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Seawater intrusion length in stratified estuaries
Water Research Pergamon Press 1970. Vol. 4, pp. 477-484. Printed in Great Britain
SEAWATER INTRUSION LENGTH IN STRATIFIED ESTUARIES Y. F. OZTURK Norwegian Institute for Water Research, Oslo, Norway (Received 10 November 1969)
Al~traet--The determination of sea-water intrusion length in stratified estuaries is very important for the solution of some estuary problems. Existing relationships for the matter do not give satifsactory results. The factors which govern the sea-water intrusion length, the estuary parameters, are evaluated and principals to estimate the length correction factor from the limited field data is demonstrated. Two new factors, estuary shape factor and Reynolds number for the river, are taken into consideration. Observations show that the general formulization of sea-water intrusion length depends on the introduction of these two factors into the related relationships.
INTRODUCTION THE INTRUSIONof seawater into estuaries is due to the density differences between the seaward flowing river-water and sea-water. In many estuaries during the seasons of reduced river discharges and in the absence of tide, separate seaward moving river-water and landward moving sea-water in the form of wedge can be observed. The density difference between river-water and sea-water produce gravitational currents in the estuaries. Such currents may create turbulant interface through which various degree of mixing may occur between river-water and sea-water layers flowing opposite to each other. Under such conditions the rate of sea-water flowing in the upstream direction decreases and sea-water moving upstream near the bottom is balanced by the same amount of its seaward flow. The decrease in the flow rate in the wedge in the upstream direction is evidence for the vertical transport of sea-water through the interface. On the other hand while river-water flows seaward its salinity, volume and velocity of flow increase and reach maximum near the estuary entrance. F o r the solution of many estuary problems such as domestic and industrial waste disposal, transportation, sedimentation, chemical and biological changes, etc. the knowledge about the estuary flow conditions is essential. At the present knowledge this matter involves the investigation of intrusion, mixing and distribution aspects of sea-water in the estuaries. The generalization of the equations in this respect has not been achieved yet and this has been due to the difliculties in the formulation of the different range of conditions observed in various estuaries. Because of the objective of this study the existing experimental data for the sea-water intrusion length phenomenon were investigated. This was to see if to make use of such data and obtain a satisfactory relationship for the matter would be possible. An extensive laboratory experiment in this relation was reported by KEULEGAN (1966) and the offered relationships for the matter are imperfect as the model estuaries 477
478
Y.F. OZTORK
did not include the influence of varying geometry of estuary beds and some other unknown factors governing the phenomenon. In the following the theoretical principals for the arrested sea-water wedge, reported experimental and some field data are examined. The evaluation of experimental and field data showed very clearly that in the evaluation of sea-water intrusion length, the influence of depth-width ratio can be considered for straight experimental estuaries of constant cross-section, but in real estuaries the estuary shape factor is the governing factor. On the other hand, it seems that transition from one type of flow to another exists for certain riverflow velocities, which shows the importance of Reynolds number to the river in this respect. At the same time two very significant independent estuary parameters and length correction factor, which depends on the estuary shape factor, and Reynolds number for the river were determined. Due to the limited experimental and field data through quantitative evaluations in this relation have not been possible, but analysis undertaken clearly show how one should approach the solution of the problem. Varification of the present observations require further laboratory and field data. SEAWATER INTRUSION LENGTH The hydrodynamical evaluation of the sea-water intrusion length is based on the theory of density currents. Although the geometry of estuaries may vary from place to place the experimental estuaries used for this purpose are usually the channels of constant shape which do not include the influence of the local variations in depth, width, velocity and salinity. For the analysis of theoretical principals of sea-water intrusion length, the estuary of rectangular cross-section containing distinctly separated river-water and sea-water layers, is considered. The properties of such an estuary system can be described by its own variables, and dimensionless analysis in this relation show that all the variables in question are functions of two parameters, FA and RA, the densimetric Froude number and densimetric Reynolds number respectively. FA --
RA --
Vr VA VA H
(1)
(2)
V
where, Vr is average velocity of riverflow opposing advancing wedge, VA is densimetric velocity, H is depth of river at river mouth, v is kinematic viscosity of riverwater, p is density of riverwater, p + A p is density of sea-water, pm is average density of two liquids and g is acceleration of gravity. Dimensionless analysis for the length Lo of the arrested sea-water wedge result in the following functional relationship Lo _ f (FA, RA, H/B). H
(4)
Seawater Intrusion Length in Stratified Estuaries
A
479
Based on the data given in FIGS. 1 and 2, the relationship between Lo/H-2 FA and RA can be expressed respectively as -
-
Lo _ A (2 FA)-" H
(5)
and A = C RA"
(6)
40C
20C
~"o
IOO
J
8O
60
40
20 0.15
0.4
0.2
0-6
0.8
1.0
Z'0
2FA
FIG. 1. Length of arrested saline wedge (KEULEOAN 1966). 200 B>>H .... IOC 80 60
[] o
40
A 20
/
/ o ~
Lo/H=2FZ~ o H=2B
10 8 6
n
H=B
4
20';
t
I'{
I
0-4 0-60"8 I
I
2
I
{ I I
4
6 8 I0
I
20
I
I I
40 60 80 IOOXIO4
R~
FIG. 2. Effect of channel width on arrested saline wedges (KEULEGAN 1966).
