= Solution
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> perturbation
$$
s=\frac{\delta p}{p_0}-\gamma\delta.
$$
Use $D=\partial_t+\mathbf u_\perp\cdot\nabla_\perp$ and $\nabla_\parallel=\partial_z+(\delta\mathbf B_\perp/B_0)\cdot\nabla_\perp$. Perpendicular force balance makes the leading total-pressure perturbation vanish:
$$
\delta p+\rho_0v_A^2b=0.
$$
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 $Du_\parallel=-\nabla_\parallel\delta p/\rho_0$. Put
$$
r=\delta+\frac{s}{\gamma},\qquad\delta p=\rho_0c_s^2r,\qquad b=-\frac{c_s^2}{v_A^2}r.
$$
The two compressive equations therefore become
$$
Dr=-\frac{v_A^2}{v_A^2+c_s^2}\nabla_\parallel u_\parallel,\qquad Du_\parallel=-c_s^2\nabla_\parallel r.
$$
Let $c_T=v_Ac_s/(v_A^2+c_s^2)^{1/2}$ be the <tube speed> and $h=c_s^2r/c_T$. Then
$$
Du_\parallel=-c_T\nabla_\parallel h,\qquad Dh=-c_T\nabla_\parallel u_\parallel.
$$
Thus the <slow and entropy fluctuations in reduced magnetohydrodynamics> have characteristic amplitudes $f^\pm=u_\parallel\pm h$ obeying
$$
\boxed{Df^\pm=\mp c_T\nabla_\parallel f^\pm,\qquad Ds=0.}
$$
The 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 $\tau_\ell\sim\mathcal E^{-1/3}\ell_\perp^{2/3}$, then
$$
\delta f_\ell^2\sim\mathcal F_f\tau_\ell,\qquad\boxed{E_f(k_\perp)\sim\mathcal F_f\mathcal E^{-1/3}k_\perp^{-5/3}.}
$$
The 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.
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