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Stability Properties and Nonlinear Mappings of Two and Three‐Layer Stratified Flows
Author(s) -
Chumakova L.,
Menzaque F. E.,
Milewski P. A.,
Rosales R. R.,
Tabak E. G.,
Turner C. V.
Publication year - 2009
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2008.00426.x
Subject(s) - baroclinity , barotropic fluid , nonlinear system , hydrostatic equilibrium , stratified flow , stratified flows , stability (learning theory) , mathematics , mechanics , flow (mathematics) , layer (electronics) , mathematical analysis , geology , geometry , physics , turbulence , materials science , computer science , quantum mechanics , machine learning , composite material
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.