12.1.1 Interesting Problems for EBMs

• The first problem for EBMs has already come up in Chapter 2 in which we found a possible ice-covered Earth solution to our global energy balance equation. The operating curve showed that if the solar brightness was dropped by only a small percentage, the planet would plunge into a deep-freeze state. Of course, the percentage depends on the details of model parameter choices.
• The second problem is the effect of the land–sea distribution over deep geological time scales where, through continental drift, the landmasses have continuously changed their configurations. It could be that summertime's warmest temperatures determine whether snow will linger over the warm season and allow the growth of an ice field and eventually a large-scale ice sheet. We know that the proximity to the poles and the shapes of landmasses can alter the summertime temperatures. Could EBMs tell us when ice sheets should have been present? The seasonal cycle is a key factor.
• The third concerns the most recent few tens of millions of years during which the Earth has shifted from a warmer state to one with large continental ice sheets, the most prominent being Antarctica and Greenland. What were the conditions for these and what might EBMs have to contribute?
• A fourth has to include the ice sheets that have waxed and waned over North America and Western Europe (but not over Siberia). Observations provide evidence suggesting that the glaciations exhibit a regularity and are in step with the theoretically calculated oscillations of orbital elements of our planet. Might EBMs be of use in understanding the glaciations over the Pleistocene (last few million years), and even the most recent 10 ka, (some authors use 12 ka) called the Holocene?

Not all problems are approachable, not to mention solvable, by EBMs. Atmospheric chemistry and ocean circulation play roles possibly as important as the changing land–sea configuration and the associated radiative-energy-balance effects on summertime temperatures. Sometimes, chemistry and aerosol effects can be inserted into an EBM, but even then the problem might not be solved or may not be interesting. For instance, if atmospheric increased because of some geological process, the EBM contribution may be large, but it is not the importantaspect of the problem (e.g., why did increase?). But if the greenhouse gas and aerosol concentrations are given, it is straightforward to provide EBM solutions. Long-lived ocean current anomalies or surface temperature aberrations (e.g., a shift in the Gulf Stream) can be included by prescribing them, but as yet no one has found a way to include them dynamically in such a simple model. Unfortunately for the EBM aficionado, the use of a general circulation model (GCM) may be far more appropriate for such a forcing as they are likely to excite interesting quasi-stationary wave patterns that affect storm paths (all this is outside the scope of an EBM). As we found in Chapter 10, we can include some oceanic effects in EBMs, but to attempt the circulation of the ocean currents is well beyond the scope of simulation at this level of the model hierarchy. Perhaps the single, most important advantage for EBMs is to spell out how the seasonal cycle can play a role in various paleoclimate problems, as the seasonal cycle responds almost linearly to forcings due to changes in orbital elements, , and aerosol concentrations, as well as the geographical distribution of land and sea. These concepts have been a common theme in the many works of Crowley and his colleagues Hyde, Baum, and Short (for a list of others see the preface of this book.).

12.3.1 Modeling Permian Glacials

Much of climate change over the long term (tens of millions of years) is governed by the greenhouse gas concentrations. This is not an especially interesting aspect of the dynamical behavior of EBMs. The dependence on concentration is logarithmic (see (2.13) and surrounding discussion). When concentration doubles, the global temperature will increase by 2–4 K (including feedbacks, and after equilibrium is attained). If the concentration is halved, the temperature will correspondingly decrease by 2–4 K. Interesting problems for EBMs come up when there is a large change in the seasonal cycle, especially when a landmass moves near the pole. When the landmass is large and its center is over the pole, summers will be too hot to initiate or sustain an ice sheet. The action comes when a landmass is near the pole where the mean annual surface temperatures will be low enough for freezing to occur, but again the key is for the summer temperatures to be below freezing. This latter happens if the landmass edge is near the pole or if there is a field of broken landmasses near the pole. The presence of smaller-scale water-covered regions can moderate the summer temperatures, keeping them below freezing. As Pangea breaks up, the chances are good for this to happen.

A series of studies by Hyde et al. (1990) with a nonlinear (snow/ice–albedo feedback) two-dimensional EBM including a seasonal cycle consider the importance of continental size and positioning with respect to the poles. This study does not include ice-sheet dynamics. There has to be some land near the poles to initiate glaciation. However, the continent covering the poles cannot be too large, otherwise the summers will be too warm to allow snow to build up into an annual ice cover. When the continent is smaller, say 3000 km across, this smaller landmass will allow penetration of the mild maritime summer to suppress the seasonal cycle in the continental interior. Under these conditions, summers will be mild enough (staying below freezing) to allow the snow to build up, eventually leading to an ice field or glacier. The upshot is that the actual geography including shoreline configuration near the pole is important.

