JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 94, NO. C12, PAGES 18,023-18,030,DECEMBER 15, 1989
ShearInstabilitiesof the Mean LongshoreCurrent
1. Theory
A. J. BOWEN
Departmentof Oceanography,DalhousieUniversity,Halifax, Nova Scotia,Canada
R. A. HOLMAN
College of Oceanography,OregonState University,Corvallis
A new classof nearshorewavesbasedon the shearinstabilityof a steadylongshorecurrentis discussed.
The
dynamicsdependon the conservation
of potentialvorticitybut with the backgroundvorticityfield, traditionally
the role of Coriolisin largerscaleflows, suppliedby the shearstructureof the longshorecurrent.The resulting
vorticitywavesarelongshore-progressive
with celeritiesroughlyequalto V0/3,whereV0 is thepeaklongshore
currentvelocity. A natural frequencyscalingfor the problem is fs, the shearof the seawardface of the
longshorecurrent.While the instabilitycan spana rangeof frequenciesand wavenumbers,a representative
frequency
isgivenby0.07fs,typically
intherange
of 10-3-10
-2 Hz (called
theFarInfragravity,
orFIG,band
because
frequencies
arejustbelowthoseof theinfragravityband).Wavelengths
areof theorderof 2x0, wherex0
isthewidthof thelongshore
current.
Growthis exponential
withane-folding
timethatis typically
halfof a
wave period.Field data, presentedin the companionpaper [Oltman-Shayet al., this issue],demonstratethe
presenceof energeticmotionsfrom a natural beach whose behavior matchesthe theory in many aspects.
Resultsfrom the model suggestthat shearinstabilitywill be more importanton barred,ratherthan monotonic
beachprofiles,a resultof the strongershearsexpectedover the bar crest.Sincevorticity waveswill probably
havea profoundeffecton cross-shore
mixing aswell as longshorecurrentdissipation,we expectthe dynamics
of barred and monotonic beaches to show fundamental
differences.
INTRODUCTION
[ 1971] for example)and dependson the conservationof potential
vorticity. For the larger scale case, a backgroundpotential
vorticity is suppliedby the Earth'srotationthrougha termf/h,
wheref is Coriolis and h is the depth.For the nearshoreCoriolis
andswellfrequencies
(O(10-•) Hz) andthemorerecently
studied is neglectedandthe backgroundvorticityis derivedfrom the shear
infragravityfrequencies
(O(10-2) Hz). The bandbetween structureof a steadylongshorecurrent.Perturbationof the current
vorticity(or shear)wave,
infragravityandmeanflowshasreceivedlittle study,anddynamics will thenleadto a longshore-progressive
have generallybeen associatedwith shelf-scaleor atmospheric where the restoringforce may be consideredto be potential
vorticity instead of the traditional gravitational acceleration.
events [Munk et al., 1964].
Wavelengthsare substantiallyshorterthan gravity wavesof the
Free gravity wave motions in the infragravity band may be
eitherleakymodes
withlongshore
wavenumber,
k,lessthan(52/g samefrequency.
In the next sectionwe develop the basic equationsfor the
(where (5- 2n/T is the frequency and g is gravitational
conservation
of potential vorticity for the case of a small
acceleration),
ortrapped
edgewaves
withlarger
wavenumber,
(52/g
perturbationto a steadylongshorecurrent.The equationsare then
_(k (_(52/g[•,where[3 is the beachslope.No gravitywave
solvedfor a simpleexamplegeometry.Under someconditionsthe
motionsexist for wavenumberslarger (shorterwavelengths)than
solutionis unstableso that the wave will grow exponentially.
this upper limit.
Dominant energy sourcesin the nearshorehave long been
consideredto be gravity wavesand meancurrentsor circulation.
Frequencybandsof gravitywavesincludedthe obviouswind wave
In the companionpaper[Oltman-Shayet al., 1989] (hereinafter
OSHB89), field measurementsof nearshorewave motions are
described
whereinwavelengths
are an orderof magnitudetoo short
to be gravitywaves.Thesemotionsoccurat frequenciestypically
The characteristic
in therange(10-3-10-2) Hz, exceeding
thetraditional
lowfrequencylimit of the infragravity band. This band has been
named the Far Infragravity (FIG) band in analogy to the
relationshipof infrared and far infrared in the electromagnetic
spectrum.
