Wind and Wind Wave Sediment Transpor t
in Large Lakes and Reservoir s
b y
W .G . McDouga l
C .P . Le e
Water Resources Research Institut e Oregon State Universit y Corvallis, Orego n
WRRI-99 September 1984
WIND AND WIND WAVE SEDIMENT TRANSPORT
IN LARGE LAKES AND RESERVOIR S
by
W .G . McDougal and C .P. Lee Department of Civil Engineerin g Oregon State University
Final Technical. Completion Repor t Project Number G864-0 8
Submitted to
United States Department of the Interior Geological . Survey Reston, Virginia 2209 2
Project Sponsored by :
Water Resources Research Institut e Oregon State University Corvallis, Oregon 9733 1
The research on which this report is based was financed in part by the Unite d States Department of the Interior as authorized by the Water Research and Development Act of 1978 (P .L . 95-467) .
Contents of this publication do not necessarily reflect the views and policies of the United States Department of the Interior ., nor does mention o f trade names or commercial products constitute their endorsement by the U .S . Government .
WRRI-99 September 1984 ABSTRAC T
A rather simple model has been proposed to examine the effect o f wind and wind waves on shoreline erosion . The technique employed fo r estimating waves is that developed by the Corps of Engineers . Th e solutions for the combined wind and wave setup is based on radiatio n stress concepts, as is the generation of the longshore current . Th e sediment transport model is based on energetics . And finally, th e shoreline evaluation model is based on conservation of sediment . The methodology and numerical results were produced for the first fou r tasks . Only the methodology was outlined for the fifth task .
Without the completion of the fifth task, it is difficult to quan- tify the effects of wind wave erosion on the shoreline . However, it i s clear that the sediment transport is increased in a narrow band alon g the water line . Whether this sediment transport is of sufficient magni- tude or duration to cause significant erosion remains unanswered . FOREWORD
The Water Resources Research Institute, located on the Oregon Stat e
University campus, serves the State of Oregon . The Institute fosters ,
encourages, and facilitates water resources research and educatio n
involving all aspects of the quality and quantity of water available fo r
beneficial use . The institute administers and coordinates statewide an d
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research . The Institute also coordinates the interdisciplinary progra m of graduate education in water resources at Oregon State University .
It is Institute policy to make available the results of significan t water-related research conducted in Oregon s universities and colleges .
The Institute neither endorses nor rejects the findings of the author s of such research . It does recommend careful consideration of the accu- mulated facts by those concerned with the solution of water-relate d problems .
ii
TABLE OF CONTENTS
Page
INTRODUCTION 1
METHODOLOGY 2
1. Wind Waves 2
2. Water Depth 4
3. Longshore Current 1 6
4. Sediment Transport 1 9
5. Shoreline Evolution 2 3
CONCLUSIONS 24
REFERENCES 2 5
iii LIST OF FIGURE S
Figure Page
1 . Effective fetch method 3
2 . Forecasting curves for wave height 5
3 . Forecasting curves for wave period 6
4 . Definition sketches for nearshore domain 8
5 . Wind stress time coefficient 1 3
6 . Setup profile 1 5
7 . Longshore current profiles 2 0
8 . Longshore sediment transport profiles 22
iv INTRODUCTIO N
There are many large bodies of fresh water in the Pacific North- west . Significant wind waves may be generated on these bodies of wate r which increase sediment transport along the shoreline . This increase d transport may result in erosion and turbidity . Higher levels of turbid- ity reduce the environmental and recreational quality of the lake .
Increased erosion may result in a terraced shoreline which has reduce d recreational and aesthetic value and is more difficult to manage as a flood control reservoir .
A model is proposed to address this problem and involves severa l specific tasks . The first task, estimating wave conditions, employ s existing methods . Next, an expression for the water depth along th e shoreline is developed including the combined effects of wind an d waves . The wind setup is due to a surface stress at the free surfac e while the wave setup is due to the onshore-onshore radiation stress .
Longshore currents are also calculated for the combined effects of win d and waves . The forcing mechanisms are similar to those in the setup , but the flow is now resisted by a bottom drag rather than a pressur e gradient . A sediment transport model is presented which is based o n energetics . This model has been widely used in the marine environmen t and the empirical transport coefficient has been determined . The fina l step, determination of the shoreline response due to the longshor e currents and waves, is discussed but no analytical results ar e presented .
