Abstract
Full Text
Reports of the Academy of Sciences of the USSR
1970. Volume 193, No. 6
UDC 533.9
PHYSICS
M. A. GINZBURG
SELF-SIMILAR WAVES IN A COLD PLASMA WITH A MAGNETIC FIELD
(Presented by Academician Ya. B. Zel’dovich on 17 XII 1970)
Nonstationary waves in a plasma with a magnetic field
The following problems are solved: at time (t = 0) there is a cylindrical plasma filament or a plane plasma layer with density decreasing with distance from the axis as (n_0/r^2), and with electric and magnetic fields decreasing as (E_0/r), (H_0/r), or a spherical plasma cloud (in the spherical case—without a magnetic field, but with an electric field (E_0/r) at (t = 0)). Determine the motion of the plasma and the field for (t > 0). The substitution
[
E_\phi = \frac{1}{r} E(\eta);\quad
H_\phi = \frac{1}{r} H(\eta);\quad
v_\phi = v(\eta);\quad
n_\phi = \frac{1}{r^2} n(\eta);\quad
\eta = t/r
\tag{1}
]
(the subscript (\phi) denotes physical field vectors) makes it possible to reduce Maxwell’s equations and the equations of two-fluid hydrodynamics
[
\operatorname{rot} H_\phi = j_\phi + \partial E_\phi/\partial t;\quad
\operatorname{rot} E_\phi = -\partial H_\phi/\partial t;\quad
\partial p_{i,e}/\partial t + (v_{i,e}\nabla)p_{i,e} =
]
[
= e\left(E + [v_{i,e}H]\right);\quad
j_\phi = e(n_i v_i - n_e v_e);\quad
p_{i,e} = m v_{i,e}/\sqrt{1 - v_{i,e}^2}.
\tag{2}
]
to a system of ordinary differential equations (*). Below, the results of solving this system on a machine are presented.
Interchanging (t) and (r) in (1), we obtain a system of equations with independent variable (\xi = r/t), describing expansion for a prescribed regime at the origin of coordinates.
In the nonrelativistic approximation there exist two limiting classes of self-similar solutions (s.s.s.) of system (1). In solutions of the longitudinal-magnetic type ((LM)), (H_z, E_\phi, u_\phi, v_\phi, u_r, v_r) are nonzero ((v) is the electron velocity, (u) the ion velocity). In solutions of the longitudinal-electric type ((LE)), (E_z, H_\phi, u_z, v_z, u_r, v_r) are nonzero. Mixed initial conditions and relativistic terms (v(Ev)) lead to more complicated s.s.s.
Let us begin with the limiting case (n_i = n_e = 0), with s.s.s. in vacuum. Then equations (2) have exact analytical s.s.s.:
[
H_z = H_{z0}/\sqrt{1-\eta^2};\quad
E_\phi = E_{\phi0} + \eta H_{z0}/\sqrt{1-\eta^2};\quad
E_z = E_{z0}/\sqrt{1-\eta^2};
]
[
H_\phi = H_{\phi0} - \eta E_{z0}/\sqrt{1-\eta^2}.
\tag{3}
]
The solution with (E_z, H_\phi) describes the process of decrease of the electromagnetic field upon sudden termination of the axial current (j_z). In the plane case:
[
H_z = (H_{z0} + \eta E_{y0})/(1-\eta^2);\quad
E_y = (E_{y0} + \eta H_{z0})/(1-\eta^2).
\tag{4}
]
The invariant
[
E^2 - H^2 = \frac{(E_{y0}^2 - H_{z0}^2)}{x^2 - t^2}
= (E_{y0}^2 - H_{z0}^2)\frac{1}{2t}
\left[\frac{1}{x+t} - \frac{1}{x-t}\right]
]
is a function only of the other invariant of Lorentz transformations—the interval (x^2 - t^2).
A self-similar solution can also be obtained by the substitution (E, H=r^\beta f(\eta)) ((\beta) arbitrary), but only for (\beta=-1) are the Poisson equations and the equations of motion of the plasma also self-similar. A self-similar solution of type (4) can be represented in the form of a superposition of two wave processes:
[
H_{z\phi}=\frac{H_{z0}}{2}\left[\frac{1+Q}{x-t}+\frac{1-Q}{x+t}\right];
]
[
E_{y\phi}=\frac{E_{y0}}{2}\left[\frac{1+Q^{-1}}{x-t}+\frac{1-Q^{-1}}{x+t}\right],\quad
Q=\frac{E_{y0}}{H_{z0}}. \tag{5}
]
Self-similar solutions for fields in vacuum were also found in (2), but there the substitution is different,
[
E\sim \frac{1}{\sqrt r}f(\eta).
