Progress in Oceanography Progress in Oceanography 61 (2004) 1–26 www.elsevier.com/locate/pocean
Review
How tides and river flows determine estuarine bathymetries D. Prandle
*
Proudman Oceanographic Laboratory, Bidston Observatory, Prenton, Merseyside CH43 7RA, UK Received 18 June 2003; received in revised form 16 February 2004; accepted 16 March 2004 Available online 13 May 2004
Abstract For strongly tidal, funnel-shaped estuaries, we examine how tides and river flows determine size and shape. We also consider how long it takes for bathymetric adjustment, both to determine whether present-day bathymetry reflects prevailing forcing and how rapidly changes might occur under future forcing scenarios. Starting with the assumption of a ‘synchronous’ estuary (i.e., where the sea surface slope resulting from the axial gradient in phase of tidal elevation significantly exceeds the gradient in tidal amplitude ^f), an expression is derived for the slope of the sea bed. Thence, by integration we derive expressions for the axial depth profile and estuarine length, L, as a function of ^f and D, the prescribed depth at the mouth. Calculated values of L are broadly consistent with observations. The synchronous estuary approach enables a number of dynamical parameters to be directly calculated and conveniently ^ , ratio of friction to inertia terms, estuarine length, illustrated as functions of ^f and D, namely: current amplitude U stratification, saline intrusion length, flushing time, mean suspended sediment concentration and sediment in-fill times. ^ Uo (Uo is the Four separate derivations for the length of saline intrusion, LI , all indicate a dependency on D2 =f U residual river flow velocity and f is the bed friction coefficient). Likely bathymetries for ‘mixed’ estuaries can be delineated by mapping, against ^f and D, the conditions LI =L < 1; EX =L < 1 (EX is the tidal excursion) alongside the Simpson–Hunter criteria D=U 3 \50 m2 s3 . This zone encompasses 24 out of 25 ‘randomly’ selected UK estuaries. However, the length of saline intrusion in a funnel-shaped estuary is also sensitive to axial location. Observations suggest that this location corresponds to a minimum in landward intrusion of salt. By combining the derived expressions for L and LI with this latter criterion, an expression is derived relating Di , the depth at the centre of the intrusion, to the corresponding value of Uo . This expression indicates Uo is always close to 1 cm s1 , as commonly observed. Converting from Uo to river flow, Q, provides a morphological expression linking estuarine depth to Q (with a small dependence on side slope gradients). These dynamical solutions are coupled with further generalised theory related to depth and time-mean, suspended sediment concentrations (as functions of ^f and D). Then, by assuming the transport of fine marine sediments approximates that of a dissolved tracer, the rate of estuarine supply can be determined by combining these derived mean concentrations with estimates of flushing time, FT , based on LI . By further assuming that all such sediments are deposited, minimum times for these deposition rates to in-fill estuaries are determined. These times range from a decade for the shortest, shallowest estuaries to upwards of millennia in longer, deeper estuaries with smaller tidal ranges. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Bathymetry; Estuaries; River flow; Saline intrusion; Sediment; Tides; Dynamics
*
Tel.: +44-151-653-1546; fax: +44-151-653-6269. E-mail address:
[email protected] (D. Prandle).
0079-6611/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.pocean.2004.03.001
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D. Prandle / Progress in Oceanography 61 (2004) 1–26
Contents 1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.
Tidal dynamics: application to a synchronous estuary with triangular cross-section derivations of bathymetry and stratification levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Synchronous estuary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Analytical solution for 1-D momentum and continuity equations (Prandle, 2003) . . . . . 2.4. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ^ ........................................... 2.4.1. Current amplitudes u 2.4.2. Role of bed friction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Bed slope SL and estuarine length L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4. Rate of funnelling in a synchronous estuary . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Stratification levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 4 4 4 6 6 6 6 9 10
3.
Saline intrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Analytical solutions for vertical profiles of residual velocities and salinities (associated with a constant axial saline gradient, Sx ). . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Lengths of saline intrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Observational data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Rates of mixing by vertical diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3. Sea-bed velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.4. Mean value for LI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Observed vs computed values of saline intrusion length, LI . . . . . . . . . . . . . . . . . . . . . 3.5. Axial location of saline intrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 11
4.
Estuarine bathymetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Estuarine bathymetry consistent with tidal dynamics and saline intrusion. . . . . . . 4.2. Estuarine depths as a function of river flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Observed vs computed estuarine bathymetries . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
16 16 18 18
5.
Sediment regime. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Suspended sediment time-series in tidal estuaries. . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Mean concentrations of SPM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Estimates of in-fill times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . .
. . . .
. . . .
. . . .
19 19 21 21
6.
Summary and conclusions . . . . . . . . . . . . . . . 6.1. Tidal dynamics – estuarine length . . . . . 6.2. Salinity intrusion, length and location . . 6.3. Bathymetry as a function of river flow . . 6.4. Sedimentation . . . . . . . . . . . . . . . . . . . 6.5. Estuarine characteristics . . . . . . . . . . . .
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24 24 24 24 24 25
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
Appendix A. Estuarine
depths (at mouth and mean river flows). . . . . . . . . . . . . . . . . . . . . . .
25
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction There is growing concern about sustainable exploitation of estuaries, in particular how economic and natural environment interests can be reconciled in the face of increasingly larger scale developments accompanied by accelerating rates of change in the global environment (Fig. 1). The limited capabilities of models to simulate associated changes to estuarine regimes is widely recognised. A particular concern is the uncertainty in predicting longer-term (decadal) bathymetric evolution, where incremental changes can depend on antecedent bathymetry, resulting in random or chaotic model outcomes. Here, the principal objective is to explain existing bathymetric conditions in terms of the balance with prevailing tides, river flow and alluvium. To obtain generic analytical solutions to the governing equations, considerable simplifications are necessary. Here, we restrict consideration to strongly tidal, funnel-shaped estuaries. The derived solutions should be regarded as characteristic rather than precise descriptions of prevailing conditions. The familiar assumption of a single predominant (M2 ) tidal constituent is utilised. The adoption (Section 2) of a ‘synchronous’ estuary with a triangular cross-section is an expedient that provides a direct relationship between localised tidal dynamics and the slope of the sea bed, SL. Integration of the latter provides an estimate of estuarine length, L. Moreover, these approximations enable salient features of estuarine tidal dynamics and related levels of stratification to be illustrated directly as functions of D and ^f. Likewise the analyses of saline intrusion (Section 3) based on the expedient assumption of a constant (in time and depth) axial density gradient Sx , provides an expression for the length of saline intrusion, LI . However, previous studies have shown that this expression does not account for observed variations in LI over either the spring to neap tidal cycle or flood to drought river flows. Here, the importance, in a funnelshaped estuary, of axial migration in determining LI is emphasised; explaining this difficulty in applying formulations for LI to observations. Adopting the criterion, noted from observed patterns of saline intrusion, that the location of saline intrusion corresponds to limiting of the landward extent of intrusion, a relationship (Section 4), linking the depth at the mouth with river flow Q is derived. Sensible correspondence is shown throughout between the
Fig. 1. Schematic of factors influencing estuarine bathymetry.
