The influence of bottom and internal friction upon tidal currents: Taylor's problem in three dimensions

~

Pergamon

Continental Shelf Research, Vol. 15, No. 10, pp. 1251-1285, 1995 Copyright © 1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0278-4343/95 $9,50 + 0.00 0278--4343(94)00076-X

The influence of bottom and internal friction upon tidal currents: Taylor's problem in three dimensions A. M. DAVIES* and J. E. JONES* (Received 25 November 1992; in revised form 17 March 1994; accepted 20 July 1994) Abstraet--ldealised calculations using a three-dimensional model of the North Sea, represented by a flat bottom rectangular basin, open at its northern end, are used to examine the influence of bottom topography, internal friction (parameterised using an eddy viscosity formulation) and bottom friction upon the position of M 2 tidal amphidromes and tidal current distributions over the North Sea. Calculations show that the bottom slope of the North Sea, from deep in the north to shallow in the south, together with the coefficient of bottom friction, influence the position of the tidal amphidromes. In a three-dimensional model, it is not just bottom friction which influences the position of tidal amphidromes, but also the magnitude of eddy viscosity. In the case of a no-slip condition applied at the sea bed, the magnitude of near-bed viscosity has a major influence upon the position of tidal amphidromes, and tidal current profiles. In a non-linear model in which bed stress and eddy viscosity magnitude depend upon the total tidal current, calculations show that enhanced levels of turbulence due to the M2 tide have a significant influence upon other tidal constituents (e.g. the S 2 tide). By using a combination of a single point model in the vertical, based upon a rotary decomposition of the tide, and the full three-dimensional model of the basin, significant insight into the parameters (e.g. water depth, bottom friction, internal friction and tidal frequency) influencing the spatial distribution of tidal ellipses, and the profile of tidal currents is obtained.

1. INTRODUCTION In order to gain a detailed appreciation of the physical mechanisms governing tides it is essential to have a combination of fully analytical models (e.g. Taylor, 1921), semianalytical (e.g. Brown, 1973, 1987, 1989; Johns and Dyke, 1971; Davies, 1990) and full numerical simulations (e.g. Davies, 1986). The problem of Kelvin wave propagation in a flat bottomed rotating channel closed at one end, in the absence of friction was investigated by Taylor (1921). The solution demonstrated the existence of amphidromic points, although they were symmetric about the centre line of the channel. Subsequently the influence of bottom friction and the generation of Poincar6 waves at the closed end of the channel was investigated by a number of authors (e.g. Hendershott and Speranza, 1971; Rienecker and Teubner, 1980) using semi-analytical methods. These methods were analytical in the sense that an *Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside, L43 7 R A , U.K. 1251

1252

A . M . Davies and J. E. Jones

analytical solution in terms of an infinite expansion was obtained for the linear equations, although subsequently a Galerkin or collocation method was used to solve a truncated expansion. These calculations clearly showed the importance of bottom friction in determining the position of the amphidromes. Frictional influence together with water depth and period of oscillatory forcing determined the generation of Poincar6 waves at the closed end of the basin, and the distance these waves could propagate. Other calculations (Brown, 1987, 1989) used a semi-enclosed channel representing the North Sea, to investigate the importance of a reflective boundary at the southern end of the North Sea, upon tides in the North Sea. In all these cases, the basin was of a constant depth, however such simple models give significant insight into full numerical model simulations using realistic bottom topography and coastlines (Davies, 1986). To date, to the author's knowledge Taylor's problem has never been investigated in three dimensions, although a number of three-dimensional tidal models have been validated against observations (Davies and Jones, 1990; Davies, 1986). As we will show in this paper, Taylor's problem in three dimensions, is ideal for demonstrating the importance of bottom and internal friction, together with the period of tidal forcing, upon both the position of the amphidromes and the spatial distribution of tidal currents and their profiles. The mathematical development of the three-dimensional model is described in Section 2, using both slip and no-slip bottom boundary conditions. Time variability in the model is retained rather than using a rotary decomposition method at the single frequency, so that the influence of a number of tidal harmonics in combination can be considered. A numerical solution using finite difference methods is employed in the horizontal, as this approach is most readily extended to realistic coastlines. Other approaches namely the finite element method which has been used very successfully in tidal simulations (e.g. Lynch and Werner, 1991; Waiters and Werner, 1989; Werner and Lynch, 1989) could also be applied in the horizontal. A semi-analytical approach is used in the vertical, namely the Galerkin method, with an expansion of functions, taken as eigenfunctions of the eddy viscosity profile. This technique can be used with an arbitrary distribution of bottom topography, and yields significant insight into the mechanisms influencing tidal current profiles. Results of calculations using an idealized semi-enclosed channel having dimensions representative of the North Sea are presented in Section 3.

2. T H E N U M E R I C A L M O D E L 2.1. Hydrodynamic equations and boundary conditions

To be consistent with other three-dimensional models of the North Sea and Continental Shelf (Davies, 1986) we will develop the equations using polar coordinates, thus

1

o Ih

1

Of h

0~ 4- v cos ¢p dz + - Ot R cos q~ O~ _~ Rcosq~0z au at

u Ou v Ou + - - - + w RcosdpOZ ROq~

- - + - -

Ou Oz

uvtanq~ R

yv

-

_~

u dz = 0

g 0~ + O__(pOu] Rcosq~Oz Oz\ Oz/

(1)

(2)

Taylor's problem in three dimensions

Ov+__u Ov ~ _ v O V + w _ _Ov+ _ _u2 + ytan ¢p Ot R cos ~b 0Z R Ocp Oz R 1

w -

- R cos 0

0 [h oZ

u

tanq~Ih dz . . . . . . v dz t¢

z

1253

u . . .gO~ . + O[..Ov] R Ocp -~z t '~ )~z +

10 -R

I'j

v dz.

(3)

(4)

However in some calculations, in order to compare results with Taylor's original problem, we will neglect the non-linear terms. Also for clarity the solution using the Galerkin method in the vertical (see later) is developed in terms of a linear set of equations. [The solution using the full non-linear equations is given in Davies (1986).] In equations (1)-(4), Z, ~ denote east longitude and north latitude, t is time, with h water depth and z vertical coordinate. Elevation above the mean level is denoted by ~, with R radius of the Earth, y = 209e sin q~ the Coriolis parameter, with wE speed of Earth's rotation and g acceleration due to gravity, u,v horizontal components of current at depth, w vertical velocity, and ~ vertical eddy viscosity. In order to use a functional approach in the vertical, it is necessary to transform equations (1)-(4) to a normalised sigma coordinate (a) given by, o = (z +

+

Extensive details of the method are given in Davies (1986) and will not be repeated here. For tides a zero surface stress boundary condition is appropriate, whereas at the bed a slip condition can be applied, namely -o

h +~

OaJ

FB = - K p U h ,

- f f - - ~ t x ~ o ) = GB = -KpVh

(5)

with FB, G8 the two components of bed stress and Uh, Vh corresponding components of velocity, where K = k I a linear slip condition or K = k2(U~ + V~) ''2, a quadratic slip condition, with kl and k2 linear or quadratic friction coefficients, and p density of water. An alternative bottom boundary condition, involves no-slip, thus,

U h = Vh = 0.

