Modes of Interannual Tropical Ocean–Atmosphere Interaction—a Unified View. Part II: Analytical Results in the Weak-Coupling Limit

The properties of the eigenmodes of the coupled tropical ocean-atmosphere system, linearized about a climatological basic state—and hence of the first bifurcation, which strongly determines the nature of the interannual variability, such as El Niño—show considerable dependence on the parameters of the coupled system. These eigenmodes are examined in a modified shallow-water model with simplified mixed-layer dynamics and a sea surface temperature (SST) equation, coupled to a simple atmospheric model. The model is designed so as to make analytical approximations feasible in various limits, as in a previous study by Neelin where the x-periodic case was analyzed. The realistic case of a finite ocean basin is treated here. An integral formulation of the eigenvalue problem is derived that provides a basis for making consistent approximations that include the effects of atmospheric and oceanic boundary conditions. We provide a scaling analysis to select parameters that give the most succinct insights into the behavior of the system, and outline the portions of this parameter space that are accessible to analytic results through the limits explored here and in Part III of this study. Important limits include the fast-wave limit, the limit where the time scale of ocean adjustment is fast compared to the time scale of SST change by coupled processes, and its converse, the fast-SST limit. The region of validity of the weak-coupling limit overlaps both of these, while that of the strong-coupling limit overlaps the fast-SST limit and approaches the region of validity of the fast-wave limit without a formal matching region. In this part, we examine the weak-coupling limit, in which one expects the modes to be most closely related to those of the uncoupled problem. Here we treat two classes of mode from the uncoupled case: the SST modes (related to the time derivative of the SST equation) and the discrete modes from the ocean-dynamics spectrum, the ocean basin modes. From the numerical results of Part I, we know that away from the weak-coupling and fast-wave limits, the continuous surfaces in parameter space formed by the eigenvalues of each type of mode are joined, so that through most of parameter space the coupled modes are best characterized as mixed SST/ocean-dynamics modes. Series solutions for the weakly coupled modes are found to have radii of convergence that extend over modest but significant ranges of coupling values. The transition from the uncoupled modes to the fundamentally coupled mixed modes is examined. For the SST modes, coupling effects come to dominate the structure of basin-scale modes even at tiny coupling values. The structure of the ocean basin modes persists over a perceptible range of coupling, but structure changes involving the SST equation enter importantly as coupling is increased and the transition to mixed-mode structure occurs at small coupling, well within the range of the weak-coupling limit. This suggests that intuition and terminology borrowed from the uncoupled system is of limited value in analyzing coupled models and that it is more productive to consider prototype modes in fully coupled regimes.