Download Electric Currents In the Ocean Induced By the Geomagnetic Sq Field

Geophys. 1. Int. (1991) 104,381-385
Electric currents in the ocean induced by the geomagnetic Sq field
and their effects on the estimation of mantle conductivity
Masahiko Takeda
Data Analysis Center for Geomagnetism and Space Magnetism, Kyoto University, Kyoto 606,Japan
Accepted 1990 September 17. Received 1990 September 17; in original form 1990 May 9
SUMMARY
Electric currents induced in the ocean by the geomagnetic Sq field are simulated for
a system which consists of a uniform geocentric sphere of finite conductivity and a
non-uniformly conductive thin-shell sphere representing the sea-land distribution.
The simulation results show that the induced currents have maximum intensity in
the northern Pacific Ocean at 22 hr UT and enhance the total induced fields by about
60 per cent. Even when the external current vortex is located above the Eurasian
continent, the internal current vortex is enhanced by about 10 per cent by the
existence of the ocean. Next we have checked the effects of the electric currents in
the ocean on the estimation of the mantle conductivity and found that ignoring the
currents induced in the shell gives a less conductive and shallower estimation of the
mantle conductor. Furthermore, an apparent dependence of the conductivity on the
depth appears because of the effects of the currents induced in the shell even if the
mantle is a uniform conductor. The conductivity and depth are estimated to be
0.1 S m-' and 500 km respectively from the data analysis for the 1 day period
variation field of the Pi mode if the effects of the currents in the shell are ignored,
but it should be 1S m-' and 780 km if the effects are considered.
Key words: mantle conductivity, oceanic currents, Sq.
1 INTRODUCTION
Since the decay time constant of the induced currents in the
Oceans is about 10 hr, it is expected that fairly large electric
currents are induced in the Oceans by the geomagnetic solar
quiet daily variation (Sq), with a the period of one or a few
cycles per day. However, currents are also induced in the
mantle at the same time and couple with those flowing in the
Oceans. Therefore, in order to examine the internal Sq field,
it is necessary to examine the response of the mantle-ocean
coupled system to the external field. Simulations for the
induced currents in the Oceans have been made using the
mantle model of the infinitely conductive sphere. For
example, Beamish et af. (1980a, b, 1983) simulated the
induction in realistically shaped oceans and examined the
contribution in each harmonic term due to the currents in
the oceans and in the mantle by the ocean currents. They
concluded that the effect on the 8 hr period p', mode Sq field
is the largest and that about 30 per cent of the internal
variation fields result from currents in the oceans.
Simulations by Hobbs & Dawes (1979) and Hobbs (1981)
showed that total induced currents are enhanced by about
50 per cent at its maximum by the presence of the oceans.
Takeda (1985) analysed the Sq field and found oceanic
effects on the internal Sq field.
Some researchers used the response of the internal to the
external Sq field for studying the mantle conductivity
assuming that the induced currents flow only in the mantle.
For example, Campbell (1987) and Campbell &
Schiffinacher (1988) estimated the mantle conductivity from
the amplitude ratio and the phase difference between the
internal and external parts of Sq. In these estimations the
phase difference is regarded to be caused purely by the
finiteness of the conductivity of the mantle. On the other
hand, Winch (1984) used a model which consists of a
uniform finite-conductivityshell representing the oceans and
a perfectly conductive core and estimated the conductivity
of the shell from the phase difference between the internal
and external parts of the Sq field. The results are
reasonable, which suggests that the phase difference may be
considered as the result of the finite electric conductivity not
of the mantle but of the oceans.
These two treatments contradict each other in that one
ignored the effect of the induced currents in the Oceans and
the other assumed that the phase difference is due to the
oceanic current. To examine the effects of the oceanic
381
M. Takeda
382
currents on the phase difference between the internal and
external Sq fields, it is necessary to simulate the electric
currents induced in the Oceans above the finitely conductive
mantle. In this study, we first simulate the induced currents
in a shell of which the conductivity distribution is
determined by the sea-land distribution. Effects of the
electromagnetic mutual coupling of the shell with the
finite-conducting mantle are considered. Next we examine
the effects of the electric currents in the shell on the
estimation of the mantle conductivity and discuss the real
mantle conductivity excluding the effects of the oceanic
currents .
2
Table 1. Complex values of I and J for N, n S 2.