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D. Prandle / Progress in Oceanography 61 (2004) 1–26
various simplified formulations and observed conditions. This capability to predict the likely range of estuarine bathymetries from the governing dynamical equations encourages us to extend the study (Section 5) to consider how quickly these derived estuarine bathymetries might in-fill under worst-case scenarios.
2. Tidal dynamics: application to a synchronous estuary with triangular cross-section derivations of bathymetry and stratification levels 2.1. Approach The essential characteristics of tidal dynamics in estuaries can be readily explained from analytical solutions (Friedrichs & Aubrey, 1994; Prandle & Rahman, 1980). These dynamics are almost entirely determined by a combination of tides at the mouth and estuarine bathymetry with some modulation by bed roughness and river flows. The present approach differs from earlier studies by exploring localised dynamics (as opposed to seeking whole-estuary solutions) expressed in terms of tidal elevation amplitude and water depth. Specifically, we derive localised dynamical-bathymetric relationships for a case of a single (predominant) tidal constituent in a triangular shaped estuary. The solutions obtained assume that, whilst the inclination of these side slopes can be asymmetric, they must remain (locally) constant. The dynamical solutions are independent of the actual value of these side slopes. The analytical solutions assume that: (i) tidal forcing predominates and can be approximated by a cross-sectionally averaged axial propagation of a single semi-diurnal tidal constituent; (ii) advective and density gradient terms can be neglected and (iii) the friction term can be sensibly linearised. 2.2. Synchronous estuary Dyer (1997) describes how frictional and energy conservation effects can combine in funnel-shaped bathymetry to produce a ‘synchronous estuary’ with constant tidal elevation amplitudes ^f. Prandle (2003) used a numerical simulation applied to the corresponding bathymetry (derived here in Section 2.4) to compare the ratio of sea surface gradients associated with: (i) change in amplitude ðo=oxÞ^f and (ii) change in phase ^fðoU=oxÞ, where U is the (axial variation in) tidal phase of ^f. The results show that the assumption that ðo=oxÞ^f ! 0 remains valid except for a small section at the tidal limit. Moreover, it is shown in Section 2.4 that the shape and lengths pertaining to a synchronous estuary are in the centre of the observed range for funnel-shaped estuaries. 2.3. Analytical solution for 1-D momentum and continuity equations (Prandle, 2003) Omitting the advective term from the momentum equation, we can describe tidal propagation in an estuary by: ou of ulul þg þf ¼ 0; ot ox H B
of o þ Au ¼ 0; ot ox
ð1Þ
ð2Þ
where u is velocity in the x-direction, f is water level, D is water depth, H is the total water depth ðH ¼ D þ fÞ, f is the bed friction coefficient ð 0:0025Þ, B is the channel breadth, A is the cross-sectional area, g is the gravitational acceleration, and t is the time.
D. Prandle / Progress in Oceanography 61 (2004) 1–26
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Concentrating on the propagation of one predominant tidal constituent ðM2 Þ, the solutions for u and f at any location can be expressed as: f ¼ ^f cos ð K1 x xtÞ;
ð3Þ
^ cos ð K2 x xt þ hÞ; u¼U
ð4Þ
where K1 and K2 are the wave numbers, x is the tidal frequency and h is the phase lag of ^u relative to ^f. The ‘synchronous estuary’ assumption is that ^f does not vary axially. In deriving solutions to (1) and (2), a ^ . The resulting solutions for U ^ indicate this additional similar approximation is assumed to apply to U assumption is valid, except in the shallowest waters (Fig. 2). Further assuming a triangular cross-section with constant side slopes, (2) reduces to: of of oD 1 ou þu ðf þ DÞ ¼ 0: ð5Þ þ þ ot ox ox 2 ox Friedrichs and Aubrey (1994) indicate that uðoA=oxÞ Aðou=oxÞ in convergent channels. Likewise assuming ðoD=oxÞ ðof=oxÞ, we adopt the following form of the continuity equation: of oD D ou þu þ ¼ 0: ot ox 2 ox
ð6Þ
The component of f ðulul=H Þ at the predominant tidal frequency M2 may be approximated by ^ ju 8 25 jU f ¼ Fu; 3p 16 D
ð7Þ
^ =D; with F ¼ 1:33f U where 8=3p derives from the linearisation of the quadratic velocity term. The factor 25=16 is derived by assuming that the tidal velocity at any transverse location is given by a balance between the quadratic
Fig. 2. Tidal current amplitudes, ^u (m s1 ) as a function of tidal elevation amplitude, ^f and water depth D.
