Instability of surface quasigeostrophic spatially periodic flows

2020 ◽  
Author(s):  
Maksim Kalashnik ◽  
Michael Kurgansky ◽  
Sergey Kostrykin
Keyword(s):  
Nonlinear Oscillations ◽  
Nonlinear Differential Equations ◽  
Classical Mechanics ◽  
Basic Research ◽  
Growth Increment ◽  
Russian Science ◽  
Periodic Flows ◽  
Galerkin Approximations ◽  
Spatially Periodic ◽  
The Galerkin Method

<p>The surface quasigeostrophic (SQG) model is developed to describe the dynamics of flows with zero potential vorticity in the presence of one or two horizontal boundaries (Earth surface and tropopause). Within the framework of this model, the problems of linear and nonlinear stability of zonal spatially periodic flows are considered. To study the linear stability of flows with one boundary, two approaches are used. In the first approach, the solution is sought by decomposing into a trigonometric series, and the growth rate of the perturbations is found from the characteristic equation containing an infinite continued fraction. In the second approach, few-mode Galerkin approximations of the solution are constructed. It is shown that both approaches lead to the same dependence of the growth increment on the wavenumber of perturbations. The existence of instability with a preferred horizontal scale on the order of the wavelength of the main flow follows from this dependence. A similar result is obtained within the framework of the SQG model with two horizontal boundaries. The Galerkin method with three basis trigonometric functions is also used to study the nonlinear dynamics of perturbations, described by a system of three nonlinear differential equations similar to that describing the motion of a symmetric top in classical mechanics. An analysis of the solutions of this system shows that the exponential growth of disturbances at the linear stage is replaced by a stage of stable nonlinear oscillations (vacillations). The results of numerical integration of full nonlinear SQG equations confirm this analysis.</p><p>The work was supported by the Russian Foundation for Basic Research (Project 18-05-00414) and the Russian Science Foundation (Project 19-17-00248).</p>

scholarly journals Instability of Surface Quasigeostrophic Spatially Periodic Flows

2019 ◽  
Vol 77 (1) ◽  
pp. 239-255 ◽  
Author(s):  
M. V. Kalashnik ◽  
M. V. Kurgansky ◽  
S. V. Kostrykin
Keyword(s):  
Continued Fraction ◽  
Nonlinear Oscillations ◽  
Nonlinear Differential Equations ◽  
Classical Mechanics ◽  
Growth Increment ◽  
Periodic Flows ◽  
Galerkin Approximations ◽  
Spatially Periodic ◽  
Zero Potential ◽  
The Galerkin Method

Abstract The surface quasigeostrophic (SQG) model is developed to describe the dynamics of flows with zero potential vorticity in the presence of one or two horizontal boundaries (Earth surface and tropopause). Within the framework of this model, the problems of linear and nonlinear stability of zonal spatially periodic flows are considered. To study the linear stability of flows with one boundary, two approaches are used. In the first approach, the solution is sought by decomposing into a trigonometric series, and the growth rate of the perturbations is found from the characteristic equation containing an infinite continued fraction. In the second approach, few-mode Galerkin approximations of the solution are constructed. It is shown that both approaches lead to the same dependence of the growth increment on the wavenumber of perturbations. The existence of instability with a preferred horizontal scale on the order of the wavelength of the main flow follows from this dependence. A similar result is obtained within the framework of the SQG model with two horizontal boundaries. The Galerkin method with three basis trigonometric functions is also used to study the nonlinear dynamics of perturbations, described by a system of three nonlinear differential equations similar to that describing the motion of a symmetric top in classical mechanics. An analysis of the solutions of this system shows that the exponential growth of disturbances at the linear stage is replaced by a stage of stable nonlinear oscillations (vacillations). The results of numerical integration of full nonlinear SQG equations confirm this analysis.

scholarly journals NONLINEAR DYNAMICS OF LONG-WAVE PERTURBATIONS OF THE INVISCID KOLMOGOROV FLOW

