Markov models are used widely in meteorology and oceanography to describe the statistical behavior of transient eddies. Often, the time evolution of the transient eddies is assumed to be diffentiable. This assumption is inconsistent with the fact that first order Markov processes are not differentiable in time. Differentiability constrains the rate at which a variable may decorrelate with itself at later times; thus Markov models, unconstrained by differentiability, produce much stronger decorrelation rates at small lags than are possible for differentiable time series. As a result, an empirical Markov model fitted to a differentiable time series will display discrepancies of a distinctive character at sufficiently small time-lags. These discrepacies are consistent with recent results obtained in fitting Markov models to quasigeotrophic models and to general circulation models. Precisely the same discrepancies occur when a Markov model is fitted to covariances that are twice differentiable with respect to lag or to spectra that decay faster than the third power of frequency, both of which arei equivalent to differentiable time series (in the mean-square sense), and neither of which are compatible with first order Markov models. It is shown in an appendix that certain effects of spatial filtering cannot account for the discrepancies, leaving differentiability as the most compelling explanation for the discrepancies. It is suggested that Markov models are still useful despite being fundamentally incompatable with differentiable time series because realistic covariances asymptote to Markov model form at sufficiently long lags. This conjecture is proven for a one-dimensional time series under very general conditions.
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Center for Ocean-Land-Atmosphere Studies
last update: 30 July 1998
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