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.
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