Geophysical Journal (1988) 92, 99-105
Transport properties of liquid metals and viscosity of the Earth’s
core
J. P. Poirier
[mtitut de Physique du Globe de Paris, 4 Place Jussieu, 75252 Paris Cedex 05, France
Accepted 1987 June 25. Received 1987 June 25; in original form 1987 April 3
SUMMARY
Coefficients expressing transport properties of liquid metals (viscosity and diffusivity) can be
scaled to the absolute melting temperature. A systematic review of the published experimental data leads to the conclusion that the effect of pressure on viscosity and diffusivity can be
taken into account through its effect on the melting temperature. Hence, the viscosity and
diffusivity at the melting temperature are constants for a given metal. The viscosity of the
Earth’s liquid core, near the inner core boundary, is probably close to that of liquid iron at
ambient pressure, namely 6 CP (centipoises). The viscosity profile in the fluid core has been
calculated. Using the same approach for the effect of impurities, it is suggested that the
cm2 s-’.
diffusivity of sulphur in the core is probably of the order of
Key words: Earth’s core, viscosity, liquid metals, diffusivity
1 INTRODUCTION
The Earth’s liquid outer core is the seat of convection
currents which eliminate residual heat as well as the latent
heat of crystallization of the inner core and generate the
Earth’s magnetic field. The viscosity of the core is,
accordingly, an important physical property of the Earth,
albeit possibly the least well known (Jacobs 1975). Apart
from theoretical estimates, the only available ‘determinations’ of the core viscosity have been obtained through
indirect geophysical means such as attenuation of seismic
waves or decay time of oscillations of the inner core (see
Jacobs 1975 for a review). These methods are fraught with
such uncertainties that it is hardly surprising that estimates
of the core viscosity range over 12 orders of magnitude.
(Table 1). It must, however, be noted that it is a
long-established practice in geomagnetism to assume that
viscous forces are negligible in comparison with the Coriolis
force and entirely to neglect viscosity in the equations of
motion of the core (e.g. Elsasser 1939). This viewpoint has
recently been challenged by Officer (1986) and it may be
timely to investigate the physical bases of the assumption
that the core has a low viscosity. It will turn out that it can
be vindicated by consideration of numerous experimental
data on liquid metals.
In the lower range of estimates of the core viscosity, the
theoretical calculations of Gans (1972) are based on
Andrade’s (1934,1952) theory of the viscosity of unassociated liquids at the melting point.
Andrade’s theory rests on the idea (well-verified since for
metals) that the structure of a liquid at the melting point is
very much like that of a solid; he accordingly approached
viscosity from a solid state viewpoint, considering that it is
due to transfer of momentum from layer to layer, through
the amplitude of vibration of atoms of the liquid. He
obtained a simple formula giving the viscosity r ] :
r] = 4mv/3a,
where m is the mass of an atom, a, the interatomic spacing
and v the relevant vibration frequency. For the viscosity at
the melting point, Andrade used the frequency given by
Lindemann’s formula vL-an
appropriate choice, since
Lindemann’s theory of melting rests on the same physical
basis, melting being seen to occur when the amplitude of
atomic vibrations exceeds a certain limit. Lindemann’s
frequency is
vL = CTLnA-1lzQ-1‘3,
(2)
where C = 3.1 x lo1’ (in CGS units) is Lindemann’s
constant, T, (in K) is the melting point, A the molar mass
and SZ the molar volume. Taking into account that P = A l p
and m l a = (A/N)’”p’”, where N is Avogadro’s number
and p the density, the viscosity can be expressed simply in
terms of the melting point and the density:
rl = $CN-2/3A-1/6Tz2p2/3.
(3)
Using p = 13 for the core density and T, = 4575 K for the
melting point of pure iron at core pressure, taken from
Higgins & Kennedy (1971), Gans (1972) finds a value of the
viscosity of liquid iron at its melting point under core
conditions, of the order of 0.1 P. The viscosity is presumably
lowered by the impurities ( S , Si . . .) present in the core,
which lower the melting point.
It must be noted that Andrade (1934,1952) finds a good
agreement between the values of the viscosity predicted by
equation (3) and experimental values for molten metals.