2019 ◽  
Vol 47 (1) ◽  
pp. 64-65
Author(s):  
M.V. Kalashnik ◽  
M.V. Kurgansky
Keyword(s):  
Nonlinear Dynamics ◽  
Nonlinear Oscillations ◽  
Initial Perturbation ◽  
Analytical Formula ◽  
Basic Research ◽  
Elliptic Functions ◽  
Structural Features ◽  
Kolmogorov Flow ◽  
Long Wave ◽  
The Galerkin Method

The nonlinear dynamics of long-wave perturbations of the inviscid Kolmogorov flow, which models periodically varying in the horizontal direction oceanic currents, is studied. To describe this dynamics, the Galerkin method with basis functions representing the first three terms in the expansion of spatially periodic perturbations in the trigonometric series is used. The orthogonality conditions for these functions formulate a nonlinear system of partial differential equations for the expansion coefficients (Kalashnik, Kurgansky, 2018). Based on the asymptotic solutions of this system, a linear, quasilinear and nonlinear stage of perturbation dynamics are identified. It is shown that the time-dependent growth of perturbations during the first two stages is succeeded by the stage of stable nonlinear oscillations. The corresponding oscillations are described by the oscillator equation containing a cubic nonlinearity, which is integrated in terms of elliptic functions. An analytical formula for the period of oscillations is obtained, which determines its dependence on the amplitude of the initial perturbation. Structural features of the field of the stream function of the perturbed flow are described, associated with the formation of closed vortex cells and meandering flow between them. The research was supported by the RAS Presidium Program «Nonlinear dynamics: fundamental problems and applications» and by the Russian Foundation for Basic Research (Projects 18-05-00414, 18-05-00831).

Optimization of Nonlinear Unbalanced Flexible Rotating Shaft Passing Through Critical Speeds

2021 ◽  
Author(s):  
Sadegh Amirzadegan ◽  
Mohammad Rokn-Abadi ◽  
R. D. Firouz-Abadi
Keyword(s):  
Degrees Of Freedom ◽  
Nonlinear Oscillations ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Angular Acceleration ◽  
Beam Theory ◽  
Rotor System ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Applied Torque

This work studies the nonlinear oscillations of an elastic rotating shaft with acceleration to pass through the critical speeds. A mathematical model incorporating the Von-Karman higher-order deformations in bending is developed to investigate the nonlinear dynamics of rotors. A flexible shaft on flexible bearings with springs and dampers is considered as rotor system for this work. The shaft is modeled as a beam and the Euler–Bernoulli beam theory is applied. The kinetic and strain energies of the rotor system are derived and Lagrange method is then applied to obtain the coupled nonlinear differential equations of motion for 6 degrees of freedom. In order to solve these equations numerically, the finite element method (FEM) is used. Furthermore, for different bearing properties, rotor responses are examined and curves of passing through critical speeds with angular acceleration due to applied torque are plotted. Then the optimal values of bearing stiffness and damping are calculated to achieve the minimum vibration amplitude, which causes to pass easier through critical speeds. It is concluded that the value of damping and stiffness of bearing change the rotor critical speeds and also significantly affect the dynamic behavior of the rotor system. These effects are also presented graphically and discussed.

Nonlinear Oscillations of Hyperelastic Annular Membranes With Varying Density

2017 ◽  
Author(s):  
Renata M. Soares ◽  
Paulo B. Gonçalves
Keyword(s):  
Hypergeometric Functions ◽  
Nonlinear Oscillations ◽  
Outer Boundary ◽  
Equations Of Motion ◽  
Mode Shapes ◽  
Element Formulation ◽  
The Galerkin Method ◽  
Stretching Ratio ◽  
Annular Membrane ◽  
Membrane Density

This research presents the mathematical modeling for the nonlinear oscillations analysis of a pre-stretched hyperelastic annular membrane with varying density under finite deformations. The membrane material is assumed to be homogeneous, isotropic, and neo-Hookean and the variation of the membrane density in the radial direction is investigated. The membrane is first subjected to a uniform radial traction along its outer circumference and the stretched membrane is fixed along the outer boundary. Then the equations of motion of the pre-stretched membrane are derived. From the linearized equations of motion, the natural frequencies and mode shapes of the membrane are obtained analytically. The vibration modes are described by hypergeometric functions, which are used to approximate the nonlinear deformation field using the Galerkin method. The results are compared with the results evaluated for the same membrane using a nonlinear finite element formulation. The results show the influence of the stretching ratio and varying density on the linear and nonlinear oscillations of the membrane.

