According to statistical turbulence theory, the ensemble averaged squared vorticity ρE is expected to grow not faster than dρE/dt ~ ρE 3/2. Solving a variational problem for maximal bulk enstrophy (E) growth, velocity fields were found for which the growth rate is as large as dE/dt ~ E3. Using numerical simulations with well resolved small scales and a quasi- Lagrangian advection to track fluid sub-volumes with rapidly growing vorticity, we study spatially resolved statistics of vorticity growth. We find that the volume ensemble averaged growth bound is satisfied locally to a remarkable degree of accuracy. Elements with dE/dt ~ E3 can also be identified but their growth tends to be replaced by the ensembleaveraged law when the intensities become too large. This joint work with Jörg Schumacher and Bruno Eckhardt was published in Physics Letters A 374, 861 (2010).

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

In the presentation we will discuss our research program concerning the study of extreme vortex events in viscous incompressible flows. These vortex states arise as the flows saturating certain fundamental mathematical estimates, such as the bounds on the maximum enstrophy growth in 3D [1]. They are therefore intimately related to the question of singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. Similar questions are in fact also relevant in the context of the 1D Burgers and 2D Navier-Stokes systems. While these systems are known not to lead to singularity formation in finite time, the question of the sharpness of their worst-case estimates is still important, as these estimates are obtained using analogous methods as in the 3D case. We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it allows one to determine whether or not certain mathematical estimates are “sharp”, in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. In the presentation we will review a number of results concerning the maximum possible growth of enstrophy or palinstrophy in the 1D Burgers problem [2], and the 2D and 3D Navier-Stokes problems [3, 4, 5]. In particular, we will show that the finite-time growth of palinstrophy in 2D corresponding to the worst-case initial data found through the solution of a variational problem (Figure 1) saturates the mathematical estimates, thus demonstrating their sharpness. In the 3D case, while the time evolution corresponding to the extreme vortex states leads to a larger growth of enstrophy than when other types of the initial data are used, it reveals no indication of singularity formation in finite time.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

We study basic problems of the Navier-Stokes equations [1] and review some blowup criteria for their solutions, stressing the scale-invariant properties. After recalling Leray's classic bounds on the enstrophy and the velocity [2], we consider the criterion with the L3-norm [3] and contrast it with the Beale-Kato- Majda criterion [4] for the 3D Euler equations. As an application, we show that a possible asymptotic behavior of the L3-norm should be a single-logarithmic function of time, excluding weaker iterated logarithms on the basis of the absence of self-similar [5, 6] and asymptotically self-similar [7, 8] blowup. We then turn our attention to the critical criteria using L-norms (e.g. vector potential for the velocity). By writing down dynamical equations for the vector potential as a non-local version of the Hamilton-Jacobi equations, we discuss possible blowup conditions with the L3-norm of the vector potential. The cases of hypo-dissipativity e.g. (-)1/2 or a linear damping (soluble) will be also addressed similarly.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

This talk is devoted to a deterministic approach, it does not entirely fit in the statistical theory of turbulence. However, the following remarks makes it closely related to this theory. First, with the only available uniform estimate (the energy balance), it uses the notion of weak convergence. Weak convergence is based on some type of average as such it shares some similarity with the statistical theory. Second, my talk is based on a theorem of Kato (in the spirit of classical functional analysis). To the best of my knowledge this is the only case where a clear cut link between anomalous energy dissipation and turbulence can be made. Third, it concerns the interaction of an obstacle (for instant the wing of an air plane) with a uid ow and one should observe that in almost all cases turbulence is generated by boundary effects. Even experiments on homogenous isotropic turbulence are made with grid effect. Of course observation is done in the wake, far away from the grid. But the grid has been essential for the generation of turbulence. And in this spirit wall law for turbulence (like the Prandlt-Von Karman wall law) involves a reference velocity u_ which appears also in a very similar way in an updated formulation of the Kato theorem.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

Extreme events can occur in a variety of fluid flows. I am particularly interested in extreme events that occur in the context of a near singularity, such as in free surface flows, quantum fluid reconnection, and possible Euler singularities. In many cases, these near singularities are associated with changed in topology (e.g. droplet pinch-off). Indeed, it seems to be the rule that changes in topology in physical systems are mediated by singularities of various sorts. I will focus on examples from capillary waves, gravity waves, reconnection of vortices in superfluid helium, plasma reconnection, and remarks on Euler flows.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

A robust energy transfer mechanism is found in nonlinear wave systems, which favours transfers towards modes interacting via triads with nonzero frequency mismatch, applicable in meteorology, nonlinear optics and plasma wave turbulence. We emphasise the concepts of truly dynamical degrees of freedom and triad precession. Transfer efficiency is maximal when the triads' precession frequencies resonate with the system's nonlinear frequencies, leading to a collective state of synchronised triads with strong turbulent cascades at intermediate nonlinearity. Numerical simulations conrm analytical predictions.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

