Abstract
The understanding of turbulent flows is one of the biggest current challenges in physics, as no firstprinciples theory exists to explain their observed spatiotemporal intermittency. Turbulent flows may be regarded as an intricate collection of mutuallyinteracting vortices. This picture becomes accurate in quantum turbulence, which is built on tangles of discrete vortex filaments. Here, we study the statistics of velocity circulation in quantum and classical turbulence. We show that, in quantum flows, Kolmogorov turbulence emerges from the correlation of vortex orientations, while deviations—associated with intermittency—originate from their nontrivial spatial arrangement. We then link the spatial distribution of vortices in quantum turbulence to the coarsegrained energy dissipation in classical turbulence, enabling the application of existent models of classical turbulence intermittency to the quantum case. Our results provide a connection between the intermittency of quantum and classical turbulence and initiate a promising path to a better understanding of the latter.
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Introduction
Vortices are manifestly the most attractive feature of fluid flows occurring in the Nature. They are highly rotating zones of the fluid that often take the form of elongated filaments, of which tornadoes are one prominent example in atmospheric flows. Such structures can travel and interact with other vortex filaments, as well as with the surrounding fluid. In fact, the dynamics of vortex filaments in fluid flows is highly nontrivial, as they can reconnect changing the topology of the flow^{1}. Their nontrivial arrangements may lead to very complex configurations and in particular to turbulence, an outofequilibrium state characterised by a largescale separation between the scales at which energy is injected and the one at which it is dissipated. In threedimensional flows, because of the inherently nonlinear character of turbulence, energy initially injected at large scales is transferred towards the small scales through a cascadelike process.
In turbulent flows, the typical thickness of a vortex filament is comparable to the smallest active scale of turbulence^{2}, itself usually much smaller than the eddies carrying most of the energy content of the flow. Vortex filaments may thus be seen as the fundamental structure of turbulence, whose collective dynamics leads to the multiscale complexity of such flows. Indeed, depending on their individual intensities and orientations, a set of vortex filaments located within a given spatial region may contribute constructively or destructively to the fluid rotation rate. In fluid dynamics, the rotation rate of a twodimensional fluid patch is commonly quantified by the velocity circulation around the closed loop \({{{{{{{\mathcal{C}}}}}}}}\) surrounding the patch,
where v is the fluid velocity field. Note that, by virtue of Stokes’ theorem, the circulation is equal to the flux of vorticity, ω = ∇ × v, through the fluid patch.
The above view of vortex filaments as the fundamental unit of fluid flows is particularly appropriate in superfluids, such as lowtemperature liquid helium and Bose–Einstein condensates (BECs). Indeed, in such fluids, vortices are welldefined discrete objects about which the circulation is quantised, taking values multiple of κ = h/m. Here h is Planck’s constant and m is the mass of the bosons constituting the superfluid^{3}. Such property arises from their quantum nature, as vortices are topological defects of the macroscopic wave function describing the system. For this reason, vortex filaments in superfluids are called quantum vortices.
One of the most striking properties of lowtemperature superfluids is their total absence of viscosity. Despite this fact, quantum vortex reconnections are possible, since Helmholtz’ theorem that forbids reconnections in classical inviscid fluids^{1} breaks down due to the vanishing fluid density at the vortex core. This picture was first suggested by Feynman in 1955^{4} and later confirmed numerically in the framework of the Gross–Pitaevskii (GP) equation^{5}. Since then, quantum vortex reconnections have been observed experimentally in superfluid helium^{6} and in BECs^{7}. They are characterised by universal scaling laws^{8,9} and have been linked to irreversibility, both in experiments^{10} and in numerical simulations^{11}. In the early vortex filament simulations by Schwarz^{12}, it was noticed that quantum vortex reconnections are a key physical process for the development of quantum turbulence, a state described by the complex interaction of a tangle of quantum vortices. Such a state is illustrated by the vortex filaments (in green and yellow) visualised in Fig. 1, obtained from the GP simulations performed in Ref. ^{13}.
Quantum turbulence is characterised by a rich multiscale physics. At small scales, between the vortex core size (about 1 Å in superfluid ^{4}He) and the mean intervortex distance ℓ (~1 μm), the physics is governed by the dynamics of individual quantised vortices^{14}. At such scales, Kelvin waves (waves propagating along vortices) and vortex reconnections are the main physical processes carrying energy along scales^{15,16}. In contrast, at scales larger than ℓ, the quantum nature of the superfluid becomes less important and a regime comparable to classical turbulence emerges. Indeed, at such scales, a Kolmogorov turbulent cascade is observed, provided that a largescale separation exists between ℓ and the largest scale of the system. In particular, the scaling law predicted by Kolmogorov’s celebrated K41 theory^{17} for the kinetic energy spectrum has been observed in superfluid helium experiments^{18,19} and in numerical simulations of quantum turbulence^{20,21,22}.
Previous studies have suggested that, in quantum turbulence, the emergence of K41 scaling laws is associated to a local polarisation of the vortex tangle^{14,23,24,25,26,27}. In other words, within a given spatial region, the orientations of nearby vortices are not independent, but instead have some degree of correlation. This phenomenon is visible in Fig. 1, where vortex bundles—regions of samecoloured vortex filaments—can be clearly identified. This local polarisation is present even in ideally isotropic flows, and should not be confused with the preferential largescale orientation of vortices, which typically occurs in anisotropic flows. A classic example of the latter is a rotating cylindrical vessel filled with superfluid helium^{4}.
In a recent work^{13}, we have shown that the quantitative similarities between classical and quantum turbulence go far beyond the Kolmogorov energy spectrum. Indeed, both systems display the emergence of extreme events that result in the breakdown of Kolmogorov’s K41 theory—a phenomenon known as intermittency. Our work was motivated by the recent study of Iyer et al.^{28}, which suggested that intermittency has a relatively simple signature on the statistics of circulation in classical turbulence. In particular, the moments of the circulation measured over fluid patches of area A ~ r^{2} follow a power law of the form
with scaling exponents λ_{p} that increasingly deviate from the K41 prediction \({\lambda }_{p}^{\,{{{{\mathrm{K41}}}}}\,}=4p/3\) as the moment order p increases. By performing simulations of a generalised GP equation, we have shown that the anomalous scaling exponents λ_{p} in the inertial scales of quantum turbulence closely match those observed in classical turbulence^{13}. Note that, up until now, most of the advances in the understanding of intermittency have been made in terms of velocity increments. However, despite many theoretical efforts^{17,29,30,31}, there is still no firstprinciples theory able to explain this phenomenon. The abovecited findings suggest that circulation may provide an alternative path towards a better understanding of turbulence (as first hinted the by pioneering theoretical work of Migdal^{32}), and eventually, to novel circulationbased theories of intermittency^{33,34}.
