Geophys. J . Int. (1989) 99,457-468
A complete spherical harmonic approach to luni-solar tides
S. R. Dickman
Department of Geological Sciences, State University of New York, Binghamton, N Y 13901, USA
Accepted 1989 April 17. Received 1Y89 March 17; in original form 1988 October 12
SUMMARY
In this work a spherical harmonic theory of ocean tides is presented. The theory is
based on Laplace tide equations modified to include turbulence with constant eddy
viscosity, linearized bottom friction, and oceanic loading and self-gravitation.
Variable bathymetry is also treated in harmonic terms, and no-flow boundary
conditions are applied at continental coastlines. The tide and boundary constraint
equations are reduced to matrix form and solved by a weighted least-squares
procedure. Five zonal luni-solar tides, ranging in period from 14 days to 18.6 yr, are
investigated using the theory; such tides have typically been difficult to compute
using traditional numerical approaches. The polar motion and changes in the length
of day induced by these long-period tides are calculated. Tidal solutions are
compared extensively with results from other tidal theories and from recent satellite
and sea-level observations. The greatest limitation to accurate prediction of zonal
tides-for any theory-appears to be the marginal failure of all tide theories to
conserve mass globally; the use of additional mass constraints may be warranted.
Key words: ocean tides, satellite perturbations, spherical harmonic theory,
INTRODUCTION
The ability of ocean tides to modify the geopotential,
satellite orbits, and Earth’s rotation depends on the
world-wide characteristics of the tides. Until recently, our
knowledge of ocean tides has been limited to observations at
a finite number of points, most of which represented coastal
locations. The noisy quality of tide data-compounded by
the distortions produced through coastal dynamics-limit
further the usefulness of such observations and make it
necessary for us to turn to tidal theory. Of the variety of
theoretical models attempting to predict deep-ocean tides,
that of Schwiderski (1978) is perhaps the most successful: his
combination of data and theory, achieved through a
technique he calls ‘hydrodynamical interpolation’, has
yielded global predictions of all major short-period tides,
and a few long-period tides, with an average accuracy of
better than 5 cm (Schwiderski 1983).
In the past 10yr or so, satellite observations have also
been used to estimate tide characteristics. Perturbations in
satellite orbits result from the global mass redistributions
accompanying oceanic, solid-earth, and meteorological tidal
motions; with assumptions made concerning the body tide,
global parameters of various oceanic (plus meteorological)
tides can be inferred. The most recent such analysis, by
Christodoulidis et al. (1988), yields estimates of 616
spherical harmonic coefficients (66 determined purely
observationally, the rest partially interpolated) for 32 major
and minor ocean tides.
Despite the impressive progress made both in global tide
theory and global tide observation, there is still a strong
need for an improved deep-ocean tide theory. Interpretation
of the satellite observations is fundamentally problematic
because of the difficulty in separating solid-earth and
meteorological effects from the oceanic ones. Furthermore,
in order to estimate ocean-wide tidal currents from satellite
observations, a theoretical connection is required between
tide velocities and the satellite-inferred tide height
harmonics. Also, finite-difference numerical tide theories
account for oceanic loading and self-gravitation parametrically, reducing their accuracy by perhaps -10 per cent (see
Dickman 1988a). Finally, finite-difference theories are not
easily extended to long periods (e.g. Schwiderski 1982;
Carton & Wahr 1986); the longest-period tides are
essentially not predictable dynamically, while moderately
long-period tides are of uncertain accuracy.
There are several reasons for basing a global tide theory
on a spherical harmonic approach. In many situations-such
as tidal effects on wobble and length-of-day, and estimates
of tidal friction-solutions to the tidal equations directly in
terms of spherical harmonics may be more useful.
Furthermore, when applied to diurnal tides, a spherical
harmonic theory can account for the effects of core
resonance (Wahr & Sasao 1981) better than a numerical
457
458
S. R. Dickman
theory can (cf. Hinderer & Legros 1988). It can facilitate the
interpretation of satellite-derived tidal coefficients; or,
combined with those coefficients, it can produce a tidal
solution constrained by global observations. And, as with
the theory presented here, it can account for oceanic loading
and self-gravitation without approximation.
Potential problems with a spherical harmonic approach,
including slow convergence and a Gibbs effect at coastlines,
are well known. However, in the work discussed below we
have encountered no difficulties with convergence, and the
Gibbs phenomenon is likely to contaminate our ‘untruncated’ solutions only slightly (cf. Dickman & Steinberg 1986;
also, Estes 1980; Schwiderski 1983). It might also be noted
that a spherical harmonic approach is intrinsically areal, just
as discrete (numerical) approaches are-the
proper,
physically meaningful way to deal with ‘continuous’ tide
equations (like the Laplace tide equations) which are
originally point-wise (Schwiderski 1980); the harmonic
equations, though, can be manipulated analytically to reveal
additional, general characteristics of the solution (Dickman
1988a).
In this article, the theory for hi-solar tides-in oceans
which are non-global, turbulent, self-gravitating and
loading; which possess bottom friction and realistic
bathymetry, and which overlie an elastic e a r t b i s
developed and formally solved in spherical harmonic terms.
Solutions are then obtained and evaluated for five major
long-period tides.
u = a {u}, u = 0 exp (iot),
and U = M , Y ~ .
The potential generates a tide of height T with horizontal
velocities u, (north-to-south) and uA (west-to-east); for
these tidal variables, we define
{u,},
=
UA
= 9% {u,},
(24
A =2
I,n
[
q q ] y ;- a u / g ;
p.9
the yielding of the elastic earth to the tidal potential
modifies the potential by (Y = 1+ k , - h,, the usual Love
number factor, while tidal loading and self-gravitation cause
the tide height coefficients to be scaled according to
In (2b), 6.. is the Kronecker delta, pw is the sea-water
density, p is mean Earth density, k ; and h; are load Love
numbers of degree I , and O = C T Y ; is the ocean function
(zero on land, unity over oceans); the triple-product integral
I
We follow the same complex spherical harmonic approach
used in Dickman (1985, 1988a,b). The luni-solar tidal
potential U, which is degree 2 order S (S = 0, 1, or 2),
complex frequency o,amplitude l&l and phase arg (&), is
replaced by a complex potential U where
Ue
and
A X = (Y;)*YzY; d s ,
THEORY
T = 9% {T},
where f = 2Q cos 8 is the Coriolis parameter; B is the mean
Earth angular velocity; g, a are surface gravity, radius; F
denotes the horizontal Laplacian; and ‘” denotes time
differentiation. Equations (1) are Laplace tide equations
(see Lamb 1932) modified to include linearized bottom
friction, with constant drag coefficient P (see Proudman
1960), and lateral turbulent dissipation, with constant eddy
viscosity A . We account exactly for mantle elasticity,
oceanic self-gravitation, and the vertical deformation
induced by tidal loads by writing
T = ‘k exp (iot), and ‘k =
VY;
u, = Be exp (iat), and i, =
U,
=
exp (iot), and ii, =
c u;Y;
2 v;Y;
where the summations are over all 1, n such that 1 2 0 ,
-1 5 n 5 1. The complex harmonic functions Y ; = Y;( 8, A)
of colatitude 8 and longitude A are fully normalized here.
