Implausibility of thermal convection in the Earth's solid inner core

Physics of the Earth and Planetary Interiors 108 Ž1998. 1–13

Implausibility of thermal convection in the Earth’s solid inner core Takesi Yukutake

)

Department of Earth and Planetary Sciences, Kyushu UniÕersity, Hakozaki, Fukuoka, 812-81, Japan Received 5 August 1997; revised 5 January 1998; accepted 10 March 1998

Abstract A question whether thermal convection occurs in the solid inner core or not is not yet completely solved. In this paper the temperature profile in the core is examined when the solid core grows, being cooled from the surface. In order for convection to occur, the temperature gradient must exceed the adiabatic gradient. In the solid inner core, temperature due to thermal conduction is calculated and compared with the adiabatic temperature. In the outer core, temperature is assumed to be adiabatic due to vigorous fluid convection. The temperature gradient in the solid inner core has been estimated under different conditions, i.e., for different growth rates and different concentration of radioactive material, but it has been found that it can never exceed the adiabatic gradient. If we assume that the inner core has been solidified to the present size in 4.5 billion years, the temperature decreases, maintained almost isothermal inside the inner core. Even if it has been formed in such a short period as 1.5 billion years, the temperature gradient is still much smaller than the adiabat. This indicates that no thermal convection can occur. Presence of radiogenic heat sources raises the temperature, but even with such a high concentration as 100 ppm of K no convection is likely to occur. q 1998 Elsevier Science B.V. All rights reserved. Keywords: Thermal convection; Temperature gradient; Inner core

1. Introduction Whether thermal convection occurs or not is an issue that has great influence on the study of the inner core. It is crucial, for example, for inferring the temperature distribution. If subsolidus convection occurred in the solid inner core as in the mantle, temperature distribution would become adiabatic, which is entirely different from that caused by heat conduction. Thermal convection could be also a ) Corresponding author. 58-63 Otsuka, Hachioji, Tokyo, 192-03, Japan.

cause of the observed anisotropy of seismic wave propagation in the inner core by producing inhomogeneous shear distribution ŽJeanloz and Wenk, 1988; Jeanloz, 1990; Romanowicz et al., 1996.. Jeanloz and Wenk Ž1988., assuming radiogenic heat sources in the inner core, considered that the Rayleigh number exceeded the critical value and claimed that thermal convection occurred in the inner core. Weber and Machetel Ž1992. have further extended their discussion to obtain flow patterns for high Rayleigh numbers. On the other hand, Stacey Ž1995. considers that no radioactive material is contained in the inner core and concludes that thermal

0031-9201r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 0 3 1 - 9 2 0 1 Ž 9 8 . 0 0 0 9 7 - 1

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T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

convection is implausible, since the thermal relaxation time is shorter than any reasonable formation time of the inner core. In this study, we calculate the time-dependence of temperature in the growing inner core, and examine the possibility of thermal convection by comparing the temperature distribution with the adiabatic temperature. The inner core is cooled from the surface with heat transported away by fluid convection in the outer core. If the surface cooling is so fast as the heat removed by the outer core exceeds the heat supplied from the inside by conduction, and the latent heat released at the surface by solidification, thermal convection is possible in the inner core. A criterion for the occurrence of thermal convection is whether the ambient temperature gradient exceeds the adiabatic one or not. In this study, we compute the temperature change inside the inner core, when heat is transported by conduction to the surface which is cooled uniformly at a constant rate, and examine whether the temperature gradient exceeds the adiabatic one. Since the inner core grows as it cools, the problem is to solve the thermal diffusion equation for a moving boundary whose temperature decreases with time. We obtain the solutions in an approximate way by increasing the inner core radius step by step. Two factors determine the temperature gradient in the inner core. One is the cooling rate of the surface of the inner core, and the other is the internal heat generation. When the cooling rate is high, the temperature gradient becomes so steep that convection may be possible. In the first place, we examine the case with no heat sources, and a low cooling rate. As the lowest cooling rate, we take a model, in which the inner core began to solidify soon after core formation, and has grown to the present size in 4.5 billion years at a constant rate. In this case thermal diffusion is effective in conducting heat to the surface to make the inside of the inner core almost isothermal. As the highest cooling rate, we assume that the inner core has grown to the present size in 1.5 billion years. Even in this case, the temperature gradient is smaller than the adiabatic one, indicating that thermal convection cannot occur. In the second place, we calculate the temperature distribution when the radiogenic heat sources are contained as suggested by Jeanloz and Wenk Ž1988.. Potassium is

assumed as the radioactive substance. Even when heat generation is taken as much as 10y9 Wrm3 , which is equivalent to the heat generation of 22 ppm content of potassium, the temperature profile does not change greatly from the case of no heat. Since the potassium content in Earth is considered to be about 120 ppm on an average, the above potassium content is not totally unrealistic. However, in contrast to Jeanloz and Wenk, this study indicates that thermal convection is unlikely to occur in the inner core, even if the radiogenic heat production is assumed.

