# Workshop Programme

## for period 23 - 27 July 2012

### Topological Fluid Dynamics (IUTAM Symposium)

23 - 27 July 2012

Timetable

Monday 23 July | ||||

09:45-10:30 | Registration | |||

10:30-11:00 | Moffatt, K | |||

Welcome & Opening Remarks | Sem 1 | |||

Chair: Yoshi Kimura | ||||

11:00-11:40 | Doering, CR (University of Michigan) |
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Ultimate state of two-dimensional Rayleigh-Bénard convection | Sem 1 | |||

Determining the transport properties of high Rayleigh number convection turbulent convection remains a grand challenge for experiment, simulation, theory, and analysis. In this talk, after a brief review of the theory and applications of Rayleigh-Bénard convection we describe recent results for mathematically rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Bénard convection between stress-free isothermal boundaries derived from the Boussinesq approximation of the Navier-Stokes equations. These bounds challenge some popular theoretical arguments regarding the nature of the asymptotic high Rayleigh number ‘ultimate regime’ of turbulent convection. This is joint work with Jared Whitehead. |
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11:45-12:05 | Schneider, K; Jacobitz, F; Bos, W; Farge, M (Aix-Marseille U; U. San Diego; Ecole Centrale Lyon; ENS, Paris) |
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On helical multiscale characterisation of homogeneous turbulence | Sem 1 | |||

The helical properties of five prototypical homogeneous turbulent flows are investigated: statistically steady forced isotropic turbulence, decaying isotropic turbulence, decaying rotating turbulence, growing sheared turbulence, and growing rotating sheared turbulence with a rotation ratio f/S = +0.5. The five turbulent flows were originally studied using direct numerical simulations. An orthogonal wavelet decomposition is used to study the scale-dependent properties of the cases. For comparison, a solenoidal uncorrelated Gaussian random field is included in the analysis as a sixth case. It was found that flows with growing turbulent kinetic energy and turbulent motion at large scales show a maximum in the velocity helicity probability distribution functions (PDFs) at zero, corresponding to a trend to local two-dimensionalization of the flow with vorticity and velocity being perpendicular. Flows with decaying turbulent kinetic energy and turbulent motion at small scales, how ever, show maxima of the velocity helicity PDFs at plus and minus one, indicating a preference for helical motion with alignment or anti-alignment of vorticity and velocity. The PDFs of vorticity helicity always assume maxima at plus and minus for all flows. Joint PDFs of relative velocity helicity and relative vorticity helicity show that the quantities tend to have the same sign for all flows including the random field, indicating that vorticity helicity dissipates velocity helicity. |
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12:10-12:30 | Chkhetiani, O; Koprov, V; Koprov, B (Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences) |
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Turbulent vorticity and helicity in stratified atmospheric boundary layer | Sem 1 | |||

To measure the spatial derivative of velocity v, it is necessary to possess the sensors, which size much less than internal scale of turbulence. In a surface layer it is estimated by size of an order 1 mm. The concept of the acoustical method of vorticity measurements and the first results of its realization are obtained in IAP [Bovsheverov et al. 1971]. Helicity (a scalar product of the velocity v and the vorticity) is one of the important characteristics of large-scale atmospheric motions [Etling 1985, Moffat, Tsinober 1992; Kurgansky 2002, Chkhetiani 2001]. Direct experiments aimed at the measurement of turbulent helicity are extremely rare. They have been carried out under laboratory conditions in turbulence beyond a grid [Kholmyansky et al. 2001]. First helicity measurements in atmospheric boundary layer were made in IAP Zvenigorod station in 2004 [Koprov et al. 2005]. Experimental estimates of the spectrum of the turbulent helicity in the atmospheric boundary layer give the spectrum slope of about -5/3. Proceeding from the helicity and energy spectra, we obtain for dissipation ? ? 0.0003 m/s3, ? ? 0.003 m2/s3 and ? ? 0.0005 m/s3, ? ? 0.001m2/s3. Helicity components in day conditions shows considerably big intermittency than circulation. Average value for Hz has made 0.2 m/s2. The correlation factor between product factors in a day series at moderately unstable stratification has made 0.344. Similar indicators for Hx: average value of 0.46 m/s2, factor of correlation 0.215. In an evening series average values of both measured helicity components had the same sign, as in the afternoon, but on 1-2 order smaller values. Probability density functions (PDF) for circulations Zz, Zx, vertical velocity w and temperature T have been calculated at unstable and stable stratification. Asymmetry of Zz changes a sign at change of a sign on parameter of stratification whereas asymmetry for Zx is small and keeps a sign at any stratification. PDF of helicity components and complex triple two-point correlations of velocity and vorticity show strong non-gaussian character. Spectrum slope for these correlations is close to f -1. This fact corresponds to the "2/15" law for the helicity cascade [Chkhetiani 1996, 2010]. The data obtained for both the values of turbulent helicity in the atmospheric boundary layer and helicity spectra indicate the existence, at least in the region of the scales that have been considered, of parallel cas-cades of the energy and helicity. The importance of the determination of actual helicity cascades in natural systems is also stimulated by the fact that numerical calculations of the Navier-Stokes equations manifest certain effects of nonzero helicity on the energy transfer over the spectrum. This emphasizes the role of helicity in the formation of large-scale structures. |
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12:30-13:30 | Lunch at Wolfson Court | |||

Chair: Mitch Berger | ||||

14:00-14:40 | Oberlack, M (TU Darmstadt) |
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New conservation laws of helical flows | Sem 1 | |||

Conservation laws in incompressible fluid dynamics, in particular inviscid motion, constitute an axiomatic basis for fluid mechanics. In 3D, mass and momentum conservation forms the fundamental basis, which is further extended by energy, vorticity and helicity conservation. Interesting enough considering reduced dimensions a much broader set of conserved quantities is observed in particularly for 2D/planar and axisymmetric flows. For the planar case it is well known that any once differential function of the vorticity is a materially conserved quantity and hence an infinite number of additional conservation laws exist. Further, the most simple one, the square of the vorticity, is named enstrophy, and is “weakly conserved” in the viscous case and constitutes a fundamental invariant for 2D turbulence. Recently we have shown that the known set of additional conservation laws may be considerably extended for helical flows which constitute a Lie symmetry induced concaten ation of planar and axisymmetric flows living on the (r, a z + b \phi; t) spatially reduced system with a^2 + b^2 > 0 and r, z, \phi are the classical coordinates in a cylinder coordinate system. Various infinity dimensional new conservations laws have been established including e.g. a generalized helicity. Even for the 2D/planar and axisymmetric flows new conservation laws have been derived not reported in the literature before. The construction of the new results is based on the “direct method”. It relies on two key theorems: (i) the Euler operator applied to a term is always zero if and only if the term is in divergence form; (ii) any non-trivial conservation law of a given set of differential equations can only be constructed by a linear combination of the given equations with some multipliers to be determined by theorem (i). This is a necessary and sufficient condition. The process of finding the new conservation laws was aided by the computer algebra system Maple employing the package GeM by A. Cheviakov. |
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14:45-15:05 | Kurgansky, MV (A.M. Obukhov Institute of Atmospheric Physics, Russian Academy of Sciences) |
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Simple models of helical baroclinic vortices | Sem 1 | |||