N
M
n
m
1
1
1
1
1
1
1
1
1
1
2
0
0
0
0
0
1
1
1
0
2
2
0
1
2
2
2
2
2
2
2
2
1
1
2
2
1
1
1
2
1
0
0
0
1
2
2
2
2
2
2
2
1
1
o
1
0
1
2
0
1
2
0
1
2
2
1
2
2
2
1
2
2
2
2
o
J
I
3.660E-03, 0.000E+00)
(-2.057E-04, 1.194E-04)
(-8.343E-05, 0.000E+00)
(-6.599E-05, 9.386E-05)
2.939E-05.-2.571E-041
i-z.o57~-oa;-i.i94E-o4j
( 2.3258-03, O.OOOE+OO)
( 8.224E-05, 2.822E-04)
( 8.897E-04, 0.000E+00)
(-2.0848-04. 5.827E-04)
( 2.824E-03, O.OOOE+OO)
(-6.902E-05,-1.419E-04)
(-1.315E-05,-2.821E-O5)
(-6.9021-05. 1.419E-041
( 1.889E-03. 0.000E+001
f-l.658E-04: 2.014E-041
i-i.315~-05; 2 . ~ 2 1 ~ - 0 5 j
(-1.658E-04,-2.014E-O4)
( 1.2718-03. 0.000E+00)
(
9.2558-03, 0.000E+00)
-2.645E-04, 1.001E-04)
6.1218-04, 0.000E+00)
-5.934E-04. 1.5778-031
1.311E-04; -1.161E-03)
-2.645E-04,-1.001E-04)
8.9661-03, O.OOOE+OO)
7.438E-04. 1.867E-03)
6.512E-03, 0.000E*00)
-1.8828-03. 4.810E-031
2.062E-02; O.OOOE+OO j
-5.631E-04,-3.050E-04)
-4.0948-04, 1.715E-04)
-5.6318-04. 3.050E-041
1.938E-02, 0.000E+00)
-1.35OE-03. 1.309E-031
-4.094~-04;-1.715~-04j
-1.350E-03,-1.309E-03)
1.4668-02. 0.000E+00)
METHOD OF SIMULATION
Winch (1989) showed the formulation of the electric
currents induced in the oceans above the finitely conductive
mantle, using the spherical harmonic expansion. He
represented the sea-land system as a infinitely thin spherical
shell and gave the following formula (his equations 28, 30,
31 and 32) for the spherical harmonic coefficients of the
currents in the oceans:
where
J(P, n, N', m,M ' )
4n
ae ae sin2 e a+ a+
x P;,'(e, 4) sin 8 do d+,
(32)
a is the radius of the Earth (m), po is the magnetic
permeability (4n x lo-' H m-'), w is the angular frequency
of the harmonics (rad), Eo is the external magnetic potential
(A m-'), p is the surface resistivity of the shell (a),8 is the
colatitude,
is the longitude, Yy are spherical functions,
W: are expanded coefficients of the current function in the
shell (A m-'), q is the radius of the mantle as a proportion
of radius of the Earth, a, j, is a spherical Bessel function,
and Aqa are the dimensionless arguments of the spherical
Bessel function. Also
+
given by Takeda (1985, 1989). The spherical harmonic
coefficients which represent the Sq field were obtained at
every universal time on each day. Then they were averaged
at every universal time through all the days and expanded as
Fourier series. The Fourier coefficients of the external
spherical harmonic coefficients field from one to four cpd
(cycles day-') are used as the inducing field.
As for the resistivity distribution at the Earth's surface,
the ETOPO-5 of NOAA model of the global bathmetry is
used to calculate the distribution. Bathmetry in the
geomagnetic coordinates obtained by the average at every
1"x 1" mesh is shown in Fig. 1. Conductivities of sea and
land are assumed to be 4 and 0.01 S m-', respectively, and
surface resistivities on the 1" x 1" mesh in geomagnetic
coordinate are calculated by integrating the conductivities
from the Earth's surface to 10 km depth. Maximum degrees
and orders of the spherical harmonics are both taken to be
the 25th by the restriction of computer capacity and
accuracy in the estimation of the spherical function. We
have checked the effects of the truncation by limiting the
degrees and orders to the 20th, but the results are almost
identical. Thus the numbers of the highest orders and
degrees seem to be large enough. The mantle is regarded as
a uniform geocentric sphere of finite conductivity of radius
qa m. We first show the result by assuming that the mantle
conductivity is 0.1 S m-' and the depth is 500 km. These
values are based on those obtained from the observed
results assuming a uniform geocentric sphere of finite
conductivity of radius qa m without the surface conductor.