2. Growth of the inner core with cooling of the core It is believed that, as the core cools, the inner core begins to solidify from the center of the Earth ŽJacobs, 1953, 1987.. This occurs when the melting temperature gradient of the core material is steeper than that of adiabatic temperature. When the core was formed, temperature distribution is supposed to have been adiabatic due to convective motions of fluid. Maintaining the adiabatic gradient, the core cools, and its temperature first reaches the melting temperature at the center. With decrease in the core temperature, the solidification point, the cross point of the adiabatic temperature curve with the melting one, moves outwards, i.e., the solid inner core grows. If the adiabatic gradient is steeper, the situation is completely different ŽHiggins and Kennedy, 1971.. The solidification occurs at the core–mantle boundary first. Therefore, it is vitally important to know the melting temperature as well as the adiabatic temperature under the core conditions. Unfortunately, our knowledge, however, is far from satisfactory. Here, we make a very rough approximation of these quantities to see the solidification process of the core. 2.1. Melting temperature model With regards to the melting temperature of the core, there is still a wide range of uncertainty. In spite of recent great progress of high pressure studies, the experiments have not yet succeeded in pro-

T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

viding the melting temperature of iron at the inner core conditions. Furthermore, diverse results are reported by the techniques employed. Shock wave experiments give significantly higher melting temperatures than static ones. Employing a static pressure technique, Boehler Ž1993. obtained the melting temperature of iron up to 200 GPa. Extrapolating the data, he estimated the iron melting point at the inner core boundary Ž330 GPa. to be 4850 " 200 K. Shock wave experiments Žfor example, Bass et al., 1990; Williams et al., 1987, 1991; Yoo et al., 1993; Saxena et al., 1994. have attained high pressures to obtain the melting temperature at 300 GPa. These shock wave experiments all give higher temperatures than the static experiments, and the estimated melting temperature at the inner core boundary ranges from 6000 to 7600 K ŽAnderson, 1995.. The actual melting temperature of the core is considered to be lower than the pure iron because of the light elements included in the core. The melting temperature depression depends on the elements and their content. The current estimates suggest it to be between 500 and 1000 K Žsee Anderson, 1995., although Boehler considers that the impurities does not significantly change the actual melting temperature due to superposition of diverse effects of different impurities. Practically the melting temperature of the actual core is supposed to be lower than those given by the shock experiments. Here, we examine the solidification of the core for two extreme cases, the case of high melting temperature based on the data from Williams et al. Ž1991. and the low melting case based on the data from Boehler Ž1993.. Since no data at the inner core pressure are available, we have to extrapolate the currently existing data. This causes another problem. Different extrapolation schemes produce different melting temperatures. In this study, we have tried three extrapolation formulas, Simon’s equation, Kraut–Kennedy’s and Lindemann’s Žsee Poirier, 1994.. The absolute value of the extrapolated melting temperature differs by the formulas, but we found that they did not influence the final conclusion of this study, since essential to the present discussion is not the absolute value of the melting temperature itself but its gradient. Here, we principally present the results obtained by use of Lindemann’s equation.

3

For the melting temperature Tm , we use the equation as a function of density r , following Lindemann’s formulation Žsee Poirier, 1991; Stacey, 1992., Tm s Tms

rs

2r3

½

ž /

exp 2g 1 y

r

rs r

5

,

Ž 1.

where g is Gruneisen parameter, Tms and rs repre¨ sent melting temperature and density at a pressure Ps . For later convenience, we rewrite Eq. Ž1. with reference to the quantities at the center of the Earth as,

Tm s Tm 0

r0

2r3

ž / r

½ ½

exp 2g 1 y exp 2g 1 y

rs r rs r0

5 5

,

Ž 2.

where Tm0 and r 0 are the melting temperature and density at the pressure of the Earth’s center P0 , and Tm 0 s Tms

rs

ž / r0

2r3

½

exp 2g 1 y

rs r0

5

.

Ž 3.

By shock wave experiments, Williams et al. Ž1991. obtained the melting temperature of iron as 4300 K, 4800 K, and 6700 K for the pressures 105 GPa, 133 GPa, and 243 GPa, respectively. In order to find densities of iron for these pressures, we used the Rankine–Hugoniot equations Žfor example, Poirier, 1991; Brown and McQueen, 1982.. When the densities and the melting temperatures are given, we can estimate the Gruneisen parameter by Eq. Ž1.. Taking ¨ the data for 243 GPa as the reference, namely Tms s 6700 K, and applying a least square fit to the data set of Williams et al.’s and a zero-pressure data ŽTm s 1791 K. in Williams et al. Ž1991., we obtain g s 1.47 " 0.241. The estimated uncertainty of g produces 512 K of root–mean-square differences of Tm . The Gruneisen parameter obtained here is for 243 ¨ GPa. This may sound somewhat higher for the high pressure value, but it is still in a range of currently inferred g , 1.27 to 1.5 ŽAnderson, 1995.. In this study, ignoring the pressure effect, we use this estimate throughout the core for extrapolation of the melting temperature. With these values, Eq. Ž2. gives our melting temperature model for the high melting case, which is plotted in Fig. 1. This model yields melting temperature of 7817 K at the Earth’s center,

