First Principles Magnetohydrodynamical Theory for the Expanding Box Model: Effects on Expansion-Induced Alfvén Wave Reflection
The Expanding Box Model (EBM) has been widely employed to simulate multiscale plasma phenomena in the expanding solar wind by transforming the MHD equations to a co-moving, non-inertial frame. However, traditional formulations have suffered from historical ambiguity regarding the physical separation between the co-moving and inertial reference frames, primarily arising from a classical approximation of an invariant magnetic field between them. To resolve this inconsistency, we reformulate the EBM from first principles using a fully covariant approach. Here, we model the expanding solar wind frame as an anisotropic expanding spacetime metric, allowing us to incorporate radial acceleration profiles and differential transverse expansion, ensuring that all physical fields are correctly transformed by expansion. We demonstrate that asymmetries identified in previous EBM-MHD literature are direct consequences of neglecting the tensorial scaling of the magnetic field. Our covariant treatment eliminates these residues, restoring symmetry in the co-moving frame. Projecting our system back into the inertial frame clarifies the distinction between local plasma dynamics and plasma expansion, revealing the anisotropy of the Parker spiral as a geometric projection. Furthermore, linear wave analysis using Elsässer variables reveals that plasma expansion induces a low-frequency cutoff and geometric damping. Numerical integration demonstrates that expansion drives the reflection of Alfvén waves, generating counter-propagating modes primarily at low frequencies relative to the expansion rate, while high frequencies converge to the WKB approximation. This provides a consistent foundation for simulations, establishing that expansion can serve as a source of counter-propagating waves necessary to drive solar wind turbulence at low frequencies.