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A universal law of estuarine mixing is derived here, combining the approaches of salinity coordinates, Knudsen relations, total exchange flow, mixing definition as salinity variance loss, and the mixing–exchange flow relation. As a result, the long-term average mixing within an estuarine volume bounded by the isohaline of salinity S amounts to M(S) = S2Qr, where Qr is the average river runoff into the estuary. Consequently, the mixing per salinity class is m(S) = ∂SM(S) = 2SQr, which can also be expressed as the product of the isohaline volume and the mixing averaged over the isohaline. The major differences between the new mixing law and the recently developed mixing relation based on the Knudsen relations are threefold: (i) it does not depend on internal dynamics of the estuary determining inflow and outflow salinities (universality), (ii) it is exactly derived from conservation laws (accuracy), and (iii) it calculates mixing per salinity class (locality). The universal mixing law is demonstrated by means of analytical stationary and one-dimensional and two-dimensional numerical test cases. Some possible consequences for the salinity distribution in real estuaries are briefly discussed. Since the mixing per salinity class only depends on the river runoff and the chosen salinity, and not on local processes at the isohaline, low-mixing estuaries must have large isohaline volumes and vice versa.

The well-known Knudsen relations and the total exchange flow (TEF) analysis framework provide quantifications of exchange flow across an open boundary to the adjacent ocean in terms of bulk values (Knudsen theory: inflow and outflow volume or salinity) or with resolution in salinity space (TEF: profiles of volume and salt flux in salinity coordinates). In the present study, these theories are extended toward mixing of salinity, defined as the decay of salinity variance due to turbulent mixing. In addition to the advective fluxes, diffusive fluxes across the boundary are also considered now. These new Knudsen and TEF relations for mixing are derived by applying Gauss’s theorem to the salinity square and salinity variance equations. As a result of the analysis, four different Knudsen relations for the mixing in estuaries are derived. The first one is exact and considers nonperiodicity as well as nonconstancy of the inflow and outflow salinities. The other three formulations are approximate only, in the sense that either nonperiodicity or nonconstancy or both are relaxed. The simplest of those formulations has recently been derived by MacCready et al. and estimates the estuarine mixing as the product of inflow salinity, outflow salinity, and time-averaged river runoff. These four mixing estimates are systematically assessed by means of a number of idealized estuarine test cases. For periodic tidal flow, the simplest estimate still predicts the effective (physical plus numerical) mixing within an error of about 10%.

A new parameterization for the estuarine turbulent eddy viscosity coefficient is developed considering the influence of wind forcing and feedback between stratification and shear. The emerging tidally averaged eddy viscosity profile A v T is parameterized as parabolic under well-mixed conditions, and is composed of a skewed-Gaussian-like form for the upper layer, and a parabolic form for the bottom layer under stratified conditions. The precise shape of the profiles depends parametrically on the bottom boundary layer thickness, the bulk Richardson number, and the Wedderburn number. The parameterized A v T profiles show excellent agreement with profiles obtained from numerical models. To explore the importance of vertically varying A v T with regard to exchange processes, an analytical model is designed. This one-dimensional model is based on a balance between frictional forces and pressure gradient. The resulting exchange flow is analyzed over the relevant parameter space that is associated with horizontal and vertical stratification through the bulk Richardson number, and the bi-directional wind stress via the Wedderburn number. Down-estuary wind enhances the up-estuary flow near the bottom and down-estuary flow near the surface driving an exchange flow pattern typically associated with gravitational circulation. Up-estuary wind results in either a two-layer inverted circulation opposing the gravitational circulation, or a three-layer flow that is up-estuarine at the surface with classical two-layer circulation underneath. Three-layer flow emerges with a weak wind. With increasing runoff velocity, three-layer flow transitions to a single layer flow under weak stratification conditions.

