Gubbins D., Herrero-Bervera E. Encyclopedia of Geomagnetism and Paleomagnetism
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F
FIRST-ORDER REVERSAL CURVE (FORC)
DIAGRAMS
Introduction
In many geomagnetic, geological, and environmental magnetic studies,
it is important to have reliable methods of characterizing the composi-
tion and grain size distribution of magnetic minerals within samples.
For example, identification of single domain (SD) magnetic grains is
important in absolute paleointensity studies because SD grains produce
the most reliable results, while larger multidomain (MD) grains yield the
least meaningful results. In paleoclimatic studies, useful environmental
information is often revealed by subtle changes in grain size distribu-
tion, as revealed by domain state, while the same grain size variations
will complicate determination of relative paleointensity from the same
sediment. It is therefore crucially important to have reliable methods
for determining the magnetic grain size distribution in a wide range of
geomagnetic, paleomagnetic, and environmental magnetic applications.
Determining the composition of magnetic minerals in a rock is rela-
tively straightforward, however, identification of the domain state is
more difficult. Conventional methods, such as measurement of mag-
netic hysteresis, can be powerful, but they can also yield ambiguous
results because various combinations of mineral composition, grain
size, internal stress, and magnetic interactions among the grains can
produce the same magnetic behavior. This is particularly true for the
commonly used Day plot (Day et al., 1977), which summarizes the
bulk magnetic hysteresis properties by plotting ratios of M
rs
/M
s
versus
H
cr
/H
c
, where M
rs
is the saturation remanent magnetization, M
s
is the
saturation magnetization, H
cr
is the coercivity of remanence, and H
c
is
the coercive force (see Rock magnetism, hysteresis measurements for a
description of how these parameters are determined). For example,
numerical studies have shown that magnetostatic interactions among
ideal SD grains can cause the bulk hysteresis parameters of samples
to plot within the MD region of the Day plot (Muxworthy et al., 2003).
In an attempt to remove some of the ambiguity inherent to conven-
tional hysteresis measurements, Pike et al. (1999) and Roberts et al.
(2000) developed a method of mineral and domain state discrimination
using a type of hysteresis curve called a first-order reversal curve
(FORC). Measurement of a suite of FORCs provides detailed informa-
tion from within the major hysteresis loop, which enables determina-
tion of the distribution of switching fields and interaction fields for
all of the particles that contribute to the hysteresis loop. The ability
to measure sufficient FORCs to construct a FORC diagram has only
recently become possible with rapid and sensitive vibrating sample
magnetometers and alternating gradient magnetometers.
Measuring and constructing FORC diagrams
A FORC is measured by progressively saturating a sample in a field
H
SAT
, decreasing the field to a value H
A
, reversing the field and
sweeping it back to H
SAT
in a series of regular field steps (H
B
)(Figure
F1a). This process is repeated for many values of H
A
, which yields a
series of FORCs, and the measured magnetization M at each step as a func-
tion of H
A
and H
B
gives M(H
A
, H
B
)(Figure F1b). M(H
A
, H
B
) can then be
plotted as a function of H
A
and H
B
in field space (Figure F1c). The field
steps are chosen such that H
A
and H
B
are regularly spaced, which means
that M(H
A
, H
B
) can be plotted on a regular grid. The FORC distribution
r(H
A
, H
B
) is defined as the mixed second derivative of the surface shown
in Figure F1c:
rðH
A
; H
B
Þ]
2
MðH
A
=H
B
Þ]H
A
=]H
B
(Eq. 1)
When r(H
A
, H
B
) is plotted as a contour plot (i.e., as a FORC diagram;
Figure F1d), it is convenient to rotate the axes by changing coordinates
from (H
A
, H
B
)toH
C
¼ (H
B
– H
A
)/2 and H
U
¼ (H
B
þH
A
)/2.
As is standard when fitting functions to experimental data, reduction
of the effect of noise on FORC distributions is achieved in a piece-
wise manner. That is, rather than fitting a function to the entire
M(H
A
, H
B
) surface and then directly differentiating this surface to
determine r(H
A
, H
B
), the FORC distribution is determined at each
point by fitting a mixed second-order polynomial of the form a
1
þ
a
2
H
A
þ a
3
H
A
2
þ a
4
H
B
þ a
5
H
B
2
þ a
6
H
A
H
B
to a local, moving grid.
r(H
A
, H
B
) is simply equal to the fitted parameter –a
6
. As the polyno-
mial is only of second order, it cannot accurately accommodate com-
plex surfaces, however, this will not commonly be a problem for
geological or environmental samples, which usually have relatively
smooth FORC distributions. The size of the local area is determined
by a user-defined smoothing factor (SF), where the size of the grid
is simply (2SFþ1)
2
. SF normally takes values between 2 and 5,
although ideally it should be 2. Taking the second derivative in
Eq. (1) magnifies the noise that is inevitably present in the magnetiza-
tion data. Increasing SF reduces the contribution of noise to the result-
ing FORC diagram, but it also removes measured data. Simple tests
can be conducted to determine suitable smoothing factors for each
sample. These tests determine when all of the noise that has been
removed by smoothing; if SF is increased above this value, the effect
will be only to remove data (Heslop and Muxworthy, 2005).
A FORC diagram is well described by 100–140 raw FORCs.
