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.
Persistent memory and tail-risk amplification in Atlantic Meridional Overturning Circulation variability: a Volterra integral framework calibrated with CMIP6
Faculty of Engineering, Universidad del Desarrollo（发展大学工程学院）
Transition to double-cell mock Walker circulations with surface warming explained by periodic convection
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Institute of Earth Sciences, The Hebrew University of Jerusalem（耶路撒冷希伯来大学地球科学学院）；Department of Geosciences, Princeton University（普林斯顿大学地质科学系）；Program in Atmospheric and Oceanic Sciences, Princeton University（普林斯顿大学大气与海洋科学项目）
Machine-learning prediction of tipping with applications to the Atlantic Meridional Overturning Circulation
School of Electrical, Computer, and Energy Engineering, Arizona State University（亚利桑那州立大学电气、计算机与能源工程学院）；Army Research Directorate, DEVCOM Army Research Laboratory（DEVCOM陆军研究实验室陆军研究局）；Department of Physics, Arizona State University（亚利桑那州立大学物理系）