Once we have obtained the induced currents in the ocean,
the effects of the currents on the estimation of the
w
0
3
+
60
H
tQ
30
_I
0
w
0
t-
w -30
Z
0
Q
t
where a, is the conductivity of the mantle (S). Complex
values of I and J for N, n S 2 are shown in Table 1.
We have adopted this formulation for the simulation of
the induced currents. Inducing external fields used in this
study are the result of the Sq analysis from 1980 March 1-22
90
-60
0
w-3UL
0
0
I
,
60
I
I
120
.
,
.
180
240
GEOMAGNETIC
L
d
3CO
360
LONGITUDE
Fiyre 1. Bathmetry in the geomagnetic coordinate obtained by the
average at every 1" X 1" mesh from ETOPOJ of NOAA. Contours
are drawn at every 2000 m.
Induced oceanic currents and mantle Conductivity
conductivity and depth of the mantle are discussed and
other values of the conductivity and depth are considered to
fit the observational result.
3 CURRENTS IN THE OCEANS A N D
RESULTANT EQUIVALENT INTERNAL
CURRENTS
383
90
3
I-
H
60
I-
4
0
H
0
30
0
0
-30
0
F
$
0
Figures 2 and 3 give the electric currents induced in the
spherical shell at 10 hr UT when the centre of the external
current external dayside vortex is above the Eurasian
continent and at 2 2 h r u ~when maximum currents flow in
the northern hemisphere, respectively. Practically no
currents flow in the dayside northern hemisphere at
10 hr UT. On the other hand strong currents flow at 22 hr UT;
the intensities of the current vortices located in the northern
and southern Pacific Ocean are 46 and 54 kA, respectively.
These values are fairly large compared with those of the
internal part of the Sq currents (-100kA), and it is
expected that the induced currents in the shell have a
significant effect on the total induced field.
Figure 4 shows the UT variation of the intensity of the
internal current vortex in the northern hemisphere in the
daytime. Thin solid and dashed lines show the UT variation
simulated for the case with and without the shell above the
mantle, respectively. For comparison, bars represent the
observational result. The thick line will be discussed later.
There is a clear effect of the currents flowing in the shell in
the UT variation. Comparing the simulated UT variation with
the observed one, it is found that the enhancement at about
18 hr UT is fairly well reproduced although the depression at
about 10hrwr found in the observational result does not
clearly appear in the simulated one. The maximum effect of
the shell is about 60 per cent enhancement at 22 hr UT when
the external current vortex is in the Pacific Ocean. A
striking feature is that the simulated current vortex is
enhanced about 10per cent by the existence of the shell
even when the external current vortex is above the Eurasian
continent. To check whether this enhancement is caused by
the finite conductivity (0.01 S m-') used for the land, we
examined the effect of the shell consisting of land only, but
the results of this simulation were same as those of the
simulation without the shell. Therefore, our introduction of
the finite conductivity for the land has no effect on the
simulated induced currents. Another feature is that there is
2
0
-60
0
W
0 - 9 4
'
'
6
.
'
.
12
'
.
-,3
,
I8
,
1
24
MAGNETIC LOCAL T I M E
Figure 3. Same as Fig. 2 but at 22 hr UT.
only slight enhancement of the induced currents when the
centre of the external current vortex is above the Atlantic
Ocean. These features seem to result from the fact that the
horizontal wavelength of the Sq field is so large that the
induced currents respond only to the gross sea-land
distribution. Even when the external current vortex comes
above the Eurasian continent, the effects of the Pacific
Ocean remain and cause the enhancement of the total
internal field, although it is possible that the enhancement
may be due to the existence of the Indian Ocean in the
southern hemisphere. On the other hand, the longitudinal
scale of the Atlantic Ocean is smaller (1/6 of the Earth)
than the wavelength of fourth harmonic of the Sq field
which is the maximum cycle wave actually appearing in the
Sq field. Inclusion of the shell enhances the internal current
vortex and it is expected that this affects the estimation of
the mantle conductivity. This is discussed in the next
section.
