Editorial Type:
Article
Article Type:
Research Article

On the Connection between Intermittency and Dissipation in Ocean Turbulence: A Multifractal Approach

Jordi Isern-Fontanet aInstitut de Ciències del Mar, ICM-CSIC, Barcelona, Spain

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https://orcid.org/0000-0002-9324-608X
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Antonio Turiel aInstitut de Ciències del Mar, ICM-CSIC, Barcelona, Spain

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Online Publication:
06 Aug 2021
Print Publication:
01 Aug 2021
DOI:
https://doi.org/10.1175/JPO-D-20-0256.1
Page(s):
2639–2653
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Abstract

The multifractal theory of turbulence is used to investigate the energy cascade in the northwestern Atlantic Ocean. The statistics of singularity exponents of horizontal velocity gradients computed from in situ measurements at 2-km resolution are used to characterize the anomalous scaling of the velocity structure functions at depths between 50 and 500 m. Here, we show that the degree of anomalous scaling can be quantified using singularity exponents. Observations reveal, on one side, that the anomalous scaling has a linear dependence on the exponent characterizing the strongest velocity gradient and, on the other side, that the slope of this linear dependence decreases with depth. Since the observed distribution of exponents is asymmetric about the mode at all depths, we use an infinitely divisible asymmetric model of the energy cascade, the log–Poisson model, to derive the functional dependence of the anomalous scaling with the exponent of the strongest velocity gradient, as well as the dependence with dissipation. Using this model we can interpret the vertical change of the linear slope between the anomalous scaling and the exponents of the strongest velocity gradients as a change in the energy cascade. This interpretation assumes the validity of the multifractal theory of turbulence, which has been assessed in previous studies.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jordi Isern-Fontanet, jisern@icm.csic.es

Abstract

The multifractal theory of turbulence is used to investigate the energy cascade in the northwestern Atlantic Ocean. The statistics of singularity exponents of horizontal velocity gradients computed from in situ measurements at 2-km resolution are used to characterize the anomalous scaling of the velocity structure functions at depths between 50 and 500 m. Here, we show that the degree of anomalous scaling can be quantified using singularity exponents. Observations reveal, on one side, that the anomalous scaling has a linear dependence on the exponent characterizing the strongest velocity gradient and, on the other side, that the slope of this linear dependence decreases with depth. Since the observed distribution of exponents is asymmetric about the mode at all depths, we use an infinitely divisible asymmetric model of the energy cascade, the log–Poisson model, to derive the functional dependence of the anomalous scaling with the exponent of the strongest velocity gradient, as well as the dependence with dissipation. Using this model we can interpret the vertical change of the linear slope between the anomalous scaling and the exponents of the strongest velocity gradients as a change in the energy cascade. This interpretation assumes the validity of the multifractal theory of turbulence, which has been assessed in previous studies.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Jordi Isern-Fontanet, jisern@icm.csic.es
Keywords: Atlantic Ocean; Mesoscale processes; Turbulence; In situ oceanic observations

1. Introduction

Predicting the evolution of a turbulent flow such as the ocean motion remains a long-standing problem in physics. Beyond the difficulties of developing a closed theory from fundamental laws, the range of interacting scales involved in oceanic motions, i.e., between 10−3 and 107 m, is too large to be resolved by ocean and climate numerical models (Ferrari and Wunsch 2009; Poje et al. 2014; Su et al. 2018). As a consequence, some kind of approximation or parameterization of the unresolved scales has to be implemented to correctly reproduce the flux of energy between scales (Pearson et al. 2017; Dubrulle 2019). But, although in three-dimensional turbulence energy is known to cascade from scales at which it is injected toward smaller scales where it is utterly converted to heat, the details of such cascade in the ocean are rather complex and still poorly understood (Müller et al. 2005; McWilliams 2016; Renault et al. 2019). This hinders its correct parameterization and becomes a source of errors in ocean and climate models (Flato et al. 2013).

In the statistical equilibrium, the global characteristics of the three-dimensional energy cascade are controlled by energy dissipation at small scales and its statistical properties completely characterize the whole statistics of the velocity field. As a consequence, the spotty nature of dissipation imprints an intermittent behavior to velocities. The statistical impact of intermittence can be seen in the so-called anomalous scaling of the velocity structure functions, i.e., the moments of velocity differences. These effects could not be explained by Kolmogorov’s 1941 theory and required some refinements (Kolmogorov 1962; Oboukhov 1962). The multifractal theory of fully developed turbulence (Benzi et al. 1984; Parisi and Frisch 1985) allowed to fill this theoretical gap. Multifractal theory assumes that dissipation and velocity are defined on multiple sets of points with a fractal dimension, each one associated to a singularity exponent in a continuum spectrum. The cornerstone of this theory is the singularity spectrum, which links the geometry of the flow (the fractal dimensions of the fractal components) with the scaling properties of the variable and flow statistics (Boffetta et al. 2008). Although the validity of the multifractal framework has been assessed for a wide range of scales, O⁡(1) m–O⁡(104) km, in the oceans (Lovejoy et al. 2001; Turiel et al. 2005; Isern-Fontanet et al. 2007; Seuront and Stanley 2014; Cabrera-Brito et al. 2017; Sukhatme et al. 2020), the choice of the relevant singularity spectrum model (Kolmogorov 1962; Schertzer and Lovejoy 1987; She and Waymire 1995) and its parameters depends on unknown underlying physical processes and remains an open question in turbulence (Dubrulle 2019).

Here, we investigate the connection between intermittency and singularity spectra in the upper layers of the ocean, below the mixed layer, at scales O⁡(1) –O⁡(100) km, and their dependence on dissipation. At these scales, rotation and stratification introduce significant changes in the energy cascade in comparison with the cascade that can be observed at scales less than O⁡(1) m. Our approach exploits recent advances in computational mathematics, which have led to a fast and robust algorithm for computing singularity exponents (Pont et al. 2013). Singularity spectra are thus obtained from the singularity exponents derived from measurements by a ship-of-opportunity in the northwestern Atlantic Ocean (Rossby et al. 2010; Wang et al. 2010; Callies et al. 2015). This allows us to avoid many numerical problems (as for instance data scarcity) (Turiel et al. 2006) and to identify the location of the most dissipative structures. Theoretical predictions from energy cascade models can be tested and used to provide a physical interpretation of the velocity intermittence.

The paper is organized as follows. Section 2 provides a coherent view of the multifractal theory of turbulence, sections 3 and 4 are about data and procedures, section 5 describes the approach used to analyze the data and outlines the key findings, and section 6 presents a theoretical justification of our findings. The general discussion of our study can be found in section 7 and conclusions in the final section (section 8).

