Attractors and frozen-in invariants in turbulent plasma

Work on turbulent equipartitions in a plasma (i. e., attractors characterized by Lagrangian invariants) is reviewed. Although such attractors also exist in the convective zone of the Sun and in atmospheres, the primary emphasis is on turbulent transport in tokamaks. By extending the hydrodynamic concept of freezing-in to Vlasov's equation, it is explained why the magnetic field topology in a collisionless plasma is conserved even though the conventional hydrodynamic description breaks down. Arguments are presented to support the conjecture that the canonical profiles of tokamak plasma are due to an attractor with a plasma frozen into the poloidal magnetic field. In fact, the exclusion from the conventional set of frozen-in integrals of the one for the toroidal magnetic field is all what is needed. The reason for the break-down of this invariant is the poloidal noninvariancy of the magnetic field, an effect to which trapped particles are particularly sensitive. The predictions of the attractor and of two attraction basin boundaries (H- mode and transport suppression by the reversed shear) are confirmed experimentally to a reasonable accuracy.