A set of numerical experiments on the periodic 3D Navier-Stokes equations has suggested that their solutions operate in a specific regime where the nonlinearity is depleted (Donzis et al 2013). The variables used are a scaled, dimensionless hierarchy of      -norms of vorticity (              ) labelled as       ,  with       proportional to the global enstrophy.  These  experiments all operate  in  the  regime
where                for a variety of initial conditions and Reynolds numbers. Mathematically two more regimes are identified, the second being                                  and the third                      ,  for
with         a constant.  In regime I there exists an absorbing ball for       when

: a good fit to the numerics is the average of these bounds. It is also shown analytically that this regime has a corresponding energy spectrum                   where                       . The estimates for the radius of the ball for       (      ), the Lyapunov dimension of the global attractor (        ), the steepness of the energy spectrum (            ) and the resolution length (           ) are uniform in     when this “average fit'' is used, suggesting a degree of universality. In regime II only weak solutions are known with              ,

whereas in regime III all the      are bounded. The talk ends with a discussion on the possibility of a phase transition between the three regimes.

Leray scaling in Navier-Stokes reconnection

Robert Kerr, Mathematics Institute, University of Warwick

Anti-parallel vortex reconnection simulations are compared to show a finite exchange of circulation in a finite time as        , along with growth of the the vorticity and the relevant velocity second-derivative consistent with the inverse powers of   predicted by singular Leray scaling. Even though none of these calculations are ever singular. The growth could be singular only in the dual limits of zero viscosity and an infinite domain. The infinite domain condition is needed because, for this initial condition, which has been shown to be regular for all times for        (Euler),  there exists a very small viscosity    below which the singular  Leray scaling as         cannot continue. A constraint that suppresses the Leray growth in these calculations and is relaxed as the domain is made bigger. What happens as the Leray scaling is relaxed might be the next step in the formation of a finite energy dissipation anomaly. That is, finite energy dissipation in a finite time as the viscosity goes to zero.

Magnetic relaxation, current sheets, and structure formation in an extremely tenuous

fluid medium

H. Keith Moffatt, DAMTP, University of Cambridge

The process of relaxation of a unidirectional magnetic field in a highly conducting tenuous fluid medium is considered. Null points of the field play a critical role in this process. During an initial stage of relaxation, variations in magnetic pressure are eliminated, and current sheets build up in the immediate neighborhood of the null points. This initial phase is followed by a long diffusive phase of slow algebraic decay of the field, during which fluid is continuously sucked into the current sheets, leading to exponential growth of fluid density and concentration of mass around the null points, which show a tendency to cluster. Ultimately, this second phase of algebraic decay gives way to a final period of exponential decay of the field. The peaks of density at the null points survive as a fossil relic of the decay process. Numerical solution of the governing equations provides convincing confirmation of this three-stage scenario. Generalizations to two- and three-dimensional fields are briefly considered. (Joint work with Konrad Bajer, University of Warsaw.)

Regimes of nonlinear depletion and regularity in the 3D Navier-Stokes equations

John D. Gibbon, Department of Mathematics, Imperial College London

Contact organisers: Simon Malham & Michal Branicki

Vortex Flow Simulations by Lagrangian Particle Methods

Robert Krasny, Department of Mathematics, University of Michigan

In this talk I'll discuss how Lagrangian particle methods are being used to investigate vortex-dominated flows. First I'll review vortex sheet computations in 2D, with emphasis on Kelvin-Helmholtz instability, the Moore singularity, spiral roll-up, and the onset of chaotic dynamics. Then I'll present vortex ring simulations in 3D with attention to the dynamics of material lines and surfaces. I'll conclude with recent simulations of vortex dynamics on a rotating sphere. Throughout these studies we face the problem of resolving nontrivial geometry and topology.

Taylor Relaxation of Braided Magnetic Fields

Gunnar Hornig, Department of Mathematics, University of Dundee

We present results on the relaxation of braided magnetic fields in plasmas of high magnetic Reynolds number. The relaxation process in such plasmas is often turbulent in nature and hence it was suggested that the relaxed state should be a Taylor state, that is a linear force-free magnetic field with the same total helicity as the initial field. We present results which show that this is not generally the case, both for line tied flux tubes like solar loops, as well as for periodic domains resembling a tokamak.  A generalised flux function can be defined for magnetic braids on such domains (Yeates and Hornig, Physics of Plasmas 18, 2011), which helps to understand the deviations from the Taylor hypotheses.  This flux function  illustrates qualitatively and quantitatively the relaxation process and can be used to predict possible relaxed states.