Download Theoretical investigation of the ocean

Physics of the Earth and Planetary Interiors, 42 (1986) 246-254
Elsevier Science Publishers B,V., Amsterdam - Printed in The Netherlands
246
Theoretical investigation of the o c e a n - c o a s t effect
at a passive continental margin
Gaston Fischer and J.T. Weaver *
Observatoire Cantonal, CH-2000 Neuchhtel (Switzerland)
(Received August 28, 1985; revision accepted December 23, 1985)
Fischer, G. and Weaver, J.T., 1986, Theoretical investigation of the ocean-coast effect at a passive continental margin.
Phys. Earth Planet. Inter., 42: 246-254.
The idea that oceanic lithosphere is thinner than continental lithosphere is widely accepted even though one would
like to see clearer evidence to support it. In fact, the very concept of lithosphere is still a matter of some debate. If there
is indeed a variation in the thickness of the lithosphere at continental margins and if this change is associated with a
lateral variation in electrical conductivity one may envisage detecting it with electromagnetic soundings methods. A
model of a passive continental margin has therefore been investigated to test whether this would be feasible. It has been
found that the well-known but strong ocean-coast effect masks the minor lithospheric effect in magnetotelluric
soundings performed on the shore. Inductive soundings, on the other hand, are highly sensitive to lateral variations in
electrical conductivity. An analysis in terms of the induction arrow has shown that such soundings carried out on land
would be perfectly suitable to reveal a changing lithospheric thickness, if the continents merely extended to the oceanic
coast. However, the presence of only a narrow continental shelf of 100 km width under 250 m of sea-water produces an
overriding coast effect ahead of the margin, and thus renders electromagnetic methods unsuited to reveal a changing
lithospheric thickness.
1. Introduction
T h e past few years have seen a n u m b e r of
papers dealing with model calculations of the
o c e a n - c o a s t effect. F o r the most part these models were t w o - d i m e n s i o n a l (2-D) a n d the ocean was
simulated with a perfectly c o n d u c t i n g semi-infinite sheet. Bailey (1977) gave a n analytic solution of the B-polarization (B-POL) response of
such a simple ocean at the surface of a u n i f o r m
half-space. Nicoll and Weaver (1977) extended
this work to include a perfect conductosphere at
typical m a n t l e depths of 5 0 - 6 0 0 kin. The Epolarization (E-POL) response of the simple model
was solved semi-analytically by Fischer et al. (1978,
1980) a n d a n analytical solution in closed form
* On leave from the University of Victoria, Victoria, B.C.,
V8W 2Y2 Canada.
0031-9201/86/$03.50
© 1986 Elsevier Science Publishers B.V.
was then given by Raval et al. (1981).
Paralleling these developments great progress
was made i n analytical a n d n u m e r i c a l calculations
with models involving superficial thin sheets of
finite conductance. These new methods are often
used to tackle 3-D problems (cf. Vasseur a n d
Weidelt, 1977; D a w s o n a n d Weaver, 1979b;
Weaver, 1979, 1982; M c K i r d y a n d Weaver, 1983;
M c K i r d y et al., 1985) in c o n n e c t i o n with regional
studies a n d are very suitable for looking at various
forms of the transition from l a n d to ocean. G r e e n
a n d Weaver (1978) considered a model simulating
a c o n t i n e n t a l m a r g i n out at sea a n d M c K i r d y a n d
Weaver (1984) generalized this m e t h o d to include
a stratified substratum. D a w s o n a n d Weaver
(1979a) a n d D a w s o n et al. (1982), respectively,
solved analytically the B-POL p r o b l e m s of two
u n i f o r m thin half-sheets a n d of two generalized
thin sheets over a u n i f o r m half-space, and Daw-
247
Thin
8beet
with
conductance "l"
×
Ef~
I~H
y
Om
o
3
I
Z
model
1 :
P3
=
model
2
:
pz=p1,
modal
3
:
P3
=
Pz
=
P3--
P2
PI
PI/1~
-- P l / I ~
Fig. 1. Sketch of the models considered. The thin conducting sheet represents the ocean whose resistivity is taken to be 0.25 f~m. The
various oceanic depths assumed correspond respectively to a continental margin (d = 250 m, ~"=1000 S) an intermediate ocean
( d = 1000 m, ~-= 4000 S) and a deep ocean (d = 4000 m, "r = 16000 S). With model 1 the perfect conductor (~" ~ o0, d --, 0) is also
considered for comparison. In addition a fourth model is studied, similar to models 2 and 3 with ~"= 16000 S, but involving a 100 km
continental shelf where ~-= 1000 S and extending over the continent from its right-hand edge.
