Development and correctness analysis of the mathematical model of transport and suspension sedimentation depending on bottom relief variation

Introduction. The paper is devoted to the study on the three-dimensional model of transport and suspension sedimentation in the coastal area due to changes in the bottom relief. The model considers the following processes: advective transfer caused by the aquatic medium motion, micro-turbulent diffusion, and gravity sedimentation of suspended particles, as well as the bottom geometry variation caused by the particle settling or bottom sediment rising. The work objective was to conduct an analytical study of the correctness of the initial-boundary value problem corresponding to the constructed model.

Materials and Methods. The change in the bottom relief aids in solution to the initial-boundary value problem for a parabolic equation with the lowest derivatives in a domain whose geometry depends on the desired function of the solution, which in general leads to a nonlinear formulation of the problem. The model is linearized on the time grid due to the “freezing” of the bottom relief within a single step in time and the subsequent recalculation of the bottom surface function on the basis of the changed function of the suspension concentration, as well as a possible change in the velocity vector of the aquatic medium.

Research Results. For the linearized problem, a quadratic functional is constructed, and the uniqueness of the solution to the corresponding initial boundary value problem is proved within the limits of an unspecified time step. On the basis of the quadratic functional transformation, we obtain a prior estimate of the solution norm in the functional space L2 as a function of the integral time estimates of the right side, and the initial condition. Thus, the stability of the solution to the initial problem from the change of the initial and boundary conditions, the right-hand side function, is established.

Discussion and Conclusions. The model can be of value for predicting the spread of contaminants and changes in the bottom topography, both under an anthropogenic impact and due to the natural processes in the coastal area.

1. Leontyev, I.O., et al. Pribrezhnaya dinamika: volny, techeniya, potoki nanosov. [Coastal dynamics: waves, flows, deposits drifts.] I.O. Leontyev, ed. Moscow: GEOS, 2001,272p. (in Russian).

2. Matishov, G.G., et al. Prirodnye katastrofy v Azovo-Chernomorskom basseyne v nachale XXI veka. [Natural disasters in the Azov-Black Sea basin at the beginning of ХХI century.] Rostov-on-Don: SRC RAS Publ. House, 2017, 160 p. (in Russian).

3. Petrov, P.G. Dvizhenie sypuchey sredy v pridonnom sloe zhidkosti. [Motion of a bed load.] Journal of Applied Mechanics and Technical Physics,1991, vol. 32, iss. 5, pp. 72–76 (in Russian).

4. Barnard, P.L., Jaffe, B.E. and Schoellhamer, D.H. A multi-discipline approach for understanding sediment transport and geomorphic evolution in an estuarine-coastal system—San Francisco Bay. Marine Geology, 2013, vol. 345, pp.1–2. DOI:https://doi.org/10.1016/j.margeo.2013.09.010

5. Xiaoying, L., Shi, Q., Yuan, H., Yuehong, C., Pengfei, D. Predictive modeling in sediment transportation across multiple spatial scales in the Jialing River Basin of China. International Journal of Sediment Research, 2015, vol. 30, iss. 3, pp. 250–255.

6. Lusher, A.L., McHugh, M., Thompson, R.C. Occurrence of microplasticsin gastrointestinal tract of pelagic and demersal fish from the English channel. Marine Pollution Bulletin, 2013, vol. 67, pp. 94-99.

7. Marchuk, G.I., et al. Matematicheskie modeli v geofizicheskoy gidrodinamike i chislennye metody ikh realizatsii. [Mathematical models in geophysical hydrodynamics and numerical methods for their implementation.] Leningrad: Gidrometeoizdat, 1987, 296 p. (in Russian).

8. Belikov, V.V., Borisova, N.M., Gladkov, G.L.Matematicheskaya model' transporta nanosov dlya rascheta zanosimosti dnouglubitel'nykh prorezey i ruslovykh kar'yerov. [Mathematical model of sediment transport for calculating the sediment accumulation indredge cuts and channel pits.] Journal of University of Water Communications, 2010, vol.2, pp.105–113 (in Russian).

