Francisco J. Beron-Vera (University of Miami):
Stability of the Malvinas Current

Deterministic and probabilistic tools from nonlinear dynamics are used to assess enduring near-surface Lagrangian aspects of the Malvinas Current. The deterministic tools are applied on a multi-year record of velocities derived from satellite altimetry data, revealing a resilient cross-stream transport barrier. This is composed of shearless-parabolic Lagrangian coherent structures (LCS), which, extracted over sliding time windows along the multi-year altimetry-derived velocity record, lie in near coincidental position. The probabilistic tools are applied on a large collection of historical satellite-tracked drifter trajectories, revealing weakly communicating flow regions on either side of the altimetry-derived barrier. Shearless-parabolic LCS are detected for the first time from altimetry data, and their significance is supported on satellite-derived ocean color data, which reveal shapes that quite closely resemble the peculiar V shapes, dubbed “chevrons,' that have recently confirmed the presence of similar LCS in the atmosphere of Jupiter. Finally, using in-situ velocity and hydrographic data, conditions for nonlinear symmetric stability are found be satisfied, suggesting a duality between Lagrangian and Eulerian stability for the Malvinas Current.

Maximilian Engel, TU München

Local phenomena in random dynamical systems: bifurcations, quasi-stationary dynamics and isochronicity

Random dynamical systems theory focuses on dynamical properties of stochastic systems, comparing trajectories with different initial conditions but driven by the same noise. A central question is the asymptotic behaviour of typical trajetories, which is often characterized by a Lyapunov spectrum and its bifurcation behavior. One part of the talk will focus on a new description of stochastic bifurcations, introducing the notion of conditioned Lyapunov exponents for trajectories that stay within a bounded domain for asymptotically long times. We show that conditioned Lyapunov exponents uncover local bifurcations that are typically destroyed in the presence of unbounded noise. A crucial ingredient for determining conditioned Lyapunov exponents is the spectral analysis of hypoelliptic Kolmogorov operators, whose properties can be exploited in computer-assisted proofs. The second part discusses a new approach for describing isochrons, which are cross-sections of limit cycles with fixed return times, in the context of (small) stochastic oscillations. We introduce stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period and discuss how this interpretation may be linked to recent physics approaches via an appropriate (S)PDE analysis.

The talk is based on joint work with Thai Son Doan (Vietnam Academy of Science and Technology), Christian Kuehn (TU Munich), Jeroen Lamb and Martin Rasmussen (Imperial College London).

John Bell, Lawrence Berkeley National Laboratory

Modeling Electrolytes at the Mesoscale

At small scales, the standard deterministic equations used for modeling fluids break down and thermal fluctuations play an important role in the dynamics. Landau and Lifshitz proposed a modifi ed verion of the Navier-Stokes equations, referred to as the fluctuating hydrodynamics equations that incorporate stochastic flux terms designed to incorporate the eff ect of fluctuations. These stochastic fluxes are constructed so that the equations are consistent with equilibrium fluctuations from statistical mechanics. In this talk, we present a generalization of fluctuating hydrodynamics to electrolytes. We then discuss some of the properties of the resulting system and show how fluctuations naturally incorporate some of the distinguishing characteristics of electrolytes. We then introduce a finite-volume method for solving the fluctuating hydrodynamics equations and present numerical results illustrating the behavior of electrolytes in some canonical flows.

Bérengère Dubrulle,
Service de Physique de l’Etat Condensé, CNRS, CEA Saclay, Université Paris-Saclay

Revisiting Kolmogorov Theory

Turbulent flows are characterized by a self-similar energy spectrum, signature of fluid movements at all scales. This organization has been described for more than 70 years by the phenomenology of "Kolmogorov cascade": the energy injected on a large scale by the work of the force that moves the fluid (e. g. a turbine) is transferred to smaller and smaller scales with a constant dissipation rate, up to the Kolmogorov scale, where it is transformed into heat and dissipated by viscosity.
I will explain why this image, which Landau questioned in the 1950s, is false. I will use recent velocity measurements obtained by very high resolution laser velocimetry to show that the energy "cascade" is in fact driven by extreme events on a very small scale, which are the signature of quasi-singularities of the Navier-Stokes equations existing under the Kolmogorov scale.

Gilles Vilmart, Université de Genève

Long time integration of stochastic differential equations: the interplay of geometric integration and stochastic integration

The preservation of geometric structures, such as the symplecticity of the flow for deterministic Hamiltonian systems, often reveals essential for an accurate numerical integration, and this is the aim of geometric integration.
In this talk we highlight the role that some geometric integration tools that were originally introduced in the deterministic setting play in the design of new accurate integrators to sample the invariant distribution of ergodic systems of stochastic ordinary and partial differential equations. In particular, we show how the ideas of modified differential equations and processing techniques permit to increase at a negligible overcost the order of accuracy of stiff integrators, including implicit schemes and explicit stabilized schemes.

This talk is based on joint works with Assyr Abdulle (EPF Lausanne), Ibrahim Almuslimani (Univ. Geneva), Charles-Edouard Béhier (Univ. Lyon), Adrien Laurent (Univ. Geneva), Gregorios A. Pavliotis (Imperial College London), Konstantinos C. Zygalakis (Univ. Edinburgh). Preprints available at http://www.unige.ch/~vilmart

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