son (1983) o b t a i n e d the a n a l y t i c solution for the
c o r r e s p o n d i n g E - P O L p r o b l e m of two u n i f o r m
half-sheets.
I n the p r e s e n t p a p e r we e x a m i n e the coast-effect when oceans of different d e p t h s are s i m u l a t e d
with u n i f o r m thin sheets of various c o n d u c t a n c e s .
F o r the s u b s t r a t u m we choose either a h o m o g e neous half-space ( m o d e l 1) or a s s u m e an a s t h e n o sphere with a ten-fold increase in conductivity. In
m o d e l 2 this a s t h e n o s p h e r e is p l a c e d at a c o m m o n
d e p t h of 100 k m a n d in m o d e l 3 its d e p t h is set at
100 k m u n d e r the l a n d a n d at 50 k m u n d e r the
ocean. These various m o d e l s are sketched in Fig.
1. A fourth m o d e l is then considered, involving a
c o n t i n e n t a l shelf b e t w e e n the coast a n d the deep
ocean.
W h a t we have in m i n d for the p r e s e n t s t u d y is
to investigate w h e t h e r it might b e p o s s i b l e with
m a g n e t o t e l l u r i c a n d / o r i n d u c t i o n s o u n d i n g s to
resolve changes in the structure of the l i t h o s p h e r e
a n d the u p p e r m a n t l e at a passive c o n t i n e n t a l
margin. T h e thickness of the c o n t i n e n t a l lithos p h e r e has in general b e e n taken to be in the range
of 7 0 - 1 0 0 k m a n d for the oceans figures as low as
2 0 - 3 0 k m (cf. Maxwell, 1984) have often b e e n
p r o p o s e d . In a recent s t u d y b a s e d on the r e s p o n s e
o f the p l a n e t to the last d e g l a c i a t i o n Peltier (1984)
c a m e f o r w a r d with c o n t i n e n t a l lithospheric thicknesses in excess o f 200 km. If there is i n d e e d an
i m p o r t a n t r e d u c t i o n of this thickness at passive
c o n t i n e n t a l margins, a n d if this change is associa t e d with a m a r k e d v a r i a t i o n in electrical resistivity, it is w o r t h verifying whether e l e c t r o m a g n e t i c
s o u n d i n g m e t h o d s could c o n f i r m it. H i t h e r t o the
o c e a n - c o a s t effect has b e e n s t u d i e d p r i m a r i l y w i t h
a view to investigating the influence of highly
c o n d u c t i n g oceans on m a g n e t o t e l l u r i c or i n d u c t i v e
responses. These effects are so large (cf. the re-
248
views by Fischer, 1979 and Parkinson and Jones,
1979) that they are likely to mask the much less
significant effect of a changing lithospheric thickness. In principle the oceanic lithosphere and upper mantle can be studied by sea-floor magnetotelluric soundings like those of Filloux (1980, 1982).
However, such experiments are technically difficult and the shielding effect of the conducting
ocean water reduces the available magnetic signal.
Measurements on land would be far easier to carry
out, but it remains to show whether they are likely
to provide the information necessary to elucidate
the structure of passive continental margins.
itself was only one cell in thickness in the case of
the smallest conductance value used, no numerical
problems arose in the finite-difference treatment
of this extreme model. The finite difference method
was also used in models 1 and 2 when computing
the deep ocean response (d = 4 km), as this depth
would violate the thin sheet approximation which
presupposes that d is much less than the skindepth.