Two distinct asymptotic solutions of inviscid Boussinesq equations for a steady helical baroclinic Rankine-like vortex with prescribed buoyant forcing are considered and critically compared. In both cases the relative distribution of the velocity components is the same across the vortex at all altitudes (the similarity assumption). The first vortex solution demonstrates monotonic growth with height of the vortex core radius, which becomes infinite at a certain critical altitude, and the corresponding attenuation of the vertical vorticity. The second vortex solution schematizes the vortex core as an inverted cone of small angular aperture. These idealized vortices are then embedded in a convectively unstable boundary layer; the resulting approximate vortex solutions have been applied to determine the maximum rotational velocity in vortices. Both models predict essentially the same dependence of the model-inferred peak rotational velocity on the local swirl ratio (the ratio of the maximum swirl velocity to the average vertical velocity in the main vortex updraft). The helicity budget of the vortex flow is analyzed in detail, where applicable. |
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15:10-15:30 | Yahalom, A (Ariel University Center of Samaria) |
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Using fluid variational variables to obtain new analytic solutions with nonzero helicity | Sem 1 | |||

Flow equations being nonlinear are notoriously difficult to solve analytically. In this work we show that through the three independent functions variational formalism for stationary barotropic flows one can obtain new analytical solutions of the flow equations. The flows are constructed such that they flow on predetermined Bernoullian surfaces from which the rest of the variational variables are derived. The flow obtained has non zero helicity. |
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15:35-15:55 | Afternoon Tea | |||

Chair: Jean-Luc Thiffeault | ||||

16:00-16:40 | Boyland, P (University of Florida) |
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Exponential growth in two-dimensional topological fluid dynamics | Sem 1 | |||

In two-dimensional multi-connected fluid regions the Thurston-Nielsen (TN) theory implies that the essential topological length of material lines grows either exponentially or linearly; the TN theory and subsequent results provide many procedures for determining which growth rate occurs. Our first application is to Euler flows. The main theorem is that there are periodic stirring protocols for which generic initial vorticity yields a solution to Euler's equations which is not periodic and further, the sup norm of the gradient of the vorticity grows exponentially in time. The second application investigates which stirring protocols maximize the efficiency of mixing in the precise, topological sense of the maximal exponential growth of per unit generator of certain push-point mapping classes on the punctured disk. Related Links http://www.math.ufl.edu/~boyland/papers.html - my paper's page |
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16:45-17:05 | Tanda, S (Hokkaido University) |
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Topological crystals as a new paradigm | Sem 1 | |||

We report the discovery of Mobius, Ring, Figure-8, and Hopf-link Crystals in NbSe3, conventionally grown as ribbons and whiskers. We also reveal their formation mechanisms of which two crucial components are the spherical selenium (Se) droplet, around which a NbSe3 bar wraps due to surface tension, and the monoclinic (P2(1)/m) crystal symmetry inherent in NbSe3, which induces a twist in the strip when bent. Our crystals provide a non-fictitious topological Mobius world governed by a non-trivial real-space topology. We class ed these topological crystals as an intermediary between condensed matter physics and mathematics. References [1]A Mobius strip of single crystals, S. Tanda et al., Nature 417, 397 (2002). [2]Formation and growth of NbSe3 topological crystals, T. Tsuneta and S.Tanda J. Cryst. Growth 267, 223 (2004). [3]Topologically linked crystals, T. Matsuura et al., J. Cryst. Growth 297, 157 (2006). [4]Topological eff ects of the superconducting vortex state in a TaSe3 ring crystal, G. Kumagai et al., Phys. Rev. B81, 184506 (2010). [5]Chiral Charge-Density waves, J. Ishioka et al., Phys. Rev. Lett 105, 176401 (2010). [6]Topology-change surgery for crystals, T. Matsuura et al., Phys. Rev. B83 174113 (2011). |
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17:10-17:30 | Ohkitani, K; Al Sulti, F (University of Sheffield) |
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Numerical and analytical study of an asymptotic equation for deformation of vortex lattices | Sem 1 | |||

It is known that when two-dimensional flows are subject to a suitable background rotation, formation of vortex lattices are observed. We can make use of critical points of the vorticity field and their connectivity (so-called, surface networks) to study reconnection of vorticity contours in 2D turbulence. In this talk we begin by noting how this method applies to the study of formation of vortex lattices. We then study a coarse-grained, asymptotic equation which describes deformation vortex lattices derived by Smirnov and Chukbar, Sov. Phys. JETP vol 93, 126-135(2001). It reads $\phi_t=\phi_{xx} \phi_{yy}-\phi_{xy}^2,$ where $\phi$ denotes displacement of vortex locations. This equation is particularly valid for geostrophic Bessel vortices with a screened interaction. Numerical results are reported which indicate an ill-posed nature of the time evolution. Self-similar blow-up solutions were already given by those authors, which have an infinite total energy. We ask whether finite-time blow-up can take place developing from smooth initial data with a finite energy. More general self-similar blow-up solutions are sought, but all are found to have infinite total energy. Finally, remarks are made in connection with the Tkachenko-type lattice. |
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17:30-18:15 | Drinks Reception | |||

18:15-19:15 | Dinner at Wolfson Court |

Tuesday 24 July | ||||

Chair: Renzo Ricca | ||||

09:00-09:40 | Peralta-Salas, D (Instituto de Ciencias Matemáticas; Madrid) |
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Knotted vortex tubes in the Euler equation | Sem 1 | |||

In this talk we will address the problem of the existence of stationary knotted and linked vortex tubes for the Euler equation. The existence of these structures is a very interesting question in Fluid Mechanics, which dates back to Lord Kelvin's studies of thin vortex tubes. We will discuss some contributions to this problem (jointly with Alberto Enciso) using Beltrami fields. This builds upon (and significantly improves) our previous work on linked vortex lines of steady solutions to the Euler equation. |
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09:45-10:05 | Enciso, A (Instituto de Ciencias Matemáticas; Madrid) |
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Knots and links in fluid mechanics | Sem 1 | |||

