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<head>1. Introduction<lb/></head>
<p>[2] Flocculation dynamics of cohesive sediment flocs<lb/> suspended in natural waters is
of particular interest for its<lb/> impact on sediment transport and deposition, and the
large-<lb/>scale morphodynamic evolution of coastal zones, estuaries,<lb/> rivers, and
water basins in general [e.g., <ref type="biblio">Dyer, 1989; Mehta,<lb/> 1989; Seminara
and Blondeaux, 2001; McAnally and<lb/> Mehta, 2001</ref>]. Among various
hydrological, biophysical,<lb/> and climatological factors [e.g., <ref type="biblio"
>Droppo et al., 1997; De<lb/> Boer et al., 2000</ref>], flocculation has a major
impact on<lb/> changes in floc size [e.g., <ref type="biblio">Van Leussen, 1994</ref>],
controlling<lb/> floc settling velocity, and vertical fluxes of sediment. At the<lb/>
smallest time and length scales, the extent of time that a floc<lb/> spends at a given
size (floc mobility) can be crucial in<lb/> determining whether that floc will remain in
suspension or<lb/> will deposit.<lb/></p>
<p>[3] Various approaches can be followed to model floccu-<lb/>lation of cohesive sediment.
One can model the rate of<lb/> change of concentration of flocs with size between L
and<lb/> L + DL by using population balance equations of the type<lb/> of <ref
type="biblio">Smoluchowski von [1917]</ref>. This can be done analytically<lb/> only
under some specific assumptions on the aggregation<lb/> and breakup kernels [e.g., <ref
type="biblio">Kokholm, 1988; Davies et al.,<lb/> 1999; Shirvani and van Roessel,
2001, 2002; Leyvraz,<lb/> 2003</ref>]. More commonly, numerical solutions can be
used<lb/> to have a higher degree of freedom in modeling aggrega-<lb/>tion and breakup
[e.g., <ref type="biblio">Rahmani et al., 2003; Serra and<lb/> Casamitjana, 1998; Flesch
et al., 1999; O'Melia, 1980;<lb/> Lick and Lick, 1988; Lick et al., 1993;
Spicer et al., 1996</ref>].<lb/> Yet, depending on the number of classes used to
discretize<lb/> the floc size distribution, and the details of aggregation and<lb/>
breakup kinematics (i.e., collision frequency, attachment<lb/> likelihood, breakup
frequency and distribution functions),<lb/> solution may require more or less demanding
computational<lb/> power, especially for parametric calibration.<lb/></p>
<p>[4] An approach complementary to Smoluchowski-based<lb/> population balance equations is
modeling the rate of change<lb/> of size of an individual,
''prototype'' floc. A successful<lb/> example of this Lagrangian
modeling approach is the one<lb/> proposed by <ref type="biblio">Winterwerp
[1998]</ref><lb/></p>
<formula>dL<lb/> dt<lb/> ¼ k a cGL 4Àd0 À k b G 3=2 ðL À L p Þ 3Àd0 L 2 ;<lb/>
<label>ð1Þ</label><lb/></formula>
<p>where L p is the primary particle size, c is the sediment mass<lb/> concentration, G is
the turbulence shear rate, d 0 is the three-<lb/>dimensional floc capacity dimension,
and k a and k b are<lb/> aggregation and breakup parameters. Although carrying<lb/>
important simplifications of aggregation and breakup<lb/> processes, and having no
explicit solution, the advantage<lb/> in using Winterwerp's model is that a steady
state (or<lb/> equilibrium) floc size is readily assessed analytically. For<lb/> d 0 =
2, value averagely observed for real flocs [e.g.,<lb/>
<ref type="biblio">Winterwerp, 1998</ref>], the equilibrium floc size L* is<lb/></p>
<formula>L* ¼ L p þ<lb/> k a c<lb/> k b G 1=2 :<lb/>
<label>ð2Þ</label><lb/></formula>
<p>None of these modeling approaches, however, is capable of<lb/> assessing the floc size
fluctuations over time because of<lb/> their deterministic nature: once the equilibrium
is reached,<lb/> a balance between aggregation and breakup will preserve<lb/> the stable
solution over time. However, if we imagine<lb/> following a primary particle during its
life, and compile a<lb/> table of the times at which its mass increases or
decreases,<lb/> we will realize that the mass will change abruptly,<lb/> remaining
subsequently at rest for a certain time Dt until<lb/> a new change will occur. The mass
variation will be a<lb/> random variable as well as the inter-event time Dt. This<lb/>
perspective suggests that a key feature in floc dynamics is<lb/> the temporal
fluctuation in floc size, calling into question<lb/> the assumptions of deterministic
flocculation models that<lb/> invoke stable solutions, that is, with no time
fluctuations.<lb/> Stochastic modeling can represent an alternative framework<lb/> when
approaching a problem in which the substantial<lb/> physics is affected by stochastic
effects.<lb/></p>
<p>[5] The aim of this work was to explore with a mathe-<lb/>matical model the temporal
(stochastic) variability of the<lb/> floc size around an equilibrium value during
flocculation.<lb/> Stochastic methods to solve the deterministic Smoluchowski<lb/>
equation have already been proposed [e.g., <ref type="biblio">Smith and<lb/> Matsoukas,
1998; Kolodko et al., 1999; Lee and Matsoukas,<lb/> 2000; Lin et al., 2002; Khelifa
and Hill, 2006b</ref>] and<lb/> compared with each other <ref type="biblio">[Zhao et
al., 2007]</ref> but none has<lb/> focused on the analysis of the dynamics and
mobility of an<lb/> individual floc in probabilistic terms. The purpose here was<lb/> to
elaborate the deterministic model of equation <ref type="formula">(1)</ref> to take<lb/>
into account stochastic effects in particle aggregation and<lb/> breakup. This approach
is complementary to the ones by<lb/>
<ref type="biblio">Smith and Matsoukas [1998]</ref> and <ref type="biblio">Khelifa and
Hill [2006b]<lb/></ref> as much as the Winterwerp equation is complementary to
the<lb/> Smoluchowski equation. Because this approach did not<lb/> require solution of
the entire population dynamics, fast<lb/> computation and calibration on experimental
data were<lb/> allowed.<lb/></p>
<p>[6] This stochastic model was used to look into specific<lb/> dynamical behaviors of
flocculation that could not be<lb/> otherwise detected by deterministic models.
