Barotropic Tides in the Northeast Atlantic Inferred from Moored Current Meter Data
Gerhard D i c k and Gerold S i e d 1 e r
UDC 551.466.75; Northeast Atlantic
Summary
Current data obtained from 7 moorings in the Northeast Atlantic in the course of many years are analysed with respect to semi-diurnal barotropic and baroclinic tides and diurnal barotropic tides. For semi-diurnal tides M2 and $2 the energy distribution is usually dominated by the barotropic mode; only in a few cases does the first-order baroclinic mode contain higher energy. Barotropic tidal ellipse orien- tations are found to be consistent with results from earlier tide gauge observations in this area. Significant deviations occur, however, in amplitudes. Results for the diurnal component K1 are also presented. With few exceptions, tides are found to be progressive waves in this area. The current ellipse pattern is similar to results obtained indirectly by C a r t w r i g h t , E d d e n , S p e n c e r et al. [1980] from tide gauge observations.
Barotrope Gezeiten im Nordostatlantik aus Daten verankerter Str6mungsmeBgeriite (Zusammenfassung)
Strrmungsdaten von 7 Verankerungen, die im nordrstlichen Atlantik iiber meh- rere Jahre ausgelegt waren, wurden im Hinblick auf halbt/igige barotrope und baro- kline sowie eint/igige barotrope Gezeiten analysiert. Bei den halbt/igigen Tiden M2 und $2 dominiert normalerweise die Energie der barotropen Eigenfunktion, nut in wenigen Ffillen enth~ilt die barokline Welle 1. Ordnung h r h e r e Energie. Die Ellipsen- orientierungen fiir die barotropen Komponenten entsprechen friiheren Ergebnissen von Tiefseepegeln in diesem Gebiet. Es gibt jedoch signifikante Abweichungen bei den Amplituden. Die Gezeiten in diesem Gebiet sind fast ausschlieglich fortschrei- tende Wellen, in einigen wenigen F/illen jedoch vom gemischten Typ. Die rfiumliche Anderung der S trrmungsellipsen entspricht weitgehend den Ergebnissen, die C a r t w r i g h t , E d d e n , S p e n c e r et al. [1980] indirekt aus Pegelbeobachtungen erhielten.
Les mar~es barotropes dans l'Atlantique Nord-Est, d~duites de mesures effectu~es l'aide de courantom~tres mouiH~s (R~sum~)
Les mesures de courants provenant de 7 mouillages sur plusieurs annres en Atlantique Nord-Est ont 6t6 analys6es en ce qui concerne les marres semi-diurnes barotropes et baroclines et les mar6es diurnes barotropes. Pour les mar6es semi-diur- nes M2 et $2 la r6partition de l'rnergie est g6n6ralement domin6e par le mode barotrope; dans peu de cas seulement, le mode barocline de premier ordre renferme la plus haute 6nergie. Les orientations de l'ellipse relevant de la marre barotrope sont en accord avec les rrsultats d'observations marrgraphiques ant6rieures dans cette rrgion. Des 6carts significatifs apparaissent toutefois en amplitude. Les rrsultats
8 Dt. hydrogr. Z. 38, 1985. H. 1. D i c k et al., Barotropic tides
concernant la composante Ki sont 6galement p r t s e n t t s . A quelques exceptions pros, les mar6es, dans cette r6gion, se sont trouv6es 6tre des ondes progressives. L e type d'ellipse des courants est semblable aux r6sultats obtenus indirectement p a r C a r t - w r i g h t , E d d e n , S p e n c e r et al. [1980] h partir des observations mar6graphi- ques.
1 Introduction
The advances in the numerical modelling of ocean tides (H e n d e r s h o t t [1977, 1981];
A c c a d and P e k e r i s [1978]) have increased the n e e d for precise m e a s u r e m e n t s of o p e n ocean tides, particularly in regions of rough t o p o g r a p h y , in o r d e r to check and improve the established models (S c h w i d e r s k i [1980]). W i t h the advent of deep-sea pressure gauges open-ocean sea level d a t a b e c a m e available ( U n e s c o [1975]; C a r t w r i g h t et al. [1980];
B a k e r [1981]). The growing n u m b e r of long-term current m e t e r moorings d e p l o y e d in some parts of the ocean now provide an additional d a t a base, providing information on barotropic and baroclinic tidal currents.
In the eastern N o r t h Atlantic, C a r t w r i g h t et al. [1980] o b t a i n e d deep-sea tide gauge data at selected positions on lines from southern Portugal to the A z o r e s and from there to Iceland and then to Scotland, on a line from the M i d - A t l a n t i c R i d g e along 54 ~ N towards Ireland and also along the shelf edge west of I r e l a n d and the British Isles. The
"response" m e t h o d of M u n k and C a r t w r i g h t [1966] was used for the analysis. Boun- dary data were interpolated by hand-drawing smooth cotidal curves, and b a r o t r o p i c tidal currents were d e d u c e d f r o m the pressure gradients in the resulting cotidal maps. Direct observations of tidal currents in this area n e a r the continental margin were p r e s e n t e d by G o u l d and M c K e e [1973] and M e i n c k e , S i e d l e r and Z e n k [1975] and for one site in the M a d e i r a A b y s s a l Plain by S a u n d e r s [1983]. F o r tidal currents in the surroun- ding area we refer to R e g a I and W u n s c h [1973] and S c h o t t [1977].
