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On Magnetohydrodynamic Waves in Gas
Recently a whole series of articles has appeared in the literature1–8 devoted to so-called magnetohydrodynamic waves, i.e., waves capable of propagating in a conducting medium located in a magnetic field. Phenomena associated with magnetohydrodynamic waves have found wide application, chiefly in astrophysical problems. In 1942 an attempt was made8 to create a theory of sunspots, according to which disturbances arising in the central regions of the Sun are transmitted to the surface by means of magnetohydrodynamic waves. In a number of theories3, 4 of the origin of cosmic
rays, magnetohydrodynamic waves play an important role*). The investigation of phenomena connected with the propagation of magnetohydrodynamic waves in the ionosphere is also of undoubted interest. It is therefore natural that clarification of the question of the nature of these waves is very topical. In this connection, the work of Alfvén, published in a JETP issue, attracts attention. In this work it was shown that magnetohydrodynamic waves are ordinary low-frequency electromagnetic waves arising in a conducting gaseous or liquid medium placed in a magnetic field, with account taken of the motion of heavy charged (ions) or neutral particles. Analogous conclusions were obtained in the work of Alfvén, in which, unlike Alfvén, collisions between particles were taken into account.
First of all let us dwell on the elucidation of the properties of magnetohydrodynamic waves on the basis of the hydrodynamic approximation \(^{1,5,6}\). To this end we consider the problem of a conducting liquid with density \(\rho\) and conductivity \(\sigma\), situated in an electromagnetic field. The equations of motion and of the field in this case have the form:
\[ \rho \frac{d\mathbf v}{dt}=-\nabla p+\frac{1}{c}[\mathbf j\mathbf H],\qquad \mathbf j=\sigma\left(\mathbf E+\frac{1}{c}[\mathbf v\mathbf H]\right), \tag{1} \]
\[ \operatorname{rot}\mathbf H=\frac{4\pi}{c}\mathbf j,\qquad \operatorname{rot}\mathbf E=-\frac{1}{c}\frac{\partial \mathbf H}{\partial t},\qquad \frac{\partial \rho}{\partial t}+\operatorname{div}\rho\mathbf v=0, \]
where \(\mathbf v\) is the velocity of the liquid, \(\mathbf j\) is the current density, \(\mathbf H\) and \(\mathbf E\) are the intensities of the magnetic and electric fields, and \(p\) is the pressure. We neglect the displacement current \(\left(\dfrac{\partial\mathbf E}{\partial t}\right)\). We shall solve the obtained system of equations in the linear approximation, i.e. we shall assume that \(\mathbf v,\mathbf j,\mathbf E,\mathbf H_1=\mathbf H-\mathbf H_0\) (\(\mathbf H_0\) is the constant external magnetic field) and \(\rho_1=\rho-\rho_0\) (\(\rho_0\) is the density of the unperturbed liquid at \(H_0=0\)) are small quantities. For simplicity we take the conductivity \(\sigma\) to be infinite (this leads to the relation \(\mathbf E=-\dfrac{1}{c}[\mathbf v\mathbf H]\)). Then the linearized system will look as follows:
\[ \rho_0\frac{\partial\mathbf v}{\partial t} =-u_0^2\nabla\rho_1+\frac{1}{c}[\mathbf j\mathbf H_0], \]
\[ \Delta\mathbf E-\nabla\operatorname{div}\mathbf E =\frac{4\pi}{c^2}\frac{\partial\mathbf j}{\partial t},\qquad \mathbf E=-\frac{1}{c}[\mathbf v\mathbf H_0], \tag{2} \]
where \(u_0^2=\dfrac{\partial p}{\partial\rho}\) is the square of the speed of sound at \(H_0=0\); the field \(\mathbf H\) has been eliminated.
We seek the solution for all the quantities entering into (2) in the form of plane waves, i.e. \(\sim \exp\{i[\omega t-\mathbf k\mathbf r]\}\).
As a result of eliminating \(\mathbf j\) and \(\mathbf E\), we obtain the following equation for determining the velocity \(u=\dfrac{\omega}{k}\) of waves that can propagate in the liquid:
\[ u^2\mathbf v =-\frac{1}{4\pi\rho_0} \left\{ [\mathbf H_0[\mathbf v\mathbf H_0]] -\left[\mathbf H_0\frac{\mathbf k}{k}\right] \left(\frac{\mathbf k}{k}[\mathbf v\mathbf H_0]\right) \right\} +\frac{u_0^2\mathbf k(\mathbf k\mathbf v)}{k^2}. \]
*) See also V. S. Vavilov, UFN 39, 612 (1949).
