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Flow of Spatiotemporal Turbulentlike Random Fields

We study the Lagrangian trajectories of statistically isotropic, homogeneous, and stationary divergence-free spatiotemporal random vector fields. We design this advecting Eulerian velocity field such that it gets asymptotically rough and multifractal, both in space and time, as it is demanded by the phenomenology of turbulence at infinite Reynolds numbers. We then solve numerically the flow equations for a differentiable version of this field. We observe that trajectories get also rough, characterized by nearly the same Hurst exponent as the one of our prescribed advecting field. Moreover, even when considering the simplest situation of the advection by a fractional Gaussian field, we evidence in the Lagrangian framework additional intermittent corrections. The present approach involves properly defined random fields, and asks for a rigorous treatment, that would justify our numerical findings, and deepen our understanding of this long lasting problem.

A powerful and physically insightful way to characterize many dynamical systems, such as those encountered in fluid mechanics, consists in studying the pathlines X(t) of a given advecting field u(x, t), at the position x ∈ R d and time t > 0, defined by In the context of fluid turbulence, where the velocity field u is governed by the Navier-Stokes equations, such Lagrangian trajectories of fluid particles have been extensively studied in laboratory and numerical flows [1][2][3][4][5][6][7][8][9][10][11][12][13].
In this situation, the three-dimensional Eulerian advecting flow u is incompressible (i.e. divergence-free) and exhibits a complex multiscale structure in both space [14] and time [15]. In particular, in the fully developed turbulent regime concerning the asymptotic limit of infinite Reynolds numbers, the velocity field gets rough (i.e. non differentiable) in both space and time, and characterized in a statistically averaged sense by an Hurst exponent of order H Eul ≈ 1/3. In the phenomenology of turbulence mostly developed by Kolmogorov [16], this can be fairly understood on dimensional grounds if it is assumed that the average dissipation by unit of mass remains finite at infinite Reynolds numbers [14]. Similarly, the Lagrangian velocity v(t) ≡ u(X(t), t), i.e. the velocity of a tracer advected by the fully developed turbulent flow u, develops small scales such that it gets rough and characterized by an Hurst exponent of order H Lag ≈ 1/2. Again, under the same assumption, this exponent can be obtained from dimensional arguments, and says that Lagrangian velocity has the same regularity as the one of a Brownian motion [15]. Whereas it remains illusive to derive these behaviors from first principles, we propose in this Letter to study the statistical properties of Lagrangian trajectories extracted from a prescribed and explicit advecting velocity field that reproduces some of the main aforementioned features of turbulence. A similar approach has been already explored by some authors for various random vector fields [17][18][19][20][21], although, as we will see, our advecting flow is more general than theirs, in particular concerning possible intermittent corrections.
In order to draw the simplest and numerically tractable picture of these phenomena, we need to come up with a proposition for the prescribed advecting velocity field u(x, t). Recall that we want it to be divergence-free at anytime to ensure statistical stationarity of induced Lagrangian velocities [22]. For this reason, we will consider henceforth a two-components vector field u = (u 1 , u 2 ) living in a two-dimensional space x = (x 1 , x 2 ) ∈ R 2 and for t ∈ R, such that ∇ · u = 0 at any time. In an asymptotic regime that we will precisely define, mimicking the behavior of turbulence at infinite Reynolds numbers, this vector field is eventually rough, governed in a statistically averaged sense by an Hurst exponent H ∈]0, 1[ (taken to be 1/3 as far as turbulence is concerned). A first step in this direction would be to consider fractional Gaussian fields, defined as linear operations on a space-time white noise (similarly to the approach developed in [23][24][25][26][27]), regularized over a small parameter ǫ > 0 ensuring differentiability in both space and time (making sense in particular to the divergence-free condition). Going beyond this Gaussian framework, we would like also to consider some intermittent (i.e. multifractal) corrections [14], and to explore their implication on the statistical behavior of Lagrangian trajectories. To make our notations lighter, without loss of generality, we consider in the sequel nondimensional space and time coordinates.
