Presentation Details 

Aly, JeanJacques 
POSTER: Lower and upper bounds on the energy of a braided forcefree magnetic field 


Amari, Tahar 
POSTER: Reconstructing the solar coronal magnetic field (Joint work with Aly JJ, Canou A, and Mikic Z) 


Bajer, Konrad 
Magnetic relaxation in lowbeta plasma 
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Relaxing magnetic fields tend to form current sheets. In a perfectly conducting medium (obeying the MHD equations) the topology of the field is preserved and tangential discontinuities are expected to be present in the ultimate magnetostatic equilibrium. The system needs inifinite time (and some energy dissipation mechanism) to reach such an equilibrium, but the first, rapid phase of the relaxation process already yields thin layers of strong current that, in a realistic medium of large but finite conductivity would then diffuse. I will discuss both phases of the relaxation process in the medium with negligible fluid pressure showing that the ideal equilibrium indeed contains tangential discontinuities and the dynamics of the diffusive phase is quite different from the incompressible analogue. 
Berger, Mitchell 
Selforganized braiding of magnetic fields 
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The Parker model for heating of the solar corona involves reconnection of braided magnetic flux elements. Much of this braiding is thought to occur at as yet unresolved scales, for example braiding of threads within an EUV or xray loop. However, some braiding may be still visible at scales accessible to TRACE or the EIS imager on Hinode. We suggest that attempts to estimate the amount of braiding at these scales must take into account the degree of coherence of the braid structure.
A randomly generated braid structure may selforganize into a more coherent structure if reconnection occurs preferentially at certain critical places. We demonstrate that simple models of braided magnetic fields which balance input of topological structure with reconnection can evolve to a selforganized critical state. An initially random braid can become highly ordered, with coherence lengths obeying power law distributions. The energy released during reconnection also obeys a power law. Our model gives more frequent (but smaller) energy releases nearer to the ends of a coronal loop. 
Birn, Joachim 
Energy release and transfer by magnetic reconnection in current sheets 
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The sudden release and transfer of energy from magnetic fields to particles by magnetic reconnection is the most important and intriguing part of dynamic processes such as solar flares and magnetospheric substorms. We review basic theoretical results as well as results from MHD and particle simulations on the partitioning of energy in large quasiopen systems as well as closed or periodic systems with and without guide field. The results stress three relevant points:
1. The importance of compressional heating (enthalpy flux) in addition to the generation of fast flows,
2. The importance of (approximate) conservation of flux tube content (mass, entropy, foot point displacement) in limiting energy release, and
3. The importance of transfer of energy in addition to release and conversion. 
Buniy, Roman 
CONTRIBUTED TALK: Covariant helicity and Beltrami fields 
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We propose the covariant generalization of the magnetic helicity, Beltrami equation and WoljterTaylor states. Covariant gauge invariance, boundary conditions, conserved currents and variational principle elucidate the subject. In particular, we prove that any extremal of the QED action $tfrac{1}{4}int_Omega F_{munu}F^{munu},d^4x$ subject to the constraint $epsilon^{munualphabeta}F_{munu}F_{alphabeta}=0$ satisfies the covariant Beltrami equation. 
Campbell, Jack 
CONTRIBUTED TALK: A geometrical understanding of helicity in a spherical geometry 
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Magnetic helicity can be understood geometrically in terms of winding numbers. Existing methods for open curves will be reviewed. Many practical applications  for instance, in the solar corona or when studying DNA  involve structures with spherical boundaries. We extend the concept of a geometrically defined helicity to spherical space. (Joint work with Mitchell Berger). 
Candelaresi, Simon 
CONTRIBUTED TALK: Topological restrictions in magnetic field dynamics and reconnection 
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The dominance of the magnetic helicity as the determining topological invariant in magnetohydrodynamics is revisited for helical and nonhelical field configurations. For fields with positive zcomponents we determine the fixed points of the field line mapping. The sum of the Poincare indices at those fixed points, the fixed pints index, gives rise to further restrictions, not captured by the magnetic helicity alone. Various braided field configurations, which correspond to helical and nonhelical knots, are simulated in fully resistive magnetohydrodynamics. We observe an evolution which confirms the importance of the fixed point index as additional restricting quantity. Moreover, a topological flux function, the field line helicity, is computed at the fixed points and monitored during turbulent relaxation.
Its rate of change gives a proxy for the magnetic reconnection rate. 
Cantarella, Jason 
Helcicity and configuration spaces 
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In this talk, we present a reformulation of the helicity integral using configuration spaces. The essential idea is that the compactified configuration space of a pair of points in R^3 is a manifold with boundary which contains a ''new'' 2dimensional homology class. We can rewrite the helicity integral as a nonsingular integral over this manifold with boundary, rather than using the usual formulation: an integral over pairs $(x,y) in mathbb{R}^3 times mathbb{R}^3$ where the integrand appears to have a singularity on the diagonal pairs in the form $(x,x)$.
The advantage of this point of view is that the helicity of a vector field is revealed to be the deRham cohomology class represented by a corresponding 2form. We can use this insight to construct some new formulae for computing helicity and to prove a theorem about how helicity can change under a diffeomorphism of the domain. The talk is joint work with Jason Parsley (Wake Forest University). 
