On an one-dimensional bi-layer shallow-water
problem
Mar#$a Luz Mu˜noz-Ruiz, Manuel Jes#us Castro-D#$az
∗, Carlos Par#es
Dpto. Analisis Matematico, Universidad de Malaga, 29080 Malaga, Spain
Received 22 March2001; accepted 15 October 2001
Abstract
This paper is concerned with the mathematical analysis and the numerical approximation of the system of partial di4erential equations governing the one-dimensional 7ow of two superposed shallow layers of immiscible viscous 7uid in a channel with variable rectangular cross-section. First, we prove the existence and uniqueness of solution for small data and some smoothness results. Next, a 9rst-order upwind scheme for numerically solving the system is proposed. We apply this scheme to the simulation of some two-layer exchange 7ows through straits with a sill and a contraction.
?2003 Elsevier Science Ltd. All rights reserved.
Keywords: Existence and uniqueness of solution for the bi-layer shallow-water equations; Finite volume methods; Upwind schemes; Two-layer exchange 7ows through straits
1. Introduction
In this paper we study the system of partial di4erential equations (PDE) governing the one-dimensional 7ow of two superposed shallow layers of immiscible 7uids. The 7uids on each layer are supposed to be viscous and homogeneous. This bi-layer 7ow is assumed to occupy a straight channel with rectangular cross-section, but bottom and breadthvariations are allowed.
Most of the paper is devoted to the mathematical analysis of the system. This analysis has been carried out by adapting the techniques developed in [7,8] for the study of the system of PDE corresponding to a shallow layer of viscous 7uid.
This research has been partially supported by the C.I.C.Y.T. (projects MAR97-1055-CO2-01,
REN2000-1162-C02-01 MAR, REN2000-1168-C02-01 MAR).
∗Corresponding author.
E-mail addresses:[email protected](M.L. Mu˜noz-Ruiz),[email protected]
(M.J. Castro-D#$az).
0362-546X/03/$ - see front matter?2003 Elsevier Science Ltd. All rights reserved. PII: S0362-546X(02)00137-2
In Section 2 we present an existence theorem for small data. The main diJculty is the need of a priori estimates in a L2 space for the thickness of the layers: in the
case of a single layer, the existence of solution can be proved with a weaker estimate for the thickness and, once the existence is proved, the L2 regularity can be obtained
assuming some further hypothesis on the data. In this case, both the existence and the
L2 regularity have to be performed at the same time.
In Section 3, some smoothness and uniqueness results are presented, under further hypothesis on the data.
In Section 4 the discretization of the system is presented: we use a 9rst-order up-wind scheme, the so-called Q-scheme of Van Leer. The treatment of the source terms corresponding to the geometry channel is performed as in [3,21,22].
Finally, in Section 5, this numerical scheme is applied to simulate some two-layer 7ows through a strait with a sill and a contraction. In order to validate the scheme, we 9rst compare some steady-state solutions with those obtained following the techniques developed in [1,2], based on the assumption of the ‘rigid-lid’ hypothesis and neglecting viscous e4ects. Then, as in [12], we impose a periodic barotropic forcing to the 7ow through the channel, obtaining periodic solutions that are qualitatively compared with the solutions obtained in the cited article.
1.1. Setting the problem
The problem under study is the following:
(P) @u1
@t −A1 @2u
1
@x2 +u1
@u1
@x +g @h1
@x +g 21
@h2
@x =g @H
@x in Q;
u1= 0 on @I ×(0; T);
u1(t= 0) =u1;0 in I;
@h1
@t + @
@x(u1h1) =− u1h1
b @b
@x in Q;
h1(t= 0) =h1;0 in I;
@u2
@t −A2 @2u
2
@x2 +u2
@u2
@x +g @h2
@x +g @h1
@x =g @H
@x in Q;
u2= 0 on @I ×(0; T);
u2(t= 0) =u2;0 in I;
@h2
@t + @
@x(u2h2) =− u2h2
b @b
@x in Q;
h2(t= 0) =h2;0 in I:
In these equations, index 1 makes reference to the deeper layer and index 2 to the upper one. The 7uid is assumed to occupy a straight channel of lengthLwitha constant rectangular cross-section. The coordinate xruns on the interval I= [0; L] and refers to
the axis of the channel. The coordinatet runs on the interval [0; T] and represents the time. Eachlayer is assumed to have a constant density, i, i= 1;2 ( 1 ¿ 2). The
unknowns ui(x; t) and hi(x; t) are de9ned on Q=I×[0; T] and they represent,
respec-tively, the mean velocity and the thickness of theithlayer at the section of coordinate
x at time t. Ai, i= 1;2, represent the constant coeJcient of viscosity; g the gravity;
H(x) the depth at x from a 9xed reference level, and b the breadth of the channel. Observe that we consider homogeneous Dirichlet boundary conditions: non-homogen-eous conditions add some diJculties to the analysis of the problem that are now under study. Nevertheless, in Sections 4 and 5 general Dirichlet conditions are considered.
1.2. Weak formulation
We will denote by (:; :) the scalar product of L2(I) and by :Wm; p the usual norm
in the space Wm;p(I).
LetV be the spaceH1
0(I) equipped withthe norm
u2
V =u2L2+ @u@x2
L2:
AsI is bounded, we can consider the equivalent norm
uV =@u@x L2;
and the bi-linear form given bya(u; v) = (@u=@x; @v=@x) is elliptic.
We consider the problem (P) under the following variational formulation:
Find (u1; h1) and (u2; h2) in [L∞(0; T;L2(I))∩L2(0; T;V)]×[L∞(0; T;L1(I))∩L2(Q)]
suchthath1¿0, h2¿0 and
(V) d
dt(u1; v) +A1
@u1 @x; @v @x +
u1@u@x1; v
−g
h1;@x@v
−g 2
1
h2;@v@x
=−g
H;@v@x
∀v∈V;
d
dt(u2; v) +A2
@u2 @x; @v @x +
u2@u@x2; v
−g
h2;@x@v
−g
h1;@v@x
=−g
H;@x@v
∀v∈V;
@h1
@t + @
@x(u1h1) =− u1h1
b @b @x; @h2 @t + @
@x(u2h2) =− u2h2
b @b @x;
u1(t= 0) =u1;0∈V; h1(t= 0) =h1;0¿0∈L2(I);
2. An existence theorem
In this section we present a global existence result with controlled data. LetC be the best constant associated with the injection of V into L∞(I):
uL∞6CuV ∀u∈V:
LetK be the least constant such that
pL∞6K@p
@x
L2 ∀p∈H
1(I) :
Ip= 0:
Letm ¿0 and M ¡∞ be two constants suchthat
m6b(x)6M ∀x∈I: (1)
Let and be positive numbers suchthat
2+
2m¡1: (2)
Then,
g1=g
1−2 −m
and g2=g
1−2 −2m− 2
1
2m
are strictly positive. We de9ne
C1=bh21|;I0|L1; C1=gT bh1;0L
1
|I|
bh1;0L1
m + 21
bh2;0L1
m +HL1
(3)
and
C2=bh22|;I0|L1; C2=gT bh2;0L
1
|I|
bh1;0L1
m +
bh2;0L1
m +HL1
: (4)
Let us assume that
A1¿ C1+C 2
2 bh1;0L1+g 1 2 +g 21
1 2;
A2¿ C2+C 2
2 bh2;0L1+g 1
Then, if we choose a positive number small enough,
B1=A1−g2 −C1−C 2
2 bh1;0L1−g 1 2 −g 21
1
2 (6)
and
B2=A2−g2 −C2−C 2
2 bh2;0L1−g 1
(7)
are strictly positive.
