Chapter
6
The
shape
of
the
Earth
When
Galileo
let
his
balls
run
down
an
inclined
plane
with
a gravity
which
he
had
chosen
himself
...
then
a
light
dawned
upon
all
natural
philosophers.
f.
Kant
Terrestrial
planets
are
almost
spherical
because
of
gravity
and
the
weakness
of
rock
in
large
masses
.
The
largest
departures
from
sphericity
are
due
to
rotation
and
variations
in
buoyancy
of
the
surface
and
interior
shells.
Otherwise,
the
overall
shape
of
the
Earth
and
its
heat
flow
are
manifestations
of
convection
in
the
interior
and
conductive
cooling
of
the
outer
layers
.
The
style
of
convection
is
uncertain.
There
are
var-
ious
hypotheses
in
this
field
that
parallel
those
in
petrology
and
geochemistry.
The
end-members
are
whole-mantle
convection
in
a
chemically
uniform
mantle,
layered
convection
with
inter-
change
and
overturns,
and
irreversible
chemi-
cal
stratification
with
little
or
no
interchange
of
material
between
layers.
Layered
schemes
have
several
variants
involving
a
primitive
lower
man-
tle
or
a
depleted
(in
U
and
Th)
lower
mantle.
In
a
convecting
Earth
we
lose
all
of
our
refer-
ence
systems.
The
mantle
is
heated
from
within,
cooled
from
above
and
experiences
secular
cool-
ing.
Global
topography
and
gravity
provide
constraints
on
mantle
dynamics
.
Topography
Although
the
Earth
is
not
flat
or
eg
g
-shaped
, as
previously
believed
at
various
times,
neither
is
it
precisely
a
sphere
or
even
an
ellipsoid
of
rev-
olution
.
Although
mountains,
ocean
basins
and
variations
in
crustal
thickness
contribute
to
the
observed
irregular
shape
and
gravity
field
of
the
Earth,
they
cannot
explain
the
long-wavelength
departures
from
a
hydrostatic
figure.
The
distribution
of
elevations
on
the
Earth
is
distinctly
bimodal,
with
a
peak
near
+ 0.1
krn
representing
the
mean
elevation
of
continents
and
a
peak
near
-4.7
krn
corresponding
to
the
mean
depth
of
the
oceans
lsee
Coogle
Imag
es
h
ypso
rn
e
tr
y
].
This
bimodal
character
contrasts
with
that
of
the
other
terrestrial
planets.
The
spherical
harmonic
spectrum
of
the
Earth's
topography
shows
a
strong
peak
for
l
=
1,
cor-
responding
to
the
distribution
of
continents
in
one
hemisphere,
and
a
regular
decrease
with
increasing
n.
The
topography
spectrum
is
simi-
lar
to
that
of
the
other
terrestrial
planets.
There
are
small
peaks
in
the
spectrum
at
l
=
3
and
I
=
9 -
10,
the
latter
corresponding
to
the
dis-
tribution
of
subduction
zones
and
large
oceanic
swells.
The
wavelength
,
in
kilometers,
is
related
to
the
spherical
harmonic
degree
I
and
the
circum-
ference
of
the
Earth
(in
km)
approximately
by
Wavelength
=
40
040
/
(l
+
0.5)
Thus,
a
wavelength
of
10
degrees
or
1100
Ian
corresponds
to
a
spherical
harmonic
degree
of
about
40.
Active
orogenic
belts
such
as
the
Alpine
and
Himalayan
are
associated
with
thick
crust,
and
high
relief
,
up
to
5 krn.
Older
orogenic
belts
such
as
the
Appalachian
and
Caledonian,
because
of
erosion
and
lower
crustal
delamination,
are
asso-
ciated
with
low
relief,
less
than
1
km,
and
thin-
ner
crusts.
Regional
changes
in
the
topography
of
the
continents
are
generally
accompanied
by
changes
in
mean
crustal
thickness
.
