Waves in Magnetized Plasmas

Consider waves propagating through a plasma placed in a uniform magnetic field, ${\bf B}_0$. Let us take the perturbed magnetic field into account in our calculations, in order to allow for electromagnetic, as well as electrostatic, waves. The linearized Vlasov equation takes the form

$$\frac{\partial f_1}{\partial t} + {\bf v}\!\cdot\!\nabla f_1... ...{e}{m}\, ({\bf E} + {\bf v}\times{\bf B})\!\cdot\!\nabla_v f_0$$ (1036)

for both ions and electrons, where ${\bf E}$ and ${\bf B}$ are the perturbed electric and magnetic fields, respectively. Likewise, $f_1$ is the perturbed distribution function, and $f_0$ the equilibrium distribution function.

In order to have an equilibrium state at all, we require that

$$({\bf v}\times{\bf B}_0)\!\cdot\!\nabla_v f_0 = 0.$$ (1037)

Writing the velocity, ${\bf v}$, in cylindrical polar coordinates, $(v_\perp, \theta, v_z)$, aligned with the equilibrium magnetic field, the above expression can easily be shown to imply that $\partial f_0/\partial \theta =0$: i.e., $f_0$ is a function only of $v_\perp$ and $v_z$.

Let the trajectory of a particle be ${\bf r}(t)$, ${\bf v}(t)$. In the unperturbed state

$$\frac{d{\bf r}}{dt} = {\bf v},$$ (1038)

$$\frac{d{\bf v}}{dt} = \frac{e}{m}\,({\bf v}\times{\bf B}_0).$$ (1039)

It follows that Eq. (1036) can be written

$$\frac{D f_1}{Dt} = -\frac{e}{m}\, ({\bf E} + {\bf v}\times{\bf B})\!\cdot\!\nabla_v f_0,$$ (1040)

where $D f_1/Dt$ is the total rate of change of $f_1$, following the unperturbed trajectories. Under the assumption that $f_1$ vanishes as $t\rightarrow -\infty$, the solution to Eq. (1040) can be written

$$f_1({\bf r}, {\bf v}, t) =-\frac{e}{m}\int_{-\infty}^t \left... ... B}({\bf r}', t')\right]\! \cdot\!\nabla_v f_0({\bf v}')\,dt',$$ (1041)

where $({\bf r}'$, ${\bf v}')$ is the unperturbed trajectory which passes through the point $({\bf r}$, ${\bf v})$ when $t'=t$.

It should be noted that the above method of solution is valid for any set of equilibrium electromagnetic fields, not just a uniform magnetic field. However, in a uniform magnetic field the unperturbed trajectories are merely helices, whilst in a general field configuration it is difficult to find a closed form for the particle trajectories which is sufficiently simple to allow further progress to be made.

Let us write the velocity in terms of its Cartesian components:

$${\bf v} = (v_\perp\cos\theta, v_\perp\sin\theta, v_z).$$ (1042)

It follows that

$${\bf v}' = \left( v_\perp\cos\!\left[{\Omega}\,(t-t')+\theta... ...\perp\sin\!\left[{\Omega}\,(t-t')+\theta\,\right], v_z\right),$$ (1043)

where ${\Omega} = e\,B_0/m$ is the cyclotron frequency. The above expression can be integrated to give

$$x'-x = -\frac{v_\perp}{{\Omega}}\,\left(\,\sin\!\left[{\Omega}\,(t-t') +\theta\,\right] -\sin\theta\right),$$ (1044)

$$y'-y = \frac{v_\perp}{{\Omega}}\,\left(\,\cos\!\left[{\Omega}\,(t-t') +\theta\,\right] -\cos\theta\right),$$ (1045)

$$z'-z = v_z\,(t'-t).$$ (1046)

Note that both $v_\perp$ and $v_z$ are constants of the motion. This implies that $f_0({\bf v}') = f_0({\bf v})$, because $f_0$ is only a function of $v_\perp$ and $v_z$. Since $v_\perp = (v_x'^{~2} + v_y'^{~2})^{1/2}$, we can write

$$\frac{\partial f_0}{\partial v_x'} = \frac{\partial v_\perp}{\partial v_x'} \frac{\partial f_0}{\parti... ...os\left[{\Omega}\,(t'-t)+\theta\,\right] \frac{\partial f_0}{\partial v_\perp},$$ (1047)

$$\frac{\partial f_0}{\partial v_y'} = \frac{\partial v_\perp}{\partial v_y'} \frac{\partial f_0}{\parti... ...in\left[{\Omega}\,(t'-t)+\theta\,\right] \frac{\partial f_0}{\partial v_\perp},$$ (1048)

$$\frac{\partial f_0}{\partial v_z'} = \frac{\partial f_0}{\partial v_z}.$$ (1049)