Cheung, Mark 
Datadriven modelling of magnetic field evolution in the solar corona 
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We present a method for performing datadriven simulations of solar active region formation and evolution. The approach is based on magnetofriction, which evolves the induction equation assuming the plasma velocity is proportional to the Lorentz force. The simulations of active region coronal field are driven by temporal sequences of photospheric magnetograms from the Helioseismic Magnetic Imager (HMI) instrument onboard the Solar Dynamics Observatory (SDO). Under certain conditions, the datadriven simulations produce flux ropes that are ejected from the modeled active region due to loss of equilibrium. Topological tools such as relative magnetic helicity and footpoint connectivity maps are used to study the topological changes experienced by the model active region.
We also present a method for the synthesis of mock coronal images based on a proxy emissivity calculated from the current density distribution in the model. This method yields mock coronal images that are somewhat reminiscent of images of active regions taken by instruments such as SDO's Atmospheric Imaging Assembly (AIA) at extreme ultraviolet wavelengths. 
Cowley, Steve 
Stochastic magnetic fields in fusion 
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The anomalous loss of heat from fusion devices is often attributed to destruction of magnetic surfaces and stochastic magnetic field. I will discuss the evidence for the stochastic magnetic field and the associated transport of heat. The mechanisms for the underlying turbulence and the breaking the magnetic fields will also be discussed. 
Daughton, William 
Tangled magnetic fields in collisionless reconnecting plasmas 
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Recent advances in computing are enabling fully kinetic simulations that span a broad range of scales, from large fluid motions down to electron kinetic scales. These studies are leading to new insights into the development of turbulence driven by magnetic reconnection and largescale flow shears. One of our key findings is that the turbulence is often dominated by coherent structures in the form of kinetic scale current sheets, which are unstable to the formation of magnetic flux ropes. The resulting magnetic field structure quickly becomes quite complicated which may affect the acceleration and transport of energetic particles. Here we show preliminary results from various techniques to quantify the magnetic topology and to explore associations between the coherent structures and the acceleration and transport of particles. 
Davies, Christina 
CONTRIBUTED TALK: Short wavelength magnetic buoyancy instability 
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We consider the magnetic buoyancy instability in the shortwavelength limit of Gilman (1970). In this limit the perturbation equations (a system of coupled ODEs) can be reduced to a single algebraic dispersion relation, with coefficients depending on height. Put otherwise it seems that, in this limit, a problem that would have been treated as an eigenvalue problem requiring a set of boundary conditions can be reduced to a single equation for which the boundary conditions are unimportant. Here I present asymptotics and numerical calculations to illustrate the link between the two systems, which can be viewed as being analogous to the more familiar problem of the quantum harmonic oscillator. 
Dritschel, David 
2D MHD turbulence at zero magnetic Prandtl number 
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While the solar interior is characterised by simultaneously high fluid and magnetic Reynolds numbers, the ratio of these numbers is very large, particularly in the solar tachocline. The inverse of this ratio is called the magnetic Prandtl number, Pm. Here we show results using a new computational method capable of efficiently studying these flows in the zero Pm limit, and examine the breaking of vorticity conservation by the Lorentz force. 
Figura, Przemysław 
POSTER: Application of Euler potentials to magnetic reconnection 


Gekelman, Walter 
Flux ropes, quasi seperatrix layers and 3D reconnection 
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Magnetic flux ropes form a dense carpet of arches on the surface of the sun. Ropes are not static, they exert mutual JXB forces causing them to twist about each other and merge. Kink instabilities cause them to violently smash into each other and reconnect at the point of contact. We report on experiments done in the large plasma device (LAPD) at UCLA (L = 17m, dia = 60 cm, 0.3<B0z<2.5 kG, n=3X1012 cm3) on three dimensional flux ropes. Two, three or more magnetic flux ropes are generated from initially adjacent pulsed current channels in a background magnetized plasma. The currents and magnetic fields form exotic shapes with no ignorable direction and no magnetic nulls. Volumetric spacetime data show multiple reconnection sites with timedependent locations. Quasiseparatrix layers (QSL), a tool to understand 3D magnetic field line reconnection without null points are introduced. Large data sets (B, n, and flow velocity, at 49,000 spatial locations, 7000 time steps) are used to derive 3D fields, currents, plasma flow and QSL’s. In our experiment the QSL is a narrow ribbonlike region(s) that twist between field lines. When the field lines are tracked they are observed to slip along the QSL during reconnection. 
Gibson, Sarah 
Magnetism and the invisible man: the mysteries of coronal cavities 
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Magnetism defines the complex and dynamic solar corona. Twists and tangles in coronal magnetic fields build up energy and ultimately erupt, hurling plasma into interplanetary space. These coronal mass ejections (CMEs) are transient riders on the everoutflowing solar wind, which itself possesses a threedimensional morphology shaped by the global coronal magnetic field. Coronal magnetism is thus at the heart of any understanding of the origins of space weather at the Earth. However, we have historically been limited by the difficulty of directly measuring the magnetic fields of the corona, and have turned to observations of coronal plasma to trace out magnetic structure. This approach is complicated by the fact that plasma temperatures and densities vary among coronal magnetic structures, so that looking at any one wavelength of light only shows part of the picture. In fact, in some regimes it is the lack of plasma that is a significant indicator of the magnetic field. Such a case is the coronal cavity: a dark, elliptical region in which strong and twisted magnetism dwells. I will elucidate these enigmatic features using the results of an international collaboration to observe coronal cavities in multiple wavelengths and from a variety of observing vantages, and show how magnetic flux rope models provide a selfconsistent picture of the cavity, its substructure, and its dynamic evolution as a CME. Moreover, I will make use of unprecedented measurements of coronal magnetism, now being obtained by the Coronal Multichannel Polarimeter (CoMP), to demonstrate the presence of twisted magnetic fields within cavities. 