2. The multifractal theory of turbulence

a. Energy budget of coarse-grained velocities

The momentum equation for the Bousinesq approximation is given by
∂u∂t+u⋅∇u=−2Ω×u−1ρ0 ∇p+bez+ν∇2u+F,
where u(x) is the velocity field; p(x) the pressure; Ω the Earth rotation; ν the viscosity; b(x) the buoyancy; F(x) an external forcing; ρ0 the reference density and ez is the vertical unit vector (Vallis 2006).1 Following Eyink (2005), the coarse-grained momentum equation at scale r is derived by applying a low-pass filter to velocities,
u¯r⁡(x)≡∫ℝddx′ 1rd G⁡(xr)u⁡(x+x′),
and the other variables, where d is the geometric dimension of the space. The filtering function G(x) is taken to be positive, normalized and G(x) → 0 rapidly for |x|→∞. Moreover, the above filtering is such that it commutes with spatial derivatives, e.g.,
∇⋅u¯r=⁡(∇⋅u)¯r.
Then, the resulting coarse-grained momentum equation is
∂u¯r∂t+u¯r⋅∇u¯r=−2Ω×u¯r−1ρ0 ∇p¯r+b¯rez+ν∇2u¯r+F¯r−∇⋅Tr.
The new term Tr⁡(x) in the filtered equations is the turbulent stress tensor defined by
Tr⁡(x)≡uu¯r−u¯ru¯r,
which is responsible of coupling large scales (>r) with small scales (>r). The difficulty to write Tr⁡(x) as a function of u¯r is what is called the closure problem in turbulence.
The equation for the conservation of energy of the large-scale flow |u¯r|2/2 is derived multiplying Eq. (4) by u¯r⁡(x), which gives
12∂|u¯r|2∂t+∇⋅Jr=−Πr−εr+b¯rez⋅u¯r+F¯r⋅u¯r.
The second term on the left-hand side of Eq. (6) is the spatial transport of energy, with
Jr⁡(x)≡12|u¯r|2u¯r+1ρ0 p¯ru¯r−ν2 ∇|u¯r|2+u¯r⋅Tr.
On the right-hand side of Eq. (6), the first term is the local flux of energy between scales
Πr⁡(x)≡−S¯r:Tr,
where S¯r is the large-scale strain tensor
S¯r⁡(x)=12 ⁡(∇u¯r+∇u¯rT).
The remaining terms on the right-hand side are source and sink terms. In particular, the second term is the energy dissipation at scale r,
εr⁡(x)≡ν|∇u¯r|2;
the third term is the conversion of potential energy −b¯r to kinetic energy |u¯r|2/2; and the last term corresponds to the exchange of energy due to external forces [see Aluie et al. (2018) for details]. These last three terms (dissipation, energy conversion, and external forces) have different scale dominance and do not mix large scales with small scales. Indeed, energy dissipation dominates at small scales; external forcing is traditionally assumed to dominate at large scales;2 and the conversion between potential and kinetic energy, which is dominated by baroclinic instability and frontogenesis, may have some dominants scales, e.g., the Rossby radius of deformation.
The focus of this study is at the range between O⁡(1) km and O⁡(100) km, i.e., at scales much larger than the dissipation scales, which implies that
|Πr⁡(x)|≫εr⁡(x).
Although energy dissipation at the scale r can be omitted in the equations at theses scales, dissipation at small scales is expected to control the flux of energy in the inertial range. Consequently, energy dissipation is at the core of theories explaining the energy cascade. In particular, according to Kolmogorov’s refined similarity hypothesis (Kolmogorov 1962), velocity differences
Δru⁡(x)≡|u⁡(x+r)−u⁡(x)|
and coarse-grained dissipation [Eq. (10)] should be related by
Δru~⁡(rεr)1/3,
where u(x) is an unspecified velocity component (Frisch 1995; Davidson 2013). Here, the symbol ~ in the above expression, and throughout the whole paper, is defined as follows. Given two functions f(x, r) and g(x, r); they are related by f(x, r) ~ g(x, r) if f(x, r) ≈ c(x)g(x, r) in the limit r → 0, where c(x) can be a function of the position x, but not of the scale r. The function c(x) has the physical dimensions necessary to balance both sides of the equation.

b. The multifractal hypothesis

A fundamental property of turbulent flows is the nonvanishing of energy dissipation in the limit of zero viscosity (or infinite Reynolds number). This empirical result requires that velocity gradients must have singularities to keep a finite dissipation at infinite Reynolds numbers (Eyink 2018). This leads to the multifractal hypothesis of turbulence (Frisch 1995). According to it, in the limit of infinite Reynolds numbers, velocity gradients are characterized by local power-law scalings in the inertial range: at each point x there is an exponent h(x) ∈ (h∞, ∞), such that3
|∇u|¯r⁡(x)~⁡(rR)h⁡(x),
where R is an integral scale and r ≪ R. The value of the exponents h(x), also known as the Hölder exponents or Hurst exponents, are a measure of the regularity of the field.
The multifractal hypothesis for velocity gradients implies that the scaling exponents of velocity differences [Eq. (12)] are
Δru⁡(x)~⁡(rR)h⁡(x)+1
[see Turiel et al. (2007) for a formal derivation]. Moreover, armed with these scalings and using heuristic arguments, it is simple to derive the scaling for the coarse-grained dissipation, i.e.,
εr⁡(x)~⁡(rR)2h⁡(x),
and the scaling for the local energy flux, i.e.,
Πr⁡(x)~⁡(rR)3h⁡(x)+2,
[see Eyink (1995, 2019) for a rigorous derivation].

c. The singularity spectrum

The singularity spectrum D(h) is a function that for each value of the singularity exponent h, it provides the fractal dimension of the associated fractal component Fh (Fh is the set of points that have the same singularity exponent h). This function determines the statistical properties of the flow. Indeed, the probability of finding a given h at the resolution r is
Pr⁡(h)∝⁡(rR)d−D⁡(h),
where d is the dimension of the space. Then, using the multifractal hypothesis [Eq. (14)] and such probability, it is possible to estimate the behavior of the moments of the velocity gradients for r ≪ R as
〈|∇u|¯rp〉~∫ℝddμ⁡(h)⁡(rR)ph⁡(rR)d−D⁡(h),
where dμ(h) is a measure that gives the weight of the different exponents (Frisch 1995). The moments of velocity gradients have the property of self-similarity
〈|∇u|¯rp〉~⁡(rR)τ⁡(p),
whose scaling function τ(p) is related to the D(h) by a Legendre transform pair
τ⁡(p)=minh[ph+d−D⁡(h)]
and
D⁡(h)=minp[ph+d−τ⁡(p)],
as shown by Parisi and Frisch (1985). This transform pair implies that the singularity exponent h and the order of the moments p are the derivatives of τ(p) and D(h), respectively,
p=dDdh,
and
h=dτdp
(e.g., Zia et al. 2009).
One consequence of the scaling of the moments of velocity gradients is that the moments of velocity differences, known as structure functions, are also self-similar
Sp⁡(r)≡〈⁡(Δru)p〉~⁡(rR)ζ⁡(p),
where ζ(p) is a continuous function of the moment order p. Using that
〈⁡(Δru)p〉~〈⁡(rR|∇u|¯r)p〉~⁡(rR)p+τ⁡(p)
and Eq. (25), the scaling of the velocity structure functions is given by
ζ⁡(p)=p+τ⁡(p)
and, thus, they are also determined by D(h). Moreover, the singularity spectrum will also determine the behavior of any variable related to velocity gradients by a linear scaling when r ≪ R (see the appendix).
Within this framework, it is usually assumed that the mean value of |∇u|¯r does not depend on the scale (because of a simple commutation between the spatial average and the coarse-grain filter). This implies that
τ⁡(1)≡0,
which becomes a constraint over the singularity exponents. Indeed, using Eq. (21), it implies that
d−D⁡(h1)=−h1,
where h1 is the singularity exponent associated to the moment of order 1, so it verifies D′(h1) = 1 (e.g., Turiel and Parga 2000).

d. Singularity spectrum models

The characteristics of the energy cascade determine the functional dependency of D(h) and, thus, the geometrical properties of the flow (fractal dimensions). As a consequence, different cascade models give rise to different singularity spectra that can be then compared to observations to get insight about the energy cascade [see Seuront et al. (2005) for a review]. These models are constrained by the nonvanishing of energy dissipation at infinite Reynolds numbers. Indeed, according to Onsager’s theory of 1949, it implies that velocity differences [Eq. (15)] should have at least some singularities such that h + 1 ≤ 1/3 (Eyink and Sreenivasan 2006; Eyink 2018). Therefore, the singularity spectrum models should have singularity exponents equal or less than
h0=−23
somewhere in the flow.
The simplest model that includes this constrain comes from Kolmogov’s theory of 1941, which corresponds to the case in which there is a single scaling exponent h0,
τ⁡(p)=h0p
and
D⁡(h)={dh=h00h≠h0.
Then, the scaling of the structure functions is linear ζ(p) = (h0 + 1)p with the slope of 1/3 predicted by Kolmogorov. Nevertheless, the observed velocity structure functions deviate from this linear scaling giving rise to the so-called anomalous scaling. As a consequence, turbulent regimes cannot have a single fractal component, i.e., a unique singularity exponent, but they must have a range of exponents implying that they are mutifractal [see Frisch (1995) for a detailed review of Kolmogorv’s theory and its connection with the multifractal framework].

It is important to recall that not only isotropic three-dimensional turbulence, but also geophysical turbulence is multifractal. Indeed, Onsager’s argument implies the existence of singularities in the flow and the observed anomalous scaling of the oceanic velocity structure functions at scales larger than the Ozmidov length (see Fig. 5 in Sukhatme et al. 2020) imply that a range of singularities must be present. In addition to this argument, it is worth mentioning that in previous studies it has already been stablished the validity of the multifractal framework in the ocean for scales covering from the largest submesoscales to the large scales (see Fig. 2 in Isern-Fontanet et al. 2007). Nevertheless, the singularity spectrum model that characterizes ocean turbulence in the submesoscale to mesoscale range still has to be determined.