4

T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

gives high temperatures. The melting temperature at ICB is estimated to be 7405–7733 K according to the formulas. This seems too high as the temperature at ICB. Since the melting temperature provided by Williams et al. is not of the actual core material but of pure iron, the melting temperature of the core may be closer to that of Boehler’s. 2.2. Adiabatic temperature model

Fig. 1. Melting and adiabatic temperatures. Thick lines represent melting temperature, and thin lines adiabatic temperature. The upper thick curve is the melting temperature extrapolated from the data of Williams et al. Ž1991., which are plotted by solid circles, while the lower thick curve is the melting temperature from the data of Boehler Ž1993.. The adiabatic temperature is depicted so that its value at the center of the Earth becomes equal to the melting temperature. The radius is non-dimensionalized by the radius of the core.

7539 K at the inner–outer core boundary ŽICB. and 5329 K at the core–mantle boundary ŽCMB.. In contrast to the shock wave experiments, static experiments provide lower melting temperature. From the data of Boehler Ž1993. we picked up 3200 K, 3600 K and 3900 K for 136 GPa, 170 GPa and 200 GPa, and used the third-order Birch–Murnaghan equation to obtain densities for these pressures. In a way similar to the high melting case, we have derived the low melting model where the data set for 200 GPa is taken as the reference, Tms s 3900 K. The Gruneisen parameters estimated for the above ¨ data-set is g s 1.45 " 0.073. This low melting model is depicted in Fig. 1 along with the high melting model. The melting temperatures are 5313 K at the center, 5048 K at ICB and 3196 K at CMB. Three types of extrapolation by using Simon’s semi-empirical formula, Kraut–Kennedy’s equation ŽKennedy and Kraut, 1973. and Lindemann’s equation yield slight differences. For example, the melting temperature at the Earth’s center is given as 7662 K, 8082 K and 7817 K by Simon, Kraut–Kennedy and Lindemann equations, respectively when they are applied to Williams et al.’s data, whereas it is given as 5078 K, 4896 K and 5313 K, respectively when applied to Boehler’s data. Whichever extrapolation equation may be taken, Williams et al.’s dataset

The adiabatic temperature gradient in the radial direction Ž r . is given Žsee Stacey, 1992. as

g Tg

dT

ž / dr

Ž 4.

sy

f

ad

where T is temperature, g Gruneisen parameter, g ¨ gravitational acceleration, and

fs

Ks

r

s Vp2 y

4 3

Vs 2 ,

Ž 5.

where K s and r denote incompressibility and density, Vp and Vs velocities of seismic longitudinal and transverse waves respectively. The gravitational acceleration g depends on the radial distance r. Here we approximate g in the core by a linear function of r as g CM B gs r Ž 6. rc where g CM B and rc are the gravitational acceleration at CMB and the radius of the core. Here we take g CM B s 10 m2rs, and rc s 3.48 = 10 6 m. Integrating Eq. Ž4., we have adiabatic temperature Tad as a function of non-dimensional radius r X Žs rrrc ., Ta s Ta0 exp yg CMB g rc u Ž r X . ,

Ž 7.

where u Ž rX . s H

rX

f Ž rX .

d rX .

Ž 8.

From a seismic model PREM Žsee also Stacey, 1992, p. 454., we approximate 1rf Ž r X . by a fourth-order polynomial of r X , and finally obtain u Ž r X . s r X 2 Ž 4.419 y 0.7394 r X q 4.023 r X 2 y4.304 r X 3 q 2.363 r X 4 . = 10y9 .

Ž 9.

T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

Taking the adiabatic temperature at the center Ta0 equal to the melting one, we plot the adiabatic temperature in Fig. 1 in addition to the melting temperatures. As the Gruneisen parameter g , we take ¨ the same value as that for the melting temperature, g s 1.47 for the high melting case and g s 1.45 for the low melting case. It is clear in the figure that the adiabatic temperature gradient is smaller than the melting gradient. It is noted, however, that they are close each other, the difference being 59 K at ICB for the high melting case and 119 K for the low melting case, although the difference becomes appreciable at CMB, 492 K and 777 K, respectively. 2.3. Growth rate of the inner core Before solidification starts, the whole core is liquid, and it cools, keeping the temperature adiabatic throughout the core. Since the adiabatic gradient is smaller than the melting point gradient, the temperature reaches the melting temperature first at the center of the Earth, and solidification begins from the center. The boundary between the solid and the liquid core is the surface where the adiabatic temperature becomes equal to the melting one as specified by the crossing point of the adiabatic curve with the melting curve. As the liquid core cools, the crossing point moves outwards from the center. The radius of the inter core is therefore determined by equating the melting temperature in Eq. Ž2. with the adiabatic temperature in Eq. Ž7..