Hesse, Michael 
The role of geometry in magnetic reconnection 
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Magnetic reconnection is arguably the most effective energy conversion and transport process in plasmas. Reconnection is subject to topological considerations in two ways. First, the process itself involves a change in topology of the combined plasmamagnetic field system. This change in topology transcends that of the magnetic field alone and accounts for flux transport relative to the motion of the plasma in the system under investigation. The second way topology is important to magnetic reconnection is through modifications of the diffusion/dissipation physics brought about by the structure of the reconnecting system. This presentation will present an overview and summary of both past and recent results pertaining to both aspects.
Michael Hesse (1), Nicolas Aunai (1), Joachim Birn (2), and Seiji Zenitani (3)
(1) Heliophysics Science Division, NASA Goddard Space Flight Center, Greenbelt, MD, USA
(2) Los Alamos National Laboratory, Los Alamos, NM, USA
(3) National Astronomical Observatory of Japan, Tokyo, Japan 
Hornig, Gunnar 
Relaxation of braided magnetic fields 
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We discuss the relaxation of magnetic fields in astrophysical plasmas of high magnetic Reynolds numbers. The relaxation process in such plasmas is often turbulent in nature and differs from the relaxation in technical plasmas in that the domain is typically nonperiodic. We present a generalised flux function which can be defined for magnetic braids on such domains (Yeates and Hornig, Physics of Plasmas 18, 2011). This flux function illustrates qualitatively and quantitatively the relaxation process and can be used to predict possible relaxed states. The theoretical considerations are backed up by number of numerical simulations which support our findings. 
Kagan, Daniel 
Three dimensional simulations of relativistic magnetic reconnection in pair plasmas 
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We investigate magnetic reconnection and particle acceleration in relativistic pair plasmas with threedimensional particleincell (PIC) simulations of a kineticscale current sheet in a periodic geometry at low magnetizations. The tearing instability is the dominant mode in the current sheet for all guide field strengths, while the linear kink mode is less important even without guide field. In its nonlinear evolution, the reconnection layer develops into threedimensional, disordered network of interconnected and interacting magnetic flux ropes. These flux ropes contain embedded magnetic substructure on plasma skin depth scales and spatially and temporally intermittent sites of dissipation. Magnetic dissipation and particle acceleration persist until the end of the simulations, with simulations with higher magnetization and lower guide field strength exhibiting greater and faster energy conversion and particle energization. At the end of our largest simulation, the particle energy spectrum attains a tail extending to high Lorentz factors that is best modeled with a combination of two additional thermal components. The primary energization mechanism is acceleration by the electric field in the Xline region. We discuss the implications of our results for macroscopic reconnection sites, and which of our results may be expected to hold in systems with higher magnetizations. 
Kephart, Thomas 
Generalized AharonovBohm effect 
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Gaussian linking of a semiclassical path of a charged particle with a magnetic flux tube is responsible for the AharonovBohm effect, where one observes interference proportional to the magnitude of the enclosed flux. We construct quantum mechanical wave functions where semiclassical paths can have second order linking to two magnetic flux tubes, and show there is interference proportional to the product of the two fluxes. 
Kida, Shigeo 
A turbulent ring and MHD dynamo in a precessing sphere 
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Recently we observed, by direct numerical simulation of MHD flow, a ring structure of highactivity both in vorticity and magnetic flux density in a precessing sphere. This ring is localized near a great circle slightly inclined from the equator. Both the magnetic and vorticity fields are activated in the vicinity of the cross sections of the ring with the equatorial plane, and their fluctuations make a prograde motion. The intensification mechanism of the magnetic and vorticity fields will be discussed in the symposium. 
Kimura, Yoshifumi 
Magnetic field generated by current filaments 
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There is a big similarity between magnetic field and vorticity. One such example is the magnetic field generated by a current density and the velocity filed generated by a vortex. In this talk we will discuss the topology of the magnetic field generated by current filaments. Our recent numerical results have demonstrated that even non parallel two straight vortex filaments in 3D space can generate a velocity field with rather convoluted structures. This suggests that the magnetic field generated by two straight current filaments may have similar rich structures. 
Kusner, Rob 
CONTRIBUTED TALK: Chirality for curves (and more?) 
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Distinguishing left and righthanded screws may be easy, but the chirality of arbitrary objects in threespace is more subtle. We'll introduce a simple matrixvalued chirality measure for space curves and discuss how configurations of quadruples of points in threespace could lead to some generalized notions of chirality.
[This is part of a project initiated with Keith Moffatt, being explored also with Giovanni Dietler, Eric Rawdon, Piotr Szymczak and possibly others at the INI this fall.] 
Kwiatkowski, Damian 
POSTER: Incoherent alphashear dynamo with colored noise 


Lathrop, Daniel 
Learning about magnetic fields and vortices from experiments in metallic sodium and liquid Helium 
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The geodynamo and solar dynamo occur at parameter values that cannot be captured entirely with theory, simulations or experiments. Experiments can offer realistic effects of rotation and turbulence on MHD and fluid flows. Our group has been exploring turbulent rotating flows in liquid sodium, liquid Nitrogen, superfluid helium, or water. After reviewing the summary observations, I'll argue three main points: (1) Systems with a very small Ekman number (very rapid rotation) are highly oscillatory. (2) The high magnetic Reynolds number states (state where resistivity is negligible) are determined by a competition of magnetic field generation and reconnection. (3) The magneto‐rotational instability and shear and instabilities are not exclusive. 
