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Estimates of the resonant period and Q in the semi-diurnal tidal band in the North Atlantic and Pacific Oceans
Deep-SeaResearch,Vol. 28A, No. 5, pp. 481 to 493, 1981. Printed in Great Britain.
0198-0149/81/050481-13 © 1981 PergamonPress Ltd.
Estimates of the resonant period and Q in the semi-diurnal tidal band in the North Atlantic and Pacific Oceans R. A. HEATH* (Received 26 October 1979 ;final revision received 1 December 1980: accepted 10 December 1980) Abstract--Fitting the deep-sea harmonic tidal constants to a single normal development of the North Atlantic semi-diurnal tides gives estimates of the resonant period based on the vectoraveraged data of between 12.6 and 12.8 h in the semi-diurnal tidal band with a Q of between 10 and 12. A two-mode development has been used in the Pacific Ocean. The percentage frequency separation between the modes has been calculated from the travel time of a Kelvin wave around the perimeter. A relationship between the resonant period and Q has been derived ; it depends on the observed spatial separation of the open-ocean amplitude maxima of the M 2, $2, and N2 tidal constituents evident in the tidal charts of the western Pacific. The ratio of the maximum open-ocean amplitude of the M2 and $2 tide was then used to estimate the relative coupling between the tidal potential and the two modes as a function of the resonant period. The two relationships are then used in calculating the frequency-dependent change in phase with position to allow comparison with the estimate made from the tidal gharts. The resonant periods near the semi-diurnal tidal band in the Pacific are estimated to be at 10.2 and 12.8 h with a Q, which has been assumed in the analysis to be the same for both modes, of about 5.
INTRODUCTION KNOWLEDGE of the n a t u r a l p e r i o d a n d rate of energy d i s s i p a t i o n o f the n o r m a l m o d e s of the o c e a n is basic to a n y u n d e r s t a n d i n g of the oceanic response to tidal forcing. GARRETT a n d M UNK (1971) e s t i m a t e d the lower b o u n d for the Q o f the n o r m a l m o d e s at frequencies in the semi-diurnal tidal band. o f 25, b a s e d o n the m e a n w o r l d - w i d e value o f the age o f the tide; the Q is a m e a s u r e of energy loss rate, the system loses 2n Q - ~ of its energy per cycle. WEBB (1973) p o i n t e d o u t t h a t m a n y of the regions with a large age of the tide are associated with p a r t i c u l a r b a t h y m e t r i c features a n d localised r e s o n a n c e s a n d are not, therefore, r e p r e s e n t a t i v e o f the o p e n ocean. H e o b t a i n e d estimates of the overall d e c a y time of the s e m i - d i u r n a l tidal energy o f 30 to 35 h (equivalent to Q - 18) from c o n s i d e r a t i o n o f b o t h the age of the tide in the o p e n ocean alone a n d the local resonances. HENDERSHOTr (1972) o b t a i n e d a Q o f 34 from direct n u m e r i c a l i n t e g r a t i o n o f L a p l a c e ' s tidal e q u a t i o n s . F o r an ocean where a tidal b a n d is d o m i n a t e d b y only a single n o r m a l m o d e , the ratio of the a m p l i t u d e s o f the tidal c o n s t i t u e n t s a n d the differences in p h a s e w o u l d be expected to have little spatial v a r i a t i o n . This a p p e a r s to be the s i t u a t i o n for the s e m i - d i u r n a l tidal b a n d in the N o r t h Atlantic (Table l, Figs 1 a n d 2) j u d g i n g from tidal c o n s t i t u e n t s c a l c u l a t e d
* New Zealand Oceanographic Institute, D.S.I.R., P.O. Box 12-346 Wellington North, New Zealand. 481
0.31 0.37
0.39 0.28
Extreme phase difference
Extreme amplitude ratio
- 23 -33
-- 23 -23
0.16 0.21
0.16 0.22
Extreme phase difference
Extreme amplitude ratio
.... 29 4
0.19 0.18
12.6 12.7
12.6 12.6
12.6
12.35 12.65
12.85 12.4
- 19 -40
- 39 29
12.55
~h~
Calculated normal period
32 7
Phase diff. M, - $2
Vector average Scalar S.D.
No. of sites 13
Western North Atlantic
0.33 0.03
Vector average Scalar S.D.
No. of sites 15
Eastern North Atlantic Ocean
~-
Amp. ratio $2
52 28
52 26
31
11.4 15
11.4 11.5
11.4
Mode Q
O.25 0.21
0.23 .J" "" ~z.
0.23 0.01
0.23 0.20
0.22 0.21
0.21 0.Ol 2
Amp. ratio N2
1.04 0.70
12,95 12.7
14 10.3 18 23
0.77 0.90
10.7 15
12.9 12.65
0.84
17 29
12
1.98 1.19
1.60 1.55
1.56 0.22
N2
Amp. ratio $2
0.10
12.8
11 10.5
8.6 12.6
10.3
Mode Q
-q
~'~
12.85 12.6
12.9 12.65
15 26
21) 23
12.7
(h)
Calculated normal period
21 3
Phase diff. M2-N 2
46 48
58 43
4
5t
62 51
66 35
53 9
Phase diff. S, - N .
12,8 12.75
" "5 1_. ,_ 12.~
12.76
12.45 12.75
12.55 12.90
12.63
(h)
Calculated normal period
17 _.9
26 21
II 4 15
~', .t~ ~.8
ii .4
Mode Q
T , b l e I. Resonant period and (,)./,r fit.s ~!! the .semi-diurnal tidal t onstituent~ m the deep North .4tlantic Ocean to a simple a~cillator response" () the ~ame h,r cat h o! the ),12, S 2, and N 2 tidal constituent.s
O~ t,o
483
Estimates of the resonant period and Q
L/ @5
(])4 6 0 -
c)6 NORTH
Q7
@8
Q9
A TLAN TIC
(31o OCEAN
@
@11 o12
27 @13
@16
/ 4/d
~)28 ~)23 1920 21,22
1
60 W
I
40
Fig. 1. Location of the pelagic tide gauge observations in the deep North Atlantic Ocean (see CARTWRIGHT et al., 1979). Site numbers 1 to 15 are referred to in the text as sited in the eastern North Atlantic and site numbers 16 to 28 as in the western North Atlantic.
from pelagic tide gauge observations in the deep North Atlantic (CARTWRIGHT,ZETLERand HAMON~ 1979). PLATZMAN (1975) obtained a period of 12.8 h for the dominant normal mode near the semi-diurnal tidal band in the North Atlantic from numerical calculations of the normal modes of the Atlantic and Indian oceans. For a Q of 18 [using WEBB'S (1973) estimate] the half-power point for the next strong mode in the North Atlantic, which has a period of 14.4 h (PLATZMAN,1975), has a spread of period either side of 14.4 h of only 0.4 h. This supports the idea that one mode is dominant in the semi-diurnal tidal band in the North Atlantic and that a single mode development of the semi-diurnal tides is meaningful. GARRETT and GREENBERG (1977) analysed the response of the semi-diurnal tidal constituents at two locations in the western North Atlantic under the assumption that the response is dominated by one mode. They calculated values of the resonant period and Q of 12.82 h and 16.4, respectively, from observations at Bermuda and 12.79 h and 17.1, respectively, from observations at Halifax. In this paper, further estimates are made of the resonant period and of Q in the semi-diurnal tidal band of the North Atlantic using the observed deep-sea tidal constituents.
