Gravitational Enstrophy: Local Geometric Origin and Inverse-Cascade Constraints

Abstract

Two-dimensional fluids conserve energy and enstrophy, driving inverse energy cascades via Fjørtoft'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 $\mathcal{Z}=\int B_{ab}{B}^{ab}\sqrt{\gamma }{d}^{3}x\approx {\sum }_k{\omega }_k^2{W}_k$ 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 Fjørtoft constraint implying 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.

41 pages (1/2 are appendices for specific details), 3 figures [a few typos fixed, added a clarifying paragraph]