulence, primarily in the region of the so-called magnetosheath. We do not consider this range of complex problems but discuss only the quasi-one-dimensional part of the inner boundary of the magnetosheath, which will be referred to below as the magnetopause. For simplicity, we assume that the perpendicular magnetic field and the hydrodynamic flow of matter are absent at this boundary and neglect the effect of turbulence or any violation of the electroneutrality of the plasma. The corresponding generalizations of the found solution, which is certainly important for application to real phenomena, can hardly be universal and completely analytical and will be considered elsewhere.

The solution to the formulated simplified magnetostatic problem in the kinetic theory of collisionless plasma is relevant because of numerous observations of the magnetopause in the Earth’s subsolar magnetosphere in recent years by specialized satellite missions, primarily by the THEMIS, Cluster, and MMS missions [1–7, 12, 17]. Each mission includes four or five satellites that are spaced by distances of 10–100 km and ensure consistent measurements of the magnetic field, current density, and momentum distribution functions of particles. In particular, velocity distribution functions of electrons were obtained by the MMS mission [2–4, 12], velocity distribution functions of ions were measured by the Cluster mission [5], and the parameters of anisotropy of various populations of particles were determined by the Cluster and THEMIS missions [6, 7]. The Parker Solar Probe, WIND, ARTEMIS, and MMS missions [1, 12, 18] provided data on current sheets for magnetopauses at the interface between regions of the stellar wind with different parameters of the plasma and different magnetic fields. The Voyager 2 spacecraft obtained data on the magnetic field [8], velocity of the plasma flow [9, 10], and the plasma density and temperature [11] near the heliopause.

These and other measurements require the development of models of the magnetopause including complex (asymmetric and multiscale) current sheets and complex distribution functions of various populations of particles in them with allowance for different degrees of magnetization. Works [12–14, 17, 18] allow the simulation of some features of the studied current sheets in the way to the interpretation of listed measurements. The review of existing particular results and possibilities of the theoretical analysis of the stated problem is beyond the scope of this work and will be given elsewhere. In this work, we focused on the description of a new class of analytical models of the magnetopause, which opens wide prospects for the use of non-Maxwellian distribution functions of particles and for the construction of very diverse self-consistent profiles of the current density in various populations of particles and the magnetic field induced by them.

The article is organized as follows. An exact solution to the problem of the magnetopause with many components and countercurrents is given in Section 2. Section 3 presents it in more detail by examples of the Maxwellian and kappa energy distributions of particles. The simplest generalization to the case of the shear of the magnetic field lines is described in Section 4. The local stability of solutions obtained is discussed in Section 5. Section 6 briefly summarizes the conclusions.

2 EXACT SOLUTION TO THE PROBLEM OF THE INTERFACE BETWEEN REGIONS OF DIFFERENTLY MAGNETIZED PLASMAS WITH DIFFERENT PARAMETERS

We consider a planar layered stationary situation, where the vector potential has a single nonzero Cartesian component \({{A}_{z}}(x)\). In this case, currents flow along the z axis, and the magnetic field B(x) = curl A is parallel to the y axis. Particles in such electroneutral current sheets move along trajectories where the m-agnitude of the total momentum p and the projection of the generalized momentum on the z axis \({{P}_{z}} = {{p}_{z}} + {{e}_{\alpha }}{{A}_{z}}{\text{/}}c\), where \({{e}_{\alpha }}\) is the electric charge of a particle of type α and c is the speed of light in vacuum, are invariants of motion.

As known (see, e.g., [15, 19]), the distribution functions \({{f}_{\alpha }}(p,{{P}_{z}})\) of electrons (α = e) and ions (α = i) depending only on these invariants describe the kinetic equilibrium of the plasma; i.e., they are exact solutions of the electrostatic Vlasov equation. We choose them in the following form taking into account a “step-like” barrier for particles, which describes the magnetopause as a sheet with a steep magnetic field transition:

$${{f}_{e}}(x,p,{{p}_{z}}) = {{N}_{{e1}}}{{F}_{{e1}}}(p){\kern 1pt} H\left( { - {{p}_{z}} + \frac{e}{c}[{{A}_{z}}(x) - {{A}_{{e1}}}]} \right)$$
$$ + \;{{N}_{{e2}}}{{F}_{{e2}}}(p){\kern 1pt} H\left( {{{p}_{z}} - \frac{e}{c}[{{A}_{z}}(x) - {{A}_{{e2}}}]} \right)$$
(1)
$$ + \;{{n}_{{e0}}}(x){{F}_{{e0}}}(p),$$
$${{f}_{i}}(x,p,{{p}_{z}}) = {{N}_{{i1}}}{{F}_{{i1}}}(p){\kern 1pt} H\left( {{{p}_{z}} + \frac{e}{c}[{{A}_{z}}(x) - {{A}_{{i1}}}]} \right)$$
$$ + \;{{N}_{{i2}}}{{F}_{{i2}}}(p)H\left( { - {{p}_{z}} - \frac{e}{c}[{{A}_{z}}(x) - {{A}_{{i2}}}]} \right)$$
(2)
$$ + \;{{n}_{{i0}}}(x){{F}_{{i0}}}(p).$$

