Variation of the predictability in a low-order spectral model of the atmospheric circulation

Temporal variations in the predictability of a simple atmospheric model are studied. The model is a spectral two-layer quasi-geostrophic hemispheric model, truncated at T5. It has chaotic properties, illustrated by 4 positive Lyapunov exponents and a smooth power spectrum of the first empirical orthogonal function (EOF). The chaotic behaviour is responsible for the eventual separation of initially nearby phase points. We use the adjoint of the tangent linear equations to obtain the few directions in which errors grow most rapidly. With the growth rates in these directions, we can estimate the distribution function for the errors. Comparison with a Monte Carlo method shows satisfactory agreement. With a 100-day run, we show how the predictability varies over the attractor. We discuss how, and under what conditions, the method can be used in a global circulation model (GCM). From a dimension derived from an EOF analysis, we obtain an estimate for the dimension of error growth. This estimate implies that information on the probability distribution of errors can be obtained at about 150 times the computational cost of a single model run.