ﬁeld and the nonuniqueness of the solution exists only up to relabeling. In the viscous case, solutions of equations describing a VSF, if they exist, are showed to be nonunique.
From our quantitative estimates, the relative error of **vortex** surfaces as **Lagrangian** surfaces is typically less than 5% in high-Reynolds-number TG and KP ﬂows prior to times typical of a potential singularity time t ∗ . The error appears to be mainly associated with the topological constraint that prevents reconnection of **Lagrangian** structures, except possibly at singularities in a strictly Euler ﬂow. This has the consequence that, for slightly viscous ﬂows, **vortex** surfaces can be well approximated as **Lagrangian** structures prior to **vortex** reconnection. In temporal evolution, initial blob-like **vortex** surfaces are progressively stretched to sheet-like shapes so that neighboring **vortex** surfaces approach each other. This ﬂattening process of **vortex** surfaces is quantiﬁed by the growth rate of the **surface** area and the averaged curvedness ⟨ ˆ C ⟩. The local topology of **vortex** surfaces is well represented by the averaged absolute value of the shape index ⟨ ˆ S ⟩, which indicates that more local hyperbolic shapes on the surfaces appear near a potential t ∗ than at the early stages of the evolution. We observed that local rolled-up shapes are predominantly located in the vicinity of high vorticity regions, which might be related to the subsequent transition to **turbulence**. Furthermore, visualization of the evolution of **vortex** surfaces shows a continuous dynamic process of **vortex** roll- up with certain symmetries, which was not observed in visualizations of iso-surfaces of vorticity magnitude.

Show more
154 Read more

(Dated: March 1, 2008)
We examine the probability distribution function (pdf) of energy injection rate (power) in nu- merical simulations of stationary two–dimensional (2D) **turbulence** in the **Lagrangian** frame. The simulation is designed to mimic an electromagnetically driven fluid layer, a well-documented system for generating two–dimensional **turbulence** in the laboratory. In our simulations, the forcing and velocity **fields** are close to Gaussian. On the other hand, the measured PDF of injected power is very sharply peaked at zero, suggestive of a singularity there, with tails which are exponential but asymmetric. Large positive fluctuations are more probable than large negative fluctuations. It is this asymmetry of the tails, which leads to a net positive mean value for the energy input despite the most probable value being zero. The main features of the power distribution are well described by Craig’s XY distribution for the PDF of the product of two correlated normal variables. We show that the power distribution should exhibit a logarithmic singularity at zero and decay exponentially for large absolute values of the power. We calculate the asymptotic behavior and express the asym- metry of the tails in terms of the correlation coefficient of the force and velocity. We compare the measured pdfs with the theoretical calculations and briefly discuss how the power pdf might change with other forcing mechanisms.

Show more
11 Read more

When applied as inlet conditions, all the methods mentioned still require, at different degrees, a zone of redeveloping of the **turbulence** inside the computational domain, and there is a strong need for improvements. A novel approach has been proposed in two recent studies, 25,26 followed on from related works. 27,28 The authors show that realistic synthetic isotropic turbulent **fields** can be generated by the so-called Multi-scale Minimal **Lagrangian** Map (MMLM), and the Multi-scale Turnover **Lagrangian** Map (MTLM). Starting from a random field, the mappings allow the fluid particles to advect freely over short time scales while maintaining incompressibility and the energy spectrum. When the advections are applied over a set of nested grids with increasing resolution, it is shown that the synthetic **fields** not only reproduce accurately the multi-scaling properties of small scale **turbulence**, but also many properties related to small-scale geometrical structures as well as the pressure field. Using the synthetic **fields** as initial conditions for simulations, more realistic time evolution can be obtained for time evolving problems, and initial transient period can be significantly shortened for stationary problems. 25 It has been generalized to the synthesis of scalar **fields**. 29 The pressure field associated with the velocity field is also further investigated. 30

