It is useful to summarize those elements which appear to be important for constructing a skillful simple model of the glacial cycles. These elements are first qualitatively discussed, after which a new simple model is constructed.
5.4.1 Qualitative features
1. Coupling of termination with orbital variations: In Chapter 4 it was found that obliquity and glacial terminations appear to be nonlinearly coupled. In particular, terminations appear to be triggered by high values of obliquity. In the Imbrie model terminations tend to occur during a positive phase of the obliquity
forcing because melting only occurs when insolation forcing is greater than negative ice-volume. Similarly, in the Paillard model the switch from a full glacial to an inter-glacial only occurs when insolation surpasses some threshold. Thus it seems that tying the triggering of a termination to changes in the Earth’s orbit and/or orientation is an effective means of setting the phase of the glacial cycles.
2. Slow accumulation: Terminations only occur every second or third obliquity cycle, suggesting the presence a climatic timescale longer than the obliquity period.
Earlier it was shown that the Imbrie model can generate a long timescale response by slowly reaccumulating sufficient ice-volume for an ablation event to occur. Similarly, the Paillard model transitions from a glacial to full glacial state only after sufficient ice-volume has accumulated, and this typically requires a period of two or three obliquity cycles. Thus one means of generating a long timescale response is to specify an amount of ice-volume required for a termination to occur and employ a relatively slow accumulation rate.
3. Episodic reglaciation: Apart from the basic structure of rapid ablation and slow accumulation, one also observes significant kinks in the reglaciation process.
That is, reglaciation seems to proceed episodically. The reglaciation after termination two provides a good example where ice-volume appears to plateau during substages five and four before bottoming out towards the Last Glacial Maximum. The plateaus are further punctuated by local maxima, but which are smaller features than the general plateau structure. These plateaus are some of the best constrained aspects of ice-volume change, being unambiguously recorded in the coral terraces of uplifting topography [e.g. at Barbados,Broecker, 1968;Gallup, 2002]. The Paillard model goes further in mimicking the detailed features in the δ18O record than does the Imbrie model. It achieves this more detailed structure by specifying climatic states so that the model output tends to re-glaciate in a series of steps. It is thus desirable for a model not to monotonically reglaciate, but rather to have a sequence of reglaciation episodes, perhaps paced by either orbital or stochastic variations.
4. Memory: The Paillard model cycles through a fixed sequence of states, requiring the system to remember its past state. The case of the Imbrie model is more involved.
First, one of the major problems with the parameterization of the Imbrie model shown in Figure 5-1b and c is that terminations occur in a single timestep. This means that
the size of the terminations are related to the time-stepping used in integrating the model. Thus the Imbrie model, as presented here, might better be thought of as a map which describes how ice-volume at one instant is related to ice-volume 1KY later. Without very rapid melting, the terminations in the Imbrie model would abort as soon as the threshold condition is crossed. If the ablation events in the Imbrie model incorporated a time delayed state dependence, termination could continue beyond this threshold crossing without resorting to such rapid timescales.
5.4.2 A quantitative expression
Each of the features discussed in the previous section can be incorporated into a simple, deterministic model using only a handful of adjustable parameters. Written in discrete form the model is
Vt = Vt−1−VtP−L× b− F
T P =
0 if F < b
p if F > b. (5.5) Here V is ice-volume, L is a time-lag, T is a time-constant, b is a threshold which also gives a bias towards positive accumulation, and P is an exponent whose value depends on whether the model is accumulating or ablating. When b > F, P = 0, making accumulation linearly dependent on insolation anomalies. Ablation occurs when the forcing is greater than the accumulation bias, and now P is some positive integer (recall, all parameters are chosen to have only one significant figure), making the rate partially dependent on ice-volumeLyears ago. IfVtis less than zero, it is reset to zero on the physical grounds that one cannot have negative ice-volume. F is given by Eq 5.2, and represents anomalies in modes of insolation forcing. If precessional effects are included in F, the model has six degrees of freedom; otherwise, if only obliquity forcing is used, it has four. To distinguish the model given by Eq 5.5 from the Imbrie and Paillard models, it will be referred to as thenew model.
Though the notation is more cumbersome, it is possible to write Eq 5.5 so that the value of P remains fixed. First define γ = (b− F)/T so that the model may be written as
Vt =Vt−1−γVtp(−L|γ|+γ)/2,
where|.|indicates the absolute value. The exponent ofVt−L goes to zero whenγ <0, and equals p for γ ≥ 0, thus playing the same role as the conditionality, but not
requiring any rules for switching the exponent.
It proves convenient to define the units of this model to be the ice-volume equiv-alent of a hundred meters of ice-volume. By this definition, one hectometer (hm) represents an important threshold in the model system. When lagged ice-volume is less than one hm, the ablation term, VtP−L, is small and little melting occurs, while values greater than one hm can induce rapid ablation. For a glacial termination to occur in the new model, two conditions must be met: the ice-volume must exceed one hm and the orbital forcing must induce a melting state. When one hm of ice-volume accumulates more slowly than the period between orbital melting states, the basic period of the glacial cycles will be controlled by the accumulation rate.
The formulation of Eq 5.5 bears two important parallels with the much more sophisticated thermomechanical ice-sheet model ofMarshall and Clark [2002]. First, the results from the thermomechanical ice-sheet model indicate that thermal enabling of basal flow is an important feedback controlling the deglaciation of North American ice-sheets, and that this basal warming requires the presence of thick, high-elevation ice-sheets. As noted, the new simple model also requires sufficient ice-volume for a deglaciation to occur. A second parallel is that basal temperatures in Marshall and Clark’s model lag surface temperatures by 10KY. The parameter L in Eq. 5.5 can thus be thought of as the timescale for heat to penetrate from the surface of an ice-sheet to its base. There are, however, some unresolved differences. Marshall and Clark [2002] conclude that at large sizes ice-sheets become independent of the orbital forcing and affect their own demise. The results of Chapter 4 imply the opposite, that the initiation of deglaciation are triggered by high obliquity states, but the physical mechanism which ties obliquity to the triggering of terminations is unclear. One possibility is that increased high-latitude insolation causes surface melting of the ice-sheet, and the associated run-off aids in lubricating basal slippage. More work should be undertaken to determine the extent to which orbital variations could influence the results of Marshall and Clark’s model and the extent to which Eq. 5.5 is a faithful simplification of the model physics, but which is beyond the scope of this thesis.
Using the simulated annealing algorithm, a best fit is achieved between the model results and observations for a time constant T = 80KY, accumulation bias b = 1, power n = 8, lag L = 9ky, precessional phase φ = 100◦, and insolation ratio α = 0.04; yielding a squared-cross-correlation of 0.60. If the model is forced only by obliquity variability, the best fit occurs for t = 90KY, b = 0.9, n = 9, and L =
9KY; and achieves a squared cross-correlation between model results and observations of 0.43. The meaning of this correlation will become more clear when the model results are compared with one another. Figure 5-5 shows the model results for the Fop (obliquity and precession) and Fo (obliquity only) forcing cases, as well as the associated periodograms.
−600 −500 −400 −300 −200 −100
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−600 −500 −400 −300 −200 −100
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Figure 5-5: Model results (red, thick line) where parameters are chosen to maxi-mize the cross-correlation with EOF1 (black, think line) a Model results using an orbital forcing comprised of obliquity and precessional variations (Fop) along with the periodogram of the model results. b Results when the model is only forced by obliquity (Fo). The vertical dashed lines at right indicate frequencies of 1/100, 1/70, 1/41, 1/27, and 1/23KY. Concentrations of variability exist at bands near each of the indicated frequencies for both model results a and b.