Evidence for the double cascade scenario in two-dimensional turbulence

Effective Transport by 2D Turbulence: Vortex-Gas Theory vs Scale-Invariant Inverse Cascade

The scale-invariant inverse energy cascade is a hallmark of 2D turbulence, with its theoretical energy spectrum observed in both direct numerical simulations (DNS) and laboratory experiments. Under this scale-invariance assumption, the effective diffusivity of a 2D turbulent flow is dimensionally controlled by the energy flux and the friction coefficient only. Surprisingly, however, we show that such scaling predictions are invalidated by numerical solutions of the 2D Navier-Stokes equation forced at intermediate wave number and damped by weak linear or quadratic drag. We derive alternate scaling-laws for the effective diffusivity based on the emergence of intense, isolated vortices causing spatially inhomogeneous frictional dissipation localized within the small vortex cores. The predictions quantitatively match DNS data. This study points to a universal large-scale organization of 2D turbulent flows in physical space, bridging standard 2D Navier-Stokes turbulence with large-scale geophysical turbulence.

Phase transitions in anisotropic turbulence

Turbulence is a widely observed state of fluid flows, characterized by complex, nonlinear interactions between motions across a broad spectrum of length and time scales. While turbulence is ubiquitous, from teacups to planetary atmospheres, oceans, and stars, its manifestations can vary considerably between different physical systems. For instance, three-dimensional turbulent flows display a forward energy cascade from large to small scales, while in two-dimensional turbulence, energy cascades from small to large scales. In a given physical system, a transition between such disparate regimes of turbulence can occur when a control parameter reaches a critical value. The behavior of flows close to such transition points, which separate qualitatively distinct phases of turbulence, has been found to be unexpectedly rich. Here, we survey recent findings on such transitions in highly anisotropic turbulent fluid flows, including turbulence in thin layers and under the influence of rapid rotation. We also review recent work on transitions induced by turbulent fluctuations, such as random reversals and transitions between large-scale vortices and jets, among others. The relevance of these results and their ramifications for future investigations are discussed.

Neural network complexity of chaos and turbulence

Chaos and turbulence are complex physical phenomena, yet a precise definition of the complexity measure that quantifies them is still lacking. In this work, we consider the relative complexity of chaos and turbulence from the perspective of deep neural networks. We analyze a set of classification problems, where the network has to distinguish images of fluid profiles in the turbulent regime from other classes of images such as fluid profiles in the chaotic regime, various constructions of noise and real-world images. We analyze incompressible as well as weakly compressible fluid flows. We quantify the complexity of the computation performed by the network via the intrinsic dimensionality of the internal feature representations and calculate the effective number of independent features which the network uses in order to distinguish between classes. In addition to providing a numerical estimate of the complexity of the computation, the measure also characterizes the neural network processing at intermediate and final stages. We construct adversarial examples and use them to identify the two point correlation spectra for the chaotic and turbulent vorticity as the feature used by the network for classification.

Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade

We examine the conjecture of equivalence of nonequilibrium ensembles for turbulent flows in two dimensions in a dual-cascade setup. We construct a formally time-reversible Navier-Stokes equation in two dimensions by imposing global constraints of energy and enstrophy conservation. A comparative study of the statistical properties of its solutions with those obtained from the standard Navier-Stokes equations clearly shows that a formally time-reversible system is able to reproduce the features of a two-dimensional turbulent flow. Statistical quantities based on one- and two-point measurements show an excellent agreement between the two systems for the inverse- and direct-cascade regions. Moreover, we find that the conjecture holds very well for two-dimensional turbulent flows with both conserved energy and enstrophy at finite Reynolds number.

