The role of sign indefinite invariants in shaping turbulent cascades

Our work answers a nearly 60-year quest to derive the turbulent spectrum of weakly interacting internal gravity waves from first principles. The classical wave-turbulence approach didn’t work, as the underlying equation, both in 2D and 3D, is an anisotropic, non-canonical Hamiltonian equation. 

A key consequence of the non-canonical Hamiltonian is the conservation of a sign-indefinite quadratic invariant alongside the sign-definite quadratic energy. In 2D, this allows us to derive a much simpler kinetic equation. We leverage this simplification into the derivation of solutions of the kinetic equation, one of which is the turbulent spectrum of weakly interacting 2D internal gravity waves. Our spectrum exactly matches the phenomenological oceanic Garrett-Munk spectrum in the limit of large vertical wave numbers and zero rotation. 

This talk is based on recent joint works with Oliver Bühler and Jalal Shatah – 

arXiv:2311.04183 (to appear soon in PRL).

arXiv:2406.06010.