Download The Effect of an Ocean on Magnetic Diurnal Variations

The Effect of an Ocean on Magnetic Diurnal Variations
Robert B. Roden*
(Received 1963 August 27)
Summary
Theoretical calculations have been made for the idealized case in
which an ocean is approximated by an infinitely long rectangular conducting sheet. The results indicate that even in the case of magnetic
variations with periods as long as 24 hr one should expect a significant
and easily observable enhancement in the vicinity of the edge of an
ocean. When the inducing field is a function of local time only, it is
found that the amplification of the variations is greater at the eastern
edge of the ocean than at the western edge.
Laboratory model studies have been used to investigate the effect of
irregularities in the shape of a coastline. Results which have been
obtained from a model of the Pacific Ocean in the neighbourhood of
Japan indicate that it is reasonable to attribute at least a part of the
anomalous diurnal variations observed in central Japan to electric
currents induced in the ocean.
Introduction
Various writers have reported the observation of anomalously large diurnal
and short-period magnetic variations in the vicinity of the edge of a large body of
sea water. At the edge of the continental shelf in the north-east Atlantic, Hill &
Mason (1962) observed diurnal variations having amplitudes almost twice as
great as those measured by land-based equipment in similar latitudes. Ponomarev
(1960) and Mansurov (1958) have noted increases in the amplitude of magnetic
variations at the shoreline near the Antarctic base Mirny, and Zhigalov (1960) has
reported similar effects in the Arctic Ocean. Anomalously large variations in the
vertical component have been recorded in Japan, e.g. at Kakioka (Nagata 1951)
and at Aburatsubo (Rikitake, Yokoyama & Hishiyama 1952).
Because of the irregularity of the oceanic coastlines, theoretical investigations
are possible only when radically simplified models are assumed. The effect on
magnetic variations of currents induced in idealized ocean models has been studied
by many authors, including de Wet (1949), Ashour (1950), Rikitake (1950, 1960,
1961a,b, 1962) and Rikitake & Yokoyama (1955). The present writer has
investigated theoretically the induction effects in an infinitely long rectangular
conducting sheet. If the thickness is very small compared to the width and to the
skin-depth corresponding to the frequency of the variations, electric currents
flowing in the vertical (i.e. perpendicular to the plane of the sheet) direction may
I.
* Present address : Texas Instruments Inc., Science Services Division, Dallas, Texas.
375
376
Robert B. Roden
be neglected and all induction effects may be assumed to be due to changes in the
vertical component of the magnetic field. Rikitake’s calculations (1961a, b, 1962)
indicated that one cannot neglect the mutual inductance between the ocean and
the highly conducting part of the Earth’s mantle. We take this into account by
assuming an infinitely conducting half-space to be situated below the conducting
sheet.
Spherical harmonic analyses of the diurnal variations on quiet days have been
done by Benkova (1940) and others. It has been found that the variations of
external origin consist of two parts. About 80 per cent of the variations result
from the rotation of the Earth relative to the magnetic field produced by an ionospheric current system which maintains a fixed aspect towards the Sun and does
not change appreciably in magnitude and pattern from day to day. This part can
be represented adequately by a function which depends only upon latitude and
local solar time. Relative to observers on the Earth‘s surface, these variations will
appear to have the form of a travelling wave. The second part is caused by distortions of the current system which are due to the non-coincidence of the rotational
axis, the geomagnetic axis, and the normal to the ecliptic. This part is mainly a
function of geomagnetic latitude and universal time
Our investigation has been primarily concerned with the more important part
of the diurnal variation, which we may represent mathematically as a function of
the form cos(ky + w t ) . T h e same treatment may be applied to problems wherein
the inducing field is a function of universal time by letting the wave-number k
approach zero. Since features such as magnetic storms or bay disturbances occur
simultaneously at all observation points on the Earth’s surface, it should be possible
to deal in a similar fashion with the induction processes which result from the
accompanying changes in the magnetic vertical component.
Rikitake, Yokoyama & Sat0 (1956) have suggested that the anomalous variations
observed in Japan are probably caused by abnormal underground electrical conductivity associated with the zone of tectonic activity, but our theoretical results
indicate that induced currents in the ocean should make an appreciable contribution to the anomaly. I n order to resolve this problem we have carried out laboratory
model studies using a conductor with its boundary shaped like the edge of the
Pacific Ocean in the neighbourhood of Japan. Using modified Helmholtz coils,
we have been able to apply both travelling and stationary variations in the magnetic
field, but we have not attempted to simulate the mantle experimentally. T h e effect
of induction in the mantle can, however, be estimated from the results of the
theoretical part of the investigation.
