Scale dependence of fractal dimension in deterministic and stochastic Lorenz-63 systems.

Many natural systems show emergent phenomena at different scales, leading to scaling regimes with signatures of deterministic chaos at large scales and an apparently random behavior at small scales. These features are usually investigated quantitatively by studying the properties of the underlying attractor, the compact object asymptotically hosting the trajectories of the system with their invariant density in the phase space. This multi-scale nature of natural systems makes it practically impossible to get a clear picture of the attracting set. Indeed, it spans over a wide range of spatial scales and may even change in time due to non-stationary forcing. Here, we combine an adaptive decomposition method with extreme value theory to study the properties of the instantaneous scale-dependent dimension, which has been recently introduced to characterize such temporal and spatial scale-dependent attractors in turbulence and astrophysics. To provide a quantitative analysis of the properties of this metric, we test it on the well-known low-dimensional deterministic Lorenz-63 system perturbed with additive or multiplicative noise. We demonstrate that the properties of the invariant set depend on the scale we are focusing on and that the scale-dependent dimensions can discriminate between additive and multiplicative noise despite the fact that the two cases have exactly the same stationary invariant measure at large scales. The proposed formalism can be generally helpful to investigate the role of multi-scale fluctuations within complex systems, allowing us to deal with the problem of characterizing the role of stochastic fluctuations across a wide range of physical systems.

Dynamical footprints of hurricanes in the tropical dynamics.

Hurricanes-and more broadly tropical cyclones-are high-impact weather phenomena whose adverse socio-economic and ecosystem impacts affect a considerable part of the global population. Despite our reasonably robust meteorological understanding of tropical cyclones, we still face outstanding challenges for their numerical simulations. Consequently, future changes in the frequency of occurrence and intensity of tropical cyclones are still debated. Here, we diagnose possible reasons for the poor representation of tropical cyclones in numerical models, by considering the cyclones as chaotic dynamical systems. We follow 197 tropical cyclones which occurred between 2010 and 2020 in the North Atlantic using the HURDAT2 and ERA5 data sets. We measure the cyclones instantaneous number of active degrees of freedom (local dimension) and the persistence of their sea-level pressure and potential vorticity fields. During the most intense phases of the cyclones, and specifically when cyclones reach hurricane strength, there is a collapse of degrees of freedom and an increase in persistence. The large dependence of hurricanes dynamical characteristics on intensity suggests the need for adaptive parametrization schemes which take into account the dependence of the cyclone's phase, in analogy with high-dissipation intermittent events in turbulent flows.

A correspondence between the multifractal model of turbulence and the Navier-Stokes equations.

The multifractal model of turbulence (MFM) and the three-dimensional Navier-Stokes equations are blended together by applying the probabilistic scaling arguments of the former to a hierarchy of weak solutions of the latter. This process imposes a lower bound on both the multifractal spectrum [Formula: see text], which appears naturally in the Large Deviation formulation of the MFM, and on [Formula: see text] the standard scaling parameter. These bounds respectively take the form: (i) [Formula: see text], which is consistent with Kolmogorov's four-fifths law ; and (ii) [Formula: see text]. The latter is significant as it prevents solutions from approaching the Navier-Stokes singular set of Caffarelli, Kohn and Nirenberg. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Characterizing most irregular small-scale structures in turbulence using local Hölder exponents.

Two scalar fields characterizing respectively pseudo-Hölder exponents and local energy transfers are used to capture the topology and the dynamics of the velocity fields in areas of lesser regularity. The present analysis is conducted using velocity fields from two direct numerical simulations of the Navier-Stokes equations in a triply periodic domain. A typical irregular structure is obtained by averaging over the 213 most irregular events. Such structure is similar to a Burgers vortex, with nonaxisymmetric corrections. A possible explanation for such asymmetry is provided by a detailed time-resolved analysis of birth and death of the irregular structures, which shows that they are connected to vortex interactions, possibly vortex reconnection.

Local estimates of Hölder exponents in turbulent vector fields.

