Quadratic Bottom Friction Research Articles

Stationary and Non-Stationary Periodic Flows Mathematical Modelling using Various Vortex Viscosity Models

Introduction. Mathematical modelling of currents is an urgent research topic in the field of hydrodynamics and oceanography. Despite ongoing research in the field of developing accurate and efficient numerical methods for solving Navier-Stokes equations that take into account vortex viscosity, the problems of accurate prediction and control of turbulence remain unresolved. The influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions also remains relevant. The aim of the study is to study the influence of linearized and quadratic bottom friction and two turbulence models on the numerical solution of stationary and non-stationary periodic flows. Special emphasis is placed on comparing numerical results with analytical solutions within the framework of using various models of bottom friction.Materials and Methods. The computational models used in this study are based on a simplified two-dimensional wave model and full three-dimensional Navier-Stokes equations. The classical model of shallow water motion and the 2D model without taking into account dynamic changes in the geometry of the reservoir surface are derived from a system of equations for a spatially inhomogeneous three-dimensional mathematical model of wave hydrodynamics of a shallow reservoir. Analytical solutions were found by linearization of the equations, which obviously has its limitations. A distinction is made between two types of nonlinear effects – nonlinearities caused by higher-order terms in the equations of motion, i. e. terms of advective acceleration and friction, and nonlinear effects caused by geometric nonlinearities, this is due, for example, to different water depths and reservoir widths, which will be important when modelling a real sea.Results. The results of modeling stationary and non-stationary periodic flows in a schematized rectangular basin using linearized bottom friction are presented. The influence of linearization on the numerical solution is investigated in comparison with analytical profiles using models calculating bottom friction in a quadratic formulation. In combination with quadratic bottom friction, two turbulence models are studied: the constant vortex viscosity and the Prandtl mixing length model. The results obtained as a result of three-dimensional modelling are compared with the results of two-dimensional modeling and analytical solutions averaged in depth.Discussion and Conclusion. New approaches to modelling and studying flows with variable vortex viscosity are proposed, including analysis of the influence of linearization and the use of various turbulence models. For the linearized and quadratic formulations of bottom friction, it is proved that the numerical results for the case of stationary flow show great similarity with analytical solutions, since the surface height is much less than the water depth and advection can be neglected. The numerical results for the unsteady flow also show a good agreement with the theory. Unlike analytical solutions, numerical modelling has minor deviations in the long run. The study of flows, within the framework of using various turbulence models, will make it possible to take into account the influence of nonlinear effects in vortex viscosity models on the accuracy of forecasts and their applicability to various flow conditions. The results obtained make it possible to better understand and describe the physical processes occurring in shallow waters. This opens up new possibilities for applying mathematical modelling to predict and analyze the impact of human activities on the marine environment and to solve other problems in the field of oceanology and geophysics.

Effects of tide-surge interactions on the temporal distribution of the peak residual in Hangzhou Bay, China

Hangzhou Bay has a trumpet-shaped geometry and is known for its extremely large tidal range. Understanding tide-surge interactions is essential for coastal defence applications during storm events. Analysis of long-term hourly sea level observation data at three sites and numerical experiments show that a remarkable cluster of peak residuals occurs 4 h before the high tide inside Hangzhou Bay (Zhapu and Ganpu Sites), while a weaker cluster occurs around the low tide outside the bay (Shengshan Site). Further analysis of various sea levels in Hangzhou Bay revealed that the non-linear level inside the bay was greater than that outside the bay, leading to the observed difference in frequency distributions between them. To understand the underlying mechanism, the contributions of each non-linear interaction factor to the tide-surge interactions were distinguished. It was found that quadratic bottom friction is the dominating factor inside Hangzhou Bay, which always peaks approximately 4 h before high tide due to wave deformation. Whereas, outside Hangzhou Bay, advective term and quadratic bottom friction are the main forces, which are much larger than wind stress, and they tend to have peaks around low tide. It is indicated that peak residual arrives synchronously with the peak non-linear interaction term.

