Waves in Magnetized Plasmas

Consider small amplitude waves propagating through a plasma placed in a uniform magnetic field, ${\bf B}_0\equiv B_0\,{\bf e}_z$. 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

$$\frac{\partial f_1}{\partial t} + {\bf v}\cdot\nabla f_1 + \frac{... ...\nabla_v f_1 = -\frac{e}{m}\, ({\bf E} + {\bf v}\times{\bf B})\cdot\nabla_v f_0$$ (7.53)

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

$$({\bf v}\times{\bf B}_0)\cdot\nabla_v f_0 = 0.$$ (7.54)

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

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

$$\frac{d{\bf r}}{dt} = {\bf v},$$ (7.55)

$$\frac{d{\bf v}}{dt} = \frac{e}{m}\,({\bf v}\times{\bf B}_0).$$ (7.56)

It follows that Equation (7.53) can be written

$$\frac{D f_1}{Dt} = -\frac{e}{m}\, ({\bf E} + {\bf v}\times{\bf B})\cdot\nabla_v f_0,$$ (7.57)

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 Equation (7.57) can be written

$$f_1({\bf r}, {\bf v}, t) =-\frac{e}{m}\int_{-\infty}^t \left[{\bf... ...+ {\bf v}'\times {\bf B}({\bf r}', t')\right] \cdot\nabla_v f_0({\bf v}')\,dt',$$ (7.58)

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

It should be noted that the previous 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, whereas in a general field configuration it is difficult to find a closed form for the particle trajectories that is sufficiently simple to allow further progress to be made.

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

$${\bf v} = \left(v_\perp\cos\theta,\, v_\perp\sin\theta,\, v_\parallel\right).$$ (7.59)

It follows that

$${\bf v}' = \left( v_\perp\cos\left[{\mit\Omega}\,(t-t')+\theta\,\... ...\, v_\perp\sin\left[{\mit\Omega}\,(t-t')+\theta\,\right],\, v_\parallel\right),$$ (7.60)

where ${\mit\Omega} = e\,B_0/m$ is the gyofrequency. The previous expression can be integrated in time to give

$$x'-x = -\frac{v_\perp}{{\mit\Omega}}\,\left(\sin\left[{\mit\Omega}\,(t-t') +\theta\,\right] -\sin\theta\right),$$ (7.61)

$$y'-y = \frac{v_\perp}{{\mit\Omega}}\,\left(\cos\left[{\mit\Omega}\,(t-t') +\theta\,\right] -\cos\theta\right),$$ (7.62)

$$z'-z = v_\parallel\,(t'-t).$$ (7.63)

Note that both $v_\perp$ and $v_\parallel$ 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 $v_\parallel$. Given that $v_\perp = (v_x'^{2} + v_y'^{2})^{1/2}$, we can write

$$\frac{\partial f_0}{\partial v_x'} = \frac{\partial v_\perp}{\partial v_x'} \frac{\partial f_0}{\par... ...eft[{\mit\Omega}\,(t'-t)+\theta\,\right] \frac{\partial f_0}{\partial v_\perp},$$ (7.64)

$$\frac{\partial f_0}{\partial v_y'} = \frac{\partial v_\perp}{\partial v_y'} \frac{\partial f_0}{\par... ...eft[{\mit\Omega}\,(t'-t)+\theta\,\right] \frac{\partial f_0}{\partial v_\perp},$$ (7.65)

$$\frac{\partial f_0}{\partial v_z'} = \frac{\partial f_0}{\partial v_\parallel}.$$ (7.66)

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 (7.58) yields

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

$$\phantom{=}\left.+(E_z + v_x'\,B_y - v_y'\,B_x)\, \frac{\partial ... ...\left[\,{\rm i}\,\{{\bf k}\cdot ({\bf r}' -{\bf r})-\omega\,(t'-t)\}\right]dt'.$$ (7.67)

Making use of Equations (7.60)–(7.66), as well as the identity (Abramowitz and Stegun 1965)

$${\rm e}^{\,{\rm i}\,a\,\sin x} \equiv \sum_{n=-\infty,\infty} J_n(a)\,{\rm e}^{\,{\rm i}\,n\,x},$$ (7.68)

where the $J_n$ are Bessel functions (Abramowitz and Stegun 1965), Equation (7.67) gives

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

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

$$\phantom{=}\times \exp\left\{\,{\rm i}\left[(n\,{\mit\Omega} + k_\parallel\,v_\parallel-\omega) \,(t'-t) + (m-n)\,\theta\,\right]\,\right\}dt',$$ (7.69)

where

$$\chi = {\mit\Omega}\,(t-t') + \theta.$$ (7.70)

Maxwell's equations yield

$${\bf k}\times{\bf E} = \omega\,{\bf B},$$ (7.71)

$${\bf k}\times{\bf B} = -{\rm i}\,\mu_0\,{\bf j} - \frac{\omega}{c^2}\,{\bf E} =-\frac{\omega}{c^2}\,{\bf K}\cdot{\bf E},$$ (7.72)

where ${\bf j}$ is the perturbed current, and ${\bf K}$ is the dielectric permittivity tensor introduced in Section 5.2. It follows that

$${\bf K}\cdot{\bf E} ={\bf E} + \frac{{\rm i}}{\omega\,\epsilon_0}... ...frac{{\rm i}}{\omega\,\epsilon_0} \sum_s e_s\int {\bf v}\,f_{1\,s}\,d^3{\bf v},$$ (7.73)

