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In the present paper, numerically robust, unconditionally positive and conservative schemes for the discretisation of stiff systems of production-destruction equations are designed. Such model systems do typically arise in geobiochemical modelling where the reproduction of these properties is vital. We suggest modified Patankar-type methods of first- and second-order in time and compare their performance by means of approximating simple linear and non-linear model problems. For the non-linear model problem, a hybrid method combining the classical Runge-Kutta scheme with a modified Patankar-type scheme gives the best numerical approximation. The classical Robertson test problem for chemical reactions which is known for its stiffness is excellently approximated with the modified Patankar-type scheme. The procedure with respect to the derivation and analysis of the modified Patankar-type schemes can be used as a guideline to develop even unconditionally positive, conservative and third-order as well as higher order methods.

A generalization of a class of differential length-scale equations typically used in second-order turbulence models for oceanic flows is suggested. Commonly used models, like the κ-e model and the Mellor-Yamada model, can be recovered as special cases of this generic model, and thus can be rationally compared. In addition, a method is proposed that yields a generalized framework for the calibration of the most frequently used class of differential length-scale equations. The generic model, calibrated with this method, exhibits a greater range of applicability than any of the traditional models. Stratified flows, plane mixing layers, and turbulence introduced by breaking surface waves are considered besides some classical test cases.

In this paper we use a combination of nu- merical modeling and data analysis to gain a better understanding of the major characteristics of the circu- lation in the East Frisian Wadden Sea. In particular, we concentrate on the asymmetry of the tidal wave and its modulation in the coastal area, which results in a com- plex pattern of responses to the sea-level forcing from the North Sea. The numerical simulations are based on the 3-D primitive equation General Estuarine Transport Model (GETM) with a horizontal resolution of 200 m and terrain-following vertical coordinates. The model is forced at its open boundaries with sea-level data from an operational model for the German Bight (German Hydrographic Office). The validation data for our model simulations include time series of tidal gauge data and surface currents measured at a pile in the back-barrier basin of the Island Langeoog, as well as several ADCP transects in the Accumer Ee tidal inlet. Circulation and turbulence characteristics are inves- tigated for typical situations driven by spring and neap tides, and the analysis is focused on dominating tem- poral and spatial patterns. By investigating the response of five back-barrier basins with rather different mor- phologies to external forcing, an attempt is made to elucidate the dominating physical balances controlling the circulation in the individual sub-basins. It is dem- onstrated that the friction at the seabed tends to slow down the tidal signal in the shallow water. This leads to the establishment of flood dominance in the shallowsea north of the barrier islands. South of the islands, where the water volume of the channels at low tide is smaller than the tidal prism, the asymmetry of the tidal signal is shifted towards ebb dominance, a feature which is par- ticularly pronounced at spring tide. At the northern open boundary, the tidal wave propagating from west to east generates a sea-level dif- ference of � 1 m along the boundary, and thereby trig- gers vigorous alongshore currents. The frictional control in the model is located in the inlets, as well as along the northern boundary. The correlation between velocity and turbulent kinetic energy tends to the establishment of a net southward transport, giving theoretical support to the observed accumulation of sediments on the in- tertidal flats. Weak turbulence along the northern shores of the barrier islands and the small magnitude of the residual currents there promote accumulation of sus- pended matter in these areas, although wave action will generally counteract this effect.

Four different two-equation turbulence models for geophysical flows are compared: The k – ϵ model, two new versions of the k – ω model, and the Mellor–Yamada model. An extension of the k – ω model for buoyancy affected and rotating flows is suggested. Model performance is evaluated for a few typical oceanic flows. First , new analytical solutions of the models for the surface layer affected by breaking surface waves are discussed. The deficiencies of earlier attempts are high-lighted, and it is demonstrated why the Mellor–Yamada model and the k – ϵ model fail. It is illustrated that only one version of the k – ω model computes correct decay rates for turbulent quantities under breaking waves. Second , it is demonstrated that all models predict almost identical mixed layer depths and profiles for the turbulent kinetic energy in a classical stratified shear-entrainment experiment if the buoyancy term in the second equation is appropriately weighted. Third , the accuracy and numerical robustness of the new k – ω model in realistic oceanic situations is confirmed by comparison with the data-set of the Ocean Weather Ship ‘Papa’.