It is still not known whether solutions to the Navier-Stokes equation can develop singularities from regular initial conditions. In particular, a classical and unsolved problem is to prove that the velocity field is Hölder continuous with some exponent h<1 (i.e., not necessarily differentiable) at small scales. Different methods have already been proposed to explore the regularity properties of the velocity field and the estimate of its Hölder exponent h. A first method is to detect potential singularities via extrema of an "inertial" dissipation D*=lim_{ℓ→0}D_{ℓ}^{I} that is independent of viscosity [Duchon and Robert, Nonlinearity 13, 249 (2000)0951-771510.1088/0951-7715/13/1/312]. Another possibility is to use the concept of multifractal analysis that provides fractal dimensions of the subspace of exponents h. However, the multifractal analysis is a global statistical method that only provides global information about local Hölder exponents, via their probability of occurrence. In order to explore the local regularity properties of a velocity field, we have developed a local statistical analysis that estimates locally the Hölder continuity. We have compared outcomes of our analysis with results using the inertial energy dissipation D_{ℓ}^{I}. We observe that the dissipation term indeed gets bigger for velocity fields that are less regular according to our estimates. The exact spatial distribution of the local Hölder exponents however shows nontrivial behavior and does not exactly match the distribution of the inertial dissipation.

Dissipation, intermittency, and singularities in incompressible turbulent flows.

We examine the connection between the singularities or quasisingularities in the solutions of the incompressible Navier-Stokes equation (INSE) and the local energy transfer and dissipation, in order to explore in detail how the former contributes to the phenomenon of intermittency. We do so by analyzing the velocity fields (a) measured in the experiments on the turbulent von Kármán swirling flow at high Reynolds numbers and (b) obtained from the direct numerical simulations of the INSE at a moderate resolution. To compute the local interscale energy transfer and viscous dissipation in experimental and supporting numerical data, we use the weak solution formulation generalization of the Kármán-Howarth-Monin equation. In the presence of a singularity in the velocity field, this formulation yields a nonzero dissipation (inertial dissipation) in the limit of an infinite resolution. Moreover, at finite resolutions, it provides an expression for local interscale energy transfers down to the scale where the energy is dissipated by viscosity. In the presence of a quasisingularity that is regularized by viscosity, the formulation provides the contribution to the viscous dissipation due to the presence of the quasisingularity. Therefore, our formulation provides a concrete support to the general multifractal description of the intermittency. We present the maps and statistics of the interscale energy transfer and show that the extreme events of this transfer govern the intermittency corrections and are compatible with a refined similarity hypothesis based on this transfer. We characterize the probability distribution functions of these extreme events via generalized Pareto distribution analysis and find that the widths of the tails are compatible with a similarity of the second kind. Finally, we make a connection between the topological and the statistical properties of the extreme events of the interscale energy transfer field and its multifractal properties.

Stochastic Chaos in a Turbulent Swirling Flow.

We report the experimental evidence of the existence of a random attractor in a fully developed turbulent swirling flow. By defining a global observable which tracks the asymmetry in the flux of angular momentum imparted to the flow, we can first reconstruct the associated turbulent attractor and then follow its route towards chaos. We further show that the experimental attractor can be modeled by stochastic Duffing equations, that match the quantitative properties of the experimental flow, namely, the number of quasistationary states and transition rates among them, the effective dimensions, and the continuity of the first Lyapunov exponents. Such properties can be recovered neither using deterministic models nor using stochastic differential equations based on effective potentials obtained by inverting the probability distributions of the experimental global observables. Our findings open the way to low-dimensional modeling of systems featuring a large number of degrees of freedom and multiple quasistationary states.

Experimental characterization of extreme events of inertial dissipation in a turbulent swirling flow.

The three-dimensional incompressible Navier-Stokes equations, which describe the motion of many fluids, are the cornerstones of many physical and engineering sciences. However, it is still unclear whether they are mathematically well posed, that is, whether their solutions remain regular over time or develop singularities. Even though it was shown that singularities, if exist, could only be rare events, they may induce additional energy dissipation by inertial means. Here, using measurements at the dissipative scale of an axisymmetric turbulent flow, we report estimates of such inertial energy dissipation and identify local events of extreme values. We characterize the topology of these extreme events and identify several main types. Most of them appear as fronts separating regions of distinct velocities, whereas events corresponding to focusing spirals, jets and cusps are also found. Our results highlight the non-triviality of turbulent flows at sub-Kolmogorov scales as possible footprints of singularities of the Navier-Stokes equation.

