I n t e r n a t i o n a l H y d r o g r a p h ie R e v ie w , M on aco, L V (2 ), J u ly 1978
OCEANIC TIDES
b y Dr. D.E. C A R T W R I G H T
In stitu te o f O ce a n o g ra p h ic Sciences,
B idston , B irk en h ea d , E n gla n d
T h is p a p e r w as f ir s t p u b lis h e d in R e p o r t s on P r o g r e s s in P h y s ics ,
is rep ro d u c e d w it h the k in d p e rm is sio n o f th e In s titu te o f P h y s ics ,
re ta in th e c o p y rig h t.
1977 (40), an d
London, w ho
ABSTRACT
T id a l research has had a lon g h is to ry , bu t the o u ts ta n d in g p rob lem s
still d e fe a t cu rren t research tech niqu es, in c lu d in g la rge-sca le co m p u ta tion .
T h e d e fin itio n o f the tid e-g e n era tin g p o te n tia l, basic to a ll research, is
re v ie w e d in m od ern term s. M o d ern usage in an alysis in tro d u ces the con cept
o f tid a l ‘ a d m itta n ce ’ fu n ction s, th ou gh lim ite d to ra th e r n a r ro w fr e q u e n c y
bands. A ‘ ra d ia tio n a l p o te n tia l ’ has also been fo u n d u sefu l in d e fin in g the
p a rts o f tid a l signals w h ich a re due d ir e c tly o r in d ire c tly to solar ra d ia tio n .
L a p la c e ’ s tid a l equ ation s ( l t e ) o m it s e v e ra l term s fr o m the fu ll d y n a ­
m ica l eq u ation s, in c lu d in g the ve rtic a l a cce lera tio n . C o n troversies about
the ju s tific a tio n fo r l t e h a ve been fa ir ly w e ll settled b y M tt.e s ’ (1974)
d e m o n stra tio n that, w h en reg a rd ed as the lo w e s t order in te rn a l w a v e m od e
in a s tr a tifie d flu id , solu tion s o f the fu ll eq u a tion s do c o n v e rg e to those o f
l t e . S o lu tion s in basins o f sim p le g e o m e try are re v ie w e d and d is tin gu ish ed
fr o m a ttem p ts, m a in ly b y P r o u d m a n , to so lve fo r the rea l oceans b y
d iv is io n in to ele m en ta ry strips, and fr o m lo c a liz e d syntheses as U s e d
b y M u n k fo r the tides o ff C a liforn ia .
T h e m o d e rn com p u ter seem ed to p ro v id e a ‘ b re a k th ro u g h ’ in s o lv in g
fo r the w o r ld ’ s oceans, but the resu lts o f in d ep en d en t w o r k e r s d iffe r,
m o s tly because o f the in a d eq u a cy in th e ir tre a tm e n t o f fr ic tio n and fo r the
elastic y ie ld in g o f the E arth . T h e d iffe re n c e b etw een es tim a tes o f the rate
o f w o r k in g o f the M oon on the tides, abou t 3.5 X 1012 W , and o f the p o w e r
loss in sh a llo w seas, about 1.7 X 1012 W , m a rk s a seriou s u n id e n tifie d sink
o f en erg y. C on version to in tern a l tides has been su ggested as a sink, but
ca lcu la tion s a re at presen t in con clu sive.
lte
C om p u ted m odels o f the g lo b a l tides do, h o w e v e r, a gree w e ll w ith recen t
estim ates o f the M o o n ’ s d ece lera tio n in lo n g itu d e. C alcu lation s based on
such a stro n o m ica l m ea su res su ggest an even h ig h er fig u r e fo r the rate o f
w o r k in g , 4.3 X 1012 W . T h is is c o n sid era b ly g rea te r than the fig u r e o f
2.7 X 1012 W accepted u n til abou t 1970, thus accen tu a tin g the p rob lem o f
the m is s in g en erg y sink.
F u tu re p rog ress w ill r e q u ire m o re ex te n siv e p ro g ra m m e s o f p e la g ic
tid a l m ea su rem en t. T h e te c h n o lo g y fo r th is is w e ll d e velop ed but slow .
A lt im e t r y o f the sea su rfa ce fr o m sa tellites cou ld p ro v id e the m uch needed
g lo b a l c o v e ra g e i f p r o p e r ly b a ck ed up b y h ig h -p re cis io n tra c k in g o f the
in s tru m e n te d sa tellite.
1. H IS T O R IC A L RETROSPECT
T id a l science as a b ra n ch o f d y n a m ic a l o c ea n o g ra p h y began w iln the
w o r k o f L a p l a c e (17 75 ), w h ic h h e la te r su m m a rized w ith subsequent w o r k
in h is M é c a n iq u e Celeste (1 8 2 4 ). O f cou rse, th e fo u n d a tio n s o f the su bject
in g r a v ita tio n a l th e o ry g o back s till fu r th e r to N e w t o n ’ s P r i n c i p i n (1687),
w h ic h in s p ire d seve ra l e x te n siv e series o f tid a l m easu rem en ts and th e ir
c o rr e la tio n w ith lu n ar m o tio n s ( D e a c o n , 1970, C a r t w r i g h t , 1972). I t is
th e r e fo r e a science o f v e n e ra b le age, as w as recog n ized b y the IA P S O (* )
w h e n th e ir ‘ C om m ittee on T id e s ’ , u n der th e late P r o fe s s o r P r o u d m a n o f
L iv e r p o o l, c o m p ile d a co m p lete b ib lio g r a p h y o f tid a l p u b lica tion s fr o m 1665
to 1939, a w o r k w h ic h has since been u p d a ted to 1969 ( I A P O 1955, 1957,
1971). T h is co m p ila tio n rev ea ls th e fo llo w in g counts o f pu b lication s o v e r
50 y e a r in te rv a ls sin ce 1670 :
1670-1719
1720-1769
1770-1819
1820-1869
1870-1919
1920-1969
27
29
37
262
1 121
2 100
T h u s , fo r a fa ir ly u n ifo rm su b ject m a tter, the q u a n tity o f research lias
been v ig o r o u s ly , i f not e x p o n e n tia lly , ris in g fo r the last 150 years. T o be
fa ir , th ere a re signs th at the ou tp u t fo r the n ext h a lf-c e n tu ry w ill not exceed
th e last fig u r e — the 1960 d ecade p rod u ced about the sam e n u m ber o f
p a p e rs (41 0) as th e 1920 d ecade — but e v id e n tly research in this old su bject
is still flo u ris h in g .
T h e above fa c ts m a y su rp rise the m a n y ed u cated p eop le w h o think o f
th e tid es as a p h en om en on o f lim ite d in terest w h ose m y steries w e re la rg e ly
re s o lv e d in the 19th cen tu ry . Such p e o p le are q u ick to p o in t out that tidetab les, w h ic h fo re ca st th e tim es and h eigh ts o f H ig h W a te r and L o w W a te r
fo r e v e r y p o rt and seaside p ier, h a ve been prod u ced s a tis fa c to r ily fo r o v er
50 y e a rs . F u rth e r, d iscrep a n cies b etw een such tables and the ob served
p h en o m e n a are m o re o fte n the resu lt o f th e less p red icta b le effects o f
w e a th e r on the sea su rfa c e than o f erro rs in the fo rm u la tio n o f the a stro­
n o m ic a lly in d u ced tid es. W h y then sh ou ld ocea n ograp h ers con tin u e to
in v e s tig a te th e m o tio n s in d u ced b y the M o on and the Sun ?
T h e reason in b r ie f is th at in te re s t has sh ifted fr o m the d e fin itio n o f
tid es in tim e to th e ir d e fin itio n in space. T id e s are u n iqu e a m o n g n atu ral
(* ) In te r n a tio n a ]
as IA P O .
A s s o c ia tio n
of
th e
P h y s ic a l
Sciences
of
the
Ocean,
then
know n
p h y s ica l processes in that one can p red ict th eir m otion s w e ll in to th e fu tu re
w ith accep tab le a ccu racy w ith o u t le a rn in g a n y th in g about th eir p h ysica l
m ech an ism . W h e n D a r w i n (1883-86) firs t fo rm u la te d the ru les o f h arm on ic
tide p re d ic tio n he v irtu a lly p rod u ced a co ok -b ook recipe fo r p ro d u cin g tidetables fo r a g ive n place w ith o u t re q u irin g a n y referen c e to L a p la c e ’s tid a l
equ ation s or any k in d o f h y d ro d y n a m ic a l la w . A ll that is needed is to
ob ta in a rec o rd o f the tide at the place in qu estion fo r a fe w lu n a r cycles
— ty p ic a lly a y e a r — and ex tra ct fr o m this record the am p litu d es and
phases o f a nu m ber o f h a rm o n ic term s at k n o w n lu n ar and solar fre q u e n ­
cies. T h ese can be ex tra p o la ted w ith m in o r a d ju stm en ts fo r m a n y decades
in to the fu tu re. T h e re have, o f course, been several advances in tech n iqu e
since D a rw in ’ s day (su m m arized b y G o d i n , 1971) and n ew approach es based
on m od ern concepts o f tim e series an alysis ( M u n k & C a r t w r i g h t , 1966),
but tid a l p red ictio n rem ains an essen tia lly e m p iric a l art. T h e search fo r
im p ro v e d accu racy in the p red ictio n o f sea le v e l and cu rren ts has been m ore
p r o fita b ly d iv erted in to the stu dy o f storm surges, a su b ject outside the
scope o f this rev iew .
But since the tim e o f L a p la ce, scientists h ave rea lized th at the tides as
a m ech an ica l system w ill n ever be fu n d a m e n ta lly u n derstood u n til w e can
d e fin e sp a tia lly the great c o m p lex o f tidal w a v es w h ich o scilla te over the
w o r ld ’ s oceans, w h ere th ey are generated. T h e g rea t m a jo r ity o f tid a l
record s in the past have been tak en at coastal ports and r iv e r estuaries,
m o stly in sh a llo w seas w h ere the tides a re dissipated. T h ese g iv e the least
possible in fo rm a tio n about the b eh aviou r o f the tides in the ocean. T o this
day no sa tisfa cto ry ‘ co tid al m ap ’ o f the oceans has y et been produ ced.
Such a m ap req u ires a co m b in a tion o f e x a c tin g m easu rem en ts fr o m the
open sea and the solu tion o f a set o f p a rtia l d iffe re n tia l equ ation s w ith
irre g u la r bou n d aries to sa tisfy the w ave m echan ics. T h e p rob lem s in v o lve d
h a ve taxed m ath em a ticia n s fo r o v er a ce n tu ry and are o n ly n o w b eg in n in g
to look possible w ith the aid o f m od ern o cea n og ra p h ic te c h n o lo g y and la rge
com puters.
It is releva n t to note that the v e r y sim p lest oceanic tid a l p ro b lem posed,
and o n ly p a r tia lly solved, b y L a p l a c e (18 24 ), n a m ely ‘ w h a t w a v es are
possible on a thin sheet o f flu id o f u n ifo rm depth on a ro ta tin g sphere ? ’
has o n ly rec en tly been co m p letely solved b y L o n g u e t - H i g g i n s (1968). E ven
the fo u n d a tion s o f L a p la c e ’s d y n a m ica l eq u ation s fo r the ocea n ic tides,
w h ich m ake ce rta in con ven ien t but qu estion a b le a p p rox im a tion s, w ere the
su bject o f ren ew ed discussion in the 1940s ( P r o u d m a n , 1942, 1948, J e f f r e y s
1943), and a ga in v e r y rec en tly ( M i l e s , 1974, P e k e r i s , 1975). E v id e n tly , the
su bject is s till u nder close scru tin y. N ew d evelop m en ts and the fa ilu re o f
old m ethods h a ve caused scientists to re-exam in e the fu n d am en ta ls.
B y 1955, despite m a n y in gen iou s m a th e m a tica l solutions fo r tides in
id ealized basins, P r o u d m a n chose as the title o f his p resid en tia l address to
the IA P O : ‘ T h e u n kn ow n tides o f the ocean ’ . C lassical m ath em a tica l
an alysis seem ed to h ave been ca rried as fa r as it w ou ld go in this fie ld : a
new app roa ch w as needed. T h is n ew a p p ro a ch cam e sh o rtly a fterw a rd s
fro m the n u m erica l tech niqu es pion eered b y H a n s e n (1949) and P e k e r i s
& D i s h o n (1961) and fr o m the deep-ocean pressu re sensors p io n ee re d b y
E y r i è s et al. (1964) and S n o d g r a s s (1968). A n ew sp irit o f o p tim is m w as
a w a k en ed in a fie ld w h ic h h ad p re v io u s ly lost its appeal to the you n g er
researchers. T h e in tern a tio n a l S c ien tific C om m ittee fo r O ceanic R esearch
(S C O R ) set up a n e w w o rk in g grou p d evoted to ocean ic tides under the
ch airm an sh ip o f W .H . M u n k (la te r succeeded by the a u th or). In 1966 this
grou p aim ed to prod u ce d e fin itiv e glob a l co tid a l m aps b y about 1972.
B u t the good in ten tion s p ro v ed over-op tim istic. A ft e r the in itia l ex­
citem en t o f new results it w as n oticed that th ey w ere n ot re a lly good enough.
R esu lts fr o m in d epen den t research ers disagreed w ith each other and w ith
natu re. N e w correction s, p re v io u s ly ign ored , w ere fo u n d to be im p ortan t
and these g rea tly co m p lica ted the solu tion o f cotid al m aps. O cean ograph ic
tid a l m easu rem ents h ave p ro v ed slo w and expen sive, and th eir rate has
se riou sly d w in d led under the presen t econ om ic stress. B y 1977, the su bject
has retu rn ed to a state o f u n certa in ty, a w a itin g n e w ideas. T h e situ ation
is id e a l fo r a rev iew .
A lth o u g h m ost o f the d evelop m en ts rev iew ed here w ill be recent ones,
it is in evita b le w ith a su bject o f this age to in clu d e som e discussion o f
o ld er, w e ll-trie d m aterial. T h is is n ecessary as a fo u n d a tion fo r the la ter
w o rk , and as an in trod u ction to the m a jo r ity o f readers w h o are not a c tiv e ly
con cern ed w ith the subject.
2. THE T ID E -G E N E R A T IN G PO T E N T IA L
T h e poten tia l o f the tid e-ra isin g fo rces on the E a rth ’s surface due to
the Sun or the M oon is fu n d am en ta l to all tidal studies. It m ay be d efin ed
as :
V = "fM/r - yM/d
(2.1)
w h e re y is the g ra v ita tio n a l constant, M is the mass o f the celestial b o d y
B, r is the distance o f its centre o f g ra v ity fro m a g ive n poin t P on the
E a r th ’s surface, and d is the co rre sp o n d in g distance BC fro m the E a rth ’ s
cen tre (fig u r e 1).
F ig. 1. — K arth, ce n tre C, N o rth p ole N. M oon o r Sun at B, d istan ce d fro m C, at ze n ith
a n gle a. r e la tiv e to p o in t P on E arth ’ s surface.
T h e firs t term in (2.1) is sim p ly the (n e g a tiv e ) poten tia l o f the N e w to n ­
ian a ttra c tio n at distance r and the second term is a con ven ien t a rb itra ry
con stan t in space. T h e v e c to r tid e -ra is in g fo rc e per u nit m ass a ctin g at P
is g ive n b y th e sp atial g ra d ien t o f V, b u t w e do not need to ex p ou n d its
w e ll-k n o w n p ro p e rtie s h ere since a ll u sefu l results m a y be ex pressed m o re
sim p ly in term s o f the scalar V. It su ffices to say that the co m p on en t o f the
tid e -ra is in g fo rc e w h ich is effe ctiv e in a cce lera tin g the w a te r m ass is th a t
ta n g e n tia l to the E a rth ’ s su rfa ce : the v e r tic a l co m p on en t is n e g lig ib le
co m p a re d w ith the E a r th ’ s o w n v e r tic a l g ra v ita tio n a l forces.
E q u a tio n (2.1) is the n ega tive o f the s tr ic tly p h ysica l p o ten tia l, and is
tak en so c o n v e n tio n a lly because a p o s itiv e in c re m en t in VT is b a lan ced b y a
p o s itiv e rise in the w a te r le v e l at P to re ta in an eq u ip o ten tia l su rface. T h e
p o te n tia l due to the c e n trifu g a l fo rc e o f the E a rth ’ s ro ta tio n :
w h e re r„ is the n orm a l fro m P to the E a r th ’ s axis NC and H is the rate o f
ro ta tio n , is m e re ly a d d itive to V. B ein g in d ep e n d e n t o f the p o sitio n o f B it
does n ot cause a tid e but a constan t eq u a to ria l b u lge in the shape o f the
E arth . (A m in o r ex cep tio n is the ‘p o le t id e ’, caused b y the v a ria tio n in
Ve due to the C han dler m o tio n o f the p o s itio n o f N, o f 1.2 y r p e r io d ic ity ).
