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Comparison of M2 tidal currents observed by some deep moored current meters with those of the Schwiderski and Laplace models
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0198-0149/91 S3.00 + 0.00 1(" J 991 Pergamon Press pic
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Comparison of M z tidal currents observed by some deep moored current meters with those of the Schwiderski and Laplace models JAMES R. LUYTEN* and HENRY M. STOMMEL * (Rt'uiVf'd 12 May 19R9; in revised form 31 May 1990; accepted 7 June 1990)
Abstract-M 2 tidal currents observed at 315 deep-sea locations (mostly in the northern hemi'phcrc) confirm thO\c computed by the numerical model of SCHWIDERSKI (1979. Naval Surface Weapons Center. Dahlgren. VA. TR 79-414). Together. they reveal large-scale geographical cohcrcncc!> similar in form to thosc of the tide-generating potential and to those of the analytical Laplace model for tidcs on a globe completely covered with water of uniform depth. By fitling the Laplace tidal ellipses to the array of current meters and to the velocities from Schwidcr~ki\ numerical model. we show that much of the Mz tide is explicable in terms of Laplacc'» simple model-with friction.
INTRODUCTION
IDEALLY a map of the global distribution of deep-ocean M2 tidal currents would be constructed from analysis of observed current data. The distribution of such data is sparse, so to make maps we must depend upon numerical model calculations. SCHWIDERSKI (1979) developed such a model. In addition to publication of tabulations of one-degree surface elevation, he also produced tapes of the current harmonic constants. We decided to compare his results with M2 tidal current determinations from data in the archives of the Moored Current Meter Group of the Woods Hole Oceanographic Institution and a few published observations. The data largely confirm Schwiderski's model results, and seem to justify accepting them as globally representative. Inspection of the fields reveals large-scale geographical coherence suggestive of the presence of a signal corresponding to that expected from Laplace's simple analytical theory of the semidiurnal tide on a water-covered globe of uniform 4000 m depth despite the complexity of the bottom topography and form of the continental coast. One hint of this is that more than half of the 5-degree velocity vectors in the Schwiderski model at time of local lunar transit lie in the quadrant between eastward and equatorward. In the inviscid Laplace model these vectors are all eastward and lag the tidal force by 3 h. Introducing linear friction in the Laplace model reduces this lag. A least squares fit of the Laplace solution with the Schwiderski model, latitude by latitude, quantifies the result. To show that this correspondence is not only a model result hut also occurs in the real ocean. the Laplace solution is compared to observed data. The observed current data are rather sparse, but in the latitude band 25°-45°N there are • Woods lIole Oceanographic Institution. Woods Hole. MA 02543. U.S.A. S573
S574
J R
Lt:YT£N
and H M
STOMMEL
sufficient observed points in both the North Atlantic and Pacific to form an east-west array to which a Laplace solution can be compared. The array indeed does pick up the Laplace signal. OBSERVED M2 CURRENTS
The archives of current meter records at Woods Hole contain observations from the Atlantic. North Pacific. the Agulhas retroflection region and the equatorial Indian Ocean. We used the SCtlUREMAN (1924) tables for the astronomical data necessary to compute the M;! harmonic constants (amplitude and Greenwich phase of the eastward and northward velocity components). The tidal constituents for the velocity components are assumed to have the form lI(f)
= u"cos(a(I-III) -
vet)
= ""cos (a(t
gu -
- I,,) - R~'
Xl
- X).
where guo gv (lre the phase at Greenwich hour 0 and X is the astronomical argument ohtained from the Schureman tahles. The procedure for estimating the amplitude and phase of the M~ tidal component is al\ follows. Thc time series for cast and north velocities are each multiplied hy cos (0' I) and sin (a· t) lind summed over the record length. where (J = ~,./28.<:JMI042 rad h '. The phase guo = tan '«I4 .. )/(u\.» + the Schureman phase for the starting time of the record. The amplitude. U 11 : : v'u? + Only long suhsurface records were chosen. Records from hoth Model R5()s and Vector.."vcraging Current Meters (VACMs) were used. Altogether some 4X4 records were analysed on a personal computer. An attcmpt was madc to ohtain gcographic:11 c()vemgc. so in many cases there was insufficient sampling in the verticnl to attempt a vertical modal analysis tn separate the harotropic from the haroclinic modes. Upon inspection it appeared that shallow records tcnded to he contaminated hy haroclinic tidal currenll\ (Hendry. 1977). and so ,,11 data from less than WOO m depth were rejected. A very few records had questionable starting times. and these also were rejected. Those used arc identifted in Tahle I. In addition to our own estimates. dctcrmin"tions of the M., tidal current constants have heen extracted from the literature. mainly in the eastern North Atlantic. Some of the tidal constants from puhlished sources represent the hafOtropic results of " modal separation (DICK .lOd SIfOl.F.R, 19R5). Several more werc provided hy Dr Peter Saunders. The numher of records reported is 11 S. A representative scatter di .. gram sh()wing R257 successive hnurly values of mcasured current from record tl2tl5 is shown in Fig. 1. The outer circle has a radius of 10 em s I. The inner eircle represents the thrcl\hold of the current metcr. at 1.8 em s I. This circle is made up of data points swept up from the center anu assigned to the threshold radius hy the algorithm employed in processing (TARBF.LL ('I al .• 198M). Four uots inside the circle are artifacts of a manual editing procedure. The current meter is a Model 850. The intermediate circle is the r.m.s. mean radius of all recorded data points, 2.7 em s 1. The ellipse is the M2 tidal ellipse as dctermincd hy harmonic analysis of this data. The position of this meter was at 20.9"N, 41 }/'W, 4015 m depth. The diagram gives an idea of the amplitude of the background of all other signals to that of the M2 signal at this station. In Fig. 2 we show the semidiurnal part of the pcriouogfllm. including the lines M N S2' K:!. The M21ine stands well ahovc the noise level. and its amplitude has n ratio to t~;at ~f
II:.
Compa rison of M2 tidal currents
S575
TQb/~ 1. V~/oc;ty amplilud~s and phases for Iht M2 1itlal compon~nt Observed
Record
name 42211. 42221. 45011. 46521. 4912L 4914L 4902L
4912L 4913L
49141. 5063L
50741. 5093L 5233L 5245L 5246L 5247L
5261L 5262L
5433B S43SA
5456A 5454B 5467A 5485B
5486B 549lX 549SC 5602B
56631. 5665L 5673L S67SL 57318
5805L 5815L 59338 59348 59SJB 59S4A
59638 59648 5975D G021A 6263A
6265A 6304A 6315£ 6361A 6371A
lat.
long.
depth 61 (h)
39.0 290.0 1027 39.0 290.0 2495 39.2 289.5 1014 39.0 290.0 2487 39.1 290.0 1019 39.1 290.0 2550 39.4 290.0 2011 39.1 290.0 1019 39.1 290.0 2030 39.1 290.0 2550 39.4 290.0 1995 39.2 290.0 2006 39.1 289.5 1987 39.4 290.0 1991 39.1 290.0 1005 39.1 290.0 20ll 39.1 290.0 2512 38.8 290.0 2006 38.8 290.0 2810 2B.O 295.0 1002 28.0 295.0 4003 27.8 304.4 4004 27.8 304.4 1996 27.9 305.1 4011 31. 0 299.9 2001 31.0 299.9 4001 34.0 300.0 1002 34.0 300.0 4002 41. 5 305. a 3994 34.9 305.0 1005 34.9 305.0 4006 31. 6 304.9 ).028 31. 6 304.9 4030 41.5 305.0 4001 · 31. 6 305 . 1 3995 34.9 304.9 3995 -0.1 50.5 1500 -0.1 50.5 3545 1.5 53.0 1500 1.5 53.0 3542 0 . 0 57.0 1550
0.0
57.0 3595 53.0 3544
0.0 41.5 305.0 3993 26.9 318.8 1514 26.9 318.8 4015
27.9 311.4 1498 27.9 311.1 4016 4.0 320.3 4256 4.0 320.7 4104
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.25 0.25 0.25 0.25 0.25 0.25 0.25 1. 00 1.00 0.25 1. 00 0.2S 0.25 0.25 0.25 1. 00 1. 00 1.00 1.00 1.00 1. 00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.25 1.00 1.00 1.00 1.00
0.25 0.2S
length 2583 2582 2444 2457 4886 4885 4846 4886
4885 4885 4214
4234 4233 23008 22969 22961 22976 22865 22841 26001 S'02 5739 27145
5701 26624
20656 25609 25610 5143 5322 5322 5155 5155 7296
gu
110
8 93 6 72 6 102
14 116 11 74 9 93 11 81 11 74 8 9 9 9
9 12 9 8 9 9
82 93 74 94 102 96 83 91 90 84
7
80 11 99 9 101 13
9S
13
96
11
94
9 130 8 B8
10 9
IS
78 94 75 93 7S 84 79 69
7272 5569 1489 5521 3289
12 9 15 7 10 12 11 22 16 21 21
5377 5377
3 300 24 164
4825
23 152
26209 8257 8257 S088 8088 34680 34688
11 101 16 16
7368
86 73
162 148 162 152
Schwiderski
"0
gv
4 28 2 243 2 282 7
28
4 347 2 349
3 4 4 347
Uo 7
17
7
11 17
'1 6 '1 7 7
3S
7 7 7 7 '1
2 357 6 42
7 7
1 339 2 349 2 0
4 4
9
2
34
2 349 14 19 9 306 '7 314 15 340 3 2
20 328 14 331 6 7 4 284 9 300 7 335 8 299 9 338 7 310 17 336 10 296 7 261 10 324 9 313
13 122 19 14 13 4 13 13 7 28
16
35
14 9
66 77
140 lSI 122 65 88 137 276 345 25 304 22 323 9 320
B
32
17 279
11
14
23 292
gu
30 19 19 19 19 19
19 19 19 17
19
7 7
19
7 6 6 7
19 30 30 97
19
7
97
8
87 87 83 87 87
8 8 8
8 S 8
83
10
83 85
9
73
9
73
8
75
8 75 10 85 8 72 9 73 21 127 21 127 21 121
21 121 22 128 22 128 21 123
"0
gv
6 237 6 237 6 ~37 5271 7271 7271 7 271 7271 7271 7 271 7271
7271
6 237 7271 7271 7271 7271 5 271 5 271 9 286 9 286 14 307 14 307 14 308 10 305 10 30S 10 306 10 306 4 287 10 309 10 309 11 309 11 309
4 287 310
12 10 14 14 15 15 11 11
309 110 110 101 101 429 429
14 lOS
10
85
16 16 11
48
2S 315
48 62
11
62
7
20 20
25 19 19 18 18
7
4 287 315 312
312 298 298
Continued
S576
J. R.
LUYfEN
and H . M. STOMMEl.
Schwiderski
Observed Record
name
lat.
~ .
67338 67348 6744C 67531. 6761C 6762N 67630 6764C 67730 67748 67'5r 6776r 6784N 67858 6786k 67938 67941. 61941. 68031. 6804l 6811C 6813D 6831! 6832' 6842C 68430 6844l 6851C 69541. 69551. 69631. 6972C 6973B 6981B 6983A 6982e 6993C 6994C 70028 70031.
70138 701410 10158 70221. 70238 7033D 70431. ?044e 704510 71838 7184A
. .
