A simple method for calculating in situ floc settling velocities based on effective density functions

Marine Geology 344 (2013) 10–18

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A simple method for calculating in situ floc settling velocities based on effective density functions Thor Nygaard Markussen ⁎, Thorbjørn Joest Andersen Department of Geosciences and Natural Resource Management, University of Copenhagen, Øster Voldgade 10, 1350 Copenhagen K, Denmark CENPERM, University of Copenhagen, Øster Voldgade 10, 1350 Copenhagen K, Denmark

a r t i c l e

i n f o

Article history: Received 7 January 2013 Received in revised form 25 June 2013 Accepted 2 July 2013 Available online 13 July 2013 Communicated by Dr. J.T. Wells Keywords: Particle settling Flocculation Effective density Cohesive sediments Sediment transport

a b s t r a c t We present a simple method to obtain settling velocity estimates of fine-grained suspended sediment using the following: (1) in situ floc volume fraction measurements, (2) calculations of in situ floc mass, (3) measurements of in situ floc size, (4) laboratory measurements of primary particle size and (5) laboratory measurements of % organic matter obtained through loss on ignition. This method is based on the mass distribution over the particle size spectra. Floc properties are incorporated by using information concerning the primary particles and the ratio between the volume concentration and the mass concentration of the in situ measurements. In this manner, the changes in effective densities caused by flocculation and biological mediation are approximated. This method relies solely on measurements conducted with a particle-sizing instrument and on an analysis of water samples. The method is applied to measurements obtained with a LISST, and data from three different estuaries in different climatic zones are used. The settling velocity estimates found using this method are compared to settling tube measurements and to mass settling method estimates, as well as to calculations using fractal dimension methods from the literature. Settling velocities calculated using the method presented in this study lie within the ranges of the three other methods, demonstrating that this simple method produces results within the ranges of settling tube velocities and those predicted by the more comprehensive, multiparameter fractal dimension models. The presented method allows for easy intercomparisons between studies by utilising instruments such as the LISST, and it is believed that this method would also produce reliable settling velocities using other instruments that yield a volume distribution as well as a proxy for total volume and mass concentration, e.g., some camera systems. © 2013 Elsevier B.V. All rights reserved.

1. Introduction The settling velocity, ws, has a significant impact on the spatial patterns of sediment deposition because it greatly influences how far a suspended particle may travel before settling. Much effort has been put into developing methods to best measure the settling velocity, which is known to be dependent on the size of the settling particles and the properties of these particles (Fettweis, 2008). The two most commonly used methods for measuring particle sizes are as follows: 1) image systems (Eisma et al., 1990; Dyer and Manning, 1999; Van der Lee, 2000) and 2) laser particle sizers such as the LISST (Laser In Situ Scattering and Transmissometry) (Agrawal and Pottsmith, 2000), as well as combinations of these two methods (Mikkelsen et al., 2005; Winter et al., 2007). The settling properties of fine-grained particles in the clay and silt size range are influenced by flocculation processes, which result in ⁎ Corresponding author at: Department of Geoscience and Natural Resource Management, University of Copenhagen, Øster Voldgade 10, 1350 Copenhagen, Denmark. Tel.: + 45 35322500. E-mail address: [email protected] (T.N. Markussen). 0025-3227/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.margeo.2013.07.002

the aggregation of particles into larger aggregates, called flocs, with lower densities than the constituent primary particles. For clarity, we use the terms flocs and flocculation in reference to particle aggregation processes in suspension and aggregates and aggregation in reference to particle aggregation processes at the bed. Flocculation processes are controlled by the rate and strength of collisions and by the biogeochemical properties of the suspended matter. The main force causing collisions is turbulence, caused by current changes due to tides and waves, which enhances flocculation by increasing collision rates until the turbulent shear exceeds the shear strength of the floc bonds, thus resulting in floc break-up (Pejrup and Mikkelsen, 2010). This threshold largely depends on the properties of the flocs. Floc properties can vary greatly between different sites, and temporal variation may even be observed at individual sites over the course of a year, e.g. caused by varying biological influence (Andersen and Pejrup, 2011). Floc properties are controlled by the floc composition and the mineral to organic ratio, as well as the aggregation process. In soil– freshwater systems, the colloidal geochemistry is known to strongly influence the particle attractive surface forces (Lead and Wilkinson, 2006). Additionally, it has been shown that the availability and

