,Journal of Coastal Research
198- 205
Royal Palm Beach. Florida
Winter 1999
A Cross-Shore Transport "Shape Function" for High
Energy Beaches
P.E. Russell and D.A. Huntley
Institute of Marine Studies
Univers ity of Plymouth
Drak e Circus
Plymouth PL4 8AA, U.K
ABSTRACT_• • • • • • • • • • • • • • • • • • • • • • • • • • • • • •_
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RUSSELL. P.E. a nd HUNTLEY. I).A.. 1!l!J9. A cross-shore tr ansport "sha pe function" for high energy beaches. Journ ol
of Coastal Research . l li( 11. 198-20fi. Royal Palm Beach r Florida r, lSSN 07 49-0208.
Field m ea suren~ en b of cro s s~ shore velocities were obtained, using electromagnetic current meters. from morphodyna mically reflective. intermediate a nd dissipative macrotidal beach sites. during high e ne rgy conditions. The dominant
velocity moment s outside th e surf zon e Were found to predict ( 11 onshore transport associated with the short wave
skewness. 121 onshore transport associated with short wave stirring. and transport hya weak mean onshore flo w. a nd
13 1 offshore tr ansport due to short wave stirr ing in long wave trou ghs. ln side the sur] zOlle th e dominant velocitv
moments predicted (11 offshore tr ansport cau sed by hoth short and long wave stirring a nd subsequent offshore tran sport by the undertow. 12 1 weak onshore tr ansport ass ociated with the short wave skewness. and l:lI at the dissipati ve
site. offshore tr ansport associate d Wi th th e long wave skewness.
By pl~ttin g the n ?rn~ ali sed velocity moment s for all the field sitps aga inst normalised depth. a second order polynomial sha pe function was produced for the tota l velocity moment . which predicts onshore tr ansport seawa rd of th e
surf zone decreasing in magnitud e toward s t he hreakpoint. and offshore tr ansport inside the sur f zone increasing in
magnitud e toward s th e shoreline. It is suggested that thi s sha pe function represent s a quasi-universal cross-shore
sediment t ran sport spatial distribut ion curve fill' high e ne rgy beaches,
ADDITIONAL INDEX WORDS: Surfeone. sed iment trn nsport, field studi es. cncruet ic» modelling, velocity moments.
INTRODUCTION
Over th e la st de cade the a u th ors h a ve undertaken field
mea surements on a variety of hi gh ene rgy beaches a rou nd
th e UK (RllSSELLet al.. 199 I : DAVI DSON e! al .. 199:3 I. Despite
th e variety of bea ches st udied a nd the vari ab ility of th e incide nt wave conditions , certain cons is te nt patterns are a pparent in th e data se ts . In parti cular, th e direction a nd rate
of cro ss-shore se d ime nt tran sport depends largely on th e
cross-s hore location of th e measu rem ents , for exa m ple.
whe t he r th e data were collected in side 0 1' ou t sid e th e s u rf
zone. Whil e th e cro ss-shore di st ribution of longshore se d iment transport ac ross th e ne arshore zone ha s been well st udied le.g. KOM AH, 1971 J, th e corresponding di stribution of
cro ss-shore se dime nt t ran sport is poorly und e rstood .
In thi s contribution . the relati ve importance of th e processes res pons ible for th e cross -s ho re sedime nt transp ort, a nd
th eir variation cross -s hore. a re in vesti gated using a n e ne rgeti cs a pproa ch iB AILARD, 1981 , 1987) whe re th e se d ime nt
transport r ate is expressed as a linear combination of te rm s
contain ing powers of the in stantaneou s velocity . Th ese t erm s
a re made up of th e various velocity 'mome nt s' s uch as th e
velocity vari ance , velocity s ke wness etc. Th ere hav e been se ve ra l pr evious studies inve stigating th e relationsh ip be t ween
velocity moments a nd sedime nt t rans por t in th e nearshore.
GUZA a nd THOHNTON (1985) applied Ball ard's mod el to
.970 10 received 15 Febru ary 1.9.97 a nd accepted ill recision 10 J uly
1.9.98.
data from t he ne ar plan ar beach a t Torrey Pines , California,
USA , and found th e wind (s hort) wav e transport com pone nt
to be maximum se aw a rd of th e s ur f zone, a nd th e s urf beat
(long wa ve or infragravit y wave) component to be maximum
a t the s horeline. Asymm etries in th e oscill atory wav e velo cit y
field tend ed to tran sport se d ime nt shorewards . whil e th e in teraction of th e offshore mean flow with waves produced an
offshore se dime n t flux . DOEI{I :-.I(; a nd BOW EN (19881 used a
hi-sp ectral a pproac h applied to various field data and found
th e s hort wav e s ke wnes s wa s di rected on shore. increa sing to
a maximum in th e br eaking region a nd th en decrea sing towards ze ro a t th e s hore line. As th e s hort wave s kew ness decr ea sed acro ss th e s ur f zone so th e total s ke wness became
dominated by offshore-directed long wa ve s kewness. ROELVI NK a nd STIVE (1989 1compared mod ell ed re sults with twodim en sion al laboratory mea surements on a barred profil e
a nd found s ig nifica nt transp ort contribut ions from t be shor t
wave velocity s ke wness , th e undertow, a nd the correl ation
between th e s hor t wave velocity variance a nd lon g wa ve velocity. T Ho lm To N et al . ( 1996 1 presented data from Duck,
North Carolina , USA, a nd found the dominant t ransport
mechanism outside the s u r f zon e to be on shore transport du e
to th e s hor t wav e velo city skewness . In sid e th e s u rf zone,
se d ime nt s us pe nde d by both th e short waves a nd th e (strong )
longshore current wa s s ubse q ue ntl y carried offshore by th e
und ertow.
