Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 69 / 2 / c / Solution

For an isotropic weak electron-magnetohydrodynamic cascade, set $l_{\parallel }\sim l_{\perp }\sim l$. Part (b) gives $\delta B_l\propto l^{1/2}$. With the one-dimensional shell convention for the magnetic energy spectrum, fluctuations on scale $l\sim k^{-1}$ satisfy $\delta B_l^2\sim k{E}_B(k)$ up to normalization constants. ThereforeThis is the shell-integrated spectrum. The three-dimensional spectral tensor coefficient has two additional powers of $k^{-1}$ and scales as $H(k)\propto k^{-4}$. These exponents apply within the assumed local, weak cascade range.