Estuarine, Coastal and Shelf Science (1987) 24,69-93

Estimates of Extreme Conditions of Tide and Surge using a Numerical Model of the North-west European Continental Shelf

R. A. Flather Institute of Oceanographic Sciences, Bidston Observatory, Birkenhead,

Merseyside, L43 7RA, U.K.

Received 17June 1985 and in revised form 10 April 1986

Keywords: tides; surges; design; statistics; models; North sea

Estimates of extreme currents and water levels due to tides, storm surges and their combination are required for the design of offshore structures, and other purposes. Techniques are described which provide such estimates for the north- west European continental shelf. The techniques employ spatial distributions of tide and surge from an established numerical sea model and existing statistical analyses of coastal sea level data. The assumptions embodied in the method are discussed and an indication given of the probable magnitude of the associated errors. The few observational estimates available provide limited verification of the results.

Introduction

The search for and development of oil and gas fields in the North Sea and other regions of the continental shelf generates a continuing requirement for environmental information for use in the design of the structures involved. In terms of oceanographic conditions, the main interest is in surface gravity waves which provide the largest contribution to the environmental forces, but a knowledge of still water levels and currents due to tides and storm surges, and in particular of their extremes, is also needed.

Traditionally, estimates of extreme still water levels have been derived from the frequency distribution of annual maxima (or minima) as described, for example by Lennon (1963). Graff (1981) discusses some of the problems in this approach as well as presenting a comprehensive analysis of the observed sea levels at 67 ports around the British Isles. For confidence in the results, observations extending over a period of order 100 years, giving a data set comprising 100 annual extremes, are desirable (Lennon, 1963). Data of this duration are available at few coastal and no offshore sites, so that the method is not applicable to the open sea.

More recently, techniques have been established which allow estimates of extremes to be derived from much shorter spans of data. In particular, the joint probability method (Pugh & Vassie, 1979, 1980), in which frequency distributions of tide and storm surge components, derived separately, are combined to produce the required probabilities of extreme high and low sea level, can give useful results from as little as one year of

69

0272-7714/87/010069+25 $03.00/O 0 1987 Academic Press Inc. (London) Limited

70 R. A. Flather

measurements. This is achieved by considering hourly values of tide and surge, of which there are N 9000 in a single year compared with only one annual maximum. Much more efficient use is thus made of the available data. Pugh and Vassie (1979,198O) applied the technique to sea levels at seven coastal sites round the British Isles, deriving estimates of extreme surges, tides and total still water levels. Special consideration was given to the analysis for Southend, where non-linear interaction between tide and surge renders invalid the fundamental assumption in the basic approach that tide and surge components are statistically independent.

A significant added advantage of the joint probability method is that its efficient use of data makes it applicable to current meter observations. Pugh (1982) demonstrated this by using it to analyse one year of current data from Inner Dowsing light tower (5319N, OO35E) in the North Sea, indicating the additional considerations arising from the treat- ment of vectors rather than scalars. Directional distributions of extreme current speeds were derived from the results. Some further aspects of the use of the technique to estimate extreme currents are discussed by Vassie and Flather (1986).

Despite these advances, the poor spatial coverage of long observational series seriously limits the usefulness of the procedures outlined above, and we are compelled to seek alternatives. One attractive possibility is to make use of hydrodynamical models to generate storm surge data. Since surge models generally cover the complete area of surge generation, data can be produced for both offshore and coastal locations. Models have been used in a number of studies of the return periods of hurricane generated surges on open coasts of the United States (see Murty, 1984, pp. 704-727), and also for Hong Kong and sections of the Australian coast. A typical procedure involves identifying all hurri- canes which have generated surges in the region of interest; computing the surge heights using the model; combining the surges with the astronomical tides under appropriate assumptions; and deriving the required extreme levels by statistical analysis of the results. There are, however, variations on these basic steps.

For the North Sea, Ishiguro (1983) has developed an electronic analogue model and used it to predict distributions of extreme surge elevation. The approach is based on the response of the model to uniform and steady wind and pressure fields (constant direction and constant intensity) suddenly applied to the sea surface over the whole region. Linear combination of the model responses to orthogonal wind (and pressure) fields gives the surge elevation at each point as a function of wind direction and time. Then, starting from an extreme wind speed, such as the hourly mean wind speed exceeded on average once in 50 years (Anon. 1977), and applying an empirical reduction factor to allow for the wind duration required to generate the peak surge, the maximum surge and the wind direction producing it are extracted at each point. Clearly, the wind direction and duration required to generate the maximum surge will vary from point to point. The spatial distribution so derived represents the extreme surge generated by a wind speed occurring once in 50 years. The results agree quite well with the largest surges observed at coastal locations round the North Sea and with 50 year surge elevations derived from observations at Lerwick, Aberdeen and Southend by Pugh and Vassie (1979). There remains, however, some uncertainty as to the influence of the non-uniformity of real wind fields, dynamical effects associated with the movement of depressions and tide-surge interaction. The prob- ability of the extreme wind speed occurring in the direction needed to produce the peak surge response is also unaccounted for.

