= Five quadratic cascade invariants of anisotropic magnetohydrodynamic turbulence
{title2=$I_A^+,I_A^-,I_S^+,I_S^-,I_e$}
For ideal <reduced magnetohydrodynamics> with periodic or vanishing-flux boundaries, the five quadratic cascade channels are $I_A^\pm=\int|\mathbf z_\perp^\pm|^2dV$, $I_S^\pm=\int(f^\pm)^2dV$ and $I_e=\int s^2dV$. 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.
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