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Comparison of observed (HF radar) and modelled nearshore velocities
ContinentalShelfResearch,Vol. 9, No. 11, pp. 941-963,1989.
0278--4343/89$3.00 + 0.00 PergamonPressplc.
Printedin GreatBritain.
C o m p a r i s o n o f o b s e r v e d ( H F radar) and m o d e l l e d n e a r s h o r e velocities D. PRANDLE* and D. K. RYDER* (Received 8 September 1987; in revised form 4 May 1989; accepted 5 July 1989) Abstract--For an enclosed shelf sea region some 18 km square, comparisons are made of the M2 and M4 tidal ellipse distributions derived from (i) month-long HF radar (OSCR) measurements and (ii) a 1 km grid numerical model. The amplitudes, phases and directions of these ellipses are shown to be in reasonable agreement; the eccentricities differ as expected since the OSCR measurements reflect surface conditions against the model's depth-averaged values. From the OSCR measurements, the magnitudes of the advective terms at the frequencies M2, M4, M6 and Zo (residual) are calculated and the origins of these terms examined. Comparisons indicate that depth-averaged models with suitably fine resolution can accurately simulate advective terms even in areas of complex bathymetry. By running the numerical model successively with various terms omitted (or linearized), the generation of M4 tidal currents by advective, shallow water and friction terms is examined. Agreement is also demonstrated between the M2 tidal component of vorticity derived from model and measurements. A "point-model" of vertical current structure is used to show that the large surface residual currents observed by the radar are consistent with forcing by horizontal density gradients. Accurate simulation of such currents in 3-dimensional models is found to be sensitive to the vertical distribution of eddy viscosity, thereby indicating that HF radar data is well suited for both the development and verification of such models.
1.
INTRODUCTION
SURFACE velocities measured by HF radar are compared against values obtained from numerical models with the aim of determining the extent of agreement by examining successively (i) the predominant semi diurnal constituents, (ii) higher harmonics (including spatial gradients) and (iii) residuals. These comparisons are used to assess the accuracy of both the measurements and models in describing the fine-scale horizontal structure of nearshore tidal currents. In the present deployment, the tidal dynamics are complicated by significant wind and density forcing and the region includes irregular topography and pronounced vertical current structure. Further deployments are proposed in more carefully selected locations (such as that examined by HAMMONDet al., 1987) using recent technical advances in the radar system that provide improved spatial and temporal resolution. It should then be possible to extend the present comparisons to examine basic theories of the interaction between topography and fine-scale tidal dynamics derived by HUTHNANCE(1973), PINGREE(1978), LODER (1980) and MAAS et al. (1987). A month-long experiment involving the measurement of surface current vectors by HF radar (OSCR-Ocean Surface Current Radar) was carried out in May 1985. An OSCR unit measures the radial components of velocity at 1.2 km intervals along each of up to 16 * Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside L43 7RA, U.K. 941
942
D PRANDI,Eand D, K. RYDER
beams out to a range of 50 km offshore (PRANDLE, 1985). By deploying two units with overlapping beams, surface current vectors can be resolved at approximately 1.8 km intervals over a region 18 km square (Fig. 1). Observations along each beam were made at hourly intervals. Further details of the experiment together with a detailed description of the relationship between observed residual currents and winds are given by PRANDLE (1987). The latter p a p e r also described details of the tidal analyses of the observational data. To assist in the evaluation of the tidal c o m p o n e n t of these O S C R measurements, a 1 km grid depth-averaged numerical model (Fig. 2) was nested within a 5 km grid Irish Sea model. An explicit finite difference scheme was used (PRANDLh, 1974). SUCCCSSlVCwetting and drying of shallow areas was incorporated. The accuracy of simulation in the Irish Sea model was demonstrated by extensive comparison with observed values of both eleva-
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tions and current ellipses for the predominant M2 tidal constituent. (The model was then extended to simulate horizontal dispersion reproducing both the annual salinity cycle and a major release of ~37Cs from Sellafield.) A final section extends the numerical model to incorporate vertical current structure, together with horizontal density gradients, to explain the large persistent residual surface currents observed by the radar. 2
COMPARISON
MEASUREMENTS
OF TIDAL VELOCITIES AND COMPUTED
AS D E R I V E D
FROM OS(R
FROM A 2-DIMENSIONAl,
MODEl..
Figure 3a and b shows current ellipse distributions for the M2 and M 4 constituents obtained from analyses of OSCR measurements; further interest will be confined to these two constituents. Contrasting characteristics are immediately evident from these two distributions. For the M 2 constituent, the smooth even distribution with rectilinear flow corresponds to the propagation of a Kelvin wave (TAYLOR, 1921), whereas the irregular pattern and more rotational flow for the M4 constituent is shown, subsequently, to be a result of localized nearshore generation. Figure 4a-c shows in contour form both OSCR and model values of amplitude, phase and direction for the Me constituent and Fig. 5a-c shows similar comparisons from M 4 .
M2 Currents" The distributions of M2 amplitude in Fig. 4a are in reasonable agreement though the surface values measured by OSCR are about 10% greater than the depth-averaged model values; respective means over the survey area are 80 and 70 cm s--~. The M2 phase comparisons in Fig. 4b are in even closer agreement, with the OSCR values in advance of the model by an average of 5 ° . Similarly good agreement is found between the direction distributions shown in Fig. 4c. The small discrepancies between OSCR and model values are within the range of variability associated with the vertical structure of tidal currents (PRANDLE, 1982). Vertical structure has a much greater influence on the eccentricity values of current ellipses and thus no comparisons are included of this parameter. M 4 Currents
The observed and modelled M 4 amplitudes in Fig. 5a indicate reasonable agreement in magnitude (less so in spatial distribution); OSCR values range from 7 to 10 cm s-~ compared with model values from 6 to 8 cm s--1. The phase values in Fig, 5b are also in reasonable agreement, the range of values being from 60 ° to 90 ° for both data sets. The spatial patterns for direction values in Fig. 5c show little agreement but the values are similar with OSCR values varying from 340 ° to 20 ° compared with model values from 340 ° to 355 °. Vertical structure may be responsible for a significant contribution to these discrepancies between model and observations for M4 currents as indicated later in Section 4, By successively omitting or linearizing those terms in the numerical model which p r o d u c e M 4 energy from the fundamental M2 propagation, it is possible to determine the origins of the M 4 constituent. Thus writing the momentum equation in the x direction in the form (by using vertically integrated transports, the continuity equation is linearized
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Fig. 3. M 2 and M4 tidal current ellipse distributions obtained from radar measurements. Length of longer line represents amplitude of the semi-major axis, shorter represents the semi-minor axis. (Reproduced from PRANDI~, 1987.)
BO
i
(o)
(b)
(c)
Fig. 4. M2 tidal current ellipse parameters (i) (thick lines) from radar measurement and (ii) from model calculations. (a) Amplitude semi-major axis, cm s-l; (b) phase; (c) direction measured anticlockwise from east.
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Fig. 5. M4 tidal current ellipse parameters (i) (thick lines) from radar measurements and (ii) from model calculations. (a) Amplitude semi-major axis, cm s-l; (b) phase; (c) direction measured anticlockwise from east.
948
D. PRANDLEand D. K. RYDER
and hence all of the non-linear terms can be represented in a single equation):
OOx Ot
- - +
002 Ox(D+ 4)
OOxOv " Oy(O+ 4)
+
+
(D
+
(i)
;)g--
04 + Ox
k
Qx(Q~+ Q~f" ~Qv 0 (1) " -= (D + 4) 2 ~ '
(ii)
(iii)
where Qx is the transport per unit width in the x direction, Qy unit transport in the y direction, D the depth below mean water, ; the surface elevation above mean water, g gravity, k the bed stress coefficient and fl the Coriolis parameter. Here and elsewhere, a co-ordinate system with the x axis positive eastward and the y axis positive northward is adopted. The M4 current ellipse amplitudes generated by: (i) advective, (ii) shallow water and (iii) friction terms are shown, respectively, in Fig. 6a and c. While advective terms predominate in the nearshore regions of high velocity shear, the shallow water term is shown to be the major source of M 4 energy in the OSCR survey region. These model results can be compared with OSCR values by considering the localized M4 generation due to products of the M2 constituent in the east-west momentum equation (now expressed in depth-averaged velocities) (M4 surface gradient and damping terms are omitted): lg
O U M 4 + /.~/O0 + i)O/'/ + _ _ _ ~ O ;
0--7
Ox
Oy
2D
~xx
1
8
2
3x
COS 01 __ _ k _ _ UI(_~/t~_:~cos 0,~ ___ (} .
