The Local Ensemble Tangent Linear Model: an enabler for coupled model 4D-Var

A leading Data Assimilation (DA) technique in meteorology is 4D-Var which relies on the Tangent Linear Model (TLM) of the nonlinear model and its adjoint. The difficulty of building and maintaining traditional TLMs and adjoints of coupled ocean-wave-atmosphere-etc. models is daunting. On the other hand, coupled model ensemble forecasts are readily available. Here, we show how an ensemble forecast can be used to construct an accurate Local Ensemble TLM (LETLM) and adjoint of the entire coupled system. The method features a local influence region containing all the variables that could possibly influence the time evolution of some target variable(s) near the centre of the region. We prove that high accuracy is guaranteed provided that (i) the ensemble perturbations are governed by linear dynamics, and (ii) the number of ensemble members exceeds the number of variables in the influence region. The approach is illustrated in a simple coupled model. This idealized coupled model has some realistic features including reasonable predictability limits in the upper atmosphere, lower atmosphere, upper ocean and lower ocean of 10, 96, 160 and 335 days, respectively. In addition, the length-scale of eddies in the ocean is about one fifth of those in the atmosphere. The easymanner in which the adjoint is obtained from the LETLM is also described and illustrated by demonstrating how the LETLM adjoint predicts the high sensitivity of oceanic boundary-layer evolution to changes in the atmosphere. Finally, the feasibility of LETLMs for 4D-Var is demonstrated. Specifically, a case is considered with a 5-day data assimilation window in which nonlinear terms play a significant role in the evolution of forecast error; it is shown that the posterior mode delivered by 4D-Var with an LETLM, its adjoint and ten outer loops approximately recovers the true state in spite of a spatially sparse observational network.