GEOPHYSICAL
RESEARCH,
VOL. JOURNAL
OF 97, NO. C4, PAGES 5623-5635, APRIL
15, 1992
A Model
for the Generation
of Two-Dimensional
LIST
Surf Beat
JEFFREY H.
Center for Coastal Geology, U.S. Geological Survey, Saint Petersburg, Florida A finite difference model predicting group-forced long waves in the nearshore is constructed with two interacting parts: an incident wave model providing time-varying radiation stressgradients across the nearshore, and a long-wave model which solves the equations of motion for the forcing imposed by the incident waves. Both shallow water group-bound long waves and long waves generated by a time-varying breakpoint are simulated. Model-generated time series are used to calculate the cross correlation between wave groups and long waves through the surf zone. The cross-correlation signal first observed by Tucker [1950] is well predicted. For the first time, this signal is decomposedinto the contributionsfrom the two mechanismsof leaky mode forcing. Results show that the cross-correlation signal can be explained by bound long waves which are amplified, though strongly modified, through the surf zone before reflection from the shoreline. The breakpoint-forced long waves are added to the bound long waves at a phase of rr/2 and are a secondarycontribution owing to their relatively small
size.
1.
INTRODUCTION
wave crests, h is the water depth averaged over the incident
wave period,and ca is the groupvelocity.A more recent
The term "surf beat" was first applied more than 40 years ago to low-frequency sea surface oscillations that correlated with incident wave groups in the nearshore [Munk, 1949; Tucker, 1950]. Tucker [1950] calculated the time-lagged correlation (cross correlation) between the incident wave envelope and low-frequency motions (periods of the order of 3 min) at a measurementlocation approximately 1000 m from the shoreline. His results, reproduced in Figure 1, showed a distinctive signal of both positive and negative correlations, which appeared at a time lag sufficient for wave travel from the recording location to the shoreline and back. These initial observations have been followed by numerous measurements of the size and structure of these low-
approachby Ottesen Hansen et al. [1981] includes the effect of group frequency by modeling the interaction of all possible pairs of incident wave frequencies. Both approaches predict a phase-locked or "bound" long wave traveling at the group speed, rr out of phase with the wave envelope. The size of the bound long wave (BLW) in deep and intermediate water is reasonably well-predicted [Sand, 1982b; Shi, 1983]. However, in shallow water this theory predicts unreasonably large wave heights and therefore cannot be applied to
the nearshore.
Another mechanism for the generation of leaky mode long waves was proposed by Symonds et al. [1982], who begin with the depth-integrated, linearized shallow water equations of momentum and continuity,
frequency motions, now generally known as infragravity waves and loosely defined by the frequency band 0.05-0.005 Hz. The significance of these waves is well recognized; during storms the spectra of run-up and inner surf zone surface elevation and velocity may be dominated by energy in this band [e.g., Guza and Thornton, 1982; Wright et al., 1982; Holman and Bowen, 1984]. These motions fall into two general categories: edge waves, which are modes trapped in the nearshore by refraction, and leaky mode waves, which are reflected from the
shoreline and radiated offshore. Studies have confirmed that
On
+=0
Ot
O(hu)
Ox
(2)
Ou --+ Ot
Orl g --= Ox
1 OSxx
ph
Ox
(3)
where u is the depth-integrated cross-shore velocity averaged over the incident wave period with positive u directed
offshore and x is the cross-shore coordinate with x = 0 at the
both types of infragravity waves are important in the nearshore [Huntley et al., 1981; Oltman-Shay and Guza, 1987]. Several models have been proposed for the generation of leaky mode waves, and these are the focus of this paper. Longuet-Higgins and Stewart [1962, 1964] showed that a second-order group-forced long wave could be described by
shoreline (Figure 3). Group-induced breakpoint variations were parameterized in terms of a Fourier series of OSxx/Ox forcing functions, in essencegenerating a time-varying setup due to incident wave height variations (groupiness). The
term OSxx/Ox varies as a function of time within the zone
r/(t) =
p La_ %2 h
1 1 [Sxx(t)
(1)
where r/(t) is the time-varying sea surface elevation averaged over the incident wave period, p is the density of water, Sxx(t) is the time-varying radiation stress perpendicular to
This paper is not subject to U.S. copyright. Published in 1992 by the American Geophysical Union.
Paper number 91JC03147.
5623
which alternately contains breaking and nonbreaking waves; this acts as a wave maker that radiates long waves at the group frequency and higher harmonics in both the shoreward and seaward directions. Inshore of the forcing zone, the superposition of the landward-radiated wave and its shoreline reflection create a standing free wave. The region offshore from the forcing region contains the superposition of two progressive waves: one radiated seaward directly from the forcing zone and one radiated seaward from the shoreline reflection. The breakpoint-forced long wave (BFLW) at the shoreline should show a positive correlation
5624
LIST: MODEL FOR GENERATION OF SURF BEAT
+40
A
12-0 6 -6-0 12-0
height gradients are accounted for. The use of random wave groups allows a direct verification of the model using field data in the form of the cross-correlation signal. Also, the methodsemployed allow the numerical separationof the two modes of leaky wave forcing, giving a first-ever look at their relative contribution to the long-wave field in the nearshore.
