  Recently a detailed correspondence was established between, on one side, four
and five-dimensional large-N supersymmetric gauge theories with $\mathcal{N}=2$
supersymmetry and adjoint matter, and, on the other side, integrable
1+1-dimensional quantum hydrodynamics. Under this correspondence the phenomenon
of dimensional transmutation, familiar in asymptotically free QFTs, gets mapped
to the transition from the elliptic Calogero-Moser many-body system to the
closed Toda chain. In this paper we attempt to formulate the hydrodynamical
counterpart of the dimensional transmutation phenomenon inspired by the
identification of the periodic Intermediate Long Wave (ILW) equation as the
hydrodynamical limit of the elliptic Calogero-Moser/Ruijsenaars-Schneider
system. We also conjecture that the chiral flow in the vortex fluid provides
the proper framework for the microscopic description of such dimensional
transmutation in the 1+1d hydrodynamics. We provide a geometric description of
this phenomenon in terms of the ADHM moduli space.
