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An approximate solution for two-dimensional estuaries
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES
H. Rasmussen Department of Applied Mathematics, University of Western Ontario London, Ontario, Canada
ABSTRACT The steady flow in a slightly stratified two-dimensionalestuary is considered. An approximate form of the salinity which contains an unknown constant and which satisfies the prescribed boundary conditions is assumed. From a simplified form of the equations of motion the streamfunction is obtained. The unknown constant is then found by applying Galerkin's method the salt continuity equation.
1.
Introduction
In many estuaries the turbulent mixing is so intense that the vertical salinity stratification is slight. In some of these cases the governing partial differential equations and associated boundary conditions can be considerably simplified. However, the obtained model is still so complicated that analytical solutions cannot be found. It is therefore necessary to consider methods for obtaining approximate solutions. In the model analysed in this paper the flow is assumed to be twodimensional so the model is only applicable to the upper parts of fairly straight rivers with little variation in the breadth. For a derivation of the model see Rasmussen and Hinwood [l]. The model has previously been studied by Rattray and Hansen [2], Hansen and Rattray 131, and Hansen [4], mainly using the concept of similarity solutions; this work has been reviewed in Rasmussen and Hinwood [5]. In these papers the estuary is divided into an outer regime, a central regime and an inner regime. Approximate solutions are then obtained for each regime. The main problem with this approach is that the solution for the central regime does not match the solutions in the other two regimes at the ends of the central regime. Another difficulty is that it is necessary to suppose that some of the eddy coefficients have particular variations with respect to the coordinates. In this paper we obtain an approximate solution valid for the inner and central parts of the estuary. The approach used does not require any particular form of the eddy coefficients. The solution is not valid for the outer regime, but since the flow is undoubtedly three-dimensionalthere, this is not a serious limitation. 415
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES 2. Formulation We shall now state the mathematical flow in a slightly
stratified
The model consists
of three parts:
an approximate
equation
system where
and z the vertical
coordinate
coordinate.
the flow can no longer be supposed to the surface.
and z positive
downward.
depth is usually
equation,
conditions.
system with x being the horizontal
to that point in the estuary
to be two-dimensional, x positive
and z = 0
in the seaward direction
Since the ratio of the length of the estuary
to the upstream
that x ranges
end of the estuary
to the
from 0 to -m, where
the
is zero.
We let I$ and s be the non-dimensional respectively.
In order
the case where
the eddy coefficients,
constant;
boundary
The origin of the coordinate
very large, we shall suppose
with x = -m corresponding salinity
We measure
two-dimensional
as it is derived [l] .
salt continuity
and appropriate
is chosen such that x = 13 corresponds
corresponds
for steady,
estuary
an approximate
of motion,
Let (x,z) be a rectangular coordinate
model
constant-depth
the method
to simplify
can be extended
Then the model can be written
breadth,
and salinity,
we shall only consider
depth and wind stress are
to treat variations
in these parameters.
in the form
a$ as ---_--_=x_+~-
ax a2
streamfunction
the presentation
a$ as
a2 ax
a2s
a2s
ax2
az2
(3)
lim *-+-CC
s = xl:m$=
0
T = rD2B/nf AR
where
X = Kx B/R 2 = Ks B/R
y = -k B g So D3/RA
The different
D
B
physical
quantities
are defined
= depth of estuary = breadth
of estuary
= horizontal
KIC = vertical K
eddy diffusivity
eddy diffusivity
AZ
= eddy viscosity
g
= acceleration
due to gravity
as follows
(5)
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES k
417
= constant (
sO
R T of
=
reference salinity
= river flow = wind stress on the surface = density of fresh water
3. Approximate Solution
The procedure consists of writing an approximation Sl to the salinity s which satisfies the boundary conditions on meters CLand C.
S
and which contains two para-
An approximation $1 to the streamfunction $ is then obtained
by substituting Sl for s in equation (2) and integrating. The unknown integrating functions are determined from the boundary conditions on $I. We now suppose that s(x,z) can be approximated by Sl(X,Z) = fo(x) + Cfl(X) cos TIZ where
(6)
fo(x) = eax/(eax + eWaz)
(7)
fl(X) = (e"lX+ e-ax)-*
(8)
a and C are unknown parameters. This approximation satisfies all the boundary conditions on s.