Stratification and destratification processes in a tidally energetic, weakly stratified inlet in the Wadden Sea (south eastern North Sea) are investigated in this modeling study. Observations of current velocity and vertical density structure show strain-induced periodic stratification for the southern shoal of the tidal channel. In contrast to this, in the nearby central region of the channel, increased stratification is already observed directly after full flood. To investigate the processes leading to this different behavior, a nested model system using GETM is set up and successfully validated against field data. The simulated density development along a cross section that includes both stations shows that cross-channel stratification is strongly increasing during flood, such that available potential energy is released in the deeper part of the channel during flood. An analysis of the potential energy anomaly budget confirms that the early onset of vertical stratification during flood at the deeper station is mainly controlled by the stratifying cross-channel straining of the density field. In contrast to this, in the shallow part of the channel, the relatively weak cross-channel straining is balanced by along-channel straining and vertical mixing. An idealized analytical model confirms the following hypothesis: The laterally convergent flood current advecting laterally stratified water masses from the shallow and wide ebb tidal delta to the deep and narrow tidal channel has the tendency to substantially increase cross-channel density gradients in the tidal channel. This process leads to stratification during flood.

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.

The Liverpool Bay Region of Freshwater Influence in the Irish Sea exhibits strong horizontal gradients which interact with the dominant tidal flow. A 25 h series of measurements of the cycle of turbulent dissipation with the FLY dissipation profiler shows a strong asymmetry between ebb and flood which is associated with a cycle of increasing stratification on the ebb and progressive mixing on the flood which results in vertical homogeneity as high water is approached. At this time strong dissipation extends throughout the water column in contrast to the ebb when there is a near shutdown of dissipation in the upper half of the column. The cycle of stratification and dissipation is closely consistent for the two semi-diurnal tidal cycles observed. We have attempted to simulate this situation, which involves a complex suite of processes including tidal straining and mixing, using a version of the k–e closure scheme in a 1-d dynamical model which is forced by a combination of the observed tidal flow and horizontal temperature and salinity gradients. The latter were measured directly at the end of the observational series but, in order to focus on the cycle of dissipation, the correct reproduction of the temperature and salinity cycle can be assured by a nudging procedure which obliges the model temperature and salinity values to track the observations. With or without this procedure, the model gives a reasonable account of the dissipation and its asymmetric behaviour on ebb and flood although nudging improves the timing of peak dissipation in the upper part of the water column near highwater. The model has also been used to examine the ratio of shear production (P/e) and buoyancy inputs to dissipation (B/e). The variation of these quantities over the tidal cycle confirms the important role of convective motions forced by tidal straining near the end of the flood phase of the tide.

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.

Dissipation rate measurements in the northern North Sea from two independent observations are compared with various numerical models. The turbulence was characterised by tidal forcing in the bottom boundary layer and atmospheric forcing in the surface boundary layer. The observations were carried out by using free-falling profilers equipped with shear probes and fast CTD sensors. The models are based on Reynolds averaging and range from simple one-equation models to two-equation models with algebraic second-moment closures. Several error measures are applied for comparison of observations and model results. It is shown that the differences between the two observations are significantly larger than the equivalent measures between the model results. This is caused by the stochastic character of turbulent microstructure in connection with under-sampling, but also by the distance between the two observational sites, the movements of the vessels, instrument errors and so forth. The models on the other hand, although closed on different levels, are all based on the same assumptions and driven by the same external forcing, thus showing only relatively small differences between each other.

A three-dimensional hydrostatic model is presented that combines a generalised vertical co-ordinate system with an efficient implicit solution technique for the free surface. The model is capable of maintaining high resolution in the surface and/or bottom boundary layers as well as dealing with steep topography. Horizontal diffusion is calculated using the Smagorinsky formulation and a k–e turbulence model is used in the vertical. In addition the model uses higher-order advection routines. An important aspect in three-dimensional models is the choice of vertical discretisation. If one is mostly interested in problems which are governed by boundary layer flows, a terrain following or sigma co-ordinate system seems attractive. This paper focuses on the development of a generalised sigma-type grid in a three-dimensional hydrostatic model. The generalised grid offers a wide range of possibilities including grid refinement toward the bed or surface, a mixed layer transformation, and a constant layer transformation where the lowermost or uppermost grid cells can be specified to have a constant height above the bed or below the surface. A number of tests are presented which show that the model is capable of simulating both shallow nearshore, estuarine flows as well as large-scale geophysical flows. These include an extreme flooding event in the shallow North Sea and the Odden ice tongue formation in the Greenland Sea.

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.

