Remember, isobars are like isotherms on an isentropic surface, and the previous description is essentially warm air and cold air advection respectively. This plays a more minor role when dealing with synoptic scale systems and outside of convection. With an approaching cold air dome and appreciable wind, one would expect substantial downward vertical motion. As a cold air dome begins to build and move toward a particular point, pressure rises at the surface correlate to strong upstream cold air advection. This cold air dome building signifies that both the slope and the height of isentropic surfaces are increasing with time.
Isentropic surfaces rise to higher levels and create a steep slope.
Recall that changes in M with time are related to changes in temperature and height of an isentropic surface with time. These changes in the relative position of isentropic surfaces produce an isallobaric wind which would accelerate air parcels down the surface in this situation. With this transport of air from a high level to a low level, it is apparent that a transfer of higher momentum air would occur and be brought down toward the surface. The first three points of caution are diabatic factors which violate the isentropic assumptions. This result is that an air parcel will move from one isentropic surface to another.