JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 87, NO. C1, PAGES 457-468, JANUARY
20, 1982
Bars, Bumps, and Holes' Models for the Generation
of Complex Beach Topography
R. A. HOLMAN AND A. J. BOWEN1
School of Oceanography, Oregon State University, Corvallis, Oregon 97331
In shallowwater, any two wavesof the samefrequencyare shownto producecomplexpatternsof
drift velocityabovethe seabed. If the longshorecomponentshi, h: of the wave numbersof the two
waves are different, these steady flow patterns exhibit a longshoreperiodicity of wave number (hi --
h:) irrespectiveof whetherthe wavespropagatein the samedirection(say hi, he positive)or in
oppositedirections'(he negative). The interaction of two edge wave modes is examined in detail. The
drift velocities are Calculatedand a simple sediment transport model is used to predict the beach
topographythat would be in equilibrium with these flow patterns. As expected, crescentic sand bars
are producedby the specialcaseof standingedgewaves (hi = --h:). Intriguingly, for all other casesa
combinationof complex transversebars plus meanderingor straightoffshorebars result, patterns that
are surprisinglyreminiscentof many publisheddescriptionsof complex, rhythmic topography.The
extensionof the model to three or more waves producestopographythat appearsto be very irregular.
Although the pattern shouldrepeat over sufficientlylong distancesalong the beach, even with only
three waves these distancesmay be very large comparedto the scale of the bars.
a standingwave is regardedas two equal waves propagating
in opposite directions, these longshore periodicities arise
from the interaction of two modes of the same frequency; a
singleprogressiveedge wave will generateonly a linear bar
system, the number of bars dependingon the modal number
of the wave [Bowen, 1980]. In the existingliterature, interest
has largely been concentratedon the rather special case of
but,surprisingly
often,thereisa strongsuggestion
Ofpattern standing edge waves, but any two modes of the same
frequency produce flow patterns or drift velocity patterns
in the offshore bar structure or the shoreline form. Of
particular interest have been shorelinefeatures that show a that have a regular longshoreperiodicity. For example, rip
regular longshoreperiodicity down the beach, variously currents may be generated by a single, normally incident
describedas beach cusps, sand waves, rhythmic topogra- wave plus a progressiveedge wave of the same frequency
[Bowen and lnrnan, 1969].
phy, or giant cusps[Hom-rnaand Sonu, 1963;Dolan et al.,
While we know that such flow patterns must repeat
1974; Komar, 1981].
periodically
alongshore,in the general case we do not know
The purposeof the presentpaperis to showhow a variety
the
actual
shape
of the velocity field within each wavelength
of regular longshorepatterns may be generatedby two
of
the
pattern.
For standing waves the pattern must be
waves of the samefrequencyand then, simplyby extending
these ideas to three waves, showhow a simpledeterministic symmetricalwithout a preferred longshoredirection but for
model produces topography that appears very irregular. two differentmodes,particularlytwo modespropagatingin
While the basic ideas of the analysis apply to any type of the same direction, some longshoreskewnessin the pattern
waves, interest will be focusedprimarily on edgewaves, the seems inevitable. The interesting question is whether the
wave motionstrapped to the shorelineon a slopingbeach. formation of a large-scale,quasi-periodicbar systemsuchas
Edge waves appear to play an important role in the that illustrated in Figure 1 can be explained in terms of a
INTRODUCTION
While the ideal beach may be conceived as a smooth
expanseof sand stretchingindefinitelyalongshore,and real
beaches do exist that approximatethis idealization, many
beachesare topographicallycomplexexhibitinga variety of
longshoreand offshorestructureat variouslengthscales.At
times the resultingmorphologyseemscompletelyirregular,
generationof severaltypes of beachmorphology.One of the
attractive featuresof edgewave modelsis the explanationof
the often observed, regular, longshoreperiodicitiesin terms
of length scales, the edge wave wavelengths,which have
well defined properties.This leadsto quantitativehypotheses that have had some successin describingthe formation
of beach cusps [Guza and lnrnan, 1975; Sallenger, 1979],
large cuspate topography [Dolan et al., 1979], crescentic
sandbars [Bowen and lnrnan, 19711,and topographyrelated
to regularly spacedrip currents [Bowen and.lnman, 1969;
Kornar, 1971].In lookingat theseexamplesit is clear that, if
simpletwo wave model. Existingexplanationsinvolve off-
shore bars, cut by rip currents, rotating around to attach to
the shoreline [Sonu, 1973]. It would be much more satisfactory to suggestthat the mechanicsof the formation of these
bars is exactly analogousto the formation of crescenticbars
but involves waves of different modal numbers. A principal
objectiveof thispaperis thereforeto lookin detailat a whole
classof possibleinteractionsbetween two waves of the same
frequency.
In the first sectionof the paper we examinethe general
conditions under which waves may perturb the bottom
topography, distinguishingthose processesthat may produce periodic longshore features from those that cannot.
Interest is then concentrated on a particular class, the
1 Permanentaddress:Departmentof Oceanography,
Dalhousie interaction between edge waves, and the drift velocity
University, Halifax, Nova Scotia, Canada.
patternassociated
with the occurrenceof anytwo edgewave
modes is derived. This defines a steady flow just above the
bottom boundary layer. A simple sedimenttransport model
Copyright ¸ 1982by the American GeophysicalUnion.
Paper number 1C1632.
0148-0227/82/001 C- 1632501.00
457
458
HOLMAN
ANDBOWEN'BARS,BUMPS,
ANDHOLES
is thenusedto estimate
thenewtopography
thatwouldbein Ursell [1952]
equilibriumwith thesetheoreticaldrift patterns.The releIXlsin(2n + 1)/3= o2/g
(la)
vanceof the modelis checkedby lookingat theformationof
crescentic
barsby standing
edgewavesfollowing
Bowenand anda setof incidentor reflected
waves,leakymodes,for
Inman [ 1971]. Results are then derived for the cooccurrence
which
of two differentmodespropagating
in the samedirectionand
in oppositedirections.Finally,theseideasare extendedto
consider
thedriftvelocities
occuring
whenthreeedgewave
modesare simultaneously
present.
sina -< oa/g
(lb)
Fora givenfrequency
tr,thereisa finitesetof edgewavesof
different
modalnumbers
n (wheren = 0, 1,2, 3 ßßßprovided
Thefreewavesona planebeachof slope/3consist
of a set (2n + 1)/3< ½r/2)
anda continuum
of leakymodes
forwhich
of edgewaveswhoselongshore
wavenumber
h is givenby X, in simplegeometries
suchasa planebeach,is determined
WAVES, CURRENTS,AND TOPOGRAPHY
Fig.1. Welded
sand
bars
onCape
Cod,
Massachusetts.
