International H ydrographie Review, M onaco, LXIII (]), January 1986
ON THE CONDITIONS FOR CLASSIFICATION
OF TIDES
by M. A M IN (*)
SUMMARY
This investigation is carried out to examine the conditions w hich are
com m only used to classify a tidal regime as diurnal or sem i-diurnal. It is observed
that the application o f C o u r t i e r ’s (1938) criterion in tidal predictions is not always
appropriate. It is show n that during som e neap tides m ost o f the m ajor tidal
harm onics o f the sem i-diurnal ban d may conspire to oppose the M 2 tide.
C onsequently, at some ports the sem i-diurnal com ponent may degenerate into a
dodge tide. U nder these circum stances, though a tide may be classified as
sem i-diurnal according to C ourtier’s criterion, the tidal profile is determ ined by the
constituents of the other species. A diurnal tide can occasionally becom e sem i­
diurnal for sim ilar reasons. A sim ple m athem atical explanation o f these conditions
and when they are likely to occur is given. F or practical purposes, new conditions
that must be satisfied for a tide to be diurnal or sem i-diurnal during the full
spring-neap cycle are suggested.
INTRODUCTION
This problem em erged from the recent com putation o f tidal predictions of
Port H edland (W estern Australia). In com putation of tim es and heights o f high and
low water, the gradient o f tidal level is evaluated at predeterm ined tim e intervals
‘ A t’ which is selected according to the type o f tide. Once the gradient changes its
sign, to find the exact time a binary search is started w ithin that interval to find
the tim e and height o f a high or low water. An optim al choice of tim e interval is
essential because it should not be too long to lose a tide and it should not be too
(*) Institute of O ceanographic Sciences, Bidston Observatory, B irkenhead, M erseyside L43 7RA,
U.K.
n
s h o r t to d o u n w a n t e d c o m p u tin g .
s u g g e s te d b y C o u r t i e r ( 1 9 3 8 ) :
3 .0
3 .0
Oi + K|
F =
In
th is
r e s p e c t, th e
c la s s if ic a t io n
o f tid e s
d iu r n a l
( 1)
m ix e d
0 .2 5
0 .2 5
M: + S2
s e m i- d iu r n a l
is w id e ly u s e d t o d e te r m i n e th e ‘ A t ’. F is c a lle d th e f o r m n u m b e r . D i e t r i c h e t al.
(1 9 6 6 , p . 4 2 7 ) c o m m e n te d , “ I t in d i c a t e s th e f o r m o f th e ti d e f o r o n e d a y . A s t r ic t
c la s s if ic a t io n b a s e d o n f o r m n u m b e r c a n n o t b e m a d e , f o r e x a m p le , in th e s e n s e t h a t
e x c lu s iv e ly a s e m i- d i u r n a l ti d e e x is ts in o n e a r e a , a n d o n ly d i u r n a l ti d e s in a n o th e r .
T h is l i m i ta t io n is e x p la i n e d b y t h e f a c t t h a t th e f o r m n u m b e r is v a lid o n ly f o r o n e
in s t a n t d u r i n g th e s p r in g - n e a p c y c le , i.e . f o r th e s p r in g t i m e ” . F ig u r e 1, h o w e v e r ,
Immingham: sem i-diurnal type
(England)
F = O-ll
(a )
2-1
m.
(b )
0San Francisco: mixed,dominant semi-diurnal type
-I - 1 (U S A )
F = 0 -9 0
I(c )
OF = 2-I5
Manila: mixed, dominant full diurnal type
(Phlilipines)
!d )
- I --------1------- 1------- 1------- 1------- 1------- 1--------1------- 1------- 1------- 1------- 1------- 1------- 1-------- 1
2
4
6
8
O
10 12 14 16 18
d a y s ----- ►
£
20
22 2 4
•
26
28
30
gives a different picture in that it suggests th at a sem i-diurnal tide has two high and
two low waters occurring each day throughout the spring-neap cycle.