Introducing the angle \(\theta\) between the vectors \(\mathbf{k}\) and \(\mathbf{H}_0\), we find that \(u\) can take the values
\[ u_1^2=\frac{H_0^2}{4\pi\rho_0}\cos^2\theta, \tag{3} \]
when the velocity \(\mathbf{v}\) is directed perpendicular to \(\mathbf{k}\) and \(\mathbf{H}_0\)—a transverse wave, and
\[ u_{2,3}^2=\frac{1}{2}\left(u_0^2+\frac{H_0^2}{4\pi\rho_0}\right) \pm \frac{1}{2}\sqrt{\frac{H_0^4}{(4\pi\rho_0)^2}+u_0^4-2\frac{H_0^2u_0^2}{4\pi\rho_0}\cos 2\theta}, \tag{4} \]
when the velocity \(\mathbf{v}\) lies in the plane of the vectors \(\mathbf{k}\) and \(\mathbf{H}_0\), and in the general case these waves are neither longitudinal nor transverse.
For \(\theta=0\) (propagation along the field),
\[ u_1^2=u_2^2=\frac{H_0^2}{4\pi\rho_0}, \qquad u_3^2=u_0^2. \tag{5} \]
Waves 1 and 2 are transverse (\(\mathbf{k}\mathbf{v}=0\)); wave 3 is longitudinal (i.e. for it \(\mathbf{v}\parallel\mathbf{k}\); see Fig. 1).
For \(\theta=\frac{\pi}{2}\) (propagation perpendicular to the field):
\[ u_1^2=u_3^2=0; \qquad u_2^2=u_0^2+\frac{H_0^2}{4\pi\rho_0}. \tag{6} \]
Here wave 2 is longitudinal (see Fig. 2).
Fig. 1.
Fig. 2.
From an analysis of the general formula (2) it follows that the velocity of transverse waves reaches its maximum value at \(\theta=0\). The correction to the velocity of longitudinal waves is \(\sim \dfrac{H_0^2}{4\pi\rho_0u_0}\), i.e. in a liquid it is always very small (for mercury at \(H_0\sim10^4\), \(\dfrac{H_0^2}{4\pi\rho_0u_0}\sim1\ \text{cm/sec}\), whereas \(u_0=1.46\cdot10^5\ \text{cm/sec}\)).
Thus, we see that the presence of conductivity at \(H_0\ne0\) leads to the appearance, for example at \(\theta=0\), of transverse waves that are usually absent in a liquid. These waves were called magnetohydrodynamic¹ and, as will be seen below from the example
gases, are ordinary electromagnetic waves of low frequency. To clarify this question, the most consistent path in the case of a gaseous medium is to solve the simultaneous kinetic equations for electrons and ions (a fully ionized gas—plasma—is considered). However, an essential simplifying circumstance in solving this problem is the fact that, for transverse waves, the kinetic treatment may be replaced by an analysis of the equations of motion of electrons and ions using certain effective “collision frequencies” \(\nu\), computed on the basis of kinetic theory\(^9\). These equations are as follows:
\[
m\frac{d\mathbf v_e}{dt}
=
-e\mathbf E-\frac{e}{c}[\mathbf v_e\mathbf H]
+m\nu(\mathbf v_i-\mathbf v_e),
\]
\[
M\frac{d\mathbf v_i}{dt}
=
e\mathbf E+\frac{e}{c}[\mathbf v_i\mathbf H]
+m\nu(\mathbf v_i-\mathbf v_e),
\tag{7}
\]
where \(m\) and \(M\) are the masses of the electron and the ion, \(-e\) and \(+e\) are the charges of the electron and the ion, \(\mathbf v_e\) and \(\mathbf v_i\) are their mean velocities, and \(\nu\) is the effective number of collisions of electrons with ions. It is assumed that the plasma is quasineutral (i.e., the concentrations of ions and electrons are equal) and that there are only singly charged ions of one mass.
To equations (7) one must add the field equations:
\[
\operatorname{rot}\mathbf H
=
\frac{4\pi\mathbf j}{c}
+
\frac{1}{c}\frac{\partial \mathbf E}{\partial t},
\qquad
\operatorname{rot}\mathbf E
=
-\frac{1}{c}\frac{\partial \mathbf H}{\partial t},
\]
\[
\Delta \mathbf E-\nabla \operatorname{div}\mathbf E
-\frac{1}{c^2}\frac{\partial^2\mathbf E}{\partial t^2}
=
\frac{4\pi}{c^2}\frac{\partial \mathbf j}{\partial t}.
