Download H-polarization induction over an ocean edge coupled to the mantle

Ceophys. J. R. astr. Sot. (1977) 49,427-441
H-polarization induction over an ocean edge coupled
to the mantle by a conducting crust
M. A. Nicoll and J. T. Weaver
Department ofPhysics, university of
Victoria, Victoria, British Columbia, Canada V8W 2 Y2
Received 1976 October 6 ; in original form 1976 July 19
Summary. The Wiener-Hopf technique is used to obtain an exact analytical
solution for the problem of H-polarization induction over the edge of a perfectly conducting thin sheet, representing an ocean, electrically connected t o
a perfectly conducting mantle through a slab of finite conductivity and
thickness, which represents the Earths crust. It is shown that the induced
currents resulting from this type of induction process are drawn up into the
sea from the cust and mantle with the greatest concentration of current
near the ocean edge. The surface impedance over the land surface, is calculated for various mantle depths and is shown to increase sharply as the
coastline is approached. The magnetic field along the ocean floor is also
plotted as a function of distance from the coastline, and the results are found
to agree very well with those calculated previously by approximate and
numerical methods.
1 Introduction
At the 2nd Workshop on Electromagnetic Induction in the Earth held in 1974 in Ottawa
papers were presented by R. C. Bailey and C. R. Brewitt-Taylor in which, for the first time,
the consequence of allowing electric currents induced in the sea to return through underlying conducting sub-strata whs examined theoretically. Interest in this pattern of current
flow arose because it appears to be the only mechanism consistent with the hypothesis of
induction by a horizontal rather than a vertical magnetic field - the oceans themselves are
just not deep enough to accommodate the required vertical current loops (Bullard & Parker
1970); and also because such a flow had been postulated as a possible explanation of the
observed orientation of Parkinson arrows around the Gulf of St Lawrence (Bailey et a/.
1974).
In the paper by Bailey a simple two-dimensional model was considered consisting of a
perfectly conducting half-plane representing an ocean edge lying on, and in electrical contact
with, the surface of a uniformly conducting half-space representing the Earth’s crust. Only
H-polarization induction gives the desired form of current flow, and Bailey was able to solve
the problem exactly in this mode, using the powerful Wiener-Hopf technique. His solution
showed how the induced currents were drawn up from the crust and into the sea near the
ocean edge.
428
M. A . Nicoll and J. T. Weaver
In the other paper, which has since been published (Brewitt-Taylor 1975), a more
detailed two-dimensional model was studied. It consisted of a perfectly conducting thin
strip representing an ocean of finite width situated above a perfectly conducting plane representing the Earth's upper mantle. The intervening region between strip and plane was
assumed to be of sufficiently high resistivity that, as a first approximation, self-induction
could be neglected. Under this assumption it was possible to obtain an approximate analytic
solution in the H-polarization mode. The induced currents were shown to flow in vertical
loops, across the ocean at the top of the loop and returning along the surface of the mantle
at the bottom. A related three-dimensional model with a rectangular ocean was also considered in order to examine the relationship between vertical and horizontal magnetic
components near the ocean edges.
Recently, Brewitt-Taylor (1976) has re-examined the two-dimensional problem using a
numerical method which avoids the necessity of having to neglect self-induction, and allows
more realistic ocean conductivities and depths to be used.
Despite the greater versatility of numerical methods for solving problems of this kind,
exact analytic solutions of certain models are always useful to have. They serve both as a
standard against which numerical solutions can be checked, and also indicate more clearly
the interdependence of the various physical parameters in the problem. In this paper we
obtain an exact analytical solution of a model which contains features of both the models
described above. As in the Bailey model, the ocean is represented by a perfectly conducting
half-plane, situated on a uniform conductor, but we also follow Brewitt-Taylor by including
a perfectly conducting plane at a finite depth so that the importance of the upper mantle in
assisting the return current flow can be examined.