OSHB89 show the FIG band waves to be energeticand very
coherentin the longshoredirection.Kinematicsare linked to the
strengthof the meanlongshorecurrent.In thispaper,we presenta
model for these waves basedon a shearinstability of the mean
longshorecurrent.While new to nearshore,the mechanismis well
studiedin larger-scalephysicaloceanography
(Niiler and Mysak
T}mo}•Y
Sincethe wavesobservedby OSHB89 are too shortto satisfy
gravity wave dynamics, we seek alternate solutions to the
equationsof motion. In particular,we will searchfor shortscale
motionsunderthe nondivergent,or rigid lid, assumption.In this
case,accelerations
of the flow will be drivenby the inertial terms
involving the mean longshorecurrent.The developmentfollows
closelythat of Niiler and Mysak [ 1971] for the caseof continental
shelf waves.
The shallow water, inviscid equations for horizontal
momentum
are
ut'+u'.Vu'=-grlx
Copyright1989by theAmericanGeophysical
Union.
Papernumber89JC01393.
0148-0227/89/89JC-01393505.00
scales of these motions are discussed and are
shown to be roughly the same as the field observationsof
OSHB89. Finally, the impact of this new class of motions to
nearshoredynamicsis discussed.
(1)
vt'+ u'. Vv'=-grly
where x andy are the horizontalcoordinates(x being positive
18,023
18,024
BOWEN AND HOLMAN: SHEARINSTABILITIESOF MEAN LONGSHORECURRENT,1
offshore from the shoreline) with corresponding velocity
componentsu' = (u', v'), rl is the seasurfaceelevation,t is time,
g is gravitationalaccelerationand V is the horizontalgradient
operator.Subscriptsindicatepartial differentiation.Coriolisforce
is neglectedsince a typical frequency,{3, is much greaterthan
Coriolis
(o/f= 102).
We assumethat the total flow consistsof a steadylongshore
current,V(x), and a small perturbation,u, suchthat u'= [u(x, y),
v(x, y)+V(x)] and u, v <<V. Substitutingand retaining only the
termsthat are linear in the perturbationvelocity, we have
Ut+Vuy
=-grlx
(2)
v t + uVx + Vvy =-grly
Under the assumption of nondivergence, Tit is assumed
negligible in comparison to horizontal fluxes, so that the
conservationof massequationbecomes
V. (hu')= (hu)
x+ (hv)y=0
VORTICITY WAVES FOR A SIMPLE TOPOGRAPHY
The essentialdynamicsof shearwaves can be illustratedwith
an extremely simple example. Considerthe caseof a flat-bottom
beachof depthh0 borderedby a coastalwall at x= 0 (Figure 1).
The longshorecurrenthaswidthx0 with a maximumcurrent,V0,
at locationx = 15x
0. This dividesthe x plane into a regionof
positivelinear shear(regionI), a regionof negativelinear shear
(region II), and a region extendingto x = ooof zero longshore
current(regionIII). Superimposedon Figure 1 is the distribution
of backgroundpotential vorticity, Vx/h. The existenceof an
extremum in region II suggeststhe possibility of unstable
(growing)wave solutions.
(3)
where the bottom is at z =-h(x), assuminga two-dimensional
beachprofile. From (3), we may representu in termsof a stream
function,•, suchthat V x • = (hu), or
(hu)
=-tISy
vorticity, Vx/h, in the interval 0 < x < oo(Rayleigh [1880] with
minor extension).
Vx
(4)
(hv ) = •x
Cross-differentiating
(2) to removerl, andsubstituting
the stream
RegionI
functionyields
II Regionll•
Recjionm'
I
•-t 0y, + T =tISY
X=Xo(•
(5)
I
x--O
•,,•
=V
O
x=xo
I
I
which is just the linearized version of the equation for the
conservationof potentialvorticity,
SWL
I
•
h
We define•s • (•+Vx)/h as the potentialvorticityof the flow,
consisting
of a relativevorticity,•/h = (Vx-Uy)/h,anda
"background" vorticity, Vx/h, arising from the shear of the
longshorecu•ent. Equation(6) just expresses
the consedationof
potential vorticity, in analogyto the conservationof potential
vorticity in largerscalegeophysicalflows, but with the shear,Vx,
taking the role of Coriolis,f. More specifically,equation(5)
statesthatchangesin the relativevorticity(left-handside)will be
causedby cross-shore
perturbations
(•y) of the background
/
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/
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/
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h--ho
'
'
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/
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/
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Fig. 1. Longshorecurrent geometryand bathymetryshowingthe three
regionsof the examplecalculations.Upper panel showsthe background
potentialvorticity distribution.