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D. Prandle / Progress in Oceanography 61 (2004) 1–26
friction term (with localised depth) and a (transversally constant) surface slope – yielding a cross-sectionally averaged velocity of 4=5 of the velocity at the deepest section. Substituting solutions (3) and (4) into Eqs. (1) and (6) with the geometric and frictional representations (6) and (7), four equations (pertaining at any specific location along the estuary) representing components of cosðxtÞ and sinðxtÞ in (1) and (2) are obtained. By specifying the synchronous estuary condition that the spatial gradient in tidal elevation amplitude is zero, we derive K1 ¼ K2 ¼ k, i.e., identical wave numbers for ^ , thence axial propagation of ^f and U tan h ¼
F SL ¼ ; x 1=2Dk
where SL ¼ oD=ox, ^ ¼ ^fgk= x2 þ F 2 1=2 ; U , k¼x
Dg 2
ð8Þ
ð9Þ
1=2 :
ð10Þ
2.4. Results A particular advantage of the above solutions is that they enable the values of a wide range of estuarine parameters to be calculated (and illustrated throughout this paper) as direct functions of D and ^f. The ranges selected for illustration here are: ^f (0–4 m) and D (0–40 m), these represent all but the deepest of estuaries. Most of these results are also sensitive to the value of the frictional parameter, f . Prandle (2003) provides an analyses of these sensitivities. ^ 2.4.1. Current amplitudes u Fig. 2 shows the solution of Eq. (9) with current amplitudes extending to 1.5 m s1 . For ^f D=10 these currents are insensitive to f . For ^f D=10, these currents change by a factor of two over the range of values ^ / ^f1=2 D1=4 , thus indicating why observed variations of U ^ are for f . Noting that for F x, (9) indicates U 1=2 ^ ^ generally smaller than for f. Conversely, for F x, (9) indicates U / fD . The contours show that ^ occur at approximately D ¼ 5 þ 10^fðmÞ, however, these are not pronounced maxima. maximum values of U 2.4.2. Role of bed friction Fig. 3 illustrates the ratio F =x of the friction to inertial terms in Eq. (1), calculated from Eqs. (8)–(10). F =x is approximately equal to unity for ^f ¼ D=10. For values of ^f D=10, tidal dynamics become frictionally dominated, whereas for ^f D=10 friction becomes insignificant. Friedrichs and Aubrey (1994) showed the predominance of the friction term in convergent channels, irrespective of depth. 2.4.3. Bed slope SL and estuarine length L In Fig. 4, utilising the values of SL from Eq. (8), the length, L, of an estuary is calculated by successively updating SL as D reduces along the estuary (assuming a constant value of ^f). By assuming F x, an equivalent simple analytical solution can be determined 0 1=2 14=5 ^f1:33f x B5 C 04=5 D¼@ ð11Þ A x ; 1=4 4 ð2gÞ
D. Prandle / Progress in Oceanography 61 (2004) 1–26
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Fig. 3. Ratios of the magnitudes of the frictional term, F , to the inertial term, x, as a function of ðD; ^fÞ.
where x0 ¼ L x or 1=4
L¼
Do5=4 4 ð2gÞ Do5=4 2460 ^f1=2 5 ð1:33f xÞ1=2 ^f1=2 o o
for f ¼ 0:0025
ð12Þ
(units m, subscripts o denote values at the mouth). The dependency on D5=4 =^f1=2 in (12) explains the distributions shown in Fig. 4 with estuarine lengths significantly more sensitive to D than to ^f. Prandle (2003) shows that this expression for estuarine length is in broad agreement with data from some 50 estuaries (randomly selected utilising previously published data, Table 1) located around the coasts of the UK and eastern USA. For the UK estuaries, estimates of
Fig. 4. Estuarine length, L (km) as a function of ðD; ^fÞ.
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D. Prandle / Progress in Oceanography 61 (2004) 1–26
Table 1 Observed estuarine parameters (m) Top USA data (Friedrichs & Aubrey, 1988), bottom UK data (Yates et al., 1996) Estuary
Depth
Tidal amplitude
Length
Mud %
N.C. Naucet MA S.C. Naucet MA Chatham MA Swan Pond MA Stony Brook NY Shark River MJ Manasquan NJ Absecon NJ Strathmere NJ Townsend NJ Delaware Bay Northam VA Wachapreague VA Rudee VA Little River SC M.C. Murrels SC O.C. Murrels SC N. Inlet SC Price SC Capers SC Dewees SC Breach SC Folly SC Duplin GA St. Marys GA Ft. George FL Ythan Montrose Eden Tyninghame Humber Breydon Blyth Alde Deben Swale Pagham Tamar Plym Hayle Swansea Bay Dyfi Artro Mawddach Glaslyn Foryd Bay Dee Lune Duddon Solway Auchencairn
1.00 2.10 2.40 0.80 1.70 1.90 1.50 2.90 2.30 2.30 5.70 1.60 4.20 4.50 2.60 1.90 1.40 2.40 3.30 3.30 4.30 3.30 3.60 4.70 5.70 2.60 2.70 2.30 2.70 2.40 24.90 1.30 1.30 2.10 4.30 4.80 2.80 7.10 5.10 4.20 4.30 2.60 2.50 3.10 2.60 2.70 6.30 7.20 4.70 11.30 3.90
1.00 1.00 1.00 0.42 0.86 0.60 0.58 0.56 0.56 0.57 0.65 0.50 0.54 0.48 0.64 0.73 0.73 0.73 0.69 0.71 0.72 0.73 0.76 0.99 0.90 0.72 1.28 1.54 1.69 1.69 2.44 0.71 0.79 0.86 1.35 1.95 1.65 1.77 1.77 2.18 3.23 1.62 1.65 1.62 1.73 1.77 2.89 3.16 2.86 3.16 2.78
3100 6800 14,000 4500 5200 4400 9200 8000 5300 5800 215,000 4700 13,000 1100 13,000 8000 4700 6500 7100 6900 7600 5200 11,000 13,000 18,000 8000 7350 5500 5050 2750 64,750 11,250 7070 18,900 16,100 18,400 2250 21,200 5370 2250 3000 8250 14,200 9050 8500 3250 20,750 11,600 14,750 40,250 4000
60 42 27 14 48 75 75 75 74 47 32 75 73 28 10 15 17 11 12 21 23 37 11 16 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
River flows (m3 s1 ) from: www.nwl.ac.uk/ih/nrfa/river_flow.
