A Matrix Approach for Assessing the Non-Local Effects of Local Feedback Processes

There are a variety of methods to assess the strength of individual feedback processes in numerical climate models. Generally these methods compute, in some form, the ratio of free- and fixed-feedback temperature anomalies when the model is subjected to forcing (e.g., by a doubling of CO2). However, these ratios (gains) are often sensitive to the overall shape of the applied forcing. For example, an ocean-atmosphere coupled model subjected to an equatorially amplified forcing will result in different gains and feedback factors (both global and local) than the same model subjected to a polar amplified forcing. We discuss a new, more general method of feedback analysis that results in n×nn times n feedback and gain matrices where nn is the resolution of the model. These matrices generalize (both globally and locally) the classically defined numerical gains and feedback factors and are independent of the applied forcing. The gain matrix, in particular, is shown to reveal, under any forcing scenario, the global pattern by which a given feedback process amplifies or dampens fixed-feedback temperature anomalies. Moreover, in the case of a feedback process that is not purely a function of local temperature, these matrices will show the degree to which this "local feedback process" depends on non-local perturbations. We apply this method in the context of a simple box model as well as a one-dimensional energy balance climate model.