  Protoplanetary disks undergo substantial mass-loss by photoevaporation, a
mechanism which is crucial to their dynamical evolution. However, the processes
regulating the gas energetics have not been well constrained by observations so
far. We aim at studying the processes involved in disk photoevaporation when it
is driven by far-UV photons. We present a unique Herschel survey and new ALMA
observations of four externally-illuminated photoevaporating disks (a.k.a.
proplyds). For the analysis of these data, we developed a 1D model of the
photodissociation region (PDR) of a proplyd, based on the Meudon PDR code and
computed the far infrared line emission. We successfully reproduce most of the
observations and derive key physical parameters, i.e. densities at the disk
surface of about $10^{6}$ cm$^{-3}$ and local gas temperatures of about 1000 K.
Our modelling suggests that all studied disks are found in a transitional
regime resulting from the interplay between several heating and cooling
processes that we identify. These differ from those dominating in classical
PDRs, i.e. grain photo-electric effect and cooling by [OI] and [CII] FIR lines.
This energetic regime is associated to an equilibrium dynamical point of the
photoevaporation flow: the mass-loss rate is self-regulated to set the envelope
column density at a value that maintains the temperature at the disk surface
around 1000 K. From our best-fit models, we estimate mass-loss rates - of the
order of $10^{-7}$ $\mathrm{M}_\odot$/yr - that are in agreement with earlier
spectroscopic observation of ionised gas tracers. This holds only if we assume
an evaporation flow launched from the disk surface at sound speed
(supercritical regime). We have identified the energetic regime regulating
FUV-photoevaporation in proplyds. This regime could be implemented into models
of the dynamical evolution of protoplanetary disks.