Shimoji & Itami (1986) using a hard sphere model theory of
99
100
J . P. Poirier
referring the reader to Shimoji and Itami (1986) for an
exhaustive, up-to-date review.
Table 1. Estimates of the viscosity of the outer core.
Reference
Method
Dynamic viscosity
q (Poises)
2.1 Viscosity
Miki (1972)
Theoretical esfimafe of Maxwellian
relaxation time of liquid core
Bukowinski 8 Knopoff
Electronic structure
?<I0
Gans (1972)
Andrade’s theory of the viscosity
of metals
0.04 < q < 0.2
Won 8 Kuo (1 973)
Decay lime of oscillations of the
inner core
q-0.1
Hide (1971)
Topography of core-mantle boundary
q < 1.3 ~ 1 0 7
Suzuki 8 Sato (1970)
Attenuation of seismic waves and
free oscillations
Toomre (1974)
Nutation of the Earlh
Officer (1986)
Damping of seismic waves
10-3 < 11 < 10-1
There exists in the literature a considerable number of
empirical or semi-empirical equations to describe the
temperature dependence of the viscosity of liquids (see
Brush 1962), but the most widely used and the one that
generally agrees rather well with experimental data for
liquid metals is the Arrhenius equation
(1 975)
3
1010< < 7
(4)
1010
q < 1.3 x lo6
q-ZxI08
liquid metals find a formula similar to Andrade’s, in good
agreement with more experimental data.
It therefore seems that the approach of Gans (1972),
leading to low values of the viscosity of the core, rests on a
firmer basis than most geophysical estimates. In the present
paper, I shall confirm and generalize this approach by
considering the transport properties of liquid metals from
the viewpoint of physical metallurgy and in the light of
recent experimental results. I shall take a view similar to
Andrade’s in the sense that I shall show that formulae useful
for studying the pressure dependence of the viscosity of
solids (creep) can be extended to the liquid state. I shall also
show that there is a systematic relation between viscosity,
diffusivity and melting temperature for liquid metals and
that the effect of pressure on the viscosity and diffusivity can
be taken into account through its effect on the melting
temperature; the approach here is similar to that of Birch
(1961) relating the sound velocity in minerals to the pressure
through their density.
It follows that for a given metal the viscosity at the
melting point remains constant when the melting point
varies with pressure since viscosity can be scaled to the
melting point. The impurity content can probably be taken
into account in the same way, through its effect on the
melting point. As a result, I submit that the viscosity of the
outer core, at least where it is close to the melting point, is
close to that of liquid iron at ambient pressure, namely
about 6 cP.
2 VISCOSITY A N D DIFFUSIVITY OF
LIQUID METALS
Viscosity r] and diffusivity D characterize transport
properties (diffusion of momentum or matter) linked in
principle by a relation of the foim rlD = const. (see below).
Although experimental measurement of r] and D in liquid
metals is difficult, there are now a number of values in the
literature, especially on metals with a low melting point.
As a background for what will follow, let us first attempt a
cursory review of viscosity and diffusion in liquid metals,
where R is the gas constant, and Q an apparent activation
energy.
Theories of the viscosity of simple liquids roughly fall into
two categories, depending on whether the liquid is
considered as solid-like or gas-like (see Brush 1962 and
Shimoji & Itami 1986). In the former case, the Arrhenius
law is theoretically justified.
Andrade (1934,1952) built his theory on the assumption
that momentum is transferred between layers of vibrating
atoms (v. supra). The temperature dependence of viscosity
is found by assuming a Boltzmann distribution for the
energy of vibrating atoms: in first approximation an
Arrhenius equation (4) is found, giving good agreement
with the experimental data. Eyring (1936) and Ewe11 and
Eyring (1937) considered viscous flow as a thermally
activated chemical reaction in which the elementary process
is the jump of one molecule into a neighbouring hole over a
potential barrier; using the theory of reaction rates, they
also obtained an Arrhenius equation.
