Hydrobiologia 235/236: 119-131, 1992 . B . T. Hart & P. G . Sly (eds), Sediment/Water Interactions . © 1992 Kluwer Academic Publishers. Printed in Belgium .
119
A model for predicting waves and suspended silt concentration in a shallow lake D . P. Vlag Ministry of Transport, Rijkswaterstaat, Institute for Inland Water Management and Waste Water Treatment, P .O . Box 17, 8200 AA Lelystad, The Netherlands
Key words :
wave model, silt model, shallow lake, bottom dynamics, silt dividing, diffusion
Abstract A wave model and a vertical silt model are developed for shallow lakes and both were applied to Lake Marken in the Netherlands . The results of the wave model serve as input for the silt model . The silt model calculates the suspended silt concentration at several depths at a certain location . The water column is divided in compartments and the suspended silt concentration can be divided into a maximum of seven fractions . It is also possible to divide the top layer of the lake bottom into 25 slices with different time-dependent soil-physical properties . The optimization procedure of the silt model showed the importance of knowledge of the soil-physical properties of the top layer of the bottom . In general, not much is known about this layer and Lake Marken is no exception . The calibrated silt model gave good results for 3 sites in Lake Marken .
Introduction During the summer, most Dutch lakes have algal blooms . However, Lake Marken is an exception . An important factor therein is though be the wind (Jagtman & Van Urk, 1988) . The dynamic character of the wind results in large variations in the concentration of suspended silt and a rapidly changing light climate in the water . To describe these variations in the light climate it is necessary to have a one-dimensional model for light and silt concentration in the vertical direction . This type of model is based on a detailed understanding of the bottom and turbulence processes . A dynamic wave model has been developed to calculate the bottom shear stress due to turbulence and waves . Fig . 1 shows the relationship between the different models . The construc-
Fig. 1 . Schematic illustration of the models .
tion, calibration and verification of the wave and silt models are discussed in this paper .
120 Lake Marken Lake Marken (Fig . 2), located in the Netherlands, is part of the former Zuiderzee . After the con-
struction of the Enclosure dike in 1932, the name of Southern Sea was changed in Lake IJssel . Since that time a number of polders were reclaimed (Wieringermeer, North East polder and
Fig. 2 . The Lake IJssel area in the Netherlands .
121 Flevoland) . In 1975 Lake Marken was separated from the Lake IJssel by closing the Houtribdijk . A plan to reclaim about 50% of the lake was recently postponed . Lake Marken is a flat and shallow lake ; nearly 50% of the lake has a depth of between 3 and 4 metres . The average depth is 3.6 metres . In the southern part of the Lake some pits have been dredged with a maximum depth of more than 30 metres . The area of the lake is 687 km 2. No rivers input to the Lake. Every year about 850 million m 3 of water enters the lake from the surrounding polders and Lake IJssel, and about 100 million m3 of water by precipitation excess . The surplus water is discharged to Lake IJssel at the sluices near Lelystad and Enkhuizen and into the North Sea Channel at the sluices near Amsterdam . The lake has been extensively monitored since its existence (e .g . Mugie et al ., 1989a, b) . Measurements of importance for the development of the models were wave parameters, suspended sediment concentration and light penetration . At site 4 (Fig . 2) only wave height is measured and at site 3 only the suspended silt concentration . The wave parameters were measured by reading the water level at a frequency of 4 times per second at sites 1 and 2, or with a frequency of 2 times per second at site 3 . The suspended sediment concentration was measured in 1 litre water samples using standard methods . The deposition of silt was measured at several locations using sediment traps and analyzed the collected material in the laboratory . The Royal Netherlands Meteorological Institute (KNMI) measures the wind parameters at 10 m above the water level at the sluices near Lelystad .