3. The magnetotelluric response
In Figs. 2, 3 and 4 the respective responses of
the three models are reproduced. Assuming a sea-
2. Method of computation
COAST EFFECT
MODEL 1
6~
A series of analytical solutions (Dawson and
Weaver, 1979a; Raval et al., 1981; Dawson, 1983)
have provided exact results for the response of
model 1 to a uniform field. However, since it has
been established (Dawson and Weaver, 1979a;
Dawson, 1983) that the numerical method of
Green and Weaver (1978) in both the E- and
B-polarization modes is extremely accurate, and
since the numerical method is much faster on the
computer than the evaluation of the closed form
integrals in the analytic solution, it was decided to
use this program to calculate the responses of
model 1. Only for a perfectly conducting ocean
were analytical results used since these have been
tabulated by Raval et al. (1981). The calculations
for model 2 were made with the aid of the computer program developed by McKirdy and Weaver
(1984) which is based on an extension of the
Green and Weaver theory to include a layered
half-space beneath the thin sheet.
The only difficulty occurred with model 3 because, at present, the thin sheet programs cannot
accommodate a substructure which includes lateral
variations of conductivity. Thus these calculations
were performed by using the full 2D finite-difference program of Brewitt-Taylor and Weaver
(1976). The thin sheet representing the ocean was
simulated by a layer of finite thickness with a
given resistivity of 0.25 £m. Even though grid
cells representing this ocean were extremely elongated in some parts of the model and the layer
/"
SB
B-POL
30
20
[,::. . . . . . . .
10
0
I
-3
"2
21
~
l
2
DISTANCE
B-POL
v
~@@@
/
J
DISTANCE
Fig. 2. M T response of model 1. To avoid crowding the graph
the curves corresponding to ~-= 16000 S have not always been
plotted. We recall that the horizontal distances are given in
terms of the skin-depth in lithospheric material. One unit
therefore amounts typically to 100 km for material of 100 ~2m
resistivity at a period of 40~r2 = 400 s. The same scale length
applies to all the following figures.
249
COASTEFFECT
COASTEFFECT
MODEL 2
66
MOOEL 3
......
70
~ttt~:::c:L[-',,I
56
60
46
50
30
40
E-POL
~.-,
_~<:222]--.,',
B POL
C~
20
]0
20
!
I
~
6
22
-]6
Z1
i0
i'
}
-
0_3
-
-2
-1
8
1
B-PDL
POL
"RL'
-,.
,,,.,
o~
E-POL
~,.,.,~,
~ " '~ I
~'<,',,I
'0,,I
"'~'--. . . . . . . .
n-.
-:
"2
Jl
2
DISTANCE
OISTANCE
0
OISTANCE
1
4¢8~
T.
loao s
s
2
'<
03
-
-'z
-'l
6
DISTANCE
1
Fig. 3. MT response of model 2. To avoid crowding the graph
some curves are left out.
Fig. 4. MT response of model 3. To avoid crowding the graph
part of the z=16000 S curve is left out in the apparent
resistivity plot.
water resistivity of 0.25 f~m the assumed ocean
conductances of ~"= (1, 4, 16) x 103 S correspond
to bathymetric depths of d = (2.5, 10, 4 0 ) × 102
m, i.e., respectively, a very shallow sea (e.g., a
typical continental margin), an intermediate and a
deep ocean. In Fig. 2 (Model 1) we have also
plotted for reference the response of a perfectly
conducting ocean (~" = oo, d = 0). The distance
scale is given in terms of the skin-depth 81 in
lithospheric material, and at the chosen period of
T = 2rr/¢0 = 40~r 2 s yields a unit of 100 km for the
assumed lithospheric resistivity of Pl = 100 ~2m.
The apparent resistivities in Figs. 2 - 4 have been
normalized with respect to their value at infinity
on the left, and the horizontal scale is in units of
skin-depth 81 - { ( 2 P l / ~ # 0
) SO that the various
graphs can, to some degree at least, be scaled to
other periods and other resistivities. For, if the
dimensions of the model measured in units of 81,
the resistivity ratio p 3 / / P l = 1/10, and the conductance of the ocean measured in units of 81/01 are
all held invariant then the curves plotted in Figs.
2 - 4 will remain unchanged for all models obtained by varying T and Pl- (Note that since the
sea-water resistivity must have the same value 0.25
S m -1 in all models, holding the dimensionless
conductance of the ocean constant causes the ocean depth to scale like ~
.)
