Random changes of flow topology in two-dimensional and geophysical turbulence

Statistics of inhomogeneous turbulence in large-scale quasigeostrophic dynamics

A remarkable feature of two-dimensional turbulence is the transfer of energy from small to large scales. This process can result in the self-organization of the flow into large, coherent structures due to energy condensation at the largest scales. We investigate the formation of this condensate in a quasigeostropic flow in the limit of small Rossby deformation radius, namely the large-scale quasigeostrophic model. In this model potential energy is transferred up-scale while kinetic energy is transferred down-scale in a direct cascade. We focus on a jet mean flow and carry out a thorough investigation of the second-order statistics for this flow, combining a quasilinear analytical approach with direct numerical simulations. We show that the quasilinear approach applies in regions where jets are strong and is able to capture all second-order correlators in that region, including those related to the kinetic energy. This is a consequence of the blocking of the direct cascade by the mean flow in jet regions, suppressing fluctuation-fluctuation interactions. The suppression of the direct cascade is demonstrated using a local coarse-graining approach allowing us to measure space dependent interscale kinetic energy fluxes, which we show are concentrated in between jets in our simulations. We comment on the possibility of a similar direct cascade arrest in other two-dimensional flows, arguing that it is a special feature of flows in which the fluid element interactions are local in space.

Two-Dimensional Turbulence with Local Interactions: Statistics of the Condensate

Two-dimensional turbulence self-organizes through a process of energy accumulation at large scales, forming a coherent flow termed a condensate. We study the condensate in a model with local dynamics, the large-scale quasigeostrophic equation, observed here for the first time. We obtain analytical results for the mean flow and the two-point, second-order correlation functions, and validate them numerically. The condensate state requires partiy+time-reversal symmetry breaking. We demonstrate distinct universal mechanisms for the even and odd correlators under this symmetry. We find that the model locality is imprinted in the small scale dynamics, which the condensate spatially confines.

Sample-path large deviations for stochastic evolutions driven by the square of a Gaussian process

Recently, a number of physical models have emerged described by a random process with increments given by a quadratic form of a fast Gaussian process. We find that the rate function which describes sample-path large deviations for such a process can be computed from the large domain size asymptotic of a certain Fredholm determinant. The latter can be evaluated analytically using a theorem of Widom which generalizes the celebrated Szegő-Kac formula to the multidimensional case. This provides a large class of random dynamical systems with timescale separation for which an explicit sample-path large-deviation functional can be found. Inspired by problems in hydrodynamics and atmosphere dynamics, we construct a simple example with a single slow degree of freedom driven by the square of a fast multivariate Gaussian process and analyze its large-deviation functional using our general results. Even though the noiseless limit of this example has a single fixed point, the corresponding large-deviation effective potential has multiple fixed points. In other words, it is the addition of noise that leads to metastability. We use the explicit answers for the rate function to construct instanton trajectories connecting the metastable states.

Irreversible energy extraction from negative-temperature two-dimensional turbulence

The formation and transition of patterns of two-dimensional turbulent flows observed in various geophysical systems are commonly explained in terms of statistical mechanics. Different from ordinary systems, for a two-dimensional flow, the absolute temperature defined for a statistical equilibrium can take negative values. In a state of negative temperature, the second law of thermodynamics predicts that energy in microscopic fluctuations is irreversibly converted to a macroscopic form. This study explores the possibility of this one-way energy conversion in a two-dimensional flow using a basic conceptual model. We consider an inviscid incompressible fluid contained in a bounded domain, the shape of which is distorted by an externally imposed force. Unlike the usual fixed boundary cases, the flow energy within the domain is exchanged with the external system via pressure work through the moving lateral boundary. Concurrently, the flow field remains constrained by vorticity conservation. Beginning from a state of Kraichnan's grand-canonical ensemble, when the domain shape is distorted from one shape to another in a finite time, the Jarzynski equality is established. This equality states that, on average, the direction of a net energy flow through the boundary during a cycle of domain distortion changes with the sign of the initial temperature of the system. Numerical experiments are carried out to verify this theoretical argument and to investigate the parameter dependence of the energy exchange rate.

