Evolution of vortex statistics in two-dimensional turbulence

Two-dimensional turbulence above topography: Vortices and potential vorticity homogenization

The evolution of unforced and weakly damped two-dimensional turbulence over random rough topography presents two extreme states. If the initial kinetic energy [Formula: see text] is sufficiently high, then the topography is a weak perturbation, and evolution is determined by the spontaneous formation and mutual interaction of coherent axisymmetric vortices. High-energy vortices roam throughout the domain and mix the background potential vorticity (PV) to homogeneity, i.e., in the region between vortices, which is most of the domain, the relative vorticity largely cancels the topographic PV. If [Formula: see text] is low, then vortices still form but they soon become locked to topographic features: Anticyclones sit above topographic depressions and cyclones above elevated regions. In the low-energy case, with topographically locked vortices, the background PV retains some spatial variation. We develop a unified framework of topographic turbulence spanning these two extreme states of low and high energy. A main organizing concept is that PV homogenization demands a particular kinetic energy level [Formula: see text]. [Formula: see text] is the separator between high-energy evolution and low-energy evolution.

Alignment rule and geometric confinement lead to stability of a vortex in active flow

Vortices are hallmarks of a wide range of nonequilibrium phenomena in fluids at multiple length scales. In this work, we numerically study the whirling motion of self-propelled soft point particles confined in circular domain, and aim at addressing the stability issue of the coherent vortex structure. By the combination of dynamical and statistical analysis at the individual particle level, we reveal the persistence of the whirling motion resulting from the subtle competition of activity and geometric confinement. In the stable whirling motion, the scenario of the coexistence of the irregular microscopic motions of individual particles and the regular global whirling motion is fundamentally different from the motion of a vortex in passive fluid. Possible orientational order coexisting with the whirling are further explored. This work shows the stability mechanism of vortical dynamics in active media under the alignment rule in confined space and may have implications in creating and harnessing macroscale coherent dynamical states by tuning the confining geometry.

Polar vortex crystals: Emergence and structure

Vortex crystals, geometric arrays of like-signed vortices, are observed in natural systems with vastly different space and time scales: at the poles of Jupiter (∼10,000-km radius and lifetime of at least 5 y) and in laboratory experiments with pure-electron plasma (∼3.5-cm radius, lifetime of about 1.7 s). We follow the adage “less is more” and show that minimal physics is required for polar vortex crystals formation and persistence. Crystals, resembling those of Jupiter, form from the free evolution of an unstratified and rapidly rotating fluid in an axisymmetric geometry. An essential ingredient in this minimal model is the decrease of the vertical component of the Coriolis force with distance from the pole. Once formed, the crystal seems to survive indefinitely.

Vortex crystals are quasiregular arrays of like-signed vortices in solid-body rotation embedded within a uniform background of weaker vorticity. Vortex crystals are observed at the poles of Jupiter and in laboratory experiments with magnetized electron plasmas in axisymmetric geometries. We show that vortex crystals form from the free evolution of randomly excited two-dimensional turbulence on an idealized polar cap. Once formed, the crystals are long lived and survive until the end of the simulations (300 crystal-rotation periods). We identify a fundamental length scale, Lγ=(U/γ)1/3, characterizing the size of the crystal in terms of the mean-square velocity U of the fluid and the polar parameter γ=fp/ap2, with f_p the Coriolis parameter at the pole and a_p the polar radius of the planet.

Intermittency of three-dimensional perturbations in a point-vortex model

Three-dimensional (3D) instabilities on a (potentially turbulent) two-dimensional (2D) flow are still incompletely understood, despite recent progress. Here, based on known physical properties of such 3D instabilities, we propose a simple, energy-conserving model describing this situation. It consists of a regularized 2D point-vortex flow coupled to localized 3D perturbations ("ergophages"), such that ergophages can gain energy by altering vortex-vortex distances through an induced divergent velocity field, thus decreasing point-vortex energy. We investigate the model in three distinct stages of evolution: (i) The linear regime, where the amplitude of the ergophages grows or decays exponentially on average, with an instantaneous growth rate that fluctuates randomly in time. The instantaneous growth rate has a small auto-correlation time, and a probability distribution featuring a power-law tail with exponent between -2 and -5/3 (up to a cutoff) depending on the point-vortex base flow. Consequently, the logarithm of the ergophage amplitude performs a Lévy flight. (ii) The passive-nonlinear regime of the model, where the 2D flow evolves independently of the ergophage amplitudes, which saturate by non-linear self-interactions without affecting the 2D flow. In this regime the system exhibits a new type of on-off intermittency that we name Lévy on-off intermittency, which we define and study in a companion paper [van Kan et al., Phys. Rev. E 103, 052115 (2021)1063-651X10.1103/PhysRevE.103.052115]. We compute the bifurcation diagram for the mean and variance of the perturbation amplitude, as well as the probability density of the perturbation amplitude. (iii) Finally, we characterize the fully nonlinear regime, where ergophages feed back on the 2D flow, and study how the vortex temperature is altered by the interaction with ergophages. It is shown that when the amplitude of the ergophages is sufficiently large, the condensate is disrupted and the 2D flow saturates to a zero-temperature state. Given the limitations of existing theories, our model provides a new perspective on 3D instabilities growing on 2D flows, which will be useful in analyzing and understanding the much more complex results of DNS and potentially guide further theoretical developments.

