INTRODUCTION
The development of models of the vertical distribution of suspended
sediments is essential for solving a number of practical tasks like the
prediction of sediment transport and estimation of water quality in
bottom layers of natural water bodies. Rouse (1937) initiated the
activity in his direction. He suggested a formula for the vertical
distribution of the concentration of suspended sediments formed jointly
by the settling of suspended particles and turbulent diffusion, with the
quadratic dependence of the coefficient of vertical turbulent diffusion
on the vertical coordinate. Rouse's idea has been followed in
several investigations (e.g. Orton & Kineke 2001; Yu & Tian
2003; Belinsky et al. 2005; Huang et al. 2008), with rather diverse
approximations of the dependence of the coefficient of vertical
turbulent diffusion on the vertical coordinate. The major disadvantage
of these approaches has been insufficient physical justification of the
adopted dependences, revealed mostly in attempts to explain the
concentration distributions with the presence of the lutocline. The
lutocline was first observed in data measured in the Severn Estuary and
Inner Bristol Channel (Kirby & Parker 1983) and was thereafter
studied in several experimental (Kirby 1986, 1992; Mehta 1988; E &
Hopfinger 1989; Wolanski et al. 1989; Mehta & Srinivas 1993) and
model investigations performed on semi-empirical background (Smith &
Kirby 1989; Gross & Nowell 1990; Noh & Fernando 1991; Toorman
& Berlamont 1993; Michallett & Mory 2004; Yoon & Kang 2005).
The model suggested in the current paper differs from the models
used in the above-mentioned publications in the following points: (a)
the theory of rotationally anisotropic turbulence (the RAT theory)
(Heinloo 1984, 1999, 2004) is applied, (b) the dominant effect in the
turbulent mixing process is attributed to the large-scale turbulence
constituent immediately interacting with the average flow and (c) the
concentration distribution with the presence of the lutocline is
explained without the necessity to include the buoyancy effect. It is
shown that for the characteristic diffusion length scale of the eddies
much smaller than the height of the Ekman bottom boundary layer the
model results in an analytic expression for the vertical distribution of
the concentration of suspended sediment, which includes both cases, with
and without the lutocline. According to this expression, the lutocline
is present for small settling velocity and/or large bottom shear stress
values. The concentration gradient in the lutocline is determined by the
scale of diffusion of turbulent eddies. No lutocline was revealed for
large settling velocity and/or small bottom shear stress. The derived
formula is tested against laboratory data reported in Coleman (1986),
where the lutocline was not observed. Data on detailed model testing for
the case with the presence of the lutocline are not available, therefore
in this case the comparison is limited to showing the qualitative
similarity between the model-predicted effects and the respective
effects observed and/or modelled in other studies (Kirby & Parker
1983; E & Hopfinger 1989; Noh & Fernando 1991; Michallett &
Mory 2004).
Dealing with the geophysical application of the suggested model,
the paper belongs to a series of works aspiring to introduce the RAT
theory into the solution of geophysical problems (Toompuu et al. 1989;
Heinloo & Vosumaa 1992; Vosumaa & Heinloo 1996; Heinloo &
Toompuu 2004, 2006, 2007, 2008, 2009; Heinloo 2006).
MODEL SETUP
Introductory notes
The model is set up in the right-hand coordinate system (x y z)
(with z directed upwards) for the area of a water body with a flat
bottom. It is assumed that the suspension concentration is sufficiently
small to not affect the constant water density. All considered fields
are assumed to depend on the vertical coordinate only. In particular,
for the average concentration of the suspension q and for the average
velocity field u we shall assume that
q = q(z)
u = ([u.sub.x](z), [u.sub.y](z), 0) (1)
The medium turbulence driving the turbulent diffusion is specified
according to the model of the Ekman bottom boundary layer, modified by
the RAT theory.
The relation between the vertical distribution of the concentration
of suspended sediments and the turbulence properties of the medium
Within the assumptions made in introductory notes the equation for
the concentration of suspended sediments q is represented as
[u.sub.g] * [[nabla].sub.q] = [nabla] * (K, [[nabla].sub.q]) (2)
where [u.sub.g] = (0, 0, -[u.sub.g]) is the settling velocity of
suspended sediments, [nab;a] = (0, 0, [delta]/[[delta].sub.z] and K is
the tensor of turbulent diffusion of suspended sediments. (K *
[[nabla[.sub.q] = h = -
, where h is the turbulent
flux of suspended sediments, q' and v' denote the
concentration fluctuation and the velocity fluctuation, respectively.)