ature distribution as the initial temperature, and assuming a uniform cooling rate, we apply the fixed boundary solution to a growing inner core model. At every time step dt, we first determine the radius of the solid sphere by equating the adiabatic temperature to the melting temperature. Then fixing the boundary for dt we calculate the temperature inside the sphere with its surface temperature specified by the assumed cooling rate. Taking the temperature thus determined as the initial temperature for the next step; we repeat the procedure to determine the temperature of the expanded solid sphere.

3.1. Temperature within a sphere with a fixed surface and a fixed surface-temperature First, we find a solution for the case where the surface of the sphere is fixed and the surface temperature is specified. Take spherical polar coordinates. Let K be the thermal conductivity, r the density, c p the specific heat, k the thermal diffusivity, and A the rate of heat generation due to uniformly distributed sources. Then k s KrŽ r c p .. We have 1 ET

k Et

1 E s

r2 Er

In this section, we formulate a procedure to determine the temperature inside the growing inner core in the case where internal heat is transported out by conduction. It is to solve the thermal diffusion equation with a moving boundary which is subject to cooling. Since it is difficult to solve analytically, we employ an approximate method. For a fixed boundary the problem has been already solved with a sphere for an arbitrary initial temperature and its surface temperature changing with time ŽCarslaw and Jaeger, 1959.. Starting with an adiabatic temper-

r2

ET

ž / Er

A q K

,

Ž 10 .

which is to be solved under the conditions, T Ž r ,0 . s To Ž r . T Ž rm ,t . s Tm

3. Thermal diffusion with a moving boundary

5

at t s 0, at r s rm ,

where To Ž r . represents an initial temperature inside the sphere, and Tm is the temperature fixed at the surface. Here we assume that the heat generation decreases exponentially with time. With the initial rate of heat generation, A o , we take A Ž t . s A o eyl t ,

Ž 11 .

where l is the decay constant. Following Carslaw and Jaeger Ž1959., we obtain the solution as, T Ž r ,t . s u Ž r ,t . q w Ž r ,t . ,

Ž 12.

T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

6

where

=

½

1 rm

rm

H0

`

2

u Ž r ,t . s Tm q

r

np r

Ý sin

rm

ns1

X

np r X

X

r To Ž r . sin

k n 2p 2

ey

2 rm

X

dr q

rm

t

Ž y1.

n

np

rm Tm

5

Ž 13 .

w Ž r ,t . s

°r

k Ao

~

ey l t

Kl

q

l

sin r

m

¢r sin r

2 rm3 A o

`

rp K

rm

y1

ž

n

l rm2 2

kp

Ta0 Ž t . s Tm 0 y ktX ,

/

k n 2p 2

ey

2 rm

t

Ž 14 .

Here uŽ r,t . satisfies the equation 1 Eu

k Et

1 E s

r2

Eu

ž /

2

r Er

for 0 F r F rm

Er

Ž 15 .

with the conditions, u s T0 Ž r . u s Tm

at t s 0, at r s rm ,

and w Ž r,t . satisfies the equation 1 Ew

k Et

1 E s

2

r Er

ž

r2

Ew Er

/

A q K

for 0 F r F rm

Ž 16 . with the conditions ws0 ws0

Ž 17 .

Assuming that the surface temperature decreases linearly with time, we take

ß

k

n n2 y

Ta s Tm 0 exp yg CMB g rc u Ž r X . .

•

1r2

Ž y1.

ns1

np r

k l

Ý

3

=sin

m

ž / ž /

¶

1r2

ing. A successive process is employed for a time step dt during which the radius of the solid sphere increases dr. Take t s 0 at the time when solidification starts at the spherical center, namely when the adiabatic temperature becomes equal to the melting temperature ŽTm0 . at the center. Then the initial temperature becomes

at t s 0, at r s rm .

3.2. MoÕing boundary approximation with its surface temperature changing with time We apply the above procedure to a moving boundary problem in which the surface of the sphere is expanding and the surface temperature is decreas-

Ž 18 .

where k is the cooling rate of the outer core. The radius of the growing sphere, rm , is determined by solving the equation that equates the adiabatic temperature at time t, TaŽ r,t . in Eq. Ž7., with the melting temperature, Tm Ž r . in Eq. Ž2., as, Ta Ž r ,t . s Tm Ž r . .

Ž 19 .

