Long-window hybrid variational data assimilation methods for chaotic climate models tested with the Lorenz 63 system

Abstract. A hybrid 4D-variational data assimilation method for chaotic climate models is introduced using the Lorenz '63 model. This approach aims to optimise an Earth system model (ESM), for which no adjoint exists, by utilising an adjoint model of a different, potentially simpler ESM. The technique relies on synchronisation of the model to observed time series data employing the dynamical state and parameter estimation (DSPE) method to stabilise the tangent linear system by reducing all positive Lyapunov exponents to negative values. Therefore, long windows can be used to improve parameter estimation. In this new extension a second layer of synchronisation is added between the two models, with and without an adjoint, to facilitate linearisation around the trajectory of the model without an adjoint. The method is conceptually demonstrated by synchronising two Lorenz '63 systems, representing two ESMs, one with and the other without an adjoint model. Results are presented for an idealised case of identical, perfect models and for a more realistic case in which they differ from one another. If employed with a coarser ESM with an adjoint, the method will save computational power as only one forward run with the full ESM per iteration needs to be carried out. It is demonstrated that there is negligible error and uncertainty change compared to the 'traditional' optimisation of full ESM with an adjoint. In a variation of the method outlined, synchronisation between two identical models can be used to filter noisy data. This reduces optimised parametric model uncertainty by approximately one third. Such a precision gain could prove valuable for seasonal, annual, and decadal predictions.