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.     

Internal waves near the turning point

Abstract

A uniformly valid solution of internal wave equations is presented, using Langer's method, for an arbitrary density profile. The dispersion relation and mean square values of the velocity components are obtained. Particular emphasis is placed on the turning point problem and on estimating the errors involved in the approximations in a recent paper by Garrett and Munk (1972). Expressions are given for various spectral ratios of interest.     

Internal waves and temperature fronts on slopes

Time series measurements from an array of temperature miniloggers in a line at constant depth along the sloping boundary of a lake are used to describe the ‘internal surf zone” where internal waves interact with the sloping boundary. More small positive temperature time derivatives are recorded than negative, but there are more large negative values than positive, giving the overall distribution of temperature time derivatives a small negative skewness. This is consistent with the internal wave dynamics; fronts form during the up-slope phase of the motion, bringing cold water up the slope, and the return flow may become unstable, leading to small advecting billows and weak warm fronts. The data are analysed to detect ‘events’, periods in which the temperature derivatives exceed a set threshold. The speed and distance travelled by ‘events’ are described. The motion along the slope may be a consequence of (a) instabilities advected by the flow (b) internal waves propagating along-slope or (c) internal waves approaching the slope from oblique directions. The propagation of several of the observed ‘events’ can only be explained by (c), evidence that the internal surf zone has some, but possibly not all, the characteristics of the conventional ‘surface wave’ surf zone, with waves steepening as they approach the slope at oblique angles.     

Internal waves in a fluid of finite Prandtl number

Abstract

It is shown, through calculation and physical arguments, that in a smoothly stratified fluid thermal diffusion plays a damping role comparable to that of viscosity only in the interior of the fluid, and not in the boundary layers.     

Internal gravity waves in a slowly varying,dissipative medium

Nonlinear internal gravity waves in a slightly dissipative, slightly compressible fluid are discussed for the case when the properties of the medium vary slowly on a scale determined by the local wave structure. A two‐timing technique is used to obtain transport equations which describe the changes in amplitude, phase and mean flow of a wave packet. Various solutions of these transport equations are discussed, with relevance to critical layer absorption.     

Internal tides and waves near the continental shelf edge

Abstract

The subject is reviewed from the viewpoints of theory, internal tide and wave structure and their implications.

A wider theoretical context suggests scope for further investigation of natural or nearly-trapped forms above the inertial frequency.

Although internal tides in many locations are observed to have first-mode vertical structure, higher modes are seen offshore from shallow shelf-break forcing and for particular Froude numbers, and may be expected locally near generation. Bottom intensification is often observed where the sea floor matches the characteristic slope. Solitons form from internal tides of large amplitude or at large changes of depth.

Internal tides and solitons are observed also at many sills and in straits, and to intensify in canyons.

Non-linear effects of the waves, especially solitons, include the conveyance of water, nutrients, ‘‘mixing potential'’ etc. away from their source to other locations, and the generation of mean currents. The waves transfer energy and possibly heat between the ocean and shelf, may be a source of medium frequency waves on the shelf (periods of minutes) and can contribute to interior mixing and overturning, bottom stirring and sediment movement.     

Internal waves in a rotating stratified fluid in an arbitrary gravitational field

Abstract

Small amplitude oscillations of a uniformly rotating, density stratified, Boussinesq, non-dissipative fluid are examined. A mathematical model is constructed to describe timedependent motions which are small deviations from an initial state that is motionless with respect to the rotating frame of reference. The basic stable density distribution is allowed to be an arbitrary prescribed function of the gravitational potential. The problem is considered for a wide class of gravitational fields. General properties of the eigenvalues and eigenfunctions of square integrable oscillations are demonstrated, and a bound is obtained for the magnitude of the frequencies. The modal solutions are classified as to type. The eigenfunctions for the pressure field are shown to satisfy a second-order partial differential equation of mixed type, and the equation is obtained for the critical surfaces which delineate the elliptic and hyperbolic regions. The nature of the problem is examined in detail for certain specific gravitational fields, e.g., a radially symmetric field. Where appropriate, results are compared with those of other investigations of waves in a rotating fluid of spherical configuration and the novel aspects of the present treatment are emphasized. Explicit modal solutions are obtained in the specific example of a fluid contained in a rigid cylinder, stratified in the presence of vertical gravity, with the buoyancy frequency N being an arbitrary prescribed function of depth.     

Internal nonlinear tidal waves generated at the Strait of Messina

Large amplitude internal waves were observed north and south of the Strait of Messina during two different periods. A comparison between the two sets of data is made. The behaviour of the shape of the waves is analysed from the physical parameters and two possible interpretations of these waves are given. The mechanism of formation related to the tide in the Strait is also discussed.     

