Dimensional freedom of an MHD spectrum ($E(k)={ϵ}^{2/3}{k}^{-5/3}\Phi (v_A{ϵ}^{-1/3}{k}^{1/3})$)

In a local isotropic hydrodynamic inertial-range model with only $ϵ$ and $k$, dimensional analysis fixes the Kolmogorov 1941 theory spectrum. A mean magnetic field supplies the independent Alfvén speed, permitting the arbitrary dimensionless function in the displayed formula even before anisotropy adds parallel scales. Cascade-time assumptions such as weak decorrelated interactions or critical balance are therefore necessary to select an MHD spectrum.