--- -40
o
2.
long w&ve
MODEL
FORMULATION
found
The model developed here contains two main parts: an incident wave model, which provides a time-varying forcing Fig. 1. Observationby Tucker [1950] of a time-lagged correlation (OSxx/Ox)and a long-wave model, which solves equations between wave groups and long waves 1000 m from the shoreline. (2) and (3) on a finite difference grid [List, 1988]. The model is run in the steady state or setup mode to provide the initial condition prior to generating long waves. The contributions with the wave envelope offshore (with an appropriatetime of the bound and breakpoint-forced models can be sepalag), contrary to that predicted for the BLW. rated numerically. Field evidence supportingeither of these mechanismsfor leaky wave generationis weak. Both Longuet-Higginsand Incident Waves Stewart [1964] and Syrnonds et al. [1982] use the classic Tucker [1950] observation (Figure 1) as support for their To model (2) and (3) on a finite difference grid in the time models; the signal shows what appear to be significant domain, an incident wave model was developed to provide correlations in both a positive and negative sense (confi- instantaneoustime series of wave heights at all cross-shore dence limits unreported). More recent observations have locations under the assumptionsof normally incident waves elaborated on Tucker' s work but have not added a great deal and two-dimensional topography (no alongshore variation). to our understanding. Guza et al. [1985] and List [1987] Because the incident wave heights are assumed to vary as a
,. () velo (.)
the
cross correlation
between
the incident
wave
steady state or probability distribution transformation models are not useful in this study. Thus many previous approaches, including those of Battjes and Janssen [1978], Thornton and Guza [1983], and Dally et al. [1985], could not be adopted here. The present approach begins with the offshore input of an observed wave height time series H(t) (twice the incident wave envelope A(t)). This group structure then progresses acrossan arbitrary bottom profile, with each wave height in several studies conducted under similar conditions have the series H(t) shoaling or breaking in appropriate water yielded incomplete and sometimesconfusingresults. Man- depths as defined below. sard and Barthel [1985] investigated the BLW shoaling A fixed distance step, suchas is commonly used in models characteristicswithout any apparent need to account for the predicting steady state wave height transformations, was potential BFLW contribution. (The presence of this wave inadequateto preserve the form of the wave group structure would have altered the results.) Conversely, Kostense [1985] during its landward progression.This is becausethe distance measured outgoing wave amplitudes which varied strongly traveled by incident waves within the model time step At as a function of frequency, a prediction of the time-varying varies with the water depth over the profile. In this case, breakpoint model, while making the assumption that the H(t) must be continually reinterpolated onto the fixed set of incident BLW is not released from group forcing and re- cross-shore grid locations. This results in a considerable flected as a free long wave in the surf zone. These lab studies amount of numerical dispersion, with a progressive loss of give perhaps a hint that both generation mechanismsmay be group structure toward the shoreline. valid, but they do not provide information on the relative or To avoid this problem, a floating grid scheme was devised
absolute sizes of the BLW and BFLW in the nearshore.
components of the long waves at a series of locations through the surf zone. Results clearly demonstrate that a negatively correlated, BLW-type signal offshoreevolves to a negative/positiveform near the surf zone, which then appears unchangedin the offshoreprogressivecomponent.List [1987] speculated that this may result from the addition of the BFLW to the BLW in the surf zone, followed by near-complete shoreline reflection. Laboratory work has not provided a much clearer picture;
envelbpe offshore the onshore offshore and and progressive represented through the regions of shoaling and breaking,
function of time and the resulting group structure must be
A basic problem with both field and laboratory studiesis the lack of an effective method of identifying the separate contributions of the BLW and BFLW. Without this ability, group-correlated long waves in the nearshore can not be conclusively attributed to either generation mechanism. In this study, a finite difference model is constructed
which simulates, in the time domain, the simultaneous generation of both bound and breakpoint-forced long waves
such that zXxwill always equal the distance traveled by a wave over the sloping bed in the At. If an initial set of grid
points is defined by x l, hi, x2, h2 as in Figure 2, then assuming shallow water phase speeds, a new grid point location x' is found as
xi = x2 +
h2- ([tan/3At(9)1/2/2] (h2)1/2)2 +
tan/3
(4)
in the nearshore. This effort differs from earlier modeling where tan/3 is the beach slope defined as negative sloping efforts [e.g., Symonds et al., 1982; Schitffer and Svendsen, seaward. Because (4) is not defined for tan/3 = 0 (such as 1989] in that random groups may be used and the forcing is over a bar), a criterionis employed suchthat when Itan/3 I
parameterized such that both broken and unbroken wave
<0.001,
LIST: MODEL FOR GENERATION OF SURF BEAT
5625
X X.' I
.
tions of continuity and momentum given by (2) and (3). The cross-shore radiation stress Sxx is found after LonguetHiggins and Stewart [1964] as
3
Sxx 16 H2 = pg
(7)
Fig. 2. Incident waves model: definition diagram for finding a new grid point, x', basedon the shallowwater travel time from x2 within the time step,/xt.
xl = x2_ At(gh21/2 )
(5)
where shallow water and linearity are assumed and H varies through time and space. $tive and Wind [ 1982] again provide evidence that the linear wave assumptionis adequate; setup predicted using (7) agrees quite well with measurements. The earlier study of Bowen et al. [1968] also substantiates the use of (7), although some overpredictions of setdown at the breakpoint were observed. Equations (2) and (3) are modeled using a finite difference schemewith a staggeredgrid as shown in Figure 3. A stable
solution is found as
which results in negligible error because the bed is near horizontal in this case. This procedurefor locating new grid points rapidly readjustsan initially constantAx grid so that the distance between all grid points represents the wave travel distance in At. Water depths may include setup and sea surfacevariationsassociated with long waves. However, (4) restricts incident waves to shallow water in which kh < ,r/10, where k is the wavenumber. With this gridding system, wave groups in the surf zone may then be modeled by defining the wave height at each positionas a function of the shoalingor breakingof the wave height from the next grid location seaward. Here the wave height H at x is found as
At
m+l (hu" 77i = n -- XX
At
huim_l)
(8)
m+l m -re+l) Ui-1 Ui-I gXX n+l-- ?i-1 -(?7'
At
phAx
(Sxx?
