Dispersion in surface waters : with special references to
particle models and synthetic eddy velocity fields
Citation for published version (APA):
Dam, van, G. C. (2009). Dispersion in surface waters : with special references to particle models and synthetic
eddy velocity fields. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR642238
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10.6100/IR642238
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1 A three-dimensional transport model
Dispersion in surface waters
With special reference to particle models and synthetic eddy velocity fields
Preface
3 A three-dimensional transport model
Dispersion in Surface Waters
With special reference to particle models and synthetic eddy velocity fields
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van
de Rector Magnificus, prof.dr.ir. C.J. van Duijn, voor een commissie aangewezen door het College
voor Promoties in het openbaar te verdedigen op maandag 30 maart 2009 om 16.00 uur
door
Gerrit Cornelis van Dam geboren te Utrecht
Preface
4
Dit proefschrift is goedgekeurd door de promotor:
prof.dr.ir. G.J.F. van Heijst
Dispersion in surface waters / G. C. van Dam
Thesis Eindhoven University of Technology - With ref. - With summary in English and Dutch ISBN/EAN: 978-90-9024084-8
NUR 930
Keywords: dispersion / surface waters / particle models / North Sea / eddy fields / kinetic energy spectrum Trefwoorden: dispersie / oppervlaktewateren / deeltjesmodellen / Noordzee / wervelvelden / energiespectrum
5 A three-dimensional transport model
Aan mijn kinderen en kleinkinderen
Dit proefschrift is goedgekeurd door de promotor:
prof.dr.ir. G.J.F. van Heijst
Dispersion in surface waters / G. C. van Dam
Thesis Eindhoven University of Technology - With ref. - With summary in English and Dutch ISBN/EAN: 978-90-9024084-8
NUR 930
Keywords: dispersion / surface waters / particle models / North Sea / eddy fields / kinetic energy spectrum Trefwoorden: dispersie / oppervlaktewateren / deeltjesmodellen / Noordzee / wervelvelden / energiespectrum
Preface
7 A three-dimensional transport model
C o n t e n t s
Preface ...9
Chapter 1...11
Introduction...13
1.1. Some historical background...13
1.2. The particle approach...14
1.3. The eddy field approach...17
1.4. Outline of this thesis ...19
Chapter 2...23
Residual currents and transport in connection with two-dimensional model computations by G. C. van Dam Report GWAO-88.042 (1988). Translated from Dutch (rpt. FA 8402) (1984) Chapter 3... 41
A three-dimensional transport model for dissolved and suspended matter in estuaries and coastal seas by G. C. van Dam and R. A. Louwersheimer In D. Prandle, Ed., Dynamics and exchanges in estuaries and the coastal zone. Coastal and Estuarine Studies, Washington, D. C., 40, 481-506 (1992) Chapter 4... 69
Model particles as representatives of particles in nature by G. C. van Dam and M. J. J. M. Geurtz Netherlands Journal of Aquatic Ecology 28 (3-4) 317-328 (1994) Chapter 5...83
Study of shear dispersion in tidal waters by applying discrete particle techniques by G. C. van Dam In: K. J. Beven, P. C. Chatwin and J. H. .Millbank, Eds., Mixing and transport in the environment. John Wiley and Sons, Ltd., 269-293 (1994) Chapter 6...111
Spectral structure of horizontal water movement in shallow seas with special reference to the North Sea, as related to the dispersion of dissolved matter by G. C. van Dam, R. V. Ozmidov, K. A. Korotenko and J. M. Suijlen Journal of Marine Systems 21, 207-228 (1999) Chapter 7...135
The spreading and dilution of waste water from production platforms in the North Sea by G. C. van Dam Aqua Systems International, Poeldijk, Netherlands, Report 2000 11.2E (2000) Chapter 8...161
Spectral representation of horizontal velocity variations as applied to particle dispersion modelling by G. C. van Dam Aqua Systems International, Poeldijk, Netherlands, Report 2003.06.1 (2003) Chapter 9...197
The energy balance of the North Sea by G. C. van Dam and G. J. F. van Heijst Submitted for publication (2008) Summary and concluding remarks...215
Samenvatting...219
Dankwoord / acknowledgements ...221
Curriculum vitae ...225
Preface
9 A three-dimensional transport model
Preface
The main purpose of this thesis is to compile information on the subject as it has been reported earlier in a scattered way. The intention is to make this information more traceable and usable. This especially regards the mathematical modelling, principles as well as details, insofar they are still of interest, so as to make this knowledge fully accessible and applicable.
The descriptions of the models contain sufficient detail to be used in up-to-date computer codes, utilising modern software and hardware facilities. Chapter 8 (2003) was even specially written for the latter purpose, commissioned by Rijkswaterstaat, The Netherlands. By including this report in this thesis, under permission of Rijkswaterstaat, technical details become available for a wider circle of users.
In the other chapters some more detail is given and a number of applications is described. In general it has been avoided to include material of merely historical significance. Such information remains accessible by means of the extensive lists of references.
Only a global historical background survey is given in the first section of the introduction. For maximum accessibility, the publications and reports selected for this thesis are all in English and so are the additional texts. In the comprehensive reference list at the end, relevant literature in Dutch is included as well.
Preface
11 Introduction
Chapter 1
Chapter 1
13 Introduction
1. Introduction
1.1. Some historical background
In the early sixties of last century, by initiative of J. C. Schönfeld (in that period head Physics Division, Rijkswaterstaat, The Netherlands) the author started the first empirical research of dispersion in surface waters in The Netherlands. This was of course related to the increased urgency of environmental problems arising in the fifties, in the western countries and Japan. Before measurements started in The Netherlands, some experimental research in natural waters had already taken place elsewhere (in the USA by e.g. Pritchard and Carpenter1), but on a rather limited scale.
Little insight existed in the influence of local factors.
In the late fifties and early sixties Schönfeld already made and reported2,3,4,5 some theoretical
studies. There were similar activities in the same period at the Deutsche Hydrographische Institut in Hamburg6,7,8. Joseph and Sendner even dared to make an estimate of a spectrum of the velocity
variations in the North Sea and the ocean6, 7, based on available flow data. Schönfeld as well as his
German colleagues were already aware of the non-classical, scale dependent character of turbulent diffusion in surface waters. They referred to earlier theoretical work by Kolmogorov, especially to his concept of transfer of turbulent energy in three-dimensional turbulent media with a high Reynolds number9, resulting in the kinetic energy spectrum E ∝ k−5/3 ("inertial range") with k the wavenumber.
However, one realised that except for scales smaller than the water depth, this relation could not hold in the shallow waters at stake, since these have a two-dimensional character at larger scales.
