= Solution
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 give
$$
\tau_{\mathrm{nl}}\sim\frac{\ell_\perp}{\delta u_\ell}.
$$
The 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 gives
$$
\mathcal E\sim\frac{\delta u_\ell^2}{\tau_{\mathrm{nl}}}\sim\frac{\delta u_\ell^3}{\ell_\perp},\qquad\boxed{\delta u_\ell\sim(\mathcal E\ell_\perp)^{1/3}.}
$$
Define the one-dimensional perpendicular <turbulent energy spectrum> so that $\int E(k_\perp)\,dk_\perp$ measures fluctuation <energy> per unit mass. Locality of the spectrum then gives
$$
\delta u_\ell^2\sim\int_{1/\ell_\perp}^{\infty}E(k_\perp)dk_\perp\sim k_\perp E(k_\perp).
$$
Hence
$$
\boxed{E(k_\perp)=C\mathcal E^{2/3}k_\perp^{-5/3},}
$$
where $C$ 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:
$$
\frac{\ell_\parallel}{v_A}\sim\mathcal E^{-1/3}\ell_\perp^{2/3},\qquad\boxed{k_\parallel\sim\frac{\mathcal E^{1/3}}{v_A}k_\perp^{2/3}.}
$$
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>.
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