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Analytic models for West Coast storm surges, with application to events of January 1976
Analytic models for West Coast storm surges, with application to events of January 1976 R. J. McIntyre Institute of Oceanographic Sciences, Bidston Observatory, L43 7RA, UK (Received July 1978; revised October 19 78)
Birkenhead,
Merseyside
Simple mathematical models describing wind-induced motion on a continental shelf are used to predict storm surges at the port of Milford Haven on the west coast of the British Isles. Attention is concentrated on surge events occurring during the stormy month of January 1976. It is demonstrated that useful first-order accuracies in prediction can be achieved. Oceanic influence on the coastal surge is shown to be small. Satisfactory results are obtained with wind input data coming either from the analysis of daily weather charts or from recorded values of wind speed and direction at a group of meteorological stations.
Introduction It has been known for some time1 that secondary depressions moving in from the Atlantic are responsible for many of the larger storm surges experienced along the west coast of the British Isles. Heaps* showed that surges at Milford Haven are produced by travelling wind fields, associated with such depressions, acting on the continental shelf to the south of Ireland (the Celtic Sea). He developed an analytic method to gain more knowledge of the mechanism of that surge generation. The method involved the investigation of the dynamic response of the water on an idealized shelf of uniform depth and width to wind stress fields with moving boundaries. Flathers extended the work carried out by Heaps, considering the idealized shelf connected to an ocean of uniform depth and width. The present work employs both the shelf model due to Heaps and the shelf-ocean model due to Flather for the investigation of the storm surges at Milford Haven during January 1976. This month was a very stormy one, with two large West Coast surge events occurring in it. The basic wind fields examined in the present work cover an area of shelf between any two lines parallel to the coast, the wind stress being constant and uniformly distributed over this area. Again, the mathematical method invokes the use of Iaplace transform techniques following the introduction of a step function wind stress suddenly created at time t = 0. The solution obtained for the shelfocean model is more general and more complex than that obtained by Flather. The latter restricted investigation to 0307-904X/79/020089-10/$02.00 0 1979 IPC Business Press
the effects of a steady wind field acting over the whole of the shelf. The observed and theoretically-derived surges at Milford Haven are found to be in remarkably satisfactory agreement - considering the simplicity of the models. A comparison of the results coming from the shelf and shelfocean models, respectively, indicates a small general reduction in surge levels when the ocean is included. A programme of numerical sea modelling is obviously required for a comprehensive attack on the problem of surge prediction on the West Coast generally. However, the present modest contribution could well prove to be practically useful on a more localized scale before a prediction system overall is established. No tation Shelf and adjoining ocean are rectangular
as indicated in The aim is to determine sea level variations at the coast x = 0 produced by wind stress, with onshore component P and longshore component Q, acting over the area of shelf between x = a and the shelf edge x = I (Figure 2). The cross-sectional geometry is illustrated more completely in Figure 3 and the notation may be summarized as follows:
Figure I, with x the offshore coordinate,
X,Y, z
rectangular Cartesian coordinates, a left-handed set with origin 0 at the coast in the undisturbed surface of the sea; Ox, Oy lie in this surface with Ox normal to the coastline; Oz is directed vertically downwards
Appt. Math. Modelling,
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West Coast storm surges: R. J. fvlclntyre x,y
components
of wind stress on sea surface
angular speed of Earth’s rotation In the mathematical analysis, subscripts s and o refer respectively to shelf and ocean while subscripts 1, 2 and 3 refer respectively to different solution regions, namely: region 1 (0 < x < a), region 2 (a < x < I), region 3 (I < x < L). Figure 7 Shelf-ocean normal to coastline
system, showing a vertical cross-section
Dynamical
equations
and their solution
We employ the linearized vertically-integrated of motion and continuity:
equations
+y+Xli-/v+gh= ax P
(1) au av ay -+-+-_=o
I I I I
ax
I
I I X=0
Coast
Oceanic edge
Figure 2 Continental shelf (plan view), showing area between and x = I over which wind stress has onshore component P and longshore component Q
x =a
ay
at
where h is a friction parameter and f (= 2w sin@) the Coriolis parameter, regarded as constants. Bottom friction has components hpU, XpV and although it might be argued that h on the shelf should be different to X in the ocean (inversely proportional to the respective depths) such a difference is not allowed for here. Assuming longshore uniformity, the U, V, { are function of x and t only and derivatives with respect to Y are zero. Solutions of (1) are obtained yielding motion in shelf and ocean induced by a uniform wind stress field imposed at time t = 0 over that part of the shelf lying between x = a and x = I (Figure 2). The field is defined by: F=G=O
O
F = -M(t)
G = QH(t)
F=G=O
a
(2) 1
l
where H(t) denotes the Heavyside unit function; P, Q are constants representing onshore and longshore stress components. Initially the water is considered to be at rest so that, everywhere: U=V={=O att=O
(3)
Much as in the earlier work by Heaps* and Flather, the equations (1) are solved analytically using the Laplace transformation and the result may be denoted by:
*ii
xl0
Oceanic edge oi shelf
coast
Figure 3 Vertical showing notation
h,>ho 1 L a c u, ‘u LJ, v
90
section and plan view of shelf-ocean
model,
depths of shelf and ocean respectively width of shelf width of shelf-ocean system distance of an arbitrary longshore line from coast, on shelf elevation of water surface components of current in directions of increasing X,Y values of U, v respectively, integrated from surface to sea bed
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u= u,
v= vi
u= u,
v= v,
u=
v=
u3
v3
<={1
O
i-=T*
a
{={3
l
(4) 1