(2)
"
At position A (Fig. 1) the values of the above parameters are: (the symbol " denotes instantaneous values of the M2 constituents Ue i"t or ;e i~t, with cs tidal frequency) D = 24m,]~ l = 2 . 5 m , g(lO¢l/Ox)= 0.7 × 10-4m s-2, 0, (phase between ¢ and O~/Ox) = 144°, k = 0.0025, l fJ I = 0.8 m s-1, 02 (phase between ~l and ¢) = 78 °. We estimate the following contributions to the amplitude UM, (with acceleration terms in (2) converted to current amplitudes by temporal integration) advective 1.4
shallow water 0.5
friction 0.2
(cm s ~t
These results for both the advective and friction terms are in accordance with the model amplitudes shown in Fig. 6a and c. The advective terms are discussed further in the following section, the friction term contributes more significantly to M 4 in shallower water (~/D 2 in equation 2). However, the OSCR shallow water current amplitude is an order of magnitude too small. Figure 7a-c shows the M4 current amplitudes derived from these three terms throughout the Irish Sea. Figure 7b shows that the M4 distribution associated with the shallow water term forms part of a large-scale motion and hence the large magnitude of this term in the survey area is due to propagation of M 4 energy originating elsewhere. Since equation (2) only considers localized generation of M4, the discrepancy can be understood. ESTIMATION OF SPA'rlAL GRADIENTS OF CURREN IS AND ADVECTI¥1: TERMS FROM OSCR DATA
Initially, data from each quadrilateral formed by the intersection of any two adjacent beams from each radar site were examined separately. A least-squares fitting technique
3
3
Fig. 6. Components of M4 current amplitude (semi-major axis) cm s-~, associated with (a) advective terms, (b) shallow water terms and (c) friction terms (survey region).
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951
Observed and modelled nearshore velocities
was used to determine planes approximating the variations in amplitude and phase of both the eastern and northern velocity amplitude components (U and V) pertaining to the tidal constituents M2, M4, M6 and Z0 (residual). It has been shown (GODIN, 1972) that these tidal constituents can be resolved to an accuracy of X/2v//Vt in amplitude and ~v/2v/ANi radians in phase, where v is the standard error of the N observations and A the tidal amplitude. For the present data set N = 700, v < 4 cm s-1, hence tidal amplitudes are resolved to within +0.2 cm s-1 and phases to within ___0.15° for M2 (A = 0.8 m s-1) and +1.5 ° for M4 (A = 0.08 m s-I). These levels of accuracy enable spatial gradients to be calculated for the major tidal constituents. Table 1 shows the mean amplitudes of U, V, OU/Ox, OU/Oy, OV/Ox and OV/Oy averaged over the survey area for each of these constituents. The average velocity gradients are of the order of 10--4 s-1. Table 1 also shows the second spatial derivatives of velocity 02U/Ox2, a2U/Oy2, 02V/Ox2 and 02V/Oy2. Assuming a sinusoidal variation in velocity along the x axis, U sin kx, k = 2rd~., where ~. represents the wavelength for U variations in the direction of current propagation. Spatial differentiation (twice) along the x axis then yields 0eu
- -k2U.
(3)
0x ~
By inserting values from Table 1 into 3 we can calculate representative wavelengths pertaining to (a) the variation of U in the x direction and (b) the variation of V in the y direction. The values obtained in km [(a) and (b), respectively] are (i) for M2:100,30; (ii) for M4:40,10; (iii) for M6:20,10 and (iv) for Z0:20,10. These results are of interest in numerical modelling where spatial resolution is determined as a fraction of parameter wavelengths. The values indicate that greater resolution is required to resolve Z0 and higher harmonics than for M2 and likewise greater resolution is required in the crossshelf (3,) direction than in the long-shelf (x) direction. In numerical models, additional terms are sometimes introduced into the x and y momentum equations to simulate dissipation due to horizontal viscosity. The terms are in the form (x and y, respectively), (a) EH([OEU/Ox 2] + [oEU/Oy2]) and (b) EH([O2V[Ox 2] + [OZV/Oy2])where Eia corresponds to a horizontal eddy viscosity coefficient. P~'~DLE (1984) assumed a value of Eli = 106 cm 2 s-1. For this value of Evi, the respective ratios of the inertial term (6U and 6V, where 6 is tidal frequency in Table 2) to the eddy viscosity terms [(a) and (b)] are (i) for M2:150,20; (ii) for M4:50,5 and (iii) for M6: 25,5, where the first value refers to x-directed accelerations and the second y-directed. Thus this horizontal viscosity term is small for all three constituents in the long-shelf direction but significant for cross-shelf motions at the M4 and M 6 frequencies. For the M2
Constituent
Table 1. Mean amplitudes of velocity gradients derived from OSCR measurements U V OU/Ox OU/Oy OV/Ox OV/Oy 02UlOx2 02U/Oy2 Ozv/ox 2 cm s-x
M2 M4 M6 Zo
82.3 7.9 2.5 1.4
16.6 3.9 2.3 6.3
× l0 s 10 4 3 3
17 5 4 5
S- 1
15 10 7 5
X
18 10 9 6
3 2 2 2
10-Ix cm -~ 5 2 2 4
02V/Oy2
S- 1
4 7 4 11
7 11 11 24
952
D. PRANDLEand D. K. RYDER
constituent which propagates as a Kelvin wave, closer to the shore the term EHO2U/Oy2 might be expected to increase due to reduction of the along-coast velocities in progressively shallow water. The minimum values (2 × 10-jl cm -l s-1) shown in Table 1 for the second spatial derivative of tidal currents may be used to estimate the standard error, e, in the constitutent current. For an average mesh size (over which the spatial differences were computed) of 1.8 km, the mean error in the calculation of the second derivative is 481180,0002, equating this with the minimum calculated derivatives yields e = 0.16 cm s-~. This value is close to the earlier estimates (PRANDLE, 1987) in which ~ was calculated from discrepancies from an assumed smooth distribution of ME current amplitudes along two specific beams. As noted previously, this value of e indicates a standard error in the radar current measurement of approximately 3 cm sl~ Table 2 shows the mean amplitudes (averaged over all quadrilaterals) of the advective terms occurring at the tidal frequencies of M2, M4, M6 and Zo where these arise due to various combinations of these four frequencies as discussed below. For all four constituents, the much larger values of U(OU/Ox) and U (OV/Ox)in comparison with V(OU/Oy) and V(OV/Oy)will be shown to be a result of the much larger value of UM2 (82 cm s-1) compared to VM2 (17 cm s-l). The table also shows the ratio of the sum of the two advective terms to the respective inertial terms (crU and ~V where ~ is tidal frequency) in the relevant momentum equations. For M2 motion in the east-west (U) direction the advective terms are shown to be negligible, whereas in the north-south (V) direction the advective terms amount to one quarter of the local acceleration term. For the higher harmonics the advective terms are important for motion in both directions, with a maximum value of 60% of the local acceleration term for M 4 motion in the north-south direction. Using the simple trignometric expansions of the form sin C cos D = ½(sin[(C + D)/2] + sin[(C - D)]/2) it can be shown that the cross products in the advective terms such as UM2(OU/OX)M~yield terms at the frequencies Ms and M4. A full listing of such terms resulting from combinations of M2, M4, M6, and Z0 constituents is shown in Table 3a. Table 3b shows the mean amplitudes of the advective terms formed by such cross products. In each of the four sections of Table 3b the largest term occurs in the top left-hand corner representing the product of M2 velocities with Mz velocity gradients, thereby contributing significantly to both the Z0 and M 4 constituents. Likewise, the next largest terms are (almost) invariably along the top lines representing the products of M2 velocity with velocity gradients due to (i) M4, (ii) M6 and (iii) Zo, and thereby contributing, respectively to (i) M6 and M2, (ii) Ms and M4 and (iii) M2. As indicated previously, the M2 contributions are small relative to the local accelerations (Table 2), whereas for both M4 and M 6 these advective contributions are significant. While it is well recognized that quadratic friction generally contributes a much larger term for M6 in shallow water, this M6 contribution from a product of M2 and M4 constituents can be significant (ZIMMERMAN, 1980). The above deductions, based on direct current measurements, may be compared with conclusions drawn from (i) theoretical analysis and (ii) detailed comparisons of model simulations using differing grid sizes. ZIMMERMAN(1978, 1980) showed that the generation of residual currents by topographic features is particularly sensitive to the ratio of
6U
1152 222 105
Constituent
M2 M4 M6 7-o
233 110 97 41
6V 24 43 19 14
UOU/Ox 10 15 7 47
VOU/Oy 27 44 19
UOU/Ox+VOU/Oy
A
0.02 0.20 0.18 62
A/6U
51 63 41 18
UOV/Ox
13 17 11 59
VOV/Oy
B
57 67 43
UOV/Ox+ VOV/Oy
Table 2. Mean amplitudes of the advective terms derived from OSCR measurements ( x 10-5 cm s-2)
0.24 0.60 0.45
B/6V
954
D. PRANDLEand D. K. RYDER Table 3a. Frequencies generated by cross products (e.g. UM2 0UM2/ax generates terms at the frequencies M4 and Zo (residual)
M2 M2
M4
M4
M6
M6
Ms
Z0
M4 Zo M2 M4
M6
Ms
Mz
Ms
M2
M12
M2 M4
Zo M4
Mlo
Z0
MlO
M2
M6
Zo M6
M2 Ma M6
Zo
Table 3b~ Mean amplitudes over survey region (cm s 2) 0 U/Ox
½U(OU/Ox)
U
M2 40 4 1 0
M4 17 2 0 0
15 4 2 1
4 1 1 2
M2 M4
M6 Zo
62 6 2 1
41 4 1 1
M2 M4 M6
16 42
8 21
Zo
6
3
M2 M4 M6
Z0
M6 12 1 0 0
Zo 13 1 0 0
3 1 0 1
4 1 1 2
x 10-5
28 3 1 0
20 2 1 0
× 10-s
8 21 3
5 11
x 10-5
OV/Oy M2
½V(OU/ay)
V
M4 M6
Zo
OV/Ox ~U(OV/Ox)
U
OV/Oy ½V(OV/Oy)
V
x 10-5
2
the tidal displacement to the length of the bed features. H e showed further that a grid size of about half the amplitude of the tidal displacement generally provides sufficient resolution. ROBINSON (1981, 1983) provides a graphic explanation of the physics that describe this coupling between tidal advection and topographic features. H o w e v e r he concludes that a smaller (than that derived by Z i m m e r m a n ) grid size may be required to resolve vorticity generation associated with topographic features and current shear transverse to the main tidal flow direction. ABRAHAM et al. (1987) examined the above theories by formulating two tidal models (depth-averaged) of the same region with grid sizes of 1.5 and 10 km, respectively. They found that the finer grid model included significant additional vorticity absent in the coarser grid model. This omission could be significant in dispersion studies where enhanced dispersion coefficients would be needed to account for such sub-grid scale motions. Vorticity
Surface gradients can be eliminated from the orthogonal m o m e n t u m equations by forming the curl of the equations as shown.
