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Sea-level fluctuations in the fleet, an English tidal lagoon
Estuarine,
Coastal and Shelf Science
Sea-Level Fluctuations English Tidal Lagoon
I. S. Robinson, Department SOg gNH,
L. Warren
Oceanography, U.K. of
Received ~4 February
16, 651-668
(I&)
in the Fleet, an
and J. F. Longbottom
University
of
Southampton,
1982 and in revised form
Southampton
8 September 1982
Keywords: lagoons; tidal heights, shallow water equations; models ; nonlinear equations ; England coast
tidal friction;
Tidal elevation data are presented for places along the length of the Fleet, which is a tidal lagoon behind Chesil Beach on the south coast of England. Harmonic analysis of the data is not able to represent the observations adequately, particularly at the inner end of the lagoon. However, careful inspection of the data shows that the tidal regime is capable of being understood in terms of the non-linear propagation of long waves in very shallow water. Distortion of the tidal wave by unequal progression speeds of high and low water, and the set-up of mean level by frictional effects, are shown to be the important physical mechanisms controlling the observed water level fluctuations. A one-dimensional numerical model which incorporates these processes is able to reproduce the observations satisfactorily. Whilst the model predicts strong effects of wind stress, the meteorological influences in the observed data appear to be largely due to external surges in the English Channel which propagate into the lagoon through its entrance.
Introduction The Fleet is a shallow coastal lagoon, approximately 12; kms in length, trapped behind Chesil Beach north west of Portland Bill on the south coast of England. Figure I shows the location of the Fleet, and indicates its topography and bathymetry, which are described by Whittaker (1980) and Ladle (1981). It is essentially a linear shape, ranging in width between IOO m and I km at its widest. Its origin is closely related to the formation of Chesil Beach, which defines its south western boundary along its entire length. The genesis of Chesil Beach is discussed in detail by Carr and Blackley (1974) and it appears that the Fleet is not a drowned river valley, but is the land below sea level enclosed behind the beach as the sea level rose during the Flandrian transgression, moving the beach North Eastwards. There is no opening to West Bay through the beach, but the Fleet is connected to the English Channel at its south eastern end through Portland Harbour. It is therefore tidal and saline. A few streams drain into it from a small catchment area of about 50 km2, although seepage through the ground into the Fleet may drain from a wider area. It is also very shallow, varying in depth from 3 to 5 m in the first 2 km between the tidal entrance at Smallmouth and The Narrows. Here it is marine in character, with strong tidal exchange to the sea. It then widens out for 2.5 km into the area known as Littlesea, which has a depth of o to 1-2 m below chart datum, intersected by narrow channels up to 4 m deep. The tide is 0272-7714/83/060651+
18 $03.00/0
0 1983 Academic
Press
Inc.
(London)
Limited
I. S. Robinson, L. Warren
652
M’J. F. Longbottom
0
Zkm
0-
1 mlk?
ABBOTSBURY
MORKHAM’S RODDEN
LAKE HIVE
LANGTON
POINT HIVE
POINT
HERBURY
English
0 0 m
Channel
-Ift 10 -4ft I1 2mi 0 D to c-lftl0 3ml > -4ft I1 2ml
SMALLMOUTH \
I
Figure
I. Map
of the Fleet.
strongly attenuated in this region and penetrates only weakly into the remaining 8 km of the lagoon, known as West Fleet which has a depth between o and 1-2 m. The weak tidal flushing and the small freshwater inflow are delicately balanced to produce a brackish water regime which is strongly tide-dependent. The circulation and mixing are studied elsewhere (Robinson 1983), but the purpose of this paper is to examine the sealevel fluctuations in detail. There are several reasons why the tidal fluctuations in this relatively small and unfrequented lagoon should be worth examining. Firstly, it appears to be the cnly lagoon of its type in the U.K. and some of the species of flora and fauna found there have not been recorded elsewhere in the country. This may be due in part to the sheltered brackish water habitat afforded by the Fleet, where summer temperatures in the shallow West Fleet rise to 30 “C on a sunny afternoon and where wave action on the bed and shore is minimized with such a short fretch. It may also be related to the interesting tidal regime which in the West Fleet tends to be more of a fortnightly than a semidiurnal period. Marine biologists investigating the ecology of the lagoon require an understanding of the basic tidal pattern. Secondly, although the tides are quite small compared with many U.K. estuaries, the penetration of salt to Abbotsbury indicates the importance of tidally driven diffusion. Sedimentation must also be controlled by tidal flows. Given the environmental pressures which are threatening the Fleet, with alterations to the mouth at Portland, and other major civil engineering developments proposed at sites along the shore of the Fleet, it is important that the tidal dynamics be understood so that any changes in tidal regime consequent upon topographic alterations may be predicted. Moreover, any prediction of the capacity of the Fleet to cope with natural or imposed pollution requires the estimation of the circulation and flushing, which will be based in part on a knowledge of the tides (see the companion paper, Robinson, 1983).
Sea-level fluctuations
in the Fleet
653
Finally the tidal regime is interesting in its own right. Being so shallow, it exhibits in its short length many of the features associated with non-linear tidal propagation in much larger estuaries. The local population who use the Fleet find great difficulty in predicting the tidal state, and it was concluded by Bird (1972) that sea-level fluctuations were dominated by wind action and by seepage through Chesil Beach, although no firm observational evidence was presented to substantiate this hypothesis. The aim of this paper is therefore to determine more confidently the mechanisms which control sea-level fluctuations in the Fleet, by analysing time-series of tidal records obtained in 1967-1968. It will be shown that the tides are by no means as irregular as previous workers have concluded from occasional observations. Indeed it will be demonstrated that it is possible to reproduce much of the tidal behaviour in a numerical model based on straightforward non-linear long-wave propagation theory, which can therefore provide the basis for further study of tidal mixing and flushing of the Fleet.
The tidal data During the summer of 1967 and winter of 1967/1968, a hydrographic survey of the Fleet was performed by Messrs. G. Wimpey and Co. Ltd. on behalf of the U.K. Central Electricity Generating Board. About ten years after the survey, the hydrographic, salinity, sediment, temperature and tidal data collected during the survey was made available by the CEGB for scientific analysis, and it is the tidal records obtained during that survey which provide the observational evidence for the understanding of the Fleet tides presented here. Tide gauges were sited at six locations along the north shore of the Fleet, indicated in Figure I. These were Monro type automatic float gauges, levelled into Ordnance Datum (Newlyn) from convenient bench marks. The tide gauge data were made available in the form of half hourly heights above ordnance datum (Newlyn). Figure 2 shows the variation of the elevation with time for each cf the tide gauges, for the period between 18 October 1967 and 28 January 1968. The only record without gaps during the 36 months was that at the Royal Engineer’s Bridging Hard. Chickerell Hive and Morkham’s Lake records have occasional gaps whilst the other stations have less complete coverage, although sufficient information is available to indicate the overall tidal distribution throughout the lagoon. The tide at Smallmouth should represent the external forcing, but as a check on possible mouth effects the tide recorded at Portland Naval Dockyard has been plotted in Figure 2. Unfortunately the permanent tide gauge at Portland was malfunctioning during the period of the survey, and the curve shown is reconstructed from the tidal heights measured at the time of predicted high and low water at Portland, using the tidal curve presented in the Admiralty Tide Tables. Some observations of tidal currents were also made at certain sites during the survey, at approximately half-hourly intervals over a 12; hour period.
Short period tidal oscillations of water level The first comment which may be made on Figure z is that although the tide records, particularly those at the inner western end of the Fleet, are not typical of coastal tide gauges, they nonetheless exhibit a regularity and a correspondence to the external English Channel tides which confirms that the water-level fluctuations in the Fleet are fundamentally driven by the astronomical tides, rather than local wind conditions. However, the tides in the West Fleet are intermittent and their regular periodicity, apparent in the time series, may easily be
654
I. S. Robinson, L. Warren &sJ. F. Longbottom
Sea-level jluctuations
in the Fleet
655
concealed from the local observer by the long-period fluctuations which increase with distance from the mouth. These will be investigated further in a later section, but first the propagation of the semidiurnal tide down the lagoon will be examined in detail. Figures 3 and 4 show the tidal wave over 24 hours for two days chosen when records were available at most of the stations, one representative of spring tides (Figure 3) and the other of neap tides (Figure 4). It is convenient to interpret these figures in terms of three effects which occur as an observer travels inward from the lagoon mouth; the progressive phase lag of high and low water, an attenuation of tidal amplitude, and a distortion of the tidal profile. Each can be accounted for broadly in terms of non-linear long-wave propagation theory. The phase lag which increases towards the head of the lagoon indicates the progressive character of the tidal wave. Short estuaries normally have such a small tidal phase change along their length that they can be considered to be standing waves (e.g. Southampton Water). However, because the Fleet is so shallow, the phase propagation speed for long
Abbotsbury I-
2 -
Moonfleet
I : E 3 ‘0 D ;
O.D. I 2-
Chickerell
I OD.,
/
\
/
O.D.
O.D. -I -2 III 0
I 4
IllI
I 8
I
I
I I2
Time Figure
3. Typical
spring
tide
record
I
I
I
I 16
(h)
17 December
1967.
I
I
1
I 20
I
I
I
I. S. Robinson, L. Warren
656
NJ.
F. Longbottom
Abbotsbury -
I0.D Morkhom’s
Loke
IL
:
2< E ’ .P 2 O.D.
