The role of sign indefinite invariants in shaping turbulent cascades.

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

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