The velocity gradient tensor characterizes the small scales of fully developed turbulence comprehensively. Its evolution equation features, besides advection with the velocity field, a local self-amplification term as well as a nonlocal pressure and viscous diffusion terms. Neglecting the pressure and viscous terms constitutes the so-called Restricted Euler model [1]. From the study of this model it is known that the local self-amplification term, considered on its own, leads to a blow-up of the dynamics infinite time [2]. This also points out its importance for the occurrence of extreme events in the velocity gradient tensor field. The nonlocal pressure and viscous terms are generally thought to mitigate the self-amplification and therefore potentially reduce extreme events in the ow, both in number as well as in amplitude. The challenge in understanding the statistical properties of the velocity gradient tensor field in terms of exact statistical evolution equations lies in specifying the nonlocal pressure and viscous effects (see [3] for a recent review of models), which represent statistically unclosed terms. In this work, we evaluate these terms under the (over-simplifying) assumption of incompressible Gaussian velocity fields [4]. While this is known to be inaccurate for turbulent flows, it allows for an exact analytical treatment of the problem and yields qualitative insights into the statistical action of pressure and viscous diffusion. The dynamics of the resulting Gaussian closure and generalizations thereof are discussed and compared to data from direct numerical simulations. The results help to explain how nonlocal pressure Hessian contributions prevent the restricted Euler singularity, and yield insights into the origin of the velocity gradient skewness related to a breaking of the time-reversal symmetry. Support from a DFG postdoctoral fellowship (WI 3544/2-1 and WI 3544/3- 1) and the US National Science Foundation (CBET 1033942) is gratefully acknowledged.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

A transport process, at large scale and long time, is typically ruled by the Fick equation, and we have the so called standard diffusion, i.e. a Gaussian probability distribution and < x2(t) >∼ t. On the other hand many situations show an anomalous behavion, i.e. ∼ t2_, with _ 6= 1/2. At variance with the most common scenario, the statistical features of anomalous diffusion, in general, are not completely characterized by a single exponent. For instance one can have the so-called “strong” anomalous diffusion, i.e. < |x(t)|q >∼ tq_(q) where _ 6= 1/2 and q_(q) is not a linear function of q. Such a feature is rather different from the “weak” superdiffusion regime, i.e. _(q) = const. > 1/2 as in some random shear flows. The strong anomalous diffusion has been observed in nontrivial chaotic dynamics, e.g. Lagrangian motion in 2d time-dependent incompressible velocity fields, 2d symplectic maps and 1d- intermittent maps. Typically the function q_(q) is piecewise linear. This corresponds to two mechanisms: a weak anomalous diffusion for the typical events and a ballistic transport for the rare excursions. In order to have strong anomalous diffusion one needs a violation of the hypothesis of the central limit theorem, this happens only in a very narrow region of the control parameters space. In the case of “strong” anomalous diffusion the probability distribution P(x, t) is not Gaussian, and cannot be rescaled in terms of a single exponent , i.e.it is not possible to find a function f( ) such that P(x, t) = t−_f(x/t_). A similar scenario holds for relative diffusion in fully developed turbulence. Although the Rinchardson law < R2(t) >∼ t3 seems to hold even in presence of intermittency, the dimensional scaling for higher moments fails, i.e. < Rq(t) >∼ t_(q) where _(q) 6= 3q/2.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

It is evident that coherent nearly singular structures play a dominant role in understanding the anomalous scaling behavior in turbulent systems. We ask the question, which role these singular structures play in turbulence statistics. More than 15 years ago, for certain turbulent systems the door for attacking this issue was opened by getting access to the probability density function to rare and strong fluctuations by the instanton approach. We address the question whether one can identify instantons in direct numerical simulations of the stochastically driven Burgers equation. For this purpose, we first solve the instanton equations using the Chernykh‐Stepanov method [2001]. These results are then compared to direct numerical simulations by introducing a filtering technique to extract prescribed rare events from massive data sets of realizations. In addition, we solve the issue why earlier simulations by Gotoh [1999] were in disagreement with the asymptotic prediction of the instanton method and demonstrate that this approach is capable to describe the probability distribution of velocity differences for various Reynolds numbers. Finally, we will present and discuss first results on the instanton solution for vorticity in 3D Navier‐Stokes turbulence.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

