Geoplzys. J. R . astr. Soc. (1980) 61, 479-488
Induction in arbitrarily shaped oceans - V. The
circulation of Sq-induced currents around land masses
D. Beamish (;eomagnetism unit, Institute of Geological Sciences,
Murchinson House, West Mains Road, Edinburgh EH9 3LA
R. c. Hewson-Browne
Department o f Applied Mathematics and
Computing Science, The University o f Sheffeld, Sheffield SI 0 2TN
P.c. Kendall Department of Mathematics, university o f Keele, Keele,
Staffordshire ST5 5BG
S . R.C . Malin Geomagnetism unit, Institute of Geological Sciences,
Murchinson House, West Mains Road, Edinburgh EH9 3LA
D.A. QUinney Department o f Mathematics, university of Keele, Keele,
Staffordshire ST5 5BG
Received 1979 August 9
Summary. The model ocean used in previous papers in this series is refined
to allow for five separate land masses. The ocean is taken to be of uniform
depth, and insulated from the mantle which is modelled by an infinitely
conducting core at uniform depth. The fast method of solution introduced in
Paper IV enables the electric currents induced by Sq to be calculated on a
2” x 2” global mesh. The ‘outer’ solution away from the coast is calculated.
The ‘inner’ solution will be a coastal correction different for every coastal
configuration. The current is allowed to circulate round isolated land masses,
giving markedly different patterns from those in Paper IV. For some
harmonics it appears necessary to differentiate even between Australia and
New Zealand, although the world-wide effect of the remote island of Spitzbergen is minimal. Current distributions are shown for periods of 6, 8, 12
and 24 hr, and a value is set on the current streaming between the various
land masses for all the Legendre harmonics considered.
1 Introduction
In Paper I (Hewson-Browne & Kendall 1978a) a problem of induction in an infinitely
conducting model hemispherical ocean above a concentric underlying perfectly conducting
sphere was solved analytically by a new method. The form of solution indicated a line of
approach to similar problems involving oceans of arbitrary shapes. This involves finding
‘inner’ and ‘outer’ solutions, as in the method of matched asymptotic expansions (Van
Dyke 1964). In Paper I1 (Hewson-Browne & Kendall 1978b) an edge correction was
D. Beamish et al.
480
developed for such infmitely conducting oceans and its accuracy validated on wellestablished models. In Paper 111 (Hewson-Browne 1978) equations were formulated for a
model ocean of finite electrical conductivity, and aligned with all previous formulations of
the problem. Meanwhile Quinney (1979) has solved the one-sided edge-effect integral
equation proposed by Hewson-Browne & Kendall (1976) for a model ocean of finite conductivity. In Paper IV (Beamish et al. 1980) we presented world curves of electric current
for the 'outer' solution on a uniform ocean, using circulation conditions round islands
equivalent to those of Bullard & Parker (1970), namely, that the current stream function has
the same value on the entire coastline. In this paper (Paper V) we present world curves
obtained by taking proper account of circulation of current around islands, and compare
these with previous results. We have been fortunate in being able to use short-cuts, as
explained in the text. These have enabled us to make all calculations in Papers IV and V on
a 2" x 2" grid. Papers IV and V also use an ocean of uniform depth 4 km,and the inland
regions have been divided into five islands: Spitzbergen, New Zealand, Australia, Antarctica
and America-Europe-Asia (Mainland). Spitzbergen might be omitted from further worldwide models, as it has a small overall effect.
2 Theory
Electric currents are supposed here to be induced in the ocean by various harmonics of Sq.
The Sq magnetic variation has been analysed recently by Malin & Gupta (1 977) and we use
their notation and analysis. Consider induction by a single harmonic whose external
potential is ( ~ / a ) ~ 3 f ' :where
,
Y denotes radial distance from the Earth's centre, a denotes the
Earth's mean radius, and the surface harmonic 3f': has the form
x aP,"(cos
e) sin mG exp (iwt).
where if t" is universal time,pt* = wt and p = 1 , 2 , 3 or 4.
