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Document 1594500
Appendix A
Weather uctuations in the forcing
The water motions in the morphodynamic model described in chapters 2 and 2 are forced by
wind stress and alongshore gradients in the mean free surface elevation. The sea bottom evolution
which is described by the model takes place during a very long time scale as the result of the
cumulative eect of many episodic storms. On such a long time scale, the forcing is essentially time
dependent mainly due to the sequence stormy weather-calm weather. As a rst step, the present
thesis has been based on taking an averaged forcing and neglecting the eects of uctuations. A
detailed description of these latter eects are considered beyond the purpose of the present thesis.
Nevertheless, it is worthwhile to give a sketch of the main ideas of such a "statistical" model.
This yields insight into the limitations of the present "fully averaged model". Also, some of the
implications of the statistical model can already be foreseen. This is the aim of this appendix.
All the variables can be decomposed into a mean plus a uctuation, where the mean refers to
an average over a time period T much larger than the typical time between two consecutive storms,
say T = O(10yr): For instance, for the velocity, v = v + v0, where v and v0 describe the mean and
uctuation, respectively. The governing equations can then be averaged as
+ f c v + grzs sD b = v0 rv0
v rv
r (D v) = 0 ; @ zb + r q = 0
@t
where the hypotheses assumed in section 2.1, namely, the quasi-steady approximation and small
Froude number have been already taken into account. Note that depth uctuations donot show
up in this model which is a consequence of the rigid lid approximation (free surface eects are
neglected) and the fact that uctuations on the topography due to storm events can be discarded.
Now, the problem is how to parametrize bottom shear stress, b; sediment transport, q; and the
"Reynolds stress" term, v0 rv0: In order to parametrize the rst two, it is convenient to consider
the instantaneous velocity near the bottom
+ v0 + v00
v ib = v
where v00 is the velocity which is not explicitly described by the shallow water model (contribution
due to waves, turbulence, etc.). Here we assume that v00 is the wave orbital velocity near the
bottom, being oscillatory with a period much smaller than the time scale of weather events and
v 00 = 0: Under the assumption that the mean current is weak in comparison to uctuations and to
wave orbital velocity, jv j jv0 + v00 j; and that the instantaneous bottom shear stress is given by
b = cd jv ib jv ib ; the mean bottom stress can be evaluated as
b = cd jv 00 jv 0 + [R] v
(A.1)
where the second order tensor [R] has components:
v00 v00
Rij = cd jv00 jÆij + cd i 00j
jv j
87
88
APPENDIX A. WEATHER FLUCTUATIONS IN THE FORCING
with Æij the Kronecker delta symbol and indices i and j can have the values 1 or 2. In a similar
way, if the instantaneous sediment ux is
v ib
q = jv ib jb (
jv j rzb )
0
ib
where b > 1 and are specied by using parameterizations which are discussed in e.g. Van Rijn
(1993) the mean sediment transport can be calculated as
jv00 jbrzb
q = jv 00 jb v 0 + [Q] v
(A.2)
where the components of the second order tensor [Q] are given by:
(A.3)
Qij = jv00 jb Æij + (b 1)jv00 jb vi00 vj00
By using (A.1) and (A.2), the basic state equations (2.6) can be rewritten as
d jv 00 jv0
V
@v0
fc V + g = sx cd x R vx0 x
(A.4)
dx D
@x
D
D
@v0
jv 00 jv0
V
(A.5)
gs = sy cd y R vx0 y
@x
D
D
D
Notice that all derivatives with respect to y vanish as there is alongshore uniformity in the basic
state. If we now dene
= sx cd jv00 jv0
= sy cd jv00 jv0
sx
sy
x
y
0
1
0
0
0
1
3
0
12
22
1
g = g + v0 2x
2
@v0
gs = gs + vx0 y
@x
equations (A.4)-(A.5) read
V
+ g ddx = sxD R DV
gs = sy R (A.6)
D
D
Given the forcing sy and s the second of these equations has the same structure than the
second of (2.6) and can be solved for V in the same manner (recall that the inuence of the free
surface elevation on the total depth D is neglected, in accordance with the small Froude number
assumption). Once V (x) is known, the rst equation can be solved for ; from where the free
surface elevation, ; can be obtained. Even though the rst equality in (A.6) has two additional
terms, sx and R V , this only produces an eect on the free surface elevation, but not on the
basic alongshore current. The mass conservation equation is satised identically while the sediment
conservation yields
d
(Q V + jv jb v0 dzb ) = 0
fc V
12
22
12
dx
12
0
00
1
x
dx
with = 0 jv00 jb: This equation would give the equilibrium topographic prole of the inner shelf,
zb (x): Observations indicate that to a rst approximation this prole is characterized by a constant
slope and this is what has been assumed in section 2.1. Therefore, a mean basic state similar to
that considered in the thesis can still be assumed even in the presence of uctuations.
An interesting issue of this "statistical" formulation, already at this simple stage, is the prediction of a decrease of the eective wind stress sy with respect to the mean value, sy : This can be
understood as follows. Let us consider a dominance of wind along the positive y-direction, so that
sy > 0 and V > 0: The longshore current uctuation will be positive during storms and negative
during fair weather, but due to the fact that fair weather occurs more often than stormy weather,
the magnitude of the uctuation is larger during00 storms than during calm weather (because
of
0 = 0). Then, since the wave orbital velocity jv j is much larger during storms, i.e., when vy0 > 0;
sy
there is a net contribution in the positive y-direction so that cdjv00jvy0 > 0: At the same time,
the eective sea surface slope, s; is not signicantly smaller or larger than its mean value. This
89
can be argued as follows. The cross-shore velocity uctuation, vx0 , has not any preferred direction
during the entire storm duration but will change sign from the initial stage to the nal stage of the
event. According to the balance between surface shear stress and bottom shear stress considered in
section 2.1, the wind driven current tends to be cross-shore uniform. Thus, the term @vy0 [email protected] will
vx0 @vy0 [email protected] may be expected to be negligible and s ' s:
be very small. Therefore,
the correlation
The fact that jsy j < jsy j while s ' s causes an increase of the a parameter in the model.