Two solutions (4) are obtained - each subject to (2) and (3). One corresponds to the complete shelf-ocean model portrayed in Figure 3, satisfying zero flow across the land boundaries at x = 0 and x = L, also continuity of flow and elevation across the edge of the wind area at x = a and across the shelf edge at x = 1. Accordingly, the following boundary conditions are satisfied: ur =o
atx=O
5‘1 =r*
u, = u,
atx=a
f2
u,
atx=Z
u,
=
13
= 0
= Ii,
atx=L
(5) I
West Coast storm surges: R. J. McIntyre
This solution is a generalization of the one due to Flatherl: the wind field now has its inner edge along the arbitrary longshore line x = a rather than along the coast x = 0 as before. The other solution obtained corresponds to the shelf model considered by Heaps2 and describes motion on the shelf alone with zero elevation imposed along its oceanic edge. The boundary conditions in this case are: u, =o
atx=O
rr = c*
u, = u,
t2 =0
atx=Z
Mathematical
atx=a
Secondly, {r for the shelf-ocean model is given by:
(12)
~lst~lo=(l-~)~
+2N
(h-U&(ha,)-fQ1 2 .=1pza,{(A-2a,)(X-a,)* +Xf2} (7) - cosm,) cos(m,N) co~i”x!.~~
(6)
(1 +MN) sinm, sin(m,N)
I
- (M +N)cosm,
analysis yields, formally, for the shelf model:
<1 = {rs(x, 6
G
f ,, = 1plJ;E;
t4N
(7)
and for the shelf-ocean model: (1 = L?rs(x, r) + r10 6, t)
(cos
(8)
Setting x = 0 in (7) or (8) gives the required wind-induced surge at the coast, {, say. Thus: c, =fr(x
= 0)
\ cos(m,N$
cosm,
cos(c,t + $:, + A:, ~ li/;)
(y) - cosm,)
cos(m,N)cos
i-:!,
r n
X
tMN)sinm,
sin(m,N)
~ (M +N) cosm, cos(m,N)
(9)
cosmz,
(15)
Clearly, since a is arbitrary, we may write: where: (10)
!L = L 0, Q)
M=
Without going into the details of the mathematical treatment the derived expressions for {rs and {rs t cl0 are now given. Thus, {r for the shelf model is given by:
t i (-l)n n=1
a) 1
41, plJ,,En
21
(
cos(~t
P{P + h +f*/(p I
tan(Mm)
+ 4, + A, - L)
(
(11) where : In
J,
= {P*-2PQf(X-pn)/d;
= (pi
En = {(A-
+f2Q2/d;}1’2
t I#/* 2pn t Af* cos2O,/d;)* + (2u, -
#n = tan-‘[(fey,/d~)/{P-f~(X A,, = tan-‘(v,/p,) $,
= tan-‘[{2v,
- (Af*/d$
Xf* sin20,/di)*} - ~J&~l
‘I2
sin20, >
IQ - 2~, + (V’/d;)
dn = {(h-/&)2t z+?$“Z 4I = tan-l{v,/(x-p,)}
cos2fMl (12)
and where: P =-&I,
-pn f ivn
(13)
are the roots of the cubic: p(p t x t f2/( p t X)) = -gh,(2n
- 1)%*/41*
+ ic,
(17)
+ 1)) = -gh,mzll*
(18)
m = m, being the nth positive root, in ascending order, of:
(2fi - 11:” - “‘) cos((2n ;;)?ix)e_‘n’
x sin
-b,
are the roots of the cubic:
(2n - l)r=\e-pnr
cos 21
n=1
P = -a,,
pz6,{(X-26,)(X-F,)*+hf2}
x sin
+ C(-l)n---
and where IA, J,!, , E,!,, c#&,AL, $L, dh,, 0; are given respectively by I,, J,,, E,, &, A,, $I,,, d,, t$, with 6, replaced by a,, pn by b, and u, by c,. Here:
20 - 42 P(A- f&z> - fQ>
(2n - l)ir(l-
(16)
(14)
t N tanm = 0
(19)
In both (11) and (15) surge elevation consists of a constant first term due to steady wind set-up, followed by an infinite series of pure exponential terms present because of the rotation of the Earth, followed by an infinite series of exponentially-damped periodic terms representing free modes of oscillation: progressively decreasing in period up the series. All the shelf modes in (11) have a node at the shelf edge: in the case of the Celtic Sea example considered in this paper, the first mode (n = 1) which predominates has a period of 10.2 h. Adding in the Atlantic Ocean, the shelf-ocean model gives a first mode of period 11.9 h, with a node in the ocean a distance 0.35 1 away from the shelf edge. Table 1 gives the values of 6,, Pi, v,, (2n ~ 1)7r/2 for the shelf model, and a,, b, , c,, m, for the shelf-ocean model, as used in this paper to investigate storm surges in the Celtic Sea.
Application of theory Using the results of the preceding section, storm surges generated by wind on the continental shelf to the south of Ireland are calculated. For theoretical purposes, a rectangular area of the shelf sea, ABCD, is considered (Figure 4). The coast line x = 0 is assumed to lie along BC and the oceanic edge, x = 1, along AD. The changing pattern of wind stress over ABCD is expressed as a linear combination of onshore and longshore stress fields, each of 1 dPa (= 1 dyn/cm*), acting at hourly intervals over the
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Table 7 Values of 6,, g,, +,, an, b,, c, (in h-’ units), along with corresponding parametric values (2% -1 )n/2, mn: used in present paper to investigate storm surges in Celtic Sea
from area 4 (Z/5 < x < 2115): { = H(t - T){,(t - T, Z/5) -H(t-T+l)~c(t-T+1,Z/5)
6,
hl
7 8 9 10
0.051078 0.083032 0.087361 0.088634 0.089169 0.089442 0.089600 0.089699 0.089765 0.089812
0.064409 0.048484 0.046319 0.045683 0.045416 0.045279 0.045200 0.045151 0.045117 0.045094
0.615328 1.45496 1 2.364946 3.287379 4.214104 5.142803 6.072570 7.002981 7.933809 8.864922
1.57080 4.71239 7.85398 10.99557 14.13716 17.27876 20.42035 23.56194 26.70353 29.84513
”
an
bn
cn
mn
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 26 27 28 29 30 31 32
0.037487 0.054336 0.071131 0.080032 0.082932 0.084456 0.086231 0.087198 0.087511 0.088053 0.088470 0.088643 0.088819 0.089027 0.089155 0.089214 0.089326 0.089415 0.089450 0.089505 0.089565 0.089598 0.089623 0.089662 0.089693 0.089705 0.089730 0.089754 0.089766 0.089780 0.089798 0.089811
0.071257 0.062832 0.054435 0.049984 0.048534 0.047772 0.046885 0.046401 0.046244 0.045973 0.045765 0.045678 0.045590 0.045487 0.045423 0‘045393 0.045337 0.045293 0.045275 0.045247 0.045218 0.045201 0.045189 0.045169 0.045154 0.045147 0.045135 0.045123 0.045117 0.045110 0.045101 0.045095
0.529872 0.642798 0.883792 1.216315 1.445689 1.631218 1.978497 2.294958 2.434995 2.753333 3.106259 3.298114 3.535959 3.894594 4.178526 4.334611 4.678280 5.022138 5.178437 5.462974 5.822816 6.061778 6.254759 6.610367 6.932046 7.073918 7.395668 7.751476 7.9446 11 8.183824 8.544168 8.829216
1.16286 1.69125 2.65294 3.86875 4.67594 5.32768 6.52850 7.61452 8.09344 9.17946 10.38028 11.03203 11.83921 13.05503 14.01671 14.54510 15.70796 16.87082 17.39922 18.36090 19.57671 20.38390 21.03565 22.23646 23.32248 23.80140 24.88743 26.08824 26.73999 27.54717 28.76299 29.72467
”
1 7
; 4
5 6
(La-lln/2
- H(t - T){,(t
< =H(t - T){,(t
Similarly,
from area 3 (21/5
(20)
(21)
c = H(t - T)S‘,(t - T, 22/5) -H(t-T+l)<&-T+1,21/5)
92
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L = 6148 km
h, = 100 m
h, = 4000 m
X =25~1O-~s-’
f
= 9.81 msC*
(25) 1
