Gravitational Enstrophy: Local Geometric Origin and Inverse-Cascade Constraints

Two-dimensional fluids conserve energy and enstrophy, driving inverse energy cascades via Fj\o rtoft's argument. We show General Relativity admits an analogous structure: for linear radiative perturbations of Petrov type D backgrounds (Kerr, Kerr--AdS), the gravitational-wave energy $W = \sum_k W_k$ and magnetic Weyl enstrophy $\mathcal{Z} = \int B_{ab} B^{ab} \sqrt{\gamma} \, d^3x \approx \sum_k \omega_k^2 W_k$ are approximately conserved in the zero-angular momentum frame, where vorticity coupling vanishes identically and curl exchange cancels mode-by-mode. This yields a gravitational Fj\o rtoft constraintt implyging nonlinear energy transfer proceeds preferentially toward lower frequencies. The constraint is dynamically active in near-extremal Kerr ($\tau_{\text{damp}} \gg \tau_{\text{nl}}$) and confined geometries (AdS), but suppressed in generic ringdown. In AdS, $\mathcal{Z}$ maps holographically to the boundary fluid enstrophy.