100,. deptb 6, (h)
length
Uo
gw
"0
'"
9529 18 37 11 342 37.0 318.0 1526 1.00 9529 18 48 14 339 3'.0 318.0 4026 1.00 4 6 4023 9433 14 45 1.00 39.S 316.1 20 7 6' 1530 16 40.3 314.7 0.25 37636 39.0 316.0 4658 0.25 37444 19 42 13 339 39.0 316.0 4697 0.25 37444 19 44 13 3H 39.0 316.0 4747 0.25 37444 19 43 12 338 39.0 316.0 4779 0.25 37444 19 45 12 34J 39.0 315.9 1521 0.25 16036 20 63 11 11 39.0 315.9 4020 0.25 37444 18 4~ 11 333 39.0 315.9 4777 0.25 37348 17 45 10 333 39.0 315.9 4857 0.25 37348 16 44 10 329 6 346 9265 13 52 38.7 314.4 3989 1.00 6 337 38.7 314.4 4835 0.25 37060 17 45 6 340 38.1 314.4 4915 0.25 37060 17 44 38.0 313.4 1504 0.25 36868 18 69 11 357 3 243 38.0 313.4 4003 1.00 1969 10 35 1 6 3961 14 82 38 . 0 313.4 4001 1.00 1 38.9 313.1 1512 0.25 36868 15 71 5 8 350 38.9 313.1 4013 1.00 9217 15 57 -33.8 320.5 3925 0.25 10864 15 326 30 77 -33.8 320.5 4275 1.00 2714 13 337 24 79 2 27 83 2715 17 -30.9 320.3 3946 1.00 2723 -30.9 320.3 4295 1.00 9 348 14 73 -31.0 320.1 2990 0.25 10958 12 34 24 89 2691 16 43 25 88 -31.0 320.1 3490 1.00 2692 18 66 22 93 -31.0 320.1 3911 1.00 -30.6 320.3 3971 0.25 10607 12 350 29 59 :2 319 41.0 152.0 3990 0.25 31204 9 356 41.0 152.0 5077 0.25 31204 3 332 9 357 7 8 39.0 152.1 3995 0.25 31012 12 39 9 46 24 5 37.5 152.0 1201 0.25 30820 9 21 5 ]54 37.5 152.0 4001 0.25 30820 50 11 6 345 36.3 152.0 507 0.25 30436 8 31 4 32 36.3 152.0 4007 0.25 30436 9 45 J0436 6 27 36.3 152.0 1207 0.25 64 '7 4 76 35.0 152.0 1211 0.25 1,092 9 26 3 341 35.0 152.0 4015 0.25 30148 67 9 25924 0.25 39 1178 5 33.8 152.0 5 31 33.8 152.0 3919 0.25 29764 10 49 8 105 7 77 32.5 152.2 1200 0.25 29572 7 1 278 32.5 152.2 4000 0.25 29572 10 6340 10 357 1 287 32.5 152.2 5540 0.25 12 35 29380 4 303 31.3 152.1 1206 0.25 4 37 31.3 152.1 4006 0.25 29380 11 36 357 14 29092 4 249 0.25 30.0 152.0 3994 43 '7 28708 1 178 0.25 1202 28.0 151.9 47 16 28708 3 295 0.25 3984 151.9 28.0 6 333 28.0 151.9 S891 0.25 28708 19 42 4 15 8978 6 125 28.0 151.9 1198 1.00 7 23 21790 7 267 28.0 151.9 3984 0.25
"0
I"
16
42
17
48
17 17
55
10 343
10 343
42
16
17 17 17
'"
"0
8 7 8 8
51 51 51 51 50
358
372 356 356
8 356
8 356
50 52 52
8 345 8 345 8 345 8 345 7 142 1 342
15
52
7 342
14 14 14
S4 S4 54 55 55
e 8 8 7
18 18
17 17
36 36 36 36 36 36
20 20 20 20 20 20
8 3S'"
..
8 357 8 359 1 I)
4 5 ..
16 16 16 16 15 15
14
14 8 8
7 7 7 7 7 7
9 9 9
50
50
1 2 :2
9
2
10
~
10
S
10 10
9 9
11 11 11
8 8 8 10 ·
12 12 12 13 1)
13 13 13
10 11 16 16 16 16 16
7
311 331 331 338 338 436 436 443 443 443 Ul 443 .... 1 319 319 327 329 329 131
4 4 4 1Jl
.. 131
.. 332 4 3J2 4 J45 4 345 3 131 3 331 3 131
326 326 317 320 320 320 :2 320 2 320
3 3 2 2 2 2
SS77
Comparison of M2 tidal currents
Table 1. Conrinued Observed
Schwiderski
Record name 71848 71850 71928 7193C 72138 12148 721SC 7222C 72438 72448 7245C 7253C 7263C 72528 72628 7272A 72838 72848 72851 72911 7292A 72928 72938 73010 73028 73038 7311D 73128 7313A 732lA 7322C 7323C 73418 7342A 77528 7764A 77658 7766A 77718 77728 77818 77918 7771Q 7804B 78058 7806B 78228 78238 7832A 78428 78438
lat.
long. depth dl (h)
length
"0
gu
"0
5 28.0 151.9 3984 0.25 10110 23 28.0 151.9 5856 0.25 35908 22 10 8977' 6 80 30.0 152.0 1202 1.00 30.0 152.0 4002 0.25 35909 21 356 8978 8 115 32.5 152.1 1214 1.00 32.5 152.1 4015 0.25 35908 13 340 32.5 152.1 5556 0.25 35909 16 336 8977 10 27 33.8 152.0 1198 1.00 8953 11 42 34.9 152.0 1204 1.00 9 356 34.9 152.0 4004 0.25 35812 9 345 34.9 152.0 5958 0.25 35813 8 358 36.3 152.0 3996 0.25 35812 37.5 152.1 3987 0.25 35813 11 355 8954 11 16 36.3 152.0 1196 1.00 8538 5 21 37.S 152.1 1186 1.00 1726 10 340 39.0 152.1 1222 1.00 8905 7 14 41.0 152.0 1237 1.00 41.0 152.0 4037 0.25 35616 10 324 41.0 152.0 5216 0.25 35621 10 321 51.0 185.1 2010 1.00 10010 22 204 804l 21 243 51.0 185.1 3010 1.00 1464 18 222 51.0 185.1 3010 1.00 8400 20 241 51.0 185.1 4510 1.00 50.5 185.2 1986 1.00 10033 16 231 50.5 185.2 2986 1.00 10033 20 241 50.5 185.2 4486 0.25 40132 25 240 49.4 185.2 2060 1.00 10033 14 252 49.4 185.2 3060 1.00 10033 20 10 6473 20 222 49.4 185.2 4560 1.00 3460 25 244 47.9 185.2 2006 1.00 47.9 185.2 3005 1.00 10033 21 250 2194 23 236 47.9 185.2 4505 1.00 46.0 185.2 2002 1.00 10033 16 220 4273 18 234 46.0 185.2 3002 1.00 41. 2 297.0 1390 1.00 12169 12 142 40.5 298.0 1512 0.25 48773 7 130 40.5 298.0 4011 1.00 12193 10 88 40.5 298.0 4789 0.25 48773 11 92 40.2 298.4 4002 1.00 7321 13 119 40.2 298.4-4871 0.25 48676 15 119 40.7 298.5 4002 1.00 12577 13 111 40.9 299.3 4005 0.25 48676 9 64 40.2 298.4 4002 1.00 12169 14 113 39.5 299.7 1009 1.00 12193 3 99 39.5 299.7 1506 0.25 48772 4 111 39.5 299.7 4004 1.00 12193 11 102 37.5 301.7 1386 1.00 12193 8 68 37.5 301.7 4008 0.25 48772 12 83 35.6 299.1 4023 0.25 48773 5 75 36.5 296.9 1407 1.00 12169 7 117 36.5 296.9 4001 0.25 48676 7 99
g"
B 263 6 289 5 63 9 270 8 69 3 273 4 241 3 345 5 20 2 278 3 289 2 7 3 300 5 331 3 39 4 322 5 5 3 282 2 275 7 37 7 228 2 23 6 257 4 318 14 222 14 201 9 274 4 355 3 131 12 192 12 207 7 156 4 119 6 158 5 332 4 296 8 306 8 315 5 17 7 13 6 309 10 262 7 2 5 266
5 278 8 339 8 308 8 305 5 283 3 345 5 316
Uo
gu
13 13 12 12 11 11 11 10 10 10 10
16 16 11 11 8
9 9 9 9
2
"0
8 8 9 5 5 5
1 2
1
8 359 8 8 8 16 16 16 16 16 16 16 17 17
357 357 357 222 222 222 222
222
222 222 230 230 17 230 14 236 14 236 14 236 13 238 13 238 16 137 10 139 10 139 10 139 B 132 B 132 8 132 8 123 8 132 7 115 7 115 7 115 B 88 8 88 7 89 5 90 5 90
g"
2 320 2 320 2 317 2 317 3 331 3 331 3 331 4 345 4 332 4 332 4 332 4331 4 329 4331 4 329 5 327 4 314 4 314 4314 3 284 3 284 3 284 3 284 3 284 3 284 3 284 3 263 3 263 3 263 2 233 2 233 2 233 4 207 4 207 8 334 6 326 6 326 6 326 5 316 5 316 5 316 5 306 5 316 5 308 5 308 5 308 7 308 7 308 7 306 6 306 6 306
Continued
S578
J . R.
LU'YTIN
TDb/~
Record name 7863A 7933A 7943A 7953A 79818 79918 80138 B0238 S0338 81539 B163A 811lA 81838 81939 82139 82238 82338 82438 82538 82738 8296L 8298L 8306L 830BL 8313L 8314L 83438 8344C 8353A 83548 83618 8364C 83738 8374C 8383A 83848 8393A 83948 8404C 84138 84148 84238 84024A 84338 8434A
and H. M.
STOMM£I.
1. Continutd Schwiderski
Observed
lit.
Ion,. depth lJt (h)
37.4 295.0 4002 39.0 208.0 4018 35.0 208.0 4016 41.0 185.0 3996 ll.0 184.9 3993 35.3 175.0 4007 41.1 165.0 4019 39.0 164.9 4004 37.0 165.0 4031 31.0 164.9 4021 33.0 165.0 3994 35.0 165.0 3995 37.0 165.0 4003 39.0 165.0 4005 41.1 165.0 4020 39.0 175.0 3998 35.3 174.9 4017 31.0 184.9 3996 35.0 185.1 4006 41.0 185.0 3986 2B.0 290.0 1062 28.0 290.0 4062 25.5 290.0 1059 25.S 290.0 4059 32.0 335.9 1071 32.0 335.9 2971 -38.0 15.S 1493 -38.0 15.5 3993 -40.1 16.6 1409 -40.1 16.6 3909 -42.0 17.8 1446 -42.0 17.8 3947 -40.1 19.7 1593 -40.1 19.7 4092 -38.0 18.5 1497 -38.0 18.5 3996 -37.9 21.1 1451 -37.9 21.1 3951 -38.6 23.1 4055 -37.2 23.0 1496 -37.2 2l.0 3995 -35.9 27.0 1512 -35.9 27.0 4011 -35.0 26.0 1505 -35.0 26.0 3503
length
0.25 0.50 0.50 0.25 1.00 1.00 1.00 1.00 1.00 0.25 0.25 0.25 0.25 0.25 0.25 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1. 00 1. 00 0.50 1. 00 0.50
1.00 0.50 1.00 0.50 1.00 0.50 1.00 O.SO 1.00 1.00 0.50 1.00 O.SO 0.50 0.50 0.50
gil
"0
48677 33842 33842 34S65 8353 7849 7489 7441 7368 37636 37541 37253 36964 36868 36580 8977 8929 8641 8617 8545 14329 14329 14185 6408 17113 17113 34622 17305 34515 17257 34514 17257 34226 17113 33843 16923 33555 4225 16777 33458 16729 33218 33219 33122 33123
7
"0
99
13 219 11 206 16 256 17 247 8 301
9 360 2 271 8 299 13 343 5 13 10 350 5 350
2 334 14 13 9 286 11 303 18 265 14 313 13 267 9 134 4 150
6 138 10 120 28 34 12 27 3 352 2 310 3 1
4 322 5 318 3 289
2 2 4 4 1 5 4
19 332 353 338 26 19 357 2 288 4 lS3 4.