In the present study current m e t e r d a t a from moorings in the eastern N o r t h Atlantic were analysed with respect to the dominating semi-diurnal and diurnal tides. The tidal currents thus o b t a i n e d are c o m p a r e d to the currents d e d u c e d from pressure-gauge d a t a by C a r t w r i g h t et al. [1980] and to S a u n d e r s ' [1983] m o o r e d current m e t e r results from one site in the southern p a r t of the a r e a under discussion. Thus new current d a t a are presented, and indirectly d e d u c e d tidal current properties are checked in an a t t e m p t to provide an extended d a t a base in the eastern N o r t h Atlantic for the verification of numerical tidal models.
2 The data set
The current m e t e r d a t a u s e d in this analysis were from moorings of the international Northeast Atlantic Dynamics Study ( N E A D S ) and also from moorings d e p l o y e d in the region by the Institut fiir M e e r e s k u n d e an d e r Universit~it Kiel (D i c k s o n [1983]; M fi 11 e r [1981]; M fi 11 e r and Z e n k [1983]). Mooring positions and the m a j o r features of b o t t o m t o p o g r a p h y are shown in Fig. 1, and a summary of the depths and durations of individual records are p r e s e n t e d in Table 1. Only d a t a from A a n d e r a a Current Meters were used to ensure a consistent d a t a set, and only records with a minimum length of 4 months were selected to allow the use of ordinary harmonic analysis to resolve M2, $2, K1 tides and inertial signals ( G o d i n [1972]). The vertical profiles of Brunt-Vfiisfil~i-frequency N requi- r e d for the modal analysis were obtained from P r i c e [1983] who had p r e s e n t e d density distributions from historic hydrographic d a t a averaged in 10 ~ x 10 ~ squares.
Dt.hydrogr. Z. 38, 1985. H. 1. D i c k et al., Barotropic tides 9
3 5 ~ 3 0 9 2 5 o 2 0 ~' "15 ~ ! . ( P 5 ~ 0 n
A o I
1;
================================================================================================================
~ ~ ~]iiiiiiiiiiiiiiiiii!iiiiii, iiiiiiiiiiiiiiiiiiiiiiiil
.. , :..g~, { ca,,,,,, !ii',iii',iiiiiiiiiiii',ii',iiiii'iii!liiiiiiiii',iii',iiiiiii'iiiil
9 .~1 o ~ i!i:ii:i:ii:ii:ii:i:ii:ii:i:ii:i:i:{:i:il :i,i;,iii:ii'ifi:ii~:,:,iiii~:l
3 5 "~ W 3 0 ~ 2 5 * 2 0 " 1,5 ~ "1-0 ~ 5 ~ 0 ~
Fig. 1. Topographic map with positions A to G of moorings, depth lines indicating 3000 m and 5000 m levels
10 Dt.hydrogr. Z. 38, 1985. H. 1. D i c k et al., Barotropic tides
T a b l e i
D e p t h s a n d o b s e r v a t i o n p e r i o d s o f m o o r e d c u r r e n t m e t e r r e c o r d s u s e d nominal year: 1977 I 1978 I 1979 1 1980 I 1981 I 1 9 ~ 1 1983 position ... I . . . I ... I . . . I ... I . . . . I ... ] and depth
IfM no : 278
A 1 203 1
31 ~ N 1 524 I
2 0 ~ W
NEADS 1 705 1
I 1139 I site 12
1 2974 1 4850 m
1 4692 1
IfM no:18~ 264 276 276-2 276-3
1 125 1 196 1 245 1
B 1 379 1 499 1 I 428 I
33 ~ N I 673 I 1-632 1 703 I 755I I 629 I
22 ~ W
READS 1 955 1 1004 1 1160 II 1032 I
11585 1 I 1608 1 I I__535 I
site I
13089 I I ~008 I I 3020 1
5260 m
I4770I 14794 I
IfM no : 277-2 277-3
C 1 255 1..277 1
I 549 I 535 I 34 ~ N
23 ~ W 1 1192 i1140 I
NEADS 1 1663 11640 I
Site 11 I 3029 i3090 I
5155 m I 4722 I
l• no. 203 229 242
O 1 780} 796 -I
38 ~ N I~6921 I 1466 1
17 ~ W 131951 3210 I 1 3098 1
READS 1 42241 I 4138-1
site 2
i50791 5550 m
E IfM no: 230
40 ~ N I 485 1
17 ~ W 1 2945 1
NEADB
site 2,5 I 4050 1
5310 m
IfM no: 266
F 44 ~ N 26 ~ W 3167 m
199 -I 806 I 2497 -I
IfM no : 265-I 265-2
G 4 8 ~ N 26 ~ W 3730 m
184 I 389 I 426 I 794 .I 830 I 2515 I 2521 I
Dt. hydrogr. Z. 38, 1985. H. 1. D i c k et al., Barotropic tides 11 3 Methods of analysis
The a r e a of investigation is north of the critical latitude where, according to linear internal wave theory, no diurnal baroclinic waves should exist. This is not so for semi-diurnal internal waves. M o d a l analysis was t h e r e f o r e applied in the case of semi-diurnal tides, discriminating between barotropic and low o r d e r baroclinic waves, while a vertical averaging scheme was selected for the study of diurnal b a r o t r o p i c tides. T h e m e t h o d used for semi- diurnal components is summarized in the diagram of Fig. 2.
