On the stability of steep gravity waves

1984 ◽  
Vol 396 (1811) ◽  
pp. 269-280 ◽  
Keyword(s):  
Phase Shift ◽  
Normal Mode ◽  
Gravity Waves ◽  
Phase Speed ◽  
Numerical Calculations ◽  
Wave Steepness ◽  
High Wave ◽  
Pure Phase ◽  
The Stability

Previous calculations of the normal mode perturbations of steep gravity waves have suggested that the lowest superharmonic mode n = 2 becomes unstable at around ak = 0.436, where 2 a is the crest-to-trough height of the unperturbed wave and k is the wavenumber. This would correspond to the wave steepness at which the phase speed c is a maximum (considered as a function of ak ). However, numerical calculations at such high wave steepnesses can become inaccurate. The present paper studies analytically the conditions for the existence of a normal mode at zero limiting frequency. It is proved that for superharmonic perturbations such conditions will occur only for a pure phase-shift (corresponding to n = 1) or when the speed c is stationary with respect to the wave steepness, that is when d c = 0. Hence the limiting form of the instability found by Tanaka ( J. phys. Soc. Japan 52, 3047-3055 (1983)) near the value ak = 0.429 must be a pure phase-shift.

The instabilities of gravity waves of finite amplitude in deep water I. Superharmonics

1978 ◽  
Vol 360 (1703) ◽  
pp. 471-488 ◽  
Keyword(s):  
Stream Function ◽  
Gravity Waves ◽  
Velocity Potential ◽  
Travelling Waves ◽  
Normal Modes ◽  
Phase Speed ◽  
Finite Amplitude ◽  
Fundamental Wave ◽  
Wave Steepness ◽  
Horizontal Scale

In this paper we embark on a calculation of all the normal-mode perturbations of nonlinear, irrotational gravity waves as a function of the wave steepness. The method is to use as coordinates the stream-function and velocity potential in the steady, unperturbed wave (seen in a reference frame moving with the phase speed) together with the time t. The dependent quantities are the cartesian displacements and the perturbed stream function at the free surface. To begin we investigate the ‘superharmonics’, i.e. those perturbations having the same horizontal scale as the fundamental wave, or less. When the steepness of the fundamental is small, the normal modes take the form of travelling waves superposed on the basic nonlinear wave. As the steepness increases the frequency of each perturbation tends generally to be diminished. At a steepness ak ≈ 0.436 it appears that the two lowest modes coalesce and an instability arises. There is evidence that this critical steepness corresponds precisely with the steepness at which the phase velocity is a maximum, considered as a function of ak. The calculations are facilitated by the discovery of some new identities between the coefficients in Stokes’s expansion for waves of finite amplitude.

Crest instabilities of gravity waves. Part 2. Matching and asymptotic analysis

1994 ◽  
Vol 259 ◽  
pp. 333-344 ◽  
Author(s):  
Michael S. Longuet-Higgins ◽  
R. P. Cleaver ◽  
M. J. H. Fox
Keyword(s):  
Growth Rate ◽  
Gravity Waves ◽  
Water Wave ◽  
Present Theory ◽  
Wave Crest ◽  
Wave Steepness ◽  
Deep Water Wave ◽  
The Stability ◽  
Rates Of Growth ◽  
Irrotational Wave

In a previous study (Longuet-Higgins & Cleaver 1994) we calculated the stability of the flow near the crest of a steep, irrotational wave, the ‘almost-highest’ wave, considered as an isolated wave crest. In the present paper we consider the modification of this inner flow when it is matched to the flow in the rest of the wave, and obtain the normal-mode perturbations of the modified inner flow. It is found that there is just one exponentially growing mode. Its rate of growth β is a decreasing function of the matching parameter ε and hence a decreasing function of the wave steepness ak. When compared numerically to the rates of growth of the lowest superharmonic instability in a deep-water wave as calculated by Tanaka (1983) it is found that the present theory provides a satisfactory asymptote to the previously calculated values of the growth rate. This suggests that the instability of the lowest superharmonic is essentially due to the flow near the crest of the wave.

