Tidal mixing in sill regions: influence of sill depth and aspect ratio

A non-hydrostatic model in cross-sectional form with an idealized sill is used to examine the influence of sill depth (h _s) and aspect ratio upon internal motion. The model is forced with a barotropic tide and internal waves and mixing occurs at the sill. Calculations using a wide sill and quantifying the response using power spectra show that for a given tidal forcing namely Froude number F _r as the sill depth (h _s) increases the lee wave response and vertical mixing decrease. This is because of a reduction in across sill velocity U _s due to increased depth. Calculations show that the sill Froude number F _s based on sill depth and across sill velocity is one parameter that controls the response at the sill. At low F _s (namely F _s ≪ 1) in the wide sill case, there is little lee wave production, and the response is in terms of internal tides. At high F _s, calculations with a narrow sill show that for a given F _s value, the lee wave response and internal mixing increase with increasing aspect ratio. Calculations using a narrow sill with constant U _s show that for small values of h _s, a near surface mixed layer can occur on the downstream side of the sill. For large values of h _s, a thick well-mixed bottom boundary layer occurs due to turbulence produced by the lee waves at the seabed. For intermediate values of h _s, “internal mixing” dominates the solution and controls across thermocline mixing.     

The effects of large and small-scale topography upon internal waves and implications for tidally induced mixing in sill regions

A free surface non-hydrostatic model in a cross-sectional form, namely, two-dimensional, in the vertical is used to examine the role of larger-scale topography, namely, sill width, and smaller scale topography, namely, ripples on the sill upon internal wave generation and mixing in sill regions. The present work is set in the context of earlier work and the wider literature in order to emphasise the problems of simulating mixing in hydrographic models. Highlights from previous calculations and references to the literature for detail, together with new results presented here with smooth and “ripple” topography, are used to show that an idealised cross-sectional model can reproduce the dominant features found in observations at the Loch Etive sill. Calculations show that on both the short and long time scales, the presence of small-scale “ripple” topography influence the mixing and associated Richardson number distribution in the sill region. Subsequent calculations in which the position and form of the small-scale sill topography is varied show for the first time that it is the small-scale topography near the sill crest that is particularly important in enhancing mid-water mixing on the lee side of the sill. Both short-term and longer-term calculations with a reduced sill width and associated time series show that as the sill width is reduced, the non-linear response of the system increases. In addition, Richardson number plots show that the region of critical Richardson number, and hence enhanced mixing, increases with time and a reduction in sill width. Calculations in which buoyancy frequency N varies through the vertical show that buoyancy frequency close to the top of the sill is primarily controlling mixing rather than its mean value. Hence, a Froude number based on sill depth and local N is the critical parameter rather than one based on total depth and mean N.     

Influence of bottom frictional effects in sill regions upon lee wave generation and implications for internal mixing

A cross-sectional nonhydrostatic model using idealized sill topography is used to examine the influence of bottom friction upon unsteady lee wave generation and flow in the region of a sill. The implications of changes in shear and lee wave intensity in terms of local mixing are also considered. Motion is induced by a barotropic tidal flow which produces a hydraulic transition, associated with which are convective overturning cells, wave breaking, and unsteady lee waves that give rise to mixing on the lee side of the sill. Calculations show that, as bottom friction is increased, current profiles on the shallow sill crest develop a highly sheared bottom boundary layer. This enhanced current shear changes the downwelling of isotherms downstream of the sill with an associated increase in the hydraulic transition, wave breaking, and convective mixing in the upper part of the water column. Both short and longer time calculations with wide and narrow sills for a number of sill depths and buoyancy frequencies confirm that increasing bottom friction modifies the flow and unsteady lee wave distribution on the downstream side of a sill. Associated with this increase in bottom friction coefficient, there is increased mixing in the upper part of the water column with an associated decrease in the vertical temperature gradient. However, this increase in mixing and decrease in temperature gradient in the upper part of the water column is very different from the conventional change in near-bed temperature gradient produced by increased bottom mixing that occurs in shallow sea regions as the bottom drag coefficient is increased.     