Along these lines, the simplest random vector field that we have in mind, which is statistically stationary, isotropic and homogeneous, and which reproduces these statistical behaviors, is given in an explicit way by where the vector kernel G ǫ,H Eul acting linearly on the random measure M ǫ,γ Eul (specified later) reads with ||x, t|| 2 ǫ = |x| 2 +t 2 +ǫ 2 a regularized spatio-temporal norm over ǫ and x ⊥ = (−x 2 , x 1 ). Note that we implicitly assume that in our non-dimensional reference-frame, the small scale ǫ plays the role of both the spatial and temporal dissipative scales. This is consistent with the similar dependence of the so-called Kolmogorov length scale η K and the sweeping time scale [15] on the Reynolds number. The scalar cutoff function ϕ ensures that this field has a finite variance,. It goes smoothly to zero as |x| gets of the order of the integral length scale L and/or t of the order of the integral time scale T . Once expressed in our non-dimensional coordinate system, we take L = T and assume ϕ(x, t) = exp − |x| 2 +t 2 2L 2 . The very form of the kernel G (Eq. 3) is inspired by the two-dimensional Biot-Savart law [28], and ensures that the velocity field defined in Eq. 2 is divergence-free for any finite ǫ > 0 and at any time. Further details and theoretical developments are provided in the Supplemental Material.
The random spatio-temporal measure M ǫ,γ Eul is chosen to be M ǫ,γ Eul (d 2 y, ds) = e γ Eul Yǫ(y,s)−γ 2 Eul Y 2 ǫ W (d 2 y, ds), (4) where W is a spatio-temporal Gaussian white noise (thus 2 + 1-dimensional) and Y ǫ a zero-average scalar Gaussian random field, logarithmically correlated in both space and time as ǫ → 0, and taken as independent of W . As we will see, the parameter γ Eul governs entirely the intermittent corrections, and M ǫ,γ Eul can be viewed as a continuous, statistically homogeneous and stationary version of the discrete cascade models [29][30][31]. Being Gaussian, the scalar field Y ǫ can be obtained as a linear operation on a independent white noise W , that Using similar technics as in Refs. [23][24][25][26][27], in particular calling for stochastic calculus methods developed in the context of multiplicative chaos theory [32], it can be shown that such a velocity field u (Eq. 2) is rough in the limit of vanishing regularizing scale ǫ → 0, such that for instance the moments of the longitudinal velocity increments δ ℓ u 1 (x, t) = u 1 (x 1 + ℓ, x 2 , t) − u 1 (x 1 , x 2 , t) (i.e. the structure functions) behave for q ≥ 1, H Eul ∈]0, 1[ and where the multiplicative factor C 2q,H,γ Eul is finite and positive. The scaling behavior entering in Eq. 5 means that the Eulerian velocity field u (Eq. 2) is intermittent and exhibits a lognormal spectrum. The respective transverse (i.e. the scale ℓ is taken along the second direction) and temporal (i.e. we look at the increment over a time τ at a fixed position) structure functions behave similarly as in Eq. 5, with the same spectrum of exponents but with different multiplicative constants. Numerical simulations of the advecting vector field u (Eq. 2) are performed in a (2 + 1)-dimensional periodic box of unit length and duration using N = 2 11 collocations points in each direction, such that dx = dt = 1/N . Convolutions of the deterministic functions G ǫ,H (Eq. 3) and H ǫ (i.e. the kernel of the log-correlated field Y ǫ entering in Eq. 4) with two independent instances W and W of variance dx 2 dt of the spatio-temporal Gaussian white noise are computed in an efficient way in the Fourier domain. We use for the large spatial and temporal scales the values L = T = 1/4. The singular kernels G ǫ,H and H ǫ are regularized over the small scale ǫ = 4dx such that, up to numerical errors, the obtained field u is differentiable in space and time, and divergence-free in particular. Finally, the trajectories X(t) of 2 14 particles, uniformly distributed in the unit square at the initial time, are computed according to Eq. 1 using a second-order Runge-Kutta time marching scheme and linear interpo-lation of the velocities, as detailed in Ref. [33]. Their respective Lagrangian velocity v(t) = dX(t)/dt and acceleration a(t) = d 2 X(t)/dt 2 are obtained using finitedifference time derivatives.
