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.

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.

On the shape and breaking of finite amplitude internal gravity waves in a shear flow

1978 ◽  
Vol 85 (1) ◽  
pp. 7-31 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Shear Flow ◽  
Internal Waves ◽  
Gravity Waves ◽  
Second Harmonic ◽  
Mixing Layer ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Finite Amplitude ◽  
Depth Interval ◽  
Plane Shear

This paper is concerned with two important aspects of nonlinear internal gravity waves in a stably stratified inviscid plane shear flow, their shape and their breaking, particularly in conditions which are frequently encountered in geophysical applications when the vertical gradients of the horizontal current and the density are concentrated in a fairly narrow depth interval (e.g. the thermocline in the ocean). The present theoretical and experimental study of the wave shape extends earlier work on waves in the absence of shear and shows that the shape may be significantly altered by shear, the second-harmonic terms which describe the wave profile changing sign when the shear is increased sufficiently in an appropriate sense.In the second part of the paper we show that the slope of internal waves at which breaking occurs (the particle speeds exceeding the phase speed of the waves) may be considerably reduced by the presence of shear. Internal waves on a thermocline which encounter an increasing shear, perhaps because of wind action accelerating the upper mixing layer of the ocean, may be prone to such breaking.This work may alternatively be regarded as a study of the stability of a parallel stratified shear flow in the presence of a particular finite disturbance which corresponds to internal gravity waves propagating horizontally in the plane of the flow.

Transport by breaking internal gravity waves on slopes

2016 ◽  
Vol 789 ◽  
pp. 93-126 ◽  
Author(s):  
Robert S. Arthur ◽  
Oliver B. Fringer
Keyword(s):  
Gravity Waves ◽  
Wave Breaking ◽  
Internal Gravity Waves ◽  
Length Scale ◽  
Two Dimensional ◽  
Turbulent Diffusivity ◽  
Nepheloid Layer ◽  
Molecular Diffusivity ◽  
Tracking Model ◽  
Offshore Transport

We use the results of a direct numerical simulation (DNS) with a particle-tracking model to investigate three-dimensional transport by breaking internal gravity waves on slopes. Onshore transport occurs within an upslope surge of dense fluid after breaking. Offshore transport occurs due to an intrusion of mixed fluid that propagates offshore and resembles an intermediate nepheloid layer (INL). Entrainment of particles into the INL is related to irreversible mixing of the density field during wave breaking. Maximum onshore and offshore transport are calculated as a function of initial particle position, and can be of the order of the initial wave length scale for particles initialized within the breaking region. An effective cross-shore dispersion coefficient is also calculated, and is roughly three orders of magnitude larger than the molecular diffusivity within the breaking region. Particles are transported laterally due to turbulence that develops during wave breaking, and this lateral spreading is quantified with a lateral turbulent diffusivity. Lateral turbulent diffusivity values calculated using particles are elevated by more than one order of magnitude above the molecular diffusivity, and are shown to agree well with turbulent diffusivities estimated using a generic length scale turbulence closure model. Based on a favourable comparison of DNS results with those of a similar two-dimensional case, we use two-dimensional simulations to extend our cross-shore transport results to additional wave amplitude and bathymetric slope conditions.

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>

A Velocity and Length Scale Approach to k–ε Modeling

1996 ◽  
Vol 118 (4) ◽  
pp. 857-863 ◽  
Author(s):  
O. Kwon ◽  
F. E. Ames
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Turbulent Flows ◽  
Dissipation Rate ◽  
Eddy Viscosity ◽  
Normal Component ◽  
Transport Equations ◽  
Length Scale ◽  
Wall Region ◽  
Wide Range

This paper describes a velocity and length scale approach to low-Reynolds-number k–ε modeling, which formulates the eddy viscosity on the normal component of turbulence and a length scale. The normal component of turbulence is modeled based on the dissipation and distance from the wall and is bounded by the isotropic condition. The model accounts for the anisotropy of the dissipation and the reduced length of mixing in the near wall region. The kinetic energy and dissipation rate were computed from the k and ε transport equations of Durbin (1993). The model was tested for a wide range of turbulent flows and proved to be superior to other k–ε based models.