River flow
3.9 280 1.35 2.14 4.27
22.5
22.8 12.0 16.8 10.7 41.7 35.2 93.4
D. Prandle / Progress in Oceanography 61 (2004) 1–26
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mud content were available, enabling some of the discrepancies between observed and estimated values of L to be reconciled by introducing an expression for f based on relative mud content. 2.4.4. Rate of funnelling in a synchronous estuary Prandle and Rahman (1980) produced a generalised framework for tidal response in funnel-shaped estuaries based on analytical solutions for Eqs. (1) and (2). Fig. 5 shows relative tidal amplitude response (dashed contours) and associated phases (full contours) across a wide range of estuarine bathymetries. The details of these solutions are omitted, the present aim is to show how the estuarine lengths and shapes derived for synchronous estuaries fit within this framework. The funnelling factor, t, derived in Prandle and Rahman is given by t¼
nþ1 ; 2m
ð13Þ
where depths are proportional to xm and breadths xn , Since Eq. (11) corresponds to m ¼ n ¼ 0:8, the synchronous estuary solution corresponds to t ¼ 1:5, i.e., close to the centre of values encountered. The vertical axis in Fig. 5 represents a transformation of the distance from the head of the estuary given by 4p y¼ 2m
!ð2mÞ=2
X ðgDÞ
1=2
P
(P tidal period), i.e., for X ¼ L; m ¼ 0:8, !0:6 D3=4 y ¼ 0:9 : ^f1=2
ð14Þ
ð15Þ
Fig. 5. Shape ðm ¼ 1:5Þ and lengths of synchronous estuaries in relation to the generalised tidal elevation response framework of Prandle and Rahman (1980). m represents degree of bathymetric funnelling and y distance from the mouth ðy ¼ 0Þ. dashed contours: relative amplitudes, full contours: relative phases.
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D. Prandle / Progress in Oceanography 61 (2004) 1–26
^ 3 m2 s3 as a function of ðD; ^fÞ. Fig. 6. Simpson–Hunter stratification criterion D=U
Taking D ¼ 5; ^f ¼ 4 m together with D ¼ 20 m, ^f ¼ 2 as representative of minimum and maximum values of y in ‘mixed’ estuaries, these correspond to y ¼ 1:22 and y ¼ 2:8. Fig. 5 shows that this range of values for y (at the mouth of the estuary; y ¼ 0 corresponds to the head of the estuary) is also representative of observed lengths, extending from a small fraction to almost a quarter wavelength for the M2 constituent. 2.5. Stratification levels The earlier derivation of tidal current amplitudes (Fig. 2) enables direct estimation of the Simpson and ^ 3 . The results, shown in Fig. 6, indicate that estuaries with tidal Hunter (1974) stratification parameter D=U
^ Fig. 7. Time for vertical mixing by diffusion: D2 =Kz ðKz ¼ f UDÞ as a function of ðD; ^fÞ.
D. Prandle / Progress in Oceanography 61 (2004) 1–26
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elevation amplitudes ^f > 1 m will generally be mixed, being less than the 55 m2 s3 demarcation separating mixed from stratified conditions. The latter value was derived for thermal stratification in shelf seas and cannot be expected to apply precisely to saline stratification in estuaries. An additional indication of stratification levels can be calculated from the time, TK , for complete vertical mixing by diffusion of a point source – estimated by Prandle (1997) as TK ¼ D2 =Kz (Kz is the ^ D, yields TK ¼ D=f U ^ , estimates of which are shown in vertical diffusivity). Approximating Kz ¼ f U Fig. 7. This approach produces results consistent with those in Fig. 6, noting that where TK > P =2 6 h, stratification is likely to persist beyond consecutive peaks of mixing on flood and ebb tides. Conversely for TK < 1 h, little stratification is likely. For intermediate values, 1 h < TK < 6 h, intratidal stratification is likely – especially via tidal straining on the flood tide (Simpson, Brown, Matthews, & Allen, 1990).
3. Saline intrusion 3.1. Approach While models of saline intrusion in estuaries exist in many forms, these often fail to accurately reproduce observed sensitivities to either the spring-neap variations in tidal range or to changes in river flows. We restrict interest to partially mixed estuaries and assume a (temporally and vertically) constant relative axial density gradient, Sx ¼ ð1=qÞðoq=oxÞ, with density linearly proportional to salinity. Tidally averaged linearised theories relating to salinity intrusion have been derived by Officer (1976), Bowden (1981) and Prandle (1985). Expressions for the length of saline intrusion are obtained from: (i) earlier flume studies, (ii) balancing the rate of mixing associated with vertical diffusion with river flow and (iii) counter-balancing at the limit of intrusion, upstream and downstream components of residual velocities associated with Sx and river flow ðUo Þ. An additional expression for stratified ‘saline wedge’ intrusion is also considered. A new expression for the axial location of intrusions is then derived, providing estimates of typical values for Uo . 3.2. Analytical solutions for vertical profiles of residual velocities and salinities (associated with a constant axial saline gradient, Sx ) Expressing the non-time-varying (residual) components in the momentum equation in the form: g
of o2 u þ gðf ZÞSx ¼ Ez 2 ðmÞ: ox oZ
ð16Þ
introducing z ¼ Z=D, the solution of (16) for residual velocities Us associated with Sx is (Prandle, 1985): D3 z3 ð17Þ Us ¼ gSx þ 0:2687z2 0:0373z 0:0293 : Ez 6 For a laterally homogeneous, partially mixed estuary, the salinity distribution is governed by a combination of advection and diffusion (Dyer, 1997; Pritchard, 1955), os os os o os o os þU þW ¼ Kz þ Kx ot ox oz oz oz ox ox
ð18Þ
(s is the salinity, Kz is the vertical eddy diffusivity, Kx is the axial diffusivity, W is the vertical velocity).
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D. Prandle / Progress in Oceanography 61 (2004) 1–26
Scaling analysis can be used to justify neglect of W os . Observations indicate the predominance of the oz os vertical diffusion term and hence we omit axial diffusion. Thence, assuming ox ¼ qSx , omitting time varying terms we obtain, for Kz constant, an expression for the time-averaged vertical structure of salinity (measured in units of relative density): Z Z Sx sðZÞ ¼ qUs dZ dZ; ð19Þ Kz i.e., from (17), defining s0 ðZÞ ¼ sðZÞ S s0 ¼ q
gSx2 D5 83z5 þ 224z4 63z3 146z2 þ 33 : Ez Kz 10000
ð20Þ
3.3. Lengths of saline intrusion 3.3.1. Observational data Prandle (1985) derived the following expression for saline intrusion length, LI , from an analysis of flume tests. LI ¼ 0:0016
D2 ðmÞ: ^ Uo fU
ð21Þ
Fig. 8 shows a comparison of observed and calculated values of LI from this flume study – indicating the robustness of (21) over a range of parameter sensitivity tests. To maintain turbulent Reynolds numbers. in hydraulic scale models, vertical exaggeration (typically by a factor of 10) is utilised. As a result, to maintain the relative magnitude of the friction term in the momentum balance (see Eq. (1)), the bed roughness, f , must be increased by the same factor. Hence, based on estimates
Fig. 8. Observed, LR , vs predicted, Lc , saline intrusion lengths (from Prandle (1985)). Results based on flume studies with changing values of: X tidal amplitude ^f, r density difference dq, m friction coefficient f , H river flow Q, j water depth D and þ flume length L.