The strong similarity between the statistics of circulation in classical and quantum turbulence is particularly striking given the very different nature of vortices in both types of fluids. This statistical equivalence opens the way for an interpretation of the intermittency of classical turbulent flows in terms of the collective dynamics of discrete vortex filaments carrying a fixed circulation. With this idea in mind, we relate in this work the intermittent statistics of velocity circulation in classical and quantum turbulence. We start by investigating in quantum turbulence how local vortex polarisation, as well as the nontrivial spatial distribution of vortex filaments, affect circulation statistics. We address the following questions: Is it possible to study both effects separately? Do they contribute in the same way to the flow intermittency? We then provide a relation between the spatial distribution of discrete vortices, and the coarsegrained energy dissipation rate in classical turbulence, a quantity at the core of existent intermittency models.
In this work, quantum and classical turbulent systems are, respectively, studied using highresolution direct numerical simulations of a generalised GP and the incompressible Navier–Stokes (NS) equations. Discrete vortices and their signs are extracted from the GP fields and then analysed. To disentangle the effects of polarisation and spatial vortex distribution, we additionally study a disordered turbulence state. Such state is generated from the discrete vortex data by randomly resetting the sign of each individual vortex while keeping its position fixed. To illustrate the differences between the turbulent (nondisordered) and the disordered turbulence states, we plot in Fig. 2 the kinetic energy spectrum associated to each vortex configuration (see “Methods” for details on the computation of the spectra from discrete vortices). First, we see that the turbulent case displays a clear k^{−5/3} range, in agreement with the energy spectra obtained from the full GP and NS fields. Note that, in the case of GP fields, we show the incompressible kinetic energy spectrum, which contains 86% of the total energy of the system—the other components being the compressible, internal and quantum energy^{20,22}. Secondly, the K41 scaling disappears once polarisation is artificially suppressed from the tangle, leading to a trivial k^{−1} scaling range for the disordered state (see “Methods” for a brief derivation). Note that this same scaling has already been observed in vortex filament simulations, once the vortex tangle has been decomposed into polarised and random components^{26}.
Results
A simple discrete model of circulation
Let us first consider a set of n discrete vortices, each of them carrying a circulation κs_{i}, where s_{i} = ±1 is the sign of each vortex. From now on we set the quantum of circulation to κ = 1 for simplicity. We propose to model the total circulation of the nvortex collection, \({{{\Gamma }}}_{n}=\mathop{\sum }\nolimits_{i = 1}^{n}{s}_{i}\), as a biased onedimensional random walk. Polarisation is naturally introduced by letting each random step s_{i} be positively correlated with the instantaneous position Γ_{i−1}, i.e. the total circulation of all previous vortices.
Concretely, we construct inductively the following toy model for the circulation. The sign of the first vortex, s_{1}, has equal probability of being positive or negative. Then, the sign of vortex n + 1 is positive with a probability p_{n+1}, which we set to depend on the total circulation at step n as p_{n+1} = [1 + f(Γ_{n}/n)]/2. Here, f(z) is a suitable function (odd, nondecreasing, taking values in [−1, 1]), such that p_{n} ∈ [0, 1] at each step n. For the sake of simplicity, we choose here f(z) = βz (see the Supplementary information for the general case), where β ∈ [0, 1] is an adjustable parameter that sets the polarisation of the system. When β = 0, one retrieves a standard random walk with scaling \({\langle  {{\Gamma }}{ }^{2}\rangle }_{n} \sim n\). Conversely, for β = 1, one recovers a fully polarised set of vortices behaving as \({\langle  {{\Gamma }}{ }^{2}\rangle }_{n} \sim {n}^{2}\).
The resulting model is a discrete Markov process, since the probability distribution of Γ_{n+1} only depends on the state {n, Γ_{n}} via the probability p_{n+1}. Concretely, the probability \({{{{{{{{\mathcal{P}}}}_n}}}}}({{\Gamma }})\) of having Γ_{n} = Γ obeys the master equation
Multiplying this equation by Γ^{2}, summing over all Γ and, for the sake of simplicity, taking the limit of continuous n, one gets a closed equation for the circulation variance,
where averages are over all realisations after n steps. For large n, this equation predicts the scaling \({\langle {{{\Gamma }}}^{2}\rangle }_{n} \sim n\) for β < 1/2 (corresponding to a set of vortices with negligible polarisation), and \({\langle {{{\Gamma }}}^{2}\rangle }_{n} \sim {n}^{2\beta }\) otherwise. In particular, choosing β = 2/3, one recovers the Kolmogorov scaling by replacing n ∝ A. This relation between the number of vortices n and the loop area A containing them is expected to hold on average under spatial homogeneity conditions, but neglects potentially important inhomogeneities in the spatial vortex distribution that may affect highorder moments of n. Besides, for the pth order moment, the model predicts the selfsimilar scaling \({\langle  {{\Gamma }}{ }^{p}\rangle }_{n} \sim {n}^{{\gamma }_{p}}\) with γ_{p} = βp. More generally, for any suitable function f(z) defining the probability p_{n} of the model, one obtains the linear scaling \({\gamma }_{p}=p\min \{\max \{1/2,f^{\prime} (0)\},1\}\) (see the Supplementary information for more details on the calculations).
The toy model introduced above shows in a very simple manner how a specific correlation (or polarisation) is responsible for the emergence of nontrivial scaling laws, as already suggested by previous works on quantum turbulence^{24,25,26}. In addition, the model yields selfsimilar statistics, suggesting that polarisation is not sufficient to reproduce the observed intermittency of circulation in classical^{28} and quantum^{13} turbulent flows. At this point, we may speculate that the lack of intermittency in the model is likely associated with the missing notion of space. Indeed, on average one expects to have a number of vortices 〈n〉 ~ A/ℓ^{2} crossing a loop of area A, where ℓ is the mean intervortex distance. Yet, fluctuations in their spatial distribution—associated to the appearance of vortex clusters and voids—may strongly influence highorder moments. As seen in Fig. 1, such effects clearly take place in turbulent flows, where they are linked to the formation of coherent structures.
Comparison with quantum turbulence data
The ideas hinted at by our toy model can be verified using actual quantum turbulence data. With this aim, we identify all vortex filaments present in our GP simulations (see “Methods” for details), and compute circulation statistics as a function of the number n of considered vortices. Crucially, groups of vortices are chosen based on their spatial proximity, which is required to preserve the correlation between vortices. On the other hand, with such a conditioning, one may expect the effect of strong spatial fluctuations of the vortex distribution to be somewhat relaxed. In practice, for each twodimensional cut of the simulation, we consider sets of n neighbouring vortices in order to compute the circulation moments \({\langle  {{\Gamma }}{ }^{p}\rangle }_{n}\). Then, to improve the statistics, we repeat such measurement for each cut and along the three Cartesian directions.
