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.

The predictable chaos of slow earthquakes

Slow earthquakes, like regular earthquakes, result from unstable frictional slip. They produce little slip and can therefore repeat frequently. We assess their predictability using the slip history of the Cascadia subduction between 2007 and 2017, during which slow earthquakes have repeatedly ruptured multiple segments. We characterize the system dynamics using embedding theory and extreme value theory. The analysis reveals a low-dimensional (<5) nonlinear chaotic system rather than a stochastic system. We calculate properties of the underlying attractor like its correlation and instantaneous dimension, instantaneous persistence, and metric entropy. We infer that the system has a predictability horizon of the order of days weeks. For the better resolved segments, the onset of large slip events can be correctly forecasted by high values of the instantaneous dimension. Longer-term deterministic prediction seems intrinsically impossible. Regular earthquakes might similarly be predictable but with a limited predictable horizon of the order of their durations.

On reversals in 2D turbulent Rayleigh-Bénard convection: Insights from embedding theory and comparison with proper orthogonal decomposition analysis

Turbulent Rayleigh-Bénard convection in a 2D square cell is characterized by the existence of a large-scale circulation which varies intermittently. We focus on a range of Rayleigh numbers where the large-scale circulation experiences rapid non-trivial reversals from one quasi-steady (or meta-stable) state to another. In previous work [B. Podvin and A. Sergent, J. Fluid Mech. 766, 172201 (2015); B. Podvin and A. Sergent, Phys. Rev. E 95, 013112 (2017)], we applied proper orthogonal decomposition (POD) to the joint temperature and velocity fields at a given Rayleigh number, and the dynamics of the flow were characterized in a multi-dimensional POD space. Here, we show that several of those findings, which required extensive data processing over a wide range of both spatial and temporal scales, can be reproduced, and possibly extended, by application of the embedding theory to a single time series of the global angular momentum, which is equivalent here to the most energetic POD mode. Specifically, the embedding theory confirms that the switches among meta-stable states are uncorrelated. It also shows that, despite the large number of degrees of freedom of the turbulent Rayleigh Bénard flow, a low dimensional description of its physics can be derived with low computational efforts, providing that a single global observable reflecting the symmetry of the system is identified. A strong connection between the local stability properties of the reconstructed attractor and the characteristics of the reversals can also be established.

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.