Role of boundary conditions in helicoidal flow collimation: Consequences for the von Kármán sodium dynamo experiment.

We present hydrodynamic and magnetohydrodynamic (MHD) simulations of liquid sodium flow with the PLUTO compressible MHD code to investigate influence of magnetic boundary conditions on the collimation of helicoidal motions. We use a simplified cartesian geometry to represent the flow dynamics in the vicinity of one cavity of a multiblades impeller inspired by those used in the Von-Kármán-sodium (VKS) experiment. We show that the impinging of the large-scale flow upon the impeller generates a coherent helicoidal vortex inside the blades, located at a distance from the upstream blade piloted by the incident angle of the flow. This vortex collimates any existing magnetic field lines leading to an enhancement of the radial magnetic field that is stronger for ferromagnetic than for conducting blades. The induced magnetic field modifies locally the velocity fluctuations, resulting in an enhanced helicity. This process possibly explains why dynamo action is more easily triggered in the VKS experiment when using soft iron impellers.

Superfluid high REynolds von Kármán experiment.

The Superfluid High REynolds von Kármán experiment facility exploits the capacities of a high cooling power refrigerator (400 W at 1.8 K) for a large dimension von Kármán flow (inner diameter 0.78 m), which can work with gaseous or subcooled liquid (He-I or He-II) from room temperature down to 1.6 K. The flow is produced between two counter-rotating or co-rotating disks. The large size of the experiment allows exploration of ultra high Reynolds numbers based on Taylor microscale and rms velocity [S. B. Pope, Turbulent Flows (Cambridge University Press, 2000)] (Rλ > 10000) or resolution of the dissipative scale for lower Re. This article presents the design and first performance of this apparatus. Measurements carried out in the first runs of the facility address the global flow behavior: calorimetric measurement of the dissipation, torque and velocity measurements on the two turbines. Moreover first local measurements (micro-Pitot, hot wire,…) have been installed and are presented.

Cross-helicity in rotating homogeneous shear-stratified turbulence.

We consider homogeneous shear-stratified turbulence in a rotating frame, that exhibits complex nonlinear dynamics. Since the analysis of relative orientation between coupled fluctuating fields helps us to understand turbulence dynamics, we focus on the alignment properties of both the velocity and gravity fields with the potential vorticity gradient. With the help of statistical mechanics, we define a vector field which plays a role in the analogous so-called cross-helicity in magnetohydrodynamics. High-resolution direct numerical simulations of developed homogeneous baroclinic turbulence are performed, and a detailed analysis of probability density functions for cross-helicity is provided. A net preference for positive cross-helicity is shown to be related to a new alignment mechanism. We argue that the analysis of cross-helicity is crucial for understanding the dynamics of buoyancy driven flows.

Evidence for forcing-dependent steady states in a turbulent swirling flow.

We study the influence on steady turbulent states of the forcing in a von Karman flow, at constant impeller speed, or at constant torque. We find that the different forcing conditions change the nature of the stability of the steady states and reveal dynamical regimes that bear similarities to low-dimensional systems. We suggest that this forcing dependence may be applicable to other turbulent systems.

Kinematic α tensors and dynamo mechanisms in a von Kármán swirling flow.

We provide experimental and numerical evidence of in-blades vortices in the von Kármán swirling flow. We estimate the associated kinematic α-effect tensor and show that it is compatible with recent models of the von Kármán sodium (VKS) dynamo. We further show that depending on the relative frequency of the two impellers, the dominant dynamo mechanism may switch from α2 to α - Ω dynamo. We discuss some implications of these results for VKS experiments.

Phase transitions and marginal ensemble equivalence for freely evolving flows on a rotating sphere.

The large-scale circulation of planetary atmospheres such as that of the Earth is traditionally thought of in a dynamical framework. Here we apply the statistical mechanics theory of turbulent flows to a simplified model of the global atmosphere, the quasigeostrophic model, leading to nontrivial equilibria. Depending on a few global parameters, the structure of the flow may be either a solid-body rotation (zonal flow) or a dipole. A second-order phase transition occurs between these two phases, with associated spontaneous symmetry breaking in the dipole phase. This model allows us to go beyond the general theory of marginal ensemble equivalence through the notion of Goldstone modes.

