scholarly journals Asymptotics of the Far Fields of Internal Gravity Waves Excited by a Source of Radial Symmetry

2021 ◽  
Vol 56 (5) ◽  
pp. 672-677
Author(s):  
V. V. Bulatov ◽  
Yu. V. Vladimirov
Keyword(s):  
Gravity Waves ◽  
Hankel Transform ◽  
Internal Gravity Waves ◽  
Radial Symmetry ◽  
Wave Mode ◽  
Buoyancy Frequency ◽  
Asymptotic Results ◽  
Model Distribution ◽  
Perturbation Source ◽  
Individual Wave

Abstract— The problem of the far field of internal gravity waves generated by a perturbation source of radial symmetry aroused at an initial instant of time is solved. The constant model distribution of the buoyancy frequency is considered and, using the Fourier–Hankel transform, an analytical solution to the problem is obtained in the form of the sum of wave modes. Asymptotics of the solutions that describe the spatial-temporal characteristics of elevation of the isopycnic lines and the vertical and horizontal velocity components far from the perturbation source are obtained. The asymptotics of the components of the wave field are expressed in terms of the square of the Airy function and its derivatives in the neighborhood of the wave fronts of an individual wave mode. The exact and asymptotic results are compared and it is shown that the asymptotic method makes it possible to calculate effectively the far wave fields at times of the order of ten and more of the Brunt–Väisälä periods.

scholarly journals Generation of Internal Gravity Waves Far from Moving Non-Local Source

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1899
Author(s):  
Vitaly Bulatov ◽  
Yury Vladimirov
Keyword(s):  
Gravity Waves ◽  
Finite Depth ◽  
Internal Gravity Waves ◽  
Radial Symmetry ◽  
Stratified Medium ◽  
Local Source ◽  
Natural Sources ◽  
Model Distribution ◽  
Different Depths ◽  
Non Local

We consider analytical solutions describing the generation of internal gravity waves far from a non-local source of disturbances. We suppose that the source moves on the surface of stratified medium of a finite depth. A model distribution of the non-local source shape with radial symmetry is used. This approximation correctly describes (qualitatively) the main spatiotemporal characteristics of natural sources of generation of internal gravity waves in the ocean. The resulting solution is the sum of wave modes. The solution is presented as a series of eigenfunctions of the spectral problem of internal gravity waves. The results of numerical calculations of internal gravity waves components at different depths are presented and discussed.

scholarly journals Momentum Flux Spectrum of Convectively Forced Internal Gravity Waves and Its Application to Gravity Wave Drag Parameterization. Part I: Theory

2005 ◽  
Vol 62 (1) ◽  
pp. 107-124 ◽  
Author(s):  
In-Sun Song ◽  
Hye-Yeong Chun
Keyword(s):  
Gravity Waves ◽  
Momentum Flux ◽  
Three Dimensional ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Spectral Structure ◽  
Two Dimensional ◽  
Resonance Factor ◽  
Flux Spectrum ◽  
Wave Momentum

Abstract The phase-speed spectrum of momentum flux by convectively forced internal gravity waves is analytically formulated in two- and three-dimensional frameworks. For this, a three-layer atmosphere that has a constant vertical wind shear in the lowest layer, a uniform wind above, and piecewise constant buoyancy frequency in a forcing region and above is considered. The wave momentum flux at cloud top is determined by the spectral combination of a wave-filtering and resonance factor and diabatic forcing. The wave-filtering and resonance factor that is determined by the basic-state wind and stability and the vertical configuration of forcing restricts the effectiveness of the forcing, and thus only a part of the forcing spectrum can be used for generating gravity waves that propagate above cumulus clouds. The spectral distribution of the wave momentum flux is largely determined by the wave-filtering and resonance factor, but the magnitude of the momentum flux varies significantly according to spatial and time scales and moving speed of the forcing. The wave momentum flux formulation in the two-dimensional framework is extended to the three-dimensional framework. The three-dimensional momentum flux formulation is similar to the two-dimensional one except that the wave propagation in various horizontal directions and the three-dimensionality of forcing are allowed. The wave momentum flux spectrum formulated in this study is validated using mesoscale numerical model results and can reproduce the overall spectral structure and magnitude of the wave momentum flux spectra induced by numerically simulated mesoscale convective systems reasonably well.

scholarly journals Phase Structure of Internal Gravity Waves in the Ocean with Shear Flows

2021 ◽  
Vol 37 (4) ◽  
Author(s):  
V. V. Bulatov ◽  
Yu. V. Vladimirov ◽  
I. Yu. Vladimirov ◽  
◽  
◽  
...  
Keyword(s):  
Shear Flow ◽  
Spectral Problem ◽  
Gravity Waves ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Flow Profile ◽  
Wave Fields ◽  
Background Shear

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

Can laboratory experiments reach regimes relevant for the oceanic dynamics?