Show more
24 Read more

of the governing Navier-Stokes (NS) equations is desirable. In this paper, we intend to provide a partial yet unified explanation for a number of observations via a simple dynamical model. To provide the background for the model, we note that it is closely related to recent research on the so-called restricted Euler approximation and several models for the small-scale dynamics of **turbulence**. In the restricted Euler (RE) approximation, the equation for the velocity gradient is truncated, and only the nonlinear term and the isotropic part of the pressure Hessian are retained [26, 27]. The velocity gradient predicted from the RE ap- proximation develops a finite time singularity. However the tensorial structure of the gradient reproduces a number of important features observed in turbu- lence, such as the preferential alignment between the vorticity vector and the intermediate eigen-direction of the strain rate tensor [26, 27, 28, 29]. Thus, the RE approximation has been used as a base model to understand the small-scale **turbulence**. A number of models for the pressure Hessian have been proposed to regularize the approximation. A useful idea is to follow the **Lagrangian** evo- lution of material elements, which has been pursued in [30, 31, 32, 33] (see also [34] for a recent model). The ideas are adopted to study the evolution of veloicty increments in [35]. A simple dynamical model for the velocity increments is de- rived by following the **Lagrangian** evolution of a linear element [35]. The model is generalized to **turbulence** in two and four spatial dimensions, and to include the increments of passive scalars in [36]. These models reproduce quite a few important observations regarding the non-Gaussian statistics of the increments, thus have helped clarify the origins of the observations from a dynamical point of view. It is also predicted that the increments of a passive scalar [36] are more intermittent in four spatial dimensions (compared with three spatial di- mensions). In this paper, we applied the ideas to study the evolution of the non-Gaussian statistics of velocity increments in rotating **turbulence**. In order to incorporate the Coriolis force, a local coordinate system attached to an evolv- ing material line is introduced. We show that, with the help of the coordinate system, a system of equations for the velocity increments over a fixed distant on the material line can be derived. The analysis of a restricted-Euler-type approx- imation of the system shows that several features of rotating **turbulence** can be reproduced, which thus provides explanations to some of the observations from a dynamical perspective.

Show more
25 Read more

theories in the frozen field (Fig. 14) is due to the fact that the statistics of the frozen field u(x) remain Gaussian. The effect of Gaussianity on the short-time decay of [r]

20 Read more

(iii) Based on **Lagrangian** statistics taken along trajectories of fluid particles, we found that the extreme events that generate steep scalar cliffs coincide with strong straining motions. Because the jumps of the scalar across these cliffs are of the order of the integral length scale times the mean scalar gradient, we argue that the large- scale straining motions have an unstable node/saddle/saddle topology of the velocity field. This is confirmed by an analysis based on the invariants of the velocity gradient tensor. In such a velocity field the separation between fluid particles reduces very rapidly from being proportional to the integral length scale to the order of the Batchelor length scale after which particles again separate. This confirms the physical validity of the **Lagrangian** stochastic two-particle models where mixing occurs only when particle pairs meet due to strain.

Show more
34 Read more

Abstract. Numerous systems of conservation laws are discretized on **Lagrangian** meshes where cells nodes move with matter. For complex applications, cells shape or aspect ratio often do not insure sufficient accuracy to provide an acceptable numerical solution and use of ALE technics is necessary. Here we are interested with conduction phenomena depending on velocity derivatives coming from the resolution of gas dynamics equations. For that, we propose the study of a mock of second order turbulent mixing model combining an elliptical part and an hyperbolic kernel. The hyperbolic part is approximated by finite-volume centered scheme completed by a remapping step see [7]. A major part of this paper is the discretization of the anisotropic parabolic equation on polygonal distorted mesh. It is based on the scheme described in [9] ensuring the positivity of the numerical solution. We propose an alternative based on the partitioning of polygons in triangles. We show some preliminary results on a weak coupling of hydrodynamics and parabolic equation whose tensor diffusion coefficient depends on Reynolds stresses.