FOOTE et al . ( 1994 ) a nalys ed data from a sing le morphodyn amicall y intermedi ate beach (S pu rn Head , UK) a nd found
199
Cro ss-S hore Tra ns port
Beach Pro file Uan ge nnith
20
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60
Beach Profi le Tei gnm out h
Beach Pro file Spurn Head
So
20
100
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Distance O ff shore ('0)
zo
100
Distan ce O ffshore (m)
Di stan ce Off sh o re ( m)
Figure L Locat ion of th e British Beach And Near shore Dyn a mics m -BANJ)) field sites a roun d the UK, wit h dept hs given in metres bel ow cha rt dat u m
(level of the lowest as tronom ical t ide ). For eac h s ite a typical beach profilc relat ive to a n a rhita ry dat u m is included, with th e inst rument pos it ion marked
I X).
that t he measured cross-sh ore velocity mom ents exhibited
consiste nt cross -shore 'shape functions' wh en plotted ag a inst
water depth , with dominan t cont ributi ons due to th e short
wa ves (directed ons hore ) a nd mean flows (di rected onshore
out sid e th e surf zone, a nd offshore in sid e th e surf zone). Th e
ter m 'sha pe fun ction' is used here to describ e th e cross-shore
dist ribution of th e cross-sh ore sedime nt t ran sport acro ss t he
nearsh ore zone. Ass uming th at longsh ore sedime nt tra nsport
gra die nts are small, converge nce s in th e cross-shore sediment
tran sport ra tes will lead to a ccre tion (ba r form ation ) and divergen ces will lead to erosion (t rough form ation ). As t he tide
advects th e sha pe funct ion over th e beach , so th e cha ra cte risti c bea ch profile is form ed .
Her e, data from three morphodynami call y different bea ch
sites (Lla nge nnith , Spurn Head a nd Teignmouth ; UK ) are
presen ted . By plott ing normali sed velocity mom ents from all
three bea ches against norm ali sed depth, consiste nt cros sshore tran sport sha pe functions are produ ced .
FIELD DATA
The field data wer e collecte d during th e Bri ti sh Bea ch And
Nea rs hore Dynamics (B-BAND) experiment (DA VID S O N et al.,
1993). Th e a im of this experiment wa s to collect and a na lys e
data from a variety of high energy, macrotid al sand bea ches
und er a ran ge of wave conditions. Three sites wer e used : a
fla t (gra dient = 0.017) dissipative beach at Llangennith on
t he west coas t of t he U K, an in ter med ia te (gra dient = 0.023)
beach a t Sp urn Head on th e east coast and a stee p (gra die nt
= 0.095) reflect ive bea ch a t Teign mouth on the south coast.
Th e loca tion of the site s and representative beach pr ofiles are
plot ted in Figure 1. All th e beaches are plan ar, un ifor m
alo ngsh ore and ha ve an a bsence of ba rs. The tidal regim e is
se mi-d iurn al a nd ma crotida l,
Th e dissipa tive beac h is a 10 k m long, h igh en er gy bea ch
with a sha llow conca ve bea ch pr ofile (gra dien t = 0.0140.020 ) consist ing offine to medium gra ined qu artz sa nds (D 50
= 0.21 rnm ). Th e gra in size sh ows little va riatio n eit her
a longs hore or cross-shore. The wa ve climate on this exposed
west facing beac h is a m ixture of high ene rgy Atl an tic swell
and locall y genera ted wind waves dr iven by th e prevailing
westerl y wind s. Th e low beach gra dient , high energy wav es
a nd la rge (u p to 9 m) tidal ran ge com bine to produce both
br oad surf (up to 350 m wide ) a nd inter ti da l (up to 500 m
wide ) zone s.
Th e in termediate bea ch site is located near th e en d of a 5
km long sand spit facing sout h-east into the North Sea. Th e
pr eva ilin g wave clima te is mild er th an that of th e dissip ative
beach , but th e coast is exposed to occas iona l viole nt stor m
wav es whi ch a pproach obliquely from th e north-east. Th e
beach profile cons ist s of a stee p h igh tid e bea ch (gra dient =
J ou rn al of Coasta l Resea rch , Vol. 15. No.1 , 1999
200
l{uss"l l a nd Hun t lev
0.097 5 ) comp ri sed of fine to medium gravels a nd a sha llow
sloping (gra d ient = 0.023 ) low t ide terrace cons ist ing of a len s
of well sorte d , me diu m quart z sa nds (Dc,,, = 0.:35 m m I. Strong
(up to 1 ms ' ) rectili ne a r t idal cu rrents (t ida l ran ge = 3 to 6
m l r un parall el t o the coast offsho re from th e field s ite flowing in a sout h -wester ly direct ion on t he flood an d a nort heaste rly directi on on th e ebb.