It is noted th at the form num ber is often used and quoted w ithout any
reference to its lim itations. The application o f this definition in the tidal predictions
may result in confusion in some cases. It is the purpose o f this research to clarify
that a tide classified as a sem i-diurnal tide may not have a sem i-diurnal com ponent
at all on certain occasions. In those circum stances, the tidal profile will depend on
constituents of the other species.
SEM I-D IU R N A L TID ES
T he condition o f equation (1) is fully justified for spring tides, but it is narrow
in the sense that it is lim ited to the spring tides only. To search for a condition
which m ust be valid for the neap tides as well as spring tides, the neap tides o f the
sm allest range m ust be investigated. By sim ple consideration, those neap tides
occur at times when all the m ajor constituents o f the sem i-diurnal band conspire
to oppose the M 2 tide. In m any respects, the problem of searching for the neaps
of the sm allest range is sim ilar to the problem o f finding the highest astronom ical
tides ( C a r t w r i g h t , 1974), or extrem e high levels in real tides ( A m i n , 1979) when
nearly all constituents are in phase.
According to the tide-generating potential, the phases of the harm onic term s
can be written as :
V = r • 0 + (p
where r is a vector o f small integer ( C a r t w r i g h t
o f jt/2 and
0
=
03
04
05
& T ayler,
(2)
1971), <p is a m ultiple
(3)
where :
9i = local m ean lunar time reduced to an angle;
02 = the m ean longitude o f the M oon;
0 3 = the m ean longitude o f the Sun;
0 4 = the m ean longitude o f the M oon’s perigee;
0 5 = the negative o f the m ean longitude of the ascending node o f the M oon,
and
06 = the longitude of the S un’s perigee.
T he longitude o f the Sun’s perigee varies only slightly over a period o f a
century and only affects the phase o f constituents T2 and R2 which are very small ;
therefore its influence will not be considered.
C onsider the harm onic term s M 2, S2, K2 and N 2, and the nodal terms
(‘satellites’) o f M2 and K2. In the equilibrium tide, their phase must satisfy the
following conditions to oppose the M2 constituent :
26,
=
0
for M2
(4)
20,
- 05 + 180 = 180
M2,_05
(5)
20, + 202 - 203
= 180
S2
(6)
20, + 202
= 180
K2
(7)
20, + 202
+ 05
= 180
K2,b5
(8)
20, - 02
+ 04
= 180
N2
(9)
The second subscript with the nam e of term indicates th at it is a nodal term
associated with the principal term represented by the first subscript. These
conditions can be satisfied by any one o f the four values 0 in table 1.
i à b Lh
i
Values of orbital elements (degrees) which can satisfy conditions in
equations (4-9).
e,
0
A
B
0
180
180
C
D
02
e,
04
90
90
90
90
0
180
0
180
90
90
90
90
05
0
0
0
0
When these conditions are satisfied the remaining principal constituents of
the sem i-diurnal species will have mixed contributions; ^ 2 and k2 will oppose M 2,
and 2N 2, u2 and L2 will be in phase with M 2.
In the diurnal species, m ain constituents which contribute to the tide are K,,
0 ,, Pi, Qi and nodal terms of K, and O, and their phases are :
01 + 02
0, + 02
0, — 02
0i
02
0, ■ 02
0, - 202
- 203
+
+ 05 +
+
- 05 +
+
+
+ 04
90
90
270
270
270
270
=
=
=
=
=
=
Vic.
V*.,e5
for K]
Ki,e5
Vo,
o,
VOt._eS
Oi, -- e5
Pi
Q.
V p,
Vo,
(10)
(ii)
(12)
(13)
(14)
(15)
By substituting the value o f 0 in equations (10-15) it can be shown that
constituents O, and K, combine together as they will be in phase, but P, and Q,
oppose them . However, the opposing effect of Q, and Pi will alm ost be com pen­
sated by nodal term s o f O, and K, and the diurnal tide can be approxim ated by :
A, ~ O, + K,
(16)
To sum m arise, the neap tides o f the smallest range occur when the following
conditions are sim ultaneously satisfied :
(a) the E arth is at one o f the equinoxes;
(b) the longitude o f the M oon is 90° away from the perigee, and
(c) the longitude o f the M oon’s ascending node is near the spring equinox.