2 The mathematical model
As shown in Fig. 1, we define a rectangular coordinate system (x, y , z ) in which the region
z < 0 representing the atmosphere is non-conducting, the region 0 < z < D representing the
Earth's crust, has a uniform conductivity u, and the plane z = D is perfectly conducting to
represent the Earth's mantle. Tne half-plane z = 0, y > O is also taken to be a perfect conductor to represent a large ocean with a coastline along the line y = 0, z = 0. Vacuum
permeability po is assumed everywhere. The problem is kept two-dimensional by postulating
that the inducing field is independent of the coordinate x. We shall assume that this field
varies in time t with an angular frequency w.
In the H-polarization mode the magnetic field B exp (iwt) has a single component in the
x-direction which is related to the y and z components of the electric field E exp (iwt) by
the equations
iwBx = aEy/aZ - aE,/ay
p o u ~=
y
as,laz,
(2.1)
yoUE, = - aB,/ay
Figure 1. The mathematical model.
(2
.a
H-polarization induction over an ocean edge
429
In the region z < 0 where u = 0, equations (2.2) give the well-known result that B, = Bo,a
constant. In the conducting region 0 < z < D , equations (2.1) and (2.2) give the familiar
induction equation
a2B,lay2+ a2B,.az2 = ia2B,
(2.3)
where (Y = ( O ~ ~ O ) ” ~ .
Equation (2.3) must be solved subject to boundary conditions imposed at z = -k 0 and
z = D.Since B, = Bo everywhere in the region z < 0, it has the same constant value along the
boundary z = - 0. This magnetic field is continuous across the surface z = 0 for y < 0, but
not for y > 0 because of surface currents flowing in the perfect conductor. However we
know that the tangential electric field vanishes at the surface of a perfect conductor so that
Ey = 0 on z = + 0 , y > 0, and on z = D.It follows from these statements and equation (2.2)
that the required boundary conditions on B, are
B,
=~,b
< 01; aB,..az = o o,> 0 )
(2 *4)
on z = t 0, and
aB,pz = o
(2.5)
onz=D.
3 Solution by the Wiener-Hopf technique
In this section we shall find it convenient to express the magnetic field as a function of the
space variables by writing B, = fb,
z). We also define the Fourier transform
where l = + iq is complex. Taking the Fourier transform of equation (2.3) and the boundary condition (2.5) we find that F satisfies the differential equation
subject to F’({, D) = 0. Here primes denote differentiation with respect to the z variable.
The appropriate solution of (3.2) is clearly
where
y=
(r2 + ia2)1’2.
In order to deal with the mixed boundary conditions on the surface z = + 0 we shall make
use of the Wiener-Hopf technique. Details of this method and the theorems to be quoted can
be found in the book by Noble (1958). We first write
fb,z) = M Y ,z) +L@,z )
(3.5 )
where f+ = 0 for y < 0, and f- = 0 for y > 0. As y -+ - m the influence of the ocean becomes
negligible and the problem approaches that of induction in a uniform slab bounded below
430
M. A. Nicoll and J. T. Werner
by a perfect conductor. Thus we can assume that as y
KO,z ) - Bo
+-
*
cosh [a(D - z ) a
coshaD4
'
When y -+ + *, however, the perfectly conducting ocean effectively screens the interior of
the Earth from the inducing field. Thus f+ + 0 as y + 00, and in view of the well-known
properties of an electromagnetic field in a conducting medium we anticipate the usual
exponential attenuation of the field. We may assert, therefore,
f+@>
z ) = 0 CeXP (- cv>l
as y -,*, where c = 4 6 2 .
The Fourier transform of equation (3.5) is
m
-
9
z)
=mi-,2) +a{,
z)
where
1
r-
is analytic in the region 77 >
-c
of the complex {-plane, and where
(3.10)
is analytic in the lower half-plane 77 < 0. It follows that F is analytic in the strip - c < 77 < 0
of the (-plane. Now the boundary conditions (2.4) give
(3.1 1)
F-((, +0) = - A / {
for 7) < 0, where A = iB&%,
and
(3.12)
Fi({, +O) = 0.
