The uniform asymptotic form of the internal gravity-wave field generated by a source moving above a smoothly varying bottom

Булатов, В. В.; Vladimirov, Yury V.

doi:10.1007/s10665-010-9388-6

Cited by 10 publications

(19 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where L is the typical scale of the horizontal variations 0 ρ , h is the typical scale of the vertical variations 0 ρ (for example, the thermocline width) [13,15]. In non-dimension coordinates the equation system (2) will be written as (index * is omitted hereafter)…”

Section: Asymptotical Analysis Of Internal Gravity Wave Dynamics 221mentioning

confidence: 99%

“…, which are assumed to be known [15] . From here on the index n will be omitted while assuming that further calculations belong to an individually taken mode.…”

Section: Asymptotic Solutionsmentioning

confidence: 99%

“…The universal way to construct the asymptotic forms of such expressions is the method of etalon integrals [13][14][15][16][17]. This paper is devoted to the systematical description of a generalization of the geometrical optics method (WKBJ method), i.e.…”

mentioning

confidence: 99%

“…The above approaches are quite general and, in principle, enable one to solve a broad spectrum of problems from the mathematical point of view; however, the problem of their practical applications and, in particular, of the visualization of the corresponding asymptotic formulas based on the Maslov canonical operator is still far from completion, and in some specific problems one can use other schemes to find the asymptotic behavior whose computer realization using software of Mathematica type is rather simple. In this paper, using the approaches developed in [1,15], we construct asymptotic solutions of the problem which is formulated as follows.…”

mentioning

confidence: 99%

See 3 more Smart Citations

Asymptotical analysis of internal gravity wave dynamics in stratified medium

Булатов¹,

Vladimirov²

2014

ams

View full text Add to dashboard Cite

In the paper taking the assumption of the slowness of the change of the parameters of the vertically stratified medium in the horizontal direction and in time, the evolution of the non-harmonic wave packages of the internal gravity waves has been analyzed. The concrete form of the wave packages can be expressed through some model functions and is defined by the local behavior of the dispersive curves of the separate modes near to the corresponding special points. The solution of this problem is possible with the help of the modified variant of the special-time ray method offered by the authors (the method of geometrical optics), the basic difference of which consists that the asymptotic representation of the solution may be found in the form the series of the non-integer degrees of some small parameter. At that the exponent depends on the concrete form of representation of this package. The obvious kind of the representation is determined from the principle of the localness and the asymptotic behavior of the solution in the stationary and the horizontallyhomogeneous case. The phases of the wave packages are determined from the corresponding equations of the eikonal, which can be solved numerically on the characteristics (rays). Amplitudes of the wave packages are determined from the laws of concervation of the some invariants along the characteristics (rays).

show abstract

Section: Asymptotical Analysis Of Internal Gravity Wave Dynamics 221mentioning

confidence: 99%

“…, which are assumed to be known [15] . From here on the index n will be omitted while assuming that further calculations belong to an individually taken mode.…”

Section: Asymptotic Solutionsmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

See 2 more Smart Citations

Asymptotical analysis of internal gravity wave dynamics in stratified medium

Булатов¹,

Vladimirov²

2014

ams

View full text Add to dashboard Cite

show abstract

“…The present issue is concerned with body motion in a two-layer fluid [6], interaction of water waves with a porous plate [7], displacement of one fluid by another in a porous medium [8], liquid-film flows over a step topography with external electric field [9], turbulent flows produced on rotating blades [10], a water-impact problem [11], the Maslov canonical operator for shallow-water equations with localized initial data [12], internal gravity waves over a bottom topography [13], a reaction-diffusion model for concrete corrosion in sewer pipes [14] and convection in a two-layer fluid with external high-frequency vibration [15]. Different methods of asymptotic analysis such as the homogenization method, the Maslov canonical operator, the method of matched asymptotic expansions, generalized methods from geometric optics, and their combinations with numerical methods are used to demonstrate their practical importance in the analysis of practical problems.…”

mentioning

confidence: 99%

Preface to the fifth special issue on practical asymptotics

Korobkin

2010

J Eng Math

View full text Add to dashboard Cite

mathematics seems, at first sight anyway, to become more and more dominated by direct numerical simulations. Admittedly, this leads to new insights which, it would seem, could not have been attained by other means" and ". . . many believe that asymptotics deals with exceptional cases which are usually outside the practical domain. However, this is a misconception!" The Guest-Editors of the following, second, third and fourth issues on practical asymptotics, argued that asymptotic solutions and asymptotic methods remain and will be valuable because they "provide important complements to numerical simulation" [2], "can offer ways to systematically study problems that may otherwise be inaccessible" [3] and "may provide quantitative and/or qualitative information that computer simulations can not" [4]. It is seen that the issues on practical asymptotics are a forum for discussion "Does the asymptotic analysis have a future in the era of supercomputers?". The answer to the question is obvious: "Yes, it has."Researchers and engineers, who use asymptotic methods and/or approximations inspired by these methods to solve practical problems, well understand that asymptotic analysis is a powerful tool, the range of applicability of which is very wide but does not cover everything we need. However, the situation is not so simple. There is a difficulty which is not about the application of asymptotic methods to practical problems but about the "public" opinion that asymptotic and, in general, mathematical tools are "old-fashioned" compared with direct numerical simulations. These days even academics sometimes ask the question "Why do we need this complicated analysis, if the problem can be easily solved by computer simulations?" Before the 90s numerical calculations were complementary to rigorous or approximate analysis of a problem. At the present time the "center of mass" in research is still moving towards computations. However, along this way, it is getting more and more clear that computer simulations being applied to practical problems is not a universal panacea. Euphoria about computers is still high but a more practical point of view is becoming at least visible these days: computation is a research tool as are many other approximate or rigorous tools of mathematics which help us to gain new knowledge, understand important phenomena and solve challenging practical problems.It would be perfect if our problems could be solved in a rigorous way. But this is an ideal situation which happens quite rarely, especially in new formulations. In order to obtain a solution, we need rigorous mathematical methods, approximate methods, with asymptotic methods being the most powerful of them, and computations. It is wrong A. Korobkin (B)

show abstract

Internal Gravity Waves in Horizontally Inhomogeneous Ocean

Булатов

Vladimirov

2018

The Ocean in Motion

View full text Add to dashboard Cite

The uniform asymptotic form of the internal gravity-wave field generated by a source moving above a smoothly varying bottom

Cited by 10 publications

References 13 publications

Asymptotical analysis of internal gravity wave dynamics in stratified medium

Asymptotical analysis of internal gravity wave dynamics in stratified medium

Preface to the fifth special issue on practical asymptotics

Internal Gravity Waves in Horizontally Inhomogeneous Ocean

Contact Info

Product

Resources

About