Th e reflect ive bea ch fa ces so uth-ea st in to th e English
Cha n nel a nd is con seque ntly shelt er ed from swe ll wav es ge nera te d in th e Atlant ic Ocean . Th e local wav e climate is domina te d by in frequent periods of wind-dri ven wa ves fro m th e
cas t. Th e upper beach profil e is ste ep (gra d ient up to 0 .110) ,
a nd th e lower profile seawa rds of th e neap low tid e level is
less steep (gradient = 0.057 ). Th e bea ch is composed of me diu m qu a rt z sand (l\ " = 0.24 m rn). Due to t he steep bea ch
pro file th e intertidal zone at th e reflect ive bea ch is compa ratively na rrow « 70 m ) in s pite of th e la rge (u p to 6 m l t idal
r a nge in th is a rea.
At eac h site , da t a collect ion ri gs consist ing of a rrays of electromagn et ic current m et ers (E MCM's) , opt ica l ba ckscatter
se ns ors (OBS's) an d a pressure t ra nsducer (PT) were deployed in a cr oss-shore t ra nsect across th e in terti da l zone a t
low tide. Th e heights of t he EMCM's used in t hi s a na lys is
we re set at 0.10 m above t h e bed . Th e da ta were rel ayed via
200 m lon g ca bles to th e to p of t h e bea ch where t he signa ls
were digit ised at 2 Hz in consecutive 17 min ute runs a s t he
t ide ro se a n d fell over th e instrumen t s, so a llowing a pr ofile
of mea surem en t s t o be mad e ac ross t he nearshore zone . For
th e present st udy , raw data wer e a na lyse d wh ich repr esented
hi gh ene rgy wave condi tio ns, a s follows:
(1) Llangenn itli (dissipati ve). Two t ides of data iLlan S f)
a n d Llan SN ) with st rong swe ll wav es were a n alysed . Signific ant br eaker heights (I-I,,) were 3.0 m a nd 2.5 m, resp ecti vely, for th e tw o tides, a nd all data were obta ined from well
within th e wid e surf zone. Th e tidal range on th ese days wa s
4.1 m , close to th e nea p part of t he cycle. Th e in struments
were position ed on th e upper part of t he pr ofile (gr a dient =
0.017 ) so that for eac h ti de data wa s collec te d for 1 to 2 h ours
eit her side of hi gh wa ter , cover ing water depths bet ween 0.4
m a nd 1.1 m . Mea sured longsh ore current s were wea k , < 0.15
ms " as wou ld be expe cted well wit hi n a wide di ssipati ve s ur f
zone.
(2) S pu rn Head (in termed ia te). Th ree t ides of data were a nal ysed . Th e first tid e (S p u rn 164pm ) consi st ed ofa stor m with
H, > 3 m, a nd data runs were obta ine d from well out side t he
s ur f zon e. Th e second t ide (S p u rn 184pm ) cons ist ed of swell
and local sea s wit h H, = 1.6 m , a nd da ta s pa nned t he nearsh ore zone . Th e third tide (S p urn 234pm ) consiste d of cle an
swell waves wit h H, = 0.7 to 1.0 m , a nd data s pa nned th e
n earsh ore zone . Th e t idal range va rie d fro m 6 m (spr ing tide )
for tide 164 pm down to 3.6 m (neap t ide ) for t ide 234 pm .
Th e in strumen t s were posit ioned on the sa n dy low tide terr ace (gradient = 0.023) a nd da t a were collec ted for betw een
1 a nd 9 hours each tid e coveri ng water depths up to 4.6 m.
Lon gsh ore curren t s pee ds were ge nerally < 0.4 m s ' .
(3 ) Teign rnoutli (reflecti ve). On e t ide of data tTeign 175 pm t
wa s a n a lysed with stee p, short period , swe ll wav es of H, =
1.0 m, and data s pa nned t h e nearsho r e zone. Waves of H, =
1.0 m do rep resent hi gh ene rgy condit ion s for this she lt ere d
bea ch environ me nt. Th e t idal ran ge on thi s day was 4 m a nd
t he in st ru me nts we re positioned on th e stee pe r upper part of
th e profile (grad ient = 0.095 ), Data was collected for j ust over
5 hours cover ing wa te r dep th s of u p to 2 m. Longsh ore cu rrent s pee ds rea ched 0.5 ms ' in t he middle of't he narrow s ur f
zone on the flood tid e. but wer e < 0.2 m s ' for th e rest of t he
t ide.