D. Prandle / Progress in Oceanography 61 (2004) 1–26
13
of a horizontal scale of 1:1000 and a vertical scale 1:100, for full scale applications we might expect the numerical coefficient in (21) to increase by up to a factor of 10. 3.3.2. Rates of mixing by vertical diffusion From (18), the mean rate of mixing MK associated with the time-averaged density structure (20) is Z D o os Sx2 D4 K dZ ¼ 0:02qg : ð22Þ MK ¼ q z oz oz Ez 0 For a stationary salinity distribution, the rate of mixing MQ to balance freshwater velocity Uo is MQ ¼ qUo Sx D:
ð23Þ
Thus for MQ to balance MK we require Uo ¼ 0:02g
Sx D 3 : Ez
ð24Þ
Approximating Sx ¼ dq=LI (with dq ¼ 0:027 representative of the density difference between sea and ^ D, then (24) may be re-written in the form: freshwater) and Ez ¼ f U LI ¼
0:005D2 ðmÞ: ^ Uo fU
ð25Þ
3.3.3. Sea-bed velocities Analogously to the derivation of Eq. (17), Prandle (1985) provided the following estimates of velocity components USO and Uoo at the sea bed: associated with axial density gradient Sx , USO ¼ 0:029
gD2 Sx ^ fU
ð26Þ
associated with depth-mean river flow Uo , Uoo ¼ 0:7Uo :
ð27Þ
A simple hypothesis for the limit of upstream intrusion of salt is the position where these upstream and downstream velocity components balance, i.e., where LI ¼
0:011D2 ðmÞ: ^ Uo fU
ð28Þ
3.3.4. Mean value for LI The above formulations (21), (25) and (28) all show identical expressions for LI but with quantitative values varying by the respective coefficients 0.0016, 0.005 and 0.011. To reconcile these values, we note that the first value might be expected to increase to reflect increased roughness in flume experiments. The ^ at the landward balancing of sea-bed velocity components in Eq. (28) should strictly use values of D and U limit of the wedge, resulting in a decrease in the coefficient shown. Thus, henceforth we adopt the value given in Eq. (25) but note that some quantitative uncertainty surrounds the expression. These expressions all relate to well or partially mixed conditions. Prandle (1985) derived, from purely dynamical considerations, the following expression for a fully stratified salt wedge:
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LI ¼
0:07D2 : ^ Uo fU
ð29Þ
This expression shares the same format as those for mixed estuaries but indicates significant extension of LI in such conditions. 3.4. Observed vs computed values of saline intrusion length, LI A major difficulty in assessing the validity of Eq. (25) is the paucity of accurate observational data. Here, we utilise the observational data provided by Prandle (1981) from six estuaries (8 data sets) summarised in Table 2. This data set provides estimates for intrusion length, LI , riverflow Q and estuarine bathymetry. The values for LI were determined by doubling the axial length measured between locations of 0.25So and 0.75So (So ocean salinity). Values for D and Uo at the centre of the intrusion were estimated from the power series ^ are estimated from Eq. (9). approximations to breadth ðxn Þ and depth ðxm Þ. The values for U ^ at successive positions along the Fig. 9 shows estimates of LI from (25) based on values for Uo ; D and U estuaries listed in Table 2. These values are shown as ratios of the observed lengths. These results are ^ at the locations XC , where the value of LI summarised in Table 3 which also indicates values of Uo , D and U from (25) equals the observed value Lo . It is encouraging to note (Table 3) that the values of XC are in reasonable agreement with the related observational values Xo . Values of Uo range from 0.17 to 0.57 cm s1 with the exception of the St. Lawrence where Uo ¼ 1:4 cm s1 . 3.5. Axial location of saline intrusion Fig. 9 also indicates the landward limits of saline intrusion Xu ¼ ðXC LI =2Þ=L corresponding to successive values of XC . We note that, with the exception of the Hudson and Delaware, the locations of XC , where observed and computed values of LI are equal, correspond or are slightly seawards of maximum values of Xu , i.e., where the landward limit of saline intrusion is a minimum. Adopting this latter result as a criterion to determine the position, xi , where the saline intrusion will be centred, requires in dimensionless terms: o ð x 0:5li Þ ¼ 0: ox
ð30Þ
Substituting li ¼ LI =L, utilising Eqs. (25) and (12) and introducing the shallow water approximation to (9), i.e., 1=2 ^ ^ 2 ¼ fxð2gDÞ U 1:33f
ð31Þ
Table 2 Estuarine parameters from Prandle (1981a): n; m breadth and depth variations ðxn ; xm Þ, k estuarine length, L observed intrusion length, Q river flow, D1 depth, B1 breadth and ^f tidal amplitude at mouth
A Hudson B Potomac C Delaware D Bristol Ch. E Bristol Ch. F Thames G Thames H St. Lawrence
n
m
k (km)
L (km)
D1 (m)
B1 (km)
Q (M3 s1 )
^f (m)
0.7 1.0 2.1 1.7 1.7 2.3 2.3 1.5
0.4 0.4 0.3 1.2 1.2 0.7 0.7 1.9
248 184 214 138 138 95 95 48
99 74 43 55 138 76 38 167
11.6 8.4 4.4 29.3 29.3 12.6 12.6 74
3.7 18 28 20 20 7 7 48
99 112 300 80 480 19 210 8500
0.8 0.7 0.6 4.0 4.0 2.0 2.0 1.5
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Fig. 9. Ratio of computed, LI , to observed, Lo , saline intrusion lengths at varying locations, XC , along estuaries; Lo from Prandle (1981), LI from Eq. (25). Vertical axis (LHS): log10 LI : Lo . Vertical axis (RHS): landward limit of intrusion, Xu ¼ ðXC LI =2Þ=L (dashed line). Horizontal axis: XC =L. X centre of observed Lo . Results for: (1) Hudson, (2) Potomac, (3) Delaware, (4) Bristol Channel, Q ¼ 80 m3 s1 , (5) Bristol Channel, Q ¼ 480 m3 s1 , (6) Thames, Q ¼ 19 m3 s1 , (7) Thames, Q ¼ 210 m3 s1 and (8) St. Lawrence. Note degree of funnelling ðn þ mÞ increases from (1) to (8).