The most striking feature of the ocean-coast
effect, as displayed in Fig. 2, is the large anisotropy of the apparent resistivity on land close to
the shore, even for a very shallow ocean, and the
abrupt change of the B-POL apparent resistivity
Os (or Pyx) at the coast. Just on the landward side
250
of the coastal boundary ( y ~ 0 - ) the electric
field has a singularity like 1/I y l 1/2, whereas on
the seaward side (y---, 0 + ) it remains finite
(Dawson and Weaver, 1979a). It follows that since
P~ = Z
(1)
the apparent resistivity exhibits a similar behaviour. These properties are easily understood. In
B-POL the surface magnetic field is uniform
whereas the current, which is distributed to depths
of order 6 a far inland, tends to accumulate at the
coast to concentrate in the ocean. This results in a
strong increase of the electric field on land near
the shore. Crossing into the sea the large discontinuity of conductivity produces a great drop in
the surface electric field. These features are further
magnified in the apparent resistivity PB as this
parameter is proportional to the square of the
electric field amplitude. As ~-~ ~ , PB becomes
smaller over the ocean, and vanishes altogether on
a perfectly conducting ocean. The phase falls below 45 °, as expected for a structure with a highly
conducting overburden (cf. Fischer, 1985).
With E-POL the surface magnetic field is no
longer uniform near the coast, but the behaviour
of Pe (or Oxy) is still controlled largely by the
electric field. Continuity of its tangential component means continuity of Ex, which must drop
steadily to the low values typical of the ocean
surface. Again this translates into stronger variations for PE. However, the ocean edge is a point
of jump discontinuity for By (Dawson, 1983) and
since PE is given by
°le
pe = - j
ture of any structural change of the lithosphere
and upper mantle which may occur at the transition associated with a passive continental margin.
4. The induction arrow
Whereas the magnetotelluric apparent resistivity and phase describe the relation between horizontal electric and magnetic fields the induction
arrow depends only on the vertical and horizontal
components of the magnetic field. Soundings of
the three magnetic components are usually performed with arrays of synchronous stations and
are called induction soundings. With Bx and B v
the horizontal and B: the downward pointing
vertical components we define the induction arrow
in the horizontal plane
e r = (ReA, ReB)
Pi = (ImA, ImB)
(3)
where A and B are given by
(4)
B, = A B x + BBy
Here we note that we have assumed a time dependence of the form exp( + iwt) (cf. Lilley and Arora,
1982; Chen and Fung, 1985).
For the simple 2-D geometry that we have
chosen in Fig. 1 the E-POL configuration involves
only the magnetic components B), and B,; so
MODEL
i
0.5
/
I
/
IZZZ
2
(2>
""
0
r,--
it experiences a jump discontinuity at the coast as
well. This is also true for the phase.
What is most striking in a comparison of Figs.
2 - 4 is their similarity. The dominant effect by far
is the presence of the ocean; variations in the
structure of lithosphere and upper mantle have
only a minor influence on the response curves of
apparent resistivity and phase. Magnetotelluric
measurements on land near an ocean coast are
therefore unlikely to shed much light on the na-
-os!
z
"2
21
ml
DISTANCE
Fig. 5. The inductive response of model 1 represented in terms
of the induction arrow.
251
MODEL 2
MODEL 3
~:-0"51
;",
0 ~-~-----~
. . . . . ........ , " " "'-, .I. . . '. . .~. . . .". .
~°\ \ , , : ~
\ \',i
c- -~.S
-t -3
-'2
{
-'1
j
3
DISTANCE
Fig. 6. The inductive response of model represented in terms of
the induction arrow. For the sake of clarity the intermediate
curve with z = 4000 S is left out.
there is no factor A. In the B - P O L configuration
there is only a B x c o m p o n e n t and thus no arrow
at all. The induction vector is therefore directed
along the y axis, i.e. it is perpendicular to our
assumed straight coastline.
Figures 5, 6 and 7 display the behaviour of the
real and imaginary parts of the induction arrow
for our three models. While there are again obvious similarities a m o n g these three figures, there
are also important differences. O n land the real
c o m p o n e n t always points away from the sea, but
the range over which there is an important c o m p o nent is m u c h larger in model 3 than in model 1.