Circulation in Quasi-2D Turbulence: Experimental Observation of the Area Rule and Bifractality

We present an experimental study of the velocity circulation in a quasi-two-dimensional turbulent flow. We show that the area rule of circulation around simple loops holds in both the forward cascade enstrophy inertial range (ΩIR) and the inverse cascade energy inertial range (EIR): When the side lengths of a loop are all within the same inertial range, the circulation statistics depend on the loop area alone. It is also found that, for circulation around figure-eight loops, the area rule still holds in EIR but is not applicable in ΩIR. In ΩIR, the circulation is nonintermittent; whereas in EIR, the circulation is bifractal: space filling for moments of the order of 3 and below and a monofractal with a dimension of 1.42 for higher orders. Our results demonstrate, as in a numerical study of 3D turbulence [K. P. Iyer et al., Circulation in High Reynolds Number Isotropic Turbulence is a Bifractal, Phys. Rev. X 9, 041006 (2019).PRXHAE2160-330810.1103/PhysRevX.9.041006], that, in terms of circulation, turbulent flows exhibit a simpler behavior than velocity increments, as the latter are multifractals.

On the Turbulent Behavior of a Magnetically Confined Plasma near the X-Point

We construct a model for the turbulence near the X-point of a Tokamak device and, under suitable assumptions, we arrive to a closed equation for the electric field potential fluctuations. The analytical and numerical analysis is focused on a reduced two-dimensional formulation of the dynamics, which allows a direct mapping to the incompressible Navier-Stokes equation. The main merit of this study is to outline how the turbulence near the X-point, in correspondence to typical operation conditions of medium and large size Tokamaks, is dominated by the enstrophy cascade from large to smaller spatial scales.

Role of Advective Inertia in Active Nematic Turbulence

Suspensions of active agents with nematic interactions exhibit complex spatiotemporal dynamics such as mesoscale turbulence. Since the Reynolds number of microscopic flows is very small on the scale of individual agents, inertial effects are typically excluded in continuum theories of active nematic turbulence. Whether active stresses can collectively excite inertial flows is currently unclear. To address this question, we investigate a two-dimensional continuum theory for active nematic turbulence. In particular, we compare mesoscale turbulence with and without the effects of advective inertia. We find that inertial effects can influence the flow already close to the onset of the turbulent state and, moreover, give rise to large-scale fluid motion for strong active driving. A detailed analysis of the kinetic energy budget reveals an energy transfer to large scales mediated by inertial advection. While this transfer is small in comparison to energy injection and dissipation, its effects accumulate over time. The inclusion of friction, which is typically present in experiments, can compensate for this effect. The findings suggest that the inclusion of inertia and friction may be necessary for dynamically consistent theories of active nematic turbulence.

Spiral chain models of two-dimensional turbulence

Reduced models, mirroring self-similar, fractal nature of two-dimensional turbulence, are proposed, using logarithmic spiral chains, which provide a natural generalization of shell models to two dimensions. In a turbulent cascade, where each step can be represented by a rotation and a scaling of the interacting triad, the use of a spiral chain whose nodes can be obtained by scaling and rotating an original wave vector provides an interesting perspective. A family of such spiral chain models depending on the distance of interactions can be obtained by imposing a logarithmic spiral grid with a constant divergence angle and a constant scaling factor and imposing the condition of exact triadic interactions. Scaling factors in such sequences are given by the square roots of known ratios such as the plastic ratio, the super-golden ratio, or some small Pisot numbers. While spiral chains can represent monofractal models of a self-similar cascade, which can span a large range of wave numbers and have good angular coverage, it is also possible that spiral chains or chains of consecutive triads play an important role in the cascade. As numerical models, the spiral chain models based on decimated Fourier coefficients have the usual problems of shell models of two-dimensional turbulence such as the dual cascade being overwhelmed by statistical chain equipartition due to an almost stochastic evolution of the complex phases. A generic spiral chain model based on evolution of energy is proposed, which is shown to recover the dual cascade behavior in two-dimensional turbulence.