In eachof thethreeregions,(8) reducesto
vorticity (right-hand side). This vorticity responseacts as a
restoringforcefor the pe•urbation.
q/xxk2q
/ =0
(10)
which has solutions
We assume a solution of the fo•
• = •e {W(x)e
i(•ot)}
wherewe assumethe longshorewavenumber,k, is real,but o and
• may be complex. •e indicates the real component.
Substituting
(7) in (5) yieldsaneigensystem
in Cp= o/kand•,
{•- k2•(V-cp)
'
-[hIx
Solutions will be vorticity waves or shear waves with phase
velocity
Cpandcross-shore
structure
•(x). Of pa•icular
interest
is
the case when the eigenvalue is complex, such that o has a
positiveimaginarycomponent,o = Ore+/Oim.The resultis an
instability in the form of a progressive wave with an
exponentiallygrowingamplitude
ß = exp{om
0 •e {•x) exp•i(ky
- •et)•}
(9)
It can be shownthat a necessaryconditionfor this instability is
the existence of an extremum of the background potential
RegionI
RegionII
•I = A1 sinh(kx)
I]/ii = A2 sinh(kx)+ B2 cosh(kx)
Region
Ill
Wm=A3e-kx
(11)
where we have taken the shoreline and far-field boundary
conditionsto be •(0) = •(oo) = 0 (no flow throughthe beachand
no mean longshoreflow associatedwith the perturbation).For
interior boundaryconditionswe require • and pressure(or sea
surfaceelevation) to be continuous.From (2), the latter condition
implies that
1 C
]
T!=- •-[(Vp)I]I
x- Vx•
(12)
must be continuous.Applying the boundaryconditionsat x =
15x
0, we have
2
B2_
SSAVx
A2 (Vo- Cp)
k - C8S
8AVx
(13)
BOWENAND HOLMAN: SHEARINSTABILITIESOFMEAN LONGSHORE
CURRENT,1
We haveusedthe shortformsS•5= sinh(kxo•5),
Ca = cosh(kxo•5),
andAVx= (Vxl-Vx2),
thedifference
between
thelinearshears
in
region of instability, the phasevelocity of the vorticity waves
will be given by
the two regions.Similarly,at x = xo, we have
B2__o+Vx2SoE
0
A2
(14)
o + Vx2CoE
0
18,025
o,8)
Cp- 9•e(o)
k --2k b - Vo
2 F(kx
(17)
Thusfora particular
geometry
of thelongshore
current,
Cpvaries
where So = sinh(kxo),C o = cosh(kxo),and E 0 = exp(-kxo).
Equating(13) and (14) yields, after considerablesimplification,a
quadraticexpression
for o
a02+ bo + c = 0
(15)
where
linearly with the strengthof the longshorecurrent,V0. Figure 3a
showsthat F(kx0, 8) is always positive and is only a weak
functionof L/xo (= 2•/kx0) and8 throughthe unstablerangeof
wavenumber.Thus vorticity waveson this geometrywill always
propagatein the directionof themeancurrentwith a speedof 1/4
to 1/2 the peak longshorecurrentvelocity.
a=l
b = -kV 0
I _
(1- e-2•xøa)
(1- e-2•xø)
+
= -kVoF(kxo,
8)
2kx08(1
- 8) 2kxo(1
- 8)
I.O
0.8
(16)
(1-e -2•xdl-b'))
22[(1-e-2•(1-e -2•cøa)
-•
•'---
0.40.6
re'x0) - a)2
0.2
TIME AND LENGTH SCALES
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For any k, equation(15) will have two roots which may either
bebothrealorcomplex
conjugates
depending
onthesignof (b2-
O--
!
4ac). The latter case, yielding exponentially growing waves
(equation9), is the caseof interest.
An exampleresultis shownin Figure 2 for longshorecurrent
•=0.8
geometry
(V0,8,x0)= (1.0ms-1, 0.5,100m),reasonable
values
for comparisonwith Figure6 of OSHB89. For k lessthana lower
limit or higher than an upper limit, there exist two real roots.
These are shearwaves that are stablebut which would rapidly
decayin the presenceof dissipation.For wavelengths,L = 2re/k,
in an intermediaterange (190 m < L < 435 m for this geometry)
o is complex and the waves grow exponentially.Through this
-2
-3
0.10
c
0.08
0.006
0.06
--0.5
0.04
'!-
0.004
0.02
' •080.6
• 0.002
i
i
8.0
1
'0.0
L/xo
•
I
Fig. 3. (a) The shapefactorF(kxo,•5)fromtheb termin equation(15).