2.
Mathematical formulation
We consider the simplified case in which the ocean is represented as a thin
rectangular sheet of non-magnetic material of conductivity a, width za, and
thickness s, where s<a. If the z axis is in the vertical direction, and the sheet is
assumed to be infinitely long, the region occupied by the sheet is : - to < x < 00,
- a <y < a, zo < x < zo + s. The effect of a highly conducting mantle at depth h is
taken into account by assuming the half-space z<zo-h to be filled with material
of zero resistivity. T h e conductivity of the continents is sufficiently low that we
may consider them to be perfect insulators.
If we assume that all field vectors are independent of x and that the sheet is
sufficiently thin that vertical electric currents may be neglected, then we may
PLATEI .-Apparatus
for the experimental investigation of induction
processes caused by the diurnal magnetic variations.
[Facing page 376
PLATE
2.-Copper
sheet model of the Pacific Ocean (length of ruler-12
in.).
The effect of an ocean on magnetic diurnal variations
377
represent the magnetic field strength, H, and the surface density of electric current,
I, as
H
=
+
[iXo +j(YO+ Yi)+ k(Zo Zz)]eiot.
1: = i1ei.t
(1)
(2)
where i,j, and k are the Cartesian unit vectors, X , Y , 2,and I are complex functions
of position, the subscripts o and i indicate inducing and induced parts and only
the real parts of H and I are to be considered. Since Maxwell's equations are
linear, we are able to treat each frequency, w , of the system separately and then
apply the principle of superposition.
Since the sheet is very thin, we may neglect the inducing effect of the horizontal
components. Our main interest is in the case where the inducing field is a function
of local time only. The inducing vertical field is then given by
which corresponds to a travelling wave moving in the negative y direction. Again,
the linearity permits us to treat separately the variations corresponding to each
wave-number K.
We attack the problem of the effect of electric currents in the mantle by considering the mantle to be an infinitely conducting half-space. The electric currents,
which must be confined to the boundary of the half-space, are made up of two
parts. The first part consists of those currents which would flow in the absence of
the conducting sheet and produces magnetic fields which contribute to YOand 20.
We need not concern ourselves with this part. The second part is induced by the
magnetic fields associated with the induced currents in the conducting sheet and
contributes to Y
z and Zt. Maxwell's (1872) image method indicates that the contribution of the second part is the same as that of a current system equal and
opposite to the current system in the sheet and flowing in the image plane,
x = xo-2h.
Applying the expression for the magnetic field due to an infinite line current
in e.m.u., and integrating to obtain the effect of the whole current system,
Since the e.m.f. around a circuit must be equal to the rate of change of magnetic
flux through the circuit,
[Zo(y")+ zi(y")]dy"
0
=
i
Cwos(eiku - ~ ) /+kiwos I(y')dy' log
-a
Letting
r = y/a, X
3
= wsca,
$.
=
ka, 9 = ah/a,
378
Robert B. Roden
We may write I(r) as the sum of two functions Ic(r)and I,(r) such that
I&)-Is(o)
= hC[cos(&)- I]/f+ih
i
I
Is(r’)dr’log
-1
I
+q2/(r’)2
+q2/(r- r’)2
The solutions of the above equations must satisfy the conditions
I&)
= --&(-r),
I&)
=
Is(- r ) .
Making use of these symmetry properties, we reduce (7) and (8) to
I&)
- Is(0) =
hC[cos(&) - 1yt+
We require that the total current crossing any plane x = const. be zero. Thus an
expression for Is(o) is obtained by integrating equation ( L O ) over the range
o < r d I and setting
s
I,(r)dr
= 0.
0
Hence :
The sum of the solutions of (9) and (11) is the electric current per unit width
flowing in the sheet when the inducing field is travelling across the sheet so that
its phases at the two edges (i.e. a t y = & a) differ by 2f radians. This corresponds
to the major part of the diurnal variations. The remainder of the diurnal variations
and such features as bays and magnetic storms do not exhibit a difference in phase
at different points. We may treat these phenomena by letting ( approach zero in
equations (9) and ( I I ) . Since the first term on the right side of ( I I ) becomes zero,
equation (11) cannot possess a non-zero solution in this case and so we need consider only the solution of (9). When t + o , this becomes
The effect of an ocean on magnetic diurnal variations
379
In this case, where the inducing field is stationary, the induced current is antisymmetrical about r = 0,and the induced magnetic field is therefore symmetrical
about r = 0. When the inducing field is not stationary, these symmetry properties
no longer hold and we are led to the conclusion that, in the case of the actual
oceans, the observed magnetic field variations should be greater at the eastern
edges than at the western edges.