Intrinsic flow structure and multifractality in two-dimensional bacterial turbulence

The active interaction between the bacteria and fluid generates turbulent structures even at zero Reynolds number. The velocity of such a flow obtained experimentally has been quantitatively investigated based on streamline segment analysis. There is a clear transition at about 16 times the organism body length separating two different scale regimes, which may be attributed to the different influence of the viscous effect. Surprisingly the scaling extracted from the streamline segment indicates the existence of scale similarity even at the zero Reynolds number limit. Moreover, the multifractal feature can be quantitatively described via a lognormal formula with the Hurst number H=0.76 and the intermittency parameter μ=0.20, which is coincidentally in agreement with the three-dimensional hydrodynamic turbulence result. The direction of cascade is measured via the filter-space technique. An inverse energy cascade is confirmed. For the enstrophy, a forward cascade is observed when r/R≤3, and an inverse one is observed when r/R>3, where r and R are the separation distance and the bacteria body size, respectively. Additionally, the lognormal statistics is verified for the coarse-grained energy dissipation and enstrophy, which supports the lognormal formula to fit the measured scaling exponent.

Instability onset and scaling laws of an auto-oscillating turbulent flow in a complex plasma

We study a complex plasma under microgravity conditions that is first stabilized with an oscillating electric field. Once the stabilization is stopped, the so-called heartbeat instability develops. We study how the kinetic energy spectrum changes during and after the onset of the instability and compare with the double cascade predicted by Kraichnan and Leith for two-dimensional turbulence. The onset of the instability manifests clearly in the ratio of the reduced rates of cascade of energy and enstrophy and in the power-law exponents of the energy spectra.

Two-dimensional Turbulence in Symmetric Binary-Fluid Mixtures: Coarsening Arrest by the Inverse Cascade

We study two-dimensional (2D) binary-fluid turbulence by carrying out an extensive direct numerical simulation (DNS) of the forced, statistically steady turbulence in the coupled Cahn-Hilliard and Navier-Stokes equations. In the absence of any coupling, we choose parameters that lead (a) to spinodal decomposition and domain growth, which is characterized by the spatiotemporal evolution of the Cahn-Hilliard order parameter ϕ, and (b) the formation of an inverse-energy-cascade regime in the energy spectrum E(k), in which energy cascades towards wave numbers k that are smaller than the energy-injection scale kin j in the turbulent fluid. We show that the Cahn-Hilliard-Navier-Stokes coupling leads to an arrest of phase separation at a length scale Lc, which we evaluate from S(k), the spectrum of the fluctuations of ϕ. We demonstrate that (a) Lc ~ LH, the Hinze scale that follows from balancing inertial and interfacial-tension forces, and (b) Lc is independent, within error bars, of the diffusivity D. We elucidate how this coupling modifies E(k) by blocking the inverse energy cascade at a wavenumber kc, which we show is ≃2π/Lc. We compare our work with earlier studies of this problem.

Anisotropic macroturbulence and diffusion associated with a westward zonal jet: From laboratory to planetary atmospheres and oceans

Turbulence with inverse energy cascade and its transport properties are investigated experimentally in a flow associated with a westward propagating jet. Turbulence and the jet were produced by an electromagnetic force in a rotating tank filled with an electrolytic saline solution. The parabolic free surface emulated the topographic β effect which evoked the zonation. The spectral and transport flow characteristics were highly anisotropic. Turbulence is diagnosed by exploring the analogy between vertical and horizontal turbulent overturns in, respectively, stably stratified and quasigeostrophic flows which gives rise to a method of potential vorticity (PV) monotonizing. The anisotropization of transport properties of the flow is investigated using the finite scale Lyapunov exponent technique. After initial exponential particle separation, radial (meridional in geophysical and planetary applications) diffusion attains a short-ranged Richardson regime which transitions to the Taylor (scale-independent diffusivity) one. The azimuthal (zonal) diffusion exhibits a double-plateau structure which attains a superdiffusive regime on large scales. The transition to the Taylor regime for the radial diffusion takes place at a scale of turbulence anisotropization. The radial eddy diffusivity in both regimes as well as the transition scale are all determined by the rate of the inverse energy cascade, ε, that can be diagnosed by the PV monotonizing. Conversely, ε can be deduced from the scale of the Richardson-Taylor regime transition in the radial eddy diffusivity which, thus, provides an additional tool of diagnosing anisotropic macroturbulence with inverse energy cascade.