T r e a tin g the E a rth as a sphere o f ra d iu s a (c o rre c tio n s fo r a sp h eroid
can be m ade, but th ey are n ot im p o r ta n t), (2 .1 ) can be ex p a n d ed in term s
o f a, the zen ith angle o f the b o d y B :
V = ga(M/E) [£2 cos a +
P 2{cos a ) + Ç*P 3(cos a ) + 0 ( £ 5) ]
(2.2)
H e re g is the g ra v ita tio n a l a cce lera tio n at P , E is the E a r th ’ s m ass, £ = a/ r
is the b o d y ’ s eq u a to ria l p a ralla x, and P „ ( c o s a ) are the L e g e n d r e p o ly n o m ials :
P 2(pi) =-^-(3/12 - 1)
= — (5/i3 - 3m)
T h e fir s t term in (2.2) co rresp o n d s to a constan t fo rc e p a r a lle l to CB :
it is b a la n ced b y the o rb ita l a ccelera tion s and has no tid a l in terest. T h e
second term is the p rin c ip a l pa rt o f the tid e -g e n e ra tin g p o te n tia l, g eo m e t­
r ic a lly a s y m m etrica l o v a l w ith m a jo r a x is a lon g CB. Since £ is abou t
1/60 fo r the M oon, the term in P 3 is c o n s id e ra b ly sm a ller th an the p r in ­
cip a l term , b u t its presen ce can be d ete cted in the ocea n ic tides. T h e
p rin c ip a l tid a l term fo r the Sun is, on a vera g e, 0.45 tim es th at o f the
M oon, but h a v in g £ o f the o rd e r o f 1 0 ~ 4, its P :i term is q u ite n e g lig ib le.
F r o m the sp h erical tria n g le N P B ', w h e r e B ' is the su b-lu nar p o in t, w e
m a y m o re c o n v e n ie n tly express a in term s o f N P = 6 = the co -la titu d e o f
the pi ace P, N B ' = D = the co -d eclin a tion o f the body, and the an gle :
w h e re X is the East lo n g itu d e o f P, R is th e R ig h t A scen sio n o f the b ody,
h is the m ean lo n g itu d e o f the Sun, and t is G reen w ich M ean T im e in days.
W e sh all res trict ou r a tten tion to the p rin c ip a l tid a l term in v o lv in g P 2
(cos a ) and express the “ e q u i l ib r i u m tide ” C = Cs + & + - = v/ g in term s
o f the u n -n orm alized sp h erical h a rm o n ics P „ m ( 9) exp (im \ ), w h e re :
W e ob tain fo r the p rin c ip a l term o f the e q u ilib riu m tid e :
2
^2 = A 2° ( t ) P 20 ( e )
+
X
[ A 2m ( t ) P 2m (6 ) cos m\ + B 2m ( t ) P 2m (6 ) sin mX] (2.3)
m= 1
here
A 2° ( t ) = a (M/E) k 2° Z 3P 2° ( D )
A 2m ( t ) 4- iB 2m ( t )
= a ( M / E )k 2m ( -
1) m £3 P 2m ( D ) exp [im { R - h -
2nt)] (2.4)
and A;,"1 is a sim p le n o r m a liz in g p a ra m e te r (n ote th a t M u n k & C a r t w r i g h t
(1966) a n d w o r k s s te m m in g fr o m that p a p er use a m o re so p h isticated n o r­
m a liz a tio n th an th at im p lie d b y (2 .3 )).
E q u a tio n (2.3) separates th e tim e va ria b le s fr o m th e g eo g ra p h ic a l
co o rd in a te s. E q u a tio n (2.4) d e fin e s the th ree ‘sp ecies’ o f tim e va ria b le
( A , B ) 2m. Species m = 0 va ries w ith the m o n th ly or y e a r ly freq u en cies o f
£ and D m u ltip le s th ereo f, ca u sin g the ‘l o n g - p e r i o d ’ tides. Species m = 1, 2
v a r y p r im a r ily at a ‘c a r r ie r ’ fre q u e n c y n ea r m cycles p e r d a y w ith lo w fr e q u e n c y m o d u la tio n fr o m £ and D, cau sin g the ‘d iu r n a l’ and ‘sem i­
d iu r n a l’ tid es re s p e c tiv e ly .
T h e fu n c tio n s *4, B are used in tw o d iffe re n t w a y s. In the fir s t w ay,
th ey are co m p u ted d ir e c tly fr o m (2.4) as a rb itr a r y fu n ction s o f tim e t fo r
d ire c t c o rr e la tio n w ith tid a l m easu rem en ts. T h e p o sitio n o f the celestia l
b o d y is c o m p u ted w ith resp ect to the e c lip tic fr o m sta n d a rd o rb ita l fo r ­
m u la e and these are s im p ly tra n s fo rm e d to g, D and R. T h u s in the case
o f the Sun, i f w e ig n o re p la n e ta r y and o th er v e r y m in or pertu rb a tion s, the
ec lip tic lo n g itu d e p and p a ra lla x £ are d e fin e d b y :
p = h + 2e sin ( h - p ’) H---- e2 sin 2 ( h — p 1) + . . .
4
£ = % [1 + e cos (h — p ' ) + e2 cos 2 ( h — p ) + . . .]
(2.5)
w h e re e is th e e c c e n tric ity o f the E a r th ’ s orb it, p ' is the lon g itu d e o f p e r i­
h e lio n , and f = 8".794 is the S u n ’ s m ean p a ra lla x, h and p ' are d e fin e d
fr o m a s tro n o m ic a l ep h em e rid es as n e a rly lin e a r fu n ction s o f tim e w ith
v e r y sm a ll a c c e le ra tiv e term s, c and e (th e o b liq u ity o f the ec lip tic ) are
n e a r ly co n stan t n u m bers w ith s lo w secu lar trends. T h e Sun’ s d eclin a tio n D
and rig h t ascension R are th en d e riv e d fr o m (2.5) by the eq u ation s :
sin D = sin p sin e
cos D cos R = cos p
cos D sin R = sin p cos e
(2.6)
and the resu lts used in (2 .4 ).
T h e case o f the M oon is m o re co m p lica te d because it has a n o n -trivia l
la titu d e v a r ia tio n abou t the eclip tic, w h ic h in v o lve s the lon gitu d es o f the
node n, and also a n u m b er o f o rb ita l term s d ep en d in g on the p o sition o f the
Sun. B esides n, we h a ve tw o a d d itio n a l secu lar va riab les, the m ean lo n g i­
tude o f th e M oon, s, and the m ean lon g itu d e o f perigee, p. T h ese are best
grou p ed in term s o f B r o w n ’ s va ria b les Q l = s — p, Q 2 = h — p', Q3 =
s — n, Q 4 = s — h. T h e M o o n ’ s p a ra lla x , lon gitu d e and la titu d e are all
exp ressed in the fo r m :
(2.7)
w h e r e M / l> are sets o f sm a ll in teg ers in c lu d in g 0, and R t are a m p litu d es
d e r iv e d fr o m the tables o f B r o w n (1919) o r as re-com p u ted b y E c k e r t et al
(1 9 5 4 ). P a r a lla x (co s) and lo n g itu d e (s in ) use the sam e secu lar a rg u m en ts :
la titu d e (s in ) use oth er sets M / l>. D a n d R are fin a lly com p u ted fr o m f o r ­
m u la e s im ila r to (2.6) bu t w ith term s in v o lv in g the M o o n ’ s la titu d e.
M o d ern tid a l w o rk e rs d iffe r in the n u m ber o f term s used in (2.7) and
h en ce in the a ccu racy o f th e ir ep h em erid es. O ne c e rta in ly does n ot need
th e s e ve ra l thou sand term s w h ich are used fo r a s tro n o m ic a l w o rk . L o n g ­
m a n (1959) used o n ly 4 h a rm o n ic a rg u m en ts.
M u n k & C a r t w r i g h t (1966)
used 7, d e rived fr o m an old er e x p a n sio n than B r o w n ' s . T h e au th or, as
d escrib ed in C a r t w r i g h t & T a y l e r (1 9 7 1 ), cu rren tly uses abou t 110 a rg u ­
m en ts fo r the M oon, g iv in g an a ccu racy o f about 10-5 in units o f rad ian s
o r m ea n p a ra lla x . Such a ccu racy is b e tte r than n ecessa ry fo r m ost p u r ­
poses. F o r o rd in a ry a n alysis o f tid a l data, about ten term s in p a ra lla x and
lo n g itu d e and six in la titu d e g ive g o o d results.
T h e second and o ld er m eth od o f r e fe r r in g to the e q u ilib riu m tid e (2.3)
is in the fo rm o f a h a rm o n ic ex p a n sion fo r A nm( t ) . (T h e h a rm o n ic ex p a n ­
sion fo r B nm( t ) is id e n tica l excep t fo r a Jvr phase ch a n g e). T h e fir s t th o ro u gh
h a rm o n ic series w as p rod u ced b y D o o d s o n (1921), w h o used a lgeb ra ic
ex p a n sion s based on B r o w n ’ s (1919) series (2 .7 ). C a r t w r i g h t & T a y l e r
(1971) and C a r t w r i g h t & E d d e n (1973) re-com pu ted th em w ith m od ern
a s tro n o m ic a l constan ts and grea ter a ccu racy, u sin g sp ectra l a n alysis o f
tim e series o f
T h e results are o f the fo rm :
( 2 .8 )
w h ere
are tab u lated am p litu d es, N / ‘> a re sets o f sm a ll in tegers, and Sj
a re the six secular a rgu m en ts (2-rrt — 77- + /1 — s), s, h, p, — n, p ' res p ec­
tiv e ly . T h ese six a rgu m en ts in crease a lm o st lin e a rly in tim e and co m p lete
one c y c le in a lu nar day, a sid erea l m on th , a tro p ic a l year, 8.85 yr,
18.61 y r and 21 X 103 y r re s p e c tiv e ly .
T h e frequ en cies o f the su m m ed a rg u m en ts in (2.8) are c o n v e n ie n tly
a rra n g ed in a h iera rch y, d ep en d in g on th e fin en ess o f sp littin g . A ll term s
w ith the saine valu e o f A’, con stitu te a ‘sp ecies’ : the species are separated
fr o m each oth er by about one cy c le p e r lu n a r day. T e r m s w ith the sam e
AV) con stitu te a g ro u p ’ : the g ro u p s are separated b y one cycle per
m on th . S im ila rly , (A r, N 3, N :t) fo rm a ‘co n stitu en t’ at on e c y c le per y e a r
sep a ration . B ecause o f the im p o rta n c e o f B r o w n ’ s a rg u m en t Q x, all term s
o f a g iv e n con stitu en t c o n v e n tio n a lly h a v e th e sam e va lu e o f A7+ also. N o n ­
ze ro va lu es o f AT3 su p p ly ‘ n od al m o d u la tio n ’ to a co n stitu en t. N o n -zero
A76 e ffe c tiv e ly im p lie s a n e a rly con stan t p h ase sh ift.
A s w ith the h a rm o n ic ex p a n sion o f th e lu n ar orbit, som e h u n d red term s
in (2.8) are n ecessary to d efin e the p o te n tia l to, say, 1 % a ccu racy, but a
few r term s have d o m in a n tly la rg e a m p litu d e s. T h e s e h ave been a lloca ted
sym b o ls (o r ig in a lly b y D a rw in ) w h ic h a re fre q u e n tly used in tid a l lite ra ­
ture. T h e fo llo w in g tab le lists p a rtic u la rs o f the fo u r la rg e st term s (c o n s ti­
tu en ts) in species 1 (d iu r n a l) and 2 (s e m i-d iu rn a l). Species 0 (lo n g -p e r io d )
co n stitu en ts are not in clu d ed because th e y are r e la tiv e ly u n im p o rta n t. F o r
th e give n term s
= A’e = 0 : cor rep re se n ts 277 tim es the listed fre q u e n c y .
Darwin
sym bol
Qi
Pi
n 2
M,
s2
k 2
N ! N2 N3 N4
Frequency cycles
per solar day
Origin
Am plitude
(m )
1 - 2 0 1
1 - 1 0
0
1
1 -2
0
1
1 0
0
0.8932441
0.9295357
0.9972621
1.0027379
Moon
Moon
Sun
Both
0.01293
0.06752
0.03142
0.09497
P2 \ e )
sin ( c j j
2 - 1 0 1
2
0
0 0
2
2 -2
0
2
2
0 0
1.8959820
1.9322736
2.0000000
2.0054758
Moon
Moon
Sun
Both
0.01558
0.08136
0.03785
0.01030
P 22 {6)
COS <Cd2 t + 2\)
G eodetic
factor
t
+ X)
A s p o in te d out b y C a r t w r i g h t & T a y l e k (1971), the am p litu des o f
the tid a l h a rm o n ic ex p a n sion are n ot constant on a lon g tim e scale but have
secu lar trends o f the o r d e r o f 10-4 per cen tu ry, due p r in c ip a lly to the trend
in o b liq u ity e. T h e 'fre q u e n c ie s ’ o f coursc also v a r y s lo w ly w ith the secular
a cce lera tio n s o f th e M o o n ’ s elem en ts, o w in g to p la n e ta ry p e rtu rb a tion s and
tid a l fr ic tio n .
3. O C E A N I C
A D M IT T A N C E S
W e need to k n o w the respon se o f the ocean to the tid e-g en era tin g
p o te n tia l. Since the la tter is c o n v e n ie n tly expan ded in a series o f low o rd er sp h eric a l h a rm o n ics (2.3) it su ffices to in vestig a te the response to
these sp a tia l h a rm o n ics in d iv id u a lly . Such a response is a fu n ction o f
tim e and p o sition . In th is section I sh all o u tlin e the n a tu re o f the response
in tim e at a g iv e n p o sitio n on the globe. T h is sort o f respon se is n a tu ra lly
d e te rm in ed b y tim e series a n a ly s is o f tid a l record s, and in fa ct has ra re ly
been trea ted b y a n a ly tic a l m ath em a tics, w h ic h are u su ally fo r con ven ien ce
res tric te d to a single p e rio d ic term such as the M., h a rm o n ic constituent.
T h e ocea n ic tid a l respon se is s tro n g ly lin ear. T h is is evid en t fro m
h y d ro d y n a m ic co n sid era tion s (§ 4) and can be ea sily d em on strated by the
fa ct th at the spectra o f tid a l rec o rd s fr o m places near the deep ocean
co n ta in s tro n g lines at the m a jo r h a rm o n ic fre q u en c ies presen t in the
p o te n tia l (fig u r e 2) and a lm ost n e g lig ib le en erg y at m u ltip les o f those
fre q u en c ies . N o n -lin e a r effects d o b e com e ev id e n t in s h a llo w seas and these
are im p o rta n t in c o m p u tin g tid e p re d ictio n s fo r coastal and estu arial ports.
H o w e v e r, th e y have little in flu e n c e on the glo b a l m ech an ics o f the oceanic
tides and w ill be n eglected in this rev iew , ex cep t in co n sid erin g the nature
o f the fr ic tio n a l fo rces (§ 5).
W e exp ect a re la tio n s h ip o f th e fo r m :
f(f)= R e ^
[ A ( t — r) + iB (t — T ) ] R ( r ) d r
(3.1)
w h e re £ is the ele va tio n o f the sea su rfa ce a b ove its m ean level at a given
place, A , B a re the tim e -d e p en d en t pa rts o f th e p o te n tia l (o r eq u ilib riu m
tide (2 .4 )) fo r a p a rtic u la r sp h eric a l h a rm o n ic, and R is a response fu n ction
c h a ra c te ris tic o f the ocean d yn a m ics. I h a ve o m itted the suffices m , n fo r
co n ven ien ce. O r e q u iv a le n tly , i f H ( f ) and G( f ) are the co m p lex ‘sp ectra’ o f
£ and .4 re s p e c tiv e ly in the d o m a in o f fre q u e n c y f, w e expect :
HQ) = Z (f)G (f)
(3.2)
w h e re Z ( f ) is the c o m p le x ‘a d m itta n c e ’ o f the ocean to the eq u ilib riu m
tid e, being the F o u rie r tra n s fo rm o f R ( t ) .
cycles/day
Fin. 2. — S pectroscopy o f p rin c ip a l p art o f g r a v ita tio n a l p o te n tia l (to p ) and o f 18 y e a rs ’
sea le v e l a t H on olu lu . B lack p ortion s o f p an els 2 and 3 denote p a rt o f spectrum c o h ere n t
w it h p o te n tia l, w h ite p o rtio n s in co h eren t p a rt. T h e lo w e s t p an els rep resen t th e c o rre s ­
p on d in g ad m ittan ce, spot e s tim a te s and sm ooth ed es tim a te s
(fr o m M u n k & C a r t w r i g h t 1966).
A co m p lete d e fin itio n o f the respon se fu n ction R ( t ) w o u ld tell us a lot
a bou t oceanic beh a viou r, but it is im p ossib le to d e te rm in e this fu n c tio n
fr o m analysis o f the tides alone. It w o u ld be eq u iva len t to -determ in ing the
a d m itta n ce Z at all frequ en cies : this is c le a rly im p ossib le since a n y one
tid a l species occu pies o n ly a n a rro w ban d o f frequ en cies, a fe w cycles p e r
m o n th w id e. It is tem p tin g to eva lu a te Z ( f ) fo r a n u m ber o f tidal species,
say d iu rn a l ( f ~ 1) and sem i-d iu rn al (/ ~ 2), and in terp o la te b etw een th em
to ob ta in w id e r co vera ge o f Z ( f ) . T h is has o c ca sion a lly been done fo r
rou gh illu s tra tiv e purposes, using the g eo d etic fa cto rs as a sort o f n o rm a l­
iza tio n (e.g. W u n s c h , 1972), but the proced u re is in v a lid because the
sp h erical h a rm on ics associated w ith th e tw o species a re d ifferen t, w ith
d iffe re n t associated a d m itta n ce fu n ctio n s. N everth eless, it is u sefu l to k n o w
w h a t these a d m itta n ces are like, even w ith in the n a rro w band o f each tidal
species. A r e th ey n e a rly constan t, sm oo th ly v a r y in g or sh arp ly p eak ed ?