(6)

Along lateral boundaries a zero normal component of flow condition is applied with a weighting term in the Coriolis parameter (Davies and Jones, submitted) whereas along open boundaries, a radiation condition is applied (Davies, 1986), which can include tidal forcing. By solving the time dependent problem with non-linear bottom friction, the non-linear interaction between the various tidal constituents imposed on the open boundary can be examined. Also by making /~ a function of the flow, interaction between the tidal constituents also occurs through the resulting non-linearity. The importance of these effects will be discussed later. By vertically integrating equations (1)-(4), omitting non-linear terms, and expressing them in Cartesian coordinates the original equations of Taylor (1921) can be readily derived. In the case of a rectangular basin of constant depth h and width A, closed at its

1254

A . M . Davies and J. E. Jones

southern end, and open at its northern end, representing the North Sea (Fig. 1), then it can be readily shown (Hendershott and Speranza, 1971 ; Brown, 1973), that tidal Kelvin waves entering through the northern boundary, are reflected at the southern boundary with the generation of a set of n Poincar6 waves which decay away from the southern end if

n2~/A 2 - (~o2 - ~ ) / g H > 0.

(7)

It is also possible to define a critical frequency coc, given by coc = (72 + :r2 gH/A2) 1/2,

(8)

such that Kelvin waves are perfectly reflected with no Poincar6 wave if co < co¢ (where co is the frequency of the open boundary input tidal wave). Also a critical depth H~, given by H~ = (cox - 72)A2/(~r2g)

(9)

such that Kelvin waves are reflected perfectly if H > H~. The influence of changes in co and H upon the distribution of co-amplitude and co-phase elevation lines has been studied (Brown, 1973; Rienecker and Teubner, 1980) in a twodimensional model. In a three-dimensional model co¢ and H,, as we will show later will also influence the profile of the tidal current.

2.2. Numerical solution In the vertical, a semi-analytical approach is employed in which the two components of velocity are expanded in terms of a set of basis functions fr(O) (modes) and coefficients in the horizontal which vary with position and time. By expressing the eddy viscosity/~ as p = ct(Z, q~, t)~(o)

(10)

with a(Z,~p,t ) taking account of spatial and temporal variations of viscosity, and ~ ( o ) a fixed profile of viscosity in the vertical, it is possible to choose the functions fr(o) as eigenfunctions of the eddy viscosity profile. Since extensive details of the method are given in Davies (1986, 1987) they will not be repeated here. It is important to note that in the case of a slip condition, the first mode is constant in the vertical and corresponds to the depth mean current, with the vertical integral of all higher modes being zero (Davies, 1987). A consequence of this is that only the first mode, which is independent of viscosity formulation contributes to sea surface elevation. In a no-slip calculation all modes contribute to changes in sea surface elevation. A standard finite difference approach using the grid shown in Fig. 1, is used in the horizontal (Davies, 1986, 1987).

2.3. Single tidal constituent, point model In order to understand the influence of coc upon tidal current structure, it is advantageous to consider a single point in the vertical, and one tidal constituent of frequency co. Defining a complex velocity

Q = u + iv

(11)

Taylor's problem in three dimensions

1255

with i = ~ - - 1 , and complex slope S-

g 0~ i- i -g 0~ R cos q$ 0 Z R Ocp

(12)

then the linear form of equations (2) and (3) can be combined, giving

0o+i, __o~ 0 !@ 0~-

- s.

(13)

Considering oscillatory flow at a single frequency m, Q and S can be written as Q =R+ e i''' + R _ e so,t

(14)

S = S+ e ia'' + S _ e i,,,q

(15)

By this means a decomposition into an anticlockwise rotating vector having amplitude R+ and a clockwise vector of amplitude R_ is achieved. Substituting (14) and (15) into (13) and writing it in terms of clockwise and anticlockwise components, gives, a

O ( . OR+]

i(}, + oJ)R+ = h~ Oo \

Oo / - S+

(16)

a 0 ( . OR_] i(7-

(o)R

= ~ 0oo\

-7o~ ] - S

.

(17)

It is apparent from (17) that if o9 is of a comparable value to • then the anti-clockwise component is dominant and R can be large. The consequences of this are discussed later with reference to the calculations.

3. E D D Y VISCOSITY F O R M U L A T I O N Initially, in the series of calculations considered later, a constant value of eddy viscosity was specified, although subsequently a flow dependent viscosity was used, since this is physically more realistic. Davies (1986), related viscosity to the flow field, using, ~t = K l ( f f 2 +

V-2)/(J.)1

(18)

with 0) 1 a characteristic frequency of tidal motion, here taken as 1.0 × 10 -4 s ~, and K~ a dimensionless coefficient, taken by them as 2.0 × 10-s, with ff and F depth mean currents. An alternative expression Bowden (1978), Bowden et al. (1959) is u = K2(ff2 + v-2)l/2h

(19)

with K 2 = 2.5 × 10 -3 a dimensionless experimentally determined coefficient [with a similar magnitude computed from a turbulence energy closure model, Davies and Jones (1990), Davies (1991)], and h water depth. An alternative expression to (18) and (19) is

1256

A.M. Davies and J. E. Jones /~ = k2(ff2 + ~-2)A

(20)

A = C U,

(2l)

where

is the bottom boundary layer thickness, and U, is the frictional velocity computed from U, =

(22)

with r B the instantaneous bed stress, with C, a constant of order 0.2-0.4 (Davies, 1990). In (20) if the computed bottom boundary layer exceeds the water depth, then the boundary layer thickness is depth limited and set to h, giving equation (19), based upon observations (Bowden, 1978; Bowden et al., 1959), in a shallow region where the bottom boundary layer was in fact depth limited. In deep water, where the bottom boundary layer thickness is rotationally limited then equation (20) is equivalent to equation (18). In the calculations described subsequently in this paper, two profiles of viscosity were used in the vertical. In the first (Profile A), viscosity was constant; a slip condition was applied at the sea bed, and no attempt was made to resolve the high shear bottom boundary layer. The second (Profile B) has a linear decrease in eddy viscosity close to the sea bed over a distance h 1, from a value p 1 applied in the upper part of the water column to a value Po at the sea bed. This profile was used with a no-slip condition at the sea bed, in order to resolve the bottom boundary layer. The modes computed with this viscosity profile show a high shear region close to the sea bed (Davies, 1993).

4. T I D A L C A L C U L A T I O N S In order to examine variations in bottom slope, eddy viscosity magnitude and formulation together with changes in bottom friction coefficient, a series of calculations were performed with a range of these parameters (Table 1). Initially only the M 2 tide was used in the model, with oJ = 28.9841 ° h - l , although subsequently the $2 tide was included ~o = 30 ° h -1 to investigate the influence of enhanced levels of friction and turbulence due to an additional tidal constituent. In a final series of calculations, ~o was doubled to examine the influence of this parameter on the spatial distribution of the tide, and tidal current structure. The North Sea rectangle used in the calculations (Fig. 1) has dimensions approximately the same as that employed by Heaps (1972) to study wind induced flow in the North Sea. Heaps (1972) used a flat bottomed rectangle of constant depth h = 65 m; here we can vary the depths linearly from h 1in the north to he in the south (see later). A finite difference grid of 41 grid points north-south and 22 grid points west-east, with grid size 1/5 ° north-south and 1/3 ° west-east was used. This is a sufficiently fine grid to represent the amphidromic distribution found by Taylor (1921), although naturally a finer grid would improve accuracy particularly for the higher harmonics which have a shorter wavelength. The rectangle was open at its northern end, where the radiation condition was applied, with an M2 tidal input appropriate to the northern North Sea. All other boundaries were closed, and in all calculations the M2 tidal input remained the same.

