Jul 13, 1984 - where ~~ is an imposed wind stress and subscript w is used to .... In certain cases the convective term should be included in equation (36) (Abbott 1960). ... The James River, Mersey and Southampton Water are sensitive to.
Estuarine,
Coastal
and Shelf
Science
(1985) 20,615-635
On Salinity Regimes and the Vertical Structure of Residual Flows in Narrow Tidal Estuaries
D. Prandle Institute of Oceanographic Sciences, Merseyside, L43 7RA, U.K. Received
15 March
1984
Bidston
and in revised
form
Keywords: tidal estuaries; salt intrusion; energy exchange
Observatory,
13July
Birkenhead,
1984
current profile;
vertical circulation;
An examination is made of the circulation in narrow estuaries subject to a predominant tidal forcing. Velocity structures are derived separately for residual flow components associated with (a) river flow, (b) wind stress, (c) a well-mixed longitudinal density gradient and (d) a fully stratified saline wedge. Dimensionless parameters are introduced to indicate the magnitude of each component and these parameters are evaluated for 9 major estuaries, thereby revealing their sensitivity to each component. For a channel of constant breadth and depth, formulae are deduced for the length of saline intrusion, L. Comparisons with observed data show that such formulae may be used with confidence to predict changes in L arising from variations in river flow, tidal range or channel depths. The level of stratification is shown to be related to a product of two parameters, one associated with velocity structure and a second involving the square of the ‘ flow ratio ’ ii/ii (i.e. residual velocity/amplitude of the tidal velocity). This relationshipprovides a simpleclassificationsystemfor estuarine
stratification which can be used to indicate the sensitivity of any particular estuaryto changingconditions. Introduction
This study arose from considering how the saline intrusion in an estuary may be modified by the construction of a tidal power scheme. The impact of such schemesmay be sufficient to completely change the nature of the mixing regime. While numerical models can be reliably used to calculate the changes in elevations and currents, simulation of changes in the mixing regime posesgreater difficulty. Here, we explore the interaction between the residual current circulation and the mixing regime. The first section of this paper examines the vertical structure of residual velocities in narrow estuaries subject to predominant tidal forcing. The second section considers how this vertical structure influences the length of saline intrusion while a third section investigates the relationship between vertical structure and the level of stratification. The objective throughout is to link coherent theoretical derivations with sensible empirical formulations so asto provide useful quantitative results. Thus at all stagesthe theoretical developments are related to observational data. The accuracy of the quantitative results presented is dependent on two basic assumptions,
the first
assumption
is of a linearised
bed-friction
formula
proportional
to a
615 0272-7714/85/050615+21
$03.00/O
0 1985 Academic
Press Inc. (London)
Limited
616
D. handle
ib)
I
o+
H
ap/ax Z
"
>x
(cl
Cd)
Figure 1. Notation for various Aow cases: (a) fresh water flow (c) well-mixed longitudinal density gradient c’p/&, (d) stratified.
Q, (b) wind
stress r,,
product of tidal current amplitiude and residual velocity. The secondassumption is of an eddy viscosity coefficient, constant through depth, and proportional to a product of tidal current amplitude and water depth (or layer thickness in stratified flow). With these assumptions, the separate components of flow (i.e. wind, density etc) are derived from linear equations and hence the net flow can be obtained by direct superposition. The treatment of a two-layer salt-wedge in the section on density currents with complete vertical mixing may be criticized in that (a) convective acceleration terms are ignored, (b) shearat the interface is underestimated and (c) the existence of such a wedge is inconsistent with the overall assumption of strong tidal forcing. However, the present approach has the merit of overall consistency involving minimal extraneous assumptions. The two-layer model developed here should not be regarded as an accurate simulation but more asa convenient extreme caseto serve as a constrast to the dynamics derived for a well-mixed system. Residual
current
profiles
In this section we examine the residual current profiles associatedwith (a) a net flow Q, (b) a surface wind stressT,, (c) a longitudinal density gradient ap/dx with complete vertical mixing and (d) a wedge-type intrusion with density difference Ap and no vertical mixing at the interface. These four casesare illustrated in Figure 1. It is assumedthat these separate motions may be described by linear equations and hence a complete flow description can be obtained by simple addition. For this assumption to be valid: (a) the residual component of the bed friction term must be effectively linear and (b) the ratio of tidal elevation to mean water depth must be small. Condition (a) is satisfied in most tidal estuaries while condition (b) tends merely to qualify the accuracy of the results in shallow estuaries.