Here L and h, are chosen to represent the North Atlantic basin to which the Celtic Sea shelf, represented by Z and h,, connects. As before, in developing the standard responses, the series in (11) are taken up to n = 10 and the series in (15) up to y1= 32. All the modes of oscillation of the shelf and the shelf-ocean are thereby included with periods greater than 0.7 h (Table 1). It is evident from (1 l), (15) and (20)-(24) that each response will diminish exponentially as time increases. Surges at x = 0 during two periods: l-3 January and 28-30 January 1976 are evaluated theoretically (as described above) and compared with observed surge heights at Milford Haven. Zero motion is assumed in the calculations at the beginning of each period. Relevant synoptic weather charts are shown in Figures 5 and 6. Theoretical models of the wind fields for each period, constructed from information given by the Daily Weather Reports of the British Meteorological Office, are given in Figures 7 and 8. From the wind velocities in Figures 7 and 8, onshore and longshore stress components are determined hourly over sub-areas l-5 using the square law: (26) rw = C&W2 where 7, is the wind stress, W the wind speed in m/s, pa the density of the air (taken as 1.25 kg m-3), and c a drag coefficient:
= -0.12
(22)
= 112.5 x 1O-6 s-l
p = 1025 kg m-’
w<5
c x lo3 = 0.565
3, April
(23)
- T, 0)
Z = 380 km
g
< 3Z/5):
- H(t - T){,(t - T, 31/5) + H(t - T + l){,(t - T + 1, 31/5)
21/5)
+H(t-T+1)[c(t-T+1,Z/5) (24) The above procedure of combining a standard set of theoretically-derived elevation responses, corresponding to unit onshore and longshore wind stress fields over subareas 1 to 5, was successfully employed by Heaps2 to simulate the observed surges at Milford Haven during periods in September 1953 and November 1954. Milford Haven lies near to the midpoint of BC and it is therefore reasonable to compare the observed surge there with the theoretical surge evaluated at x = 0. Heaps used the shelf model exclusively; in the present work, adopting the same procedure, we use the shelf model and also the shelf-ocean model described earlier, In each case, only surges arising from wind action on the shelf are considered. Specifically, the aim here is to investigate the surges which occurred at Milford Haven during the stormy periods of January 1976. Numerical parameters used by Healq2 and later by Flather, are again employed. Thus:
from area 2 (31/5 < x < 41/5):
f = H(t - T)<,(t - T, 31/5) -H(r-T+I){C(t-T+1,3Z/5) - H(t - T)<,(t - T, 41/5) +H(t-T+1)
- T+1,
-H(tT+l){,(tT+l,O) - H(t - T)f,(t - T, Z/5)
five constituent areas shown in the figure. The elevation responses at the coast x = 0 corresponding to these fields, responses derived on the basis of (9) are combined linearly in the same way to yield the coastal surge. Thus, due to wind stress acting between t = T and t = T + 1 hours say, the contribution to the surge coming from area 1 (41/S < x < 1) takes the form: 4 = H(t - T){,(t - T, 41/5) -H(tT+l)f,(t-T+1,41/5)
- T, 21/5)
+ ff(t - T+1)c,(t and from area 5 (0 < x < Z/5):
= 2.513
t 0.137W
5 < W< 19.22 W Z 19.22
(27)
West Coast storm surges: R. J. McIntyre
Figure 4 Shelf area (ABCD) considered in theory, showing sub-areas over which five constituent wind fields act; (0) meteorological used for wind velocity; (------I 100 fathom depth contour; O.W.S. ‘R’ - Ocean Weather Ship ‘R’; V - Valencia; R - Roches Point; S - Stilly; SM - St Mawgan; M - Milford Haven
Input data for the surge calculations are thus obtained. An alternative set of stress input data (of the same form as the above) is obtained by allocating hourly wind speeds and directions recorded at the meteorological stations Ocean Weather Ship R, Valencia, Roches Point, Stilly and St Mawgan to areas 1, 2, 3, 4 and 5 respectively (Figure 4). The theoretical wind surge at x = 0, incremented by the local barometric surge, is compared with the observed surge at Milford Haven in Figure 9. The results shown are obtained from the shelf model using meteorological data deduced on the one hand from the Daily Weather Charts and on the other hand from recorded winds as just described. Manifestly, theory and observation agree remarkably well considering the simplicity of the theory. The heights of the surge peaks are reproduced quite satisfactorily, a feature noted in earlier applications of the method. It seems to matter little whether the meteorological data comes from weather charts or directly from recorded wind values. This is an important finding since it means that if the winds at the meteorological stations can be predicted, the surge at Milford Haven can be estimated in advance fairly directly. For the period 1-3 January, the first peak in elevation observed at about 21 .OOh on the 1st (Figure 9) may be attributed to the passage of a small depression across the British Isles on that day bringing strong south-westerly winds to bear on the Celtic Sea shelf (Figures 5 and 7). The higher peak experienced at about 20.00 h on the 2nd is mainly due to the very strong wind fields of a more intense secondary depression which passed over Scotland after noon on that day. The period 28-30 January is marked by southerly-type winds of progressively increasing strength over the western side of the British Isles (Figures
station
6 and 8). Due to geostrophic flow, the surge at Milford Haven increases steadily and only starts to fall at the end of the 29th when the winds begin to moderate. Surges at x = 0 calculated from the shelf model and the shelf-ocean model are compared in Figure IO. Generally the shelf model gives values a few centimetres higher than the shelf-ocean combination but these differences are not really significant in the context of the overall surge profile. Thus, in determining the coastal surge, for most practical purposes the oceanic influence may be ignored and replaced by the condition of zero elevation at the shelf edge (see equation (6)). The surge at the shelf edge, as determined from the shelf-ocean model, is plotted in Figure II. The variations are small, similar in form to those at the coast but tending to be opposite in sign ~ indicating a small seiche effect across the width of the shelf. Surges at x = 0 from the shelf model and from the shelf-ocean model, incremented by the barometric surge, are compared with the observed surge at Milford Haven for the whole of January 1976 (Figure 12). This is our most important result, for the theory is not only seen to reproduce the main surge heights well on the 2nd and the 29th. but is also seen to successfully yield the average trends of height Variation throughout the whole period. The theory is linear and therefore cannot be expected to reproduce the semi-diurnal and higher-order fluctuations evident in the observed surges: assuming that these fluctuations arise from nonlinear interaction between surge and tide. Arguing along these lines, the extent of such interaction is demonstrated in Figure 12. Residuals at Bristol Channel ports are frequently characterized by semi-diurnal oscil. lations such as are displayed in this figure.