1257

Taylor's problem in three dimensions

Table 1. Summary of numerical calculations Viscosity Water depth Calculation

hi

Friction Const (m2s -1)

h2

Kl]tOl

(a) Influence of bottom topography 1 65 65 0.05 2 97.5 32.5 0.05 3 110 20 0.05 4 120 10 0.05 5 120 10 0.05

----.

K2

k1

-----

-----

0.005 0.005 0.005 0.005

----0.005

Yes Yes Yes Yes --

----Yes

M 2 only M 2 only M 2 only M 2 only M 2 only

-----__

0.005 0.005 0.005 0.001 0.025 0.025

-------

Yes Yes Yes Yes Yes Yes

M 2 only M 2 only M 2 only M 2 only M e only M 2 only

0.005 0.005

---

Yes Yes

S 2 only

0.005

--

Yes

M 4 only ~l:fl0ratio 1:0.00002 1:0.02

.

.

.

(b) Influence of eddy viscosity and bottom friction 6 120 10 -0.2 --7 120 10 --0.0025 -8 120 10 ---0.0025 9 120 10 ---0.0025 10 120 10 ---0.0025 11 120 10 ---2.5x10 5 (c) lnfluence ofothertidalconstituen~ 12A 120 10 -12B 120 10 --

---

---

0.0025 0.0025

--

--

0.0025

m

k 2 Linear

Nonlinear

14,2

Comments

M2 + $2

(d) Influeneeofdoubling 13

120

10

--

(e) lnfluence of a no-slipbottomboundarycondi6on 14

120

10

--

--

--

0.0025

--

--

--

Yes

15

120

10

--

--

--

0.0025

--

--

--

Yes

4.1. Influence o f sea bed t o p o g r a p h y In an initial time series of calculations (500 cm: s-l),

an appropriate

mean

a constant eddy viscosity coefficient of 0.05 m 2 s-

North

Sea value (Heaps,

1972) was employed,

1

with

water depth constant at hi = h2 = h = 65 m, again a mean North Sea value (Heaps, 1972). A b o t t o m f r i c t i o n c o e f f i c i e n t k l = 0 . 0 0 5 m s - 1, w i t h a l i n e a r f o r m u l a t i o n o f b o t t o m f r i c t i o n and other

non-linear

integrated

through

forcing appropriate After

terms

to the northern

five tidal cycles a periodic

harmonically Figure

analysed

2, Chart

land boundary

of this amphidrome land boundaries,

a tidal wave of decreasing

through

time

tidal amphidrome

was

situated

to the east, rather

that

Scotland

was then

(a)].

close to the eastern to the west of this

than midway

between

is d u e t o t h e t i d a l i n p u t a l o n g t h e o p e n between

in the northern

North

and Norway

(Davies,

also with a slight offset to the east occurs in the southern Bight.

the

boundary

Sea, namely 1986). This

as the one which occurs to the west of the Norwegian

t o t h a t f o u n d in t h e S o u t h e r n

was

with M2

open boundary.

obtained,

of the tidal amplitude

the true tidal distribution

amplitude

can be regarded

A second amphidrome

1). T h e m o d e l and motion

to give the M2 co-tidal chart shown in Fig. 2 [Chart

which was taken to represent

basin, corresponding

1, T a b l e

North Sea applied along the northern solution

(a), shows a northern

and eastern

amphidrome

was used (Calculation

of the model, with an intensification

point. The positioning western

omitted,

time from an initial steady state of zero elevation

Coast.

part of the

1258

A.M. Davies and J. E. Jones 59'N

--

--59°N

58ON - -

--58ON

57ON --

--57°N

56ON

--

--56°N

55ON

--

--55ON

54°N

--

--54°N

53ON - -

--53°N

52ON - -

--52ON

51ON_

IX, I

IYI -- 51ON

I

I'E

2°E

3°E

4°E

5"E

6°E

7"E

Fig. 1. Finite difference grid of the North Sea basin, open at its northern end, showing location of points X and Ywhere current profiles arc examined in detail.

In reality the North Sea does not have a constant depth, but is significantly shallower at its southern end than the northern end. The influence of changes in water depth upon the co-tidal chart and hence upon the position of these amphidromes was considered in a subsequent series of calculations in which a linear north-south variation in water depth from hi = 97.5 m in the north to h2 = 32.5 m in the south was examined, with mean water depth h = 0.5 (hi + h2) = 65 m (Table 1, Calculation 2). Calculations with a sloping bottom (hi = 97.5 m, hx = 32.5 m), showed that the northern amphidrome moved to the southeast and became degenerate. A similar southeastward movement occurred with the southern amphidrome. This more realistic depth distribution gives rise to an increase in tidal amplitude along the northwest coast of the rectangle and in the southwest corner. This arises from the fact that in deeper water less energy is lost by bottom friction, and tidal amplitude increases due to a continuity effect as the tidal wave propagates along the west coast of the model. However in the shallower southern part of the North Sea, more tidal energy is dissipated as the water is shallow and tidal amplitudes were reduced below those found previously (Calculation 1). A further increase in bottom slope, hi = 110 m, h 2 = 20 m (Table 1, Calculation 3), produces a further southeast movement of both the northern and southern amphidromes, with the northern amphidrome merging with the eastern boundary. Increasing the north south gradient further, with ht = 120 m, h: = 10 m (Table 1, Calculation 4), produces a

Taylor's problem in three dimensions

1259

further southeast movement of the amphidromes which now move onto the land regions [Fig. 2, Chart (b)]. The neglect of non-linear terms in the shallow southern region of the rectangle with a water depth h2 = 10 m, is a severe approximation, and when the non-linear terms are retained (Table 1, Calculation 5), the amphidrome in the southern part of the rectangle is again present with a degenerate amphidrome along the eastern side [Fig. 2, Chart (c)]. 4.2. Influence of eddy viscosity formulation In the previous series of calculations, eddy viscosity was constant at 0.05 m 2 s ~. The application of a constant viscosity coefficient in a region of significant water depth variation, and changes in current is not physically realistic (Davies, 1986). To investigate the influence of viscosity formulation the previous calculation was repeated using viscosity determined from ~-2 with K1/~o = 0.2 [equation (18)], hff with 1(2 = 0.0025 [equation (19)], and A~ [equation (20)] with K2 -= 0.0025 (Table 1, Calculations 6, 7 and 8). The co-tidal chart computed with a if2 viscosity was not significantly different in shallow water from that determined previously with viscosity constant at 0.05 m e s -1 [Fig. 2, Chart (c)[. No significant differences were found in co-tidal charts computed with a A~ or h~ viscosity with the same values of K2 and K2 (Calculations 7 and 8), and only that derived using a Aft viscosity is shown in Fig. 2" ) [Chart (d)]. Although this co-tidal chart is similar in shallow water to that computed with u ~, some slight differences are evident in deep water. However it is evident from Fig. 2, that changing viscosity formulation only produces slight differences in the computed co-amplitude and co-phase lines. This is to be expected, since from the analysis presented in Davies (1987) and discussed earlier, it is evident that in a linear model with a slip condition changes in sea surface elevation are determined purely from the first mode, the magnitude of which is not determined by eddy viscosity. The influence of eddy viscosity upon the first mode in a linear model only occurs through bottom friction effects, in so far as eddy viscosity determines the contribution of each mode to the current profile, and hence bed currents and bed stress which does influence the first mode and hence the distribution of elevation gradients and the co-tidal chart. The spatial distribution of surface and bed tidal current ellipses (major and minor axis) associated with Fig. 2, Chart (d) (viscosity proportional to Aft) are shown in Fig. 3(a) and (b). (The magnitude of the semi-axes is indicated by the length and multiplicity of the scale bars; magnitudes above 60 cm s-1 are shown by a blank rectangle for clarity.) In order to understand why the spatial distribution of current ellipses [Fig. 3(a) and (b)] shows a strong recti-linear flow along the western edge of the basin, with much weaker, near circular current ellipses in the southeast corner [Fig. 3(a) and (b)], it is necessary to examine contours of S+/(o9 + ~) and S_/(~o - /) computed (see Appendix) from the co-tidal chart Fig. 2(d). From the analysis using the single point model [equations (16) and (17)] presented in the Appendix, it is evident that rectilinear flow occurs when IR+[ and ]R ] are identical, with the magnitude of the flow directly proportional to the sum of IR+I and [R I. Circular current ellipses occurred when JR+ I + JR_ I = IR+ I - ]R l, see Appendix. Surface contours [Fig. 3(c)] of S+/(oJ + 7) and S /(~o - 7) for the inviscid case clearly show that [R_~Iis much larger than IR+[, particularly in the southern part of the basin where the rotary components of the tidal gradients are largest, giving rise to the strongest tidal currents [Fig. 3, Chart (a)]. In regions where the co-tidal lines are widely spaced (the northeast corner), the