480
Y.F. OZTLrRK
and functional relationship for the length of arrested sea-water wedge Lo _ C RA m (2 F/X)-" H
(7)
where A is a variable, C, m and n can be called as length correction factor and independent estuary parameters respectively. As is obvious, for known R/X and F/X values the evaluation of Equation 6 depends on the accurate determination of C, m and n. INDEPENDENT ESTUARY PARAMETERS AND LENGTH CORRECTION FACTOR Independent estuary parameter n It is obvious from FIG. 1 that the trend of experimental data representing the relationship given by Equation (2) does not change for varying RA and 2 FA values, and it is independent of other factors involved in this relation.
Independent estuary parameter m With reference to FIG. 2 which gives the relationship between the variable A in Equation 5 and RA, C varies with H/B and RA. From the scatter and limited range of data given in FIG. 2, it is difficult to obtain satisfactory values of C and m and their extrapolation to actual estuaries do not give satisfactory results. Using the experimental data given in FIG. 1, and instead of considering the relationship between A and RA, the relationship between 2 FA and RA was established. As can be seen from FIG. 3 such a plot results in a family of parallel lines, each line representing a particular length--depth ratio Lo/H. In the figure some scatter of data can be observed for increasing length--depth ratios. If this was not due to the inability to establish the required flow conditions in the experimental estuary channels, it 0,85 I
o.a!
0"6 0'5
<3 0"4
/
f.- •
0"3
*
//
/ Az/ I 0"; 9XIO ~ 104 .
f
T
2X104
3X10~
RA
FIG. 3. 2FA -- RA relationship.
r 4×EO4
5×104
Seawater Intrusion Length in Stratified Estuaries
481
could be attributed to the increasing flow velocity of the river. The double logarithmic plot of 2 FA and RA in Fie. 3 yield values of constant k and gradient m', which can be expressed by 2 FA = k RAm'
(S)
k, in Equation (8) is governed by the parameter Lo/H, but m' is independent of H/B which was third variable in the relationship represented by Equation (5). It seems that data obtained from actual estuaries would have the same trend as the data obtained from experimental estuaries. The difference might be observed only between k values of experimental and actual estuaries. Such a difference should be due to the various geometry of estuary beds and the certain range of Reynolds number for the river. The combination of Equations (5 and 8) yield the following relationship Lo _ C RA=', (2 FA)-" H
(9)
where, m'n = m, and Equation (9) is the same as Equation (7). As is obvious, n and m' are independent of H]B, the geometry of estuary beds and other dominant factors. This means that estuary parameter m is constant and does not vary.
Length correction factor C Obtaining independent n and m values, the solution of the problem now depends on the accurate determination of length correction factor. The necessary process was developed by plotting Lo/H vs. k, where k is the intercept in Fxo. 3 on 2 FA-axis for RA : 1 for each line of constant Lo/H. 2000
\
I000
\
\
\
\
\\
~",,
\\ \\
\
o.~
\\
,
\
N'. "+
.,,
-r"
_P
0
1.73
+"
2.42 3.28
• t:l
3.68
•
1.3~
I00
,,
\
0.97X10
•
~4ll
x
~
~
\ x \
i,
e=tuory
t
\
\
\ ~
,~
E
)
I
~x
llltl
10-3
!
10-z
FIG. 4.
