I n te rn a tio n a l H ydrographic R e v ie w , M onaco, LV I (2), J u l y 1979.
A NOTE ON EXTREME TIDAL LEVELS
b y M. AM IN
In s titu te of O c e a n o g ra p h ic Sciences,
B id sto n O b serv ato ry , B irk e n h e a d , U.K.
SUMMARY
A m eth o d of p re d ic tin g ex tre m e p e a k s of p re d o m in a n t s e m i-d iu rn a l
tid e s on th e b asis of 'H ig h est A stro n o m ic a l T id e ’ (H A T) is in v estig ated .
C a r t w r i g h t ’s ( 1 9 7 4 ) view s a n d c o m m e n ts h o ld in g en e ral, b u t som e
v a ria tio n s are observed betw een th e e x tre m e p h y sic a l tid e s a n d H A T ’s.
T h ese v a ria tio n s are u n d e rs ta n d a b ly d u e to n o n lin e a ritie s in th e re sp o n se
fu n c tio n of p h ase lags. T h e c o n s titu e n ts of fric tio n a l o rig in te n d to m in i­
m ise th e ra n g e o f e x tre m e p h y sical tid e s . T h e o rd e r of th e s e m a rg in a lly
re d u ced levels a t v a rio u s n e a r-e x tre m e s can be ea sily re v e rse d b y n o n lin e a ritie s of th e re sp o n se fu n c tio n . A lth o u g h ex tre m e p e a k s follow th e
m o o n ’s p erigee th e se ex tre m es o cc u r w h e n th e lo n g itu d e of th e m o o n ’s
no d e is n e a r a u tu m n equ in o x . It a p p e a rs to be im p o ssib le to specify a
sim p le ru le fo r a b so lu te d e te rm in a tio n of ex tre m e tid a l levels, b u t if a
to le ra n c e in a c c u ra c y o f a few c e n tim e tre s is allow ed th e n th e m eth o d
as p re s e n te d can w o rk ad e q u ately .
THEORETICAL BACKGROUND OF THE METHOD
T h e o b ject of th is in v e stig a tio n is to h e lp find a m o re econom ical w ay
of co m p u tin g e x tre m e tid a l levels t h a n th e c u s to m a ry one of p re d ic tin g
all tid es over a p e rio d of 19 y e a rs a n d selectin g th e la rg e st. It is obviously
w a ste fu l to co m p u te n e a p tid es a n d tid e s n e a r th e tim e s of apogee, b u t it
is n o t obvious how m a n y o th e r p e rio d s m ay be safely ig n o re d o r even
w h ich 19-year p e rio d sh o u ld be covered. C a r t w r i g h t (1974) gave a p re cise
a c c o u n t of th e p ea k tid e -ra isin g fo rces fro m p u re ly a s tro n o m ic a l re aso n in g .
T h e ex tre m es of p h y sic a l tid es ca n be easily p re d ic te d a t p lace s w h e re th e
re sp o n se of sea to th e tid e -g e n e ra tin g forces is sim p ly lin e a r (th a t is,
lac k in g s h a rp re so n a n c e s o r a n ti-re s o n a n c e s ) as b o th a re exp ected to be
close on tim e scale. T h e differences g ro w in less sim p le lin e a r reg im es or
as n o n lin e a ritie s becom e m ore p ro n o u n c e d . H ere, th e p o ssib ility of e sta b ­
lish in g a ru le w h ich m a y h e lp to p re d ic t e x tre m es of p h y sic a l tid e s in m o re
g en e ral, p re d o m in a n tly s e m i-d iu rn a l n o n lin e a r regim es fro m p e a k s of
tid e -ra is in g fo rces is ex a m in e d . In th e h a rm o n ic m eth o d th e tid a l eleva­
tio n s of a given place arc p re d ic ted as:
i : = ^ ^ c°s (v „ - g n)
( l)
n
s u m m e d over all significant h a rm o n ic s,
w h ere
a is th e a m p litu d e of a h a rm o n ic ,
g is th e p h a se lag re la tiv e to p h a se o f a stro n o m ic a l te rm ,
V =
+ k 202 + k 30 3 + k 404 -+- k s0 5 -)- kg0g
(2)
w ith
6i = local m e a n lu n a r tim e re d u c e d to an g le;
th e m ean lo n g itu d e o f th e m o o n ;
02 =
(7 3 —
m e
m e a n
lu u g n u u c
u i
u it
04 =
05 =
th e m ean lo n g itu d e o f th e m o o n ’s perig ee;
th e negative of th e m e a n lo n g itu d e of th e asc e n d in g node
of th e m o o n ;
06 = th e m ean lo n g itu d e of p e rih e lio n ; a n d
k ’s a re sm all in te g e rs k n o w n as a rg u m e n t n u m b e rs.