484
R
A sin
,k.
tiEA1"it
¢ N 2 M 2 tidal constituents
010
16 ,
: +~
,, ~'
2/
20~
3
~9
,
~9 ~) 2 3 !(z'h~ l q a 2 ; !
I 0'05 ~--
0 . 0i 5
005
- -
0 1I 0
- - - -
015 i
-----
020 I
0125 ...........
030 ~ - -
-
A 4;O~ ¢,
$2 M 2 t i d a l constituents e) 28 ~)23
~25 24~
~26
-01
(~27
22
~E)16 19
20~21
'
i:) 1c,
C, 14
015
2
0 20 (Z) 8
06
695
-0,25
Fig. 2.
Scatter diagram of the vector components A cos q~ and A sin 4, A the amplitude ratio of the ~b the phase difference, phase b-phase a. In the upper quadrant a is the N2 tidal constituent and b the M 2 constituent. In the lower quadrant a is the S 2 tidal constituent and b the M 2 constituent. Locations of the observations are shown in Fig. 1 ; 1 to 15 in the eastern North Atlantic and 16 to 28 in the western North Atlantic.
a/b tidal constituents and
In the Pacific Ocean, in contrast to the North Atlantic, amphidromes of the main semidiurnal tidal constituents are clearly separated spatially (see, e.g., LUTHER and WuNscn, 1975 ; ACCAD and PEKERIS,1978) and there are substantial spatial variations in the ratio of the amplitudes of the tidal constituents and phase difference between them (Figs 3 and 4). This indicates that at least two normal modes are present. The Pacific is sufficiently large
Estimates of the resonant period and Q
jvr
J
I'.V.X" .. ,
v
,X "',/
x
< /-
• "~ ' ' ~ \
,-,u,
.~.->~-,,
~f
.__L
.~~
~,...-
ll,~
I L(
!/;
~
"\
',
~"~"
~-~--__,__.y
' "
,, -,
I ,,o..
Io
'. ~ /I
,,., o 7 •
~.
,,....
~,
',,~ "
,~'~,'".e~-'-C-/4~L"Z,-I
I60"E
Y"k-,,
' I
%,--..
, ~o.~.,,_,<.
I: )eE
""'-
\
-_
Ms, X\\\ ~.Ix X
~- -.J_,t -.---~='~,.:r-', ~'. % ~
'[
485
.
} ~ ',/ ,
\
",,,
180'
140"W
Fig. 3. M 2 co-tidal chart for the central Pacific Ocean by LUTHER and WUNSCH (1975, Fig. 2). Solid curves represent equal Greenwich epoch G in solar hours and dashed curves are equal amplitude H in centimetres. (Used by permission.)
120"E
140*E
i60*E
180"
I60"W
140aW
Fig. 4. S 2 co-tidal chart for the central Pacific Ocean by LUTHER and WUNSCH (1975, Fig. 5). Solid curves represent equal Greenwich epoch G in solar hours and dashed curves are equal amplitude H in centimetres. (Used by permission.)
that the difference in period between the normal modes is small enough that spatial differences between the phase and amplitude distributions of the two main semi-diurnal tidal constituents, the principal lunar (M 2) and solar ($21 semi-diurnal tides, are evident even though they are separated in frequency by only 3.5'),,. In the Pacific Ocean. maxima of the M 2, S 2, and N 2 tidal constituents are observed in the western equatorial region (LUTHEr and WUNSCH, 1975. Figs 3 and 41 and minima in the Southern Ocean (A(:(al) a . d PEKErlS. 1978). The Q and resonant periods of the assumed two dominant normal modes in the semi-diurnal band m the western Pacific are estimated from the observed spatial separation of the area of maximum amplitude for M e, $2, and N 2 tidal constituents, the associated relative amplitude of the maxima, and the spatial variation of the age of the tidc WEUB (1973) argued that a large age of the tide is often associated with a localized resonance. However. there is a need for caution in using this interpretation in the westeHl South Pacific. There the observed difference in phase and amplitude of the coastal tides may be associated both with differences in the spatial distribution of the constituents offshore as well as a differing frequency response associated with resonance. For example. the decrease in the S2/M 2 amplitude ratio around southern New Zealand and the large variation in the age of the tide on the east coast of New Zealand are consequences of the differing coastal response to the differing spatial distribution of the $2 and M, tidal constituents on the west coast of New Zealand. f:or this reason. 1 choose to use tidal observations away from the area of extensive continental shelves.
IHI NORMAl
MODES
Following GARRETT and MINK (1971) the response of the oceans to tidal forcing ;it frequency ¢o is expressed as t
~
((x.t) :-- R~ Y,, 4.(!~)b~(x)e
itol
where .~ is the surface elevation at position x and time t. ~,;, is the frequency of the nth normal mode with associated eigenfunction S . ( x ) , which dissipates 2rcQ.-~ of its energy per cycle. Alternatively, Q. can be determined from the response curve of the mode for it gives the ratio of the resonant frequency .~,. to the bandwidth 2A.~. at the half-power points" Q = t,>./2ku),..
SINGLE MODE D E V E L O P M E N T FOR THE N O R T H ATLANTI(" OC E A N
For a single mode development the surface elevation reduces to ( = aoSo(x ) Ro(e)) e i°f'').
where Ro(t*~) = [ ( 0 1 0 - ¢ ' o ) 2 - k ( ½ Q o
l too) 2] ! -
0(¢0) = tan - ' [~ eoo/Qo(oJ o - e))]. ~o0. Qo are the resonant frequency and Q of the mode which are to be determined. The
Estimatesof the resonantperiod and Q
487
amplitude ratio of constituents a and b is given by
aoRo(cOa) = L(¢Oo- ~ob)2 + (½Oo - 10~o)2J a o The ratio bo/ao depends on the relative forcing of the constituents which, because the forcing function has the same spatial distribution for the constituents in the semi-diurnal band, reduces to the ratio of the equilibrium value of the constituents. Values of coo and Qo have been computed to give simultaneously the observed amplitude ratio and phase difference for the M2, $2, and N 2 tidal constituents. Pelagic tidal observations from the North Atlantic in water depths greater than 500 m have been used (Figs 1 and 2). Following CARTWRIGHTet al. (1979) the North Atlantic tidal data have been grouped into eastern and western sets. Vectorial means of the amplitude ratio and phase, and the standard deviations are given in Table 1 along with the estimates of the normal mode period and Q calculated for the vector means and the extreme values of amplitude ratio and phase. The observations in the eastern North Atlantic are within the ocean-wide amphidromic system; the spatial variations in the phase differences and amplitude ratios for the three constituents N2, M 2, and $2 are small. The estimated resonant period and Q of the normal mode calculated from the vector average are 12.55h and 11.4, respectively, from comparison of $2 and M2, and 12.70 h and 10.3, respectively, from comparison of N / a n d M2. The small difference between the periods and Q values for the observations exhibiting variations from the means (Table 1) lends confidence to the estimate using a single mode. The NE/M 2 amplitude ratio and phase difference for the western North Atlantic data are similar to those for the eastern North Atlantic (Table 1), as are the calculated resonant period and Q. However, the estimates involving the $2 constituent in the western North Atlantic differ substantially from the other estimates. There is a marked decrease in the $2/M2 amplitude ratio in the western North Atlantic compared to the eastern North Atlantic, accompanying a general decrease in all three constituents. The observed tidal constituents contain both radiational and gravitational contributions, the strongest input being in the S 2 tide (see, e.g., CARTWRIGHT, 1977). The spatial structure of the radiational and gravitational tidal forcing will differ and, therefore, even if the observed tides were dominated by a single mode, variations in the observed tidal ratios, particularly those involving the $2 tide, are expected. ZETLER(1971) showed that on the east coast of the United States the radiational $2 tide has an amplitude 0.16 that of the gravitational $2 tide and lags the gravitational $2 tide by 185 °. Neglecting any radiation contribution to the M s tide, Zetler's figures indicate that the $2/M2 gravitational tide amplitude ratio in the western North Atlantic is 19To larger than that calculated from the observations (which contain both the gravitational and radiational tide). The results of CARTWRIGHT(1968) on the north and east coast of Britain give a value of 0.17 for the ratio of the radiational to gravitational $2 tide with the phase of the radiational tide 226° greater than that of the gravitational tide. Again neglecting any radiational M 2 input, the above values give an estimate that the S2/M 2 gravitational tide is llTo larger than that observed. Using the above estimate of the influence of the $2 radiational tide gives an estimate of an expected 7To decrease in the observed $2/M2 tidal amplitude ratio from the eastern to the western North Atlantic. This is substantially smaller
