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Alfvénic turbulence is the turbulent interaction of fluctuations polarized primarily as Alfvén waves. In Elsässer variables ${\mathbf{z}}^{\pm }={\mathbf{u}}_{\perp }\pm \delta {\mathbf{B}}_{\perp }/\sqrt{4\pi {\rho }_0}$, each wave population is nonlinearly distorted by the oppositely propagating population. Equal populations give a balanced cascade; unequal populations give an imbalanced cascade. The absence of one population removes the leading nonlinear interaction in ideal reduced magnetohydrodynamics.

For smooth ideal magnetohydrodynamics of constant mass density, define $I_{\pm }={\int }_V|{\mathbf{z}}^{\pm }{|}^{2}dV$. Taking the dot product of the Elsässer variable equation with $2{\mathbf{z}}^{\pm }$ givesTherefore each integral is constant when its outward boundary flux vanishes, for example when both velocity and magnetic field are tangent to the fixed boundary, or for periodic boundary conditions. The kinetic energy plus magnetic energy is $\rho (I_{+}+I_{-})/4$, while the cross-helicity is $\sqrt{{\mu }_0\rho }(I_{+}-I_{-})/4$.

For an incompressible flow with constant mass density, the Elsässer variables are ${\mathbf{z}}^{\pm }=\mathbf{u}\pm {\mathbf{v}}_a$, where ${\mathbf{v}}_a$ is the Alfvén velocity. Adding and subtracting the Euler momentum equation and ideal magnetohydrodynamic induction equation yields ${\partial }_t{\mathbf{z}}^{\pm }+({\mathbf{z}}^{\mp }\cdot \nabla ){\mathbf{z}}^{\pm }=-\nabla \psi$, with $\psi =p/\rho +\Phi +|{\mathbf{v}}_a{|}^2/2$ and $\nabla \cdot {\mathbf{z}}^{\pm }=0$. Thus each field is transported by the other, a useful way to expose interactions of oppositely directed Alfvén waves.

For ideal reduced magnetohydrodynamics with periodic or vanishing-flux boundaries, the five quadratic cascade channels are $I_A^{\pm }=\int |{\mathbf{z}}_{\perp }^{\pm }{|}^{2}dV$, $I_S^{\pm }=\int (f^{\pm }{)}^{2}dV$ and $I_e=\int s^{2}dV$. The perpendicular Elsässer variables, slow-wave amplitudes $f^{\pm }$, and entropy fluctuation $s$ each obey an equation with divergence-free advection. Multiplying each equation by its own field and integrating makes both advection and propagation terms boundary fluxes, while the perpendicular pressure term vanishes by solenoidality. Each integral is therefore separately conserved. The two Alfvén cascades distort one another without exchanging their integrated invariants; the other three are passively cascaded by the Alfvénic motion without feedback at this order. This count concerns quadratic fluctuation-energy channels: an advected scalar also conserves arbitrary integrals $\int F(s)dV$, which are not extra independent quadratic modal channels.

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.

In the ideal, leading anisotropic system, there are five separately conserved quadratic cascade quantities: the two oppositely propagating Alfvén populations, the two slow-wave populations, and the entropy fluctuation. With the amplitudes of part (b), one convenient normalization iswhere ${\mathbf{z}}_{\perp }^{\pm }={\mathbf{u}}_{\perp }\pm \delta {\mathbf{B}}_{\perp }/\sqrt{4\pi {\rho }_0}$. Constant background factors convert the velocity variances into modal energies and the entropy variance into an entropy-related free-energy contribution. These are the five quadratic cascade invariants of anisotropic magnetohydrodynamic turbulence.

To verify conservation rather than just count polarizations, the perpendicular Elsässer variables obeyDot each equation with $2{\mathbf{z}}_{\perp }^{\pm }$ and integrate. The propagation and nonlinear terms become surface fluxes of $|{\mathbf{z}}_{\perp }^{\pm }{|}^2$, and the pressure term becomes a surface flux because the corresponding field is solenoidal. For periodic boundaries, or boundaries with vanishing invariant flux, all these terms vanish and $d{I}_A^{\pm }/dt=0$. Their sum is proportional to perpendicular kinetic plus magnetic energy, while their difference is proportional to perpendicular cross-helicity; the two Elsässer energy invariants contain both pieces of information.

