Applying the Fluctiation-Dissipation Theorem to a 2-layer QG Model

The Fluctuation-Dissipation Theorem (FDT) provides a means of calculating the response of a dynamical system to a small force by constructing a linear operator which depends only on data from the internal variability of the unperturbed system. Given the complexity of many climate systems, the possibility of generating linear operators for their response to small forces without knowledge of the underlying dynamics is intriguing. Here we use the quasi-Gaussian formulation of the FDT to estimate the response of a baroclinically unstable jet in a two-layer quasi-geostrophic model to two forms of forcing: one forcing the same sign of zonal mean zonal wind perturbation in each layer (the barotropic perturbation) and one forcing with the opposite sign in each layer (the baroclinic perturbation). The FDT is a linear theory and so we ensure that our perturbations are small enough that the model's response is a linear function of the strength of the perturbation.

We apply these perturbations for a range of supercriticalities to test how the FDT fares as this parameter is varied. This provides a broad test of the FDT under different conditions, as varying the supercriticality changes the model's behavior in ways which are relevant for the FDT calculations. Most notably, the model's eddies play an increasing role in determing the model's response as the supercriticality is decreased; the FDT must capture this behavior. The flow also becomes more persistent as the supercriticality is decreased, with significant correlations even at lags equivalent to 400 Earth days for the smallest values of the supercriticality. Finally, tests for multivariate Gaussianity show that decreasing the supercriticality results in increasing non-Gaussian behavior, though even for the largest supercriticalities there is significant non-Gaussian behavior.

To perform the FDT calculations we decompose the data onto Empirical Orthogonal Functions (EOFs), and find that the number of well resolved EOFs proves to be a good guide for how many EOFs to retain to produce the best FDT estimates. Using fewer EOFs than this means that not enough variability is captured to produce accurate estimates, while including poorly resolved EOFs can induce large errors. Our results are particularly sensitive to this due to the low dimensionality of the system: only the first five EOFs are well resolved and so these are the only EOFs included in our calculations. For larger systems it is possible that including poorly resolved EOFs would not affect the FDT estimates as much.

In the barotropic case we obtain good qualitative estimates for all values of the supercriticality, though the FDT consistently overestimates the response, perhaps due to the non-Gaussian behavior present in the model. Sensitivity tests were conducted to ensure that these estimates are robust to the choices made in the calculations. Despite the fact that the agreement between the FDT estimates and the model responses is more qualitative than quantitiative, we believe that this provides further support for the hypothesized connection between the autocorrelation time-scale of the annular modes and the response of the mid-latitudes to small perturbations. The baroclinic case is more challenging for the FDT. However, by constructing different bases with which to calculate the EOFs we show that the issue in this case is that the baroclinic variability is poorly sampled, not the failure of the FDT itself. The strategies we have developed in order to generate these new bases may be applicable to situations in which the FDT is applied to larger systems.