# Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

Transitions of Spherical Thermohaline Circulation to Multiple Equilibria The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and $$l_c$$ l c , which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for $$l_c=1$$ l c = 1 and $$l_c=2$$ l c = 2 cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for $$\text {Le}<1$$ Le < 1 but only continuous transition for $$\text {Le}>1$$ Le > 1 . In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2 $$l_c$$ l c dimensional sphere and consists entirely of degenerate steady state solutions. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Mathematical Fluid Mechanics Springer Journals

# Transitions of Spherical Thermohaline Circulation to Multiple Equilibria

, Volume 20 (2) – Jun 19, 2017
17 pages

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer International Publishing AG
Subject
Physics; Fluid- and Aerodynamics; Mathematical Methods in Physics; Classical and Continuum Physics
ISSN
1422-6928
eISSN
1422-6952
D.O.I.
10.1007/s00021-017-0331-8
Publisher site
See Article on Publisher Site

### Abstract

The main aim of the paper is to investigate the transitions of the thermohaline circulation in a spherical shell in a parameter regime which only allows transitions to multiple equilibria. We find that the first transition is either continuous (Type-I) or drastic (Type-II) depending on the sign of the transition number. The transition number depends on the system parameters and $$l_c$$ l c , which is the common degree of spherical harmonics of the first critical eigenmodes, and it can be written as a sum of terms describing the nonlinear interactions of various modes with the critical modes. We obtain the exact formulas of this transition number for $$l_c=1$$ l c = 1 and $$l_c=2$$ l c = 2 cases. Numerically, we find that the main contribution to the transition number is due to nonlinear interactions with modes having zero wave number and the contribution from the nonlinear interactions with higher frequency modes is negligible. In our numerical experiments we encountered both types of transition for $$\text {Le}<1$$ Le < 1 but only continuous transition for $$\text {Le}>1$$ Le > 1 . In the continuous transition scenario, we rigorously prove that an attractor in the phase space bifurcates which is homeomorphic to the 2 $$l_c$$ l c dimensional sphere and consists entirely of degenerate steady state solutions.

### Journal

Journal of Mathematical Fluid MechanicsSpringer Journals

Published: Jun 19, 2017

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