Stability problems of algebraic non-equilibrium second-moment closure models have given rise to the so-called quasi-equilibrium versions in which turbulence equilibrium is used as an additional constraint. In this paper, we investigate reasons for the failure of the G.L. Mellor, T. Yamada [Reviews of Geophysics 20 (1982) 851] level 2.5 closure model and suggest a remedy for this. We further discuss a new non-equilibrium closure model by V.M. Canuto, A. Howard, Y. Cheng, M.S. Dubovikov (Journal of Physical Oceanography, 2000, accepted for publication) which has proven to allow for stable calculations. All models are then numerically tested with a simple wind entrainment experiment motivated by the H. Kato, O.M. Phillips [Journal of Fluid Mechanics 37 (1969) 643] laboratory experiment, with the aid of which the instability of the Mellor and Yamada (1982) and the stability of the Canuto et al. (2000) model are confirmed. The Canuto et al. (2000) model has three advantages compared to the Mellor and Yamada (1982) which are (i) the symmetry of stability functions, (ii) a higher critical Richardson number, and (iii) that the normalised shear stress increases with normalised shear for turbulence equilibrium. The latter advantage of the new model causes its high physical and numerical stability.

A method to improve the behavior of the numerical discretization of a rotated diffusion operator such as, for example, the isopycnal diffusion parameterization used in large-scale ocean models based on the so-called z-coordinate system is presented. The authors then focus exclusively on the dynamically passive tracers and analyze some different approaches to the numerical discretization. Monotonic schemes are designed but are found to be rather complex, while simpler, linear schemes are shown to produce unphysical undershooting and overshooting. It is suggested that the choice of an appropriate discretization method depends on the importance of the rotated diffusion in a given simulation, whether the field to be diffused is dynamically active or not.

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.

Another reason why simple discretizations of rotated diffusion operators cause problems in ocean models : Comments on Isoneutral diffusion in a z-coordinate ocean model

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.

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...

Estuarine convergence (landward reduction of width and/or depth) is known to have the potential to significantly enhance estuarine circulation, a result theoretically derived under the assumption of constant eddy viscosity. Recent studies of longitudinally uniform energetic tidal channels indicate that tidal straining, a process driven by tidally varying eddy viscosity, is a major driver of estuarine circulation. The combined effect of estuarine convergence and tidal straining is investigated, for the first time, in this paper. The present idealized numerical study shows that estuarine convergence is reducing or even reversing tidal straining circulation in such a way that estuarine circulation can be weakened. This is a counterintuitive hydrodynamic effect of estuarine convergence, which may reduce (rather than increase) up-estuary particulate matter transport in estuaries and tidal inlets.

Berdalet, Elisa ... et. al.-- Special issue Harmful Algal Blooms in Stratified Systems.-- 17 pages, 12 figures, The Global Ecology and Oceanography of Harmful Algal Blooms (GEOHAB) program of the Scientific Committee on Oceanic Research (SCOR) and the Intergovernmental Oceanographic Commission (IOC) of UNESCO, was created in 1999 to foster research on the ecological and oceanographic mechanisms underlying the population dynamics of harmful algal blooms (HABs). The ultimate goal of this research is to develop observational systems and models that will eventually enable the prediction of HABs and thereby minimize their impact on marine ecosystems, human health and economic activities. In August of 2012, a workshop was held under the umbrella of the GEOHAB program at the Monterey Bay Aquarium Research Institute (MBARI). The over arching goal of this workshop was to review the current understanding of the processes governing the structure and dynamics of HABs in stratified systems, and to identify how best to couple physical/chemical and biological measurements at appropriate spatial and temporal scales to quantify the dynamics of HABs in these systems, paying particular attention to thin layers. This contribution provides a review of recent progress in the field of HAB research in stratified systems including thin layers, and identifies the gaps in knowledge that our scientific community should strive to understand in the next decade, We are also grateful to the organizations that contributed financially to the meeting, specifically the Scientific Committee on Oceanic Research (SCOR), the Intergovernmental Oceanographic Commission (IOC) of UNESCO, and the Monterey Bay Aquarium Research Institution (MBARI. EB was funded by the Spanish Government through the ECOALFACS project (CTM2009-09581). ONR wishes to acknowledge financial support from the European Union Seventh Framework Programme (FP7/2007–2013)and Marie Curie IntraEuropean Fellowship for Career Development under Grant agreement no. 255396

It is discussed in this paper how the pressure gradient error in general vertical co-ordinate models (in which the σ-transformation is a special case) can be reduced by means of hybrid models. For a better understanding, the derivation of such a general vertical co-ordinate model from the Cartesian co-ordinate model is given. Two types of hybridization between σ- and z-co-ordinate models, each using one parameter specifying the degree of hydridization, are presented: (i) the mixed layer transformation with a constant number of layers which are refined near the surface and (ii) the z/σ-transformation which introduces steps near the bottom. In order to achieve good results with the models using other than σ-co-ordinates, a profile-conserving momentum advection discretization is developed. The different co-ordinate transformations are tested with 2D barotropic and baroclinic flows over a topographic bump. Those models with nearly horizontal co-ordinate surfaces in the stratified area give the best correspondence with an isopycnal reference solution. © 1997 John Wiley & Sons, Ltd.