METHODS
An ene rget ics a pproac h (e.g . BAlLARD, 1981 , 1987 ) is followed where by th e se d ime nt tran sp ort ra te is assu me d to be
proportional to th e dissip ation of e ne rgy nea r th e bed , whi ch
can be ex pressed as a lin ea r com binat ion of term s conta in ing
powers of t he in stan taneou s velocity . Th ese Buila rd-ty pe
mod els ha ve been judged a mongs t t he 'bes t' cross-shore
t ra ns por t mod el s ( SC HO ON ~;ES a nd T HERON, 1995 ), and good
ex a m ples of their a pplica t ion a re giv en in GUZA a nd T HORNTON (1985), DOERING a nd BOWEN (1988 ), ROELVI NK an d
STIVE (1989), ROELVI NK a nd BROKER (1993) , NAIRN a nd
SOUTHGATE (1993 ) a nd THORNTON et a l, ( 1996 ).
In order to determin e th e roles of t he d ifferen t time-dependent sedime nt t ransp or t processes , t he instantaneou s nearbed cross-shore velocit ies were deco mposed into th eir mea n ,
sh ort (incident) wav e a nd long wav e com ponents, a s follows:
u
= u + Us -+- u.,
(1 1
wh ere u is th e t ot al cross-sho re velocity , u is t he mean flow
com ponent , Us is t he velocity associa te d with th e short (incident) waves, a nd u ., is th e velocity associa te d wit h t he long
period wa ves. Th e su m of Us a nd u ., gives th e oscilla to ry velocity compon en t u. Th e u da t a wa s dem eaned a nd detrended
to give u a nd u, Fro m exa m ining s ur face eleva tion s pectra
obtained a t th e differ ent sites , t he ii time-seri es was filte red
a t ~ 0. 0 5 Hz to give t im e-seri es of Us a nd U". Th e va r ious
vel ocity moment terms were t hen ca lcu la te d using t hese velocity data .
Most sediment t ra ns port models reduce to a u" dep endenc e
for high t ran sp ort rates ie.g. Dn : R, 1986 ). Many ot her workers h ave found t he (u") ter m (whe re th e ( ) sy mbol den ot es
t ime-av eraging ) to be t he cruc ia l velocity mome nt in det erm ining t he net se dime nt transp or t rate (RIBflERINK a nd ALSALEM, 1995; WILSON a nd SHAW, 1995 ). Simply tak ing t he
cube of equat ion (1) a nd t im e-a veragin g, gives te n velocity
moment terms, eac h of wh ich t hen represents th e sedime nt
t ransp ort du e to a differen t hydrodyn amic pr ocess:
Term 1:
u"
mea n velocity cubed
Term 2:
(us ")
ske wne ss of sh ort waves
Term 3:
(u . :')
ske wness of th e long wav es
Term 4:
3(u s ' )u
stirring by th e sho rt wa ves and t ransport by th e mean flow
Term 5:
3(u ,,' )u
sti rri ng by th e long wav es and transport by th e mean flow
Term 6:
6(u s u ,) u a three way corre la t ion expected to be
near zero
J ourn al of Coas ta l Resea rch, Vol. 15. No. 1, 1999
20 1
Cro ss-Shore Tran sport
8
'---------::
7
6
Q;
[II Teign 175pm I
o Spurn 234pm
8
lo s purn 184pm
o SEurn 1_~4pm
7
I
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::J
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6
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• L1an SN
I
• L1an SO
El Teign 175pm I
o Spurn 234pm I
~O Sp...'Jrn 184P~J
Q;
~
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E 4
E 4
I-
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...
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3
3
2
2
----, ~
~
'i3
-0 .2 Offshore
0 Onshore
0.2
0.4
0.6
08
-0.8
-0 .6
Normalis ed value
Term 7:
3(U L2U S )
correlation of long wave variance a nd
short wav e velocity
Term 8:
3(u s2u.)
cor relation of shor t wav e varianc e a nd
long wave velocit y
3(US)(12
Term 10: 3(Ur)(12
-0 .2
Onshore
0
0 .2
0.4
Normali sed value
Figu re 2 (a) Average va lues of nor mali sed velocity momen ts outside the
su rf zone. Posit ive va lues indi ca te onshore tra ns port. Th e defini tion of th e
diffe rent velocity moment term s is given in th e text.
Term 9:
Offshore
-0.4
time-average of oscillatory component
~ zero
tim e-average of oscillatory compone nt
~ zero
The se terms wer e normali sed by (U2)3/2 followin g BA l LARD
and INMAN (1981) a nd DOERI NG a nd BOWEN (198 7) who define th e norma lised velocity ske wness as ( U 3)/( U ~ ) 3/2. FOOTE et
al . (1994) exp la in th at this normali sation re su lt s in va lue s
that are relatively in sensitive to variations in incident wave
he igh t. For exa mple, at the intermedi at e (Spurn Head ) site ,
tide 164 pm had incid ent wav es twice as large as tide 184 pm
and values for (US3 ) in 4 m water depth differed by an ord er
of magnitude for t he un -normalised mom ents, but were simila r for the norm ali sed mom ents.
Th e normalised velocity mom ent terms were t hen plotted
against normalised depth (dept h/b reaker depth , h/h .}, wh ere
th e local depth wa s taken from the pr essure transdu cer record , a nd the br eaker depth wa s determined from the location
of ma xim um wav e heig ht a nd th e position of t he br eaker zone
not ed in t he field . Th ese plots gi ve th e charac te rist ic sha pe
functions whi ch describe th e cross-shore distribution of the
cross-sh ore sed iment transport across th e nearsh ore zone.
