Instantons and the path to intermittency in turbulent flows

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Instantons and the path to intermittency in turbulent flows

André Fuchs, Corentin Herbert, Joran Rolland, Matthias Wächter, Freddy Bouchet, Joachim Peinke

To cite this version:

André Fuchs, Corentin Herbert, Joran Rolland, Matthias Wächter, Freddy Bouchet, et al.. Instantons and the path to intermittency in turbulent flows. 2021. �hal-03263664�

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Instantons and the path to intermittency in turbulent flows

A. Fuchs¹, C. Herbert², J. Rolland³, M. W¨achter¹, F. Bouchet², J. Peinke¹

1Institute of Physics and ForWind, University of Oldenburg, K¨upkersweg 70, 26129 Oldenburg, Germany

2Univ. Lyon, Ens de Lyon, Univ Claude Bernard, CNRS, Laboratoire de Physique, F-69364 Lyon, France and

3Univ. Lille, ONERA, Arts et métiers institute of technology, Centrale Lille, CNRS UMR 9014 -LMFL, Laboratoire de mécanique des Fluides de Lille Kampé de Fériet, 59000 Lille, France

(Dated: June 14, 2021)

The physical processes leading to anomalous fluctuations in turbulent flows, referred to as intermittency, are still challenging. Here, we use an approach based on instanton theory for the velocity increment dynamics through scales. Cascade trajectories with negative stochastic thermodynamics entropy exchange values lead to anomalous increments at small-scales. These trajectories concentrate around an instanton, which is the minimum of an effective action produced by turbulent fluctuations. The connection between entropy from stochastic thermodynamics and the related instanton provides a new perspective on the cascade process and the intermittency phenomenon.

The well-known picture of the Richardson-Kolmogorov cascade, according to which energy is transferred down- scale in a turbulent flow by the self-similar dynamics of eddies does not account for the phenomenon of intermittency [1, e.g.], in which large fluctuations of velocity gradients are much more frequent than predicted by the self-similar theory. While it has long been known that these non-Gaussian statistics are related to the multi- fractal character of the cascade [2], the physical processes leading to intermittency are still not fully understood. In this Letter, we present a novel approach to this problem by characterizing the dynamics through scales of velocity increments predominantly leading to anomalous small- scale fluctuations.

To do so, we adopt a different description of the turbulent energy cascade [3] and we find an observable, stochastic thermodynamics entropy exchange [4], characterizing trajectories. Starting from experiment data, we compare the statistics of trajectories which are conditioned on an atypical value of entropy, with a theoretical computation of optimal path through scales, based on instanton theory. The statistics, conditioned on entropy exchange, show that there is a preferential path through the cascade leading to the tails of the velocity increment PDF. This is a common property: in many systems, the dynamics leading to a rare event follows a predictable path. From a theoretical point of view, it is desirable to compute such paths as instantons, i.e. minimizers of an appropriate action functional [5,6].

This approach, successful across many fields of physics, has been used in fluid dynamics in several contexts [7].

A first series of applications include the prediction of the asymptotic scaling of the right [8] and left [9] tails of the PDF of the velocity gradient in the Burgers equation, the tails of the PDF of a scalar advected by a random velocity field [10–13], and the tails of the vorticity PDF in the direct cascade of 2D turbulence [14]. Instantons have also been computed in shell model analogs of the Navier- Stokes equations [15, 16] to relate anomalous scaling to

singular structures. More recently, the pdf of the compo- nents of the velocity gradient tensor has been predicted by computing instantons for a Lagrangian model of velocity gradient dynamics [17,18]. Besides the study of cascade intermittency, instantons have been used to study different rare events that are of crucial importance, for instance to compute transition paths between bistable con- figurations in 2D [19] or geostrophic turbulence [20,21], or the shape of rogue waves [22,23].

A key difference of the present work, compared to previous instantons computations for studying intermittency, is that we do not use the bare action related to large scale force fluctuations. Indeed, for an instanton computation to be relevant for describing the dynamics through the cascade leading to intermittency, it must take into account the nonlinear interactions from which the small-scale fluctuations arise. To achieve this goal, we use an effective diffusion model of the cascade learned from the experimental data. We then compute instantons which characterize rare increment trajectories through scales, defined as trajectories conditioned on entropy exchange. This result opens the way for a better under- standing of the connection between anomalous scaling and coherent nearly singular structures like strong vor- tex tubes, correlated over distances much larger than the dissipation length scale, resulting in non-Gaussian velocity fluctuations. Comparison of rare trajectories from experimental data on one hand, and the learned effective stochastic dynamics on the other hand, is a very strin- gent test for the validity of the effective model. While the effective model has been leaned from typical increments, it is thus tested for very rare trajectories. This is ex- tremely enlightening. For instance it shows that a small modification of our effective cascade model, corresponding to log-normal statistics [24,25], does not capture the correct increment dynamics through scales upon conditioning.