Show more
20 Read more

All applications discussed in the present study focus on the three-dimensional motion of an isolated **vortex** ring in the limit of high Reynolds number. Selection of this physical setting is motivated by the fact that the **vortex** ring is unstable to azimuthal perturbations 19] whose rapid growth leads to the formation of a complex \turbulent" vortical structure (see e.g. Fig. 6.56 in 15]). Thus, the setting proves ideal for our present purpose, since one of the primary objectives of **Lagrangian** LES is to be able to model the formation of complex vortical structures and to capture their large-scale features while using very coarse computational grids. Unfortunately, one of the disadvantages of the setting is that smallscale **turbulence** and the behavior of Helmholtz stresses are hard to characterize. This is the case because (1) the isotropy and/or homogeneity assumptions which form the basis of most analytical or semi-empirical predictions do not apply, and (2) experimental measurements of the 3D vector elds are scarce and interpretation of the data is dicult, in part due to the extreme sensitivity of the ow to perturbations and initial conditions. Briey, we have sacriced detailed quantitative analysis of the performance of subgrid- scale **turbulence** models in order to examine the behavior of the scheme in a complex, challenging environment.

Show more
16 Read more

3.4.1 Flow Visualization Study
Due to the potential for errors in the experimental FTLE calculations, which used a sin- gle plane of two-component velocity data, a three-dimensional flow visualization study of the near-wake was conducted for the 𝑅𝑒 = 9, 000 case. This study sought to determine if the von Kármán vortices were nearly perpendicular to the PIV plane. The **surface** of the cylinder was painted with highly concentrated fluorescein dye, and then immersed into still water. A continuous wave laser operating at 473 nm, which was near the peak excitation wavelength of fluorescein, was used to illuminate the fluorescein as it was en- trained into the von Kármán vortices after the flow was introduced. A 12 megapixel color camera was used to capture images. The high freestream velocity and three-dimensional **turbulence** resulted in the majority of the dye being removed from the cylinder **surface** soon after the transient effects due to starting the water tunnel died out, making it dif- ficult to capture high contrast images of the wake structure. The 𝑅𝑒 = 19, 000 case was not included in this study, as the further increase in Reynolds number would degrade the clarity of images produced by this technique.

Show more
128 Read more

A mixed **Lagrangian**-Eulerian description of the evolution of the discretized vorticity eld is used for the numerical prediction of incompressible, uniformly sheared turbulent ows. In this new application a combination of several adjacent mixing layers simulates the initial condition used to generate the uniformly sheared ow as done in some of the experimental methods in wind tunnels. The two dimensional **vortex** dynamics method which has been extensively used for mixing layer analysis is adopted due to the similarities in the structure and the method of generating mixing layers and uniformly sheared ows. The governing equations are the vorticity and the continuity equations. The vorticity eld is represented by a set of **vortex** elements which moves by the induced velocity eld. To treat the large number of **vortex** elements at a reasonable computer cost, the velocity eld is calculated by the **vortex**-in-cell (VIC) method, in which 48000 vortices move through a xed Eulerian mesh system with 160 16 grid points.

Show more
14 Read more

Included in this online material is a simple routine for estimating parameters of time series as a Mat´ ern and/or complex-OU process, which can be adjusted accordingly for other stochastic processes.
4.1 Numerical Data
We start by investigating the performance of our proposed inference method on output from an idealised numerical model of ocean circulation. Synthetic **surface** drifter trajectories were generated using a wind- forced S-coordinate Primitive Equation Model (SPEM 5.2), with settings very similar to those of Danioux et al. (2008). The spatially homogenous **surface** wind forcing was based on an observed wind time series that has previously been used as the forcing field in the numerical study of Klein et al. (2004). The simulation is run without wind until a near steady-state in the background **turbulence** is achieved, after which the wind forcing is turned on. At this point the model is seeded with 200 near-**surface** particles, spread uniformly in the active region of the domain. The trajectories are then subsampled at two hour resolution resulting in 200 trajectories each with length 1,200 time points. A more detailed description of the numerical model can be found within the online code.