Longshore
spacing
isseveral
hundred
meters
(Photo
courtesy
of David S. Aubrey, WHOI).
HOLMAN AND BOWEN' BARS, BUMPS, AND HOLES
by the angle of approachin deep water, a. The modulusis
necessaryas the signof X indicatesthe direction of longshore
propagation.
The wide range of longshorewave numbersgiven by (la)
and (lb) obviously suggeststhat the cooccurrenceof two or
more waves with different X but the same frequency is the
normal situation. For example, the assumptionof angular
spreadof energy in the incident spectrumleads, from (lb), to
a correspondingrange of X for the incident waves. Interactions between two waves of the same frequency are therefore likely to be commonplace.However, for the interaction
to lead to steady patterns in time or space requires some
phase locking between the waves. Such phase locking is
implicit in the explanationof rip currentsas beinggenerated
by the interaction between an edge wave and an incident
wave [Bowen lnrnan, 1969] or two incident waves from
opposite directions [Dalryrnple, 1975] or, in fact, any two
incident waves of the same frequency from different directions. Such phase locking does not have to be perfect, but
the phases must not drift totally randomly (as is often
assumedin spectral wave theory). If the phasesare entirely
random, each position alongshoreeventually experiencesall
the possible combinationsof the two modes and the longshore variability disappearsfrom the time-averagedconditions. However, this argumentleads to the rather encouraging conclusionthat, in so far as the phasesare not entirely
random, contributionsto longshorevariability will be possible; the phase locking does not have to be perfect. In the
subsequent discussion, attention will be concentrated on
only the coherent part of the two wave motions. This should
bring out the essentialphysicsof the problem and avoid the
complication of carrying a set of unknown time integrals of
the phasesthroughoutthe calculations.
There are at least two important types of interaction
between two waves that may have significantmorphological
consequences. The first is the generation of nearshore
currents and rip currents that in turn become the primary
mechanismfor rearrangingthe sediment[Bowenand lnman,
1969; Komar, 1971]. Here cross terms between the two
waves are found in the radiation stresses;the breaking of the
incident waves forces circulationpatterns. A secondinteraction arises in the calculationof the mass transport velocity
above the seabed.Again crossproductsoccur leadingto the
possibility of longshorevariability. Drift velocities associated with a single wave, either an incident wave reflected at
the shoreline or an edge wave, produce patterns that are
uniform in the longshoredirection, possiblyforming linear
bars [Suhayda, 1974; Short, 1975; Bowen, 1980] but not
rhythmic topography.
In some cases, particularly the formation of beach cusps,
the wave processes have been considered in increasingly
usesthe drift velocity as the primary input to the sedimentary dynamics. Conditions very close to the shore will not be
considered. In principle, the model is applicable to conditions outside a small surf zone and should apply to large
scale structures.In practice, the model is not very sensitive
to the preciseform of the incidentwave field and may apply
qualitatively to conditions inside a wide surf zone. Here,
however, the sedimenttransportdue to drift velocitiesmust
compete with that associatedwith the currents driven by
wave-breaking. The relative importance of these two processesin different environments is far from clear. However,
the fairly common occurrence of crescentic bars of the
general shapepredictedby Bowen and lnrnan [1971] suggest
that drift velocitiesfrequently play a significantrole.
DRIFT VELOCITIES
In shallow water, close to the shore, the standard linear
shallowwater equations[Stoker, 1957]provide a reasonable
approximationfor the wave motion. For waves of frequency
or,the velocity potential • is given by
(h•x)x + (h(I)y)y+
= 0
(2)
where x is positive seaward from the shoreline, y is the
longshorecoordinate, and h the water depth. On a plane
beach of slope /•, Eckart (1951) showedthat solutionsfor
edge waves propagatingin the y direction are of the form
(I)n --
ang
(•n(X)exp [-i(X•
- o't)]
(3)
possible values of the longshorewave number being determined by the dispersionrelation
o2 = glXnl(2n + 1)/3
(4)
the shallow water approximation to the exact dispersion
relation (equation(la)). On the low slopestypical of natural
beachesthe approximationis fairly good provided n is not
too large.
The offshorestructureof the modesis given by
•gn(X)-' Ln (2XnX)exp (--XnX)
(5)
where Ln is the Laguerre Polynomial of order n and n is the
numberof zero crossingsof •x), so highermodesare more
complex and extend further seaward. Figure 2 shows •x)
for the four lowest modes as a function of the nondimen-
I.O -0.8
1981], but the actual sediment transport mechanism is far
from clear. Bowen and lnman [1971] show that the cuspate
0.6
0.4
shoreline features that sometimes match offshore crescentic
0.2
0
-0.2
- 0.4
- 0.6
rig. •.
In the present paper we will concentrateon a model that
ABOVE THE BOTTOM BOUNDARY
LAYER
elaborate models [Guza and lnman, 1975; Guza and Bowen,
bars might be explained in terms of the drift velocities
associatedwith a standingedge wave. However, they recognized that the concept of drift velocity relative to some
boundary layer must necessarily become meaninglessin
very shallow water. It seemslikely that the differential runup alongshore may be the most significant factor at the
459
uU•,lU, C ucpclmCn•c, Ln
wave modes.
, •.c +t... 1...... + four
460
HOLMANAND BOWEN:BARS,BUMPS,AND HOLES
sionaldistance• givenby
0=M
• =
g•
= X0x
Ox
1[ •OV* OU*
4•
V
(3 +5i)+
V•(2+3i)
(6)
]
+ U•
(1 + 2/)
(9)
Ox
where X0is the wave numberof the lowestmode,n = 0.
