PHASE RELATIONSHIPS OF PROXY VARIABLES

Following the brief introduction to time series analysis three weeks ago, a few points deserve additional emphasis:

• For valid spectral analysis, the time series must be statistically stationary (i.e. not evolving through time). This is not strictly true for oxygen isotope series (see Figure PH-1 for example), but can be handled to some extent by detrending and prewhitening. Where nonstationarity is one of the phenomena under study, the series can be analyzed by wavelet analysis or in (overlapping) segments.

Figure PH-1. Oxygen isotope record for benthic foraminifera from ODP site 849 (eastern Pacific) showing that neither the mean nor the variance of the data has been constant (stationary) over the past 4.5 million years.

• Filtering reveals the structure of a time series in a particular frequency band. Because the filters are of finite length, however, information is lost at the ends of the series. Inasmuch as we are particularly interested in the youngest end of most paleoceanographic and paleoclimatic time series (i.e. the present), various techniques are used to fill in the ends. These depend on assumptions about the properties of the time series and should be viewed with reservations.
• As Russ mentioned, the autocorrelation function of a geologic time series, rather than the series itself, yields a more robust spectrum of amplitudes, while preserving all the spectral information.
• Substitution of a second time series yields the analogous cross correlation function, which can also be fourier transformed to yield a coherence spectrum and a phase spectrum (which indicates the degree to which shared spectral peaks are in phase (Figure PH-2)).

Figure PH-2. Typical information provided by cross spectral analysis. The top panel shows the separate spectra of the two time series, the middle panel shows the cross spectral coherence of the two time series (with significance levels), and the bottom panel shows the phase relation of the two spectra (note that for the significantly coherent peaks, there is no phase lag in this case).

• Both a significant peak amplitude and a stable phase spectrum over the same frequency band are necessary to provide confidence in the reality of a cross spectral peak.

As discussed earlier, the SPECMAP group (Imbrie et al., 1992, 1993) developed an averaged del¹⁸O curve for benthic foraminifera, which they compared to the 65º N June history of incoming solar radiation (Figure OG-8) and to numerous time series of proxy variables, a number of which were discussed during the last two lectures. The results of their comparisons in the frequency domain are summarized in Figure PH-3.

Figure PH-3. Amplitude-density specta of 20 long climatic time series showing the dominance of variance in bands centered at about 100, 41, and 23 thousand years. The cross hatched areas are the coherence spectra of the time series with the SPECMAP del¹⁸O stack, whereas the solid lines are the spectra for the individual time series. The key to the series illustrated is:

• 1. Milankovitch orbital parameters
• 2. SPECMAP del¹⁸O against 65º N June radiation
• 3. Loess (dust) record from China against del¹⁸O
• 4. Dust record in the Arabian Sea
• 5. SST at 50º N in the North Atlantic
• 6. SST at 41º N in the North Atlantic
• 7. SST in the southern Indian Ocean
• 8. Delta del¹³C between equatorial Pacific surface and deep waters
• 9. Delta del¹³C between equatorial Atlantic surface and deep waters
• 10. Delta del¹³C between equatorial Pacific intermediate and deep waters
• 12. Fragmentation of foraminifera in the Caribbean
• 13. Calcite preservation in the southern Indian Ocean
• 14. Proportions of Atlantic and Pacific deep water at 41º N in the Eastern Atlantic
• 15. Cd/Ca ratio at 31º S in the South Atlantic
• 17. Cd/Ca ratio at 42º N in the north Atlantic
• 20. Delta del¹³C between the Antarctic and equatorial Pacific deep waters

For both the 41ky (tilt) and 23ky (precession) peaks the coherence is strong and persistent enough for Imbrie et al. to propose a simple linear forcing of climate by variations in these two orbital parameters. We will return to their chain of logic, which is largely self consistent and persuasive. However, one should always bear in mind the old statistical axiom that correlation does not prove causation. Cross spectral analysis is a correlation technique.

For the eccentricity period (100ky), there is strong coherence between del 18-O and the other variables, and between del¹⁸O and the eccentricity of the earth's orbit. Because eccentricity contributes so little to the variation in incoming radiation at 65º N (or any other latitude), however, there is no significant coherence between this variable and del¹⁸O in the 100ky band. Thus, the simple linear forcing attributed to precession and tilt cannot apply to eccentricity. This lack of a clear correspondence between forcing and response has led to the development of models of non-linear response, as well as suggestions that resonant oscillations within the earth's atmosphere-ocean-ice system or passage of the earth through an interplanetary dust ring about every 100,000 years are responsible for the major glacial-interglacial frequency.

The phase relations between the various time series is illustrated by the filtered records in Figures PH-4 (23ky band - eccentricity) and PH-5 (41ky band - tilt). The key is the same as above. The slopes of the dashed lines indicate that the series are not in phase with the presumed forcing. They have been arranged so that the phase lag increases from top to bottom. The lags in degrees from the del¹⁸O stack are shown on the left.

Figure PH-4. The 23-ky climate cycles, bandpassed filtered from Figure PH-3. Vertical arrangement is according to phase difference relative to del¹⁸O stack. Dashed lines show systematic phase offsets relative to 65º N radiation. Superimposed, phase-aligned records show high coherence with orbital forcing.

Figure PH-5. The 41-ky climate cycles, prepared as the 23-ky cycles (Figure PH-4).

Note that lags in a phase spectrum are expressed in degrees (or radians) rather than in years, as the lag is a function of frequency. Figure PH-6 shows the degree of coherence and phase relation of the Milankovitch peaks in proxy records versus the SPECMAP del¹⁸O stack and of the stack versus orbital parameters and June incoming radiation at 65º N.