A starting point to generate a multifractal turbulent cascade is to describe it as a multiplicative process of independent random variable identically distributed (Frisch 1995). Different probability density functions used to characterize their variability generate different τ(p) and D(h). Here, we focus on the two most extended models: the lognormal model (Kolmogorov 1962), which is still widely used in geophysical and three-dimensional turbulence (e.g., Pearson and Fox-Kemper 2018; Dubrulle 2019); and the log–Poisson model (She and Waymire 1995). Mathematically, the lognormal model assumes that the logarithm of the random variables has a normal distribution while for the log–Poisson model they follow a Poisson distribution. Although we do not present it here, it is worth mentioning the log–Lévy model (Schertzer and Lovejoy 1987), which is a generalization of the lognormal model.

The lognormal model (Fig. 1) is characterized by two parameters: the mode hd and the standard deviation σh given by
τ⁡(p)=hdp−12σh2p2,
and
D⁡(h)=d−12⁡(h−hdσh)2.
The singularity spectra derived from the lognormal model is symmetric about the mode, as it is evident from Eq. (34).
Fig. 1.
Fig. 1.

(a) Singularity spectra corresponding to the lognormal and log–Poisson models. (b) Scaling of the velocity structure functions predicted by the lognormal and log–Poisson models [see Eq. (27)]. The Legendre transform [Eqs. (21) and (22)] imply a one-to-one relation between both functions. This is indicated in the log–Poisson model with dots: (h∞, D∞) correspond to the the behavior of ζ(p) for p → ∞, (hm, Dm) correspond to (pm, ζm), and (hd, d) correspond to (0, 0). Black dots identify the key points used in this study. Red dots identify an arbitrary value. The red line has the same slope than the log–Poisson model at that point [see Eqs. (23) and (24)]. From this plot it is clear that D(h) has positive slopes for h < hd, which correspond to p > 0, while for h > hd it has negative slopes, which correspond to p < 0.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

The log–Poisson model (Fig. 1) is characterized by three parameters: the most singular exponent h∞, its fractal dimension D∞, and the intermittency parameter β,
τ⁡(p)=h∞p+(d−D∞)⁡(1−βp),
from where
D⁡(h)=D∞−h−h∞lnβ[1−ln⁡(−h−h∞⁡(d−D∞)lnβ)].
The condition given by Eq. (29) imposes an additional constraint:
β=1+h∞⁡(d−D∞),
which allows to reduce the parameters of this model to only two. An important feature of the log–Poisson model is that their parameters have a physical interpretation (She and Waymire 1995), rather than being ad hoc parameters to be fitted. According to this model, β measures the fraction of energy retained along the cascade (She and Leveque 1994; She and Waymire 1995; Turiel and Parga 2000). This model, contrary to the lognormal model, has a singularity spectrum that is asymmetric about the mode.

3. Data

a. ADCP measurements

In this study we analyzed 11 years (2005–16) of measurements provided by the Oleander project (Rossby et al. 2010), which consists of 360 weekly profiles at approximately 2-km resolution of horizontal velocities between New Jersey and Bermuda in the North Atlantic across the Gulf Stream measured by a ship-of-opportunity (see Fig. 2). The original velocity measurements can be downloaded at http://po.msrc.sunysb.edu/Oleander/. Data prior to 2005 were discarded because they correspond to a different instrument with lower performance. Velocities were decomposed into a longitudinal u1(x, y), along ship track and transverse ut(x, z), perpendicular to ship track component (with positive values to the northeast). These measurements covered a range of depths between 40 and 800 m. However, due to the scarcity of data below 500 m (Fig. 3a), the study was restricted to the range of depths between 40 and 500 m.

Fig. 2.
Fig. 2.

(left) Oleander’s cruises between Bermuda and New Jersey with the mean velocity at 50-m depth superimposed. (right) Mean transverse velocity with the positive direction to the northeast.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

Fig. 3.
Fig. 3.

(a) Number of observations at each depth. (b) Monthly mean vertical profiles of density derived from the WOA18 climatology using the TEOS-10 Equation of State for Sea Water (gray) and temporal evolution of the mean MLD (blue). The red band corresponds to the range of depths analyzed in this study.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

b. Ancillary data

Climatological values of vertical density variations were calculated from the World Ocean Atlas 2018 (WOA18) at 1/4° resolution produced by NOAA (https://www.nodc.noaa.gov/OC5/woa18/). We used monthly salinity and temperature profiles to first generate Absolute Salinity and Conservative Temperature and then compute the potential density anomaly with reference pressure of 0 dbar using the Python-GSW package (https://github.com/TEOS-10/python-gsw/). Resulting density fields were horizontally averaged over the box [31°, 39°N] × [76°, 62°W] to produce the vertical profiles shown in Fig. 3b.

This dataset was complemented with the mixed layer depth (MLD) climatology (de Boyer Montégut et al. 2004) produced by Ifremer (http://www.ifremer.fr/cerweb/deboyer/mld/home.php). Original data were also averaged over the same box as the WOA18 climatology to obtain the mean MLD shown in Fig. 3b.

Additional altimetric data were used to help interpret ADCP data (see the example in Fig. 4). In particular, sea surface heights were obtained from SSALTO/DUACS Delayed-Time Level-4 data measured by multisatellite altimetry observations over Global Ocean. Data are available at http://marine.copernicus.eu.

Fig. 4.
Fig. 4.

(a) Sea level map obtained from the measurements of multiple altimeters for 3 May 2008. The thick dotted line corresponds to the ship’s track. (b) Singularity exponents derived from transverse velocity around the date of the sea level field. White areas correspond to missing data. Black lines correspond to the transverse velocity (m s−1; positive northeastwards).

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

4. Methods

a. Retrieval of singularity exponents

Singularity exponents were computed from velocity gradients using Eqs. (2) and (14), where G(x) is the mother wavelet, which does not have the requirement of the admissibility condition (Turiel and Parga 2000). Since Oleander data consist of vertical transects, i.e., two-dimensional sections of velocities, the dimension of the space is d = 2. Then, from the coarse-grained field at the resolution of observation r0 [Eq. (14)], which is the smallest accessible scale, the singularity exponents were obtained as
h⁡(x)≈log|∇u|¯r0−log〈|∇u|¯r0〉log⁡(r0/R),
where 〈 〉 denotes a spatial average (see Turiel et al. 2007).
In this study, we used the algorithm described in Pont et al. (2013) to compute singularity exponents with two modifications. First, the anisotropy of data bins in the horizontal and vertical directions was taken into account by computing gradients in physical coordinates rather than pixel coordinates. Second, the smallest accessible scale r0 was computed as
r0R≈1Nobsd,
where r0d is the volume of each bin; Rd the total volume covered by the data; Nobs is the number of valid observations in a single transect; and d the dimension of the space. Notice that the existence of gaps in real data increases the smallest accessible resolution. The application of the algorithm proposed by Pont et al. (2013) requires, at least, on the order of 100 points.

b. Estimation of singularity spectra

Approximating the probability of finding an exponent h at the resolution of observation r0 [Eq. (18)] by its histogram, the singularity spectrum associated to a given sample of data can be estimated as
D⁡(hi)≈d−logNi−logNmaxlog⁡(r0/R),
where Ni is the number of observations within the bin [hi−(δh/2),hi+(δh/2)] and Nmax = maxi{Ni}. The largest fractal dimension is the dimension of the whole space d,
D⁡(hd)=max[D⁡(h)]≡d,
where hd is the mode of the singularity exponents. Moreover, the error bars of the singularity spectrum were estimated as
δDi=3|log⁡(r0/R)|Ni1/2
(see Turiel et al. 2006).

c. Singularity analysis of ADCP data

Vertical profiles of singularity exponents for the longitudinal and transverse components of velocity, hl(x, z) and ht(x, z), respectively, were computed for each velocity transect. In the limit for r ≪ R, the coarse-grained dissipation will be dominated by the most singular exponent and, thus,
h⁡(x,z)=min[hl⁡(x,z),ht⁡(x,z)].
Nevertheless, we have retained throughout the paper the values of the singularity exponents of both velocity components.4 Singularity spectra were obtained from the associated singularity exponents south of the continental shelf, which is located at 39°N (Fig. 4b), using Eq. (40). Singularity exponents at the borders of the domain or a data gap were removed, as the estimate of its value is of lower quality. To impose the translational invariance and correct any shift that may exist in the singularity exponents,5 two-dimensional singularity spectra were calculated for each transect (d = 2 and δh = 0.05) and Eq. (29) was used. Then, one-dimensional singularity spectra (d = 1, δh = 0.1) were computed for each transect and at each depth in order to explore the vertical variations of intermittency.