Irreversibility of the two-dimensional enstrophy cascade

We study the time irreversibility of the direct cascade in two-dimensional turbulence by looking at the time derivative of the square vorticity along Lagrangian trajectories, a quantity called metenstrophy. By means of extensive direct numerical simulations we measure the time irreversibility from the asymmetry of the probability density function of the metenstrophy and we find that it increases with the Reynolds number of the cascade, similarly to what is found in three-dimensional turbulence. A detailed analysis of the different contributions to the enstrophy budget reveals a remarkable difference with respect to what is observed for the energy cascade, in particular the role of the statistics of the forcing to determine the degree of irreversibility.

Janus Spectra in Two-Dimensional Flows

In large-scale atmospheric flows, soap-film flows, and other two-dimensional flows, the exponent of the turbulent energy spectra, α, may theoretically take either of two distinct values, 3 or 5/3, but measurements downstream of obstacles have invariably revealed α=3. Here we report experiments on soap-film flows where downstream of obstacles there exists a sizable interval in which α transitions from 3 to 5/3 for the streamwise fluctuations but remains equal to 3 for the transverse fluctuations, as if two mutually independent turbulent fields of disparate dynamics were concurrently active within the flow. This species of turbulent energy spectra, which we term the Janus spectra, has never been observed or predicted theoretically. Our results may open up new vistas in the study of turbulence and geophysical flows.

Multilocality and fusion rules on the generalized structure functions in two-dimensional and three-dimensional Navier-Stokes turbulence

Using the fusion-rules hypothesis for three-dimensional and two-dimensional Navier-Stokes turbulence, we generalize a previous nonperturbative locality proof to multiple applications of the nonlinear interactions operator on generalized structure functions of velocity differences. We call this generalization of nonperturbative locality to multiple applications of the nonlinear interactions operator "multilocality." The resulting cross terms pose a new challenge requiring a new argument and the introduction of a new fusion rule that takes advantage of rotational symmetry. Our main result is that the fusion-rules hypothesis implies both locality and multilocality in both the IR and UV limits for the downscale energy cascade of three-dimensional Navier-Stokes turbulence and the downscale enstrophy cascade and inverse energy cascade of two-dimensional Navier-Stokes turbulence. We stress that these claims relate to nonperturbative locality of generalized structure functions on all orders and not the term-by-term perturbative locality of diagrammatic theories or closure models that involve only two-point correlation and response functions.

Role of helicity for large- and small-scale turbulent fluctuations

The effects of the helicity on the dynamics of turbulent flows are investigated. The aim is to disentangle the role of helicity in fixing the direction, the intensity, and the fluctuations of the energy transfer across the inertial range of scales. We introduce an external parameter α that controls the mismatch between the number of positive and negative helically polarized Fourier modes. We present direct numerical simulations of Navier-Stokes equations from the fully symmetrical case, α=0, to the fully asymmetrical case, α=1, when only helical modes of one sign survive. We found a singular dependency of the direction of the energy cascade on α, measuring a positive forward flux as soon as only a few modes with different helical polarities are present. Small-scale fluctuations are also strongly sensitive to the degree of mode reduction, leading to a vanishing intermittency already for values of α∼0.1. If the analysis is restricted to sets of modes with the same helicity sign, intermittency is vanishing for the modes belonging to the minority set, and it is close to that measured on the original Navier-Stokes equations for the other set.

Disentangling the triadic interactions in Navier-Stokes equations

We study the role of helicity in the dynamics of energy transfer in a modified version of the Navier-Stokes equations with explicit breaking of the mirror symmetry. We select different set of triads participating in the dynamics on the basis of their helicity content. In particular, we remove the negative helically polarized Fourier modes at all wave numbers except for those falling on a localized shell of wave number, |k| ~ k(m). Changing k(m) to be above or below the forcing scale, k(f), we are able to assess the energy transfer of triads belonging to different interaction classes. We observe that when the negative helical modes are present only at a wave number smaller than the forced wave numbers, an inverse energy cascade develops with an accumulation of energy on a stationary helical condensate. Vice versa, when negative helical modes are present only at a wave number larger than the forced wave numbers, a transition from backward to forward energy transfer is observed in the regime when the minority modes become energetic enough.

