Singular Vectors and Time-Dependent Normal Modes of a Baroclinic Wave-Mean Oscillation

2008 ◽  
Vol 65 (3) ◽  
pp. 875-894 ◽  
Author(s):  
Christopher L. Wolfe ◽  
Roger M. Samelson
Keyword(s):  
Degrees Of Freedom ◽  
Channel Model ◽  
Normal Modes ◽  
Time Dependent ◽  
Inner Product ◽  
Time Interval ◽  
Baroclinic Wave ◽  
Singular Vectors ◽  
Mean Oscillation ◽  
Local Attractor

Abstract Linear disturbance growth is studied in a quasigeostrophic baroclinic channel model with several thousand degrees of freedom. Disturbances to an unstable, nonlinear wave-mean oscillation are analyzed, allowing the comparison of singular vectors and time-dependent normal modes (Floquet vectors). Singular vectors characterize the transient growth of linear disturbances in a specified inner product over a specified time interval and, as such, they complement and are related to Lyapunov vectors, which characterize the asymptotic growth of linear disturbances. The relationship between singular vectors and Floquet vectors (the analog of Lyapunov vectors for time-periodic systems) is analyzed in the context of a nonlinear baroclinic wave-mean oscillation. It is found that the singular vectors divide into two dynamical classes that are related to those of the Floquet vectors. Singular vectors in the “wave dynamical” class are found to asymptotically approach constant linear combinations of Floquet vectors. The most rapidly decaying singular vectors project strongly onto the most rapidly decaying Floquet vectors. In contrast, the leading singular vectors project strongly onto the leading adjoint Floquet vectors. Examination of trajectories that are near the basic cycle show that the leading Floquet vectors are geometrically tangent to the local attractor while the leading initial singular vectors point off the local attractor. A method for recovering the leading Floquet vectors from a small number of singular vectors is additionally demonstrated.

Normal-Mode Analysis of a Baroclinic Wave-Mean Oscillation

2006 ◽  
Vol 63 (11) ◽  
pp. 2795-2812 ◽  
Author(s):  
Christopher L. Wolfe ◽  
Roger M. Samelson
Keyword(s):  
Normal Mode Analysis ◽  
Normal Modes ◽  
Generalized Solution ◽  
Horizontal Resolution ◽  
Basic Flow ◽  
Mode Analysis ◽  
Baroclinic Wave ◽  
Mean Oscillation ◽  
The Stability ◽  
Time Periodic

Abstract The stability of a time-periodic baroclinic wave-mean oscillation in a high-dimensional two-layer quasigeostrophic spectral model is examined by computing a full set of time-dependent normal modes (Floquet vectors) for the oscillation. The model has 72 × 62 horizontal resolution and there are 8928 Floquet vectors in the complete set. The Floquet vectors fall into two classes that have direct physical interpretations: wave-dynamical (WD) modes and damped-advective (DA) modes. The WD modes (which include two neutral modes related to continuous symmetries of the underlying system) have large scales and can efficiently exchange energy and vorticity with the basic flow; thus, the dynamics of the WD modes reflects the dynamics of the wave-mean oscillation. These modes are analogous to the normal modes of steady parallel flow. On the other hand, the DA modes have fine scales and dynamics that reduce, to first order, to damped advection of the potential vorticity by the basic flow. While individual WD modes have immediate physical interpretations as discrete normal modes, the DA modes are best viewed, in sum, as a generalized solution to the damped advection problem. The asymptotic stability of the time-periodic basic flow is determined by a small number of discrete WD modes and, thus, the number of independent initial disturbances, which may destabilize the basic flow, is likewise small. Comparison of the Floquet exponent spectrum of the wave-mean oscillation to the Lyapunov exponent spectrum of a nearby aperiodic trajectory suggests that this result will still be obtained when the restriction to time periodicity is relaxed.

scholarly journals Evolution to Mirror-Symmetry in Rotating Systems

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1189
Author(s):  
Ferdinand Verhulst
Keyword(s):  
Mirror Symmetry ◽  
Initial Phase ◽  
Degrees Of Freedom ◽  
Galactic Plane ◽  
Normal Modes ◽  
Time Dependent ◽  
Model Formulation ◽  
Remarkable Feature ◽  
Rotating Systems ◽  
Slow Evolution

A natural example of evolution can be described by a time-dependent two degrees-of-freedom Hamiltonian. We choose the case where initially the Hamiltonian derives from a general cubic potential, the linearised system has frequencies 1 and ω>0. The time-dependence produces slow evolution to discrete (mirror) symmetry in one of the degrees-of-freedom. This changes the dynamics drastically depending on the frequency ratio ω and the timescale of evolution. We analyse the cases ω=1,2,3 where the ratio’s 1,2 turn out to be the most interesting. In an initial phase we find 2 adiabatic invariants with changes near the end of evolution. A remarkable feature is the vanishing and emergence of normal modes, stability changes and strong changes of the velocity distribution in phase-space. The problem is inspired by the dynamics of axisymmetric, rotating galaxies that evolve slowly to mirror symmetry with respect to the galactic plane, the model formulation is quite general.