Theorem 1. Let H∈L2(I) and letu
i;0∈V and hi;0∈L2(I)such thathi;0¿0; i= 1;2.
If hypotheses (2) and (5) are satis7ed, then the problem (V) has a solution
{(u1; h1);(u2; h2)} that satis7es the following estimate:
u1L∞(0;T;L2(I))+u2L∞(0;T;L2(I))+u12L2(0;T;V)+u22L2(0;T;V)
+h12L2(Q)+h22L2(Q)6C; (8) where C ¿0 depends on the initial data.
The proof of this theorem is split into several steps: 9rst, we give some a priori estimates, then we build a sequence of approximated solutions that satisfy these estimates, and, 9nally, we pass to the limit in the continuity and momentum equations.
2.1. A priori estimates
Lemma 1. If {(u1; h1);(u2; h2)} is a classical solution of the problem (V) and if the
relations(2) and (5) are satis7ed, then the following relations hold:
h1¿0; h2¿0; (9)
Ibh1=
Ibh1;0;
Ibh2=
Ibh2;0; (10)
1
4u12L∞(0;T;L2(I))+1
4u22L∞(0;T;L2(I))+B1u12L2(0;T;V)+B2u22L2(0;T;V) +g1mh12L2(Q)+g2mh22L2(Q)
6 2
i=1
1
2ui;02L2+Ai
Ibhi;0logbhi;0−Aibhi;0logbhi;0|I|
+C
i+
Ipi(0)bhi;0+K
2bh
i;02L1
+gT
1+M
H2
Proof. First, notice that equations (V)3 and (V)4 can be written under the form
@
@t(bhi) + @
@x(ui(bhi)) = 0; i= 1;2: (12)
Now, (9) is easily deduced from the identity
(bhi)(Xi(t); t) = (bhi;0)(x0;0)e−
t
0(@ui=@x)(Xi(s);s) ds; (13)
whereXi(t) is the solution of the problem
dXi
dt =ui(Xi(t); t); Xi(0) =x0
for i= 1;2. Using that b ¿0 and hi;0¿0 we obtain hi¿0.
The relations (10) are obtained by integration over I of the respective continuity equations. In particular, this implies the estimate
hi∈L∞(0; T;L1(I)); i= 1;2:
To obtain (11), we take v=u1 in (V)1 andv=u2 in (V)2:
1 2
d
dtu12L2+A1u12V −g
h1;@u@x1
−g 2
1
h2;@u@x1
=−g
H;@u@x1
; (14)
1 2
d
dtu22L2+A2u22V −g
h2;@u@x2
−g
h1;@u@x2
=−g
H;@u@x2
: (15)
(Notice that (ui@ui=@x; ui) = 0 for i= 1;2.)
The terms on the right-hand side are bounded using Young’s inequality
−g
H;@ui @x
6g
2ui2V +
g
2 H2L2; i= 1;2:
Adding (14) and (15), and integrating in (0; t), we get 1
2u12L2+12u22L2+
A1−g2
u12L2(0;T;V)+
A2−g2
u22L2(0;T;V)
612u1;02L2+12u2;02L2+gT H2L2
+g
T
0
h1;@u@x1+g
T
0
h2;@u@x2
+g 2
1
T
0
h2;@u@x1+g
T
0
The second step consists in obtaining estimates forh1 andh2 inL2(Q). To do this, let
considerp1 and p2 suchthat
u1=@p@x1 and u2=@p@x2:
These functions pi can be chosen such that Ipi= 0; i= 1;2.
Now, the equations (P)1 and (P)6 can be written as follows:
@ @x
@p1
@t −A1 @2p
1
@x2 +
1
2u21+gh1+g 21 h2−gH
= 0;
@ @x
@p2
@t −A2 @2p
2
@x2 +
1
2u22+gh2+gh1−gH
= 0:
Then,
@p1
@t −A1 @2p1
@x2 +
1
2u21+gh1+g 2
1h2−gH=$1; (17)
@p2
@t −A2 @2p
2
@x2 +
1
2u22+gh2+gh1−gH=$2; (18)
where$1 and$2 are functions that only depend on time.
Now we de9ne, for %∈[0; T], the function
$%(t) =
1; 06t6T−%; T −t
% ; T−%6t6T:
Multiplying (17) by $%bh1 and (18) by $%bh2 and integrating over Q, we obtain
g
Q$%bh
2 1−A1
Q
@2p 1
@x2 $%bh1+
1 2
Qu
2
1$%bh1+g 2 1
Q$%bh1h2
=
Q$1$%bh1−
Q
@p1
@t $%bh1+g
QH$%bh1; (19)
g
Q$%bh
2 2−A2
Q
@2p 2
@x2 $%bh2+
1 2
Qu
2
2$%bh2+g
Q$%bh1h2
=
Q$2$%bh2−
Q
@p2
@t $%bh2+g
QH$%bh2: (20)
The L2-estimates for h
1 and h2 will be performed by estimating the terms in these
equations and then passing to the limit when % → 0. First, note that Q$%h1h2¿0
Then, the second terms on the left-hand side are treated by formally writing
−
I
@2p
i
@x2 bhi=−
@ui
@x; bhi
=
@
@x(bhi); ui
=
1
bhi
@
@x(bhi); uibhi
=
@
@x(logbhi); uibhi
=−
logbhi;@x@ (uibhi)
:
Using the continuity equations and (10) we h ave
−
I
@2pi
@x2 bhi=
logbhi;@t@ (bhi)
=ddt
I(bhilogbhi−bhi)
= ddt
Ibhilogbhi; i= 1;2:
Then,
−Ai
Q
@2pi
@x2 $%bhi=−Ai
T
0 $%
I
@2pi
@x2 bhi=Ai
T
0 $%
d dt
Ibhilogbhi
=Ai
T
0
d dt
I$%bhilogbhi−Ai
T
0
@$%
@t
Ibhilogbhi
=−Ai
Ibhi;0logbhi;0+
Ai
%
T
T−%
Ibhilogbhi:
Using the convexity inequality
bhilogbhi¿bhilogbhi+ (logbhi+ 1)(bhi−bhi);
wherebhi represents the averaged bhi; i= 1;2, and (10), we obtain
Ibhilogbhi¿bhilogbhi|I|=bhi;0logbhi;0|I|:
Then,
Ai
%
T
T−%
Ibhilogbhi¿
Ai
%
T
T−%bhi;0logbhi;0|I|=Aibhi;0logbhi;0|I|;
and so
−Ai
Q
@2p
i
@x2 $%bhi¿−Ai
Ibhi;0logbhi;0+Aibhi;0logbhi;0|I|; i= 1;2:
To estimate the 9rst terms on the right-hand side we look for an expression for$1 and
we arrive at
$1=|1I|
I
1
2u21+gh1+g 21h2−gH
; (21)
$2=|1I|
I
1
2u22+gh2+gh1−gH
: (22)
Then,
Q$i$%bhi6Ciui
2
L2(0;T;V)+Ci
withCi and Ci; i= 1;2, given by (3) and (4).