T.N. Markussen, T.J. Andersen / Marine Geology 344 (2013) 10–18

concentration of iron alter the flocculation patterns of the freshwater constituent particles (Hassellov and von der Kammer, 2008). However, in most natural freshwater and estuarine waters, biological influences may completely dominate the geochemical differences and are highly influential in flocculation processes (Droppo et al., 1997; Droppo, 2001; Mosley et al., 2003). For instance, organic secretions, such as extracellular polymeric substances (EPS), coat particles and can strongly influence the flocculation efficiency and enhance flocculation in estuarine environments (Decho, 2000; Andersen and Pejrup, 2011). Faecal pellets and pseudo-faeces are other examples of highly biologically influenced flocs that have been shown to have much higher settling velocities than their constituent particles (Andersen and Pejrup, 2002). Such bio-aggregates are eroded or resuspended from the bed, indicating that the origin and formation of flocs vary greatly. Furthermore, aggregates created by biological activity are found in a wide range of sizes and forms and may completely encompass the inorganic components in the form of transparent exopolymer particles (TEP) in freshwater, estuarine and marine systems (Logan et al., 1995; Ayukai and Wolanski, 1997; Passow et al., 2001). The various physical and biogeochemical factors influencing floc composition are very diverse (Droppo, 2001), and the shapes of flocs are highly variable and range from a more or less rounded form to elongated, odd-sized forms (Eisma, 1986; Bainbridge et al., 2012). The result is that the shear strength of floc bonds differs according to composition as well as size, and estuarine flocs have been divided into microflocs, which are relatively small (b 100 μm), dense and stable, and more fragile macroflocs (Eisma, 1986). Others have suggested adding the order flocculi to distinguish between single particles and very small flocs (10–20 μm) that are never broken by turbulent shear (Francois and Vanhaute, 1985; Lee et al., 2012). However, we use the classic terminology of Eisma, in which primary particles are the primary constituents of microflocs, and both primary particles and microflocs are the building blocks of the macroflocs. Thus, using this terminology, both inorganic mineral particles and organic matter may be considered to be primary particles, and the mean primary particle represents the average primary particle, which is a combination of organic and inorganic matter. Typically, the settling properties of flocs are calculated using floc size and floc effective density, as incorporated in Stokes' Law, and using the relative influences of the forces acting on the settling flocs. Several authors have relied on the theory of fractal dimensions to explain the relationship between floc size and floc effective density (Kranenburg, 1994; Winterwerp, 1998; Ferguson and Church, 2004; Khelifa and Hill, 2006; Maggi et al., 2007; Kumar et al., 2010; Strom and Keyvani, 2011). The resulting methods are quite comprehensive as they aim to take into account all of the physical forces (viscous, drag and gravitational) influencing the settling flocs. However, these methods are based on a simplification of the evolution of floc size by assuming self-similarity of flocs and fixed values for a given dataset. This simplification limits the ability to take into account the potentially large variation in floc properties within a given dataset. Therefore, there seems to be a discrepancy between the comprehensiveness of the methods based on the theory of fractal dimensions and the necessary simplifications in these models. The main motivation for this paper is to present a simple method that seeks to describe the influence of varying floc densities on floc settling velocity and to include a simple proxy for the biological influence on floc settling velocity. The aim of this work is to limit the number of assumptions and necessary equipment so that this method may be used to give reliable, first order estimates of floc settling velocities in situations where more detailed information on floc characteristics (e.g., EPS-coatings or TEPs) is not available. Thus, the method presented in this study is based solely on measurements using an in situ particle sizer and water samples to determine the suspended sediment concentration and primary particle characteristics. The instrument used in this study is a LISST. However, this method is believed to be applicable to other particle sizing instruments, including imaging systems,