The resulting secondorder moment \({\langle {{{\Gamma }}}^{2}\rangle }_{n}\) is shown in Fig. 3a, along with the moment \({\langle {{{\Gamma }}}^{2}\rangle }_{A}\) measured for different loop areas A (data from Müller et al.^{13}). At small scales, \({\langle {{{\Gamma }}}^{2}\rangle }_{A} \sim {A}^{1}\) due to the discrete nature of vortices^{13}. In contrast, within the inertial range, both moments clearly exhibit the expected Kolmogorov scaling. In particular, \({\langle {{{\Gamma }}}^{2}\rangle }_{n} \sim {n}^{{\gamma }_{2}}\) with γ_{2} = 4/3. This result allows us to use the extended selfsimilarity (ESS) framework^{35} to determine the scaling properties of higherorder moments via the relation \({\langle  {{\Gamma }}{ }^{p}\rangle }_{n} \sim {n}^{{\gamma }_{p}} \sim {\langle {{{\Gamma }}}^{2}\rangle }_{n}^{{\gamma }_{p}/{\gamma }_{2}}\). Remarkably, as shown in Fig. 3b–c, the moments display a clear selfsimilar behaviour with γ_{p} = 2p/3, thus obeying Kolmogorov scaling for all orders. The selfsimilarity is also observed in the normalised probability density functions (PDFs) of Γ for different values of n (Fig. 3d), which nearly collapse and are close to Gaussian. This behaviour should be contrasted with the noncollapsing PDFs of Γ for different loop areas A (Fig. 3e). Note that, in both cases, the chosen values of A and n lay within the inertial range, represented by a grey background in Fig. 3a–c.
Disentangling polarisation and spatial vortex distribution
The fitted scaling exponents for the turbulent case \(2{{\gamma }^{{{{{\mathrm{turb}}}}}}_{p}}\), discussed above, are plotted in Fig. 4 (bluefilled stars) as a function of the moment order p. These exponents are compared to the measured values of \({{\lambda }^{{{{{\mathrm{turb}}}}}}_{p}}\) (bluefilled circles) obtained according to Eq. (2), where averages are performed for different loop areas A. The latter are the same as in Ref. ^{13}. The factor 2 in front of \({{\gamma }^{{{{{\mathrm{turb}}}}}}_{p}}\) comes from considering the relation 〈n〉 ~ A ~ r^{2}. As discussed earlier, the moments averaged for different n closely follow the selfsimilar K41 scaling \(2{{\gamma }^{{{{{\mathrm{turb}}}}}}_{p}}\approx 4p/3\) (blue solid line), while the \({{\lambda }^{{{{{\mathrm{turb}}}}}}_{p}}\) exponents—affected by the spatial vortex distribution—show signs of intermittency^{13}.
To further distinguish the effects of polarisation and spatial vortex distribution on circulation statistics, we perform the following numerical experiment. We recompute the circulation in the quantum turbulent flow, but before doing this, we randomise the sign of each vortex on each analysed twodimensional cut while keeping its position fixed. By doing this, we get rid of the system polarisation, while maintaining the nontrivial spatial distribution of vortices. We refer to this system as disordered turbulence. In our nonintermittent toy model, this setting would correspond to the unpolarised value β = 0, yielding the selfsimilar circulation scaling \({\langle  {{\Gamma }}{ }^{p}\rangle }_{n} \sim {n}^{p/2}\). In Fig. 4, we display the corresponding measured exponents of the disordered state \({{\lambda }^{{{{{\mathrm{diso}}}}}}_{p}}\) and \(2{{\gamma }^{{{{{\mathrm{diso}}}}}}_{p}}\) (red unfilled markers). Remarkably, even after suppressing vortex polarisation, \({{\lambda }^{{{{{\mathrm{diso}}}}}}_{p}}\) also presents intermittency deviations. In contrast, the scaling exponents \({{\gamma }^{{{{{\mathrm{diso}}}}}}_{p}}\) satisfy the expected selfsimilar behaviour \(2{{\gamma }^{{{{{\mathrm{diso}}}}}}_{p}}\approx p\) (red solid line).
The previous results suggest that the nontrivial polarisation of vortices, while being responsible for Kolmogorov scalings, has no major influence on the intermittency of the system. Furthermore, they indicate that the latter originates from fluctuations of the spatial distributions of vortices. From our above observations, one may therefore expect the scaling exponents of the circulation to be given by a composition of the polarisation and spatial distribution effects. That is, we may conjecture that the scaling exponents λ_{p} and γ_{p} are related by
where g is some yet unknown function.
In order to check this idea, we can try to relate the scaling exponents of the turbulent and disordered turbulent systems. If relationship (5) were to hold true, one should have that \({{\lambda }^{{{{{\mathrm{diso}}}}}}_{p}}={{\lambda }^{{{{{\mathrm{turb}}}}}}_{3p/4}}\). Using this relation with the measured exponents of the turbulent case, one indeed recovers the intermittency exponents \({{\lambda }^{{{{{\mathrm{diso}}}}}}_{p}}\) of the disordered case, as shown by the green squared markers in Fig. 4. This result strongly highlights the importance of the fluctuations of vortex concentration on the intermittency of circulation.
Spatial vortex distribution and OK62 theory
As a first step towards relating the intermittency of classical and quantum turbulence, we now quantify the spatial distribution of vortices in the latter system. If vortices were homogeneously distributed in space, then the number n of vortices within loops of area A would be expected to follow a Poisson distribution with mean value 〈n〉_{A} ∝ A. In that case, the moments of n would scale as \({\langle {n}^{p}\rangle }_{A} \sim {A}^{p}\) for sufficiently large A. Equivalently, the number of vortices per unit area Z_{A} = n/A would follow the trivial scalings \({\langle {Z}_{A}^{p}\rangle }_{A} \sim 1\) for all p > 0. As shown in Fig. 5a, this is clearly not the case, indicating that the spatial distribution of vortices is nontrivial in quantum turbulence (as may be inferred from the visualisation of Fig. 1). Indeed, while the firstorder moment recovers a constant (consistently with the relation 〈n〉_{A} ~ A), higherorder moments of Z_{A} follow a different scaling with a negative exponent—a sign of anomalous behaviour. This is confirmed by the PDFs of n displayed in Fig. 5b, which are longtailed and strongly differ from a Poisson distribution (dashed line).
In classical turbulence, it is today well accepted that the intermittency of velocity fluctuations is linked to the emergence of violent events, characterised by strong spatial fluctuations of the kinetic energy dissipation rate ε(x). Such idea led Obukhov and Kolmogorov in 1962 to develop a refined similarity theory of turbulence, commonly referred to as OK62 theory, where such fluctuations are taken into account^{17,36,37}, unlike K41 theory which only deals with the mean value of ε(x). This refined theory considers the scaleaveraged (or coarsegrained) energy dissipation rate \({\varepsilon }_{r}({{{{{{{\bf{x}}}}}}}})=\frac{3}{4\pi {r}^{3}}{\int}_{B({{{{{{{\bf{x}}}}}}}},r)}\varepsilon ({{{{{{{\bf{x}}}}}}}}^{\prime} )\ {{{{{{{{\rm{d}}}}}}}}}^{3}{{{{{{{\bf{x}}}}}}}}^{\prime}\), where B(x, r) is a ball of radius r centred at x. When applied to the spatial velocity increments δv_{r} over a distance r, OK62 theory states that the statistics of \(\delta {v}_{r}/{({\varepsilon }_{r}r)}^{1/3}\) is selfsimilar and universal. Most intermittency models use ε_{r} to predict the anomalous scaling of velocity increment statistics^{17}. Some early experiments in classical turbulence showed that, when velocity increments are conditioned on the coarsegrained dissipation, their statistics becomes Gaussian^{38,39}, proving that the intermittency of velocity fluctuations is hidden behind the distribution of energy dissipation. This observation was later confirmed by numerical simulations^{40}.
