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**A three-dimensional numerical internal tidal model involving adjoint method.**
*(English)*
Zbl 1253.86001

Summary: A three-dimensional internal tidal model involving the adjoint method is constructed based on the nonlinear, time-dependent, free-surface hydrodynamic equations in spherical coordinates horizontally, and isopycnic coordinates vertically, subject to the hydrostatic approximations. This model consists of two submodels: the forward model is used for the simulation of internal tides, while the adjoint model is used for optimization of modal parameters. Mode splitting technique is employed in both forward and adjoint models. In this model, the adjoint method is employed to estimate model parameters by assimilating the interior observations. As a preliminary feasibility study, a set of ideal experiments with the model-generated pseudo-observations of surface currents are performed to invert the open boundary conditions (OBCs). In the ideal experiments, 14 kinds of bottom topographies and six kinds of predetermined distributions of OBCs are considered to examine their influence on experiment results. The inversion obtained satisfying results and all the predetermined distributions were successfully inverted. Analysis of results suggests the following: in the case where the spatial variation of the OBC distribution is great or the open boundary is close to a rough topography, the results will be comparatively poor, but still satisfactory; both the tidal elevations and currents can be simulated very accurately with the surface currents at several observation points; the assimilation precision could be reliable and able to reflect both of the inversion and simulation results in the whole field. The performance and results of ideal experiments give a preliminary indication that the construction of this model is successful.

### MSC:

86-08 | Computational methods for problems pertaining to geophysics |

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

86A05 | Hydrology, hydrography, oceanography |

76B15 | Water waves, gravity waves; dispersion and scattering, nonlinear interaction |

### Keywords:

parameter optimization; adjoint method; 3D internal tidal model; optimization; marine hydrodynamics; ocean flows; fluid; structure interaction; shallow water; Lagrangian; free surface
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\textit{H. Chen} et al., Int. J. Numer. Methods Fluids 69, No. 10, 1584--1613 (2012; Zbl 1253.86001)

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