scholarly journals On the Discrete Normal Modes of Quasigeostrophic Theory

2021 ◽  
Keyword(s):  
Rossby Wave ◽  
Degrees Of Freedom ◽  
Normal Modes ◽  
Wave Interaction ◽  
Linear Wave ◽  
Pontryagin Space ◽  
Wave Problem ◽  
Space Setting ◽  
Geostrophic Turbulence ◽  
Baroclinic Modes

Abstract The discrete baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple modes with no internal zeros, a finite number of modes with negative norms, and their vertical structures form a basis capable of representing any quasigeostrophic state with a differentiable series expansion. These properties are a consequence of the Pontryagin space setting of the Rossby wave problem in the presence of boundary buoyancy gradients (as opposed to the usual Hilbert space setting). We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. A natural application of these modes is the development of a weakly non-linear wave-interaction theory of geostrophic turbulence that takes topography into account.

Discrete Normal Mode Decompositions in Quasigeostrophic Theory

2021 ◽  
Author(s):  
Houssam Yassin ◽  
Stephen Griffies
Keyword(s):  
Normal Mode ◽  
Rossby Wave ◽  
Degrees Of Freedom ◽  
Wave Interaction ◽  
Linear Wave ◽  
Wave Problem ◽  
Dirac Delta ◽  
Complete Basis ◽  
Geostrophic Turbulence ◽  
Baroclinic Modes

<p>The baroclinic modes of quasigeostrophic theory are incomplete and the incompleteness manifests as a loss of information in the projection process. The incompleteness of the baroclinic modes is related to the presence of two previously unnoticed stationary step-wave solutions of the Rossby wave problem with flat boundaries. These step-waves are the limit of surface quasigeostrophic waves as boundary buoyancy gradients vanish. A complete normal mode basis for quasigeostrophic theory is obtained by considering the traditional Rossby wave problem with prescribed buoyancy gradients at the lower and upper boundaries. The presence of these boundary buoyancy gradients activates the previously inert boundary degrees of freedom. These Rossby waves have several novel properties such as the presence of multiple equivalent barotropic modes, a finite number of modes with negative norms, and their vertical structures form a basis capable of representing any quasigeostrophic state. Using this complete basis, we are able to obtain a series expansion to the potential vorticity of Bretherton (with Dirac delta contributions). We compare the convergence and differentiability properties of these complete modes with various other modes in the literature. We also examine the quasigeostrophic vertical velocity modes and derive a complete basis for such modes as well. In the process, we introduce the concept of the quasigeostrophic phase space which we define to be the space of all possible quasigeostrophic states. A natural application of these modes is the development of a weakly non-linear wave-interaction theory of geostrophic turbulence that takes prescribed boundary buoyancy gradients into account.</p>

Rossby wave interaction in a shear flow with critical levels

1996 ◽  
Vol 323 ◽  
pp. 317-338 ◽  
Author(s):  
Jacques Vanneste
Keyword(s):  
Shear Flow ◽  
Rossby Wave ◽  
Critical Level ◽  
Equations Of Motion ◽  
Normal Modes ◽  
Wave Interaction ◽  
Asymptotic Results ◽  
Linear Shear ◽  
Truncated Normal ◽  
Analytical Expressions

An approach is presented for studying Rossby wave interaction in a shear flow with both regular and singular modes (i.e. those possessing a critical level). The approach relies on a truncated normal mode expansion of the equations of motion. Such an expansion remains valid in the presence of singular modes, provided that these modes are not considered individually, but that complete packets are taken into account in the truncated system. Mathematically, this means that the interaction equations need to be integrated with respect to the phase velocity (or, equivalently, the critical level position) of the singular modes.The action of two regular modes on a packet of singular modes is treated in detail; in particular, asymptotic results are deduced for the long-term behaviour of the packet. The case of a linear shear is considered as an illustration: analytical expressions are derived for the normal modes and their pseudomomentum, and they are used to present explicit results for the evolution of the packet of singular modes.

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.

Dissipation Improvement of MUSCL Scheme for Computational Aeroacoustics

2001 ◽  
Vol 17 (1) ◽  
pp. 39-47
Author(s):  
San-Yin Lin ◽  
Sheng-Chang Shih ◽  
Jen-Jiun Hu
Keyword(s):  
Wave Interaction ◽  
Sine Wave ◽  
Numerical Dissipation ◽  
Finite Volume Scheme ◽  
Linear Wave ◽  
Riemann Solver ◽  
Two Dimensional ◽  
Order Accuracy ◽  
Upwind Schemes ◽  
Continuous Waves