High‐Resolution Numerical Modeling of Barotropic Global Ocean Tides for Satellite Gravimetry

AbstractThe recently upgraded barotropic tidal model TiME is employed to study the influence of fundamental tidal processes, the chosen model resolution, and the bathymetric map on the achievable model accuracy, exemplary for the M2 tide. Additionally, the newly introduced pole‐rotation scheme allows to estimate the model’s inherent precision (open ocean rms: 0.90 cm) and enables studies of the Arctic domain without numerical deviations originating from pole cap handling. We find that the smallest open ocean rms with respect to the FES14‐atlas (3.39 cm) is obtained when tidal dissipation is carried out to similar parts by quadratic bottom friction, wave drag, and parametrized eddy‐viscosity. This setting proves versatile to obtaining high accuracy values for a diverse ensemble of additional partial tides. Using the preferred model settings, we show that for certain minor tides it is possible to obtain solutions that are more accurate than results derived with admittance assumptions from data‐constrained tidal atlases. As linear admittance derived minor tides are routinely used for de‐aliasing of satellite gravimetric data, this opens the potential for improving gravity field products by employing the solutions from TiME.

Nonlinear tide-surge interactions in the Río de la Plata Estuary

The nonlinear tide-surge interaction (NTSI) can significantly affect the levels associated with storm surges that threaten coastal areas. In this work, water level observations and numerical simulations are used to investigate those interactions in the large, mighty and socio-economically important Río de la Plata (RdP) estuary. We introduce a novel objective approach to seek evidence of nonlinear interactions in hourly water level records from six tide gauges collected along the coast. Results indicate that NTSI occurs in the RdP and suggest that it becomes more important upstream and along the southern coast of the estuary. CROCO ocean numerical model is used to quantify the interactions, to determine the areas where they are stronger, and to identify their sources. The amplitude of the interaction accounts for 16% of the total water level at the upper RdP, being comparable to tidal amplitude. The quadratic bottom friction is the principal cause of NTSI, and is modulated by the tidal current; therefore the nonlinearity is present all the time and is independent of the wind speed and direction. This is a particularity of this estuary and occurs because, as a consequence of its large width, surge currents are weak in spite of the big volume of water mobilized by the storms. Due to its nature, the interaction attenuates and smoothes the level anomalies due to the surge. Results highlight the need of utilizing a complete NTSI model for the forecast of the surge in the RdP; otherwise, the estuarine dynamic would be misrepresented generating significant forecasts errors (10%–20% at the upper RdP).

Tide-Surge Interaction in the Pearl River Estuary: A Case Study of Typhoon Hato

In this study, the characteristics and mechanisms of tide-surge interaction in the Pearl River Estuary (PRE) during Typhoon Hato in August 2017 are studied in detail using a 3D nearshore hydrodynamic model. The wind field of Typhoon Hato is firstly reconstructed by merging the Holland parametric tropical cyclone model results with the CFSR reanalysis data, which enables the model to reproduce the pure astronomical tides and storm tides well; especially the distinctive oscillation pattern in the measured water levels due to the passage of the typhoon has been captured. Three different types of model runs are conducted in order to separate the water level variations due to the astronomical tide, storm surge and tide-surge interactions in the Pearl River Estuary. Results show the strong tidal modulation of the surge level, as well as alteration of the phase of surge which also changes the peak storm tidal level, in addition to the tidal modulation effects. In order to numerically assess the contributions of three nonlinear processes in the tide-surge interaction and quantify their relative significance, the widely used ``subtraction" approach and a new “addition" approach are tested in this study. The widely used “subtraction'' approach is found unsuitable for the assessment due to the ``rebalance" effect and thus a new “addition'' approach is proposed along with a new indicator to represent the tide-surge interaction, from which more reasonable results are obtained. Detailed analysis using the ``addition'' approach indicates that the quadratic bottom friction, shallow water effect and nonlinear advective effect play the first, second and third most important roles in the tidal-surge interaction in the estuary, respectively.

