arXiv subjects

Joran Rolland

Publications and source records attributed to Joran Rolland.

At least 19 recordsLinked to original sources

arXiv2026

The prediction of extreme uncertainty-production events in three-dimensional Navier-Stokes turbulence

We investigate the exponential growth of uncertainty energy in 3D Navier-Stokes turbulence, emphasising the intermittent and highly localized amplification/production of uncertainty, a critical factor in understanding the predictability of turbulent systems. From the Navier-Stokes equations one can identify some key fields contributing to the growth/decay of uncertainty-production term $P_{\Delta}$: strain rate, vorticity, and vortex deformation. The dynamics of these fields are examined in the $Q-R$ plane, where $Q$ and $R$ are the second and third invariants of the velocity gradient tensor, to understand their role in the evolution of uncertainty-production term $P_{\Delta}$. We proceed by estimating committor functions across the entire spatiotemporal domain of direct numerical simulations (DNS) of turbulence in a periodic domain at different Reynolds numbers. Our estimates of the probability of rare extreme events of local uncertainty-production term as a function of uncertainty energy, strain rate, vorticity, and vortex deformation confirm the role of strain rate in driving uncertainty. Where strain rate and vorticity are too close to their space-average values, stable probabilistic forecasts appear impossible solely on the basis of the fields considered here.

physics.flu-dyn↗

arXiv2026

Characterisation of a multistable turbulent wake: application of an improved regime identification with analytical model training

This article presents the experimental study and the modelling of the multistable jet in the wake of two side by side square bars separated by a distance $G$ at Reynolds number $R=U_\infty H/\nu=10000$ (with $U_\infty$ the velocity of the incoming flow, $H$ the bar side and $\nu$ the kinematic viscosity). The velocity field downstream of the bars is measured by means of two dimensional two components Particle Image Velocimetry (PIV). We use the weighted transverse position of the jet $Y_m$ and the jet width $w$ to characterise the regimes of multistability as the gap ratio $G/H$ is increased. Three main regimes of multistability are possible: tristability, bistability, and monostability. In our wind tunnel, tristability is observed for $G/H\in [1.15,1.25]$, bistability is observed for $G/H\in [1.5,2.65]$ and monostability is observed for $G/H\in [3.0,3.5]$. Within the first two ranges of $G/H$, there exists gap ratios for which multistability is more complex. In order to analyse the simple and complex multistability regimes as well as the transition from bistable to monostable, we construct an analytical stochastic differential equation (SDE) modelling $Y_m$ for each gap ratio. These SDEs are written with polynomial drift and diffusion. For this matter we use a data based method that finds an trade--off between simplicity of the model (smaller number of monomials) and precision. A first key advantage of the use of the data-based model fitting method is that when the flow is tristable, bistable or monostable, we recover the drifts expected from the theory of bifurcations, but we are now able to correct it with the right multiplicative noise expressed by the diffusion. The second key advantage is that we can also fit atypical drift expressions when the multistability regime is complex that help us make sense of the jet behaviour.

physics.flu-dyn↗

arXiv2025

Eulerian-Lagrangian scaling of the Lyapunov exponent in homogeneous turbulence

We present a heuristic derivation of the maximal Lyapunov exponent $γ$ of homogeneous turbulence which yields two new relations, a sweeping relation and the scaling of the uncertainty field's integral length $L_Δ$. These relations and the maximal Lyapunov exponent's scaling that they imply are confirmed by periodic turbulence simulations. As the Reynolds number $Re_\lammbda$ increases, $L_Δ$ and $γ^{-1}$ decrease towards values smaller than the Kolmogorov length and time scales.

physics.flu-dyn↗

arXiv2025

The interscale behaviour of uncertainty in three-dimensional Navier-Stokes turbulence