11
2 292 J 5 3 356
gv
S 335 21 175 17 154
5
6 16 16 7 13 10 5
92
9 222 8 233 12 246
8 194
12 197 '} 217 7 274 1 32 6 212 B 248 3 262 7 250 5 257 2 350 11 289 4 249 13 224 12 219 15 237 5 210 7 343 4 257 7 311 7 313 27 348 24 31 1 359 3 286 5 328 5 320 8 335 8312 5 327 4317 4 354 4 341
4 318 3 336 5 325 9 332 6317 4 357
7 313 3 354 5 321
11 265 10 287 10 317 10 324 10 321 9341 9 336 9 333 10 321 10 324 10 317 11 278 10 289 11 265
12 258 12 248 6 103 6 103 7 116 7 116 19 18 19 18 3 307 3 l07 2 305
2 305 1 296 1 296 43 43 1 8 1 8 3 55 3 55 2 59 2 45 2 45 3 76 J 76 1 76 1 76
o o
317 174 173
197 200 211 243 6 236 6 235 6 233 6 234 6 235 6 215 6 236 5 243 8 213 10 214 13 200 12 198 7 198 8 265 8 265 8264 8 264 24 348 24 348 6 325 6 325 7 320 7 320 8 317 8 317 7 308 7 J08 6 303 6 J03 4 284 4 284
6 5 5 5 5
301 302 l02 257 257
5 257 5 257
Polymode Clu.ter C Hoorlnga(P.Nil1er, T917L T918L
15.2 306.8 2508 15.2 306.8 4008
1.00 1.00
8471 8471
11 161 14 194
20 20
4 :2
12 120 12 120
23 309 23 309
("o,.,i"II('(/
S579
Comparison of M2 tidal currents
Table 1.
Continued
Schwiderski
Observed Record
name
tat.
long. depth 61 (h)
length
gu
Uo
"0
g"
Uo
gu
gv
"0
Pequod Mooring8(C. Eriksen, J. Richman) PI02$ P103$ P202$ p203$ E103$ £105$ £203$ £205$ 0107$ 0109$ 0111$ 0205$ 0207$ 0209$ 0211$ U105$ UI07$ UI09$ U111$ U205$ U207$ U209$ U211$ 0103$ 010S$ 0203$ 0205$ 0102$ 0202$ 0203$
0.0 212.1 1526 0.0 212.1 3026 0.0 212.1 1465 0.0 212.1 2992 -0.1 214.9 1565 -0.1 214.9 3069 0.0 215.0 1441 0.0 215.0 2953 0.0 215.3 1615 0.0 215.3 3015 0.0 215.3 3115 0.0 215.3 1482 0.0 215.3 1582 0.0 215.3 2982 0.0 215.3 3082 0.3 215.5 1492 0.3 215.5 1592 0.3 215.5 2992 0.3 215.5 3092 0.3 215.5 1496 0.3 215.5 1596 0.3 215.5 2996 0.3 2l5.5 3096 -0.0 216.0 1554 -0.0 216.0 3051 0.0 216.0 1453 0.0 216.0 2974 0.0 221.9 1547 0.0 222.0 1445 0.0 222.0 2958
1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
9834 8694 8795 8796 9955 9805 8987 8987 8834 8802 9248 6592 3600 9283 9271 9095 9037 9199 9198 8979 8979 8979 8979 8340 9295 9271 9271 8739 9270 9270
11 315
30B 321 313 315 320 315 351 11 317 13 313 12 309 11 304 11 321 12 311 10 309 10 324 12 306 11 307 11 308 12 316 13 318 12 329 10 314 12 328 12 316 11 310 10 298 B 353 7 326 6 324
12 15 11 13 8 15 11
5 15 4 347 6 20 3 344 7 5 2 352 3 352 4 87 4 25 3 42 4 36 4 337 4 31 4 12 4 31 2 356 3 336 3 341 4 336 3 347 2 348 2 24 2 12 3 12 3 87 2 30 3 21 4 92 3 62 3 70
15 289 15 2B9 15 289 15 289 15 292 15 292 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 295 15 295 14 295 14 295 12 300 12 300 12 300
4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
333 333 333 333 344 344 360 360 360 360 360 360 360 360 360 360 360 360 360 360 360 360 360 3 357 3 357 2 373 2 373 4 421 4 421 4 421
the N2 tide that corresponds to the ratio of the amplitudes of the astronomical force assigned to these two frequencies, as might be expected for two components of the force with such close frequencies and identical form in latitude and longitudinal wavenumber. The clarity of the M2 signal is superior, however, and for this reason (as well as the fact that gridded current values of the Schwiderski model are available only for the M2 component) we have limited most of our analyses to the M2 component. The K2 and S2 cannot be separated. COMPARISON OF OBSERVED CURRENTS WITH THOSE OF THE SCHWIDERSKI MODEL
Figures 3-6 exhibit the consistency ofthe M2 tides as determined (a) from current meter data with those (b) determined by the Schwiderski model. Small open circles indicate
S580
J. R . LUYTEN and II . M. SroWMU.
Fig. 1. Scatter diagram of H2:-;7 hourly readlOltli for Model X~o current meter numher fl2MA. The outer circle i'l 10 cm 'i I . The inner Circle 110 the thrc'ihold ve\C)C1ty of \ .Hem" 1 hencath which the rotor doe!! not re5pond . It cOo!"!lt~ nf (l~rvatl(Jn!l ",.hcl'iC direction I' mea.. ured . hut ",.hme amplitude is set arhitranly at I Hem ,, - I . The clhJl"C 1'1 the eC1mputed M ~ tidal elhf"lC for this record.
,
I
l",j
*
N2
M2
S2 K2
t
Fig. 2. Energy spectrum of record fl2MA. ",.ith inset lIhuwmg expanded View in the scmidlurnal range. The M2 tide isc\early well ahuve hackgrnund lev('l : the duminantline in the entire lipcctrum.
exceptional vectors of amplitude greater than 5 em 5 I that were not plotted. Both show the currents for () h Greenwich lunar time. The agreement is fairly striking. Figure 7a shows the sense of rotation of M2 tidal velocities dctermined from the data; Fig. 7b shows the sense of rotation for the Schwiderski model. The sense of rotation of the tidal forces is clockwise in the northern hemisphere. counter-clockwise in the southern hemisphere. rna gross way. the tidal currents tcnd to rotate in the !lame sense ClS that of the force from the equator to mid-latitude. Regions of current rotation opposite to that of the force are nearer to coasts and to the poles. Out there are notahlc exceptions.
S581
Comparison of M2 tidal currents
......
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ow
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----
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Fig. 3. M2 tidal currents in the North Atlantic at time of Greenwich transit of fictive moon: (a) observed on current meter records, (b) as calculated by the Schwiderski model. Small open circles indicate locations where Schwiderski amplitudes exceed 5 em S- I. Solid circles indicate origin of vectors.
S582
J R
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and If M. StoM,.m
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FIg . 4 M ~ IIdill curren" in the North Pacltic ill lIme IIf (irccn~kh Ir:.o..11 (,,' Ikllvc mn(IO' (ill on..cf\'cd on current meter rc('onl" (hI ;" Cilil-UI;llcd ny Ih\· 'il'hwldcf,kl mudd Small open I:Ircic'l tndll:atc IIXl\llnn' where ~hw\(\cr,l" .Imph"I(\c' (',ecl'" , \ 'm .. I \1\\i,1 Clrl' \c' mdit:al\' orlllln 01 "~dllr'
Compa rison of M2 tidal current s
S583
(0)
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60· Fig . 5. M2 tidal currents in the Indian Ocean at time of Greenw ich transit of fictive moon : (a) ohserve d on current meter records . (b) as calcula ted hy the Schwid erski model. Small open circles indicate IO~dlions where SchwidcTski amplitu des exceed 5 em S- l. Solid circles indicate orlgm of vectors .
S584
J, R Ll'YT[1't and tI M
SWMWU
(bJ
(0/
I
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M: (id,.1 ,'Urrenl\ In !he Snuth ""anti\.' al lime (If (ircenw\<:h Irlln!lll of fktivc moun: la I uMerved nn current melc:r r("(Or"~, (h) a.. l:ilkulaltd hy Ihe ~hwIIJcr\kl model. Small opcn circle, In,lIcale h:alilln .. where: "il:hwlder~kl amp"Imll.... \""eec,1 ~ em' I, Sulid clrc\C!I mda:"tc IIrljtln of vel'Int, . r:i~ fl
COHERENCE WITII I.API.ACE·S MODEl.
Because observation confirms the gross velocity patterns of the Schwidcrski model, the laller model deserves serinus considcmtion. The glohal distrihution of Schwidcrski's M2 tidal elipses is depicted in Fig. tt The radial segment indicates direction and amplitude when the fictive monn is over Greenwich. The last J h of the tidal ellipses are omitted so that the direction of rotation can he dearly secn. In mid-ocean the ellipses are noticeahly coherent on a scale comparahle to that of the equilihrium tide. To demonstrate this we offer in Fig. Mh. for comparison, the corresponding tidal ellipses for the semidiurnal L;lplace solution for a rotating globe covered with water of uniform depth of 4(XX) m (LAMB. 1()32, section 221).ln Fig.Rh the continents do not exist. but are shown for reference purp()~es. In the neighborhood of 40(XJ m depth (350H-4500 m) the Laplace solution is not very sensitive to depth, the overall amplitude of the ellipses diminishes somewhat with increase in depth. When we compare Fig. Ra to 8b we are comparing what we think represents the actu~,1 oceanic distrihution of M2 tides in the real ocean to those given hy the simplest possihle dynamical model (with flat hottom and no land), Figure Ma and h shows that over much of the deep ocean there is an approximate agreement in phase and amplitude. and that deep Atl.tntic and Pacific Schwiderski velocities tend to lead the (frictionless) Lapla<.:c solution hy an hour or so, ;'IS might be
Comparison of M2 tidal currents
S585
(0)
c
o
0 0
o
o
D
...