]
The substitution
[
E,H\sim \frac{1}{r}f(\eta)
]
was obtained in (3), but for another system of equations, different from (2).
Fig. 1. Oscillations of a plasma column with a longitudinal electron current and with an azimuthal initial field (H_\phi), (\rho=n_i-n_e)
Let us return to the self-similar solutions with plasma. Let the plasma be nonrelativistic. Then at (\eta=0) ((t=0)) in the cylindrical case the plasma is quasineutral, and the currents are small, i.e., the fields are close to the self-similar solutions in vacuum (3)—(5). As (\eta) increases, plasma effects grow.
Figure 1 shows oscillations of a cylindrical plasma column with a longitudinal current in the field (H_\phi). The units of measurement in equations (1) (for (r=1) cm) are as follows: for the fields (E,H), (Mc^2/e); for the density, (Mc^2/4\pi e^2); for the velocity, the speed of light (c). The initial conditions of Fig. 1 are: (v_{z0}=0.02), (n_0=1), (H_{\phi0}=10^{-3}), i.e., for (r=1) cm, (n=5\cdot10^{14}\ \text{cm}^{-3}), (H_\phi=3) kOe. The Alfvén velocity
[
v_A=\frac{H}{\sqrt n}\ll 1,
]
so the motion is nonrelativistic. The current (nev_z) creates the variable field (H_\phi). The field (H_\phi) generates a vortex electric field (E_z), which sets the electrons in motion and gives rise to ((v_z'=-\lambda E_z n/n_0;\ E_z'=nv_z)) Langmuir oscillations of the inhomogeneous plasma:
[
v_z=v_{z0}\cos\sqrt{\lambda n_0}\eta
= v_{z0}\cos\omega_0 t;
]
[
\omega_0^2(r)=\frac{4\pi e^2 n_0(r)}{m};
]
[
\lambda=\frac{M}{m},
]
their period
[
T=\frac{2\pi}{\sqrt{\lambda n_0}}=0.15
]
agrees with the results of Fig. 1.
Fig. 2. Spreading of a plasma column in a longitudinal magnetic field (H_z)
Figure 2 shows the evolution of the column in an initial longitudinal field ((H_{z0}=0.2), (u_{z0}=0.02), (n_0=0.7;\ v_A\lesssim 1)). The field (H_z) generates, according to the self-similar solution in vacuum (3), the component (E_\phi), and this, in turn, generates the electron current (v_z). For (\eta>0.3), charge-separation effects are substantial, and a maximum also arises
of electron density, moving outward. Oscillations still occur, but they are of a different nature—not Langmuir oscillations ((E_\varphi' = H_z + n v_\varphi,) but (H_z > n v_\varphi), i.e., for (\eta < 1) the plasma is still quasineutral). The particles are accelerated to relativistic energies.
Fig. 3. Self-similar wave in a plasma column in a radial electric field (relativistic case)
Expansion of a plasma column in an electric field (E_r) (Fig. 3). At (t = 0), (E_r = 0.04,\ v_{r0} = 0.005). As can be seen, the electrons also reach relativistic velocities. For (\eta < 1) and small (v), the oscillations become Langmuir oscillations ((n = n_0,\ E_r' = n v_r,\ v_r' - \lambda \dfrac{n}{n_0} E_r)).
Fig. 4. Stationary two-dimensional flow of plasma with a magnetic field. (H_{z0} = 2 \cdot 10^{-4};\ n_0 = 0.1;\ u_{z0} = v_{z0} = 0.001)
Unlike in Figs. 1–2, along the abscissa axis there is plotted not the time (\eta), but the distance (r) at a fixed instant (t_0). With distance from the axis, the wavelength of the self-similar wave increases; its amplitude does not change for the velocity, and decreases for the density.