However, with more recent statistical theories an
Arrhenian dependence cannot be found. Variants of the
Chapman-Enskog theory that consider the liquid as a dense
gas of hard spheres and use the Boltzmann transport
equation find a temperature dependence of the viscosity in
771’2
2.2
Diffusion
Although in crystals the mechanism of diffusion by a
vacancy mechanism with an Arrhenian temperature
dependence is well attested and theoretically well studied,
there are many objections to its transfer to the liquid state,
even if liquids are thought to possess short-range order. The
‘hole and activated state’ theories like Eyring’s (1936) now
seem little credible for diffusion, in that they assume a
localization of the energy and of the hole.
The theory of Cohen & Turnbull (1951) involving
fluctuations of free volume gives the diffusion coefficient a
dependence in T’” as do theories based on hard-sphere
models (Shimoji & Itami 1986). Swalin’s (1959) theory of
continuous fluctuations gives a dependence in T 2 , while
Nachtrieb’s (1967) model assumes a state of continuous
diffusive movement on a time scale coarser than the
vibrational period and leads to a dependence in T.
However, despite the fact that no ‘good’ physical theory
predicts an Arrhenian dependence, most experimental
results on metals can often satisfactorily be described with it
(Nachtrieb 1967).
Transport properties of liquid metals
good empirical correlation between the viscosity 9 and the
self-diffusion coefficient D,of the form:
2.3. Relation between viscosity and diffusion
As seen above, modern statistical theories of the two closely
related transport properties, diffusion and viscosity (e.g.
Rahman & Bhuiyan 1986), do not lead to an Arrhenian
temperature dependence. However, in first approximation
and for many metals, at least close to the melting point, an
Arrhenius law (4) fits the experimental data in a rather
satisfactory manner. As a consequence, it is widely used to
analyse experimental data, even by authors who propose
other models (e.g. Nachtrieb 1967; Hsieh & Swalin 1974;
Shimoji & Itami 1986).
I will follow this custom here, because it allows a practical
way of systematizing the experimental data in terms of an
apparent activation energy and of introducing the pressure
dependence in the activation energy, as it is done with
diffusion and creep in solid (see Poirier 1985).
Reviewing the relation between viscosity and atomic
mobility in liquid metals, Saxton & Sherby (1962) found a
Table 2. Apparent activation energies for viscosity (Q,) and
diffusivity (Q,) in liquid metals. The data are taken from
Shimoji & Itami (1986) for viscosity and diffusion and Nachtrieb
(1967) for diffusion. The values for Fe are taken from Lucas
(1964) and for Ni and Co from Cavalier (1963).
Metal
T,(K)
Q,(KJ/mol)
Qd (KJ/mol)
M
234
234
234
234
302
303
303
312
337
337
371
371
430
430
459
505
505
505
544
544
544
577
594
594
601
601
693
693
904
904
913
923
933
983
1097
1123
1193
1193
1208
1234
1234
1234
1336
1357
1357
1406
1725
1768
1809
2,51
4.23
4.81
4,86
5.94
cs
ca
Rb
K
Na
In
Li
Sn
Bi
TI
cd
Pb
Zn
sb
Pu
Mg
Al
Ba
Yb
ca
La
Pr
&
Au
cu
U
Ni
ca
Fe
481
4,02
5,15
5,02
5.23
6.65
5,56
5,44
6.44
8.12
10,88
8.62
10,88
21.97
12,89
30,54
16,53
17,45
23.77
27,20
25,24
26,24
11.17
22,18
15.90
30.54
30.46
33,05
35.15
41,42
101
4,69
4.81
8.28
8.46
10,67
9.79
10,17
10,67
10.17
11.84
8,37
11,59
10,80
7.45
10.51
17.71
15,15
13,85
18,50
13.06
18.63
21,31
23,44
17,70
23.36
qD a kTla,
where k is Boltzmann’s constant, T is the absolute
temperature and a an atomic size parameter. Saxton and
Sherby noted that this type of relation could be derived
from a number of various theories applying to different
physical cases:
hard, uninteracting spheres, the classical Stokes-Einstein
case;
metallic ions in a solvent (Sutherland 1905);
hole theory of viscosity of liquids (Eyring 1936); see also
Egelstaff (1967), p. 174);
Nabarro-Herring diffusion creep of crystals (see Poirier
1985).