Starting with the initial temperature given by Eq. Ž17., and specifying the temperature at rm to be Tm Ž rm ., we can estimate the temperature inside the sphere. Taking the temperature thus determined as the initial temperature for the next step, we determine the temperature inside the expanded sphere. This procedure is repeated to obtain the temperature of the growing solid core. Let subscript i designate the i-th step quantities. Then ri is the i-th step radius of the solid sphere. For t i s t iy1 q dt, we have T Ž r ,t i . s u Ž r ,t i . q w Ž r ,t i . , u Ž r ,t i . s Tm Ž ri . q

q

Ž y1. np

2 r

`

Ý ns1

½

1 ri

Ž 20. a Ž n,t iy1 .

n

5

ri Tm Ž ri . sin

np r ri

k n 2p 2

ey

ri

dt

,

Ž 21 .

T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

°r sin r

yl t iy 1

w Ž r ,t i . s

k Ao e

Kl

q

~

ey ld t

¢r sin r

2 ri3A o ey l t iy 1 r p 3K

=sin

np ri rm

i

`

ž / ž / k l

r i2

•

1r2

Ž y1.

ž

y1

ß

k

Ý ns1

k n 2p 2

ey

i

¶

1r2

l

n n2 y

a

l ri2 kp 2

/

dt

Ž 22 .

where a Ž n,t iy1 . s

ri

H0

r X T Ž r X ,t iy1 . sin

np r X ri

d rX .

7

stances, U and Th as well as K, might have entered the core at the time of core formation, which are now emanating substantial heat. Here, we first examine the temperature distribution in the case of no radioactivity, and then extend to the case of radioactive heat sources contained. At the surface of the inner core, latent heat of solidification is released as it solidifies. This could influence the temperature distribution in the solid core, unless the heat is transported away from the surface quickly enough. Here, we assume that the heat is immediately removed from the core surface by vigorous convection of the liquid outer core to have no effect on the temperature distribution in the solid core.

Ž 23 .

Repeating this process, we obtain the temperature distribution in the solid core for every dt. It is noted that, for the initial temperature of the i-th step the temperature obtained at the Ž i y 1.-th step for 0 F r F riy1 was extrapolated to cover the i-th step range 0 F r F ri by a Fourier series with coefficients given in Eq. Ž23..

4. Inner core cooling by thermal conduction Two cases are examined. One is the case when no internal heat source exists in the core, and the other is the case when a radiogenic heat source is contained. The question whether radioactive materials exist in the core or not is not yet completely solved. It is widely believed that such lithophile elements as U and Th are contained in the crust and mantle and that only K is the candidate that can enter the core. It is known that K is depleted in the crust and mantle in comparison with the solar abundance ratio. Verhoogen Ž1973. considered that this was because substantial part of the depleted K had entered the core, and significant amount of thermal energy was generated. On the other hand, from consideration of no K content in iron meteorites, and of high volatility of K, some argue that K has dissipated into space at an early stage of the Earth’s history, and therefore K is unlikely to exist in the core Žsee Stacey, 1992.. Jeanloz and Wenk Ž1988., however, does not exclude the possibility that a trace of radioactive sub-

4.1. Cooling in the absence of internal heat sources Taking Eq. Ž17. as the initial temperature with an assumption that the initial temperature in the liquid core was adiabatic, and taking A s 0, we calculate the change in temperature inside the core. Temperature distribution in the solid core and its time variation are highly dependent on how fast the solid core surface is cooled. If the cooling rate is high, the temperature gradient in the solid core is supposed to become large and in extreme cases solid convection would be expected to occur. On the other hand, if the cooling rate is low, temperature gradient becomes small, and the solid core could be nearly isothermal. Here we calculate two cases, a slow-cooling case and a fast-cooling case. As the slow-cooling case, we adopt a model in which the inner core takes 4.5 billion years to grow to the present size, whereas, as the fast-cooling case, it takes 1.5 billion years. For numerical calculation we take, as the physical quantities of the inner core, thermal conductivity K s 50 Wrm K ŽStacey, 1992., density r i s 13 = 10 3 kgrm3 , specific heat c p s 640 Jrkg K ŽStacey, 1992., thermal diffusivity k i s 6.0 = 10y6 m2rs and the radius of the present time inner core ri s 1.22 = 10 6 m. There are many other suggestions about K and c p Žsee Anderson, 1995.. Braginsky and Roberts Ž1995., for example, take K s 40 Wrm K and c p s 826 Jrkg K, which lead to k i s 3.7 = 10y6 m2rs. The result of the low cooling rate is shown in Fig. 2 for the low Ža. and the high Žb. melting model. Starting with the adiabatic temperature, the tempera-

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T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

between the center and the inner core boundary is only 40 K for the present size of the solid core, whereas the temperature difference is 143 K for the adiabatic temperature distribution. The estimated temperature at the center differs from the adiabatic temperature by 103 K. This result implies that thermal convection is impossible, since the temperature gradient is far below the adiabatic one. Fig. 3Ža. shows the result of the high cooling rate in the low temperature model, where the temperature

