Kinetic interchange advection cancellation ($\delta f=-{\xi }_x{\partial }_x{f}_0$)

In the steep-gradient, low-plasma beta limit of kinetic magnetohydrodynamics, magnetic induction gives $\delta b_x=i{k}_{\parallel }{\xi }_x$ and $u_x=-i\omega {\xi }_x$. Therefore the forcing of the drift-kinetic equation is $(u_x+v_{\parallel }\delta b_x)f_0^{\prime }=(-i\omega +i{k}_{\parallel }{v}_{\parallel }){\xi }_x{f}_0^{\prime }$. It cancels the streaming operator on $\delta f=-{\xi }_x{f}_0^{\prime }$, leaving no leading kinetic phase mixing on this displacement branch. Freezing the parallel derivative to the equilibrium direction would lose this cancellation.