Oceanic vortex mergers are not isolated but influenced by the β-effect and surrounding eddies

Oceanic vortices are ubiquitous in the ocean. They dominate the sub-inertial energy spectrum, and their dynamics is key for the evolution of the water column properties. The merger of two like-signed coherent vortices, which ultimately results in the formation of a larger vortex, provides an efficient mechanism for the lateral mixing of water masses in the ocean. Understanding the conditions of such interaction in the ocean is thus essential. Here, we use a merger detection algorithm to draw a global picture of this process in the ocean. We show that vortex mergers are not isolated, contrary to the hypothesis made in most earlier studies. Paradoxically, the merging distance is well reproduced by isolated vortex merger numerical simulations, but it is imperative to consider both the β-effect and the presence of neighbouring eddies to fully understand the physics of oceanic vortex merger.

The vortex gas scaling regime of baroclinic turbulence

The mean state of the atmosphere and ocean is set through a balance between external forcing (radiation, winds, heat and freshwater fluxes) and the emergent turbulence, which transfers energy to dissipative structures. The forcing gives rise to jets in the atmosphere and currents in the ocean, which spontaneously develop turbulent eddies through the baroclinic instability. A critical step in the development of a theory of climate is to properly include the eddy-induced turbulent transport of properties like heat, moisture, and carbon. In the linear stages, baroclinic instability generates flow structures at the Rossby deformation radius, a length scale of order 1,000 km in the atmosphere and 100 km in the ocean, smaller than the planetary scale and the typical extent of ocean basins, respectively. There is, therefore, a separation of scales between the large-scale gradient of properties like temperature and the smaller eddies that advect it randomly, inducing effective diffusion. Numerical solutions show that such scale separation remains in the strongly nonlinear turbulent regime, provided there is sufficient drag at the bottom of the atmosphere and ocean. We compute the scaling laws governing the eddy-driven transport associated with baroclinic turbulence. First, we provide a theoretical underpinning for empirical scaling laws reported in previous studies, for different formulations of the bottom drag law. Second, these scaling laws are shown to provide an important first step toward an accurate local closure to predict the impact of baroclinic turbulence in setting the large-scale temperature profiles in the atmosphere and ocean.

Emergent Non-Eulerian Hydrodynamics of Quantum Vortices in Two Dimensions

We develop a coarse-grained description of the point-vortex model, finding that a large number of planar vortices and antivortices behave as an inviscid non-Eulerian fluid at large scales. The emergent binary vortex fluid is subject to anomalous stresses absent from Euler's equation, caused by the singular nature of quantum vortices. The binary vortex fluid is compressible, and has an asymmetric Cauchy stress tensor allowing orbital angular momentum exchange with the vorticity and vortex density. An analytic solution for vortex shear flow driven by anomalous stresses is in excellent agreement with numerical simulations of the point-vortex model.

Resonance phenomena in a two-layer two-vortex shear flow

The paper deals with a dynamical system governing the motion of two point vortices embedded in the bottom layer of a two-layer rotating flow experiencing linear deformation and their influence on fluid particle advection. The linear deformation consists of shear and rotational components. If the deformation is stationary, the vortices can move periodically in a bounded region. The vortex periodic motion induces stirring patterns of passive fluid particles in the both layers. We focus our attention on the upper layer where the bottom-layer singular point vortices induce a regular velocity field with no singularities. In the upper layer, we determine a steady-state regime featuring no closed fluid particle trajectories associated with the vortex motion. Thus, in the upper layer, the flow's streamlines look like there is only external linear deformation and no vortices. In this case, fluid particles move along trajectories of almost regular elliptic shapes. However, the system dynamics changes drastically if the underlying vortices cease to be stationary and instead start moving periodically generating a nonstationary perturbation for the fluid particle advection. Then, we demonstrate that this steady-state regime transits to a perturbed state with a rich phase portrait structure featuring both periodic and chaotic fluid particle trajectories. Thus, the perturbed state clearly manifests the impact of the underlying vortex motion. An analysis, based on comparing the eigenfrequencies of the steady-state fluid particle rotation with the ones of the vortex rotation, is carried out, and parameters ensuring effective fluid particle stirring are determined. The process of separatrix reconnection of close stability islands leading to an enhanced chaotic region is reported and analyzed.