Particular<lb/> attention was given to the correlation between the fluctua-<lb/>tion
distribution of L(t) and the probabilities of aggregation<lb/> and breakup, the
intermittency patterns raising at steady<lb/> state, and the impact of the population
distribution on the<lb/> floc size transition toward steady state. We analyzed how<lb/>
these features strongly depended on the particle concentra-<lb/>tion, turbulent shear
rate, and the fractal model used to<lb/> describe floc geometry, i.e., with either
constant or variable<lb/> capacity dimension.<lb/></p>
<head>2. Stochastic Model<lb/></head>
<head>2.1. Theory<lb/></head>
<p>[7] Prescribed that flocs can be modeled as fractal units,<lb/> the number of primary
particles n within a floc of size L can<lb/> be written as <ref type="biblio">[Vicsek,
1992]<lb/></ref></p>
<formula>n ¼ ðL=L p Þ d0 ;<lb/>
<label>ð3Þ</label><lb/></formula>
<p>with d 0 the capacity (fractal) dimension of the flocs. Solving<lb/> equation (3) for L
and substituting it throughout<lb/> equation (1), with the differential dL = ðL p n 1=d
0 À1 =d 0 Þ dn,<lb/> we can equivalently re-write equation (1) in terms of n as<lb/></p>
<formula>dn<lb/> dt<lb/> ¼<lb/> d 0 G<lb/> L d0À3<lb/> p<lb/> ck a n<lb/> 3<lb/> d 0 À
G<lb/> 1<lb/> 2 k b n<lb/> d 0 þ1<lb/> d 0 L p n<lb/> 1<lb/> d 0 À 1<lb/> 3Àd0<lb/>
!<lb/> :<lb/>
<label>ð4Þ</label><lb/></formula>
<p>The factors k a and k b are defined as<lb/></p>
<formula>k a ¼ k a 0<lb/> 1<lb/> L 3Àd0<lb/> p<lb/> d 0 r s<lb/> ; k b ¼ k b 0 1<lb/> L
3Àd0<lb/> p<lb/> d 0<lb/> m<lb/> F y<lb/> 1<lb/> 2<lb/> ;<lb/>
<label>ð5Þ</label><lb/></formula>
<p>with k a<lb/> 0 and k b<lb/> 0 dimensionless calibration parameters, r s the<lb/>
sediment density, m the water dynamic viscosity, and F y the<lb/> floc strength.
equation <ref type="formula">( 4)</ref> can be re-written as<lb/></p>
<formula>dnðtÞ<lb/> dt<lb/> ¼ f a ðnÞn À f b ðnÞn;<lb/> ð6Þ<lb/></formula>
<p>where f a (n) and f b (n) are the aggregation and breakup<lb/> frequency functions (or
rates of gain and loss) written as<lb/></p>
<formula>f a ðnÞ ¼ k a 0 cG<lb/> r s<lb/> n<lb/> 3<lb/> d 0<lb/> À1 ;<lb/>
<label>ð7Þ</label><lb/></formula>
<formula>f b ðnÞ ¼ k b 0 L p G<lb/> 3<lb/> 2<lb/> m<lb/> F y<lb/> 1<lb/> 2<lb/> ðn<lb/>
1<lb/> d 0 À 1Þ 3Àd0 n<lb/> 1<lb/> d 0 :<lb/>
<label>ð8Þ</label><lb/></formula>
<p>Given the initial value n(0) = n 0 , the steady state solution n*<lb/> occurs when f a
(n) = f b (n) and corresponds to the number of<lb/> primary particles within the
equilibrium floc of size L* as in<lb/> equation <ref type="formula">(2)</ref>.<lb/></p>
<p>[8] The Lagrangian deterministic model of equation <ref type="formula">(6)</ref><lb/> has
the mathematical advantage of being written in the same<lb/> form as birth-death
dynamics where aggregation and break-<lb/>up frequency functions f a (n) and f b (n) are
the analogue of<lb/> birth and death rates. Stochastic models for these types of<lb/>
equations and for jDnj = 1 are well known as continuous-<lb/>time discrete-state
Markov-chain models. The principles of<lb/> the mathematical theory of continuous-time
discrete-state<lb/> Markov-chain models is used within this context [e.g.,<lb/>
<ref type="biblio">Allen, 2003; Allen and Allen, 2003; Matis and Kiffe,<lb/> 2004;
Novozhilov et al., 2006</ref>], but we introduce new<lb/> elements to take into
account that stochastic aggregation<lb/> and breakup imply discrete changes jDnj ! 1 as
in<lb/> stochastic methods to solve the Smoluchowski equation<lb/> [e.g., <ref
type="biblio">Smith and Matsoukas, 1998</ref>].<lb/></p>
<p>[9] A stochastic modeling of the time evolution of the<lb/> number of primary particles n
implies that n is not known<lb/> with certainty but with some probability. Let the
primary-<lb/>particle number n(t) 2 N be the floc state at time t 2 R, and<lb/> the
probability that the random variable X(t) equals n at time<lb/> t be defined as p n (t)
= Prob{X(t) = n}. To relate the random<lb/> variables {X(t)} in time we make the
following assump-<lb/>tions:<lb/></p>
<p>[10] (i) the system has a fixed number of primary particles<lb/> N ) 1;<lb/></p>
<p>[11] (ii) the probability Prob{X(t) = n 0 } equals 1 at time<lb/> t = 0, i.e., the
initial state is know with certainty;<lb/></p>
<p>[12] (iii) in a suitably small time Dt only one event can<lb/> occur, either aggregation
or breakup;<lb/></p>
<p>[13] (iv) in a sufficiently small time Dt the probability for<lb/> aggregation to occur
is approximately f a (n) Dt, while the<lb/> probability for breakup to occur is
approximately f b (n) Dt;<lb/></p>
<p>[14] (v) when aggregation occurs, the probability to gain<lb/> m primary particles in a
single aggregation event is p m<lb/> a (t)<lb/> with 1 m N À n;</p>
<p>[15] (vi) when breakup occurs, the probability to lose m<lb/> primary particles in a
