The thermal regime of Venus

ICARUS 84, 280-295 (1990)

The Thermal Regime of Venus V. S. S O L O M A T O V AND V. N. Z H A R K O V Department of Theoretical Physics, Institute of Physics of the Earth, Academy of Sciences of the USSR, B.Gruzinskaya 10, Moscow 123810, USSR Received July 6, 1988; revised May 22, 1989

Models for the thermal evolution of Venus are constructed numerically. The mantle is proposed to consist of two independently convecting layers (upper and lower mantles); the floating crust keeps the temperature of the upper surface of the convecting mantle near 1200°C and the core may solidify at temperatures below the liquidus. Previously, these models were calculated with the usual parameterization of convection based mostly on investigations of constant-viscosity convection. Now this parameterization is modified to take into account new numerical investigations of convection in media with complicated rheology. The thermal evolution of Venus is assumed to have begun 4.6 billion years (b.y.) ago. It can be divided into three periods: (1) adjustment of the upper mantle to the thermal regime of the lower mantle (~0.5 b.y.); (2) transition of the entire mantle to the asymptotic regime (~3-4 b.y.); (3) asymptotic regime. Parameters of the convecting planet in the asymptotic regime do not depend on initial conditions (the planet "forgets" its initial state) and are found analytically. The present-day heat flux is equal to ~50 erg cm-2 s-1. The results are close to those obtained previously. Comparison of the present thermal models of Venus and Earth with the recent studies of melting of the FeFeS system suggest that the absence of a dipole magnetic field is closely connected to the absence of solidification of the Venusian core at present. The position of the iron triple point (~-c-liquid) makes this conclusion very probable. The thermal regime of the Venusian crust is also discussed. Convection in the lower part of the crust is shown to play a role in certain regions with specific crustal composition. The prevailing mechanism of heat transfer to the surface is advection by magmas produced by melting the lower layers of the crust. ©1990AcademicPress,Inc. I. INTRODUCTION

Models for the thermal evolution of Venus based on the approximation of parameterized convection (APC) have been considered previously by Schubert (1979), Turcotte et al. (1979), and Stevenson et al. (1983). In these models, whole-mantle convection was assumed. Models with doublelayered mantle convection were investigated by Solomatov et al. (1986, 1987). All of the previous models are based mostly on the results of investigations of constant-viscosity fluid convection and on the assumption that these results are valid in the case of a more complicated rheology. Recent numerical studies of convection in a medium with a rheology closer to that of E a r t h ' s

mantle rock indicate the necessity to modify APC. In this paper the results of numerical experiments (Parmentier et al. 1976, Parmentier 1978, Christensen 1984a,b, 1985a,b) are approximated by simple analytic expressions, which are then used to parameterize convection in Venus' interior. These expressions can be derived qualitatively from the equations of thermal convection. The work reported here is based on that of Solomatov et al. (1986, 1987) but makes use of the new parameterization of convection. The models consist of a subdivision of the mantle into an upper region depleted in radioactive elements, a lower undepleted region, and a nonsubducting crust which keeps the temperature at the surface of the 28O

THERMAL REGIME OF VENUS convective upper mantle near 1200°C. The core is allowed to solidify at temperatures below liquidus. Modified APC (MAPC) is used to estimate the present thermal regime of the Earth. Comparing the present core temperatures of Earth and Venus with the new data on melting of the F e - F e S system (Brown and McQueen 1986, Anderson et al. 1987, Williams et al. 1987) and studying the phase diagram of iron (Brown and McQueen 1986, Anderson 1986) allows us to discuss the problem of the absence of an intrinsic magnetic field on Venus. In this paper the role of convection in the lower part of the crust in transporting heat from the mantle to the planet's surface, is also studied. The following mechanisms of heat transport have previously been discussed: mantle convection reaching the planet's surface (plate tectonics), " hot spot" mechanism, conductive heat transfer, and advective heat transfer by magmas penetrating the crust (Zharkov 1986, Solomatov et al. 1986, 1987, Anderson 1980, Solomon and Head 1982, Head and Crumpier 1987, Morgan and Phillips 1983, Phillips et al. 1981). II. M O D I F I E D A P P R O X I M A T I O N O F PARAMETERIZED CONVECTION

The usual parameterization of convection in plane fluid layers is based on the simple dependence of the Nusselt number, Nu, on the Rayleigh number, Ra (see, for example, Sharpe and Peltier 1979, Schubert 1979, Shubert et al. 1979, Turcotte et al. 1979, McKenzie and Richter 1981, Stevenson et al. 1983, Richter 1984, Solomatov et al. 1986, 1987): Nu = a Ra#, Nu -

d Fd - - 28 2aeAT1'

Ra - °tgpATd3,

(1) (2) (3)

Here, a and/3 are constants that depend on the type of heating and the boundary condi-

281

tions; 8 = aeAT~/F is the thickness of the thermal boundary layers, which are supposed to be same near the upper and lower boundaries of the convecting layer; F is average heat flux at the surface of the layer; d is the thickness of the layer; ae is the coefficient of thermal conductivity; o~is the coefficient of thermal expansion; g is gravity; p is the density; AT is the average difference in temperatures across the layer; AT1 is the average temperature drop in the upper thermal boundary layer; X is the coefficient of thermal diffusivity; and ~1 is the viscosity. In the case o f ~ = const., Eqs. (1) to (3), with sufficient accuracy, give the uniform parameterization for different types of heating. In the case of variable viscosity, it is usually supposed that the parameterization (1)-(3) is conserved. The experiments of Booker (1976), Booker and Stengel (1978), Nataf and Richter (1982), and Richter et al. (1983) are usually quoted as providing the basis for this assumption; however, in the last two papers the validity of the usual parameterization was questioned. Investigations by Christensen (1984a,b, 1985a,b) have shown that to describe evolutionary models it is necessary to modify APC and by introducing additional nondimensional combinations, to allow for the new parameters in the viscosity law. Let viscosity obey the law (Zharkov 1983, 1986)

~9 = ~

- ~~

exp

exp ~-7

(4)

3 where r = (½ ~i,j=l Y2)1/2 is the second invariant of the tangential stress tensor; ~ is the second invariant of the strain rate tensor; p is the density; b is a weak function of T in comparison with the exponent, so that b ~ const.; rn ~ 3; A0 is the activation enthalpy for the self-diffusion (in K) at the reference value of the density, p = O0. The coefficient L is similar to the Grtineisen pa-

282

SOLOMATOV AND ZHARKOV

rameter; it is a weakly diminishing function of the density, L = L(p), 0 In H V*Kr L = = O In p E* '

(6)

T = Ta(z) + 0 = ro(l\ + D z_)/ + O, (7) d where D = c~g___dd, cn

(8)

pgd D Kr - F'

(9)

c~Kr

(10)

oG

is the Grtineisen p a r a m e t e r , Cp is the specific heat capacity at constant pressure. It is supposed here that s 0 ~ 1, to < 1, D < 1, and F - 1-2. Assuming that O/To < I, we can transform Eq. (5) to f exp(Ao/To) 7 ~

Trn-I

exp - T--~° +

To

~¢ .