2021 ◽  
Author(s):  
Costanza Rodda ◽  
Clement Savaro ◽  
Antoine Campagne ◽  
Miguel Calpe Linares ◽  
Pierre Augier ◽  
...  
Keyword(s):  
Gravity Waves ◽  
Scaling Laws ◽  
Spatial Scales ◽  
Scaling Law ◽  
Internal Gravity Waves ◽  
Energy Spectra ◽  
Finite Size Effect ◽  
Wave Turbulence ◽  
Common Mechanism ◽  
Buoyancy Frequency

<p>Atmospheric and oceanic energy spectra are characterized by global scaling laws, suggesting a common mechanism driving the energy route to dissipation. Although several possible theories have been proposed, it is not clear yet what the phenomena contributing the most to the energy at the different spatial scales are. One possible scenario is that internal gravity waves, which can be ubiquitously found in the atmosphere and the ocean and play a fundamental role in the energy transfer, cause the observed spectral slopes at the mesoscales in the atmosphere and submesoscales in the oceans. In the context of this open field of investigation, we present an experimental study where internal gravity waves are forced at a given frequency by the oscillating walls of a large pentagonal-shaped domain filled with a stably stratified fluid. The setup is built inside the 13-meters-diameter tank at the Coriolis facility in Grenoble, where geophysical regimes (with high Reynolds number and low Froude) can be achieved and rotation can also be added. The purpose of our investigation is to determine whether it is possible to induce a wave turbulence cascade by forcing internal waves at the large scales. Following a previous study<sup>1</sup>, where instead of the pentagonal a square domain was utilized, we obtained the velocity field employing time-resolved particle image velocimetry and then calculated the energy spectra. The previous study inside a square domain showed some evidence of a cascade, but it was strongly affected by 2D modes that sharpened the spectrum. Therefore, we changed the domain shape to a pentagon to reduce this finite-size effect. When the waves are forced at frequency <em>ω<sub>F</sub>=0.4 N</em>, our data shows that the spectra follow the scaling law <em>ω<sup>-2</sup></em> at frequencies larger than the forcing frequency and extending beyond <em>N</em>. The experimental spectra strikingly resemble the characteristic Garret-Munk spectrum measured in the ocean. As the interaction of weakly non-linear waves dominates the dynamics at frequencies smaller than the buoyancy frequency <em>N</em>, we can conclude that the experimental spectra are generated by weak internal wave turbulence driving the turbulent cascade at the high-frequency end of the spectrum. </p><p> </p><p>1 "<em>Generation of weakly nonlinear turbulence of internal gravity waves in the Coriolis facility", C. Savaro, A. Campagne, M. Calpe Linares, P. Augier, J. Sommeria, T. Valran, S. Viboud, and N. Mordant, PRF 2020</em></p>

On the normal modes of parallel flow of inviscid stratified fluid

1976 ◽  
Vol 75 (1) ◽  
pp. 149-171 ◽  
Author(s):  
W. H. H. Banks ◽  
P. G. Drazin ◽  
M. B. Zaturska
Keyword(s):  
Velocity Profile ◽  
Gravity Waves ◽  
Richardson Number ◽  
Normal Modes ◽  
Internal Gravity Waves ◽  
Stratified Fluid ◽  
Parallel Flow ◽  
Marginal Stability ◽  
Mean Flow ◽  
Asymptotic Results

The overall pattern of normal modes of parallel flow of inviscid stratified fluid is examined. For a given flow and wavenumber the modes are divided into five classes, some of which may be empty: (i) a finite class of non-singular unstable modes; (ii) a conjugate finite class of non-singular damped stable modes; (iii) a finite class of singular stable modes, each of these having a branch point and being the limit of unstable modes; (iv) a discrete class of modified internal gravity waves, these being non-singular stable modes (if the density decreases with height everywhere); (v) a continuous class of singular stable modes. The modified internal gravity waves are described asymptotically for large values of the Richardson number. These asymptotic results are related to and extended by numerical calculations for a sinusoidal basic velocity profile and a Bickley jet. The wave speeds for small values of the Richardson number are found to depend only upon the local behaviour of the mean flow near an overall simple maximum or minimum of the velocity profile. Finally some difficulties in the use of the Howard formula for perturbation at a curve of marginal stability are elucidated.