single breakup event is p m<lb/> b (t), with 1<lb/> m n À 1.<lb/><lb/></p>
<p>[16] Based on assumptions (iii) to (vi), for a floc to have n<lb/> primary particles at
time t + Dt, either it is at state n À m at<lb/> time t and aggregation of a floc with m
primary particles<lb/> occurs during Dt, or it is at state n + m at time t and loss
of<lb/> m primary particles due to breakup occurs during Dt, or it is<lb/> at state n at
time t and neither mass gain nor mass loss occur<lb/> during Dt. This can be formulated
as the conditional<lb/> probability<lb/></p>
<formula>p n ðDtÞ ¼ Prob DX ðtÞ ¼ mjX ðtÞ ¼ n<lb/> f<lb/> g<lb/> ¼<lb/> ðn À mÞf a ðn À mÞp
a<lb/> m ðtÞDt;<lb/> ðn þ mÞf b ðn þ mÞp b<lb/> m ðtÞDt;<lb/> 1 À ½ðn À mÞf a ðn À mÞp
a<lb/> m ðtÞ þ ðn þ mÞf b ðn þ mÞp b<lb/> m ðtÞDt;<lb/> 8<lb/> <<lb/> :<lb/>
<label>ð9Þ</label><lb/></formula>
<p>with DX(t) = X(t + Dt) À X(t). In terms of difference<lb/> equations we have<lb/></p>
<formula>p n ðt þ DtÞ ¼ ðn À mÞp nÀm ðtÞf a ðn À mÞp a<lb/> m ðtÞDt<lb/> þ ðn þ mÞp nþm ðtÞf
b ðn þ mÞp b<lb/> m ðtÞDt<lb/> þ p n ðtÞ 1 À ½ðn À mÞf a ðn À mÞp a<lb/> m ðtÞ<lb/>
È<lb/> þ ðn þ mÞf b ðn þ mÞp b<lb/> m ðtÞDt<lb/> É<lb/> :<lb/></formula>
<p>Subtracting p n (t) from both sides and dividing by Dt ! 0<lb/> we obtain the forward
Kolmogorov differential equation<lb/>
<ref type="biblio">[Allen, 2003]</ref><lb/> dp n ðtÞ<lb/> dt<lb/></p>
<formula>¼ ðn À mÞp nÀm ðtÞf a ðn À mÞp a<lb/> m ðtÞ<lb/> þ ðn þ mÞp nþm ðtÞf b ðn þ mÞp
b<lb/> m ðtÞ<lb/> À ½ðn À mÞf a ðn À mÞp a<lb/> m ðtÞ<lb/> þ ðn þ mÞf b ðn þ mÞp b<lb/>
m ðtÞ;<lb/>
<label>ð10Þ</label><lb/></formula>
<p>which holds 8n 6 ¼ 1, N. When the floc state is n = 1 (the<lb/> primary particle) no
breakup is allowed and all terms with<lb/> p m<lb/> b (t) drop, whereas if n = N (all
particles in the system have<lb/> integrated into a single floc) no further aggregation
is<lb/> possible, and all terms with p m<lb/> a (t) drop.<lb/></p>
<p>[17] Finding the probability p n (t) in equation <ref type="formula">(10)</ref>
solves<lb/> the stochastic modeling problem as much as finding the<lb/> solution n(t) in
equation <ref type="formula">(6)</ref> solves the deterministic prob-<lb/>lem. However,
the analytical solution to equation (10) can<lb/> be difficult to find when the rates f
a (n) and f b (n) are<lb/> functions of the floc state n. Alternatively, a
numerical<lb/> approach that mirrors the assumptions made to write the<lb/> forward
Kolmogorov differential equation can be used. To<lb/> accomplish this, we must define
the inter-event time Dt, the<lb/> probability p m<lb/> a (t), and the probability p
m<lb/> b (t).<lb/></p>
<head>2.2. Inter-event Time<lb/></head>
<p>[18] In the case of aggregation only, it can be easily<lb/> shown [e.g., <ref
type="biblio">Allen, 2003</ref>] that the inter-event time Dt a for<lb/> aggregation
to occur is a continuous random variable with<lb/> exponential distribution. Hence for a
floc with n primary<lb/> particles, the time t to the next event has cumulative<lb/>
distribution<lb/></p>
<formula>Pðt ! Dt a Þ ¼ e ÀfaðnÞnDta ;<lb/>
<label>ð11Þ</label><lb/></formula>
<p>with f a (n) the rate parameter defined in equation <ref type="formula">(7)</ref>.
The<lb/> inter-event time is calculated by substituting a random<lb/> number between 0
and 1, extracted from a uniform<lb/> distribution, to P(t ! Dt a ) in equation <ref
type="formula">(11)</ref>, which is next<lb/> solved for Dt a . As f a is a
monotonically increasing function<lb/> of n and c, the time Dt a will decrease with n
and c<lb/> increasing. Similarly, a negative exponential distribution<lb/> function can
be written for the inter-event time Dt b for<lb/> breakup to occur by exchanging f a (n)
with f b (n), and Dt a<lb/> with Dt b in equation <ref type="formula">(11)</ref>.
Following assumption (iii) that<lb/> exclusively one process between aggregation and
breakup<lb/> occurs, we extract the aggregation and breakup inter-event<lb/> times Dt a
and Dt b from their cumulative distributions, and<lb/> determine Dt by taking<lb/></p>
<formula>Dt ¼ minfDt a ; Dt b g:<lb/>
<label>ð12Þ</label><lb/></formula>
<p>This procedure also serves to determine which of the two<lb/> events is to happen, that
is, if Dt a < Dt b aggregation will<lb/> occur, vice versa otherwise.<lb/></p>
<head>2.3. Stochastic Aggregation<lb/></head>
<p>[19] For Dt a < Dt b , a gain of m primary particles will<lb/> occur with a
probability p m<lb/> a (t) that we can define as<lb/></p>
<figure>p a<lb/> m ðtÞ ¼ p m ðtÞa n;m ðtÞ;<lb/>
<label>ð13Þ</label><lb/></figure>
<p>where p m (t) is the probability for a floc with n primary<lb/> particles to collide with
a floc with m primary particles, and<lb/> a n,m (t) is the probability for these to
attach.<lb/></p>
<p>[20] The probability p m (t) is, in essence, the size distri-<lb/>bution of the floc