(b)Thequantity
-(b2-4ac)fromwhichthemagnitude
of theimaginary
0.004
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frequencyis calculated.Only the unstableregionis shown.(c) Frequencies
of theinstabilitynormalizedby theoffshoreshear,fs.
0.002-
The existenceand strengthof the instabilitydependon the sign
andmagnitude
of thequantity
(b2-4ac).Figure3b,a plotof the
quantity
-(b2-4ac)versus
L/xofortheinstability
regime,
shows
-
øo
0.002
0.004
'
0.006
' O.o'o
8
O.OLO
Fig. 2. Frequency-wavenumberplots of the shear instability for the
examplegeometry:Figure (a) real and (b) imaginarycomponents.
Wave
motionsfor which(Jim> 0 are unstableandwill grow exponentiallyat a
rateproportional
to (Jim'The motionfor which(Jimis a maximumis used
to determine characteristic
scales of the waves.
that the wavenumberrangeof the instabilitydependson •5,with
the mechanismbeingnarrow-banded
for small8 but quitebroad
for large8. Figure3c showsthecorresponding
wavefrequencies,
f
(= l/T), normalizedby fs, the shearon the seawardside of the
longshorecurrent,
fs =
Vo
Xo(1- 8)
(18)
18,026
BOWEN AND HOLMAN: SHEARINSTABILITIESOF MEAN LONGSHORECURRENT, 1
The instability exists from a high-frequency limit of approx-
imately 0.1 fs to a low frequencylimit that varieswith 15.If
scientificinterestis focussedon conditionsof large 15andlarge
shearfs (yielding the fastest growth rates), then the span of
frequencies
canbe quitelarge.
While the instability occurs over a range in frequency and
wavenumber,there will be a singlefastestgrowingwave that we
can use to determine characteristictime and length scales.This
corresponds
to the point of maximum{Jim(Figure2b). Figure4a
showsthefrequency,
ff, of thefastestgrowingvorticitywave,
normalized
byfs.Figure4bshows
themaximum
growth
rate,fg,
alsonormalizedbyfs, while Figure4c showsthe corresponding
wavelengthnormalizedby x0, the width of the entire longshore
celeritymotionsobservedin OSHB89, and substantiallyshorter
thanthe O(10 km ) wavelengthof the shortestgravitywave (mode
0 edgewave)of thisfrequency.
Thegrowthtimescale(= {Jim
-1)
gives a characteristic,e-folding time scalefor the instability of
300 s, substantiallyfaster than one period. Clearly, this is a
strong instability and the assumption of linearity in the
perturbationwould not be valid for long.
SPATIAL STRUCTURE OF SHEAR WAVES
Knowing o, we may now calculatethe spatialstructureof the
wave
motions.
The
coefficients
are found
from
the interior
boundaryconditions,where
A1S6= A2S6+ B2C6
current.
A3S
0 = A2S
0 + B2C0
0.10
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B2
A2
i
0.08
(20)
• + Vx2SoE
0
• + Vx2CoEo
If we choseA2 = a, thenwe seefrom (20) that for frequenciesin
the unstablerange(0 complex),B2 will be complex.In this case
A 1 and A 3 will also be complex. Since fluid motions are
associated
only with the real componentof •, we canincorporate
the complexcoefficientsinto a phaseshift, 0, suchthat
0.06
0.04
0.02
RegionsI andIII
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0.3
(21)
RegionII
0 = tan-1
0.2
Bim
cosh(kx
)
a sinh(kx)+ Brecosh(kx)
That is, the phasesin region I and in region III will be constant
but not necessarilythe same.In region II, the phasewill vary
with cross-shore
position.Figure5a showsthe real andimaginary
0.1
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60
:5.0
Region
'rrr
40
o
20
•.o-
01
-20
-• 1.o
-40
-60
øo
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,.o
-80
Fig.4. (a) Characteristic
frequencies,
d•,(b) growthrates,
fg, and(c)
wavelengths,
Lf, of the fastestgrowingwaveas a functionof the
Region
•
m
,
m
! ,•
Region
I Region
17 ••
197
•
longshorecurrentgeometry.
,
b
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8 =0.5
•'
For the geometryunderconsideration,
it is apparentthatfs and
x0 provideappropriate
temporalandspatialscaling,respectively.
