JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 90, NO. B14, PAGES 12,633-12,644, DECEMBER 10, 1985

High Rayleigh Number Convection With Strongly Variable Viscosity' A Comparison Between Mean Field and Two-Dimensional Solutions

FRANCESCA QUARENI, 1 DAVID A. YUEN, 2'3'4 GRANVILLE SEWELL, 5 AND ULRICH R. CHRISTENSEN 6

We have studied single-mode mean field equations associated with variable viscosity convection for both steady state and time-dependent situations. Steady state mean field solutions can be obtained by treating the governing mean field equations as two coupled fourth-order differential systems, which are solved as a series of coupled two-point boundary value problems with underrelaxation. Steady solutions can be achieved in which there can exist viscosity contrasts exceeding 108 and interior Rayleigh numbers of 0(109). For comparison with two-dimensional solutions we have employed results derived from two finite element methods. Both temperature-dependent and temperature- and pressure-dependent viscosity with the surface viscosity fixed have been studied. Our results show that the differences between the two approaches grow with both viscosity contrast and convective vigor. The steady state Nusselt numbers and interior temperatures are greater in the case of mean field solutions. The power law index fl governing the relationship between the interior Rayleigh number and the Nusselt number is larger for the mean field. Comparison of time-dependent solutions shows that one can monitor rather faithfully the evolution of the averaged interior temperature and surface heat flow over a long time scale with the mean field method. Initially, thermal instabilities originating from the mean field boundary layers are found to be correlated with two-dimensional boundary layer instabilities and are much more violent in character and cause large oscillations of the surface heat flow. The time scales associated with the secular variations of the Urey number, representing the ratio of the heat production and the surface heat loss, agree well between the two approaches. These results suggest that the mean field equations may serve as an efficient vehicle for studying planetary convection with complicated physics.

1. INTRODUCTION

Recent developments in the study of the earth's thermal history have drawn attention to the role played by variable viscosity, since this aspect of the problem is central to deter- mining the thermal evolution of the earth [Tozer, 1967]. Most of the work in the past [McKenzie and Weiss, 1975; Sharpe and Peltier, 1978; Schubert et al., 1979; Spohn and Schubert, 1982] has been based on the concept of parameterized convec- tion, which relies basically on the heat transfer relationship drawn from boundary layer theory for constant viscosity con- vection [Turcotte and Oxburgh, 1967]. However, there have been some recent studies [Christensen, 1984a, b, 1985a, b; Flei- tout and Yuen, 1984a] which indicate that the basic assump- tion made in parameterized convection regarding the heat flux and the strength of convection may require revision for vari- able viscosity fluids, since this dependence has been shown to be considerably weaker for variable viscosity. This would imply that the present thermal state is still influenced by the initial thermal conditions and that the plate velocities have remained nearly constant throughout most of the earth's his- tory [Christensen, 1984b, 1985a, b].

•Dipartimento di Fisica, Settore Geofisica, Universitfi di Bologna, Italy.

2Department of Geology, Arizona State University, Tempe. 3Now at Department of Geology and Geophysics and Supercom-

puter Institute, University of Minnesota, Minneapolis. 4Also at CIRES, University of Colorado, Boulder. 5Department of Mathematical Sciences, University of Texas at E1

Paso.

6Max-Planck-Institut ffir Chemie, Abteilung Geochemie, Mainz, Federal Republic of Germany.

Copyright 1985 by the American Geophysical Union.

Paper number 5B5428. 0148-0227/85/005B-5428505.00

Since the numerical simulation of convective processes by means of the complete two-dimensional equations can be fairly time consuming, a simpler and faster approach would be desired. A simpler method, which has been devised by astro- physicists for studying problems in stellar convection [Gough et al., !975; Latour et al., 1976; Toomre et al., 1982] is the mean field approximation, in which the planform is assumed. This approach has enabled the investigation of numerous con- vective processes in stellar interiors, such as compressible and penetrative convection, for Rayleigh numbers far exceeding the numerical capabilities of two-dimensional solutions [Latour et al., 1976; Toomre et al., 1977].