and assuming Q ¼ Uo D2i =a where ‘a’ is the side slope of the triangular cross-section, we obtain: x2i ¼
333Qa 5=2
Do
:
ð32Þ
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Table 3 Observed and computed estuarine parameters for saline intrusion: Xo (fraction of k) centre of observed intrusion, XC centre of cal^ depth, residual and tidal currents at XC culated intrusion when L(Eq. (20)) ¼ Lo , D; Uo ¼ Q/area, U
A Hudson B Potomac C Delaware D Bristol Ch. E Bristol Ch. F Thames G Thames H St. Lawrence
Xo
XC
Uo (cm s1 )
^ (m s1 ) U
D (m)
0.80 0.60 0.80 0.30 0.55 0.60 0.75 0.60
0.70 0.50 1.0 0.40 0.65 0.55 0.80 0.65
0.35 0.20 0.29 0.223 0.28 0.17 0.57 1.4
0.59 0.54 0.47 1.53 1.70 1.01 1.06 0.80
9.7 6.3 4.4 9.5 17.3 8.0 10.2 30.0
Noting that the depth, Di , at xi is Do x0:8 i , we find: 1=2
Uo ¼ Di =333 m s1 :
ð33Þ
This expression for the residual component of river flow at the centre of the intrusion is independent of ^f; f and a. It yields values of Uo of: 0.006 m s1 at D ¼ 4 m, 0.012 m s1 at D ¼ 16 m. Noting that (32) corresponds to li ¼ 2=3xi , these values for Uo will increase by a factor of 2 at the upstream limit and decrease by 40% at the downstream limit. Noting also the inaccuracy inherent in measurements of Uo , we conclude that these estimates of residual velocity associated with river flow in the saline intrusion region are reasonably consistent with the observed values shown in Table 3. If in proceeding from Eqs. (30)–(32), in Section 3.5, we introduce estuarine bathymetry of the from: breadth Bo xn and depth Do xm , we obtain the following alternative form for (32) !1=ð11=4Þmþn1 855Q : ð34Þ xi ¼ 3=2 Do Bo 114 m þ n 1 An especially interesting feature of the results for the axial location of saline intrusion (Eqs. (32) and (34)) and the expression for residual river flow current (Eq. (33)) is their independence of both tidal amplitude and bed friction coefficient. (Although there is an implicit requirement that tidal amplitude is sufficient to maintain partially mixed conditions.) Eqs. (32) and (34) emphasise how the centre of the intrusion adjusts for changes in river flow Q. This ‘axial migration’ can severely complicate the sensitivity of saline intrusion beyond the anticipated direct responses apparent from the expression (25) for the length of intrusion, LI .
4. Estuarine bathymetry 4.1. Estuarine bathymetry consistent with tidal dynamics and saline intrusion Noting the above result that the riverine component of velocity in the saline intrusion region approximates 1 cm s1 , Fig. 10 illustrates typical values of the lengths of saline intrusion obtained from Eq. (25). Moreover, combining this latter result with that for estuarine length, L, from Eq. (12), Fig. 11(a) shows the ratio LI =L. Similarly Fig. 11(b) shows the ratio of tidal excursion EX as a fraction of L for a tracer released at the mouth on the flood tide. These values for EX include compensation for the reduction in tidal velocity with decreasing (upstream depths) but ignore axial variations in tidal phase.
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17
Fig. 10. Saline intrusion length, LI Eq. (25), in km as a function of ðD; ^fÞ values for Uo ¼ 0.01 m s1 , multiply by 0.01/Uo for other values of Uo .
(a)
(b)
Fig. 11. (a) Ratio of saline intrusion length, LI Eq. (25): estuarine length, L Eq. (12), as a function of ðD; ^fÞ LI for Uo ¼ 0:01 m s1 as Fig. 10. (b) Ratio of tidal excursion EX : estuarine length, L Eq. (12), as a function of ðD; ^fÞ.
Introducing the requirements that: (i) LI =L < 1, (ii) EX =L < 1 and ^ 3 \50 m2 s3 . (iii) D=U Fig. 12 indicates the corresponding zone of estuarine bathymetry. This zone is shown to be reasonably consistent with the related distribution of D and ^f values from the 25 UK estuaries listed in Table 1.
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^ 3 > 50 m2 s3 . O indicates observed values of ðD; ^fÞ from 25 UK Fig. 12. Bathymetric zone delineated by: LI =L < 1; EX =L < 1 and D=U estuaries shown in Table 1.
4.2. Estuarine depths as a function of river flow The results for xi and Di (Eqs. (32) and (33)) in Section 3.5 can be used to obtain estimates of the depth Do at the mouth of the estuary. Noting that for li ¼ 2=3xi , for the intrusion to be confined to the estuary, the maximum value for xi ¼ 0:75 and mixing will occur in the seaward half of the estuary. Inserting this value for xi into Eq. (32) we obtain 0:4
Do ¼ 12:8ðQaÞ : Combining this result with Eq. (12), the estuarine length, L, is given by !1=2 Qa ; L ¼ 2980 f ^f
ð35Þ
ð36Þ
where estuarine bathymetry was established under historical conditions with much larger (glacial melt) values of Q, we might expect saline mixing to start landwards of the mouth. Conversely where saline mixing involves an offshore plume, it might be postulated that bathymetric erosion to balance existing river flow is hindered. The results for Uo , Eq. (33) and Do , Eq. (35) are independent of both the friction coefficient, f , and the tidal amplitude, ^f. However, the two expressions for estuarine length, Eqs. (12) and (36), are dependent on the inverse square root of both of these parameters. 4.3. Observed vs computed estuarine bathymetries Examination of the UK observational data set shown in Table 1 indicates that: 0:02 > a > 0:002, Eq. (35) then corresponds to 2:68Q0:4 > Do > 1:07Q0:4 :
ð37Þ
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19
Fig. 13. Depths at the mouth, Do (m), vs river flow, Q (m3 s1 ), Eq. (35). Lines show theoretical values (Eq. (37)). Observed values from Table 1 and Appendix A.