ABSTRACTAn upwind finite-volume scheme is studied for solving the solutions of two dimensional Euler equations. It based on the MUSCL (Monotone Upstream Scheme for Conservation Laws) approach with the Roe approximate Riemann solver for the numerical flux evaluation. First, dissipation and dispersion relation, and group velocity of the scheme are derived to analyze the capability of the proposed scheme for capturing physical waves, such as acoustic, entropy, and vorticity waves. Then the scheme is greatly enhanced through a strategy on the numerical dissipation to effectively handle aeroacoustic computations. The numerical results indicate that the numerical dissipation strategy allows that the scheme simulates the continuous waves, such as sound and sine waves, at fourth-order accuracy and captures the discontinuous waves, such a shock wave, sharply as well as most of upwind schemes do. The tested problems include linear wave convection, propagation of a sine-wave packet, propagation of discontinuous and sine waves, shock and sine wave interaction, propagation of acoustic, vorticity, and density pulses in an uniform freestream, and two-dimensional traveling vortex in a low-speed freestream.

Peculiarities of linear wave interaction with nonlinear media interface

2018 ◽  
Vol 32 (30) ◽  
pp. 1850371 ◽  
Author(s):  
S. E. Savotchenko
Keyword(s):  
Guided Waves ◽  
Mathematical Formulation ◽  
Wave Interaction ◽  
Linear Wave ◽  
Stationary Waves ◽  
One Dimensional ◽  
Nonlinear Properties ◽  
Dimensional Boundary ◽  
Nonlinear Interface ◽  
Plane Defect

We analyze guided waves in the linear media separated nonlinear interface. The mathematical formulation of the model is a one-dimensional boundary value problem for the nonlinear Schrödinger equation. The Kerr type nonlinearity in the equation is taken into account only inside the waveguide. We show that the existence of nonlinear stationary waves of three types is possible in defined frequency ranges. We derive the frequency of obtained stationary states in explicit form and find the conditions of its existence. We show that it is possible to obtain the total wave transition through a plane defect. We determine the condition for realizing of such a resonance. We obtain the reflection and transition coefficients in the vicinity of the resonance. We establish that complete wave propagation with nonzero defect parameters can occur only when the nonlinear properties of the defect are taken into account.

Fully Nonlinear Potential/RANSE Simulation of Wave Interaction With Ships and Marine Structures

2008 ◽  
Author(s):  
Pierre Ferrant ◽  
Lionel Gentaz ◽  
Bertrand Alessandrini ◽  
Romain Luquet ◽  
Charles Monroy ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Stokes Equations ◽  
Wave Interaction ◽  
Irregular Waves ◽  
Navier Stokes ◽  
Viscous Effects ◽  
Navier Stokes Equations ◽  
Fully Nonlinear ◽  
Nonlinear Potential ◽  
6 Degrees Of Freedom

This paper documents recent advances of the SWENSE (Spectral Wave Explicit Navier-Stokes Equations) approach, a method for simulating fully nonlinear wave-body interactions including viscous effects. The methods efficiently combines a fully nonlinear potential flow description of undisturbed wave systems with a modified set of RANS with free surface equations accounting for the interaction with a ship or marine structure. Arbitrary incident wave systems may be described, including regular, irregular waves, multidirectional waves, focused wave events, etc. The model may be fixed or moving with arbitrary speed and 6 degrees of freedom motion. The extension of the SWENSE method to 6 DOF simulations in irregular waves as well as to manoeuvring simulations in waves are discussed in this paper. Different illlustative simulations are presented and discussed. Results of the present approach compare favorably with available reference results.

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.

Response Characteristics of Serially Connected Semisubmersibles

1999 ◽  
Vol 43 (04) ◽  
pp. 229-240
Author(s):  
H. R. Riggs ◽  
R. C. Ertekin
Keyword(s):  
Energy Loss ◽  
Natural Frequencies ◽  
Normal Modes ◽  
Induced Response ◽  
Linear Wave ◽  
Response Characteristics ◽  
Response Modes ◽  
Wave Induced ◽  
The Impact ◽  
Large Aircraft

One design for a mobile offshore base is to link serially as many as five large semisubmersibles to form a platform long enough to support large aircraft. This paper investigates the linear, wave-induced response characteristics of serially-connected semisubmersibles. A major motivation of this study is to understand more completely the forces required to link semisubmersible modules. The impact of connector strategy and damping on the response, especially the connector forces, is investigated, and the response "modes" which contribute to the connector forces are evaluated in detail. It is shown that the response characteristics can be impacted significantly by the connection strategy, and that connector damping can be a significant source of energy loss when compared to radiation damping. The wet natural frequencies and normal modes are also determined and used to explain the response characteristics of different connection strategies. Although the analyses are based on a specific semisubmersible design, the results provide insight on how other systems of connected semisubmersibles would likely behave.

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.

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