(11)

A law o f this form with m = 1; 3 was used by Christensen. In the equations there appear explicitly three parameters: the " s u r f a c e " Rayleigh n u m b e r Rao -

otgpAOd 3 X7o

,

(12)

the " t e m p e r a t u r e " p a r a m e t e r p~-

AoAO To2 ,

ao(Lto - D) To

(14)

H e r e , A0 is the superadiabatic t e m p e r a t u r e

p ~- pa(z)~- p0(1 + t o z ) d'

r -

P2=

(5)

where H is the activation enthalpy for selfdiffusion; E is the activation energy at P = 0; V* = V(p) is activation volume; and K r = KT(p) is the isothermal modulus of c o m p r e s s i o n (Zharkov and Kalinin 1971). L e t us expand Eq. (4) with respect to the adiabat (Pa; Ta) with the initial point (p0; To) on the surface z = 0:

t0-

and the " p r e s s u r e " p a r a m e t e r

(13)

difference

at the boundaries

70=7(0=0;z=0;~ -

=

60)

fVm Ao ~(0m j)/,~exp mTo'

(15)

where

6o = x / d 2

(16)

is the scaling value of the strain rate. The scaling value of the tangential stress is T0 = 70~0.

(17)

Allowance for a z d e p e n d e n c e of viscosity m a y be included for the thick layers (D < 1) if T is thought of as the superadiabatic temperature and 1)2 is defined in the linear approximation by Eq. (14). Instead of Pl, it is convenient in M A P C to introduce a new Rayleigh n u m b e r (Christensen 1984b, 1985a)

agpAOd3, X7r

Rat-

7r=7

(0 =

(18)

.

0;z=~,r

= 7oexp -p~-~

=~'0 +~p2

) , (19)

where 0 is the mean value of 0 in the layer. The Nusselt n u m b e r at a given A0 is defined by the formula Nu -

Fd ~eA0

(20)

where F is the average surface heat flux. The definition o f Nu (2) supposes equality of the thermal b o u n d a r y layer thicknesses near the boundaries of the layer and thus is not valid in all cases. The cases m = 1, Pl # 0, P2 = 0; m = 1, Pl # 0 , p 2 # 0 ; a n d m = 3, p t # 0 , p2 = 0, and in part the case rn =3, P l # 0, P2 # O, were studied by Christensen (1984b,

THERMAL REGIME OF VENUS 1985a). In all cases the boundaries are free and the temperature difference at the layer boundaries is fixed. If 7/is a function of T and m = 1; 3, p2 = 0, the two regimes can be distinguished on the basis of the viscosity contrasts in the layer. At moderate contrasts, the flow pattern is almost symmetric, the thermal b o u n d a r y layers have approximately the same thickness, and the velocities at the layer boundaries are also nearly equal. The relationship of Nu (Rao, Rat) can be approximated by the formula Nu = a Ra0~oRa~~.

(21)

Data for parameterization of the same problem but at fixed b o u n d a r y conditions can be found in Parmentier et al. (1976) and Parmentier (1978), for the case Pl = Pz = 0 (i.e., Ra0 = RAT), m = 3. Approximating the results gives a = 0.134, fl0 + /3~ = 0.50. Supposing [3o/[3T to be the same as for the free boundaries, we obtain a = 0.134, fl0 = 0.35, fit = 0.15.

(22)

which coincides with Eqs. (1) to (3) for "0 = const. In the case of m = 3 the following approximation can be suggested: a = 0.29, f l 0 = 0.37, f l T = 0.16.

(

fv Tij OUi d V = - J r ctpOgiuidV,

(31)

The velocity field can be approximated by the cylindric rolls:

ui =

(- u

sin ~ x " cos ~ z; O;

(23)

Large viscosity contrasts cause the viscous thermal boundary layer to become stagnant and the flows to concentrate in the lower regions where viscosity is small. The flow pattern is asymmetric and the Nusselt number depends mainly on Ra0: N u ' ~ a'Ra~ ~.

(30)

In this case, Eq. (29) can be taken as the transition criterion. Relationship (21) can be derived from the following integral expression:

In the case of m = 1 (Christensen 1984b) a = 0.27, fl0 = 0, fiT = 0.32.

283

(24)

usin~z-cos~x

.

(32)

Supposing that the fluid is well mixed (0 ~ const. ~ 0), we obtain

Oui f v "gij-~jXj dO -~(Tru]~"'+"/'b'/mexp( ~ A

• V,

(33)

In the case of m = 1 (Christensen 1984b) a = 0.4, fl~ = 0.35.

(25)

In the case of m = 3, the following approximation is suggested: a = 0.34,/3~ = 0.7.

Ra 0(06 /30)/87.

~

(~-2)1/2

(27)

Thus we have RaT,t~ ~ 4Rao,

m = I,

(28)

RaT.t~ ~ 2Ra~,

m = 3.

(29)

=

7ru

d '

(26)

The transition criterion for both the m = 1 case and m = 3 case may be found by the requirement that Nu = N u ' [(21) and (24)]: RaT,tr --

and insert into Eq. (33) the following estimates:

Ao(p) L -~-

A

ao(ffatL

-~ - ~ , ~ /

(34)

,

(35)

= nTu,

(36)

A0 (p.~tL = -n- \~0/ "

(37)

Tu is the temperature at the base o f the upper thermal boundary layer (the beginning of the adiabat in the core of the cell), and n is the ratio of the mean adiabatic tempera-

284

SOLOMATOV AND ZHARKOV

ture in the core of the cell to the surface value. Supposing that the entire superadiabatic temperature difference and the main buoyancy force are contained in the thermal boundary layers of the same thickness 8(d), we obtain the estimate --

fv otpOgiuidV ~ 2ap ~0 ug ---d-8(d) V.