Liu, Xin 
POSTER: Tackling fluid structures complexity by the Jones polynomial 


Longcope, Dana 
A topological bound on the energy stored in the tangled magnetic field of a coronal active region 
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In a simplified model of solar coronal magnetic fields, each field line is anchored at both ends to a rigid conducting boundary: the photosphere. A nonlinear forcefree equilibrium is that magnetic field which minimizes the total magnetic energy subject to constraints from linetying all footpoints. The fact that photospheric field appears concentrated in relatively isolated patches suggests a simplification whereby only the net flux linking concentrations is constrained. Subjecting the minimization to this subset of constraints results in a forcefree field with less magnetic energy than the fullyconstrained field. It also provides an estimate of the magnetic energy which could be released if topological changes could be made by some localized process. The coronal magnetic field will naturally build up magnetic energy as a result of changes occurring in the photospheric concentrations to which it is anchored. The simplified constraints have the practical advantage of being derivable from observations. The method has been used, in numerous cases, to place bounds on the energy stored up, over the course of several days, prior to a solar flare. 
Malapaka, Shiva Kumar 
CONTRIBUTED TALK: Inverse cascade of magnetic helicity 
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Inverse cascade of magnetic helicity is studied in high resolution (1024^3) direct numerical simulations of isotropic, homogeneous, incompressible 3DMHD turbulence, with two setups driven and decaying. It is shown that for these two systems the spectral power in the inertial range is not k^(2) but is 3.3 for the driven and 3.6 for the decaying case respecively. These new power laws are understood using dimensional analysis of the magnetic helicity equation from the MHD equations under the EDQNM approximation. From this analysis we obtain a relation which shows that spectral behavior of magnetic helicity would depend on a) squared ratio of the energies(magnetic to kinetic) b) inverse square of the wave number and c) the kinetic helicity. This relation and the lack of uniformity in the power law value for magnetic helicity in decaying and driven cases point to 1) probable lack of universality in the power of magnetic helicity, which is an ideal invariant in MHD and 2) that in order to understand a phenomenon in MHD turbulence, one could explore the interrelationship(s) among different physical quantities of the flow. 
Mazur, Grzegorz 
POSTER: Outflows from GRMHD simulations of accretion disks 


McCormick, Julie 
POSTER: The structure and collapse of magnetic separators 
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Magnetic reconnection plays a key role in many plasma processes on the Sun and in the magnetosphere. Reconnection in three dimensions is known to be profoundly different from that in two dimensions. One of the main places where 3D reconnection can occur is in the vicinity of separators, special field lines that link pairs of null points and lie along the intersection of four topologically distinct domains. Although separators are key locations for reconnection, little is known about their local magnetic field structure, but such knowledge is essential to understand fully the nature of separator reconnection.
A general expression for the local magnetic field about a straight potential separator of unit length will be presented. The magnetic separator configurations that this generates will be presented, explaining the relation of the local and global field structures within them. We will also discuss the effects of currents on the separator structures and their stability. 
Moffatt, Keith 
Relaxation to magnetostatic equilibria of complex topology: a review 
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Magnetic field relaxation in a fluid medium, assumed perfectly conducting, is constrained by the topology of the field, leading to stable magnetostatic equilibria compatible with this topology, in which the magnetic energy is minimal with respect to isomagnetic perturbations. In the context of ideal MHD, a similar relaxation process that respects invariance of both magnetic helicity and cross helicity can be similarly constructed, leading to steady solutions of the ideal MHD equations of complex topology. These processes will be reviewed in the light of recent results of Enciso & PeraltaSalas concerning the existence of Beltrami fields of knotted topology. 
Monastyrsky, Michael 
CONTRIBUTED TALK: Classification of phases and linked defects in superfluids and neutron stars 
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We consider the topological structures in condensed matter, mainly textures and line defects(vortices). The main examples are superfluid He3 and neutron stars. We introduce high order topological invariants and give some applications in the problem of reconnection of magnetic spread lines in magnetohydrodynamics. 
Oberti, Chiara 
POSTER: Winding number effects on the induction of a field in the shape of a torus knot: preliminary results 
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Preliminary results on the effects of the winding number on the induction due to a stationary field in the shape of a torus knot are presented and discussed for the first time. The induction is computed by solving the BiotSavart integral on torus knots and comparative results for different winding numbers are shown. Work to extend this study further is in progress, including possible applications to solar coronal magnetic fields, electric fields and astrophysical flows. (Coauthor Renzo L Ricca) 
Panagiotou, Eleni 
POSTER: A study of the linking number in systems employing periodic boundary conditions 


Parnell, Clare 
The topological evolution of complex 3D magnetic fields 
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In order to understand the structure and evolution of magnetic fields it is useful to identify topological structures such as magnetic null points and their associated lines/spines and separatrix surfaces, as well as special field lines, called separators, that lie along the intersection of two separatrix surfaces and thus, lie at the edge of four topologically distinct regions. By means of various numerical experiments, which model key phenomena on the sun and in the magnetosphere, I will explain the main mechanisms by which globally complex magnetic fields may evolve. In particular, the topological features and associated magnetic reconnection processes that are central to both the tangling and the untangling of the magnetic fields found in the numerical experiments will be explained. 