The approach adopted in the present work is to use a numerical model of the north-west European continental shelf to reproduce the water movements associated with the tides

Estimates of tide and surge using a numerical model 71

and with a number of actual storms. The complex bathymetry and shape of the North Sea and neighbouring areas make it difficult to identify a priori a comprehensive set of storms which can be expected to account for ail the largest surges throughout the region. There is, on the other hand, a large and comprehensive body of observational information on tides, storm surges and total still water levels and useful analyses of their extremes, as discussed earlier. Making use of reasonably representative model simulations together with data derived from the observations leads to the required estimates of extreme conditions.

The model employed here was developed at 10s and used operationally for storm surge prediction at the Meteorological Office from 1978 to 1982 when it was replaced by an improved and extended version (Proctor & Flather, 1983). It is a two-dimensional model based on non-linear depth-averaged hydrodynamical equations in spherical polar co- ordinates, which are solved numerically to give surface elevation and east and north components of current on the grid shown in Figure 1 (Davies & Flather, 1978).

In the following sections we consider, in turn, tides, storm surges and their combi- nation, giving results from the model and indicating how, when combined with analyses of coastal observations, estimates can be derived of the distribution over the shelf of the associated extremes of elevation and depth mean current. The paper concludes with a discussion.

Tides

Spatial distributions of the M, and S, constituents of tidal elevation, derived from harmonic analysis of results of a 29-day calculation using the model, are shown in Figure 2. The distributions are in good agreement with published charts based on observations (see, for example, Proudman & Doodson, 1924; Doodson & Corkan, 1932; Robinson, 1979) and other models (e.g. Flather, 1976; Pingree & Griffiths, 1981). A comparison of computed amplitudes with those observed at a number of coastal locations in Great Britain and Ireland is given in Table 1. Typical errors are of order 1001,. Larger errors occur at a few places, for example Dover and Kings Lynn, probably because of inadequate model resolution. The large error at Portland is associated with the incorrect representation of the degenerate amphidrome on the south coast of England, which in turn could be due to resolution problems.

The M, and S, constituents combine to give a spring-neap cycle in the model tide. Adding the amplitudes, Hhl and H,, of these constituents at any point gives a useful and easily obtained indication of the tidal amplitude at springs. Similarly, subtracting the amplitude of S,, H,, from that of M,, HM, gives an approximation for the tidal amplitude under neap conditions.

Corresponding results for depth-mean currents have been used to construct the distri- butions in Figure 3, which give currents associated with spring tides. Harmonic analyses of 29 days of model generated data gave the amplitudes and phases of the M, and S, constituents of the east-directed and north-directed components of the depth-mean current at each grid point. The parameters describing the M, and S, tidal current ellipses were then derived. These parameters are:

a the semi-major axis of the ellipse; b the semi-minor axis; a positive value was used to indicate anticlockwise rotation of

the current vector, a negative value for clockwise rotation;

72 R. A. Flather

50-N SO'N

1OW 0"E 10'E

Figure 1. Finite difference grid of the continental shelf model (CSM) with locations of some points referred to in the text.

ga

a

the phase lag relative to high water of the equilibrium tide on the Greenwich meridian of the first current maximum (of which there are two corresponding to the two major axes of the ellipse) (0

Estimates of tide and surge using a numerical model 73

expressed as a percentage. Positive or negative values, arising from the sign of b, indicate the sense of rotation of the tidal current vector. Large values, of either sign, indicate that the ellipse is fat, + 100 corresponding to a circle. Small values indicate that the current is nearly rectilinear.

The distributions derived from the model, plotted in Figure 3, are generally in very good agreement with previously published charts, e.g. Sager and Sammler (1975). The latter distribution corresponding to Figure 3a contains a number of local maxima associated with small scale features which are not resolved by the model, e.g. Spurn Point. This suggests that tidal currents can be substantially influenced by local bathymetry and consequently their distribution contains more small scale variability than tidal elevations. Clearly the model cannot resolve such small scale features. It should also be noted that on the Hebridean shelf, diurnal tidal currents associated with continental shelf waves are known to be significant (Cartwright et al., 1980). In this area, the model based distributions will be unreliable. For similar reasons, the model results should be treated with caution near other bathymetric features which might support such motions.

The overall dominance of the semi-diurnal constituents over the region suggests that the spatial distributions for spring tides should generally be similar to those corresponding to extreme tides. In particular, for elevations, the spring tidal amplitude distribution (HM + H,) should be similar to the distributions of highest and lowest astronomical tides (HAT and LAT) measured relative to mean sea level. Clearly, this will not be true everywhere. In particular, close to the centres of amphidromic regions in Figure 2, the semi-diurnal tides are small and the extremes will be determined to a greater extent by constituents of other species (diurnal and long-period). Since the tides will, in any case, be relatively small in these regions, this restriction may not be of any practical importance. Also, in some nearshore areas, the distortion of the tidal curve associated with shallow water constituents may become significant.

Empirical factors, T + and T -, relating HAT and LAT (measured from mean sea level) to the spring tidal amplitude for eight coastal reference ports [the seven ports analysed by Pugh and Vassie (1979), with the addition of Immingham, data for which has been provided by G. A. Alcock and D. L. Blackman (1985)] are given in Table 2. Some variability can be seen, in particular,s+ varies from 1.24 to 1.61 withameanof 1,37andz- varies from - 1.17 to - 1.49 with a mean of - 1.37. Nevertheless, it is suggested that reasonable estimates of HAT and LAT at any location can be obtained by extracting the spring tidal amplitude there from Figure 2, and multiplying it by r or r- from the nearest reference port or by interpolated values from two reference ports, as appropriate.