It is w orth noting here that the conditions required for the diurnal tide to be
near its peak include the condition that the Earth should be near perihelion. This
contradicts condition (a) in centuries aro u n d the present time. It is difficult for all
these elem ents to satisfy the conditions in table 1 sim ultaneously. The dates w hen
they ap p ro ach closest to these values in this century are listed in table 2. F or exact
value o f 0i, some adjustm ents in other elem ents will also be essential. Therefore
some tolerance, say e, is allowed to relax these conditions.
TABLE 2
The dates when 0 approaches nearest to the values in table 1.
1969 M a rc h 2 2 .............
1986 S e p te m b e r 25 ....
1987 S e p te m b e r 16 ....
02
03
04
05
90.0
90.0
90.0
2.86
18443
174.43
271.29
263.50
303.16
0.21
21.63
2.83
In the real tide, the phase o f any tidal constituent lags with respect to its
phase in the equilibrium tide. However, 0 can be com puted for a particu lar place
by introducing the phase lags o f constituents in equations (4-9) as :
20,
=
0 for m 2
(17)
- gM2
20,
— 05 + 180 — gM2 _ e5 = 180
m 2, _ e5
(18)
20, + 20, - 20J
= 180
S2
(19)
- gs2
20, + 20,
= 180
k2
(20)
— gic2
20, + 202
= 180
(21)
+
- gK.2,e5
K2,e5
20, - 02
= 180
n2
(22)
+ 04
— gN2
Substituting the values o f g’s in equations (17-22), and solving for 0, tim es can
be com puted. W hen these conditions are satisfied the am plitude o f sem i-diurnal
tide can be approxim ated as :
A2 ~ 0.93 M 2 - (S2 + K 2 + N 2)
The factor 0.93 is to account for the nodal tides o f M 2 and K 2.
(23)
The reduction in the size o f the sem i-diurnal com ponent depends on the
am plitudes o f S2, N 2 and K2. The relationship betw een the phases o f S2 and K 2 in
the observed tide is nearly the sam e as in the equilibrium tide, and relationships
betw een M2 and its nodal term , K 2 and its nodal term in the observed tide are
nearly the same as in the equilibrium tide. It is suggested th at conditions (a) and
(c) for the equilibrium sem i-diurnal tide o f the lowest range are expected to rem ain
valid for the observed tide.
A change in the solution o f 0 (or tim es) will occur due to the phase lags o f
constituents in equations (17-22) for the observed tide as com pared w ith th at
com puted from equations (4-10) for the equilibrium tide. The diurnal tides Oi and
Ki m ay not be in phase at that time, and the am plitude o f diurnal com ponent m ay
be m uch sm aller than anticipated by equation (23).
Example
C onsider the tidal regime o f Port H edland (A ustralia), the tidal constituents
o f w hich are listed in table 3. Using the criterion o f C ourtier, the form num ber
(F = 0.145) is well within the limit to classify the tide as sem i-diurnal. Substituting
the values o f g in equations (17-22), the solution for 0 is as given in table 4.
TABLE 3
Tidal constituents of Port Hedland (W. Australia).
N am e
Qi
o,
p,
K,
2N 2
H2
N,
v2
m 2
X2
l2
s2
k
2
A m p litu d e
(m )
P h ase lag
(deg)
0.033
0.150
0.072
0.244
0.028
0.047
0.276
0.062
1.681
0.026
0.064
1.021
0.282
265.12
272.71
288.10
299.39
215.56
294.72
276.16
272.05
304.86
307.34
323.73
14.04
13.28
TABLE 4
Values of orbital elements (degrees) which can satisfy conditions in
equations (11-16) for the tide of Port Hedland.
