  In previous work we used magnetic SU(N) gauge theory with adjoint
representation Higgs scalars to describe the long distance quark-antiquark
interaction in pure Yang-Mills theory, and later to obtain an effective string
theory. The empirically determined parameters of the non-Abelian effective
theory yielded $Z_N$ flux tubes resembling those of the Abelian Higgs model
with Landau-Ginzburg parameter equal to $ 1/\sqrt{2}$, corresponding to a
superconductor on the border between type I and type II. However, the physical
significance of the differences between the Abelian and the $Z_N$ vortices was
not elucidated and no principle was found to fix the value of the
'Landau-Ginzburg parameter' $\kappa$ of the non-Abelian theory determining the
structure of the $Z_N$ vortices. Here we reexamine this point of view. We
propose a consistency condition on $Z_N$ vortices underlying a confining
string. This fixes the value of $\kappa$. The transverse distribution of
pressure $p(r)$ in the resulting $Z_N$ flux tubes provides a physical picture
of these vortices which differs essentially from that of the vortices of the
Abelian Higgs model. We speculate that this general picture is valid
independent of the details of the effective magnetic gauge theory from which it
was obtained. Long wavelength fluctuations of the axis of the $Z_N$ vortices
lead from an effective field theory to an effective string theory with the
Nambu-Goto action. This effective string theory depends on a single parameter,
the string tension $\sigma$. In contrast, the effective field theory has a
second parameter, the intrinsic width 1/M of the flux tube.
