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Upwelling diffusion (UD' models are currently used to estimate the transient response of the climate system to perturbations in atmospheric composition. We linearize a global mean vertical upwelling diffusion model of the coupled ocean‐atmosphere system to obtain a solution for the transient response of the climate system to a given perturbation. For an N‐box model, the exact solution for the transient response of the surface air temperature to a given step change in radiative forcing is shown to be related to the equilibrium change in surface temperature (Δ T eq,1 ' by Δ T eq,1 Σ k a 1, k (1‐ e −t/τ k'. The time constants (τ k ' and the contribution of each mode (a 1,k ' can be expressed in terms of the matrix characterizing the energy exchange between the boxes representing the atmosphere and different layers of the ocean. We find that a minimum of 5‐boxes (one atmosphere and four ocean levels' is needed to adequately reproduce the response of a 20‐box UD model. The transient response of the atmospheric temperature to a step change in radiative forcing is dominated by three modes with characteristic time constants of 1.7, 7.2, and 40.2 years and contributions of 22%, 47%, and 27% respectively to the equilibrium response of the system.
MacKay et al. (Sat,) studied this question.