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West Coast storm surges: R. J. McIntyre
1200
h.,
1200h., Figure 5
Synoptic
weather
I/l/76.
OOOOh.,
2/l/76.
charts for l-3
0000 January
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The observed surges in Figures 9 and 12 are those from Milford Haven incremented by 25 cm. Such an increment was carried out to achieve the best agreement between observation and theory. This datum correction might be explained in terms of a change in mean sea level produced by meteorological influences external to the area ABCD. Alternatively, the correction might reflect a mean sea level difference, due perhaps to wave action, between the port of Milford Haven and the open sea. Observed surge values in Figures 9 and 12, respectively, were derived using tidal predictions based on harmonic constants from different tidal analyses. This explains some minor differences between the observed surge curves in these respective figures.
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2/I/76.
Conclusions
and possible extensions
It has been shown that rather simple idealized models of shelf and shelf-ocean can simulate, to first-order accuracy at least, the storm surges observed at Milford Haven during the month of January 1976. Based on these models, a surge prediction system for this port, employing predicted winds at certain meteorological stations around the shores of the Celtic Sea to the south of Ireland, seems to be a possibility. Only surge reproduction for Milford Haven has been tested. However, it seems likely that this result will give a general indication of the surges at the mouth of the Bristol Channel and at the southern boundary of the Irish Sea. A number of possible extensions to the theory presented here can be envisaged. First of all, the longshore
West Coast storm surges: R. J. McIntyre
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Figure 6
h.,
1200h., Synoptic
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28/l/76.
0000
29/l/76.
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h., 29/i/76.
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variations neglected in this paper could be included by following Voit4 and using a Fourier transform with respect toy together with the Laplace transform with respect to t. The blocking effect of the coastline of southern Ireland could then be investigated, although it is an interesting and as yet unexplained fact that the omission of this effect at present does not appear to spoil our results. Secondly, the effect of wind stress over the ocean could be included, although the vast area to be covered by this could cause some problems. Thirdly, the inclusion of the Bristol Channel as a one-dimensional exponentially-varying estuary in the mathematical model might lead to a more detailed understanding of the physical situation. Finally, it appears feasible that improved accuracy could be attained by dividing the sea area under investigation into a greater
number of sub-areas for wind stress specification. These are tentative suggestions and only further work would show whether their consideration would lead to improved results.
Acknowledgements The author is grateful to Mr D. L. Blackman for determining observed surge residuals for Milford Haven, and to Mr R. A. Smith for preparing the figures. The work described in this paper was funded by a Consortium consisting of the Natural Environment Research Council, the Ministry of Agriculture, Fisheries and Food, and the Departments of Energy, Environment and Industry. The paper, in its present form, was prepared by Dr N. S. Heaps on the basis of a first draft by the author.
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NH ,
NW I II
I 40 NW I 35NW 30 NW 25 NW
NW I...
, 30-31 . 31-32 32-33 33-34 I
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25 25
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23 22 21 30
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Figure 7 Theoretical model of wind fields over shelf area ABCD from 12.00 h on 1 January to 06.00 h on 3 January shelf area with its oceanic edge lying to the right. Wind speeds in knots are followed by wind direction
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West Coast storm surges: R. J. McIntyre
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30 30 30 30 20 20
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25 S 25 S
I
t3000,
4-5
IO SW IO SW
20 20
Figure8
25’S
IO SW
X=//S
28/l/76
l-2
20 SW IO SW
I
1200, O-l
20 S 25OW 20 s
20 20
X=0
NW
x=3//5
model of wind fields over shelf area ABCD
28-29 30-31 31-32 32-33 33 -34 34-35 35-36 36 -37 37-38 38-39 39-40 40-4
1
x=4//5
x= [
from 12.00 h on 28 January to 06.00
Appl.
I
41-42
Math.
h on 30 January
Modelling,
1979,
1976
Vol 3, April
97
West Coast storm surges: R. J. McIntyre
I
,l
: ’ : ! , 1 c. , : ‘: I .‘# : \ : ( :I: : .,./‘I, ., :/ ...,8‘$
6 ‘~
60.-
4~. 7” 2 ; L o-5
6
?\, .,..
,”
:
,.”
..“‘..,2 18 24 . .. ,.__,--__ ... ,,__\‘...‘.‘..,_,,
,I’ ;;
30
‘..,,‘.<..
_..‘.
_: ‘“,’
/,....., \\ 5,. ‘\\ \\ I, ‘\
:I .!:: :: ‘\ ‘i ‘\ :, ‘.. .“,,,. “,I ‘. .: ...-
‘.I’ .....< \ :’ ,’ ,’ \.\,/’ -4.-6r
a
36
.,
4 i
t 2.. Y U,.‘..._ d
._:. \c-‘-. ‘____\ 6
Time in ~CIUE bettor 1200h
-40
‘.,-,;:,
‘...
18 ., .... ,I\ ,__-A.,> .,‘.’ ‘L_*“,f’
24 . . . . ..
..“. ..,.,
L-,
~ I’ I__’
30
;’
/ ., : ,, :,.,,_ Z.‘.\\_:, ......d
,*“i.
28 Jan 1976 -4
-60 i
Figure
9
estimated from
,200h28
Time I” hours after Surge from
at x = 0 from daily
meteorological
variations
are
shelf
weather
charts
stations
shown
compared
with
model,
using
(------) in Figure
observed
either
surge
4 f..
Figure
winds
or recorded
winds
17
shelf-ocean
.). These
charts
(------)
Haven
shown
in
at Milford
Surge
elevation
model
using
or recorded
Figure4
at shelf either
edge,
winds
winds
Jan 1976
x = 1, as determined
estimated
from
from
meteorological
daily
from weather
stations
(......)
