Abstract
A highly idealized model for the oceanic haline circulation is studied. Specifically, loops filled with salty water and subjected to either the natural boundary condition, the virtual salt flux condition, or salinity relaxation are considered. It is shown that the characteristics of the solutions, especially the transition between steady and unsteady convection, depend critically on the applied boundary conditions. It is found that the relaxation condition generally modifies the location of the Hopf bifurcation so highly that models based on it should always remain in the regime of steady convection. On the other hand, the location of the Hopf bifurcation for models based on flux conditions is much less extreme. Thus, in these models, limit cycles or chaotic behavior can easily be excited.
Further, the nature of the Hopf bifurcation depends sensitively on the boundary condition. For example, if the frictional parameter is gradually reduced, the model based on the natural boundary condition goes through a supercritical Hopf bifurcation, while the model based on virtual salt flux goes through a subcritical Hopf bifurcation. Similar dependencies are found when other parameters are varied. Beyond the Hopf bifurcations, windows of limit cycle solutions alternate with windows of chaos. In addition, for a given set of parameters, the system can have multiple solutions, such as a limit cycle and a chaotic solution, or limit cycles which have distinctively different structure.
These results comment on the types of behavior that more complicated three-dimensional models may exhibit.
