Person:

Hassanzadeh, Pedram

A linear response function (LRF) determines the mean response of a nonlinear climate system to weak imposed forcings, and an eddy flux matrix (EFM) determines the eddy momentum and heat flux responses to mean-flow changes. Neither LRF nor EFM can be calculated from first principles owing to the lack of a complete theory for turbulent eddies. Here the LRF and EFM for an idealized dry atmosphere are computed by applying numerous localized weak forcings, one at a time, to a GCM with Held–Suarez physics and calculating the mean responses. The LRF and EFM for zonally averaged responses are then constructed using these forcings and responses through matrix inversion. Tests demonstrate that LRF and EFM are fairly accurate. Spectral analysis of the LRF shows that the most excitable dynamical mode, the neutral vector, strongly resembles the model’s annular mode. The framework described here can be employed to compute the LRF and EFM for zonally asymmetric responses and more complex GCMs. The potential applications of the LRF and EFM constructed here are (i) forcing a specified mean flow for hypothesis testing, (ii) isolating/quantifying the eddy feedbacks in complex eddy–mean flow interaction problems, and (iii) evaluating/improving more generally applicable methods currently used to construct LRFs or diagnose eddy feedbacks in comprehensive GCMs or observations. As an example for (iii), in Part II, the LRF is also computed using the fluctuation–dissipation theorem (FDT), and the previously calculated LRF is exploited to investigate why FDT performs poorly in some cases. It is shown that dimension reduction using leading EOFs, which is commonly used to construct LRFs from the FDT, can significantly degrade the accuracy owing to the nonnormality of the operator.

A linear response function (LRF) relates the mean response of a nonlinear system to weak external forcings and vice versa. Even for simple models of the general circulation, such as the dry dynamical core, the LRF cannot be calculated from first principles owing to the lack of a complete theory for eddy–mean flow feedbacks. According to the fluctuation–dissipation theorem (FDT), the LRF can be calculated using only the covariance and lag-covariance matrices of the unforced system. However, efforts in calculating the LRFs for GCMs using FDT have produced mixed results, and the reason(s) behind the poor performance of the FDT remain(s) unclear. In Part I of this study, the LRF of an idealized GCM, the dry dynamical core with Held–Suarez physics, is accurately calculated using Green’s functions. In this paper (Part II), the LRF of the same model is computed using FDT, which is found to perform poorly for some of the test cases. The accurate LRF of Part I is used with a linear stochastic equation to show that dimension reduction by projecting the data onto the leading EOFs, which is commonly used for FDT, can alone be a significant source of error. Simplified equations and examples of 2 × 2 matrices are then used to demonstrate that this error arises because of the nonnormality of the operator. These results suggest that errors caused by dimension reduction are a major, if not the main, contributor to the poor performance of the LRF calculated using FDT and that further investigations of dimension-reduction strategies with a focus on nonnormality are needed.

A linear response function (LRF) that relates the temporal tendency of zonal-mean temperature and zonal wind to their anomalies and external forcing is used to accurately quantify the strength of the eddy–jet feedback associated with the annular mode in an idealized GCM. Following a simple feedback model, the results confirm the presence of a positive eddy–jet feedback in the annular mode dynamics, with a feedback strength of 0.137 day−1 in the idealized GCM. Statistical methods proposed by earlier studies to quantify the feedback strength are evaluated against results from the LRF. It is argued that the mean-state-independent eddy forcing reduces the accuracy of these statistical methods because of the quasi-oscillatory nature of the eddy forcing. Assuming the mean-state-independent eddy forcing is sufficiently weak at the low-frequency limit, a new method is proposed to approximate the feedback strength as the regression coefficient of low-pass-filtered eddy forcing onto the low-pass-filtered annular mode index. When time scales longer than 200 days are used for the low-pass filtering, the new method produces accurate results in the idealized GCM compared to the value calculated from the LRF. The estimated feedback strength in the southern annular mode converges to 0.121 day−1 in reanalysis data using the new method. This work also highlights the significant contribution of medium-scale waves, which have periods less than 2 days, to the annular mode dynamics. Such waves are filtered out if eddy forcing is calculated from daily mean data. The present study provides a framework to quantify the eddy–jet feedback strength in GCMs and reanalysis data.