3D numerical computation of the tidally induced Lagrangian residual current in an idealized bay

A numerical model that solves 3D first-order Lagrangian residual velocity (uL) equations is established by modifying the HAMSOM model. With this model, uL is studied in a wide, idealized bay. The results show that the vertical eddy viscosity term of Stokes’ drift (π1) in the tidal body force determines the overall flow state of uL, and the contribution of the advection term (π2) is responsible for the small correction. In addition, two types of Coriolis effects introduced into the residual current system not only enhance the lateral flow and break the symmetry of the flow regime in the bay but also slightly correct the flow state driven by the entire tidal body force. It is also found by numerical sensitivity experiments that the increase in the aspect ratio δ, implying a decrease in the topographic gradient, can simplify the residual flow state. The increase in tidal amplitude at the open boundary significantly enhances the intensity of uL and causes the residual flow regime to be more complicated in the bay. This can be ascribed to the disproportionate increase in the tidal body force. The proportion of the vertical eddy viscosity term of Stokes’ drift in the tidal body force also varies with the vertical eddy viscosity coefficient, which leads to different residual current states. Compared with the influence of incoming tidal strength on the residual current, the effect of the bottom friction coefficient on the residual current is relatively mild. An increase in the quadratic bottom friction coefficient induces an unbalanced decrease in the tidal body force. Therefore, uL decreases, but the flow regime is more complex. The influence of the nonlinear effect of the bottom friction decreases from the bay head towards the bay mouth. The residual current only changes in magnitude near the bay mouth but changes in pattern near the bay head for different bottom friction coefficients. By keeping the bottom friction coefficient in the zeroth-order tidal equations constant, the sensitivity experiment shows that uL is insensitive to the change in bottom friction coefficient in the governing equations of uL.

Model Development and Hindcast Simulations of NOAA’s Integrated Northern Gulf of Mexico Operational Forecast System

NOAA’s National Ocean Service is upgrading three existing northern Gulf of Mexico (GOM) operational nowcast/forecast systems (OFS) by integrating them into one single system (INGOFS) and developing additional domain coverage to encompass the lower Mississippi River, Lake Pontchartrain, Texas coastal embayments, and Mexican coastal waters. The system will produce real-time nowcast and short-range forecast guidance for water levels, 3-dimensional currents, water temperature, and salinity. INGOFS will be implemented using the Finite Volume Community Ocean Model (FVCOM). This paper describes the model configuration and results from a one-year (2 August 2016–1 August 2017) hindcast simulation. The model grid is composed of about 300,000 nodes and 600,000 elements, and has a spatial resolution ranging from 45 m near the coast to around 10 km on the open ocean boundary. It uses the FVCOM wetting and drying feature, the quadratic bottom friction scheme, and the two-equation model of the Mellor-Yamada Level 2.5 turbulence closure scheme. The hindcast results of water levels, surface temperatures, and salinity were verified by comparing the model time series with in situ observations. The root-mean-squared errors are about 0.08 m for water levels, about 1.1 °C for temperatures, and about 3.7 psu for salinity. The hindcast configuration will be further tested in a nowcast/forecast environment for a one-year period. The upgraded system is anticipated to be in operational production in mid-2020.

The nonlinear effects of the eddy viscosity and the bottom friction on the Lagrangian residual velocity in a narrow model bay