3. Solution of the integral equations
I n Section 2, we have reduced the conditions governing the electric current
densities to integral equations of the form
i
+ix
I(r) = f(Y)
K(Y,Y‘)I(Y’)dY’.
0
(13)
Using the trapezium formula for numerical integration with n intervals, and
letting r k = K/n, we obtain a system of linear algebraic equatibns in the unknowns
I ( ~ o )., . - 7 I(rn):
n
I(?,$)- ix
2’K(rg,rk)I(rk)/n= f(rz).
k=O
(14)
Here the prime indicates that the first and last terms must be multiplied by
If (ri,r k ) is a singular point of K , then K(ri, r k ) must be replaced by
1
i.
rk+lI 2 n
n
K(ri,Y‘)dY’.
rk-1/2n
T o calculate the magnetic fields, we approximate the current flowing in the
interval Y ~ - I < r < rk by a line current at the centre of the interval with strength
Hence
where
d
=
r-rk+t.
4. Computed solutions
Since this investigation was originally prompted by the observations of Hill &
Mason (1962) at the edge of the continental shelf in the western approaches to the
English Channel, we have computed solutions on the EDSAC I1 computer of the
Cambridge University Mathematics Laboratory using parameters which correspond roughly to the dimensions of the Atlantic Ocean in that neighbourhood.
Using the following dimensions :
depth of ocean = 5 km
width of ocean = 4 ooo km = 55 degrees longitude
depth of mantle = 600 km
conductivity of sea water = 4 x 10-11 e.m.u.,
380
Robert B. Roden
we obtain for the non-dimensional parameters :
X
5
=
o.3N
=o-~N
9 = 0.6
where N is the frequency of the inducing field in cycles per day. We have obtained
solutions of (9) and (11) for the 24, 12, 8 and 6 hr harmonics. Figures I and z
give the induced horizontal magnetic field Variations and the total vertical magnetic
field variations corresponding to these solutions when the inducing field is of unit
amplitude. The effect of the highly conducting mantle can be seen by comparing
Figures 2 and 3. T o obtain the data presented in Figure 3, we have neglected the
conducting mantle by assuming it to be an infinite distance below the sheet.
Since 7 is proportional to the depth to the mantle, this is achieved by letting
approach infinity in equations (9) and ( I I).
Lahiri & Price (1939) investigated the relationship between the diurnal variations of external and internal origin and concluded that the electrical conductivity
in the mantle must be negligibly small in the upper regions and must begin to
increase rapidly with depth in the neighbourhood of 600 km depth. I n the present
calculations we have assumed that the conductivity suddenly becomes infinite at
600 km depth, and so our results must represent an upper limit to the effectiveness
of the currents induced in the mantle. We should therefore expect that, in the
real case, the solutions for the vertical force variations must lie between the solutions plotted in Figures z and 3 and that the induced horizontal force variations
must be somewhat less than those given by Figure I. The importance of the
value assumed for the depth to the top of the highly-conducting part of the mantle
can be seen in Figure 4. Here, we have assumed the same ocean dimensions as in
the previous calculations and calculated the total vertical force variations associated
with a stationary inducing field of period 24 hr when the mantle depth is 100,
300, 500 and 10 ooo km. The solution for h = 10 ooo is almost identical to the
solution obtained when the electric currents in the mantle are entirely neglected.
Rikitake (1961a, b, 1962) calculated the effect of a semi-cylindrical and a
hemispherical ocean on the diurnal variations, taking into account the presence of
a highly-conducting mantle. His solutions, expressed as a series of harmonics,
indicated that induction in the mantle should cause an overall reduction in the
anomaly to such an extent that the maximum anomaly should hardly exceed
z gamma, considerably less than one-tenth of the normal diurnal variations. Our
computed solutions show that one should take into account the mutual inductance
between the ocean and the mantle, but they also indicate that the currents in the
mantle can have little effect within distances from the edge of the ocean which are
comparable to the depth at which the currents are flowing. T h e termination of
the series in Rikitake’s solutions causes one to overlook the fact that an important
edge effect persists even when the mantle is brought quite close to the ocean.
