Compatibility Conditions, Complex Frequency, and Complex Vertical Wave Number for Models of Gravity Waves in the Thermosphere

2020 ◽  
Vol 125 (7) ◽  
Author(s):  
Dave Broutman ◽  
Harold Knight ◽  
Stephen D. Eckermann
Keyword(s):  
Gravity Waves ◽  
Wave Number ◽  
Complex Frequency ◽  
Compatibility Conditions ◽  
Models Of Gravity

Wave Motion of a Compressible Viscous Fluid Contained in a Cylindrical Shell

1993 ◽  
Vol 115 (3) ◽  
pp. 302-312 ◽  
Author(s):  
J. H. Terhune ◽  
K. Karim-Panahi
Keyword(s):  
Velocity Field ◽  
Viscous Fluid ◽  
Imaginary Part ◽  
Fluid Velocity ◽  
Wave Number ◽  
Mode Shapes ◽  
Infinite Length ◽  
Complex Frequency ◽  
Compressible Viscous Fluid ◽  
Fluid Velocity Field

The free vibration of cylindrical shells filled with a compressible viscous fluid has been studied by numerous workers using the linearized Navier-Stokes equations, the fluid continuity equation, and Flu¨gge ’s equations of motion for thin shells. It happens that solutions can be obtained for which the interface conditions at the shell surface are satisfied. Formally, a characteristic equation for the system eigenvalues can be written down, and solutions are usually obtained numerically providing some insight into the physical mechanisms. In this paper, we modify the usual approach to this problem, use a more rigorous mathematical solution and limit the discussion to a single thin shell of infinite length and finite radius, totally filled with a viscous, compressible fluid. It is shown that separable solutions are obtained only in a particular gage, defined by the divergence of the fluid velocity vector potential, and the solutions are unique to that gage. The complex frequency dependence for the transverse component of the fluid velocity field is shown to be a result of surface interaction between the compressional and vortex motions in the fluid and that this motion is confined to the boundary layer near the surface. Numerical results are obtained for the first few wave modes of a large shell, which illustrate the general approach to the solution. The axial wave number is complex for wave propagation, the imaginary part being the spatial attenuation coefficient. The frequency is also complex, the imaginary part of which is the temporal damping coefficient. The wave phase velocity is related to the real part of the axial wave number and turns out to be independent of frequency, with numerical value lying between the sonic velocities in the fluid and the shell. The frequency dependencies of these parameters and fluid velocity field mode shapes are computed for a typical case and displayed in non-dimensional graphs.

scholarly journals Gravity wave propagation in the realistic atmosphere based on a three-dimensional transfer function model

2007 ◽  
Vol 25 (9) ◽  
pp. 1979-1986 ◽  
Author(s):  
L. Sun ◽  
W. Wan ◽  
F. Ding ◽  
T. Mao
Keyword(s):  
Transfer Function ◽  
Gravity Wave ◽  
Gravity Waves ◽  
Minimum Energy ◽  
Three Dimensional ◽  
Wave Number ◽  
Transfer Function Model ◽  
Wave Band ◽  
Function Model ◽  
Ionospheric Height

Abstract. In order to study the filter effect of the background winds on the propagation of gravity waves, a three-dimensional transfer function model is developed on the basis of the complex dispersion relation of internal gravity waves in a stratified dissipative atmosphere with background winds. Our model has successfully represented the main results of the ray tracing method, e.g. the trend of the gravity waves to travel in the anti-windward direction. Furthermore, some interesting characteristics are manifest as follows: (1) The method provides the distribution characteristic of whole wave fields which propagate in the way of the distorted concentric circles at the same altitude under the control of the winds. (2) Through analyzing the frequency and wave number response curve of the transfer function, we find that the gravity waves in a wave band of about 15–30 min periods and of about 200–400 km horizontal wave lengths are most likely to propagate to the 300-km ionospheric height. Furthermore, there is an obvious frequency deviation for gravity waves propagating with winds in the frequency domain. The maximum power of the transfer function with background winds is smaller than that without background winds. (3) The atmospheric winds may act as a directional filter that will permit gravity wave packets propagating against the winds to reach the ionospheric height with minimum energy loss.