Triatomic molecules

2018 ◽  
Author(s):  
John H. D. Eland ◽  
Raimund Feifel
Keyword(s):  
Degrees Of Freedom ◽  
Normal Modes ◽  
Electronic States ◽  
Triatomic Molecules ◽  
Doubly Charged Ions ◽  
Spectral Bands ◽  
Valence Electrons ◽  
Doubly Charged ◽  
Charged Ions ◽  
Double Ionisation

Double ionisation of the triatomic molecules presented in this chapter shows an added degree of complexity. Besides potentially having many more electrons, they have three vibrational degrees of freedom (three normal modes) instead of the single one in a diatomic molecule. For asymmetric and bent triatomic molecules multiple modes can be excited, so the spectral bands may be congested in all forms of electronic spectra, including double ionisation. Double photoionisation spectra of H2O, H2S, HCN, CO2, N2O, OCS, CS2, BrCN, ICN, HgCl2, NO2, and SO2 are presented with analysis to identify the electronic states of the doubly charged ions. The order of the molecules in this chapter is set first by the number of valence electrons, then by the molecular weight.

Introduction

2018 ◽  
Author(s):  
Niels Engholm Henriksen ◽  
Flemming Yssing Hansen
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Degrees Of Freedom ◽  
State Level ◽  
Boltzmann Distribution ◽  
Theoretical Description ◽  
Time Dependent ◽  
Nuclear Dynamics ◽  
Molecular Reaction ◽  
Oppenheimer Approximation

This introductory chapter considers first the relation between molecular reaction dynamics and the major branches of physical chemistry. The concept of elementary chemical reactions at the quantized state-to-state level is discussed. The theoretical description of these reactions based on the time-dependent Schrödinger equation and the Born–Oppenheimer approximation is introduced and the resulting time-dependent Schrödinger equation describing the nuclear dynamics is discussed. The chapter concludes with a brief discussion of matter at thermal equilibrium, focusing at the Boltzmann distribution. Thus, the Boltzmann distribution for vibrational, rotational, and translational degrees of freedom is discussed and illustrated.

The Natural Frequencies of Free and Constrained Non-Uniform Beams

1960 ◽  
Vol 64 (599) ◽  
pp. 697-699 ◽  
Author(s):  
R. P. N. Jones ◽  
S. Mahalingam
Keyword(s):  
Degrees Of Freedom ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Normal Modes ◽  
Iteration Process ◽  
Conservative System ◽  
Basic System ◽  
Orthogonal Functions ◽  
Added Masses ◽  
The Given

The Rayleigh-Ritz method is well known as an approximate method of determining the natural frequencies of a conservative system, using a constrained deflection form. On the other hand, if a general deflection form (i.e. an unconstrained form) is used, the method provides a theoretically exact solution. An unconstrained form may be obtained by expressing the deflection as an expansion in terms of a suitable set of orthogonal functions, and in selecting such a set, it is convenient to use the known normal modes of a suitably chosen “ basic system.” The given system, whose vibration properties are to be determined, can then be regarded as a “ modified system,” which is derived from the basic system by a variation of mass and elasticity. A similar procedure has been applied to systems with a finite number of degrees of freedom. In the present note the method is applied to simple non-uniform beams, and to beams with added masses and constraints. A concise general solution is obtained, and an iteration process of obtaining a numerical solution is described.

scholarly journals THE DYNAMIC RESPONSE OF OFFSHORE STRUCTURES TO TIME-DEPENDENT FORCES

1964 ◽  
Vol 1 (9) ◽  
pp. 29
Author(s):  
William S. Gaither ◽  
David P. Billington
Keyword(s):  
Degrees Of Freedom ◽  
Wind Waves ◽  
Offshore Structures ◽  
Time Dependent ◽  
Wave Forces ◽  
Ocean Bottom ◽  
Single Wave ◽  
Single Degree Of Freedom ◽  
The Many ◽  
Floating Objects

This paper is addressed to the problem of structural behavior in an offshore environment, and the application of a more rigorous analysis for time-dependent forces than is currently used. Design of pile supported structures subjected to wave forces has, in the past, been treated in two parts; (1) a static analysis based on the loading of a single wave, and (2) a dynamic analysis which sought to determine the resonant frequency by assuming that the structure could be approximated as a single-degree-of-freedom system. (Ref. 4 and 6) The behavior of these structures would be better understood if the dynamic nature of the loading and the many degrees of freedom of the system were included. A structure which is built in the open ocean is subjected to periodic forces due to wind, waves, floating objects, and due occasionally to machinery mounted on the structure. To resist motion, the structure relies on the stiffness of the elements from which it is built and the restraints of the ocean bottom into which the supporting legs are driven.

