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Critical balance sets ${\tau }_{\mathrm{nl}}\sim {\tau }_w$, hence $l_{\parallel }\sim l_{\perp }/b_l$. A local strong cascade with constant energy flux has $ϵ\sim v_A^3{d}_i{b}_l^3/l_{\perp }^2$, giving the displayed exponents and perpendicular magnetic energy spectrum $E_B(k_{\perp })\propto k_{\perp }^{-7/3}$. The anisotropy becomes stronger towards small perpendicular scales.

Invert the two-dimensional Fourier transform in the long-path projection of a magnetic correlation, then perform the angular integration in polar coordinates. The angular integral is $2\pi J_0(kr)$, a Bessel function of the first kind, giving the displayed Hankel transform with the stated Fourier-Hankel normalization for an isotropic spectrum. The result recovers the scalar tensor coefficient $H$, whose shell-integrated magnetic energy spectrum has an additional factor $k^2$.

Invert the two-dimensional Fourier transform from part (c):In polar coordinates, the angular integral is $2\pi J_0(ks)$, where $J_0$ is a Bessel function of the first kind. The Fourier-Hankel normalization for an isotropic spectrum therefore yieldsThis Hankel inversion of a rotation-measure correlation recovers the scalar spectral tensor coefficient. To obtain the shell-integrated magnetic energy spectrum, take the trace of the three-dimensional tensor:With magnetic energy density $B^2/(8\pi )$ in Gaussian units, the one-dimensional convention ${\int }_0^{\infty }{E}_B(k)dk=\mathbb{E}|\mathbf{B}{|}^2/(8\pi )$ gives $\boxed{E_B(k)=k^{2}H(k)/(8{\pi }^3)}$. A spectrum normalized instead to $\mathbb{E}|\mathbf{B}{|}^2/2$ has coefficient $k^{2}H(k)/(2{\pi }^2)$; the physical normalization must be stated.

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.

Critical balance makes the nonlinear interaction time comparable to the wave period:An order-one interaction transfers energy in ${\tau }_l\sim {\tau }_{\mathrm{nl}}$, so constant energy flux gives $ϵ\sim v_A^3{d}_i{b}_l^3/l_{\perp }^2$. Thus the critically balanced electron-magnetohydrodynamic cascade hasFor the corresponding perpendicular magnetic energy spectrum, $\delta B_l^2\sim k_{\perp }{E}_B(k_{\perp })$ gives $E_B(k_{\perp })\propto k_{\perp }^{-7/3}$. The increasing ratio $l_{\parallel }/l_{\perp }\propto l_{\perp }^{-2/3}$ describes progressively more anisotropic fluctuations at smaller perpendicular scales, within the range in which the model and critical balance assumptions hold.

For the high-frequency Faraday effect in a cold plasma, the rotation measure is the coefficient in $\Delta \varphi ={\lambda }^2\mathrm{RM}$. It is a density-weighted line integral of the line-of-sight magnetic field; the coefficient $a_0$ depends on the units. A sign convention for the line of sight fixes its sign. Correlations across an extended source can constrain a foreground magnetic energy spectrum, subject to statistical assumptions.

For $b_l{l}_{\parallel }/l_{\perp }\ll 1$, the weak wave cascade estimate gives ${\tau }_{\mathrm{cas}}\sim l_{\perp }^3/(v_A{d}_i{b}_l^2{l}_{\parallel })$. Constant energy flux $ϵ\sim v_A^2{b}_l^2/{\tau }_{\mathrm{cas}}$ yields the displayed amplitude. An additionally isotropic local cascade has $b_l\propto l^{1/2}$ and shell-integrated magnetic energy spectrum $E_B(k)\propto k^{-2}$. These statements are phenomenological consequences of the chosen locality and decorrelation assumptions.