Lo/H
--
I
't
\
x \
,,, I
\ '~
\
~
GISmoesluory •
tO
x
\ \
Ix \ ~ ~'
4"67 • Otra estuary ¢ Mi.lssippl
\
,,,
I I I ITll lO-t
Rz~m, mF~-L
R%~(2 FA) -1 relationship.
\
\
\
\
\
\
\ ,
: \
T
I
+ I I I II
482
Y.F. 0ZT0aK
As can be observed the trend of plotted data in FIG. 4 has the same trend of data plotted in FIG. 1, and such a relationship makes possible the satisfactory determination of C for the experimental estuary. Concerning the influence of the geometry of estuaries instead of H]B, which will include the influence of depth and width at the estuary mouth as well, some of the data obtained from actual estuaries were plotted in FIG. 4. Assuming the independent estuary parameter n being the same for all estuaries, to obtain C values for the actual estuaries in question, to draw lines through the related data at a gradient ofn is sufficient. From the examination of the results of experimental and actual data given in FIG. 4, it was observed that there could not be a suitable correlation between C and H[B. This obviously showed that C is not governed only by H]B, but the geometry of estuary bed. On the other hand, Lo/H vs. RAm (2 FA)" were plotted in FIG. 5. It can be observed
1000
/9/
fi"
.,/" --~
I00
f
•
0.97X104 o 1.55 • 1.73 /,
,~
Aft I
2-42 3'28
/ AO/A~ ]
,
IG 103
,
,
,
•
Q • •
3',68 4'67
rlr,I
t
f
~
,
,~,1
t04
'
'
~
'
I I Irl
10 ~
106
Rt~(ZFZX) "z'5 FIG. 5. Lo/H -- RA°'s2 (2FA)-a'5 relationship.
that transition occurs for certain velocities of river flow, although the range of data is not sufficient enough for the varification of the matter. If such a transition exists in the flow conditions in estuaries, the length correction factor C of a particular estuary is variable and obviously such variation depends on the Reynolds number for the river. As a result, length correction factor should be represented by the following functional relationship C = f(/3, Rr)
where/3 is estuary shape factor and Rr is Reynolds number for the river.
(10)
Seawater Intrusion Lengthin StratifiedEstuaries
483
The above given investigation suggests the functional relationship for the length of sea-water wedge to be in the following form
Lo -- f (FA, RA, fl, Rr). H
(11)
Length of sea-water wedge equations of some estuaries Based on the above given principals the length of sea-water wedge equations of some estuaries were estimated as follows; Experimental rectangular cross-section estuaries Lo _ 0"000708 RAT M FA -2's. H
(12)
Otra estuary, Norway Lo _
1"062 R A0"82 g A - 2 ' s .
(13)
H
Gldma estuary, Norway Lo _ 2"665 R A °'a2 F A-2"5.
(14)
H
Mississippi estuary, U.S.A. Lo - - 6.018 RA°'a2 F A-2"5. H
(15)
As can be seen from FIG. 4, data representing Otra and Gfftma estuaries fit well the related relationships as was assumed.
RESULTS OF INVESTIGATION AND CONCLUSION The validity of the experimental material presented in this paper rests on the simplified estuary models, in which the effect of tide, estuary geometry and Reynolds number for the river have been neglected during the experiments. The re-evaluation of the experimental data have clearly shown that, m and n are independent estuary parameters. After making sure about the independent estuary parameters in question, the determination of length correction factor has great significance, and as mentioned previously it has to be evaluated as a function of estuary shape factor and Reynolds number for the river. The effect of estuary shape can be observed in the velocity distribution of flow in both sea-water wedge and seaward flowing brackish water. The distortion in the velocity of flow caused by the configuration of the estuary bed should influence the friction existing between flow and estuary bed, between brackish water and sea-water flowing opposite to each other. From this investigation it is obvious that by some developed model estuaries or sufficient data obtained from the real estuaries, the determination and introduction of
484
Y.F. OZTORI~
estuary shape factor and Reynolds number for the river into the sea-water intrusion length equation may be possible. For the practical evaluation of arested sea-water wedge lengths in any stratified estuary, the knowledge about the densities of sea-water and river-water, kinematic viscosity of river-water, depth of river at the estuary mouth, velocity of river flow at the tip of wedge and corresponding length of sea-water wedge is sufficient. REFERENCE KEULEGANG. H. (1966) The mechanismof an arrested saline wedge. Estuary and Coastline Hydrodynamics (Editedby A. T. IPPEN).McGraw-Hill,New York.