In the case of lowtemperature quantum turbulence, such as the one studied here, energy is taken away from the inertial range and transferred towards small scales by the Kelvin wave cascade and vortex reconnections^{9,41}. Furthermore, the velocity field diverges at the vortex core, and thus the definition of the dissipation field is delicate. Nevertheless, we can give a phenomenological interpretation of the dissipation by assuming that the system is well represented as a dilute pointvortex gas. Such a picture was recently used by Apolinário et al.^{33} to model the velocity circulation in classical turbulence, and becomes particularly pertinent in quantum fluids. Although the superfluid is inviscid, one can model smallscale physics by some effective viscosity ν_{eff}^{23,42}, whose value is not important here. This approach allows us to directly estimate the coarsegrained dissipation field by using its classical definition in terms of velocity gradients and a Diraclike supported vorticity field (see Supplementary information). Given a disk of radius r crossed by n vortices, a straightforward calculation gives the estimate
where ε_{r} is the average of the local dissipation rate ε(x) over the disk, A = πr^{2} is the disk area, ξ the typical vortex thickness and κ the quantum of circulation. The number of vortices per unit area Z_{A} = n/A would then be the quantum analogous of the coarsegrained dissipation ε_{r}. Remarkably, and similarly to ε_{r}—which is known to exhibit lognormal statistics in classical turbulence^{43}—the normalised PDFs of \({{{{{{\mathrm{log}}}}}}}\,({Z}_{A})\) almost collapse and are close to Gaussian in the bulk (Fig. 5c), reinforcing the pertinence of relation (6).
To make a stronger connection between classical and quantum turbulence, we recall that the classical coarsegrained energy dissipation rate is a highly fluctuating quantity that presents anomalous scaling laws traditionally denoted by \(\langle {\varepsilon }_{r}^{p}\rangle \sim {r}^{\tau (p)}\). It follows from Eq. (6) that the number of vortices should satisfy
Note that, because of homogeneity, τ(1) = 0, which translates as 〈n〉_{A} ~ A for the mean number of vortices. In the classical turbulence literature, there are several multifractal models for the anomalous exponents τ(p) that are able to reproduce experimental and numerical measurements^{17}. Among those, the She–Lévêque model^{44}
has one adjustable parameter D_{∞} corresponding to the fractal dimension of the most singular structures of the system. In the original model, which closely matches existent turbulence measurements^{35,44,45,46}, these structures are assumed to be vortex filaments, hence D_{∞} = 1. The combination of prediction (7) with the original She–Lévêque model, represented by the blue dashed lines in Fig. 5a, is in good agreement with our quantum turbulence data for sufficiently large A, although some deviations due to the limited scaling range may be present.
Classical turbulence and conditioned circulation
We now apply some of the previous ideas to classical turbulence. We perform a direct numerical simulation of the NS equations in a statistically steady state at a Taylorbased Reynolds number of Re_{λ} = 510. The simulation is performed using 2048^{3} collocation points. We then compute the velocity circulation over planar square loops of area A = r^{2}, and, following the framework of the OK62 refined similarity hypothesis, we condition its statistics on the coarsegrained dissipation field ε_{r}. The latter is obtained by averaging the local dissipation ε over the interior of each loop. See “Methods” for details on the numerical simulations and the data analysis.
We first consider the unconditioned velocity circulation PDFs, shown in Fig. 6a. The PDFs display heavy tails (associated with intermittency) which depend on the considered scale r/λ, with λ the Taylor microscale. This is consistent with the classical turbulence simulations of Iyer et al.^{28,47}. The PDF tails are strongly suppressed when the statistics is conditioned on low values of the local coarsegrained dissipation, ε_{r}/〈ε〉 ∈ [0.5, 1], as seen in Fig. 6b. The suppression of intermittency is also manifest in Fig. 6c, where the scaling exponents of circulation are displayed after conditioning on different intervals of ε_{r}. With no conditioning (black crosses), the scaling exponents match those of Iyer et al.^{28}, whereas when conditioning on low values of ε_{r} the K41 selfsimilar scaling is recovered.
Note that the above conditioning is slightly different from the one presented in Fig. 3, as here we are conditioning both on the loop area A and on the value of ε_{r} within such loops. In the case of quantum turbulence, the equivalent would be to study \({\langle {{{\Gamma }}}^{p} n\rangle }_{A}\), i.e. to consider only loops of area A having n vortices. Such a double conditioning is very restrictive, as it requires a very large amount of statistics. Nevertheless, we perform a similar analysis, considering loops having a low, average and high number of vortices relative to the mean. The respective scaling exponents are displayed in Fig. 6d. We find that, for loops with low and average number of vortices, the selfsimilar K41 scaling is recovered, whereas for loops having large vortex concentrations the statistics is still intermittent. The lack of selfsimilarity in regions of high dissipation (in classical flows) or high vortex concentration (in quantum flows) hints at the idea that not all such events contribute equally to circulation statistics.
Can OK62 theory describe circulation intermittency?
Considering the relation introduced in Eq. (6) and the fact that the number of vortices per unit area follows the same intermittent behaviour as ε_{r}, one could try to apply OK62 theory to relate scaling exponents of circulation λ_{p} with those of dissipation τ(p), as traditionally done for velocity increments. Within this reasoning, \({{\Gamma }} \sim {\varepsilon }_{r}^{1/3}{r}^{4/3}\), yielding a OK62based relation λ_{p} = 4p/3 + τ(p/3). However, such a relation is in strong disagreement with our data (classical and quantum turbulence, see Supplementary information) and with early NS studies^{48}. Nevertheless, this disagreement is not in contradiction with the fact that the anomalous scaling of the number of vortices is well described by standard multifractal dissipation models (see Fig. 5). Indeed, if one considers a vortex dipole (two vortices of same magnitude and opposite sign), their contribution to large fluctuations of the local dissipation field and to velocity increments may be very important. On the other hand, for the circulation, the dipole contribution is exactly zero due to vortex cancellation. This fact suggests that not all extreme dissipation events result in extreme circulation values. In particular, intense circulation events would be correlated to those highly dissipative structures in turbulence which carry a strong vortex polarisation, such as vortex sheets or bundles (at scales r ≫ ℓ). Note that, in classical fluids, the idea of vortex filaments organising into groups forming vortex sheets is consistent with the recently proposed sublayers’ vortex picture of dissipation^{49}.
