One Aspect of the Dynamics of a Coast. Partly Protected by a Row of Groynes

ONE ASPECT OF THE DYNAMICS OF A COAST. PARTLY PROTECTED BY A ROW OF GROYNES

Ir. W.T. Bakker RIJKSWATERSTAAT DIRECTIE WATERHUISHOUDING EN WATERBEWEGING AFD. KUSTONDERZOEK 's-GRAVENHAGE KONINGSKADE 25 TELEFOON 18 32 80

Aan de V/eledelgestrenge Heer Ir. J.A. Battjes Afd. Weg- en Y/aterbouwkunde van de Technische Hogeschool Oostplantsoen 25 DELFT

67.162B uw KENMERK: UW BRIEF VAN: ONS KENMERK: 's-GRAVENHAGE

TERUG: BIJLAGEN 3 29 mei 196? ONDERWERP:

Waarde Jurrien, , Overeenkomstig mijn belofte je op de hoogte te houden met wat ik bestudeer, zend ik je hierbij een exemplaar van het rapport: "The Coastal Dynamics of Sandwaves and the • Influence of Breakwaters and Groynes" en een exemplaar van het voorlopig rapport: "One Aspect of the Dynamics of a Coast, partly protected by a Row of Groynes". Het laatste rapport, dat van 17 april stamt, hen ik op het ogenblik aan het omwerken, met wat betere aannamen. V/at meer hierover vind je in bijlage 3. De definitieve versie van het laatstgenoemde rapport zal ik je te zijner tijd sturen.

Met hartelijke groeten,

(ir. W.T. Bakker)

Verzoeke bij Uw antwoord (ln voud) kenmerk en datum dezes te vermelden en slechts één onderwerp in een brief te behandelen. Bij brief nr. 67.162B, d.d.

Opmerking betreffende:

"ONE ASPECT OF TPIE DYNAMICS-OF A COAST. PARTLY PROTECTED BY A ROW OF GROYT^TES"

Een betere benadering van het transportmechanisme bij strandhoofden kan worden gegeven door een profiel aan te nemen volgens fig. 1 en aan te. nemen dat een aeewaarts transpoi't optreedt als > y^ (te steil profiel) en een landwaarts transport als y.| < yg» Hierbij is de gemiddelde afstand van de .vooroever tot het referentievlak XZ gelijk aan y^ + V/; W moet worden gezien als de afstand die in een evenwichts- profiel bestaat tussen strand en vooroever.

- y

Het is mogelijk een theorie te ontwikkelen, gebaseerd op de transport vergelijkingen:

Qy = ^^y^^i - ~ dwars trans port

Q. = Q„., - q. T-— - langstransport bij het strand I O I 1 oX

Q„ = Q „ - q„ -— - langstransport bij de vooroever

Met de aanname, dat het langstransport over het strand nul is bij een strandhoofd, vindt men de kustvormen (fig. 3). - 2 -

K X + . —:—;—~r-r- sinh K X Si O smh K^i, • 0 K L o 1 + q^ tgh K^L

K L sinh K X ''o"" sinh K 1 0 ^0^2 ^ K L 1 + 0 tgh K^L

fig. 3

Van belang blijkt een referentie-lengte 1.

Als de afstand 2L tussen de strandhoofden kleiner is dan 2L^, dan stelt het strand (y^) zich vrijwel loodrecht op de golfrichting in, terwijl de vooroevex" (yg) geen merkbare invloed ondergaat. Als de afstand tussen de hoofden groter wordt, gaat de theorie gelden, zoals ontwikkeld in het bovengenoemde rapport. De transportvergelijking wordt:

Q = 0 - q^ ^x waarin y op dezelfde manier gedefinieerd is als in het rappoi-t;

tgh K^L 1 - ÏTL O '^02 ^ ^0 1 tgh K^L 1 +

^1 ^ ^2 tgh K^L en P 1 + q = q^ tgh K^L q q K L O - 3 -

Voor grote v/aarden van K^L (L » L^) wordt p;

1 + ^2 ^o^

De V/aarde " ^ " uit het rapport is dus gelijk aan:

Voor kleine waarden van K^L (L« L^) wordt q gelijk aan q^.

