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The modelling of the approach of bores to a shoreline
Coastal Engineering, 3 (1980) 207--219
207
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
THE MODELLING OF THE APPROACH OF BORES TO A SHORELINE
B. JOHNS
Department of Meteorology, University of Reading, 2 Earley Gate, Whiteknights, Reading RG6 2A U (Great Britain) (Received March 9, 1979; revised version accepted October 2, 1979)
ABSTRACT
Johns, B., 1980. The modelling of the approach of bores to a shoreline. Coastal Eng., 3: 207--219. Turbulent processes beneath bores approaching a shoreline over a sloping beach are modelled using a turbulence energy closure scheme. The procedure incorporates the simulation of turbulence production at the face of each bore together with shear-generated turbulence in the bottom boundary layer at the beach. The effect of these two sources of turbulence is represented by production terms in the turbulent energy quation and the solution of this determines the value of exchange coefficients which are then used to calculate the Reynolds-averaged flow conditions. Numerical experiments are described that relate to the approach of both a single bore and a train of periodic bores to the shoreline. The bottom stress is calculated and it is shown that it is strongly influenced by turbulence production at the face of the bore.
INTRODUCTION
The approach of surface waves to a shoreline is characterized by propagation through three different regimes. During the initial phase, when the waves are on the surface of relatively deep water, the dynamical processes may be described in terms of well-established theory relating to sinusoidal nonbreaking waves. During the second phase, when the waves have moved into shallower water, their structure is increasingly influenced by the beach topography and ultimately they steepen and begin to break. Subsequently, the wave disturbances propagate through the surf zone in the form of a sequence of periodic bores (Phillips, 1966; p. 56). These are characterized by a steep-fronted turbulent face and a relatively smooth non-turbulent back. During this final phase beach processes, such as the generation of long-shore currents and bedload transport, are strongly dependent on the b o t t o m stresses associated with the incoming periodic bores. In particular, a theoretical treatment of long-shore current generation necessarily involves an appropriate parameterization of these b o t t o m stresses. In Longuet-Higgins (1970) and Bowen (1969), for example, the b o t t o m stress is represented by an empirical
208 relation involving the near-bottom orbital velocity induced by the periodic disturbances. However, the validity of this procedure for bore-like disturbances is not clear. In such circumstances, the b o t t o m stress is determined by the intensity of turbulence in the system supporting the Reynolds stresses. Moreover, in the surf zone, there are two turbulence generation mechanisms. Firstly, there is the contribution from shear-generated turbulence at the beach and secondly, the generation of turbulence at the face of the bore itself. This secondary source does not appear to have been included in any existing theoretical treatments and, in shallow water, the turbulence intensity and corresponding b o t t o m stress must be expected to be influenced by this process. In the present work, a model is developed for the approach of bores to a shoreline which is based upon the shallow water equations {Peregrine, 1972). Using the technique developed by Johns (1977, 1978), the turbulence processes are included by the application of an equation for the turbulence energy density. Source terms in this simulate turbulence production at the beach and the face of the bore. In the first numerical experiment, the b o t t o m stress is determined that results from a single bore-like disturbance. In the vicinity of the face of the bore, the calculations show that the distribution of turbulence energy is strongly influenced by the contribution from the face of the bore. A calculation of the corresponding b o t t o m stress indicates the importance of accounting for this source of turbulence production in any assessment of beach processes. This point is substantiated by determining the value of the friction coefficient in an empirical b o t t o m stress law so as to obtain the best overall representation of the b o t t o m stress. A comparison of the b o t t o m stress evaluated from this with a direct evaluation from the turbulence models reveals a gross underestimation by the former beneath the face of the bore. In the second experiment, a train of incoming periodic bore-like disturbances is considered. An evaluation is made of the mean b o t t o m stress beneath these periodic disturbances. On approaching the shoreline it is found that the mean b o t t o m stress changes from being on-shore to off-shore. It is suggested that this may result in the production of an off-shore region of bedload transport convergence with an associated sand-bar development. FORMULATION A Cartesian frame of reference is chosen in which the axis 0x is normal to the shoreline and which is fixed in the equilibrium level of the sea-surface. The origin 0 is located at the seaward extremity of the analysis area and 0z points vertically upwards. The equilibrium position of the shoreline is at x = L and the equilibrium depth of the water over a plane sloping beach is given by: h(x)
=
ho(1-x/L)
where ho is the undisturbed depth at x = O.