4 EFFECT OF THE CURRENTS IN THE
OCEANS ON THE ESTIMATION OF
MANTLE CONDUCTIVITY
Now we have got the electric currents induced in the shell
representing the sea-land distribution, we can discuss their
effects on the internal Sq field and on the estimation of the
mantle conductivity. We first estimated the conductivity and
the depth of the conductor from the amplitude ratio of the
100
3>
tl-l
9
50
W
t-
z
OO
MAGNETIC LOCAL T I M E
12
18
U N I V E R S A L T I ME
24
ur variation of northern internal current intensity
obtained by the simulations with (thin solid line) and without
(dashed line) the shell representing the sea-land distribution above
the uniformly conducting mantle of 500 km depth and 0.1 S m-'.
The thick solid line shows the UT variation of the simulation with
the shell above the uniformly conducting mantle of 780 km depth
and 1 Sm-'.
Figwe 4.
Flgwe 2. Current function representing the calculated induced
currents in the shell at 1 0 h r m . Ordinate and abscissae are
geomagnetic latitude and geomagnetic local time at the equinox,
respectively. Contours are drawn at every 10 kA.
6
384
M. Takeda
2
0.0
,
0
,o.
,
,
1
2 0 0 4 0 0 6 0 0 a00 1000
E (
:
I
34,
(a)
I
I
l 4 $ 1 ,
0
60.0
0
,
,
1
2 0 0 4 0 0 600 800 1000
(b)
ment of the total induced currents by the existence of the
shell makes the estimated depth of the sphere shallower.
A more clear feature appears in this figure. An apparent
distribution of the conductivity with depth is found. That is,
the shorter the period of the Sq harmonics, the smaller
estimated conductivity, and the shallower the depth. This is
also caused by the response of the shell to the inducing field.
For the external fields of shorter period, more of the
induced currents flow in the finitely conductive shell and the
effects of the shell become larger. Thus, it is possible that
the previous conductivity distribution by depth obtained
from the response of the Sq harmonics does not reflect the
real conductivity distribution in the mantle, but only the
effect on the shell or oceans.
How can the real conductivity distribution from the
observed Sq field be obtained? For this purpose, we
calculated the conductivity and depth of the uniform
geocentric conductor by using the fields which are obtained
by subtracting the effects of the currents in the shell from
the total observed fields. The effects of the currents in the
uniformly conductive sphere induced by the fields generated
by the currents flowing in the shell are also taken into
consideration. This procedure should be iterated until the
estimated conductivity and depth agree with those of the
conductive sphere used for the calculation of the currents in
the shell. It was found that for the l d a y period variation
field of Pi mode, a uniformly conductive sphere of 1S m-'
below the depth of 780km can explain the observed
response if the effects of the shell are considered. This depth
of 780 km is an interesting value, because from the study of
the seismic precursors it is shown that there is a transition
region of the Earth's upper mantle at about 670 km depth. It
is possible that the estimated depth is related to the sudden
increase of the electric conductivity accompanied by this
transition. However, we do not have enough data to discuss
this problem further.
Figure 5(c) gives the conductivity and depth of the
conductor estimated from the field generated by currents
induced in both the shell and the conductive sphere on the
assumption that the total induced fields are generated by a
uniformly conductive sphere only. This figure resembles Fig.
5(a) not only in the estimated conductivity and depth for the
1 day period variation field of the P: mode but also in the
apparent dependence of the conductivity and depth on the
period as the shorter period gives smaller conductivity and
shallower depth, except that the 114 day period variation
field of the
mode gives a shallower depth than the 1/3
day period variation field of the
mode in Fig. 5(a).
Therefore, it is possible that the depth variation of the
mantle conductivity obtained by the geomagnetic induction
fields of various periods is not realistic but reflects the effects
of the currents induced in the oceans.
This change of the mantle parameters varies the induced
currents. However, it is found that the currents induced in
the shell are almost the same as those for the previous
parameters (the difference is less than 10 per cent) and are
not shown here. Newly estimated UT variation of the
intensity of the internal current vortex in the northern
hemisphere in the daytime is presented by the thick line in
Fig. 4. Comparing the thick and thin solid lines, it is clear
that the general overestimation of the internal current
intensity is removed by the change of the mantle
o'21
O.]
tI . : ' ,
o.oo
,
,
1
2 0 0 4 0 0 6 0 0 8 0 0 1000
DEPTH ( k m l
(C)
Figare 5. (a) Conductivity profile estimated by the application of
the uniform-conductivity sphere model to the result of the data
analysis by Takeda (1985, 1989). The result of simulation in which
the effects of the shell are considered in addition to the uniformly
conducting mantle of (b) 500km depth and 0.1Sm-' and (c)
780 km depth and 1 S m-'.
internal field to the external field and their phase difference
obtained by data analysis on the assumption that the
induced currents flow in a uniformly conductive sphere, by
using the method shown in Campbell & Anderssen (1983).