We consider next the second terms on the right-hand side. We 9rst integrate by parts
−
Q
@pi
@t $%bhi=−
Q
@
@t(pi$%bhi) +
Qpi
@ @t($%bhi)
=
Ipi(0)bhi;0+
Qpi
@$%
@t bhi+
Qpi$%
@ @t(bhi);
and then, we use the continuity equation
−
Q
@pi
@t $%bhi=
Ipi(0)bhi;0+
Qpi
@$%
@t bhi−
Qpi$%
@
@x(uibhi)
=
Ipi(0)bhi;0+
Qpi
@$%
@t bhi+
Qu
2
i$%bhi:
Now we bound the term
Qpi
@$%
@t bhi=
T
0
@$%
@t
Ipibhi
by using that
Ipibhi6piL
∞bhiL16K @pi
@x
L2bhiL
1=KuiL2bhi;0L1;
and we obtain
Q
@$%
@t pibhi6
1
4ui2L∞(0;T;L2(I))+K2bhi;02L1:
To estimate the term Qu2
i$%bhi we take into account the presence of 12Qu2i$%bhi in
the left-hand side of the Eqs. (19) and (20), so that we only must bound 1
2
Qu
2
i$%bhi612
T
0 ui 2
L∞bhiL16C
2
The estimate of the terms in the right-hand side of (19) and (20) concludes withthe estimate of gQH$%bhi:
g
QH$%bhi6g
2
Q$%bh
2
i +g21
Q$%bH
26g
2
Q$%bh
2
i +gT 2MH2L2:
Then, (19) and (20) yield, respectively,
g−g2
Q$%bh
2 16A1
Ibh1;0logbh1;0−A1bh1;0logbh1;0|I|
+C1u12L2(0;T;V)+C1+
Ip1(0)bh1;0
+14u1L∞(0;T;L2(I))+K2bh1;02L1
+C2
2 bh1;0L1u12L2(0;T;V)+gT M
2H2L2 (24) and
g−g2
Q$%bh
2 26A2
Ibh2;0logbh2;0−A2bh2;0logbh2;0|I|
+C2u22L2(0;T;V)+C2+
Ip2(0)bh2;0
+14u2L∞(0;T;L2(I))+K2bh2;02L1
+C22bh2;0L1u22L2(0;T;V)+gT M
2H2L2: (25) The next step is adding Eq. (16) to Eqs. (24) and (25). Before this, we take the supremum in terms of the left-hand side of (16) and make % tend to zero in (24) and (25).
Thus, we have 1
4u12L∞(0;T;L2(I))+1
4u22L∞(0;T;L2(I))
+
A1−g2−C1−C 2
2 bh1;0L1
u12L2(0;T;V)
+
A2−g2−C2−C 2
2 bh2;0L1
u22L2(0;T;V)
+
g−g
2
√bh12L2(Q)+
g−g
2
6
2
i=1
1
2ui;02L2+Ai
Ibhi;0logbhi;0−Aibhi;0logbhi;0|I|
+C
i+
Ipi(0)bhi;0+K
2bh
i;02L1
+gT
1+M
H2
L2
+g
T
0
h1;@u@x1+g
T
0
h2;@u@x2
+g 2
1
T
0
h2;@u@x1+g
T
0
h1;@u@x2:
Using that mhi2L2(Q)6
√
bhi2L2(Q) and the estimates.
g
T
0
h1;@u@x16g2h12L2(Q)+g 1
2u12L2(0;T;V);
g
T
0
h2;@u@x26g2h22L2(Q)+g 1
2u22L2(0;T;V);
g 2
1
T
0
h2;@u@x16g 2 1
2h22L2(Q)+g 2
1
1
2u12L2(0;T;V);
g
T
0
h1;@u@x26g2h12L2(Q)+g 1
2u22L2(0;T;V);
we obtain (11).
Remark 1. We have obtained a priori estimates that avoid the appearance of a stability space.
If instead of (23) the following estimate was used 1
2
Qu
2
i$%bhi64
Q$%bh
2
i +MC
2
4 ui2L∞(0;T;L2(I))ui2L2(0;T;V);
then the coeJcients of ui2L2(0;T;V) in (11) would be
A1−g2 −C1−g21 −g 2 1
1 2 −
MC2
4 u12L∞(0;T;L2(I)) and
A2−g2 −C2−g1−MC 2
4 u22L∞(0;T;L2(I)) instead ofB1 and B2, given by (6) and (7).
In this case, the hypothesis on A1,A2 could be less restrictive than (5), but a small
data condition for the initial velocities would be required.
2.2. Approximated solutions
Let us introduce a basis for V denoted by {&1; : : : ; &n; : : :}, whose elements belong
toH3(I). Let Vn be the set of linear combinations of the n 9rst elements of the basis.
We consider the problem:
Find (u1;n; h1;n) and (u2;n; h2;n) in [L∞(0; T;L2(I))∩L2(0; T;Vn)]×C1( SQ) suchthat
(Vn)
@u1;n
@t ; &
+A1
@u1;n
@x ; @& @x +
u1;n@u@x1;n; &
−g
h1;n;@&@x
−g 2
1
h2;n;@&@x
=−g
H;@x@&
∀&∈Vn;
@u2;n
@t ; &
+A2
@u2;n
@x ; @& @x +
u2;n@u@x2;n; &
−g
h2;n;@&@x
−g
h1;n;@&@x
=−g
H;@&@x
∀&∈Vn;
@
@t(bh1;n) + @
@x(u1;nbh1;n) = 0; @
@t(bh2;n) + @
@x(u2;nbh2;n) = 0;
u1;n(t= 0) =u1;0;n∈Vn; u2;n(t= 0) =u2;0;n∈Vn;
h1;n(t= 0) =h1;0;n¿0∈C1(I); h2;n(t= 0) =h2;0;n¿0∈C1(I);
where the data and the constants satisfy the conditions of Theorem1. Then we have:
Lemma 2. The problem (Vn) has a solution {(u1;n; h1;n);(u2;n; h2;n)} in
[[L∞(0; T;L2(I)2)∩L2(0; T;V
n)]×C1( SQ)]2;
which satis7es
u1;nL∞(0;T;L2(I))+u2;nL∞(0;T;L2(I))+u1;n2L2(0;T;V)
+u2;n2L2(0;T;V)+h1;n2L2(Q)+h2;n2L2(Q)6C: (26)
Proof. To prove this lemma we apply the second Schauder 9xed point theorem [23] as it is made in [7]. We obtain approximated solutions that satisfy the a priori estimates.
In fact, due to the regularity of the basis, we have: u1;n; u2;n∈H1(0; T;H3(I)).
Therefore,u1;n; u2;n∈C0([0; T];C2( SI)) and, using (13) and the positivity of initial data
h1;0; h2;0, we h ave h1;n; h2;n∈C1( SQ) andh1;n; h2;n¿0.