Chickerell
2Bridging
2
Hord
Smallmouth
I O.D. -I
. 1 0
I
I
I
I 4
I
I
I
I 6
I
I
I
I 12 Time
Figure
4. Typical
neap tide record
I
I
I
I 16
I
I
I
I 20
I
I
I
( h1
25 November
1967.
waves is sufficiently slow for the phase variations in a few kilometres to be appreciable. Moreover, the frictional dissipation is strong in such shallow water, so that little energy is likely to be reflected at the head of the lagoon and there is consequently virtually no standing component to the tidal wave. In both these respects the Fleet is equivalent to much longer and deeper estuaries which would have the same non dimensional response to tidal forcing as illustrated by Robinson and Perry (1980). Table I indicates the time lag of high and low water behind high and low water times at Smallmouth, for the representative spring and neap tidal cycles. Given that the average depth between Smallmouth and Bridging Hard is about z m, the propagation speed of small amplitude waves (g*ha) is expected to be 4.5 ms-’ which suggests a propagation time over this section of only 7.5 min. It appears that non-linear finite amplitude, and frictional effects are causing the slowing of the phase propagation. Even more marked is the apparently different propagation speed of high and low water. However, because of the changed profile of low water as it propagates along the Fleet, it is not meaningful to identify the lowest elevation
With
point
with
a single
phase
such a large amplitude/depth
predicts
a phase speed ofg% i
(I
point
of the semidiurnal
ratio, non-linear
cycle
long-wave
travelling
theory
+i i) where a is the local tidal elevation.
along
the estuary.
(e.g. Lamb,
1932)
For a spring ampli-
Sea-level fluctuations
in the Fleet
I. Phase lag of the tide
TABLE
relative Spring
(17
Location Bridging Chickerell Moonfleet Morkham’s Abbotsbury
High
Hard Hive
Lake
657
Dec.
water
to Smallmouth tide
Neap tide (25 Nov. 1967)
1967)
_____ Low
water
High
water
30 min
3 h 40 mina
15 min
I h ao min
4 h 5 min”
I h 5 min
zh 4h 5hr5min
5h 7h 8h
2
h
20
-
min
Low
water
3 h 30 mina (15 min) 4hr5mina (1 h) -
a This is the time lag of the lowest point of the changed profile behind the first low at Smallmouth. The bracketed figure is the time lag of the first low when it is lower than the second.
tude of 0.8 m and a neap amplitude of 0.4 m this would indicate that spring tide travel times of high and low water between Smallmouth and Bridging Hard would respectively be 4.7 min and 184 min and at neap tides 5.8 min and 10.7 min. Although the profile changes do not allow a direct comparison with the observed data to be made, it is clear that the observed travel times are considerably larger than simple long-wave theory predicts. The discrepancy is probably due to the non-uniform cross-section of the channel. The wave speed has been estimated roughly from the mean depth of the main channel along which the wave propagates. However, there are areas of tidal flats exposed at low water, and other large areas which are too shallow to permit propagation of the tidal wave, but which have a capacity to be filled by the incoming tidal wave, thus impeding its progression. This physical process may be simply expressed analytically by adopting a one-dimensional continuity equation of the form
au ac b, - = -6, h - where
at
ax
the depth h and breadth
b, of the deep channel
through
which
the
wave propagates, and the breadth b, of the actual water surface, are assumed to be independent of x and t. Hence the linear solution for long waves in this case indicates a phase speed of (g&/b,) I. Given the bathymetry of this section of the Fleet b,/b, may be estimated to be 0.1, which would increase the travel time from Smallmouth to Bridging Hard to about 24 min which is much closer to that which is observed. Beyond Bridging Hard, where the Fleet opens out into Littlesea this effect is even more pronounced and is the likely cause of the long time taken for the tide to penetrate to Chickerell Hive Point. Furthermore, because of the extensive drying flats, the cross sectional area (hb,) available for high-water propagation is very much greater than that for low tide, so that b,/b, is much less at low water, contributing to the significant difference between the travel time of high-water and low-water spring tides. The change in shape of the tidal curve as it progresses along the Fleet is related to the different travel speeds of high and low water. As high water catches up the preceding low, the tidal rise is rapid and its fall much slower, producing the saw-tooth shape characteristic of much larger estuaries such as the Severn where a bore is eventually formed at Spring tides. Associated with the elevation asymmetry is a short rapid flood and a long slow ebb, which may be seen in the model results of Figure 9 which are confirmed by current meter measurements. It appears that the flood tide sweeps relatively easily over the shallow Littlesea into the West Fleet, but the ebb is impeded by the shallows and only slow drainage can occur through the few deeper channels. This has the further consequence of ‘setting-up’ mean sea level into
658
I. S. Robinson, L. Warren
My.
F. Longbottom
the West Fleet aswill be discussedlater. The flow regime in the East Fleet provides conditions which will tend to trap sedimentin the Fleet with high energy inflow and lower energy outflow. It is alsointeresting to note that the double low water, which characterizesthe tides in the English Channel around Weymouth, is much reduced at Bridging Hard and has virtually disappearedby Chickerell. Presumably both the friction and the capacitative effect of the shallowstend to smooth out the higher frequency sea-levelfluctuations more rapidly. This too is a non-linear effect, being more pronounced at spring than at neap tides, when double low watersare still detectableat Chickerell. The attenuation of the tidal amplitude is alsovery marked in soshort an estuary. The most significant reduction in amplitude is between Chickerell and Moonfleet, from which it is inferred that the wide, shallow region of Littlesea and the narrow shallowsby Moonfleet dissipatemuch of the tidal energy, and act to control the tidal penetration into the West Fleet where the attenuation of such a small signal is very much less,as evidenced by comparison of the Abbotsbury and Morkham’s Lake tidal curves. Indeed, at neap tides the tide is virtually non-existent in the West Fleet. Only at spring tide doesit penetrate in recognizable form to Abbotsbury. The reasonfor this marked difference betweenspring and neap behaviour of the West Fleet is to be found in the change in mean sealevel. At spring tide, the strong dissipation of the progressivelong-wave energy results in a set up of mean sealevel which raisesthe water level in West Fleet sufficiently to permit the tide to penetrate to Abbotsbury. At neap tide the mean sealevel falls to such an extent that the tide gauge at Moonfleet dried out, with the resulting gaps in the data shown in Figure 2. The water remainsin West Fleet, which is deeperthan the narrow stretch adjacentto Moonfleet, and for several days is virtually isolated from the tide. This must clearly have consequencesfor the circulation and flushing of the West Fleet. Harmonic analysis of the full-length records confirms these conclusions drawn from inspecting sampledays of the record. Table z gives the harmonic constantsof the major tidal constituents obtained from a least squares harmonic analysis algorithm, based on the computer algorithm TIRA usedby the Bidston Laboratory of the Institute of Oceanographic Sciences,U.K. The analysismethod is able to copewith gapsin the time series,so that there was effectively at least 30 days record of half-hourly values for all the stations. The total TABLE 2. Harmonic phase angle
Bridging Hard
Smallmouth H(ft) Mean Mm Msf MI I(1 M2 S2 M4 MO 3MS,
S.D.(obs.) S.D.(res.) Record length
constants
Go
H(ft)
0'54 0.28 0.16 0'30
46 38 338 IO3
1.81 0.92 0.51 0.19 o-013 I ‘292
190 241 2 206
0’430 42 days in five blocks
0'21
G”s
0.61
-49
0’32 0’12
0.23 I’32 0.61 0.31 0’10
tidal
Chickerell Hive -_____H(ft) Go,
0’49
0’21
of major
8 29 23 20 22
6 20
0.043 43 I-199 0.472 I oo days
0.16 0’45 0’10
H(ft)
-2
Go,
-87
0.18
52 56
0.19
45 69 59
0’31 0.046 0.085 0.23 0.15 0.084
101 76 88 83 131 -37 46
0'022
0.970 0.482
50 and z8 days
I2
0-014
0’472 0’300 77 days in
four
the amplitude
Markham’s Lake
Moonfleet
0'10
0’9.5 0’43
0'021
His
1.24
-45 60 44
0'02
coast.
