## FANDOM

176 Pages

Let us consider a plasma in a strong uniform magnetic field $B_0$ and a weak random field $B_0 \gg \delta B$. At each moment the particle rotates with a perturbed Larmor radius. The unperturbed component of the radius is, assuming the particle is ultra - relativistic

$r_l \approx \frac{\varepsilon}{q B}$

where $\varepsilon$ is the energy of the particle and $q$ is its charge. The amplitude of the perturbation to the Larmor radius is

$\delta r_l \approx \frac{\varepsilon}{q B_0^2} \delta B$

How long will it take the net displacement to be comparable to the unperturbed Larmor radius $r_l$ ?

$c \tau \left(\frac{\delta r_l}{r_l} \right)^2 \approx r_l \Rightarrow \tau \approx \frac{r_l}{c} \left(\frac{B}{\delta B}\right)^2$

In a realistic situation, the random field has a certain spectrum $\delta B^2 \propto k^{m-1}$, where $m<1$ is a constant. Magnetic fields at wavelengths larger than $r_l$ vary the orbit in an almost coherent way, so their contribution to scattering is small. Magnetic fields on wavelengths much smaller than $r_l$ are very weak, so their contribution to scattering is also small. Hence, the range of wavelengths that contributes the most to scattering is the same order of magnitude as $r_l$. Substituting $k \approx r_l^{-1}$ yields $\tau \propto r_l^m$. The diffusion coefficient is

$D \approx c^2 \tau \propto r_l^m \propto \varepsilon^m$.

For magnetic fields in a Kolmogorov spectrum

$\delta B^2 \propto \delta v^2 \propto k^{-2/3} \Rightarrow m=\frac{1}{3}$.