Experimental observation of spatially localized dynamo magnetic fields.

We report the first experimental observation of a spatially localized dynamo magnetic field, a common feature of astrophysical dynamos and convective dynamo simulations. When the two propellers of the von Kármán sodium experiment are driven at frequencies that differ by 15%, the mean magnetic field's energy measured close to the slower disk is nearly 10 times larger than the one close to the faster one. This strong localization of the magnetic field when a symmetry of the forcing is broken is in good agreement with a prediction based on the interaction between a dipolar and a quadrupolar magnetic mode.

Experimental evidence of a phase transition in a closed turbulent flow.

We experimentally study the susceptibility to symmetry breaking of a closed turbulent von Kármán swirling flow from Re=150 to Re≃106. We report a divergence of this susceptibility at an intermediate Reynolds number Re=Re(χ)≃90,000 which gives experimental evidence that such a highly space and time fluctuating system can undergo a "phase transition." This transition is furthermore associated with a peak in the amplitude of fluctuations of the instantaneous flow symmetry corresponding to intermittencies between spontaneously symmetry breaking metastable states.

Slow decay of concentration variance due to no-slip walls in chaotic mixing.

Chaotic mixing in a closed vessel is studied experimentally and numerically in different two-dimensional (2D) flow configurations. For a purely hyperbolic phase space, it is well known that concentration fluctuations converge to an eigenmode of the advection-diffusion operator and decay exponentially with time. We illustrate how the unstable manifold of hyperbolic periodic points dominates the resulting persistent pattern. We show for different physical viscous flows that, in the case of a fully chaotic Poincaré section, parabolic periodic points at the walls lead to slower (algebraic) decay. A persistent pattern, the backbone of which is the unstable manifold of parabolic points, can be observed. However, slow stretching at the wall forbids the rapid propagation of stretched filaments throughout the whole domain, and hence delays the formation of an eigenmode until it is no longer experimentally observable. Inspired by the baker's map, we introduce a 1D model with a parabolic point that gives a good account of the slow decay observed in experiments. We derive a universal decay law for such systems parametrized by the rate at which a particle approaches the no-slip wall.

Chaotic dynamos generated by a turbulent flow of liquid sodium.

We report the observation of several dynamical regimes of the magnetic field generated by a turbulent flow of liquid sodium (VKS experiment). Stationary dynamos, transitions to relaxation cycles or to intermittent bursts, and random field reversals occur in a fairly small range of parameters. Large scale dynamics of the magnetic field result from the interactions of a few modes. The low dimensional nature of these dynamics is not smeared out by the very strong turbulent fluctuations of the flow.

Walls inhibit chaotic mixing.

We report on experiments of chaotic mixing in a closed vessel, in which a highly viscous fluid is stirred by a moving rod. We analyze quantitatively how the concentration field of a low-diffusivity dye relaxes towards homogeneity, and we observe a slow algebraic decay of the inhomogeneity, at odds with the exponential decay predicted by most previous studies. Visual observations reveal the dominant role of the vessel wall, which strongly influences the concentration field in the entire domain and causes the anomalous scaling. A simplified 1D model supports our experimental results. Quantitative analysis of the concentration pattern leads to scalings for the distributions and the variance of the concentration field consistent with experimental and numerical results.

Generation of a magnetic field by dynamo action in a turbulent flow of liquid sodium.

We report the observation of dynamo action in the von Kármán sodium experiment, i.e., the generation of a magnetic field by a strongly turbulent swirling flow of liquid sodium. Both mean and fluctuating parts of the field are studied. The dynamo threshold corresponds to a magnetic Reynolds number R(m) approximately 30. A mean magnetic field of the order of 40 G is observed 30% above threshold at the flow lateral boundary. The rms fluctuations are larger than the corresponding mean value for two of the components. The scaling of the mean square magnetic field is compared to a prediction previously made for high Reynolds number flows.