Crisis-induced flow reversals in magnetoconvection

We report the occurrence of flow reversals induced by the attractor-merging crisis in Rayleigh-Bénard convection of electrically conducting low-Prandtl-number fluids in the presence of a uniform external horizontal magnetic field. The simultaneous collision of two coexisting chaotic attractors with an unstable fixed point and its associated stable manifold takes place in the higher-dimensional phase space of the system, leading to a single merged chaotic attractor. The effect of strength of the magnetic field on the flow reversal phenomena is also explored in detail.

Multiple States in Turbulent Large-Aspect-Ratio Thermal Convection: What Determines the Number of Convection Rolls?

Wall-bounded turbulent flows can take different statistically stationary turbulent states, with different transport properties, even for the very same values of the control parameters. What state the system takes depends on the initial conditions. Here we analyze the multiple states in large-aspect ratio (Γ) two-dimensional turbulent Rayleigh-Bénard flow with no-slip plates and horizontally periodic boundary conditions as model system. We determine the number n of convection rolls, their mean aspect ratios Γ_{r}=Γ/n, and the corresponding transport properties of the flow (i.e., the Nusselt number Nu), as function of the control parameters Rayleigh (Ra) and Prandtl number. The effective scaling exponent β in Nu∼Ra^{β} is found to depend on the realized state and thus Γ_{r}, with a larger value for the smaller Γ_{r}. By making use of a generalized Friedrichs inequality, we show that the elliptical shape of the rolls and viscous damping determine the Γ_{r} window for the realizable turbulent states. The theoretical results are in excellent agreement with our numerical finding 2/3≤Γ_{r}≤4/3, where the lower threshold is approached for the larger Ra. Finally, we show that the theoretical approach to frame Γ_{r} also works for free-slip boundary conditions.

Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields

We introduce a variant of the Hybrid Monte Carlo (HMC) algorithm to address large-deviation statistics in stochastic hydrodynamics. Based on the path-integral approach to stochastic (partial) differential equations, our HMC algorithm samples space-time histories of the dynamical degrees of freedom under the influence of random noise. First, we validate and benchmark the HMC algorithm by reproducing multiscale properties of the one-dimensional Burgers equation driven by Gaussian and white-in-time noise. Second, we show how to implement an importance sampling protocol to significantly enhance, by orders of magnitudes, the probability to sample extreme and rare events, making it possible to estimate moments of field variables of extremely high order (up to 30 and more). By employing reweighting techniques, we map the biased configurations back to the original probability measure in order to probe their statistical importance. Finally, we show that by biasing the system towards very intense negative gradients, the HMC algorithm is able to explore the statistical fluctuations around instanton configurations. Our results will also be interesting and relevant in lattice gauge theory since they provide unique insights into reweighting techniques.

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 nonequilibrium statistical mechanics.

A Hybrid Approach for Model Order Reduction of Barotropic Quasi-Geostrophic Turbulence

We put forth a robust reduced-order modeling approach for near real-time prediction of mesoscale flows. In our hybrid-modeling framework, we combine physics-based projection methods with neural network closures to account for truncated modes. We introduce a weighting parameter between the Galerkin projection and extreme learning machine models and explore its effectiveness, accuracy and generalizability. To illustrate the success of the proposed modeling paradigm, we predict both the mean flow pattern and the time series response of a single-layer quasi-geostrophic ocean model, which is a simplified prototype for wind-driven general circulation models. We demonstrate that our approach yields significant improvements over both the standard Galerkin projection and fully non-intrusive neural network methods with a negligible computational overhead.

Connection between nonlinear energy optimization and instantons

How systems transit between different stable states under external perturbation is an important practical issue. We discuss here how a recently developed energy optimization method for identifying the minimal disturbance necessary to reach the basin boundary of a stable state is connected to the instanton trajectory from large deviation theory of noisy systems. In the context of the one-dimensional Swift-Hohenberg equation, which has multiple stable equilibria, we first show how the energy optimization method can be straightforwardly used to identify minimal disturbances-minimal seeds-for transition to specific attractors from the ground state. Then, after generalizing the technique to consider multiple, equally spaced-in-time perturbations, it is shown that the instanton trajectory is indeed the solution of the energy optimization method in the limit of infinitely many perturbations provided a specific norm is used to measure the set of discrete perturbations. Importantly, we find that the key features of the instanton can be captured by a low number of discrete perturbations (typically one perturbation per basin of attraction crossed). This suggests a promising new diagnostic for systems for which it may be impractical to calculate the instanton.

