  In the cuprate superconductors, the spatial coherence of the charge density
wave (CDW) state grows rapidly below a characteristic temperature
$T_\mathrm{CDW}$, the nature of which is debated. We have combined a set of
x-ray scattering techniques to study La$_{1.88}$Sr$_{0.12}$CuO$_{4}$
($T_\mathrm{CDW}$~$\approx$~80\,K) to shed light on this discussion. We observe
the emergence of a crystal structure, which is consistent with the CDW
modulation in symmetry, well above $T_\mathrm{CDW}$. This global structural
change also induces strong fluctuations of local atomic disorder in the
intermediate temperature region. At $T_\mathrm{CDW}$, the temperature
dependence of this structure develops a kink, while the atomic disorder is
minimized. We find that the atomic relaxation dynamics cross over from a
cooperative to an incoherent response at $T_\mathrm{CDW}$. These results reveal
a rich interplay between the CDWs and atomic fluctuations of distinct
spatio-temporal scales. For example, the CDW coherence is enhanced on
quasi-elastic timescales by incoherent atomic relaxation.