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2

⌘ L

h✏i l_ch

injection

h✏i

dissipation transferh✏i

FIG. 1. Illustration of the turbulent energy cascade process.

Shown in grey dashed lines are different cascade trajectories [u(·)] taken by the process between integralLand Taylor scale λ. The black solid trajectory represents the preferential path through the cascade, called instanton, that is predicted by computing the minimizers of an action functional. It should be kept in mind that this is simply a schematic representation of the phenomenologically inspired Richardson energy cascade model [30]. Therfore, a single trajectory should not be taken as one isolated large eddy evolving down-scale.

Experimental and numerical data.— We construct an ensemble of realizations of the cascade process from hotwire velocity measurements of fractal grid flow. Based on velocity time series v(t) (component in direction of the mean flow), we obtain the longitudinal velocity incre- mentsur=v(t+τ)−v(t), relying on the Taylor hypoth- esis [26]r=−τhvi(the turbulence intensity is less than 20 %) [27]. The velocity increments ur are computed in units ofσ∞ =√

2σv = 3.5 m/s (σv is the standard deviation of velocity timeseriesv(t)).

One classically studies the structure functions [28]

Sn(r) =E[uⁿ_r] or the PDFs [29]P(u, r) =E[δ(ur−u)] as a function of scaler. While these quantities provide es- sential descriptions of the statistical properties of homo- geneous isotropic turbulence, they do not convey infor- mation about the dynamics of the energy cascade. Here, we instead consider the stochastic dynamics of velocity increments ur as they go through the cascade process, from large to small scales. We introduce the change of variables=−ln(r/L), which provides a natural description of the cascade, as it increases as the cascade process unfolds, fromsi= 0 at the integral scaleLto a positive value sf = −ln(λ/L) at the Taylor scale λ [3]. Cas- cade trajectories defined as [u(·)] = {u0, . . . , uf}, with u(si) =u0,u(sf) =uf are illustrated in Fig.1.

Previous studies have shown that the stochastic process us can be considered as Markovian to a reasonable approximation [3,31, 32], at least down to a scale close

to the Taylor scale (∆EM ≈0.9λ; discrete-scale Markov model [32,33]). We model it as a diffusion process [34]

across scales

dus=D⁽¹⁾(us, s)ds+ q

2D⁽²⁾(us, s)◦dWs, (1) whereW is the Wiener process. We use the Stratonovitch convention because we will consider in the sequel scale- reversed (s→sf−s) trajectories.

For the estimation of the drift and diffusion coefficients D^(1,2)(u, s) from experimental data an optimization procedure, based on approximating the solution of the cor- respondingFokker-Planck equation using the short time propagator [35], has been worked out [36–40] and imple- mented in an open-source Matlab package [41].

Next, two data sets of trajectories are generated by numerical integration of the stochastic differential equation (SDE) (1) using a simple continuous diffusion model of the cascade

D⁽¹⁾(u, s) =−(α+γ)u, D⁽²⁾(u, s) =β+γu². (2) The initial condition is a centered Gaussian distribution with variance σ² = 0.082, which fits well the PDF at the integral scale observed in experimental data.

The Kolmogorov-Obukhov theory [24, 25], hereafter referred to as K62, can be recovered with the choice α= (3 +µ)/9, β= 0 and γ=µ/18, where µ= 0.234 is the standard intermittency parameter [3]. In this case the SDE (1) becomes a standard stochastic process known as geometric Brownian motion which has the analytical solutionus=u0exp

−(α+γ)s+√2γ(Ws−W0) , from which structure functions can be computed di- rectly. This simple model can be extended by considering jumps [42, 43], leading to the She-Leveque scaling [44].

Such extensions are not considered here.

In Fig. 2(a) the evolution of the PDF P(u, s) of velocity increments through scales, for both experimental and numerical trajectories generated from the diffusion model of the cascade for the two valuesβ= 0 (K62) and β= 0.044 (chosen empirically) is shown as a contour plot in (u, s) space.

