Dynamic transition theory for thermohaline circulation

Abstract

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.

Approximation of Stochastic Invariant Manifolds

“…The theory of invariant manifolds for deterministic dynamical systems has been an active research field for a long time, and is now a very well-developed theory; see, e.g., [6,7,8,9,10,30,43,44,52,62,74,79,80,88,89,92,94,105,117,118,119,121,132,143,147,148,149,150,152,153,154]. Over the past two decades, several important results on random invariant manifolds for stochastically perturbed ordinary as well as partial differential equations (PDEs) have been obtained; these results often extend those found in the deterministic setting; see, e.g., [1,2,3,12,20,22,25,26,29,51,57,65,66,113,123,124,…”

Phase transitions for Belousov-Zhabotinsky reactions

“…To this end, we need to determine the third eigenvalue 13 and eigenvector of M 1 at = 0 . We know that 13 …”

“…Let u = y e k 0 + (y), y∈ R 1 and (y) is the center manifold function. Let = y 2 = o(2); then by the approximation formula of center manifolds (see (A.10) in [13]), satisfies…”

Dynamic transition and pattern formation in Taylor problem

“…Then it is easy to check that G(u 0 ) ∈ E ⊥ 0 . Hence, by the center manifold approximation formula in [12,11], we find that Ψ = φ(x, y) + o(|x| 2 + |y| 2 ) + O(β 1 (λ)(|x| 2 + |y| 2 )), −L λ φ = G(ψ 1 , ψ 1 )x 2 + G( ψ 1 , ψ 1 )y 2 + (G(ψ 1 , ψ 1 ) + G( ψ 1 , ψ 1 ))xy. (6.4) On the center manifold, u = u 0 + Φ(u 0 ).…”