Fig. 7. Regions of clockwise (solid) and counterc:lockwise (gray) rotation of M2 tidal ellipses at (a) location of current meters, (b) I-degree grid of Schwiderski model.
expected in the presence of dissipation. North of 45°N the Schwiderski ellipses have opposite rotation, unlike the Laplace solution . This reversal of direction of ellipses at high latitudes can be reproduced in the Laplace model if a zonal coastal boundary is placed upon a high latitude circle, and the latitudinal factor of the separ ated Laplace equat ion integrated numerically from the equat or by "shooting" to get zero merid ional velocity at
S586
J R Lt ..-ns and II M
STOWMf I
(b)
-. - .... ~
•
•
0'
.
, ~
•
hg x Sc.:hwldcr..1r.1 M" Iidal e:lhP'C" RadII .. how dlfc\.'tl
""',,Ie
•
Comparison of M2 tidal currents
S587
that boundary. However, the reversal is very sensitive to the precise placement of the boundary, due to resonances, and the position of the natural zonal boundaries at Iceland and the Aleutians is too irregular to specify a representative latitude for the formal zonal boundary to be useful as a quantitative explanation of the reversal. There is little agreement between the two models in the Indian Ocean. For an ocean of uniform depth of 4000 m the Laplace solution gives an inverted tide at the equator. At local transit of the fictive moon the M2 tidal current of the Laplace solution flows eastward, lagging the direction of the tidal force by 3 h. Friction reduces this lag, turning the current more toward the direction of the tide-producing force. Accordingly, we might expect that over much of the ocean, the M2 tidal current flows in the quadrant between eastward and equatorward when the fictive moon is on local transit. A test of this supposition can be made using Schwiderski M2 current vectors on a 5° grid at local transit of the fictive moon. There are 1512 vectors, of which 763 lie in the indicated quadrant with a lag of ~90° behind the tidal force at local transit of the fictive moon. In the theory of forced linear shallow water waves it is standard practice to construct a complete solution for forced waves in bounded basins by first finding a particular integral that satisfies the forced equation. and then adding to it free solutions of the homogeneous equation to satisfy the boundary conditions. The sum is the complete solution. The Laplace solution for the semidiurnal tide on a rotating sphere completely covered by water of uniform depth is a convenient particular integral. For a depth of 4000 m the Laplace solution lies between two resonant depths at about 2000 and 8000 m depth, so the solution is reasonably insensitive to variations of the depth between 3000 and 6000 m such as actually occur in the real ocean. It is not difficult to introduce body friction into the Laplace solution by numerical integration of the complex amplitude in latitude, in this way finding the dependence of amplitude and phase of the variables upon friction. In the real ocean, and in the Schwiderski model, friction is not so uniform. If dissipation is so strong near the coasts that these free waves do not penetrate far into the interior. or if the complicated geometry of the coasts leads to interference patterns in the free waves, then it seems possible that the Laplace particular integral might dominate the velocity pattern in the deep ocean, as approximately observed. Despite the sparseness of current meter data a workable antenna can be constructed from it in the latitude band between 25° and 45°N. for comparison with the Schwiderski and Laplace models. In this particular latitude band, we construct averages of all of the direct values of the M2 tidal velocity, the Schwiderski model results (excluding the near coastal) and the Laplace model, with and without friction. The averages are constructed at the time of local lunar time, so that the Laplace velocity is the same for all longitudes. Figure 9 shows velocity ellipses for a tidal period of the data and model averages. A lower value of friction, k/20 = 0.2 makes the phase of F the same as that of D at the expense of increasing the amplitude of F. As k/20' is varied, the "closest approach" of F to D in the diagram is at k/2Q = 0.4. REGRESSION OF OBSERVED M2 VELOCITIES ONTO THE LAPLACE SOLUTIONS
The field of M2 current vectors tabulated in the Schwiderski model is uniformly distributed over the ocean in 1-degree squares. Therefore it is possible to determine how geographically coherent the vectors are. Exploration of the field indicates that values must be separated by about 20° to be regarded as independent. This means that the number of
J R.
S588
'I
Fig.
Q.
lUYfIN
and H. M.
STOMMFI
J
Tidal ClllP'C'i computed from avcra~c .. of veln":I!\, OIl hX',lllunat time
In
the lalilUde hand
2Y'-4~"N. The 'iolid radial hnc \Odicate' the VChlClh Vl'l'Ior at lern In\'allunar hnllt" All rotate dnckwi!'C. The 400n m Laplace ellipo.e (1., ~lllll 'lJIIIITe .. ' hll'l ca.. twl1td vchx·' ...· lit Icr.' I(x'allunar time. With hudy fnctlOn of kl2U - 11.4 the clhP'C 1'1 reduced ttl ,.. ( ...,lId ItI'IOltk .. ), with VCllll'lIY 1OVl'ICld one hy BhoYI 2 h. The ~h""ldcr.. kl menn ell1pooc t.\, npcn Inanltlc .. ) h." mtermedlate ampiliude and phil"':. The dala from Ihe .mil} of ";4 curn:nl mch:t!i YIeld .. the cUip .. c () ( ... ,lld l'In:Ic .. ). ~tUltc dl''-\' II •. \
Icading the
degrees of freedom for determining the precision of our average vector over a hand of latitude is rather small-at most 5 in the North Pacific. for example. In this l'mme ocean the range of longitude sampled by current meter data is more restricted. so that in the latitude hand 25°-45°N there arc at most J degrees of freedom. ,lOd in the North Atlantic only 2 degrees of freedom. Errors in determination of the M2 harotropic tidal current vector from individual current meter records arise from many sources: instrumentation. editing llnd processing procedures. contamination due tn baroc1inic tides. finite duration of record. the numerical harmonic analysis. local hottom topographic effects. yeltr·to-year variations in some of the slowly varying astronomical conshtnts. The sum of these errors can he estimated hy comparing groups of records obtained within a small area at different depths nnd different years. For example. in the Pacific p~trt of the array there arc seven records. over two different years in the box bounded by 2T'-2H"'N. 1~1"'-15J>E. The mean vector at local lunar transit is 1.3 em s I. leading Laphlcc hy too. with probahle error vectnr 0.6 em s- I. the prohahle error being computed on the assumption that we have madc seven independent measurements of the same vector at different times and depths. In the Atlantic portion of the army there arc 22 independent determinations in the hox 38°-4(fN, 2St.r'-292°E. over three different years. The mean vector ilt local lunar tmnsit is 0.6 em s ' I • leading Laplace hy I HO. with It probahle error vectm 0.2 em s I. Dy contrast. outside the array. in the equatorial Pacific. there ure 27 independent measurements at different depths in the hox lOS-ION. 212()-217°E whose mean vector itt local lunar transit is 1.03 cm s- I. lagging Laplace by 2(). with a proh.,hle error vector of 0.2 em s I
Comparison of M2 tidal currents
S589
The data array in the latitude band 25°N-45°N is very uneven, with large ranges of longitude occupied by land. In the Pacific there are 62 records that can be compared with vectors from the Schwiderski model from the same I-degree squares. At local lunar transit the mean data vector is 0.61 cm S-1 leading Laplace by 14°, whereas Schwiderski's model vector is 0.56 em S-1 leading Laplace by 23°. There is no significant statistical difference. The corresponding numbers from the 90 records in the Atlantic part of the band are: 1.01 cm S-1 leading by 22°, Schwiderski, 0.08 em S-1 leading by 40°, again substantially the same. The weaker amplitude in the Atlantic is due to the preponderance of current meter records in the weak current western portion of that ocean. The near equality of phase lead in the two oceans informs us that the integer longitudinal wavenumber s = 2 (or O!), not 1 or 3. A simple linear regression of the observed M2 phase on longitude [g~) = s(Lon + phase) + f], gives s = -1.8 ± 0.3, phase = 29° ± 23° at 95% confidence, assuming 5 degrees of freedom. Observed values are taken from the 154 current meter records between 25° and 45°N. Note that since the phases are defined modulo 360°, S = 0 is also a possible solution. The analysis of the data is not adequate to determine s within a single ocean alone. The mean phase lead of 18°-31° informs us that the Laplace-type model should have a moderate value of body friction : 0.2 < kl2n < 0.4 . CONCLUSION
The Schwiderski model has M2 tidal currents that conform well with M2 tidal currents observed in 314 one- to two-year moored current meter records. Most of these long records deeper than 999 m exhibit sharp spectral M2 lines that probably represent the barotropic tidal current. The M2 currents exhibit a geographical pattern that appears to be coherent with the astronomical tidal force field. The signal of the Laplace particular integral can be extracted from both the model and observed current fields. Acknowledgements-We are deeply grateful to Dr Peter Saunders who permitted us to make use of several of his unpublished determinations of the semi diurnal tidal currents in the northeastern Atlantic Ocean , and who corre(;ted an error in interpretation of phase from the Schwiderski tapes that we had made in an earlier version of our tables. He also offered numerous useful comments. Dr George Platzman kindly discussed some of the dynamical ideas with us. Susan Tarbell and Ann Spencer of the Woods Hole Buoy Group prepared edited versions of the current meter data records from the archives. Barbara Gaffron prepared the manuscript. The work was done with support from contract NOOOI4-84-C-0134, NR 083-400 with the Office of Naval Research (JRL) and grants OCE86-13810. OCE87-15735 and OCE88-16165 from the National Science Foundation (lIMS) . Contribution no. 7067 from the Woods Hole Oceanographic Institution . REFERENCES DICK G . and G. SIEDLER (1985) Barotropic tides in the Northeast Atlantic inferred from moored current meter data. Deutsche lIydrographische Zeitschrift. 38. 7-22. HENDRY R. M. (1977) Observations of the semdiurnal internal tide in the western North Atlantic Ocean. Philosophical Transactions. 286. 24. LAMB H. (1932) Hydrodynamics. Cambridge University Press. 738 pp. SCHUREMAN P. (1924) A manual of the harmonic analysis and prediction o/tides. Department of Commerce. U .S. Coast and Geodetic Survey , Special Publication No . 98.416 pp. SCHWIDERSKI E. W. (1979) Global Ocean Tides. Part II . The semidiurnal principal lunar tide (l\1z). Atlas of Tidal Charts and Maps . Naval Surface Weapons Center. Dahlgren. Virginia. Technical Report TR 79-414. 87 pp. TAltBELt S. A .• A. SPENCElt and E . T. MONTGOMElty (l988) The Buoy Group data processing system . Woods Hole Oceanographic Institution Technical Memorandum No . 3-88.209 pp.
rr
0198-0149/91 S3.00 + 0.00 1(" J 991 Pergamon Press pic
""7'-\'K". IlJCIl
Comparison of M z tidal currents observed by some deep moored current meters with those of the Schwiderski and Laplace models JAMES R. LUYTEN* and HENRY M. STOMMEL * (Rt'uiVf'd 12 May 19R9; in revised form 31 May 1990; accepted 7 June 1990)
Abstract-M 2 tidal currents observed at 315 deep-sea locations (mostly in the northern hemi'phcrc) confirm thO\c computed by the numerical model of SCHWIDERSKI (1979. Naval Surface Weapons Center. Dahlgren. VA. TR 79-414). Together. they reveal large-scale geographical cohcrcncc!> similar in form to thosc of the tide-generating potential and to those of the analytical Laplace model for tidcs on a globe completely covered with water of uniform depth. By fitling the Laplace tidal ellipses to the array of current meters and to the velocities from Schwidcr~ki\ numerical model. we show that much of the Mz tide is explicable in terms of Laplacc'» simple model-with friction.
