A three dimensional hydrodynamic model of tides off the north-west coast of Scotland

J 0 LJ K N A L 0 F

MARINE SYSTEMS

ELSEVIER

Journal of Marine Systems 7 (1996) 43-66

A three dimensional hydrodynamic model of tides off the north-west coast of Scotland R. Proctor

*, A.M. Davies

Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside L43 7RA, UK

Received 11 April 1994; accepted 14 December 1994

Abstract A three dimensional hydrodynamic model using a fine l/12” x l/12” finite difference grid in the horizontal with a spectral approach in the vertical is used to examine the spatial distributions of the M, and 0, tidal elevations and currents at the shelf edge. The region chosen for the study is the Malin-Hebrides shelf off the west coast of Scotland, where a set of offshore elevation and current meter data exists. Calculations show that M, tidal elevations and currents increase rapidly from ocean to shelf having the largest amplitude in the near shore regions. In contrast 0, elevations and currents are intensified at the shelf edge showing spatial changes corresponding to a first mode shelf wave. Computed 0, current ellipses show significant small scale horizontal spatial variability in the shelf edge region. Comparison of computed and observed M, tidal currents, shows that away from abrupt changes in topography, the model can adequately reproduce the currents, although in regions of rapid change in water depth a finer grid is probably required. Calculations show that with the finite difference grid used here the intensity and the spatial variability of the 0, tidal currents in the shelf edge region is strongly influenced by the magnitude of horizontal eddy viscosity, the exact formulation of which is poorly known.

1. Introduction

Two dimensional vertically integrated hydrodynamic models have been used extensively over the past thirty years to study both ocean tides and shallow sea tides separately. Recently with increased computer power, models which were originally used to study tides on the shelf (e.g. Flather, 1976) have been extended to cover re-

* Corresponding author.

gions of both shelf and ocean (Flather, 1987, Flather, 1988; Flather et al., 199.5). Also finite element models (Lynch and Werner, 1987,1991) with their ability to grade the mesh have been used very successfully in two dimensions to study shelf edge tides (Foreman and Walters, 1990; Foreman et al., 1993). Although three dimensional tidal models have been used extensively in understanding the physics of tidal current structure in shallow water (e.g. Davies and Furnes, 1980; Proctor, 1981,1987; Davies, 1986; Gordon and Spaulding, 1987; Aldridge and Davies, 1993) these models have

0924-7963/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved L!?DI0924-7963(94)00036-O

44

R. Proctor, A.M. Davies/Journal

of Marine Systems 7 (1996) 43-66

not been used significantly in deep water or shelf edge regions. In shallow water with strong tidal currents, bottom generated turbulence (Davies and Jones, 1990; Baumert and Radach, 1992) can penetrate through the whole water column, which remains well mixed. Consequently tidal current structure is determined in these regions by the retarding force of the sea bed, and internal frictional forces. In deep water, with lower tidal currents, turbulence intensity is insufficient to maintain a well mixed water column, and stratification occurs at depth. When this stratification intersects a sloping region (e.g. the continental shelf edge) then internal tides can be generated, and internal pressure differences associated with them, together with the influence of internal density upon turbulence levels in the water column,

Fig. 1. Geographical

location

are responsible for determining tidal current structure. In order to develop a model to simulate the internal tide, it is necessary first to produce an accurate model of the barotropic tide, since this is the forcing mechanism giving rise, with bottom topography and density effects, to the baroclinic tide. The region chosen for this model development is the north west Scottish continental shelf between 55”N and 59”N extending from the coast offshore to 12”W (see Fig. 1). Bathymetry (Fig. 2) varies from < 100 m on shelf to > 2000 m off shelf and the shelf slope has gradients between 0.015 at 58.5”N and 0.06 at 57”N. Tidal structure in the region has been well explained by Cartwright et al. (1980). They

of the various

regions.

R. Proctor, A.M. Davies /Journal

45

of Marine Systems 7 (1996) 43-66

together with a discussion of the model accuracy and the importance of bottom topography and horizontal viscosity. The final section of the paper summarises the important results and considerations which must be taken into account in future calculations.

2. Mathematical analysis and numerical solution 2.1. Hydrodynamic equations

1PW

9DW

7w

5sw

Fig. 2. Bottom topography of the region covered by the shelf edge model. Contours are in lo-m-intervals.

demonstrate that the semi-diurnal tide is essentially a northward propagating Kelvin wave whereas the diurnal tide is a combination of a Kelvin wave and a shelf wave, both propagating northwards but out of phase. The diurnal combination gives rise to the observed elevations which increase away from the coast and the strong clockwise-polarised currents at the shelf edge. This is the only section of the continental shelf slope between Brittany (France) and Norway where such a diurnal anomaly is evident (Cartwright, 1976). In this paper we present results from a full three dimensional shelf edge model, covering the area off the west coast of Scotland. A temporal and spatially evolving eddy viscosity is used in the model, related to the flow field. Although density effects are not included within the model, the Galerkin approach can be extended to include a time varying density field (Davies, 1982,1983). This paper therefore represents a first stage towards developing a prognostic shelf edge model including internal tides, namely the accurate determination of the barotropic tide. The mathematical formulation of the model is described in Section 2. Its application to the determination of the M, and 0, tides off the west coast of Scotland is presented in Section 3,

By using a splitting approach, whereby the current is expressed in terms of its depth averaged part, U, and the deviation from this u’ [details of which are standard and can be found in Davies (19871, Davies and Proctor (1990)], the three dimensional hydrodynamic equations neglecting the advective terms can be written as

al

z+&

i

$(h+i)ii+$(h+i)Fcos4)

= 0

(1)

au -

at

-2wsin4ij

a5 A**

=_--++ g

Rcos4

ax

1

a2u

x

---tan4--++ i

~0~~4

ax2

R2

au

a2u

a4

a4 1

FL3

(2)

- P(h +G aa at +2wsin@

= --- gal

A,

Racp+R' X

1 a2fi ---tan4--+f

1

~0~~4ax2 GB

P(h

+ 43

au

a22j

a4

a4

(31

R. Proctor, A.M. Davies /Journal

46

and hi

-

at

-22wsinbu’= 43

+

P(h + 0 thl’ at

1

+2wsin+‘=

-

(h+l)* +

a

aa i %

ad i

GB P(h + s>

Eqs. (2) and (3) are in essence the two dimensional vertically integrated hydrodynamic equations, containing the elevation gradient terms and bed stress terms F, and G, which in this case relate the bed stress to the bottom current. Eqs. (4) and (5) are prognostic equations for current structure, with FB and G, included with the opposite sign to Eqs. (2) and (3). By this means

of Marine Systems 7 (1996) 43-66

when the equations are combined we obtain the original set of three-dimensional equations. [Details are given in Davies (1987) and will not be repeated here]. The justification for omitting the advective terms in these equations is that in subsequent calculations we will only be concerned with the primary tides M, and 0, and not the overtides and compound tides which are influenced by the advection terms (Xing and Davies, 1994a), or wind driven flows where advection can have some influence, (Xing and Davies, 1994b). In these equations (T= (z + 5)/(/z + t) is a normalised coordinate varying from 0 at the surface (z = -4’) to + 1 at the sea bed (z = +h), with x, C#Jlongitude and latitude respectively. Other terms are, t time, 5 free surface elevation, h water depth, R radius of earth, w angular speed of rotation of the Earth, g acceleration due to gravity. Horizontal eddy viscosity, A, was initially taken as lO*m*s-‘, although in some

a Fig. 3. Finite difference in the comparison.

grid of the model, with (a) location

of tide gauges

used in the comparison

and (b) location

of currents

used

R. Proctor, A.M. Davies /Journal

P”U

of Marine Systems 7 (1996) 43-66

47

S!U

S’U

Fig. 3 (continued).

calculations it depended on water depth (see later). Here p denoted the vertical eddy viscosity. The formulation of the bed stress FB, G, will be considered later. Depth mean currents, II, b are given by 1

i-j=

/0

uda,

1

V=

/0

vda

By separating the equations into essentially the “standard” depth averaged hydrodynamic Eqs. (l)-(3), which can be solved using forward time stepping, subject to the CFL stability condition for the free surface, and the set Eqs. (4) and (5) which are not restricted by this condition, a highly efficient and vectorizable algorithm can be developed which is optimal on pipeline vector processors such as the CRAY X-MP. Details of the method will not be given here but can be found in Davies and Proctor (1990). In these equations, bed stresses FB, G, are

given by a quadratic currents uh, vh FB=

- &

p; [

law, in terms of sea bed

1

1 = Kpu,(u;

+v;)~‘~

(7)

1

+ ~2)~‘~

(8)

and G,=

-

,_L;1 = Kpv,(u;

& [

with K a coefficient of bottom friction. At the sea surface in the absence of wind effects the surface stress is zero. 2.2. Numerical solution The numerical solution of these equations is accomplished using a standard finite difference grid scheme in the horizontal (Fig. 3a, b). The

R. Proctor, A.M. Davies /Journal

48

of Marine Systems 7 (1996) 43-66

Galerkin approach is used in the vertical (see Davies, 1986,1987 for details), in which the two components of velocity U, u are expanded in terms of basis functions f,(r) extending from sea surface to sea bed and sets of coefficients A,.(x, 4, t) and B,(x, 4, t) varying with horizontal position and time, thus

to be eigenfunctions defined by

(12) where E, are the eigenvalues. The advantage of using a basis set of eigenfunctions is that in the linear case the equations are uncoupled, reducing computational effort and yielding code which is ideally suitable for integration on the new generation of multiprocessor vector computers (Davies and Proctor, 1990; Davies et al., 1992; Ashworth and Davies, 1993). By solving Eq. (12) subject to surface and bed conditions of the form,

By expressing the vertical eddy viscosity as a function of horizontal position and time a(~, 4, t), and a fixed function @(a) representing the vertical profile of viscosity, thus CL(x&$,a,t) = Q(x&)@(a>

(13)

(11)

the first mode is constant in the vertical (in essence the depth mean current) with higher

it is possible to choose the basis functions f,(a)

1O"W

of the eddy viscosity profile,

6OW

6’W

4OW

:--. .*: 1.: 2oook

:‘

:..