(1)
209 The Reynolds-averaged components of velocity ( u , w ) then satisfy: au -at
au
au
+ u-ax
+ w
-
a~
1 a
ax
+ - -pax
g
az
1 a rxx
+p
-az
rzx
(2)
where [ is the sea-surface elevation above its equilibrium value, the pressure has been assumed hydrostatic and r x x a n d r z x are components of the Reynolds stress. The equation of continuity is: au/ax
+ aw/az
(3)
= 0
which has the equivalent form (cf. Phillips, 1966, p.45): a~ a -- + -at ax
(
Jh
udz = 0
(4)
_
The Reynolds-averaged turbulent energy density E satisfies: aE --
at
aE + u--
ax
aE + w
az
-
rxx au
p
rzx au
+----+~-e ax p az
(5)
where the first two terms on the right-hand side represent the production of turbulence energy by transfer from the Reynolds averaged flow. They simulate the production of turbulence energy at the face of the bore and the production of turbulence energy by shearing in the b o t t o m layers. ¢ represents the re-distribution of turbulence energy by the eddy motion and e represents the dissipation of turbulence energy in the system. Until hypotheses are made about r x x , r z x , ¢ and e the only assumption in eqs. 2--5 is that of a hydrostatic balance. An absence of fluid slippage at the beach requires: u = w = 0
at z = - h ( x )
(6)
and the kinematical surface condition leads to: = 0
a~/at+ua~/ax-w
a t z = ~'(x,t)
(7)
A condition must also be satisfied by the Reynolds stress at the free surface. An absence of applied surface stress will be prescribed which is represented by the requirement: rzx
= 0
at z = ~ ( x , t )
(8)
This may n o t be an entirely satisfactory condition in the neighbourhood of a steep-fronted surface disturbance but a localized breakdown of eq. 8 is t h o u g h t unlikely to have an effect on the overall flow conditions. Assuming that the turbulent transfer of m o m e n t u m is diffusive in character, we write: rzx
= p K au/az
,
rxx
= p N au/ax
(9)
210
where K and N are vertical and horizontal exchange coefficients. Turbulence closure is achieved as in Johns (1978) and we take: K
(10)
= c 1/~ IE 1/~
where c = 0.08 and the length scale l is determined from: (E1/2 /l)
l -
, l = Kz0
as z - ~ - h
(11)
d-d-(El/,ll ) dz
The quantity K is von Karman's constant 0.4 and z0 is the roughness length of the b o t t o m elements. Additionally, the horizontal exchange coefficient is related to the turbulence energy density by: N
= a A x E 1/:
(12)
where AX is the grid increment in the final discretization scheme and ~ is an empirical factor which must be prescribed. Hence, the length scale associated with the horizontal mixing is aAx. It might be added that the inclusion of a horizontal mixing term is essential to maintain the computational stability of the numerical scheme of solution. Such an artifice is often incorporated implicitly by use of a Lax-Wendroff integration scheme to smooth out any developing discontinuities. A distinguishing feature of the present work is that the energy thus extracted from the mean flow is retained in a turbulence energy budget. Turbulence closure in eq. 5 is achieved by assuming the re-distribution of turbulence energy to be diffusive with the same exchange coefficients as for m o m e n t u m transfer. Accordingly: (P = -
3x
N -
Ox
+
~z
K -
(13)
~z
The dissipation e is parameterized as in Johns (1977, 1978) and we write: (14)
e = c3/~E~h/l
Eq. 5 then becomes:
aE - -
-+
~t
U
~x
+
+K_
:
~z
\ ~x ]
a "
\3z
1
÷ _
~x
_
÷
~x
-
~z
(15
_
l
An absence of turbulent energy transfer across the beach and free surface is then approximated by: OE/Oz
= 0
at z = - h and z =
In the absence of the horizontal production term, further details of the turbulence closure scheme, and its justification for steady flows, are given by Launder and Spalding (1972). Its application to oscillatory oceanic flows has been considered by Vager and Kagan {1971) and Weatherly (1975).
211 COORDINATE TRANSFORMATION
As in Johns (1978), it is convenient to introduce the coordinate transformation: (16)
o = (z + h ) / ( ~ + h )
so t h a t the beach and free surface correspond to o = 0 and o = 1 respectively. In comparison with Johns (1978), the presence of horizontal mixing terms complicates the form of the transformed equations in which x, o and t are taken as independent variables. In particular, eq.2 becomes:
ou --
ou
ou
Ox
ao
+u--+co--
Ot
=
~
+
-g~x
1 (o,xx p\
+
ax
o,x~ ~ ax
ao !
oz a,z,
+--
p
(17)
ao
where: =
(18)
ot+UOx+Waz
and the subscripts denote differentiations. However, the retention of the additional terms associated with the horizontal mixing appears hardly worthwhile with the assumptions made in the representation of the physical process. Accordingly, the transformed forms of eqs. 2, 4 and 5 lead to: au au au -- +u-+co-=-g
at
ax
aH
ao
ax
gho a /_au\ -- + --~N~x ~+
L
ax
1
a
H 2 ao
/__au \ ~K~--
(19)
ao
1
--
°[-s
+
at
ax
udo 0
]
=
(20)
0
and:
--
or
+u--
a~
+co--
ao
= H2
~aol
~s
+~
tNo~i
---H ~ ~o
~
-e
(21)
where H is the total depth ~ + h. Additionally, co is determined diagnostically from: t
Hw
=
°[- Su4 -° [-7 u o]
Oax
~X
0
0
The accompanying boundary conditions then yield: u=O
ate
= 0
co = 0
a t : o = 0 and o = 1
au/ao
= o
ate
aE/ao
= o
at o = 0 a n d
(23)
= 1 o = 1
212 NON-DIMENSIONALIZATION Non-dimensional variables are defined by: H
= hoI:I ,
u = (gho)V'fi,
x
= Lf,
g
= [ho2(gho)V3/L]k,
~ = [(gho) v'/Ll&
,
E = gh(~
(24)
t = [L/(gho)V']~ N
(25)
= L(gho)V'lV,
e =[(gho)~4/L]~,
l = ho[
(26)
This scaling then reduces eqs. 19--22 to: -: + ~ - - + & - - + - - + l 8t 8k 80
8-~- + - 82 --
--
o
= -8k
82
~do
+c5--
+
--
~
80
/~-80
(27)
= 0
(28)
-
/:/~ = o 8k 8 /~
~-
+
~da
+--
-82 - /~
o
--
--
fide
-~
+
(30)
0
Non-dimensionalization of the closure relations eqs. 10, 12 and 14 yields:
/~ = c l / ' / / ~ l / ' ( ~ o )
N = ~ , ~ 2 / ~ v' '
~ = c3/,(L~ '
\ho]
j~3/2_ 1
(31)
whilst eq.11 leads to: /'=
k
o\~!