Us ing this ene rgetics a pproach limits th e a pplica tion of th e
results to cases where th e sa nd moves in the direction of the
st rongest velocity, no bedforms a re pr esent, a nd th ere is a n
instantaneous response of sediment to velocity throughout
th e water column. On high en ergy bea ches th e bed is usu ally
fiat, a nd th e high degree of turbulen ce ca uses rapid mixin g
of sus pended sediment throughout th e water column so th at
(b ) Ave rag e va lues of norm ali sed velocity momen ts inside the surf zone.
Positi ve va lues ind ica te ons hore tr a nsport. Th e definiti on of the di fferent
velocity mom ent t erm s is given in the text.
ti me lags in t he vert ical a re sma ll. Thus th e energetics a pproach is cons idere d re asonable in th ese env ironments .
RESU LTS
Th e avera ge normalised values of the eight non- zero velocity mom en t terms are plott ed for th e data runs outside the
surf zone (Figure 2a ) and inside the surf zone (F igu re 2b). In
F igure 2a, Teign 175 pm represents a n ave ra ge of all the 13
data runs (eac h ru n is 17 minutes ) collect ed outsid e th e s urf
zone on t ha t tid e, similarly, Spurn 234 pm represents 20 data
runs , Spurn 184 pm is 20 data runs a nd Spurn 164 pm is 4
data run s. In Figure 2b, Llan SN represents all th e 9 data
runs collecte d in side th e surf zone on t ha t tid e at the dissipative site, simil arly, Llan SD is 8 data runs a t th e dissipativ e site , Teign 175 pm is 7 data r uns a t th e reflective s ite ,
a nd Spurn 234 pm a nd Spurn 184 pm a re 12 and 3 data runs
respecti vely a t th e inter mediate site . Th e differential data
cover age is largely a result of t he different wav e conditions
on t he differ ent tid es at th e different sites. Th ere is some
variation in th e a vera ge value s plotted in Figure 2. For exa m ple, in Figure 2a (outside the surf zone ), t erm 2 for tide
Spurn 234 pm has an a verage valu e of 0.67 with a standa rd
deviation of 0.1 3. Nev erthele ss, even with this variation, th e
dominant terms insid e a nd out sid e th e surf zone a nd th e sign
(direction) of th ose terms a re not changed.
Despite th e ran ge of wav e a nd beach conditi ons represented, Figures 2a a nd 2b reveal a remarkabl e degr ee of consiste ncy in t he mom ent terms. For th e intermediate (Spurn) a nd
reflectiv e (Te ign l beaches th e magnitudes of th e terms generally ag ree to much better than a factor of two , both outsid e
a nd insid e t he surf zone . Even for th e dissip ative beach (Lla n l
there are simila rities , particularly in th e dominant terms.
Out side th e surf zone (Figure 2a ), th e three dominant terms
are terms 2, 4 a nd 8, with terms 2 and 4 being direct ed onshore (positive va lues ) a nd term 8 bein g directed offshore
Journal of Coast a l Resear ch , Vol. 15 , No. 1, 1999
202
Ru ssell a nd Huntl ey
(nega t ive values). Overall , the pr edi cted transport is direct ed
onshore. Term 2 is the short wav e skewness, with th e large
positive values indicating th e strongest velociti es a re occurring under th e shor t wave cre sts, as would be expecte d for
shoa ling wav es . Term 4 repres ents sedime n t s us pe nde d by
th e short wav es and transported onshore by th e weak onshore-d irec te d mean flow, and term 8 represents th e neg ative
cor re la t ion that would be expecte d between th e shor t wav e
variance and long wav e velocity, for bound long wav es se award of t he s urf zon e.
In side th e surf zone (F igu re 2b ), th e pattern is some wha t
different. Terms 4, 5 and 2 are dominant, with th e ove ra ll
transport being direct ed offshore. Terms 4 and 5 a re th e largest, representing sediment suspended by th e sho rt and long
wav es resp ectively , and transported offshore by th e mean
flow (u nde rtow ). Term 2 shows the short wav e velocit ies a re
st ill ske wed onshore, but with a much sma lle r magnitude
than ou ts ide th e su rf zone. The velocity mom ent terms at th e
dissipative si te sho w se vera l interesting features . Th e largest
term is term 5 (long wav e st ir r ing and transport by th e mean
flow) with addit iona l cont r ibut ions coming from term 4 (sho rt
wa ve st irr ing and transport by th e mean flow ), term 3 (long
wav e ske wness) and term 1 (mea n velocity cubed). Th ese res ults sho w th at well insid e th e surf zone it is th e cont r ibutions of lon g wa ves and mean flow (u nde r tow) that are most
important (a lso see RUSSELL, 1993 ). Term 8 a t th e dissipativ e
s ite becom es positive, as expected in a sa t u ra te d s u rf zone
where depth cont rols wav e height, so th e largest short wav es
will occur a t th e long wav e crests .