In Paper 111 (1978) Hewson-Browne showed that the current stream function $ for the
outer solution approximately satisfies
V , . ( p t i y n ) V s $ = -ip;'ynn(2n t l)flnm
(2)
where 0,is the dimensionless surface gradient
O, = [o, alae, (sin e)-'a/a~l
in spherical polar coordinates r, 8 , G and
y n = ( l - k Z n + ' ) r / ( 2 nt I),
(3)
where r = w ~ ~ p Here
~ a k. g is a constant scale value for K , the integrated surface conductivity of the ocean, and po is the permeability of free space; also, p = K ~ / K Equation
.
(2)
should be solved subject to the following conditions derived by Hewson-Browne (1978)
in Paper 111:
around every separate land mass C,, where u = 1 , 2 , . . . ,M - 1 and a/aN denotes differentiation along the direction of the normal in S to C., As $ will then be unique apart from an
arbitrary additive constant, it is convenient to think of ,
C as the 'mainland' on which
$ = 0. The total number of land masses or islands is also M .
Induction in arbitrarily shaped oceans - V
Consider now the case of uniform surface conductivity
becomes
K
=K
~ Then
.
48 1
equation ( 2 )
This formulation has the advantage that we can solve
e)
0," 6: = P," (COS cos m4
V : G ~= P," (cos
e ) sin m4
giving
For convenience in solving equations (7a) and (7b) we substitute
n(n t 1 ) 0; = c r
P," (cos e) cos m@
(94
P," (cos e ) sin m@.
(9b)
-
and
n(n t 1)6; = s r
-
Then the basic problem becomes one of solving
v:cr
=
v:s,"
=0
with the boundary conditions on the coastline
c," = P," cos mq5
and
s," = P," sin mq5.
(11)
I)= 0 on every coastline, dealt with in Paper N.
To satisfy the circulation conditions equation (4) we compute functions Hq(e, q5), q =
I , 2,. . . ,M - 1, such that
This would solve the restricted problem of
Q2Hq = 0
(12)
with Hq = 6,, on C,, u = 1,2,. . .,M , where 6,, denotes the Kronecker delta. Define
Then equation (4) will be satisfied by computing constants X y n , p y n for q = 1,2,. . .,M - 1
and all relevant m and n as follows. Solve the sets of linear equations
1 9 u ~ M - 1
q=1
482
D.Beamish et al.
and
M- 1
PY"f"(H, 1 = f&,"
1 G u < M - 1.
)
q=1
If we now define
n(n + 1) u;(e, 4) = c,"(e, @) -~,"(cos
M-
e ) cos m@- C
1
h,mnHq(e,$)
(15 4
q=l
and
n(n + 1) 2.3;(e, $) = s,"(e, 4) -P,"(COSe ) sin rn4 -
M-l
C
g n H q ( e ,@),
q=l
equations (7) and (8a, b) provide the stream function $, provided that 0; is replaced by
U r and G; by .
:
3
2 Thus
3 Results and discussion
Calculations have been made for all the Legendre coefficients obtained by Malin & Gupta
(1977). The matrix of coefficients f,(H,) is common to all and is shown in Table 1.
According to Green's reciprocal theorem f,(H,) = f q ( H , ) , so the matrix should be
symmetric. The closeness to symmetry offers a useful check upon the calculations. Table 2
shows the calculated values of
and by" for the various harmonics.
Table 2 permits comparison of the effects of circulation of electric currents around the
various land masses. The various entries "A:
and p r n represent the values of the stream
function $ on the islands, as compared with 9 = 0 on the mainland. Increments in 9 are
proportional to the total current flowing between the land masses. Thus, as might have been
expected, Spitzbergen has a negligible world-wide effect. Otherwise, it seems that on this
model even Australia and New Zealand have been justifiably separated. It will be of great
interest to compare the cases when the conductivity is taken to be non-uniform in due
course.