Regrettably, there is little published data for offshore locations based on observations against which to check the estimates derived from the model. There is, however, a set of values of HAT and LAT for a number of coastal tide gauge sites published in the Admiralty Tide Tables (Anon., 1984). Although these values cannot easily be related to mean sea level, they do provide a check on the accuracy with which the model estimates the extreme range of the tide. For example, estimates of HAT - LAT from the model (with the equivalent value from Anon. 1984 in brackets) are:

(a) North Shields using the mean of factors for reference ports Aberdeen and Immingham-563 cm (570 cm) error - 1 lo;

(b) Lowestoft using the mean of Immingham and Southend- cm (280 cm) error 20,,; (c) Esbjerg using Immingham-232 cm (230 cm) error N 19,; (d) Bergen using Lerwick-149 cm (170 cm) error 127;.

74 R. A. Flather

Figure 2. Charts of (a) the M, and (b) the S, tidal constituents derived from the model (- amplitude, cm; - -- phase, ).

The estimates for North Shields and Esbjerg are excellent, and that for Bergen is reason- ably good. The larger error at Lowestoft probably reflects a significant contribution from other tidal species close to the amphidromic point in the Southern Bight of the North Sea.

Similar procedures, applied to the current distributions in Figure 3, can be used to give estimates of extreme tidal currents. Unfortunately, there are very few estimates of extreme tidal currents based directly on observations, so that scaling factors equivalent to those for elevations, given in Table 2, but giving the ratio of extreme to spring tidal current speeds can only be obtained at a few locations. Since this ratio may be sensitive to the small scale variations in current mentioned earlier, and may also depend on the height above the sea bed at which the measurements were taken, they would, in any case, be of limited value.

An alternative is to make the basic assumption that current is roughly proportional to elevation, so that scaling based on elevations may be applied. Clearly this assumption implies

Estimates of tide and surge using a numerical model 75

b)

Figure 2 continued

greater uncertainty in the resulting estimates of extreme tidal currents than was the case for elevations. The appropriate factor here would be the ratio of extreme to spring tidal range i(HAT - LAT)/(H, + Hs), listed as tC in Table 2. Values of rc range from 1.21 to 155 with a mean of 1.37. It is suggested that an estimate of the extreme tidal current at any location can be obtained by extracting the semi-major axis of the spring tide ellipse there from Figure 3(a)

and multiplying by the appropriate value of rc from Table 2, in a similar manner to that applied to elevations. It may be assumed that the ratio b/a and orientation a of the extreme tidal current ellipse are the same as those for spring tides, given in Figures 3(b) and (c).

The only readily available data for verification of the model based estimates comes from the analysis by Pugh (1982) of data from Inner Dowsing. This is not ideal because of the known influence of the Wash, not represented in the model, on the currents in the region (see Pugh & Vassie, 1976). Nevertheless, taking values from Figure 3, a=80 cm s- ,

b=25cms- and a G 200. Equivalent values based directly on the analysis given by Pugh andVassiearea=94cms-,b=24cms- , a = 186. So that the model underestimates the

TABL

E 1.

Tid

e,

extre

me

surg

e an

d st

ill

wate

r le

vel

data

de

rived

fro

m

obse

rvat

ions

and

from

th

e m

odel

. Un

less

ot

herw

ise

indi

cate

d,

the

obse

rvat

iona

l va

lues

qu

oted

fo

r S>

, an

d L:

,, ba

sed

on

join

t pr

obab

ilitie

s ar

e fro

m

the

work

of

Pugh

an

d Va

ssie

(1

979)

an

d fo

r L

:s,,

base

d on

an

nual

m

axim

a of

Gr

aff

(198

1).

The

mod

el

base

d va

lues

of

S:

,, ar

e th

ose

base

d on

th

e m

axim

um

com

pute

d su

rge

Tida

l am

plitu

de

loo-y

ear

leve

l L:

,, 50

-year

+

ve

surg

e S

&

4

(relat

ive

to

MSL

)

Ht.4

s2

Hs

~

HM+H

s+S:

o ob

serv

ed

mod

el

obse

rved

m

odel

ob

serv

ation

m

odel

an

nual

m

ax.

join

t pr

ob.

