The Linear Response Function of an Idealized Atmosphere

The linear response function M relates the response x of a nonlinear system (such as the atmosphere) to weak external forcings f and tendencies x_t via x_t= M x + f and can be useful in studying climate change and internal variability. However, M cannot be determined from the first principles because it not only contains known terms such as advection by the mean flow, but it also contains eddy feedback for which an analytical theory does not exist. We find M for an idealized atmosphere using a simple GCM, a dry dynamical core with Held-Suarez forcing. 400 zonally-symmetric small-amplitude localized forcings of zonal wind and temperature (i.e. f) are added one at a time at various latitudes and pressure levels, and the zonal- time mean responses of zonal wind and temperature (i.e. x) are calculated from 45000 day integrations. Given that x_t vanishes due to time averagings, M can be found from x and f by matrix inversion. In a number of tests, for a target response y, running the GCM with a forcing calculated from –M y successfully produced y as the response, hence validating M as an accurate response function. The singular vectors with the smallest singular values (also known as neutral vectors) and eigenmodes of several of the slowest decaying modes closely resemble the annular mode.

We calculate the part of M that is due to the eddy feedbacks, explore using M to produce specific mean flows in the GCM useful for hypothesis testing, and compare M with the response function constructed from the fluctuation-dissipation theorem.