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.
Deep Dive into Dynamic Transition Theory for Thermohaline Circulation.
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 b
One of the primary goals in climate dynamics is to document, through careful theoretical and numerical studies, the presence of climate low frequency variability, to verify the robustness of this variability's characteristics to changes in model parameters, and to help explain its physical mechanisms. The thorough understanding of this variability is a challenging problem with important practical implications for geophysical efforts to quantify predictability, analyze error growth in dynamical models, and develop efficient forecast methods.
Oceanic circulation is one of key sources of internal climate variability. One important source of such variability is the thermohaline circulation (THC). Physically speaking, the buoyancy fluxes at the ocean surface give rise to gradients in temperature and salinity, which produce, in turn, density gradients. These gradients are, overall, sharper in the vertical than in the horizontal and are associated therefore with an overturning or THC.
The thermohaline circulation is the global density-driven circulation of the oceans, which is so named because it involvse both heat, namely “thermo”, and salt, namely “haline”. The two attributes, temperature and salinity, together determine the density of seawater, and the defferences in density between the water masses in the oceans cause the water to flow.
The thermohaline circulation is also called the great ocean conveyer, the ocean conveyer belt, or the global conveyer belt. The great ocean conveyer produces the greatest oceanic current on the planet. It works in a fashion similar to a conveyer belt transporting enormous volume of cold, salty water from the North Atlantic to the North Pacific, and bringing warmer, fresher water in return. Figure 1.1 gives a simplified map of the great ocean conveyer and Figure 1.2 gives a diagram of oceanic currents of thermohaline circulation.
In oceanography, the procedure of the Conveyer is usually described by starting with what happens in the North Atlantic, under and near the polar region see ice. There warm, salty water that has been northward transported from tropical regions is cooled to form frigid water in vast quatities, which results in a bigger density of seawater (unlike fresh water, saline water does not have a density maximum at 4 • C but gets denser as it cools all the way to its freezing point of approximatively -1.8 • C). When this seawater freezes, its salt is excluded (see ice contains almost no salt), increasing the salinity of the remaining, unfrozen water. This salinity makes the water denser again. The dense water then sinks into the deep basin of the sea to form the North Atlantic Deep Water (NADW), and it drives today’s ocean thermohaline ciculation.
Eurasia N. America S.America fresher water upwells, while a shallow-water counter-current has been generated. This counter-current moves southward and westward, through the Indian Ocean, still heading west, and rounding southern Africa, then crosses through the South Atlantic, still on the surface (though it extends a kilometer and a half deep). It then moves up along the east coast of the North America, and on across to the coast of Scandinavia. When this warmer, less salty water reaches high northern latitudes, it chills, and naturally becomes North Atlantic Deep Water, completing its circuit.
The THC varies on timescales of decades or longer, as far as we can tell from instrumental and paleoclimatic data [Martinson et al., 1995]. There have been extensive observational, physical and numerical studies. We refer the interested readers to [3,2] for an extensive review of the topics; see also among others [33,29,11,9,27,28,34,35,36,4,5,6,7].
The main objective of this series of articles is to study dynamic stability and transitions in large scale ocean circulations associated with THC. A crucial starting point of this theory is that the complete set of transition states are described by a local attractor, rather than some steady states or periodic solutions or other type of orbits as part of this local attractor. Following this philosophy, the dynamic transition theory is recently developed by the authors to identify the transition states and to classify them both dynamically and physically. The theory is motivated by phase transition problems in nonlinear sciences. Namely, the mathematical theory is developed under close links to the physics, and in return the theory is applied to the physical problems, although more applications are yet to be explored. With this theory, many long standing phase transition problems are either solved or become more accessible. In fact, the study of the underlying physical problems leads to a number of physical predictions. For example, the study of phase transitions of both liquid helium-3 and helium-4 [21,23,24] leads not only to a theoretical understanding of the phase transitions to superfluidity observed by experiments, but also to such physical predictions as the existence o
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