While the scale dependence ofD^(1,2)(u, s) observed in experimental data is more complex (see Supplemental Material [45] for a detailed presentation) than assumed by the simplified diffusion model, the results shown in Fig.2(a)-(b) indicate that for bothβ = 0 andβ= 0.044 this model correctly reproduces the statistics of the velocity increments including the fingerprint of intermittency expressed by the kurtosis. To obtain a nearly perfect accordance between numerical and experimental data, which is not our aim here, a scale dependence of the coefficientsα(s), β(s) and γ(s) has to be taken into account [27, 39]. For this paper, we take the small deviations when approaching the Taylor scale as an error for this simplified model approximation.

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3

10^-3 10^-2 10^-1 10⁰

FIG. 2. (a) PDF of velocity incrementsus as a function of scale sfor the experimental data (colors) and the numerical data generated by direct sampling of the Markov process with β= 0 andβ= 0.044 (dotted resp. solid black contours). (b) Standard deviation and kurtosis of velocity incrementsusas a function of scales.

Action and entropy.— Next we set the cascade trajectories in the context of the path integral formalism [5].

The main idea is that the probability of a trajectory is proportional to the factor e^−A[u(·)], where A is the action. Hence, the expectation value of any observableO can be computed as a weighted-integral over all the possible trajectories

E[O] = 1 Z

Z

D[u(·)]O[u(·)]e^−A[u(·)], (3) whereD[u(·)] is the measure over the space of increment trajectories and Z a normalization factor. The action A depends on the discretization. For the Stratonovitch process described by (1), the action is:

A[u(·)] = Z sf

si

"

u˙s−D⁽¹⁾+D⁰⁽²⁾/2²

4D⁽²⁾ +D⁰⁽¹⁾ 2

# ds,(4) interpreted in the Stratonovitch convention [46–48]. Here we dropped the arguments ofD^(1,2)(u, s).

In the spirit of non-equilibrium stochastic thermodynamics [49] it is possible to associate with every cascade trajectory a total entropy varia- tion [4, 39, 49–51], given by the sum of two terms

∆Stot[u(·)] = ∆Smed[u(·)] + ∆Ssys[u(·)], which are related to heat, work and internal energy [49–51]. The system entropy ∆Ssys[u(·)] =−ln_p(u

f,sf) p(u0,si)

is the change in entropy associated with the change in state of the system. If u^r_s = us_f−s denotes the scale-reversed path associated with the stochastic process us, the entropy exchanged with the surrounding medium is the functional [52, 53]

∆Smed[u(·)] =−lnP[u^r_s=us]

P[us=us], (5)

= Z sf

si

u˙s

D⁽¹⁾−D⁰⁽²⁾/2 D⁽²⁾

ds, (6) which measures the irreversibility of the trajectories of the velocity increment through scales.

FIG. 3. (a) PDF of the normalized action estimated using experimentally acquired and numerically generated cascade trajectories. The grey dashed resp. solid line corresponds to a Gaussian fit and a generalized extreme value distribution fit to the experimental data, withk = 0.1. (b) PDF of the entropy exchanged with the surrounding medium.

Figure 3(a) shows the PDF of the action estimated using (4), which is essentially Gaussian for the cascade trajectories generated numerically (for both β = 0 and β = 0.044) while the experimental data (β = 0.044) is characterized by a heavy tail on the right. The solid grey line belongs to a generalized extreme value distribution fit to the experimental data, with the shape param- eterk = 0.1 (type II extreme value distributions, i.e. a Fr´echet distribution).

In Figure 3(b) the comparison of the PDF of the entropy exchanged with the surrounding medium ∆Smed, using (6), also indicates discrepancies. In particular, K62 (β = 0) is characterized by distinct deviations: besides a broader distribution, the relative frequency of entropy consuming trajectories (i.e. ∆Smed <0) is much lower.

Forβ = 0.044 the model correctly reproduces the statistics of the negative and positive medium entropies in a good approximation. In the Supplemental Material [45]

the influence of β on the integral fluctuation theorem, which is a rigorous law of non-equilibrium stochastic thermodynamics, is presented.

Cascade trajectories conditioned on ∆Smed.— Next,

∆Smedis used as an observable, which allows characterizing the increment trajectories. In Fig.4, the contours of the PDF of velocity increment conditioned on the entropy exchange are presented as a function of scale for experimental and numerical data withβ= 0.044. Conditioning trajectories on ∆Smedreveals two distinct characteristics.