Figure PH-6. Statistical summary of cross spectra of climatic time series. Signs have been changed for consistency with Figure PH-3. The del¹⁸0 stack is compared to orbital indices; other variables are compared to the del¹⁸O stack. Positive phase means that the variable lags the reference variable. k is coherence (in brackets if not significant at 80% level), phase angles are shown with 80% confidence intervals

To present an enormous amount of data in a compact form, Imbrie et al. make use of "phase clocks" for each frequency band (Figure PH-7) The vectors on such clocks make it easy to visualize relative leads (counterclockwise offsets) and lags (clockwise offsets) as well as groupings within multiple data sets.

Figure PH-7. Phases of orbital, radiation, and climatic cycles. As shown in Figure PH-6, some proxy variables lead del¹⁸O, others lag. In most cases, the behavior is the same for all three frequency bands

At all three Milankovitch frequencies, certain sets of proxies tend to group together. In the case of the 100ky (eccentricity) band, the appropriate "zero point" in the eccentricity cycle is unclear. Imbrie et al. chose "maximum eccentricity," even though there is no a priori reason to suppose that this is the key point in the cycle. This zero point occurs after a number of important environmental changes, suggesting that minimum eccentricity or some intermediate point in the cycle may be more significant.

The events that lead minimum ice volume are:

• Minimum deep-water Cd/Ca at 31º S in the South Atlantic
• Maximum SST at 44º S in the Indian Ocean
• Maximum del¹³C depletion in the surface E equatorial Pacific and tropical Atlantic
• Maximum del¹³C enrichment in the Antarctic

Those that lag minimum ice volume are:

• Maximum Caribbean intermediate water carbonate dissolution
• Maximum oceanic del¹³C enrichment
• Minimum dust fluxes
• Maximum SST at 50º N in the North Atlantic
• Maximum ventilation at 3.5km at 41º N

The leads and lags, in years, are the same (within measurement uncertainties) for the 23ky (precession) and 41ky (tilt bands). Note that :

• Lag time (years) = Period of the spectral peak * Lag (degrees)/360.

Thus, for the precession peak at 23ky, a lag of 40 degrees converts to 2,560 years. The same lag in years would be 22 degrees for the tilt peak at 41ky.

Imbrie et al. concluded that the pattern of lags requires a four state system to explain it. These states they label:

• Interglacial
• Preglacial
• Glacial
• Postglacial

Based on the phase relations from the frequency domain analyses, and the nature and magnitude of the changes at critical boundaries (such as the terminations) in the time domain records, Imbrie et al. proposed the following attributes for these four states:

Interglacial (Figure PH-8)

Figure PH-8. First stage of a generic Milankovitch gl;aciation cycle, according to Imbrie et al. (1992).Symbols for Figures PH-8 to 11shown below. AA = Antarctic ocean, NOR = Nordic ocean. For the 23-ky and 41-ky bands, the forcing is 65º N radiation, for the 100-ky band, the forcing is the mode of ocean overturning.

During this state, both sea ice and northern hemisphere continental ice caps are at their minimum extents. Deep water is formed in the North Atlantic both by cooling of relatively saline water in the Nordic (Norwegian + Iceland + Greenland) Seas ("Nordic Heat Pump") and by cooling in the Labrador Sea and open Atlantic ("Boreal Heat Pump"). The deep water moves south to the Antarctic, where sea ice is also at a minimum and the westerlies are close to the tip of South America. There the NADW and southern deep water masses mix to create the deep and bottom waters of the Pacific and Indian Oceans. These waters upwell in the North Pacific and Indian Oceans and return to the North Atlantic via surface currents which transport heat and salt to the northern Atlantic source regions where the "grand tour" began.

Preglacial (Figure PH-9)

Figure PH-9. Second stage of a generic Milankovitch glaciation cycle (see Figure PH-8 caption).

This state is triggered by decreased northern hemisphere radiation, which leads to freshening of the Nordic Seas, thereby terminating the formation of NADW and shutting down the Nordic Heat Pump. The reduction if heat from the north leads to growth in Antarctic sea ice, which pushes the wind field and water masses north. This appears to reduce the rate of overturn of Antarctic waters, which in turn could draw down atmospheric carbon dioxide values (to be discussed in more detail later).

Glacial (Figure PH-10)

Figure PH-10. Third stage of a generic Milankovitch glaciation cycle (see Figure PH-8 caption).

This state results from the continuing growth of the northern ice caps which steer northern hemisphere winds. These produce more convection in the boreal Atlantic, and possibly Pacific (analogous to that occurring in the Labrador Sea today) thereby providing "younger" NAIW and NPIW. The loss of water to the ice caps eventually lowers sea level to the point where ice sheets are grounded on large areas of the continental shelf.

Deglacial (Figure PH-11)

Figure PH-11. Fourth (final) stage of a generic Milankovitch glaciation cycle (see Figure PH-8 caption).

As high latitude northern hemisphere radiation increases, the atmosphere and surface waters warm, evaporation increases, and sea ice retreats from the Nordic Seas. The initial retreat of the glaciers modifies the winds to drive warm saline Atlantic water northward. This accelerates glacial melting and provides the buoyancy flux necessary to turn on the Nordic Heat Pump and form NADW, apparently to a greater extent than during full interglacials, thereby transporting heat to Antarctica to accelerate sea ice melting there and enhanced ventilation of the southern deep water, thereby releasing trapped carbon dioxide and setting the stage for a catastrophic collapse of the continental ice sheets.

The evidence for these states and a model to describe the chain of events in a full glacial-interglacial cycle will be revisited after the productivity, and NADW sub-systems have been examined in more detail.