Upscale energy transfer in three-dimensional rapidly rotating turbulent convection

Rotating Rayleigh-Bénard convection exhibits, in the limit of rapid rotation, a turbulent state known as geostrophic turbulence. This state is present for sufficiently large Rayleigh numbers representing the thermal forcing of the system, and is characterized by a leading order balance between the Coriolis force and pressure gradient. This turbulent state is itself unstable to the generation of depth-independent or barotropic vortex structures of ever larger scale through a process known as spectral condensation. This process involves an inverse cascade mechanism with a positive feedback loop whereby large-scale barotropic vortices organize small scale convective eddies. In turn, these eddies provide a dynamically evolving energy source for the large-scale barotropic component. Kinetic energy spectra for the barotropic dynamics are consistent with a k-3 downscale enstrophy cascade and an upscale cascade that steepens to k-3 as the box-scale condensate forms. At the same time the flow maintains a baroclinic convective component with an inertial range consistent with a k-5/3 spectrum. The condensation process resembles a similar process in two dimensions but is fully three-dimensional.

Scaling properties and intermittency of two-dimensional turbulence in pure electron plasmas

When the cold nonrelativistic guiding center approximation is valid, the transverse dynamics of highly magnetized electron plasma columns confined in Penning-Malmberg traps is analogous to that of an incompressible, inviscid, two-dimensional (2D) fluid whose vorticity corresponds, up to a constant of proportionality, to the axially averaged electron plasma density. In this work intermittency phenomena in the freely decaying 2D electron plasma turbulence are investigated through scaling properties of the probability density functions and flatness of spatial vorticity increments, computed by analyzing the results of experiments performed in the Penning-Malmberg trap ELTRAP. It is shown that the intermittency properties of the turbulence strongly depends on the initial conditions and the relation of these results to the dynamics of the system is discussed.

Effective merging dynamics of two and three fluid vortices: application to two-dimensional decaying turbulence

We present a kinetic theory of two-dimensional decaying turbulence in the context of two-body and three-body vortex merging processes. By introducing the equations of motion for two or three vortices in the effective noise due to all the other vortices, we demonstrate analytically that a two-body mechanism becomes inefficient at low vortex density n<<1. When the more efficient three-body vortex mergings are considered (involving vortices of different signs), we show that n~t(-ξ), with ξ=1. We generalize this argument to three-dimensional geostrophic turbulence, finding ξ=5/4, in excellent agreement with direct Navier-Stokes simulations [McWilliams et al., J. Fluid Mech. 401, 1 (1999)].

Instability of a uniformly collapsing cloud of classical and quantum self-gravitating Brownian particles

We study the growth of perturbations in a uniformly collapsing cloud of self-gravitating Brownian particles. This problem shares analogies with the formation of large-scale structures in a universe experiencing a "big-crunch" or with the formation of stars in a molecular cloud experiencing gravitational collapse. Starting from the barotropic Smoluchowski-Poisson system, we derive a new equation describing the evolution of the density contrast in the comoving (collapsing) frame. This equation can serve as a prototype to study the process of self-organization in complex media with structureless initial conditions. We solve this equation analytically in the linear regime and compare the results with those obtained by using the "Jeans swindle" in a static medium. The stability criteria, as well as the laws for the time evolution of the perturbations, differ. The Jeans criterion is expressed in terms of a critical wavelength λ(J) while our criterion is expressed in terms of a critical polytropic index γ(4/3). In a static background, the system is stable for λ<λ(J) and unstable for λ>λ(J). In a collapsing cloud, the system is stable for γ>γ(4/3) and unstable for γ<γ(4/3). If γ=γ(4/3), it is stable for λ<λ(J) and unstable for λ>λ(J). We also study the fragmentation process in the nonlinear regime. We determine the growth of the skewness, the long-wavelength tail of the power spectrum and find a self-similar solution to the nonlinear equations valid for large times. Finally, we consider dissipative self-gravitating Bose-Einstein condensates with short-range interactions and show that, in a strong friction limit, the dissipative Gross-Pitaevskii-Poisson system is equivalent to the quantum barotropic Smoluchowski-Poisson system. This yields new types of nonlinear mean-field Fokker-Planck equations, including quantum effects.