Shape Functions
In ord er to inv esti ga te th e distribution of th ese mom ents
acro ss th e nearshore, normali sed val ues of th e mom ents wer e
plotted aga ins t normali sed depth (local depth/break er depth )
to produce th e s ha pe function s. Positive values of th e nor mali sed moment values represent onshore tran sp ort a nd nega t ive va lues represent offsh ore transport. Values of (de pt h/
breaker depth ) < 1 a re insid e the surf zone , so th at 0 on thi s
a xis represents t he shore line and 1 represents th e 'breakpoin t' . Th e br eak er depth s used here are 3.85 m (Lla nS D) a nd
3.21 m (JJanSN ) for th e di ssip ative site, 3.85 m (Spurn 164
prn ), 1.67 m (Spu rn 184 pm i and 1.37 m (Spu rn 2:34 pm ) for
th e intermedi ate s it e a nd 1.71 m (Te ign 175 pm J for th e reflectiv e s ite .
Results for th e four most important mom ents (Term s 2, 4,
5 a nd 8 ) are shown in Figures 3a-d. It is acknowl ed ged that
th ere is a fair degr ee of sca tter in these Figures . Thi s may
be expecte d with field data collected a t different tim es at differ ent sites, though some of the scatter will be cau sed by th e
anal ysis technique whi ch requires a defin ition of th e breakpoint, wh ere, in reality, a breaker zone will exist.
Term 2 (short wave skeuiness ) (F igu re 3a) sho ws a cha rac t eri sti c positi ve (ons ho re l s ke wness , with a maximum just
se a wa rd of th e br eaker zone, gradually decrea sin g s horeward s. Term 4 (stirring by (he short waves and tran sport by
th e mean flow ) (F igu re 3b) shows a bal an ce between onshore
t ra ns port seawa rd of th e su rf zone a nd offsh ore tran sport ins ide th e su rf zone, indicating that the decrea se in th e s hort
wave variance s hore wa rds is balanced by an increa se in th e
st re ngt h of th e und ertow. Tern! .5 (long wave stirring and
tran sport by the mean flow ) (F igure 3c) is very s ma ll outsid e
th e surf zone, but increa ses st rong ly shore wa rds insid e th e
su rf zone, becoming th e dominant term in the inner su rf zone
at the dissipative s ite . Term 8 (correlation betw een short wave
varia nce and long wave velocity) (F igu re 3d ) shows th e expected negative correlation outside th e su rf zone , consist ent
wit h forced long wav e motion . As decoupling occurs shoreward of th e br eakpoint, th e negative correlation reduces in
magnitude and becom es positive in th e inner surf zone, where
th e largest short wav es ca n exist at the long wave cre sts.
Summing a ll eigh t terms gives a total shape function (Figure 4 J, that is characterised by a second order polynomial fit
with the normali sed depth . Th at is, the main factor a ffect ing
the cros s-shore transport rates a nd directions is th e crossshore location relati ve to th e br eakpoint. The mathematical
form of thi s sha pe fun ction em pir ica lly fitt ed to th es e data is:
(u")
- =
(u" )"'"
(h)"+ (h)
- 0.52 h.,
2.27 - 1.58
h.,
lR" = 0.85 ) (21
Th e sha pe function pr edicts onshore tran sport outside th e
surf zone which de cr ea ses in magnitude tow ards th e br eakpoint, a nd offshore transport insid e th e s urf zone whi ch in creases in magnitude s hore wa rds . For exa mple, during
st or ms the su rf zone will be wide and offshore transport will
dominate, whil st during ca lme r conditions the surf zone will
be narrow and onshore transport will dominate the nearsh or e
zone . The convergen ce of th e tran sport directions a rou nd th e
breaker zone would favour th e form ation of a 'break point bar'
(SALLENCEH and H OWl) , 1989 ), but a ny s uch bars were obscured on the mea sured bea ch profil es a t these sites du e t o
th e effects oftidal s moot h ing. Th e form of th e sha pe function
in Figure 4 ma y be comp ared with th at used in a model by
DALLY and DEAN (1984) which pr edicts onshore transport
se a wa rd of th e break er' zone a nd st rong offshore transport
just shorewa rd of th e br eak er zone. which then decreases
shore wa rds so th at t he transport a pproac hes zero a t th e
s hore line .
DISCUSSION
The suggest ion, reveal ed by th ese data , that there are consistent cross-shore patterns of velocit y mom ents, dep endent
only on the normali sed mean water depth and ind ep endent
of beach slop e. is intrigu ing . In particul ar it impli es a surpri sing lack of dep enden ce on th e wav e br eaker-type. which
wa s spilling for th e dissipative bea ch. a nd spilling/plu nging
for the intermedi ate and reflect ive s ites. Th ere are indication s of some differenc es not abl y th e much larger lon g wave
terms (t erms :3 a nd 5, Figure 2b J in th e su rf zone of th e diss ipa t ive beach. as might be expected, but s uch differen ces
appear t o be relatively unimpor tant compa red to th e simila riti es of the larger mom ent terms .
It should be emphasise d th at th e s ha pe fun cti ons dedu ced
here represent high energy cond itions wh en th e beaches are
lik ely to be continually out of equilibr iu m with the incoming
wav es. In low en ergy condit ions, sma ll wave s ma y be expected to cause a ccret ion a long th e shore line . and the develop-
.Juu rna l of" Coa st ul R,'sl'arl"h . Vol. J s, No. J.