Let us first focus on the statistical analysis of the trajectories in an advecting Gaussian velocity field u(x, t) (Eq. 2). To do so, we consider the non-intermittent case γ Eul = 0, and the particular value H Eul = 1/3 to mimic the regularity of turbulence. We display in Fig. 1(a) the trajectories of particles initially uniformly distributed in the unit square represented by thick black lines. We indeed observe strong chaotic mixing, and notice that during the unit duration of the simulation, particles have traveled a distance of order unity, as expected. We show in Fig. 1(c) a typical time series of velocity v(t) and acceleration a(t) over the duration of the simulation. We can see that these series are statistically stationary, which give a meaning to subsequent analysis that we will be performing. Also, v is correlated over the large integral time scale T , whereas a gets correlated over the small time scale ǫ, which is consistent with the phenomenology of turbulence. A trained eye would see that a deviates from Gaussianity, we will come back to this point latter.
At this stage, it is tempting to explore the statistics of the trajectories obtained while advecting the tracers by a frozen-in-time velocity field, say u(x, 0). We represent in Fig. 1(b) the respective trajectories. Mixing is there much less efficient than for the time evolving velocity field ( Fig. 1(a)). In particular, many of them have closed orbits. Typical times series of v and a on a closed orbit are shown in Fig. 1(d), displaying an expected periodicity.
Let us now estimate the regularity of v(t) obtained from a Gaussian velocity field u(x, t) (Eq. 2 with γ Eul = 0), and quantify its dependence on H Eul . To do so, we perform simulations using 10 values for H Eul between 0.1 and 0.9. Subsequent statistics are obtained using 2 14 trajectories from 10 independent realizations of the random Eulerian field. To quantify the regularity of v, we estimate the moments of the velocity time increments , and define the respective Lagrangian Hurst exponent H Lag and intermittency coefficient γ Lag as such that H Lag can be estimated while fitting in the inertial range (i.e. for ǫ ≪ τ ≪ T ) the power-law exponent of the second-order structure function (δ 1 τ v) 2 ∝ τ 2HLag . We display in Fig. 2(a) the dependence on the scale τ of the second-order structure function in a logarithmic representation, for the 10 values of the Eulerian Hurst exponent H Eul . We indeed observe a power-law behavior between the dissipative range τ ≪ ǫ, where (δ 1 τ v) 2 ∝ τ 2 and the large scales τ ≫ T for which we get a saturation towards 2 v 2 . We proceed with the fit of the power-law exponent (represented by solid black lines) and gather our results in Fig. 2(c) (using •). We can see that the estimated regularity of Lagrangian trajectories H Lag is very close to the imposed Eulerian regularity H Eul , that is H Lag ≈ H Eul , as it was observed in the synthetic threedimensional, slowly evolving in time, flow of Ref. [21] and in the frozen Navier-Stokes field of Ref. [34]. We superimpose with a dashed-line such a prediction, showing that is does reproduce some of our estimations when H Eul is smaller than 1/2. Since the level of regularity is high, it is tempting to check whether similar results are obtained with the second-order increment, that is the increments of the increments δ 2 which is not only orthogonal to constants, but also to local linear trends, allowing in particular to estimate Hurst exponents greater than unity. We display in the inset of Fig.  2(a) the behavior of their second moment as a function of the scale τ . Once again, we observe a power-law behavior between the dissipative range, where (δ 2 τ v) 2 ∝ τ 4 and the large scales τ ≫ T for which we get a saturation towards 6 v 2 . We fit the obtained exponents and reproduce our results in Fig. 2  and H Lag = 1/2 (represented by a solid black line). We performed the same analysis using the third-order in- , and obtain same results as with δ 2 τ v (data not shown). We report in Figs. 2(b) and (d) a similar study, but with a frozen-intime advection velocity field u(x, 0), as it is illustrated in 1(b) and (d). The very same conclusions as in the time-evolving case can be drawn. In the Supplemental Material, we perform additional numerical simulations, using larger resolutions up to N = 2 16 collocation points of purely spatial advecting fields, that allow to get rid in an unambiguous manner of the effects of regularization at