It is easy
to show that -l
co.
(9)
When the approximation Sl(x,z) is substituted into equation (2), we obtain the following differential equation for $1, the approximation to Ji, a4*1 -= az4
yf~(X) + Cyfi(X) COS TfZ .
By integrating four times we get
@,(x,zf
=
-I._f'(x)z4 +
+
gzwz2+ gl(x)z
24
$f;(x, lr
*
cos AZ + g3(x)z3
+ g,(x).
(10)
Expresssions for the integrating functions go(x), g,(x), g,(x) and g3(x) are obtained by application of the boundary conditions for q at z = 0 and 1; hence
g,(x)
= 1 -
$
f;(X),
II 2 g,(x) = -Y+&
f;(x) + 12-n 4x4
y Cfi(x),
01) g,(x)
=$T+L
2n2
fi(X)*
AN APPROXIMATESOLUTION FOR TWO-DIMENSIONALESTUARIES 2 g,(x)= y
-&
f;(x) -=y 4a4
Cfi(x)
We note that Jll(x,z)satisfies the boundary condition (5) at the end of the estuary. If we substitute $l,and Sl for JIand s in the salt continuity equation (l), we obtain a residual R(x,z) given by
a+, as1
-_-.--+
ax
a2
wl
a2
as1
ax
02)
If $1 and Sl were the exact solutions, R would be identically zero. We now use Galerkin's method to obtain relationship between a and C from R.
The
method consists of multiplying R by the term fl(xf cos IIZand then insisting that the integral of this expression over the complete domain vanishes, i.e. we insist that 0
1 fl(x) cos ?TZR(x,z) dz = 0 .
dx I -co
i 0
This leads to the following equation for C F2 C2 + Fl C + Fo = 0
(13)
where F,
= -
ya :+ 10T + 60
1
240 772 2 XCr+-g-p&+ F1 = 7s
3(2 - T) 512n (14)
_ 2x5 + 27774- 1921r2+ 2304
Yo f
38,864x6 F
= (n2 + 4)(77+ 1) Ya 2
1120 I?
For given values of the different parameters this equation can then be solved for C; the solution must satisfy condition (9). The other parameter CIis determined from the assumed length of the estuary; this is discussed in the next section.
4. Results We shall now derive some characteristicvalues of the physical parameters determining the flow. Let us suppose that all the parameters are constant. It is reasonable to assume that the eddy diffusivities and viscosity are given by Kx =
10 u*D, Kx = 0.1 u*D, A = 0.1 u,D
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONAL ESTUARIES where
419
"* = 0.05 g R
= river flow ,
D
= depth of the estuary ,
B
= breadth of the estuary .
Thus we have KB x = x R
= 0.5,
KB 2 = -?-- = 0.005 . R
-4 In the equation of motion the parameter y appears. Let k S = 10 , O4 B = lo4 cm, D = lo3 cm, R = lo7 cm3/sec. Then y = - 2 x 10 . In [l] it was shown that the model is valid provided the Froude number F = !l/Jg k So D is substantially less than unity; here cl is the horizontal mean velocity. If we set Fl = R/B
D, we get that F = 0.1, so the condition is satisfied.
The value of a is obtained by supposing that the two-dimensionalpart of the estuary is lo6 cm long and that f (x) there is only 10% of fo(0). The result O-3 is that a is approximately 16 x 10 . We shall now show representative results using the above values of the parameters together with different values for T which represents the effect of wind stress on the surface. In Table 4.1 the two solutions Cl and C2 of equation (13) are given for T varying between -50 and 50.