We derive the scale-by-scale uncertainty energy budget equation and demonstrate theoretically and computationally the presence of a self-similar equilibrium cascade of decorrelation in an inertial range of scales during the time range of power law growth of uncertainty in statistically stationary homogeneous turbulence. This cascade is predominantly inverse and driven by compressions of the reference field's relative deformation tensor and their aligments with the uncertainty velocity field. Three other subdominant cascade mechanisms are also present, two of which are forward and also dominated by compressions and one of which, the weakest and the only non-linear one of the four, is inverse. The uncertainty production and dissipation scalings which may follow from the self-similar equilibrium cascade of decorrelation lead to power law growths of the uncertainty integral scale and the average uncertainty energy which are also investigated. Compressions are not only key to chaoticity, as previously shown, but also to stochasticity.

physics.flu-dyn↗

arXiv2024

Does rare, noise-induced, bypass transition in plane Couette flow bypass instantons ?

We study rare noise induced paths that go all the way from stable laminar to transitional turbulent plane Couette flow and investigate whether these paths share the properties of classical noise induced transitions. The rare paths from laminar to turbulent flow are computed using Adaptive Multilevel Splitting, a rare event simulation method, and are validated against Direct Numerical Simulations at moderately small energy injection rate. The flow is forced outside its natural scales and redistribute energy to the unforced large scales so that the reactive trajectories display forced streamwise velocity tubes at the natural scale of velocity streaks. As the trajectory proceeds, these tubes gradually grow in amplitude until they cross the separatrix between laminar and turbulent flow. Streamwise vortices are visible after velocity tubes have reached near turbulent amplitude. As the domain size is increased from Lx * Lz=6 *4 to 36*24, spatial localisation then extension of the generated coherent streaks and vortices in the spanwise direction is observed. The paths computed in MFU display many of the characteristics of instantons, that often structure noise induced transitions: concentration of trajectories, exponentially increasing waiting times before transition, and Gumbel distribution of trajectory durations. However, bisections started on the reactive trajectories indicate that the trajectory lack two key ingredients of instantons. They do not visit the neighbourhood of the nearest saddle point and do not display the natural relaxation path from that saddle to transitional wall turbulence. The reactive paths do not concentrate more and more around the same trajectory as energy injection rate is decreased, but instead gradually move in phase space. They might reconnect with instantons at very small energy injection rate and exceedingly long waiting times.

physics.flu-dyn↗

arXiv2023

The production of uncertainty in three-dimensional Navier-Stokes turbulence

We derive the evolution equation of the average uncertainty energy for periodic/homogeneous incompressible Navier-Stokes turbulence and show that uncertainty is increased by strain rate compression and decreased by strain rate stretching. We use three different direct numerical simulations (DNS) of non-decaying periodic turbulence and identify a similarity regime where (a) the production and dissipation rates of uncertainty grow together in time, (b) the parts of the uncertainty production rate accountable to average strain rate properties on the one hand and fluctuating strain rate properties on the other also grow together in time, (c) the average uncertainty energies along the three different strain rate principal axes remain constant as a ratio of the total average uncertainty energy, (d) the uncertainty energy spectrum's evolution is self-similar if normalised by the uncertainty's average uncertainty energy and characteristic length and (e) the uncertainty production rate is extremely intermittent and skewed towards extreme compression events even though the most likely uncertainty production rate is zero. Properties (a), (b) and (c) imply that the average uncertainty energy grows exponentially in this similarity time range. The Lyapunov exponent depends on both the Kolmogorov time scale and the smallest Eulerian time scale, indicating a dependence on random large-scale sweeping of dissipative eddies. In the two DNS cases of statistically stationary turbulence, this exponential growth is followed by an exponential of exponential growth, which is in turn followed by a linear growth in the one DNS case where the Navier-Stokes forcing also produces uncertainty.

