Compressible Helical Beltrami Flow

Consider the case of a compressible fluid undergoing helical (Beltrami) flow with nonuniform vorticity and velocity. This is a generalization of the ABC (Arnold–Beltrami–Childress) flow for the compressible case [1] in which chaotic dynamics appear [2] that involve a "Q-flow" (flows with quasisymmetry).

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The Q-flows have a Hamiltonian ; then
,
,
,
where is a perturbation parameter (when the advection equations are integrable) and
with and .
Now, introducing a soliton-like function related to the Beltrami flow condition for a scalar function (condition for compressibility), in this case of the form
where the variable is used instead of and are related according to , is a constant, and is a scale parameter along . Consider a generating function of the type
;
then, rewriting the equations, we have
,
,
.
References
[1] A. Morgulis, V. I. Yudovich, and G. M. Zaslavsky, "Compressible Helical Flows," Communications on Pure and Applied Mathematics, 48(5), 2006 pp. 571–582. doi:10.1002/cpa.3160480505
[2] G. M. Zaslavsky, Hamiltonian Chaos and Fractional Dynamics, New York: Oxford University Press, 2005.
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