F ig u r e 2 show s a d ire ct sp ectra l co m p arison b etw een the d iu rn a l tide
at H o n o lu lu and the co rre s p o n d in g p o te n tia l fu n c tio n A 21( t ) , evalu a ted by
M u n k and C a r t w r i g h t (1966) — one o f m an y s im ila r analyses w h ic h have
since been c a rrie d out fo r oth er places. T h e top pan el show s a spectru m
G ( f ) at 1 c y c le p e r y ea r resolu tion , com p u ted as a F o u rie r tra n s fo rm o f
A aK f) fo r a span o f tim e sim u ltan eou s w ith the sea-level data. T h e
‘g ro u p s ’ o f tid a l con stitu en ts d escrib ed in § 2 are ob viou s. T h e second pan el
show s the sp ectru m H ( f ) o f £(/) — top lev el o f each sm a ll co lu m n — and
also that p a rt o f £(/) w h ich is ‘c o h eren t’ w ith A .^ it) in the sta tistica l sense
— low er, b la ck level. T h e th ird pan el show s th e sam e data as the second
but w ith the ‘in c o h e re n t’ (w h ite ) p o rtio n o f //(/) u n dern eath . T h e d iffe r ­
ences b etw een the w h ite and b ia ck levels (lo g a rith m ic scale) sh ow the
im p o rta n t fa ct, still ig n o re d b y m a n y tid a l a n alysts, o f a noise con tin u u m
in the sea -level spectru m , fillin g th e gaps b e tw ee n the tid a l grou ps and
settin g a lim it to the r e s o lv a b ility o f the w e a k e r constitu en ts. It is the
co n tin u u m w h ic h p reven ts us g ettin g an y m o re in fo r m a tio n ou tside the
fr e q u e n c y r a n g e b etw een grou p s (1 ± 4) sh ow n in the fig u re, a lth o u gh the
d iu rn a l p o te n tia l con ta in s w e a k sp ectral e n e rg y ou tsid e this ran ge. T h e
co n tin u u m le v e l rises at lo w e r frequ en cies, as in m ost geo p h y sic a l p ro ­
cesses. T h e lo n g -p erio d tides are th e re fo re a lm o st co m p le te ly sw am p ed
b y it, unless record s o f v e r y lo n g d u ration are an alysed o r sp ecial tech niqu es
a re em p lo y ed to red u ce th e noise lev el ( W i t n s c h , 1967 ; C a r t w r i g h t , 1968).
T h e l o w e s t t w o p a n e l s o f f i g u r e 2 s h o w th e e s t i m a t e s o f Z ( f )
derived
f r o m th e s p e c t r a l a n a l y s i s , f i r s t l y a s re a l a n d i m a g i n a r y p a r t s , a n d s e c o n d ly
as
am p litu d e
ratio
JZ | a n d
phase
lead
( Z ).
arg
Evidently,
the
function
is n o t c o n s t a n t , b u t n e i t h e r d o e s it s h o w a n y d e t a i l e d s t r u c t u r e w i t h i n the
given
frequency
band.
It
is
b es t
described
as
‘s m o o t h l y
v a r i a b l e ’.
M ost
t i d a l a d m i t t a n c e s w h i c h h a v e b e e n e v a l u a t e d c o u l d b e s i m i l a r l y d e s c r ib e d ,
alth o u gh som e
s h o w a sin gle
peak
or trough
in
| Z |, s u g g e s t i n g
reson an ce o r an ti-re so n a n c e o f a p a rtic u la r ocean basin .
a broad
T h i s is in te re s tin g ,
because c alcu latio n s of n o r m a l w a v e m odes on a rotatin g sp h ere ( L o n g u e t H i g g i n s , 1968) o r r e a l is t i c o c e a n ( P l a t z m a n , 1975) s h o w a d e n s e r a n g e o f
s u c h m o d e s at f r e q u e n c i e s o c c u p i e d b y th e t i d a l b a n d s .
T h e s e sh o u ld p ro ­
d u c e a c o r r e s p o n d i n g n u m b e r o f p e a k s in a d m i t t a n c e f u n c t i o n s , w i t h r a p i d
‘Q f a c t o r ’ is h i g h e n o u g h . T h e f a c t that
s u g g e s t s t h a t the o c e a n i c Q m u s t
b e r a t h e r l o w a n d d i s s i p a t i o n r e l a t i v e l y h i g h ( G a r r e t t & M u n k , 1971). O n
th e o t h e r h a n d , W e b b (1973 a ) h a s i d e n t i f i e d a v e r y n a r r o w p e a k w i t h
changes
in
phase,
provided
the
s u c h p e a k s a r e not in g e n e r a ]
associated
phase
change
at
observed
Cairns,
NE
A u stralia,
indicating
l i g h t l y d a m p e d r e s o n a n c e o f the C o r a l S e a (se e a ls o M c M u r t r e i :
1975).
The
m ost
fam ou s
tidal
resonance,
in
the
Bay
of
a
rather
& W ebb,
Fundy,
Nova
S co tia, w h i c h p r o d u c e s th e l a r g e s t r e c o r d e d t i d a l a m p l i t u d e s in th e w o r l d ,
h a s a m u c h b r o a d e r a d m i t t a n c e p e a k , a f f e c t i n g the w h o l e s e m i - d i u r n a l spe­
cies.
T h is
a p p e a r s to be a c o m b i n a t i o n
o f a gen eral m agn ification
s e m i - d i u r n a l tid e s in the N o r t h A t l a n t i c O c e a n a n d
i n v o l v i n g t h e G u l f o f M a i n e a n d t h e B a y it s e lf ( G a r r e t t ,
G reenberg,
1974).
C a rtw rig h t
(1971)
has
of
the
m o re local resonances
id e n t i f i e d
an
1973 ; H e a p s &
anti-resonance
o f the diu rn a l tides in the southw est A tla n tic , b rin g in g the a d m itta n ce
a m p litu d e to zero near the fr e q u e n c y o f
at Sim ons B ay.
CYCLES
PER
DAY
Fie,.
Log a m p litu d e s and phases o f a d m itta n ce s (C<t) d iu rn a l tid es, ( b ) s e m i-d iu rn a l
tid e s ) a t fo u r w id e ly spaced loc a tio n s in th e B ritis h Is le s ; L e r w ic k (S h e tla n d ), S to rn o w a y
(L e w is ), M alin H ead (E ire ) and S c illy Isless (C o r n w a ll). A r ro w s denote the fre q u en c ies
o f the p rin cip a l h a rm o n ic con stitu ents.
F ig u re 3 gives an im pression o f h o w little the a d m itta n ce fu n c tio n
v a rie s in ch aracter o v er a distance o f the order o f 1 000 km . T h e n a tu ral
lo g a rith m o f Z is d ra w n fo r diu rn a l and sem i-diu rn al tides fo r fo u r places
on the oceanic coast o f the U K , fro m St M a ry ’s, S cilly Isles to L e rw ic k ,
Shetland. In Z = In |Z j -f- i a rg ( Z ) has the ch a ra cteristic that a gen eral
m a g n ific a tio n or change o f phase lead o v e r the w h o le tid a l species m e re ly
sh ifts the curves rep resen tin g the real and im a g in a ry parts up or dow n,
w ith o u t a lterin g th eir general shape. In fa ct, one sees that the shapes
are rem a rk a b ly sim ila r at the fo u r places. T h is a pplies to the tides o f the
w h o le o f the n orth w est E u ropean seaboard, w hen a llow a n ces are m ade fo r
n on -lin ea r effects induced b y sh allow w a ter.
O ne unusual ch ara cteristic show n in fig u re 3 is the pron ou n ced n ega ­
tive slope o f diu rnal phase lead, n early 2tt across the species b a n d w id th .
T h is ph enom enon is observed o v er the northeastern A tla n tic fro m the
A zo re s to the F aeroes, but vanishes on th e N o rth A m e rica n coast. It has
n ever been p h ys ica lly explained. A less p ron ou n ced phase slope is co m m o n
in the sem i-diu rn al tides th rou gh ou t the w orld w ith som e exceptions, and
is responsible fo r the lon g-ob served ph en om en on o f tidal ‘a ge’ w h ereb y the
sp rin g tides lag the tim es o f F u ll and N e w M oon by a day or so, due to
the d ifferen ce in phase o f M 2 and S2. T id a l age is now fir m ly a ttribu ted to
the effect o f tidal fric tio n ( P r o u i i m a n , 1941 ; G a r r e t t & M u n k , 1971 ;
W e b b , 1973 b) but the m ech an ism w h e re b y it is produced is too co m p lica ted
fo r a sim ple heu ristic explan ation .
On a m ore tech n ica l level, th ere is som e interest in m eth od s o f tra n s­
fo rm in g a b a n d -lim ited tid a l a d m itta n ce fu n ction as discussed above in to a
tim e rela tio n b etw een £ (0 and A (/), B ( t ) . A lth o u g h w e cannot fu lly d efin e
a response fu n c tio n R ( t ) as it appears in (3.1), th ere are va riou s w ays o f
d e fin in g a fin ite series o f term s lik e :
K
f(0=
X wk Lk [A{t) + ifi(r )]
fc=o
(3.3)
w h e re L k is a series o f sim p le lin e a r op erators and wk is a set o f com p lex
‘w e ig h ts ’ , w h ose F o u rie r tra n s fo rm fits the a d m itta n ce (3.2) o p tim a lly
w ith in its v a lid band o f frequ en cies. M u n k & C a r t w r i g h t (1966) used a
series o f tim e -la g g ed term s :
Lk[C (t)] =C (t
kA t )
w ith A t = 2 days, eq u iva len t to fittin g a r oui ici series to Z ( f ) o f p erio d ­
ic ity 0.5 cycles p er day in fre q u en c y . Z e t l e r & M u n k (1975) h ave im ­
p ro ved on this s lig h tly b y w r it in g the righ t-h an d side as C ( t — T — k A r ) ,
w h ere T is the tid a l ‘a g e ’ a p p ro p ria te to the species. W e b b (1974) p ro­
posed :
£ ki c ( 0 ] = exp ( i 0Jmt ) ( - id/dt)k [exp ( - icom t) C * ( t ) ]
e q u iva len t to a sim ple p o w e r series d evelop m en t o f Z ( f ) about a central
fre q u e n c y / = œm/ 2
‘ ir. Y e t a n oth er schem e has been proposed b y G r o v e s
& R e y n o l d s (1975), in w h ich L k is a sequence o f sp ecia lly constructed
o rth o g o n a l fu n c tio n s o f tim e, ca lled b y the authors ‘ o rth o tid es ’ . T h e ir
ad va n tage is that th eir associated ‘ orth ow eigh ts ’ w k are, u n lik e in the
oth er schem es m en tion ed, n u m erica lly stable, w ith a ten d en cy to dim in ish
in m agn itu d e as k is increased.
3.1 Radiational tides
W h e n adm itta n ces to the g ra v ita tio n a l tid e-g e n era tin g p oten tia l are
evalu ated fr o m a set o f tid a l data, it is g e n e ra lly found that a sm a ll but
d e fin ite a n o m a ly appears at the fre q u en c ies o f the solar tides, near 1 and
2 cycles p e r so la r day and at 1 c y c le p er year. T h ese an om alies h a ve been
k n o w n fo r m an y yea rs to tid a l an alysts and g e n e ra lly ascribed to ‘ m eteoro­
lo g ica l effects ’ w ith o u t in s p irin g m uch cu rio sity . N o w the tides in the
a tm osp h ere a re alm ost e n tir e ly at the solar frequ en cies, due to the therm al
effect o f ra d ia tio n ( C h a p m a n & L i n d z e n , 1970). T h e lu nar a tm osp h eric tides
are e x tre m e ly sm all in co m p a ris o n w ith the solar, sh ow in g that the g ra v i­
tation a l tid a l fo rces are in e ffe c tiv e : pa rt o f the lu n ar tide is even d erived
fro m c o u p lin g w ith the ocean ( H o l l i n g s w o r t h , 1971). M u n k & C a r t ­
w r i g h t (1966) postu lated that the solar a n om alies in the ocean w ere also
the resu lt o f so la r rad iation , w ith o u t s p e c ify in g the p recise p h ysica l m echan­
ism . T h e r e are va riou s possible m echan ism s : d irect rad iation pressure
(s m a ll), d ire ct su rface h eatin g and co olin g, and onshore w in d s caused
b y h e a tin g and c o o lin g o f coastal land, co u p lin g w ith the atm osp h eric tide
th rou gh su rfa ce pressu re. In o rd e r to deal w ith a n y o f these in a general
w a y , M u n k d efin e d a ‘ ra d ia tio n a l p oten tia l ’, w h ose g ra d ien t equals the
am ou nt o f ra d ia tio n received per unit area at a given p oin t o f the E a rth ’s
surface, n e g le ctin g tran sien t a tm o sp h eric losses. T h u s :
cos a
(0 < a < V* n, day)
0
(½ it < a < n, night)
£/(0, X, 0 =
(3.4)
(
w h e re S is th e solar constant, tak en as u n ity fo r con ven ien ce, g is the Sun’ s
p a ra lla x and £ its m ean valu e, and a is th e zen ith an gle o f the Sun at the
p o in t P as in fig u r e 1.
U m a y be expan ded in sp h erical h a rm o n ics as w as the g ra v ita tio n a l
p o te n tia l in § 2. T h e p rin c ip a l fea tu res are the appearan ce o f h a rm o n ics
P|°, P ^ o w in g to th e a s y m m e try o f the fu n c tio n (3 .4 ), and the slo w c o n v e r­
gence o f the h a rm o n ics o f d egree 2, 4, 6, ... o w in g to the n ig h t-tim e cu toff.
T h e zon a l h a rm o n ic P / ’ has a stro n g a n n u al term , oth erw ise w e a k ly r e p re ­
sented in the g ra v ita tio n a l tide. T h is p ro v id e s w h a t is m ore u su a lly called
the ‘ seasonal v a ria tio n in m ean sea le v e l
w h ose g lo b a l d is trib u tio n w as
in te n s iv e ly stu died b y P a t u l l o et al. (1955). Its p h ys ica l cause is c e rta in ly
a co m b in a tio n o f the last th ree o f the fo u r m ech an ism s m en tion ed ea rlier,
in co m p a ra b le p rop ortio n s.
T h e I V term , p ro p o rtio n a l to e u cos 8 (0 is the latitu d e, \ is th e lo n g i­
tu d e) p ro v id e s a term o f so la r d a ily p e rio d w h ich is a ga in v e r y w e a k ly
rep resen ted in the g ra v ita tio n a l tide. T h e term s P 3°, P 2\ P ,,2 are s im ila r to
the co rre sp o n d in g g ra v ita tio n a l term s but m a y be d istin gu ish ed fr o m them
in the ocean ic tid e by the fa ct that th ey lack co n trib u tion s fr o m the M oon
at th eir resp ec tiv e frequ en cies, e s p e c ia lly in the h a rm o n ic con stitu en ts K !
and K 2. T h e fu ll h a rm o n ic d e v e lo p m e n t o f all the lea d in g term s in the
ra d ia tio n a l p o ten tia l has been tabu lated b y C a r t w r i g h t & T a y l k r (1971).
A n a ly s e s o f tid a l record s e m p lo y in g both g ra v ita tio n a l and ra d ia tio n a l
tid a l p o ten tia ls ( (2.2) and (3.4) ) b y C a r t w r i g h t (1968), Z e t l k r (1971) and
C a r t w r i g h t & E d d e n (1977) s tr o n g ly su ggest that the m a jo r c o m p o n e n t o f
the ra d ia tio n a l a n o m a ly is fo r c in g fr o m the a tm o s p h e ric tide. In the firs t
place, the d iu rn a l a n o m a ly (S ^ is a lw a y s m u ch w e a k e r than the sem i­
d iu rn a l (S 2). T h is m a y sound stra n ge fo r a respon se to the d iu rn a l ra d ia tio n
p a ttern , but it is a w e ll-k n o w n fe a tu re o f the tid e in the su rfa ce a tm o sp h eric
pressure. A s ex p la in ed b y C h a p m a n & L i n d z e n (1970), th e tide in the u pper
a tm osp h ere is in d eed p r in c ip a lly d iu rn a l w ith h ig h e r h a rm o n ics, bu t the
ve rtic a l w a v e fo rm is such that at grou n d level the d iu rn a l co m p o n e n t is
su ppressed and the firs t h a rm o n ic (S 2) p red om in a tes. In the ocean the
a m p litu d e o f S, is ty p ic a lly 10 m m or so (a n d in som e cases cou ld be
p a r tia lly a ttrib u ted to a diu rn a l d e fe ct in the re c o rd in g m ech an ism o f the
co n ve n tio n a l tide g a u g e ), w h ile that o f S2 is about 17 % o f the g ra v ita tio n a l
S2 co m p on en t, ty p ic a lly 50-100 m m .
A n o th e r fea tu re o f the S2 ra d ia tio n a l tid e su p p ortin g its d e riv a tio n
fr o m the a tm osp h ere is the fa c t th at its phase tends to lead that o f the g r a ­
v ita tio n a l S3 co m p on en t b y an a n g le n e a r 240°. T h is ta llies w ith the w e llk n o w n p ro p e rty o f the se m i-d iu rn a l a tm o s p h e ric tide, th at its m in im u m
(c o rre s p o n d in g to a p o sitive sta tic rise in sea le v e l) occu rs at abou t 4 o ’clo ck ,
local tim e, e v e ry w h e re on the globe. H o w e v e r, since the a m p litu d e o f the
a tm o s p h e ric tid e is o n ly o f the o rd e r o f 1 m bar, co rre sp o n d in g s ta tis tic a lly
to about a 10 m m rise in sea level, the d y n a m ics o f the tra n s fe r m ech an ism
ca n n o t be said to be p r o p e r ly u n d erstood .