+ EV'/tf
(32) 0
where: Zo = z o / h o
(33)
and Eb is the value of f: at a = 0. Apart from the parameters describing the wave input at the seaward boundary 2 = 0, the solution depends upon the parameter given in eq. 33 and a slope parameter ~ = h o / L . The solution is obtained by having ten unequally spaced computational levels in the vertical and by selecting a staggered grid having 99 points in the horizontal in which elevations are recorded at odd points and currents at even points. The technique is identical to that described by Johns (1978) and is n o t repeated here. A disturbance enters the analysis area at 2 = 0 at time t = 0 and propagates towards the shoreline. If, during the integration, the distur-
(29
213 bance does not reach the shoreline there is no need for a shoreline boundary condition. If the disturbance reaches the shoreline, we define } to be such that ~ = 1 + } gives the disturbed position of the shoreline and then apply:
/~=i+~ = 0 fido
}
=
~=1+~
0
Eqs. 34 are readily shown to be consistent with mass conservation and, in the numerical evaluation, are approximated for small } by: =
fido
+} ,~ =1
fido o
(35) ~ =1
and:
l~= 1
__
-}
xa~ ~=1
(36)
Eq. 35 provides a m e t h o d of updating ~ using values of fi extrapolated from interior current points. Using this value of }, eq. 36 yields an evaluation o f / ~ at the equilibrium position of the shoreline. More precisely, replacing (0H)~ ~-- =1 in eq. 36 by the one-sided difference approximation:
4~ [3it/x=1- 4/~"~=-1-2hx + /~.,~=1--4A;~]
O~ ]~=1
(37)
it follows that: 4A~ + 3} NUMERICAL
(38)
EXPERIMENTS
Two sets of numerical experiments have been performed. In the first of these, a single bore is simulated by prescribing that: f/~= o =
l l for t < 0 2 for t > 0
(39)
The slope factor/~ is 0.01 and the roughness parameter Zo = 0.001. The first integration was performed with ~ = 10.0 and this was continued until the effect of the disturbance was first felt at a distance of about L/4 from the shoreline. An interesting feature of the induced velocity distribution beneath the bore
214 is its uniformity with depth. The bulk of the shear is concentrated into an extremely thin layer adjacent to the beach as shown in Fig. 1 in which are given characteristic profiles of ft. In fact, on defining the depth-averaged value of a quantity of q~ by: 1
(40)
= f Cdo 0
it is found that u2/u 2 < 1.04, which is indicative of an almost total absence of vertical structure in the Reynolds-averagedflow field. The distribution of turbulence energy is given in Fig.2. That generated at the face of the bore spreads downwards (whilst decaying) to affect the entire depth of the flow and completely dominates the shear-produced turbulence adjacent to the beach. This must, of course, contribute to the bottom stress which will not therefore be expressible by a simple law. A frequently used parameterization of the bottom stress is given by: Tb --
=
Cf ~[al
(41)
P where Cf is a friction coefficient. This may be compared with:
P
=-h
and a value deduced for Cf. The calculations show Cf to be extremely variable h0
I
~
E SURFACE
z ho
0.0
~7--;
~7 -b0
.
0.25
.
.
.
.
.
L
0.5
-
~
0.75
Fig. I. Characteristic profiles of fi in the flow behind the face of the advancing bore. The superimposed scale represents a unit value of ft.
215 |.0
_z ho
65
-1.0 0.25
0.5
~:
0.75
Fig. 2. C o n t o u r s o f e q u a l v a l u e s o f 104~: b e n e a t h a s i n g l e b o r e : ~ = 1 0 . 0 .
but an optimized constant value may be defined by: L
f (~'b/P).lal Cf -
0
(43)
L
f
dx
~4 dx
0
This yields Cf = 5.56 × 10 -3 and the exact formulation may be compared with one based on eq. 41. This is shown in Fig. 3 from which it is apparent that the empirical law cannot reproduce the peaking of the b o t t o m stress beneath the face of the bore. This enhancement of the b o t t o m stress by turbulence production at the face of the bore is likely to be of considerable importance in any evaluation of beach processes such as the ability of waves in the surf zone to initiate bedload transport. In order to assess the dependence of these results on the value used for a, the integration has also been performed with a = 5.0. The results for the turbulence energy distribution are shown in Fig. 4 and an optimized value of the friction coefficient is given by Cf = 6.02 X 10 -3. The corresponding distribution of the b o t t o m stress is shown in Fig. 5 and, except at the face of the bore, the similarity between these results and those based on a = 10.0 is apparent. For a = 5.0, the turbulence energy density at the face of the bore is greater than that for a = 10.0 and it is clear that the associated production term is playing an important role in the turbulent energy budget. The length scale of the horizontal mixing process is now reduced and, consequently, the tur-
216
FREE SURFACE
6.0
-- ~ ' l ~ / ( g h ° ) x l O '
4.0
--
2.0 ~..C. ~
~
~
\
o
L
0.25
~
0.5
X
~c
0 75
Fig. 3. Distribution of bottom stress, Tb, beneath a single bore. Continuous curve is from eq.42. Broken curve is from eq.41 with Cf = 5.56 × 10-3: ~ = 10.0. L0
1~10 Z ho
-1°o.25
I00
200
75
z,g0
0.s
~
07s
Fig. 4. C o n t o u r s o f equal values o f 104E beneath a single bore: ~ = 5.0.