9. Sanne, L.N. Modelling of sand dunes in steady and tidal flow. Denmark: Technical University of Copenhagen, 2003, 185 p.

10. Ballent, A., Pando, S., Purser, A., Juliano, M., Thomsen, L. Modelled transport of benthic marine microplastic pollution in the Nazaré Canyon. Biogeosciences, 2013, vol. 10, pp. 7957-7970. https://doi.org/10.5194/bg-10-7957-2013

11. Miles, J. Wave shape effects on sediment transport. J. Coastal Res., 2013, vol. 2, iss. 65, pp. 1803–1808. DOI:https://doi.org/10.2112/SI65-305.1

12. Popkov, V.I.Strukturnye osobennosti i genezis dislokatsiy dna Azovskogo morya.[Structural features and genesis of dislocations of the Sea of Azov bottom.] Geology, Geography and Global Energy, 2008, no. 1, pp. 77–90 (in Russian).

13. Sidoryakina, V.V., Sukhinov, A.I.Issledovanie korrektnosti i chislennaya realizatsiya linearizovannoy dvumernoy zadachi transporta nanosov. [Well-posedness analysis and numerical implementation of a linearized twodimensional bottom sediment transport problem.] Computational Mathematics and Mathematical Physics, 2017, vol. 57, no.6, pp. 985–1002 DOI: https://doi.org/10.7868/S0044466917060138 (in Russian). DOI:https://doi.org/10.7868/S0044466917060138

14. Sukhinov, A.I., Sidoryakina, V.V. O skhodimosti resheniya linearizovannoy posledovatel'nosti zadach k resheniyu nelineynoy zadachi transporta nanosov. [Convergence of linearized sequence tasks to the nonlinear sediment transport task solution.] Mathematical Models and Computer Simulations, 2017, vol.29, no.11, pp. 19–39 (in Russian). http://mi.mathnet.ru/mm3905

15. Sukhinov, A.I., Sidoryakina, V.V., Sukhinov, A.A. Dostatochnye usloviya skhodimosti polozhitel'nykh resheniy linearizovannoy dvumernoy zadachi transporta nanosov. [Sufficient conditions for convergence of positive solutions to linearized two-dimensional sediment transport problem.] Vestnik of DSTU, 2017, vol.17, no.1, pp.5–17 (in Russian). DOI: https://doi.org/10.23947/1992-5980-2017-17-1-5-17

16. Sukhinov, A.А., Sukhinov, A.I. 3D Model of Diffusion-Advection-Aggregation Suspensions in Water Basins and Its Parallel Realization. Parallel Computational Fluid Dynamics, Mutidisciplinary Applications, Proceedings of Parallel CFD 2004 Conference, Las Palmas de Gran Canaria, Spain, ELSEVIER, Amsterdam-Berlin-London-New York-Tokyo, 2005, pp. 223-230. DOI: https://doi.org/10.1016/B978-044452024-1/50029-4

17. Protter, M.H., Weinberger, H.F.Maximum Principles in Differential Equation. Springer-Verlag New York, Inc. 1984, 276 p. DOI https://doi.org/10.1007/978-1-4612-5282-5

18. Ladyzhenskaya, O.A., et al. Lineynye i kvazilineynye uravneniya parabolicheskogo tipa. [Linear and quasilinear parabolic equations.] Moscow:Nauka, 1967, 736 p. (in Russian).

19. Vladimirov, V.S., et al. Uravneniya matematicheskoy fiziki. [Equations of mathematical physics.] Moscow: Nauka, 1981,512 p . (in Russian).

20. Tikhonova, A.N., et al. Uravneniya matematicheskoy fiziki. [Equations of mathematical physics.] Moscow: Nauka, 1977,735 p. (in Russian).