Optimal Perturbation of a Coupled Lorenz Model

The conditional nonlinear optimal perturbation (CNOP) technique is a useful tool for studying the limits of predictability in numerical weather forecasting and climate predictions. The CNOP is the optimal combined mode of the initial and model parameter perturbations that induce the largest departure from a given reference state. The CNOP has two special cases: the CNOP-I is linked to initial perturbations and has the largest nonlinear evolution at the time of prediction, while the other case, CNOP-P, is related to the parameter perturbations that cause the largest departure from a given reference state at a given future time. Solving the CNOPs of a numerical model is a mathematical problem. In this paper, we calculate the CNOP, CNOP-I, and CNOP-P of a coupled Lorenz model and study the properties of these CNOPs. We find that the CNOP, CNOP-I, and CNOP-P always locate the boundary of their respective constraints. This property is also demonstrated analytically for the model whose solutions depend continuously on the initial and parameter perturbations, which provides a theoretical basis for testing the accountability of the numerically computed CNOPs. In addition, we analyze the features of the CNOPs for the coupled Lorenz model and explain their structures