1260

A . M . D a v i e s a n d J. E . J o n e s

I

- -

Amplltudo

........

Phase

I

I

(a)

Fig. 2. C o m p u t e d co-tidal charts d e t c r m i n e d using the l i n e a r m o d c l with viscosity c o n s t a n t at 0.05 m 2 s - I , for a r a n g c o f s l o p e s h I and h 2, n a m e l y (a) h I = h2 = 65 m, (b) ht = 1 2 0 r e , h : = 1 0 m , and d e t e r m i n e d using thc n o n - l i n e a r m o d e l with h I = 120 m, h 2 = 10 m and a n u m b e r of viscosity f o r m u l a t i o n s , n a m e l y (c)/~ c o n s t a n t at 0.05 m 2 s J and (d) ~ oc A g.

rotary components are much weaker and tidal currents are reduced [Fig. 3, Chart (a)]. At positions between the amphidromes where there is significant curvature in the coamplitude and co-phase lines, near circular current ellipses occur. Inclusion of viscous effects [Fig. 3(d)] significantly reduces the surface magnitudes of dR_ I and IR+ ]in shallow

Taylor's problem in three dimensions I

I

1261

I A M a

I - -

amplitude

........

Phase

I

I

(b) Fig. 2.

(Continued)

water regions [compare Fig. 3(c) and (d)] although not deeper water due to the l / h 2 factor in the viscous term [equations (16) and (17)]. The fact that the ]R ] boundary layer is much thicker than the ]R+] layer means that in shallow water the ]R_] layer can intersect the surface and hence viscous effects have a much larger influence in shallow water upon surface values of ] R I than ]R+] [compare Fig. 3(c) and (d)]. This suggests that the most sensitive test of a three-dimensional hydrodynamic model's ability to correctly include

1262

A . M . Davies and J. E. Jones

!11 k I

-

-

........

I

Amplitude Phase

Fig. 2.

(Continued)

viscous effects is its capability to reproduce the JR_ ] component of any observed currents, particularly if these observations were only made in the upper part of the water column. Due to the influence of frictional effects, tidal currents at the bed are significantly smaller than those at the surface with rotational effects aligning the major axis to the right of that found at the surface. (Compare surface and bed tidal current ellipses in Fig. 3.) Current profiles of the M 2 tide in deep water are not significantly different using the various viscosity parameterizations. However, in shallow water, where tidal velocities are

Taylor's problem in three dimensions

I -

-

........

1263

I

A mpllt~,ade

Phase

(d) Fig. 2.

(Continued)

larger (points X, Y, Fig. 1), currents are significantly larger (of order 15 cm s -1) with the Au than the ~2 velocity [Fig. 4(a)], although in this region there is no difference between the Au and hu formulations of viscosity, since in this shallow area the boundary layer thickness is depth limited. [Only the amplitude of the dominant component of current is given in Fig. 4(a), namely the v component at X and u component at Y.] A significantly larger shear through the vertical is evident with the Au, than the u 2 velocity, associated

1264

A.M. Davies and J. E. Jones I

I

t

I I . ~ k

Itll

Itll 1111 1111 II11 1111 1111 1111 1111 111 III

IIIII II111 IIIII IIIll I III1~ I IIIII I IIIII I IIIII I IIIII II IIIII IIIIIIE IIIIll IIIIlll

I I II lIl'ttt't' ~ IIIII!tttt

CM.SEC --

IIIII~~

15 30 45

I

--

I

I

Fig. 3. Current ellipses (a) at sea surface, (b) at sea bed computed with h I = 120 m, he = 10 m with k = 0.005, R 2 = 0.0025 (Calculation 8). Also shown are (c) contours of [R~+i (. . . . ) and IR~_I ( ), and (d) contours of IR+l ( ) and [ R I (.... ) for the viscous case. w i t h t h e l o w e r viscosity d e t e r m i n e d f r o m a Au p a r a m e t e r i z a t i o n t h a n a//2 p a r a m e t e r i z a tion. T h e classic w o r k of T a y l o r (1921), a n d t h e p a p e r s o f H e n d e r s h o t t a n d S p e r a n z a (1971), R i e n e c k e r a n d T e u b n e r (1980), c l e a r l y s h o w e d t h a t in a t w o - d i m e n s i o n a l m o d e l w i t h a flat bottom, the coefficient of bottom friction influenced the position of the amphidromes with r e d u c i n g f r i c t i o n c a u s i n g a w e s t w a r d m o v e m e n t o f t h e a m p h i d r o m e . T h e s a m e effect was f o u n d in t h e t h r e e - d i m e n s i o n a l m o d e l w i t h t h e s l o p i n g b o t t o m , in t h a t w h e n t h e c o e f f i c i e n t

Taylor's problem in three dimensions

1265

(b) If I I l I

It

It

t / t t t t t t t t /

f f ~ / / I / l l / / / t / ~ f f f ¢ [ / / / / I I I I i i i ~ r ~ / / l l l l l / / J / J f f t ~ / / / / / I / / / / / J

l l l l ~ r / / l l l l l ¢ / i ~ I f l f I I//III I I I / I I

I f dnnn

~/III//

i ~

I f f f f ~ f / l l l l /

~Iflt flfff

,

~ i ~

f//llll~1~ f//ll/l~J~

l]Ifflf//llll/J~ l f f f f f ~//I

II

/I~,

I I | f l l l ~ / / t t t / ~ l l l l | I t l ~ / l l l l / ~

llllllt~,lll//~ llll~llt~llll/~ l l l l l l l l t IIIII IIII

t ~I II/ / / ~tl t / I

CM. --

~ l l ' l l ' ' ' ~ t / ' '

~;EC 15 30 45

Ii

lllf//~#~+xx IIIII//#~*~xX lt//S~/

lllI/s~--

\ .....

