Why ripples become sweiis,and swells in shallow water decay rapidly

Abstract

In an ocean with a horizontal bottom where no wind is blowing it is shown that the spin (angular momentum) of the ocean is conserved. Thus, when energy is dissipated, at least one of three things will happen: i) Wave spectra may move towards lower frequencies. ii) The directional distribution may be changed towards long-crested waves. iii) Shear currents may be generated. By neglecting ii) and iii), the frequency shift of a spectrum is calculated due to molecular dissipation. When all energy transforming phenomena as e.g. wave breaking and turbulence generation are taken into account, the conservation of spin seems to be able to explain the frequency shift of wave spectra. In shallow water it is shown that there is energy transfer from the waves to shear currents.     

Convection in a rotating,laterally heated annulus pattern velocities and amplitude oscillations

Abstract

The velocities of the wave patterns relative to the rotating annulus have been measured with either increasing or decreasing positive radial temperature gradients and different rotation rates, with the fluid in thermal equilibrium and in contact with a rigid lid. The pattern velocities are dependent on initial conditions except in the unique areas of the stability diagram, where the velocities observed with either increasing or decreasing ΔT, overlap. The pattern velocities change discontinuously with each wave number transition, with a particularly large discontinuity at the transition from two to one wave. The frequency of the amplitude oscillations of the waves has been measured also. It has been found that the period of the oscillation of the three wave pattern is inversely proportional to the period of the pattern velocity, which means that in this case the ratio of the frequency of amplitude oscillation and the frequency of the pattern revolution is incommensurate.     

REFRACTION LE LONG DES BANCS MINCES RAPIDES ET EFFET D'ECRAN POUR LES MARQUEURS PROFONDS *

Refraction along thin high velocity layers and along basement is investigated in two cases. a) high velocity layer just on the basement. b) high velocity layer higher above. Period and attenuation of refracted waves are givers as a function of the layer thickness H. Refracted arrivals along thin high velocity layers are visible at significant distances if the layer thickness is not smaller than A/6, where A is the longitudinal wavelength in high velocity medium. The pseudoperiod is proportional to the layer thickness H. The attenuation at large distance follows an x^-ne^-k1x law, where n is close to I and k₁ is inversely proportional to H. Refracted arrivals along the basement are observable even in the case of thin high velocity layers situated in the overburden; their intensity is smaller and their pseudo-period larger than when no layer exists in the overburden. The intensity of the basement arrival decreases and the pseudoperiod increases with increasing layer thickness. The pseudoperiod and the attenuation of refracted arrivals along high velocity layers and along the basement are also highly dependent on acoustic contrasts. Both arrivals from a high velocity layer and from the basement can be recorded simultaneously, provided the frequency spectrum of the seismic chain is sufficiently broad. In all cases layer arrivals show a character very different from basement arrivals.     

Microseismic field affected by local geological heterogeneities and microseismic sounding of the medium

Experiments and numerical model studies have shown that heterogeneities of the Earth’s crust distort the spectrum of the low frequency microseismic field, decreasing spectral amplitudes of a specific frequency f at the Earth’s surface over high velocity heterogeneities and increasing them above low velocity heterogeneities. The frequency f is connected with the depth of a heterogeneity H and the velocity of the fundamental mode of Rayleigh waves V _R (f) through the relation H = 0.5 V _R (f)/f. The low frequency microseismic field is considered as the superposition of trains of Rayleigh fundamental modes with different frequency spectra. The paper proposes an experimentally tested technology enabling the determination of the deep structure of complex geological objects using data on the microseismic background field.     

Edge‐, bottom‐, and Rossby waves in a rotating stratified fluid

Abstract

It is found that in a rotating stratified fluid bounded by a single rigid wall, edge waves may occur at all frequencies less than or equal to N sin a (a is the angle of the wall from the horizontal and N the Brunt‐Vaisala frequency). These decay exponentially away from the boundary, in a distance of O(S) wavelengths, for α = O(1), or O(S ^‐1) wavelengths, for αS ≤ O(1), where S is the ratio of N to the Coriolis parameter f, taken for illustration to be large. The phase and energy both move with a component to the left, facing shallow water. The waves could, for example, appear as an internal tide at the continental rise or as baroclinic meandering of currents over a slope.