Salinity
and
structure
617
ojverticalflows
To provide useful quantitative results from the theory developed here, two basic assumptions are made. First we adopt a value for the bed stresscoefficient k=O.O025. Second we assumethe eddy viscosity coefficient E to be constant and given by (Prandle 1982): E=kuH,
(1)
where u is the amplitude of the tidal velocity and H the water depth. Netflow
Q
For steady flow along one horizontal axis x (positive downstream) we obtain (2) where g is gravitational acceleration, 6 surface elevation, i, the surface gradient associated with Q, Z the vertical axis measuredupwards from the bed. By integrating equation (2) twice w.r.t. Z we obtain the current profile
(31 where the constants of integration were determined from the two boundary conditions: : stressat the surface FZzH = p Ez = 0, (4) ?Z StressatthebedF,,,
=pEg=p/lu,=,
CT
Condition (5) applies when zi>>U, in which caseBowden (1953) showedthat 8_% u. 71
Q 1 Finally, introducing the depth-averaged velocity U = H = G
u d Z, equation (3) gives
(6) and (7)
11 = ii
with z = Z/H. In quantitative terms from equation (l), equations (6) and (7) reduce to 20
= -0.89 ‘2 gH
(8)
and (9) The current profile (9) is illustrated in Figure 2, further discussion follows in the section Summary of results
D. Prandle
618
‘r 0.8 t 0.6 b 0.4 c
0
0.4
0.8
1.2
-0.2
u/.0
0
0.2
-0.02
U,/(W.u/f)
U,/LslJ/f)
(b)
(cl
(al
Figure 2. Vertical structure for residual flow components: (a) freshwater flow Q =U H, observed data from three positions, see Prandle (1982) for sources; (b) wind stress r,, (c) well-mixed longitudinal density gradient. W, F and S defined in equation (37).
Surface willd stress T,,
Proceeding asfor net flow Q, but with U, = 0 and the surface boundary condition (10) where ~~ is an imposed wind stress and subscript w is used to denote wind-driven components, we obtain
(11) and (12)
or in quantitative terms from equation (1) % = .L
pkzi
jO.574 z2 - 0.149 .z - 0.117)
(13)
i, = 1.15 T,/pgH.
(14)
and
The factor 1.15 in equation (14) lies between the value 1.5 obtained for a no-slip bed condition
and
1.0 for
a full-slip
condition
(Rossiter,
1954).
Density currents with complete vertical
mixing
A well-mixed longitudinal density gradient ap/Sx introduces an additional pressure gradient so that equation (2) now takes the form
Salinity
and structure of vertical
619
flows
(15) to give
To isolate the influence of the density gradient (subscript Ml we define i 17) then substracting tiQ given by equation (3) from equation (16) and applying (a) the condition i&, = 0 and (b) the empirical condition (1) we obtain g 6~ H2 l&= - p
8x
kti
.z3
- ;
+0.269z2
-O.O37z-0.029
and (19’)
Density
currents with complete vertical
mixing
Using the notation shown in Figure l(d) we note that the pressure at height Z is (a) in the top layer 23 D, (bj in the bottom layer Z < D,
P=Pg
(2%)
K-z), p=pg
K-D)+(P+
bk(D-z).
(21)
In addition to the earlier boundary conditions (4) and (5), at the interface (subscript I) we assumecontinuity of both velocity and stress,thus at Z= D u,=u~=u~
(22)
and :23:,
where subscripts T and B denote values in the top and bottom layers, respectively. We further assume: (a) p + Ap %p in masscalculations not involving buoyancy effects, (b) zero net flow in the lower layer and net flow Q in the top layer (c) eddy viscosity in the lower layer given by a modified form of condition (l), namely E,=ktiD,
(24)
(d) eddy viscosity in the top layer given by E,=yE,.
~25)
620
D.
Pratde
The preceding asin earlier sections we obtain. Qc 1 & = - Hd ii ;’ u n=
--~
z2 --z+d-f 2
QE (1 - d) H
d’
-0.308d(l
-d)
.