:’

.: 200m

’ . . . - . . . .. :‘ ..

. .. . ..

.

: .* -. .’

Fig. 4. Computed M, co-tidal chart, (--_) amplitude in cm, c----j phase in degrees.

t

57”N

R. Proctor, A.M. Davies/Journal

modes which are orthogonal to it giving current structure in the vertical. This is in essence the basis of the time split method used here, see Davies (1987) for details. Based upon earlier calculations (Davies, 1992,1993) an expansion of

of Marine Systems 7 (1996) 43-66

49

thirty functions was used in the vertical. This is much larger than the number used previously in shallow water, typically ten, and is due to the slower rate of convergence of the modes in deep water (see Davies, 1987,1992).

Table 1 Comparison of observed (Ohs) and modelled (Mod) amplitude hkm) and phase gfdeg) of the M, tidal elevation. Also given is the percentage error (AH%) and difference in phase AC

No. 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 33 34 35 36 37 38 39 40 41

Lat.

58.78 58.45 58.28 58.20 58.20 58.03 58.00 57.82 57.77 57.85 57.67 57.43 57.37 57.32 57.23 57.30 57.30 57.28 56.92 56.95 56.50 56.32 56.55 56.30 56.10 56.02 55.87 55.67 55.25 55.17 55.38 55.43 55.63 55.68 55.72 55.97 55.55 55.47 55.20 55.32 55.25

Long.

-

7.50 5.05 6.78 6.38 9.97 8.58 6.42 8.57 7.03 5.63 7.48 5.80 7.27 9.87 - 10.07 - 9.32 - 7.65 - 5.72 - 8.58 - 7.48 - 6.80 - 6.38 - 5.42 - 5.58 - 5.78 - 8.58 - 6.33 - 10.82 - 10.72 - 6.77 - 7.40 - 7.23 - 6.18 - 5.73 - 5.30 - 4.82 - 4.68 - 6.17 - 6.07 - 5.63 - 4.87

I

54 83 62 69 25 41 68 42 59 77 55 74 58 26 24 33 53 75 41 56 63 68 79 77 75 41 68 15 16 53 56 58 71 76 81 87 88 70 72 77 86

J

3 7 9 10 10 12 12 15 15 14 16 19 20 21 22 21 21 21 25 25 30 33 30 33 35 36 38 40 45 45 44 43 41 40 40 37 42 43 46 45 45

Obs

Mod

h

g

h

AH% g

104 140 119 139 96 106 140 107 129 149 120 153 119 102 101 105 118 156 109 113 118 114 113 65 106 112 101 104 107 57 108 97 16 19 107 116 109 7 43 66 105

191 208 190 197 175 178 193 176 185 199 177 195 180 169 168 169 171 195 168 232 166 165 163 135 163 163 163 159 157 198 178 175 92 81 342 342 340 125 309 337 340

94 141 115 135 90 101 135 105 127 149 122 155 131 98 97 102 118 156 108 122 129 117 114 109 109 108 106 100 103 119 115 92 23 33 104 111 104 19 31 70 100

194 212 193 198 175 179 197 176 184 200 180 196 180 168 168 169 171 196 168 172 166 155 153 153 155 163 150 159 157 169 173 175 101 104 359 358 357 129 317 5 355

- 9.1 1.4 - 2.8 - 2.6 - 5.3 - 4.1 - 3.0 - 1.2 - 1.4 0.5 2.5 1.4 10.3 - 3.5 - 3.0 - 2.8 0.7 0.4 - 0.8 8.7 10.0 3.3 1.8 68.6 3.7 - 2.7 5.0 - 2.9 - 3.6 109.2 7.3 - 4.9 48.5 76.1 - 2.1 - 3.5 - 4.3 178.3 - 26.7 6.4 - 4.7

AC

2.5 4.0 2.8 1.3 0.1 1.1 4.2 0.3 - 1.1 1.5 2.9 0.9 - 0.2 - 0.7 - 0.5 0.0 - 0.5 0.7 - 0.2 - 59.7 - 0.1 - 10.2 - 9.9 17.8 - 7.7 0.4 - 13.5 - 0.2 0.1 - 28.7 - 5.2 - 0.2 9.0 22.5 16.9 16.0 16.7 3.8 7.9 27.6 15.4

R. Proctor, A.M. Davies /Journal

50

of Marine Systems 7 (1996) 43-66

2.3. Lateral boundaries and tidal input

where C = (g/r)‘/*, with

At lateral coastal boundaries the normal component of current is set to zero, for all t 2 0; thus

qr=

A, cos $ + B, sin Cc,= 0, r = 1,2 ,..., m

and

(14)

where tc, denotes the inclination of the normal to the direction of increasing x. At open boundaries (Fig. 3a, b) where tidal forcing occurs, a radiation condition is applied (see for example Proctor, 1981,1987; Davies, 1986; Davies and Aldridge, 1993; Flather et al., 1995). This condition involves a relationship between the total normal components of depth mean current q and total elevation 5 given by,

(15)

5

Qi(Xd)cos[~i’

i=l

-

Yi( X,4)1

(16)

Gi( X,4>]

(17)

In Eqs. (16) and (17) oi denotes the frequency of the various (M) tidal constituents, here just the M, and 0, tide. The amplitude and phase of the current are denoted by Qi and yi, with Hi and Gi the amplitude and phase of the elevation. The tidal input terms Qi, yi, Hi and Gi were interpolated from a large area two dimensional model of the North East Atlantic (Flather et al., 1995).

.5Qk

57’N

a Fig. 5. Distribution of major and minor axis of M, current ellipses (a) at surface and (b) at sea bed.

51

56’N

b

55’N I

Y

Fig. 5 (continued).

2.4. Form of vertical eddy viscosity

In the following calculations eddy viscosity was a function of the flow field using the parameterisation of Davies and Furnes (1980), thus j.L= K,( E2 + -2 u j/s, with S, = 1.0 x 10-4s-’ a characteristic frequency of motion and K, = 2.0 X lo-’ a dimensionless coefficient. This formulation has been successful in reproducing tidal current profiles in a number of three dimensional models using a constant viscosity in the vertical and a slip condition at the sea bed (Proctor, 1981, 1987; Lynch and Werner, 1991; Davies and Jones, 1992; Davies and Aldridge, 1993) and in deep water corresponds in essence to a viscosity scaled with the friction velocity and thickness of the bottom boundary layer (see Davies and Jones, 1992; Davies and Aldridge, 1993).

3. Tidal calculations 3.1. Topography of the region and tidal forcing

The area covered by the model (Fig. 1) spans a range of water depths from the order of 2000 m in the ocean, to 100 m in the Hebrides and Malin shelf areas (Fig. 11, with the two regions connected by a region of rapidly changing topography, the shelf slope (Fig. 2). Close to the Scottish and Irish coasts, water depths are the order of 20 m. The model’s finite difference grid together with the location of tide gauges and current meters used in the comparison is shown in Fig. 3a, b. The grid has a resolution of l/12” x l/12”, which although it is sufficiently fine to resolve the wavelength of the surface tidal waves considered here, as we will show, is probably not fine enough to resolve changes in tidal current in the shelf edge region.