The previous observations motivate us to introduce a modified OK62 theory for the circulation (“mOK62” in the following), where the most relevant singular structures are not vortex filaments but structures of higher fractal dimension. To check this idea, we adapt the She–Lévêque model τ_{SL}(p) (Eq. (8)) by setting D_{∞} = 2.2 instead of 1. The chosen dimensionality exactly corresponds to the monofractal fit obtained by Iyer et al.^{28} and Müller et al.^{13} for the highorder circulation moments (p > 3) in classical and quantum turbulence, and, as suggested in the former work, it may be linked to the effect of wrinkled vortex sheets. Note that, for large p, our mOK62 model simplifies to \({\lambda }_{p}\approx \frac{10}{9}p+(3{D}_{\infty })\), which is equivalent to the monofractal fit by Iyer et al.^{28}. In Fig. 4, it is shown that the adapted model matches strikingly well the anomalous exponents of circulation both in the turbulent and in the disordered cases for p > 3 (dashed lines), while for p < 3 there are some deviations.
Our mOK62 model can be generalised to an arbitrary degree of polarisation, which is fully determined by the exponent γ_{1} ∈ [1/2, 1]. Using dimensional analysis and reintroducing the fundamental quantum of circulation κ, we have \({{\Gamma }} \sim {\varepsilon }_{r}^{{\gamma }_{1}/2}{r}^{2{\gamma }_{1}}{\kappa }^{13{\gamma }_{1}/2}\), leading to λ_{p} = 2pγ_{1} + τ(pγ_{1}/2). Accordingly, the conjecture stated in Eq. (5) would be fulfilled with g(x) = 2x + τ(x/2). We recall that K41 turbulence corresponds to γ_{1} = 2/3, in which case the dependence on κ consistently disappears. This model also accurately reproduces disordered turbulence data (see Fig. 4), which corresponds to γ_{1} = 1/2. In this case, λ_{p} = p + τ(p/4), and intermittency corrections thus vanish at p = 4 (instead of p = 3 in the turbulent case).
The previous results provide a possible interpretation for the difference between the intermittency of velocity fluctuations and of circulation, based on the different topologies of the dissipative structures contributing to extreme events. We shall notice that an alternative interpretation is also possible, based on the recent works by Apolinário et al.^{33} and Moriconi^{34}. In this framework, the circulation should scale as \({\varepsilon }_{r}^{1/2}\), instead of \({\varepsilon }_{r}^{1/3}\), namely \({{\Gamma }} \sim {\varepsilon }_{r}^{1/2}{\nu }_{r}^{1/2}{r}^{2}\), where ν_{r} is Kraichnan’s eddy viscosity^{50}. The latter is found by assuming that the energy spectrum takes the form E(k) ~ ε^{2/3}k^{−5/3+α} (where α is an intermittency correction), yielding ν_{r} ~ r^{4/3+α}. Note that this phenomenological approach mixes a meanfield approximation for determining ν_{r} with the fluctuations arising from \({\varepsilon }_{r}^{1/2}\). Moreover, in its present form, it does not directly account for vortex cancellations. Nevertheless, when combined with the standard She–Lévêque model (with D_{∞} = 1), this model provides an expression for the exponents λ_{p} as accurate as our mOK62 model in the turbulent case. There is certainly a need to pursuit further investigations to understand how both models differ and complement each other.
Discussion
In this work, we have attempted at providing an interpretation for the intermittent statistics of velocity circulation in turbulent flows. We have done so by viewing turbulent flows as a polarised tangle of discrete and thin vortex filaments, each carrying a constant circulation. While this view is a priori only appropriate in lowtemperature quantum fluids, we expect it to be a very pertinent model of classical turbulence, considering the strong similarities recently unveiled between both systems^{13}.
By introducing and solving a simple toy model and by analysing data of GP quantum turbulence simulations, we have shown that, in discretevortex systems, the Kolmogorov selfsimilar scalings result from a partial polarisation of the vortices (in agreement with previous quantum turbulence studies), while the intermittency of circulation statistics is linked to the nontrivial (nonPoissonian) spatial distribution of vortices. In fact, within fluid patches of varying area A in the inertial range of scales, the number of vortices n is found to be the quantum equivalent of the coarsegrained dissipation ε_{r} in classical turbulence, as they both follow the approximately lognormal distribution first hypothesised by the celebrated Obukhov–Kolmogorov OK62 theory for ε_{r}^{43}. Quantitatively, we show that the intermittency of n is well described by the She–Lévêque model for ε_{r}, confirming the strong equivalence between both observables.
It is important to remark that the quantum turbulence simulations presented in this work have been performed on periodic domains, and are based on the GP equation describing an ideal superfluid at very low temperature. In contrast, most superfluid turbulence experiments using liquid helium are performed in confined systems and at finite temperatures^{18,19}, in a regime that may be described by a twofluid model^{51}. Early experimental studies showed that the signature of intermittency on velocity increments is nearly independent of the temperature, matching observations in classical fluids^{52,53,54}. These observations were later contradicted by a recent experimental investigation, which showed an enhancement of velocity intermittency in the twofluid regime compared to classical turbulence^{55}, in agreement with previous numerical simulations of related models^{56,57}. Compared to velocity increments, we expect the circulation to be a much more robust observable in quantum fluids, as it does not display singular behaviour in the vicinity of vortices^{13}. For this reason, measuring the scaling properties of circulation in future experiments may help disambiguate existent contradictions, and provide a clearer answer on the intermittency of finitetemperature quantum turbulence. Recent experiments have made initial attempts at reconstructing Eulerian velocity fields from Lagrangian particle tracking measurements in turbulent superfluid helium^{55}. Such a technique could be used in principle to measure the velocity circulation in superfluid helium, although addressing highorder statistics might still be challenging. However, note that such an approach is delicate because, due to the twofluid nature of finitetemperature superfluid helium, particles may fail to capture important Eulerian flow features^{58,59}, and further work is needed to determine its suitability.
Finally, using data from NS and GP simulations, we have confirmed that the classical OK62 theory does not fully account for the intermittency of the circulation in classical and quantum turbulence. We have provided an explanation based on the presumed topology of the turbulent structures that most contribute to extreme circulation events. We have then proposed a modified OK62 description of circulation, where relevant singular structures have a fractal dimension D_{∞} ≈ 2.2 associated to vortex sheets^{28}. This value differs from the dimensionality D_{∞} = 1 of isolated vortex filaments, used in the modelling of velocity increment statistics^{44}. Using this idea, we have shown that the intermittency of circulation is well reproduced by a modified version of the She–Lévêque model, bringing support to the vortex sheet interpretation first proposed by Iyer et al.^{28}. All the previous ideas were additionally tested by introducing a disordered turbulence state, obtained by artificially suppressing vortex polarisation from a GP numerical simulation.
