The Condition that a Long-Period Tide shall follow the
Equilibrium-Law
J. Proudman
(Received 1959 December 7)
Summary
It is only when friction is taken into account that the limiting form of
a long-period tidal constituent always follows the equilibrium-law.
Hence the question arises as to how long the period of a constituent
must be in order that its distribution may be given approximately by
the equilibrium-law. In this paper reasons are given for believing that:
(i) the constituent whose period is nearly 19 years will certainly
follow the equilibrium-law ;
(ii) the semiannual and annual constituents will probably follow
the equilibrium-law ;
(iii) the fortnightly and monthly constituents will probably not follow
the equilibrium law.
Introduction
Because, in the ocean, it is possible for currents to exist with a stationary oceansurface, the limiting forms of frictionless tides of long period are not always given
by the equilibrium-law.
In a paper “On the dynamical theory of the tides of long period”, G. H. Darwin
(1886) said, “In treating these oscillations Laplace . . seeks to show that friction
suffices to make the ocean assume at each instant its form of equilibrium. His
conclusion is no doubt true, but the question remains as to what amount of friction
is to be regarded as sufficing to produce the result, and whether oceanic tidal
friction can be great enough to have the effect which he supposed it to have.
.
The quickest of the tides of long period is the fortnightly tide, hence for the
applicability of Laplace’s conclusion the modulus of decayWmust be short compared with a week. Now it seems practically certain that the friction of the ocean
bed would not much affect the velocity of a slow ocean current in a day or two.
Hence we cannot accept Laplace’s hypotheses as to the effect of friction.
There
is one tide, however, of long period of which Laplace’s argument from friction
must be true. In consequence of the regression of the nodes of the Moon’s orbit
there is a minute tide with a period of nearly nineteen years and in this case
friction must be far more important than inertia.”
I n his Tfiorie des narkes (1910)Poincard termed tides which follow the equilibrium-law “mardes statiques de la premikre sorte” and the other limiting forms
“mardes statiques de la deuxikme sorte”. Then, following S. S. Hough (1896),
I.
.
..
.. .
* 2/k in the notation of Section 2 below
244
The condition &at a long-pedod tide &all follow the equilibrium-law
24s
he said: “NOUSverrons qu’il faut une dizaine d’annks pour que le frottement
puisse se faire sentir; par consequent, les mar& annuelles et de periode plus
courte seront bien de la deuxikme sorte; au contraire, la mar& ayant pour periode
18 ans serait une mark de premikre sorte, qu’il devrait calculer par la thdorie de
l’equilibre”.
The object of this Note is to show that the conclusion of Darwin and Poincark
regarding the tidal constituent with a period of nearly 19 years is correct, but that
Poincare‘s statement regarding the annual constituent is probably not correct.
Hough and Poincar6 supposed that tidal friction arose through molecular
viscosity and followed a linear law. In the present Note it is supposed that total
friction arises through eddy-viscosity near the ocean-bed and follows a quadratic
law. But by supposing that the long-period constituent is superposed on a much
larger short-period constituent, we arrive at a linear law for the long-period
constituent with a coefficient which is proportional to the maximum value of
the short-period current (Bowden 1953).
2.
Law of bottom-friction
Let the bottom-currents of the long- and short-period constituents be denoted
respectively by
0,
vcosuot,
where t denotes the time, V, GO are constants and w is treated as a constant over a
number of short periods. Then the force of bottom-friction is taken as
uot+w),
(1)
o.oo2~p~vcos
u~t+o~(Vcos
p denoting the density, and Bowden’s paper shows that the frictional term in the
ordinary equation of motion for the long-period constituent will be
kv, where k
=
I V
-I-h’
One would expect that when the period of this constituent is long compared
with the frictional period 2 4 k the constituent will follow the equilibrium-law,
and that when the period is short compared with 2r/k the constituent will follow
the limiting form with currents. This criterion is supported by the analysis of
the following section.
3. Dynamical equations
In 1916 I transformed the dynamical equations of the tides, for any sea or
ocean, into a form using an infinite number of discrete coordinates. I then
neglected friction, but, if k of ( 2 ) is treated as uniform, friction may easily be taken
into account.
For a tidal system with two degrees of freedom, the equations are
j+kj+pp+vy = P
(3)
q+kQ-pp=o,
(4)
where p, q, P are functions of time and k, p, v2 are constants. Here p determines
’
the elevation of the sea-surface,
4 determines that part of the current which is
246
J. Proudman
possible with a stationary elevation, P represents the tide-generating forces, k is
the coefficient of friction of (2), j? a geostrophic coefficient due to the rotation of
the earth, and 9a coefficient of stability depending on the Earth's gravity.
The equilibrium-hw is given by
p
so that
On neglecting friction, the equation (4) gives
=
q =o
q
=
v2p =
P.
(5)
pp
and integration of this equation gives
4 = pp+constant.
(6)
When the motion is periodic, the constant of (6) vanishes. The limiting form of
long-peuiod tides is then given by (3) with
q=pp
p=o,
so that
P.