The low‐frequency limit, αS ? 1, is studied in detail. To allow for large scales of motion other rigid boundaries and variations in f are included. The edge (actually “bottom") waves then merge with topographic‐planetary waves as the wavelengths increase; the familiar depth‐independent mode is found to be possible in the sea for wavelengths exceeding about 450 km. The ß‐effect introduces modes complementary to that trapped at the bottom, which instead are isolated from it.     

On strong ground motion synthesis with k −2 slip distributions

This study deals with the methodical aspects of k ^–2(Bernard et al., 1996) kinematic strong motions modelling: (1) it is shown how to incorporate the k-dependent rise time for 2D fault geometry in the strong motion synthesis according to the representation theorem, (2) it is suggested how to produce realistic k ^–2 slip models including asperity(ies), (3) modifications are introduced concerning the typeof used slip velocity function and the corner wave number in the slip distribution. High frequency effects of these generalized models are discussed.It is shown that, assuming the rise time proportional to the spatial slip wavelength at high wave numbers, the spectral decay of displacement at frequencies higher than the corner frequency is given just by the decay ofthe slip distribution spectrum, regardless of the type of slip velocity function. It is shown numerically that this model provides -squared source spectrum even in a vicinity of a 2D normal fault buried in 1D structure, which is an agreement with previous studies.     

SEISMIC WAVES IN ANELASTIC NON-LINEAR MEDIA—A THEORETICAL CONTRIBUTION *

Numerical solutions of the wave equation for a particular type of non-linear “constant Q” medium were carried out. These solutions were obtained after the transformation of the space derivatives in finite differences; power series of the time are used to express the solutions. The medium is characterized by a not single valued stress-strain relation; the stresses are greater when the absolute values of strain are increasing (passive work), and are less when they are decreasing (active work). A loss of energy follows which is constant for every cycle and independent of frequency. This model represents the simplest type of medium in agreement with the laboratory data on rock samples. For a similar medium the stress’values do not depend only on the instantaneous value of the strain, but also on the previous strain values, i.e. the history of the medium. All these characteristics are not compatible with linearity and require particular types of computing procedures similar to the one shown in this paper. The results of calculations here shown refer both to the propagation of an isolated wave and to the generation of a wave in a spherical hole by change of pressure. They refer particularly to the shape, the amplitude and the width of the isolated wave along the propagation path. The most important results for this type of medium are the following: a) for a plane single isolated wave, the displacement amplitude wave does not change along the propagated distance. The width increases linearly as function of the distance; b) the corresponding particle velocity decreases in amplitude; c) for single isolated spherical waves the displacement amplitude decreases with propagated distance only due to the geometric factor, i.e. inversely proportional to the propagated distance; its width increases in the same way as for plane waves. The comparison between these theoretical results with the field and seismological data show a sufficiently good agreement as far as the value interval of wave width and frequencies is concerned. Less satisfactory is the comparison regarding a linear dependence of the wave width on the distance. This fact happens probably because in the field often long trains of waves and not isolated waves occur. In effect, for trains of waves the behaviour is different than that of an isolated wave; particularly, for the former the frequency variations along the travelled path is less and the displacement variations greater. However, it seems likely that a further similar theoretical research for trains of waves propagating in this type of non-linear medium might be carried out to complete the present research.     

Energy spectra associated with long's model for steady finite-amplitude flow over orography

Abstract

The process of wave steepening in Long's model of steady, two-dimensional stably stratified flow over orography is examined. Under conditions of the long-wave approximation, and constant values of the background static stability and basic flow, Long's equation is cast into the form of a nonlinear advection equation. Spectral properties of this latter equation, which could be useful for the interpretation of data analyses under mountain wave conditions, are presented. The principal features, that apply at the onset of convective instability (density constant with height), are:

i) a power spectrum for available potential energy that exhibits a minus eight-thirds decay, in terms of the vertical wavenumber k _z ^-;

ii) a rate of energy transfer across the spectrum that is inversely proportional to the wavenumber for large k _z ^-;

iii) an equipartition between the kinetic energy of the horizontal motion and the available potential energy, under the longwave approximation, although all the disturbance energy is kinetic at the point where convective instability is initiated. It is also shown that features i) and ii) apply to more general conditions that are appropriate to Long's model, not just the long-wave approximation. Application to fully turbulent flow or to conditions at the onset of shearing instability are not considered to be warranted, since the development only applies to conditions at the onset of convective instability.     

