January 13, 2004
14:37
WSPC/101-CEJ
00097
519
Analytical Model of Incipient Breaching of Coastal Barriers
means of assuring robust results though imposition of morphologic constraints -- in
this situation, of equilibrium values.
Insertion of Eq. (3) into Eqs. (1) and (2) and taking the limit yields the following
coupled, first-order nonlinear governing equations for breaching:
,
1
QS
x
a
dx
a=
=
, x(0) = x0 > 0
(4)
-
dt
z
xe
L
and
,
1
dz
b
QB
z
=
,
z(0) = z0 > 0
(5)
b=
-
dt
x
ze
L
Equations (4) and (5) cannot be solved without specifying a non-zero perturbation
of the barrier (representing an indentation or "pilot channel" through the barrier),
as given by a finite initial width x0 and initial depth z0. Equation (4) describes a
one-sided breach such as constrained by a jetty in nature or a wall in a flume. If both
sides of the breach can move, then the value of QS in Eq. (4) should be doubled.
3.1. Solution for xe and ze → ∞
This special case corresponds to very short elapsed time of incipient breaching, for
which the governing Eqs. (4) and (5) reduce to dx/dt = a/z and dz/dt = b/x. The
solution of this simplified set of coupled non-linear equations is,
b/a
x
a
a+b
x
a+b
x = 0 a + (a + b) 0 t
(6)
z0
and
a/b
z
b
a+b
z0
a+b
z=
t
(7)
+ (a + b)
b
0
x0
If the initial perturbation or pilot channel is small, the cross-sectional area of
the breach is found to grow as
QS + QB
t
(8)
xy ∼ (a + b)t =
L
indicating a linear increase in area immediately after the breach occurs. For the
special case of a = b, Eqs. (6) and (7) simplify to,
1/2
x
x0
2
(9)
t
x=
0 + 2a
z0
and
1/2
z
z0
2
z=
+ 2a t
(10)
0
x0
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