A reexamination of methods for evaluating the predictability of the atmosphere

Pioneering work by Lorenz (1965, 1968, 1969) developed a number of methods for exploring the limits of predictability of the atmosphere. One method uses an integration of a realistic numerical model as a surrogate for the atmosphere. The evolution of small perturbations to the integration are used to estimate how quickly errors resulting from a given observational error distribution would grow in this perfect model context. In reality, an additional constraint must be applied to this predictability problem. In the real atmosphere, only states that belong to the atmosphere's climate occur and one is only interested in how such realizable states diverge in time. Similarly, in a perfect model study, only states on the model's attractor occur. However, a prescribed observational error distribution may project on states that are off the attractor, resulting in unrepresentative error growth. The 'correct' error growth problem examines growth for the projection of the observational error distribution onto the model's attractor. Simple dynamical systems are used to demonstrate that this additional constraint is vital in order to correctly assess the rate of error growth. A naive approach in which this information about the model's 'Climate' is not used can lead to significant errors. Depending on the dynamical system, error doubling times may be either underestimated or overestimated although the latter seems more likely for more realistic models. While the magnitude of these errors is not large in the simple dynamical systems examined, the impact could be much larger in more realistic forecast models.