R. Proctor, A.M. Davies /Journal

52

Along the open boundaries of the model, tidal forcing was specified using the radiation condition (Eq. 15) with tidal elevation and current

of Marine Systems 7 (1996) 43-66

input at both the M, and 0, tidal frequencies. The model was started from a state of zero elevation and current, and due to the lack of frictional

Table 2 Comparison of observed (Ohs) and modelled (Mod) amplitude h kms -‘) tidal current Place

A A B B B C C C D E F G G H I I J J K K W W L L L M M M N N N 0 0 P Q R R R s S S T T U v

Lat

55.50 55.50 55.42 55.42 55.42 55.52 55.52 55.52 55.47 56.00 55.88 56.48 56.48 56.92 57.02 57.02 57.00 57.00 56.98 56.98 57.13 57.13 57.47 57.47 57.47 57.32 57.32 57.32 57.35 57.35 57.35 57.58 57.58 58.02 58.03 57.93 57.93 57.93 58.00 58.00 58.00 58.03 58.03 58.78 58.72

Long

- 8.02 -8.02 -7.52 -7.52 -7.52 -6.85 -6.85 -6.85 -6.17 -8.57 -6.55 -7.98 -7.98 -8.58 -9.03 -9.03 -9.00 -9.00 -9.02 -9.02 -9.00 -9.00 -11.07 -11.07 -11.07 -9.67 -9.67 -9.67 -9.45 -9.45 -9.45 -8.17 -8.17 -9.15 -9.00 -8.85 -8.85 -8.85 -8.73 -8.78 -8.73 -8.17 -8.17 -7.50 -5.88

I

48 48 54 54 54 62 62 62 70 41 65 48 48 41 36 36 36 36 36 36 36 36 11 11 11 28 28 28 31 31 31 46 46 34 36 38 38 38 39 39 39 46 46 54 73

J

42 42 43 43 43 42 42 42 42 36 37 30 30 25 24 24 24 24 24 24 22 22 18 18 18 20 20 20 20 20 20 17 17 12 12 13 13 13 12 12 12 12 12 3 3

u

and phase g(deg) of the u and u components of the M, V

Obs

Mod

Obs

Mod

h

g

h

i?

h

g

h

g

17.6 14.1 46.8 45.2 33.6 82.9 82.2 69.0 46.9 13.7 26.8 11.2 15.6 16.5 25.4 18.8 24.7 19.9 18.4 22.9 19.1 17.5 5.4 2.2 2.9 1.0 1.7 1.8 17.1 11.2 13.9 5.7 9.5 11.2 8.5 7.9 9.1 10.3 5.7 9.2 8.9 3.7 6.6 22.0 17.1

170 158 202 202 192 234 234 233 242 140 91 159 116 113 127 139 126 131 119 131 153 150 271 207 196 30 83 230 161 186 151 163 145 203 193 232 201 162 234 194 188 191 178 140 152

15.0 16.2 34.4 34.1 29.7 82.5 81.9 67.2 44.9 12.1 21.4 8.7 12.1 15.2 14.4 14.6 16.6 16.4 13.3 13.9 19.2 17.9 6.3 6.1 5.3 1.2 1.2 1.1 3.7 3.5 3.5 11.1 7.6 8.8 9.6 8.6 8.4 8.7 7.6 9.2 8.7 6.7 6.3 14.7 23.1

159 171 206 204 190 236 236 234 234 117 73 118 132 119 141 129 139 124 119 137 153 136 258 258 258 87 99 84 155 151 151 160 148 213 209 204 205 209 216 218 215 194 193 160 170

4.7 3.4 10.1 6.7 10.8 50.8 56.1 41.4 31.6 7.3 70.3 8.7 12.2 11.3 18.2 14.1 16.1 13.2 11.5 16.8 16.0 14.5 11.9 8.1 8.0 6.9 5.3 8.3 14.8 13.5 11.3 14.7 11.3 17.8 12.9 17.1 14.3 8.8 14.4 12.7 11.2 14.6 12.8 8.0 14.2

80 357 187 183 239 77 75 73 22 63 70 88 41 46 53 69 54 63 56 64 82 84 188 185 180 200 198 180 103 151 105 159 147 168 172 182 171 144 192 172 172 195 188 354 237

4.6 1.4 17.7 17.2 15.1 65.7 65.0 50.1 29.5 7.0 64.8 6.4 10.9 8.5 11.0 9.5 12.5 9.7 7.3 10.4 16.0 12.1 13.3 13.2 12.3 7.2 7.2 7.2 8.0 7.7 7.7 16.7 14.3 15.2 15.5 16.3 16.3 16.4 17.0 17.7 17.1 17.2 17.3 7.1 11.4

353 123 203 206 218 81 80 70 17 27 69 92 77 65 94 81 89 73 72 89 98 84 186 187 187 188 187 188 162 164 164 152 167 182 181 180 180 181 189 185 186 197 200 263 275

Dep w

Dep m

73 73 54 54 54 58 58 58 110 136 44 170 170 126 134 134 139 139 138 138 140 140 589 589 589 1614 1614 1614 145 145 145 135 135 194 177 155 1.52 155 146 142 146 142 142 110 114

41 10 30 28 11 39 37 11 6 25 11 140 50 25 98 28 95 25 20 90 75 25 300 100 50 1405 1104 503 90 25 25 75 25 151 25 100 100 25 105 28 25 75 25 25 11

R. Proctor, A.M. Davies /Journal

dissipation in the deep water it was necessary to run the model for a period of the order of ten days, before separating the M, and 0, tide by harmonic analysis. Energy dissipation at the sea bed was parameterised using the quadratic friction law (Eqs. 7, 8), with a bottom friction coefficient K = 0.005 (a value used by Davies, 1986), in essence representing a C,,, value (the coefficient relating the bottom stress to the current lm above the bed) typical of shelf seas (Channon and Hamilton, 1971). 3.2. M, tidal elevations and currents The computed M, co-tidal chart (Fig. 4), shows tidal elevations increasing significantly from ocean to shelf, with tidal amplitudes off the north west coast of Scotland, in the region of The Minch (Fig. 11, reaching the order of 1.40 m. The phase of the M, tide increases from south to north, approximately corresponding to a propagating Kelvin wave (Cartwright et al., 1980), with an M, amphidromic point to the north of the North Channel. The position of the amphidromic point, and the distribution of co-amplitude and co-phase lines is in good agreement with observations (Howarth, 1990) and results from larger area models (Flather et al., 19951, and more limited area finer grid models (Davies and Jones, 1992). A comparison of observed and computed M, tidal elevation amplitude and phase at the 41 points shown in Fig. 3a is given in Table 1. From this Table it is evident that there are some major errors at certain locations. For example the model overpredicts the tides at location (241, although at adjacent locations (23) and (25) there is no significant error. This suggests that there may be a local feature at (24) which the model cannot resolve or an error in the observations. Similarly there are large errors at location (30), although model results are in good agreement with observations at (31) and (32). Although the percentage error is large at (331, (341, (38) and (39) the tidal amplitude in this region is small due to the presence of the amphidromic point, and these errors probably reflect the inability of the model to locate the exact position of this point since only the nearest grid point is taken, instead of interpo-

of Marine Systems 7 (1996) 43-66

53

Table 3 Comparison of observed (0) and modelled (M) semi-major axis (ems - ’ 1,semi-minor axis (ems-‘J, orientation alpha (deg) and rotation of the M, tidal current ellipse at a number of depths and locations Place

Major

Minor

CM A A B B B C C C D E F G G H I I J J K K W W L L L M M M N N N 0 0 P Q R R R S S S T T U v

17.60 14.46 47.81 45.64 34.44 95.69 98.08 79.52 53.70 13.83 74.69 11.87 16.26 17.40 26.27 19.88 25.47 20.82 19.40 24.42 20.59 19.29 11.92 8.36 8.48 6.97 5.35 8.38 19.87 16.76 16.56 15.76 14.76 20.26 15.23 17.93 16.48 13.38 15.05 15.42 14.17 15.06 14.37 23.02 17.24

15.69 16.25 38.77 38.25 32.75 103.12 102.32 83.14 51.43 12.10 68.27 10.61 14.60 16.25 16.81 16.32 19.03 17.81 14.39 16.05 22.34 19.86 13.56 13.39 12.52 7.21 7.20 7.27 8.82 8.53 8.53 20.02 16.10 17.18 17.86 18.21 18.12 18.24 18.42 19.46 18.84 18.53 18.48 14.86 23.37

0

Alpha ___ MOMOM

4.70 1.08 2.56 2.16 7.71 17.20 16.85 12.29 17.74 7.04 9.04 7.76 11.31 9.87 16.91 12.53 14.85 11.70 9.72 14.50 14.04 12.02 5.35 0.80 0.75 0.17 1.53 1.37 10.80 5.17 6.82 0.37 0.25 5.64 2.58 5.77 3.95 2.09 3.65 2.84 1.94 0.25 1.02 4.28 14.03

1.08 1.08 0.82 0.50 6.48 22.26 21.61 11.29 15.66 7.02 1.38 2.37 7.44 6.49 6.95 6.40 8.34 6.95 4.99 6.71 11.33 8.60 5.97 5.77 5.03 1.20 1.22 1.13 0.45 0.72 0.72 1.21 2.17 4.00 4.05 3.25 3.13 3.71 3.22 4.70 3.83 0.29 0.67 6.90 10.92

0 347 12 8 13 149 146 150 149 9 70 26 23 23 20 25 18 21 21 26 31 33 86 76 71 98 98 82 37 51 37 69 50 60 57 71 59 40 73 55 52 76 63 163 12

Rot 343 4 27 27 25 + 142 142 144 149 + 1 72 35 40 22 34 28 33 24 24 33 36 29 80 80 80 92 + 9t++ 92 65 66 66 56 63 + 62 60 63 64 64 68 65 65 69 + 70 + 172 351 +

+ + + + + + + + + + + + +

R. Proctor, A.M. Davies /Journal

54

lation to actual measurement sites. In a region where amplitude is changing rapidly this can be significant. A significant phase error occurs at position (20), (Fig. 3a) located at the southern end of the Hebrides (Fig. 11, although at nearby locations, positions (19) and (21) the model accurately reproduces the observed phase, suggesting a local

of Marine Systems 7 (1996) 43-66

effect or an observational error at position (20). Some large phase errors also occur in the region of the amphidromic point, presumably associated with errors in locating the exact position of the amphidrome. The computed RMS error in amplitude and phase was 0.133 m and 13”, with the large RMS error being in part caused by the very significant errors at a few locations (of order 6).