The basic distinguishing feature of the current model consists in
specification of flux h according to the RAT theory (Heinloo 2004). Let
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
where e = v'/v' and s is the length of the ? v streamline
curve, denote the curvature radius of the velocity fluctuation
streamline. Using identity v' = R x [[OMEGA].sup.*] in which
[[OMEGA].sup.8] = v' x R/[R.sup.2], we have for : h
h = [h.sub.0] + [OMEGA] x (3)
where [OMEGA] = <[[OMEGA].sup.*]> and [h.sub.0] =
<[[OMEGA].sup.'] x (Q'R)'> ([OMEGA]' =
[OMEGA]* - [OMEGA]). The first and the second term on the right side of
Eq. (3) describe the transport of the suspended sediment by the
turbulence constituents not contributing and contributing to, [OMEGA]
respectively. The quantity, [OMEGA] having the dimension of angular
velocity, quantifies the average effect of orientation of eddy rotation.
Within Kolmogorov's (Kolmogorov 1941) complement to
Richardson's conception about cascading turbulence (Richardson
1922) [OMEGA] is interpreted as a characteristic of the large-scale
turbulence constituent interacting immediately with the average flow.
Unlike this large-scale turbulence constituent, the turbulence
constituent which does not contribute to [OMEGA] is interpreted as the
small-scale turbulence constituent. Proceeding from the latter
interpretation, we shall assume for [h.sub.0] that
[h.sub.0] = [k.sub.0][[nabla].sub.q], (4)
where [k.sub.0] is constant. The expression (Heinloo 2008)
= [k.sub.1][[nabla].sub.q]x[OMEGA] +
[k.sub.2][[nabla].sub.q] (5)
where [k.sub.1], [k.sub.2] > 0 are constants, presents a
vanishing for [[nabla].sub.q] = 0 approximation of
linear on [[nabla].sub.q] and [OMGA] Using Eqs (3)-(5), we shall have
for K
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)
where [??] and [??] are the unit and the Levi-Civita tensors and
[OMEGA] = [absolute value of [OMEGA]] It follows from h = K *
[[nabla].sub.q] and Eq. (6) that for q depending on the vertical
coordinate only the vertical component of [OMEGA] does not influence the
vertical diffusion process, therefore we shall assume further that
[OMEGA] = ([[OMEGA].sub.x](z), [[OMEGA].sub.y](z), 0),
due to which Eq. (2) simplifies to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)
Further discussion is restricted to the constant settling velocity
[u.sub.g] and to [k.sub.0] negligibly small compared with
[k.sub.1][[OMEGA].sup.2]. For the boundary conditions of Eq. (7),
specified as
q = [q.sub.0] at z=0,
and
q [right arrow] 0 for z >> H,
where H is the characteristic thickness of the layer containing
suspended sediments, from (7) we shall have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)
Equation (8) relates the vertical distribution of q to the
particular specification of the dependence ( ) z ? ? = and
[u.sub.g]/[k.sub.1].
Determination of the [OMEGA]-field
To determine [OMEGA], we shall assume, in addition to the
assumptions adopted in introductory notes, that [absolute value of
[OMEGA]] >> [[omega].sup.0] = [absolute value of [[omega].sup.0]
where [[omega].sup.0] is the normal projection of the angular velocity
of the Earth's rotation. The equations of the RAT theory (Heinloo
2004), corresponding to the model conditions adopted in introductory
notes and to [OMEGA] specified above, are represented as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)
Equation (9) differs from the corresponding equation in Heinloo
(2004) by an additional (Coriolis) term. In Eqs (9) and (10):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is the coefficient
of turbulence shear viscosity; [gamma] is the coefficient of turbulence
rotational viscosity; [kappa] is the coefficient quantifying the
suppression of the average effect of prevailing orientation of eddy
rotation by the cascading process; [theta]/J is the diffusion
coefficient of the angular momentum J[OMEGA] where J is the effective
moment of inertia the square root of which determines the characteristic
spatial scale of eddies contributing to [OMEGA]. Above the Ekman layer
Eq. (10) vanishes and Eq. (9) reduces to the geostrophic balance
condition 2[rho]U x [[omega].sup.0] = [[nabla].sub.h]p, where U is the
geostrophic velocity.
We shall specify the boundary conditions for u and [OMEGA] as
follows:
for z >> [H.sub.E]
u [right arrow] and [OMEGA] [right arrow] 0, (11)
for z = 0
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)
Equation (12) states that the stress at 0 z = is determined by the
turbulent shear stress only.