On the influence of stratification and tidal forcing upon mixing in sill regions

A cross-sectional non-hydrostatic model with idealized topography was used to examine the processes influencing tidal mixing in the region of sills. Initial calculations with appropriate parameters for the sill at the entrance to Loch Etive showed that the model could reproduce the main features of the observed mixing in the region. In particular, the hydraulic jump in the sill region was reproduced, as was an intense mid-water jet that was observed to separate from the lee side of the sill. Shear instabilities associated with the jet appeared to be a source of mixing within the thermocline. In addition, internal lee waves were generated on the lee side of the sill, with the observed amplification because of trapping during the flood stage. Their magnitude and hence the mixing increased with increasing Froude number (F _r). In the case of vertically varying buoyancy frequency, its value near the sill top determined the F _r number, with its value below influencing internal waves magnitude at depth. At high F _r values particularly with strong currents, short waves and overturning occurred.     

On the importance of non-hydrostatic processes in determining tidally induced mixing in sill regions

The importance of using a non-hydrostatic model to compute tidally induced mixing and flow in the region of a sill is examined using idealized topography representing the sill at the entrance to Loch Etive. This site is chosen since detailed measurements were recently made there. Calculations are performed with and without the inclusion of non-hydrostatic dynamics using a vertical slice model for a range of sill widths corresponding to typical sill regions. Initial non-hydrostatic calculations showed that the model could reproduce the observed flow characteristics in the region. However, when calculations were performed using the model in hydrostatic form, the significant artificial convective mixing that occurred in order to remove density inversions led to excessively high vertical mixing. This influenced the computed temperature field and the intensity of the current jet that separated from the sill on its lee side. In addition it affected the magnitude and spatial characteristics of the lee waves generated on the lee side of the sill. Calculations with a range of sill widths, showed that as the sill width decreased the difference between the solution computed with the non-hydrostatic and hydrostatic model increased.     

Effect of water depth and the bottom boundary layer upon internal wave generation over abrupt topography

The role of water depth and bottom boundary layer turbulence upon lee-wave generation in sill regions is examined. Their effect upon vertical mixing is also considered. Calculations are performed using a non-hydrostatic model in cross-section form with a specified tidal forcing. Initial calculations in deeper water and a sill height such that the sill top is well removed from the surrounding bed region showed that downstream lee-wave generation and associated mixing increased as bottom friction coefficient k increased. This was associated with an increase in current shear across the sill. However, for a given k, increasing vertical eddy viscosity A _v reduced vertical shear in the across sill velocity, leading to a reduction in lee-wave amplitude and associated mixing. Subsequent calculations using shallower water showed that for a given k and A _v, lee-wave generation was reduced due to the shallower water depth and changes in the bottom boundary layer. However, in this case (unlike in the deepwater case), there is an appreciable bottom current. This gives rise to bottom mixing which in shallow water extends to mid-depth and enhances the mid-water mixing that is found on the lee side of the sill. Final calculations with deeper water but small sill height showed that lee waves could propagate over the sill, thereby reducing their contribution to mixing. In this case, bottom mixing was the major source of mixing which was mainly confined to the near bed region, with little mid-water mixing.     

Numerical modelling of stratified tidal flow over a fjord sill

Field observations of tidally driven stratified flow in the sill area of Knight Inlet (British Columbia) revealed a very complicated structure, which includes solitary waves, upstream bifurcation, hydraulic jump and mixing processes. Recent observations suggest that the flow instabilities on the plunging pycnocline at the lee side of the sill may contribute to solitary wave generation through a subharmonic interaction. The present study reports on a series of numerical experiments of stratified tidal flow in Knight Inlet performed with the help of a fine resolution fully non-linear non-hydrostatic numerical model. The model reproduces all important stages of the baroclinic tidal dynamics observed in Knight Inlet. Results demonstrate that solitary waves are generated apart from the area of hydrodynamic instability. Accelerating tidal flux forms a baroclinic hydraulic jump just above the top of the sill, whereas the bifurcations and zones of shear instabilities are formed downstream of the sill. The first baroclinic mode having the largest velocity escapes from the generation area and propagates upstream, disintegrating further into a packet of solitary waves reviling the classical “non-subharmonic” mechanism of generation. The remaining part of the disturbance (slow baroclinic modes) is arrested by tidal flow and carried away to the lee side of the obstacle, where shear instability, billows and mixing processes are developed. Some sensitivity runs were performed for different value of tidal velocity.     