Examples of extreme events will be presented. In particular we discuss the impact of wind gusts an wind energy, the appearance of rough waves and extreme stock market uctuations. The aim of this presentation is to show evidence that there is a class of systems characterized by building up extreme events which are due to hierarchical cascade processes. For the case of turbulence we show that the turbulent cascade process satisfy a generalized 2nd law of thermodynamics for non-equilibrium conditions, namely the integral uctuation theorem. The nding of Markow properties of velocity increments statistics conditioned on dierent scales opened up the possibility to describe the cascade process by stochastic equations, like Fokker-Planck or the Kolmogorov equations. In this framework it is even possible to get access to the general n-point statistics of [1]. The stochastic cascade process is evolving in an instationary way with the scale. Thus the statistics, expressed by probability density functions of velocity increments, are changing with the scale too, which is the central feature of intermittency and producing extreme events. The common multifractal cascade models for turbulence will be expressed in terms of such instationary cascade processes. Using concepts of non-equilibrium thermodynamics an integral uctuation theorem for the entropy production associated with the stochastic evolution of velocity increments along the cascade has been proposed [4], which demonstrates that the instationarity of the process appears to be crucial for the correct modeling of the intermittency found in turbulent ows. The integral uctuation theorem allows to rule out which cascades. Here we show how this concept of the integral uctuation theorem can be used as a test of the validity of multifractal models for turbulence and to validate dierent features of the cascade, like for example scaling behavior, or log normal statistics. Finally we show how based on the stochastic description of the cascade a model for synthetic data can be set up. We show that the extreme events can be modeled correctly, thus give evidence that for these systems extreme events are multi-point quantities, which is equivalent to the saying the the extreme events are caused by cascade process.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

In three-dimensional turbulent flows energy is supplied at large scales and cascades down to the smallest scales where viscosity dominates. The generation of small scales from larger ones results in a flux of energy through scales and implies the irreversibility fundamental to the dynamics of turbulent flows. As we have shown recently, this irreversibility manifests itself by an asymmetry of the probability distribution of the instantaneous power p ≡ u·a of the forces acting on fluid elements, where u and a are the fluid velocity and acceleration, respectively. In particular, the third moment of p was found to be negative. Establishing a physical connection between the negative third moment of p and the energy flux or small-scale generation is the main result of this work. With analytical calculations and support from numerical simulation of fully developed turbulence we connect the asymmetry in the power distribution, i.e., the negativity of ⟨p3⟩, directly to the generation of small scales, or more precisely, to the amplification (stretching) of vorticity in turbulent flows. This work is joined with: Alain Pumir (Ecole Normale Superieure de Lyon), Haitao Xu (Max Planck Institute for Dynamics and Self-Organization), Rainer Grauer (Ruhr University Bochum)

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

The dynamical effects of mode reduction in Fourier space for three dimensional turbulent flows is studied. We focus on fully resolved numerical simulations of Navier-Stokes equations with Fourier modes constrained to live on a fractal set of dimension D [1]. The robustness of the forward energy cascade and vortex stretching mechanisms is tested at changing D, from the standard, fully resolved field, corresponding to fractal dimension D=3, to a strongly decimated field where only up to a 3% of the Fourier modes interact, at D=2.5. The direct energy cascade persist, but deviations from the Kolmogorov scaling are observed in the kinetic energy spectra. A model in terms of a correction with a linear dependency on the codimension of the fractal set explains the results. At small scales, the intermittent behavior due to the vorticity production is strongly modified by the fractal decimation, leading to an almost Gaussian statistics already at D=2.98. These effects are connected to a genuine modification in the triad-to-triad nonlinear energy transfer mechanism as proven by the fact that when the fractal mode-reduction is applied a posteriori to configurations obtained from fully resolved Navier-Stokes equations the reduction in the fluctuations is much smaller.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