METHODOLOGY..
The determination of the wind wave effects on sediment transpor t
involves several specific tasks . These include : 1) estimating th e
waves from the wind speed and duration and the lake depth and length ; 2 )
determining the influence of the wind and waves on water depth at th e
shoreline (setup) ; 3) calculating the wind and wave-induced currents ;
4) estimating the sediment transport ; and 5) calculating the change i n
shoreline configuration . Each of these tasks is discussed in greate r
detail in this section .
1 . Wind Waves
The wind waves which are generated in the lake or reservoir are a
function of the wind speed and duration, as well as the fetch length an d
depth of the lake or reservoir . The fetch length may be determine d
using the effective fetch method (U .S . Army,, 1962) . The application o f
this method is shown in Fig . 1 .
The generation of wind waves for a given wind speed is eithe r
limited by the duration of the storm event or by the size of the body o f
water . Wave generation is, therefore, referred to as being fetch, o r
duration, limited . For wind speeds of interest (greater than 30 mph), a
fetch length of 30 statute miles will become fully arisen in a duratio n
of less than 4 hours . A body of water with a fetch length of 10 statut e
miles will require a duration of less than 2 hours . Therefore, i n
smaller bodies of water, it is reasonable to assume that the generatio n
of wind waves is fetch limited . Bretschneider and Reid (1953) propose d
a model considering the effects of bottom friction and percolation in
_ 155 .46 F" -II 51 Units e . 13 .51 2 Where based on map scal e One Unit =1714 fee t
eff =11 .51 X 52Ig~=3.7MIies
X i = Projected radial distanc e (ie X1 = r Cos-c) 0 6000 12000 I 1 Scale In fee t
Fig. 1 . Effective fetch method .
3
the bottom sediments . This technique was modified by Bretschneide r
(1965) and Ijima and Tang (1966) yielding the relationships given i n
Figs . 2 and 3 for wave height and period, respectively . In Figs . 2 and
3, g is the acceleration due to gravity (ft/s 2 ) ; H is the wave heigh t
(ft) ; U is the wind speed (f t/s) ; F is the fetch length (ft) ; and d i s
the water depth (ft) .
The forecasting technique assumes that the bottom is flat and tha t
the lake is of constant depth . These assumptions may be acceptabl e
because the length scale associated with the wind waves is small wit h
respect to the length scale of the lake . Also, if the water depth i s
more than twice the wave length, the waves are deep water waves and are
insensitive to the depth .
2 . Water Depth
In the central parts of the lake or reservoir, where the waves ar e
generated, the waves tend to be somewhat insensitive to the botto m
profile . However, as the waves approach the shoreline and ultimatel y
break, they are very sensitive to the bottom profile and water depth .
Because of this sensitivity, special care must be taken when estimatin g
the water depth near the shoreline .
As the wind blows over the surface of the water, a shear stress i s
developed . This stress is balanced by a pressure gradient associate d
with a setup of water at the shoreline . This wind setup, sometime s
referred to as a storm tide, may significantly increase the water dept h
at the shoreline, relative to the wave height .
There is a second mechanism, which is associated with the win d
waves, that also produces a setup of shoreline water levels . This wave
4
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5
setup produces a pressure gradient which balances the change in momentu m
flux associated with the waves as they break approaching the shore -
line. This breaking zone, or surf zone, is the region in which th e
effects of the waves are most important in generating turbulence, cur -
rents, and sediment transport .
Definition sketches of the nearshore region are shown in Fig . 4, in
which
waves . The brackets denote time averaging over the wave period . This
is done to remove the fluctuating component of motion at the wave fre-
quency . The MWL (mean water level) is the time-averaged free surfac e
while the SWL (still water level) would be the location of the fre e
surface if no wind or waves were present . The SWL is assumed to b e
known for the design conditions and it is necessary to determine the
MWL . The still water depth is termed h and the mean water depth (tota l
depth, actual depth) is denoted as d and is given by the sum of th e
still water depth plus the setup .