Parsley, Jason 
The geometry of the Taylor problem 
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Plasma injected into a toroidal container loses energy rapidly until it reaches a quasistable state while its helicity (an average linking number of its field lines) remains essentially constant. J.B. Taylor showed that by also fixing the flux of the field  assumed divergencefree and tangent to the boundary  through a crosssectional disk, the resulting minimal energy field well approximates experimental results. We consider the problem of Taylor on arbitrary subdomains in R^3. We show a solution always exists and investigate the role of geometry on the problem. 
Platten, Sarah 
POSTER: Global 3D coronal magnetic skeleton configurations through the course of a solar cycle 
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The Sun's photospheric magnetic field is constantly evolving and exhibits significant changes in behaviour throughout a solar cycle. The magnetic field structure in the corona above follows suit and shows striking differences in configuration as the cycle progresses. Specific field structures are required to provide the right conditions for phenonena driven by reconnection to occur, such as CMEs and solar flares, and also provide the appropriate environment to accelerate the solar wind. The magnetic skeleton enables us to detect the key topological structures of the magnetic field and is a clear and robust way of analysing a magnetic field's structure. Using a potential magnetic field construction of the global solar corona, from KittPeak Magnetogram data (van Ballegooijen et al, 1998), we look at the changes in the magnetic field structure of the corona over 27 years through a study of their magnetic skeletons. This poster will look at the 3D magnetic field structure configurations we have found using this model, as well as analysing trends we have observed through the solar cycle. We will then consider which topological structures found in extrapolated coronal fields are important for specific phenomena found observed in the corona. 
Plihon, Nicolas 
vonKàrmàn sodium dynamos: dynamics, flow topology and boundary conditions 
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The features of the magnetic field generated by dynamo instability in the fully turbulent liquid sodium vonKàrmàn flow first will be introduced. The flow is generated by the counterrotation of impellers in a cylinder and displays dynamo action above a threshold. When the flow is driven asymmetrically, dynamical regimes similar to those encountered in astrophysical situations have been observed and characterized. The role of boundary conditions and flow topology on the dynamo instability and the dynamics of the magnetic field will be discussed. 
Pontin, David 
CONTRIBUTED TALK: Current singularities at 3D magnetic null points and the implications for the solar corona 
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We present evidence that singularities of the electric current naturally occur during the ideal relaxation of a magnetic field containing threedimensional null points. In a real plasma with finite dissipation this implies that such nulls are preferential sites for the formation of intense but finite current layers, and thus magnetic reconnection. The implications for relaxation processes in the solar corona will be discussed. 
Proctor, Michael 
Meanfield electrodynamics at moderate and large magnetic Reynolds numbers 
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Meanfield electrodynamics has been a popular method of representing the effects of smallscale turbulent flow on magnetic fields so as to produce evolution equations for the averaged magnetic field. The method works well when the magnetic Reynolds number of the flow is small. But when this is not the case problems may arise. These are of two types; firstly the mean electromotive force can be a small fraction of that expected on order of magnitude grounds due to cancellation between induced flux elements at different locations. Secondly if the magnetic Reynolds number is sufficiently large, the mean field dynamo can interact with a smallscale dynamo, driven principally by stretching, and so cannot be described by the induction equation alone. I will discuss both these problems with reference to recent work. 
Rachmeler, Laurel 
CONTRIBUTED TALK: What can we learn about the magnetic field from coronal linear polarization measurements? 
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We still have no reliable way to directly measure the coronal magnetic field. The linear and circular polarization of certain coronal emission lines has been known for decades to contain information about the magnetic field, but this information is not widely used. The Coronal Multichannel Polarimeter (CoMP) has been taking daily measurements of linear polarization in the Fe XIII 10747 Å line since October 2011. I will describe the information contained in this line and how it can be used to understand the coronal field. Although it is not a full measurement of the 3D field, it still provides previously unavailable information and can shed light on many unanswered coronal physics questions. 
Ricca, Renzo L 
Topological bounds on the energy and complexity of magnetic fields 
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New results on topological bounds on the energy and structural complexity of magnetic fields are presented. By using classical results by Arnold, Moffatt and Freedman & He, we show how the topology of a magnetic system of knots and links in ideal magnetohydrodynamics (MHD) provides lower bounds to the magnetic energy minima. In presence of dissipation new lower bounds on structural complexity of magnetic fields, given by the average crossing number and its change in time, are provided by the magnetic helicity of the system. These results may find useful applications in the study of dynamical systems and in astrophysical flows, particularly as regards coronal structures on the Sun, and in the diagnostics of direct numerical simulation of MHD flows. 
