Geophys. J. astr. SOC.(1977) 48, 385-392
Electromagnetic induction over the edge of a perfectly
conducting ocean: the H-polarization case
R. c. Bailey Department of Physics, University of Toronto, Toronto, Ontario,
Canada
Received 1976 August 20
Summary. Using the Wiener-Hopf technique, a solution is obtained for the
problem of H-polarization electromagnetic induction at an ocean edge. The
ocean is represented by a perfectly conducting thin sheet. The implications
of the solution for magnetotelluric and geomagnetic sounding are discussed.
Introduction
The electromagnetic response of ocean edges to natural geomagnetic variations is a problem
of some interest in electromagnetic studies of earth structure. The simplest models to
examine are those which represent the ocean by a thin conducting sheet. In Cartesian
coordinates, Parker (1968) and Wiedelt (1971) have obtained analytic solutions for the
E-polarization case, in which the horizontal exciting magnetic field is normal to the ocean
edge. Analogous solutions for oceans on a spherical earth have also been obtained (see
Ashour 1973, for a review). In all of these models, currents induced in the oceans remain
confined to the oceans. None of these solutions examine the H-polarization case in which
currents are driven across the oceanic boundary into a conducting substratum. BrewittTaylor (1975) has obtained a solution which does examine these currents, but neglects
higher order, self-induction terms.
This paper therefore presents an analytic solution for the H-polarization case. Although
H-polarization induction produces no perturbation of the uniform magnetic field observed at
the surface in a two-dimensional structure, it is nevertheless of interest. For example,
magnetotelluric apparent resistivities obtained for this polarization are perturbed near a
coastline. Perhaps more important, coastline irregularities and conductivity irregularities near
the coast can perturb the otherwise horizontally uniform currents driven across the ocean
edge by channelling them according to Ohm’s law. In interpreting experimental results of
magnetic variometer surveys, it is important to know what regional H-polarization currents
are available for such channelling.
The model used in this paper for an ocean edge is two-dimensional. It represents the earth
by a uniform substratum of conductivity u and permeability p occupying the half space
z < 0. The ocean is represented by a perfectly conducting half sheet covering the right-hand
half (x > 0) of the uniform half space and in electrical contact with it. A cross-section of
the model is shown in Fig. 1. A uniform horizontal magnetic field directed in the y direction
R . C.Bailey
386
r
Figure 1. The conductivity model.
(parallel to the ocean edge) oscillating with angular frequency o is applied by sources above
the earth.
A perfectly conducting, thin sheet is not a realistic model of an ocean. That is, there is no
frequency at which the thickness of a real ocean is both very much greater than the skin
depth in the ocean and very much less than the skin depth in the solid earth. Nevertheless,
the model represents a useful limiting case which puts bounds on the behaviour to be
expected from real oceans.
Mathematical formulation
In the case of H-polarization, H, is taken as the only non-vanishing component of the
magnetic field, and
EX = - -I- aHy
az’
I aH,
E =-u ax
as the non-vanishing components of the electric field. In the substratum, H, obeys the
magnetic diffusion equation
HY
V2Hy= pu a-.
at
The conductivity and permeability of the uniform substratum are denoted by u and p. No
is assumed for the fields. The above
quantities vary with y and a time dependence exp (iot)
equation reduces to
a”,- azH, = ia2Hy
+
ax2
aZ2
where for convenience wpu has been set equal to a?.
The solution must satisfy certain boundary conditions. That the sources of the fields lie
above the surface of the Earth requires
lim
Hy=0.
z+-m
The perfectly conducting sheet cannot support a tangential electric field on its surface.
Hence
The H-polarization case
387
The notation z = 0 - denotes the underside of the sheet. Finally, the vertical current density
at the exposed surface of the substratum must be zero, or
Jz=3=O;
z=O,
x<O.
ax
It is an elementary exercise to obtain the solution of (1) as a Fourier integral
=j--d k @(k) exp [zd(k' +icy')]
+-
H,,(x, z)
(4)
exp(ikx).
Note that complex roots are to be interpreted here as principal roots.
As both boundary conditions (3) and (4) for Hy involve derivatives of H, and are
homogeneous, H, will be determined by these and (2) except for a multiplicative and an
additive constant. These can be determined afterwards from
lim H,, = H o ;
X-+--W
lim H,, = O
X'+W
z=0-
z=0
where H, is the uniform field observed at the surface.
The root of k2+ ia2 with positive real part is taken in this equation in order to satisfy
boundary condition (2). Presuming uniform convergence of the required integrals (which
may be shown to hold except at the point x = 0, z = 0), Hy may be differentiated through
the integral sign to obtain
+-
Ex = -
+ ia')
dk d(k'
@(k) exp [zd(k' + icy')]
U
exp(ikx)
(7)
and
+-
dk ik
@(k) exp [zd(k' +ia')] exp(ikx).
It remains to determine @(k) so as to satisfy the remaining boundary conditions (3) and
(4).
The Wiener-Hopf solution
The above problem is a textbook exercise in the application of the Wiener-Hopf technique
(cf. Noble 1958). Wiedelt (1971) has already applied the technique to a related E-polarization problem.
In accordance with this method, we note that (3) implies that the Fourier transform of
Ex(x) at z = 0 is an analytic and bounded function in the upper half complex k-plane, i.e.
-
d(k'
+ ia2)
@(k) = A+(k).
U
Similarly, (4) implies
ik
- @(k) =B_(k)
U
where B(k) is analytic and bounded in the lower half complex k-plane. That Ez tend to zero
as x tends to minus infinity implies that B-(k)is also analytic and bounded on the real k axis.