Here, \(H(\xi )\) is the Heaviside step function, which is 1 if \(\xi > 0\) and 0 otherwise, e is the elementary charge, ions are assumed single-charged for simplicity, the last terms do not contribute to the current density and are introduced for the electroneutrality of constructed current sheets, and the constants Nαs and Aαs are generally different for different components α = e, i and s  = 1, 2, where subscript s specifies populations of components with countercurrents. In the general case, where electrons or ions are separated into populations, subscript α can have more than two values; in this case, the first two terms \({{f}_{{\alpha s}}}\) (\(s = 1,2\)) in distribution functions (1) and (2) should be replaced by the corresponding sums \(\sum\nolimits_{\alpha s} {{f}_{{\alpha s}}}\) of such distribution functions of particles for these populations. This possibility is implied everywhere below, but sums are not written to avoid the elongation of formulas.

The self-consistent magnetic field B(x) and the corresponding vector potential A(x) appearing in Eqs. (1) and (2) are determined according to the Ampere law by the total current density of all components of the plasma, \(\mathbf{j}(x) = \sum\limits_{\alpha s} {{e}_{\alpha }}\int \mathbf{v}{{f}_{{\alpha s}}}(x,\mathbf{p}){{d}^{3}}\mathbf{p}\):

$${\text{curl}}\mathbf{B} = \frac{{4\pi }}{c}\mathbf{j}(x) = \frac{{4\pi e}}{c}\int \mathbf{v}\left( {{{f}_{i}} - {{f}_{e}}} \right){{d}^{3}}\mathbf{p},$$
(3)

where \(\mathbf{v} = ({{m}_{\alpha }}{{\gamma }_{\alpha }}{{)}^{{ - 1}}}\mathbf{p}\) is the velocity of the particle and \({{\gamma }_{\alpha }}\) is its Lorentz factor. The local density of an individual component is \({{n}_{{\alpha s}}}(x) = \int {{f}_{{\alpha s}}}{{d}^{3}}\mathbf{p}\).

The substitution of the definition of the vector potential \({{B}_{y}} = - d{{A}_{z}}{\text{/}}dx\) into Eq. (3) gives the Grad–Shafranov equation [16]

$$\frac{{{{d}^{2}}{{A}_{z}}}}{{d{{x}^{2}}}} = - 4\pi \frac{d}{{d{{A}_{z}}}}{{P}_{{xx}}}({{A}_{z}}),$$
(4)

where

$${{P}_{{xx}}}({{A}_{z}}) = \sum\limits_{\alpha s} \int {{p}_{x}}{{v}_{x}}{{f}_{{\alpha s}}}{\kern 1pt} {{d}^{3}}\mathbf{p} + {\text{const}}$$
(5)

is the component of the pressure tensor of the plasma along the x axis of inhomogeneity (with an accuracy to a constant chosen below).

The total current density for the distribution functions given by Eqs. (1) and (2) has the form

$$\begin{gathered} {{j}_{z}}({{A}_{z}}) = \sum\limits_{\alpha ;{\kern 1pt} s = 1,2} {{( - 1)}^{{s + 1}}}2\pi \frac{e}{{{{m}_{\alpha }}}}{{N}_{{\alpha s}}}\int\limits_{|{{a}_{{\alpha s}}}|{{p}_{{\alpha s}}}}^{ + \infty } p{{F}_{{\alpha s}}}(p) \\ \times \;\frac{{{{p}^{2}} - a_{{\alpha s}}^{2}p_{{\alpha s}}^{2}}}{2}{\kern 1pt} dp, \\ \end{gathered} $$
(6)

where \({{a}_{{\alpha s}}}(x) = [{{A}_{z}}(x) - {{A}_{{\alpha s}}}]{\kern 1pt} e{\text{/}}({{p}_{{\alpha s}}}c)\) and pαs = \({{(2{{m}_{\alpha }}{{T}_{{\alpha s}}})}^{{1/2}}}\) is the “thermal” momentum and \({{T}_{{\alpha s}}} = {{\langle {{p}^{2}}\rangle }_{{\alpha s}}}{\text{/}}(3{{m}_{\alpha }})\) is the effective temperature of \(\alpha s\) particles in the region of their isotropy, i.e., at \({{A}_{z}}{{( - 1)}^{{s + 1}}} \to + \infty \) and \({{\langle \mathbf{p}\rangle }_{{\alpha s}}} = 0\). Here, \({{\langle \ldots \rangle }_{{\alpha s}}} = \) \(\int ( \ldots ){{f}_{{\alpha s}}}{{d}^{3}}\mathbf{p}{\text{/}}{{n}_{{\alpha s}}}\).

