The spectral continuum

When it comes to the spectra of paleoclimate variability, the peaks receive much more attention than the back-ground continuum. In a sense, this focus on the peaks is at odds with modern climate studies. For the atmospheric sciences, weather holds greater interest than the annual temperature cycle; in oceanography, mixing garners more interest than tidal cycles. This focus could be because the annual cycle and tides are largely solved problems. A better comparison might be made between the quasi-periodic ice-ages and the North Atlantic Oscillation or El Ni˜no variability.

Regardless, just as the modern spectral continuum embodies a rich set of physics, one should expect the spectral continuum at low-frequencies to provide insight into the processes which govern long-term climate variability.

A useful description of the background spectrum observed in Figures 1-1 and 1-2 is the spectral power-law, q, which relates power-density, Φ, to frequency, s,

Φ = As^−q.

A is a multiplicative factor which sets the level of the background spectrum. Spectra

with a positive q are referred to as red, in analogy with visible light being red at its lowest-frequencies. For the same reason a negative q is blue, and an approximately zero q indicates a white spectrum. Because of uncertainties in the proxy measure-ments, spatial variability, and temporal nonstationarity (discussed in Appendix A and the subsequent sections) these power-laws estimates should be thought of as in-dicators of the scaling relationships in temperature variability, not physical constants.

A particular concern, detailed in Appendix C, is that aliased higher-frequency energy will bias the power-law estimates towards being too red. While establishing how ap-plicable the observed scaling laws are to the global temperature variability will require much further work, the relatively simple power-law behavior and the agreement be-tween multiple different proxies found here suggests the power-law estimates provide a useful description of tropical SST and high-latitude SAT variability. The influence of age-model errors and orbital-tuning on spectral power-laws is largely unknown;

pending further study, it is assumed that the effect is small.

In keeping with most geophysical records, tropical SSTs have red spectra. In this case, the power-law is remarkably stable, with a value of one between frequencies of 1/100KY and 1/1yr — five orders of magnitude. This behavior is well replicated in multiple proxies, giving some confidence in its accuracy (but see Appendix C for other ways of producing such a power-law). At frequencies above the annual cycle, the spectrum falls off more quickly with frequency, with q ≈ 2. There is a bulge of energy centered on 1/100KY so that for slightly lower frequencies, the power-law is briefly blue. Imbrie and Shackleton [1990] find that at even lower frequencies, a power-law near one resumes.

The power-density spectrum of high-latitude SATs [see Figure 1-2], has a more complicated power-law behavior than tropical SSTs. From the highest resolved fre-quencies to 1/200yr, the spectrum has a power-law relationship of 0.4, while frequen-cies between 1/200yr and 1/100KY are more red with a power-law near two. At the millennial timescales there are marked differences between the power-laws de-rived from the three ice-cores records included here. The most energetic millennial scale variability is observed in temperatures estimated from the Greenland Ice-sheet Project 2 (GISP2) ice-core, followed by the Antarctic Byrd and then Vostok records.

As one approaches the 100KY timescale, the power-density of the three ice-cores con-verges. This touches on the topic of spatial changes in temperature variability: in the following section the spatial and temporal shifts in temperature variability are

discussed in more detail.

Comparison of the power-law structures shown in Figures 1-1 and 1-2 indicates roughly equal decadal variability in tropical SSTs and high-latitude SATs, greater centennial SST variability, and for periods longer than centuries greater high-latitude SAT variability. The greater SAT variability at lower frequencies agrees with es-timates indicating tropical SSTs underwent relatively small glacial to interglacial changes relative to high-latitude temperatures [e.g. CLIMAP Project Members, 1981].

Apart from tropical/high-latitude differences, the spectral structure probably also re-flect differences between atmospheric and sea-surface temperature variability and/or inaccuracies in the proxy measurements.

Figure 1-3 shows the composite spectra from Figures 1-1 and 1-2 after multiplying the power-density by frequency. This representation has the virtue of making the area under a log-linear plot proportional to the variance contained within each band.

Another effect of multiplying by frequency is to remove a power-law of one from each composite spectra. That is, the area preserving plots scale as,

Φ = As^−q×s=As^−q+1.

Because q ≈ 1 for the SAT and SST variability, the area preserving representation removes the trend in the background continuum and makes the detailed structure more evident. High-latitude SAT shows a relative minimum in scaled energy near frequencies of 1/100year (this minimum was identified as a change in the spectral slope in Figure 1-2.) The scaled tropical SST spectra also shows a weak minimum at the same 1/100year band. Qualitatively, this spectral structure suggests that the mechanisms responsible for climate variability change near the 1/100year timescale.