4. L A P L A C E ’S T ID A L E Q U ATIO N S A N D SOLU TION S
FO R ID E A L IZ E D GE O M ETR Y
W e now have to consider the dynamical equations on which so much
analytical tidal w ork has been based over the past 150 years. Laplace’ s
equations for continuity and m om entum are physically plausible and have
been verified in several limited seas. However, they do not perfectly repre­
sent the m echanical system. T he neglected terms are small, but it is not
at all obvious at what stage of solution of the complete equations they
should be set equal to zero, when oceans of global dim ensions are consider­
ed. Scepticism was first raised by B j e r k n e s et al. (1933), who pointed out
that in neglecting vertical acceleration, Laplace’ s equations did not permit
vertical cellular m otions which could he im portant features of tidal m otion
at certain critical latitudes, especially for the diurnal tides in the atm os­
phere. There followed a period of controversy in which the principal defen­
der of the conventional approxim ations was P r o u d m a n (1942). He showed
that in certain spherical basins for which simple solutions to both the
‘ complete ’ and Laplace’s equations were possible, Laplace’s equations gave
a very good approxim ation everywhere except near the poles for the Iv.,
constituent and near the equator for long-period tides. Later, following
some work of H y l l k r a a s (1939), P r o u d m a n (1948) showed that when den­
sity stratification is allowed for, even these restrictions are removed. Most
recently, the limiting processes in ignoring vertical acceleration and other
terms in Laplace’ s equations have been rigorously analysed by M i l e s (1974),
who arrived at a justification which, when expressed in simple terms, is
fairly similar to that of P r o u d m a n (1948).
In this review it would be impossible to give an adequate account of
analysis, but it is appropriate to point out the ways in which
Laplace’s equations differ from a complete mechanical description and
under what circum stances the relevant terms m ay be justifiably neglected.
W e take fixed local coordinates (x, y, z) in the south, east and vertically
upwards directions respectively, and corresponding velocity components
(u, v, w). Let also v denote the two-dim ensional horizontal vector with
com ponents (u, v) and £ the elevation of the surface above its mean level
z = 0. W e assum e the bottom, z = — h(x, y ), to be fixed at this stage ; the
elastic yielding of the Earth will be considered in g 6.1.
M il e s ’
In the first place, we ignore products of the variables, such as occur in
the distinction between Eulerian acceleration dv/d t and the particle accel­
eration D r /D t = dv/dt -)- v grad . v ; we also ignore horizontal viscosity.
Both may be shown to be very sm all in comparison with the leading terms
in the oceanic tides, and m oreover they do not radically alter the character
of the m otion. Vertical viscosity vd2u>/dz2 may enter the equations when
integrated over the depth, giving a resultant horizontal stress from bottom
friction, but even this is negligible in the main body of the ocean with which
we are concerned at present. The equation of volum e continuity is :
du/dx + dvjdy + div/dz = — p —1dp/dt
where p is the density.
(4.1)
If the fluid is taken to he homogeneous, the only
contribution to p ~ 1dp /d t is through com pressibility. This term m ay be
shown to be o f the order of (g h / c 2) tim es typical term s on the left, where
c is the speed of sound in water (1400 km s - 1). Since h is typically 4 km ,
the factor quoted is of the order of 1 /5 0 and may be neglected because
again it does not alter the character of the motion. Putting the right-hand
side o f (4.1) equal to zero and integrating vertically, we obtain the usual
expression of continuity :
d(hü)/dx + d(hv)/dy + 9 f/0 f = 0
(4.2)
where (ü, v) are the depth-averaged horizontal components of velocity.
W ith no more than the above-mentioned restrictions the equations of
horizontal and vertical acceleration are respectively :
d\>fdt + 2SiAt> + 2 S iA \ u = — grad (p/p —
(4.3)
dujfot - 2SiA\> = ( — p ~ l dp[dz — g )k
(4.4)
where S i', S i " are the vectors representing the local vertical and horizontal
components o f the E arth’s rotation, of magnitudes fi cos 6, fl sin 0 (0 is the
co-latitude) respectively, w, k are the vertical vectors of m agnitude w, 1,
and p is the pressure. g% is the generating potential as discussed in § 2. The
terms involving w and the second term o f (4.4) are in general sm all com ­
pared with the leading terms and are often omitted, as they are in Laplace’ s
equations. However, it is just these om issions which have raised scepticism .
T o illustrate the difficulties, in the sim plified case o f periodic m otions
in constant depth h and uniform density, Laplace’s equations (4.2) and (4.9)
reduce to a freeware equation in £ :
V2 r + ( ^ ) - 1 (w 2 _ 4 S Y 2K =
0
(4.5)
where 2-77/oj is the period and f T = |Si'| = f l cos 0 (2Ii' is often denoted by
the symbol f). Equation (4.5) is elliptic and has solutions which change
character from trigonometric to exponential at the ‘ critical co-latitudes ’
given by cos 0 = w /2 { l (near 3 0 “ latitude for diurnal periods, near the poles
for sem i-diurnal). The full equations (4 .1 ), (4.3) and (4.4),\however, reduce
to an equation which is hyperbolic if co2 < 4 O2, and their solutions near the
critical latitudes do not reduce to those of (4.5) as h approaches zero. A s
P r o u d m a n (1942) showed, the solutions of the full and the approxim ate
equations approach each other uniform ly only in certain zones remote from
the critical latitudes.
By allowing for the realistic com plication of density stratification the
dilem m a is resolved ( P r o u d m a n , 1948, M i l e s , 1974). Equations (4.1), (4.3)
and (4.4) still stand with p = / ) ( - , /), and in addition we have the equation
for m ass continuity :
dp/dt — w N 2p/g = 0
(4.6)
where :
N 2 = ~ g p - 1 dp/dz
(4 . 7 )
(N is the ‘buoyancy frequency’ which is the natural frequency of internal
oscillations o f the fluid, typically about 10 times the sem i-diurnal tidal
frequency). The non-static terms in the equation for vertical m otion (4.4)
are now balanced by buoyancy forces through the term p - 1. The solution
of (4.1), (4.3), (4.4) and (4.6) allows an infinite set of wave modes with
different vertical structure according to their mode number. The lowest
order mode, having motion nearly uniform with respect to the vertical, is
found to agree closely with the solution to the simplified barotropic (i.e.
Laplace’ s) equations at all latitudes. Near the critical latitudes the previous
discrepancy is replaced by a weak coupling between the barotropic motion
and the internal motions represented by the highest order modes.
Confidence in the traditional approximation for obtaining global tidal
solutions is thus restored, since the ocean is always well stratified. La­
place’ s equations m ay be derived by setting p = p, the mean density of the
sea, assumed constant, and ignoring the term in w in (4.3) and the left-hand
side of (4.4). Vertical integration of (4.4) then gives the ‘ hydrostatic’
approximation :
(4.8)
P = P g (Ç -z)+ P 0
where p 0 is the atmospheric pressure, assumed uniform.
in (4.3) and again averaging in the vertical gives :
Substituting (4.8)
dü/dr + 2SÎA\> — — g grad (f — f )
(4.9)
Equation (4.9) combined with the continuity equation (4.2) and expressed
if necessary in terms of co-latitude and longitude constitute Laplace’s tidal
equations for the independent variables ü, v, £ ( P e k e r i s (1975) has recently
given a more form al derivation).
M uch
m a th e m a tica l
1950
about
e ffo rt
w as
in
d eriv in g
solu tion s
u n ifo rm
depth
cov e rin g
a co m p le te
R ev iew s
of
the
(1958).
H ere
b e a rin g
on
solu tion
d iv id e s
m ay
o n ly
w ork.
the
tw o
a so cia te d
and
su rface
force
2 S i'A i>
the
w ith
n,
elev a tion .
correspond
a r y w a v e s ’, in w h ic h
: and
w 'ith
m a in
or
fou n d
(1897)
in
the
tim e
eq u a tio n s
sim p le
p o rtio n s
fa r
such
tid a l
m erge
those
tend
w h ich
W aves
o f cla ss
we
restorin g
as
th ey
have
a
of
s h o w e d th at th e fa m ily o f e ig e n so lu tio n s
in to
to
ii
now
know
force
is th e
( L o n g u e t -H ig g in s ,
‘ co n tin e n ta l
freq u en cies
ord in a ry
steady
ex ist
as
o n ly
v a ria tio n
1964).
sh e lf
g ra v ity
cu rren ts
at
‘R o s s b y ’ o r
the
at
as
sphere.
D oodson
L a p la ce ’s m e th o d
(v a ry in g
/i)
a
as
topograp h y
ex ist
of
and
to
of
on
a s b e in g a s s o cia te d w ith
to
ocean
resu lts
sin ce b een id en tified
show n
L a p la ce
an
im p rov ed
w aves have
been
of
in
(1932)
L amb
th o se w h ic h
w hat
la titu d e
fro m
tid a l
p rin cip a l
H ough
c l a s s e s : i,
ten d s to ze ro
the
sphere
be
fo r a co m p le te sp h ere, a n d
q u e n cie s
lis
w ork
ou tlin e
m odern
itse lf in to
w a ves as fi
little
I
old e r
devoted
fo r
of
1969,
fre ­
C orio-
cla ss
ii
changes
in
w aves ’ w h ich
(C a r t w r i g h t ,
w ith
‘p l a n e t ­
the
O ther
abru pt
low
have
H uthnance,
1974).
The complete range of possible eigenfunctions for the spherical ocean
was computed by L o n g u e t - H i g g i n s (1968). Figure 4 shows a plot of the
eigenfrequencies as a function of (g h /a 2)1/2, both normalized with respect
to 2fi (a is the radius) for spherical harmonic waves denoted by parameters
n and s, with in this case 5 = 2. They are divided firstly into a group of
modes travelling eastward (a), in which all waves are of class I, and those
travelling westward (b) which are further subdivided into a highfrequencies group, again of class I, and a lower frequency group which are of
class II. A numerical value of the abscissa for typical oceanic depths is
0.2 and the semi-diurnal tidal frequency gives about ± 1 in the ordinate.
W e see that these values fall well within the range of normal modes, so
it is not surprising that oceanic resonances to the tide-generating forces
occur. L o n g u e t - H i g g i n s also identified eigensolutions corresponding to
negative values of h. These are not possible physically as free waves, of
course, but they would become relevant in matching a given forcing field
with a complete set of normal modes.
F ig . 4. — Eigenfrequencies o f free inodes o f oscillation on a sphere p r o p o r t io n a l to
exp (2iX). (a) Modes travelling eastwards, (b) m odes travelling westwards. In the abscissa,
h is the u n ifo r m depth o f the flu id in units o f the Earth’ s radius. The ordin ate gives
frequency in units o f the inertial frequency at the pole (from L o n g u et -H iggins 1968).
Solutions for a bounded portion of a sphere come a little nearer to
representing the effects of the continental boundaries to the real oceans.
The necessity to set the normal velocity to zero along the boundaries greatly
complicates the m athematics. In an im portant series of papers, P r o u d m a n
(1917, 1931) set out general theorems by which the normal m odes and
forced solutions m ay be calculated for oceans bounded by arbitrary coasts.
G o l d s b r o u g h (1927) solved for the forced K 2 tidal constituent in a sea
bounded by two meridians 60° apart, som ewhat like the Atlantic Ocean.
He found that resonance occurred at depths rather less than that of the
actual Atlantic Ocean. P r o u d m a n & D o o d s o n (1936, 1938) solved for both
the Ki and K 2 tidal constituents in an ocean bounded by m eridians 1 80 ”
apart, somewhat like the Pacific. A s the assum ed mean depth was reduced,
successive resonances were passed, at each of which a new node of zero
elevation appeared in the solution, more nodes for K 2 than for K ,. It is
now known that at least 2 nodes exist in the Pacific Ocean for K , and 6
for K ,. The complete set of normal modes for such a hemispherical ocean
has been computed by L o n g u e t - H i g g i n s & P o n d (1970), in a m anner similar
to L o n g u e t - H i g g i n s ’ treatment of the spherical ocean. All these calcula-
tion s
co n firm
som ew hat
that
a fter
the
e lev a tion s
m anner
ten d
of
to
the
be
h ig h e st
w e ll-k n o w n
near
the
‘K e lv in
b o u n d in g
w ave’
in
coast,
sim p ler
geom etry.
5. T ID A L C A LC U L A T IO N S IN OCEAN OF M ORE G E N ER A L SH APE
Exact solutions for simple portions o f a sphere aid our understanding
of howT tides behave in the real ocean, but do no more than that. Irregular
boundaries and depth topography greatly complicate the situation and
require other techniques for solution. W e should distinguish here between
oceanic solutions and solutions for shallow semi-enclosed seas. In the
latter case, the gravitational forcing function represented by g£ in equa­
tion (4.9) can be ignored and the tides can be treated as free waves driven
by the wave form at the sea’s openings to the ocean, which in many cases
can be determined from measurement. The elevations round the coast are,
in any case, determinable from measurement. If the tidal currents are also
known these also provide boundary gradients of £ through equation (4.9).
Even simpler is the case of an elongated gulf or channel like the
Adriatic or Red Seas. There we m ay take the r axis along the gulf, and
assum e that transverse currents v are much sm aller than u (a procedure
justified with some rigour by P r o u d m a n (1925 a)). The equations of
continuity and acceleration ((4.2) and (4.9)) reduce to :
d(hu)/dx = — 9 f/9 1
and
du/dr = — gd£/dx — r(u)/ph
respectively, where r(u) is the frictional stress on the bottom, and h = h(x)
is the mean depth of each transverse section of the sea. W ith a linear form
r = pcü for the frictional stress, which is usually adequate at this degree
of approximation, and a tidal constituent of period 2ir/oo for which ü =
U exp <— iajt), £ = Z exp (— icot), the above equations yield :
iccZ = d{hU)jdx
(to2 — icoc) U + g(b/dx)\b(hU)jdx] = 0
These equations m ay easily be solved numerically for a given sequence of
depths h (x ), with the boundary values Z =
at the open sea end(s) of
the gulf and U = 0 at a closed end. Finally, the gradient of elevation
across the width of the gulf may be obtained from the transverse equation
involving simply the ‘Coriolis’ or ‘geostrophic’ stress :
9 f/av = -
2n 'v(x)ig
D e f a n t (1961) shows several examples of cotidal maps for elongated seas
derived essentially by this method with hand computation.
But such methods, even when extended to two dimensions, cannot deal
directly with seas of oceanic scale because of the complications of spherical
geometry, the tide-generating forces, and the relative lack of boundary data.
W h ile the full problem can only be tackled by finite-difference methods
with large modern computers, it is worth briefly considering some of the
semi-analytical methods which have been applied to limited portions of
open oceans. Several approaches were developed by P r o u d m a n and his
associates in pre-computer days. His ‘tidal theorem’ ( P r o u d m a n , 1925 b)
relates surface integrals of the forced tides over the area of any sea on a
sphere to line integrals round its boundaries, after the m anner of Green’s
potential theorem. In its more practical form , if a certain fam ily o f free
waves can be defined in the sea area A P A ' (figure 5 (« )) but unrestricted by
the boundaries, then the elevations and currents of forced tides along the
parallel o f latitude A A ' can be determined by a series of explicit integrals
involving the fam ily of free waves, the tide-generating potential, and the
(coastal) tidal elevations round the boundary A P A '. The process can then
be repeated for the zone B A A 'B ', and so continued to cover possibly an
extensive sea area. Unfortunately, however, the free-wave solutions can
only be solved tractably for a sea o f constant depth, so at least each latitu­
dinal strip must be approximated by a uniform depth, which is sometimes
unrealistic. The method appears to have been applied seriously only in one
case, to part of the Indian Ocean ( F a i r b a i r n , 1954).
p
Fie.. 5.
(a) Illustrating P h o u iim an ’ s tidal th eorem. ( h) Geometry o f P h o u d m a n ’ s (1946)
general expan sion for tides in a tw o -d im e n s io n a l la tit udinal section.
A method which involves dividing an oceanic area into a sm all number
of rectangles, each o f constant depth, was applied by G r a c e (1932, 1935)
to the Gulf of Mexico and the Bay of Biscay. It has the disadvantage that
the tidal currents m ust be assum ed constant along each of the rectangle’s
sides, which tend to be rather long for such a property to be realistic.
P r o u d m a n (1946) developed a more powerful approach. A narrow
strip of ocean is bounded by two parallels of latitude, R R ' and SS' (figure
5 (6 )). W ith in the strip, the depth topography m ay be any function of
longitude, but m ust be constant along each meridian. An important touch
of realism is added by allowing the tides within the strip to radiate outwards
into adjacent shallow water at R and R' at an arbitrary rate determined
by the depths at those ends (possibly zero) ; this allows for frictional loss
of energy. Given any distribution o f tidal elevation and current along RR',
P r o u d m a n (1946) derived explicit form ulae for the tides in the interior,
including the other boundary SS'. He went on to apply this form alism to
the central Atlantic Ocean between 35° S and 45° N, by dividing the whole
area between the two continents into 5° strips, each of appropriate depth
topography. Starting with an arbitrary functional form for the tides along
the southern boundary, he calculated its effect propagating through each
successive strip to the northern boundary. He did this for a range of
starting functions, and then found a linear combination which best fitted
the known tidal elevations along the continental edges. A s related in his
George Darwin Lecture ( P r o u d m a n , 1944) (*) the result was only partly
successful, but could have been improved by taking a longer series of
boundary functions which were essentially a series of free Kelvin and
Poincaré wave form s, with an added wave forced by the generating poten­
tial. However, it would appear that the promise of the computer-based
solutions produced soon after by H a n s e n (1949) discouraged P r o u d m a n
from pursuing his more analytical method any further.
