In 2016, at the Liège Colloquium, we presented preliminary evidence that mixed-layer instabilities modify the multifractal properties of Sea Surface Temperature (SST) fields. Ten years later, those initial ideas have matured and merged with recent advances in turbulence theory, leading to a coherent framework for analyzing submesoscale dynamics. Over the past decade, we have also refined the methodological tools required to apply this framework to both observational datasets and high‑resolution numerical simulations.

In this colloquium, we first outline the theoretical basis that links the governing dynamical equations to local singularity exponents, as well as practical methods used to estimate these exponents from observations and numerical models. We then demonstrate the applicability of this approach to both oceanic and atmospheric flows, and compare the resulting singularity exponents with those obtained from Direct Numerical Simulations (DNS) of isotropic turbulence. Finally, we use this framework to explore how upper‑ocean fronts modulate the interscale transfer of energy.

Within the multifractal framework, a coarse‑grained turbulent variable $s(\vec{x})$ exhibits local power‑law behavior at sufficiently small scales, characterized by the singularity exponents $h(\vec{x})$: \[ \overline{|\nabla s|}_{\ell} (\vec{x})\equiv\int_{{\rm I\!R}^d} d\vec{x}'G_\ell(\vec{x}') |\nabla s|(\vec{x}+\vec{x}')\sim\left(\frac{\ell}{\ell_0}\right)^{h(\vec{x})} \] where $\vec{x}$ is the position; $d$ the dimension of the space; $\nabla$ the $d$-dimensional gradient operator; $G_\ell(\vec{x})$ the low-pass filter at $\ell$, the scale of analysis; and $\ell_0$ an integral scale. As shown by Parisi and Frisch (1985), such local scaling behavior underlies the global scaling laws of structure functions and, consequently, turbulent energy spectra. The central quantity in this framework is the singularity spectrum $D(h)$, which gives the fractal dimension of the set of points sharing a given exponent.

Here, we assess the validity of this approximation -- often referred to as the multifractal conjecture -- and evaluate the robustness of our method for computing singularity exponents. Using numerical simulations and large enough datasets, we verify that: (1) coarse‑grained gradients indeed display power‑law behavior, and (2) the scaling of structure functions can be reconstructed from the singularity spectrum. By coarse‑graining the governing equations, we derive an expression for the large‑scale energy budget and demonstrate how the energy flux depends on the local singularity exponent. Comparisons between high‑resolution ocean models and the corresponding singularity spectra of the velocity field show strong agreement.

In an oceanic context, singularity exponents of tracers, such as temperature or density, can serve as proxies for frontal intensity. This offers a physically grounded perspective on the role of fronts in the energy cascade and the statistical organization of flow. Through theoretical and experimental analysis, we demonstrate that the smallest exponents, which serve as a measure of the intensity of the strongest fronts, are related to the anomalous scaling of the structure functions. This relationship provides a unifying explanation for the results initially reported at the Liège Colloquium a decade ago.