Table 4 As Table 1, but for 0, elevations No.

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 33 34 35 36 37 38 39 40 41

Lat.

58.78 58.45 58.28 58.20 58.20 58.03 58.00 57.82 57.77 57.85 57.67 57.43 57.37 57.32 57.23 57.30 57.30 57.28 56.92 56.95 56.50 56.32 56.55 56.30 56.10 56.02 55.87 55.67 55.25 55.17 55.38 55.43 55.63 55.68 55.72 55.97 55.55 55.47 55.20 55.32 55.25

Long.

- 7.50 - 5.05 - 6.78 - 6.38 - 9.97 - 8.58 - 6.42 - 8.57 - 7.03 -5.63 - 7.48 - 5.80 - 7.27 -9.87 - 10.07 - 9.32 - 7.65 -5.72 - 8.58 - 7.48 - 6.80 - 6.38 -5.42 -5.58 - 5.78 - 8.58 - 6.33 - 10.82 - 10.72 -6.77 - 7.40 - 7.23 -6.18 -5.73 -5.30 - 4.82 - 4.68 -6.17 - 6.07 -5.63 - 4.87

I

54 83 62 69 25 41 68 42 59 77 55 74 58 26 24 33 53 75 41 56 63 68 79 77 75 41 68 15 16 53 56 58 71 76 81 87 88 70 72 77 86

J

3 7 9 10 10 12 12 15 15 14 16 19 20 21 22 21 21 21 25 25 30 33 30 33 35 36 38 40 45 45 44 43 41 40 40 37 42 43 46 45 45

Obs

Mod

h

g

h

g

8 8 6 9 8 7 9 7 6 8 3 9 7 7 7 7 3 8 6 7 5 5 8 7 8 7 5 6 6 8 8 7 7 8 10 9 11 7 8 9 10

4 350 331 350 4 357 349 353 10 345 353 330 15 356 357 359 16 352 3 44 29 34 351 44 16 351 25 355 355 30 15 356 43 37 44 45 37 44 37 54 40

6 8 8 8 7 7 8 6 4 7 3 6 6 7 7 7 2 7 6 5 4 5 5 5 5 6 5 6 6 6 6 6 6 6 8 8 8 6 7 7 8

12 344 326 0 6 354 0 352 324 348 309 352 23 1 1 0 22 351 0 35 33 50 44 48 46 3 45 354 352 28 28 32 56 57 60 58 59 53 50 60 58

AH%

AG

- 19.0 2.0 45.7 -4.8 - 13.5 -2.1 -6.4 -6.1 - 34.0 -11.3 9.6 - 24.7 -8.1 -3.2 -0.9 1.8 - 38.4 - 12.5 1.5 - 17.1 - 19.2 - 13.1 -31.4 - 28.0 - 36.6 -9.0 9.0 -9.4 - 11.8 - 15.9 - 16.1 -4.6 - 13.2 - 16.2 - 18.0 -11.3 - 25.1 - 14.2 - 14.0 - 13.9 - 20.0

7.9 - 6.5 -4.9 9.2 1.3 -2.9 11.4 - 1.2 - 45.5 3.4 - 44.2 22.3 7.8 4.1 3.2 1.2 6.8 -0.7 - 2.8 - 8.7 4.2 15.5 52.5 4.3 30.4 11.4 20.5 -0.3 - 2.9 - 2.6 12.9 36.0 12.9 19.9 15.7 13.4 21.5 8.8 13.6 5.8 18.2

R. Proctor, A.M. Davies /Journal

of Marine Systems 7 (1996) 43-66

Excluding these major errors, the model appears to be able to reproduce M, tidal amplitude in general to within 0.05 m, although it has a tendency to slightly underpredict the tide, possibly due to some inaccuracy in the tidal input which was interpolated from the coarser grid model of Flather et al., 1995. Although it is tempting to omit comparisons from regions which cannot be resolved with the present finite difference grid, we have included them to show that in some locations the results are inaccurate due to the modelling method adopted i.e. a uniform grid. Other approaches, namely the finite element method (e.g. Foreman et al., 1993), may be able to overcome this problem or may still show discrepancies due to physical deficiencies of the model or errors in the observations. The spatial distribution of the major and minor axis of the M, tidal current ellipse at the sea surface (Fig. 5a), shows a region of strong (of order 1 ms-‘) tidal currents in the area to the north of the North Channel (Fig. l), with the

10°W I

Fig. 6. Computed

0, co-tidal

55

major axis of the ellipse aligned with the topography. Similarly within The Minch (Fig. 1) tidal currents are stronger and aligned with the topography, whereas in the deep water beyond the shelf edge (the 200 m contour) tidal currents are small. The spatial distribution of bottom currents is not significantly different from those at the sea surface, although frictional effects tend to reduce the magnitude of the near bed currents (Fig. 5b). A detailed comparison of the amplitude and phase of the u and u components of current at the locations shown in Fig. 3b is given in Table 2. Amplitude and phase rather than rotary components of the tide are presented in this table to be consistent with other model results (Xing and Davies, 1994a). Also rotary components or any tidal ellipse parameters can be readily derived from this information. Referring to Table 2, it is clear that the model can accurately reproduce the strong u component of current at location C, and its decrease from 0.82 ms-’ at the surface to 0.69 ms-’ at the bed, with the model slightly underes-

cc”W I

chart, (-_) amplitude

6”W I

in cm, (---I

j9’N

phase in degrees.

R. Proctor, A.M. Davies/Journal

56

of Marine Systems 7 (1996) 43-66

timating the bed current. The u component of current in the model at this location is overestimated in the model, although at locations D and E the amplitude of both the u and u components are correctly reproduced. Further north at location M in deep water, the model correctly reproduces the small u component of tidal velocity, and the larger u component with the phase of the u component agreeing well with observations. The intensification in tidal currents from deep (location M) to shallow, location N is reproduced by the model. At positions I, J and K, the model underestimates the tidal current, with the u component of phase in general agreeing to within 10 degrees, although the error in the u component is larger. Further north at P, Q, R and S, where the water depth is larger, the model correctly reproduces the decrease in the u component of velocity compared with positions I, J and K. The u component

12% Fig. 7. Distribution

1

o”w

of velocity at locations P, Q, R and S is on average reproduced by the model to within 0.03 ms-‘, although the model shows little vertical variation in currents, compared with the observations which show a major change, possibly due to stratification effects, which are not included in the model. Current ellipse parameters (Table 3) at location C show that the model can reproduce the major and minor axis of the ellipse on average to within 0.05 ms-’ and its orientation to better than lo”, with similar good agreement at D, and the model shows the change in ellipse rotation between C and D. The decrease in the major axis in going north to G is reflected in the model which appears to underestimate the major axis by the order of 0.02 ms-‘, although the decrease in current magnitude of 0.044 ms-’ with depth is reasonably accurately reproduced by the model which gives a decrease of 0.04 ms-‘. The orienta-

6%

of major and minor axis of 0, current

6%

ellipses (a) at sea surface

4Ow

and (b) at sea bed.

