Interpreting Forecast Error Growth and Saturation Curves

Forecast errors arise from initial errors as well as model errors. Determining their relative importance has implications for forecast system development and potential predictability. To this end the short-range forecast error is often split into a part attributed to the exponential growth of initial error and a part attributed to model error, as originally proposed by Kalnay and colleagues in the 1980s and performed in many studies since. More recent diagnosis of operational NWP error growth curves suggests, however, that depending upon how one defines the forecast error and performs the split, such attributions can be ambiguous and lead to wrong conclusions. The basic difficulty is the assumption that in all chaotic systems the short-range errors of perfect-model forecasts grow exponentially due to linearly unstable perturbation dynamics. This ignores the fact that in general such error growth is not of pure modal form. It also ignores the fact that in a system with many degrees of freedom, the effect of the unresolved chaotic dynamics on the resolved dynamics is often indistinguishable from that of a stochastic forcing, and the resolved perturbation dynamics are not unstable in all contexts and for all system components. The combined effect of these features can lead to deviations from pure exponential forecast error growth even for a perfect model. Indeed a perfect model with linearly stable perturbation dynamics can display all the varieties of forecast error growth encountered in practice depending upon the singular values of the associated linear evolution operator. It is also generally easier to explain the almost universally encountered exponential saturation of forecast error with a linearly stable and stochastically forced error model than with a linearly unstable and nonlinearly saturating error model. An exciting possibility raised by this linear dynamical reinterpretation of forecast error growth and saturation is to perform detailed diagnosis of weather and climate forecasting systems using their error statistics at both short and long forecast ranges.