In-situ microseism noise generation measured from distributed acoustic sensing on seafloor optical cable

2020 ◽  
Author(s):  
Diane Rivet ◽  
Gauthier Guérin ◽  
Daniel Mata ◽  
Itzhak Lior ◽  
Anthony Sladen ◽  
...  
Keyword(s):  
Solid Earth ◽  
Gravity Waves ◽  
Ambient Noise ◽  
Wave Number ◽  
Apparent Velocity ◽  
Noise Generation ◽  
Linear Interaction ◽  
Frequency Wave ◽  
Acoustic Sensing ◽  
Scholte Waves

<p>Measuring seismic and acoustic signals on seafloor telecom cables has proven recently its very high potential for earthquake monitoring but also for beter understanding the interaction between the oceans and the solid earth. A consequence of these interactions is the generation of the primary and secondary microseismic noise on coastal regions and in the deep ocean respectively. These seismic noises that propagate across continents are central to a large fraction of todays' seismic imagery and monitoring campaigns. Compared to previous studies and instrumentation setups, acoustic sensing over oceanic telecom cables offer the unique ability to measure in a very dense manner waves that are generated on the seafloor. We analyse a week long record of ambient noise measurements on the 41.5 km-long seafloor telecom cable offshore Toulon, south of France. At shallow depth, close to the coast, we measure the pressure changes caused by the oceanic gravity waves. The bottom pressure is then compared to an oceanographic buoy located a few kilometers away from the cable. The amplitude and frequency of the pressure are modulated by the gravity waves height and dominant periods. This observation opens the way for a distributed measurement of the oceanic waves characteristics over several kilometers. At depth larger than a 1km, we observe Scholte waves at the ocean-solid earth interface produced by the non-linear interaction of gravity waves. These waves have the double frequency of the gravity waves seen at the coast. We find that the amplitude and frequency change over time, as do the gravity waves observed near the coast. The frequency-wave number decomposition of the signal reveals that the apparent velocity of the Scholte waves does not depend of the azimuth of the fiber. These observations confirm that these deep Scholte waves are secondary microseismic noise, generated locally from the interaction of landward gravity waves with oceanward gravity wave reflected on the coast. Spatially distributed monitoring of the ambient noise wave field at the ocean-solid earth interface provides a better understanding of the noise generation and therefore will allow a better modeling of the ambient noise in the future.</p>

scholarly journals Analysis of vertical wave number spectrum of atmospheric gravity waves in the stratosphere using COSMIC GPS radio occultation data

2011 ◽  
Vol 4 (8) ◽  
pp. 1627-1636 ◽  
Author(s):  
T. Tsuda ◽  
X. Lin ◽  
H. Hayashi ◽  
Keyword(s):  
Gravity Waves ◽  
Radio Occultation ◽  
Temperature Fluctuations ◽  
Ground Level ◽  
Wave Number ◽  
Radiosonde Data ◽  
Small Scale ◽  
Vertical Resolution ◽  
Full Spectrum ◽  
Gps Radio Occultation

Abstract. GPS radio occultation (RO) is characterized by high accuracy and excellent height resolution, which has great advantages in analyzing atmospheric structures including small-scale vertical fluctuations. The vertical resolution of the geometrical optics (GO) method in the stratosphere is about 1.5 km due to Fresnel radius limitations, but full spectrum inversion (FSI) can provide superior resolutions. We applied FSI to COSMIC GPS-RO profiles from ground level up to 30 km altitude, although basic retrieval at UCAR/CDAAC sets the sewing height from GO to FSI below the tropopause. We validated FSI temperature profiles with routine high-resolution radiosonde data in Malaysia and North America collected within 400 km and about 30 min of the GPS RO events. The average discrepancy at 10–30 km altitude was less than 0.5 K, and the bias was equivalent with the GO results. Using the FSI results, we analyzed the vertical wave number spectrum of normalized temperature fluctuations in the stratosphere at 20–30 km altitude, which exhibits good consistency with the model spectra of saturated gravity waves. We investigated the white noise floor that tends to appear at high wave numbers, and the substantial vertical resolution of the FSI method was estimated as about 100–200 m in the lower stratosphere. We also examined a criterion for the upper limit of the FSI profiles, beyond which bending angle perturbations due to system noises, etc., could exceed atmospheric excess phase fluctuations. We found that the FSI profiles can be used up to about 28 km in studies of temperature fluctuations with vertical wave lengths as short as 0.5 km.

scholarly journals Analysis of vertical wave number spectrum of atmospheric gravity waves in the stratosphere using COSMIC GPS radio occultation data

2011 ◽  
Vol 4 (2) ◽  
pp. 2071-2097
Author(s):  
T. Tsuda ◽  
X. Lin ◽  
H. Hayashi ◽  
Keyword(s):  
Gravity Waves ◽  
Radio Occultation ◽  
Temperature Fluctuations ◽  
Ground Level ◽  
Wave Number ◽  
Radiosonde Data ◽  
Small Scale ◽  
Vertical Resolution ◽  
Full Spectrum ◽  
Gps Radio Occultation

