More in detail...

More in detail...	

Vitaly Bulatov, Yury Vladimirov 
Wave motions of stratified mediums: 
theory and applications.  
Saarbrucken: Palmarium Academic 
Publishing, 2012, 577 pp. 
(in Russian)   

ANNOTATION

In monograph fundamental problem of internal gravity waves dynamics are considered. Analytical and numerical algorithms for calculating the wave fields for the real ocean parameters are presented. Obtained mathematical models can describe the internal gravity wave dynamics, taking into account the actual physical characteristics of sea water, bottom topography, etc. The history of studying the internal gravity waves in the ocean, as is known, originated in the Arctic Region after F. Nansen had described a phenomenon called "Dead Water". Nansen was the first man to observe the internal gravity waves in the Arctic Ocean. The notion of internal waves involves different oceanic phenomena such as "Dead Water", internal tidal waves, large scale oceanic circulation, and powerful pulsating internal waves. Such natural phenomena exist in the atmosphere as well; however, the theory of internal waves in the atmosphere was developed at a later time along with progress of the aircraft industry and aviation technology. Studying the oceanic currents of the Arctic Ocean became the principal objective of the Fram expedition in 1893-1986 and was continued in the years to follow. At that time such a voyage was an equivalent of a travel to the Moon. In the process of the expedition the scientists made a lot of observations and collected many data sheets and measurements in the Arctic which had been essentially unexplored at the time. During his arctic journey F. Nansen was the first scientist to classify the manner in which the "Dead Water" phenomenon occurs. This phenomenon comes about from the internal gravity waves generated by a slow moving vessel. The first theoretical work dedicated to internal gravity waves was the thesis work by V.W. Ekman, who provided a detailed definition of dead water and systematized the data obtained by F. Nansen. The "Dead Water" effect from internal gravity waves has been long known to sailors. Sailing vessels after being caught in the thermocline (a density contrast layer) suddenly brought down to a complete stop. This phenomenon resulted from the internal gravity waves generated by the vessel. But since the sailors saw no waves on the surface behind the ship this enormous water resistance seemed to be inexplicable whatsoever, and they blamed the bewitched drowned for holding the ship in place and not letting her go. Up to the 1960-s of the 20 century the research was for the most part focused on tidal waves, however, in the middle of the 1950-s some theoretic developmental studies and laboratory investigations were undertaken that involved the internal pulsating waves. As early as in 1950 there appeared the first definition for a superficial wake of internal gravity waves in the ocean. In 1965 the first scientific observations were made concerning the oceanic large amplitude internal waves and solitons. The interest to investigations involving the internal gravity waves grew up after the WW2 when the US Navy lost a few of its most advanced at the time submarines. After those accidents there were assumptions made that the disaster might have been caused by the internal gravity waves. As is known, the submarines often move along the thermocline (a density contrast layer) to avoid detection since the thermocline surface reflects the acoustical signals of active sonars and sea vessels . The most notorious incident involved the US Navy Thresher submarine that was lost at sea in 1963 with the crew of 129 on board. The US Thresher submarine was a most advanced boat in the world in the 1960-s and she could descend to depths and move at velocities that were inconceivable just a few years before she was constructed. It might be that the Thresher submarine was going along the thermocline and a large internal wave took her down to a depth pressure that she could not survive. There were no failures reported in operation of the submarine instrumentation, and no severe storms were detected in the area where the submarine was lost. It all might happen very quickly since the crew was not able to prevent the boat from falling down to deep water The first scientific explanation of what might be happening with the submarines appeared in 1965. It was the year that in the Andaman Sea for the first time ever discovered were large internal waves which happened to be a real sensation. Moving along the thermocline the wave could go 80 meters down. The oceanographers in the world until then believed no such waves existed. However, the Russian and US space programs allowed the scientists to take a look at our planet from the space. The panoramic photographs made from the orbit showed multiple wakes of waves. The point is that internal waves can create rather strong currents on the ocean surface. The flow is changed depending on the wave extension: its velocity is greater at the wave crest and wave trough, and is slower where thermocline oscillations are little. If several wave packets follow each other this pattern on the ocean surface is repeated. These surface flows are getting stronger or weaker when