It is easily seen that the current density \({{j}_{{\alpha sz}}}\) of each of the components in the sum in Eq. (6) has the same sign for all \({{A}_{z}}\) values, has a single extremum reached at \({{a}_{{\alpha s}}} = 0\), and tends to 0 at \({{A}_{z}} \to \pm \infty \). We note that, although we consider below only the nonrelativistic case, Eq. (6) is generalized to the relativistic plasma by the substitution \({{F}_{{\alpha s}}}(p) \to \gamma _{\alpha }^{{ - 1}}(p){{F}_{{\alpha s}}}(p)\) in the integrand, which does not change a hump-like shape of the function \({{j}_{{\alpha sz}}}({{a}_{{\alpha s}}})\) in the sum in Eq. (6) and expands the solution below to the relativistic case.

As seen from the comparison of Eqs. (3) and (4), since \({{j}_{z}}({{A}_{z}}) = cd{{P}_{{xx}}}{\text{/}}d{{A}_{z}}\), the corresponding contribution \({{P}_{{\alpha s}}}({{A}_{z}})\) from each component in Eq. (5) is a monotonic function of the vector potential \({{A}_{z}}\) and tends to two different constants at \({{A}_{z}} \to \pm \infty \). The general expression for the pressure \({{P}_{{xx}}}({{A}_{z}})\) at arbitrary energy factors \({{F}_{{\alpha s}}}(p)\) of the components is easily obtained by integrating the current density \({{j}_{z}}({{A}_{z}})\) or directly from Eq. (5):

$${{P}_{{xx}}}({{A}_{z}})$$
$$ = \sum\limits_{\alpha ,s} {{( - 1)}^{{s + 1}}}\frac{\pi }{{{{m}_{\alpha }}}}{{N}_{{\alpha s}}}\left[ {{\text{sgn}}\left( {{{a}_{{\alpha s}}}} \right)\int\limits_0^{|{{a}_{{\alpha s}}}|{{p}_{{\alpha s}}}} p{{F}_{{\alpha s}}}(p)\frac{2}{3}{{p}^{3}}dp} \right.$$
(7)
$$\left. { + \;{{p}_{{\alpha s}}}\int\limits_{|{{a}_{{\alpha s}}}|{{p}_{{\alpha s}}}}^{ + \infty } p{{F}_{{\alpha s}}}(p){{a}_{{\alpha s}}}\left( {{{p}^{2}} - \frac{{a_{{\alpha s}}^{2}}}{3}p_{{\alpha s}}^{2}} \right)dp} \right] + {\text{const}}.$$

The densities of particles of types αs \({{n}_{{\alpha s}}}({{a}_{{\alpha s}}}) = \int {{f}_{{\alpha s}}}{{d}^{3}}\mathbf{p}\) are calculated using the relation \(\partial H\left( \xi \right){\text{/}}\partial \xi = \delta \left( \xi \right)\), where δ is the Dirac delta function, and have the same behavior, being monotonic functions:

$$\begin{gathered} {{n}_{{\alpha s}}} = {{N}_{{\alpha s}}}\left[ {H\left( {{{{( - 1)}}^{{s + 1}}}{{a}_{{\alpha s}}}} \right) - {{{( - 1)}}^{{s + 1}}}{\text{sgn}}\left( {{{a}_{{\alpha s}}}} \right)} \right. \\ \left. { \times \;2\pi \int\limits_{|{{a}_{{\alpha s}}}|{{p}_{{\alpha s}}}}^{ + \infty } p{{F}_{{\alpha s}}}(p)\left( {p\; - \;{\text{|}}{{a}_{{\alpha s}}}{\text{|}}{{p}_{{\alpha s}}}} \right)dp} \right]. \\ \end{gathered} $$
(8)