Sxxi_1
(9)
Hx,, min Hx2 =
, hl
(6)
where , is the breaking wave height to depth ratio, the first term within the bracketsrepresentsshoaling,and the second term represents depth-limited breaking Equation (6) is a rather simplistic approximation, including the assumptions of linear shoaling,a constantbreaker height criterion, and the absence of refraction, diffraction, or frictional dissipation. However, there is ample evidence to suggestthat the approximationgiven by (6) is adequatefor the presentstudy. For example, in a laboratory study of radiation stress gradientsin the nearshore, Stive and Wind [1982] compare the measuredradiation stressacrossa shoalingand breaking region with two nonlinear theories [Cokelet, 1977; James, 1974] and linear theory following (6). Depending on the choice of % Stive and Wind found that linear theory fits the observations equally as well as do nonlinear theories. With this incident wave model, Sxx gradientsassociated with unbroken waves may progressto the inner part of the nearshorewhen H(t) is small. Conversely, when groupsof large waves are incident (H(t) large), the zone of Sxx gradients associatedwith broken waves expands seaward. Thus Sxx gradientsassociated with breaking and nonbreaking waves may alternately occupy the same region, underscoringthe need for a long-wavemodel that can accountfor both breakpoint-forced and bound long waves in the surf
zone.
where rn representsthe time step, i is the distance step, and the water depth h includes the setup (or setdown) and may include ?7if this mode of long/short wave interaction is desired. Note that this scheme is semi-implicit in that u is calculatedusing ?7 the current time step but ?7 calculated at is using u at the previous time step. Unlike the incident wave model, &r is constant here. Incident wave heights are therefore regridded at each time step to provide the radiation stress at each of the ?7grid points in Figure 3. Since the incident wave model operates with a floating grid independently of the long-wave model, the regridding of incident wave heightsto the long-wave grid at each time step does not result in cumulative numerical dispersionof Sxx gradients Equations (8) and (9) are subject to the Courant condition, in which stability is ensured when
Ax/At> (gh)1/2
(10)
Uo:O u, .:.-:..... ': . .
.
Equations of Motion
.
Following Symondset al. [1982], the equationsmodeled here are the depth-integrated,linearized shallow water equa-
Fig. 3.
Long waves model: staggered grid scheme. Here x is defined as positive offshore.
5626
LIST: MODEL FOR GENERATION OF SURF BEAT
where(qh)1/2represents shallow the water,linearphase
speed. Numerical dispersionis also a concern here. Following Roache [1972], the numerical dispersion is found using Hirt's stability analysis as
T/max-- T BLW + T/max(Off)
(16)
(tlhAt/2)(O2u/Ox 2)
(11)
where TBLW is the incoming bound wave. This method is similar in concept to the "matched impedance condition" [Shaw, 1970; Vemulakonda et al., 1988] in which reflection is suppressedat the boundary when Oq/Ot= -c(Oq/Ox) (17)
which can be shown to have the units of kinematic viscosity.
Since 02U/OXis small thecase longwaves, should 2 in of (11)
also be small. This may be conveniently tested by varying At and observing the effect on the solution. In this study it was found that cutting At in half had no discernible effect on the results, suggesting that the numerical dispersionwas negligible. Finally, two boundary conditions are required in (8) and (9). At the shoreline, long-wave reflection occurs at some predetermined minimum water depth measuredat an T grid point. This condition was used to avoid stability and continuity problems with a continually moving boundary. Although Reid and Bodine [1968] have proposed a movingboundary scheme in a numerical model and Carrier and Greenspan [1958] have theoretically derived expressionsfor wave motion over a sloping bed, the condition
However, this is extended here to include the addition of the incoming BLW. If the proposed method is successfulat suppressingopen boundary reflections, the shoreline amplitude of the long waves generatedshouldbe independentof the standingwave node/antinodestructure relative to the open boundary. Tests with monochromatic groups of widely varying frequencies
demonstrated that this was the case [List, 1988]. Other tests,
including an examination of cross-correlation signals between groups and long waves, showed that a negligible
amount open of boundary reflection occurred [List,1988].
By using a higher-order extrapolation than (15), a smaller degreeof reflectionmay be possible,but was not considered necessary here.
huo = 0
(12)
Setup
where u0 is the most landward u grid point, has been used
here for simplicity. For boundaries such as a steepforeshore this is a reasonable approximation in that the wave travel time is not significantly affected. This condition has also been used in numerous previous studies, including those of Dube et al. [1985], Vernulakonda et al. [1988], and many
others.