11

provided that they give information on the particle size distribution (PSD) as well as proxies for the volume and mass concentration. A second aim of this method is to use the entire PSD determined by the instrument rather than using only the mean values. In this way, the varying effective density over the size range of the instrument will be incorporated. Such alterations have been carried out by some researchers (Mikkelsen et al., 2005; Curran et al., 2007), but these alterations lack of a thorough description that can be applied by other authors. The goal of this work is that the simple settling velocity method presented in this paper can be used by other authors to allow for easier intercomparisons between studies. 2. Instruments and theory The LISST (version 100C) is a widely used instrument for in situ measurements of the size distribution of suspended matter in estuarine and marine waters (Mikkelsen and Pejrup, 2001; Voulgaris and Meyers, 2004; Fettweis, 2008; Andrews et al., 2010). This instrument makes measurements by emitting a laser beam through a volume of water over a path length of a maximum of 5 cm. Forward scattering by particles in the water is measured by 32 ring detectors, with a transmissometer measuring the optical transmission in the middle of the ring detectors. Lower transmissions indicate higher concentrations of particles that diffract and/or absorb the beam. The photosensitive ring detectors are logarithmically spaced, which allows for the capture of diffraction angles at distinct ranges. To convert the scattering pattern to a PSD, the signal is inverted using a kernel matrix. This matrix defines how the volume of particles influences the attenuation of light at scattering angles corresponding to the rings. In this work, the updated matrix (“natural particles matrix”), supplied by the manufacturer Sequoia Sci., is used (Agrawal et al., 2008). Every particle scatters at roughly all angles within the range, but the maximum scattering angles change according to the particle size. After the signal is inverted according to the kernel matrix, the result is a volume distribution over the size range of the LISST (roughly 2–400 μm with the applied kernel matrix), and the sum of this distribution is the volume concentration, VC. It has been shown that the volume concentration obtained using the LISST can be used as a direct measure of the volume concentration of flocs, Vf, (Mikkelsen and Pejrup, 2001; Voulgaris and Meyers, 2004). Therefore, if the floc mass concentration, Mf, can be estimated, the floc density, ρf, can be obtained using the equation, ρf = Vf/Mf. According to this definition, the floc mass concentration is defined as:
 Mp M f ¼ M p þ M w ¼ M p þ ρw V f −V p ¼ M p þ ρw V f − ρp

! ð1Þ

where M is mass per unit volume, V is the volume of flocs or particles per unit volume and suffixes p, w and f denote primary particle, water and floc, respectively (Fettweis, 2008). Mp is the suspended particulate mass concentration (SPMC) and ρp is the primary particle density. Mp and ρp can be estimated based on the analysis of water samples (see Section 4). The fractal dimension theory methods also seek to calculate floc density. Fractal dimension theory dictates that primary particles and flocs are statistically self-similar at different scales, meaning that self-similarity exists between a floc of size df and the primary particles of size dp. As df increases, the fractal dimension, F, decreases according to the following equation: 1

d f ¼ dp  N F

ð2Þ

provided that the amount of primary particles within the floc, N, is constant (Kranenburg, 1994). F can vary from 1 to 3, with 3 being a sold sphere. According to Winterwerp (1998), very fragile marine

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flocs have a fractal dimension of close to 1.4, whereas the fractal dimension is approximately 2.2 for strong estuarine flocs. However, Strom and Keyvani (2011) found that a better fit to most estuarine floc settling velocity data can be obtained by setting F equal to 2 to 2.5 using the following explicit equation: F−1

ws ¼

g  Rp  d f qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F−3 b1  v  dp þ b2  g  Rp  d f F  dp F−3

ð3Þ

where Rp is the submerged specific density of the primary particle (equal to Δρp/ρw), g is gravitational acceleration, ν is kinematic viscosity and b1 and b2 are constants (Strom and Keyvani, 2011). The first term of the denominator accounts for drag under pure viscous settling, whereas the second term in the denominator accounts for drag in the inertial settling range. Thus, b1 and b2 are essentially shape and permeability parameters. Strom and Keyvani (2011) tested this expression using 25 published datasets, and they found that b2 is only relevant for floc sizes above approximately 500 μm. Note that when b2 = 0, b1 = 18 and F = 3, Eq. (3) is identical to Stokes' Law. The authors demonstrated that by neglecting the inertial drag (b2 = 0), increasing the shape coefficient b1 and allowing F to equal 2 to 2.5, good agreement for most estuarine flocs was established. They found that a b1 of 120 is typically best at capturing the measured trends when F is greater than 2 and that a b1 in the range of 18–20 is best for very fragile flocs (F b 2). The constants b1 and b2 can be estimated based on pre-investigations involving imaging methods or by testing various guessed values (Strom and Keyvani, 2011). In typical settling velocity calculations for flocs (e.g., those resembling Eq. (3)), the fractal dimension is a key parameter. This parameter can be taken as a constant or modelled as a function of floc and primary particle size (variable fractal dimension). Maggi et al. (2007) proposed a simple power-law function based on Eq. (2) to model F: F ¼δ

df dp

!γ ð4Þ

where δ is the fractal dimension of the primary particles (approximately 3) and γ represents the degree of deviation from self-similarity in the geometry of the floc. The authors tested their model on kaolinite minerals and found the self-similarity parameter to be −0.1 for the minerals in a flocculated form. Kumar et al. (2010) proposed a model to calculate F based on the characteristic floc size, dfc, and the fractal dimension, Fc: −kdf

