Symmetries and Casimir invariants for perfect fluid
Abstract
We investigate the relations between symmetries of an adiabatic inviscid fluid and Casimir invariants of Eulerian Poisson bracket. It is shown that there exist two types of inner symmetries in addition to external symmetries. Noether's conserved quantities due to these inner symmetries are Casimir invariants when they are written in terms of Eulerian fields only. We construct the most general form for the two types of such quantities that correspond to these symmetries. However, these are shown to be equivalent. The most general Casimir of the system is shown to be the wellknown Casimir functional. For a zero potential vorticity fluid, helicity is also allowed for the independent Casimir.
 Publication:

Fluid Dynamics Research
 Pub Date:
 March 1990
 DOI:
 10.1016/01695983(90)90023R
 Bibcode:
 1990FlDyR...5..273K
 Keywords:

 Ideal Fluids;
 Inviscid Flow;
 Symmetry;
 Adiabatic Conditions;
 Boundary Conditions;
 Conservation Laws;
 Entropy;
 Lagrange Multipliers;
 Fluid Mechanics and Heat Transfer