Normalized m o d e s are calculated by numerically integrating the internal wave equation d2W N 2 d W N 2 - 092
- - + - - + - - ~ 2 W = 0 d z 2 g d z ~o2 _ f2
for the given Brunt-V/iis/il/i-ffequency N, using the R u n g e - K u t t a m e t h o d , with vertical velocity amplitude W, vertical coordinate z , gravitational acceleration g, angular frequency
~o and horizontal wave n u m b e r g. Vertical velocity equals zero at the surface and the bottom. The horizontal velocities can b e o b t a i n e d from the vertical derivative of the vertical velocity for each m o d e ( K r a u s s [1966]). The F o u r i e r coefficients of horizontal tidal currents, d e t e r m i n e d by harmonic analysis O f the current m e t e r time series, can then b e least-squares a p p r o x i m a t e d by a set of modes. The zero-order m o d e is equated to the barotropic tide. A s an example, the m o d e s of the M2 tide resulting from our analysis are presented in Fig. 3.
current ellipse
[
~, b, 0, t o
t
harraonle constants [~ [ astronomical argument and
r I
~1
approximationby [~
--I nOrmal modes 1--
numerical
integration N
t
internal waves S,
T, P
tidal currents I corrected for mooring motion
ll ta I
U~ V.... l__m
harmonle analysis
data
P Fig. 2. Method of semi-diurnal tide analysis
12 Dt.hydrogr. Z. 38, 1985. H. 1. D i c k et al., Barotropic tides
0
rTl
1 0 0 0
2 0 0 0 Z
3 0 0 0
4 0 0 0
5 0 0 0 - 2
/.t ~ V
-1 0 1 crr'~ 1 2 -2 -1
' ' ; ' / ' / I/
I n = 3
,/
0
m
1 0 0 0
2 0 0 0
i3000
4 0 0 0
5 0 0 0
n=21
0 1 cm r 2
Fig. 3. Normal modes of the east (u) and north (v) component of the M2 tide at position B, mooring no. 264-1.
Full circles indicate levels of current meters used for determining modal amplitudes
Mooring motions can not be neglected when determining the tidal currents, and typical errors due to this effect will be discussed here. The moorings usually contained pressure sensors near the top, and their records can be used to determine the changes in instrument depths due to the drag forces of currents acting on the mooring. With a current of a single period alone, the pressure will change with half that period. If superimposed on slower varying and stronger currents, the tidal period will be found in the pressure record. The pressure spectra contain strong semi-diurnal tidal peaks, indicating the latter conditions.
According to F o f o n o f f [1966], resonant modes of deep-sea mooring motions will have periods which are several orders of magnitude below tidal periods. The response of the pendulum mode of a mooring to changing tidal current drag will therefore be almost in phase with the forcing current. The data indicate a decrease of instrument depth changes with increasing depth, suggesting a rigid inverted pendulum mode as a useful approximation.
Assuming the mooring to move in the plane of the major tidal ellipse, the 180 ~ phase uncertainty in relating pressure and current records can be removed. The procedure is included in the schematic diagram of Fig. 2, and results for one mooring will be presented later.
Phases are referred to Greenwich Mean Time (GMT) [Universal Time Co-ordinated (UTC)]. From the amplitude and phase information for the east and north components, tidal current ellipses can be determined, with the major axis 2a, the minor axis 2b, the angle 0 of the major axis with respect to north, and the time to of the current maximum referred to Greenwich (D o o d s o n [1941]).
Dt. hydrogr. Z. 38, 1985. H. 1. D i c k et al., Barotropic tides 13 4 Tidal currents
Results from the M2 analysis are presented in Fig. 4 and Tables 2, 3 and 4. Results from mooring 264-1 (B in Fig. 1) will be used to discuss the accuracy of the M2 tidal parameters.