F ¼ ð3−F c Þ  e

þ Fc

ð5Þ

where d⁎f is the non-dimensional floc size, (df − dp)/dp, and k is found using the following equation:   0:025F c  −1 k ¼ −ln  dfc 3− F c

ð6Þ

where d⁎f c is the non-dimensional characteristic floc size. The Kumar model follows the same trends as those of the Maggi model (Eq. (4)) at values of df up to approximately 100–300 μm (Strom and Keyvani, 2011). At larger floc sizes, F-values based on Eq. (4) continue to decrease, whereas F-values based on the Kumar model (Eqs. (5) and (6)) approach the characteristic fractal dimension. 3. The md-method The proposed method for calculating settling velocities utilises the entire size spectrum of the LISST rather than merely the mean size

values. Essentially, this method provides a simple way to recalculate a volume distribution into a mass distribution and to use that mass distribution to calculate a settling velocity distribution and a mean settling velocity for each measurement. Because this method is based on mass distributions, the term md-method is used. This method is only applied to measurements performed with a LISST but should be applicable to other particle sizing instruments that provide volume distributions, e.g. camera systems. This section presents a step-by-step explanation of the proposed method so that this method may be easily used by other researchers. The necessary input data is listed as follows: • Volume distributions and total volume of suspended matter • Simultaneous CTD-measurements • A representative amount of water samples ○ A number of these samples filtered on glass microfiber filters to determine loss on ignition (LOI) ○ The remaining samples filtered on cellulose filters for dispersion and primary particle size measurements The procedure is divided into three steps that must be performed for each LISST measurement: 1. Filter water samples (only once per dataset) 2. Calculate a density function 3. Recalculate the density function into a settling velocity distribution a. Calculate a mass distribution b. Recalculate the frequency distribution c. Calculate a settling velocity distribution and a mean ws These calculations are mathematically simple but may benefit from being performed through a program such as Matlab. 3.1. Step 1: Water samples Water samples are collected and filtered in accordance with available sampling opportunities and laboratory equipment. The sampling and filtration steps described here serve as an example. Two water samples are collected at a number of representative sites and depths. The first set of samples are filtered through pre-filtered Whatman GF/F glass microfiber filters (0.7 μm nominal retention diameter), dried for 2–3 h at 60 °C, weighed and burned for 2–3 h at 550 °C to determine the loss on ignition (LOI). The second set of samples are filtered through pre-filtered Millipore cellulose filters (0.45 μm nominal retention diameter), weighed, dispersed with ultrasonic sound in a sodium pyrophosphate-solution and analysed on a Malvern Mastersizer E/2000 laser particle sizer. The LOI-results and Malvern-PSD are used to describe the mean density and size of primary particles (see below). The SPMC from all of the filtered water samples is used to calibrate the beam attenuation signal of the LISST and of the OBS-sensor. 3.2. Step 2: Density functions The calculations of density functions are an essential part of this method because they determine the weight of each group of the PSD. Thus, based on our understanding of how large macroflocs have smaller densities than smaller microflocs, these larger flocs are given relatively less weight in the calculation of mean settling velocity. Note that we use mean values calculated using the method of moments rather than using median values. The density functions are calculated using power laws because it is well established that power laws describe the relationship between effective density and mean floc size (Dyer and Manning, 1999; Curran et al., 2007; Fettweis, 2008). The density functions are calculated based on two known or estimated conditions: 1) the measured mean size and estimated mean density of primary particles and 2) an estimation of the mean floc size and density. For condition (1), the mean primary particle size, dp, and density, ρp, are assumed to be constant for each dataset, unless

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the water samples show a different pattern. The ρp value is estimated based on the LOI because it is assumed that the LOI is a proxy for the mean organic matter content of the primary particles making up the flocs. The solid density of organic matter is normally in the range of 900–1400 kg m−3 (Brady and Weil, 2007), and has for the calculations been estimated to be 1300 kg m−3, whereas the mineral particulates are given a density of 2650 kg m−3. For instance, if filtered and dispersed primary particles from a water sample have a LOI of 22%, they are composed of 78% minerals and 22% organic matter and the mean density of primary particles of size dp, ρp, is estimated to be 2350 kg m−3 (0.22 × 1300 kg m−3 + 0.78 × 2650 kg m−3). The fraction of the PSD that represents particles less than dp is assumed to have a density of ρp. For condition (2), the mean floc size, df, is assumed to be the mean of the PSD, and the mean floc density, ρf, is calculated using Eq. (1). Thus, df and ρf change for each measurement. The particulate mass, Mp, in Eq. (1) is the calibrated beam attenuation of the LISST or the OBS-sensor. In summary, the density of the ith group of the PSD is calculated as follows:

where di is the size represented by the ith group. Using the two known conditions of a measurement as points in a power law, the density of the ith group is found as follows: a