The nonlinear effects of the eddy viscosity and the bottom friction on the Lagrangian residual velocity (LRV) are studied numerically in a narrow model bay. Three groups of the experiments with different eddy viscosity and different forms of the bottom friction are designed and are carried out in the three kinds of the topography. When the eddy viscosity is obtained from a two-equation turbulence closure model, the pattern of the LRV is more complex than that of the time invariant eddy viscosity case and the intensity is from more than 1.3 times to one order smaller than that of the linear eddy viscosity condition. The LRV are also acquired when the eddy viscosity varies from the flood-averaged one to the ebb-averaged one. It is found that when the flood-averaged eddy viscosity is bigger than the ebb-averaged eddy viscosity (flood-dominated asymmetry), the direction of the breadth-averaged LRV and the 3D LRV is nearly opposite to that when the eddy viscosity asymmetry is reverse (ebb-dominated asymmetry). However, the intensity of the LRV for the ebb-dominated case decreases toward the flood-dominated case as the ratio of the maximum depth in the deep channel and the minimum depth on the shoal increases. The different forms of the bottom friction also play a role in the LRV. The structures of the 3D LRV and the depth-integrated LRV are simpler, and the intensity of the LRV is two times smaller when the linear bottom friction is used than those when the quadratic bottom friction is used.

Coastal tomographic mapping of nonlinear tidal currents and residual currents

Depth-averaged current data, which were obtained by coastal acoustic tomography (CAT) July 12–13, 2009 in Zhitouyang Bay on the western side of the East China Sea, are used to estimate the semidiurnal tidal current (M2) as well as its first two overtide currents (M4 and M6). Spatial mean amplitude ratios M2:M4:M6 in the bay are 1.00:0.15:0.11. The shallow-water equations are used to analyze the generation mechanisms of M4 and M6. In the deep area, where water depths are larger than 60m, M4 velocity amplitudes measured by CAT agree well with those predicted by the advection terms in the shallow water equations, indicating that M4 in the deep area is predominantly generated by the advection terms. M6 measured by CAT and M6 predicted by the nonlinear quadratic bottom friction terms agree well in the area where water depths are less than 20m, indicating that friction mechanisms are predominant for generating M6 in the shallow area. In addition, dynamic analysis of the residual currents using the tidally averaged momentum equation shows that spatial mean values of the horizontal pressure gradient due to residual sea level and of the advection of residual currents together contribute about 75% of the spatial mean values of the advection by the tidal currents, indicating that residual currents in this bay are induced mainly by the nonlinear effects of tidal currents. This is the first ever nonlinear tidal current study by CAT.

Mapping tidal residual circulations in the outer Xiangshan Bay using a numerical model

Xiangshan Bay, which is elongated and semi-enclosed, is characterized by strong tides. The original understanding of the tidal residual circulation in the bay was based on the Eulerian time-mean method. However, it has been theoretically proved that the Lagrangian time-mean method rather than the Eulerian one should be employed to detide. This knowledge motivated us to remap the tidal residual circulation in the bay. A three-dimensional numerical model with a Lagrangian particle-tracking module was used to simulate tides, from which the Lagrangian and Eulerian tidal residual circulations were produced. The Lagrangian residual circulation exhibited a two-branch pattern that connected the southern and northern outer seas of Xiangshan Bay, whereas the Eulerian residual circulation was characterized by eddies that are not favorable for water exchange. Through comparing to the observed salinity distribution, the Lagrangian residual current presented a reasonable spatial pattern which could explain the inter-tidal mass transport. Furthermore, the Lagrangian residual current performed better in sustaining the mass conservation as predicted by theory. A series of sensitivity experiments were conducted to investigate the effect of nonlinear mechanisms on the Lagrangian residual circulation. The results showed that nonlinear advection is dominant while the time-varying width and depth play a minor role, and the quadratic bottom friction has no influence with only M2 tide driving the model.

Indirect evidence for substantial damping of low‐mode internal tides in the open ocean