Model development Wave model
Waves generated by wind are the most important force for resuspending silt from the bottom sediment . A dynamic model has been developed to
describe to waves generated as a result of the recorded wind patterns . To obtain a proper description of the wave parameters dynamics, it was necessary to average the wind speed (collected hourly) over several hours, with a weight factor for each hour . The equation for the average weighted wind speed is: N
0,5
E (W * Vwi )2 i= 1
(1)
Vw = N
E Wz 1= 1
where l ..n are the hours before the time of the calculation, W; is the weight factor for hour i, Vwi is the averaged wind speed over the past hour i (m s -1 at the hour i), Vw is the weight-averaged wind speed (m s - 1 at the hour of calculation). For wind direction, effective fetch and depth, the same type of weight function was used . Thus, the weight-averaged wind direction (R w in degrees) is given by : N
E Wi * Vwi *Rw i i= 1
(2)
R w = N
E Wi i=1
where Rw i is the wind direction at the hour i. Rw and Rwi are replaced by Fe and Fe; in case of the effective fetch (m) and D and Di in case of depth (m) . Two different depths are used in the model : (a) for calculation of wind-induced currents, the average depth (D) over the fetch is used ; and (b) for calculating the wave parameters, an average depth over the vicinity (D,,) is used . The vicinity of a point in Lake Marken is about 2 kilometres . The fetch and depth can either be computed by the model from a regular grid of the lake using data on the average depth for each grid element, or derived from a map and then fed into the model . For calculation of the significant wave height (HS , m) and the wave period (TS , s), the equations from CERC (1975) were used (Vlag, 1990) . The
122 equations for significant wave height and wave period in shallow water are : g*HS
g*Dn = Cl * tanh DI
VwZ
D2
2
( Vw ) F2 g * Fe Fl( Vw2 )
*tank D2 g*Dn tank Dl ( Vw2 ) g*TS
g*Dn
(3)
ac at +
D4
= C2 * tanh D3
Vw
( Vw2 (g* Fe
) F4
F3 Vw 2 ) * tanh D4 g*Dn tank D3 ( ) Vw 2
In a stagnant lake with a bottom covered with silt, the net horizontal transport will be small . Almost the whole bottom of Lake Marken is covered with a silt layer with a thickness ranging from about 1 mm to more than 10 cm . Because the horizontal silt gradient is very small, no important errors are made in short-term simulations by neglecting horizontal transport . The transport equation can be written as (Teeter, 1984) :
(4)
where g is the gravity (m s -2), C1, C2, D1 . .D4 and F1 . .F4 are numerical coefficients, and Dn is the vicinity depth (m). The model was calibrated for location 1 (Fig . 2) . A 15 day period of the calibration is shown in Fig . 3 . The agreement between measured and computed significant wave heights and wave periods seems reasonable . The verification for another period gave similar results . The model with the optimal set of coefficients for site 1 was then used for two other sites in the lake, 2 and 4 (Fig . 2) . Since the differences between the model results and measured wave parameters were small, it was concluded that the calibrated wave model could be used reliably to describe the waves over the total lake .
Vertical silt model The silt model can be used to describe the sus-
pended silt concentration at a location during the time and for the distribution of silt particles in the bottom . Once the silt concentration is known at all depths, it is possible to calculate the extinction coefficient at each site and then use this extinction coefficient to predict the growth of algae .
aw' C
a
8C
(5)
8h (E`` 8h )
8h
where C is the silt concentration (kg m - 3 ), h is the vertical coordinate (m), t is the time (s), Ws the settling velocity (m s -1 ), and Eh the turbulent diffusion in the vertical (m2 s -1 ). The model allows the silt to be divided into a maximum of seven fractions, with each fraction characterized by its own settling velocity . For simulating erosion and sedimentation the water column can be divided into a maximum of 15 compartments (Fig . 4) . The minimum height of a compartment depends on the maximum settling velocity of a silt fraction and on the time step used in the calculations . A silt particle is not allowed to pass two compartment boundaries in one time step . It is possible to vary the time step from 1 second up to one hour . Figure 5 shows the transport through a compartment . The equations governing the transport of each fraction are : (6)
W1 = Ws * Ck El = Ehk
*
Ck - I (0,5 * (hk
W2 = - Ws * Ck + E2 =
Ehk
1
-
Ck hk -
(7) 2))
(8)
Ck-Ck+ I (9) * (0,5 * (hk + I - hk -1))
where Ck + 1, Ck and Ck -1 are the concentrations (kg m - 3 ) of the compartments k + 1, k and k - 1 . W1 and W2 are transports (kg m - 2 s -1 ) caused by gravity and the concentration of the silt fractions . El and E2 are the transport caused by
123
15
i. .