Let us assume an $\exp[\,{\rm i}\,({\bf k}\!\cdot\!{\bf r} - \omega\,t)]$ dependence of all perturbed quantities, with ${\bf k}$ lying in the $x$-$z$ plane. Equation (1041) yields

$$f_1 = -\frac{e}{m}\int_{-\infty}^t \left[ (E_x + v_y'\,B_z - v_z'\,B_y)... ...} +(E_y + v_z'\,B_x - v_x'\,B_z)\, \frac{\partial f_0}{\partial v_{y}'} \right.$$

$$\left.+(E_z + v_x'\,B_y - v_y'\,B_x)\, \frac{\partial f_0}{\parti... ...\,{\rm i}\,\{{\bf k}\!\cdot\! ({\bf r}' -{\bf r})-\omega\,(t'-t)\}\right]\,dt'.$$ (1050)

Making use of Eqs. (1043)-(1049), and the identity

$${\rm e}^{\,{\rm i}\,a\,\sin x} \equiv \sum_{n=-\infty}^{\infty} J_n(a)\,{\rm e}^{\,{\rm i}\,n\,x},$$ (1051)

Eq. (1050) gives

$$f_1 = -\frac{e}{m}\int_{-\infty}^t \left[ (E_x - v_z\,B_y)\,\cos\chi\,\... ...rp} + (E_y + v_z\,B_x)\,\sin\chi\,\frac{\partial f_0}{\partial v_\perp} \right.$$

$$\left.+(E_z+ v_\perp\,B_y\,\cos\chi - v_\perp\,B_x\,\sin\chi)\,\f... ...,v_\perp} {{\Omega}}\right)J_m\!\left(\frac{k_\perp\,v_\perp} {{\Omega}}\right)$$

$$\times \exp\left\{\,{\rm i}\left[(n\,{\Omega} + k_z\,v_z-\omega) \,(t'-t) + (m-n)\,\theta\,\right]\,\right\}\,dt',$$ (1052)

where

$$\chi = {\Omega}\,(t-t') + \theta.$$ (1053)

Maxwell's equations yield

$${\bf k}\times{\bf E} = \omega\,{\bf B},$$ (1054)

$${\bf k}\times{\bf B} = -{\rm i}\,\mu_0\,{\bf j} - \frac{\omega}{c^2}\,{\bf E} =-\frac{\omega}{c^2}\,{\bf K}\!\cdot\!{\bf E},$$ (1055)

where ${\bf j}$ is the perturbed current, and ${\bf K}$ is the dielectric permittivity tensor introduced in Sect. 4.2. It follows that

$${\bf K}\!\cdot\!{\bf E} ={\bf E} + \frac{{\rm i}}{\omega\,\e... ...ega\,\epsilon_0} \sum_s e_s\int {\bf v}\,f_{1\,s}\,d^3{\bf v},$$ (1056)

where $f_{1\,s}$ is the species-$s$ perturbed distribution function.

After a great deal of rather tedious analysis, Eqs. (1052) and (1056) reduce to the following expression for the dielectric permittivity tensor:

$$K_{ij} = \delta_{ij} + \sum_s \frac{e_s^{~2}}{\omega^2\,\eps... ...ac{S_{ij}}{\omega - k_z \,v_z - n\,{\Omega}_s} \,\,d^3{\bf v},$$ (1057)

where

$$S_{ij} = \left( \begin{array}{ccc} v_\perp\,(n\,J_n/a_s)^2\,... ...m i}\,v_z\,J_n\,J_n'\,U & v_z\,J_n^{~2}\,W\end{array} \right),$$ (1058)

and

$$U = (\omega-k_z\,v_z)\,\frac{\partial f_{0\,s}}{\partial v_\perp} + k_z\,v_\perp\,\frac{\partial f_{0\,s}}{\partial v_z},$$ (1059)

$$W = \frac{n\,{\Omega}_s\,v_z}{v_\perp} \,\frac{\partial f_{0\,s}} {\p... ...al v_\perp} + (\omega -n\,{\Omega}_s)\,\frac{\partial f_{0\,s}} {\partial v_z},$$ (1060)

$$a_s = \frac{k_\perp\,v_\perp}{{\Omega}_s}.$$ (1061)

The argument of the Bessel functions is $a_s$. In the above, $'$ denotes differentiation with respect to argument.

