Analysis of Forecast Error Growth in Atmospheric Multiscale Lorenz Systems

In classical low‑dimensional chaotic systems, small initial‑condition errors grow exponentially on average in the tangent‑linear regime, with a rate set by the leading Lyapunov exponent, before entering a nonlinear regime in which the growth follows a quadratic law and saturates at a finite error amplitude. In systems with coupled temporal and spatial scales, the growth of initial‑condition errors is scale‑dependent and is most appropriately described by a power‑law behavior. We demonstrate how the parameters of the power law are linked to the intrinsic properties of individual scales and to the coupling between them. In systems where the model does not perfectly represent reality due to the omission of small temporal and spatial scales, the mean growth of model error (in the absence of initial‑condition error) can be approximated by a quadratic law with an additional parameter characterizing model error. To describe this process, we extend Orrell’s definition of drift by interpreting its generation at each time step, within our hypothesis, as an effective initial‑condition error that evolves according to classical chaotic growth. Based on this hypothesis, we explain the values of the parameters governing the model‑error growth law. The interpretations of the parameters and the underlying hypotheses are tested using multiscale atmospheric Lorenz systems.