In the same period Schönfeld wrote a small numerical code for the recently introduced digital computer. In those days digital computers were still very rare. Schönfeld used the X8 in Amsterdam. His computer code enabled the calculation of concentrations in discrete points, as caused by a continuous source at some distance. So in principle a whole field of concentrations could be computed at various points of time. The calculation was based on the addition of a series of a large number of analytical distributions originating from "instantaneous point sources". For the time dependency of the top values of these basic concentration distributions, Schönfeld had programmed three options: cmax ∝
t-−3 corresponding to Kolmogorovs inertial range, c
max ∝ t−2 (proposed by Joseph and Sendner) and cmax
∝ t−1 ("classical" diffusion). The experiments should reveal what option was best.
In the years following, dispersion experiments in the North Sea were performed by several countries, including The Netherlands10-27. In 1965 a large international experiment was carried out in
the central part of the North Sea with participation of Germany, the United Kingdom and The Netherlands, the RHENO-experiment (Rhodamine Experiment North Sea)19. After that, at
Rijkswaterstaat (The Netherlands) the author and his team carried out many other experiments of the same kind12,13,17,18,20,22-27. As far as regards the North Sea, most of these experiments took place at
shorter distances to the coast than RHENO; some were performed in the Dutch estuaries24, in the
IJsselmeer (Lake IJssel)22 and in some rivers. The tests at sea continued through the eighties, two final
experiments were performed in the early nineties.
In the same period similar activities took place in other parts of the world. The direct field studies of dispersion made use of tracers, mainly rhodamine-B. Like in the North Sea, the majority of these tests started with a short ("instantaneous") release of small extent, so as to approach a point release in space and time as closely as possible.
Everywhere it was found that none of the three above mentioned approximations of time behaviour with a constant exponent (-3, -2 or -1) fitted the observations, except sometimes for a short duration. When it was tried to fit the data from longer experiments or from a group of experiments globally to a t−β curve with a constant β-value, β-values somewhat larger than 2 gave the best
fit13,17,23,25,28-31,37,38,40. Okubo and Ozmidov32 fitted such a group of experiments by a cascade of three
successive t−3 curves on different levels, reasoning that this was in agreement with the hypothetical
energy cascade proposed by Ozmidov33 in 1965 for the (mostly horizontal!) eddies in the ocean. Also
here, reference was made to Kolmogorov, neglecting that, except for the smaller scales, the turbulence did not have the 3D character required by Kolmorov's theory. It is remarkable that Okubo and Ozmidov in their paper contributed the horizontal dispersion at all scales to eddies, while in other publications Okubo extensively dealt with shear dispersion as an important dispersion
Chapter 1
14
mechanism34,35,36. It seems that Okubo did not draw the consequence that the shear between velocities
at different heights was playing an important part in the processes shaping the dilution curves for seas and lakes, although this could be concluded from the quantitative effects Okubo calculated in his publications on shear dispersion.
It is true though that a complete evaluation of the joint effect of eddies and shear can only be obtained by a model in which both are explicitly taken into account. This was accomplished by the author38,39,44 in his model that simulates the dispersing matter by particles driven by a vertically
structured velocity field simultaneously with a field of eddies of all relevant scales. It should be added here that for practical applications the effect of the eddies can be fairly well approximated by a refined mode of the computationally less intensive method of "scaled random walk" 41.
The particle approach and the modelling of the effect of spectral eddy structure (directly or indirectly) are both essential for a correct model simulation of dispersion by the joint effect of velocity shear and eddy structure in a three-dimensional system.
The particle concept was introduced at the Rijkswaterstaat by the author after the example of Ernst Maier-Reimer42 in his Ph.D. thesis (1973). Maier-Reimer was probably the first to combine a
particle technique with a hydrodynamic model. While Maier-Reimer after taking his Ph.D. used this concept only a few times, it was worked out in detail and used in practice at Rijkswaterstaat in the eighties for quite a few practical purposes and soon it was also implemented at the Delft Hydraulics Laboratory. An applied particle model for calamitous releases at sea was developed in a joint project (called MARS) of Delft Hydraulics and Rijkswaterstaat, initially in two dimensions, soon in three, to account for the simultaneous action of shear dispersion and spectral structure explicitly.
Although from 1980 the author performed numerical experiments with particles in explicit spectrally structured eddy fields40,46,47 and studied the dispersion characteristics of these fields using a
super-computer40, in the eighties the concept was not yet used in combination with hydrodynamic
models. One of the practical reasons was the required computing time. In addition, research capacity was limited and largely consumed by the tracer experiments.
Therefore, in this period, "scaled random walk" was used to account for spectral effects. This approach was by that time restricted to spectra approximated by a k−α function of the wave number k.
It should be mentioned that Maier-Reimer in his thesis already mentions the possibility of using the related t−β functions for concentration decrease in time. As far as we know, he did not use the scaling
idea in applications.
In 1990, at the IUTAM Symposium on Fluid Mechanics of Stirring and Mixing38 the author
presented some first results obtained with a model of a shearing (tidal) velocity field combined with an eddy field having a spectral structure different from the "first order" k−α approximation. In an abstract
in Physics of Fluids39 this was summarised as follows. "By adjusting the spectrum of the horizontal
velocity structure to the rather extensive set of field data, a theoretical envelope was obtained from the model, enclosing the set of measured curves quite accurately". The relevant graph can be found in this thesis (ch.4 fig.6, ch.5 fig.14.13d). This was a crucial stage of development in the project. Finally, in 1999 (chapter 6 of this thesis) an explicit and more detailed North Sea energy spectrum could be published, based upon a larger set of data, compiled by Suijlen and Van Dam in the meantime44; orally
presented at the 29th Liège Colloquium on Ocean Hydrodynamics, "Turbulence Revisited" (1997). The computed dilution curves in the concentration vs. time graph afterwards seem to contain some small imperfections (the relevant computer codes have become inaccessible). It is most satisfactory though that new computations in 2007 with an improved code (chapter 9 of this thesis) fully support the spectrum as it was published in the late nineties. No indications for adjustments were found.
1.2 The particle approach
This subject, in the context of this thesis, refers to particle models for the simulation of dispersion of matter in surface waters. Yet it would be an omission not to refer to the first particle model in literature, the one that Albert Einstein43 is using in one of his famous 1905 papers for
describing Brownian motion and at the same time showing that a simple random walk approach is equivalent to classical diffusion (strictly in the limit for particle numbers going to infinity and random
15 Introduction
mechanism34,35,36. It seems that Okubo did not draw the consequence that the shear between velocities
at different heights was playing an important part in the processes shaping the dilution curves for seas and lakes, although this could be concluded from the quantitative effects Okubo calculated in his publications on shear dispersion.
It is true though that a complete evaluation of the joint effect of eddies and shear can only be obtained by a model in which both are explicitly taken into account. This was accomplished by the author38,39,44 in his model that simulates the dispersing matter by particles driven by a vertically
structured velocity field simultaneously with a field of eddies of all relevant scales. It should be added here that for practical applications the effect of the eddies can be fairly well approximated by a refined mode of the computationally less intensive method of "scaled random walk" 41.