Substituting equation (3.1 1) into equation (3.8) evaluated at z = +0, and differentiating
equation (3.3) we obtain
sinh y (D - z )
cosh y D
(3.13)
By differentiating equation (3.8) at z = + O and substituting the boundary condition (3.12)
we see at once that F '( {, +0) = FJ ({, + 0). Thus equation (3.13) evaluated at z = + 0 can be
put in the form
(3.14)
(3.15)
Now we can write
y2 =
c2 + i d = [{
- QI
exp(3ni/4)] [{ - (11exp(- 4 4 ) J
(3.16)
43 1
H-polarization induction over an ocean edge
where the first factor is non-vanishing in the lower half, 71 < 0, of the complex {-plane and
the second factor is non-vanishing in the upper half-plane r) > -c. We assume further that
the factorization
K(f) = K+(f>K 4 f )
(3.17)
can be made, where K+ and K- are non-vanishing and analytic in the half-planes 7) > - c and
q < 0 respectively. The details of this factorization are given in the Appendix. Equation
(3.14) can now be rearranged in the form
( - Q exp(-ni/4)
D [f
(3.18)
K+(0
- a exp(3ni/4)]
Making use of the identity
1
-=
SK+(0
-
K+(0) - K+(0
CK+(0) K+(f)
+-
(3.19)
1
SK+(O)
we can decompose equation (3.18) still further to obtain
K-({) F:((, +o)
D [f - 01 exp(3ni/4)]
+
~a exp(-ni/4)
fK+(O)
(3.20)
= G(f)
where
The left-hand side of equation (3.20) is analytic in the lower half-plane 7) < 0 while the righthand side of equation (3.21) is analytic in the upper half-plane ?) > -c. Thus the two expressions for G are equal only in the strip - c < q < 0, but they can be regarded separately as
analytic continuations of G into the lower and upper half-planes respectively. The function
G is therefore analytic in the entire {-plane. Moreover, as f m in the appropriate halfplanes we have (i) K&) = O(f”2) as shown in the Appendix, and (ii) F,(f, +0) = O(l/f)
and F$’, +0) = o(l/f) by the well-known asymptotic property of one-sided Fourier transforms plus the fact that continuity of B, and Ey at (0, +0) requires f,(+O, +0) = Bo and
f-’(-O, +0) = 0. Thus C(()+O as f - + m in equations (3.20) and (3.21), and since C is
analytic everywhere we deduce from Liouville’s theorem that C({)=O. It follows from
equation (3.21) that
-+
A
F+(f,+0) =-
.f
+
A(Yexp(-ni/4) K+(f)
f K - 01 exp ( - n W l
(3.22)
K+(O)
and hence, by equations (3.3), (3.9) evaluated at z = +0, and (3.1 l), that
F(f, z) =
A a exp(-ni/4)&({)
coshy(D
-
z)
{ [( - 01 exp(-ni/4)] K+(O) coshyD
(3.23)
’
Since F is an analytic function of { in the strip - c < 7) < 0, the inverse of the Fourier
transform (3.1) is
(3.24)
432
M A. Nicoll and J. T. Weaver
for any b, 0 < b < c. Substituting equation (3.23) in (3.24) and recalling that A=iB,,/fi,
we obtain as the final solution for the magnetic field in the region 0 < z < D,
s
---ib
B, = ___
2nK+(O) ---it,
K + ( n e x p ( - ~ f ) coshy(D-z)
{[f
- 01exp(-ni/4)] coshyD
df .
(3.25)
By virtue of equations (3.15) and (3.17) we may substitute
K+(f) K+(O)
yK-(O) coth yD
(3.26)
a d K - ( { )cothaD-\/;
in equation (3.25) to express the solution in the alternative form
--ib
B, =
y exp(-&f) coshy(D - z ) df f [f - a exp(-ni/4)] K ( { )sinhyD'
(3.27)
We shall use this form of the solution when y > 0; equation (3.25) will be used when y < 0.