Our governing equations in spherical coordinates are
where ‘*’ denotes complex conjugation, is over the unit
sphere. As discussed in Dickman (1988b), equation (2)
represents a generalization of previous expressions for
oceanic loading and self-gravitation (Dickman 1988a). In the
absence of self-gravitation, loading, and mantle elasticity, A
would reduce to T - U / g , and A would reduce to ‘k - U/g.
Pointwise mass conservation (‘continuity’) is given by (see
Lamb 1932)
1
[sin
a sin2 e
T=--
a a
e[hue]+ -[I;..])
ae
dA
where h = h ( 8 , A) is the Ocean depth and
6 = h sin 8.
The usual approach to solving tide equations (e.g. Lamb
1932) involves combining equations (1) and (3) into a single
equation for T. Because of the turbulent friction terms in
(l), we have found it necessary to first express the equations
in matrix form. Thus, the tide equations (1) are re-written
using complex tidal variables, and their harmonic expansions are substituted in, leading to
where p = cos 8,D l a p = (1 - p’) aiap, and
A
2~ cos e au,
-____
UA+----a2sin2 8
a2sin28 a
(3)
w 1= A + (ili + P)a2(1 - p2) +AI(I + 1)(1 - p 2 ) ;
w, =fa2(1 - p’) - 2iAnp.
Harmonic approach to h i - s o l a r tides
In deriving (4) we have used derivative properties of
spherical harmonic functions (see Dickman 1985) and also
-dp"'Y; = I(1
+ 1)Y;
Equations (6b) can be solved jointly to determine the tide
velocity coefficients as functions of the tide height and tidal
forcing. We find
(7)
(Arfken 1970).
Then, using the product-to-sum relations in Dickman
(1988a) for pY;, p2Y;, and p3Y;, equations (4) can be
written
where
C u;[dny; +sin YF+2 +2, '7-21
+ C. V;[~1~nY;+l+i1~nY;-~
+g31nYF+3+ i 3 / ~ - ~ 1
and
=gadiqP-
2
-2 u;[gl~ny;+1+i11ny;-l
= w + 5.0-1.5;
g=w-'.(G-5-j);
n-1
i=N. ( 5 . v - l . n'+ i);
E=N-(V+~.W-'.~);
F = 0-' . ( 9 - 5. E)
DA
aP
459
(5)
V;[dnY;+dlnk;l+2+i1nY;-21
+93/nY;+, +k3/nY;-31
= - g a V i q 7 %ad
where now d,, sin, ,i gl,, ill,,, g3,, $3,,-defined in
Appendix l - d e p e n d on A , P, and u but not p or A.
The next step is to substitute expression (2) for A into
equations (5). After using harmonic function derivative
relations, the right-hand sides of (5) will involve T;, &,
and various Y;; because of the d
m factor, however,
orthogonality cannot be directly invoked. We therefore
define
using the 'elegant' alternative product-to-sum relation
mentioned in Dickman (1988a), and Abramowitz and
Stegun (1964), it can be shown that
It might be noted that the large number of matrix operations
already encountered is ideally suited to a vector computer
facility, on which they end up executing more rapidly than,
say, the less complicated but less algebraic code from
non-turbulent oceans (Djckman-1988a).
The expressions for U and V are to be substituted into
continuity, equation (3). To deal with continuity, we have
m rather than the ocean depth
chosen to expand = h
h in spherical harmonics:
The $ were constructed directly from recent bathymetry
data (to be discussed later), though it is also possible to
estimate them from h;, using the Q functions. Using the
expansions for $, f,, fA and 6,and applying the harmonic
derivative and product-to-sum relations then orthogonality,
continuity can be written (for each 1, n)
XOT;
+ X(-2)T+Z + X Z T - 2
1
uh'C i;s C
+
L,N
j,s
ark A n/ (( sj ++ rk))L( N - k )
r.k=-I
j=even
In short, after substituting (2) into ( 5 ) , using harmonic
derivative relations, multiplying by (Y;)*, and integrating
over the unit sphere, the result is
where the x and a are defined in Appendix 1. Finally, when
equations (7) are substituted into (8), we find the
tide-governing equation for turbulent, loading and selfgravitating oceans to be
2 BZT = &by
P.9
for each 1, n, or
2tv :;V;
- 2 s:u;
where n
,:
r,?zand which depend on QFl, and
are defined in Appendix 1. Note that n; and are
=
2 VE ';qpT+ (Ybb1;
..
v::, ;5:
t:,
vz,
1;
functions of S, the harmonic order of the imposea ti&
potential, and that (see Appendix 1) 3; = 0 for zonal tides.
Considering all possible L, N the momentum equations can
be written
!. d + 2. 3 = 9 . [g?.
F]+ &&ii
.!! 3 - 2 . d =r=.
[g. F]+ an&;
(6b)
where f = {T;},d = {u;}, and 3 = {$'} are the collections
of unknown tide height and velocity coefficients.
460
S. R . Dickman
As discussed in Dickman (1988a,b), no-flow boundary
conditions at coastlines are rephrased as
ii. G H = O
where H = h,O. Using the same procedures followed for the
tide equations, this can be written
a
whose amplitude was roughly the mean of all those
possible for the range of h , (the variation in
was roughly
f7 per cent). Similar experiments for non-turbulent
oceans-not
incorporated into Dickman (1988a,b),
however-would
yield h, = 2.4 km. We might anticipate
that these 'optimal' values of h, would lead to slightly more
dynamic tidal solutions, compared to those with lg/@l,
because of relative down-weighting of the no-flow coastal
boundary conditions.