Out-of-equilibrium phase transitions in the Hamiltonian mean-field model: a closer look

We provide a detailed discussion of out-of-equilibrium phase transitions in the Hamiltonian mean-field (HMF) model in the framework of Lynden-Bell's statistical theory of the Vlasov equation. For two-level initial conditions, the caloric curve β(E) only depends on the initial value f(0) of the distribution function. We evidence different regions in the parameter space where the nature of the phase transitions between magnetized and nonmagnetized states changes: (i) For f(0)>0.10965, the system displays a second-order phase transition; (ii) for 0.109497<f(0)<0.10965, the system displays a second-order phase transition and a first-order phase transition; (iii) for 0.10947<f(0)<0.109497, the system displays two second-order phase transitions; and (iv) for f(0)<0.1047, there is no phase transition. The passage from a first-order to a second-order phase transition corresponds to a tricritical point. The sudden appearance of two second-order phase transitions from nothing corresponds to a second-order azeotropy. This is associated with a phenomenon of phase reentrance. When metastable states are taken into account, the problem becomes even richer. In particular, we find another situation of phase reentrance. We consider both microcanonical and canonical ensembles and report the existence of a tiny region of ensemble inequivalence. We also explain why the use of the initial magnetization M(0) as an external parameter, instead of the phase level f(0), may lead to inconsistencies in the thermodynamical analysis. Finally, we mention different causes of incomplete relaxation that could be a limitation to the application of Lynden-Bell's theory.

Unifying scaling theory for vortex dynamics in two-dimensional turbulence

We present a scaling theory for unforced inviscid two-dimensional turbulence. Our model unifies existing spatial and temporal scaling theories. The theory is based on a self-similar distribution of vortices of different sizes A. Our model uniquely determines the spatial and temporal scaling of the associated vortex number density which allows the determination of the energy spectra and the vortex distributions. We find that the vortex number density scales as n(A,t)-t(-2/3)/A, which implies an energy spectrum E-k(-5), significantly steeper than the classical Batchelor-Kraichnan scaling. High-resolution numerical simulations corroborate the model.

Something old, something new

Some thoughts on the development of theoretical fluid dynamics in this century and some reflections on the developments of the past century were provided as an introduction to a broad-ranging conference. A written synopsis of these remarks is provided here.

Interacting scales and triad enstrophy transfers in generalized two-dimensional turbulence

The local and nonlocal characteristics of triad enstrophy transfer in the enstrophy inertial range of generalized two-dimensional turbulence, so-called alpha turbulence, are investigated using direct numerical simulations, with a special emphasis on alpha=1 , 2, and 3. The enstrophy transfer via nonlocal triad interactions dominates the transfer dynamics in the enstrophy inertial range, irrespective of alpha . However, the contributions from more local interactions to the total enstrophy transfer increase as alpha decreases. The results are discussed in connection with the local and nonlocal transition of the enstrophy transfer at alpha=2 expected from the phenomenological scaling theory. The specific nature of the enstrophy transfer in surface quasigeostrophic turbulence (alpha=1) is also discussed.

Formation and relaxation of two-dimensional vortex crystals in a magnetized pure-electron plasma

Systematic examinations are carried out experimentally about the contribution of background vorticity distributions (BGVD's) to the spontaneous formation and decay of ordered arrays (vortex crystals) composed of strong vortices (clumps) by using a pure-electron plasma. It is found that the BGVD level needs to be higher for an increasing number of clumps to form vortex crystals and that the number of the clumps constituting the crystal decreases in time as proportional to gamma lnt in contrast to proportional to t (-xi) with xi approximately 1 as accepted well in turbulence models. The decay rate gamma increases with the BGVD level. The observed configurations of the clumps cover the theoretically predicted catalogue of vortex arrays in superfluid helium, suggesting a possible relaxation path of the crystal states.

