Phase Alignment, Orthogonality Forcing, and the Absence of Blow-Up in a Vector Shell Model of Turbulence

The paper presents a mathematical and numerical investigation of how Fourier-phase alignment affects finite-time singularity formation in turbulent shell models. As stated in the abstract, “in the scalar GOY model, perfect phase alignment forces the enstrophy to blow up in finite time while energy remains bounded,” whereas in the vector L’vov model “the enstrophy stays uniformly bounded, and no finite-time singularity is observed.” Three forcing protocols—no forcing, global phase alignment, and orthogonality forcing—are tested. All preserve energy exactly, and none produce blow-up. The geometric explanation is that the triple cross-product structure suppresses coherent forward energy transfer, suggesting that the Navier–Stokes equations may be protected by intrinsic geometric frustration.