P r o u d m a n would have been more successful if he had had some direct
measurements along his initial boundary. More recently M u n k et al (1970)
have used a method related to P r o u d m a n ’ s (1944) to rationalize the tidal
system in the Pacific Ocean off California using deep-ocean tidal measure­
m ents (see § 9) as well as those from ordinary coastal tide gauges. They
treated the coastline as straight (the x axis) and the ocean as semi-infinite
with a uniform underwater profile independent of x, stretching in uie
y direction from a narrow shelf structure to a uniform ly deep region some
distance from the coast. The effect of varying latitude on the vertical com ­
ponent of the E arth’s rotation Cl' was ignored. Over this simplified topo­
graphy they considered all possible wave forms :
/ 0 ) exP [i(/3x - cor)]
which satisfy Laplace’s tidal equations. The frequency co w7as taken as
positive, so positive values of /3 represent waves travelling along the coast
with shallow water on their left, and vice versa. The principal solutions
divide into two groups according to whether f(y) behaves at a distance like
exp (— fiy) or like exp (ifxy). The first case represents ‘trapped’ waves,
of which the ‘Kelvin’ wave, for which /j, = 2CI' ( g h ) ~ w~, is a classic example
( L a m b , 1932) and these, like the Kelvin wave, are associated with a unique
wavenumber /3 for a given frequency &>. If the frequency is less than the
inertial frequency (e.g. the diurnal tides polewards of 30° latitude), then /3
m ust be negative to maintain geostrophic balance in the northern hemi­
sphere (positive in the southern). At higher frequencies, positive but
numerically smaller wavenumbers J3 are also possible.
The other principal fam ily of waves is oscillatory in the offshore direc­
tion and represents the ie a k y ’ modes, of which the classic example is the
‘Poincaré’ wave ( L a m b , 1932). These waves exist only at frequencies a>
higher than the inertial frequency 2Cl' and their wavenumbers, positive or
negative, are generally lower than the trapped waves at the same fre­
quency. Also, since ji is arbitrary in this case, the wavenumbers /3 may
take any value between certain limits for a given frequency. There is
thus a ‘continuum ’ region of permissible leaky wave modes in the (/3, a>)
plane.
The complete range of solutions for the shelf geometry considered by
et al is represented in figure 6, in which /3 is scaled to L ~ 1 where L
is the shelf width, and the vertical &> axis is scaled to the inertial frequency
2Cl'. The trapped waves as described above are represented by the loci
named ‘discrete edge waves, n = O’ . Similar loci for n = 1 and 2 are higher
mode trapped waves possessing oscillations in f(y) between the shelf edge
M
unk
(*) This
in 1942.
paper
logically
fo llo w s
P r oc dm an
(1946),
which
was actually
presented
and the coast. The leaky m ode ‘continuum ’ is shown bounded by a hyper­
bola with apex at <d/ 2 Q ' = 1. W ith in this continuum , the authors identified
certain discrete waves for w hich the am plitudes near the coast are m agni­
fied (full circles) or reduced (open circles). The form er are considered to
be more likely to represent the observed tidal conditions. From these freewave solutions, to which they added a waveform directly forced by the
generating potential, the authors m anaged to select a com bination which
convincingly fitted a num ber of onshore and offshore tidal m easurem ents
in amplitude and phase. Both diurnal and sem i-diurnal tides were treated.
The semi-diurnal tidal w aveform indicated an am phidrom e (node) about
2 000 km offshore, which was later confirm ed by supplementary m easure­
ments.
F ig . 6.
Dispersion dia gram f o r lo n g waves off a m o d e l oceanic shelf. The abscissa
represents lon g-shore w a v en u m b e r w ith positiv e valu es progressin g w ith sh allo w w a te r to
th e left (northern hemis phere). The ordin ate represents frequ ency in units o f the local
in ertia l frequency 2n'. F u ll curves denote perm issible (trapped) m odes w ho se am p litu d e
decreases e x p on en tia lly tow ards the deep ocean. W aves o f P o in ca ré-typ e (periodic in
all directions) are possible a n y w h e re inside the inner h y p e r b o la but are likely
to be large near the coast at the full circles, sm all at the open circles
(from M u n k et al., 1970).
At the bottom left of figure 6, another species of freewave is described
as a ‘ discrete shelf w a v e ’ . T h is is otherwise named a ‘ continental shelf
wave ’ or ‘ topographical R ossby wave ’, and was referred to in § 4 as a
new ly discovered ( R o b i n s o n , 1964) exam ple o f H o u g h ’ s ‘class II’ w aves,
consisting principally of currents. As figure 6 suggests, shelf waves exist
only at frequencies below the inertial, and travel only with shallow water
on their right (northern hem isphere). Although M u n k et al measured tidal
currents at their pelagic sites, they did not find any evidence for the
existence of such waves at tidal frequencies off California. C a r t w r i g h t
(1969, 1974), however, found that their properties matched those of some
unusually strong diurnal tidal currents off the west o f Scotland, a pheno­
m enon which had been virtually forgotten since first publicized in an early
volum e of the Royal Society’s transactions by Sir Robert Moray. H u t h n a n c e (1974) has also exam ined som e enhanced diurnal currents on Rockall
Bank in terms of shelf wave theory. In both areas mentioned the sem i­
diurnal tidal frequency is higher than 2Cl', so cannot include continental
shelf waves or leaky modes of Poincaré-type. They appear to have the pro­
perties of a trapped Kelvin-like wave, although the complicated form of
the coast prevents any sim ple interpretation.
6.
M O DERN G L O B A L SO LU TIO N S
A N D A T T E M P T S T O M O D E L D IS S IP A T IO N
The development of autom atic computers in the 1950s at last opened
up the possibility of finding realistic solutions of Laplace’ s tidal equations
over all oceans and seas w ithout any idealization of geometry. In principle,
the procedure is straightforward. One divides the ocean by a network of
elem entary areas, sm all enough to define the tidal structure. In each area
specified by (/, j) (figure 7) the depth hif is specified from a good bathy­
metric chart, depth-m ean current üijt
are assum ed at the central point
(cross), and elevation £i+1 /2 j and £,jJ + i /2 at the edges. In the continuity
equation (4.2), d(hli)/dx is approxim ated by :
Ar/
l ( h a u if
— h i —i j u i —
i,/)
where A (j is the m esh length and is associated with 3 (£ ,_ 1/2. f)/dt. In the
acceleration equation (4.9), 9£/9.r is approximated by :
and associated with d(u()) /d t and 2
t he frequency 2fl' being defined of
course by the latitude, as is the generating potential g£. At land boundaries
L- 1
-
+
0
+
0
4-
O -
-f
O ---- V — o
I
- 0 - 4 ----- 0 - + - 0 -
V
X
y -'
J
y+1
Fig. 7. — Basic n e tw ork c o m m o n ly used f o r fin ite-difference tidal m odels,
elevations calculated at th e circles and current c o m p o n e n t s at crosses.
with
the component o f velocity norm al to the coast (not necessarily parallel to
a mesh side) is equated to zero. At open sea boundaries, either £ or some
linear combination of v and £ m ust be specified by empirical data.
If the linearized form of the equations is used, as in (4.2) and (4 .9 ), a
com m on time factor exp (— ia>t) may be removed, and the problem is
expressed as an array of ordinary linear equations determined by the
boundary conditions (a ‘boundary-value problem ’ ). Such a method was
followed by P e k e r i s & A c c a d (1969). More usually, the time factor is
retained and the solution approached by tim e-stepping from an assumed
condition of rest everywhere until a steady state of oscillation is reached.
This enables non-linear terms, such as a quadratic friction law, to be
retained ( Z a h e l , 1970). There are also questions of numerical stability,
choice of time increment, etc, which are outside the scope of this review.
If we exclude various attempted solutions for individual oceans,
which invariably suffer from lack of reliable data along the open boundaries,
at least five independent global solutions have been proposed for the M.,
tide alone during the last decade. T o these we should add the very credit­
able empirically drawn cotidal m aps by D i e t r i c h (1944), later modified by
V i l l a i n (1952), which are still invoked in com parisons with pelagic m easu­
rements ( Z e t l e r et al, 1975). H e n d e r s h o t t (1973) is an excellent com pa­
rative review : see also Z a h e l (1977). Unfortunately, the solutions differ
considerably from each other and from the measured tides in various parts
of the ocean. It is therefore a matter of considerable importance to try to
understand where each method has gone wrong. In the first place, the
m ethods and types o f solution can be divided into two classes, according
to whether the solution is derived solely from the tide-generating potential
and oceanic topography, or whether it is constrained to agree with tidal
m easurem ents along the coastal boundaries. The m aps of P e k e r i s & A c c a d
(1969) and Z a h e l (1970, 1977) are of the form er class : those of B o g d a n o v
& M a g a r i k (1967), T i r o n et al (1967) and H e n d e r s h o t t (1972) are of the
latter class. A typical example from each class is shown in figures 8 and 9.
A ‘ cotidal map ’ as exhibited here defines the oceanic loci of equal
phase lag (co-phase lines), here typically in steps of ,30°, and the loci of
equal amplitude (co-am plitude lines) of a harmonic constituent of the tide,
in this case M.,. The nodal points, traditionally known as ‘ am phidrom es ’,
stand out as centres of zero am plitude round which the co-phase lines
rotate, usually but not invariably clockwise in the southern hemisphere
and anticlockwise in the northern hemisphere. Less obvious, but perhaps
more important, are the ‘anti-am phidrom es’, zones of locally m axim um
amplitude with little change of phase, such as shown in both figures 8
and 9 south of India, though with different am plitude. T he two maps agree
roughly in some other gross features ; the am phidrom es of the North
Atlantic and Central Pacific, and anti-am phidrom es somewhere in the
equatorial Pacific, and the rotatory system (‘virtual am phidrom e’ ) centred
on New Zealand. But the resemblances are clearly outnumbered by the
m any differences, some quite radical. T he two am phidrom es west of the
North American continent in figure 8 rotate in opposite direction from
those in figure 9, although both contrive to show7 a northward progression
o f phases along the Californian coast. (The map of P e k e r i s & A c c a d
(1969) gives a southward propagation there, contrary to observation).
Figure 8 agrees with P e k e r i s & A c c a d in placing a prom inent anti­
clockwise am phidrom e in the central south Atlantic, which is absent from
figure 9 and from all maps constrained to agree with coastal observations.
P e k e r i s i'’' A c c a d (1969) showed that this interpretation was consistent
w ith data from the four islands in this ocean area — a point further inves­
tigated by C a r t w h i g h t (1971) — but it is difficult to reconcile it with the
observed northward progression of phase along the Brazilian coast.
Comparing free and coastally constrained tidal solutions naturally
tends to emphasize differences, although it is unfortunate that such differ­
ences even exist. But the constrained solutions also differ from each other,
indicating errors in physical representation. In fact, all the authors m en­
tioned differ in their treatment of friction, which is a factor o f fundam ental
importance as shown in § 7. Basically, the force per unit m ass due to
friction m ay be added to equation (4.9) in the form :
— cü)ui/h
(6.1)
where c is a non-dim ensional constant known from direct m easurem ent in
shallow water to be about 0.0025 ( B o w d e n & F a i r b a i r n , 1956). v should
properly represent a velocity about 1 m above the bottom though the depthm ean velocity v is often assumed. The direct effects of (6.1) are therefore
practically confined to the shallow seas, where v is large and h is sm all.
However, global tidal com putations are m ostly confined to deep-water zones
for practical reasons (e.g. shallow-water tides require a much finer mesh
owing to their smaller scale of horizontal variation), and so dissipation can
only be properly represented by radiation of energy out of the m odel into
bounding seas.
The im portance of boundary radiation was first pointed out seriously
by P r o u d m a n (1941) when he showed that the analytical solution for a
frictionless hemispherical ocean was radically altered when the usual
boundary condition of zero norm al velocity was replaced by a radiation
condition. In essence, if a tidal wave of elevation £ and m ean norm al
velocity vn propagates from an ocean of depth h to a shelf of depth h' where
its energy is completely absorbed by friction, the following relations hold at
the shelf edge :
Ç =ï
hvn = h'vn' =
(6.2)
h v j s = (gh’y i 2
(6.3)
that is :
where the prime denotes the wave parameters on the shelf. Thus (6.3) is
the appropriate boundary relation instead of vn = 0. It im plies normal
velocity in phase with the tidal elevation. In general, not all the energy
propagated across a given stretch of boundary is absorbed, so a relation
like :
hvn = X£
(6.4)
where X is a complex parameter, is appropriate.
usually only be assigned empirically.
In the real ocean, X can
Figure 9 was computed by H e n d e r s h o t t (1972) with boundary rela­
tions like (6.4) where possible, the body of the ocean being assumed friction-
BO
IN TERN ATION AL
H YD R O G R A PH IC
R E V IEW
less. A. = 0 w as, o f course, assum ed round continents like Africa whose
shelf zone is very narrow. Figure 8 was com puted by Z a h e l (1977) using
direct friction ( 6 . 1 ) in the field equations which, unlike H e n d e r s h o t t ’ s ,
extended to depths as sm all as 50 m . M ost dissipation occurs in seas m uch
shallower than 50 m , but follow ing T r e p k a (1 96 7 ), Z a h e l added an arbit­
rary term representing horizontal shear stress, approxim ately :
A d 2v/dx2
(6-5)
The physical significance of such a term as (6.5) is hardly known from
oceanographic m easurem ents, but it serves as a stabilizing parameter which
also accounts for a good deal of dissipation not directly represented in the
num erical m odel. In fact, the arbitrary factor A was adjusted so that the
cotidal map agreed as nearly as possible with coastal data.
P e k e r i s & A c c a d (1969) restricted their area o f com putation to depths
o f a kilometre or more and, like Z a h e l , used reflecting boundaries. Some
dissipation had to be introduced artificially, and since they had an interest
in keeping the equations linear, they adopted a form — a.ctiv(h{J h )- where
h 0 = 1 km, and a took a sequence o f trial values in the range 0.1-0.5.
Physically, this frictional term was m uch greater than the true frictional
stress at any given point of the ocean, but was intended to com pensate for
the lack o f genuine shallow -w ater friction. The resulting cotidal maps were
only qualitatively successful, but from P e k e r i s & A c c a o ’ s account, it
appears that they regarded this work as a pioneering exercise, to be im ­
proved later.
The earlier published global maps by B o g d a n o v & M a g a r i k (1967) and
by T i r o n et at. (1967) apparently were derived from frictionless equations,
on the assum ption that the effects of friction are incorporated in the
empirical boundary data to which their solutions were constrained. This
assum ption is partly true, but with later knowledge one sees that the use of
reflective boundary conditions everywhere m akes for solutions which fall
short of realism . Their results agree w ith neither figure 8 nor figure 9.
However, both groups of Russian authors also give cotidal maps for S2,
K , and O j. Otherwise only Z a h e l (1973) has computed for a constituent
(K ,) other than M^.
Apart from this troublesom e question o f friction, the diversity of results
has led some scientists to investigate other neglected physical factors which
m ight prove surprisingly im portant on a global scale. One such factor
whose importance has recently been dem onstrated is the elastic yielding of
the E arth’s crust to tidal loading. This introduces the Earth tide into our
discussion and requires a separate sectional heading.
6.1 The Earth tide
It has long been know n from precise m easurem ents o f gravity (still
longer in theory) that the E arth ’ s crust itself yields elastically to the tidal
forces of the M oon and Sun. The normal m odes o f oscillation of the Earth,
unlike those of the ocean, have frequencies at least an order of m agnitude
higher than the tidal frequencies, so the response is virtually static. If £,.
is the tidal elevation of a portion of the Earth’s crust above its mean posi­
tion, due to direct yielding :
Se(6
(6.6)
= h j ( 0 ,\ ,t)
where £ is Ibe ‘ equilibrium tide ’ defined in g 2 and h., is the elastic constant
( ‘ Love number ’) now known to have the value 0.61. Further, as a result
of this yielding the tidal potential on the Earth’s surface is increased by
k 2g £, where k 2 = 0.30 is another Love number.
£,, as defined by (6.6) is known as the ‘ body tide ’, the rise and fall of
the Earth’s crust which would occur in the absence of the oceans. Since the
equilibrium tide due to the Moon has an average amplitude near the equator
of about 0.24 in, the corresponding amplitude of the body tide is about
0.15 m . The actual rise and fall is, however, considerably modified by the
variable loading of the oceanic tide, which produces additional vertical
crustal m ovements o f typical amplitude 0.05 m , as discussed below. Such
m ovem ents are sensed by gravimeters, the amplitudes of the change in
gravity associated with the last two figures being 45 and 15 ^iGals, respect­
ively. The tilt and strain of the E arth’s crust associated with these com ­
ponents of the Earth tide are also of interest, but these cannot be pursued
in the present context.