Now it is quite possiblefor two or moremodesof the same
wherethe asteriskindicatesthe complexconjugate.If the
frequencyto occur together. For two modes,the surface
wavefieldconsists
of two modesasin (7), then,denoting
the
elevation,takingthe real part of (3), wouldbe of the form
drift velocitiesassociated
with the edgewavesas u•, o•, (9)
can be expressedas
1 c9•
g Ot
= amr•m(X)sin (h•v -- at)
+ arC•r(X)sin(hry- at + O)
where 0 is the phaseanglebetweenthe waves. As can be
seenin Figure2 therearesubstantial
regions
in which•bm(X)
is approximately
equalto •br(X)particularlynearthe shoreline. Here (7) could be rewritten as
2
Y-at+ •)
'sin(
(hm
+xr)
2
4a 3
2
• = (am- ar)•X)sin (kroy- at) + 2arqb(x)
ß cos
(2m
+ 1)2&m•m'
g2•ø3{am213qbm'q
1
]
+ar2
[3&r'&r"
-•(2r+I1)2rkrrkr'
]
(2r+ 1)2c•rn'C•r
+amar[COSlX(3r•m'r•r"
3rkm"rkr'(2rn
+ l)2C•mC•r'+
(2m
(2r1
+ ' (o"or'
+ qbm'qbr)
1+ 1)'1-••
)
+sin
•x
(5•9
m'
•)r"
--5•)rn"l•r'
--(2r+ 1)2C•rn'C•r
Hi --
(7)
2
y -
(8)
The wave motionappearsas simpleprogressive
componentproportionalto am- ar plusa complicated
progressive
componentwith wave numberequalto the averageof the
two basicwaves,stronglymodulatedin the longshore
direc-
3
3
2
(2rn
+ 1)2l•)ml•)rt+
(2rn
+ 1)(2r
+ 1)
tion at half the differencein wave numbers.This modulated
(10)
componentwill havenodesin elevationat regularlongshore
intervals,provided 0 is constantßThis illustratesan interest-
where•b'represents
thederivativeof •bwithrespectto • and
ing sampling
problem.Fieldmeasurements
of spectralener/x describesthe longshoredependence
of the solutionwhere
gy will showa longshoremodulation,suggestive
of a standingwavecomponent
of wavenumber[Xm- Xr]/2.However,
/J•= -- (Xm -- Xr)y + 0
(11)
crossspectrawill show the wave form to be progressiveß
Combinations
of threeor moremodesproduceverycompliThetermsin (10)proportional
to am2andar2represent
the
cated spatial patterns that can only be resolvedby an
self interactionof the two wavesand have no longshore
extensivelongshorearray of instruments[Huntley et al.,
variation.The termin amar,whichmayproduce
interesting
1981].
1ongshore•
variabilityin the drift velocities,arisesfrom the,
So far we have consideredonly a linear combinationof
crossinteractionof the two modes.The wavelength
of the
wave modes. This indicates how the overall motion will
longshorevariabilityis now 2rd(•.m- •.r), half the wave-
appearto an observer.Of moreinterestfor the generationof lengththat appearsas a modulationin the linearcombination
beach morphologyis the way in which any second-order of the solutionsin (8).
terms includenot only the self interactionof each modebut
also their cross interactions.FollowingHunt and Johns
[1963],whenthe velocityfieldabovethe bottomboundary
layer is definedby
2-- 3 •gm•gm"
g2•k03{
[ 5 •gm
o•= 4ø
• am
2 (2rn+
U = U(x, y)ei•
v = V(x, y)ei•
the masstransportvelocity(t•, 0) at the topof the boundary
layer is given by
a=•
Oy
1[ -•xOU* OV*
4•
U
(3 +5i)+
Thelongshore
component
of thedriftvelocityis givenby
U•
(2 + 3i)
(2rn+ 1)
+(2m
+1)4•'"'2
+ar2(2r+1)
3qbr2
-(2r+1)•br•r"
(2r
+1)•kr'2
+amar
cos/.6
(51•)ml•)
r (2m
+1)(2r
+1)
2
OU*
+V
1)3
m
Oy
+(2rn
+ 1)2
(1
+2/)]
1(2r+ 1))- 3((2rn1+ 1)4•m&r"
HOLMAN AND BOWEN: BARS, BUMPS, AND HOLES
461
1
, )
)
+sin/x
(-3•bm•br
(
+2((2m
,+ 1)•m{•r"(2r+, 1)•m"•r
)
•(2r +
1)
dgm"dgr
-•'2C•m'
dgr'
(2r+ 1)
(2m + 1)
1
1
(2m+ 1) (2r + 1)2 (2m + 1)2(2r + 1)
- &m'&r'
(2r+1)-(2m+1)
A nondimensional
drift velocity(up,vp)is thengivenby
0.0
2.0
4.0 •, 6.0
8.0
I0.0
Fig.3b. Perturbation
profile
h•(J,9)fort.heinteraction
(1,- 1)
803
with K = 1. Stippledareasrepresenterosion(h• < 0) andplainareas
Up
= g2X03
(am2
+ ar•)'Ul
(13)
accretion.
The longshorepatterngivenby (10) and (12) hastermsin patternon nearshoretopography.Clearlya more quantitacos/x and sin/x bothmultipliedby rathercomplexfunctions tive model is preferable.
of •. In generalthe patternswill not be symmetricaboutany
SEDIMENT TRANSPORT MODELS
offshoreprofile. The particularcaseof standingedgewaves,
Xm = - Xr, studied by Bowen and Inrnan [1971] which
To try to make an objectiveassessment
of the possible
clearly has to be symmetricalis a rather specialcase for importanceof the drift velocity,a simplesedimenttransport
which the coefficientof sin/x vanisheseverywhere.
modelbasedon the ideasof Bagnold[1963]hasbeenusedto
One wavelength of the drift velocity pattern for the calculatethe equilibriumbeach slope. For the purposeof
standingwave of moden = 1, the casern = 1, r = - 1 (or (1, illustration, it is assumedthat the sedimenttransport is
-1) in our notation)is shownin Figure 3a, where up is dominatedby suspendedload, but usinga bedloadmodel
multipliedby a factor•2. (A negativemodenumberis a
would result in similar shapesfor the bars.
convenientway to indicatea wave progressivein the negaThe transportof suspendedsedimentis assumedto be
tive y direction.Howeverit is importantto rememberthat Xr given by is where
-- (r/Irl) (X0/(21rl+ 1))). Thiscanbe regardedas scalingupby
•. ('r' u)u
is =
(14)
the drift velocityof the incidentwaves.Not surprisingly,the
w -u's
drift pattern is very similar to that previouslyshownby
Bowen and Inrnan [1971], the pattern further offshore is where •s is an efficiency, T the bottom stress, w the fall
more noticeable due to the particular scaling. However velocity of the sediment,and s = (•h/•x), (•h/•y) the local
Bowen and Inrnan [1971] were forced to make a rather bottom slope.