Nonrobustness of the two-dimensional turbulent inverse cascade

The inverse energy cascade in two-dimensional Navier-Stokes turbulence is examined in the quasisteady regime, with small-scale, band-limited forcing at scale kf-1, with particular attention to the influence of forcing Reynolds number Re on the energy distribution at large scales. The strength of the inverse energy cascade, or fraction of energy input that is transferred to larger scales, increases monotonically toward unity with increasing Re proportional, variantkmax2kf2, where kmax is the maximum resolved wave number. Moreover, as Re increases beyond a critical value, for which a direct enstrophy cascade to small scales is first realized, the energy spectrum in the energy-cascading range steepens from a k-53 to k-2 dependence. The steepening is interpreted as the result of a greater tendency for coherent vortex formation in cases when forcing scales are adequately resolved. In spectral space, it is associated with nonlocality of the inverse energy transfer.

Effect of the deformation radius on the evolution of vortex properties in geostrophic turbulence

Coherent vortices in geostrophic turbulence grow in size and become fewer as they merge. It is shown that the deformation radius L(D) has no effect on the growth of the average vortex radius a as a grows from a L(D) to a>> L(D) . Its growth is algebraic, given by a proportional to t(xi/4) , where xi=0.72 . However, the deformation radius does have an effect on the decay of the number of vortices or vortex density rho, given by rho proportional to t(-xi) for a <<L(D) and rho t(-xi/2) for a>> L(D) . Thus the decay of rho becomes slower once a has grown to a size comparable to that of L(D) . One scaling theory for the entire range from a<< L(D) to a>> L(D) is presented and verified by numerical experiments. A special method for quadruplicating the numerical domain when the vortices become too few is proposed, which keeps the computation inexpensive. This work generalizes and agrees with previous work, in which the two special cases a<<L(D) and a>> L(D) are independently investigated.

Large scale dissipation and filament instability in two-dimensional turbulence

Coherent vortices in two-dimensional turbulence induce far-field effects that stabilize vorticity filaments and inhibit the generation of new vortices. We show that the large-scale energy sink often included in numerical simulations of statistically stationary two-dimensional turbulence reduces the stabilizing role of the vortices, leading to filament instability and to continuous formation of new coherent vortices. This counterintuitive effect sheds new light on the mechanisms responsible for vortex formation in forced-dissipated two-dimensional turbulence, and it has significant impact on the temporal evolution of the vortex population in freely decaying turbulence. The time dependence of vortex statistics in the presence of a large-scale energy sink can be approximately described by a modified version of the scaling theory developed for small-scale dissipation.

Crossover in the enstrophy decay in two-dimensional turbulence in a finite box

The numerical simulation of two-dimensional decaying turbulence in a large but finite box presented in this Letter uncovered two physically different regimes of enstrophy decay. During the initial stage, the enstrophy Omega, generated by a random Gaussian initial condition, decays as Omega(t) proportional variant t(-gamma) with gamma approximately 0.7-0.8. After that, the flow undergoes a transition to a gas or fluid composed of distinct vortices. Simultaneously, the magnitude of the decay exponent crosses over to gamma approximately 0.4. A theory predicting N(t) proportional variant t(-xi) and the magnitudes of exponents gamma=2/5 and xi=4/5 is presented.

Maximum entropy theory and the rapid relaxation of three-dimensional quasi-geostrophic turbulence

Turbulent flow in a rapidly rotating stably stratified fluid (quasi-geostrophic turbulence) commonly decays toward a stable pattern of large-scale jets or vortices. A formula for the most probable three-dimensional end state, the maximum entropy state (MES), is derived using a form of Lynden-Bell statistical mechanics. The MES is determined by a set of integral invariants, including energy, as opposed to a complete description of the initial condition. A computed MES qualitatively resembles the quasistationary end state of a numerical simulation that is initialized with red noise, and relaxes for a time on the order of 100 (initial) eddy turnovers. However, the potential enstrophy of the end state, obtained from a coarsened potential vorticity distribution, exceeds that of the MES by nearly a factor of 2. The estimated errors for both theory and simulation do not account for the discrepancy. This suggests that the MES, if ever realized, requires a much longer time scale to fully develop.

Formation of a vortex crystal cell assisted by a background vorticity distribution

A vortex crystal is a quasistationary, symmetric array of intense vortices (clumps). A low level of background vorticity is experimentally observed to assist three clumps in forming an equilateral triangle starting from initial positions on a linear array. The triangle constitutes a unit cell of a crystal in a many-vortex system. The background vortex curbs the orbital motion of the clumps with unequal strengths to arrest them at the vertices of an equilateral triangle by wrapping them with different sized belts of depleted vorticity (ring holes). We characterize the contributions of a low-level background vorticity distribution on the formation of ordered states of clumps in light of the experimental results and existing theories.