Consider now the effect of the body tide on Laplace’s tidal equations
for the ocean ( (4.2) and (4.9) ), remembering that the water elevation £ is
strictly the relative elevation of the surface with respect to the local E arth’s
crust if theory is to be compared w ith conventional m easurements (*). The
equation of cpntinuity (4.2) is unaltered but since the horizontal gradient
of pressure is strictly referred to geocentric elevation (£ + £,.) and the
potential is increased by the factor k.,. the right-hand side of (4.9) should
now read :
- g grad [r + r e - ( 1
+ * 2) ? ] = - f grad K - ( 1
+ k2 - h2) f ] .
(6.7)
Thus, from mere consideration of the body tide, the oceanic tidegenerating potential should be multiplied by (1 + k.2 — h 2) = 0.69 in solving
L aplace’s equations. This adjustm ent was first applied practically by G r a c e
(1930), who attempted to estimate the elastic constant from an analysis of
the tides in the Red Sea. It was also used in the calculations of Munk
et al. (1970) and of C a r t w r i g h t (1971). Surprisingly, it seems to have been
ignored by all computers of global tides considered in § 6 prior to 1972.
Presum ably the constraints to fit boundary data and the adjustm ent of
arbitrary parameters have to some extent compensated for the error.
H e n d e r s h o t t (1972), however, went further than the body-tide adjust­
ment (6.7) ; he allowed also for the ‘loading tide’. T his is the additional
yielding of the Earth’s crust due to the direct loading o f the ocean tide
itself. The deformation o f the sea floor due to elastic yielding to a point
load £S ( # — 0', \ — A') at (0', A') can be expressed as :
f e'( 0 ,X ) = rCW)
(6.8)
where G is a function only of the angular distance (/} between (0, A) and
(*) Sa te llite a lti m e te r s
t i d a l e l e v a t i o n s , (Ç + Ç. +
in t r o d u c e
see §9.
th e
possibility
of
directly
m e a s u r in g
geocentric
{O', A/)- G depends on the elastic structure of the Earth, but F a r r e l l
(1972) has shown that several plausible m odels based on seismic observa­
tions yield approxim ately the same function, at least for values of
greater
than a few degrees. By applying certain harmonic integrals to G (<f>) one
can equivalently express the Earth deform ation to a spherical harmonic in
ocean elevation of degree n, nam ely £n as :
(6.9)
where an is a normalized density ratio (w ater/earth ), and h n’ is a ‘loading
Love num ber’ ( L a m b , 1932). The com bination of loading harm onic and
corresponding crustal deform ation also increase the effective potential by :
( 6 .10 )
(1 + k „ ') g a nÇn
The quantities h n', k n\ a n have been evaluated to high order and m ay
be found listed for exam ple in H e n d e r s h o t t (1972). Since positive sea
elevation depresses the crust in the m ain, h n' and kn' are negative.
W ith both body tide and loading included, the equation for horizontal
acceleration now becom es :
dxïdt + 2 l Î A ÿ = - g grad [ f - ( 1
+ k2 - h2) f + £
(i + kn’ - hn')a n f J
(6 .11)
n
Since the expansion of £ into spherical harmonic coefficients £„ implies
an integral process, this is an integro-differential equation. H e n d e r s h o t t
(1972) attempted to solve it by iteration, starting with the straightforward
solution which ignores the loading tide (as represented in figure 9), using
this to evaluate the spherical harmonic series £„, then re-solving (6.11 ) with
this value of £„. The loading tide has am plitudes of several centimetres so
the correction is not entirely trivial. Unfortunately, H e n d e r s h o t t found
that the second solution to (4.2) and (6.11) differs considerably from the
first, suggesting that an iterative procedure on the sam e lines would not
converge. At a sym posium on ‘ Tidal Interactions ’ held during the IUGG
General Assem bly at Grenoble in 1975, H e n d e r s h o t t reported some further
investigations of iterative procedures for solving these equations but
without a w holly satisfactory result (*).
Thus, no solution has yet been found to the equations which most
com pletely represent the physical m echanism of the global oceanic tides.
A ll we have is a num ber of differing solutions to sets of equations which
only partially represent the physical m echanism . In § 9 we shall consider
other possible approaches for the future, involving a more coordinated
approach to com putation combined with direct m easurem ent of tides in the
open sea. W h ile still under the heading ‘ Earth tides
m ention should also
be made of some recent attempts by K u o & J a c h e n s (1977) to deduce the
ocean tides in limited areas entirely from land-based m easurem ents o f the
loading tide. Given sufficiently wide coverage o f accurate m easurem ents of
tidal gravity and tilt over the world’s continents, the loading tide m ay be de­
duced by subtraction o f the known body tide. In principle it m ay be possible
to deduce £(0, \, t) from £ /(6 , \, t) by inverting an integral equation derived
(*) P e ke b is ( ]9 7 7 ,
b o u n d a r y data.
le c t u r e )
has
r e c e n t ly
solved
(6 .1 1 )
when
uncon strain ed
by
from (6 .8 ), and a sim ilar one involving erustal tilt. There are, however,
problem s of precision of m easurem ent, especially tilt ( B a k e r & L e n n o n ,
1973) and the influence of near-shore tidal system s. Some o f the results
reported by Kuo & J a c h e n s (1977) for the north-eastern Pacific appear
prom ising, but the possibility o f global coverage is highly controversial.
7. P O W E R B U D G E TS A N D IN T E R N A L TIDES
In the previous section m uch was made o f the way friction has been
represented in various global tidal m odels. This is not m erely a matter of
com putational technique : ultim ately it has a bearing on a fundam ental
geophysical quantity, the total rate of dissipation of energy by the oceanic
tides. In this and the follow ing section we shall consider present know ­
ledge of total dissipation and its bearing on the celestial m echanics o f the
Earth-M oon system.
Viewing the oceans as a m echanical entity, the first energetic quantity
which deserves attention is the total (mean) energy of the tidal system,
evidently ‘ c o n sta n t’ on the time scale of a century or so (*). Calling this
E o , wTe have :
Eo = Potential Energy (PEo) + Kinetic Energy (KEo)
where
PEo
Here the sym bols < . . . > denote time averages of the enclosed quantities and
the integrals are supposed taken over the surface of the ocean. Curiously
few authors have carried out these integrals. H e n d e r s h o t t ’ s (1972) values
for the Mo tide, nam ely 2.1, 5.2, 7.3 X 1017 J for PEo, KEo and Eo
respectively, m ust be taken as the standard estimates (calculations based
on the equilibrium tide give about an order of magnitude less).
More im portant are the rates at which the gravitational forces feed
energy into the tidal system and at which energy is lost from it through
friction. These m ust of course be equal but we have to distinguish at least
three types o f estimate o f power transfer which unfortunately do not
agree, owing to the inadequacy of our knowledge of the mechanical sys­
tem. These are represented by P u P., and P :i in figure 10.
P , is the power input from the Moon and the Sun and from the
Earth-tidal m ovem ent of the sea bed. The input from the Earth tide is
ignored by all authors except H e n d e r s h o t t (1972), who shows that it con­
tributes about 30 % of the total ; authors of rigid-Earth calculations may,
however, compensate by some artificial means of dissipation. It may be
shown to be expressible as :
1%
(*) C^nTwitJtfHT
f o u n d the a m p l i t u d e
p e r c e n t u r y o v e r a b o u t 240 y e a r s .
of
M., at
B resl
to
lu ac
decreased
by
For the M 2 tide, H e n d e r s h o t t (1972) derived P l = 3.0 X 1012 W , while
from simpler but roughly equivalent expressions M u n k & M a c D o n a l d
(1960) derived 3.2 from D i e t r i c h ’ s (1944) cotidal m a p , P e k e r i s & A c c a d
(1969) derived 6.3 and Z a h e l (1977) 3.8 X 1012 W . W e should n o t place
too much weight o n P e k e r i s & A c c a d ’ s figure since their model lacks
realism, so a fair average for P 1 is 3.5 ± 0.5 X 1012 W from the M ., tide
only. A n estimate including M a, S2 a n d other tidal constituents is 5 ± 1 X
1012 W . The M , tide thus has its mean energy replaced in E „ / P x =
(7 X 1017) /( 3 .5 X 1012) = 2.0 X 10r>s = 2.3 d. The corresponding ‘ Q fac­
tor’ for the ocean tides is Q ~ a)E0/ P , = 28.
SHALLOW S E A S
AND EST U A R IES
P ir.. 10. — D ia g ra m of p o w e r in p u t to the o c e a n ? , f r o m .M o o n anti S u n and o u t p u t
a cross the c o n t i n e n t a l s h e lf b o u n d a r i e s to t h e s h a l l o w - w a t e r d i s s i p a t i o n areas.
At this point, mention should be made of a loose relationship between
the ocean’ s Q factor and the ‘age’ of the tide. Age is a 19th century term
for the time lag of the peak of spring tides after the peak of the generating
forces at Full and New Moons. In most parts of the ocean it is 1-2 d,
although a few places have much larger or negative ages. It is easily seen
that positive age implies an increase in phase lag with frequency from the
lunar Mj constituent to the higher frequency solar
constituent. P h o u d m a n (1941) showed analytically that energy dissipation in coastal seas was
necessary to produce predominantly positive age in a hemispherical frictionless ocean. G a r r e t t & M u n k (1971) showed how the phase gradient
could be roughly related to the response curve of the normal modes of the
ocean near resonance, the gradient being proportional to Q. They deduced
from a worldwide survey of tidal ages that Q must be greater than 15, an
interesting result to obtain from such simple observations. Using an esti­
mate for P l somewhat lower than that quoted in the previous paragraph,
and a value of E 0 for the equilibrium tide of 3 X 101(Î J, they also deduced
that the spatial coupling between the tidal forcing function and the dom i­
nant normal mode m ust be rather weak, a fact which is qualitatively
evident from figures 8 or 9. W e b b (1937 b) is also interested by the Q/a g e
relationship and points out that the parts of the oceans where the tidal age
is large are also those w'here dissipation is known to be concentrated ( M i l ­
l e r , 1966)
and in some cases local resonance is suspected (e.g. Gulf of
Patagonia, Coral Sea, Hudson Bay). Because of this localized nature of the
strongly resonating areas, W e b b criticizes G a r r e t t & M u n k ’ s (1971) use
of an overall Q factor. He prefers to discuss the mechanics of ocean tides
in terms of ‘decay tim es’, roughly equal to the age, and the time needed
for the energy fed into mid-ocean to be transported to the dissipating areas
of resonance. But this is a controversial subject to which it is difficult to
apply precise arguments.
Returning again to figure 10, P 3 and P 2 are respectively the power
dissipated directly by friction on the bottom of the shallow seas and the
power transmitted across their boundaries with the ocean. They are given
by
where F is the frictional stress, integrated over shallow seas, and vn is the
inward normal component o f mean velocity across their oceanic boundaries.
Since F = — cp |v 12/ h , the integral for P a involves the cube o f the current
speed which is certainly not known to any precision over m ost of the
w orld’ s seas. T a y l o r (1919), who was the first to examine these relations
seriously, showed that P 2 and P 3 were fairly comparable in the isolated case
o f the Irish Sea, for which the total P., principally from the southern
entrance, was 6.4 X 1010 W , for Spring tides. Reduced to the power
dissipated by the M2 tide alone, this comes to 5 X 1010 W , about a hun­
dredth of the total astronomical input to the oceans.
Soon after T a y l o r ' s ( 1 9 1 9 ) paper, two geophysicists independently
estimated the total dissipation in all the world’s shallow seas — essentially
by P 3 calculations using the known areas and depths of such seas and
what figures were available for the strength of their tidal currents. J e f ­
fre ys
( 1 9 2 0 ) obtained 1 . 1 and H e i s k a n e n ( 1 9 2 1 ) 1 . 9 X 1 0 12 W for the M 2
tide. But the dependence of these estimates on the cube of roughly known
current speeds made them appear dubious. M i l l e r ( 1 9 6 6 ) repeated the
calculation using the more reliable P2 integral and up-to-date information
on the amplitudes and phases of currents and elevations, and obtained
1 . 7 X 1 0 12 W . He showed that about two-thirds of this total comes from
the Bering Sea, the Okhotsk Sea, the seas north of Australia, the seas
surrounding the British Isles, the Patagonian Shelf, and Hudson Bay, in
order of importance, while the narrow shelves surrounding m ost of Africa
and along the west coast of the Am ericas south of Canada dissipate hardly
any power at all. ( J e f f r e y s and H e i s k a n e n had named m ostly the same
im portant seas as M i l l e r , but differed greatly in their individual contribu­
tions ).
M i l l e r ’ s estimate for P., seems fairly reliable, although the data on
w hich the dissipation rates for individual seas are based are very variable
in quality and quantity. Figure 11 shows a detailed distribution of the P 3
dissipation over the various parts of the seas surrounding the British Isles
and the corresponding P., power transfers from a recent computational
model by F l a t h e r (1975). The tidal elevations and currents are known
in great detail over most of this sea area, and the m ajor dissipation figures
are corroborated by other estim ates. The total P 2 input is 20.5 X 1010 W>
whereas M i l l e r (1966) assumed 17.3, no great difference. On the other
hand, M i l l e r ’ s figure of 13 X 1010 W for the Patagonian Shelf was based
on very sparse current data. W e b b (1976) suggests by a sim plified model
that the Patagonian Shelf may be a very strong absorber of tidal energy.
T his region therefore seems to merit a thorough investigation based on
m easurem ent and computation. The result may (or may not) add signi­
ficantly to M i l l e r ’ s total P 2, and incidentally may shed some light on the
problem of the existence or non-existence of a M ^ amphidrome in the South
Atlantic Ocean (cf. figures 8 and 9).
Fig. 11. — (a) Cotidal map for the M2 tide in the shelf seas surrounding the British Isles based on measured data along the open oceanic
boundary, zero normal flow at the coast, and suitable frictional laws in the interior. (b) Rate of flow of energy into the area across the
oceanic boundaries and frictional losses in the interior of the same model (from F lather 1975).
At all events, the salient point in all this is that the current estimate
of about 1.7 X 1012 W for shallow-sea dissipation falls short of the esti­
mated input power, P lt from the gravitational forces by 50 % . This is a
very puzzling gap in the power budget, for which neither oceanographers
nor solid-Earth geophysicists can at present suggest a likely explanation.
From a certain point of view it m ay be seen as the principal motivation for
continuing the study of oceanic tides. The problem has been appreciated
for several years and was discussed at length by M u n k & M a c D o n a l d
(1960) and by M u n k (1968). Their discussions are, however, based on an
estimate of P , of only 2.7 X 1012 W derived from observations of the
angular deceleration of the M oon’s orbit. Recent advances by astronomers
(see § 8) now suggest a figure for P j which is even larger than the estimate
of 3.5 X 1012 W suggested earlier in this section. The gap in the budget is
therefore even more serious than when discussed by M u n k and his
colleagues.
W h ere else is tidal energy being dissipated ? There are of course many
hundreds o f coral atolls and other island features where large tidal currents
are set up, which could cause dissipation on a localized scale. M i l l e r
(1966) considered these in a general way but did not think they could contri­
bute significantly. He also estimated that the tidal currents of the order of
1 cm s _1 in the deep oceans would contribute less than 10!) W despite their
large area. J e f f r e y s (1968) seems to be alone in considering the tides as
setting up a small perturbation to ordinary sea waves which is dissipated
as those waves break on a coastline in the usual way. No rigorous calcula­
tions have been carried out, but intuitively it would seem to be a minor
effect. Another possible sink is in the internal viscosity of the ocean-loading
Earth tide (for dissipation in the Earth as a whole the viscous dissipation
in the body tide also has to be included, but the results for both are sm all).
This is discussed by M u n k & M a c D o n a l d (1960) and by J e f f r e y s (1976 and
earlier editions). From argum ents based on the Q of the 14-m onth ‘ Chand­
ler wobble ’ of the North Pole, M u n k & M a c D o N A L D (1960) deduce
0.3-0.6 X 1012 W from the total Earth tide, which is about 100 times greater
than J e f f r e y s ’ estimate. Allow ing 0.3 X 1012 W for the loading tide may
bring the ocean dissipation up to 2 X 1012 W , but this is still a long way
short of the accepted P t figure. There does, however, remain one plausible
but as yet uncertain m echanism , the conversion of ordinary ocean tides to
‘ internal tides ’ in the density layering of the ocean. An account of this
m echanism requires a separate subsection.
7.1
Internal tides
Continuous measurem ents of temperature at fixed depths in the interior
of the ocean show wave m otions of large vertical amplitude (tens of metres)
known as internal waves. These waves cover a wide spectrum of frequen­
cies, m ostly between the frequencies of ordinary surface waves and the
tides. But peaks do occur at the tidal frequencies, characterizing ‘ internal
tides
Records of current in the deep ocean also show tidal components
much larger than the few cm s—1 associated with the barotropic tide, and
these too are due to the internal tides. Internal tides generally have much
lower signal/noise ratio than the ordinary barotropic or surface tides, and
their phases are not so closely locked to the tide-generating potential. The
random element in their phases,, which can also be described as ‘ line
broadening
is caused by the dependence of internal tidal mechanics on
the density structure of the ocean which varies with the large-scale structure
of oceanic currents. A comprehensive review of the internal tides has
recently been made by W u n s c h (1975). Here I review briefly only those
features which are relevant to the global tidal energetics.
