Nambu-bracket for three-dimensional ideal fluid dynamics and magnetohydrodynamics

The ideal magnetohydrodynamics (MHD) as well as the ideal fluid dynamics is governed by the Hamilton equation with respect to the Lie-Poisson bracket. The Nambu bracket manifestly represents the Lie-Poisson structure in terms of derivative of the Casimir invariants. We construct a compact Nambu-bracket representation for the three-dimensional ideal MHD equations, with use of three Casimirs for the second Hamiltonians, the total entropy and the magnetic and cross helicities, whose coefficients are all constant. The Lie-Poisson bracket induced by this Nambu bracket does not coincide with the original one, but supplemented by terms with an auxiliary variable. The supplemented Lie-Poisson bracket qualifies the cross-helicity as the Casimir. By appealing to Noether’s theorem, we show that the cross-helicity is the integral invariant associated with the particle-relabeling symmetry. Employing the Lagrange label function, as the independent variable in the variational framework, facilitates implementation of the relabeling transformation. By incorporating the divergence symmetry, other known topological invariants are put on the same ground of Noether’s theorem.