In other words, even if wind stress may be dominant during storms, the sea surface slope is much
more eective in driving the long term mean alongshore current. Some simple computations have
been done by considering the following forcing sequence: 1 day of constant wind stress (always
in the same direction), 9 days of calm weather. The sea surface slope has been assumed to be
present all the time. According to observations o the central Dutch coast, this distribution is not
unrealistic. Then, the computations predict an increase in parameter a of the order 40%:
Once the statistical formulation and the basic state have been established, the next step would
be to write down the linear stability equations. Now, this requires to evaluate the rst order variation of the uctuation on the ow, v0, due to the presence of the small topographic perturbation,
h: This will not be further pursued in the present thesis.
90
APPENDIX A. WEATHER FLUCTUATIONS IN THE FORCING
Appendix B
Solution procedure: Steady model
B.1 Equations
The three linear dierential equations (2.12) can be reduced to a single equation for the cross{shore
velocity component of the form:
U uxx + U ux + U u = H hx + H h
(B.1)
where
U =
^
U = x + f
^ + r + ikV
U = x f
H
H = ^
H = x + f
2
1
0
2
2
1
2
0
3
1
3
2
1
3
1
1
0
1
0
1
with the and coeÆcients given below. Solving (B.1) for u as a function of h, under the
boundary conditions u = 0 at x = 0 and for x ! 1, denes the linear operator U. Then, back
substitution into (2.12) yields for the linear operators V and E:
V = I + Ux + U
(B.2a)
1
2
3
E = 1 I + 2 Ux + 3 U
(B.2b)
where I denotes the identity operator.
The coeÆcients in (B.1) and (B.2) are
1 =
i
k
= i Hx
2 =
3
V
H
kH
V2 i
V
Æ
1 =
+ k r2 H 2 H
H
r
iV
2 = 2 2
kH k rH
i V Hx Vx
3 = 22 2x
kH
k
H
91
f^
92
APPENDIX B. SOLUTION PROCEDURE: STEADY MODEL
Once the operators U and V are known, operator B can be computed as
(m 1) VxU + Ux + ikmV
m
B = jV j
1
^jV j
V
m d
d2
Vx + 2
V dx dx
(B.3)
k2 I
Then, the solution of (2.15) with (B.3) yields the eigenvalues and eigenfunctions.
B.2
x{discretization
The diferential equations are solved by means of spectral method based upon truncated expansions
in Chebyshev polynomials, and are solved at collocations points (see Canuto et al. 1988 and Gottlieb
& Orszag 1977). Here a brief outline of the spectral numerical method used is gived. In order to
use Chebyshev polynomials, the [0; 1) interval is transformed into the ( 1; 1] interval. The both
intervals are related by the maps:
z z0
( 1; 1]
(z) = az + (1
( )=
! ( 1; 1]
(z0 )=x
! [0; 1)
(1 z0) ;
(z 0 ) = l
(1 + z0)
a)z
3
and N Gaus-Lobatto nodes are chosen as collocations points
zi = cos(
i
)
N
i=0N:
At the transformation , the l parameter is the distance where half of the collocation points are
located. The tranformation approach the collocation points to 0 in the ( 1; 1] interval and,
therefore, to l in the [0; 1) interval. Note that x = 0 and xN = 1.
The aproximate solution f (x) of a function F (x) on [0; 1) is expanded as a truncated Chebyshev
series
N
X
f (x) = f^j Tj (z ) ;
0
j =0
Here N is the order of expansions, Tj (z) the Chebyschev polynomials of rst kind and order j (ie.
Tj (z ) = cos(jcos z )) and f^j the projection of the F (x) function in Tj (z ).
Expansions for rst and second order derivaties at the collocation points from the values of the
funtion at these points, f (xj ) = fj , are
1
df dx x
x
= i
=
N
X
j =0
d f dx x
2
Dij fj
1
2
x
= i
=
N
X
j =0
Dij2 fj
The derivaties operators Dkl and Dkl follow from the derivaties of the Chebyschev polynomials
and the map z 7! x, and read
1 D kl
1 D d x=dz D kl Dkl =
Dkl =
dx=dz
(dx=dz) kl dx=dz
dx = d d
d x = d d + d d :
dz dz dz0
dz dz dz0 dz dz0
1
2
1
2
2
2
2
2
2
2
2
2
2
2
2
93
B.3. DISCRETIZATED EQUATIONS
The elements D kl are dened as
1
D = (2N + 1)
6
zk
D kk =
2(1 zk )
c ( 1)k l
D kl = k
cl zj zl
c =cN = 2
cj =1
and the elements D kl as
= 61 (2N + 1)
k =1 N 1
D NN
2
00
2
+
2
k; l =0 N; j 6= k
0
j =1 N
1
2
2
D kl
=
N
X
m=0
D km D ml
k; l =0 N :
In case that a funtion H (x) veries boundary conditions
H (x = 0) = 0
H (x ! 1) ! 0 ;
a linear combination of Chebyschev polynomials, which veries the same boundary conditions, is
used. That read as:
1 (1 + ( 1)j )T (z) + (1 + ( 1)j )T (z) :
gj (z ) = Tj (z )
2
+1
0
1
The aproximation of H at the collocation points, hi, is
hi =
N
X
j =2
G0ij h^ j
G0ij
= gj (zi )
i = 0 N; j = 2 N :
Expansions for rst and second order derivaties, at the collocation points, are
N
X
dh =
G1 h^
dx x=xi j=2 ij j
G1ij
=
N
X
k=0
N
X
d2 h =
G2 ^h
dx2 x=xi j=2 ij j
G2ij =
1
Dik
G0kj
N
X
k=0
2
Dik
G0kj :
Note that at the last expansions, values at the collocation points come from the projections of
H (x) in the basis gj (z ).