The basic dynamical equations for internal tides have already been
given in § 4 in connection with Laplace’s tidal equations. In the simplified
case of a constant (exponential) density gradient, and hence constant
buoyancy-frequency N and constant depth h, it may be sim ply shown that
the vertical dependence of w and the other variables of the internal waves
are defined by the functions :
sin N yn (z + h)
7„ ^ nn/Nh
n = 1,2,3,...
which, it will be seen, give zero vertical velocity at the bottom and the
surface of the sea, and rt — 1 reversals of sign in between. In fact, it is
found that internal tides, unlike internal waves of high frequency, are
concentrated in the first few vertical modes (low values of n). Further, the
horizontal velocities and pressure gradients are governed by equations
identical in form to the barotropic (Laplace’s) tidal equations but with
equivalent depths d„ = (h N /m r)2/g , which are very m uch smaller than the
actual depth h. For example, with typical values N = 0.002 s —1, ft = 4 000 m ,
we have dx = 0.64 m, d.> = 0.16 m. W h ile a surface tide of ‘ Kelvin ’ type
travels with a natural speed (gh )A/2 = 200 m s _1, the first mode of the
internal tide of similar form travels with speed (gdi)1/2 = 2.5 m s - 1 .
W e thus have at first sight a complete mismatching in both vertical
waveform and speed of the internal tides on one hand, the surface tide and
the tide-generating stress on the other, both the latter being essentially con­
stant over the water depth. From this point of view it is hard to see how
the observed internal tides are generated at all. However, it is now well
established that the irregularities in the ocean bottom permit a transference
of energy from the surface tides to the internal tides, and hence providing a
source of generation for the latter and of energy loss to the former. The
gradient of the bottom imparts a vertical component on the otherwise
uniform ly horizontal motion of the surface tide. This deflects the surfaces
of constant density and hence internal waves are generated of tidal period,
but with wavelength determined by the bottom topography.
In mathematical terms, suppose for simplicity that the bottom irregu­
larity is small compared with the general depth of water and that it is
uniform in the y direction (geographically arbitrary). If ü exp (— icot) is
the horizontal velocity due to the surface tide, assumed uniform on the
length scale of the irregularities, and h is the mean depth, then ü exp
(— iojf) dh/dx must equal w (— ft), the vertical velocity at the bottom.
Dropping the time factor, the last expression may be expanded in a Fourier
series in x :
/* oo
udh/dx =
j^
A(Jc) exp (\kx) dk
The vertical velocity field w (x , z), which essentially defines the internal
tide, is then given by :
<
x
w ( x ,z )
/* "
, /)N sin [Nkz(cj2 — 4f2,2)- 1 ^2l
A(k) - — ------ — - — ----;2 - .,, exp (ikx) d£
sin [ — Nkh (co2 — 4£2 ) ^ ]
This integral is dominated by the values of the bottom spectrum .4 (A) for
which :
k2 = (w 2 - 4 n.'2)[gd„
dn being the equivalent depth for the internal wave of nth-order mode in a
constant depth h, as previously defined.
The full linearized theory of internal tide generation by bottom scatter­
ing was first expounded by Cox & S a n d s t r o m (1962), who calculated A (A)
for typical sections of the Atlantic Ocean, üy a similar calculation, M u n k
(1966) estimated the rate of energy transfer from the M 2 tide into the first
20 internal modes in the global ocean to be 0.5 X 1012 W . This is by no
means negligible, but it is only some 30 % of the gap in the power budget
discussed in the main part of § 7.
The restriction to small irregularities compared with h in the theory
discussed above eliminates m any m ajor features such as seamounts and
the edges of continental shelves. Strong local generation of internal tides
near such features is confirmed by measurement, but theoretical assessment
of their rate of transfer is more difficult. The most amenable theories for
finite topography ( R a t t r a y et al., 1969, B a i n e s , 1974) treat the internal
motion not as a superposition of modes with continuous vertical waveforms
but as a series of beams of concentrated energy whose direction in the
vertical may be calculated for a given density structure and frequency by
the method of characteristics. Energy transfer occurs most intensively at
those points of the topography where its slope coincides with the direction
of a characteristic. However, the beam theory is strictly two-dimensional
(horizontal and vertical) and it is not clear how a general three-dimensional
shelf edge will reflect. Observational evidence for the existence of such
beams is controversial ( W u n s c h , 1975), and a modal decomposition is
usually m ore meaningful.
W h a t is needed from the point of view of energetics is either a tractable
means of calculating the energy transfer to internal motions at all the shelf
boundaries and other large topographical features in the ocean, or else an
estimate of the rate at which observed internal tides lose their energy by
turbulent viscosity. The transfer rate was estimated by W u n s c h & H e n d r y
(1972) at the intensively measured ‘ site D ’ off the New England shelf, but
worldwide extrapolation of the result gave only about 10° W . S c h o t t (1977)
also arrives at an insignificantly small figure compared to the known loss
to the surface tide. The weakness of any such extrapolation is the sensiti­
vity of any tractable solution to the detailed density structure. S a n d e r s t r o m
(1976) has, however, recently suggested that the shelf profile is not a
sensitive factor and that a worldwide calculation based only on figures for
stratification and using vertical-wall shelf edges would be viable.
Calculations of the dissipation rate of internal tides are even more
controversial, and probably involve the full non-linear theory of interactions
between the whole continuum of internal waves in the ocean. Some appr­
oaches are discussed briefly by W u n s c h (1975) ; on the whole they seem
to point again to a rather low figure, of the order of 10 - 1 of the required
power loss.
Yet another possible mechanism for internal tidal generation is at the
critical latitudes where a tidal frequency equals the inertial frequency 2ft'.
Here, theory tells us that the wavelengths of internal tides become ‘ infi­
nite ’ and the possibility of direct coupling with the surface tide becomes
more plausible. M i l e s (1974) has further shown that such coupling takes
place at the critical latitudes through the terms involving the horizontal
component f t " of the Earth’ s rotation, usually neglected in the full dyna­
mical equations for the tides. This possibility requires further investigation,
but if it were important we should expect significant generation of diurnal
internal tides near 30° latitude, at least in the Pacific Ocean where the
diurnal surface tides are fairly large. However, observations of internal
tides in such regions do not differ greatly from observations elsewhere. The
problem remains open, but without any hopeful pointers towards a likely
sink of the required magnitude.
8. TH E M O O N ’S O R B IT A N D TH E E A R T H ’S R O T A T IO N
The emphasis on energy dissipation in §§ 6 and 7 m ay seem exaggerated
to the uninvolved reader in view of the m ore obvious controversial features
o f tidal definition in particular oceans. Nevertheless, we have to pursue the
implications of global tidal friction still further in order to explain its
importance to our understanding of the history of the M oon’s orbit and of
astronomical time. In the briefest terms, the average bulge of the tide
(strictly, its P 2 (cos a ) harmonic) is retarded by terrestrial friction so that
its mean axis points some tens of degrees east o f the Moon (*). The gravita­
tional forces acting on this displaced second-harm onic bulge constitute a
couple in opposite sense to the Earth’ s rotation, so the day lengthens. But
the total angular m om entum of the Earth-Moon system m ust be constant,
so the decrease in the Earth’s rotation is accompanied by an increase in the
m omentum of the M oon’s orbit. From Kepler’ s third law this can only be
brought about by a steady increase in the M oon’s m ean distance combined
with a related deceleration in the mean angular m otion of its longitude.
The deceleration of the Moon’ s longitude, originally estimated at
about 1 0 " century- 2 , was first detected by Hailey in com paring records of
ancient eclipses with contemporary observations of the M oon’s orbit. About
a century later Laplace, in his one negative contribution to tidal theory,
thought he had explained it in terms of planetary perturbations. Adam s
showed that Laplace’s calculations were inaccurate and that the full calcu­
lation gave only 5 " c e n tu r y -2, thus re-opening the suggestion that tidal
friction was an important factor, first m ade apparently by the philosopher
(*) The u su a l d ia g r a m o f th is s itu a tio n (M un k & M a c D o n a ld
1976, p 316)
tid e th e o r y .
ca n
g iv e
th e
m is le a d in g
im p r e s s io n
of
a
s lig h t ly
1960, p 199, J e f f r e y s
m o d if ie d e q u ilib r iu m
Kant. The role of tidal friction in the evolution of the lunar orbit was taken
up seriously by Kelvin and more notably by G.H. Darwin ; details of its
magnitude have been a subject of active discussion to the present day
(references to the earlier literature may be found in M vnk & MacDoNALD
(1960) — see also Munk (1968) for a somewhat later assessment).
As mentioned in § 7, the subject has taken a new turn in the 1970s
with the agreement among several independent astronomers on a radically
increased figure for the longitudinal deceleration of about 4 0 " century- 2 .
Discussions earlier than 1970 usually accept a figure of 22" .4 century- derived by S p e n c e r J o n e s in 1939 from an apparently exhaustive study of
‘ modern ’ (1670-1930) astronomical observations of the Moon. S p e n c e r
J o n e s ’ figure could not be reconciled with records of solar eclipses going
back to 1000 b c , but there was so much scatter and uncertainty in interp rêtü îî'
tïié
m iC iciil
tfcLuicts
tliâ l
tliis
W as
nut
i'ê g a r u c d
as
iiiip u rla ix l.
ïl
would be beyond the scope of this review and the author’ s experience to
give an extended account of the astronomy, but a brief sum m ary of the
main concepts which are involved seems worthwhile.
If T represents astronomical time in centuries from a certain epoch of
origin, the mean longitude of the Moon (that is, the observed longitude
with removal of all periodic terms) can be expressed as :
= *0 + 11 T + « 2
l^ T 3
(8 . 1 )
where the ln are established partly by observation and partly (in particular
l2 and /s) by calculation from the Newtonian theory of rigid gravitational
bodies. In the absence of planetary and tidal perturbations, only the uni­
form m otion described by the first two terms would apply. Planetary
perturbations supply the sm all accelerative terms L and /3. ii, to use the
invariable notation of the literature, is the non-Newtonian acceleration
(numerically negative) caused by tidal deformation with friction, and is
one of the quantities which concern us. However, if T is expressed in the
so-called ‘Universal T im e’ system , as is effectively the case for all observ­
ations before very recent times, T itself is subject to the rotation of the
Earth, which also accelerates negatively.
The difference between T and the strictly uniform ‘ Ephemeris Tim e
which is based on the Earth’s orbit round the Sun, may be expressed as :
A T = r 0 + r l T + ^ ÙT 2 + /(7 1
(8.2)
where the coefficients are here entirely based on observation of stellar
transits, and f(T) is an apparently random function with variations in a
time scale of less than a century. As in (8.1), the first two terms describe
the uniform motion which was unquestioned until the late 19th century,
while the last two terms, whose causes are still only partially understood,
represent the mean acceleration and its fluctuations observed in modern
precise measurements and by their comparison with ancient records. As
will be shown later, Ù is partly related to n by conservation of angular
m om entum : the two quantities are rather difficult to separate from each
other from purely astronomical observations. The effect of the Earth’ s
acceleration Ù on estimates of ri when L M is measured from, say, times of
occultations of stars by the M oon may be removed by comparing with
observations of the longitude of the Sun or of an inner planet, which are
affected in a sim ilar way, proportionally to the mean motions of the Moon
and Sun respectively. But solar eclipses of mediaeval and ancient times
were not accurately timed : when correctly identified, the only usable
inform ation from them is the geographical position o f the place of observ­
ation. This depends on both ri and Ù.
M u l l e r & S t e p h e n s o n (1975) seem to have cleared up a long history
o f research and controversy about interpretation of historical solar eclipses
in which the m ost notable authority had previously been Fotheringham .
They introduced newly translated Chinese records to the Babylonian and
European eclipses used by others, and applied rigorous tests for reliability
before any record w ould be included in calculations. Out of more than
1000 historical accounts of eclipses, only 28 passed these tests, ranging
from 1375 b c at Ugarib to some observations organized by Hailey him self
in 1715. Adding the ‘ m o d e rn ’ observations of lunar phenomena from 1650
to 1970, M u l l e r & S t e p h e n s o n evaluated all the variables r0, r,, n , n inde­
pendently, with the final result :
ft = — 37.5 ± 5" century- 2
= — 91.6 ± 10 s century- 2 ,
where the tolerances quoted are actual bounds to the data, not standard
deviations. The figure for fi can also be expressed as an increase in the
length of the day by 2.4 ± 0.3 m s century- 1 . N e w t o n (1970) obtained
2.1 ± 0.3, re-interpreted by M u l l e r & S t e p h e n s o n as 2.3 ± 0.3. The figures
for n, the acceleration in lunar longitude, compare well with — 41.5 ± 4.3
( N e w t o n , 1970 ; ancient eclipses), — 38 ± 8 ( O e s t e r w i n t e r & C o h e n , 1972;
20th century data), and other modern results.
The corresponding value o f h, the rate of change of the sem i-m ajor axis
of the Moon’ s orbit, is, by differentiating Kepler’s law :
à = — 2ahj3n = 0.058 m yr_1
(8.3)
Theories of the implied state of affairs som e 10!! yr ago, when the Moon m ay
have been drastically closer to the Earth and the tides enorm ous, are outside
the scope of this review. They are discussed by M u n k (1968) but w ith som e­
w hat different num erical parameters.
How far do the new estimates o f the angular accelerations tie up
with oceanographic estimates of the oceanic tides discussed in earlier sec­
tions ? L a m b e c k (1975) has outlined a direct method of calculating d from
the perturbing gravitational potential corresponding to a given global cotidal
m ap. The tidal elevation for a given harm onic constituent, s, is expanded in
spherical harm onics (as in the elastic Earth problem discussed in § 6.1)
to give :
r,(0 , x ) =
X p nm{ COS 6) Cm „ (s) cos (co s t + m X - e ra „ « )
(8.4)
m,n
’
’
By using an expression due to K a u l a (1964) for the perturbing potential
due to £„ applying it to the dynamical equations for the orbit, and extracting
terms of zero frequency, the secular rate of increase of the sem i-m ajor
axis can be expressed in the form :
à = Ka 5>2(l + k2') Y , A s(a ,e ,i )
S
sin emJ s)
(8.5)
where A' is a combination of physical constants, k,' is the loading Love
number (§ 6.1) for the Earth tide, and A , is a rational function of a, and
the eccentricity e and inclination i of the orbit. Expressions similar in form
to (8.5) can also be written for d e /d t and dr/d<. For the semi-diurnal tides,
only the second-order harmonic m = n = 2 need be considered. M 2 is the
principal contributor to à, but the function A s and analogous functions
E s and I8 in the expressions for d e /d f,-d i/d f vary considerably for different
m ajor tidal constituents, so that for example N2 has the m ajor influence on
e, and K 2 on di/dt.
(1975) applied these calculations to the cotidal m aps of H e n (1972), P e k e r i s & A c c a d (1969) and B o g d a n o v & M a g a r i k (1967),
with the following numerical results for n, derived from à through equalions (8.3) :
L am beck
d e rs h o tt
PEKERIS & ACCAD (1969).........
HENDERSHOTT (1972)..............
BOGDANOV & MAGARIK (1967)
Average h for M2 ........................
Addition for S2, etc...................
C(s>
*<*>
h
0.044 m
0.051 m
0.043 m
70°
46°
48°
- 33" century-2
- 29" century-2
- 26'' century-2
—29" century —2
_2
- 6 century
Total...........................................
- 35'' century- 2 (*)
The values listed as C<s), z<‘ > are the arguments in equation (8.4) for
m = n = 2. The agreement between the three values of n from three
very different looking cotidal m aps is remarkable, suggesting that the
details of such m aps are not really very important in their effect on the
second harmonic. Equally rem arkable is the very good agreement between
the final result and M u l l e r & S t e p h e n s o n ’ s estimate for n quoted earlier,
derived independently from entirely different data and procedures. The
agreement greatly strengthens one’s confidence in both schemes of
reasoning.
M 2,
Having established that the observed acceleration in longitude is
accounted for by the observed ocean tides, we may proceed to calculate the
corresponding rate of change of the Earth’s rotation by using the m om en­
tum equation. The angular m om entum of the M oon’s orbit resolved in the
plane of the Earth’ s equator is :
= M E (M + E ) ~ l c2 n( 1 — e2) ‘/2 cos i
(8.6)
where M and E are the m asses of Moon and Earth respectively (to use the
notation of § 2). A similar expression H E gives the m om entum of the
Earth’s orbit about the Sun, and w e must have :
H m = H f + CCI = constant
(8.7)
(*) K. L am beck (1977, private communication) sa ys th a t h is fig u r e s a n d th e a s tr o ­
n o m ic a l e s tim a te f o r ri h a v e sin ce b e e n re d u ce d t o a b o u t 2 8 " ce n tu ry —2. P 4 (see 8.8)
is a ffe c t e d p r o p o r t io n a t e ly .
where C is the E arth’s principal m om ent of inertia.
Differentiating
(8.7) with respect to time and taking C as constant (an assum ption which
is nowadays seriously questioned), we get a fairly simple relationship
between ft and the time derivates of a (or n), e and i, which can be calcu­
lated in term s o f the tides as explained above. L a m b e c k (1975) thus obtained
the follow ing components of — ft, expressed here in units of m s d —1
century- 1 ;
Mean of three solutions for M 2 ...............................................
Estim ates for N2, L 2, O j, Kj ...................................................
F rom B o g d a n o v & M a g a r i k (1967) for S2 ......................
Atm ospheric S2 t i d e .......................................................................
T otal increase in length of day per c e n t u r y ......................
2.4
0.4
0.5
— 0.1
3.2 ms
The negative increment for the atmospheric tide reflects the fact that its
phase lead of 2 h causes a slight acceleration in the E arth’s rotation.
