Itzhak Fouxon

Bound on viscosity and the generalized second law of thermodynamics

We describe a new paradox for ideal fluids. It arises in the accretion of an ideal fluid onto a black hole, where, under suitable boundary conditions, the flow can violate the generalized second law of thermodynamics. The paradox indicates that there is in fact a lower bound to the correlation length of any real fluid, the value of which is determined by the thermodynamic properties of that fluid. We observe that the universal bound on entropy, itself suggested by the generalized second law, puts a lower bound on the correlation length of any fluid in terms of its specific entropy. With the help of a new, efficient estimate for the viscosity of liquids, we argue that this also means that viscosity is bounded from below in a way reminiscent of the conjectured Kovtun-Son-Starinets lower bound on the ratio of viscosity to entropy density. We conclude that much light may be shed on the Kovtun-Son-Starinets bound by suitable arguments based on the generalized second law.

Distribution of particles and bubbles in turbulence at a small Stokes number.

The inertia of particles driven by the turbulent flow of the surrounding fluid makes them prefer certain regions of the flow. The heavy particles lag behind the flow and tend to accumulate in the regions with less vorticity, while the light particles do the opposite. As a result of the long-time evolution, the particles distribute over a multifractal attractor in space. We consider this distribution using our recent results on the steady states of chaotic dynamics. We describe the preferential concentration analytically and derive the correlation functions of density and the fractal dimensions of the attractor. The results are obtained for real turbulence and are testable experimentally.

Formation of density singularities in ideal hydrodynamics of freely cooling inelastic gases: A family of exact solutions

We employ granular hydrodynamics to investigate a paradigmatic problem of clustering of particles in a freely cooling dilute granular gas. We consider large-scale hydrodynamic motions where the viscosity and heat conduction can be neglected, and one arrives at the equations of ideal gas dynamics with an additional term describing bulk energy losses due to inelastic collisions. We employ Lagrangian coordinates and derive a broad family of exact nonstationary analytical solutions that depend only on one spatial coordinate. These solutions exhibit a new type of singularity, where the gas density blows up in a finite time when starting from smooth initial conditions. The density blowups signal formation of close-packed clusters of particles. As the density blow-up time tc is approached, the maximum density exhibits a power law ∼(tc−t)−2. The velocity gradient blows up as ∼−(tc−t)−1 while the velocity itself remains continuous and develops a cusp (rather than a shock discontinuity) at the singularity. The gas ...

Formation and evolution of density singularities in hydrodynamics of inelastic gases.

We use hydrodynamics to investigate clustering in a gas of inelastically colliding spheres. The hydrodynamic equations exhibit a finite-time density blowup, where the gas pressure remains finite. The density blowup signals the formation of close-packed clusters. The blowup dynamics is universal and describable by exact analytic solutions continuable beyond the blowup time. These solutions show that dilute hydrodynamic equations yield a powerful effective description of a granular gas flow with close-packed clusters, described as finite-mass pointlike singularities of the density. This description is similar in spirit to the description of shocks in ordinary ideal gas dynamics.

Superdiffusive Dispersals Impart the Geometry of Underlying Random Walks

It is recognized now that a variety of real-life phenomena ranging from diffusion of cold atoms to the motion of humans exhibit dispersal faster than normal diffusion. Lévy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient.

Clustering of particles in turbulence due to phoresis.

We demonstrate that diffusiophoretic, thermophoretic, and chemotactic phenomena in turbulence lead to clustering of particles on multifractal sets that can be described using one single framework, valid when the particle size is much smaller than the smallest length scale of turbulence l_{0}. To quantify the clustering, we derive positive pair correlations and fractal dimensions that hold for scales smaller than l_{0}. For scales larger than l_{0} the pair-correlation function is predicted to show a stretched exponential decay towards 1. In the case of inhomogeneous turbulence we find that the fractal dimension depends on the direction of inhomogeneity. By performing experiments with particles in a turbulent gravity current we demonstrate clustering induced by salinity gradients in conformity to the theory. The particle size in the experiment is comparable to l_{0}, outside the strict validity region of the theory, suggesting that the theoretical predictions transfer to this practically relevant regime. This clustering mechanism may provide the key to the understanding of a multitude of processes such as formation of marine snow in the ocean and population dynamics of chemotactic bacteria.Lukas Schmidt, Itzhak Fouxon, Dominik Krug, Maarten van Reeuwijk, and Markus Holzner 1 ETH Zurich, Stefano Franscini-Platz 5, 8093 Zurich, Switzerland 2 Department of Computational Science and Engineering, Yonsei University, Seoul 120-749, South Korea 3 Department of Mechanical Engineering, The University of Melbourne, Parkville, VIC 3010, Australia and 4 Department of Civil and Environmental Engineering, Imperial College London, London SW7 2AZ, UK