.9
10
5
I 3
1 .0
1 .0
E
0.8
Wr~11~ C~ , ;
0.6
It 1
02
0 .0 1
2
3
4
5
6
7
8
9
Days in 1Wy +
measixed wave height
0
0 10 11 12 13 14 15
1982
measured wave period
- model wave height model wave period
Fig. 3 . Results of the wave model .
diffusion. For the top water column compartment, W2 and E2 are zero . For the compartment at the bottom, WI stands for sedimentation (S) or resuspension (R) and El is zero . The resuspension equation (given later), provides for transport from the bottom into the bottom compartment . Bottom shear stress . The bottom shear stress (tb)
can be caused in three ways : (a) by wind currents, (b) by waves, and (c) by discharge flows . As Lake
Marken has no major discharges, the bottom shear stress from discharge flows is not included in the model . (a) By wind current . Calculation of the weighted
average wind speed and wind direction was carried out using Eqns . 1 and 2, and assuming there is always a steady-state situation . This latter is reasonable because the steady state-situation in Lake Marken is reached after 2 to 3 hours .
124 (b) by waves . The maximum orbital velocity at
the bottom caused by shallow water waves is (CERC, 1975) : h
n
UbW b. =
Compartm . N h
1 ( 12 )
T,.
2ir*D
sinh
(L w
,r 1
Compartm N-1
)
where Ubw is the bottom current caused by the waves in (m s - 1 ) and L,v is the wave length (m) . The bottom shear stress due to waves during a wave period is :
h rr2
h 3 'r"'(0
Compartm 3
= P", *fw * [u bW
*
cos
2 7r * t)1 2 T ( T,
(13)
h 2
for t = 0 ± 27r, zw is maximal . Where r,, (t) is the bottom shear stress (N m -2 ) caused by waves at time t (s) and fw is the friction coefficient . The value of fw depends on the Reynolds number and on the roughness of the bottom (Blondeaux, 1987) . The Reynolds number (Re) is given as :
Corrpartm 2 h 1
Corrpartm. 1 h 0
1 metre
Re =
Fig . 4 . Schematization of the water column .
The magnitude of the bottom current is computed from the wind speed with the following equation (Nortier, 1980) :
where Ubc is the bottom current (m s -1 ) and Fk is a coefficient . The value of Fk for rectangular lakes with a constant depth is about 0 .01 (Nortier, 1980). The bottom shear stress through wind current can be written as : *
U2 bc
1
HS
Ab
c = Pw * fc
v
where v is the kinematic viscosity (m 2 s -1 ) . A b is the amplitude of the wave height (m) near the bottom and can be written as (CERC, 1975) :
UbC = Fk * Vw
,r
Ubw*Ab
(11)
where is is the bottom shear stress (N m -2 ), pw _ 3) is the density of water (kg m and f, is the friction coefficient for current . The value of fc is about 0.004 . Tamminga (1987) uses 0 .0045 for Lake IJssel and Sheng & Lick (1979) use the value of 0 .004 in Lake Erie. The direction of r c is opposite to the wind direction .
=
-* 2
(15)
2n*D„ sink i Lw
Blondeaux (1987) gives two equations for calculating of fw , one for laminar flow and one for turbulent flow . laminar :
fw = 0 .10 *Re -
turbulent :
fw = 0 .71 *
0.23
(16)
0.31
(A I
(17)
b
where k is the bottom roughness (m) and Re is derived from Eqn . 14 . The value of k is a natural lake with a top layer of silt will vary with the wind force. When there is no wind, k will be very small
125 is given for four different values of k and for laminar flow . Because the value of k is uncertain in all practical applications, Eqns . 16 and 17 were not used . One value of fx, (0 .015) covers all bottom conditions in Lake Marken . In calculating the resuspension, it was assumed that the acting force during a wave period was equal to the average bottom shear stress over the wave period: T
a
b
1
= -
f
I
T. + 1 ;,
(18)
dt
TS 0
where 'rb is the averaged bottom shear stress (N m -2 ). From Eqns . 11, 13 and 18, the explicit equation for rb can be written as : Schematic illustration of the transports through a compartment .
Fig . 5.