In Paper IV we estimated the contribution of the ocean currents to the total induced
part of the geomagnetic variations by comparing the modulus of the Malin & Gupta (1 977)
internal (Z) coefficients of the dominant spherical harmonic at each frequency with the
corresponding ocean-current coefficient (O), deduced by spherical harmonic analysis of the
oceanic stream function and converted to nT using the relations given by Chapman &
Bartels (1940). We also calculated 01,the contribution due to currents induced in the core
by the ocean currents. In Table 3 we give the corresponding quantities for the five-island
model, together with the one-island results in parenthesis. It is clear that the inclusion of
Ayn
Table 1. The matrix of vducs fv(Hq),
u, q = 1,
f,(H,)= 0 (v + 1) as Spitzbergen is so remote.
uls
1
2
3
4
1
-4.32904
0.00000
0.00000
0.00000
2
0.00000
-2.16427
1.48485
0.18736
2 , . . .,4. Note that
3
0.00000
1.48432
- 6.5 1I81
1.03901
4
0.00000
0.18548
1.03908
-5.13433
Induction in arbitrarily shaped oceans
Table 2. The coefficients A T n and
entries are not recorded.
Spitzbergen
p y
New Zealand
-
V
for equations (15a, b). Zero
q=2
Australia
4=3
q=4
0.01241
- 0.46265
-0.30702
-
- 0.00141
- 0.49390
0.00773
0.03593
0.06155
0.13964
-0.36895
0.19159
- 0.06921
-0.14265
0.04703
0.76807
q=
1
0.00588
0.01555
-0.00077
0.00179
0.06555
0.75143
Polar Cap
0.51888
0.54912
-0.00824
0.24785
0.41999
-0.38745
0.26751
- 0.36131
0.31186
0.02420
0.02647
0.00095
0.00527
- 0.00014
0.00015
0.09845
0.21559
0.09964
-0.52481
0.03647
-- 0.32358
- 0.00972
-0.27692
0.18055
0.06468
0.03710
0.42894
0.05496
0.33383
-0.08322
0.11860
0.09534
-0.03279
-0.1 1135
0.00083
-0.07366
0.66034
0.05511
0.04132
0.0071 1
0.01209
0.00029
0.00080
. 0.00009
-0.000
I3
0.00218
0.00287
-0.00002
-0.00023
. -0.0001
1
-0.00008
0.00051
-0.00006
0.23808
0.0824
0.42177
0.03835
0.26650
0.08120
0.14791
-0.13457
-0.46748
- 0.14315
--0.397
52
0.07604
-0.05361
-0.02824
0.69723
-0.05410
0.32784
0.32639
0.16317
0.00056
-0.37121
-0.14258
-0.29494
- 0.27001
0.16042
-~ 0.19032
0.25973
-0.00062
0.37993
-0.07422
0.08897
0.16075
-0.181 24
0.02478
0.02151
0.15438
-0.13443
0.18824
- 0.01230
-0.1 6352
0.10323
-0.23337
- 0.04286
0.1 7933
0.08989
-0.05984
0.11398
-0.1 1643
- 0.1 7469
-
0.13789
0.13257
0.12105
0.05078
-0.76057
-
Table 3. Moduli (in units of nT) and ratios of harmonic coefficients.
Symbols are described in the text.
Period
24 hr
12 hr
8 hr
6 hr
Harmonic
Pi
p:
p:
p:
I
3.61
0.73 (0.47)
0.36(0.24)
2.31
0.90
0.16
0.20 (0.29)
0.11 (0.16)
0.17 (0.28)
0.05 (0.03)
0.20 (0.13)
0.10 (0.07)
0.09(0.13)
0.05 (0.07)
0.09(0.14)
0.19 (0.32)
0.10(0.16)
0.02(0.01)
0.30(0.17)
0.13 (0.07)
0
OI
OII
0111
483
484
D. Beamish et al.
=
.
a
9s'w*
0)
Figure 1. (a) World curves of the in-phase part of the electric current induced in the oceans for 24 hr
period. Circulation conditions and the five islands are taken into account as described in the text. 2000 A
intervals between contours. In all diagrams zero value contours are shown as broken lines. (b) The inquadrature part of induced currents for 24 hr period. 2000 A intervals between contours.
islands has a marked effect, changing the ocean contribution by typically 50 per cent.