B.3 Discretizated equations
B.3.1 FOT-problem
Application of the collocation method to the equation for u, equation (B.1), yields
NX1
j =1
U (xi )Dij + U (xi )Dij + U (xi )Æij uj =
2
2
N X
j =2
1
1
0
H (xi )Gij + H (xi )Gij h^ j
1
1
0
0
i=1N
1
Note that, because of the boundary conditions (u = uN = 0), the index for the collocation points
is running from 1 to N 1. Same boundary conditions veries h and they are included in basis
gj (x) of the appendix B.2. Solving this system of (N 1) (N 1) equations a matrix Uij which
0
94
APPENDIX B. SOLUTION PROCEDURE: STEADY MODEL
denes the cross-shore velocity component u in the collocation points xi as a linear combination
of h^ j , ie. the bottom perturbation, is found:
ui =
N
X
j =2
Uij h^ j
i =0 N
1:
The boundary condition for u at x = 0 gives U j = 0.
Back substitution into equations (B.2), yields
Vij = (xi )Gij + ( (xi )Dik + (xi )Æik ) Ukj
i; j =0 N 1
Eij = (xi )Gij + ( (xi )Dik + (xi )Æik ) Ukj
i; j =0 N 1 ;
and from these operators, the values of v and at the collocation points are
0
0
1
2
0
1
vi =
i =
2
N
X
j =2
N
X
j =2
3
3
Vij h^ j
i =0 N
1
Eij h^ j
i =0 N
1:
B.3.2 Bottom evolution equation
Application of the collocation method to the equation for h and using the solutions of the FOTproblem, the equation (B.3) become
!
N X
N
X
j =2
G0ij h^ j =
NX1 j =2
D0x
V
1
(
m 1) x jV jm 1 Æik + (m 1)jV jm 1 Dik
Ukj
D
V
0
i
k=0
jV jm 1 V G0
ikm
ij
D0
i
+ ^ jV jmG2ij + m VVx jV jmG1ij k2jV jmG0ij h^ j :
i
m
The result of write this equations in the collocation points (i = 1 N 1) is a generalized eigenvalue
problem
! Ah^ = Bh^
The generalized eigenvectors, h^ i , are the components in gj (x) of the bottom perturbation.
Appendix C
Analytical approximation for m = 1
A simple analytical approximation of the eigenvalue problem discussed in section 2.2 can be obtained for the parameter values ^ = f^ = r = F = 0; m = a = 1 and 1. In essence this reduces
the model to the system studied by Trowbridge
(1995), but heo did not persue this method. By
n
^
assuming normal mode solutions h = Re h(x) exp (iky + !t) it follows that the bed evolution
equation (2.17) reduces for 0 x 1 to
Uh = h
H
where hats have been dropped for simplicity, = ! ik and operator U is dened in appendix B.
The boundary conditions are h = 0 at both x = 0 and x = 1: due to the absence of slope eects
in the sediment transport h = 0 at the outer shelf. Next we assume the expansions
h(x) =h0 (x) + h1 (x) + : : :
=1 + 2 2 + : : :
U =U0 + U1 + : : :
so that at the lowest order we will have the eigenproblem for U :
U h = h
Operator U involves solving equation (B.1) for horizontal at bottom
current, which, owing to r = 0; f^ = 0, reduces to
d u k u = ik dh
dx
dx
where u = U h . Therefore, solving (C.1) is equivalent to solving
d h + 2s dh k h = 0 ; h (0) = h (1) = 0
dx
dx
with
k
s=i
2
The solutions of (C.2) are easily found to be
h (x) = ei1 x ei2 x
where
p
p
s +k )
;
= i(s + s + k )
= i(s
and where the boundary condition at x = 1 requires
= 2n
;
n = 1; 2; 3; : : :
0
0
0
1
0
0
2
0
and uniform basic
0
2
2
(C.1)
0
2
0
2
0
2
0
0
0
1
0
2
1
2
2
2
1
95
2
2
(C.2)
(C.3)
(C.4)
(C.5)
(C.6)
96
APPENDIX C. ANALYTICAL APPROXIMATION
4.0
4.0
3.5
3.5
3.0
3.0
2.5
2.5
2.0
2.0
1.5
1.5
1.0
1.0
0.5
0.5
0.05
0.04
V
0.03
σ
numerical
0.02
analytical
0.01
0.00
0
2
4
6
8
10
k
0.0
0.0
0.5
1.0
0.0
0.0
0.5
1.0
Figure C.1: Left: Comparison of the instability curves obtained either by the numerical model or
by the analytical approximation, for ^ = f^ = r = 0; m = a = 1 and = 0:1. Right: Perturbation
in the topography given by the analytical model, modes 1 and 2.
From (C.5) , (C.6) and (C.3) the eigenvalue spectrum is found to be
k
1
= p
;
n = 1; 2; 3; : : :
2 k +n From this it follows that the growth rates to rst order in the slope, , are given by
1
2
! = ik
2
2
2 p k
n = 1; 2; 3; : : :
;
k2 + n2 2
(C.7)
Figure C.1 shows ! as a function of k for the dominant mode (n = 1) and for = 0:1, computed
both by the MORFO20 numerical model and by equation (C.7). Note the striking resemblance
between the analytical and the numerical model. This even applies in case = 1, see gure 4 in
Trowbridge (1995) for a comparison. From (C.4) and taking into account the dependence on the
alongshore coordinate, the sea bed perturbation at any time can be written as proportional to
h(x; y) = cos(ky + x) cos(ky + x)
(C.8)
Each term is wave-like with straight crests parallel to ky + x = 0. Therefore, recalling that the
basic current is V < 0, the topographic features are downcurrent oriented if > 0 and upcurrent
oriented if < 0. In the case of growing bedforms, that is, > 0, (C.5) and (C.6) imply
p
p
;
= ( k + n n)
= ( k + n + n)
which both are negative. So, growing bedforms are upcurrent oriented. In contrast, decaying
bedforms yield positive values for and and thus they are downcurrent oriented. This is in
agreement with the numerical computations in section 2.3 and with the eld observations described
in the introduction. Finally, the topographic contours computed by means of (C.8) for = 0:1,
k = are shown in gure C.1 for the rst and second growing modes (n = 1, n = 2). Also, a
remarkable similarity with the bedforms computed with the numerical model is found (see, for
instance, gure 2.2 A and B).