This paper quantifies spurious dissipation and mixing of various advection schemes in idealised experiments of lateral shear and baroclinic instabilities in numerical simulations of a re-entrant Eady channel for configurations with large and small Rossby numbers. In addition, a two-dimensional barotropic shear instability test case is used to examine numerical dissipation of momentum advection in isolation, without any baroclinic effects. Effects of advection schemes on the evolution of background potential energy and the dynamics of the restratification process are analysed. The advection schemes for momentum and tracers are considered using several different methods including a recently developed local dissipation analysis. As highly accurate but computationally demanding schemes we apply WENO and MP5, and as more efficient lower-order total variation diminishing (TVD) schemes we use among others the SPL-max-View the MathML source13 and a third-order-upwind scheme. The analysis shows that the MP5 and SPL-max-View the MathML source13 schemes provide the most accurate results. Following our comprehensive analysis of computational costs, the MP5 scheme is approximately 2.3 times more expensive in our implementation. In contrast to the configuration with a small Rossby number, in which significant differences between schemes are apparent, the different advection schemes behave similarly for a larger Rossby number. Regions of high numerical dissipation are shown to be associated with low grid Reynolds numbers. The major outcome of the present study is that generally positive global numerical dissipation and positive background potential energy evolution delay the restratification process.

We have developed a Lagrangian model to investigate a potential mechanism based on phototaxis behavior of phytoplankton cells for the formation of thin layers. We assume that all cells follow a time-regulated diurnal vertical migration during which they experience photo-acclimation based on the Denman and Marra (1986) model. When a cell experiences stress due to strong light that exceeds a threshold level, the cell swims downward, away from the light. We applied the Lagrangian model to a one dimensional second order turbulence closure model that generates a realistic surface mixing condition for a given set of physical parameters, such as wind and optical water type. For the chosen swimming velocities and prescribed behavior, we found that, in coastal water type and Jerlov III type, thin layer formation takes place up to 5 m s−1 winds, while 10 m s−1 winds cause sufficiently strong mixing to prevent the formation of thin layer. We have also investigated the effects of changing the irradiance threshold for the onset of the photoinhibition, the initial density profile and random walk swimming. In conclusion, thin layer formation due to photoinhibition may be possible for a low value of photoinhibition threshold that may occur either due to upwelling or strong light exposure.

In this study a model system consisting of the three-dimensional General Estuarine Transport Model (GETM) and the third generation wind wave model SWAN was developed. Both models were coupled in two-way mode. The effects of waves were included into the ocean model by implementing the depth-dependent Radiation stress formulation (RS) of Mellor (2011a) and the Vortex force formulation (VF) presented by Bennis et al. (2011) . Thus, the developed model system offers a direct comparison of these two formulations. The enhancement of the vertical eddy viscosity due to the energy transfer by white capping and breaking waves was taken into account by means of injecting turbulent kinetic energy at the surface. Wave-current interaction inside the bottom boundary layer was considered as well. The implementation of both wave-averaged formulations was validated against three flume experiments. One of these experiments with long period surface waves (swell), had not been evaluated before. The validation showed the capability of the model system to reproduce the three-dimensional interaction of waves and currents. For the flume test cases the wave-induced water level changes (wave set-up and set-down) and the corresponding depth-integrated wave-averaged velocities were similar for RS and VF. Both formulations produced comparable velocity profiles for short period waves. However, for large period waves, VF overestimated the wave set-down near the main breaking points and RS showed artificial offshore-directed transport at the surface where wave shoaling was taking place. Finally the validated model system was applied to a realistic barred beach scenario. For RS and VF the resulting velocity profiles were similar after being significantly improved by a roller evolution method. Both wave-averaged formulations generally provided similar results, but some shortcomings were revealed. Although VF partly showed significant deviations from the measurements, its results were still physically reasonable. In contrast, RS showed unrealistic offshore-directed transport in the wave-shoaling regions and close to steep bathymetry.

Interleaving motions on a wide, baroclinic front are modeled using a second-moment closure to represent unresolved fluxes by turbulence and salt fingering. A linear perturbation analysis reveals two broad classes of unstable modes. First are scale-selective modes comparable with interleaving as observed in oceanic fronts. These correspond well with observations in some respects but grow by a very different mechanism, which ought to be easily distinguished in hydrographic profiles. The second mode type is the so-called ultraviolet catastrophe, which is expected to lead to steppy profiles even in the absence of interleaving. Both modes are driven by positive feedbacks between interleaving and the underlying small-scale mixing processes. Contrary to expectations, use of the second-moment closure in place of earlier empirical mixing models does not lead to improved agreement with observations.

