Material dispersion by oceanic internal waves

Internal gravity waves that are generated in the open ocean have a universal frequency spectrum, called Garrett–Munk spectrum. By initializing internal waves that satisfy the Garrett–Munk spectrum in a non-hydrostatic numerical model, we investigate the material dispersion produced by these internal waves. Three numerical experiments are designed: Exp.-1 uses a linearly stratified fluid, Exp.-2 has an upper mixed layer, and Exp.-3 incorporates a circular front into the upper mixed layer. Resorting to neutrally buoyant particles, we investigate the dispersion in terms of metrics of the relative dispersion and finite-scale Lyapunov exponent (FSLE). Exp.-1 shows that the dispersion regime produced by these internal waves is between ballistic and diffusive based on relative dispersion, and is however ballistic according to FSLE. The maximum FSLE at scales of 100 m is about 5 day\(^{-1}\), which is comparable to that calculated using ocean drifters. Exp.-2 demonstrates that internal waves can generate flows and material dispersion in an upper mixed layer. However, when mixed layer eddies are present, as in Exp.-3, the dispersion in the mixed layer is controlled by the eddies. In addition, we show that inertial oscillations do not affect the relative dispersion, but impact FSLE at scales of inertial oscillations.     

Statistical properties of concentration fluctuations in two merging plumes

The statistics of the fluctuating concentration field within a plume is important in the analysis of atmospheric dispersion of toxic, inflammable and odorous gases. Previous work has tended to focus on concentration fluctuations in single plumes released in the surface layer or at ground level and there is a general lack of information about the mixing of two adjacent plumes and how the statistical properties of the concentration fluctuations are modified in these circumstances. In this work, data from wind tunnel experiments are used to analyse the variance, skewness, kurtosis, intermittency, probability density function and power spectrum of the concentration field during the mixing of two identical plumes and results are compared with those obtained for an equivalent single plume. The normalised variance, skewness and kurtosis on the centre-lines of the combined plume increase with distance downwind of the stack and, in the two-source configuration, takes lower values than those found in the single plumes. The results reflect the merging process at short range, which is least protracted for cases in which the sources are in-line or up to 30 \(^{\circ }\) off-line. At angles of 45 \(^{\circ }\) and more, the plumes are effectively side-by-side during the merging process and the interaction between the vortex pairs in each plume is strong. Vertical asymmetry is observed between the upper and the lower parts of the plumes, with the upper part having greater intermittency (i.e. the probability that no plume material is present) and a more pronounced tail to the concentration probability distribution. This asymmetry tends to diminish at greater distances from the source but occurs in both buoyant and neutral plumes and is believed to be associated with the ‘bending-over’ of the emission in the cross-flow and the vortex pair that this generates. The results allowed us to identify three phases in plume development. The first, very near the stack, is dominated by turbulence generated within the plume and characterised by concentration spectra with distinct peaks corresponding to scales comparable with those of the counter-rotating vortex pair. A second phase follows at somewhat greater distances downwind, in which there are significant contributions to the concentration fluctuations from both the turbulence internal to the plume and the external turbulence. The third phase is one in which the concentration fluctuations appear to be controlled by the external turbulence present in the ambient flow.     

The similarity solution for turbulent mixing of two-layer stratified fluid

Experiments are reviewed in which a two-layer salt-stratified tank of water was mixed by turbulence. The density profile began as a single step and evolved to a smooth mixed profile. The turbulence was generated by many excursions of a horizontally moving vertical rod with Richardson number Ri >  0.9 and Reynolds Number Re  >  600. There was almost perfect collapse of all the profiles to one universal profile as a function of a similarity variable. We develop a theoretical model for a simple mixing law with a buoyancy flux that is a function of internal Richardson number Rii. A similarity equation is found. A flux law that increases with small Rii and decreases with large Rii is considered next. Since no analytical solution is known, the similarity concept is tested by numerically integrating the equations in space and time. With buoyancy flux monotonically increasing with internal Richardson number, the similarity approach is valid for a profile starting from a slightly smoothed step. However, a shock forms for a mixing law with higher initial Rii (so that buoyancy flux decreases with Richardson number) and the similarity approach is invalid for those initial conditions.     