mod

el

Dubl

in

134

123

40

37

111

266

271

Belfa

st

120

101

29

23

116

270

240

Ardr

ossa

n 10

7 12

6 27

31

20

6 31

9 36

3 He

ysha

m

316

I 28

9 10

3 I

89

218

629

6001

59

6 Ba

rrow

308

100

Hilb

re

Isla

nd

292

281

95

87

190

547

548

Holyh

ead

181

164

60

51

132

336

347

Fish

guar

d 13

6 14

4 53

53

10

5 10

5 31

5 33

3 30

2 M

ilford

Ha

ven

224

212

82

73

127

436

412

Swan

sea

314

313

113

104

156

564

573

Avon

mou

th

425

1 42

5 15

0 1

144

171

837*

3

740

Card

iff 39

2 14

6 73

6 Ne

wlyn

17

1 17

3 58

56

87

87

31

5 34

3 31

6 De

vonp

ort

169

167

60

57

89

299

313

Portl

and

58

96

30

42

103

198

241

Ports

mou

th

139

158

43

46

125

286

329

Newh

aven

22

4 20

7 73

60

13

7 40

3t

404

Dove

r 22

4 26

7 71

78

20

8 44

8 55

3 M

arga

te 16

4 16

0 48

44

23

2 35

5 43

6 Sh

eern

ess

202

I 19

3 58

I

54

264t

27

5 43

5 1

522

Sout

hend

20

4 59

27

5 44

6 46

01

Harw

ich

133

137

38

37

248t

23

8 36

5t

412

Lowe

stof

t 70

68

21

22

23

5t

239

283t

32

9 Ki

ngs

Ly

nn

201

239

60

77

227

547

543

Imm

ingha

m

228

213

75

69

207t

22

5 46

0t

507

North

Sh

ield

s 15

8 15

9 53

52

15

0 33

6 36

1 Ro

syth

19

0 18

4 65

62

14

7 36

0 39

3 Ab

erde

en

130

129

45

43

127

127

280

293

299

Lerw

ick

59

58

21

22

76

76

158

168

156

Ulla

pool

15

1 15

6 59

58

11

5 33

5 32

9 St

omow

ay

139

144

55

53

101

101

- 31

8 29

8 M

alin

Head

10

9 11

4 42

45

11

3 11

3 -

280

272

*Fro

m

Blac

kman

(1

985)

. tG

. A.

Al

cock

an

d D.

L.

Bl

ackm

an

(per

s. co

mm

unica

tion)

, 19

85.

IMea

n of

5

and

9 ba

nd

resu

lts

from

Pu

gh

and

Vass

ie

(198

0),

Tabl

e 6.

Estimates of tide and surfe usin,y a numerical model 77

value of a for spring tide conditions by about 157;. Using the factor rc for Immingham leads to estimates of extreme tidal current based on the model of

a=103cms~, b=32cms-,

and based on the actual spring tide values of

a=121 cms~, b=31 ems-,

These may be compared with the extreme tidal current speed from Pugh (1982) of 130cms- based on tidal current harmonic analyses. Clearly, given an accurate value for spring tides, the scaling gives a good estimate for the extreme value, only 7yb low. The model based value is 2 1 I 0 low, but this is seen to be due largely to the underestimated spring tide value of a. The underestimate may be due to deficiencies of the model or to the difference between the tidal current speed at the level of the observations and the depth mean value. Clearly, further estimates of extreme current based on observations are required for model verification.

Surges

Table 3 lists the storm periods simulated by the model from which data were included in the present analysis. The choice was based on, first, the availability of data from earlier simula- tions. Thus, storms A-D belong to a period of 44 days in November and December 1973 investigated by Davies and Flather (1978), in which a careful comparison of computed and observed surge elevations at coastal sites round the North Sea was made. Similarly, an intensive investigation of storm J was carried out by Flather and Davies (1978), with special emphasis on the specification of meteorological forcing. The best results were obtained by using a combination of data from the IO-level atmospheric forecast model, operated by the Meteorological Office, with analyses of wind and pressure fields over the North Sea based on observations (Harding & Binding, 1978). These latter analyses were carried out to provide meteorological data for use in the North Sea Wave Modelling Project-NORSWAM. For this event, in addition to the verification of computed elevations, some comparisons between computed and observed currents off the north-east coast of England were made. The agree- ment was very good. Storms L and M produced substantial surges in the Irish Sea, investi- gated by Flather (198 l), and storm N (Flather, 1979) produced some of the highest recorded sea levels on the north-east coast of England. For storm 0, a somewhat unusual event associated with easterly gales over the southern North Sea, use was made of some unpub- lished simulations.

A second criterion was the availability of the required meteorological data. The use of existing data from the Meteorological Office atmospheric model together with NORSWAM wind and pressure data for the North Sea permitted new simulations of storms E-I and K specifically for the present purpose. Finally, in view of the special signifi- cance of the great surge of 1953, storm P, an investigation of this event was carried out. The results are reported in Flather (1984).

All of the storm simulations involved the same procedure, requiring two model runs. In the first, the motion due to the M, and S, tidal constituents and an approximate surge based on the hydrostatic law, introduced on the open boundaries of the model, together with forcing by surface wind stress and gradients of atmospheric pressure, applied over the interior of the sea, was computed. This yielded the total motion due to tide and surge. A second run was then carried out to determine the model tide without meteorological forcing

78 R. A. Flather

55-N

50 N

io*w 5*w 0 5-E I I I 1

a)

Figure 3. Parameters defining the spring tidal ellipse derived from the model: (a) semi-major axis, a (cm s-l); (b) ratio of minor/major axes, 100 b/a (%), +ve/-ve for anticlockwise/clockwise rotation of the current vector; (c)orientation of the major axis, a ( true).

or surge boundary input. The difference between the two solutions gives the surge residual, including effects of surge-tide interaction generated by computing the two components of the motion together in the first run. The resulting values of elevation, east- directed and north-directed components of the depth mean current were saved at hourly intervals for every grid point of the model.

