Three-vortex quasi-geostrophic dynamics in a two-layer fluid. Part 1. Analysis of relative and absolute motions

2013 ◽  
Vol 717 ◽  
pp. 232-254 ◽  
Author(s):  
M. A. Sokolovskiy ◽  
K. V. Koshel ◽  
J. Verron
Keyword(s):  
Vortex Motion ◽  
Point Vortex ◽  
Bottom Layer ◽  
Phase Portraits ◽  
Fundamental Result ◽  
Barotropic Fluid ◽  
Oscillation Frequencies ◽  
Trilinear Coordinates ◽  
Analytical Expressions

AbstractThe results presented here examine the quasi-geostrophic dynamics of a point vortex structure with one upper-layer vortex and two identical bottom-layer vortices in a two-layer fluid. The problem of three vortices in a barotropic fluid is known to be integrable. This fundamental result is also valid in a stratified fluid, in particular a two-layer one. In this case, unlike the barotropic situation, vortices belonging to the same layer or to different layers interact according to different formulae. Previously, this occurrence has been poorly investigated. In the present work, the existence conditions for stable stationary (translational and rotational) collinear two-layer configurations of three vortices are obtained. Small disturbances of stationary configurations lead to periodic oscillations of the vortices about their undisturbed shapes. These oscillations occur along elliptical orbits up to the second order of the Hamiltonian expansion. Analytical expressions for the parameters of the corresponding ellipses and for oscillation frequencies are obtained. In the case of finite disturbances, vortex motion becomes more complicated. In this case we have made a classification of all possible movements, by analysing phase portraits in trilinear coordinates and by computing numerically the characteristic trajectories of the absolute and relative vortex motions.

The Nielsen-Thurston Classification

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Hyperbolic Geometry ◽  
Mapping Class Groups ◽  
Classification Theorem ◽  
Mapping Class ◽  
Fundamental Result ◽  
Class Groups ◽  
Mapping Torus ◽  
Higher Genus ◽  
Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

Reduction of Arbitrary Dynamical Systems by Wavelet Bases

2001 ◽  
Author(s):  
Roger Ghanem ◽  
Francesco Romeo
Keyword(s):  
Dynamical Systems ◽  
Reduction Process ◽  
Inner Product ◽  
Scaling Functions ◽  
Time Varying ◽  
Governing Equations ◽  
Wavelet Bases ◽  
Input And Output ◽  
Analytical Expressions

Abstract A procedure is developed for the identification and classification of nonlinear and time-varying dynamical systems based on measurements of their input and output. The procedure consists of reducing the governing equations with respect to a basis of scaling functions. Given the localizing properties of wavelets, the reduced system is well adapted to predicting local changes in time as well as changes that are localized to particular components of the system. The reduction process relies on traditional Galerkin techniques and recent analytical expressions for evaluating the inner product between scaling functions and their derivatives. Examples from a variety of dynamical systems are used to demonstrate the scope and limitations of the proposed method.

scholarly journals Some estimates on the space scales of vortex pairs emitted from river mouths

2001 ◽  
Vol 8 (1/2) ◽  
pp. 1-7 ◽  
Author(s):  
V. P. Goncharov ◽  
V. I. Pavlov
Keyword(s):  
Point Vortex ◽  
Vortex Pair ◽  
Vortex Structures ◽  
Shelf Zone ◽  
Vortex Pairs ◽  
Polygonal Boundary ◽  
Dipole Vortex ◽  
Qualitative Classification ◽  
River Mouths

Abstract. Two-dimensional vortex pairs are frequently observed in geophysical conditions, for example, in a shelf zone of the ocean near river mouths. The main aims of the work are to estimate the space scales of such vortex structures, to analyze possible scenarios of vortex pair motion and to give the qualitative classification of their trajectories. We discuss some features of the motion of strong localized vorticity concentrations in a given flow in the presence of boundaries. The analyses are made in the framework of a 2D point vortex mo-del with an open polygonal boundary. Estimations are made for the characteristic parameters of dipole vortex structures emitted from river mouths into the open ocean.

The Usage of Solar Power in the Heating Treatment of Concrete

2021 ◽  
pp. 44-50
Author(s):  
Alexander M. Ibragimov ◽  
Lyubov Y. Gnedina ◽  
Svetlana V. Gerasimova
Keyword(s):  
Heat Treatment ◽  
Reinforced Concrete ◽  
Solar Energy ◽  
Alternative Energy ◽  
Heat Transfer Process ◽  
Reinforced Concrete Structure ◽  
Heating Treatment ◽  
Modern Level ◽  
Analytical Expressions

This article proposes the usage of alternative energy instead of traditional, as safer, and environmentally friendly for the planet and all humankind in the manufacture of monolithic concrete and reinforced concrete products. Monolithic concreting in the conditions of the construction site is now widespread in the construction industry. The subject of the article is considered of the issues of heat treatment of concrete and acceleration of its hardening processes using solar energy. A brief review and classification of the most well-known and common energy sources used to harden acceleration of the concrete mix such as wind power, biofuels, solar energy, alternative hydropower is given. For consideration in the article offers the solar energy – the energy received from solar radiation. The sun as a source is an inexhaustible and affordable type of energy that supplies energy several times higher than traditional sources. In this regard, attention is paid to this field of alternative energy when using it in various spheres of the national economy, especially in the field of construction. Already existing methods of heat treatment of concrete with the help of solar technology are considered, the modern level of development of these methods is defined and further ways of their development are planned. As an example, a physical and mathematical model of a non-stationary heat transfer process in a flat reinforced concrete structure during its heat treatment using solar energy, considering the hydration of cement, is considered. Analytical expressions are obtained that allow us to model the process at any stage and actively intervene and adjust the external parameters to create rational and comfortable conditions for increasing the strength of concrete.

scholarly journals The Motion of a Point Vortex in Multiply-Connected Polygonal Domains

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1175
Author(s):  
El Mostafa Kalmoun ◽  
Mohamed M. S. Nasser ◽  
Khalifa A. Hazaa
Keyword(s):  
Cross Sections ◽  
Single Point ◽  
Vortex Motion ◽  
Point Vortex ◽  
Multiply Connected Domains ◽  
Polygonal Domains ◽  
Multiply Connected ◽  
Contour Lines ◽  
Circular Domains ◽  
Prime Function

We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion.

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