Alfvén wave mixing and non-JWKB waves in stellar winds

Alfv?n wave mixing equations used in locally incompressible turbulence transport equations in the solar wind contain as a special case, non-Jeffreys?Wentzel?Kramers?Brouillon (non-JWKB) wave equations used in models of Alfv?n wave driven winds. We discuss the canonical wave energy equation; the physical wave energy equation, and the JWKB limit of the wave interaction equations. Lagrangian and Hamiltonian variational principles for the waves are developed. Noether?s theorem is used to derive the canonical wave energy equation which is associated with the linearity symmetry of the equations. A further conservation law associated with time translation invariance of the action, applicable for steady background wind flows is also derived. In the latter case, the conserved density is the Hamiltonian density for the waves, which is distinct from the canonical wave energy density. The canonical wave energy conservation law is a special case of a wider class of conservation laws associated with Green?s theorem for the wave mixing system and the adjoint wave mixing system, which are related to Noether?s second theorem. In the sub-Alfv?nic flow, inside the Alfv?n point of the wind, the backward and forward waves have positive canonical energy densities, but in the super-Alfv?nic flow outside the Alfv?n critical point, the backward Alfv?n waves are negative canonical energy waves, and the forward Alfv?n waves are positive canonical energy waves. Reflection and transmission coefficients for the backward and forward waves in both the sub-Alfv?nic and super-Alfv?nic regions of the flow are discussed.