Stationary two-dimensional flow of a plasma jet with a magnetic field. The following problem was also solved: in the plane (z = 0) there are prescribed electric and (or) magnetic fields decreasing with distance from the axis as (1/r), as well as the velocity vector of a plasma jet flowing through this plane. At (z = 0) the density is (n_\varphi = n_0/r^2). It is required to find a stationary ((\partial/\partial t \equiv 0)) flow in the half-space (z > 0) satisfying these boundary conditions. In substitution (1) we replace (t) by (z) and obtain a system of self-similar equations, which we solve numerically (Fig. 4). The substitution (\eta = z/r) makes it possible to find stationary self-similar solutions also in vacuum:
[
\begin{gathered}
H_z=H_{z0}\sqrt{1+\eta^2};\qquad
H_r=H_{r0}-\eta H_{z0}\sqrt{1+\eta^2};\
H_x=(H_{x0}-\eta H_{z0})/(1+\eta^2);\qquad
H_z=(H_{z0}+\eta H_{x0})/(1+\eta^2).
\end{gathered}
\tag{6}
]
At small angles (\eta \ll 1), it is precisely the “vacuum” self-similar solutions (6) that are realized. (H_z) decreases, (-H_r) increases—the lines of force bend toward the axis. The plasma also moves toward the axis ((u_r<0,\ v_r<0)). At large angles, space charge and the fields (E_r, E_z) and (H_\varphi) appear; the plasma changes the structure of the vacuum field.
Equations (2) are no longer applicable for (\eta>c/v_{\tau,e}). In an analogous way, the region of applicability of the self-similar solutions for a plasma with hot electrons ((n_e\sim e^\varphi), see (1)) is restricted by the conditions (\sqrt{T_e/T_i}>\eta>\sqrt{m/M}); these inequalities make it possible to find the integral of the total number of particles and to determine the constant of the self-similar motion.
Thermal motion can be taken into account by a self-similar kinetic equation. For simplicity, we shall carry out all calculations for the planar case. We seek the electron distribution functions (\tilde f_e) and ion distribution functions (\tilde f_i) in the form (\tilde f_{e,i}=\dfrac{1}{x^2}f_{e,i}).
Maxwell’s equations can be integrated with respect to the variable (\eta):
[
E_x=E_{x0}+\iint (f_i-f_e)v_xv^2\,dv\,d\eta,\qquad
E_y=\frac{1}{1-\eta^2}\left{E_{y0}+\eta H_{z0}
-\iint (f_i-f_e)v_yv^2\,dv\,d\eta\right},
]
[
H_z=\frac{1}{1-\eta^2}\left{H_{z0}+\eta E_{y0}
-\eta\iint (f_i-f_e)v_yv^2\,dv\,d\eta\right}.
]
Substituting these expressions into the kinetic equation
[
\frac{\partial \tilde f}{\partial t}
+\frac{\mathbf p}{m\gamma}\frac{\partial f}{\partial x}
+\frac{e}{m}\left(\mathbf E+\frac{1}{\gamma c}[\mathbf p\mathbf H]\right)
\frac{\partial \tilde f}{\partial \mathbf p}=0;\qquad
\gamma^2=\left[\frac{p^2}{p^2+1}\right]^{-1},
]
we obtain a system of two equations for (f_i) and (f_e) with independent variables (\eta) and (\mathbf p).
Applying Maxwell’s equations in integral form, we are convinced that in problems with cylindrical symmetry charges are present on the axis. In a two-dimensional flow of a cold plasma (Fig. 4), these are magnetic charges (for example, permanent magnets). In a nonstationary plasma flow with hot electrons ((1), (n_e=e^\varphi)), an electric charge (\varkappa T_e/e) is present on the axis; its field prevents the electrons from escaping.
The author first became acquainted with the general properties of self-similar solutions and with advice on how to seek them in the theoretical department of L. V. Ovsyannikov at the Institute of Hydrodynamics, Siberian Branch of the USSR Academy of Sciences, in 1966, and thanks L. V. Ovsyannikov, N. Kh. Ibragimov, and V. V. Pukhnachev for this.
Institute of Terrestrial Magnetism,
Ionosphere and Radio-Wave Propagation
P.O. Akademgorodok, Podolsk District, Moscow Region.
Received
15 XII 1970
CITED LITERATURE
- M. A. Gintsburg, Astr. Zhurn., 46, 1019 (1969).
- E. I. Zababakhin, ZhETF, 33, 432 (1957).
- V. P. Korobeinikov, Magnetohydrodynamics, 3, 55 (1967).