On the convergence of the normal form transformation in discrete wave turbulence theory for the Charney-Hasegawa-Mima (CHM) equation*

A crucial problem in discrete wave turbulence theory concerns extending the validity of the normal form transformation beyond the weakly nonlinear limit. The main difficulty is that even if the transformation converges in a given domain around the origin, there is no assurance that all orbits starting in the domain will remain there at all times. Therefore a situation could arise whereby the original system exhibits behavior that is not captured by the normal form system evolution, regardless of the order of the transformation.

We demonstrate this for the CHM equation, Galerkin-truncated to 4 Fourier modes. By calculating the transformation to 7th order (keeping all resonances up to 8-wave), we perform numerical simulations of both the original and mapped equations to find that the problems occur precisely when the initial conditions lead to precession resonance, a finite-amplitude phenomenon characterized by strong energy transfers across Fourier modes [1].

We use the dynamical systems approach to extend this result to complex wave-turbulent regimes in the CHM equation, leading to a working definition of convergence radius for normal transformations in terms of invariant manifolds.

[1] Bustamante MD et al. (2014) PRL 113, 084502