.....

~\\ ~\\

1 I Fig. 3. (Continued) of bottom friction k 2 was reduced from 0.005 to 0.001 (Calculation 9, Table 1) the degenerate amphidrome at the eastern edge of the rectangle moved westward. This change in position of the amphidromic points, affects the co-amplitude and co-phase lines, giving rise to a slightly different distribution of IR~I and IR_~I. Consequently the pattern of tidal current ellipses also changes. A five fold increase in bottom friction coefficient from 0.005 to 0.025 (Calculation 10, Table 1) producing a major shift in position of the amphidromes towards the southeast [Fig. 5(a)], with an associated decrease in tidal currents along the eastern edge of the rectangle, compare Figs 5(a) and 2(d).

1266

A . M . Davies and J. E. J o n e s

(c)

I"

I"

t

IR-

Fig. 3.

(Continued)

The physical nature of tidal currents is that they are truly three-dimensional, and hence it is not just bottom friction, but internal friction which has a role to play in determining the co-tidal chart. This effect can be readily seen when the viscosity coefficient Ka is reduced from 2.5 x 10 -3 to 2.5 x 10 -5 (Calculation 11) with bottom friction coefficient maintained at k2 --- 0.025. The computed co-tidal chart shows two amphidromes [Fig. 5(b)], in close proximity to each other, with a region of zero elevation gradient in the centre of the basin, in marked contrast to Fig. 5(a). Surface current ellipses [Fig. 5(c)], associated with this cotidal distribution, exhibit strong recti-linear flows in regions of rapid change in the co-

T a y l o r ' s p r o b l e m in t h r e e d i m e n s i o n s

(d)

1267

_ ._.............__~

t R+

I-

-t

------

R" Fig. 3.

(Continued)

amplitude lines (the southern part of the basin), with near zero tidal currents associated with the area of zero elevation gradient. The effect of the large coefficient of bottom friction is to dramatically decrease tidal currents at the bed [compare Fig. 5(d), with Fig. 3(b)]. However because eddy viscosity in this calculation is low, the influence of bed friction is restricted to the near-bed layer, producing a significant vertical variation in the tidal current [Fig. 4(b)]. Profiles of the amplitude of the dominant component of current namely the v component at X and u component at Y (Fig. 1) computed with R 2 = 2.5 × 1O- 2, k 2 = 0.001 and

1268

A . M . Davies and J. E. J o n e s

(a)

hv (mr1)

hu (roll -1)

0 0

0.5 =

i

,

i

0.5

0

I

1

i

o

"'¢'"

I

'1' I I

I I

X /

/

0.5-

Y

/ 0.5

/

-

/ /

,/

1.0

(b)

1.0

hv (ms-l) 0

hu (rns-1)

1

0

0

2

0

I

t I

I

I I

X

I

/ I o.5-

/

I I

I

Y

I

0.5-

/

/

/

I

/ /

J 1.0

/" ~

(C) 0

o

1.0-hv (mr1) 0.5

I

hu (mfl-1) 1.0

0

o

X

0.5

1.0

I

I

Y 0o5 --

1.0

Fig. 4.

/f

J

f

0.5-

1.0-

Profiles of the a m p l i t u d e of the d o m i n a n t c o m p o n e n t of the M2 tidal current, n a m e l y h,, at X a n d h , atYcomputed(a)withl~acu2( ) a n d / ~ o c h u ( . . . . ), and (b) with R2 = 2.5 x l0-2 k2 = 0.001 ( ) and K2 = 2.5 x 10 -5, k 2 = 0.025 ( . . . . ), and (c) for the M 4 c o m p o n e n t of the tide. Also s h o w n (d) profiles for the M 2 current c o m p u t e d using a no-slip condition with Pl :/~ 1:0.00002 ( ), and/~l:P0 ratio 1:0.02 ( . . . . ).

Taylor's problem in three dimensions

'j/

1269 hu (nrw-1) 0.5 j I

hv (mr1)

(d)

0 0

X O°S --

1.0

0.5

,//

1.0 I

0 0

I I I

J 0,5

--

1.0 I

Y

/

1.0-

Fig. 4. (Continued) K2 = 2.5 x 10-5, k 2 = 0.025 [Fig. 4(b)] besides showing significant differences in current

magnitude due to frictional effects, exhibit significant differences in current shear. In the case of K2 = 2.5 x 10 -2 in shallow water vertical eddy viscosity is high, and bottom friction is low, giving a nearly uniform current profile from sea surface to sea bed. In the case of Calculation 11, K2 = 2.5 x 10 -5, k 2 = 0.025, frictional retardation at the sea bed is high; however because of the low value of viscosity this retarding force is not transmitted through the water column, and shear in the vertical is large. This series of calculations clearly illustrates that in a three-dimensional model, it is not just the coefficient of bottom friction that determines the position of the amphidromes, but also the intensity of turbulence in the water column, since this is important in transmitting the frictional retarding force of the bed into the upper parts of the water column. In nature, other tidal constituents are present, in particular the $2 component of the tide having a period of 12 h. The quadratic nature of bottom friction and the flow dependence of the eddy viscosity means that enhanced levels of turbulence will be present in a model with both M2 and $2. This problem is investigated in the next section. 4.3. Influence of the S 2 tide In this series of calculatic;ns the bed slope was maintained at h 1 = 120 m, h 2 = 10 m, with ~'2 = 0 . 0 0 2 5 , k 2 = 0.005. Hence results of running the model with M2 and $2 combined could be compared directly with Calculation (8). In an initial calculation (Calculation 12A) the model was run with the $2 tide only. The co-tidal chart [Fig. 6(a)], shows two amphidromic points situated in the southern part of the basin, with tidal elevation amplitude about one third of that for the m 2 tide [compare Fig. 6(a), and Fig. 2(d)], a direct consequence of the reduced tidal forcing at the northern end of the basin. In the case of the $2 tide, current magnitudes are also about a third lower [Fig. 6(c)] and hence bed frictional effects are reduced giving the amphidrome off the eastern side of the basin, which is not present in the case of the M 2 tide [Fig. 2(d)]. Running the model with a combined M2 and $2 tide (Calculation 12B) requires a much longer simulation in order to separate the M 2 and S 2 tidal constituents by harmonic analysis (of order 30 days). The $2 co-tidal chart [Fig. 6(b)] is significantly different from

1270

A . M . Davies and J. E. Jones

(a)

m

I

.

.

.

.

.

.

.

I

I

Amplitude

.

Phase

Fig. 5. Computed co-tidal charts determined using (a) k_~ = 2 . 5 × 10-3,k2 = 0 . 0 2 5 , ( b ) K2 = 2 . 5 × 10 -5, k 2 = 0.025, and currcnt cllipscs (c) sca surfacc and (d) sca bed ellipses computed with K2 = 2.5 × 10 5 k2=0.025.

that produced with the $2 tide alone [compare Fig. 6(a) and (b)] due to the higher levels of turbulence (increased bottom stress and eddy viscosity) due to the presence of the M2 tidal currents. Changes to the M2 co-tidal chart due to the presence of the $2 current are significantly less [compare Fig. 6(e) with Fig. 2(d)]. Similarly the magnitudes of the $2 tidal currents at all depths are significantly reduced [compare Fig. 6(c) with Fig. 6(d)] due to the

Taylor's problem in three dimensions

1271

(b)

I

I

Amplitude ........ Phase Fig. 5.

