Power spectrum

The power spectrum describes how the variance of a random field or stationary process is distributed over spatial wavenumber or temporal frequency.

For a statistically homogeneous vector random field, its spectral tensor is the Fourier transform of the componentwise two-point correlation $C_{ij}(\mathbf{r})=\mathbb{E}[B_i(\mathbf{x}+\mathbf{r})B_j(\mathbf{x})]$. It distributes cross-correlations between components over wavevectors. With $\hat{B}(\mathbf{k})=\int B(\mathbf{r})e^{-i\mathbf{k}\cdot \mathbf{r}}{d}^{d}r$, homogeneity gives $\mathbb{E}[{\hat{B}}_i(\mathbf{k}){\hat{B}}_j({\mathbf{k}}^{\prime })]=(2\pi {)}^d\delta (\mathbf{k}+{\mathbf{k}}^{\prime }){\hat{C}}_{ij}(\mathbf{k})$, understood as an identity of distributions for an infinite stationary field.

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.

A random field is statistically homogeneous when all of its joint distributions are invariant under a common spatial translation of their arguments. Its momentum-space correlation functions consequently contain a Dirac delta function imposing total-momentum conservation.

A random field is statistically isotropic when all of its joint distributions are invariant under a common spatial rotation of their arguments. A statistically isotropic scalar power spectrum depends only on the magnitude of its wavevector.

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.