Rankine-Hugoniot conditions in Q-variables: a wave-aligned formulation of MHD discontinuities
The recently developed Q-variable formalism generalises the Els\"asser representation by providing a wave-aligned representation applicable to a broad class of magnetohydrodynamic disturbances, including Alfv\'enic, fast, slow, and kink waves. While this framework has proven useful for the study of wave dynamics and turbulence, its behaviour in the presence of plasma discontinuities has not yet been established. In this work, we derive the complete set of Rankine-Hugoniot jump conditions in terms of the Q-variables by rewriting the ideal MHD equations in a form suitable for shock-frame jump analysis. This yields explicit jump relations for mass, momentum, magnetic flux, and energy. We then demonstrate analytically that these relations are exactly equivalent to the classical MHD Rankine-Hugoniot conditions. This reformulation provides a wave-aligned representation of MHD discontinuities and offers a natural framework for discussing directional wave content and branch-restricted limits when $\alpha$, the wave-branch parameter entering the Q-variable definition, is chosen consistently with the relevant characteristic speed. The resulting formulation is well suited for the analysis of wave-shock interactions in magnetised plasmas, with potential applications to the solar wind, magnetospheric systems, and large-scale models of structured plasma environments such as UAWSOM.