Stability Properties and Nonlinear Mappings of Two and Three‐Layer Stratified Flows

Abstract

Two and three-layer models of stratified flows in hydrostatic balance are studied. For the former, nonlinear transformations are found that map [baroclinic] two-layer flows with either rigid top and bottom lids or vertical periodicity, into [barotropic] single-layer, shallow water free-surface flows. We have previously shown that two-layer flows with Richardson number greater than one are nonlinearly stable, in the following sense: when the system is well-posed at a given time, it remains well-posed through the nonlinear evolution. Here, we give a general necessary condition for the nonlinear stability of systems of mixed type. For three-layer flows with vertical periodicity, the domains of local stability are determined and the system is shown not to satisfy the necessary condition for nonlinear stability. This means that there are wave-motions that evolve into shear unstable flows.

Dispersive dam-break and lock-exchange flows in a two-layer fluid

“…(b) A remarkable result has been noted by Chumakova et al (2009). The canonical form (2.17) appears to be exactly identical to that of the single-layer SWE.…”

“…However, the relationship cannot be as straightforward as a one-to-one mapping, as suggested by Chumakova et al (2009). Physically, the reason for this is that there is no single-layer counterpart to a 'lock-exchange' flow in a two-layer fluid, in which the initial interface profile is a step spanning the centre of the fluid domain.…”

“…Flows which are everywhere in a single state are analogous to flows in the single-layer system. For these flows, a one-toone mapping to the single-layer system can be used following the arguments of Chumakova et al (2009). Flows which switch between states, however, are unique to the two-layer system.…”

Simple waves do not avoid eigenvalue crossings

“…One can translate these thoughts into a local characterization of stability [1,2,7], describing the system as stable (well-posed) where it is hyperbolic, and unstable (ill-posed) where it is elliptic (i.e., some eigenvalues become complex). Since all the points on the boundary between the hyperbolic and elliptic domains (later referred to as the HE boundary) have repeated eigenvalues, studying such points helps us understand the system's stability properties.…”

Nonlinear stability in three-layer channel flows

“…develop complex conjugate eigenvalues). This phenomenon has been encountered in different fluid applications - (Bürger et al 2002;Jackson & Blunt 2002;Talon et al 2004;Chumakova et al 2009;Boonkasame & Milewski 2012) -and has raised concern regarding the physical relevance of the mathematical models. Due to the absence of dissipative mechanisms in systems of conservation laws, the existence of ellipticity introduces a catastrophic instability in the sense of Hadamard - (Joseph & Saut 1990).…”