In this talk I will discuss the existence of steady solutions to the incompressible Euler equations that have stream and vortex lines of any prescribed knot (or link) type. More precisely, I will show that, given any locally finite link L in R^3, one can transform it by a smooth diffeomorphism F, close to the identity in any C^p norm, such that F(L) is a set of periodic trajectories of a real analytic steady solution u of the Euler equations in R^3. If the link is finite, we shall also see that u can be assumed to decay as 1/|x| at infinity, so that u is in L^p for all p>3. This problem is motivated by the well-known analysis of the structure of steady incompressible flows due to V.I. Arnold and K. Moffatt, among others. Time permitting, we will also very recent results on the topology of potential flows, that is, of steady fluids whose velocity field is the gradient of a harmonic function in R^3. These results are closely related to classic questions in potential theory that were first considered by M. Morse and W. Kaplan in the first half of the XX century and have been revisited several times after that, by Rubel, Shiota and others. The guiding principle of the talk will be that a strategy of "local, analysis-based constructions" + "global approximation methods", fitted together using ideas from differential topology, can be used to shed some light on the qualitative behavior of steady fluid flows. Most of the original results presented in this talk will be based on the papers: A. Enciso, D. Peralta-Salas, Knots and links in steady solutions of the Euler equation, Ann. of Math. 175 (2012) 345-367. A. Enciso, D. Peralta-Salas, Submanifolds that are level sets of solutions to a second-order elliptic PDE, arXiv:1007.5181. A. Enciso, D. Peralta-Salas, Arnold's structure theorem revisited, in preparation. |
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10:10-10:30 | Maggioni, F; Alamri, SZ; Barenghi, CF; Ricca, RL (Univ. of Bergamo; Taibah Univ.; Newcastle Univ.; Univ. of Milano-Bicocca) |
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Velocity, energy and helicity of vortex knots and unknots | Sem 1 | |||

In this talk we examine the effect of several geometric and topological aspects on the dynamics and energetics of vortex torus knots and unknots. The knots are given by small-amplitude torus knot solutions [1] to the Localized Induction Approximation (LIA) law. Vortex evolution is thus studied in the context of the Euler equations by direct numerical integration of the Biot-Savart law. Earlier stability results on vortex knots and unknots [2] are here extended [3]-[4], and the velocity, helicity and kinetic energy of different vortex knots and unknots are presented for comparison. Vortex complexity is parametrized by the winding number w given by the ratio of the number of meridian wraps to that of longitudinal wraps. We find that for w < 1 vortex knots and toroidal coils move faster and carry more energy than a reference vortex ring of same size and circulation, whereas for w > 1 knots and poloidal coils have approximately same speed and energy of the reference vortex ring. Kinetic helicity is dominated by writhe contributions and increases with knot complexity. The stabilizing effect of the Biot-Savart law for all knots and unknots tested is also confirmed. Our results provide information on relationships between geometry, topology and dynamics of complex vortex systems and apply to quantized vortices in superfluid 4He. References [1] Ricca, R.L. (1993) Torus knots and polynomial invariants for a class of soliton equations. Chaos 3, 83-91. [1995 Erratum. Chaos 5, 346.] [2] Ricca, R.L., Samuels, D.C. & Barenghi, C.F. (1999) Evolution of vortex knots. J. Fluid. Mech. 391, 29-44. [3] Maggioni, F., Alamri, S.Z., Barenghi, C.F. & Ricca, R.L. (2009) Kinetic energy of vortex knots and unknots. Il Nuovo Cimento C, 32(1), 133–142. [4] Maggioni, F., Alamri, S., Barenghi, C.F. & Ricca R.L. (2010) Velocity, energy and helicity of vortex knots and unknots. Phys. Rev. E., 82(2), 026309–026317. |
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10:30-11:00 | Morning Coffee | |||

Chair: Edriss Titi | ||||

11:00-11:40 | Gibbon, JD (Imperial College London) |
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Intermittency and conditional regularity of solutions of the 3D Navier-Stokes equations | Sem 1 | |||

Both numerical and experimental evidence suggests that solutions of the three-dimensional Navier-Stokes equations display a high degree of intermittency, which is manifest in spiky excursions away from averages in the vorticity field. This phenomenon may be intimately bound up with the enduring problem of regularity and is addressed by discussing two new conditional (& unusual) regularity assumptions. |
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11:45-12:05 | Barenghi, CF (Newcastle University) |
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Quantum vortex reconnections | Sem 1 | |||

In superfluid helium and in atomic Bose-Einstein condensates, quantum mechanics constrains the rotational motion to discrete filaments of fixed circulation which is equal to Planck's constant divided by the mass of the relevant boson. Because of their simplicity (no viscosity, vorticity confined to vortex lines, fixed circulation), quantum fluids are ideal systems where to study the topology of vortex flows. In this talk I shall report results on the motion of vortex rings perturbed by Kelvin waves (a classical problem first studied by Lord Kelvin), vortex bundles, vortex knots and turbulent tangles of such discrete vortices. In the case of turbulence, I shall focus on its properties, and the relation between kinetic energy and vortex length. |
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12:10-12:30 | Luzzatto-Fegiz, P; Williamson, CHK (Woods Hole Oceanographic Institution & Cornell University) |
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An accurate and efficient method to compute steady vortices without symmetry | Sem 1 | |||

When considering steady solutions of the Euler equations, it is often of interest to find isovortical flows, that is, solutions that can be obtained from rearrangements of a given vorticity distribution. Since inviscid transitions between such flows are, in principle, possible, these solutions may act as attractors in the unsteady dynamics (e.g. Dritschel 1986, 1995; Flierl & Morrison 2011). The computation of such steady vortex flows still presents some challenges. Existing Newton iteration methods become inefficient as the vortices develop fine-scale features; in addition, these methods do not, in general, find solutions from isovortical rearrangements. On the other hand, available relaxation approaches are more affordable, but their convergence is not guaranteed. In this work, we consider flows that may be approximated by a collection of uniform vortices, and overcome the limitations outlined above by using a discretization, based on an inverse-velocity mapping, which radically increases the efficiency of Newton iteration methods. In addition, we introduce a procedure to enforce the isovortical constraint in the solution method. We illustrate our methodology by exploring the solution structure of a wide range of unbounded flows. We uncover several families of lower-symmetry vortices. While asymmetric point vortex flows have been found by Aref & Vainchtein (1998), it appears that this is the first time that nonsingular, asymmetric steady vortices have been computed. In addition, we discover that, as the limiting vortex state for each flow is approached, each family of solutions traces a clockwise spiral in a bifurcation plot consisting of a velocity-impulse diagram. By the recently introduced ‘‘IVI diagram’’ stability approach (Luzzatto-Fegiz & Williamson 2010, 2011), each turn of this spiral is associated with a loss of stability. Such spiral structure is suggested to be a universal feature of steady, uniform-vorticity Euler flows. |
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12:30-13:30 | Lunch at Wolfson Court | |||

Chair: Eckart Meiburg | ||||

14:00-14:40 | Friedlander, S (University of Southern California) |
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Instabilities and ill-posedness for the magnetogeostrophic equation | Sem 1 | |||

We discuss an active scalar equation that is motivated by a model presented by Keith Moffatt for the geodynamo and magnetogeostrophic turbulence in the Earth's fluid core. We prove that the non-difusive equation is Lipshitz ill-posed in Sobolev spaces. In contrast, the diffusive equation is globally well-posed. In this case we give an example of a steady state that is nonlinearly unstable, and hence produces a dynamo effect in the sense of an exponentially growing magnetic field. This is joint work with Vlad Vicol. |
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14:45-15:05 | Kerr, RM (University of Warwick) |
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Dissipation and enstrophy statistics in turbulence: are the simulations and mathematics converging? | Sem 1 | |||

