Parameterizations

The sensitivity of the Imbrie and Paillard model results to changes in parameterization were discussed earlier (see Figures 5-2 and 5-4). Figure 5-7 shows that the sensitivity of the new model to perturbations. Changes in the time-constant, the time-lag, and the forcing function of the new model all strongly influence the cross-correlation between EOF1 and model results. Generally speaking, the Imbrie model is least sensitive to perturbation in the parameters, followed by the Paillard model, and then the new model. According to the rules of the simple model building game outlined by Saltzman [2002] the new model must be discounted to some degree due to the sensitivity of the results to small perturbations. Later, the model is shown to be capable of chaotic behavior, and it is not surprising that small changes in its parameterization can cause large changes in its behavior.

Just as model robustness to small changes in parameterizations is desirable in terms of achieving a simple result, so is robustness to perturbations in the forcing func-tion. It is expected that the external climate forcing contains not only low-frequency shifts, but also relatively rapid perturbations due to solar variability, cosmic dust, and high frequency perturbations to the earth’s orbit [e.g. Muller and MacDonald, 2000].

Internal variability should also be expected in the form of changes in albedo (e.g.

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Figure 5-6: Sample trajectories of each model using randomized initial conditions:

a the Imbrie model, b Paillard mode, and c the new model. Each of the models converge onto a single trajectory. The last realization from each model is plotted in red.

clouds, sea-ice, and snow cover variability), variations in atmospheric composition and aerosol loading (e.g. dust, forest fires, and volcanic eruptions), and variations in the atmospheric and oceanic fluxes of moisture and heat, to name but a few of the expected sources of variability. Figure 5-8 shows how the squared-cross-correlation between model results and EOF1 falls off with the addition of increasing amounts of band-limited white noise to the insolation forcing. Plotted are the average squared-cross-correlation between EOF1 and a hundred model results obtained from random realizations of the forcing with the prescribed noise level. The sensitivity to noise

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Figure 5-7: Similar to Figures 5-2 and 5-4, but now the sensitivity of the squared-cross-correlation between the new model and EOF1 to parameter perturbations.

Starting from the best fit, the cross-correlation between model results and EOF1 is computed over the last 650KY after varying one of the parameters over a range of plausible values while holding the other fixed. From left to right are the time-constant T measured in KY, the accumulation bias b, the power-law exponent n, the time-lag L measured in KY, and the precessional phase φ and amplitude α.

initially appears to scale with the degrees of freedom available to each model. Thus, while the unperturbed Paillard model results describe the most variance, when the fraction of noise to orbital forcing variance is 0.1, all three models have an average squared-cross-correlation with EOF1 of roughly 0.3. By the time the noise variance equals the insolation forcing variance, all the models describe only a minor fraction of EOF1 with the new model preforming the worst. The forcing perturbation results indicate that the skill of these models strongly depends on how important stochastic variability is at long timescales.

In calculating the response of the simple models to stochastic forcing perturba-tions, it is assumed the system is discrete and the perturbations are uncorrelated, thus corresponding to the Ito calculus. For a discussion of Ito versus other forms of calculus see Penland [2003]. A more detailed investigation would incorporate the likely time-correlated nature of the stochastic perturbations, the continuous (i.e. not

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cross−correlation

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Figure 5-8: Model sensitivity to the addition of band-limited white noise to the forcing. Plotted is the squared-cross-correlation between the model results and EOF1 using the F^op parameterizations. The x-axis indicates the ratio of noise to orbital forcing variance — a value of one indicates equal noise and insolation variance. Note the x-axis is logarithmic. Each value is the average of a hundred noisy model runs.

Results assume a discrete system corresponding to the Ito calculus.

discrete) nature of ice-volume variability, and possibly such factors as the integrated response of the climate system to higher-frequency annual or even diurnal insolation variability. The more simple stochastic response calculated here suffices to make the point that the response of these models critically depends on the degree of stochastic variability. If at timescales of tens of thousands of years, most of the climate sys-tem is controlled by low-frequency insolation forcing, then the results of these simple models can be interpreted at face value. Indeed, the coupling of obliquity with the terminations provides evidence that low-frequency shifts in insolation are important for controlling the long-term evolution of the climate system. On the other hand, the presence of precession band variability in a variety of climate indicators indicates the importance of the annual cycle (see the discussion regarding rectification in Chap-ter 2), and suggests that high-frequency variability also contributes to long-Chap-term

cli-mate change. Separating the stochastic from the deterministically forced components of climate variability remains a challenging problem in dynamical paleoclimatology.