Particle transport by internal breathers

The processes of particle transport induced by internal breathers in quasi three-layer fluid are investigated in the framework of weakly nonlinear theory. We use the Gardner equation to describe the displacement at the maximum of vertical baroclinic mode. To determine the fields of vertical and horizontal velocities, which are used for calculating of Lagrangian particle trajectories, we apply three versions of the wave fields' vertical structure: linear mode, weakly nonlinear approximation (with taking into account the first nonlinear correction to linear mode) and weakly nonlinear-weakly dispersive approximation (with taking into account both the first nonlinear correction and the first dispersive correction to linear mode). The processes of particle transport are investigated for different initial configurations of breathers, as the velocity fields induced by nonlinear wave packets are substantially different for breathers with different phases. The comparison of the form of particle trajectories for different horizons and different breathers' configurations is made. It is shown that the use of the weakly nonlinear model is sufficient to determine the trajectories of fluid particles. Taking into account the first dispersive correction to the modal function almost does not affect particles' displacements, neither qualitatively, nor quantitatively. A significant difference between solutions of the problem of fluid particles' trajectories for two types of nonlinear wave motions in a stratified fluid-solitons and breathers-is revealed

Internal breathers' loads on marine facilities

We study interaction of long internal breathers in the idealized conditions, which correspond to realistic summer density stratification in the western part of the Cyprus coastal waters near the offshore gas fields. According to the maps of nonlinear and kinematic parameters of internal waves, which have been produced in our earlier studies, in this region the coefficient of cubic nonlinearity in the Gardner equation is positive. Therefore, breathers of internal waves can be generated in such a medium, for example, during the degeneration of long disturbances (from tidal or atmospheric sources). A series of numerical experiments on the interaction of breathers in the framework of the Gardner equation was carried out. Statistical moments up to the fourth order were calculated, and spectral analysis of the interaction process was provided. The structure of the velocity field induced by internal breather as well as the corresponding loads upon underwater engineering constructions are determined

Particle transport by internal breathers

The processes of particle transport induced by internal breathers in quasi three-layer fluid are investigated in the framework of weakly nonlinear theory. We use the Gardner equation to describe the displacement at the maximum of vertical baroclinic mode. To determine the fields of vertical and horizontal velocities, which are used for calculating of Lagrangian particle trajectories, we apply three versions of the wave fields' vertical structure: linear mode, weakly nonlinear approximation (with taking into account the first nonlinear correction to linear mode) and weakly nonlinear-weakly dispersive approximation (with taking into account both the first nonlinear correction and the first dispersive correction to linear mode). The processes of particle transport are investigated for different initial configurations of breathers, as the velocity fields induced by nonlinear wave packets are substantially different for breathers with different phases. The comparison of the form of particle trajectories for different horizons and different breathers' configurations is made. It is shown that the use of the weakly nonlinear model is sufficient to determine the trajectories of fluid particles. Taking into account the first dispersive correction to the modal function almost does not affect particles' displacements, neither qualitatively, nor quantitatively. A significant difference between solutions of the problem of fluid particles' trajectories for two types of nonlinear wave motions in a stratified fluid-solitons and breathers-is revealed

Impact of IW-induced currents on submersed pillars

We study the influence of intensive internal gravity waves, which are generated from baroclinic tidal wave and disturbances related to the variations in atmospheric pressure, on the hypothetic pillars of marine structures. We use temperature and salinity vertical profiles from GDEM climatology in a region near the Zohr offshore natural gas field located in the Egyptian sector of the Mediterranean Sea. Investigation of internal waves' dynamics was carried out in the framework of program complex intended for numerical modelling of propagation and transformation of such waves in the ocean, that implements procedure of numerical integration of fully nonlinear two-dimensional (vertical plane) system of equations of hydrodynamics of inviscid incompressible stratified fluid in the Boussinesq approximation bearing in mind the forcing by barotropic tide. The variations of the horizontal and vertical velocities are studied. Morison's formula is used to calculate the pressure force on the lateral surface of the elongated cylinder (pile). In the framework of this approach flow pressure contains the inertial (linear, depending on the acceleration of fluid particles in the wave) and the drag (non-linear resistance force, quadratic in velocity) components. It is shown that the inertial force is mainly smaller in absolute value than the drag force. A distribution of the loads and torques upon unit length of the pillar in time is considered

Variability of baroclinic rossby radius of deformation

In this study, we investigate variability of the first baroclinic Rossby radius (LR) of deformation for the Baltic Sea and the basin of the Mediterranean Sea. The baroclinic Rossby radius of deformation was calculated as the ratio of the phase speed of long linear internal waves of the lowest mode (which is the solution of the Sturm-Liouville problem for the vertical structure of the mode function and depends on the depth and stratification of the water density at the specific geographic location) and the Coriolis parameter. We use data from the GDEM V.3.0 climatology for the considered region. The seasonal and spatial variability of the baroclinic Rossby radius is discussed in detail. Its values for open areas do not exceed 10 km in the Baltic Sea, 20 km in the Black Sea and 15-18 km in the Mediterranean Sea. For the shelf zones of these seas, the values of LR are 2-3 times smaller. Seasonal differences are moderate, most pronounced in shallow areas, and can reach for most 5-9 km in the coastal regions of the Mediterranean Sea