small ǫ and large L scales, which confirm that Let us finally quantify intermittent corrections on the trajectories (i.e. the dependence of H Lag and γ Lag on H Eul and γ Eul ). To do so, we repeat former simulations for 5 values of the parameter γ Eul . Recall that in a 3d turbulent field, γ 2 Eul ≈ 0.025 [14]. We start by performing a similar study as presented in Fig. 2, but with a varying γ Eul , and found no differences with former conclusions: H Lag ≈ H Eul , independently of γ Eul (data not shown). This is a non trivial property. Furthermore, higher order statistics of trajectories extracted from a Gaussian field (i.e. γ Eul = 0) are intermittent. To see this, we display in Fig. 3(a) the Probability Density Functions (PDFs) of Lagrangian velocity increments at various scales (using H Eul = 1/3 and γ Eul = 0). We indeed observe the continuous shape deformation of the PDFs, which is characteristic of the intermittency phenomenon [35]. Actually, these non-Gaussian behaviors were already seen on the typical time series of acceleration in Fig. 1(c). In the same line, we represent in Fig. 3(b) the acceleration PDFs for varying γ Eul , and for H Eul = 1/3. We see that as γ Eul increases, the acceleration PDF develops larger and larger tails, which shows that γ Lag increases in a monotonic way with γ Eul . To quantify more precisely this dependence, we estimate the Lagrangian velocity Flatness that is expected to behave, according to Eq. 6, as τ −4γ 2 Lag in the inertial range. We represent in Fig. 3(c) the behavior of the flatness for the six increasing values of γ Eul and H Eul = 1/3. We see that flatness is close to 3 at large scales τ ∼ T , i.e. the value for a Gaussian process, and increases, all the more as γ Eul gets bigger, as the scale decreases. The overall dependence of γ Lag on both H Eul and γ Eul is illustrated in Fig. 3(d), where the estimation of γ Lag is based on both the flatness of the first-order (solid lines) and secondorder (dashed lines) increments. We can conclude to a complex dependence of γ Lag on the parameters of the advecting Eulerian field. Interestingly, the Lagrangian intermittency coefficient in experimental and numerical 3d flows has been found compatible with γ 2 Lag ≈ 0.085 [7], a value which is of the order of what is found presently when we focus on the particular value H Eul ≈ 1/3 and γ 2 Eul = 0.025. To summarize, we have built an incompressible statistically homogeneous, isotropic and stationary spatiotemporal Eulerian advection field (Eq. 2). It is asymptotically rough and multifractal (Eq. 5), governed at small scales by the parameters H Eul and γ Eul . We have then estimated, based on numerical simulations, the statistical properties of its Lagrangian trajectories. We find that they are also asymptotically rough and multifractal (Eq. 6), and relate their parameters H Lag and γ Lag to those of the advecting Eulerian field. In particular, we estimate with good accuracy, at second-order from a statistical point of view, that the regularity of the trajectories follows closely the one of the Eulerian field. Furthermore, we evidence unambiguous intermittent corrections, even when the advecting field is prescribed to be Gaussian. These are new and non trivial results that are calling for new theoretical developments. To this regard, great progresses have been made in the mathematical description of pathlines of some rough advecting fields [36,37]. Also, the proposed velocity field could be used to investigate related important situations, such as the passive advection of scalars [22], and the relative dispersion of particle pairs [38,39]. Advecting fields of Ref. [20], among some of which get rid of the sweeping by large scales, are explored in the Supplemental Material, and lead for some aspects to similar conclusions. Finally, including the in-trinsically asymmetrical nature of the distributions of the advecting field (i.e. the skewness phenomenon), as it is proposed in Refs. [26,40], may allow to reproduce the observed values H Eul = 1/3 and H Lag = 1/2, possibly on the line H Lag = H Eul + 1 6 . We warmly thank K. Gawedzki for many crucial discussions, M. Adda-Bedia and S. Ciliberto for a critical proofreading of the manuscript. L.C. is supported by the Simons Foundation. We now provide a short introduction to fractional Gaussian fields (fGfs) on which our random vector field (Eq. 2) is based on. A detailed presentation of these fields is proposed for instance in [23][24][25][26][27]. At this