Boundary conditions control for a shallow-water model. (English) Zbl 1397.76026
Summary: A variational data assimilation technique was used to estimate optimal discretization of interpolation operators and derivatives in the nodes adjacent to the rigid boundary. Assimilation of artificially generated observational data in the shallow-water model in a square box and assimilation of real observations in the model of the Black Sea are discussed. It is shown in both experiments that controlling the discretization of operators near a rigid boundary can bring the model solution closer to observations as in the assimilation window and beyond the window. This type of control also allows to improve climatic variability of the model.
MSC:
| 76B75 | Flow control and optimization for incompressible inviscid fluids |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 76B10 | Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing |
| 86A05 | Hydrology, hydrography, oceanography |
References:
| [1] | Le Dimet FX A general formalism of variational analysis 1982 |
| [2] | Le Dimet, Variational algorithm for analysis and assimilation of meteorological observations. theoretical aspects, Tellus 38A pp 97– (1986) · doi:10.1111/j.1600-0870.1986.tb00459.x |
| [3] | Lions, Contrôle optimal de systèmes gouvernés pas des équations aux dérivées parielles (1968) |
| [4] | Marchuk, Formulation of theory of perturbations for complicated models, Applied Mathematics and Optimization 2 pp 1– (1975) · Zbl 0324.65053 · doi:10.1007/BF01458193 |
| [5] | Losch, Bottom topography as a control variable in an ocean model, Journal of Atmospheric and Oceanic Technology 20 pp 1685– (2003) · doi:10.1175/1520-0426(2003)020<1685:BTAACV>2.0.CO;2 |
| [6] | Heemink, Inverse 3d shallow water flow modelling of the continental shelf, Continental Shelf Research 22 pp 465– (2002) · doi:10.1016/S0278-4343(01)00071-1 |
| [7] | Kazantsev, Identification of optimal topography by variational data assimilation, Journal of Physical Mathematics 1 pp 1– (2009) · Zbl 1264.86002 · doi:10.4303/jpm/S090702 |
| [8] | Shulman, Local data assimilation in specification of open boundary conditions, Journal of Atmospheric and Oceanic Technology 14 pp 1409– (1997) · doi:10.1175/1520-0426(1997)014<1409:LDAISO>2.0.CO;2 |
| [9] | Shulman, Optimized boundary conditions and data assimilation with application to the m2 tide in the yellow sea, Journal of Atmospheric and Oceanic Technology 15 (4) pp 1066– (1998) · doi:10.1175/1520-0426(1998)015<1066:OBCADA>2.0.CO;2 |
| [10] | Taillandier, Controlling boundary conditions with a four-dimensional variational data-assimilation method in a non-stratified open coastal model, Ocean Dynamics 54 (2) pp 284– (2004) · doi:10.1007/s10236-003-0068-1 |
| [11] | ten Brummelhuis, Identification of shallow sea models, International Journal for Numerical Methods in Fluids 17 pp 637– (1993) · Zbl 0793.76066 · doi:10.1002/fld.1650170802 |
| [12] | Zou, An optimal nudging data assimilation scheme using parameter estimation, Quarterly Journal of the Royal Meteorological Society 118 pp 1163– (1992) · doi:10.1002/qj.49711850808 |
| [13] | Panchang, On the Determination of Hydraulic Model Parameters Using the Strong Constraint Formulation Modeling Marine Systems 1 pp 5– (1988) |
| [14] | Chertok, Variational data assimilation for a nonlinear hydraulic model, Applied Mathematical Modelling 20 (9) pp 675– (1996) · Zbl 0859.93041 · doi:10.1016/0307-904X(96)00048-0 |
| [15] | Verron, The no-slip condition and separation of western boundary currents, Journal of Physical Oceanography 26 (9) pp 1938– (1996) · doi:10.1175/1520-0485(1996)026<1938:TNSCAS>2.0.CO;2 |