Thus, any solution to Eq. (4) corresponds to a self-consistent current sheet with the current density in the form of the sum (6) of hump-like contributions of all components, which can have different signs and can be spatially separated. Equation (4) has the form of the equation of motion of a material point in a potential; therefore, the types of current configurations [16] can be classified just as the classification of motion in mechanics. This equation has the first integral \(B_{y}^{2} + 8\pi {{P}_{{xx}}} = 8\pi {{P}_{0}} \equiv {\text{const}}\) expressing the balance of the magnetic field pressure and the thermal pressure of particles in equilibrium planar layered configurations. Using it, one can write the solution for the total magnetic field:

$${{B}_{y}} \equiv - \frac{{d{{A}_{z}}}}{{dx}} = - {{(8\pi )}^{{1/2}}}{{\left[ {{{P}_{0}} - {{P}_{{xx}}}({{A}_{z}})} \right]}^{{1/2}}}.$$
(9)

Here and below, to construct current sheets, we take the constant \({{P}_{0}}\) such that the expression in the square brackets is positive, which can be ensured by choosing \({{P}_{0}} \geqslant \max {{P}_{{xx}}}\) (since the last quantity is finite). Then, the magnetic field does not change sign at the passage through the current sheet, varying from one constant far to the left from the sheet to another constant far to the right, which corresponds to the magnetopause. For definiteness, the direction of the y axis is chosen such that this sign is negative, as in Eq. (9).

Finally, the coordinate dependence of the vector potential is determined from the expression

$$x({{A}_{z}}) = \int\limits_0^{{{A}_{z}}} \frac{{dA{\kern 1pt} '}}{{ - {{B}_{y}}(A{\kern 1pt} ')}},$$
(10)

where the limits of integration are consistent with the absolute value of the vector potential so that \({{A}_{z}} = 0\) in the conditional center of the sheet at \(x = 0\). Since the magnetic field has the same sign everywhere, the dependence \(x({{A}_{z}})\) is monotonic and reversible, and \({{A}_{z}} \to \pm \infty \) usually in the limits \(x \to \pm \infty \). This is obvious for realistic positive distribution functions \(F(p) > 0\). If \(F(p \geqslant {{p}_{*}}) \equiv 0\), where \({{p}_{*}}\) is a certain threshold momentum, with increasing distance from the current sheet in the positive x direction toward the vanishing magnetic field, we asymptotically have \({{A}_{z}}(x \to \infty ) \to {{A}_{*}}\) = \({{p}_{*}}c{\text{/}}e\) and \({{B}_{y}}(x \to \infty ) \to 0\) because \({{B}_{y}}({{A}_{z}} = {{A}_{*}}) = 0\); in the opposite direction, where the magnetic field is nonzero, we still have \({{A}_{z}} \propto x\) at \(x \to - \infty \).

Expressions (6), (9), and (10) parametrically specify the exact solution of the problem of kinetic equilibrium in the collisionless plasma, which corresponds to the magnetopause, including the case with zero magnetic field at one (right) edge.

This solution determines the properties of a partially magnetized plasma with arbitrary distribution functions \({{f}_{{\alpha s}}}\). The current density \({{j}_{{\alpha sz}}}({{a}_{{\alpha s}}}(x))\) of the component in the general magnetic field keeps a hump-shaped profile, though distorted by a slightly nonlinear dependence of \({{A}_{z}}(x)\) determined from Eq. (10), and the spatial dependence of the anisotropy parameter \(\tau = 1 - \left( {\langle p_{z}^{2}\rangle - {{{\langle {{p}_{z}}\rangle }}^{2}}} \right){\text{/}}\langle p_{x}^{2}\rangle \) is easily determined:

$${{\tau }_{{\alpha s}}} = {{\left[ {{{a}_{{\alpha s}}}\frac{d}{{d{{a}_{{\alpha s}}}}}\ln \langle p_{x}^{2}\rangle - \frac{{\langle p_{x}^{2}\rangle }}{{p_{{\alpha s}}^{2}}}\frac{{{{d}^{2}}}}{{da_{{\alpha s}}^{2}}}\ln n\langle p_{x}^{2}\rangle } \right]}_{{\alpha s}}}.$$
(11)

Below, we describe a current sheet where the plasma density decreases in the direction of increasing magnetic field, as in the magnetospheres of planets. However, the developed model is not limited to this case. Indeed, according to Eqs. (7) and (9), the total change in the field energy density at the passage through the sheet is determined by the energies stored in the components of the plasma:

$$\begin{gathered} \frac{1}{{8\pi }}\left[ {B_{y}^{2}(x = + \infty ) - B_{y}^{2}(x = - \infty )} \right] \\ = {{P}_{{xx}}}( - \infty ) - {{P}_{{xx}}}( + \infty ) = - \sum\limits_{\alpha ,s} {{( - 1)}^{{s + 1}}}{{N}_{{\alpha s}}}{{T}_{{\alpha s}}}. \\ \end{gathered} $$
(12)