It is noticed that
Table 4.1 Values of Cl and C2 for different values of T with x = 0.500, z = 0.005, y = -20000, o = 0.0014. T -50.0
c1 -569.178
c2 0.232
-40.0
-556.647
0.187
-30.0
-544.113
0.139
-20.0
-531.577
0.089
-10.0
-519.038
0.037
0.0
-506.497
-0.018
10.0
-493.953
-0.076
20.0
-481.407
-0.136
30.0
-468.857
-0.200
40.0
-456.303
-0.268
50.0
-443.745
-0.339
no acceptable solution exists for T 2 - 10, and a simple analysis shows that = 0 for T = -3.2. This means that if the windstress is too large in the c2 upstream direction, the approximate solution is no longer satisfactory. It is not clear if this result also holds for the exact solution. A possible physical explanation is that as T is changed from the value zero, the values of the turbulent parameters X, 2, y must also be changed. Since it is not clear how they should be changed, this idea has not been studied any further, and in the remaining part of the section we shall only consider T positive. In Tables 4.2 and 4.3 we present the salinity and horizontal velocity component at different depths and positions for T = 0. We notice 'thatthe
420
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES Table 4.2 Salinity at different depths and positions with X = 0.500, Z = 0.005, y = -20000, a = 0.0014, T = 0.0. Depth
Distance upstream
0.0
0.25
0.50
0.75
1.00
0
0.4955
0.4968
0.5000
0.5032
0.5045
-200
0.3594
0.3606
0.3635
0.3665
0.3677
-400
0.2427
0.2437
0.2460
0.2484
0.2493
-600
0.1547
0.1554
0.1571
0.1588
0.1595
-800
0.0947
0.0951
0.0962
0.0973
0.0978
0.0564
0.0566
0.0573
0.0580
0.0583
-1000
salinity changes by almost a factor ten over a distance of 1000 times the depth. From Table 4.3 it is seen that there is no reversal of horizontal velocity for T = 0. Table 4.3 Mrizontal velocity at different depths and positions with X = 0.500, Z = 0.005, y = -20000, a = 0.0014, T = 0.0. Distance upstream
Depth 0.0
0.25
0.50
0.75
1.00
0
1.7917
1.5703
1.0521
0.4557
0.0
-200
1.7696
1.5580
1.0577
0.4708
0.0
-400
1.7159
1.5279
1.0712
0.5076
0.0
-600
1.6540
1.4930
1.0866
0.5502
0.0
-800
1.6011
1.4632
1.0998
0.5866
0.0
-1000
1.5628
1.4416
1.1094
0.6130
0.0
Some values of the horizontal velocity component for different values of T are given in Table 4.4.
It is seen that velocity reversal takes place
for T = 10 at a depth of approximately 0.62, and that the depth at which this happens decreases as T increases. This is in agreement with the physics Table 4.4 Horizontal velocity for different values of T and at different depths at the downstream end with x = 0.500, z = 0.005, y = -20000, o = 0.0014. T Depth .
0
10
20
30
0.0
1.7917
4.2917
6.7917
9.2917
0.2
1.6453
2.4453
3.2453
4.0453
4.8453
5.6453
0.4
1.2810
0.9810
0.6810
0.3810
0.0810
-0.2190
0.6
0.8107
0.0107
-0.7893
-1.5893
-2.3893
-3.1893
0.8
0.3463
-1.0537
-I.7537
-2.4537
-3.1537
1.0
0.0
-0.3537 0.0
0.0
0.0
40 11.7917
0.0
50 14.2917
0.0
AN APPROXIMATE SOLUTION FORTWO-DIMENSIONS
ESTUARIES
42f
of the problem. In Table 4.5 we show the effects of increasing the windstress on the surface salinity. It shows, as expected, that as the windstress increases, the surface salinity decreases. Table 4.5 Surface salinity at different positions and for different values of T with X = 0.500, Z = 0.005, y = -20000, a = 0.0014. Distance upstream 500
1000
0.4955
0.1950
0.0564
10
0.4811
0.1858
0.0532
20
0.4659
0.1762
0.0500
30
0.4500
0.1661
0.0465
40
0.4331
0.1554
0.0429
50
0.4153
0.1440
0.0390
T 0
0
5.
