Wave turbulence and intermittency in directional wave fields

The evolution of surface gravity waves is driven by nonlinear interactions that trigger an energy cascade similarly to the one observed in hydrodynamic turbulence. This process, known as wave turbulence, has been found to display anomalous scaling with deviation from classical turbulent predictions due to the emergence of coherent and intermittent structures on the water surface. In the ocean, waves are spread over a wide range of directions, with a consequent attenuation of the nonlinear properties. A laboratory experiment in a large wave facility is presented to discuss the sensitivity of wave turbulence on the directional properties of model wave spectra. Results show that the occurrence of coherent and intermittent structures become less likely with the broadening of the wave directional spreading. There is no evidence, however, that intermittency completely vanishes.     

Hierarchical structures in a turbulent pipe flow

A hierarchical structure (HS) analysis (β-test and γ-test) is applied to a fully developed turbulent pipe flow. Velocity signals are measured at two cross sections in the pipe and at a series of radial locations from the pipe wall. Particular attention is paid to the variation of turbulent statistics at wall units 10<y⁺<3000. It is shown that at all locations the velocity fluctuations satisfy the She–Leveque hierarchical symmetry (Phys. Rev. Lett. 72 (1994) 336). The measured HS parameters, β and γ, are interpreted in terms of the variation of fluid structures. Intense anisotropic fluid structures generated near the wall appear to be more singular than the most intermittent structures in isotropic turbulence and appear to be more outstanding compared to the background fluctuations; this yields a more intermittent velocity signal with smaller γ and β. As turbulence migrates into the logarithmic region, small-scale motions are generated by an energy cascade and large-scale organized structures emerge which are also less singular than the most intermittent structures of isotropic turbulence. At the center, turbulence is nearly isotropic, and β and γ are close to the 1994 She–Leveque predictions. A transition is observed from the logarithmic region to the center in which γ drops and the large-scale organized structures break down. We speculate that it is due to the growing eddy viscosity effects of widely spread turbulent fluctuations in a similar way as in the breakdown of the Taylor vortices in a turbulent Couette–Taylor flow at high Reynolds numbers.     

A study of type I intermittency of a circular differential equation under a discontinuous right-hand side

In this paper we study a circular differential equation under a discontinuous periodic input, developing a quadratic differential equations system on S¹ and a linear differential equations system in the Minkowski space M³. The symmetry groups of these two systems are, respectively, PSOo(2,1) and SOo(2,1). The Poincaré circle map is constructed exactly, and a critical value αc of the parameter is identified. Depending on α of the input amplitude the equation may exhibit periodic, subharmonic or quasiperiodic motions. When α varies from α>αc to α<αc, there undergoes an inverse tangent bifurcation; consequently, the resultant Poincaré circle map offers one route to the quasiperiodicity via a type I intermittency. In the parameter range of α<αc the orbit generated by the Poincaré circle map is either m-periodic or quasiperiodic when n/m is a rational or an irrational number.     

Intermittency as a transition to turbulence in pipes: A long tradition from Reynolds to the 21st century

Intermittencies are commonly observed in fluid mechanics, and particularly, in pipe flows. Initially observed by Reynolds (1883), it took one century for reaching a rather full understanding of this phenomenon whose irregular dynamics (apparently stochastic) puzzled hydrodynamicists for decades. In this brief (non-exhaustive) review, mostly focused on the experimental characterization of this transition between laminar and turbulent regimes, we present some key contributions for evidencing the two concomittant and antagonist processes that are involved in this complex transition and were suggested by Reynolds. It is also shown that a clear explicative model was provided, based on the nonlinear dynamical systems theory, the experimental observations in fluid mechanics only providing an applied example, due to its obvious generic nature.     

Intermittency at critical transitions and aging dynamics at the onset of chaos

We recall that at both the intermittency transitions and the Feigenbaum attractor, in unimodal maps of non-linearity of order ζ > 1, the dynamics rigorously obeys the Tsallis statistics. We account for theq-indices and the generalized Lyapunov coefficients λ^q that characterize the universality classes of the pitchfork and tangent bifurcations. We identify the Mori singularities in the Lyapunov spectrum at the onset of chaos with the appearance of a special value for the entropic indexq. The physical area of the Tsallis statistics is further probed by considering the dynamics near criticality and glass formation in thermal systems. In both cases a close connection is made with states in unimodal maps with vanishing Lyapunov coefficients.     