At the same time, a change in the plasma density at the passage through the sheet is

$$\sum\limits_{\alpha ,s} \left[ {{{n}_{{\alpha s}}}(x = + \infty ) - {{n}_{{\alpha s}}}(x = - \infty )} \right] = \sum\limits_{\alpha ,s} {{( - 1)}^{{s + 1}}}{{N}_{{\alpha s}}}.$$
(13)

Thus, using components with countercurrents (\(s = 1,2\) in Eqs. (1) and (2)) and independently setting the densities \({{N}_{{\alpha s}}}\) and different (!) temperatures \({{T}_{{\alpha s}}}\) of each component, one can ensure an increase, though nonmonotonic, in the magnetic field and the total plasma density at the passage through the current sheet. This corresponds to, e.g., the Voyager 2 data at the passage through the heliopause [8, 11].

We also note that the minimum thickness of magnetopauses of this type is determined by the typical ion gyroradius of the component with the highest stored energy. However, the total thickness can be larger than this value if the current in the magnetopause is generated by different particle populations shifted from each other in space.

3 EXAMPLES OF MAGNETOPAUSES WITH TRUNCATED MAXWELLIAN OR KAPPA DISTRIBUTIONS OF PARTICLES

Let the distribution functions \({{F}_{{\alpha s}}}(p)\) be Maxwellian with fixed temperatures of particles \({{T}_{{\alpha s}}}\):

$${{F}_{{\alpha s}}}(p) = {{\left( {2\pi {{m}_{\alpha }}{{T}_{{\alpha s}}}} \right)}^{{ - 3/2}}}\exp \left( { - \frac{{{{p}^{2}}}}{{p_{{\alpha s}}^{2}}}} \right).$$
(14)

Then, the current density (6) and magnetic field (9) in the self-consistent layer are given by the expressions

$${{j}_{z}}({{A}_{z}}) = \sum\limits_{\alpha ,s} \frac{{{{{( - 1)}}^{{s + 1}}}}}{{2{{\pi }^{{1/2}}}}}e{{N}_{{\alpha s}}}\frac{{{{p}_{{\alpha s}}}}}{{{{m}_{\alpha }}}}\exp \left( { - a_{{\alpha s}}^{2}} \right),$$
(15)
$$\begin{gathered} {{B}_{y}} = - 2{{\pi }^{{1/2}}}\left[ {2{{P}_{0}} - \sum\limits_{\alpha ,s} {{{( - 1)}}^{{s + 1}}}{{N}_{{\alpha s}}}{{T}_{{\alpha s}}}} \right. \\ {{\left. { \times \;{{{\left( {{\text{erf}}{\kern 1pt} {{a}_{{\alpha s}}} - {\text{erf}}{{{\left. {{{a}_{{\alpha s}}}} \right|}}_{{x = 0}}}} \right)}}_{{\begin{subarray}{l} {\kern 1pt} \\ {\kern 1pt} \end{subarray}} }}} \right]}^{{1/2}}}, \\ \end{gathered} $$
(16)

where \({\text{erf}}\left( \xi \right) = 2{{\pi }^{{ - 1/2}}}\int_0^\xi \exp \left( { - {{t}^{2}}} \right)dt\). The densities of particles (8) and their anisotropy parameter (11) are also easily determined [20]:

$${{n}_{\alpha }}({{A}_{z}}) = \sum\limits_s \frac{1}{2}{{N}_{{\alpha s}}}\left[ {1 + {{{( - 1)}}^{{s + 1}}}{\text{erf}}\left( {{{a}_{{\alpha s}}}} \right)} \right],$$
(17)
$${{\tau }_{\alpha }}({{A}_{z}}) = \frac{{{{e}^{{ - 1}}}{{m}_{\alpha }}n_{\alpha }^{{ - 1}}j_{\alpha }^{2} + \sum\limits_s {{p}_{{\alpha s}}}{{a}_{{\alpha s}}}{{j}_{{\alpha s}}}}}{{e\sum\limits_s {{{( - 1)}}^{{s + 1}}}{{N}_{{\alpha s}}}{{T}_{{\alpha s}}}{\kern 1pt} {\text{erf}}\left( {{{a}_{{\alpha s}}}} \right)}},$$
(18)

where \({{j}_{\alpha }} = \sum\nolimits_s {{j}_{{\alpha s}}}\) and \({{j}_{{\alpha s}}}\) are the terms in sum (15). It is easily seen that the distribution function for each component \(\alpha s\) is isotropic at \(x \to + \infty \); i.e., \({{\tau }_{{\alpha s}}} \to 0\); the anisotropy parameter in the center at \({{a}_{{\alpha s}}} = 0\) is \(2{\text{/}}\pi \); at \(x \to - \infty \), where \({{a}_{{\alpha s}}} \to - \infty \) and the density of particles (17) is very low \({{n}_{\alpha }} \to 0\), the anisotropy parameter is maximal \({{\tau }_{\alpha }} \to 1\) because all particles have high z components of the velocity.