As normalized,thevariablesdependonly on •5,andeventhen,the
x•,
:/00
dependence
isweak(particularly
forthewavefrequency,
ff).From
Figure4, roughscalesfor the fastestgrowinginstabilitieswill be
ff -- O.07 fs
Oim
= (0.1- 0.2)fs= (1- 3)ff
(19)
Lf = (1 - 3)x 0
O0
m
I
I .o
I
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2 .o
!
3.0
x/x o
For the examplelongshorecurrentgeometryof Figure 2, we find
Fig. 5. (a) Real and imaginarycomponentsof the streamfunction,W,
as a function of cross-shoredistance.(b) Cross-shorephaseshift of the
Tf= 1/j•= 753s andLf= 250m, clearlyin therangeof thelow-
stream function.
BOWENANDHOLMAN:SHEARINSTABILITIES
OFMEANLONGSHORE
CURRENT,
1
18,027
components
of •t(x) andFigure5b showsthephase,0(x), tor the
fastestgrowinginstabilityfor thelongshore
currentgeometry(V0,
relationships
measured
fromfield datamaybe usefulin mapping
fi,x0)= (1.5ms-1, 0.5, 100m).Theseplotsconfirm
therather
Figure 7 showsthe spatial form of •t(x, y) of the fastest
growingwavefor thisparticulargeometryandwith depth,h0 = 1
m. The patternconsists
of a seriesof highsandlowswith each
offsetoverregionII by the complexphaseshift.Circulationof
the vorticity wave will be along streamlines,going clockwise
aroundhighswith a velocityproportionalto the local gradient.
The entirepatternis progressive
in thelongshore
direction.Figure
8 showsthe spatialpatternof the total velocity,u'= V+u. The
selectedmagnitudeof theperturbation
vorticitywaveis arbitrary;
the equationsonly showthat the motionis unstable.For this
complicated
natureof themotionsin regionII. Moreover,it can
be seenthat•tx (hencev) is not continuous.
This non-physical
behaviorstemsfrom the non-physical
discontinuities
in Vx in the
originalgeometry.In contrast,cross-shore
velocitieswill be
continuous.The surfaceelevationsignalof the vorticity waves
may be foundby equation(12). Magnitudestend to be much
smaller
than
forgravity
waves,
scaling
roughly
asVo/2g
I u,vI =
O(5cm).
Measurements
of the local phaserelationshipsbetweenfluid
variablesare often usedto distinguishclassesof wave motions
(standingversusprogressive
motions,for instance).Figure 6
showsthelocalphasedifferencebetweenu andv (Figure6a) and
betweenu andrl (Figure6b) for this geometry.Interestingly,u
the shearstructureof naturally-occurring
longshorecurrents.
figure,
thewave
motion
wasscaled
such
thatIv(x0fi)
I = V0,a
magnitude
ratherlargerthanoursmallamplitudeassumption,
but
useful for illustration.The obviousfeatureis a meanderingof the
longshore
currentaboutits positionof maximumstrength.Small
re-circulationcells are seenin the nearshoreand offshoreregions.
Again, this patternis progressive
in the directionof the mean
longshore
current,sothata fixedcurrentmeterwouldseestrong
fluctuationsovertheperturbationperiod.
Finally,the validityof the originalrigid lid assumption
canbe
examined. The ratio of the vertical acceleration and the advective
termsin thefull continuity
equation
scales
as(V-cre)2/gh
•
•=1.5
$ =0.5
O(0.52/10)= 0(0.025).Thustherigidlid approximation
is valid
-
for these motions.
% =/00
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DISCUSSION
The previous results demonstratethe mechanism of shear
instability of a mean longshorecurrent in the nearshore.The
generatedwaveshavephasevelocities,lengthand time scalesthat
are similar to thoseobserved[OSHB89], lending supportto the
model. Yet the example geometry of a flat bottom beach and
linear shearscould be consideredonly a grossapproximationof
:5"I'1'"
the natural situation. As discussed earlier, the source of the
0
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instability lies in the structure of the background potential
vorticity,Vx/h(equations(6) and (8)). Figures9a and 9b showan
actual bathymetry from SUPERDUCK (the field experiment at
which the OSHB89 data were collected) and a more realistic form
for the longshorecurrent,given analyticallyby
V(x)= V' x exp[-(c•x)
n]
(22)
whichhasa peakvelocity,
V0= V'/(ot(n
e)TM)atlocation
x•5=
1/(Or
nun).Reasonable
values
ofV0,x•5,andn were1.2ms-1,90
m, and 3, respectively.Figure 9c showsthe potential vorticity
profile for this more realisticcase.While the profile is smoother
thanthat of the example,the basicstructureis the same.Potential
vorticity still goes from positive near the shore, through a
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negativeminimum,thento zero offshore.The necessarycondition
-7/'0
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for an instability,the existenceof an extremum,is still satisfied,
x/x o
and we would expect the instability to scale roughly as the
Fig. 6. Relative phasebetween(a) cross-shoreand longshorevelocity
example.