Riols, Antoine 
CONTRIBUTED TALK: Of tangles and magnetic fields: global bifurcations to subcritical magnetorotational dynamo action 
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Instabilitydriven dynamos are very interesting candidates to explain sustained magnetic activity and turbulence in a variety of differentially rotating astrophysical bodies such as stellar interiors or accretion disks. Our understanding of the physical mechanisms and conditions of existence of these unusual subcritical, nonkinematic dynamos remains very crude though. To make progress on these questions, we have been investigating the nature of the transition to longlived dynamics in the prototypical problem of magnetorotational dynamo action in keplerian shear flow. In this talk, we present several new numerical results that reveal the fascinating complexity of this dynamo transition with unprecedented detail. First, we show that the system starts to exhibit features typical of a chaotic repeller such as fractal patterns in maps of turbulence lifetime as the magnetic Reynolds number is increased. We then attempt to identify what causes this transition by studying in detail the structure of the dynamics in the neighbourhood of the many nonlinear MRI dynamo cycles that we identify in the transitional regime. We show that the emergence of dynamo cycles through saddlenode bifurcations is quickly followed by a sequence of global heteroclinic and homoclinic bifurcations resulting in the creation of Poincare tangles linking these cycles and the emergence of chaos. The importance of individual dynamo cycles in the transition process opens new theoretical perspectives to understand the conditions of emergence of the MRI dynamo in Nature, such as its dependence on the magnetic Prandtl number. 
Schekochihin, Alexander 
Plasmoid reconnection and saturated state of the turbulent dynamo 
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In the last few years, we have learned that in most interesting situations (and certainly in resistive MHD), magnetic reconnection occurs via unstable current sheets that turn into
stochastic plasmoid chains. It appears also that magnetic folds generated by the turbulent dynamo turn, upon dynamo saturation, into such unstable current sheets and so the statistically steady nonlinear state of the dynamo is dominated by these structures. I will outline the mechanism of the formation of stochastic plasmoid chains and show
how they appear (or appear to appear) in the numerical simulations of turbulent dynamo.
Joint work with A. Schekochihin (Oxford), N. Loureiro (Lisbon), D. Uzdensky (UC Boulder), R. Samtaney (KAUST), A. Beresnyak (Los Alamos), A. Iskakov (Moscow), S. Cowley (Culham)
References:
Loureiro et al., Phys. Plasmas 19, 042303 (2012)
Uzdensky, Loureiro & Schekochihin, Phys. Rev. Lett. 105, 235002 (2010)
Samtaney et al., Phys. Rev. Lett. 103, 105004 (2009)
Loureiro, Schekochihin & Cowley, Phys. Plasmas 14, 100703 (2007)
Schekochihin et al., Astrophys. J. 612, 276 (2004) 
Shonkwiler, Clayton 
CONTRIBUTED TALK: Generalized Gauss maps and triple linking integrals 
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I will describe a new approach  not based on Massey products  to defining an integral for Milnor's triple linking number. This comes from a generalized Gauss map which can be associated to any threecomponent link, the homotopy invariants of which are the link homotopy invariants of the link. An adaptation of Whitehead's integral formula for the Hopf invariant yields a geometrically natural integral formula for the triple linking number. The hope is that this approach will point the way towards definitions of higher order helicities which can provide lower bounds on the field energy even when the usual helicity vanishes. 
Spineanu, Florin 
The ideal versus the reality: topology and turbulence in the current density in tokamak 
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Models for the current density profile in magnetically confined plasma have been proposed starting from first principles, basically in analogy with the selforganization of the vorticity in the twodimensional Euler fluid. The principle of maximum entropy has led to the derivation of the Liouville equation, under some criticism regarding the possibility of a statistical approach. Without claiming that this will solve the problem we show that the coupled evolution of the vorticity and the current density in twodimensions has strange analogy with the problem of axial anomaly and baryon asymmetry in the Standard Model. In the latter the topological aspects are more pronounced and we can benefit of suggestions for the 2D MHD problem. We identify a class of asymptotic stationary states for the coupled fields of currentdensity and vorticity, using the Backlund transform. For this class of asymptotic states the maxima of the vorticity and current density must coincide. For tokamak this is important: the Hmode layer is simultaneously a vorticity and a current density sheet. This is unstable to a modulation due to an instability called "anomalous polytropic" and it can be shown that it leads to breakup and filamentation, offering a possible explanation to an observed dangerous edge mode. Due to its relevance for the current density distribution in tokamak (and more general problems) we will examine the motion of a current filament on a gradient of current density. 
Stremler, Mark 
POSTER: Topological chaos in the braiding of almost cyclic sets 


Sumners, De Witt 
CONTRIBUTED TALK: Flux tube reconnection and helicity conservation 
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Reconnection is an important event in many areas of science, including sitespecific DNA recombination and the reconnection of field lines in astro and plasma physics. Using the theorem of Moffatt and Ricca (Proc. Roy. Soc. 1992), the helicity of a magnetic flux tube can be calculated in terms of the writhe of the centerline and the twist of the ribbon determined by the centerline and one of the other field lines in the flux tube tube. This talk will present the proof of the following: Theorem: Given a reconnection event between a pair of magnetic flux tubes of identical flux, suppose that the twist of the reconnected tube is the sum of the twists of the individual tubes. Then helicity is conserved by the reconnection event — the helicity of the reconnected flux tube is the sum of the helicities of the individual tubes. Any deviation from helicity conservation is entirely due to twist inserted locally at the reconnection site (the writhe component of helicity is always conserved). This is joint with Christian Laing and Renzo Ricca. 
Thiffeault, JeanLuc 
The structure of random braids 
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Braid descriptions have been used by many to understand how magnetic field lines are tangled, in an attempt to understand phenomena such as magnetic reconnection. These are useful both in solar applications, where the lines form tubes, or in tokamaks, where they form tori. I will discuss the application of tools from lowdimensional topology, the ThurstonNielsen theory, to determining the degree of entanglement of field lines, as well as the existence of invariant regions. Both of these aspects benefit from recent improvements in the mathematical tools available, in particular the use of equivalence classes of loops to probe the topological structure. 