The generation of residual circulation in a tidally energetic estuary with constant longitudinal salinity gradient and parabolic cross section is examined by means of a two-dimensional cross-sectional numerical model, neglecting river runoff and Stokes drift. It is shown how the longitudinal and lateral residual circulation can be decomposed into contributions from various processes such as tidal straining circulation, gravitational circulation, advectively driven circulation, and horizontal mixing circulation. The sensitivity of the residual circulation and its components from various processes to changes in forcing is investigated by varying the Simpson number (nondimensional longitudinal buoyancy gradient) and the unsteadiness parameter (nondimensional tidal frequency), as well as the bed roughness and the width of the estuary. For relatively weak salinity gradient forcing, the tidal straining circulation dominates the residual exchange circulation in support of classical estuarine circulation (up-estuary flow near the bed and down-estuary flow near the surface). The strength of the longitudinal estuarine circulation clearly increases with increased salinity gradient forcing. However, when the Simpson number exceeds 0.15, the relative contributions of both gravitational circulation and advectively driven circulation to estuarine circulation increase substantially. Lateral residual circulation is relatively weak for small Simpson numbers and becomes flood oriented (divergent flow near the bed and convergent flow near the surface) for larger Simpson numbers because of increasing contributions from gravitational and advectively driven circulation. Increasing the unsteadiness number leads to decreased longitudinal and lateral residual circulation. Although changes in bed roughness result in relatively small changes in residual circulation, results are sensitive to the width of the estuary, mainly because of changes in residual exchange circulation driven by tidal straining.

The interaction of shear, stratification, and turbulence in boundary layers on sloping topography is investigated with the help of an idealized theoretical model, assuming uniform bottom slope and homogeneity in the upslope direction. It is shown theoretically that the irreversible vertical buoyancy flux generated in the boundary layer is directly proportional to the molecular destruction rate of small-scale buoyancy variance, which can be inferred, for example, from microstructure observations. Dimensional analysis of the equations shows that, for harmonic boundary layer forcing and no rotation, the problem is governed by three nondimensional parameters (slope angle, roughness number, and ratio of forcing and buoyancy frequencies). Solution of the equations with a second-moment closure model for the turbulent fluxes reveals the periodic generation of gravitationally unstable boundary layers during upslope flow, consistent with available observations. Investigation of the nondimensional parameter space with the help of this model illustrates a systematic increase of the bulk mixing efficiencies for (i) steep slopes and (ii) low-frequency forcing. Except for very steep slopes, mixing efficiencies are substantially smaller than the classical value of Γ = 0.2.

The physics of shallow gravity currents passing through a rotating channel at subcritical Froude number is investigated here with a series of idealized numerical experiments. It is found that the combined effects of friction and rotation set up a complex transverse circulation that has some crucial implications for the entrainment process. A key component of this secondary circulation is a geostrophically balanced transverse jet in the interface that laterally drains fluid from the interface. This effect is shown to result in a strong cross-channel asymmetry and a spatial separation of the entrainment process: drained interfacial fluid is partly replaced by entrained ambient fluid on the deep side of the gravity current, whereas the downward mixing of buoyant fluid occurs on the shallow side. These results, closely corresponding to recent measurements in a shallow, channelized gravity current in the western Baltic Sea, illustrate that the description of entrainment as a strictly vertical mixing process with the help of local bulk parameters like the Froude number is not generally applicable in rotating gravity currents.