Numerical studies of flow over a sill: sensitivity of the non-hydrostatic effects to the grid size

A non-hydrostatic terrain-following model in cross sectional form is applied to study the processes in the lee of a sill in an idealized stratified fjord during super-critical tidal inflow. A sequence of numerical studies with horizontal grid sizes in the range from 100 to 1.5625 m are performed. All experiments are repeated using both hydrostatic and non-hydrostatic versions of the model, allowing a systematic study of possible non-hydrostatic pressure effects and also of the sensitivity of these effects to the horizontal grid size. The length scales and periods of the internal waves in the lee of the sill are gradually reduced and the amplitudes of these waves are increased as the grid size is reduced from 100 down to 12.5 m. With a further reduction in grid size, more short time and space scale motions become superimposed on the internal waves. Associated with the internal wave activity, there is a deeper separation point that is fairly robust to all parameters investigated. Another separation point nearer to the top of the sill appears in the numerical results from the high-resolution studies with the non-hydrostatic model. Associated with this shallower separation point, an overturning vortex appears in the same set of numerical solutions. This vortex grows in strength with reduced grid size in the non-hydrostatic experiments. The effects of the non-hydrostatic pressure on the velocity and temperature fields grow with reduced grid size. In the experiments with horizontal grid sizes equal to 100 or 50 m, the non-hydrostatic pressure effects are small. For smaller grid sizes, the time mean velocity and temperature fields are also clearly affected by the non-hydrostatic pressure adjustments.     

Influence of open boundary conditions and sill height upon seiche motion in a gulf

A cross-sectional model of an idealised constant depth gulf with a sill at its entrance, connected to a deep ocean, is used to examine the barotropic and baroclinic response of the region to wind forcing. The role of the oceanic boundary condition is also considered. Calculations show that in the case of a tall sill, where the pycnocline intersects the sill, the baroclinic response of the gulf is similar to that of a lake, and internal waves cannot radiate energy out of the gulf. The barotropic response shows free surface oscillations, with nodes located close to the centre of the oceanic basin and entrance to the gulf, with associated barotropic resonant periods. As the sill height is reduced, baroclinic wave energy is radiated from the gulf into the ocean, and the form of the baroclinic response changes from a standing wave (tall sill) as in a lake to a progressive wave (no sill). The location of sea surface elevation nodes and resonant periods changes as the sill height is reduced. Calculations of the barotropic resonant periods with and without stratification could not determine if they were influenced by the presence of stratification, although published analytical theory suggests that they should be able to when energy is lost from the gulf by internal wave radiation. This inability to detect changes in barotropic resonant period due to stratification effects is due to the small change in resonant frequency produced by baroclinic effects, as shown by analytical results, and the broad peak nature of the computed resonant frequency. In the case of a closed offshore boundary (an offshore island), there is a stronger and narrower energy peak at the resonant frequency than when a barotropic radiation condition is applied. However, the influence of stratification upon the resonant frequency could not be accurately determined. Although the offshore boundary was well removed from the gulf to such an extent that any baroclinic waves reflected from it could not reach the gulf within the integration period, it did, however, slightly influence the gulf baroclinic response due to its influence on the barotropic response.     

An explanation for salinity- and SPM-induced vertical countergradient buoyancy fluxes

Measurements of turbulent fluctuations of velocity, salinity, and suspended particulate matter (SPM) are presented. The data show persistent countergradient buoyancy fluxes. These countergradient fluxes are controlled by the ratio of vertical turbulent kinetic energy (VKE) and available potential energy (APE) terms in the buoyancy flux equation. The onset of countergradient fluxes is found to approximately coincide with larger APE than VKE. It is shown here that the ratio of VKE to APE can be written as the square of a vertical Froude number. This number signifies the onset of the dynamical significance of buoyancy in the transport of mass. That is when motions driven by buoyancy begin to actively determine the vertical turbulent transport of mass. Spectral and quadrant analyses show that the occurrence of countergradient fluxes coincides with a change in the relative importance of turbulent energetic structures and buoyancy-driven motions in the transport of mass. Furthermore, these analyses show that with increasing salinity-induced Richardson number (Ri), countergradient contributions expand to the larger scales of motions and the relative importance of outward and inward interactions increases. At the smaller scales, at moderate Ri, the countergradient buoyancy fluxes are physically associated with an asymmetry in transport of fluid parcels by energetic turbulent motions. At the large scales, at large Ri, the countergradient buoyancy fluxes are physically associated with convective motions induced by buoyancy of incompletely dispersed fluid parcels which have been transported by energetic motions in the past. Moreover, these convective motions induce restratification and enhanced settling of SPM. The latter is generally the result of salinity-induced convective motions, but SPM-induced buoyancy is also found to play a role.     