affected by wind waves depending on the set of wind, and can be defined as variations in light reflecting capacity of the ocean surface by remote radar sensing . The Apollo-Soyuz Test Project in 1975 was the first joint Russian-US enterprise in the space. The NASA researchers asked the crew to monitor the internal waves and photograph them. John Apel, who was a pioneer in studying the internal waves of the World Ocean, in 1978, wrote in the general scientific report of the expedition the following : "At least three photographs made by the "Apollo-Soyuz" crewmen have revealed obvious signs of internal gravity waves in the ocean, which is evident from periodically changing optical reflections from the ocean surface positioned above those waves. The wave packet (or the wave group) observed at Cadiz in Spain had the characteristics similar to those of internal waves shown by satellite photos taken close to the East Coast of the USA. In the Andaman Sea near the Malay Peninsula observed were several wave groups with the wave-lengths of 5 to 10 km and separations between the groups of 70 to 115 km. If these are really surface wakes of internal waves, these waves are one of the largest and fastest to this day. The measurements made earlier from aboard a sea vessel indicated presence in that area of large amplitude internal waves". The destruction of submarines gives evidence of the force of internal gravity waves. The internal waves generally move along the thermocline (a density contrast layer) positioned at a certain depth which separates by rather a weak at the ocean surface from deep waters, and their oscillation vector is directed either downwards or upwards. Once occurred these waves are propagating while maintain their form and force, and are capable of covering long distances. The internal waves also function as a carrier vehicle by transferring biomass and nourishment from one place to another. The underwater waves traveling upwards the shelf take the nourishment from the ocean deep water to the more salty shallow waters with ideal living conditions for larvae and fingerling. The wave motion in this case may be compared to a pumping action . The amplitude of internal gravity waves is generally comparable to the depth of the near-surface ocean. However, there was reported an occurrence when the wave was five times higher than the thermocline height. Since the sea water always contains layers positioned above each other with different temperature and salinity characteristics the internal gravity waves are generally in existence everywhere within the ocean thickness, but reach their maximum amplitudes typically near the thermocline. In equatorial areas the thermocline is located at the depth of 200 to 300 m, in the region of the Ormen Lange gas-field (Norway, Arctic basin) it is at 550-m depth, and in the Norwegian fiords with flowing in fresh water the thermocline is just 4 to 10 m deep . The industrial activities on the continental shelf involving crude-oil and gas production and other mining works have become an important factor for beginning the research on internal gravity waves with large amplitudes. The vessels and rigs for drilling and underwater constructions use long tubes connecting them to the sea bottom. The builders of underwater structures in equatorial areas have experienced the effect of large underwater waves and strong surface flows that can be shaped as a steep waterfall. Some time ago, when the phenomenon of internal wave was not known yet there were times when the builders got their equipment lost. Such losses are quite costly and make it clear that to protect and keep safe the fixed structures at sea we have to control the effect of internal gravity waves . The construction of sea platforms such as, for example, the Ormen Lange gas-field (Norway, Arctic basin) and other constructions at the sea bottom have stipulated many scientific studies including the fundamental research. Thus, for instance, the thermocline at the Ormen Lange gas-field (Norway, Arctic basin) is located at the depth of 500 m. It separates the Atlantic warm water of some 70C from the polar cold water of about 10C. The additionally accumulated warm current of the Atlantic Ocean can drop the thermocline even lower. The measurements in the region of Ormen Lange have registered once the current to lower the thermocline down from its regular depth to 550 m where it stayed for three days.It went down to the platform positioned at 850-m depth. After that the water was flowing back and upslope. In the beginning its motion velocity was half a meter per second which was very fast for a near-bottom current. Gradually the velocity dropped down, but the oscillations continued for a surprisingly long time of full 24 hours . The special interest to the research involving internal gravity waves is attributed also to intensive development of the Arctic and its natural resources. The internal waves are still poorly studied in the Arctic region since they are moving below the ice and practically are not visible from above. However, the available information on the movement of underwater objects indicates their presence. Yet, there may be exceptions when the internal gravity waves reach the ice cover lifting it up or down with certain periodicity, which