Fig. 5(a) shows the conductivity and the depth of the
conductor. Symbols 1, 2, 3 and 4 represent the values
estimated from the 1cpd term of the Pi mode, the 2cpd
term of the P', mode, the 3 cpd term of the Pj mode and the
4cpd term of the
mode, respectively. Those terms are
chosen because they are the principal terms of each
frequency and have large amplitudes.
Figure 5(b) shows the same profile calculated from the
current distribution obtained by the present simulation,
assuming that the calculated internal field, which is affected
by the shell in reality, is generated by the induced currents
in a uniformly conductive sphere without the shell. If the
effects of the shell representing the sea-land is negligible,
the resultant conductivity and depth for all terms should be
0.1 S m-I and 500 km, respectively because these are the
parameters of the uniformly conductive sphere used in the
present simulation. However, the results show clear effects
of the shell. The 1 day period of the Pi mode gives the result
of 0.05Sm-' and 240km depth, and the higher harmonics
gives less conductivity and a shallower depth. The estimated
conductivity and depth from the 1 day period variation field
of the Pi mode are 0.05S m-' and 240 km, respectively, and
conductivities and depths from higher modes are smaller
and shallower, respectively. This is caused by the currents
induced in the shell, almost all in the oceans. That is, the
phase difference of the currents in the shell due to the finite
conductivity makes the total phase difference larger and
estimated conductivity of the sphere smaller, and enhance-
e
Induced oceanic currents and mantle conductivity
parameters. The amplitude of the UT variation becomes
larger and more similar to the observational result, although
inclusion of the shell itself does not improve the agreement
with the observational result so conspicuously. Disparity
between the observational and simulated results may be due
to the insufficient number of the observatories in the oceanic
region.
We have shown that electric currents in the oceans have a
significant effect on the estimation of the mantle
conductivity for the Sq field. Usually, variation with depth
of the electric conductivity of the mantle was mainly
estimated by using the response of geomagnetic variation
field of the
mode of the period from a few to several
hundred days (e.g. Banks 1%9; Campbell & Anderssen
1983) ignoring the effects of the oceans. However, Roberts
(1984) found that the Oceans have a considerable effect on
the response even at the periods of 2-200 days, by
comparing of the responses at different stations. It will be
necessary to simulate the effects of the electric currents
induced in the Oceans for the Pi mode up to the period of
several hundred days for the precise estimation of the
mantle conductivity. We will discuss this problem in a
subsequent paper.
5
CONCLUSIONS
We have simulated the electric currents induced in the
Ocean by using the model of a shell conductor representing
the sea-land distribution and a finitely conductive sphere
representing the mantle. This study reveals the following
major points.
(1) The simulation results show that at 22 hr UT currents
in the ocean have maximum intensity of 54kA in the
northern Pacific Ocean and enhance the total induced fields
by about 60 per cent. This shows that the induced currents
in the Oceans cannot be neglected for the estimation of the
conductivity in the mantle.
(2) Even when the external current vortex is located
above the Eurasian continent, the internal current vortex is
enhanced by about 10 per cent by the existence of the shell.
(3) Electric currents induced in the ocean cause
less-conductive and shallower estimation of the mantle
conductor and apparent depth dependence of the conductivity. The conductivity and depth estimated from the data
analysis for the 1 day period variation field of the P i mode
without the consideration of the oceanic currents are
0.1 S m-' and 500 km, respectively, but it would be 1S m-'
and 780km if the effects of the currents in the shell
representing the sea-land are considered. Apparent
dependence of the conductivity and depth on the period can
be made by the effect of the oceanic currents even if the
mantle is a uniform conductor.
385
ACKNOWLEDGMENTS
The bathmetry data used in this study are E T O P O J of
NOAA presented through Japan Oceanographic Data
Center of Maritime Safety Agency, Japan. The author
wishes to thank Professor T. Araki for valuable discussions.
The geomagnetic data used in the present study were
obtained through the World Data Center C2 for
Geomagnetism, and the data processing was performed
using facilities at the Data Processing Center, both at the
Kyoto University.
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