Propagation regimes of interfacial solitary waves in a three-layer fluid

Long weakly nonlinear finite-amplitude internal waves in a fluid consisting of three inviscid layers of arbitrary thickness and constant densities (stable configuration, Boussinesq approximation) bounded by a horizontal rigid bottom from below and by a rigid lid at the surface are described up to the second order of perturbation theory in small parameters of nonlinearity and dispersion. First, a pair of alternatives of appropriate KdV-type equations with the coefficients depending on the parameters of the fluid (layer positions and thickness, density jumps) are derived for the displacements of both modes of internal waves and for each interface between the layers. These equations are integrable for a very limited set of coefficients and do not allow for proper description of several near-critical cases when certain coefficients vanish. A more specific equation allowing for a variety of solitonic solutions and capable of resolving most near-critical situations is derived by means of the introduction of another small parameter that describes the properties of the medium and rescaling of the ratio of small parameters. This procedure leads to a pair of implicitly interrelated alternatives of Gardner equations (KdV-type equations with combined nonlinearity) for the two interfaces. We present a detailed analysis of the relationships for the solutions for the disturbances at both interfaces and various regimes of the appearance and propagation properties of soliton solutions to these equations depending on the combinations of the parameters of the fluid. It is shown that both the quadratic and the cubic nonlinear terms vanish for several realistic configurations of such a fluid

Impact of IW-induced currents on submersed pillars

We study the influence of intensive internal gravity waves, which are generated from baroclinic tidal wave and disturbances related to the variations in atmospheric pressure, on the hypothetic pillars of marine structures. We use temperature and salinity vertical profiles from GDEM climatology in a region near the Zohr offshore natural gas field located in the Egyptian sector of the Mediterranean Sea. Investigation of internal waves' dynamics was carried out in the framework of program complex intended for numerical modelling of propagation and transformation of such waves in the ocean, that implements procedure of numerical integration of fully nonlinear two-dimensional (vertical plane) system of equations of hydrodynamics of inviscid incompressible stratified fluid in the Boussinesq approximation bearing in mind the forcing by barotropic tide. The variations of the horizontal and vertical velocities are studied. Morison's formula is used to calculate the pressure force on the lateral surface of the elongated cylinder (pile). In the framework of this approach flow pressure contains the inertial (linear, depending on the acceleration of fluid particles in the wave) and the drag (non-linear resistance force, quadratic in velocity) components. It is shown that the inertial force is mainly smaller in absolute value than the drag force. A distribution of the loads and torques upon unit length of the pillar in time is considered

Internal breathers' loads on marine facilities

We study interaction of long internal breathers in the idealized conditions, which correspond to realistic summer density stratification in the western part of the Cyprus coastal waters near the offshore gas fields. According to the maps of nonlinear and kinematic parameters of internal waves, which have been produced in our earlier studies, in this region the coefficient of cubic nonlinearity in the Gardner equation is positive. Therefore, breathers of internal waves can be generated in such a medium, for example, during the degeneration of long disturbances (from tidal or atmospheric sources). A series of numerical experiments on the interaction of breathers in the framework of the Gardner equation was carried out. Statistical moments up to the fourth order were calculated, and spectral analysis of the interaction process was provided. The structure of the velocity field induced by internal breather as well as the corresponding loads upon underwater engineering constructions are determined

Variability of baroclinic rossby radius of deformation

In this study, we investigate variability of the first baroclinic Rossby radius (LR) of deformation for the Baltic Sea and the basin of the Mediterranean Sea. The baroclinic Rossby radius of deformation was calculated as the ratio of the phase speed of long linear internal waves of the lowest mode (which is the solution of the Sturm-Liouville problem for the vertical structure of the mode function and depends on the depth and stratification of the water density at the specific geographic location) and the Coriolis parameter. We use data from the GDEM V.3.0 climatology for the considered region. The seasonal and spatial variability of the baroclinic Rossby radius is discussed in detail. Its values for open areas do not exceed 10 km in the Baltic Sea, 20 km in the Black Sea and 15-18 km in the Mediterranean Sea. For the shelf zones of these seas, the values of LR are 2-3 times smaller. Seasonal differences are moderate, most pronounced in shallow areas, and can reach for most 5-9 km in the coastal regions of the Mediterranean Sea

Sea water density parameterization for baltic sea

For modelling of internal waves in the ocean, an important task is to estimate the density stratification in the computational domain for the initialization of the models. In this case, a particular difficulty arises: data from various sources can have different resolutions, be based on different sets of initial measurement results, and have different averaging and smoothing algorithms. In this regard, the purpose of this study is to assess the sea density stratification from hydrology data (GDEM and WOA) from the point of view of the applicability of analytical models. Simplified layered models of stratified fluid are the most attractive for theoretical studies: The number of parameters of the medium is small, and analytical solutions can be constructed quite easily. The simplest layered model is a two-layer fluid. A large number of analytical, numerical, experimental laboratory and field studies have been devoted to the study of internal wave motions, and the properties of waves at the interface in such a model. Physical processes in two-layer fluid have been investigated from all possible aspects in the framework of linear, weakly and fully nonlinear theories. Cases of a multilayered medium are intermediate between a two-layer and continuously stratified, while allowing some analytical results. In the present research zoning of the Baltic Sea by the type of density stratification is carried out. Areas, in which the vertical profile of the sea water density is well approximated by a two-or a three-layer model, are marked out. Such zoning makes possible to simplify the preliminary investigation of internal gravity waves in the region, effectively reducing it to the use of known analytical results for waves propagating at the interface between the layers of constant densities, and it is easy to make preliminary estimates of their kinematic and nonlinear characteristics. Maps of distributions of parameters' values for layered models in winter and summer seasons are given. The seasonal variability of these zones is discussed. Hydrological data for calculating the density field are taken from the GDEM climatology. In addition, we use the calculated temperature and salinity data from the RCO model for the Baltic Sea. Areas with density stratifications close to symmetric with respect to the half-depth are particularly allocated. For such regions, the nonlinear properties of wave processes are subject to further refinement because lower-order nonlinearities vanish in weakly nonlinear models