Using notations [??] = [u.sub.x] + [iu.sub.y], [??] =
[[OMEGA].sub.x] + i[[OMEGA].sub.y] and [??] = [partial
derivative]/]partial derivative]x + i[partial derivative]/[partial
derivative]y instead of u, [OMEGA] and [[nabla].sub.h] (i is the
imaginary unit), we can rewrite Eqs (9) and (10) as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (14)
where f = 2[[omega].sup.0] > 0 is the Coriolis parameter and
[??] = -i[??]p/[rho]f. In Eqs (13) and (14) and hereafter the prime
denotes derivative with respect to the vertical coordinate z. The
solution of Eqs (13) and (14) for constant [mu], [gamma], [kappa]
[theta] and [rho] satisfying conditions (11) read as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (15)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (16)
where [[lambda].sub.1] and [[lambda].sub.2] are the roots of the
biquadratic equation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)
with negative real parts. In (16) and (17)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (18)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
where [[mu].sub.ef] = [mu] + [gamma][kappa]/([gamma] + [kappa]).
Boundary conditions in Eq. (12), written in terms of [??] and [??]
as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] determine
[C.sub.1] and [C.sub.2] in Eqs (15) and (16) as
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (19)
where
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (20)
Using Eq. (19), we have from Eqs (15) and (16)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (21)
and
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (22)
Equations (21) and (22) represent the solution for the Ekman bottom
boundary layer, generalized by the RAT theory. Equation (8) together
with Eq. (22) determines q = q(z).
APPROXIMATE FORMULA FOR q = q(z)
Consider the situation described by Eqs (8) and (22) for l
restricted to l << [l.sub.1], [l.sub.2]. In particular this
inequality has been found holding for turbulent flows in plane channels,
round tubes and between rotating cylinders (Heinloo 1999, 2004), where
instead of [l.sub.1] and [l.sub.2] the characteristic transverse length
scale of the flow region had been used. For l << [l.sub.1].
[l.sub.2], Eq. (17) simplifies to
[([lambda]l).sup.4] - [(lambda]l).sup.2] + i[[epsilon].sup.2] = 0,
where [epsilon] = l/[l.sub.2] from which we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (23)
Equation (23) determines the flow in the Ekman bottom boundary
layer through two characteristic lengths: l, characterizing the
turbulent diffusion, and [l.sub.2] [equivalent to] [H.sub.E],
determining the thickness of the Ekman bottom boundary layer for the
turbulence viscosity identified with [[mu].sub.ef]. Using Eq. (23), we
have for [a.sub.1], [a.sub.2], [b.sub.1] and [b.sub.2] in Eqs (18) and
(20)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
and, consequently,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (24)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (25)
For restriction [mu] << [gamma], justified by the
Richardson-Kolmogorov turbulence conception about the cascading nature
of turbulence, and for [gamma] of the order of [kappa] Eq. (25)
simplifies to
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (26)
From Eq. (26) it follows for [OMEGA] = [absolute value of [??]]
that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
where [[tau].sub.0] = [absolute value of [[??].sub.0] and,
according to Eq. (8),
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (27)
In Eq. (27)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]
Equation (27) determines q = q(z) as depending on [q.sub.0] and on
two parameters D and l.
If [u.sub.g] is relatively large or [[tau].sub.0] sufficiently
small so that dl > 1, then according to Eq. (27), we have [[partial
derivative].sup.2]q/[partial derivative][z.sup.2] for all z and
[absolute value of [partial derivative]/[partial derivative]z] decreases
monotonously with z increasing. This situation is typical of
coarse-grained sediments like sand. For Dl < 1 the entire suspension
layer can be divided into two layers, one with [[partial
derivative].sup.2]q/[partial derivative][z.sup.2] < 0 (located
immediately next to the boundary) and the other with [[partial
derivative].sup.2]q/[partial derivative][z.sup.2] < 0 separated by an
inflection point at z = -0.5l ln (Dl).
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
There is a concentration jump (lutocline) in the concentration
distribution for Dl << 1, explaining the lutocline as emerging for
a relatively large bottom shear [[tau].sub.0] and/or small settling
velocity [u.sub.g], typical of fine-grained sediments like clay or mud.