(-_) 60
Time ,n hours after 1200h
1 Jan 1976
Day
Figure
72
Surge
or shelf-ocean Milford are in
Haven
based
on
at x = 0 obtained
model (recorded
~3 month
from
,), compared
(. ): for
whole
winds
from
either with
of January meteorological
shelf
observed 1976.
model
(------)
surge Model
stations
at results
shown
Figure 4
References 1 2
3 Figure
70
Surge
(-
) or
shelf-ocean
meteorological
98
Appl.
at x = 0 obtained model
stations
Math.
shown
Modelling,
from
(------I,
either
using
in figure
1979,
shelf
recorded
4
Vol 3, April
model winds
Lennon, G. W. Quart. J. R. Met. Sot. 1963,89,381 Heaps, N. S. Phil. Trans. R. Sot., A, 1965, 257, 351 Flather, R. A. In ‘Mathematical Models in Geophysics’, Proc. MOSCOW Symp., August 197 1. IAHS-AISH Publication No. 116,215
from 4
pp,
1976
Voit, S. S. Oceanology
1969,9,
12
Birkenhead,
Merseyside
Simple mathematical models describing wind-induced motion on a continental shelf are used to predict storm surges at the port of Milford Haven on the west coast of the British Isles. Attention is concentrated on surge events occurring during the stormy month of January 1976. It is demonstrated that useful first-order accuracies in prediction can be achieved. Oceanic influence on the coastal surge is shown to be small. Satisfactory results are obtained with wind input data coming either from the analysis of daily weather charts or from recorded values of wind speed and direction at a group of meteorological stations.
Introduction It has been known for some time1 that secondary depressions moving in from the Atlantic are responsible for many of the larger storm surges experienced along the west coast of the British Isles. Heaps* showed that surges at Milford Haven are produced by travelling wind fields, associated with such depressions, acting on the continental shelf to the south of Ireland (the Celtic Sea). He developed an analytic method to gain more knowledge of the mechanism of that surge generation. The method involved the investigation of the dynamic response of the water on an idealized shelf of uniform depth and width to wind stress fields with moving boundaries. Flathers extended the work carried out by Heaps, considering the idealized shelf connected to an ocean of uniform depth and width. The present work employs both the shelf model due to Heaps and the shelf-ocean model due to Flather for the investigation of the storm surges at Milford Haven during January 1976. This month was a very stormy one, with two large West Coast surge events occurring in it. The basic wind fields examined in the present work cover an area of shelf between any two lines parallel to the coast, the wind stress being constant and uniformly distributed over this area. Again, the mathematical method invokes the use of Iaplace transform techniques following the introduction of a step function wind stress suddenly created at time t = 0. The solution obtained for the shelfocean model is more general and more complex than that obtained by Flather. The latter restricted investigation to 0307-904X/79/020089-10/$02.00 0 1979 IPC Business Press
the effects of a steady wind field acting over the whole of the shelf. The observed and theoretically-derived surges at Milford Haven are found to be in remarkably satisfactory agreement - considering the simplicity of the models. A comparison of the results coming from the shelf and shelfocean models, respectively, indicates a small general reduction in surge levels when the ocean is included. A programme of numerical sea modelling is obviously required for a comprehensive attack on the problem of surge prediction on the West Coast generally. However, the present modest contribution could well prove to be practically useful on a more localized scale before a prediction system overall is established. No tation Shelf and adjoining ocean are rectangular
as indicated in The aim is to determine sea level variations at the coast x = 0 produced by wind stress, with onshore component P and longshore component Q, acting over the area of shelf between x = a and the shelf edge x = I (Figure 2). The cross-sectional geometry is illustrated more completely in Figure 3 and the notation may be summarized as follows:
Figure I, with x the offshore coordinate,
X,Y, z
rectangular Cartesian coordinates, a left-handed set with origin 0 at the coast in the undisturbed surface of the sea; Ox, Oy lie in this surface with Ox normal to the coastline; Oz is directed vertically downwards
Appt. Math. Modelling,
1979,
Vol 3, April
89
West Coast storm surges: R. J. fvlclntyre x,y
components
of wind stress on sea surface
angular speed of Earth’s rotation In the mathematical analysis, subscripts s and o refer respectively to shelf and ocean while subscripts 1, 2 and 3 refer respectively to different solution regions, namely: region 1 (0 < x < a), region 2 (a < x < I), region 3 (I < x < L). Figure 7 Shelf-ocean normal to coastline
system, showing a vertical cross-section
Dynamical
equations
and their solution
We employ the linearized vertically-integrated of motion and continuity:
equations
+y+Xli-/v+gh= ax P
(1) au av ay -+-+-_=o
I I I I
ax
I
I I X=0
Coast
Oceanic edge
Figure 2 Continental shelf (plan view), showing area between and x = I over which wind stress has onshore component P and longshore component Q
x =a
ay
at
where h is a friction parameter and f (= 2w sin@) the Coriolis parameter, regarded as constants. Bottom friction has components hpU, XpV and although it might be argued that h on the shelf should be different to X in the ocean (inversely proportional to the respective depths) such a difference is not allowed for here. Assuming longshore uniformity, the U, V, { are function of x and t only and derivatives with respect to Y are zero. Solutions of (1) are obtained yielding motion in shelf and ocean induced by a uniform wind stress field imposed at time t = 0 over that part of the shelf lying between x = a and x = I (Figure 2). The field is defined by: F=G=O
O
F = -M(t)
G = QH(t)
F=G=O
a
(2) 1
l
where H(t) denotes the Heavyside unit function; P, Q are constants representing onshore and longshore stress components. Initially the water is considered to be at rest so that, everywhere: U=V={=O att=O
(3)
Much as in the earlier work by Heaps* and Flather, the equations (1) are solved analytically using the Laplace transformation and the result may be denoted by:
*ii
xl0
Oceanic edge oi shelf
coast
Figure 3 Vertical showing notation
h,>ho 1 L a c u, ‘u LJ, v
90
section and plan view of shelf-ocean
model,
depths of shelf and ocean respectively width of shelf width of shelf-ocean system distance of an arbitrary longshore line from coast, on shelf elevation of water surface components of current in directions of increasing X,Y values of U, v respectively, integrated from surface to sea bed
Appl.
Math.