4 Evaluation of the integrals
Consider first the case y > 0. The integral (3.27) is evaluated in the usual manner by
examining
and where rl is the closed contour shown in Fig. 2, consisting of the line q = - b ,
-R < E < R , and a semi-circular return path, f = R exp (@),- n < 4 < 0. The semi-circle is
F i g = 2. The contours rl and
r2in the complex 5 plane.
H-polarization induction over an ocean edge
433
drawn in the lower half-plane so that, for y > 0 , the factor exp ( j R sin 4) in the integrand of
I, ensures that the integral along the return path vanishes as R -+m.
Since K-(c) is analytic and non-zero for q < 0 g ( { ) is analytic on and within the contour
rl, and although y has branch points at aexp(-ni/4) and aexp(3ni/4) the functions
coshy(D - z ) and y-' sinhyD are analytic in the entire { plane. Thus the only singularities
enclosed by rl are the simple pole 01 exp (- ni/4) and other simple poles corresponding to
the zeros of y-'sinhyD in the lower half-plane. These zeros are located at the points
5 = - iy, (n = 1, 2,3, . . .) where
y, = (ui
+ ia2)'I2, u,
= nn/D
(4.3)
all lying on the rectangular hyperbola Eq = - $$a2as shown in Fig. 2. The principal branch
of the square root in equation (4.3) must be taken so that Re 7, 2 0. We note that
The residue B0at the pole a exp (-ni/4) is immediately found to be
&$=g(a!exp(-ni/4))
exp(-yafi)/D.
(4.5)
The residue 9,at the pole -iy, (n = 1 , 2 , 3 , . . .) is given by
which can be simplified with the aid of (4.3) and (4.4) to give
-
As R + 00 all the poles become interior to
equation (3.27) gives
rl and
the residue theorem together with
m
Bx=-2ni
1 g,,,( j > O ) .
n=O
Substituting the expressions (4.5) and (4.6) into this result and using equation (4.2) we can
express the magnetic field in the region y > 0 in the form
(4.10)
the infinite product being the expression for K-(O)/K-(-iy,) derived as equation (A16) in
the Appendix. Equation (4.9) will be recognized as a Fourier cosine series expansion of B,
in the region 0 < z < D.
For the case y < 0, we consider the integral
(4.1 1)
(4.17)
and where 1; is the closed contour returning via the upper half-plane as shown i n Fig. 7. The
calculation follows exactly the same line as before except that the poles are now at
5 = iy, - y 2 ( n = 1 , 2, 3, . . .). where yn - % is defined as in equation (4.3). The residue at
iy, - y2 is
(4.13)
There is an additional pole at ( = 0 with residue
w =h ( 0 ) cosh [a(D
~
z)4
1
(4.1 4)
cosh aD&
When R
-+ m
equation ( 3 . 2 5 ) and the residue theorem give
(4.15)
Substituting from equations (4.12), (4.13) and (4.14) in this result we obtain for the magnetic field in the regiony < 0,
(4.1 6)
44.1 7 )
and where K+(iy,, - yZ)/K+(O)has been expressed as an infinite product according to equation
(A15) derived in the Appendix. The first term in equation (4.16) is the asymptotic form of
the solution as y + - m as defined in (3.6). The other term is a Fourier sine series expansion
in 0 < z < D of the remainder field.
5 Mantle a t infinite depth
As D + 00 our model approaches the one investigated by R. C. Bailey (see Section 1) in
which there is no perfectly conducting mantle. It is of interest to obtain the solutions for
Bailey’s model as a limiting case of the solutions (4.9) and (4.16).