From the 53 days of model data at each point, two sets of parameters were extracted. The first consisted of the maximum and minimum computed surge elevations, SL,, and S,,,, respectively, and the maximum computed surge current speed q& with its associated direction, us. Since the sample of storms analysed included a number of extreme events, the distributions of S La, and q,fax should approximate those of the required extreme surge elevation and current. The accuracy of the approximation could be assessed by comparing

Estimates of tide and surge using a numerical model 79

IOW O0 5OE IOE

60N

5PN

5dN

Figure 3 continued

TABLE 2. Factors relating extreme levels due to tide and surge to values derived from model results

Newlyn Fishguard Malin Hd. Stomoway Lerwick Aberdeen Immingham Southend

I+ 1.27 1.48 1.44 1.36 1.61 1.30 1.24 1.24 T - 1.28 - 1.29 -1.41 -1.49 - 1.49 -1.47 - 1.33 - 1.17

TL 1.28 1.38 1.43 1.43 155 1.39 1.29 1.21 2.33 1.17 1.44 1.20 1.41 1.18 0.92 0.96 7.63 5.59 4.56 3.69 3.33 4.19 3.70 3.96

r + = HAT/& + If,),

I?+ = s:,/s;,,, mean sea level, T - = LAT/(HM + H,), rc = +(HAT - LAT)/(H, + H,),

(HAT and LAT are measured relative to

the computed value of SL,, with the positive surge elevation with return period, say, 50 years, S;,, extracted from the analyses of Pugh and Vassie (1979). Values of the ratio

CT+ = s:,/sim

80 R. A. Flather

55N

50N

Figure 3 continued

are given in Table 2. In the southern and central North Sea, o+ N 1, indicating that the computed surges were comparable with one in 50 year events. Further north, as on the Hebridean shelf and in the Irish Sea, of increases to 1.2-1.4, so that the computed surges still give a fair approximation to the extremes. At Newlyn, however, G+ is 2.33, reflecting the absence of any large surge affecting the Celtic Sea and western English Channel in our sample of storms. Because the sample of storms favoured positive rather than negative surges, the distribution of S,,, was not considered to be a good approximation for extreme negative surges.

Although the maximum computed value in the sample is, in a sense, an extreme value in its own right, it must be borne in mind that it will be very sensitive to the data included. In particular, omitting one extreme event may change its value substantially. A more stable alternative is obtained by computing

m 11= , mz2= , mJ3= -c(>

Estimates of tide and surge using a numerical model 81

TABLE 3. List of storms simulated by the CSM and included in the sample.

Storm Period NORSWAM

No.

A 1800 14/11-1800 17/11/73 - B 1800 17/11-000021/11/73 30 C 1800 05/12-0000 09/12/73 - D 0600 12/12-0000 16/12/73 33 E 1800 16/12-1200 20/12/74 19 F 0600 04/01-1800 07/01/75 59 G 0600 21/01-1800 25/01/75 60 H 1800 26/11-120029/11/75 61

f 0000 0000 01/12-0000 OZjOl-1200 04/12/75 04/01/76 62 63 K 0000 19/01-0600 23/01/76 64 L 0600 lO/ll-1800 13/11/77 M 1800 13/11-0000 16/11/77 N 0600 09/01-1800 12/01/78 0 1800 28/12/78- 1200 01/01/79 - P 0000 31/01-0000 02/02/53 -

and

m 12= .

Here, < is the surge elevation, u and v the east and north directed components of the

depth-mean surge current, and < > indicates that the mean of all hourly values in the sample is taken. These parameters may be more representative of average surge conditions

than of extremes. Values of

o;= S:,/(m,,)i

are given in Table 2. They contain almost as much scatter as do the values of o-+ . Better estimates of, say, S:, for some offshore point may be obtained by taking the

computed value of SL,, there and multiplying by the appropriate value of cr+ from the nearest reference port, or a value interpolated between two reference ports, as pro-

posed for the tides. An alternative estimate can be obtained from the mean square surge parameters using 0:. Essentially, the spatial distributions derived from model results are used to interpolate the 50 year surge values derived from observations at coastal sites.

Spatial distributions of S:, can be obtained by carrying out these procedures for each model point and contouring the results. Figure 4(a) shows the distribution obtained from Sf max using o+ from the seven ports analysed by Pugh and Vassie (1979) (those in Table 2 less Immingham, which analysis had not then been carried out). The interpolation scheme used for g+ was the simplest possible, assuming either a linear variation in one direction or taking simply the value from the nearest reference port. The equivalent distribution derived from (m,$ using f~: is plotted in Figure 5(a). It can be seen that the two distri- butions are broadly similar, perhaps the most noticeable differences being in the southern central North Sea, where the estimate based on mean square surge values is about 50 cm higher than that based on Si,,. The new joint probability analyses carried out by G. A. Alcock and D. L. Blackman (1985) for East Anglian ports provide an independent check of the computed values. It can be seen from Table 1 that the model values based on S&,

82 R. A. Flather

55N

50N

IOW

a)

Figure 4. (a) Storm surge elevation, S:,

(ems-7, (cm), and (b) storm surge current speed, q&,

with return period 50 years derived from the distributions of maximum computed elevation and current.