A general form for the depth- and time-averaged equation of motion
in the surf zone is (cf . Liu and Mei, 1974 )
pwd =
Vh - p wgd V
(1 ) + T s IVs l + T b
in which D/Dt is the material derivative, V is the two-dimensiona l
horizontal gradient operator, V . is the divergence operator, <•> is a
temporal averaging operation over one wave period, is the depth- an d
time-averaged horizontal velocity vector,
is the time mean pressure ,
7 a) C a) a) 0 0 Q) L c, X
PLA N
Z
PROF/L E
Fig . 4 . Definition sketches for nearshore domain .
8 presence of the wind and waves, p w is the water density, g is th e acceleration due to gravity, d is the total depth, s is the radiatio n stress tensor, t r is the integrated Reynolds stress tensor, t m is th e integrated viscous stress tensor, T s and T b are the surface and botto m stresses, respectively, and s and b are the material coordinates for th e surface and bottom, respectively . Seeking steady-state solution s
(8/8t = 0) and assuming no longshore gradients (3/3y = 0), that the flo w is sufficiently turbulent to neglect the viscous transport of momentum
(t m = 0), that the time mean pressure, bottom gradients, and mean fre e surface gradients vary slowly (
Vh = 0, IVbl = 1, Vs = 1), and tha t the turbulent stresses are related to the time and depth mean flows b y an eddy viscosity model (t r. = p e d «u>0 + VJ), Eq . (1) reduces t o
T sx = 0 (2a) Pwgd dx
-dx S xy +T by +Tsy+dx (ueddxv) = 0 (2b)
in which S xx is the onshore-onshore components of the radiation stres s tensor, S xy is the onshore-longshore component, T sx is the onshor e component of surface stress, T sy is the longshore component of surfac e stress, T by is the longshore component of bottom stress, ue is an edd y viscosity, and v is the mean longshore current .
The radiation stress terms, the excess flux of momentum due to th e presence of the waves, were given by Longuet-Higgins and Stewart (1960 ) for a coordinate system relative to the wave . Transforming to coordi- nates relative to the beach (see Fig. 4), these terms are given by
9
Sxx = E [ (2n-1) cos 20 + (n-1/2)sin 20 ] (3a)
Sxy = -E [nsinOcosO ] (3b)
in which E is the wave energy density (E = 1/8 pgH 2 ), a is the loca l
wave height, n is the ratio of the group velocity to the local wav e
celerity (n = 1/2[1+2kh/sinh 2kh]) provided that k, the wave number, i s
determined by 2ir/T = (2k tanh kh ) 1/2 , T is the wave period, and 0 is the
wave angle as defined in Fig . 4 .
Inside the breaker line (x < x B ), the water is assumed to be shal-
low and, due to refraction, the wave angle is small . It is furthe r
assumed that the breakers are of the spilling type such that the break-
ing wave height is related to the local water depth by H = icd . Insid e
the breaker line, the radiation stress terms may then be given a s
Sxx = 3/16 ic 2d 2 (4a) pwg
Sxy = 1/16 p wg K 2d 2 sine (4b )
To determine the total depth profile, Eq . (2a) must be integrated ,
employing an appropriate value for the surface stress . The only surfac e
stress that will be considered is that due to the wind . This stress i s
taken to be
2 (5 ) T sx = Spa ca w cosy
10
in which p a is the density of air, ca is a wind stress coefficient, w i s
the wind speed, and y is the wind direction relative to the shoreline .
The wind stress coefficient was given by Van Dorn (1953) a s
1 .1 x 10-6 ; w < 16 mph (6a )
6 1 .1x1D 6 + (2 .5x10 )(1 - 6w ) ; w > 16 mph (6b)
The a term accounts for the unsteadiness in the development of the win d
current . In the simplification of the equation of motion it was assume d
that only steady-state solutions would be sought . This is a reasonable
assumption for the wave-generated current because it forms rapidly .
However, the wind current develops more slowly . The S term accounts fo r
this development and estimates for S may be determined from the one -
dimensional equation for a wind-generated current without waves ,
dv_ 1 pw dt d (TS T b ) (7 )
in which v is the mean fluid velocity . The wind stress is given by Eq .