(38)

The results obtained for convection with fixed temperatures at the boundaries of the layer can be generalized for other variants of heating if we suppose approximate equality of the thermal boundary layers at the top and bottom o f the layer. With the help of formula (34) the average tangential stresses and the average viscosity in the layer can be estimated:

(rub

From thermal boundary layer theory (Richter 1984, Turcotte and Schubert, 1982, Solomatov et al. 1986, 1987) it is known that

~ \ d

a ~ lira

exp-~!

,

b A ~ ~-~-i exp ~-~.

(45) (46)

7r X___dd ~'X tt = ~- 82 = --d-- N u 2 '

(39)

where 8 = (X/-~rr/4)8(d) is the average thickness o f the thermal boundary layer, defined by (2). As a consequence we obtain dependence (22) with the following parameters:

a

=

7"l'-[(3rn+4)/2(m+2)];

2.

(40)

1, a = 0.26,

/3o= 0,

/3r = 0.33;

(41)

(47)

dT2 "~'(R]z - R3c)p2Cp2 dt

4 Ir(R~2 - R3)Q2 3 -4rrRZ2F21 + 47rR~Fc,

(48)

4 dT¢ dm 3 rrR3cCpcPc ~ + Qc dt

a t m = 3: a = 0.23,/30 = 0.40,

/3r = 0.20.

(42)

These values are in good agreement with the numerical results [see Eqs. (22) and (23)]. Probably the effective value of Pa in (35) should be taken near the bottom of the cell. Indeed, the dependence on the third parameter p2 investigated in partial calculations by Christensen (1985a) at m = 3 can be estimated by the formula Nu -~ Nu(p~ = 0) • e x p ( - 0 . 0 8 p2) = Nu(p~ = 0 ) - e x p ( -

/3r

-~-p2).

(43)

This formula corresponds to Eq. (21) without explicit dependence on p2, if the viscosity at the bottom of the layer is defined to be r/r

4_rr(R 3 _ R~2)plCR, dT~ 3 dt 4

1

Atm=

The heat balance equations for the upper mantle, the lower mantle, and the core are written in the form

= 41rR22Fl2 - 4rrR2FL,

m-1 /30 = m + 2 ,

~3r-m+

IIl. D E S C R I P T I O N O F T H E M O D E L

rl(T = T; z

d; r

z0).

(44)

= 4~rR2Fc,

(49)

where the indices I, 2, and C refer respectively to the upper mantle, the lower mantle, and the core; T is the average temperature of the layer; and FL is the heat flux beneath the crust. We assume that all the radioactive elements of the upper mantle have been transferred to the crust during its melting. F n = F2! are the heat fluxes respectively for the upper and lower mantles at R n , which is the radius of the boundary between the upper and lower mantle. It is assumed that the boundaries o f the upper mantle are effectively rigid and the boundaries of the lower mantle are effectively free. We shall see that in the mantle of Venus, the first type of convective regime is

THERMAL REGIME OF VENUS realized. Therefore, the upper mantle convection is parameterized by Eqs. (21) and (30) for the fixed boundaries and the lower mantle convection is parameterized by Eqs. (21) and (23) for the free boundaries. In a c c o r d a n c e with Zharkov (1983) and Solomatov et al. (1986, 1987), the parameters in the viscosity law [(4) and (37)] are bl = 4.3 x 1015 dyn 3 cm 6 s, × 104 K,

AI = 7.5

A01 = 6.9 × 104 K;

(50) b2 = 1.1 × 1017 dyn 3 cm -6 s, × 105K,

Tc, is determined as in Stevenson et al. (1983) by the simple expression Tc = ncTcM,

TM(p) = T0(1 - (~x)

Ao2 = 1.3 × 105K.

TL = nLTu.

(51)

For each of the mantle regions, To is the temperature at the base of the upper thermal boundary layer, and TL is the temperature at the top of the lower thermal boundary layer. The heat release, Q2, is supposed to be equal to the heat release in the undifferentiated silicate reservoir of the Earth. In accordance with O ' N i o n s et al. (1979) the concentrations of the radioactive elements are taken to be 20 ppb for U, 1 0 4 ppb for K / U , and 3.8 for Th/U. In Eq. (3) the term Qc d m / d t describes the energy released during solidification of the core after the core adiabat falls below the liquidus; d m / d t is the rate of solidification (Stevenson et al. 1983, Solomatov et al. 1986, 1987). The solid inner core is supposed to consist of pure iron; the liquid outer core includes sulfur, which, during solidification remains in the melt. The average temperature of the convective adiabatic outer core,

(52)

where nc = 1.2, TcM is the temperature at the c o r e - m a n t l e boundary, the average core density Pc = 10.5 g cm 3, and Cpc = 4.7 x 106 erg g-i K-1 (Zharkov and Trubitsyn 1980). Equations for the melting curve and the adiabat are written in the form (Zharkov 1983, Solomatov et al. 1986, 1987)

A2 = 1.45

Values for the other parameters are RE Ro = 6.05 × 108 cm, R 1 2 = 5.30 × 108 cm, Rc = 3.21 × 108 cm, Cpl = Cp2 = 1.2 × 107 e r g g - 1 K - I , g l = g z = 9 0 0 c m 2s - l , a l = 3 X 10 -5 K l, pl = 3 . 7 g c m 3, X1 = 10 -2 cm 2 s l, nl = 1.15, riLl = 1.3, e~Z = 1.5 × 10 -5 K -1, 02 = 4.9 g cm -3, X2 = 3 × 10 -2 cm z s -I, n2 = 1.13, nL2 = 1.26. The parameters n and nL link T and TL with Tu by the formulas = nT.,

285

p )2.24 "P-~M" '

(53)

/ p \1.45

rao(O)-- rcM P PCM

)

,

(54)

,,,4_000940,

(U

- 0.05672

(U

,

(55)

where To is the melting temperature of pure iron at the core boundary, c~ = const. ~ 2, OCM = 9.59 g cm 3 is the core density at the c o r e - m a n t l e boundary, and x is the mass fraction of the light c o m p o n e n t (sulfur) in the fluid core: x(Ri)

xoR3c

=-

R3 _ Ri3 .

