scholarly journals The Effect Of Randomness On The Stability Of Capillary Gravity Waves In The Presence Of Air Flowing Over Water

2015 ◽  
Vol 20 (4) ◽  
pp. 835-855
Author(s):  
D.P. Majumder ◽  
A.K. Dhar
Keyword(s):  
Growth Rate ◽  
Evolution Equation ◽  
Gravity Waves ◽  
Fourth Order ◽  
Order Term ◽  
Spectral Width ◽  
Higher Order ◽  
Order Contribution ◽  
Fourth Order Term ◽  
The Stability

Abstract A nonlinear spectral transport equation for the narrow band Gaussian random surface wave trains is derived from a fourth order nonlinear evolution equation, which is a good starting point for the study of nonlinear water waves. The effect of randomness on the stability of deep water capillary gravity waves in the presence of air flowing over water is investigated. The stability is then considered for an initial homogenous wave spectrum having a simple normal form to small oblique long wave length perturbations for a range of spectral widths. An expression for the growth rate of instability is obtained; in which a higher order contribution comes from the fourth order term in the evolution equation, which is responsible for wave induced mean flow. This higher order contribution produces a decrease in the growth rate. The growth rate of instability is found to decrease with the increase of spectral width and the instability disappears if the spectral width increases beyond a certain critical value, which is not influenced by the fourth order term in the evolution equation.

scholarly journals The effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline

1998 ◽  
Vol 40 (2) ◽  
pp. 190-206
Author(s):  
Sudebi Bhattacharyya ◽  
K. P. Das
Keyword(s):  
Growth Rate ◽  
Evolution Equation ◽  
Gravity Waves ◽  
Fourth Order ◽  
Deep Water ◽  
Water Surface ◽  
Spectral Width ◽  
Surface Gravity ◽  
Surface Gravity Waves ◽  
The Stability

AbstractThe effect of randomness on the stability of deep water surface gravity waves in the presence of a thin thermocline is studied. A previously derived fourth order nonlinear evolution equation is used to find a spectral transport equation for a narrow band of surface gravity wave trains. This equation is used to study the stability of an initially homogeneous Lorentz shape of spectrum to small long wave-length perturbations for a range of spectral widths. The growth rate of the instability is found to decrease with the increase of spectral widths. It is found that the fourth order term in the evolution equation produces a decrease in the growth rate of the instability. There is stability if the spectral width exceeds a certain critical value. For a vanishing bandwidth the deterministic growth rate of the instability is recovered. Graphs have been plotted showing the variations of the growth rate of the instability against the wavenumber of the perturbation for some different values of spectral width, thermocline depth, angle of perturbation and wave steepness.

Higher‐order moveout spectra

Geophysics ◽  
1979 ◽  
Vol 44 (7) ◽  
pp. 1193-1207 ◽  
Author(s):  
Bruce T. May ◽  
Donald K. Straley
Keyword(s):  
Fourth Order ◽  
Seismic Reflection ◽  
Order Term ◽  
Higher Order ◽  
Second Order ◽  
Equation Modeling ◽  
Fourth Order Term ◽  
Higher Order Terms ◽  
Velocity Spectra

Higher‐order terms in the generalized seismic reflection moveout equation are usually neglected, resulting in the familiar second‐order, or hyperbolic, moveout equation. Modeling studies show that the higher‐order terms are often significant, and their neglect produces sizable traveltime residuals after correction for moveout in such cases as kinked‐ray models. Taner and Koehler (1969) introduced velocity spectra for estimating stacking velocity defined on the basis of second‐order moveout. Through the use of orthogonal polynomials, an iterative procedure is defined that permits computation of fourth‐order moveout spectra while simultaneously upgrading the previously computed, second‐order spectra. Emphasis is placed on the fourth‐order term, but the procedure is general and can be expanded to higher orders. When used with synthetic and field recorded common‐midpoint (CMP) trace data, this technique produces significant improvements in moveout determination affecting three areas: (1) resolution and interpretability of moveout spectra, (2) quality of CMP stacked sections, and (3) computation of velocity and depth for inverse modeling.

Non-normal stability analysis of a shear current under surface gravity waves

2008 ◽  
Vol 609 ◽  
pp. 49-58
Author(s):  
D. AMBROSI ◽  
M. ONORATO
Keyword(s):  
Growth Rate ◽  
Modal Analysis ◽  
Gravity Waves ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Surface Gravity ◽  
Horizontal Shear ◽  
Surface Gravity Waves ◽  
Shear Current ◽  
The Stability

The stability of a horizontal shear current under surface gravity waves is investigated on the basis of the Rayleigh equation. As the differential operator is non-normal, a standard modal analysis is not effective in capturing the transient growth of a perturbation. The representation of the stream function by a suitable basis of bi-orthogonal eigenfunctions allows one to determine the maximum growth rate of a perturbation. It turns out that, in the considered range of parameters, such a growth rate can be two orders of magnitude larger than the maximum eigenvalue obtained by standard modal analysis.