A
B
C
D
0,
02
03
04
05
! 52
152
332
332
125
125
125
125
0
180
0
180
277
277
277
277
0
0
0
0
F o r conjugate values o f 02 and 04, four additional values o f 0, which can
satisfy equations (17-22), are obtained. The years o f this century when 02, 03, 0 4 and
05 will be nearest to the values given in table 4 are 1969, 1986 and 1987. 0, can be
ad justed to the required value at the cost o f some small perturbations in the other
elem ents.
The am plitude o f the sem i-diurnal com ponent, A2, when the above conditions
are satisfied, will becom e :
A2 ~ 0.93 M 2 - (S2 + N 2 + K2) ~ 0
(24)
This shows th at the sem i-diurnal com ponent can virtually disappear and the tidal
profile will be determ ined by the constituents o f the other species. The predicted
tides, show n in figure 2, confirm th at the procedure based on C ourtier’s criterion
for classification o f a tide as ‘sem i-diurnal’ can break down. The actual dates when
the tide becam e diurnal, com puted using the full set o f constituents, are 1969
February 27 and 1986 A ugust 30; estim ated dates, using equations (17-22), are 1969
M arch 28 and 1986 Septem ber 28 respectively. These displacem ents are due to
non-linear tides which becom e im portant in the determ ination of such events which
are very close otherw ise.
F ig . 2. — Predicted
tides of Port Hedland
(W. A ustralia).
TABLE 5
The estimated dates when 6 is nearest to the values in table 4.
1969 M a rc h 2 8 .............
1986 S e p te m b e r 28 ....
1987 S e p te m b e r 18 ....
02
03
04
05
125.0
125.0
125.0
5.48
186.97
177.05
271.59
263.82
303.39
0.05
21.72
2.72
It is im portant to note th at relaxation o f tolerance £i in an elem ent 0j depends
on the am plitude o f constituents w hose phase depends on it. F o r example, 0s is
involved in the phase o f nodal term s. Since their am plitudes are small, a large
tolerance is possible for it.
It is show n thât orbits! elem ents do conic vçry closç to ysiIuçs w hich ceü
substantially reduce the sem i-diurnal com ponent of tide. By representing diurnal
an d sem i-diurnal com ponents by the nom inal speeds o f Mi and M2 constituents
respectively, it can be shown th at th at the tidal profile will be sem i-diurnal if
A2 > A i/4 ( D o o d s o n & W a r b u r g , 1941, p. 221). However, the tide will be similar
to the m ixed dom inant sem i-diurnal type shown in figure 1(b). F or a tide to be
sem i-diurnal on such occasions so th at all low waters are below m ean sea level and
all high w aters are above m ean sea level, it m ust satisfy :
A2 > A,
(25)
It can be show n, by substituting the values o f 0 from table 4 in equations (10,
12) for the observed tide, th at phases of constituents Ki and Oi differ by 30° when
the sem i-diurnal com ponent of Port H edland is minimum. The am plitude o f the
diurnal com ponent should be com puted by vector sum o f constituents Oi and Ki
and it will be sm aller than th at given by equation (16). The phase lags o f the
observed tides ten d to relax condition (25) because Ai decreases. In m arginal cases
it will be essential to calculate 0 from equations (17-22) and phases of constituents
Oi and Ki to com pute the exact am plitude of the diurnal com ponents.
DIURNAL TIDES
In the diurnal species, m ajor constituents w hich determ ine the profile o f
tide are Qi, Oi, Pi and K |. Assum ing that Ki is the dom inant constituent o f
species, the diurnal com ponent will be near its m inim um when it is opposed by
Pi and Qi. This w ould imply th at phases o f these constituents should satisfy
following conditions :
0 i + 02
0 1 - 02
0, + 02 - 203
0, - 202
+ 04
+ 90 — gKi
+ 270 - go,
+ 270 - gP,
+ 270 - gQl
=
=
=
=
V k,
Vo,
V P,
V Ql
=
=
=
=
0
180
180
180
the
the
Oi,
the
(26)
(27)
(28)
(29)
For the equilibrium tide, g = 0, the possible solutions of system o f equations
(26-29) are given in table 6.