(about l5 µm), but when the wind increases, the current and waves will make ripple-marks and the value of k increases . In Fig . 6 the value of f„
4*t* 2b = ~~*
-
1 +
pw*fw*
*
1 -
C
2*t* TS
1 4*n*t* - - *sin (19) it
TS
0 .100 Laminar
Transitional
Turbulent
20 .010
-1 1 10 3 10 4 Reynolds number
0.001 10 2
k= k= k= . 0.002 0.001 0.005
Fig. 6 .
Ubw*
C TS
k= 0.0001
Effect of bottom roughness (k) on fw .
10°
Laminor
126 with S = t* =
s s Cos 2 2*n
\ 1
1/
be * ~UbW
0,5
(20)
w/
Ws; * C,,. * p
(22)
i= 1
with p = 1 -
ib (is >_ tb
)
( 23)
TS
This expression for ib is comparable to the one used by Sheng & Lick (1979) The exchange of silt particles between the compartments of the water column is driven by diffusion . Normally the turbulent diffusion coefficient Eh , from Eqn . 5, is described by a parabolic distribution over the water depth (Van Rijn, 1984 ; Tsanis, 1989) : Diffusion .
(21) Eh =K*UbC *hl1-h) l D where K is a numerical constant . Teeter (1984) and Van Rijn (1984) use for K the value of 0 .4,
the Von Karman constant . Tsanis (1989) used different values of K in his laboratory experiments, varying from 0 .25 to 0 .60 for different bottom roughnesses . For Lake Marken a value of 0.4 was used. Lake bottom . The lake bottom in the model was
divided vertically into 25 thin layers . Each slice had the same initial height . The initial height of the thin layers depends on the maximum rate of erosion . With this model a good description of the bottom consolidation and change in critical shear stress can be made . At the start of the computations, values of the distribution of the particle fractions, the critical shear stress after consolidation and the dry weight of silt must be given for each slice . During the simulation, the actual state of the fraction distribution in each slice and the momentary critical shear stress - as a function of consolidation - are updated regularly . Because there is no net horizontal transport, there will be no net sedimentation or erosion during the shortterm simulations . Sedimentation . The total sedimentation flux is the
sum of the sedimentation fluxes of all silt fractions . The sedimentation equation is given by Krone (1962):
where S is the sedimentation of all fractions (kg m - 2 s -1 ), Ws; is the settling velocity of fraction i (m s -1 ), C 1i the concentration of fraction i in compartment 1 (kg m - 3), is is the critical shear stress for sedimentation (N m -2), and Nf is the number of silt fractions . p gives the probability that a particle reaching the bottom will deposit and remain on the bottom. With a high value of is , the value of p will be close to 1, and free settling can occur . For the Lake Marken model the effect of different values of is were small. The reason of this relative insensitivity is the small timestep (10 s) of the computations . A small timestep is necessary because in shallow lakes the turbulent diffusion overrules the sedimentation process . If 'rb is high enough, particles deposited during one time step will be resuspended during the next time step . The simulations for Lake Marken were done with a value of 10 N M-2 for LS. During sedimentation the slices of the bottom are filled and each slice is subjected to consolidation from the moment the slice is filled . Consolidation . After sedimentation pore water
will be retained in the deposits . However, this will be forced out by the weight, and the density and critical shear strength will increase . The time scale of the consolidation process is given in Fig. 7 (from Hayter, 1984) . Recently tests with silt from another semi-stagnant Dutch lake, the Hollandsch Diep, give the same relationship between time and consolidation rate (Kuijper et al., 1990). In the model newly deposited silt is given a low critical shear stress value, but this shear stress is allowed to increase with the rate of consolidation . The relationship between time after sedimentation and consolidation rate as shown in Fig. 7 was used in the silt model .
127 value of c,, (z) was assumed constant within each slice of the bottom . 1 .0
Lake Marken model 0.8
w
a
0.6
0.2
0 .0 0
40
80
120
160
Hours after sedimentation
Fig . 7 . Consolidation of silt (Hayter, 1984) .