Continents
stand
high
because
of
thick,
low-density
crust,
compared
with
oceans.
TI1ere
is
a
sharp
cut-off
in
crustal
thickness
at
about
50
km,
probably
due
to
delamination
of
over-thickened
crust
at
the
gabbro-eclogite
phase
change
boundary.
As
the
dense
root
grows,
the
surface
subsides,
form-
ing
sedimentary
basins.
Upon
delamination,
the
surface
pops
up,
forming
a
swell,
often
accom-
panied
by
magmatism.
Many
continental
flood
basalt
provinces
(CFB)
erupt
on
top
of
sedimen-
tary
basins
and
the
underlying
crust
is
thinner
than
average
for
the
continents.
The
long-wavelength
topography
of
the
ocean
floor
exhibits
a
simple
relationship
to
crustal
age,
after
averaging
and
smoothing
.
The
system-
atic
increase
in
the
depth
of
the
ocean
floor
away
from
the
midocean
ridges
can
be
explained
by
simple
cooling
models
for
the
evolution
of
the
oceanic
lithosphere
.
The
mean
depth
of
ocean
ridges
is
2.5
Ian
below
sealevel
although
regional
variations
off
1
lm1
around
the
mean
are
observed.
Thermal
subsidence
of
the
seafloor
is
well
approximated
by
an
empirical
relationship
of
the
form
d(t)
=
d
0
+
At
112
where
d
is
seafloor
depth
referred
to
sea-level
and
positive
downward,
do
is
mean
depth
of
mid-
ocean
ridges
and
t
is
crustal
age.
TI1e
value
of
A
is
around
350
m/(my)
1
1
2
if
d
and
d
0
are
expressed
in
meters
and
t
in
my.
Depth
anomalies
or
r
es
id-
ual
depth
anomalie
s
refer
to
oceanfloor
topog-
raphy
minus
the
expected
thermal
subsidence.
Although
there
is
a
large
literature
on
the
inter-
pretation
of
positive
depth
anomalies
-
swells
-
it
should
be
kept
in
mind
that
in
a
convect-
ing
Earth,
with
normal
variations
in
temperature
and
composition,
the
depth
of
the
seafloor
is
not
expected
to
be
a
simple
function
of
time
or
age.
Geophysical
anomalies,
both
positive
and
nega-
tive,
are
well
outside
the
normal
expected
varia-
tions
for
a
uniform
isothermal
mantle.
Data
from
the
western
North
Atlantic
and
central
Pacific
Oceans
,
for
seafloor
ages
from
TOPOGRAPHY
63
0
to
70
Myr,
topography
are
described
by
d(t)
=
2500
+
350t
1
1
2
where
t
is
crustal
age
in
Myr
and
d(t)
is
the
depth
in
meters
.
Older
seafloor
does
not
follow
this
simple
relationship,
being
shallower
than
predicted,
and
there
is
much
scatter
at
all
ages
.
Slightly
different
relations
hold
if
the
seafloor
is
subdivided
into
tectonic
corridors
.
There
are
large
portions
of
the
ocean
floor
where
depth
cannot
be
explained
by
simple
thermal
models;
these
include
oceanic
islands,
swells,
aseismic
ridges
and
oceanic
plateaus
as
well
as
other
areas
where
the
effects
of
surface
tectonics
and
crustal
structure
are
not
readily
apparent.
Simple
cool-
ing
models
assume
that
the
underlying
mantle
is
uniform
and
isothermal
and
that
all
of
the
variation
in
bathymetry
is
due
to
cooling
of
a
thermal
boundary
layer
(
TBL)
.
TI1e
North
Atlantic
is
generally
too
shallow
for
its
age,
and
the
Indian
Ocean
between
Australia
and
Antarctica
is
too
deep.
Continental
insulation,
a
chemically
heterogenous
mantle
and
accumulated
slabs
at
depth
may
explain
these
anomalies.
There
is
no
evidence
that
shallow
regions
are
caused
by
par-
ticularly
hot
mantle.