(26)
(- 0.5742’ + 0,149zd + 0.1 17d2),
ci; i,=-=
_dX
Qk&
(28)
gH’
(1 - d) + 0.308d
,
a=0.149-1.149d-‘,
(30)
where (31) From expressions(30) and (31), the slope of the interface, dh/dx, is always greater and opposite in sign to the product of surface slope and density difference, (Ap/p x d 10 or 1W/F 1>3 we observe that the longitudinal density gradient will significantly influence (i.e. by >30°0) residual current structure in 4 of the 10 estuaries. While a steady wind of 10 knots (equivalent to rw = 1 dyne cm -2) significantly influences current structure in 3 cases.The JamesRiver, Mersey and Southampton Water are sensitive to both while the Bristol Channel, Tay, Tees, Thames and Vellar are insensitive to either. These results must be qualified by recognising (a) the assumption of complete mixing in equation (36) and (b) the difficulty in specifying representative hydrographic parameters in Table 2. In most cases,the values of S, F and W will vary both along the estuary and in time due to both seasonaland spring-neap tidal cycles. Salinity
intrusion
The intrusion length in an estuary varies with time due to changes in tidal amplitude, river flow, wind forcing etc. Likewise the intrusion length may be changed by dredging, barrier construction or upstream regulation. These changesmay have important implications for factors such aswater quality, sedimentation and dispersion of pollutants.
Salinity
and structure of vertical
flaws
623
624
D. handle
2. Estuary and flume data for salinity intrusion. Observed data: L&p fractional change in density over length L, H depth, ii amplitude of tidal velocity, ti residual velocity, ss/Sdifference in salinity between bed and surface:depth-averaged salinity. Computed values: S/F, IV/F defined by equation (37), S, defined by equation (52). Values of ware for a wind stress of 1 dyne cm-? directed downstream. Data sources: (a) Dyer (1973), (b) Bassindale (1943), (c) West & Williams (1972), (d) Inglis & Allen (1957), (e) Rigter (1973) and Winterwerp (1983), (fl Ippen & Harleman (1961).
TABLE
Data source
Estuary Vellar Columbia James Tees Southampton way Tay Thames Mersey Bristol Channel
H
(a) tat ;a) (bi
0.03 0.03 0.017 0.03
1.5 20 7.5 8
04
0.8
1.0 o-35 1.5
0.06 0.005 0.02
10 20 24 25
0.1 40 90 10
0.1 0.7 23 1.3
2.0 0.060 0,014 0.013
3 6 47 476
(a) Cc) td) (a) (b:)
0.004 0.03 0.03 O-002 0.03
10 8 6 20 6 (ml
0.6
i.9 1.9 2.0 (ms -‘1
0.002 0.02 0,015 0.002 0.007 (ms ‘1
(e)
0.021
2l.b
11.7
0.021 0,021
21.6 21.6
0,017 0.017 0,019 0.020
15.2 15.2 15.2 15.2 (cm)
2 0.5 0.25 0.2
13 30 65 10 80 (km>
100 6 5 83 4
33 1.0 3 10 3
0.003 0.010 0.017 0.001 0.003
760 1354 634 9286 18338
1.99
22.0
7
1.6
0.170
4 2
16.9 11.9
1.99 1.49
10.3 30.0
10 7
1.1 2.1
0.118 0.125
6 8
13.1 18.0 13.1 21.3 (ems-‘)
1.71 1.71 0,61 0.61 (ems-I)
29.3 18.9 381 41.1 fm)
1.8 2.0 4.3 2.6
1.3 1.0 3.8 2.3
0.131 0.095 0.047 0.029
28 47 91 403
Water-
Rotterdam Waterway Flume TlO .I‘ests T20 T30 WES 11 10 16 14
-1) 11
(fi
0.1 0.1 0.05 11.05 O-02
1-5 1 1.30 0.79 0.74 0.24
Even with the aid of sophisticated numerical models it is often difficult to reproduce the observed intrusion in an estuary. Consequently, in a study such as a major tidal power scheme, the prediction of the change in the salinity regime posesa major challenge. Here, we approach the problem of predicting intrusion lengths by extending the theory relating to residual velocities described in the previous section. A first section considers salinity intrusion in stratified water and a secondsection develops an empirical formula for predicting intrusion lengths in mixed conditions. The results cited apply to channels of constant breadth and depth. This restriction stems from reasons of (aj mathematical simplicity and (b) availability of observational data. Salinity
intrusionfor
fully stratifiedfiaw
The theory developed in the section on density current with complete vertical mixing, may be simply extended for the special caseof a channel of constant breadth and water depth If, and a level bed. For the latter conditions, referring to Figure l(d) and equations (31), dh - =a--.dC P dx dx LIP From equations (30) and (33), equation (39) simplifies to