There are still some questions that remain open for future works. In particular, further investigation on the topology of relevant structures for the intermittency of circulation is required. We have argued that the smallest structures significant for circulation are vortex sheets, as simpler structures are irrelevant due to vortex cancellation. One way of approaching this topic is by use of cancellation exponents^{60,61,62}, method that exploits the fact that circulation can take either negative or positive values. An alternative approach, based on recent works by Apolinário et al.^{33} and Moriconi^{34}, suggests that the most relevant singular structures for velocity circulation should still be vortex filaments. Further investigations on the fractal dimension of circulation would help develop more accurate models of intermittency.
Our findings hint at the existence of a coarsegrained quantity different from ε_{r}, which may better encapsulate the intermittency of circulation in classical turbulence in the spirit of an OK62like theory. Furthermore, it may be appropriate to investigate the relevance of quantities, such as the local vorticity magnitude (or enstrophy) or the local strain. Such coarsegrained quantity would be expected to display intermittent statistics with extreme values associated to the presence of quasitwodimensional structures such as vortex sheets.
More generally, our present results reinforce the strong equivalence between classical and quantum turbulence, and constitute an attempt at providing an explicit connection between the intermittency of both systems. We expect such a connection to provide a possible path to a simplified description of the intermittency of classical turbulence, a highly challenging topic from a modelling standpoint, yet extremely relevant to the understanding of fluid flows occurring in the Nature.
Methods
Numerical simulations
We study the dynamics of quantum turbulence in the framework of a generalised GP model
where ψ is the condensate wave function describing the dynamics of a compressible superfluid at zero temperature. Here, m is the mass of the bosons, μ is the chemical potential, n_{0} the particle density and g = 4πℏ^{2}a_{s}/m is the coupling constant proportional to the swave scattering length. The dimensionless parameters χ and γ correspond to the amplitude and order of beyond mean field corrections. The nonlocal interaction between bosons is given by the potential V_{I}(x − y) which is chosen, together with χ and γ, to reproduce the roton minimum in the excitation spectrum and the equation of state of superfluid helium. Details on the chosen parameters can be found in Ref. ^{22}. The use of a standard or a generalised GP model does not affect the statistics of velocity circulation^{13}.
The hydrodynamic interpretation of Eq. (9) stems from the Madelung transformation \(\psi =\sqrt{\rho /m}{e}^{im\phi /\hslash }\), where ρ is the local density and ϕ the phase of the complex wave function. The velocity field is then given by v = ∇ϕ. Note that ϕ is not defined at the locations where ψ vanishes, which implies that the velocity field is singular along quantum vortices^{63}.
The generalised GP equation (9) is solved in a threedimensional periodic cube by direct numerical simulations using the Fourier pseudospectral code FROST, with an explicit fourthorder Runge–Kutta method for the time integration^{22}. The quantum turbulent regime is studied in a freely decaying Arnold–Beltrami–Childress (ABC) flow^{13,21} with 2048^{3} collocation points. To reduce acoustic emissions, the initial condition is prepared using a minimisation process^{20}. The box has a size L = 1365ξ and the intervortex distance is ℓ ≈ 28ξ, with ξ the healing length.
We also perform direct numerical simulations of the incompressible NS equations
using the Fourier pseudospectral code LaTu^{64} in a periodic cubic domain. The temporal advancement is performed with a thirdorder RungeKutta scheme. Above, p is the pressure field, ν the fluid kinematic viscosity and f an external forcing stirring the fluid. The latter acts at large scales within a spherical shell of radius ∣k∣ ≤ 2 in Fourier space. The turbulent regime is studied once the simulation reaches a statistically steady state. The simulation is performed using 2048^{3} collocation points at a Taylorbased Reynolds number of Re_{λ} = 510.
Evaluation of circulation and coarsegrained dissipation
To obtain the circulation from GP and NS simulation data, we take advantage of the spectral nature of both solvers, and compute the circulation from the Fourier coefficients of the velocity fields. Namely, over a given Lperiodic 2D cut of the physical domain, we write the circulation over a square loop of side r, centred at a point x = (x, y), as the convolution
where \(\omega =({{{{{{{{\boldsymbol{\nabla }}}}}}}}}_{{{{{\mathrm{2D}}}}}}\times {{{{{{{\bf{v}}}}}}}})\cdot \hat{{{{{{{{\bf{z}}}}}}}}}\) is the outofplane vorticity field and B_{r}(x) is a square of side r centred at x. The convolution kernel can be written as the product of two rectangular functions, H_{r}(x) = Π(x/r) Π(y/r), where Π(x) = 1 for \( x \, < \, \frac{1}{2}\) and 0 otherwise. Note that we have used Stokes’ theorem to recast the contour integral (1) as a surface integral of vorticity. The convolution in Eq. (12) can be efficiently computed in Fourier space using the Fourier transform of the rectangular kernel, which may be written in terms of the normalised sinc function as \({\widehat{H}}_{r}({k}_{x},{k}_{y})={(r/L)}^{2}{{{{{{\mathrm{sinc}}}}}}}\,({k}_{x}r/2\pi )\ {{{{{{\mathrm{sinc}}}}}}}\,({k}_{y}r/2\pi )\).
As mentioned earlier, the GP velocity field diverges at vortex locations. To minimise the numerical errors resulting from such singularities, we first resample each twodimensional cut of the GP wave function field ψ(x) into a very fine grid of resolution 32768^{2}, using Fourier interpolation. The velocity field is then evaluated in physical space using the Madelung transformation. This resampling procedure is described in more detail in Ref. ^{13}.
In NS simulations, the above algorithm is also applied to compute the coarsegrained dissipation ε_{r}(x) over squares of side r. Instead of the vorticity, the convoluted quantity is in this case the dissipation field ε(x) = 2νs_{ij}s_{ij}, where s(x) = [∇v + (∇v)^{T}]/2 is the threedimensional strainrate tensor.
Vortex detection from GP simulations
For a given twodimensional cut of a GP velocity field, we identify the signs and locations of the quantum vortices crossing the cut as follows. First, the circulation is computed on a discrete grid following the procedure described above, taking small square loops of side r ~ ξ ≪ ℓ. The result is a discrete circulation field, where each circulation value is either zero if no vortex crosses the small loop centred at that position, or ±κ if a single vortex crosses it. For very small loop sizes, the former case is much more likely than the latter. As a result, the vortex distribution can be sparsely described by storing the locations and signs of the nonzero circulation values. By repeating this procedure over different cuts of the simulation, one can reconstruct the threedimensional vortex structure, as visualised in Fig. 1.