Door het bouwen van strandhoofden is de tijdschaal t dus maximaal te verlengen tot maal de oorspronkelijke; dit gebeurt reeds bij ^^2 hoofden op een onderlinge afstand van 2L^.

Het leggen van hoofden op kortere afstand van elkaar dan 2L heeft dus O weinig zin.

Den Haag, 24 mei 196?. SYMBOIENLIJST

2_ L o halve afstand tussen de hoofden

^1 ^2 evenredigheidsconstante in vgl. van het langstransport bij het strand.

" " " " " langstransport bij de voor oever.

'* " " " dwarstransport tussen strand en vooroever.

" " " " " langstransport bij de schijnbare kustlijn. langstransport langs de kust langstransport langs het strand langstransport langs de vooroever dwarstransport langstransport langs het strand, als = O langstransport langs de vooroever, als —— = O t langstransport langs de kust, als = O afstand tussen strand en vooroever bij een evenwiohtsprofiel referentielijn, ongeveer in kustrichting. over de diepte gemiddelde afstand uit het referentievlak XZ van het strand over de diepte gemiddelde afstand uit het referentievlak XZ van de vooroever, min W y-coördinaat van de schijnbare kustlijn *

ONE ASPECT OP THE DYNAMICS OF A COAST, PARTLY PROTECTED BY A ROW OF GROYNES

1. Abstract

A mathematical theory will be given about the phenomena that occur if on a coastal area groynes are constructed. The theory is a simplification: it just deals-ahottt one aspect. The starting point is the coastal equation of Pelnard-Considère [1_ and a linearisation between the transport along the groyne and the size of the "step" in the coastline at the groyne. The change of the coastline, caused by changing boundary conditions, by stationary transport and by the -straightening" of the coast are considered. The conclusion is drawn, that in the middle of a row of groynes the same processes occur as without groynes, but on a larger timescale. Hear the edges of the row there are edge efrects. that cause bigger erosion, and accretion than without groynes.

2. Definitions and assumptions

m the theory of Felnard-Oonsidère only the influence of waves is

taken into account. According to this theory, the profile of the coast is schemalised

(fig. 1) as a horizontal area A, at a depth D, where no transport takes place, because the waves are not able to move the material at this depth, and an area B, which moves to and fro We define the "coastline" y as the line that links all points along the coast with the mean y^i where y^ is the distance to a vertical reference line

(fig. 1): Pig. 1 D 'If

- 6 -

5. Influence of a row of groynes

We consider a coast, where over a long stretch groynes are constructed

at time t = 0. What is the influence of the groynes on the coast? We will consider three influences, which can give an impression about

the processes that occur (fig. 6):

1° the influence of the boundary conditions (fig. 6b,c);

2° the influence of the stationary transport (fig. 6d,e); 3° the influence of the sWeWhing of the coast (6f). 5.1 The Influence of the boundary conditions 0 (y = 0), • We consider a coastline ^¥^\s straight at time t without stationary transport (Q^ = 0).(!^ 'Jl )

NO STATIONARY NO STATIONARY TRANSPORT TRANSPORT Y

MOVING 7-U REST

BOUNDARY

FIC.7^ coastline at tiive t

SAME BOUNDARY CONDITION

FIG . 7" coastline at time t, if no groynes would have been constructed

We assume, that the row of groynes begins at x - 0, and that the

motion of the coast is given in x = - oo. We will show, that the formation of the coastline ( yi for x > 0. y^ for X < of c'a!n be found from the coastline y that' would exist at the same time, if no groynes would have been constructed, by the following operation: i a the coastline of the protected part^'ban be found by reducing the x-scale of y by a factor i and by multiplying the y-scale ^-ith a factor (1 + r). in which:

