Low-Frequency Wave Propagation

Consider 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., $\omega\ll {{\mit\Omega}}_i, {{\mit\Pi}}_i$), we have [see Equations (5.28)–(5.30)]

$$S \simeq 1 +\frac{{{\mit\Pi}}_i^{2}}{{{\mit\Omega}}_i^{2}},$$ (5.74)

$$D \simeq 0,$$ (5.75)

$$P \simeq -\frac{{{\mit\Pi}}_e^{2}}{\omega^2}.$$ (5.76)

Here, use has been made of ${{\mit\Pi}}_e^{2}/({{\mit\Omega}}_e\,{{\mit\Omega}}_i) =-{{\mit\Pi}}_i^{2}/ {{\mit\Omega}}_i^{2}$. Thus, the eigenmode equation (5.42) reduces to

$$\left(\!\begin{array}{ccc} 1+{{\mit\Pi}}_i^{2}/{{\mit\Omega}}... ...begin{array}{c} E_x\\ E_y \\ E_z \end{array}\!\right) = {\bf0}.$$ (5.77)

The solubility condition for Equation (5.77) yields the dispersion relation

$$\left\vert \!\begin{array}{ccc} 1+{{\mit\Pi}}_i^{2}/{{\mit\Omega}... ...& -{{\mit\Pi}}_e^{2}/\omega^2 - n^2\,\sin^2\theta \end{array}\!\right\vert = 0.$$ (5.78)

Now, in the low-frequency ordering, ${{\mit\Pi}}_e^{2}/\omega^2\gg {{\mit\Pi}}_i^{2}/{{\mit\Omega}}_i^{2}$. 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 (Cairns 1985). In this manner, we obtain two roots:

$$n^2\,\cos^2\theta = 1 + \frac{{{\mit\Pi}}_i^{2}}{{{\mit\Omega}}_i^{2}},$$ (5.79)

and

$$n^2 = 1 + \frac{{{\mit\Pi}}_i^{2}}{{{\mit\Omega}}_i^{2}}.$$ (5.80)

It is fairly easy to show, from the definitions of the plasma and cyclotron frequencies [see Equations (5.16)–(5.19)], that

$$\frac{{{\mit\Pi}}_i^{2}}{{{\mit\Omega}}_i^{2}} = \frac{c^2}{B_0^{2}/(\mu_0\,\rho)} = \frac{c^2}{V_A^{2}}.$$ (5.81)

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

$$V_A = \sqrt{\frac{B_0^{2}}{\mu_0\,\rho}}$$ (5.82)

is known as the Alfvén velocity. Thus, the dispersion relations (5.79) and (5.80) can be written

$$\omega = \frac{k\,V_A\,\cos\theta}{\sqrt{1+V_A^{2}/c^2}}\simeq k\,V_A\,\cos\theta \equiv k_\parallel\,V_A,$$ (5.83)

and

$$\omega = \frac{k\,V_A}{\sqrt{1+V_A^{2}/c^2}}\simeq k\,V_A,$$ (5.84)

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

The dispersion relation (5.83) corresponds to the slow or shear-Alfvén wave, whereas the dispersion relation (5.84) corresponds to the fast or compressional-Alfvén wave. The fast/slow terminology simply refers to the relative magnitudes of the phase-velocities of the two waves. The shear/compressional terminology refers to the velocity fields associated with the waves. In fact, it is clear from Equation (5.77) that $E_z=0$ for both waves, whereas $E_y=0$ for the shear wave, and $E_x=0$ for the compressional wave. Both waves are, in fact, MHD modes that satisfy the linearized MHD Ohm's law [see Equation (4.196)]

$${\bf E} + {\bf V} \times {\bf B}_0 = {\bf0}.$$ (5.85)

Thus, for the shear wave

$$V_y = - \frac{E_x}{B_0},$$ (5.86)

and $V_x=V_z=0$, whereas for the compressional wave

$$V_x = \frac{E_y}{B_0},$$ (5.87)

and $V_y=V_z=0$. Now, $\nabla\cdot{\bf V} = {\rm i}\,{\bf k}\cdot{\bf V} = {\rm i}\,k\,V_x\,\sin\theta$. Thus, the shear-Alfvén wave is a torsional wave, with zero divergence of the plasma flow, whereas the compressional wave involves a non-zero flow divergence. In fact, the former wave bends magnetic field-lines without compressing them, whereas the latter compresses magnetic field-lines without bending them (Hazeltine and Waelbroeck 2004). It is important to realize that the physical entity that resists compression in the compressional wave is the magnetic field, not the plasma, because there is negligible plasma pressure in the cold-plasma approximation.

It should be noted that the thermal velocity is not necessarily negligible compared to the Alfvén velocity in a conventional plasma. Thus, we would expect the dispersion relations (5.83) and (5.84), for the shear- and compressional-Alfvén waves, respectively, to undergo considerable modification in a “warm” plasma. (See Section 8.4.)