Discussion
The present work indicates that Galerkin's method can be used to obtain approximate solutions for slightly stratified estuaries and thus be used to study the effects of different assumptions concerning the form of the turbulent eddy coefficients. Because of the lack of experimental data it is, however, difficult to estimate the accuracy of the solution when only a twoterm expansion is used. It would be of interest to expand the present work to a four-term expansion. Then the nonlinear transformatian 2 s n+l.Sn-l - 'n 'I(%) = Sn+l + Sn_l - 2s n which extraqts from any three successive terms of a sequence Sn the estimated limit kl(Sn) for n -+m, see Van Dyke [6], can be used to estimate the accuracy of the expansion. However, in this case the integrations must be carried out numerically.
Acknowledgements The author wishes to thank Or. P. Sullivan, University of Western Ontario and Or. J.B. Hinwood, Monash University for many helpful discussions concerning flow in estuaries. The work was supported by National Research Council of Canada (Grant No. A9251).
422
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES References
[l]
RASMUSSEN, H. & HINWOOD, J.B., On flow in estuaries, Part III. La Houille Derivation of general and breadth integrated models. Blanche: 212, 319, 1973.
[Z]
RATTRAY, M., Jr. & HANSEN, D.V., A similarity solution in an estuary. J. Mar. Res.: 0, 121, 1962.
[3]
HANSEN, D.V. & RATTRAY, M., Jr., Gravitational circulation 104, 1965. and estuaries, J. Mar. Res.: 3,
[4]
HANSEN, D.V., Salt balance and circulation in partially mixed estuaries. In Estuaries, ed. G.H. Lauff, Publication No. 83, American Association for the Advancement of Science, Wash., p. 45, 1967.
[5]
RASMUSSEN, H. & HINWOOD, J.B., On flow in estuaries, Part I. A critical review of some studies of slightly stratified estuaries. La Houille Blanche: 209, 377, 1972.
[6]
VAN DYKE, M., Analysis and improvement of perturbation Vol. XXVII, 21, 1974. Mech. Appl. Math.:
for circulation
in straits
series.
Q. Jl.
H. Rasmussen Department of Applied Mathematics, University of Western Ontario London, Ontario, Canada
ABSTRACT The steady flow in a slightly stratified two-dimensionalestuary is considered. An approximate form of the salinity which contains an unknown constant and which satisfies the prescribed boundary conditions is assumed. From a simplified form of the equations of motion the streamfunction is obtained. The unknown constant is then found by applying Galerkin's method the salt continuity equation.
1.
Introduction
In many estuaries the turbulent mixing is so intense that the vertical salinity stratification is slight. In some of these cases the governing partial differential equations and associated boundary conditions can be considerably simplified. However, the obtained model is still so complicated that analytical solutions cannot be found. It is therefore necessary to consider methods for obtaining approximate solutions. In the model analysed in this paper the flow is assumed to be twodimensional so the model is only applicable to the upper parts of fairly straight rivers with little variation in the breadth. For a derivation of the model see Rasmussen and Hinwood [l]. The model has previously been studied by Rattray and Hansen [2], Hansen and Rattray 131, and Hansen [4], mainly using the concept of similarity solutions; this work has been reviewed in Rasmussen and Hinwood [5]. In these papers the estuary is divided into an outer regime, a central regime and an inner regime. Approximate solutions are then obtained for each regime. The main problem with this approach is that the solution for the central regime does not match the solutions in the other two regimes at the ends of the central regime. Another difficulty is that it is necessary to suppose that some of the eddy coefficients have particular variations with respect to the coordinates. In this paper we obtain an approximate solution valid for the inner and central parts of the estuary. The approach used does not require any particular form of the eddy coefficients. The solution is not valid for the outer regime, but since the flow is undoubtedly three-dimensionalthere, this is not a serious limitation. 415
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES 2. Formulation We shall now state the mathematical flow in a slightly
stratified
The model consists
of three parts:
an approximate
equation
system where
and z the vertical
coordinate
coordinate.
the flow can no longer be supposed to the surface.
and z positive
downward.
depth is usually
equation,
conditions.
system with x being the horizontal
to that point in the estuary
to be two-dimensional, x positive
and z = 0
in the seaward direction
Since the ratio of the length of the estuary
to the upstream
that x ranges
end of the estuary
to the
from 0 to -m, where
the
is zero.