SOLUTIONS A N D DISCUSSION
The a , , are defined in Appendix 1. Substituting equations
(7) into (10) leads to
for each 1, n, or
for the collection of all 1, n, where
In earlier work, h,, represented the mean ocean depth (or
lh{;/@$) and was viewed as a natural way to scale the
boundary constraint equations so that they were comparable
in magnitude to the tide equations. Our goal is the solution
to both the tide and constraint equations; we have
proceeded by combining equations (9) and (11) into
% . %MJi
(12)
where !B = (BE)' and 6 = (6;)'. But there is no reason for !B
to be a square matrix. It is desirable to use as many
equations as possible, even when solving for only a few
low-degree harmonics, so that in general, !B will be
rectangular. This corresponds to an 'overdetermined'
situation, and the solution will be obtained by generalized
inversion (see Menke, 1984):
where '*" denotes the transpose of the complex conjugate.
Equation (13) is equivalent to a least-squares approach, and
because of the h,, factor amounts to weighted least-squares
(see Lawson & Hanson 1974).
h, thus represents an additional parameter whose value
may be chosen to yield an 'improved' solution. Because a
variety of improvements are conceivable-smaller residual
error in (12), smaller pi (see Dickman 1988a), selected T;
equal to specified values (or within a specified r a n g e b t h e
'optimal' boundary constraint weighting factor h, is
somewhat arbitrary. Based on a number of computational
experiments involving the lunar fortnightly tide, in which a
wide range of h,, (0-70 km) was considered, we have chosen
h0=2.3km. Such a value of h,, produced the smallest rms
residual to equation (13) (pre-inversion), at the same time
yielding a slightly smaller fl than did larger h, and yielding
Spherical harmonic coefficients of the ocean function 0 and
modified ocean depth
were computed by the author to
degree and order 24 based on digital world-wide bathymetry
data kindly provided by J. Marsh at NASA/Goddard Space
Flight Center (J. Marsh, private communication 1987). The
DBDBS data, originally from NOAA and Washington
University, is specified at 5-min increments of latitude and
longitude; topographic heights are given to the nearest
metre although true accuracies are probably appreciably
worse (D. Sandwell, private communication 1988). The data
were averaged over 0.5" X 0.5" areas; straightforward
integration (double precision, no recursions for Legendre
functions) yielded S;., @ and also hf. Table 1 lists these
coefficients through degree and order 8, renormalized for
comparison with Balmino et al. (1973); such comparison
reveals that our ocean model covers about 1.5 per cent
more of Earth's surface, and has less pronounced
short-wavelength coastline variability, than the model
described by the Balmino et al. coefficients.
Results were obtained for five long-period tides: lunar
fortnightly (Mf)
and monthly (Mm), solar semi-annual (Ssa)
and annual (Sa), and lunar nodal (18.6yr). Such tidal
predictions are especially useful because of the numerical
difficulty in carrying out finite-difference schemes over long
time spans (see, e.g., Schwiderski 1982, section on Mf
Ocean Tide Parameters; Carton & Wahr 1986); they are
especially important because long-period tidal signals are
now detectable, using space-geodetic techniques, in a
variety of rotational datasets [length-of-day, e.g. Merriam
(1985); nutation, e.g. Herring et al. (1986) (long-period but
non-tidal core mode); possibly rapid polar motion, cf.
Dickey & Eubanks (1986)l.
Each type of tide is, of course, actually a tidal 'band' of
signals (e.g. Doodson 1922). In this work, only the major
tide in each band is investigated: the 13.66-day tide in the
fortnightly band, the 27.56-day tide in the monthly band,
the 182.62-day semi-annual tide, the 365.26-day annual tide,
and the 6798.36-day nodal tide. The amplitudes,
of
their forcing potentials were taken from Lambeck (1980,
table 6.3) and appropriately renormalized. Work is currently
in progress to determine the nature of the oceanic response
within each tidal band; in the absence of results, it might be
assumed (cf. Christodoulidis et al. 1988; Wahr & Sasao
1981; see also Schwiderski 1983) that the oceanic response is
flat, so that the ratio of a tide to the major tide within the
same band would simply be equal to the ratio of their
forcing amplitudes.
As discussed in Dickman (1988a), spherical harmonic
solutions to tidal equations, whether static or dynamic, are
non-zero on land-a fortunate situation, since constraining
w,
Harmonic approach to luni-solar tides
Table 1*
a6
B"
L
0.71316
-020272
-909309
-.12soa
-.02587
0.01222
0.12973
0.02535
0.01105
-.00071
-008075
0.01708
0.00965
-*00147
0.00017
0.34602
-*00116
0.00407
-.oooso
.
-.00037
0 00000
-1 1750
0-00233
0.00086
0.00002
-
-.oooos
0.00001
0.00000
0.21605
-*00046
--00138
0-00006
0.00003
0.00000
0.00000
0.00000
0.02878
0007 3
-.00051
-.00006
0.00000
0.00000
0.00000
0.00000
0.00000
-
0.
0.
A-
I
t
-.
0.
461
0 5 1 17
-a03941
--00055
-0.-.OlS14
021 50
-.00574
0.
0.01205
-.002S6
0.00015
-.00106
0.
0.00477
0.00212
-.00020
0.00012
-.00006
0.
0.00669
-.00015
-.00028
0.00005
0.00001
0.00000
- 2 1720 94
692.94
426.62
1761.2C
181.79
-96.48
-149.39
-80.34
-98.10
7.86
473.64
-94.60
-20.26
8.29
-0.90
-719.17
-9.16
2-14
1.25
1.62
-0.0s
163.81
17.61
1-51
-0.03
0.
-.01334
00023
-.00009
-.00001
0.00000
0.00000
0 00000
0.
0.00945
0.00023
-.00006
-
.
0.22
-0.03
0.01
-207.51
-4 25
3.41
-0.56
-0.28
0.00
0.00
0.00
-182.62
-10.79
4.30
0.11
-0.01
0.00
0.00
0.00
0.00
-.00001
0.00000
0.00000
0.00000
0.00000
-2623.81
1092.96
465-61
1022.92
216.35
0.