Observed and modelled nearshore velocities
-~x
955
+ V-~x + V-~y + g -oy- + - -OR V + ~ U
=0
(4) 0 {0_~ OU __OU _0; --KRu+E~U} = 0 -~y + U"~x + V oy + g ox + D (R = (u 2 + v2)
.
Then defining the vertical component of vorticity co =
dt (i)
-
Ot
+ U--+
Ox
V--=
Oy
(o+
f~)
~x (ii)
+
-
(OV/Ox) - (OU/Oy) we --D (iii)
o3---D
\
obtain
V OD OD o x - V Oy
.(5)
(iv)
Figure 8 shows the vorticity amplitude at the M2 frequency as (i) calculated from the O S C R observations and (ii) obtained from the numerical model. The O S C R values were calculated by fitting planes through the M2 current components for each quadrilateral as described previously. The spatial distributions for the calculated and observed vorticity amplitudes are in reasonable agreement and the amplitudes agree within a factor of two. The computed values are of the same order as those calculated by ABRAHAM et al. (1987) from a 1.5 km grid model of the southern North Sea.
Fig. 8. Amplitudeof M 2 tidal vorticity (i) (thick lines) from radar measurements and (if) from model calculations. Contour units 10-5 s-1.
956
D. PRANDLE and D. K. RYDER
The maximum observed amplitude of co ~ 3 x 10-5 s-1 is smaller than the planetary vorticity I} = 10-4 s-1. For the M2 constituent in the above equation (5), ([OU/Ox] + [OV/Oy]) ~ 10 -5 S-1, kR/O ~ 104 s-1, 1/D[V(OO/Ox) - U(OD/Oy)] ~ 3 x 10-6 s-1. Thus the magnitude of the respective terms is approximately: (i) 3 x 10-1°, (ii) 10- 9, (iii) 3 x 10-9 and (iv) 3 x 10-9. Hence we see that the main M 2 vorticity balance is determined by the influence of bed friction, with term (iii) representing direct dissipation and term (iv) generation via differences in bed topography (PINGREE, 1978). F r o m (i) and (iii) a time constant associated with dissipation of viscosity (of any frequency) can be determined, namely T = (D/kR) ~ 3 h, indicating the rapid dissipation of vorticity in this area (ROBINSON, 1983). The maximum amplitude of the observed residual vorticity is approximately 10-5 s-1, i.e. small in relation to f] and insufficient to generate bed features (PINGREE, 1978). From the estimated accuracy of determination of the tidal constituents (0.16 cm s-l), by taking spatial gradients over an average of 1.8 km, this residual vorticity magnitude is close to the noise level of the calculation. 4.
RESIDUAL
SURFACE CURRENTS AND VERTICAL CURRENT STRUCTURE
Figure 9 shows the time-averaged (over 30 days) residual surface currents measured by OSCR, these currents range from 17 cm s-1 in the shallower southern area to 3 cm s-1 in the deeper northern area. By assuming these surface currents persist through depth, / I 0 era s -~
/
'
/
i
/
/o-2
X
~//I
,.....,.~c
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./o
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I
I
/
'
..
T'
.
'
X /P
-'"
I "(d
x
4" 20'
4*
Fig. 9. Arrowsindicate time-average (over 30 days) residual surface currents measured by H F radar. Lines at positions A to F indicate direction of density contour. Shorter parallel lines indicate spacing between density contours, at.
53*20'
957
Observed and modelled nearshore velocities
Table 4. Tidal ellipse measurements at position A (cm s-l). 1, OSCR near-surface; 2, current meter moored at mid-depth; 3, bottom-mounted current meter. All values determined from approximately 30 days data (not concurrent). Direction measured a--c from east Constituent
Anticlockwise Amplitude Phase
Clockwise Amplitude
Phase
M2
1 2 3
34.7 28.8 15.8
125.0 124.0 129.0
45.1 29.4 11.6
217.0 216.0 211.0
Sz
1 2 3
12.9 8.9 4.5
89.0 85.0 89.0
13.8 9.4 4.5
245.0 249.0 255.0
M4
1 2 3
2.0 2.8 1.3
270.0 248.0 259.0
5.6 2.6 0.9
90.0 96.0 189.0
7-o
1 2 3
Amplitude 9.7 3.0 3.8
Direction 90° 194° 254°
PRANDLE (1987) estimated a net northerly flow through this region of 40 × 103m3s-1. This value is clearly unrealistic and thus one deduces that some compensatory flow exists nearer the bed. Observational evidence for such a flow reversal is shown in Table 4, where current data obtained close to position A (Fig. 1) are shown for: (i) surface values measured by OSCR, (ii) mid-depth values measured by a moored current meter and (iii) a near bed (+ 1 m) value measured from a bottom-mounted current meter. All three data sets extended over about 30 days but the recordings were not concurrent. The residual currents (associated with non-linearities in tidal propagation) produced by the depth-averaged model are generally much smaller and are more clearly related to topographic features. For detailed comparison with these observed residuals a fully three-dimensional model is required. However, for the present, a simple model of vertical current structure at a single point is used with depth-integrated transports, surface elevation and a (depth-averaged) horizontal density gradient specified directly.
4.1.
Point model of vertical current structure
Neglecting advective terms, the following equations of motion were assumed:
and
OU --+ Ot
O~ g -~x
OV
O~
Ot
~y
--+g
[~-z\Op O + ~:-~-) - - - f ~ V = - - N g Ox Oz [~-z\Op
Oy
Oz
O
OV
Oz
Oz'
+ ~-~)--+fIU=--N-g
OU
(6)
where N is a vertical eddy viscosity coefficient and z is measured up from the bed. The vertical boundary conditions imposed were:
OR (i) stress at the surface z = S: zs = N = 0, Oz where R = U + iV (ii) stress at the bed z = 0, x0 = pkRlRI .
(7)
958
[), PRANDLE and D. K, RYDER
An (alternating direction) implicit finite difference numerical scheme was formulated to solve the above equations in a time-stepping mode (surface gradients determined by vertical integration). The scheme involved 20~ layers, with velocities calculated at the mid-depth of each layer and the lowest velocity at the b e d By depth-integrating the observed current data shown in Table 4, M2 and $2 transports were prescribed as
Q(t) = Re D{0.271 exp iloM:t~ 126~') -~- 0.292 exp iteM::, '~:~ + 0.089 exp il'~s~t+ss~3 + 0.(/94 exp -~(~sJ+249'~t],
(8)
where the mean depth D = 24 m and ~(t) = 2.5 COS(CYMJ -- 308 °) + 0.83 COS(~s~ - 34T'~
(9)
The model was run over a 15 day M2 - $2 cycle and the computed tidal constituents at the appropriate depths compared with the observed values shown in Table 4. 4.2.
M2, Sz and Zo current structure
Assuming N to be time invariant, reasonable agreement between model and observed current structures were obtained for both the M2 and Sz constituents with values of N = 2 0 0 c m 2 s-1 and k = 0.01 [BOWDEN et al. (1959) deduced values of 130 < N < 270 cm 2 s-1 and k = 0.0035 from measurements of current profiles in this area]. Examination of the effect of varying N with z showed that these tidal current profiles were relatively insensitive to three cases, namely: (i) N = 200, (ii) N(z) = 400z/D, (iii) N(z) = 400(D - z)/D. However, when horizontal density gradients were introduced (with the assumption of no net flow), the resultant residual current profiles were found to be much more sensitive to these vertical variations in N. A reasonable fit to the observed residual current structure could only be obtained by using the relationship (iii), i.e. eddy viscosity decreasing linearly with height above the bed (to zero at the surface). Figures 10 and 11 show the current profiles for M2, $2, Z~ and M4 for this eddy viscosity formulation. The residual profile mainly results from the specification of a northerly density gradient Op/pOy of 0.23 × 10-6 m -1 (or ~t increasing northwards by 0.23%0 km-1). Assuming the same values for N and k, the above process was repeated at five additional locations shown in Fig. 9 and values of Op/pOx and Op/pOy determined such that the model produced the observed surface residual current. Then, using the requisite density gradients from all six locations an overall distribution of horizontal density gradients was sketched. Comparison of this distribution with an observed distribution (Fig. 12) shown by MAa~rHEWS et al. (1988) indicates reasonable comparability. The surface residuals computed by the model increase approximately linearly with both increasing density gradient and decreasing eddy viscosity, while the residuals increase with depth squared. In the above simulations the formulation for N was fixed; since N generally increases with water depth the inferred density distribution can only be regarded as approximate.