Wave-driven circulation patterns in the lee of groynes

Surf zone drifters and a current meter were used to study the nearshore circulation patterns in the lee of groynes at Cottesloe Beach and City Beach in Western Australia. The circulation patterns revealed that a persistent re-circulation cell was present in the lee of the groyne which was driven by changes in wave set-up resulting from lower wave heights in the lee of the groyne. The re-circulation consisted of a longshore current directed towards the groyne which was deflected offshore due to groyne resulting in a rip current along the groyne face. The offshore-flowing rip current and the incoming waves converged at the offshore extent of this circulation cell, with the deflection of the rip current parallel to the shoreline and then completing the recirculation through an onshore component. The Eulerian measurements revealed that 55% of the currents on the lee side of the groyne were directed offshore and that these currents had a maximum speed of 2 m s^?1. Spectral analysis of the wave heights and the currents revealed several corresponding peaks in the measured spectral densities with timescales between 12 s and 50 min. Numerical simulations of an idealised beach with a shore-normal groyne were conducted using a circulation model driven by waves, and confirmed the formation of a persistent eddy in the lee of the groyne. Sensitivity studies indicated that the incident wave angle, wave period, and especially the wave height controlled the circulation. The eddy vorticity, a measure of an eddy's strength, increased roughly proportional to an increase in the incident wave energy flux.     

Energy of acoustic signals in a borehole

In the present article, the dependencies of the acoustic signal total energy and the energies of the wave packets of different types of the waves on the elastic parameters and permeability of rocks have been studied. We have considered traditional logging tools containing acoustical monopole source. Calculations were performed in a frequency range of dozens of kilohertz, typical for acoustic well logging. It was shown that in a typical high-velocity formation (v_s > v_f, where v_s and v_f are the velocities of the shear wave in the rock and of the compressional wave in the borehole fluid, respectively), the pseudo-Rayleigh waves, whose elastic properties depend mainly on the shear modulus of the rock, contributed significant energy to the total signal energy in the borehole. The energies of different wave packets depend on the permeability in different ways. The greatest sensitivity to permeability change has been shown by the acoustic signal total energy and the energy of the low-velocity part of the pseudo-Rayleigh wave packet. The theoretical analysis was illustrated by real sonic log data.     

Nearshore circulation over transverse bar and rip morphology with oblique wave forcing

There is a paucity of field data to describe the transition in nearshore circulation between alongshore, meandering and rip current systems. A combination of in‐situ current meters and surf zone drifters are used to characterize the nearshore circulation over a transverse bar and rip morphology at Pensacola Beach, Florida in the presence of relatively low energy oblique waves. Current speeds vary in response to the relative wave height ratio (H_s/h), which defines the degree and extent of breaking over the shoal. In the absence of wave breaking the nearshore circulation was dominated by an alongshore current driven by the oblique waves. As waves begin to break across the shoal (0.2<H_s/ h<0.5) the nearshore circulation is characterized by a meandering alongshore current. As conditions became more dissipative (H_s/h>0.5), the meandering current is replaced by an unsteady rip circulation that moves offshore between the shoals before turning alongshore in the direction of wave advance outside the surf zone. The increase in wave dissipation is associated with an increase in very low frequency (VLF) variations in the current speed across the shoal and in the rip channel that caused the circulation to oscillate between an offshore and an alongshore flow. The unsteady nature of the nearshore circulation is responsible for 55% of all surf zone exits under these more dissipative conditions. In contrast, only 29% of the drifters released from the shoal exited the surf zone and bypassed the adjacent shoal with the alongshore‐meandering current. While the currents had a low velocity (maximum of ~0.4 m s^‐1) and would not pose a significant hazard to the average swimmer, the results of this study suggest that the transverse bar and rip morphology is sufficient to create an alongshore variation in wave dissipation that forces alongshore meandering and low‐energy rip circulation systems under oblique wave forcing. Copyright © 2013 John Wiley & Sons, Ltd.     