2.3. Passage to the limit
In this section, we present a lemma that is used to pass to the limit in the approxi-mated equations and to conclude the proof of the theorem. The passage to the limit is done by adapting the procedure developed in [7]. In that case, the most diJcult point was to pass to the limit in the continuity equation. Now, this can be done in an easier way, because we have obtained an estimate for hi in L2(Q).
Lemma 3. For each n∈N, let
{(u1;n; h1;n);(u2;n; h2;n)} ∈[[L∞(0; T;L2(I))∩L2(0; T;Vn)]×C1( SQ)]2 be the solution of (Vn) given by Lemma 2.
Then we have, for i= 1;2,
ui;nhi;n is bounded in L2(0; T;L1(I)); (27)
@ui;n
@t is bounded in L4=3(0; T;H−1(I)); (28)
and we can extract from ui;n and hi;n subsequences still denoted ui;n and hi;n such that
ui;n→ui in L2(0; T;V) weakly; (29)
ui;n→ui in L∞(0; T;L2(I)) weakly-star; (30)
hi;n→hi in L2(Q) weakly; (31)
ui;nhi;n→uihi in L4=3(Q) weakly; (32)
ui;n@u@xi;n →ui@u@xi in L4=3(Q) weakly: (33)
Proof. Results (27), (29), (30) and (31) are a direct consequence of (26). To prove (28), (32) and (33) we 9rst notice that
u4
L46u2L∞u2L26C2u2L2u2V ∀u∈V:
This implies thatui;n is bounded in L4(Q). Thenui;nhi;n andui;n(@ui;n=@x) are bounded
inL4=3(Q), and we can extract subsequences from u
i;n andhi;n suchthat
and
ui;n@u@xi;n →-i in L4=3(Q) weakly:
In order to obtain,i=uihi and-i=ui(@ui=@x) we need an estimate for @ui;n=@t.
Notice that @hi;n=@x is bounded in L2(0; T; H−1(I)). By estimating the other terms
in the momentum equations we deduce that @ui;n=@t is bounded in L4=3(0; T; H−1(I)).
Now, using Aubin’s compacity theorem [13] with
A0=V; A1=L2(I); A2=H−1(I);
p= 2; q=4 3;
we have
ui;n→ui in L2(Q) and a:e: in Q: (34)
Let’∈D(Q). Then,
|(ui;nhi;n−uihi; ’)|6|(ui;nhi;n−uihi;n; ’)|+|(uihi;n−uihi; ’)|
6ui;n−uiL2(Q)hi;nL2(Q)’L∞(Q)+|(hi;n−hi; ui’)| and
ui;n@u@xi;n−ui@u@xi; ’6
ui;n@u@xi;n −ui@u@xi;n; ’
+
ui @u@xi;n−ui @u@xi; ’
6ui;n−uiL2(Q) @ui;n
@x
L2(Q) ’L ∞(Q)
+
@ui;n
@x − @ui
@x; ui’
:
So,
ui;nhi;n→uihi in D(Q)
and
ui;n@u@xi;n →ui@u@xi in D(Q);
which implies that,i=uihi and-i=ui(@ui=@x).
2.4. Proof of the theorem
Let{u1;0;n} and {u2;0;n} be two sequences withelementsui;0;n∈Vn suchthat
u1;0;n→u1;0 and u2;0;n→u2;0 in V:
Also, let{h1;0;n} and {h2;0;n} be two sequences inC1(I) suchthat
h1;0;n→h1;0 and h2;0;n→h2;0 in L2(I):
For each n∈N, set
{(u1;n; h1;n);(u2;n; h2;n)} ∈[[L∞(0; T;L2(I))∩L2(0; T;Vn)]×C1( SQ)]2;
a solution of (Vn) given by Lemma 2. This satis9es the estimate (26).
Using Lemma3, we can extract two subsequences to {u1;n} and{u2;n}, also denoted
by {u1;n} and{u2;n}, suchthat
u1;n→u1 and u2;n→u2 in L∞(0; T; L2(I))∩L2(0; T;V) weakly-star;
and two subsequences toh1;n andh2;n also denoted by h1;n andh2;n suchthat
h1;n→h1 and h2;n→h2 in L2(Q):
Then,
u1;nh1;n→u1h1 and u2;nh2;n→u2h2 in L4=3(Q) weakly:
Now we can deduce from the previous result that (@=@x)(uibhi) belongs to L4=3
(0; T;W−1;4=3(I)) and so hi;t. We also havehi(t= 0) =hi;0.
And we can pass to the limit in momentum equations and obtain ui(t= 0) =ui;0.
This concludes the proof that {(u1; h1);(u2; h2)} is solution of the weak
problem (V).
Having shown the existence of solutions to the problem (V), we are going to prove the uniqueness of the solution. In order to do this, 9rst we have to prove some smoothness results for (ui; hi); i= 1;2.
3. Some smoothness and uniqueness results
3.1. Further estimates
The following result concerns the functions$i appearing in (17) and (18), and it is
easily deduced from (21) and (22):
Lemma 4. The functions $i verify
$1∈L∞(0; T); $2∈L∞(0; T) (35)
and
$1;t=|1I|
I
u1u1;t+gh1;t+g 2
1h2;t
$2;t=|1I|
I(u2u2;t+gh1;t+gh2;t): (37)
Now we give a lemma that will allow us to prove our smoothness theorem.
Lemma 5. Let Qt be equal to I×(0; t) for anyt∈[0; T]. IfH∈L4(I) and h
i;0∈L3(I), then the relation
bhi3L∞(0;t;L3(I)2)+bhi4L4(Qt)6C(1 +pi;t4L4(Qt)) (38)
holds for i∈ {1;2}, and also the relation
@2pi
@x2
4
L4(Qt)6C 1 + 2
j=1
pj;t4L4(Qt)
: (39)
In both cases, C ¿0 is a constant that depends only on the data.
Proof. To prove (38) we 9rst multiply the continuity equations (12) by 3(bhi)2. We
obtain
@(bhi)3
@t + @
@x(ui(bhi)3) + 2(bhi)3 @2pi
@x2 = 0:
Now we only have to replace@2pi=@x2 withtheir values given by Eqs. (17) and (18),
integrate the result over Qt and use Young’s inequality to obtain the estimate (38).
Asm ¡ b(x)¡ M, we also have
hi3L∞(0;t;L3(I)2)+hi4L4(Qt)6C(1 +pi;t4L4(Qt)):
Finally, to prove (39) we multiply Eq. (17) by (@2p
1=@x2)3, Eq. (18) by (@2p2=@x2)3,
and then we integrate over Qt.
Using Gagliardo–Nirenberg’s inequality [4]
u2
L86Cu7L=25 @u@x3=5
L4 ;
again Young’s inequality and (38), we obtain (39).
Observe that, if we prove thatp1;t andp2;t belong toL4(Q), then we will also have
that hi and @ui=@x are in L4(Q); i= 1;2. This is the goal of the next result.
