The interaction between a stable zonal jet and a vortex is studied numerically with two one-and-a-half layer models, one with quasi-geostrophic dynamics, the other with shallow-water equations. In both models, simulations on the f-plane evidence three regimes occuring with increasing vortex strength: (regime 1) weak vortices do not cross the jet and steadily drift along it; (regime 2) stronger vortices cross the jet, tear an opposite-sign meander from the jet with which they pair as a dipole; the trajectory of this dipole depends on the strength of the initial vortex; since most dipoles are asymmetric, they veer back towards the jet axis where they are split apart in the ambient shear; (regime 3) even stronger vortices cross the jet and tear a vorticity filament without dipole formation. The influence of various physical parameters on jet-vortex interaction is studied. In particular, beta-effect is not sufficient to drive all vortices through the jet. Numerical simulations show that jet crossing occurs when the maximum velocity of the vortex is larger than, and opposite to, that of the jet. This allows the mathematical derivation of an analytical criterion for jet crossing in both models, which relates the potential vorticity anomalies of the jet and vortex, the vortex and internal deformation radii. In the shallow-water model, an asymmetry is observed between anticyclones north of the jet and cyclones south of it. The role of a spatially varying deformation radius and of vortex cyclostrophy on this asymmetry is discussed.