## FANDOM

148 Pages

Let us consider an incompressible, turbulent medium embedded in a uniform magnetic field $B$. We denote the Alfven velocity by $b \approx B/\sqrt{\rho}$, where $\rho$ is the mass density. We are interested in the regime where the Larmor radius is smaller than the size of the eddies (in the case where it's the other way around, we simply get the Kolmogorov turbulence). In this regime energy diffuses from one eddy to another with the timescale

$\tau \approx \frac{m c}{q B} \frac{b^2}{\delta v^2} \propto \frac{b}{\delta v^2}$

where $m$ is the mass of the particles, $q$ is their charge, $c$ is the speed of light and $\delta v$ is the amplitude of the turbulent velocity. Suppose that energy per unit mass dissipates at a rate $\varepsilon$. In stead state

$\delta v^2 \approx \tau \varepsilon \Rightarrow \delta v^4 \propto b \varepsilon$

The velocity spectrum is given by $E \approx \delta v^2 / k$, and by dimensional analysis

$E \approx \left(b \varepsilon\right)^{1/2} k^{-3/2}$