scholarly journals Analytical Approximations of Dispersion Relations for Internal Gravity Waves Equation with Shear Flows

Symmetry ◽  
2020 ◽  
Vol 12 (11) ◽  
pp. 1865
Author(s):  
Vitaly Bulatov ◽  
Yury Vladimirov
Keyword(s):  
Bessel Function ◽  
Dispersion Equation ◽  
Gravity Waves ◽  
Shear Flows ◽  
Internal Gravity Waves ◽  
Stratified Medium ◽  
Buoyancy Frequency ◽  
Modified Bessel Function ◽  
Analytical Approximations ◽  
Background Shear

The problem of internal gravity waves fields in a stratified medium of finite depth is considered for model distributions of background shear currents. For the analytical solution of the problem, a constant distribution of the buoyancy frequency and various linear dependences of the background shear current on depth were used. The dispersion dependences are obtained, which are expressed in terms of the modified Bessel function of the imaginary index. Under the Miles–Howard stability condition and large Richardson numbers, the Debye asymptotics of the modified Bessel function of the imaginary index were used to construct analytical solutions. The dispersion equation is solved using the proposed analytical approximation. The properties of the dispersion equation are studied and the main analytical characteristics of the dispersion curves are investigated depending on the parameters of background shear flows.

scholarly journals Acoustic Tunnelling of Internal Gravity waves in the stratified fluids

2019 ◽  
Vol 15 ◽  
pp. 6121-6137
Author(s):  
Gangamani Hv
Keyword(s):  
Gravity Waves ◽  
Acoustic Waves ◽  
Internal Gravity Waves ◽  
Wave Number ◽  
Small Scale ◽  
Buoyancy Frequency ◽  
Frequency Variation ◽  
Infrasonic Waves ◽  
Barrier Region ◽  
Density Barrier

This paper focuses on the study of acoustic propagation of internal gravity waves which generates small scale variations through propagation and hence can obtain transmission co-efficients using N2 buoyancy frequency variation of a compressible stratified fluid for a small regions. We have also analysed the results using the asymptotic expansions for large compressible limits. The reduction of the transmission in the N2-barrier region for the density layers sandwiched along with acoustic waves is obtained through graphs for different density barrier regions. The dispersion characteristics shows the contours of the transmission in the wave number plane. The curves for ! < N0 are hyperbolic, representing internal gravity waves as these become the dispersionwaves for an incompressible fluid and the curve with ! > N0 are ellipsoids which represent the acoustic gravity or infrasonic waves for the cut off frequency

scholarly journals Phase Structure of Internal Gravity Waves in the Ocean with Shear Flows

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
V. V. Bulatov ◽  
Yu. V. Vladimirov ◽  
I. Yu. Vladimirov ◽  
◽  
◽  
...  
Keyword(s):  
Shear Flow ◽  
Spectral Problem ◽  
Gravity Waves ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Internal Gravity Waves ◽  
Buoyancy Frequency ◽  
Flow Profile ◽  
Wave Fields ◽  
Background Shear

Purpose. The description of the internal gravity waves dynamics in the ocean with background fields of shear currents is a very difficult problem even in the linear approximation. The mathematical problem describing wave dynamics is reduced to the analysis of a system of partial differential equations; and while taking into account the vertical and horizontal inhomogeneity, this system of equations does not allow separation of the variables. Application of various approximations makes it possible to construct analytical solutions for the model distributions of buoyancy frequency and background shear ocean currents. The work is aimed at studying dynamics of internal gravity waves in the ocean with the arbitrary and model distributions of density and background shear currents. Methods and Results. The paper represents the numerical and analytical solutions describing the main phase characteristics of the internal gravity wave fields in the stratified ocean of finite depth, both for arbitrary and model distributions of the buoyancy frequency and the background shear currents. The currents are considered to be stationary and horizontally homogeneous on the assumption that the scale of the currents' horizontal and temporal variability is much larger than the characteristic lengths and periods of internal gravity waves. Having been used, the Fourier method permitted to obtain integral representations of the solutions under the Miles – Howard stability condition is fulfilled. To solve the vertical spectral problem, proposed is the algorithm for calculating the main dispersion dependences that determine the phase characteristics of the generated wave fields. The calculations for one real distribution of buoyancy frequency and shear flow profile are represented. Transformation of the dispersion surfaces and phase structures of the internal gravitational waves’ fields is studied depending on the generation parameters. To solve the problem analytically, constant distribution of the buoyancy frequency and linear dependences of the background shear current on depth were used. For the model distribution of the buoyancy and shear flow frequencies, the explicit analytical expressions describing the solutions of the vertical spectral problem were derived. The numerical and asymptotic solutions for the characteristic oceanic parameters were compared. Conclusions. The obtained results show that the asymptotic constructions using the model dependences of the buoyancy frequency and the background shear velocities’ distribution, describe the numerical solutions of the vertical spectral problem to a good degree of accuracy. The model representations, having been applied for hydrological parameters, make it possible to describe qualitatively correctly the main characteristics of internal gravity waves in the ocean with the arbitrary background shear currents.

