**3. Method**

#### *3.1. Singularity Analysis*

Singularity analysis is the key stone of the so-called Microcanonical Multifractal Formalism (MMF) [10]; where the fluid is understood as a hierarchical arrangement of different fractal components, each own with its own fractal dimension and characterized by a particular value of the singularity exponent. Singularity exponents are related to the ocean circulation and thus they are not specific to any particular scalar under study. In fact, it has been verified that with a good approximation singularity lines coincide with the streamlines of the flow [20], confirming that singularity exponents are characterized by the flow and not scalar-specific. The emergence of such a singular structure is the

result of the advective forces acting on a quasi-2D turbulence regime, and can be thus observed for scales ranging from kilometers to the planetary scale.

The calculation of singularity exponents in a given noisy, discretized signal requires the use of an appropriate interpolation scheme, the most usual one being wavelet projections. Let *s* be an arbitrary 2D scalar signal. It will be said that *s* has a singularity exponent *h*(*x*) at the point *x* if, for any wavelet function Ψ the following relation holds:

$$\begin{split} T\_{\Psi} |\nabla s|(\mathbf{x}, r) &\equiv \int d\mathbf{x}' \, |\nabla s|(\mathbf{x}') \, \frac{1}{r^2} \Psi \left( \frac{\mathbf{x} - \mathbf{x}'}{r} \right) \\ &= a(\mathbf{x}) \, r^{h(\mathbf{x})} + o\left(r^{h(\mathbf{x})}\right) \end{split} \tag{1}$$

where *r* stands for an arbitrary scale parameter (normalized by the integral scale so it is dimensionless and smaller than 1) and *o rh*(*x*) is a term that becomes negligible compared to *rh*(*x*) when *r* goes to zero. The amplitude function *α*(*x*) does not depend on the particular scale at which the wavelet projection is calculated and has the same units as the gradient of the scalar.

The left hand side, i.e., *T*Ψ|∇*s*|(*x*,*r*), is called the wavelet projection of the gradient modulus of *s* over the wavelet Ψ, and represents a local zoom of variable size around the point *x*. What is important in Equation (1) is the function *h*(*x*), which is called the singularity exponent of the function *s* at the point *x*. The singularity exponent is, by construction, a dimensionless measure of the regularity or irregularity of the function *s* around the point *x*.