We note first that according t o the results (A24) and (A25) derived in the Appendix,
-
K - ( o ) / K - ( - - ~ ~ , ) a%(y, t ad)-”’ exp(nil8)
-
K+(i yn - y2 )/K+(O) a-
yz (y,
- yz
+ ad)”’ exp( --ni/8)
(5.1)
(5.2)
as D -+ 00. Thus referring to the definitions (4.10) and (4.17) we see that a, - a , and
b , b, where
-
H-polurixtiotr
i / i J t i i . / i o / i o w r uti o c e m
edge
43 5
II' we write I ( , , + I = u , ~+ A//( 1 1 = I . 2 . 3. . . .). tlien accot-ding to tlic definition (4.3)
Sir = r/U. I'liLiS tlie liniiting l'oi-ni o l ' t l i u soluiiori (4.0) can be written as
i n the litnil. tlic Fouriei- series passes into a Fourier cosine integral i n the variable I I . Therefore we Ieplace ti,, by t i in equation (5.5). which means that y,*becomes y i n tlie expression
for B , l . t o obtain the solution for tlie magnetic licld in the I-egion -1) > 0
corresponding to a niaritle at infinite depth.
When D --t m in equation (4.16). the Fourier series approaches. in similar fashion. a
Fourier sine integral in which u, - y2 is replaced the variable u , while the first term reduces t o
B0 exp( ( Y Z ~ ) Thus
.
the solution in the region.pj < 0 for the mantle at infinite depth is
(5.7)
6 Numerical calculatioiis
For nuinerical work it is convenient t o scale all lengths in terms of the skin depth of the
conductor by defining the dimensionless variables
y ' = y a / f i , z' = z a / f l ,
D'
= Dcy/fi. ii;, = nn/D',
-yA = (u:
+ 2i)'".
(6.1)
The solutions (4.9), (4.16). (5.6) and (5.7) for the dimensionless magnetic field Bi = Bx/Bo
can then be expressed in terms of dimensionless quantities by simply substituting the primed
variables defined in (6.1) for the corresponding unprinied ones, and by replacing a J w i t h
1 + i everywhere. According to equations (2.2) the dimensionless electric field
E' = (2p0o/w)"* E/Bo
(6.2)
is given at once by the derivatives
E; = a&:iaz',
~ ,=t
~
agiay'.
(6.3)
except at the singular pointyo, 0).
The dimensionless magnetic and electric fields were computed at selected points b',z ' )
for D' = 0.0 1, 0.1 and 1.O. For a fixed conductivity and frequency, these values correspond
to different depths of the mantle, but if the mantle depth is regarded as fmed, they correspond t o inducing fields of different periods. The period o f the inducing field corresponding
t o the three values of D' are given below for different mantle depths D in km. A crustal
conductivity of 10-smmho/m has been assumed, b u t the periods can easily be converted to
correspond to other conductivities because period and conductivity change together by the
same factor.
D'
0.0 1
0.1
1
6 0 0 km
39.5 hr
23.7 niin
14.2 s
Mantle depth D
300 km
9.9 hr
5.9 min
3.6 s
50 k m
16.4 min
9.9 s
0.1 s
43 6
M.A . Nicoll and J. T.Weaver
..
..
Y (skin
I
-2.0
0.0
- 1.0
depths
0.0
..o
1
I.o
T
0.25
c
c
a
W
0
c
.-
-
0.5
2
v)
IV
0.75
I .o
I N PHASE CURRENTS
( L I N E S of CONSTANT MAGNETIC F I E L D )
(a)
- 2.0
(b)
Y
-1.0
(skin
0.0
OUT OF PHASE CURRENTS
( L I N E S of CONSTANT MAGNETIC F I E L D )
Figure 3. (a) In-phase currents and (b) quadrature currents for D ' = 1 . The figures denote the constant
values of B,/B,, along the current lines.
The period also varies as (D/D'j2so that its values for other values o f D and D' are easily
found from the figures tabulated above.
Fig. 3 illustrates the pattern o f induced current flow for the case D' = 1 . Both the
currents in phase with the inducing field and the quadrature currents 90" out o f phase are
shown. The diagrams illustrate quite clearly how electric currents in the ocean can arise as a
result of induction by a horizontal magnetic field. They are drawn u p into the ocean from
the crust and mantle with the greatest concentration near the ocean edge. Such a flow is
only possible because a crust with non-vanishing, albeit small, conductivity has been
included in the model.