\tag{8}
\]
We seek the solution of the simultaneous system (7)—(8) in the form of a plane wave propagating along the field \(\mathbf H_0\), in which all quantities are proportional to \(e^{i(\omega t-kr)}\).
After substitution into (7) and (8), we obtain a system of homogeneous linear algebraic equations for the unknown quantities, which has a solution different from zero under the condition that the determinant of the system vanish. This condition leads to the following relation determining the wave velocity \(u\):
\[ \frac{k^2c^2}{\omega^2} = \frac{c^2}{u^2} = 1- \frac{4\pi e^2N} {m\omega\left(\omega-i\nu \mp \omega_H-\omega_H\dfrac{\Omega_H}{\omega}\right)}, \tag{9} \]
where
\[ \omega_H=\frac{eH}{mc} \quad\text{and}\quad \Omega_H=\frac{eH}{Mc} \]
are the gyromagnetic frequencies of electrons and ions. The two signs in the denominator correspond to two different transverse waves, circularly polarized respectively to the left and to the right. In deriving this relation, terms \(\sim m/M\) were discarded. We note that the system (7)—(8) has one more solution, corresponding to the ordinary longitudinal plasma wave, which for \(\nu=0\) has the frequency
\[ \omega^2=\frac{4\pi e^2N}{m}. \]
Let us consider the two most interesting limiting cases of high and low frequencies. For \(\omega\gg\Omega_H\) we obtain the usual formula for elec-
electromagnetic ionospheric wave:
\[ \frac{k^2 c^2}{\omega^2}=\frac{c^2}{u_\pm^2} =1-\frac{4\pi e^2 N}{m\omega(\omega \mp \omega_H-i\nu)}. \]
For \(\Omega_H \gg \omega;\ \Omega_H\omega_H \gg \omega\nu\), formula (9) takes the form
\[ \frac{c^2}{u_\pm^2} =1+\frac{4\pi e^2 N}{m\omega_H\Omega_H} \simeq \frac{4\pi MNc^2}{H_0^2}. \tag{10} \]
It is easy to see that the last expression is equivalent to formula (5) of the present note, and also to formula (5) in paper \(^{1}\) for the velocity of a magnetohydrodynamic wave, since \(MN\) is precisely the density \(\rho_0\). In the case when (10) is satisfied, the initial equations (6) and (6) also pass into the hydrodynamic equations of an incompressible fluid (without pressure). Thus it has been shown that magnetohydrodynamic waves are nothing other than ordinary electromagnetic waves whose frequency is considerably less than the gyromagnetic frequency of the ions.
In the last section of paper \(^{6}\), in addition to the ions, the motion of neutral particles is taken into account. In this more complicated case the expressions for the wave velocity at sufficiently low frequencies also coincide with the hydrodynamic formula (5).
Let us note in conclusion that recently \(^{5}\) the first attempt was made at an experimental investigation of magnetohydrodynamic waves in a liquid (mercury).
The results obtained in this experiment are not in very good agreement with the theory, developed as applied to the conditions that took place in the experiment. The author explains this discrepancy by the presence of a constant disturbance on the surface of the mercury.
Recently there have also appeared a number of works devoted to solving the magnetohydrodynamic equations in the nonlinear approximation, in the presence of discontinuities and turbulence \(^{10,11}\).
V. F.
CITED LITERATURE
- H. Alfven, Nature 150, 405 (1942); Arkiv f. Mat. Astr. o. Fys. 29B, No. 2 (1942).
- C. Walen, Arkiv f. Mat. Astr. o. Fys. 30A, No. 15 (1941); 31B No. 3 (1944).
- E. Fermi, Phys. Rev. 75, 1169 (1949).
- R. D. Richtmyer and E. Teller, Phys. Rev. 75, 1729 (1949).
- S. Lundquist, Phys. Rev. 76, 1805 (1949).
- V. L. Ginzburg, ZhETF, 21, 788 (1951).
- E. Astrom, Arkiv f. Fysik 2, 443 (1951).
- H. Alfven, Arkiv f. Mat. Astr. o. Fys. 29A, No. 12 (1943).
- V. L. Ginzburg, Theory of the Propagation of Radio Waves in the Ionosphere, Gostekhizdat, 1949.
- S. Chandrasekhar, Proc. Roy. Soc. 204A, 435 (1950); Proc. Roy. Soc. 207A, 301 (1951).
- Batchelor, Proc. Camb. Phil. Soc. 47, 359 (1951).