This numerical modeling study quantifies for the first time the contribution of various processes to estuarine circulation in periodically stratified tidal flow under the impact of a constant horizontal buoyancy gradient. The one-dimensional water column equations with periodic forcing are first cast into nondimensional form, resulting in a multidimensional parameter space spanned by the modified inverse Strouhal number and the modified horizontal Richardson number, as well as relative wind speed and wind direction and the residual runoff. The along-tide momentum equation is then solved for the tidal-mean velocity profile in such a way that it is equated to the sum of the contributions of tidal straining (resulting from the temporal correlation between eddy viscosity and vertical shear), gravitational circulation (resulting from the depth-varying forcing by a constant horizontal buoyancy gradient), wind straining, and depth-mean residual flow (resulting from net freshwater runoff). This definition of tidal straining does not only account for tidal asymmetries resulting from horizontal buoyancy gradients but also from wind straining and residual runoff. For constant eddy viscosity, the well-known estuarine circulation analytical solution with polynomial residual profiles is directly obtained. For vertically parabolic and constant-in-time eddy viscosity, a new analytic solution with logarithmic residual profiles is found, showing that the intensity of the gravitational circulation scales with the horizontal Richardson number. For scenarios with realistic spatially and temporally varying eddy viscosity, a numerical water column model equipped with a state-of-the-art two-equation turbulence closure model is applied to quantify the individual contributions of the various processes to estuarine circulation. The fundamental outcome of this study is that, for irrotational flow with periodic stratification and without wind forcing and residual runoff, the tidal straining is responsible for about two-thirds and gravitational circulation is responsible for about one-third of the estuarine circulation, proportionally dependent on the horizontal Richardson number, and weakly dependent on the Strouhal number. This new and robust result confirms earlier estimates by H. Burchard and H. Baumert, who suggested that tidal straining is the major generation mechanism for estuarine turbidity maxima. However, a sensitivity analysis of the model results to details of the turbulence closure model shows some uncertainty with respect to the parameterization of sheared convection during flood. Increasing down-estuary wind straining and residual runoff reduce the quantitative contribution of tidal straining. For relatively small horizontal Richardson numbers, the tidal straining contribution to estuarine circulation may even be reversed by down-estuary wind straining.

The aim of this one-dimensional water column study is to combine modified versions of three characteristic parameters for periodic tidal flow under the influence of a longitudinal buoyancy gradient—the horizontal Richardson number, the inverse Strouhal number, and the inverse Ekman number—into a parameter space study, including constant wind forcing from various directions. It is shown how the underlying dynamical equations can be cast into nondimensional form, depending mainly on these three nondimensional parameters plus the relative wind speed and the wind direction. Idealized model simulations are carried out for the whole realistic range of horizontal Richardson and inverse Strouhal numbers for various latitudes, showing the amplitude of the tidally induced stratification for a wide range of scenarios. It is found that classical threshold values for the horizontal Richardson number, indicating the switch from periodic stratification to permanent stratification, are valid only for special cases, and that this switch also strongly depends on the inverse Strouhal number, the inverse Ekman number, and the wind vector. The transverse residual flow is close to the thermal wind balance for a variety of parameters, and nondimensionalized longitudinal residual flow shows the classical estuarine exchange flow pattern with little variation in the near-bed onshore component. Wind straining is confirmed as an important estuarine and coastal process, enhancing estuarine circulation for offshore (down estuary) winds and vice versa. Agreement with field data from Liverpool Bay is good, including the explanation of a flood tide local maximum of the dissipation rate in the upper half of the water column. An equation for the second time derivative of the potential energy anomaly is derived for quantifying the dynamical processes leading to stratification due to the straining of the horizontal density gradient.

In this paper, a time-dependent dynamic equation for the potential energy anomaly, ϕ , is rigorously derived from dynamic equations for potential temperature and salinity, the continuity equation and the equation of state for sea water. The terms locally changing ϕ are (A) the ϕ -advection, (B) the depth-mean straining, (C) the non-mean straining, (D) the vertical advection, (E) the vertical mixing, (F) surface and bottom density fluxes, (G) inner sources of density e.g. due to absorption of solar radiation and the non-linearity of the equation of state, and (H) horizontal divergence of horizontal turbulent density fluxes. In order to derive the equation in concise form, a vertical velocity (linearly varying with depth) with respect to depth-proportional vertical coordinates had to be defined. The evaluation of the terms in the ϕ -equation is then carried out for a one-dimensional tidal straining study and a two-dimensional estuarine circulation study. Comparisons to empirical estimates for these terms are made for the one-dimensional study. It is concluded that the ϕ -equation provides a general reference for empirical bulk parameterisations of stratification and mixing processes in estuaries and coastal seas and that it is a tool for complete analysis of the relevant terms from numerical models.