Internal gravity waves in atmospheres with realistic dissipation and temperature part I. Mathematical development and propagation of waves into the thermosphere

Abstract

In this, the first part of a three part study, we develop a model for the theoretical analysis of 3‐dimensional internal gravity waves in atmospheres with arbitrary distributions of basic temperature, molecular weight, viscosity and conductivity (both eddy and molecular), Newtonian cooling, anisotropic hydromagnetic (ion) drag, and thermal excitation. Attention is given to the physical bases for our models, and a detailed outline is given of the numerical method used to solve the resulting system of equations.

As an application of the above development, we study the ability of five particular gravity waves (chosen for their observed importance in the neighborhood of 90–100 km—four of the waves are simulated tidal modes) to propagate from 90 km into the thermosphere. We choose to define the thermosphere as that portion of the atmosphere above the turbopause (ca. 110 km). Among the most significant results to emerge are the following: (i) the effects of molecular viscosity and conductivity appear to be more significant than the effects of hydromagnetic drag, and (ii) while most waves considered are significantly attenuated somewhere between 90 Ian and the thermosphere, the main semidiurnal tidal mode is not. In fact, semidiurnal temperature oscillations of only a few degrees amplitude at 90 km can give rise to oscillations of over a hundred degrees amplitude in the exosphere.     

Internal waves over the shelf in the western Bay of Bengal: a case study

Generation and propagation of internal waves (IWs) in the coastal waters of the extended shelf of the western Bay of Bengal are investigated for late winter by using the Massachusetts Institute of Technology General Circulation Model (MITgcm). The model is forced with astronomical tides and daily winds. Monthly climatological temperature and salinity fields are used as initial conditions. The simulations are compared with time series observations of temperature and currents from acoustic Doppler current profiler (ADCP) and conductivity-temperature-depth (CTD) moored at three locations south of Gopalpur: two at a local depth of 100 m and another at 400-m depth during 19–21 February 2012. The comparison of the spectral estimates for the time series of temperature from the model and observations are in reasonable agreement for the near-tidal frequency waves. The peak of temperature spectra is always found near the shelf break region which steadily lost its intensity over the continental shelf. The calculation of Richardson number reflected the presence of local mixing due to density overturning in the shelf region. To understand further the generation and propagation of internal tides in the region, energy flux and conversion of barotropic-to-baroclinic M₂ tidal energy are examined. The model simulations suggest that the internal tide is generated all along the shelf slope. The energy flux analysis shows that the internal tides propagate to either side of the generation sites.     

Internal waves,baroclinic energy fluxes and mixing at the European shelf edge

The energy flux in internal waves generated at the Celtic Sea shelf break was estimated by (i) applying perturbation theory to a week-long dataset from a mooring at 200 m depth, and (ii) using a 2D non-hydrostatic circulation model over the shelf break. The dataset consisted of high resolution time-series of currents and vertical stratification together with two 25-h sets of vertical profiles of the dissipation of turbulent kinetic energy. The observations indicated an average energy flux of 139 W m⁻¹, travelling along the shelf break towards the northwest. The average energy flux across the shelf break at the mooring was only 8 W m⁻¹. However, the waves propagating onshelf transported up to 200 W m⁻¹, but they were only present 51% of the time. A comparison between the divergence of the baroclinic energy flux and observed dissipation within the seasonal thermocline at the mooring showed that the dissipation was at least one order of magnitude larger. Results from a 2D model along a transect perpendicular to the shelf break showed a time-averaged onshelf energy flux of 153–425 W m⁻¹, depending on the magnitude of the barotropic forcing. A divergence zone of the energy flux was found a few kilometre offshore of the location of the observations in the model results, and fluxes on the order of several kW m⁻¹ were present in the deep waters further offshelf from the divergence zone. The modelled fluxes exhibited qualitative agreements with the phase and hourly onshelf magnitudes of the observed energy fluxes. Both the observations and the model results show an intermittent onshelf energy flux of 100–200 W m⁻¹, but these waves could only propagate ∼20–30 km onshore before dissipating. This conclusion was supported by a 25-h dataset sampled some 180 km onto the shelf, where a weak wave energy flux was found going towards the shelf break. We therefore conclude that shelf break generated internal waves are unlikely to be the main source of energy for mixing on the inner part of the shelf.     