b u l e n c e e n e r g y g e n e r a t e d at the face o f the b o r e is c o n c e n t r a t e d into a smaller v o l u m e . T h e series o f e x p e r i m e n t s was c o n t i n u e d b y c o n t i n u o u s l y r e d u c i n g (~ = 2.5 leads t o Cf = 6 . 4 8 × 10-3). U l t i m a t e l y , o f course, t h e familiar t y p e o f s t e e p e n i n g develops t h a t is associated with the s o l u t i o n o f the shallow w a t e r e q u a t i o n s a n d the n u m e r i c a l s c h e m e breaks d o w n . In such c i r c u m s t a n c e s , it is suggested t h a t the r e d u c e d h o r i z o n t a l m i x i n g in the m o d e l is insufficient to
217
/(gho) xlO ~ 8.0
FREE SURFACE
8
/~
~.o
i
2.0 --
00.25
0-5
~c
0.75
Fig. 5. D i s t r i b u t i o n o f b o t t o m s t r e s s , r b , b e n e a t h a single b o r e . C o n t i n u o u s c u r v e is f r o m eq. 42. B r o k e n c u r v e is f r o m eq. 41 w i t h Cf = 6 . 0 2 X 10-3: ~ = 5.0.
simulate the sub-grid scale physical processes leading to turbulence production and a consequential drain of energy from the Reynolds-averaged flow. Clearly, the choice of a suitable value of a must depend, for example, o~n laboratory or field experiments. These might be based on the reproduction in the model of the observed decrease in the height of the bore as it approaches the shoreline. The present study, however, indicates the importance of retaining the energy loss from the bore in an overall turbulent energy budget. In the second set of experiments, an input of periodic bores is simulated by: 1 + 4A
/~e= 0
I(+;
---
=
1 for r < t < r , +
,
0
T
(44)
(b
This represents an incoming train of steep-fronted disturbances having a m a x i m u m height A above the equilibrium level of the free surface. Again, we take 13 = 0.01, z0 = 0.001, a = 10.0 and select A = 0.5. Additionally, r = 0.05 and r, = 0.5. With h0 = 2 m, this corresponds to waves with a period of approximately 11.3 s, the roughness length being 2 mm. The integration is started from an initial state of rest and is continued until an effectively oscillatory response is obtained. A diagnostic study may then be made of the final cycle of integration. As for the single bore, the friction coefficient in eq. 41 may be determined
218 by comparison with eq. 42. In this case, Cf varies bot h spatially and temporally. However, an optimized constant value may be defined by:
tp L o
Cf =
L f0tP f0 U4 dxdt
(45)
where tp is the dimensional wave period. This then yields Cf = 1.41 × 10 -2. Although the m a x i m u m input height of the periodic bores is only half t h a t of the single bore, Cf is markedly different in the two cases. For the periodic bores the friction coefficient is greater by a factor of more than 2.5 than that for the single bore. F o r effective practical application, Cf should be independ e n t of the t y p e o f wave m o t i o n generating the b o t t o m stress. The present model indicates th at this is n o t so. Clearly, in the case of periodic bores, the background o f residual turbulence persisting from one cycle to the n e x t will enhance the b o t t o m stress (with an effective increase in Cf) in comparison with th at beneath a bore advancing into turbulence-free water. It is informative to determine the mean b o t t o m stress beneath the waves during the final cycle of integration when the response is oscillatory. Provided that the instantaneous b o t t o m stresses are sufficient to produce bedload movement, the mean b o t t o m stress should indicate any t e n d e n c y to induce a mean bedload transport. The calculation shows there to be a continuous reduct i on in the value o f a shoreward directed mean b o t t o m stress (%) from the seaward limit at k = 0 to a poi nt a b o u t 10 grid increments from ~ = 1. At this position:
\ ghop/
1.23 X 10 -6
(46)
/
which, with h0 = 2 m, corresponds to a mean b o t t o m stress of about 0.24 dynes cm -2. This compares with an instantaneous b o t t o m stress at this position of a b o u t 9 dynes cm -2. Shoreward of this position, a reversal occurs in the sign o f and, at five grid increments from ~ = 1, there is a seaward directed mean b o t t o m stress equal in magnitude to t hat given by eq. 46. It is suggested, therefore, t ha t the incoming periodic bores will lead to a convergence o f bedload transport with a consequential sand-bar f o r m a t i o n in the intervening zone. With h0 = 2 m, this zone would be of order 4 m from the equilibrium position o f the shoreline. The actual position of the shoreline changes during the wave cycle (and there is a wave set-up) and is displaced f r o m the equilibrium position by as m uc h as 2 m in the landward direction.
219 REFERENCES Bowen, A.J., 1969. The generation of longshore currents on a plane beach. J. Mar. Res., 27: 2 0 6 - 2 1 5 . Johns, B., 1977. Residual flow and boundary shear stress in the turbulent b o t t o m layer beneath waves. J. Phys. Oceanogr., 7: 733--738. Johns, B., 1978. The modelling of tidal flow in a channel using a turbulence energy closure scheme. J. Phys. Oceanogr., 8: 1042--1049. Launder, B.E. and Spalding, D.B., 1972. Mathematical Models of Turbulence. Academic Press, New York, N.Y., 169 pp. Longuet-Higgins, M.S., 1970. Longshore currents generated by obliquely incident sea waves, 1. J. Geophys. Res., 75: 6778--6789. Peregrine, D.H., 1972. Equations for water waves and the approximations behind them. In: R.E. Meyer (Editor), Waves on Beaches and Resulting Sediment Transport. Academic Press, New York, N.Y., pp. 95--121. Phillips, O.M., 1966. The Dynamics of the Upper Ocean. Cambridge University Press, 261 pp. Vager, B.J. and Kagan, B.A., 1971. Vertical structure and turbulent regime in a stratified boundary layer of a tidal flow. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 7: 766--777. Weatherly, G.L., 1975. A numerical study of time-dependent turbulent Ekman layers over horizontal and sloping bottoms. J. Phys. Oceanogr., 5: 288--299.
