Kinetic equation of weakly nonlinear internal gravity waves: Theory and Numerics

Speaker: Vincent Labarre

Abstract: Internal gravity waves propagating in stratified flows are an important
component of geophysical and astrophysical systems, encompassing a wide range of
scales. In the weakly nonlinear regime, these waves interact through nonlinear resonant
triadic interactions, leading to a mixing of density, momentum, and energy. Representing
such wave dynamics is of paramount importance, particularly in climate modelling.
We present a new derivation of the kinetic equation for weak internal gravity wave
turbulence. This equation aligns with the one obtained by Caillol and Zeitlin, but it is
expressed in a more compact form. We demonstrate that it conserves energy without
relying on the frequency resonance condition, and we investigate steady, scale-invariant

We parameterize the resonant manifold, corresponding to the set of resonant triadic
interactions among internal gravity waves. It enables us to numerically verify that the
interaction coefficients are fully symmetric under the permutation of wave vectors on the
resonant manifold, simplify the collision integral, and evaluate the transfer coefficients of
all triadic interactions.

In the hydrostatic limit, our equation is equivalent to the Hamiltonian description of Lvov
and Tabak. The steady, bi-homogeneous, nonequilibrium spectrum associated with
energy conservation, the Pelinovsky-Raevsky spectrum, is derived without employing the
frequency resonance condition, an uncommon approach in Weak Wave Turbulence
theory. We also present preliminary results from the numerical simulation of the kinetic