Theorem 2. Let b verifying (1) and @b=@x∈L∞(I). Let H∈L4(I)and let hi;0∈L3(I)
and ui;0∈H2(I)2 for i= 1;2. Then we have
pi∈W1;4(Q); (40)
Proof. The proof of this theorem is very technical and uses the techniques developed in [20]. To show thatp1∈W1;4(Q), it is enoughto prove that p1;t∈L4(Q). To prove
this, we 9rst di4erentiate Eq. (17) withrespect to the independent variables, and obtain the system
@u1
@t −A1 @2u
1
@x2 +
1 2
@u2 1
@x +g @h1
@x +g 21
@h2
@x −g @H
@x = 0; (41)
@p1;t
@t −A1 @2p
1;t
@x2 +u1u1;t+gh1;t+g 21 h2;t=$1;t: (42)
Next, we multiply (41) by 4u3
1, (42) by 2bp1;t, and integrate over I:
d dt
I|u1|
4+ 12A 1
I
u1@u@x1
2 = 4 I 1
2u21+gh1+g 2
1h2−gH
@u3 1
@x; (43)
d dt
I|
√
bp1;t|2+ 2A1
I
√b@p@x1;t2+ 2A1
I
@p1;t
@x p1;t @b @x
= 2
I($1;t−u1u1;t)bp1;t−2g
Ibh1;tp1;t−2g
2 1
Ibh2;tp1;t: (44)
Now, some of the terms of previous equations are rewritten. The last term in (43) can be expressed as
4
I
1
2u21+gh1+g 2
1h2−gH
@u3 1 @x = 12 I 1
2u21+gh1+g 2
1h2−gH
u2 1@u@x1;
and substituting g(h1+ ( 2= 1)h2) by its value in (17) we h ave
4
I
1
2u21+gh1+g 2
1h2−gH
@u3 1
@x
6 u1@u@x1
2
L2+C
I
|$1|2+|p1;t|2+
@u1 @x 2
|u1|2:
Then, Eq. (43) yields
d
dtu14L4+ (12A1− )u1@u@x1 2 L2 6C I
|$1|2+|p1;t|2+@u@x1
2
And the last two terms in (44) can be treated using the continuity equations (12), as follows:
−2g
Ibh1;tp1;t= 2g
I
@
@x(u1bh1)p1;t=−2g
Iu1bh1
@p1;t
@x
and
−2g 2
1
Ibh2;tp1;t= 2g
2 1
I
@
@x(u2bh2)p1;t=−2g 21
Iu2bh2
@p1;t
@x :
Now, notice that we can give an expression forh2 in which h1 does not appear if we
subtract Eq. (18) from Eq. (17). In a similar way, an expression forh1 (not containing
h2) is obtained.
Using these, we have that
−2g
Ibh1;tp1;t62
I
√b@p1;t @x
2+C
IS1;2|u1|
2
and
−2g 2
1
Ibh2;tp1;t62
I
√b@p@x1;t2+C
IS1;2|u2|
2;
where
S1;2=
i=1;2
|$i|2+|pi;t|2+@u@xi
2
+|ui|4+|H|2
:
Then, (44) yields d
dt
√
bp1;t2L2+ (2A1− )
√
b@p@x1;t2
L26−2A1
I
@p1;t
@x p1;t @b @x
+ 2
I($1;t−u1u1;t)bp1;t+C
IS1;2(|u1|
2+|u
2|2): (46)
Adding (45) and (46) we obtain
d
dt[u14L4+
√
bp1;t2L2] + (2A1− )
u1@u@x1
2
L2
+√b@p1;t @x
2
L2
6C
IS1;2(|u1|
2+|u
2|2) + 2A1
I
@p1;t
@x
|p1;t|@b@x
+ 2
I($1;t−u1u1;t)bp1;t: (47)
We start withthe terms IS1;2|u1|2:
As$i∈L∞(0; T) and u1∈L∞(0; T; L2(I)2),
I|$i|
2|u 1|26C:
We use the embedding ofV into L∞(I) andu1∈L∞(0; T;L2(I)2) again to obtain
I|pi;t|
2|u
1|26pi;t2L4u12L46pi;tL∞pi;tL2u1L∞u1L2
6C2p
i;tL2 @pi;t
@x
L2u1L 2u1V
6 √b@p@xi;t2
L2+Cu1
2
V
√
bpi;t2L2: (48)
We also have
I
@ui
@x
2 |u1|26@u@xi
L2 @ui
@x
L2 u
2 1L4
6C1=2u2 11L=22
@u21
@x
1=2
L2 @ui
@x
L2 @ui
@x
L4
6 u1@u@x1 2
L2+
u
iV@u@xi
2
L4+C
ui2Vu14L4: (49)
The termI|ui|4|u1|2 is bounded by
I|ui|
4|u
1|26C2u1|2Vui4L4: (50)
AsH andu1 are in L4(Q), we also have
I|H|
2|u
1|26HL24u12L4
withthe second term integrable in [0; T].
The termIS1;2|u2|2 is estimated in the same way.
The second term on the right-hand side of (47) is estimated using that @b=@x∈
L∞(I):
2A1
I
@p1;t
@x
|p1;t|@b@x6
√
b@p1;t @x
2
L2+C
√
bp1;t2L2:
Using (36) and the continuity equations (P)4 and (P)9 we have that
I$1;tbp1;t6C
u1L2 @p1;t
@x
L2+u1L
∞h1L1+u2L∞h2L1
p1;tL2
6 √b@p@x1;t2
L2+C(1 +u1
2
V +u22V)
√
bp1;t2L2: (51)
Next, the term Iu1; u1;tbp1;t is bounded by
Iu1u1;tbp1;t6u1L ∞
√
b@p1;t @x
L2
√
bp1;tL2
6 √b@p@x1;t2
L2+Cu1
2
V
√
bp1;t2L2: (52)
Now we de9ne
y1(t) = sup
3∈(0;t)(u1(3) 4
L4+
√
bp1;t(3)2L2);
z1(t) =u1@u@x1(t) 2
L2+
√b@p1;t @x (t)
2
L2;
y2(t) = sup
3∈(0;t)(u2(3) 4
L4+
√
bp2;t(3)2L2);
z2(t) =u2@u@x2(t) 2
L2+
√b@p@x2;t(t)2
L2;
and
51;2(t) = 1 +u1(t)2V +u2(t)2V:
Integrating (47) in (0; t) we 9nd that, by virtue of previous inequalities,
y1(t) = (2A1− )
t
0 z1(3) d36C
1 +
t
0 z2(3) d3
+ t
0 u1V
@u1
@x
2
L4+
t
0 u2V
@u2
@x
2
L4
+ t
0 51;2(3)y1(3) d3+
t
0 51;2(3)y2(3) d3
: (53)
We can obtain the analogous result for i = 2 after di4erentiating Eq. (18) with respect to all independent variables. Adding the results for i = 1;2 it
follows that
y1(t) +y2(t) + (2A1− )
t
0 z1(3) d3+ (2A2− )
t
0 z2(3) d3
6C
1 + t
0 u1V
@u1
@x
2
L4+
t
0 u2V
@u2
@x
2
L4
+ t
0 51;2(3)y1(3) d3+
t
0 51;2(3)y2(3) d3
: (54)
Setting Qt=I×(0; t) and using Lemma5 we have
t
0 uiV
@ui
@x
2
L46uiL 2(0;t;V)
@ui
@x
2
L4(Qt) 6C(1 +pi;t2L4(Qt)+pj;t2L4(Qt)) for i; j∈ {1;2}; j=i.