blocks
H(ft) 0’93 0’11 0.40
Abbotsbury Gas
0.047 0.116 0.088 0.031 0’009 0’004 0’401 0.259 75 days
H(ft) 0.88 0.19
-115
0'055
and Gso the
21
176 141 I53 I47 217 -
0.28 0.016 0.037
0’010 0’024 0’004 0’004 0’003
Gas
-154 40 -31
I42 97 -
0’353 0.286 41 days in two blocks
Sea-level fluctuations
in the Fleet
659
length of data, minus any gaps, on which each analysis is based is shown in Table 2. With this length of record, 36 constituents could be resolved, as well as eight further constituents whose relation to a major constituent was determined from the equilibrium tide. The amplitude H in Table 2 is in feet, and the phase angle G” is relative to the equilibrium tidal phase of the constituent in the case of Smallmouth, and for all the other stations G”s is given relative to the phase at Smallmouth, to demonstrate the progression of the tide along the Fleet. The analysis for Smallmouth, Bridging Hard and Chickerell Hive, is quite satisfactory, as measured by a comparison (see Table 2) of the standard deviations of the tidal record, and of the residual signal which remains after removal of the signal accounted for by the 44 harmonic constituents. The record becomes noisier away from the mouth, but is basically of a tidal harmonic form. The amplitude of the diurnal, semidiurnal and higher harmonic components are seen to decrease and the phase lag increase away from the mouth. The semidiurnal phase lag may be interpreted in terms of travel times along the Fleet, and compared with those obtained from sample records, in Table I (e.g. iU2 phases indicate 40 min to Bridging Hard, I h 48 min to Chickerell and 3 h to Moonfleet). However, by Moonfleet the residuals are of comparable magnitude to the original record. In this case the tidal signal is intermittent over the spring neap cycle, and in the case of Morkham’s Lake and Abbotsbury, when it does penetrate it is half-wave rectified. Both these effects preclude a satisfactory harmonic analysis, although the phase lag of M2 is probably of some significance as far as Moonfleet. The harmonic constants for Abbotsbury and Morkham’s Lake are in fact virtually meaningless. Inspection of the list of constituents derived from the analysis shows that those selected for inclusion in Table 2 as being the dominant lines of their species at Smallmouth are often smaller than nearby constituents. It is concluded that apart from Smallmouth and Bridging Hard, the predicton of tides in the Fleet using the harmonic method is unsatisfactory, because of the gross distorting effect of the non-linear propagation processes. The most satisfactory means of approximate tidal prediction is to use Figures 3 and 4 in conjunction with the tidal predictions for Portland. Other apparently non-tidal, irregular fluctuations with dominant frequency less than a day are seen in the records, but only in isolated instances, e.g. 31 October-r November 1967, 4 December 1967. Although there are no obvious correlations with the wind record at Portland (Figure 6) these fluctuations are most likely to be related to wind effects, but very often these anomalies occur on all the tide records, including the mouth, suggesting that they are in fact storm surge elevation peaks in the English Channel whose presence is transmitted like the tide into the Fleet. Spectral analysis of the individual records and the residuals after harmonic analysis, revealed no important peaks of energy at non-tidal frequencies, and cross-spectral analysis between pairs of stations gave no indication of coherence between the residuals at different locations. It is concluded that seiching due to direct wind action on the waters of the Fleet is not an important feature of the water-level fluctuations, contrary to what might have been expected in such an enclosed body of water. Indeed, the winds do not appear to have much direct influence at all, despite the subjective conclusions of local observers (see e.g. Bird, 1972).
Long-period fluctuations of sea level The sea level records were low-pass filtered by taking a simple 25 h running mean, and the results are plotted in Figure 5. The most striking feature to be revealed is the fortnightly fluctuation of water level, related to the spring-neap cycle, which is strongest towards the western end. The same feature is revealed by the harmonic analysis, where Msf is seen to
660
I. S. Robinson, L. Warren
3 2 I
&“J. F. Longbottom
,
1i +
0.0.
Morkham’s
3
Lake
2 I 0.0. Moonfleet
3 z ; n‘;
2 1 0.0.
f
3
4 .-6 6 1 0) 6
2
EI ; ,” ,”
Chickerell
I 0.0. 3
Bridging
Hard
2 I 1
0.0. -I 2
Smallmouth
I 0.0. -I
Figure
i 2 _
Portland
5. zs-hour
running
mean
of the time
series in Figure
2.
increase between Smallmouth and Chickerell Hive, and remains unattenuated as far as Abbotsbury. In fact at Moonfleet and beyond it becomes the dominant harmonic and at Abbotsbury is an order of magnitude greater than the other constituents as resolved by the least squares analysis. Table 2 also shows an increase in the mean sea level as far as Moonfleet. (Beyond there, the estimation of the mean is unreliable because of the inadequacy of the harmonic method for analysing the intermittent and half-wave rectified record). The set-up in mean sea level must be related to the bottom friction acting asymmetrically over the tidal cycle. The average depth during the inflowing tide is greater than during the ebb. Frictional effects are less on the flood, therefore, and time averaging of the frictional forces leads to a strong mean force acting inwards which is balanced by a tidal mean surface slope. During neap tides the velocities and thus the friction forces are weaker than at springs, and hence the mean set-up fluctuates over the spring-neap cycle, as reflected in the large amplitude Msf. The same effect is present in many estuaries, and has been discussed in detail for the St Lawrence by LeBlond (1979).
Sea-level
fluctuations
661
in the Fleet
What is striking about the Fleet is that the effect should occur over a length scale two orders of magnitude smaller than the St Lawrence. The phase of Msf indicated by Table 2 shows a delay of about a day at Morkham’s Lake, and nearly two at Abbotsbury, for the spring-neap modulation of mean level to penetrate to the end of the Fleet. Figure 5 also shows up other long period fluctuations of mean sea level. It is significant that these can usually be identified at all the tide gauges, including Portland, unlike the fortnightly oscillation which only becomes obvious at Chickerell and beyond. It is concluded that these are lower frequency surges of meteorological origin, generated in the English Channel rather than the Fleet itself. A weak correlation can be discerned between these surges and strong winds in the record of Figure 6.
Across SW4
the Fleei NE
NW4
SE ,
I Ott
Nov
0I
I Oec
1967
1968
Jon
Figure 6. Wind speed recorded daily at Portland. Because the long-period fluctuations are able to penetrate to Abbotsbury with little attenuation, unlike the higher frequency semidiurnal tides, they have considerable impact on the water level regime in the West Fleet. The fortnightly tidally driven fluctuations are regular and predictable, but to be able to estimate the non-tidal fluctuations, an understanding of the English Channel storm surge dynamics is required rather than a simple mechanism of local wind set-up in the Fleet. This is perhaps what has confused so many of the local observers in their attempts to comprehend the causes of sea level variations in the West Fleet.
A one-dimensional, non-linear numerical model of tidal propagation in the Fleet If, as is suggested earlier, the tidal regime can be explained qualitatively in terms of non-linear long-wave propagation, then it should be possible to demonstrate quantitative agreement with theory using a numerical model.
662
Such a model
I. S. Robinson, L. Warren
has been constructed,
&J.
F. Longbottom
based on the one-dimensional
equation
of motion: (1)
and continuity: h ;
+ ;
[b,u(h+
a]
= o,
where 5 is the tidal elevation above the mean level at the mouth, h is the water depth below this datum, x is the distance along the lagoon measured from the mouth, u is the cross sectionally averaged velocity, 6, is the breadth of the channel at the sea surface and b, the mean breadth such that b, (h+ 5) is the local instantaneous cross-sectional area. k is a coefficient of quadratic friction, normally taken in shallow sea dynamics to be 0~002-0~0025. The Fleet is represented as 12+ sections each of length I km. Even numbered grid points from 2 to 26 correspond to section boundaries. The mouth is taken at grid point I, the centre of the first section, and the head at point 26. If 5 is defined at odd numbered and u at even numbered grid points, equations (I) and (2) can be represented in finite difference form: nun -zz nt
uL-u2n-~
-p+1-
2k 44
L-1) ax
4ax
- zhn+
L+l+
L-1
(3)
and $$=&{u,i,b+,
(ha+, hn-I
+
+ 5n:,+in)-un-lbz.n-1
Inf5n-2 2
)
(4)
)>
where ax is the distance between xn and xns2, i.e. I km, and At is chosen to be IO s, well within the stability limits of the model. By means of equations (3) and (4), u and &’ can be predicted for time tfnt given the value of variables at t. The boundary condition corresponding to the head at Abbotsbury is uaa = o. Evaluation of n {a5 by use of equation (4) assumes that [a7 = [as. The mouth boundary condition is determined by specifying [r. For spring tide simulation, a typical spring cycle at Smallmouth obtained from the records of Figure 2 is used to define [i, by use of the half-hourly observed values with linear interpolation for the intermediate time steps. For neap tide simulation, exactly half the spring tide amplitude at Smallmouth provides a suitable boundary condition. Initially, the elevation throughout the Fleet is zero, and the model is run with the same mouth forcing for several cycles until repeatability of the response is obtained. In practice this usually occurs after two cycles. Fitting of the model simulation to the observed distribution of tidal elevation along the Fleet can be achieved by using R and b, as tunable parameters. h and 6, have been determined as objectively as possible by means of the hydrographic information from the CEGB survey, assuming that in the Littlesea area these should correspond with the deeper channels through which the water flows along the Fleet, rather than the wide shallow areas which provide storage capacity. The capacitance effect is parameterized by assigning b,. It is found that variation of K does not vary the tidal regime very much, whereas using the time-varying depth hf 5 in the friction term instead of the simple h improves the fit considerably. After some trial and error, the appropriate parameters have been selected as k = 0.002 and b,, b, and h as shown in Table 3.