4.3.
M4 current structure
The combination of specified M2 variations in flow and elevation produce M 4 currents (but not flow). These currents are shown in Fig. l lb. Of particular interest is the
959
Observed and modelled nearshore velocities
/ 0
20 cm $-i
I
I
40
120*
'~ 140"
o-c
0
20 cm s-I
(o)
0.5
I
I*
I
40
210"
230*
c-w
•
I 0
I0 cm$-i
I
I
20 70* O-c
~
I
90*
0
(b)
I0 cms-I
20
240*
"1 260*
C-W
Fig. 10. Vertical current profiles at position A for (a) M 2 and (b) $2 tidal constituents. Arrows and * indicate values observed by (i) radar at the surface, (ii) a m o o r e d current m e t e r at middepth and (iii) a b o t t o m - m o u n t e d current m e t e r at z = + 1 m. C o n t i n u o u s curves indicate values calculated from a numerical model of vertical structure.
pronounced vertical structure of this constituent that clearly complicates the comparisons of (depth-averaged) model vs observed (surface) currents described in earlier sections. A second feature shown in this figure and also for the residual current distribution in Fig. l l a , are currents pertaining to specific heights above the bed (as measured by moored current meters). These differ significantly from values pertaining to "fractional" heights above the bed. In general, this difference increases towards the surface and again emphasizes a problem in comparing OSCR surface measurements with moored current meter recordings. For the predominant M2 and $2 constituents this discrepancy is much less significant.
960
D . PRANDLE and D. K, RYDER
\
,
)
' I
o
1 io
5
I
---ioo*
___
I
~l__~.
14o*
too*
J zzo °
Z6o°
c m S- I
amplitude
.
(Q)
direction
/
c-
J i
/
0"5
-
'
//
°
/
/
/.
-
/ 0
2
4
330 °
350 °
0
C rrl S - I
2 cm
a-c
(b)
4 290 °
510'
330 °
550'
$-I
c-w
Fig. 11. Vertical current profiles at position A for (a) Zo and (b) M4 (residual) tidal constituents. Arrows and * indicate values observed by (i) radar at the surface; (ii) a moored current m e t e r at mid-depth and (iii) a b o t t o m - m o u n t e d current meter z = + 1 m. Continuous curves indicate values calculated from a numerical model of vertical current structure. T h e s e values pertain to fixed fractional heights above the bed; the points are the s a m e m o d e l values interpolated to represent conditions at fixed heights above the bed.
5. CONCLUSIONS A comparison was made of tidal current ellipse parameters derived (i) from OSCR measurements and (ii) from a 1 km grid depth-averaged numerical model. For the predominant M2 constituent, amplitude, phase and direction values are in reasonably good agreement. The surface amplitudes measured by OSCR exceed the model value by approximately 10% as expected from standard theory on the vertical structure of tidal currents. (This vertical structure precludes a comparison of eccentricities of the ellipses.)
961
Observed and modelled nearshore velocities
Cumbt~
I
4*W
i
3*W
I
Fig. 12. Density distribution for May 1978, 6t reproduced from MATHEWSet al. (1988).
For the major higher harmonic constituent M4, broad agreement is evident but poorer than for M2. The contributions from each of the non-linear terms [namely (i) advective, (ii) shallow water and (iii) friction] responsible for the generation of M4 (from the fundamental ME) are quantified from both the model and the OSCR measurements. Reasonable agreement is shown for the contributions arising from the advective and friction terms. A discrepancy in the magnitudes associated with the shallow water terms is related to a large-scale motion throughout the Irish Sea, the effect of which was not included in the OSCR calculations. By calculating mean values of the spatial gradients of the tidal current amplitudes and relating these to mean current amplitudes [assuming U(x) = U cos 27r.r/E], wavelengths, k, associated with each tidal frequency were calculated. For M2, L = 0(100 kin), whereas for M4, M6 and Z0, ~. may be as small as 10 km, thus accurate resolution of these latter
962
D. PRANDLE and D. K, RYDER
constituents in numerical models requires much finer grid resolution than for M2. This latter deduction is widely recognized by modellers (ABRAHAM et al., 1987) but this quantification from observations is valuable. The second spatial derivatives of current calculated from the O S C R measurements are used to estimate the influence of "horizontal eddy viscosity". The O S C R data are also used to examine the magnitude of the separate advective terms averaged over the survey region. In the east-west m o m e n t u m equation for Me, the advective terms amount to only 2% of the inertial terms, ~U, whereas for the north-south m o m e n t u m equations for M 4 and M6, respectively, this ratio is 60 and 45%. In a detailed analysis of the generation of these advective terms, these large north-south advective terms are shown to be due to large values of (i) at M4: UM2(OVMJOX) and UM~(OV~/Ox) and (ii) at M~: UM2(OVM,/OX). These spatial gradients of tidal currents were also used to calculate vorticity. The amplitude of the ME vorticity component, obtained in this way from O S C R observations, shows a similar spatial distribution to the corresponding pattern calculated from the numerical model, though the observed amplitude was approximately half the model value. In the present survey region, it is shown that vorticity is rapidly dissipated by bed friction. The deployment of O S C R in other regions where dissipation is less rapid offers exciting prospects, particularly where topographic scales produce flow separation and the shedding of discrete eddies. A simplified " p o i n t - m o d e l " of vertical current structure was developed to explain the large time-averaged surface residuals measured by OSCR. These residuals were shown to be compatible with forcing by horizontal density gradients, the required spatial distribution of which was shown to be in qualitative agreement with related observations. This model was also used to indicate how, for M4 and residual currents, significant discrepancies can arise in comparing currents taken at fixed heights above the bed (as with m o o r e d current meters) and values pertaining to fractional heights (e.g. the surface as measured by OSCR). The residual vertical current structure calculated by the model was found to be sensitive to the specified vertical distribution of eddy viscosity. Thus both the tidal and residual currents obtained from O S C R measurements are well suited for the development and verification of 3-dimensional models.
Acknowledgement--Iam indebted to the referees for detailed constructive criticism ot this paper. REFERENCES
ABRAHAMG., H. GERR1TSENand G. J. H. LINDIJER(1987) Sub-grid tidally induced residual circulations. Continental Shelf Research, 7,285-303. BOWDENK. F., L. A. FAIRBAIRNand P. HUGHES(1959). The distribution of shearing stresses in a tidal current. Geophysical Journal of the Royal Astronomical Society, 2, 288-305. GODING. (1972) The analysis of tides. Liverpool University Press, 264 pp. HAMMONDT. M., C. B. PAITIARATCHI,D. ECCLES,M. J. OSBOm,~E,L. A. NASHand M. B. COLLINS(1987) Ocean Surface Current Radar (OSCR) vector measurements on the inner continental shelf. Continental Shelf Research, 7, 411-431. HUTHNANCEJ. M. (1973) Tidal current asymmetries over the Norfolk sandbanks. Estuarine and Coastal Marine Science, 1, 89-99. LODER J. W. (1980) Topographic rectification of tidal currents on the sides of Georges Bank. Journal o] Physical Oceanography, 10, 1399-1416. MAASL. R. M., J. T. F. Z~MEI~AN and N. M. TEMME(1987) On the exact shape of the horizontal profile o1 a topographically rectified tidal flow. Geophysical and Astrophysical Fluid Dynamics, 31t, 10.5-129.
Observed and modelled nearshore velocities
963
MATHIEWSJ. P., J. H. SIMPSONand J. BROWN(1988) Remote sensing of shelf sea current using the OSCR HF Radar. Journal of Geophysical Research, 93, 2303-2310. PINGREE R. (1978) The formation of the shambles and other banks by tidal stirring of the seas. Journal of the Marine BiologicalAssociation of the United Kingdom, 58, 211-226. PRANDLE D. (1974) A numerical model of the southern North Sea and River Thames. Institute of Oceanographic Sciences, Report No. 4, 25 pp. PRANDLE D. (1982) The vertical structure of tidal currents and other oscillatory flows. Continental Shelf Research, 1, 191-207. PRANDLE D. (1984) A modelling study of the mixing of 137Cs in the seas of the European Continental Shelf. Philosophical Transactions of the Royal Society, London, A310, 407-436. PRAttLE D. (1985) Measuring currents at the sea surface by H. F. Radar (OSCR). Journal of the Societyfor Underwater Technology, 11, 24-27. PRANDLE D. (1987) The fine-structure of near-shore tidal and residual circulations revealed by H. F. Radar surface current measurements. Journal of Physical Oceanography, 27, 231-245. ROBINSONI. S. (1981) Tidal vorticity and residual circulation. Deep-SeaResearch, 28, 195-212. ROBINSONI. S. (1983) Tidally induced residual flows. In: Physical oceanography of coastal and shelf seas, B. JOHNS, editor, Elsevier Oceanography Series, 35, pp. 321-355. TAYLORG. I. (1921) Tidal oscillations in gulfs and rectangular basins. Proceedingsof the London Mathematical Society, ser. 2, XX, 148-181. ZtMMEmaAN J. T. F. (1978) Topographic generation of residual circulation by oscillatory (tidal) currents. Geophysical and Astrophysical Fluid Dynamics, 11, 35-47. ZIMMERMANJ. T. F. (1980) Vorticity transfer by tidal currents over an irregular topography. Journal of Marine Research, 38, 601-630.
0278--4343/89$3.00 + 0.00 PergamonPressplc.