Recent improvements in system hardware have made it possible to
measure 100 FORCs in approximately 1 h. The exact time depends
on the field-step size, the aspect ratio of the FORC diagram, H
SAT
,
and the averaging time. The effects of these measurement variables
depend as follows:
1. The chosen field-step size depends ultimately on the mineralogy of
the sample under investigation. For a simple SD-dominated sample
as in Figure F1d, the main peak of the FORC diagram is directly
related to the coercive force H
c
. It is necessary to choose suitable
boundary values for H
U
and H
C
to fully depict the FORC distribu-
tion. The field-step size will be larger for FORC diagrams with lar-
ger boundary values, which then increases the measurement time.
That is, it will take longer to measure a FORC diagram dominated
by high coercivity minerals such as hematite compared to low coer-
civity minerals such as magnetite.
2. The aspect ratio of the FORC diagram also affects the measurement
time. The overall measurement time will decrease as the ratio of the
length of the axes, i.e., (H
U
axis)/(H
C
axis), increases. A square FORC
diagram therefore is optimal in terms of measurement time (but the
characteristics of the measured sample should dictate the selected axis
lengths). Also, in order to rigorously calculate a FORC distribution, it
is necessary to measure points outside the limits of the FORC diagram
(depending on the value of SF). This will also (marginally) increase
the measurement time.
3. H
SAT
should be sufficient to magnetically saturate the sample.
However, making H
SAT
unnecessarily large will increase measure-
ment time because of the finite time taken to sweep the magnet
down from high values of H
SAT
.
4. The averaging time is the amount of time taken to measure each
data point, and is usually 0.1–0.25 s. Typically, the averaging time
is set to 0.15 s. Increasing the averaging time increases the total
measurement time, but improves signal to noise ratios. In systems
that magnetically relax on the same time scale as the averaging
time, the averaging time can be critical to the resulting FORC dia-
gram. The averaging time should therefore always be stated in the
accompanying text or figure caption.
FORC and Preisach diagrams
The FORC method originates in the phenomenological Preisach-Néel
theory of hysteresis (Preisach, 1935; Néel, 1954). There has been
much unnecessary confusion in the paleomagnetic community over
the relationship between Preisach diagrams and FORC diagrams.
Much of this confusion has arisen due to the lack of understanding
as to what a Preisach diagram is. Put simply, there are many ways of
measuring a Preisach diagram, of which the FORC method is one.
FORC diagrams are essentially a new well-defined algorithm or
method for rapidly generating a particular class of Preisach diagram.
Characteristic FORC diagrams
Noninteracting SD behavior
To help in the understanding of FORC diagrams, consider a FORC
diagram for a SD grain, with uniaxial anisotropy, that is aligned in
the direction of the applied field. Such an ideal SD grain will have a
perfect square hysteresis loop (Figure F2a). When plotting raw FORC
Figure F1 Illustration of how FORC diagrams are constructed. (a) After applying a field at positive saturation (H
SAT
), the field is
reversed to H
A
and is then progressively increased at a range of H
B
values in the direction of H
SAT
. The magnetization is denoted by
M(H
A
, H
B
). The dashed line represents the major hysteresis loop and the solid line represents a single FORC. (b) A set of consecutive
FORCs. (c) The M(H
A
, H
B
) surface plotted in nonrotated field space (H
A
, H
B
). (d) The resulting FORC diagram for the data shown in
(b) and (c) for SF ¼ 4. The FORC data shown in (b) to (d) are for a numerical model of randomly orientated uniaxial SD grains.
FIRST-ORDER REVERSAL CURVE (FORC) DIAGRAMS 267
data for such a particle in nonrotated field space, M(H
A
, H
B
) can take
one of two values, i.e., þM
s
or M
s
(Figure F2b). For H
A
> H
SW
(where H
SW
is the switching field), M(H
A
, H
B
)isþM
s
,forH
A
< H
SW
and H
B
< þH
SW
, M(H
A
, H
B
)isM
s
, and for H
A
< H
SW
and
H
B
> þH
SW
, M(H
A
, H
B
)isþM
s
. On differentiating the surface with
respect to H
A
and H
B
,onlyatH
A
¼ H
B
¼ H
SW
is M(H
A
,H
B
)nonzero.
That is, the FORC distribution should be a perfect delta function, normal to
the FORC plane; however, due to the smoothing factor, r(H
A
, H
B
) will
have finite width (Figure F2c). An ideal noninteracting uniaxial SD particle
with field applied along the easy axis of magnetization will therefore have
a FORC distribution that lies at H
c
¼ H
SW
and H
U
¼ 0.
For assemblages of randomly orientated, noninteracting, identical,
uniaxial SD grains, there are three main features in the FORC diagram
(Figure F3): first, there is a central peak; second, the main peak dis-
plays an asymmetric “boomerang” shape; and third, there is a negative
region near the bottom left-hand corner of the FORC diagram. The
central peak is due to the switching of the magnetization at H
SW
. From
a more mathematical point of view, the positive peak is associated with the
increase in @M/@H
B
with decreasing H
A
as highlighted in Figure F3.The
lower left-hand arm of the boomerang is related to FORCs near the rela-
tively abrupt positive switching field. The right-hand arm of the boomer-
ang is related to more subtle contours, which are due to the FORCs having
different return paths, as highlighted in Figure F3. The shape of the return
paths is controlled by the orientation of the grains with respect to the
applied field, which results from the fact that the orientation controls the
coercivity. Initially, return curve behavior is dominated by grains oriented
45
to the field. As H
A
decreases, grains with orientations closer to 90
and 0
will start contributing to the hysteresis curve, so each time H
A
is
decreased the return path includes grains with slightly differently shaped
hysteresis loops. The right-hand arm is therefore a result of moving from
the return path for a 45
assemblage in the first instance, into the return
path for a randomly orientated assemblage. This effect is particularly
enhanced for an assemblage of identical grains. The origin of the negative
region is related to sections of the FORCs where H
B
< 0 (Newell, 2005).