Extremely rare collapse and build-up of turbulence in stochastic models of transitional wall flows

This paper presents a numerical and theoretical study of multistability in two stochastic models of transitional wall flows. An algorithm dedicated to the computation of rare events is adapted on these two stochastic models. The main focus is placed on a stochastic partial differential equation model proposed by Barkley. Three types of events are computed in a systematic and reproducible manner: (i) the collapse of isolated puffs and domains initially containing their steady turbulent fraction; (ii) the puff splitting; (iii) the build-up of turbulence from the laminar base flow under a noise perturbation of vanishing variance. For build-up events, an extreme realization of the vanishing variance noise pushes the state from the laminar base flow to the most probable germ of turbulence which in turn develops into a full blown puff. For collapse events, the Reynolds number and length ranges of the two regimes of collapse of laminar-turbulent pipes, independent collapse or global collapse of puffs, is determined. The mean first passage time before each event is then systematically computed as a function of the Reynolds number r and pipe length L in the laminar-turbulent coexistence range of Reynolds number. In the case of isolated puffs, the faster-than-linear growth with Reynolds number of the logarithm of mean first passage time T before collapse is separated in two. One finds that ln(T)=A_{p}r-B_{p}, with A_{p} and B_{p} positive. Moreover, A_{p} and B_{p} are affine in the spatial integral of turbulence intensity of the puff, with the same slope. In the case of pipes initially containing the steady turbulent fraction, the length L and Reynolds number r dependence of the mean first passage time T before collapse is also separated. The author finds that T≍exp[L(Ar-B)] with A and B positive. The length and Reynolds number dependence of T are then discussed in view of the large deviations theoretical approaches of the study of mean first passage times and multistability, where ln(T) in the limit of small variance noise is studied. Two points of view, local noise of small variance and large length, can be used to discuss the exponential dependence in L of T. In particular, it is shown how a T≍exp[L(A^{'}R-B^{'})] can be derived in a conceptual two degrees of freedom model of a transitional wall flow proposed by Dauchot and Manneville. This is done by identifying a quasipotential in low variance noise, large length limit. This pinpoints the physical effects controlling collapse and build-up trajectories and corresponding passage times with an emphasis on the saddle points between laminar and turbulent states. This analytical analysis also shows that these effects lead to the asymmetric probability density function of kinetic energy of turbulence.

Predictability of escape for a stochastic saddle-node bifurcation: When rare events are typical

Transitions between multiple stable states of nonlinear systems are ubiquitous in physics, chemistry, and beyond. Two types of behaviors are usually seen as mutually exclusive: unpredictable noise-induced transitions and predictable bifurcations of the underlying vector field. Here, we report a different situation, corresponding to a fluctuating system approaching a bifurcation, where both effects collaborate. We show that the problem can be reduced to a single control parameter governing the competition between deterministic and stochastic effects. Two asymptotic regimes are identified: When the control parameter is small (e.g., small noise), deviations from the deterministic case are well described by the Freidlin-Wentzell theory. In particular, escapes over the potential barrier are very rare events. When the parameter is large (e.g., large noise), such events become typical. Unlike pure noise-induced transitions, the distribution of the escape time is peaked around a value which is asymptotically predicted by an adiabatic approximation. We show that the two regimes are characterized by qualitatively different reacting trajectories with algebraic and exponential divergences, respectively.

Statistical theory of reversals in two-dimensional confined turbulent flows

It is shown that the truncated Euler equation (TEE), i.e., a finite set of ordinary differential equations for the amplitude of the large-scale modes, can correctly describe the complex transitional dynamics that occur within the turbulent regime of a confined two-dimensional flow obeying Navier-Stokes equation (NSE) with bottom friction and a spatially periodic forcing. The random reversals of the NSE large-scale circulation on the turbulent background involve bifurcations of the probability distribution function of the large-scale circulation. We demonstrate that these NSE bifurcations are described by the related TEE microcanonical distribution which displays transitions from Gaussian to bimodal and broken ergodicity. A minimal 13-mode model reproduces these results.