If we can demonstrate that the errors incurred by the mean field approximation are tolerable by geophysical standards, then we will have in hand a very facile means for exploring the vast parameter space in geophysical problems. A limited study for variable viscosity convection has been conducted by Flei- tout and Yuen [1984a] in comparing these two methods. An extensive comparison for constant viscosity convection has been carried out by Quareni and Yuen [1984]. Our purpose here is to conduct a very thorough investigation of the differ- ences between these two approaches for a commonly used Newtonian viscosity law which depends strongly on temper- ature and also on both temperature and pressure.

In section 2 the equations are formulated for the full two- dimensional model and for the single-mode mean field ap- proximation; the numerical techniques employed for solving these two types of problems are described. We then discuss the rheological laws in section 3. In section 4 we compare the steady state solutions obtained by the mean field approxi- mation with those obtained from the two dimensional set of

equations. Section 5 deals with the comparison of time- dependent solutions, which are of importance in studying the earth's thermal history. The potential applications of the mean field approximation to geophysical problems are discussed in section 6 along with the major findings of this work.

12,633

12,634 QUARENI ET AL.' HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY

2. MATHEMATICAL FORMULATION AND

NUMERICAL METHODS

2.1. Equations

Let us consider a rectangular box of aspect ratio a = w/h (where w is the width and h is the height) filled with an infinite Prandtl number fluid. A two-dimensional Cartesian coordi-

nate system is considered, where the coordinates (x, z) are defined with z pointing downward and the origin taken at the top boundary and with x aligned with the horizontal. The gravitational acceleration vector q thus aligns with the z axis. The temperature at the top surface is fixed and the lateral boundaries have reflecting boundary conditions. At the bottom boundary we specify either the temperature or an adiabatic condition.

For the Bousinnesq approximation the conservation of mass, momentum, and energy is given by the following set of nondimensional equations'

V. v = 0 (la)

-Vp' + V. r + RaoTœ= 0 (lb)

c3T • + v. vr = V2T + H(t) (lc) c•t

where v is the velocity, p' the perturbation pressure, T the temperature, H(t) a time-dependent heat source, assumed here to be of the form Hof(t ). We shall consider a Newtonian rheology, as the mean field formulation can only handle linear stress versus strain rate relationship. In a Newtonian fluid the deviatoric stress tensor r is related to the velocity field through the dynamic viscosity r/as

f c•u, c•u• = + (2) All variables are nondimensionalized according to the

scheme given by Quareni and Yuen [1984] and the value of the viscosity at the top surface, r/o, is used for scaling the viscosity [Christensen, 1984b]. The Rayleigh number appearing in (lb) is given by

pg•ATh 3 Ra o = • (3a)

tcr/o

in the case of fixed temperature at the bottom boundary, or by

pg•Ho h5 Ra = • (3b)

tc2r/o

if internal heat sources are present and the bottom heat flux is set to zero. In (3a) and (3b), • is the coefficient of thermal expansion, and AT is the temperature difference across the layer.

In the last few years the mean field method has been applied to mantle convection problems for constant viscosity by Olson [1981] and Quareni and Yuen [1984] and for variable vis- cosity by Fleitout and Yuen [1984a, b] and by Yuen and Flei- tout [1985]. According to the mean field approximation [Her- ring, 1963' Roberts, 1965' Gough et al., 1975] the temperature field can be decomposed into a mean and a fluctuating com- ponent, which are in turn expanded in terms of a complete set of functions along the horizontal coordinate x. The velocity field is also expanded in terms of the same set of functions. The single-mode mean field approximation consists in re- taining only the first term of these expansions. For planforms consisting of rolls we may decompose the total temperature

temperature field as the sum of its horizontal mean t and a fluctuating component ,9, which varies sinusoidally with the horizontal direction. The single-mode mean field theory then consists in keeping only the first term of the expansion of the fluctuating variable. The temperature field may be written as

r(x, z, t) = •(z, t) + ,9(z, t) cos kx (4)

where k is the dimensionless wave number, given by 2•rh/,•, ,• being the wavelength. The aspect ratio of mean field cells is then given by a = rc/k.