(56)

It is worth noting that the effect of the nonzero linear term in Eq. (15) is negligible and Eq. (15) approximates the distributions of both density and its derivative in most of the core at r > 100 km with sufficient accuracy. The core model described here is similar to the model constructed by Stevenson et al. (1983). IV. NUMERICAL RESULTS AND ASYMPTOTIC SOLUTIONS OF THE MAPC EQUATIONS In our models it is necessary to give three initial temperatures Tul(0), Tu2(0), TCM(0). Thanks to the small thermal inertia of the upper mantle, it quickly adjusts to the regime of the lower mantle. Thus, T~I is not important after --0.5 billion years (b.y.) of

286

SOLOMATOV AND ZHARKOV

evolution. Any period when the core is heated cannot be included in our scheme. F o r the present we consider only models without a core. Thus, there remains only the p a r a m e t e r Tu2(0), which is chosen to be equal to 2500, 3000, and 3500 K. T h e numerical results are shown in Fig. 1. During the first 0.5 b.y. (the characteristic time for thermal diffusion of the upper mantle), the u p p e r mantle adjusts to the thermal regime of the lower mantle. Then, for - 3 b.y. (the characteristic time for thermal diffusion of the whole mantle), the whole mantle changes to the a s y m p t o t i c regime independent of the initial conditions; but, for Tu2(0) = 2500 K the evolutionary curves remain s o m e w h a t different from the a s y m p t o t i c at present. On the whole, the results are similar to those obtained by Sol o m a t o v et al. (1986, 1987). The present-day model p a r a m e t e r s are

3500

1-To2 (0) = 2500K 2-Tu2 (0)= 3 0 0 0 K

3000

250G w

200( Tul

~5oot

, ~

3

.

1

2 AGE t, b.y.

3

12

4

4.6

b

7

100

\\',,

0 0 uJ

x* d

50 \

, .......

I-. uJ

0L

i

_

7ul = 1700-1720 K,

t

_ _

A G E t, b.y.

81 = 28-30 kin, Tl2 = 2500-2530 K, 82 = 130-140 kin, Tu2 = 2840-2870 K, ut = 2.1-2.4 c m year -1,

(57)

FL = 35-40 erg cm s, Uz = 0.8-1.0 cm year -1, rl ~ 5 - 6 bar, r2 ~ (I 10-120)bar, */1 ~ 1-2 × 1021 P, r12 ~ 3-10 × 1022 P. T1z is the t e m p e r a t u r e at the b o u n d a r y between the u p p e r and lower mantles. The evolution is such that the criteria (28) for the applicability of the parameterization are satisfied. At present,

RaTI ~ 7 - 9 × 104, Ra0t --~ 4 × 103, RaT2 ~ 6 - 7 × 102,

Rao2 ~- 4 - 5 × 10:. (58)

The criteria for quasi-stationarity of convection are also satisfied (Solomatov e t al.

FIG. 1. (a) Evolution temperatures Tul and Tu2 of the base of the upper thermal boundary layers of the upper and lower mantles and the temperature T12 of the boundary between the upper and lower mantles are shown for different initial conditions. The core is not included and TcM= TLz. (b) Evolution of the heat flux beneath the lithosphere is shown for different initial conditions. The surface heat flux is obtained by adding -11 erg cm 2 s-~ produced by radioactivity in the crust. The dashed line represents the heat flux generated by radioactivity in the lower mantle.

1987) which demand that the characteristic circulation time should be small in comparison with the characteristic time for thermal diffusion. S o l o m a t o v e t al. (1986, 1987) give the asymptotic analysis of the evolutionary equations for APC. If the characteristic time for thermal diffusion of the system is small in c o m p a r i s o n with the characteristic time of radioactive d e c a y (Solomatov e t al. 1986, 1987), tr = - \

( d In Q2]-' dt / ,

(59)

THERMAL REGIME OF VENUS which is equal to 5.67 × 109 years at present, then the asymptotic expression for heat flux in the first approximation is FL = FQ(1 + t i tn#] '

(60)

where 4 3 ~'(R12

FQ =

-

-

R3)pzQ2

4~.RZL

(61)

is the heat flux due to radioactive elements in the lower mantle (Fig, lb, dashed line); (62)

/in = tinl + tin2(1 + ~),

287

tional parameterization of convection; the upper value corresponds to its slight modification. The mantle and core temperatures are somewhat lower than those in Solomatov et al. (1986, 1987). At the c o r e - m a n t l e boundary this difference amounts to - 3 0 0 K. Allowance for the core in the absence o f solidification results in TCM = 3720 K, Fc = 15 erg cm -2 s -l, and FL = 44 erg cm -2 s -1. The initial temperature TcM(0) is unimportant after t > 1 b.y., so the temperature distribution for models with a core is chosen to be that of the melting curve of the mantle.

where /inl --

MInlCplATI /~1 M2Q2

+/31 +

~IA1ATI~ -1

~

/'

AT1 = Tuj - TB,

tin2 =

n2Cp2A T2 ( Q2 1 + f12 +

(63)

fl2A2AT2]-1 T22

AT2 = T u 2 -

/

T12,

,

(64)

AT1 1 + riLlS-

AT2 I + flo + f12 - 2(flo - 2f12)AD2AT2/3T22

1 + fll + flI(AIAT1/T2uO (65) Here, /inl and tin2 are the characteristic times for thermal diffusion (related to thermal inertia) for each layer; tin is the corresponding characteristic time for a system consisting of two layers; M~ and ME are the masses of the layers; and S = R22/R~. Expressions (62) to (65) must be calculated in the zeroth approximation (FL = FQ). Equations (59) to (65) explain why the new parameterization results in no essential change in the behavior of the evolutionary curves. The characteristic time for thermal diffusion is one of the most important parameters of the models and is equal to - 3 . 5 b.y. That places this model among the models considered by Solomatov et al. (1986, 1987), for which tin ~ 2.5--4 b.y. The lower value of tin corresponds to the tradi-