1^ 1 + 1

(9) 4

- 9-

The above mentioned procedure can be put into formula ae:

if X ^ 0 then y^'(x) = (1 + r) y (px) (10a)

The redttction of-the x-scale is reasona))le: the^/time-scale of , ihe / boundary condition x = 0 for tHe par^/of the coast with x> Ö remains

same^,dr y /^^^^^^^^^^ (5)/the x-scale must be ^^ "00' b the coastline of the unprotected part y^ can be found as the sum of the original y plus a "reflected y"; the latter one being the reflection of the original y (for x > O) with respect to the y-axis and multiplied with the reflection factor r, given in eq. (9):

if X <0 then y^ (x) - y(x) + r.y (-x) (lObj

In fig. 8 the method of construction is visualised.

Y,Yi

fig. 8 The following proof can be given. It will be seen that y^ suffices the coastal equation (4), as it is a superposition of two solutions of this equation, -i-t—hete-alread-y-been' - shown, that y' suffices the coastal equation with coastal constant , so We 3*94—have- to show that the boundary conditions at x = 0 are fulfilled. 4

These conditions are:

1° y^iO, t) = y.|(0, t), which is correct: (0, t) = y\{0, t) - (1 + r) y (0, t )..(a)

2° Q(x - - 0) - Q(x = + 0)

As » 0, this gives, according to (1) and (8):

^ iir " ^ 5T- ^or X . 0 (b)

Prom (10a) and (10b) one finds respectively:

^ PTrL;^' P(1 * , so U-1 . p(i ^^^/x-O

4=0 v^xy^^Q Ux;^.o

Substituting this in (b), one finds:

1 - r q' < . y " P Q • which is correct, according to (9). j'V.ii'--: ^. ^ We will show now, that the coastline of the protected part is y' given in (10a).

The protected part begins at x . 0. According to (a) the time scale of the boundary condition at this place remains the same for y (the coastline if np groynes were constructed) and for y^'. The coastal constant pf y' is a factor ~ smaller tha the coastal P constant of y, according to (9), and so the x-scale must be a factor - smaller, according to (5). ^ Therefore, if y(px) is a solution of the coastal equation for the unprotected coast, then y(x) is a solution of the coastal equation for the protected coast. As mentioned in chapter 3, the y-scale can be chosen auxiliary and may be multiplied, for instance with a factor (1 + r). Then one gets yj(x) - (1 + r)y(px) according to (10a), which must be a solution of the coastal equation (s) for the protected coast.

As the unprotected part y^, the protected part y.,', and the point x - 0 all suffice the required conditions, the coastline given in eq. (10a) and (10b) must be correct. . - 10 - . . _ _

5.2 The Influence of the stationary transport

We consider a coastline, which is a straight line at time t = 0 (y « o), and remains at rest for x » oo .

All along the coast the stationary transport is Q^, before the construc tion of the groynes at t = 0.

Consider first the situation of fig. 9a (computed situation Ila of fig. 6),

1 11 //////// y //// y // ) / y ) / /' // I // /': / I

Coastline at time t = 0 Pig. 9a

Coastline at time t Pig. 9b - li»

On the left-hand side at infinity the transport is Q^, on the right-hand side it is ; therefore the sedimentation per unit time must be Q - Q' . With the same method as given in 5.1 one can proof, that the o 0 following "delta" will occur (fig. 9b). ïhe branch y^ is the branch, that would occur, if a river would debouch at X » 0 on an unprotected shore, bringing to the shore per unit of time a quantity of sediment Qj*'

• ^o • putting all its sediment on the left-hand side of the river-mouth. The branch y ' is the branch that would occur if that river debouched H q' 3E on a protected shore with coastal constant ^ , bringing a quantity Q^j ,