The modal structure and the resulting partition of energy in those modes will depend on the actual profile of the Brunt-V/iisfilfi frequency N. In Table 2 we compare results obtained when using the P r i c e [1983] mean N profile and when using N calculated from a single CTD-profile obtained near the site of 264-1 in April 1982. Although deviations in the barotropic current component amplitudes are only 3 to 4 % , the energy partition is affected considerably, with a change of about 7% in the barotropic tidal energy. The effect of mooring motion is presented in Table 3. Here, deviations in the current component ampli- tudes are of order 10%, while the partition of energy in modes is less affected.
Additional errors will be d u e to the deviations between actual tidal currents and the Gaussian fit using a limited number of normal modes. D u e to different numbers of instru- ments on the moorings, the quality of the fit varies. Results for moorings 264-1 (B) and 242-1 .(D) are given in Table 4. The deviations are in the range of 6 to 22% of current component amplitudes. Resulting errors are up to 15% for the major ellipse axis and up to 70% for the minor axis. In the case of narrow ellipses the sense of rotation becomes uncertain.
We will now discuss the results of the M2 tide. According to Table 5, a strong dominance of mode 0 is usually found, and mode 1 dearly dominates among the baroclinic modes. In two cases, however, the energy of the first-order mode is larger than the energy of the barotropic mode (moorings no. 229-1 and 230-1). These two moorings were in place at the same time in late 1978, both to the north of the Azores Fracture Z o n e ( D a n d E in Fig. 1). One might speculate that bottom topography in that area was responsible for the generation of baroclinic tides and an increased transfer of energy from barotropic to baroclines waves. Harmonic constants and tidal ellipse parameters for the barotropic M2 tide are given in Table 6, and the ellipses (excluding moorings no. 229-1 and 230-1) are compared to the results obtained by C a r t w r i g h t et al. [1980] and S a u n d e r s [1983] in Fig. 4.
At positions in the Northern Canary Basin, to the southeast of the Azores (A, B and C), we find the major axes of the current ellipses oriented towards the northeast, with clockwise rotation. While the major axis orientation is similar to that in the Iberian Basin, to the east of the Azores (D), anti-clockwise motion is found there. At the two positions north of the Azores Rise (F and G) the major axes are oriented approximately east-west, with anti-clockwise rotation. The trend in the regional variation of the major axis orientation is consistent with the results of C a r t w r i g h t et al. [1980] and S a u n d e r s [1983], but major axis amplitudes differ up to about 30%. The differences between the results of this study and S a u n d e r s? [1983] much higher values at position B ( N E A D S Site 1) are particularly noteworthy.
It will be attempted to differentiate between progressive and standing tidal waves by inter-comparing our current phases with the phases of surface elevation from the C a r t- w r i g h t et al. [1980] observations. In the case of progressive waves, current and surface elevation should be in phase, while standing waves will result in a 90 ~ phase shift. S a g e r [1963] proposed a classification of tides by allowing a deviation +1/16 of a period from the in-phase or the out-of-phase condition to define progressive or standing waves, respectively, with all other waves being of mixed type. Cotidal lines and corange lines taken from C a r t w r i g h t et al. [1980] are included in Fig. 4. When interpolating linearly b e t w e e n cotidal lines and directly comparing the phase information of the coastal gauges given in the figure with the phase data obtained here from current meter data (standard deviation +0.4 h) progressive waves are found in most cases. Only at site G is a mixed-type M2 wave determined.
14 Dt. hydrogr. Z.38, 1985. H. 1. D i c k et al., Barotropic tides
3 5 ~ 3 0 ~ 2 5 o 2 0 ~ 1 5 "
4 0 6 0 8 0
\ \ \
k \ \
I I
100 5 ~ O"
I \ ~ ":'::::?
I I 6 0 ~ g
r \ \ \ ~ 3- 8 .i:~i:~ilili i.... !~
t
I \
I \
I \ ... :~37;2r
,...
.... 'NN
0.2
!)iiiiiiiiiiiiiiiiiiiii!i
_
I 5 " 10 ~ 5 D 0 ~
I , ~ / I .