In general,it will be true that the structureof potential
components,
and (b) cross-shore
velocityand seasurfaceelevationas a
vorticity will be basedmore on the velocity structurethan on the
functionof cross-shore
position,with (c) showingthephasedifferencesin
thecross-shore
velocitysignalbetweenthe shorelineandpointsoffshore.
beachprofile; the velocityshearcan changesign,whereasdepthis
alwayspositive.
The shearof the seawardface of the current,fs, appearsto
andv are in quadrature
in bothregionsI andIII, althoughphase
providea naturalscalingfor the frequenciesof the instability.We
differencechangesfrom a lag (u lagsv by n/2) in regionI to a
leadin regionIII. The phaserelationship
in regionII is spatially expect the greatestscientific interest to focus on conditionsof
large offshoreshearsincegrowth rateswill be fastest(strongest
complex: u andrl showphasesthat are relativelyconstantin
regionsI andIII (althoughnot multiplesof n/2) with a transition instability)and wave frequencieswill be largestand most easily
in regionII. Figure6c showstherelativephasechanges
in cross- resolved.This suggeststhatthe shearwave instabilitymechanism
shorevelocity,u, betweentheshorelineanddifferentcross-shore may be moreimportanton barredbeaches(wherethe concentration
locations.Again, a phaseshift occursin regionII, the offshore of wave breakingon the bar crestshouldforce a large shear)than
shearregionof thelongshore
current.It is conceivable
thatphase on monotonicprofiles.
,028
BOWENAND HOLMAN: SHEARINSTABILITIESOFMEAN LONGSHORECURRENT,1
/ // ////,5
2.0
_•/
/
/•
/
••• /•
/
_
/
''
•
/
/
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• ; 1/1'1111'//
;•,.', ',
!
Y/X0
I t • • • • • // / / / /
1.0
'\x
,
;/ ! ///"'
0.5
0.0
•
0.0
i
0.5
1.0
1.5
x/x o
Fig.7. Stream
function
pattern
for onewavelength
of a shearwave.Themotiondecays
exponentially
offshore.
Thepattern
is
progressive
in thepositivey direction.
The presentmodelis inviscid,showingonly the growthof an
instability. The proper inclusionof viscositymay be difficult
since advective terms will also be required. As a rough
approximation
we caninvestigate
theconsequences
of includinga
linearizedfrictionof the form -3•u. In thiscase,dissipationcanbe
incorporated
into the imaginarycomponentof frequency,{Jim,SO
thatgrowthwill occurasexp[(Oim-30t].
Thustheinstabilitywill
be dampedunless{Jimis greaterthandissipation.However,once
Oim> •, theinstabilitywill suddenly
startto grow.If we take3•=
CdVh
-], thentypical
values
of3•willbeO(10-2-10-3),
sogrowth
will not occurunless{Jimis greaterthan this value. Thus shear
waves are theoretically expected to be less important on
monotonicprofileswherethe offshoreshear,fs, is not largeand
{Jimis expectedto be small.
In contrastto edgewavesor leakymodes,for anyfrequencyof
shearwave there is only a singlewavenumber(for a particular
longshorecurrentgeometry).Thuslag coherence
plotsfor shear
wave dominated bands should be more coherent than those where a
suiteof edgewavesandleakymodesareimportant.Increases
of
wavenumberbandwidthbeyondthe expectedresolutionof the
analysiswouldresultonly from fluctuationsin meanlongshore
currentstrength
(overtimescaleslongerthana shearwaveperiod)
andwouldhavea relativemagnitudeAk/k-- AV/V, whereAV is
theunsteadiness
of the meanlongshorecurrent.Wavesfrom days
of stationaryand homogeneous
longshorecurrentmay be very
coherentin the longshore.