1
2
1
1
2
2
2
1
2
2
2
2
2
Appendix D
Derivation of the sediment
transport parametrization
Sediment transport is a very complex process so that for practical applications it is necessary to
model it by means of gross parametrizations. The latter are based on a combination of basic
physics, dimension analysis and observations in the eld and in the laboratory, see Dyer (1986)
and Fredsoe & Deigaard (1993) for further details.
Now consider noncohesive sediment with a uniform grainsize which is transported as bedload.
Then, in case of a at bed, a frequently used parametrization for the dimensional volumetric ux
per unit width is
q =u ;
i.e., the transport is proportional to the cube of the total velocity u near the bed. Here it is assumed
that the critical velocity for erosion is much smaller than the actual velocity. The foundation of this
dates back to Bagnold (1956) and has been conrmed by many experimental data. A characteristic
value for the parameter is 10 s m .
In order to apply this parametrization to the depth-averaged model used in this paper it is
important to realize that
u = v + v0 :
Here v is the depth-averaged velocity which consists of both a steady and tidal components.
Furthermore, v0 is the part of the velocity eld that is not explicitly accounted for by the depthaveraged model; in particular waves and small-scale turbulent motions determine the behaviour of
v0 .
During storms we assume that the currents induced by waves and turbulence are much larger
than the steady and tidal currents, in other words jv0j jvj. Moreover it is assumed that jv0j is
independent of the location and that waves donot induce any net sediment transport over a tidal
cycle. On the inner shelf these conditions seem reasonable because water depths are generally too
large to cause signicant wave steepening and wave breaking. Then it follows by straightforward
means that the tidally averaged volumetric ux per unit width during storms can be approximated
by
<q>
= hjv0j i < v >;
where the proportionality factor is the so-called wave stirring factor. This result can be easily
generalized to a sloping bed and motivates the use of the dimensionless ux q in equation (3.7),
where = hjv0j i=U if we choose [q] = U . Note that by denition 1.
Likewise the situation during mild weather conditions can be analyzed, during which we assume
that jv0 j jvj. In that case it follows for the dimensional ux
< q > = h v i:
Its dimensionless analogon is the ux q in equation (3.7) with = 1.
3
5
2
1
2
storms
1
2
2
1
3
1
quiet
3
3
3
97
98
APPENDIX D. SEDIMENT TRANSPORT PARAMETRIZATION
Appendix E
Solution procedure: Tidal model
In chaper 3 equations for u, v and h were found, in this appendix the numerical procedure to solve
those equations is gived.
E.1 Equations
The equation for the cross-shore velocity component u(x; t) reads
U utxx + U utx + U ut + U uxx + U ux + U u = H hx + H h
where
U =1
U = HHx
12
11
10
02
01
00
1
0
12
11
x
U = k + HHxx H
H
U = Hr + ikV
10
2
2
2
02
U =ikV HHx
01
U =r Hk + HHxx 2HHx
2
2
00
2
3
!
2
^
+ ik k V Vxx + VxHx + V HHxx + (f=)Hx VHH2x
k2 V 2
Vt r(2V U0 )
ikÆ
H1 = H + ik H +
H2
H
!
2
^
H0 =k2 2V Vx H (f=)V + VHH2 x + ik VHtx VHt H2x
Æ
sx
0 )Hx
+
H
+
ik
+ ik r 2Vx H 2U0x 2(2V HU
3
H2 x
H2
2
The equation for the perturbation in the longshore ow v(x; t) reads
v=
V
i
H
h+
u + xu
H
k x H
And the equation for the bottom perturbation h(x; t) is
@h
@
=
~ ~q11 i=1
q hr
99
(1
~ ~q31 i=2
) hr
100
APPENDIX E. SOLUTION PROCEDURE: TIDAL MODEL
where hi denotes the average over a tidal period
r~ ~qm1 = m jV jm
H
m x
H
(m 1) VVx u + (m 1)ux
V
Vx
2
ikm h + jV j^ hxx + m hx k h
H
V
1
E.2 Discretization
To solve the linear stability problem the Chebyschev collocation method of the appendix B in x
and a Fourier Galerkin method in t has been used. The approximate solution are expanded as
M
X
u(xn ; t; ) =e!
p= M
!
e hn
v(xn ; t; ) =e!
upn eipt
h(xn ; ) =
M
X
p= M
vnp eipt
The expansions for the derivatives of u and h at the collocation points are
X
@ 3 u 2
ipDnj
upj eipt
=
e!
2
@tx xn
M pM
1j N 1
X
@ 2 u 1
=
e!
ipDnj
upj eipt
@tx xn
M pM
1j N 1
X
@u =
e!
ipÆnj upjeipt
@t xn
M pM
1j N 1
X
@ 2 u 2
Dnj
upj eipt
=
e!
2
@x xn
M pM
1j N 1
X
@u 1
=
e!
Dnj
upj eipt
@x xn
M pM
1j N 1
1
N
X
@ 2 h !
=
e
G2nj h^ j
@x2 xn
j =2
X
ujxn = e!
N
X
@h !
=
e
G1nj h^ j
@x xn
j =2
M pM
jN 1
Ænj upj eipt
hjxn = e!