We let I$ and s be the non-dimensional respectively.
In order
the case where
the eddy coefficients,
constant;
boundary
The origin of the coordinate
very large, we shall suppose
with x = -m corresponding salinity
We measure
two-dimensional
as it is derived [l] .
salt continuity
and appropriate
is chosen such that x = 13 corresponds
corresponds
for steady,
estuary
an approximate
of motion,
Let (x,z) be a rectangular coordinate
model
constant-depth
the method
to simplify
can be extended
Then the model can be written
breadth,
and salinity,
we shall only consider
depth and wind stress are
to treat variations
in these parameters.
in the form
a$ as ---_--_=x_+~-
ax a2
streamfunction
the presentation
a$ as
a2 ax
a2s
a2s
ax2
az2
(3)
lim *-+-CC
s = xl:m$=
0
T = rD2B/nf AR
where
X = Kx B/R 2 = Ks B/R
y = -k B g So D3/RA
The different
D
B
physical
quantities
are defined
= depth of estuary = breadth
of estuary
= horizontal
KIC = vertical K
eddy diffusivity
eddy diffusivity
AZ
= eddy viscosity
g
= acceleration
due to gravity
as follows
(5)
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES k
417
= constant (
sO
R T of
=
reference salinity
= river flow = wind stress on the surface = density of fresh water
3. Approximate Solution
The procedure consists of writing an approximation Sl to the salinity s which satisfies the boundary conditions on meters CLand C.
S
and which contains two para-
An approximation $1 to the streamfunction $ is then obtained
by substituting Sl for s in equation (2) and integrating. The unknown integrating functions are determined from the boundary conditions on $I. We now suppose that s(x,z) can be approximated by Sl(X,Z) = fo(x) + Cfl(X) cos TIZ where
(6)
fo(x) = eax/(eax + eWaz)
(7)
fl(X) = (e"lX+ e-ax)-*
(8)
a and C are unknown parameters. This approximation satisfies all the boundary conditions on s.
It is easy
to show that -l
co.
(9)
When the approximation Sl(x,z) is substituted into equation (2), we obtain the following differential equation for $1, the approximation to Ji, a4*1 -= az4
yf~(X) + Cyfi(X) COS TfZ .
By integrating four times we get
@,(x,zf
=
-I._f'(x)z4 +
+
gzwz2+ gl(x)z
24
$f;(x, lr
*
cos AZ + g3(x)z3
+ g,(x).
(10)
Expresssions for the integrating functions go(x), g,(x), g,(x) and g3(x) are obtained by application of the boundary conditions for q at z = 0 and 1; hence
g,(x)
= 1 -
$
f;(X),
II 2 g,(x) = -Y+&
f;(x) + 12-n 4x4
y Cfi(x),
01) g,(x)
=$T+L
2n2
fi(X)*
AN APPROXIMATESOLUTION FOR TWO-DIMENSIONALESTUARIES 2 g,(x)= y
-&
f;(x) -=y 4a4
Cfi(x)
We note that Jll(x,z)satisfies the boundary condition (5) at the end of the estuary. If we substitute $l,and Sl for JIand s in the salt continuity equation (l), we obtain a residual R(x,z) given by
a+, as1
-_-.--+
ax
a2
wl
a2
as1
ax
02)
If $1 and Sl were the exact solutions, R would be identically zero. We now use Galerkin's method to obtain relationship between a and C from R.
The
method consists of multiplying R by the term fl(xf cos IIZand then insisting that the integral of this expression over the complete domain vanishes, i.e. we insist that 0
1 fl(x) cos ?TZR(x,z) dz = 0 .
dx I -co
i 0
This leads to the following equation for C F2 C2 + Fl C + Fo = 0
(13)
where F,
= -
ya :+ 10T + 60
1
240 772 2 XCr+-g-p&+ F1 = 7s
3(2 - T) 512n (14)
_ 2x5 + 27774- 1921r2+ 2304
Yo f
38,864x6 F
= (n2 + 4)(77+ 1) Ya 2
1120 I?
For given values of the different parameters this equation can then be solved for C; the solution must satisfy condition (9). The other parameter CIis determined from the assumed length of the estuary; this is discussed in the next section.