Ground surface response induced by shallow buried explosions

Based on the spherical cavity expansion theory in the elastic half space,the ground surface movement characteristics of shallowly buried explosions are analyzed.The results show that the induced seismic wave is a longitudinal wave in the near zone and a Rayleigh wave in the far zone.The maximum displacement(velocity) of the longitudinal wave and the Rayleigh wave are inversely proportional to the scaled distance,and can be described by exponential function with exponents equal to 1.4 and 0.5,respectively.The vibration frequencies of the waves have almost no change.The vibration frequency of the longitudinal wave approximates the natural vibration frequency of the cavity in the broken area,and the vibration frequency of the Rayleigh wave is about half that of the longitudinal wave.On the same reduced buried depth and reduced distance,the particle displacement is directly proportional to the product of the boundary loading and cavity radius,and is inversely proportional to the transversal wave velocity.Meanwhile,the particle velocity is directly proportional to the boundary loading and inversely proportional to the wave velocity ratio.In the far zone,the buried depth of the explosive only has a slight effect on the longitudinal wave,but has a larger effect on the Rayleigh wave.     

Internal waves in a randomly stratified fluid

Abstract

We discuss the propagation of internal waves in a rotating stratified unbounded fluid with randomly varying stability frequency, N. The first order smoothing approximation is used to derive the dispersion relation for the mean wave field when N is of the form N ² = N _o ²(1 + ?μ), where μ is a centered stationary random function of either depth (z) or time (t), N _o = constant and O < ?² ≦ 1. Expressions are then derived for the change in phase speed and growth rate due to the random fluctuations μ; in particular, attention is focused on the behaviour of these expressions for short and long correlation lengths (case μ = μ(z)) and times (case μ = μ(t)). For the case μ = μ(z), which represents a model for the temperature and salinity fine-structure in the ocean, the appropriate statistics of the fluctuations observed at station P (50°N, 145°W) have been incorporated into the theory to estimate the actual importance of the effects due to these random fluctuations. It is found that the phase speed of the mean wave decreases significantly if (i) the wavelength is short compared to g/N_o ² or (ii) the wave number vector is essentially horizontal and the wave frequency is very close to N _o. Also, the random fluctuations cause a significant growth (decay) in the amplitude of a wave propagating upwards (downwards) through a depth of a few kilometers. However, in the direction of energy propagation, the kinetic energy is conserved. Finally, it is shown that the average effect of the depth dependent fluctuations at station P is to slightly decrease the stability frequency and the magnitude of the group velocity.     

Determination of the dispersion of low frequency waves downstream of a quasiperpendicular collisionless shock

A method of wave mode determination, which was announced in Balikhin and Gedalin, is applied to AMPTE UKS and AMPTE IRM magnetic field measurements downstream of supercritical quasiperpendicular shock. The method is based on the fact that the relation between phase difference of the waves measured by two satellites, Doppler shift equation, the direction of the wave propagation are enough to obtain the dispersion equation of the observed waves. It is shown that the low frequency turbulence mainly consists of waves observed below 1 Hz with a linear dependence between the absolute value of wave vector |k| and the plasma frame wave frequency. The phase velocity of these waves is close to the phase velocity of intermediate waves V_int = V_acos().     