The particle approach and the modelling of the effect of spectral eddy structure (directly or indirectly) are both essential for a correct model simulation of dispersion by the joint effect of velocity shear and eddy structure in a three-dimensional system.
The particle concept was introduced at the Rijkswaterstaat by the author after the example of Ernst Maier-Reimer42 in his Ph.D. thesis (1973). Maier-Reimer was probably the first to combine a
particle technique with a hydrodynamic model. While Maier-Reimer after taking his Ph.D. used this concept only a few times, it was worked out in detail and used in practice at Rijkswaterstaat in the eighties for quite a few practical purposes and soon it was also implemented at the Delft Hydraulics Laboratory. An applied particle model for calamitous releases at sea was developed in a joint project (called MARS) of Delft Hydraulics and Rijkswaterstaat, initially in two dimensions, soon in three, to account for the simultaneous action of shear dispersion and spectral structure explicitly.
Although from 1980 the author performed numerical experiments with particles in explicit spectrally structured eddy fields40,46,47 and studied the dispersion characteristics of these fields using a
super-computer40, in the eighties the concept was not yet used in combination with hydrodynamic
models. One of the practical reasons was the required computing time. In addition, research capacity was limited and largely consumed by the tracer experiments.
Therefore, in this period, "scaled random walk" was used to account for spectral effects. This approach was by that time restricted to spectra approximated by a k−α function of the wave number k.
It should be mentioned that Maier-Reimer in his thesis already mentions the possibility of using the related t−β functions for concentration decrease in time. As far as we know, he did not use the scaling
idea in applications.
In 1990, at the IUTAM Symposium on Fluid Mechanics of Stirring and Mixing38 the author
presented some first results obtained with a model of a shearing (tidal) velocity field combined with an eddy field having a spectral structure different from the "first order" k−α approximation. In an abstract
in Physics of Fluids39 this was summarised as follows. "By adjusting the spectrum of the horizontal
velocity structure to the rather extensive set of field data, a theoretical envelope was obtained from the model, enclosing the set of measured curves quite accurately". The relevant graph can be found in this thesis (ch.4 fig.6, ch.5 fig.14.13d). This was a crucial stage of development in the project. Finally, in 1999 (chapter 6 of this thesis) an explicit and more detailed North Sea energy spectrum could be published, based upon a larger set of data, compiled by Suijlen and Van Dam in the meantime44; orally
presented at the 29th Liège Colloquium on Ocean Hydrodynamics, "Turbulence Revisited" (1997). The computed dilution curves in the concentration vs. time graph afterwards seem to contain some small imperfections (the relevant computer codes have become inaccessible). It is most satisfactory though that new computations in 2007 with an improved code (chapter 9 of this thesis) fully support the spectrum as it was published in the late nineties. No indications for adjustments were found.
1.2 The particle approach
This subject, in the context of this thesis, refers to particle models for the simulation of dispersion of matter in surface waters. Yet it would be an omission not to refer to the first particle model in literature, the one that Albert Einstein43 is using in one of his famous 1905 papers for
describing Brownian motion and at the same time showing that a simple random walk approach is equivalent to classical diffusion (strictly in the limit for particle numbers going to infinity and random
step size going to zero). The latter is a general result that also applies to our subject, especially to the vertical mixing where the classical diffusion concept is applied.
Also Maier-Reimer42 refers to the very early and classical paper by Einstein when introducing
particle simulation in a hydrodynamic model. In first instance Maier-Reimer proposes to apply the same simple random walk approach as used by Einstein for complementing the deterministic horizontal velocities given by the hydrodynamic model, but at the end of his thesis he rightly observes that for realistic results, a scaling of the random steps is needed, so as to account for the presence of eddies of various sizes. It may be added here that the proposed increase of random step size with increasing length scale must gradually be reduced and finally ended, some time after the dispersing system reaches the scale of the mesh of the hydrodynamic model, since from a certain scale beyond the size of the mesh, the model will gradually start to generate the larger eddies. It is not so strange that this latter point is not mentioned by Maier-Reimer: by the time he wrote his thesis, the grid size of hydrodynamic models was so coarse that in particle simulations a particle cloud size of several meshes was in fact beyond feasibility. An illustration of this is figure 1a of chapter 4, where an example from Maier-Reimer's thesis is reproduced. The example nicely illustrates that the whole phenomenon occurs within a range of less than three meshes of the hydrodynamic model (the dispersion scale is even much smaller: it corresponds to the plume width!), which means that the latter is still far from reproducing the larger eddies. In fact, eddies need to have a size of some 5 grid lengths for being properly generated by the deterministic model25,45. The precise scale where this happens depends upon
the character of the model (2D or 3D, accounting for density gradients or not), the geometry of the area and the quality of topography representation by the model.
The figure is a nice illustration of an important potential of particle modelling. One is seeing a good global representation of the size and the structure of the compound plume. The plume results from a continuous release of dissolved matter over about 20 tidal periods. The global structure shown is generated by the residual and tidal currents in the area and mainly caused by the time dependency of these currents. This happens within a sea region narrower than one grid cell of the hydrodynamic model. A finite difference transport model on the same grid would give an entirely different result: just three numbers, giving the average concentration in each of three neighbouring grid cells, without any further information on structure. The global shape of the plume axis shown in figure 1a is a subgrid feature of the particle distribution obtained without account of the subgrid velocity structure. About the latter it is not quite certain what Maier-Reimer has applied, probably just a simple random walk, equivalent to classical diffusion. The relevant detail is not well distinguishable in the figure.
The example illustrates that in a particle model certain important subgrid characteristics of a distribution are already obtained without a gauged subgrid velocity model. The subgrid characteristics shown in the figure are mainly determined by the time dependency of the deterministic current field, but in addition, an effect of gradual increase of tidal velocity amplitude from south to north is noticeable.
After the introduction of the particle modelling principle by the author at the end of the seventies, as inspired by Maier-Reimer, its implementation was gradually improved and extended and used in various applications, some of which can be found in the papers and reports compiled in this thesis. Most of this work was done at Rijkswaterstaat, or in co-operation with or under commission of the Rijkswaterstaat.
In this thesis it is illustrated how the particle approach enables
(a) Accurate computation of advection by deterministic velocity fields in order to investigate and compare their characteristics in transport calculations (chapter 2: comparison of tidal and tidally averaged velocity fields).
(b) Computation of concentration distributions due to instant or continuous release in environmental studies or in the management of calamitous events (chapter 3, 4, 7). (c) The analysis of the dispersion properties of longitudinal flows (rivers and estuaries) such
as shearing currents in a horizontal or vertical plane or jointly in a 3D simulation (chapter 5) with flow profile and transverse exchange rate as determining factors.