scholarly journals Scale-invariant scalar spectrum from the nonminimal derivative coupling with fourth-order term

2015 ◽  
Vol 24 (14) ◽  
pp. 1550095 ◽  
Author(s):  
Yun Soo Myung ◽  
Taeyoon Moon
Keyword(s):  
Fourth Order ◽  
Order Term ◽  
Order Derivative ◽  
Scalar Perturbation ◽  
De Sitter ◽  
Scale Invariant ◽  
Derivative Coupling ◽  
Sitter Spacetime ◽  
Fourth Order Term ◽  
Derivative Term

In this paper, an exactly scale-invariant spectrum of scalar perturbation generated during de Sitter spacetime is found from the gravity model of the nonminimal derivative coupling with fourth-order term. The nonminimal derivative coupling term generates a healthy (ghost-free) fourth-order derivative term, while the fourth-order term provides an unhealthy (ghost) fourth-order derivative term. The Harrison–Zel’dovich spectrum obtained from Fourier transforming the fourth-order propagator in de Sitter space is recovered by computing the power spectrum in its momentum space directly. It shows that this model provides a truly scale-invariant spectrum, in addition to the Lee–Wick scalar theory.

scholarly journals GAUGED LIFSHITZ MODEL WITH CHERN–SIMONS TERM

2013 ◽  
Vol 28 (09) ◽  
pp. 1350025 ◽  
Author(s):  
GUSTAVO S. LOZANO ◽  
FIDEL A. SCHAPOSNIK ◽  
GIANNI TALLARITA
Keyword(s):  
Fourth Order ◽  
Order Term ◽  
Equations Of Motion ◽  
Nontrivial Solutions ◽  
Oscillatory Solutions ◽  
Modulated Phases ◽  
Fourth Order Term ◽  
Spatial Derivatives ◽  
Chern Simons ◽  
Derivatives Of

We present a gauged Lifshitz Lagrangian including second- and fourth-order spatial derivatives of the scalar field and a Chern–Simons term, and study nontrivial solutions of the classical equations of motion. While the coefficient β of the fourth-order term should be positive in order to guarantee positivity of the energy, the coefficient α of the quadratic one need not be. We investigate the parameter domains and find significant differences in the field behaviors. Apart from the usual vortex field behavior of the ordinary relativistic Chern–Simons–Higgs model, we find in certain parameter domains oscillatory solutions reminiscent of the modulated phases of Lifshitz systems.

THE NAMBU-JONA-LASINIO SOLITON WITH GENERALIZED SCALAR INTERACTIONS

1993 ◽  
Vol 08 (01) ◽  
pp. 79-88 ◽  
Author(s):  
C. WEISS ◽  
R. ALKOFER ◽  
H. WEIGEL
Keyword(s):  
Fourth Order ◽  
Scalar Meson ◽  
Order Term ◽  
Soliton Solutions ◽  
Meson Field ◽  
Scale Invariant ◽  
Scalar Glueball ◽  
Fourth Order Term ◽  
Scalar Meson Field

Soliton solutions are studied as a generalization of the bosonized Nambu-Jona-Lasinio model with a fourth order term in the scalar meson field. Such an interaction arises in the context of a scale-invariant modification of the Nambu-Jona-Lasinio action, in which the scalar meson field is coupled to a scalar glueball field. It is shown that a fourth order term in the scalar meson field is crucial for the existence of stable solitons. We investigate the dependence of soliton properties on the scalar-glueball coupling.

scholarly journals On pion mass and decay constant from theory

2021 ◽  
Author(s):  
Ole Lynnerup Trinhammer ◽  
Henrik G. Bohr
Keyword(s):  
Fourth Order ◽  
Decay Constant ◽  
Order Term ◽  
Pion Mass ◽  
Higgs Field ◽  
Energy Scale ◽  
Configuration Spaces ◽  
Neutron Decay ◽  
Coordinate Space ◽  
Fourth Order Term

Abstract We calculate the pion mass from Goldstone modes in the Higgs mechanism related to the neutron decay. The Goldstone pion modes acquire mass by a vacuum misalignment of the Higgs field. The size of the misalignment is controlled by the ratio between the electronic and the nucleonic energy scales. The nucleonic energy scale is involved in the neutron to proton transformation and the electronic scale is involved in the related creation of the electronic state in the course of the electroweak neutron decay. The respective scales influence the mapping of the intrinsic configuration spaces used in our description. The configuration spaces are the Lie groups U(3) for the nucleonic sector and U(2) for the electronic sector. These spaces are both compact and lead to periodic potentials in the Hamiltonians in coordinate space. The periodicity and strengths of these potentials control the vacuum misalignment and leads to a pion mass of 135.2(1.5) MeV with an uncertainty mainly from the fine structure coupling at pionic energies. The pion decay constant 92 MeV results from comparing the fourth order self-coupling in an effective pion field theory with the corresponding fourth order term in the Higgs potential. We suggest analogies with the Goldberger-Treiman relation.

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