The complex argument of WX(s,t) can be interpreted as the instantaneous phases of X{f 1..., fN} at the scale s. We utilize the strength of the instantaneous phase angle difference between two series (X and Y), also known as the mean phase coherence, p(X,Y) (Mokhov and Smirnov, 2006). We are interested in causative relations, so it is appropriate to measure p between the instantaneous phases f and 0 of the two time series

We vary the relative phase delay between the two series by lagging f relative to 0 by a phase lag, A. Significance testing of p is done by Monte Carlo methods against 1000 realizations of a red noise background (Grinsted, Moore and Jevre-jeva, 2004), and the results can be visualized in a two-dimensional plot of p in l-A space analogous to the wavelet frequency-time space plot. As a further refinement in the utility of such a plot we find it useful to contour the strength of linear regression of the wavelet filtered time series as a function of l and A, so that the color scale bar corresponds to the value of m in the equation of WY(1,t+ A) = m WX (1,t). The phase relationship over the range multi-year to decadal periods was examined by filtering both time series with a Paul wavelet with l between the Nyquist frequency and 40 years with six l per octave of scale.

Results

It is well known that the TC time series is not Normally distributed but follows a Poisson distribution (Solow and Moore, 2000). However here we are interested to see how the non-Normality affects the novel statistical techniques we use. There is also a question as to how discrete data such as TC can be used in methods that were developed for continuously distributed data. One approach to providing a more continuous time series could be smoothing by running averaging the TC rate over a variety of scales, though any particular length of the running average would create data that would still be rational numbers. The smoothing window would naturally tend to produce a more Normal distribution via the Central Limit Theory. The CWT method is superior to running means as it effectively smoothes the data by the particular wavelet filter used, and this creates a much less discreet set of data. For both the modified and raw TC time series a Bera-Jarque test of Normality is rejected (p = 0.02) (Fig. 2), however, the data are acceptably Lognormal (p = 0.15). Clearly this is due to the TC being non-negative with a long tail.

We can remove the lack of Normality from the TC distribution completely by making use of a Normalization procedure (Jevrejeva, Moore and Grinsted, 2003). We transform the original data using a data adaptive transformation function. The transformation operator is optimally chosen so that the new probability density function is Normal, has zero mean and unit variance. This is calculated by making

Number

Fig. 2 Distribution of TC (black) and modified TC (grey), and their best fit Normal distributions

Number

Fig. 2 Distribution of TC (black) and modified TC (grey), and their best fit Normal distributions

the inverse normal cumulative distribution function of the percentile distribution of the original distribution (Fig. 3). We refer to this procedure as Normalization and it can be a rather drastic operation to use on a time series. However, Jevrejeva, Moore and Grinsted, 2003 have shown that the results from even grossly non-Normal distributions, that would not produce reliable results with the wavelet method, do give results after Normalization that are consistent with alternative methods of signal extraction such as Singular Spectrum Analysis. Henceforth we denote the Normalized modified TC series as TC.

Figure 4 shows at first glance, quite large differences in significant regions. However, the differences in the actual values of coherence are rather slight, the coherence being quite close to the 95% value that marks the border. There are quite small differences in the time derivative dSSTC plots. The differences become smaller if the simple normalized times series or the simple modified time series are compared with the original TC. The largest differences are in the 25-30 year band, with no significant region in the raw TC curve but a quite large region in the normalized modified TC data. Again at first glance this may seem to offer support for the low frequency AMO oscillation, but there should be a number of cautions. The largest region of significance is in the rather dubious physical region of the graph whereby TC determines SSTC at rather long lead times of a decade or more.

An alternative complimentary method of examining the data is using wavelet coherence. Figure 5 shows that there are very slight differences between the TC'' Normalized modified TC time series and the raw TC series.