(d) The study of 2D turbulent fields, the latter as simulated by the synthetic eddy field concept described in section 1.3 and in parts of chapters 3, 4, 6 and 8).
Chapter 1
16
(e) The determination of the spectrum of spatial variation of horizontal velocities when a set of dispersion data is available over a sufficient range of length scales. In chapter 6 is described how this succeeded for the North Sea on the basis of a large set of tracer data over a long period of time.
(f) The analysis and proper simulation of the separate and the joint dispersive effects of the shear dispersion in the vertical plane and the spectral velocity structure in the horizontal plane, demonstrating what part each of the two mechanisms is taking at various time and length scales.
(g) The demonstration of how vertical shear, vertical mixing and horizontal spectral velocity structure jointly determine the total phenomenon of dispersion in three-dimensional systems of surface waters, in particular in tidal flow regimes.
For a correct simulation of the transport and dispersion phenomena, various provisions had to be made at different stages of the model development. In course of time these have included the instructions for particle behaviour at open and closed model bounds and/or at the bottom, the option of positive or negative buoyancy properties, sedimentation and erosion criteria and rules for particles caught on drying areas, all as far as applying in the type of model or application. When hydrodynamic models with depth variations were applied, all 2D horizontal dispersion algorithms (random walk, scaled random walk, eddy fields) being basically isotropic, needed a provision to avoid that equal numbers of particles were sent upgrade and downgrade (see chapter 8). Provisions were made for sufficient small time steps to resolve spatial details, and on the other hand avoiding unnecessary small steps to prevent long computations of no need. In many cases this resulted in time steps growing in length during a computation, mostly up to a maximum to prevent insufficient sampling of time dependent flow or fast vertical exchange. In case of eddies, a minimum number of steps is required during one revolution of the smallest eddy, a minimum that can be increased when eddies are "dropped" at the lower edge of a spectral window (chapter 8).
Model particles represent real particles in nature. In Einstein's Brownian motion study, the model particles represent suspended particles larger than molecules. In the applications in this thesis the model particles may represent molecules like in the case of dissolved chemical pollutants or radioactive elements, larger particles in cases of organic matter (domestic waste, certain organic industrial waste products) or suspended sediment, in all cases accounting for buoyancy effects when applying. The numbers of model particles are usually many orders of magnitude smaller than of the particles in the prototype, to keep simulations tractable. This is never a problem as long as the set of model particles gives a good statistical representation of the phenomena. For a detailed reproduction of the concentration distribution in the prototype, pretty large numbers of model particles are sometimes useful. Delft Hydraulics has sometimes used numbers of the order of 105 but this only
makes sense with a correspondingly fine model structure. With the present day computers such large sets of particles do not cause serious problems but normally the numbers can be kept in the order of 104 (for an example see the distributions discussed and pictured in chapter 7). Some more local detail,
e.g. near a source, can usually be obtained with comparable numbers of particles in separate computations in a subregion.
Model particles in our context are nothing but sets of co-ordinates changing in small time steps dt, moving according to the instructions for the displacements ds. In our natural waters applications, the displacement instructions for the horizontal directions x and/or y always have a "deterministic" component ds = v⋅dt where v is a horizontal velocity that sometimes only depends on time (homogeneous velocity field, an example is chapter 7) or is given by a simplified (usually analytic) spatial distribution (e.g. the horizontal and/or vertical velocity profiles for the estuaries and rivers of chapter 5). In this thesis one can find some examples of merely deterministic displacements, such as in the comparison of tidally averaged and time dependent velocity fields in chapter 2. Another example is the illustration in chapter 4 (figure 2) of the important role in the simulated dispersion process played by the deterministic (i.e. hydrodynamically computed) velocities in an estuary model. The patterns formed by the computed field in this case have great similarity with the patterns formed by synthetic eddy fields (see figure 8 of chapter 3).
17 Introduction
(e) The determination of the spectrum of spatial variation of horizontal velocities when a set of dispersion data is available over a sufficient range of length scales. In chapter 6 is described how this succeeded for the North Sea on the basis of a large set of tracer data over a long period of time.
(f) The analysis and proper simulation of the separate and the joint dispersive effects of the shear dispersion in the vertical plane and the spectral velocity structure in the horizontal plane, demonstrating what part each of the two mechanisms is taking at various time and length scales.
(g) The demonstration of how vertical shear, vertical mixing and horizontal spectral velocity structure jointly determine the total phenomenon of dispersion in three-dimensional systems of surface waters, in particular in tidal flow regimes.
For a correct simulation of the transport and dispersion phenomena, various provisions had to be made at different stages of the model development. In course of time these have included the instructions for particle behaviour at open and closed model bounds and/or at the bottom, the option of positive or negative buoyancy properties, sedimentation and erosion criteria and rules for particles caught on drying areas, all as far as applying in the type of model or application. When hydrodynamic models with depth variations were applied, all 2D horizontal dispersion algorithms (random walk, scaled random walk, eddy fields) being basically isotropic, needed a provision to avoid that equal numbers of particles were sent upgrade and downgrade (see chapter 8). Provisions were made for sufficient small time steps to resolve spatial details, and on the other hand avoiding unnecessary small steps to prevent long computations of no need. In many cases this resulted in time steps growing in length during a computation, mostly up to a maximum to prevent insufficient sampling of time dependent flow or fast vertical exchange. In case of eddies, a minimum number of steps is required during one revolution of the smallest eddy, a minimum that can be increased when eddies are "dropped" at the lower edge of a spectral window (chapter 8).
Model particles represent real particles in nature. In Einstein's Brownian motion study, the model particles represent suspended particles larger than molecules. In the applications in this thesis the model particles may represent molecules like in the case of dissolved chemical pollutants or radioactive elements, larger particles in cases of organic matter (domestic waste, certain organic industrial waste products) or suspended sediment, in all cases accounting for buoyancy effects when applying. The numbers of model particles are usually many orders of magnitude smaller than of the particles in the prototype, to keep simulations tractable. This is never a problem as long as the set of model particles gives a good statistical representation of the phenomena. For a detailed reproduction of the concentration distribution in the prototype, pretty large numbers of model particles are sometimes useful. Delft Hydraulics has sometimes used numbers of the order of 105 but this only
makes sense with a correspondingly fine model structure. With the present day computers such large sets of particles do not cause serious problems but normally the numbers can be kept in the order of 104 (for an example see the distributions discussed and pictured in chapter 7). Some more local detail,
e.g. near a source, can usually be obtained with comparable numbers of particles in separate computations in a subregion.