II. P 0

putting all its material on the right-hand side of the river-mouth. The branch y^'^ is a reflection of branch y^. with respect to the y-axis, but on a smaller x-scale (fig. 10):

Pig. 10

The formulas of y^. and y^'j can be found with the theory of Pelnard-Considere and are given in annex 1. In annex 2 the shape of the delta is given as a function of — .1, in which 1 is the distance between the groynes. The maximum accretion at x = 0 is relative to ^^t:

. t (11) Tt D - 12 -

The formulas for the transport along the coast can be found with the aid of (1) and (8). They are given in annex 1. The transport along the unprotected coast at a long distance of the beginning of the row of groynes is Q^, along the protected coast at a long distance of the beginning - S| , and just at the beginning it is ^ . P ^ If one considers the real coastline y^^. instead of the virtual coastline ^II* ^^^^ finds for the step at the first groyne:

a ^0 (12) ''^V.l — ^V r m M. w p-t—t V ^1

This step does not change in the course of time.

Q'o Qo

1 1 1 1 1 1 1 1 1 1 1 1. J ///////////////////// / / / y / ////////

Yff_____

Pig. 11b

The computation of the scourhole at the lee-side of a row of groynes .(fig. 11) is quite the same as the'computation of the "delta** on the other side. Branch y^^j and y^^ are the same as branch y^'^ and y^ respectively, reflected with respect to the x- and the y-axis (see annex 1). The maximum scour is given by y^^^Q in (11).

By enlarging the distance between the groynes at the end of the row, or their permeability, one will be able to spread the erosion more evenly over a larger area. Considering the influence of the stationary transport and of the boundary conditions, it was assumed, that the phenomena near A in fig. 6a did not affect the phenomena near B. The coastline was schematised as one semi-infinite unprotected and one semi-infinite protected peo't (fig. 6). The influence of the phenomena which take place near A will attenuate quick on the protected part from A to B and therefore their influence can be neglected near B. - 15 -

The influence of the stationary transport at A wili be, that after some time the groynes are covered by sand in which case the theory does not hold. It is assumed, that the groynes are lengthened before they are covered and are shortened in case of heavy erosion. The final situation in the case of stationary transport would be *

original coastline

final coastline

the situation given in fig. 12, where everywhere the transport is Q^. It is very unrealistic to consider this casein detail, because long before the situation will have been altered by human interference (building more groynes and so on).

5.3 The influence of the stretching of the coastline

In 5.1 and 5.2 it was assumed, that the coastline was a straight line at the moment of construction of the groynes. We will now first compare the dynamics of a protected coast y* and an unprotected coast y, if the boundaries remain at rest and if there is no stationary transport. At t « 0 we assume y » y (fig. 13a and 13t))

no stationary no stationary same cunve (t transport transport no stationary transport coastline at rest' rest time t r t(

Pig. 13a Pig, 13b

The coasts y and y have a different coastal constant (.q

The time-scale is related with the coastal constant, according to

equation (5): n t n constant. cc n As the coastline is given at time t <= 0 as a function of x, the x-scale of y and y is the same and therefore the time-scale n is equal to —^ * "cc

Conclusion:

At a protected coast the same processes of the unprotected coast will occur, at a longer time-scale t , in which:

t' - p^t . (1 + ,^)t

However, mostly only a certain part of the coastline is protected.

Pig. 14

Prom the foregoing it will be clear, that this longer time-scale only occurs in area IV of fig. 14, that there will be no influence in the areas I and VII, and that there will be disturbances near A and B in the areas II, III, V and VI. The last-mentioned areas will increase in course of time (proportional to ft).

A grafical method for computing the coastline in this case is the method of Schmidt, about which will deal another paper.

6, Conclusions

1. It depends of the case whether groynes will work favourably or not. One has to pay the accretion on one place with the erosion on another. With the aid of groynes one can more or less choose these places.