As illustrated in Figure F3, @M/@H
B
decreases with decreasing H
A
at H
B1
,
which gives rise to negative values for r(H
A
, H
B
). The decrease in
@M/@H
B
with H
A
is not as pronounced for H
B
< 0; consequently the nega-
tive region is significantly smaller than the large central peak near H
B2
.If
the SD grains have a distribution of switching fields, this causes the FORC
diagram to stretch out in the H
C
direction (Figure F4). A distribution of
coercivities (e.g., Figure F4) is much more typical of natural samples than
an assemblage of identical grains as depicted in Figure F3.
Interacting SD behavior
FORC distributions are highly sensitive to magnetostatic interactions.
As a starting point to understanding the contribution of interactions
to the FORC diagram, it is simplest to consider Néel's (1954) interpre-
tation of the Preisach (1935) diagram. Néel (1954) showed that for
interacting SD grains, H
C
corresponds to the coercive force of eachSD
loop in the absence of interactions and that H
U
is the local interaction
field H
I
. It follows in the Preisach-Néel interpretation that r(H
a
, H
b
)
is the product of two distributions: the coercivity distribution g(H
C
)
and the interaction field distribution f(H
U
). In a simple visualization,
when the ideal SD grain in Figure F2a is affected by magnetic interac-
tions, H
I
will shift the switching field (Figure F5a). This asymmetry
between the forward and backward switching fields gives rise to spread-
ing in the H
U
direction on a FORC diagram (Figure F5b). Micromag-
netic models for FORC diagrams have shown that Néel's (1954)
approximation is correct for moderately interacting systems, but that it
breaks down when the grains are separated by less than a grain width
(Muxworthy et al., 2004). Highly interacting systems produce FORC
diagrams that appear to be essentially the same as those for MD grains
(see below and compare Figure F5 with Pike et al., 2001a).
MD and pseudosingle domain grains
FORC diagrams for large MD grains (Pike et al., 2001a) produce a
series of contours that run parallel or nearly parallel to the H
U
axis
(Figure F6a). This spreading of the FORC distribution is essentially
the same as for interacting SD grains; instead of the spreading being
due to inter-grain magnetostatic interaction fields, it is due to internal
demagnetizing fields. Pseudo-single domain (PSD) grains display beha-
vior intermediate between true MD and true SD behavior (Figure F6b).
That is, they display both the closed peak structures observed for SD
grains and the more open contours that become increasingly parallel to
the H
U
axis with coarser grain sizes (Roberts et al., 2000).
Superparamagnetic behavior
Superparamagnetic (SP) grains will only manifest themselves on the
FORC diagram if their relaxation time is of the same order as the aver-
aging time (Roberts et al., 2000; Pike et al., 2001b). If the grains have
shorter relaxation times, i.e., H
SW
! 0, then r(H
A
, H
B
) ¼ 0 at all
values. SP grains with relaxation times of 0.1–0.25 s have FORC
diagrams similar to those for MD grains, i.e., r(H
A
,H
B
) plots as a
Figure F2 (a) Schematic square hysteresis loop for an ideal noninteracting SD particle, with uniaxial anisotropy, aligned along the
direction of the applied field (H
SW
is the switching field). (b) The raw hysteresis data for a series of FORCs for the grain shown in (a)
plotted in nonrotated field space (H
A
, H
B
). (c) FORC diagram for data shown in (a) and (b). The FORC distribution for this grain lies
at H
c
¼ H
SW
and H
u
¼ 0.
268 FIRST-ORDER REVERSAL CURVE (FORC) DIAGRAMS
series of contours running almost parallel to the H
U
axis (Figure F6c).
Although SP behavior may initially seem similar to MD behavior, the
coercivity spectra of MD and SP samples and the shapes of the FORC
distributions can be visually distinguished (Figure F6a,c). Cooling a
sample by a few degrees can also easily enable identification of ther-
mal relaxation. Due to the exponential nature of SD relaxation times,
small temperature variations are sufficient to increase relaxation times,
which effectively make the grains thermally stable on the time scale of
a measurement. Upon cooling, the FORC diagram for a SP sample
would then resemble that of a stable SD sample (cf. Figure F4). In con-
trast, FORC diagrams for MD samples would not display such var-
iations with temperature.
Further featur es of FORC diagrams
The H
U
axis
Due to the use of the local grid when calculating a FORC diagram for
any value of SF, it is necessary to fit the polynomial surface using
points outside the plotted FORC diagram. For the upper, lower, and
right-hand bounds of the FORC diagram, this is readily achievable
by simply measuring extra data points. For the H
U
axis, however, this
is not possible. This means that points on the H
U
axis are not directly
Figure F4 FORC diagram (SF ¼ 3; averaging time ¼ 0.15 s) for an
assemblage of noninteracting ideal SD grains for a tuff sample
from Yucca Mountain, Nevada (from Roberts et al. (2000)).