Figures 1 and 2 illustrate the presence of the indicated two types of
vertical distributions of sediment concentrations in terms of
q/[q.sub.0] and q/Q, where Q = [[integral].sup.[infinity].sub.0]qdz as
functions of [zeta] = z/l for fixed l and for different values of Dl For
fixed D the lutocline appears steeper for smaller " (Fig. 3). In
Fig. 4 a modelled vertical distribution of normalized suspension
concentration q/Q is compared with the laboratory data from Coleman
(1986) for two series of 20 (Fig. 4a) and 10 (Fig. 4b) individual flume
flow runs, each run injected with a different amount of suspended
matter. The suspension was formed of sand particles with the diameter of
0.105 and 0.210 mm in the first (20 runs) and the second (10 runs)
series, respectively. Due to missing lutocline data suitable for testing
Eq. (27), we just refer to the qualitative resemblance of the
distributions depicted in Fig. 2 with the similar observed and/or
modelled distributions published in Kirby & Parker (1983), E &
Hopfinger (1989), Noh & Fernando (1991) and Michallett & Mory
(2004).
[FIGURE 3 OMITTED]
[FIGURE 4 OMITTED]
CONCLUSIONS
A stationary model of the vertical distribution of the
concentration of suspended sediments in a turbulent boundary flow over a
flat bottom is presented. The distinguishing feature of the suggested
model consists in considering the concentration distribution formed by
the large-scale turbulence, which, according to the applied theory of
rotationally anisotropic turbulence (the RAT theory), is considered
having a prevailing orientation of eddy rotation. The model results in
an analytic expression covering both observed types of the vertical
distribution of concentration--with and without the presence of the
lutocline. The generation of turbulence driving the vertical mixing of
suspended sediment is considered within the Ekman bottom boundary layer
model generalized by the RAT theory.
For the model parameters l and D, estimated from the observed data,
the derived formula for the vertical distribution of the concentration
of suspended sediments can be applied in circulation models for
calculation of the water quality and for sediment transport. In these
calculations the vertical distribution of the concentration of sediments
is treated as quasi-stationary, and the horizontal distribution as
quasi-homogeneous.
doi: 10.3176/earth.2010.3.05
Acknowledgements. The authors thank Prof. E. Wolanski and Prof. A.
J. Mehta for useful advice, Prof. R. Kirby for providing his works on
the topic discussed, Prof. V. Zhurbas and anonymous referees for
critical comments.
Received 27 May 2009, accepted 6 November 2009
REFERENCES
Belinsky, M., Rubin, H., Agnon, Y., Kit, E. & Atkinson, J. F.
2005. Characteristics of resuspension, settling and diffusion of
particulate matter in water column. Environmental Fluid Mechanics, 5,
415-441.
Coleman, N. L. 1986. Effects of suspended sediment on the open
channel velocity distribution. Water Resources Research, 22, 1377-1384.
E, X. & Hopfinger, E. J. 1989. Stratification by solid particle
suspension. In Proceedings of the 3rd International Symposium on
Stratified Flows, pp. 1-8. Caltech, Pasadena.
Gross, T. F. & Nowell, A. R. M. 1990. Turbulent suspension of
sediments in deep sea. Philosophical Transactions of the Royal Society
A, 331, 167-181.
Heinloo, J. 1984. Fenomenologicheskaya mekhanika turbulentnykh
potokov [Phenomenological Mechanics of Turbulent Flows]. Valgus,
Tallinn, 245 pp. [in Russian].
Heinloo, J. 1999. Mekhanika turbulentnosti [Turbulence Mechanics].
Estonian Academy of Sciences, Tallinn, 270 pp. [in Russian].
Heinloo, J. 2004. The formulation of turbulence mechanics. Physics
Review E, 69, 056317.
Heinloo, J. 2006. Eddy-driven flows over varying bottom topography
in natural water bodies. Proceedings of the Estonian Academy of
Sciences, Physics, Mathematics, 55, 235-245.
Heinloo, J. 2008. The description of externally influenced
turbulence accounting for a preferred orientation of eddy rotation. The
European Physical Journal B, 62, 471-476.
Heinloo, J. & Toompuu, A. 2004. Antarctic Circumpolar Current
as a density-driven flow. Proceedings of the Estonian Academy of
Sciences, Physics, Mathematics, 53, 252-265.
Heinloo, J. & Toompuu, A. 2006. Modeling a turbulence effect in
formation of the Antarctic Circumpolar Current. Annales Geophysicae, 24,
3191-3196.
Heinloo, J. & Toompuu, A. 2007. Eddy-to-mean energy transfer in
geophysical turbulent jet flows. Proceedings of the Estonian Academy of
Sciences, Physics, Mathematics, 56, 283-294.
Heinloo, J. & Toompuu, A. 2008. Modelling turbulence effect in
formation of zonal winds. The Open Atmospheric Science Journal, 2,
249-255.
Heinloo, J. & Toompuu, A. 2009. A model of average velocity in
oscillating turbulent boundary layers. Journal of Hydraulic Research,
47, 676-680.