(included in the Table) are in good agreement with observational values, with errors ranging from + 9% at Immingham to - 4% at Harwich. At Lowestoft, in particular, the agreement is excellent, whereas the value based on the RMS surge distribution [Figure 5(a)] is about 40 cm (17%) high. This suggests that the distribution based on the maximum computed surge may be the more accurate, though clearly observational estimates from other areas are required to confirm this. Both of the present distributions give larger values of S

Estimates of tide and surge using a numerical model a3

50N

10-w 5OW O0 5E IOE

L-w

-L.

I . /

b,-.

Figure 4 continued i --- -+ 4

therefore implied for currents than for surge elevations. An estimate of the surge current speed with return period 50 years, qSo, based on the maximum computed surge current, q&,,

is obtained by taking q&, = o+qz,,. The resulting distribution is plotted in Figure 4(b). In a similar fashion, it may be assumed that the mean square surge parameters, m, 1, m, 2 and mz2 define an elliptical distribution of surge current with major axis, minor axis and orientation given by

a2=ml, sins +m,, cosa +m,, sin2a

b2= m,, cos 2 a fm,, sin2a -ml2 sin2a

tan2a=2m,,/(m2,-m,,).

Then the required estimate of q&, is aLa. The resulting distribution is plotted in Figure 5(b). As was the case for elevations, there are broad similarities between the two distributions. The maximum centred on Dogger Bank, for example, is nearly the same. There are strong currents close to the Danish coast in both distributions, but those in Figure 5(b) are larger by a factor of almost 2. This may be due to a persistent current into the

84 R. A. Flather

a)

Figure 5. (a) Storm surge elevation (cm), and (b) storm surge current speed (cm s-l), with return period 50 years derived from the mean square surge parameters.

Skaggerak induced by the condition applied on the open boundary there. Such a flow would contribute substantially to the mean square parameters. The distribution based on maxi- mum computed surge, Figure 4(b), gives generally slightly stronger currents in the northern North Sea. Since the observational evidence suggests that its equivalent for elevations is the more accurate, it may be appropriate to take Figure 4b as the better distribution of 50 year surge current speed.

The computed surge currents have also been examined from the point of view of determining their directional properties. The ellipse parameters, defined above, provide directional distributions directly. In terms of maximum computed currents, an equivalent is to consider the maximum current speeds in directional sectors. This was done by taking 24 sectors, each of angle 19, the first centred on O, i.e. directed due north. Taking the maxi- mum computed current speed in each sector from the complete sample of storms, and scaling

Estimates of tide and surge usinn a numerical model 85

b)

LI Figure 5 continued

this by (T+ gives a directional distribution of extreme surge current. However, it is essential to note that the division of an already limited data set into 24 parts, each of which, on average, contains only about 50 samples implies great uncertainty in the reliability of the results.

Directional distributions of & based on both maximum computed and mean square surge estimates are plotted for five locations in the North Sea in Figure 6. There is broad agree- ment in the directions of strongest and weakest surge currents predicted by the alternative approaches, but the range of speeds is greater in the estimates based on maximum computed values than in the elliptical distributions. At some locations, for example B, the current speed does not vary much with direction, whereas at others, E, for example, the strongest currents are clearly directed either to the SE or NW. The nature of the directional distribution is indicated, for example, by the ratio of minor to major axes, b/a, of the elliptical distribution. Small ratios indicate a strong directional preference; large ratios suggest that the strongest surge currents can occur in any direction. Figure 7 shows the distribution of surge current characteristics over the shelf obtained in this way, using b/a < 0.3 to indicate strong direc- tional preference and b/u > 0.7 to indicate weak directional preference. In the North Sea

86 R. A. Flather

:I I I \ \ OO

\ \

ki

OO 2 I_

\ I

/I

/I I

-2

-3

-tn

Estimates of tide and surge using a numerical model 87

SO F

5o h

3-

b/a > 0.7

0.7 >/ b/a > O-3 m

b/a 4 0.3 unshadec

Figure 7. Properties of the directional distribution ofextkme surge current based on the elliptical distribution derived from mean square surge parameters.

there appear to be preferred directions only in a coastal band, probably because of the constraining influence of the coastlines. In the English Channel and St Georges Channel the surge currents tend to flow along the channel axis. In many offshore areas of the North Sea and Celtic Sea, the strongest surge currents can occur in any direction. The arrows on Figure 4(b) indicate the direction of the maximum surge current in regions which show a preference.

There are no independent estimates of extreme surge current for direct verification of the model results. A very limited comparison is possible for Inner Dowsing using information presented by Pugh (1982). From the model results, Figure 4(b), & = 65 cm s- I, directed towards 163, i.e. SSE. The directional distributions suggest that the strongest currents occur towards the SSE or NNW, with b/a + 0.27 at the nearest model grid point. The joint frequency distribution of surge current components based on observations (Pugh, 1982) shows that the maximum surge current to occur in the period covered was about 50 cm s- . It appears that currents directed along the north-south axis are more probable than those in

88 R. A. Flather

Boo-

700 -

6OQ-

500 -

L+loo

(observed) cm. 400 -

300 -

0 loo 200 300 400 500 600 700 600 900

HM+ H,+S,, (model) cm.