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

After a great deal of rather tedious analysis, Equations (7.69) and (7.73) reduce to the following expression for the dielectric permittivity tensor (Harris 1970: Cairns 1985):

$$K_{\alpha\beta} = \delta_{\alpha\beta} + \sum_s \frac{e_s^{2}}{\o... ...\beta}}{\omega - k_\parallel \,v_\parallel - n\,{\mit\Omega}_s} \,\,d^3{\bf v},$$ (7.74)

where

$$S_{\alpha\beta} = \left( \begin{array}{ccc} v_\perp\,(n\,J_n/... ...lel\,J_n\,J_n'\,U, & v_\parallel\,J_n^{2}\,W\end{array}\right),$$ (7.75)

and

$$U =(\omega-k_\parallel\,v_\parallel)\,\frac{\partial f_{0\,s}}{\par... ..._\perp} + k_\parallel\,v_\perp\,\frac{\partial f_{0\,s}}{\partial v_\parallel},$$ (7.76)

$$W =\frac{n\,{\mit\Omega}_s\,v_\parallel}{v_\perp} \,\frac{\partial ... ...+ (\omega -n\,{\mit\Omega}_s)\,\frac{\partial f_{0\,s}} {\partial v_\parallel},$$ (7.77)

$$a_s = \frac{k_\perp\,v_\perp}{{\mit\Omega}_s}.$$ (7.78)

The argument of the Bessel functions is $a_s$. In the previous formulae, $'$ denotes differentiation with respect to argument, and ${\mit\Omega}_s = e_s\,B_0/m_s$.

The warm-plasma dielectric tensor, (7.74), can be used to investigate the properties of waves in just the same manner as the cold-plasma dielectric tensor, (5.37), was employed in Chapter 5. Note that our expression for the dielectric tensor involves singular integrals of a type similar to those encountered in Section 7.2. 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, of the form

$$f_{0\,s} = \frac{n_s}{(2\pi\,T_s/m_s)^{3/2}}\exp\left[-\frac{m_s\,(v_\perp^{2}+v_\parallel^{2})}{2\,T_s}\right],$$ (7.79)

we can explicitly perform the velocity-space integral in Equation (7.74), making use of the identities (Watson 1995)

$$\int_0^\infty x\,J_n^{2}(s\,x)\,{\rm e}^{-x^2}\,dx =\frac{1}{2} \,{\rm e}^{-s^2/2}\,I_n(s^2/2),$$ (7.80)

$$\int_0^\infty x^3\left[J_n'(s\,x)\right]^2{\rm e}^{-x^2}\,dx =\frac{1}{4} \,{\rm e}^{-s^2/2}\left[2\,n^2\,I_n(s^2/2)/s^2 + s^2\,I_n(s^2/2) -s^2\,I_n'(s^2/2)\right],$$ (7.81)

where $I_n$ is a modified Bessel function (Abramowitz and Stegun 1965). We obtain

$$K_{\alpha\beta} = \delta_{\alpha\beta} + \sum_s\frac{{\mit\Pi}_s^... ...\rm e}^{-\lambda_s}}{k_\parallel\,v_s} \sum_{n=-\infty,\infty} T_{\alpha\beta},$$ (7.82)

where ${\mit\Pi}_s = (n_s\,e_s^{2}/\epsilon_0\,m_s)^{1/2}$, $v_s=(2\,T_s/m_s)^{1/2}$, and (Harris 1970; Cairns 1985)

$$T_{\alpha\beta} = \left( \begin{array}{ccc} n^2\,I_n\,Z/\lambda_s... ...,\lambda_s^{1/2}\,(I_n'-I_n)\,Z'/2^{1/2}, & -I_n\,Z'\,\xi_n \end{array}\right).$$ (7.83)

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

$$\lambda_s = \frac{k_\perp^{2}\,v_s^{2}}{2\,{\mit\Omega}_s^{2}},$$ (7.84)

whereas $Z$ and $Z'$ represent the plasma dispersion function and its derivative, both functions being evaluated with the argument

$$\xi_n = \frac{\omega - n\,{\mit\Omega}_s}{k_\parallel\,v_s} .$$ (7.85)

Let us consider the cold-plasma limit, $v_s\rightarrow 0$. It follows from Equations (7.84) and (7.85) that this limit corresponds to $\lambda_s\rightarrow 0$ and $\xi_n\rightarrow \infty$. According to Equation (7.47),

$$Z(\xi_n) \rightarrow -\frac{1}{\xi_n},$$ (7.86)

$$Z'(\xi_n) \rightarrow \frac{1}{\xi_n^{2}}$$ (7.87)

as $\xi_n\rightarrow \infty$. Moreover, (Abramowitz and Stegun 1965)

$$I_n(\lambda_s) \rightarrow \left(\frac{\lambda_s}{2}\right)^{\vert n\vert}$$ (7.88)

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

$$K_{11} = K_{22} = 1-\frac{1}{2}\sum_s\frac{{\mit\Pi}_s^{2}}{\omega^2} \l... ...\omega}{\omega-{\mit\Omega}_s} + \frac{\omega}{\omega + {\mit\Omega}_s}\right),$$ (7.89)

$$K_{12} = -K_{21} = -\frac{{\rm i}}{2} \sum_s\frac{{\mit\Pi}_s^{2}}{\omeg... ...\omega}{\omega-{\mit\Omega}_s} - \frac{\omega}{\omega + {\mit\Omega}_s}\right),$$ (7.90)

$$K_{33} = 1-\sum_s\frac{{\mit\Pi}_s^{2}}{\omega^2},$$ (7.91)

and $K_{13} = K_{31} = K_{23}=K_{32}=0$. It is easily seen, from Section 5.3, that the previous expressions are identical to those found 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 derived in Chapter 5.