  In this paper we discuss conservation laws in ideal magnetohydrodynamics
(MHD) and gas dynamics associated with advected invariants. The invariants in
some cases, can be related to fluid relabelling symmetries associated with the
Lagrangian map. There are different classes of invariants that are advected or
Lie dragged with the flow. Simple examples are the advection of the entropy S
(a 0-form), and the conservation of magnetic flux (an invariant 2-form advected
with the flow). The magnetic flux conservation law is equivalent to Faraday's
equation. We discuss the gauge condition required for the magnetic helicity to
be advected with the flow. The conditions for the cross helicity to be an
invariant are discussed. We discuss the different variants of helicity in fluid
dynamics and in MHD, including: fluid helicity, cross helicity and magnetic
helicity. The fluid helicity conservation law and the cross helcity
conservation law in MHD are derived for the case of a barotropic gas. If the
magnetic field lies in the constant entropy surface, then the gas pressure can
depend on both the entropy and the density. In these cases the conservation
laws are local conservation laws. We obtain nonlocal conservation laws for
fluid helicity and cross helicity for non-barotropic fluids using the Clebsch
variable formulation of gas dynamics and MHD. Ertel's theorem and potential
vorticity, the Hollman invariant, and the Godbillon Vey invariant for special
flows for which the magnetic helicity is zero are also discussed.
