Valery Yu. Liapidevskii

A three-layer shallow water model in the Boussinesq approximation with allowance for nonlinearity, dispersion, and mixing is used to describe the propagation and breaking of large-amplitude internal waves interacting with uneven bottom topography. The proposed equations of motion are solved numerically by applying the Godunov method with additional inversion of an elliptic operator at each time step. Stationary solutions in the form of mode-1 solitary waves are constructed. Mixing processes induced by breaking internal solitary waves due to their interaction with a single or combined obstacle are modeled. It is shown that the numerical results are in good agreement with experimental data and direct numerical simulation.

Unsteady one-dimensional shear flows of a viscoelastic fluid are considered. A general approach is formulated for fluids with several relaxation times, which allows the known models of viscoelastic flows to be presented as evolutionary systems of first-order equations. Conditions of hyperbolicity of flow classes considered are found for the Johnson–Segalman, Giesekus, and Rolie-Poly models. The equations of motion of the viscoelastic fluid are presented in the form of a full nonlinear system of conservation laws. A method of calculating unsteady discontinuous flows within the framework of the models under consideration is proposed. The class of unsteady Couette flows in the gap between the cylinders used in rheological tests is studied numerically. The process of shear banding and its influence on the structure of steady flows are investigated. The numerical results obtained are compared with experimental data.

Two-layer flow of a density-stratified fluid with mass transfer between the layers is considered. In the Boussinesq approximation, the equations of motion are reduced to a homogeneous quasilinear system of partial differential equations of mixed type. The flow parameters in the intermediate mixed layer are determined from the equilibrium conditions in a more general model of three-layer flow of a miscible fluid. In particular, the equilibrium conditions imply the constancy of the interlayer Richardson number in velocity-shift flows. A self-similar solution to the problem of breakdown of an arbitrary discontinuity (the lock-exchange problem) in the domain of hyperbolicity of the system under consideration is constructed. The transcritical flow regimes over a local obstacle are studied and the conditions under which the obstacle determines the upstream flow are determined. A comparison of steady-state and time-dependent solutions with the solutions obtained for the original three-layer models of miscible fluid flow is carried out.

This paper presents a discussion on observations of nonlinear internal waves (NLIWs) in the coastal zone of the Sea of Japan, based on the mooring of thermostring clusters in different seasons of 2022. For statistical evaluation of the frequency of event occurrence and determination of NLIW movement direction, we use our observations of the past 12 years. We present the NLIW structures, observed in spring, summer, and autumn of 2022, which are typical for this shelf area. Two types of nonlinear waves are described—solitary and undular bores, with or without strong vertical mixing behind the front. We demonstrate spatial transformation of an undular bore as it moves over the shelf. A mathematical model based on the second-order shallow water approximation is proposed for numerical simulation. To simplify calculations, the authors limit themselves to two- and three-layer shallow water models. We investigate the possibility of spatiotemporal reconstruction of internal nonlinear structures between thermostrings using experimental data and proposed models. The authors show that at distances of up to several kilometers between thermostrings, the wave fields of strongly nonlinear and nonstationary structures can be successfully reconstructed. Water flow induced by NLIWs can be reconstructed from the data of even one thermostring.

A mathematical model is proposed that describes the formation of internal hydraulic jumps and the mixing of codirectional flows of an ideal stratified fluid in the Boussinesq approximation. The model is based on a three-layer representation of the flow taking into account the entrainment of fluid from the outer layers into the intermediate vortex layer and is represented as a system of inhomogeneous conservation laws. The rate of entrainment is given by the equilibrium condition within the framework of a more general model for the evolution of the mixing layer. The velocity of propagation of disturbances and the concepts of subcritical and supercritical flows are formulated. It is shown that the model is suitable for describing flow mixing and splitting in deep-sea currents. Solutions were constructed that correspond to the flow around an obstacle with the formation of an internal hydraulic jump and a region of intense mixing. The results of numerical simulation were shown to be in good agreement with experimental data.

The characteristic feature of stratified flows in large bodies of water is generation of intense short-period internal waves at the front of long-wave disturbances. The nonlinear processes are most pronounced during the propagation of bottom and near-surface disturbances. The effective tool for studying wave processes in the ocean is the theory of multilayer shallow water with regard for the effects of nonlinearity and dispersion. It is shown that the developed mathematical models are suitable for describing the transformation of nonlinear internal waves both in the shelf zone of the sea and in the deep freshwater reservoirs. In particular, the structure of near-bottom internal waves in the shelf zone of the Sea of Japan and recently discovered near-surface internal waves in Lake Teletskoye are compared. The mechanism of generation of intense internal waves during excitation of seiche oscillations in narrow water reservoirs is discussed. Traveling waves in a multilayer fluid are constructed and numerical solutions to the nonstationary problem of generating internal waves are found. A comparison with the laboratory experiments on generation of a packet of short-period internal waves during seiche oscillations of a two-layer fluid in a long channel, as well as with the recorded near-surface internal solitary wave in Lake Teletskoye, is carried out.

The flow of a stratified fluid over a slope is considered. In the one-layer shallow water approximation, a mathematical model is constructed for a turbulent flow of a denser fluid over a uniform slope, with the entrainment of the ambient fluid at rest and the sediment entrainment at the wave front taken into account. The main focus is on analyzing the structure of a self-sustaining wave (underwater avalanche) and on estimating its propagation velocity. The mathematical model arises from the equilibrium conditions in a more complete three-parameter model and contains only one numerical parameter that represents a combination of the parameters of the original model characterizing the slope, vortex energy dissipation rate, and entrainment rate. The structure of traveling waves is studied, exact self-similar solutions are constructed, and transition of the flow to a self-similar regime is analyzed numerically. It is shown that depending on the thickness and initial density of the sediment layer, self-similar solutions have different structures and front propagation velocities.

We propose a system of first-order balance laws that describe the propagation of internal solitary waves in a multilayer stratified shallow water with non-hydrostatic pressure in the upper and lower layers. The construction of this model is based on the use of additional variables, which make it possible to approximate the Green–Naghdi-type dispersive equations by a first-order system. In the Boussinesq approximation, the governing equations allow one to simulate the propagation of non-linear internal waves, taking into account fine density stratification, a weak velocity shear in the layers, and uneven topography. We obtain smooth steady-state soliton-like solutions of the proposed model in the form of symmetric and non-symmetric waves of mode-2 adjoining to a given multilayer constant flow. Numerical calculations of the generation and propagation of large-amplitude internal waves are carried out using both the proposed first-order system and Green–Naghdi-type equations. It is established that the solutions of these models practically coincide. The advantage of the first-order equations is the simplicity of numerical implementation and a significant reduction in the calculation time. We show that the results of numerical simulation are in good agreement with the experimental data on the evolution of mode-2 solitary waves in tanks of constant and variable height.

The propagation and disintegration of nonlinear internal waves at the hydrophysical test site of the Pacific Oceanological Institute FEB RAS on the shelf of the Sea of Japan have been studied in the summer-autumn period for a number of years. The mechanisms of generation and propagation of nonlinear internal wave trains due to internal tide decay have been determined by continuous measurements of the vertical temperature and velocity distribution at depths of 20–60 m using bottom stations and by a regular study of the distributions of the main hydrodynamic characteristics along selected paths. In addition, other short-period sources of thermocline deformation have been detected that can be attributed to the propagation of bottom and surface vortex structures. The high spatiotemporal resolution of the temperature field in the neighborhood of bottom stations made it possible to reveal the fine structure of wave perturbations of various types. Multilayer shallow-water models for wave-train propagation have been developed and verified using the results of field and laboratory experiments.