5. Analysis of the empirical singularity spectra

a. An example

Figure 4 shows an example of a vertical profile of transverse singularity exponents ht(x, z). Recall that singularity exponents measure the degree of continuity of velocity gradients in the limit r0/R → 0. In particular, negative values imply the existence of singularities. Sea surface height shown in Fig. 4a provides a view of the flow topology simultaneous to ship measurements. In this example, the ship crossed the Gulf Stream (between 37° and 38°N) and a mesoscale vortex centered at 36°N. Singularity exponents (Fig. 4b) exhibit a strong vertical coherence of those singularities associated to the Gulf Stream and those related to the vortex, which are located within the maxima of velocities.

The scaling of the local fluxes of energy implies that they are related at different scales by
Πr2⁡(x)≈⁡(r2r1)3h⁡(x)+2Πr1⁡(x),
with r2 < r1. Based on the above relation, the spatial distribution of singularity exponents (for instance, those in Fig. 4) maps the pathways through which energy is transferred toward smaller scales. Contrary to the computation of the local energy flux Πr(x), it is not necessary to know the velocity at small scales and can be used to analyze real observations [see Aluie et al. (2018) and the discussion there]. For sure, singularity exponents cannot quantify the flux of energy nor even estimate its sign. Nevertheless, at the resolution of the data (~2 km) we expect a direct energy cascade (McWilliams 2016). It is worth mentioning that, in contrast with isotropic three-dimensional turbulence, at coarser resolutions (>10 km) there is an inverse cascade and singularity exponents may not be the same. To test for this change, we would need more data than a velocity transect to keep the accuracy and precision of h reasonable (Pont et al. 2013).

The singularity spectrum at 100 m (Fig. 5) shows a characteristic convex curve, from which two features are evident: the asymmetry of this singularity spectrum and the existence of a minimum singularity h∞ with a fractal dimension D(h∞) > min[D(h)]. Recall that, we are analyzing a one-dimensional transect of a three-dimensional flow, implying that negative dimensions have to be interpreted as occurrence probabilities (Mandelbrot 1990). Indeed, a filament in a 3D space (d = 3) will have a dimension D = d − 2 = 1 while a surface will have a dimension of D = d − 1 = 2. Then, if we take an ADCP transect (d = 2) the dimension of the filament will be D = 0 and for a surface D = 1. If we take a fixed depth of the ADCP transect (d = 1) will give D = −1 for the filament or D = 0 for the surface. Moreover, the singularity spectrum does not change with the change of the spatial resolution, although as resolution is getting coarser the statistics is reduced and therefore the resulting spectra get noisier and the error bars larger. The independence of resolution of the singularity spectrum is a key characteristic of multifractal systems. In Turiel et al. (2007) you can find an extensive discussion on the validity of the multifractal approach in many different contexts.

Fig. 5.
Fig. 5.

Singularity spectrum at 100-m depth (dots) for the same transect with the corresponding error estimations (vertical lines) and bin width (horizontal lines). The light green interval corresponds to Δh−, bounded by h∞ to the left and by hd to the right. Dashed line corresponds to the log–Poisson model obtained using the observed h∞ and the β obtained at that depth. The dotted line corresponds to a parabola fitted to the data.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

b. Characterization of singularity spectra

The singularity spectrum shown in the example of Fig. 5 is representative of the spectra observed at different depths and times. A systematic characterization of each singularity spectrum, i.e., for each depth and transect, has been done using four shape parameters:

  1. Pearson’s first skewness coefficient or mode skewness sd, i.e., the skewness about the mode hd,
    sd⁡(z,t)≡〈h〉−hdσh,
    where 〈h〉 is the mean value and σh the standard deviation. The variable z refers to the depth and t to the central time of the transect from where the singularity exponents h were calculated.
  2. The most singular exponent h∞,
    h∞⁡(z,t)≡min⁡(h).
  3. The fractal dimension of its set D∞,
    D∞⁡(z,t)≡D⁡(h∞).
  4. The width of the singularity spectrum Δh−, defined as
    Δh−⁡(z,t)≡hd−h∞,
    which is a measure of the anomalous scaling.

Indeed, the value of singularity exponent hd associated to the maximum of D(h) corresponds to p = 0 [horizontal tangent line to D(h)] and h∞ corresponds to the largest (in absolute value) negative slope of τ(p) and thus to the largest moments p → ∞. Recall that D(h) is a convex function and p and h are the slopes of D(h) and τ(p), respectively [Eqs. (23) and (24), see also Fig. 1]. Consequently, Eq. (48) can be rewritten using Eqs. (24) and (27) as
Δh−=dτdp|p=0−dτdp|p→∞
=dζdp|p=0−dζdp|p→∞
that is, Δh− is a measure of the difference between the slopes of ζ(p) at the origin and at the largest orders of the structure function. In our calculations, the mode hd was estimated by adjusting a parabola in a small region around the maximum of D(h), and then calculating its maximum from the adjusted curve.

c. Analysis of the shape parameters

Mode skewness sd [Eq. (45)] is a measure of the symmetry around the mode and can be used to test the symmetry conjecture, i.e., the conjecture stating that the singularity spectrum is symmetric (Dubrulle 2019). Figure 6a shows the histogram of sd at all depths for both the transverse and the longitudinal velocity components, revealing that sd has a negligible probability of being negative or even zero, thus evidencing that the singularity spectra are almost always asymmetric.

Fig. 6.
Fig. 6.

(a) Observed distributions of mode skewness sd and (b) fractal dimension of the most singular set D∞. (c) Time-averaged vertical variation of anomalous scaling 〈Δh−〉 (solid lines) and most singular exponent 〈h∞〉 (dashed lines). Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

The most singular exponent h∞ [Eq. (46)] measures the strength of the most intense velocity gradient. Its value ranges from −0.6 to −0.1, with a tendency to decrease with depth; transverse and longitudinal components have different h∞, with the first being more singular (smaller h, Fig. 6c). Moreover, the histograms for h∞ are dominated by the contribution of the Gulf Stream and the mesoscale vortices detached from it, as it can be seen in the example of Fig. 4b.

The fractal dimension of the most singular set D∞ [Eq. (47)] spreads between −0.5 and 0.25 (Fig. 6b), which would correspond to values between 1.5 and 2.25, if the singularity spectrum would have been computed in three dimensions. The distribution of D∞ shows a peak around D∞ ≈ − 0.4 due to those sets composed of a single point, which is indicative of the need to have more observations to accurately estimate this dimension [see Eq. (42) and the error bars in Fig. 4c]. Besides, dissipative structures defined by h∞ are not filamentary as in She and Leveque (1994), i.e., have fractal dimensions larger than one (D∞ > −1 for d = 1, Fig. 6b).

The width of singularity spectrum spreads from 0.2 to 0.7 approximately, suggesting that they have significant variability, with the spectra of the transverse velocity component being wider (Fig. 7). The time-averaged amplitude of the singularity spectrum 〈Δh−〉 is maximum at depths between 100 and 250 m (Fig. 6c). Moreover, the comparison between all the observed Δh− and h∞ at each depth shows a decreasing linear dependence between them (Fig. 7):
Δh−≈m⁡(z)h∞,
with m(z) < 0. We estimate the value of m(z) at each depth z by means of a least squares regression of Δh−(z) versus h∞(z) obtained from both, longitudinal and traverse velocity components. Results are shown in Fig. 8a, along the linear correlation parameter between Δh− and h∞. Largest correlations, on the order of −0.8, are found in the upper layers of the ocean, where the range of values for h∞ is larger, and reduce to values on the order of −0.75 below 150 m. In the lower layers, where the ranges of values of both Δh− and h∞ are smaller, the linear correlations are on the order of −0.6.
Fig. 7.
Fig. 7.

Scatterplot between the anomalous scaling measure Δh− and the singular exponents h∞ at 50, 200, and 350 m, respectively. The dashed line corresponds to the fitted straight line. Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

Fig. 8.
Fig. 8.