Fig. 4. Wavelet lag coherence plots showing: (a) TC sensitivity on SSTC (WY(p,t+A) = m WX(p,t), m in number per °C is shown on the colour bar, as a function of Paul wavelet filtered period (p) and phase lag (A), solid black contour is 95% confidence interval of mean phase coherence (p) contours The arrow notation in Y^X etc. denotes that Y leads X in lag space. (b) TC and the d SSTC. (c) Normalized modified TC (TC'') and SSTC and (d) TC'' and the dSSTC

Fig. 4. Wavelet lag coherence plots showing: (a) TC sensitivity on SSTC (WY(p,t+A) = m WX(p,t), m in number per °C is shown on the colour bar, as a function of Paul wavelet filtered period (p) and phase lag (A), solid black contour is 95% confidence interval of mean phase coherence (p) contours The arrow notation in Y^X etc. denotes that Y leads X in lag space. (b) TC and the d SSTC. (c) Normalized modified TC (TC'') and SSTC and (d) TC'' and the dSSTC

1880 1900 1920 1940 1960 1980 2000

Year

Fig. 5 (a) Squared wavelet coherence between SSTC and TC (dark high values, light low values). The 5% significance level against red noise is shown as a thick contour. The relative phase relationship is shown as arrows (with in-phase pointing right, anti-phase pointing left, and SSTC leading TC by 90° pointing straight down), the curved lines with no colouring delineate the region affected by data boundaries (Grinsted, Moore and Jevejeva, 2004); (b) As for (a) but with the Normalized modified TC and SSTC.

1880 1900 1920 1940 1960 1980 2000

Year

Fig. 5 (a) Squared wavelet coherence between SSTC and TC (dark high values, light low values). The 5% significance level against red noise is shown as a thick contour. The relative phase relationship is shown as arrows (with in-phase pointing right, anti-phase pointing left, and SSTC leading TC by 90° pointing straight down), the curved lines with no colouring delineate the region affected by data boundaries (Grinsted, Moore and Jevejeva, 2004); (b) As for (a) but with the Normalized modified TC and SSTC.

It has been suggested that the large scale atmosphere is impacted by cyclone activity for some considerable period after the cyclone has dies away. This memory may be expected to manifest itself on seasonal snow cover in the Northern Hemisphere. We investigate this using the long series of snow cover estimates from Brown (2000). Figs. 6 and 7 show the behaviour of Northern Hemisphere spring snow coherence and sensitivity with TC and TC''. Perhaps most surprising is that Fig. 6 shows that the relationship is basically in-phase, so that more spring snow implies greater numbers of TC. However, Fig. 7 shows that the relationship is not significant except at rather long positive and negative lags of about a decade. Particular mechanisms for interactions with snow cover have been proposed by Hart, Maue and Watson (2007). In particular they suggest that autumnal snow cover may be influenced by TC. Figs. 8 and 9 examine October snow cover extent 1922-1997 in Eurasia—time series for the whole Northern Hemisphere not being available. In contrast with Figs. 6 and 7, we see that the relationship is consistently anti-phase, with zero or small lag times, but significant only at decadal periods. Thus we see that the spring and autumn snow covers react in quite different ways. We also

Fig. 6 (a) Squared wavelet coherence between Northern Hemisphere spring snow cover and TC. Contours and arrows as for Fig. 5. (b) coherence between Northern Hemisphere spring snow and TC''

Fig. 6 (a) Squared wavelet coherence between Northern Hemisphere spring snow cover and TC. Contours and arrows as for Fig. 5. (b) coherence between Northern Hemisphere spring snow and TC''

tested the Eurasian spring snow cover relationship with TC (not shown here) and found the wavelet coherence to be very similar as for the Northern Hemisphere as a whole (see Fig. 6), but the lag coherence had no areas of significance.

Discussion and Conclusions

It has already been argued (Mann et al., 2007) that the modifications suggested by Landsea (2007) and others, do not affect the main results of trend and correlation analysis between SSTC and TC. We show quite clearly that this is also true of analysis of the coherence and lag regressions of the modified time series, which even largely survive the gross manipulation of the time series to ensure complete Normality. One reason why the wavelet methods we use are not particular sensitive to the actual distribution of the data is that the Paul—and indeed most if not all wavelets, use more data points in their filter than required by simple Nyqust frequency considerations. This means that we smooth the data by a series of filters of different lengths. While the wavelet filters are infinitely long, the minimum scale used here is 2 which for the Paul wavelet of order 4 corresponds to a shortest period of about 2.8 years for annual data. The longer the filter, the more smoothed the data and the closer the distribution of data within that sample length will be to Normally distributed by the Central Limit Theory.