Modelling,
1979,
Vol 3, April
u= u,
v= vi
u= u,
v= v,
u=
v=
u3
v3
<={1
O
i-=T*
a
{={3
l
(4) 1
Two solutions (4) are obtained - each subject to (2) and (3). One corresponds to the complete shelf-ocean model portrayed in Figure 3, satisfying zero flow across the land boundaries at x = 0 and x = L, also continuity of flow and elevation across the edge of the wind area at x = a and across the shelf edge at x = 1. Accordingly, the following boundary conditions are satisfied: ur =o
atx=O
5‘1 =r*
u, = u,
atx=a
f2
u,
atx=Z
u,
=
13
= 0
= Ii,
atx=L
(5) I
West Coast storm surges: R. J. McIntyre
This solution is a generalization of the one due to Flatherl: the wind field now has its inner edge along the arbitrary longshore line x = a rather than along the coast x = 0 as before. The other solution obtained corresponds to the shelf model considered by Heaps2 and describes motion on the shelf alone with zero elevation imposed along its oceanic edge. The boundary conditions in this case are: u, =o
atx=O
rr = c*
u, = u,
t2 =0
atx=Z
Mathematical
atx=a
Secondly, {r for the shelf-ocean model is given by:
(12)
~lst~lo=(l-~)~
+2N
(h-U&(ha,)-fQ1 2 .=1pza,{(A-2a,)(X-a,)* +Xf2} (7) - cosm,) cos(m,N) co~i”x!.~~
(6)
(1 +MN) sinm, sin(m,N)
I
- (M +N)cosm,
analysis yields, formally, for the shelf model:
<1 = {rs(x, 6
G
f ,, = 1plJ;E;
t4N
(7)
and for the shelf-ocean model: (1 = L?rs(x, r) + r10 6, t)
(cos
(8)
Setting x = 0 in (7) or (8) gives the required wind-induced surge at the coast, {, say. Thus: c, =fr(x
= 0)
\ cos(m,N$
cosm,
cos(c,t + $:, + A:, ~ li/;)
(y) - cosm,)
cos(m,N)cos
i-:!,
r n
X
tMN)sinm,
sin(m,N)
~ (M +N) cosm, cos(m,N)
(9)
cosmz,
(15)
Clearly, since a is arbitrary, we may write: where: (10)
!L = L 0, Q)
M=
Without going into the details of the mathematical treatment the derived expressions for {rs and {rs t cl0 are now given. Thus, {r for the shelf model is given by:
t i (-l)n n=1
a) 1
41, plJ,,En
21
(
cos(~t
P{P + h +f*/(p I
tan(Mm)
+ 4, + A, - L)
(
(11) where : In
J,
= {P*-2PQf(X-pn)/d;
= (pi
En = {(A-
+f2Q2/d;}1’2
t I#/* 2pn t Af* cos2O,/d;)* + (2u, -
#n = tan-‘[(fey,/d~)/{P-f~(X A,, = tan-‘(v,/p,) $,
= tan-‘[{2v,
- (Af*/d$
Xf* sin20,/di)*} - ~J&~l
‘I2
sin20, >
IQ - 2~, + (V’/d;)
dn = {(h-/&)2t z+?$“Z 4I = tan-l{v,/(x-p,)}
cos2fMl (12)
and where: P =-&I,
-pn f ivn
(13)
are the roots of the cubic: p(p t x t f2/( p t X)) = -gh,(2n
- 1)%*/41*
+ ic,
(17)
+ 1)) = -gh,mzll*
(18)
m = m, being the nth positive root, in ascending order, of:
(2fi - 11:” - “‘) cos((2n ;;)?ix)e_‘n’
x sin
-b,
are the roots of the cubic:
(2n - l)r=\e-pnr
cos 21
n=1
P = -a,,
pz6,{(X-26,)(X-F,)*+hf2}
x sin
+ C(-l)n---
and where IA, J,!, , E,!,, c#&,AL, $L, dh,, 0; are given respectively by I,, J,,, E,, &, A,, $I,,, d,, t$, with 6, replaced by a,, pn by b, and u, by c,. Here:
20 - 42 P(A- f&z> - fQ>
(2n - l)ir(l-
(16)
(14)
t N tanm = 0
(19)
In both (11) and (15) surge elevation consists of a constant first term due to steady wind set-up, followed by an infinite series of pure exponential terms present because of the rotation of the Earth, followed by an infinite series of exponentially-damped periodic terms representing free modes of oscillation: progressively decreasing in period up the series. All the shelf modes in (11) have a node at the shelf edge: in the case of the Celtic Sea example considered in this paper, the first mode (n = 1) which predominates has a period of 10.2 h. Adding in the Atlantic Ocean, the shelf-ocean model gives a first mode of period 11.9 h, with a node in the ocean a distance 0.35 1 away from the shelf edge. Table 1 gives the values of 6,, Pi, v,, (2n ~ 1)7r/2 for the shelf model, and a,, b, , c,, m, for the shelf-ocean model, as used in this paper to investigate storm surges in the Celtic Sea.
Application of theory Using the results of the preceding section, storm surges generated by wind on the continental shelf to the south of Ireland are calculated. For theoretical purposes, a rectangular area of the shelf sea, ABCD, is considered (Figure 4). The coast line x = 0 is assumed to lie along BC and the oceanic edge, x = 1, along AD. The changing pattern of wind stress over ABCD is expressed as a linear combination of onshore and longshore stress fields, each of 1 dPa (= 1 dyn/cm*), acting at hourly intervals over the
Appl. Math. Modelling, 1979, Vol 3, April
91
West Coast storm
surges:
R. J. McIntyre
Table 7 Values of 6,, g,, +,, an, b,, c, (in h-’ units), along with corresponding parametric values (2% -1 )n/2, mn: used in present paper to investigate storm surges in Celtic Sea
from area 4 (Z/5 < x < 2115): { = H(t - T){,(t - T, Z/5) -H(t-T+l)~c(t-T+1,Z/5)
6,
hl
7 8 9 10
0.051078 0.083032 0.087361 0.088634 0.089169 0.089442 0.089600 0.089699 0.089765 0.089812
0.064409 0.048484 0.046319 0.045683 0.045416 0.045279 0.045200 0.045151 0.045117 0.045094
0.615328 1.45496 1 2.364946 3.287379 4.214104 5.142803 6.072570 7.002981 7.933809 8.864922
1.57080 4.71239 7.85398 10.99557 14.13716 17.27876 20.42035 23.56194 26.70353 29.84513
”
an
bn
cn
mn
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 26 27 28 29 30 31 32
0.037487 0.054336 0.071131 0.080032 0.082932 0.084456 0.086231 0.087198 0.087511 0.088053 0.088470 0.088643 0.088819 0.089027 0.089155 0.089214 0.089326 0.089415 0.089450 0.089505 0.089565 0.089598 0.089623 0.089662 0.089693 0.089705 0.089730 0.089754 0.089766 0.089780 0.089798 0.089811
0.071257 0.062832 0.054435 0.049984 0.048534 0.047772 0.046885 0.046401 0.046244 0.045973 0.045765 0.045678 0.045590 0.045487 0.045423 0‘045393 0.045337 0.045293 0.045275 0.045247 0.045218 0.045201 0.045189 0.045169 0.045154 0.045147 0.045135 0.045123 0.045117 0.045110 0.045101 0.045095
0.529872 0.642798 0.883792 1.216315 1.445689 1.631218 1.978497 2.294958 2.434995 2.753333 3.106259 3.298114 3.535959 3.894594 4.178526 4.334611 4.678280 5.022138 5.178437 5.462974 5.822816 6.061778 6.254759 6.610367 6.932046 7.073918 7.395668 7.751476 7.9446 11 8.183824 8.544168 8.829216
1.16286 1.69125 2.65294 3.86875 4.67594 5.32768 6.52850 7.61452 8.09344 9.17946 10.38028 11.03203 11.83921 13.05503 14.01671 14.54510 15.70796 16.87082 17.39922 18.36090 19.57671 20.38390 21.03565 22.23646 23.32248 23.80140 24.88743 26.08824 26.73999 27.54717 28.76299 29.72467
”
1 7
; 4
5 6
(La-lln/2
- H(t - T){,(t
< =H(t - T){,(t
Similarly,
from area 3 (21/5
(20)
(21)
c = H(t - T)S‘,(t - T, 22/5) -H(t-T+l)<&-T+1,21/5)
92
Appl.