scholarly journals ANALYSIS OF THE TIME‐DEPENDENT BEHAVIOUR OF COMPOSITE CROSS‐SECTIONS BY LAPLACE‐TRANSFORM

2005 ◽  
Vol 11 (3) ◽  
pp. 203-209
Author(s):  
Erich Raue ◽  
Thorsten Heidolf
Keyword(s):  
Laplace Transform ◽  
Cross Sections ◽  
Composite Structures ◽  
Time Dependent ◽  
Time Interval ◽  
Final State ◽  
Long Time ◽  
Time Dependent Behaviour ◽  
Linear Differential ◽  
Dependent Behaviour

Composite structures consisting of precast and cast in‐situ concrete elements are increasingly common. These combinations demand a mechanical model which takes into account the time‐dependent behaviour and analysis of the different ages of the connected concrete components. The effect of creep and shrinkage of the different concrete components can be of relevance for the state of serviceability, as well as for the final state. The long‐time behaviour of concrete can be described by the rate‐of‐creep method, combined with a discretisation of time. The internal forces are described for each time interval using a system of linear differential equations, which can be solved by Laplace‐transform.

The Double Seesaw Oscillator in a Windfield: Theory and Experiments

1997 ◽  
Author(s):  
A. H. P. van der Burgh ◽  
T. I. Haaker ◽  
B. W. van Oudheusden
Keyword(s):  
Equilibrium State ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Typical Result ◽  
Resonant Case ◽  
Accurate Model ◽  
Codimension Two ◽  
Model Equations ◽  
Theory And Experiments ◽  
Codimension Two Bifurcation

Abstract In this paper the dynamics of an oscillator with two degrees-of-freedom of a double seesaw type in a windtunnel is studied. Model equations for this aeroelastic oscillator are derived and an analysis of these equations is given for the non-resonant case. A typical result is a (local codimension two) bifurcation which describes the transfer from the unstable equilibrium state to one of the two normal modes of the oscillator. Some experimental results are presented from which on may conclude that more accurate model equations should be developed.

Model Reduction of a Nonlinear Rotating Beam Through Nonlinear Normal Modes

1999 ◽  
Author(s):  
E. Pesheck ◽  
C. Pierre ◽  
S. W. Shaw
Keyword(s):  
Differential Equations ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Nonlinear Partial Differential Equations ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Single Equation ◽  
Rotating Beam ◽  
Model Size ◽  
Nonlinear Normal

Abstract Equations of motion are developed for a rotating beam which is constrained to deform in the transverse (flapping) and axial directions. This process results in two coupled nonlinear partial differential equations which govern the attendant dynamics. These equations may be discretized through utilization of the classical normal modes of the nonrotating system in both the transverse and extensional directions. The resultant system may then be diagonalized to linear order and truncated to N nonlinear ordinary differential equations. Several methods are used to determine the model size necessary to ensure accuracy. Once the model size (N degrees of freedom) has been determined, nonlinear normal mode (NNM) theory is applied to reduce the system to a single equation, or a small set of equations, which accurately represent the dynamics of a mode, or set of modes, of interest. Results are presented which detail the convergence of the discretized model and compare its dynamics with those of the NNM-reduced model, as well as other reduced models. The results indicate a considerable improvement over other common reduction techniques, enabling the capture of many salient response features with the simulation of very few degrees of freedom.

scholarly journals Study of a Nonlinear Membrane Absorber Applied to 3D Acoustic Cavity for Low Frequency Broadband Noise Control

Materials ◽  
2019 ◽  
Vol 12 (7) ◽  
pp. 1138 ◽  
Author(s):  
Jianwang Shao ◽  
Tao Zeng ◽  
Xian Wu
Keyword(s):  
Degrees Of Freedom ◽  
Noise Control ◽  
Normal Modes ◽  
Low Frequency ◽  
Harmonic Balance Method ◽  
Nonlinear Normal Modes ◽  
Broadband Noise ◽  
Frequency Noise ◽  
Acoustic Cavity ◽  
Nonlinear Membrane

As a new approach to passive noise control in low frequency domain, the targeted energy transfer (TET) technique has been applied to the 3D fields of acoustics. The nonlinear membrane absorber based on the TET can reduce the low frequency noise inside the 3D acoustic cavity. The TET phenomenon inside the 3D acoustic cavity has firstly investigated by a two degrees-of-freedom (DOF) system, which is comprised by an acoustic mode and a nonlinear membrane without the pre-stress. In order to control the low frequency broadband noise inside 3D acoustic cavity and consider the influence of the pre-stress for the TET, a general model of the system with several acoustic modes of 3D acoustic cavity and one nonlinear membrane is built and studied in this paper. By using the harmonic balance method and the numerical method, the nonlinear normal modes and the forced responses are analyzed. Meanwhile, the influence of the pre-stress of the nonlinear membrane for the TET is investigated. The desired working zones of the nonlinear membrane absorber for the broadband noise are investigated. It can be helpful to design the nonlinear membrane according the dimension of 3D acoustic cavity to control the low frequency broadband noise.

Sign in / Sign up

Export Citation Format

Share Document