Turbulent energy spectrum ($E(k)$)

The turbulent energy spectrum $E(k)$ distributes kinetic energy over wavenumber shells, normalized by ${\int }_0^{\infty }E(k)dk=⟨|\mathbf{u}{|}^2⟩/2$. For isotropic incompressible flow its tensor form is ${\Phi }_{ij}=E(k)({\delta }_{ij}-k_i{k}_j/k^2)/(4\pi k^2)$. Its low-wavenumber behavior records large-eddy momentum statistics.