N
X
j =2
G0nj h^ j
E.2.1 FOT-problem
Application of the collocation method to the equation for u yields
X
M pM
jN 1
2
ipU12 (xn ; t)Dnj
+ ipU11(xn ; t)Dnj1 + ipU10(xn ; t)Ænj
1
+ U (xn ; t)Dnj + U (xn; t)Dnj + U (xn ; t)Ænj
2
02
N
X
j =2
1
01
H (xn ; t)Gnj + H (xn ; t)Gnj h^ j
1
1
0
0
Galerkin projection of the various terms results in
X
M pM
jN 1
2
ipU12 (xn ; t)Dnj
upj eipt ; e iqt
1
X
M pM
1j N 1
ipU11 (xn ; t)
Dnj upj eipt ; e iqt
=
upj eipt =
n=1N
X
=
1
1
00
M pM
jN 1
1
2
ipIp0 q Dnj
upj
H
1
ip x Ip0 q Dnj
upj
H
n
M pM
1j N 1
X
101
E.2. DISCRETIZATION
X
ipU10 (xn ; t)Ænj upj eipt ;e iqt
M pM
1j N 1
M
X
p= M
X
M pM
1j N 1
X
M pM
1j N 1
X
1
M pM
jN 1
U (xn ; t)Dnj upj eipt ;e
iqt
2
02
=
=
rIp0 q
H
M pM
1j N 1
X
U (xn ; t)
Dnj upj eipt ; e iqt
01
U (xn ; t)
Ænj upj eipt ; e iqt
00
+ ik
1
=
=
+ ikIn;p
1
j =2
1
iqt
=
N X
N
X
j =2
H (xn )Gnj h^ j ; e
0
0
iqt
+ ik
+r
+
=
2
Dnj
upj
1
M X
r
p= M
+ HHxx2
k2
H
2Hx 2
I0
H3 n p q
1
2 In;p
q Hx
upn
H2
n
2
k2 In;
q
+
H
j =2
K
ik n; q
H
0
n
0 1 q Hx + In;p
1
^ )Ip0 q Hx
q Hxx + (f=
H
H (xn )Gnj h^ j ;e
q
I 00 1p q
1
k2 In;p
q
1
H
1
ikIn;p q x Dnj
upj
H
n
M pM
1j N 1
X
+Ip
X
N
Hx2
I 0 up
H2 n p q n
H
k2 + xx
H
ip
(2In;
+r
1
N
X
j =2
1
Kn;
q
H
2I 01n; q
k2
q
U0 I 0 q )
!
H2
2I 0n;
0
Kn;
q Hx
H2
U0x I 0 q
2
q
Æ 0
I
H q
n
G1nj h^ j
^ )In; q In; q Hx (f=
+ H
H
1
2
2
2(2In;
1
H2
Hx
sx 0
Æ 2 + ik 2 I q
G0nj ^hj
H
H
n
q
0
U0 In;
q )Hx
3
H
!
Thus the values of the indices are M q M and 0 i N 1. The 'I; J; K '{coeÆcients will
be specied later.
This denes a system of (2M +1) (N 1) equations which can be solved by standard methods.
The results are summarized in a matrix Upnj , which denes the cross-shore velocity component u
in the collocation points and for the dierent Fourier modes. It is dened as
X
n =0 N 1
un =e!
Upnj eipt h^ j
M pM
2j N
The boundary conditions for u at x = 0 gives Upj = 0.
0
102
APPENDIX E. SOLUTION PROCEDURE: TIDAL MODEL
Substitution in the equation for v(x; t) yields
p
Vnj
=
!
1
In;
i NX1 0 Hx
p
p
0
2H nGnj + k k=0 Dnk + H Ænk Ukj
n
Then the values of v in the collocation points are
X
p eipt h
^j
vn =e!
Vnj
n =0 N
M pM
2j N
1
In particular the tidally averaged velocity components read
hun i =e!
N
X
j =2
U0nj h^ j
hvn i =e!
N
X
j =2
0 ^
Vnj
hj
E.2.2 Bottom evolution equation
Application of the collocation method to the equation for h results in
N
X
!
j =2
G0nj h^ j =
N
X
j =2
f q hr~ ~q i
11
~ ~q31 i=2 ;j gh^ j
)hr
(1
1 ;j
=
at the n = 1 N 1 collocation points.
Using the solutions of the FOT-problem the terms hr~ ~qm i
1
M
X
p= M
M
X
p= M
mm jV jm
Dx
Up eipt
D n nj
1
1
2
1
become
=
M
X
D
mm x
I m 1 Up
D n p= M n;p nij
(m 1)mjV jm
1=2 ;j
=
Vx
Up eipt
V n nj
=
M
1 (m 1)m X
p
I 0m
n;p Unj
2
p M
1
=
X
M pM
1kN 1
(m 1)m
jV jm
1
Dnk Upkj eipt
n
1 (m 1)m
2
V
G0
D n nj
1
X
=
m 1 D1 Up
In;p
nk kj
M pM
kN 1
1 ikm m I m G1
2
D n n;0 nj
1
=
1 ^mJ m G
^m jV jm Gnj =
n; nj
2
n
Vx
1
m
^mm jV j
G
= 2 ^mm J 0mn; Gnj
V n nj
1 ^mk J m G
^m jV jm k Gnj =
n; nj
2
n
Having calculated the operator Upnj for = and for = , determines hr~ ~qm i
the term hr~ ~qm i 2 ;j , respectively.
ikmm
jV jm
1
2
0
2
1
2
0
2
1
1
=
1
0
2
0
0
1
1 ;j
=
and
103
E.3. GALERKIN INTEGRALS
E.3 Galerkin integrals
The Galerkin method and the tidaliptaverages give rise the 'I; J; K '{coeÆcients. These coeÆcients
arise from integrals which involve e and the basic velocity V (x; t). In this section these coeÆcients
are written explicily. First, in order to compute the integrals with absolute values of the velocity
are involved, sign functions, (x; t) and (x; t), are dined. Afterwards the 'I; J; K '{coeÆcients
are written by means of some auxiliar coeÆcients, which are given at the end of this section.