Sea-level
fluctuations
663
in the Fleet
TABLE 3. Model parameters, in metres. n
hn I 2 3 4
b I,*
b 2,n
2.6
Smallmouth
272 1'2
:
1'1
7 8
0.7
*3 14 I5 16 I7 18 19 20 21 22 23 24
25
location
74 74 99
9 IO II 12
Approximate
Bridging
Hard
148 642 III 741
0.3
88
793 0.15
146 290
0.15
91 366
0'2
Moonfleet
122 457 195
o-3 42" 0.32
173 207 146
0.4 198 0.45
IS9 272
Abbotsbury
Figures 7 and 8 display the model predictions for the tidal elevation curves at locations corresponding to the tide gauge observations, for spring and neap conditions respectively. Remarkably good correspondence between model predictions and observations is achieved. Whilst this is the result of tuning the model for spring conditions, the fact that the time lag of high and low water along the lagoon, the shape of the tidal curve, and the amplitude are all modelled realistically, suggests that the correct physical processes are represented in the model. The agreement of the neap tide response, for which the model was not specifically tuned, reinforces this conclusion. There is certainly no need to invoke the effect of seepage through Chesil beach as a mechanism necessary to account for the observed tidal regime, as has been suggested by some authors (Bird, 1972). The most serious shortcoming of the model is in the overestimation of the neap tide in the West Fleet. Although the model predicts a very low amplitude, it is in practice non-existent under certain conditions. This is probably due to an overestimation of the effective depth in the narrows near Moonfleet, which act to control propagation into the West Fleet. It may also be that friction is underestimated in the West Fleet where Spartina grass fills the channel during much of the year (Whittaker, 1980). The effect is stronger at neap tides when mean water level is low than at springs when the surface layers are clear of the grass. Examination of the model results shows that the mean level set-up is also reproduced quite well. This is important if the model is to reproduce effectively the strong control which mean water-depth exercises over the tidal propagation. However, the model in its present simple form, with time invariant topography, cannot fully represent the fact that not only do 6, and b, vary within the tidal cycle, but more importantly, vary over a spring-neap cycle with the fortnightly variation of the tidal mean water level.
664
I. S. Robinson, L. Warren
M’J. F. Longbottom
3 T
AbbotsburY
2 I O.D. 2
0
Moonfleet
Chickerell
Bridging
Hard
/
I O.D. -I
i
-2 -3
Figure 7, Elevation observed high-water
Applications
predicted by model-spring and low-water positions
tide. @ indicates (Figure 3).
the corresponding
of the Model
Apart from this limitation, the model as tuned appears to represent the tide propagation dynamics quite well, and can therefore be used to predict further properties of the water movements. Thus in Figure g the water transport across different sections is presented, as predicted by the model. This is difficult to measure in the field, except where the velocities are strong and the channel narrow. Comparison with observations made at Bridging Hard show quite good agreement with the model results, which provide useful information on which to base an understanding of circulation and flushing of the Fleet to be described in a further paper (Robinson, 1583). The salient feature of the volume transport is the asymmetry between the strong short flood and the long weak ebb, particularly noticeable in the West Fleet. The corresponding asymmetry of the tidal streams is potentially capable of acting as a sediment trap, able to transport sediments into but not out of the Fleet. There is no evidence at present for appreciable deposition of sediments by this mechanism, but given an adequate source of material, the Fleet could rapidly silt up and its character be changed. Certainly any anticipated civil engineering works which involve dredging in or near the mouth of the Fleet, and the deposition of spoils into the Fleet, must be carried out with great caution. Even a fairly small amount of sediment deposition in the channel by Moonfleet might seriously reduce the tidal propagation into the West Fleet. Exclusion of tidal energy could accelerate
Sea-level fluctuations
I
in the Fleet
665
Moonfleet
Bridging
Hard
\,~ --
O.D.
Time Figure
8. Elevation
predicted
by model-neap
(h) tide.
the deposition of sediment and buildup of detritus. Wind stirring of the West Fleet would help to prevent this, and it is not possible to predict with certainty how rapidly siltation would occur if material were available. The effect on the tides of siltation in certain parts, or the effect of the reduction in crosssectional area produced by civil engineering operations can be estimated from the model. Figure IO compares the height of spring tidal high and low water levels at locations along the Fleet, as predicted by the Model, (a) with the present topography; (b) with the breadth and depth at the mouth reduced to half their present value; and (c) with h and b, reduced to half their present value in the narrows by Moonfleet. The effect of constricting the mouth, whilst it reduces the tidal range in the Narrows and Littlesea, has little effect on the range in the West Fleet, although the mean level is slightly lowered. Reducing the cross-section at Moonfleet, on the other hand, almost eliminates the semidiurnal tidal range in the West Fleet, although it would not affect the fortnightly variation of water level. Finally, it is possible to use the model to explore the direct influence of winds on the water level and tidal propagation. Although in practice the surface wind stress must induce fairly complex three-dimensional circulation patterns even in shallow water, its overall effect on the one-dimensional depth-averaged flow can be represented by adding a wind stress term kp,v,Iv,l/[p(h+.z)] to the right-hand side of equation (I), where pa is the air density and ~1~ the wind speed component along the Fleet, a few metres above the surface of the water.
666
I. S. Robinson, L. Warren
&J.
F. Longbottom
200 100 0
‘.
b
.
5El
0
!
-
12 Time
,
6.5km
I
IO-5km
24 (h)
Figure 9. Volume transport rate across sections, predicted by the model for spring tide conditions for various distances from the mouth: (a) dotted line-observed volume flow at Smallmouth, 3 December 1967; (b) dotted line -observed volume flow at Bridging Hard, 4 November 1967.
In finite difference term
form, this requires
the addition
to the right-hand
side of equation
(3) of a
It is assumed that va does not vary over a length scale the size of the Fleet. To determine the response to the wind, the model was run for two cycles from its initial state without wind, to reproduce the normal tidal oscillation and then a wind was suddenly introduced. The most noticeable effect of the wind is the change in mean level, in the West Fleet particularly, while the actual tidal propagation does not appear to be directly influenced by the wind except through the change in mean level. As an example Figure I I shows the lowering of mean level at different locations along the Fleet, predicted by the model for a wind speed of 6.3 m s m1 from the north-west, for the neap cycle modelled earlier. The wind has a much stronger influence under neap tide, low mean level, conditions than at spring tide. Its influence is strongest in the West Fleet. With the wind blowing from the North IVest, it is found that a relatively weak IO m s --I wind is able to dry out parts of the model. The parameterization of depths in the model, and the simple representation of wind stress is clearly not adequate for the confident prediction of this extreme situation to be made from the model. However, the indications of the model are that the wind appears to have a strong direct influence on the
Sea-level
fluctuations
Distance
in the Fleet
from
667
Smollmouth
(km)
Figure IO. High-water and low-water levels for spring tide conditions predicted by the model at Bridging Hard (2 km from Smallmouth), Chickerell (4 km), Moonfleet (7 km) and Abbotsbury (IZ km): (a) model parameters tuned to match observations; (b) with the breadth and depth at Smallmouth halved; (c) with the breadth and depth near Moonfleet (between 6 and 8 km) halved.
E 3 ;
0,2-
e s E
O.I-
.F 0” f3
O
0
0
0
0 0
0
0
Y--al-o--P 0
4 Distance
from
I 8 mouth
I
I 12
(km)
Figure I I. The change in the tidal mean water level along the Fleet, predicted the model for a N.W. wind of 6.3 m s-l blowing in neap tide conditions.
by
water levels only at the west end of the Fleet. Wind-induced changes in mean level at spring tide are approximately half those shown for neap tide. Furthermore, it takes the model between three and four tidal cycles to reach the steady mean levels indicated, and a very rapid response to wind forcing is not obvious in the model forecasts. This may account for the apparent lack in the sea-level observations of any evidence of seiching driven directly by wind stress in the Fleet itself. Conclusion Although the sea level variations in the Fleet do not have the regular tidal oscillation pattern typical of most U.K. ports and estuaries, it has been possible to account for the observed pattern in terms of the non-linear propagation of long waves. The tidal regime in the East
668
I. S. Robinson,
L.
Warren
&J.
F. Longbottom
Fleet is typical of a shallow estuary with extreme tidal assymetry resulting from the large range to depth ratio. The strong frictional effects result in a much stronger set-up of mean level at spring tides than at neaps, and consequently the tides of West Fleet have a strong fortnightly component. The semidiurnal tide propagates into the West Fleet only weakly when the mean level is high at spring tides, and not at all when the mean level is low at neaps. The very shallowness of the lagoon appears to damp all high frequency oscillations, both tidal and wind driven, and acts as a low-pass filter enabling the longer period tides and external surges to penetrate to Abbotsbury. Viewed in this light, the apparently irregular tidal regime is much more comprehendable and more readily predicted.
Acknowledgements The author gratefully acknowledges the cooperation of the Central Electricity Generating Board in making available the tidal records for 1967/1968. The work was funded by a research grant from the University of Southampton.
References E. C. F. 1972 The physiography of the Fleet. Proceedings of the Dorset Natural History and Archaeological Society 93, I 25-13 I. Carr, A. P. & Blackley, M. W. 1974 Ideas on the origin and development of Chesil Beach, Dorset. Proceedings of the Dorset Natural History and Archaeological Society 95, 9-17. Lamb, H. 1932 Hydrodynamics. Cambridge University Press, Cambridge. Ladle, M. (ed.) 1981. The Fleet and Chesil Beach. .4 scientific account compiled by the Fleet Study Group. Dorset County Council, Dorchester, August 1981. LeBlond, P. H. 1979 Forced fortnightly tides in shallow rivers. Atmosphere-ocean. 17, 253~264. Robinson, I. S. & Perry, J. 1980 Tidal power from rectangular estuaries: tidal dynamics constraints. Proceedings of A.S.C.E.,~ournal of Hydraulics Division 106, No. HYII, 1915-1934. Robinson, I. S. 1983 A tidal flushing model of the Fleet-an English tidal lagoon. Estuarine Coastal and Shelf Science (forthcoming). Whittaker, J. E. 1980 The Fleet, Dorset-A seasonal study of the watermass and its vegetation. Proceedings of the Dorset Natural History and Archaeological Society, 100, 73-99.