Figure 1 shows a current sheet formed by two anisotropic populations of electrons responsible for countercurrents and two populations of protons; all populations have truncated Maxwellian distributions. It is noteworthy that the x scales in the left and right panels of Fig. 1 differ by one and a half orders of magnitude because of the corresponding difference between the gyroradii of thermal electrons and protons.

Fig. 1.
figure 1

(Color online) Current sheet formed by two populations of protons and two populations of electrons at \({{N}_{{e1,i1}}} = 3{{N}_{{e2,i2}}}\), \({{N}_{{e1}}} = {{N}_{{i1}}}\), \({{T}_{{e2,i2}}} = 2{{T}_{{e1,i1}}}\), \({{T}_{{i1}}} = 1.5{{T}_{{e1}}}\), \({{P}_{0}} = 0.3\sum {{N}_{{\alpha s}}}{{T}_{{\alpha s}}}\), \({{A}_{{i2}}}{\kern 1pt} e{\text{/}}c = {{p}_{{Ti2}}}\), and \({{A}_{{e2}}}{\kern 1pt} e{\text{/}}c = 2{{p}_{{Te1}}}\) \(({{A}_{{i1,e1}}} = 0)\). (a) Profiles of the electron density in the (blue line) first and (black line) second populations, as well as (green line) the difference of densities of all protons and all electrons divided by \(\sum {{N}_{{\alpha s}}}\). (b) Profiles of the proton density in the (red line) first and (violet line) second populations divided by \(\sum {{N}_{{\alpha s}}}\); the green line is the same as in panel (а). (c) Profiles of (black line) the magnetic field and (blue line) the total density of electrons of both populations in units of their maximum absolute values. (d) Profiles of (black line) the magnetic field and (green line) the total density of protons of both populations in units of their maximum absolute values. The x coordinate on the horizontal axis is given in the gyroradius rL0 of the thermal electron in the field \({{B}_{y}}(0)\).

Full size image

To simulate the excess of suprathermal particles often present in the solar wind, the following kappa energy distribution is used instead of the Maxwellian distribution [21]:

$${{F}_{{\alpha s}}}(p) = \frac{{{{M}_{\kappa }}}}{{{{\pi }^{{3/2}}}p_{{\alpha s}}^{3}}}{{\left( {1 + \frac{{{{p}^{2}}}}{{\left( {\kappa - 3{\text{/}}2} \right)p_{{\alpha s}}^{2}}}} \right)}^{{ - \kappa - 1}}},$$
(19)

where \(\kappa > 3{\text{/}}2\), \({{p}_{{\alpha s}}} = {{\left( {2{{m}_{\alpha }}{{T}_{{\alpha s}}}} \right)}^{{1/2}}}\), and

$${{M}_{\kappa }} = {{\left( {\kappa - \frac{3}{2}} \right)}^{{ - 3/2}}}\frac{{\Gamma (\kappa + 1)}}{{\Gamma (\kappa - 1{\text{/}}2)}}.$$

The integrals for the current density and density are easily calculated [21]:

$$\begin{gathered} {{j}_{{\alpha s,z}}}({{A}_{z}}) = \frac{{{{{( - 1)}}^{{s + 1}}}}}{{2{{\pi }^{{1/2}}}}}e{{N}_{{\alpha s}}}\frac{{{{p}_{{\alpha s}}}}}{{{{m}_{\alpha }}}}{{M}_{\kappa }} \\ \times \;\frac{{{{{\left( {\kappa - 3{\text{/}}2} \right)}}^{2}}}}{{\kappa (\kappa - 1)}}{{\left( {1 + \frac{{a_{{\alpha s}}^{2}}}{{\kappa - 3{\text{/}}2}}} \right)}^{{ - \kappa + 1}}}, \\ \end{gathered} $$
(20)
$$\frac{{d{{n}_{{\alpha s}}}}}{{d{{a}_{{\alpha s}}}}} = \frac{{{{{( - 1)}}^{{s + 1}}}}}{{{{\pi }^{{1/2}}}}}{{N}_{{\alpha s}}}{{M}_{\kappa }}\frac{{\kappa - 3{\text{/}}2}}{\kappa }{{\left( {1 + \frac{{a_{{\alpha s}}^{2}}}{{\kappa - 3{\text{/}}2}}} \right)}^{{ - \kappa }}}.$$
(21)

As a result, the density and pressure of the plasma as functions of the vector potential \({{A}_{z}}\) are expressed in terms of the hypergeometric function, and the profile of the magnetic field is determined from the balance of pressures (9) taking into account implicit relation (10) between the vector potential and coordinate. According to [21], profiles of the magnetic field and current for kappa and Maxwellian distributions for similar sets of populations of particles with the identical corresponding temperatures \({{T}_{{\alpha s}}}\) and densities \({{N}_{{\alpha s}}}\) are qualitatively similar but can be significantly different quantitatively.

