hydrophysical parameters. In this contribution, we generalize a method of geometrical optics, i.e., the space-time ray method, which permits solving the problem
of mathematical modeling of IGW dynamics in the horizontally inhomogeneous
and vertically stratified ocean. The ray representations agree well with the intuitive
and empirical concepts of IGW propagation in the real ocean. This method is
sufficiently universal, and in many cases, this is the only possible method for
approximate calculations of wave fields in the ocean. The most typical horizontal
inhomogeneities of the real ocean are the variations in the bottom topography of the
ocean, horizontal inhomogeneities of the density field, and unsteady ocean currents.
An exact analytic solution can be obtained, for example, using the method of
separation of variables only if the density distribution and the bottom topography
can be described by sufficiently simple model functions. If the bottom topography
and the ocean stratification are arbitrary, then one can construct only the asymptotic
representations of the solution or solve the problem numerically. But the numerical
solution does not permit obtaining and analyzing the qualitative characteristics of
the wave field at large distances, which is necessary, for example, when solving the
IGW detection problem by remote methods including, for example, radar imaging
[8, 10, 12, 13, 15].
The mathematical modeling of IGW wave dynamics in the horizontally inhomogeneous and vertically stratified ocean is possible on the basis of a modified
version of the space-time ray method (a method of geometrical optics). The specific
form of asymptotic representations can be determined by solving the problems,
which describe the IGW dynamics in the vertically stratified, horizontally homogeneous, and steady-state ocean. As a rule, when studying the evolution of IGW
packets in the ocean with slowly varying and unsteady parameters, it is assumed
that this wave packet is locally harmonic. In contrast to the majority of works, in
which this problem has been studied, the proposed modified method of geometrical
optics allows one to describe the structure of wave packets near singular surfaces
such as caustics and wave fronts [3, 4, 6, 7].
The term “geometrical optics” has different meanings in the scientific literature.
The geometrical optics understood in the narrow (or ray) sense deals only with the
methods for constructing images by using the rays, while the geometrical optics
understood in the wider (or wave) sense is a method for obtaining approximate
descriptions of wave fields. In the wave interpretation, which is used in this paper,
the rays, as a rule, form only the geometric skeleton, on which the wave filed
is “sewn on”. According to the two previous interpretations of the geometrical
optics, two periods in its development exist. The first ray period was ideologically
completed by Hamilton’s fundamental works, which significantly influenced the
development of the classical mechanics. The construction of rays underlies the
instrumental optics, which is mainly oriented to design various optical devices. The
contemporary wave period originates from the Debye’s works, which decisively
influenced the formation of ray concepts in the wave theory [5].
The asymptotic representation of the solutions of wave packet propagation in the
ocean with horizontally inhomogeneous density and numerical computations at the
typical oceanic parameters testify that the horizontal inhomogeneity significantly
110
V. V. Bulatov and Y. V. Vladimirov
of mathematical modeling of IGW dynamics in the horizontally inhomogeneous
and vertically stratified ocean. The ray representations agree well with the intuitive
and empirical concepts of IGW propagation in the real ocean. This method is
sufficiently universal, and in many cases, this is the only possible method for
approximate calculations of wave fields in the ocean. The most typical horizontal
inhomogeneities of the real ocean are the variations in the bottom topography of the
ocean, horizontal inhomogeneities of the density field, and unsteady ocean currents.
An exact analytic solution can be obtained, for example, using the method of
separation of variables only if the density distribution and the bottom topography
can be described by sufficiently simple model functions. If the bottom topography
and the ocean stratification are arbitrary, then one can construct only the asymptotic
representations of the solution or solve the problem numerically. But the numerical
solution does not permit obtaining and analyzing the qualitative characteristics of
the wave field at large distances, which is necessary, for example, when solving the
IGW detection problem by remote methods including, for example, radar imaging
[8, 10, 12, 13, 15].
The mathematical modeling of IGW wave dynamics in the horizontally inhomogeneous and vertically stratified ocean is possible on the basis of a modified
version of the space-time ray method (a method of geometrical optics). The specific
form of asymptotic representations can be determined by solving the problems,
which describe the IGW dynamics in the vertically stratified, horizontally homogeneous, and steady-state ocean. As a rule, when studying the evolution of IGW
packets in the ocean with slowly varying and unsteady parameters, it is assumed
that this wave packet is locally harmonic. In contrast to the majority of works, in
which this problem has been studied, the proposed modified method of geometrical
optics allows one to describe the structure of wave packets near singular surfaces
such as caustics and wave fronts [3, 4, 6, 7].
The term “geometrical optics” has different meanings in the scientific literature.
The geometrical optics understood in the narrow (or ray) sense deals only with the
methods for constructing images by using the rays, while the geometrical optics
understood in the wider (or wave) sense is a method for obtaining approximate
descriptions of wave fields. In the wave interpretation, which is used in this paper,
the rays, as a rule, form only the geometric skeleton, on which the wave filed
is “sewn on”. According to the two previous interpretations of the geometrical
optics, two periods in its development exist. The first ray period was ideologically
completed by Hamilton’s fundamental works, which significantly influenced the
development of the classical mechanics. The construction of rays underlies the
instrumental optics, which is mainly oriented to design various optical devices. The
contemporary wave period originates from the Debye’s works, which decisively
influenced the formation of ray concepts in the wave theory [5].
The asymptotic representation of the solutions of wave packet propagation in the
ocean with horizontally inhomogeneous density and numerical computations at the
typical oceanic parameters testify that the horizontal inhomogeneity significantly
110
V. V. Bulatov and Y. V. Vladimirov