  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 an off-axis region of the core. In
the $H$-mode, the invariance requires the accumulation of the current density
in the rotation layer at the edge. 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.

Plasma Physics