However, the changes are not systematic, with the 24 and 6 hr terms showing an increase
and the 12 and 8hr terms showing a decrease. Overall, the ocean-current contribution
remains at about 20 per cent of the total induced part. The 8 hr term no longer dominates,
and it appears that streaming of electric current between the southern oceans is more
important than the resonance obtained previously.
Figs I(a, b)-4(a, b) show the electric current streamlines for the Sq periods of 24, 12,
8 and 6 hr ( p = 1, 2, 3, 4) in pairs of in-phase (t = 0) and quadrature [t = -n/(2p)] on a
2" x 2" grid. The current flows parallel to the streamlines in the direction indicated by
486
D. Beamish et al.
~ , , , , , , , , r , , , l , , i , , l , , , , , l l , , , , l , , , , , l , , , , , i , , , , , l , , , , , l , , , , , , , , , , , I , , , , ,asl
,_*I
?Em+
im
9J.v
w
IM/
m
mf
9x
>znE
I S
IWL
cb)
Figure 3. (a) In-phase part of induced currents for 8 hr period. 2000 A intervals between contours. (b)
In-quadrature part of induced currents for 8 hr period. 2000 A intervalsbetween contours.
are modified. The more realistic constraints applied in the present analysis allow current to
circulate around Antarctica and between New Zealand, Australia and SE Asia. The extent
of the modification over the one-island model (Paper IV) is a function of the morphology
of the inducing field, i.e. its period, amplitude and phase. Hence we observe significant
current flow between South America and Antarctica at a period of 24 hr (Fig. la, b) while
such current flow is absent at a period of 12 hr (Fig. 2a, b). As a general result, a major
modification of current circulation takes place in the oceans abutting Australasia where the
constraints provided by Antarctica, New Zealand, Australia and SE Asia perform a complex
coupling of current between the Indian and Pacific oceans.
We see from the large difference between the current patterns of the one- and five-island
models that it is quite unrealistic to constrain the islands to have the Same potential as the
48 7
Induction in arbitrmily shaped oceans - V
,’
,o
1
= ma
L
T
T
-
,
1
*m
” ’
,
,
mA
I
,
I
-
. , -,-----e,
n
1M
- -
-ta
- .- - ___._
~
9x
7 , ;
7
-1.-
@)
Figure 4. (a) In-phase part of induced currents for 6 hr period. 1000 A intervals between contours. (b)
In-quadraturepart of induced currents for 6 hr period. 1000 A intervals between contours.
mainland (America-Europe), and the present results represent a major step towards the
delination of the true ocean currents.
We hope to refine the model to take account of the variable depth of the ocean and the
finite conductivity of both crust and mantle.
References
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arbitrarily shaped oceans - IV. Sq for a Simple case, Geophys. J. R. astr. SOC.,60,435-443.
Bullard, E. C. & Parker, R. L., 1968. Electromagnetic induction in the. oceans, in The Sea, vol. IV, part I,
eds Bullard, Sir Edward & Worzel, J. L., 696-730.
488
D.Beamish e t al.
Chapman, S . & Bartels, J., 1940. Geornagnetisnz, Oxford University Press.
Ilewson-Browne, R. C., 1978. Induction in arbitrarily shaped oceans 111. Oceans of finite conductivity,
Geophys. J . R . astr. Soc., 55,645- 654.
Ilewson-Browne, R. C. & Kendall, P. C., 1976. Magneto-telluric modelling and inversion in three
dimensions, Acta Geodaet. Gcophys. Montanist, Acad. Sci. Ilung., 11 (3-4),427-446.
IlewsonBrowne, K. C. & Kendall, P. C., 1978a. 1: Some new ideas on induction in infinitely-conducting
occans of arbitrary shapes, Geophys. J. R. astr. SOC.,53,431-444.
llewson-Browne, R. C. & Kendall, P. C., 1978b. Induction in arbitrarily shaped oceans - 11: edge correction for the case of infinite conductivity, J. Geonnagn. Geoeiect., in press.
Malin, S. K. C. & Gupta, J . C., 1977. The Sq system during the International Gcophysical Year, Geoplzys.
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Geophys. J. R. astr. Soc.. 56, 119-126.
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