Bird,
Coastal and Shelf Science
Sea-Level Fluctuations English Tidal Lagoon
I. S. Robinson, Department SOg gNH,
L. Warren
Oceanography, U.K. of
Received ~4 February
16, 651-668
(I&)
in the Fleet, an
and J. F. Longbottom
University
of
Southampton,
1982 and in revised form
Southampton
8 September 1982
Keywords: lagoons; tidal heights, shallow water equations; models ; nonlinear equations ; England coast
tidal friction;
Tidal elevation data are presented for places along the length of the Fleet, which is a tidal lagoon behind Chesil Beach on the south coast of England. Harmonic analysis of the data is not able to represent the observations adequately, particularly at the inner end of the lagoon. However, careful inspection of the data shows that the tidal regime is capable of being understood in terms of the non-linear propagation of long waves in very shallow water. Distortion of the tidal wave by unequal progression speeds of high and low water, and the set-up of mean level by frictional effects, are shown to be the important physical mechanisms controlling the observed water level fluctuations. A one-dimensional numerical model which incorporates these processes is able to reproduce the observations satisfactorily. Whilst the model predicts strong effects of wind stress, the meteorological influences in the observed data appear to be largely due to external surges in the English Channel which propagate into the lagoon through its entrance.
Introduction The Fleet is a shallow coastal lagoon, approximately 12; kms in length, trapped behind Chesil Beach north west of Portland Bill on the south coast of England. Figure I shows the location of the Fleet, and indicates its topography and bathymetry, which are described by Whittaker (1980) and Ladle (1981). It is essentially a linear shape, ranging in width between IOO m and I km at its widest. Its origin is closely related to the formation of Chesil Beach, which defines its south western boundary along its entire length. The genesis of Chesil Beach is discussed in detail by Carr and Blackley (1974) and it appears that the Fleet is not a drowned river valley, but is the land below sea level enclosed behind the beach as the sea level rose during the Flandrian transgression, moving the beach North Eastwards. There is no opening to West Bay through the beach, but the Fleet is connected to the English Channel at its south eastern end through Portland Harbour. It is therefore tidal and saline. A few streams drain into it from a small catchment area of about 50 km2, although seepage through the ground into the Fleet may drain from a wider area. It is also very shallow, varying in depth from 3 to 5 m in the first 2 km between the tidal entrance at Smallmouth and The Narrows. Here it is marine in character, with strong tidal exchange to the sea. It then widens out for 2.5 km into the area known as Littlesea, which has a depth of o to 1-2 m below chart datum, intersected by narrow channels up to 4 m deep. The tide is 0272-7714/83/060651+
18 $03.00/0
0 1983 Academic
Press
Inc.
(London)
Limited
I. S. Robinson, L. Warren
652
M’J. F. Longbottom
0
Zkm
0-
1 mlk?
ABBOTSBURY
MORKHAM’S RODDEN
LAKE HIVE
LANGTON
POINT HIVE
POINT
HERBURY
English
0 0 m
Channel
-Ift 10 -4ft I1 2mi 0 D to c-lftl0 3ml > -4ft I1 2ml
SMALLMOUTH \
I
Figure
I. Map
of the Fleet.
strongly attenuated in this region and penetrates only weakly into the remaining 8 km of the lagoon, known as West Fleet which has a depth between o and 1-2 m. The weak tidal flushing and the small freshwater inflow are delicately balanced to produce a brackish water regime which is strongly tide-dependent. The circulation and mixing are studied elsewhere (Robinson 1983), but the purpose of this paper is to examine the sealevel fluctuations in detail. There are several reasons why the tidal fluctuations in this relatively small and unfrequented lagoon should be worth examining. Firstly, it appears to be the cnly lagoon of its type in the U.K. and some of the species of flora and fauna found there have not been recorded elsewhere in the country. This may be due in part to the sheltered brackish water habitat afforded by the Fleet, where summer temperatures in the shallow West Fleet rise to 30 “C on a sunny afternoon and where wave action on the bed and shore is minimized with such a short fretch. It may also be related to the interesting tidal regime which in the West Fleet tends to be more of a fortnightly than a semidiurnal period. Marine biologists investigating the ecology of the lagoon require an understanding of the basic tidal pattern. Secondly, although the tides are quite small compared with many U.K. estuaries, the penetration of salt to Abbotsbury indicates the importance of tidally driven diffusion. Sedimentation must also be controlled by tidal flows. Given the environmental pressures which are threatening the Fleet, with alterations to the mouth at Portland, and other major civil engineering developments proposed at sites along the shore of the Fleet, it is important that the tidal dynamics be understood so that any changes in tidal regime consequent upon topographic alterations may be predicted. Moreover, any prediction of the capacity of the Fleet to cope with natural or imposed pollution requires the estimation of the circulation and flushing, which will be based in part on a knowledge of the tides (see the companion paper, Robinson, 1983).
Sea-level fluctuations
in the Fleet
653
Finally the tidal regime is interesting in its own right. Being so shallow, it exhibits in its short length many of the features associated with non-linear tidal propagation in much larger estuaries. The local population who use the Fleet find great difficulty in predicting the tidal state, and it was concluded by Bird (1972) that sea-level fluctuations were dominated by wind action and by seepage through Chesil Beach, although no firm observational evidence was presented to substantiate this hypothesis. The aim of this paper is therefore to determine more confidently the mechanisms which control sea-level fluctuations in the Fleet, by analysing time-series of tidal records obtained in 1967-1968. It will be shown that the tides are by no means as irregular as previous workers have concluded from occasional observations. Indeed it will be demonstrated that it is possible to reproduce much of the tidal behaviour in a numerical model based on straightforward non-linear long-wave propagation theory, which can therefore provide the basis for further study of tidal mixing and flushing of the Fleet.
The tidal data During the summer of 1967 and winter of 1967/1968, a hydrographic survey of the Fleet was performed by Messrs. G. Wimpey and Co. Ltd. on behalf of the U.K. Central Electricity Generating Board. About ten years after the survey, the hydrographic, salinity, sediment, temperature and tidal data collected during the survey was made available by the CEGB for scientific analysis, and it is the tidal records obtained during that survey which provide the observational evidence for the understanding of the Fleet tides presented here. Tide gauges were sited at six locations along the north shore of the Fleet, indicated in Figure I. These were Monro type automatic float gauges, levelled into Ordnance Datum (Newlyn) from convenient bench marks. The tide gauge data were made available in the form of half hourly heights above ordnance datum (Newlyn). Figure 2 shows the variation of the elevation with time for each cf the tide gauges, for the period between 18 October 1967 and 28 January 1968. The only record without gaps during the 36 months was that at the Royal Engineer’s Bridging Hard. Chickerell Hive and Morkham’s Lake records have occasional gaps whilst the other stations have less complete coverage, although sufficient information is available to indicate the overall tidal distribution throughout the lagoon. The tide at Smallmouth should represent the external forcing, but as a check on possible mouth effects the tide recorded at Portland Naval Dockyard has been plotted in Figure 2. Unfortunately the permanent tide gauge at Portland was malfunctioning during the period of the survey, and the curve shown is reconstructed from the tidal heights measured at the time of predicted high and low water at Portland, using the tidal curve presented in the Admiralty Tide Tables. Some observations of tidal currents were also made at certain sites during the survey, at approximately half-hourly intervals over a 12; hour period.
Short period tidal oscillations of water level The first comment which may be made on Figure z is that although the tide records, particularly those at the inner western end of the Fleet, are not typical of coastal tide gauges, they nonetheless exhibit a regularity and a correspondence to the external English Channel tides which confirms that the water-level fluctuations in the Fleet are fundamentally driven by the astronomical tides, rather than local wind conditions. However, the tides in the West Fleet are intermittent and their regular periodicity, apparent in the time series, may easily be
654
I. S. Robinson, L. Warren &sJ. F. Longbottom
Sea-level jluctuations
in the Fleet
655
concealed from the local observer by the long-period fluctuations which increase with distance from the mouth. These will be investigated further in a later section, but first the propagation of the semidiurnal tide down the lagoon will be examined in detail. Figures 3 and 4 show the tidal wave over 24 hours for two days chosen when records were available at most of the stations, one representative of spring tides (Figure 3) and the other of neap tides (Figure 4). It is convenient to interpret these figures in terms of three effects which occur as an observer travels inward from the lagoon mouth; the progressive phase lag of high and low water, an attenuation of tidal amplitude, and a distortion of the tidal profile. Each can be accounted for broadly in terms of non-linear long-wave propagation theory. The phase lag which increases towards the head of the lagoon indicates the progressive character of the tidal wave. Short estuaries normally have such a small tidal phase change along their length that they can be considered to be standing waves (e.g. Southampton Water). However, because the Fleet is so shallow, the phase propagation speed for long
Abbotsbury I-
2 -
Moonfleet
I : E 3 ‘0 D ;
O.D. I 2-
Chickerell
I OD.,
/
\
/
O.D.
O.D. -I -2 III 0
I 4
IllI
I 8
I
I
I I2
Time Figure
3. Typical
spring
tide
record
I
I
I
I 16
(h)
17 December
1967.
I
I
1
I 20
I
I
I
I. S. Robinson, L. Warren
656
NJ.
F. Longbottom
Abbotsbury -
I0.D Morkhom’s
Loke
IL
:
2< E ’ .P 2 O.D.