population at time t, and is a model input<lb/> that can be provided in various ways,
including the distri-<lb/>bution measured in real experiments, or a polynomial<lb/>
expression as in <ref type="biblio">Khelifa and Hill [2006b]</ref>. In this
modeling<lb/> approach we will give preference to an analytical descrip-<lb/>tion of p m
(t) to conveniently take into account the effect of<lb/> time and shear rate. From
experimental optical recording<lb/> carried out in settling column tests <ref
type="biblio">[Maggi, 2007]</ref>, the<lb/> steady state floc primary-particle
distribution can be<lb/> described with a lognormal probability distribution
function<lb/></p>
<formula>p m ð1Þ ¼<lb/> 1<lb/> ½mðLÞ<lb/>
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi<lb/> 2ps 2 ð1Þ<lb/> p<lb/>
e<lb/> À 1<lb/> 2<lb/> fln½mðLÞÀmð1Þg 2<lb/> s 2 ð1Þ<lb/> ;<lb/>
<label>ð14Þ</label><lb/></formula>
<p>where m(L) is the number of primary particles in a floc of<lb/> size L. The distribution
parameters for the steady state floc<lb/> populations<lb/></p>
<formula>s 2 ð1Þ ¼ log½1 þ s 2<lb/> m =m 2 ;<lb/>
<label>ð15Þ</label><lb/></formula>
<formula>mð1Þ ¼ log½m À s 2 ð1Þ=2;<lb/><label>ð16Þ</label><lb/></formula>
<p>are computed from the average number of primary-particle<lb/> in a floc, m, and variance,
s m<lb/> 2 , obtained form the same<lb/> experimental data <ref type="table">(Table
1)</ref>. Finally, p m is normalized such<lb/> that Sp m Dm = 1. Equation <ref
type="formula">(14)</ref> reproduces reasonably well<lb/> empirically acquired floc
distributions <ref type="figure">(Figure 1)</ref>. Although<lb/> a first approximation,
the use of lognormal distributions has<lb/> been shown in earlier works [e.g., <ref
type="biblio">Lambert et al., 1981;<lb/> McCave, 1984; Burban et al., 1989</ref>] to
agree well with real<lb/> distributions. Equation <ref type="formula">(14)</ref> is
therefore used as a starting<lb/> point to introduce in this stochastic model a floc
distribution<lb/> in a mathematically simple way, and to allow numerical<lb/>
computation with ease compared to other techniques, e.g.,<lb/> by solution of the
Smoluchowski equation.<lb/></p>
<p>[21] In order to take into account the effect of time on p m<lb/> during the transition
toward steady state we introduce the<lb/> timescale T. For 0 < t < T, we can write
s 2 and m of<lb/> equation <ref type="formula">(14)</ref> as functions of t as
follows<lb/></p>
<formula>mðtÞ ¼ mð0Þ þ ½mð1Þ À mð0Þ<lb/> t<lb/> T<lb/> ;<lb/>
<label>ð17Þ</label><lb/></formula>
<formula>s 2 ðtÞ ¼ s 2 ð0Þ þ ½s 2 ð1Þ À s 2 ð0Þ<lb/> t<lb/> T<lb/> :<lb/>
<label>ð18Þ</label><lb/></formula>
<p>The parameters m(0) and s 2 (0) describe the initial lognormal<lb/> distribution p m (0),
while m(1) and s 2 (1) describe the<lb/> steady state p m (1) for t > T.<lb/></p>
<p>[22] The probability of effective attachment a n,m (t)<lb/> between two interacting flocs
in equation <ref type="formula">(13)</ref> depends on<lb/> their hydrodynamic
interaction. This interaction can be<lb/> expressed in terms of floc size using the
formulation by<lb/>
<ref type="biblio">Pruppacher and Klett [1978]</ref> later extended to include<lb/>
hydrodynamic effects due to the flocs porosity via their<lb/> capacity dimensions <ref
type="biblio">[Maggi, 2007]</ref><lb/> a n;m ðtÞ ¼<lb/> 9<lb/></p>
<formula>d 3 ðLðnÞÞd 3 ðLðmÞÞ<lb/> R 2<lb/> 2½1 þ R 2 ;<lb/>
<label>ð19Þ</label><lb/></formula>
<p>with R ¼ minfLðnÞ;LðmÞg<lb/> maxfLðnÞ;LðmÞg and 0 < a n,m < 1.<lb/></p>
<p>[23] The product p m (t)a n,m (t) is normalized such that the<lb/> cumulative<lb/> P<lb/>
p m (t)a n,m (t) Á Dm equals 1; next, the cumu-<lb/>lative is used to determine the gain
m following aggregation<lb/> and the new floc state n(t + Dt) = n(t) + m.<lb/></p>
<p>[24] Finally, because of assumption (i), we assume that<lb/> any change in floc state
n(t) ! n(t + Dt) will not affect the<lb/> population distribution p m (t).<lb/></p>
<head>2.4. Stochastic Breakup<lb/></head>
<p>[25] Breakup occurs in our model when Dt a ! Dt b .<lb/> Determining the loss m during
fragmentation is not a<lb/> straightforward task as it presupposes some knowledge
of<lb/> the number of fragments and their mass distribution while,<lb/> at present,
there is no general consensus on this aspect or<lb/> direct, visual evidence such as
imaging recording.<lb/></p>
<p>[26] Many works have been devoted to numerical analy-<lb/>sis of the effect of various
breakup kinematics on the floc<lb/> size distribution. Breakup into binary fragments
with equal<lb/> mass was one of the first hypothesis, and was later extended<lb/> to
ternary breakup (e.g., one floc with half mass and two<lb/> flocs with one fourth of the
mass of the originating floc<lb/>
<ref type="biblio">[Spicer and Pratsinis, 1996; Flesch et al., 1999]</ref>).