2. If anywhere erosion occurs, where it is not tolerable, one first has to classify the cause of erosion. area II 3. One cause of erosion can be a gully or a river-mouth, which withdraws sand from the coast. With the aid of some groynes, one can stop the erosion on the side of

gully Pig. 15 - 15 - •I

the row of groynes, off the gully (area II in fig, 15). One increases the erosion in area I, near the gully.

4. Another cause of erosion can be the lee-side scour on a row of groynes. One can spread this erosion over a larger area by increasing the distance between the groynes or by increasing their permeability (expressed in the permeability factor x^, (2)) at the end of the row.

5. A third cause can be, that the coastline has not found yet its equilibrium. One can stop the erosion by protecting all this part with groynes and by additional artificial nourishment. One can find the economical optimum between the number of groynes and the amount of artificial nourishment.

6. If a coast accretes, because it has not found yet its equilibrium or because of a delta, supplying sand, the effect of a row of groynes can be: diminishing the accretion in the area of the groynes. This is the inverse of the cases, mentioned in conclusions 3 and 5.

The Hague, 17th of April I967. LIST OF SYMBOLS

D » water depth at horizontal part of the bottom (area A in fig. 1).

1 = distance between two successive groynes. n ^ scale of toastal constant" . cc D

- time scale, n^ = scale in x-direction. ny. = scale in y-direction. n„ « vertical scale. z p = \| 1 + ^ ' x-scale of phenomena which occur at an unprotected scale is p times the x-scale of the phenomena at the protected coast, if the boundary conditions are the same. q - proportionality constant in the transport equation (1). Large q means large "sensibility" of the coast for changement of the coastal direction. * q q - = proportionality constant in the transport equation (8); with p respect to the virtual coastline. Q - littoral drift.

^ stationary transport » transport at places where the coastal direction is parallel to the x-axis ( = 0). Qo Q = —T' = transport at places where the virtual coastline is parallel 0 p^; I to the X-axis ( §^ = 0). r = P " , ; r is a reflection coefficient. P + < t = time. ' 2 t * p t

X ^ reference line, about in coastal direction. y = mean distance from reference line of area B (fig. 1). y y-coordinate of "virtual coastline" (the line that links the middle between the groynes). . 2 -

y-j^ « y on lefthand side of a groyne,

y^, =^ y on righthand side of a groyne.

y^ « horizontal distance of contour line of depth z from reference line.

Ai * proportionality constant. Large M- means large "permeability" of the groyne. stationary stationary transport Q transport Qo

• X

branch 3JL

X < 0 branch (real coastline) ^ ^)

Byi X > 0 (x=0) branch y|j (virtual coastline) yn('<) = y,(-px: = -pYo(^{l-erf(pxy:^)) ^x ^ - ^ -Hpx\U,) branch y^ (real coastline) y Aqt •dx x< L (x=L) branch y' (virtual coastline) y;(x) = -y^(L-x) = -y, |p(x-L)|

branch yj^^ (real coastline )

Jiynr. X > L branch y^^ (real coastline ) ytv^x) = -yi (L-x) y,/n [werf((L-x)\|:^); Bx •

in which

1 P-1 QQ INFLUENCE OF STATIONARY TRANSPORT ANNEX 1 •IK P ^ q DEVELOPMENT OF COASTLINE erf X /e-'^dt R'JKSWATER STAAT Getek. Gewijz Gezien Acc erfc X = 1 - erf X DIRECTIE y^.zn W. AFD. KUSTONDERZOEK ft 5^ A3 67.10A Y,Y'

-1.0

INFLUENCE OF STATIONARY TRANSPORT ANNEX 2 SHAPE OF COASTLINE AS FUNCTION OF DISTANCE BETWEEN THE GROYNES RIJKSWATERSTAAT Gtlck. Gtwijz. Gezien Acc. DIRECTIE W. en W. AFD. KUSTONDERZOEK Nr. 67.106