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A Detailed Comparison of a Range of Three-Dimensional Models of the M2 Tide in the Faeroe–Shetland Channel and Northern North Sea

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  • 1 Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside, United Kingdom
  • | 2 Korea Ocean Research and Development Institute, Seoul, Korea
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Abstract

A three-dimensional hydrodynamic model of the Faeroe–Shetland Channel and northern North Sea is used to investigate the spatial variability of M2 tidal elevations and currents in the region. This area is chosen because it covers a range of water depths. Also, there is a significant database of tidal elevations (namely 41 gauges) and current meters (namely 89 observations) with which comparisons can be made. With the exception of a couple of measurements made at the shelf edge, which may be influenced by the internal tide, namely a 180° phase shift across the thermocline due to a first mode internal tide, the observations correspond to those of a barotropic tide. Two different approaches are used to represent the profile of tidal currents in the vertical. In the first a spectral/functional method is used, while in the second a finite difference grid is applied. A range of parameterizations of vertical eddy viscosity (suitable for deep water regions) are used, from ones in which viscosity is related to the flow field and water depth, to the flow field only, with a final calculation involving a Prandtl mixing length formulation. Calculations with the flow and depth dependent viscosity model show that in deep water, this parameterization leads to an artificially high viscosity and hence to a boundary layer thickness that is too large. Both the Prandtl mixing length model and the one in which viscosity is related to only the flow field give low viscosity in deep water, with tidal current profiles showing a high sheared bottom boundary layer with little shear above this. In shallow water comparable viscosity values and current profiles are computed with all the various parameterizations of eddy viscosity.

On average the mean error in tidal elevation amplitude was 3.6 cm, with a phase error of −8 deg, although there was a bias to underpredict tidal elevations. For tidal currents in general there was a slight bias to overpredict currents, with the magnitude of the semimajor axis being reproduced on average with an rms error of 2.3 cm s−1. A calculation in which the open boundary input was adjusted to give a mean elevation amplitude error of zero, with a phase error of −1.9°E, and no significant bias in the elevations, did however show a bias to underpredict tidal currents, with an rms error of 3.6 cm s−1 in the semimajor axis.

Model calculations showed that the most sensitive test of the model's accuracy was a detailed comparison of tidal current profiles, in the near-bed region. Also the accuracy of computed tidal currents in deep water was determined more by the uncertainty in the boundary forcing to the model than the exact form of the eddy viscosity parameterization.

Corresponding author address: Dr. Alan M. Davies, Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead L43 7RA, United Kingdom. Email: amd@pol.ac.uk

Abstract

A three-dimensional hydrodynamic model of the Faeroe–Shetland Channel and northern North Sea is used to investigate the spatial variability of M2 tidal elevations and currents in the region. This area is chosen because it covers a range of water depths. Also, there is a significant database of tidal elevations (namely 41 gauges) and current meters (namely 89 observations) with which comparisons can be made. With the exception of a couple of measurements made at the shelf edge, which may be influenced by the internal tide, namely a 180° phase shift across the thermocline due to a first mode internal tide, the observations correspond to those of a barotropic tide. Two different approaches are used to represent the profile of tidal currents in the vertical. In the first a spectral/functional method is used, while in the second a finite difference grid is applied. A range of parameterizations of vertical eddy viscosity (suitable for deep water regions) are used, from ones in which viscosity is related to the flow field and water depth, to the flow field only, with a final calculation involving a Prandtl mixing length formulation. Calculations with the flow and depth dependent viscosity model show that in deep water, this parameterization leads to an artificially high viscosity and hence to a boundary layer thickness that is too large. Both the Prandtl mixing length model and the one in which viscosity is related to only the flow field give low viscosity in deep water, with tidal current profiles showing a high sheared bottom boundary layer with little shear above this. In shallow water comparable viscosity values and current profiles are computed with all the various parameterizations of eddy viscosity.

On average the mean error in tidal elevation amplitude was 3.6 cm, with a phase error of −8 deg, although there was a bias to underpredict tidal elevations. For tidal currents in general there was a slight bias to overpredict currents, with the magnitude of the semimajor axis being reproduced on average with an rms error of 2.3 cm s−1. A calculation in which the open boundary input was adjusted to give a mean elevation amplitude error of zero, with a phase error of −1.9°E, and no significant bias in the elevations, did however show a bias to underpredict tidal currents, with an rms error of 3.6 cm s−1 in the semimajor axis.

Model calculations showed that the most sensitive test of the model's accuracy was a detailed comparison of tidal current profiles, in the near-bed region. Also the accuracy of computed tidal currents in deep water was determined more by the uncertainty in the boundary forcing to the model than the exact form of the eddy viscosity parameterization.

Corresponding author address: Dr. Alan M. Davies, Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead L43 7RA, United Kingdom. Email: amd@pol.ac.uk

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