Math.
Modelling,
1979,
Vol
L = 6148 km
h, = 100 m
h, = 4000 m
X =25~1O-~s-’
f
= 9.81 msC*
(25) 1
Here L and h, are chosen to represent the North Atlantic basin to which the Celtic Sea shelf, represented by Z and h,, connects. As before, in developing the standard responses, the series in (11) are taken up to n = 10 and the series in (15) up to y1= 32. All the modes of oscillation of the shelf and the shelf-ocean are thereby included with periods greater than 0.7 h (Table 1). It is evident from (1 l), (15) and (20)-(24) that each response will diminish exponentially as time increases. Surges at x = 0 during two periods: l-3 January and 28-30 January 1976 are evaluated theoretically (as described above) and compared with observed surge heights at Milford Haven. Zero motion is assumed in the calculations at the beginning of each period. Relevant synoptic weather charts are shown in Figures 5 and 6. Theoretical models of the wind fields for each period, constructed from information given by the Daily Weather Reports of the British Meteorological Office, are given in Figures 7 and 8. From the wind velocities in Figures 7 and 8, onshore and longshore stress components are determined hourly over sub-areas l-5 using the square law: (26) rw = C&W2 where 7, is the wind stress, W the wind speed in m/s, pa the density of the air (taken as 1.25 kg m-3), and c a drag coefficient:
= -0.12
(22)
= 112.5 x 1O-6 s-l
p = 1025 kg m-’
w<5
c x lo3 = 0.565
3, April
(23)
- T, 0)
Z = 380 km
g
< 3Z/5):
- H(t - T){,(t - T, 31/5) + H(t - T + l){,(t - T + 1, 31/5)
21/5)
+H(t-T+1)[c(t-T+1,Z/5) (24) The above procedure of combining a standard set of theoretically-derived elevation responses, corresponding to unit onshore and longshore wind stress fields over subareas 1 to 5, was successfully employed by Heaps2 to simulate the observed surges at Milford Haven during periods in September 1953 and November 1954. Milford Haven lies near to the midpoint of BC and it is therefore reasonable to compare the observed surge there with the theoretical surge evaluated at x = 0. Heaps used the shelf model exclusively; in the present work, adopting the same procedure, we use the shelf model and also the shelf-ocean model described earlier, In each case, only surges arising from wind action on the shelf are considered. Specifically, the aim here is to investigate the surges which occurred at Milford Haven during the stormy periods of January 1976. Numerical parameters used by Healq2 and later by Flather, are again employed. Thus:
from area 2 (31/5 < x < 41/5):
f = H(t - T)<,(t - T, 31/5) -H(r-T+I){C(t-T+1,3Z/5) - H(t - T)<,(t - T, 41/5) +H(t-T+1)
- T+1,
-H(tT+l){,(tT+l,O) - H(t - T)f,(t - T, Z/5)
five constituent areas shown in the figure. The elevation responses at the coast x = 0 corresponding to these fields, responses derived on the basis of (9) are combined linearly in the same way to yield the coastal surge. Thus, due to wind stress acting between t = T and t = T + 1 hours say, the contribution to the surge coming from area 1 (41/S < x < 1) takes the form: 4 = H(t - T){,(t - T, 41/5) -H(tT+l)f,(t-T+1,41/5)
- T, 21/5)
+ ff(t - T+1)c,(t and from area 5 (0 < x < Z/5):
= 2.513
t 0.137W
5 < W< 19.22 W Z 19.22
(27)
West Coast storm surges: R. J. McIntyre
Figure 4 Shelf area (ABCD) considered in theory, showing sub-areas over which five constituent wind fields act; (0) meteorological used for wind velocity; (------I 100 fathom depth contour; O.W.S. ‘R’ - Ocean Weather Ship ‘R’; V - Valencia; R - Roches Point; S - Stilly; SM - St Mawgan; M - Milford Haven
Input data for the surge calculations are thus obtained. An alternative set of stress input data (of the same form as the above) is obtained by allocating hourly wind speeds and directions recorded at the meteorological stations Ocean Weather Ship R, Valencia, Roches Point, Stilly and St Mawgan to areas 1, 2, 3, 4 and 5 respectively (Figure 4). The theoretical wind surge at x = 0, incremented by the local barometric surge, is compared with the observed surge at Milford Haven in Figure 9. The results shown are obtained from the shelf model using meteorological data deduced on the one hand from the Daily Weather Charts and on the other hand from recorded winds as just described. Manifestly, theory and observation agree remarkably well considering the simplicity of the theory. The heights of the surge peaks are reproduced quite satisfactorily, a feature noted in earlier applications of the method. It seems to matter little whether the meteorological data comes from weather charts or directly from recorded wind values. This is an important finding since it means that if the winds at the meteorological stations can be predicted, the surge at Milford Haven can be estimated in advance fairly directly. For the period 1-3 January, the first peak in elevation observed at about 21 .OOh on the 1st (Figure 9) may be attributed to the passage of a small depression across the British Isles on that day bringing strong south-westerly winds to bear on the Celtic Sea shelf (Figures 5 and 7). The higher peak experienced at about 20.00 h on the 2nd is mainly due to the very strong wind fields of a more intense secondary depression which passed over Scotland after noon on that day. The period 28-30 January is marked by southerly-type winds of progressively increasing strength over the western side of the British Isles (Figures
station
6 and 8). Due to geostrophic flow, the surge at Milford Haven increases steadily and only starts to fall at the end of the 29th when the winds begin to moderate. Surges at x = 0 calculated from the shelf model and the shelf-ocean model are compared in Figure IO. Generally the shelf model gives values a few centimetres higher than the shelf-ocean combination but these differences are not really significant in the context of the overall surge profile. Thus, in determining the coastal surge, for most practical purposes the oceanic influence may be ignored and replaced by the condition of zero elevation at the shelf edge (see equation (6)). The surge at the shelf edge, as determined from the shelf-ocean model, is plotted in Figure II. The variations are small, similar in form to those at the coast but tending to be opposite in sign ~ indicating a small seiche effect across the width of the shelf. Surges at x = 0 from the shelf model and from the shelf-ocean model, incremented by the barometric surge, are compared with the observed surge at Milford Haven for the whole of January 1976 (Figure 12). This is our most important result, for the theory is not only seen to reproduce the main surge heights well on the 2nd and the 29th. but is also seen to successfully yield the average trends of height Variation throughout the whole period. The theory is linear and therefore cannot be expected to reproduce the semi-diurnal and higher-order fluctuations evident in the observed surges: assuming that these fluctuations arise from nonlinear interaction between surge and tide. Arguing along these lines, the extent of such interaction is demonstrated in Figure 12. Residuals at Bristol Channel ports are frequently characterized by semi-diurnal oscil. lations such as are displayed in this figure.