The velocityfield consistsof the incidentwaves,assumed
subjectivedecisionaboutthe influenceof the drift velocity
to be the dominantprocess,plus the edge waves. If the
incidentwavesapproachwith crestsapproximatelyparallel
to the shoreline, the velocity field is of the form
u = (u0icos•rit + u0ecosfret+ t•
+ ul, V0e
COS
•ret+ Vi)
(15)
whereUo,
andu0•aretypicalorbitalvelocities
fortheincident
and edgewaves, a is the drift velocity associatedwith the
incident waves, and u•, v• are the edgewave drift velocities
from (10) and (12). For this case, definingthe bottom stress,r
by
'r = CD plulu
(16)
the analysisof Bowen[1980]canbe readilyextendedto both
o.o-
0.0
horizontal
components.
If weassume
thatu0•,a, u•, v•areall
muchsmallerthanUo,but maythemselves
be of the same
>
2'.0
4.0
6.O
•
8.0
I0.0
9•
.• ¾:z.o
Fig. 3a. Nondimensional
drift velocityassociated
with a standing mode 1 edgewave, the interaction(1, -1) in our notation.The
magnitudeof the drift vectorshas been normalizedby the drift
velocityassociated
with shoaling
incidentwaves.A scaleis provided at the bottom right. The pattern repeats in the longshore
direction.
order, then to lowest order the two componentsof suspended transport are
•X....
w
15•r
5 //03(
a + //1)+
EsCDp
4
iY= W ß__.
3•r //03/91
I
w dx
(h + h•)
(17)
462
HOLMAN AND BOWEN: BARS, BUMPS, AND HOLES
We takefi to bethedepthprofilewhichisin equilibrium
with
the general qualitative trends as to how many bars, their
position, and general appearance are not sensitive to the
particular transport model used.
the drift velocity associatedwith the incident waves (hence
those two terms cancel in equation (17)), and h' to be the
perturbation profile which would be produced by the edge
wave drift velocities. Interestingly, longshoreslopedoesnot
As onlythe amplitude
of h• is a functionof K, h• would
seem to be a useful description of the effects of the edge
waves. The physical meaningis clear, it is the local departure from the plane beach, slope/3, for which the edgewave
enter into fx sincethe term, (l10311101/W)
dh/dy will be of
higher order than other terms.
The criterion for the beach to be in equilibrium is that the
divergence of the sediment transport is zero everywhere.
Orx
--
Ox
+-
Oy
= 0
solutions
werederived.Figure3b showsh• for thestanding
(18)
and, substituting (17), and integrating in from deep water
where h• will vanish
dx
u02
Oy
dh
• 5wu
• 4u05
5w
Ij //03
O0
=
•.
dx
(19)
Now in the region of interest, the regime seawardof the surf
zone, the orbital velocity is given by shallow water theory as
UO
= h3/4 = 'y/2
' (gh)b
1/2
(20)
wave (1, -1) calculatedfrom (23). The longshoreperiodicity
of the depth perturbation is immediately evident. However,
the crescenticnature of the systemis not obvious. Instead,
isolatedmoundsof sedimentappearat zonesof convergence
of the drift velocity. This illustratesa generalproblem.What
is observed on casual inspection, particularly on aerial
photographs,is not the perturbationtopography,but is the
total water depth, equal to the sum of the perturbationand
mean beach profiles. (In fact, it is unclear how one could
estimate the mean beach profile from field data, even given
detailed surveys.It is not just the averageover a longshore
period of onshore-offshoredepth profiles since those could
involve linear bars generatedby the self-interactionterms.)
To appreciate how the theoretical sediment distributions
appear to a casualobserver, it proved necessaryto construct
a measureof the total depth and to displaycontoursof equal
depth. Here, •0 is used as an arbitrary origin offshoreand
where ao is the incident wave amplitude, hothe water depth,
and 3' the ratio of wave height to water depth, all defined at
thebreakpoint.Defining
a nondimensional
depth/• as
$t•
- •h•
Xoh•
g/32
fir = (•0 - •) + h•
(24)
so fir is elevationabovea submarine
datuma distance
x0
offshore.Figure4 showscontours
of fir generated
by the
standing edge wave (1, -1) for the case K equal to 1.0.
(21)Contour
valuesare actuallyfi7/4,a simplescalingfor the
purpose of avoiding double digit values in the contour
program. To create a standing wave, the two constituent
waves have equal amplitudes,the scaledamplitudesbeing •i•
= d_• = 1.0. The crescentic pattern is now obvious. The
(22) profile given below the contour plot in Figure 4 showsthe
and writing in terms of the scaled coordinates g, y, (19)
becomes
•
d•
= -K
+
4
g-9/4
dg
linearbar producedby eitherof thesemodesactingseparate-
where
K=1-•'
[ •? '
8.0-
(22) can be integrated numerically by usingthe conditionthat
the drift velocities and h• die away seaward. So
6.0-
ßd.•
(23)
4.0-
The final result depends on only one free parameter K
which definesthe amplitude of the perturbed topography.It
containsthe terms one would expect, the settlingvelocity in
the dimensionlessform frequently encountered[Dean, 1973;
Dalrymple and Thompson, 1977], the width of the surf zone
and the relative size of the edge waves. In integrating (22)
and (23), it became clear that the second term in (22)
containing the longshore divergence of O is generally very
small compared to the first term (althoughit was retained in
the calculation). The primary balance is between the onshore-offshore component of the drift velocity and the
gravitational effects due to the beach slope.