We could add that equation (5) can also be derived from
vibrational theories of diffusion (Nachtrieb 1967) or viscosity
(Andrade 1934) of liquids, as well as from time correlation
functions (Zwanzig 1983). In view of the wide applicability
of (5) one expects a good correlation between the apparent
activation energies of viscosity Q, and diffusivity QD,
defined by:
Q, = R d In q l a ( l / T )
QD = -R d In D/d(l/T).
The experimental values of Q , and QD of various liquid
metals, from Shimoji & Itami (1986) are listed in Table 2,
and plotted in Fig. 1. One indeed sees that there is a rather
good correlation between Q, and Q, for liquid metals.
3 A P P A R E N T ACTIVATION ENERGYMELTING TEMPERATURE SYSTEMATICS
When dealing with thermally activated properties of
condensed matter, it has long been a general practice to
look for (and usually to find) a systematic empirical
relationship between apparent activation energy and melting
temperature (e.g. Grosse 1963).
In solids, a linear relationship between Q and T, has been
found for self-diffusion (see Adda & Philibert 1966) and
viscosity in high-temperature creep (see Poirier 1985):
Q = gRT,,
Q,(J mol-l) = 26.91, - 2347
40,63
40.64
(7)
where g is an empirical constant.
A linear regression analysis for the 30 liquid metals for
which we have found values of Q, and the 16 for which we
have found values of QD (Table 2) gives:
Q,(J mol-’) = 22.1T, - 3013
33.07
34,ll
32,06
(5)
r = 0.92
r = 0.96.
It does not seem unreasonable, in view of equation (7) to
write these empirical relations as:
= 22T, = 2.6RTm
Q , = 27T, = 3.2RT,.
Q,
Plots of Q, and Q , vs T, are shown in Fig. 2.
Saxton and Sherby plotted log D, calculated from
viscosity data with the help of equation ( 5 ) , vs T,/T. They
102
J . P. Poirier
40
Table 3. Coefficients g of the linear empirical relation Q =gRT,
between the apparent activation energies of transport processes and
melting temperature.
F
Transport process
g = QIRT,
Viscosity of liquid metals
2.9
Diffusivify of liquid metals
3.6
Diffusivily of liquid metals
3
Diffusivity of liquid semi-metals
2.8
High T creep of solid metals
18
Diffusivily of solid metals
18
High T creep of ceramics and minerals
0
10
20
30
Qd (kJ/mol)
40
30
Reference
Data from Table 2
,,
,,
,.
Saxton 8 Sherby (1962)
Poirier (1985
Adda 8 Philiberl (1966)
Poirier (1985)
50
Figure 1. Correlation between apparent activation energy of
viscosity and apparent activation energy of self-diffusion for 16
liquid metals (from left to right: Hg, Ga, Rb, K, Na, In, Li, Sn, Bi,
TI,Cd, Pb, Zn, Sb, Ag, Cu).
found that for metals QD=3RTm, and for semi-metals,
Q D = 2.75RTm.
Coefficients of empirical linear relations between Q and
T, are gathered in Table 3 for the various cases. Such
simple relations are normally susceptible of an interpretation in rather general terms. Nachtrieb (1967), for instance,
proposes a model for diffusion in liquid metals in which
atoms vibrate in cages of their neighbours with average
energy E = 3kTm at melting temperature, equivalent to the
‘activation energy’ if the entire system is considered as
activated (even though the concept of activation practically
loses all meaning in that case). For the case of the
high-temperature creep of solids, Poirier & Liebermann
(1985) have shown that the assumption that the Gibbs free
energy of the activation process is elastic strain energy leads
to the same conclusions as assuming that the apparent
activation energy is proportional to the melting
temperature.
The fact that the proportionality factor g is considerably
larger in solids than in liquids may be seen as reflecting the
fact that transport necessarily occurs via formation and
migration of point defects in solids whereas in liquids the
concept of localized defect somewhat loses its meaning and
migration is closely linked to vibration of the atoms.