SEAWATER INTRUSION LENGTH IN STRATIFIED ESTUARIES Y. F. OZTURK Norwegian Institute for Water Research, Oslo, Norway (Received 10 November 1969)
Al~traet--The determination of sea-water intrusion length in stratified estuaries is very important for the solution of some estuary problems. Existing relationships for the matter do not give satifsactory results. The factors which govern the sea-water intrusion length, the estuary parameters, are evaluated and principals to estimate the length correction factor from the limited field data is demonstrated. Two new factors, estuary shape factor and Reynolds number for the river, are taken into consideration. Observations show that the general formulization of sea-water intrusion length depends on the introduction of these two factors into the related relationships.
INTRODUCTION THE INTRUSIONof seawater into estuaries is due to the density differences between the seaward flowing river-water and sea-water. In many estuaries during the seasons of reduced river discharges and in the absence of tide, separate seaward moving river-water and landward moving sea-water in the form of wedge can be observed. The density difference between river-water and sea-water produce gravitational currents in the estuaries. Such currents may create turbulant interface through which various degree of mixing may occur between river-water and sea-water layers flowing opposite to each other. Under such conditions the rate of sea-water flowing in the upstream direction decreases and sea-water moving upstream near the bottom is balanced by the same amount of its seaward flow. The decrease in the flow rate in the wedge in the upstream direction is evidence for the vertical transport of sea-water through the interface. On the other hand while river-water flows seaward its salinity, volume and velocity of flow increase and reach maximum near the estuary entrance. F o r the solution of many estuary problems such as domestic and industrial waste disposal, transportation, sedimentation, chemical and biological changes, etc. the knowledge about the estuary flow conditions is essential. At the present knowledge this matter involves the investigation of intrusion, mixing and distribution aspects of sea-water in the estuaries. The generalization of the equations in this respect has not been achieved yet and this has been due to the difliculties in the formulation of the different range of conditions observed in various estuaries. Because of the objective of this study the existing experimental data for the sea-water intrusion length phenomenon were investigated. This was to see if to make use of such data and obtain a satisfactory relationship for the matter would be possible. An extensive laboratory experiment in this relation was reported by KEULEGAN (1966) and the offered relationships for the matter are imperfect as the model estuaries 477
478
Y.F. OZTORK
did not include the influence of varying geometry of estuary beds and some other unknown factors governing the phenomenon. In the following the theoretical principals for the arrested sea-water wedge, reported experimental and some field data are examined. The evaluation of experimental and field data showed very clearly that in the evaluation of sea-water intrusion length, the influence of depth-width ratio can be considered for straight experimental estuaries of constant cross-section, but in real estuaries the estuary shape factor is the governing factor. On the other hand, it seems that transition from one type of flow to another exists for certain riverflow velocities, which shows the importance of Reynolds number to the river in this respect. At the same time two very significant independent estuary parameters and length correction factor, which depends on the estuary shape factor, and Reynolds number for the river were determined. Due to the limited experimental and field data through quantitative evaluations in this relation have not been possible, but analysis undertaken clearly show how one should approach the solution of the problem. Varification of the present observations require further laboratory and field data. SEAWATER INTRUSION LENGTH The hydrodynamical evaluation of the sea-water intrusion length is based on the theory of density currents. Although the geometry of estuaries may vary from place to place the experimental estuaries used for this purpose are usually the channels of constant shape which do not include the influence of the local variations in depth, width, velocity and salinity. For the analysis of theoretical principals of sea-water intrusion length, the estuary of rectangular cross-section containing distinctly separated river-water and sea-water layers, is considered. The properties of such an estuary system can be described by its own variables, and dimensionless analysis in this relation show that all the variables in question are functions of two parameters, FA and RA, the densimetric Froude number and densimetric Reynolds number respectively. FA --
RA --
Vr VA VA H
(1)
(2)
V
where, Vr is average velocity of riverflow opposing advancing wedge, VA is densimetric velocity, H is depth of river at river mouth, v is kinematic viscosity of riverwater, p is density of riverwater, p + A p is density of sea-water, pm is average density of two liquids and g is acceleration of gravity. Dimensionless analysis for the length Lo of the arrested sea-water wedge result in the following functional relationship Lo _ f (FA, RA, H/B). H
(4)
Seawater Intrusion Length in Stratified Estuaries
A
479
Based on the data given in FIGS. 1 and 2, the relationship between Lo/H-2 FA and RA can be expressed respectively as -
-
Lo _ A (2 FA)-" H
(5)
and A = C RA"
(6)
40C
20C
~"o
IOO
J
8O
60
40
20 0.15
0.4
0.2
0-6
0.8
1.0
Z'0
2FA
FIG. 1. Length of arrested saline wedge (KEULEOAN 1966). 200 B>>H .... IOC 80 60
[] o
40
A 20
/
/ o ~
Lo/H=2FZ~ o H=2B
10 8 6
n
H=B
4
20';
t
I'{
I
0-4 0-60"8 I
I
2
I
{ I I
4
6 8 I0
I
20
I
I I
40 60 80 IOOXIO4
R~
FIG. 2. Effect of channel width on arrested saline wedges (KEULEGAN 1966).