V. ESTIMATE OF THE PRESENT THERMAL STATE OF THE EARTH It is interesting to estimate the present thermal state of the Earth with the help of MAPC equations. Earth loses heat mostly by convection, with the heat reaching the surface in oceanic regions (through the mechanism of plate tectonics). The rheology of the Earth near the surface is more complicated than that considered in this study. For this reason we estimate the temperature T~2 at the boundary between the lower and upper mantles from the estimate of temperature at - 1 0 0 km depth from petrological data (1300-1400°C) (Basaltic Volcanism Study Project 1981) and from the estimate of temperature at the depth of the first phase transition zone ( - 4 2 0 km; 1600 -+ 50°C) (Zharkov and Trubitsyn 1980, Zharkov 1986). Letting Tu~ = 1623 K (1350°C) and continuing this temperature to the lower boundary (670 km) along the mantle adiabat, we obtain T~: ~ 2100 K. The temperature jump in the lower thermal boundary layer of the upper mantle can be estimated if the small-scale convection under plates is considered (Richter and Parsons 1975, Parsons and McKenzie 1978). If the transition boundary between effectively viscous and effectively elastic behavior is determined by the isotherm Tb ~ 1200°C, then in the upper thermal boundary layer of

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SOLOMATOV AND ZHARKOV

the u p p e r mantle, AT = 150°C. As the upper estimate of Tl2 we take TI2 =-

2300-2500 K.

(66)

The heat flux on this boundary, F12 ~ R~-----~2 F - Fcr + CpM1 dt

~ 80 erg cm -2 s -1,

(67)

where F --~ 80 erg cm -2 s-i is the total average heat flux from Earth, F - Fcr ~- 70 erg cm -2 s -I is the heat flux without radiogenic heat production in the crust [both estimates are f r o m Sclater et al. (1980)1, and d T t / d t ~ - 1 0 0 K / b . y . is the cooling rate of the upper mantle (Basaltic Volcanism Study Project, 1981). We choose the following values of viscosity p a r a m e t e r s for the lower mantle of the Earth: b~ = 6.6 x 1016dyn 3 c m - 6 s -l, x 105K,

As = 1.5

Ao2 = 1.3 x 105K;

(68)

and p a r a m e t e r i z e convection by Eqs. (21) and (23). Core heat flux is estimated by the adiabatic value - 3 0 erg cm -2 s -1 (Zharkov 1986). The values for the physical parameters o f the lower mantle are g2 = 103 cm S 2 ot2 = 1.5 X 10 -5 K -1,

Cp2=

1.2 × 10 7 e r g g - 1

K i, P2 -= 5 g cm -3, X2 = 3 × 10 -2 cm 2 s -l, n2 = 1.15, and nc2 = 1.3. As a result, the following estimates are obtained: T,2 = 2800-2900 K, TL2 = 3600-3800 K,

(69) TcH = 3800-4000 K, ~2 ~

100 km.

Thus, the t e m p e r a t u r e at the c o r e - m a n t l e b o u n d a r y of the Earth, TCME, exceeds that of Venus by TCME -- TCMV = 100--300 K.

(70)

The lower value of the difference in Eq. (70) c o r r e s p o n d s to the lower t e m p e r a t u r e TI2 of the Earth (66). It is worth noting that the large heat flux from the Earth and the high t e m p e r a t u r e at

the c o r e - m a n t l e b o u n d a r y are connected mainly with the mass of the lower mantle. It is - 1 . 3 times larger than the Venusian lower mantle mass and thus generates a greater amount of heat for the same concentration of radioactive elements ( - 3 0 erg cm -z s -j at the E a r t h ' s surface). This factor also m a k e s the thermal inertia of the planet greater and increases the a m o u n t of heat released from the cooling of the planet [Eqs. (58)-(65)]. There is a contribution by the core to the heat flux from the Earth ( - 9 erg cm -2 s ~ at the E a r t h ' s surface). As a result, the ratio of heat flux produced by radioactive elements to heat flux from the mantle without heat production in the crust (the U r e y number), U, ~ 0 . 6 for Venus and ~ 0 . 4 for Earth. VI. M A G N E T I S M AND T H E R M A L R E G I M E OF E A R T H ' S A N D V E N U S ' CORES

The p r e s e n c e of a magnetic field on Earth is associated with the convective flows in the conductive, mainly iron liquid core. An a b s e n c e of such flows and thus the a b s e n c e of a dipole magnetic field on Venus m a y be due to the difference between the thermal regimes of the cores of Venus and Earth (Stevenson et al. 1983). Three alternatives are possible: (1) the Venusian core does not solidify and the core heat flux is subadiabatic; (2) the core has solidified completely or nearly completely; (3) the planet passed through a period of heating and, because of the large thermal inertia of the core (which is liquid or partially solidified), the planet has not yet reached the regime of core cooling and solidification (Solomatov et al. 1987). Alternative 3 is possible if Venus and Earth reach essentially different asymptotic thermal regimes. H o w e v e r , the estimates done in the preceding section show that the t e m p e r a t u r e s in the interior of Earth and Venus are similar. Therefore, alternative 3 seems unlikely. We must also abandon alternative 2, which implies a completely solidified core, and in this case the t e m p e r a t u r e near the core surface must be close to the eutectic