Shear waves may have several important consequencesto
nearshore
dynamics.Previously,dissipationof longshorecurrents
was assumed to be simply through bottom friction. This
mechanismshowsthat stronglongshorecurrentscanbe unstable.
The dissipationof the instability would undoubtedlybe only
poorlymodelledby traditionaldissipationterms.Moreover,the
dissipationwill dependnotjust on the magnitudeof the current,
but also on the maximum shear, a characteristic that will
distinguish
barredfrommonotonicbeachprofiles.
As shownin Figure 8, shearwaves can also act as a strong
mechanismfor mixing in the cross-shore.If we parameterize
horizontalmixing with a horizontaleddydiffusivity,AH = UX,
where U andX are characteristiccross-shorevelocity and crossshore displacementscalesof the mixing agent, then we can
BOWENANDHOLMAN: SHEARINSTABILITIES
OFMEANLONGSHORE
CURRENT,1
2.0
, • • • • *' • •' \ •' ........
1.5.
' '
•
....
•
....
.......
,,•X•
.
N X X •
•
• ..........
•
•
,
.
.
,
• • t ....
• • • • • • , .........
.....
, .....
.
. . , • • I • t t • ,
.
' '
Y/Xo
•
18,029
. . , t • t t t t • • .....
' '
t
t t t t t • • ,
. -
0.5-
,,
• t t t t •••,,,
..............
, , + t t t t t t t .......
, t t t t t t t • t .........
2.Sin
sec
-1
0.0 , , , t t t t t
0.0
0.s
•.0
•.s
x/x o
Fig.•. Totalv•locitypatternfor on• wavelength
of th• wav• motion.Th• shearwav• hasb•n scaledsothatits p•ak
magnitud•
•qualsth• p• longshor•
cu•nt, Vo.Th• motionisprogr•ssiv•
in th• direction
of th• longshor•
cu•nt.
comparethe relativerolesof incidentand shearwavesin mixing.
Both will have similar velocities,but an excursionscaleof the
shearwavesmay be 50 m (Figure8), comparedto approximately
5 m for incident wave. Thus shearwaves may at times be the
predominant
mechanism
forhorizontal
mixing.
Given the apparentstrengthof thisinstability,it may seemodd
that these motions do not dominate in more circumstances,
particularlyin wave tank experimentswhere steadylongshore
currentsaregenerated.
Total velocityfigures(similarto Figure8)
were calculated for a variety of shear wave magnitudes.
Surprisingly,
for shearwavemagnitudes
of up to 50% of thepeak
longshorecurrent,the resultingpatternappearedstraightand
steady,indiscernibleby eye from the steadyflow alone.Thus
wave tanksexperimentsmay in fact be contaminatedwith shear
wave motions that, in the limited length of the tank, have not
grown to sufficientsize to be detectableby rudimentarymeans.
Contaminationmay becomea problemfor long tanks(relativeto
thewidth of thelongshorecurrent)or for a longshorecurrentsin a
spiralwavetank.
CONCLUSIONS
It has been shown that a steady longshorecurrent in the
nearshorecan supportoscillationsthat are not gravity waves,but
which are much shorterscaleshearwaves (or vorticity waves).
The underlying mechanismis the conservationof potential
vorticity, the sameas for larger scaleoceanographicflows, but
with the shearof the longshorecurrent,Vx,playingthe traditional
role of Coriolis. Thus a perturbationacrossthe background
potentialvorticity, Vx/h,will be balancedby a relativevorticity
in the watercolumnwhichactsasa restoringforce.
The mechanismmay be unstableif an extremumof background
potential vorticity exists in the region (true for all longshore
18,030
BOWENANDHOLMAN:SHEARINSTABllJTIES
OFMEANLONGSHORE
CURRENT,1
1.5 *
*
* a
Spatialpatternsof velocitydependon therelativemagnitudeof
the shearwave and the meancurrent.For shearwave magnitudes
lessthan 1/2 the peak longshorecurrent,the vorticity wave may
•
instruments).For shearwaves that are comparableto V0, the
resultingpatternis oneof a clearlymeanderinglongshorecurrent.
The relativephasebetweenwave variablesu, v, andq vary with
cross-shore
position;theseexpectedvariationsmay proveuseful
in inferring the shearstructureof naturallyoccurringlongshore
currentsfrom measuredphaserelations.It is expectedthatthevery
simple geometry of the model will reproduce the important
featuresof morerealisticbathymetriesandcurrentstructures
since
it mimics the essential elements of the potential vorticity
1.0
appear
negligible
to theeye(although
clearlydetectable
to
0.5
0
-2
structure.
