 |
Jason Goodman
Program in Atmospheres, Oceans and Climate
Department of Earth, Atmospheric and Planetary Sciences
Massachusetts Institute of Technology
The statistical significance of the lag-correlation and spectrum of a timeseries of the wintertime strength of the North Atlantic Oscillation is examined. The NAO timeseries has significant lag-correlation, but no spectral features significantly different from a simple autoregressive process. The NAO pattern must interact with a system posessing interannual memory (such as the ocean) to produce this lag-correlation, but there is apparently no need to invoke resonant atmosphere-ocean modes or other coupled feedbacks involving active ocean dynamics to explain peaks in the NAO spectrum. The decorrelation timescale of the NAO is consistent with the usual lifetime of oceanic mixed-layer temperature anomalies.
Despite its importance, authors have differing ideas about what is meant by `decadal variability'. Some, like Latif & Barnett (1996) and Goodman and Marshall (1997) describe a resonant oscillatory mode: this would generate a spectral peak in a specific frequency band. Others, like Frankignoul et al (1996), describe a `knee' in the oceanic response to stochastic atmospheric forcing at decadal scales, separating the red spectrum on short timescales from a white signal at long timescales. Many researchers have noted the ``strong decadal fluctuations'' in the low-pass-filtered version of Hurrell's (1995) wintertime NAO timeseries (figure 1), and many have talked about ``decadal peaks in the NAO signal''. Hurrell and Van Loon's (1997) NAO spectrum shows a peak in the 6-10 year range, and Deser & Blackmon's (1993) power spectra of empirical orthogonal functions of atmospheric and oceanic fields which resemble the NAO's variability pattern show spectral peaks with a 12-year period.
|
The atmosphere generates chaotic fluctuations intrinsically. Thermal and pressure anomalies in the atmosphere are damped by radiative loss and surface drag in a few weeks' time. This lack of memory suggests that the NAOI should have no correlation from year to year if its forcing remains unchanged, generating a flat white spectrum. If the NAO pattern is coupled to an ocean which retains a simple memory of the previous year's NAO state, we would expect to observe lag-correlation in the NAOI, and a red spectrum. If the NAO couples to oceanic processes involving resonant oscillatory modes or other complicated dynamics, we should observe spectral peaks, or other spectral features inconsistent with a red or white spectrum.
Which of these possibilities (white, red, or complicated) is the NAO timeseries consistent with? We must know the statistical significance of lag-correlation and spectral estimates to answer this question, a topic not well considered by previous studies.
Hurrell's winter NAO index (figure 1) has a one-year
lag-correlation of .148; it gradually increases from -.08 in the first 60 years
of the record to .37 in the last 60 years (Hurrell and Van Loon, 1997). Are
these correlations likely to be produced by white noise? Suppose we take N
samples of a Gaussian random variable. The lag-1 autocorrelation of this
`timeseries' will have an asymptotically Gaussian probability distribution,
with standard deviation 
(Press, 1992). Since 
,there is only a 2% chance such a large lag-correlation could be produced by a
white-noise process. The .37 autocorrelation in the last half of the record is
even more significant.
The NAO signal is definitely not purely random; there is significant persistence and memory over one year. However, only 14% of the signal persists into the following year: prediction based on simple persistence will not be very good.
We now consider a simple autoregressive ``red-noise'' model with the same variance and lag-correlation as the NAO signal. Is the NAO timeseries power spectrum consistent with the output of such a model?
Figure 2 shows a spectrum of the NAO signal, estimated using the Thomson (1990a,b) multitaper method. This method provides a high-resolution spectrum with minimal spectral leakage and bias, as well as a rigorous estimate of statistical significance. The NAO index shows a predominantly white spectrum, with some extra energy at long timescales, a square-topped mesa for periods between 7 and 10 years, a second peak at around 2.3 years, and reduced power at 10-20 years and at 3 years. The spectrum closely resembles that computed by Hurrell and Van Loon. The dashed line represents the spectrum of a red-noise process with the same variance and lag-1 autocovariance, and the thin line paralleling it is the 95% confidence limit for this red-noise process. However, this is an a priori confidence limit: if we choose a specific frequency beforehand, 95% of spectra sampled from a red-noise process will lie below this line at that frequency. However, we're looking for peaks anywhere in the NAO spectrum. What is the probability that a red-noise process will generate peaks this high anywhere in the spectrum?
 |
There is a 5% probability that a sample of red noise will exceed the 95% curve at any given frequency, so a red-noise sample would have an average of 5% of the points above that curve. This is entirely consistent with the NAO index spectrum in figure 2.
This is not just statistical sanity-checking: it is a crucial point. The NAO timeseries spectrum is spectrally indistinguishable from a red-noise process. This fact was verified by visual examination of a number of sample red-noise spectra (not shown): almost every one had peaks of similar height and shape to figure 2. An unpublished note by Wunsch (1998) agrees with this study: artificial timeseries of weakly colored noise show many of the properties of the NAO timeseries. Occam's razor suggests that there is no need to propose elaborate atmosphere-ocean feedback mechanisms involving ocean dynamics to explain the decadal variability of the NAO: a simple autoregressive process will suffice. However, this is not at all a proof that such dynamical feedback mechanisms do not exist.
Even if ocean dynamics is responsible for the decadal peak, that peak accounts for only a small part of the variance of the NAO index. The fraction of variance explained by a spectral peak equals the area beneath that peak (plotted on a linear scale, not shown). The decadal peak explains only about 5-10% of the NAOI variance: even complete understanding of a hypothetical feedback mechanism producing enhanced decadal variability would not lead to much predictability of the NAO signal. This estimate agrees with Wunsch (1998): he shows that the linear predictive skill of the NAO index is less than 3% of the variance.
Robust spectral estimation techniques like the multitaper method must smooth the raw spectrum to reduce spurious peaks. If the amount of smoothing is reduced, the decadal peak in the NAO spectrum becomes narrower and taller. This suggests the presence of a narrowband or harmonic component to the spectrum; however, note that the peak becomes no more statistically significant. Interestingly, Thomson's f-test for harmonic components within a timeseries (Thomson, 1990a) suggests that a significant `spectral line' exists at 7.5 years period. However, this result is not robust: a slightly different NAO index (Iceland-Azores pressure difference), which is nearly identical to the Iceland-Lisbon index, shows no hint of a spectral line. The Iceland-Azores timeseries spectrum also shows a decadal peak, but is also consistent with a red-noise process. The possibility of a pure sinusoidal signal in the Iceland-Lisbon NAO index is intriguing, but since it is not robust and is difficult to justify theoretically it is probably a fluke.
Decadal fluctuations are readily apparent in the filtered timeseries in figure 1. Hurrell (1995) applies a 7-point filter to his timeseries data to remove variability on timescales shorter than four years. While useful, such filters can be misleading. Convoluting data in the time domain is equivalent to multiplying by a window in the spectral domain, and vice versa. The effect of Hurrell's filter is to multiply the raw spectrum by the shape in figure 3. It's clear that Hurrell's filter acts strongly on frequencies longer than 4 years as well. 10-year oscillations are reduced by a factor of 2.5, while 7-year oscillations are down a factor of 10; after filtering, the 7-year periods have 4 times less energy than 10-year cycles. This reddening is small beyond the 10-year band. Such a filter artificially accentuates decadal variability compared to slightly shorter timescales. One must not read too much into the easily-visible decadal oscillations in the filtered NAO timeseries; a spectrum of the unfiltered data provides more reliable information.
 |