/ 2.1
13
/
/%
.5, /
3 5 ~ W 3 0 ~ ~ 5 ~ g 0 ~
I ~ 1 cm s -1
Fig. 4. M2 tidal ellipses from this study (heavy lines) with c denoting clockwise rotation and from C a r t w r i g h t et al. [1980] and S a u n d e r s [1983] (narrow line). Phase values of moored data and coastal data give hours referred to equilibrium tide maximum at Greenwich. Cotidal lines (solid)
and corange lines (broken, in millibar) were taken from C a r t w r i g h t et al. [1980]
D t . h y d r o g r . Z . 3 8 , 1985. H . 1. D i c k et al., B a r o t r o p i c tides 15
T a b l e 2
Mz tidal ellipse data and energy partition on modes for Brunt-Vfiisfilii frequency 57 obtained from mean density data and N* from density data of one individual CTD station at the site (Mooring no. 264-1 (Site B)). East velocity component u, north velocity component v, major ellipse axis 2a, minor axis 2b, angle 0 counted clockwise between north and major axis, and phase to with respect
to equilibrium tide maximum at Greenwich. Negative sign of b denotes clockwise rotation H a r m o n i c c o n s t a n t s Tidal ellipse
H g a b 0 to
c m s -I c m s -1 c m s -I h
u : 1.5 29.2 ~
57 2.6 - 0 . 2 34.5 ~ 0.8
v : 2.2 22.1 ~
u : 1.5 23.5 ~
N* 2.6 - 0 . 0 36.5 ~ 0.8
v : 2.1 23.2 ~
K i n e t i c e n e r g y p a r t i t i o n
M o d e M o d e M o d e M o d e
0 1 2 3
57 7 6 . 9 % 1 6 . 9 % 4 . 8 % 1 . 4 %
N* 8 3 . 0 % 1 2 , 5 % 3 . 2 % 1 . 3 %
T a b l e 3
M2 ellipse data and energy partition with mooring motion considered (Mooring no. 264-1 (Site B)), and deviations from respective data for N in Table 2. For symbols see Table 2
H a r m o n i c c o n s t a n t s Tidal ellipse
H g a b 0 to
c m s -1 c m s -1 c m s -1 h
values a n d u : 1.3(+0.2) 40.7~ ~
d e v i a t i o n s 2.4(+0.2) -0.3(+0.1) 33.2~ ) 1.1(-0.3)
v : 2.0(+0.2) 26.60(-4.5 ~
K i n e t i c e n e r g y p a r t i t i o n
M o d e M o d e M o d e M o d e
0 1 2 3
57 7 4 . 2 % 1 8 . 9 % 5 . 7 % 1 . 2 %
16 D t . h y d r o g r . Z . 3 8 , 1985. H . 1. D i c k et al., B a r o t r o p i c t i d e s
T a b l e 4
M2 ellipse data for two selected moorings and deviations between actual Mz tidal currents and Gaussian fit using modes 0 to 3. For symbols see Table 2
H a r m o n i c c o n s t a n t s T i d a l ellipse I d e n t i f i c a t i o n
H g a b 0 to
c m s -1 c m s -1 c m s -1 h
264-1 u : 1.5 • 0.3 29.20 •
2.6 - 0.2 34.5 ~ 0.8
+0.4 • • •
(Site B ) v : 2.2 • 0.5 22.10 • 12.30
242-1 u : 2.3• 34.90• ~
3.2 1.5 38.2 ~ 2.3
• • • •
( S i t e D ) v : 2.6• 87.1~177 ~
T a b l e 5
Energy partition on modes for M2 tide
I d e n t i f i c a t i o n K i n e t i c e n e r g y p a r t i t i o n N u m b e r o f M o d e 0 M o d e i M o d e 2 M o d e 3 d e p t h s
S i t e A 278-1 9 5 . 5 % 3 . 4 % 0 . 7 % 0 . 4 % 6
Site B
184-1 9 3 . 7 % 6 . 2 % 0 . 1 % - - 4
264-1 7 6 . 9 % 1 6 . 9 % 4 . 8 % 1 . 4 % 6
~ 6 - 1 7 8 . 9 % 1 8 . 2 % 0 . 1 % 2 . 8 % 6
~ 6 - 2 9 0 . 8 % 9 . 0 % 0 . 2 % - - 5
~ 6 - 3 7 7 . 5 % 1 6 . 1 % 6 . 4 % - - 5
277-2 64.0 % 29.3 % 4.2 % 2.5 % 6
Site C
277-3 72.3 % 26.9 % 0.8 % - - 5
203-1 7 1 . 5 % 2 8 . 5 % 0 . 0 % - - 4
Site D 229-1 46.4 % 53.6 % - - - - 3
242-1 6 6 . 4 % 3 3 . 6 % - - - - 3
Site E 230-1 32.7 % 67.3 % - - - - 3
Site F 266-2 81.0 % 19.0 % - - - - 3
Site G 265-2 85.5 % 14.5 % - - - - 3
D t , h y d r o g r . Z. 38, 1985. H . 1. D i c k et al., B a r o t r o p i c tides 17
T a b l e 6
Harmonic constants and tidal ellipse parameters for barotropic M2 fide.