THERMAL REGIME OF VENUS Teu t ~ 2000-3000 K ( U s s e l m a n 1975a,b, A n d e r s o n e t al. 1987). Consequently, the t e m p e r a t u r e at the c o r e - m a n tle b o u n d a r y of the Earth must have been at least 1000 K higher than that of Venus for Venus to possess a small solid inner core. Only a nearly complete absence of sulfur in the Venusian core (x < 10 -2 to 10 -3) would result in the c o m p l e t e solidification of the core. But according to c o s m o c h e m i c a l and cosmogonical theories (Basaltic Volcanism Study Project 1981, Z h a r k o v 1983; Stevenson e t al. 1983), such sulfur concentrations in V e n u s ' core are of very low probability. L e t us now consider alternative 1: no solidification of the core. I f the entire pressure range PCMV < P < PIE (CM denotes c o r e - m a n t l e boundary, I denotes inner core b o u n d a r y ; E denotes Earth and V, Venus) belongs to the region of y - Fe stability, then, irrespective of the absolute value o f the melting t e m p e r a t u r e of iron, there exists a difference TCME -- Tcrv b e t w e e n the C M E b o u n d a r y t e m p e r a t u r e , TCME, and the C M V b o u n d a r y t e m p e r a t u r e , Tcrv, for the beginning of solidification. This t e m p e r a ture difference is obtained f r o m the intersection of the adiabat (54) with the liquidus (53) at the points P = P1E and P = Pcv (C is the center of a planet). In the linear approximation we h a v e

value,

TCME --

Tcrv

= TCME

+ 1.45 ApCM

[0.79 k

Aplc

289

In the preceding section, we estimated that the C M E b o u n d a r y t e m p e r a t u r e exceeds the t e m p e r a t u r e at the C M V b o u n d a r y by 100-300 K. Thus, if Xv > XE, then V e n u s ' core does not begin to solidify, e v e n w h e n it is cooler than E a r t h ' s core. If XE -- Xv > 0.02, then TCME -- T c r v < 100 K and V e n u s ' core m a y solidify, resulting in generation of a magnetic field. In the absence of solidification the present-day heat flux, Fc -~ 15 erg c m -2 s -1, is less than the surface adiabatic value, - 3 0 erg cm -2 s -l (Zharkov 1986). Thus, a magnetic field cannot be generated. During the first 2 b.y., a magnetic field might h a v e b e e n generated by intensive core cooling, a conclusion similar to that of S t e v e n s o n e t al. (1983). L e t us now discuss the absolute value of the melting t e m p e r a t u r e of iron TEe to compare it with estimates of t e m p e r a t u r e s of the E a r t h ' s core. The two values of TFe (2.43 Mbar) were suggested by B r o w n and M c Q u e e n (1986) and by Williams e t al. (1987). Corresponding values of TEe at the C M E b o u n d a r y are To = 4050 -+ 300 K (this value was obtained by extrapolation of B r o w n and M c Q u e e n ' s (1986) data by the L i n d e m a n n formula) and To = 4800 -+ 200 K (Williams e t al. 1987) (Fig. 2). The intersection of the adiabat (54) and the liquidus (53) at the I E b o u n d a r y for x = 0.09-0.12 (Ahrens 1979) and at a = l; 2 [Eq. (62)] gives

PIE

a(XE -- X v ) [ ,

PCME

J

(71)

w h e r e PXE and PCME are the iron densities at the I E and C M E boundaries; Ap~c and ApcM are the differences in the iron densities between I E and CV boundaries and b e t w e e n C M E and C M V boundaries; XE and Xv are the mass fractions of sulfur in liquid parts of E a r t h ' s and V e n u s ' cores. At TCME ~ 4000 K, Aplc = 0.399 g c m -3, PIE = 12.139 g c m -3, ApCM = 0.318 g c m -3, PCME = 9.909 g c m 3, a = 2, a n d x E -- Xv = 0-0.7, we have TCM E -- Tcr V "~ ( d - 3 0 0 ) -~- ( - 3 0 0 )

K.

(72)

TCME = (3800--4000) -- 300 K,

Ot~2,

(73) TCM E =

(4300-4400) - 300 K, (74)

for To = 4050 K; and TCME = (4500--4800) -- 200 K,

a=2 (75)

TCME = (5100-5200) --- 200 K,

or----l,

(76) for To = 4800 K. a -- 2 c o r r e s p o n d s to ideal solution (x 0) (Stevenson e t al. 1983). a = 1 for x - 0.1

290

SOLOMATOV AND ZHARKOV

8000

I I

7000

r ~1

i

Xrl 8o00

°1°1

S

J

i4000

I

1

ooo,t / / r// ~ooo~" J t / 1000~ 0

It l' , 1

i

II ix I

, 2

II,

3

PRESSURE,M B A R

FIG. 2. Melting curves o f pure iron from the different sources and the b o u n d a r y of the y-Fe --o e-Fe transition. (1) F r o m Williams et al. (1987). (2) F r o m Anderson (1986) [with the experimental data of Brown and M c Q u e e n (1986)]. T h e 7 ~ e transition b o u n d a r y , obtained by A n d e r s o n (1986), together with his melting curve 2 locates the triple point y-e-1. The vertical bars s h o w errors in determination o f temperature for c u r v e s 1 and 2. CM, c o r e - m a n t l e b o u n d a r y ; C, center of a planet; I, b o u n d a r y o f inner core. E denotes the Earth and V, Venus. y-e-I indicates the position of the triple point.

melting temperature of iron, TFe(P), and for the temperature of the y - e transition, Tr_~(P), are shown in Fig. 2. Note that Pcv = 2.9 Mbar, PIE = 3.3 Mbar, and PCE = 3.6 Mbar, and Anderson's value of P t p = 2.8 Mbar is near these values. P t p cannot be accurately determined yet. The difficulties are that the angle between the curves TFe(P) and Tv_~(P) at their intersection is small and extrapolations of these curves may result in large errors. For example, with the new data on the y - e transition at a pressure of 190-360 kbar (Mao and Hadidiacos 1987), the Anderson (1986) method of extrapolation results in Pip > P C E . Near the triple point the liquidus curve follows the path shown in Fig. 3. The ratio o f the decrease in liquidus temperature due to the presence of sulfur for y - Fe to that for e - Fe near the triple point is T~=Fe -- Tm.}, ~ o~, _

AS~

T~=Fe -- Tm~

ASv'

(77)

where a is the coefficient in (53): R ASFeMFe

is an intermediate value between this value and the value o f 0.5-0.6 obtained from the estimates of Anderson et al. (1987) for a eutectic solution (x - 0.25). Estimate (73) agrees well with estimate (69) where TCM E = 3800-4000 K. If the remark concerning the pressure dependence of heat transport parameters (43) and (44) is taken into account, then estimate (69) should be increased by 300 K (the same increase is obtained in the case of Venus). Thus, TCME = 4100--4300 K, which is in agreement with both Eqs. (73) and (74). The position of the triple point y - e - I in the phase diagram of iron may play a role in the explanation of the absence of a dipole magnetic field on Venus. Anderson (1986) has estimated the pressure and the temperature at the triple point: P t p -- 280 Mbar, Ttp = 5760 K. In his calculations, he used the experimental data of Brown and McQueen (1986). His extrapolated curves for the

a~

MFe

Ms " (78)