Abstract. GPS radio occultation (RO) is characterized by high accuracy and excellent height resolution, which has great advantages in analyzing atmospheric structures including small-scale vertical fluctuations. The vertical resolution of the geometrical optics (GO) method in the stratosphere is about 1.5 km due to Fresnel radius limitations, but full spectrum inversion (FSI) can provide superior resolutions. We applied FSI to COSMIC GPS-RO profiles from ground level up to 30 km altitude, although basic retrieval at UCAR/CDAAC sets the sewing height from GO to FSI below the tropopause. We validated FSI temperature profiles with routine high-resolution radiosonde data in Malaysia and North America collected within 400 km and about 30 min of the GPS RO events. The average discrepancy at 10–30 km altitude was less than 0.5 K, and the bias was equivalent with the GO results. Using the FSI results, we analyzed the vertical wave number spectrum of normalized temperature fluctuations in the stratosphere at 20–30 km altitude, which exhibits good consistency with the model spectra of saturated gravity waves. We investigated the white noise floor that tends to appear at high wave numbers, and the substantial vertical resolution of the FSI method was estimated as about 100–200 m in the lower stratosphere. We also examined a criterion for the upper limit of the FSI profiles, beyond which bending angle perturbations due to system noises, etc, could exceed atmospheric excess phase fluctuations. We found that the FSI profiles can be used up to about 28 km in studies of temperature fluctuations with vertical wave lengths as short as 0.5 km.

Waves a heavy, viscous, incompressible, electrically conducting fluid of variable density, in the presence of a magnetic field

1955 ◽  
Vol 233 (1194) ◽  
pp. 376-396 ◽  
Keyword(s):  
Magnetic Field ◽  
Variational Principle ◽  
Gravity Waves ◽  
Vertical Direction ◽  
Finite Depth ◽  
Variable Density ◽  
Initial Disturbance ◽  
Wave Number ◽  
Stable Case ◽  
Hydromagnetic Waves

The equilibrium of a horizontal layer of a heavy incompressible fluid of variable density p 0 in the vertical direction is stable or unstable according as d p o /dz (z being the upward vertical) is everywhere negative or is anywhere positive. In the unstable case, the rate n at which the system departs from equilibrium depends on the total wave number k of the initial disturbance, and there is, in general, one mode characterized by n m and k m which grows more rapidly than any other. In the stable case, after an initial disturbance the equilibrium may be restored either periodically or aperiodically, depending on the value of k . The periodic type of motion gives rise to horizontally propagated ‘gravity waves’. It is the purpose of this paper to examine the influence of viscosity and hydromagnetie forces on the hydrodynamical motion produced by a small disturbance of the aforementioned equilibrium situation. The appropriate perturbation theory is developed initially for any density field p 0 (z) and kinematical viscosity v ( z ) for a fluid of constant electrical conductivity cr e.m.u. and magnetic permeability k in the presence of a uniform magnetic field of strength H 0 in the direction of gravity, acceleration g . The solution is expressed in the form of integrals, and is shown to be characterized by a variational principle. Based on the variational principle an approximate solution is obtained for the special case of a fluid of finite depth d stratified according to the law p 0 = p 1 exp ( βz ), and for which v is constant. It is shown that if n and k are measured in suitable units, they are related by an equation involving three dimensionless parameters: R = ( ηπs /2 dV A ), and B = ( gβd 2 2 dV A ) and B = ( gβd 2 /π 2 s 2 V 2 A ), where η = (4πKo) -1 , V 2 A = (kH 2 0 /4πP 1 ), and s is an integer involved in the description of the velocity field. Explicit solutions may be obtained in three cases, namely, (a) R ->oo (b) B = 0, (c) R = 0. Case (a) has been discussed in a previous paper. Other eases will be discussed in a future paper. Only cases (b) and (c) are considered here. In case (6) we have the problem of hydromagnetic waves damped by viscosity and electrical resistance; the properties of these waves are described. In case (c), we limit ourselves to an ideal conductor. When B > 0 the equilibrium is unstable; the influence of viscosity and hydromagnetic forces on the mode of maximum instability is briefly discussed. When B < 0, the equilibrium is stable. If — B < 2, S 2 > 1, or — B > 2, S 2 > (4(1 — B) 3 /21 B 2 ), the equilibrium is always restored aperiodically. Otherwise, waves can be generated, but only when the wavelength lies within a certain range. These weaves combine the properties of gravity waves and hydromagnetic waves.

The damping of surface gravity waves in a bounded liquid

1973 ◽  
Vol 59 (2) ◽  
pp. 239-256 ◽  
Author(s):  
C. C. Mei ◽  
L. F. Liu
Keyword(s):  
Boundary Layer ◽  
Gravity Waves ◽  
Side Wall ◽  
The Other ◽  
Wave Number ◽  
Viscous Damping ◽  
Alternative Derivation ◽  
Surface Gravity Waves ◽  
Damping Rate ◽  
Side Walls

In deducing the viscous damping rate in surface waves confined by side walls, Ursell found in an example that two different calculations, one by energy dissipation within and the other by pressure working on the edge of the side-wall boundary layers, gave different answers. This discrepancy occurs in other examples also and is resolved here by examining the energy transfer in the neighbourhood of the free-surface meniscus. With due care near the meniscus a boundary-layer–Poincaré method is employed to give an alternative derivation for the rate of attenuation and to obtain in addition the frequency (or wave-number) shift due to viscosity. Surface tension is not considered.