Boundary-reflected waves and ultrasonic coda waves in rock physics experiments

Ultrasonic coda waves are widely used to study high-frequency scattering. However, ultrasonic coda waves are strongly affected by interference from by boundary-reflected waves. To understand the effect of boundary-reflected waves, we performed ultrasonic experiments using aluminum and shale samples, and the rotating staggered-mesh finite-difference method to simulate the wavefield. We analyzed the wavefield characteristics at the different receiving points and the interference characteristics of the boundary-reflected waves with the ultrasonic coda wave, and the effect of sample geometry on the ultrasonic coda waves. The increase in the aspect ratio of the samples delays the interference effect of the laterally reflected waves and reduces the effect on the ultrasonic coda waves. The main waves interfering with the ultrasonic coda waves are laterally reflected PP-, PS-, PPP-, and PPS-waves. The scattering and attenuation of the high-frequency energy in actual rocks can weaken the interference of laterally reflected waves with the ultrasonic coda waves.     

Nonlinear waves in rotating magnetohydrodynamics

We consider an electrically conducting fluid in rotating cylindrical coordinates in which the Elsasser and magnetic Reynolds numbers are assumed to be large while the Rossby number is assumed to vanish in an appropriate limit. This may be taken as a simple model for the Earth's outer core. Fully nonlinear waves dominated by the nonlinear Lorentz forces are studied using the method of geometric optics (essentially WKB). These waves are assumed to be of the form of an asymptotic series expanded about ambient magnetic and velocity fields which vanish on the equatorial plane. They take the form of short wave, slowly varying wave trains. The first-order approximation is sinusoidal and basically the same as in the linear problem, with a dispersion relation modified by the appearance of mean terms. These mean terms, as well the undetermined amplitude functions, are found by suppressing secular terms in a “fast” variable in the second-order approximation. The interaction of the mean terms with the dispersion relation is the primary cause of behaviors which differ from the linear case. In particular, new singularities appear in the wave amplitude functions and an initial value problem results in a singularity in one of the mean terms which propagates through the fluid. The singularities corresponding to the linear ones are shown to develop when the corresponding waves propagate toward the equatorial plane.     

Rayleigh waves in granular medium

Summary The wave velocity equation in the form of a ninth order determinantal expression is derived appropriate to Rayleigh type waves in a granular half-space supporting a different granular layer. The calssical frequency equation when both media are elastic has been deduced as a particular case by limiting process.     

Thermoelastic waves in non-simple media

Summary The present paper is an attempt to investigate the propagation of thermoelastic waves propagated in non-simple media and study the distinction between propagation in simple and nonsimple media. Plane progressive waves and Rayleigh waves have been discussed.     

Highly nonlinear wind waves in Currituck Sound: dense breather turbulence in random ocean waves

Ocean Dynamics - We analyze surface wave data taken in Currituck Sound, North Carolina, during a storm on 4 February 2002. Our focus is on the application of nonlinear Fourier analysis (NLFA)...     

Multiple scattering of seismic waves in fractured media: Velocity and effective attenuation of the coherent components ofP waves andS waves

We show that the multiple scattering by small fractures of seismic waves with wavelengths long compared to the fracture size and fracture spacing is indistinguishable from multiple-scattering effects produced by regular porosity, except for an orientation factor due to fracture alignment. The fractures reduce theP-wave andS-wave velocities and produce an effective attenuation of the coherent component of the seismic waves. The attenuation corresponds to 1000/Q of about unity for a Gaussian spectrum of fractures, and it varies with frequencyf asf ³. For a Kolmogorov spectrum of fractures of spectral index the attenuation is an order of magnitude or so larger and varies with frequency asf ^3-v The precise degree of attenuation depends upon the matrix properties, the fracture porosity, the degree of fracture anisotropy, the type of fluid filling the fractures, and the incidence angle of the wave.For fracture porosities less than about 15% theP-wave andS-wave velocities are decreased by the order of 5–10% with a lesser dependence on the type of fluid filling the fractures (gas, oil, or brine) and with a dependence on both the degree of anisotropy and the incident angle made by the wave. The tendency of fractures to occur perpendicularly to bedding suggests that the best way to measure seismically fractured rock behavior in situ is by using the travel-time delay and reflection amplitude. As both the offset and the azimuth of receivers vary from a shot, the travel-time delay and reflection amplitude should both show an elliptical pattern of behavior—the travel-time delay in response to the varying seismic speed, and the reflection amplitude in response to angular variations in the multiple scattering. Observations of attenuation at several frequencies should permit (a) determination of the spectrum of fractures (Gaussian versus Kolmogorovian) and (b) determination of the contribution of viscous damping to the effective attenuation.