Usingpi;t2L46Cpi;tL2@pi;t=@xL2 again we can write
pi;t2L4(Qt)6
C2
m2
t
0
√
bpi;t2L2
√b@p@xi;t2
L2 1=2
6Cmyi(t)1=2
t
0 zi(3) d3
1=2
6 t
0 zi(3) d3+Cyi(t)
for i= 1;2.
Hence, choosing and small enough, (54) yields
y1(t) +y2(t) +
t
0 z1(3) d3+
t
0 z2(3) d3
6C
1 +
t
0 51;2(3)y1(3) d3+
t
0 51;2(3)y2(3) d3
: (55)
As51;2 is integrable in [0; T], we can apply Gronwall–Bellman’s lemma to conclude
that
u14L∞(0;T;L4(I))+u24L∞(0;T;L4(I))+p1;t2L∞(0;T;L2(I)2)+p2;t2L∞(0;T;L2(I)2)
+u1 @u@x1 2
L2(Q)+
u2@u@x2 2
L2(Q)+ @p1;t
@x
2
L2(Q)
+@p@x2;t2
L2(Q)6C: (56)
3.2. A uniqueness theorem
From the previous estimate we deduce that ui∈L4(0; T;W1;4(I)2), and using this
regularity ofui the following result can be shown as in [8]:
Lemma 6. If hi;0¿0;loghi;0∈L∞(I) and ui;0∈H1(I)2, then we have
hi and h1 i∈L
∞(Q) (57)
for i= 1;2.
Now we state the uniqueness theorem:
Theorem 3. If ui;0 and hi;0 verify the hypotheses of Theorem 2 and Lemma 6, then
the problem (V) has a unique solution {(u1; h1);(u2; h2)} such that
{(u1; h1);(u2; h2)} ∈[L4(0; T;W1;4(I))×L∞(Q)]2: (58)
Proof. Let {( Su1;hS1);( Su2;hS2)} and {( ˜u1;h˜1);( ˜u2;h˜2)} be two solutions of (V), with
S
u1=@@xpS1; uS2=@@xpS2 and u˜1=@@xp˜1; u˜2=@@xp˜2:
Then, the functions (u1; h1) = ( Su1−u˜1;hS1−h˜1) and (u2; h2) = ( Su2−u˜2;hS2−h˜2), with
u1=@p@x1; u2=@p@x2;
p1= Sp1−p˜1; p2= Sp2−p˜2;
verify
@p1
@t −A1 @2p
1
@x2 +
1
2u1( Su1+ ˜u1) +gh1+g 21h2= S$1−
˜
$1; (59)
@(bh1)
@t + @
@x(u1bhS1+ ˜u1bh1) = 0 (60)
and
@p2
@t −A2 @2p
2
@x2 +
1
2u2( Su2+ ˜u2) +gh2+gh1= S$2−$˜2; (61)
@(bh2)
@t + @
We de9ne the auxiliary functions 1 and 2 as the solutions of the problems
@2 1
@x2 =bh1 in I
@ 1
@x = 0 on @I
and
@2 2
@x2 =bh2 in I;
@ 2
@x = 0 on @I:
Now, multiplying (59) by bp1, (60) by 1 and integrating over I, we obtain
1 2
d dt
√
bp12L2+A1
√
b@p1 @x
2
L2+A1
I
@p1
@x p1 @b @x
+1 2
Ibp1u1( Su1+ ˜u1) +g
Ip1
@2 1
@x2 +g 21
Ip1
@2 2
@x2 =
Ibp1( S$1−
˜
$1);
1 2
d dt
@ 1
@x
2
L2+
I 1
@
@x(u1bhS1+ ˜u1bh1) = 0:
Adding these equations we have
1 2
d dt
√bp12L2+ @ 1
@x
2
L2
+A1
√
b@p@x12
L2
=
Ibp1( S$1−
˜
$1)−A1
I
@p1
@x p1 @b @x −
1 2
Ibp1u1( Su1+ ˜u1)
+g
I
@p1
@x @ 1
@x +g 21
I
@p1
@x @ 2
@x +
I
@ 1
@x u1bhS1+
I
@ 1
@x u˜1bh1: (63)
We de9ne the quantities
y1=
√
bp12L2+ @ 1
@x
2
L2 and y2=
√
bp22L2+ @ 2
@x
2
L2:
All the terms of the right-hand side of (63), except the last one, are easily estimated by
√b@p@x12
L2+Cy1 or
√b@p@x12
L2+Cy2
The last term of (63) is integrated by parts. We obtain
I
@ 1
@x u˜1 @2
1
@x2 =−
1 2
I
@ 1
@x
2@u˜1
@x :
By using Lemma 6, h1 is a bounded function and then @ 1=@x is also
a bounded function. Thus, if |@ 1=@x|6M a.e., we can estimate the previous
term by
M2%
@ 1
@x
2(1−%)
L1=(1−%) @u˜1
@x
L1=%=M
2%@ 1
@x
2(1−%)
L2 @u˜1
@x
L1=%
with%∈(0;1). Multiplying Eq. (17) by (@2p1=@x2)(1−%)=% and integrating over I gives
the estimate @u˜1
@x
L1=%6C:
Then,
I
@ 1
@x u˜1 @2
1
@x2 6Ky11−%:
Now, if we choose suJciently small, equality (63) gives dy1
dt 6C1y1+K1y11−%+C2y2: (64)
Multiplying (61) by p2 and (62) by 2 we can obtain the analogous result for
i= 2: dy2
dt 6C2y2+K2y21−%+C1y1; (65)
and adding (64) and (65) we obtain d(y1+y2)
dt 6C(y1+y2) +K(y1+y2)1−%;
and consequently,
d((y1+y2)%)
dt 6%C(y1+y2)%+%K:
Gronwall–Bellmann’s lemma gives the estimate
and thus,
y1(t) +y2(t)6(%K)1=%eCt:
The term on the right-hand side converges to zero as % tends to zero. This proves that y1(t) and y2(t) are equal to zero and concludes the proof of the
theorem.
4. The numerical scheme
In this section we present a 9rst-order upwind numerical scheme for solving the system (P). As in [15], both the advective and the viscous terms are treated explicitly, using a volume 9nite method. The advective term is discretized using the Q-scheme of Van-Leer described in [6], where the techniques used in [16] and [17] are used. The viscous term is discretized as in [9]. The source terms due to depth and breadth variations are upwinded using the techniques developed in [21,22].