207
© Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
THE MODELLING OF THE APPROACH OF BORES TO A SHORELINE
B. JOHNS
Department of Meteorology, University of Reading, 2 Earley Gate, Whiteknights, Reading RG6 2A U (Great Britain) (Received March 9, 1979; revised version accepted October 2, 1979)
ABSTRACT
Johns, B., 1980. The modelling of the approach of bores to a shoreline. Coastal Eng., 3: 207--219. Turbulent processes beneath bores approaching a shoreline over a sloping beach are modelled using a turbulence energy closure scheme. The procedure incorporates the simulation of turbulence production at the face of each bore together with shear-generated turbulence in the bottom boundary layer at the beach. The effect of these two sources of turbulence is represented by production terms in the turbulent energy quation and the solution of this determines the value of exchange coefficients which are then used to calculate the Reynolds-averaged flow conditions. Numerical experiments are described that relate to the approach of both a single bore and a train of periodic bores to the shoreline. The bottom stress is calculated and it is shown that it is strongly influenced by turbulence production at the face of the bore.
INTRODUCTION
The approach of surface waves to a shoreline is characterized by propagation through three different regimes. During the initial phase, when the waves are on the surface of relatively deep water, the dynamical processes may be described in terms of well-established theory relating to sinusoidal nonbreaking waves. During the second phase, when the waves have moved into shallower water, their structure is increasingly influenced by the beach topography and ultimately they steepen and begin to break. Subsequently, the wave disturbances propagate through the surf zone in the form of a sequence of periodic bores (Phillips, 1966; p. 56). These are characterized by a steep-fronted turbulent face and a relatively smooth non-turbulent back. During this final phase beach processes, such as the generation of long-shore currents and bedload transport, are strongly dependent on the b o t t o m stresses associated with the incoming periodic bores. In particular, a theoretical treatment of long-shore current generation necessarily involves an appropriate parameterization of these b o t t o m stresses. In Longuet-Higgins (1970) and Bowen (1969), for example, the b o t t o m stress is represented by an empirical
208 relation involving the near-bottom orbital velocity induced by the periodic disturbances. However, the validity of this procedure for bore-like disturbances is not clear. In such circumstances, the b o t t o m stress is determined by the intensity of turbulence in the system supporting the Reynolds stresses. Moreover, in the surf zone, there are two turbulence generation mechanisms. Firstly, there is the contribution from shear-generated turbulence at the beach and secondly, the generation of turbulence at the face of the bore itself. This secondary source does not appear to have been included in any existing theoretical treatments and, in shallow water, the turbulence intensity and corresponding b o t t o m stress must be expected to be influenced by this process. In the present work, a model is developed for the approach of bores to a shoreline which is based upon the shallow water equations {Peregrine, 1972). Using the technique developed by Johns (1977, 1978), the turbulence processes are included by the application of an equation for the turbulence energy density. Source terms in this simulate turbulence production at the beach and the face of the bore. In the first numerical experiment, the b o t t o m stress is determined that results from a single bore-like disturbance. In the vicinity of the face of the bore, the calculations show that the distribution of turbulence energy is strongly influenced by the contribution from the face of the bore. A calculation of the corresponding b o t t o m stress indicates the importance of accounting for this source of turbulence production in any assessment of beach processes. This point is substantiated by determining the value of the friction coefficient in an empirical b o t t o m stress law so as to obtain the best overall representation of the b o t t o m stress. A comparison of the b o t t o m stress evaluated from this with a direct evaluation from the turbulence models reveals a gross underestimation by the former beneath the face of the bore. In the second experiment, a train of incoming periodic bore-like disturbances is considered. An evaluation is made of the mean b o t t o m stress beneath these periodic disturbances. On approaching the shoreline it is found that the mean b o t t o m stress changes from being on-shore to off-shore. It is suggested that this may result in the production of an off-shore region of bedload transport convergence with an associated sand-bar development. FORMULATION A Cartesian frame of reference is chosen in which the axis 0x is normal to the shoreline and which is fixed in the equilibrium level of the sea-surface. The origin 0 is located at the seaward extremity of the analysis area and 0z points vertically upwards. The equilibrium position of the shoreline is at x = L and the equilibrium depth of the water over a plane sloping beach is given by: h(x)
=
ho(1-x/L)
where ho is the undisturbed depth at x = O.