4. Results We shall now derive some characteristicvalues of the physical parameters determining the flow. Let us suppose that all the parameters are constant. It is reasonable to assume that the eddy diffusivities and viscosity are given by Kx =
10 u*D, Kx = 0.1 u*D, A = 0.1 u,D
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONAL ESTUARIES where
419
"* = 0.05 g R
= river flow ,
D
= depth of the estuary ,
B
= breadth of the estuary .
Thus we have KB x = x R
= 0.5,
KB 2 = -?-- = 0.005 . R
-4 In the equation of motion the parameter y appears. Let k S = 10 , O4 B = lo4 cm, D = lo3 cm, R = lo7 cm3/sec. Then y = - 2 x 10 . In [l] it was shown that the model is valid provided the Froude number F = !l/Jg k So D is substantially less than unity; here cl is the horizontal mean velocity. If we set Fl = R/B
D, we get that F = 0.1, so the condition is satisfied.
The value of a is obtained by supposing that the two-dimensionalpart of the estuary is lo6 cm long and that f (x) there is only 10% of fo(0). The result O-3 is that a is approximately 16 x 10 . We shall now show representative results using the above values of the parameters together with different values for T which represents the effect of wind stress on the surface. In Table 4.1 the two solutions Cl and C2 of equation (13) are given for T varying between -50 and 50.
It is noticed that
Table 4.1 Values of Cl and C2 for different values of T with x = 0.500, z = 0.005, y = -20000, o = 0.0014. T -50.0
c1 -569.178
c2 0.232
-40.0
-556.647
0.187
-30.0
-544.113
0.139
-20.0
-531.577
0.089
-10.0
-519.038
0.037
0.0
-506.497
-0.018
10.0
-493.953
-0.076
20.0
-481.407
-0.136
30.0
-468.857
-0.200
40.0
-456.303
-0.268
50.0
-443.745
-0.339
no acceptable solution exists for T 2 - 10, and a simple analysis shows that = 0 for T = -3.2. This means that if the windstress is too large in the c2 upstream direction, the approximate solution is no longer satisfactory. It is not clear if this result also holds for the exact solution. A possible physical explanation is that as T is changed from the value zero, the values of the turbulent parameters X, 2, y must also be changed. Since it is not clear how they should be changed, this idea has not been studied any further, and in the remaining part of the section we shall only consider T positive. In Tables 4.2 and 4.3 we present the salinity and horizontal velocity component at different depths and positions for T = 0. We notice 'thatthe
420
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES Table 4.2 Salinity at different depths and positions with X = 0.500, Z = 0.005, y = -20000, a = 0.0014, T = 0.0. Depth
Distance upstream
0.0
0.25
0.50
0.75
1.00
0
0.4955
0.4968
0.5000
0.5032
0.5045
-200
0.3594
0.3606
0.3635
0.3665
0.3677
-400
0.2427
0.2437
0.2460
0.2484
0.2493
-600
0.1547
0.1554
0.1571
0.1588
0.1595
-800
0.0947
0.0951
0.0962
0.0973
0.0978
0.0564
0.0566
0.0573
0.0580
0.0583
-1000
salinity changes by almost a factor ten over a distance of 1000 times the depth. From Table 4.3 it is seen that there is no reversal of horizontal velocity for T = 0. Table 4.3 Mrizontal velocity at different depths and positions with X = 0.500, Z = 0.005, y = -20000, a = 0.0014, T = 0.0. Distance upstream
Depth 0.0
0.25
0.50
0.75
1.00
0
1.7917
1.5703
1.0521
0.4557
0.0
-200
1.7696
1.5580
1.0577
0.4708
0.0
-400
1.7159
1.5279
1.0712
0.5076
0.0
-600
1.6540
1.4930
1.0866
0.5502
0.0
-800
1.6011
1.4632
1.0998
0.5866
0.0
-1000
1.5628
1.4416
1.1094
0.6130
0.0
Some values of the horizontal velocity component for different values of T are given in Table 4.4.