Heinloo, J. & Vosumaa, U. 1992. Rotationally anisotropic
turbulence in the sea. Annales Geophysicae, 10, 708-715.
Huang, S., Sun, Z., Xu, D. & Xia, S. 2008. Vertical
distribution of sediment concentration. Journal of Zhejiang
University-Science A, 9, 1560-1566.
Kirby, R. 1986. Suspended Fine Cohesive Sediment in the Severn
Estuary and Inner Bristol Channel. U.K. Report to United Kingdom Atomic
Energy Authority under contract No: E/5A/CON/4042/1394. Ravensrodd
Consultants Ltd., Taunton Somerset.
Kirby, R. 1992. Detection and transport of high concentration
suspensions. In Proceedings of International Conference on the Pearl
River Estuary in the Surrounding Area of Macao, Macao, 19-23 October
1992, pp. 67-84. Macao.
Kirby, R. & Parker, W. R. 1983. Distribution and behavior of
fine sediment in the Severn Estuary and Inner Bristol Channel, U.K.
Canadian Journal of Fisheries and Aquatic Sciences, 40, 83-95.
Kolmogorov, A. N. 1941. The local structure of turbulence in
incompressible viscous fluids for very large Reynolds numbers. Doklady
Akademii Nauk SSSR, 30, 376-387 [in Russian; reprinted in English:
Proceedings of the Royal Society, London A, 1991, 434, 9-13].
Mehta, A. J. 1988. Laboratory studies on cohesive sediment
deposition and erosion. In Physical Processes in Estuaries (Dronkers, J.
& van Leussen, W., eds), pp. 427-445. Springer-Verlag, Berlin.
Mehta, A. J. & Srinivas, R. 1993. Observations on the
entrainment of fluid mud by shear flow. In Nearshore and Estuarine
Cohesive Sediment Transport, Coastal and Estuarine Studies, Vol. 42
(Mehta, A. J., ed.), pp. 224-246. AGU, Washington.
Michallett, H. & Mory, M. 2004. Modelling of sediment
suspensions in oscillating grid turbulence. Fluid Dynamics Research, 35,
87-106.
Noh, Y. & Fernando, H. J. S. 1991. Dispersion of suspended
particles in turbulent flow. Physics of Fluids A, 3, 1730-1740.
Orton, P. M. & Kineke, G. C. 2001. Comparing calculated and
observed vertical suspended-sediment distributions from a Hudson River
turbidity maximum. Estuarine, Coastal and Shelf Science, 52, 401-410.
Richardson, L. F. 1922. Weather Prediction by Numerical Process.
Cambridge University Press, Cambridge, 226 pp.
Rouse, H. 1937. Modern conceptions of the mechanics of fluid
turbulence. Transactions of the American Society of Civil Engineers,
102, 463-541.
Smith, T. J. & Kirby, R. 1989. Generation, stabilization and
dissipation of layered fine sediment suspensions. Journal of Coastal
Research, 5, 63-73.
Toompuu, A., Heinloo, J. & Soomere, T. 1989. Modeling the
Gibraltar salinity anomaly. Oceanology, 29, 698-702 [English edition,
published by the American Geophysical Union in June 1990].
Toorman, E. A. & Berlamont, J. E. 1993. Mathematical modeling
of cohesive sediment settling and consolidation. Observations on the
entrainment of fluid mud by shear flow. In Nearshore and Estuarine
Cohesive Sediment Transport, Coastal and Estuarine Studies, Vol. 42
(Mehta, A. J., ed.), pp. 167-184. AGU, Washington.
Vosumaa, U. & Heinloo, J. 1996. Evolution model of the vertical
structure of the active layer of the sea. Journal of Geophysical
Research, 101, C11, 25,635-25,646.
Wolanski, E., Asaeda, T. & Imberger, J. 1989. Mixing across a
lutocline. Limnology and Oceanography, 34, 931-938.
Yoon, J.-Y. & Kang, S.-K. 2005. A numerical model of
sediment-laden turbulent flow in an open channel. Canadian Journal of
Civil Engineering, 32, 233-240.
Yu, D. & Tian, C. 2003. Vertical distribution of suspended
sediment at the Yangtze river estuary. In Proceedings of the
International Conference on Estuaries and Coasts, November 9-11, 2003,
pp. 214-220. Hangzhou, China.
Jaak Heinloo and Aleksander Toompuu
Marine Systems Institute, Tallinn University of Technology,
Akadeemia tee 21, 12618 Tallinn, Estonia; heinloo@phys.sea.ee,
alex@phys.sea.ee