Exploring the influence of spatio-temporal scale differences in coupled data assimilation

Identifying the optimal strategy for initializing coupled climate prediction systems is challenging due to the spatio-temporal scale separation and disparities in the observational network. We aim to clarify when strongly coupled data assimilation (SCDA) is preferable to weakly coupled data assimilation (WCDA). We use a two-components coupled Lorenz-63 system, mimicking the atmosphere and the ocean, and the Ensemble Kalman Filter (EnKF) to compare WCDA and SCDA for diverse spatio-temporal scale separations and observational networks - only in the atmosphere, the ocean, or both components. In the fully observed scenario, SCDA and WCDA yield similar performances. However, little differences are present, and we conjecture these are due to the SCDA being more sensitive to the approximations at the basis of the EnKF present in the cross-update - linear analysis update and sampling error, and how they impact the cross-update between ocean and atmosphere. This sensitivity increases as the temporal scale separation increases and is stronger on the slow and large-scale components. When observations are only in one of the components, the spatio-temporal scale separation influences SCDA's performance. In this scenario, the largest improvements are found when the observed component has a smaller spatial scale. The fast-to-slow update has a larger benefit with a larger temporal scale separation. Meanwhile, with the slow-to-fast update, the improvement is limited to instances when the temporal scale separation is less than one-half. This suggests that SCDA of fast atmospheric observations can potentially improve the large and slow ocean component. Conversely, observations of the fine ocean can improve the large atmosphere at a comparable temporal scale. However, when both components are highly chaotic, and the observed component's spatial scale is the largest, SCDA does not improve over WCDA. In such a case, the cross-updates may become too sensitive to data assimilation approximations. We further validated that WCDA systematically outperforms uncoupled data assimilation (UCDA) in both components, legitimizing the transition toward WCDA.