This presentation will be based upon the Focus on Fluids article with this title to appear in JFM 700 (2012). The Focus will be on: Yeung, Donzis, & Sreenivasan, 2012 Dissipation, enstrophy and pressure statistics in turbulence simulations at high Reynolds numbers. J. Fluid Mech. 700 and the two themes of the FoF are that Yeung et al resolves a remaining question about the convergence of higher-order statistics and that this result is related to new mathematics on temporal intermittency in turbulence in Gibbon, J.D. 2009 Estimating intermittency in three- dimensional Navier-Stokes turbulence. J. Fluid Mech. 625. What Yeung et al. finds is that even if the fluctuations of the higher-order vorticity and strain statistics are so large that they do not converge individually, their ratios do converge. Gibbon (2009) shows that this type of behaviour is expected and Gibbon (TODW01) will present specific predictions for the ordering of these statistics at any given time and the t ype of maximum growth during the most intermittent periods. However, Yeung et al does not give time variations, so a direct comparison is not possible. My new results are from simulations of the reconnection of anti-parallel vortex tubes, an example of the events assumed by Gibbon (2009), where this time-dependent analysis has been done. This simulation develops, after just two reconnection steps, most of the properties associated with fully-developed turbulence, including a -5/3 spectrum with the proper coefficient and the expected enstrophy production skewness, and the intermittency ratios are consistent with Yeung et al. Turbulence develops after reconnections by: Forming orthogonal vortices, which wrap up as in the Lundgren spiral vortex model. The temporal ordering and growth of the higher-order vorticity statistics obey the bounds of the new mathematical predictions exactly. Thus the connection between the latest high Reynolds number calculations and the latest mathematics is demonstrated. |
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15:10-15:30 | Kolomenskiy, D; Moffatt, HK; Farge, M; Schneider, K (CERFACS, Toulouse; Univ. DAMPT, Cambridge; Paris ENS; Aix-Marseille U) |
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Fluid dynamics of flapping wings associated with change of domain topology | Sem 1 | |||

We re-examine the clap-fling-sweep mechanism employed by some insects to increase lift. As argued by Lighthill (J Fluid Mech 60(1):1-17, 1973), this mechanism can create a circulatory motion even in a totally inviscid fluid, due to a topological change of the solid boundary that represents the wings surfaces. During the stroke, the wings first clap together behind the insect's back, then open in a fling motion around the `hinge' formed by the two trailing edges, and finally separate at the hinge and sweep apart. In a two-dimensional approximation, we use two different conformal mappings in simply and doubly connected domains, respectively, to calculate the complex potential at all stages of the process. The results indicate that circulation (equal in magnitude and opposite round the two wings) can be generated in an inviscid fluid, and that this circulation appears when a solid body immersed in the fluid breaks into two pieces (when fling gives way to sweep). Bound vortex sheets produced during fling are still carried by the just-separated wings. This is accompanied by a continuous time evolution of the velocity everywhere in the fluid, although the pressure field jumps instantaneously at the moment of wing separation. In a viscous fluid, the flow during the break is essentially different because, locally, the Reynolds number is very low near the hinge point. We describe it by local similarity solutions to the Stokes equation (J Fluid Mech 676:572-606, 2011). Three-dimensional effects are present in the flow. We study them by performing numerical simulations of the Navier-Stokes equations using a Fourier spectral method with volume penalization. The flow before the break is found to be in a good agreement with the two-dimensional approximation. After the wings move farther than one chord length apart, the three-dimensional nature of the flow becomes essential (J Fluids Struct 27(5-6):784-791, 2011). |
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15:35-15:55 | Petrosyan, A (Space Research Institute, Russian Academy of Sciences) |
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Vortical dynamo in turbulent multiphase flows | Sem 1 | |||

Magnetic disturbances are known to be amplified by helical turbulence. The possibility of amplification of large-scale hydrodynamic fields by small-scale helical turbulence is considered. The important difference between hydrodynamic and magnetic theories is that the latter describe the evolution of magnetic field on the background of a given hydrodynamic flow (kinematic dynamo), whereas in hydrodynamics such a situation is more complex. The hydrodynamic problem is self-consistent and non-linear. A generation of large-scale helical vortices resulting from the instability of small-scale helical turbulence with respect to two-scale disturbance is considered. In order to investigate such instability, we consider two cases: (1) an incompressible fluid containing rigid particles; (2) an incompressible fluid containing gas babbles. An equation describing the evolution of mean disturbances is derived and the instability increment is obtained. The analysis revealed that helical turbulence in an incompressible fluid with rigid particles and in incompressible fluid with gas babbles is unstable against vortical disturbances. The generation terms formally coinciding with those in the theory of hydromagnetic dynamo are contained in Reynolds averaged equations derived at the scale of mean motions. It should be noted that only helicity is enough for the process of generation in magnetohydrodynamics. In hydrodynamic theory, because of the mentioned differences, it is also necessary to take into account additional factors. In this paper two such additional factors are the presence of rigid particles or gas babbles whose motions provide the existence of divergence at a turbulent scale and thus provide a non-zero value of the Reynolds stresses in the averaged equations. |
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15:55-16:05 | Libin, A | |||

Coherent Beltrami Structures | Sem 1 | |||

16:00-16:40 | Afternoon Tea & Poster Session | |||

Chair: Keith Moffatt | ||||

16:45-17:30 | Brøns, M (Technical University of Denmark) |
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Relative equilibria of point vortices. (Aref Memorial Lecture) | Sem 1 | |||

A relative equilibrium of a system of point vortices is a configuration which rotates with constant angular velocity around its centre of vorticity. It is easy to write down the equations for the vortex positions and many simple configurations with symmetry are known. Several asymmetric states have been found numerically, including some surprising ones with some of the vortices being very close. Very little is known analytically about the general problem. Here we consider the case where the vortices are identical and placed on two perpendicular lines which we choose to be the axes of a coordinate system. We define two polynomials p(z) and q(z) whose roots are the vortex positions on each line in the complex plane, and derive a differential equation for p for given q. We discuss how the general solution to the differential equation relates to physical vortex configurations. The main result is that if q has m solutions symmetrically placed relative to the real axis and p is of degree n, it must have at least n-m+2 real roots. For m=2 this is a complete characterisation, and we obtain an asymptotic result for the location of the two vortices on the imaginary axis as the number of vortices on the real axis tends to infinity. |
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18:15-19:15 | Dinner at Wolfson Court |

Wednesday 25 July | ||||

Chair: Yasuhide Fukumoto | ||||

09:00-09:40 | Khesin, B (University of Toronto) |
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The filament equation in any dimension | Sem 1 | |||

We show that the LIA approximation of the incompressible Euler equation describes the skew-mean-curvature flow on vortex membranes in any dimension. This generalizes the classical binormal, or vortex filament, equation in 3D. We present a Hamiltonian framework for higher-dimensional vortex filaments and vortex sheets as singular 2-forms with support of codimensions 2 and 1, respectively. This framework, in particular, allows one to define the symplectic structures on the spaces of vortex sheets. |
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09:45-10:05 | Liu, X; Ricca, RL (University of Sydney & University of Milano-Bicocca) |
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Tackling structural complexity by Jones' polynomials | Sem 1 | |||