(4) without the S term and a simple, linear bottom stress is given b y
T b = ap w cwv (8 )
in which a is a dimensional linearizing coefficient and c w is the fluid
stress coefficient . As a first approximation, a may be taken as 3% o f
the wind speed . This is based on the observation that the wind drift i s
about this fraction of the wind speed (Wu, 1975) . Substitution of Eqs .
(5) and (8) into (7) and integrating yield s
ac t p c w2 - w v= pW cw 1 (9 ) d cosy [1-e d 11
11
in which it is assumed that the water is initially at rest . The maximum
current, v., occurs at very large times . The ratio of the time-
dependent velocity to v. is
ac t w v - d v = [1- e (10)
This ratio defines the B given in Eq . (5) . Figure 5 shows this term fo r
dimensionless time defined as acwt/d .
With all of the terms in the cross-shore equation of motion speci -
fied, Eq . (2a) may be writte n
+ pacaw2 cosy = 0 ; x < xB
A solution to this equation for a planar beach is given by
d=ax+b [1n(b+ 1) - 1] + c (12a)
in which
3 a = m (12b) 2 8+3K 2 8 p acaw cosy b = m apwg (12c)
and c is an integration constant .
Seaward of the breaker line, the shallow water assumptions may no t
be invoked . However, seaward of the breaker line, energy is conserve d
12
rn
U 0 .2 0 .4 0 .6 0 . 8 DI ENSIONLESS TH E
Fig. 5 . Wind stress time coefficient .
13
(friction and percolation are small) so that an energy conservatio n
equation may be written . The time-averaged Bernoulli equation is
2 2 P+u 2 +w2 Bpacaw cosy <>- = ~ + u 2gw + < > (13) p wg 2g n p wg n ; x x B
in which ( ) denotes a value in deep water . Noting that the wave-
induced mean free surface displacement goes to zero in deep water ,
assuming that the wind setup vanishes at great distances offshore, an d
employing linear wave theory yield s
2 ap H k acaw2cosy x (14)
The integration constant in Eq . (12) may be determined by equating
(12) and (14) at the breaker line, x = x B ,
K2 (16 - K 40 - 3K2 2 )h B c = hB-b[1n( +1)- 1] (15) 168+3K2 16b
Typical setup profiles are given in Fig . 6 . These setup profile s
are very nearly linear . Therefore, assume
d = b x + b 2 1 (16a )
in which
b1 =b (16b)
1/2dB +b K2 40- 3K 2 b2=bin( (16c) dB +b ~ + 16 8+3K 2 hB
14
It is convenient to define a coordinate transformation such that the
origin is located at the point on the beach where the depth goes t o
zero. This is given by
d=b 1 x (17a )
where
x = x + b1 /b2 (17b )
It is also convenient to express the depth in a dimensionles s
form. Using upper case letters to denote dimensionless variables defin e
D = d/xB (18a)
X = x/xB (18b )
B1 = b 1 (18c )
The dimensionless depth profile, including wind and wave setup, is give n
by
D= B 1 X (19 )
3 . Longshore Curren t
The longshore current due to wind and waves is determined from Eq .
(2b) . The divergence of the radiation stress term is determined employ-
ing Snell s Law for refraction and assuming conservation of energy i n
the offshore .
1 6
- ~ (d) x < x B (20a) 5 6 pwgK2 sine d) 1/2 d 1 B dx
0 ; x > x B (20b )
The wind stress term is the same as for the onshore equation of motio n
except that the sin(y) component is taken rather than cos(y) component .
The bottom stress term may be approximated using a linear shallo w
water, small wave angle relationship for the waves (cf . Liu and
Dalrymple, 1978) and a linear relationship for the wind current (see Eq .
(8)) .
T by = - - K p w(gd) 1/2 v ap wcwv (21 )
The eddy viscosity is assumed to be proportional to a characteristi c
velocity, density, and length . The form proposed by Longuet-Higgin s
(1970b) is employed .
u e = Pw N(gd) 1/2 x (22)
in which N is a numerical constant .
