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Pedram Hassanzadeh and Zhiming Kuang

Abstract

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.

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Pedram Hassanzadeh and Zhiming Kuang

Abstract

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.

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Pedram Hassanzadeh and Zhiming Kuang

Abstract

The linear response function (LRF) of an idealized GCM, the dry dynamical core with Held–Suarez physics, is used to accurately compute how eddy momentum and heat fluxes change in response to the zonal wind and temperature anomalies of the annular mode at the quasi-steady limit. Using these results and knowing the parameterizations of surface friction and thermal radiation in Held–Suarez physics, the contribution of each physical process (meridional and vertical eddy fluxes, surface friction, thermal radiation, and meridional advection) to the annular mode dynamics is quantified. Examining the quasigeostrophic potential vorticity balance, it is shown that the eddy feedback is positive and increases the persistence of the annular mode by a factor of more than 2. Furthermore, how eddy fluxes change in response to only the barotropic component of the annular mode, that is, vertically averaged zonal wind (and no temperature) anomaly, is also calculated similarly. The response of eddy fluxes to the barotropic-only component of the annular mode is found to be drastically different from the response to the full (i.e., barotropic + baroclinic) annular mode anomaly. In the former, the eddy generation is significantly suppressed, leading to a negative eddy feedback that decreases the persistence of the annular mode by nearly a factor of 3. These results suggest that the baroclinic component of the annular mode anomaly, that is, the increased low-level baroclinicity, is essential for the persistence of the annular mode, consistent with the baroclinic mechanism but not the barotropic mechanism proposed in the previous studies.

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Sandro W. Lubis and Pedram Hassanzadeh

Abstract

The variability of the zonal-mean large-scale extratropical circulation is often studied using individual modes obtained from empirical orthogonal function (EOF) analyses. The prevailing reduced-order model of the leading EOF (EOF1) of zonal-mean zonal wind, called the annular mode, consists of an eddy–mean flow interaction mechanism that results in a positive feedback of EOF1 onto itself. However, a few studies have pointed out that under some circumstances in observations and GCMs, strong couplings exist between EOF1 and EOF2 at some lag times, resulting in decaying-oscillatory, or propagating, annular modes. Here, we introduce a reduced-order model for coupled EOF1 and EOF2 that accounts for potential cross-EOF eddy–zonal flow feedbacks. Using the analytical solution of this model, we derive conditions for the existence of the propagating regime based on the feedback strengths. Using this model, and idealized GCMs and stochastic prototypes, we show that cross-EOF feedbacks play an important role in controlling the persistence of the annular modes by setting the frequency of the oscillation. We find that stronger cross-EOF feedbacks lead to less persistent annular modes. Applying the coupled-EOF model to the Southern Hemisphere reanalysis data shows the existence of strong cross-EOF feedbacks. The results highlight the importance of considering the coupling of EOFs and cross-EOF feedbacks to fully understand the natural and forced variability of the zonal-mean large-scale circulation.

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Bryn Ronalds, Elizabeth Barnes, and Pedram Hassanzadeh
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Ding Ma, Pedram Hassanzadeh, and Zhiming Kuang

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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.

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Bryn Ronalds, Elizabeth Barnes, and Pedram Hassanzadeh

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Previous studies have found that the most consistent response of the eddy-driven jet to sea ice loss and Arctic amplification in fully coupled general circulation models (GCMs) is a broad region of anomalous easterlies on the poleward flank. In this study, a similar response is noted in a dry dynamical core GCM with imposed surface heating at the pole, and it is shown that in both a fully coupled GCM’s North Atlantic basin and the dry dynamical core, the anomalous easterlies cause an asymmetrical narrowing of the jet on the poleward flank of the climatological jet. A suite of barotropic model simulations run with polar forcing shows decreased jet positional variability consistent with a narrowing of the jet profile, and it is proposed that this narrowing decreases the distance Rossby waves can propagate away from the jet core, which drives changes in jet variability. Since Rossby wave propagation and dissipation is intrinsic to the development and maintenance of the eddy-driven jet, and is tightly coupled to a jet’s variability, this acts as a meridional constraint on waves’ ability to propagate outside of the jet core, leading to the decreased variability in zonal-mean jet position. The results from all three models demonstrates that this relationship is present across a model hierarchy.

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