480
Y.F. OZTLrRK
and functional relationship for the length of arrested sea-water wedge Lo _ C RA m (2 F/X)-" H
(7)
where A is a variable, C, m and n can be called as length correction factor and independent estuary parameters respectively. As is obvious, for known R/X and F/X values the evaluation of Equation 6 depends on the accurate determination of C, m and n. INDEPENDENT ESTUARY PARAMETERS AND LENGTH CORRECTION FACTOR Independent estuary parameter n It is obvious from FIG. 1 that the trend of experimental data representing the relationship given by Equation (2) does not change for varying RA and 2 FA values, and it is independent of other factors involved in this relation.
Independent estuary parameter m With reference to FIG. 2 which gives the relationship between the variable A in Equation 5 and RA, C varies with H/B and RA. From the scatter and limited range of data given in FIG. 2, it is difficult to obtain satisfactory values of C and m and their extrapolation to actual estuaries do not give satisfactory results. Using the experimental data given in FIG. 1, and instead of considering the relationship between A and RA, the relationship between 2 FA and RA was established. As can be seen from FIG. 3 such a plot results in a family of parallel lines, each line representing a particular length--depth ratio Lo/H. In the figure some scatter of data can be observed for increasing length--depth ratios. If this was not due to the inability to establish the required flow conditions in the experimental estuary channels, it 0,85 I
o.a!
0"6 0'5
<3 0"4
/
f.- •
0"3
*
//
/ Az/ I 0"; 9XIO ~ 104 .
f
T
2X104
3X10~
RA
FIG. 3. 2FA -- RA relationship.
r 4×EO4
5×104
Seawater Intrusion Length in Stratified Estuaries
481
could be attributed to the increasing flow velocity of the river. The double logarithmic plot of 2 FA and RA in Fie. 3 yield values of constant k and gradient m', which can be expressed by 2 FA = k RAm'
(S)
k, in Equation (8) is governed by the parameter Lo/H, but m' is independent of H/B which was third variable in the relationship represented by Equation (5). It seems that data obtained from actual estuaries would have the same trend as the data obtained from experimental estuaries. The difference might be observed only between k values of experimental and actual estuaries. Such a difference should be due to the various geometry of estuary beds and the certain range of Reynolds number for the river. The combination of Equations (5 and 8) yield the following relationship Lo _ C RA=', (2 FA)-" H
(9)
where, m'n = m, and Equation (9) is the same as Equation (7). As is obvious, n and m' are independent of H]B, the geometry of estuary beds and other dominant factors. This means that estuary parameter m is constant and does not vary.
Length correction factor C Obtaining independent n and m values, the solution of the problem now depends on the accurate determination of length correction factor. The necessary process was developed by plotting Lo/H vs. k, where k is the intercept in Fxo. 3 on 2 FA-axis for RA : 1 for each line of constant Lo/H. 2000
\
I000
\
\
\
\
\\
~",,
\\ \\
\
o.~
\\
,
\
N'. "+
.,,
-r"
_P
0
1.73
+"
2.42 3.28
• t:l
3.68
•
1.3~
I00
,,
\
0.97X10
•
~4ll
x
~
~
\ x \
i,
e=tuory
t
\
\
\ ~
,~
E
)
I
~x
llltl
10-3
!
10-z
FIG. 4.
Lo/H
--
I
't
\
x \
,,, I
\ '~
\
~
GISmoesluory •
tO
x
\ \
Ix \ ~ ~'
4"67 • Otra estuary ¢ Mi.lssippl
\
,,,
I I I ITll lO-t
Rz~m, mF~-L
R%~(2 FA) -1 relationship.