(39)
Salinity
625
and structure of vertical flows
dh
1 dD
ds=i
dx
1.149
1.56
Qkli
p
gH3
Ap'
0.149 - ~
(40) d
> (1 -d)'
By letting y = 1.149- O.l49d, equation (40) may be integrated to give .y=-1300---
gH2
a/.,
klz
p
+ 1.574~’ - 3.298~ + 1.1491ogp - s1
(41)
where xi is an arbitrary constant. The profile of d versus x obtained from equation (41) is effectively linear throughout the range d = 0 to 1. This is in general agreement with the profiles for salinewedges deduced elsewhere(Ippen, 1966, ch 11) except closeto the two limits (d=O and 1) where the simplified physics considered here may be inadequate. Neglecting these ‘ end effects ‘, by substituting d= 0 and d= 1 in equation (41) we obtain the following estimate for the total length L of the wedge, (42)
This result may be compared with the following expression for the length, L,, of an arrested salinewedge given by Keulegan (Ippen, 1966, chl 1) (43)
where A is a parameter which varies with river conditions. Since Keulegan’s expression was derived from observational data, the correspondence between equations (42) and (43) adds useful support to the present theoretical result. Officer (1976, ch 4) provides alternative derivations for the length of an arrested salt wedge, the results obtained obviously depend on the particular dynamical assumptions. A comparison of the relative merits of these dynamical assumptions is beyond the scope of the present paper. Intrusion
length for partly
mixed conditions
Rigter (1973) carried out an extensive study of intrusion lengths both in a laboratory flume and in Rotterdam Waterway, conditions in the flume being dynamically ‘ similar ’ to those in the Waterway. Ippen and Harleman (1961) studied the mechanics of salinity intrusion by analysing results from a seriesof experiments carried out at the Waterways Experimental Station, Vicksburg, these data are subsequently referred to as the WES tests. In all of these casesthe channels were of constant depth and breadth. Some of these data (where the level of stratification, &/Sdefined in the next, were available) are shown in Table 2. By introducing experimental results into a dimensional analysis approach, Rigter deduced that LiR
=f
zi+5i KAddgtil’i2
u i( ’
[(A~l~Wll’*
j ’
(44)
where Li is the minimum length of salinity intrusion and f indicates a function. The dimensionlessgroupings on the r.h.s. are modified Froude numbers. Since the scaling laws governing the flume tests are derived from the requirement to maintain similarity of the Froude numbers, the origin of these groupings is evident. The occurrence of klH on
626
D. Pratldle
the 1.h.s. of equation (44) may be understood from the scale ratio for k which, for a ‘ Froude model ‘, is equal to the vertical scaledivided by the horizontal scale. By making the approximation li +U = li in equation (44), we now suggestthe following functional form
[(Ad~kH)11’2 22
~(&d~kN)11’2 3 u >
(45)
with 6 a constant. The functional form of equation (45) is then in precise agreement with the theoretical result (42) derived for a fully stratified intrusion. We now examine the validity of equation (45) by comparison with the observational data cited earlier. Thus, we further assume LIP gH” Li = 6, ^ p kuu
+dz i
146)
where i, = g:Hf P (P is the tidal period and the wavelength i is an appropriate horizontal dimension) and the additional term 6,A is introduced to allow for variations in the degree of mixing at the mouth of the channel. A least squares fitting procedure was used to determine the constants 6, and d, in equation (46). For the 39 data setsgiven by Rigter (in his tables 3 and 8), comprising 32 flume tests and 7 for the Rotterdam Waterway, the values 6, = 0.187 and d2= - 0.006 were obtained. A comparison of observed and computed values for Li is shown in Figure 5(a), the excellent agreement is confirmed by a correlation coefficient calculated as r=0.97.
