The Multifractal Theory of Turbulence on the Oceanic Energy Flux Between Scales

How the energy propagates between different oceanic scales is a fundamental problem that is still far from being completely understood. Not only is it important for theoretical considerations but it is also critical for improving the parameterization of oceanic models. Using the coarse-grain method, one can perform an energetic study on a turbulent flow such as the ocean that is local in both space and scale (Contreras et al. (2023)). Thus, through the application of a mathematical filter operation, the spatial and scaling properties of the local flux of energy between scales (Π_r(x,t)) were investigated in oceanic simulations. To this end, this work exploits the behaviour of coarse-grained fields at small scales. When the filter scale is small, the coarse-grained velocity verifies the multifractal hypothesis:

The singularity exponents (h(x,t)) constitute a local measure of the degree of continuity of the underlying turbulent flow and define a multifractal decomposition into universality classes. From the multifractal hypothesis, a scaling law can be derived theoretically for Π_r (see Isern-Fontanet and Turiel (2021) and references therein):

One month of data was analysed from a numerical simulation of the circulation in the North Atlantic Ocean generated using the Coastal and Regional Ocean COmmunity (CROCO) model, with a 6-7 km spatial resolution. Π_r was computed through the application of a low-pass filter using a top-hat kernel. h(x,t) were extracted from the velocity field using the approach developed by Pont et al. (2013). The spatial analysis supports the existence of a connection between singularity and intensity of Π_r. The scaling analysis found that Π_r obeyed a scaling law governed by the exponent 2h+1 rather than the theoretical prediction of 3h+2. This finding is supported by the computation of the critical exponent in the singularity spectrum, h_∞≈-0.5, under a numerical error of O(0.1). Such discrepancy in the representation of the scaling of Π_r suggests that the Reynolds tensor scales as h+1 rather than 2h+2. Furthermore, it implies that the parameterization utilized by this model affects the representation of the turbulent energy cascade in the simulation and requires a compensation.

M. Contreras, L. Renault and P. Marchesiello, Understanding Energy Pathways in the Gulf Stream, Journal of Physical Oceanography, 53, pp 719-736, 2022.
J. Isern-Fontanet and A. Turiel, On the connection between intermittency and dissipation in ocean turbulence: A multifractal approach, Journal of Physical Oceanography, 51, pp. 2639–2653, 2021.
O. Pont , A. Turiel and H. Yahia, Singularity analysis of digital signals through the evaluation of their unpredictable point manifold, International Journal of Computer Mathematics, 90:8, 1693-1707, 2013.