Fig. 2. Time variations in the temperature profiles in the core in the case of low cooling rate where the inner core has grown to the present size in 4.5 billion years. Starting with the adiabatic temperature, the temperature profile is plotted at the interval of dt s 0.9=10 9 years. The lowest curve gives the present-day temperature distribution. The dashed curve represents the melting curve. The radius is non-dimensionalized by the core radius. Ža. The low melting temperature model. Žb. The high melting temperature model.

ture profile is plotted at an interval of dt s 0.9 = 10 9 years. The lowest curve gives the present-day temperature distribution in this slow-cooling model. As is clearly seen when compared with the initial adiabatic temperature curve, the temperature gradient in the solid core is much smaller than the adiabat in both models. In this case the slope of the temperature profile is almost flat, indicating that the inner core is close to the isothermal state, irrespective of the melting temperature model. The low melting model, for example, predicts that the temperature difference

Fig. 3. Time variations in the temperature profiles in the core for the low melting temperature model. The case of high cooling rate where the inner core growth time is taken to be 1.5 billion years. Ža. The case with no heat source. Žb. The case with a heat source of q s1.0=10y9 Wrm3. Starting with the adiabatic temperature, the temperature profile is plotted at an interval of dt s 0.3=10 9 years. The lowest curve gives the present-day temperature distribution. The dashed curve represents the melting curve. The radius is non-dimensionalized by the core radius. Ža. The low melting temperature model. Žb. The high melting temperature model.

T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

profile is given at an interval of dt s 0.3 = 10 9 years. It is clear that the gradient is steeper than in the case of the low cooling rate. Nevertheless it is still less than the adiabatic gradient. The temperature difference between the center and the inner core boundary is 113 K for the present size of the solid core, which is smaller than that for the adiabatic temperature, 143 K. Temperature at the center differs from that of the adiabat by 30 K. Even with this high cooling rate thermal convection is difficult to occur. The above results indicate that in the absence of internal heat no convection can occur in the growing solid inner core. This is foreseeable from estimation of the thermal diffusion time as discussed in Stacey Ž1995.. Let t i be the thermal time constant for the inner core. Then we have

t i s ri2r Ž p 2k i . ,

Ž 24 .

where ri and k i are the radius and the thermal diffusivity of the inner core respectively. Then t i is estimated to be 8.0 = 10 8 years. This is much shorter than the inner core growth time, that is 1.5 = 10 9 years in the fast-cooling model. If we take Braginsky and Roberts’ values for K and c p , we obtain 1.3 = 10 9 years for t i , which is still shorter. This implies that heat is transported out very effectively by conduction, and the temperature inside the solid core easily follows the temperature decrease at the surface. As a consequence the solid core becomes almost isothermal in the slow-cooling case, where the growth time is 4.5 = 10 9 years, more than five times the diffusion time. No convection is needed to transport away the internal heat. 4.2. Cooling when an internal heat source is present Assuming that the radiogenic heat sources are contained in the inner core, Jeanloz and Wenk Ž1988. consider that thermal convection occurs. They estimate the Rayleigh number in the presence of internal heat by Ra H s q a gri5rŽ3 r c p k i2n ., where q is the rate of heat generation and n the kinematic viscosity. They assume q s 10y9 Wrm3, which, they say, is a rather moderate estimate, since the heat production by the radioactive decay for the present-day bulk Earth is 2.5 = 10y8 Wrm3 Žsee for instance Verhoogen, 1980.. With n s 10 9 m2rs, they estimate Ra H to be of the order of 10 10 . Since it far

9

exceeds the critical Rayleigh number for the size of the inner core, 3 = 10 3, they claim that thermal convection takes place in the inner core. Solving the diffusion equation that includes the internal heat term, we calculate the temperature in the growing solid core. As in the previous section, the adiabatic temperature is taken as the initial temperature. We first assume the same rate, q s 1.0 = 10y9 Wrm3, for the present time heat generation. We take potassium as the representative of radioactive substances with a decay constant of 5.544 = 10y1 0ryear. In the case of the high cooling rate, the temperature profiles are obtained at the interval of dt s 0.3 = 10 9 years, and shown in Fig. 3Žb. for the low melting temperature model. The results are very similar to the case of no internal heat shown in Fig. 3Ža.. The difference is less than 3 K. The temperature gradient is smaller than the adiabatic gradient. It follows that thermal convection is difficult to occur even if the inner core contains a heat source with a generation rate of q s 1.0 = 10y9 Wrm3. Next, we calculate temperatures when q s 4.5 = 10y9 Wrm3. This corresponds to 100 ppm of K as the present-day concentration in the inner core. Stacey Ž1992. estimates the mean K concentration to be about 120 ppm, when the average is taken of all K in the Earth, including the crust. Therefore 100 ppm is a high concentration for the inner core. Temperatures are shown in Fig. 4 for q s 4.5 = 10y9 Wrm3 together with that for q s 1.0 = 10y9 Wrm3 and for no heat source. The temperature for q s 1.0 = 10y9 Wrm3 is, however, indistinguishable from that of no heat source. Fig. 4Ža. is for the fast-cooling model, whereas Fig. 4Žb. for the slow-cooling model. The initial temperature is the adiabatic temperature, which is shown by A in the figure. At t s 4.5 billion years, when the inner core has grown to the present size, the temperature is significantly higher for the internal heating of q s 4.5 = 10y9 Wrm3 ŽC. than for the case of no heat source ŽD.. The difference, however, is not very large, 16 K at the center for the fast-cooling model, and 29 K for the slow-cooling model. As clearly seen, the gradient is more gradual than the adiabatic one, even for this large internal heating. When the thermal instability of the Earth’s interior is discussed in terms of Rayleigh number, the adiabatic temperature gradient is a crucial parameter.