488
R. A
tt~.ArH
than the 42% decrease observed. Presumably the excessive decrease in the Sz amplitude in the western North Atlantic is due to a localized response. Forcing the S2/M, data in this area to a single mode response leads to an excessive estimate of Q. In the above estimates Q has been assumed to be the same for each of the three tidal constituents. The frictional dissipative contribution to Q is non-linear however, and as pointed out by GaRRFTT (1972) if a tidal current is composed of one large constituent and several much smaller constituents the dissipative Q of the larger constituent is 50% greater than that of the smaller constituent. For QR the radiative Q (assumed to be the same for each constituent and thus neglecting differences in the radiative tidal input), QD, t.he dissipative Q for the largest tidal constituent, the Me tide, and Qt~:the dissipative Q for the smaller tidal constituents N 2 or $2 then Q~) = 1~5 Qo, and i
o,,,,,
l
1
I
o,, +
1
os =
1
1.5
1
o,,
Q s
Defining C = Qns/'QM~< I, the value of Qys and the resonant period (7'), which give the observed vector average amplitude ratio and phase difference between N 2 and M 2 tidal constituents, have been computed for values of C in the range 0.66 to I. In the eastern North Atlantic these Q and T values are closest to those calculated from the N 2 and $2 averages for C = 1, indicating that the frictional dissipation is small. A similar situation exists in the western North Atlantic (Fig. 5). The fit of a single normal mode to the deep-sea tidal data in the North Atlantic gives best estimates of the resonant period in the semi-diurnal tidal band of between 12.6 and 12.8 h with a Q of between 10 and 12.
FWO MODE DEVELOPMENT
FOR THE PACII-IC O C F A N
An estimate of the frequency separation ,~,÷ 1 -~,o,, of the normal modes near the semidiurnal band in the Pacific Ocean can be made by considering the time for a Kelvin wave to travel around the boundaries and along the equator. LONGUFT-HIGGINSand POND (1970) considered the free oscillations of fluid on a hemisphere bounded by meridians 180 apart. They showed that the normal mode frequencies can be calculated by considering the path length around the land boundaries and along the equator with allowance for a change in phase where the equatorial section meets the coast. In a real ocean, however, the meridians of the coastal boundaries are not continuous at the pole. Further, in a real ocean the perimeter north of the equator generally would not be the same as that south of the equator and there would be some difficulty in matching the phases at the equator. The phase change along the coasts has been calculated by assuming a Kelvin wave passes smoothly clockwise around the South Pacific and anticlockwise around the North Pacific. The distance eastwards along the equator, southwards along the west coast of South America to Palmer Land, westwards along Antarctica to about longitude 150'E and northwards to the equator is L ~ 36.2x 105 kin. For a mean depth of 3891 m (BuLE!K, 1966) the time for a long wave to travel around the perimeter L, neglecting phase changes where the equator meets the coast, is 51.5 h. The appropriate phase change at the equatorial corners is 0.2875 n (S = 4 in Fig. 6 LON~UET-H1GGINSand POND, 1970). The period of the normal mode (T) for n = 4 is then given by n T = 51.5+0.2875T, i.e.,
Estimates of the resonant period and Q
489
+
@ Vector Mean 4-
20' 4X X07 X [o
X09
08
IC
North-West Atlantic
12t6 12'.8 Period (hours)
12"~-
2O
13A'O
t+
Vector Mean(~)
o,
,z'4
tz'6
Xq0 XO 9Xo 8 X07 +
t2'e
North-East Atlantic
13!o
Period (hours)
Fig. 5. Resonant period (T) and QNs of a single normal mode fit to the deep-sea $2 and N 2 tidal constants in the western North Atlantic (upper) and eastern North Atlantic (lower). The encircled dot (C)) is the calculated values for the vector mean of the observations; the plus signs (+) are the values for the two extreme amplitude ratios and the two extreme phase differences; and crosses ( x ) are the calculated T and QSN for the vector mean of the S2 and M 2 observations for values of C = QsN/QM2 in the range 0.66 to 1.
T = 13.9 h. Similarly for n = 3 and n = 5 the resonant periods are 18.7 and 11 h, respectively. At or near the equator the phase of the m o d e in the South Pacific must match that in the N o r t h Pacific. N o r m a l m o d e periods for the N o r t h Pacific estimated as above are, for a m e a n depth of 3859 m (B1ALEK, 1966), 18.3 (n = 3), 13.6 (n = 4), and 10.7 (n = 5 ) h . Satisfactory matching of the m o d e s will result in the 'equator' for the m o d e s being shifted only about half a degree of latitude south of the geographic equator. The normal m o d e periods will then be 18.5, 13.7, and 10.9 h, with the latter two m o d e s separated by 25%. For the Q estimated by WEBB (1973) of 18 the half bandwidth at the half-power point is 3% of the resonant frequency, indicating that probably only two m o d e s contribute substantially to the observed tide. 21
490
R '~. ]-|,f:ArI-!
Using a t w o - m o d e development the semi-diurnal tidal elevation is given by ((w, .',-, t ) = R c [fl) S l( .\ )Rj 0o )e i'' -+-d 2S2(-'()R 2(1!)tc i0 } e i.... with normal m o d e frequencies for R,{w). R2(o~} of eq and "~2, respectively. Expressing the spatial dependence in terms of Kelvin waves travelling a r o u n d the b o u n d a r y with normal mode wave numbers k~ and k: the M 2 tidal elevation is given by ~M: = (" cos ((or~- !)), where
C = [A2+A2+2A1A2cos[(k t - k 2 ) x - t - O m - q ~ M . ] ] ~
[-AI
If)
sin (klX + O M ) + A 2 sin (k2x +q~M) ]
A 1 = alRl(WM ):
A2 = a2R2(C,)M )
and the spatial dependence perpendicular to the coast is incorporated in the coefficients u~ and a2. Similarly the $2 tidal elevation is given by ~s, = D cos (Wst-@), where
D = [B~+B~+2BIB 2 cos [(k I -k2jxq'-Os-4)s]] ~-
!3)
1[ Blsin(k,x+Os)+B2sin(kzx+(Ps)] d0 = tan
B~ cos ( k - 5 +-0~s)+i~2 ~ s T k z x + ~ s i /
B1 = thRl(ws,):
B2 =
(4j
b2Rz{eOs~L
Extreme tidal elevations in the open ocean (not associated with coastal amplification) occur for the M 2 tide at x = x~ where ?~M/PX = 0 and for the S 2 tide at x = x~ + A x where ?~s/?X = 0. This gives the relationships
(k~ -k2)x~ +OM--gaM = 0 {k 1--kzj(X I + A X ) + 0 S - ~ S
= 0,
t5! [6)
which in turn give an expression for the separation between the areas of m a x i m u m elevation for $2 and Mz:
Ax = (0u +(a M-Os-qas)/(k, -k2).