In
fact,
there
is
evidence
for
moderate
mantle
temperature
anomalies
associated
with
hotspot
volcanism
.
Residual
depth
anomalies,
the
depar-
ture
of
the
depth
of
the
ocean
from
the
value
expected
for
its
age,
in
the
ocean
basins
h ave
dimensions
of
order
2000
km
and
amplitudes
greater
than
1
km
.
Part
of
the
residual
anoma-
lies
are
due
to
regional
changes
in
crustal
thick-
ness.
This
cannot
explain
all
of
the
anomalies.
Positive
(shallow)
depth
anomalies
-
or
swells
-
are
often
associated
with
volcanic
regions
such
as
Bermuda,
Hawaii,
the
Azores
and
the
Cape
Verde
Islands.
These
might
be
due
to
thinning
of
the
plate,
chemically
buoyant
material
in
the
shal-
low
mantle,
or
the
presence
of
abnormally
hot
upper
mantle.
Patches
of
eclogite
in
the
man-
tle
are
dense
when
they
are
colder
than
ambi-
ent
mantle
,
but
they
melt
at
temperatures
some
200
°C
colder
than
peridotite
and
can
therefore
be
responsible
for
elevation
and
melting
anomalies.
Shallow
areas
often
exceed
1200
m
in
hei
g
ht
above
the
expected
depth
and
occupy
almost
the
entire
North
Atlantic
and
most
of
the
western
64
THE
SHAPE
OF
THE
EARTH
Pacific.
Almost
every
volcanic
island
,
seamount
or
seamount
chain
surmounts
a
broad
topographic
swell.
The
swells
generally
occur
directly
beneath
the
volcanic
centers
and
extend
along
fracture
zones.
Small
regions
of
anom
.
alously
shallow
depth
occur
in
the
northwestern
Indian
Ocean
south
of
Pakistan,
in
the
western
North
Atlantic
near
the
Caribbean,
in
the
Labrador
Sea
and
in
the
southernmost
South
Pacific.
They
are
not
associated
with
volcanism
but
are
slow
regions
of
the
upper
mantle
as
determined
from
seismic
tomography.
Shallow
re
gio
ns
probably
associated
with
plate
flexure
border
the
Kurile
Trench,
the
Aleutian
Trench
and
the
Chile
Trench.
Major
volcanic
lineaments
without
swells
include
the
northern
end
of
the
E
mperor
Seamount
chain,
the
Cobb
Seamounts
off
the
west
coast
of
North
America
and
the
Easter
Island
trace
on
the
East
Pacific
Rise.
Bermuda
and
Vema,
in
the
south-
east
Atlantic,
are
isolated
swells
with
no
associ-
ated
volcanic
trace.
For
most
of
the
swells
expla-
nations
based
on
sediment
or
crustal
thickness
and
plate
flexure
can
be
ruled
out.
They
seem
instead
to
be
due
to
variations
in
lithospheric
composition
or
thickness,
or
abnormal
upper
mantle
. Dike
and
sill
intrusion
,
underplating
of
the
lithosphere
by
basalt
or
depleted
peridotite
,
serpentinization
of
the
lithosphere,
delamina-
tion,
or
reheating
and
thinnin
g
the
lithosphere
are
mechanisms
that
can
decrease
the
density
or
thiclmess
of
the
lithosphere
and
cause
uplift
of
the
seafloor.
A
higher
temperature
astheno-
sphere,
greater
amounts
of
partial
melt,
chem-
ical
inhomogeneity
of
the
asthenosphere
and
upwelling
of
the
asthenosphere
are
possible
sub-
lithospheric
mechanisms.
A
few
places
are
markedly
deep
,
notably
the
seafloor
between
Australia
and
Antarctica
-
the
Australian-Antarctic
Discordance
or
AAD
-
and
the
Argentine
Basin
of
the
South
Atlantic.