Figure F5 (a) Schematic diagram depicting the effect of a local
interaction field H
I
on a square hysteresis loop for an ideal SD
particle, with uniaxial anisotropy, aligned along the direction
of the applied field, as shown in Figure F2a. (b) Numerical
simulation for an assemblage of 1000 evenly spaced, magnetite-
like ideal SD grains with oriented uniaxial anisotropy. The
distance between grain is 1.5 times the grains size.
Figure F3 FORCs and FORC diagram for a numerical model of
1000 identical noninteracting SD grains with randomly
distributed uniaxial anisotropy (modified after Muxworthy et al.
(2004)). The origins of the negative and positive regions in the
FORC diagram are highlighted (SF ¼ 4). The negative region is
due to a decrease in @M/@H
B
with decreasing H
A
for negative
values of H
B
(H
B
1
). The large positive peak is associ ated with the
increase in @M/@H
B
with decreasing H
A
for positive values of H
B
,
near the switching field (H
B
2
). The different return paths give
rise to the positive region of the FORC distribution to the right of
the main peak as illustrated.
FIRST-ORDER REVERSAL CURVE (FORC) DIAGRAMS 269
measured, and there will be a gap of size 2SFþ1 multiplied by the
field spacing between r(H
A
, H
B
) and the H
U
axis on the FORC dia-
gram. Three approaches have been made to deal with this problem.
1. Do not plot the FORC distribution for the region where the polyno-
mial cannot be rigorously calculated. This results in a “truncated”
FORC diagram (Figure F7a).
2. Relax the calculation of r(H
A
, H
B
) back on to the H
U
axis by redu-
cing the smoothing near the H
U
axis. Such “relaxed fit” diagrams
(Figure F7b ) will produce distortions, which can give misleading
results if the field step size is too large, but they have the advantage
that it is at least possible to observe low coercivity magnetization
components. Such components are usually important in natural sam-
ples, so relaxing the fit is considered by some workers to be prefer-
able to truncation.
3. Reversible-ridge method (Figure F7c) (Pike, 2003). This approach,
which has been developed from Preisach theory, accommodates the
reversible component of the magnetization data in addition to
the irreversible part. With this approach, FORCs are extrapolated
beyond H
A
< 0, which enables rigorous calculation of r(H
A
, H
B
)
back on to the H
U
axis. For SD assemblages this approach can be
successful, however, for natural samples containing PSD or MD
material the reversible ridge can often swamp the signal from the
irreversible component of magnetization.
Asymmetry of FORC diagrams
The FORC method is a highly asymmetric method of measuring a Pre-
isach diagram. The asymmetry originates from the measurement
method in which each FORC starts from a saturated state. In interact-
ing SD and MD systems, this means that there is bias in the magneto-
static interaction and internal demagnetizing fields in the direction of
H
SAT
. The only FORCs that do not contain a history of the initial H
SAT
state are those that start from a sufficiently large negative H
A
value that
saturates the sample in the opposite direction. Asymmetry is therefore
inherent to FORC diagrams.
Figure F6 (a) FORC diagram (SF ¼ 4; averaging time ¼ 0.2 s) for a MD magnetite sample (mean grain size of 76 mm). (b) FORC
diagram (SF ¼ 3; averaging time ¼ 0.15 s) for a PSD magnetite sample (mean grain size of 1.7 mm). (c) FORC diagram (SF ¼ 3;
averaging time ¼ 0.2 s) for tuff sample CS014 from Yucca Mountain, Nevada, containing SP magnetite (from Roberts et al. (2000)).
Figure F7 Illustration of three methods for fitting a FORC diagram. (a) A “truncated” FORC diagram, with no extrapolation of data onto
the H
U
axis; (b) a “relaxed-fit” FORC diagram with extrapolation onto the H
U
axis; and (c) “reversible-ridge” fitting (following Pike,
2003). The sample is a PSD magnetite (mean grain size of 0.3 mm; SF ¼ 4; averaging time ¼ 0.1s).
270 FIRST-ORDER REVERSAL CURVE (FORC) DIAGRAMS
Applications of the FORC diagram
FORC diagrams provide much more detailed information about mag-
netic assemblages than standard hysteresis measurements. Thermal
relaxation, variations in domain state, and magnetostatic interactions
all produce characteristic and distinct manifestations on a FORC dia-
gram. Here we illustrate four applications of FORC diagrams in paleo-
magnetic and environmental magnetic studies.