Obviously, individual populations in the found general solution given by Eqs. (6)–(10) can have not only different effective temperatures \({{T}_{{\alpha s}}}\) and densities \({{N}_{{\alpha s}}}\) but also different energy distributions \({{F}_{{\alpha s}}}\), e.g., Maxwellian and kappa distributions. Combinations of different populations with countercurrents and spatial separation allow one to construct very diverse analytical models of magnetopauses with complex profiles of the current density and magnetic field.

The constructed current sheets are not completely electroneutral because the sum of the densities of all components \(\sum\nolimits_{\alpha s} {{n}_{{\alpha s}}}\) is nonzero in a narrow region near the center of the sheet (see, e.g., the green dashed lines in Figs. 1a and 1b). This occurs because the spatial scales of currents formed by particles with different masses are different, being about the gyroradius of particles. This charge can be compensated by isotropic components of the plasma \({{n}_{{\alpha 0}}}\), which should be chosen such that

$${{n}_{i}} + {{n}_{{i0}}} - {{n}_{e}} - {{n}_{{e0}}} = 0.$$
(22)

If their effective temperatures specified by the distributions \({{F}_{{\alpha 0}}}(p)\) are low, these isotropic components do not significantly violate the self-consistency of anisotropic distribution functions (1) and (2) and the corresponding currents and magnetic fields.

4 GENERALIZATION TO THE SHEAR OF MAGNETIC FIELD LINES

In addition to currents flowing along the z axis and magnetic fields that are induced by them and are parallel to the y axis, structures of the magnetopause with orthogonal orientations of currents and magnetic fields along the y and z axes, respectively, similar to Eqs. (6) and (9) can be taken. For such conjugated structures, the magnetic fields of one structure do not affect the motion of particles in the other at a chosen functional dependence of the distributions (1) and (2), whose arguments are the total momentum p and only one of the projections of the generalized momentum, \({{P}_{z}} = {{p}_{z}} + {{e}_{\alpha }}{{A}_{z}}{\text{/}}c\) or \({{P}_{y}} = {{p}_{y}} + {{e}_{\alpha }}{{A}_{y}}{\text{/}}c\).

According to the preceding sections, the distribution function of particles in the form of the sum of such pairs of orthogonal distributions for each component,

$${{f}_{{\alpha s}}}(x,\mathbf{p}) = f_{{\alpha s}}^{{(y)}}(x,p,{{P}_{y}}) + f_{{\alpha s}}^{{(z)}}(x,p,{{P}_{z}}),$$
(23)

is also a solution of the electrostatic Vlasov equation in the general self-consistent field \({\mathbf{B}}(x)\), but describes a current sheet with the shear of magnetic field lines. Indeed, substituting Eq. (23) into Eq. (3), we obtain two independent Grad–Shafranov-type equations for two components of the vector potential \({{A}_{y}}(x)\) and \({{A}_{z}}(x)\) and the corresponding components of the magnetic field \({{B}_{z}}(x)\) and \({{B}_{y}}(x)\).

Consequently, combining found current sheets without shear, one can simulate the structure of planar layered magnetopauses with the shear of magnetic field lines often occurring, e.g., in the solar wind. Figure 2 presents such a current sheet composed of two sheets with identical Maxwellian distribution functions, temperatures, and densities of particles at \(x \to \infty \), but with different \({{P}_{0}}\) values. Other examples of such magnetostatic structures can be found in [22].

Fig. 2.
figure 2

(Color online) Current sheet with the shear of (black straight lines) magnetic field lines, which is composed of (blue and orange straight lines) two sheets with planar fields \({{B}_{z}}(x)\) and \({{B}_{y}}(x) \approx - {{B}_{z}}( - x)\), respectively, each formed by the electron and proton components with the distributions given by Eq. (14). For both sheets, \({{P}_{0}} = 0.55\sum\nolimits_\alpha {{N}_{\alpha }}{{T}_{\alpha }}\), \({{T}_{i}} = 2{{T}_{e}}\), \({{N}_{i}} = {{N}_{e}}\), and \({{A}_{{e,i}}} = 0\). The magnetic field strength and the x coordinate are given in units of \({{B}_{y}}(0) = (8\pi {{P}_{0}}{{)}^{{1/2}}}\) and the gyroradius rL0 of the thermal electron in the field \({{B}_{y}}(0)\).

