A note on using the accelerated convergence method in climate models

Application of the accelerated convergence (or asynchronous integration) method of Bryan (1984) to climate problems with time-dependent forcing is investigated using an ocean general circulation model (GCM), with an idealized box ocean and forced with an idealized seasonal restoring surface temperature. Numerical experiments consist of a control experiment, which is integrated synchronously for 9000 years, and 2 experiments with asynchronous integration, one with depth dependent acceleration and one with depth independent acceleration. The latter 2 cases were integrated synchronously for 9000 years after asynchronous equilibria are reached. It is found that a few thousand years of synchronous integration is needed to reach a new equilibrium after asynchronous equilibrium is obtained, consistent with a scaling argument. However, at the new equilibrium, temperature in the deep ocean only differs from that of the early stage of synchronous adjustment by about 0.01°C. So for practical purposes, 50 years of synchronous integration beyond asynchronous equilibrium is sufficient. A simple interpretation of the accelerated convergence method of Bryan is also presented.