# Graduate School in Nonlinear Science

### Sponsored by the Danish Research Academy

**
MIDIT OFD CATS
Modelling, Nonlinear Dynamics Optics and Fluid Dynamics Chaos and Turbulence Studies
and Irreversible Thermodynamics Risø National Laboratory Niels Bohr Institute and
Technical University of Denmark Building 128 Department of Chemistry
Building 321 P.O. Box 49 University of Copenhagen
DK-2800 Lyngby DK-4000 Roskilde DK-2100 Copenhagen Ø
Denmark Denmark Denmark
**

**TWO LECTURES**

by** Luis Garcia Gonzalo**

Departamento de Fisica

Universidad Carlos III

Madrid, Spain

**Wednesday May 5, 1999, 13.00 h**

and Thursday May 6, 1999 at 10.00

at Meeting Room of the Physics Department Fys-108

Risø National Laboratory;

**Lecture 1: TURBULENCE IN TOROIDAL DEVICES**

**Abstract**: Fusion plasmas are far from thermodynamic equilibrium. Many forms of
random, non-growing oscillations are observed experimentally in a plasma.
The interest in these random fluctuations is that they may drastically
modify the transport properties of the plasma. On the other hand, according
to plasma theory, almost all plasmas are unstable to the growth of a
variety of waves. The main sources of free energy that are available for
driving instabilities are of two types: Gradients of density and
temperature or velocity space anisotropy. Linearly, these instabilities
give rise to exponentially growing modes. The nonlinear interaction of
these modes brings about a turbulent saturated state. The main theoretical
task is to identify for each case the saturation mechanism and to calculate
the steady-state fluctuation levels in order to compare with the
experimental results. The importance of different mode types (interchange,
ballooning, ITGs, etc.) will be discussed. Also, the theory for a
particular type of instability will be developped.

**Lecture 2: ROTATION AND ENHANCED CONFINEMENT IN TOKAMAKS**

**Abstract**: One of the common features that are seen in all devices which achieve
enhanced confinement is the formation of a transport barrier in the plasma.
The results obtained in many magnetic configurations show that the
formation of transport barriers is closely related to increased shear in
ExB velocity. During the last years, several theoretical models have been
developed to explain the transition to enhanced confinement regimes. The
models are based on the generation of averaged flows. Linearly, a shear
flow in a plasma confined by a sheared magnetic field has generally a
strong stabilizing effect. Nonlinearly, the effective turbulence
decorrelation time is reduced, leading to a reduction in fluctuation levels
and induced transport. For ExB shear decorrelation of turbulence, the
shearing rate must be comparable to the nonlinear decorrelation rate in the
absence of ExB shear. There are several possible sources for the generation
of the ExB velocity shear. One of these is the plasma turbulence. The
turbulent fluctuations modify the mean flow and pressure profiles that
simultaneously control the level of fluctuations. Therefore,
self-consistent calculations of plasma turbulence in the presence of flows
are needed. The more complete models describe the simultaneous evolution of
the fluctuation spectrum and the ExB velocity shear.