(Continued

increased frictional effects of adding the M 2 tide, although changes in M2 tidal current ellipses are significantly less. The changes in turbulence intensity and its influence upon tidal current strength and position of the amphidrome, clearly shows that there will be a spring neap variation in the harmonic analysis of currents. Observational evidence (Baker and Aicock, 1983; Pugh and Vassie, 1976), exists to support this. Also Pugh (1981) found variations in the position of the M2 amphidrome in the Irish Sea. These calculations

1272

A.M. Davies and J. E. Jones

(c)

I

1-

I

I I ~ ~ ÷ k ~ t I I I I 1 | t t t l t I I ! 1 1 1 t ! t l t I I I I ! t t l l It IIi~it11111 IIiit!!1111 III!111111t I I I I l t i l l l f | ! 1 | | I I I I ! t t!1111 ! t ~

/tt tll III Iit I I I ! t I I t ! I

t I ! t t

ttt tFt

P D P P ~ P t l I t ,

I | l l t | l l

Ilttlt11~tll I l t t l t l Itt111ttlt1111

. . . .

~ ~1 I I I I I!

I I ~ ~

pl~PPPJ I m~ t t t ! PItPPt I ! o i ~ ~ Bi ! DrDDPl~oooDIt DalfltP~ttto! I i i ~l/~IPw

t#÷

~t

C M . S E C

t t t

--

15

=

30 45

~

7 I

Z

I

•

•

~ ~

--__L._ J . - - . - I - . - - . J

--1--

I

Fig. 5. (Continued) support this variation in amphidromic position over the spring-neap cycle and suggest that it may possibly be larger for the $2 tide. These calculations clearly show that when simulating a single tidal constituent in isolation it is necessary to take account of enhanced turbulence and frictional effects of other constituents (Le Provost and Fornerino, 1985). 4.4. Influence of changes in co The effect of doubling ~o was examined (Calculation 13, Table 1) by repeating the calculations with hi = 120 m, h e = 10 m, with k 2 = 0.0025 and k 2 = 0.005. The resulting

Taylor's problem in three dimensions (d)

I

I

I

+

+ + + +

+

+

÷

+

•

•

•

4

+ + +

+

+

+

÷

+

*

*

.

¢

+ + +

+

+

+

+

,

,

.

.

+

+ + + +

.

k

+ +

k

~ + +

~ + + +

+

+

÷

÷

,

.

.

+

+

÷

÷

*

~

,

+

+

+4

1273

. . . .

.

.

.

.

.

.

.

.

.

.

.

.

. .

.

.

,

,

÷

÷

+

.

÷

.

¢

.

.

.

.

.

÷

.

.

.

.

.

÷

+

+

+

~

¢

¢

¢

+

÷

*

÷

4

¢

~

+

+

+

4

+

¢

¢

+

+

+

+

+

+

~

CM,SE~ --

1S 30 45

I-. . . .

Fig.

5.

I

(Continued)

co-tidal chart [Fig. 7(a)] shows significant horizontal spatial variability, due to the fact that w now exceeds 0% [equation (8)] and hence Poincar6 waves are generated at the southern end of the channel (Hendershott and Speranza, 1971; Brown, 1973). The large horizontal variability shown in Fig. 7(a), produces significant spatial changes in IR~I and IR ~_[, giving greater horizontal change in the tidal current ellipses [Fig. 7(b)], than that found with the M 2 tide. Again with strongest currents occurring in regions where the co-tidal lines are closest. Profiles of the amplitude h,, hv of the u and v currents [Fig. 4(c)], show similar vertical variations to those found previously with to set at the M 2 frequency, namely a frictionally

1274

A . M . Davies and J. E. Jones

(a)

I

I

I

I

I

I

Amplitude ........

Phase

Fig. 6. Computed S 2 co-tidal chart (a) computed running the S2 tide alone (b) derived by running the model with M 2 and S2 tides. Also shown (e) is the M2 co-tidal chart derived from the M2 and S, run. Surface S2 current ellipses derived (c) running the S2 tide along (d} extracted from running the model with M 2 and S 2 tides.

retarded bottom boundary layer, with a more gradual change in the upper part of the water column. Current magnitude of the v component at position X is larger than that found previously [Fig. 4(a)], due to the fact that doubling a), changes the co-tidal chart, giving rise to increases in the rotary component of current S+ and S_ at positions X and Y. Also

1275

Taylor's problem in three dimensions

(b)

+ 38

•I

340 °

-

-

~

--

15

61.. "i" -

-

........

1"

1"

A m p U t u d e

Phase

Fig. 6.

(Continued)

since o) has increased to 2a), then the magnitudes of ]R+I and IR~1 are now obtained from, S+/(2w + y) and S_/(2~o - ?,), and hence the relative importance of R+ and R_ change producing changes in the magnitude of the major axis of the current ellipse and the ratio of major to minor axis. These calculations clearly demonstrate that increasing ~, beyond ~oc, produces a significant change in the spatial distribution of the co-amplitude and co-phase lines, giving rise to a spatial distribution of the current ellipses markedly different from that when ~o <

1276

A.M. Davies and J. E. Jones

(c)

e

,

.

.

p , t

w

.

. .

.

,

.

. .

. .

i I

I I

t t t t

I I I III

CM.SEC

t

T

t

--

15

=

30

-~-

45

?

Fig. 6. (Continued)

o)c. Not only is it the change in co-tidal chart that produces this change in current ellipse patterns (due to spatial variability in S+ and S_), the change in the frequency terms (o) - 7) and (~o + 7) are also responsible. These changes in frequency terms also change the relative proportions of the clockwise and anticiockwise current spirals, producing changes in current profiles. Obviously any change in current profile, and spatial variability of current, feeds into the viscosity term which is related to current magnitude, subsequently influencing frictional dissipation and hence the position of the amphidromes. Consequently the influence of changing a~ in a three-dimensional model, in which

1277

Taylor's problem in three dimensions

(d)

i~

~

II

~{llll

i, I,

,Illl~ olllll

II |i II I!

~'tttt ~'tttt ,,tttt ~'tttt

~t It

II

I

~

! 1 ! I

~ t t t t ~,tttt ~ t t t / t ~ t t /

I l I

~ 1 1 t Iltl/t

I~ttt CM.REC

Ill,l/

--

II#t// llltt/

15

30 45

Illt,/t

I ! II/~//xx

~ "

I t !1111//~//~-." I I I t t/till~

..... I

Fig. 6.

1 (Continued)

viscous and frictional effects are related to tidal current profile, has a more complex effect than in a two-dimensional model. 4.5. Influence of a no-slip condition In the previous series of calculations, a slip condition was applied at the sea bed, with a quadratic law of bottom friction. Such a parameterization does not resolve the near-bed high shear layer which is important in a number of physical processes, e.g. sediment transport. In this series of calculations a no-slip condition was considered, again with the

1278

A.M.

D a v i e s a n d J. E . J o n e s

(e)

I

--AmpUl~tde . . . . . . . . Phase Fig. 6.