A diagnostic method for calculating physical and numerical mixing of tracers in ocean models is presented. The physical mixing is defined as the turbulent mean tracer variance decay rate. The numerical mixing due to discretisation errors of tracer advection schemes is shown to be the decay rate between the advected square of the tracer variance and the square of the advected tracer and can be easily implemented into any ocean model. The applicability of the method is demonstrated for four test cases: (i) a one-dimensional linear advection equation with periodic boundary conditions, (ii) a two-dimensional flat-bottom lock exchange test case without mixing, (iii) a two-dimensional marginal sea overflow study with mixing and entrainment and (iv) the DOME test case with a dense bottom current propagating down a broad linear slope. The method has a number of advantages over previously introduced estimates for numerical mixing.

A 19-h time series of dissipation, stratification, and horizontal velocities has been obtained for a dense gravity current flowing into the Arkona Basin in the western Baltic Sea. The observations are compared with one-dimensional, quasi-steady theory, in which the gravity component in the flow direction is balanced by bottom friction, while that in the cross-flow direction is balanced by the Coriolis force. The observations deviate from the theory in that the bottom shear stress is more than 2 times as large as that required to balance the gravity. Several reasons for this discrepancy are discussed. A 1D turbulence model is also compared with the observations. Profiles of velocity, stratification, and dissipation rates generally show similar variations with depth as the observations, although the observed dissipation rates are somewhat larger than the modeled and the modeled transverse velocities are much larger than the observed. Subsequently, the model is used to investigate the variation of the entrainment parameter for a large range of Ekman and Froude numbers. Within the modeled parameter space, the entrainment parameter can be collapsed onto a curve that is an increasing function of both the Froude and the Ekman numbers. There is one puzzling result of the observations that differs from the model results and earlier observations: the observed entrainment rate increases dramatically during the observation period, where the Froude number decreases slightly. Some reasons for this increase are discussed.

The dependency of the estuarine circulation on the depth-to-width ratio of a periodically, weakly stratified tidal estuary is systematically investigated here for the first time. Currents, salinity, and other properties are simulated by means of the General Estuarine Transport Model (GETM) in cross-sectional slice mode, applying a symmetric Gaussian-shaped depth profile. The width is varied over four orders of magnitude. The individual along-channel circulation contributions from tidal straining, gravitation, advection, etc., are calculated and the impact of the depth-to-width ratio on their intensity is presented and elucidated. It is found that the estuarine circulation exhibits a distinct maximum in medium-wide channels (intermediate depth-to-width ratio depending on various parameters), which is caused by a maximum of the tidal straining contribution. This maximum is related to a strong tidal asymmetry of eddy viscosity and shear created by secondary strain-induced periodic stratification (2SIPS): in medium channels, transverse circulation generated by lateral density gradients due to laterally differential longitudinal advection induces stable stratification at the end of the flood phase, which is further increased during ebb by longitudinal straining (SIPS). Thus, eddy viscosity is low and shear is strong in the entire ebb phase. During flood, SIPS decreases the stratification so that eddy viscosity is high and shear is weak. The circulation resulting from this viscosity–shear correlation, the tidal straining circulation, is oriented like the classical, gravitational circulation, with riverine outflow at the surface and oceanic inflow close to the bottom. In medium channels, it is about 5 times as strong as in wide (quasi one-dimensional) channels, in which 2SIPS is negligible.

Recent contributions to second-order turbulence modelling are reviewed with an emphasis on models of the coastal ocean. Classical and recent turbulence models are reformulated using a unifying notation, making differences in model structure and model parameters more transparent. The essential characteristics of a number of models are compared. It is further demonstrated that the mathematical stability and realisability of the models is crucial for practical applications. The parameter constraints to insure stable computations are discussed. Model performance is evaluated for a number of idealised entrainment scenarios that are typical for shelf seas: entrainment in linearly stratified and two-layer fluids caused by (a) surface wind stress (b) bottom stress due to water motion driven by a barotropic pressure gradient (c) bottom stress due to water motion in a bottom jet. It is shown that the predicted entrainment velocities are in good agreement with available laboratory data, provided the models have been calibrated according to a theoretical constraint. In addition, model behaviour in free convection is re-analysed with the help of recent data from large eddy simulations (LES). It is shown that even the most simple models, which do not solve a differential equation for the turbulent kinetic energy (TKE), can approximately reproduce the entrainment depth, however, they fail completely in reproducing the correct budget of the TKE-balance. In contrast, second-order models solving transport equations for the TKE and a length scale determining variable are in reasonable agreement with the TKE-budgets from LES.