17)
Assuming that Q1 = Q2, with k,-/;2
\,g~\l-.~,!
where w 1 and ca 2 are related by the estimated frequency separation of the normal modes.
Estimates of the resonant period and Q
O
491
O0'
x
70
t x
60
Ix
sa C=l
I x
M~
50 40
I
S~
x x
....
$2 C=09 M'Z
30 20 I0
I(3"
/
x
Period (hours)
9 8"
7'. QI a"2
65" 43" 2
\ I0
4' 5"
8x8~
~i 8
Ill
1'2
Period (hours)
/ Period (hours)
Fig. 6. Plots of the possible values of QsN, relative coupling of the modes ( a l / a 2 ) , and spatial variation of the change in phase with frequency, all as a function of the resonant period T for a twomode development of the M2 and $2 semi-diurnal tides in the Pacific for C = 1. Also shown are the possible values ofQsN and T calculated from the $2 and N2 tides for C = 1 and from the $2 and M2 tide with C = QsN/QM2= 0.9.
and Ax by observation, equation (7) gives a relationship between the two u n k n o w n s Q1 and 091 . C o m p a r i s o n of Luther and W u n s c h ' s co-tidal charts for the western Pacific gives a value o f approx. Ax = 450 km for $2 and M2 and 580 k m for N2 and $2. Plots of T1 ( = 2n/~ol ) vs Q for these values of Ax are shown in Fig. 6. The value of Q for the M 2 tides (QM) was allowed to differ from that for the other two constituents (QNs) in the range C(QNs/QM)= 0.66 to 1; the curves of T1 vs Q for the separation of the m a x i m a of the three tidal constituents gives the closest agreement for QM = QNS ( Fig. 6).
492
J< .~, It~:~u
Tile ratio of the maxima of the $2 to M2 tidal amplitude, with equations 15i and Its} satisfied, is given by D
B 1 +B~
c max
=
=
b z / a 2 = bl/'U ~
.~i, + ,t~
F"
(/2 / ( l
The ratio
=
1
.
1
i8)
is the relative amplitude of the $2 to M 2 equilibrium tide O.e,,
0.465) for, as mentioned above, the spatial dependence of the forcing function is the same for both tidal constituents. For values of ~,~ and Q~ satisfying equation (7), equation (8) gives an estimate of the ratio h~/h 2 = a l i a z, which is a measure of the relative coupling of the separate modes (R~, R z ) to the forcing function. Comparison of LUTHER and WUNSCWS (1975) co-tidal charts for S 2 and M 2 in the western Pacific give a value of approx~ D / C = 0.4. Values ofbt/b2 = a~/'0 2 a r e possible only for the higher period arm of the T vs Q curve satisfying equation (7), that is for T between 9.6 and 10.2 h (Fig. 6). With two modes, the change in phase between tidal constituents is not constant in space, that is referring to equation (2), C2~I/'~'Xg(O ~ O. The spatial change in the variation of phase with frequency is a function of c,), Q~, ¢0~, c,~2, al/a2, and position x. A very approximate estimate of c?2O/?x?o9 in the vicinity of the area where the M 2 and S z tidal constituents have their largest open ocean amplitudes in the western Pacific can also be made from Luther and Wunsch's co-tidal charts along the equator between the amplitude maxima. This gives
(Os, - - ( 0 M:
~ .
AX
/
Values of 02¢,/(?x at/, computed for integer values ofQ~, with e~t specified by equation (7), a~/a2 by equation (8), and x given by equation (5), are shown in Fig. 6. A value of
- 5 x 10 -2 s m -~ is consistent with a resonant period T~ of 10.2 h and a Q of about 5 with approximately equal spatial coupling between the tidal potential and the two modes. The second mode has a period of 12.8 h. The (2 of 5 is small compared with other estimates for the world tides, G g r a E r r and M uNK's (1971) of 25, WEaWS (1973) of 18 and HENDERSHOTT'S(1972) of 34. However, if we accept the calculation of the frequency separation of the modes, the two modes need to be evenly distributed either side of the semi-diurnal tidal band or the Q has to be small to allow contribution to the semi-diurnal tides from both modes. Acknowledgements- The author is grateful to the Nuffield Foundation for a Nuflield TravellingFellowship and to
the New Zealand Department of Scientificand Industrial Research for provision of a Study Award, which allowed this research to be undertaken at the Bidston Observatory of the Institute of Oceanographic Sciences. I am indebted to the staff of the Bidston Observatory for making my stay both beneficial and enjoyable and in particular to Dr J. M. HUTHNANCEfor helpful discussion. Mrs J. CaMPaELLof the Bidston Observatory kindly drew Figs 5 and 6, and I am grateful to Drs LUTHERand WUNSCr~for allowing repreduction of Figs 3 and 4. REFERENCES
ACCADY. and C. L. PEKERIS(1978) Solution of the tidal equations for the M 2 and $2 tides in the world's oceans from a knowledge of the tidal potential alone. Philosophical Transactions of the Royal Society of London. A290 (1368), 235-266.
Estimates of the resonant period and Q
493
BIALEK E. L. (1966) Handbook of oceanographic tables. United States Naval Oceanographic Office, Special Publication SP68, 427 pp. CARTWRmHT D. E. (1968) A unified analysis of tides and surges round north and east Britain. Philosophical Transactions of the Royal Society of London, A2,63, !-55. CARTWRIGHT D. E. (1977) Oceanic tides. Reports on Progress in Physics. 40, 665-708. CARTWRIGHT D. E., B. D. ZETLERand B. V. HAMON, editors (1979) Pelagic tidal constants, (compiled by IAPSO Advisory Committee on Tides and Mean Sea Level) Publication Scientifique, International Association for the Physical Sciences of the Ocean, IUGG, No. 30, 65 pp. GARRETr C. J. R. (1972) Tidal resonance in the Bay of Fundy and Gulf of Maine. Nature, London, 238, 441-443. GARRETT C. J. R. and D. GREENBERG(1977) Predicting changes in tidal regime: The open boundary problem. Journal of Physical Oceanography, 7, 171-181. GARRETT C. J. R. and W. H. MUNK (1971) The age of the tide and the "Q" of the oceans. Deep-Sea Research, 18, 493 504. H ENDERSHOXTM. C. (1972) The effects of solid earth deformation on global ocean tides. Geophysical Journal o/ the Royal Astronomical Society. 29, 389-403. LONGUET-HIGGINS M. S. and G. S. POND (1970) The free oscillations of fluid on a hemisphere bounded by meridians of longitude. Philosophical Transactions of the Royal Society of London, A266, 193-223. LUTHER D. S. and C. WUNSCH(I 975) Tidal charts of the central Pacific Ocean. Journal of Physical Oceanography, 5, 222-230. PLATZMANG. W. (1975) Normal modes of the Atlantic and Indian oceans. Journal of Physical Oceanography, 5, 117-128. WEBB D. J. (1973) On the age of the semi-diurnal tide. Deep-Sea Research, 20, 847-852. ZETLER B. D. (1971) Radiational ocean tides along the coasts of the United States. Journal o/ Physical Oceanography, 1, 34-38.