Interaction between mean flow and turbulence in two dimensions

This short note is written to call attention to an analytic approach to the interaction of developed turbulence with mean flows of simple geometry (jets and vortices). It is instructive to compare cases in two and three dimensions and see why the former are solvable and the latter are not (yet). We present the analytical solutions for two-dimensional mean flows generated by an inverse turbulent cascade on a sphere and in planar domains of different aspect ratios. These solutions are obtained in the limit of small friction when the flow is strong while turbulence can be considered weak and treated perturbatively. I then discuss when these simple solutions can be realized and when more complicated flows may appear instead. The next step of describing turbulence statistics inside a flow and directions of possible future progress are briefly discussed at the end.

Population-dynamics method with a multicanonical feedback control

We discuss the Giardinà-Kurchan-Peliti population dynamics method for evaluating large deviations of time-averaged quantities in Markov processes [Phys. Rev. Lett. 96, 120603 (2006)PRLTAO0031-900710.1103/PhysRevLett.96.120603]. This method exhibits systematic errors which can be large in some circumstances, particularly for systems with weak noise, with many degrees of freedom, or close to dynamical phase transitions. We show how these errors can be mitigated by introducing control forces within the algorithm. These forces are determined by an iteration-and-feedback scheme, inspired by multicanonical methods in equilibrium sampling. We demonstrate substantially improved results in a simple model, and we discuss potential applications to more complex systems.

Large-deviation statistics of vorticity stretching in isotropic turbulence

A key feature of three-dimensional fluid turbulence is the stretching and realignment of vorticity by the action of the strain rate. It is shown in this paper, using the cumulant-generating function, that the cumulative vorticity stretching along a Lagrangian path in isotropic turbulence obeys a large deviation principle. As a result, the relevant statistics can be described by the vorticity stretching Cramér function. This function is computed from a direct numerical simulation data set at a Taylor-scale Reynolds number of Re(λ)=433 and compared to those of the finite-time Lyapunov exponents (FTLE) for material deformation. As expected, the mean cumulative vorticity stretching is slightly less than that of the most-stretched material line (largest FTLE), due to the vorticity's preferential alignment with the second-largest eigenvalue of strain rate and the material line's preferential alignment with the largest eigenvalue. However, the vorticity stretching tends to be significantly larger than the second-largest FTLE, and the Cramér functions reveal that the statistics of vorticity stretching fluctuations are more similar to those of the largest FTLE. In an attempt to relate the vorticity stretching statistics to the vorticity magnitude probability density function in statistically stationary conditions, a model Kramers-Moyal equation is constructed using the statistics encoded in the Cramér function. The model predicts a stretched-exponential tail for the vorticity magnitude probability density function, with good agreement for the exponent but significant difference (35%) in the prefactor.

Violent relaxation in two-dimensional flows with varying interaction range

Understanding the relaxation of a system towards equilibrium is a long-standing problem in statistical mechanics. Here we address the role of long-range interactions in this process by considering a class of two-dimensional flows where the interaction between fluid particles varies with the distance as ∼r(α-2) for α>0. We find that changing α with a prescribed initial state leads to different flow patterns: for small α, a coarsening process leads to the formation of a sharp interface between two regions of homogenized α-vorticity; for large α, the flow is attracted to a stable dipolar structure through a filamentation process. Assuming that the energy E and the enstrophy Z are injected at a typical scale smaller than the domain scale L, we argue that convergence towards the equilibrium state is expected when the parameter (2π/L)(α)E/Z tends to one, while convergence towards a dipolar state occurs systematically when this parameter tends to zero. This suggests that weak long-range interacting systems are more prone to relax towards an equilibrium state than strong long-range interacting systems.

