Adjoint models are used for atmospheric and oceanic sensitivity studies in order to efficiently evaluate the sensitivity of a cost function (e.g., the temperature or pressure at some target time t f , averaged over some region of interest) with respect to the three-dimensional model initial conditions. The time-dependent sensitivity, that is the sensitivity to initial conditions as function of the initial time t i , may be obtained directly and most efficiently from the adjoint model solution. There are two approaches to formulating an adjoint of a given model. In the first (''finite difference of adjoint''), one derives the continuous adjoint equations from the linearized continuous forward model equations and then formulates the finite-difference implementation of the continuous adjoint equations. In the second (''adjoint of finite difference''), one derives the finite-difference adjoint equations directly from the finite difference of the forward model. It is shown here that the time-dependent sensitivity obtained by using the second approach may result in a very strong nonphysical behavior such as a largeamplitude two-time-step leapfrog computational mode, which may prevent the solution from being used for time-dependent sensitivity studies. This is an especially relevant problem now, as this second approach is the one used by automatic adjoint compilers that are becoming widely used. The two approaches are analyzed in detail using both a simple model and the adjoint of a primitive equations ocean general circulation model. It is emphasized that both approaches are valid as long as they are used for obtaining the gradient or sensitivity at a single time, as needed in data assimilation, for example. Criteria are presented for the choice of the appropriate adjoint formulation for a given problem.

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One of the major factors determining the strength and extent of ENSO events is the instability state of the equatorial Pacific coupled ocean-atmosphere system and its seasonal variations. This study analyzes the coupled instability in a hybrid coupled model of the Indo-Pacific region, using the adjoint method for sensitivity studies. It is found that the seasonal changes in the ocean-atmosphere instability strength in the model used here are related to the outcropping of the thermocline in the east equatorial Pacific. From July to December, when the thermocline outcrops over a wide area in the east Pacific, there is a strong surface-thermocline connection and anomalies that arrive as Kelvin waves from the west along the thermocline can reach the surface and affect the SST and thus the coupled system. Conversely, from February to June, when the thermocline outcropping is minimal, the surface decouples from the thermocline and temperature anomalies in the thermocline depth range do not affect the surface and dissipate within the thermocline. The role of vertical mixing rather than upwelling in linking vertical thermocline movements to SST changes is emphasized. It is therefore suggested that the seasonal ocean-atmosphere instability strength in the equatorial Pacific is strongly influenced by the thermocline outcropping and its seasonal modulation, a physical mechanism that is often neglected in intermediate coupled models and that can be represented properly only in models that employ the full dynamics of the mixed layer.