Wave-branch classification of ideal MHD discontinuities in Q-variables

Abstract

The $Q$-variable formalism combines velocity and magnetic field into branches aligned with a selected MHD wave. For homogeneous ideal MHD, appropriate choices of the branch parameter $α$ correspond to the fast, Alfvén, and slow characteristic families, with the background variables $\boldsymbol{Q}_0^{\pm}$ representing the propagation direction and speed of the corresponding characteristics. In structured plasmas, $α$ may instead be selected from the dispersion relation of a global mode such as a kink wave. Recent work reformulated the ideal-MHD Rankine-Hugoniot conditions in $Q$-variables and established their exact equivalence with the classical conservative jump conditions. Here, we apply this formulation to the standard ideal-MHD discontinuity catalogue. We determine how the two discontinuity-frame branches $\boldsymbol{Q}^{\prime+}$ and $\boldsymbol{Q}^{\prime-}$, their sums, and their $α$-normalised differences are constrained across contact, tangential, and rotational discontinuities and across the principal shock families. Material interfaces are distinguished by their normal branch content, rotational discontinuities by an Alfvénic tangential relation, and fast, slow, and intermediate shocks by the magnitude or orientation of the $α$-normalised tangential branch difference. Perpendicular, parallel, switch-on, and switch-off shocks appear as geometrical or limiting cases. The resulting classification expresses the classical discontinuity conditions in a wave-aligned form and, through the characteristic interpretation of $\boldsymbol{Q}_0^{\pm}$, shows how ideal-MHD discontinuities constrain changes in characteristic propagation direction and speed across the interface. It therefore establishes the background interface structure required for subsequent studies of reflection, transmission, coupling, and mode conversion.