CC BY
22.P = c2(.
� .0), .0 = 1. Even though its change is described by the continuity equation this parameter does not
coincide with the density. Parameter c is an artificial sound speed. It is determined by the condition that the
parameter . does not deviate from the initial value by more than one percent. The density equation is a
transport type.
The pressure force acting on the free surface returning it to the state of static equilibrium. It can be called
"soft cover" mechanism. This pressure depends on the magnitude of deviation of the free surface from the
equilibrium position. A standard kinematic condition is also set at the liquid boundary.
The explicit-implicit CABARET scheme is used to solve the hyperbolized system, explicit in horizontal direc-
tions and implicit in depth. It makes it possible to use large time steps on grids with thinning of layers to a free
surface. The proposed non-hydrostatic model is verified by laboratory experiments, in which the formation
and propagation of internal bottom waves formed during the rapid extraction of a lock gate separating a dens-
er part of the liquid with a controlled vertical salinity distribution from a homogeneous liquid in a long channel
are investigated. The results are compared with laboratory measurements and calculations using a hydrostatic
model.
Validation of the CABARET-MFSH hydrostatic model for modeling the flows of stratified fluids with a free
surface in laboratory experiments
V. M. Goloviznin1, Pavel A. Maiorov1,2, Petr A. Maiorov1, A. V. Solovjov2
1Lomonosov Moscow State University
2Nuclear Safety Institute, Moscow
Email: gol@ibrae.ac.ru, pavel.a.mayorov@gmail.com, maiorov.peter@gmail.com, solovjev@ibrae.ac.ru
DOI 10.24412/cl-35065-2021-1-01-26
This paper is devoted to the results of validation on two series of laboratory experiments of the new low-
dissipative multilayer hydrostatic model CABARET-MFSH, which describes the dynamics of a fluid with variable
density and a free surface. The computational algorithm of the new model is based on representing a multi-
layer environment as separate layers interacting across interfaces. We call this method hyperbolic decomposi-
tion. An explicit CABARET scheme is used to solve the system of hyperbolic equations in each layer. The
scheme has a second order of approximation and is time reversible. To regularize the multilayer hydrostatic
model, mass and momentum exchange between the layers and the filtration of flux variables are used. The
filtration parameters are determined empirically from the condition of the stability of the algorithm and the
minimality of the scheme viscosity.
In laboratory experiments, the emergence and propagation of bottom internal waves is investigated. They
are formed when the gate is quickly pulled out between a liquid with a controlled vertical salinity distribution
and a homogeneous liquid.
The calculation results are in good agreement with the experimental data.
Changes in the state of the Siberian Arctic seas in the last two decades
E. N. Golubeva, M. V. Kraineva, G. A. Platov, D. F. Yakshina
The Institute of Computational Mathematics and Mathematical Geophysics SB RAS
Email: elen@ommfao.sscc.ru
DOI 10.24412/cl-35065-2021-1-01-27
This report discusses issues related to climatic changes in the ice and water state of the shelf seas of the
Siberian sector of the Arctic. The research is based on analyzing observational data and modeling results using
a numerical three-dimensional model of the Arctic Ocean. Changes in the timing of the formation and melting
of ice, an increase in heat input into surface sea waters, an increase in the heat content of the seas are ana-
lyzed. The extreme rise in surface water temperature in shelf seas can be classified as the existence of marine
heatwaves. To identify marine heatwaves, the method described in (1) was applied.
This work was supported by the Russian Science Foundation (19-17-00-154).
References:
1. A.J. Hobday et al. A hierarchical approach to defining marine heatwaves // Progress in Oceanography � 2016. 141.
P. 227�238.
Simulation of the interaction of long surface waves with semi-submerged structures with an uneven bilge
O. I. Gusev, G. S. Khakimzyanov, L. B. Chubarov
Federal Research Center for Information and Computational Technologies
Email: gusev_oleg_igor@mail.ru
DOI 10.24412/cl-35065-2021-1-01-28
The study is devoted to the problem of the interaction of long surface waves with semi-submerged fixed
structures with an uneven bilge. Numerical algorithms [1] are constructed for one-dimensional nonlinear shal-
low water models with and without frequency dispersion [2]. The influence of bilge irregularities on the char-
acteristics of the reflected and transmitted waves is investigated. The obtained numerical solutions are com-
pared with the results of other authors [3].
References
1. Khakimzyanov G.S., Dutykh D. Long wave interaction with a partially immersed body. Part I: Mathematical models
// Communications in Computational Physics. 2020. V. 27, N. 2. P. 321-378.
2. Khakimzyanov G., Dutykh D., Fedotova Z., Gusev O. Dispersive Shallow Water Waves. Theory, Modeling, and
Numerical Methods. Lect. Notes in Geosystems Mathematics and Computing. Basel, Birkhauser, 2020.
3. Chang C.-H., Wang K.-H., Hseih P. -C. Fully nonlinear model for simulating solitary waves propagating through a
partially immersed rectangular structure // J. of Coastal Research. 2017. V. 33, N. 6. P. 1487-1497.
Some nonlinear models of transport theory
A. V. Kalinin1,2, A. A. Tyukhtina1 A. A. Busalov1, O. A. Izosimova1
1Lobachevsky State University of Nizhny Novgorod
2Institute of Applied Physics RAS
Email: avk@mm.unn.ru
DOI 10.24412/cl-35065-2021-1-01-29
A wide class of problems in physics and engineering leads to the study of integro-differential equations of
transport theory. The foundations of mathematical and numerical modeling of particle transport processes
were laid in the works [1-4]. We consider stationary and non-stationary problems for nonlinear systems of in-
tegro-differential equations of transport theory [5]. The issues of correctness of statements of the correspond-
ing mathematical problems, properties of their solutions and algorithms for numerical solution are discussed.
The theoretical study of the problems uses the methods of ordered function spaces.
This work was supported by the Scientific and Education Mathematical Center "Mathematics for Future Technolo-
gies" (Project No. 075-02-2020-1483/1).
References
1. Marchuk G. I. Methods for nuclear reactor calculations. M.: Gosatomizdat, 1961.