Chickerell
2Bridging
2
Hord
Smallmouth
I O.D. -I
. 1 0
I
I
I
I 4
I
I
I
I 6
I
I
I
I 12 Time
Figure
4. Typical
neap tide record
I
I
I
I 16
I
I
I
I 20
I
I
I
( h1
25 November
1967.
waves is sufficiently slow for the phase variations in a few kilometres to be appreciable. Moreover, the frictional dissipation is strong in such shallow water, so that little energy is likely to be reflected at the head of the lagoon and there is consequently virtually no standing component to the tidal wave. In both these respects the Fleet is equivalent to much longer and deeper estuaries which would have the same non dimensional response to tidal forcing as illustrated by Robinson and Perry (1980). Table I indicates the time lag of high and low water behind high and low water times at Smallmouth, for the representative spring and neap tidal cycles. Given that the average depth between Smallmouth and Bridging Hard is about z m, the propagation speed of small amplitude waves (g*ha) is expected to be 4.5 ms-’ which suggests a propagation time over this section of only 7.5 min. It appears that non-linear finite amplitude, and frictional effects are causing the slowing of the phase propagation. Even more marked is the apparently different propagation speed of high and low water. However, because of the changed profile of low water as it propagates along the Fleet, it is not meaningful to identify the lowest elevation
With
point
with
a single
phase
such a large amplitude/depth
predicts
a phase speed ofg% i
(I
point
of the semidiurnal
ratio, non-linear
cycle
long-wave
travelling
theory
+i i) where a is the local tidal elevation.
along
the estuary.
(e.g. Lamb,
1932)
For a spring ampli-
Sea-level fluctuations
in the Fleet
I. Phase lag of the tide
TABLE
relative Spring
(17
Location Bridging Chickerell Moonfleet Morkham’s Abbotsbury
High
Hard Hive
Lake
657
Dec.
water
to Smallmouth tide
Neap tide (25 Nov. 1967)
1967)
_____ Low
water
High
water
30 min
3 h 40 mina
15 min
I h ao min
4 h 5 min”
I h 5 min
zh 4h 5hr5min
5h 7h 8h
2
h
20
-
min
Low
water
3 h 30 mina (15 min) 4hr5mina (1 h) -
a This is the time lag of the lowest point of the changed profile behind the first low at Smallmouth. The bracketed figure is the time lag of the first low when it is lower than the second.
tude of 0.8 m and a neap amplitude of 0.4 m this would indicate that spring tide travel times of high and low water between Smallmouth and Bridging Hard would respectively be 4.7 min and 184 min and at neap tides 5.8 min and 10.7 min. Although the profile changes do not allow a direct comparison with the observed data to be made, it is clear that the observed travel times are considerably larger than simple long-wave theory predicts. The discrepancy is probably due to the non-uniform cross-section of the channel. The wave speed has been estimated roughly from the mean depth of the main channel along which the wave propagates. However, there are areas of tidal flats exposed at low water, and other large areas which are too shallow to permit propagation of the tidal wave, but which have a capacity to be filled by the incoming tidal wave, thus impeding its progression. This physical process may be simply expressed analytically by adopting a one-dimensional continuity equation of the form
au ac b, - = -6, h - where
at
ax
the depth h and breadth
b, of the deep channel
through
which
the
wave propagates, and the breadth b, of the actual water surface, are assumed to be independent of x and t. Hence the linear solution for long waves in this case indicates a phase speed of (g&/b,) I. Given the bathymetry of this section of the Fleet b,/b, may be estimated to be 0.1, which would increase the travel time from Smallmouth to Bridging Hard to about 24 min which is much closer to that which is observed. Beyond Bridging Hard, where the Fleet opens out into Littlesea this effect is even more pronounced and is the likely cause of the long time taken for the tide to penetrate to Chickerell Hive Point. Furthermore, because of the extensive drying flats, the cross sectional area (hb,) available for high-water propagation is very much greater than that for low tide, so that b,/b, is much less at low water, contributing to the significant difference between the travel time of high-water and low-water spring tides. The change in shape of the tidal curve as it progresses along the Fleet is related to the different travel speeds of high and low water. As high water catches up the preceding low, the tidal rise is rapid and its fall much slower, producing the saw-tooth shape characteristic of much larger estuaries such as the Severn where a bore is eventually formed at Spring tides. Associated with the elevation asymmetry is a short rapid flood and a long slow ebb, which may be seen in the model results of Figure 9 which are confirmed by current meter measurements. It appears that the flood tide sweeps relatively easily over the shallow Littlesea into the West Fleet, but the ebb is impeded by the shallows and only slow drainage can occur through the few deeper channels. This has the further consequence of ‘setting-up’ mean sea level into
658
I. S. Robinson, L. Warren
My.
F. Longbottom
the West Fleet aswill be discussedlater. The flow regime in the East Fleet provides conditions which will tend to trap sedimentin the Fleet with high energy inflow and lower energy outflow. It is alsointeresting to note that the double low water, which characterizesthe tides in the English Channel around Weymouth, is much reduced at Bridging Hard and has virtually disappearedby Chickerell. Presumably both the friction and the capacitative effect of the shallowstend to smooth out the higher frequency sea-levelfluctuations more rapidly. This too is a non-linear effect, being more pronounced at spring than at neap tides, when double low watersare still detectableat Chickerell. The attenuation of the tidal amplitude is alsovery marked in soshort an estuary. The most significant reduction in amplitude is between Chickerell and Moonfleet, from which it is inferred that the wide, shallow region of Littlesea and the narrow shallowsby Moonfleet dissipatemuch of the tidal energy, and act to control the tidal penetration into the West Fleet where the attenuation of such a small signal is very much less,as evidenced by comparison of the Abbotsbury and Morkham’s Lake tidal curves. Indeed, at neap tides the tide is virtually non-existent in the West Fleet. Only at spring tide doesit penetrate in recognizable form to Abbotsbury. The reasonfor this marked difference betweenspring and neap behaviour of the West Fleet is to be found in the change in mean sealevel. At spring tide, the strong dissipation of the progressivelong-wave energy results in a set up of mean sealevel which raisesthe water level in West Fleet sufficiently to permit the tide to penetrate to Abbotsbury. At neap tide the mean sealevel falls to such an extent that the tide gauge at Moonfleet dried out, with the resulting gaps in the data shown in Figure 2. The water remainsin West Fleet, which is deeperthan the narrow stretch adjacentto Moonfleet, and for several days is virtually isolated from the tide. This must clearly have consequencesfor the circulation and flushing of the West Fleet. Harmonic analysis of the full-length records confirms these conclusions drawn from inspecting sampledays of the record. Table z gives the harmonic constantsof the major tidal constituents obtained from a least squares harmonic analysis algorithm, based on the computer algorithm TIRA usedby the Bidston Laboratory of the Institute of Oceanographic Sciences,U.K. The analysismethod is able to copewith gapsin the time series,so that there was effectively at least 30 days record of half-hourly values for all the stations. The total TABLE 2. Harmonic phase angle
Bridging Hard
Smallmouth H(ft) Mean Mm Msf MI I(1 M2 S2 M4 MO 3MS,
S.D.(obs.) S.D.(res.) Record length
constants
Go
H(ft)
0'54 0.28 0.16 0'30
46 38 338 IO3
1.81 0.92 0.51 0.19 o-013 I ‘292
190 241 2 206
0’430 42 days in five blocks
0'21
G”s
0.61
-49
0’32 0’12
0.23 I’32 0.61 0.31 0’10
tidal
Chickerell Hive -_____H(ft) Go,
0’49
0’21
of major
8 29 23 20 22
6 20
0.043 43 I-199 0.472 I oo days
0.16 0’45 0’10
H(ft)
-2
Go,
-87
0.18
52 56
0.19
45 69 59
0’31 0.046 0.085 0.23 0.15 0.084
101 76 88 83 131 -37 46
0'022
0.970 0.482
50 and z8 days
I2
0-014
0’472 0’300 77 days in
four
the amplitude
Markham’s Lake
Moonfleet
0'10
0’9.5 0’43
0'021
His
1.24
-45 60 44
0'02
coast.