In this paper we present new results based on applications of knot theory to tackle and quantify structural complexity of vortex dynamics. In the ideal case of Euler equations we show that the topology of any vortex tangle, made by knots and links, can be identified and described by the Jones’s polynomials of the tangle, expressed in terms of kinetic helicity. Explicit calculations of the Jones polynomial for the left-handed and right-handed trefoil knots and for the Whitehead link via the figure-of-eight knot are worked out for illustration. This novel approach contributes to establish a fundamentally new paradigm in topological fluid mechanics, by extending the former interpretation of helicity in terms of linking numbers to the much richer context of knot polynomials, and by opening up new directions of work both in mathematical fluid dynamics and in numerical diagnostics of vortex flows. |
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10:10-10:30 | Mitsumatsu, Y (Chuo University) |
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Helicity in differential topology | Sem 1 | |||

The helicity plays many intresting roles in 3-dimensinal diffential topology. One of its earliest appearences in was the question by Dennis Sullivan, asking to express the Godbilon-Vey invariant in terms of linking of fluids. Here the Godbillon-Vey is an invariant for codimension1 foliations which lives in the 3rd de Rham cohomology. The question is well-understood and we know which fluid motion should be taken. If we think of helicity as a quadratic function on the space of incompressible fluids, namely the space of divergence free vector fields, it goes down to a symmetric bilinear form. Some ideas concerning this bilinear form for studies of foliations and contact structures are introduced. For example, in the case of codimmension one foliations the 1st foliated cohomology will appear. If the foliation is deofrmed to contact structures, unexpectedly phenomena which might be related to a quatization procedure is found. |
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10:30-11:00 | Morning Coffee | |||

Chair: John Gibbon | ||||

11:00-11:40 | Titi, ES (University of California, Irvine and The Weizmann Institute of Science) |
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How much do we know about the problem of global regularity for the three-dimensional Navier-Stokes and Euler equations? | Sem 1 | |||

In this talk I will survey the status of, and the most recent advances concerning, the questions of global regularity of solutions to the three-dimensional Navier-Stokes and Euler equations of incompressible fluids. Furthermore, I will also present recent global regularity results concerning certain three-dimensional geophysical flows, including the three-dimensional viscous "primitive equations" of oceanic and atmospheric dynamics. |
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11:45-12:05 | Ricca, RL (Università degli Studi di Milano - Bicocca) |
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Impulse of vortex knots from diagram projections | Sem 1 | |||

By using methods based on the analysis of standard plane projections of complex tangles, we can extract geometric information and use it to determine the impulsive forces associated with vortex knots and links. This method relies on the interpretation of linear and angular momentum of ideal vortex filaments in terms of projected areas [1]. An immediate application of this method allows one to make predictive estimates of the evolution and dynamical features of vortex knots and links [2]. This will be illustrated by a number of examples, some related to well-known results from laboratory and numerical experiments, where vortex ring collisions and vortex linking have been studied, and some others, such as the production of a trefoil vortex knot, proposed as "thought" experiments [3]. [1] Ricca, R.L. (2008) Momenta of a vortex tangle by structural complexity analysis. Physica D 237, 2223-2227. [2] Ricca, R.L. (2009) Structural complexity and dynamical systems. In Lectures on Topological Fluid Mechanics (ed. R.L. Ricca), pp. 169-188. Springer-CIME Lecture Notes in Mathematics 1973. Springer-Verlag. [3] Ricca, R.L. (2009) New developments in topological fluid mechanics. Nuovo Cimento C 32, 185-192. |
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12:10-12:30 | Goldstein, RE; Moffatt, HK; Pesci, AI (University of Cambridge) |
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A Soap-Film Mobius Strip Changes Topology with a Twist Singularity | Sem 1 | |||

It is well-known that a soap film spanning a looped wire can have the topology of a Möbius strip and that deformations of the wire can induce a transformation to a two-sided film, but the process by which this transformation is achieved has remained unknown. In this talk I will disucss recent experimental and theoretical work [Goldstein, Moffatt, Pesci, and Ricca, PNAS 107, 21979 (2010)] that has uncovered the dynamics of this transition. We find that this process consists of a collapse of the film toward the boundary that produces a previously unrecognized finite-time twist singularity that changes the linking number of the film's Plateau border and the centerline of the wire. We conjecture that it is a general feature of this type of transition that the singularity always occurs at the surface boundary. The change in linking number is shown to be a consequence of a viscous reconnection of the Plateau border at the moment of the singularity. High-speed imaging of the collapse dynamics of the film's throat, similar to that of the central opening of a catenoid, reveals a crossover between two power laws. Far from the singularity, it is suggested that the collapse is controlled by dissipation within the fluid film surrounding the wire, whereas closer to the transition the power law has the classical form arising from a balance between air inertia and surface tension. Analytical and numerical studies of minimal surfaces and ruled surfaces are used to gain insight into the energetics underlying the transition and the twisted geometry in the neighborhood of the singularity. A number of challenging mathematical questions arising from these observations are posed. |
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12:30-13:30 | Lunch at Wolfson Court | |||

14:00-19:00 | Social Activity / Free Afternoon | |||

19:30-22:00 | Conference dinner at Emmanuel College |

Thursday 26 July | ||||

Chair: Mark Stremler | ||||

09:00-09:40 | Thiffeault, J-L (University of Wisconsin-Madison) |
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Topological approaches to problems of stirring and mixing | Sem 1 | |||

I will review two topological approaches to stirring and mixing. The first involves constructing systems such that the fluid motion is topologically complex, usually by imposing a specific motion of rods. I will discuss optimization strategies that can be implemented. The second is diagnostic, where flow characteristics are deduced from observations of periodic or random orbits and their topological properties. |
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09:45-10:05 | Hornig, G; Wilmot-Smith, A; Pontin, D (University of Dundee) |
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Relaxation of braided magnetic and vorticity fields | Sem 1 | |||

In this talk we will first report on a series of numerical MHD experiments on the turbulent relaxation of braided magnetic fields in plasmas of high magnetic Reynolds numbers (Wilmot-Smith et al. 2009, 2010). These experiments have produced relaxed states which in some cases differ drastically from the predictions of the Taylor hypothesis, that is the assumption that the final state of a turbulent relaxation is a linear force-free field with the same total helicity as the initial state. We present a method to determine the topological degree of the field line mapping which shows that there are further constraints on the relaxation process beyond the conservation of the total helicity (A. Yeates et al., Phys. Rev. Lett. 105, 2010). These constraints can prevent the system from relaxing to a Taylor state and hence limit the energy which can be released. We will then report on a new series of experiments where we test whether similar constraints hold in the hydrodynamic case, that is we investigate the relaxation of incompressible flows with braided vorticity field lines. |
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10:10-10:30 | Velasco Fuentes, O (CICESE, Mexico) |
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Stirring vortices with vorticity holes | Sem 1 | |||