Employing the above relationship in the equation of motion yield s
NTrbl d "5/2 dv "1/2 an px ) - (x + v = 1/2 K K cw dx dx (gb l )
p c Tr b 5 B) 1/2 sineB b1 3/ 2 w2siny x < xb K bl(g d dB w 16 cw x 1/2 Pw c (gb)~1 (23a)
p c 2siny ; x > xB (23b) x ) 1/2 pw cw w (gb l
17
Again, it is convenient to define dimensionless variable s
NitB 1 P 1 (24a) K C w
7t __ a (24b) P2 K 2 (gB xb ) I/ 1
B 1 P 3 (24c ) DB
ca aP = 01 w2 i (24d ) P4 Bl l/2 Pw cw nY (g xB)vBL s
V = = bl (gdB ) 1/2 sin6 B] (24e ) v/vBL v/[516 c w K
The term vBL is the longshore current due to waves alone which occurs a t
the breaker line for the case of no turbulent lateral mixing (Longuet -
Higgins, 1970a) . It is common to use this for scaling longshor e
currents .
In dimensionless variables the equation is written a s
- P3 X3/2 - P4 ; X < 1 (25a ) d X5/2 dV 1/2 P T.( 757,) _ + P ) V = 1 ( (x 2 P4 ; X > 1 (25b )
Boundary conditions are that the longshore current velocity i s
bounded at the shoreline and far offshore, and that the velocity an d
gradient of velocity match at the breaker line . Equation (25) has bee n
integrated analytically in the course of this study . Unfortunately, th e
resulting solution is rather complicated and it is more convenient t o
18
just use a numerical integration . Several example profiles are shown i n
Fig . 7 for different wind and wave conditions .
4 . Sediment Transport
Bagnold (1963) proposed a sand transport model which assumes tha t
the sediment is mobilized by wave motion and wave power is expended t o
maintain the motion of the sediment . The presence of a mean curren t
transports the sediments . The immersed-weight transport rate per uni t
width of surf zone is (McDougal and Hudspeth, 1983 )
i = K dx (26) (2 pga2Cg) u m
where K is a dimensionless coefficient which depends on the degree o f
the transport mechanism developed, C g is the wave group velocity . The
coefficient can be treated as a constant for a particular coast from th e
view of long-term climate . Furthermore, it can be considered as a
constant in a season or in a storm (Hou, Lee, and Lin, 1980) .
Assuming linear shallow-wave conditions exist and that the wav e
energy flux and the bottom orbital velocity outside the breaker line ca n
be evaluated at the breaker line, Eq . (26) become s
pg K d dr (d) v x < xb (27a) dx
K S pgK [d d" (d)]- v ; x > xb (27b) dx
In terms of nondimensional variables, the immersed-weight transport rat e
becomes
19
In nI
r • E a C OO rn O In 0,
N
ofO f q - N f f fn L. CL E a+ rn v N. w 0
N O. 0 en en 0 fn f7 0, N O O O
CO 0
0
0
0
I i I 1 I I 1 I I 0 c3 (q (~ {i) ua v KJ. cV 0 0 0 0 0 0 0 C? 0 0
A
20
(EL) ) D b dX V (x) X < 1 (28a
d dx rr V (X) ; X > 1 (28b) b X= 1
where
i I (28c) 1BL
5 (28d ) 1BL __ 8 R p s gKdbSbvBL
d b Sb xb (28e )
and p s is the density of the sediment .
For the linear total depth profile given in Eq . (19), the sedimen t
transport is
XV X < 1 (29a)
V X > 1 (29b)
Dimensional transport profiles are shown in Fig . 8 for the long-
shore current profiles given in Fig . 1 .
The total sediment transport is given b y
co 1 co IT = 0J I dX = f XV dX + 1f VdX (30 ) o
This integral is numerically evaluated .
21
N
tr)
X
rn 0
N 0
LC) 0
:r1 0
c.) <
0 U-) h ) 0 0 CD 0 c 0 0 0 O
H
22
5 . Shoreline Evolutio n
Shoreline evolution is based on the net flux of sediment along a
given beach . The shoreline is broken into a number of segments o f
length AS . The immersed weight dynamic transport given in Eq . (30) may
be converted to a volume transport rate by dividing by the porosity, n ,
and the immersed weight density .