For symbols see Table 2
H a r m o n i c c o n s t a n t s Tidal ellipse Identification
H g a b 0 to
e m s -~ c m s -~ e m s - t h
Site A 278-2 u : 2.0 11.6 ~ 2.8 - 0 . 2 43.6 ~ 0.2
v : 2.1 2.9 ~
Site B
184-1 u : 1.2 36"0~ 1.7 - 0 , 1 46.4 ~ 0.8
v : 1.2 19.7 ~
264-1 u : 1.5 29'2~ 2.6 - 0 . 2 34.5 ~ 0.8
v : 2.2 22.1 ~
276-1 u : 2.2 63"1~ 2.7 - 1 . 1 52.2 ~ 1.6
v : 1.9 17.5 ~
276-2 u : 2.0 29"2~ 2.8 - 0 . 7 45.5 ~ 0.5
v : 2.0 0.5 ~
276-3 u : 2.5 41"3~ 3.0 " 1 . 0 52.4 ~ 0.9
v : 2.0 1.6 ~
Site C
277-2 u : 1.5 39"8~ 2.2 - 0 . 3 42.8 ~ 1.1
v : 1.6 26.5 ~
277-3 u : 2.0 51"7~ 3.0 - 0 . 8 38.9 ~ 1.2
v : 2.4 22.1 ~
Site D
203-1 u : 1.3 55"8~ 1.9 0.2 40.8 ~ 2.1
v : 1.5 66.0 ~
229-1 u : 1.2 343:2 ~ 1.2 0.1 75.20 11.9
v : 0.3 7.9 ~
242-1 u : 2.3 34"9~ 3.2 1,5 38.2 ~ 2.3
v : 2,7 87.1 ~
Site E 230-1 , u : 0.7 344"9~ 0.8 0.4 68.1 ~ 12.3
v: 0.4 44.2 ~
Site F 266-2 u : 1.8 42.1 ~ 1.9 0.3 71.4 ~ 1.6
v : 0.7 72.1 ~
Site G 265-2 u : 2.0 60"0~ 2.0 0.3 103.9 ~ 2.0
v : 0.5 210.9 ~
18 Dt.hydrogr. Z.38, 1985. H. 1. D i c k et al., Barotropic tides
35~ 3 0 o 25~ 2 0 ~
1 5 0 ~ - -
1 5 o -tO o 5 a 0 o
~ ' " " ~ " ~ - : ~ : i : ! : ! : : i : i : : - i ~
~!iiii!iii!iiiiiii ~ ~iiiiiiii!ii!~i
\
~ ~ :::~:~:~:~:[:!:!:~:[:~:i:~:~:~:~:i" ~ ~
~4
I I
t\
\ \
\\l
(
!:iil iiiiiiiiiiii '
15 ~ 10 ~
5 o 0 o1 2 0 ~
9 0 c
2 0 3 0
\ \ \
\ \
4.2 I
I/ / -
I I ~.~ I
\ , / ,
/ 2 . 0 [
~~ / ~
' i
2 . 2
3 5 ~ W 3 0 ~ 2 5 o 2 0 ~
I I 0 . 3 c m s - 1
Fig. 5. $2 tidal ellipses from this study with c denoting clockwise rotation. Phase values of moored data and coastal data give hours referred to equilibrium tide maximum at Greenwich. Cotidal lines
(solid) and corange lines (broken, in millibar) were taken from C a r t w r i g h t et al. [1980]
Dt. hydrogr. Z. 38, 1985. H. 1. D i c k et al., Barotropic tides 19
R e s u l t s for t h e $2 tide a r e g i v e n in Fig. 5 a n d T a b l e s 7 and 8. D u e t o l o w e r a m p l i t u d e s c o m p a r e d to M2, t h e r e l a t i v e e r r o r s a r e l a r g e r h e r e , a n d t h e sense of r o t a t i o n is t h e r e f o r e u n c e r t a i n in this case. E x c e p t for m o o r i n g 229-1, t h e z e r o - o r d e r m o d e d o m i n a t e s . Ellipses are o r i e n t e d t o w a r d s the n o r t h e a s t t h r o u g h o u t t h e area. N o c u r r e n t ellipse d a t a f r o m e a r l i e r analyses are a v a i l a b l e for c o m p a r i s o n o f $2. W i t h i n t h e accuracy a v a i l a b l e h e r e (_+ 0.3 h) all w a v e s a r e f o u n d to b e progressive.
A s e x p l a i n e d e a r l i e r , a v e r t i c a l a v e r a g i n g s c h e m e was u s e d for d e t e r m i n i n g diurnal b a r o t r o p i c tides. R e s u l t s for K1 a r e p r e s e n t e d in Fig. 6. M a j o r axis o r i e n t a t i o n varies considerably. P r o g r e s s i v e w a v e s are f o u n d h e r e e x c e p t f o r site A w h e r e a m i x e d t y p e of t i d e is d e t e r m i n e d . B e c a u s e o f large r e l a t i v e e r r o r s d u e to small a m p l i t u d e s , O1 tides will n o t b e c o n s i d e r e d in this study. W e c o n c l u d e t h a t t h e analysis p r o v i d e s tidal c u r r e n t i n f o r m a t i o n t h a t can b e u s e d for t h e testing of n u m e r i c a l m o d e l s o f t h e M2, $2 a n d K1 tide in t h e N o r t h e a s t A t l a n t i c . D e v i a t i o n s are f o u n d f r o m t h e results of C a r t w r i g h t et al. [1980]
a n d S a u n d e r s [1983] w i t h r e s p e c t to M2 tidal c u r r e n t a m p l i t u d e s , b u t a similar c u r r e n t p a t t e r n is o b t a i n e d .