A vorticity hole is a region with, in absolute value, significantly lower vorticity than its surroundings. Here we discuss the dynamics of a Rankine vortex with either one elliptic hole or two equal circular holes. If we assume symmetry and null vorticity within the holes, the evolution only depends on the hole size and either the aspect ratio of the elliptic hole or the separation of the circular holes. We computed the evolution with a contour-dynamics model and found that it is analogous to either that of the Kirchhoff vortex or that of the vortex pair, but the vorticity holes are additionally affected by their interaction with the boundary of the Rankine vortex. To quantify the stirring of fluid particles, both inside and outside the vortex, we analysed the set of hyperbolic trajectories and associated manifolds of the time-evolving velocity field. The strongest stirring always occurred in the areas of highest vorticity, which contradicts the generally accepted notion that vortices are regions of null to weak stirring. |
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10:30-11:00 | Morning Coffee | |||

Chair: Andrew Gilbert | ||||

11:00-11:40 | Stremler, MA (Virginia Polytechnic Institute and State University) |
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Identifying topological chaos using set-oriented methods | Sem 1 | |||

Identifying topological chaos using the Thurston-Nielsen classification theorem (TNCT) is a powerful approach to quantifying and predicting chaos in a variety of fluid systems. This approach is most easily applied to systems stirred by physical rods, since the rods can be prescribed to move on sufficiently complex space-time trajectories. In many cases, however, an analysis based solely on the motion of physical rods cannot capture the full complexity of the flow. Consideration of 'ghost rods', or material particles that 'stir' the fluid, can provide the missing information needed for an accurate topological representation. Unfortunately, even when such low-order periodic orbits exist, they can be difficult to identify. We will discuss the use of set-oriented, or mapping-based, statistical methods for identifying periodic regions in the domain having high local residence time. These 'almost-cyclic sets' can reveal the underlying topology of the s ystem, enabling application of the TNCT even in the absence of low-order periodic orbits. Viscous flow examples show that this approach can provide a good representation of system behavior over a range of parameters. |
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11:45-12:05 | Kimura, Y (Nagoya University) |
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Mass transport by vortex motions | Sem 1 | |||

Mass transport by the motion of 3D vortex filaments is studied. Perhaps the simplest example of such a transport is one by a vortex ring. As another example, we recently demonstrated that a 3D vortex soliton (Hasimoto soliton) can also transport mass along with its propagation. The common features of the transport, one by a vortex ring and the other by a vortex, is that some fluid particles move in closed orbits which make knots with the vortices. For the analysis of the mechanism of this trapping, a model composed of two straight vortex filaments in the 3D space, which is called the chopsticks model, is presented. We report mass transport by this model. |
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12:10-12:30 | Sadowski, W (University of Warsaw) |
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On the regularity of Lagrangian trajectories in the 3D Navier-Stokes flow | Sem 1 | |||

The paper considers suitable weak solutions of the 3D Navier-Stokes equations. Such solutions are defined globally in time and satisfy local energy inequality but they are not known to be regular. However, as it was proved in a seminal paper by Caffarelli, Kohn and Nirenberg, their singular set S in space-time must be ``rather small'' as its one-dimensional parabolic Hausdorff measure is zero. In the paper we use this fact to prove that almost all Lagrangian trajectories corresponding to a given suitable weak solution avoid a singular set in space-time. As a result for almost all initial conditions in the domain of the flow Lagrangian trajectories generated by a suitable weak solution are unique and C^1 functions of time. This is a joint work with James C. Robinson. |
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12:30-13:30 | Lunch at Wolfson Court | |||

Chair: Marie Farge | ||||

14:00-14:40 | Kida, S (Kyoto University) |
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Instability by weak precession of the flow in a rotating sphere | Sem 1 | |||

The linear stability analysis is performed of the steady flow in a weakly precessing sphere of rapid rotaion. It is well-known that all the disturbances damp with decay rate proportional to Re^{-1/2} without precession, where Re is the Reynolds number defined by the sphere radius, the the spin angular velococity, and the kinematic viscosity of fluid. We show by an asymptotic analysis for large Re and small Gamma, the ratio of the precession and spin angular velocities, that with weak precession of Gamma of order Re^{-1/2} destabilizes the disturbances by the coupling between an symmetric (with respect to the spin axis) mode and (2,1,1) mode through "the conical shear layers" emanating from the critical circles along the sphere boundary. It is found the critical curve for the instability behaves as Gamma = $7.9 Re^{-0.5}$ asymptotically, which agrees well with an observation in an precessing spheroid of ellipticity $0.9$ by Goto {\it et al.} (2011). |
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14:45-15:05 | Elimelech, Y; Kolomenskiy, D; Moffatt, HK; Dalziel, SB (Israel Institute of Technology; CERFACS Toulouse; DAMTP,Cambridge) |
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Evolution of the leading edge vortex over an accelerating rotating wing | Sem 1 | |||

Flapping flight is a subject of interest for more than two decades. During this time it has been found that a stable leading edge vortex is responsible for the high lift that flapping and revolving wings can produce. However, many of these studies were limited to Reynolds numbers of few hundred, which characterize insects. Recently, the interest on designing and realizing miniature hovering vehicles requires expanding our understanding of the basic flow mechanism which govern such wing maneuvers at higher Reynolds numbers. In this study the flow field over an accelerating rotating wing model is analyzed in various Reynolds numbers ranging from 250 to 2000 using particle image velocimetry. These experimental results are compared with three-dimensional and time-accurate Navier-Stokes flow simulations. The study depicts the characteristic size and time scales of the leading-edge vortex. The results show that the topology of the leading-edge vortex is Reynolds number dependent; i n comparison to a diffused and detached leading-edge vortex at Reynolds number 250, at Reynolds number 2000 the leading-edge vortex is not stationary and can cover up to about 75 percent of the local wing chord. Furthermore, it is shown that the spanwise velocity component increases considerably at Reynolds number of 1000 and above. Moreover, in Reynolds number 250 the circulation within the leading-edge vortex during wing acceleration exceeds its asymptotic value which develops over steadily revolving wings. At Reynolds number 1000 and above, on the other hand, the circulation within the leading-edge vortex evolves much slower. These findings shed new insights about the differences between the aerodynamic characteristics of steady revolving wings and flapping ones and will be utilized to investigate the stability of the leading-edge vortex in wider range of Reynolds numbers. |
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15:10-15:30 | Kudela, H; Kosior, A (Wrocław University of Technology , Poland) |
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Parallel computation of vortex tube reconnection using graphics card and vortex particle methods | Sem 1 | |||