E q u a tio n (1) ca n be w ritte n a s:
£ = am cos (Vm - gm) + £ ar cos (Vr - g,)
(3)
r
w h e re s u b s c rip t m den o tes M., tid e a n d Llie su m m a tio n is over th e r e ­
m a in in g lines.
S u b s titu tin g
<t> = V m ~ V r “ ( g m “ & )
(4)
3
qs =
— sin
ar
(5)
e q u a tio n (3) fo r p re d o m in a n t s(Miii-diurn;il lido, borom cs
w h e re
f
=
3m F
COS
(7)
(Vm
F = (qs2 + qcV
(8)
Ÿ = t g - 1 (qs/qc)
(9)
T h e c o n d itio n s for p ea k s of se m i-d iu rn a l tide g en e ra tin g forces given by
C a r t w r i g h t (1974) ca n be in te rp re te d in te rm s of o rb ita l elem e n ts ( 0 lt
02 • • • • Oq) as
0 2 , e 3 , 9 4 = 0,7T
e
- ,
\
(10)
It is d ifficu lt fo r all th ese e lem e n ts to s a tis fy c o n d itio n s in e q u a ­
tio n s ( 1 0 ) s im u lta n e o u s ly ; th e re fo re som e to le ra n c e , say s, is allow ed for
re la x a tio n of th ese co n d itio n s. T im es w h e n th ese c o n d itio n s are satisfied
w ith in th e to leran ce lim it d e te rm in e th e p ea k s of se m i-d iu rn a l tid e -ra isin g
fo rces, a n d ex trem es of th e p h y sical tid e s a re ex pected to o ccu r n e a r th e se
p o in ts w ith som e p e rtu rb a tio n d ue to p h ase lags of p rin c ip a l h a rm o n ic s,
h a rm o n ic s of y ea rly cycle Sa, a n d th e seaso n ally m o d u la tin g h a rm o n ic s
s u c h as MA 2 a n d T2. T h e n ec essary p ro c e d u re re q u ire s one to find tim e s
w h e n c o n d itio n s in e q u a tio n s (10) a re satisfied a n d th e n to co m p u te F fo r a
few sp rin g tid es a ro u n d th is e s tim a te d tim e.
APPLICATION OF THE METHOD AND CONSIDERATION
OF OTHER FACTORS AFFECTING THE EXTREME LEVELS
In sh allo w w a te r a re a s, b o th th e advective a n d fric tio n a l te rm s in
h y d ro -d y n a m ic a l eq u a tio n s of p ro g ressiv e tid es can be sig n ifican t a n d
c a n n o t be ig n o red. T h e tid a l s p e c tru m is seen to c o n ta in a large n u m b e r
of sig n ifican t shallow w a te r c o n stitu e n ts. T he d ev elo p m en t of th ese con­
s titu e n ts is su ch th a t th e p h ase lags of som e of th e se te n d to re d u ce th e
ra n g e of ex trem e tid es w hile o th e rs h e lp to in cre ase th e ran g e. T o il­
lu s tra te th is fact, co n sid er th e p rin c ip a l c o n s titu e n ts of S o u th e n d tid es
Table 1
Values of Solar and Lunar orbital elem ents on d a tes favourable
to generate extrem e tides
Year
Day
lim e
(h)
®2
^3
*4
(deg)
(deg)
(deg)
(deg)
(deg)
(deg)
1905
21 March
0054
180.0
178.12
357.85
186.61
201.72
281.31
1922
21 Sept.
23 March
1207
180.0
181.319
179.59
178.92
180.25
281.61
171.14
158.79
281.91
1940
1150
0.0
177.95
0.818
(A m in , 1976), g iv en in ta b le 1. A t th e tim e o f a n e x tre m e tid e o n e c a n
e x p e c t t h a t th e fo llo w in g c o n d itio n s s h o u ld h o ld :
VN2 ~ 8 n 2 = 0
^M 2 ~ 8 m2 = 0
V S2 - S S 2 = 0
VK2 —SK2 = 0
T h ese c o n d itio n s re q u ire th a t:
2 = 8n 2 = — 30.72
^M 2 = SM2 = — ^.48
^S2 = §S2 = 49.41
V K2 = 8k2= 49.90
I
W h en th e se c o n d itio n s are satisfied, th e p h a se s of sh allo w w a te r con­
stitu e n ts 2MKo, 2MS.,, 2M N,.......... can be a p p ro x im a te d as :
V2MK2 - 82MK2 = 2VM2 - V K2 _ §2MK2
= - 168.32°
(C)
V2MS2 - ê2MS2 = - 167.82°
^2MN2 — 82m n 2 = — 160.94
S im ilarly for A voiinioutli :
V2MK2 _ S2MK2 = ~ 125.1
^
— g-iMc.