ANALYSIS OF DIFFRACTION PATTERN FILTERING OF VARIABLY DENSITY AND VARIABLY AREA RECORDINGS

When a small, transparent replica of a seismic section is illuminated by a homogeneous beam of coherent, monochromatic parallel light a diffraction pattern is created that is representation of the double Fourier spectrum of the recorded seismic waves, i.e. their spectrum in terms of frequency, f, and apparent wave number, k. Masking selected parts of this diffraction pattern causes the spectrum to be filtered: the recomposition of the filtered spectrum then provides a filtered section. The ideal seismic section for this purpose would be a continuous variable density section obtained from recording made at every point of the seismic line. The light transmission coefficient (in terms of light amplitude) at each spot of the replica should be linearly related to the instantaneous seismic signal strength at the spot on the seismic section to which it refers. Unfortunately we cannot make recordings at every point of a seismic line and in our practically realisable recordings we have to be content with sampling in the direction of the location coordinate x. This means that with variable density recordings aliasing will be present and evident in the spectrum obtained in the direction of k; furthermore, the aliased spectrum is also multiplied by a sine function of k because the recording obtained at a given station is not presented on the seismic section as a single line along the time axis, but occupies the entire width of the trace. The diffraction patterns created by variable density recordings of dipping sine waves, including clipped recordings, and of the effect of dip filtering in such sections are discussed. The efficiency of dip rejection is shown to decrease with increasing dip. The diffraction pattern of a variable density recording is found to be characterised by a relationship between point pairs in the pattern. No such simple relationship has been found for the diffraction pattern of a variable area section; the spectra of such VAR sections belong to a very special class, because the amplitude transmission coefficient has only two values, viz. 0 and 1. Consequently, selective masking of the diffraction pattern of a VAR section may give rise to a filtered profile that does not look like a VAR section at all. General statements about the diffraction pattern of VAR sections are hard to give, because the transmission coefficient at a given point in the replica is not proportional to the signal level in the seismic section at the relevant point. In the case of VAR presentation of harmonic waves it was found that, as well as the aliasing effect in the k direction, higher harmonics of the frequency are also introduced. Some synthetic examples are given that show dip filtering to be less effective with VAR than with variable density recordings. Some arguments are advanced in favour of the opinion that high-pass filtering of VAR sections will have less success than low-pass filtering. This is demonstrated by two synthetic examples.     

Signal reconstruction of surface waves on SASW measurement using Gaussian Derivative wavelet transform

Surface wave method consists of measurement and processing of the dispersive Rayleigh waves recorded from two or more vertical transducers. The dispersive phase data are inverted and the shear wave velocity versus depth is obtained. However, in case of residual soil, the reliable phase spectrum curve is difficult to be produced. Noises from nature and other human-made sources disturb the generated surface wave data. In this paper, a continuous wavelet transform based on mother wavelet of Gaussian Derivative was used to analyze seismic waves in different frequency and time. Time-frequency wavelet spectrum was employed to localize the interested seismic response spectrum of generated surface waves. It can also distinguish the fundamental mode of the surface wave from the higher modes of reflected body waves. The results presented in this paper showed that the wavelet analysis is able to determine reliable surface wave spectrum of sandy clayey residual soil.     

Scaling relations for seismic events induced by mining

The values of seismic moment andS-wave corner frequency from 1575 seismic events induced in South African, Canadian, Polish, and German underground mines were collected to study their scaling relations. The values ofP-wave corner frequency from 649 events were also available. Seismic moments of these events range from 5*10³ to 2*10¹⁵ N·m (moment magnitude is from –3.6 to 4.1), theS-wave corner frequency ranges from 0.7 to 4438 Hz, and theP-wave corner frequency is between 5 and 4010 Hz. The slope of a regression line between the logarithm ofS- andP-wave corner frequencies is equal to one, and the corner frequencies ofP waves are higher than those ofS waves on the average by about 25 percent. In studies of large and moderate earthquakes it has been found that stress drop is approximately independent of the seismic moment, which means that seismic moment is inversely proportional to the third power of corner frequency. Such a behavior was confirmed for most of the data considered here. A breakdown in the similarity betwen large and small events seems to occur for the events with moment magnitude below –2.5. The average values of seismic moment referred to the same range of corner frequency, however, are vastly different in various mining areas.     