AbstractA global high‐resolution ocean circulation model forced by atmospheric fields and the M2 tidal constituent is used to explore plausible scenarios for the damping of low‐mode internal tides. The plausibility of different damping scenarios is tested by comparing the modeled barotropic tides with TPXO8, a highly accurate satellite‐altimetry‐constrained tide model, and by comparing the modeled coherent baroclinic tide amplitudes against along‐track altimetry. Five scenarios are tested: (1) a topographic internal wave drag, argued here to represent the breaking of unresolved high vertical modes, applied to the bottom flow (default configuration), (2) a wave drag applied to the barotropic flow, (3) absence of wave drag, (4) a substantial increase in quadratic bottom friction along the continental shelves (with wave drag turned off), and (5) application of wave drag to the barotropic flow at the same time that quadratic bottom friction is substantially increased along the shelves. Of the scenarios tested here, the default configuration (1) yields the most accurate tides. In all other scenarios (2–5), the lack of damping on open ocean baroclinic motions yields baroclinic tides that are too energetic and travel too far from their sources, despite the presence of a vigorous mesoscale eddy field which can scatter and decohere internal tides in the model. The barotropic tides are also less accurate in the absence of an open ocean damping on barotropic motions, that is, in scenarios (3) and (4). The results presented here suggest that low‐mode internal tides experience substantial damping in the open ocean.

Fortnightly tides and subtidal motions in a choked inlet

Amplitudes of semi-diurnal tidal fluctuations measured at an ocean inlet system decay nearly linearly by 87% between the ocean edge of the offshore ebb-tidal delta and the backbay. A monochromatic, dynamical model for a tidally choked inlet separately reproduces the evolution of the amplitudes and phases of the semi-diurnal and diurnal tidal constituents observed between the ocean and inland locations. However, the monochromatic model over-predicts the amplitude and under-predicts the lag of the lower-frequency subtidal and fortnightly motions observed in the backbay. A dimensional model that considers all tidal constituents simultaneously, balances the along-channel pressure gradient with quadratic bottom friction, and that includes a time-varying channel water depth, is used to show that that these model-data differences are associated with nonlinear interactions between the tidal constituents that are not included in non-dimensional, monochromatic models. In particular, numerical simulations suggest that the nonlinear interactions induced by quadratic bottom friction modify the amplitude and phase of the subtidal and fortnightly backbay response. This nonlinear effect on the low-frequency (subtidal and fortnightly) motions increases with increasing high-frequency (semi-diurnal) amplitude. The subtidal and fortnightly motions influence water exchange processes, and thus backbay temperature and salinity.

Modeling tidal stream energy extraction and its effects on transport processes in a tidal channel and bay system using a three-dimensional coastal ocean model

This paper presents a numerical modeling study for simulating in-stream tidal energy extraction and assessing its effects on the hydrodynamics and transport processes in a tidal channel and bay system connecting to coastal ocean. A marine and hydrokinetic (MHK) module was implemented in a three-dimensional (3-D) coastal ocean model using the momentum sink approach. The MHK model was validated with the analytical solutions for tidal channels under one-dimensional (1-D) conditions. Model simulations were further carried out to compare the momentum sink approach with the quadratic bottom friction approach. The effects of 3-D simulations on the vertical velocity profile, maximum extractable energy, and volume flux reduction across the channel were investigated through a series of numerical experiments. 3-D model results indicate that the volume flux reduction at the maximum extractable power predicted by the 1-D analytical model or two-dimensional (2-D) depth-averaged numerical model may be overestimated. Maximum extractable energy strongly depends on the turbine hub height in the water column, which reaches a maximum when turbine hub height is located at mid-water depth. Far-field effects of tidal turbines on the flushing time of the tidal bay were also investigated. Model results demonstrate that tidal energy extraction has a greater effect on the flushing time than volume flux reduction, which could negatively affect the biogeochemical processes in estuarine and coastal waters that support primary productivity and higher forms of marine life. ► Model validation of tidal energy extraction against an analytical solution. ► 3-D effects on velocity, power extraction and volume flux. ► Volume flux reduction predicted by 1-D and 2-D models may be overestimated. ► Effects of tidal energy extraction on flushing time.