ρi ¼ b  d i log a¼

ρf ρp

df log dp

ð7Þ ! ! ; and b ¼

ρp dp a

ð8Þ

The effective density distribution is multiplied by the volume distribution (vd) of the LISST, yielding a mass distribution (md) per unit volume (Fig. 1b). The md is used to calculate a new frequency distribution (Fig. 1c), which is thus weighted by the densities rather than having an equal weight over the size range of the LISST. The resulting increased frequencies of small floc sizes in the md-frequency distribution compared with those of the vd-frequency distribution (Fig. 1c) are in accordance with the understanding that denser microflocs will have a larger relative influence on the mean settling velocity than more porous macroflocs. The settling velocity distribution for one measurement is calculated using Stokes' Law: 2

1400 1200 800 600 400 200

(b)

12 10 8 6 4 2 0

0 10

0.1

ð9Þ

14

(a)

1000

0.12

g  Δρi  d i 18  μ

where i again is the size group (1–32 for the LISST), μ is dynamic viscosity, Δρi is the effective density of the ith group and di is the midpoint size represented by that group. The resulting settling velocity distribution (Fig. 1d) is again shifted towards smaller values in the md-version because this distribution is based on the weighted mdfrequency distribution. By recalculating the settling velocity distribution into phi-units (logarithmical transformation), the mean settling velocity, ws, is found using the method of moments. The measurement used for the example in Fig. 2 has a dp of 6 μm and a ρp of 2350 kg m−3, and the change in frequency between the md and vd is seen in Fig. 2 (blue line). If the density is raised to 2650 kg m−3, the frequencies of size classes with sizes close to dp naturally increase (Fig. 2, green line). Second, if the density is 2350 kg m−3, but dp is increased to 10 μm, all particles below this size are given a constant high density and their weight is even higher than that with a dp of 6 μm (Fig. 2, red line). These results are satisfactory; the overall implication is that the md-method, in relative terms, gives the larger flocs less weight in the calculated ws.

Mass (mg L-1)

Effective density (kg m-3)

where a and b are constants of the power law. Finally, all densities are recalculated into effective densities, Δρ, by subtracting the density of water, calculated using the standard polynomials described by Fofonoff and Millard (1983). The result is an effective density distribution over the range of the PSD of the LISST, with a constant

Frequency

3.3. Step 3: Recalculations

wsi ¼

≤ dp, ρi = ρp N dp, ρi is calculated using the density function = df, ρi = ρf N df, ρi is calculated using the density function

For di For di For di For di

density until the mean primary particle size is reached and a gradual decrease in density over the rest of the size spectrum (Fig. 1a).

100

10

(c)

0.08 0.06 0.04 0.02 0 10

100

Particle size (μm)

1000

Settling velocity (mm s-1)

➣ ➣ ➣ ➣

13

100

1000

100

1000

1.2

(d) 0.9 0.6 0.3 0 1

10

Particle size (μm)

Fig. 1. Examples of the functions that are calculated for the md-method. (a) Effective density, (b) mass distribution, (c) frequency distributions, (d) settling velocity distributions. Dashed lines are calculated based on the volume distribution, solid lines are calculated based on the modified mass distribution. The dots in (a) show the two boundary conditions, the example is with dp = 6 μm and ρp = 2350 kg m−3.

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T.N. Markussen, T.J. Andersen / Marine Geology 344 (2013) 10–18