Quasi-two-dimensional turbulence in shallow fluid layers: the role of bottom friction and fluid layer depth

The role of bottom friction and the fluid layer depth in numerical simulations and experiments of freely decaying quasi-two-dimensional turbulence in shallow fluid layers has been investigated. In particular, the power-law behavior of the compensated kinetic energy E0(t)=E(t)e(2lambda t), with E(t) the total kinetic energy of the flow and lambda the bottom-drag coefficient, and the compensated enstrophy Omega(0)(t)=Omega(t)e(2lambda t), with Omega(t) the total enstrophy of the flow, have been studied. We also report on the scaling exponents of the ratio Omega(t)/E(t), which is considered as a measure of the characteristic length scale in the flow, for different values of lambda. The numerical simulations on square bounded domains with no-slip boundaries revealed bottom-friction independent power-law exponents for E0(t), Omega(0)(t), and Omega(t)/E(t). By applying a discrete wavelet packet transform technique to the numerical data, we have been able to compute the power-law exponents of the average number density of vortices rho(t), the average vortex radius a(t), the mean vortex separation r(t), and the averaged normalized vorticity extremum omega(ext)(t)/square root E(t). These decay exponents proved to be independent of the bottom friction as well. In the experiments we have varied the fluid layer depth, and it was found that the decay exponents of E0(t), Omega(0)(t), Omega(t)/E(t), and omega(ext)(t)/square root E(t) are virtually independent of the fluid layer depth. The experimental data for rho(t) and a(t) are less conclusive; power-law exponents obtained for small fluid layer depths agree with those from previously reported experiments, but significantly larger power-law exponents are found for experiments with larger fluid layer depths.

Vortex statistics from Eulerian and Lagrangian time series

Coherent vortices are an important component of the dynamics of geophysical turbulence, but direct estimates of the properties of the vortex population from measured data is usually difficult. Motivated by this problem, we propose a new method for determining the statistical properties of coherent vortices in two-dimensional turbulence based on a small number of Lagrangian and Eulerian time series. The method provides reliable estimates of the mean vortex size and vortex number density.

Jets, stickiness, and anomalous transport

Dynamical and statistical properties of the vortex and passive particle advection in chaotic flows generated by 4- and 16-point vortices are investigated. General transport properties of these flows are found to be anomalous and exhibit a superdiffusive behavior with typical second moment exponent mu approximately 1.75. The origin of this anomaly is traced to the presence of coherent structures within the flow, the vortex cores, and the region far from where vortices are located. In the vicinity of these regions stickiness is observed and the motion of tracers is quasiballistic. The chaotic nature of the underlying flow dictates the choice for thorough analysis of transport properties. Passive tracer motion is analyzed by measuring the mutual relative evolution of two nearby tracers. Some tracers travel in each other's vicinity for relatively long times. This is related to a hidden order for the tracers, which we call jets. Jets are localized and found in sticky regions. Their structure is analyzed and found to be formed of a nested set of jets within jets. The analysis of the jet trapping time statistics shows a quantitative agreement with the observed transport exponent.

Experiments on free decay of quasi-two-dimensional turbulent flows

Decaying quasi-two-dimensional turbulence in a thin-layer flow is explored in laboratory experiments. We report the presence of power-law interval in the enstrophy decay law, in agreement with earlier experiments by Cardoso et al. [Phys. Rev. E 49, 454 (1994)] and Hansen et al. [Phys. Rev. E 58, 7261 (1998)]. The decay exponent proves sensitive to the way in which the energy decay is compensated. For the range of initial microscale Reynolds numbers between 35 and 95, the decay exponent is close to -0.4 for the ratio of enstrophy to energy, and to -0.75 for the enstrophy multiplied with a compensating factor of exp(-2lambda(t)), where lambda is the bottom-drag coefficient and t the decay time. The vorticity behavior does not comply with the theory of Carnevale et al. [Phys. Rev. Lett. 66, 2735 (1991)]: robust vortices are not observed in the vorticity field and the vorticity kurtosis is less than the Gaussian value.

Scaling laws and vortex profiles in two-dimensional decaying turbulence

We use high resolution numerical simulations over several hundred of turnover times to study the influence of small scale dissipation onto vortex statistics in 2D decaying turbulence. A scaling regime is detected when the scaling laws are expressed in units of mean vorticity and integral scale, like predicted in Carnevale et al., Phys. Rev. Lett. 66, 2735 (1991), and it is observed that viscous effects spoil this scaling regime. The exponent controlling the decay of the number of vortices shows some trends toward xi=1, in agreement with a recent theory based on the Kirchhoff model [C. Sire and P. H. Chavanis, Phys. Rev. E 61, 6644 (2000)]. In terms of scaled variables, the vortices have a similar profile with a functional form related to the Fermi-Dirac distribution.

