= Slow and entropy fluctuations in reduced magnetohydrodynamics
{title2=$Df^\pm=\mp c_T\nabla_\parallel f^\pm$}
In ideal <reduced magnetohydrodynamics> with constant background sound and <Alfvén speeds>, perpendicular total-pressure balance gives $\delta p+\rho_0v_A^2\delta B_\parallel/B_0=0$. Put $s=\delta p/p_0-\gamma\delta\rho/\rho_0$, $r=\delta\rho/\rho_0+s/\gamma$, $c_T=v_Ac_s/(v_A^2+c_s^2)^{1/2}$ and $h=c_s^2r/c_T$. Then the parallel momentum, continuity and induction equations yield
$$
Du_\parallel=-c_T\nabla_\parallel h,\qquad Dh=-c_T\nabla_\parallel u_\parallel,\qquad Ds=0.
$$
Thus $f^\pm=u_\parallel\pm h$ satisfy $Df^\pm=\mp c_T\nabla_\parallel f^\pm$. 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.
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