Only the electric field shows any variation along the surface of the earth o n the landward
side of the ocean edge. In practice, the usual field measurement is the surface impedance
I poE,,/Bo I. Graphs showing the variation of the closely related dimensionless quantity
IE; I along the surface z = 0 for negative values of y are given in Fig. 4 for the different
values of D ' . The expected sharp increase in the surface impedance as the coastline is
H-polarization induction over an ocean edge
y'
Figure 4. Variation of the magnitude 1;"
Earth.
(skin depths)
I of the dimensionless electric field along the surface of the
y'
1.0
0
:
437
I .o
(skin depths)
2.0
3
I
y ' (skin depths)
3
Figure 5. Variation o f (a) the in-phase (real part) and (b) the quadrature (imaginary part) dimensionless
magnetic field B ; along the o e a n floor.
438
M. A. Nicoll and J. 7: Weaver
approached is clearly depicted by these graphs. F o r a given conductivity and frequent-y, the
surface impedance at any given point becomes larger as the mantle depth increases through
the values 0.01, 0.1 and 1 skin depths. The trend does not continue indefinitely however.
F o r larger values of D' the surface impedance starts t o decrease again, and eventually
approaches its limiting value corresponding t o D' = 00, which can be calculated using the
solutions obtained in Section 5 . The variation of the surface impedance for D' = ~0 is given
b y the curve with the broken line in Fig. 4. The type o f behaviour described above is already
familiar from the well-known magneto-telluric curves for a two-layer earth with a highly
conducting second layer (Keller & Frischknecht 1966, p. 2 19), where the apparent resistivity first increases with increasing thickness of the t o p layer t o a maximum greater than its
limiting value before descending t o approach the resistivity of the top layer as the thickness
becomes infinite. Note that as y + - 03 the curves in Fig. 4 are asymptotic to the value
(2op,)"* where pa is the apparent resistivity. Thus the D' = w curve tends t o fias y + ~0
since pa = l / o in this case.
The only other field component in this problem which can be measured in practice is the
magnetic field parallel to the coastline recorded o n the ocean bottom. In Fig. 5 we have
plotted the real and imaginary parts of the dimensionless magnetic field Bx/Bo along the
surface z = +0, y > 0, for the different values o f D ' . The general behaviour of these curves
compares very favourably with the numerical solutions obtained b y Brewitt-Taylor (1976)
for an ocean whose half-width is large compared with the mantle depth. The rapid attenuation of the magnetic field as it diffuses out under the ocean is clearly evident from the
diagrams. When D' lies between 0.01 and 0.1, which covers a wide range of periods and
mantle depths likely t o occur in practice, the magnetic field at any point does not change
appreciably. However it is substantially different for D' = ~0 which shows that the presence
of a highly conducting mantle has an important effect on the results.
~
Acknowledgment
Financial support from the National Research Council of Canada in the form of a scholarship (MAN) and a research grant (JTW) is gratefully acknowledged.
References
Bailey. R . C., Edwards, R. N., Garland. G. D.. Kurtz, K. & Pitcher, D.. 1974. I'lectrical conductivity
studies over a tectonically active area in 1,:astern Canada, J. (;eoniag. Geoelrctr.. 26, 125.- 146.
Brewitt-Taylor. C. K., 1975. A model tor the coast ctfect, Ph-ys. Earfli planet. h i t . . 10. 151 - 158.
Brewitt-'l'aylor, C. R., 1976. A model for the coast-effect further results, Phys. Earrh planer. f n t . , 13,
9 - 14.
Bullard. 1 . C. & Parker, K . L., 1970. Electromagnetic induction in the oceans, Tl1c sea, ed. A. 1;. hla\wcll.
vol. 4 , pp. 695 730, Wiley, New York.
Keller. (;. V. & I:rischknecht, 1;. C., 1966. Electrical tncthods it1 geophysical prospectitig, Pergamon.