The basic morphologycould therefore be producedby any
2.0-
0.0
0.0
I010
hT0.0
0.0
2.0
4.0
6.0
B.O
I0.0
Fig. 4. (Top)Onelongshore
cycleof the totalbeachprofilefir
(,•,:) fortheinteraction
(1, - 1)withK = 1. Valuesplottedarefi•4.
model in which the slopewas assumedproportionalto up (Bottom)Linearsandbarprofile,fi•4, for a single,mode1, edge
(provided the coefficient is a reasonable function of •) and
wave (i.e., ar = 0), K = 2.0.
HOLMAN AND BOWEN: BARS, BUMPS, AND HOLES
2.0
463
systemas it would be seenby a casualobserver.The point of
usingfir becomeseven clearerwhen K becomessmall.
0.0
0.0
Fig. 5.
2'.0
,,'.o
6'.0
8'.0
,o'.o
'Low tide terrace' for a singlemode 1 edgewave with small
K - 1.1.Valuesplottedarehz/4.
ly (ti• = 1.0, ti_• = 0, K = 2). It is interestingto seethat the
bar crest in the linear profile and the maximum offshore
distanceof the crest in the crescenticpattern occur at ;½=
3.0. Bowen and Inrnan [1971] suggestedthat the crest would
be located at ;½= 4.5, the point of drift convergence(Figure
When the perturbationis small, there is no obviousbar. The
crest disappears;the profile is variable in slope, containinga
low tide terrace that is virtually fiat at some value of K
(Figure 5). At still smaller values of K the profile shows a
gentle slopeinshoresteepeningoffshore.Figure 5 showsthe
profile producedby a singleprogressivewave, as in Figure
4b but for a smaller value of K. There is clearly a twodimensionalanalogywhere the amplitudeof a crescenticbar
system is not large enoughto generateobviousbars (points
whereOhr/O.f
= 0), andtheresulting
structure
appears
asa
complexterracingin the profiles.
'
An apparent difficulty in Figure 4 is that the integration
3a),andthisisindeedtheposition
whereh•islargest
(Figure does not produce a shoreline. However, the model is not
3b). However,whenthe meanprofileis addedto h•, our expected to apply in very shallow water, and conditionsas ;½
ideas of what constitutesa crest clearly change.If we take --> 0 become increasinglyunrealistic. Nevertheless, it is clear
the usual definition of the bar crest as the position where that the general pattern of a crescenticbar is producedas
(dhr/d.O
= 0, thenthevaluepredicted
bythemodelis;½= 3. expected.
However, this position is a function of the amplitudeof the
INTERACTION BETWEEN TWO DIFFERENT MODES
perturbation topography,which in turn is a function of K. As
K becomes
small,thebartendsto ;½= 4.5. As expected,
fir
The casesthat have not been previouslyexaminedare the
seemsto provide a better insightinto the structureof the bar
interaction between two different modes. Figure 6 (left)
45.0 -
40.0 -
35.0 -
30.0 -
25.0 -
20.0 -
•5.0 -
i
5.0--
0.0-
o.o
0.0
5[.0
]
10.0
A
X
Fig. 6.
•
15.0
20b •
5.o
20'.0
io.o
A
•u;
---2.0
u•
(Left) Normalized drift velocity for the interaction(1,2).
(Right)Totaldepthprofile,fiz/4for theinteraction
(1,2)withK = 1.
Note the prediction of welded sand bars.
X
464
HOLMAN AND BOWEN: BARS, BUMPS,AND HOLES
TABLE 1. WavelengthL, of the Topographyfor the Case(m, r) in the NormalizedForm XoL/,r=
A-• WhereA is the AspectRatioof the InshoreBar
Y
m
-3
-2
-1
-0
0
1
2
-3
-2
- 1
-0
0
I
2
3
4
X
35.0
X
10.5
15.0
X
2.33
2.50
3.00
X
1.75
1.67
1.50
1.00
X
4.20
3.75
3.00
1.50
3.00
X
5.83
5.00
3.75
1.67
2.50
15.00
X
Where L(m, r) = L(r, m); L(-m, r) = L(m, -r); L(-m,
3
7.00
5.83
4.20
1.75
2.33
10.50
35.00
X
4
7.88
6.43
4.50
1.80
2.25
9.00
22.50
63.00
X
-r) = L(m, r).
shows the drift velocity pattern for a mode 1 and mode 2
edgewave propagatingin the samedirection, the case(1, 2).
The periodicity of the motion is evident now on a larger
length scale. As the longshore wave number is km -- kr,
waves propagatingin the same direction produce patterns
beach slopesoffshore, accentuatesthe bar structurein this
region.The weak offshoredrift near] = 45 leadsto a more
gentle offshore slope, and the bar appears to truncate
alongshore.
in opposite directions (Table 1). In Figure 6 (left), the
nearshore drift velocity is basically offshoreout to k = 4.5,
although its magnitudevaries very noticeablyalongshore.
Offshore the flow is dominatedby a meanderinglongshore
flow, rather reminiscent of the meanderingcurrents describedby Sonu [ 1972].However, this is really the flow just
above the boundary layer and only partially reflects the
pattern of the flow over the whole water depth. Figure 6
the aspectratio A of the pattern, the offshoredistanceto the
first bar divided by the longshorewavelength,varies as a
Other combinations
of modesproduceessentiallythe
with a longerlengthscalethanthe samewaves'propagatingsame morphologicalpattern as seenin Figure 6 (right), but
function of the modal numbers. As the offshore structure for
all the highermodesis very similarcloseto shore(Figure2),
the maximum offshore distance of the first bar crest tends to
be of the order of k = 3. The aspect ratio is therefore
approximately given by
(right)showsthedepthcontours
fit for thiscase.Intriguing-
A=
2rn+ 1 2r+ 1
(25)
ly, the model predicts the generationof periodic, oblique
sand bars welded to the shoreline.The prediction of this
Figure7, showingthe topographythat resultsfrom the (2,
morphologycameas somewhatof a surprisebut canperhaps
-3) interaction, illustratesthis point; the length scale is
bestbe understoodby lookingat the drift velocitypatternin
relatively short as the waves propagatein oppositedirecconjunctionwith (22) and (23). Equation(22) is dominatedby
tions. However, the mostlikely combinationof modesof the
up, the contributionfrom the termsin vpbeingsmall.The samefrequencyis probablytwo consecutivemodespropanearshorepattern of up appearsto be simplyperiodicin gatingin the samedirection,for example(0, 1) (seeDiscusFigure 6 leadingto the expectationof a crescenticsystem.In
sion). Interestingly,this pair also generatesa welded bar,
fact, a plot of the location of the bar, independentof the
even though n = 0 has no zero crossingsin its offshore
height of the crest, is crescentic. However, equation (23)
structure.It is clear that the aspectratio of suchconsecutive
containsan offshoreintegral, and it is the behavior offshore
pairsof modesrapidlybecomessmallwith increasing
n. For
that producesthe asymmetry observed. The weak onshore
example, (2, 3) has an aspectratio of 1:35, a very long
componentof drift velocity near • = 30, producingsteeper
longshorelength scale.