It is seen that velocity reversal takes place
for T = 10 at a depth of approximately 0.62, and that the depth at which this happens decreases as T increases. This is in agreement with the physics Table 4.4 Horizontal velocity for different values of T and at different depths at the downstream end with x = 0.500, z = 0.005, y = -20000, o = 0.0014. T Depth .
0
10
20
30
0.0
1.7917
4.2917
6.7917
9.2917
0.2
1.6453
2.4453
3.2453
4.0453
4.8453
5.6453
0.4
1.2810
0.9810
0.6810
0.3810
0.0810
-0.2190
0.6
0.8107
0.0107
-0.7893
-1.5893
-2.3893
-3.1893
0.8
0.3463
-1.0537
-I.7537
-2.4537
-3.1537
1.0
0.0
-0.3537 0.0
0.0
0.0
40 11.7917
0.0
50 14.2917
0.0
AN APPROXIMATE SOLUTION FORTWO-DIMENSIONS
ESTUARIES
42f
of the problem. In Table 4.5 we show the effects of increasing the windstress on the surface salinity. It shows, as expected, that as the windstress increases, the surface salinity decreases. Table 4.5 Surface salinity at different positions and for different values of T with X = 0.500, Z = 0.005, y = -20000, a = 0.0014. Distance upstream 500
1000
0.4955
0.1950
0.0564
10
0.4811
0.1858
0.0532
20
0.4659
0.1762
0.0500
30
0.4500
0.1661
0.0465
40
0.4331
0.1554
0.0429
50
0.4153
0.1440
0.0390
T 0
0
5.
Discussion
The present work indicates that Galerkin's method can be used to obtain approximate solutions for slightly stratified estuaries and thus be used to study the effects of different assumptions concerning the form of the turbulent eddy coefficients. Because of the lack of experimental data it is, however, difficult to estimate the accuracy of the solution when only a twoterm expansion is used. It would be of interest to expand the present work to a four-term expansion. Then the nonlinear transformatian 2 s n+l.Sn-l - 'n 'I(%) = Sn+l + Sn_l - 2s n which extraqts from any three successive terms of a sequence Sn the estimated limit kl(Sn) for n -+m, see Van Dyke [6], can be used to estimate the accuracy of the expansion. However, in this case the integrations must be carried out numerically.
Acknowledgements The author wishes to thank Or. P. Sullivan, University of Western Ontario and Or. J.B. Hinwood, Monash University for many helpful discussions concerning flow in estuaries. The work was supported by National Research Council of Canada (Grant No. A9251).
422
AN APPROXIMATE SOLUTION FOR TWO-DIMENSIONALESTUARIES References
[l]
RASMUSSEN, H. & HINWOOD, J.B., On flow in estuaries, Part III. La Houille Derivation of general and breadth integrated models. Blanche: 212, 319, 1973.
[Z]
RATTRAY, M., Jr. & HANSEN, D.V., A similarity solution in an estuary. J. Mar. Res.: 0, 121, 1962.
[3]
HANSEN, D.V. & RATTRAY, M., Jr., Gravitational circulation 104, 1965. and estuaries, J. Mar. Res.: 3,
[4]
HANSEN, D.V., Salt balance and circulation in partially mixed estuaries. In Estuaries, ed. G.H. Lauff, Publication No. 83, American Association for the Advancement of Science, Wash., p. 45, 1967.
[5]
RASMUSSEN, H. & HINWOOD, J.B., On flow in estuaries, Part I. A critical review of some studies of slightly stratified estuaries. La Houille Blanche: 209, 377, 1972.
[6]
VAN DYKE, M., Analysis and improvement of perturbation Vol. XXVII, 21, 1974. Mech. Appl. Math.:
for circulation
in straits
series.
Q. Jl.