On the transition between axisymmetric and non-axisymmetric flow in a rotating liquid annulus subject to a horizontal temperature gradient: Hysteresis effects at moderate Taylor number and baroclinic waves beyond the eady cut-off at high Taylor number

Abstract

The transition between axisymmetric and non-axisymmetric régimes of flow in a rotating annulus of liquid subject to horizontal temperature gradient is known from previous experimental studies to depend largely on two dimensionless parameters. These are Θ, which is proportional to the impressed density contrast Δρ and inversely proportional to the square of the angular speed of rotation ω, and  (Taylor number), which is proportional to ω² /v² where v is the coefficient of kinematic viscosity. At moderate values of , around 10⁷, the critical value of Θ above which axisymmetric flow is found to OCCUT and below which non-axisymmetric fully-developed baroclinic waves (sloping convection) occur, is fairly insensitive to . Though sharp, the transition exhibits marked hysteresis when the upper surface of the liquid is free (but not when the upper surface is in contact with a rigid lid), and it is argued on the basis of the experimental evidence supported by various results of baroclinic instability theory that both the sharpness of the transition and the hysteresis phenomenon are consequences of the combined effects of potential vorticity gradients and viscosity on the process of sloping convection.

We also present some new experiments on fully-developed baroclinic waves, conducted in a large rotating annulus using liquids of very low viscosity (di-ethyl ether), thus attaining values of  as high as 10⁹ to 10¹⁰. The transition from axisymmetric to non-axisymmetric flow is found to lose its sharpness at such high values of , and it is argued that this occurs because viscosity is no longer able to inhibit instabilities at wavelengths less than the so-called ‘Eady short-wave cut-off’, which owe their existence to potential vorticity gradients in the main body of the fluid.     

Numerical analysis of one-dimensional nonlinear acoustic wave

Numerical investigations on one-dimensional nonlinear acoustic wave with third and fourth order nonlinearities are presented using high-order finite-difference (HFD) operators with a simple flux-limiter (SFL) algorithm. As shown by our numerical tests, the HFDSFL method is able to produce more stable, accurate and conservative solutions to the nonlinear acoustic waves than those computed by finite-difference combined with the flux-corrected-transport algorithm. Unlike the linear acoustic waves, the nonlinear acoustic waves have variable phase velocity and waveform both in time-space (t-x) domain and frequency-wavenumber (f-k) domain; of our special interest is the behaviour during the propagation of nonlinear acoustic waves: the waveforms are strongly linked to the type of medium nonlinearities, generation of harmonics, frequency and wavenumber peak shifts. In seismic sense, these characteristics of nonlinear wave will introduce new issues during such seismic processing as Normal Moveout and f-k filter. Moreover, as shown by our numerical experiment for a four-layer model, the nonlinearities of media will introduce extra velocity errors in seismic velocity inversion.     

Several effects of a baroclinic current on the cross‐stream propagation of inertial‐internal waves†

Several effects of a baroclinic current on inertial‐internal waves at constant frequency are investigated, primarily through use of the method of characteristics. The special case of waves propagating transverse to a baroclinic current is considered. When the slope of an isopycnal is of the same order of magnitude as the slope of the characteristics, appreciable asymmetries are induced in the characteristics, the phase and group velocities, and the solution itself. These asymmetric effects are especially significant for waves at the low frequency end of the passband for free waves. Also, modifications occur to the passband, resulting in anomalously high and low frequency bands. The effective local inertial frequency, σ_f = [f(f+v_x )]^1/2, separates the normal and anomalously low frequency bands. Hence, the low frequency limit of the normal frequency band increases or decreases depending upon whether the horizontal shear in the mean flow is cyclonic or anticyclonic. In the anomalous frequency bands, the slopes of both characteristics have the same sign, causing various refraction and reflection phenomena. If the absolute value of the slope, s, of an isopycnal exceeds its critical value, s_c = effective local inertial frequency/Väisälä‐Brunt frequency, the anomalously low frequency band extends to imaginary frequencies. If s ? 0, the reflection of waves from a boundary is modified, the effective wavelength is increased, and the lines of constant phase are tilted from the vertical. For the general solution, discontinuities in the first‐order partial derivatives of the velocity field occur across certain characteristics. The nonseparable normal modes do not exhibit these discontinuous derivatives, but they only satisfy one of the two pairs of kinematic boundary conditions in rectangular regions.     