Based
onsimple
scaling
arguments,
shear
waves
areexpected
to
-3
-4
-5
0.03
½
0.02
O.Ol
o
-o.o
1
I
o
,oo
,,,oo
x(m)
Fig.9. (a) Morerealistic
profileof lon•gshore
current.
Analytical
form
is given by (22) with V 0 = 1.2 ms-1 , xfi = 90 m, and n = 3. (b)
Representativebathymetryfrom the barredbeachat Duck during the
SUPERDUCK experiment.(c) Resultingpotentialvorticityprofile.
providean effectivehorizontaleddyviscositythatcanbe up to 10
timesgreaterthanthatfor incidentbandgravitywaves.Similarly,
the ultimate dissipation of longshore currents may depend
critically on the shearinstability,with the traditionaldissipation
models providing only a crude approximation of the true
mechanism.Theselatter conclusionsdependon the existenceof
the instability,which in tum dependson the offshoreshear,fs.
Thus barred beaches,where longshorecurrent shear can be
concentrated
nearthebarcrest,may exhibitfundamentally
different
dynamicsthanmonotonicbeaches.
Acknowledgments.Much of this work was carriedout while R.A.H.
was on sabbaticalat DalhousieUniversity, andhe would like to thankthe
oceanography
groupfor theirkind hospitality,stimulatingenvironment
anddonuts.The entirestudyresultedfrom FIG bandobservations
madeby
Joan Oltman-Shay, Peter Howd, and Bill Birkemeier during
SUPERDUCK and has benefitedfrom lively discussions
with them. The
authorswould also like to thank Ed Thornton,Tony Dalrymple, Jurgen
Battjes,and Dave Hebert for useful discussions
on the impactof FIG
waves.As always,we would like to thankthe staffof the Field Research
Facility andCurt Mason,withoutwhoseeffortsnoneof this wouldhave
happened.
Continuingandappreciated
supportfor R.A.H. wassupplied
by
Office of Naval Research,CoastalSciencesprogram,undercontract
N00014-87-K0009. A.J.B. is fundedby National ScienceandEngineering
currentson naturalbeaches).In thatcase,theperturbation
may ResearchCouncil, Canada.
growwithaninitialgrowthratethatis exponential.
Fora simple
REFERENCES
examplegeometry,
thecelerityof thesemotionsis shownto vary
asapproximately
1/3 V0, whereV0 is thepeaklongshore
current. Munk, W., F. Snodgrass,andF. Gilbert, Long waveson the continental
shelf: An experimentto separateleaky andtrappedmodes,J. Fluid
A naturalfrequency
scalingof themotionsisfs, theshearon the
Mech., 20(4), 529-544, 1964.
seaward
sideof thelongshore
current.Theinstability
occursfor a
Niiler, P. P., and L. A. Mysak, Barotropicwaves along an eastern
rangeof frequencies
fromabout0.1fs to a lowerlimitdepending
continentalshelf,Geophys.Fluid Dyn., 2,273-288, 1971.
on fi, the relativelocationof the peakvelocityin the mean
longshorecurrent.If we take the fastestgrowingwavesas
representative,
thecharacteristic
longshore
wavelength
is of order
2x0, where x 0 is the width of the longshorecurrent. A
characteristic
frequency
will be0.07fs andgrowth(e-folding)
time
scalewill beabouthalfof oneperiod.Obviously
theinstability
is
strongand assumptions
of linearitywould soonfail. Inclusionof
a simpleRayleighdamping
factor,)•, suggests
thattheinstability
will not growunlessthe calculatedgrowthrateis fasterthan)•,
whichshould
beO(10-•-10-3)Hz.
Oltman-Shay,J., P. A. Howd, and W. A. Birkemeier, Shearinstabilities
of the mean longshorecurrent,2, Field Observations,J. Geophys.
Res., this issue.
Rayleigh,Lord (J. W. Strutt), On the stability,or instability,of certain
fluid motions,Proc. London Math. Soc., 11, 57-70, 1880.
A. J. Bowen, Departmentof Oceanography,DalhousieUniversity,
Halifax, Nova Scotia, CanadaB3H-4J1.
R. A. Holman, College of Oceanography,OregonStateUniversity,
Corvallis, OR 97331.
(Received March 1, 1989;
acceptedMay 6, 1989.)