4 EFFECT OF HYDROSTATIC PRESSURE
O N DIFFUSIVITY A N D VISCOSITY
In the framework of an Arrhenian dependence of q and D
on temperature, the effect of pressure manifests itself
though a pressure dependence of the apparent activation
energy; it is embodied in an apparent activation volume
AV:
q = qoexp [(Qov + PAVv)/RTI
or
D = Do exp [-(QoD
(8)
+ PAV,)/RT]
with AV, an AVD respectively defined as:
AV, = R T ( d In q/dP).
and
AVD = - R T ( d In q/dP).
(9)
In the case of solids where there is no doubt that thermal
activation controls the transport properties, the activation
volumes have been well investigated (see e.g. Sammis et al.
50
40
-
40
30
-
30
20
-
20
10
-
10
n
0
1000
2000
2000
Tm (K)
Figure 2. Correlation between the melting temperature and apparent activation energy of viscosity (a) and self-diffusivity (b), for liquid metals
(data from Table 2). A linear regression analysis (straight line) gives Q, = 0.75 Qd - 0.64 with r = 0.75.
Transport properties of liquid metals
103
1981; Poirier 1985). Poirier and Liebermann (1985) have
shown that for the viscosity of solids the relations Q = gRT,
leads to a value of the activation volume:
analysis of his results gives:
AV, = gR(dT,/dP) = Q(d In Tm/dP)
with a correlation coefficient r=0.998. Hence from
equation (9) AV = 0.62 cm3 mol-'. The empirical value
AV, = 0.59 cm3 mol-' was calculated from equation (lo),
using Q,=2510Jmol-1
from Table 2. The value of
dTm/dP = 5 , s X 10-8 K Pa-'
was determined using
Clapeyron's formula:
which can be expressed in terms of Griineisen parameter if
Lindemann's law is assumed to be valid.
Since the apparent activation energy for transport in
liquid metals also scales with the absolute melting
temperature, we will use the same approach and define an
activation volume.
From (7) we see that if AV, is the activation volume for
transport (diffusion or creep) in the solid state, and AI.; the
activation volume for transport (diffusion or viscous flow) in
the liquid state, we have:
A ~ / A ~ = g ~0.1,
I g s ~
where g , and g, are the empirical constants for liquid and
solid metals respectively (see Table 3). The apparent
activation volume is roughly one order of magnitude smaller
in liquids than in solids, which can be attributed to the fact
that in solids most of the activation volume about one
atomic value is contributed by the formation volume of
point defects, with the migration volume being about one
tenth of the activation volume; in liquids, where, due to the
existence of free volume there is no need to form vacancies,
the activation volume is mostly migration volume, about
one-tenth of atomic volume (Nachtrieb & Petit 1956). It is
interesting to note that apparent activation volumes
calculated from hard sphere models are of the same order of
magnitude, even though the theory does not involve
activation.
As an example, we will examine the activation volumes
(experimental and calculated from equation 10) for
mercury, the most investigated liquid metal.
Table 4 shows the apparent activation volumes for
diffusivity and viscosity of liquid mercury. The experimental
value AV, = 0.59 cm3 mol-' was measured by Nachtrieb &
Petit (1956), the theoretical value AVD = 0.30 cm3 mol-I
was calculated with a hard sphere model by Shimoji (1977).
The experimental value of the activation volume for
viscosity was derived from Bridgman's (1958) measurements
of the viscosity of liquid mercury at 30°C under pressures
ranging from 1to 12 kg cmP2 (his units). A linear regression
Table 4. Experimental, empirical and theoretical determination of
the activation volume for viscosity and difisivity of liquid mercury.
Determination
Viscosity, experimental
Activation volume
cm3Imol
Fraction ofat VOI
0.62
0.04
Reference
Data from Bridgrnan
(1958)
Viscosity. empirical
0.59
0.04
This work
Diffusion, experimental
0.59
0.04
Nachtrieb 8 Petit (1956)
Diffusion, empirical
0.99
0.07
This work
Diffusion, theoretical
0.30
0.02
Shimoji (1977)
In q(poise) = 2.46 X lo-' P - 4.2
dT,/dP = AVm/ASm
and taking the melting volume: AV, = 0.54 an3mol-' and
the melting entropy AS, = 9.92 J K-' from Ubbelohde
(1978). The empirical value AV, = 0.99 cm3 mol-' was
calculated in the same way, using Q, = 4200 J mol-'
(Nachtrieb & Petit 1956).