According to the data of Spiliopoulos and Stacey (1984) and Anderson (1986) ASz,=FeMFe/R = 0.86, AS~-FeMFe/R = 1.7, so that a~ ~ 2, and av/a~ ~- 2. I f x ~ 0. I then T~F¢ -- Tmv ~ 1200 K and T~Fe -- Tm~ -'~ 600 K. At dT~,JdP --~ 17 K / GPa (Anderson 1986) the transition between Tm~ and Tmv (Fig. 3) lies in the narrow pressure range P 2 - PJ ~ P i p - P 2 ~- 0.35 Mbar. I f P l < PIE and P2 > Pcv then the effect o f the triple point is that the critical adiabat for the beginning of solidification of Venus' core is lower and the difference (71) is larger. The maximum decrease in the critical adiabat is - 6 0 0 K and is reached if PI Pcv and P 2 ~ P I E ( P I - P2 "~ Pcv - PIE 0.4 Mbar). Thus, there are possible positions of the iron triple point that could prevent the solidification of Venus' core and the generation of a magnetic field.

THERMAL REGIME OF VENUS

I

Drw T < l-r"

T2

"'

T.

uJ I.-

t

~ .

.

.

TeFel'/

.

.

.

e

.

2Y I I

P1 P2 PRESSURE

Ptp

FIG. 3. Schematic liquidus curve near the iron triple point y - ~ - l . T~.~(p) and T~_F~(p)are the melting temperatures of y-Fe and ~-Fe, Tm~(p)and Tm+(p)are the liquidus temperatures of y - F e - F e S and e - F e - F e S , T~_~(p) is the y - ~ transition curve. Liquidus curve: Tm(p) = Tm+(p)at p < Pt, Tin(p) = T+_+(p)at Pl < P < P2, TIn(P) = Tm~.at p > P2.

vii. THERMALREGIME OF VENUS' CRUST The question of the mechanism of heat transport from Venus' interior to its surface has been the subject of several recent discussions. Plate tectonics similar to those on Earth seems to be absent on Venus (Zharkov 1983, Solomon and Head 1982). Some authors believe that in certain regions of Venus there exist structures characteristic of plate tectonics (Head and Crumpler 1987), but this cannot be the dominant heat

291

transport mechanism. Heat can be partially transported by the hot spot mechanism (Morgan and Phillips 1983). Conductive heat transport through the crust seems to play an important role. However, the thick crust, as indicated by the data of several authors (Zharkov 1983, Anderson 1980, Solomatov et al. 1986, 1987), together with heat flux at the base of the crust [40-45 erg cm 2 s-i (Section III)], results in melting the lower layers of the crust, and the heat transported by the melt in effusive or intrusive ways amounts to about one-half of the entire heat flux (Solomatov et al. 1986, 1987). However, before the melting begins, the thick hot basaltic crust can become convectively unstable and heat can be transported by subsolidus convection. We now consider this possibility. Table I summarizes data on the rheological parameters of albite, anorthosite, quartz diorite, and diabase taken from Kirby and Kronenberg (1987a,b). We take these data to be representative of Venus' crust. The crust thickness, dcr, is set equal to 70 km. Below 70 km, the gabbro-eclogite transition occurs and eclogite sinks because of its high density. The values for crustal parameters taken from Zharkov et al. (1969) are: p = 2.8 g cm -3,

a = 2 x 10-5 K -t,

andx=

6 x 10 3cm 2s-l.

(79)

Because of the strong dependence of rl on temperature, the crust can be separated in an elastic layer with thickness de and a

TABLE I RHEOLOGICAL PARAMETERS OF TYPICAL CRUSTAL ROCKS "0 =

(b/z'~-t) exp(A/T)"

Parameter

Albite

Anorthozite

Quartz diorite

Diabase

b (gin s I z,~ cm m) m A (K) A (K) a t l = 3 5 k m b

4 × 1032 3.9 2.82 × 104 3.0 x 104

4 × 1025 3.2 2.86 x 104 3.0 × 104

4 X 1019 2.4 2.63 × 104 2.8 x 104

3 X 1027 3.4 3.13 × 104 3.3 x 104

o F r o m Kirby and Kronenberg (1987a,b). b The activation volume is taken to be V* = 13.4 cm 3 mole 1.

292

SOLOMATOV AND ZHARKOV TABLE

I1

P A R A M E T E R S OF T H E C O N V E C T 1 N G C R U S T FOR E L ~ 4 0 e r g c m

Parameter de (km) To (K) Tb (K)

Rao RaT Nu RaT;tr

Albite 10 933 1830 4 1 x 105 1.2 10-102

20 1133 1930 10 1 x 104 1.2 102-103

Anorthozite 30~ 1333 2030 17 2 x 103 1.1 102-103

10 933 1690 8 8 x 104 1.5 102-103

2 s 1

Quartz diorite 20a 1133 1730 30 8 x 103 1.7 103-104

30 1333 1730 47 9 x 10z 1.4 103-104

10 933 1630 18 7 x 104 1.9 102-103

20" 1133 1630 85 7 × 103 2.3 104-105

Diabase 30 1333 1680 160 I x 103 2.5 104-105

10 933 1830 3 3 x 105 1.3 10-102

20 1133 1880 12 2 x 104 1.3 102-103

30" 1333 1930 24 2 x 103 1.3 102-103

" The elastic crust thickness that satisfies the condition R a t - RaT.tr.

viscous layer with thickness dv = d c r - d e . The boundary between these layers is determined by the condition that convection is effective throughout the viscous layer. The transition to convection with a stagnant upper layer occurs for m -- 3 and free boundaries when RaT exceeds Rar.tr (29). We suppose that the boundary of the effectively convective layer is determined at a given heat flux by the equation RaT = R a T , t r . For FL = 40 erg cm -2 s -I, the temperature at this boundary is To = (733 + 20)(de in km) K,

