Strong Solvability of a Variational Data Assimilation Problem for the Primitive Equations of Large-Scale Atmosphere and Ocean Dynamics

• Published in: Journal of nonlinear science Vol. 31; no. 3
• Main Author: Korn, Peter
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2021; Springer Nature B.V
• Subjects: Algorithms; Analysis; Classical Mechanics; Data assimilation; Economic Theory/Quantitative Economics/Mathematical Methods; Initial conditions; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Minima; Norms; Ocean dynamics; Primitive equations; Theoretical; Boussinesq equations; 49J20; Primitive equations; 76D55; 35Q35; Atmosphere-ocean dynamics; Adjoint method; 35Q86; 49N15; 93C20; Data assimilation
• ISSN: 0938-8974; 1432-1467; 1432-1467
• DOI: 10.1007/s00332-021-09707-3

For the primitive equations of large-scale atmosphere and ocean dynamics, we study the problem of determining by means of a variational data assimilation algorithm initial conditions that generate strong solutions which minimize the distance to a given set of time-distributed observations. We suggest a modification of the adjoint algorithm whose novel elements is to use norms in the variational cost functional that reflects the H 1 -regularity of strong solutions of the primitive equations. For such a cost functional, we prove the existence of minima and a first-order adjoint condition for strong solutions that provides the basis for computing these minima. We prove the local convergence of a gradient-based descent algorithm to optimal initial conditions using the second-order adjoint primitive equations. The algorithmic modifications due to the H 1 -norms are straightforwardly to implement into a variational algorithm that employs the standard L 2 -metrics.