Why is data assimilation important? Dynamical models of atmosphere and ocean describe the change of the respective flow based on physical principles such as conservation of mass or momentum. For a real-world simulation one needs to incorporate the actual state of the real atmosphere and ocean that is described by observational data. The scientific discipline that integrates dynamical models and observations is called “data assimilation”. It is highly relevant for ocean state estimation and weather prediction. The improvements of the weather forecast over the last decades is to a large extend due to the improvements of data assimilation algorithms.

Variational data assimilation formulates a measure of the distance between dynamical model and observations and aims to minimize this distance. The minimizer of such a distance measure provides an optimal initial state for a prediction. The key element in variational data assimilation is how the model-observation-distance is measured, this decides for example if an optimal initial condition exists at all or not.

The study by Korn (2021) considers the hydrostatic Boussinesq equations, that form the basis of actual global ocean general circulation models such as ICON-O and of the previous generation of atmospheric models such as ECHAM. The paper demonstrates that if certain mathematical properties of the underlying dynamical model are reflected in the definition of the distance between model and observations, then one can establish that an optimal initial state exists. Furthermore, the paper proves the computationally highly-relevant fact that the new definition of the model-observation distance a classical optimization algorithm actually converges to this optimal initial state. For nonlinear systems such as atmosphere/ ocean equations such statements are notoriously difficult and only a few theoretical results exist.

The novelty of the paper that enables this fundamental insight was possible by measuring the distance between model and observation not only in terms of the squared difference of the velocity or tracer fields but also in terms of the squared difference of their derivatives of velocity and tracers. The derivative information is usually not part of variational data assimilation algorithms. Including the derivative information made this fundamental statement about variational data assimilation for the hydrostatic Boussinesq equations possible. Instrumental for this result were fundamental insights from Cao and Titi (2007) into the mathematical structure of these equations.

This work continues the line of research that has started in Korn (2019), and extends it to a class of models that are of practical relevance for atmosphere/ocean data assimilation.

### Original publication:

Korn, P. (2021) Strong Solvability of a Variational Data Assimilation Problem for the Primitive Equations of Large-Scale Atmosphere and Ocean Dynamics. *Journal of Nonlinear Science*, 31-56. https://link.springer.com/article/10.1007/s00332-021-09707-3

Korn, P. (2019) A Regularity-Aware Algorithm for Variational Data Assimilation of an Idealized Coupled Atmosphere–Ocean Model. *Journal of Scientific Computing*, 79, 748–786. https://link.springer.com/article/10.1007/s10915-018-0871-y

Cao, C. & E. S. Titi (2007) Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. *Ann. Math*. 166, 245–267. https://annals.math.princeton.edu/wp-content/uploads/annals-v166-n1-p07.pdf

### Contact:

Dr. Peter Korn

Max Planck Institute for Meteorology

Phone: +49 (0) 40 41173 470

Email: peter.korn@ mpimet.mpg.de