Resuspension . There are many equations to de-
scribe the resuspension of silt . Winterwerp (1989) carried out a literature survey out on the topic of silt erosion . One conclusion of this survey is the importance of several physico-chemical parameters for resuspension . However, this conclusion is based on experiments in small apparatus on heavily compacted samples . The effects of the physico-chemical parameters in nature are not very well known . The erosion equation of Parchure (Parchure and Metha, 1985) was chosen, for its empirical nature and simplicity . R -M(1
'r",
W)
(R >_ 0)
Not all parameters in the model need to be calibrated, because some are directly related to one another . For instance, the dry weight of the silt is related with M in Eqn . 24 and M is also related with i,, (z) . In the model, it is possible to give every bottom slice a different value of dry weight of the silt . However, the dry weight is kept constant in the depth with a value of 225 kg M -3. This value is in good agreement with the laboratory tests in Kuijper et al. (1990). The silt was split up in four particle size fractions (Table 1) . Fraction 1 had a very low settling velocity, 2 .5 x 10-6 m S - 1 ( = 0 .2 m day - 1) and is responsible for the background concentration of the suspended sediment concentration . From Winkels et al. (1989) and others (yet unpublished) measurements in Lake Marken, the proportion of the silt in the low fractions was fraction 1-2 .5% ; fraction 2-23% ; fraction 325 .5 % and fraction 4-49 % . Although fraction 4 contains a relatively large proportion of this total silt content it was relatively unimportant for the extinction coefficient, since the settling velocity is so high that within a few hours after the wind drops most of this fraction settles out . Table 1 shows a first value for the settling velocities (Ws ;) as derived from laboratory experiments (Kuijper et al., 1990 ; Van Duin & Kuypers, 1989). The first simulation was made with a homogeneously mixed bottom and a wind input so strong that the whole bottom was resuspended .
(24)
Zb
where R is the resuspension (kg m- 2 s -1 ), M is a numerical constant (kg m - 2 s -1 ), and ic , (z) is the critical shear stress (N m -2 ) at the depth z in the bottom . All the uncertain factors which can have an effect on resuspension have been taken into account in M. The value of M varies from 0.01 10-3 to 0 .4 10-3 (Winterwerp, 1989) . The
Table 1 . Settling velocities for Lake Marken Dimensions [µm]
From experiments [m s -1 ]
Fraction Fraction Fraction Fraction
1 2 3 4
<2 2-6 7-18 > 18
2 .5 20 90 420
10 -6 10 -6 10 -6 10-6
After calibration [m s -1 ]
2 .5 21 96 440
10 -6 10 -6 10 -6 10-6
128 After the wind dropped the bed was recomposed by settling of the silt . The result was a distribution of the particle size fractions for each slice . This distribution of the fractions was the input for the calibration . Figure 8 clearly shows that the bottom is sieved out and that the smallest particles are in the top slices of the bed . Calibration Calibration of the model was carried out for sites 1 and 2 . The water column was divided into compartments for describing the resuspension and sedimentation and the concentration vertically . Major gradients in suspended silt concentration are caused by a fast decrease in wind speed . These
0
fraction 1
~'
1
fraction 2
/fraction 3
-,
;
gradients can last for several hours . In order to get a proper description of these gradients, the water column at site 1 is divided into four compartments and for site 2 the water column is divided into three compartments . The height of the compartments was just less than one metre . The following parameters were optimized in the indicated order, 'ce, (z), Wsi, Fk, 're, for newly deposited silt (Xnew) and M. The calibration is an iterative process . The settling velocity Ws(I) for each fraction is assumed to be the same for the whole lake . The optimization of the settling velocity was done on site 1 . The resulting settling velocities for the fraction 2, 3 and 4 are respectively 21 x 10-6, 95 x 10-6 and 440 x 10 - 6m s - t . After calibration, the parameters Fk, ineW and M appear to have the same value at both sites . The results of the calibration given in Table 2 show that the optimum value for Fk is lower than generally found in the literature . There could be two reasons for this . First, Lake Marken is not a rectangular lake, and second the depth of the lake is not constant. The model was not calibrated for site 3, because according to Winkels et al. (1989) site 1 and 3 have the same type of bottom .
Results and discussion
2 The model results of site 1 for the calibration period are given in Fig . 9 . The flat parts in the concentration curve are caused by the differences in Tc, of the slices of the bottom . In these cases the value of ib is smaller than T,, of the next slice so that erosion does not proceed . Figures 10 and 11 show results of the model verification at site 1 for other periods with the same set of parameters . Most of the model results are in good agreement with the measured data . However, the results
C s . 0 3 fraction 4
Table 2 . Calibrated parameters for site 1 and 2
4 0
20
40
60
80
100 Fk [-]
tinew [N M-21
M [-]
0 .006
0 .025
0 .04
Percents of the bottom Fig. 8 . Distribution of the silt fractions at site 1 .