The second boundary condition is T at the open end, for which two requirements must be satisfied. First, there must be an incomingbound long wave as a function of the incident group structure. Second, any long waves radiated seaward from the surf zone must pass freely through the boundary
without reflection.
To satisfy both boundary requirements, a scheme was devised to isolate the outgoingwave at the boundary, which is then added to the incoming B LW to give the total boundary condition. The two T grid points shorewardof the outside boundary, Timax-1and Timax-2(Figure 3), are decomposedinto the shoreward and seaward progressivecomponents T(on) and V(off) following Guza et al. [1985] as
Before solving for long waves, the model is first run in setupor steadystatemode, providinga mean water level and mean wave height profile to the long-wave model. In effect, this results in a separationof the solution into the mean and fluctuatingparts very similar to the analytical separationof Symonds et al. [1982]. This separation is necessary here to determine the position of the shorelineboundary condition; setup may result in a significantshift in the x position of the shoreline as defined by the minimum depth criterion. The setup solution is therefore designed to allow for a moving boundary. To find the steady state solution, the mean of the wave height time series, H(t), is input at the seaward boundary. As the setup develops, the shoreline is allowed to migrate
landward. After a sufficient time for stabilization, the steady state wave height and setup/setdownprofile results, with u
approachingzero at all points. In addition, water depths in
the incident wave model are modified by the evolving setup
and setdown.
n(on) =
q- (h/#)1/2//
2
(13)
(off) = /
n + (h/#)1/2U
2
(14)
Figure 4 gives an example of the solution for setup and steady state wave height for the simple case of a linear profile. Although the model works equally well over irregular and bar-troughprofiles, the linear case shownhere can be tested directly against the setup prediction of Bowen et al.
[ 1968]:
For the case of model generated data and shallow water groups this separation works extremely well, without the problems associated with field data (obliquely incident waves, presenceof edge waves, etc.). The outgoingcomponent at the boundary is linearly extrapolated at each time
step as
dT
dx
= -(1 + 2.67y-2)-1 tan/3
(18)
Over a linear profile from the break point x 2 to the shoreline x l, this can be written as
/max(Off) T imax-l(Ofq- (T imax-l(Of --- T imax-2(Off))
The total boundary T is then found as
(15)
ATxx,x2 + 2.67y-2) tan = --(1 -1 /3(x2-xl) (19)
within the breakingwaves zone. In Figure 4 the changein sea surfaceelevation from the point of maximum setdownto the shoreline AT = 0.260 m; equation(19) givesAT = 0.262 m, is
LIST: MODEL FOR GENERATION OF SURF BEAT
5627
Tg = 61.6
s..
t = 240.
BLW
Hrn
H o
o.d
-
ie
(e) o. o:
oo 200 300 400
o
D'rST
2.
(c)
,,,,,
lOO
200
3
o
4
o
DXST
o oo 2 o 300 oo
D'rST
Fig. 5. Long-wave generation by incident wave groupiness, usingthe setup solution shown in Figure 4. Incident waves with GF
Fig. 4. Example of the setup solution using the long-wave equations of motion taken to the steady state. A constant wave height of H = 0.8 m enters the systemat x = 360 m with tan/3 = 0.025 and 3' = 0.7.
= 0.5 aredefined Hi,ha = 0.8 q-0.4 sin[(2r/Tg)nAt] with Tg by x m,
= 61.6 s, and 3' - 0.7. The input BLW height is found using(22) with B = 0.02. The solution is shown at t = 240 s for (a) the constant and fluctuating wave height profiles, (b) the forcing term, given by (21), and (c) the instantaneous sea surface and current across the nearshore (u defined as positive offshore).
demonstratingthat the model proposed here is in excellent agreementwith Bowen et al. [1968].
Long Waves
Long waves are generated by allowing a time-varying wave height to enter the model with the setup solution as an initial condition. For each model run, the mean of the input wave height time series must be the constant value used in the setup solution. The water depths in the long-wave model are the profile depths plus the steady setup (or setdown)
solution. Radiation stress is found as the difference between the
The incoming BLW at the boundary, ,/BLW, can be set equal to zero, the analytical prediction of Longuet-Higgins and Stewart [1964] or Ottesen Hensen et al. [1981], or some empirical function of the wave height squared such as
vBLW;na _B[(Hinax_ (Hnax)2] x __ )2
(22)
where B is a constant coefficient. With (22), an input B LW
maybegenerated correlates H(t)2, aspredicted that with by
(1), but with any desired size. Figures 5 and 6 give examples of the long-wave solution at times correspondingto the maximum and minimum breakpoint positions, respectively, for the simplified case of monochromaticgroups with a groupinessfactor GF of 0.5 as defined by List [1991] (pure sinusold;GF = 0.0, two beating sinusoldsof equal amplitude; GF = 1.0). For this model run, the steady state setup and wave height profiles are shown in Figure 4. The incoming BLW was generated using (22) with
instantaneousand steady state values at each location:
3
Sxx? =
_
2]
(20)
where Hf is the steady state wave height. The magnitude of Sxx is unimportant here; only Sxx gradients affect the forcing term,
At
phAx
(Sxx? - Sxxim_l)
(21)
B = 0.02, givingHBLw = 0.026 m, a small incomingwave of the order of that observed by List [1987]. For this example and other model results presented below, the size of the incoming BLW at the offshore boundary is an order of magnitudesmaller than the long waves generatedtoward the
5628
LIST: MODEL FOR GENERATION OF SURF BEAT
Tg = 61.6
s.,
t = 285.
