# Tutorials

The following are the tutorial or review talks:

## Sadruddin BENKADDA

Patterns, intermittent transport and universality in convective turbulence in magnetic fusion plasmas

Buoyancy-driven flows such as thermal convection are of great importance for a wide range of phenomena in geophysical, astrophysical and fusion plasmas [1,2,3]. We consider here intermittent aspects of convective turbulence and transport in magnetized plasma of magnetic fusion machines such as tokamaks or stelerators. These investigations use Direct Numerical Simulation (DNS) of Ion Temperature Gradient instability (ITG) which is identical to the Rayleigh-Bénard thermal convection problem in neutral fluids [4,5,6]. Rayleigh-Bénard convection in particular is a fundamental paradigm for nonlinear dynamics including instabilities and bifurcations, pattern formation, chaotic dynamics and developed turbulence.

Using a weakly non-linear analysis, we show that the back-reaction on the OAmean profile is the natural mechanism for saturation and suggest that it will stay the main non-linear coupling mechanism in the turbulent state. We also will review some basic aspects of the interaction between convective cells and a mean flow [7,8]. In particular and still along the line of the "defreezing" assumption we study the behavior of a model for shear flow instability : transient bursts of vorticity flux are generated in this model. We briefly discuss the advantage of this kind of formulation compared to the "non-normal" operator approach where the mean velocity is also assumed frozen.

An extension of Herring model is derived. It takes into account the self-consistent generation of a mean flow. It is shown that our model has substantially richer dynamics than the one of Herring. In particular the interaction between the convective modes and the mean flow leads in the turbulent state to a transition in the statistical properties of the transport. This bifurcation is analogous the so-called soft to hard turbulence transition in convection. In the strongly turbulent state, intermittent bursts of thermal transport are observed in both cases. For the latter regime, the reduced model as well as DNS show that the Nusselt number Nu (normalized heat flux) scales with the normalized ion pressure gradient Ki as Nu ~ Ki^{1/3}[6]. Since the Rayleigh number for ITG turbulence is proportional to Ki, the Nusselt number scaling for ITG turbulence is thus similar to the classical Globe & Dropkin scaling for Rayleigh-Bénard convection in neutral fluids.

[1] S. Benkadda, P. Beyer, N. Bian et al, Nuclear Fusion 41, 997 (2001)

[2] P. Beyer, S. Benkadda, X. Garbet and P.H. Diamond, Phys. Rev Letters 85, 4892 (2000).

[3] Amita Das, Abhijit Sen, and Predhiman Kaw, S. Benkadda and Peter Beyer, Physics of Plasmas 12, 032302 (2005).

[4] K. Takeda, S. Benkadda, S. Hamaguchi and M. Wakatani, Physics of Plasmas 11, 3561-3571 (2004).

[5] K. Takeda, S. Benkadda, S. Hamaguchi and M. Wakatani, J. Plasma Fusion Res. 6 570 (2004).

[6] K. Takeda, S. Benkadda, S. Hamaguchi and M. Wakatani, Physics of Plasmas 12, 052309 (2005).

[7] N. Bian, S. Benkadda, X. Garbet, O. Garcia and J. Paulsen, Physics of Plasmas, 10, 1382, (2003).

[8] O. Garcia, N. Bian, J. Paulsen , S. Benkadda and K. Rypdal, Plasma Physics and Controlled Fusion 45, 919-932, (2003).

## Jay FINEBERG

A review of fast fracture: Where are we now and where are we going?

The physics of crack propagation are critically linked to our fundamental understanding of material strength and stability. Fracture is a process in which a putatively singular stress field, formed at the tip of a crack, preferentially breaks the bonds ahead of the crack's tip. Once the fracture process begins, cracks in brittle materials will rapidly accelerate to velocities on the order of material sound speeds. Thus the fracture process is characterized by a singular stress field that is propagating at nearly the speed of information within a given material. We present a brief review of the field, highlighting both the central open questions and a number of new experimental and theoretical approaches to their solution.

## Peter JUNG

The role of spatial organization for biologic excitable systems

Ion channels in membranes of biological cells are often organized in clusters. The role of clustering for function, however, is poorly understood. I will focus on a specific cell-signaling mechanism, i.e. intracellular calcium signaling, to elucidate the possible roles of ion channel clustering for signaling-function based on mathematical and computational modeling. While calcium dynamics belongs into the realm of excitable systems, the clustering of the key-signaling effectors generates novel dynamic behavior associated with the spatial inhomogeneities, fluctuations due to the small size of the clusters and discreteness effects. For example, clustered signaling arrangement allows for global cellular oscillations for the same (physiologically relevant) parameters homogeneous systems do not. Such effects may be important for the biologic cell where calcium signals with different spatiotemporal shape in general signal a different function.

## Arkady PIKOVSKY

Synchronization of coupled oscillators: from Huygens clocks to chaotic systems and large ensembles

I introduce basic features of the synchronization phenomenon, starting with classical examples and an elementary theory for coupled periodic oscillators. Then an extension to chaotic systems is discussed. In large ensembles the synchronization appears as a nonequilibrium phase transition. Together with classical results of Kuramoto I discuss specific features appearing due to nonlinear coupling.