Flow structure and turbulence characteristics downstream of a spanwise suspended linear array

Laboratory experiments are conducted to quantify the mean flow structure and turbulence properties downstream of a spanwise suspended linear array in a uniform ambient water flow using Particle Tracking Velocimetry. Eighteen experimental scenarios, with four depth ratios (array depth to water column depth) of 0.35, 0.52, 0.78, and 0.95 and bulk Reynolds number (length scale is the array depth) from 11,600 to 68,170, are investigated. Three sub-layers form downstream of the array: (1) an internal wake zone, where the time-averaged velocity decreases with increasing distance downstream, (2) a shear layer which increases in vertical extent with increasing distance downstream of the array, and the rate of the increase is independent of the bulk Reynolds number or the depth ratio, and (3) an external wake layer with enhanced velocity under the array. The location of the shear layer is dependent on the depth ratio. The spatially averaged and normalized TKE of the wake has a short production region, followed by a decay region which is comparable to grid turbulence decay and is dependent on the depth ratio. The results suggest that the shear layer increases the transfer of horizontal momentum into the internal wake zone from the fluid outside of the array and that the turbulence in the internal wake zone can be modeled similarly to that of grid turbulence.     

Stratified flow interactions with a suspended canopy

Field observations of the interactions between a stratified flow and a canopy suspended from the free surface above a solid boundary are described and analysed. Data were recorded in and around the canopy formed by a large long-line mussel farm. The canopy causes a partial blockage of the water flow, reducing velocities in the upper water column. Deceleration of the approaching flow results in a deepening of isopycnals upstream of the canopy. Energy considerations show that the potential for an approaching stratified flow to be diverted beneath a porous canopy is indicated by a densimetric Froude number. Strong stratification or low-velocities inhibit vertical diversion beneath the canopy, instead favouring a horizontal diversion around the sides. The effect on vertical mixing is also considered with a shear layer generated beneath the canopy and turbulence generated from drag within the canopy. In the observations, stratification is shown to be of sufficient strength to limit the effectiveness of the first mixing process, while the turbulence within the canopy is likely to enhance vertical exchange. Velocity and temperature microstructure measurements are used to investigate the effect of the canopy on turbulent dissipation and show that dissipation is enhanced within the canopy.     

Simulation of Ekman Boundary Layers by Large Eddy Model with Dynamic Mixed Subfilter Closure

Theoretical analysis of boundary layer turbulence has suggested a feasibility of sufficiently accurate turbulence resolving simulations at relatively coarse meshes. However, large eddy simulation (LES) codes, which employ traditional eddy-viscosity turbulence closures, fail to provide adequate turbulence statistics at coarse meshes especially within a surface layer. Manual tuning of parameters in these turbulence closures may correct low order turbulence statistics but severely harms spectra of turbulence kinetic energy (TKE). For more than decade, engineering LES codes successfully employ dynamic turbulence closures. A dynamic Smagorinsky turbulence closure (DSM) has been already tried in environmental LES. The DSM is able to provide adequate turbulence statistics at coarse meshes but it is not completely consistent with the LES equations. This paper investigates applicability of an advanced dynamic mixed turbulence closure (DMM) to simulations of Ekman boundary layers of high Reynolds number flows. The DMM differs from the DSM by explicit calculation of the Leonard term in the turbulence stress tensor. The Horizontal Array Turbulence Study (HATS) field program has revealed that the Leonard term is indeed an important component of the real turbulence stress tensor. This paper presents validation of a new LES code LESNIC. The study shows that the LES code with the DMM provides rather accurate low order turbulence statistics and the TKE spectra at very coarse meshes. These coarse LES maintain more energetic small scale fluctuations of velocity especially within the surface layer. This is critically important for success of simulations. Accurate representation of higher order turbulence statistics, however, requires essentially better LES resolution. The study also shows that LES of the Ekman boundary layer cannot be directly compared with conventionally neutral atmospheric boundary layers. The depth of the boundary layer is an important scaling parameter for turbulence statistics.     