stage, to keep the discussion simple, we consider a scalar field u a (x) in a d-dimensional space, i.e. x ∈ R d . We furthermore assume this field Gaussian, statistically homogeneous, isotropic and of zero average, thus fully defined by its covariance function C a (|ℓ|) = u a (x)u a (x + ℓ) . Given these assumptions, the Gaussian field u a (x) can be equivalently and conveniently written as the following stochastic integral, where W is a Gaussian white noise of variance d d y, and g a deterministic function (i.e. the filtering kernel) that remains to be determined. This kernel g is related to the covariance C as | g| 2 = C, where . stands for the Fourier transform. We choose it such that u a (i) is a finite-variance process, and (ii) has locally the same regularity as the fractional Brownian motion [41] of parameter H Eul ∈]0, 1[. For these reasons, and statistical isotropy, we choose g to be where ||x|| 2 ǫ = |x| 2 + ǫ 2 is a regularized norm over ǫ which ensures that, at a given ǫ > 0, the field u a is differentiable. The regularizing parameter ǫ plays the role of the dissipative length scale of turbulence, that goes to 0 as the Reynolds number increases. The cutoff function ϕ is also chosen as an isotropic function of the vector x and allows to introduce the decorrelation length L (i.e. the integral length scale in the vocabulary of turbulence). As we will see, its precise shape has no impact on the small scale structure of the field u a , besides ensuring that u a has a finite variance and decorrelates over a given length scale L. For these reasons, we choose the isotropic function as ϕ(x) = ϕ(|x|) ∝ exp(−|x| 2 /(2L 2 )).
Following the lines developed for instance in [23][24][25][26][27], it is straightforward to get for the variance which is finite for any H Eul > 0. To investigate the regularity of this field, consider the velocity increment δ ℓ u a (x) = u a (x + ℓ) − u a (x) over a given scale ℓ, and get where c 2 is finite for any 0 < H Eul < 1 and reads, for any unit vector e, The statistical behaviors given in Eqs. S3 and S4 fulfill the constraints of (i) finite variance and (ii) local regularity of parameter H Eul ∈]0, 1[. Higher-order structure functions are straightforward to get since u a and its increments are Gaussian. For these reasons, odd-order moments vanish, and even-order ones are given by showing that asymptotically the process u a is monofractal of parameter H Eul .

II. Multiplicative chaos as a model of the intermittency phenomenon
As reviewed in Refs. [23][24][25][26][27], a way to incorporate intermittent corrections to fractional Gaussian fields is to perturb the white noise measure W entering in Eq. S1 by a positive and independent random weight taken as the exponential of a log-correlated Gaussian field Y . Such a procedure requires some care because involved fields are necessarily of infinite variance. In few words, following a well-posed regularizing procedure, we can give a meaning to the exponential of such a field (see the review [32] on mathematical developments of multiplicative chaos theory) that can be viewed as a continuous, statistically homogeneous and/or stationary version of the discrete cascade models [29][30][31] used to model intermittency.
Following the lines leading to the Gaussian fractional field u a (Eq. S1), we now propose an intermittent version that reads where Y is assumed to be Gaussian, and independent on W , and given by with s d = 2π d/2 /Γ(d/2) the surface of the unit sphere in dimension d (Γ standing for the usual Gamma function) and W an independent white noise measure. The field Y can be seen as a regularized version (over ǫ) of a fGf of vanishing Hurst exponent H Eul = 0. It has a vanishing average and its variance can be computed as Whereas the variance diverges as ǫ → 0, its covariance remains bounded over a finite scale |ℓ|, and we get Because we assumed that the fields Y and W are independent, it is easy to show that the covariance C b (|ℓ|) = u b (x)u b (x + ℓ) is unchanged and equal to the covariance C a of u a . Same conclusions can be drawn for the variance (Eq. S3) and second-order structure function (Eq. S4). Concerning the behavior at small scales of high-order structure functions, we obtain, for an integer q ≥ 1, H Eul ∈]0, 1[ and γ 2 Eul < H Eul /(q − 1), where the positive multiplicative factor c ′ 2q can be computed, showing that the process u b is asymptotically multifractal, its spectrum of exponents being quadratic, of parameter H Eul and γ 2 Eul .

