Probabilistic Stability of an Atmospheric Model to Various Amplitude Perturbations

Abstract

Every forecast should include an estimate of its likely accuracy, as a measure of predictability. A new measure, the first passage time (FPT), which is defined as the time period when the model error first exceeds a predetermined criterion (i.e., the tolerance level), is proposed here to estimate model predictability. A theoretical framework is developed to determine the mean and variance of FPT. The low-order Lorenz atmospheric model is taken as an example to show the robustness of using FPT as a quantitative measure for prediction skill. Both linear and nonlinear perspectives of forecast errors are analytically investigated using the self-consistent Nicolis model. The mean and variance of FPT largely depends on the ratio between twice the maximum Lyapunov exponent (σ) and the intensity of attractor fluctuations (q2), λ = 2σ/q2. Two types of predictability are found: λ > 1 referring to low predictability and λ < 1 referring to high predictability. The mean and variance of FPT can be represented by the e-folding timescales in the low-predictability range, but not in the high-predictability range. The transition between the two predictability ranges is caused by the variability of the attractor characteristics along the reference trajectory.