0.
246.60
0.
138.34
-35.89
0.
3-60
59. I*
34-60
0.
-64.56
9.03
-2.30
-115.94
-63- 26
-65.71
-59.47
8 . 15
aaa.80
-96.77
-32.37
9.99
-0.85
4.17
-1325.71
0.
-37.29
-12.93
0.61
-0.07
0.21
0.
-3.49
-5 62
1.60
-0.30
-0.08
0.00
0.
37.62
2.09
-0.10
0.02
-0.01
0.00
0.00
0.
-0.33
2.35
-0.01
0.05
0.00
0.00
0.00
0.00
-23.05
-0.28
1.52
1- 74
-0.05
459*06
5. 64
0174
0.50
0.2s
-0.03
0.01
-943.90
-6.86
-
5.03
- 0 - 67
-0.29
0.00
0.00
0.00
-175.90
-10.58
4-40
0.14
-0.01
0.co
0.00
0.00
0.00
0.
0.
287.03
0.
162.95
-36.92
0.
33.93
69.a~
37.40
0.
-74.86
7.84
-1.94
4.28
0.
-26.69
-10.78
1-61
-0.06
0.22
0.
-24.17
-6.86
2.06
-0.28
-0.09
0.00
0.
99-86
2.53
0.00
0.04
-
-0.01
0.00
0.00
0.
13.78
1.35
0.12
0.06
0.00
0.00
0.00
0.00
*Cosine and sine coefficients (commonly denoted A; and BY) o f the ocean
function 8,modified ocean d e p t h x , and ocean depth h, respectively, in the
Balmino et al. (1973) normalization, listed consecutively through degree and
non-negative order 8.
the tide to vanish on land would have contaminated the T;
through a Gibbs effect (see Dickman & Steinberg 1986;
Dahlen 1976; Estes 1980). In order to determine the total
tide height at any ocea?ic location, we need merely compute
the harmonic sum for T at that point. In contrast, harmonic
coefficients inferred from tide-gauge data, finite-difference
tide theories, or satellite observations represent the tide's
effects integrated over the oceans. For comparison with such
coefficients our solution T must be 'truncated', through
multiplication by the ocean function 6.The truncation is
easily achieved computationally using our spherical harmonic approach; we have
(Dickman 1988a).
Equation (14) points to a general difficulty with
theoretical prediction of tide coefficients. Most tidal theories
share the common problem of generating non-zero G,
though for differing reasons (see e.g., Agnew 1983;
Schwiderski 1980); yet,
should vanish if the tide
conserves mass globally. Clearly the existence of a net tidal
mass will contaminate the determination of truncated tide
coefficients, whether through (14) or implicitly, as when
numerical solutions are integrated over the oceans.
Incidentally, the discrepancy found between two finitedifference predictions of the principal semi-diurnal tide
harmonic (discussed in Schwiderski 1983) may be a result of
such contamination. Table 2 illustrates the effect in our
solutions for the case of the principal fortnightly tide. It
should be noted that zonal tides approaching equilibrium
(even only roughly) should have largest Ti coefficients, and
TY should be larger than T:. From Table 2 we see that the
original solution does not follow this rule but the truncated
Table 2
Truncation Examples for M
f
*
Truncated Amplitude (cm)
rz
excluding '
l
'
:
T"
Amplitude (cm)
To
-2
1.41
0.91
0.96
L;
0.24
0.19
0. I 4
0.15
0.28
0.25
--I
including
-
*Solutions, converted t o Lambeck [19801 norqalization, are for oceans
with friction coefficients A = 1.5~10' cm Isec, P
1.5xlO-'
6ec-l.
462
S. R. Dickman
e
longer. Curiously, inclusion of
in the truncated
coefficients causes the tide to appear more dynamic at all
periods (see also Table 8).
Given the differences between our theoretical model and
numerical tide m o d e l e t h e latter include more detailed
bathymetry, but can only account for oceanic loading and
self-gravitation approximately, and they implicitly include
%--one might hesitate to compare solutions. Furthermore,
as discussed in Schwiderski (1983), harmonics derived from
truncated solutions are difficult to interpret reliably.
Nevertheless, our theoretical results and the numerical
results summarized in Table 3, from Schwiderski (1982) and
Carton (1983), seem to be in reasonable agreement for Mf
and M,. There are discrepancies for S,, but it is not clear
why the pure ocean tide should exceed equilibrium so
strongly at a semi-annual period, as implied by
Schwiderski’s solution.
Differences in the amplitude predictions of all three
theories suggest an intrinsic accuracy of roughly 10 per cent
or better, except for Ssa.
Our results compare poorly with satellite-based inferences
of @ at seasopal periods, which is to be expected since a
variety of meteorological phenomena with semi-annual and
annual periods contribute to the satellite signals (see
Christodoulidis et al. 1988); in fact, the differences could be
used to estimate the total atmospheric and groundwater
zonal seasonal excitation. There is also a discrepancy at
monthly periods; most likely the satellite determination is at
fault, since even with dynamical behavior M, should be
much more than 30 per cent of static. The satellite estimate
of Mf is, within its uncertainty, compatible with all
theoretical predictions.
The average Mf and M , tidal admittances for the Pacific
basin, excluding low-graded data, were taken from Luther’s
(1980) extensive data analysis and multiplied times the static
solutions do. However, the truncated solutions can differ
significantly depending on whether
is excluded from (14).
Non-zero
also leads to ambiguity in the determination
of total tide heights, depending on whether it has been
included in the harmonic sum. In this work, our preferred
solutions when truncation is necessary or total tide heights
are computed are those in which
has been excluded. A
more rigorous resolution of this problem-possible only
with a spherical harmonic approach-would be to include
constraints on
within the solution. This possibility is
currently being explored.
