Normal modes and the transient response of the climate system

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.