(a) Slope [red; m(z) in Eq. (51)] and linear correlation parameter between h∞ and Δh− (blue). (b) Intermittency parameter [red; β in Eq. (55)] and normalized RMSE [orange and green; erms in Eq. (59)]. Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively. The gray band indicates the depths that can be within the mixed layer.

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

On the other hand, the vertical variation of the shape parameters (h∞ and Δh− in Fig. 6) and the histogram of singularity exponents (Fig. 9) imply changes in the energy transferred between scales. This is evident from Eq. (44) for the changes of singularity exponents. The Kolmogorov singularity exponent h0 = −2/3 (see section 2) would imply that all energy is transferred through scales without loses. Nevertheless, as shown in Fig. 9, h0 rarely takes the value −2/3 although the errors in the estimation of h difficult to univocally confirm this observation. The interpretation of m(z), i.e., the slope between h∞ and Δh−, in terms of the energy cascade requires the introduction of a model for the energy transfer.

Fig. 9.
Fig. 9.

Empirical histograms of singularity exponents at each depth. The range of values for the singularity exponents go from the Kolmogorov exponent h0 = −2/3 to 1 in bins of 0.008. The histogram relative frequencies N/Ntotal are computed during the whole period analyzed and are logarithmically represented with a color code. The solid line corresponds to the vertical variation of the mean singularity exponent while the dashed line corresponds to the vertical variation of the most probable value (mode).

Citation: Journal of Physical Oceanography 51, 8; 10.1175/JPO-D-20-0256.1

6. Analytical model

Our results show that singularity spectra are asymmetric (Fig. 6a), implying that those cascade models generating symmetric distributions of singularity exponents should be discarded. This is the case of the lognormal model, widely applied to three-dimensional turbulence (Dubrulle 2019). The log–Poisson model, on the contrary, is asymmetric and verifies D∞ > min[D(h)]. Moreover, it is based on a representation of the energy cascade and, thus, its parameters have a physical meaning rather than being set ad hoc as in the lognormal model or its generalization, the log–Lévy model. Figure 5c shows an example of the better fitting of the log–Poisson model to the observed singularity spectrum than the lognormal. Consequently, we used it to derive the analytical relationship between h∞ and Δh− [Eq. (51)].

a. Slope between Δh− and h∞

Using that D(hd) ≡ d, the log–Poisson model [Eq. (36)] gives
d=D∞−hd−h∞lnβ{1−ln[−hd−h∞⁡(d−D∞)lnβ]}.
Then, introducing the definition of Δh− [Eq. (48)] and reordering terms; we obtain
−⁡(d−D∞)lnβΔh−=1−ln[−Δh−⁡(d−D∞)lnβ].
The solution to this equation is
Δh−=−(d−D∞)lnβ.
and, inserting Eq. (37), it becomes
Δh−=lnβ1−β h∞.
Notice that the lognormal model does not provide any prediction for this relation because in this model h∞=−∞. Then, comparing Eqs. (51) and (55) we get that
m⁡(z)=lnβ1−β,
which reveals that the vertical decrease of observed slopes results in a reduction of the intermittency parameter (Fig. 8b). As a consequence, the dependence of intermittency with depth (Fig. 8b) is interpreted as vertical changes of the processes underlying the cascade, whose competition with changes in h∞ (Fig. 6c) leads to a maximum anomalous scaling of velocity structure functions between 100 and 250 m (Fig. 6c).
The relationship between anomalous scaling and dissipation can then be retrieved from Eq. (55) [or equivalently Eq. (51)], introducing the relation between the amplitude of the singularity spectrum and the change of slope in the structure functions scaling [Eq. (50)] on one side and singularity exponents of dissipation
h˜⁡(x)≡2h⁡(x),
on the other side;
dζdp|p=0−dζdp|p→∞=12lnβ1−β min⁡(h˜),
which states that anomalous scaling is a function of the intermittency parameter and the most singular exponent of dissipation. Recall that h˜∞≡min⁡(h˜). Notice that the above relation is independent from Kolmogorov’s Refined Similarity Hypothesis [Eq. (13)].

b. Model assessment

To get further evidence on the validity of the log–Poisson model, the normalized root-mean-square error (RMSE) has been defined as the RMSE of the differences between the model [Eq. (36)] and observations [Eq. (40)] divided by the standard deviation of observational error estimates [Eq. (42)]:
erms={∑i[Dobs⁡(hi)−Dtheory⁡(hi)]2∑iδDi2}1/2.
We have found that it is smaller than one only for depths below 100 m (Fig. 8b), implying that above this depth the deviations of our fit of the log–Poisson model from observations does not fall within the estimated errors. The existence of such discrepancies may be explained by the presence of the mixed layer (ML), only partly captured by Oleander data. Indeed, observations are within the ML only during the period October–March approximately for depths shallower than 100 m (Fig. 3b). The lack of an instantaneous determination of the ML prevents to separately investigate the energy cascade within it. Besides, the maximum amplitude of the singularity spectra i.e., the strongest anomalous scaling, corresponds to depths dominated by the seasonal variation of the pycnocline (40–200 m, Fig. 3b). It is worth mentioning, however, that even at these depths, singularity spectra are still asymmetric and h∞ and Δh− tend to be linearly related (see Fig. 8a).

7. Discussion

The multifractal theory of turbulence outlined in the section 2 is based on three main contributions: Onsager’s theory of ideal turbulence, as presented by Eyink (2018); the multifractal formalism described in Frisch (1995); and different models of turbulent cascade, mainly from Kolmogorov (1962) and She and Leveque (1994). Nevertheless, we have chosen to build it from the singularity exponents of velocity gradients |∇u|¯r instead of velocity differences Δrυ or the coarse-grained dissipation εr as it is usually done (Frisch 1995). The motivation is twofold. First, the observation of finite energy dissipation for infinite Reynolds numbers point to the existence of singularities in velocity gradients. Second, the existing method to compute singularities is based on the gradients of the turbulent variable to overcome the influence of long-range correlations (Turiel and Parga 2000). Notice, however, that it can be built based on any variable whose singularity exponents are a linear combination of the singularity exponents of velocity (see the appendix).

Since the seminal work of Parisi and Frisch (1985), 35 years ago, a large theoretical body has been developed concerning the multifractal theory of turbulence (see the reviews by Boffetta et al. 2008; Dubrulle 2019). Nevertheless, the lack of an appropriate method to compute singularity exponents has constrained its application mainly to the analysis of statistical quantities such as the moments of given variables or structure functions (e.g., Verrier et al. 2014; de Bruyn Kops 2015; Poje et al. 2017; Sukhatme et al. 2020). This has had two major consequences. First, some controversies, such as the validity of the lognormal or the log–Poisson models, have remained open (Dubrulle 2019). Second, the geometrical properties of singularity exponents have not been analyzed with detail. The publication of a fast and robust method to compute singularity exponents by Pont et al. (2013) should contribute to change this situation. Indeed, this study is one of the firsts attempts to use the multifractal theory of turbulence based on the direct computation of singularity exponents to get insight about the energy cascade in the ocean.

In this study, we have defined simple shape parameters that allow to characterize and discriminate among cascade models. Of particular importance is the test on the symmetry of singularity spectra. Such a test is difficult to apply to the structure functions or the moments of a turbulent variable because it would be required to compute the moments of negative order (see Fig. 1) that are usually ill behaved. On the contrary, using an approach based on the statistics of singularity exponents, it is rather simple. In particular, the asymmetry of the observed singularity spectra (Fig. 6) implies that the lognormal model should be discarded. Moreover, our results have shown that below the mixed layer the log–Poisson is able to reproduce the observed distribution of singularity exponents, within the observational errors. Furthermore using the log–Poisson model it has been possible to analytically derive the observed relation between the amplitude of the singularity spectrum and the most singular exponent. Although we cannot discard the idea that other models may yet fit the observations, particularly if better observations become available, the found evidence makes a strong case in favor of the log–Poisson model.

