| 1. |
- Ibragimov, Nail H., et al.
(författare)
-
Applications of Lie group analysis in Geophysical fluid dynamics
- 2011
-
Bok (övrigt vetenskapligt/konstnärligt)abstract
- This book introduces an effective method for seeking local and nonlocal conservation laws and exact solutions for nonlinear two-dimensional equations which provide a basic model in describing internal waves in the ocean. The model consists of non-hydrostatic equations of motion which uses the Boussinesq approximation and linear stratification. The Lie group analysis is used for constructing non-trivial conservation laws and group invariant solutions. It is shown that nonlinear equations in question have remarkable property to be self-adjoint. This property is crucial for constructing physically relevant conservation laws for nonlinear internal waves in the ocean. The comparison with the previous analytic studies and experimental observations confirrms that the anisotropic nature of the wave motion allows to associate some of the obtained invariant solutions with uni-directional internal wave beams propagating through the medium. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams are considered. The behavior of the invariant solutions near the critical latitude is investigated.
|
|
| 2. |
- Ibragimov, Nail H., et al.
(författare)
-
Approximate symmetries and solutions of the Kompaneets equation
- 2014
-
Ingår i: Communications in nonlinear science & numerical simulation. - : MAIK NAUKA/INTERPERIODICA/SPRINGER. - 1007-5704 .- 1878-7274. ; 55:2, s. 220-224
-
Tidskriftsartikel (refereegranskat)abstract
- Different approximations of the Kompaneets equation are studied using approximate symmetries, which allows consideration of the contributions of all terms of this equation previously neglected in the analysis of the limiting cases.
|
|
| 3. |
- Ibragimov, Nail H., et al.
(författare)
-
Integration by quadratures of the nonlinear Euler equations modeling atmospheric flows in a thin rotating spherical shell
- 2011
-
Ingår i: Physics Letters A. - : Elsevier. - 0375-9601 .- 1873-2429. ; 375:44, s. 3858-3865
-
Tidskriftsartikel (refereegranskat)abstract
- We study the nonlinear incompressible non-viscous fluid flows within a thin rotating atmospheric shell that serve as a simple mathematical description of an atmospheric circulation caused by the temperature difference between the equator and the poles. The model is also superimposed by a particular stationary flow which, under the assumption of no friction and a distribution of temperature dependent only upon latitude, models the zonal west-to-east flows in the upper atmosphere between the Ferrel and Polar cells. Owing to the Coriolis effects, the resulting achievable meteorological flows correspond to the asymptotical stable flows that are being translated along the equatorial plane. The exact solutions in terms of elementary functions are found by using Lie group methods.
|
|
| 4. |
- Ibragimov, Nail H., et al.
(författare)
-
Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics
- 2010
-
Ingår i: Communications in nonlinear science & numerical simulation. - : Elsevier. - 1007-5704 .- 1878-7274. ; 15:8, s. 1989-2002
-
Tidskriftsartikel (refereegranskat)abstract
- It is shown that Lie group analysis of differential equations provides the exact solutions of two-dimensional stratified rotating Boussinesq equations which are a basic model in geophysical fluid dynamics. The exact solutions are obtained as group invariant solutions corresponding to the translation and dilation generators of the group of transformations admitted by the equations. The comparison with the previous analytic studies and experimental observations confirms that the anisotropic nature of the wave motion allows to associate these invariant solutions with uni-directional internal wave beams propagating through the medium. It is also shown that the direction of internal wave beam propagation is in the transverse direction to one of the invariants which corresponds to a linear combination of the translation symmetries. Furthermore, the amplitudes of a linear superposition of wave-like invariant solutions forming the internal gravity wave beams are arbitrary functions of that invariant. Analytic examples of the latitude-dependent invariant solutions associated with internal gravity wave beams that have different general profiles along the obtained invariant and propagating in the transverse direction are considered. The behavior of the invariant solutions near the critical latitude is illustrated. © 2009 Elsevier B.V. All rights reserved.
|
|
| 5. |
- Ibragimov, Nail H., et al.
(författare)
-
Invariant solutions as internal singularities of nonlinear differential equations and their use for qualitative analysis of implicit and numerical solutions
- 2009
-
Ingår i: Communications in nonlinear science & numerical simulation. - : Elsevier Science. - 1007-5704 .- 1878-7274. ; 14:9-10, s. 3537-3547
-
Tidskriftsartikel (refereegranskat)abstract
- Lie group analysis of nonlinear differential equations reveals existence of singularities provided by invariant solutions and invisible from the form of the equation in question. We call them internal singularities in contrast with external singularities manifested by the form of the equation. It is illustrated by way of examples that internal singularities are useful for analyzing a behaviour of solutions of nonlinear differential equations near external singularities.
|
|
| 6. |
- Ibragimov, Nail H., et al.
