David Prandle

The vertical structure of tidal currents and other oscillatory flows

Abstract Assuming a linearized equation of motion in the absence of stratification, the vertical structure of tidal currents is shown to be a function of two dimensionless parameters. The first of these, Y = (ω/E)1/2D, is analogous to an Ekman height with ω the tidal frequency, D the depth, and E the (constant) vertical eddy viscosity. The second parameter, J = (ωE)1/2[(8/3π)kŪ], introduces the effect of a quadratic bed stress through the bed-stress coefficient k and the depth-averaged velocityU. From these it is possible to illustrate the full range of possible vertical structure and to understand the basic scaling laws involved. By assuming E =aU¯D good agreement between theory and observation was found. With this assumption vertical structure reduces to a function of just one parameter, namely kS, where S =Ū2π/Dω is the Strouhal number. By resolving tidal current ellipses into clockwise and anticlockwise rotating components the original theory developed for recti-linear flow can be applied to fully three-dimensional flow. In this way, many of the observed characteristics of current structure in shallow seas may be explained.

The vertical structure of tidal currents

Abstract The accuracy to which the vertical structure of tidal currents can be predicted is examined. Theoretical models for current structure are developed employing (a) a constant eddy viscosity E = e and (b) an eddy viscosity varying linearly with height above the sea bed z; E(z)=βz. By requiring these models to satisfy the commonly accepted quadratic friction law, the condition e>½k is deduced where k is the bed friction coefficient, W a representative velocity and D the depth. The sense of rotation of a current ellipse is shown to be related to the configuration of co-tidal charts. The vertical structure of the current ellipse is illustrated from the theoretical models and the sensitivity of this structure is examined for the following variables: (a) eddy viscosity e or βz, (b) the bed friction parameter kW, (c) rotation of the prescribed pressure gradients and (d) tidal period. While reasonable agreement between observed and calculated current profiles may often be reported, precise agreement is sho...

A new view of near-shore dynamics based on observations from HF Radar

Abstract Since 1984 the OSCR HF Radar system has been used in over 50 deployments to measure near-shore surface currents for both scientific and engineering applications. The enhanced scope, resolution and accuracy of these measurements have yielded new insights into the tidal, wind and density driven dynamics of the near-shore zone. Tidal current ellipses obtained from these radar measurements have been shown to be in good aggrement with values calculated by numerical models both for the predominant constituents and also for higher harmonics. Coherent patterns of wind-forced currents ahve been determined with strong evidence of a “slab-like” surface response. In one deployment, with offshore winds blowing over relatively deep water, this “slab” rotated clockwise at near-inertial frequency. Strong (up to 20cm s−1), persistent surface residual currents are commonly observed, these are almost certainly generated by (small) horizontal density gradients. These observed surface residuals provide ideal data for rigorous testing of 3-D numerical models. With a threatened rise in sea level, HF Radar is well-suited for observing the expected changes in the dynamics of near-shore regions. Continuing development of these radar systems offers exciting prospects of remote sensing of both surface waves and currents. Future applications may extend beyond the near-shore region to measurements along the shelf-edge, in oceanic gyres and for “beach-processes”.

Simple theory for designing tidal power schemes

Abstract Basic parameters governing the design of tidal power schemes are identified and converted to dimensionless form by reference to (i) the mean tidal range and (ii) the surface area of the enclosed basin. Optimum values for these dimensionless parameters are derived and comparison made with actual engineering designs. A theoretical framework is thus established which can be used (i) to make a rudimentary design at any specific location or (ii) to compare and evaluate designs for various locations. Both one-way (flood or ebb) and two-way (flood and ebb) schemes are examined and, theoretically, the two-way scheme is shown to be more efficient. However, in practice, two-way schemes suffer disadvantages arising from (i) two-way flow through both turbines and sluices and (ii) lower average turbine heads. An important dimensional aspect of tidal power schemes is that, while energy extracted is proportional to the tidal amplitude squared, the requisite sluicing area is proportional to the square root of the tidal amplitude. In consequence, sites with large tidal amplitudes are best suited to tidal power development whereas for sites with low tidal amplitudes sluicing costs may be prohibitive.