The velocity field is given in terms of the vertical velocity w(x, z, t), a fluctuat.ing variable, with the form of W(z, t) cos kx. We note that within the quasi-linear flamework of the mean field theory the planforms are specified to within an arbitrary constant.

By this Fourier decomposition we reduce the dimensional- ity of (la)-(lc) by one to a system of partial differential equa- tions with the time and the depth as the independent vari- ables. They take the form

c3 '•W 2k 2 •2W k, • 2 dr/(•3W k2 c3W) c•z, • • + W +- -

1 d2r/f•2W ) + • •z • • c•z• + k2W + Rao k2 - = O (Sa)

- - (wo)

63t -- C3Z 2 -- k2,9 -- W 63-• (5c) The boundary conditions associated with the mean field

equations are as follows. The top temperature is fixed'

rl:=o = 0 ,91:=0 = 0 (6)

For a fluid heated from below the temperature is also constant at the bottom'

TI:= 1 : 1 ,91:= 1 = 0 (7a)

For an internally heated fluid the adiabatic bottom is gov- erned by

c•' = 0 •zz = 0 (7b) •ZZ z=l z=l As there is no mass flux out of the boundaries, the appropriate boundary conditions for the mean vertical velocity are

Wl=o - o Wl= 1 : 0 (8)

Here we will study only stress-free boundaries, which demand that

I •2W = 0 • = 0 (9) •Z2 z=O z= 1

Two different approaches have been employed for solving the full system of two-dimensional equations (1 a)-(1 c).

Christensen [1984a, b] has cast (la)-(lc) in terms of a stream function and temperature formulation in which the stream function ½ is related to v by

ß = . , The resulting set of momentum and energy equations is

solved by means of a B spline finite element method, using bicubic splines for ½ and biquadratic splines for the temper-

QUARENI ET AL.' HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY 12,635

ature field. The Ritz-Rayleigh variational principle is applied to the momentum equation (lb), leading to a set of linear equations for the coefficients of the spline representation of the stream function ½. The energy equation (lc) is solved by the upwind weighted residual technique. Further details can be found in the work by Christensen [-1984a-I.

Another finite element formulation has been employed, using a general purpose two-dimensional code TWODEPEP which has been designed for solving a wide variety of partial differential equations [Sewell, 1981], ranging from quantum mechanics to elasticity. The convection equations have been cast in the primitive variables involving the velocity and tem- perature fields, together with a penalty function approach in which the pressure is associated with the Lagrange multiplier. A penalty function parameter value of 10 6 has been found sufficient to achieve the same degree of accuracy in the Nuss- elt numbers found tabulated by Christensen [-1985a, b-I.

A third code NACHOS [Gartlinq, 1978] has also been used by M. Koch to check the results of the Nusselt numbers ob- tained by the other two methods. This code is based on the three primitive variables of temperature, velocity and pertur- bation pressure. For viscosity contrasts of 0(10 3 ) and fairly vigorous convection with Nu 0(10), an agreement of the Nuss- elt number out to three significant digits has been found among these three methods. Christensen's code is found to be faster by a factor of about 5 over the two others because of the fewer number of unknowns per nodal point, a consequence of the spline formulation.

2.2. Numerical Methods for Mean Field Equations

We have devised two numerical methods for solving the mean field equations different from those used by Fleitout and Yuen [1984a, b].

For solving the mean field equations (5a)-(5c) we have de- fined a new variable as follows'

•3W 1 dr/(•2W k2 Y--•--•z3 +•zz[,-•-z2 + W (11) Rewriting the momentum equation in terms of this variable,

we obtain an equation in which only the first derivatives of the viscosity appear. This transformation facilitates greater nu- merical speed and accuracy in the computations. The new momentum equation takes the form

r3Y ldr/( r3W) •2W k4 r3-•- + • •zz Y - 3 k 2 •z - 2 k 2 -•-•z 2 + W + R a o k 2 - = O (12)