Turbulence structure and coherent vortices in open-channel flows with wind-induced water waves

When wind-induced water waves appear over the free-surface flows such as natural rivers and artificial channels, large amounts of oxygen gas and heat are transported toward the river bed through the interface between water and wind layers. In contrast, a bed region is a kind of turbulent boundary layer, in which turbulence generation and its transport is promoted by the production of bed shear stress. In particular, coherent hairpin vortices, together with strong ejection events toward the outer part of the layer, promote mass and momentum exchanges between the inner and outer layers. It is inferred that such a near-bed turbulence may be influenced significantly by these air–water interfacial fluctuations accompanied with free-surface velocity shear and wind-induced water waves. However, these wind effects on the wall-turbulence structure are less understood. To address these exciting and challenging topics, we conducted particle imagery velocimetry (PIV) measurements in open-channel flows combined with air flows, and furthermore the present measured data allows us to investigate the effects of air–water interactions on turbulence structure through the whole depth region.     

Experimental analysis of Froude number effect on air entrainment in the hydraulic jump

A hydraulic jump is characterized by strong energy dissipation and mixing, large-scale turbulence, air entrainment, waves, and spray. Despite recent pertinent studies, the interaction between air bubbles diffusion and momentum transfer is not completely understood. The objective of this paper is to present experimental results from new measurements performed in a rectangular horizontal flume with partially developed inflow conditions. The vertical distributions of the void fraction and the air bubbles count rate were recorded for inflow Froude number Fr ₁ in the range from 5.2 to 14.3. Rapid detrainment process was observed near the jump toe, whereas the structure of the air diffusion layer was clearly observed over longer distances. These new data were compared with previous data generally collected at lower Froude numbers. The comparison demonstrated that, at a fixed distance from the jump toe, the maximum void fraction C _max increases with the increasing Fr ₁. The vertical locations of the maximum void fraction and bubble count rate were consistent with previous studies. Finally, an empirical correlation between the upper boundary of the air diffusion layer and the distance from the impingement point was derived.     

Modelling hydraulic jump using the bubbly two-phase flow method

Hydraulic jumps have complex flow structures, characterised by strong turbulence and large air contents. It is difficult to numerically predict the flows. It is necessary to bolster the existing computer models to emphasise the gas phase in hydraulic jumps, and avoid the pitfall of treating the phenomenon as a single-phase water flow. This paper aims to improve predictions of hydraulic jumps as bubbly two-phase flow. We allow for airflow above the free surface and air mass entrained across it. We use the Reynolds-averaged Navier–Stokes equations to describe fluid motion, the volume of fluid method to track the interface, and the k–ε model for turbulence closure. A shear layer is shown to form between the bottom jet flow and the upper recirculation flow. The key to success in predicting the jet flow lies in formulating appropriate bottom boundary conditions. The majority of entrained air bubbles are advected downstream through the shear layer. Predictions of the recirculation region’s length and air volume fraction within the layer are validated by available measurements. The predictions show a linear growth of the shear layer. There is strong turbulence at the impingement, and the bulk of the turbulence kinetic energy is advected to the recirculation region via the shear layer. The predicted bottom-shear-stress distribution, with a peak value upstream of the toe of the jump and a decaying trend downstream, is realistic. This paper reveals a significant transient bottom shear stress associated with temporal fluctuations of mainly flow velocity in the jump. The prediction method discussed is useful for modelling hydraulic jumps and advancing the understanding of the complex flow phenomenon.     