Development of a model to reproduce observed suspended sediment distributions in the southern North Sea using Principal Component Analysis and Multiple Linear Regression

As part of the NERC North Sea Project (1987–1992), concentrations of fine suspended particulate matter were determined at over 100 locations on 15 monthly survey cruises. Each of these monthly data sets were interpolated to provide continuous “synoptic” representations over the southern North Sea (south of 56°N). Here, statistical techniques are used in conjunction with numerical model simulations to interpret these data. Principal Component Analysis is performed on these monthly series of observations to locate statistically significant sources and sinks of suspended particulate matter. This analysis reveals the month-to-month variability of a primary source, accounting for around 80% of the total variation in the observations, and located in the vicinity of The Wash estuary and adjacent coast. A numerical dispersion model is then developed to simulate the erosion, settling and transport of fine suspended sediment from this source. The formulations and associated coefficients used to describe re-erosion and settling rates are then fine-tuned by comparison with observations. In the second part of this study, the sediment model developed is used to simulate dispersion from discrete sources. Thence a Multiple Linear Regression technique, as described by McManus and Prandle (1994) (in Marine Pollution Bulletin28, 451–455), is used to fit these modelled dispersion patterns to the original observations to determine rates of sediment supply from these discrete sources. This technique reveals that (i) riverine sources are not statistically significant and (ii) the East Anglian sources are small in magnitude compared with the Dover Strait and northern North Sea sources. Further analysis explains this apparent contradiction with the first part of the study by illustrating how the longer effective flushing times of the two coastal sources amplifies their net contribution to the suspended particulate matter distribution in the southern North Sea. Estimates of mean annual supply from the statistically significant sources (in 106 t) are Dover Strait, 44.4, northern North Sea, 41.7, The Wash, 3.2, and Suffolk Coast, 0.7. Comparable estimates published in the 1993 North Sea Quality Status Report (North Sea Task Force, Olsen and Olsen, Denmark) are generally in reasonable agreement. Likewise, both the location and rates of supply from the sources determined in this study are in broad agreement with the earlier estimates of McCave (1987) (in Journal of the Geological Society144, 149–152). Thus, the success achieved provides encouragement towards the goal of developing robust suspended sediment models of shelf seas.

The dynamics of nearshore surface currents generated by tides, wind and horizontal density gradients

Abstract Nearshore surface currents measured over 23 days by HF Radar are analysed. These data are equivalent to 100 current meter deployments spaced approximately 1.5 km apart over an area 15 km square. The semidiurnal tidal constituents predominate and exhibit an even coherent propagation through the region. The higher harmonic tides are small, indicating little non-linear interaction—a consequence of the relatively deep water (20–50 m). This low level of interaction also enables the wind-driven component to be clearly separated, thereby revealing dynamics never previously identified with such clarity. The time-varying (non-tidal) residual currents are shown to be primarily due to wind forcing, with the surface currents veering up to 45° to the right of the wind at a magnitude of between 1 and 2% of the wind speed. To reproduce these features with a simple steady-state analytical model requires a vertical eddy viscosity of approximately 0.01 m 2 s −1 . Using an empirical orthogonal function analysis, these wind-driven residuals are shown to move in a uniform slab-like manner with the direction rotating clockwise at a rate close to the inertial frequency (as suggested by the analytical model). The persistence of winds towards the northeast over the survey period generated significant net residual surface currents. An additional (non-wind-driven) net residual was also evident and this was shown to be consistent with forcing by (depth and time-averaged) horizontal density gradients with a vertical eddy viscosity varying linearly from 0.01 m 2 s −1 at the bed to zero at the surface. Since this eddy viscosity formulation is close to that derived for the wind-driven currents, the possibility exists of using HF Radar measurements of surface currents to determine appropriate vertical eddy viscosity coefficients for 3-D models.