Likewise the slow-mode equations giveHere ${\mathbf{u}}_{\perp }$ is divergence-free and ${\nabla }_{\parallel }$ differentiates along the divergence-free vector $\hat{\mathbf{z}}+\delta {\mathbf{B}}_{\perp }/B_0$, so both integrals are boundary fluxes. Finally $Ds=0$ gives $d{I}_e/dt=-\int {\mathbf{u}}_{\perp }\cdot {\nabla }_{\perp }{s}^{2}dV=0$. This proves all five quadratic conservation laws under the same boundary assumptions.

The two Alfvén cascades interact dynamically: each population strains the other and is needed for its nonlinear transfer between scales. Their integrated invariants are nevertheless conserved separately, so this interaction does not transfer one population's invariant into the other. The slow and entropy cascades are driven by the Alfvénic mixing and field-line motion, with no feedback on the perpendicular cascade at the retained order. Their injection rates set independent fluxes; those fluxes need not be equal, even when the resulting perpendicular spectra share the $-5/3$ scaling.

The count is a count of independent quadratic modal cascade channels. Ideal passive advection also preserves higher moments, such as $\int F(s)dV$ for arbitrary smooth $F$, and analogous integrals for the slow characteristic scalars. These are not additional quadratic modal energies. Forcing and dissipative terms make the corresponding invariants source-and-sink balances; a stationary inertial range conserves their flux through scales rather than their total content in a continually forced domain.

Let the constant mass density be $\rho >0$ and put ${\mathbf{v}}_a=\mathbf{B}/\sqrt{{\mu }_0\rho }$, the Alfvén velocity. Since both the velocity and the magnetic field have zero divergence, the magnetic tension and magnetic pressure decomposition of the Lorentz force givesThe Newtonian gravitational potential $\Phi$ remains in $\psi$; uniform mass density does not justify dropping a prescribed gravitational acceleration. Adding and subtracting the two equations proves the Elsässer variable equationsHere ${\mathbf{z}}^{\pm }=\mathbf{u}\pm {\mathbf{v}}_a$; each Elsässer variable is transported by the other.

Define $I_{\pm }={\int }_V|{\mathbf{z}}^{\pm }{|}^{2}dV$. Taking the dot product with $2{\mathbf{z}}^{\pm }$ gives the Elsässer energy invariant balanceThe divergence theorem proves $d{I}_{\pm }/dt=0$ whenever the net boundary flux vanishes. In particular, $\mathbf{u}\cdot \mathbf{n}=\mathbf{B}\cdot \mathbf{n}=0$ makes ${\mathbf{z}}^{\pm }\cdot \mathbf{n}=0$, so both fluxes vanish pointwise. Periodic boundary conditions or decay at infinity are also sufficient. These boundary conditions matter: fixed volume alone gives no conservation.

The kinetic energy plus magnetic energy isExpanding $|\mathbf{u}\pm {\mathbf{v}}_a{|}^2$ and using the cross-helicity definition yieldsBoth energy and cross-helicity therefore follow from the two Elsässer energy invariants. The quantity $E$ here is exactly the requested kinetic energy plus magnetic energy, without an additional gravitational term.

Reduced magnetohydrodynamics is the leading ideal-fluid description of small, anisotropic fluctuations about a strong uniform guide field, with $k_{\parallel }/k_{\perp }\sim \delta B_{\perp }/B_0\sim \delta u_{\perp }/v_A\ll 1$. Its solenoidal perpendicular Elsässer variables obeyThe parallel magnetic derivative includes bending of field lines: ${\nabla }_{\parallel }={\partial }_z+(\delta {\mathbf{B}}_{\perp }/B_0)\cdot {\nabla }_{\perp }$. Slow compressive and entropy mode fluctuations are passively transported at this order. The system excludes the rapidly propagating fast magnetosonic wave sector rather than describing all possible MHD perturbations.