(Continued)

sloping rectangle with h I = 120 m, h2 = 10 m, to examine the influence of viscosity, in particular near-bed viscosity upon the position of the amphidromic points, and tidal current profile. In all calculations the Au formulation of viscosity was applied with K2 = 0.0025. Observations and boundary layer theory (Bowden et al., 1959; Bowden, 1978), suggest that eddy viscosity in the near-bed layer should decrease linearly from ¢q to kt0 over a

Taylor's problem in three dimensions

1279

distance hi = 0.2 h. In an initial calculation (Table 1, Calculation 14) a/~1:/~0 ratio of 1:0.00002 was used, a physically realistic ratio (Davies, 1992). The computed co-tidal chart, Fig. 8(a), is similar to that found previously [Fig. 2(d)] using a slip condition, with an amphidromic point at the Southern end of the basin and a degenerate amphidrome along the western edge of the basin. Calculated spatial distribution of surface and near-bed (o = 0.9) tidal current ellipses were similar to those found previously [Fig. 3(a) and (b)]. Tidal current profiles in the upper part of the water column [Fig. 4(d)], at positions X and Yare similar to those found using a slip condition [Fig. 4(a)] with surface currents only the order of 2 cm s-1 different. However in the near-bed region the tidal current falls rapidly to zero, due to the no-slip bottom boundary condition. Increasing near-bed eddy viscosity (Calculation 15, Table 1) giving a ~1:/~0 ratio of 1:0.02, increases the effect of near-bed viscosity, producing increased frictional damping, causing the amphidromic points to move towards the coastlines [Fig. 8(b)]. Increased frictional effects also reduce surface currents with the dominant components of surface current at positions X and Y reduced by over 20 cm s- 1 [Fig. 4(d)] due to this increase in bed viscosity. Currents in the near-bed region are significantly lower than those obtained previously, with current profiles at points X and Y [Fig. 4(d)] no longer exhibiting a high shear bottom boundary layer. These calculations clearly show that increasing near-bed eddy viscosity in a no-slip model produces a similar effect to that found in a slip model, namely a movement of the amphidromic points, with a corresponding change in the distribution of current ellipses. Using a physically realistic value of bed viscosity (Calculation 14), gives similar current profiles above the near-bed layer to those found with a slip condition. However the no-slip model can resolve the bottom logarithmic layer. Also in reality the application of a no-slip bottom boundary condition is physically more realistic than the use of a slip condition with an empirical friction coefficient, and in nature the changing position of the amphidromes is controlled by near-bed turbulence and not a frictional coefficient. Calculations using the single point model, equations (16) and (17) with specified external gradients S+ and S_ also showed that increasing near-bed viscosity reduced shear in the near-bed region, and reduced the surface current. 5. C O N C L U D I N G R E M A R K S In this paper we have re-examined Taylor's problem (Taylor, 1921) of tidal oscillations in semi-enclosed basins using a three-dimensional model. The work is aimed at filling the gap between analytical models (Taylor, 1921; Brown, 1973, 1987, 1989; Hendershott and Speranza, 1971; Rienecker and Teubner. 1980) which by necessity have to deal with idealized systems (although providing significant insight), and full three-dimensional numerical simulation models (e.g. Davies, 1986; Davies and Jones, 1990) which although able to reproduce observed tidal currents, do not give detailed insight into the mechanisms influencing the position of tidal amphidromes or current structure. Initial calculations clearly show the importance of water depth, particularly the shallowing of the North Sea at its southern end, in determining the position of amphidromes, the spacing of co-amplitude and co-phase lines, and hence the intensity of tidal currents, and the nature of the current ellipse (circular or recti-linear). An examination of the rotary components of the tidal currents, suggests that the most sensitive test of a

1280

A . M . Davies and J. E. Jones

I

(a) ]

I

9.

I

- -

80

T

I

~Lmpll~Lde

. . . . . . . .

Fig. 7.

I

320 °

Phase

Computed (a)

M4

co-tidal chart, (b)

M 4 surface

current ellipses.

model's ability to correctly include viscous effects is a comparison of the R_ component from model and observations. Subsequent calculations with a slip condition, illustrate the movement of amphidromes in response to changes in bottom friction coefficient, with increasing bottom friction coefficient moving the amphidrome to the southeast. In a twodimensional model for a given topography and open boundary input, bottom friction is the only parameter controlling the position of the amphidromes. In a three-dimensional model, with a slip condition at the bed, calculations clearly show that internal friction also

1281

Taylor's problem in three dimensions

(b)

-----~

........

~

4 - - -

l l l l l ~ ~ \ \ \ / ~

l i l t ] t / f l i t t i n g / I l l l l l l l l l l l l I I I I I I I I I I I I I I I I l l l l l l l l l l I I I I I l l l l l l ~ t I I I I I I I 1 ~ ~ I III I I I 1 ~ ~/// I I I I I I 1 ~ / / / / / III lllt ~ t ~///II I I I I I I I I 1 ~ / 1 1 1 1

IIIII I II1~ ~ ~111tt111,, I I I I t t t t ~ I I I I I 1 ~ t t t t ~ I I I I I I l l t t t

t ~ ~ ~ ~ I ~ ~ ~ ~ / ~/i I I

~ I I I ,, I ~ ~ I t ~ I I

CM.SEC --15 --

30 45

I I I I I / / t * ~ f f

t

I I IItlllt~tlt~] !1~tt I

t

Fig. 7.

I

(Continued)

has an influence not only on current profiles but also upon the position of amphidromes. With a no-slip condition the magnitude of eddy viscosity at the sea bed is particularly important in determining both the position of the amphidromic points, and current shear in the near-bed region. Calculations using the single point model with an imposed gradient forcing, showed that increasing bed viscosity, and viscosity higher in the water column, reduces surface currents. In the rectangular basin the same mechanism applies. However because nearbed viscosity determines tidal energy dissipation which influences the position of the amphidromes, and the distribution of co-amplitude and co-phase lines, it determines local

1282

A . M . Davies and J. E. J o n e s

(a)

I

i

Amplitude ........ Phase Fig. 8.

C o m p u t e d M 2 co-tidal chart determined using viscosity profile (B), (a) with ~¢1 :Po ratio 1 : 0.00002, and (b) with u 1 ,u0 rat o 1: 0.02.

gradients which influence tidal current magnitude. Consequently the role of eddy viscosity in a shelf sea region is more complex than that revealed with a single point model, due to these various "feed back" mechanisms. Similarly changes in bed stress and eddy viscosity magnitude due to a combination of tidal harmonics, can significantly influence the co-tidal chart and magnitude of the currents

Taylor's problem in three dimensions

1283

(b)

I

AmpZltude ........

Phase

Fig. 8.