physics.flu-dyn↗

arXiv2022

Instantons and the path to intermittency in turbulent flows

Processes leading to anomalous fluctuations in turbulent flows, referred to as intermittency, are still challenging. We consider cascade trajectories through scales as realizations of a stochastic Langevin process for which multiplicative noise is an intrinsic feature of the turbulent state. The trajectories are conditioned on their entropy exchange. Such selected trajectories concentrate around an optimal path, called instanton, which is the minimum of an effective action. The action is derived from the Langevin equation, estimated from measured data. In particular instantons with negative entropy pinpoint the trajectories responsible for the emergence of non-Gaussian statistics at small-scales.

physics.flu-dyn↗

arXiv2022

Coupling rare event algorithms with data-based learned committor functions using the analogue Markov chain

Rare events play a crucial role in many physics, chemistry, and biology phenomena, when they change the structure of the system, for instance in the case of multistability, or when they have a huge impact. Rare event algorithms have been devised to simulate them efficiently, avoiding the computation of long periods of typical fluctuations. We consider here the family of splitting or cloning algorithms, which are versatile and specifically suited for far-from-equilibrium dynamics. To be efficient, these algorithms need to use a smart score function during the selection stage. Committor functions are the optimal score functions. In this work we propose a new approach, based on the analogue Markov chain, for a data-based learning of approximate committor functions. We demonstrate that such learned committor functions are extremely efficient score functions when used with the Adaptive Multilevel Splitting algorithm. We illustrate our approach for a gradient dynamics in a three-well potential, and for the Charney-DeVore model, which is a paradigmatic toy model of multistability for atmospheric dynamics. For these two dynamics, we show that having observed a few transitions is enough to have a very efficient data-based score function for the rare event algorithm. This new approach is promising for use for complex dynamics: the rare events can be simulated with a minimal prior knowledge and the results are much more precise than those obtained with a user-designed score function.

math.DS↗

arXiv2021

Collapse of transitional wall turbulence captured using a rare events algorithm

This text presents one of the first successful applications of a rare events method for the study of multistability in a turbulent flow without stochastic energy injection. The trajectories of collapse of turbulence in plane Couette flow, as well as their probability and rate of occurrence are systematically computed using \emph{Adaptive Multilevel Splitting} (AMS). The AMS computations are performed in a system of size $L_x\times L_z=24\times 18$ at Reynolds number $R=370$ with an acceleration by a factor $\mathcal{O}(10)$ with respect to DNS and in a system of size $L_x\times L_z=36\times 27$ at Reynolds number $R=377$ with an acceleration by a factor $\mathcal{O}(10^3)$. The AMS results are validated with a comparison to DNS in the system of size $L_x\times L_z=24\times 18$. Visualisations in both systems indicate that turbulence collapses because the self sustaining process of turbulence fails locally. The streamwise vortices decay first in streamwise elongated holes, leaving streamwise invariant streamwise velocity tubes that experience viscous decay. These holes then extend in the spanwise direction. The examination of more than a thousand of trajectories in the $(E_{c,x}=\int u_x^2/2\,{\rm d}^3\mathbf{x},E_{c,y-z}=\int (u_y^2/2+u_z^2/2)\,{\rm d}^3\mathbf{x})$ plane in the system of size $L_x\times L_z=24\times 18$ confirms the faster decay of streamwise vortices and shows concentration of trajectory. This hints at an instanton phenomenology in the large size limit. The computation of turning point states, beyond which laminarisation is certain, confirms the hole formation scenario and shows that it is more pronounced in larger systems. Finally, the examination of non reactive trajectories, where a hole opens then closes, indicates that hole opening and closing are distinct processes. Both the vortices and the streaks reform concomitantly when the laminar holes close.

physics.flu-dyn↗

arXiv2021

Multistability and rare spontaneous transitions in barotropic $β$-plane turbulence