6.0
•
4.0
2.0
,
0,0
5.0
I0.0
15.0
20.0
o.o
o.o
Fig. 7.
Welded sandbar predictedfor the interaction(2, -3), with
K=I.
Fig. 8a. hz/4forthecrescentic
sandbarcase(1, - 1)butwitham-1.0, ar = 0.5, K - 1. Compare with Figure 4 (top).
HOLMANAND BOWEN:BARS,BUMPS,AND HOLES
465
i
appearwhenevertheamplitudes
of themodesareroughlyof
,z,'
i!i
,xi i
0.0
0.0
i 'I
4.•
2.0
•'.o
•'.o
the sameorder. The linear bar shownin Figures4 and 5 only
occurs if one mode is totally dominant.
INTERACTION OF THREE OR MORE MODES
More complicatedmorphologies
are producedif more
than two modesinteract.The perturbationtopographyis the
sumof that generatedby eachpairof modestakenseparate-
ly. However, the phaseanglesbecomeimportant.While
changing0 in thetwo modecasesimplydisplaces
thepattern
alongshore,
with moremodesthephasesdeterminewhether
the different modal combinations tend to reinforce one
another or tend to cancel. Figure 9 shows the patterns
producedseparately
by the interactions
(0, 1), (0, 2), (1, 2),
andfinallythe sumof all three,withphaseangleszeroandK
io'.o
Fig. 8b. Same as Figure 8a but with ar = 0.25. Note that the
= 1. For this case the wavelengthsare simplyrelated, the
essential crescentic nature is maintained.
wavelengthgivenby (1, 2) is exactlyfivewavelengths
of (0,
1) andsixof (0, 2) (Table1). Thepatternshownby (0, 1, 2) in
with the waveFigure8 showsthe effectof varyingthe relativeamplitude Figure 9 will thereforerepeatalongshore
However,the patternshownby
of the two modesby usingthe standingwave (1, -1) as an lengthof (1, 2) interaction.
illustration;d• is held at 1.0, andd_• is setto 0.5 (Figure8a) the (0, 1, 2) interactionseemsvery irregular at scales
to (0, 1) and(0, 2). As canbe seenfromTable1,
and 0.25 (Figure8b), and K = 1.0 in all cases.Compared appropriate
with Figure 4, the morphologyis virtually unchangedin the triple interactiondoes not necessarilyrepeat in the
of thelargestinteraction;
the(-0, 0, 1)does,but
Figure8a, and even in Figure8b the essentialcrescentic wavelength
of (2, 3) to repeatand
nature is retained. This insensitivityof the basicstructureto the(1, 2, 3) requiresthreewavelengths
of (3, 4). It seemslikely
the precisevaluesof d suggests
thatrhythmicpatternswill (1, 3, 4) requiresfive wavelengths
I
i
/2
40.0
30.0
a
20.0
15.0
5.0
0.0
'l•
0.0 2D 4.06.08.010.0 OD2.04.06.08D 10.0 0.0 2'.04.06.0 '0 I•3.00.0 2.0 4,06.08.0 10.0
A
A
A
A
X
X
X
X
(0,1)
(0,2)
(I,2)
(0,1,2)
Fig.9. fid4foronelongshore
cycleofthetripleinteraction
(0,1,2)andforeachofitsconstituent
interactions
(0, 1),
(0, 2), and(1, 2). Thetripleinteraction
isjusttheaverage
of itsconstituents.
466
HOLMAN AND BOWEN: BARS, BUMPS, AND HOLES
that the irregularity of the pattern at the small scale will be
much more apparent than the repetition at long intervals
alongshore. Patterns involving more than three modes became increasingly complex although many repeat quite
simply, (-0, 0, 1, 2) also repeats exactly the (1, 2) wavelength.
OFFSHORE BARS
the primary interest here is to indicatethe generalprinciples
involved in the generationof topographyby differentmodes
of the same frequency and to sketch some of the possible
morphologiesthat might result.
RESIDUAL
TRANSPORT
Theperturbation
topography
fil isgenerated
byanequilib-
rium model that assumesthat the beach has adjusted comFor the higherwave modes,zerosin upexistoffshoreand pletely to the imposed velocity field. However, the sediment
give rise to the possibilityof multiple bars, either crescentic, transport does not vanish everywhere. The condition for no
meandering, or linear. The present model does not resolve further accumulationor erosionis that the divergenceof the
these structures at all well. This seems to be inevitable if the
transport vector V'is is zero, equation (18). Patterns of
original beach slope is assumed to be plane. The drift sedimenttransport exist over the new topographyin essenvelocity solutionsdie away rapidly offshore, and reasonable tially two forms, longshoretransport down the beach and
perturbations at the location of the offshore bars would be closedcirculationof sedimentwithin a local area. Longshore
accompanied by very large changesindeed in the inshore transport is produced by a singlewave and is consequently
bar. Scale analysis [Bowen, 1980] suggeststhat this problem pronounced when two modes propagate in the same direcwould not arise so severely on the concave profile more tion. This is suggestedin the drift velocity patternsin Figure
typical of actual beaches;the perturbationsin the slopedue 6 (left). However, as the primary balancein the topographic
to the drift velocity patterns would be comparedto a much model is between the transport due to drift velocity and the
steeper local slope near the shore and a gentler slope gravitational effects of the bottom slope, the transport up
offshore.
and down the slopeis everywhere almost in equilibrium. The
An intriguing result of the present model is the generation unbalanced part of the transport, the residual transport,
of morphology, which is periodic alongshorenear the shore therefore tends to run along depth contours, perpendicular
but linear offshore. This occurs when one of the waves has a
to the slope.
relatively short offshorelengthscale.For example, a mode 1
When there is no net longshoretransport, the transport
and mode 3 edge wave interact to produce a welded bar patterns are weaker but still nonzero. Figure 10 showsthe
system inshore, but offshorethe mode 3 is dominant (Figure residual transportin the dimensionless
form ip for the
2) and the morphologywould be almostthat producedby the standingedgewave (1, - 1). The total transportiT is given by
mode 3 alone. In general, modesof adjacentmodal numbers
are not sufficiently different in offshore scale that the longEsCD,y3
pg2 (am
2 + ar2)
iT:
(.•b)15/4
shore periodicity will disappear altogether. However, if a
24•r W .