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T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

adiabat between the top and the bottom into DT. Then, taking the same value for viscosity as Jeanloz and Wenk, n s 10 9 m2rs, we have Ra AD s 6 = 10 11 . This is a very large value, exceeding the critical one. However this does not mean that convection occurs. In order for convection to occur, the nominal Rayleigh number estimated with a simple temperature difference must first exceeds Ra AD . For instability to occur it is further required that the difference between the nominal Rayleigh number and the Rayleigh number for the adiabatic temperature must exceeds the critical Rayleigh number. The same discussion applies to the Rayleigh number expressed in terms of the internal heat. Because the heat source Rayleigh number Ra H is equivalent to the ordinary Rayleigh number Ra, of which the temperature difference DT is replace by that produced by the heat source of q, DT s qd 2rŽ2 r c p k i .. Ra H of 10 10 is much smaller than Ra AD of 6 = 10 11 , indicating that convection is impossible.

5. Discussions 5.1. The growth rate of the inner core

Fig. 4. Temperature distributions with and without radiogenic heat sources for the low melting temperature model. Ža. The fast-cooling model. Žb. The slow-cooling model. The melting temperature is shown by a dashed line ŽB.. The initial temperature is the adiabatic temperature ŽA.. The temperature distributions at t s 4.5 billion years, when the inner core has grown to the present size, are shown by solid lines, C and D. Curve C is the case with a heat source of q s 4.5=10y9 Wrm3 , which corresponds to 100 ppm concentration of K, whereas curve D is the case without heat source. D is almost identical with the case of a low heat generation of q s1.0=10y9 Wrm3.

For the Benard type convection, the Rayleigh num´ ber is defined as Ra s a g DTri3rk i n , where DT is usually taken as the temperature difference between the top and the bottom of the convection layer. However, in the present case, where the compressibility is unignorable, the deviation from the adiabat must be taken Žfor example, Jeffreys, 1930; Busse, 1989.. Assume temperature inside the inner core is adiabatic, and put the temperature difference of the

If the inner core is cooled at a very fast rate from its surface, it is not necessarily impossible for thermal convection to occur. Since the diffusion time is so short as 8.0 = 10 8 years that the inner core must grow much faster than this. Unfortunately, however, the growth rate is not well determined, since the time is not known when the inner core began to solidify. There are several estimates of the growth time ŽBuffett et al., 1992, 1996; Sumita et al., 1994; Labrosse et al., 1997; Stevenson et al., 1983.. These studies discuss the growth rate by assuming the total heat flux out of the core to the mantle, which is contributed from radioactivity, latent heat of solidification, heat lost due to temperature drop, and release of gravitational potential energy by compositional convection. If the heat flux is high, the time required for solidification of the inner core becomes short. Once the heat flux at the core–mantle boundary is known, one can estimate the growth rate of the inner core. For the heat flux of 2 = 10 12 W to 1 = 10 13 W, the estimated growth time ranges from 4 = 10 9 years to

T. Yukutaker Physics of the Earth and Planetary Interiors 108 (1998) 1–13

1.5 = 10 9 years. For the generally accepted value of 3–4 = 10 12 W, the growth time is obtained to be 2.8–2.0 = 10 9 years ŽBuffett et al., 1996; Sumita et al., 1994.. Labrosse et al. Ž1997. give a shorter growth time, claiming that, if the growth time is longer than 1.7 = 10 9 years, the inner core must be larger than the present size. The minimum estimate of these studies is 1.5 = 10 9 years. This is still much longer than the thermal diffusion time, 8.0 = 10 8 years, implying that the thermal diffusion is so fast and effective that no convection is needed. This is clear from the results of the previous section that the diffusive temperature gradient for the growth time of 1.5 = 10 9 years is less steep than the adiabatic one. 5.2. RadioactiÕe substances in the core A question whether radioactive materials exist in the inner core or not is a speculative issue at the present time. It is widely accepted that, in the outer core, in addition to iron some lighter elements should exist such as O, S, C and H Žsee, for example, Brett, 1976; Poirier, 1994.. If sulfur is present, it is considered that potassium could enter the core due to its chalcophilic nature Žfor example, Murthy and Hall, 1972; Goettel, 1976.. Then significant amount of heat would be generated by radioactive decay of 40 K ŽVerhoogen, 1973, 1980.. There are, however, strong arguments against this view that K is not necessarily chalcophilic from measurements of partition of potassium between, for example, Fe–FeS and basaltic silicate melts, and moreover a large amount of K is likely to have been lost by outgassing at the early stage of the Earth’s history due to its high volatility Žsee, for example, Ringwood, 1977.. Even with this small rate of partition, however, Murthy Ž1991. suggests that potassium of about 10% of its present-day concentration in the Earth could enter the core. Examining reactions between molten iron and silicate melt under high pressures, Ito et al. Ž1993. have obtained the partition coefficients for K and Si to suggest that 5 ppm K could be in the core. Regarding the inner core similar arguments will be stirred up. Recent high-pressure studies have confirmed that the density of pure iron, whatever it may be, ´ , g or some other phase, is higher than that of the inner core predicted by seismology, suggesting again existence of lighter elements in the inner core Žfor