INTRODUCTION
IDEALLY a map of the global distribution of deep-ocean M2 tidal currents would be constructed from analysis of observed current data. The distribution of such data is sparse, so to make maps we must depend upon numerical model calculations. SCHWIDERSKI (1979) developed such a model. In addition to publication of tabulations of one-degree surface elevation, he also produced tapes of the current harmonic constants. We decided to compare his results with M2 tidal current determinations from data in the archives of the Moored Current Meter Group of the Woods Hole Oceanographic Institution and a few published observations. The data largely confirm Schwiderski's model results, and seem to justify accepting them as globally representative. Inspection of the fields reveals large-scale geographical coherence suggestive of the presence of a signal corresponding to that expected from Laplace's simple analytical theory of the semidiurnal tide on a water-covered globe of uniform 4000 m depth despite the complexity of the bottom topography and form of the continental coast. One hint of this is that more than half of the 5-degree velocity vectors in the Schwiderski model at time of local lunar transit lie in the quadrant between eastward and equatorward. In the inviscid Laplace model these vectors are all eastward and lag the tidal force by 3 h. Introducing linear friction in the Laplace model reduces this lag. A least squares fit of the Laplace solution with the Schwiderski model, latitude by latitude, quantifies the result. To show that this correspondence is not only a model result hut also occurs in the real ocean. the Laplace solution is compared to observed data. The observed current data are rather sparse, but in the latitude band 25°-45°N there are • Woods lIole Oceanographic Institution. Woods Hole. MA 02543. U.S.A. S573
S574
J R
Lt:YT£N
and H M
STOMMEL
sufficient observed points in both the North Atlantic and Pacific to form an east-west array to which a Laplace solution can be compared. The array indeed does pick up the Laplace signal. OBSERVED M2 CURRENTS
The archives of current meter records at Woods Hole contain observations from the Atlantic. North Pacific. the Agulhas retroflection region and the equatorial Indian Ocean. We used the SCtlUREMAN (1924) tables for the astronomical data necessary to compute the M;! harmonic constants (amplitude and Greenwich phase of the eastward and northward velocity components). The tidal constituents for the velocity components are assumed to have the form lI(f)
= u"cos(a(I-III) -
vet)
= ""cos (a(t
gu -
- I,,) - R~'
Xl
- X).
where guo gv (lre the phase at Greenwich hour 0 and X is the astronomical argument ohtained from the Schureman tahles. The procedure for estimating the amplitude and phase of the M~ tidal component is al\ follows. Thc time series for cast and north velocities are each multiplied hy cos (0' I) and sin (a· t) lind summed over the record length. where (J = ~,./28.<:JMI042 rad h '. The phase guo = tan '«I4 .. )/(u\.» + the Schureman phase for the starting time of the record. The amplitude. U 11 : : v'u? + Only long suhsurface records were chosen. Records from hoth Model R5()s and Vector.."vcraging Current Meters (VACMs) were used. Altogether some 4X4 records were analysed on a personal computer. An attcmpt was madc to ohtain gcographic:11 c()vemgc. so in many cases there was insufficient sampling in the verticnl to attempt a vertical modal analysis tn separate the harotropic from the haroclinic modes. Upon inspection it appeared that shallow records tcnded to he contaminated hy haroclinic tidal currenll\ (Hendry. 1977). and so ,,11 data from less than WOO m depth were rejected. A very few records had questionable starting times. and these also were rejected. Those used arc identifted in Tahle I. In addition to our own estimates. dctcrmin"tions of the M., tidal current constants have heen extracted from the literature. mainly in the eastern North Atlantic. Some of the tidal constants from puhlished sources represent the hafOtropic results of " modal separation (DICK .lOd SIfOl.F.R, 19R5). Several more werc provided hy Dr Peter Saunders. The numher of records reported is 11 S. A representative scatter di .. gram sh()wing R257 successive hnurly values of mcasured current from record tl2tl5 is shown in Fig. 1. The outer circle has a radius of 10 em s I. The inner eircle represents the thrcl\hold of the current metcr. at 1.8 em s I. This circle is made up of data points swept up from the center anu assigned to the threshold radius hy the algorithm employed in processing (TARBF.LL ('I al .• 198M). Four uots inside the circle are artifacts of a manual editing procedure. The current meter is a Model 850. The intermediate circle is the r.m.s. mean radius of all recorded data points, 2.7 em s 1. The ellipse is the M2 tidal ellipse as dctermincd hy harmonic analysis of this data. The position of this meter was at 20.9"N, 41 }/'W, 4015 m depth. The diagram gives an idea of the amplitude of the background of all other signals to that of the M2 signal at this station. In Fig. 2 we show the semidiurnal part of the pcriouogfllm. including the lines M N S2' K:!. The M21ine stands well ahovc the noise level. and its amplitude has n ratio to t~;at ~f
II:.
Compa rison of M2 tidal currents
S575
TQb/~ 1. V~/oc;ty amplilud~s and phases for Iht M2 1itlal compon~nt Observed
Record
name 42211. 42221. 45011. 46521. 4912L 4914L 4902L
4912L 4913L
49141. 5063L
50741. 5093L 5233L 5245L 5246L 5247L
5261L 5262L
5433B S43SA
5456A 5454B 5467A 5485B
5486B 549lX 549SC 5602B
56631. 5665L 5673L S67SL 57318
5805L 5815L 59338 59348 59SJB 59S4A
59638 59648 5975D G021A 6263A
6265A 6304A 6315£ 6361A 6371A
lat.
long.
depth 61 (h)
39.0 290.0 1027 39.0 290.0 2495 39.2 289.5 1014 39.0 290.0 2487 39.1 290.0 1019 39.1 290.0 2550 39.4 290.0 2011 39.1 290.0 1019 39.1 290.0 2030 39.1 290.0 2550 39.4 290.0 1995 39.2 290.0 2006 39.1 289.5 1987 39.4 290.0 1991 39.1 290.0 1005 39.1 290.0 20ll 39.1 290.0 2512 38.8 290.0 2006 38.8 290.0 2810 2B.O 295.0 1002 28.0 295.0 4003 27.8 304.4 4004 27.8 304.4 1996 27.9 305.1 4011 31. 0 299.9 2001 31.0 299.9 4001 34.0 300.0 1002 34.0 300.0 4002 41. 5 305. a 3994 34.9 305.0 1005 34.9 305.0 4006 31. 6 304.9 ).028 31. 6 304.9 4030 41.5 305.0 4001 · 31. 6 305 . 1 3995 34.9 304.9 3995 -0.1 50.5 1500 -0.1 50.5 3545 1.5 53.0 1500 1.5 53.0 3542 0 . 0 57.0 1550
0.0
57.0 3595 53.0 3544
0.0 41.5 305.0 3993 26.9 318.8 1514 26.9 318.8 4015
27.9 311.4 1498 27.9 311.1 4016 4.0 320.3 4256 4.0 320.7 4104
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.25 0.25 0.25 0.25 0.25 0.25 0.25 1. 00 1.00 0.25 1. 00 0.2S 0.25 0.25 0.25 1. 00 1. 00 1.00 1.00 1.00 1. 00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.25 1.00 1.00 1.00 1.00
0.25 0.2S
length 2583 2582 2444 2457 4886 4885 4846 4886
4885 4885 4214
4234 4233 23008 22969 22961 22976 22865 22841 26001 S'02 5739 27145
5701 26624
20656 25609 25610 5143 5322 5322 5155 5155 7296
gu
110
8 93 6 72 6 102
14 116 11 74 9 93 11 81 11 74 8 9 9 9
9 12 9 8 9 9
82 93 74 94 102 96 83 91 90 84
7
80 11 99 9 101 13
9S
13
96
11
94
9 130 8 B8
10 9
IS
78 94 75 93 7S 84 79 69
7272 5569 1489 5521 3289
12 9 15 7 10 12 11 22 16 21 21
5377 5377
3 300 24 164
4825
23 152
26209 8257 8257 S088 8088 34680 34688
11 101 16 16
7368
86 73
162 148 162 152
Schwiderski
"0
gv
4 28 2 243 2 282 7
28
4 347 2 349
3 4 4 347
Uo 7
17
7
11 17
'1 6 '1 7 7
3S
7 7 7 7 '1
2 357 6 42
7 7
1 339 2 349 2 0
4 4
9
2
34
2 349 14 19 9 306 '7 314 15 340 3 2
20 328 14 331 6 7 4 284 9 300 7 335 8 299 9 338 7 310 17 336 10 296 7 261 10 324 9 313
13 122 19 14 13 4 13 13 7 28
16
35
14 9
66 77
140 lSI 122 65 88 137 276 345 25 304 22 323 9 320
B
32
17 279
11
14
23 292
gu
30 19 19 19 19 19
19 19 19 17
19
7 7
19
7 6 6 7
19 30 30 97
19
7
97
8
87 87 83 87 87
8 8 8
8 S 8
83
10
83 85
9
73
9
73
8
75
8 75 10 85 8 72 9 73 21 127 21 127 21 121
21 121 22 128 22 128 21 123
"0
gv
6 237 6 237 6 ~37 5271 7271 7271 7 271 7271 7271 7 271 7271
7271
6 237 7271 7271 7271 7271 5 271 5 271 9 286 9 286 14 307 14 307 14 308 10 305 10 30S 10 306 10 306 4 287 10 309 10 309 11 309 11 309
4 287 310
12 10 14 14 15 15 11 11
309 110 110 101 101 429 429
14 lOS
10
85
16 16 11
48
2S 315
48 62
11
62
7
20 20
25 19 19 18 18
7
4 287 315 312
312 298 298
Continued
S576
J. R.
LUYfEN
and H . M. STOMMEl.
Schwiderski
Observed Record
name
lat.
~ .
67338 67348 6744C 67531. 6761C 6762N 67630 6764C 67730 67748 67'5r 6776r 6784N 67858 6786k 67938 67941. 61941. 68031. 6804l 6811C 6813D 6831! 6832' 6842C 68430 6844l 6851C 69541. 69551. 69631. 6972C 6973B 6981B 6983A 6982e 6993C 6994C 70028 70031.
70138 701410 10158 70221. 70238 7033D 70431. ?044e 704510 71838 7184A
. .