T a b l e 7
Energy partition on modes for $2 tide Kinetic energy partition Identification
Mode 0 M o d e 1 M o d e 2 Mode 3
Site A 278-2 87.9 % 10.7 % 1.1% 0.3 %
Site B
184-1 88.6 % 11.4 % 0.0 % - -
264-1 49.3% 14.7% 28.6% 7.4%
276-1 84.5 % 13.2 % 0.1% 2.2 %
276-2 91.9% 8.0% 0.1% - -
276-3 47.8 % 41.9 % 10.3 % - -
277-2 91.9 % 1.4 % 4.9 % 1.8 %
Site C
277-3 95.8 % 3.7 % 0.5 % - -
203-1 86.5 % 13.5 % 0.0 % - -
Site D 229-1 42.0 % 58.0 % - - - -
242-1 65.8% 34.2% - - - -
Site E 230-1 67.6% 32.4% - - - -
Site F 266-2 74.7 % 25.3 % - - - -
Site G 265-2 57.7 % 42.3 % - - - -
20 Dt. h y d r o g r . Z. 38, 1985. H . 1. D i c k et al., B a r o t r o p i c tides
T a b l e 8
Harmonic constants and tidal ellipse parameters for barotropic $2 tide.
For symbols see Table 2
H a r m o n i c c o n s t a n t s Tidal ellipse Identification
H g a b 0 to
c m s -1 c m s -1 cm s -1 h
Site A 278-2 u : 0.6 34.3 ~ 0.9 0.0 40.1 ~ 1.3
v : 0.7 39.9 ~
Site B
184-1 u : 0.5 57"4~ 0.6 0.0 52.3 ~ 2.0
v : 0.4 63.0 ~
264-1 u : 0.5 61"6~ 0.6 - 0 . 0 56.0 ~ 2.1
v : 0.3 61.0 ~
276-1 u : 0.5 41"0~ 0.7 0.3 37.3 ~ 2.3
v : 0.6 84.3 ~
276-2 u : 0.6 52"5~ 0.8 - 0 . 0 46.2 ~ 1.7
v : 0.6 51.4 ~
276-3 u : 0.7 71"2~ 0.8 - 0 . 3 48.0 ~ 1.7
v : 0.6 25.0 ~
Site C
277-2 u : 0.6 58"5~ 0.8 - 0 . 1 49.2 ~ 1.8
v: 0.6 45.5 ~
277-3 u : 0.5 78"5~ 0.9 - 0 . 1 33.6 ~ 2.2
v : 0.7 39.9 ~
Site D
203-1 u : 0.5 75"7~ 0.8 0.1 41.1 ~ 2.9
v : 0.6 96.5 ~
229-1 u : 0.4 51"9~ 0.5 0.2 58.4 ~ 2.1
v : 0.3 91.3 ~
242-1 u : 1.5 55"7~ 1.5 - 0 . 5 79.8 ~ 1.7
v : 0.6 347.9 ~
Site E 230-1 u : 0.6 100"7~ 0.9 0.0 48.6 ~ 3.4
v : 0.6 101.8 ~
Site F 266-2 u.' 0.4 129.6 ~ 0.6 - 0 . 0 40.7 ~ 4.2
v : 0.5 125.0 ~
Site G 265-2 u : 0.6 107"7~ 0.6 0.2 71.5 ~ 3.8
v : 0.3 153.5 ~
Dt. hydrogr. Z. 38, 1985. H. 1. D i c k et al., B a r o t r o p i c tides 21
35~ 30 ~ 250 Z O ~
8 ~
9.2 ? " " , !
6! / "
1 2 0 ~ - - \ ~
7.3 , \ ~
1 0 5 ~
9 0 . / ~
.~=. \ \
75 o
60~ ~/
35 ~ W 3 0 ~
3.5 o " i 0 o 50 0 ~
. / ~ . ~e s i - x + :
.: .. .! .iiii!iigi!iiiii.iiiiiN
25~ 2 0 o 15o I 0 o 5 o Or,
i I 0 . 1 c m s - I
Fig. 6. K~ tidal ellipses from this study with counter-clockwise rotation. Phase values of m o o r e d data and coastal data give hours referred to equilibrium tide m a x i m u m at Greenwich. Cotidal lines
(soIid) and corange lines (broken, in millibar) were t a k e n from C a r t w r i g h t et al. [1980]
22 Dt. hydrogr. Z. 38, 1985. H. 1. D i c k et al., Barotropic tides
A c k n o w l e d g e m e n t s
T h e authors h a v e b e n e f i t e d f r o m discussions w i t h H a r t m u t P e t e r s o n this subject. T h e study was partly f u n d e d by t h e D e u t s c h e F o r s c h u n g s g e m e i n s c h a f t , B o n n .