With the deep ocean (~ = 16000 S) for example
one has - P, > 0.5 for - y < 0.30, 0.36 and 0.65
for models 1 to 3, respectively, and - P, > 0.2 for
- y < 0.78, 0.82 and 1.44. At sea the real c o m p o nent reverses direction at some distance from the
coast, except for T --- o0 when there is no induction
arrow over the perfect conductor. Since B z = 0 on
the surface of a perfect conductor, it is to be
expected that the greater the c o n d u c t a n c e of the
ocean, the smaller the length of the induction
arrow. In other words we expect to find a longer
induction arrow over a shallow sea than over a
deep one.
The magnitude of the imaginary c o m p o n e n t of
-:
-2
-1
8
1
2
3
DISTANCE
Fig. 7. The inductive response of model 3 represented in terms
of the induction arrow. For the sake of clarity the intermediate
curve with z = 4000 S is left out.
the induction vector is, in general, smaller on land,
and larger at sea, than the magnitude of the
corresponding real component. F o r a deep ocean
it points toward the coast on land, except for a
short range close to the coast where it reverses
direction and becomes very large. This reversed
range widens for shallower seas, but the c o m p o nent then weakens. A nearly antisymmetric behaviour occurs at sea: the imaginary vectors generally point away from shore, except in a narrow
range close to the coast. For shallow seas this
range is only slightly wider, but here it exhibits a
stronger amplitude. Beyond the narrow range of
reversed imaginary c o m p o n e n t s the arrows at first
are stronger for deep oceans, but far out to sea the
arrows for shallow oceans again predominate. For
a perfect c o n d u c t o r the width of the coastal ranges
narrows to zero and the imaginary arrow completely vanishes over the ocean.
As with the real c o m p o n e n t the imaginary arrow
undergoes an important change between model 2
(Fig. 6) and model 3 (Fig. 7), suggesting that
induction soundings m a y indeed be suitable to
resolve variations in the structure of the lithosphere and upper mantle at a passive continental
margin.
252
5. The effects of a submerged continental margin
In the m o d e l s discussed so far we have set the
edge of the c o n t i n e n t at the s e a - c o a s t . As is well
known, however, these two edges d o n o t coincide,
especially at passive margins. I n m o s t instances
the c o n t i n e n t a l m a r g i n is situated several h u n d r e d
kilometres on the s e a w a r d side of the coast. O n l y
a l o n g the shores of A f r i c a can long coastal sections be f o u n d where the c o n t i n e n t a l shelf is less
t h a n a b o u t 100 k m in width. But the coast effect
of even a shallow ocean is so strong that a subm e r g e d c o n t i n e n t a l shelf of only 100 k m c o u l d
well d e s t r o y any h o p e of observing a changing
lithosphere at the true m a r g i n of the continent.
T o test this hypothesis we have a n a l y s e d a
fourth m o d e l involving a 100 k m shelf. M o d e l 4 is
therefore identical with models 2 a n d 3, except
that now the thin c o n d u c t i n g sheet is p r o l o n g e d a
d i s t a n c e of 100 k m to the left. F r o m its new left
edge the sheet c o n d u c t a n c e is set at ~- = 1000 S for
the first 100 k m a n d then at • = 16000 S to the
r i g h t - h a n d m o d e l b o u n d a r y . In the actual 2-D
MEASUREMENTS
MODEL 4
ON S H E L F
m o d e l the ocean resistivity is m a i n t a i n e d at p =
0.25 t i m b u t the b a t h y m e t r i c d e p t h is taken as
250 m over the shelf a n d 4000 m a b o v e the oceanic crust.
Magnetotelluric or induction m e a s u r e m e n t s
w o u l d be difficult to p e r f o r m at the surface of the
ocean. A t the same time, however, these measurements will have to be carried out as close to the
c o n t i n e n t a l m a r g i n as possible if one hopes to
derive any i n f o r m a t i o n a b o u t the oceanic lithosphere. W e have, therefore, c o m p u t e d the m o d e l
responses in the c o n t i n e n t a l shelf area b o t h at the
ocean surface a n d at the ocean floor. O u r c o m p u tations show, as expected, that magnetotelluric
soundings would be totally i n a d e q u a t e to reveal a
changing lithospheric thickness, regardless of
whether the m e a s u r e m e n t s in the shelf area are
carried out at the ocean b o t t o m or at its surface.
