Hans Burchard

Comparative Analysis of Four Second-Moment Turbulence Closure Models for the Oceanic Mixed Layer

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...

Comparing the performance of the Mellor‐Yamada and the κ‐ε two‐equation turbulence models

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.

The Formation of Estuarine Turbidity Maxima Due to Density Effects in the Salt Wedge. A Hydrodynamic Process Study

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...

Simulating the Wave-Enhanced Layer under Breaking Surface Waves with Two-Equation Turbulence Models

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–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 ...

Quantifying the Contributions of Tidal Straining and Gravitational Circulation to Residual Circulation in Periodically Stratified Tidal Estuaries

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 definitio...

Application of k‐ϵ turbulence models to enclosed basins: The role of internal seiches

[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

Impact of Density Gradients on Net Sediment Transport into the Wadden Sea

Abstract This study tests the hypothesis that horizontal density gradients have the potential to significantly contribute to the accumulation of suspended particulate matter (SPM) in the Wadden Sea. It is shown by means of long-term observations at various positions in the Wadden Sea of the German Bight that the water in the inner regions of the Wadden Sea is typically about 0.5–1.0 kg m−3 less dense than the North Sea water. During winter this occurs mostly because of freshwater runoff and net precipitation; during summer it occurs mostly because of differential heating. It is demonstrated with idealized one-dimensional water column model simulations that the interaction of such small horizontal density gradients with tidal currents generates net onshore SPM fluxes. Major mechanisms for this are tidal straining, estuarine circulation, and tidal mixing asymmetries. Three-dimensional model simulations in a semienclosed Wadden Sea embayment with periodic tidal forcing show that SPM with sufficiently high se...

A three-dimensional hydrostatic model for coastal and ocean modelling using a generalised topography following co-ordinate system

Abstract 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.

Gravity Current Dynamics and Entrainment—A Process Study Based on Observations in the Arkona Basin

Abstract 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 t...

Microstructure of turbulence in the northern North Sea: a comparative study of observations and model simulations

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.