On the Conservation of Cross Helicity and Wave Action in Solar-wind Models with Non-WKB Alfvén Wave Reflection

The interaction between Alfvén-wave turbulence and the background solar wind affects the cross helicity (\int {d}³x {\boldsymbol{v}}\cdot {\boldsymbol{B}}) in two ways. Non-WKB reflection converts outward-propagating Alfvén waves into inward-propagating Alfvén waves and vice versa, and the turbulence transfers momentum to the background flow. When both effects are accounted for, the total cross helicity is conserved. In the special case that the background density and flow speed are independent of time, the equations of cross-helicity conservation and total-energy conservation can be combined to recover a well-known equation derived by Heinemann and Olbert that has been interpreted as a non-WKB generalization of wave-action conservation. This latter equation (in contrast to cross-helicity and energy conservation) does not hold when the background varies in time.