The set of equations (5b), (5c), and (12) with the associated boundary conditions (equations (6)-(9)) has been integrated as an initial value problem to steady state or directly solved for the steady state as a system of nonlinear boundary value problems (equations (5b) and (5c) with r3/r3t = 0 and equation (12)). In the time-dependent integration we must first supply the initial temperature fields for T and • into the elliptic mo- mentum equation (12). The initial profiles for T and • in the case of base-heated convection [Fleitout and Yuen, 1984a] read

T(z, t•) = a• erf I2(ti• •/2] + (a• -- a2) { 1 -- erf 4(t•)•/2 (13a)

and

2(tt)•/2 exp -- (13b) where t• is the initial time of integration and is taken to be 0(10 -3) in dimensionless units, a• and a 2 are the initial in- terior and bottom temperatures, respectively, and a3 is a scal- ing parameter of the initial finite amplitude perturbation, a 3 being of 0(10-2). The momentum equation becomes then a linear two-point boundary value problem for W. This is solved by a variable order, variable step-size finite difference method [Pereyra, 1978]. For most of the runs about 60 une- venly spaced points were deemed sufficient. We note that this particular linear two-point boundary value problem is nu- merically unstable when treated with the multiple-shooting method because of local parasitic solutions encountered in the regions with sharp viscosity gradients in the formulation of equation (5a). The energy equations (5b) and (5c) are suc- cessively integrated using the values of the velocity, obtained previously, and having been interpolated locally by Hermite cubic splines. The solution of these time-dependent equations is obtained by the method of lines, where the solution vector has been expanded in terms of a series of Hermite basis func- tions. A system of nonlinear ordinary differential equations is then obtained for the coefficients of this collocation expansion and time stepping is done through the Gear ordinary differ- ential equation (ODE) integrator [Sewell, 1982' Gear, 1971' Hindmarsh, 1974], since the overall discretized system, having about 100 nodal points, is inherently stiff. This method is only first-order correct in time, since the velocity field is not updat- ed again before it is substituted into the two energy equations.

In order to reduce the computational time for obtaining steady state solutions, we have approached (5b), (5c) and (12) as a coupled system of two-point boundary value problems. The entire set of mean field equations constitutes an eighth- order system. We have found that it is not possible to solve this entire system all at once. It is necessary to break the system up into two coupled fourth-order systems. The mo- mentum equation (12) and the two energy equations (5b) and (5c), with the time derivative terms dropped, thus constitute two coupled fourth-order ODE's. These are solved by the same finite difference technique described above. The equa- tions are treated sequentially as individual two-point bound- ary value problems, with the momentum equation leading.

The initial profiles for '9 and T, given by (13a) and (13b) are used for the first iteration in solving for W in (12). The system of the energy equations then employs this velocity solution in deriving the temperature profiles for this particular iteration cycle. The boundary conditions of the two-point boundary value problems are satisfied to one part in 105 . These values of '9 and T, interpolated by Hermite cubic splines, are then sub- stituted back in (12) for the new cycle of the next iteration. This procedure is repeated until convergence to the steady state is attained. This is defined by a variation in the Nusselt number of one part in 104 between the successive iterations. This procedure usually requires 6-10 iterations.

We find that the usage of underrelaxation methods is neces- sary at times. In the underrelaxation procedure one supplies at the (n + 1)th iteration in the momentum equation the values of the two temperature fields obtained at the previous nth and (n - 1)th iteration, which read

Zn+ • -- (1 - fi•)Zn- • + fi• Zn (14a)

'9,,+ 2' - (1 - ,82),9n_ • + ,82,9 . (14b)

12,636 QUARENI ET AL.: HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY

In the same way, the value of the velocity fed into the (n + 1)th iteration of the energy boundary value problems as- sumes the form

W,+ •' = (• -/h)W,- • +/hW, (•5)

The values of the underrelaxation parameters /•,/•2, and/•3 range from 0 to 1. We have found that the optimal values of the underrelaxation parameters are around 0.5. Sometimes convergent solutions could only be achieved with/•i •- 0.5. A typical steady state run on the CRAY-IS at NCAR requires a few seconds for the profiles displayed in Figures 2 and 3.