Forced two-dimensional turbulence in spectral and physical space

Two-dimensional (2D) turbulence in the energy range exhibits nonuniversal features, manifested in the departure (at low k) from the k(-5/3) energy spectrum law, variable energy flux, and irregular, nonlocal transfers. To unravel the underlying mechanism we conducted a detailed study of the 2D turbulence in spectral and physical space. It revealed complex multiscale organization of vorticity field and dynamic processes, ranging from large-scale meandering jets to strong localized vortices. The latter bear prime responsibility for the nonuniversal behavior of 2D turbulence, and we examined their statistical features and the growth mechanism. Our results are based on the numeric simulation of 2D turbulence on the 512 grid under different forcing-dissipation conditions.

Effect of viscosity in the dynamics of two point vortices: exact results

An exact, unstationary, two-dimensional solution of the Navier-Stokes equations for the flow generated by two point vortices is obtained. The viscosity nu is introduced as a Brownian motion in the Hamiltonian dynamics of point vortices. The point vortices execute a stochastic motion whose probability density can be computed from a Fokker-Planck equation, equivalent to the original Navier-Stokes equation. The derived solution describes, in particular, the merging process of two Lamb vortices, and the development of the characteristic spiral structure in the topology of the vorticity. The viscous effects are thoroughly investigated by an asymptotic analysis of the solution. In particular, the selection mechanism of a specific pattern among the infinity satisfying the nu=0 (Euler) equation is discussed.

Chaotic advection near a three-vortex collapse

Dynamical and statistical properties of tracer advection are studied in a family of flows produced by three point-vortices of different signs. Tracer dynamics is analyzed by numerical construction of Poincaré sections, and is found to be strongly chaotic: advection pattern in the region around the center of vorticity is dominated by a well developed stochastic sea, which grows as the vortex system's initial conditions are set closer to those leading to the collapse of the vortices; at the same time, the islands of regular motion around vortices, known as vortex cores, shrink. An estimation of the core's radii from the minimum distance of vortex approach to each other is obtained. Tracer transport was found to be anomalous: for all of the three numerically investigated cases, the variance of the tracer distribution grows faster than a linear function of time, corresponding to a superdiffusive regime. The transport exponent varies with time decades, implying the presence of multi-fractal transport features. Yet, its value is never too far from 3/2, indicating some kind of universality. Statistics of Poincaré recurrences is non-Poissonian: distributions have long power-law tails. The anomalous properties of tracer statistics are the result of the complex structure of the advection phase space, in particular, of strong stickiness on the boundaries between the regions of chaotic and regular motion. The role of the different phase space structures involved in this phenomenon is analyzed. Based on this analysis, a kinetic description is constructed, which takes into account different time and space scalings by using a fractional equation.

Vortex statistics for turbulence in a container with rigid boundaries

The evolution of vortex statistics for decaying two-dimensional turbulence in a square container with rigid no-slip walls is compared with a few available experimental results and with the scaling theory of two-dimensional turbulent decay as proposed by Carnevale et al. Power-law exponents, computed from an ensemble average of several numerical runs, coincide with some experimentally obtained values, but not with data obtained from numerical simulations of decaying two-dimensional turbulence with periodic boundary conditions.

Statistics of velocity fluctuations arising from a random distribution of point vortices: the speed of fluctuations and the diffusion coefficient

This paper is devoted to a statistical analysis of the fluctuations of velocity and acceleration produced by a random distribution of point vortices in two-dimensional turbulence. We show that the velocity probability density function PDF behaves in a manner which is intermediate between Gaussian and Levy laws, while the distribution of accelerations is governed by a Cauchy law. Our study accounts properly for a spectrum of circulations among the vortices. In the case of real vortices (with a finite core), we show analytically that the distribution of accelerations makes a smooth transition from Cauchy (for small fluctuations) to Gaussian (for large fluctuations), probably passing through an exponential tail. We introduce a function T(V) which gives the typical duration of a velocity fluctuation V; we show that T(V) behaves like V and V-1 for weak and large velocities, respectively. These results have a simple physical interpretation in the nearest neighbor approximation, and in Smoluchowski theory concerning the persistence of fluctuations. We discuss the analogies with respect to the fluctuations of the gravitational field in stellar systems. As an application of these results, we determine an approximate expression for the diffusion coefficient of point vortices. When applied to the context of freely decaying two-dimensional turbulence, the diffusion becomes anomalous and we establish a relationship nu=1+(xi/2) between the exponent of anomalous diffusion nu and the exponent xi which characterizes the decay of the vortex density.

