Electrostatic Waves

It is instructive to consider the propagation of electrostatic waves through a magnetized plasma. Such waves have purely electrostatic perturbed electric fields of the form

$${\bf E} = -\nabla{\mit\Phi} =-{\rm i}\,{\mit\Phi}\,{\bf k}.$$ (7.114)

Equation (7.8) can be generalized to give

$$k^2\,{\mit\Phi} =\sum_s \frac{e_s}{\epsilon_0}\int f_{1\,s}\,d^3{\bf v}.$$ (7.115)

Moreover, it follows from Equation (7.71) that

$${\bf B} = \omega^{-1}\,{\bf k}\times {\bf E} = {\bf0}.$$ (7.116)

In other words, there is no perturbed magnetic field associated with an electrostatic wave. Equation (7.69) yields

$$f_{1\,s} = {\rm i}\,{\mit\Phi}\,\frac{e_s}{m_s}\int_{-\infty}^t\left(k_\pe... ...}{{\mit\Omega}_s}\right)J_m\left(\frac{k_\perp\,v_\perp}{{\mit\Omega}_s}\right)$$

$$\phantom{=} \times\exp\left\{{\rm i}\,\left[(n\,{\mit\Omega}_s+k_\parallel\,v_\parallel-\omega)\,(t'-t)+(m-n)\,\theta\right]\right\}dt'.$$ (7.117)

Here, ${\mit\Omega}_s = e_s\,B_0/m_s$ and $\chi=-{\mit\Omega}_s\,(t'-t)+\theta$, whereas the Cartesian components of ${\bf k}$ and ${\bf v}$ are written $(k_\perp,\,0,\,k_\parallel)$ and $(v_\perp\,\cos\theta,\,v_\perp\,\sin\theta,\,v_\parallel)$, respectively. The equilibrium magnetic field takes the form ${\bf B}_0=(0,\,0,\,B_0)$. Equations (7.115) and (7.117) can be combined to give

$$k^{2} = {\rm i}\sum_s\frac{e_s^{2}}{\epsilon_0\,m_s}\int_{-\infty}^t\in... ...}{{\mit\Omega}_s}\right)J_m\left(\frac{k_\perp\,v_\perp}{{\mit\Omega}_s}\right)$$

$$\phantom{=} \times\exp\left\{{\rm i}\,\left[(n\,{\mit\Omega}_s+k_... ...allel\,v_\parallel-\omega)\,(t'-t)+(m-n)\,\theta\right]\right\}d^3{\bf v}\,dt'.$$ (7.118)

After some tedious analysis, the previous expression reduces to the so-called Harris dispersion relation (Harris 1961)

$$1+ \sum_s \frac{e_s^{2}}{k^{2}\,\epsilon_0\,m_s}\sum_{n=-\infty,\... ...k_\parallel\,\frac{\partial f_{0\,s}}{\partial v_\parallel}\right)d^3{\bf v}=0.$$ (7.119)

For Maxwellian distribution functions of the form (7.79), we can explicitly perform the velocity-space integrals in the Harris dispersion relation to give

$$1+\sum_s\frac{2\,{\mit\Pi}_s^{2}}{(k_\parallel^{2} + k_\perp^{2})... ...{\rm e}^{-\lambda_s}\!\sum_{n=-\infty,\infty}I_n(\lambda_s)\,Z(\xi_n)\right]=0,$$ (7.120)

where ${\mit\Pi}_s = (n_s\,e_s^{2}/\epsilon_0\,m_s)^{1/2}$, $v_s=(2\,T_s/m_s)^{1/2}$, $\lambda_s=k_\perp^{2}\,v_s^{2}/(2\,{\mit\Omega}_s^{2})$, and $\xi_n = (\omega-n\,{\mit\Omega}_s)/(k_\parallel\,v_s)$. Here, the $I_n$ are modified Bessel functions (Abramowitz and Stegun 1965), whereas $Z$ is a plasma dispersion function. (See Section 7.4.) In deriving the previous expression, use has been made of the identity (Watson 1995)

$$\sum_{n=-\infty,\infty} {\rm e}^{-\lambda_s}\,I_n(\lambda_s) = 1.$$ (7.121)

Consider electrostatic waves propagating parallel to the equilibrium magnetic field. In this case, $k_\perp\rightarrow 0$ and $\lambda_s\rightarrow 0$, so the dispersion relation (7.120) reduces to

$$1 + \sum_s\frac{2\,{\mit\Pi}_s^{2}}{(k_\parallel\,v_s)^2}\left[1+\xi_0\,Z(\xi_0)\right]=0,$$ (7.122)

with the eigenvector $(0,\,0,\,E_z)$. (Recall that ${\bf E}\propto {\bf k}$ for an electrostatic wave.) It can be seen that this expression is identical to the dispersion relation (7.97) for longitudinal plasma waves. Consider electrostatic waves propagating perpendicular to the equilibrium magnetic field. In this case, $k_\parallel\rightarrow 0$ and $\xi_n\rightarrow \infty$, so the dispersion relation (7.120) reduces to

$$1+\sum_s \frac{2\,{\mit\Pi}_s^{2}}{(k_\perp\,v_s)^2}\left[1-\xi_0... ... e}^{-\lambda_s}\!\sum_{n=-\infty,\infty}\frac{I_n(\lambda_s)}{\xi_n}\right]=0,$$ (7.123)

with the eigenvector $(E_x, \,0, \,0)$. Making use of the identity (7.121), as well as the fact that $I_{-n}(\lambda_s)=I_n(\lambda_s)$ (Abramowitz and Stegun 1965), the previous expression can be rearranged to give

$$1-\sum_s\frac{{\mit\Pi}_s^{2}}{\omega}\,\frac{{\rm e}^{-\lambda_s... ...}\sum_{n=-\infty,\infty}\frac{n^2\,I_n(\lambda_s)}{\omega-n\,{\mit\Omega}_s}=0.$$ (7.124)

It can be seen that this expression is identical to the dispersion relation (7.111) for Bernstein waves. Thus, we can now appreciate that plasma waves and Bernstein waves are merely different aspects of a more general type of electrostatic wave. This wave takes the form of a plasma wave when propagating parallel to the equilibrium magnetic field, of a Bernstein wave when propagating perpendicular to the magnetic field, and takes an intermediate form when propagating obliquely to the magnetic field.