Most of the studies of the energy cascade in the ocean have focused on the spectral transfer of energy (Scott and Wang 2005; Capet et al. 2008; Khatri et al. 2018; Soh and Kim 2018). Nevertheless, the capability to compute the singularity exponents unveils the location of the regions where the energy cascade is stronger. As discussed in previous sections, singularity exponents give information about the local energy flux without the need to measure the velocity field below the resolution of observations (Aluie et al. 2018; Schubert et al. 2020). In particular, the spatial distribution of singularities shows that the smallest singularity exponents (h < 0) are mainly located in the Gulf Stream and mesoscale vortices (e.g., Fig. 4b) thus linking large- and small-scale intermittency (Feraco et al. 2018; Pouquet et al. 2019). Interestingly, in situ observations have shown that three-dimensional turbulence is enhanced in ocean fronts (D’Asaro et al. 2011). This picture is consistent with the interpretation of singularity exponents as a measure of the intensity of the flux of energy toward small scales suggested by Eq. (44). On the other side, the statistical properties of singularity exponents are similar to those provided by Lagrangian methods such as the finite-size Lyapunov exponents (Hernández-Carrasco et al. 2011), which provide information about frontal structures and the turbulent cascade (García-Olivares et al. 2007; d’Ovidio et al. 2009). Besides, the analysis based on singularity exponents allows to easily segmentate the domain into subareas and explore the geographical dependence of energy dissipation pathways. In particular, the analysis of the global model used by Pearson and Fox-Kemper (2018), or the model used by Su et al. (2018), should allow one to investigate the geographical variability of our findings. It is worth mentioning that our multifractal approach is also very well suited to validate ocean models due to its scale invariance which allows to compare data and models of different spatial resolutions (e.g., Skákala et al. 2016).

As a final remark, classical theories of turbulence assume that dissipation controls the energy cascade and thus it is a key variable to characterize velocities in the inertial range such as in Eq. (13), which is the so-called Kolmogorov’s refined similarity hypothesis. Nevertheless, the introduction of the scaling of velocity differences and coarse-grained dissipation [Eqs. (15) and (16)] into Eq. (13) implies that h has to be a constant, contradicting the observations here presented (Δh− > 0), or ν has to be a multifractal variable instead of a physical constant (at constant temperature). On the contrary, Kraichnan (1974) argued that, in the inertial range, the local energy flux Πr(x) should be used instead and proposed
Δru~⁡(rΠr)1/3,
what is known as the Kraichnan’s refined similarity hypothesis. Such relation has consistent scaling in both sides [see Eqs. (15) and (17)], which indeed points out to the use of the local energy flux Πr as the central variable in the inertial range instead of energy dissipation εr. However, following the classical approach in turbulence, we wrote Eq. (58) as a function of the singularity exponents of dissipation but, it is worth noting that, it could be written as well using any other variable whose singularity exponents are linearly related to h(x) (see the appendix).

8. Conclusions

A central concept of the multifractal theory of turbulence are the singularity exponent of velocity gradients. Although singularity exponents cannot quantify the flux of energy between scales nor its sign, they unveil the location of the regions where the energy cascade is stronger, with smaller values corresponding to more intense energy cascade. Observations have shown that, within the margins of errors, the singularity exponents are always larger than −2/3, the Kolmogorov exponent. On the other hand, the geometrical properties of these exponents, contained in the singularity spectra, also provide key information about the overall properties of the energy cascade.

Here, singularity spectra have been derived directly from singularity exponents thanks to recent advances in computational mathematics instead of deriving them using the canonical approach based on the scaling of the structure functions. This approach, which has been validated in previous studies, has shown that singularity spectra are asymmetric about the mode and therefore it dismisses the symmetry conjecture in the ocean. As a consequence, the log–Poisson model has been identified as a better candidate to explain observations than the widely used lognormal model. Moreover, ocean observations have unveiled a link between the anomalous scaling of the structure functions and the strong velocity gradients associated to mesoscale vortices and the Gulf Stream. Such an empirical relation has also been derived analytically using the log–Poisson model, and it has been possible to relate it to energy dissipation.

Armed with these results, and assuming the validity of the multifractal theory of turbulence, it is possible to interpret the vertical changes in the amplitude of the singularity spectra, which corresponds to changes in the anomalous scaling of the velocity structure functions, as changes of the energy cascade. The reported results underline the importance of investigating the intermittency properties of turbulent flows to get insight about the energy cascade and also the central role played by the singularity spectrum. Indeed, the singularity spectrum provides the connection between the arrangement of velocity gradients, anomalous scaling, intermittency, and the intensity of energy dissipation.

Acknowledgments

We thank the insightful comments done by Emilio García-Ladona and Joaquim Ballabrera-Poy and some anonymous reviewers. This work was supported by the Ministry of Economy and Competitiveness, Spain; the Ministry of Science and Innovation Spain; and FEDER, EU, through the National R+D Plan under TURBOMIX (CGL2015-73100-EXP) and L-BAND (ESP2017-89463-C3-1-R) projects and by CSIC-funded project reference 2020AEP185. We also acknowledge support from Fundación General CSIC (Programa ComFuturo). This work acknowledges the “Severo Ochoa Centre of Excellence” accreditation (CEX2019-000928-S).

Data availability statement

Singularity exponents derived from velocity measurements can be downloaded from the PANGAEA repository https://doi.pangaea.de/10.1594/PANGAEA.904616.

APPENDIX

The Multifractal Formalism for Derived Turbulent Variables

Given the singularity spectrum D(h) and the scaling of the moments τ(p) of |∇u|¯r⁡(x), it is possible to derive the corresponding functions for any other turbulent variable γ¯r⁡(x) related to |∇u|¯r⁡(x) by scaling relations of the form
γ¯r~rb|∇u|¯ra,
with
〈γ¯rp〉~rζ˜⁡(p).
Indeed, from Eqs. (20), (A1), and (A2) it can be shown that
ζ˜⁡(p)=pb+τ⁡(ap),
which is also related to the singularity spectrum of this new variable D˜⁡(h) by a Legendre transform pair:
ζ˜⁡(p)=minh[ph+d−D˜⁡(h)],
D˜⁡(h)=minp[ph+d−ζ˜⁡(p)].
Furthermore, the singularity exponents of the new variable h˜⁡(x) are related to those of |∇u|¯r⁡(x) by
h˜⁡(x)=b+ah⁡(x).
The scaling according to the log–Poisson model for the variable γ¯r⁡(x) related to |∇u|¯r⁡(x) by a relation of the type given by Eq. (A1) will be given by
ζ˜⁡(p)=pb+ah∞p+(d−D∞)⁡(1−βap)
and, using the Legendre transform [Eq. (A5)], the corresponding singularity spectrum is
D˜⁡(h)=D∞−h−ah∞−blnβa{1−ln[−h−ah∞−b⁡(d−D∞)lnβa]}.
From the above definition it is possible to define two new parameters h˜∞≡ah∞+b and β˜≡βa from which the original form of the log–Poisson model [Eq. (36)] is recovered. Moreover, the relation between the anomalous scaling of the structure functions and the singularity exponents of γ¯r⁡(x)
dζdp|p=0−dζdp|p→∞=1alnβ1−β [min⁡(h˜)−b],
notice that h˜∞≡min⁡(h˜).

As a final remark, if translational invariance [Eq. (28)] is not imposed to the data, the above equations should be used with a = 1 and b = τ(1).

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  • Pearson, B., B. Fox-Kemper, S. Bachman, and F. Bryan, 2017: Evaluation of scale-aware subgrid mesoscale eddy models in a global eddy-rich model. Ocean Modell., 115, 42–58, https://doi.org/10.1016/j.ocemod.2017.05.007.

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  • Poje, A. C., and Coauthors, 2014: Submesoscale dispersion in the vicinity of the deepwater horizon spill. Proc. Natl. Acad. Sci. USA, 111, 12693–12698, https://doi.org/10.1073/pnas.1402452111.

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  • Poje, A. C., T. M. Özgökmen, D. J. Bogucki, and A. D. Kirwan, 2017: Evidence of a forward energy cascade and kolmogorov self-similarity in submesoscale ocean surface drifter observations. Phys. Fluids, 29, 020701, https://doi.org/10.1063/1.4974331.

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  • Pont, O., A. Turiel, and H. Yahia, 2013: Singularity analysis of digital signals through the evaluation of their unpredictable point manifold. Int. J. Comput. Math., 90, 1693–1707, https://doi.org/10.1080/00207160.2012.748895.

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  • Pouquet, A., D. Rosenberg, and R. Marino, 2019: Linking dissipation, anisotropy, and intermittency in rotating stratified turbulence at the threshold of linear shear instabilities. Phys. Fluids, 31, 105116, https://doi.org/10.1063/1.5114633.