An example of conditions that lead to bifurcations (or tipping points) is shown in Figure 12.7 where Crowley and Baum (1993) considered the continental configuration near the South Pole around 300 Ma. In this study, they used the two-dimensional, seasonal EBM similar to the one discussed in Chapter 8. They examined the areas of ice cover for a continuous range of values of the solar luminosity (we have used the more modern term total solar irradiance, TSI 4) from 327 to 333 W m. A sharp bifurcation is found as the ice cap suddenly decreases in area.

12.6.1 EBMs in the Pleistocene: Short's Filter

The study of EBMs leads to some interesting results for the Pleistocene. We start with the 2-D seasonal model of North et al. (1983) (see also North and Crowley, 1985). The model was run with the present TSI, then small percentage changes in the TSI were used. Ice cover was prescribed when the summer temperature fell below freezing. The model was iterated until a steady state was found. Figure 12.17 shows that a bifurcation occurs at a value between 1 and 2 C. Below this value, the “ice sheet” extends completely across the Arctic Ocean and includes Alaska and much of western Canada, even some of Siberia. This “glaciation” is asymmetric between the Eastern and Western hemispheres—there is more ice cover in the Western hemisphere. But the big ice sheet does not resemble the Laurentide Ice Sheet much. There could be some problems with the model perhaps due to its poor spatial resolution. Experience with previous nonlinear, one-dimensional models with ice caps (Chapter 7 and especially Figures 7.10 and 7.12, both taken from Mengel et al., 1988) suggest that the transition to extensive ice cover might be rather rapid.

A comprehensive chapter by Short et al. (1991) shows a number of linear responses to orbital element changes by the 2-D seasonal model (North et al., 1983) discussed in Chapter 8. In this section, we present results from this paper. As referred to in the title of the paper, the responses are the result of filtering the changes in orbital element forcing through the model being thought of as a “filter.” It is useful to note that the model is solved for its mean annual solution, its annual harmonic solution, and its semiannual solution. Then these Fourier components are composed into the complete solution for the seasonal cycle of the temperature field. But it is interesting to examine the effect on the Fourier components directly both in the insolation and in the responses.

First consider the latitude dependence of the forcing (perturbation of the insolation components). Figure 12.18 shows how the insolation as function of latitude changes for extreme values of the obliquity (a) and for the precession varying from perihelion to aphelion (b). It is interesting that the insolation in both (a) and (b) show that some harmonics of the insolation dip to negative values near the equator.

Next consider the geographical distribution of the response in maximum NH summer temperatures (C) due to a change in obliquity from its known extremes 22.1 (solid) to 24.4 (see Figure 12.19). In this case, the eccentricity is set to zero. The continents are shown with blocklike edges to emphasize the course resolution of the model. The tropical response in this figure shows a negative response (180 out of phase with the extratropical response). Short et al. (1991) point out that this is in agreement with results of core RC24-30 taken in the tropical Atlantic (Imbrie et al., 1992).

Figure 12.20 shows the response pattern of northern summer temperatures as the summer solstice moves from aphelion to perihelion. Figure 12.21 combines the difference of extremes for obliquity and maximum eccentricity (0.06). The Southern Hemisphere is also depicted at summer solstice by combining data from the two hemispheres at six-month intervals into one map.

Next are considerations of how the changes in insolation are filtered into the thermal responses especially over oceans where deep-sea cores are collected. We are always interested in the maximum summer temperature in this series of simulations. While the summer temperature field is a linear response to the forcing, the maximum summer temperature is not a linear function of the forcing because finding the maximum is the result of finding the root of the equation after we have found , where is the and is the time after winter solstice in the repeating seasonal cycle. In the center, the great continent of Asia where the relaxation timescale is much less than a year, the response (a) mirrors that of the forcing (b) at each of the orbital signals (see Figure 12.22).

As we look at different points in the Atlantic, the response as filtered from the forcing (and the maximum found) presents different results from points in the center of a large continental landmass. Figures 12.22 and 12.23 focused on the north Atlantic shows the time series and amplitude spectra of the maximum summer temperature and the insolation. Here we see strong peaks at the obliquity and precession periods, but also several low-frequency peaks. These are related to the eccentricity and its effect on the other orbital elements.

