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**Open boundary control problem for Navier-Stokes equations including a free surface: adjoint sensitivity analysis.**
*(English)*
Zbl 1118.49022

Summary: This paper develops the adjoint sensitivities to the free-surface barotropic Navier- Stokes equations in order to allow for the assimilation of measurements of currents and free-surface elevations into an unsteady flow solution by open-boundary control. To calculate a variation in a surface variable, a mapping is used in the vertical to shift the problem into a fixed domain. A variation is evaluated in the transformed space from the Jacobian matrix of the mapping. This variation is then mapped back into the original space where it completes a tangent linear model. The adjoint equations are derived using the scalar product formulas redefined for a domain with variable bounds. The method is demonstrated by application to an unsteady fluid flow in a one-dimensional open channel in which horizontal and vertical components of velocity are represented as well as the elevation of the free surface (a 2D vertical section model). This requires the proper treatment of open boundaries in both the forward and adjoint models. A particular application is to the construction of a fully three-dimensional coastal ocean model that allows assimilation of tidal elevation and current data. However, the results are general and can be applied in a wider context.

### MSC:

49K40 | Sensitivity, stability, well-posedness |

49N99 | Miscellaneous topics in calculus of variations and optimal control |

### Keywords:

Navier-Stokes equations; free surface; open boundary; optimal control; adjoint equations; sensitivity analysis; ocean; waves### Software:

SOLA
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\textit{I. Yu. Gejadze} and \textit{G. J. M. Copeland}, Comput. Math. Appl. 52, No. 8--9, 1243--1268 (2006; Zbl 1118.49022)

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### References:

[1] | Couch, S. J.; Copeland, G. J.M., Tidal straining, mixing and Lagrangian flow residuals around headlands, J. of Marine Environmental Engineering, 7, 2, 25-45 (2003) |

[2] | Lin, B.; Falconer, R. A., Three-dimensional layer-integrated modelling of estuarine flows with flooding and drying, Estuarine, Coastal and Shelf Science, 44, 737-751 (1997) |

[3] | Ezer, T.; Mellor, G. L., Simulations of the Atlantic Ocean with a free surface sigma coordinate ocean model, J. Geophys. Res., 102, 15, 647-657 (1997) |

[4] | Casulli, V.; Zanolli, P., Semi-implicit numerical modelling of non-hydrostatic free-surface flows for environmental problems, Mathl. Comput. Modelling, 36, 1131-1149 (2002) · Zbl 1027.76034 |

[5] | Namin, M. M.; Lin, B.; Falconer, R. A., An implicit numerical algorithm for solving non-hydrostatic free- surface flow problems, Int. J. Numer. Meth. Fluids, 35, 341-356 (2001) · Zbl 1008.76065 |

[6] | Sasaki, Y., Some basic formalism in numerical variational analysis, Monthly Weather Rev., 98, 875-883 (1970) |

[7] | Marchuk, G. I., Numerical Solution of the Problems of the Dynamics of the Atmosphere and Oceans (1974), Gidrometeoizdat: Gidrometeoizdat Leningrad |

[8] | Cacuci, D. G., Sensitivity theory for non-linear systems, J. Math. Phys., 22, 12, 2794-2812 (1981) |

[9] | Zou, X.; Navon, I. M.; Berger, M.; Phua, M. K.; Schlick, T.; Le-Dimet, F. X., Numerical experience with limited- memory, quasi-Newton methods for large-scale unconstrained nonlinear minimization, SIAM J. Optimization, 3, 582-608 (1993) · Zbl 0784.90086 |

[10] | Sanders, B. F.; Katopodes, N. D., Adjoint sensitivity analysis for shallow-water wave control, J. Eng. Mech., ASCE, 909-919 (2000) |

[11] | Lardner, R. W., Optimal control of open boundary conditions for a numerical tidal model, Comput. Meth. Appl. Meth. Eng., 102, 367-387 (1992) · Zbl 0767.76036 |

[12] | Zou, J.; Hsieh, W. W.; Navon, I. M., Sequential open-boundary control by data assimilation in a limited-area model, Month. Weath. Review, 123, 9, 2899-2909 (1995) |

[13] | Li, Z.; Navon, I. M.; Hussaini, M. Y.; Le Dimet, F. X., Optimal control of cylinder wakes via suction and blowing, Computers and Fluids, 32, 2, 149-171 (2003) · Zbl 1151.76443 |

[15] | Gejadze, I. Yu.; Copeland, G. J.M., Adjoint sensitivity analysis for fluid flow with free surface, Int. J. Numer. Meth. Fluids, 47, 1027-1034 (2005) · Zbl 1134.86006 |

[16] | Moore, A. M.; Arango, H. G.; Miller, A. J.; Cornuelle, B. D.; Di Lorenzo, E.; Neilson, D. J., A comprehensive ocean prediction and analysis system based on the tangent linear and adjoint components of a regional ocean model, Ocean Modelling, 7, 227-258 (2004) |

[17] | Blayo, E.; Debreu, L., Revisiting open boundary conditions from the point of view of characteristic variables, Ocean Modelling, 9, 3, 231-252 (2005) |

[18] | Mohammadi, B.; Pironneau, O., Shape optimization in fluid mechanics, Annu. Rev. Fluid Mech., 36, 11, 1-25 (2004) |

[19] | Sokolowski, J.; Zochowski, A., On the topological derivative in shape optimization, SIAM J. Control Optim., 37, 1251-1272 (1999) · Zbl 0940.49026 |

[21] | Van Brummelen, E. H.; Segal, A., Numerical solution of steady free-surface flows by the adjoint optimal shape design method, Int. J. Numer. Meth. Fluids, 41, 3-27 (2003) · Zbl 1025.76039 |

[22] | Wedi, N. P.; Smolarkiewicz, P. K., Extending Gal-Chen Somerville terrain-following coordinate transformation on time-dependent curvilinear boundaries, J. Comp. Phys., 193, 1-20 (2003) · Zbl 1117.76307 |

[23] | Hedstrom, G. W., Nonreflecting boundary conditions for nonlinear hyperbolic system, J. Comp. Phys., 30, 222-237 (1979) · Zbl 0397.35043 |

[24] | Sedenko, V. I., Solvability of initial-boundary value problems for the Euler equations of flows on an ideal incompressible nonhomogeneous fluid and ideal barotropic fluid that are bounded by free surfaces, English translation, Russian Acad. Sci. Sb. Math., 83, 2, 347-368 (1995) |

[25] | Solonnikov, V. A., Solvability of the problem of dynamics of viscous incompressible flow bounded by a free surface, Continuous Media Dynamics, 23, 123-128 (1975) |

[26] | Oliger, J.; Sandstrom, A., Theoretical practical aspects of some initial boundary value problems in fluid mechanics, SIAM J. Appl. Math., 35, 419-446 (1978) · Zbl 0397.35067 |

[27] | Lions, J. L.; Temam, R.; Wang, S., On the equations of the large-scale ocean, Nonlinearity, 5, 1007-1053 (1992) · Zbl 0766.35039 |

[28] | Temam, R.; Tribbia, J., Open boundary conditions for the primitive and Boussinesq equations, J. Atmos. Sci., 60, 2647-2660 (2003) |

[30] | Navon, I. M.; Zou, X.; Derber, J.; Sela, J., Variational data assimilation with an adiabatic version of the NMC spectral model, Mon. Weather Rev., 120, 1433-1446 (1992) |

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