- - , = —
—m -■j
[,
- ,
^ 2MN2 — S2mn 2 ~
133.6
rrȕ
1
T h ese sh allo w w a te r c o n s titu e n ts, clearly, oppose th e a stro n o m ic a l
tid e -g e n e ra tin g forces a t th e tim e w hen th e y a re n e a rly in p h a se an d
th e re fo re ex p ected to g e n e ra te ex tre m e tid es. H ow ever, th e re are som e
c o n s titu e n ts su c h as M 4 a n d Me w h ic h are never ex actly in p h ase w ith M 2
a t th e tim e of h ig h w a te rs b u t n ev e rth eless alw a y s m a k e som e c o n trib u ­
tio n , w h ic h m ay be po sitiv e o r negative.
F u rth e rm o re , th e p h a se lags o f all p rin c ip a l c o n stitu e n ts w h ic h are
re sp o n sib le fo r e x tre m e tid es can in te ra c t to sh ift th e tim e of ex trem e
tid es, in a fa s h io n w h ich m ay n o t be easily p re d ic tab le. In th o se cases
w h ere th e re sp o n se fu n c tio n of p h a s e lags
g =
/ ( 0 -)
(11)
is a lin e a r fu n c tio n , th e sh ift in tim e s of h ig h w a te rs ‘t 8’ can be easily
a p p ro x im a te d b y th e slope of the line a s :
ts = (gj - gj) I (oj - ffj)
( 12 )
w h ere (g„ o-i) a n d (gj, o y a re tw o p o in ts of th e fu n c tio n or, sim ply, p h ase
lags a n d sp eed s of tw o p rin c ip a l c o n stitu e n ts.
O bserved tid e s in sh allo w w a te r a re a s a re n o t
sponse fu n c tio n s are, gen erally , n o n lin e a r as sh o w n
no sim p le w a y of c o m p u tin g tim e sh ift ‘t 8’, b u t one
o p tim a l valu e. F o r o p tim isa tio n , it is suggested t h a t
m ay be re p re se n te d by a s tra ig h t line.
g = a + Pa
sim ple, an d th e ir re ­
in figure 1. T h ere is
can a p p ro x im a te th e
th e re sp o n se fu n c tio n
(1 3 )
su c h th a t
wc(gi - gj)2 = rnin
(14)
T h e w e ig h ts w sh o u ld be re la te d to the a m p litu d e of th e asso ciated
c o n s titu e n ts , i.e. m ay be ch osen p ro p o rtio n a l to th e am p litu d e of the con­
stitu e n ts . T h e p h ase lags te n d to d isp lace th e tim in g s of observed high
w a te rs, an d n o n lin e a ritie s in th e re sp o n se fu n c tio n s of p h a se lags have
a n in v e rse effect on th e p re d ic tib ility of th e tim e shift.
F
ig
.
p h ase la g s of s e m id iu rn a l tid e s : (a) S o u th e n d ,
(b) A v o n m o u th .
x----------x = as in observed tid e w ith d is to rtio n due to th e presen ce of sh a llo w w a te r
c o n stitu e n ts ;
------------- = sm ooth fu n c tio n s.
1.
—
R esponse
fu n c tio n s
of
T h u s we see th a t th e s e a re tw o m a in fa c to rs w h ic h c a n o b s tru c t th e
sim p le p re d ic tio n of ex tre m e tid e s:
(a)
Som e sh allo w w a te r c o n s titu e n ts te n d to c o m p re ss th e ra n g e of
ex tre m e h ig h tid e s an d re d u c e th e difference in ra n g e o f v a rio u s
tid es w h ic h a re ex tre m e o r n e a r ex trem e.
(b)
P h a s e lags of p rin c ip a l c o n stitu e n ts ca n sh ift th e tim in g of
e x tre m e tid es a n d a n o n lin e a r re s p o n se m ay c o n sp ire w ith u n ­
fa v o u ra b le astro n o m ic a l co n d itio n s to p ro d u ce tid es h ig h er th a n
th o se g en e rated u n d e r a s tro n o m ic a lly fa v o u rab le co n d itio n s.