Model particles in our context are nothing but sets of co-ordinates changing in small time steps dt, moving according to the instructions for the displacements ds. In our natural waters applications, the displacement instructions for the horizontal directions x and/or y always have a "deterministic" component ds = v⋅dt where v is a horizontal velocity that sometimes only depends on time (homogeneous velocity field, an example is chapter 7) or is given by a simplified (usually analytic) spatial distribution (e.g. the horizontal and/or vertical velocity profiles for the estuaries and rivers of chapter 5). In this thesis one can find some examples of merely deterministic displacements, such as in the comparison of tidally averaged and time dependent velocity fields in chapter 2. Another example is the illustration in chapter 4 (figure 2) of the important role in the simulated dispersion process played by the deterministic (i.e. hydrodynamically computed) velocities in an estuary model. The patterns formed by the computed field in this case have great similarity with the patterns formed by synthetic eddy fields (see figure 8 of chapter 3).
For a more complete and realistic simulation of the horizontal dispersion in computed velocity fields, the deterministic displacements need to be supplemented by an additional displacement due to the "subgrid" velocities representing the scales smaller than those resolved in the hydrodynamic computation. The most common way to do this is by a displacement of random direction and prescribed average size (see for example chapter 5 under 14.2.1. Method and definitions). For the horizontal directions in natural water bodies, simple random walk (such as used for Brownian motion by Einstein) does not satisfy because of the existence of horizontal eddies at widely varying scale, briefly discussed in the preceding section and in the next one. Better is the (improved) scaled random walk method mentioned earlier and described in more detail in chapter 8. As mentioned above and as will be discussed in more detail later (e.g. chapter 8), this method has imperfections that can be avoided by using synthetic eddy velocity fields instead. One might see this as is an illustration of the statement "diffusion is advection" which of course does not refer to molecular diffusion or the related Brownian motion.
Although the use of a three-dimensional synthetic eddy field has been considered (thus including vertical mixing), in this thesis the vertical mixing is treated by the classical diffusion concept only. The use of eddy fields is restricted to the horizontal water movement. There are no indications that in any of the questions at stake a better result might be possible by the use of explicit vertical eddies for simulating vertical mixing.
1.3 The eddy field approach
Numerical experiments with artificial eddy fields were started by the author around 1980 46,47,
briefly after the introduction of the particle modelling concept at Rijkswaterstaat in the late seventies. It was realised that the particle technique enabled simulation of the (horizontal) dispersion of matter in an eddy field in a very direct way. Since it was already clear that in nature the eddy spectra can be very different, due to topographic and other local factors, a code was constructed allowing a free choice of spectral energy distribution for horizontal eddies over the whole relevant scale range. It was not considered to do this for three dimensions, but only for the horizontal direction, so strongly dominant for the shallow waters at stake.
The simplest solution was believed to be the construction of an eddy velocity field by means of simple addition of a series of harmonic functions of space with discrete regularly spaced wavelengths, in two independent directions. By letting the harmonic components not depend on the direction of the velocity, each component and thus the entire field obeys the law of continuity for an incompressible fluid. When in a later stage of the project the method was implemented in models with varying depth, a factor inversely proportional to depth had to be added to guarantee continuity.
In the eighties the properties of the eddy fields were investigated by means of a super-computer (CRAY-XMP)40. For the standard computers of that time the computations were considered
to be too time consuming. The latter circumstance was also one of the reasons that it lasted rather long before the method was implemented in a code to be used in field applications. Another reason was that the limited research capacity was largely consumed by the tracer experiments (section 1.1; chapter 6).
In the super-computer investigation, the role of various factors was investigated such as the influence of the number of discrete eddy sizes per unit wavelength, the influence of the finite lifetime of eddies (assumed to be proportional to eddy size) and the random procedures. The time behaviour of patch size was checked for some well-known cases such as the inertial range spectrum and classical diffusion. Some time after the tests had been successful, the method was gradually implemented in systems for field applications and research. For details on the applied algorithms we refer to chapters 6 and 8.
After the eddy field had been combined with routines for tidal flow computation including vertical shear dispersion in an effectively three-dimensional simulation set-up, the extensive set of data from dispersion experiments in the North Sea could be used for an iterative approximation of an eddy energy density spectrum of the North Sea. A preliminary result could be presented in 199038,39; an
explicit spectrum, based on a more complete data set, was obtained in 1997 and published44 in 1999
Chapter 1
18
version of one of the computer codes. Satisfactorily no indications for adjustment were found (chapter 9).
It is of interest to explain in some detail why the two envelopes (fast and slow vertical mixing) computed in 2007 (chapter 9, figures 2 and 3) are fitting more tightly around the data cloud than the envelope published in 1999 (chapter 6, fig.14), computed in 1997.
In 1997, two different computer codes had to be applied. For the upper bound in fig.14 (ch.6), the case without shear (and consequently without tidal current), a simplified code could be used, dealing with the eddy field only, and programmed for a PC. The computation time needed for a single run was relatively short. So for each case the average of several runs (order 10) could be taken (for each run the program chooses a different random seed from which the various random quantities during the computation are derived). This averaging was desirable because of the rather great variability of the results, due to the random elements in the simulation. For the case with shear and consequently the inclusion of tidal currents, vertical velocity distributions and vertical mixing, the only adequate software available could be run only on a mainframe computer of Rijkswaterstaat which was difficult to access (the author had to retire from the Rijkswaterstaat organisation a few years earlier because of reaching the age of 65). It was quite an effort to realise just a very few runs on the mainfraime computer. Therefore the lower bound in figure 14 of chapter 6 is showing some fluctuations in time. The damping effect of the shear dispersion (which does not contain random elements) has kept the fluctuation within acceptable limits so that a sufficiently convincing picture for the publication could be obtained. The better result of 2007 was reached by extending the PC program from 2D to 3D, with tides, vertical velocity shear and vertical exchange. The runtimes remained acceptable, also since computers had become must faster in the meantime. It should be added though that the code with eddies only remains very much faster than a properly handled code including tidal currents and shear. This is so because the quickly changing tidal current and the often short vertical mixing time both set rigorous limits to the time step. With eddies only the timestep can keep growing during the entire computation since the smaller eddies successively become irrelevant while the scale of the dispersing cloud keeps increasing.
The question may be raised if it had not been possible to reduce the randomness of the eddy field code, so as to reduce the variability. Surely there are ways to do so. So far this has not been realised.
We will now summarise the existing random procedures, referring to chapters 6 and 8 for details.
The finite lifetime of eddies was already mentioned. It is obtained by multiplying eddy velocity amplitudes by a function of the type (1 + cos ωt). Each time the amplitude of an eddy component of a particular wavelength (covering the entire plane with individual eddies of the same size, see fig. 17 of chapter 6) becomes zero, its phases ϕx and ϕy in space are changed with random
amounts Δϕx and Δϕy , which simply means that the picture of identical eddies shown in fig. 17 of
chapter 6, is shifting over random distances in x- and y-directions at the moment all velocities of the entire picture (pertaining to only one eddy size!) have become zero and at the same moment start a new life cycle in the shifted position. The only phase in time is that of the periodic function letting the velocity amplitude move between zero and maximum amplitude. This phase is fixed at the start of the computation, randomly, but different for each wavelength.
