A MHD invariant with effects on the confinement regimes in Tokamak

Fundamental Lagrangian, frozen-in and topological invariants can be useful to explain systematic connections between plasma parameters. At high plasma temperature the dissipation is small and the robust invariances are manifested. We invoke a frozen-in invariant which is an extension of the Ertel's theorem and connects the vorticity of the large scale motions with the profile of the safety factor and of particle density. Assuming ergodicity of the small scale turbulence we consider the approximative preservation of the invariant for changes of the vorticity in an annular region of finite radial extension (i.e. poloidal rotation). We find that the ionization-induced rotation triggered by a pellet requires a reversed-$q$ profile. In the $H$-mode, the invariance requires a accumulation of the current density in the rotation layer. Then this becomes a vorticity-current sheet which may explain experimental observations related to the penetration of the Resonant Magnetic Perturbation and the filamentation during the Edge Localized Modes.