TABLE 6
Values of orbital elements (degrees) which satisfy condi­
tions in equation (26-29) for diurnal component to be
minimum.
0
,
270
270
90
90
6
:
0
0
180
180
03
04
180
180
0
180
0
0
0
180
The longitude of the M oon’s node is not im portant because nodal term s of
constituents Oi and Ki will cancel each other. On those occasions the am plitude
o f the diurnal com ponent can be approxim ated as :
A ' i ~ Ki — (Oi 4- Pi + Qi)
(30)
Substituting the values of 0 from table 6 in equations (4-9), it can be shown
that constituents M 2, S2, N 2 and K2 are in phase and the sem i-diurnal com ponent
reaches near to its m axim um value :
A '2 — Mi + S2 + K2 -(- N 2
(31)
This is in accordance with the fact th at the sem i-diurnal com ponent
m axim ises at the expense o f the diurnal com ponent ( C a r t w r i g h t , 1975). F or a tide
to be diurnal all the time it m ust satisfy the condition ( D o o d s o n & W a r b u r g ,
1941):
A', > 4AS
(32)
However, in the real tide the sem i-diurnal com ponent m ay not be near its
m axim um value when the diurnal com ponent is minimum. Thus the condition (32)
can be relaxed to some extent.
Example
The form num bers o f tides o f D o-Son (Vietnam) and Poeloe Langkotas
(Sum atra) calculated from their constituents in table 7 are 19 and 25, respectively;
therefore, according to C ourtier’s criteria, these tides should be classified as
diurnal. Do-Son tide is classified as diurnal by D i e t r i c h (1966), see figure 1. The
values o f 0, when K, is opposed by other constituents, are given in table 8 an d dates
when 0 approaches these values are listed in table 9. The diurnal com ponent can
easily vanish on these dates and the tidal profile will be determ ined by the
constituents of the sem i-diurnal species as is confirm ed by predictions of 18 O cto­
ber 1948 in figure 3.
In the case o f Poeloe Langkotas, condition (32) is not satisfied if A 2 is
evaluated simply by equation (31). W hen the diurnal tide approaches n ear its
m inim um , i.e. 0 satisfies equations (26-29), constituents M 2 and S2 differ in phase
by 116°. For the observed tide A2 should be calculated by vector sum of
constituents M2, S2, ...in stead o f by a sim ple sum as in equation (31) w hich is
applicable for in phase constituents o f the equilibrium tide. A2 com puted in this
way is sm aller than that given by equation (31) and it satisfies condition (32) and
tidal profile rem ains diurnal as show n in figure 4. Clearly, phase lags o f consti­
tuents play an im portant role in the determ ination o f profile o f the tide.
TABLE 7
Constituents of Do-Son, Vietnam (Long. 106°47' E,
Lat. 20°43' N) and Poeloe Langkotas, Sumatra
(Long. 107°37' E, Lat. 2°32' S).
Time zones are - 7 , - 8 hours respectively.
D o -S o n
N am e
Q.
o,
p,
K,
n 2
m 2
s2
k
2
P o e lo e L an g k o tas
A m p litu d e
(m )
P h ase lag
(d eg )
0.136
0.700
0.240
0.720
0.008
0.044
0.030
0.010
354.0
26.0
89.0
89.0
85.0
102.0
136.0
137.0
A m p litu d e
(m )
—
P h ase lag
(deg)
—
0.390
0.130
0.620
76.0
132.0
139.0
—
—
0.020
0.030
225.0
11.0
—
—
TABLE 8
Values of orbital elements (deg.) when diurnal component becomes
minimum.
D o -S o n
P o elo e L angkotas
0,
02
03
04
0.
02
03
90
90
212
32
0
180
153
153
90
270
211
31
0
180
04
TABLE 9
Dates when orbital elements approach nearest to values in table 8
diurnal tides close to their minimum.