How turbulence regulates biodiversity in systems with cyclic competition

Cyclic, nonhierarchical interactions among biological species represent a general mechanism by which ecosystems are able to maintain high levels of biodiversity. However, species coexistence is often possible only in spatially extended systems with a limited range of dispersal, whereas in well-mixed environments models for cyclic competition often lead to a loss of biodiversity. Here we consider the dispersal of biological species in a fluid environment, where mixing is achieved by a combination of advection and diffusion. In particular, we perform a detailed numerical analysis of a model composed of turbulent advection, diffusive transport, and cyclic interactions among biological species in two spatial dimensions and discuss the circumstances under which biodiversity is maintained when external environmental conditions, such as resource supply, are uniform in space. Cyclic interactions are represented by a model with three competitors, resembling the children's game of rock-paper-scissors, whereas the flow field is obtained from a direct numerical simulation of two-dimensional turbulence with hyperviscosity. It is shown that the space-averaged dynamics undergoes bifurcations as the relative strengths of advection and diffusion compared to biological interactions are varied.

Dimensional transition in rotating turbulence

In this work we investigate, by means of direct numerical hyperviscous simulations, how rotation affects the bidimensionalization of a turbulent flow. We study a thin layer of fluid, forced by a two-dimensional forcing, within the framework of the "split cascade" in which the injected energy flows both to small scales (generating the direct cascade) and to large scale (to form the inverse cascade). It is shown that rotation reinforces the inverse cascade at the expense of the direct one, thus promoting bidimensionalization of the flow. This is achieved by a suppression of the enstrophy production at large scales. Nonetheless, we find that, in the range of rotation rates investigated, increasing the vertical size of the computational domain causes a reduction of the flux of the inverse cascade. Our results suggest that, even in rotating flows, the inverse cascade may eventually disappear when the vertical scale is sufficiently large with respect to the forcing scale. We also study how the split cascade and confinement influence the breaking of symmetry induced by rotation.

Lattice Boltzmann simulation for forced two-dimensional turbulence

The direct numerical simulations of forced two-dimensional turbulent flow are presented by using the lattice Boltzmann method. The development of an energy-enstrophy double cascade is investigated in the two cases of external force of two-dimensional turbulence, Gaussian force and Kolmogorov force. It is found that the friction force is a necessary condition of the occurrence of a double cascade. The energy spectrum k(-3) in the enstrophy inertial range is in accord with the classical Kraichnan theory for both external forces. The energy spectrum of the Gaussian force case in an inverse cascade is k(-2); however, the Kolmogorov force drives the k(-5/3) energy in a backscatter cascade. The result agrees with Scott's standpoint, which describes nonrobustness of the two-dimensional turbulent inverse cascade. Also, intermittency is found for the enstrophy cascade in two cases of the external force form. Intermittency refers to the nonuniform distribution of saddle points in the two-dimensional turbulent flow.

Statistics of the inverse-cascade regime in two-dimensional magnetohydrodynamic turbulence

We present a detailed direct numerical simulation of statistically steady, homogeneous, isotropic, two-dimensional magnetohydrodynamic turbulence. Our study concentrates on the inverse cascade of the magnetic vector potential. We examine the dependence of the statistical properties of such turbulence on dissipation and friction coefficients. We extend earlier work significantly by calculating fluid and magnetic spectra, probability distribution functions (PDFs) of the velocity, magnetic, vorticity, current, stream-function, and magnetic-vector-potential fields, and their increments. We quantify the deviations of these PDFs from Gaussian ones by computing their flatnesses and hyperflatnesses. We also present PDFs of the Okubo-Weiss parameter, which distinguishes between vortical and extensional flow regions, and its magnetic analog. We show that the hyperflatnesses of PDFs of the increments of the stream function and the magnetic vector potential exhibit significant scale dependence and we examine the implication of this for the multiscaling of structure functions. We compare our results with those of earlier studies.

