Geology Reference
In-Depth Information
over-relaxation
terms
depending
from
the
V.x;y C h/ C V.x;y h/ D 2V .x;y/
weighting factor œ and
the deviation
of
the
C h 2 @ 2 V
@y 2 C O h 4
neighbor average from the central value:
(3.33)
V nC1 .x;y/ D V n .x;y/ C . C 1/ n .x;y/
(3.36)
Now we add expressions ( 3.32 )and( 3.33 ).
Applying the equation of Laplace ( 3.30 ), we
have:
where the deviation factor 4 n ( x , y )at( x , y )atstep
n is given by:
1
4 ŒV .x C h;y/ C V.x h;y/
C V.x;y C h/ C V.x;y h/
C O h 4
V.x;y/ D
1
4 ŒV n .x C h;y/ C V n .x h;y/
C V n .x;y C h/ C V n .x;y h/
V n .x;y/
n .x;y/ D
(3.34)
(3.37)
This expression shows that the potential at
any point ( x , y ) equals, up to high-order terms,
the average of V over a neighbor of ( x , y ). This
is a general property of the harmonic functions,
which will be proved rigorously in Chap. 4 .
Therefore, V cannot have maxima or minima
within the region R . The numerical approach
also allows to calculate easily the values of V
in R , starting from the boundary values. This
algorithm assigns the initial value of each grid
cell internal to R to an arbitrary constant value
V D V 0 , while the points along the frontier of
R ,( x , y ) 2 ( R ), are set through the boundary
conditions (Fig. 3.8 ). At the next step, for each
point ( x , y ) 2 R ( R ) we calculate iteratively
more precise values of the field by the following
assignment:
This procedure, which can be easily gener-
alized to the three-dimensional space, does not
require an explicit analytic solution of Laplace's
equation. It is a practical method to find the
values of V in R when it is not possible to
determine an exact solution. However, in the next
chapter we shall consider a general class of exact
solutions of this equation that are commonly used
to represent the geomagnetic potential. Then,
a complete analytical procedure for finding the
solutions of Laplace's equation in spherical coor-
dinates will be described. Finally, in Chap. 14 we
shall see that also the Earth's gravity potential is
a harmonic function outside the Earth's surface.
3.4
Magnetization
1
4 ŒV n .x C h;y/ C V n .x h;y/
C V n .x;y C h/ C V n .x;y h/
(3.35)
V nC1 .x;y/ D
All ordinary materials, when placed in a magnetic
field, acquire a magnetization , which is a man-
ifestation of the presence of a large number of
magnetic dipoles at atomic scale. These dipoles
result from microscopic currents, associated with
the motion of electrons within the atoms and
the with the intrinsic magnetic moments of the
elementary particles (spin). The magnetic field
generated by these microscopic sources adds to
that produced by macroscopic currents (flows of
electrons in conductors, motion of electrically
charged fluids, etc.). For many substances, the
net magnetic moment of the individual atoms
The algorithm terminates when the standard
deviation of the values over the neighbor of
each point, with respect to the central value,
falls below an assigned threshold. To reduce
the number of iterations, it is possible to use
a recurrence formula more sophisticated than
( 3.35 ), which includes an over - relaxation factor
0 œ<1. In this approach, the update expression
( 3.35 )
is
modified
by
the
addition
of
an
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