R. Proctor, A.M. Dauies /Journal

of Marine Systems 7 (1996) 43-66

57 59’N

J

5S’N

r’ 55’N

6%

dOW

Fig. 7 (continued).

tion of the current ellipse appears to be adequately represented at this location. In the shelf edge region at positions I, J and K, the model appears to underestimate both the major and minor axis of the ellipse, although its orientation is accurately reproduced. Depth variations in this region are very poorly resolved in the model, with changes from west to east in the location of I, J and K from 806 m to 364 m to 134 m over three adjacent grid boxes. Further north at positions P, Q, R and S where the topography changes more regularly, although water depths still change significantly from 240 m at P to 153 m at S, (five grid boxes away), the model on average reproduces the major axis to within 0.03 ms-‘, although its change with depth is underestimated. These comparisons suggest that although the model has a relatively fine grid (l/12” X 1/12O 1, it cannot adequately resolve the abrupt changes in topography that occur in locations such as I, J

and K, and even further north at positions P to S the model is probably not adequately resolving the topography. In general the model is able to represent the magnitude of the M, tidal current to within 10% at the majority of the locations and there does not appear to be a strong bias to over-predict or under-predict the current magnitude. The orientation of the current is reproduced to within 10” at all but a small number of locations (Table 3). The model clearly shows the observed reduction of current magnitude with depth found at the majority of the locations although the model tends to underestimate the vertical variation of current magnitude in the observations. This may be due to the fact that the model does not include stratification effects which are known to be important in generating the internal tide in the shelf edge region off the west coast of Scotland (Sherwin and Taylor, 1989).

R. Proctor, A.M. Daaoies/Journal

58

The model does, however, provide an explanation for the apparent mismatch in M, currents described in Cartwright et al. (1980) where cur-

of Marine Systems 7 (1996) 43-66

rents appear to approximate a northward propagating Kelvin wave at 58” N (stations 0 to T, Fig. 3b, Tables 2, 3) but not at 57” N (stations H to K,

Table 5 As Table 2, but for 0, currents Place

A A B B B C C C D E F G G H I I J J K K W W L L L M M M N N N 0 0 P

Q R R R S S s T T U V

Lat.

55.50 55.50 55.42 55.42 55.42 55.52 55.52 55.52 55.47 56.00 55.88 56.48 56.48 56.92 57.02 57.02 57.00 57.00 56.98 56.98 57.13 57.13 57.47 57.47 57.47 57.32 57.32 57.32 57.35 57.35 57.35 57.58 57.58 58.02 58.03 57.93 57.93 57.93 58.00 58.00 58.00 58.03 58.03 58.78 58.72

Long.

- 8.02 - 8.02 - 7.52 - 7.52 - 7.52 - 6.85 - 6.85 - 6.85 -6.17 -8.57 - 6.55 - 7.98 - 7.98 - 8.58 - 9.03 - 9.03 - 9.00 -9.00 - 9.02 - 9.02 - 9.00 -9.00 - 11.07 - 11.07 - 11.07 - 9.67 - 9.67 - 9.67 - 9.45 - 9.45 - 9.45 -8.17 -8.17 -9.15 - 9.00 - 8.85 - 8.85 - 8.85 - 8.73 - 8.78 - 8.73 -8.17 -8.17 - 7.50 - 5.88

I

48 48 54 54 54 62 62 62 70 41 65 48 48 41 36 36 36 36 36 36 36 36 11 11 11 28 28 28 31 31 31 46 46 34 36 38 38 38 39 39 39 46 46 54 73

J

42 42 43 43 43 42 42 42 42 36 37 30 30 25 24 24 24 24 24 24 22 22 18 18 18 20 20 20 20 20 20 17 17 12 12 13 13 13 12 12 12 12 12 3 3

u

Dep w

V

Obs

Mod

Obs

Mod

h

g

h

g

h

g

h

g

1.10 1.20 0.80 0.80 1.40 2.90 3.30 2.90 1.70 0.90 0.80 1.70 2.10 1.90 2.50 2.40 2.70 2.00 2.90 3.00 2.30 2.50 1.10 1.70 1.40 0.20 0.30 0.30 1.50 2.60 1.60 1.80 0.90 3.10 2.30 2.60 3.30 2.20 2.90 2.30 3.00 3.70 3.50 5.60 3.10

186 143 306 309 289 324 324 314 330 144 204 260 290 271 241 253 238 253 276 258 258 266 229 235 229 252 246 186 276 246 266 196 193 240 212 227 232 229 218 232 233 216 229 284 273

2.61 1.86 0.52 0.49 0.54 2.25 2.24 2.00 1.44 1.39 0.65 1.33 1.06 2.09 11.74 11.66 11.84 11.84 10.19 9.94 12.68 12.57 0.89 0.80 0.58 1.35 1.48 1.60 5.11 5.59 5.59 1.79 2.63 5.12 4.84 2.06 2.06 2.00 3.75 3.74 4.05 6.83 7.89 10.94 1.97

130 151 74 101 137 327 327 326 321 177 161 264 354 235 262 267 246 253 244 234 313 323 55 54 43 105 103 113 117 111 111 211 181 83 115 108 109 116 159 151 164 216 220 54 350

2.50 1.50 1.00 0.60 0.70 1.20 1.80 2.00 1.00 3.20 2.30 3.40 3.80 3.80 3.70 3.90 3.00 3.60 4.40 4.60 3.20 3.40 0.20 0.90 0.80 0.30 0.30 0.60 3.20 1.70 3.10 4.20 3.70 1.90 1.90 2.80 3.00 2.90 1.80 2.10 2.80 3.40 2.80 5.00 1.60

75 64 59 48 83 163 165 190 131 88 143 116 148 154 164 157 151 150 170 151 153 151 211 141 143 325 293 306 169 164 164 145 165 153 153 166 160 176 173 168 173 171 174 208 212

2.84 3.26 1.44 1.38 1.28 2.67 2.65 2.27 0.31 3.81 2.71 4.33 2.16 5.35 12.04 12.42 13.45 13.70 11.57 11.05 12.16 12.03 1.25 1.18 1.07 0.90 0.88 0.68 2.14 1.91 1.91 5.63 5.10 5.94 4.92 2.97 2.96 2.70 1.88 2.47 1.91 3.59 4.53 12.72 1.10

98 119 136 131 119 160 160 162 85 114 144 149 158 149 181 187 165 174 172 163 215 227 332 334 330 206 209 192 78 53 53 186 176 325 347 312 312 314 15 5 26 164 166 294 285

73 73 54 54 54 58 58 58 110 136 44 170 170 126 134 134 139 139 138 138 140 140 589 589 589 1614 1614 1614 145 145 145 135 135 194 177 155 155 152 146 142 146 142 142 110 114

Dep m

41 10 30 28 11 39 37 11 6 25 11 140 50 25 98 28 95 25 20 90 75 25 300 100 50 1405 1104 503 90 25 25 75 25 151 25 100 100 25 105 28 25 75 25 25 11

R. Proctor, A.M. Davies /Journal

Fig. 3b, Tables 2,3X It can be seen (Fig. 5) that at 57” N the local shelf topography tends to enhance the cross-shelf component of the wave and re-orient the major axis along the direction of the bathymetry. Values computed by the model are in good agreement with those observed. Acoustic Doppler profiler measurements in the Minch (see Fig. 1) show the model performs well in this shallow water area (Simpson et al., 1990). 3.3. 0, tidal elevations and currents The 0, cotidal chart (Fig. 61, shows a region of maximum 0, tidal amplitude at the shelf edge to the north-west of the Hebrides (Fig. 1) with other local maxima along the shelf edge to the south of this. The 0, amplitude increases from 4 cm in the Sea of the Hebrides (Fig. 1) to the order of 8 cm in The Minch (Fig. 1) and further north (Fig. 6) with a maximum of 10 cm at the shelf break. The phase changes from 30” in the region to the north of the North Channel to 0” at the shelf break. A detailed comparison with observations (Table 4) reveals that on average the model can reproduce the 0, elevations to within 2 cm (the order of the error in tidal observations, Pugh and Vassie, 1976), although since the observations are small this appears as a large percentage error. There appear to be local intensifications at positions 35 and 37, probably not resolved in the model, although the model accurately reproduces the tide at position 36. Phase errors are quite large at some locations for example at positions 9 and 11 (errors of order 45”) possibly due to local effects associated with the channel through the north and south Hebrides Islands. Similarly there are large phase errors (of order 50” and 30”) at locations 23 and 25, again possibly due to local effects, although at location 24 the model accurately reproduces the 0, elevation. The spatial distribution of 0, current ellipses Fig. 7a shows regions of strong, near circular ellipses in the shelf edge area to the north-west and south-west of the Hebrides. In the shallow water regions, the 0, currents are very weak (< 0.05 ms-‘1, and the spatial distribution is significantly different from that found for the M,

of Marine Systems 7 (1996) 43-66

59

tide. Current ellipses at the sea bed (Fig. show a similar spatial variability to those at sea surface, although current magnitudes are duced by frictional effects. A detailed comparison of computed and

7b) the reob-

Table 6 As Table 3 but for 0, tidal ellipses Place

ii-A B B B C C C D E F G G H I I J J K K W W L L L M M M N N N 0 0 P Q R R R S S S T T U v

Major

Minor

Alpha

0

M

0

M

2.54 1.54 1.09 0.81 1.54 3.12 3.72 3.18 1.95 3.24 2.34 3.69 4.18 3.92 3.77 3.91 3.02 3.64 4.52 4.73 3.30 3.66 1.12 1.70 1.40 0.31 0.30 0.62 3.24 2.62 3.12 4.37 3.79 3.10 2.62 3.30 3.63 3.30 3.21 2.65 3.56 4.65 4.01 5.97 3.22