Elliptical tracers in two-dimensional, homogeneous, isotropic fluid turbulence: the statistics of alignment, rotation, and nematic order

We study the statistical properties of orientation and rotation dynamics of elliptical tracer particles in two-dimensional, homogeneous, and isotropic turbulence by direct numerical simulations. We consider both the cases in which the turbulent flow is generated by forcing at large and intermediate length scales. We show that the two cases are qualitatively different. For large-scale forcing, the spatial distribution of particle orientations forms large-scale structures, which are absent for intermediate-scale forcing. The alignment with the local directions of the flow is much weaker in the latter case than in the former. For intermediate-scale forcing, the statistics of rotation rates depends weakly on the Reynolds number and on the aspect ratio of particles. In contrast with what is observed in three-dimensional turbulence, in two dimensions the mean-square rotation rate increases as the aspect ratio increases.

Transition of the scaling law in inverse energy cascade range caused by a nonlocal excitation of coherent structures observed in two-dimensional turbulent fields

We numerically investigate the inverse energy cascade range of two-dimensional Navier-Stokes turbulence. Our focus is on the universality of the Kolmogorov's phenomenology. In our direct numerical simulations, two types of forcing processes, the random forcing and the deterministic forcing, are employed besides the systematically varied numerical parameters. We first calculate the two-dimensional Navier-Stokes equations and confirm that results in the quasi steady state are consistent with the classical phenomenology for both types of forcing processes. It is also found that the difference in forcing process appears after the inverse energy cascade range reaches the system size; the dipole coherent vortices emerge and grow only when the random forcing is adopted. Then we add a large-scale drag term to the Navier-Stokes equations to obtain the statistically stationary state. When the random forcing is used, the scaling exponent of the energy spectrum in the stationary state starts to differ from the predicted -5/3 in the inverse energy cascade range as the infrared Reynolds number Re(d) increases, where Re(d) is defined as k(f)/k(d) with the forcing wave number k(f) and the large-scale drag wave number k(d). That can be interpreted as a transition phenomenon in which the local maximum vorticity grows like an order parameter caused by excitation of strong coherent vortices. Strong coherent vortices emerge and grow after the quasi steady state and destroy the scaling law when Re(d) is over a critical value. These coherent vortices are not due to the finite-size effect, unlike the dipole coherent vortices. On the other hand, when the deterministic forcing is adopted, strong coherent vortices are hardly seen and the -5/3 scaling law holds independently of Re(d). We examine the cases of the combination of both types of forcing processes and find that formation of such coherent vortices is sensitive to the mechanism of the external forcing process as well as the numerical parameters. Several types of large-scale drag terms are also tested and their insignificant influence on these qualitative properties is revealed.

Inverse energy cascade in three-dimensional isotropic turbulence

We study the statistical properties of homogeneous and isotropic three-dimensional (3D) turbulent flows. By introducing a novel way to make numerical investigations of Navier-Stokes equations, we show that all 3D flows in nature possess a subset of nonlinear evolution leading to a reverse energy transfer: from small to large scales. Up to now, such an inverse cascade was only observed in flows under strong rotation and in quasi-two-dimensional geometries under strong confinement. We show here that energy flux is always reversed when mirror symmetry is broken, leading to a distribution of helicity in the system with a well-defined sign at all wave numbers. Our findings broaden the range of flows where the inverse energy cascade may be detected and rationalize the role played by helicity in the energy transfer process, showing that both 2D and 3D properties naturally coexist in all flows in nature. The unconventional numerical methodology here proposed, based on a Galerkin decimation of helical Fourier modes, paves the road for future studies on the influence of helicity on small-scale intermittency and the nature of the nonlinear interaction in magnetohydrodynamics.

Turbulence in noninteger dimensions by fractal Fourier decimation

Fractal decimation reduces the effective dimensionality D of a flow by keeping only a (randomly chosen) set of Fourier modes whose number in a ball of radius k is proportional to k(D) for large k. At the critical dimension D(c)=4/3 there is an equilibrium Gibbs state with a k(-5/3) spectrum, as in V. L'vov et al., Phys. Rev. Lett. 89, 064501 (2002). Spectral simulations of fractally decimated two-dimensional turbulence show that the inverse cascade persists below D=2 with a rapidly rising Kolmogorov constant, likely to diverge as (D-4/3)(-2/3).