4. Experimental data

5. Method comparisons and discussion

The proposed method has been applied to three datasets from estuarine environments in three distinct climatic zones: 1) the temperate climate of the Hjerting tidal channel in the Danish Wadden Sea (DK dataset), 2) the arctic climate of the Mittivakkat glacier delta in Southeast Greenland (GR dataset) and 3) the tropical climate of the Nha Phu Bay in Vietnam (VN dataset) (see Fig. 3). All data series were collected during periods with calm weather and little or no wind. The DK dataset was collected by vertical profiling (casts) spanning five days in September 2011 (DK1) and by eulerian (moored) deployment in the lower part of the water column spanning four days in January 2012 (DK2). The GR data was collected by casts over a three-day period in July 2010, and the VN data was collected by casts over a four-day period in November 2010. Water samples were gathered for all three campaigns to characterise the suspended particulate matter concentration (SPMC), which was used to calibrate the beam attenuation signal of the LISST or OBS for each individual campaign. The dispersed primary particles in the DK and VN campaigns both had consistent mean sizes, dp, of 6 μm, whereas the mean size of the primary particles in the GR campaign was 12 μm. The LOI of the DK sediment was 11% on average, whereas the average LOI of the VN sediment was more than double that of the DK sediment at 27.6%. It has not been possible to burn the GR samples, so no LOI has been measured for that campaign. However, a large number of samples were taken in the same region as that of the GR dataset the summer of the previous year (2009), and the LOI of these samples was between 1–2%. This LOI value is believed to be usable for the GR samples. Settling tube experiments using an Owen settling tube (Owen, 1976) were performed in the GR campaign. Of twelve experiments, eight provided a median settling velocity, w50. The df value was measured with a LISST and varied from 20 to 65 μm (very small microflocs), and w50, measured with a settling tube, was consistently low, at approximately 0.02–0.05 mm/s. The decreases in SPMC in the vicinity of slack water in the Danish eulerian deployment (DK2) were used to calculate mass settling velocities based on half-lives of the concentration as described by Amos and Mosher (1985), see Fig. 4. The df value was in the range of 60–120 μm for these periods, and the calculated mass settling velocities of the six slack periods ranged from 0.20 to 0.55 mm/s.

The effective density relations calculated for the md-method are within the range of calculated relations for estuarine flocs found in the literature (Fig. 5). These relations are based on various calculation methods which e.g. cause the densities of small flocs to be in the high end compared to those found in the literature, as small flocs approach those of the primary particles in the md-method. The settling velocities calculated using the md-method have been plotted with calculated velocities using fractal theory methods to assess the performance of the md-method against established methods (Fig. 6). The VN dataset has been split into two groups to exemplify differences within individual datasets. The orange dots in the VN dataset designate data from the estuary head (VN1,1), whereas the purple dots represent data sampled from further out in the bay (VN1,2). The md-method velocities of the GR data are almost all below 0.1 mm/s and are thus in the range of those measured using the settling tube. The md-method velocities in the DK2 dataset range roughly from 0.05 to 0.3 mm/s, which also corresponds well with those calculated using the mass settling method. The two subsets in Fig. 6 show significantly different calculated settling velocities, which are dependent upon the applied fractal dimension theory method. A constant F of 3 (spherical solid particles) inevitably yields high velocities compared with a situation with a constant F of 2 (estuarine flocs according to Winterwerp (1998)), and this estuarine floc assumption of F = 2 yields velocities well below those calculated with the md-method in almost all situations (Fig. 6a). An F of 2.5, the upper limit of estuarine flocs, according to Strom and Keyvani (2011), is instead used as the standard constant F-value. Strom and Keyvani (2011) recommended using a b1 value of 120 but found that in some situations, the b1 value can be as low as 18 (as set forth by Stokes) if the fractal dimensions used are between 1.5 and 2 (see the black lines in Fig. 6a for examples of the influence of changing the b1 value). Settling velocities calculated based on varying fractal dimensions, found using Eqs. (4) and (5), produce highly varying velocities depending on the values used (Fig. 6b). Velocities calculated based on the Kumar model are within the range of the md-method, but the increase in velocities with floc size is highly dependent on the specified critical floc size and on the fractal dimension. Kumar et al.

0.02

Frequency change

0.01

0

-0.01

-0.02 1

10

100

1000

Particle size (μm) Fig. 2. (a) Change in frequency between the standard volume distribution method and the md-method. A positive frequency change means higher frequencies with the md-method. Blue is dp = 6 μm and ρp = 2,350 kg m−3, green is dp = 6 μm and ρp = 2,650 kg m−3, red is dp = 10 μm and ρp = 2,350 kg m−3.

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Fig. 3. Location of the three study sites from which data is gathered for testing the method. Upper right corner: location of Ho Bugt, the DK dataset, lower right corner: Nha Phu Bay, the VN dataset, and lower left corner: the Mittivakkat glacier delta in Sermilik Fjord, the GR dataset.