Other
deep
regions
occur
in
the
central
Atlantic
and
the
eastern
Pacific
and
others,
most
notably
south
of
India,
are
not
so
obvious
because
of
deep
sedimentary
fill.
Most
of
the
negative
areas
are
less
than
400
m
below
the
expected
depth
,
and
they
comprise
a
relatively
small
fraction
of
the
seafloor
area.
They
represent
cold
mantle
,
lower
melt
contents,
dense
lower
crust
or
an
underlying
and
sinking
piece
of
subducted
sl
ab
or
delaminated
lower
crust.
Dynamic
topography
The
long-wavelength
topography
is
a
dynamic
effect
of
a
convecting
mantle.
It
is
difficult
to
determine
because
of
other
effects
such
as
crustal
thiclmess
.
Density
and
thermal
variations
in
a
convecting
mantle
deform
the
surface,
and
thi
s
is
known
as
the
mantle
dynamic
topography
.
The
long
-wave
length
geoid
of
the
Earth
is
controlled
by
density
variations
in
the
deep
man-
tle
and
has
been
explained
by
circulation
models
involving
whole
mantle
flow.
However,
the
rela-
tionship
of
long-wavelength
topography
to
man-
tle
circulation
has
been
a
puzzling
problem
in
geodynamics
.
Dynamic
topography
is
mainly
due
to
density
variations
in
the
upper
mantle.
Lay-
ered
mantle
convection,
with
a
shallow
origin
for
surface
dynamic
topography,
is
consistent
with
the
spectrum,
small
amplitude
and
pattern
of
the
topography.
Layered
mantle
convection,
with
a
barrier
near
1000
km
depth
provides
a
self-consistent
geodynam
ic
model
for
the
amplitude
and
pattern
of
both
the
lon
g-wave
lengt
h geoid
and
surface
topography.
The
geoid
The
centrifuga
l
effect
of
the
Earth's
rotation
causes
an
equatorial
bulge,
the
principal
depar-
ture
of
the
Earth's
surface
from
a
spherical
shape
.
If
the
Earth
were
covered
by
oceans
then,
apart
from
winds
and
internal
currents,
the
surface
would
reflect
the
forces
due
to
rotation
and
the
gravitational
attraction
of
external
bodies,
such
as
the
Sun
and
the
Moon
,
and
effects
arising
fi·om
the
interior.
When
tidal
effects
are
removed,
the
shape
of
the
surface
is
due
to
density
variations
in
the
interior.
Mean
sea
level
is
an
equipotential
surface
called
the
geoid
or
figure
of
the
Earth.
Crustal
features,
continents,
mountain
ranges
and
midoceanic
ridges
rep
resent
departures
of
the
actual
surface
from
the
geoid,
but
mass
com-
pensation
at
depth,
isostasy
,
minimizes
the
influ-
ence
of
surface
features
on
the
geoid.
To
first
PACIFIC
PLATE
~~lilill·
Geoid
lows
are
concentrated
in
a
narrow
polar
band
passing
through
Antarctica
,
the
Canadian
Shield
and
Siberia
.
Most
of
the
continents
and
smaller
tectonic
plates
are
in
this
band
. Long-wavelength
geoid
highs
and
the
larger
plates
(Africa,
Pacific)
are
antipodal
and
are
centered
on
the
equator
.
The
geoid
highs
control
the
location
of
the
axis
of
rotation.
Large-scale
mass
anomalies
in
the
deep
mantle
control
the
long-wavelength
geoid.
These
in
turn
can
affect
the
stress
in
the
surface
plates
.
order,
near-surface
mass
anomalies
that
are
com-
pensated
at
shallow
depth
have
no
effect
on
the
geoid.
The
shape
of
the
geoid
is
now
known
fairly
well,
particularly
in
oceanic
regions,
because
of
the
contributions
from
satellite
geodesy
[see
geoid
images
].
