Strong solvability of a variational data assimilation problem for the primitive equations of large-scale atmosphere and Ocean dynamics. (English) Zbl 1462.35406
Summary: 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.
MSC:
| 35Q86 | PDEs in connection with geophysics |
| 35Q35 | PDEs in connection with fluid mechanics |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 49N15 | Duality theory (optimization) |
| 76D55 | Flow control and optimization for incompressible viscous fluids |
| 93C20 | Control/observation systems governed by partial differential equations |
| 86A05 | Hydrology, hydrography, oceanography |
| 86A10 | Meteorology and atmospheric physics |
| 35D35 | Strong solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A15 | Variational methods applied to PDEs |
Keywords:
data assimilation; adjoint method; primitive equations; Boussinesq equations; atmosphere-ocean dynamicsReferences:
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