London.
Noble. B.. 1958. Methods based on the Wiener--Hop/ technique fi)r rlicp sohition of partial dijjercntial
eyuarioris, Pergamon, London.
~
~
Appendix
Defhing y = (C2 + '")rci
and tinl = m n / D its in equations (3.4) and (4.3) we niay quote a
well-known formula ( e g Noble 1958, p. 40) in the form
H-polarization induction over an ocean edge
439
This result is readily established by using the standard procedure for representing an entire
function as a infinite product. We now write S = S+S- where
m =1
and where ym = (u& + ia2)"' as defined in equation (4.3). Defined in this way S+ has no
zeros in the upper half-plane 77 > - c, while S- has no zeros in the lower half-plane 77 < 0.
The exponential factors ensure the absolute and uniform convergence of the infinite products, and hence the analyticity of s+and s-.Since
we can write equation (3.17) in the form
K+(<)K-(<)= W, 2 D ) / [ S ( <D>12.
,
(A41
I t follows that K+ and K - will have the required properties of being analytic and non-zero in
the half-planes 7) > - c and 77 < 0 respectively if we define
K,
(0= s, (5, 2 0 ) / [ S ,(5, a]'.
(AS)
Noting that m n / 2 D = uxm we can rearrange equation (A2) to obtain
Substitution of equations (A2) and (A6) in equation (AS) yields explicit expressions for K+
and K-, namely
{
--f
It remains only to verify that the functions K , ({) have the desired algebraic behaviour as
w. As { + m in the lower half-plane q < 0.
%
and defining 0 = i{U/n we can write this in the form
I n this expression, the numerator and denominator can be regarded as separate infinite
products each of which can be evaluated in terms of the factorial function as follows (Noble
1958. p. 41)
440
M. A. Nicoll and J. T Weaver
where Cis Euler’s constant. It follows from (A9)that
as f + CQ, the last result following from Stirling’s formula, and one readily derived from it,
namely
(0
-
I) !
- (2n/~)”*(P/e)P,
(0
!
~
- (2n)”*
(P/e)@,
(A13)
both valid for large I P I Similar conclusions can be drawn about the asymptotic behaviour of
K.. in the upper half-plane. We deduce, therefore, that K + (f) = O(f”’)as f + 00.
Substituting f = 0, in equation (A7) we obtain at once
whence the further substitutions f = iy,, -Y2, and f = -- iy,, in (A7) give
The ratios (A15) and (A16) appear in the solutions for the magnetic field obtained in
Section 3 .
Finally, we examine the asymptotic behaviour of the ratio
K-(O)
=n
K-(C)
~
1 + i f / y m-
m =I
1 +(firm
-
.~
‘/2
as D + m. This can be written as the ratio P/Q where
ffi
p
=
m -1
Q
=
[( 1 t iD2/u2 ) l i 2
-
(1
+ ifD/au,] exp(--x/m)
+ iD2/uL)”*exp(-Xo/m)
ifD/aum
exp(,-x/m)
rI [( 1 + iD2/uk+ iD2/uk+ Y2)li2
exp(-xo/m)
.% ) l ”
- 1/,]
_-.___.A.
rn=l
(1
and where we have written u , = mn/a. The functions x and xo are chosen to be
Using the identity (A1 1 ) and some simple algebra we can show that
H-polarization induction over an ocean edge
44 1
The infinite product in equation (A2 1) is clearly convergent and in fact it is uniformly convergent when regarded as a product of functions of D.Thus
since Z - t O as D + m. A similar result holds for the infinite product Q , except that the
identity (A10) is used. When D -+ 00, therefore, we have
where x and xo are defined in (A20).We may again userhe asymptotic formulas in (A13) to
simplify the above expression to
Since K ( c ) = y D coth y D
K+({) - [{
lim - D-00 K+(O)
15
- (Y
- y D as D
-+
e~p(-ni/4)]"~
& exp(3ni/8)
00,
it follows immediately from (A24) and (3.17) that