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