HEIGHT
( METRES
)
( o ) SO UT HE N D
Y EARS
( b)
AVONMOUTH
2. — T h e p lo ts o f h ig h e s t s p rin g a n d a u tu m n e q u in o x tid e s in y e a r s 1 9 2 1 — 1999.
-------------- = s p r in g e q u in o x .
-------------- = a u tu m n e q u in o x .
T h e d a tu m a d d e d i s 3 . 0 24 m [ 2 . 9 m b e lo w O rd n a n c e D a tu m (N e w ly n )] fo r S o u th ­
en d , a n d 6 . 9 22 m [ 6 . 5 0 m b e lo w O rd n a n c e D a tu m (N e w ly n )] f o r A v o n m o u th .
F
ig.
DISCUSSION ON RESULTS
To find a feasible p ro c ed u re , p re d ic tio n s w e re c o m p u te d fo r S o u th ­
u sin g th e h a rm o n ic c o n sta n ts d eriv e d in A m i n (1976). T h e ir ex tre m e
v alu es a re p lo tte d in fig u re 2. T h e H ighest A stro n o m ic a l T ide (H A T)
ex p ected on 21 S ep tem b er 1922, C a r t w r i g h t (1974) w a s sh ifte d to 23
S ep tem b er 1922 a n d th e a c tu a l e x tre m e h ig h tid e o c c u rre d a m o n th la te r
on 22 O cto b er 1922. T h e tid es of 29 O ctober 1905 a n d 3 O cto b er 1940
exceeded th e h ig h e st tid e of 1922 by 1 cm a n d 2 cm resp ectiv ely . A stro ­
n o m ical co n d itio n s on th e la te r d a te s w ere not fa v o u ra b le , as co m p ared
to th o se o n 21 S ep tem b er 1922, (see ta b le 2) to g e n e ra te h ig h tid es, b u t
th e se c o n d itio n s co n sp irin g w ith p h a se lags p ro d u c e d th e h ig h e st tid e
o n 3 O cto b er 1940. T h is suggests th a t n o n lin e a ritie s in th e re sp o n se of
end,
Table 2
L is t of p rin c ip a l sem i-diurnal co n stitu en ts and other m a jo r con stitu en ts
w hich p la y an im p o rta n t role in d eterm in in g the p e a k tides. Phases of
2M S 2 a n d 2MN., are a ss u m e d to be the sam e as /x2 an d L 2 respectively.
Constant term s in a rg u m e n t n u m b e r are neglected.
Southend
Avonmouth
Argument
number
Constituent
Amplitude
(m)
Phase lag
(deg)
Amplitude
(m)
Phase lag
(deg)
00 1000
Sa
0.055
215.42
0.098
195.32
1-1 0 0 0 0
o,
0.131
187.74
0.084
8.81
1 10000
0.111
11.22
0.062
136.58
2-20000
K,
2MK2
0.049
105.46
0.135
269.41
2-20200
2N 2
0.041
305.54
0.095
177.48
2-22000
( /½
( 2MS2
0.053
0.173
310.20
103.40
0.098
0.445
183.40
276.78
2-10100
n2
0.349
329.28
0.730
187.83
0.108
320.43
0.197
152.18
2-1 2-1 0 0
2 0-1 0 0 0
m a2
0.038
306.13
0.056
138.62
200000
m2
2.044
353.52
4.221
201.89
{ l2
( 2MN2
0.054
0.093
20.10
359.70
0.116
0.242
225.56
169.60
0.590
49.41
1.477
260.63
220000
S2
k2
0.172
49.90
0.434
254.34
4 00000
m4
0.097
8.67
0.337
346.69
4 2 -2 0 0 0
m s4
0.035
72.37
0.298
26.09
2 10-100
2 2-2000
the sea m a y d isp lace th e e x tre m e tid es on a tim e scale m e a su re d by single
or m u ltip le p erio d s of o rb ita l elem e n ts 02,
04 a n d 05. It w ill n o t be easy
to m e a s u re su ch a d isp la c e m e n t b ecau se of d iffe ren t ra te s of in cre ase of
o rb ita l elem e n ts. T he seq u en ces of ex tre m e h ig h tides, show n in figure 2,
are in close ag re e m e n t w ith C a r t w r i g h t (1974) in th a t th e lo n g itu d e of
p erig ee of th e m oon p la y s a m o re im p o rta n t role th a n th e node of the
m oon in d e te rm in a tio n of th e cycle of e x tre m e levels. H ow ever, years
of e x tre m e tid es a re n o t ex a ctly as expected in th e g en e ral fo rm as su g ­
gested by C a r t w r i g h t (1974). H ig h p e a k s a t S o u th e n d are p re d ic te d for
