Interfacial long traveling waves in a two-layer fluid with variable depth
Long wave propagation in a two-layer fluid with variable depth is studied for specific bottom configurations, which
allow waves to propagate over large distances. Such configurations are found within the linear shallow-water theory
and determined by a family of solutions of the second-order ordinary differential equation (ODE) with three arbitrary
constants. These solutions can be used to approximate the true bottom bathymetry. All such solutions represent
smooth bottom profiles between two different singular points. The first singular point corresponds to the point
where the two-layer flow transforms into a uniform one. In the vicinity of this point nonlinear shallow-water theory is
used and the wave breaking criterion, which corresponds to the gradient catastrophe is found. The second bifurcation
point corresponds to an infinite increase in water depth, which contradicts the shallow-water assumption.
This point is eliminated by matching the “nonreflecting” bottom profile with a flat bottom. The wave transformation
at the matching point is described by the second-order Fredholm equation and its approximated solution is then
obtained. The results extend the theory of internal waves in inhomogeneous stratified fluids actively developed by
Prof. Roger Grimshaw, to the new solutions types.
In this paper for the explain of the mechanism of formation of smooth strips (slicks) on the sea surface under the action of internal waves are used the film of surface-active substances, attendees everywhere in the sea. Experimental data on the real characteristics of marine films of surface-active substances are used for the calculation of histograms of contrast in the spectrum of wind ripples in the centimeter range for various parameters of the internal wave and wind wave lengths within the "film" mechanism of the effects of internal waves on the spectrum of wind-generated waves. It is shown that the ripple in the wavelength range 2-3 cm contrast weakly depends on the parameters of the internal waves (although with increasing internal wave amplitude), and the average number of 6-7 dB. For greater lengths ripple contrast is strongly dependent on the ratio of the rate of flow of water particles in the internal waves to the phase velocity of the internal wave. This dispersion deviations from average contrast values around the average value, which indicates a strong variation of contrast in each case. Nevertheless, it can be concluded relatively low sensitivity of "film" mechanism of action internal waves on the sea surface to a particular type of surface-active substances.
Simulation of abnormally large internal waves generated by the baroclinic tide is now quite important due to the increased number of offshore platforms installed on offshore oil and gas fields. The height of the internal waves in many areas of the oceans can be up to 100 m, and these waves become really dangerous. All of this points to the need to study the possible dynamic and catastrophic phenomena accompanying the propagation of internal waves of large amplitude. In terms of the computational analysis of transient wave motion is a very complex task. There was designed widely used numerical code MIT, decisive full hydrodynamic equations with the real bottom topography, the Earth's rotation and turbulent processes. However, this model requires a lot of computing resources, a number of viable solutions to the practical problems of Oceanology. However, even such a complete model is not yet take into account the current practice relatively stable background horizontal - nonuniform stratification, which is typical for the real ocean conditions. That is why the research and analysis of new phenomena is widely used asymptotic model based on the Korteweg–de Vries equation and its generalizations. In this paper we use the asymptotic model for the analysis of the two most important effects in the bottom layer, induced by internal waves: the change in pressure at the bottom and sediment transport. Moving the sediment and erosion of the bottom near the supports are well known to storm surges in the coastal zone. The specifics of the internal waves is their greater length (a few kilometers), so that such waves are always "get" to the bottom, even far from the coast and can lead to erosion of supports oil platforms in deep water. Particularly dangerous abnormally large waves, since the characteristics of erosion is proportional to the cube of the amplitude of the internal wave. In our country, yet the effects of internal waves on the stability of marine structures are not regulated. What is needed is an active study of this problem in theoretical terms, and the accumulation of data on the hazards of internal waves in areas where there are oil and gas platforms.
Purpose: Numerical modeling of internal baroclinic disturbances of different shapes in a model lake with variable depth, analysis of velocity field of wave-induced current, especially in the near-bed layer.
Approach: The study is carried out with the use of numerical full nonlinear nonhydrostatic model for stratified fluid.
Findings: The full nonlinear numerical modeling of internal wave dynamics in a stratified lake is carried out. The calculated distributions of near-bed velocities are analyzed; the significance of 3D effects for the velocity fields is emphasized; the regions of maximal (where internal waves are the main driving factor for sediment resuspension and erosion processes on the bed) and minimal velocities are marked out.
Originality: The results are new and can have practical application for many applied problems, especially ecological and economical, concerned with the processes of propagation of natural and anthropogenic pollutions in natural basins and the investigation of water quality, as well as with influence upon engineering structures and sediment transport.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.
Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.
Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.