Printedin GreatBritain.
C o m p a r i s o n o f o b s e r v e d ( H F radar) and m o d e l l e d n e a r s h o r e velocities D. PRANDLE* and D. K. RYDER* (Received 8 September 1987; in revised form 4 May 1989; accepted 5 July 1989) Abstract--For an enclosed shelf sea region some 18 km square, comparisons are made of the M2 and M4 tidal ellipse distributions derived from (i) month-long HF radar (OSCR) measurements and (ii) a 1 km grid numerical model. The amplitudes, phases and directions of these ellipses are shown to be in reasonable agreement; the eccentricities differ as expected since the OSCR measurements reflect surface conditions against the model's depth-averaged values. From the OSCR measurements, the magnitudes of the advective terms at the frequencies M2, M4, M6 and Zo (residual) are calculated and the origins of these terms examined. Comparisons indicate that depth-averaged models with suitably fine resolution can accurately simulate advective terms even in areas of complex bathymetry. By running the numerical model successively with various terms omitted (or linearized), the generation of M4 tidal currents by advective, shallow water and friction terms is examined. Agreement is also demonstrated between the M2 tidal component of vorticity derived from model and measurements. A "point-model" of vertical current structure is used to show that the large surface residual currents observed by the radar are consistent with forcing by horizontal density gradients. Accurate simulation of such currents in 3-dimensional models is found to be sensitive to the vertical distribution of eddy viscosity, thereby indicating that HF radar data is well suited for both the development and verification of such models.
1.
INTRODUCTION
SURFACE velocities measured by HF radar are compared against values obtained from numerical models with the aim of determining the extent of agreement by examining successively (i) the predominant semi diurnal constituents, (ii) higher harmonics (including spatial gradients) and (iii) residuals. These comparisons are used to assess the accuracy of both the measurements and models in describing the fine-scale horizontal structure of nearshore tidal currents. In the present deployment, the tidal dynamics are complicated by significant wind and density forcing and the region includes irregular topography and pronounced vertical current structure. Further deployments are proposed in more carefully selected locations (such as that examined by HAMMONDet al., 1987) using recent technical advances in the radar system that provide improved spatial and temporal resolution. It should then be possible to extend the present comparisons to examine basic theories of the interaction between topography and fine-scale tidal dynamics derived by HUTHNANCE(1973), PINGREE(1978), LODER (1980) and MAAS et al. (1987). A month-long experiment involving the measurement of surface current vectors by HF radar (OSCR-Ocean Surface Current Radar) was carried out in May 1985. An OSCR unit measures the radial components of velocity at 1.2 km intervals along each of up to 16 * Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, Merseyside L43 7RA, U.K. 941
942
D PRANDI,Eand D, K. RYDER
beams out to a range of 50 km offshore (PRANDLE, 1985). By deploying two units with overlapping beams, surface current vectors can be resolved at approximately 1.8 km intervals over a region 18 km square (Fig. 1). Observations along each beam were made at hourly intervals. Further details of the experiment together with a detailed description of the relationship between observed residual currents and winds are given by PRANDLE (1987). The latter p a p e r also described details of the tidal analyses of the observational data. To assist in the evaluation of the tidal c o m p o n e n t of these O S C R measurements, a 1 km grid depth-averaged numerical model (Fig. 2) was nested within a 5 km grid Irish Sea model. An explicit finite difference scheme was used (PRANDLh, 1974). SUCCCSSlVCwetting and drying of shallow areas was incorporated. The accuracy of simulation in the Irish Sea model was demonstrated by extensive comparison with observed values of both eleva-
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944
D. PRANDLE and D. K. RYDER
tions and current ellipses for the predominant M2 tidal constituent. (The model was then extended to simulate horizontal dispersion reproducing both the annual salinity cycle and a major release of ~37Cs from Sellafield.) A final section extends the numerical model to incorporate vertical current structure, together with horizontal density gradients, to explain the large persistent residual surface currents observed by the radar. 2
COMPARISON
MEASUREMENTS
OF TIDAL VELOCITIES AND COMPUTED
AS D E R I V E D
FROM OS(R
FROM A 2-DIMENSIONAl,
MODEl..
Figure 3a and b shows current ellipse distributions for the M2 and M 4 constituents obtained from analyses of OSCR measurements; further interest will be confined to these two constituents. Contrasting characteristics are immediately evident from these two distributions. For the M 2 constituent, the smooth even distribution with rectilinear flow corresponds to the propagation of a Kelvin wave (TAYLOR, 1921), whereas the irregular pattern and more rotational flow for the M4 constituent is shown, subsequently, to be a result of localized nearshore generation. Figure 4a-c shows in contour form both OSCR and model values of amplitude, phase and direction for the Me constituent and Fig. 5a-c shows similar comparisons from M 4 .
M2 Currents" The distributions of M2 amplitude in Fig. 4a are in reasonable agreement though the surface values measured by OSCR are about 10% greater than the depth-averaged model values; respective means over the survey area are 80 and 70 cm s--~. The M2 phase comparisons in Fig. 4b are in even closer agreement, with the OSCR values in advance of the model by an average of 5 ° . Similarly good agreement is found between the direction distributions shown in Fig. 4c. The small discrepancies between OSCR and model values are within the range of variability associated with the vertical structure of tidal currents (PRANDLE, 1982). Vertical structure has a much greater influence on the eccentricity values of current ellipses and thus no comparisons are included of this parameter. M 4 Currents
The observed and modelled M 4 amplitudes in Fig. 5a indicate reasonable agreement in magnitude (less so in spatial distribution); OSCR values range from 7 to 10 cm s-~ compared with model values from 6 to 8 cm s--1. The phase values in Fig, 5b are also in reasonable agreement, the range of values being from 60 ° to 90 ° for both data sets. The spatial patterns for direction values in Fig. 5c show little agreement but the values are similar with OSCR values varying from 340 ° to 20 ° compared with model values from 340 ° to 355 °. Vertical structure may be responsible for a significant contribution to these discrepancies between model and observations for M4 currents as indicated later in Section 4, By successively omitting or linearizing those terms in the numerical model which p r o d u c e M 4 energy from the fundamental M2 propagation, it is possible to determine the origins of the M 4 constituent. Thus writing the momentum equation in the x direction in the form (by using vertically integrated transports, the continuity equation is linearized
I 0 0 cm $-t 53"30'
--
~
235"
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\~
Z
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~
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--
_
j~
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BO
i
(o)
(b)
(c)
Fig. 4. M2 tidal current ellipse parameters (i) (thick lines) from radar measurement and (ii) from model calculations. (a) Amplitude semi-major axis, cm s-l; (b) phase; (c) direction measured anticlockwise from east.
i
(a)
(b)
~
_=-
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./
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.--
Fig. 5. M4 tidal current ellipse parameters (i) (thick lines) from radar measurements and (ii) from model calculations. (a) Amplitude semi-major axis, cm s-l; (b) phase; (c) direction measured anticlockwise from east.
948
D. PRANDLEand D. K. RYDER
and hence all of the non-linear terms can be represented in a single equation):
OOx Ot
- - +
002 Ox(D+ 4)
OOxOv " Oy(O+ 4)
+
+
(D
+
(i)
;)g--
04 + Ox
k
Qx(Q~+ Q~f" ~Qv 0 (1) " -= (D + 4) 2 ~ '
(ii)
(iii)
where Qx is the transport per unit width in the x direction, Qy unit transport in the y direction, D the depth below mean water, ; the surface elevation above mean water, g gravity, k the bed stress coefficient and fl the Coriolis parameter. Here and elsewhere, a co-ordinate system with the x axis positive eastward and the y axis positive northward is adopted. The M4 current ellipse amplitudes generated by: (i) advective, (ii) shallow water and (iii) friction terms are shown, respectively, in Fig. 6a and c. While advective terms predominate in the nearshore regions of high velocity shear, the shallow water term is shown to be the major source of M 4 energy in the OSCR survey region. These model results can be compared with OSCR values by considering the localized M4 generation due to products of the M2 constituent in the east-west momentum equation (now expressed in depth-averaged velocities) (M4 surface gradient and damping terms are omitted): lg
O U M 4 + /.~/O0 + i)O/'/ + _ _ _ ~ O ;
0--7
Ox
Oy
2D
~xx
1
8
2
3x
COS 01 __ _ k _ _ UI(_~/t~_:~cos 0,~ ___ (} .
(2)
"
At position A (Fig. 1) the values of the above parameters are: (the symbol " denotes instantaneous values of the M2 constituents Ue i"t or ;e i~t, with cs tidal frequency) D = 24m,]~ l = 2 . 5 m , g(lO¢l/Ox)= 0.7 × 10-4m s-2, 0, (phase between ¢ and O~/Ox) = 144°, k = 0.0025, l fJ I = 0.8 m s-1, 02 (phase between ~l and ¢) = 78 °. We estimate the following contributions to the amplitude UM, (with acceleration terms in (2) converted to current amplitudes by temporal integration) advective 1.4
shallow water 0.5
friction 0.2
(cm s ~t
These results for both the advective and friction terms are in accordance with the model amplitudes shown in Fig. 6a and c. The advective terms are discussed further in the following section, the friction term contributes more significantly to M 4 in shallower water (~/D 2 in equation 2). However, the OSCR shallow water current amplitude is an order of magnitude too small. Figure 7a-c shows the M4 current amplitudes derived from these three terms throughout the Irish Sea. Figure 7b shows that the M4 distribution associated with the shallow water term forms part of a large-scale motion and hence the large magnitude of this term in the survey area is due to propagation of M 4 energy originating elsewhere. Since equation (2) only considers localized generation of M4, the discrepancy can be understood. ESTIMATION OF SPA'rlAL GRADIENTS OF CURREN IS AND ADVECTI¥1: TERMS FROM OSCR DATA
Initially, data from each quadrilateral formed by the intersection of any two adjacent beams from each radar site were examined separately. A least-squares fitting technique
3
3
Fig. 6. Components of M4 current amplitude (semi-major axis) cm s-~, associated with (a) advective terms, (b) shallow water terms and (c) friction terms (survey region).