Classification of Eastward Propagating Waves on the Spherical Earth

Observational evidence for an equatorial non‐dispersive mode propagating at the speed of gravity waves is strong, and while the structure and dispersion relation of such a mode can be accurately described by a wave theory on the equatorial β‐plane, prior theories on the sphere were unable to find such a mode except for particular asymptotic limits of gravity wave phase speeds and/or certain zonal wave numbers. Here, an ad hoc solution of the linearized rotating shallow‐water equations (LRSWE) on a sphere is developed, which propagates eastward with phase speed that nearly equals the speed of gravity waves at all zonal wave numbers. The physical interpretation of this mode in the context of other modes that solve the LRSWE is clarified through numerical calculations and through eigenvalue analysis of a Schrödinger eigenvalue equation that approximates the LRSWE. By comparing the meridional amplitude structure and phase speed of the ad hoc mode with those of the lowest gravity mode on a non‐rotating sphere we show that at large zonal wave number the former is a rotation‐modified counterpart of the latter. We also find that the dispersion relation of the ad hoc mode is identical to the n = 0 eastward propagating inertia–gravity (EIG0) wave on a rotating sphere which is also nearly non‐dispersive, so this solution could be classified as both a Kelvin wave and as the EIG0 wave. This is in contrast to Cartesian coordinates where Kelvin waves are a distinct wave solution that supplements the EIG0 mode. Furthermore, the eigenvalue equation for the meridional velocity on the β‐plane can be formally derived as an asymptotic limit (for small (Lamb Number)‐1/4) of the corresponding second order equation on a sphere, but this expansion is invalid when the phase speed equals that of gravity waves i.e. for Kelvin waves. Various expressions found in the literature for both Kelvin waves and inertia–gravity waves and which are valid only in certain asymptotic limits (e.g. slow and fast rotation) are compared with the expressions found here for the two wave types.

Zooplankton can actively adjust their motility to turbulent flow

Significance Zooplankton possess narrow swimming capabilities, yet are capable of active locomotion amid turbulence. By decoupling the relative velocity of swimming zooplankton from that of the underlying flow, we provide evidence for an active adaptation that allows these small organisms to modulate their swimming effort in response to background flow. This behavioral response results in reduced diffusion at substantial turbulence intensity. Adjusting motility provides fitness advantage because it enables zooplankton to retain the benefits of self-locomotion despite the constraints enforced by turbulence transport. Vigorous swimming and reduced diffusion oppose turbulence advection, can directly affect the dispersal of zooplankton populations, and may help these organisms to actively control their distribution in dynamic environments. Calanoid copepods are among the most abundant metazoans in the ocean and constitute a vital trophic link within marine food webs. They possess relatively narrow swimming capabilities, yet are capable of significant self-locomotion under strong hydrodynamic conditions. Here we provide evidence for an active adaptation that allows these small organisms to adjust their motility in response to background flow. We track simultaneously and in three dimensions the motion of flow tracers and planktonic copepods swimming freely at several intensities of quasi-homogeneous, isotropic turbulence. We show that copepods synchronize the frequency of their relocation jumps with the frequency of small-scale turbulence by performing frequent relocation jumps of low amplitude that seem unrelated to localized hydrodynamic signals. We develop a model of plankton motion in turbulence that shows excellent quantitative agreement with our measurements when turbulence is significant. We find that, compared with passive tracers, active motion enhances the diffusion of organisms at low turbulence intensity whereas it dampens diffusion at higher turbulence levels. The existence of frequent jumps in a motion that is otherwise dominated by turbulent transport allows for the possibility of active locomotion and hence to transition from being passively advected to being capable of controlling diffusion. This behavioral response provides zooplankton with the capability to retain the benefits of self-locomotion despite turbulence advection and may help these organisms to actively control their distribution in dynamic environments. Our study reveals an active adaptation that carries strong fitness advantages and provides a realistic model of plankton motion in turbulence.

Evolution of collision numbers for a chaotic gas dynamics.

We put forward a conjecture of recurrence for a gas of hard spheres that collide elastically in a finite volume. The dynamics consists of a sequence of instantaneous binary collisions. We study how the numbers of collisions of different pairs of particles grow as functions of time. We observe that these numbers can be represented as a time integral of a function on the phase space. Assuming the results of the ergodic theory apply, we describe the evolution of the numbers by an effective Langevin dynamics. We use the facts that hold for these dynamics with probability one, in order to establish properties of a single trajectory of the system. We find that for any triplet of particles there will be an infinite sequence of moments of time, when the numbers of collisions of all three different pairs of the triplet will be equal. Moreover, any value of difference of collision numbers of pairs in the triplet will repeat indefinitely. On the other hand, for larger numbers of pairs there is but a finite number of repetitions. Thus the ergodic theory produces a limitation on the dynamics.

The mixed Rossby-gravity wave on the spherical Earth: PALDOR et al..

This work revisits the theory of the mixed Rossby–gravity (MRG) wave on a sphere. Three analytic methods are employed in this study: (a) derivation of a simple ad hoc solution corresponding to the MRG wave that reproduces the solutions of Longuet‐Higgins and Matsuno in the limits of zero and infinite Lambs parameter, respectively, while remaining accurate for moderate values of Lambs parameter, (b) demonstration that westward‐propagating waves with phase speed equalling the negative of the gravity‐wave speed exist, unlike the equatorial β‐plane, where the zonal velocity associated with such waves is infinite, and (c) approximation of the governing second‐order system by Schrödinger eigenvalue equations, which show that the MRG wave corresponds to the branch of the ground‐state solutions that connects Rossby waves with zonally symmetric waves. The analytic conclusions are confirmed by comparing them with numerical solutions of the associated second‐order equation for zonally propagating waves of the shallow‐water equations. We find that the asymptotic solutions obtained by Longuet‐Higgins in the limit of infinite Lambs parameter are not suitable for describing the MRG wave even when Lambs parameter equals 104. On the other hand, the dispersion relation obtained by Matsuno for the MRG wave on the equatorial β‐plane is accurate for values of Lambs parameter as small as 16, even though the equatorial β‐plane formally provides an asymptotic limit of the equations on the sphere only in the limit of infinite Lambs parameter.