The basic state velocity eld reads
V (x; t) =U0 (x) + (1
jj)U (x) sin (t + '(x))
1
where
H
U1 (x) = p
'(x) =arctan
H 2 + r2
r
H
Now dene the following sign-functions:
x; t)j
(x; t) = jVV ((x;
t)
(x; t) =
jj
It appears than they are related according to
= (1 2((t
t1 ) (t t2 )))
where is the Heavyside function and
j
jU
t = ' + arcsin
(1 jj)U jjU
t =2 ' arcsin
(1 jj)U
0
1
1
0
2
1
The subsindex n refers to the numerical value of the functions in a collocation point xn .
E.3.1 'I; J; K '-coeÆcients
'I; J; K '-coeÆcients are computed by means of the following auxiliar coeÆcients
kl =
Xn;
q
Yn;kl q
=
kl =
Zn;
q
Z 2
0
Z 2
0
(xn ; t) sink (t + 'n) cosl(t + 'n)e
sink (t + 'n) cosl(t + 'n)e
Z 2 n 'n
+n 'n
iqt dt
sink (t + 'n ) cosl (t + 'n )e
They will be computed in the next subsection.
iqt dt
iqt dt
= (Yn;kl q 2Zn;kl q )
104
APPENDIX E. SOLUTION PROCEDURE: TIDAL MODEL
'I '-coeÆcients
I0 q =
1
In;
Z 2
0
00
e iqt dt
=Y q
Z =
V (xn ; t)e iqt dt
q
=U nY q + (1 jj)U n Yn; q
Z V (xn ; t)e iqt dt
q=
= U nY q + 2jj(1 jj)U n U nYn; q + (1 jj) U nYn;
Z =
V (xn ; t)e iqt dt
q
= U nYn; q + 3 (1 jj)U n U nYn; q
+ 3(1 jj) U nU nYn; q + (1 jj) U nYn; q
2
0
00
0
2
In;
3
In;
2
0
2
2
2
0
00
2
0
3
0
Z 2
0
3
0
00
2
2
0
2
1
0
2
1
20
q
10
1
20
3
3
1
30
=U 0 nY q + (1 jj)fU 0 n Yn; q + U n'0nYn; q g
Z =
V (xn ; t)Vx (xn ; t)e iqt dt
q
= U nU 0 nY q
+ jj(1 jj)f(U 0 n U n + U nU 0 n)Yn; q + U nU n'0n Yn; q g
+ (1 jj) fU nU 0 nYn; q + U n'0n Yn; q g
Z Vxx (xn ; t)e iqt dt
q=
=U 00nY q
+ (1 jj)f(U 00n U n('0n ) )Yn; q + (2U 0 n'0n + U n'00n )Yn; q g
00
10
1
01
1
2
0
2
0
00
0
1
0
2
I 00 1n;
2
Vx (xn ; t)e iqt dt
0
I 0 2n;
10
1
3
2
I 0 1n; q =
10
1
1
0
20
1
10
1
2
1
0
01
1
11
2
0
00
0
2
1
1
10
01
1
1
'J '-coeÆcients
Jn; q =
0
1
Jn;
Z 2
0
00
Jn;3
iqt dt
V (xn ; t)
=Xn; q
Z jV (xn ; t)je iqt dt
q=
=U n Xn; q + (1 jj)U n Xn; q
Z =
V (xn ; t)jV (xn ; t)je iqt dt
q
= U nXn; q + 2jj(1 jj)U nU nXn; q + (1 jj) U nXn;
Z =
V (xn ; t)jV (xn ; t)je iqt dt
q
= U nXn; q + 3 (1 jj)U n U nXn; q
+ 3(1 jj) U nU nXn; q + (1 jj) U nXn; q
2
0
00
0
Jn;2
jV (xn ; t)j e
1
10
2
0
2
2
0
2
0
3
00
0
10
1
2
2
3
0
00
2
2
0
2
0
2
1
20
1
10
3
3
1
30
2
1
20
q
105
E.3. GALERKIN INTEGRALS
= jVV ((xxn ;; tt))j Vx(xn ; t)e iqt dt
n
=U 0 nX q + (1 jj)fU 0 nXn; q + U n'0nXn; q g
Z =
Vx (xn ; t)jV (xn ; t)je iqt dt
q
= U nU 0 nX q
+ jj(1 jj)f(U 0 n U n + U nU 0 n)Xn; q + U nU n'0nXn; q g
+ (1 jj) fU nU 0 nXn; q + U n'0nXn; q g
Z V (xn ; t)Vx (xn ; t)jV (xn ; t)je iqt dt
q=
= U nU 0 nX q
+ (1 jj)f(2U nU 0 nU n + U nU 0 n)Xn; q + U nU n'0nXn; q g
+ (1 jj) f(U 0 nU n + 2U nU nU 0 n)Xn; q + 2U nU n'0n Xn; q g
+ (1 jj) fU nU 0 nXn; q + U n'0n Xn; q g
Z 2
J 01
n; q
0
00
0
J 0 2n;
01
1
2
0
2
0
00
0
1
0
2
J 0 3n;
10
1
1
0
20
1
10
1
2
1
0
01
1
11
2
0
3
2
0
0
00
2
0
2
2
1
0
3
2
1
2
0
1
0
0
30
1
1
10
1
20
1
3
1
2
0
1
0
2
1
01
11
21
'K '-coeÆcients
Kn; q =
0
1
Kn;
q=
Z 2
0
Z 2
0
Vt (xn ; t)e iqt dt = (1
Vtx (xn ; t)e iqt dt = (1
jj)U n Yn;
01
1
q
jj)fU 0 n Yn;
01
1
U1n '0n Yn;10 q g