1.0
Hm
BLW '
Hc
, 0.0
0.5
t=O
-0.5
t=Tg/2
0.
-! o0
O.
i i
I O0.
200.
300.
400.
DISTANCE
OFFSHORE
(.)
Fig. 7.
Sum of the first three time-varying forcing terms from
Syntonds al. [1982] t = 0 andTa/2.GF = 0.5. et at
continuously varies. In Figure 5 an incident wave height maximum has just progressed to the breakpoint, and the BFLW forcing zone is defined as the region in which the instantaneousbreaking wave height exceeds the steady state wave height. In Figure 6 a wave height minimum has 0oe progressedto the nearshore, and the BFLW zone is where the breakingwave heightin the steady state solutionexceeds the nonbreaking wave height in the instantaneous solution. Inshore of the BFLW region there is no forcing, and offshore unbroken wave height gradients force the bound long wave. The model is run in BFLW mode by zeroing the forcing -2 term (equation (21)) outside the BFLW zone and assuming 1 o 200 3 4 0 no incoming BLW at the open boundary. Alternately, the DTST model is run in BLW mode by zeroing the forcing term Fig. 6. Same as Figure 5 except with a solutiontime at t = 285 s. within the B FLW zone, and including an input B LW. The division between BFLW and BLW forcing zones varies between the maximum and mean break point and must be shoreline. Figures 5 and 6 show the instantaneousand steady redefined at every time step. With a long-wave-dependent state wave height profiles, the forcing magnitude, and the water depth as described in the previous section, the separation of breakpoint-forced and bound long waves is not resulting long wave expressedby /and u. exactly linear. However, because of the restriction here to very weak model nonlinearities, the summationof the indeNonlinearities pendently generatedBLW and BFLW is nearly identical to As was mentioned above, ,/variations due to long waves the long-wave-generatedin total forcing mode. At this point a comparisoncan be made between the form may be included in the water depths used to find incident wave heights as well as /itself. In the later case, however, of the breakpoint forcing used here and the analytical the small-amplitude assumptionsof linear wave theory are forcing function of Syntonds et al. [1982]. The sum of the violated if q/his not small. For the caseof a large-amplitude first three time-varying forcing terms of Symonds et al. is long wave traveling over a shallow bed, the model generates given by
-
i
i
!
i
I
bores.
Nonlinear
effects
such as these are not within
the
scope of this study; the model runs presented below generally have q/h-< 0.1, for which these nonlineareffectswere not observed. This small amplitude assumptionhas been used in numerous studies, including those of Holntan and Bowen [1979] and Syntonds et al. [1982].
Separation of Forcing Modes
To assessthe relative contributions from the time-varying breakpoint and bound long-wave models, a forcing separation scheme was devised. In Figures 5 and 6 the forcing regions attributed to the breakpoint-forced and bound longwave models are delineated by vertical lines. Note that the BFLW region is defined within the zone of breaking waves, whether in the steady state or time-varying solution, and that the boundary between the BFLW and BLW forcing zones
3[ n,r (nooat) a(x*) [sin(nr)] (23) =n__ cos
r = cos -
GF
(24)
where istheradian % group frequency (2,r/Ta)and isthe x*
offshore distance normalized by the position of the mean breakpoint. This function is shown in Figure 7 at t = 0 and
t = Ta/2withGF = 0.5 andx scaled in Figures and6. as 5
Figure 7 shows that the analytical forcing function of Syntonds et al. [1982] has a form similar to the BFLW forcing schemeadoptedhere (equation (21), Figures 5 and 6). At t = 0 in Figure 7, a(x) shows a positive peak which is shifted
seaward the negative of peakoccurring t = Ta/2. Figure at
LIST: MODEL FOR GENERATION OF SURF BEAT
5629
5 shows a positive peak within the BFLW forcing zone which is shifted seaward of a negative peak occurring
(A)
0.6 0.5
approximately s laterin Figure6. Tg/2
However, several differences between Symonds et al. [1982] and the present formulation are evident. First, the forcing here is not symmetricabout zero, with larger positive than negative values. As can be seen in Figure 6, when the break point is at a minimum, the wave height gradient in unbroken waves nearly matches the gradient in the steady state solution, resulting in a lower forcing term than in the maximum break point case (Figure 5). Symondset al. did not
account for this.
0.4
0.$
0.2
0.1
&MODEL
(B)
r .04 .0
(c)
.081
.0 1254 5 6 7
The second feature not included in the theoretical approach is the time asymmetry of the minimum and maximum breaker positions. The time difference between Figure 5 (maximum break point) and Figure 6 (minimum break point)
.4
0.3
is 45 s, which is 14 s longerthan Tg/2. This extra 14 s
represents the short-wave travel time from the maximum to minimum break point positions. Although this effect was noted by Symonds et al., the forcing term (23) neglects this time delay, restricting the analysis to low GF. It is interesting to note that Figures 5 and 6 show a larger forcing magnitude within the BFLW zone than in the BLW zone, leading to an expectation that breakpoint-forced waves should be more energetic than the BLW. The relative magnitudesof break point-forced and bound long waves are examined below for model runs simulating a field situation.