A similar’ exercise for 10 WES tests (Rigter, table 5) produced the values n‘, = 0.134 and 6, = 0.026. The comparison of observed and computed values for Li are shown in Figure 5(b), good agreement is again indicated with r = 0.84. [The observed values for L, shown by Rigter are rounded to units of 10ft, in the above comparisons more accurate values were substituted basedon the figures shown by Ippen and Harleman (1961).] In both data sets the term in equation (46) involving 6, accounts for over SO’%,of the variance in Li. Comparing the calculated values for 6, with the value 0.26 derived for the fully stratified case(42), we find the intrusion length for partly mixed systemsis reduced by 30”:, and 50(x,. The greater reduction for the WES tests may be related to the enhanced mixing relative to Rigter’s data (seethe section on mixing related to energy balance). Thus these results are in agreement with the observation that, in general, intrusion will increaseas stratification increases(Bowden, 1965; Festa & Hansen, 1976). The excellent correlation found for Rigter’s data suggeststhat equation (46) could be used with confidence to estimate the change in intrusion length in the Rotterdam Waterway arising from small modifications to: (a) the depths or roughness of the channel, (b) river flow or (c) tidal range. However, in determining the change in intrusion length in, say, the Bristol Channel due to a tidal barrier, a small change indicated by a formula such as equation (46) may be amplified by the effects of variable cross-sectional areas. Moreover, where such a barrier leadsto a significant change in the level of stratification, the present approach may be invalidated. Mixing
in estuaries
At the level of turbulent motions, the degree of mixing in an estuary is determined by the Richardson no. R,, where
Salinity and structure ofverticalpows
628
D. Prandle
d
O-25
m
0.50 0.75 I.0
0.001
L
I IO
I (2)
I 100
I 1000
I 104
105
us/u II I I IO (24)
I 100
I 1000
I 104
I 105
I 106
S/F
Figure 6. Hansen and Rattray stratification diagram: SsF salinity difference between bed and surface/depth-averaged salinity, u,/ii residual velocity at the surface/ depth-averaged value, d fractional height above the bed of the interface in a stratified system, S/F ratio of saiinity:frictional term, index of gravitational circulation [equation (3711.
(47) represents the ratio of buoyancy forces to vertical turbulent forces. For Ri < 0.25 the turbulence is sufficient to overcome density layering and vertical mixing occurs. However, for many estuaries, data on the finer structure of motions is not available and hence the following section examines the degree of mixing in terms of the gross estuarine parameters used in Table 2. Effectively, we resort to classification systems which can provide a useful insight into the dynamics of a system even when scant data are available. Hanserl and Rattray stratification diagram Figure 6 shows the Hansen and Rattray stratification diagram. The two basic parameters which form the orthogonal axes are (1) &/S, the salinity difference between bed and surface divided by the depth averaged salinity (mean values over a tidal cycle), (2) us/U, residual velocity at the surface divided by the depth mean value. Four estuarine types are identified with sub-divisions into (a) mixed and (b) stratified according to whether bs/Sis less or greater than 0.1, respectively for full details [see Hansen & Rattray, (1966)].