(Continued)

associated with the minor tidal constituents. This was clearly evident in the case of adding the M 2 tide to an $2 solution. Frictional effects, both internal friction (eddy viscosity) and in the case of a slip condition the coefficient of bottom friction are responsible for determining the position of tidal amphidromes. In essence low viscosity and a high coefficient of bottom friction can give similar tidal elevations to high viscosity (in the limit of infinite viscosity, a twodimensional model) and a low coefficient of bottom friction. This explains why friction

1284

A.M. Davies and J. E. Jones

coefficients used in t w o - d i m e n s i o n a l m o d e l s are lower t h a n those in t h r e e - d i m e n s i o n a l m o d e l s (Davies, 1986). A l t h o u g h similar e l e v a t i o n d i s t r i b u t i o n s can b e o b t a i n e d for a r a n g e of viscosity a n d b o t t o m friction coefficients, c u r r e n t profiles are very different. A high viscosity with low friction coefficient giving a u n i f o r m c u r r e n t profile in the vertical, w h e r e a s a low viscosity with high b o t t o m friction p r o d u c i n g a high shear b o t t o m b o u n d a r y layer. W i t h a no-slip c o n d i t i o n the viscosity profile in the n e a r - b e d r e g i o n is crucial in d e t e r m i n i n g the level of shear close to the bed. T h e calculations p r e s e n t e d here, clearly show that e l e v a t i o n d i s t r i b u t i o n s are n o t a rigorous test of a t h r e e - d i m e n s i o n a l m o d e l , with the critical test b e i n g its ability to r e p r o d u c e c u r r e n t shear. T h e f r e q u e n c y to is particularly i m p o r t a n t in d e t e r m i n i n g the spatial variability of the cotidal chart. A l s o , as was clearly e v i d e n t from the p o i n t m o d e l , to was i m p o r t a n t in d e t e r m i n i n g the m a g n i t u d e of the clockwise a n d c o u n t e r - c l o c k w i s e c u r r e n t spirals. A g a i n the a d d i t i o n a l complexity of the co-tidal chart c h a n g i n g in r e s p o n s e to v a r i a t i o n s in to a n d this i n f l u e n c i n g the local g r a d i e n t a n d h e n c e c u r r e n t m a g n i t u d e , t o g e t h e r with its influence o n tidal c u r r e n t structure is clearly e v i d e n t from a c o m b i n a t i o n of p o i n t m o d e l a n d threed i m e n s i o n a l r e c t a n g u l a r basin m o d e l . F u r t h e r calculations to e x a m i n e the influence of e n h a n c e d t u r b u l e n c e due to wind i n d u c e d c u r r e n t s , a n d e n h a n c e d b e d stress d u e to wind wave effects, u p o n tidal c u r r e n t profiles are p r e s e n t l y in progress a n d results will be r e p o r t e d in d u e course. Acknowledgements--The authors are indebted to Mrs J. Hardcastle for typing the text and Mr R.A. Smith for preparing the diagrams. REFERENCES Baker T. F. and Alcock G. A. (1983) Time variation of ocean tides. In: Proceedings of the Ninth International Symposium on Earth Tides, E. Schweizerbart'sche Verlagsbuchhandlung, D-7000 Stuttgart, pp. 341-350. Bowden K. F. (1978) Physical problems in the benthic boundary layer. Geophysical Surveys, 3,255-296. Bowden K. F., L. A. Fairbairn and P. Hughes (1959) The distribution of shearing stresses in a tidal current. Geophysical Journal of the Royal Astronomical Society, 2,288-305. Brown P. J. (1973) Kelvin wave reflection in a semi-infinite canal. Journal of Marine Research, 31, 1-10. Brown T. (1987) Kelvin wave reflection at an oscillatingboundary with applications to the North Sea. Continental Shelf Research, 7, 351-365. Brown T. (1989) On the general problem of Kelvin wave reflection at an oscillating boundary. ContinentalShelf Research, 9, 931-937. Davies A. M. (1986) A three-dimensional model of the Northwest European Continental Shelf with application to the M4 tide. Journal of Physical Oceanography, 16,797-813. Davies A. M. (1987) Spectral models in continental shelf sea oceanography. In: Three-dimensional coastalocean models, Vol. 4, N. S. Heaps, editor, AGU Washington D.C., pp. 76-106. Davies A. M. (1990) On extracting tidal current profiles from vertically integrated two-dimensional hydrodynamic models. Journal of Geophysical Research, 95, 18,317-18,342. Davies A. M. (1991) On using turbulence energy models to develop spectral viscositymodels. ContinentalShelf Research, 11, 1313-1353. Davies A. M. (1993) A bottom boundary layer-resolving three-dimensional tidal model: a sensitivity study of eddy viscosityformulation. Journal of Physical Oceanography, 23, 1437-1453. Davies A. M. and J. E. Jones (1990) Application of a three-dimensional turbulence energy model to the determination of tidal currents on the Northwest European Shelf. Journal of Geophysical Research, 95, 18,143-18,162. Davies A. M. and J. E. Jones (submitted) The influenceof spurious lateral boundary layers in three-dimensional hydrodynamic models upon the interior flow. Heaps N. S. (1972) On the numerical solution of the three-dimensional hydrodynamic equations for tides and storm surges. Memoires de la Societe Royal des Sciences de Liege, 6(2), 143-180.

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Hendershott M. C. and A. Speranza (1971) Co-oscillating tides in long, narrow bays: the Taylor problem revisited. Deep-Sea Research, 18,959-980. Johns B. and P. P. G. Dyke (1971) On the determination of the structure of an off-shore tidal stream. Geophysical Journal of the Royal Astronomical Society, 23,287-297. Le Provost C. and M. Fornerino (1985) Tidal spectroscopy of the English Channel with a numerical model. Journal of Physical Oceanography, 15, 1009-1031. Lynch D. R. and F. E. Werner (1991) Three-dimensional velocities from a finite element model of English Channel/Southern Bight tides. In: Tidal hydrodynamics, B. B. Parker, editor, John Wiley and Sons Inc., New York, pp. 183-200. Pugh D. T. (1981) Tidal amphidrome movement and energy dissipation in the Irish Sea. Geophysical Journal of the Royal Astronomical Society, 67, 515-527. Pugh D. T. and J. M. Vassie (1976) Tide and surge propagation offshore in the Dowsing Region of the North Sea. Deutsche Hydrographische Zeitschrift, 29,163-213. Rienecker M. M. and M. D. Teubner (1980) A note on frictional effects in Taylor's problem. Journal of Marine Research, 38, 183-191. Taylor G. I. (1921) Tidal oscillations in gulfs and rectangular basins. Proceedings of the London Mathematical Society, 20,148-181. Waiters R. and F. E. Werner (1989) A comparison of two finite element models of tidal hydrodynamics using a North Sea data set. Advances in Water Resources, 12, 184--193. Werner F. E. and D. R. Lynch (1989) Harmonic structure of English Channel/Southern Bight tides from a waveequation simulation. Advances in Water Resources, 12,121-142.

APPENDIX Here we briefly consider the determination of the inviscid rotary components of the tidal currents R~+ and R~ from a computed co-tidal chart. The rotary components of sea surface elevation gradient St and S_ can be readily derived from (12) by expressing ~ as = ~'o cos (wt - &)

(A.I)

with ~o amplitude and g¢ phase, giving

S+ = gg [(a x + by) + i(ay - bx) ]

(A.2)

S

(A.3)

= ~ [(ax - by) + flay + b,)]

with l

8

1

a ~ - Rcos(pO Z(~°c°sgg)"

la.~

ay= ~(ocosg¢),

0

b x - R c o s ~ 0 Z(~°sing~)

la

by = ~ - ( ~ o s i n g g )

giving from (16) and (17) in the inviscid case

R+ = -S+/i(y + ~)

(A.4)

R ~ = -S_/i(y - a)).

(A.5)

Consequently once the co-tidal chart has been computed the terms/~+ and R~ can be readily derived from local gradients. The semi-major axis (A) and semi-minor axis (B) of the current ellipse are given by A = [R+I +

IR-I

B = IR+I-

IR-I

with recti-linear flow being given by B = 0 and circular ellipses when A = B.