In equatorial Atlantic (0N, 30E), as seen in Figure 12.24, the obliquity signal is weak, but the eccentricity frequencies corresponding to periods of 400 and 100 ky are very strong. Note also a first harmonic of the two precession frequencies (periods of and 10 ky). These harmonics are explained in Figure 12.25 (see also Short and Mengel, 1986). Figure 12.25 shows the response as function of lag (days) after the winter solstice. At 86 ky, the temperature curve (indicated by +s) is symmetric about the summer solstice as the Sun passes over the equator twice during the year. There are two maxima of equal magnitude at this time. Then a quarter of the precession cycle later at 90 ky (indicated by s), there is one absolute maximum. Next consider 94 ky (indicated by s), when there are two (nearly) equal maxima. Finally, in the cycle at 98 ky (indicated by △s), there is a single global maximum. This sequence shows that there are two global maxima occurring at a separation of about one half of the precession period. Finding the absolute maxima as a function lag in the seasonal cycle is a nonlinear procedure from the direct forcing amplitude: hence, the first harmonic of the forcing.

Before moving away from spectra, we draw attention to the latitude dependence of observed spectra. Figure 12.26 shows two spectral densities of temperatures estimated from microfossils along the Mid-Atlantic Ridge. The right-most spectrum from a latitude of 41 shows a prominent peak at 23 ky and the leftmost shows a similar spectrum but taken at 54N, where the 23 ky peak has disappeared and a peak at 41 ky appears prominently. This is close to what the 2-D model would suggest. As we leave Short's work, we see how the linear model is able to produce spectra comparable to our expectation and even some data in the middle and lower latitudes. Moreover, the nonlinearity of the maximum summer temperatures as driven by linear responses of the temperature field invites the possibility of under- and overtones appear in the spectra along with evidence of the eccentricity cycles.

An interesting set of paleoclimatological experiments with the linear seasonal 2-D EBM were recently conducted by Zhuang (private communication, 2015). He employed the same physical model as was used in North et al. (1983), but as modified by Stevens who used a finite-difference numerical scheme based on multigrid relaxation as discussed in Bowman and Huang (1991), Huang and Bowman (1992), Stevens and North (1996). No parameter changes were employed in his work. Among his experiments were the following cases shown in Figure 12.27. Panel (a) shows a simulation of the linear (no ice feedback) model wherein the full Last Glacial Maximum (LGM) were employed, including the orbital parameters, the concentration of , and the placement of ice in the Laurentide Ice Sheet, the Greenland Ice Sheet, and the Fenno-Scandian Ice Sheet. As usual, no topography and no dynamical ice volume model were introduced in the EBM. It can be seen that the summertime 5 C line hugs the equatorward edge of the ice sheet on both continents. While he did not iterate the solution to include the ice–albedo feedback, it appears that this would be a solution if the ice line were to be close to the 5 C line. Figure 12.27b shows the simulated temperature (linear model) when the ice albedo of the Laurentide Ice Sheet is removed and replaced by bare land, while the concentration of and the orbital elements were unaltered from their LGM settings. As the figure shows, the 5 C line jumps to leave only Greenland and northwestern Eurasia with ice cover. It appears that the ice albedo is dominant over the other conditions, which seem to play a rather minor role. This is a rather puzzling result. It suggests that there are equilibrium solutions for the big ice sheet and for a small one. An attractive possibility is that if the ice is placed there it will stay. If remove it will remain removed. This may be a kind of neutral stability. Perhaps a large fluctuation could make it jump, aided by the orbital forcing. Another result that has been known for some time is that the orbital perturbation alone is not enough to start an ice sheet in North America. Might a large fluctuation do the trick? We leave this to the imagination of our readers.

12.6.2 Last Interglacial

Figure 12.28 shows a time series going back to 420 ka from the ice core data derived from the Vostok site in Antarctica (Petit et al., 2001). Estimates of temperature departures from a modern long-term average are plotted versus time, increasing from right to left. The last interglacial is the peak between roughly 110 and 140 ka. Other information from this core including other proxies, concentrations, and spectral estimates can be found in Petit et al. (1999).

Crowley and Kim (1994) employed the two-dimensional, seasonal EBM (described in Chapter 8) to study temperatures and inferred estimates of sea level during the last interglacial (LIG). They found model indications of 3–C July (mid-summer) increases at high latitudes from 140 to 130 ka. In these simulations, they kept fixed but included the forcing from obliquity and precession changes over that period. The three sites are (75N, 120W), Northern Canada, (75N, 30E) Arctic Ocean north of Scandinavia, and (45N, 90W), US Canadian border near Lake Superior. They note the following:

Although it is at the same latitude as the Barents Sea site, peak warming at the Canadian Arctic site occurred 4000 years later. This response reflects the greater importance of the precession-controlled continentality effect on the Canadian site. However, all sites were warmer than they are now by 130 000 years ago.