Salinity
629
andstructure ofverticalflows
In estuaries of type 1, residual flow is seawardsat all depths and consequently mixing of salt is entirely by diffusion. For type 2, residual flow reverseswith depth and mixing is due to both advection and diffusion. For type 3, the vertical structure of the residual flow is so pronounced that advection accounts for over 99’!;, of the mixing process; in type 3(b) mixing is confined to the near-surface region. Type 4 exhibits maximum stratification and approximates a salt wedge. The upper demarkation line shown in Figure 6 for estuaries of type 3 and 4 may be explained directly from the results obtained for a stratified system in the section on density currents with complete vertical mixing. Thus from Table 1, 11,
1.26
(48)
;=(1while from Figure l(d), assumingp = I+ aS where a is a constant, 6s
cp --=-
-s -
Apd
1
(491
d’
Hence we obtain (SS UJ -= s (us/u- 1.26) ’
1.50)
and this expression agreesprecisely with the upper demarkation line. Moreover, expression (48) and (49) provide alternative axes parameters in terms of d as shown in Figure 6. Using these alternative axes, the stratification diagram then shows that a stratified estuary with an interface at d GO.75 must be of type 4, whereas a stratified estuary of type 3(b) must have the interface at d-* 1. A seconddemarkation line in Figure 6, namely that which separatesestuariesof types 1 and 2, may also be explained from the present theory. Thus from Table 1, in the absenceof wind forcing, z+/U= 1.14+0.036 S/F,
at the surface, (51)
u,/U = 0.70 - 0.029 S/F,
at the bed
The distinction between estuariesof types 1 and 2 is basedon the absenceof upstream flow in the former. From equations @I), the present theory requires that for flow reversal S/F> 24 or u,/U > 2 and this value of up/%again agreesclosely with the demarkation line shown in the stratification diagram. Moreover, the relationship (51) provides an alternative axes parameter asshown in Figure 6. Mixing
related
to residual
currents
One requirement for a system to change from being well-mixed to stratified (or vice versa) might be an equivalence between the two states of (a) surface velocities, (b) bed velocities or (c) surface gradients. From Table 1 such equivalences involve specific relationships between the parameter S/F in a mixed system and the fractional depth d in a stratified system. However, data shown in Table 2 do not support any such relationships. A further relationship between stratification and residual current structure may be derived by assuming (Bowden, 1963; Officer, 1976, ch 5) that mixing is primarily a
630
D. handle
balance between horizontal advection and vertical diffusion. The resulting equation can then be integrated to give an expression for the vertical variation in density as a function of vertical current structure. Again attempts to verify such a relationship using the data from Table 2 proved unsuccessful and hence we conclude that residual velocity structure, characterized by S/F, doesnot directly determine the level of stratification. Mixing
related to energy balance
Ippen and Harleman (1961) and Simpson and Hunter (1974) demonstrated that vertical mixing could be related to energy considerations. Ippen and Harleman defined a stratification number, S, by s,=“=
energy dissipated J
gain in potential energy’
(52)
The rate of gain in potential energy of the fresh water due to an increasein density L/I is 4A y g HZ U. Tidal energy is dissipated over the mixing length, L, at a rate (4/37c)k pG3L (seeSimpson and Hunter (1974) for related energy derivations). Hence S, = 0.85
kiiL $g
H’ii
’
or from expression (37)
Thus, energy arguments indicate that the ‘ flow ratio ’ parameter U/C must be introduced to augment the parameter S/F in determining stratification levels. The significance of this additional parameter was recognized by Pritchard (1955), while Schultz and Simmons (1957) suggestedthat stratification will occur when ii/C > l/n whereascomplete vertical mixing will occur for ii/C < 0.1/n. Differences in the definitions of G andJ in equation (53) from the expressionsused by Ippen and Harleman make present values approximately double those calculated by the latter authors. Allowing for this discrepancy, these authors showed that S, < 100 corresponds to significant stratification while S, > 400 indicates strong mixing. Table 2 showsvalues of E/z& S/F, S, and 6s/?for 9 major estuariestogether with similar data from 3 flume tests simulating conditions in Rotterdam Waterway and 4 WES tests. The sources for these data are indicated, however, for the estuarine data the values adopted may show significant variations both in time and space. Using these data, Figure 7 showsa plot of Gs/SversusS,. The points lie closeto the line
and thus confirm the results of Ippen and Harleman. The demarkation values of S, = 100 and 400 are equivalent to the definitions of &/S-C0.15 as well mixed with Ss/S> 0.32 as stratified. Figure 8 usesthe sameestuarine and flume data but plots the location of each estuary from its respective values of S/F and U/C. In addition, by making use of expressions(54) and (55) contour lines for &/Fare constructed. This figure can then be used to provide an immediate estimate of the level of stratification in any estuary. Moreover, the separate influences of the two axes parameters are evident and hence the sensitivity of the existing
Salinity and structure of vertical
flows
--
631
l OStratIfled
5-
.
'\
'\
2-
Well-maxed
x 3. '\ 0 '\
I.13X ‘? * co
+ '\
'\
t '\ X
'. ‘\
‘.
.
0 .I -
0.0
I I
I
I
I
ii
5
IO
Figure 7. Level of stratification (6s/?) TJSstratification no. (S,) (log-log scales): ~S/S salinity difference between bed and surface/depth-averaged salinity, S, = G/3= energy dissipated/gain in potential energy during mixing [equation (52)], x estuarine values, 0 Rigter’s (1973) flumedata, + WES tests.