(2010) found reasonable values to be Fc = 1.95 and dfc = 200 μm, which indeed yield velocities that fit perfectly with the estuary head data from Vietnam (VN1,1) but are higher than all of the other values calculated using the md-method presented here (Fig. 6b). Lowering the critical fractal dimension yields lower velocities because flocs reach the critical value later and are thus modelled with a smaller

(a)

SPMC (mg L-1)

100

density. The opposite situation is the case when the critical floc size is lowered. All of the three examples using the Kumar model are high compared with the md-model calculations. Velocities calculated using the Maggi model and a self-similarity parameter, γ, of − 0.05 appear to be more comparable to the values calculated using the md-method (Fig. 6b). Maggi et al. (2007) recommended a γ of − 0.1,

5.5

(b)

5

120

4.5 4

80

Depth (m)

Mean floc size (μm)

10

3.5 40

3 12

18

24

30

36

42

48

54

Hours from start Fig. 4. Campaign DK2-data. (a) SPMC over time and (b) depth and mean floc size over time. The solid black lines in (a) are the half-life functions that are used to calculate the mass settling velocities as described by Amos and Mosher (1985).

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which produces rather slow settling velocities over the entire floc size range. Settling velocities are even lower when using a γ of −0.2. The settling velocities calculated using the md-method form groupings of data within individual datasets. These groupings are designated by different colours in Fig. 6. The VN dataset shows differences between the inner and middle/outer estuarine areas, such that the flocs in the area close to the estuary head (VN1,1) have faster calculated settling velocities relative to their sizes than those further from the estuary head (VN1,2). Essentially, these differences among the VN data are caused by a difference in the relationship between VC and SPMC, which is likely caused by lower floc densities further from the estuary head. This conclusion agrees with expectations because flocs should be more susceptible to biological mediation and should experience lower turbulent shear levels down estuary, both of which likely lead to lower-density flocs. The DK dataset shows differences between the autumn and winter campaigns, DK1 and DK2, respectively, which again are likely caused by varying floc densities. The results from the DK and VN datasets indicate a significant variability in floc characteristics (e.g., effective density) both temporally and spatially, even on a short temporal or spatial scale. These results support the utility of the md-method because this method estimates the changes and differences in effective densities of the suspended flocs in a very simple way. The suspended load in estuarine waters is typically a mixture of resuspended fragments of bio-aggregates formed at the bed, mineral particles and organic rich aggregates supplied from terrestrial sources and mixed flocs formed by flocculation in the water column. The self-similarity of aggregate morphology across the observed size-range that is assumed with a constant fractal dimension is, in this respect, undesirable. 5.1. Method implications Fractal dimension theories may be quite useful in taking into account the physical factors and forces influencing a settling floc (e.g., drag, gravitation and viscous forces). However, although fractal dimension theories are comprehensive, they can only incorporate the well-known floc mediations caused by biological agents to a limited degree. Fettweis (2008) reported standard deviations in calculated settling velocities that were often above 100%. These high standard deviations must be caused by the multitude of forces and factors that influence sediment properties and dynamics in nature. The

concept of essentially constant floc properties over the entire floc size range at a given site, which is assumed in fractal dimension models, seems contradictory to the documented differences in floc composition and shape. It is possible that the self-similarity of flocs is higher in high turbidity estuaries where the main aggregation process is flocculation in the water column, with a certain mean floc evolution, shape and composition. However, in low turbidity estuaries, for instance, floc shape and composition differences can strongly influence settling velocities for example because of the relatively larger influence of biogenic aggregates created by a range of biological processes. These differences in floc properties limit the utility of fractal dimension theories. Thus, there appears to be a disparity between the comprehensiveness of fractal dimension methods and the ability to describe actual settling velocities. The proposed md-method is much simpler and yields results within the range of examples from settling tube measurements, within the range of mass settling velocities calculated on the basis of SPMC-half-life observations in the vicinity of slack water, and within the range of settling velocities predicted using fractal dimension methods. Comprehensive studies and field measurements would be required to take into account all of the factors influencing the properties of settling flocs. For example, to understand floc composition, one would need detailed microscope images of flocs, together with particulate samples for the determination of biogenic matter (chlorophyll content, biogenic minerals, dissolved and particulate carbon, TEP and EPS, etc.) and inorganic matter (lithogenic minerals, clay mineral composition, etc.) (see e.g., Bainbridge et al., 2012). Second, a model of the settling velocity, which incorporates all of these factors, would have to be developed specifically for the investigated area and would need to be calibrated against in situ measurements, e.g., with a video camera. Researchers are most often limited to broader approximations when calculating floc settling velocities, and typically only the floc size distribution (e.g., LISST measurements), SPMC and LOI are measured. The md-method can be used to give first order estimates of floc settling velocities in these situations in which more detailed information concerning floc characteristics is not available. Although the LOI is a crude number that does not describe the composition, activity and types of organic matter, it serves as a simple proxy for the relative biological influence on the altered effective densities between sites. There are indications that the md-method can take spatial and temporal variations, which are also caused by