Energy spectrum computation from discrete vortices
For each twodimensional cut, once the positions r_{i} and the signs s_{i} of each vortex crossing the plane are determined, we first compute a regularised twodimensional vorticity field \(\omega ({{{{{{{\bf{r}}}}}}}})=\kappa \mathop{\sum }\nolimits_{i = 1}^{N}{s}_{i}{\delta }_{\eta }({{{{{{{\bf{r}}}}}}}}{{{{{{{{\bf{r}}}}}}}}}_{i})\), where N is the number of vortices on the 2D cut. Here, \({\delta }_{\eta }({{{{{{{\bf{r}}}}}}}})=\exp ( {{{{{{{\bf{r}}}}}}}}{ }^{2}/2{\eta }^{2})/2\pi {\eta }^{2}\), and η is the scale of the regularisation (we have used η = ξ in Fig. 2). Then, the energy spectra are computed by noting that \( \widehat{{{{{{{{\bf{v}}}}}}}}}({{{{{{{\bf{k}}}}}}}}){ }^{2}= \widehat{\omega }({{{{{{{\bf{k}}}}}}}}){ }^{2}/ {{{{{{{\bf{k}}}}}}}}{ }^{2}\), where \(\widehat{{{{{{{{\bf{v}}}}}}}}}\) and \(\widehat{\omega }\) are the Fourier transforms at the wavevector k of the velocity field and of ω, respectively. Finally, by averaging over all 2D cuts and integrating over a shell ∣k∣ = k, the energy spectrum reads
where the integral is performed over all angles Ω. Note that the largewavenumber range in Fig. 2 is determined by the regularised Dirac function δ_{η} and has no physical meaning.
For disordered turbulence, as there is no correlation between the signs and the vortex positions, it is easy to show that \(\left\langle {\sum }_{i,j}{s}_{i}{s}_{j}{e}^{i{{{{{{{\bf{k}}}}}}}}\cdot ({{{{{{{{\bf{r}}}}}}}}}_{i}{{{{{{{{\bf{r}}}}}}}}}_{j})}\right\rangle =\langle N\rangle\), from where it follows E(k) ~ k^{−1}.
Data availability
Processed data used in the other figures are available from the corresponding authors upon request. Source data are provided with this paper.
Code availability
Code used to process solution fields from GP and NS simulations is openly available at https://github.com/jipolanco/Circulation.jl and on Zenodo^{65}, along with detailed installation instructions and a complete set of examples. The software is licensed under the opensource Mozilla Public License 2.0.
References
Kida, S. & Takaoka, M. Vortex reconnection. Annu. Rev. Fluid Mech. 26, 169 (1994).
Jiménez, J., Wray, A. A., Saffman, P. G. & Rogallo, R. S. The structure of intense vorticity in isotropic turbulence. J. Fluid Mech. 255, 65 (1993).
Barenghi, C. F., Skrbek, L. & Sreenivasan, K. R. Introduction to quantum turbulence. Proc. Natl Acad. Sci. USA 111, 4647 (2014a).
Feynman, R. P. Application of quantum mechanics to liquid helium. In Progress in Low Temperature Physics, Vol. 1 (ed. Gorter, C. J.) 17–53 (Elsevier, 1955).
Koplik, J. & Levine, H. Vortex reconnection in superfluid helium. Phys. Rev. Lett. 71, 1375 (1993).
Bewley, G. P., Lathrop, D. P. & Sreenivasan, K. R. Visualization of quantized vortices. Nature 441, 588 (2006).
Serafini, S. et al. Vortex reconnections and rebounds in trapped atomic BoseEinstein condensates. Phys. Rev. X 7, 021031 (2017).
Villois, A., Proment, D. & Krstulovic, G. Universal and nonuniversal aspects of vortex reconnections in superfluids. Phys. Rev. Fluids 2, 044701 (2017).
Galantucci, L., Baggaley, A. W., Parker, N. G. & Barenghi, C. F. Crossover from interaction to driven regimes in quantum vortex reconnections. Proc. Natl Acad. Sci. USA 116, 12204 (2019).
Švančara, P. & La Mantia, M. Flightcrash events in superfluid turbulence. J. Fluid Mech. 876, R2 (2019).
Villois, A., Proment, D. & Krstulovic, G. Irreversible dynamics of vortex reconnections in quantum fluids. Phys. Rev. Lett. 125, 164501 (2020).
Schwarz, K. W. Threedimensional vortex dynamics in superfluid ^{4}He: homogeneous superfluid turbulence. Phys. Rev. B 38, 2398 (1988).
Müller, N. P., Polanco, J. I. & Krstulovic, G. Intermittency of velocity circulation in quantum turbulence. Phys. Rev. X 11, 011053 (2021).
Barenghi, C. F., L’vov, V. S. & Roche, P.E. Experimental, numerical, and analytical velocity spectra in turbulent quantum fluid. Proc. Natl Acad. Sci. USA 111, 4683 (2014b).
Boué, L. et al. Exact solution for the energy spectrum of Kelvinwave turbulence in superfluids. Phys. Rev. B 84, 064516 (2011).
Krstulovic, G. Kelvinwave cascade and dissipation in lowtemperature superfluid vortices. Phys. Rev. E 86, 055301 (2012).
Frisch, U. Turbulence: The Legacy of A.N. Kolmogorov 1st edn (Cambridge University Press, 1995).
Maurer, J. & Tabeling, P. Local investigation of superfluid turbulence. Europhys. Lett. 43, 29 (1998).
Salort, J., Chabaud, B., Lévêque, E. & Roche, P.E. Investigation of intermittency in superfluid turbulence. J. Phys. Conf. Ser. 318, 042014 (2011).
Nore, C., Abid, M. & Brachet, M. E. Decaying Kolmogorov turbulence in a model of superflow. Phys. Fluids 9, 2644 (1997a).
Clark di Leoni, P., Mininni, P. D. & Brachet, M. E. Dual cascade and dissipation mechanisms in helical quantum turbulence. Phys. Rev. A 95, 053636 (2017).
Müller, N. P. & Krstulovic, G. Kolmogorov and Kelvin wave cascades in a generalized model for quantum turbulence. Phys. Rev. B 102, 134513 (2020).
Vinen, W. F. & Niemela, J. J. Quantum turbulence. J. Low Temp. Phys. 128, 167 (2002).
L’vov, V. S., Nazarenko, S. V. & Rudenko, O. Bottleneck crossover between classical and quantum superfluid turbulence. Phys. Rev. B 76, 024520 (2007).
Roche, P.E. & Barenghi, C. F. Vortex spectrum in superfluid turbulence: interpretation of a recent experiment. EPL (Europhys. Lett.) 81, 36002 (2008).
Baggaley, A. W., Laurie, J. & Barenghi, C. F. Vortexdensity fluctuations, energy spectra, and vortical regions in superfluid turbulence. Phys. Rev. Lett. 109, 205304 (2012).
Baggaley, A. W. The importance of vortex bundles in quantum turbulence at absolute zero. Phys. Fluids 24, 055109 (2012).
Iyer, K. P., Sreenivasan, K. R. & Yeung, P. K. Circulation in high Reynolds number isotropic turbulence is a bifractal. Phys. Rev. X 9, 041006 (2019).
Zybin, K. P. & Sirota, V. A. Vortex filament model and multifractal conjecture. Phys. Rev. E 85, 056317 (2012).
Zybin, K. P. & Sirota, V. A. Multifractal structure of fully developed turbulence. Phys. Rev. E 88, 043017 (2013).