3. RHEOLOGICAL LAWS

There are various ways of describing the thermally activated process of mantle viscosity. The first would be to write • as being proportional to exp (Q/RT), where Q is the activation enthalpy, R is the gas constant, and T is the absolute temper- ature. Another way is to expand the argument of the ex- ponential in a Taylor series about a reference temperature and keeping just the first term of this expansion in the argument of the exponential yielding a viscosity proportional to exp (-AT). This approximation is good for describing the dynamics of the upper thermal boundary layer, but suffers greatly as the flow penetrates deeper. The viscosity law studied in this work is based on this assumption. For a temperature- dependent viscosity this type of formulation has been used in laboratory [Natal and Richter, 1982; Richter et al., 1983], analytical [Fowler, 1985; Morris and Canright, 1984], and nu- merical [McKenzie, 1977; Christensen, 1984a, b, 1985a] stud- ies. This viscosity is given by

•(T) = •o exp (-AT) (16)

One may generalize this to a temperature- and pressure- dependent viscosity [Christensen, 1985a] and write

•(r, p) = •(r, z) = •o exp (--AT + Cz) (17)

For both of the above creep laws, •o denotes the viscosity at the surface. The quantity exp (--A) measures the viscosity variation across the entire layer from the temperature depen- dence, while exp (C) represents the increase of the viscosity from top to bottom due to the pressure dependence. We denote the total viscosity contrast • across the layer by

• - - exp (A - C) (18)

The particular form of the laws (16) and (17) has the partic- ular attribute that raising the parameter A would lower the interior viscosity, thus making the medium softer and leading, as a consequence, to a more vigorous convection in the in- terior. For this reason, we will henceforth call this rheology "molle," meaning soft in Italian.

Christensen [1984b, 1985a] has argued that the "molle" rheology should be used in modeling thermal evolution of the earth, since the viscosity of the earth's surface should remain constant throughout most of its history. But this is not the only viscosity law for modeling geodynamical problems. Alter- natively, one can constrain the interior viscosity to a certain • at a certain reference temperature T• and pressure, as done in studying secondary convection beneath an immobile litho- sphere by Yuen and Fleitout [1985] and Fleitout and Yuen [1984a, b]. This form of the creep law reads

•(T,J) • exp [B(• •) ] = -- + Cz (19)

where B is proportional to the activation energy. In this case, increasing B in the rheological law (19) would increase the viscosity at the surface, making the lithosphere harder. We call this rheology "duro," which means hard in Italian, to dis- tinguish it from the "molle" rheology, given by (16) and (17). By the same token, the presence of T in the denominator of equation (19) would cause big differences in the thermomecha- nical structure of the lower half of the convective circulation.

Fleitout and Yuen [1984a] have conducted a limited compara- tive study of the two-dimensional and mean field equations for the "duro" rheology. They found good agreement between these two approaches.

The "molle" rheology, however, behaves quite differently from the "duro" creep law, since there is still a substantial amount of deformation in the lithosphere (cf. Figures 2 and 3). In this work we direct our attention on the "molle" rheology, which has already been investigated extensively [Nataf and Richter, 1982; Richter et al., 1983; Christensen, 1984a, b; Morris and Canright, 1984; Fowler, 1985].

4. COMPARISON OF STEADY STATE RESULTS

In this section we will compare the steady state results be- tween the mean-field and two-dimensional calculations. The

bulk of the two-dimensional results can be found described in

greater details by Christensen [1984, 1985a].