For positive entropy values (see Fig.4(a, c)) a splitting of the probabilities is seen for large scale incrementsu0, which have the tendency to decay with decreasing scale, as it fits well to the traditional picture of turbulence (ur≈r^1/3≈e^−s/3). On the other hand, for negative entropy values (see Fig.4(b, d)), the conditioned PDF markedly splits into two branches when approaching the Taylor scale, corresponding to atypical values of the velocity incrementuf. This splitting up can be interpreted

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4

FIG. 4. PDF of velocity incrementusas a function of scales (contours), conditioned on positive (a, c) and negative (b, d) entropy exchange, for the experimental (top row) and numerical data (bottom row) with β= 0.044. The entropy value is, from (a) to (d): ∆Smed = 15,−5,13 and −5. On top of the PDF are represented randomly chosen cascade trajectories characterized by the same entropy consumption (resp.

production), color-coded using the value of their action (in- creasing from white to red). The black solid resp. dotted lines represent the instanton trajectories for the diffusion model with β = 0.044 and β= 0 computed by solving the variational problem given by (15).

as the dynamic fluctuations of small-scale intermittency, which is a fundamental feature of the turbulent cascade.

These two cases suggest that entropy producing trajectories typically go from large to small velocity increments (u0> uf)|∆Smed>0, whereas entropy consuming trajectories feature (u0< uf)|∆Smed<0.

To illustrate this further, a few randomly selected individual trajectories conditioned on ∆Smed, color-coded by the value of the corresponding action are presented on top of the PDF contour lines in Fig. 4. Trajectories with small action are located closer to the ridges of the probability density functions, while trajectories with large action depart further from it. It can be noted that while the qualitative behavior described above is captured by the experimental data and by the diffusion model with β = 0.044, differences between the two subsist: experimental increment trajectories with large action have endpoints in the expected regions but tend to display large fluctuations away from the typical dynamics, while large action trajectories for the diffusion model fluctuate less but may start and end farther away from the high-probability regions. Another difference

between the experimental data and the diffusion model is the asymmetry between negative and positive increments in the former, which is absent in the latter. Note that a symmetric distribution is assumed for the model at the initial condition (the inertial scale PDF) and this simple model conserves skewness.

Instanton formalism.— The conditioned probability density for velocity increments has revealed the existence of two qualitatively different statistical dynamics. We now ask whether the ridge of the PDF corresponds to an optimal trajectory that could be predicted by computing the minimizers of the action functional, calledinstantons.

Such trajectories are important because when their contribution dominates the path integral, they allow for computing the statistical properties of any observable using saddle-point approximations in (3). This is for instance the case in the context of large deviation theory, in the weak-noise limit for noise-perturbed vector fields [6]. For known initial and final points,u0 anduf, instantons are trajectories [u(·)] which satisfy the following variational problem

A(u0, uf) = inf

u {A[u(·)] |u(si) =u0, u(sf) =uf}.(7) Variational problems of this form can be solved using the standard machinery of analytical mechanics. For instance, instanton trajectories are governed by the Hamil- ton equations, for the Hamiltonian

H =D⁽²⁾p²+

D⁽¹⁾−D⁰⁽²⁾ 2

p−D⁰⁽¹⁾

2 , (8)

so that the equations of motion read du

ds = ∂H

∂p = 2 β+γu²

p−(α+ 2γ)u, (9) dp

ds =−∂H

∂u =−2γup²+ (α+ 2γ)p. (10) Here, it is natural to assume that the large-scale velocity increment u0 is not fixed, but random with a given probability distributionp0(u0). At the same time we shall leave the final value of the increment unconstrained. The probability of a path becomes weighted by p0(u0)e^−A[u(·)], and the variational problem (7) becomes infu{A[u(·)] +f(u0)}, wheref(u0) =−lnp0(u0).

A natural choice is a centered Gaussian distribution f(u0) =u²0/(2σ²). Minimizing this modified action yields the Hamilton equations (9)–(10) with boundary condi- tionsp(si) =f⁰(u0) andp(sf) = 0.

Next, the connection between entropy from stochastic thermodynamics and the related instanton formalism to the turbulent cascade process is discussed. There- fore, the statistics conditioned on the entropy production

∆Smed[u(·)] associated with a path is computed. The path integral formalism presented above still applies, re- stricting the integrals to the fields satisfying the imposed

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5 condition. Hence, the instantons conditioned on entropy

satisfy the following variational problem A(S) = inf

u {A[u(·)] +f(u0)| ∆Smed[u(·)] =S}. (11) For the coefficients given by (2), this constrained variational problem can be reduced to an unconstrained variational problem. Indeed, the entropy exchange can be computed analytically as a stochastic integral in the Stratonovitch convention [34]:

∆Smed[u(·)] =− Z sf

si

α+ 2γ

β+γu²_sus◦dus (12)

=−α+ 2γ

2γ ln β+γu²_f β+γu²₀

!

