(0)8. 0
With a 110 year data set, it is possible to estimate low-frequency variabilities with periods up to about 30 years. To accomplish this, we employ the maximum-entropy method of spectral estimation and SSA methods following Press et al.[15], Vautard and Ghil [16], Penland et al.[17], and Vautard et al.[18]. We present some of the details in the context of one of the time series involved in the comparison. The time series is the domain-averaged potential energy, sampled 10 times a year over 110 years, for the shallow water case SW1. We estimate the lag-covariance matrix, first using a maximum lag of 8 years and again using a maximum lag of 30 years. The eigenvalue-spectra of the two lag-covariance matrices is shown in Fig. 5a (the `*' and `+' symbols and the y-axis on the left), in the decreasing order of the variance they measure. The spectrum plotted using the symbol `*' is for the maximum lag of 8 years and the spectrum plotted using `+' is for the 30 year maximum lag. The error bars are calculated using the formula , where M is the window length and N is the number of data points [19]. [Note that these error bars are very conservative and that the error bars calculated using the formula , where L is estimated to be the inverse of the natural logarithm of the lag-one autocorrelation coefficient [19] are essentially negligible.] While the 8-year lag spectrum seems to show a break at mode 10, signifying a natural truncation level, signs of a break in the 30-year lag spectrum are less obvious.
We digress here momentarily to emphasize the data-adaptive nature of the basis into which the time series is decomposed. Presently, the basis vectors (called temporal Empirical Orthogonal Functions or t-EOFs) are the eigenvectors of the lag-covariance matrix, and thus are not pure sines and cosines as in the usual Fourier analysis. In the spectra of Fig. 5a, the eigenvalues are arranged in a manner such that the t-EOF, the projection of the time series on to which captures the most variance is mode 1, and so on in a descending order. So the low-modes are not necessarily pure low frequency components and the high-modes are not necessarily pure high frequency components.
In neither of the spectra in Fig. 5a were we able to fit a noise model to satisfactorily explain the higher (low-variance) modes. We tried both a white-noise model and a more typical red-noise model after fixing various high-variance modes to account for the effects of the low-variance modes. This failure seems to indicate that a large number of the degrees of freedom are significant and need to be incorporated in explaining the variability of the system. In light of this, we presently use SSA purely as a scheme for data-adaptive filtering [17]. so that the spectrum of the time series may be estimated by the ME method using only a small number of poles.
Fig. 5a also shows the cumulative percentage of variance explained (the open triangle and diamond symbols and the y-axis on right) by the first M modes (x-axis). While the first ten modes are enough to explain 90% of the variance of potential energy in the 8-year lag case, the first thirty modes must be considered in the 30-year lag case to reach the same level of variance. Finally, the spectra of the reconstruction of the time series for the 30-year lag and 8-year lag cases, both accounting only for the first 90% of the variance are shown in Fig. 5b. The significant peaks in the spectra of the reconstructed series are now captured by just a few poles. Given the close correspondence between the features of the two spectra we use a reconstruction consisting of the first 30 modes in the 30-year maximum lag analysis to estimate the low frequency variability of the different systems.
We point out that the estimate of the frequency spectrum of the potential energy of case SW1 in Fig. 3b does not exhibit peaks as in Fig. 5b, mainly because we have filtered at the 90% variance level using the t-EOFs in Fig. 5a. Since the y-axis is logarithmic in Fig. 5b, the actual contributions beyond the first decade fall off very rapidly and thus the remaining 10% of the variance which was filtered out would fill in the spectrum at frequencies greater than 0.5 cycles per year in Fig. 5b and make it flatter. Also, considering the number of data points in Fig. 3b, the number of poles used is too small to be able to resolve the peaks seen in Fig. 5b.
Fig. 6 shows the low frequency spectral estimates of the domain-averaged potential energy for the five cases. Eliminating variability due to the higher modes (those accounting for the last 10% of the variability) exposes the remnants of dominant periodicities in the system. This is both an artifact of the diagnostics and an emergent feature of the dynamics at this level of description. Again, at this level of analysis, we note that the differences in the structure of the low-frequency variability due to the nonhydrostatic terms (i.e., the differences between (a) & (d) and between (a) & (e)) is at a level comparable to that obtained by large variations of parameters in the purely hydrostatic case (i.e., the differences between (a) & (b) and (a) & (c)).
With the aid of animations of the flow field, we have concluded that the sharp peak occuring between 0.8 cpy and 1.0 cpy corresponds to the disruption of the WBCs by sufficiently energetic eddies. Since eddies of different strengths are shed by the unstable jet over a wide range of east-west locations, this peak does not correspond to the mean life of an eddy. Note that although this peak seems visually dominant in the spectra, it contains between 5 and 10 times less energy than the much lower frequency variations, and thus does not represent the most energetic frequencies.