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