U n f o r t u n a t e l y , as Fig. 8 shows, the same conclusion is also o b t a i n e d for i n d u c t i o n soundings when
m e a s u r e m e n t s are carried out at the ocean floor
over the shelf area (the answer is also negative for
sea-surface soundings over the shelf).
FLOOR
l
i
CI_ L
IMAG
-1_:
-'2 --
0
2
3
DISTANCE
Fig. 8. The inductive response of model 4 represented in terms of the induction arrow. Over the continental shelf the arrow is
computed at the ocean floor. The full line corresponds to an extension of model 3, i.e. with a changing lithospheric thickness. The
dashed line arises from an extension of model 2. Note the minute difference between the two curves.
253
6. Conclusions
C o n t i n e n t a l l i t h o s p h e r e is u s u a l l y t a k e n to b e
thicker than oceanic lithosphere, although even
t o d a y t h e r e are h i g h l y d i v e r g e n t o p i n i o n s as to
w h a t c o n s t i t u t e s the l i t h o s p h e r e a n d h o w t h i c k it
r e a l l y is ( M a x w e l l , 1984; Peltier, 1984). If the
c h a n g i n g l i t h o s p h e r i c t h i c k n e s s at a p a s s i v e c o n t i n e n t a l m a r g i n is a s s o c i a t e d w i t h a m a r k e d varia t i o n o f t h e d e p t h p r o f i l e of t h e e l e c t r i c a l resistivity, e l e c t r o m a g n e t i c s o u n d i n g m e t h o d s m a y b e
d e e m e d c a p a b l e o f r e v e a l i n g this m a r g i n s t r u c t u r e .
H o w e v e r , the s i m u l t a n e o u s o c c u r r e n c e o f the
s t r o n g o c e a n - c o a s t e f f e c t m a s k s the l i t h o s p h e r i c
effect. T h i s is e s p e c i a l l y t r u e for t h e m a g n e t o t e l l u r i c r e s p o n s e w h i c h is t h e r e f o r e ill-suited to reveal the changing lithospheric thickness through
m e a s u r e m e n t s o n land. T h e i n d u c t i v e r e s p o n s e , o n
t h e o t h e r h a n d , is well k n o w n for its s e n s i t i v i t y to
lateral changes within conductive structures and
w e h a v e s h o w n t h a t in t h e a b s e n c e o f a c o n t i n e n tal shelf u n d e r t h e o c e a n s u r f a c e it c o u l d c o n s t i t u t e a s u i t a b l e m e t h o d to r e v e a l a c h a n g i n g
l i t h o s p h e r i c thickness. H o w e v e r , a n a r r o w subm e r g e d shelf of o n l y 100 k m w i d t h u n d e r 250 m of
sea-water produces such a strong coast effect ahead
o f t h e c o n t i n e n t a l m a r g i n , t h a t it also r e n d e r s
i n d u c t i v e m e t h o d s u n s u i t e d to r e v e a l a c h a n g i n g
l i t h o s p h e r i c thickness, e v e n w h e n t h e s e m e a s u r e m e n t s are c a r r i e d o u t at sea o n the shelf f l o o r
f i g h t up to the e d g e o f the c o n t i n e n t .
Acknowledgements
This research project received financial support
f r o m the Swiss N a t i o n a l S c i e n c e F o u n d a t i o n a n d
the Natural Sciences and Engineering Research
C o u n c i l of C a n a d a . It was b e g u n w h i l e o n e of us
( J . T . W . ) was a v i s i t o r at the O b s e r v a t o i r e C a n t o n a l
o f N e u c h ~ t e l . T h e a u t h o r s are also g r a t e f u l to B.V.
L e Q u a n g w h o c a r r i e d o u t m u c h o f the c o m p u t a t i o n a l work.
References
Bailey, R.C., 1977. Electromagnetic induction over the edge of
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