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  • Renault, L., P. Marchesiello, S. Masson, and J. C. McWilliams, 2019: Remarkable control of western boundary currents by eddy killing, a mechanical air-sea coupling process. Geophys. Res. Lett., 46, 2743–2751, https://doi.org/10.1029/2018GL081211.

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  • Rossby, T., C. Flagg, and K. Donohue, 2010: On the variability of gulf stream transport from seasonal to decadal timescales. J. Mar. Res., 68, 503–522, https://doi.org/10.1357/002224010794657128.

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  • Schertzer, D., and S. Lovejoy, 1987: Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes. J. Geophys. Res., 92, 9693–9714, https://doi.org/10.1029/JD092iD08p09693.

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  • Schubert, R., J. Gula, R. J. Greatbatch, B. Baschek, and A. Biastoch, 2020: The submesoscale kinetic energy cascade: Mesoscale absorption of submesoscale mixed layer eddies and frontal downscale fluxes. J. Phys. Oceanogr., 50, 2573–2589, https://doi.org/10.1175/JPO-D-19-0311.1.

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  • Scott, R. B., and F. Wang, 2005: Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry. J. Phys. Oceanogr., 35, 1650–1666, https://doi.org/10.1175/JPO2771.1.

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  • Seuront, L., and H. E. Stanley, 2014: Anomalous diffusion and multifractality enhance mating encounters in the ocean. Proc. Natl. Acad. Sci. USA, 111, 2206–2211, https://doi.org/10.1073/pnas.1322363111.

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  • Soh, H. S., and S. Y. Kim, 2018: Diagnostic characteristics of submesoscale coastal surface currents. J. Geophys. Res. Oceans, 123, 1838–1859, https://doi.org/10.1002/2017JC013428.

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  • Su, Z., J. Wang, P. Klein, A. F. Thompson, and D. Menemenlis, 2018: Ocean submesoscales as a key component of the global heat budget. Nat. Commun., 9, 775, https://doi.org/10.1038/s41467-018-02983-w.

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  • Sukhatme, J., D. Chaudhuri, J. MacKinnon, S. Shivaprasad, and D. Sengupta, 2020: Near-surface ocean kinetic energy spectra and small-scale intermittency from ship-based ADCP data in the Bay of Bengal. J. Phys. Oceanogr., 50, 2037–2052, https://doi.org/10.1175/JPO-D-20-0065.1.

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  • Turiel, A., and N. Parga, 2000: The multifractal structure of contrast changes in natural images: From sharp edges to textures. Neural Comput., 12, 763–793, https://doi.org/10.1162/089976600300015583.

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  • Turiel, A., J. Isern-Fontanet, E. García-Ladona, and J. Font, 2005: A multifractal method for the instantaneous evaluation of the stream-function in geophysical flows. Phys. Rev. Lett., 95, 104502, https://doi.org/10.1103/PhysRevLett.95.104502.

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  • Turiel, A., C. J. Pérez-Vicente, and J. Grazzini, 2006: Numerical methods for the estimation of multifractal singularity spectra on sampled data: A comparative study. J. Comput. Phys., 216, 362–390, https://doi.org/10.1016/j.jcp.2005.12.004.

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  • Turiel, A., H. Yahia, and C. J. Pérez-Vicente, 2007: Microcanonical multifractal formalism—A geometrical approach to multifractal systems: Part I. Singularity analysis. J. Phys. A: Math. Theor., 41, 015501, https://doi.org/10.1088/1751-8113/41/1/015501.

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  • Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 745 pp.

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  • Verrier, S., M. Crépon, and S. Thiria, 2014: Scaling and stochastic cascade properties of nemo oceanic simulations and their potential value for GCM evaluation and downscaling. J. Geophys. Res. Oceans, 119, 6444–6460, https://doi.org/10.1002/2014JC009811.

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  • Wang, D., C. Flagg, K. Donohue, and H. Rossby, 2010: Wavenumber spectrum in the gulf stream from shipboard adcp observations and comparison with altimetry measurements. J. Phys. Oceanogr., 40, 840–844, https://doi.org/10.1175/2009JPO4330.1.

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  • Zia, R., E. Redish, and S. McKay, 2009: Making sense of the legendre transform. Amer. J. Phys., 77, 614–622, https://doi.org/10.1119/1.3119512.

1

The time dependence of all variables has been dropped to alleviate the notation.

2

Notice, however, that the coupling at the ocean surface between the ocean and the atmosphere may span a wide range of scales in contrast with the standard assumptions in homogeneous three-dimensional turbulence.

3

Notice that r must be also larger than the dissipation scale.

4

In this paper, two classes of subindex are used for singularity exponents. On one side, exponents h0, h1, hd, and h∞ refer to specific values. To refer to an arbitrary specific value the exponent hm is used. On the other side, hl and ht refer to the set of components of the longitudinal and traverse components, respectively. If necessary, they can be combined, i.e., ht∞ is the most singular exponent of the transverse velocity component. Finally, singularity exponents obtained from the linear transform of those of velocity gradients are written as h˜ [see Eq. (57) and the appendix].

5

If translational invariance is not imposed to the data, a potential shift of singularity exponents should be explicitly taken into account (see the appendix).

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  • Rossby, T., C. Flagg, and K. Donohue, 2010: On the variability of gulf stream transport from seasonal to decadal timescales. J. Mar. Res., 68, 503–522, https://doi.org/10.1357/002224010794657128.

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  • Schertzer, D., and S. Lovejoy, 1987: Physical modeling and analysis of rain and clouds by anisotropic scaling multiplicative processes. J. Geophys. Res., 92, 9693–9714, https://doi.org/10.1029/JD092iD08p09693.

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  • Schubert, R., J. Gula, R. J. Greatbatch, B. Baschek, and A. Biastoch, 2020: The submesoscale kinetic energy cascade: Mesoscale absorption of submesoscale mixed layer eddies and frontal downscale fluxes. J. Phys. Oceanogr., 50, 2573–2589, https://doi.org/10.1175/JPO-D-19-0311.1.

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  • Scott, R. B., and F. Wang, 2005: Direct evidence of an oceanic inverse kinetic energy cascade from satellite altimetry. J. Phys. Oceanogr., 35, 1650–1666, https://doi.org/10.1175/JPO2771.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Seuront, L., and H. E. Stanley, 2014: Anomalous diffusion and multifractality enhance mating encounters in the ocean. Proc. Natl. Acad. Sci. USA, 111, 2206–2211, https://doi.org/10.1073/pnas.1322363111.

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    • Search Google Scholar
    • Export Citation

  • Seuront, L., H. Yamazaki, and F. Schmitt, 2005: Intermittency. Marine Turbulence: Theories, Observations, and Models. Results of the CARTUM Project, Cambridge University Press, 66–78.

  • She, Z., and E. Leveque, 1994: Universal scaling laws in fully developed turbulence. Phys. Rev. Lett., 72, 336–339, https://doi.org/10.1103/PhysRevLett.72.336.

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    • Search Google Scholar
    • Export Citation

  • She, Z.-S., and E. C. Waymire, 1995: Quantized energy cascade and log-poisson statistics in fully developed turbulence. Phys. Rev. Lett., 74, 262–265, https://doi.org/10.1103/PhysRevLett.74.262.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Skákala, J., P. W. Cazenave, T. J. Smyth, and R. Torres, 2016: Using multifractals to evaluate oceanographic model skill. J. Geophys. Res. Oceans, 121, 5487–5500, https://doi.org/10.1002/2016JC011741.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Soh, H. S., and S. Y. Kim, 2018: Diagnostic characteristics of submesoscale coastal surface currents. J. Geophys. Res. Oceans, 123, 1838–1859, https://doi.org/10.1002/2017JC013428.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Su, Z., J. Wang, P. Klein, A. F. Thompson, and D. Menemenlis, 2018: Ocean submesoscales as a key component of the global heat budget. Nat. Commun., 9, 775, https://doi.org/10.1038/s41467-018-02983-w.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Sukhatme, J., D. Chaudhuri, J. MacKinnon, S. Shivaprasad, and D. Sengupta, 2020: Near-surface ocean kinetic energy spectra and small-scale intermittency from ship-based ADCP data in the Bay of Bengal. J. Phys. Oceanogr., 50, 2037–2052, https://doi.org/10.1175/JPO-D-20-0065.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turiel, A., and N. Parga, 2000: The multifractal structure of contrast changes in natural images: From sharp edges to textures. Neural Comput., 12, 763–793, https://doi.org/10.1162/089976600300015583.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turiel, A., J. Isern-Fontanet, E. García-Ladona, and J. Font, 2005: A multifractal method for the instantaneous evaluation of the stream-function in geophysical flows. Phys. Rev. Lett., 95, 104502, https://doi.org/10.1103/PhysRevLett.95.104502.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turiel, A., C. J. Pérez-Vicente, and J. Grazzini, 2006: Numerical methods for the estimation of multifractal singularity spectra on sampled data: A comparative study. J. Comput. Phys., 216, 362–390, https://doi.org/10.1016/j.jcp.2005.12.004.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Turiel, A., H. Yahia, and C. J. Pérez-Vicente, 2007: Microcanonical multifractal formalism—A geometrical approach to multifractal systems: Part I. Singularity analysis. J. Phys. A: Math. Theor., 41, 015501, https://doi.org/10.1088/1751-8113/41/1/015501.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press, 745 pp.