| [16] | Chen, Estimation of two-sided boundary conditions for two dimensional inverse heat conduction problems, International Journal of Heat and Mass Transfer 45 pp 15– (2002) · Zbl 0986.80005 · doi:10.1016/S0017-9310(01)00138-7 |
| [17] | Gillijns, Modelling, Identification, and Control (2007) |
| [18] | Le Dimet, Retrieval of balanced fields: an optimal control method, Tellus A 45 (5) pp 449– (1993) · doi:10.1034/j.1600-0870.1993.00009.x |
| [19] | Kazantsev, Identification of an optimal boundary approximation by variational data assimilation, Journal of Computational Physics 229 pp 256– (2010) · Zbl 1375.65115 · doi:10.1016/j.jcp.2009.09.018 |
| [20] | Kazantsev, Optimal boundary discretisation by variational data assimilation, International Journal for Numerical Methods in Fluids (2010) · Zbl 1375.65115 |
| [21] | Leredde, On initial, boundary conditions and viscosity coefficient control for burgers’ equation, International Journal for Numerical Methods in Fluids 28 (1) pp 113– (1998) · Zbl 0928.76035 · doi:10.1002/(SICI)1097-0363(19980715)28:1<113::AID-FLD702>3.0.CO;2-1 |
| [22] | Lellouche, Boundary control on burgers equation. a numerical approach, Computers and Mathematics with Applications 28 pp 33– (1994) · Zbl 0807.76065 · doi:10.1016/0898-1221(94)00138-3 |
| [23] | Gill, Atmosphere-Ocean Dynamics (1982) |
| [24] | Pedlosky, Geophysical Fluid Dynamics (1987) · doi:10.1007/978-1-4612-4650-3 |
| [25] | Arakawa, Methods in Computational Physics 17 pp 174– (1977) |
| [26] | Arakawa, A potential enstrophy and energy conserving scheme for the shallow water equations, Monthly Weather Review 109 pp 18– (1981) · doi:10.1175/1520-0493(1981)109<0018:APEAEC>2.0.CO;2 |
| [27] | Ide, Unified notation for data assimilation: operational, sequential and variational, Journal of the Meteorological Society of Japan 75 (1B) pp 181– (1997) · doi:10.2151/jmsj1965.75.1B_181 |
| [28] | Gilbert, Some numerical experiments with variable storage quasi-Newton algorithms, Mathematical Programming 45 pp 407– (1989) · Zbl 0694.90086 · doi:10.1007/BF01589113 |
| [29] | Le Provost, Wind-driven ocean circulation transition to barotropic instability, Dynamics of Atmospheres and Oceans 11 pp 175– (1987) · doi:10.1016/0377-0265(87)90005-4 |
| [30] | Simmonet, Low-frequency variability in shallow-water models of the wind-driven ocean circulation, Journal of Physical Oceanography 33 pp 729– (2003) · doi:10.1175/1520-0485(2003)33<729:LVISMO>2.0.CO;2 |
| [31] | Munk, On the wind-driven ocean circulation, Journal of Meteorology 7 pp 3– (1950) · doi:10.1175/1520-0469(1950)007<0080:OTWDOC>2.0.CO;2 |
| [32] | Bryan, A global ocean-atmosphere climate model. Part ii. The oceanic circulation, Journal of Physical Oceanography 5 (1) pp 30– (1975) · doi:10.1175/1520-0485(1975)005<0030:AGOACM>2.0.CO;2 |
| [33] | Griffies, Spurious diapycnal mixing associated with advection in a z-coordinate ocean model, Monthly Weather Review 128 (3) pp 538– (2000) · doi:10.1175/1520-0493(2000)128<0538:SDMAWA>2.0.CO;2 |
| [34] | Kazantsev, Sensitivity of the attractor of the barotropic ocean model to external influences: approach by unstable periodic orbits, Nonlinear Processes in Geophysics 8 pp 281– (2001) · Zbl 0994.37043 · doi:10.5194/npg-8-281-2001 |
| [35] | Dymnikov, Modelling of the climatic system response to small external forcing, Russian Journal of Numerical Analysis and Mathematical Modelling 15 (1) pp 15– (2000) · Zbl 0977.76506 · doi:10.1515/rnam.2000.15.1.15 |
| [36] | Korotaev, Seasonal, interannual, and mesoscale variability of the black sea upper layer circulation derived from altimeter data, JGR 108 pp 3122– (2003) · doi:10.1029/2002JC001508 |
| [37] | Korotaev, Satellite altimetry observations of the black sea level, JGR 106 pp 917– (2001) · doi:10.1029/2000JC900120 |
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