y ea rs 1922, 1940, 1958, 1976, 1998 a n d 2015, a n d a t A v o n m o u th fo r years
1922, 1944, 1962, 1980, 1997 a n d 2015. A v o n m o u th tid e s are sh ifte d by
h a lf a p erig e e cycle since 1940 fro m th e y e a rs of HAT. A sim ila r sh iftin g
ta k e s p la c e at S o u th en d fro m v e a r 1993 to y e a r 1998; T'his a p p e a rs to
re s u lt fro m the in te ra c tio n of p h a se lags of in d iv id u a l c o n stitu e n ts of th e
p h y sical tid e a n d th e 186 y ea rs cycle (a p p ro x im a te ly 21 cycles of #4 or
10 cycles of 05). In th e case of S o u th en d , ex trem e tid es g en e rally to o k
place in th e sam e y ea rs as HAT, b u t tid es exp ected in y ea rs 1975 a n d 1997
o c c u rre d (a n d should occu r) in y e a rs 1976 a n d 1998 respectively. A n n u a l
e x tre m es a t S o u th e n d are alw ays in a u tu m n w h e re a s a t A v o n m o u th they
o ccur n e a r b o th eq u in o x es, w ith o u t a n y obvious p a tte rn . T h is m a y be a
co n se q u en ce of th e fa c t th a t th e re sp o n se fu n c tio n of p h a se lags is alm o st
lin ear fo r S o u th e n d a n d th e re fo re th e w eig h t of c o n s titu e n ts Sa a n d MA2,
w h ich com e in p h ase w ith M2 in a u tu m n , m a y have co m p en sa ted or
exceeded a n y sm all difference d u e to tid e -ra isin g forces. A co m p arativ ely
h ig h n o n lin e a rity in th e re sp o n se of A v o n m o u th u p sets th is system as th e
p h ase lags of p rin c ip a l c o n s titu e n ts are c o n sid e ra b ly m ore effective th a n
Sa a n d M A2. It is also possible th a t th e d iu rn a l co m p o n en ts affect th e tw o
p a rts d iffe re n tly a t th e tim e s of larg e se m i-d iu rn a l tides. H ow ever, it could
be co n c lu d e d fro m re su lts in lig u re 2 th a t th e m ost su itab le co n d itio n s
o ccu r w h e n :
9 s =* *
\
0 3 ~ 0 or 7T
\
03 ~ 0 or 7T
)
(E )
If th e se c o n d itio n s are su itab le, th e n the m e a n lu n a r tim e 'd\ a n d 02 can
be a d ju s te d to zero or 77■ v alu e w h ich e v er is su itab le, at th e cost of som e
div erg en ce of
a n d 04 fro m th e ir o p tim a l v alu es. A lth o u g h th e observed
e x tre m e v alue m a y n o t be ex a ctly in th e sam e m o n th or y e a r as th a t p re ­
d icted, th e p re d ic te d v alu e is exp ected to be w ith in 5 cm of th e n e a re st
ex tre m e value.
In co n clu sio n , it a p p e a rs to be im p o ssib le to specify a sim ple ru le fo r
th e a b so lu te d e te rm in a tio n of e x tre m e (high o r low) tid a l levels, b u t if a
to le ra n c e in a c c u ra c y of a few c e n tim e tre s is allow ed, th e n p re d ic tio n s
a ro u n d th e eq u in o x es of a few c e rta in y e a rs w ould a p p e a r a d e q u a te fo r
p re d o m in a n tly s e m i-d iu rn a l reg im es.
ACKNOWLEDGEMENTS
S pecial th a n k s are d u e to D r. D. E.
a n d h e lp fu l co m m ents.
Ca r tw
r ig h t
fo r h is su g g estio n s
T h e w o rk d esc rib ed in th is p a p e r w as fu n d e d b y a c o n so rtiu m of
th e N a tu ra l E n v iro n m e n t R esea rch C ouncil, th e M in istry of A g ric u ltu re ,
F is h e rie s a n d F ood, a n d th e D e p a rtm e n ts of In d u s try a n d E nergy.
REFERENCES
Am
M. (1976) : The fine resolution of tidal harm onics. Geophys. J.R. Astr. Soc.,
44, pp. 293-310.
in
D.E. (1974) : Years of peak astronom ical tides. Nature, 248, No. 5450,
pp. 656-657.
C a r t w r ig h t