Appl. Math. Modelling, 1979, Vol 3, April
93
West Coast storm surges: R. J. McIntyre
1200
h.,
1200h., Figure 5
Synoptic
weather
I/l/76.
OOOOh.,
2/l/76.
charts for l-3
0000 January
Appl.
Math.
Modelling,
1979,
Vol 3, April
h., 3/l /76.
1976
The observed surges in Figures 9 and 12 are those from Milford Haven incremented by 25 cm. Such an increment was carried out to achieve the best agreement between observation and theory. This datum correction might be explained in terms of a change in mean sea level produced by meteorological influences external to the area ABCD. Alternatively, the correction might reflect a mean sea level difference, due perhaps to wave action, between the port of Milford Haven and the open sea. Observed surge values in Figures 9 and 12, respectively, were derived using tidal predictions based on harmonic constants from different tidal analyses. This explains some minor differences between the observed surge curves in these respective figures.
94
2/I/76.
Conclusions
and possible extensions
It has been shown that rather simple idealized models of shelf and shelf-ocean can simulate, to first-order accuracy at least, the storm surges observed at Milford Haven during the month of January 1976. Based on these models, a surge prediction system for this port, employing predicted winds at certain meteorological stations around the shores of the Celtic Sea to the south of Ireland, seems to be a possibility. Only surge reproduction for Milford Haven has been tested. However, it seems likely that this result will give a general indication of the surges at the mouth of the Bristol Channel and at the southern boundary of the Irish Sea. A number of possible extensions to the theory presented here can be envisaged. First of all, the longshore
West Coast storm surges: R. J. McIntyre
1200
Figure 6
h.,
1200h., Synoptic
weather
28/l/76.
0000
29/l/76.
charts for 28-30
h., 29/i/76.
OOOOh., January
30/l/76.
1976
variations neglected in this paper could be included by following Voit4 and using a Fourier transform with respect toy together with the Laplace transform with respect to t. The blocking effect of the coastline of southern Ireland could then be investigated, although it is an interesting and as yet unexplained fact that the omission of this effect at present does not appear to spoil our results. Secondly, the effect of wind stress over the ocean could be included, although the vast area to be covered by this could cause some problems. Thirdly, the inclusion of the Bristol Channel as a one-dimensional exponentially-varying estuary in the mathematical model might lead to a more detailed understanding of the physical situation. Finally, it appears feasible that improved accuracy could be attained by dividing the sea area under investigation into a greater
number of sub-areas for wind stress specification. These are tentative suggestions and only further work would show whether their consideration would lead to improved results.
Acknowledgements The author is grateful to Mr D. L. Blackman for determining observed surge residuals for Milford Haven, and to Mr R. A. Smith for preparing the figures. The work described in this paper was funded by a Consortium consisting of the Natural Environment Research Council, the Ministry of Agriculture, Fisheries and Food, and the Departments of Energy, Environment and Industry. The paper, in its present form, was prepared by Dr N. S. Heaps on the basis of a first draft by the author.
Appl.
Math.
Modelling,
1979,
Vol 3, April
95
West Coast storm surges: R. J. McIntyre
1200,
IO SW
30 30
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Appl.
Math.
Modelling,
Vol 3, April
3/l/76
39-40 40-4
x=141/5
1979,
0000,
37-38
I
x=;l
Figure 7 Theoretical model of wind fields over shelf area ABCD from 12.00 h on 1 January to 06.00 h on 3 January shelf area with its oceanic edge lying to the right. Wind speeds in knots are followed by wind direction
96
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1976.
Boxes represent
West Coast storm surges: R. J. McIntyre
20
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1
x=4//5
x= [
from 12.00 h on 28 January to 06.00
Appl.
I
41-42
Math.
h on 30 January
Modelling,
1979,
1976
Vol 3, April
97
West Coast storm surges: R. J. McIntyre
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Figure
9
estimated from
,200h28
Time I” hours after Surge from
at x = 0 from daily
meteorological
variations
are
shelf
weather
charts
stations
shown
compared
with
model,
using
(------) in Figure
observed
either
surge
4 f..
Figure
winds
or recorded
winds
17
shelf-ocean
.). These
charts
(------)
Haven
shown
in
at Milford
Surge
elevation
model
using
or recorded
Figure4
at shelf either
edge,
winds
winds
Jan 1976
x = 1, as determined
estimated
from
from
meteorological
daily
from weather
stations
(......)
(-_) 60
Time ,n hours after 1200h
1 Jan 1976
Day
Figure
72
Surge
or shelf-ocean Milford are in
Haven
based
on
at x = 0 obtained
model (recorded
~3 month
from
,), compared
(. ): for
whole
winds
from
either with
of January meteorological
shelf
observed 1976.
model
(------)
surge Model
stations
at results
shown
Figure 4
References 1 2
3 Figure
70
Surge
(-
) or
shelf-ocean
meteorological
98
Appl.
at x = 0 obtained model
stations
Math.
shown
Modelling,
from
(------I,
either
using
in figure
1979,
shelf
recorded
4
Vol 3, April
model winds
Lennon, G. W. Quart. J. R. Met. Sot. 1963,89,381 Heaps, N. S. Phil. Trans. R. Sot., A, 1965, 257, 351 Flather, R. A. In ‘Mathematical Models in Geophysics’, Proc. MOSCOW Symp., August 197 1. IAHS-AISH Publication No. 116,215
from 4
pp,
1976
Voit, S. S. Oceanology
1969,9,
12