III. Final comments on the structure of the proposed advection field
The proposed advecting spatiotemporal Eulerian vector field u(x, t) = (u 1 , u 2 ) (Eq. 2) can be seen as a generalization of the scalar field u b (Eq. S5) in dimension d = 3, two dimensions being used for space and one dimension for time. In this case, the area of the unit-sphere is s 3 = 4π. The incompressible nature of the vector field u is fulfilled while introducing the vector x ⊥ = (−x 2 , x 1 ) in the picture, properly normalized such that its has a unit norm as ǫ → 0.

IV. Alternative propositions and their temporal behavior
For the sake of generality, and to make a connection with the propositions of Ref. [20], let us now consider d dimensions for space x ∈ R d , t ∈ R, and vector fields u = (u 1 , . . . , u d ) ∈ R d . Also, to simplify the discussions, let us assume u to be a zero-average Gaussian random vector field, and thus neglect additional intermittent corrections. In this case, assuming furthermore statistical isotropy, homogeneity and stationarity, the vector field u is fully characterized by its correlation function C ij that has a rather simple expression in the Fourier space [20]. It reads where D 2 is a multiplicative constant taken such that |u| 2 = C ii (0, 0) = 1 (we adopt Einstein's convention of sum over repeated indices), P ij (k) = δ ij − kikj |k| 2 the Fourier transform of the Leray's projector on divergence-free vector fields (δ ij being the Kronecker symbol), and g a scalar function that depends only on the norm of the wave vector k and frequency ω.
Once the correlation function C ij is imposed (Eq. S7), the corresponding vector field u(x, t) has a unique expression in terms of linear filtering of a Gaussian white noise vector measure W (d d x, dt) = W 1 (d d x, dt), . . . , W d (d d x, dt) , each W i being independent copies of the (d + 1)-dimensional scalar white noise, and we note by W j (d d k, dω) their Fourier transform. This expression reads

IV.a. Considerations on three different random vector fields
Let us now study three different incompressible random vector fields, call them u a , u b and u c , which spatio-temporal structure is governed by the kernel g(|k|, ω) entering in Eq. S8.
We first consider a kernel leading to similar behaviors as the field u used in the first part of this Letter (Eq. 2), a situation for which, roughly speaking, space and time are treated indifferently. Such a kernel would read, where D 3 is a constant that has dimension of a velocity (i.e. a length over a time). There, ǫ and L play the same roles as in Eq. 2, corresponding to respectively a small scale regularization ensuring differentiability and a large scale cut-off that warrants a finite variance. We discard any further multiplicative factor that is eventually included in the constant D 2 such that the corresponding velocity field is of unit variance (Eq. S8). When D 3 = 1 and d = 2, the main difference between the field given in Eq. 2 and the one governed by Eq. S9 originates from these small and large scales regularizations, and we expect very similar behaviors at small scales as those considered in Figs. 2 and 3 when ǫ → 0.