The eddy field method is more realistic than a scaled random walk method. It does not use the unphysical assumption that an individual particle possesses information about its "age" or about size or other characteristics of the cloud or group of particles it belongs to. All unrealistic effects originating from such assumptions are avoided. Like in nature, the only information given to the particle is the local flow velocity and possibly its arrival at the bottom, the water surface or a dry spot. If needed, additional information on intrinsic properties of the individual particle itself can be added, such as fall velocity in the case of sediment (chapter 3) or a decay rate in case of radioactivity or decaying organic matter.
If the eddy field is combined with a flow model, the eddies large enough to be generated by the flow model have to be omitted from the synthetic field. The eddies taken in account are called "subgrid". The quotation marks indicate that the eddies that still have to be taken in account are several times larger than the mesh of the grid of the flow model. In the models where eddies were
19 Introduction
version of one of the computer codes. Satisfactorily no indications for adjustment were found (chapter 9).
It is of interest to explain in some detail why the two envelopes (fast and slow vertical mixing) computed in 2007 (chapter 9, figures 2 and 3) are fitting more tightly around the data cloud than the envelope published in 1999 (chapter 6, fig.14), computed in 1997.
In 1997, two different computer codes had to be applied. For the upper bound in fig.14 (ch.6), the case without shear (and consequently without tidal current), a simplified code could be used, dealing with the eddy field only, and programmed for a PC. The computation time needed for a single run was relatively short. So for each case the average of several runs (order 10) could be taken (for each run the program chooses a different random seed from which the various random quantities during the computation are derived). This averaging was desirable because of the rather great variability of the results, due to the random elements in the simulation. For the case with shear and consequently the inclusion of tidal currents, vertical velocity distributions and vertical mixing, the only adequate software available could be run only on a mainframe computer of Rijkswaterstaat which was difficult to access (the author had to retire from the Rijkswaterstaat organisation a few years earlier because of reaching the age of 65). It was quite an effort to realise just a very few runs on the mainfraime computer. Therefore the lower bound in figure 14 of chapter 6 is showing some fluctuations in time. The damping effect of the shear dispersion (which does not contain random elements) has kept the fluctuation within acceptable limits so that a sufficiently convincing picture for the publication could be obtained. The better result of 2007 was reached by extending the PC program from 2D to 3D, with tides, vertical velocity shear and vertical exchange. The runtimes remained acceptable, also since computers had become must faster in the meantime. It should be added though that the code with eddies only remains very much faster than a properly handled code including tidal currents and shear. This is so because the quickly changing tidal current and the often short vertical mixing time both set rigorous limits to the time step. With eddies only the timestep can keep growing during the entire computation since the smaller eddies successively become irrelevant while the scale of the dispersing cloud keeps increasing.
The question may be raised if it had not been possible to reduce the randomness of the eddy field code, so as to reduce the variability. Surely there are ways to do so. So far this has not been realised.
We will now summarise the existing random procedures, referring to chapters 6 and 8 for details.
The finite lifetime of eddies was already mentioned. It is obtained by multiplying eddy velocity amplitudes by a function of the type (1 + cos ωt). Each time the amplitude of an eddy component of a particular wavelength (covering the entire plane with individual eddies of the same size, see fig. 17 of chapter 6) becomes zero, its phases ϕx and ϕy in space are changed with random
amounts Δϕx and Δϕy , which simply means that the picture of identical eddies shown in fig. 17 of
chapter 6, is shifting over random distances in x- and y-directions at the moment all velocities of the entire picture (pertaining to only one eddy size!) have become zero and at the same moment start a new life cycle in the shifted position. The only phase in time is that of the periodic function letting the velocity amplitude move between zero and maximum amplitude. This phase is fixed at the start of the computation, randomly, but different for each wavelength.
The eddy field method is more realistic than a scaled random walk method. It does not use the unphysical assumption that an individual particle possesses information about its "age" or about size or other characteristics of the cloud or group of particles it belongs to. All unrealistic effects originating from such assumptions are avoided. Like in nature, the only information given to the particle is the local flow velocity and possibly its arrival at the bottom, the water surface or a dry spot. If needed, additional information on intrinsic properties of the individual particle itself can be added, such as fall velocity in the case of sediment (chapter 3) or a decay rate in case of radioactivity or decaying organic matter.
If the eddy field is combined with a flow model, the eddies large enough to be generated by the flow model have to be omitted from the synthetic field. The eddies taken in account are called "subgrid". The quotation marks indicate that the eddies that still have to be taken in account are several times larger than the mesh of the grid of the flow model. In the models where eddies were
combined with flow models, the consequent "cut off" of the spectrum is performed in a gradual way. So far, the applied flow models were of the type of a two-dimensional velocity field with a space and time dependent water depth.
In some cases it is sufficient to use a simpler approach, namely when the computation regards a relatively uniform area where it is acceptable to let the "deterministic" velocities depend upon time only (the velocity histories can be taken from a model or from observations). Then it is not necessary to cut off a part of the eddy spectrum; some very large wavelengths that do not cause noticeable velocity gradients in the computation area can be left out, just for efficiency reasons. In fact, one is using a flow model with an infinite mesh width. An example of this approach is the project described in chapter 7. The spectrum used is that of the entire North Sea, as derived in chapter 6. It is of interest to observe that in this way the effect of velocity variations typical for the North Sea is taken in account in a statistical way, but it is not neglected. In practical terms this means that, if we repeat the simulation a couple of times, we will have a good impression of the impact of this kind of variations. It is even so that since the randomness of the eddy field simulation seems to be somewhat exaggerated (see above), we expect that the variability will certainly not be underestimated in case we derive it from repetition of computer runs. This statement has so far not been verified.
1.4. Outline of this thesis
This thesis deals with the investigation and modelling of dispersion in surface waters. Emphasis is on the development of the mathematical models and the application of those in practice. The experiments in nature, however, are at the basis of these activities and there is great emphasis on the empirical validation and calibration of the models.
In chapter 1, after a brief historical survey (1.1), the two main modelling principles applied, model particles and synthetic eddy fields, are briefly described in sections 1.2 and 1.3. In the following chapters relevant activities are presented in a strictly chronological order.
In chapter 2 (on residual currents), a first important application of particle modelling is presented. The particle technique appears to be a most adequate tool for investigating in how far the averaging of velocities over the periods in periodic velocity fields such as tidal fields, is allowable in transport computations.