We see that there is an excellent agreement between the
experimental value of AV, and the value calculated using
equation (10) as well as between the experimental values of
AV,, and AV, (about 4 per cent of the atomic volume).
Unfortunately, mercury is, to our knowledge, the only
metal for which both diffusivity and viscosity have been
measured under pressure. The activation volume for
diffusion was also measured for potassium and rubidium
(Hsieh & Swalin 1974) and found to be respectively about 3
and 4 per cent of the atomic volume, The values calculated
from equation (10) are somewhat larger (10 per cent of the
atomic volume) but still in reasonable agreement. Thus, it
does not seem unreasonable to state that the scaling relation
between apparent activation energy for viscosity and
diffusivity and melting temperature not only holds for
different liquid metals but also for the same metal at
different pressures. As with solids, we are somewhat
justified in taking into account the effect of pressure through
the melting temperature:
7 = 90 ~ X (gTm/T).
P
(11)
5 EFFECT OF THE IMPURITY CONTENT
ON VISCOSITY
Following the same line of reasoning as in the case of the
effect of pressure, we can account at least qualitatively, for
the effect of dilute impurities on viscosity by their effect on
the melting point. Indeed, simple inspection of the initial
slope of the liquidus of dilute binary alloys should allow a
qualitative prediction of the effect of alloying on viscosity, in
cases where the addition of a small quantity of the impurity
element does not significantly alter the structure of the
liquid. Here again, due to ease of experimentation or to
industrial importance, the two most investigated metals are
mercury and iron. Kasama et al. (1975) found that addition
of 1 atom per cent of various impurities to liquid mercury
increased its viscosity: the largest increase was for gold and
silver, whereas the other metals caused much smaller
increases (Table 5). Inspection of the known binary phase
diagrams (Hansen & Anderko 1958) shows that in all cases
the melting point of mercury increases with addition of the
metals used by Kasama et al. and that there is qualitative
agreement between the increase in viscosity and the increase
in melting point (Table 5). Kasama et al. also report Russian
J. P. Poirier
104
Table 5. Variation of viscosity and of the freezing point of
liquid iron and mercury with addition of 1 atom per cent of
impurities (viscosity data from Kasama ei al. 1975).
Solvent
Solute
Hs
Viscosity
(Cenlipoises)
1.186
Freezing point
(K)
234
Zil
1.192
233
cd
1.199
236
Sil
1.206
284
Pb
1.224
282
ps
1.279
414
Au
1.247
481
6.27
1807
Fe
Al
4.12
1806
Mn
6.15
1800
Si
5.12
1800
La
6.27
1798
S
3.15
1790
P
3.15
1790
and Japanese results showing a decrease in the viscosity of
iron at 16OO0C, with the addition of 1 atom per cent of
various elements (La, Mn, Si, Al, S, P). Here too, the
reported decrease in viscosity is in agreement with the
negative slope of the liquidus of dilute iron-based solid
solutions of these elements. It is, however, difficult to go
much further than these qualitative consideration. A linear
regression between In and T,/T for the alloys quoted in
Table 5 yields values of the slope much higher than g = 3 for
iron alloys and much lower for mercury alloys. This is
probably due to the fact that the chemical interactions
between metal and solute atoms are very specific and, even
though their gross effect on the liquidus slope and the
viscosity is qualitatively similar, a general quantitative
correlation would not make sense. Nevertheless, I believe it
is not unreasonable for practical purposes to assume that
sulphur and silicon act on the viscosity of iron through their
effect on its melting point; at any rate no better assumption
can be provided at this time.
6
equal to the melting temperature at the inner core
boundary;
(ii) The melting temperature can be calculated for
decrements in pressure AP, starting from the inner core
boundary (ICB), by
Tm(P - A P ) = T, - AP(dT,/dP);
(iii) T, at the inner core boundary and dT,JdP are
calculated for extreme reasonable values of the
Griineisen parameter (Poirier 1986): for y = 1.1
TgcBt = 5122 K
dTm/dP = 7 K GPa-’,
for y = 1.5
TgcB’ = 4766 K
dTm/dP = 10 K GPa-’;
(iv) The viscosity variation is given by equation (11) with
g = 3 and r ] , = 0.06/exp 3 = 0.003 P;
(v) The variation of pressure with depth is given by the
PREM model (Dziewonski & Anderson 1981).