Multiple<lb/> fragmentation with binomially distributed mass was shown<lb/> to give
better results for the time evolution of the floc size<lb/> distribution [e.g., <ref
type="biblio">Serra and Casamitjana, 1998; Mietta et<lb/> al., 2005</ref>], but it
is not supported by direct evidence, and<lb/> had the disadvantage of making the
computation very<lb/> onerous when solving the Smoluchowsi equation. Binary<lb/>
fragmentation into flocs with mass extracted from a uniform<lb/> distribution was
recently shown by <ref type="biblio">Khelifa and Hill [2006b]</ref><lb/> to achieve an
apparent success while retaining model<lb/> simplicity. However, binary breakup with
either equal mass<lb/> or mass extracted from a uniform distribution may combine<lb/>
various breakup mechanisms with no chance to distinguish<lb/> surface erosion and
fracture from one another [e.g., <ref type="biblio">Mietta<lb/> et al., 2005; Khelifa
and Hill, 2006b</ref>]. To address this, an<lb/> interesting insight came from 2D
and 3D mechanistic<lb/> (rheologic) breakup models by <ref type="biblio">Higashitani and
Iimura<lb/> [1998]</ref> and <ref type="biblio">Higashitani et al. [2001]</ref>,
which took into<lb/> account various attractive forces among the particles, and<lb/> the
drag force from the fluid. Their results suggested that<lb/> fragmentation produces a
spectrum of daughter flocs dis-<lb/>tributed with a power law (i.e., many small
fragments, and<lb/> relatively few large fragments). They also suggested that a<lb/>
floc with high fractal dimension disintegrates into a smaller<lb/> number of flocs with
high fractal dimension, while a floc<lb/> with low fractal dimension produces a larger
number of<lb/> flocs with a broader distribution.<lb/></p>
<p>[27] To best approximate these observations, this model<lb/> describes breakup as a
binary process that satisfies assump-<lb/></p>
<figure>Figure 1. Steady state floc primary-particle distributions<lb/> measured in settling
column tests at various turbulence<lb/> shear rates G, and for mass concentration c =
0.5 kg m À3 .<lb/> Values of m were determined via equation (3) by measuring<lb/> the
size L of individual flocs and estimating their capacity<lb/> dimension d 0 from optical
recordings <ref type="biblio">[Maggi and Winterwerp,<lb/> 2004]</ref>. The sediment used
was kaolinite with L p % 5 mm and<lb/> density r s = 2650 kg m À3 .<lb/></figure>
<figure type="table">Table 1. Statistical Parameters of the Floc Distributions of Figure
1<lb/> Expressed in Terms of Primary-Particle Number and Floc Size a<lb/> G<lb/> [ s À1
]<lb/> 5<lb/> 10<lb/> 20<lb/> 40<lb/> Average<lb/> m<lb/> 89.1<lb/> 73.1<lb/> 72.2<lb/>
58.6<lb/> % Std. deviation<lb/> s m<lb/> 241.4<lb/> 228.5<lb/> 191.2<lb/> 152.1<lb/>
Median<lb/> m 50<lb/> 30.8<lb/> 22.2<lb/> 25.5<lb/> 21.1<lb/> m<lb/> 3.4<lb/> 3.1<lb/>
3.2<lb/> 3.0<lb/> s 2<lb/> 2.1<lb/> 2.3<lb/> 2.1<lb/> 2.0<lb/> Average<lb/> L<lb/>
[mm]<lb/> 60.6<lb/> 54.9<lb/> 54.5<lb/> 49.1<lb/> % Std. deviation<lb/> s L<lb/>
[mm]<lb/> 39.4<lb/> 38.3<lb/> 35.1<lb/> 31.2<lb/> Median<lb/> L 50<lb/> [mm]<lb/>
30.8<lb/> 22.3<lb/> 25.5<lb/> 21.1<lb/> a<lb/> These quantities are related to each
other via equation (3) for a value of<lb/> capacity dimension d 0 = 2.<lb/></figure>
<p>tion (iii). The mass loss m is modeled here with a power law<lb/> probability
distribution<lb/></p>
<formula>p b<lb/> m ¼ m<lb/> À<lb/> d 0<lb/> 3Àd 0<lb/> b ;<lb/>
<label>ð20Þ</label><lb/></formula>
<p>where m is in the range 1 m n À 1 with n ! 2, and b is a<lb/> calibration parameter. p
m<lb/> b is normalized such that<lb/> P<lb/> p m<lb/> b Dm =<lb/> 1. Breakup is
described in equation <ref type="formula">(20)</ref> in terms of capacity<lb/> dimension
d 0 in a manner consistent with surface erosion of<lb/> single primary particles or
detachment of small fragments<lb/> when d 0 % 3, and fragmentation into floc with
broader<lb/> distribution when d 0 < 3. In addition, multiple fragmentation<lb/> can
be interpreted as a sequence of binary breakup events<lb/> that extend over a larger
time extent.<lb/></p>
<p>[28] Finally, m is randomly extracted from the cumulative<lb/> probability<lb/> P<lb/> p
m<lb/> b Dm, so that n(t + Dt) = n(t) À m.<lb/></p>
<head>3. Results<lb/></head>
<head>3.1. Model Calibration<lb/></head>
<p>[29] The deterministic and stochastic models of equation <ref type="formula"
>(6)</ref><lb/> and equation <ref type="formula">(10)</ref> share the aggregation and
breakup param-<lb/>eters k a<lb/> 0 and k b<lb/> 0 . Calibration of k a<lb/> 0 and k
b<lb/> 0 was performed by<lb/> using the deterministic model with the experimental
median<lb/> floc size L 50 of the four data sets of <ref type="figure">Figure 1</ref>.
For these<lb/> experiments we used sediment with primary-particle size L p =<lb/> 5 mm,
density r s = 2650 kg/m 3 , concentration c = 0.5 kg/m 3 ,<lb/> estimate floc strength F
y = 3 Á 10 À11 N, and floc capacity<lb/> dimension d 0 = 2. Calibration resulted in k
a<lb/> 0 = 0.30 and k b<lb/> 0 =<lb/> 1.5 Á 10 À6 . Determination of k a<lb/> 0 and k
b<lb/> 0 also allowed for the<lb/> use of the deterministic model to calculate with a
tolerance<lb/> criterium the timescale T required in equations <ref type="formula"
>(17)</ref> and <ref type="formula">(18)</ref><lb/> of the stochastic model. Next,
the lognormal floc distributions<lb/> p m (t) of <ref type="figure">Figure 1</ref> were
used in the stochastic model, and the<lb/> breakup exponent b = À0.41 of equation <ref
type="formula">(20)</ref> was suitably<lb/> calibrated in a way that the average
value of L(t) computed<lb/> with the stochastic model approximated L* ' L 50 at
steady<lb/> state.<lb/></p>
<p>[30] Initial conditions n(0) = 1 and p m=1 (0) = 1 were<lb/> applied to all simulations
by using m(0) and s(0) 2 suffi-<lb/>ciently smaller than 1 so that the lognormal
distribution<lb/> collapsed into a Dirac distribution with p m = 1 (0) = 1.<lb/></p>
<p>[31] Simulations comparing experimental observations of<lb/> the four data sets and
predictions of L 50 from deterministic<lb/> and stochastic modeling are represented in
<ref type="figure">Figure 2</ref>.<lb/> Although the median size predicted by the
two models<lb/> overestimated the experimental one for low G-values<lb/>
<ref type="figure">(Figures 2a and 2b)</ref>, and underestimates it for high
G-<lb/>values <ref type="figure">(Figures 2c and 2d)</ref>, the difference was small and
in<lb/> the order of magnitude of 5 mm. Floc growth, as already<lb/> observed by <ref
type="biblio">Lick and Lick [1988]</ref>, <ref type="biblio">Burban et al.
