Collisional Damping

Let us now consider a real-life damping mechanism. Equation (5.15) specifies the linearized Ohm's law in the collisionless cold-plasma approximation. In the presence of collisions, this expression acquires an extra term (see Section 4.12), such that

$${\bf E} = - {\bf V}\times{\bf B}_0 + \frac{{\bf j}\times{\bf B}_0... ...,\frac{\omega\,m_e}{n_e\,e^2}\,{\bf j} + \frac{\nu_e\,m_e}{n_e\, e^2}\,{\bf j},$$ (6.62)

where $\nu_e \equiv\tau_e^{\,-1}$ is the electron collision frequency. Here, for the sake of simplicity, we have neglected the small difference between the parallel and perpendicular plasma electrical conductivities. When Equation (6.62) is used to calculate the dielectric permittivity for a right-handed wave, in the limit $\omega\gg {\mit\Omega}_i$ , we obtain

$$R \simeq 1- \frac{{\mit\Pi}_e^{\,2}}{\omega\,(\omega+{\rm i}\,\nu_e -\vert{\mit\Omega}_e\vert)}.$$ (6.63)

A right-handed circularly polarized wave, propagating parallel to the magnetic field, is governed by the dispersion relation (see Section 5.9)

$$n^{\,2} = R \simeq 1+ \frac{{\mit\Pi}_e^{\,2}}{\omega\,(\vert{\mit\Omega}_e\vert -\omega - {\rm i}\,\nu_e)}.$$ (6.64)

Suppose that $n=n(z)$ . Furthermore, let

$$\vert{\mit\Omega}_e\vert = \omega + \vert{\mit\Omega}_e\vert'\,z,$$ (6.65)

so that the electron cyclotron resonance is located at $z=0$ . We also assume that $\vert{\mit\Omega}_e\vert'>0$ , so that the evanescent region corresponds to $z<0$ . It follows that, in the immediate vicinity of the resonance,

$$n^2 \simeq \frac{b}{z+ {\rm i}\,\epsilon},$$ (6.66)

where

$$b = \frac{{\mit\Pi}_e^{\,2}}{\omega\,\vert{\mit\Omega}_e\vert'},$$ (6.67)

and

$$\epsilon = -\frac{\nu_e}{\vert{\mit\Omega_e}\vert'}.$$ (6.68)

It can be seen that $\epsilon<0$ , which is consistent with the absorption of incident wave energy by the resonant layer. The approximate width of the resonant layer is

$$\delta \sim \vert\epsilon\vert = \frac{\nu_e}{\vert{\mit\Omega_e}\vert'}.$$ (6.69)

Note that the damping mechanism--in this case collisions--controls the thickness of the resonant layer, but does not control the amount of wave energy absorbed by the layer. In fact, in the simple theory outlined previously, all of the incident wave energy is absorbed by the layer.