S = I (31) n( ps-pw) g
in which S is the volume transport and p s is the density of the sedi-
ment . From conservation of the sediment mass, the change sediment in
depth is given a s
aE 1 al M at - n(p s-p w)g ax As h Np (32)
in which E is the normal to the bottom and h Np is the depth to the nul l
point . The null point is the maximum depth of wave-induced sedimen t
transport. For lakes and reservoirs, this depth may be on the order o f
five feet . The change in depth also results in a change in the positio n
of the shoreline . This change is given by
aR m2 + I ) D E at - 2 ) ta (33 ) m
in which R is the shoreline position relative to the initial conditio n
and m is the bottom slope .
The change in shoreline may be computed using the solution obtaine d
for total longshore transport and then solving Eq . (33) numerically . An
implicit finite difference scheme may be used in space while the solu-
tion is explicitly marched in time .
23 It should be noted that at each change in shoreline configuration , the breaker angle must be recomputed for each segment . This may b e rapidly, but rather crudely, done employing Snell s Law .
CONCLUSIONS
A rather simple model has been proposed to examine the effect o f wind and wind waves on shoreline erosion . The technique employed fo r estimating waves is that developed by the Corps of Engineers . The solution for the combined wind and wave setup is based on radiation stress concepts, as is the generation of the longshore current . The sediment transport model is based on energetics . And, finally, the shoreline evolution model is based on conservation of sediment . The methodology and numerical results were developed for the first fou r tasks . Only the methodology was outlined for the fifth task .
Without the completion of the fifth task, it is difficult to quan- tify the effects of wind wave erosion on the shoreline . However, it i s clear that the sediment transport is increased in a narrow band along the water line . Whether this sediment transport is of sufficient magni- tude or duration to cause significant erosion remains unanswered .
24 REFERENCE S
Bagnold, R .A . (1963), " Beach and Nearshore Processes, Part 1 : Mechanic s of Marine Sedimentation ." In : M .N . Hill (Editor), The Sea, pp . 507-528 .
Bretschneider, C .L . and Reid, R .O . (1953), "Change in Wave Height Due to Bottom Friction, Percolation and Refraction," 34th Annual Meetin g of the AGU .
Bretschneider, C .L . (1965), "Generation of Waves by Wind ; State of the Art," Report No . SN-134-6, National Engineering Science Co . , Washington, D.C .
Hou, H.S, Lee, C .P ., and Lin, L .H . (1980), " Relationship Between Along- shore Wave Energy and Littoral Drift in the Mid-West Coast a t Taiwan," Proc . 17th Coastal Eng ., Chap . 76, pp . 1255-1274 .
Ijima, T . and Tang, F .L.W . (1966), " Numerical Calculation of Wind Wave s in Shallow Water," Proc . 10th Conf . on Coastal Engeering, pp . 38- 45 .
Liu, P .L-F and Mei, C .C . (1974), " Effects of a Breakwater on Nearshor e Currents Due to Breaking Waves," Ralph M . Parson Laboratory , Department of Civil Engineering, MIT, Report No . 12 .
Liu, P .L-F and Dalrymple, R .A . (1978), " Bottom Frictional Stresses an d Longshore Currents Due to Waves with Large Angles of Incidence , " J . Mar . Res ., 36 :357-375 .
Longuet-Higgins, M .S . and Stewart, R .W. (1960), " Changes in the Form of Short Gravity Waves on Long Waves and Tidal Currents , " J . Mar . Res ., 36 :357-375 .
Longuet-Higgins, M .S . (1970a), " Longshore Currents Generated b y Obliquely Incident Sea Waves, 1," J . Geophysical Res ., 75 :6778- 6789 .
Longuet-Higgins, M .S . (1970b), "Longshore Currents Generated by Obliquely Incident Sea waves, 2 , " J . Geophysical Res ., 75 : 6790- 6801 .
McDougal, W .G . and Hudspeth, R .T . (1983), " Longshore Sediment Transpor t on Non-Planar Beaches," Coastal Eng ., 7 :119-131 .
U .S . Army, Corps of Engineers (1962), "Waves in Inland Reservoirs , (Summary Report on Civil Works Investigation Projects CW-164 an d CW-165)," TM-132, Beach Erosion Board, Washington, D .C .
Van Dorn, W .C . (1953), "Wind Stress on an Artificial Pond, " Jour . o f Marine Res ., Vol. 12 .
25 Wu, Jin (1975), "Wind-Induced Drift Currents," J . Fluid Mech ., 68 :49-70 .
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