It will be instructive to compare our tidal solutions to the
ideal, static tide. Static solutions to the tide equations (1)
are given by A = 0 or
e
c
As discussed in Dickman (1988a), these solutions differ from
traditional static constructs for non-global, loading and
self-gravitating oceans (e.g. Dahlen 1976), in that they do
not include a small correction (the ‘Darwin’ term) to insure
global mass conservation. On the other hand, actual
dynamic solutions also do not include additional, global
mass conservation constraints (their
fail to vanish), so the
comparison will be fair.
c
Comparisons
through degree and order 5 using 36
We have solved for
tide equations and 36 constraint equations. Our solutions for
1-1
are presented in Table 3; for consistent comparisons,
twice
are actually listed, following an apparent
tradition regarding zonal tide coefficients (MERIT standards
1983; Christodoulidis et at. 1988). As expected, the tide
becomes more nearly static as the forcing period becomes
Table 3
Amplitude Comparisons*
9-2,
tide
Hf
‘m
‘sa
‘a
Nodal
static
solution
this
work**
n=O
Christodoulidis
et al. [19881
Schwiderski
Carton
Luther 119801
2.21
.92
(1.82)
1.80
2
0.41
1.70
1.91
1.75
L 0.38
1.17
.ll
(1.06)
0.36
2
0.50
1.06
1.04
1.16
5 0.55
1.03
.02
(0.98)
1.68
2 0.72
1.24
-_
--_
0.17
0.17
(0.16)
2.44
0.77
-_
-_
_--
0.93
0.93
(0.88)
-_
__
--_
-__
lx;l,
*All amplitudes are twice
presented in the Lambeck [1980]
normalization; units are cm. Values from Christodoulidis et al. [1988] are
derived from satellite observations, while those from Luther 119801
represent sea level data (Pacific only); values from Schwiderski [C. Goad,
H. Eubanks, pers. comm. 19881 and Carton [1983; see Herriam 19841 are
predictions from finite-difference theories.
**using same friction coefficients as in Table 2. These truncated harmonics
our preferred
are computed excluding To (numbers without parentheses
estimates) and i n c l u d i n 2 T O (numbers within parenthesea); harmonics found by
the other researchers l i s z d here implicitly include
--
2.
Harmonic approach to luni-solar tides
463
Table 4
Phase Comparisons*
P-2, n-0
tide
this
work**
Mf
-1.5
(-1.6)
Mul
'sa
Nodal
-24.6
i4. I
-2.1
Schwiderskj
Carton
2 13.1
-18.0
-7
-15.0
2
-11.1
-4
-17.9
-48.3
--
---
--
--
---
--
---
76.6
Luther 119801
13.6
2
28.1
(-2.3)
-0.5
-46.9
5 26.4
(-0.5)
-0.2
(-0.3)
a'
Christodoulidis
et al. [19881
-239.0
2
18.8
-0.0
(-0.0)
"Phases are in degrees, relative t o equilibrium phase; negative values
indicate phase lags. See Table 3 for references.
**See Table 3 for detalls.
l@l
listed in Table 3 to yield estimates of the actual Mf and
M, Iml.Within the uncertainties of the observations, all
three tidal theories appear to generate acceptable
predictions for those tides. It might be noted that some of
Luther's results were among the tide gauge analyses
hydrodynamically incorporated into Schwiderski's (1982)
predictions of M, and M,,
As might be expected, phase predictions fare much less
successfully (see Table 4). There is no indication of what
the real phases might be. Once again, the satellite values at
seasonal periods reflect the predominant meteorological
rather than tidal forcing. Interestingly, there is a suggestion
that Schwiderski's semi-annual tide represents a meteorological rather than gravitational tide (see Table 3 also),
perhaps a result of his incorporation of actual tide data
through hydrodynamical interpolation. For all tides, the
phase lags predicted by our spherical harmonic theory are
unusually small; since our accurate accounting of oceanic
self-gravitation and loading should not cause the phase lags
to be reduced so drastically, the predicted lags may indicate
some secondary problems with our theory or its
implementation (see discussion of Tables 8 and 9 below).
Complete assessment of a tidal theory should, of course,
not be based solely on the value of a single harmonic
component. Although the degree-2 zonal tide height
coefficient will dominate in all long-period tides, the other
coefficients may not be very small and may better indicate
the extent of a dynamic response. Table 5 lists various other
coefficients of our tidal solutions for M,,M,, and S,,, plus
the total tide height at two selected locations, mid-Pacific
( 8 = 80", A = 200": point A) and northern Pacific (8 = 40°,
A = 200": point B). In a personal communication (1988),
E. Schwiderski has stated a reluctance to place confidence
limits on his Mf tide predictions, because of their small
amplitudes. On the other hand, to the extent his predictions
depend on actual tide data, we might estimate conserva-
Table 5
Extended Comparisons with Schwiderski Tide Predictions
T"
tide type
Mf
m'
'sa
this work*
-*Schwiderski**
total tide***
this work
Schwiderski
(1.82)
(0.38)
(0.56)
(1.00)
1.70
0.19
0.60
0.59
A : 1.35
1.11 (1.06)
0.08 (0.13)
0.26 (0.29)
0.40 (0.55)
1.02
0.03
0.23
0.37
1.92
0.29
0.50
0.75
(0.98)
(0.08)
(0.26)
(0.50)
(0.60)
1.1
B: 1.38 (2.13)
1.5
1.06
0.06
0.20
0.46
A: 0.78 (0.36)
0.6
B: 0.69 (1.11)
1.2
1.24
0.06
0.33
0.52
A : 0.71
(0.34)
0.8
B: 0.59 (0.97)
2.0
*See Table 3 footnotes for details. Truncated coefficients listed are n-0;
1-2,3,4,5 respectively.
**as computed by C. Goad [M. Eubanks, personal communication 19881,
presented in the Lambeck 119801 normalization.
1-200';
B is 40° north of A.
***Point A is central Pacific: 8-80',
Schwiderski magnitudes for Mf were estimated from his tide map [Schwiderski
19821; for M and Ssa they were kindly provided by E. Schwiderski [personal
communicatioz 1989 I.
464
S. R . Dickman
tively that error bars for Mfwould be at least 0.2-0.3cm
(see Luther 1980, table 3.3). We see from Table 5 that, on
an absolute scale, differences between our coefficients and
Schwiderski’s are typically -0.11 cm (0.12cm if q; is
included) in amplitude; relatively speaking, -30 per cent
(24 per cent). Clearly, our predictions and Schwiderski’s are
in the same ‘ball park’; the larger coefficients are reasonably
similar but smaller coefficients like lC@l are not. Significant
differences appear in the total tide height; curiously, our
results agree better when we exclude
(differences are
-0.2cm or -14 per cent) than when we include it
(differences are -0.6cm or -56 per cent).