q
E.3.2 Auxiliary integrals
Now dening
Yn;j q =
Z 2
eij(t+'n ) e iqt dt = eij'n 2Æ0;j q
0
Then the Y -integrals can be written as
Yn; q =Yn; q
i
Yn; q = Yn; q Yn; q
2
1
Yn; q = Yn; q + Yn; q
2
1
1 Y + Y Yn; q = Yn; q
2
4 n; q n; q
i
Yn; q = Yn; q Yn; q
4
1
1
Yn; q = Yn; q + Yn; q + Yn; q
2
4
i
3
i
Yn; q Yn; q
Yn; q = Yn; q Yn; q
8
8
1
1
Yn; q = Yn; q + Yn; q
Yn; q + Yn; q
8
8
00
0
10
1
+1
01
1
+1
20
0
2
11
02
1
Finally, introducing
j
Zn;
q=
Z 2 n 'n
+n 'n
2
+2
+1
1
21
+2
+2
0
30
2
+1
3
3
+3
+3
eij(t+'n ) e iqt dt
=eij'n ( 2n)Æ ;j
0
q
i
eiq'n n i(j q)n
e
j q
( 1)j
q ei(j q)n
o
106
APPENDIX E. SOLUTION PROCEDURE: TIDAL MODEL
Then the Z -integrals can be written as
Zn; q =Zn; q
i
Zn; q = Zn; q Zn; q
2
1
Zn; q = Zn; q + Zn; q
2
1
1 Z + Z Zn; q = Zn; q
2
4 n; q n; q
i
Zn; q = Zn; q Zn; q
4
1
1
Zn; q = Zn; q + Zn; q + Zn; q
2
4
i
3
i
Zn; q Zn; q
Zn; q = Zn; q Zn; q
8
8
1
1
Zn; q = Zn; q + Zn; q
Zn; q + Zn; q
8
8
00
0
10
1
+1
01
1
+1
20
0
2
11
02
30
21
2
+2
2
+2
+2
0
1
+1
1
+1
3
3
+3
+3
Since n = arcsin( jjjUj0Un1n ), the phases of the solution are in the interval [0; ]. In the points
where jjUj0ij > U i it follows that n = , hence in these points Zn; q ,Zn; q ,Zn; q y Zn; q are
zero.
(1
(1
)
1
)
2
0
1
2
2
3
Appendix F
F.1 Adjoint operator
Dening an inner product hji as:
hf jgi =
and
4
X
i=1
(fi; gi)
1 Z 0
Z
(fi ; gi) =
fi (x; y)gi (x; y)dxdy ;
from the denition of adjoint operator L+, ie. hjLi = hL+ji, it follows that
(!knk !k+ nk )h+k nk jS knk i = 0 ;
where knk (resp. +k nk ) are the eigenvectors of L (resp. L+) with eigenvalue !knk (resp. !k+ nk ),
or
(!knk !k+ nk )(h+k nk e k y ; hknk e ky ) = 0
Under the hypothesis that the domain of L is the same+than the domain of L+, the relations written
above mean that the set of eigenvectors knk and k nk are a bi{orthogonal set under the inner
product hSji, ie.
h+k nk jS knk i = Ækk Ænk nk
or that
(
+
+
k
y
ky
(hk nk e ; hknk e ) =6= 00 ifif !!knk 6== !!k+ nk :
0
0
0
0
0
0
0
0
0
0
0
0
i
0
i
0
0
0
0
i
0
0
0
0
0
0
0
i
knk
0
0
k nk 0
h+k0 n0k
0
In particular, given hk nk ; we always can assume that there exists a
which is non-orthogonal
to it.
The adjoint operator of Lk reads
0
1
d
d
r
^
V
+
f
H
i
kV
+
x
B
C
H
dx
dx
r
B
C
^
B
C
f
i
kV
+
i
kH
i
k
B
C
H
L+k = B
C
d
B
C
i
k
0
0
B
C
dx
@
A
Æ
Vx d
d
0
i
kV
^jV j(
+
k)
H
V dx d x
2
2
2
The pertinent boundary conditions are that u+ and h+ should vanish at x = 0 and x ! 1 for
each solution knk .
F.2 Nonlinear system
In this appendix equations (4.10) are explicitly written.
107
108
{
u knk
APPENDIX F. NONLINEAR TOOLS
equation
Equation (4.10a) gives
0=
X
nk
0
(L (k; nk ; n0k )^ukn k + L (k; nk ; n0k )^vkn k
0
11
0
12
+ L (k; nk ; n0k )^kn k ) + N (k; nk ) ;
0
13
and these coeÆcients read
L (k; nk ; n0 k ) =
11
L12 (k; nk ; n0 k ) =
L (k; nk ; n0 k ) =
1
Z
1
0
Z0 1
u+knk vkn k ( f^)dx
0
0
Z
1
u+knk (kn k )xdx
0
0
u+knk Fk [email protected] + [email protected] u +
where Fk is the k{component of the Fourier expansion.
