  We present a general approach to the modelling of the brightness and
polarization structures of adiabatic, decelerating relativistic jets, based on
the formalism of Matthews & Scheuer (1990). We compare the predictions of
adiabatic jet models with deep, high-resolution observations of the radio jets
in the FR I radio galaxy 3C 31. Adiabatic models require coupling between the
variations of velocity, magnetic field and particle density. They are therefore
more tightly constrained than the models previously presented for 3C 31 by
Laing & Bridle (2002). We show that adiabatic models provide a poorer
description of the data in two crucial respects: they cannot reproduce the
observed magnetic-field structures in detail, and they also predict too steep a
brightness decline along the jets for plausible variations of the jet velocity.
We find that the innermost regions of the jets show the strongest evidence for
non-adiabatic behaviour, and that the adiabatic models provide progressively
better descriptions of the jet emission at larger distances from the galactic
nucleus. We briefly discuss physical processes which might contribute to this
non-adiabatic behaviour. In particular, we develop a parameterized description
of distributed particle injection, which we fit to the observed total
intensities. We show that particles are preferentially injected where bright
X-ray emission is observed, and where we infer that the jets are
over-pressured.