OISTAtVCE OFFSHORE
Fig. 8. Cross-shore variations in incident wave parameters for field model and generated data: (a) groupiness factor, (b) standard deviation of the envelope, and (c) Hrms.
optimized the cross correlations in the field data. Both field and model A(t), q,and u are thus restricted to the 0.007- to 3. MODEL CALIBRATION: DUCK85 0.03-Hz band in the analysis presented below. A(t) progresses landward at the shallow water group List [1987] examined the relation between the incident speedand shoalsuntil the wave height to depth criterion y is wave groupinessand nearshore long waves for a cross-shore exceeded for each discrete segment of record. The criterion array of current and pressure sensors from the DUCK85 y is the primary parameter used to calibrate the incident experiment on September 9, 1500 eastern standard time. In wave model to match the degree of wave height variability this section the model is designed to simulate the field observed in the field data through the surf zone. Because the conditions during this experiment. generation of long waves is primarily related to the degree of incident wave height variability, y was adjusted suchthat the Incident Waves standard deviation of A(t), trA , in model data deviated Observed incident waves had a narrow-banded spectrum minimally from trA in field data. Figure 8 compares model with a peak period of 12 s and Hrms = 0.42 m. The nearshore and field-observed cross-shore variations in GF, trA , and morphology lacked bars (Figure 8) and was nearly two Hrms using y-- 0.4. Within the breaking-wave region, model dimensional, although some minor longshore irregularities and field observed trA differ by an average of 5%, although existed close to the shoreline. Although directional data maximum deviations reach +-25%. Differences for GF and were not obtained, the incident waves were visually close to Hrms are greater. normally incident and are assumed to be so here. The clear cross-correlationsignalbetween the incident group structure offshore and both the onshore and offshore progressive Long Waves components throughout the nearshore observed by List The long-wave model was run with At = 0.5 s and two [1987] supportsthis assumption.Days with oblique incident main controls: the boundary condition BLW and the point of waves showed a rapid decay in the group/long wave corre- shoreline reflection. Since the seaward boundary of the lation as the spatial separation between sensorsincreased. model (location 9, Figure 8) is considered shallow water for However, as was noted by Sand [1982a], the directional the incident wave groups, theoretical BLW predictions spread of the incident waves may have a significant influence could not be used for an input condition. Using the crosson the size of the bound wave, at least in deep water. How correlation signals, List [1987] roughly estimated the BLW this may affect the two-dimensional, shallow water formulasize at location 9 to be Hrms -- 0.01 m. Here an input BLW tion used here is unclear. of this size is generated by letting B = 0.032 in (22). As Incident waves are simulated here when the wave envementioned previously, this initial long wave is an order of lope, A(t), enters the model at the seaward boundary and magnitude smaller than long waves generated closer to progresses over the nearshore profile shown in Figure 8. shore; the model results are not very sensitive to the exact choice of B. A (t) was derived through filtering methods [List, 1991] from Long waves are reflected from the shoreline at the x field measurements of pressure (number of samples n = 4800, At = 0.5 s) at profile location 9. The frequency content position where h = 0.63 m, including the setup contribution of A(t) was chosen to match the band of frequencies which (0.03 m). This results in a reflection point on the lower
5630
LIST: MODEL FOR GENERATION OF SURF BEAT
foreshore or step of the beach (location 0, Figure 8). The choice of this reflection point is constrainedby the model profile, the distancestep (Ax = 5 m here), and the shoreline size of the long waves (Hrms 0.10 m here). The resulting reflectionpoint is 6 m from the profile location at which h = 0, suggesting the possibility that the reflected waves inmodel-generateddata could precede the reflected waves in field data by a few seconds.However, this lag was not observedin model/fieldcomparisons (Figure 9).
4. RESULTS
1 ON
A(t)at
W orr (t)at {
Model generated time series of wave envelope, crossshore current, and sea surface elevation are treated identi-
cally to the field data examined by List [1987] at each of the locationsin Figure 8. After a comparisonbetweenobserved and model group/long wave cross-correlationsignals, the contributions the boundlong wave and breakpoint-forced of long wave are examined separately.
Cross-Correlation Signals
The cross correlation between series, and Y, is rj two X
found as
MODEL
............ OBSERVATIONS
E (Xi--)(Yi-j
i=1
$ROUP TRAVEL TIME (MODEL)
i_ E (Yi-j- --< rjnjl n- )2 J0 -I (X)2I/I ]1/2
Li=I i=1
(25)
E (Xi+j-- x)(r
i=1
i -- y)
Fig. 9. Cross correlationsbetween the wave envelope at location 8 and long-wavecomponents (onshoreand offshoredecomposition of /) at six cross-shorelocations for field observationsand model generated data. A(t) and /were band-passfiltered between 0.007 and 0.03 Hz. Solid dotsindicatethe incidentwave grouptravel time from the position of A(t) measurement (location 8) to the position of /measurement. The 95% C.I. on r = 0 for field and model signalsis approximately +0.15.