1. Unraveling mixed magnetic assemblages. A key problem in many
paleomagnetic and environmental studies is identifying and isolat-
ing specific minerals and/or grain size distributions within a sam-
ple. This can be difficult in mixed samples if the signal from one
mineral is magnetically swamped by that of another mineral. FORC
diagrams have been shown to be successful at identifying magnetic
signals due to minerals with low intrinsic magnetizations when they
are present along with more strongly magnetic minerals (Roberts
et al., 2000; Muxworthy et al., 2005). For example, consider a mix-
ture of PSD magnetite and PSD hematite. The hysteresis loop in
Figure F8a provides no evidence for the presence of hematite,
whereas in the FORC diagram the presence of hematite is clearly
visible by the high-coercivity component that extends to high
values of H
C
(Figure F8b). Although the FORC diagram is more
successful than standard magnetic hysteresis measurements at iden-
tifying weaker magnetic phases in mixtures with more strongly
magnetic phases, there are still concentration thresholds beyond
which the magnetic signal of the strongly magnetic phase domi-
nates and the weakly magnetic phase is no longer detectable in
FORC diagrams (Muxworthy et al., 2005). In such cases, other mag-
netic methods may be more suitable than FORC analysis. For exam-
ple, in mixtures of MD magnetite and hematite, low-temperature
techniques can be more sensitive than FORC diagrams at identifying
hematite in the presence of magnetite.
2. Source discrimination. Due to the detailed nature of FORC dia-
grams, they can be used to identify subtle differences between
source materials that are not so readily observed using magnetic
hysteresis data. For example, in a study of a loess/paleosol
sequence from Moravia, two different source materials could be
clearly distinguished and identified using FORC diagrams (van
Oorschot et al., 2002). There are many other apparent uses of
FORC diagrams in environmental magnetic applications.
3. Assessing leaching. Sequential leaching techniques are commonly
used by soil chemists to remove iron oxides from samples to facil-
itate clay mineral analysis. FORC analysis has been shown to help
in elucidation of the leaching process where magnetic hysteresis
alone was less diagnostic (Roberts et al., 2000).
4. Pre-selection for paleointensity studies. In absolute paleointensity
studies, SD assemblages should provide reliable results, while sam-
ples with strong magnetostatic interactions and/or MD grains
should yield unacceptable results. Domain state and magnetostatic
interactions are both readily identifiable on a FORC diagram,
which should make them useful for sample screening and selection
for absolute paleointensity studies. Likewise, strict criteria apply to
the mineralogy, domain state, and concentration of magnetic parti-
cles in sedimentary sequences in terms of their suitability for rela-
tive paleointensity studies. FORC diagrams therefore also have
potential for screening sediments for relative paleointensity studies.
Adrian R. Muxworthy and Andrew P. Roberts
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Cross-references
Rock Magnetism, Hysteresis Measurements
FISHER STATISTICS
When describing the dispersion of paleomagnetic directions about
some mean direction it is a standard practice within paleomagnetism
to employ Fisher (1953) statistics. The theory of Fisher statistics
assumes a mean direction, defined by a Cartesian unit vector
^
x
m
, and
data, each defined by Cartesian unit vectors
^
x. The off-axis angle y
between the mean direction and a datum is defined by
cos y ¼
^
x
^
x
m
: (Eq. 1)
With this, then, the Fisher distribution gives the probability that a par-
ticular directional datum
^
x falls between y and y þ dy as
P
f
ðyjkÞ¼
Z
yþdy
y
p
f
ðy
0
jkÞdy
0
; (Eq. 2)
where the probability-density function is
p
f
ðyjkÞ¼
k
2 sinh k
sin y expðk cos yÞ; (Eq. 3)
and where k is a parameter that measures the directional dispersion of the
data about the mean direction. If one considers the dispersion of directions
over the unit sphere, then the probability of particular a datum falling onto
a unit differential area
dA ¼ sin ydydf; (Eq. 4)
where f is the azimuthal angle symmetrically-distributed about
^
x
m
,is
just
P
f
ðAjkÞ¼
Z
fþdf
f
Z
yþdy
y
p
f
ðA
0
jkÞ sin y
0
dy
0
df
0
; (Eq. 5)
where the corresponding density function is
p
f
ðAjkÞ¼
k
4p sinh k
expðk cos y Þ: (Eq. 6)
As with all probability-density functions, that a particular datum is rea-
lized is certain, we are, after all, describing data that exist, and therefore
Z
p
0
p
f
ðy
0
jkÞdy
0
¼
Z
2p
0
Z
p
0
p
f
ðA
0
jkÞ sin y
0
dy
0
df
0
¼ 1: (Eq. 7)
In Figure F9 we show examples of the Fisher density function.
Application
Let us consider now a set of paleomagnetic data, with the ith paleo-
magnetic direction
^
x
i
defined by an inclination-declination pair
(I
i
, D
i
), such that the Cartesian components are
x
i
¼ cos I
i
cos D
i
; y
i
¼ cos I
i
sin D
i
; z
i
¼ sin I
i
: (Eq. 8)
The mean unit direction
x given by N data is
x ¼
1
R
X
N
i¼1
x
i
;
y ¼
1
R
X
N
i¼1
y
i
; z ¼
1
R
X
N
i¼1
z
i
; (Eq. 9)
where
R
2
¼
X
N
i¼1
x
i
!
2
þ
X
N
i¼1
y
i
!
2
þ
X
N
i¼1
z
i
!
2
: (Eq. 10)
The mean vector
x is an estimate of the true mean
^
x
m
corresponding to
the underlying distribution p
f
, and with this estimated mean direction
we can measure the off-axis angle y
i
of each datum, where
cos y
i
¼
x
^
x
i
: (Eq. 11)
Figure F9 Examples of the Fisher probability-density function
p
f
(y) for a variety of k dispersion parameters: 1, 4, 16, 64. Note
that as k is increased the dispersion decreases.