In chapter 3 (on a three-dimensional transport model), it is shown how three-dimensional dispersion modelling can be reached by extending two-dimensional velocity fields (like for example obtained by solving the shallow-water equations) by means of analytical vertical velocity distributions and by supplementing the deterministic velocity fields by using various techniques of 'subgrid' modelling. The two techniques described for the supplementation of the horizontal velocity fields are the scaled random walk technique and the synthetic eddy field technique. In this chapter it is also shown that particle techniques are adequate for the modelling of suspended sediment and its (temporary or periodical) sedimentation and erosion.
In chapter 4, "Model particles as representatives of particles in nature", the principles of particle techniques are briefly summarised. Some more examples and applications are given, especially for estuaries. Examples include time dependent vertical distributions of suspended and settled sediment and of solved matter distributions in regions with large areas which are periodically flooded.
A first presentation is given of a graph that quantitatively shows the importance of vertical
shear dispersion aside of horizontal spatial variability of velocities and how the combined action of
these two main dispersion mechanisms can be well reproduced by the 3D model that was presented in chapter 3.
Chapter 5, "Study of shear dispersion in tidal waters by applying discrete particle techniques"
Chapter 1
20
longitudinal systems (estuaries and tidal and non-tidal rivers). It is shown that in these longitudinal systems the transverse analogue of the shear dispersion in the water column is equally or even more important than the latter. It is seen that the large values of empirically observed longitudinal dispersion coefficients in these longitudinal systems can be fully explained by the models.
In chapter 6, "Spectral structure of horizontal water movement in shallow seas with special reference to the North Sea, as related to the dispersion of dissolved matter", various aspects and details of the experimental and modelling work are given. This chapter presents and uses the most extensive data set for the North Sea so far. The aforementioned synthetic eddy field technique is described in more detail and the algorithms of the essential computational procedures are presented. By combining the extensive Sea data set with model simulations (again with vertical shear dispersion and eddy field simulation in one model) the kinetic energy density spectrum of the southern North Sea is determined by iteration.
In chapter 7, "The spreading and dilution of waste water from production platforms in the North Sea", a practical application is described of the 3D modelling technique presented in foregoing chapters, using the energy spectrum derived in chapter 6.
In chapter 8, "Spectral representation of horizontal velocity variations as applied to particle dispersion modelling", the various approaches are summarised in greater detail. As a new element, the relation between spectral structure and generalised diffusion coefficient (as related to scaled random walk) is worked out and quantitatively presented.
After the energy spectrum for the velocity variations in the North Sea had been derived and used in practice, it was found an intriguing question what proportion the amount of energy implied would bear to the energy in tidal and wind waves and also what the rates of input and output of the various components would be. Global answers to these questions are derived and presented in chapter
9, "The energy balance of the North Sea". With an improved computer code the energy spectrum
derived in chapter 6, could be critically analysed. It was most satisfying that no indications for adjustment resulted from this analysis.
In most chapters, computed velocity fields from two-dimensional hydrodynamic models are mentioned in the description of methods or applications. The computations by these models are based on solving the shallow-water equations: the equation of motion (essentially Newton's law applied to water parcels) and the fluid continuity equation, implemented for topographies (bathymetry and coastlines) of particular regions. In practice the name "model" is usually given to these particular implementations of the computational method. The geometry is "schematised" on a horizontal grid which may have coarse meshes (like in chapter 2) or relatively fine (such as the Ems estuary model used in illustrations of chapter 4). Apart from the topography, a region bound property is the bottom friction, which is often assumed to be uniform for the whole model region. The model computed velocity fields used in this thesis are all given on rectangular grids with square meshes. If the flow model to be used has a curvilinear grid, co-ordinate transformations are required. These are not dealt with in this thesis.
The two-dimensional flow models (in x and y) are, to some extent, also figuring in the z-direction: the bathymetry is given in z-co-ordinates and the height of the water column for each grid cell is computed for every timestep. But the computed flow velocities are only given as horizontal velocity components in (x,y)-co-ordinates (one value per cell and per timestep) and have to be seen as vertical averages. If the horizontal velocities are computed as a function of three spatial dimensions the model is named a three-dimensional flow model. Such models have not been used in the particle modelling described in this thesis. Using them would certainly have advantages; these are briefly touched in the summary and concluding remarks at the end of this thesis.
The flow models actually used in this thesis are all of the two-dimensional WAQUA type (references in the relevant chapters)
21 Introduction
longitudinal systems (estuaries and tidal and non-tidal rivers). It is shown that in these longitudinal systems the transverse analogue of the shear dispersion in the water column is equally or even more important than the latter. It is seen that the large values of empirically observed longitudinal dispersion coefficients in these longitudinal systems can be fully explained by the models.
In chapter 6, "Spectral structure of horizontal water movement in shallow seas with special reference to the North Sea, as related to the dispersion of dissolved matter", various aspects and details of the experimental and modelling work are given. This chapter presents and uses the most extensive data set for the North Sea so far. The aforementioned synthetic eddy field technique is described in more detail and the algorithms of the essential computational procedures are presented. By combining the extensive Sea data set with model simulations (again with vertical shear dispersion and eddy field simulation in one model) the kinetic energy density spectrum of the southern North Sea is determined by iteration.
In chapter 7, "The spreading and dilution of waste water from production platforms in the North Sea", a practical application is described of the 3D modelling technique presented in foregoing chapters, using the energy spectrum derived in chapter 6.
In chapter 8, "Spectral representation of horizontal velocity variations as applied to particle dispersion modelling", the various approaches are summarised in greater detail. As a new element, the relation between spectral structure and generalised diffusion coefficient (as related to scaled random walk) is worked out and quantitatively presented.
After the energy spectrum for the velocity variations in the North Sea had been derived and used in practice, it was found an intriguing question what proportion the amount of energy implied would bear to the energy in tidal and wind waves and also what the rates of input and output of the various components would be. Global answers to these questions are derived and presented in chapter
9, "The energy balance of the North Sea". With an improved computer code the energy spectrum
derived in chapter 6, could be critically analysed. It was most satisfying that no indications for adjustment resulted from this analysis.
In most chapters, computed velocity fields from two-dimensional hydrodynamic models are mentioned in the description of methods or applications. The computations by these models are based on solving the shallow-water equations: the equation of motion (essentially Newton's law applied to water parcels) and the fluid continuity equation, implemented for topographies (bathymetry and coastlines) of particular regions. In practice the name "model" is usually given to these particular implementations of the computational method. The geometry is "schematised" on a horizontal grid which may have coarse meshes (like in chapter 2) or relatively fine (such as the Ems estuary model used in illustrations of chapter 4). Apart from the topography, a region bound property is the bottom friction, which is often assumed to be uniform for the whole model region. The model computed velocity fields used in this thesis are all given on rectangular grids with square meshes. If the flow model to be used has a curvilinear grid, co-ordinate transformations are required. These are not dealt with in this thesis.