24
Figure 8. S/F vs ’ flow ratio ‘, liti (log-log scales). Dashed lines represent contours of 6s/F; n‘s/S salinity difference between bed and surface/depth-averaged salinity; S/F ratio of salinity: frictional term, index of gravitational circulation [equation (37)]; ii depth-averaged residual velocity; li amplitude of tidal velocity; 0 Rigter’s 1’1973) flume data, +WES tests; x estuaries (listed in Table 2 and labelled with first two letters).
632
D. Pratrdle
level of stratification to changesin parameters such as fresh water flow U or tidal velocity li may be appreciated. Spatial changes in the adopted parameters may be particularly important. For example, one anomaly in Figure 8 is the Mersey Narrows which appears to be wellmixed whereas Dyer (1973) suggeststhe estuary is only partially mixed. Abbott (1960) shows that values of S/F at a point about 6 km upstream of the Narrows are an order of magnitude greater than in the Narrows, hence stratification may be induced in this region. Conversely, temporal changesin stratification over the usual 14 day spring-neap cycle are seldom as pronounced as might be indicated by expression (54) from direct substitution of the changes in ti (Simpson & Bowers, 1979). However, longer term changes in 1( may significantly modify the estuarine classification [see data for Mersey and Vellar shown by Dyer (1973)]. Conclusions
Velocity structures are derived separately for residual flow components associatedwith (a) a river flow Q, (b) a wind stressrW,(c) a well-mixed longitudinal density gradient and (d) a fully stratified saline wedge. The relative magnitudes of each of these components are defined in terms of dimensionlessparameters (37). The principal results, namely surface gradients and velocities at surface and bed, are summarisedin Table 1. It is shown that, in the steady state, both wind and density forcing are mainly balanced by surface gradients with only a small fraction of the forcing effective in maintaining a vertical circulation. The above theoretical results were applied to nine major estuaries to indicate their relative sensitivity to wind and density forcing. The circulation for a salinewedge is deduced and is shown to accord with the generally perceived pattern. Thus with the interface at a fraction d of the depth above the bed, a maximum landward velocity occurs at 0,13d, flow reversal occurs at 0.59d and a maximum seawardflow occurs at the surface. The slope of the interface is opposite in sign to that of the surface gradient and is greater in magnitude by somefactor in excessof the density ratio ny/p. For channels of constant depth and breadth, a formula for the total length of the saline intrusion, L, is deduced. This formula is used to examine data for observed intrusion lengths in laboratory flumes and, subject to evaluation of 2 empirical coefficients, the formula provides an excellent prediction of changes in L arising from variations in: river flow, tidal range or channel depth and roughness. The velocity structure results are used to explain basic features of the stratification diagram presented by Hansen and Rattray (1966). An attempt to relate the level of stratification, 6s/S, to velocity structure proved unsuccessful, thus confirming the latter authors’ conclusion that ds!?is not a ‘ dynamical indicator ‘. The level of stratification is subsequently related to energy considerations, specifically to a stratification number, S,, defined as the ratio of energy dissipation to the gain in potential energy arising from the mixing process. This relationship was verified using observed levels of stratification from nine estuaries and seven flume tests. Moreover the value of S, is shown to be inversely proportional to S/F (ii/C)‘, where S/F is the ‘dynamical indicator ’ derived from the velocity structure results while E/G is a ‘flow ratio ’ relating freshwater flow to the volume of tidal inflow. The above relationships may be distilled to provide a stratification diagram for
Salinity
and structure
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633
estuaries with S/F and U/C as orthogonal axes and Ss/S displayed as contour lines. This diagram provides a clear illustration of the level of mixing in any estuary and also an understanding of the likely sensitivity to changing conditions. In practice any changes in intrusion length caused by dredging, barrier construction etc. may have serious implications for water quality, sedimentation or dispersion of pollutants. Thus further research of the governing mechanisms is required. Modelling of these mechanisms is generally limited by uncertainties in defining eddy viscosities and eddy diffusivities. While field measurements in various estuaries continuously add to existing knowledge, there seems a strong case for more work in laboratory flumes. Such work could extend the range of conditions covered by previous investigators (Figure g), in addition channels with variations in depth and breadth should be studied. These flume studies could also indicate the magnitude of various diffusive and advective mixing processes and might also examine the adjustment times associated with the neap-spring cycle or changes in river flow.