Nonlinear superposition of direct and inverse cascades in two-dimensional turbulence forced at large and small scales

We inquire about the properties of 2D Navier-Stokes turbulence simultaneously forced at small and large scales. The background motivation comes by observational results on atmospheric turbulence. We show that the velocity field is amenable to the sum of two auxiliary velocity fields forced at large and small scale and exhibiting a direct enstrophy and an inverse-energy cascade, respectively. Remarkably, the two auxiliary fields reconcile universal properties of fluxes with positive statistical correlation in the inertial range.

Effects of friction on forced two-dimensional Navier-Stokes turbulence

Large-scale dissipation mechanisms have been routinely employed in numerical simulations of two-dimensional turbulence to absorb energy at large scales, presumably mimicking the quasisteady picture of Kraichnan in an unbounded fluid. Here, "side effects" of such a mechanism--mechanical friction--on the small-scale dynamics of forced two-dimensional Navier-Stokes turbulence are elaborated by both theoretical and numerical analysis. Given a positive friction coefficient α, viscous dissipation of enstrophy has been known to vanish in the inviscid limit ν→0. This effectively renders the scale-neutral friction the only mechanism responsible for enstrophy dissipation in that limit. The resulting dynamical picture is that the classical enstrophy inertial range becomes a dissipation range in which the dissipation of enstrophy by friction mainly occurs. For each α>0, there exists a critical viscosity ν(c), which depends on physical parameters, separating the regimes of predominant viscous and frictional dissipation of enstrophy. It is found that ν(c)=[η'(1/3)/(Ck(f)(2))]exp[-η'(1/3)/(Cα)], where η' is half the enstrophy injection rate, k(f) is the forcing wave number, and C is a nondimensional constant (the Kraichnan-Batchelor constant). The present results have important theoretical and practical implications. Apparently, mechanical friction is a poor choice in numerical attempts to address fundamental issues concerning the direct enstrophy transfer in two-dimensional Navier-Stokes turbulence. Furthermore, as relatively strong friction naturally occurs on the surfaces and at lateral boundaries of experimental fluids as well as at the interfaces of shallow layers in geophysical fluid models, the frictional effects discussed in this study are crucial in understanding the dynamics of these systems.

Double cascade turbulence and Richardson dispersion in a horizontal fluid flow induced by Faraday waves

We report the experimental observation of Richardson dispersion and a double cascade in a thin horizontal fluid flow induced by Faraday waves. The energy spectra and the mean spectral energy flux obtained from particle image velocimetry data suggest an inverse energy cascade with Kolmogorov type scaling E(k) ∝ k(γ), γ ≈ -5/3 and an E(k) ∝ k(γ), γ ≈ -3 enstrophy cascade. Particle transport is studied analyzing absolute and relative dispersion as well as the finite size Lyapunov exponent (FSLE) via the direct tracking of real particles and numerical advection of virtual particles. Richardson dispersion with <ΔR(2)(t)> ∝ t(3) is observed and is also reflected in the slopes of the FSLE (Λ ∝ ΔR(-2/3)) for virtual and real particles.

Vorticity statistics in the direct cascade of two-dimensional turbulence

For the direct cascade of steady two-dimensional (2D) Navier-Stokes turbulence, we derive analytically the probability of strong vorticity fluctuations. When ϖ is the vorticity coarse-grained over a scale R, the probability density function (PDF), P(ϖ), has a universal asymptotic behavior lnP~-ϖ/ϖ(rms) at ϖ≫ϖ(rms)=[Hln(L/R)](1/3), where H is the enstrophy flux and L is the pumping length. Therefore, the PDF has exponential tails and is self-similar, that is, it can be presented as a function of a single argument, ϖ/ϖ(rms), in distinction from other known direct cascades.