Baroclinicity is proportional to the cross product of the gradients of pressure and density . Both gradients are perpendicular to their respective level sets, meaning surfaces of constant density and pressure. In a baroclinic fluid these are parallel, so the cross product vanishes. A baroclinic motion is a motion caused by the misallignment of these two surfaces, so this term is nonzero and effects the dynamics of the vorticity equation.
An example from the baroclinity entry of Wikipedia: "Divers may be familiar with the very slow waves that can be excited at a thermocline or a halocline; these are internal waves. Similar waves can be generated between a layer of water and a layer of oil. When the interface between these two surfaces is not horizontal and the system is close to hydrostatic equilibrium, the gradient of the pressure is vertical but the gradient of the density is not. Therefore the baroclinic vector is nonzero, and the sense of the baroclinic vector is to create vorticity to make the interface level out. In the process, the interface overshoots, and the result is an oscillation which is an internal gravity wave. Unlike surface gravity waves, internal gravity waves do not require a sharp interface. For example, in bodies of water, a gradual gradient in temperature or salinity is sufficient to support internal gravity waves driven by the baroclinic vector."