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Turbulence Closure, Steady State, and Collapse into Waves

A new simple two-equation turbulence closure is constructed by hypothesizing that there is an extra energy sink in the turbulent kinetic energy (k) equation representing the transfer of energy from k to internal waves and other nonturbulent motions. This sink neither contributes to the buoyancy flux nor to dissipation, the nonturbulent mode being treated as inviscid. The extra sink is proportional to the squared ratio between the turbulent time scale t ; k/«, with turbulent dissipation rate «, and the buoyancy period T 5 2p/N. With a focus on high‐Reynolds number, spatially homogeneous, stably stratified shear flow away from boundaries, the turbulence is described by equations for a master length scale L ; k3/2/« and the master time scale t. It is assumed that the onset of the collapse of turbulence into nonturbulence occurs at t 5 T. The new theory is almost free of empirical parameters and compares well with laboratory and numerical experiments. Most remarkable is that the model predicts the turbulent Prandtl number, which is generally s 5 s 0/[1 2 (t/T) 2], with s 0 5 1/2, and hence is not a unique function of mean flow variables. Only in structural equilibrium ( 5 0) is the Prandtl t ˙ number a unique function of the gradient Richardson number Rg: s 5 s 0/(1 2 2Rg). These forms of the Prandtl number function immediately determine the flux Richardson number R f 5 Rg/s. Steady state occurs at 5 1/ s Rg 4 with R f 5 1/4, and within structural equilibrium the collapse of turbulence is complete at Rg 5 1/2.

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

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-Hérnandez, R., Hargreaves, J.C., Carretero Albiach, J.C., Luo, W., Sclavo, M., Günther, 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., Sündermann, 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 (<15 m), tidal and wave dynamics interact with significant mutual adjustments and with first-order influence on stress at the sea bed and thereby erosion and suspension processes. 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.

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Modelling and numerical simulation of turbulence, waves and suspended sediments for pre-operational use in coastal seas

The role of small-scale processes in models of coastal seas is reviewed, and the respective uses of vertically integrated and vertically resolving models are described. Although applied with heavily tuned empirical parameters to the Holderness coast a vertically integrated model shows the importance of surface waves for predicting suspended particulate matter (SPM) and their distributions. In formulating a generic vertically resolving module, as kernel, the k– ε turbulence closure has been selected. On a uniform vertical grid this model gave reasonably accurate results for a neutrally stratified channel flow forced by an M 2 tidal wave (Elbe estuary) as well as on a nonuniform grid highly refined in the high-dissipation near-bed region for short-period (8 s) surface waves in a laboratory flume. The model was completed with modules accounting for the effect of waves on the turbulent kinetic energy (TKE) influx at the surface and on the apparent roughness at the bottom. It was finally coupled with different versions of vertically high resolving SPM models. In test applications to the English Channel and to the Sylt–Rømø Bight (Germany/Denmark) the generic model versions performed with sufficient accuracy. However, in both cases: (i) fine tuning of erosion and deposition terms was necessary thus underlining the need for further experimental research towards an improved data base on erodible sediments and SPM; (ii) the parameters of the submodel for the TKE injection by surface waves could not be determined consistently and indicate the existence of a further still hidden parameter; (iii) the technical basis for in situ observations of small-scale processes in the coastal zone needs further improvements and consolidation.

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