We demonstrate that turbulent zonal jets, analogous to Jovian ones, which are quasi-stationary, are actually metastable. After extremely long times, they randomly switch to new configurations with a different number of jets. The genericity of this phenomenon suggests that most quasi-stationary turbulent planetary atmospheres might have many climates and attractors for fixed values of the external forcing parameters. A key message is that this situation will usually not be detected by simply running the numerical models, because of the extremely long mean transition time to change from one climate to another. In order to study such phenomena, we need to use specific tools: rare event algorithms and large deviation theory. With these tools, we make a full statistical mechanics study of a classical barotropic beta-plane quasigeostrophic model. It exhibits robust bimodality with abrupt transitions. We show that new jets spontaneously nucleate from westward jets. The numerically computed mean transition time is consistent with an Arrhenius law showing an exponential decrease of the probability as the Ekman dissipation decreases. This phenomenology is controlled by rare noise-driven paths called {\it instantons}. Moreover, we compute the saddles of the corresponding effective dynamics. For the dynamics of states with three alternating jets, we uncover an unexpectedly rich dynamics governed by the symmetric group ${\cal S}_3$ of permutations, with two distinct families of instantons, which is a surprise for a system where everything seemed stationary in the hundreds of previous simulations of this model. We discuss the future generalization of our approach to more realistic models.

physics.ao-ph↗

arXiv2019

Machine learning of committor functions for predicting high impact climate events

There is a growing interest in the climate community to improve the prediction of high impact climate events, for instance ENSO (El-Ni{ñ}o-Southern Oscillation) or extreme events, using a combination of model and observation data. In this note we explain that, in a dynamical context, the relevant quantity for predicting a future event is a committor function. We explain the main mathematical properties of this probabilistic concept. We compute and discuss the committor function of the Jin and Timmerman model of El-Ni{ñ}o. Our first conclusion is that one should generically distinguish between states with either intrinsic predictability or intrinsic unpredictability. This predictability concept is markedly different from the deterministic unpredictability arising because of chaotic dynamics and exponential sensibility to initial conditions. The second aim of this work is to compare the inference of a committor function from data, either through a direct approach or through a machine learning approach using neural networks. We discuss the consequences of this study for future applications to more complex data sets.

physics.ao-ph↗

arXiv2019

A rare event algorithm links transitions in turbulent flows with activated nucleations

Many turbulent flows undergo drastic and abrupt configuration changes with huge impacts. As a paradigmatic example we study the multistability of jet dynamics in a barotropic beta plane model of atmosphere dynamics. It is considered as the Ising model for Jupiter troposphere dynamics. Using the adaptive multilevel splitting, a rare event algorithm, we are able to get a very large statistics of transition paths, the extremely rare transitions from one state of the system to another. This new approach opens the way for addressing a set of questions that are out of reach through direct numerical simulations. We demonstrate for the first time the concentration of transition paths close to instantons, in a numerical simulation of genuine turbulent flows. We show that the transition is a noise-activated nucleation of vorticity bands. We address for the first time the existence of Arrhenius laws in turbulent flows. The methodology we developed shall prove useful to study many other transitions related to drastic changes for the turbulent dynamics of climate, geophysical, astrophysical and engineering applications. This opens a new range of studies impossible so far, and bring turbulent phenomena in the realm of non-equilibrium statistical mechanics.