Fig. 13 shows results from UK estuaries listed in Table 1 and from a wider area (Appendix A). For the steeper side slopes, (37) yields values for Do of: 2.7 m for Q ¼ 1 m3 s1 , 6.7 m for Q ¼ 10 m3 s1 , 16.9 m for Q ¼ 100 m3 s1 and 42.4 for Q ¼ 1000 m3 s1 . Comparable figures for the smaller side slopes are depths of 1.1, 2.7, 6.7 and 16.9 m. Fig. 13 shows that the envelope described by (37) encompasses almost all of the observed values of D and Q. The mean discharge of the world’s largest river, the Amazon, is 200,000 m3 s1 , representing 20% of net global freshwater flow. Moreover, the cumulative discharge of the next nine largest rivers amounts to a similar total (Schubel & Hirschberg, 1982). Outside of these 10 largest rivers, Q 10 m will often involve freshwater plumes extending seawards.
5. Sediment regime Simulation of the processes of sediment erosion, transport and deposition is much less accurate than for the preceding dynamical and mixing processes. Uncertainties abound in determining: bed stress, associated erosion rates, turbulent suspension levels and the complications of ‘settling’ and ‘capture’ at the bed. Thus, the following attempt to explore likely sedimentation rates in estuaries is of a much more general character. 5.1. Suspended sediment time-series in tidal estuaries Fig. 14 shows characteristic model-generated spring-neap cycles of suspended particulate matter, SPM, associated with localised resuspension in a tidally dominated estuary (Prandle, 1997). The degree of vertical mixing is determined by: (i) D2 =Kz , i.e., time taken for vertical mixing by diffusion and (ii) D=Ws , i.e., time taken for settling by fall velocity Ws . The associated times in suspension then determine the relative predominance of the quarter-diurnal, M4 , and fortnightly, MSf, constituents. Prandle (1997) shows that for: Kz < 0:1Ws D, particles will scarcely reach the surface and with a short half-life in suspension of OðD=Ws Þ,
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Fig. 14. Model simulations of SPM over a spring-neap tidal cycle. C1; C2; . . . ; C10 indicate concentrations at fractional heights above the bed of Z 1=2 ¼ 0:15 to 0.95 (Prandle, 1997). Top Kz =DWs ¼ 0:1. Bottom Kz =DWs ¼ 10Kz . Eddy diffusivity, Ws fall velocity.
the quarter-diurnal constituent will predominate. Conversely for Kz > 10Ws D, particles are evenly distributed through the water column and the half-life of OðD2 =Kz Þ amplifies the MSf constituent relative to M4 . Prandle, Lane, and Wolf (2001) showed that the relative concentration amplitude, Cx , associated with a unit of erosion at a tidal frequency x can be approximated by Cx ¼ ða2 þ x2 Þ
1=2
;
ð38Þ
where a1 is the half-life in suspension given by the lesser of: a ¼ 0:7Ws2 =Kz
ð39Þ
a ¼ 0:1Kz =D2
ð40Þ
or
with (39) predominating for coarse particles with large settling velocities and (40) for finer particles. ^ D, Figs. 15(a) and (b) show estimates of a1 , i.e., half-life, of sediments in Approximating Kz ¼ f U suspension as functions of ^f and D for: (a) Ws ¼ 0:005 m s1 and (b) Ws ¼ 0:0005 m s1 . The important result is that for the finer sediment, the half-life in suspension is almost always of 0(6 h) or greater. This implies that the estuarine sedimentary regime associated with the influx of marine sediments approximates that of a conservative tracer such as salt.
D. Prandle / Progress in Oceanography 61 (2004) 1–26
(a)
21
(b)
Fig. 15. Half-lives of SPM in suspension, Eqs. (39) and (40), as a function of ðD; ^fÞ. Left: Ws ¼ 0:005 m s1 and right: Ws ¼ 0:0005 m s1 .
5.2. Mean concentrations of SPM Erosion formulae generally take the form ERðtÞ ¼ qcf ðU ðtÞ Uc ÞP :
ð41Þ
For a tidally dominated regime, ERðtÞ is relatively insensitive to either the power P (over the commonly ^ ). Hence for simplicity we adopt encountered range 2–5) or the critical erosion threshold Uc (for Uc < 0:5U P ¼ 2;
c ¼ 0:0001 m1 s
and
Uc ¼ 0:
The corresponding time- and depth-averaged concentrations are given by (Prandle et al., 2001) , ^2 U C ¼ cqf Da: 2
ð42Þ
ð43Þ
^ . This figure shows that the Fig. 16 shows corresponding mean concentrations as a function of Ws and U mean concentrations of fine sediments overwhelms that of coarse sediments. Moreover, for Ws < 104 m s1 this concentration is independent of the actual value of Ws . These latter results enable estimates to be made of maximum depth- and time-averaged concentrations in estuaries where a plentiful supply of such fine sediments exists. Fig. 17 shows such concentrations as functions of ^f and D. For these fine sediments, from ^ . West and Sangodoyin (1991) showed (43) and (40), the mean concentration is directly proportional to U ^ similar results from observations in three estuaries with C 0:04U (C in mg l1 and U in m s1 ). The range of values shown is consistent with commonly observed estuarine concentrations in muddy coastal regions. 5.3. Estimates of in-fill times In UK estuaries, measurements of present-day fluvial SPM loads indicate minimal impact on overall estuarine bathymetry. Likewise, sedimentary cores indicate a corresponding history of overwhelming
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^ Fig. 16. Mean SPM concentrations in mg l1 , Eq. (43), as a function of fall velocity Ws and tidal current amplitude U.
Fig. 17. Time- and depth-averaged mean SPM concentrations in mg l1 (for fine sediment) as a function of ðD; ^fÞ.
influence of marine sediments. Hence, we estimate minimum in-fill times for estuaries assuming a marine source of fine sediment with mean concentrations given by Fig. 17 and with a rate of supply determined from the flushing time FT for salt. For the conditions of ðoS=oxÞ ¼ Sx (constant) assumed in Section 3, and Eq. (25) for LI , we can estimate FT ¼
0:5ðLI =2Þ 0:0013D2 ¼ ; ^ U2 Uo fU o
ð44Þ
i.e., the time to replace 0.5 of the salinity content by freshwater over the intrusion length LI . Fig. 18 indicates values of FT as functions of ^f and D. These values are consistent with commonly encountered estimates (see Balls, 1994; Dyer, 1997). Hence, assuming all sediment is deposited and ignoring changes in the sedimentation rate as bathymetry evolves, in-fill times are given by I ¼ qFT =0:69C:
ð45Þ
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23
Fig. 18. Flushing time, FT , (days) Eq. (44) as a function of ðD; ^fÞ values of FT are for Uo ¼ 0:01 m s1 , multiply by 0.01/Uo for other values of Uo .