Are breaking waves,bores, surges and jumps the same flow?

The flow structure in the aerated region of the roller generated by breaking waves remains a great challenge to study, with large quantities of entrained air and turbulence interactions making it very difficult to investigate in details. A number of analogies were proposed between breaking waves in deep or shallow water, tidal bores and hydraulic jumps. Many numerical models used to simulate waves in the surf zone do not implicitly simulate the breaking process of the waves, but are required to parameterise the wave-breaking effects, thus relying on experimental data. Analogies are also assumed to quantify the roller dynamics and the energy dissipation. The scope of this paper is to review the different analogies proposed in the literature and to discuss current practices. A thorough survey is offered and a discussion is developed an aimed at improving the use of possible breaking proxies. The most recent data are revisited and scrutinised for the use of most advanced numerical models to educe the surf zone hydrodynamics. In particular, the roller dynamics and geometrical characteristics are discussed. An open discussion is proposed to explore the actual practices and propose perspectives based on the most appropriate analogy, namely the tidal bore.     

Physical and numerical modelling of tsunami generation by a moving obstacle at the bottom boundary

This paper presents a study of the waves generated by a solid block landslide moving along a horizontal boundary. The landslide was controlled using a mechanical system in a series of physical experiments, and laser-induced fluorescence measurements resolved both spatial and temporal variations in the free surface elevation. During its constant-velocity motion, the landslide transferred energy into ‘trapped’ offshore-propagating waves within a narrow frequency band. The wave trapping is demonstrated by investigating the wave dispersion characteristics using a two-dimensional Fourier Transform. The first of the trailing waves broke at Froude numbers greater than or equal to 0.625. The parametric dependence of the largest-amplitude waves and the potential energy within the wave field are discussed. The experimental results were compared to the predictions of an incompressible Navier–Stokes solver with and without turbulence models. The numerical model under-predicted the measured wave amplitudes, although it accurately predicted the measured wave phasing. The turbulent model more accurately predicted the shapes of the trailing waves. Both experimental and numerical results confirmed that investigations into wave generation by submerged objects moving at constant velocity should also consider the initial acceleration of the object, as this affects the overall evolution of the wave field. The applicability of the horizontal-boundary results to more realistic field scenarios is discussed.     

Growth and damping of interfacial waves on a diffuse interface

The resonant interaction of surface and internal waves produces a nonlinear mechanism for energy transfer among wave components in oceans, lakes, and estuaries. In many field situations, the stratification may be well approximated by a two-layer fluid with a diffuse interface. The growth and damping rates of sub-harmonic interfacial waves generated by a surface wave through a three-wave resonant interaction are measured in the laboratory. These measurements are compared with theoretical predictions. A diffuse interface reduces the damping rate and increases the growth rate. The predicted growth rate provides excellent comparison with the laboratory measurements. The inclusion of the effects of a diffuse interface significantly improve the comparison.     

Model of turbulence penetration into a suspension layer on a sediment bed and effect on vertical solute transfer

The vertical diffusional mass (solute) transfer through a suspended sediment layer, e.g. at the bottom of a lake, reservoir or estuary, by the propagation of velocity fluctuations from above was investigated. The attenuation of the velocity fluctuations in the suspension layer and the associated effect on solute transfer through the suspension layer was simulated. To represent large eddies traveling downstream in water over a high-concentration suspended sediment layer, a streamwise velocity fluctuation moving in downstream direction was imposed along the upper boundary of the suspension layer. Velocity fluctuations and downstream velocity were normalized by the shearvelocity (U_*) at the top of the suspension layer. Streamwise and vertical velocity components inside the suspension layer, were obtained from the 2-D continuity and the Navier–Stokes equations. The persistence of turbulence with depth—as it penetrates from the overlying water into the suspension layer—was found to depend on its amplitude, its period, and on the apparent viscosity of the suspension. The turbulence was found to propagate efficiently into the suspension layer when its frequency is low, and the apparent viscosity of the suspension is high. Effects on vertical mass transfer were parameterized by penetration depth and effective diffusion coefficient, and related to apparent viscosity of the suspension, Schmidt number and shear velocity on top of the suspension layer. The enhancement of turbulence penetration by viscosity is similar to the flow near an oscillating flat plate (Stokes’ second problem), but is opposite to turbulence penetration into a stationary porous and permeable sediment bed. The information is applicable to water quality modeling mear the sediment/water interface of lakes, river impoundments and estuaries.     