Apart
from
the
geoid
highs
associated
with
subduction
zones,
there
is
little
correlation
of
the
long-wavelength
geoid
with
such
features
as
continents
and
1nidocean
ridges.
The
geoid
reflects
temperature
and
density
vari-
ations
in
the
interior,
but
these
are
not
simply
related
to
the
surface
expressions
of
plate
tectonics.
The
largest
departures
of
the
geoid
from
a
radially
symmetric
rotating
spheroid
are
the
equatorial
and
antipodal
geoid
highs
centered
on
the
central
Pacific
and
Africa
(Figures
6.1
and
6.2).
The
complementary
pattern
of
geoid
lows
lie
in
a
polar
band
that
contains
most
of
the
large
shield
regions
of
the
world.
The
largest
geoid
highs
of
intermediate
scale
are
associated
with
subduc-
tion
zones.
The
most
notable
geoid
high
is
cen-
tered
on
the
subduction
zones
of
the
southwest
Pacific
near
New
Guinea,
again
near
the
equa-
tor.
The
equatorial
location
of
geoid
highs
is
not
accidental;
mass
anomalies
in
the
mantle
control
the
moments
of
inertia
of
the
Earth
and,
there-
fore,
the
location
of
the
spin
axis
and
the
equa-
tor.
TI1e
largest
intermediate-wavelength
geoid
THE
GEOID
65
lows
are
found
south
of
India,
near
Antarctica
(south
of
New
Zealand)
and
south
of
Australia.
The
locations
of
the
mass
anomalies
responsible
for
these
lows
are
probably
in
the
lower
man-
tle.
Many
shield
areas
are
in
or
near
geoid
lows,
some
of
which
are
the
result
of
deglaciation
and
incomplete
rebound.
The
thick
continental
crust
would,
by
itself,
raise
the
center
of
gravity
of
con-
tinents
relative
to
oceans
and
cause
slight
geoid
highs
.
The
thick
lithosphere
(
~
150
km)
under
continental
shields
is
cold,
but
the
seismic
veloc-
ities
and
xenoliths
from
kimberlite
pipes
suggest
that
it
is
olivine-rich
and
garnet-poor;
the
temper-
ature
and
petrology
have
compensating
effects
on
density.
TI1e
lon
gte
rm
stability
of
shields
indi-
cates
that
,
on
a
ver
age,
the
crust
plus
its
under-
lying
lithosphere
is
buoyant.
Midocean
ridges
show
mild
intermediate-wavelength
geoid
hi
g
hs
,
but
they
occur
on
the
edges
of
long-wavelen
gt
h
highs.
Hotspots,
too,
are
associated
with
geoid
highs.
The
lon
g-wave
length
features
of
the
geoi
d
a
re
probably
due
to
density
variations
in
the
lower
mantle
and
the
resulting
deformations
of
the
core-mantle
boundary
and
other
boundaries
in
the
mantle
(Richards
and
Hager,
1984).
Geoid
anomalies
are
expressed
as
the
differ-
ence
in
elevation
between
the
measured
geo
id
and
some
reference
shape
.
The
reference
shape
is
usually
either
a
spheroid
with
the
observed
flat-
tening
or
the
theoretical
hydrostatic
flattening
associated
with
the
Earth's
rotation,
the
equilib-
rium
form
of
a
rotating
Earth.
TI1e
latter,
used
in
Figure
6.2,
is
the
appropriate
geoid
for
geo-
physical
purposes
and
is
known
as
the
nonhy-
drostatic
geoid.
TI1e
geometric
flattening
of
the
Earth
is
1 /2
98.26.
The
hydrostatic
flattening
is
1/299.64
.
The
maximum
geo
id
anomalies
are
of
the
order
of
100
m.
This
can
be
compared
with
the
21
Jan
difference
between
the
equatorial
and
polar
radii.
To
a
good
approximation
the
net
mass
of
all
columns
of
the
crust
and
mantle
are
equal
when
averaged
over
dimensions
of
a
few
hundred
kilometers.
