  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.