As it is considered in Ref. [20], let us now consider a kernel that treats differently time and space. The proposition of Ref. [20] consists in assuming an exponential correlation in time of characteristic duration given by a power-law of the wave number |k|. Equivalently, it reads in the wave vector and frequency domains where now D 3 has dimension of a length to the power 2β over time, as argued in Ref. [20]. We can recognize in Eq. S10 a Lorentzian term, reminiscent of an exponential correlation in time. As we will see, the free parameter β entering in Eq. S10 governs the temporal structure of the field. As it is proposed in Ref. [20], in order to illustrate the temporal behaviors of the random velocity fields induced by the kernels of Eqs. S9 and S10, we consider the correlation time τ c (|ℓ|) of the velocity differences δ ℓ u(x, t) = u(x + ℓ, t) − u(x, t), defined by  TABLE I. Definition of the three incompressible random fields u a , u b and u c , based on Eq. S8. Case a: we use the kernel g(|k|, ω) which square is provided in Eq. S9. Case b (resp. c), we use the kernel given in Eq. S10 with β = 1 2 (resp. β = 1−H Eul 2 ). In all cases, we consider any space dimension d ≥ 2 and HEul ∈]0, 1[. We then provide the behaviors of the second moment of the spatial δ ℓ u(x, t) = u(x + ℓ, t) − u(x, t) and temporal δτ u(x, t) = u(x, t + τ ) − u(x, t) velocity increments. All behaviors are understood taking first the limit ǫ → 0, and only then the respective length |ℓ| or temporal τ scales going to zero. Similarly, in the same double limit, we give the scale dependence of the two characteristic durations τc(|ℓ|) (Eq. S11) and τe(|ℓ|) (Eq. S12). and the eddy turnover time scale τ e (|ℓ|) defined by (S12) We define in table I three incompressible random fields u a , u b and u c which spatio-temporal structure differ, depending on the choice of the kernel (Eq. S9 or Eq. S10). Whereas u a is very similar to the one we have defined in the core of this Letter (Eq. 2), the main differences lying in the methods of regularization at small (over ǫ) and large (over L) length scales, the temporal structures of u b and u c are of different nature. In all cases, the spatial structure is similar, as it can be seen from the behavior at small scales of the spatial velocity increment (Third column of table I): the parameter H Eul governs completely the regularity in space. The fields u a and u b share also a similar temporal regularity, as evidenced by the behavior at small time scales of the temporal velocity increment (Fourth column of table I), the same parameter H Eul characterizing the temporal regularity. Also, the time correlation τ c (|ℓ|) (Eq. S11) of the velocity increments over ℓ (Fifth column of table I) is always much smaller than the eddy turnover time scale τ e (|ℓ|) (Eq. S12). Let us notice that τ c for the field u b undertake a transition when H Eul = 1/2 that is related to the existence of the integral entering in Eq. S11, without changing the fact that τ c ≪ τ e as ℓ → 0. Note also that having τ c proportional to |ℓ| is characteristic of the sweeping of the small scales by the large scales. The time regularity of u c is rather different from the one of the two other fields, and is always smoother whatever the value of H Eul . Furthermore, τ c is of the same order as τ e , as it is discussed in Ref. [20], and is, in this sense, not affected by the sweeping by the large scales. Again, note a transition in the behavior of the temporal increments as H Eul crosses 1/2, which is again due to existence of some underlying integrals.

IV.b. Numerical simulations
We perform numerical simulations of the fields u a , u b and u c for d = 2, and extract their respective induced Lagrangian trajectories, in a very similar manner as in the beginning of the article. We use N = 2 11 collocation points in each spatial or temporal directions, using dx = dt = 1/N , ǫ = 2dx, L = 1/2 and D 3 = 1. Recall that D 2 is chosen such that |u a,b,c | 2 = 1. Again, we track 2 14 particles, initially uniformly distributed in the unit square, and display the results of our statistical analysis of velocity along of trajectories in Fig. S1.
Indeed, as expected and mentioned before, statistical properties of Lagrangian velocity extracted from u a are displayed in Figs. S1(a) and (d) and are found to be very similar to those extracted from u (Eq. 2), although the proportionality of H Lag to H Eul is not as clear as in Fig. 2. This is very probably due to the different choices that have been made to define regularizations over small and large length scales. Again, and we will come back to this point later in this Appendix, we obtain very similar behaviors if we consider a frozen-in-time version of u a to advect the particles (data not shown).