Capillary–gravity waves of solitary type on deep water

1989 ◽  
Vol 200 ◽  
pp. 451-470 ◽  
Author(s):  
Michael S. Longuet-Higgins
Keyword(s):  
Free Surface ◽  
Lower Bound ◽  
Solitary Waves ◽  
Gravity Waves ◽  
Deep Water ◽  
Phase Speed ◽  
Numerical Calculations ◽  
Maximum Angle ◽  
Monotonically Decreasing Function

On physical grounds it was recently suggested that limiting capillary–gravity waves of solitary type may exist on the surface of deep water (Longuet-Higgins 1988). This paper describes accurate numerical calculations which support the conjecture. The limiting wave has a phase speed c = 0.9267 (gτ)¼. It is one of a family of solitary waves having speeds c [les ] 1.30 (gτ)¼. The maximum angle of inclination αmax of the free surface is a monotonically decreasing function of the speed c. Physical arguments suggest that αmax has a positive lower bound.

On resonant over-reflexion of internal gravity waves from a Helmholtz velocity profile

1979 ◽  
Vol 90 (1) ◽  
pp. 161-178 ◽  
Author(s):  
R. H. J. Grimshaw
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Weakly Nonlinear ◽  
Velocity Discontinuity ◽  
Interface Displacement ◽  
Boussinesq Fluid

A Helmholtz velocity profile with velocity discontinuity 2U is embedded in an infinite continuously stratified Boussinesq fluid with constant Brunt—Väisälä frequency N. Linear theory shows that this system can support resonant over-reflexion, i.e. the existence of neutral modes consisting of outgoing internal gravity waves, whenever the horizontal wavenumber is less than N/2½U. This paper examines the weakly nonlinear theory of these modes. An equation governing the evolution of the amplitude of the interface displacement is derived. The time scale for this evolution is α−2, where α is a measure of the magnitude of the interface displacement, which is excited by an incident wave of magnitude O(α3). It is shown that the mode which is symmetrical with respect to the interface (and has a horizontal phase speed equal to the mean of the basic velocity discontinuity) remains neutral, with a finite amplitude wave on the interface. However, the other modes, which are not symmetrical with respect to the interface, become unstable owing to the self-interaction of the primary mode with its second harmonic. The interface displacement develops a singularity in a finite time.

scholarly journals On strongly nonlinear gravity waves in a vertically sheared atmosphere

2020 ◽  
Vol 6 (1) ◽  
pp. 63-74
Author(s):  
Mark Schlutow ◽  
Georg S. Voelker
Keyword(s):  
Gravity Waves ◽  
Lower Layer ◽  
Vertical Shear ◽  
Horizontal Wind ◽  
Necessary Condition ◽  
Mean Flow ◽  
Individual Layer ◽  
Strongly Nonlinear ◽  
The Mean ◽  
The Stability

Abstract We investigate strongly nonlinear stationary gravity waves which experience refraction due to a thin vertical shear layer of horizontal background wind. The velocity amplitude of the waves is of the same order of magnitude as the background flow and hence the self-induced mean flow alters the modulation properties to leading order. In this theoretical study, we show that the stability of such a refracted wave depends on the classical modulation stability criterion for each individual layer, above and below the shearing. Additionally, the stability is conditioned by novel instability criteria providing bounds on the mean-flow horizontal wind and the amplitude of the wave. A necessary condition for instability is that the mean-flow horizontal wind in the upper layer is stronger than the wind in the lower layer.

scholarly journals Observation of Kelvin–Helmholtz instabilities and gravity waves in the summer mesopause above Andenes in Northern Norway