3.71 3.66 1.46 1.44 1.38 3.47 3.45 3.00 1.45 3.87 2.78 4.37 2.39 5.35 12.85 13.18 13.85 14.26 12.61 12.18 13.23 13.06 1.26 1.20 1.09 1.36 1.51 1.61 5.38 5.68 5.68 5.86 5.74 6.76 6.18 3.54 3.54 3.33 4.06 4.32 4.32 7.24 8.43 14.64 2.04

1.01 1.15 0.68 0.58 0.28 0.36 0.57 1.51 0.28 0.74 0.69 0.92 1.18 1.64 2.39 2.38 2.68 1.93 2.72 2.79 2.15 2.11 0.06 0.90 0.80 0.19 0.30 0.25 1.42 1.67 1.55 1.35 0.41 1.90 1.43 1.93 2.60 1.55 1.15 1.64 2.04 1.91 2.00 4.55 1.35

1.08 0.87 0.45 0.24 0.16 0.39 0.39 0.40 0.26 1.22 0.19 1.19 0.27 2.09 10.85 10.79 11.36 11.16 8.88 8.52 11.55 11.49 0.88 0.78 0.55 0.87 0.83 0.66 1.29 1.59 1.59 0.72 0.22 3.97 3.07 0.72 0.69 0.49 1.04 1.20 1.19 2.68 3.43 8.20 0.96

Rot

___ 0

101 70 120 166 155 158 152 152 150 81 80 114 116 106 75 96 77 100 106 107 109 117 10 177 3 72 0 107 100 9 98 73 78 3 35 49 36 57 27 39 41 42 34 32 17

MO 48 62 79 73 68 130 130 131 173 80 77 98 115 88 49 54 65 64 56 54 144 145 80 78 78 167 167 5 19 11 11 74 63 126 134 124 124 126 156 149 159 21 23 127 17

M + + + + + + + -

+ + + + + + + -

R. Proctor, A.M. Davies /Journal

60

of Marine Systems 7 (1996) 43-66

served amplitudes and phases of the u and u components of velocity, and current ellipse parameters are given in Tables 5 and 6. Referring to these Tables, it is clear that the observed 0, tidal currents are the order of 2 cm s-l, reaching a maximum of approximately 5 cm s-’ at locations K and U. Comparing computed and observed amplitudes and phases, it is clear that in shallow water, (e.g. at location C) the model reproduces the observed amplitude and phase. However at positions I, J, K and W (at the top of the shelf edge) the model significantly overestimates the 0, tidal current amplitude. As discussed previously water depths change very rapidly in this region, and are poorly resolved in the model. The physical nature of the 0, tide is that its period of 25.82 h (13.943” hh’) is close to the resonance period of the first mode shelf wave, in the area to the west of the Hebrides, with the period of the shelf wave depending critically upon the cross shelf slope (Huthnance, 1986; Huthnance et al., 1986). Consequently in areas where the cross shelf slope is such that the computed period is close to 25.82 h then resonance occurs and the 0, tide is amplified, and this appears to occur in the model to the west of the Hebrides. For a number of cross-shelf sections normal to the bathymetry, model barotropic wave forms (Kelvin and shelf waves for 0,) were calculated using the method described in Cartwright et al. (1980) (and subsequently used in Heaps et al., 1988) and a search was made for the maximum (resonant) frequency. Grid spacing was taken as the model cell size (- 5 km) and depths were

taken from the model. Boundary conditions for the calculations were a) no normal flow at the coast, and b) exponentatial decay offshore at a depth of N 2000 m. (Change of boundary condition to no normal flow offshore, and increasing the grid resolution to N 1 km (by spline interpolation of the model depths) did not substantially alter the results). It is important to note that although the computed 0, tide in the cross shelf model did not change on the finer 1 km grid, the depths used were based upon those on the original 5 km grid. In essence the wavelength of the 0, is sufficiently long that it can be resolved on a 5 km grid, but the detailed structure of the cross shelf topography which is responsible for determining the intensification of the 0, tide needs to be resolved as accurately as is possible. From Table 7 it can be seen that the Kelvin wave remains consistent over the model domain with a wavenumber of N -3”/100 km (negative indicates propagation northwards) but decreasing to the south. The shelf wave component has a wave number of - 60”/100 km. The maximum frequency at each section shows that 58.5”N is closest to resonance with 0, whilst 57” N is furthest. This reflects the bathymetric profiles at these points, the profile at 58.5” N has a smaller gradient than at 57“ N. The consequence is that maximum 0, amplitudes occur (Fig. 6) at approximately 58.5” N. Cross shelf structure of the waveforms are shown in Fig. 8 for sections at 57” N and 58.5” N. For each section the figure shows (a> depth profile, (b) normalised amplitudes of elevation (z), along shelf current (a> and cross shelf current (c)

Table 7 Barotropic wave forms from the model 0, Kelvin period CT) = 25.84 h

58.5” N 58” N 57.5” N 57” N 56.5” N 56”N

0, shelf

maximum frequency

h uzln)

k (o/100 km)

A

k

A

k

T(h)

11135 11826 11138 12083 13174 13613

-3.2 -3.0 -3.2 -3.0 - 2.7 -2.6

379 655 711 672 586 645

-95 -55 -51 -54 -61 -56

240 295 262 238 220 236

- 1.50 -122 - 138 -151 - 163 - 153

23.7 20.4 19.0 18.4 19.8 22.6

R. Proctor, A.M. Davies /Journal

Fig. 8. Semi-analytical calculations of barotropic UC,,, = Kelvin wave, S, = shelf wave) at (b) 0, (c) maximum frequency (S,) at 57”N and elevation, a = alongshelf current, c = cross shelf the depth profile.

wave profiles frequency and 58S”N. Z= current. (a) is

of Marine Systems 7 (1996) 43-66

61

for the Kelvin wave (kw) and shelf wave (SW) at 0, frequency, and (c) normalised amplitudes of z, a and c for the maximum shelf wave frequency. Negative amplitudes indicate a phase change of 180” relative to positive amplitudes. At 58.5” N the shelf wave form of 0, closely resembles that at the maximum frequency. Here, the alongshelf wave component of current is seen to have its largest values over the shelf, diminishing rapidly as the depth increases. The cross shelf component however shows a maximum at the shelf edge, reducing to zero both onshore and offshore. This leads to the increasingly rotary current structure seen on approaching the shelf slope and the sudden decay offshore (Fig. 7a). Currents have a maximum at approximately 40 km offshore (200 m depth) whereas elevations have a maximum 70 km offshore (500 m depth). The Kelvin wave shows consistent alongshore current and almost no cross shelf component (as expected). At 57” N, however, shelf wave elevation and current maxima are more coincident at the top of the slope (200 m depth). The Kelvin wave exhibits a different behaviour (to 58.5” N) in which the alongshore component of current has a minimum (near zero value) at the top of the slope (at 90 km) where the cross shore component has a maximum. A theoretical calculation of the 0, co-tidal chart using the wavenumbers calculated for 58.5” N (Fig. 9a) agrees well with the model computed 0, at this latitude (Fig. 61, correctly positioning the maximum amplitude (70 km offshore at the top of the slope) and agreeing in magnitude. The relative contributions of Kelvin wave and shelf wave are shown in Fig. 9b and c, respectively. The shelf wave of Fig. 9c is seen to contribute most of the structure in Fig. 9a. The maximum elevation occurs where the two waves are in phase. Diminishing near shore amplitudes are the result of the shelf wave being out of phase with the Kelvin wave. The resonant period of shelf waves however is influenced by stratification, which is not included in the model, although the ocean thermocline is at the order of 500 to 1000 m. Also horizontal viscous effects, which are a major source of

R. Proctor, A.M. Davies /Journal

62

of Marine Systems 7 (1996) 43-66

Table 8 As Table 5, Ot currents but computed with increased horizontal eddy viscosity Place

Lat

Long

I

Ju

” Obs

A A B B B C C C D E F G G H I I J J K K W W L L L M M M N N N 0 0 P

Q R R R S s s T T U V

55.50 55.50 55.42 55.42 55.42 55.52 55.52 55.52 55.47 56.00 55.88 56.48 56.48 56.92 57.02 57.02 57.00 57.00 56.98 56.98 57.13 57.13 57.47 57.47 51.47 57.32 57.32 57.32 57.35 57.35 57.35 57.58 57.58 58.02 58.03 57.93 57.93 57.93 58.00 58.00 58.00 58.03 58.03 58.78 58.72

Mod

Obs

Dep w

Dep m

73 73 54 54 54 58 58 58 110 136 44 170 170 126 134 134 139 139 138 138 140 140 589 589 589 1614 1614 1614 145 145 145 135 135 194 177 155 152 155 146 142 146 142 142 110 114

