Dynamic Transition Theory for Thermohaline Circulation

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.

References in corpus (5)

• Cahn-Hilliard Equations and Phase Transition Dynamics for Binary Systems
• Dynamic Model and Phase Transitions for Liquid Helium
• Tropical Atmospheric Circulations: Dynamic Stability and Transitions
• Superfluidity of Helium-3
• Phase Transition and Separation for Mixture of Liquid He-3 and He-4

Cited by in corpus (10)

• Phase Transitions for Belousov-Zhabotinsky Reactions
• Phase Transitions for the Brusselator Model
• Dynamical transition of hydromagnetic convection in a rotating fluid layer
• Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows
• Transitions of Spherical Thermohaline Circulation to Multiple Equilibria
• Remarks on the Rayleigh-Benard Convection on Spherical Shells
• Dynamic Transition and Pattern Formation for Chemotactic Systems
• Baroclinic Instability and Transitions in a Two-Layer Quasigeostrophic Channel Model
• Dynamic Transitions of Surface Tension Driven Convection
• Dynamic Transition and Pattern Formation in Taylor Problem