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Goldreich–Sridhar turbulence combines critical balance, balanced counterpropagating Alfvén waves, a local perpendicular cascade, and constant energy flux per unit mass $\mathcal{E}$. With ${\tau }_{\mathrm{nl}}\sim {\ell }_{\perp }/\delta u_{\ell }$ and $\mathcal{E}\sim \delta u_{\ell }^2/{\tau }_{\mathrm{nl}}$, one obtains $\delta u_{\ell }\sim (\mathcal{E}{\ell }_{\perp }{)}^{1/3}$. Defining the one-dimensional perpendicular turbulent energy spectrum by $\delta u_{\ell }^2\sim k_{\perp }E(k_{\perp })$ gives $E(k_{\perp })\sim {\mathcal{E}}^{2/3}{k}_{\perp }^{-5/3}$. Critical balance also gives $k_{\parallel }\sim {\mathcal{E}}^{1/3}{k}_{\perp }^{2/3}/v_A$. This scaling argument neglects scale-dependent alignment and intermittency corrections, which are additional physical effects rather than consequences of the stated assumptions.

Use incompressible ideal MHD at scales above the ion skin depth, a strong uniform mean field, and a statistically steady local energy cascade with mean flux $ϵ$ per unit mass. Assume balanced counterpropagating wave populations, comparable velocity and magnetic amplitudes in velocity units, and no additional scale-dependent alignment or intermittency correction. Write $\mathbf{b}=\delta \mathbf{B}/\sqrt{4\pi \rho }$ and introduce the fluctuation Elsässer variables ${\mathbf{z}}^{\pm }=\mathbf{u}\pm \mathbf{b}$. Their equations areOnly opposite populations distort one another. A single population has no leading nonlinear self-interaction, so the balance assumption is essential to a one-amplitude cascade estimate.

Let $\delta z_l$ denote the typical balanced amplitude and let $l_{\perp },l_{\parallel }$ be coherence lengths relative to the local mean field. The Alfvén wave time and eddy turnover time areFor $\chi \ll 1$, one counterpropagating encounter changes the packet fractionally by order $\chi$. Assume phases decorrelate between encounters. After $N$ encounters, random accumulation is of order $\sqrt{N}\chi$; an order-one change requires $N\sim {\chi }^{-2}$. The weak wave cascade time is thereforeWeakness by itself does not prove this random walk estimate; phase decorrelation and the interaction geometry are additional assumptions.

For the Iroshnikov-Kraichnan spectrum, impose isotropy, $l_{\parallel }\sim l_{\perp }\sim l$, in addition to $\delta z_l\ll v_A$. Then ${\tau }_A=l/v_A\ll {\tau }_{\mathrm{nl}}=l/\delta z_l$ andHence $\delta z_l\sim (ϵv_{A}l{)}^{1/4}$. A local shell-integrated turbulent energy spectrum has $\delta z_l^2\sim kE(k)$, $k\sim l^{-1}$, soThe isotropy is imposed phenomenologically; a strong guide field actually distinguishes directions and does not force this isotropic picture. The spectrum is not an automatically valid asymptotic result for every strongly magnetized cascade.

For the anisotropic weak Alfvénic cascade, keep $l_{\parallel }$ approximately fixed while cascading to smaller $l_{\perp }$, and require $\chi \ll 1$. This geometry follows from resonant three-wave Alfvén interactions: frequency conservation together with parallel-wavenumber conservation makes one member of an oppositely propagating resonant triad have zero parallel wavenumber, so the propagating modes retain their parallel wavenumber in the leading resonant transfer. Thus there is no leading parallel cascade in this ideal weak-wave ordering. Using the fixed parallel scale,Therefore $\delta z_l\sim (ϵv_A/l_{\parallel }{)}^{1/4}{l}_{\perp }^{1/2}$. For the one-dimensional perpendicular spectrum, $\delta z_l^2\sim k_{\perp }{E}_{\perp }(k_{\perp })$ givesUnlike the isotropic estimate, the parallel length does not decrease with the perpendicular length. Since $\chi =\delta z_l{l}_{\parallel }/(v_A{l}_{\perp })\propto l_{\perp }^{-1/2}$, the interactions become stronger down the cascade. If ${\chi }_0\ll 1$ at outer perpendicular scale $L_{\perp }$, the weak approximation breaks at $l_{\perp ,\ast }\sim L_{\perp }{\chi }_0^2$. This is the weak-to-strong transition of an Alfvénic cascade.