41 10 30 28 11 39 37 11 6 25 11 140 50 25 98 28 95 25 20 90 75 25 300 100 50 1405 1104 503 90 25 25 75 25 151 25 100 100 25 105 28 25 75 25 25 11

Mod

h

g

h

g

h

g

h

g

- 8.02

48

42

1.10

186

2.06

135

2.50

- 8.02 - 7.52 - 7.52 - 7.52 - 6.85 - 6.85 - 6.85 -6.17 - 8.57 - 6.55 - 7.98 - 7.98 - 8.58 - 9.03 - 9.03 - 9.00 - 9.00 - 9.02 - 9.02 -9.00 - 9.00 - 11.07 - 11.07 - 11.07 - 9.67 - 9.67 -9.67 - 9.45 - 9.45 - 9.45 - 8.17 -8.17 -9.15 - 9.00 - 8.85 - 8.85 - 8.85 - 8.73 - 8.78 - 8.73 -8.17 -8.17 - 7.50 - 5.88

48 54 54 54 62 62 62 70 41 65 48 48 41 36 36 36 36 36 36 36 36 11 11 11 28 28 28 31 31 31 46 46 34 36 38 38 38 39 39 39 46 46 54 73

42 43 43 43 42 42 42 42 36 37 30 30 25 24 24 24 24 24 24 22 22 18 18 18 20 20 20 20 20 20 17 17 12 12 13 13 13 12 12 12 12 12 3 3

1.20 0.80 0.80 1.40 2.90 3.30 2.90 1.70 0.90 0.80 1.70 2.10 1.90 2.50 2.40 2.70 2.00 2.90 3.00 2.30 2.50 1.10 1.70 1.40 0.20 0.30 0.30 1.50 2.60 1.60 1.80 0.90 3.10 2.30 2.60 3.30 2.20 2.90 2.30 3.00 3.70 3.50 5.60 3.10

143 306 309 289 324 324 314 330 144 204 260 290 271 241 253 238 253 276 258 258 266 229 235 229 252 246 186 276 246 266 196 193 240 212 227 232 229 218 232 233 216 229 284 273

1.67 0.35 0.25 0.27 2.42 2.41 2.12 1.53 1.35 0.57 1.02 0.80 2.70 5.14 5.37 5.15 5.62 4.78 4.12 5.96 6.33 1.17 1.12 1.04 0.77 0.88 0.87 2.46 2.82 2.82 2.09 2.90 3.11 3.15 1.25 1.26 1.22 3.28 3.12 3.57 6.64 7.30 10.24 1.97

148 74 99 190 327 327 326 323 178 166 291 307 248 262 264 251 257 251 243 300 309 28 19 4 132 135 137 138 125 125 206 195 92 130 158 158 165 181 175 185 221 225 47 348

1.50 1.00 0.60 0.70 1.20 1.80 2.00 1.00 3.20 2.30 3.40 3.80 3.80 3.70 3.90 3.00 3.60 4.40 4.60 3.20 3.40 0.20 0.90 0.80 0.30 0.30 0.60 3.20 1.70 3.10 4.20 3.70 1.90 1.90 2.80 3.00 2.90 1.80 2.10 2.80 3.40 2.80 5.00 1.60

75 64 59 48 83 163 165 190 131 88 143 116 148 154 164 157 151 150 170 151 153 151 211 141 143 325 293 306 169 164 164 145 165 153 153 166 160 176 173 168 173 171 174 208 212

2.64 2.81 1.07 1.06 1.08 2.84 2.82 2.28 0.58 3.56 2.52 3.65 2.95 5.74 5.84 6.11 6.49 6.91 6.01 5.42 6.23 6.35 1.69 1.63 1.53 0.63 0.55 0.61 1.14 0.89 0.89 5.69 5.76 3.85 2.92 1.66 1.66 1.60 0.48 0.89 0.73 4.06 4.82 12.46 1.07

98 117 135 131 118 163 163 162 68 111 145 146 143 151 172 173 161 167 166 160 193 204 315 308 302 235 235 232 109 79 79 181 173 322 345 287 287 282 58 19 77 172 175 289 282

Fig. 9. Semi-analytical calculation for 0, co-tidal chart using parameters obtained at 58.5”N section (Fig. 8) with (a) combined wave, (b) Kelvin wave and (c) Shelf wave (- amplitude in mm, -phase in degrees).

R. Proctor, A.M. Davies/Journal

01

(a) 150. j

1

63

of Marine Systems 7 (1996) 43-66

CO-TIDAL

CHART

’

2

50 Distance

01

1

CO-TIDAL ’

0 oIong

CHART

-50 coast

(km)

-

KELVIN 1_11-

-100

WAVE

15’-

08

,

I I 50

T v6 ,9

TOO-

80

I / I I I ,350

I I / 355: I

E ! 8

-

: 2

50 -

/

I --------+-85-

85

/ I I I , I

/ I I

F I90 0 100

I1, 90

: 1

I

I 50

a Dtstance

01 (4

‘““I

CO&TIDAL

CHART

/I

100

50

Distance

: 0 aIonq

8 -50

SHELF

:

-100

.%-I.

WAVE ’ I

1

-50 coast

II

(km)

360 1,111

,’

olv.

-

1 I II

1, 0 don9 coast

(km)

.:

J

-100

64

R Proctor, A.M. Davies /Journal

damping in the ocean, are not included within the model, and this will certainly influence the magnitude of any small scale feature.

of Marine Systems 7 (1996) 43-66 Table 9 As Table 6, 0, tidal current ellipses but computed increased horizontal eddy viscosity Place

3.4. Influence of horizontal viscous effects To examine the extent to which horizontal viscosity influences the tide, the previous calculation was repeated with A,(m2s-‘) = Ah(m), with A = 1.0 ms-’ giving a horizontal diffusion coefficient which is larger in deep water. A comparison of M, and 0, tidal elevations and current, revealed that the only substantial changes (i.e. changes larger than 0.01 m in elevation, and 0.01 ms-’ in current and 2” in phase) occurred for the 0, component of the tidal current (Tables 8 and 9). Comparing Tables 8 and 9 with Tables 5 and 6, it is clear that over the majority of the region adding a horizontal viscosity leads to a small change in the 0, currents, except at locations I, J, K and W where the amplitude of the u and v components of current decrease from the earlier value of order 0.12 ms-’ to values of approximately 0.06 ms-’ (see Table 8) with some small (less than 10”) changes of phase, although at location W the v-component of phase changes by the order of 20”. The change in amplitude of the 0, components of current is reflected in the reduced magnitude of the major and minor axis of the current ellipses at these locations which are now in better agreement with the observations (Table 9). The alignment of the current ellipse also changes, although it is not clear that this leads to any improvement in accuracy. This calculation clearly shows that the magnitude of the horizontal eddy viscosity has an important influence upon features which show significant spatial variability in the shelf edge region, such as the 0, tidal current off the west coast of the Hebrides. Although large scale motion such as the M, tide is not affected by the horizontal viscosity. These different responses are to be expected since horizontal viscosity acts as a spatial smoothing operator which smoothes short wave small scale processes much more than the longer wavelength effects. Although a simple parameterization of the co-

A A B B B C C C D E F G G H I I .I J K K W W L L L M M M N N N 0 0 P

Q R R R s s s T T U V

Major

Minor

Alpha -

0

M

0

MO

2.54 1.54 1.09 0.81 1.54 3.12 3.72 3.18 1.95 3.24 2.34 3.69 4.18 3.92 3.77 3.91 3.02 3.64 4.52 4.73 3.30 3.66 1.12 1.70 1.40 0.31 0.30 0.62 3.24 2.62 3.12 4.37 3.79 3.10 2.62 3.30 3.63 3.30 3.21 2.65 3.56 4.65 4.01 5.97 3.22

3.18 3.17 1.08 1.08 1.08 3.70 3.67 3.08 1.54 3.61 2.57 3.74 3.05 5.75 5.84 6.11 6.49 6.91 6.05 5.47 6.93 7.13 1.75 1.70 1.64 0.80 0.89 0.88 2.67 2.89 2.89 6.01 6.38 4.51 4.09 1.90 1.90 1.75 3.30 3.23 3.58 7.28 8.09 13.98 2.03

1.01 1.15 0.68 0.58 0.28 0.36 0.57 1.51 0.28 0.74 0.69 0.92 1.18 1.64 2.39 2.38 2.68 1.93 2.72 2.79 2.15 2.11 0.06 0.90 0.80 0.19 0.30 0.25 1.42 1.67 1.55 1.35 0.41 1.90 1.43 1.93 2.60 1.55 1.15 1.64 2.04 1.91 2.00 4.55 1.35

1.04 0.77 0.30 0.13 0.26 0.53 0.53 0.43 0.56 1.22 0.20 0.58 0.20 2.67 5.14 5.37 5.15 5.62 4.74 4.05 5.14 5.44 1.08 1.02 0.86 0.59 0.54 0.60 0.51 0.63 0.63 0.85 0.97 2.03 1.30 0.85 0.87 0.99 0.40 0.35 0.69 2.76 3.34 8.04 0.94

with

Rot MOM

101 70 120 166 155 158 152 152 150 81 80 114 116 106 75 96 77 100 106 107 109 117 10 177 3 72 0 107 100 9 98 73 78 3 35 49 36 57 27 39 41 42 34 32 17

54 61 80 78 85 130 130 133 174 81 78 103 105 94 92 93 91 90 80 78 131 135 71 70 65 157 170 173 23 13 13 71 64 126 138 122 123 120 175 165 176 26 28 124 16

+ + + + + + + -

+ + + + + + + + + -

efficient of horizontal eddy viscosity A,, has been used in these calculations, in order to examine its influence, more complex forms based upon low

R Proctor, A.M. Davies /Journal

order turbulence models exist in the literature (Smagorinsky, 1963). These models suggest that A, values will increase and hence the horizontal diffusion term will become more important in regions of large horizontal shear and vorticity, such as the shelf edge, and provide a physical justification for the use of this term in shelf edge models.