Thornton, Andrew 
CONTRIBUTED TALK: The effect of tangled magnetic fields on instabilities in tokamak plasmas 
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The high pressure gradients in the edge of a tokamak plasma can lead to the formation of explosive plasma instabilities known as edge localised modes (ELMs). The control of ELMs is an important requirement for the next generation of fusion devices such as ITER. Experiments performed on the Mega Amp Spherical Tokamak (MAST) at Culham have shown that the application of nonaxisymmetric magnetic perturbations can be used to mitigate ELMs. During the application of the RMPs, clear structures are observed in visiblelight imaging of the Xpoint region. These lobes, or tangles, have been observed for the first time and their appearance is correlated with the mitigation of ELMs. Tangle formation is seen to be associated with the RMPs penetrating the plasma and may be important in explaining why the ELM frequency increases during ELM mitigation. Whilst the number and location of the tangles can be explained by vacuum magnetic field modelling, obtaining the correct radial extent of the tangles requires the plasma response to be taken into account. 
Title, Alan 
Solar collective behaviour 
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The Atmospheric Imaging Assembly on the Solar Dynamics Observatory provides 24/7 full Sun coverage with a 12 second cadence with images that span the temperature range from 6000 to 20,000,000 K with arc second resolution. The Heliospheric and Magnetic imager provides doppler data, lineofsight magnetograms every 45 seconds, and vector magnetograms every 5 minutes. With this data set and observations with the pair of STEREO satellites it has become apparent that many flares, filament eruptions, and CME’s have causal connections. These connections often span a hemisphere or more. Numerical simulations indicates there is at least one mechanism for triggering of remote events. Maps of the magnetic topology constructed from the LOS field and a PFSS model often indicates both how the regions that are connected and their boundaries. Movies of everts and numerical simulations will be presented as well as topological mappings that indicated the zones of connectivity. 
Tobias, Steven 
Measurement of the turbulent diffusivity of a magnetic field: the Turbulent Angstrom Method and the Method of Oscillatory Sines 
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We argue that a method developed by Angstrom (1861) to measure the thermal conductivity of solids can be adapted to determine the effective diffusivity of a largescale magnetic field in a turbulent electrically conducting fluid. The method consists of applying an oscillatory source and measuring the steadystate response. We illustrate
this method in a twodimensional system. This geometry is chosen because it is possible to compare the results with independent methods that are restricted to twodimensional flows. We describe two variants of this method; one (the "Turbulent Angstrom Method") that is better suited to laboratory experiments the second (the "Method of Oscillatory Sines'') that is effective for numerical experiments. We show that, if correctly implemented, all methods agree. Based on these results we argue that these
methods can be extended to threedimensional numerical simulations and laboratory experiments. 
Tronko, Natalia 
CONTRIBUTED TALK: Magnetorotational instability and generalised energy principle 
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The Magnetorotational Instability (MRI) is believed to play a crucial role in transferring angular momentum, particularly in the context of accretion discs where it is important for allowing star formation.Because of this crucial role, previous work has paid special attention to the study of nonlinear mechanisms for MRI saturation. Therefore, conditions for marginal stability of general equilibria are important to investigate.Here we do so by using the noncanonical Hamiltonian approach [1], since it provides variational principles for equilibria that can be used to assess stability. In this work we show that a reduced system of equations derived in [2] is an infinitedimensional noncanonical Hamiltonian system. The noncanonical Poisson bracket is identified and shown to obey the Jacobi identity, at the same time the families of Casimir invariants are also obtained.
Explicit sufficient conditions for the energy stability of two classes of equilibria are identified by means of the EnergyCasimir method. The comparison between the stability conditions obtained in the two cases indicate that the presence of an equilibirum magnetic field along the direction of the ignorable coordinate does not introduce destabilizing effects. An analogy is found between terms in the expression of the second variation of the free energy and terms appearing in the energy principle stability analysis of compressible reduced magnetohydrodynamics for tokamaks. 
Van Compernolle, Bart 
CONTRIBUTED TALK: Morphology and dynamics of kinkunstable flux ropes 
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The dynamics of rotating kinkunstable flux ropes is studied in a laboratory magnetoplasma (L = 16 m, Wpe/Wce ~ 20, helium fill gas). The flux ropes are created by fieldaligned current channels emitted from a lanthanum hexaboride cathode and collected by a remote anode 11 m away. The ropes are linetied at the cathode but are nonline tied at the anode. The poloidal field of the ropes is a few percent of the background field and the current density is high enough to make the ropes kink unstable. A three dimensional data set of three interacting flux ropes was acquired. The helical field line structure of the ropes and their kinking is visualized. The ropes are observed to rotate along the guide field with increasing radius of rotation away from the cathode. The three dimensional data set of the three flux ropes is supplemented by parameter studies of the dynamics of a single flux rope. Temperature, density, field aligned current density and rotation properties of the flux rope are measured as a function of guide field and background plasma parameters. 
Vlad, Madalina 
CONTRIBUTED TALK: Magnetic field line diffusion and helicity in stochastic magnetic fields 
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Magnetic field line diffusion in 2dimensional stochastic magnetic fields is studied using a semianalytical statistical approach, the decorrelation trajectory method (DTM). This method [1], developed for fusion plasma studies, is able to account for the nonlinear process of formation of helical segments on the magnetic field lines (stochastic solenoids). The aim of this study is to determine the statistical characteristics of the solenoids and their effects on magnetic line diffusion. We determine the solenoid average number and size as functions of the parameters of the stochastic magnetic field. 1. M. Vlad, F. Spineanu, J.H. Misguich, R. Balescu, Physical Review E 67, 02640612, 2003. 