199~)
203
Cro ss-Shore T ransport
o
0.9
o
o
o
o
o
0.7
~ 0.6
.~ 0.5
CJ
L1an
q,o
o
- --
o
Onshore
>
n
o
o
o
o
0.2
Onshore
o
Offshore
Q)
o
o
z
o
-0.6
o
3
-0.8
2
-1
y = -0 .08x + 0.12i' + 0.38x
R =0.7 1
O -t-""""=-----t----t----t--- -- + - - - --\--- ---l
o
0.5
1
1.5
2
2.5
3
o
o
o
..
JoSPu
rJ
I
y = -0.38x 3 + 1.41i' - 0.95x - 0.21
CJ :Ian
l~e i g~
R2 =0.7 3
o
-1.2
o
1.5
0.5
(Depth I Breaker depth )
2.5
2
3
(Depth I Brea ker depth )
0.2
0.1
Onsho re
0.1
Offshore
o
0
Onshore
o
o
-0.2
:::J
00
o
~ -0.3
<ii -0.1
>
~
n
o
n
<ii
..
..
<ii
o
-0.6
y = 0.09 x
3
E
-
o -0 3
0.59 i' + 1.22x - 0 .78
R2 =0 .81
z
· spurn ]
L1an
I~ Te
i9.n :
-0.7
-0 .8
-0.9
o
0.5
1.5
2
2.5
3
.
o Offsh ore
..
00
o
s
o
e
o
-0.4
0
CJ
o
Oq.
e
.~ -0 2
.~ -0 .4
E -0.5
o
o
Q)
:::J
o
o
o
.!!? -0.2
<ii
E -0.4
o
0.1
z
- - -- - - - - - ,
o
:::J
0.2
Q)
-
o
CJ
-0 .1
- --
0.4
<ii
0.3
o
-
0.6
Q)
<ii
E 0.4
z
0.8 ,--- --
,
o
:::J
n
Spurn I
~e ig n
0.8
Q)
o
-0. 5
-06
r
o
O sp~
CJ
L1an
..T~gn
y = -0.05x 3 + 0.26x 2 - 0.47x + 0.11
I
o
l
0.5
1
R2=0.53
1.5
2
2.5
3
(Depth I Brea ker depth )
(Depth I Brea ker depth)
Figure 3 (a, top left ) Sha pe fun ction : norm ali sed term 2 (short wave skew ness) ag a inst norm alised depth . (b, top righ t ! Shap e fun ct ion: nor ma lised term
4 (short wave stirring and transport by th e mean flow ). (c, bottom left) Sh a pe functi on: norm alised term .5 (fang wave sti rring and tran sp ort by th e mean
(low ). (d, botto m right ) Shap e fu nction : norm a lised term 8 (short wave stirring and tran sport by th e long wave oelocityv.
Posi t ive values of the norm al ised moment val ues indica te onsh ore tra nsport. Va lue s of (dept h/b reaker dept h ) < 1 a re inside t he sur f zone.
ment of ripp led beds offs hore would provid e a n offsh ore
tran spor t mechanism outs ide t he surf zone. In add ition , all
three field sites described here hav e simple linea r pr ofiles
a nd a n a bse nce of bars. THORNTON et aZ. (996) pr esented
data from Duck, Nort h Ca rolina , USA, over a ten day peri od
whe n t he bathym etry evolved from a th ree-di men sion al terra ce to a well developed lin ear bar. Despite th ese complexities th eir resul ts br oadl y agree with th ose pr esented here in
th at outsi de the surf zone th ey found t he dom inant tran sport
mechan ism to be ons hore transport du e to t he s hor t wa ve
velocity skewness (our term 2) a nd inside the surf zone th ey
found t he t ra ns port to be directed offsh ore, parti a lly du e to
sediments mobilised by inci de nt s hort waves a nd tran sp ort ed
offsh ore by th e undertow (our term 4). Their data was collected a t a ti me whe n longshore current spee ds were exce ptiona lly st rong (1.5 ms - 1) so that sedime nt mobilised by th is
longshore cur re nt and t ra ns por te d offshore by th e mean undertow provided an other imp ortan t offshore tran sp ort mech-
a nis m ins ide t he surf zone for th eir data . Th e longsh ore currents mea sured at th e field site s pr esen ted here were mu ch
weak er «<0.5 ms - 1) a nd, for simplicity, longsh ore processes
ha ve been neglected from our a na lysis. T HORNTON et al.
( 1996) a lso found th a t th eir energetics model perform ed bet ter in stor m conditions th a n in mild wa ve conditions. Th e
broad agreeme nt in th e domin an t mecha nisms for cross-sho re
t ra nsp ort betw ee n th e results presen ted here a nd th ose pr esen te d from differen t sites by G U ZA and THORNTON ( 985 ),
DOEHING a nd BOWEN (198 8), ROELVI NK an d STIVE (1989)
a nd THORNTON et aZ. (1996) is encouraging a nd suggests th at
th is sha pe function a pproac h may be widely a pplicable.