References
A c c a d , Y. and C. L. P e k e r i s , 1978: Solu- tion of the tidal equations for the M2 and $2 tides in the World Oceans from a knowledge of the tidal potential alone. Philos. Trans.
Roy. Soc. (A) 290, 235-266.
B a k e r , D . J . , 1 9 8 1 : Ocean instruments and experiment design. In: W a r r e n , B. A.
and C. W u n s c h (eds.), Evolution of physi- cal oceanography. Cambridge, Mass.: M I T Pr. pp. 396-433.
C a r t w r i g h t , D. E., A. C. E d d e n , R.
S p e n c e r et al., 1980: The tides of the Northeast Atlantic Ocean. Philos. Trans.
Roy. Soc. (A) 298, 87-139.
D i c k s o n , R. R., 1983: Global summaries and intercomparisons: Flow statistics from long- term current meter moorings. In: R o b i n - s o n , A. R.: Eddies in marine science. Berlin [usw.]: Springer. pp. 278-353.
D o o d s o n , A. T., 1941: Admiralty manual of tides. London: His Majesty's Stationary Office.
F o f o n o f f , N. P., 1966: Oscillation modes of a deep-sea mooring. Geo-mar. Technol. 2, 13-17.
G o d i n , G., 1972: The analysis of tides. Liver- pool: Liverpool University Press. 264 pp.
G o u l d , W. J. and W. D. M c K e e , 1973:
Vertical structure of semi-diurnal tidal cur- rents in the Bay of Biscay. Nature. 244, No. 5411, 88-91.
H e n d e r s h o t t , M. C., 1977: Numerical mo- dels of ocean tides. In: G o 1 d b e r g, E. D., I. N. M c C a v e , J. J . O ' B r i e n et al.
(eds.), The sea. Vol. 6: Marine modeling.
New York: Wiley. pp. 4 7 - 9 5 .
H e n d e r s h o t t , M. C., 1981: Long waves and ocean tides. In: W a r r e n , B. A. and C.
W u n s c h (eds.): Evolution of physical oce- anography. Cambridge, Mass.: M I T Pr. pp.
292-341.
K r a u s s, W., 1966: Methoden und Ergebnisse der theoretischen Ozeanographie. Band 2:
Interne Wellen. Stuttgart: Borntraeger.
248 pp.
M e i n c k e , J., G. S i e d l e r and W. Z e n k , 1975: Some current observations near the continental slope off Portugal. " M e t e o r "
Forsch.-Ergebn. (A) No. 16, 15-22.
M ii 11 e r , T. J., 1981: Current and temperature measurements in the North-East Atlantic during N E A D S . Bet. Inst. Meeresk. Kiel.
No. 90, 100 pp.
M i i l l e r , T. J. a n d W . Z e n k , 1983: Some Eulerian current measurements and XBT-sta- tions from the North East Atlantic, October 1980 - March 1982 - A data report. Ber.
Inst. Meeresk. Kiel. No. 114, 145 pp.
M u n k , W. H. and D. E. C a r t w r i g h t , 1966: Tidal spectroscopy and prediction. Phi- los. Trans. Roy. Soc. (A) 259, 533-581.
P r i c e , J. M., 1983: Historic hydrographic and meteorological data from the North Atlantic and some derived quantities. Ber. Inst. Mee- resk. Kiel. No. 117, 91 pp.
R e g a l , R. and C. W u n s c h , 1973:M2 tidal currents in the western North Atlantic. Deep- Sea Res. 20, 493-502.
S a g e r , G.: 1963: Atlas der Elemente des Tidenhubes und der Gezeitenstrrme fiir Nordsee, Kanal und Irische See. Rostock:
Deutsche Akademie der Wissenschaften zu Berlin, Inst. Meeresk. 59 pp.
S a u n d e r s, P. M., 1983: Benthic observations on the Madeira Abyssal Plain: Currents and dispersion. J. Phys. Oceanogr. 13, 1416-1429.
S c h o t t , F., 1977: On the energetics of barocli- nic tides in the North Atlantic. Ann. Geo- phys. 33, 4 1 - 6 2 .
S c h w i d e r s k i , E. W., 1980: On charting glo- bal ocean tides. Rev. Geophys. Space Phys.
1 8 , 2 4 3 - 2 6 8 .
U n e s c o , 1975: A n intercomparison of open s e a tidal pressure sensors. Report of S C O R working group 27: "Tides of the open sea."
U N E S C O Techn. Pap. Mar. Sci. No. 21, 67 pp.
Eingegangen am 7. M/irz 1985 A n g e n o m m e n am 18. Mfirz 1985 Anschrift der Verfasser:
Dipl. Oz. Gerhard Dick, Dr. Gerold Siedler
Institut ffir Meereskunde an der Universitgt Kiel, Diisternbrooker Weg 20, 2300 Kiel 1