(författare)
-
Nonlinear Whirlpools Versus Harmonic Waves in a Rotating Column of Stratified Fluid
- 2013
-
Ingår i: Mathematical Modelling of Natural Phenomena. - : EDP SCIENCES. - 0973-5348 .- 1760-6101. ; 8:1, s. 122-131
-
Tidskriftsartikel (refereegranskat)abstract
- Propagation of nonlinear baroclinic Kelvin waves in a rotating column of uniformly stratified fluid under the Boussinesq approximation is investigated. The model is constrained. by the Kelvin's conjecture saying that the velocity component normal to the interface between rotating fluid and surrounding medium (e.g. a seashore) is possibly zero everywhere in the domain of fluid motion, not only at the boundary. Three classes of distinctly different exact solutions for the nonlinear model are obtained. The obtained solutions are associated with symmetries of the Boussinesq model. It is shown that one class of the obtained solutions can be visualized as rotating whirlpools along which the pressure deviation from the mean state is zero, is positive inside and negative outside of the whirlpools. The angular velocity is zero at the center of the whirlpools and it is monotonically increasing function of radius of the whirlpools.
|
|
| 7. |
- Ibragimov, Nail H., et al.
(författare)
-
Rotationally symmetric internal gravity waves
- 2012
-
Ingår i: International Journal of Non-Linear Mechanics. - : Elsevier. - 0020-7462 .- 1878-5638. ; 47:1, s. 46-52
-
Tidskriftsartikel (refereegranskat)abstract
- Many mathematical models formulated in terms of non-linear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating non-linear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is to provide the group theoretical modeling of internal waves in the ocean. The approach is based on a new concept of conservation laws that is utilized to systematically derive the conservation laws of non-linear equations describing propagation of internal waves in the ocean. It was shown in our previous publication that uni-directional internal wave beams can be obtained as invariant solutions of non-linear equations of motion. The main goal of the present publication is to thoroughly analyze another physically significant exact solution, namely the rotationally symmetric solution and the energy carried by this solution. It is shown that the rotationally symmetric solution and its energy are presented by means of a bounded oscillating function.
|
|
| 8. |
- Ibragimov, Nail H., et al.
(författare)
-
Symmetries and Conservation Laws of a Spectral Nonlinear Model for Atmospheric Baroclinic Jets
- 2014
-
Ingår i: Mathematical Modelling of Natural Phenomena. - : EDP SCIENCES S A.. - 0973-5348 .- 1760-6101. ; 9:5, s. 111-118
-
Tidskriftsartikel (refereegranskat)abstract
- In this paper, we shall obtain the symmetries of the mathematical model describing spontaneous relaxation of eastward jets into a meandering state and use these symmetries for constructing the conservation laws. The basic eastward jet is a spectral parameter of the model, which is in geostrophic equilibrium with the basic density structure and which guarantees the existence of nontrivial conservation laws.
|
|
| 9. |
- Ibragimov, Nail H., et al.
(författare)
-
Three-dimensional non-linear rotating surface waves in channels of variable depth in the presence of formation of a small perturbation of atmospheric pressure across the channel
- 2009
-
Ingår i: Communications in nonlinear science & numerical simulation. - : Elsevier Science. - 1007-5704 .- 1878-7274. ; 14:11, s. 3811-3820
-
Tidskriftsartikel (refereegranskat)abstract
- We consider three-dimensional free-boundary problem on the propagation of incompressible, homogeneous and inviscid fluid with zero surface tension confined in a channel of variable depth. Since for large-scale flows the fluid motion is affected by the rotation of the earth, the model is considered in rotating reference frame. Additionally, small atmospheric pressure variations across the channel are taken into account. It is shown that the non-trivial solution to the problem represents three-dimensional solitary wave which is given by the rotation modified Korteweg-de Vries equation (fKdV): b(1)xi(xxx) + b(2)xi xi(x) + b(3)(f)xi(x) = 0, where x is the down-channel coordinate and the coefficients b(i) (i = 1,2,3) of the resulting fKdV equation depend on the transverse topography of the channel and, additionally, b(3) depends on the Coriolis parameter f. It is also shown that if the vertical profile of the channel is symmetric about the vertical axis, the small atmospheric variations will not appear in the resulting fKdV equation. The effects of channel's cross-sectional geometry on the shape of the resulting three-dimensional wave profile in a longitudinal direction are studied numerically. Additionally, to better understand the effects of the Earth rotation, the above analysis is performed at different latitudes. (C) 2008 Elsevier B.V. All rights reserved.
|
|
| 10. |
- Ibragimov, Nail H., et al.
(författare)
-
Utilization of photon orbital angular momentum in the low-frequency radio domain
- 2007
-
Ingår i: Physical Review Letters. - : American Physical Society. - 0031-9007 .- 1079-7114. ; 99:8
-
Tidskriftsartikel (refereegranskat)abstract
- We show numerically that vector antenna arrays can generate radio beams that exhibit spin and orbital angular momentum characteristics similar to those of helical Laguerre-Gauss laser beams in paraxial optics. For low frequencies (1 GHz), digital techniques can be used to coherently measure the instantaneous, local field vectors and to manipulate them in software. This enables new types of experiments that go beyond what is possible in optics. It allows information-rich radio astronomy and paves the way for novel wireless communication concepts.
|
|