(7)
For complex harmonic motion with time-factor et"t, the equations (3), (4)
become
("2+p2)p =
so that, on eliminating #,
iu
- u2 + iuk + v2 + - P ] p
=
zu+k
P.
This indicates that the form of the long-period tides depends on the value of
iu
ia+k
when
Q
is small. When
k#o,
U = O
the equation (10) gives the equilibrium-form ( 5 ) ; when
u
+0, k / u + o
the equation (10) gives the limiting form with currents (7).
For four degrees of freedom the equations, with an obvious extension of the
notation of (3), (4), are
+k$l+ pp2 +p1,1q1+ 81,242 +v12p1 = Pl
$2 + k j z -/?PI +&141+ flz2,242+ vz2pz = P2
2%
ql+~l+r61q2-pl,lpl-j32,lp2
= o
(12)
(13)
(14)
q2+k42-B'ql-B1,2pl-p2,2p2
= 0.
(15)
For stationaq motion
p 1 = p 2 = 0,
q1
=
q2
=0
Tbe condition that a long-period tide sball follow the equilibrium-law
247
so that the equations ( 1 2 x 1 5 ) become
+ = Pl
8 2 , 1 4 1 + 8 2 , 2 4 2 +v 2 2 p 2 = P 2
8 1 , 1 4 1 +81,242
V12pl
=0
k41+/3’42
= 0.
k42-8’41
From (IS), (19) it follows that, unless
= 0,
k2+8’2
then
41 = 4 2 = 0,
and then ( 1 6 ) ~(17) give
Ply v 2 2 p 2 = P 2
(20)
which is the equilibrium-form. The condition that the limiting form of frictionless tides shall not be the equilibrium-form is thus
v12p1 =
8’
(21)
When the condition (21) is satisfied and the motion is complex harmonic, the
equations (12)-(15) give
= 0.
( - uz + id + V I ~ ) P I + ia8pz +81,141 +81,242 = PI
( - o2+ iak + ~ 2 ~ 1 p - 2ioPp1+192,141+
82,2@ = P 2
+
(io k)Q1 = ia(pl,lpl+ 8 2 , 1 p 2 )
(ia+ 4
so that the elimination of
4 2 = i4%,2p1+/32,2p2)
41, q 2 gives
ia
+ia&2 +-(/31,182,1+81,282,2)p2
io+k
ia
- iajIpl+ -(/32,181,1
ia+k
=
Pl
+j32,2/31,2)~1 = P 2 -
Again, the single factor
ia
io+k
is a criterion as to how far the solution approximates to the equilibrium-form.
When
k#o, U = O
(22), (23) become respectively
v12p1 =
Pl,
v22p2 = P 2
J. Proudman
248
which give the equilibrium-form. When
k/u +O
u +o,
(22), (23) become respectively
(VL2+&,P+81,22)p1+
(181,182,1+/312/92,2)p2
+82,221p2 +@2,181,1+
(v2%+P2,12
8 2 281,2)pl
=
s
= p2
(25)
(26)
which determine the limiting form with currents.
For any number of degrees of freedom it is clear that a sufficient condition for
the limiting form not to be the equilibrium-form is that all the geostrophic coefficients between pairs of 4’s are zero. When this condition is satisfied, it is also
clear that the factor
za
iu+k
will enter the solution exactly as in the cases of two and four degrees of freedom.
4. Numerical estimates
From ( 2 )
2w
_
-- 2004-h
k
V
h
V
= 2 x 103-
very approximately. But h/V varies enormously over the ocean, while the formulae of Section 4 imply a uniform value of k. It is therefore only possible to make a
rough estimate of the effective value of 21r/k.
In shallow water, we might have
h = morn,
V
=
Socm/s
so that
h
V
-=
mom.
= 200s
Socm/s
and
2n
_
-- 4 x 105s = 46d.
k
In mid-ocean, we might have
h = 4000m,
V
=
5 cm/s
so that
h
V
- --
400om
= 8x
scm/s
104s
and
2T
-- 16x 107s = ~ ‘ ~ y e a r s .
k
249
The effective value of the frictional period 27r/k will be between 4-6d and
5.1 years.
If we suppose that shallow water extends over one-tenth of the ocean, and take
k one-tenth of the value for shallow water, we have
The condition that a long-period tide shall follow the equilibrium-law
27r
_
-- 46d.
k
If this estimate is valid, it will follow that the fortnightly and monthly constituents
will not follow the equilibrium-law, but that the semiannual and constituents of
longer periods will follow the equilibrium-law. The conclusion about the 19 yearly
constituent appears to be quite safe, as its period is long, even compared with the
larger frictional period of 5.1 years.
Edgemoor,
Verwood,
Dorset:
1959 December.
References
Bowden, K. F., 1953. Proc. Roy. SOC.
A , 219,426.
Darwin, G. H., 1886. Proc. Roy. SOC.
A , 41, 337.
Hough, S. S., 1896. Proc. London Math. SOC.,
28,264.
PoincarC, H., 1910. TMorie des markes, 182.
Proudman, J., 1916. Proc. London Math. SOC.,18, 21, 51.