272 FISHER STATISTICS
The precision parameter k is given approximately by (McFadden, 1980):
k ¼
N 1
N R
; (Eq. 12)
As R ! N the precision parameter k increases, and the distribution of
directions becomes more tightly clustered about the mean direction.
As an example of a fit of the Fisher distribution to real data, in
Figure F10 we show a histogram of off-axis angles corresponding to
a compilation of Brunhes-age paleomagnetic data collected at or near
the island of Réunion (Love and Constable, 2003). Note that the Fisher
distribution fitted using the procedure outlined here captures most of
the actual distribution of the data, but that the fit is also not perfect.
Indeed, although the Fisher distribution is often used in paleomagnet-
ism, only rarely does paleaomagnetic data actually show a strict Fisher
distribution. Both the secular variation of the geomagnetic field and
the process by which rocks obtain their paleomagnetic signatures are
extremely complicated, and it is, therefore, not too surprising that there
is some misfit to a Fisher distribution. Although the Fisher distribution
does arise from first principles in the context of the Langevin theory of
paramagnetism, more generally, there often is very little reason to
expect a set of paleomagnetic data to exhibit perfect Fisher statistics.
The real utility of the Fisher distribution comes as a benchmark for
comparison, the deviation from its relatively simple mathematical form
that is of interest. Further review material on Fisher statistics can be
found in the books by Butler (1992) and Tauxe (1998).
Jeffrey J. Love
Bibliography
Butler, R.F., 1992. Paleomagnetism, Cambridge, MA: Blackwell
Scientific Publications.
Fisher, R.A., 1953. Dispersion on a sphere. Proceedings of the Royal
Society of London, Series A, 217: 295–305.
Love, J.J., and Constable, C.G., 2003. Gaussian statistics for palaeo-
magnetic vectors. Geophysical Journal International, 152:515–565.
McFadden, P.L., 1980. The best estimate of Fisher's precision para-
meter. Geophysical Journal of the Royal Astronomical Society,
60: 397–407.
Tauxe, L., 1998. Paleomagnetic Principles and Practice. Dordrecht,
The Netherlands: Kluwer Academic Publishers.
Cross-references
Magnetization, Natural Remanent (NRM)
Paleomagnetic Secular Variation
Statistical Methods for Paleovector Analysis
FLEMING, JOHN ADAM (1877–1956)
Few individuals influenced geophysics in the 20th century more
profoundly than John A. Fleming (Figure F11): geophysicist, engineer,
scientific organizer, and administrator. He devoted his life to promoting
the study of geomagnetism and building its professional organizations,
and played a leading role in organizing magnetic and electric surveys
of the Earth during the first half of the 20th century. Yet, as Sydney
Chapman (q.v.) wrote of him, “He was so self-effacing that only those
who knew and worked with him can properly assess... what he did
for geophysics in USA and in the world at large” (Chapman, 1957).
Fleming was born in Cincinnati, Ohio on January 28, 1877. Educated
as a civil engineer at the University of Cincinnati (1895–1899), he
worked briefly in construction after receiving his B.S. degree, then
joined the U.S. Coast and Geodetic Survey's Division of Terrestrial
Magnetism. He advanced steadily at the Survey from 1899 to 1903
and was involved with the planning and construction of magnetic obser-
vatories in Alaska, Hawaii, and Maryland. In 1904, he was appointed
“Chief Magnetician” at the newly established Department of Terrestrial
Magnetism (DTM, q.v.) of the Carnegie Institution of Washington.
He would be associated with the Institution for the next 50 years.
Figure F10 Comparison of a Fisher distribution fit to a histogram
of Re
´
union data recording secular variation during the Brunhes.
Figure F11 John Adam Fleming (photograph: Carnegie Institution,
Department of Terrestrial Magnetism).
FLEMING, JOHN ADAM (1877–1956) 273
Working under the direction of DTM's founder, Louis A. Bauer (q.v.)
(with whom he had previously worked at the Coast and Geodetic
Survey), Fleming proved himself an able manager of DTM's world
magnetic survey—a far-flung enterprise of overland magnetic expedi-
tions and ocean research vessels. He developed new and improved mag-
netic instruments for use on land and at sea, and made important
contributions to establishing and maintaining the world's magnetic
standards. He designed the department's geophysical observatories at
Huancayo, Peru and Watheroo, Australia. Appointed acting director
in 1929, he ran the department for several years during Bauer's incapa-
citation and after his death. In 1935, bolstered by honorary degrees
from the University of Cincinnati (D.Sc., 1933) and Dartmouth College
(D.Sc., 1934), Fleming was finally named director of DTM.
Fleming substantially broadened DTM's focus to include studies of
the ionosphere, cosmic rays, Earth currents, physical oceanography,
marine biology, and atomic physics. He was instrumental in initiating
in 1935 the highly productive Washington Conferences on Theoretical
Physics. In 1941, in conjunction with the University of Alaska, he estab-
lished a cooperative research facility for arctic geophysics (College
Observatory) in Fairbanks—then later helped lobby successfully for
creation of the Geophysical Institute there. During World War II,
Fleming oversaw all of DTM's defense contracts, ranging from radio
propagation predictions for the armed forces to the development of
magnetic compasses and odographs.