A reasonably good agreement with Schwiderski’s tide
predictions is probably the best that can be expected,
because his predictions are influenced by actual, predominantly coastal, tide data. Although his approach
guarantees a match between theory and observation at
prescribed coastal locations, such a guarantee does not
extend to the tide world-wide. For the semidiurnal (M2)
tide, comparisons with deep-sea tide data that had been
excluded from hydrodynamical interpolation revealed a
remarkably good agreement, to within a few per cent in both
amplitude and phase (Schwiderski 1983). But in general,
tidal dynamics should differ from coast to mid-ocean, and
the mid-ocean tide need not closely resemble any coastal
tide. Furthermore, tidal dynamics may vary with frequency
differently at coasts than in mid-ocean. Evidence of the
disparity between coastal and deep-ocean tides at long
periods is described by Dickman & Preisig (1986), who
argued that successful theoretical prediction of the North
Sea pole tide may require consideration of non-linear tidal
dynamics within shallow seas; and by Dickman (1988a), who
concluded that local coastal effects must be responsible for
apparent enhancement of the pole tide at Honolulu, and
that globally the tide must be much less enhanced.
Schwiderski (1983) concludes that ‘locally confined errors’,
due for example to coastal distortions, average out and do
not significantly affect his harmonic tide coefficients.
Nevertheless, for accurate prediction of long-period
luni-solar tides in mid-ocean, and accurate determination of
tidal effects on Earth’s rotation and satellites, alternatives to
hydrodynamical interpolation may be preferable.
Rotational effects
Ocean tides can modify the Earth’s rotation, through the
changes they generate in Earth’s inertia tensor and the
relative angular momentum their tidal currents create. Any
tide, zonal or otherwise, can affect all three components of
the rotation vector in non-global oceans. We take the
change in the angular velocity vector to be from Szi to
shii Q2; m = m,+ im, represents the dimensionless
wobble amplitude, while m3 relates to the change in the
length of day. Because the tides possess an e x p ( i d ) time
dependence, each component of 6 will be proportional to
exp ( i o t ) and/or exp (-iu*t). For polar motion we can write
m = mp exp (id)+ m R exp ( - i a * t ) , while for the axial
component we have m3 = mp exp (id)
+ m Rexp (-ia*t)
with m p and m R complex conjugates. Appendix 2 details the
relations between m, and the tidal spherical harmonic
coefficients [6T]:,
V;, and u;, and explains how rotational
decoupling of the fluid core and mantle loading are
accounted for. Table 6 lists the coefficients mp, m R of the
polar motion and axial speed increment induced by tidal
inertia; Table 7 lists the corresponding values resulting from
tidal relative momentum.
Overall, the magnitude of the tides’ inertia effects on the
length of day appear to scale according to the forcing
this is to be expected when tidal dynamics
amplitude &I
play a minor role. Similarly, we find the axial response to be
nearly instantaneous, i.e. the phase lag in m3 is small
(because 9 m
<< %k
As zonal tides approach
equilibrium, we would expect their T:’ components to
diminish significantly (but not, in non-global oceans, to
zero); however, the induced wobble is also affected by the
nearness of the tidal frequency to the Chandler resonance
+
{e} {e}).
Table 6
Effects of Tidal Inertia on Rotation*
axial
tide type
equatorial
%
3
ZP
“f
-2.6 - 0. li
(2.2 - 0.11)
-0.8 + 0 . 2 1
(0.4 + 2.11)
1.0 - 0.31
(0.0 - 1.8i)
Mm
-1.5 - 0. li
(1.2 - 0.lf)
-0.9 + 0.01
(0.5 + 2.11)
1.0 + 0.01
(-0.1 - 1.7i)
-1.4 - 0.01
0.01)
(1.0
-8.8
0.9i
(4.2 + 18.8i)
-
-
sa
-0.2 - 0.01
(0.2 - 0.01)
‘a
-1.3
(0.9
Nodal
-10.9
-
1.21
( 5 . 0 + 22.9i)
- 0.01
- 0.01)
11.8 + 1.4i
(-5.2 - 24.41)
3.7
(-1.5
+
-
0.51
7.4i)
0.9 + 0.li
(-0.4 - 1.91)
10.4
(-4.5
+ 1.31
-
21-51)
*effects, including mantle loading and core de-coupling, efeduced when tidegeneratlng body is over Creenvich meridian. Units are 10
radians. Polar
motion prograde and retrograde amplitudes in mi$liarcseconda may be found by
multiplying the equatorial values times 2.06~10 ; ’prograde’ changes i n the
length of day $n milliseconds may be found by multiplying the axial values
times -8.62~10
Numbers in parentheses include To in truncation.
.
-0
Harmonic approach to Euni-solar tides
465
Table 7
Effects of Tidal Momentum on Rotation*
axial
tide type
id
m'
5sa
a'
Nodal
equatorial
-0.2
+
0.11
-0.1
3
+ 0.6i
-0.0
+
0.01
-0.1
+
-0.0
+
0.
ZP
oi
-0.0 + 0. 01
0
+ 0.01
0.41
SR
-
0.
oi
0.5
0.3 + 0.21
+ 0.71
0.1
+
0.31
0.0 + 0.41
0.0
+
0.01
-
0.0
+ 0.01
0.0
-0.0
0.01
*See table 6 for comments; note, however, that for tidal relative angular
momentum there is no mantle loading, and truncation of the velocity
coefficients is unnecessary.
(see Appendix 2). For the five long-period tides considered,
the resonant effects alone are in the ratio -15: -31: 319: -2376: +463 (Mf:M,:S,,:S,:nodal).
Considering also
the relative forcing amplitudes, it is not surprising that the
nodal tide produces the largest wobble of all zonal tides.
With the exception of the annual tide, the induced polar
motions have roughly equal prograde and retrograde
components, and are thus highly elliptical.
The effects of tidal currents on the length of day are
negligible for all five tides. For all, u; are generally smaller
than v;; nevertheless-presumably because of the Chandler
resonance-the polar motions induced by fortnightly and
monthly tidal currents represent a significant fraction of the
total induced wobbles. Tidal velocities decrease drastically
at very long periods, so V; and u; generate negligible polar
motion for semi-annual, annual, and nodal forcing.
