We consider a nonlinear climatic oscillator, which includes radiation balance, oceanic thermal inertia, a highly simplified hydrological cycle, the mass balance and the plastic flow of ice sheets, and the elasticity of the Earth’s lithosphere and the viscosity of its mantle, as well as their various inter-actions. The study of the unforced behaviour of the system (Reference Källén, Crafoord and GhilKä11én and others 1979, Reference Ghil and Le TreutGhil and Le Treut 1981, Reference Ghil, Ghil, Benzi and ParisiGhil and Ravantzis in press) has shown the existence of a self-sustained periodic oscillation with amplitude of a few degrees Celsius, in the absence of any periodic forcing. The free period of the oscillator, depending on model parameters, lies roughly between 5 and 15 ka.
The forced oscillations of this climatic oscillator is studied next (Reference Le Treut and GhilLe Treut and Ghil 1983, Reference Ghil, Ghil, Benzi and ParisiGhil in press): the model is subjected to forcing at the astronomical periodicities of precession 19 and 23 ka, obliquity 41 ka, and eccentricity 100 and 400 ka. The forcing is assumed to act on the climatic system by variations in mean annual isolation, in the case of eccentricity, as well as by its effects on the ice-mass balance through the nonlinear precipitation-temperature feedback.
The effects investigated cause only small changes in ice-mass V and global temperature T when self-sustained oscillations are absent. In their presence, a nonlinear resonant response to the forcing leads to large changes in T and V. The systematic study of the Fourier power spectrum for various conditions of forcing gives an insight into the dynamics of the model. For low values of forcing, the response is at the frequency of the free oscillation, at the frequency of the forcing and at combination tones of these.
For higher values of the forcing, due to the mechanism of frequency locking (entrainment), the frequency of the free oscillation disappears from the spectrum and is replaced by the harmonic of a forcing frequency. In all cases, the nonlinear character of the response also leads to combination tones, i.e. to linear combinations of the forcing frequencies with integer coefficients. Among these frequencies, the largest peaks occur near 100 and 10 ka. The peak near 100 ka corresponds to a re-synthetization of the eccentricity period when the model is forced at the precessional periods.
Detrainment (loss of frequency locking) leads to a change in the spectrum of the model: the sharp peaks in spectral density at the forcing frequencies and at their combination tones become superimposed on a continuous background. The spectral power in the background decreases with increasing frequency, like random red noise. The deterministic aperiodic behaviour associated with this frequency-dependent background leads to a loss of predictability which is studied first by comparing model solutions with different initial conditions. Small differences in the initial conditions produce errors as large as the total amplitude of model solutions in a time of the order of 100 ka.
We have also investigated the properties of the lagged-correlation function of the model (Reference Le Treut and GhilLe Treut and Ghil 1983, Reference Ghil, Ghil, Benzi and ParisiGhil in press). In the case of a perfectly periodic solution, the autocorrelation is perfectly periodic itself with an amplitude equal to 1. In the case of a red noise-like signal, the amplitude of the lagged correlation decreases exponentially with a characteristic decay time corresponding to the relaxation time of the model. For those cases in which a line spectrum is superimposed on a continuous spectrum, we observe first an exponential decay of the lagged correlations, leaving a residual periodic or quasi-periodic part of amplitude considerably smaller than 1.
These results suggest that the ability to simulate accurately the palaeoclimatic history of the Earth decreases with the length of simulated time. Complete lack of correlation between a simulated history and one reconstructed from data is likely to occur over a time interval of 100 to 1000 ka.
Spectral information, on the other hand, is considerably more robust (Reference Hays, Imbrie and ShackletonHays and others 1976, Reference OerlemansOerlemans 1982, Reference Le Treut and GhilLe Treut and Ghil 1983, Reference Ghil, Ghil, Benzi and ParisiGhil in press). Changes in the spectrum in different proxy records between early and late Pleistocene, for instance, might be statistically significant. Further modelling efforts are desirable to ascertain whether these are due to changes internal to the climatic system (Reference SerginSergin 1980) or to changes in orbital frequencies and amplitudes over the time scale of 1 OOO ka. (Reference Buys, Ghil, Berger, Hays, Imbrie and SaltzmanBuys and Ghil in press).