Because of the small net rates of energy flow involved in very long-term changes in ice volume (10−1 W m−2) it will be impossible to proceed in a purely deductive manner to develop a theory for these changes. An inductive approach will be necessary entailing the formulation of multi-component stochastic-dynamical systems of equations governing the variables and feedbacks thought to be relevant from qualitative physical reasoning (e.g. “conceptual models”). The output of such models should be required to conform as closely as possible to all lines of observational evidence on climatic change and, in addition, should have a predictive quality in the search for new observational evidence. Moreover, the models themselves should be required to satisfy the general conservation laws and all the results of physical measurement of the fast response (high energy flow) processes in the system that generally lead to diagnostic relationships. General discussions of these questions are given by Reference Saltzman and HechtSaltzman (1983 and in press) and Saltzman and Sutera (in preparation Footnote †).
A prototype of such an inductive model, recently developed by Saltzman and Sutera (in preparation Footnote †), is described. The model is formulated by considering the feedbacks that are likely to dominate, in the form of a nonlinear dynamical system governing three prognostic components; continental ice mass ς, marine 1ce mass x, and mean ocean temperature Θ (see Fig.1). The dynamical climatic system is the following:
Fig. 1 Schematic meridional cross-section of the climatic system showing the three main prognostic variables treated in this study: ς (total continental ice mass extending to the grounding line), χ (total marine Ice mass; ice shelves, icebergs and pack ice beyond the grounding line), and Θ (mean ocean temperature). Other prognostic variables not considered in this study are isostatic depression e and ice thickness above sea-level h.
where the primes denote departures from an equilibrium, F denotes external deterministic forcing, R denotes stochastic forcing, and the coefficients are positive constants (e.g. c2 −1, the only linear damping time constant in the system, 1s taken to be 10 ka).
The free, unforced, solution is shown in Figures 2(a) and (b) without and with stochastic perturbation, respectively. It can be seen to have several features In common with the l8/l60-der1ved records of ς, e.g. a period of nearly 100 ka with rapid deglaciations, and the solution also predicts concomitant variability in χ and Θ. The distributions of ς’, χ’ and θ’ corresponding to six consecutive points of interest in this solution cycle are shown in Figure 3.
Fig. 2 Solution obtained for the ç’, χ’, Θ’) system of Equations (1) (2) and (3): (a) with no deterministic or stochastic forcing and (b) with stochastic forcing included in Equation (3). The curves shown are for the non-dimensional values ς* = 1.45Ζ−1ζ’, χ* = 0.75X−1 x’, and θ* = 0.960−1Θ’, which can be inverted to give ς’, χ’, and Θ’ for any choice of characteristic ranges of fluctuations of ς, χ, and 0 denoted by Θ, X and Θ, respectively.
Fig. 3 Schematic snapshot representations of the solution shown in Figure 2(a), at selected consecutive points in time starting with maximum continental glaciation. The first three points (1,2,3) represent the rapid deglaciation phase and the next three points (4,5,6) represent the slower glacial buildup.