In all cases we find the rotational effects of long-period
tides to be small. The prograde and retrograde wobble
amplitudes are each about 1/4 marcsec (1/2 marcsec if
calculations include
for the nodal tide, smaller for the
other tides. The maximum change (real part) in the length
of day, for the fortnightly tide, is only +0.04msec
(-0.04msec). Such effects, which for lod are roughly
10-15 per cent as great as those of the solid body tides, are
at present marginally detectable using space-geodetic
observational techniques. It is interesting to note that our
predicted theoretical effects can differ significantly, both in
magnitude and sign, depending on whether
is included in
truncation; predictions from other, numerical, tidal theories
would be similarly uncertain.
c)
effect (static tides, of course, are independent of
bathymetry).
The ability of bottom topography to generate dynamic
behaviour is also demonstrated in _thetide current velocities,
computed from the solutions for T using equations (7). The
maximum zonal velocity coefficient (v;), for example, is
1.7 x lO-'cm s-' for the fortnightly tide in flat oceans,
3.0 x lO-'cm s-' in realistic oceans; for the nodal tide
bathymetry causes max {IV;l} to increase from 9 x lop6 to
4x
cm s-'.
From the tide-governing equations, equation ( l ) , we can
for large j , s) to
expect high-harmonic bathymetry
influence low-degree tidal characteristics (e.g.,
though
with diminishing effect as the degree and order ( j , s) are
increased. Our solutions, as summarized in Tables 3-5, are
based on an ocean model that does not include bathymetry
of degree 25 or greater; conceivably this might partially
explain why they tend to be slightly closer to equilibrium
than the numerical solutions, both in amplitude and phase.
Table 9 presents our solutions for Mffor a variety of eddy
viscosity values ranging from A = 0-A = 1013cm2 s-'; note
that estimates of A (horizontal) in the real oceans range
from -105--109cm2s-'
(Pedlosky 1979). It is clear that
the solutions for tide height coefficients are quite insensitive
to A, and change significantly only for unrealistically large
eddy viscosity. This result, which at first appears to be in
contrast to other tidal studies (see Schwiderski 1980),
actually follows from consideration of the original tidal
equations (1). The two frictional forces in the momentum
equations are essentially
(6
c),
-Pii+AV%
OTHER INV ESTIGATIONS A N D FINAL
COMMENTS
We close this article with a brief examination of the effects
of bathymetry and turbulence on long-period luni-solar
tides. Table 8 compares zonal tide coefficient amplitudes for
Mf and the nodal tide in oceans with realistic bathymetry
and no bathymetry (flat basins); for comparison the static
tide solutions are also shown. Clearly, for the fortnightly
tide bathymetry causes the tide to depart further from
equilibrium; for the 18.6-yr tide, dynamical behaviour is
already minimal and bathymetry creates no discernible
which become, for the degree-I order-n spherical harmonic
component,
[P + AI(I + l)/a2]Y;
times u; or V; [using the relation given between equations
(4) and ( 5 ) ] . For bottom friction P = 1.5 x
s-', as we
have considered here, the two frictional terms are
comparable in magnitude only if A 3 x 10" cm2 s-' (for
I = l ) , A 0.5 X lo9cm2 s-l (for 1 = lo), or A - 6 x lo6 (for
I = 100). Thus, an insensitivity to turbulent drag should be
expected for low-degree harmonics, which are the focus of
-
-
466
S. R. Dickman
Tnble 8
E f f e c t s of Bathymetry*
s t a t i c solution
t i d e type
Mf
Nodal
f l a t basins**
actual hasins
2.21
0.06
0.46
0.77
(2.22)
(0.05)
(0.45)
(0.74)
2.04
0.10
0.46
0.71
(1.94)
(0.20)
(0.52)
(0.97)
1.92
0.29
0.50
0.75
(1.82)
(0.38)
(0.56)
(1.00)
0.93
0.03
0.20
0.33
(0.94)
(0.02)
(0.19)
(0.31)
0.93
0.03
0.20
0.33
(0.88)
(0.08)
(0.24)
(0.46)
0.93
0.03
0.20
0.33
(0.88)
(0.07)
(0.23)
(0.45)
*See Tables 3 and 5 f o r d e t a i l s .
**Depth w i t h i n b a s i n s is 3.679 itm b u t 8818 weight f a c t o r as i n " a c t u a l
b a s i n s " solution (ho-2.3 km) is used f o r boundary c o n d i t i o n s .
Table 9
E f f e c t s of Turbulence*
A-1.5~10~
A-1.5~10~
A-0
Non-turb.
**
P-2, n-0
2.122
(2.058)
1.916
(1.823)
1.917
(1.824)
1.917
(1.824)
2.138
(2.313)
1-3.
0.081
(0.136)
0.287
(0.383)
0.286
(0.382)
0.286
(0.382)
0.078
(0.119)
n-0
0.525
(0.551)
0.499
(0.559)
0.499
(0.559)
0.499
(0.559)
0.361
(0.248)
1 - 5 , n-0
0.790
(0.966)
0.749
(1.005)
0.748
( 1.004 )
0.748
(1.004)
0.695
(0.210)
9-4,
n-0
*See T a b l e s 3 and 5 f o r d e t a i l s .
**Nan-turbulent t h e o r y adapted from Dickman 119888, b l , w i t h same bottom
f r i c t i o n , bathymetry, and w e i g h t i n g a s i n t h e t u r b u l e n t s o l u t i o n s .
the present work; but the value(s) chosen for the eddy
viscosity may be crucial for high-resolution studies, such as
those based on finite-difference algorithms.
Table 9 also presents the solution for Mf in non-turbulent
oceans, using the tide theory developed in Dickman
(1988a,b) but with all numerical parameters excluding A the
same as at present. It is uncertain to the author why this
solution fails to match the turbulent solution for A = 0. At
worst it indicates an accuracy of the turbulent code no better
than -0.2cm (0.5cm if computations include g).
Conceivably the poor tide phase predictions in Table 4 also
reflect inaccuracies to some extent, in the smaller, imaginary
parts of
However, it is possible that the disparity is
rather to be expected, as a result of the different structure of
the tide equations depending on whether turbulence is
included or not; thus, the case of A-+O may represent
'weak turbulence' rather than a true non-turbulent state.
