Plan V. Rozhansky, E. Kaveeva St.Petersburg State Polytechnical University, , Polytechnicheskaya 29, St.Petersburg, Russia Poloidal and Toroidal Rotations near Magnetic Islands and Transport Barrier Formation 1)Similarity between ETB and ITB formation. Neoclassical approach and role of anomalous viscosity. 2)Plasma fluxes near the magnetic island. 3)Electric field for stationary and rotating island. 4)Conclusions.

Radial electric field near the edge separatrix Far from the separatrix the radial electric field is close to the neoclassical field. In the separatrix vicinity the radial field differs from neoclassical field due to the anomalous transport processes. Modeling results for ASDEX-Upgrade. Radial electric field at the outer midplane for discharge without NBI. Normal direction of magnetic field, n=2·10 19 m -3, T i =98 eV at the distance 1cm inside the separatrix, I=1 MA, B=2 T.

Different components of parallel momentum balance equation Averaged over the flux surface different components of the parallel momentum balance equation: 1-inertia and perpendicular anomalous viscosity, 2-neoclassical parallel viscosity. No NBI; n=2·10 19 m -3, T i =42 eV at the distance 1cm from the separatrix; I=1 MA, B=2 T.

Modification of electric field profile by the island The radial electric field shear depends on the width of the transition layer:

Model The radial perturbation of the magnetic field : The island is formed near the rational flux surface The island width is larger than poloidal ion gyroradius, and larger than the radial scale. of toroidal velocity variation. (parameter is radial scale of radial electric field variation),

Fluid equations 1) Ion continuity equation where 2) The sum of momentum balance equations where is the classical Braginskii parallel viscosity tensor The anomalous viscosity is taken in the simplest form where D is anomalous diffusion coefficient 3) The energy balance is not significant when

Plasma fluxes inside the island 1)yields: In contrast to the situation in the axisymmetric tokamak, where the coefficient in front of equals unity, inside the island the term is small, since the averaging is performed over both sides of the island. yields: 2) To keep the net current zero, the drift velocity much smaller than the poloidal projection of the parallel velocity is necessary. The characteristic radial scale is

The electrostatic potential is a flux surface function inside the island. Corresponding radial electric field causes the poloidal rotation in the opposite directions at the different sides of the island. The Pfirsch-Schlueter flows and pressure perturbation, both caused by drifts, also have different signs at the different sides of the island. The difference between the current caused by and. flowing in the radial direction into the island and the current flowing out of the island is proportional to the radial electric field and does not contain the small factor. The Pfirsch-Schlueter flows and radial currents corresponding to the average parallel velocity are almost the same at the inner and outer sides of island.

Rotation profiles inside the island Toroidal rotation decreases inside the island with the scale. Radial electric field and poloidal rotation decrease inside the island at the scale. At the different sides of island the signs of the radial electric field are different.

Plasma fluxes outside the island Coordinates where is orthogonal to the perturbed flux surface. 2) Zero current condition where The poloidal rotation changes at the scale from the value at the separatrix to the neoclassical value: 1) Parallel momentum balance

Combination of poloidal rotation profiles inside and outside of the island The neoclassical rotation on both sides of the island is almost the same. The poloidal rotation at the two sides of the island is different. The only symmetric solution is Hence everywhere inside the island

Rotating islands The non-potential electric field is induced in the parallel to direction. The potential field arises to make the total parallel electric field zero: Plasma rotates in the poloidal direction with the velocity of an island: The magnetic field perturbation is not stationary:

The structure of electric field near the rotating island Plasma rotates in the poloidal direction with the velocity of an island: a) New equations for toroidal velocity inside the island: b) The radial electric field decreases at the scale from neoclassical value to the field at the separatrix.

Conclusions The radial electric field and the toroidal rotation profiles are calculated in the presence of the magnetic island. The simplest case of fluid regime is considered. 1) It is demonstrated, that the radial profiles of the toroidal and poloidal rotation in the vicinity of magnetic island are determined by the parameter. depending on the collisionality and on the value of anomalous viscosity coefficient. 2) For the non-rotating island the radial electric field is zero inside the island. Outside the separatrix the radial electric field changes from the zero value at the separatrix towards the neoclassical value with the scale. 3) The surface averaged toroidal rotation decreases inside the island from its value at the separatrix towards zero with a typical scale. The toroidal rotation profile outside the island is roughly the same as without island. 4)The rise of the drift shear in the vicinity of the island may be sufficient to initiate transport barrier formation near the rational flux surface where the magnetic island is formed.