Nomenclature Upper
case
depth of the lower layer vertical eddy viscosity coefficient defined by equation (37) frictional stress at level z depth of water length of saline intrusion tidal period river flow defined by equation (37) stratification no. defined by equation (52) wind speed defined by equation (37) vertical axis measured upwards from the bed
D E F
FZ H L
P
Q S St v W
Z
Lower
case
constant = D/H
function gravitational constant elevation of the interface surface gradient bed stress coefficient ( = 0.0025) pressure salinity horizontal velocity surface value of u horizontal axis, positive downstream = Z/H
Greek 11
linearized bed stresscoefficient constant salinity difference between surface and bed defined by equation (29) surface elevation wavelength 3.14159 density wind stress
ii n‘s
t‘ r i 71 1) rw Subscripts
These denote component associatedwith: B bottom-layer well-mixed horizontal density gradient river flow u” S completely stratified system T top layer W wind stress An over bar (e.g. i2) indicates a depth-averaged value A circumflex (e.g. 12)indicates a tidal amplitude References Abbott, M. R. 1960 Salinity effects in estuaries. Journal of Marine Research 18, 101-l 11. Bassindale, R. 1943 A comparison of the varying salinity conditions on the Tees and Severn estuaries. Journal of Animal Ecology 12, l-10. Bowden, K. F. 1963 The mixing processes in a tidal estuary. Internarkmaf ‘Journal of Air and Warer Pollution
7, 343-356.
Bowden, K. F. 1965 Horizontal mixmg in the sea due to a shearing current. rournal of Fluid Mechanics 21, 83-95. Dyer, K. R. 1973 Estuaries: a Physical Ixtroduction. John Wiley, New York. 140~~. Festa J. F. & Hansen, D. V. 1976 A two-dimensional numerical model of estuarine circulation. The effects of altering depth and river discharge. Estuarine and Coastal Marine Science 4,309-323. Hansen, D. V. & Rattray, M. J. 1966 New dimensions in estuary classification. Limnology Oceanography 11, 319-326. Inglis, C. C. & Allen, F. H. 1957 The regimen of the Thames estuary as affected by currents, salinities, and river flows. Proceedings of the Institution of Civil Engzneers 7,827-868. Ippen, A. T. (ed). 1966 Estuary and Coastline Hydrodynemics. McGraw-Hill, New York. 744~~. Ippen, A. T. & Harleman, D. R. F. 1961 One-dimensional analysis of salinity intrusion in estuaries. Technical Bulletin no. 5, Committee on Tidal Hydraulics Waterways Experiment Station, Vicksburg, Mississippi. Officer, C. B. 1976 Physical Ocearrography of Estuaries. John Wiley, New York. 465~~. Prandle, D. 1982 The vertical structure of tidal currents and other oscillatory flows. Conri?renral Sheu Research 1 (2), 191-207. Pritchard, D. W. 1955 Estuarine circulation patterns. Proceedings of the American Society of Civrl Engineers 81,71711-717111. Rigter,.B. P. 1973 Minimum length of salt intrusion in estuaries. Proceedings of the American Society of Ctevl Engineers, your& of the Hydraulics Division 99, (HY9), 1475-1496. Rossiter, J. R. 1954 The North Sea Storm Surge of 31 January and 1 February 1953. Philosophzcal Transactions of the Royai Society of London A246,317+30. Schultz, E. A. & Simmons, H. B. 1957 Fresh water-salt water density currenrs, a major cause of siltation in estuaries. Technical Bulletin no. 2, Communication on Tidal Hydraulics, U.S. Army, Corps of Engineers. 28pp.
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and structure
ojverticalflows
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Simpson, J. H. & Bowers, D. 1979 Shelf sea fronts adjustments revealed by satellite IR imagery. Nature 280,648-651. Simpson, J. H. & Hunter, J. R. 1974 Fronts in the Irish Sea. Nature 250,404-406. West, J. R. &Williams, D. J. A. 1972 An evaluation of mixing in the Tay Estuary. American Society of Civil Engineers. Proceedings 13th Conference on Coastal Engineering, pp. 2153-2169. Winterwerp, J. C. 1983 Decomposition of the mass transport in narrow estuaries. Esruarine, Coastal md Shelf Science 16,627-638.