E.H. Vestine observed that Fleming possessed “a most unusual gift
in judging the future possibilities and capabilities of the young men
he brought to DTM” (Vestine, 1956, p. 532). Among these were staff
members Merle Tuve, Lloyd Berkner, and Scott Forbush; research
associates Julius Bartels and Sydney Chapman; and the department's
first postdoctoral fellows, among them James Van Allen.
In addition to authoring more than 130 papers, primarily on magnetic
surveys and secular variation, Fleming was a dedicated editor. (A biblio-
graphy of Fleming's work was published by Tuve in 1967.) He edited
and published the journal Terrestrial Magnetism and Atmospheric
Electricity—the forerunner of the Journal of Geophysical Research—
for two decades (1928–1948). He contributed to, and edited, an ency-
clopedic volume on Terrestrial Magnetism and Electricity (1939) in
the National Research Council's landmark “Physics of the Earth” series.
But Fleming's most enduring legacy was probably as an organizer and
administrator of geophysics. In the 1890s Bauer had begun a crusade
to professionalize the study of geomagnetism and to set it on an equal
footing with established disciplines such as astronomy and meteorology
(Good, 1994). Fleming shared Bauer's zeal, and as General Secretary of
the American Geophysical Union (AGU) from 1925 to 1947 he devoted
himself tirelessly to fostering the community of geophysicists. In the
days before AGU had paid administrative staff, Fleming contributed
long hours in correspondence, organizing meetings, and editing thou-
sands of pages of the AGU Transactions, with only a clerical assistant
or two.
In 1941 he was awarded the William Bowie Medal, AGU's highest
honor, and when he stepped down as general secretary he was elected
“honorary president” (a title created for him) for life. Other noteworthy
leadership positions of Fleming's included the presidency of the Asso-
ciation of Terrestrial Magnetism and Electricity (now the International
Association of Geomagnetism and Aeronomy, q.v.) of the International
Union of Geodesy and Geophysics (1930–1948), and of the Interna-
tional Council of Scientific Unions (1946–1949). He was a member
of the National Academy of Sciences as well as numerous foreign aca-
demies and learned societies. In 1945 the Physical Society (London)
honored him with its Charles Chree Medal and Prize.
In 1946 Fleming retired from DTM, but continued to serve the
Carnegie Institution as Adviser in International Scientific Relations
until 1954. His last major project consisted of chronicling the origin
and development of the AGU in characteristically detailed and quanti-
tative fashion.
Fleming died in San Mateo, California on July 29, 1956. The AGU
posthumously established the John Adam Fleming Medal in his honor.
The medal recognizes original research and technical leadership in
geomagnetism, atmospheric electricity, aeronomy, space physics, and
related sciences. Fleming's colleagues considered him modest, suppor-
tive, and scientifically broad-minded, but also demanding, strict, and
strong-willed —“a man of enormous energy with a most notable capa-
city for getting things done. His work was his life and his hobby, but
above all he enjoyed companionship and discussions with his many
friends and colleagues” (Vestine, 1956, p. 532).
Shaun J. Hardy
Bibliography
Carnegie Institution of Washington, Department of Terrestrial
Magnetism. Archives Files, “John A. Fleming” folder. Personal
communication, Sydney Chapman to Merle A. Tuve, March 9,
1957.
Good, G.A., 1994. Vision of a Global Physics. In Good, G.A. (ed.) The
Earth, the Heavens, and the Carnegie Institution of Washington.
Washington, DC: American Geophysical Union, pp. 29–36.
Tuve, M.A., 1967. John Adam Fleming, 1877– 1956. Biographical
Memoirs, National Academy of Sciences, 39: 103–140.
Vestine, E.H., 1956. John Adam Fleming. Transactions, American
Geophysical Union, 37: 531–533.
Cross-references
Bartels, Julius
Bauer, Louis Agricola (1865–1932)
Carnegie Institution of Washington, Department of Terrestrial
Magnetism
Carnegie, Research Vessel
Chapman, Sydney (1888–1970)
Geomagnetism, History of
IAGA, International Association of Geomagnetism and Aeronomy
Instrumentation, History of
FLUID DYNAMICS EXPERIMENTS
Over the last half century, fluid dynamics experiments have provided
researchers with a means of simulating processes relevant to planetary
cores. These experiments utilize actual fluids, such as water and liquid
metals, to replicate various aspects of core flows, including rotating
convection, rotating magnetoconvection, precession, and dynamo gen-
eration (Nataf, 2003). In comparison to numerical simulations, labora-
tory experiments can presently simulate conditions that are closer to
those in planetary cores. In addition, experiments can contain small-
scale flow structures that are not resolved in numerical models. Thus,
experimental studies supply results that are essential for improving our
understanding of planetary core dynamics and dynamo action.
In a planetary dynamo, kinetic energy of iron-rich core fluid
motions is converted into large-scale magnetic field energy. On the
basis of the assumption that buoyancy driven motions dominate pre-
cessionally driven motions in Earth's core (see Precession and core
dynamics), the majority of investigations have focused on understand-
ing how core convection can generate dynamo action. Thus, most
laboratory research has recently followed two main tacks: buoyancy-
driven convection experiments and mechanically driven dynamo
experiments. These approaches provide distinct but complementary
ways of studying core processes. The mechanically driven dynamo
experiments impose a strong velocity field in order to produce
dynamo-generated magnetic fields (see Dynamos, experimental ). These
mechanically forced flows, which are typically driven by pumps or
rapidly rotating impellers, may not be realistic analogs to flows in plane-
tary cores. Yet in successful experiments where dynamo generation
274 FLUID DYNAMICS EXPERIMENTS
occurs, the results provide an excellent opportunity to study flows that
generate dynamo action and the process by which dynamo-generated
magnetic fields equilibrate (see Dynamo, Gailitis ).