The behavior of particles in the fields u b and u c are found to be different from those seen in u a . Concerning trajectories extracted from u b (resp. u c ), we show the statistical properties of Lagrangian velocity in Figs. S1(b) and (e) (resp. Figs. S1(c) and (f)). As it can be seen in Figs. S1(e) and (f), H Lag does not behave linearly with  Table I As far as u b is concerned, this phenomenon is very probably due to the transition undertaken by the correlation time τ c (Eq. S11) of increments over ℓ, as it is stated in Table I. Similarly, concerning u c , we interpret the weakening of the dependence of H Lag on H Eul by the transition undertaken by the temporal velocity increment δ τ u 2 at H Eul = 1/2, as it is recalled in Table I. We also estimated the statistical behaviors of Lagrangian velocity in frozen-in-time versions of the advecting fields u b and u c , and found behaviors similar to the ones observed for u a (data not shown). Hence, frozen versions of u b and u c lead to Lagrangian velocities that are different from those obtained from time-evolving ones, and this makes a clear difference with fields u (Eq. 2) and u a .

IV.c. Additional numerical simulations at high resolutions of frozen-in-time advecting fields
We have seen that statistics of Lagrangian trajectories are very similar for the fields u (Eq. 2) and u a , as it can be seen in Figs. 2(c) and S1(d). As we have shown in Fig. 2(d), very similar behaviors are observed for a frozen version of u. In the same way, a frozen version of u a would also give similar results (data not shown). The fact that statistics of Lagrangian velocity are the same in evolving and frozen-in-time versions of these fields is consistent with the treatment on the same foot of space and time in u and u a . Recall also that frozen versions of u b and u c give also similar behaviors to those obtained with evolving or frozen versions of u and u a (data not shown), whereas time evolving versions of u b and u c give different behaviors, as it is shown in Figs. S1(e) and (f). In particular, we evidence a transition when H Eul crosses 1/2. Again, it is expected since time is taken into account in a different way than space, as it can be seen in the functional form of their kernel (Eq. S10).
As a final numerical study, we would like now to take the opportunity to perform highly resolved simulations of d = 2 dimensional, purely spatial, advecting fields, to clarify whether the observed Lagrangian Hurst exponent H Lag is closer to its Eulerian counterpart H Eul or to H Eul + 1 6 . Performing simulations at a much higher resolution, as we  Fig. 2, but in (a) and (c) for the field u 2D (Eq. S13) and in (b) and (d) for u 2D,abc (Eq. S14) using N = 2 16 collocation points in each spatial directions.
will eventually do, would allow us to decipher between genuine inertial range scaling behaviors (for ǫ ≪ τ ≪ L) and bottleneck effects related to regularizations at small ǫ and large L scales. Assuming d = 2 and discarding the temporal dimension, the advecting velocity field u (Eq. 2) reduces to and Eq. S8 reduces to u 2D,abc (x) = D 2 e −2πǫ|k| |k| 2 + L −2 1+H Eul P (k)e 2iπk·x W (d 2 k), where (P ) ij = P ij is acting on the white noise vector W . Note that the 2D versions of u a , u b and u c coincide with u 2D,abc (Eq. S14). A quick look of the expressions provided in Eqs. S13 and S14 confirms that u 2D and u 2D,abc share similar statistics, besides the regularization procedures at small and large length scales. Once again, the multiplicative constant D 2 is defined such that respective fields are of unit-variance. We perform numerical simulations of the fields u 2D and u 2D,abc , using for ǫ and L the values given respectively for u (Eq. 2) and in the former section, and extract from them Lagrangian velocity. Working with only bidimensional versions of these fields allows to use N = 2 16 collocation points in each spatial directions. We display the results of the statistical analysis of Lagrangian velocity in Fig. S2.
As expected, we obtain statistics that are very similar for the two fields, and clearly observe a Lagrangian Hurst exponent H Lag that is closer to H Eul than to H Eul + 1 6 . Moreover, differences between estimations of H Lag based on first and second velocity increments disappear. Thus, if indeed frozen-in-time advecting fields give the same picture as evolving fields u and u a , then we can conclude that observations made in Fig. 2 are subject to the influence of cut-off methods at small and large length scales, and that the Lagrangian Hurst exponent is given in a good approximation by H Lag ≈ H Eul .