2018 ◽  
Vol 18 (9) ◽  
pp. 6721-6732 ◽  
Author(s):  
Gunter Stober ◽  
Svenja Sommer ◽  
Carsten Schult ◽  
Ralph Latteck ◽  
Jorge L. Chau
Keyword(s):  
Gravity Waves ◽  
Middle Atmosphere ◽  
Phase Speed ◽  
Velocity Measurements ◽  
Wind Variability ◽  
Short Period ◽  
Strong Shear ◽  
Summer Echoes ◽  
Period Wave ◽  
Horizontal Wavelength

Abstract. We present observations obtained with the Middle Atmosphere Alomar Radar System (MAARSY) to investigate short-period wave-like features using polar mesospheric summer echoes (PMSEs) as a tracer for the neutral dynamics. We conducted a multibeam experiment including 67 different beam directions during a 9-day campaign in June 2013. We identified two Kelvin–Helmholtz instability (KHI) events from the signal morphology of PMSE. The MAARSY observations are complemented by collocated meteor radar wind data to determine the mesoscale gravity wave activity and the vertical structure of the wind field above the PMSE. The KHIs occurred in a strong shear flow with Richardson numbers Ri < 0.25. In addition, we observed 15 wave-like events in our MAARSY multibeam observations applying a sophisticated decomposition of the radial velocity measurements using volume velocity processing. We retrieved the horizontal wavelength, intrinsic frequency, propagation direction, and phase speed from the horizontally resolved wind variability for 15 events. These events showed horizontal wavelengths between 20 and 40 km, vertical wavelengths between 5 and 10 km, and rather high intrinsic phase speeds between 45 and 85 m s−1 with intrinsic periods of 5–10 min.

Combined use of COSS and S.COSY for recording of pure-phase shift-scaled spectra

1988 ◽  
Vol 76 (2) ◽  
pp. 218-223 ◽  
Author(s):  
R.V Hosur ◽  
Anu Sheth ◽  
Ananya Majumdar
Keyword(s):  
Phase Shift ◽  
Combined Use ◽  
Pure Phase

Stability of parallel flow of a multi-component cylindrical plasma to electrostatic perturbations

1981 ◽  
Vol 26 (2) ◽  
pp. 369-383
Author(s):  
R. Lucas
Keyword(s):  
Normal Mode ◽  
Axial Velocity ◽  
Sufficient Conditions ◽  
Parallel Flow ◽  
Zeroth Order ◽  
Unperturbed State ◽  
Plasma Component ◽  
Cylindrical Plasma ◽  
The Stability ◽  
System Variables

Sufficient conditions for the stability of parallel flow of a warm N-component cylindrical plasma to electrostatic perturbations are obtained. In the unperturbed state the jth plasma component is assumed to have axial velocity Vj0(r), r being the radial co-ordinate, and the equilibrium quantities are permitted to be arbitrary functions of r consistent with the zeroth-order equations. The L2-norms of certain system variables are shown to be bounded uniformly in time. Circle theorems are obtained for the complex eigenfrequencies of any normal mode.

Calculation of Transient Flow in Circulating Fluidized Beds With Straight and Double Abrupt Outlet Configuration: Relation With the Inelasticity of the Particle-Particle Collisions

2001 ◽  
Author(s):  
Juray De Wilde ◽  
Jan Vierendeels ◽  
Geraldine J. Heynderickx ◽  
Erik Dick ◽  
Guy B. Marin
Keyword(s):  
Kinetic Theory ◽  
Gravity Waves ◽  
Granular Flow ◽  
Fluidized Beds ◽  
Transient Flow ◽  
Industrial Scale ◽  
Circulating Fluidized Beds ◽  
Restitution Coefficient ◽  
Particle Collisions ◽  
The Stability

Abstract Gas-solid flow in industrial scale Circulating Fluidized Beds (CFB’s) is calculated in 3D using the Eulerian-Eulerian approach and the Kinetic Theory of Granular Flow (KTGF). Two outlet configurations are used: a straight top outlet and a double abrupt T-outlet. The effect of the value of the restitution coefficient for particle-particle collisions on the stability of the flow is investigated. Oscillations appearing in CFB’s, are shown to be gravity waves.

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