However, the im portant fact which arises is that the final figure for the
tidal effect, 3.2 m s, is significantly greater than the value 2.4 ± 0.3 ms
observed by M u l l e r & S t e p h e n s o n (1975) or the similar figures from
N e w t o n (1 97 0 ). This implies some geophysical factor which is positively
increasing the E arth’s rotation, causing the day to decrease by some 0.8 ms
century- 1 , in opposition to the stronger retarding effect of the tides.
The non-tidal acceleration, as it is called, has been appreciated for
several years, even when m uch lower figures were accepted for n and ft.
It is extensively discussed by M u n k & M a c D o n a l d (1960) and by other
authors since then. Several explanations have been suggested, too diverse
to sum m arize here, but no one cause has yet been positively identified.
Most probably, the acceleration is due to a steady decrease in the m om ent
of inertia C, caused by some change in the distribution of the E arth’s m ass.
M u n k & M a c D o n a l d rule out the transfer of water between the oceans and
polar ice, pointing out that there are large fluctuations in the length of day
on a m uch shorter time scale, which are clearly not correlated with the
known variations in mean sea level. The m ass transfer process m ay be
occurring in the atmosphere or in the liquid core (a popular hunting ground
for theorists). W e cannot proceed further with this m anifestly non-tidal
subject here.
Finally, we m ay calculate the rate of energy dissipation corresponding
to the tidally induced part of ft. The rate of loss of rotational energy due
to both lunar and solar tides is — Cftft, but the M oon’s orbit gains rota­
tional energy to give a small correction. The correction m ay most sim ply
be derived by observing that Cft m ust equal the total tidal couple acting
on the Earth, and its rate of working is the product of this couple and
the angular velocity of the Earth relative to the Moon, i.e. the total dissi­
pation rate is :
PA = - C ft(ft - « )
(8.8)
The tidal lengthening of the day of 3.2 m s century—1 is equivalent to
ft = — 8.9 X 1 0 - 22 rad s ~ 2. W ith (ft — n) = 70 X 1 0 ~ 6 rad s -> , and C =
81 X 1036k g m 2, (8.8) gives P4 = 5.0 X 1012 W . Alternatively, if we restrict
the calculations to those terms which are due to the M oon only, with a
proportional allowance for the lunar part of K t, we get ftM = — 7.6 X 1 0 ~ 22,
P 4 — 4.3 X 1012 W . This last figure is comparable with the estimate of the
work done by the Moon on the tides as derived by direct calculation ( § 7 ) ,
nam ely P x = 3.5 X 1012 W , with an uncertainly of about 1 X 1012. This
m a y be taken to confirm the validity of the rather roundabout chain of
arguments used and suggests that no important factor has been omitted.
T h e two m ajor gaps in our understanding of this geophysical system are
the cause of the non-tidal acceleration of the Earth and the m issing sink
of energy in the oceanic tidal dynam ics.
9. N EW R E S E A R C H T E C H N IQ U E S A N D T H E F U T U R E
I have now surveyed the principal areas of our knowledge about the
oceanic tides and identified its shortcom ings. It is appropriate to conclude
with some speculations about the techniques which will probably be used
in the future towards overcom ing the present difficulties. This is not
easy because, as I stated in the Introduction, the subject has entered a
period of recession, following a wave o f renewed interest in the 1965-1975
decade. The previous recession in oceanic tidal research, for which the
definitive review paper was D o o d s o n (1958), marked a feeling of exhaustion
of the field of analytical tidal solutions, while the hope for the future was
in numerical methods. Now num erical methods have been extensively
explored and have been found to lack physical realism in one w ay or
another. Coupling with the Earth tide has been shown to be an indis­
pensable factor, raising the com putational problem into a still higher class
of difficulty. W e need to know m ore about tidal dissipation, internal tides,
and the actual definition of the surface tide in regions remote from tho
land. These call for a vigorous program m e o f tidal measurement on a
worldwide scale. In m y view, the next phase of progress will only come
after such a program m e.
A t a few points in this review I have mentioned the recent facility for
recording tides in the open sea — the ‘pelagic tide recorder’ — without
describing any outstanding results from its use. In fact, the development
o f the pelagic tide recorder in the 1960s was a m ajor ‘breakthrough’ in
oceanographic technology, desired for decades by previous generations of
researchers. It has not yet figured im portantly in this review because the
theoreticians have advanced too rapidly to allow any interaction with the
technologists. Occasionally, a researcher computing the global tides has
pointed to a few spots on the globe where some measurements w ould be
useful for verification, usually zones of locally m axim um amplitude. But
none of the few laboratories equipped with the new tidal technology (in
England, France and the USA) have felt justified in sending a ship thous­
ands of miles for this express purpose. Rather, they have deployed their
instrum ents where logistically feasible (usually not too far from hom e),
or where research cruises were directed for other objectives. The experi­
m ents have sometimes provided interesting papers in their own right, but
w ithout any obvious impact on the global tidal scene (e.g. C o l l a r & C a r t ­
w r i g h t , 1972 ; I r i s h
& S n o d g r a s s , 1972 ; F i l l o u x , 1973 ; Z e t l e r et al,
1975). Could a more globally oriented program m e of m easurem ents be
mounted ? To answer this we m ust first consider what a pelagic tide
recorder consists of and how it is deployed.
The principle is to record the variations of pressure at a capsule sitting
on the sea bed for a period o f the order o f one month. A precision of 1 m m
equivalent head of water at a depth o f some kilometres is less than one
part per million, while the capsule itself has to withstand an ambient pres­
sure of the order 5 X 106 k g m ~ 2. Pressure sensors are inevitably sensitive
also to temperature, which contains tidal variations, so this also has to be
recorded and compensated for. Mechanical creep causes the pressure signal
to drift, requiring another correction. Fused quartz crystals have the
best all-round properties but are expensive. Good results have also been
obtained from cheaper but carefully engineered m etallic strain gauges.
Some designs use a compressed nitrogen backing pressure, applied when
the capsule reaches full depth and then held ‘ constant ’ , so that the
sensor has only to deal with the small dynam ic range of the tide itself, but
this requires knowledge of the sensitivity o f the com pressed gas to tem per­
ature variations. The pressure is usually recorded as a total count of
vibrations of a variable oscillator over a fixed period, of the order of
102 — 103 s, and so a timing m echanism precise to about 1 in 10(i is also
required. Low power consumption is obviously desirable, and the battery
power supply, com puting circuits and data logger, all inside the pressurized
capsule (usually a metal sphere) have to be robust and reliable at tem per­
atures near 0 °C. Another important feature is the acoustic transponding
system , used for interrogation, hom ing-in and releasing of ballast when the
capsule is finally recalled to the surface.
Altogether, these requirements have been compared with those of an
instrumented space satellite. The capsules are expensive and have to be
tended by specialized scientists. Not least in cost is the ship itself, which
either has to make two return journeys between hom e-port and recording
site or have another research program m e in the same sea area to occupy
the m onth or so between laying and recovering the capsule(s). Obviously
this activity is possible only to a scientifically developed nation with an
interest in tidal research. The small num ber of deploym ents is therefore
not surprising.
I have not quoted any references in the last two paragraphs because
most of the instrum ents which have been developed since the first experi­
ments in the early 1960s are summarized in an account of an intercalibration experiment (Unesco, 1975) which w as organized by the international
SCOR W ork in g Group No 27 in late 1973 in the North Biscay area. Seven
capsules designed in various laboratories in Canada, France, UK and USA
were laid in close proxim ity from the British RRS Discovery and their
results compared. Out of a great variety of techniques and independent
calibrations, estim ates of the M2 tide were within ± 1 % in amplitude and
± 1" in phase. The experiment also brought out at least two other interest­
ing facts. Nine different methods of analysing one-m onth tidal records,
individually favoured by various authorities, were compared in application
to a single record : their results differed by more than the various instru­
ments. Tidal analysis is not such a perfected art as some authorities would
have us believe. The other interesting fact was that four out of the seven
capsules submitted were restricted to shelf-sea depths (200 m or less). This
emphasizes the somewhat easier technique required for shallow-water
work, and a tendency for institutes concerned with tides to restrict their
activity to their national continental shelf seas.
The intercalibration experiment marked the culmination of activity of
SCOR W orkin g Group No 27. It disbanded shortly after the publication.
During its ten years of office it did much to encourage development of
pelagic instruments and methods of analysis, and generally to re-awaken
interest in ocean tides. Under its influence, more than 600 station-days
o f pelagic tidal records were made at places in the Atlantic, Pacific and
Southern oceans where the tides were previously unknown. Yet it failed
to stimulate a genuinely worldwide activity or to organize any campaign
of measurement COOrdi*"!0*0*^
r * t t h u+1»t*
ncfin t iH rppr^rHiricf
remains the pursuit of three or four independent national groups, all in the
northern hemisphere, and some expressly confined to national waters. The
reasons are basically expense, slowness of the method to achieve significant
results, and a shortage o f oceanographers who are really dedicated to the
study of tides.
The British (IOS) program m e now seems to be the most active and
deserves some description, since it represents the only attempt to simulate
part of a plan suggested to SCOR W orkin g Group No 27 by H a n s e n (1966)
of Hamburg (a pioneer of numerical tidal modelling). The plan is to
surround a sizeable area of ocean with a chain of tidal measurements, then
compute the tides in the interior o f the area by one of the usual methods.
By extending the chain across to important land boundaries, it can be
slowly extended to cover whole oceans and eventually the globe. Concen­
tration on a relatively small well-measured area enables one to test the
various formulations and choose an optimum method in a way inaccessible
to global tidal modellers. Processes at the shelf edge and internal tidal
generation can be directly studied. The area chosen by IOS is bounded by
the western European coastline and the m id-Atlantic Ridge between the
latitudes of Iceland and the Azores, and measurements at about 500 km
separation are now nearly complete. In addition, the tides along the edge
of the continental shelf from Brittany to Norway have been measured
with associated currents at 100 km separation. The latter measurements
have already provided the necessary input to a shelf-sea model (figure 11)
and the direct evaluation of power loss over this important oceanic bound­
ary. Tidal models corresponding to the entire oceanic area are being pre­
pared at IOS. The completion o f such an area may encourage others to
extend it to the west as far as Canada and the USA, while the possibility of
isolating the whole of the North Atlantic Ocean by measurements between
the Guinea Coast and northern Brazil is now within reach.
Even so, the possibility of parcelling out all the world’s oceans by such
combinations of pelagic measuring technology and computation looks rather
remote. It would undoubtedly require generous cooperation from oceano­
graphic institutes in the southern hemisphere and the countries bordering
the Pacific Ocean. Perhaps equally important is to obtain reliable measure­
ments of the m ajor dissipating regions such as the Okhotsk Sea and the
shelf sea off Argentina. In these days of extended national m aritime bound­
aries this has to be left largely to the national authorities them selves and
the hope that they are interested in tides.
The difficulties in the above scenario m ake one turn to other techno­
logies than oceanographic which m ight be stretched to give useful m eas­
urements of ocean tides. The possibility o f using widespread m easurem ents
of Earth-tidal gravity was mentioned in § 6.1 — the inverse loading prob­
lem . T his m ay be reliable for areas o f ocean well surrounded by points
of land and islands, but the large tidal am plitudes localized in shallow seas
tend to distort the loading picture. Large areas of the southern oceans are
sparsely furnished with islands. If one has to put gravimeters on the ocean
floor, one m ay as well use direct tide recorders as discussed previously.
A totally different technique, of tantalizing prom ise for the future, is
the high-precision altimetry of the sea surface from Earth satellites. After
several years’ focusing on meteorology and Earth im agery, instrum ented
satellites are becoming more oriented towards geophysical and geodetic
m easurem ents, particularly those designed in France and the U SA . One
o f the latest developments in the USA is a com pressed-pulse vertical radar
which can time the reflection from the sea surface to nanosecond or better
accuracy. W ith reasonably known corrections for atm ospheric trans­
m ission properties and bias from surface waves, it can essentially m easure
the height o f separation to 0.1 — 0.5 m precision. The basic principles of
its use are discussed in G r e e n w o o d et al (1969). Prelim inary results from
the radar altimeter mounted in the Skylab satellite are very prom ising
( M c G o o g a n et al, 1975) ; improved models are now in use in G E O S-3 and
planned for S E A S A T -A in 1978.
For the results to be useful for tidal and other studies, however, the
absolute geocentric height of the orbit m ust also be know n to decimetre
accuracy. T his is the more critical limitation at present. For accurate
calculations of the orbit, the spacecraft m ust be well above the atmosphere
to avoid drag and the smaller scale variations of the E arth’s gravitational
field, but too great a height would strain the accuracy of the altimeter and
increase the area of its ‘target’ on the ocean surface. 800 km above the
m ean surface is typical of current plans. In order to fix the orbit to the
required precision, laser-ranging stations are set up at strategic points on
the E arth’ s surface, but it is not yet know n whether this precision w ill
extend more than 1 000 km or so from the ranging stations. A t present
the laser-ranging stations are concentrated near the USA, but a consortium
o f European laboratories is also planning a high-precision tracking exercise
on their side of the Atlantic Ocean.
Given the altimeter data and adequate ranging — and there are political
difficulties even here — extraction of tidal inform ation is still far from
straightforward. Ideally, one would like estimates o f the height of each
given portion of ocean surface, say a 500 km square, at 100 different times,
effectively at random . But satellite tracks are never precisely repeated, and
the m ean geocentric height of the sea surface can vary spatially by several
metres over such an area due to the shape of the geoid ( K i n g - H e l e , 1975).
Indeed, the prim e purpose of satellite altim etry to m an y scientists is the
improved determination of the geoid, with the tides as a mere ‘noise factor’
to be removed by averaging. In any case, evaluation of the tidal signals
will almost certainly have to be done in conjunction with computer m odels
which aim to simulate the tides by the methods discussed earlier. Certainly,
the disentanglement of the com plex tissue of data in time, latitude and
longitude which will arise from these altimeters will require the coopera­
tion of experts from a range of scientific disciplines. It should also be
noted that the altimeter detects the total tide, the sum of the oceanic tide
as usually defined, the body tide, and the loading tide of the solid Earth.
Separation of these components is a secondary problem ( C a r t w r i g h t , 1977).
Despite these difficulties, the prospect of nearly global coverage without
the expense and logistics of research ships makes satellite altimetry, in the
author’s opinion, the main hope for the eventual solution of the w orld’ s
cotidal map. Meanwhile a continuing programme of ship-based oceanocHniilrl
O' " I " --- - *rpcparnh
--------—- ----------— -ennrpntrntp
----- -------- rm îhp
- - n n t s t a n H i n p0’ rïhvsical
jL
^
*D r o b l e ï ï l S
associated with tidal dissipation.
ACKNOW LEDGEM ENTS
Ï am grateful to the Editors o f the Institute of Physics for persisting
over some five years in trying, and finally succeeding, to persuade me to
lay aside an autumn of weekends and dark evenings to writing this paper.
Although the writing was done alm ost entirely outside normal working
hours, m y employers, the Natural Environm ent Research Council, supported
by funds from the Ministry of Agriculture, Fisheries and Food, and the
Departm ents of Industry, Energy and of the Environment are to be thanked
for providing an institutional background in which tides can rightly be
considered a viable and useful subject for nationally funded scientific
research.
Postscript (February 1978)
The re-publication of this article by the International Hydrographic
Organisation allows me to note som e important developments during 1977.
According to recent lectures by the respective authors, both C. P e k e r i s
and W . Z a h e l have independently solved the global tidal equations (4.2),
(6.11) which allow for elastic crustal yielding and self-attraction. See also
G o r d e e v et al (1977). The iteration processes apparently converge in the
absence of constraints from boundary data, whereas they diverged in
H e n d k r s h o t t ’ s (1972) form ulation. Z a h e l , using a 4° grid, obtains signifi­
cantly closer agreement with observed coastal data than in his previous
solution with the same grid size for a rigid Earth, ( Z a h e l , 1970), but the
general pattern remains qualitatively similar to that o f Figure 8.
K. L a m b i -x k (1977) has published an extensive new review of tidal
dissipation in the Earth-Moon system , demonstrating even better agreement
between astronomically based estim ates and those based on computed coti­
dal maps than in the figures quoted in this article. A typical agreed
figure for the dissipation at the M., frequency is 3.20 ± 0.15 X 1012 W , a
little lower than as quoted above. L a m b e c k (1977) suggests that the Bering
Sea has been grossly over-rated as a major sink of energy, and this is
confirmed independently by a computer model of that sea by S i t n d e r m a n n
(1977). Removing most of the contribution from the Bering Sea reduces
M i l l e r ’ s (1966) estimate of M ., bottom-frictional losses from 1.7 to 1.5 X
1012 W a tts, thus widening the gap still further.
As additional confirmation of the higher figure for total tidal dissipa­
tion, L a m b e c k (1977) discusses the important recent estimates from the
perturbations of satellite orbits ( C a z e n a v e , D a i l l e t & L a m b e c k , 1977). The
tidal masses distort the gravitational field to give periodic perturbations
to the inclination and nodal longitude of artificial satellites, with am pli­
tudes of order 0 ".0 5 . These can now be detected with some accuracy, and
give in effect independent estimates of (L ., and e..2 2 in (8.4), namely 0.031 m
and 123° respectively. These and the equivalent
dissipation of 2.7 X
1012 W are lower than the figures from cotidal maps, but there is some
uncertainty associated with the influence of the C 4 u harmonic on the
satellite results.
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