rj= n I
[i=1
n-IJl
i=1
/2 J-> 0
clearly reproducesthe observedsignal, althoughthe magnitude of the field signal is lower, most likely owing to the where j is the range of time lags to be considered.Here the presence of other long-wave components, such as edge waves, and the assumptionof a normally incident, unidirectwo seriescorrelated are A(t) and r/(t). As adjacent points in a periodic time series are not tional incident wave field. All major features of the observed independent, the 95% confidence interval on zero correlation signalsare reproducedby the model data. The model signal was found usinga reducednumberof points, n*, given by changesfrom a single negative peak at the input boundary (location 9, not shown in Figure 9) to the often observed Garrett and Toulany [ 1981] as positive/negative form in the nearshore. This signal then nt appearslargely unchangedin the outgoingcomponentof r/, 1 1 2 suggesting shorelinereflection of the incominggroup-forced (26) = tl + tl E (n- j)Rxy(J) - it* waves. Also, the incomingsignalshowsa progressivephase j=l shift such that the negative peak, likely due to the BLW, where is thenumber samples thetimeseries, xy(J) increasinglylags behind the group structure toward shore. n of in R is the laggedautocorrelationof the product of the two series This observation has also been made by Elgar and Guza to be correlated, n' is the number lagsuntilRxy [1985] in an examination of the biphase. and of experiencesa zero crossing.Here n* varied between 163and Figure 9 substantiatesthe model's ability to simulate the 266 for field data (n = 4800) and between 104 and 159 for observed data and generate group-forced long waves, at model-generated data (n = 4000). Thus the 95% confidence least under one set of conditions. However, many questions
and BFLW to the signal. Although the cross correlation betweenA(t) and ,/(on) beyondthe break point is clearly due location8 (Figure 8) and the onshoreand offshorecompo- to the BLW, the altered form of the correlation with r/ nents of r/(obtained with (13) and (14)) at six cross-shore elsewhere remains unexplained. Speculations have attriblocations. Figure 9 compares these observed signalswith uted this to the addition of the break-point-forced composignals calculated using model-generated data. The model nent [List, 1987]; this is investigated here using separate
interval (C.I.) on r = 0 is -+0.15 or less for field data and -+0.20 or less for model-generated data. List [1987] found the cross correlation between A(t) at
remain. Foremost is the relative contribution of the BLW
LIST: MODEL FORGENERATION SURF BEAT OF
5631
} ON at
1o0
vs. A (t) at
,
} OFF at () vs. A (t) at ()
1,0 , ,
I
\
',
/\
R 0,0
TORIG
1,; ',,F\ .,,/ . ,,,
TOTAL
BFLW
R 0.0
-o.
..............
BLW
-o.5 FORCING \,./ TOTAL
BFLH/
..............
-I .0
,,
:' ',.:
'
-50.0
GROUPTRAVELTIME LOCATION to 1 8
0.0
LA (eee)
BLW
50,0
100,0
0 0
50.0
LAO (eeo)
100.0
150.0
i. 1. ro
colation btwn () at location
run in total, BW, an BW o. h
Fig. 11. (:rosscorrelation between A(t) and W(off) location at 8
for model runs in total, BLW, and BFLW modes. The 95% (:.I. on
r = 0 is the same as in Figure 10.
location 1 for o1
oal-praicta roup travel ti ffo location to location1 hown by a olia aot. h .I. on r: i approximately.1
BFEW curve.
model runs in breakpoint-forced and bound long-wave
modes.
similar to the BLW signal, owing to the smaller size and phaserelation of the BFLW contribution. Several points are strongly suggested these model by results.The cross-correlation signalfirst observedby Tucker [1950]is largely due to the presenceof a modified, shallow
water bound wave which is released as a free long wave in the surf zone and radiated offshore following shoreline
reflection. These shallow water modifications include a
Figure 10 shows the cross correlation between A(t) at location8 and /(on) at location 1 for three separatemodel runs using total forcing, time-varying breakpoint forcing, and boundlongwave forcing. As expected,the BFLW signal shows a simple, positive correlation with wave groups: larger waves produce a larger setup. Quite unexpectedly, however, the BLW signal has an asymmetricform with a positive and negative peak almost identical to the total signal. Analytical solutionswhich predict a simple bound wave, r out of phasewith wave groups,are not supported by
this model observation.
phaselag behindwave groupsand a change the bound in
wave form such that the cross-correlation signal becomes
asymmetric. Examples time series of presented belowallow
a visualization of these changes.
Time Series Examples
Synchronous time series of envelope and ,/(on) at 8 cross-shore locationsare presented Figures 12a, 12b, and in
12c for model runs in total, BLW, and BFLW modes
The total signalis nearly identical to the BLW signalfor two reasons. First, the B FLW contribution is small com-
respectively. inputA (t) structure location is subject The at 9
to initial breakingat location 5 (some of the A(t) peaks are truncated),and is almostentirely depth limited by station 1. In the total and BLW forcing modes, ,/(on) at location 9 is
paredto the BLW, as will be shownbelow. (Note that while
the BFLW correlation is higher than the BLW correlation in
Figure 10, this does not indicate the relative size of the associatedlong waves since these correlationswere calcu- the input BLW. In the BFLW modethe boundary,/(on) is oscillation resulting lated usingseparatetime series).Second,the BFLW signal near zero, with only a barely discernible is addedto the BLW signalat a phasenear r/2, which results from open boundary reflection. As in the cross-correlation signal,the BLW in Figure 12b in neither constructive nor destructive interference. As can lags behind wave groups and developsan be seenfrom Figure 10, without the shallow water phaselag progressively formtowardshore; risein waterlevelprecedes a of the B LW signal (negative part) behind the group struc- asymmetric in ture, the BFLW would be addedat a phaseof r, resultingin the wave group maxima and a depression water level follows the wave group maxima. Although incident waves purely destructive interference. The cross-correlation signalin the outgoing componentoff- start breakingat location 5, the BLW continuesto increase shore morecomplicated, is shown Figure11.According in size almost to the shoreline. is as in In the BFLW mode run (Figure 12c), r/(on) is negligible to the model of Symonds et al. [1982], breakpoint-forced longwavesin this regionshould the superposition long until wave begin to break at location 5, then develops be of with A(t) peaksoffshore correlating with elewavesradiated directly seawardfrom the forcing region and predictably, long waves traveling seawardafter being radiatedlandward vated water levels near the shoreline.