{
v knk
+ Hr )dx
u+knk ukn k (ikV
0
13
N (k; nk ) =
1
Z
1
ruh
gdx
H (H h)
equation
Equation (4.10b) gives
X
0 = (L (k; nk ; n0k )^ukn k + L (k; nk ; n0k )^vkn k
nk
0
21
0
22
0
+ L (k; nk ; n0k )^kn k + L (k; nk ; n0 k )h^ kn k ) + N (k; nk ) ;
and these coeÆcients read
Z 1
+ u
0
L (k; nk ; n k ) = vkn
(V + f^)dx
k kn k x
0
23
L (k; nk ; n0 k ) =
0
Z
L23 (k; nk ; n0 k ) =
+ v
vkn
(ikV
k kn k
0
Z
1
+ Hr )dx
+ vkn
(ik)dx
k kn k
0
Z
0
1
Æ
)dx
H
0
Z 1
+ F [email protected] v + [email protected] v + (rv Æh)h gdx :
N2 (k; nk ) = vkn
x
y
k k
H (H h)
0
L (k; nk ; n0 k ) =
24
equation
1
0
22
{
2
0
21
knk
0
24
Equation (4.10c) gives
0=
nk
0
(L (k; nk ; n0k )^ukn k + L (k; nk ; n0k )^vkn k
0
31
0
32
+ L (k; nk ; n0k )h^ kn k ) + N (k; nk ) ;
0
L31 (k; nk ; n0 k ) =
L (k; nk ; n0 k ) =
1
Z
L (k; nk ; n0 k ) =
Z
N (k; nk ) =
0
+ (H u
kn
x kn k + H (ukn k )x )dx
k
1
0
+ v
kn
(ikH )dx
k kn k
0
Z
0
1
+ h
kn
( ikV )dx
k kn k
0
34
Z
3
0
0
32
3
0
X
34
and these coeÆcients read
+ h
vkn
(
k kn k
1
0
+ F f @ (uh) @ (vh)gdx :
kn
x
y
k k
109
F.3. TIME INTEGRATION SCHEME
{
h knk
equation
Equation (4.10d) gives
X
n0 k
^
S (k; nk ; n0 k ) dhdknt k =
44
0
X
(L (k; nk ; n0k )^ukn k + L (k; nk ; n0k )^vkn k
nk
+ L (k; nk ; n0k )h^ kn k ) + N (k; nk ) ;
0
41
0
42
0
0
44
4
and these coeÆcients read
Z 1
0
S (k; nk ; n k ) = h+knk hkn k dx
0
44
L (k; nk ; n0 k ) =
Z
0
Z
0
L (k; nk ; n0 k ) =
Z
0
44
N (k; nk ) =
4
1
Z
0
h+knk ( ukn k )x dx
1
h+knk vkn k ( ik)dx
0
42
L (k; nk ; n0 k ) =
1
0
41
1
V
h+knk (^ jV j( x (hkn k )x + (hkn k )xx k2 hkn k )dx
V
0
0
h+knk Fk [email protected](^ (jv j
0
0
jV j)@x h) + @y (^ (jvj jV j)@y h)gdx :
F.3 Time integration scheme
A stiy{stable type scheme (see Karniadakis, Israeli & Orszag, 1991, sec 4.2) is then used to carry
out time integration of the system (4.11):
0 = L U n + M hn +
1
0 Shn+1
PJi
q
1
=0
t
q Shn q
+1
1
+1
= L U n + M hn +
2
+1
2
+1
JX
e 1
q=0
JX
e 1
q=0
q f (U n q ; hn q )
q g(U n q ; hn q )
The values of the coeÆcients from Karniadakis et al. (1991) are reproduced in table F.1. Note
that an Euler{forward/backward integration rule corresponds to the frist-order scheme. For higher
orders, the scheme is implicit for the linear terms and explicit for the nonlinear terms. The nonlinear
part is computed as an extrapolation atn + 1 time from the previous Je steps.
CoeÆcient 1st Order 2nd Order 3rd Order
1
3/2
11/6
1
2
3
0
-1/2
-3/2
0
0
1/3
1
2
3
0
-1
-3
0
0
1
Table F.1: Stiy{Stable Scheme CoeÆcients
0
0
1
2
0
1
2
F.4
k=0
mode
The nonlinear self{interaction of any mode with wavenumber k excite a component with k = 0,
ie. alongshore uniform, in the expansion (4.9). However, this was not taken into account in the
computations where modes with k = 0 have been neglected. The reason for this is that this self{
interaction was found to be very weak. For this propouse the dynamics of an alongshore uniform
110
APPENDIX F. NONLINEAR TOOLS
botom perturvation has been study. This can be shown by using the analytical approximation to
the linear modes in case of m = 1, = 0 and r = 0 developped in appendixC.
As start, let us split the perturbation of the basic state, in a longshore mean and a periodic
function in a longshore length 2L:
u =hui + u0
=hi + 0
0
v =hvi + v
h =hhi + h0
where hi = (1=2L) R LL dy. hu0 i = hv0i = h0 i = hh0i = 0 and hui, hvi, hi, and hhi refer to k = 0
modes. Using this notation, the nality of this appendix is to show that hhi = 0.
From mass conservation (4.6) and sediment conservation (4.7) equations in case = 0,
@x ((H h)u) + @y ((H h)(V v)) = 0
@t h + @x u + @y v = 0
and making averages in the longshore direction, hi, equations for hhi and hui are found:
@x ((H hhi)hui hh0 u0 i) = 0 ;
hence,
0 0
+
hui / Hhh uhhi i ;
and
@t hhi = @x hui :
Now, we consider self-intercations of fk; ng modes h0 = h^ (x) eiky
apporximated modes of appendixC:
h^ (x) = e 1 x e 2 x
and
i
Therefore,
Then
1;2 = n
i
d u^
dx
2
2
u^ =
1 k
ei1 x
+ k2
2
1
k2 u^ =
i
p
k2 + n2 2
^
ik ddxh
2 k
ei2 x
+ k2
1
and u0 = u^(x) e ky . From the
2
2
1
hh0 u0 i = h[h^ (x) e ky + h^ (x) e ky ][^u(x) e ky + u^(x) e
= h^(x)^u(x) + c.c. + h2k-modesi
i
and
h^ (x)^u(x) =
i
i
i
ky
]i
1 k 2i1 x i1 x i(1 +2 )x i2 x e
e
e
+e
21 + k2
2 k i(1 +2 )x i1 x 2i2 x i2 x e
e
e
+e
:
22 + k2
Sustituting expressions for ; in h^(x)^u(x) all terms cancel: self-interaction of modes fk; ng only
forces 2k-mode whose mean vanishes. Hence hh0u0i = 0 and then hui = 0 and hhi = 0.
12
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