0198-0149/81/050481-13 © 1981 PergamonPress Ltd.
Estimates of the resonant period and Q in the semi-diurnal tidal band in the North Atlantic and Pacific Oceans R. A. HEATH* (Received 26 October 1979 ;final revision received 1 December 1980: accepted 10 December 1980) Abstract--Fitting the deep-sea harmonic tidal constants to a single normal development of the North Atlantic semi-diurnal tides gives estimates of the resonant period based on the vectoraveraged data of between 12.6 and 12.8 h in the semi-diurnal tidal band with a Q of between 10 and 12. A two-mode development has been used in the Pacific Ocean. The percentage frequency separation between the modes has been calculated from the travel time of a Kelvin wave around the perimeter. A relationship between the resonant period and Q has been derived ; it depends on the observed spatial separation of the open-ocean amplitude maxima of the M 2, $2, and N2 tidal constituents evident in the tidal charts of the western Pacific. The ratio of the maximum open-ocean amplitude of the M2 and $2 tide was then used to estimate the relative coupling between the tidal potential and the two modes as a function of the resonant period. The two relationships are then used in calculating the frequency-dependent change in phase with position to allow comparison with the estimate made from the tidal gharts. The resonant periods near the semi-diurnal tidal band in the Pacific are estimated to be at 10.2 and 12.8 h with a Q, which has been assumed in the analysis to be the same for both modes, of about 5.
INTRODUCTION KNOWLEDGE of the n a t u r a l p e r i o d a n d rate of energy d i s s i p a t i o n o f the n o r m a l m o d e s of the o c e a n is basic to a n y u n d e r s t a n d i n g of the oceanic response to tidal forcing. GARRETT a n d M UNK (1971) e s t i m a t e d the lower b o u n d for the Q o f the n o r m a l m o d e s at frequencies in the semi-diurnal tidal band. o f 25, b a s e d o n the m e a n w o r l d - w i d e value o f the age o f the tide; the Q is a m e a s u r e of energy loss rate, the system loses 2n Q - ~ of its energy per cycle. WEBB (1973) p o i n t e d o u t t h a t m a n y of the regions with a large age of the tide are associated with p a r t i c u l a r b a t h y m e t r i c features a n d localised r e s o n a n c e s a n d are not, therefore, r e p r e s e n t a t i v e o f the o p e n ocean. H e o b t a i n e d estimates of the overall d e c a y time of the s e m i - d i u r n a l tidal energy o f 30 to 35 h (equivalent to Q - 18) from c o n s i d e r a t i o n o f b o t h the age of the tide in the o p e n ocean alone a n d the local resonances. HENDERSHOTr (1972) o b t a i n e d a Q o f 34 from direct n u m e r i c a l i n t e g r a t i o n o f L a p l a c e ' s tidal e q u a t i o n s . F o r an ocean where a tidal b a n d is d o m i n a t e d b y only a single n o r m a l m o d e , the ratio of the a m p l i t u d e s o f the tidal c o n s t i t u e n t s a n d the differences in p h a s e w o u l d be expected to have little spatial v a r i a t i o n . This a p p e a r s to be the s i t u a t i o n for the s e m i - d i u r n a l tidal b a n d in the N o r t h Atlantic (Table l, Figs 1 a n d 2) j u d g i n g from tidal c o n s t i t u e n t s c a l c u l a t e d
* New Zealand Oceanographic Institute, D.S.I.R., P.O. Box 12-346 Wellington North, New Zealand. 481
0.31 0.37
0.39 0.28
Extreme phase difference
Extreme amplitude ratio
- 23 -33
-- 23 -23
0.16 0.21
0.16 0.22
Extreme phase difference
Extreme amplitude ratio
.... 29 4
0.19 0.18
12.6 12.7
12.6 12.6
12.6
12.35 12.65
12.85 12.4
- 19 -40
- 39 29
12.55
~h~
Calculated normal period
32 7
Phase diff. M, - $2
Vector average Scalar S.D.
No. of sites 13
Western North Atlantic
0.33 0.03
Vector average Scalar S.D.
No. of sites 15
Eastern North Atlantic Ocean
~-
Amp. ratio $2
52 28
52 26
31
11.4 15
11.4 11.5
11.4
Mode Q
O.25 0.21
0.23 .J" "" ~z.
0.23 0.01
0.23 0.20
0.22 0.21
0.21 0.Ol 2
Amp. ratio N2
1.04 0.70
12,95 12.7
14 10.3 18 23
0.77 0.90
10.7 15
12.9 12.65
0.84
17 29
12
1.98 1.19
1.60 1.55
1.56 0.22
N2
Amp. ratio $2
0.10
12.8
11 10.5
8.6 12.6
10.3
Mode Q
-q
~'~
12.85 12.6
12.9 12.65
15 26
21) 23
12.7
(h)
Calculated normal period
21 3
Phase diff. M2-N 2
46 48
58 43
4
5t
62 51
66 35
53 9
Phase diff. S, - N .
12,8 12.75
" "5 1_. ,_ 12.~
12.76
12.45 12.75
12.55 12.90
12.63
(h)
Calculated normal period
17 _.9
26 21
II 4 15
~', .t~ ~.8
ii .4
Mode Q
T , b l e I. Resonant period and (,)./,r fit.s ~!! the .semi-diurnal tidal t onstituent~ m the deep North .4tlantic Ocean to a simple a~cillator response" () the ~ame h,r cat h o! the ),12, S 2, and N 2 tidal constituent.s
O~ t,o
483
Estimates of the resonant period and Q
L/ @5
(])4 6 0 -
c)6 NORTH
Q7
@8
Q9
A TLAN TIC
(31o OCEAN
@
@11 o12
27 @13
@16
/ 4/d
~)28 ~)23 1920 21,22
1
60 W
I
40
Fig. 1. Location of the pelagic tide gauge observations in the deep North Atlantic Ocean (see CARTWRIGHT et al., 1979). Site numbers 1 to 15 are referred to in the text as sited in the eastern North Atlantic and site numbers 16 to 28 as in the western North Atlantic.
from pelagic tide gauge observations in the deep North Atlantic (CARTWRIGHT,ZETLERand HAMON~ 1979). PLATZMAN (1975) obtained a period of 12.8 h for the dominant normal mode near the semi-diurnal tidal band in the North Atlantic from numerical calculations of the normal modes of the Atlantic and Indian oceans. For a Q of 18 [using WEBB'S (1973) estimate] the half-power point for the next strong mode in the North Atlantic, which has a period of 14.4 h (PLATZMAN,1975), has a spread of period either side of 14.4 h of only 0.4 h. This supports the idea that one mode is dominant in the semi-diurnal tidal band in the North Atlantic and that a single mode development of the semi-diurnal tides is meaningful. GARRETT and GREENBERG (1977) analysed the response of the semi-diurnal tidal constituents at two locations in the western North Atlantic under the assumption that the response is dominated by one mode. They calculated values of the resonant period and Q of 12.82 h and 16.4, respectively, from observations at Bermuda and 12.79 h and 17.1, respectively, from observations at Halifax. In this paper, further estimates are made of the resonant period and of Q in the semi-diurnal tidal band of the North Atlantic using the observed deep-sea tidal constituents.
484
R
A sin
,k.
tiEA1"it
¢ N 2 M 2 tidal constituents
010
16 ,
: +~
,, ~'
2/
20~
3
~9
,
~9 ~) 2 3 !(z'h~ l q a 2 ; !