Numerical renormalization group of vortex aggregation in two-dimensional decaying turbulence: the role of three-body interactions

We introduce a numerical renormalization group procedure which permits long-time simulations of vortex dynamics and coalescence in a two-dimensional turbulent decaying fluid. The number of vortices decreases as N approximately t(-xi), with xi approximately 1 instead of the value xi=4/3 predicted by a naive kinetic theory. For short time, we find an effective exponent xi approximately 0.7 consistent with previous simulations and experiments. We show that the mean square displacement of surviving vortices grows as <x(2)> approximately t(1+xi/2). Introducing effective dynamics for two- and three-body collisions, we justify that only the latter become relevant at a small vortex area coverage. A kinetic theory consistent with this mechanism leads to xi=1. We find that the theoretical relations between kinetic parameters are all in good agreement with experiments.

Characteristics of two-dimensional turbulence that self-organizes into vortex crystals

Experiments have found that freely relaxing turbulence in inviscid, incompressible two-dimensional Euler flows can self-organize into ordered structures-vortex crystals-in which a number N(c) approximately 2-20 of strong vortices form stable, rigidly rotating patterns in a low vorticity background. In this paper we show that N(c) can be roughly predicted from properties of the flows in the early stage of the turbulent relaxation.

Ultrasound scattering and the study of vortex correlations in disordered flows

In an idealized way, some turbulent flows can be pictured by assemblies of many vortices characterized by a set of particle distribution functions. Ultrasound provides a useful, nonintrusive, tool to study the spatial structure of vorticity in flows. This is analogous to the use of elastic neutron scattering to determine liquid structure. We express the dispersion relation, as well as the scattering cross section, of sound waves propagating in a "liquid" of identical vortices as a function of vortex pair correlation functions. In two dimensions, formal analogies with ionic liquids are pointed out.

Critical behavior of vorticity in two-dimensional turbulence

We point out some similarities between the statistics of high Reynolds number turbulence and critical phenomena. An analogy is developed for two-dimensional decaying flows, in particular by studying the scaling properties of the two-point vorticity correlation function within a simple phenomenological framework. The inverse of the Reynolds number is the analog of the small parameter that separates the system from criticality. It is possible to introduce a set of three critical exponents; for the correlation length, the autocorrelation function, and the so-called susceptibility, respectively. The exponents corresponding to the well-known enstrophy cascade theory of Kraichnan and Batchelor are, remarkably, the same as the Gaussian approximation exponents for spin models. The limitations of the analogy, in particular the lack of universal scaling functions, are also discussed.

Anisotropic geophysical vortices

A survey is made of many types of coherent vortices in the Earth's ocean and atmosphere. These vortices often occur with strong, environmentally induced anisotropy in their velocity and vorticity fields. We propose a definition of the essential characteristics of coherent vortices and formulate hypotheses concerning their dynamical role in complex, anisotropic fluid motions. Finally, we analyze numerical solutions both for uniformly rotating, stably stratified three-dimensional flow and for two-dimensional flow for the phenomena of enstrophy cascade and dissipation, intermittency, isotropy in the appropriate coordinate frame, coherent vortex emergence, vortex population dynamics, and approach to a nonturbulent end state.

The coherent structures of shallow-water turbulence: Deformation-radius effects, cyclone/anticyclone asymmetry and gravity-wave generation

Over a large range of Rossby and Froude numbers, we investigate the dynamics of initially balanced decaying turbulence in a shallow rotating fluid layer. As in the case of incompressible two-dimensional decaying turbulence, coherent vortex structures spontaneously emerge from the initially random flow. However, owing to the presence of a free surface, a wealth of new phenomena appear in the shallow-water system. The upscale energy cascade, common to strongly rotating flows, is arrested by the presence of a finite Rossby deformation radius. Moreover, in contrast to near-geostrophic dynamics, a strong asymmetry is observed to develop as the Froude number is increased, leading to a clear dominance of anticyclonic vortices over cyclonic ones, even though no beta effect is present in the system. Finally, we observe gravity waves to be generated around the vortex structures, and, in the strongest cases, they appear in the form of shocks. We briefly discuss the relevance of this study to the vortices observed in Jupiter's atmosphere.