Sreenivasan, K. R. & Yakhot, V. Dynamics of threedimensional turbulence from NavierStokes equations. Phys. Rev. Fluids 6, 104604 (2021).
Migdal, A. A. Loop equation and area law in turbulence. Int. J. Mod. Phys. A 09, 1197 (1994).
Apolinário, G. B., Moriconi, L., Pereira, R. M. & Valadão, V. J. Vortex gas modeling of turbulent circulation statistics. Phys. Rev. E 102, 041102(R) (2020).
Moriconi, L. Multifractality breaking from bounded random measures. Phys. Rev. E 103, 062137 (2021).
Benzi, R. et al. Extended selfsimilarity in turbulent flows. Phys. Rev. E 48, R29 (1993).
Oboukhov, A. M. Some specific features of atmospheric tubulence. J. Fluid Mech. 13, 77 (1962).
Kolmogorov, A. N. A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 82 (1962).
Gagne, Y., Marchand, M. & Castaing, B. Conditional velocity pdf in 3D turbulence. J. Phys. II Fr. 4, 1 (1994).
Naert, A., Castaing, B., Chabaud, B., Hébral, B. & Peinke, J. Conditional statistics of velocity fluctuations in turbulence. Phys. D. 113, 73 (1998).
Homann, H., Schulz, D. & Grauer, R. Conditional Eulerian and Lagrangian velocity increment statistics of fully developed turbulent flow. Phys. Fluids 23, 055102 (2011).
Fonda, E., Meichle, D. P., Ouellette, N. T., Hormoz, S. & Lathrop, D. P. Direct observation of Kelvin waves excited by quantized vortex reconnection. Proc. Natl Acad. Sci. USA 111, 4707 (2014).
Boué, L. et al. Energy and vorticity spectra in turbulent superfluid ^{4}He from T = 0 to T_{λ}. Phys. Rev. B 91, 144501 (2015).
Dubrulle, B. Beyond Kolmogorov cascades. J. Fluid Mech. 867, P1 (2019).
She, Z.S. & Lévêque, E. Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72, 336 (1994).
Anselmet, F., Gagne, Y., Hopfinger, E. J. & Antonia, R. A. Highorder velocity structure functions in turbulent shear flows. J. Fluid Mech. 140, 63 (1984).
Boffetta, G., Mazzino, A. & Vulpiani, A. Twentyfive years of multifractals in fully developed turbulence: a tribute to Giovanni Paladin. J. Phys. A 41, 363001 (2008).
Iyer, K. P., Bharadwaj, S. S. & Sreenivasan, K. R. The area rule for circulation in threedimensional turbulence. Proc. Natl Acad. Sci. USA 118, e2114679118 (2021).
Cao, N., Chen, S. & Sreenivasan, K. R. Properties of velocity circulation in threedimensional turbulence. Phys. Rev. Lett. 76, 616 (1996).
Elsinga, G. E., Ishihara, T. & Hunt, J. C. R. Extreme dissipation and intermittency in turbulence at very high Reynolds numbers. Proc. R. Soc. A 476, 20200591 (2020).
Kraichnan, R. H. Eddy viscosity in two and three dimensions. J. Atmos. Sci. 33, 1521 (1976).
Donnelly, R. J. The twofluid theory and second sound in liquid helium. Phys. Today 62, 34 (2009).
La Mantia, M. & Skrbek, L. Quantum turbulence visualized by particle dynamics. Phys. Rev. B 90, 014519 (2014).
Švančara, P. & La Mantia, M. Flows of liquid ^{4}He due to oscillating grids. J. Fluid Mech. 832, 578 (2017).
Rusaouen, E., Chabaud, B., Salort, J. & Roche, P.E. Intermittency of quantum turbulence with superfluid fractions from 0% to 96%. Phys. Fluids 29, 105108 (2017).
Tang, Y., Bao, S., Kanai, T. & Guo, W. Statistical properties of homogeneous and isotropic turbulence in He II measured via particle tracking velocimetry. Phys. Rev. Fluids 5, 084602 (2020).
Boué, L., L’vov, V., Pomyalov, A. & Procaccia, I. Enhancement of intermittency in superfluid turbulence. Phys. Rev. Lett. 110, 014502 (2013).
Biferale, L. et al. Turbulent statistics and intermittency enhancement in coflowing superfluid He 4. Phys. Rev. Fluids 3, 024605 (2018).
Duda, D., La Mantia, M., Rotter, M. & Skrbek, L. On the visualization of thermal counterflow of He II past a circular cylinder. J. Low Temp. Phys. 175, 331 (2014).
Outrata, O. et al. On the determination of vortex ring vorticity using Lagrangian particles. J. Fluid Mech. 924, A44 (2021).
Ott, E., Du, Y., Sreenivasan, K. R., Juneja, A. & Suri, A. K. Signsingular measures: fast magnetic dynamos, and highReynoldsnumber fluid turbulence. Phys. Rev. Lett. 69, 2654 (1992).
Imazio, P. R. & Mininni, P. D. Cancellation exponents in helical and nonhelical flows. J. Fluid Mech. 651, 241 (2010).
Zhai, X. M., Sreenivasan, K. R. & Yeung, P. K. Cancellation exponents in isotropic turbulence and magnetohydrodynamic turbulence. Phys. Rev. E 99, 023102 (2019).
Nore, C., Abid, M. & Brachet, M. E. Kolmogorov turbulence in lowtemperature superflows. Phys. Rev. Lett. 78, 3896 (1997b).
Homann, H., Kamps, O., Friedrich, R. & Grauer, R. Bridging from Eulerian to Lagrangian statistics in 3D hydro and magnetohydrodynamic turbulent flows. N. J. Phys. 11, 073020 (2009).
Polanco, J. I., Müller, N. P. & Krstulovic, G. Circulation.jl: tools for computing velocity circulation statistics from periodic 3D Navier–Stokes and Gross–Pitaevskii fields. Zenodo https://doi.org/10.5281/zenodo.5578953 (2021).
Acknowledgements
We acknowledge useful scientific discussions with L. Galantucci and S. Thalabard. This work was supported by the Agence Nationale de la Recherche through the project GIANTE ANR18CE30002001. G.K. was also supported by the Simons Foundation Collaboration grant “Wave Turbulence” (Award ID 651471). This work was granted access to the HPC resources of CINES, IDRIS and TGCC under the allocation 2019A0072A11003 made by GENCI. Computations were also carried out at the Mésocentre SIGAMM hosted at the Observatoire de la Côte d’Azur.
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Navier–Stokes and Gross–Pitaevskii simulations were performed by J.I.P. and N.P.M., respectively. J.I.P. and N.P.M. postprocessed data. J.I.P., N.P.M. and G.K. equally contributed to theoretical developments and writing the paper.
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Polanco, J.I., Müller, N.P. & Krstulovic, G. Vortex clustering, polarisation and circulation intermittency in classical and quantum turbulence. Nat Commun 12, 7090 (2021). https://doi.org/10.1038/s41467021273826
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DOI: https://doi.org/10.1038/s41467021273826
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