In this study we compare turbulence parameters from field observations and model simulations specifically under the influence of weak to moderate wind forcing and breaking short waves. The experiment was performed during 12 days under very weak stratification at a fetch-limited lake in Switzerland. The near surface observations were obtained by using a quasi-free rising profiler which measured small scale shear and temperature fluctuations. We used a two-equation k-epsilon. turbulence model with an algebraic second-moment Closure scheme. The one-dimensional numerical model was extended to consider breaking waves by a shear-dependent parameterisation. The agreement of observed and simulated turbulence quantities is very promising. Especially well simulated is the enhanced turbulence level in the wave-affected-surface-layer (WASL) of a few din thickness. The logarithmic slope of the turbulent dissipation rate in this WASL was found to vary between -2.1 and -1.7. Below the WASL the classic law-of-the-wall was well reproduced by the data and the model.

The aim of this study is to intervalidate observations and numerical simulation results for the turbulent dissipation rate under strong wind conditions in the Northern North Sea during one week in October 1998. The observations were obtained by spatially and temporally averaging measurements of small-scale shear with a free-falling shear probe. The 1D numerical model used for this study is based on a state-of-the-art two-equation k − e turbulence model with an algebraic second-moment closure scheme. It is discussed by means of annual and seasonal model simulations how the influence of heat and salt advection and internal waves can be accounted for. After these precautions, the agreement between observations and simulations of the turbulent dissipation rate are fairly good. Remaining differences cannot only be explained by problems such as undersampling and noise level, but also by idealising model assumptions.

In this study we aim at comparing turbulence parameters from field observations and model simulations under convective conditions. The comparison is focused on the depth dependence and temporal dynamics of viscous and diffusive dissipation rates ɛ and χ. The near-surface observations were obtained by using a quasi-free-rising profiler which measured small-scale shear and temperature fluctuations to within the vicinity of the water surface. Convective conditions during the experiment are characterized by low wind speeds (between 0 and 4 m s−1) and a typical heat loss of about 150 Wm−2. We applied a state-of-the-art two-equation k-ɛ turbulence model with an algebraic second-moment closure scheme. The qualitative agreement of the turbulence quantities resulting from observations and simulations is rather good. The temporal dynamics of the temperature field is simulated correctly, whereas in the spatial dynamics some deficiencies of the model due to its local character can be seen. It is concluded that such models realistically reproduce convective turbulence and therefore represent a reasonable compromise between complexity and simplicity, so that they can be used with acceptable costs in large-scale models.

An energy-conserving discretisation of the shear and buoyancy production terms for turbulent kinetic energy is derived. In contrast to `ad hoc' numerical schemes, this guarantees that all mean kinetic and potential energy which is lost or gained due to vertical mixing is exactly subtracted or added to the turbulent kinetic energy budget. It is further shown in a numerical wind-entrainment experiment that this new methods results in significantly more stable numerical solutions.

The purpose of this paper is to modify two-equation turbulence models such that they are capable of simulating dynamics in the wave-enhanced layer near the surface. A balance of diffusion of turbulent kinetic energy (TKE) and dissipation is assumed as the surface boundary condition for TKE following the suggestion of Craig and Banner. It is shown that this theory, originally developed under the assumption of a macro length scale linearly increasing down from the surface, fails for two-equation models such as the well-known k–e model. Suggestions are made how to modify such models for overcoming this deficiency. The basic idea is to insert the analytic solution of a model problem suggested by Craig into the dissipation rate equation and solve for the turbulent Schmidt number of the dissipation rate equation, which may be formulated as a function of the production/dissipation ratio. With this modification, the linear behavior of the macro length scale is properly reproduced by the k–e model. It is ...

In this comparative study, four different algebraic second-moment turbulence closure models are investigated in detail. These closure schemes differ in the number of terms considered for the closure of the pressure–strain correlations. These four turbulence closures result in the eddy-diffusivity principle such that the closure assumptions are contained in dimensionless so-called stability functions. Their performance in terms of Prandtl number, Monin–Obukhov similarity theory, and length scale ratios are first tested against data for simple flows. The turbulence closure is then completed by means of a k–ϵ two-equation model, but other models such as the two-equation model by Mellor and Yamada could also be used. The concept of the steady-state Richardson number for homogeneous shear layers is exploited for calibrating the sensitivity of the four models to shear and stable stratification. Idealized simulations of mixed layer entrainment into stably stratified flow due to surface stress and due to...