I 0'05 ~--
0 . 0i 5
005
- -
0 1I 0
- - - -
015 i
-----
020 I
0125 ...........
030 ~ - -
-
A 4;O~ ¢,
$2 M 2 t i d a l constituents e) 28 ~)23
~25 24~
~26
-01
(~27
22
~E)16 19
20~21
'
i:) 1c,
C, 14
015
2
0 20 (Z) 8
06
695
-0,25
Fig. 2.
Scatter diagram of the vector components A cos q~ and A sin 4, A the amplitude ratio of the ~b the phase difference, phase b-phase a. In the upper quadrant a is the N2 tidal constituent and b the M 2 constituent. In the lower quadrant a is the S 2 tidal constituent and b the M 2 constituent. Locations of the observations are shown in Fig. 1 ; 1 to 15 in the eastern North Atlantic and 16 to 28 in the western North Atlantic.
a/b tidal constituents and
In the Pacific Ocean, in contrast to the North Atlantic, amphidromes of the main semidiurnal tidal constituents are clearly separated spatially (see, e.g., LUTHER and WuNscn, 1975 ; ACCAD and PEKERIS,1978) and there are substantial spatial variations in the ratio of the amplitudes of the tidal constituents and phase difference between them (Figs 3 and 4). This indicates that at least two normal modes are present. The Pacific is sufficiently large
Estimates of the resonant period and Q
jvr
J
I'.V.X" .. ,
v
,X "',/
x
< /-
• "~ ' ' ~ \
,-,u,
.~.->~-,,
~f
.__L
.~~
~,...-
ll,~
I L(
!/;
~
"\
',
~"~"
~-~--__,__.y
' "
,, -,
I ,,o..
Io
'. ~ /I
,,., o 7 •
~.
,,....
~,
',,~ "
,~'~,'".e~-'-C-/4~L"Z,-I
I60"E
Y"k-,,
' I
%,--..
, ~o.~.,,_,<.
I: )eE
""'-
\
-_
Ms, X\\\ ~.Ix X
~- -.J_,t -.---~='~,.:r-', ~'. % ~
'[
485
.
} ~ ',/ ,
\
",,,
180'
140"W
Fig. 3. M 2 co-tidal chart for the central Pacific Ocean by LUTHER and WUNSCH (1975, Fig. 2). Solid curves represent equal Greenwich epoch G in solar hours and dashed curves are equal amplitude H in centimetres. (Used by permission.)
120"E
140*E
i60*E
180"
I60"W
140aW
Fig. 4. S 2 co-tidal chart for the central Pacific Ocean by LUTHER and WUNSCH (1975, Fig. 5). Solid curves represent equal Greenwich epoch G in solar hours and dashed curves are equal amplitude H in centimetres. (Used by permission.)
that the difference in period between the normal modes is small enough that spatial differences between the phase and amplitude distributions of the two main semi-diurnal tidal constituents, the principal lunar (M 2) and solar ($21 semi-diurnal tides, are evident even though they are separated in frequency by only 3.5'),,. In the Pacific Ocean. maxima of the M 2, S 2, and N 2 tidal constituents are observed in the western equatorial region (LUTHEr and WUNSCH, 1975. Figs 3 and 41 and minima in the Southern Ocean (A(:(al) a . d PEKErlS. 1978). The Q and resonant periods of the assumed two dominant normal modes in the semi-diurnal band m the western Pacific are estimated from the observed spatial separation of the area of maximum amplitude for M e, $2, and N 2 tidal constituents, the associated relative amplitude of the maxima, and the spatial variation of the age of the tidc WEUB (1973) argued that a large age of the tide is often associated with a localized resonance. However. there is a need for caution in using this interpretation in the westeHl South Pacific. There the observed difference in phase and amplitude of the coastal tides may be associated both with differences in the spatial distribution of the constituents offshore as well as a differing frequency response associated with resonance. For example. the decrease in the S2/M 2 amplitude ratio around southern New Zealand and the large variation in the age of the tide on the east coast of New Zealand are consequences of the differing coastal response to the differing spatial distribution of the $2 and M, tidal constituents on the west coast of New Zealand. f:or this reason. 1 choose to use tidal observations away from the area of extensive continental shelves.
IHI NORMAl
MODES
Following GARRETT and MINK (1971) the response of the oceans to tidal forcing ;it frequency ¢o is expressed as t
~
((x.t) :-- R~ Y,, 4.(!~)b~(x)e
itol
where .~ is the surface elevation at position x and time t. ~,;, is the frequency of the nth normal mode with associated eigenfunction S . ( x ) , which dissipates 2rcQ.-~ of its energy per cycle. Alternatively, Q. can be determined from the response curve of the mode for it gives the ratio of the resonant frequency .~,. to the bandwidth 2A.~. at the half-power points" Q = t,>./2ku),..
SINGLE MODE D E V E L O P M E N T FOR THE N O R T H ATLANTI(" OC E A N
For a single mode development the surface elevation reduces to ( = aoSo(x ) Ro(e)) e i°f'').
where Ro(t*~) = [ ( 0 1 0 - ¢ ' o ) 2 - k ( ½ Q o
l too) 2] ! -
0(¢0) = tan - ' [~ eoo/Qo(oJ o - e))]. ~o0. Qo are the resonant frequency and Q of the mode which are to be determined. The
Estimatesof the resonantperiod and Q
487
amplitude ratio of constituents a and b is given by
aoRo(cOa) = L(¢Oo- ~ob)2 + (½Oo - 10~o)2J a o The ratio bo/ao depends on the relative forcing of the constituents which, because the forcing function has the same spatial distribution for the constituents in the semi-diurnal band, reduces to the ratio of the equilibrium value of the constituents. Values of coo and Qo have been computed to give simultaneously the observed amplitude ratio and phase difference for the M2, $2, and N 2 tidal constituents. Pelagic tidal observations from the North Atlantic in water depths greater than 500 m have been used (Figs 1 and 2). Following CARTWRIGHTet al. (1979) the North Atlantic tidal data have been grouped into eastern and western sets. Vectorial means of the amplitude ratio and phase, and the standard deviations are given in Table 1 along with the estimates of the normal mode period and Q calculated for the vector means and the extreme values of amplitude ratio and phase. The observations in the eastern North Atlantic are within the ocean-wide amphidromic system; the spatial variations in the phase differences and amplitude ratios for the three constituents N2, M 2, and $2 are small. The estimated resonant period and Q of the normal mode calculated from the vector average are 12.55h and 11.4, respectively, from comparison of $2 and M2, and 12.70 h and 10.3, respectively, from comparison of N / a n d M2. The small difference between the periods and Q values for the observations exhibiting variations from the means (Table 1) lends confidence to the estimate using a single mode. The NE/M 2 amplitude ratio and phase difference for the western North Atlantic data are similar to those for the eastern North Atlantic (Table 1), as are the calculated resonant period and Q. However, the estimates involving the $2 constituent in the western North Atlantic differ substantially from the other estimates. There is a marked decrease in the $2/M2 amplitude ratio in the western North Atlantic compared to the eastern North Atlantic, accompanying a general decrease in all three constituents. The observed tidal constituents contain both radiational and gravitational contributions, the strongest input being in the S 2 tide (see, e.g., CARTWRIGHT, 1977). The spatial structure of the radiational and gravitational tidal forcing will differ and, therefore, even if the observed tides were dominated by a single mode, variations in the observed tidal ratios, particularly those involving the $2 tide, are expected. ZETLER(1971) showed that on the east coast of the United States the radiational $2 tide has an amplitude 0.16 that of the gravitational $2 tide and lags the gravitational $2 tide by 185 °. Neglecting any radiation contribution to the M s tide, Zetler's figures indicate that the $2/M2 gravitational tide amplitude ratio in the western North Atlantic is 19To larger than that calculated from the observations (which contain both the gravitational and radiational tide). The results of CARTWRIGHT(1968) on the north and east coast of Britain give a value of 0.17 for the ratio of the radiational to gravitational $2 tide with the phase of the radiational tide 226° greater than that of the gravitational tide. Again neglecting any radiational M 2 input, the above values give an estimate that the S2/M 2 gravitational tide is llTo larger than that observed. Using the above estimate of the influence of the $2 radiational tide gives an estimate of an expected 7To decrease in the observed $2/M2 tidal amplitude ratio from the eastern to the western North Atlantic. This is substantially smaller
