Covariant Helmholtz-Hodge Decomposition: Resolving Spurious Vorticity via Acoustic Geometry

The separation of acoustic and vortical fluctuations in compressible turbulence becomes ambiguous in thermodynamically inhomogeneous media, where refraction by entropy gradients and shocks can be misclassified as solenoidal content by Euclidean post-processing. We introduce a covariant Helmholtz--Hodge decomposition (CHHD) with respect to an effective acoustic metric, which identifies the irrotational (potential) component with the exact part of the metric-dual velocity one-form. Thermal refraction and shock-induced bending are absorbed into the induced curvature, ensuring that such geometric variations are not misidentified as physical vorticity. For canonical entropy-spot refraction and normal-shock discontinuities, Euclidean Helmholtz--Hodge and momentum-potential post-processing produce significant leakage in the refracting/discontinuous region, whereas the covariant splitting remains at the numerical noise floor (typically $\lesssim 10^{-12}$) throughout the domain, demonstrating robustness even at the sonic horizon, where the Euclidean metric singularity typically causes catastrophic error amplification. This geometric framework for velocity fields resolves the ambiguity of irrotational motion in inhomogeneous media and establishes a necessary foundation for future generalizations to full thermodynamic state vectors.