The resulting viscosity profile is given in Fig. 3. We see
that it is quite insensitive to the value of the Gruneisen
parameter and that the viscosity decreases by about a factor
of 2 between the inner core boundary and the core-mantle
boundary, from 6 CP to 3 cP. It seems, therefore, probable
that the viscosity of the Earth’s liquid core is hardly larger
than that of water at the Earth’s surface (1 cP).
The relative importance of viscous forces and the Coriolis
force is expressed by Ekman’s number: E = v/QL2 where
Y is the kinematic viscosity, Q, the angular velocity of the
Earth and L a characteristic length of the core. Taking the
density of the core p = 11g ~ m - the
~ , kinematic viscosity is
Y = r ] / p = 2.7 X loT3cm2 s-l at the core-mantle boundary.
With L=1000km, Ekman’s number, E, is found to be
approximately equal to 4 x
Viscosity is indeed
negligible compared to the Coriolis force. The half thickness
of Ekman’s viscous boundary layer is A = (v/Q)ln (Tritton,
1977). The viscous boundary layer at the core mantle
boundary is therefore less than 1m thick.
Self diffusivity of liquid iron in the core is probably not an
important parameter and anyway the self-diffusion
coefficient of liquid iron has not been measured, even at
ambient pressure; however, diffusion of sulphur away from
the crystallizing inner core is an important phenomenon; the
CONSEQUENCES F O R THE EARTH’S
CORE
From the above considerations, it follows that the viscosity
of liquid iron, containing dilute impurities, at its melting
temperature can be thought of as independent of pressure
and its impurities. The Earth’s liquid core is iron with a few
per cent nickel and lighter elements such as sulphur and
silicon (Jacobs 1975); the temperature at the inner core
boundary is the melting temperature, which can be
estimated to be close to 5000K (Poirier 1986). It
immediately follows therefore that the viscosity of the liquid
core at the inner core boundary is, in all probability, quite
close to that of iron at its melting point under atmospheric
pressure, namely 0.06 P.
The viscosity profile in the core can be deduced using the
following assumptions:
(i) The temperature varies adiabatically in the core, and is
0.07
t
ICB
0 02
1000
CMB
I
I
2000
3000
4000
R (km)
Figure 3. Viscosity profile of the outer core, calculated for y = 1.1
(open symbols) and y = 1.5 (closed symbols) (see text).
Transport properties of liquid metals
diffusion coefficient of sulphur in the outer core is unknown
and a value is usually chosen without much justification (e.g.
D = lop3cmz s-l, in Morse 1986). I suggest that it might be
better to use the value of the diffusion coefficient of sulphur
in liquid iron at ambient pressure. An average value of
several results in the literature has been calculated by Bester
& Lange (1972): D = 6.365 x
exp (-8520Ca’/RT)
cmz s-’. Indeed, we note that g = 8520/RTm= 2.4 which is
not far from the value of g for the self diffusion of liquid
metals. It is not improbable that the diffusion coefficient of
dilute solutes might also scale with the melting temperature
if the chemical interaction between solute and solvent is not
too strong, although we have no experimental support for
this assumption. If this is the case then the diffusion
coefficient of sulphur in the core at the inner core boundary
might be taken as D = 6 X
cm2 s-’.
7
CONCLUSION
A scaling relationship between melting temperature of
metals and the viscosity and diffusivity of the melts has been
established. The viscosity of the outer core is probably close
to 3 CP at the core mantle boundary and the coefficient
diffusion of sulphur at the inner core boundary is probably
close to 6 X lop5cm2 s-’.
105
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, I
ACKNOWLEDGMENTS
I warmly thank J. L. Le Moiiel, V. Courtillot, P. Richet and
C. V. Voorhies for reading the manuscript and making
many useful comments. This work was partly supported by
CNRS (UA 734). This is IPG contribution no. 969.
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