The flux of any conserved quantity with a nontrivial spectrum can be considered as representing a turbulent cascade, since it has to ‘traverse’ the different scales before the quantity is dissipated. Note that such fluxes are the natural objects in which to study the transfer, since, for example, production or dissipation tend to represent only one of the end-points. Well-known examples are the energy cascade in two- or three-dimensional turbulence, whose flux is  = -ijSij, or the enstrophy cascade in two-dimensional flows. Less attention has been paid in the cascade literature to the transfer of momentum in shear flows, although the corresponding flux, the tangential Reynolds stress  = - , is an object of intense interest in engineering. In a wall bounded turbulent flow, streamwise momentum is typically injected across the flow section by the mean pressure gradient, and dissipated at the wall by viscosity. Away from the immediate near-wall viscous region, most of the tangential stress resides in a hierarchy of intense structures often known as ‘sweeps’ and ‘ejections’. They are roughly stratified by their distance to the wall, and are the means by which momentum is transferred towards the wall by a multiscale process involving eddies of different sizes. The presence of the wall organizes the transfer but does not appear to be necessary. An eddy hierarchy with very similar properties is found in homogeneous shear turbulence, where streamwise momentum is transferred but there is no wall. There is, of course, no spatial stratification in this case. Our group has been studying the intense uv-carrying structures for some time. They are defined as connected regions in which (-uv) is larger than a threshold that depends weakly on the distance from the wall [3]. Geometrically, they are fractal objects (D ≈ 2.2) with self-similar aspect ratios that span the whole range of sizes from the Kolmogorov to the integral scales. Between those limits there is a range of sizes that can be characterized as ‘inertial’ in the sense that neither the channel width nor the viscosity are important. Note that this characterization, in which geometry is studied as a function of intensity, is dual to the more usual one in which intensity is studied as a function of geometry (size). Both form a pair by which the multiscale momentum transfer can be studied in more detail than by any one of them alone. In particular, once eddies are characterized as individual objects their evolution can be tracked in time. When this is done, the result is a complex connectivity graph of periods of smooth growth, mergers and splits [4]. The latter increase or decrease discontinuously the eddy size, and can be interpreted as inverse and direct cascade steps. Both are required in a statistically steady system, but a surprising result is their approximate symmetry. There is very little difference between both processes, and they proceed simultaneously. There is not a period of inverse cascade in which eddies grow, followed by a direct one in which they decay. Mergers and splits alternate with very little mutual memory, and the evolution of an eddy can be described as a random walk in scale space with small corrections that are more procedural than physical. The final dissipation of the eddy is determined by the ruin of the walk. It is unknown whether such a description can be extended to the transfer of kinetic energy. The dynamics of both cascades is probably different, because  is known to be intermittent [2] while  is known not to be [1]. The eddies of  are under current active investigation.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)

Tropical cyclones in the Earth’s atmosphere are amongst the most dangerous and mighty weather events. Despite considerable efforts of modern science their genesis remains one the most intricate enigmas of meteorology as well as no a clear consensus of opinion has yet emerged concerning physical mechanisms contributing to it. In this contribution, a role of helical turbulence in the genesis and further evolution of tropical cyclones (TCs) is discussed. Our first finding of non-zero helicity in a real natural system [1], namely, the tropical atmosphere of the Earth during TC formation gave us an impetus to try and further characterize the large-scale vortex instability. In works [2-6], we proposed a helical scenario of TC formation based on the fundamental ideas on self-organization in turbulence. Building on the known cases of large-scale alpha-like instabilities – the alpha-effect in magnetohydrodynamics (Steenbeck et al., 1966), hydrodynamic alpha-effect (Moiseev et al., 1983), and anisotropic kinetic alpha (AKA)-effect (Frisch et al., 1987) – we are developing an interpretation for TC formation as a threshold extreme event in the helical atmospheric turbulence of a vorticity-rich environment of a pre-depression cyclonic recirculation zone in the tropical atmosphere. To trace and analyze processes of self-organization in the tropical atmosphere, spanning convective clouds with horizontal dimensions of 1-5 km to mesoscale vortices of hundreds of kilometers, we use data of near cloud-resolving numerical simulations [7]. Helicity is the scalar product of velocity and vorticity vectors. It characterizes the degree of linkage of vortex lines and is also a measure of departure from the mirror symmetry of turbulence [8]. In a case of TC formation, we define the helicity of the associated flow by the corresponding integral [8] being over all space of mesoscale vortex core, approximately 300x300x20 km in horizontal and vertical directions, respectively. Our research approach developed and applied in [2-6] allows diagnosis of WHEN a nascent large vortex becomes energy self-sustaining. For purposes of quantitative diagnosis of TC genesis we analyze the evolution of structure and energetics of the forming vortex. It has been found that the onset of large-scale vortex instability requires a special topology of the vortex velocity field – the newly forming mesoscale vortex becomes energy-self-sustaining when a helical structure of the system-scale circulation organizes and interacts with the moisture rich boundary layer. Such helical mesoscale organization is only possible due to the linkage of TC-scale tangential and transverse circulation which is realized through rotating convective structures of cloud scales, which were first found in [9] and dubbed ‘vortical hot towers’ (VHTs). Thus, we use a pseudoscalar – helicity of the velocity field (helicity density, integral helicity as well as its horizontal and vertical contributions) to quantitatively analyze the topology, and the integral kinetic energy of tangential and transverse circulation to diagnose the onset of large-scale vortex instability. The moment in time when the mutual intensification of both circulations starts can be considered as a beginning of tropical cyclogenesis. The chosen quantitative criterion has a clear physical motivation. This criterion contributes to the development of a universally accepted definition of tropical cyclogenesis, which does not currently exist. The presented contribution suggests the usefulness of combining the fundamental ideas on self-organization in turbulence and the most advanced modern tools of numerical analysis of atmospheric processes. As a practical perspective, we consider applications of our approach to analyze data of real observations and field experiments in order to implement the diagnosis and forecasting of TC genesis by means of operational weather models.

• Thematic program: Turbulence and Fusion (2014)
• Event: Workshop on "Basic issues of extreme events in turbulence" (2015)