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## Abstract

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 *q*
^{2} 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 *q*
^{2}
*l.* In a simple wind entrainment experiment, the failure of the *q*
^{2}
*l* equation in the case of unlimited *l* is shown. With the new estimate for the buoyancy production contribution to *q*
^{2}
*l,* physically sound results are obtained even for the unlimited case. In applications of the modified *q*
^{2}
*l* equation to three oceanic test cases in the northern North Sea and the northern Pacific, the results of the wind entrainment experiment are confirmed.

## Abstract

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 *q*
^{2} 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 *q*
^{2}
*l.* In a simple wind entrainment experiment, the failure of the *q*
^{2}
*l* equation in the case of unlimited *l* is shown. With the new estimate for the buoyancy production contribution to *q*
^{2}
*l,* physically sound results are obtained even for the unlimited case. In applications of the modified *q*
^{2}
*l* equation to three oceanic test cases in the northern North Sea and the northern Pacific, the results of the wind entrainment experiment are confirmed.

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## Abstract

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.

## Abstract

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.

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## Abstract

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*) = *S*
^{2}
*Q*
_{
r
}, where *Q*
_{
r
} is the average river runoff into the estuary. Consequently, the mixing per salinity class is *m*(*S*) = ∂_{
S
}
*M*(*S*) = 2*SQ*
_{
r
}, 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.

## Abstract

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*) = *S*
^{2}
*Q*
_{
r
}, where *Q*
_{
r
} is the average river runoff into the estuary. Consequently, the mixing per salinity class is *m*(*S*) = ∂_{
S
}
*M*(*S*) = 2*SQ*
_{
r
}, 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.

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## Abstract

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*–ε 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*–ε model. It is shown how near-surface dissipation rate measurements under breaking waves can be simulated by an extended *k*–ε model considering a shear-dependent closure for the second moments. Finally, the overall performance of this new model approach is tested with a typical upper mixed layer scenario in the northern North Sea.

## Abstract

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*–ε 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*–ε model. It is shown how near-surface dissipation rate measurements under breaking waves can be simulated by an extended *k*–ε model considering a shear-dependent closure for the second moments. Finally, the overall performance of this new model approach is tested with a typical upper mixed layer scenario in the northern North Sea.

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## Abstract

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.

## Abstract

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.

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## Abstract

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 free convection are carried out. For the latter experiment, comparison to recent large eddy simulation data is made. Finally, the well-known temperature profile data at OWS Papa are simulated for an annual cycle. The main result of this paper is that the overall performance of the new second-moment closure model by Canuto et al.—expressed as nondimensional stability functions—is superior compared to the others in terms of physical soundness, predictability, computational economy, and numerical robustness.

## Abstract

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 free convection are carried out. For the latter experiment, comparison to recent large eddy simulation data is made. Finally, the well-known temperature profile data at OWS Papa are simulated for an annual cycle. The main result of this paper is that the overall performance of the new second-moment closure model by Canuto et al.—expressed as nondimensional stability functions—is superior compared to the others in terms of physical soundness, predictability, computational economy, and numerical robustness.

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## Abstract

By means of a numerical model of an idealized flat-bottom estuary, the paper studies the hydrodynamic control of the turbidity zone by the combined effect of the salt wedge and tidal movements. The model is of two- dimensional (*x, z*) finite-difference type with high resolution in time and space. It computes momentum, surface elevation, salinity, suspended particulate matter (SPM), turbulent kinetic energy, and dissipation rate as prognostic state variables. At the seaward boundary a tidal forcing is applied. At the landward boundary a weir is situated where a constant freshwater discharge is prescribed. The initial SPM concentration is horizontally homogeneous. After simulating a few tidal periods the model results exhibit the evolution of a stable SPM peak (the estuarine turbidity maximum or ETM) at the tip of the salt wedge. An inspection of the tidal mean velocity profiles around the ETM shows that this trapping of SPM is due to a residual near-bottom upstream current in the region of the salt wedge. Three physical causes for this residual countercurrent are investigated in greater detail by numerical experiments, namely, (i) the *residual gravitational circulation,* (ii) the *tidal velocity asymmetry,* and (iii) *the tidal mixing asymmetry.* The first mechanism is related to the baroclinic part of the longitudinal pressure gradient. The second and third mechanism are based on the differences between the vertical profiles of velocity and SPM, respectively, at flood and ebb tide. For the macrotidal estuary considered here, the consideration of both (i) and (ii) could be shown to be *necessary* for the establishment of an ETM in the considered idealized estuary. It could further be shown that (iii) affects the ETM formation only quantitatively but not qualitatively and appears to be not necessary for the existence of an ETM.

## Abstract

By means of a numerical model of an idealized flat-bottom estuary, the paper studies the hydrodynamic control of the turbidity zone by the combined effect of the salt wedge and tidal movements. The model is of two- dimensional (*x, z*) finite-difference type with high resolution in time and space. It computes momentum, surface elevation, salinity, suspended particulate matter (SPM), turbulent kinetic energy, and dissipation rate as prognostic state variables. At the seaward boundary a tidal forcing is applied. At the landward boundary a weir is situated where a constant freshwater discharge is prescribed. The initial SPM concentration is horizontally homogeneous. After simulating a few tidal periods the model results exhibit the evolution of a stable SPM peak (the estuarine turbidity maximum or ETM) at the tip of the salt wedge. An inspection of the tidal mean velocity profiles around the ETM shows that this trapping of SPM is due to a residual near-bottom upstream current in the region of the salt wedge. Three physical causes for this residual countercurrent are investigated in greater detail by numerical experiments, namely, (i) the *residual gravitational circulation,* (ii) the *tidal velocity asymmetry,* and (iii) *the tidal mixing asymmetry.* The first mechanism is related to the baroclinic part of the longitudinal pressure gradient. The second and third mechanism are based on the differences between the vertical profiles of velocity and SPM, respectively, at flood and ebb tide. For the macrotidal estuary considered here, the consideration of both (i) and (ii) could be shown to be *necessary* for the establishment of an ETM in the considered idealized estuary. It could further be shown that (iii) affects the ETM formation only quantitatively but not qualitatively and appears to be not necessary for the existence of an ETM.

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## Abstract

In straight tidal estuaries, residual overturning circulation results mainly from a competition between gravitational forcing, wind forcing, and friction. To systematically investigate this for tidally energetic estuaries, the dynamics of estuarine cross sections is analyzed in terms of the relation between gravitational forcing, wind stress, and the strength of estuarine circulation. A system-dependent basic Wedderburn number

## Abstract

In straight tidal estuaries, residual overturning circulation results mainly from a competition between gravitational forcing, wind forcing, and friction. To systematically investigate this for tidally energetic estuaries, the dynamics of estuarine cross sections is analyzed in terms of the relation between gravitational forcing, wind stress, and the strength of estuarine circulation. A system-dependent basic Wedderburn number

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## Abstract

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.

## Abstract

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.

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## Abstract

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.

## Abstract

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.