Th e sha pe function is potentiall y of consi derable va lue for
sim plifyi ng t he modelling of sa nd transpo rt on bea ches a nd
th e evolution of bea ch profiles du e to cross-shore t ra ns port .
For exa mple, FOOTE et aZ. (994) ha ve devised a conce pt ual
model of th e developm ent of macrotidal bea ch pr ofiles by advecting a tra ns port sha pe function of t hi s kind across an in i-
-Iourna l of Coa sta l Research, Vol. 15, No. 1, 1999
Russell an d Hun tley
204
1.5
0
0
0
0
Ql
0.5
0
0
0
0
0
00
o
0
0
'1.
<f~ 0
0
n
Ql
.!!?
ro
E
<P
Onshore
0
"0
"
0
g
-0.5
Offshore
"
(;
z
0
~
::J
ro>
0
"
-1
"
-1.5
Y = -0.52,(' + 2.27x - 1.58
_"J eign
-2
0
0.5
1.5
I ~ spurn
c L1an
R2=0 .85
2
2.5
3
(Depth I Breaker depth)
Figure 4. Shape function: normalised total velocity moment term aga inst
normalised depth . Posit ive values of the normalised moment values indicate onshore tra nsport. Values of (depth/breaker depth ) < I are inside
the surf zone.
ti ally pla ne bea ch pr ofile. The resu lt s show simila rities to
real pro files, a nd further work is in progr ess to ma ke quan titative predicti ons, (FISHER a nd O'HARE, 1997 l. Further
measur ements will obviously be valua ble to pr ovide con firma tio n of t hese qu asi-un iversal sha pe function s, particul a rly
for dissip ative beac hes where t he pr esen t da ta set is very
lim ited in depth ra nge (0.7 rn). Da ta from th e innermost surf
zone a nd swas h zone is a lso vita l so t hat th e sha pe fu nction
ca n be exte nde d to th e shore line.
It must al so be recogni sed t ha t th e link betw een the velocit y mo ment s a nd actual sa nd tran sport needs to be better
tested in t he field. In part icul a r , th e effects of varia ble gra in
size s, a nd va ry ing bedfor ms, a re not incl uded in t he sha pe
fun ct ion . Compa risons between th e va rious cross-shore velocity moment s a nd t he mea sured cross-shore se dime nt tran spor t rates for t he se data ca n be foun d in RUSSELL et al.
(1996 ). The se compa risons were some wha t inconclusive as
the measured se dim ent transport rates were only obt ained at
discrete points above t he bed a nd were a ffecte d by cha nges
in t he bed rough ness (fla t bed/ rippled ). Thi s resul ted in occasi ona l poor agreem ent between the velocity mom ent pr edictions a nd the me asured tran sports for t he shor t wav e componen t due to 'reverse' t ra nsport over ripp les (DAVIDSON et
al., 1993), as th is type of tra nspo rt ca nnot be infe rred from
energetic s expressions. However t here was a t least qua lit ative ag ree me nt between velocity mom ent pr edictions a nd
measured tra nsports assoc ia te d wit h the mea n a nd long wave
components.
CONCLUSIONS
1. Da ta from a range of high energy ma crotidal beach sites
show t hat velocit y mom en ts pr ed ict net onshore transport
outside the surf zone, due to: ( i) ons hore-di rected short wave
skewness, (ii) onshore tran spo rt associated wit h short wa ve
stirring , a nd t ra ns por t by a weak onshore mean flow, a nd (iii )
offshore tra nsport asso ciated with short wave stir ring a nd
offsho re transpor t by long wa ve velociti es. Velocity momen ts
pr ed ict a ne t offshore transport ins ide the surf zone, du e to:
( i ) offshore tra nsport ca used by both short a nd long wave stirring a nd su bsequent offsh ore t ransport by the und ertow, (ii)
wea k ons hore t ra nspor t associa te d wit h t he short wave skewness, a nd (iii) at th e dissip at ive site, offsh ore tra ns port associate d wit h t he long wav e ske wness .
2. By plot ting the norma lised velocity momen ts from all th e
field sites agains t depth/b reak er depth , a univer sal second
order polynomial 'sha pe fun ction' is produced for th e total
velocity moment, whi ch pred icts onshor e t ran sport seaward
of t he surf zone decreasin g in magni tude towa rd s th e breakpoint , a nd offs hore t ra nspo rt insi de t he surf zone increasing
in mag ni tude towards th e shore line . It is sugges te d that t his
sha pe function repre se nts a qu asi-un iversal cross-s hore
t ra nsp ort spa tia l distr ibu ti on curve for high ene rgy beaches.
ACKNOWLEDGEMENTS
Th e a uthors would like to tha nk t he Ocean Science division's
computing research as sista nt, Pauline Fram ingham , for her
pa tience and help with ma ny aspects of th e work. The assista nce of Yolan da Foote wit h some of th e initial da ta processin g
dur ing her post-doctoral gra nt NE RC GR9/1484 is also gr at efully a cknowledged. Th e help ful comment s of Dr. Mar k Davidson markedly improved the data pr esen ta tion. Tha nk -you also
to Darren Stevens for ass isting with Figure 1.
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