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

Abstract

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.

Figures (5)

• Figure 1: Thermodynamic stabilization of the acoustic metric. The streamwise metric component $\gamma_{xx}$ diverges at the sonic horizon ($U=c_0$). Crucially, entropy injection (higher $c_0$) shifts this horizon, effectively stabilizing the metric in the region of interest.
• Figure 2: Visualizing separation in a thermal entropy spot. (a) Conceptual ray tracing showing geodesic bending of acoustic paths due to $\nabla c_0$. (b) Euclidean HHD misinterprets this refraction as spurious vorticity $|\boldsymbol{\omega}|$ (bright structures). Inset: Doak's MPT diagnostic. (c) The covariant residual $\zeta_{\text{cov}}$ remains at the numerical floor, consistent with geometric absorption of thermal refraction.
• Figure 3: Entropy-spot validation (centerline leakage). Centerline $\mathcal{L}(x,y=0)$ defined by Eq. \ref{['eq:leakage_metric']}. Euclidean HHD exhibits high-amplitude leakage across the entire wave packet, and Doak's MPT exhibits pronounced, spatially distributed artifacts inside the thermal lens despite improved far-field behavior, whereas the covariant splitting remains at the numerical noise floor ($\mathcal{L} \lesssim 10^{-12}$) throughout the domain.
• Figure 4: Separation across a normal shock ($M=1.5 \to 0.7$). (a) Background Mach number showing the discontinuity at $x=3.0$. (b) Euclidean Helmholtz--Hodge post-processing produces a vortex-sheet-like artifact at the shock and contaminates the downstream region. Inset: Doak's MPT diagnostic. (c) The covariant formulation yields negligible residuals throughout the domain, demonstrating robustness against shock discontinuities.
• Figure 5: Shock-wave validation (centerline leakage). Centerline $\mathcal{L}(x,y=0)$ defined by Eq. \ref{['eq:leakage_metric']}. Euclidean HHD produces a persistent error plateau downstream of the shock, and Doak's MPT exhibits spurious oscillatory spikes at the shock location, whereas the covariant residual maintains negligible residuals ($\mathcal{L} \lesssim 10^{-12}$) even at the shock location, eliminating the spurious artifacts observed in other methods.