Random transitions described by the stochastic Smoluchowski-Poisson system and by the stochastic Keller-Segel model

We study random transitions between two metastable states that appear below a critical temperature in a one-dimensional self-gravitating Brownian gas with a modified Poisson equation experiencing a second order phase transition from a homogeneous phase to an inhomogeneous phase [P. H. Chavanis and L. Delfini, Phys. Rev. E 81, 051103 (2010)]. We numerically solve the N-body Langevin equations and the stochastic Smoluchowski-Poisson system, which takes fluctuations (finite N effects) into account. The system switches back and forth between the two metastable states (bistability) and the particles accumulate successively at the center or at the boundary of the domain. We explicitly show that these random transitions exhibit the phenomenology of the ordinary Kramers problem for a Brownian particle in a double-well potential. The distribution of the residence time is Poissonian and the average lifetime of a metastable state is given by the Arrhenius law; i.e., it is proportional to the exponential of the barrier of free energy ΔF divided by the energy of thermal excitation kBT. Since the free energy is proportional to the number of particles N for a system with long-range interactions, the lifetime of metastable states scales as eN and is considerable for N≫1. As a result, in many applications, metastable states of systems with long-range interactions can be considered as stable states. However, for moderate values of N, or close to a critical point, the lifetime of the metastable states is reduced since the barrier of free energy decreases. In that case, the fluctuations become important and the mean field approximation is no more valid. This is the situation considered in this paper. By an appropriate change of notations, our results also apply to bacterial populations experiencing chemotaxis in biology. Their dynamics can be described by a stochastic Keller-Segel model that takes fluctuations into account and goes beyond the usual mean field approximation.

Emergence of large scale structure in barotropic β-plane turbulence

In this Letter, we use a nonequilibrium statistical theory, the stochastic structural stability theory (SSST), to show that an extended version of this theory can make predictions for the formation of nonzonal as well as zonal structures (lattice and stripe patterns) in forced homogeneous turbulence on a barotropic β plane. Comparison of the theory with nonlinear simulations demonstrates that SSST predicts the parameter values for the emergence of coherent structures and their characteristics (scale, amplitude, phase speed) as they emerge and at finite amplitude. It is shown that nonzonal structures (lattice states or zonons) emerge at lower energy input rates of the stirring compared to zonal flows (stripe states) and their emergence affects the dynamics of jet formation.

Phase transitions in wave turbulence

We consider turbulence within the Gross-Pitaevsky model and look into the creation of a coherent condensate via an inverse cascade originating at small scales. The growth of the condensate leads to a spontaneous breakdown of statistical symmetries of overcondensate fluctuations: First, isotropy is broken, then a series of phase transitions marks the changing symmetry from twofold to threefold to fourfold. We describe respective anisotropic flux flows in the k space. At the highest level reached, we observe a short-range positional and long-range orientational order (as in a hexatic phase). In other words, the more one pumps the system, the more ordered the system becomes. The phase transitions happen when the system is pumped by an instability term and does not occur when pumped by a random force. We thus demonstrate nonuniversality of an inverse-cascade turbulence with respect to the nature of small-scale forcing.

Structures in magnetohydrodynamic turbulence: detection and scaling

We present a systematic analysis of statistical properties of turbulent current and vorticity structures at a given time using cluster analysis. The data stem from numerical simulations of decaying three-dimensional magnetohydrodynamic turbulence in the absence of an imposed uniform magnetic field; the magnetic Prandtl number is taken equal to unity, and we use a periodic box with grids of up to 1536³ points and with Taylor Reynolds numbers up to 1100. The initial conditions are either an X -point configuration embedded in three dimensions, the so-called Orszag-Tang vortex, or an Arn'old-Beltrami-Childress configuration with a fully helical velocity and magnetic field. In each case two snapshots are analyzed, separated by one turn-over time, starting just after the peak of dissipation. We show that the algorithm is able to select a large number of structures (in excess of 8000) for each snapshot and that the statistical properties of these clusters are remarkably similar for the two snapshots as well as for the two flows under study in terms of scaling laws for the cluster characteristics, with the structures in the vorticity and in the current behaving in the same way. We also study the effect of Reynolds number on cluster statistics, and we finally analyze the properties of these clusters in terms of their velocity-magnetic-field correlation. Self-organized criticality features have been identified in the dissipative range of scales. A different scaling arises in the inertial range, which cannot be identified for the moment with a known self-organized criticality class consistent with magnetohydrodynamics. We suggest that this range can be governed by turbulence dynamics as opposed to criticality and propose an interpretation of intermittency in terms of propagation of local instabilities.