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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


Exact Solutions and Stability Analysis of a Nonlinear Model of Open-Ocean Deep Convection that Allows Multiple Steady States

Discontinuity, Nonlinearity, and Complexity 8(2) (2019) 169--186 | DOI:10.5890/DNC.2019.06.005

Igor L. Bashmachnikov$^{1}$,$^{2}$, DmitryV. Kovalevsky$^{3}$

$^{1}$ The Saint Petersburg State University, Department of Oceanography, Universitetskaya emb. 7-9, 199034 St. Petersburg, Russia

$^{2}$ Nansen International Environmental and Remote Sensing Centre, 14th Line 7, office 49, Vasilievsky Island, 199034 St. Petersburg, Russia

$^{3}$ Climate Service Center Germany (GERICS), Helmholtz-Zentrum Geesthacht, Fischertwiete 1, 20095 Hamburg, Germany

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We present analytical solutions of the two-basin model of open-ocean deep convection. Originally suggested by Whitehead [Whitehead, J.A. (2000), Stratified convection with multiple states, Ocean Modelling, 2(3-4), 109-121], the model allows regimes with multiple steady states (multiple equilibria). We provide the full analytical description of the steady states for the particular case of constant surface heat flux from the ocean to the atmosphere, and explore analytically stability of the equilibria within the Lyapunov theory. The results show that, for this particular case, the steady state is unique and stable for all dynamic flow regimes. We also present analytical expressions for dependence of critical values of sea-surface heat flux, at which transitions between the dynamic regimes occur, on the model parameters.


The research was supported by Russian Science Foundation – RSF (project No. 17-17-01151).


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