In this idealized numerical modeling study, the composition of residual sediment fluxes in energetic (e.g., weakly or periodically stratified) tidal estuaries is investigated by means of one-dimensional water column models, with some focus on the sediment availability. Scaling of the underlying dynamic equations shows dependence of the results on the Simpson number (relative strength of horizontal density gradient) and the Rouse number (relative settling velocity) as well as impacts of the Unsteadiness number (relative tidal frequency). Here, the parameter space given by the Simpson and Rouse numbers is mainly investigated. A simple analytical model based on the assumption of stationarity shows that for small Simpson and Rouse numbers sediment flux is down estuary and vice versa for large Simpson and Rouse numbers. A fully dynamic water column model coupled to a second-moment turbulence closure model allows to decompose the sediment flux profiles into contributions from the transport flux (product of subtidal velocity and sediment concentration profiles) and the fluctuation flux profiles (tidal covariance between current velocity and sediment concentration). Three different types of bottom sediment pools are distinguished to vary the sediment availability, by defining a time scale for complete sediment erosion. For short erosion times scales, the transport sediment flux may dominate, but for larger erosion time scales the fluctuation sediment flux largely dominates the tidal sediment flux. When quarter-diurnal components are added to the tidal forcing, up-estuary sediment fluxes are strongly increased for stronger and shorter flood tides and vice versa. The theoretical results are compared to field observations in a tidally energetic inlet.

Floser, G., Nauw, J., Burchard, H., Riethmuller, R., 2013. Observational evidence for the inward transport of suspended matter by estuarine circulation in the Wadden Sea. Observational evidence is presented that corroborates the hypothesis that estuarine circulation represents a generic process in the Wadden Sea, independent of (the amount of) river runoff. Long-term current velocity data from three locations in the Wadden Sea, taken on moored acoustical current meters, onboard anchored ships and across-channel ship-of-opportunity platforms show in all cases features concurring with the predictions of the theory: a tidally residual outflow of Wadden Sea waters in the upper part and a residual inflow of seawater in the lower part of the water column. This may be regarded as a further confirmation for a generic process to import suspended sediments from the German Bight / North Sea into the Wadden Sea against the prevailing concentration gradient.

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.

This paper describes an online coupling between a 3D discontinuous Galerkin finite element marine model and a 1D vertical turbulence closure model based on finite differences. The coupling exploits the topology of the 3D mesh, that is formed by stacking layers of prisms in the vertical direction. A robust mapping between the finite difference grid and the finite element function space is designed, taking into account the discontinuities in the latter. The coupling is tested with two horizontally homogeneous flows and an idealised 3D estuary simulation. The results are in good agreement with those obtained with a finite difference model using the same turbulence closure, indicating that the coupling does not deteriorate the performance of the turbulence model.

[1] This study presents, for the first time, micro-structure observations in tidally energetic, weakly stratified regimes obtained in the Wadden Sea, a tidal shallow coastal area in the South-Eastern North Sea characterised by barrier islands separated by tidal gulleys. The tidal currents are typically overlaid by a weak horizontal density gradient due to freshwater run-off from land. The observations in an energetic tidal channel clearly show the expected effects of tidal straining: destratification during flood and increased stratification during ebb. Microstructure observations are consistent with the tidal straining dynamics: during the flood relatively high values of viscous dissipation are observed whilst during ebb the values are substantially smaller. It is also shown that the tidal cycle of stratification and destratification depends on the position in the tidal channel. In parts of the channel, increased stratification occurs during full flood, a phenomenon which can only be explained by advection of stratified water masses formed outside the tidal channel. The observations presented here show the general significance of the tidal straining process for tidally energetic weakly stratified regimes.

Global coupled climate models are generally capable of reproducing the observed trends in the globally averaged atmospheric temperature. However, the global models do not perform as well on regional scales. Here, we present results from two 140-year, high-resolution regional ocean model experiments for the Western Baltic Sea. The forcing is taken from a regional atmospheric model and a medium scale ocean model. The model runs with two greenhouse gas emission scenarios (each for 100 years), A1B and B1, for the period 2000–2100. A control run (C20) from 1960 to 2000 is used for validation. For both scenarios, the results show the expected warming, with an increase of 0.5–2.5 K at the sea surface and 0.7–2.8 K below 40 m. The simulations further indicate a decrease in salinity, a change in stratification, and an increase of the return period of storm surges.