100,. deptb 6, (h)
length
Uo
gw
"0
'"
9529 18 37 11 342 37.0 318.0 1526 1.00 9529 18 48 14 339 3'.0 318.0 4026 1.00 4 6 4023 9433 14 45 1.00 39.S 316.1 20 7 6' 1530 16 40.3 314.7 0.25 37636 39.0 316.0 4658 0.25 37444 19 42 13 339 39.0 316.0 4697 0.25 37444 19 44 13 3H 39.0 316.0 4747 0.25 37444 19 43 12 338 39.0 316.0 4779 0.25 37444 19 45 12 34J 39.0 315.9 1521 0.25 16036 20 63 11 11 39.0 315.9 4020 0.25 37444 18 4~ 11 333 39.0 315.9 4777 0.25 37348 17 45 10 333 39.0 315.9 4857 0.25 37348 16 44 10 329 6 346 9265 13 52 38.7 314.4 3989 1.00 6 337 38.7 314.4 4835 0.25 37060 17 45 6 340 38.1 314.4 4915 0.25 37060 17 44 38.0 313.4 1504 0.25 36868 18 69 11 357 3 243 38.0 313.4 4003 1.00 1969 10 35 1 6 3961 14 82 38 . 0 313.4 4001 1.00 1 38.9 313.1 1512 0.25 36868 15 71 5 8 350 38.9 313.1 4013 1.00 9217 15 57 -33.8 320.5 3925 0.25 10864 15 326 30 77 -33.8 320.5 4275 1.00 2714 13 337 24 79 2 27 83 2715 17 -30.9 320.3 3946 1.00 2723 -30.9 320.3 4295 1.00 9 348 14 73 -31.0 320.1 2990 0.25 10958 12 34 24 89 2691 16 43 25 88 -31.0 320.1 3490 1.00 2692 18 66 22 93 -31.0 320.1 3911 1.00 -30.6 320.3 3971 0.25 10607 12 350 29 59 :2 319 41.0 152.0 3990 0.25 31204 9 356 41.0 152.0 5077 0.25 31204 3 332 9 357 7 8 39.0 152.1 3995 0.25 31012 12 39 9 46 24 5 37.5 152.0 1201 0.25 30820 9 21 5 ]54 37.5 152.0 4001 0.25 30820 50 11 6 345 36.3 152.0 507 0.25 30436 8 31 4 32 36.3 152.0 4007 0.25 30436 9 45 J0436 6 27 36.3 152.0 1207 0.25 64 '7 4 76 35.0 152.0 1211 0.25 1,092 9 26 3 341 35.0 152.0 4015 0.25 30148 67 9 25924 0.25 39 1178 5 33.8 152.0 5 31 33.8 152.0 3919 0.25 29764 10 49 8 105 7 77 32.5 152.2 1200 0.25 29572 7 1 278 32.5 152.2 4000 0.25 29572 10 6340 10 357 1 287 32.5 152.2 5540 0.25 12 35 29380 4 303 31.3 152.1 1206 0.25 4 37 31.3 152.1 4006 0.25 29380 11 36 357 14 29092 4 249 0.25 30.0 152.0 3994 43 '7 28708 1 178 0.25 1202 28.0 151.9 47 16 28708 3 295 0.25 3984 151.9 28.0 6 333 28.0 151.9 S891 0.25 28708 19 42 4 15 8978 6 125 28.0 151.9 1198 1.00 7 23 21790 7 267 28.0 151.9 3984 0.25
"0
I"
16
42
17
48
17 17
55
10 343
10 343
42
16
17 17 17
'"
"0
8 7 8 8
51 51 51 51 50
358
372 356 356
8 356
8 356
50 52 52
8 345 8 345 8 345 8 345 7 142 1 342
15
52
7 342
14 14 14
S4 S4 54 55 55
e 8 8 7
18 18
17 17
36 36 36 36 36 36
20 20 20 20 20 20
8 3S'"
..
8 357 8 359 1 I)
4 5 ..
16 16 16 16 15 15
14
14 8 8
7 7 7 7 7 7
9 9 9
50
50
1 2 :2
9
2
10
~
10
S
10 10
9 9
11 11 11
8 8 8 10 ·
12 12 12 13 1)
13 13 13
10 11 16 16 16 16 16
7
311 331 331 338 338 436 436 443 443 443 Ul 443 .... 1 319 319 327 329 329 131
4 4 4 1Jl
.. 131
.. 332 4 3J2 4 J45 4 345 3 131 3 331 3 131
326 326 317 320 320 320 :2 320 2 320
3 3 2 2 2 2
SS77
Comparison of M2 tidal currents
Table 1. Conrinued Observed
Schwiderski
Record name 71848 71850 71928 7193C 72138 12148 721SC 7222C 72438 72448 7245C 7253C 7263C 72528 72628 7272A 72838 72848 72851 72911 7292A 72928 72938 73010 73028 73038 7311D 73128 7313A 732lA 7322C 7323C 73418 7342A 77528 7764A 77658 7766A 77718 77728 77818 77918 7771Q 7804B 78058 7806B 78228 78238 7832A 78428 78438
lat.
long. depth dl (h)
length
"0
gu
"0
5 28.0 151.9 3984 0.25 10110 23 28.0 151.9 5856 0.25 35908 22 10 8977' 6 80 30.0 152.0 1202 1.00 30.0 152.0 4002 0.25 35909 21 356 8978 8 115 32.5 152.1 1214 1.00 32.5 152.1 4015 0.25 35908 13 340 32.5 152.1 5556 0.25 35909 16 336 8977 10 27 33.8 152.0 1198 1.00 8953 11 42 34.9 152.0 1204 1.00 9 356 34.9 152.0 4004 0.25 35812 9 345 34.9 152.0 5958 0.25 35813 8 358 36.3 152.0 3996 0.25 35812 37.5 152.1 3987 0.25 35813 11 355 8954 11 16 36.3 152.0 1196 1.00 8538 5 21 37.S 152.1 1186 1.00 1726 10 340 39.0 152.1 1222 1.00 8905 7 14 41.0 152.0 1237 1.00 41.0 152.0 4037 0.25 35616 10 324 41.0 152.0 5216 0.25 35621 10 321 51.0 185.1 2010 1.00 10010 22 204 804l 21 243 51.0 185.1 3010 1.00 1464 18 222 51.0 185.1 3010 1.00 8400 20 241 51.0 185.1 4510 1.00 50.5 185.2 1986 1.00 10033 16 231 50.5 185.2 2986 1.00 10033 20 241 50.5 185.2 4486 0.25 40132 25 240 49.4 185.2 2060 1.00 10033 14 252 49.4 185.2 3060 1.00 10033 20 10 6473 20 222 49.4 185.2 4560 1.00 3460 25 244 47.9 185.2 2006 1.00 47.9 185.2 3005 1.00 10033 21 250 2194 23 236 47.9 185.2 4505 1.00 46.0 185.2 2002 1.00 10033 16 220 4273 18 234 46.0 185.2 3002 1.00 41. 2 297.0 1390 1.00 12169 12 142 40.5 298.0 1512 0.25 48773 7 130 40.5 298.0 4011 1.00 12193 10 88 40.5 298.0 4789 0.25 48773 11 92 40.2 298.4 4002 1.00 7321 13 119 40.2 298.4-4871 0.25 48676 15 119 40.7 298.5 4002 1.00 12577 13 111 40.9 299.3 4005 0.25 48676 9 64 40.2 298.4 4002 1.00 12169 14 113 39.5 299.7 1009 1.00 12193 3 99 39.5 299.7 1506 0.25 48772 4 111 39.5 299.7 4004 1.00 12193 11 102 37.5 301.7 1386 1.00 12193 8 68 37.5 301.7 4008 0.25 48772 12 83 35.6 299.1 4023 0.25 48773 5 75 36.5 296.9 1407 1.00 12169 7 117 36.5 296.9 4001 0.25 48676 7 99
g"
B 263 6 289 5 63 9 270 8 69 3 273 4 241 3 345 5 20 2 278 3 289 2 7 3 300 5 331 3 39 4 322 5 5 3 282 2 275 7 37 7 228 2 23 6 257 4 318 14 222 14 201 9 274 4 355 3 131 12 192 12 207 7 156 4 119 6 158 5 332 4 296 8 306 8 315 5 17 7 13 6 309 10 262 7 2 5 266
5 278 8 339 8 308 8 305 5 283 3 345 5 316
Uo
gu
13 13 12 12 11 11 11 10 10 10 10
16 16 11 11 8
9 9 9 9
2
"0
8 8 9 5 5 5
1 2
1
8 359 8 8 8 16 16 16 16 16 16 16 17 17
357 357 357 222 222 222 222
222
222 222 230 230 17 230 14 236 14 236 14 236 13 238 13 238 16 137 10 139 10 139 10 139 B 132 B 132 8 132 8 123 8 132 7 115 7 115 7 115 B 88 8 88 7 89 5 90 5 90
g"
2 320 2 320 2 317 2 317 3 331 3 331 3 331 4 345 4 332 4 332 4 332 4331 4 329 4331 4 329 5 327 4 314 4 314 4314 3 284 3 284 3 284 3 284 3 284 3 284 3 284 3 263 3 263 3 263 2 233 2 233 2 233 4 207 4 207 8 334 6 326 6 326 6 326 5 316 5 316 5 316 5 306 5 316 5 308 5 308 5 308 7 308 7 308 7 306 6 306 6 306
Continued
S578
J . R.
LU'YTIN
TDb/~
Record name 7863A 7933A 7943A 7953A 79818 79918 80138 B0238 S0338 81539 B163A 811lA 81838 81939 82139 82238 82338 82438 82538 82738 8296L 8298L 8306L 830BL 8313L 8314L 83438 8344C 8353A 83548 83618 8364C 83738 8374C 8383A 83848 8393A 83948 8404C 84138 84148 84238 84024A 84338 8434A
and H. M.
STOMM£I.
1. Continutd Schwiderski
Observed
lit.
Ion,. depth lJt (h)
37.4 295.0 4002 39.0 208.0 4018 35.0 208.0 4016 41.0 185.0 3996 ll.0 184.9 3993 35.3 175.0 4007 41.1 165.0 4019 39.0 164.9 4004 37.0 165.0 4031 31.0 164.9 4021 33.0 165.0 3994 35.0 165.0 3995 37.0 165.0 4003 39.0 165.0 4005 41.1 165.0 4020 39.0 175.0 3998 35.3 174.9 4017 31.0 184.9 3996 35.0 185.1 4006 41.0 185.0 3986 2B.0 290.0 1062 28.0 290.0 4062 25.5 290.0 1059 25.S 290.0 4059 32.0 335.9 1071 32.0 335.9 2971 -38.0 15.S 1493 -38.0 15.5 3993 -40.1 16.6 1409 -40.1 16.6 3909 -42.0 17.8 1446 -42.0 17.8 3947 -40.1 19.7 1593 -40.1 19.7 4092 -38.0 18.5 1497 -38.0 18.5 3996 -37.9 21.1 1451 -37.9 21.1 3951 -38.6 23.1 4055 -37.2 23.0 1496 -37.2 2l.0 3995 -35.9 27.0 1512 -35.9 27.0 4011 -35.0 26.0 1505 -35.0 26.0 3503
length
0.25 0.50 0.50 0.25 1.00 1.00 1.00 1.00 1.00 0.25 0.25 0.25 0.25 0.25 0.25 1.00 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1. 00 1. 00 0.50 1. 00 0.50
1.00 0.50 1.00 0.50 1.00 0.50 1.00 O.SO 1.00 1.00 0.50 1.00 O.SO 0.50 0.50 0.50
gil
"0
48677 33842 33842 34S65 8353 7849 7489 7441 7368 37636 37541 37253 36964 36868 36580 8977 8929 8641 8617 8545 14329 14329 14185 6408 17113 17113 34622 17305 34515 17257 34514 17257 34226 17113 33843 16923 33555 4225 16777 33458 16729 33218 33219 33122 33123
7
"0
99
13 219 11 206 16 256 17 247 8 301
9 360 2 271 8 299 13 343 5 13 10 350 5 350
2 334 14 13 9 286 11 303 18 265 14 313 13 267 9 134 4 150
6 138 10 120 28 34 12 27 3 352 2 310 3 1
4 322 5 318 3 289
2 2 4 4 1 5 4
19 332 353 338 26 19 357 2 288 4 lS3 4.