2
ip (26)
mode 2 interacts with a mode 1 wave of considerablysmaller
amplitude, the offshorestructuremight be dominatedby the where
effects of the mode 2 wave.
These results are not evident in the figures. To resolve the
ip = -.•-9/4
(27)
offshore bars adequately we would need a numerical model
for calculating the drift velocities on any specifiedtopography. The plane beach solutionsrepresent only a first step; As can be seen in Figure 10 the transport in the presenceof
the crescenticbar shown in Figure 4 is seaward over the
central portion of the crescentand landward over the cusp.
This is exactly the pattern of sedimenttransport noted by
• d•,•-9/4Up
Greenwood and Davidson-Arnott [1979]. However, Greenwood and Davidson-Arnott ascribed the transport to the
nearshore circulation pattern, observing that rip currents
tend to flow seawards over the central area of the bar.
DISCUSSION
To estimate the practical importance of the interaction
between two or more edge waves of the samefrequency, we
need some insight into the likelihood of forcing particular
sets of modes. Bowen and Guza [1978] discussthe generation of edge waves by an incident spectrumapproachingthe
beach at an angle a with beam width 2 Aa. They showthat as
Aa becomes small a sequence of edge wave modes are
forced at frequencies trnwhere
O'n= 20'i sin a sin (2n + 1)•
Fig. 10. Residual sediment transport over the crescenticbar
systemshownin Figure 4 (top). Transportis offshoreover the center
portion of the crescent and landward over the cusp. The dashed
arrows indicate
that the 'circulation'
will be closed.
(28)
where •ri is the central frequency of the incoming waves.
Figure 11 shows schematically the resulting edge wave
spectrum. The presenceof a small angular spreadAa widens
the resonantbands. Overlap occurs, first between adjacent
HOLMAN AND BOWEN: BARS, BUMPS, AND HOLES
/
log
energy
//
frequency
Fig. 11. Conceptual spectrumof edge waves forced a spect•m
of incidentwaves with central angleof incidencea and a beamwidth
2Aa. For Aa = 0 a sequenceof modes will be forced (solid lines).
For smallbut finite Aa, a spreadof the modal peaks (dashedlines) is
predicted.
modes and eventually between several neighboringmodes if
Aa continues to increase. This suggests that the most
common interactions would occur between adjacent modes
or adjacent sets of modeswith consecutivemodal numbers.
Interaction between nonconsecutive modes might be more
important if the incident spectrum contains two separate
wave trains approachingfrom different directions.
In addition to the forcing by incident waves, we have the
reflection of waves from headlands,piers, or groins providing the appropriate negative wave numbers and standingor
partially standingwaves. Cartwright and Young[1974]used
the modal combination (-1, -0, 0, 1) to satisfy nearshore
boundary conditionssuggestedby the headlandsand islands
on the east coast of Shetland. This indicates a more general
aspect of forcing; the responseto any pattern of forcing of
given frequency can be constructedfrom the eigensolutions,
the normal modes of the problem. For complex forcing
patternsor complicatedgeometry(a bay or, as in Cartwright
and Young, a series of headlands or groins), the general
solutionmay consistof a combinationof a number of modes
of the same frequency. In these casesthe phaserelationship
between various modesmay emergeas an integral part of the
solution, for 'example, if the total longshorevelocity must
467
edge waves is extended to include these skewed bars of
various aspect ratios.
Although the model makes quantitative predictionsof the
relative scalesof the bars, the existence of a large number of
possible modes leads to a very large selection of possible
length scalesfor eachfrequency (Table 1) unlessconsecutive
modes are assumed.The uncertainty is compoundedby the
fact that the frequency of interest will not normally be
known. There are further difficulties in estimating the beach
slope which also enters the scaling. Patterns shown in the
literature, such as Figure 12, can only provide very qualitative support for this hypothesis.
CONCLUSIONS
The interaction of two modes of the same frequency may
generate topography of three somewhat distinct types: (1)
shore parallel bars only (either the waves are not coherentor
one mode is dominan0; (2) rhythmic patterns nearshore,
parallel bars offshore (two modes are at least partially
coherent, they could be very coherent if the modal numbers
are quite different or if the wave of lower modal number is of
relatively small amplitude); (3) totally rhythmic patterns (the
extreme caseis the crescenticbar but highly coherentmodes
of adjacent model numbersproduce other complex patterns
vanish at a headland.
Usually, the problem is less well defined, and the phase
relations are not known. When we speakof the amplitude or
phase in an interaction we are referring only to the coherent
part of the motion. The relative magnitudeof the coherence
provides another factor in the balance between the generation of periodic features as opposedto linear features. The
longshoreperiodicity dependson the coherent motion; two
modes of random phase produce a set of linear bars.
The analysisin this paper is based on linear wave theory
for a plane beach. We ignorethe effectsof the changesin the
beach profile on both the edge waves and incident waves. In
reality, as the bars grow they will increasinglyinfluencethe
velocity field. The detailed structureof the bar must result in
part from theselate interactionsandwe wouldnot expectthe
present model to predict exactly the structureseenin Figure
1. However, we can suggestthat the general pattern of the
bars is very reminiscent of the mature bars seen in nature
(Figures 1, 12) and advance the hypothesisthat the major
features of such systems,the wavelength,the location of the
major depositional features, and major troughs, are determined by the interaction of two wave modes of the same
frequencies.The analogyto crescenticbars is complete,and
the idea that major bar systemsmay be generatedby two
o
IOO
Fig. 12. Nearshore welded bar structurefor January 13, 1972, for
Durras Beach, N. S. W. Australia [after Chappel and Eliot, 1979].
