On Spurious Vorticity Generation in Vector-Coupled Metric Damping for Geophysical CFD: Theoretical Analysis of the Parasitic Curl and Cauchy Tensor Regularization

In ultra-high-resolution mesoscale atmospheric simulations (sub-kilometer grid spacing), extreme velocity gradients across the eyewall of intense tropical cyclones induce severe grid cell deformations, frequently triggering Courant-Friedrichs-Lewy (CFL) numerical instabilities and vertical velocity (W) blow-ups. To mitigate such singularities, an intuitive approach consists in introducing a prognostic scalar metric strain variable J(x, t) and applying a direct vector drag restoring force. In this technical note, we demonstrate analytically and numerically that this intuitive vector formulation introduces a fatal pathology: it acts as an artificial generator of vertical vorticity through an anomalous, non-conservative cross-gradient curl term. We trace the root cause to the violation of Galilean invariance and Cauchy's stress theorem, and formulate the mathematically rigorous remedy: replacing the vector force with the divergence of a rank-2 symmetric stress tensor.