The dielectric tensor (1057) can be used to investigate the properties of waves in just the same manner as the cold plasma dielectric tensor (485) was used in Sect. 4. Note that our expression for the dielectric tensor involves singular integrals of a type similar to those encountered in Sect. 6.2. In principle, this means that we ought to treat the problem as an initial value problem. Fortunately, we can use the insights gained in our investigation of the simpler unmagnetized electrostatic wave problem to recognize that the appropriate way to treat the singular integrals is to evaluate them as written for ${\rm Im}(\omega)>0$, and by analytic continuation for ${\rm Im}(\omega)\leq 0$.

For Maxwellian distribution functions, we can explicitly perform the velocity space integral in Eq. (1057), making use of the identity

$$\int_0^\infty x\,J_n^{~2}(s\,x)\,{\rm e}^{-x^2}\,dx= \frac{{\rm e}^{-s^2/2}}{2} \,I_n(s^2/2),$$ (1062)

where $I_n$ is a modified Bessel function. We obtain

$$K_{ij} = \delta_{ij} + \sum_s\frac{\omega_{p\,s}^{~2}}{\omeg... ...rac{{\rm e}^{-\lambda_s}}{k_z} \sum_{n=-\infty}^\infty T_{ij},$$ (1063)

where

$$T_{ij} = \left( \begin{array}{ccc} {\scriptstyle n^2\,I_n\,Z... ...'/2^{1/2}} &{\scriptstyle -I_n\,Z'\,\xi_n} \end{array}\right).$$ (1064)

Here, $\lambda_s$, which is the argument of the Bessel functions, is written

$$\lambda_s = \frac{T_s\,k_\perp^{~2}}{m_s\,{\Omega}_s^{~2}},$$ (1065)

whilst $Z$ and $Z'$ represent the plasma dispersion function and its derivative, both with argument

$$\xi_n = \frac{\omega - n\,{\Omega}_s}{k_z} \left(\frac{m_s}{2\,T_s}\right)^{1/2}.$$ (1066)

Let us consider the cold plasma limit, $T_s\rightarrow 0$. It follows from Eqs. (1065) and (1066) that this limit corresponds to $\lambda_s\rightarrow 0$ and $\xi_n\rightarrow \infty$. From Eq. (1030),

$$Z(\xi_n) \rightarrow -\frac{1}{\xi_n},$$ (1067)

$$Z'(\xi_n) \rightarrow \frac{1}{\xi_n^{~2}}$$ (1068)

as $\xi_n\rightarrow \infty$. Moreover,

$$I_n(\lambda_s) \rightarrow \left(\frac{\lambda_s}{2}\right)^{\vert n\vert}$$ (1069)

as $\lambda_s\rightarrow 0$. It can be demonstrated that the only non-zero contributions to $K_{ij}$, in this limit, come from $n=0$ and $n=\pm 1$. In fact,

$$K_{11} = K_{22} = 1-\frac{1}{2}\sum_s\frac{\omega_{p\,s}^{~2}}{\omega^2} \... ...( \frac{\omega}{\omega-{\Omega}_s} + \frac{\omega}{\omega + {\Omega}_s}\right),$$ (1070)

$$K_{12} = -K_{21} = -\frac{{\rm i}}{2} \sum_s\frac{\omega_{p\,s}^{~2}}{\ome... ...( \frac{\omega}{\omega-{\Omega}_s} - \frac{\omega}{\omega + {\Omega}_s}\right),$$ (1071)

$$K_{33} = 1-\sum_s\frac{\omega_{p\,s}^{~2}}{\omega^2},$$ (1072)

and $K_{13} = K_{31} = K_{23}=K_{32}=0$. It is easily seen, from Sect. 4.3, that the above expressions are identical to those we obtained using the cold-plasma fluid equations. Thus, in the zero temperature limit, the kinetic dispersion relation obtained in this section reverts to the fluid dispersion relation obtained in Sect. 4.