Stratified turbulence forced with columnar dipoles: numerical study

2015 ◽  
Vol 769 ◽  
pp. 403-443 ◽  
Author(s):  
Pierre Augier ◽  
Paul Billant ◽  
Jean-Marc Chomaz
Keyword(s):  
Reynolds Number ◽  
Energy Spectrum ◽  
Kinetic Energy ◽  
Numerical Simulations ◽  
Numerical Study ◽  
Stationary Flow ◽  
Length Scale ◽  
Computational Domain ◽  
Stratified Turbulence ◽  
Kinetic Energy Spectrum

This paper builds upon the investigation of Augier et al. (Phys. Fluids, vol. 26 (4), 2014) in which a strongly stratified turbulent-like flow was forced by 12 generators of vertical columnar dipoles. In experiments, measurements start to provide evidence of the existence of a strongly stratified inertial range that has been predicted for large turbulent buoyancy Reynolds numbers $\mathscr{R}_{t}={\it\varepsilon}_{\!K}/({\it\nu}N^{2})$, where ${\it\varepsilon}_{\!K}$ is the mean dissipation rate of kinetic energy, ${\it\nu}$ the viscosity and $N$ the Brunt–Väisälä frequency. However, because of experimental constraints, the buoyancy Reynolds number could not be increased to sufficiently large values so that the inertial strongly stratified turbulent range is only incipient. In order to extend the experimental results toward higher buoyancy Reynolds number, we have performed numerical simulations of forced stratified flows. To reproduce the experimental vortex generators, columnar dipoles are periodically produced in spatial space using impulsive horizontal body force at the peripheries of the computational domain. For moderate buoyancy Reynolds number, these numerical simulations are able to reproduce the results obtained in the experiments, validating this particular forcing. For higher buoyancy Reynolds number, the simulations show that the flow becomes turbulent as observed in Brethouwer et al. (J. Fluid Mech., vol. 585, 2007, pp. 343–368). However, the statistically stationary flow is horizontally inhomogeneous because the dipoles are destabilized quite rapidly after their generation. In order to produce horizontally homogeneous turbulence, high-resolution simulations at high buoyancy Reynolds number have been carried out with a slightly modified forcing in which dipoles are forced at random locations in the computational domain. The unidimensional horizontal spectra of kinetic and potential energies scale like $C_{1}{\it\varepsilon}_{\!K}^{2/3}k_{h}^{-5/3}$ and $C_{2}{\it\varepsilon}_{\!K}^{2/3}k_{h}^{-5/3}({\it\varepsilon}_{\!P}/{\it\varepsilon}_{\!K})$, respectively, with $C_{1}=C_{2}\simeq 0.5$ as obtained by Lindborg (J. Fluid Mech., vol. 550, 2006, pp. 207–242). However, there is a depletion in the horizontal kinetic energy spectrum for scales between the integral length scale and the buoyancy length scale and an anomalous energy excess around the buoyancy length scale probably due to direct transfers from large horizontal scale to small scales resulting from the shear and gravitational instabilities. The horizontal buoyancy flux co-spectrum increases abruptly at the buoyancy scale corroborating the presence of overturnings. Remarkably, the vertical kinetic energy spectrum exhibits a transition at the Ozmidov length scale from a steep spectrum scaling like $N^{2}k_{z}^{-3}$ at large scales to a spectrum scaling like $C_{K}{\it\varepsilon}_{\!K}^{2/3}k_{z}^{-5/3}$, with $C_{K}=1$, the classical Kolmogorov constant.

scholarly journals On the possibility of wave-induced chaos in a sheared, stably stratified fluid layer

1994 ◽  
Vol 1 (4) ◽  
pp. 219-223 ◽  
Author(s):  
W. B. Zimmermann ◽  
M. G. Velarde
Keyword(s):  
Reynolds Number ◽  
Gravity Waves ◽  
Thermal Energy ◽  
Richardson Number ◽  
Fluid Layer ◽  
Fluid Velocity ◽  
Internal Gravity Waves ◽  
Temperature Wave ◽  
Perturbation Scheme ◽  
Turbulent Burst

Abstract. Shear flow in a stable stratification provides a waveguide for internal gravity waves. In the inviscid approximation, internal gravity waves are known to be unstable below a threshold in Richardson number. However, in a viscous fluid, at low enough Reynolds number, this threshold recedes to Ri = 0. Nevertheless, even the slightest viscosity strongly damps internal gravity waves when the Richardson number is small (shear forces dominate buoyant forces). In this paper we address the dynamics that approximately govern wave propagation when the Richardson number is small and the fluid is viscous. When Ri << 1, to a first approximation, the transport equations for thermal energy and momentum decouple. Thus, a large amplitude temperature wave then has little effect on the fluid velocity. Under such conditions in the atmosphere, a small amplitude "turbulent burst" is observed, transporting momentum rapidly and seemingly randomly. A regular perturbation scheme from a base state of a passing temperature wave and no velocity disturbance is developed here. Small thermal energy convection-momentum transport coupling is taken into account. The elements of forcing, wave dispersion, (turbulent) dissipation under strong shearing, and weak nonlinearity lead to this dynamical equation for the amplitude A of the turbulent burst in velocity: Aξ = λ1A + λ2Aξξ + λ3Aξξξ + λ4AAξ + b(ξ) where ξ is the coordinate of the rest frame of the passing temperature wave whose horizontal profile is b(ξ). The parameters λi are constants that depend on the Reynolds number. The above dynamical system is know to have limit cycle and chaotic attrators when forcing is sinusoidal and wave attenuation negligible.

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