Full size image

5 LOCAL STABILITY OF THE SIMPLEST MODEL OF THE MAGNETOPAUSE

When developing models of stationary current sheets, the problem of their stability is important. Global (and hydrodynamic) stability in the case of the magnetopause can be ensured by a strong magnetic field on one of the sides of the sheet and is not discussed below. In this short article, we discuss only local stability because proposed sheets certainly have anisotropic distribution functions of particles and, thereby, the Weibel instability with a high growth rate is possible in them.

For simplicity, we analyze the fastest electron Weibel instability for the Maxwellian energy distribution of particles (14) (the analysis of ion instability is similar). Although this instability depends on the form of the anisotropic velocity distribution [16, 23, 24], we make estimates with a bi-Maxwellian distribution and use the known linear theory [25, 26], disregarding the magnetic field of the current sheet (external for Weibel perturbations) in the dispersion relation.

For the distribution given by Eq. (1), perturbations with wave vectors along the current (the z axis), i.e., in the direction of the minimum dispersion of velocities of particles, grow most rapidly (see, e.g., [25]). Only perturbations with wavelengths shorter than a certain value \({{\lambda }_{{\alpha ,{\text{min}}}}}\) can grow; the highest growth rate at the anisotropy parameter \({{\tau }_{\alpha }}\) (18) much smaller than unity is reached at the wavelength \({{\lambda }_{\alpha }} \approx {{3}^{{ - 1/2}}}{{\lambda }_{{\alpha ,{\text{min}}}}}\). Comparing this wavelength to the gyroradius of the thermal electron and taking into account only one component in (1), we obtain

$${{\varkappa }_{\alpha }} \equiv {{\left( {\frac{{2{\kern 1pt} {{r}_{{{\text{L}}\alpha }}}}}{{{{\lambda }_{{\alpha {\text{,min}}}}}}}} \right)}^{2}} = \frac{{{{\theta }_{\alpha }}}}{{{{\pi }^{2}}}}\frac{{8\pi {{N}_{\alpha }}{{T}_{\alpha }}}}{{B_{y}^{2}}},$$
(24)

where \({{\theta }_{\alpha }} = ({{n}_{\alpha }}{\text{/}}{{N}_{\alpha }}){\kern 1pt} {{\tau }_{\alpha }}{{(1 - {{\tau }_{\alpha }})}^{{ - 1}}}\). Since \(2{{P}_{0}} \geqslant {{N}_{\alpha }}{{T}_{\alpha }}\) according to Eq. (16), we have \({{\varkappa }_{\alpha }} \lesssim 1\) in the region of the current sheet's localization, \({\text{|}}{{a}_{\alpha }}{\text{|}} \leqslant 2\). Therefore, electrons in the sheet are magnetized and the instability cannot develop. In the general case of the presence of several components, particularly separated in space and with countercurrents, the absence of the local Weibel instability in the considered class of models of the magnetopause is not guaranteed.

At the same time, estimates for complex multicomponent sheets of this type also show that the development of the Weibel instability is strongly complicated because of the localization of currents of individual components in regions with a thickness of about the gyroradius of typical particles, i.e., because of the magnetization of particles in the self-consistent field. This statement is confirmed by our numerical calculations by the method of macroparticles using the EPOCH code [26] with a wide set of the parameters of two- and three-component models of the magnetopause.

6 CONCLUSIONS

A class of current sheets separating two regions of the anisotropic collisionless plasma with different parameters and different magnetic fields have been constructed analytically. They allow one to simulate complex magnetopauses. The spatial profiles of the current density in them can have more than one maximum and can be asymmetric and sign-alternating. They can include several spatially separated populations of ions and electrons with countercurrents localized in layers with strongly different scales. They also allow the shear of magnetic field lines, which often occurs in magnetopauses in the solar wind.

The possibility of using arbitrary energy distributions of particles, in particular, non-Maxwellian, and sufficiently strong and different anisotropies of the velocity distributions of certain particle populations, in particular, suprathermal, which are typical of current sheets in the space plasma is particularly important. Although the plasma is nonequilibrium, there are reasons against development of the local Weibel instability and global instability of considered current sheets because of their pronounced localization and strong magnetization by a sufficiently high self-consistent magnetic field. Thus, it is reasonable to apply the found exact solution for interpreting such current configurations in magnetopauses of space objects.