[1989]</ref>,<lb/>
<ref type="biblio">Oles [1992]</ref>, <ref type="biblio">Winterwerp [1998]</ref>, and
<ref type="biblio">Khelifa and Hill<lb/> [2006b]</ref>, showed a first exponential
growth followed by<lb/> a gentle approach to equilibrium.<lb/></p>
<p>[32] An attempt to replicate Burban's observations was<lb/> performed by using the
parameters calibrated here. However,<lb/> the results (omitted here) did not match well
with his<lb/> measurements. This observation can be ascribed to differ-<lb/>ences in the
test material. The sediment collected in site by<lb/> Burban was made of a mixture of
many minerals (e.g.,<lb/> calcite, chlorite, dolomite, illite, kaolinite,
montmorillonite,<lb/> potassium feldspar, and quartz), while the experiments used<lb/>
to calibrate the deterministic and stochastic models presented<lb/> here were carried
out with only kaolinite minerals. This test<lb/> suggested that the parameters
calibrated here could not be<lb/> considered of general validity, but rather, sediment
specific.<lb/></p>
<head>3.2. Effect of Mass Concentration and Turbulence<lb/> Shear Rate<lb/></head>
<p>[33] Changes in c and G impact the shape of the steady<lb/> state floc distribution p m
(1) [e.g., <ref type="biblio">Dyer, 1989</ref>] and, in turn,<lb/> the median floc size.
To integrate their effect into p m (t), we<lb/> elaborated further on the parameter m(1)
as follows. The<lb/> median m 50 = e m(1) of the steady state lognormal floc<lb/>
distribution p m (1) can be written in terms of L 50 as<lb/> m 50 ¼ ðL 50 =L p Þ d 0 ¼ e
mð1Þ by using the fractal scaling in<lb/> equation (3). Next, having at our disposal the
solution of<lb/> L* ' L 50 from the deterministic model for any c and G in<lb/>
equation <ref type="formula">(2)</ref>, we can obtain m(1) as<lb/></p>
<formula>mð1Þ ¼ d 0 ln 1 þ<lb/> k a c<lb/> k b L p G 1=2<lb/> !<lb/> :<lb/>
<label>ð21Þ</label><lb/></formula>
<figure>Figure 2. Deterministic and stochastic modeling results<lb/> compared to
experimental L 50 for sediment mass concen-<lb/>tration c = 0.5 kg m À3 and various
shear rates G. The curve<lb/> corresponding to stochastic model is the average of
100<lb/> realizations.<lb/></figure>
<p>Substituting equation <ref type="formula">(21)</ref> into equation <ref type="formula"
>(17)</ref> we obtain a<lb/> mass concentration-and shear rate-dependent
parameter<lb/> m(t) for the population distribution p m (t) for use in the<lb/>
stochastic model. The value s 2 (1) = 2.2 from <ref type="table">Table 1</ref> was<lb/>
instead kept invariant.<lb/></p>
<p>[34] The stochastic and deterministic models matched<lb/> well for various combinations
of c and G <ref type="figure">(Figure 3)</ref>.<lb/> Predictions of L 50 qualitatively
agreed with prior theoretical<lb/> and experimental works <ref type="biblio">[Dyer,
1989]</ref>, i.e., decreasing for<lb/> G increasing and increasing for c increasing
within the<lb/> range 0À10 kg/m 3 . The model is expected to depart from<lb/> real
observations for c exceeding this range, as further<lb/> increases in mass concentration
were suggested to lead to<lb/> lower L 50 <ref type="biblio">[Dyer,
1989]</ref>.<lb/></p>
<head>3.3. Correlation Between p n (t), p m (t) and p m<lb/> b<lb/></head>
<p>[35] The mobility of a floc of size L(t) within the<lb/> population size spectrum is
related to the population and<lb/> breakup probability distributions p m (t) and p
m<lb/> b . Their com-<lb/>bined effect determines the fluctuations of L(t) around
its<lb/> average L* ' L 50 , and, consequently, the shape of p n (t).<lb/></p>
<p>[36] Limiting this analysis to the steady state, the fluctua-<lb/>tions p n (t) of L(t)
and the population distribution p m (t) are<lb/> highly correlated for various values of
G, while the corre-<lb/>lation between p n (t) and p m<lb/> b is low <ref type="figure"
>(Figure 4)</ref>. The strong<lb/> correlation of p n (t) with p m (t) was found to
hold for a wide<lb/> range of c as well (data not shown).<lb/></p>
<p>[37] The correlation of p n (t) with p m (t) and p m<lb/> b drops<lb/> drastically if we
consider a suspension that is uniformly<lb/> distributed and binary breakup is modeled
as in <ref type="biblio">Khelifa and<lb/> Hill [2006b]</ref> by uniformly distributed
fragments <ref type="figure">(Figure 5a)</ref>.<lb/> A similar drop in correlation is
obtained if the suspension is<lb/> Dirac-distributed at the primary particle size and
breakup<lb/> consists of erosion of primary particles <ref type="figure">(Figure
5b)</ref>.<lb/></p>
<p>[38] These results show that when stochastic processes of<lb/> aggregation and breakup
are either white noises with jDnj !<lb/> 1 or colored noises with jDnj = 1 there still
remains a<lb/> characteristic distribution of the fluctuations of L(t) at steady<lb/>
state, hence, a characteristic shape of p n (1).<lb/></p>
<head>3.4. Intermittency<lb/></head>
<p>[39] Three single realizations (i.e., no repetition and<lb/> averaging is applied) of the
time evolution of L(t) respec-<lb/>tively computed for increasing sediment mass
concentration<lb/> c and turbulent shear rate G are represented in <ref type="figure"
>Figures 6a–<lb/> 6c</ref>. The floc size L(t) passed through bursts characterized
by<lb/> alternation of high and low rates of aggregation or breakup.<lb/> During a
burst, L(t) varied largely in time. Visual compar-<lb/>ison of panels (a) through (c)
suggests that, in general, the<lb/> burst frequency sensibly increased with c and G,
while it<lb/> shortened in duration. Re-scaling the time with (cG) À1<lb/> qualitatively
showed that burst frequency was nearly inde-<lb/>pendent from c and G, suggesting a more
general timescale-<lb/></p>
<figure>Figure 3. Comparison between stochastic and determinis-<lb/>tic modeling for various
values of mass concentration c and<lb/> shear rate G.<lb/></figure>
<figure>Figure 4. Comparison between the probabilities p n (1),<lb/> p m (1), and p m<lb/> b
calculated, respectively, from equations (10),<lb/> (14), and (20) for two combinations
of mass concentration c<lb/> and shear rate G.<lb/></figure>
<figure>Figure 5. Comparison between probability distribution<lb/> p n (1) from equation
(10), floc population distribution<lb/> p m (1), and breakup probability distribution p
m<lb/> b for , (a),<lb/> p m (1) = p m<lb/> b uniform, and (b), p m (1) = p m<lb/> b
