**Speaker**: Michal Shavit

**Abstract**: Strongly dispersive waves in geophysical fluid dynamics occur on scales from

centimetres to thousands of kilometres and contribute in an essential way to the long-

term nonlinear dynamics of the climate system. Examples include internal inertia–gravity

waves and Rossby waves, all of which owe their existence to some combination of

gravity, rotation, and curvature of the Earth. Many of these waves are far too small in

scale to be resolvable numerically, making their study a pressing issue for theoretical

modelling. For small-amplitude waves, wave turbulence theory can play an important

part. Arguably, progress has been hampered by the extremely cumbersome form taken

by the relevant anisotropic equations when attempting to shoe-horn them into classical

wave turbulence theory, which was formulated in canonical variables for

Hamiltonian isotropic systems. But the underlying fluid equations are non-canonical

Hamiltonian systems, as is made obvious by the fact that the Euler equations are

nonlinear yet their energy function is quadratic.

I will present a reformulation of kinetic wave theory for a number of two-dimensional fluid

systems with quadratic energies based on a particular choice of non-canonical

variables. The practical utility of thesevariables, derives from the existence of a second

quadratic invariant in these systems, which, albeit not sign-definite, greatly simplifies the

wave interaction equations. I will leverage these simplifications into a derivation of

scaling laws for the isotropic component of wave spectra, which gains a small correction

with respect to the Kolmogorov-Zakharov scaling of isotropic systems. I will present

evidence for the importance of these second invariants in shaping the overall wave

spectra, including the possibility of driving an inverse energy cascade of internal gravity

waves.

This talk is based on a joint work with Oliver Bühler and Jalal Shatah.