  (Abridged) Through spectrally unresolved observations of high-J CO
transitions, Herschel-PACS has revealed large reservoirs of warm (300 K) and
hot (700 K) molecular gas around low-mass protostars. We aim to shed light on
the excitation and origin of the CO ladder observed toward protostars, and on
the water abundance in different physical components using spectrally resolved
Herschel-HIFI data. Observations are presented of the highly excited CO line
J=16-15 with Herschel-HIFI toward 24 low-mass protostellar objects. The
spectrally resolved profiles show two distinct velocity components: a broad
component with an average FWHM of 20 km/s, and a narrower component with a FWHM
of 5 km/s that is often offset from the source velocity. The average rotational
temperature over the entire profile, as measured from comparison between CO
J=16-15 and 10-9 emission, is ~300 K. A radiative-transfer analysis shows that
the average H2O/CO column-density ratio is ~0.02, suggesting a total H2O
abundance of ~2x10^-6. Two distinct velocity profiles observed in the HIFI line
profiles suggest that the CO ladder observed with PACS consists of two
excitation components. The warm component (300 K) is associated with the broad
HIFI component, and the hot component (700 K) is associated with the offset
HIFI component. The former originates in either outflow cavity shocks or the
disk wind, and the latter in irradiated shocks. The ubiquity of the warm and
hot CO components suggests that fundamental mechanisms govern the excitation of
these components; we hypothesize that the warm component arises when H2 stops
being the dominant coolant. In this scenario, the hot component arises in
cooling molecular H2-poor gas just prior to the onset of H2 formation. High
spectral resolution observations of highly excited CO transitions uniquely shed
light on the origin of warm and hot gas in low-mass protostellar objects.