Blockage of saline intrusions in restricted,two-layer exchange flows across a submerged sill obstruction

Results are presented from a series of large-scale experiments investigating the internal and near-bed dynamics of bi-directional stratified flows with a net-barotropic component across a submerged, trapezoidal, sill obstruction. High-resolution velocity and density profiles are obtained in the vicinity of the obstruction to observe internal-flow dynamics under a range of parametric forcing conditions (i.e. variable saline and fresh water volume fluxes; density differences; sill obstruction submergence depths). Detailed synoptic velocity fields are measured across the sill crest using 2D particle image velocimetry, while the density structure of the two-layer exchange flows is measured using micro-conductivity probes at several sill locations. These measurements are designed to aid qualitative and quantitative interpretation of the internal-flow processes associated with the lower saline intrusion layer blockage conditions, and indicate that the primary mechanism for this blockage is mass exchange from the saline intrusion layer due to significant interfacial mixing and entrainment under dominant, net-barotropic, flow conditions in the upper freshwater layer. This interfacial mixing is quantified by considering both the isopycnal separation of vertically-sorted density profiles across the sill, as well as calculation of corresponding Thorpe overturning length scales. Analysis of the synoptic velocity fields and density profiles also indicates that the net exchange flow conditions remain subcritical (G < 1) across the sill for all parametric conditions tested. An analytical two-layer exchange flow model is then developed to include frictional and entrainment effects, both of which are needed to account for turbulent stresses and saline entrainment into the upper freshwater layer. The experimental results are used to validate two key model parameters: (1) the internal-flow head loss associated with boundary friction and interfacial shear; and (2) the mass exchange from the lower saline layer into the upper fresh layer due to entrainment.     

OHP–APT 2002 gravity wave campaign: waves,turbulence and forecasts

In July 2002, a multi-sensor campaign was conducted in southern France to investigate the hypothesized connection between gravity waves and optical turbulence. A generalized scidar (GS) was mounted on the 1.93 m diameter telescope at l’Observatoire de Haute-Provence (OHP). The GS provides continuous profiles of optical turbulence with 300 m vertical resolution from telescope altitude up to 25 km. Thermosondes, which provide in situ measurement of optical turbulence by measuring temperature variance, were launched at the OHP site and at a site approximately 20 km west-northwest of OHP. Gravity wave activity was deduced from temperature and wind velocity measured by radiosondes, which are part of the thermosonde system. In this paper, gravity waves were analyzed using techniques for simple two-dimensional mountain waves, with only fair results. Mesoscale models were run at moderately high resolution for the period. The forecasts were analyzed for wave activity, and a post processor model was used to diagnose the optical turbulence. Mountain waves were evident in the forecast, but quantitative comparison showed the forecast to be inadequate in predicting wave strength. The forecast optical turbulence was in fair agreement with measurements with notable exceptions. This paper is declared a work of the U.S. Government and is not subject to copyright protection in the United States. The U.S. Government’s right to retain a non-exclusive, royalty-free license in and to any copyright is acknowledged. Presented at the Turbulence and Waves in Stably Stratified Atmospheric Shear Flows: Measurements, High Resolution Simulations and Numerical Prediction Challenges Conference, 13–15 September 2004, Lighthill Institute of Mathematical Sciences, University College London, London, UK.     