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0~o ,a
.-,,
,~
,,, o
• [~ { n
"13
_.'9
',
:.i " - ~ " . . ~
"~. . . .
~,
," ...-."~o,
.i ;~<',::"- -~ ',,_
951
Observed and modelled nearshore velocities
was used to determine planes approximating the variations in amplitude and phase of both the eastern and northern velocity amplitude components (U and V) pertaining to the tidal constituents M2, M4, M6 and Z0 (residual). It has been shown (GODIN, 1972) that these tidal constituents can be resolved to an accuracy of X/2v//Vt in amplitude and ~v/2v/ANi radians in phase, where v is the standard error of the N observations and A the tidal amplitude. For the present data set N = 700, v < 4 cm s-1, hence tidal amplitudes are resolved to within +0.2 cm s-1 and phases to within ___0.15° for M2 (A = 0.8 m s-1) and +1.5 ° for M4 (A = 0.08 m s-I). These levels of accuracy enable spatial gradients to be calculated for the major tidal constituents. Table 1 shows the mean amplitudes of U, V, OU/Ox, OU/Oy, OV/Ox and OV/Oy averaged over the survey area for each of these constituents. The average velocity gradients are of the order of 10--4 s-1. Table 1 also shows the second spatial derivatives of velocity 02U/Ox2, a2U/Oy2, 02V/Ox2 and 02V/Oy2. Assuming a sinusoidal variation in velocity along the x axis, U sin kx, k = 2rd~., where ~. represents the wavelength for U variations in the direction of current propagation. Spatial differentiation (twice) along the x axis then yields 0eu
- -k2U.
(3)
0x ~
By inserting values from Table 1 into 3 we can calculate representative wavelengths pertaining to (a) the variation of U in the x direction and (b) the variation of V in the y direction. The values obtained in km [(a) and (b), respectively] are (i) for M2:100,30; (ii) for M4:40,10; (iii) for M6:20,10 and (iv) for Z0:20,10. These results are of interest in numerical modelling where spatial resolution is determined as a fraction of parameter wavelengths. The values indicate that greater resolution is required to resolve Z0 and higher harmonics than for M2 and likewise greater resolution is required in the crossshelf (3,) direction than in the long-shelf (x) direction. In numerical models, additional terms are sometimes introduced into the x and y momentum equations to simulate dissipation due to horizontal viscosity. The terms are in the form (x and y, respectively), (a) EH([OEU/Ox 2] + [oEU/Oy2]) and (b) EH([O2V[Ox 2] + [OZV/Oy2])where Eia corresponds to a horizontal eddy viscosity coefficient. P~'~DLE (1984) assumed a value of Eli = 106 cm 2 s-1. For this value of Evi, the respective ratios of the inertial term (6U and 6V, where 6 is tidal frequency in Table 2) to the eddy viscosity terms [(a) and (b)] are (i) for M2:150,20; (ii) for M4:50,5 and (iii) for M6: 25,5, where the first value refers to x-directed accelerations and the second y-directed. Thus this horizontal viscosity term is small for all three constituents in the long-shelf direction but significant for cross-shelf motions at the M4 and M 6 frequencies. For the M2
Constituent
Table 1. Mean amplitudes of velocity gradients derived from OSCR measurements U V OU/Ox OU/Oy OV/Ox OV/Oy 02UlOx2 02U/Oy2 Ozv/ox 2 cm s-x
M2 M4 M6 Zo
82.3 7.9 2.5 1.4
16.6 3.9 2.3 6.3
× l0 s 10 4 3 3
17 5 4 5
S- 1
15 10 7 5
X
18 10 9 6
3 2 2 2
10-Ix cm -~ 5 2 2 4
02V/Oy2
S- 1
4 7 4 11
7 11 11 24
952
D. PRANDLEand D. K. RYDER
constituent which propagates as a Kelvin wave, closer to the shore the term EHO2U/Oy2 might be expected to increase due to reduction of the along-coast velocities in progressively shallow water. The minimum values (2 × 10-jl cm -l s-1) shown in Table 1 for the second spatial derivative of tidal currents may be used to estimate the standard error, e, in the constitutent current. For an average mesh size (over which the spatial differences were computed) of 1.8 km, the mean error in the calculation of the second derivative is 481180,0002, equating this with the minimum calculated derivatives yields e = 0.16 cm s-~. This value is close to the earlier estimates (PRANDLE, 1987) in which ~ was calculated from discrepancies from an assumed smooth distribution of ME current amplitudes along two specific beams. As noted previously, this value of e indicates a standard error in the radar current measurement of approximately 3 cm sl~ Table 2 shows the mean amplitudes (averaged over all quadrilaterals) of the advective terms occurring at the tidal frequencies of M2, M4, M6 and Zo where these arise due to various combinations of these four frequencies as discussed below. For all four constituents, the much larger values of U(OU/Ox) and U (OV/Ox)in comparison with V(OU/Oy) and V(OV/Oy)will be shown to be a result of the much larger value of UM2 (82 cm s-1) compared to VM2 (17 cm s-l). The table also shows the ratio of the sum of the two advective terms to the respective inertial terms (crU and ~V where ~ is tidal frequency) in the relevant momentum equations. For M2 motion in the east-west (U) direction the advective terms are shown to be negligible, whereas in the north-south (V) direction the advective terms amount to one quarter of the local acceleration term. For the higher harmonics the advective terms are important for motion in both directions, with a maximum value of 60% of the local acceleration term for M 4 motion in the north-south direction. Using the simple trignometric expansions of the form sin C cos D = ½(sin[(C + D)/2] + sin[(C - D)]/2) it can be shown that the cross products in the advective terms such as UM2(OU/OX)M~yield terms at the frequencies Ms and M4. A full listing of such terms resulting from combinations of M2, M4, M6, and Z0 constituents is shown in Table 3a. Table 3b shows the mean amplitudes of the advective terms formed by such cross products. In each of the four sections of Table 3b the largest term occurs in the top left-hand corner representing the product of M2 velocities with Mz velocity gradients, thereby contributing significantly to both the Z0 and M 4 constituents. Likewise, the next largest terms are (almost) invariably along the top lines representing the products of M2 velocity with velocity gradients due to (i) M4, (ii) M6 and (iii) Zo, and thereby contributing, respectively to (i) M6 and M2, (ii) Ms and M4 and (iii) M2. As indicated previously, the M2 contributions are small relative to the local accelerations (Table 2), whereas for both M4 and M 6 these advective contributions are significant. While it is well recognized that quadratic friction generally contributes a much larger term for M6 in shallow water, this M6 contribution from a product of M2 and M4 constituents can be significant (ZIMMERMAN, 1980). The above deductions, based on direct current measurements, may be compared with conclusions drawn from (i) theoretical analysis and (ii) detailed comparisons of model simulations using differing grid sizes. ZIMMERMAN(1978, 1980) showed that the generation of residual currents by topographic features is particularly sensitive to the ratio of
6U
1152 222 105
Constituent
M2 M4 M6 7-o
233 110 97 41
6V 24 43 19 14
UOU/Ox 10 15 7 47
VOU/Oy 27 44 19
UOU/Ox+VOU/Oy
A
0.02 0.20 0.18 62
A/6U
51 63 41 18
UOV/Ox
13 17 11 59
VOV/Oy
B
57 67 43
UOV/Ox+ VOV/Oy
Table 2. Mean amplitudes of the advective terms derived from OSCR measurements ( x 10-5 cm s-2)
0.24 0.60 0.45
B/6V
954
D. PRANDLEand D. K. RYDER Table 3a. Frequencies generated by cross products (e.g. UM2 0UM2/ax generates terms at the frequencies M4 and Zo (residual)
M2 M2
M4
M4
M6
M6
Ms
Z0
M4 Zo M2 M4
M6
Ms
Mz
Ms
M2
M12
M2 M4
Zo M4
Mlo
Z0
MlO
M2
M6
Zo M6
M2 Ma M6
Zo
Table 3b~ Mean amplitudes over survey region (cm s 2) 0 U/Ox
½U(OU/Ox)
U
M2 40 4 1 0
M4 17 2 0 0
15 4 2 1
4 1 1 2
M2 M4
M6 Zo
62 6 2 1
41 4 1 1
M2 M4 M6
16 42
8 21
Zo
6
3
M2 M4 M6
Z0
M6 12 1 0 0
Zo 13 1 0 0
3 1 0 1
4 1 1 2
x 10-5
28 3 1 0
20 2 1 0
× 10-s
8 21 3
5 11
x 10-5
OV/Oy M2
½V(OU/ay)
V
M4 M6
Zo
OV/Ox ~U(OV/Ox)
U
OV/Oy ½V(OV/Oy)
V
x 10-5
2
the tidal displacement to the length of the bed features. H e showed further that a grid size of about half the amplitude of the tidal displacement generally provides sufficient resolution. ROBINSON (1981, 1983) provides a graphic explanation of the physics that describe this coupling between tidal advection and topographic features. H o w e v e r he concludes that a smaller (than that derived by Z i m m e r m a n ) grid size may be required to resolve vorticity generation associated with topographic features and current shear transverse to the main tidal flow direction. ABRAHAM et al. (1987) examined the above theories by formulating two tidal models (depth-averaged) of the same region with grid sizes of 1.5 and 10 km, respectively. They found that the finer grid model included significant additional vorticity absent in the coarser grid model. This omission could be significant in dispersion studies where enhanced dispersion coefficients would be needed to account for such sub-grid scale motions. Vorticity
Surface gradients can be eliminated from the orthogonal m o m e n t u m equations by forming the curl of the equations as shown.
