Reduced magnetohydrodynamics ($k_{\parallel }/k_{\perp }\ll 1$)

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.

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.

In ideal reduced magnetohydrodynamics with constant background sound and Alfvén speeds, perpendicular total-pressure balance gives $\delta p+{\rho }_0{v}_A^2\delta B_{\parallel }/B_0=0$. Put $s=\delta p/p_0-\gamma \delta \rho /{\rho }_0$, $r=\delta \rho /{\rho }_0+s/\gamma$, $c_T=v_A{c}_s/(v_A^2+c_s^2{)}^{1/2}$ and $h=c_s^{2}r/c_T$. Then the parallel momentum, continuity and induction equations yieldThus $f^{\pm }=u_{\parallel }\pm h$ satisfy $D{f}^{\pm }=\mp c_T{\nabla }_{\parallel }{f}^{\pm }$. The two slow-wave populations propagate along bent magnetic field lines and are mixed by the perpendicular Alfvénic turbulence, while the entropy fluctuation is simply advected. They do not force the perpendicular Alfvénic system at this leading order.