Dirac-<lb/> distributed. These simulations refer to mass concentration<lb/> c = 0.5 kg m
À3 and shear rate G = 50 s À1 .<lb/></figure>
<p>invariant burst behavior during dynamical equilibrium (data<lb/> not shown).<lb/></p>
<p>[40] This behavior in floc dynamics was rather unexpected,<lb/> and, together with the
results in section 3.3, gave a very<lb/> different picture of floc mobility compared to
deterministic<lb/> models, in which steady state floc size, once reached,<lb/> remains
invariant. Emergence of more or less rapid bursts<lb/> with highly variable L values can
be explained as a result of<lb/> the combined effects of the rates f a (n) and f b (n)
and<lb/> probabilities p m<lb/> a (t) and p m<lb/> b of stochastic aggregation and<lb/>
breakup. For small L, f a (n) and f b (n) determine large Dt,<lb/> retaining L constant
for relatively long time extents. For<lb/> large L, instead, f a (n) and f b (n)
determine small inter-event<lb/> times Dt. L(t) and the right-adjacent inter-event time
Dt are<lb/> therefore negatively correlated <ref type="figure">(Figure 6d)</ref>, with
Dt span-<lb/>ning more than four orders of magnitude for floc sizes<lb/> ranging less
than two orders. Variability in L(t), instead, is<lb/> regulated by the probabilities p
m<lb/> a (t) and p m<lb/> b to gain or lose m<lb/> primary particles following
stochastic aggregation and<lb/> breakup.<lb/></p>
<p>[41] Correlation time lag (computed from the autocorrelo-<lb/>gram) for the fluctuating
component of L(t) of the three time<lb/> sequences in <ref type="figure">Figures
6a–6c</ref> was found to increase first, and<lb/> decrease for higher c and G (data
not shown). A Fourier<lb/> analysis also showed that there was no specific
frequency<lb/> associated with the bursts, and that the power spectral<lb/> density
could be described with a power law with constant<lb/> slope comprised between À2 and À1
(data not shown). No<lb/> further interpretation was attempted, but we believe that<lb/>
more mathematical analysis in this direction, supported by<lb/> experimental data, could
lead to additional clarifying<lb/> aspects of floc dynamics and mobility within the
size<lb/> spectrum.<lb/></p>
<head>3.5. Transition Toward Steady State<lb/></head>
<p>[42] The floc size L(t) predicted by deterministic and<lb/> stochastic modeling evolved
with similar shape and floccu-<lb/>lation timescale T for p m (t) evolving from the
initial Dirac<lb/> probability p m=1 (0) = 1 toward the lognormal probability<lb/> p m
(1) (e.g., <ref type="figure">Figures 2 and 3</ref>).<lb/></p>
<p>[43] By relaxing the prescribed initial condition, two<lb/> cases emerge that are worthy
of study to understand the<lb/> effect of p m (t) on L(t) and T: the first is when a
floc grows<lb/> within a steady state population (p m (t) = p m (1) 8t), while<lb/> the
second is when a floc grows within a Dirac-distributed<lb/> population (p m (t) = p m=1
(0) = 1 8t), case typical of<lb/></p>
<figure>Figure 6. (a) – (c) single-realization stochastic modeling of the time evolution of
the floc size L(t) for<lb/> three combinations of mass concentration c and shear rate G.
(d) relation between L(t) and the right-<lb/>adjacent inter-event time Dt for G = 20 s
À1 and c = 0.2 kg m À3 .<lb/></figure>
<p>diffusion-limited aggregation processes <ref type="biblio">[Meakin, 1991]</ref>. In<lb/>
both cases, p m (t) should feed back some dynamical patterns<lb/> to L(t) not capturable
by the deterministic model neither in<lb/> terms of equilibrium size nor in terms of
timescales, as the<lb/> latter does not account for the population probability<lb/>
distribution p m . To reveal these aspects with the stochastic<lb/> model, the following
two initial conditions were used; (i) for<lb/> m(t) m(1) and s 2 (t) s 2 (1) in
equations <ref type="formula">(17)</ref> and <ref type="formula">(18)</ref><lb/> the
probability p m (t) is permanently at steady state p m (1);<lb/> (ii) for m(t) m(0) and
s 2 (t) s 2 (0) the probability p m (t)<lb/> remains equal to the initial distribution p
m = 1 (0) = 1.<lb/></p>
<p>[44] When a floc evolved within a steady state popula-<lb/>tion, the stochastic and
deterministic models predicted the<lb/> same equilibrium (median) floc size, which
reached steady<lb/> state within identical timescale T <ref type="figure">(Figure
7)</ref>. Condition (i)<lb/> (i.e., p m (t) = p m (1) ) represented the case for the
floc to<lb/> potentially grow most rapidly, and produced the upper<lb/> boundary for the
time evolution of L(t) during transition<lb/> toward steady state. Conversely, when a
floc evolved within<lb/> a Dirac-distributed population centered at the primary<lb/>
particle size, the flocculation timescale for the individual<lb/> floc became larger
than that of the deterministic model, and<lb/> the steady state floc size never reached
the value foretold by<lb/> the deterministic model. This can be explained by the
trade-<lb/>off between aggregation with only primary particles and<lb/> breakup into
power law distributed fragments, the balance<lb/> of which turns into an averagely
smaller floc. Condition (ii)<lb/> (i.e., p m = 1 (t) = 1) produced the lower boundary
for the time<lb/> evolution of an individual floc. As a result, the transition of<lb/>
L(t) within an evolving population (solid thin line in<lb/>
<ref type="figure">Figure 7</ref>) was always comprised between the limit cases<lb/>
depicted by the dashed and dotted lines of the stochastic<lb/> model. It is possible to
find analogue boundaries to the<lb/> symmetric problem, that is, for the initial value
L(0) larger<lb/> that the equilibrium one (data not shown).<lb/></p>
<head>3.6. Constant Fractal Dimension<lb/></head>
<p>[45] The deterministic and stochastic models responded<lb/> differently to changes in
capacity dimensions d 0 <ref type="figure">(Figures 8a,<lb/> 8b)</ref>. In the
deterministic model, T increased slightly with d 0<lb/> increasing, while the
equilibrium floc size decreased. In the<lb/> stochastic model, instead, we observed the
opposite behav-<lb/>ior: both T and the equilibrium floc size increased with d 0<lb/>
increasing.<lb/></p>
<p>[46] The behavior of the deterministic model can be<lb/> explained in terms of
aggregation and breakup rates; in<lb/> fact, while aggregation scales with f a ðnÞ / n