Stationary gravity waves on non-uniform free streams: jet-like streams

1975 ◽  
Vol 77 (2) ◽  
pp. 415-438 ◽  
Author(s):  
D. H. Peregrine ◽  
Ronald Smith
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Small Amplitude ◽  
Wave Number ◽  
Explicit Solutions ◽  
Stationary Waves ◽  
Two Dimensional ◽  
Large Wave ◽  
Surface Gravity Waves ◽  
Large Wave Number

AbstractThe basic state considered in this paper is a parallel flow of a jet-like character with the centre of the jet being at or near a free surface which is horizontal. Stationary surface gravity waves may exist on such a flow, and a number of examples are looked at for small amplitude waves. Explicit solutions are given for ‘top-hat’ profile jets and for two-dimensional flows. Asymptotic solutions are developed for stationary waves of large wave-number.

On the mean motion induced by internal gravity waves

1969 ◽  
Vol 36 (4) ◽  
pp. 785-803 ◽  
Author(s):  
Francis P. Bretherton
Keyword(s):  
Gravity Waves ◽  
Wave Energy ◽  
Wave Train ◽  
Three Dimensional ◽  
Internal Gravity Waves ◽  
Wave Number ◽  
Mean Flow ◽  
Mean Velocity ◽  
Mean Motion ◽  
The Mean

A train of internal gravity waves in a stratified liquid exerts a stress on the liquid and induces changes in the mean motion of second order in the wave amplitude. In those circumstances in which the concept of a slowly varying quasi-sinusoidal wave train is consistent, the mean velocity is almost horizontal and is determined to a first approximation irrespective of the vertical forces exerted by the waves. The sum of the mean flow kinetic energy and the wave energy is then conserved. The circulation around a horizontal circuit moving with the mean velocity is increased in the presence of waves according to a simple formula. The flow pattern is obtained around two- and three-dimensional wave packets propagating into a liquid at rest and the results are generalized for any basic state of motion in which the internal Froude number is small. Momentum can be associated with a wave packet equal to the horizontal wave-number times the wave energy divided by the intrinsic frequency.

On the trapping of long-period waves round islands

1969 ◽  
Vol 37 (4) ◽  
pp. 773-784 ◽  
Author(s):  
M. S. Longuet-Higgins
Keyword(s):  
Gravity Waves ◽  
Pressure Field ◽  
Critical Radius ◽  
Wave Number ◽  
Coriolis Parameter ◽  
Long Period ◽  
Short Period ◽  
Zero Radius ◽  
Inertial Currents ◽  
The Waves

The trapping of short-period gravity waves by islands and seamounts has been studied by Chambers (1965) and by Longuet-Higgins (1967). It was shown by the latter that in the absence of rotation, or when the wave frequency σ is large compared with the Coriolis parameter f, these waves cannot be perfectly trapped; some energy must always leak away to infinity. Very long-period oscillations in the presence of a sloping shelf surrounding an island, with σ [Lt ] f, have been studied by Mysak (1967) and Rhines (1967, 1969). Here perfect trapping is possible. However, as pointed out in Longuet-Higgins (1968), the rotation itself exerts a strong trapping effect not only when |σ| [Lt ] f, but also whenever a |σ| < f. It seems not to have been noticed that this effect is capable of trapping waves round an island in an ocean of uniform depth, in the absence of any shelf or sloping region offshore.The existence of such waves is demonstrated for a circular island in § 1 of the present paper. It is shown that the waves exist only when the azimuthal wave-number n is at least 1. The waves always progress round the island in a clockwise sense in the northern hemisphere. At large distances r from the island, the wave amplitude decays exponentially, but this exponential trapping occurs only if the radius a of the island exceeds the critical value (n(n − 1)gh)½/f. When n = 1, this critical radius is zero, so that in theory the waves exist for any island of non-zero radius.The application of these results to the ocean is discussed in § 2. Except possibly for baroclinic motions, it appears that only the waves corresponding to n = 1 could exist in fact, and that their frequency would be nearly equal to the inertial frequency f. It is unlikely that f could be regarded as constant over a sufficiently wide area for the model to apply without qualification. Nevertheless, the oscillations may be regarded as the local adjustment of the pressure field to inertial currents in the neighbourhood of the island. It is possible that the peak at about 0·73 c.p.d. in the spectrum of sea-level at Oahu, as found by Miyata & Groves (1968), can be interpreted in this way.

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