# Abstracts

**Dwight Barkley.*** The onset of turbulence in pipe flow*

* Abstract*: More than a century ago Osborne Reynolds launched the quantitative study of turbulent transition as he sought to understand the conditions under which fluid flowing through a pipe would be laminar or turbulent. Since laminar and turbulent flow have vastly different drag laws, this question is as important now as it was in Reynolds' day. Reynolds understood how one should define ''the real critical value'' for the fluid velocity beyond which turbulence can persist indefinitely. He also appreciated the difficulty in obtaining this value. For years this critical Reynolds number, as we now call it, has been the subject of study, controversy, and uncertainty. I will discuss recent developments in experiments, simulations, and modeling that show a deep connection both to statistical phase transitions (directed percolation) and to the dynamics of action potentials in a nerve axons. From these insights, we at last have an accurate estimate of the real critical Reynolds number for the onset of turbulence in pipe flow, and with it, an understanding of the nature of transitional turbulence.

**Freddy Bouchet. ***Stochastic averaging, jet formation and multistability in geostrophic turbulence*

** Abstract: ** J. Laurie, C. Nardini, T. Tangarife, O. Zaboronski have given contributions to one or several of the results discussed during this talk.

We consider the formation of large scale structures (zonal jets and vortices), in geostrophic turbulence forced by random forces, within the barotropic quasi-geostrophic model. We study the limit of a time scale separation between inertial dynamics on the one hand, and the effect of forces and dissipation on the other hand. We prove that stochastic averaging can be performed semi-explicitly in this problem, which is unusual in turbulent systems. It is then possible to integrate out all fast turbulent degrees of freedom, and to get an equation that describes the slow evolution of zonal jets.

The equation for this slow evolution, is a one dimensional stochastic differential equation with multiplicative noise. The average is described by a non-linear Fokker-Planck equation. It describes the attractors for the dynamics (alternating zonal jets, whose number depend on the force spectrum), and the relaxation towards those attractors. We describe regimes where the system has multiple attractors for the same physical parameters.

We discuss possible transitions between attractors with either, three, four or more pairs of zonal jets in models of turbulent atmosphere dynamics. Those transitions are extremely rare, and occur over time scales of centuries or millennia. They are extremely hard to observe in direct numerical simulations, because this would require on one hand an extremely good resolution in order to simulate accurately the turbulence and on the other hand simulations performed over an extremely long time. Their study through numerical computations is inaccessible using conventional means. We present an alternative approach, based on instanton theory and large deviations. We discuss preliminary results on the computation of such instantons in the framework of the 2D Navier-Stokes equations and discuss briefly instantons for atmosphere jet dynamics.

**Colm Connaughton. ***Feedback of zonal flows on wave turbulence driven by small scale instability in the Charney-Hasegawa-Mima model*

* Abstract:* We demonstrate theoretically and numerically the zonal-flow/drift-wave feedback mechanism in an idealised 2--dimensional model of plasma turbulence driven by a small scale instability. Zonal flows are generated by a secondary modulational instability of the modes which are directly driven by the primary instability. The zonal flows then suppress the small scales thereby arresting the energy injection into the system, a process

which can be described using nonlocal (in scale) wave turbulence theory. Finally, the arrest of the energy input results in saturation of the zonal flows at a level which can be estimated from the theory and the system reaches stationarity without large scale dissipation.

**Petr Denissenko. ***TBA*

** Abstract:** TBA

**Bogdan Hnat.**

*Turbulence and Zonal Flows in Magnetically Confined Fusion Plasma*

* Abstract:* Understanding turbulent transport in magnetically confined plasmas is one of the key aspects in developing future fusion reactors. Turbulence mediates the flux of energy from small scale instabilities to large scales flows which in turn influence global confinement properties. A generic class of drift instabilities is believed to provide a driving mechanism for the turbulence at the edge of a confinement device where steep density and temperature gradients are present. These are often referred to as a primary instabilities. Upon achieving certain amplitudes, turbulent fluctuations become unstable and can self-organise into zonal flows via, for example, a local cascade mechanisms or a non-local modulation instability. These zonal flows are linearly stable and thus act as a sink of energy for turbulence.

We discuss recent results regarding zonal flow generation via modulation instability for finite Larmor radius plasma as well as experimental evidence of interaction between turbulence and zonal flows.

**Stefan Grosskinsky. ***Scale invariant growth processes in expanding space*

* Abstract:* Many nonequilibrium growth processes lead to intriguing stochastic patterns and complex fractal structures which exhibit local scale invariance properties. Such structures can often be described effectively by space-time trajectories of interacting particles, and their large scale behaviour depends on the overall growth geometry. We establish an exact relation between statistical properties of structures in uniformly expanding and fixed geometries, which preserves the local scale invariance and is independent of other properties such as the dimensionality. This relation generalizes standard conformal transformations as the natural symmetry of self-affine growth processes. We illustrate our main result numerically for various structures of coalescing Levy flights and fractional Brownian motions, including also branching and finite particle sizes. One of the main benefits of this new approach is a full, explicit description of the asymptotic statistics in expanding domains, which are often non-trivial and random due to amplification of initial fluctuations.

This is joint work with Adnan Ali, Robin Ball and Ellak Somfai.

**Robert Kerr. ***Cascades in Quantum, Classical and Stratified Turbulence*

**Abstract:**** **Numerical results showing the generation of energy cascades to small scales from the interaction and reconnection of very long anti-parallel vortices for three types of flows, superfluids, Navier-Stokes turbulence and stably stratified fluids, that are known to generate an energy cascade to small scales and a -5/3 kinetic energy spectrum.

**Sergey Nazarenko.** *Quadratic invariants for clusters of interacting waves.*

* Abstract.* We consider clusters of interconnected resonant triads arising from the Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a linearly independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix A with entries 1, -1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ≡ N − R ≥ N − M, where R is the number of linearly independent rows in A. We formulate an algorithm for decomposing large clusters of complicated topology into smaller ones and show how various invariants are related to certain parts and linking types of a cluster, including the basic structures leading to R < M. We illustrate our findings by examples taken from the Charney-Hasegawa-Mima wave model.

**Daniel Ueltschi.** *Random loop models and quantum spin systems*

* Abstract:* I will describe random loop models that are related to quantum lattice spin systems

at equilibrium. They yield useful information about quantum correlations. They also

seem to be related to classical interacting particle systems, but this is less clear.