This short paper focuses on the problems caused by the equal contribution of shear and buoyancy production in the length scale equation transporting the product of turbulent kinetic energy q2 and macro length scale l as suggested by Mellor and Yamada. It is shown that this model has no steady-state solutions for homogeneous shear flows. The concept of the steady-state Richardson number is used for estimating the adequate contribution of the buoyancy production to the equation for q2l. In a simple wind entrainment experiment, the failure of the q2l equation in the case of unlimited l is shown. With the new estimate for the buoyancy production contribution to q2l, physically sound results are obtained even for the unlimited case. In applications of the modified q2l equation to three oceanic test cases in the northern North Sea and the northern Pacific, the results of the wind entrainment experiment are confirmed.

A new method for forcing numerical one-dimensional (1D) water column models of turbulent tidal flow is presented. It needs as an input time series of water depth and velocity at one point in the water column. These time series are here interpreted as indirect measurements of the surface slope. The method is based on an operator splitting technique as discretization in time in order to guarantee exact reproduction of the input data. For simple two-dimensional (2D) flow in the xz plane, a parametrization of horizontal advection is given which slightly improves the results. This new method is validated against the performance of a three-dimensional (3D) estuarine model from which forcing data for the 1D model have been extracted. The advantage of this method in relation to the classical method (prescription of surface slopes) is demonstrated. As a demonstration of its potential, the method is finally applied to turbulent tidal flow in the Eastern Scheldt, The Netherlands.

The aim of this paper is to systematically compare κ-e and Mellor-Yamada two-equation turbulence models. Both models include prognostic equations for turbulent kinetic energy and a length scale related parameter which are used to calculate eddy viscosities and vertical diffusivities. The results from laboratory experiments, using mixed and stratified flows, are simulated in order to systematically compare and calibrate the models. It is shown that the Monin-Obukhov similarity theory is well represented in both models. The models are used to simulate stratified tidal flow in the Irish Sea, and the results show that the κ-e models generally predict a larger phase lag between currents and turbulent dissipation, in the bottom boundary layer, than the Mellor-Yamada models. The comparison between the model results and field measurements, of the rate of dissipation of turbulent kinetic energy, shows that both models require modification through the inclusion of an internal wave parameterization in order that they are able to correctly predict the observed levels of turbulent dissipation. As the main result, it is shown that the choice of the stability functions, which are used as proportionality factors for calculating the eddy viscosity and diffusivity, has a stronger influence on the performance of the turbulence model than does the choice of length scale related equation.

In this study, the method of numerical mixing analysis is presented for three-dimensional ocean models with general vertical coordinates. Numerical mixing of a scalar is defined as the decay of the square of the scalar due to the three-dimensional advection discretisation. It is shown that for any advection scheme the numerical mixing can be calculated as the difference between the advected square of the scalar and the square of the advected tracer, divided by the time step. Special emphasis on directional-split advection schemes is made. It is shown that for those directional-split schemes the numerical analysis method is exact only when the involved advection of the square of the scalar is carried out individually for each split step. As applications, an idealised meso-scale eddy test scenario without any explicit mixing is calculated. It is shown that only for high-order advection schemes for the scalar (salinity in that case) and the momentum, a physically reasonable solution is obtained. Finally, the method is demonstrated for a fully realistic application to the dynamics of the Western Baltic Sea. Here it becomes clear that physical and numerical mixing depend on each others (increased physical mixing leads to decreased numerical mixing) with the dynamically most relevant mixing being the effective mixing, i.e., the sum of the physical and the numerical mixing.