blocks
H(ft) 0’93 0’11 0.40
Abbotsbury Gas
0.047 0.116 0.088 0.031 0’009 0’004 0’401 0.259 75 days
H(ft) 0.88 0.19
-115
0'055
and Gso the
21
176 141 I53 I47 217 -
0.28 0.016 0.037
0’010 0’024 0’004 0’004 0’003
Gas
-154 40 -31
I42 97 -
0’353 0.286 41 days in two blocks
Sea-level fluctuations
in the Fleet
659
length of data, minus any gaps, on which each analysis is based is shown in Table 2. With this length of record, 36 constituents could be resolved, as well as eight further constituents whose relation to a major constituent was determined from the equilibrium tide. The amplitude H in Table 2 is in feet, and the phase angle G” is relative to the equilibrium tidal phase of the constituent in the case of Smallmouth, and for all the other stations G”s is given relative to the phase at Smallmouth, to demonstrate the progression of the tide along the Fleet. The analysis for Smallmouth, Bridging Hard and Chickerell Hive, is quite satisfactory, as measured by a comparison (see Table 2) of the standard deviations of the tidal record, and of the residual signal which remains after removal of the signal accounted for by the 44 harmonic constituents. The record becomes noisier away from the mouth, but is basically of a tidal harmonic form. The amplitude of the diurnal, semidiurnal and higher harmonic components are seen to decrease and the phase lag increase away from the mouth. The semidiurnal phase lag may be interpreted in terms of travel times along the Fleet, and compared with those obtained from sample records, in Table I (e.g. iU2 phases indicate 40 min to Bridging Hard, I h 48 min to Chickerell and 3 h to Moonfleet). However, by Moonfleet the residuals are of comparable magnitude to the original record. In this case the tidal signal is intermittent over the spring neap cycle, and in the case of Morkham’s Lake and Abbotsbury, when it does penetrate it is half-wave rectified. Both these effects preclude a satisfactory harmonic analysis, although the phase lag of M2 is probably of some significance as far as Moonfleet. The harmonic constants for Abbotsbury and Morkham’s Lake are in fact virtually meaningless. Inspection of the list of constituents derived from the analysis shows that those selected for inclusion in Table 2 as being the dominant lines of their species at Smallmouth are often smaller than nearby constituents. It is concluded that apart from Smallmouth and Bridging Hard, the predicton of tides in the Fleet using the harmonic method is unsatisfactory, because of the gross distorting effect of the non-linear propagation processes. The most satisfactory means of approximate tidal prediction is to use Figures 3 and 4 in conjunction with the tidal predictions for Portland. Other apparently non-tidal, irregular fluctuations with dominant frequency less than a day are seen in the records, but only in isolated instances, e.g. 31 October-r November 1967, 4 December 1967. Although there are no obvious correlations with the wind record at Portland (Figure 6) these fluctuations are most likely to be related to wind effects, but very often these anomalies occur on all the tide records, including the mouth, suggesting that they are in fact storm surge elevation peaks in the English Channel whose presence is transmitted like the tide into the Fleet. Spectral analysis of the individual records and the residuals after harmonic analysis, revealed no important peaks of energy at non-tidal frequencies, and cross-spectral analysis between pairs of stations gave no indication of coherence between the residuals at different locations. It is concluded that seiching due to direct wind action on the waters of the Fleet is not an important feature of the water-level fluctuations, contrary to what might have been expected in such an enclosed body of water. Indeed, the winds do not appear to have much direct influence at all, despite the subjective conclusions of local observers (see e.g. Bird, 1972).
Long-period fluctuations of sea level The sea level records were low-pass filtered by taking a simple 25 h running mean, and the results are plotted in Figure 5. The most striking feature to be revealed is the fortnightly fluctuation of water level, related to the spring-neap cycle, which is strongest towards the western end. The same feature is revealed by the harmonic analysis, where Msf is seen to
660
I. S. Robinson, L. Warren
3 2 I
&“J. F. Longbottom
,
1i +
0.0.
Morkham’s
3
Lake
2 I 0.0. Moonfleet
3 z ; n‘;
2 1 0.0.
f
3
4 .-6 6 1 0) 6
2
EI ; ,” ,”
Chickerell
I 0.0. 3
Bridging
Hard
2 I 1
0.0. -I 2
Smallmouth
I 0.0. -I
Figure
i 2 _
Portland
5. zs-hour
running
mean
of the time
series in Figure
2.
increase between Smallmouth and Chickerell Hive, and remains unattenuated as far as Abbotsbury. In fact at Moonfleet and beyond it becomes the dominant harmonic and at Abbotsbury is an order of magnitude greater than the other constituents as resolved by the least squares analysis. Table 2 also shows an increase in the mean sea level as far as Moonfleet. (Beyond there, the estimation of the mean is unreliable because of the inadequacy of the harmonic method for analysing the intermittent and half-wave rectified record). The set-up in mean sea level must be related to the bottom friction acting asymmetrically over the tidal cycle. The average depth during the inflowing tide is greater than during the ebb. Frictional effects are less on the flood, therefore, and time averaging of the frictional forces leads to a strong mean force acting inwards which is balanced by a tidal mean surface slope. During neap tides the velocities and thus the friction forces are weaker than at springs, and hence the mean set-up fluctuates over the spring-neap cycle, as reflected in the large amplitude Msf. The same effect is present in many estuaries, and has been discussed in detail for the St Lawrence by LeBlond (1979).
Sea-level
fluctuations
661
in the Fleet
What is striking about the Fleet is that the effect should occur over a length scale two orders of magnitude smaller than the St Lawrence. The phase of Msf indicated by Table 2 shows a delay of about a day at Morkham’s Lake, and nearly two at Abbotsbury, for the spring-neap modulation of mean level to penetrate to the end of the Fleet. Figure 5 also shows up other long period fluctuations of mean sea level. It is significant that these can usually be identified at all the tide gauges, including Portland, unlike the fortnightly oscillation which only becomes obvious at Chickerell and beyond. It is concluded that these are lower frequency surges of meteorological origin, generated in the English Channel rather than the Fleet itself. A weak correlation can be discerned between these surges and strong winds in the record of Figure 6.
Across SW4
the Fleei NE
NW4
SE ,
I Ott
Nov
0I
I Oec
1967
1968
Jon
Figure 6. Wind speed recorded daily at Portland. Because the long-period fluctuations are able to penetrate to Abbotsbury with little attenuation, unlike the higher frequency semidiurnal tides, they have considerable impact on the water level regime in the West Fleet. The fortnightly tidally driven fluctuations are regular and predictable, but to be able to estimate the non-tidal fluctuations, an understanding of the English Channel storm surge dynamics is required rather than a simple mechanism of local wind set-up in the Fleet. This is perhaps what has confused so many of the local observers in their attempts to comprehend the causes of sea level variations in the West Fleet.
A one-dimensional, non-linear numerical model of tidal propagation in the Fleet If, as is suggested earlier, the tidal regime can be explained qualitatively in terms of non-linear long-wave propagation, then it should be possible to demonstrate quantitative agreement with theory using a numerical model.
662
Such a model
I. S. Robinson, L. Warren
has been constructed,
&J.
F. Longbottom
based on the one-dimensional
equation
of motion: (1)
and continuity: h ;
+ ;
[b,u(h+
a]
= o,
where 5 is the tidal elevation above the mean level at the mouth, h is the water depth below this datum, x is the distance along the lagoon measured from the mouth, u is the cross sectionally averaged velocity, 6, is the breadth of the channel at the sea surface and b, the mean breadth such that b, (h+ 5) is the local instantaneous cross-sectional area. k is a coefficient of quadratic friction, normally taken in shallow sea dynamics to be 0~002-0~0025. The Fleet is represented as 12+ sections each of length I km. Even numbered grid points from 2 to 26 correspond to section boundaries. The mouth is taken at grid point I, the centre of the first section, and the head at point 26. If 5 is defined at odd numbered and u at even numbered grid points, equations (I) and (2) can be represented in finite difference form: nun -zz nt
uL-u2n-~
-p+1-
2k 44
L-1) ax
4ax
- zhn+
L+l+
L-1
(3)
and $$=&{u,i,b+,
(ha+, hn-I
+
+ 5n:,+in)-un-lbz.n-1
Inf5n-2 2
)
(4)
)>
where ax is the distance between xn and xns2, i.e. I km, and At is chosen to be IO s, well within the stability limits of the model. By means of equations (3) and (4), u and &’ can be predicted for time tfnt given the value of variables at t. The boundary condition corresponding to the head at Abbotsbury is uaa = o. Evaluation of n {a5 by use of equation (4) assumes that [a7 = [as. The mouth boundary condition is determined by specifying [r. For spring tide simulation, a typical spring cycle at Smallmouth obtained from the records of Figure 2 is used to define [i, by use of the half-hourly observed values with linear interpolation for the intermediate time steps. For neap tide simulation, exactly half the spring tide amplitude at Smallmouth provides a suitable boundary condition. Initially, the elevation throughout the Fleet is zero, and the model is run with the same mouth forcing for several cycles until repeatability of the response is obtained. In practice this usually occurs after two cycles. Fitting of the model simulation to the observed distribution of tidal elevation along the Fleet can be achieved by using R and b, as tunable parameters. h and 6, have been determined as objectively as possible by means of the hydrographic information from the CEGB survey, assuming that in the Littlesea area these should correspond with the deeper channels through which the water flows along the Fleet, rather than the wide shallow areas which provide storage capacity. The capacitance effect is parameterized by assigning b,. It is found that variation of K does not vary the tidal regime very much, whereas using the time-varying depth hf 5 in the friction term instead of the simple h improves the fit considerably. After some trial and error, the appropriate parameters have been selected as k = 0.002 and b,, b, and h as shown in Table 3.