Figure 8. Comparison of computed values of HM+ H,+ S:, and observed values of L:oo, relative to mean sea level, for British ports. 0 and X indicate that the value of L :o,, was based on the generalized extreme value method, or the joint probability method, respectively.

the east-west direction, but the largest observed currents show no clearly favoured direc- tion. Many more analyses of long series of current observations are required for adequate verification of the model based estimates.

Combination of tides and surges

Although model solutions were obtained for the total motion due to tide and surge for the 16 storms, it is probably better to consider the contributions separately and then to combine them. In this way it is implicitly assumed, as in the joint probability method, that the surge and tide may occur in any combination. The starting point, therefore, is to consider those combinations of tide and surge which approximate the required total level. Clearly there are many possible combinations. The easiest to compute from the information already pre- sented is the sum of the amplitudes, HM and H,, of the M, and S, tidal constituents and the 50-year positive surge, S:,. The values based on model results are given in the last column

of Table 1 with observational estimates of the loo-year water level, L: oo, for comparison. Plotting the observed values of L:,, a g ainst the model estimates of H, + H, + S:,, as in Figure 8, it is clear that almost all points lie between the lines L Too = 1-l (H, + H, + S &) and L+ ioo = 0.8 (HM+ H,+ S:,). Those points which lie close to or outside these limits include Belfast (BF) and Portland (PL), where the model tide does not agree well with that observed

Estimates of tide and surge using a numerical model 89

Figure 9. The distribution of H,+ H,+ S:,, from the model.

(Table l), and Avonmouth (AV), Dover (DV) and Margate (MG) which locations may be affected by poor model resolution or be subject to significant tide-surge interaction. Overall, it appears that the combination of a spring tide and a 50-year surge gives an estimate of the

loo-year level which should be accurate to within about loo/,, with a slight tendency to overestimation.

Figure 9 shows the spatial distribution of the model based HM + H, + S:,, which should therefore approximate that of L: ,,o. A slightly more refined estimate of L ror, at any required location may be obtained by taking an appropriate scaling factor from nearby reference ports, using the data given in Table 1, and using it to adjust the value for the position of interest read from Figure 9. This procedure is analogous to that used earlier for tides and

surges. In seeking to extend the procedure to estimating extreme total currents due to tide and

surge, further complications arise because of the directional properties of the two contributions. Various possibilities might be adopted:

90 R. A. Flather

(a) Ignoring the directional properties of the tide and surge currents-simplest would be to combine the major axis of the spring tidal current ellipse (Figure 3) and the 50-year surge current (Figure 4) as though they were scalars, using the same method as suggested for elevations. The resulting current might be assumed to occur in any direction giving a circular distribution of extreme total current. This would be a fairly conservative assumption which might lead to high estimates of extreme total currents in cases where the orientation of the major axis of the tidal ellipse was significantly different from the preferred direction of surge currents.

(b) Taking account of the directional properties of the tidal current but ignoring directional variations of the extreme surge current-Since the directional distribution of the tidal currents can be estimated with reasonable confidence (as plotted in Figure 3), a slightly less conserva- tive assumption would be that the extreme surge current might occur in any direction. This is actually the case in many areas, as discussed above. Then, for any chosen direction, the appropriate scalar combination of the spring tide current in that direction and &, (irrespec- tive of direction) would be taken as the estimate for the loo-year total current, qf;,,. The direction of the strongest total current under this assumption would be the same as the orientation of the major axis of the spring tide ellipse, with weaker currents in other directions.

(c) Taking account of the directional properties of both tide and surge currents-The final possibility would be to take the appropriate scalar combination of the components of tidal current and surge current in each chosen direction. The uncertainties in the directional properties of the surge currents, discussed in the preceding section, suggest that limited significance should attach to the results so obtained.

Consider, now, how the estimates of extreme total current obtained from the model compare with those based on observations at Inner Dowsing (Pugh, 1982). Values for tides and surges have already been quoted. They were for the spring tide ellipse a = 80 cm s- 1 (94), b= 25 cm s- (24), a=200 (186), where the bracketed values are those based on observations, and q&, = 65 cm s- in the direction as = 163. The basic estimate of extreme total current is thus qkoO = a + do = 145 cm s - from the model. The loo-year current speed based on observations is 163 cm s- towards 188 (Pugh, 1982). In other words, the model estimate is N 11 y0 low. As pointed out earlier, the model underestimates the spring tidal currents by about 15% at this location. Using the observed value of a gives qkoo = 159 cm s-l, which is remarkably close to the observed value. Accepting the direc- tional distribution of tidal current, the extreme total current perpendicular to the direction of the strongest current is estimated to be

4400 = IbI+q&=90cms- The value from Pugh (1982) is u 72 cm s - , so again the agreement is fair.

Consulting Table 1, both Immingham and Kings Lynn might be considered appropriate reference ports for Inner Dowsing. The relevant factors, L :oo/(HM + Zf, + S:,) are 0.91 and 1.01, respectively. On this basis, the model estimates, given above, might be reduced by about 5 Oh.

