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, they 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 injection rates $\epsilon_j(k)$ and KE transfers $F(k)$ across scales using sparse data. This approach requires inverse methods to convert between separation $r$ and wavenumber $k$ space. A previous study employed the $D3(r)$ framework to estimate $F(k)$ and $\epsilon_j(k)$ using non-negative least squares (NNLS), assuming that $F(k)$ is an increasing function of $k$, an assumption not always satisfied. In this study, an improved methodology is presented to estimate $F(k)$ and $\epsilon_j$ using regularized least-squares (RLS), where the inclusion of prior uncertainty in $D3(r)$ and $\epsilon_j$ reduces overfitting. Moreover, the improved methodology allows for estimating both positive and negative $\epsilon_j$ and makes no assumptions about the shape of $F(k)$. RLS quantitatively diagnoses the structure of $F(k)$ in an isotropic quasi-geostrophic turbulence simulation, including both positive and negative $\epsilon_j(k)$, an aspect unattainable with NNLS. This improved methodology is also applied to data from two drifter experiments in the Gulf of Mexico. The analysis reveals the presence of bi-directional energy transfers, with a KE inverse cascade at mesoscales in both seasons and a forward cascade at submesoscales that is stronger in winter than summer. Unlike NNLS, RLS fits $D3(r)$ better as the method detects wavenumbers where $\epsilon_j<0$. This improved methodology allows for a more refined analysis of KE transfers from sparse observations.