Tide, wave and suspended sediment modelling on an open coast — Holderness

Abstract An intensive series of observations off the Holderness coast was followed by a related set of modelling applications. Observations included: aircraft and satellite remote sensing, H.F. and X-band radar, ship surveys and in situ instruments on the sea bed and at the sea surface. These observations aimed to monitor, over three successive winter periods, the dynamics and sediment distributions in the vicinity of this rapidly eroding coastline. Associated modelling applications included components simulating: (i) tides and surge currents; (ii) wave evolution; (iii) vertical distributions of turbulence and SPM (suspended particulate matter) and (iv) resulting spatial patterns of sediment transport in the region. Simulations of tidal currents confirmed the accuracy of such models, given accurate fine-resolution bathymetry and appropriate boundary conditions. New developments of WAM, the spectral wave model required for fine-resolution applications in shallow water (described by Monbaliu et al. [Monbaliu, J., Padilla-Hernandez, R., Hargreaves, J.C., Carretero Albiach, J.C., Luo, W., Sclavo, M., Gunther, H., 2000. The spectral wave model WAM adapted for applications with high spatial resolution. This volume.]) are tested here. A number of additional features pertaining to shallow water are revealed including the sensitivity to specification of wind directions and the excessive temporal spreading of short-lived distant events. Likewise, the application of the generic single-point models for vertical profiles of turbulence and SPM (described by Baumert et al. [Baumert, H., Chapalain, G., Smaoui, H., McManus, J.P., Yagi, H., Regener, M., Sundermann, J., Szilagy, B., 2000. Modelling and numerical simulation of turbulence, waves and suspended sediment for pre-operational use in coastal seas. This volume]), are tested and also shown to be appropriate for simulating localised resuspension of SPM. This simulation also illustrates how, in shallow water ( Some preliminary simulations of net sediment movement are included, involving an integration of the above effects. These simulations emphasise how, in all but the shallowest water, the mobility of coarse grain sediments is limited to occasions of extreme waves. By contrast, the movement of fine sediments follows that of the residual tidal current streamlines, i.e., primarily longshore with attendant cross-shore dispersion. However, significant variation between closely-spaced observations indicates the irregularity and complexity of such distributions. It is concluded that because of the inability to prescribe the spatial distribution of available surficial sediments (including size distributions) such simulations can only be expected to reproduce the essential statistical characteristics of SPM concentrations. The availability of extensive remote sensing or in situ data can help to circumvent this problem.

Modelling approaches for coastal simulation based on cellular automata: the need and potential

The paper summarizes the theoretical and practical needs for cellular automata (CA)-type models in coastal simulation, and describes early steps in the development of a CA-based model for estuarine sedimentation. It describes the key approaches and formulae used for tidal, wave and sediment processes in a prototype integrated cellular model for coastal simulation designed to simulate estuary sedimentary responses during the tidal cycle in the short-term and climate driven changes in sea-level in the long-term. Results of simple model testing for both one-dimensional and two-dimensional models, and a preliminary parameterization for the Blackwater Estuary, UK, are shown. These reveal a good degree of success in using a CA-type model for water and sediment transport as a function of water level and wave height, but tidal current vectors are not effectively simulated in the approach used. The research confirms that a CA-type model for the estuarine sediment system is feasible, with a real prospect for coupling to existing catchment and nearshore beach/cliff models to produce integrated coastal simulators of sediment response to climate, sea-level change and human actions.

The influence of bed friction and vertical eddy viscosity on tidal propagation

Abstract Single-point solutions to the horizontal momentum equation (describing tidal propagation) are used to illustrate the influence of both bottom friction and vertical eddy viscosity. These influences determine whether a simplified vertically-averaged, two-dimensional model will adequately simulate tidal propagation, or where a fully three-dimensional model is required. The effective influences of these two terms is sensitive to both latitude and tidal frequency, in addition to current speed and water depth - thus complicating the a-priori selection of a 2-D or 3-D model. The present study shows that, for applications concerned primarily with prediction of water levels and depth-integrated tidal fluxes, 2-D models are generally entirely adequate in water depths exceeding 50 m—a consequence of small frictional influence. However, exceptions to this conclusion may arise at latitudes corresponding to the inertial frequencies, i.e. λ>70° for semi-diurnal constituents and 27°≤λ≤30° for diurnal constituents. Analytical solutions can be used to augment 2-D model results providing reliable qualitative descriptions of variations in current profiles. However, for applications where detailed current profiles are important, including accurate representation of the near-bed velocity and associated erosional stress, 3-D models with detailed descriptions of turbulence structure are required. The use of 3-D models for shallow estuaries and bays should avoid the necessity of empirical adjustment to the dissipation processes that are poorly represented by a quadratic-type law based on depth-averaged velocities. The above conclusions apply to the propagation of tides in the absence of pronounced effects of wind forcing or density gradients.

Introduction: Operational oceanography in coastal waters

Abstract Operational oceanography embraces hindcasting, nowcasting and forecasting of parameters from physics to ecology on scales from global to coastal. While coastal engineers are familiar with everyday requirements for forecasting tides, surges and waves, the wider linkages are less immediately evident. This introduction aims to articulate these linkages — highlighting both the common interests of oceanographers and engineers and the diversities in methodologies employed to address these on global to localised scales. The European Commission MAST III project, PROMISE (PRe-Operational Modelling In the Seas of Europe), aimed to expedite the development of operational oceanography. This introduction provides the background to this project, explaining the formulation of its objectives and indicating where descriptions of the related approaches are described in subsequent chapters.