A Process Study of the Tidal Circulation in the Persian Gulf

A homogeneous shallow-water model with free surface is used to model the tidal circulation in the Persian Gulf. The numerical finite-difference model includes harmonic diffusion of horizontal momentum and quadratic bottom friction, it has a 9 km mesh size and it is forced by 7 tidal components at its southern boundary. High precision bathymetric data are used to obtain the bottom topography. The numerical model is run for more than a year. The results are the following: 1) The model accurately reproduces the tidal phase and amplitude observed at 42 tidal gauges in the region. This accuracy is attributed to the presence of the 7 components which are able to interact nonlinearly; 2) The amphidromic points are also well positioned by the model due to a proper choice of bathymetry. This was checked also with a simpler geometry of the domain; 3) The tidal currents can be strong in the Straits of Hormuz and in shallow areas; thus they will have an effect of the hydrology of the region. The residual currents are weak so that they will be negligible for the large-scale circulation on long periods; 4) Finally, the sea-surface elevation forecast by the model is in close agreement with in-situ measurements of pressure in the Straits, performed during the GOGP99 experiment.

Analytical and numerical analysis of tides and salinities in estuaries; part I: tidal wave propagation in convergent estuaries

Analytical solutions of the momentum and energy equations for tidal flow are studied. Analytical solutions are well known for prismatic channels but are less well known for converging channels. As most estuaries have a planform with converging channels, the attention in this paper is fully focused on converging tidal channels. It will be shown that the tidal range along converging channels can be described by relatively simple expressions solving the energy and momentum equations (new approaches). The semi-analytical solution of the energy equation includes quadratic (nonlinear) bottom friction. The analytical solution of the continuity and momentum equations is only possible for linearized bottom friction. The linearized analytical solution is presented for sinusoidal tidal waves with and without reflection in strongly convergent (funnel type) channels. Using these approaches, simple and powerful tools (spreadsheet models) for tidal analysis of amplified and damped tidal wave propagation in converging estuaries have been developed. The analytical solutions are compared with the results of numerical solutions and with measured data of the Western Scheldt Estuary in the Netherlands, the Hooghly Estuary in India and the Delaware Estuary in the USA. The analytical solutions show surprisingly good agreement with measured tidal ranges in these large-scale tidal systems. Convergence is found to be dominant in long and deep-converging channels resulting in an amplified tidal range, whereas bottom friction is generally dominant in shallow converging channels resulting in a damped tidal range. Reflection in closed-end channels is important in the most landward 1/3 length of the total channel length. In strongly convergent channels with a single forward propagating tidal wave, there is a phase lead of the horizontal and vertical tide close to 90o, mimicking a standing wave system (apparent standing wave).

Study on linear and nonlinear bottom friction parameterizations for regional tidal models using data assimilation

Data assimilation technique (adjoint method) is applied to study the similarities and the differences between the Ekman (linear) and the Quadratic (nonlinear) bottom friction parameterizations for a two-dimensional tidal model. Two methods are used to treat the bottom friction coefficient (BFC). The first method assumes that the BFC is a constant in the entire computation domain, while the second applies the spatially varying BFCs. The adjoint expressions for the linear and the nonlinear parameterizations and the optimization formulae for the two BFC methods are derived based on the typical Largrangian multiplier method. By assimilating the model-generated ‘observations’, identical twin experiments are performed to test and validate the inversion ability of the presented methodology. Four experiments, which employ the linear parameterization, the nonlinear parameterizations, the constant BFC and the spatially varying BFC, are carried out to simulate the M 2 tide in the Bohai Sea and the Yellow Sea by assimilating the TOPEX/Poseidon altimetry and tidal gauge data. After the assimilation, the misfit between model-produced and observed data is significantly decreased in the four experiments. The simulation results indicate that the nonlinear Quadratic parameterization is more accurate than the linear Ekman parameterization if the traditional constant BFC is used. However, when the spatially varying BFCs are used, the differences between the Ekman and the Quadratic approaches diminished, the reason of which is analyzed from the viewpoint of dissipation rate caused by bottom friction. Generally speaking, linear bottom friction parameterizations are often used in global tidal models. This study indicates that they are also applicable in regional ocean tidal models with the combination of spatially varying parameters and the adjoint method.