(1)
209 The Reynolds-averaged components of velocity ( u , w ) then satisfy: au -at
au
au
+ u-ax
+ w
-
a~
1 a
ax
+ - -pax
g
az
1 a rxx
+p
-az
rzx
(2)
where [ is the sea-surface elevation above its equilibrium value, the pressure has been assumed hydrostatic and r x x a n d r z x are components of the Reynolds stress. The equation of continuity is: au/ax
+ aw/az
(3)
= 0
which has the equivalent form (cf. Phillips, 1966, p.45): a~ a -- + -at ax
(
Jh
udz = 0
(4)
_
The Reynolds-averaged turbulent energy density E satisfies: aE --
at
aE + u--
ax
aE + w
az
-
rxx au
p
rzx au
+----+~-e ax p az
(5)
where the first two terms on the right-hand side represent the production of turbulence energy by transfer from the Reynolds averaged flow. They simulate the production of turbulence energy at the face of the bore and the production of turbulence energy by shearing in the b o t t o m layers. ¢ represents the re-distribution of turbulence energy by the eddy motion and e represents the dissipation of turbulence energy in the system. Until hypotheses are made about r x x , r z x , ¢ and e the only assumption in eqs. 2--5 is that of a hydrostatic balance. An absence of fluid slippage at the beach requires: u = w = 0
at z = - h ( x )
(6)
and the kinematical surface condition leads to: = 0
a~/at+ua~/ax-w
a t z = ~'(x,t)
(7)
A condition must also be satisfied by the Reynolds stress at the free surface. An absence of applied surface stress will be prescribed which is represented by the requirement: rzx
= 0
at z = ~ ( x , t )
(8)
This may n o t be an entirely satisfactory condition in the neighbourhood of a steep-fronted surface disturbance but a localized breakdown of eq. 8 is t h o u g h t unlikely to have an effect on the overall flow conditions. Assuming that the turbulent transfer of m o m e n t u m is diffusive in character, we write: rzx
= p K au/az
,
rxx
= p N au/ax
(9)
210
where K and N are vertical and horizontal exchange coefficients. Turbulence closure is achieved as in Johns (1978) and we take: K
(10)
= c 1/~ IE 1/~
where c = 0.08 and the length scale l is determined from: (E1/2 /l)
l -
, l = Kz0
as z - ~ - h
(11)
d-d-(El/,ll ) dz
The quantity K is von Karman's constant 0.4 and z0 is the roughness length of the b o t t o m elements. Additionally, the horizontal exchange coefficient is related to the turbulence energy density by: N
= a A x E 1/:
(12)
where AX is the grid increment in the final discretization scheme and ~ is an empirical factor which must be prescribed. Hence, the length scale associated with the horizontal mixing is aAx. It might be added that the inclusion of a horizontal mixing term is essential to maintain the computational stability of the numerical scheme of solution. Such an artifice is often incorporated implicitly by use of a Lax-Wendroff integration scheme to smooth out any developing discontinuities. A distinguishing feature of the present work is that the energy thus extracted from the mean flow is retained in a turbulence energy budget. Turbulence closure in eq. 5 is achieved by assuming the re-distribution of turbulence energy to be diffusive with the same exchange coefficients as for m o m e n t u m transfer. Accordingly: (P = -
3x
N -
Ox
+
~z
K -
(13)
~z
The dissipation e is parameterized as in Johns (1977, 1978) and we write: (14)
e = c3/~E~h/l
Eq. 5 then becomes:
aE - -
-+
~t
U
~x
+
+K_
:
~z
\ ~x ]
a "
\3z
1
÷ _
~x
_
÷
~x
-
~z
(15
_
l
An absence of turbulent energy transfer across the beach and free surface is then approximated by: OE/Oz
= 0
at z = - h and z =
In the absence of the horizontal production term, further details of the turbulence closure scheme, and its justification for steady flows, are given by Launder and Spalding (1972). Its application to oscillatory oceanic flows has been considered by Vager and Kagan {1971) and Weatherly (1975).
211 COORDINATE TRANSFORMATION
As in Johns (1978), it is convenient to introduce the coordinate transformation: (16)
o = (z + h ) / ( ~ + h )
so t h a t the beach and free surface correspond to o = 0 and o = 1 respectively. In comparison with Johns (1978), the presence of horizontal mixing terms complicates the form of the transformed equations in which x, o and t are taken as independent variables. In particular, eq.2 becomes:
ou --
ou
ou
Ox
ao
+u--+co--
Ot
=
~
+
-g~x
1 (o,xx p\
+
ax
o,x~ ~ ax
ao !
oz a,z,
+--
p
(17)
ao
where: =
(18)
ot+UOx+Waz
and the subscripts denote differentiations. However, the retention of the additional terms associated with the horizontal mixing appears hardly worthwhile with the assumptions made in the representation of the physical process. Accordingly, the transformed forms of eqs. 2, 4 and 5 lead to: au au au -- +u-+co-=-g
at
ax
aH
ao
ax
gho a /_au\ -- + --~N~x ~+
L
ax
1
a
H 2 ao
/__au \ ~K~--
(19)
ao
1
--
°[-s
+
at
ax
udo 0
]
=
(20)
0
and:
--
or
+u--
a~
+co--
ao
= H2
~aol
~s
+~
tNo~i
---H ~ ~o
~
-e
(21)
where H is the total depth ~ + h. Additionally, co is determined diagnostically from: t
Hw
=
°[- Su4 -° [-7 u o]
Oax
~X
0
0
The accompanying boundary conditions then yield: u=O
ate
= 0
co = 0
a t : o = 0 and o = 1
au/ao
= o
ate
aE/ao
= o
at o = 0 a n d
(23)
= 1 o = 1
212 NON-DIMENSIONALIZATION Non-dimensional variables are defined by: H
= hoI:I ,
u = (gho)V'fi,
x
= Lf,
g
= [ho2(gho)V3/L]k,
~ = [(gho) v'/Ll&
,
E = gh(~
(24)
t = [L/(gho)V']~ N
(25)
= L(gho)V'lV,
e =[(gho)~4/L]~,
l = ho[
(26)
This scaling then reduces eqs. 19--22 to: -: + ~ - - + & - - + - - + l 8t 8k 80
8-~- + - 82 --
--
o
= -8k
82
~do
+c5--
+
--
~
80
/~-80
(27)
= 0
(28)
-
/:/~ = o 8k 8 /~
~-
+
~da
+--
-82 - /~
o
--
--
fide
-~
+
(30)
0
Non-dimensionalization of the closure relations eqs. 10, 12 and 14 yields:
/~ = c l / ' / / ~ l / ' ( ~ o )
N = ~ , ~ 2 / ~ v' '
~ = c3/,(L~ '
\ho]
j~3/2_ 1
(31)
whilst eq.11 leads to: /'=
k
o\~!
