The Geomagnetic Field - Quantitative Plate Tectonics - page 124

Fig. 4.19 Magnetic
polarity boundaries of the
oceanic crust, based on
Tivey ( 1996 ). Arrows are
directions of
magnetization. The black
and grey regions have
respectively normal and
reversed polarity

important to note that in so far as the new oceanic
crust moves away from the ridge, the cooling
rate of rocks at any given depth decreases, as we
shall see in Chap. 14 . Therefore, the geometry of
the boundaries of blocks with opposite polarity
is determined by the balance between cooling
rate and spreading velocity, so that they are not
vertical but curved towards or away from the
ridge. In most cases, the magnetic boundaries of
the rapidly cooling extrusive 2A layer dip toward
the spreading axis. Then, the boundaries become
nearly vertical in the 2C dikes layer and gently
dip away from the spreading center in the gabbro
section (3A-3B) (Fig. 4.19 ).
An important feature of the magnetization
pattern in the oceanic crust is represented by
the strong magnetization of the normal polar-
ity axial blocks associated with the present day
chron (C1n), which is generally two to three
times that of adjacent blocks (10-15 Am 1 in-
stead of 5-6 Am 1 ). McElhinny and McFad-
den ( 2000 ) report a time constant of magneti-
zation decay (the time for the magnetization to
reduce to 1/ e of its initial value) of 20 kyrs.
Towards the continental margins, the magnetiza-
tion becomes less intense, with typical values
of M D 3 4Am 1 . Furthermore, some of the
anomalies observed close to the COBs may be as-
sociated with upper mantle serpentinization and
not with sea floor spreading (see Sect. 1.3 ) , a
common situation along non-volcanic continental
margins such as the western Iberian margin. In
the next sections, we shall consider in detail the

mathematical properties of the geomagnetic field,
a necessary step to create crustal field models that
match the observed data.

4.7 The Geomagnetic Potential

According to Gauss' law, the Earth's magnetic
field, B , is solenoidal (Eq. 3.25 ) . Outside the
Earth's surface, it is also irrotational, because
j D 0 in ( 3.27 ) almost everywhere. Therefore,
there exists a scalar field V such that ( 3.28 )
holds. We say that the magnetic field is a po-
tential field in the region outside the Earth. In
this instance, the scalar geomagnetic potential V
satisfies Laplace's equation ( 3.29 ) . Now we are
going to describe some important mathematical
properties of the potential, which will be helpful
to fully understand the meaning of geophysical
models of crustal magnetization and mass den-
sity distributions. These properties are known as
Green identities .

Green's First Identity
Let ¥ D ¥( r ) and § D §( r ) be two scalar fields ,
defined in a closed region R with boundary S ( R ).
Then :
I Z

¥.r/ @§

@n dS D

r ¥ r §dV

R

S. R /

Z

2 §dV (4.61)

C ¥.r/ r

R