(80)

because the contribution of the radioactivity of the crust to the heat flux is small. Because m = 3 (Table I), the dependence of Nu on Ra0 and R a t is taken to be that given by Eqs. (21) and (30) which are valid for fixed boundary conditions. Results of the calculations for de --- 10, 20, and 30 km are presented in Table II. Crustal temperatures are seen to be greater than melting temperature ( - 1 5 0 0 K) in all cases, indicating that convection does not prevent the crust from melting and that heat is transported by the melt. The heat transported by convection can be estimated. For this purpose, let the temperature at the base of the crust be equal to TB = 1500 K and let us evaluate the Nusselt number by formulas (21) and (30). Of all the rocks (Table I), only for quartz diorite is Nu ~ 1.7; for the other rocks, Nu < 1. From Christensen's (1984b, 1985a) results it follows that for the onset o f convection,

Eq. (21) gives Nu - 1.5-2, as for convection at r/ = const. We suppose that the onset of convection can be estimated with the heat transport law for finite-amplitude convection. Quartz diorite is the only rock that may be convecting on Venus at present. In that event, 25-30 erg cm -2 s -r is transported by convection, whereas the remaining 10-20 erg cm -2 s -l is transported by melt. The velocity of the convective flow is about 0.3-0.5 cm year -1 [by Eq. (39)]. The estimates presented here show that convection in the crust can play a role in certain regions having favorable compositions. Most of the heat flux in the models considered here is transported by melt. VIII. SUMMARY

AND DISCUSSION

Four conclusions are made on the basis of this study: 1. Modification of the approximation of parameterized convection results in no marked changes in the main characteristics of models such as the characteristic time scale for thermal diffusion of the planet (34 b.y.), contribution of planet cooling to heat flux ( - 4 0 % ) , and asymptotic regime in the thermal evolution, which does not depend on the initial state of the planet. Insufficient knowledge of the rheological parameters of the mantle and of the parameters o f heat transport laws can influence the distribution of mantle temperatures but not the heat flux. The thermal regime of the planet is determined mainly by the heat released in

THERMAL REGIME OF VENUS the l o w e r mantle, as was suggested in Solom a t o v e t al. (1986, 1987). 2. E s t i m a t e s of the p r e s e n t - d a y thermal state of the E a r t h , t o g e t h e r with the results of modeling the t h e r m a l e v o l u t i o n of Venus, show that Venus was h e a t e d to app r o x i m a t e l y the same t e m p e r a t u r e s as the Earth. T e m p e r a t u r e s at the c o r e - m a n t l e b o u n d a r y of E a r t h are higher than those of Venus by 100-300 K. 3. The a b s e n c e of core solidification should be c o n s i d e r e d as the m o s t p r o b a b l e e x p l a n a t i o n for the p r e s e n t a b s e n c e of a magnetic field on Venus. This situation t a k e s place when the sulfur c o n t e n t in Venus' core is close to or higher than that in E a r t h ' s core, e v e n though E a r t h ' s core is s o m e w h a t h o t t e r than V e n u s ' core. This arg u m e n t is even stronger if the triple point y - e - l i q u i d of iron lies at p r e s s u r e s near t h o s e found at the c e n t e r s of these planets. In this case, the sulfur content in V e n u s ' core can be several times l o w e r than that in E a r t h ' s core. 4. H e a t t r a n s p o r t from V e n u s ' mantle through the thick crust to the surface seems to be i m p o s s i b l e by heat c o n d u c t i o n for the mantle heat flux calculated: 40-45 erg cm -~ c 1. Our analysis shows that most of the heat flux is t r a n s p o r t e d by basaltic m a g m a p r o d u c e d by the melting of the l o w e r layers o f the crust; the rest is t r a n s p o r t e d c o n d u c tively. In some regions, the subsolidus conv e c t i o n in the crust m a y play a role. In c o n c l u s i o n , it is worth r e m a r k i n g that the m o d e l s d e s c r i b e d here are a v e r a g e d . On both E a r t h and Venus, mantle c o n v e c t i o n is actually a v e r y c o m p l i c a t e d nonstat i o n a r y p h e n o m e n o n with flow p a t t e r n s that change during the evolution. This follows from the e s t i m a t e s of the time for establishm e n t of a s t e a d y - s t a t e regime, a time which is c o m p a r a b l e to the age of the planet ( Z h a r k o v 1983, 1986), and from the evolution of the global p a t t e r n of plate t e c t o n i c s o b s e r v e d on Earth. This n o n s t a t i o n a r i t y is c a u s e d by the instability o f the c o n v e c t i v e s y s t e m and in some cases is u n a v o i d a b l e [as, for e x a m p l e , in cases of large internal

293

heating and c o m p l i c a t e d r h e o l o g y ( M c K e n zie e t al. 1974, Busse 1978, C h r i s t e n s e n 1984a,b, 1985a)]. The c h a r a c t e r i s t i c p e r i o d of such oscillations is d e t e r m i n e d by t d / U , w h e r e d and U are the c h a r a c t e r i s t i c d i m e n s i o n and velocity. At d - 108-109 cm, U - 1 cm y e a r - l , we o b t a i n t - 108-109 years. Such time scales c o r r e s p o n d to the " l i f e t i m e " of regional fluctuations of heat flux, volcanic activity, etc. In principle, in some local regions c o n v e c t i o n m a y reach the surface (as in the case of plate tectonics), in a c c o r d a n c e with the geological arguments of H e a d and C r u m p l e r (1987). ACKNOWLEDGMENTS The authors are grateful to Dr. S. V. Gavrilov for helping to improve the English, to Dr. W. B. Hubbard and the unknown referee for valuable comments on improving the manuscript, and to Dr. D. J. Stevenson for useful remarks concerning the Fe-FeS phase diagram. REFERENCES AHRENS, T. J. 1979. Equations of state of iron sulfide and constraints on the sulfur content of the Earth. J. Geophys. Res. 84, 985-999. ANDERSON,D. L. 1980. Tectonics and composition of Venus. Geophys. Res. Lett. 7, 101-102. ANDERSON, O. L. 1986. Properties of iron at the Earth's core conditions. Geophys. J. R. Astron. Soc. 84, 561-579, ANDERSON, W. W., T. J. AHRENS, AND B. SVENDSEN

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