+ EV'/tf
(32) 0
where: Zo = z o / h o
(33)
and Eb is the value of f: at a = 0. Apart from the parameters describing the wave input at the seaward boundary 2 = 0, the solution depends upon the parameter given in eq. 33 and a slope parameter ~ = h o / L . The solution is obtained by having ten unequally spaced computational levels in the vertical and by selecting a staggered grid having 99 points in the horizontal in which elevations are recorded at odd points and currents at even points. The technique is identical to that described by Johns (1978) and is n o t repeated here. A disturbance enters the analysis area at 2 = 0 at time t = 0 and propagates towards the shoreline. If, during the integration, the distur-
(29
213 bance does not reach the shoreline there is no need for a shoreline boundary condition. If the disturbance reaches the shoreline, we define } to be such that ~ = 1 + } gives the disturbed position of the shoreline and then apply:
/~=i+~ = 0 fido
}
=
~=1+~
0
Eqs. 34 are readily shown to be consistent with mass conservation and, in the numerical evaluation, are approximated for small } by: =
fido
+} ,~ =1
fido o
(35) ~ =1
and:
l~= 1
__
-}
xa~ ~=1
(36)
Eq. 35 provides a m e t h o d of updating ~ using values of fi extrapolated from interior current points. Using this value of }, eq. 36 yields an evaluation o f / ~ at the equilibrium position of the shoreline. More precisely, replacing (0H)~ ~-- =1 in eq. 36 by the one-sided difference approximation:
4~ [3it/x=1- 4/~"~=-1-2hx + /~.,~=1--4A;~]
O~ ]~=1
(37)
it follows that: 4A~ + 3} NUMERICAL
(38)
EXPERIMENTS
Two sets of numerical experiments have been performed. In the first of these, a single bore is simulated by prescribing that: f/~= o =
l l for t < 0 2 for t > 0
(39)
The slope factor/~ is 0.01 and the roughness parameter Zo = 0.001. The first integration was performed with ~ = 10.0 and this was continued until the effect of the disturbance was first felt at a distance of about L/4 from the shoreline. An interesting feature of the induced velocity distribution beneath the bore
214 is its uniformity with depth. The bulk of the shear is concentrated into an extremely thin layer adjacent to the beach as shown in Fig. 1 in which are given characteristic profiles of ft. In fact, on defining the depth-averaged value of a quantity of q~ by: 1
(40)
= f Cdo 0
it is found that u2/u 2 < 1.04, which is indicative of an almost total absence of vertical structure in the Reynolds-averagedflow field. The distribution of turbulence energy is given in Fig.2. That generated at the face of the bore spreads downwards (whilst decaying) to affect the entire depth of the flow and completely dominates the shear-produced turbulence adjacent to the beach. This must, of course, contribute to the bottom stress which will not therefore be expressible by a simple law. A frequently used parameterization of the bottom stress is given by: Tb --
=
Cf ~[al
(41)
P where Cf is a friction coefficient. This may be compared with:
P
=-h
and a value deduced for Cf. The calculations show Cf to be extremely variable h0
I
~
E SURFACE
z ho
0.0
~7--;
~7 -b0
.
0.25
.
.
.
.
.
L
0.5
-
~
0.75
Fig. I. Characteristic profiles of fi in the flow behind the face of the advancing bore. The superimposed scale represents a unit value of ft.
215 |.0
_z ho
65
-1.0 0.25
0.5
~:
0.75
Fig. 2. C o n t o u r s o f e q u a l v a l u e s o f 104~: b e n e a t h a s i n g l e b o r e : ~ = 1 0 . 0 .
but an optimized constant value may be defined by: L
f (~'b/P).lal Cf -
0
(43)
L
f
dx
~4 dx
0
This yields Cf = 5.56 × 10 -3 and the exact formulation may be compared with one based on eq. 41. This is shown in Fig. 3 from which it is apparent that the empirical law cannot reproduce the peaking of the b o t t o m stress beneath the face of the bore. This enhancement of the b o t t o m stress by turbulence production at the face of the bore is likely to be of considerable importance in any evaluation of beach processes such as the ability of waves in the surf zone to initiate bedload transport. In order to assess the dependence of these results on the value used for a, the integration has also been performed with a = 5.0. The results for the turbulence energy distribution are shown in Fig. 4 and an optimized value of the friction coefficient is given by Cf = 6.02 X 10 -3. The corresponding distribution of the b o t t o m stress is shown in Fig. 5 and, except at the face of the bore, the similarity between these results and those based on a = 10.0 is apparent. For a = 5.0, the turbulence energy density at the face of the bore is greater than that for a = 10.0 and it is clear that the associated production term is playing an important role in the turbulent energy budget. The length scale of the horizontal mixing process is now reduced and, consequently, the tur-
216
FREE SURFACE
6.0
-- ~ ' l ~ / ( g h ° ) x l O '
4.0
--
2.0 ~..C. ~
~
~
\
o
L
0.25
~
0.5
X
~c
0 75
Fig. 3. Distribution of bottom stress, Tb, beneath a single bore. Continuous curve is from eq.42. Broken curve is from eq.41 with Cf = 5.56 × 10-3: ~ = 10.0. L0
1~10 Z ho
-1°o.25
I00
200
75
z,g0
0.s
~
07s
Fig. 4. C o n t o u r s o f equal values o f 104E beneath a single bore: ~ = 5.0.