Wave propagation in two layered medium under initial stresses

Summary In this paper, the frequency equation for phase velocity of waves propagated in a laminated medium consisting of two eleastic layers of finite thickness under initial stresses, has been obtained. It has been shown that when wave length becomes very small compared to the thickness of each layer, the wave approaches two Rayleigh waves at the two outer surfaces with the possibility of Stoneley waves at the interface. The propagation ofSH-waves in the composite medium under initial stresses has also been discussed. A particular case has been taken to find the velocity of Love wave in the homogeneous half space under initial compressive stresses.Biot's incremental deformation theory has been used.     

Finite amplitude holmboe waves

Abstract

We investigate the evolution of a parallel shear flow which has embedded within it a thin, symmetrically positioned layer of stable density stratification. The primary instability of this flow may deliver either Kelvin-Helmholtz waves or Holmboe waves, depending on the strength of the stratification. In this paper we describe a sequence of numerical simulations which reveal for the first time the behavior of the Holmboe wave at finite amplitude and clarify its structural relationship to the Kelvin-Helmholtz wave.

The flows investigated have initial profiles of horizontal velocity and Brunt-Vaisala frequency given in nondimensional form by U = tanhζ and N ²=J sech² RCζ, respectively, in which ζ is a nondimensional vertical coordinate, J is the value of the gradient Richardson number N ²/(dU/dζ)² at ζ=0, and R = 3. Linear stability theory predicts that the flow will develop Holmboe instability when J exceeds some critical value J_c' and Kelvin-Helmholtz instability when J is less than J_c; J_c being approximately equal to 0.25 when R=3. We simulate the evolution of flows with J=0.9, J=0.45, and J = 0.22, and find that the first two simulations yield Holmboe waves while the third yields a Kelvin-Helmholtz wave, as predicted.

The Holmboe wave is a superposition of two oppositely propagating disturbances, a right-going mode whose energy is concentrated in the region above the centre of the shear layer, and a left-going mode whose energy is concentrated below the centre of the shear layer. The horizontal speed of the modes varies periodically, and the variations are most pronounced at low values of J. If J ζ J_c' the minimum horizontal speed of the modes vanishes and the modes become phase-locked, whereupon they roll up to form a Kelvin-Helmholtz wave as predicted by Holmboe (1962). When J is moderately greater than J_c' the Holmboe wave ejects long, thin plumes of fluid into the regions above and below the shear layer, as has often been observed in laboratory experiments, and we examine in detail the mechanism by which this occurs.     

On recent developments in E-region irregularity simulationsand a summary of related theory

Theoretical and simulation approaches to E-region irregularities (gradient drift and Farley-Buneman instabilities) are reviewed, and an account is given of some relevant observations. A new hybrid linear dispersion relation is also derived and presented. The most important problem that cannot be explained by more straightforward theories is the saturation of the phase velocity to the ion acoustic speed (C_s saturation). This phenomenon is well-known from equatorial electrojet radar observations. Recent particle simulations have yielded an interesting new explanation for the (C_s saturation, which has been named flow angle stabilization: the phase velocity is not actually (C_s saturated, but the flow angle distribution of the spatial power spectrum is highly asymmetric. The asymmetry is such that the most intense waves propagate at the k⋅E < 0 edge of the linearly unstable sector, and thus the phase velocity of the most intense waves is close to (C_s. Depending on the level of larger scale turbulence, the radar observes varying degrees of (C_s saturation. If the larger scale turbulence level is high (equatorial electrojet case), the local flow angle fluctuates, and there are always subregions within the scattering volume with local flow angles favorable for the detection of the most intense waves. Under these conditions, the spectra show (C_s saturation. If the larger scale turbulence level is lower, there will not always be enough mixing of the flow angle for even the most intense waves to be observed. In these cases, the mean Doppler shift will be proportional to the electric fied, i.e. it will obey the linear theory.