Type-I intermittency with noise versus eyelet intermittency

In this Letter we compare the characteristics of two types of the intermittent behavior (type-I intermittency in the presence of noise and eyelet intermittency taking place in the vicinity of the chaotic phase synchronization boundary) supposed hitherto to be different phenomena. We show that these effects are the same type of dynamics observed under different conditions. The correctness of our conclusion is confirmed by the consideration of different sample systems, such as quadratic map, Van der Pol oscillator and Rössler system. Consideration of the problem concerning the upper boundary of the intermittent behavior also confirms the validity of the statement on the equivalence of type-I intermittency in the presence of noise and eyelet intermittency observed in the onset of phase synchronization.     

Laminar length and characteristic relation in Type-I intermittency

A method to investigate systems showing Type-I intermittency phenomenon is presented. This method is an extension of the procedure we have recently established to study the Type-II and Type-III intermittencies. With this approach, new accurate analytical expressions for the reinjection and the laminar phase length probability densities are obtained. The new theoretical formulas are tested by numerical computation, showing an excellent agreement between analytical models and numerical results. In addition, our method fully generalizes the well-known classical characteristic relations, in such a way that it properly characterizes those systems showing Type-I intermittency.     

Columnar vortices in isotropic turbulence

The structure of the intense vorticity regions is studied in numerically simulated homogeneous, isotropic, equilibrium turbulent flow fields at four different Reynolds numbers, in the rangeRe =35–170, and is found to be organized in coherent, cylindrical or ribbon-like, vortices (worms). At the Reynolds numbers studied, they are responsible for much of the extreme intermittent tails observed in the statistics of the velocity gradients, but their importance seems to decrease at higherRe . Their radii scale with the Kolmogorov microscale and their lengths with the integral scale of the flow, while their circulation increases monotonically withRe . An explanation is offered for this latter scaling, based in the assumed presence of axial inertial waves along their cores, excited by a random background strain of the order of the root mean square vorticity. This explanation is consistent with the presence of comparable amounts of stretching and compression along the vortex cores.

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Sommario La struttura di regioni ad intensa vorticità in campi di flusso turbolento omogenei, isotropi ed in equilibrio, simulati numericamente, viene studiata per quattro differenti numeri di Reynolds nell'intervalloRe =35÷170, e si trova che tali regioni si organizzano in vortici coerenti, cilindrici o a forma di nastro (vermi). Con rifermento ai numeri di Reynolds studiati, si vede che tali vortici sono responsabili per gran parte delle code estreme ed intermittenti, osservate nelle statistiche dei gradienti di velocità, ma la loro importanza sembra decrescere a più altiRe . I loro raggi scalano con la microscala di Kolmogorov e le loro lunghezze con la scala integrale del flusso, mentre la loro circolazione cresce monotonicamente conRe . Per quest'ultimo riscalamento viene offerta una spiegazione basata sull'assunzione della presenza di onde inerziali assiali lungo i loro nuclei, eccitate da una deformazione di fondo casuale dell'ordine della radice quadrata della velocità media. Questa spiegazione è consistente con la presenza di incrementi paragonabili di allungamenti e compressioni lungo i nuclei dei vortici.

     

Coherent noise, scale invariance and intermittency in large systems

We introduce a new class of models in which a large number of “agents” organize under the influence of an externally imposed coherent noise. The model shows reorganization events whose size distribution closely follows a power law over many decades, even in the case where the agents do not interact with each other. In addition, the system displays “aftershock” events in which large disturbances are followed by a string of others at times which are distributed according to a t⁻¹ law. We also find that the lifetimes of the agents in the system possess a power-law distribution. We explain all these results using an approximate analytic treatment of the dynamics and discuss a number of variations on the basic model relevant to the study of particular physical systems.