129
0 .8
0.8 E 4-
a
0.6 0.4
0 .2 150
150 n award G
0)
modal
100
0
n
maasvad
U)
0 18
20
22
24
Days in April
26
28
30
1988
8
10
12
14
16
Days in august
18
20
1988
Fig. 9. Results silt model - calibration site 1 .
Fig. 10 . Results silt model - verification site 1 .
shown in Fig. 11 do not fit very well . Probably a net horizontal transport appears here, due to a different wind direction . The most common direction of the wind in the Netherlands is southwest, but, during the period shown in Fig . 11, the wind blew from the north . To the north of site 1 the bottom of the lake rises and becomes sandy . These factors can be the reason that the concentration predictions of the model are too high . Figure 12 gives the model results of the calibration for site 2. As indicated before it turned out that the parameters are the same as for site 1 except for r,, (z) (see Fig . 14) . For verification, the model was also used for site 3 with the same set of parameters as for site 1 . Only the thickness of the bottom slices was adjusted to the depth of the water . As a result ic , (z) is stretched out compared to site 1 (Fig . 14). The results of this verification with the adjusted slices are shown in Fig . 13 . The predictions of the model are good, although systematically a little too low . Two reasons can be given for this : (a) the fraction of smaller particles in the silt at site 3 is
larger than at site 1, or (b) the adjusted 'r,,. (z) of the slices are not comparable with site 1 . Due to the lack of data the true reason could not be assessed . However this verification suggest that the silt model, calibrated at site 1, is suitable for the whole area where sedimentation dominates . Figure 14 shows the i,, (z) of the three sites . The differences between the bottoms is clear to see . Site 2 (depth 2 .7 m) seems to be under erosion conditions and in the deeper part of the lake sedimentation dominates . This is in good agreement with the experiences of Hakanson (1982) . In lakes or parts of lakes where wind-wave action dominates, the rate of net sedimentation has a maximum in the deepest part of the lake . The samples of the Lake Marken bottom as described in Winkels et al. (1989) confirm this assumption. The vertical differences in suspended silt concentration in the water column are small . Only when the wind suddenly decreased, the suspended silt concentration in the compartment just above the bottom will be 5 to l0 g m - 3 higher than near the water surface .
130
0 .8 E
0 .6
c,
0.4
t
0.2
m
3 150 C) E model
1
°
100
v
a
U) 0 13
15
17
19 21 23
Days in june
25
1988
Fig . 11 . Results silt model - verification site 1 .
Fig. 12 . Results silt model - calibration site 2.
Conclusions 0 .8 E
The developed wave model has been successfully applied to Lake Marken, using only one set of calibrated parameters . Although less accurate, the
0 .6 0 .4
11 ro
3
0 .2 150 0
,~eeaved
100
50
0 18 20 22 24 26 28 30 Days in april
1988
-4 ∎ 0.0 0 .5
∎
1 .0
1 .5
2 .0
Tcr after consolidation [N m2 ] Fig . 13 . Results silt model - verification site 3 .
Fig. 14 . 2i, as a function of the depth in the bottom .
131 suspended silt concentration model is also applicable for this large shallow lake with one set of parameters . To obtain better model results it will be necessary to know more about the properties of the lake bottom, e .g . the particle size distribution and the critical shear stress . A bottom module that includes the consolidation process is necessary to obtain a correct description of the resuspension process . It is a useful tool to describe the distribution of silt particles as a function of the depth of the bottom . The results of the suspended silt concentration model are accurate enough to be used in a light/algae model . The relation between the light extinction due to algae and that due to silt in Lake Marken is : Ext = 0 .021 10 3 * Chl + 0 .046 * C + 0 .759 (25)
where Ext is the extinction coefficient (m -1 ), Chl is the chlorophyll-a concentration (g m -3) and C is the silt concentration (g m -3). The coefficients of Chl and C are in m2 g -1 . Investigations are underway into the dispersion of light by the different silt fractions, so that in the future a better description of the extinction coefficient will be possible with the results of the silt model .
Acknowledgement The author wishes to thank the directorate Flevoland of Rijkswaterstaat for the use of their measurements .
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