Watson, Stephen 
Evolving nanofaceted crystals: from emergent symmetries, via the principle of maximal dissipation a la Onsager, to universal morphometrics 
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We theoretically uncover an emergent spatiotemporal scaling symmetry of the edgenetwork statistics governing thermokinetically nanofaceting interfaces. A novel asymptotic expansion of a global reformulation of the dynamics (), which is obtained via the Principle of Maximal Dissipation (a la Onsager), provides the key analytical insight here; this novel analytical approach applies more generally to any singularly perturbed purely dissipative evolution (gradient flow). We thereby characterise the emergent nonlocal facet dynamics and associated symmetry group uderpinning our theory. Our scaling theory is then validated through direct simulations using a novel computational geometry tool, which is able to simulate the evolution and coarsening dynamics of millionfacet ensembles. We also report on some of the resulting universal morphometrics (shape statistics). 
WilmotSmith, Antonia 
Heating of braided coronal loops 
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Parker's notion of topological dissipation views current sheet formation as an inevitable consequence of field line braiding via photospheric footpoint motions. We discuss a model that develops this idea. We argue that braiding leads to the development of nonsingular current layers, with the threshold for the development of instabilities being rather low (occurring at just a few percent free magnetic energy) when motions with high topological entropy are taken into account. We consider the viability of braiding as a mechanism for coronal heating, showing how the resultant heating depends on the nature of the photospheric driver as well as the coronal resistivity. 
Yahalom, Asher 
CONTRIBUTED TALK: Variational principles of MHD with non trivial topology 
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Variational principles for magnetohydrodynamics were introduced by previous authors both in Lagrangian and Eulerian form. Yahalom & LyndenBell [1] have introduced a simpler Eulerian variational principles from which all the relevant equations of magnetohydrodynamics can be derived. The variational principle is given in terms of six independent functions for nonstationary flows and three independent functions for stationary flows. This is less then the seven variables which appear in the standard equations of magnetohydrodynamics. Later Yahalom [2] has shown that four functions will suffice for nonstationary flows.
Some concerns have been raised by the community regarding the validity of the method suggested for the case that the magnetic field lines have a nontrivial topology, in particular for the case that the field lines are knotted. In such a case the flow has a topological constants of motion denoted cross and magnetic helicity. I will show that there is no place for concern and that the variational approach is valid for this case as well, provided that some of the functions used are non single valued. Moreover, the law of cross helicity conservation can be obtained through the Noether theorem using a simple symmetry group of one of our variables. 
Yeates, Anthony 
CONTRIBUTED TALK: Unique topological characterisation of magnetic braids 
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We introduce a topological flux function to quantify the topology of magnetic braids, which are nonzero linetied magnetic fields whose field lines all connect between two boundaries. This scalar function is an ideal invariant defined on a crosssection of the magnetic field, and measures the average poloidal magnetic flux around any given field line. Moreover, its integral over the crosssection yields the relative magnetic helicity. Using the fact that the flux function is also an action in the Hamiltonian formulation of the field line equations, one can prove that it uniquely characterises the field line mapping and hence the magnetic topology. (Joint work with Gunnar Hornig.) 
Yokoi, Nobumitsu 
Cross helicity effect in dynamo and reconnection 
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Cross helicity, velocitymagneticfield correlation, is expected to play a role in turbulent transport. In the presence of the largescale vortical motion, the turbulent crosshelicity (density), <u'.b'> (u': fluctuation velocity, b': fluctuation magnetic field), induces a component of the turbulent electromotive force aligned with the largescale vorticity. Since largescale rotational motion is ubiquitous in astrophysical and plasma phenomena, it is suggested that the cross helicity plays some important role in dynamos. As an example, an application to the effect to the solar dynamo is presented with special emphasis on the crosshelicity evolution. Another important feature of the crosshelicity effect is seen in the turbulent momentum transport. Unlike the usual alpha or helicity effect, the meanfield configuration induced by the cross helicity in general leads to a nonvanishing meanfield Lorentz force. With fully utilizing this feature, the turbulent magnetic reconnection is considered. It is shown that the cross helicity plays very important role in confining the reconnection region in a tiny region, which is favourable for the fast reconnection. 
Yokoyama, Tomoo 
CONTRIBUTED TALK: Word representation of streamline topologies for structurally stable vortex flows in multiply connected domains 
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The instantaneous flow field of the inviscid and incompressible fluids in twodimensional multiply connected exterior domains is described by a Hamiltonian vector field whose Hamiltonian becomes the stream function, which is the imaginary part of the complex potential. The flow is characterized topologically by the streamline pattern, which consists of the contour lines of the stream function. The present research provides us with a new topological classification procedure of the structurally stable streamline patterns generated by many vortex points in the presence of the uniform flow. The procedure allows us to assign a word representation to each structurally stable Hamiltonian vector field in the multiply connected domains. In the present talk, I will explain how to assign the word to the structurally stable streamline patterns and show all the streamline patterns in multiply connected domains of low genus with their word representations. 
Zharkov, Sergei 
POSTER: Geometry of acoustic wavefronts generated by a point source in the Sun: effects of cutoff frequency and caustics 