02
03
04
D o-S on
22 S e p te m b e r 1948 .............
23 M a rc h 1966 ....................
32
32
180
1
157
149
P oeloe L an g k o tas
23 S e p te m b e r 1979 .............
24 M a rc h 1985 ....................
211
31
182
1
338
202
(m )
HEIGHT
F i g . 3. — Tidal predictions of Do-Son (O ctober 1-30, 1948) when diurnal com ponent becomes m inim um .
C O N C L U SIO N S
It is shown that there are times when m ost of the m ajor tidal harm onics of
the sem i-diurnal species can conspire to oppose the M 2 tide. Similarly, there are
times when the m ain constituents o f the diurnal species oppose the K.i tide.
The procedure o f classification o f tides, based on C ourtier’s criterion, is valid
for the limited duration o f spring tides. M oreover, the two assum ptions, i.e.
M2 = 3S2 and Oi = Ki, are apparently inferred from typical tides o f E uropean
waters. Therefore, its application to the o ther parts o f the world m ust be treated
with caution. It is overstretched when it is extended to a full spring-neap cycle. The
sem i-diurnal com ponent o f the tide is substantially reduced w hen the M2 tide is
o pposed by the other m ajor sem i-diurnal constants. It is shown m athem atically and
by illustrating with an exam ple th at such events can occur. On those days the tidal
profile is determ ined by the constituents o f the other species, and a tide classified
as sem i-diurnal may behave as a mixed tide or diurnal tide.
Similarly a tide classified as diurnal u n d er C ourtier’s criterion is show n to
become sem i-diurnal or mixed.
In view o f the analysis presented in this paper, tides can be divided into the
following categories :
(i) diurnal always (Poeloe Langkotas), if condition (31) is satisfied;
(ii) diurnal usually, with occasional sem i-diurnal wave (D o-Son), if F > 3;
(iii) mixed (San Francisco and M anilla), if 3 > F > 0.25;
(iv) sem i-diurnal usually with occasional diurnal wave (P ort H edland) if
F < 0.25, and
(v) sem i-diurnal always, (Im m ingham ), if condition (31) is satisfied.
C riteria for tides to be always diurnal o r sem i-diurnal are very restricted but
can be safely applied to a full spring-neap cycle.
172
IN T ER N A T IO N A L H Y D R O G R A P H IC REVIEW
F ig . 4. -
Predicted tides of Poeloe Langkotas (1979, 1985) for years when diurnal component becomes
1985
Acknowledgments
I
w o u ld lik e to th a n k D r. D.E. C a r t w r ig h t a n d G .A . A l c o c k f o r th e i r
h e lp f u l c o m m e n ts to im p r o v e th is m a n u s c r ip t a n d M rs. V.L. D o o d s o n f o r b rin g in g
th e p r o b le m t o m y n o tic e .
REFERENCES
A m i n , M. (1979) : A no te on ex trem e tid al levels.
Int. Hydrogr. Rev., LV1(2), pp. 133-141.
D .E . (1 9 7 4 ): Y ears o f p eak astronom ical tid es
pp. 656-657.
C a rtw rig h t.
N nturp
248. N n 5450,
C a r t w r i g h t , D .E . & T a y l e r , R.A. (1971) : N ew co m p u tatio n s o f tid e-g en eratin g p o ten tial.
Geophys. J.R. astr. Soc., 23, p p . 45-74.
C o u r t i e r , A. (1938) : M arées. Service h y d ro g ra p h iq u e de la M arine, Paris.
G ., K a l l e , K ., K r a u s s , W . & S i e d l e r , G. ( 1 9 7 5 ) : G e n e ra l o ce an o g ra p h y - an
in tro d u ctio n . J o h n W iley & S ons, N ew York.
D
ie trich,
D
oodson,
A.T. & W
arburg,
H . D . (1941) : A dm iralty m an u a l o f tides, H M S O , L ondon.