Show more
21 Read more

Received: 22 February 2019 – Discussion started: 1 March 2019
Revised: 13 June 2019 – Accepted: 27 June 2019 – Published: 17 July 2019
Abstract. Empirical flow field data evaluation in a well- studied ocean region along the US west coast revealed a surprisingly strong relationship between the **surface** integrals of kinetic energy and enstrophy (squared vorticity). This re- lationship defines a single isolated Gaussian super-**vortex**, whose fitted size parameter is related to the mean eddy size, and the square of the fitted height parameter is proportional to the sum of the square of all individual eddy amplitudes obtained by standard **vortex** census. This finding allows very effective coarse-grained eddy statistics with minimal com- putational efforts. As an illustrative example, the westward drift velocity of eddies is determined from a simple cross- correlation analysis of kinetic energy integrals.

Show more
and different nonlinearities such as those of the FCSH and the singly resonant OPO, thus demonstrating the universality of the phenomenon.
In conclusion, we demonstrate a mechanism for pro- ducing RWs in the transverse area of externally driven nonlinear optical devices via **vortex** **turbulence**. Given the universality of our model, this mechanism should be observable in a large variety of systems. Models of lasers with injected signal, where the invariance of the Adler limit cycle is well known [11, 24], can be easily extended to semiconductor media [25] and to class B lasers, thus including the largest majority of solid state lasers. Out- side optics, **vortex**-mediated **turbulence** without driving has been observed in nematic liquid crystals [27], chemi- cal reactions [28] and fluid dynamics [29]. In the unlocked regime of these systems with driving, **vortex** **turbulence** can excite RWs and lead to the formation of highly inho- mogeneous **fields** with non-Gaussian statistics.

Show more
3 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91103, USA Received: 1 July 2002 – Revised: 17 May 2003 – Accepted: 20 May 2003
Abstract. A model of **vortex** with embedded discontinuities in plasma flow is developed in the framework of ideal MHD in a low β plasma. **Vortex** structures are considered as a re- sult of 2-D evolution of nonlinear shear Alfv´en waves in the heliosphere. Physical properties of the solutions and vector **fields** are analyzed and the observational aspects of the model are discussed. The ratio of normal components to the discon- tinuity B r /V r can be close to − 2. The alignment between velocity and magnetic field vectors takes place. Spacecraft crossing such vortices will typically observe a pair of dis- continuities, but with dissimilar properties. Occurrence rate for different discontinuity types is estimated and agrees with observations in high-speed solar wind stream. Discontinu- ity crossing provides a backward rotation of magnetic field vector and can be observed as part of a backward arc. The Ulysses magnetometer data obtained in the fast solar wind are compared with the results of theoretical modelling.

Show more
(d) the same with τ = 10 days.
field of a purely advected quantity. The right-hand column shows the M function at the **surface** z = 31.3 km obtained from particle trajectories in full 3-D calculations. Similarly to the second column, the displayed information is **Lagrangian**, but here we obtain more fundamental information in this re- gard. This figure provides feedback for characterizing the time evolution of any purely advected scalar field, while the previous one displays just the realization of one particular initial datum. More specifically, the third column highlights the position and evolution of two hyperbolic points in the outer part of the **vortex**, as well as the **vortex** itself. As dis- cussed by García-Garrido et al. (2017), hyperbolic points are responsible for filamentation processes. Whether or not these filaments are eventually observed depends on the distribution of the scalar field. For instance, if the scalar field is com- pletely uniform in the whole domain, then its time evolution will show nothing about the features highlighted by M. How the features of the M field are visible in a scalar field depends on how the initial distribution of the advected field is with re- spect to the features of M. Figure 5 illustrates these facts in

Show more
14 Read more

75 When the porosity decreases such that it enters the range of 0.67 ≤ φ ≤ 0.8, flow is
entrained by the **vortex** interaction in a single direction that causes a shift in the stagnation point.
Close to the transition point to deviatory flow (φ ~ 0.80), two peaks in the pressure distribution on the wake incident **surface** of the solid obstacle are observed. Each peak is associated with stagnation caused by the impingement of a partially dissipated **vortex**. With regard to the mean flow, one peak clearly dominates the other resulting in a single stagnation point at the location of the stronger peak. In the transition region, characteristics from both regimes are observed with the deviatory flow being localized in the wake alone while the remainder of the flow remains unaffected. Given that the case with a porosity of 0.8 is close to transition, the behaviour is easily upset by small disturbances such as the numerical error associated with the discretized periodic boundary conditions. Far from the influence of the boundaries, the flow pattern is identical throughout the domain.