Sea-level
fluctuations
663
in the Fleet
TABLE 3. Model parameters, in metres. n
hn I 2 3 4
b I,*
b 2,n
2.6
Smallmouth
272 1'2
:
1'1
7 8
0.7
*3 14 I5 16 I7 18 19 20 21 22 23 24
25
location
74 74 99
9 IO II 12
Approximate
Bridging
Hard
148 642 III 741
0.3
88
793 0.15
146 290
0.15
91 366
0'2
Moonfleet
122 457 195
o-3 42" 0.32
173 207 146
0.4 198 0.45
IS9 272
Abbotsbury
Figures 7 and 8 display the model predictions for the tidal elevation curves at locations corresponding to the tide gauge observations, for spring and neap conditions respectively. Remarkably good correspondence between model predictions and observations is achieved. Whilst this is the result of tuning the model for spring conditions, the fact that the time lag of high and low water along the lagoon, the shape of the tidal curve, and the amplitude are all modelled realistically, suggests that the correct physical processes are represented in the model. The agreement of the neap tide response, for which the model was not specifically tuned, reinforces this conclusion. There is certainly no need to invoke the effect of seepage through Chesil beach as a mechanism necessary to account for the observed tidal regime, as has been suggested by some authors (Bird, 1972). The most serious shortcoming of the model is in the overestimation of the neap tide in the West Fleet. Although the model predicts a very low amplitude, it is in practice non-existent under certain conditions. This is probably due to an overestimation of the effective depth in the narrows near Moonfleet, which act to control propagation into the West Fleet. It may also be that friction is underestimated in the West Fleet where Spartina grass fills the channel during much of the year (Whittaker, 1980). The effect is stronger at neap tides when mean water level is low than at springs when the surface layers are clear of the grass. Examination of the model results shows that the mean level set-up is also reproduced quite well. This is important if the model is to reproduce effectively the strong control which mean water-depth exercises over the tidal propagation. However, the model in its present simple form, with time invariant topography, cannot fully represent the fact that not only do 6, and b, vary within the tidal cycle, but more importantly, vary over a spring-neap cycle with the fortnightly variation of the tidal mean water level.
664
I. S. Robinson, L. Warren
M’J. F. Longbottom
3 T
AbbotsburY
2 I O.D. 2
0
Moonfleet
Chickerell
Bridging
Hard
/
I O.D. -I
i
-2 -3
Figure 7, Elevation observed high-water
Applications
predicted by model-spring and low-water positions
tide. @ indicates (Figure 3).
the corresponding
of the Model
Apart from this limitation, the model as tuned appears to represent the tide propagation dynamics quite well, and can therefore be used to predict further properties of the water movements. Thus in Figure g the water transport across different sections is presented, as predicted by the model. This is difficult to measure in the field, except where the velocities are strong and the channel narrow. Comparison with observations made at Bridging Hard show quite good agreement with the model results, which provide useful information on which to base an understanding of circulation and flushing of the Fleet to be described in a further paper (Robinson, 1583). The salient feature of the volume transport is the asymmetry between the strong short flood and the long weak ebb, particularly noticeable in the West Fleet. The corresponding asymmetry of the tidal streams is potentially capable of acting as a sediment trap, able to transport sediments into but not out of the Fleet. There is no evidence at present for appreciable deposition of sediments by this mechanism, but given an adequate source of material, the Fleet could rapidly silt up and its character be changed. Certainly any anticipated civil engineering works which involve dredging in or near the mouth of the Fleet, and the deposition of spoils into the Fleet, must be carried out with great caution. Even a fairly small amount of sediment deposition in the channel by Moonfleet might seriously reduce the tidal propagation into the West Fleet. Exclusion of tidal energy could accelerate
Sea-level fluctuations
I
in the Fleet
665
Moonfleet
Bridging
Hard
\,~ --
O.D.
Time Figure
8. Elevation
predicted
by model-neap
(h) tide.
the deposition of sediment and buildup of detritus. Wind stirring of the West Fleet would help to prevent this, and it is not possible to predict with certainty how rapidly siltation would occur if material were available. The effect on the tides of siltation in certain parts, or the effect of the reduction in crosssectional area produced by civil engineering operations can be estimated from the model. Figure IO compares the height of spring tidal high and low water levels at locations along the Fleet, as predicted by the Model, (a) with the present topography; (b) with the breadth and depth at the mouth reduced to half their present value; and (c) with h and b, reduced to half their present value in the narrows by Moonfleet. The effect of constricting the mouth, whilst it reduces the tidal range in the Narrows and Littlesea, has little effect on the range in the West Fleet, although the mean level is slightly lowered. Reducing the cross-section at Moonfleet, on the other hand, almost eliminates the semidiurnal tidal range in the West Fleet, although it would not affect the fortnightly variation of water level. Finally, it is possible to use the model to explore the direct influence of winds on the water level and tidal propagation. Although in practice the surface wind stress must induce fairly complex three-dimensional circulation patterns even in shallow water, its overall effect on the one-dimensional depth-averaged flow can be represented by adding a wind stress term kp,v,Iv,l/[p(h+.z)] to the right-hand side of equation (I), where pa is the air density and ~1~ the wind speed component along the Fleet, a few metres above the surface of the water.
666
I. S. Robinson, L. Warren
&J.
F. Longbottom
200 100 0
‘.
b
.
5El
0
!
-
12 Time
,
6.5km
I
IO-5km
24 (h)
Figure 9. Volume transport rate across sections, predicted by the model for spring tide conditions for various distances from the mouth: (a) dotted line-observed volume flow at Smallmouth, 3 December 1967; (b) dotted line -observed volume flow at Bridging Hard, 4 November 1967.
In finite difference term
form, this requires
the addition
to the right-hand
side of equation
(3) of a
It is assumed that va does not vary over a length scale the size of the Fleet. To determine the response to the wind, the model was run for two cycles from its initial state without wind, to reproduce the normal tidal oscillation and then a wind was suddenly introduced. The most noticeable effect of the wind is the change in mean level, in the West Fleet particularly, while the actual tidal propagation does not appear to be directly influenced by the wind except through the change in mean level. As an example Figure I I shows the lowering of mean level at different locations along the Fleet, predicted by the model for a wind speed of 6.3 m s m1 from the north-west, for the neap cycle modelled earlier. The wind has a much stronger influence under neap tide, low mean level, conditions than at spring tide. Its influence is strongest in the West Fleet. With the wind blowing from the North IVest, it is found that a relatively weak IO m s --I wind is able to dry out parts of the model. The parameterization of depths in the model, and the simple representation of wind stress is clearly not adequate for the confident prediction of this extreme situation to be made from the model. However, the indications of the model are that the wind appears to have a strong direct influence on the
Sea-level
fluctuations
Distance
in the Fleet
from
667
Smollmouth
(km)
Figure IO. High-water and low-water levels for spring tide conditions predicted by the model at Bridging Hard (2 km from Smallmouth), Chickerell (4 km), Moonfleet (7 km) and Abbotsbury (IZ km): (a) model parameters tuned to match observations; (b) with the breadth and depth at Smallmouth halved; (c) with the breadth and depth near Moonfleet (between 6 and 8 km) halved.
E 3 ;
0,2-
e s E
O.I-
.F 0” f3
O
0
0
0
0 0
0
0
Y--al-o--P 0
4 Distance
from
I 8 mouth
I
I 12
(km)
Figure I I. The change in the tidal mean water level along the Fleet, predicted the model for a N.W. wind of 6.3 m s-l blowing in neap tide conditions.
by
water levels only at the west end of the Fleet. Wind-induced changes in mean level at spring tide are approximately half those shown for neap tide. Furthermore, it takes the model between three and four tidal cycles to reach the steady mean levels indicated, and a very rapid response to wind forcing is not obvious in the model forecasts. This may account for the apparent lack in the sea-level observations of any evidence of seiching driven directly by wind stress in the Fleet itself. Conclusion Although the sea level variations in the Fleet do not have the regular tidal oscillation pattern typical of most U.K. ports and estuaries, it has been possible to account for the observed pattern in terms of the non-linear propagation of long waves. The tidal regime in the East
668
I. S. Robinson,
L.
Warren
&J.
F. Longbottom
Fleet is typical of a shallow estuary with extreme tidal assymetry resulting from the large range to depth ratio. The strong frictional effects result in a much stronger set-up of mean level at spring tides than at neaps, and consequently the tides of West Fleet have a strong fortnightly component. The semidiurnal tide propagates into the West Fleet only weakly when the mean level is high at spring tides, and not at all when the mean level is low at neaps. The very shallowness of the lagoon appears to damp all high frequency oscillations, both tidal and wind driven, and acts as a low-pass filter enabling the longer period tides and external surges to penetrate to Abbotsbury. Viewed in this light, the apparently irregular tidal regime is much more comprehendable and more readily predicted.
Acknowledgements The author gratefully acknowledges the cooperation of the Central Electricity Generating Board in making available the tidal records for 1967/1968. The work was funded by a research grant from the University of Southampton.
References E. C. F. 1972 The physiography of the Fleet. Proceedings of the Dorset Natural History and Archaeological Society 93, I 25-13 I. Carr, A. P. & Blackley, M. W. 1974 Ideas on the origin and development of Chesil Beach, Dorset. Proceedings of the Dorset Natural History and Archaeological Society 95, 9-17. Lamb, H. 1932 Hydrodynamics. Cambridge University Press, Cambridge. Ladle, M. (ed.) 1981. The Fleet and Chesil Beach. .4 scientific account compiled by the Fleet Study Group. Dorset County Council, Dorchester, August 1981. LeBlond, P. H. 1979 Forced fortnightly tides in shallow rivers. Atmosphere-ocean. 17, 253~264. Robinson, I. S. & Perry, J. 1980 Tidal power from rectangular estuaries: tidal dynamics constraints. Proceedings of A.S.C.E.,~ournal of Hydraulics Division 106, No. HYII, 1915-1934. Robinson, I. S. 1983 A tidal flushing model of the Fleet-an English tidal lagoon. Estuarine Coastal and Shelf Science (forthcoming). Whittaker, J. E. 1980 The Fleet, Dorset-A seasonal study of the watermass and its vegetation. Proceedings of the Dorset Natural History and Archaeological Society, 100, 73-99.
Bird,