Local and Nonlocal Advected Invariants and Helicities in Magnetohydrodynamics and Gas Dynamics II: Noether's Theorems and Casimirs

Conservation laws in ideal gas dynamics and magnetohydrodynamics (MHD) associated with fluid relabelling symmetries are derived using Noether's first and second theorems. Lie dragged invariants are discussed in terms of the MHD Casimirs. A nonlocal conservation law for fluid helicity applicable for a non-barotropic fluid involving Clebsch variables is derived using Noether's theorem, in conjunction with a fluid relabelling symmetry and a gauge transformation. A nonlocal cross helicity conservation law involving Clebsch potentials, and the MHD energy conservation law are derived by the same method. An Euler Poincar\'e variational approach is also used to derive conservation laws associated with fluid relabelling symmetries using Noether's second theorem.