ð3Àd 0 Þ=d 0 , break-<lb/>up scales with f b ðnÞ / n ð4Àd 0 Þ=d 0 , the balance being in
favor<lb/> of breakup for increasing d 0 . This scaling holds for the<lb/> stochastic
model, too. However, in this case, the time to<lb/> reach steady state and the
equilibrium floc size depends also<lb/> on the probabilities a n,m and p m<lb/> b ,
which are not taken into<lb/> account in the deterministic model. For d 0 increasing, a
n,m<lb/> decreases, and so does the average rate of aggregation.<lb/> However, an
increasing d 0 causes breakup to occur more<lb/> likely as erosion of small particles
rather than fracture. This<lb/> behavior would yield an overall balance in favor of
aggre-<lb/>gation, hence producing larger flocs. By letting d 0 vary over<lb/> the range
1.7 – 2.3, the equilibrium floc size increased<lb/> proportionally to d 0 and the
sediment mass concentration,<lb/> behavior that repeated for various combinations of c
and<lb/></p>
<figure>Figure 7. Time evolution of the floc size L(t) computed<lb/> with permanently steady
state (p m (t) = p m (1) À case i) and<lb/> permanently Dirac-distributed (p m (t) = p m
= 1 (0) À case ii)<lb/> floc populations. These are compared with L(t) computed<lb/>
with evolving p m (t) and with the deterministic model. Mass<lb/> concentration c = 0.8
kg/m 3 and turbulence shear rate G =<lb/> 20 s À1 were used.<lb/></figure>
<figure>Figure 8. (a)-(b) comparisons between the time evolution of the floc size L(t)
computed with the<lb/> stochastic and deterministic models for capacity dimensions d 0 =
1.7, 2.2, and for two combinations of<lb/> mass concentration c and shear rate G. (c)
relation between the median floc size computed with the<lb/> stochastic model for
increasing d 0 and four combinations of c and G.<lb/></figure>
<p>G <ref type="figure">(Figure 8c)</ref>. High values of G produced small flocs<lb/>
regardless of d 0 .<lb/></p>
<p>[47] Although the author is unaware of specific experi-<lb/>mental data sets to be used
for comparison, it appears<lb/> physically sound to expect larger equilibrium flocs
for<lb/> higher capacity dimension. In fact, a more compact particle<lb/> network and a
higher number of contacts per particle would<lb/> make flocs stronger against breakup in
turbulently agitated<lb/> environments, while lower capacity dimensions would<lb/>
result in fragile, smaller flocs.<lb/></p>
<head>3.7. Variable Fractal Dimension<lb/></head>
<p>[48] Recent experimental and modeling investigations<lb/> have shown that d 0 may not be
constant during floc growth<lb/> but, rather, changing with the floc size at constant
hydraulic<lb/> and sedimentological conditions, the behavior successfully<lb/> described
with a power law function of L as <ref type="biblio">[Maggi, 2005,<lb/> 2007; Khelifa
and Hill, 2006a]</ref></p>
<formula><lb/> d 0 ðLÞ ¼ d<lb/> L<lb/> L p<lb/> x<lb/> ;<lb/>
<label>ð22Þ</label><lb/></formula>
<p>where d is the primary particle capacity dimension and x is<lb/> the rate of change of d
0 with L. If we use equation <ref type="formula">(22)</ref> to<lb/> describe variations
in capacity dimension, the rates f a (n) and<lb/> f b (n), and aggregation and breakup
probabilities p m<lb/> a (t) and<lb/> p m<lb/> b will be affected. To explore this
effect, we implemented<lb/> equation <ref type="formula">(22)</ref> in both
deterministic and stochastic models,<lb/> and a second calibration was performed (data
not shown)<lb/> resulting in k a<lb/> 0 = 0.4, k b<lb/> 0 = 1.87 Á 10 À4 , b = 0.08, and
with<lb/> d = 3 and x = À0.1 taken from <ref type="biblio">Maggi [2005]</ref>. Though
not<lb/> shown here, variable capacity dimension improved pre-<lb/>dictions of the
experimental median size L 50 in both<lb/> deterministic and stochastic models with
respect to using a<lb/> constant capacity dimension. This improvement is consistent<lb/>
with the one stemming from application of equation <ref type="formula">(22)</ref><lb/>
in deterministic <ref type="biblio">[Maggi et al., 2007]</ref> and stochastic <ref
type="biblio">[Khelifa<lb/> and Hill, 2006b]</ref> Smoluchowski-based population
balance<lb/> equations to model the floc size distribution.<lb/></p>
<p>[49] The primary-particle capacity dimension d and the<lb/> exponent x can vary with the
crystal structure of the<lb/> minerals and the sediment mixture. An increasing x
caused<lb/> the average equilibrium floc size L e to increase <ref type="figure">(Figure
9a)</ref>.<lb/> Similarly, an increasing values of x produced larger median<lb/>
floc sizes <ref type="figure">(Figure 9b)</ref>. Increasing values of c caused
higher<lb/> values of L e while increasing values of G produced lower L e<lb/> as
already observed in <ref type="figure">Figure 3</ref> (data not shown).<lb/></p>
<head>4. Conclusions<lb/></head>
<p>[50] The stochastic model for flocculation of cohesive<lb/> sediment presented here has
been addressed to analyze the<lb/> dynamical behavior of the floc size around an
average<lb/> value, calibrated on the modal floc size observed in real<lb/>
suspensions.<lb/></p>
<p>[51] Model results have highlighted interesting aspects of<lb/> floc mobility within the
population size spectrum: (i) the<lb/> probability distribution of the fluctuations of
L(t) largely<lb/> correlated with the population floc size distribution, and<lb/>
minimally depended on the breakup probability distribution;<lb/> (ii) floc dynamics were
strongly marked by burst events<lb/> repeating in time rather irregularly; (iii)
flocculation time-<lb/>scales and floc size transition toward steady state were<lb/>
limited by upper and lower bounds determined by the initial<lb/> conditions p m (0);
(iv) the equilibrium floc size and floccu-<lb/>lation timescales were largely impacted
by various, but<lb/> constant, values of floc capacity dimension; (v) a variable<lb/>
floc capacity dimension improved predictions of the median<lb/> floc size and gave a
higher degree of freedom in taking into<lb/> account the mineralogic characteristics of
the sediment at<lb/> the primary particle scale.<lb/></p>
<figure>Figure 9. (a) median floc size computed with the<lb/> stochastic model for various
values of x of equation (22).<lb/> (b) median floc size computed for various values of
primary-<lb/>particle capacity dimension d. Simulations were run with c =<lb/> 0.2 kg m
À3 and G = 20 s À1 .</figure>
</text>
</tei>