Richardson number profiles in laboratory experiments applied to shallow seas

Abstract

A unified analysis is given of the critical conditions for the onset of stratification due to either a vertical or a horizontal buoyancy flux, with tidal or wind stirring.

The critical conditions for the onset of stratification with a horizontal buoyancy flux are found to be of the form of ratios of the tidal slope, or wind setup, to the equivalent surface slope due to the lateral density gradient. These ratios, which are easily determined from sea data, indicate that the profiles of critical flux Richardson Number, averaged over the stirring cycle, are similar to those inferred from the laboratory experiments of Hopfinger and Linden (1982) in which there is zero mean shear turbulence with a stabilising buoyancy flux, and also that the efficiency for the conversion of kinetic energy to potential energy for tidal stirring is similar to that for wind stirring.

The observed much greater efficiency for wind stirring, compared with tidal stirring with a vertical buoyancy flux, is also consistent with the existence of flux Richardson Number profiles in the sea similar to those occurring in the corresponding laboratory experiments. Using the solution of the turbulent kinetic energy equation for the water column, the relative importance of the production of turbulent kinetic energy, and its diffusion by turbulence are assessed, and the critical conditions for the onset of stratification with a vertical buoyancy flux are shown to reduce the classical Simpson—Hunter form.     

Decreasing trend of sediment transfer function of the Upper Yellow River,China, in response to human activity and climate change

Abstract

An index (F_s) for sediment transfer function is introduced, based on the sediment budget at the channel scale. The purpose of this study is two-fold: to gain a deeper insight into how F_s is influenced by natural and human factors, and to provide some new knowledge for decision making in the management of the Upper Yellow River, China. Since 1960, the F_s of the Lanzhou to Toudaoguai reach of the Upper Yellow River shows a decreasing trend. At the drainage basin level, the decreased F_s can be explained by changes in precipitation and air temperature, as well as by a number of variables describing human activity, such as reservoir regulation, water diversion, and soil and water conservation. The higher temperature reduces the transfer function, while the larger runoff coefficient increases it. At the channel level, the decreased F_s can be explained by a number of variables of flow and sediment input. Three countermeasures for restoration of the F_s are suggested.

Editor Z.W. Kundzewicz     

Wave impacts on coastal cliffs: Do bigger waves drive greater ground motion?

Coastal cliff erosion is caused by a combination of marine forcing and sub-aerial processes, but linking cliff erosion to the environmental drivers remains challenging. One key component of these drivers is energy transfer from wave–cliff interaction. The aim of this study is to directly observe cliff ground motion in response to wave impacts at an individual wave scale. Measurements are described from two coastal cliff sites: a 45-minute pilot study in southern California, USA and a 30-day deployment in Taranaki, New Zealand. Seismometers, pressure sensors and video are used to compare cliff-top ground motions with water depth, significant wave height (H_s) and wave impact types to examine cliff ground motion response. Analyses of the dataset demonstrate that individual impact events can be discriminated as discrete events in the seismic signal. Hourly mean ground motion increases with incident H_s, but the largest hourly peak ground motions occurred across a broad range of incident H_s (0.9–3.7 m), including during relatively calm conditions. Mean hourly metrics therefore smooth the short-term dynamics of wave–cliff interaction; hence, to fully assess wave impact energy transfer to cliffs, it is important also to consider peak ground motion. Video analyses showed that the dominant control on peak ground motion magnitude was wave impact type rather than incident H_s. Wave–cliff impacts where breaking occurs directly onto the cliff face consistently produced greater ground motion compared to broken or unbroken wave impacts: breaking, broken and unbroken impacts averaged peak ground motion of 287, 59 and 38 μm s⁻¹, respectively. The results illustrate a novel link between wave impact forcing and cliff ground motion response using individual wave field measurements, and highlight the influence of wave impact type on peak energy transfer to coastal cliffs. © 2019 John Wiley & Sons, Ltd. © 2019 John Wiley & Sons, Ltd.     