. (13) Thus the entropy production is fixed by the value of the increments at the Taylor and integral scale (uf resp.u0).

For a given entropy productionS, (13) can be inverted

u˜f(S, u0) = sβ

γ

e⁻^α+2γ^2γS −1

+u²₀e⁻^α+2γ^2γS , (14) and the constrained variational problem simplifies to

A(S) = inf

u0 {A(u0,±u˜f(S, u0)) +f(u0)}. (15) To compute the instantons conditioned on entropy for β 6= 0, the variational problem given by (15) can be solved numerically. For the values of the entropy used in Fig. 4, two solutions with the same action are found in each case: one with positive u0 and one with negative u0. For the data generated numerically from the diffusion model (Fig. 4(c, d)), the instantons match quite well the ridge of the PDF of velocity increments as a function of scale and many individual trajectories are within the vicinity of the instanton. In the K62 case (β = 0), the variational problem can be solved analytically by exploiting the fact that (9)–(10) conserve u×p. It is found that if the most probable increment at the integral scale is u0 = 0, then the instanton, represented by the black dotted lines in Fig.4, vanishes everywhere regardless of the imposed value of entropy (see Supplemental Material [45]). Finally and impor- tantly, this explains why the K62 theory fails to capture the two distinct behaviors of the statistics conditioned on entropy shown for the cascade trajectories in Fig. 4.

In combination with the analysis of ∆Smed in Fig.3(b), this can be explained by the well known fact that K62 underestimates the frequency of large fluctuations on small scales [1, 28]. Although significant differences between the instantons and the ridge of the PDF are evident for the experimental data (Fig.4(a, b)), we emphasize that the overall behavior of entropy dependent dynamics are reproduced qualitatively correct by the instantons for β = 0.044. Note, we have

chosen a simplified stochastic process with constant coef- ficientsα(s),β(s) andγ(s) to obtain this principal result.

Conclusion.— We have identified a criterion, entropy exchange, that separates the statistics of trajectories for which a typical velocity increment at large scales devel- ops into an atypical velocity increment at small scales (negative entropy) and trajectories for which atypical large-scale velocity increments are brought back to typical values at small-scale (positive entropy). Investigat- ing the contribution of negative entropy trajectories to the probability ofuf, we find for the experimental data that up to≈90% of the heavy tailed statistics is given by these trajectories (see Supplemental Material [45]).

The connection between the statistical quantity entropy and the structure related instanton formalism is worked out and may be referred to as an entropon. The findings presented in this paper indicate that the entropy dependent dynamics through scales are qualitatively reproduced by entropons, given by the analytical expres- sion (15), if the K62 model is modified by adding the constant β value to the diffusion. Further, the entropons for negative entropy confirm the observation of zero probability of uf ≈0|∆Smed<0 at Taylor length. Ac- cordingly, theβterm, which breaks the scaling symmetry of the increment structure functions, takes an important structural effect on entropons paths.

These findings provide a new perspective on the intermittency phenomenon, by pinpointing the trajectories to instantons responsible for the emergence of non-Gaussian statistics at small-scales, as has been proposed more gen- erally for turbulence [7, 11]. The introduced approach, based on instanton theory, provides an interesting test for models of the energy cascade that exceed the classical procedure to show how well anomalous scaling is reproduced. This dynamical constraint on models was never considered before. Furthermore, entropons could be relevant for a wide range of problems in fluid dynamics as this connection provides an alternative way to common analysis incorporating the results into the statistical theory of turbulence and nonequilibrium thermodynamics.

Perspective for future works include the computation of effective action and instantons, from the hydrody- namic equation, as has been considered for instance for problems in geostrophic turbulence [54].

We acknowledge financial support by Volkswagen Foundation and by the Laboratoire d’Excellence LANEF in Grenoble (ANR-10- LABX-51-01). This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sk lodowska-Curie Grant Agreement 753021. This publication was supported by a Subagreement from the Johns Hopkins University with funds provided by Grant No. 663054 from Simons Foundation (F. Bouchet). Its contents are solely the responsibility of the authors and

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6 do not necessarily represent the official views of Simons

Foundation or the Johns Hopkins University.

We acknowledge helpful discussions with A. Girard, J.

Friedrich, J. Ehrich, A. Engel, G. G¨ulker, S. Kharche, D.

Nickelsen and T. Wester.

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