[1] Wind is shown to have an important influence on water column structure and residual circulation through a straining mechanism in the Liverpool Bay region of freshwater influence. By combining a long-term data set of water column observations collected from a mooring we estimate the horizontal gradients in temperature and salinity that are used to drive a one-dimensional turbulence model. Various combinations of the processes that impact the stratification are investigated, including the wind speed and direction, horizontal buoyancy gradient, and the tidal amplitude. From this we assess the significance of these processes and find that the effect of wind direction is important in determining water column structure. Both observations and a modeling study show that an offshore wind accelerates the development of stratification on the ebb tide and decelerates the destruction of stratification on the flood tide due to a “wind straining” process. The reverse situation occurs for an onshore wind.

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.

Biochemical systems are bound by two mathematically-relevant restrictions. First, state variables in such systems represent non-negative quantities, such as concentrations of chemical compounds. Second, biochemical systems conserve mass and energy. Both properties must be reflected in results of an integration scheme applied to biochemical models. This paper first presents a mathematical framework for biochemical problems, which includes an exact definition of biochemical conservation: elements and energy, rather than state variable units, are conserved. We then analyze various fixed-step integration schemes, including traditional Euler-based schemes and the recently published modified Patankar schemes, and conclude that none of these deliver unconditional positivity and biochemical conservation in combination with higher-order accuracy. Finally, we present two new fixed-step integration schemes, one first-order and one second-order accurate, which do guarantee positivity and (biochemical) conservation. © 2005 IMACS.

In this paper, we apply recently developed positivity preserving and conservative Modified Patankar-type solvers for ordinary differential equations to a simple stiff biogeochemical model for the water column. The performance of this scheme is compared to schemes which are not unconditionally positivity preserving (the first-order Euler and the second- and fourth-order Runge–Kutta schemes) and to schemes which are not conservative (the first- and second-order Patankar schemes). The biogeochemical model chosen as a test ground is a standard nutrient–phytoplankton–zooplankton–detritus (NPZD) model, which has been made stiff by substantially decreasing the half saturation concentration for nutrients. For evaluating the stiffness of the biogeochemical model, so-called numerical time scales are defined which are obtained empirically by applying high-resolution numerical schemes. For all ODE solvers under investigation, the temporal error is analysed for a simple exponential decay law. The performance of all schemes is compared to a high-resolution high-order reference solution. As a result, the second-order modified Patankar–Runge–Kutta scheme gives a good agreement with the reference solution even for time steps 10 times longer than the shortest numerical time scale of the problem. Other schemes do either compute negative values for non-negative state variables (fully explicit schemes), violate conservation (the Patankar schemes) or show low accuracy (all first-order schemes).

A new numerical model for simulating estuarine dynamics is introduced here. This model, called General Estuarine Transport Model (GETM), has been specifically designed for reproducing baroclinic, bathymetry-guided flows where the tidal range may exceed the mean water depth in large parts of the domain such that drying and flooding processes are relevant. Several physical and numerical features of the model support exact and stable results for such domains. For the physics, high-order turbulence closure schemes guarantee proper reproduction of vertical exchange processes. Among the specific numerical features, generalised vertical coordinates, orthogonal curvilinear horizontal coordinates, high-order TVD advection schemes and stable drying and flooding algorithms have been implemented into GETM. The model is applied here to simulate the dynamics of estuarine turbidity maxima (ETMs), a complex feature present in most tidal estuaries. First, idealised simulations for a two-dimensional domain in the xz space will be shown to reproduce the basic generation mechanisms for ETMs. Then, a realistic three-dimensional simulation of the Elbe estuary in Northern Germany will be carried out. It is demonstrated that for a given forcing situation the model reproduces a stable ETM at the correct location.

Publisher Summary This chapter describes the application of hydrodynamic circulation models for simulating oceanic dispersion of radioactivity. It reviews the application of numerical hydrodynamic models to the transport of anthropogenic radioactivity in the ocean. There is a large number of such applications. The intention is to summarize different approaches and model configurations, to evaluate strengths and weaknesses and to give guidelines to users. The review looks at HD models and transport models separately, but with the focus equally weighted on both model types. HD and transport models have been applied for several years now. Model physics and numerical methods have been validated in many applications and both model types present well established tools in marine environmental assessment studies. In addition, the chapter mentions that numerical models benefit in general from increasing computer power. This allows better spatial resolutions which improve the results in nearly all aspects. However, higher resolution is not a universal remedy that cures all problems in numerical ocean models.

2 pages.-- Issue title: "Impact of Small-scale Physics on Marine Biology - Selected papers from the 2nd Warnemunde Turbulence Days"

[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.