librium. Prior to equilibrium, greater D ( x ) occurred at most of
the locations except at the two peaks. Modest improvement in
uniformity across the surf zone occurred after the beach reached
equilibrium.
For the plunging case, D ( x ) patterns were significantly differ-
ent before 0 40 min and after average between 280 and 630
min equilibrium Fig. 4 B . Before equilibrium, D ( x ) varied
considerably across the surf zone, with an overall landward-
decreasing trend. At equilibrium, the D ( x ) value was nearly one
order of magnitude greater at the main breaker line than those at
the rest of the surf zone, where D ( x ) was reasonably uniform.
Based on the analyses of Dean 1977 , equilibrium energy
dissipation per unit volume, D , can be calculated as
*
g 2
5 A 3/2 g
D
(3)
* 24
breaker index. The value, determined here as H mo / h
where
ranged from 0.6 to nearly 1, and varied across the surf zone Fig.
1 . The average value was 0.66 for the spilling case and 0.64 for
the plunging case. Because H rms was used to calculate the wave
energy Eq. 2 , the breaker index corresponding to H rms( rms
H rms / h ) was 0.47 for the spilling case and 0.45 for the plung-
ing case.
The calculated D using rms Eq. 3 is on average 480%
*
excluding the near-shore peak and the breaker-line peak greater
than the measured D ( x ) for the spilling case and 250% exclud-
ing the breaker-line peak greater for the plunging case Fig. 4 .
At the breaker line, D was 30% less than the measured D ( x ) for
*
the spilling case and 270% less for the plunging case.
Fig. 5. Profiles of undertow through the water column
The derivation of Dean 1977 was conducted under the as-
sumption of spilling breakers. The similar dissipation patterns be-
fore and after equilibrium for the spilling case were probably
Osborne and Greenwood 1992 found that in different parts of
because the initial beach was close to equilibrium. The signifi-
the surf zone, the direction and magnitude of cross-shore trans-
cantly greater D ( x ) at the plunging breaker line corresponds with
port were dominated by different terms. In general, offshore-
the local deviation from the power function. The region having
directed transport is dominated by the undertow while the asym-
relatively uniform energy dissipation coincided with the portion
metrical oscillatory motions dominated onshore-directed
of the surf zone that is dominated by surf bores.
transport. The transport by oscillatory motions was not considered
in Larson et al. 1999 analysis; offshore-directed transport was
Nonlocal Balance of Onshore and Offshore Sediment
assumed to be driven by undertow.
Transport
Vertical profiles of cross-shore current were measured at the
Larson et al. 1999 derived the equilibrium profile Eq. 1 by
ten cross-shore locations after the beaches reached equilibrium
balancing the gradient of near-bottom undertow-driven offshore
Fig. 5 . Undertow was measured through most of the water col-
transport with a vertical transport due to net sedimentation or
umn below the wave trough. Time-averaged sediment-
suspension . Cross-shore sediment flux, F cs( x , z , t ) , is calculated
concentration profiles demonstrated rapidly upward-decreasing
as
trends over nearly four orders of magnitude. One exception oc-
u x,z,t
c x,z,t
(4)
F cs x , z , t
curred at the main plunging breaker line, where the variation
spanned only one order of magnitude Wang et al. 2002 . If os-
where u cross-shore current, and c sediment concentration. To
cillatory transport is neglected, the time-averaged cross-shore-flux
simplify the complex temporal variations of the surf-zone sedi-
profile obtained from the product of time-averaged profiles of
ment motion, current velocity and sediment concentration were
current and sediment concentration is dominantly offshore di-
often partitioned as e.g., Osborne and Greenwood 1992; Thorn-
rected and exhibits a steep upward-decreasing trend Fig. 6 .
u
ton et al. 1996
Larson et al. 1999 balanced the gradient of offshore transport
by undertow with a net vertical sedimentation or suspension as
u u
u
~ low ~ high
(5)
d
q undertowc 0
dF offshore
c c
c
c ~ ~ low ~ high
(6)
d
wc0
(8)
dx
x
where and time-averaged velocity and sediment concentra-
c
u
where F offshore depth-integrated offshore flux, q undertow
u
tion, respectively. ~ and ~ are oscillatory components of velocity
c
discharge of undertow, c 0 characteristic sediment concentra-
and concentration. The subscripts high and low indicate high- and
empirical coefficient, and w sediment fall speed. A
tion,
low-frequency components. The time-averaged sediment flux at
location x, z is determined as
nonlocal balance implies, as illustrated in the Fig. 1 of Larson
et al. 1999 , that an offshore-increasing undertow transport can
F cs u c ~ low~ low ~ high~ high
u c u c
u c
(7)
be nonlocally balanced by sedimentation from net onshore trans-
44 / JOURNAL OF WATERWAY, PORT, COASTAL AND OCEAN ENGINEERING / JANUARY/FEBRUARY 2003