scholarly journals Solving nonlinear boundary value problems by the Galerkin method with sinc functions

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Sinc Approximation ◽  
Sinc Functions ◽  
The Galerkin Method

AbstractIn this paper, the sinc-Galerkin method is used for numerically solving a class of nonlinear differential equations with boundary conditions. The importance of this study is that sinc approximation of the nonlinear term is stated as a new theorem. The method introduced here is tested on some nonlinear problems and is shown to be a very efficient and powerful tool for obtaining approximate solutions of nonlinear ordinary differential equations.

The stability of spatially periodic flows

1981 ◽  
Vol 108 ◽  
pp. 461-474 ◽  
Author(s):  
D. N. Beaumont
Keyword(s):  
Velocity Profile ◽  
Phase Speed ◽  
Neutral Curve ◽  
Long Waves ◽  
Inviscid Fluid ◽  
Periodic Flows ◽  
A Value ◽  
Parallel Flows ◽  
Spatially Periodic ◽  
The Stability

The stability characteristics for spatially periodic parallel flows of an incompressible fluid (both inviscid and viscous) are studied. A general formula for the determination of the stability characteristics of periodic flows to long waves is obtained, and applied to approximate numerically the stability curves for the sinusoidal velocity profile. The neutral curve for the sinusoidal velocity profile is obtained analytically. The stability of two broken-line velocity profiles in an inviscid fluid is studied and the results are used to describe the overall pattern for the sinusoidal velocity profile in the case of long waves. In an inviscid fluid it is found that all periodic flows (other than the trivial flow in which the basic velocity is constant) are unstable to long waves with a value of the phase speed determined by simple integrals of the basic flow. In a viscous fluid it is found that the sinusoidal velocity profile is very unstable with the inviscid solution being a good approximation to the solution of the viscous problem when the value of the Reynolds number is greater than about 20.

Correlation between wave activities in different layers of the atmosphere

2020 ◽  
Author(s):  
Konstantin Ratovsky ◽  
Irina Medvedeva ◽  
Anna Yasyukevich ◽  
Boris Shpynev ◽  
Denis Khabituev
Keyword(s):  
Correlation Coefficients ◽  
Internal Gravity Wave ◽  
Basic Research ◽  
Temporal Variations ◽  
Wave Activity ◽  
Electron Content ◽  
Russian Science ◽  
Period Range ◽  
Total Electron ◽  
Mesopause Temperature

<p>We study the correlation between wave activities in different layers of the atmosphere. The variability of the measured characteristic in the range of internal gravity wave periods is used as a proxy of wave activity. In the case of ground-based measurements, we consider temporal variations with periods less than ~ 6 hours; while in the case of satellite measurements we take into account spatial variations with periods less than ~ 1000 km. The wave activity is calculated as the standard deviation of variations in the indicated period range with averaging over one day. The aim of the study is to detect a correlation between day-to-day variations of wave activity in different layers of the atmosphere. Correlation coefficients are calculated for various intervals from one month to one year. Correlation analysis reveals the potential relationship between wave phenomena in the stratosphere, mesosphere and ionosphere. The study uses the following characteristics. The ionospheric characteristics are the peak electron density from the Irkutsk ionosonde (52.3 N, 104.3 E) and the total electron content from the Irkutsk GPS receiver. The characteristic of the mesosphere is the mesopause temperature from spectrometric measurements of the OH emission (834.0 nm, band (6-2)) near Irkutsk (51.8 N, 103.1 E, Tory). The stratospheric characteristic is the vertical gas velocity at 1 hPa from the ERA-Interim reanalysis (apps.ecmwf.int/datasets/data/).</p><p>This study was supported by the Grant of the Russian Science Foundation (Project N 18-17-00042). The observational results were obtained using the equipment of Center for Common Use «Angara» http: //ckp-rf.ru/ckp/3056/ within budgetary funding of Basic Research program II.12.</p>

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