For Goldreich–Sridhar turbulence, abandon the weak-collision estimate and impose critical balance, ${\tau }_A\sim {\tau }_{\mathrm{nl}}$. The local cascade is strong, with transfer time of order one eddy turnover time. ThenThe parallel scale is set dynamically, not held fixed:At smaller scales $l_{\parallel }/l_{\perp }$ grows, so the eddies are increasingly elongated along the local magnetic field. Strong turbulence does not mean $\delta z_l\sim v_A$: small transverse amplitudes can still be strong when the eddies are sufficiently anisotropic to satisfy critical balance. Imbalance, alignment or nonlocal interactions would require different cascade reasoning.

Finally, in the usual isotropic hydrodynamic Kolmogorov 1941 theory inertial-range argument, the only relevant dimensional quantities are $ϵ$ and $k$. With $[ϵ]=L^2{T}^{-3}$ and $[E]=L^3{T}^{-2}$, writing $E\propto {ϵ}^a{k}^b$ forces $a=2/3$, $b=-5/3$. This uses the physical assumptions that outer-scale, viscous and intermittency effects do not supply another leading dependence; dimensions alone are not a proof of those assumptions.

A magnetic guide field supplies the independent speed $v_A$, and anisotropy adds independent parallel lengths. Even before adding anisotropy, dimensions permitwith arbitrary dimensionless $\Phi$. For example $\Phi (x)\propto x^{1/2}$ reproduces the isotropic Iroshnikov-Kraichnan spectrum estimate. This dimensional freedom of an MHD spectrum explains why propagation, decorrelation, anisotropy and cascade-time assumptions must select among the spectra: purely dimensional considerations do not select a unique MHD exponent.

Use the Goldreich–Sridhar turbulence model in its balanced form. Assume a uniform strong guide field; anisotropic fluctuations $k_{\parallel }\ll k_{\perp }$; comparable counterpropagating Alfvén wave amplitudes; a statistically steady inertial range with negligible forcing and dissipation within it; transfer mainly between comparable perpendicular scales; and a scale-independent energy flux per unit mass $\mathcal{E}$. The estimate also assumes no extra scale-dependent alignment factor in the nonlinear interaction and neglects intermittency corrections.

Let ${\ell }_{\perp }$ and ${\ell }_{\parallel }$ be an eddy's perpendicular and parallel scales, and $\delta u_{\ell }\sim \delta B_{\ell }/\sqrt{4\pi {\rho }_0}$ its Alfvénic velocity amplitude. The nonlinear interaction is supplied by the oppositely propagating Elsässer variable, so comparable populations giveThe Alfvén propagation time is ${\tau }_A\sim {\ell }_{\parallel }/v_A$. For strong turbulence, critical balance makes these times comparable; the decorrelation and cascade time is therefore of order ${\tau }_{\mathrm{nl}}$, not a much longer time obtained by adding many weak encounters.

A constant perpendicular energy flux givesDefine the one-dimensional perpendicular turbulent energy spectrum so that $\int E(k_{\perp })d{k}_{\perp }$ measures fluctuation energy per unit mass. Locality of the spectrum then givesHencewhere $C$ is a dimensionless normalization constant not determined by the scaling argument. The magnetic and kinetic spectra have the same leading scaling under the assumed Alfvénic amplitude relation.

Critical balance also determines the scale-dependent anisotropy:Thus the eddies become relatively more elongated along the guide field at smaller perpendicular scales. The $-5/3$ exponent follows under the listed cascade assumptions; it is not established for every imbalanced or aligned MHD state solely by invoking critical balance.