They suggest that the Milankovitch forcing might be responsible for the decaying ice sheet (see Figure 12.29).

The problem of temperature and sea level (ice volume) during the LIG has recently received attention especially because of its similarity to the present interglacial. Some studies suggest that sea level might have been much higher during the LIG.

12.6.3 EBMs and Ice Volume

Since the two-dimensional-seasonal EBM (North et al., 1983) simulates the seasonal cycle surface temperature well to first order, it was natural that Peltier and colleagues use it in their modeling of the glacial–interglacial cycles of the Pleistocene (DeBlonde and Peltier, 1991; Hyde and Peltier, 1987). Also, Pollard (1982) has tried including an ice model to an EBM comparable to the beachball model (see Section 8.1). Others, including Paillard (2001), tried combining ice models to EBMs. As noted by Huybers and Wunsch (1994), this combination always leads to a number of new tunable parameters, as there are only eight or so glaciations at the 100 ky timescale. The problem of tuning comes up immediately, as one tries to model glacial growth by inserting precipitation in the form of snow. But this is only the one of the problems encountered as one contemplates the flow and decay of ice sheets. This is not to discourage experimentation wherein much might be learned, but it is to recognize that this is a very big step fraught with potential errors of overfitting.

Crowley and Hyde (2008) re-plotted the Lisiecki and Raymo data in a way that emphasizes the increase of amplitude and the lower frequencies of the fluctuations as a function of time. They note the striking resemblance between Figure 2.10 and Figure 12.30 that have been reproduced from Crowley and North (1988). Crowley and Hyde (2008) argue that a series of threshold crossings are in progress. First is a small ice sheet in North America, then a transition to a larger one. Next is a smaller ice cover in northeastern Europe. Finally, to come is a large ice sheet in Eurasia. Their model goes beyond the simple EBMs of this book in employing a dynamical ice sheet.

An unsolved puzzle is the transition from the so-called 41 ky world to the 100 ky world. This is clearly shown in Figures 12.13 and 12.30 wherein the peaks in global ice volume suddenly change from a dominant obliquity forcing to that associated with the less well understood 100 ky forcing. Interesting attempts at explaining the transition are reviewed in Raymo and Huybers (2008). One mystery is: Why are not the 21 ky peaks evident in the “41 ky world?” Raymo notes that the precession forcing is antisymmetric between the hemispheres and she posits the ice volume forcing may be canceling. Huybers has been interested in the possibility of integrating the insolation over the entire summer season, whose length changes with the precession phase and is modulated by the eccentricity.

12.6.4 What Can Be Done without Ice Volume

Our use of EBMs in paleoclimatology has been to partially answer questions about the inception of glaciation as in Sections 12.4 and 12.5. EBMs might be able to address these issues without any new adjustable parameters. This is as opposed to trying to map the detailed time series in order to compare with a corresponding time series of data. We are guided by the results shown in Figure 7.10 where a latitude-only seasonal EBM was run with a disklike continent centered at the pole. This simulation shows how as a threshold of some control parameter is crossed there is a qualitatively different steady state. Figure 7.12 shows that as the threshold is crossed, it may take tens if not hundreds of years to adjust to the new steady state. The long adjustment time is a result of the nonlinear snow-albedo effect. This time as illustrated in that figure is sensitively dependent on the initial conditions in the problem. This is done without any noise forcing in the problem. One wonders how this bizarre behavior would be affected by background noise as in Chapter 9. Figure 7.14 shows another case where a zonal band of land replaces the pole-centered disk. In this band-of-land geography there are two thresholds. This kind of simulation is tricky because one must be very careful to use high latitudinal resolution. One cannot help but wonder what happens if there is longitudinal dependence in this nonlinear seasonal EBM. And what if noise is included? Might there be quasi-periodic jumping? Some seasonal simulations of two dimensional, seasonal EBMs including nonlinear snow feedback and with realistic and idealized two-dimensional geography (in their case, the late Permian geography) have been conducted by Baum and Crowley (1991). The two-dimensional seasonal simulations seem to confirm the one-dimensional works of Mengel et al. (1988) and Lin and North (1990), and Huang and Bowman (1992) that bifurcations exist when (not-too-large, or appropriately fragmented) landmasses lie near or at the poles when forced by a seasonal cycle with an interactive snowline. Then Crowley et al. (1994) used to Genesis GCM to investigate the effect. They showed a comparison between the EBM and the GCM results. The GCM seems to confirm the bifurcationand the corresponding catastrophic change in ice cover as the TSI is varied.