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Statistical isotropy and statistical reflection symmetry allow the spectral tensor to contain only ${\delta }_{ij}$ and $k_i{k}_j$:The solenoidal magnetic-field constraint is essential here. Taking its Fourier transform gives $k_i{\hat{B}}_i=0$, hence $k_i{\hat{C}}_{ij}=0$ and $A+D{k}^2=0$. Thus the spectral tensor is a scalar multiple of the transverse projector of a vector field. Its trace is $2A$, so the prescribed normalization givesThis is the solenoidal isotropic spectral tensor. Without the solenoidal condition, isotropy and reflection symmetry would leave two scalar functions. Without reflection symmetry, an antisymmetric term proportional to $i{ϵ}_{ijm}{k}_m$ could additionally encode magnetic helicity.

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.

A random field is a family of random variables indexed by points of a spatial domain, or more generally by a multi-dimensional parameter set. Its componentwise covariance function and spectral tensor describe second-order dependence. A Gaussian random field has jointly Gaussian finite-dimensional distributions; a general random field need not be Gaussian.

For a statistically homogeneous, isotropic, reflection-symmetric solenoidal vector random field in three dimensions, the nonzero-wavevector spectral tensor is a scalar multiple of the transverse projector of a vector field. Isotropy gives $A(k){\delta }_{ij}+D(k)k_i{k}_j$, while zero divergence gives $A+D{k}^2=0$. Its trace is $2A$, giving the displayed normalization. Without reflection symmetry an additional antisymmetric, helical term can occur.

Statistical reflection symmetry means that the joint distributions of a random field are unchanged by a simultaneous spatial reflection, with each field component transformed according to its vector or pseudovector character. For an isotropic magnetic field, it excludes a parity-odd magnetic helicity contribution to the two-point spectral tensor.