b u l e n c e e n e r g y g e n e r a t e d at the face o f the b o r e is c o n c e n t r a t e d into a smaller v o l u m e . T h e series o f e x p e r i m e n t s was c o n t i n u e d b y c o n t i n u o u s l y r e d u c i n g (~ = 2.5 leads t o Cf = 6 . 4 8 × 10-3). U l t i m a t e l y , o f course, t h e familiar t y p e o f s t e e p e n i n g develops t h a t is associated with the s o l u t i o n o f the shallow w a t e r e q u a t i o n s a n d the n u m e r i c a l s c h e m e breaks d o w n . In such c i r c u m s t a n c e s , it is suggested t h a t the r e d u c e d h o r i z o n t a l m i x i n g in the m o d e l is insufficient to
217
/(gho) xlO ~ 8.0
FREE SURFACE
8
/~
~.o
i
2.0 --
00.25
0-5
~c
0.75
Fig. 5. D i s t r i b u t i o n o f b o t t o m s t r e s s , r b , b e n e a t h a single b o r e . C o n t i n u o u s c u r v e is f r o m eq. 42. B r o k e n c u r v e is f r o m eq. 41 w i t h Cf = 6 . 0 2 X 10-3: ~ = 5.0.
simulate the sub-grid scale physical processes leading to turbulence production and a consequential drain of energy from the Reynolds-averaged flow. Clearly, the choice of a suitable value of a must depend, for example, o~n laboratory or field experiments. These might be based on the reproduction in the model of the observed decrease in the height of the bore as it approaches the shoreline. The present study, however, indicates the importance of retaining the energy loss from the bore in an overall turbulent energy budget. In the second set of experiments, an input of periodic bores is simulated by: 1 + 4A
/~e= 0
I(+;
---
=
1 for r < t < r , +
,
0
T
(44)
(b
This represents an incoming train of steep-fronted disturbances having a m a x i m u m height A above the equilibrium level of the free surface. Again, we take 13 = 0.01, z0 = 0.001, a = 10.0 and select A = 0.5. Additionally, r = 0.05 and r, = 0.5. With h0 = 2 m, this corresponds to waves with a period of approximately 11.3 s, the roughness length being 2 mm. The integration is started from an initial state of rest and is continued until an effectively oscillatory response is obtained. A diagnostic study may then be made of the final cycle of integration. As for the single bore, the friction coefficient in eq. 41 may be determined
218 by comparison with eq. 42. In this case, Cf varies bot h spatially and temporally. However, an optimized constant value may be defined by:
tp L o
Cf =
L f0tP f0 U4 dxdt
(45)
where tp is the dimensional wave period. This then yields Cf = 1.41 × 10 -2. Although the m a x i m u m input height of the periodic bores is only half t h a t of the single bore, Cf is markedly different in the two cases. For the periodic bores the friction coefficient is greater by a factor of more than 2.5 than that for the single bore. F o r effective practical application, Cf should be independ e n t of the t y p e o f wave m o t i o n generating the b o t t o m stress. The present model indicates th at this is n o t so. Clearly, in the case of periodic bores, the background o f residual turbulence persisting from one cycle to the n e x t will enhance the b o t t o m stress (with an effective increase in Cf) in comparison with th at beneath a bore advancing into turbulence-free water. It is informative to determine the mean b o t t o m stress beneath the waves during the final cycle of integration when the response is oscillatory. Provided that the instantaneous b o t t o m stresses are sufficient to produce bedload movement, the mean b o t t o m stress should indicate any t e n d e n c y to induce a mean bedload transport. The calculation shows there to be a continuous reduct i on in the value o f a shoreward directed mean b o t t o m stress (%) from the seaward limit at k = 0 to a poi nt a b o u t 10 grid increments from ~ = 1. At this position:
\ ghop/
1.23 X 10 -6
(46)
/
which, with h0 = 2 m, corresponds to a mean b o t t o m stress of about 0.24 dynes cm -2. This compares with an instantaneous b o t t o m stress at this position of a b o u t 9 dynes cm -2. Shoreward of this position, a reversal occurs in the sign o f
219 REFERENCES Bowen, A.J., 1969. The generation of longshore currents on a plane beach. J. Mar. Res., 27: 2 0 6 - 2 1 5 . Johns, B., 1977. Residual flow and boundary shear stress in the turbulent b o t t o m layer beneath waves. J. Phys. Oceanogr., 7: 733--738. Johns, B., 1978. The modelling of tidal flow in a channel using a turbulence energy closure scheme. J. Phys. Oceanogr., 8: 1042--1049. Launder, B.E. and Spalding, D.B., 1972. Mathematical Models of Turbulence. Academic Press, New York, N.Y., 169 pp. Longuet-Higgins, M.S., 1970. Longshore currents generated by obliquely incident sea waves, 1. J. Geophys. Res., 75: 6778--6789. Peregrine, D.H., 1972. Equations for water waves and the approximations behind them. In: R.E. Meyer (Editor), Waves on Beaches and Resulting Sediment Transport. Academic Press, New York, N.Y., pp. 95--121. Phillips, O.M., 1966. The Dynamics of the Upper Ocean. Cambridge University Press, 261 pp. Vager, B.J. and Kagan, B.A., 1971. Vertical structure and turbulent regime in a stratified boundary layer of a tidal flow. Izv. Akad. Nauk SSSR, Fiz. Atmos. Okeana, 7: 766--777. Weatherly, G.L., 1975. A numerical study of time-dependent turbulent Ekman layers over horizontal and sloping bottoms. J. Phys. Oceanogr., 5: 288--299.