  We analyzed 200 ks of Chandra ACIS observations of the merging galaxy cluster
A2142 to examine its prominent cold fronts in detail. We find that the southern
cold front exhibits well-developed Kelvin-Helmholtz (KH) eddies seen in the sky
plane. Comparing their wavelength and amplitude with those in hydrodynamic
simulations of cold fronts in viscous gas, and estimating the gas tangential
velocity from centripetal acceleration, we constrain the effective viscosity to
be at most 1/5 of Spitzer isotropic viscosity, but consistent with full
Braginskii anisotropic viscosity for magnetized plasma. While the northwestern
front does not show obvious eddies, its shape and the structure of its
brightness profile suggest KH eddies seen in projection. The southern cold
front continues in a spiral to the center of the cluster, ending with another
cold front only 12 kpc from the gas density peak. The cool peak itself is
displaced ~30 kpc from the BCG (the biggest such offset among centrally-peaked
clusters), while the X-ray emission on a larger scale is still centered on the
BCG, indicating that the BCG is at the center of the gravitational potential
and the cool gas is sloshing in it. The specific entropy index of the gas in
the peak ($K\approx49$ keV cm$^2$) makes A2142 a rare "warm core"; apparently
the large displacement of the cool peak by sloshing is the reason. Finally, we
find a subtle narrow, straight channel with a 10% drop in X-ray brightness,
aligned with the southern cold front --- possibly a plasma depletion layer in
projection.
