Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 658345, 9 pages http://dx.doi.org/10.1155/2013/658345 Research Article On the Stability of an Intermediate Coupled Ocean-Atmosphere Model Tianxu Zhao 1 and Guang-an Zou 2,3 1 Department of Mathematics, Baoji University of Arts and Sciences, Baoji 721013, China 2 Key Laboratory of Ocean Circulation and Wave, Institute of Oceanology, Chinese Academy of Sciences, Qingdao 266071, China 3 University of Chinese Academy of Sciences, Beijing 100049, China Correspondence should be addressed to Guang-an Zou; zouguangan@gmail.com Received 24 April 2013; Revised 24 September 2013; Accepted 20 October 2013 Academic Editor: Driss Mehdi Copyright © 2013 T. Zhao and G.-a. Zou. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Te explicit fnite diference scheme for solving an intermediate coupled ocean-atmosphere equations has been proposed and discussed. Te discrete Fourier analysis within Gerschgorin circle theorem is applied to the stability analysis of this numerical model. Te stability criterion that we obtained includes advection, rotation, dissipation, and friction terms, without any assumptions, which is also including the Courant-Friedrichs-Lewy (CFL) condition as a special case. Numerical sensitivity experiments are also carried out by varying the model parameters. 1. Introduction Te geophysical motions in both the atmosphere and ocean can be described by partial diferential equations (PDEs), in recent years, which have become the very popular and impor- tant tools in the study of climate change, weather forecasting, and climate prediction. However, both the coupling process of the atmosphere to the ocean and the corresponding PDEs are very complicated; it is almost impossible to fnd the exact solution of coupled ocean-atmosphere equations. Research on numerical simulation of the ocean-atmosphere system has aroused many scientists and engineers’ interest; a great variety of numerical methods (especially for fnite diference method) have been developed to solve this PDEs system [1– 23]. Nevertheless, as we know, there are few people who gives the stability analysis of these complicated numerical models from the mathematical point of view. Te main purpose of this study is to introduce and solve an intermediate coupled ocean-atmosphere PDEs. Te stability analysis of numerical method has been taken into consideration by the discrete Fourier analysis combined with Gerschgorin circle theorem. Compared with the time step allowed by the CFL stability criterion, our stability bounds are more accurate and efective. Numerical examples are also presented to test the sensitivity of model. Tis paper is organized as follows. In the next section, the brief description of an intermediate coupled ocean-atmosphere mathematical model has been introduced. In Section 3, the explicit fnite diference scheme is used to solve the PDEs. Te stability conditions are given by the numerical analysis in Section 4. Ten the results of experiments are presented and discussed in Section 5. Finally, conclusions are drawn in Section 6. 2. The Mathematical Model Te intermediate coupled ocean-atmosphere model used here includes the model of ocean fuid dynamics, the mixed-layer thermodynamics, and the empirical atmospheric model. Tis coupled ocean-atmosphere model is a modifed version of the intermediate coupled model (ICM) developed by Chang [24] and Wang et al. [25], which is an extension of 1.5-layer reduced gravity system that includes the physics of the surface mixed layer and allows the prediction of the sea surface temperature. ICM had been successfully used to the study of El Ni˜ no-Southern Oscillation (ENSO), to simulate