488
R. A
tt~.ArH
than the 42% decrease observed. Presumably the excessive decrease in the Sz amplitude in the western North Atlantic is due to a localized response. Forcing the S2/M, data in this area to a single mode response leads to an excessive estimate of Q. In the above estimates Q has been assumed to be the same for each of the three tidal constituents. The frictional dissipative contribution to Q is non-linear however, and as pointed out by GaRRFTT (1972) if a tidal current is composed of one large constituent and several much smaller constituents the dissipative Q of the larger constituent is 50% greater than that of the smaller constituent. For QR the radiative Q (assumed to be the same for each constituent and thus neglecting differences in the radiative tidal input), QD, t.he dissipative Q for the largest tidal constituent, the Me tide, and Qt~:the dissipative Q for the smaller tidal constituents N 2 or $2 then Q~) = 1~5 Qo, and i
o,,,,,
l
1
I
o,, +
1
os =
1
1.5
1
o,,
Q s
Defining C = Qns/'QM~< I, the value of Qys and the resonant period (7'), which give the observed vector average amplitude ratio and phase difference between N 2 and M 2 tidal constituents, have been computed for values of C in the range 0.66 to I. In the eastern North Atlantic these Q and T values are closest to those calculated from the N 2 and $2 averages for C = 1, indicating that the frictional dissipation is small. A similar situation exists in the western North Atlantic (Fig. 5). The fit of a single normal mode to the deep-sea tidal data in the North Atlantic gives best estimates of the resonant period in the semi-diurnal tidal band of between 12.6 and 12.8 h with a Q of between 10 and 12.
FWO MODE DEVELOPMENT
FOR THE PACII-IC O C F A N
An estimate of the frequency separation ,~,÷ 1 -~,o,, of the normal modes near the semidiurnal band in the Pacific Ocean can be made by considering the time for a Kelvin wave to travel around the boundaries and along the equator. LONGUFT-HIGGINSand POND (1970) considered the free oscillations of fluid on a hemisphere bounded by meridians 180 apart. They showed that the normal mode frequencies can be calculated by considering the path length around the land boundaries and along the equator with allowance for a change in phase where the equatorial section meets the coast. In a real ocean, however, the meridians of the coastal boundaries are not continuous at the pole. Further, in a real ocean the perimeter north of the equator generally would not be the same as that south of the equator and there would be some difficulty in matching the phases at the equator. The phase change along the coasts has been calculated by assuming a Kelvin wave passes smoothly clockwise around the South Pacific and anticlockwise around the North Pacific. The distance eastwards along the equator, southwards along the west coast of South America to Palmer Land, westwards along Antarctica to about longitude 150'E and northwards to the equator is L ~ 36.2x 105 kin. For a mean depth of 3891 m (BuLE!K, 1966) the time for a long wave to travel around the perimeter L, neglecting phase changes where the equator meets the coast, is 51.5 h. The appropriate phase change at the equatorial corners is 0.2875 n (S = 4 in Fig. 6 LON~UET-H1GGINSand POND, 1970). The period of the normal mode (T) for n = 4 is then given by n T = 51.5+0.2875T, i.e.,
Estimates of the resonant period and Q
489
+
@ Vector Mean 4-
20' 4X X07 X [o
X09
08
IC
North-West Atlantic
12t6 12'.8 Period (hours)
12"~-
2O
13A'O
t+
Vector Mean(~)
o,
,z'4
tz'6
Xq0 XO 9Xo 8 X07 +
t2'e
North-East Atlantic
13!o
Period (hours)
Fig. 5. Resonant period (T) and QNs of a single normal mode fit to the deep-sea $2 and N 2 tidal constants in the western North Atlantic (upper) and eastern North Atlantic (lower). The encircled dot (C)) is the calculated values for the vector mean of the observations; the plus signs (+) are the values for the two extreme amplitude ratios and the two extreme phase differences; and crosses ( x ) are the calculated T and QSN for the vector mean of the S2 and M 2 observations for values of C = QsN/QM2 in the range 0.66 to 1.
T = 13.9 h. Similarly for n = 3 and n = 5 the resonant periods are 18.7 and 11 h, respectively. At or near the equator the phase of the m o d e in the South Pacific must match that in the N o r t h Pacific. N o r m a l m o d e periods for the N o r t h Pacific estimated as above are, for a m e a n depth of 3859 m (B1ALEK, 1966), 18.3 (n = 3), 13.6 (n = 4), and 10.7 (n = 5 ) h . Satisfactory matching of the m o d e s will result in the 'equator' for the m o d e s being shifted only about half a degree of latitude south of the geographic equator. The normal m o d e periods will then be 18.5, 13.7, and 10.9 h, with the latter two m o d e s separated by 25%. For the Q estimated by WEBB (1973) of 18 the half bandwidth at the half-power point is 3% of the resonant frequency, indicating that probably only two m o d e s contribute substantially to the observed tide. 21
490
R '~. ]-|,f:ArI-!
Using a t w o - m o d e development the semi-diurnal tidal elevation is given by ((w, .',-, t ) = R c [fl) S l( .\ )Rj 0o )e i'' -+-d 2S2(-'()R 2(1!)tc i0 } e i.... with normal m o d e frequencies for R,{w). R2(o~} of eq and "~2, respectively. Expressing the spatial dependence in terms of Kelvin waves travelling a r o u n d the b o u n d a r y with normal mode wave numbers k~ and k: the M 2 tidal elevation is given by ~M: = (" cos ((or~- !)), where
C = [A2+A2+2A1A2cos[(k t - k 2 ) x - t - O m - q ~ M . ] ] ~
[-AI
If)
sin (klX + O M ) + A 2 sin (k2x +q~M) ]
A 1 = alRl(WM ):
A2 = a2R2(C,)M )
and the spatial dependence perpendicular to the coast is incorporated in the coefficients u~ and a2. Similarly the $2 tidal elevation is given by ~s, = D cos (Wst-@), where
D = [B~+B~+2BIB 2 cos [(k I -k2jxq'-Os-4)s]] ~-
!3)
1[ Blsin(k,x+Os)+B2sin(kzx+(Ps)] d0 = tan
B~ cos ( k - 5 +-0~s)+i~2 ~ s T k z x + ~ s i /
B1 = thRl(ws,):
B2 =
(4j
b2Rz{eOs~L
Extreme tidal elevations in the open ocean (not associated with coastal amplification) occur for the M 2 tide at x = x~ where ?~M/PX = 0 and for the S 2 tide at x = x~ + A x where ?~s/?X = 0. This gives the relationships
(k~ -k2)x~ +OM--gaM = 0 {k 1--kzj(X I + A X ) + 0 S - ~ S
= 0,
t5! [6)
which in turn give an expression for the separation between the areas of m a x i m u m elevation for $2 and Mz:
Ax = (0u +(a M-Os-qas)/(k, -k2).
