Derivation of dispersion relation of electromagnetic waves in a magnetized hot plasma

Question

I look to solve analytically the Vlasov-Maxwell equations for a magnetized hot plasma $$\frac{\partial f_{s1}}{\partial t}+\vec{v}\cdot\frac{\partial f_{s1}}{\partial \vec{r}}+\frac{q_s}{m_s}\Big(\vec{v}\wedge\vec{B}_0 \Big)\cdot\frac{\partial f_{s1}}{\partial \vec{v}} =-\frac{q_s}{m_s}\Big(\vec{E}_1+\vec{v}\wedge\vec{B}_1 \Big)\cdot \frac{\partial f_{s0}}{\partial \vec{v}}$$ $$\vec{\nabla}\cdot\vec{E}_1=\dfrac{1}{\varepsilon_0}\displaystyle\sum_s q_s \int f_{s1}\,\mathrm{d^3}v$$ $$\vec{\nabla}\cdot\vec{B}_1=0$$ $$\vec{\nabla}\wedge\vec{E}_1=-\frac{\partial \vec{B}_1}{\partial t}$$ $$ \vec{\nabla}\wedge\vec{B}_1=\mu_0 \displaystyle\sum_s q_s \int \vec{v}\; f_{s1}\,\mathrm{d^3}v+\frac{1}{c^2}\frac{\partial \vec{E}_1}{\partial t}$$

With $f_{s0}(\vec{v})\equiv f_{s0}(v_\perp,v_z)=$ the zero order distribution function in cylindrical geometry.

$(f_{s1}, \vec{E}_1, \vec{B}_1)=$ the perturbed parameters of order 1 , $B_{0}=$ the constant magnetic field.

In order to calculate the perturbed distribution function $f_{s1}$, resolution of Vlasov equation is done by the method of characteristics with a cylindrical geometry which involves very difficult calculations. (see Swanson p.91-95)

Instead of this method, other works use the development of $f_{s1}$ on the spherical harmonics $$f_{s1}= \displaystyle\sum_l\sum_m f_l^m(v)\;Y_l^m (\theta,\varphi)$$ then by projection of the $Y_l^m$ on Vlasov equation and the relation of orthogonality of $Y_l^m$ they form an infinit system of coupled equations in $f_l^m$.

So I wanted to apply this method of spherical harmonics in this problem but I couldn't solve the system of coupled equations in $f_l^m$. I therefore concluded that spherical harmonics cannot be applied.

Please, do you know of any other alternative methods to calculate the perturbed distribution function $f_{s1}$ in the same cylindrical geometry?

Answer 1

I therefore concluded that spherical harmonics cannot be applied.

I think this is a premature assessment. Dum et al. [1980] provides a nice derivation of the dispersion relation using spherical harmonics. Let's first start with some background: $$ \begin{align} Y_{l}^{m}\left( \theta, \phi \right) & = N_{l}^{m} \ P_{l}^{m}\left( \cos{\theta} \right) \ e^{i \ m \ \phi} \tag{0a} \\ N_{l}^{m} & = \left[ \frac{ \left( 2 l + 1 \right) \left( l - m \right)! }{ 4 \ pi \left( l + m \right)! } \right]^{1/2} \tag{0b} \\ Y_{l}^{-m} & = \left( -1 \right)^{m} \ \left( Y_{l}^{m} \right)^{*} \tag{0c} \\ P_{l}^{m}\left( x \right) & = \left( -1 \right)^{m} \ \left( 1 - x^{2} \right)^{m/2} \ \frac{ 1 }{ 2^{l} \ l! } \frac{ d^{l + m} }{ dx^{l + m} } \left( x^{2} - 1 \right)^{l} \tag{0d} \\ P_{l}\left( \hat{\mathbf{k}} \cdot \hat{\mathbf{v}} \right) & = \frac{ 4 \pi }{ 2 l + 1 } \sum_{m = -l}^{l} \ Y_{l}^{m}\left( \hat{\mathbf{k}} \right) \ \left( Y_{l}^{m} \right)^{*}\left( \hat{\mathbf{v}} \right) \tag{0e} \\ Q_{l}\left( z \right) & = \frac{ 1 }{ 2 } \int_{-1}^{+1} dt \ P_{l}\left( t \right) \frac{ 1 }{ z - t } = \left( -1 \right)^{l + 1} Q_{l}\left( -z \right) \tag{0f} \\ P_{l}\left( z \right) & = \frac{ 1 }{ 2^{l} \ l! } \ \frac{ d^{l} }{ dz^{l} } \left( z^{2} - 1 \right)^{l} \tag{0g} \\ W_{l - 1}\left( z \right) & = \sum_{m = 1}^{l} \ \frac{ 1 }{ m } \ P_{m - 1}\left( z \right) \ P_{l - m}\left( z \right) \tag{0h} \\ Q_{0}\left( z \right) & = \frac{ 1 }{ 2 } \ln{ \frac{ z + 1 }{ z - 1 } } \tag{0i} \\ Q_{l}\left( z \right) & = P_{l}\left( z \right) \ Q_{0}\left( z \right) - W_{l - 1}\left( z \right) \tag{0j} \end{align} $$

where the $P_{l}\left( z \right)$ are Legendre polynomials and $\mu = \hat{\mathbf{k}} \cdot \hat{\mathbf{v}}$ is the cosine of the wave normal angle relative to particle velocity.

We can now write the dispersion relation as: $$ \begin{align} \epsilon_{j}\left( \mathbf{k}, \omega \right) & = \sum_{l,m} Y_{l}^{m}\left( \hat{\mathbf{k}} \right) \left( \frac{ \omega_{j} }{ k } \right)^{2} \ \int_{0}^{\infty} \ dv \ 4 \pi \ v^{2} \Biggl\{ \frac{ 1 }{ 2 } \int_{-1}^{+1} d\mu \\ & \frac{ 1 }{ \hat{\omega} - \mu } \ \frac{ k }{ k \ v } \left[ \mu \frac{ \partial }{ \partial v } + \left( 1 - \mu^{2} \right) \frac{ 1 }{ v } \frac{ \partial }{ \partial \mu } \right] \ P_{l}\left( \mu \right) \ f_{l}^{m}\left( v \right) \Biggl\} \tag{1} \end{align} $$

where $\hat{\omega} = \tfrac{ \omega }{ k \ v }$, and the $f_{l}^{m}\left( v \right)$ is expressed as:

$$ \sum_{l,m} Y_{l}^{m}\left( \hat{\mathbf{k}} \right) \ f_{l}^{m}\left( \frac{ \omega }{ k } \right) = f\left( \frac{ \omega }{ k } \ \hat{\mathbf{k}} \right) \tag{2} $$

This is specific to an unmagnetized plasma, yes, but it allows for electromagnetic waves (so long as they propagate along the magnetic field). This can be generalized to a magnetized plasma with more work.

You can also look at Vinas and Gurgiolo [2009] for a discussion of generating velocity moments from spherical harmonic expansions of the velocity distribution function.