Modelling the propagation of an internal solitary wave across double ridges and a shelf-slope

Evolution of the internal solitary waves (ISWs) in the northern South China Sea (SCS) has recently attracted the attention of many oceanographers in Taiwan and the United States. These ISWs are believed to have been induced by a branch of the Kuroshio current over Luzon Strait, which propagates westward over two ridges in the Luzon Strait between Taiwan and the Philippines, and further onto the continental margin with a shelf-slope in the SCS. This paper presents some preliminary results for the evolution of a depression ISW across two triangular obstacles using numerical modelling and laboratory experiments. The experimental results confirm that the intervals and relative height between the two obstacles are important factors in the interaction of an ISW with the obstacles. However, in the case of the movement of an ISW of depression-type across the Luzon Strait, the effect of the two ridges on the characteristics of the ISW might be less significant than that from the shelf-slope, due to the variations in relative water depth. Results from numerical experiments also show that the amplitude of an ISW can be augmented once the wave commences its contact with a shelf-slope, where an internal hydraulic jump and wave breaking with vortex motion are evident in the laboratory experiments. Eventually, an ISW of depression-type could become an elevation-type at the edge of the continental shelf landwards beyond the turning point, where the upper layer is larger than the bottom layer in a stratified water column.     

The growth response of a model Gymnodinium splendens in stationary and wavy water columns

A computer model is used to investigate the simulated growth of a theoretical dinoflagellate resembling Gymnodinium splendens in response to a variety of field conditions. Literature data on G. splendens are combined with probable estimates of organism response (where direct data are lacking) to yield light-and temperature-dependent production curves. These production curves are superimposed on a physical model characterized by a diurnally variable light cycle, by a two-layered water column (16°C water overlaying 12°C water) of variable layer thicknesses, and by variable extinction coefficients in the upper layer. The water column is either stationary or perturbed by a semidiurnal (12.4 h) internal wave. Organism behavior ranges from the continuous occupation of selected strata (stationary or wavy) to diurnal vertical migrations within the upper layer or across the thermocline. In stationary water columns, species patchiness depends on spatial differences in the depth preferences of nonmigrating organisms or in the details of the behavior of migrating organisms. In water columns perturbed by a semidiurnal internal wave, spatial differences in the phase relationship between the wave form and daylight supplement organism behavior as a source of patchiness. The models result in their most complex spatial patterns when a population migrates through a thermocline perturbed by a semidiurnal internal wave.University of Texas Marine Science Institute Contribution No. 280.     

A Simple Model to Determine In Situ Mixing Height Growth from Surface Measurements

A generic In Situ Mixing Height Growth (IMG) model, capable of predicting the real-time growth of the mixed layer and its diurnal evolution from routinely observed simple surface meteorological is developed. The algorithm for the determination of temporally growing daytime mixing height includes both convective and mechanical turbulence contributions. It accounts for neutral as well as height varying potential temperature gradients above the mixed layer. For thermally stable and mechanically dominated unstable night time Atmospheric Boundary Layer (ABL) the module uses similarity formulae based on the wind velocity [1]), the Monin—Obukhov length [2], and the Coriolis parameter. In the convective case simple slab model is integrated, based on initial lapse rate and the surface heat flux. The lapse rate is evaluated on the basis of vertical atmospheric stability, surface type and surface temperature. This differentiates the IMG model from other existing mixing height models that need initial measured lapse rate for calculation. IMG model is site specific as it calculates the radiative incoming heat flux depending on the solar declination estimates based on-site latitude and longitude. The IMG model is applied to calculate mixing height for India by using surface data (viz. wind speed, surface temperature, surface type) from 152 surface meteorological stations. Results have been evaluated with radiosonde mixing height data procured from 18 upper air stations. Sensitivity analysis of the model with respect to various parameters is performed. The model is formulated after reviewing presently available radiosonde mixing height data in India and can satisfactorily provide an alternative means of estimating mixing height for air pollution dispersion models.