In order to introduce the numerical scheme, the system (P) is rewritten under the form
@U @t (x; t) +
@F
@x(U(x; t)) + @G
@x(U(x; t)) =:(x; U(x; t)); (66)
where
U(x; t) =
u1(x; t)
h1(x; t)
u2(x; t)
h2(x; t)
; (67)
and, given UT= [u
1; h1; u2; h2], and r= 2
1, we de9ne
F(U) =
u2 1
2 +gh1+rgh2
u1h1
gh1+u 2 2
2 +gh2
u2h2
; (68)
G(U) =
−A1@u@x1
0
−A2@u@x2
0
and 9nally,
:(x; U) =
gH(x)
−u1bh1b(x)
gH(x)
−u2bh2b(x)
: (70)
Using this notation, the numerical scheme is written as follows:
Un+1
i =Uin+VVtx(Fin−1=2−Fin+1=2) +VVtx(Gin−1=2−Gni+1=2)
+Vt
Vx(PR+n;i−1=2:˜ni−1=2+ PR−n;i+1=2:˜ni+1=2); (71)
where
Fn
i+1=2=12(F(Uin) +F(Uin+1))−12|A˜ni+1=2|(Uin+1−Uin); (72)
Gn i+1=2=
−A1u
n
1;i+1−un1;i
Vx
0
−A2u
n
2;i+1−un2;i
Vx
0
: (73)
Here, Vt represents the chosen time step; Vx=xi+1−xi; i=0; : : : ; N−1 the size of the
cells,xi; i=0; : : : ; N, the nodes of the mesh; the size of the cells;Uin the approximation
of U(xi; nVt) given by the numerical scheme; and ˜Ani+1=2=A( ˜Uni+1=2), where A(U) is
the Jacobian matrix of the 7ux F(U):
A(U) =@U@F(U) =
g u1 rg 0
u1 h1 0 0
g 0 g u2
0 0 u2 h2
and
˜
Un
i+1=2=U
n i +Uin+1
2 :
Finally, ˜
:n
and
PR±
n;i+1=2=12Pni+1=2(Id+ sgn(Dni+1=2))(Pin+1=2)−1; (74)
whereId is the identity matrix and
sgn(Dn i+1=2) =
sgn(n
i+1=2;1) · · · 0
...
0 · · · sgn(n i+1=2;4)
:
In the formula above, n
i+1=2;l; l= 1; : : : ;4 represent the eigenvalues of ˜Ani+1=2 and
Pn
i+1=2 is a matrix whose columns are eigenvectors corresponding to the eigenvalues
n
i+1=2;l. Observe that source terms due to channel geometry are treated as in
[3,22].
A CFL-like requirement has to be imposed in order to ensure the numerical stability. In practice, we impose the following condition:
Vt6?min
Vx
|n
l;i+1=2|;16l64; 16i6N
;
(Vx)2
2Aj ; j= 1;2 (75)
witha chosen?∈(0;1].
On the other hand, to prevent the numerical viscosity of theQ-scheme from vanishing when any of the eigenvalues of the matrices |An
i+1=2| are zero, we apply the Harten
regulation [11].
The application of the previous numerical scheme to (66) needs the calculation of the eigenvalues and eigenvectors of the matrices |A˜n
i+1=2|. The 9rst step consists in
solving the algebraic equation
det( ˜An
i+1=2−Id) = 0: (76)
This has been done as follows: 9rst, Newton’s method is applied to (76) using, as initial guess, the 9rst-order approximation of eigenvalues of A(U) given in [18]:
±=u1h2+u2h1
h1+h2 ±(g(h1+h2))
1=2: (77)
Once the 9rst two roots are found, the fourth-degree polynomial is de7ected. The two roots of the resulting second-degree polynomial (if they are real) are used again as an initial guess for the Newton’s method applied to (76). With this choice of initial guess, Newton’s method converges rapidly (one or two iteration are needed in the
numerical tests performed). Once the eigenvalues are approximated, the calculation of their associated eigenvectors is performed. The matrix-vector and inverse of matrix computations are made by using the C + +-library newmat09.
5. Numerical results
In this section we present some numerical results obtained with the numerical scheme (71)–(74) applied to (66) in a channel with a simple geometry.
In Section5.1, the scheme is used to obtain some steady-state solutions correspond-ing to decreascorrespond-ing values of the viscosity coeJcient. These steady-state solutions are compared with the approximations found by Armi and Farmer (A&F hereafter) in [1,2,10] under the hypothesis of rigid lid and neglecting the viscous e4ects.
In Section 5.2, a periodic barotropic forcing is imposed to the two-layer exchange 7ow, as in [12] and we obtain some periodic solutions.
In both cases, the geometry of the channel is given by the functions (see Fig. 1)
H(x) = 2− 1
cosh2(3:75x); x∈[−1;2];
b(x) =−0:5 + 1:5(1−e−a2(x−1)2
); x∈[−1;2]; a=
0:637 if x61;
1:273 if x ¿1:
They represent a channel with a sill placed at x= 0 and a contraction placed at
x= 1.
5.1. A&F steady solutions
A&F obtained the stationary solutions of two-layer exchange 7ows through con-tractions (in [1,2]) and over sills (in [1,10]). Their model is based on the Bernoulli equations under the assumption of rigid lid. They parameterize the 7ows in terms of the internal Froude numbers for each layer Fi given by
F2
i = u
2
i
ghi:
The steady-state solutions are then found as curves in the Froude-number plane (F2 1; F22).
This analysis, based on the dimensionless Bernoulli equation expressed in terms ofF2 1
and F2
2, has been extended recently by Mac#$as to channels with a contraction and a
sill (see [14] for more details).
The 9rst experiment performed consists in taking
h1(0; x) =H(x)−0:7;
h2(0; x) = 0:7;
-2 -1.5 -1 -0.5 0
-1 -0.5 0 0.5 1 1.5 2 Bottom
0 0.5 1 1.5 2
-1 -0.5 0 0.5 1 1.5 2 Breadth of the channel
(a)
(b)
Fig. 1. Geometry of the channel (depth and breadth). (a) Depth of the channel and (b) breadth of the channel.
-2 -1.5 -1 -0.5 0
-1 -0.5 0 0.5 1 1.5 2
Free Surface Interface Bottom
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4
-1 -0.5 0 0.5 1 1.5 2
A&F solution Viscosity=0.0 Bottom
(a)
(b)
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4
-1 -0.5 0 0.5 1 1.5 2
Viscosity=0
0 Viscosity=0.001
1 Viscosity=0.01
2 Bottom
(c)
Fig. 2. A&F solution and stationary solution corresponding toA1=A2=0:0; A1=A2=0:01 andA1=A2=0:001:
(a) initial condition; (b) stationary solution withA1=A2=0 and comparison withA&F solution; (c) stationary
-2 -1.8 -1.6 -1.4 -1.2 -1 -0.8 -0.6 -0.4
-1 -0.5 0 0.5 1 1.5 2 0->t=0
0 1->t=200
1 2->t=400
2 3->t=600
3
Bottom
Fig. 3. Intreface evolution during a periodic barotropic 7ow.
as initial conditions and
h1(t;2) = Sh1;
h2(t;−1) = Sh2;
u1(t;−1) =uL1; u1(t;2) =uR1;
u2(t;−1) =uL2; u2(t;2) =uR2
as boundary conditions, where Sh1; hS2; uL1; uR1; uL2 anduR2 are the values corresponding
to a given stationary A&F solution at the boundary of the channel.