\
\
\
\
\
\
\ ,
: \
T
I
+ I I I II
482
Y.F. 0ZT0aK
As can be observed the trend of plotted data in FIG. 4 has the same trend of data plotted in FIG. 1, and such a relationship makes possible the satisfactory determination of C for the experimental estuary. Concerning the influence of the geometry of estuaries instead of H]B, which will include the influence of depth and width at the estuary mouth as well, some of the data obtained from actual estuaries were plotted in FIG. 4. Assuming the independent estuary parameter n being the same for all estuaries, to obtain C values for the actual estuaries in question, to draw lines through the related data at a gradient ofn is sufficient. From the examination of the results of experimental and actual data given in FIG. 4, it was observed that there could not be a suitable correlation between C and H[B. This obviously showed that C is not governed only by H]B, but the geometry of estuary bed. On the other hand, Lo/H vs. RAm (2 FA)" were plotted in FIG. 5. It can be observed
1000
/9/
fi"
.,/" --~
I00
f
•
0.97X104 o 1.55 • 1.73 /,
,~
Aft I
2-42 3'28
/ AO/A~ ]
,
IG 103
,
,
,
•
Q • •
3',68 4'67
rlr,I
t
f
~
,
,~,1
t04
'
'
~
'
I I Irl
10 ~
106
Rt~(ZFZX) "z'5 FIG. 5. Lo/H -- RA°'s2 (2FA)-a'5 relationship.
that transition occurs for certain velocities of river flow, although the range of data is not sufficient enough for the varification of the matter. If such a transition exists in the flow conditions in estuaries, the length correction factor C of a particular estuary is variable and obviously such variation depends on the Reynolds number for the river. As a result, length correction factor should be represented by the following functional relationship C = f(/3, Rr)
where/3 is estuary shape factor and Rr is Reynolds number for the river.
(10)
Seawater Intrusion Lengthin StratifiedEstuaries
483
The above given investigation suggests the functional relationship for the length of sea-water wedge to be in the following form
Lo -- f (FA, RA, fl, Rr). H
(11)
Length of sea-water wedge equations of some estuaries Based on the above given principals the length of sea-water wedge equations of some estuaries were estimated as follows; Experimental rectangular cross-section estuaries Lo _ 0"000708 RAT M FA -2's. H
(12)
Otra estuary, Norway Lo _
1"062 R A0"82 g A - 2 ' s .
(13)
H
Gldma estuary, Norway Lo _ 2"665 R A °'a2 F A-2"5.
(14)
H
Mississippi estuary, U.S.A. Lo - - 6.018 RA°'a2 F A-2"5. H
(15)
As can be seen from FIG. 4, data representing Otra and Gfftma estuaries fit well the related relationships as was assumed.
RESULTS OF INVESTIGATION AND CONCLUSION The validity of the experimental material presented in this paper rests on the simplified estuary models, in which the effect of tide, estuary geometry and Reynolds number for the river have been neglected during the experiments. The re-evaluation of the experimental data have clearly shown that, m and n are independent estuary parameters. After making sure about the independent estuary parameters in question, the determination of length correction factor has great significance, and as mentioned previously it has to be evaluated as a function of estuary shape factor and Reynolds number for the river. The effect of estuary shape can be observed in the velocity distribution of flow in both sea-water wedge and seaward flowing brackish water. The distortion in the velocity of flow caused by the configuration of the estuary bed should influence the friction existing between flow and estuary bed, between brackish water and sea-water flowing opposite to each other. From this investigation it is obvious that by some developed model estuaries or sufficient data obtained from the real estuaries, the determination and introduction of
484
Y.F. OZTORI~
estuary shape factor and Reynolds number for the river into the sea-water intrusion length equation may be possible. For the practical evaluation of arested sea-water wedge lengths in any stratified estuary, the knowledge about the densities of sea-water and river-water, kinematic viscosity of river-water, depth of river at the estuary mouth, velocity of river flow at the tip of wedge and corresponding length of sea-water wedge is sufficient. REFERENCE KEULEGANG. H. (1966) The mechanismof an arrested saline wedge. Estuary and Coastline Hydrodynamics (Editedby A. T. IPPEN).McGraw-Hill,New York.