It has been suggested that our significance tests done on the wavelet data may be misinterpreted. That is small areas of significance at the 95% level could occur purely randomly some of the time, and so if the significant region is a small part of the whole figure, it may be there purely by chance. However this does not take into account that the tests are on phase relationships not measures of common power. Hence the significance will not be inflated simply by a few common bits of high power in the two series. This is borne out by testing of many series where we find absolutely no region of significant coherence regardless of how large the plot is made in lag-period space. The significance test uses the most conservative red noise model available, i.e. matching the original series mean, standard deviation and lag-1 autocorrelation, so the Monte Carlo common coherence thresholds found will be more conservative than simply random noise would give. This follows as red-noise that does not possess the same characteristics as the data would be less correlated with the data and hence provide a lower significance threshold in Monte Carlo testing than given by noise matching the data characteristics. However, since the procedure is essentially band pass filtering, the type of noise distribution is not very critical for significance testing. Similarly as the coherence is a phase matching rather than common power finding method, the relative power distribution is not important in frequency space. Therefore the actual noise model e.g. red noise auto regressive (AR1) or fractional Gaussian (self-similar scaling), is less important for significance testing than would be case for many other statistical methods.

The results presented in Figs. 6-9 are rather curious. The differences between spring and autumn snow cover are somewhat suggestive of the differences seen at 5 year and decadal periods in TC and SSTc which Moore, Grinsted and Jevrejeva (2008) interpreted by ENSO and Gulf Stream/NAO variability. The decadal power seen in autumn snow is consistent with the ideas suggested by Hart, Maue and Watson, 2007 regarding the extra-tropical impact of tropical cyclones. The larger the number of tropical cyclones, the less autumn snow cover appears to be logical given the energy transport from tropics mediated by the cyclones. The surprising feature is that this effect is only apparent at decadal periods. This suggests a common causal factor with SSTc decadal variability (Figs. 4 and 5) ascribed to NAO/Gulf Stream variability at 7.8 years. NAO phase is known to strongly impact precipitation in Europe and the Middle East, so this observation is consistent with ideas that NAO plays a useful role in predicting TC. Positive NAO phase has been related to decreased sea level pressures (SLP) over the Arctic region - with a minimum over Iceland- and a northeastward extension of the Atlantic storm track to Greenland, Iceland, Norway and Barents Seas, causing major increases in cyclone activity in the area and thus increased heat flux over the region (Serreze et al., 1997; Alexandersson et al., 1998). Such situations enhance southerly warm winds over the western Nordic Seas, causing 1) compaction and reduced freezing in the ice margin (Vinje 2001), 2) warm air advection (Deser, Walsh and Timlin, 2000), and 3) enhanced flow of warm and saline Atlantic water (Grotefendt et al., 1998; Morison, Aagaard and Steele, 2000; Polyakov et al., 2004). Persistent positive NAO phase is predicted by climate models as a consequence of global warming (e.g. Gillett, Graf, and Osborn, 2003). Regardless of this, NAO relationships with Arctic environment are far from stationary. Surface air temperatures (SAT), SST, and SLP over the North Atlantic during the period 1873-2000 have alternated decades of strong negative with decades of strong positive correlations with NAO (Polyakova et al., 2006). Likewise, NAO and SAT records from Europe showed significant non-stationarities on decadal time-scales (Slonosky, Jones and Davies 2001). Suggested mechanisms for such non-stationarities are the co-occurrence or otherwise of several NAO-related SLP patterns (Maslanik et al., 200), or the planetary-scale SLP wave (Cavalieri, 2002).

The spring snow cover—in the northern hemisphere, but not in Eurasia, has significant common coherence with TC in the 5 year band. If this is an ENSO feature then it is entirely plausible given the impact of ENSO on the Pacific Decadal Oscillation (PDO) and the observed large impact that the PDO has on North American climate (Biondi, Gershunov, and Cayan, 2001). The long lags seen (Fig.7) may in fact be a reflection of the dominant bi-decadal periodicity of the PDO (Minobe, 1999) on the fundamental ENSO impact on TC that has been observed for many years (Gray, 1984; Moore Grinsted and Jevrejeva 2008.

Acknowledgments We thank the Finnish Academy, the Thule Institute and the Natural Environmental Research Council for financial support. An anonymous referee gave many valuable comments.

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