cond-mat.stat-mech↗

arXiv2018

Finite size analysis of a double crossover in transitional wall turbulence

This article presents the finite size analysis of two consecutive crossovers leading laminar-turbulent bands to uniform wall turbulence in transitional plane Couette flow. Direct numerical simulations and low order modeling simulations of the flow are performed. The kinetic energy $E$ of the turbulent flow and the order parameter $M$, a measure of the spatially organised modulation of turbulence, are sampled and processed in view analytical results from the phenomenology of phase transitions. The first crossover concerns the loss of spatial organisation of turbulence in the flow. In the band phase, the order parameter $M$ decreases continuously with the Reynolds number $R$ toward a small value, while its response function $χ_M$ displays a maximum at the crossover. In the uniform phase, the order parameter $M$ and its variance $σ$ decrease toward zero following mean field field scalings $M,σ\propto 1/\sqrt{L_xL_z(R-R_c)}$ as $R$ is increased. The kinetic energy $E$ is an affine function of $R$ except in a small range where a sharp increase is detected, which corresponds to the second crossover. In this range, spatial and temporal coexistence of the uniform turbulence phase and laminar-turbulent bands phase is observed. This sharp increase is concomitant with a maximum of the response function of the kinetic energy. The finite size analysis reveals that the jump does not steepen and that the maximum of response function of $E$ saturates as size is increased. The first crossover is formally identical to a critical phenomenon in condensed matter. The second crossover is in agreement with a first order phase transition smeared by finite noise. The analytical analysis of this phenomenon assuming a non interacting gas of fronts between domain of the two phases provides a scaling of the response function consistent with that of $E$.

physics.flu-dyn↗

arXiv2015

Formation of spanwise vorticity in oblique turbulent bands of transitional plane Couette flow, part 2: modelling and stability analysis

This article presents a modelling of the formation of spanwise vorticity in the turbulent streaks of the oblique bands and spots of transitional plane Couette flow. A functional model is designed to mimic the coherent flow in the streaks. The control parameters of the model are extracted from Direct Numerical Simulations (DNS) statistical data. A Reynolds stress is proposed to study the effect on the instability of this additional force maintaining the baseflow. Local (quasi-parallel) temporal stability analysis is performed on that model to investigate the linear development of the spanwise vorticity. Results show that average profiles, even if they have an inflection, are stable: the shear layers inside the velocity streaks are responsible for the vorticity formation. Emphasis is put on the convective or absolute nature of the instability, depending on the location in the band. This shows that a transition from a convective to an absolute instability occurs in the zone in between fully turbulent and laminar flow. The group velocity of the most unstable modes compare well to the advection velocity of spanwise vorticity measured in DNS. This investigation is completed by the global (non-parallel) stability analysis of a typical band case. Eventually, the possible cycles of sustainment of localised low Reynolds number turbulence of shear flows are discussed in the light of these results.

physics.flu-dyn↗

arXiv2015

Stochastic analysis of the time evolution of Laminar-Turbulent bands of plane Couette flow

This article is concerned with the time evolution of the oblique laminar-turbulent bands of transitional plane Couette flow under the influence of turbulent noise. Our study is focused on the amplitude of modulation of turbulence. In order to guide the numerical study of the flow, we first perform an analytical and numerical analysis of a Stochastic Ginzburg-Landau equation for a complex order parameter. The modulus of this order parameter models the amplitude of modulation of turbulence. Firstly, we compute the autocorrelation function of said modulus once the band is established. Secondly, we perform a calculation of average and fluctuations around the exponential growth of the order parameter. This type of analysis is similar to the Stochastic Structural Stability Theory. We then perform numerical simulations of the Navier-Stokes equations in order to confront these predictions with the actual behaviour of the bands. Computation of the autocorrelation function of the modulation of turbulence shows quantitative agreement with the model: in the established band regime, the amplitude of modulation follows an Ornstein-Uhlenbeck process. In order to test the S3T predictions, we perform quench experiments, sudden decreases of the Reynolds number from uniform turbulence, in which modulation appears. We compute the average evolution of the amplitude of modulation and the fluctuations around it. We find good agreement between numerics and modeling. The average trajectory grows exponentially, at a rate clearly smaller than that of the formation of laminar holes. The actual time evolution remains in a flaring envelope, centred on the average, and expanding at the same rate. These results provide further validation of the stochastic modeling for the time evolution of the bands for further studies. They stress on the difference between the oblique band formation and the formation of laminar holes.

physics.flu-dyn↗

arXiv2015

Computing transition rates for the 1-D stochastic Ginzburg--Landau--Allen--Cahn equation for finite-amplitude noise with a rare event algorithm

