Low-Frequency EM Waves in Magnetized Plasmas

Consider electromagnetic wave propagation through a magnetized plasma at frequencies far below the ion cyclotron or plasma frequencies, which are, in turn, well below the corresponding electron frequencies. In the low-frequency limit (i.e., $\omega\ll {\mit\Omega}_i, \omega_{p\,i}$), we have [see Equations (9.124), (9.131), (9.138), (9.170), and (9.171)]

$$S \simeq 1 +\frac{\omega_{p\,i}^{\,2}}{{{\mit\Omega}}_i^{\,2}},$$ (9.172)

$$D \simeq 0,$$ (9.173)

$$P \simeq -\frac{\omega_{p\,e}^{\,2}}{\omega^{\,2}}.$$ (9.174)

Here, use has been made of $\omega_{p\,e}^{\,2}/({\mit\Omega}_e\,{\mit\Omega}_i) =\omega_{p\,i}^{\,2}/ {\mit\Omega}_i^{\,2}$. Thus, the eigenmode equation (9.168) reduces to

$$\left(\!\begin{array}{ccc} 1+\omega_{p\,i}^{\,2}/{\mit\Omega}... ...ft(\begin{array}{c}0\\ [0.5ex] 0\\ [0.5ex] 0\end{array}\right).$$ (9.175)

The solubility condition (Riley 1974) for the homogeneous matrix equation (9.175) yields the dispersion relation

$$\left\vert \!\begin{array}{ccc} 1+\omega_{p\,i}^{\,2}/{\mit\Omega... ...{p\,e}^{\,2}/\omega^{\,2} - n^{\,2}\,\sin^2\theta \end{array}\!\right\vert = 0.$$ (9.176)

Now, in the low-frequency limit, $\omega_{p\,e}^{\,2}/\omega^{\,2}\gg 1$, $\omega_{p\,i}^{\,2}/{\mit\Omega}_i^{\,2}$. Thus, we can see that the bottom right-hand element of the previous determinant is far larger than any of the other elements. Hence, to a good approximation, the roots of the dispersion relation are obtained by equating the term multiplying this large factor to zero. In this manner, we obtain two roots:

$$n^{\,2}\,\cos^2\theta = 1 + \frac{\omega_{p\,i}^{\,2}}{{\mit\Omega}_i^{\,2}},$$ (9.177)

and

$$n^{\,2} = 1 + \frac{\omega_{p\,i}^{\,2}}{{\mit\Omega}_i^{\,2}}.$$ (9.178)

It is fairly easy to show, from the definitions of the plasma and cyclotron frequencies [see Equations (9.117)–(9.120)], that

$$\frac{\omega_{p\,i}^{\,2}}{{\mit\Omega}_i^{\,2}} = \frac{c^{\,2}}{B_0^{\,2}/(\mu_0\,\rho)} = \frac{c^{\,2}}{V_A^{\,2}}.$$ (9.179)

Here, $\rho\simeq n_e\,m_i$ is the plasma mass density, and

$$V_A = \sqrt{\frac{B_0^{\,2}}{\mu_0\,\rho}}$$ (9.180)

is known as the Alfvén speed. Thus, the dispersion relations (9.177) and (9.178) can be written

$$\omega = \frac{k\,V_A\,\cos\theta}{\sqrt{1+V_A^{\,2}/c^{\,2}}}\simeq k\,V_A\,\cos\theta,$$ (9.181)

and

$$\omega = \frac{k\,V_A}{\sqrt{1+V_A^{\,2}/c^{\,2}}}\simeq k\,V_A,$$ (9.182)

respectively. Here, we have made use of the fact that $V_A\ll c$ in a conventional plasma.

The dispersion relation (9.181) corresponds to the so-called shear-Alfvén wave, whereas the dispersion relation (9.182) corresponds to the compressional-Alfvén wave. The shear-Alfvén wave bends magnetic field-lines without compressing them, whereas the compressional-Alfvén wave compresses magnetic field-lines without bending them. Likewise, the shear-Alfvén wave does not compress the plasma, whereas the compressional-Alfvén wave does (Hazeltine and Waelbroeck 2004).

The shear-Alfvén wave is analogous to a wave on a string in tension, which propagates at the phase velocity $v=(T/\rho)^{\,1/2}$, where $T$ is the tension, and $\rho$ the linear mass density. (See Section 6.3.) At low frequencies, the plasma and the magnetic field are “tied” (i.e., if one moves then so must the other), so it is possible to consider a magnetic field-line to be “loaded” with a plasma of density $\rho$ (Fitzpatrick 2015). Furthermore, in terms of the Maxwell stress tensor, the field-line is under a tension $B_0^{\,2}/\mu_0$ (Fitzpatrick 2008). Hence, $v = (B_0^{\,2}/\mu_0\,\rho)^{\,1/2} = V_A$. We, thus, obtain the correct result for waves propagating along the magnetic field. The compressional-Alfvén wave is similar to a conventional sound wave (see Section 5.4), except that the restoring force emanates from magnetic pressure, rather than the thermal pressure of the plasma (which has actually been neglected in the present analysis) (Fitzpatrick 2015).