Understanding the dynamics and mutual interaction among various types of vortical motions is a key ingredient in clarifying and controlling fluid motions . One of the most fundamental 3D vortical interactions related to the vortex tube reconnection. In the paper will be present the numerical results of the vortex tube reconnections for different initial configurations like reconnection o the vortex tube with counter-rotating vortices that are initially parallel and was sinusoidal perturbed (Crow instability), the vortex reconnection of the initially straight offset tube and reconnection of the vortex rings . We try to find and demonstrate some universal process for core reconnections. It will be shown the effect of mixing of the fluid by the reconnection. It was done by tracing the passive markers that were initially placed near the space where the reconnection took place. For numerical simulations we use the vortex particle methods. Due to the large time consuming at single processor unit we constructed the numerical code for multiprocessor unit of graphics card. It was proved that vortex particle method in version Vortex -in Cell are very good suited for parallel computation. We carefully tested the method by comparing the numerical results with some theoretical results (the motion of the vortex ring) and with results that were published in literature. The speed-up which we obtained was nearly 50 times grater with comparison to the single processor. |
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15:35-15:55 | Lipniacki, T; Bajer, K; Kursa, M (Univ. of Warsaw & Institute of Fundamental Technological Research, Poland) |
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Cascade of vortex loops initiated by a single reconnection of quantum vortices | Sem 1 | |||

We demonstrate that a single reconnection of two quantum vortices can lead to the creation of a cascade of vortex rings. Our analysis involves localized induction approximation, high-resolution Biot-Savart and Gross-Pitaevskii simulations. The latter showed that the rings cascade starts on the atomic scale, with rings diameters orders of magnitude smaller than the characteristic line spacing in the tangle. Vortex rings created in the cascades may penetrate the tangle and annihilate on the boundaries. This provides an efficient decay mechanism for sparse or moderately dense vortex tangle at very low temperatures. |
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15:55-16:05 | Boatto, S (Universidade Federal do Rio de Janeiro (UFRJ) |
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Point-vortex dynamics and stability of relative equilibria on surfaces | Sem 1 | |||

16:00-16:40 | Afternoon Tea & Poster Session | |||

Chair: Timothy Pedley | ||||

16:45-17:30 | Córdoba, D (ICMAT-CSIC) |
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Finite time singularities for the free boundary incompressible Euler equations | Sem 1 | |||

We prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity. Moreover, we show a stability result together with numerical evidence that there exist solutions of the 2D water wave equation that start from a graph, turn over and collapse in a splash singularity (self intersecting curve in one point) in finite time. Joint work with A. Castro, C. Fefferman, F. Gancedo and J. Gomez-Serrano. |
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18:15-19:15 | Dinner at Wolfson Court |

Friday 27 July | ||||

Chair: Konrad Bajer | ||||

09:00-09:40 | Meiburg, EH (University of California, Santa Barbara) |
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Three-dimensional vorticity dynamics in miscible Hele-Shaw displacements | Sem 1 | |||

We perform three-dimensional DNS simulations of the transient, variable viscosity Navier-Stokes equations in the Boussinesq approximation, coupled to a convection-diffusion equation for a concentration field, to simulate miscible viscous fingers in Hele-Shaw cells. The three-dimensional problem allows for new instabilities and patterns that cannot be captured by traditional gap-averaged modeling. For constant density displacements, the simulations reveal the mechanism by which the initial spanwise vorticity of the base flow, when perturbed, gives rise to the cross-gap vorticity that drives the fingering instability in the classical Darcy sense. Cross-sections at constant streamwise locations reveal the existence of a streamwise vorticity quadrupole that induces fluid transport from the walls of the cell to its center, thereby leading to a new hydrodynamic instability, termed 'inner splitting' that had not been previously reported. If gravity is included, the nature of the two-dimensional base flow and its subsequent instability changes dramatically. The interaction between Saffman-Taylor and Rayleigh-Taylor instabilities can lead to additional splitting events, and it can significantly enhance the mixing rates of the two fluids, thereby altering the overall displacement efficiency. |
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09:45-10:05 | Hattori, Y; Llewellyn-Smith, SG (Tohoku University & University of California, San Diego) |
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Motion of axisymmetric magnetic eddies with swirl | Sem 1 | |||

We consider the motion of axisymmetric magnetic eddies with swirl in ideal MHD (magnetohydrodynamics) flow. The magnetic field is assumed to be toroidal, while the velocity field has both toroidal and poloidal components. First, the contour-dynamics formulation by Hattori and Moffatt (2006) for the case without swirl is extended to include swirl velocity so that the cross helicity does not vanish in general. The strength of vortex sheets which appear on the contours varies with time under the influence of the centrifugal force due to swirl and the magnetic tension due to the Lorentz force. Numerical simulation using the contour-dynamics formulation shows that there exist counter-propagating dipolar structures at the radius of balance between the centrifugal force and the magnetic tension; these structures are well described by the steady solutions obtained by perturbation expansion. The effects of vorticity inside the eddy on the motion of the eddies are also investigated. Next, we seek exact steady spherical solutions of magnetic eddies with swirl. A generalized family of exact solutions is found; it includes Hill’s spherical vortex, the Hicks-Moffatt family which is a non-MHD vortex with swirl and the family of a MHD vortex without swirl found by Hattori and Moffatt (2006). |
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10:10-10:30 | Hirota, M; Morrison, PJ; Ishii, Y; Yagi, M; Aiba, N (Japan Atomic Energy Agency & University of Texas at Austin) |
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Variational method for estimating nonlinear acceleration of collisionless magnetic reconnection | Sem 1 | |||

Magnetic reconnection in collisionless plasma is studied by applying a technique that utilizes variational principle. Following the fact that a model of collisionless (namely, dissipationless) plasma constitutes a Hamiltonian system, the corresponding variational principle is formulated, where the displacement field of fluid elements is the dynamical variable to be varied. The effect of so-called "electron inertia" is a singular perturbation that modifies the topological invariant of plasma and, hence, allows magnetic reconnection to occur without any dissipation mechanism. If the potential energy of this variational principle decreases for some displacement field (under the modified topological constraint), such the fluid motion turns out to grow with the release of free energy. A rather simple fluid motion is enough to prove the occurrence of spontaneous magnetic reconnection. Decrease of potential energy in the nonlinear regime (where the magnetic island is larger than the width of boundary layer) is found to be steeper than in the linear regime, resulting in acceleration of the reconnection. |
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10:30-11:00 | Morning Coffee | |||

Chair: Chas Williamson | ||||

11:00-11:40 | Yoshida, Z (University of Tokyo) |
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Singular Casimir elements: their mathematical justification and physical implications | Sem 1 | |||

