# On the convergence of the normal form transformation in discrete Rossby and drift wave turbulence

## TL;DR

This paper investigates the convergence of the normal form transformation in discrete Rossby and drift wave turbulence, revealing that it cannot fully explain finite-amplitude phenomena like precession resonance, indicating the need for a more general theory.

## Contribution

It numerically analyzes the convergence of normal form transformations in a reduced wave turbulence model, linking divergence to finite-amplitude effects like precession resonance.

## Key findings

- Normal form transformation diverges near precession resonance amplitudes.
- Divergence of the transformation correlates with strong energy transfers.
- Standard normal form methods are insufficient for intermediate nonlinearities.

## Abstract

We study numerically the region of convergence of the normal form transformation for the case of the Charney-Hasagawa-Mima (CHM) equation to investigate whether certain finite amplitude effects can be described in normal coordinates. We do this by taking a Galerkin truncation of four Fourier modes making part of two triads: one resonant and one non-resonant, joined together by two common modes. We calculate the normal form transformation directly from the equations of motion of our reduced model, successively applying the algorithm to calculate the transformation up to $7^\textrm{th}$ order to eliminate all non-resonant terms, and keeping up to $8$-wave resonances. We find that the amplitudes at which the normal form transformation diverge very closely match with the amplitudes at which a finite-amplitude phenomenon called $precession$ $resonance$ (Bustamante $et$ $al.$ 2014) occurs, characterised by strong energy transfers. This implies that the precession resonance mechanism cannot be explained using the usual methods of normal forms in wave turbulence theory, so a more general theory for intermediate nonlinearity is required.