Effective density (kg m-3)

1000 This study Gibbs (1985) Al Ani et al. (1991) Fennessy et al. (1994) Manning and Dyer (1999) Mikkelsen and Pejrup (2001)

100

10

1

1

10

100

1000

10000

Floc size (µm) Fig. 5. Calculated relations for the effective density as a function of floc size for estuarine flocs. The bold line is the overall mean of the effective densities for the DK1 dataset, calculated using the method described in the paper, and other lines show functions from (Gibbs, 1985; Al Ani et al., 1991; Fennessy et al., 1994; Manning and Dyer, 1999; Mikkelsen and Pejrup, 2001).

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Settling velocity (mm s-1)

1

(a)

Δρ =1,650

F=3.0

b1=18

17

b1=70

0.8

0.6

b1=120

0.4

0.2

F=2.0

0

Settling velocity (mm s-1)

1

(b)

dfc =200, Fc =1.95

dfc =100, Fc =2.5

Δρ =1,650

0.8

dfc =200, Fc =1.8 γ=-0.05

0.6

0.4

γ= -0.1

0.2

γ= -0.2

0

0

40

80

120

160

200

Mean floc size (μm) Fig. 6. Settling velocities calculated with the md-method (dots) and methods from literature using fractal dimensions (lines). Green dots represent the autumn DK campaign (DK1), dark green is the winter DK campaign (DK2), orange is estuary head measurements from VN (VN1,1), red is middle/outer bay measurements in the VN dataset (VN1,2), and purple is GR data. The dash-dotted line shows calculations based on Stokes' Law and a constant effective density of 1650 kg m−3. All other lines show velocities based on fractal theory, which are calculated using Eq. (3) and a b1 of 120 if nothing else is stated. In (a) constant fractal dimensions are used, solid lines are F = 2.5 and show the influence of varying b1-values, and dashed lines show the influence of varying F (b1 = 120). (b) shows variable fractal dimension methods and a b1 of 120, solid lines are calculated using Eq. (4) of Maggi et al. (2007), dashed lines are calculated using Eqs. (5) and (6) of Kumar et al. (2010).

differences in effective densities, into account as well. Thus, effective density calculations are quite important for the accurate determination of settling velocities. This method is highly dependent on the assumption that the volume concentration of the LISST is a measure of the volume concentration of flocs (Mikkelsen and Pejrup, 2001; Voulgaris and Meyers, 2004) and uses the method for calculating floc mass (Eq. (1)) set forth by (Fettweis, 2008). If volume concentrations are known and the mean primary particle size and organic/mineral ratio have been measured or can be estimated, the effective density functions can be calculated, and the md-method can be used. A future study will focus on using the md-method with a camera system and an OBS-sensor to test how the method applies to other instruments in addition to the LISST and if effective density functions can still be calculated. This future work may further enable a direct comparison between settling velocities calculated using the md-method and settling velocities measured using a non-intrusive in situ camera system. 6. Conclusion A number of fractal dimension methods that demonstrate ways to calculate the settling velocity of flocs, essentially based on their effective density and size, are present in literature. Although these methods may be physically sound, they do not sufficiently explain the diversity of floc morphology and composition, particularly the diversity caused by

variations in the biological influence over the measured size spectrum. These difficulties regarding floc morphology and composition have been exemplified in the significantly different ranges observed in calculated settling velocities using fractal dimensions, which are dependent on the applied values or constants. The proposed md-method does not explain the comprehensiveness of floc properties, nor is it an entirely new way to calculate settling velocities. However, there are clear indications that this method can take into account the varying effective densities of flocs within a single dataset or from an individual measurement campaign without having to change certain constants or variables manually. Furthermore, this method is extremely simple and relies on only two factors: 1) measurements of floc size and floc volume using a LISST, e.g., and 2) water samples to characterise the mean properties of the primary particles that make up the flocs and to calibrate the attenuation of the measuring instrument. The water samples also aid in providing a proxy for the biological influence through the LOI. Initial tests of the md-method show that the calculated settling velocities agree very well with measurements from a settling tube and a mass settling method. Additionally, the calculated settling velocities show the same size dependency and are within the range of those predicted by fractal dimension models. Thus, the md-method is believed to yield reliable, first order estimates of floc settling velocities that are particularly useful when detailed information about the floc composition is not known.

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