The two-dimensional flow models (in x and y) are, to some extent, also figuring in the z-direction: the bathymetry is given in z-co-ordinates and the height of the water column for each grid cell is computed for every timestep. But the computed flow velocities are only given as horizontal velocity components in (x,y)-co-ordinates (one value per cell and per timestep) and have to be seen as vertical averages. If the horizontal velocities are computed as a function of three spatial dimensions the model is named a three-dimensional flow model. Such models have not been used in the particle modelling described in this thesis. Using them would certainly have advantages; these are briefly touched in the summary and concluding remarks at the end of this thesis.
The flow models actually used in this thesis are all of the two-dimensional WAQUA type (references in the relevant chapters)
References to Chapter 1
1 Pritchard, D. W. and J. H. Carpenter, 1960. Measurements of turbulent diffusion in estuarine and inshore waters.
Symposium on tidal rivers, Helsinki, July 1960.
2 Schönfeld, J. C., 1957. Hydrodynamische aspecten van de lozing van radioactief afval in de Noordzee. Rijkswaterstaat, Centrale Studiedienst, nota CSD 57-6.
3 Schönfeld, J. C., 1959. Diffusion by homogeneous isotropic turbulence. Rijkswaterstaat, Directie Waterhuishouding en Waterbeweging, Fysische Afdeling, nota FA-1959-1.
4 Schönfeld, J. C., 1962. Integral diffusivity. J. Geoph. Res., 67(8), 3187-3199 (presented in 1961 at the International Symposium on Fundamental Problems in Turbulence and their relation to Geophysics, Marseille).
5 Schönfeld, J. C., 1964. Turbulente diffusie. Lozing vanuit een bron in zee. Rijkswaterstaat, Directie Waterhuishouding en Waterbeweging, Mathematisch-Fysische Afdeling, nota MFA 6411.
6 Joseph, J. and H. Sendner, 1958. Über die horizontale Diffusion im Meere. Dt. Hydrogr. Zt. 11, 49-77.
7 Joseph, J. and H. Sendner, 1962. On the spectrum of the mean diffusion velocities in the ocean. J. Geophys. Res. 67( 8), 3201-3205.
8 Joseph, J., H. Sendner and H. Weidemann, 1964. Untersuchungen über die horizontale Diffusion in der Nordsee. Dt. Hydrogr. Zt. 17, 57-75.
9 Kolmogorov, A. N, 1941. Local turbulent structure in incompressible fluids at very high Reynolds number. Doklady Ak.
Nauk SSSR 30, 299-303.
10 Van Dam, G. C., 1963. Preliminary result of first measurement and computationsof diffusive spread in the North Sea at short distances from the coast of Holland. Report FA 1963-1, Physics Division, Rijkswaterstaat, The Netherlands 11 Van Dam, G. C., 1965. Horizontal diffusion in the North Sea near the Netherlands' coast in connection with waste
disposal. 11th Int. Congress IAHR, Leningrad.
12 Van Dam, G. C. and J. A. G. Davids, 1966. Radioactive waste disposal and investigations on turbulent diffusion in the Netherlands' coastal areas. Proc. Symp. Disposal of Radioactive Wastes into Seas, Oceans and Surface Waters, IAEA, Vienna, 233.
13 Van Dam, G. C., 1968. Dispersie van opgeloste stoffen in zee gebracht ter hoogte van Wijk aan Zee op 3 km uit de kust.
Rijkswaterstaat, Neth., Rept. MFA 6812 (in Dutch; with English summary)
14 Barrett, M. J., D. Munro and A.R. Agg, 1969. Radiotracer dispersion studies in the vicinity of a sea outfall. Proc. 4th Int. Conf. on Water Pollution, Pergamon Press, New York, 863.
15 Meerburg, A. J., 1970. Een diffusie-experiment voor de Nederlandse kust met behulp van Rhodamine-B. Royal Netherlands Meteorological Institute, Scientific Report W. R. ,70-71.
16 Meerburg, A. J., 1970. A diffusion experiment near an amphidromic point in the North Sea. ICES Hydrographic Committee, C. M. 1970 / C:22.
17 Van Dam, G. C., J. S. Sydow and J. W. Westhoff, 1970. A diffusion experiment near the Dutch Coast 10 km off Ter
Heijde. Rijkswaterstaat, Neth. (Math. Phys. Div.), Rpt. MFA 7003 (in Dutch).
18 Westhoff, J. W., G. C. van Dam and J. A. G. Davids, 1971. Dispersieproeven met continue injectie ter hoogte van Petten
op 3,5 km uit de kust. Rijkswaterstaat (Math. Phys. Div.), Rept. MFA 7101 (in Dutch).
19 Weidemann, H. (Ed.), 1973. The ICES diffusion experiment RHENO 1965. Rapp. P.v. Réun. Cons. Int. Explor, Mer
163.
20 Van Dam, G. C., 1974, The Hague Outfall. In: A. L. H. Gameson (Ed), Proc. Conf. Discharge of Sewage from Sea Outfalls (London). Pergamon Press, p.393-401.
21 Talbot, J.W. and G. H. Talbot,1974. Diffusion in shallow and in English coastal and estuarine waters. Rapp. P. V. Réun. Cons. Int. Explor. Mer, 167, 93-110.
22 Suijlen, J. M., 1975. Turbulent diffusion in the IJsselmeer near Medemblik, measured by means of Rhodamine-B.
Rijkswaterstaat (Phys. Div.), Rept. FA 7501 (in Dutch).
23 Suijlen, J. M., 1980. A dye experiment in the North Sea up to a very large scale. ICES Hydrography Committee, C.M. 1980 / C:29
24 Suijlen, J. M., 1981. Meting van turbulente diffusie in het Eems-estuarium met behulp van rhodamine-B. Deel 1: Scheepsmetingen. Rijkswaterstaat, Directie Waterhuishouding en Waterbeweging, Fysische Afdeling, rapport 04 81-FA. 25 Van Dam, G. C., 1989. Dispersion of dissolved matter in the North Sea. ICES, Hydrography Committee, C. M. 1989 /
C:13.
26 Suijlen, J. M., J. S Sydow, C. Heins, P. C. Beukenkamp, 1990. Measurements of turbulent diffusion and residual displacements by dye experiments in the southern North Sea in 1971, 1973, 1978 and 1979 (data report). Rijkswaterstaat, Tidal Waters Division, The Hague, report GWAO-90.022.
27 Suijlen, J. M., J. S Sydow, C. Heins, P. C. Beukenkamp, 1990. Measurements of turbulent diffusion and residual displacements by dye experiments in the southern North Sea in 1982 (data report). Rijkswaterstaat, Tidal Waters Division, The Hague, report GWAO-90.023.