and reflected from the shoreline. The resulting BFLW signal
The ,/(on)generated total forcingmodeis nearlyidenin
in in Figure 11 is asymmetric, consistingof a negative peak tical to the ,/(on)generated BLW mode,as seenin Figure of followed by a positive peak. The BLW signal, on the other 12a. A comparison r/(on)in BLW mode(Figure12b) with that thisis dueto hand, is the same as in the incoming component, unchanged ,/(on)in BFLW mode(Figure12c) shows after shoreline reflection. Again, the total signal is very the =r r/2 phaserelation between these two series, as was
5632
LIST: MODEL FOR GENERATION OF SURF BEAT
(w) NO G
......
O O OO
g
/
'
;
'
'
0
..
o
(w) () v
LIST: MODEL FOR GENERATION OF SURF BEAT
0.10
5633
1,}OFF
0.05
m
rn TOTAL.
o.oo 0.10
- ..... - BLW
well-constrained, rate. This growth rate is far less than that predicted by available analytical solutions [e.g. LonguetHiggins and Stewart, 1964; Ottesen Hansen et al., 1981] and does not stop or reverse upon incipient incident wave breaking. Second, waves generated by the time-varying break point mechanism are secondary in height to shallow water bound waves, at least for the set of conditions defining this experiment.
5. DISCUSSION
0.00
O.
'......-e....... -e........ -e ..............
1. \013 5
'
6
v
7
v
8
9
O
w -$.
100. 200.
DISTANCE
300.
OFFSHORE
400.
500.
Fig. 13. Cross-shore variationsin the Hrms of on) and off)
from model-generated data in total, BLW, and BFLW forcing
modes.
also the case for the cross-correlation signals (Figure 10), and because ,/(on) generated in BFLW mode is relatively
small, as is shown in more detail below.
Shoaling Characteristics
The cross-shore variations in long-wave height for the onshore and offshore componentsof ,/are shown in Figure
13 for model runs in total, BLW, and BFLW modes. The incoming B LW increases toward shore at a rate much greater than simple long-wave shoaling, but much less than predicted by the analytical solution (i.e., equation (1)). The BLW continues to grow through the region of intermittent incident wave breaking. The breakpoint-forced long wave grows rapidly after incident wave breaking beginsat location 5, but it only reaches 50% of the BLW size at the shoreline. Again, the total forcing solution is almost identical to the
BLW solution.
In the outgoing waves, the BFLW solution increases seaward toward the maximum breakpoint (location 5), as the component radiated seaward from the forcing region is added to the component radiated landward and reflected from the shoreline. Still, the BFLW size offshoreis only 66% of the BLW, which decreasesoffshore at a rate following
simpleinverse shoalingof a free long wave (H2 = H(h/
One of the primary results of this study is that long waves generatedby the time-varying breakpoint model are secondary in size to shallow water bound waves, at least for one set of model conditions including assumptionsand simplifications similar to those of Symonds et al. [1982]. This result is supportedby field observations of a cross-correlation signal that is very well predicted by model-generated data. If the breakpoint-forced component were more dominant in the field data than is predicted by the model, the form of the cross-correlation signal in the outgoing waves would be closer to the model-predicted breakpoint-forced signal (Figure 11). Additional support for the relatively small size of the breakpoint-forced wave comes indirectly from the absence of field observations documenting a frequency-dependent amplitude in the long waves offshore, a primary prediction of the Symonds et al. model. The development of a method for separating the breakpoint-forced and bound long waves in field data would help resolve this issue. It may seem odd that breakpoint-forced long waves are secondaryin light of the observationsmade previously that the radiation stress gradients generating these waves are larger than those associated with the bound long wave (Figures 5 and 6). The answer may lie in the fact that radiation stressgradientsin the breakpoint-forcing zone are very transient, alternating sign at the group period, while the gradients associated with unbroken waves maintain their sign and continue to force long waves during their entire travel history across the nearshore. The assumption of an ample responsetime for breakpoint-forced waves made by Symondset al. [1982] may not be universally valid. Some of the most interesting and unexplained results of this study concern the shallow water behavior of the bound long wave. In deep water the bound long wave is theoretically constructed as a second-order Stokes-type expansion of the incident wave velocity potential [Longuet-Higgins and Stewart, 1962]; some have suggestedthat removal of the incident waves (e.g., by breaking) should result in removal of the bound waves. In contrast, this study predicts and documents a bound long wave that is released and reflected from near the shoreline as a free long wave. Apparently, the free long-wave dispersion relation is satisfied before incident wave breaking begins, and there is simply no mechanismfor subsequent damping of this wave. Longuet-Higgins and Stewart [1962, 1964] first speculated that this could explain
Tucker [1950] observation.
h2)/4).Theoutgoing waves thetotalsolution slightly the in are
higher than the BLW solution; appare...