R . C. Bailey
388
Elimination of @(k) between the last two equations yields, after some manipulation
-d(k - i3l4a)B-(k)
=
ikA+(k)
,/(k t i3I4aj
Both sides of this equation represent a function C(k). As the left-hand side is analytic in the
lower half k-plane and on the real k axis, and the right-hand side is analytic in the upper
half k-plane, C(k) is analytic for all k. Furthermore, C(k) must be bounded in order that
B_ and A + be bounded in the appropriate half planes. The only bounded analytic function
is a constant. Thus
C(k) = C = constant.
Therefore Q(k) can be reconstructed as
@(k)=
-OC
ikt/(k - i 3J4(Yj
The solution
The integrals (5), (7) and (8) for Hy, Ex and E, may be evaluated analytically at z = 0- and
numerically for z < 0. If the contour of integration along the real axis is taken over the pole
at k = 0, rather than through it, and Cis set equal to 4(-i3‘4a)Ho/u, the correct multiplicative and additive constants implied by ( 6 ) are obtained.
The results are, for z = 0 and x < 0:
Hy = Ho
E, = 0.
F o r z = O - a n d x > 0:
H, = H , [ I -erfd(axi’’’)]
Ex = 0
The solution for the fields above the sheet (x > 0, z = 0+)is separately obtained as
H, = Ho
Ex = 0
E, = 0.
Finally, the surface current density in the sheet is
j , = H~ erf d(ili2ax).
The solutions for Ex,E,, and Hy at z = 0- are shown as functions of x in Fig. 2. & is taken
to be unity, and x is in units of a skin depth (W/JO)-”~.
The corresponding apparent resistivity and phase of Ex with respect to Hy on the landward side (x < 0) are shown in Fig. 3. The
current system as a function of depth and horizontal position is displayed in Figs 4 and 5. In
The H-polarization case
389
Figure 2. The fields at the Earth's surface. Solid lines are in-phase components and dashed lines quadrature.
The inducing field is the phase reference. For x > 0 values are those on the underside of the sheet. Horizontal units are skin depths; vertical units are H, for Hy and H,J(wfi/o) for the electric components.
-2
I0"
-2
Figure 3. Log,, of the apparent resistivity
normalized by multiplication by a.
o
p
-I
0
-I o
0
L
and the EJH,, phase @. Units of
@
are degrees and p is
particular, the current streamlines (or equivalently, the contours of equal Ify) are decomposed into in-phase (real) and quadrature (imaginary) parts, with respect to the exciting
magnetic field, in Fig. 4. Fig. 5 shows the real-time current system in steps of 1/16 of a
cycle.
Discussion
Far to the seaward side, the induced current is purely real and confined to the ocean. Far
to the landward side, the induced current is distributed over several skin depths and has
a suLface phase of n/4 with respect to the inducing magnetic field. The electric field has a
l/dx singularity at the sheet edge and a phase of n/8 there with respect to the inducing
390
R. C. Bailey
REAL
I
IMAG
Figure 4. A sectional view of the real (in phase) and imaginary (quadrature) induced current streamlines.
These are also contours of equal Hyat intervals of 0.05 Ho. The diagram is four skin depths wide.
magnetic field. As expected, the transition zone is of the order of a skin depth in width. The
absence of imaginary current far to the seaward side requires that near-surface imaginary
currents coming from the landward side turn around and return at depth upon meeting the
ocean. An interesting corollary of this is that the imaginary current flowing in the ocean
reaches a maximum about 0.6 skin depths out to sea.
Values for apparent resistivity and Ex/Hy phase on the landward side are interesting in
their relative lack of sensitivity to the ocean. A magnetotelluric survey made one skin depth
from a perfectly conducting thin ocean, with typical errors of 10 per cent, would not detect
the ocean. A thin ocean of finite conductivity would be even less detectable.
Both real and imaginary currents are available for channelling by any shallow conductivity anomalies striking across the coastline. If magnetic variation data in the form of
transfer functions or the related induction arrows (Parkinson 1959; Wiese 1962; Schmucker
1964, 1970) evaluated on the landward side are examined, one would expect to see the
largest channelling perturbations in the real part of the transfer function, as the real currents
are stronger near the surface. However, real coastlines are irregular, and perturbations of the
predominantly real current along LL: coastline will mask the above effect. It seems reasonable to suppose that channelling effects of conductive anomalies striking across a coastline
39 1
me H-polarization case
1
I
t = O/l6
t = 4/16
t = 1/16
t = 5/16
5
t
t=2/16
r-----t = 3/16
t =6/16
t = 7/16
Figure 5. The evolution of the current streamlines in time in steps of 1/16 of a cycle. These are also
contours of equal H,, at intervals of 0.05 H,,. The diagram is four skin depths wide.
will be smaller but less obscured in the imaginary part of the induction arrows than in the
real part.
Finally, it is worth noting that the Wiener-Hopf method will work with a number of
extensions of the problem dealt with here. An arbitrarily layered substratum may be used,
and the ocean may be taken as having finite integrated conductivity. The orthogonal or
E-polarization case can also be done. Non-uniform source fields can also be dealt with. These
392
R. C. Bailey
problems are more difficult, however, because they generally require that the separation of
the Fourier transform into parts analytic in the upper and lower half k-planes be performed
numerically rather than analytically.
Acknowledgment
This work was financially supported by the National Research Council of Canada.
References
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