Observed and modelled nearshore velocities
-~x
955
+ V-~x + V-~y + g -oy- + - -OR V + ~ U
=0
(4) 0 {0_~ OU __OU _0; --KRu+E~U} = 0 -~y + U"~x + V oy + g ox + D (R = (u 2 + v2)
.
Then defining the vertical component of vorticity co =
dt (i)
-
Ot
+ U--+
Ox
V--=
Oy
(o+
f~)
~x (ii)
+
-
(OV/Ox) - (OU/Oy) we --D (iii)
o3---D
\
obtain
V OD OD o x - V Oy
.(5)
(iv)
Figure 8 shows the vorticity amplitude at the M2 frequency as (i) calculated from the O S C R observations and (ii) obtained from the numerical model. The O S C R values were calculated by fitting planes through the M2 current components for each quadrilateral as described previously. The spatial distributions for the calculated and observed vorticity amplitudes are in reasonable agreement and the amplitudes agree within a factor of two. The computed values are of the same order as those calculated by ABRAHAM et al. (1987) from a 1.5 km grid model of the southern North Sea.
Fig. 8. Amplitudeof M 2 tidal vorticity (i) (thick lines) from radar measurements and (if) from model calculations. Contour units 10-5 s-1.
956
D. PRANDLE and D. K. RYDER
The maximum observed amplitude of co ~ 3 x 10-5 s-1 is smaller than the planetary vorticity I} = 10-4 s-1. For the M2 constituent in the above equation (5), ([OU/Ox] + [OV/Oy]) ~ 10 -5 S-1, kR/O ~ 104 s-1, 1/D[V(OO/Ox) - U(OD/Oy)] ~ 3 x 10-6 s-1. Thus the magnitude of the respective terms is approximately: (i) 3 x 10-1°, (ii) 10- 9, (iii) 3 x 10-9 and (iv) 3 x 10-9. Hence we see that the main M 2 vorticity balance is determined by the influence of bed friction, with term (iii) representing direct dissipation and term (iv) generation via differences in bed topography (PINGREE, 1978). F r o m (i) and (iii) a time constant associated with dissipation of viscosity (of any frequency) can be determined, namely T = (D/kR) ~ 3 h, indicating the rapid dissipation of vorticity in this area (ROBINSON, 1983). The maximum amplitude of the observed residual vorticity is approximately 10-5 s-1, i.e. small in relation to f] and insufficient to generate bed features (PINGREE, 1978). From the estimated accuracy of determination of the tidal constituents (0.16 cm s-l), by taking spatial gradients over an average of 1.8 km, this residual vorticity magnitude is close to the noise level of the calculation. 4.
RESIDUAL
SURFACE CURRENTS AND VERTICAL CURRENT STRUCTURE
Figure 9 shows the time-averaged (over 30 days) residual surface currents measured by OSCR, these currents range from 17 cm s-1 in the shallower southern area to 3 cm s-1 in the deeper northern area. By assuming these surface currents persist through depth, / I 0 era s -~
/
'
/
i
/
/o-2
X
~//I
,.....,.~c
53°50 ,
./o
....../>,,i I,E
/
I
I
/
'
..
T'
.
'
X /P
-'"
I "(d
x
4" 20'
4*
Fig. 9. Arrowsindicate time-average (over 30 days) residual surface currents measured by H F radar. Lines at positions A to F indicate direction of density contour. Shorter parallel lines indicate spacing between density contours, at.
53*20'
957
Observed and modelled nearshore velocities
Table 4. Tidal ellipse measurements at position A (cm s-l). 1, OSCR near-surface; 2, current meter moored at mid-depth; 3, bottom-mounted current meter. All values determined from approximately 30 days data (not concurrent). Direction measured a--c from east Constituent
Anticlockwise Amplitude Phase
Clockwise Amplitude
Phase
M2
1 2 3
34.7 28.8 15.8
125.0 124.0 129.0
45.1 29.4 11.6
217.0 216.0 211.0
Sz
1 2 3
12.9 8.9 4.5
89.0 85.0 89.0
13.8 9.4 4.5
245.0 249.0 255.0
M4
1 2 3
2.0 2.8 1.3
270.0 248.0 259.0
5.6 2.6 0.9
90.0 96.0 189.0
7-o
1 2 3
Amplitude 9.7 3.0 3.8
Direction 90° 194° 254°
PRANDLE (1987) estimated a net northerly flow through this region of 40 × 103m3s-1. This value is clearly unrealistic and thus one deduces that some compensatory flow exists nearer the bed. Observational evidence for such a flow reversal is shown in Table 4, where current data obtained close to position A (Fig. 1) are shown for: (i) surface values measured by OSCR, (ii) mid-depth values measured by a moored current meter and (iii) a near bed (+ 1 m) value measured from a bottom-mounted current meter. All three data sets extended over about 30 days but the recordings were not concurrent. The residual currents (associated with non-linearities in tidal propagation) produced by the depth-averaged model are generally much smaller and are more clearly related to topographic features. For detailed comparison with these observed residuals a fully three-dimensional model is required. However, for the present, a simple model of vertical current structure at a single point is used with depth-integrated transports, surface elevation and a (depth-averaged) horizontal density gradient specified directly.
4.1.
Point model of vertical current structure
Neglecting advective terms, the following equations of motion were assumed:
and
OU --+ Ot
O~ g -~x
OV
O~
Ot
~y
--+g
[~-z\Op O + ~:-~-) - - - f ~ V = - - N g Ox Oz [~-z\Op
Oy
Oz
O
OV
Oz
Oz'
+ ~-~)--+fIU=--N-g
OU
(6)
where N is a vertical eddy viscosity coefficient and z is measured up from the bed. The vertical boundary conditions imposed were:
OR (i) stress at the surface z = S: zs = N = 0, Oz where R = U + iV (ii) stress at the bed z = 0, x0 = pkRlRI .
(7)
958
[), PRANDLE and D. K, RYDER
An (alternating direction) implicit finite difference numerical scheme was formulated to solve the above equations in a time-stepping mode (surface gradients determined by vertical integration). The scheme involved 20~ layers, with velocities calculated at the mid-depth of each layer and the lowest velocity at the b e d By depth-integrating the observed current data shown in Table 4, M2 and $2 transports were prescribed as
Q(t) = Re D{0.271 exp iloM:t~ 126~') -~- 0.292 exp iteM::, '~:~ + 0.089 exp il'~s~t+ss~3 + 0.(/94 exp -~(~sJ+249'~t],
(8)
where the mean depth D = 24 m and ~(t) = 2.5 COS(CYMJ -- 308 °) + 0.83 COS(~s~ - 34T'~
(9)
The model was run over a 15 day M2 - $2 cycle and the computed tidal constituents at the appropriate depths compared with the observed values shown in Table 4. 4.2.
M2, Sz and Zo current structure
Assuming N to be time invariant, reasonable agreement between model and observed current structures were obtained for both the M2 and Sz constituents with values of N = 2 0 0 c m 2 s-1 and k = 0.01 [BOWDEN et al. (1959) deduced values of 130 < N < 270 cm 2 s-1 and k = 0.0035 from measurements of current profiles in this area]. Examination of the effect of varying N with z showed that these tidal current profiles were relatively insensitive to three cases, namely: (i) N = 200, (ii) N(z) = 400z/D, (iii) N(z) = 400(D - z)/D. However, when horizontal density gradients were introduced (with the assumption of no net flow), the resultant residual current profiles were found to be much more sensitive to these vertical variations in N. A reasonable fit to the observed residual current structure could only be obtained by using the relationship (iii), i.e. eddy viscosity decreasing linearly with height above the bed (to zero at the surface). Figures 10 and 11 show the current profiles for M2, $2, Z~ and M4 for this eddy viscosity formulation. The residual profile mainly results from the specification of a northerly density gradient Op/pOy of 0.23 × 10-6 m -1 (or ~t increasing northwards by 0.23%0 km-1). Assuming the same values for N and k, the above process was repeated at five additional locations shown in Fig. 9 and values of Op/pOx and Op/pOy determined such that the model produced the observed surface residual current. Then, using the requisite density gradients from all six locations an overall distribution of horizontal density gradients was sketched. Comparison of this distribution with an observed distribution (Fig. 12) shown by MAa~rHEWS et al. (1988) indicates reasonable comparability. The surface residuals computed by the model increase approximately linearly with both increasing density gradient and decreasing eddy viscosity, while the residuals increase with depth squared. In the above simulations the formulation for N was fixed; since N generally increases with water depth the inferred density distribution can only be regarded as approximate.