5. Model experiments
Our theoretical results have led us to question the explanations which have
been suggested for the Japanese anomaly. Rikitake, Yokoyama & Sat0 (1956)
have reported the observation of anomalous diurnal variations at Kakioka Observatory near Tokyo, but they have stated that they do not think that electric currents
The effect of an ocean on magnetic diurnal variations
I
1
I
-1
0
1
DISTANCE, ?'
FIG.I a.-Amplitude
of the anomalous horizontal magnetic field produced
by diurnal variations of unit amplitude.
DISTANCE,
Y
FIG.1b.-Phase relative to universal time of the anomalous horizontal
magnetic field produced by diurnal variations with phase zero at r = 0.
Robert B. Roden
0
-2
I
-1
DISTANCE,
I
1
0
1
r
FIG.aa.-Amplitude
of the total vertical magnetic field produced by
diurnal variations of unit amplitude.
FIG.zb.-Phase
relative to the inducing field of the total vertical magnetic
field produced by diurnal variations.
The effect of an ocean on magnetic diurnal variations
-2
1
0
DISTANCE,
1
r
FIG.ga.-Amplitude
of the total vertical magnetic field produced by
diurnal variations of unit amplitude, when the mantle effect is neglected.
t9l
-2
DISTANCE,
r
FIG.gb.-Phase relative to the inducing field of the total vertical magnetic
field produced by diurnal variations, when the mantle effect is neglected.
384
Robert B. Roden
2-
3
t
e
............
..................
05,
0
....
I
I
I
I
05
1
15
2
DISTANCE,
Y
FIG.4a.-Amplitude of the total vertical magnetic field produced by a
stationary inducing field of period 24 hr and unit amplitude calculated
for various depths h (km) to the top of the conducting mantle.
--0
~
0.5
1
1.5
2
DISTANCE, ?'
FIG.4b.-Relative phase of the total vertical field produced by a stationary
inducing field of period 24 hr calculated for various depths h (km) to
the top of the conducting mantle.
The effect of an ocean on magnetic diurnal variations
385
induced in the ocean should make any significant contribution. Since the effect
is apparently a local one, covering an area somewhat smaller than that of the Japanese islands, we concluded that, if it is an ocean effect, it must somehow be related
to the shape of the coastline in that region. Because our theoretical methods are
severely limited to very simple models, we have pursued this problem by means of
model experiments in the laboratory.
It is well known that electromagnetic induction mechanisms can be modelled
exactly, as long as the dimensionless constants (which we have introduced in
Section 2) are correct. Thus, using a copper sheet of thickness 0.022 in. to
represent the ocean, and applying an inducing field of mains frequency (50 c.P.s.),
we have investigated the z4-hour and Iz-hour variations by using horizontal
scalings of I :IO million and I :5 million. Stationary inducing fields were obtained
using a set of square Helmholtz coils with 6-ft. sides and 3-ft. separation. For
travelling fields, a somewhat more complicated arrangement was required. Two
sets of coils were designed, one producing an A.C. field proportional to the cosine
of the east-west distance and the other producing an A.C. field proportional to the
sine. By means of a suitable connection of resistances to the terminals of a threephase supply, these fields were superimposed with a phase difference of 90' to
obtain a travelling wave. Induction effects of short-period disturbances were also
investigated by applying the signal from an audio-frequency oscillator to the
Helmholtz coils. The total vertical field was in all cases found by measuring the
e.m.f. induced in a circular coil of radius in.
We did not attempt to simulate the conducting part of the mantle in our
experiments, because of restrictions which would have been placed on the choice
of scaling factors. We would not be able to use the same scaling factors as before,
since in order to model the part of the mantle below 900 km depth, we would
require a slab of material with conductivity higher than that of silver at 10OK.
Since the linear dimensions of the model cannot be changed very much, it would
be necessary to increase the frequency of the applied field and decrease the conductivity of the sheet which we are using to represent the ocean. This is not
prohibitively difficult, but we had neither the time nor the amplifying equipment
which would be needed to produce the required power at frequencies other than
50 C.P.S.
The first step in the experimental investigation was to compare the results
obtained using rectangular sheets of copper with the theoretical solutions. These
were found to agree within the limits of observational accuracy except at distances
somewhat less than I in. from the edge of the sheet, where the field changes so
rapidly with distance that the value at a point cannot be measured accurately
using a coil of finite size. The east-west asymmetry in the case of travelling
inducing fields was observed experimentally.
