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It is shown that the universal steady Euler flow field, independent ofboundary shape or symmetry, in a toroidal domain with fixed boundary obeys anonlinear Beltrami equation, with the nonlinearity arising from aBoltzmann-like, velocity-dependent factor. Moreover, this is a relaxed velocityfield, in the sense that it extremizes the total kinetic energy in the domainunder free variations of the velocity field, constrained only by tangentialvelocity and vorticity boundary conditions and conservation of total fluidhelicity and entropy. This is analogous to Woltjer-Taylor relaxation of plasmamagnetic field to a stationary state. However, unlike the magnetic field case,attempting to derive slow, quasi-relaxed dynamics from Hamilton's actionprinciple, with constant total fluid helicity as a constraint, fails to agree,in the static limit, with the nonlinear Beltrami solution of the Eulerequations. Nevertheless, an action principle that gives a quasi-relaxeddynamics that does agree can be formulated, by introducing a potentialrepresentation of the velocity field and defining an analogue of the magnetichelicity as a new constraint. A Hamiltonian form of quasi-relaxed fluiddynamics is also given.

Relaxation of Compressible Euler Flow in a Toroidal Domain