Dynamic Transition Theory for Thermohaline Circulation

TL;DR

This paper develops a mathematical transition and stability theory for thermohaline circulation, analyzing how large-scale ocean flows change states or oscillate based on physical parameters and domain geometry.

Contribution

It introduces a general transition framework for the Boussinesq system governing ocean circulation, including criteria for different transition types and the effects of friction terms.

Findings

First transition can lead to multiple steady states or oscillations.

Type-I (continuous) and Type-II (jump) transitions are characterized by parameters.

Hysteresis phenomena and metastable states are identified.

Abstract

The main objective of this and its accompanying articles is to derive a mathematical theory associated with the thermohaline circulations (THC). This article provides a general transition and stability theory for the Boussinesq system, governing the motion and states of the large-scale ocean circulation. First, it is shown that the first transition is either to multiple steady states or to oscillations (periodic solutions), determined by the sign of a nondimensional parameter $K$, depending on the geometry of the physical domain and the thermal and saline Rayleigh numbers. Second, for both the multiple equilibria and periodic solutions transitions, both Type-I (continuous) and Type-II (jump) transitions can occur, and precise criteria are derived in terms of two computable nondimensional parameters $b_1$ and $b_2$. Associated with Type-II transitions are the hysteresis phenomena, and the physical reality is represented by either metastable states or by a local attractor away from the basic solution, showing more complex dynamical behavior. Third, a convection scale law is introduced, leading to an introduction of proper friction terms in the model in order to derive the correct circulation length scale. In particular, the dynamic transitions of the model with the derived friction terms suggest that the THC favors the continuous transitions to stable multiple equilibria. Applications of the theoretical analysis and results to different flow regimes will be explored in the accompanying articles.