can be monitored radar sensing equipment. The effect of waves of all types can result in breaking the Arctic ice cover. In addition the waves provoke iceberg displacement and move various pollutants. This is why the research of wave dynamics in the Arctic shelf region appears to be an important scientific and practical task to ensure safety in construction and operation of sea platforms . To make a detailed description of a wide range of physical phenomena that belong to wave dynamics of stratified, horizontally non-uniform and non-stationary mediums one should proceed from rather advanced mathematical models which usually become quite complex non-linear and multivariate, and can be fully and effectively explored only if using numerical methods. In certain situations, however, an adequate initial representation of the explored phenomena circle can be obtained when using more simple asymptotic models and analytical methods. For that matter there are as quite characteristic the problems of mathematically modeling the dynamics for non-harmonic packets of internal gravity waves, and even within the bounds of linear models they offer rather specific solutions that provide along with nontrivial physical effects for a self-sustained mathematical interest Now in connection with the new problems arising in geophysics, oceanology, physics of atmosphere, usage of the cryogenic liquids in the engineering sphere, as well as the problems of protection and study of the medium, operation of the complex hydraulic engineering facilities, including the marine oil producing complexes, and a number of other actual problems facing the science and engineering we can observe the growth of interest to the research of the dynamics of the wave movements of the different inhomogeneous liquids and, in particular, the stratified liquids. This interest is caused not only by the practical needs, but also by the need to have the solid theoretical base to solve the arising problems . It is necessary to note, that solution of the problems of the mechanics of continua and hydrodynamics always served as the stimulus of new directions in mathematics and mathematical physics. As the illustration to the above may serve the stream of the new ideas in the theory of the nonlinear differential equations, and also the discovery of the startling dependencies between the can be appearing the different branches of mathematics, that has followed after exploration of Cartevega de Vriza equation for the waves on the shallow water. Certainly, for the detailed description of the big amount of the natural phenomena connected with the dynamics of the stratified non-uniform in the horizontal direction and the non-stationary mediums, it is necessary to use the sufficiently developed mathematical models, which as a rule are the rather complex nonlinear multiparametric mathematical models and for their full-size research only the numerical methods are effective. The interest to the internal gravity waves is caused by their wide presence in the nature. Both the air atmosphere, and the oceans (Arctic basin) are stratified. Reduction of the air pressure and its density at the increase of the elevation are well known. But the sea water is also stratified. Here the raise of the water density with the increase of its depth is determined, mainly, not by the rather small compressibility of the water, but by the fact, that with the increase of the depth, as a rule, the temperature of the water is decreasing, and its saltiness grows. In the capacity of the stratified medium, as a rule, one considers the medium, the physical characteristics (density, dynamic viscosity and others) of which in the medium stationary status are changing only along some concrete direction. Stratification of the natural mediums (the ocean, the atmosphere) can be caused by the different physical reasons, but the most often by the gravity. This force creates in the stratified medium such a distribution of the particles of the dissolved in it salts and suspensions, at which it forms the heterogeneity of the medium along the direction of the gravity field in the stratified medium. This heterogeneity is called the density stratification. The stratification of density, as the experimental and natural observation show, renders the most essential influence, as compared with other kinds of stratification, on the dynamic properties of the medium and on the processes of distribution in the medium of the wave movements. Consequently at consideration of the wave generations in the stratified mediums usually neglect all other kinds of wave stratification, except for the density stratification, and in the capacity of the stratified medium they consider the medium with density stratification caused by the gravity. In the real oceanic conditions (Arctic basin) the density changes are small, the periods of oscillations of the internal waves are changing from several minutes (in the layers with rather fast change of the temperatures and the depth) up to the several hours. Such great periods of the fluctuations means, that even at the big amplitude of the internal waves, but they can achieve dozens of meters along the vertical direction , the speeds of the particles in the internal wave are low - for the vertical components