Show more
118 Read more

A MATLAB code is written to solve the equations of motion of the inertial and gas particles and which immediately calculates the corresponding FTLE **fields** at that time step.
The code is tested for a simple case and is used to analyse the **vortex** flow. For low Stokes numbers the accumulation is very slow, but when this is increased the accumulation becomes much quicker. Above a critical value of the Stokes number, where the inertial particles will not accumulate anymore and all will move to the wall. There is no attractor or repeller found which should lead to the accumulation of the particles in the FTLE **fields**. The particles follow the structures that can be seen, but these are not leading to the accumulation.

Show more
67 Read more

Djath et al., 2014; Gaultier et al., 2014). It has also been shown previously that the **surface** quasi-geostrophic model is a simple, but eﬃcient model, for the representation of **surface** intensified vortices.
This study has shown that **surface** temperature vortices can merge only at closer range than their counterparts in incompressible, two-dimensional fluids. This weaker ability of vor- tices to merge in SQG can explain part of the observed diﬀerence in energy spectra between SQG and classical geostrophic **turbulence**. Indeed, it implies a less eﬃcient upscale energy transfer.

Show more
28 Read more

Using the Fully **Lagrangian** Approach (FLA) for the dispersed phase coupled with the DNS solver for the carrier phase, an axially symmetric transient particle-laden flow in a **vortex** ring has been investigated.
Two problem formulations have been considered: injection of a two-phase jet into a **vortex** ring field and interaction of a **vortex** ring with a cloud of droplets. In the case when droplets are identified with fuel droplets, these zones of particle accumulation are expected to lead to the formation of zones of high fuel vapour concentration. It has been observed that in cases when the dispersed medium forms folds, caustics with singularities and local zones of particle accumulation in the concentration **fields** appear. Accurate calculations of the number density in these zones could not be performed if the analysis was based on the conventional rather than the Fully **Lagrangian** Approach.

Show more
by Grant and Belcher (2009). These modifications include imposing a Stokes drift, which affects the momentum budgets through the **vortex** force, the Coriolis–Stokes force, and a modified pressure (Craik and Leibovich 1976; McWilliams et al. 1997). The Stokes drift profile is given by u s 5 u s ^x 5 u s0 exp(z/d)^x, where u s0 is the **surface** Stokes drift; z is depth (negative); d is the Stokes pene- tration length (positive), which is related to the wavelength of **surface** waves by d 5 l/(4p); and ^x is a unit vector aligned with the x axis. The turbulent Langmuir number La t 5 (u */u s0 ) 1/2 , where u* is the **surface** friction velocity of the water (McWilliams et al. 1997), can be used to charac- terize the **turbulence** in the presence of wind and wave forcing. For equilibrium wind seas, La t 5 0.3 (Li et al. 2005). The model domain is 256 m 3 256 m in the horizontal (x, y) and 90 m in the vertical (z), with resolutions of 2 and 0.6 m in the horizontal and vertical, respectively. The forcing for the following simulations was chosen so that the boundary layer **turbulence** was resolved, with the shallowest simulated boundary layers being just over 10 m deep. The resolution probably begins to have an effect in the shallowest mixed layers, but the resolved turbulent fluxes are much greater than the subgrid fluxes in all the simulations. The domain is horizontally doubly periodic. A damping layer is imposed below 65 m to damp gravity waves and prevent their reflection off the lower boundary.

Show more
15 Read more