17)
Assuming that Q1 = Q2, with k,-/;2
\,g~\l-.~,!
where w 1 and ca 2 are related by the estimated frequency separation of the normal modes.
Estimates of the resonant period and Q
O
491
O0'
x
70
t x
60
Ix
sa C=l
I x
M~
50 40
I
S~
x x
....
$2 C=09 M'Z
30 20 I0
I(3"
/
x
Period (hours)
9 8"
7'. QI a"2
65" 43" 2
\ I0
4' 5"
8x8~
~i 8
Ill
1'2
Period (hours)
/ Period (hours)
Fig. 6. Plots of the possible values of QsN, relative coupling of the modes ( a l / a 2 ) , and spatial variation of the change in phase with frequency, all as a function of the resonant period T for a twomode development of the M2 and $2 semi-diurnal tides in the Pacific for C = 1. Also shown are the possible values ofQsN and T calculated from the $2 and N2 tides for C = 1 and from the $2 and M2 tide with C = QsN/QM2= 0.9.
and Ax by observation, equation (7) gives a relationship between the two u n k n o w n s Q1 and 091 . C o m p a r i s o n of Luther and W u n s c h ' s co-tidal charts for the western Pacific gives a value o f approx. Ax = 450 km for $2 and M2 and 580 k m for N2 and $2. Plots of T1 ( = 2n/~ol ) vs Q for these values of Ax are shown in Fig. 6. The value of Q for the M 2 tides (QM) was allowed to differ from that for the other two constituents (QNs) in the range C(QNs/QM)= 0.66 to 1; the curves of T1 vs Q for the separation of the m a x i m a of the three tidal constituents gives the closest agreement for QM = QNS ( Fig. 6).
492
J< .~, It~:~u
Tile ratio of the maxima of the $2 to M2 tidal amplitude, with equations 15i and Its} satisfied, is given by D
B 1 +B~
c max
=
=
b z / a 2 = bl/'U ~
.~i, + ,t~
F"
(/2 / ( l
The ratio
=
1
.
1
i8)
is the relative amplitude of the $2 to M 2 equilibrium tide O.e,,
0.465) for, as mentioned above, the spatial dependence of the forcing function is the same for both tidal constituents. For values of ~,~ and Q~ satisfying equation (7), equation (8) gives an estimate of the ratio h~/h 2 = a l i a z, which is a measure of the relative coupling of the separate modes (R~, R z ) to the forcing function. Comparison of LUTHER and WUNSCWS (1975) co-tidal charts for S 2 and M 2 in the western Pacific give a value of approx~ D / C = 0.4. Values ofbt/b2 = a~/'0 2 a r e possible only for the higher period arm of the T vs Q curve satisfying equation (7), that is for T between 9.6 and 10.2 h (Fig. 6). With two modes, the change in phase between tidal constituents is not constant in space, that is referring to equation (2), C2~I/'~'Xg(O ~ O. The spatial change in the variation of phase with frequency is a function of c,), Q~, ¢0~, c,~2, al/a2, and position x. A very approximate estimate of c?2O/?x?o9 in the vicinity of the area where the M 2 and S z tidal constituents have their largest open ocean amplitudes in the western Pacific can also be made from Luther and Wunsch's co-tidal charts along the equator between the amplitude maxima. This gives
(Os, - - ( 0 M:
~ .
AX
/
Values of 02¢,/(?x at/, computed for integer values ofQ~, with e~t specified by equation (7), a~/a2 by equation (8), and x given by equation (5), are shown in Fig. 6. A value of
- 5 x 10 -2 s m -~ is consistent with a resonant period T~ of 10.2 h and a Q of about 5 with approximately equal spatial coupling between the tidal potential and the two modes. The second mode has a period of 12.8 h. The (2 of 5 is small compared with other estimates for the world tides, G g r a E r r and M uNK's (1971) of 25, WEaWS (1973) of 18 and HENDERSHOTT'S(1972) of 34. However, if we accept the calculation of the frequency separation of the modes, the two modes need to be evenly distributed either side of the semi-diurnal tidal band or the Q has to be small to allow contribution to the semi-diurnal tides from both modes. Acknowledgements- The author is grateful to the Nuffield Foundation for a Nuflield TravellingFellowship and to
the New Zealand Department of Scientificand Industrial Research for provision of a Study Award, which allowed this research to be undertaken at the Bidston Observatory of the Institute of Oceanographic Sciences. I am indebted to the staff of the Bidston Observatory for making my stay both beneficial and enjoyable and in particular to Dr J. M. HUTHNANCEfor helpful discussion. Mrs J. CaMPaELLof the Bidston Observatory kindly drew Figs 5 and 6, and I am grateful to Drs LUTHERand WUNSCr~for allowing repreduction of Figs 3 and 4. REFERENCES
ACCADY. and C. L. PEKERIS(1978) Solution of the tidal equations for the M 2 and $2 tides in the world's oceans from a knowledge of the tidal potential alone. Philosophical Transactions of the Royal Society of London. A290 (1368), 235-266.
Estimates of the resonant period and Q
493
BIALEK E. L. (1966) Handbook of oceanographic tables. United States Naval Oceanographic Office, Special Publication SP68, 427 pp. CARTWRmHT D. E. (1968) A unified analysis of tides and surges round north and east Britain. Philosophical Transactions of the Royal Society of London, A2,63, !-55. CARTWRIGHT D. E. (1977) Oceanic tides. Reports on Progress in Physics. 40, 665-708. CARTWRIGHT D. E., B. D. ZETLERand B. V. HAMON, editors (1979) Pelagic tidal constants, (compiled by IAPSO Advisory Committee on Tides and Mean Sea Level) Publication Scientifique, International Association for the Physical Sciences of the Ocean, IUGG, No. 30, 65 pp. GARRETr C. J. R. (1972) Tidal resonance in the Bay of Fundy and Gulf of Maine. Nature, London, 238, 441-443. GARRETT C. J. R. and D. GREENBERG(1977) Predicting changes in tidal regime: The open boundary problem. Journal of Physical Oceanography, 7, 171-181. GARRETT C. J. R. and W. H. MUNK (1971) The age of the tide and the "Q" of the oceans. Deep-Sea Research, 18, 493 504. H ENDERSHOXTM. C. (1972) The effects of solid earth deformation on global ocean tides. Geophysical Journal o/ the Royal Astronomical Society. 29, 389-403. LONGUET-HIGGINS M. S. and G. S. POND (1970) The free oscillations of fluid on a hemisphere bounded by meridians of longitude. Philosophical Transactions of the Royal Society of London, A266, 193-223. LUTHER D. S. and C. WUNSCH(I 975) Tidal charts of the central Pacific Ocean. Journal of Physical Oceanography, 5, 222-230. PLATZMANG. W. (1975) Normal modes of the Atlantic and Indian oceans. Journal of Physical Oceanography, 5, 117-128. WEBB D. J. (1973) On the age of the semi-diurnal tide. Deep-Sea Research, 20, 847-852. ZETLER B. D. (1971) Radiational ocean tides along the coasts of the United States. Journal o/ Physical Oceanography, 1, 34-38.