11
2 292 J 5 3 356
gv
S 335 21 175 17 154
5
6 16 16 7 13 10 5
92
9 222 8 233 12 246
8 194
12 197 '} 217 7 274 1 32 6 212 B 248 3 262 7 250 5 257 2 350 11 289 4 249 13 224 12 219 15 237 5 210 7 343 4 257 7 311 7 313 27 348 24 31 1 359 3 286 5 328 5 320 8 335 8312 5 327 4317 4 354 4 341
4 318 3 336 5 325 9 332 6317 4 357
7 313 3 354 5 321
11 265 10 287 10 317 10 324 10 321 9341 9 336 9 333 10 321 10 324 10 317 11 278 10 289 11 265
12 258 12 248 6 103 6 103 7 116 7 116 19 18 19 18 3 307 3 l07 2 305
2 305 1 296 1 296 43 43 1 8 1 8 3 55 3 55 2 59 2 45 2 45 3 76 J 76 1 76 1 76
o o
317 174 173
197 200 211 243 6 236 6 235 6 233 6 234 6 235 6 215 6 236 5 243 8 213 10 214 13 200 12 198 7 198 8 265 8 265 8264 8 264 24 348 24 348 6 325 6 325 7 320 7 320 8 317 8 317 7 308 7 J08 6 303 6 J03 4 284 4 284
6 5 5 5 5
301 302 l02 257 257
5 257 5 257
Polymode Clu.ter C Hoorlnga(P.Nil1er, T917L T918L
15.2 306.8 2508 15.2 306.8 4008
1.00 1.00
8471 8471
11 161 14 194
20 20
4 :2
12 120 12 120
23 309 23 309
("o,.,i"II('(/
S579
Comparison of M2 tidal currents
Table 1.
Continued
Schwiderski
Observed Record
name
tat.
long. depth 61 (h)
length
gu
Uo
"0
g"
Uo
gu
gv
"0
Pequod Mooring8(C. Eriksen, J. Richman) PI02$ P103$ P202$ p203$ E103$ £105$ £203$ £205$ 0107$ 0109$ 0111$ 0205$ 0207$ 0209$ 0211$ U105$ UI07$ UI09$ U111$ U205$ U207$ U209$ U211$ 0103$ 010S$ 0203$ 0205$ 0102$ 0202$ 0203$
0.0 212.1 1526 0.0 212.1 3026 0.0 212.1 1465 0.0 212.1 2992 -0.1 214.9 1565 -0.1 214.9 3069 0.0 215.0 1441 0.0 215.0 2953 0.0 215.3 1615 0.0 215.3 3015 0.0 215.3 3115 0.0 215.3 1482 0.0 215.3 1582 0.0 215.3 2982 0.0 215.3 3082 0.3 215.5 1492 0.3 215.5 1592 0.3 215.5 2992 0.3 215.5 3092 0.3 215.5 1496 0.3 215.5 1596 0.3 215.5 2996 0.3 2l5.5 3096 -0.0 216.0 1554 -0.0 216.0 3051 0.0 216.0 1453 0.0 216.0 2974 0.0 221.9 1547 0.0 222.0 1445 0.0 222.0 2958
1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
9834 8694 8795 8796 9955 9805 8987 8987 8834 8802 9248 6592 3600 9283 9271 9095 9037 9199 9198 8979 8979 8979 8979 8340 9295 9271 9271 8739 9270 9270
11 315
30B 321 313 315 320 315 351 11 317 13 313 12 309 11 304 11 321 12 311 10 309 10 324 12 306 11 307 11 308 12 316 13 318 12 329 10 314 12 328 12 316 11 310 10 298 B 353 7 326 6 324
12 15 11 13 8 15 11
5 15 4 347 6 20 3 344 7 5 2 352 3 352 4 87 4 25 3 42 4 36 4 337 4 31 4 12 4 31 2 356 3 336 3 341 4 336 3 347 2 348 2 24 2 12 3 12 3 87 2 30 3 21 4 92 3 62 3 70
15 289 15 2B9 15 289 15 289 15 292 15 292 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 294 15 295 15 295 14 295 14 295 12 300 12 300 12 300
4 4 4 4 4 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
333 333 333 333 344 344 360 360 360 360 360 360 360 360 360 360 360 360 360 360 360 360 360 3 357 3 357 2 373 2 373 4 421 4 421 4 421
the N2 tide that corresponds to the ratio of the amplitudes of the astronomical force assigned to these two frequencies, as might be expected for two components of the force with such close frequencies and identical form in latitude and longitudinal wavenumber. The clarity of the M2 signal is superior, however, and for this reason (as well as the fact that gridded current values of the Schwiderski model are available only for the M2 component) we have limited most of our analyses to the M2 component. The K2 and S2 cannot be separated. COMPARISON OF OBSERVED CURRENTS WITH THOSE OF THE SCHWIDERSKI MODEL
Figures 3-6 exhibit the consistency ofthe M2 tides as determined (a) from current meter data with those (b) determined by the Schwiderski model. Small open circles indicate
S580
J. R . LUYTEN and II . M. SroWMU.
Fig. 1. Scatter diagram of H2:-;7 hourly readlOltli for Model X~o current meter numher fl2MA. The outer circle i'l 10 cm 'i I . The inner Circle 110 the thrc'ihold ve\C)C1ty of \ .Hem" 1 hencath which the rotor doe!! not re5pond . It cOo!"!lt~ nf (l~rvatl(Jn!l ",.hcl'iC direction I' mea.. ured . hut ",.hme amplitude is set arhitranly at I Hem ,, - I . The clhJl"C 1'1 the eC1mputed M ~ tidal elhf"lC for this record.
,
I
l",j
*
N2
M2
S2 K2
t
Fig. 2. Energy spectrum of record fl2MA. ",.ith inset lIhuwmg expanded View in the scmidlurnal range. The M2 tide isc\early well ahuve hackgrnund lev('l : the duminantline in the entire lipcctrum.
exceptional vectors of amplitude greater than 5 em 5 I that were not plotted. Both show the currents for () h Greenwich lunar time. The agreement is fairly striking. Figure 7a shows the sense of rotation of M2 tidal velocities dctermined from the data; Fig. 7b shows the sense of rotation for the Schwiderski model. The sense of rotation of the tidal forces is clockwise in the northern hemisphere. counter-clockwise in the southern hemisphere. rna gross way. the tidal currents tcnd to rotate in the !lame sense ClS that of the force from the equator to mid-latitude. Regions of current rotation opposite to that of the force are nearer to coasts and to the poles. Out there are notahlc exceptions.
S581
Comparison of M2 tidal currents
......
-...
II
"..
•
~¥"
(
ow
I
...
-
J
4
.... r'
//
J
30·N
'\
'\
"\ ~
o • em/s
90·W
...
../
.,.,..
/'
./
./
I
/
I
/ /
/ /
/
/
.--'"
/
--........-
/
/
//
./
/
/
/
/
/ /
/
/
\
I
/
/
/
......
I
/
/
/'
/
,.-
....... 60·
----
-
... ............
"30·
"
..........
"'--..
--.--
Fig. 3. M2 tidal currents in the North Atlantic at time of Greenwich transit of fictive moon: (a) observed on current meter records, (b) as calculated by the Schwiderski model. Small open circles indicate locations where Schwiderski amplitudes exceed 5 em S- I. Solid circles indicate origin of vectors.
S582
J R
Ll 'YTE~
and If M. StoM,.m
so· ~
...... -., -- ... ;;
~
-- ~
!
,
"-
"\
•
--.
,
•
/
30·N
0
H 0
,
em/.
" ,
__ .. __
J !!lO·E
l~
•_ _ ,;r -
-
l~O"W
o~/;~rl~ I
'" '"
.-
\
0
-
----
...
.. r-
I
I
I
....
..... .....
"
"-
,
"
"
\
\
0/ \
\
",•
I
I
30·N
..........
,-
/
/
f
!
/
/ /
/
SO·
......
_
,/
"
0
~
......
I
...-'"'"
-- - --
, -~~ ~~
--
-;./" :l
"-
....
O·
_ ___ _ ______. _ .1
160·
(b)
/
...
r-"..
/
I
1---------__ --1-.____ _ 1~O·E teo·
l..!- - - -. . . . . . .
I
\
\
\
\
"
\
,
--
/
,-
__ _ _ _ _-....1___ 1!lO.W
/
FIg . 4 M ~ IIdill curren" in the North Pacltic ill lIme IIf (irccn~kh Ir:.o..11 (,,' Ikllvc mn(IO' (ill on..cf\'cd on current meter rc('onl" (hI ;" Cilil-UI;llcd ny Ih\· 'il'hwldcf,kl mudd Small open I:Ircic'l tndll:atc IIXl\llnn' where ~hw\(\cr,l" .Imph"I(\c' (',ecl'" , \ 'm .. I \1\\i,1 Clrl' \c' mdit:al\' orlllln 01 "~dllr'
Compa rison of M2 tidal current s
S583
(0)
1--1
o •
em/,
0°
30 0 E
60·
Cb)
I
"-
,
"
-- ---"'-
--
--------.
-
-------
--
----- ------,,/
.,,--
...-- ------ / / ,,/
/'
.,-
/'
/'
'"
/
./
/ /
/'
/
/
I
/
/
I
o·
\
,,/
",-
"~----0..___
-----::---- -"
, \
'\
....
" ,
"-
"'"
"'-
0
0
----- --...,...,
...-
".
-
,/
30·E
O·
'-..
(j -- -- -----
."
/
//' /' / / .--" / /
0
(--
--- -
",-
"
........
30·5
/
,./
/
60· Fig . 5. M2 tidal currents in the Indian Ocean at time of Greenw ich transit of fictive moon : (a) ohserve d on current meter records . (b) as calcula ted hy the Schwid erski model. Small open circles indicate IO~dlions where SchwidcTski amplitu des exceed 5 em S- l. Solid circles indicate orlgm of vectors .
S584
J, R Ll'YT[1't and tI M
SWMWU
(bJ
(0/
I
\
'"
\
/
/
/
/
/
/
/
./
'-
-'0
/
........ o I
( ( ,"
)
\
./
cm/~
/
I
j
-~ ~
I,
...
/
"/,/,,
\
0
" ...... " ---
- ---...-
M: (id,.1 ,'Urrenl\ In !he Snuth ""anti\.' al lime (If (ircenw\<:h Irlln!lll of fktivc moun: la I uMerved nn current melc:r r("(Or"~, (h) a.. l:ilkulaltd hy Ihe ~hwIIJcr\kl model. Small opcn circle, In,lIcale h:alilln .. where: "il:hwlder~kl amp"Imll.... \""eec,1 ~ em' I, Sulid clrc\C!I mda:"tc IIrljtln of vel'Int, . r:i~ fl
COHERENCE WITII I.API.ACE·S MODEl.