Internal gravity waves generated by a turbulent bottom Ekman layer

2007 ◽  
Vol 590 ◽  
pp. 331-354 ◽  
Author(s):  
JOHN R. TAYLOR ◽  
SUTANU SARKAR
Keyword(s):  
Boundary Layer ◽  
Mixed Layer ◽  
Gravity Waves ◽  
Outer Layer ◽  
Wave Amplitude ◽  
Internal Gravity Waves ◽  
Ekman Layer ◽  
Buoyancy Frequency ◽  
Viscous Effects ◽  
Outer Flow

Internal gravity waves excited by the turbulent motions in a bottom Ekman layer are examined using large-eddy simulation. The outer flow is steady and uniformly stratified while the density gradient is set to zero at the flat lower wall. After initializing with a linear density profile, a mixed layer forms near the wall separated from the ambient stratification by a pycnocline. Two types of internal wave are observed. Waves with frequencies larger than the free-stream buoyancy frequency are seen in the pycnocline, and vertically propagating internal waves are observed in the outer layer with characteristic frequency and wavenumber spectra. Since a signature of the pycnocline waves is observed in the frequency spectrum of the mixed layer, these waves may affect the boundary-layer turbulence. The dominant outer-layer waves have a group velocity directed 35-60° from the vertical axis, which is consistent with previous laboratory studies. The energy flux associated with the radiated waves is small compared to the integrated dissipation in the boundary layer, but is of the same order as the integrated buoyancy flux. A linear model is proposed to estimate the decay in wave amplitude owing to viscous effects. Starting from the observed wave amplitudes at the bottom of the pycnocline, the model prediction for the spectral distribution of the outer layer wave amplitude compares favourably with the simulation results.

A localized turbulent mixing layer in a uniformly stratified environment

2018 ◽  
Vol 849 ◽  
pp. 245-276 ◽  
Author(s):  
Tomoaki Watanabe ◽  
James J. Riley ◽  
Koji Nagata ◽  
Ryo Onishi ◽  
Keigo Matsuda
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Gravity Waves ◽  
Potential Vorticity ◽  
Mixing Layer ◽  
Internal Gravity Waves ◽  
Core Region ◽  
Buoyancy Frequency ◽  
Length Scale ◽  
Turbulent Region

Localized turbulence bounded by non-turbulent flow in a uniformly stratified environment is studied with direct numerical simulations of stably stratified shear layers. Of particular interest is the turbulent/non-turbulent interfacial (TNTI) layer, which is detected by identifying the turbulent region in terms of its potential vorticity. Fluid near the outer edge of the turbulent region gains potential vorticity and becomes turbulent by diffusion arising from both viscous and molecular effects. The flow properties near the TNTI layer change depending on the buoyancy Reynolds number near the interface,$Re_{bI}$. The TNTI layer thickness is approximately 13 times the Kolmogorov length scale for large$Re_{bI}$($Re_{bI}\gtrsim 30$), consistent with non-stratified flows, whereas it is almost equal to the vertical length scale of the stratified flow,$l_{vI}=l_{hI}Re^{-1/2}$(here$l_{hI}$is the horizontal length scale near the TNTI layer, and$Re$is the Reynolds number), in the low-$Re_{bI}$regime ($Re_{bI}\lesssim 2$). Turbulent fluid is vertically transported towards the TNTI layer when$Re_{bI}$is large, sustaining the thin TNTI layer with large buoyancy frequency and mean shear. This sharpening effect is weakened as$Re_{bI}$decreases and eventually becomes negligible for very low$Re_{bI}$. Overturning motions occur near the TNTI layer for large$Re_{bI}$. The dependence on buoyancy Reynolds number is related to the value of$Re_{bI}$near the TNTI layer, which is smaller than the value deep inside the turbulent core region. An imprint of the internal gravity waves propagating in the non-turbulent region is found for vorticity within the TNTI layer, inferring an interaction between turbulence and internal gravity waves. The wave energy flux causes a net loss of the kinetic energy in the turbulent core region bounded to the TNTI layer, and the amount of kinetic energy extracted from the turbulent region by internal gravity waves is comparable to the amount dissipated in the turbulent region.

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