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example, Anderson, 1995; Jephcoat and Olson, 1987; Saxena et al., 1994; Anderson and Ahrens, 1994.. In discussing the thermal instability in the inner core, Jeanloz and Wenk Ž1988. have assumed the rate of production of the radiogenic heat as 10y9 Wrm3 from consideration of the radiogenic heat for the present-day bulk Earth being 2.5 = 10y8 Wrm3. If heat were produced entirely by the decay of 40 K, this would be equivalent to the potassium content of 22 ppm by weight in the inner core. Although this production rate appears moderate when compared with the present-day concentration of K for the whole Earth, 119 ppm by weight ŽStacey, 1992., it is still about five times as high as the upper-bound estimates by the high-pressure experiments, 5 ppm ŽIto et al., 1993.. With this rate of heat production, 10y9 Wrm3 , our calculation shows that the inner core is not sufficiently heated for the temperature gradient to exceed the adiabatic one ŽFig. 4.. Even if we take such a high value as 4.5 = 10y9 Wrm3 , equivalent to 100 ppm content of K, the difference in temperature from the case of no heat source is not great. The temperature gradient is much smaller than the adiabat. If we take 2 = 10y8 Wrm3, about 900 ppm of K that is equivalent to the concentration in an ordinary chondrite, the temperature exceeds the melting point and the inner core cannot be solid. 5.3. Melting and adiabatic temperatures in the core A serious problem in discussing the inner core growth is the fact that the melting temperature as well as the adiabatic one are not well determined with the core materials. Fortunately, however, as long as we use the same Gruneisen parameter for the ¨ melting and the adiabatic temperature, the melting temperature gradient is steeper than the adiabatic one, and we can avoid such a trouble as the ‘core paradox’ suggested by Kennedy and Higgins Ž1973.. Although there are diverse melting temperature models, whichever model we may take, the difference between the melting and the adiabatic temperature is small, particularly near the center of the Earth. This implies a possibility that solidification of the inner core proceeds quickly at its beginning stage. In this study, we have examined the inner core solidification when it is cooled from the surface uniformly. However, if the cooling is laterally inho-

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mogeneous, and the surface temperature is not uniform as discussed in Yoshida et al. Ž1996., the cooling process will become more complicated and need further investigation.

Acknowledgements I thank two anonymous referees for their comments, particularly on the Gruneisen parameter, ¨ which were very useful in revising the manuscript.

6. Conclusion References If the inner core were cooled very rapidly from the surface, or the inner core contained large amount of heat sources, thermal convection would not be impossible in the solid inner core. However, the present study suggests that, under the currently acceptable conditions, the cooling rate is not so rapid nor the radioactive material so abundant as to cause thermal convection. The first requisite for convection to occur is that the temperature gradient exceed the adiabatic gradient. Solving the thermal conduction equation for the growing solid core which is cooled from the surface, we have examined the temperature profile and its time variation in the inner core for several cases, for the high and low cooling rates, in the absence or existence of radiogenic heat sources, and for the high or low melting temperature cases. All the results indicate that the temperature gradient is smaller than the adiabat. In the case of low cooling rate, where the inner core is assumed to have taken 4.5 billion years to grow to the present size, the inner core cools keeping the whole volume almost isothermal. This is because thermal diffusion is so effective that the diffusion time is much shorter than the inner core growth time. For the rapid cooling case, where the present size inner core has been formed during only 1.5 billion years, cooling of the center retards behind the surface, but the temperature gradient is more gradual than the adiabatic one. Even if such a high content of heat source as 100 ppm of K is assumed, the temperature does not differ appreciably. Since the temperature gradient is the crucial parameter, the melting temperature itself does not influence the result, whether we may take the high melting model or the low melting one. We conclude that, as far as the inner core is cooled from the surface uniformly, solid convection is impossible to occur in the inner core.

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