In contrast, buoyancy-driven experiments study the fundamental
dynamics and characteristics of core-type flows. Because the labora-
tory convective flows are far less energetic than flows in the mechani-
cally-driven experiments, the magnetic Reynolds number, which is the
ratio of magnetic induction to diffusion, is small. Thus, convection-
driven induction processes are weak, and dynamo action cannot be
achieved in such experiments. However, these experiments are critical
for simulating the details of planetary core convection (q.v.).
In addition to buoyancy forces, core flows are also strongly affected
by Coriolis forces, due to planetary rotation, and Lorentz forces, due
to planetary magnetic fields. The Elsasser number, L, is the ratio of
Lorentz and Coriolis forces. In cases where L 1, Coriolis forces
dominate the Lorentz forces and the rotationally constrained fluid
motions tend to become two-dimensional along the direction of the
rotation axis (see Proudman-Taylor theorem). From planetary mag-
netic field observations, including that of the Earth, L is estimated to
be O(1) (see Dynamos, planetary and satellite). However, numerical
simulations of magnetoconvection and dynamo action have found that
fluid motions still tend to remain nearly two-dimensional and aligned
along the rotation axis for even relatively large Elsasser numbers,
i.e., L
<
10 (see Magnetoconvection). Thus, both rotating convection
experiments, in which Lorentz forces are not present, and rotating
magnetoconvection experiments in the L 1 regime can yield critical
insights into the dynamics of planetary cores.
Rotating convection experiments
Rotating convection experiments study fluid motions within a spheri-
cal shell that mimics the geometry of the Earth's outer core. In the
majority of these devices the outer spherical shell, which simulates
the Earth's core-mantle boundary, is made of a transparent plastic
such as polycarbonate and has a typical radius measuring between
5 and 15 cm. This allows for visualization studies of the convection
pattern in transparent fluids like water. By rapidly rotating the shell,
a centrifugal acceleration is produced that acts as an effective gravity
within the fluid. This centrifugal gravity varies linearly with cylindri-
cal radius, similarly to gravity within a planetary core. However, it
points outward, opposite to the direction of a planet's gravity. In order
to account for this reversal in gravity's direction, the inner spherical
shell, which represents the Earth's inner core boundary, must be held
at a lower temperature than the outer shell for thermal convection to
occur. Because the direction of gravity and the sign of the temperature
difference across the shell are both reversed, the buoyancy forces in
the experiment act in the same way as in a planetary core.
The first such experiments studied flows near the onset of nonmag-
netic rotating convection (q.v.) in spherical shells (Busse and Carrigan,
1976). These studies showed that convective motions first manifest
as thin cylindrical columns aligned parallel to the axis of rotation (Fig-
ure F12a). The two-dimensionality of the flow demonstrates
the applicability of the Proudman-Taylor theorem. Further experi-
ments studied stronger convection occurring at higher rotation rates
(Cardin and Olson, 1994). In such studies, it was found that strong
thermal or compositional buoyancy forces drive fluid motions on
columnar structures that fill the entire volume of the spherical shell
(Figure F12b). It was also determined that the typical width of the con-
vection columns decreases with the shell's rate of rotation. Sumita
and Olson (2000) carried out rotating convection experiments
using even stronger thermal buoyancy forces, and found that the con-
vective motions occurred on thin two-dimensional sheets that stretch
between the inner and outer spherical boundaries (Figure F12c). Retro-
grade azimuthal zonal flows were also found to develop in the vast
majority of these studies (Aubert et al., 2001; Aurnou et al., 2003;
Shew and Lathrop, 2005). Numerical simulations (q.v.) verify that
the well-organized columnar motions and the large-scale zonal flows
that are inherent in rotating convection are important components in
generating dynamo action.
Recent rotating convection experiments have employed ultrasonic
Doppler velocimetry techniques to make precise, quantitative flow
velocity measurements (Aubert et al., 2001). They have produced
velocity scaling laws as a function of thermal forcing and shell rotation
rate, using water and low viscosity liquid gallium as working fluids.
Using the same experimental device, Brito et al. (2004) have been able
to measure changes in water's effective viscosity in a set of turbulent
rotating convection experiments. These studies demonstrate how
improved experimental diagnostics allow characterizations of the
detailed flow properties that are crucial for parametrization and better
understanding of core processes.
Figure F12 (a) Sideview of the onset of rotating spherical
shell convection. Adapted from Busse and Carrigan (1976).
(b) Sideview of moderately supercritical rotating spherical shell
convection showing convection columns filling the entire shell
volume. Adapted from Cardin and Olson (1992). (c) Planview of
fully-developed rotating convection in a hemispherical shell
showing spiralling sheets that remain aligned with the rotation
axis. Adapted from Sumita and Olson (2000).
FLUID DYNAMICS EXPERIMENTS 275