The problem of singular Poisson operator, which occurs in the noncanonical Hamiltonian formulation of equations describing ideal fluid and plasma dynamics, is studied in the context of a Casimir deficit, where Casimir elements constitute the center of the Lie-Poisson algebra underlying the Hamiltonian formulation. The nonlinearity of the evolution equation makes the Poisson operator inhomogeneous on phase space (the function space of the state variable), and it is seen that this creates a singularity where the nullity of the Poisson operator (the "dimension" of the center) changes. Singular Casimir elements stemming from this singularity are unearthed using a generalized functional derivative (which may be regarded as hyperfunctions generated by an "infinite-dimensional partial differential operator" with a singularity). A singular perturbation (introduced by finite dissipation) destroys the leaves foliated by Casimir elements, removing the topological constraint and allowing the state vector to move towards lower-energy state in unconstrained phase space. The dynamics of an ideal plasma is constrained by the helicity as well as helical-flux Casimir elements pertinent to resonant singularities. A finite-resistivity singular perturbation gives rise to negative-energy "tearing modes" by destroying the helical-flux Casimir leaves. |
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11:45-12:05 | Fukumoto, Y; Sakuma, H (Kyushu University & Japan Agency for Marine-Earth Science and Technology) |
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A unified view of topological invariants of barotropic & baroclinic fluids & their application to formal stability analysis of three-dimensional ideal gas flows | Sem 1 | |||

Integrals of an arbitrary function of the vorticity, two-dimensional topological invariants of an ideal barotropic fluid, take different guise from the helicity. Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invariants of a barotropic fluid are variants of the cross helicity. Baroclinic fluid flows admit, as the Casimir invariants, a class of integrals including an arbitrary function of the entropy and the potential vorticity. A consideration is given to them from the view point of Noether's theorem. We then develop a new energy-Casimir convexity method for a baroclinic fluid, and establish a novel linear stability criterion, to three-dimensional disturbances, for equilibria of general rotating flows of an ideal gas without appealing to the Boussinesq approximation. By exploiting a larger class of the Casimir invariants, we have succeeded in ruling out a term including the gradient of a dep endent variable from the energy-Casimir function. For zonally symmetric flows, the resulting criterion is regarded as an extended Richardson number criterion for stratified rotating shear flows with compressibility taken into account. |
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12:10-12:30 | Nguyen-van-yen, R; Klein, R; Waidman, M (Freie Universität Berlin) |
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Non-stationary boundary layers and energy dissipation in incompressible flows | Sem 1 | |||

In fully turbulent incompressible flows, the two presumed culprits for energy dissipation are thin boundary layers on the one hand, and three-dimensional stretching of vortex tubes on the other hand. Although both suspects seem equally important, much more theoretical effort has been invested to study the latter, due to the emphasis that has been put on homogeneous turbulence since the 1930s. We argue that the same effort should be dedicated to interrogate the former, especially since it abides the simpler two-dimensional (2D) setting. Indeed, while the shortcomings of the perfect fluid model, which lead to the d'Alembert paradox, are well understood, the failure of the Prandtl viscous boundary layer theory to predict the right scaling of energy dissipation still lacks a complete explanation. Several possibilities have been explored in recent years, like finite time singularities in the Prandtl equations, ill-posedness of the Prandtl equations, or even earlier breakdown of the asymptotic expansion itself. What happens beyond the Prandtl regime is even more mysterious. According to a criterion proven by Kato, energy should then be dissipated at scales as fine as the inverse of the Reynolds number, but the process by which this could happen remains elusive. We review the main issues at hand, using a 2D dipole-wall interaction as illustrative test case. First we re-derive the Euler-Prandtl equations in the vorticity formulation, solve them numerically, and pinpoint the origin of the discrepancy with a reference Navier-Stokes solution. We then proceed to explore alternative asymptotic expansions, allowing for a localized collapse of the boundary layer to finer scales, emphasizing in particular the importance of the topology of the vorticity field, as well as the consequences regarding the scaling of energy dissipation. |
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12:30-13:30 | Lunch at Wolfson Court | |||

Chair: Charlie Doering | ||||

14:00-14:40 | Childress, S (Courant Institute of Mathematical Sciences) |
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How fast can vorticity stretch itself? | Sem 1 | |||

We consider some examples of computing or bounding the long-time growth of vorticity in Euler flows under various assumptions. Estimates for axisymmetric flow without swirl, obtained under constraints on vorticity volume and and kinetic energy, are discussed. Axisymmetric flow with swirl is treated under a truncation equivalent to a axisymmetric "stretched" version of the Taylor-Green problem. The close relation of flows with swirl to 2D Boussinesq convection is used to give some examples of periodic flows developing arbitrarily large vorticity in a bounded domain. Finally, a moving line problem is introduced which produces finite time singularities. It is noted that locally 2D vortex structures of the kind obtained in the growth estimates without swirl motion could be consistent with the equations of motion of the line, but no computable examples of this are known. |
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14:45-15:05 | Hormoz, S; Brenner, MP (Harvard University & KITP) |
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Absence of singular stretching of interacting vortex filaments | Sem 1 | |||

A promising mechanism for generating a finite-time singularity in the incompressible Euler equations is the stretching of vortex filaments. Here, we argue that interacting vortex filaments cannot generate a singularity by analyzing the asymptotic dynamics of their collapse. We use the separation of the dynamics of the filament shape, from that of its core to derive constraints that must be satisfied for a singular solution to remain self consistent uniformly in time. Our only assumption is that the length scales characterizing filament shape obey scaling laws set by the dimension of circulation as the singularity is approached. The core radius necessarily evolves on a different length scale. We show that a self similar ansatz for the filament shapes cannot induce singular stretching, due to the logarithmic prefactor in the self interaction term for the filaments. More generally, there is an antagonistic relationship between the stretching rate of the filaments and the requ irement that the radius of curvature of filament shape obeys the dimensional scaling laws. This suggests that it is unlikely that solutions in which the core radii vanish sufficiently fast to maintain the filament approximation exist. |
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15:10-15:30 | Protas, B (McMaster University) |
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Probing fundamental bounds in hydrodynamics using variational optimisation methods | Sem 1 | |||

This work demonstrates how the modern methods of PDE-constrained optimization can be used to assess sharpness of a class of fundamental functional estimates in fluid mechanics. These estimates concern bounds on the instantaneous rate of growth and finite-time growth of quadratic quantities such as the enstrophy and palinstrophy in viscous incompressible flows. Sharpness of such estimates is inherently related to the problem of singularity formation in the 3D Navier-Stokes system. In our presentation we will first review earlier results of Lu & Doering (2008) and Ayala & Protas (2011) concerning the maximum growth of enstrophy the 1D Burgers equation. We will then present several new results regarding the maximum growth of palinstrophy in 2D flows and will discuss some questions concerning sharpness of the corresponding analytical estimates. While it is well known that solutions of 1D Burgers equations and 2D Navier-Stokes equation evolving from smooth initial data remain smooth for all times, the question whether the best available estimates for the maximum growth of enstrophy and palinstrophy are sharp is both interesting and relevant. One reason is that such estimates are derived using similar mathematical techniques as in the 3D case where blow-up cannot be ruled out. We will show how new insights regarding these problems can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable adjoint--based gradient descent (or ascent) methods. In particular, we will discuss certain topological features of the families of vorticity fields maximizing the instantaneous rate of growth of palinstrophy in 2D. In offering a systematic approach to finding flow solutions closest to saturating a given analytical bound, the proposed approach provides a bridge between theory and computation. |
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15:35-15:55 | Closing Comments | |||

18:15-19:15 | Dinner at Wolfson Court |