This possibility is speculative and, despite favourable
comparisons with other tide theories, the true accuracy of
our spherical harmonic approach remains uncertain.
Even more important, perhaps, are the uncertain effects
of non-zero
on our results and on those of all other tidal
theories. As shown in this work, inclusion or exclusion of T",
causes individual truncated T; to vary in magnitude by at
least 0.1-0.2 cm; the total long-period tide height may differ
by -0.5-1.0 cm. Rotational effects can differ by 0.05 ms in
time or 0.5marcsec in polar motion amplitude. At these
r.
e,
looms threateningly in the
levels of accuracy, the role of
background of all long-period tidal theories.
ACKNOWLEDGMENTS
Discussions with S. Klosko and D. Christodoulidis were
invaluable. Conversations with M. Eubanks, Y.-S. Nam, D.
Sandwell and E. Schwiderski, also J. Barker, R. Dickman,
C. Goad, J. Preisig, D. Rapp, D. Robertson, I. Schapiro,
and D. Steinberg, were appreciated. I am indebted to J.
Marsh for providing the DBDB5 digital bathymetry tape.
Computations were performed on the SUNY-Binghamton
IBM 3090 vector facility; for bathymetry harmonics the
Cornell NSF IBM 3090 was also used. The consulting staffs
at both institutions were helpful as usual. This research was
supported by NASA grant NAG 5-145, with additional
computational support by a SUNY-Binghamton Provost
grant.
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Wahr, J. M. & Sasao, T., 1981. A diurnal resonance in the ocean
tide and in the Earth's load response due to the resonant free
'core nutation', Geophys. 1. R. astr. SOC., 64, 747-765.
Wahr, J. M., Sasao, T. & Smith, M. L.. 1981. Effects of the fluid
core on changes in the length of day due to long-period tides,
Geophys. 1. R. astr. SOC.,64, 635-650.
APPENDIX 1
D e t a i l s of matrixed t i d e equations
In the derivation of the tide equations, we require the
following definitions.
For equation (5):
d,, = A + {a2[ia+ P ] + Af(I + 1)}(1- C;+',,C; - eC;-,)
il,, = {a2[ici+ PI + AI(I + l)}(-C;C;+,)
zl,, = {az[ia+ P I + AI( I+ ~ ) } ( - e c - ~ )
gl,
+
+ c;+,c;+ c;c;-,- l)]
= [Cy][2iAn ~ Q u ~ ( ~ + ~ C ; + ,
ill,, = [c][2iAn + 2Qa2(C;+,C;
+ c;c;-,+ e7-1c;-2 - l)]
93, = 2 s ~ a ~ C ~ c ~ + , c ~ + ~
,+3!,,= 2 ~ a ' C ; e _ , C ; _ ,
For equation (6):
S. R . Dickman
468
For equation (8):
d"
- cp eior
+
e-ru*r
R
~n=iau[l-C~+l~;-~;~~_ll
x2= - i a a ~ ; _ , ~ ; - ,
x(-*) =
ark = 0 if r = 0
-i~e+~e+~;
a,, = E - ( L , N)D:;
@in=
a , ( - , ) = --E+(L,N)D]-"
- & + ( j , S ) D ; ~ - '+ [-Eo(j, s)
+ E()(L,N)]q+ L ( j , s)D;-l
a - l l = ~ - ( ~ , ~ ) a~- , ;( - l , = - - E + ( L , N ) B ; s
a - l , = - - ~ + ( j ,~ ) l j ~ - +
~ -[ E', ( j , s)
+ E(,(L, N
) I +~c ( j , s)Bj-'
Definitions of C, D, and E may be found in Dickman (1985,
1988a). In the derivation of the boundary constraint
equations, we require the following definitions.
For equation (10):
Length-of-day is affected by the change c j 3 in the oceanic
polar moment of inertia and by the axial angular momentum
/3 associated with tidal currents. Using orthogonality and the
definitions of these quantities from Lambeck (1980), we
have
The rotationally symmetric, low-viscosity fluid core does not
participate in a spin-up of the mantle. Letting C, denote the
mantle's polar moment of inertia, the dimensionless tidal
increase m, in axial angular velocity is then m 3 =
m eexp (iat) (mp)*exp (-iu*t) where
+
( g a [ \ / 3 (1 + k & ) e
c m
3
- (1
+ k;)c]
+ 2sz
$(h;)*)
and where the load Love numbers k&, k; account for the
change in mantle inertia caused by tidal loadi.1g.
Polar motion is affected by the oceanic products of inertia
c = c I 3 ic,, and equatorial tidal angular momentum
e = + i/2. We find
+
or l + p = e v e n
n +q
and 1 + p =odd
=
-1
are defined in Dickman (1988a); the
and where 'CJ,
derivation for ePfollows the same procedures, described in
the text, employed for Q. Note that t', involves h rather
than h. The expression for eR is the same as that for ,'t
except u;, V; are replaced by their complex conjugates and
Qm is replaced by (- l)nQ'-n)".
Wobble of the mantle engenders a slight response of the
fluid core due to the non-zero flattening f, of the
core-mantle boundary. Including mantle loading and core
near-decoupling, and also accounting for the rotational
deformation of the Earth accompanying the tidal forcing,
the resulting polar motion is described by m =
mp exp ( i d ) mRexp (-io't) where
+
d3 = pwa3Re {eimC $[$*}.
3
A;;""
n + q # -1
N ) ~ J
T i d a l effects o n rotation
l p a3
..
X , ( m )
APPENDIX 2
me=
lo
I
a,+ = a10 - &n(L
wq
u l - = a-lo - E,,(L,
where
and where K = 1 -f,, A,,,, A, are mantle, core equatorial
inertia, and 00 is the observed Chandler frequency (taken to
correspond to a Chandler period of 433.2 days).
It should be remembered that, in all the above
expressions, truncated versions of T; should be employed.
Finally, Merriam (1980) and Wahr et al. (1981; see also
Sasao et al. 1977) have discussed the effect of core
decoupling on the solid-earth Love numbers. Because of the
relatively greater deformation of the fluid core, for example,
the load Love number k; is reduced in magnitude by 20 per
cent (Y.-S. Nam, pers. comm. 1988); however, the
modifications of h; and k; in general are not known. For the
sake of consistency with the tide solution, the rotational
effects derived here were computed using the non-decoupled
load Love numbers.