  We have used resonant x-ray scattering to determine the momentum dependent
charge correlations in YBa$_2$Cu$_3$O$_{6.55}$ samples with highly ordered
chain arrays of oxygen acceptors (ortho-II structure). The results reveal
nearly critical, biaxial charge density wave (CDW) correlations at in-plane
wave vectors (0.315, 0) and (0, 0.325). The corresponding scattering intensity
exhibits a strong uniaxial anisotropy. The CDW amplitude and correlation length
are enhanced as superconductivity is weakened by an external magnetic field.
Analogous experiments were carried out on a YBa$_2$Cu$_3$O$_{6.6}$ crystal with
a dilute concentration of spinless (Zn) impurities, which had earlier been
shown to nucleate incommensurate magnetic order. Compared to pristine crystals
with the same doping level, the CDW amplitude and correlation length were found
to be strongly reduced. These results indicate a three-phase competition
between spin-modulated, charge-modulated, and superconducting states in
underdoped YBa$_2$Cu$_3$O$_{6+\delta}$.