Regional ocean tide loading modelling around the Iberian Peninsula

Abstract We developed a new 1/12° resolution oceanic tide model in the complex region that surrounds the Iberian Peninsula. The model, named IBER01, allows us to obtain more accurate tidal loading computations for precise geodetic and gravimetric observations in this area. The modelling follows the scheme of data assimilation (coastal tide gauge, bottom pressure sensors and TOPEX/Poseidon altimetry) into a hydrodynamical model, which is based on two-dimensional barotropic depth averaged shallow-water equations. Detailed bathymetry data and quadratic bottom friction with a specific drag coefficient for the region have been considered. Improved ocean load maps for the Iberian Peninsula are obtained for eight harmonic constituents (Q1, P1, O1, K1, N2, M2, S2 and K2), after computing the load effect (Newtonian attraction and elastic contribution) using IBER01 and six present-day global oceanic tide models for comparison. The results achieved verify the quality of the new model. Our ocean loading computations reduce considerably the discrepancies between the theoretical Earth tide parameters and those from observations at the level of 0.3%.

A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows

A set of weakly dispersive Boussinesq-type equations, derived to include viscosity and vorticity terms in a physically consistent manner, is presented in conservative form. The model includes the approximate effects of bottom-induced turbulence, in a depth-integrated sense, as a second-order correction. Associated with this turbulence, vertical and horizontal rotational effects are captured. While the turbulence and horizontal vorticity models are simplified, a model with known physical limitations has been derived that includes the quadratic bottom friction term commonly added in an ad hoc manner to the inviscid equations. An interesting result of this derivation is that one should take care when adding such ad hoc models; it is clear from this exercise that (1) it is not necessary to do so – the terms can be included through a consistent derivation from the viscous primitive equations – and (2) one cannot properly add the quadratic bottom friction term without also adding a number of additional terms in the integrated governing equations. To solve these equations numerically, a highly accurate and stable model is developed. The numerical method uses a fourth-order MUSCL-TVD scheme to solve the leading order (shallow water) terms. For the dispersive terms, a cell averaged finite volume method is implemented. To verify the derived equations and the numerical model, four cases of verifications are given. First, solitary wave propagation is examined as a basic, yet fundamental, test of the models ability to predict dispersive and nonlinear wave propagation with minimal numerical error. Vertical velocity distributions of spatially uniform flows are compared with existing theory to investigate the effects of the newly included horizontal vorticity terms. Other test cases include comparisons with experiments that generate strong vorticity by the change of bottom bathymetry as well as by tidal jets through inlet structures. Very reasonable agreements are observed for the four cases, and the results provide some information as to the importance of dispersion and horizontal vorticity.

Derivation of a new two-dimensional viscous shallow water model with varying topography, bottom friction and capillary effects

Resume Considering the three-dimensional Navier–Stokes equations with a free moving surface boundary condition and hydrostatic approximation, we study the derivation, with asymptotic analysis, of a new two-dimensional viscous shallow water model in rotating framework, with irregular topography, linear and quadratic bottom friction terms and capillary effects. A new formulation of the viscous effects, consistent with a previous one-dimensional analysis, is obtained. Finally, we propose some simple numerical experiments in order to validate the proposed model.

M 2 tidal dissipation around Vancouver Island: an inverse approach

A representer approach is used to assimilate M 2 tide gauge harmonics into a high-resolution two-dimensional finite element model of Puget Sound and the waters surrounding Vancouver Island. A preliminary model solution which employs a conventional drag coefficient for quadratic bottom friction is shown to produce elevation amplitudes and phases that differ significantly from the harmonics computed from an analysis of tide gauge observations. The assimilation procedure not only rectifies this problem but also shows that this better fit implies significant momentum equation residuals in regions where strong turbulent mixing and internal tide generation are known to occur. Thus, the assimilation identifies regions where dissipation beyond conventional bottom friction is required. Computed dissipation rates arising from these residuals reveal the largest energy sinks to be in the Gulf and San Juan Islands, Juan de Fuca Strait, and the channels off northeast Vancouver Island.