Tidal straining, which can mathematically be described as the covariance between eddy viscosity and vertical shear of the along-channel velocity component, has been acknowledged as one of the major drivers for estuarine circulation in channelized tidally energetic estuaries. In this paper, the authors investigate the role of lateral circulation for generating this covariance. Five numerical experiments are carried out, starting with a reference scenario including the full physics and four scenarios in which specific key physical processes are neglected. These processes are longitudinal internal pressure gradient forcing, lateral internal pressure gradient forcing, lateral advection, and the neglect of temporal variation of eddy viscosity. The results for the viscosity–shear covariance are correlated across different experiments to quantify the change due to neglect of these key processes. It is found that the lateral advection of vertical shear of the along-channel velocity component and its interaction with the tidally asymmetric eddy viscosity (which is also modified by the lateral circulation) is the major driving force for estuarine circulation in well-mixed tidal estuaries.

Occasionally, numerical simulations using local turbulence closure schemes to estimate vertical turbulent fluxes exhibit small-scale oscillations in space, causing the eddy coefficients to vary over several orders of magnitude on short distances. Theoretical developments suggest that these spurious oscillations are essentially due to the way the eddy coefficients depend on the vertical gradient of the model's variables. An instability criterion is derived based on the assumptions that the artefacts under study are due to the development of small-amplitude, small time- and space-scale perturbations of a smooth solution. The relevance of this criterion is demonstrated by applying it to a series a closure schemes, ranging from the Pacanowski-Philander formulas to the Mellor-Yamada level 2.5 model.

[1] Synoptic, high-resolution, measurements of turbulent kinetic energy dissipation, current velocity and water column stratification across a fast (up to 0.7 m s -1 ) oceanic saline gravity current are presented. Our data provide, for the first time, a detailed two-dimensional picture of the turbulence structure inside a gravity current. Strong boundary-layer and interfacial turbulence can be distinguished from a quiet core, and a strong asymmetry of mixing near the outer edges of the gravity current is apparent. This asymmetry is mirrored by the computed entrainment velocities, varying approximately by a factor of 5 across the gravity current. It is argued that the asymmetry is due to rotational effects that can be clearly identified also in the velocity and density fields.

[1] A numerical model was developed for the prediction of the density stratification of lakes and reservoirs. It combines a buoyancy-extended k-� model with a seiche excitation and damping model to predict the diffusivity below the surface mixed layer. The model was applied to predict the seasonal development of temperature stratification and turbulent diffusivity in two medium-sized lakes over time periods ranging from 3 weeks to 2 years. Depending on the type of boundary condition for temperature, two or three model parameters were optimized to calibrate the model. The agreement between the simulated and the observed temperature distributions is excellent, in particular, if lake surface temperatures were prescribed as surface boundary condition instead of temperature gradients derived from heat fluxes. Comparison of different model variants revealed that inclusion of horizontal pressure gradients and/or stability functions is not required to provide good agreement between model results and data. With the aid of uncertainty analysis it is shown that the depth of the mixed surface layer during the stratified period could be predicted accurately within ±1 m. The sensitivity of the model to several parameters is discussed. INDEX TERMS: 4211 Oceanography: General: Benthic boundary layers; 4255 Oceanography: General: Numerical modeling; 4568 Oceanography: Physical: Turbulence, diffusion, and mixing processes; KEYWORDS: lake, turbulence model, seiche, stratification, simulation, turbulence kinetic energy

This paper investigates the interaction between stratification and turbulence by means of turbulence models. The standard and the advanced turbulent kinetic energy - dissipation (κ-e) model are derived theoretically, including algebraic stress relations. It is shown that a certain empirical constant in the standard model turns out to be a complicated implicit function in the advanced model, namely, a function of the turbulent shear number, the turbulent buoyancy number, and a wall correction. For a better understanding and physical interpretation of the κ-e models, an analysis is carried out for a simplified case where diffusive fluxes are neglected. For this idealization it is shown that (1) the flux Richardson number Rf has a certain lower bound Rƒ− due to the establishment of convection, (2) a steady state flux Richardson number Rƒst (which is defined here for this purpose) labels the borderline between the tendency of turbulence to decrease or collapse (Rƒ > Rƒst) or to increase (Rƒ < Rƒst), and (3) the well-known upper limit for turbulent shear flow, Rƒ+ ≈ 0.25, fits our theory. Using the standard model, the advanced model, a modified version of the level-2 model of Mellor and Yamada and a modified version of Kochergin's model, the evolution of thermal stratification in the northern North Sea during the Fladen Ground Experiment (FLEX'76) is simulated numerically and compared with the measurements. In this specific application, the two κ-e models performed best.