This
is
one
definition
of
isostasy
.
Smaller-scale
anomalies
can
be
sup-
ported
by
the
strength
of
the
crust
and
litho-
sphere
.
TI1e
geo
id
anomaly
is
nonzero
in
such
cases
and
depends
on
the
distribution
of
mass.
A
negative
t,p,
caused
for
example
by
thermal
66
THE
SHAPE
OF
THE
EARTH
Geoid
undulations
(to
degree
180)
referred
to
a
hydrostatic
shape,
flattening
of
1 /
299.
638
[called
the
non-hydrostatic
flattening
of
the
geoid
].
Contour
interval
is
5 m
(after
Rapp
.
1981
).
expansion,
will
cause
the
elevation
of
the
surface
to
increase
(t-.p
=
positive)
and
gives
a
positive
geoid
anomaly
because
the
center
of
mass
is
closer
to
the
Earth's
surface.
The
mass
deficiency
of
the
anomalous
material
is
more
than
canceled
out
by
the
excess
elevation.
All
major
subduction
zones
are
characterized
either
by
geoid
highs
(Tonga
and
Java
through
Japan,
Central
and
South
America)
or
by
local
maxima
(Kuriles
through
Aleutians).
The
long-
wavelength
part
of
the
geoid
is
about
that
expec-
ted
for
the
excess
mass
of
the
cold
slab.
The
shorter
wavelength
geo
id
anomalies,
however,
are
less,
indicating
that
the
excess
mass
is
not
simply
rigidly
supported.
There
is
an
excellent
correlation
between
the
geoid
and
slabs;
this
can
be
explained
if
the
viscosity
of
the
mantle
increases
with
depth
by
about
a
factor
of
3
0.
The
high
viscosity
of
the
mantle
at
the
lower
end
of
the
slab
partially
supports
the
excess
load
.
Phase
boundaries
and
chemical
boundaries
may
also
be
involved
.
The
deep
trenches
rep-
resent
a
mass
deficiency,
and
this
effect
alone
would
give
a
geoid
low.
The
ocean
floor
in
back
-
arc
basins
is
often
deeper
than
equivalent-age
normal
ocean,
suggesting
that
the
mass
excess
associated
with
the
slab
is
pulling
down
the
sur-
face.
A
thinner-than-average
crust
or
a
colder
or
denser
shallow
mantle
could
also
depress
the
seafloor.
Cooling
and
thermal
contraction
of
th
e
oceanic
lithosphere
cause
a
depression
of
the
seafloor
with
age
and
a
decrease
in
the
geoid
height.
Cooling
of
the
lithosphere
causes
the
geoid
height
to
decrease
uniformly
with
increas-
ing
age,
symmetrically
away
from
the
ridge
crest.
The
change
is
typically
5-10
m
over
distances
of
1000-2000
km.
The
elevation
and
geoid
offset
across
fracture
zones
is
due
to
the
age
differences
of
the
crust
and
lithosphere
.
The
long-wavelength
topographic
highs
in
the
oceans
generally
corre-
late
with
positive
geoid
anomalies,
giving
about
6-9
meters
of
geoid
per
kilometer
of
relief.
There
is
a
good
correlation
between
inter-
mediate-wavelength
ge
oid
anomalies
and
seismic
velocities
in
the
upper
mantle;
slow
regions
are
geoid
highs
and
vice
versa.
Subduction
zones
are
slow
in
the
shallow
mantle
,
presumably
due
to
the
hot,
partially
molten
mantle
wedge
under
back-arc
basins.
In
subduction
regions
the
total
geoid
anomaly
is
the
sum
of
the
positive
effect
of
the
dense
sinker
and
the
negative
effects
caused
by
bound-
ary
deformations.
For
a
l
ayer
of
uniform
vis-
cosity,
the
net
dynamic
geoid
anomaly
caused
by
a
dense
sinker
is
negative;
the
effects
from
the
deformed
boundaries
overwhelm
the
effect