Ideal shocks in 2-layer flow - Part II: Under a passive layer

In this second part of the study, ideal shock theory in two-layer stratified flow is extended to include a third passive layer (i.e., a two and a half layer system). With the presence of a passive layer, two linear wave modes and “viscous tail modes” exist, complicating the solubility conditions and uniqueness proofs for two layer shocks. It is found however, that shocks can be unambiguously classified as external or internal based on the states of criticality that they connect. The steepening condition, while still necessary, provides a less restrictive constraint than it did with a rigid lid. Thus, we have to rely more on solutions to the full viscous shock equations to establish shock existence. The detailed structure, momentum exchange, and Bernoulli loss in a viscous shock are examined using an analytical weak shock solution and a set of numerical solutions for shocks with finite amplitudes. A shock regime diagram (F₁ by F₂) is constructed based on the numerical integration of the full viscous shock equations. For strong external jumps, a cusp region (i.e., in the sense of catastrophe theory) is identified on the regime diagram. For pre-shock states within the cusp, three end states are possible and two of these are realizable. The cusp has several physical implications. It indicates that an equal distribution of dissipation between the two layers in shocks is mathematically possible but physically inaccessible. It also allows hysteresis in time varying flows, and promotes the occurrence of double shocks (i.e., closely spaced shocks of different character). The results are compared with classical shock solutions and a set of time dependent numerical experiments.