468
HOLMAN AND BOWEN: BARS, BUMPS, AND HOLES
if they are of roughly the same amplitude. The inshorebar
tends to appear as a welded bar, its aspect ratio being a
function of the two modes involved).
The
extension
of these ideas to three or more modes
Cartwright, D. E., and C. M. Young, Seichesand tidal ringingin the
sea near Shetland, Proc. R. $oc. London, $er. A, 338, 111-128,
1974.
Chappell, J., and I. G. Eliot, Surf-beach dynamics in time mad
spacemAn Australian case study, and elements of a predictive
model, Mar. Geol., 32, 231-250, 1979.
Dalrymple, R. A., A mechanismfor rip current generation on an
open coast, J. Geophys. Res., 80, 3485-3487, 1975.
Dalrymple, R. A., and W. W. Thompson, Study of equilibrium
beach profiles, in Proceedings15th Conferenceon Coastal Engineering, pp. 1277-1296, American Society of Coastal Engineering, New York, 1977.
Dean, R. G., Heuristic modelsof sandtransportin the surf zone, in
Conference on Engineering Dynamics in the Surf Zone, Sydney,
Australia, 1973.
Dolan, R., L. Vincent, and B. Hayden, Crescentic coastal landforms, Z. Geomorph., 18, 1-12, 1974.
Dolan, R., B. Hayden, and W. Felder, Shorelineperiodicitiesand
edge waves, J. Geol., 87, 175-185, 1979.
Eckart, C., Surface waves in water of variable depth, Wave Rep.
100, ScrippsInst. Oceanogr., Ref. 51-12, La Jolla, Calif., 1951.
producesdeterministicbut apparentlyirregular topography
alongshoreif the waves are coherent. Any random fluctuation in phase will encouragethe formation of linear bars;
they will, in a sense,make the systemappear more regular.
The discussionof bar formation does not introduce any
physicsthat dependson the barsamplitude.In practice,this
would change the velocity field of both the incomingand
edge waves, but this has not been taken into accountin the
first order solution. However, the way in which the bar is
perceived on the profile is very much a function of the
amplitude both in terms of the positionof the bar and the
definition of a bar. At low amplitude the bar looks exactly
like a terrace, perhaps the classical 'low tide' terrace.
Nomenclature is also interestingfor the channelinside the Greenwood, B., and R. G. D. Davidson-Arnott, Sedimentation and
equilibrium in wave formed bars: A review and case study,
welded bar, particularly obvious in Figures 6 (right) and 7.
Canadian J. Earth $ci., 16, 312-332, 1979.
This is a classicalrip channel.There is no doubt that a rip
Guza, R. T., and A. J. Bowen, On the amplitude of beach cusps,J.
Geophys. Res., 86, 4125-4132, 1981.
Guza, R. T., and D. L. Inman, Edge waves and beach cusps, J.
Geophys. Res., 80, 2997-3012, 1975.
Hom-ma, M., and C. Sonu, Rhythmic patterns of longshorebars
related to sediment characteristics,in Proceedings8th Conferassumptionsabout the form of the sedimenttransport.It is
ence on Coastal Engineering, pp., 248-278, American Society of
important to emphasizethat the generaltrend of the results
Civil Engineering, New York, 1963.
does not dependon the preciseform of theseassumptions. Hunt, J. N., and B. Johns, Currents producedby tides and gravity
waves, Tellus, 15, 343-351, 1963.
Given these drift velocity patterns, interesting rhythmic
topographywill be generatedby any reasonablemodel. The Huntley, D. A., R. T. Guza, and E. B. Thornton, Field observations
of surf beat, I, Progressiveedge waves, J. Geophys. Res., 86,
interestingpoint is that someof thesepatternslook familiar.
6451-6466, 1981.
Komar, P. D., Nearshore cell circulation and the formation of giant
Acknowledgments. This work was supportedby the Office of
cusps, Geol. Soc. Am. Bull., 82, 2643-2650, 1971.
Naval Research, GeographyBranch, under contract NR 388-168. Komar, P. D., Rhythmic shoreline features and their origins, in
We would like to thank Dave Aubrey for allowing us to use his
Large-scale Geomorphology,edited by R. Gardner et al., Oxford
photographfor Figure 1. We would alsolike to thank Gail Davis and
University Press, New York, in press, 1981.
Bonnie Hommel for their patiencein typing the manuscript.
Sallenger, A. H., Beach cusp formation, Mar. Geol., 29, 23-37,
current would flow in this channel, but it is clearly not
necessarily the causal mechanism.
To look at the sedimentary patterns in a more or less
objective way, it is necessary to introduce a number of
1979.
REFERENCES
Bagnold, R. A., Mechanics of marine sedimentation,in The Sea,
vol. 3, edited by M. N. Hill, pp. 507-528, Interscience, New
York, 1963.
Bowen, A. J., Simple modelsof nearshoresedimentation;beach
profilesand longshorebars, in The Coastlineof Canada, editedby
S. B. McCann, pp. 1-11, GeologicalSurvey of Canada, Ottawa,
1980.
Bowen, A. J., and R. T. Guza, Edge waves and surf beat, J.
Geophys. Res., 83, 1913-1920, 1978.
Bowen, A. J., and D. L. Inman, Rip currents, 2, Laboratory and
field observation,J. Geophys.Res., 74, 5479-5490, 1969.
Bowen, A. J., and D. L. Inman, Edge waves and crescenticbars, J.
Geophys. Res., 76, 8662-8671, 1971.
Short, A.D.,
Multiple offshore bars and standing waves, J.
Geophys. Res., 80, 3838-3840, 1975.
Sonu, C., Field observationsof nearshorecirculation and meandering currents, J. Geophys. Res., 77, 3232-3247, 1972.
Sonu, C., Three dimensionalbeach changes,J. Geol., 81, 42-64,
1973.
Stoker, J. J., Water Waves, Interscience, New York, 1957.
Suhayda, J. N., Standingwaves on beaches,J. Geophys.Res., 72,
3065-3071, 1974.
Ursell, F., Edge waves on a slopingbeach,Proc. R. Soc. London,
$er. A, 214, 79-97, 1952.
(Received July 6, 1981;
revised October 5, 1981;
accepted Obtober 5, 1981.)