Discussion

Procedures have been suggested by means of which spatial distributions of model generated tide and surge parameters may be combined with estimates of extreme tide, surge and total water levels at a few coastal tide gauge sites to give estimates of extreme conditions of

Estimates of tide and surge using a numerical model 91

tide, surge and their combination at any offshore location. Essentially, the model results provide the spatial variation, and the analysis of coastal observations gives the probability distribution.

The probable accuracy of the estimates produced will depend on a number of factors, including the accuracy with which the model reproduces the motion of interest, the validity of the various assumptions made, and the distance from the location of interest to the nearest

reference port. Model errors may be associated with the approximations used in its formula- tion, assumptions about open boundary conditions, and its lack of spatial resolution. Some of these were discussed earlier. A simple and useful indication of the error associated with limited model resolution is obtained by taking half the difference between the predicted values at points separated by one grid length ( N 30 km). The most comprehensive verifica- tion of the model is provided by comparisons of computed and observed elevations due to

tides and surges at coastal sites. These suggest errors of order 100/b. Since the changes in level are brought about by transports represented by the model depth-mean currents, this sug- gests indirectly that the depth-mean currents should also be accurate to within about loo,. Direct verification of model currents presents many difficulties. Perhaps the most significant problem is that of relating currents measured at specific levels in the water column to the

depth-mean value. This requires an understanding of the vertical structure of currents associated with tides and storm surges which probably goes beyond our present knowledge. Even then there remains the uncertain but important influence of small scale differences in currents, associated with local bathymetry and coastal orientation.

In estimating the extreme elevations associated with tides, storm surges and their combi-

nation, it was assumed that the spatial distribution of the required extreme was similar to that of some parameter derived from the model results. For extreme tides, HAT and LAT (measured relative to mean sea level), the model parameter chosen was the spring tidal amplitude obtained simply by summing the amplitudes of the M, and S, constituents. Except near amphidromic points of M, and S, and in shallow water, where the tidal curve becomes strongly distorted, this assumption appears to be justified. The equivalent assumption for surge elevations was that the spatial distribution of extreme surge was similar to that of the maximum computed surge in the sample of storms considered. The values of

the empirical factor g, given in Table 2, suggest that the assumption is justified for much of the North Sea, the Irish Sea and the shelf areas to the north-west of Scotland. The justifica- tion is less clear for the northern North Sea and the shelf north of Ireland. The large value of G+ at Newlyn indicates that the storm sample did not contain any event giving a large surge

in that region, so the results for the Celtic Sea and western English Channel must be treated with caution.

The results presented in Section 4 suggest that the combination of the model based distribution of spring tidal amplitude plus 50-year surge gives quite a good approximation to

an extreme total still water level, with probable errors of order loo,, . The use of data from a reference port to scale the basic value obtained from the model

distribution means that the model distributions are adjusted to fit the observations at those locations. As the point of interest shifts farther from a reference port then the estimated extreme value becomes increasingly dependent on the accuracy of the model distribution. For this reason, it would be valuable to have comprehensive joint probability (or other) analyses of data from more coastal tide gauges. Such results for some ports on the continent of Europe would be particularly valuable since then the model would be effectively inter- polating extremes at interior locations from known values on either side of the North Sea rather than, as now, extrapolating from the British coast.

92 R. A. Flather

The use of the procedures to estimate extreme currents involves further assumptions. Most fundamental is the assumption that current is approximately proportional to elevation. If, for example, the elevation at a reference port is dominated by a Kelvin wave and the current at the offshore location of interest is substantially due to the same Kelvin wave, then the assumption will be justified. If, on the other hand, the offshore current comprises mainly a continental shelf wave, or some other motion physically unrelated to the coastal water level, then the justification is less clear. The link then must presumably be through the intensity of the forcing causing the motion. Clearly, much greater uncertainty attaches to the estimates of extreme current than to those of extreme elevation.

Further complication arises in considering the directional properties of the extreme currents. The information contained in the model distributions of tidal ellipse parameters is probably reasonably reliable, but for the surge currents the small size of the sample examined permits only a general indication of the nature of the directional properties. A useful and conservative assumption is that the extreme surge current can occur in any direction. Throughout, it must be remembered that the model estimates are for depth- mean currents. The current at a particular depth may differ substantially from the depth-mean value.

In making use of the information presented here for making practical estimates of extreme elevations and currents, it is important to take account of the various assumptions and approximations discussed above, since they will influence the degree of confidence to be attached to the estimates obtained. Wherever possible use should be made of offshore measurements, either at the position of interest or at some representative nearby location, to check the model results.

During the last few years, the data and procedures described above have been used extensively by the Marine Information and Advisory Service in 10s to provide estimates of extreme currents and/or elevations at numerous offshore and several coastal locations. In many instances no alternative was available. For a few sites, subsequent observations and analysis have confirmed the model estimates. Overall, the results appear to be of consider- able value in a variety of applications.

Acknowledgements

The author is much indebted to Dr N. S Heaps for critical comments on a draft manuscript, to Mrs Linda Parry for her care and patience with the typing, and to Robert Smith for his work on the figures.

Some of the work was funded by the Ministry of Agriculture, Fisheries and Food.

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