On proceeding with the treatment of a model of a real ocean, we are met with
the question of where the effective edge of the ocean is located. Since the induction
by the diurnal variations is essentially a deep-water effect, it seems that the edge
of the continental shelf should give more realistic results than the actual coastline.
A comparison of records from Nantes Observatory (on the French coast and remote from the shelf edge) and from Coimbra Observatory (on the Portuguese coast
and near the shelf edge) supports this suggestion. I n our model experiments, we
have used models in which the effective edges are taken as the coastline (Figure 5 )
and the 2-km depth contour (Figure 6). The most reasonable solution is probably
some smeared-out average of the two.
+
Robert B. Roden
FIG. ga.-Ratio of the experimentally observed amplitude of the total
vertical magnetic field to the amplitude of the inducing field when the
coastline was used as the effective boundary of the ocean.
FIG. 5b.-Experimentally observed phase (hours) of the total vertical
magnetic field when the coastline was used as the effective boundary of
the ocean (phase of inducing field indicated by broken contours).
FIG.6a.-Ratio of the experimentally observed amplitude of the total
vertical magnetic field to the amplitude of the inducing field when the
a-km depth contour was used as the effective boundary of the ocean.
FIG. 6b.-Experimentally observed phase (hours) of the total vertical
magnetic field when the 2-km depth contour was used as the effective
boundary of the ocean (phase of inducing field indicated by broken
contours).
The effect of an ocean on magnetic diurnal variations
387
Rikitake, Yokoyama & Sat0 observed that the main features of the diurnal
variations in the vertical component occur about one or two hours earlier at
Kakioka (+ = 36"14'N, X = 140~11'E)than at other observatories in the Far
East and that the amplitude is greater at Kakioka than the world-wide average for
that latitude. Our experimental results for both the zq-hour and 12-hour (not
illustrated here, but similar to the z4-hour case) variations agree qualitatively with
these observations, even after allowances have been made for the effect of electric
currents induced in the mantle.
6. Conclusions
By means of theoretical calculations and experimental model studies, we have
shown that diurnal magnetic variations should induce electric currents in the
oceans and that these currents should produce anomalous magnetic variations
which are significant and easily detectable, especially in the vicinity of the edge
of the ocean. We have not been able to solve the equations analytically, but the
solutions appear to be sufficiently well-behaved that numerical methods are
adequate for our purposes. The effect of neglecting the Earth's curvature and the
latitude dependence of the inducing field should not be very important. Our
most serious approximation has been to assume that our ocean possesses a sharp
boundary, rather than a sloping one. It is possible to construct models in which
the ocean depth is copied more faithfully, but this would involve a great deal of
extra labour. It seems reasonable that the edge effect would lose some of its
sharpness if such a model were used, but Ponomarev's (1960) calculations for a
uniformly sloping bottom show that the edge effect persists. We have not included the conducting mantle in our model experiments, but our theoretical
results indicate that the importance of the mantle has been grossly overestimated
by others who have attempted this problem.
In this study, we have not considered the induction effects which are due to
tidal motions of sea water across the lines of force of the main geomagnetic field.
Although this is a separate problem, it is worth mentioning here that LonguetHiggins (1949) has shown that such effects are measureable only in places where
a very large volume of water is forced to flow through a narrow strait (such as in
the English Channel) and that, in any case, the induced magnetic fields do not
contribute appreciably to the measured variations, except below the Earth's
surface.
We have succeeded in finding a qualitative agreement between the observations
of Rikitake, Yokoyama & Sat0 (1956) and the results of our experimental investigation of the process of electromagnetic induction in the ocean by diurnal magnetic
variations. Although it would be foolhardy to say that the Japanese anomaly is
entirely the result of electric currents in the ocean, it is clear that there is no
justification for assuming that it must be explained entirely in terms of the conductivity distribution beneath Japan.
7. Acknowledgments
The writer wishes to thank Sir Edward Bullard for his suggestion of the problem and for his assistance and encouragement. Support was provided by the
Board of Trade in the form of an Athlone Fellowship.
Department of Geodesy and Geophysics,
University of Cambridge.
I 963 August.
388
Robert B. Roden
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