4. Concluding

remarks

In this paper we have described the major stages in developing a high resolution shelf edge model for the region off the west coast of Scotland, using a functional representation in the vertical. In deep water off the continental shelf, the expansion converges slowly requiring the order of thirty functions in the vertical. An enhanced rate of convergence can in theory be obtained using a mixed basis set (Davies, 19921, although this method has not been proved in a realistic application. Calculations have shown that the model can reproduce M, tidal elevations over the region, with currents accurately determined in regions of gradually changing bathymetry, although at the shelf edge where model depths change rapidly, large errors in the computed currents can occur. In deep water, tidal currents and elevations are small, with amplitudes increasing rapidly in a uniform manner in shallow water, with the strongest M, tidal currents in the region to the north of the North Channel. The 0, tidal elevations are an order of magnitude smaller than M,, with multiple maxima occurring along the shelf edge region where 0, tidal elevations and currents are intensified, because the period of 0, is close to that of the first mode shelf wave. The 0, tidal current ellipses show significant spatial variability in the shelf edge region, and calculations have clearly shown that their magnitude is very sensitive to the value of horizontal eddy viscosity. Semi-analytical calculations of shelf wave profiles suggest that 0, elevation is adequately modelled and is not significantly affected by increasing the resolution of the bathymetry.

of Marine Systems 7 (1996) 43-66

65

Although these calculations have been for the barotropic tide, an exact determination of this tide is important since it is the forcing mechanism producing the baroclinic tide. Calculations have clearly shown the importance of accurately determining and representing the shelf edge topography and the influence of horizontal eddy viscosity upon the spatial variability of tidal currents in the shelf edge region. Future calculations aimed at simulating shelf edge processes in this region will certainly require a finer grid in the cross shelf region and if possible a physically realistic representation of horizontal eddy viscosity.

Acknowledgements

Partial support for this work from the Defence Research Agency is gratefully acknowledged. The authors are indebted to Mr. R.A. Smith for preparing figures and Mrs. J. Hardcastle for typing the text.

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west European continental shelf with application to the M, tide. J. Phys. Oceanogr., 16: 797-813. Davies, A.M., 1987. Spectral models in continental shelf sea oceanography. In: N.S. Heaps (Editor), Three-dimensional Coastal Ocean Models. Am. Geophys. Union. Coastal and Estuarine Sciences, 4: 71-106. Davies, A.M., 1992. Modelling currents in highly sheared surface and bed boundary layers. Cont. Shelf Res., 12: 189-211. Davies, A.M., 1993. Numerical problems in simulating tidal flows with a friction velocity dependent eddy viscosity and the influence of stratification. Int. J. Num. Meth. Fluids, 16: 105-131. Davies, A.M. and Furnes, G.K., 1980. Observed and computed M, tidal currents in the North Sea. J. Phys. Oceanogr., 10: 237-257. Davies, A.M. and Aldridge, J.N., 1993. A numerical model study of parameters influencing tidal currents in the Irish Sea. J. Geophys. Res., 98: 7049-7067. Davies, A.M., Grzonka, R.B. and Stephens, C.V., 1992. Implementation of a three dimensional hydrodynamic numerical sea model using parallel processing on a CRAY X-MP series computer. In: D.J. Evans (Editor), Advances in Parallel Computing, Vol 2. JAI Press, pp. 145-185. Davies, A.M. and Jones, J.E., 1990. Application of a three-dimensional turbulence energy model to the determination of tidal currents on the northwest European shelf. J. Geophys. Res., 95: 18,143-18,162. Davies, A.M. and Jones, J.E., 1992. A three dimensional model of the M,, S,, N,, K, and 0, tides in the Celtic and Irish Seas. Prog. Oceanogr., 29: 197-234. Davies, A.M. and Proctor, R., 1990. Developing and optimizing a 3D-spectral/finite difference hydrodynamic model for the CRAY X-MP. Comput. Fluids, 18: 259-270. Flather, R.A., 1976. A tidal model of the north west European continental shelf. Mem. Sot. R. Sci. Liege, 10: 141164. Flather, R.A., 1987. A tidal model of the northeast Pacific. Atmos. Ocean., 25: 22-45. Flather, R.A., 1988. A numerical model investigation of tides and diurnal period continental shelf waves along Vancouver Island. J. Phys. Oceanogr., 18: 1155139. Flather, R.A., Proctor, R. and Wolf, J., 1995. A tidal model of the northeast Atlantic Ocean (in prep.) Foreman, M.G.G. and Walters, R.A., 1990. A finite-element tidal model for the southwest coast of Vancouver Island. Atmos. Ocean., 18: 261-287. Foreman, M.G.G., Henry, R.F., Walters, R.A. and Ballantyne, V.A., 1993. A finite element model for tides and

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resonance along the north coast of British Columbia. J. Geophys. Res., 98: 2509-2531. Gordon, R.B. and Spaulding, M.L., 1987. Numerical simulations of the tidal- and wind-driven circulation in Narragansett Bay. Estuarine Coastal Shelf Sci., 24: 611-636. Heaps, N.S., Huthnance, J.M., Jones, J.E. and Wolf, J., 1988. Modelling tthe storm-driven shelf waves north of Scotland - I. Idealised Model. Cont. Shelf Res., 8: 1187-1210. Howarth, M.J., 1990. Atlas of Tidal Elevations and Currents around the British Isles. Dep. Energy, Offshore Technol. Rep., OTH 89 293, 16 pp. Huthnance, J.M., 1986. The Rockall slope current and shelf edge processes. Proc. R. Sot. Edinburgh, 88B: 83-101. Huthnance, J.M., Mysak, L.A. and Wang, D.-P., 1986. Coastal trapped waves. In: C.N.K. Mooers (Editor), Baroclinic Processes on Continental Shelves. Am. Geophys. Union. Coastal and Estuarine Science, 3: 1-18. Lynch, D.R. and Werner, F.E., 1987. Three-dimensional hydrodynamics on finite elements. Part 1: Linearized harmonic model. Int. J. Numer. Methods Fluids, 7: 871-909. Lynch, D.R. and Werner, F.E., 1991. Three-dimensional velocities from a finite-element model of English Channel/Southern Bight tides. pp. 183-200 In: B.B. Parker (Editor), Tidal Hydrodynamics. Wiley, New York, 883 pp. Proctor, R., 1981. Tides and Residual Circulation in the Irish Sea: A Numerical Modelling Approach. PhD Thesis, Liverpool Univ. Proctor, R., 1987. A three-dimensional numerical model of the eastern Irish Sea. In: J. Noye (Editor), Numerical Modelling. Application to Marine Systems. Elsevier, Amsterdam, pp. 25-45. Pugh, D.T. and Vassie, J.M., 1976. Tide and surge propagation off-shore in the Dowsing region of the North Sea. Dtsch. Hydrogr. Z., 29: 163-213. Sherwin, T.J. and Taylor, N., 1989. The application of a finite difference model of internal tide generation to the NW European Shelf. Dtsch. Hydrogr. Z., 42: 151-167. Simpson, J.H., Mitchelson-Jacob, E.G. and Hill, A.E., 1990. Flow structure in a channel from an acoustic Doppler current profiler. Cont. Shelf Res., 10: 589-603. Smagorinsky, J., 1963. General circulation experiments with the primitive equations. I. The basic experiment. Mon. Weather Rev., 91: 99-164. Xing, J. and Davies, A.M., 1994a. Application of turbulence energy models to the computation of tidal currents and mixing intensities in shelf edge regions (submitted). Xing, J. and Davies, A.M., 1994b. A numerical model of the long term flow along the Malin-Hebrides shelf (submitted).