Because observation confirms the gross velocity patterns of the Schwidcrski model, the laller model deserves serinus considcmtion. The glohal distrihution of Schwidcrski's M2 tidal elipses is depicted in Fig. tt The radial segment indicates direction and amplitude when the fictive monn is over Greenwich. The last J h of the tidal ellipses are omitted so that the direction of rotation can he dearly secn. In mid-ocean the ellipses are noticeahly coherent on a scale comparahle to that of the equilihrium tide. To demonstrate this we offer in Fig. Mh. for comparison, the corresponding tidal ellipses for the semidiurnal L;lplace solution for a rotating globe covered with water of uniform depth of 4(XX) m (LAMB. 1()32, section 221).ln Fig.Rh the continents do not exist. but are shown for reference purp()~es. In the neighborhood of 40(XJ m depth (350H-4500 m) the Laplace solution is not very sensitive to depth, the overall amplitude of the ellipses diminishes somewhat with increase in depth. When we compare Fig. Ra to 8b we are comparing what we think represents the actu~,1 oceanic distrihution of M2 tides in the real ocean to those given hy the simplest possihle dynamical model (with flat hottom and no land), Figure Ma and h shows that over much of the deep ocean there is an approximate agreement in phase and amplitude. and that deep Atl.tntic and Pacific Schwiderski velocities tend to lead the (frictionless) Lapla<.:c solution hy an hour or so, ;'IS might be
Comparison of M2 tidal currents
S585
(0)
c
o
0 0
o
o
D
...
Fig. 7. Regions of clockwise (solid) and counterc:lockwise (gray) rotation of M2 tidal ellipses at (a) location of current meters, (b) I-degree grid of Schwiderski model.
expected in the presence of dissipation. North of 45°N the Schwiderski ellipses have opposite rotation, unlike the Laplace solution . This reversal of direction of ellipses at high latitudes can be reproduced in the Laplace model if a zonal coastal boundary is placed upon a high latitude circle, and the latitudinal factor of the separ ated Laplace equat ion integrated numerically from the equat or by "shooting" to get zero merid ional velocity at
S586
J R Lt ..-ns and II M
STOWMf I
(b)
-. - .... ~
•
•
0'
.
, ~
•
hg x Sc.:hwldcr..1r.1 M" Iidal e:lhP'C" RadII .. how dlfc\.'tl
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Comparison of M2 tidal currents
S587
that boundary. However, the reversal is very sensitive to the precise placement of the boundary, due to resonances, and the position of the natural zonal boundaries at Iceland and the Aleutians is too irregular to specify a representative latitude for the formal zonal boundary to be useful as a quantitative explanation of the reversal. There is little agreement between the two models in the Indian Ocean. For an ocean of uniform depth of 4000 m the Laplace solution gives an inverted tide at the equator. At local transit of the fictive moon the M2 tidal current of the Laplace solution flows eastward, lagging the direction of the tidal force by 3 h. Friction reduces this lag, turning the current more toward the direction of the tide-producing force. Accordingly, we might expect that over much of the ocean, the M2 tidal current flows in the quadrant between eastward and equatorward when the fictive moon is on local transit. A test of this supposition can be made using Schwiderski M2 current vectors on a 5° grid at local transit of the fictive moon. There are 1512 vectors, of which 763 lie in the indicated quadrant with a lag of ~90° behind the tidal force at local transit of the fictive moon. In the theory of forced linear shallow water waves it is standard practice to construct a complete solution for forced waves in bounded basins by first finding a particular integral that satisfies the forced equation. and then adding to it free solutions of the homogeneous equation to satisfy the boundary conditions. The sum is the complete solution. The Laplace solution for the semidiurnal tide on a rotating sphere completely covered by water of uniform depth is a convenient particular integral. For a depth of 4000 m the Laplace solution lies between two resonant depths at about 2000 and 8000 m depth, so the solution is reasonably insensitive to variations of the depth between 3000 and 6000 m such as actually occur in the real ocean. It is not difficult to introduce body friction into the Laplace solution by numerical integration of the complex amplitude in latitude, in this way finding the dependence of amplitude and phase of the variables upon friction. In the real ocean, and in the Schwiderski model, friction is not so uniform. If dissipation is so strong near the coasts that these free waves do not penetrate far into the interior. or if the complicated geometry of the coasts leads to interference patterns in the free waves, then it seems possible that the Laplace particular integral might dominate the velocity pattern in the deep ocean, as approximately observed. Despite the sparseness of current meter data a workable antenna can be constructed from it in the latitude band between 25° and 45°N. for comparison with the Schwiderski and Laplace models. In this particular latitude band, we construct averages of all of the direct values of the M2 tidal velocity, the Schwiderski model results (excluding the near coastal) and the Laplace model, with and without friction. The averages are constructed at the time of local lunar time, so that the Laplace velocity is the same for all longitudes. Figure 9 shows velocity ellipses for a tidal period of the data and model averages. A lower value of friction, k/20 = 0.2 makes the phase of F the same as that of D at the expense of increasing the amplitude of F. As k/20' is varied, the "closest approach" of F to D in the diagram is at k/2Q = 0.4. REGRESSION OF OBSERVED M2 VELOCITIES ONTO THE LAPLACE SOLUTIONS
The field of M2 current vectors tabulated in the Schwiderski model is uniformly distributed over the ocean in 1-degree squares. Therefore it is possible to determine how geographically coherent the vectors are. Exploration of the field indicates that values must be separated by about 20° to be regarded as independent. This means that the number of
J R.
S588
'I
Fig.
Q.
lUYfIN
and H. M.
STOMMFI
J
Tidal ClllP'C'i computed from avcra~c .. of veln":I!\, OIl hX',lllunat time
In
the lalilUde hand
2Y'-4~"N. The 'iolid radial hnc \Odicate' the VChlClh Vl'l'Ior at lern In\'allunar hnllt" All rotate dnckwi!'C. The 400n m Laplace ellipo.e (1., ~lllll 'lJIIIITe .. ' hll'l ca.. twl1td vchx·' ...· lit Icr.' I(x'allunar time. With hudy fnctlOn of kl2U - 11.4 the clhP'C 1'1 reduced ttl ,.. ( ...,lId ItI'IOltk .. ), with VCllll'lIY 1OVl'ICld one hy BhoYI 2 h. The ~h""ldcr.. kl menn ell1pooc t.\, npcn Inanltlc .. ) h." mtermedlate ampiliude and phil"':. The dala from Ihe .mil} of ";4 curn:nl mch:t!i YIeld .. the cUip .. c () ( ... ,lld l'In:Ic .. ). ~tUltc dl''-\' II •. \
Icading the
degrees of freedom for determining the precision of our average vector over a hand of latitude is rather small-at most 5 in the North Pacific. for example. In this l'mme ocean the range of longitude sampled by current meter data is more restricted. so that in the latitude hand 25°-45°N there arc at most J degrees of freedom. ,lOd in the North Atlantic only 2 degrees of freedom. Errors in determination of the M2 harotropic tidal current vector from individual current meter records arise from many sources: instrumentation. editing llnd processing procedures. contamination due tn baroc1inic tides. finite duration of record. the numerical harmonic analysis. local hottom topographic effects. yeltr·to-year variations in some of the slowly varying astronomical conshtnts. The sum of these errors can he estimated hy comparing groups of records obtained within a small area at different depths nnd different years. For example. in the Pacific p~trt of the array there arc seven records. over two different years in the box bounded by 2T'-2H"'N. 1~1"'-15J>E. The mean vector at local lunar transit is 1.3 em s I. leading Laphlcc hy too. with probahle error vectnr 0.6 em s- I. the prohahle error being computed on the assumption that we have madc seven independent measurements of the same vector at different times and depths. In the Atlantic portion of the army there arc 22 independent determinations in the hox 38°-4(fN, 2St.r'-292°E. over three different years. The mean vector ilt local lunar tmnsit is 0.6 em s ' I • leading Laplace hy I HO. with It probahle error vectm 0.2 em s I. Dy contrast. outside the array. in the equatorial Pacific. there ure 27 independent measurements at different depths in the hox lOS-ION. 212()-217°E whose mean vector itt local lunar transit is 1.03 cm s- I. lagging Laplace by 2(). with a proh.,hle error vector of 0.2 em s I
Comparison of M2 tidal currents
S589
The data array in the latitude band 25°N-45°N is very uneven, with large ranges of longitude occupied by land. In the Pacific there are 62 records that can be compared with vectors from the Schwiderski model from the same I-degree squares. At local lunar transit the mean data vector is 0.61 cm S-1 leading Laplace by 14°, whereas Schwiderski's model vector is 0.56 em S-1 leading Laplace by 23°. There is no significant statistical difference. The corresponding numbers from the 90 records in the Atlantic part of the band are: 1.01 cm S-1 leading by 22°, Schwiderski, 0.08 em S-1 leading by 40°, again substantially the same. The weaker amplitude in the Atlantic is due to the preponderance of current meter records in the weak current western portion of that ocean. The near equality of phase lead in the two oceans informs us that the integer longitudinal wavenumber s = 2 (or O!), not 1 or 3. A simple linear regression of the observed M2 phase on longitude [g~) = s(Lon + phase) + f], gives s = -1.8 ± 0.3, phase = 29° ± 23° at 95% confidence, assuming 5 degrees of freedom. Observed values are taken from the 154 current meter records between 25° and 45°N. Note that since the phases are defined modulo 360°, S = 0 is also a possible solution. The analysis of the data is not adequate to determine s within a single ocean alone. The mean phase lead of 18°-31° informs us that the Laplace-type model should have a moderate value of body friction : 0.2 < kl2n < 0.4 . CONCLUSION
The Schwiderski model has M2 tidal currents that conform well with M2 tidal currents observed in 314 one- to two-year moored current meter records. Most of these long records deeper than 999 m exhibit sharp spectral M2 lines that probably represent the barotropic tidal current. The M2 currents exhibit a geographical pattern that appears to be coherent with the astronomical tidal force field. The signal of the Laplace particular integral can be extracted from both the model and observed current fields. Acknowledgements-We are deeply grateful to Dr Peter Saunders who permitted us to make use of several of his unpublished determinations of the semi diurnal tidal currents in the northeastern Atlantic Ocean , and who corre(;ted an error in interpretation of phase from the Schwiderski tapes that we had made in an earlier version of our tables. He also offered numerous useful comments. Dr George Platzman kindly discussed some of the dynamical ideas with us. Susan Tarbell and Ann Spencer of the Woods Hole Buoy Group prepared edited versions of the current meter data records from the archives. Barbara Gaffron prepared the manuscript. The work was done with support from contract NOOOI4-84-C-0134, NR 083-400 with the Office of Naval Research (JRL) and grants OCE86-13810. OCE87-15735 and OCE88-16165 from the National Science Foundation (lIMS) . Contribution no. 7067 from the Woods Hole Oceanographic Institution . REFERENCES DICK G . and G. SIEDLER (1985) Barotropic tides in the Northeast Atlantic inferred from moored current meter data. Deutsche lIydrographische Zeitschrift. 38. 7-22. HENDRY R. M. (1977) Observations of the semdiurnal internal tide in the western North Atlantic Ocean. Philosophical Transactions. 286. 24. LAMB H. (1932) Hydrodynamics. Cambridge University Press. 738 pp. SCHUREMAN P. (1924) A manual of the harmonic analysis and prediction o/tides. Department of Commerce. U .S. Coast and Geodetic Survey , Special Publication No . 98.416 pp. SCHWIDERSKI E. W. (1979) Global Ocean Tides. Part II . The semidiurnal principal lunar tide (l\1z). Atlas of Tidal Charts and Maps . Naval Surface Weapons Center. Dahlgren. Virginia. Technical Report TR 79-414. 87 pp. TAltBELt S. A .• A. SPENCElt and E . T. MONTGOMElty (l988) The Buoy Group data processing system . Woods Hole Oceanographic Institution Technical Memorandum No . 3-88.209 pp.