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At leading anisotropic order, perpendicular Alfvénic turbulence evolves independently of the compressive fluctuations. The slow modes and the entropy mode are transported by that turbulence. The fast magnetosonic frequency is of order $k_{\perp }(v_A^2+c_s^2{)}^{1/2}$, much larger than the reduced Alfvén frequency; the low-frequency reduced magnetohydrodynamics ordering filters out that fast-wave sector.

The passive evolution can be made explicit for a uniform ideal adiabatic plasma. Let $c_s^2=\gamma p_0/{\rho }_0$, $\delta =\delta \rho /{\rho }_0$, $b=\delta B_{\parallel }/B_0$, and define the normalized entropy perturbationUse $D={\partial }_t+{\mathbf{u}}_{\perp }\cdot {\nabla }_{\perp }$ and ${\nabla }_{\parallel }={\partial }_z+(\delta {\mathbf{B}}_{\perp }/B_0)\cdot {\nabla }_{\perp }$. Perpendicular force balance makes the leading total-pressure perturbation vanish:The ideal entropy equation gives $Ds=0$. The leading continuity and parallel-induction equations have the same compressive divergence term; subtracting them gives $D(\delta -b)=-{\nabla }_{\parallel }{u}_{\parallel }$. Parallel momentum gives $D{u}_{\parallel }=-{\nabla }_{\parallel }\delta p/{\rho }_0$. PutThe two compressive equations therefore becomeLet $c_T=v_A{c}_s/(v_A^2+c_s^2{)}^{1/2}$ be the tube speed and $h=c_s^{2}r/c_T$. ThenThus the slow and entropy fluctuations in reduced magnetohydrodynamics have characteristic amplitudes $f^{\pm }=u_{\parallel }\pm h$ obeyingThe two slow-wave populations propagate in opposite directions along the bent field lines while being mixed across them by ${\mathbf{u}}_{\perp }$. The entropy perturbation is carried by the fluid without a restoring-wave frequency. A pure entropy perturbation has $r=b=u_{\parallel }=0$ and $\delta =-s/\gamma$, so its density variation is accompanied by a temperature change with no leading pressure variation.

The coefficients and advecting fields in these three scalar equations come from the Alfvénic cascade; the scalars do not enter the leading perpendicular Alfvén equations. They therefore develop small perpendicular scales through passive mixing. If a scalar $f$ has constant variance flux ${\mathcal{F}}_f$ and is mixed on the Goldreich–Sridhar time ${\tau }_{\ell }\sim {\mathcal{E}}^{-1/3}{\ell }_{\perp }^{2/3}$, thenThe flux and amplitude need not equal the Alfvénic energy flux. This passive-spectrum statement uses the same locality and stationary inertial-range assumptions as part (a), and ignores additional damping. If a passive mode is absent initially and is not forced, its homogeneous leading equation keeps it absent; the Alfvénic cascade does not have to generate every compressive mode.