The numerical scheme can be used without viscosity termsA1=A2=0. Nevertheless,
it gives bad results where discontinuities develop. As it is well known [19], this is a consequence of the choice of the formulation of the equations: when shocks develop, a formulation based on the use of the conservative unknowns qi =uihi and hi is
required. A numerical scheme based on this type of formulation has been presented in [5]. Nevertheless, in the present case, the 7ow does not develop any discontinuity and the numerical results obtained with both schemes are the same. Figs.2(a) and (b) show, respectively, the initial and the 9nal stationary state reached in the experiment performed with A1=A2= 0. In the latter 9gure, this 9nal state is compared with the
The same experiment has been run for di4erent values of the viscosity coeJcients (A1; A2). Fig. 2(c) shows the interface for di4erent values of A1 and A2. The curves
denoted by 0;1;2 correspond to the interface corresponding to the steady state computed for the choices of coeJcients:A1=A2=0; A1=A2=0:01 andA1=A2=0:001, respectively.
As it could be expected, viscous solutions converge to the inviscid one as the viscosity coeJcients tend to zero.
5.2. Periodic barotropic forcing
In [12] a model is presented for the study of time-dependent two-layer hydraulic 7ows through straits, under the rigid-lid hypothesis. The model is used to simulate 7ows forced by a periodic barotropic (tidal) 7ow. We present here a numerical experiment similar to one of those presented by this author. We take as initial condition the steady-state solution corresponding the experiments presented in the previous section withviscosity coeJcients A1=A2= 0:005. The boundary conditions needed for this
experiment are computed using the model developed by Castro–Macias–Par#es where a periodic barotropic transport is imposed (see [5,6] for more details).
Fig. 3 depicts the interface along the channel at four di4erent times through the forcing period. The behavior of the obtained solution is similar to that described in [12]: the interface moves back-downwards and forth-upwards with the barotropic 7ow but still similar in shape to the steady solution. The periodic solution is obtained from the initial state without any prior adjustment period.
6. Concluding remarks
The analysis and the numerical solution of the system of partial di4erential equations governing the one-dimensional 7ow of two superposed shallow layers of immiscible viscous 7uid have been carried out.
For its mathematical analysis, we have adapted the techniques developed by Orenga and Chatelon for the case of a single layer of shallow water. The main diJculty of this adaptation was related to the need of some stronger estimates. We have solved this diJculty by obtaining L2 a priori estimates for the thickness of the layers. Moreover,
the regularity properties of Sobolev spaces in dimension 1 avoid the appearance of stability spaces in the existence theorem.
The case of non-homogeneous Dirichlet boundary conditions is now under study. The next step will be the study of the two-dimensional case: the proofs on this article do not apply to that case, because of the lack of regularity of Sobolev spaces in two-dimensional domains.
Concerning the approximation of the solution, we have presented an explicit numeri-cal scheme and we have tested it by comparing its results with the steady-state approx-imated solutions obtained by using the techniques developed by Armi and Farmer. We have also considered a solution on a strait with a sill and a contraction, and a periodic barotropic forcing, as in [12]. In all the cases, the numerical solutions provided by the scheme behave well.
The 9nal goal of this work is to obtain a two-dimensional numerical model, well suited to analyse the relationship between the water exchange through the Strait of Gibraltar and the generation of internal waves induced by tidal e4ects.
References
[1] L. Armi, The hydraulics of two 7owing layers with di4erent densities, J. Fluid Mech. 163 (1986) 27–58.
[2] L. Armi, D. Farmer, Maximal two-layer exchange through a contraction with barotropic net 7ow, J. Fluid Mech. 164 (1986) 27–51.
[3] A. Berm#udez, M.E. V#azquez, Upwind methods for hyperbolic conservation laws with source terms, Comput. Fluids 23 (8) (1994) 1049–1071.
[4] H. Brezis, Analyse Fonctionnelle, th#eorie et application, Collection appliqu#ee pour la maˆ$trise, Masson, 1983.
[5] M.J. Castro, J. Mac#$as, C. Par#es, Simulation of two-layer exchange 7ows through a contraction witha 9nite volume shallow water model, in: Actas de las II Jornadas de An#alisis de Variables y Simulaci#on Num#erica del Intercambio de Masas de Agua a trav#es del Estrecho de Gibraltar, C#adiz, 2000, pp. 205–221.
[6] M.J. Castro, J. Mac#$as, C. Par#es, AQ-scheme for a class of system of coupled conservation laws with source terms, Application to a two-layer 1D shallow water system, Math. Model. Numer. Anal. 35 (1) (2001) 107–127.
[7] F.J. Chatelon, P. Orenga, On a non homogeneous shallow water problem, Mod#elisation Math. Anal. Num#er. 31 (1) (1997) 27–55.
[8] F.J. Chatelon, P. Orenga, Some smoothness and uniqueness results for a shallow water problem, Adv. Di4erential Equations 3 (1) (1998) 155–176.
[9] R. Eymard, T. GallouYet, R. Herbin, Finite volume methods, Pr#epublication no. 97-19 du LATP, UMR 6632, Marseille, 1977, in: P.G. Ciarlet, J.L. Lions (Eds.), Handbook of Numerical Analysis, to appear.
[10] D. Farmer, L. Armi, Maximal two-layer exchange over a sill and through a combination of a sill and contraction withbarotropic 7ow, J. Fluid Mech. 164 (1986) 53–76.
[11] A. Harten, P. Lax, A. van Leer, On upstream di4erencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev. 25 (1983) 35–61.
[12] K.R. Helfrich, Time-dependent two-layer hydraulic exchange 7ows, J. Phys. Oceanogr. 25 (3) (1995) 359–373.
[13] J.L. Lions, Quelques m#ethodes de R#esolution des Probl[emes aux Limites non Lin#eaires, Dunod, Paris, 1969.
[14] J. Mac#$as, Two layer exchange over straits with barotropic 7ow, Internal Journal 01-15, group on “Di4erential Equations, Numerical Analysis and Applications”, University of M#alaga, Spain, 2001, in progress.
[15] B. Mohammadi, Fluid dynamics computation with NSC2KE an User-Guide Release 1.0, Rapport technique, INRIA, 1994, p. 164.
[16] P.L. Roe, Approximate Riemann solvers, parameter vectors and di4erence schemes, J. Comput. Phys. 43 (1981) 357–371.
[17] P.L. Roe, Upwinding di4erenced schemes for hyperbolic conservation laws with source terms, in: Carasso, Raviart, Serre (Eds.), Proceedings of the Conference on Hyperbolic Problems, Springer, Berlin, 1986, pp. 41–51.
[18] J.B. Schijf, J.C. Schonfeld, Theoretical considerations on the motion of salt and fresh water, in: Proceedings of the Minn. Int. Hydraulics Conv., Joint meeting IAHR and Hyd. Div. ASCE., September, 1953, pp. 321–333.
[19] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. A Practical Introduction, Springer, Berlin, 1997.
[20] V.A. Vaigant, A.V. Kazhikhov, Global solutions to the potential 7ow equations for a compressible viscous 7uid at small Reynolds numbers, Di4erential Equations 30 (6) (1994) 935–947.
[21] M.E. V#azquez-Cend#on, Estudio de Esquemas Descentrados para su Aplicaci#on a las leyes de Conservaci#on Hiperb#olicas con T#erminos Fuente, Ph.D. Thesis, Universidad de Santiago de Compostela, 1994.
[22] M.E. V#azquez-Cend#on, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. Comp. Phys. 148 (1999) 497–526.
[23] E. Zeidler, Nonlinear Functional Analysis and its Applications, Fixed Point Theorems, Vol. I, Springer, Berlin 1986.