Fig. 19 indicates these minimum in-fill times, ranging from around 25 years in shallow estuaries with high tidal ranges up to thousands of years in deep estuaries with small tides. Hutchinson and Prandle (1994) estimated capture rates of available marine sediments in a ‘rapidly’ accreting estuary of 3%, suggesting these minimum in-fill times might be increased by at least an order of magnitude.
Fig. 19. Minimum in-fill times (years), Eq. (45), as a function of ðD; ^fÞ. Values are for Uo ¼ 0:01 m s1 , multiply by 0.01/Uo for other values of Uo . (Assumes 100% sedimentation of marine source.)
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6. Summary and conclusions For the case of partially mixed, tidally dominated, funnel-shaped estuaries, new theories have been developed and translated into characteristic responses for: tidal propagation, saline intrusion and sedimentation. Moreover, by inter-relating these responses new expressions are derived linking bathymetry to dynamics. 6.1. Tidal dynamics – estuarine length The dynamics of tidal propagation are examined for the particular case of a synchronous (see Section 2) estuary with a triangular section and a predominant ðM2 Þ tidal constituent. Incorporating these approximations into the cross-sectionally averaged governing equations, produces, for prescribed values of ^f and D, an estimate of bed slope. Axial integration of this slope enables both the shape and length of the estuary to be determined. These lengths are shown to be consistent with observed data. 6.2. Salinity intrusion, length and location The related dynamics of saline intrusion were examined for the case of a vertically and temporally constant axial salinity gradient. An expression for the length of saline intrusion was derived. Application of this derivation highlighted the complications arising from the uncertainty of the axial location of the intrusion zone. In examining the fit between theory and observations, a condition based on minimum landward intrusion of salinity was derived. Intrusions respond to changes in tidal range and river flows by adjustment both to length and by axial migration. These twin adjustments explain some of the difficulties in reconciling observations and theory. 6.3. Bathymetry as a function of river flow Utilising this derived criterion of minimum landward intrusion and incorporating the earlier derivations for both estuarine length and saline intrusion length, an expression for Uo , the river flow component of residual velocity at the centre of the intrusion, was derived in terms of the depth, Di . The expression indicates values of Uo ranging from 0.006 to 0.012 m s1 for Di from 4 to 16 m. These estimates for Uo are independent of: tidal range, bed friction coefficient and steepness of side slopes Moreover, these values for Uo accord well with observed values which are invariably around 1 cm s1 . These latter analyses were extended further to provide an expression for estuarine depth, Do , at the estuarine mouth in terms of the river flow and side slope of the triangular cross section. By incorporating typical observed ranges for this side-slope, an estimate for Do is obtained directly in terms of Q (again independent of either tidal range or bed friction coefficient). Comparison of the associated envelope of D as a function of Q again accords sensibly with observed values. Thus, this study provides a complete description of estuarine bathymetry (i.e., depth, length and axial shape) in terms of Q, ^f and f , i.e., the obvious ‘natural’ boundary conditions. 6.4. Sedimentation Utilising existing theories on the nature of locally resuspended sediment regimes in tidally dominated regimes, it was shown that significant estuarine siltation is most likely to occur via the entrainment of fine marine sediments. Moreover, the transport of such sediments in estuaries (of the kind considered) approximates that of a conservative tracer such as salt. Thence by combining theoretical estimates of time-
D. Prandle / Progress in Oceanography 61 (2004) 1–26
25
and depth-averaged sediment concentrations with estimate of salinity flushing times (based on LI and Uo ), maximum rates of estuarine in-filling were estimated. Factoring these results from observed rates of sediment capture, it was concluded that bathymetric changes over decadal time scales are likely to be minor for all but the shortest and shallowest of estuaries.
6.5. Estuarine characteristics Frameworks of characteristic estuarine values, as functions of depth, D, and tidal amplitude, ^f were illustrated for the following parameters: ^ indicating a dependency on ^f1=2 D1=4 f 1=2 in shallow water (see (b)). (a) tidal current amplitude U (b) ratio of friction: inertia terms, shown to be proportional to 10 ^f=D, the latter providing a demarcation of ‘shallow water’ in terms of friction predominating. (c) estuarine length, L, proportional to D5=4 =ðf 1=2^f1=2 Þ with depth increasing axially at a rate x0:8 , i.e., midrange in shape and length in the Prandle and Rahman (1980) generalised response theory. (d) stratification, based on both the Simpson–Hunter criteria D=U 3 or on the time for vertical mixing by diffusion D2 =Kz , the latter giving an indication of intra-tidal stratification. ^ based on four independent approaches. (e) salinity intrusion, LI , proportional to D2 =fUo U (f) bathymetry, definition of zone bounded by LI =L < 1; EX =L < 1 (EX tidal excursion) and D=U 3 < 55 m2 s3 . time- and depth-averaged values for fine sediments, proportional to U ^. (g) mean sediment concentration, C (h) flushing time, FT proportional to LI =Uo . (i) In-fill time I, proportional to FT = C.
Acknowledgements This study was supported by the UK Natural Environment Research Council, DEFRA and the Environment Agency through the EstProc Contract FD1905/TR1.
Appendix A. Estuarine depths (at mouth and mean river flows) See Table 4.
Table 4 Estuary
Depth (m)
Flow (m3 s1 )
Reference
Delaware Elbe Ems Hudson Potomac Louisa Creek Seine Thames Loire Weser
18 16 10 15 10 3 10 9.7 12 13.5
552 785 100 550 112 5 450 52 800 322
Galperin and Mellor (1990) Schroder and Siedler (1989) de Jonge (1992) Geyer, Signell, and Kineke (1998) Prandle (1981) Lessa (1996) Brenon and Hir (1988) Rossiter and Lennon (1965) Normant, Peltier, and Teisson (1988) Grabemann and Krause (1988)
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