4.3.
M4 current structure
The combination of specified M2 variations in flow and elevation produce M 4 currents (but not flow). These currents are shown in Fig. l lb. Of particular interest is the
959
Observed and modelled nearshore velocities
/ 0
20 cm $-i
I
I
40
120*
'~ 140"
o-c
0
20 cm s-I
(o)
0.5
I
I*
I
40
210"
230*
c-w
•
I 0
I0 cm$-i
I
I
20 70* O-c
~
I
90*
0
(b)
I0 cms-I
20
240*
"1 260*
C-W
Fig. 10. Vertical current profiles at position A for (a) M 2 and (b) $2 tidal constituents. Arrows and * indicate values observed by (i) radar at the surface, (ii) a m o o r e d current m e t e r at middepth and (iii) a b o t t o m - m o u n t e d current m e t e r at z = + 1 m. C o n t i n u o u s curves indicate values calculated from a numerical model of vertical structure.
pronounced vertical structure of this constituent that clearly complicates the comparisons of (depth-averaged) model vs observed (surface) currents described in earlier sections. A second feature shown in this figure and also for the residual current distribution in Fig. l l a , are currents pertaining to specific heights above the bed (as measured by moored current meters). These differ significantly from values pertaining to "fractional" heights above the bed. In general, this difference increases towards the surface and again emphasizes a problem in comparing OSCR surface measurements with moored current meter recordings. For the predominant M2 and $2 constituents this discrepancy is much less significant.
960
D . PRANDLE and D. K, RYDER
\
,
)
' I
o
1 io
5
I
---ioo*
___
I
~l__~.
14o*
too*
J zzo °
Z6o°
c m S- I
amplitude
.
(Q)
direction
/
c-
J i
/
0"5
-
'
//
°
/
/
/.
-
/ 0
2
4
330 °
350 °
0
C rrl S - I
2 cm
a-c
(b)
4 290 °
510'
330 °
550'
$-I
c-w
Fig. 11. Vertical current profiles at position A for (a) Zo and (b) M4 (residual) tidal constituents. Arrows and * indicate values observed by (i) radar at the surface; (ii) a moored current m e t e r at mid-depth and (iii) a b o t t o m - m o u n t e d current meter z = + 1 m. Continuous curves indicate values calculated from a numerical model of vertical current structure. T h e s e values pertain to fixed fractional heights above the bed; the points are the s a m e m o d e l values interpolated to represent conditions at fixed heights above the bed.
5. CONCLUSIONS A comparison was made of tidal current ellipse parameters derived (i) from OSCR measurements and (ii) from a 1 km grid depth-averaged numerical model. For the predominant M2 constituent, amplitude, phase and direction values are in reasonably good agreement. The surface amplitudes measured by OSCR exceed the model value by approximately 10% as expected from standard theory on the vertical structure of tidal currents. (This vertical structure precludes a comparison of eccentricities of the ellipses.)
961
Observed and modelled nearshore velocities
Cumbt~
I
4*W
i
3*W
I
Fig. 12. Density distribution for May 1978, 6t reproduced from MATHEWSet al. (1988).
For the major higher harmonic constituent M4, broad agreement is evident but poorer than for M2. The contributions from each of the non-linear terms [namely (i) advective, (ii) shallow water and (iii) friction] responsible for the generation of M4 (from the fundamental ME) are quantified from both the model and the OSCR measurements. Reasonable agreement is shown for the contributions arising from the advective and friction terms. A discrepancy in the magnitudes associated with the shallow water terms is related to a large-scale motion throughout the Irish Sea, the effect of which was not included in the OSCR calculations. By calculating mean values of the spatial gradients of the tidal current amplitudes and relating these to mean current amplitudes [assuming U(x) = U cos 27r.r/E], wavelengths, k, associated with each tidal frequency were calculated. For M2, L = 0(100 kin), whereas for M4, M6 and Z0, ~. may be as small as 10 km, thus accurate resolution of these latter
962
D. PRANDLE and D. K, RYDER
constituents in numerical models requires much finer grid resolution than for M2. This latter deduction is widely recognized by modellers (ABRAHAM et al., 1987) but this quantification from observations is valuable. The second spatial derivatives of current calculated from the O S C R measurements are used to estimate the influence of "horizontal eddy viscosity". The O S C R data are also used to examine the magnitude of the separate advective terms averaged over the survey region. In the east-west m o m e n t u m equation for Me, the advective terms amount to only 2% of the inertial terms, ~U, whereas for the north-south m o m e n t u m equations for M 4 and M6, respectively, this ratio is 60 and 45%. In a detailed analysis of the generation of these advective terms, these large north-south advective terms are shown to be due to large values of (i) at M4: UM2(OVMJOX) and UM~(OV~/Ox) and (ii) at M~: UM2(OVM,/OX). These spatial gradients of tidal currents were also used to calculate vorticity. The amplitude of the ME vorticity component, obtained in this way from O S C R observations, shows a similar spatial distribution to the corresponding pattern calculated from the numerical model, though the observed amplitude was approximately half the model value. In the present survey region, it is shown that vorticity is rapidly dissipated by bed friction. The deployment of O S C R in other regions where dissipation is less rapid offers exciting prospects, particularly where topographic scales produce flow separation and the shedding of discrete eddies. A simplified " p o i n t - m o d e l " of vertical current structure was developed to explain the large time-averaged surface residuals measured by OSCR. These residuals were shown to be compatible with forcing by horizontal density gradients, the required spatial distribution of which was shown to be in qualitative agreement with related observations. This model was also used to indicate how, for M4 and residual currents, significant discrepancies can arise in comparing currents taken at fixed heights above the bed (as with m o o r e d current meters) and values pertaining to fractional heights (e.g. the surface as measured by OSCR). The residual vertical current structure calculated by the model was found to be sensitive to the specified vertical distribution of eddy viscosity. Thus both the tidal and residual currents obtained from O S C R measurements are well suited for the development and verification of 3-dimensional models.
Acknowledgement--Iam indebted to the referees for detailed constructive criticism ot this paper. REFERENCES
ABRAHAMG., H. GERR1TSENand G. J. H. LINDIJER(1987) Sub-grid tidally induced residual circulations. Continental Shelf Research, 7,285-303. BOWDENK. F., L. A. FAIRBAIRNand P. HUGHES(1959). The distribution of shearing stresses in a tidal current. Geophysical Journal of the Royal Astronomical Society, 2, 288-305. GODING. (1972) The analysis of tides. Liverpool University Press, 264 pp. HAMMONDT. M., C. B. PAITIARATCHI,D. ECCLES,M. J. OSBOm,~E,L. A. NASHand M. B. COLLINS(1987) Ocean Surface Current Radar (OSCR) vector measurements on the inner continental shelf. Continental Shelf Research, 7, 411-431. HUTHNANCEJ. M. (1973) Tidal current asymmetries over the Norfolk sandbanks. Estuarine and Coastal Marine Science, 1, 89-99. LODER J. W. (1980) Topographic rectification of tidal currents on the sides of Georges Bank. Journal o] Physical Oceanography, 10, 1399-1416. MAASL. R. M., J. T. F. Z~MEI~AN and N. M. TEMME(1987) On the exact shape of the horizontal profile o1 a topographically rectified tidal flow. Geophysical and Astrophysical Fluid Dynamics, 31t, 10.5-129.
Observed and modelled nearshore velocities
963
MATHIEWSJ. P., J. H. SIMPSONand J. BROWN(1988) Remote sensing of shelf sea current using the OSCR HF Radar. Journal of Geophysical Research, 93, 2303-2310. PINGREE R. (1978) The formation of the shambles and other banks by tidal stirring of the seas. Journal of the Marine BiologicalAssociation of the United Kingdom, 58, 211-226. PRANDLE D. (1974) A numerical model of the southern North Sea and River Thames. Institute of Oceanographic Sciences, Report No. 4, 25 pp. PRANDLE D. (1982) The vertical structure of tidal currents and other oscillatory flows. Continental Shelf Research, 1, 191-207. PRANDLE D. (1984) A modelling study of the mixing of 137Cs in the seas of the European Continental Shelf. Philosophical Transactions of the Royal Society, London, A310, 407-436. PRAttLE D. (1985) Measuring currents at the sea surface by H. F. Radar (OSCR). Journal of the Societyfor Underwater Technology, 11, 24-27. PRANDLE D. (1987) The fine-structure of near-shore tidal and residual circulations revealed by H. F. Radar surface current measurements. Journal of Physical Oceanography, 27, 231-245. ROBINSONI. S. (1981) Tidal vorticity and residual circulation. Deep-SeaResearch, 28, 195-212. ROBINSONI. S. (1983) Tidally induced residual flows. In: Physical oceanography of coastal and shelf seas, B. JOHNS, editor, Elsevier Oceanography Series, 35, pp. 321-355. TAYLORG. I. (1921) Tidal oscillations in gulfs and rectangular basins. Proceedingsof the London Mathematical Society, ser. 2, XX, 148-181. ZtMMEmaAN J. T. F. (1978) Topographic generation of residual circulation by oscillatory (tidal) currents. Geophysical and Astrophysical Fluid Dynamics, 11, 35-47. ZIMMERMANJ. T. F. (1980) Vorticity transfer by tidal currents over an irregular topography. Journal of Marine Research, 38, 601-630.