An Improved Methodology to Estimate Cross-Scale Kinetic Energy Transfers from Third-Order Structure Functions using Regularized Least-Squares

Manuel Othon Gutierrez-Villanueva , Bruce Cornuelle, Sarah T. Gille , Matthew R, Mazloff, Dhruv Balwada

Several methods exist for estimating cross-scale kinetic energy (KE) transfers; however, most are ill-adapted for sparse ocean observations, hindering the study of oceanic KE transfers. A newly developed third-order structure function, $D3(r)$, framework allows estimation of KE injections $\xi_j(k)$ and spectral flux $F(k)$ across scales using sparse data. This approach requires inverse methods to convert between separation $r$ and wavenumber $k$ space. A previous study employed the structure-function framework to estimate $F(k)$ and $\xi_j(k)$ using non-negative least squares (NNLS), assuming that the spectral flux is an increasing function of wavenumber, an assumption not always satisfied. Here, an improved methodology is presented to estimate $F(k)$ and $\xi_j$ using regularized least-squares (RLS), where the inclusion of prior uncertainty in $D3(r)$ and $\xi_j$ reduces overfitting. Moreover, the improved methodology allows for estimating both positive and negative injections without making assumptions about the shape of the spectral flux. As a proof of concept,  the improved methodology was implemented in an eddy-rich quasi-geostrophic simulation output. RLS quantitatively diagnoses the structure of $F(k)$, including both positive and negative $\xi_j(k)$, an aspect unattainable with NNLS. The improved methodology was then applied to data from two drifter experiments in the Gulf of Mexico. The analysis reveals the presence of bidirectional energy transfers, with a KE inverse transfer at mesoscales in both seasons and a forward transfer at submesoscales that is stronger in winter than in summer. Unlike NNLS, RLS fits $D3(r)$ better as the method detects wavenumbers where $\xi_j<0$ while preserving smoothness. This improved methodology allows for a more refined analysis of KE transfers from sparse observations.