    • Crossref
    • Export Citation

  • Verrier, S., M. Crépon, and S. Thiria, 2014: Scaling and stochastic cascade properties of nemo oceanic simulations and their potential value for GCM evaluation and downscaling. J. Geophys. Res. Oceans, 119, 6444–6460, https://doi.org/10.1002/2014JC009811.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Wang, D., C. Flagg, K. Donohue, and H. Rossby, 2010: Wavenumber spectrum in the gulf stream from shipboard adcp observations and comparison with altimetry measurements. J. Phys. Oceanogr., 40, 840–844, https://doi.org/10.1175/2009JPO4330.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zia, R., E. Redish, and S. McKay, 2009: Making sense of the legendre transform. Amer. J. Phys., 77, 614–622, https://doi.org/10.1119/1.3119512.

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    Fig. 1.

    (a) Singularity spectra corresponding to the lognormal and log–Poisson models. (b) Scaling of the velocity structure functions predicted by the lognormal and log–Poisson models [see Eq. (27)]. The Legendre transform [Eqs. (21) and (22)] imply a one-to-one relation between both functions. This is indicated in the log–Poisson model with dots: (h∞, D∞) correspond to the the behavior of ζ(p) for p → ∞, (hm, Dm) correspond to (pm, ζm), and (hd, d) correspond to (0, 0). Black dots identify the key points used in this study. Red dots identify an arbitrary value. The red line has the same slope than the log–Poisson model at that point [see Eqs. (23) and (24)]. From this plot it is clear that D(h) has positive slopes for h < hd, which correspond to p > 0, while for h > hd it has negative slopes, which correspond to p < 0.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    (left) Oleander’s cruises between Bermuda and New Jersey with the mean velocity at 50-m depth superimposed. (right) Mean transverse velocity with the positive direction to the northeast.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    (a) Number of observations at each depth. (b) Monthly mean vertical profiles of density derived from the WOA18 climatology using the TEOS-10 Equation of State for Sea Water (gray) and temporal evolution of the mean MLD (blue). The red band corresponds to the range of depths analyzed in this study.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) Sea level map obtained from the measurements of multiple altimeters for 3 May 2008. The thick dotted line corresponds to the ship’s track. (b) Singularity exponents derived from transverse velocity around the date of the sea level field. White areas correspond to missing data. Black lines correspond to the transverse velocity (m s−1; positive northeastwards).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    Singularity spectrum at 100-m depth (dots) for the same transect with the corresponding error estimations (vertical lines) and bin width (horizontal lines). The light green interval corresponds to Δh−, bounded by h∞ to the left and by hd to the right. Dashed line corresponds to the log–Poisson model obtained using the observed h∞ and the β obtained at that depth. The dotted line corresponds to a parabola fitted to the data.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    (a) Observed distributions of mode skewness sd and (b) fractal dimension of the most singular set D∞. (c) Time-averaged vertical variation of anomalous scaling 〈Δh−〉 (solid lines) and most singular exponent 〈h∞〉 (dashed lines). Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Scatterplot between the anomalous scaling measure Δh− and the singular exponents h∞ at 50, 200, and 350 m, respectively. The dashed line corresponds to the fitted straight line. Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    (a) Slope [red; m(z) in Eq. (51)] and linear correlation parameter between h∞ and Δh− (blue). (b) Intermittency parameter [red; β in Eq. (55)] and normalized RMSE [orange and green; erms in Eq. (59)]. Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively. The gray band indicates the depths that can be within the mixed layer.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Empirical histograms of singularity exponents at each depth. The range of values for the singularity exponents go from the Kolmogorov exponent h0 = −2/3 to 1 in bins of 0.008. The histogram relative frequencies N/Ntotal are computed during the whole period analyzed and are logarithmically represented with a color code. The solid line corresponds to the vertical variation of the mean singularity exponent while the dashed line corresponds to the vertical variation of the most probable value (mode).

Fig. 1.

(a) Singularity spectra corresponding to the lognormal and log–Poisson models. (b) Scaling of the velocity structure functions predicted by the lognormal and log–Poisson models [see Eq. (27)]. The Legendre transform [Eqs. (21) and (22)] imply a one-to-one relation between both functions. This is indicated in the log–Poisson model with dots: (h∞, D∞) correspond to the the behavior of ζ(p) for p → ∞, (hm, Dm) correspond to (pm, ζm), and (hd, d) correspond to (0, 0). Black dots identify the key points used in this study. Red dots identify an arbitrary value. The red line has the same slope than the log–Poisson model at that point [see Eqs. (23) and (24)]. From this plot it is clear that D(h) has positive slopes for h < hd, which correspond to p > 0, while for h > hd it has negative slopes, which correspond to p < 0.
Fig. 2.

(left) Oleander’s cruises between Bermuda and New Jersey with the mean velocity at 50-m depth superimposed. (right) Mean transverse velocity with the positive direction to the northeast.
Fig. 3.

(a) Number of observations at each depth. (b) Monthly mean vertical profiles of density derived from the WOA18 climatology using the TEOS-10 Equation of State for Sea Water (gray) and temporal evolution of the mean MLD (blue). The red band corresponds to the range of depths analyzed in this study.
Fig. 4.

(a) Sea level map obtained from the measurements of multiple altimeters for 3 May 2008. The thick dotted line corresponds to the ship’s track. (b) Singularity exponents derived from transverse velocity around the date of the sea level field. White areas correspond to missing data. Black lines correspond to the transverse velocity (m s−1; positive northeastwards).
Fig. 5.

Singularity spectrum at 100-m depth (dots) for the same transect with the corresponding error estimations (vertical lines) and bin width (horizontal lines). The light green interval corresponds to Δh−, bounded by h∞ to the left and by hd to the right. Dashed line corresponds to the log–Poisson model obtained using the observed h∞ and the β obtained at that depth. The dotted line corresponds to a parabola fitted to the data.
Fig. 6.

(a) Observed distributions of mode skewness sd and (b) fractal dimension of the most singular set D∞. (c) Time-averaged vertical variation of anomalous scaling 〈Δh−〉 (solid lines) and most singular exponent 〈h∞〉 (dashed lines). Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively.
Fig. 7.

Scatterplot between the anomalous scaling measure Δh− and the singular exponents h∞ at 50, 200, and 350 m, respectively. The dashed line corresponds to the fitted straight line. Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively.
Fig. 8.

(a) Slope [red; m(z) in Eq. (51)] and linear correlation parameter between h∞ and Δh− (blue). (b) Intermittency parameter [red; β in Eq. (55)] and normalized RMSE [orange and green; erms in Eq. (59)]. Green and orange colors correspond to transverse (ht) and longitudinal (hl) singularity exponents, respectively. The gray band indicates the depths that can be within the mixed layer.
Fig. 9.

Empirical histograms of singularity exponents at each depth. The range of values for the singularity exponents go from the Kolmogorov exponent h0 = −2/3 to 1 in bins of 0.008. The histogram relative frequencies N/Ntotal are computed during the whole period analyzed and are logarithmically represented with a color code. The solid line corresponds to the vertical variation of the mean singularity exponent while the dashed line corresponds to the vertical variation of the most probable value (mode).
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