In this paper we compute and analyse the transition rates and duration of reactive trajectories of the stochastic 1-D Allen-Cahn equations for both the Freidlin-Wentzell regime (weak noise or temperature limit) and finite-amplitude white noise, as well as for small and large domain. We demonstrate that extremely rare reactive trajectories corresponding to direct transitions between two metastable states are efficiently computed using an algorithm called adaptive multilevel splitting. This algorithm is dedicated to the computation of rare events and is able to provide ensembles of reactive trajectories in a very efficient way. In the small noise limit, our numerical results are in agreement with large-deviation predictions such as instanton-like solutions, mean first passages and escape probabilities. We show that the duration of reactive trajectories follows a Gumbel distribution like for one degree of freedom systems. Moreover, the mean duration growths logarithmically with the inverse temperature. The prefactor given by the potential curvature grows exponentially with size. The main novelty of our work is that we also perform an analysis of reactive trajectories for large noises and large domains. In this case, we show that the position of the reactive front is essentially a random walk. This time, the mean duration grows linearly with the inverse temperature and quadratically with the size. Using a phenomenological description of the system, we are able to calculate the transition rate, although the dynamics is described by neither Freidlin--Wentzell or Eyring--Kramers type of results. Numerical results confirm our analysis.

physics.flu-dyn↗

arXiv2015

Mechanical and statistical study of the laminar hole formation in transitional plane Couette flow

This article is concerned with the numerical study and modelling of two aspects the formation of laminar holes in transitional turbulence of plane Couette flow (PCF). On the one hand, we consider quenches: sudden decreases of the Reynolds number R which force the formation of holes. The Reynolds number is decreased from featureless turbulence to the range of existence of the oblique laminar-turbulent bands [Rg;Rt]. The successive stages of the quench are studied by means of visualisations and measurements of kinetic energy and turbulent fraction. The behaviour of the kinetic energy is explained using a kinetic energy budget: it shows that viscosity causes quasi modal decay until lift-up equals it and creates a new balance. Moreover, the budget confirms that the physical mechanisms at play are independent of the way the quench is performed. On the other hand we consider the natural formation of laminar holes in the bands, near Rg. The Direct Numerical simulations (DNS) show that holes in the turbulent bands provide a mechanism for the fragmented bands regime and orientation fluctuations near Rg. Moreover the analysis of the fluctuations of kinetic energy toward low values demonstrates that the disappearance of turbulence in the bands can be described within the framework of Large Deviations. A Large Deviation function is extracted from the Probability Density Function of the kinetic energy.

physics.flu-dyn↗

arXiv2014

Statistical behavior of adaptive multilevel splitting algorithms in simple models

Adaptive multilevel splitting algorithms have been introduced rather recently for estimating tail distributions in a fast and efficient way. In particular, they can be used for computing the so-called reactive trajectories corresponding to direct transitions from one metastable state to another. The algorithm is based on successive selection-mutation steps performed on the system in a controlled way. It has two intrinsic parameters, the number of particles/trajectories and the reaction coordinate used for discriminating good or bad trajectories. We investigate first the convergence in law of the algorithm as a function of the timestep for several simple stochastic models. Second, we consider the average duration of reactive trajectories for which no theoretical predictions exist. The most important aspect of this work concerns some systems with two degrees of freedom. They are studied in details as a function of the reaction coordinate in the asymptotic regime where the number of trajectories goes to infinity. We show that during phase transitions, the statistics of the algorithm deviate significatively from known theoretical results when using non-optimal reaction coordinates. In this case, the variance of the algorithm is peaking at the transition and the convergence of the algorithm can be much slower than the usual expected central limit behavior. The duration of trajectories is affected as well. Moreover, reactive trajectories do not correspond to the most probable ones. Such behavior disappears when using the optimal reaction coordinate called committor as predicted by the theory. We finally investigate a three-state Markov chain which reproduces this phenomenon and show logarithmic convergence of the trajectory durations.

math.NA↗