the speeds of the particles have the order of mm/s, and for the horizontal - cm/s. Therefore the dissipative losses - the losses caused by of the liquid viscosity in the internal waves are very small, and the waves propagation can propagate practically without fading within the big distances,. At that the speed of propagation of the internal waves in the ocean is low - the order of dozens of cm/s . These properties of the internal gravity waves mean, that they can keep the information about the sources of their generations for the long time. Unfortunately, it is very difficult to orientate in this information because the internal waves pass the dozens and hundreds of kilometers from the source the generations up to the place of supervision; and practically everywhere, where there is the stratification of the ocean takes place, we can observe the internal waves, but simultaneously we can "hear" the "voices" of the most different sources. At that the qualitative (and the quantitative) properties of the internal waves, caused by that or other concrete source depend not only on its physical nature, and also on its spatial and time distribution, but also depends on the properties of the medium located between the source of the waves and the place of the observation . The internal waves represent the big interest not only from the point of view of their applications. They are of the interest to the theorists occupied with the problem of propagation ща the waves, as the internal waves properties in many respects differ from the properties of the accustomed to us the acoustic or electromagnetic waves. For example, for short harmonious internal waves of the following kind , where ) - the rays are directed not perpendicularly to the wave fronts - to the surfaces of the equal phase , but along these surfaces . The stratification, or the layered structure of the natural mediums (oceans and the air atmosphere) causing formation of the internal gravity waves plays then appreciable role in different oceanic and atmospheric processes and influences on the horizontal and vertical dynamic exchanges. The periods of the internal waves can make from several minutes up to several hours, the lengths of the waves can to achieve up to dozens of kilometers, and their amplitudes can exceed dozens of meters. The physical mechanism of formation of the internal waves is simple enough: if in the steadily stable stratified medium has appeared a generation, which has caused the particle out its balance state, then under action of gravity and the buoyancy the particle will make fluctuations about its balance position . The theory of the wave movements of the stratified mediums being the section of the modern hydrodynamics is quickly developing recently and rather interesting in the theoretical aspect as well as it is connected with the major applications in the engineering field (hydraulic engineering, shipbuilding, navigation, energy) and in geophysics (oceanology, meteorology, hydrology, preservation of the environment). Now the majority of the applied problems, concerning the waves generation caused by various generations are solved just in the linear aspect, that is considering the assumption, that the amplitude of the wave movements is small in comparison with length of the wave. The relative simplicity of the solution of the linear equations as compared with the solution of the complete nonlinear problem, the modern development of the corresponding mathematical tools and the computer engineering allows to meet many challenges of practice . Initially the theory of wave movements of the stratified medium was developing as the theory of superficial waves describing the behavior of the free surface of the liquid being in the gravity field. Later it has been understood, that the superficial waves represent the special type of the waves existing on the border of the separation of the various mediums densities, which in turn represent the special case of the internal waves in the medium non-uniform (stratified) in density. In the real ocean (Arctic basin) the non-uniform distribution of density may take place both in the vertical, and in the horizontal directions. At that considering the existing heterogeneity of the medium both in the vertical the horizontal directions, and also its nonstationarity at research of the distribution of the internal gravity waves require to use the special mathematical tools. As a rule it is supposed, that the density distribution is steady, that is the density does not decrease with the change of the depth . The reasons of initiation of the superficial and internal waves in the real ocean are very different: the fluctuations of the atmospheric pressure, the flow past of the bottom asperities, movement of the surface or the underwater ship, deformation in the density field, the turbulent spots formed by any reasons, the bottom shift or the underwater earthquake, the surface or underwater explosions, etc. One of the mechanisms of generation of the internal gravity waves may be excitation of the wave fields caused by, for example, at movement (flow past) of the non-local sources (underwater vessels, sea platforms), the turbulent spots, the water lenses and the other non-wave formations with the abnormal characteristics.