As insolation forcing is weak between the annual and secular periods of variability, it is tempting to identify the structure of the climatic background continuum with high and low-frequency responses to the insolation forcing. Other possibilities are that slow temperature fluctuations associated with the deep oceans and cryosphere only become important at the 1/100year timescale. Determining the cause of this apparent change in slope, however, awaits further investigation into the mechanisms controlling the background variability of the climate spectrum.

The estimates presented here generally agree with previous studies of the back-ground spectrum. At periods shorter than 200yr, Pelletier [1998] finds nearly the same SAT spectral structure, including a greater decline in subannual SAT

variabil-ity for stations near marine environments, in qualitative agreement with the SST spectrum. One difference is that, on the basis of a Lomb-Scargle periodogram anal-ysis [e.g. Press et al, 1999] of the Vostok deuterium record, Pelletier [1998] suggests the spectral continuum of SAT is white at frequencies below 1/40 KY. Given the bulge of energy expected near the 100KY periods, and that the Vostok record cannot resolve frequencies below 1/420KY, it is difficult to draw inferences regarding such low-frequency behavior from analysis of the ice-core record. It appears safer to assume that at long periods SATs behave like SSTs, and continue to have a red power-law behavior.

The power-laws of climate records are also discussed by Wunsch [2003b]. For ice-core δ¹⁸Oice and deuterium records his results agree with those shown in Fig-ure 1-2. Wunsch [2003b] does find a steeper power-law for marine sediment-core δ¹⁸O, but this is not an unexpected result. In focusing on the 100KY variability, Wunsch [2003b] estimated the power-law behavior of δ¹⁸O over frequencies of 1/100 to roughly 1/10KY. This band of variability is steeper than other parts of the δ18O spectrum [seeImbrie and Shackleton, 1990; Figure 1-1] and, as argued in Appendix A, is probably strongly influenced by ice-volume variability. In support, note that the Mg/Ca estimates shown in Figure 1-1 are not sensitive to ice-volume and maintain a power-law relationship much closer to one. Note Wunsch [2003b] interpreted the δ¹⁸O record as indicative of climate, not temperature, variability.

The origins of the background climate continuum remain an important ques-tion. One possibility is for the climate system to have a long memory, causing high-frequency variations to accumulate into progressively larger and longer period variability. Wunsch [2003b] has presented a simple random walk model of ice accu-mulation which is driven at all frequencies (a white forcing spectrum) but generates an energetic quasi-100KY variability and, at higher-frequencies, a background contin-uum with a power-law of two. Generalizing this idea to temperature, the power-law relationship observed in Figures 1-1 and 1-2 could represent the organization of high frequency temperature variability into progressively larger and longer timescale vari-ations — similar to the SST variability modeled byHasselmann [1976], but extending over longer timescales. In a recent paper, Pelletier [2003] has suggested an explana-tion for the overall spectral shape of the temperature record in terms of a coherence resonance model incorporating radiative, ice-sheet, and lithospheric deflection pro-cesses. Apart from creating an excess of energy near the 100KY timescale, the most

notable feature of the modeled temperature variability is a transition in the spectral power-law relationship from q= 2 to q= 0.5 near 1/2 KY, similar to observations.

The possibility also exists that the background variability observed in Figures 1-1 and 1-2 is related to the annual cycle, at least in part. Chapter 2 discusses how rectification of the annual cycle causes precession period variability to appear. Such rectification also causes a transfer of energy to the background continuum between 1/100 and 1 cycle per KY. The background continuum of rectified insolation has a steep power-law at low-frequencies and a transition to a more white spectrum at higher-frequencies, in qualitative agreement with Figure 1-2. Alternatively, low fre-quency insolation forcing could drive a low-frefre-quency temperature response which cascades towards higher frequency temperature variations. In certain regimes, such as Kolmogorov’s turbulent spectra, this flow of energy from low to high frequencies is well known. A larger scale example is the conversion of potential energy, supplied by the meridional insolation gradient, into synoptic scale variations by baroclinic insta-bility [e.g. Eady, 1949; Charney and Stern, 1962]. A variety of plausible mechanisms exist to explain the background spectrum of climate variability; further observations and dynamical research are needed to quantify and understand the mechanisms re-sponsible for the continuum energy at these broad range of frequencies.