Secondary airflow and sediment transport in the lee of a reversing dune

Lee-side windspeed and sediment transport were measured over a small (1·2 m) transverse ridge in the Silver Peak dunefield, west-central Nevada, USA, using an intensive array of 25 cup anemometers and seven total flux traps. During crest-transverse and transporting flow conditions (u_0·3crest ≈ 8·4 m s⁻¹), windspeed near the surface of the lee slope averaged half (48 per cent) that of crest speeds. Dimensionless speeds in the separation zone ranged from 0·2 to 0·8 that of the outer flow (u₁₂). Along the boundary of the separation cell, windspeed increased by 10 per cent of the crest speed before separation. Equilibrium of upper and lower wake regions was not observed by the documented eight dune heights, suggesting that wake recovery may not occur over closely spaced dunes. Sediment transport measured directly on both the lee slope and interdune surfaces averaged approximately 15 per cent of crest inputs. This suggests that a significant amount (c. 70–95 per cent) of sediment transported over the crest moved as fallout. For this data set, flux was approximately proportional to the cube of the near-surface windspeed (u_0·3) and in general there was an order of magnitude difference between flux measured at the crest and that measured within the separation zone. Transport direction in the separation zone was acutely oblique to the incident direction owing to secondary flow deflection. Beyond the interdune, transport direction progressed from oblique to crest-transverse. This indicates that an appreciable amount of sediment may move laterally along the lee slope and interdune corridor under crest-transverse flows. Regarding the grain size and sorting properties of transported sediment, there was no significant difference in mean grain size over the dune, although in general particles were finer and more poorly sorted in the lee. Copyright © 1999 John Wiley & Sons, Ltd.     

Acoustic reflection logging for open and cased holes

In the present work, the waveforms of reflected wave sonic log for open and cased boreholes are calculated. Calculations are performed for a borehole containing an acoustic multipole source (monopole, dipole, or quadrupole). A reflected wave is more efficiently excited at resonant frequencies. These frequencies for all source types are close to the frequencies of oscillations of a fluid column located in an absolutely rigid hollow cylinder. It is shown that the acoustic reverberation is controlled by the acoustic impedance of the rock Z = V_p ρ_s for fixed parameters of the borehole fluid, where V_p is the compressional wave velocity in the rock, and ρ_s is the rock density. This result is correct for all types of acoustic sources (monopole, dipole, or quadrupole). Methods of the waveform processing for determining parameters characterizing the reflected wave are discussed.     

Hydromagnetic waves in a differentially rotating annulus. II. Resistive instabilities

Abstract

In part I of this study (Fearn, 1983b), instabilities of a conducting fluid driven by a toroidal magnetic field B were investigated. As well as confirming the results of a local stability analysis by Acheson (1983), a new resistive mode of instability was found. Here we investigate this mode in more detail and show that instability exists when B(s) has a zero at some radius s=s _c. Then (in the limit of small resistivity) the instability is concentrated in a critical layer centered on s _c . The importance of the region where B is small casts some doubt on the validity of the simplifications made to the momentum equation in I. Calculations were therefore repeated using the full momentum equation. These demonstrate that the neglect of viscous and inertial terms when the mean field is strong does not lead to spurious results even when there are regions where B is small.     

Modelling the influence of small-scale effects upon the larger scale: an oceanographic challenge

The problem of resolving or parameterising small-scale processes in oceanographic models and the extent to which small-scale effects influence the large scale are briefly discussed and illustrated for a number of cases. For tides and surges in near-shore regions, the advantages of using a graded mesh to resolve coastal and estuarine small-scale features are demonstrated in terms of a west coast of Britain unstructured mesh model. The effect of mesh resolution upon the accuracy of the overall solution is illustrated in terms of a finite element model of the Irish Sea and Mersey estuary. For baroclinic motion at high Froude number, the effect of resolving small-scale topography within a non-hydrostatic model is illustrated in terms of tidally induced mixing at a single sill, or two closely spaced sills. The question of how to parameterise small-scale non-linear interaction processes that lead to significant mixing, in a form suitable for coarser grid hydrostatic models, is briefly considered. In addition, the importance of topographically induced mixing that occurs in the oceanic lateral boundary layer, namely, the shelf edge upon the large-scale ocean circulation is discussed together with the implications for coarse grid oceanic climate models. The use of unstructured grids in these models to enhance resolution in shelf-edge regions in a similar manner to that used in storm surge models to enhance near coastal resolution is suggested as a suitable “way forward” in large-scale ocean circulation modelling.