4.1. Constant Viscosity Comparison

Recently, Schubert and Anderson [1985] have carried out by finite elements an extensive set of constant viscosity calcula- tions for very high Rayleigh numbers, close to 0(109). Treating the mean field as a series of two-point boundary value prob- lems proves to be numerically much superior to the differ- ential algebraic (DAE) method [Petzold, 1982] used by Quareni and Yuen [1984]. It is now possible to go up to a Rayleigh number of 10 •ø by using, as a starting solution, the result from a lower Rayleigh number. Altogether we need about 10 calculations to construct the solid curve shown in

Figure 1. The total computational time of the mean field solu- tions shown in Figure 1 took less that 4 min of CPU time on a CDC 7600. In Figure 1 we compare the Nu versus Ra re- lationship for constant viscosity, base-heated convection be- tween the results of Schubert and Anderson [1985] and our mean field calculations. For vigorous convection a power law relationship can be found relating the Nusselt number Nu to the Rayleigh number Ra. For constant viscosity it is given by

Nu = 3,Ra • (20)

where 3' is a coefficient of 0(1) and fl is the power law ex- ponent. The results of Schubert and Anderson [1985] yield 3, = 0.27 and fl = 0.32; our mean field results give 3, = 0.25 and fl = 0.36. Thus the value of the logarithmic slope fl, defined henceforth to be c3 In (Nu)/c3 In (Ra), is only 10% greater for the mean field calculation.

4.2. Comparison of Temperature and Velocity Profiles for Variable Viscosity

At this point it is convenient for us to introduce another definition of the Rayleigh number, different from (3a) which measures better the vigor of convection in a variable viscosity medium [Nataf and Richter, 1982]. This interior Rayleigh number Rat is determined by the viscosity of the globally averaged temperature (T) for r/(T) and is taken at middepth for evaluating the depth in r/(T, z). It is given by [Christensen, 1985a]

QUARENI ET AL.' HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY 12,637

1o

I I I i I I

This work

_ • Mean field••-.• _. _

10 5 10 6 10 7 10 8 10 9 1010 Ra

Fig. 1. Nu versus Ra relationship in the steady state for constant viscosity convection obtained by mean field results (solid curve) and two-dimensional solution (dashed curve) from Schubert and Anderson [1985]. An aspect ratio of 1 is taken.

pgo•A Th 3 RaT -- r/((T), z = 0.5) (21)

In Figure 2 we show for both the mean-field (solid curves) and two-dimensional (dashed curves) results for the averaged horizontal velocity •, which is nondimensionalized by •c/h, and the mean temperature T as functions of depth. Two different viscosity contrasts (Ar/= 250, 1000) have been considered for r/(T). Convective strength is fairly strong for Ra = 5 x 10 '•, since the corresponding interior Rayleigh number, RaT, lies between 10 6 and 10 ?. The interior temperatures predicted by mean field theory are lower than the two-dimensional results, with the differences becoming greater with increasing Ar/, which enhances the vigor of convection in the interior. For At/-- 10 3 the difference in the interior temperature is about 20%. The horizontal velocities for the mean field solutions are

greater at the top and smaller in the hot, bottom thermal boundary layer. We can also see that the thermal boundary layers from mean field are thinner than the two-dimensional ones, as have been found for constant viscosity [Quareni and Yuen, 1984]. This same trend is also preserved for temperature- and pressure-dependent viscosity (cf. Figure 3).

Figure 3 shows the horizontal velocity, mean temperature, and viscosity profiles for a temperature- and pressure- dependent viscosity. The total viscosity contrast between the

top and the bottom is 500, while the viscosity increase from depth dependence amounts to 10 3 . The mean field solution is unable to reproduce the stably stratified temperature gradient exhibited by the two-dimensional solution, thus accounting for a marked difference in the viscosity structure. For the "molle" rheology the thermal structure for r/(T, z) is nearly symmetric which behaves quite differently from that derived for a viscosity with a simple temperature dependence. In con- trast to the situation for r/(T) the horizontal velocity is greater everywhere for the full solution. However, the mean field Nusselt number is about 30% greater.

The behavior of the globally averaged temperature (T) as a function of the viscosity contrast is illustrated in Figure 4. Without the pressure dependence the average temperature is higher than 0.5, but as the value of C is increased beyond one, T falls below 0.5, as shown in Figure 3. This facet is due to the peculiarity of the "molle" rheology and does not occur for the "duro" rheology. The averaged temperature with r/(T, z) for the "molle" rheology is different from the (T) derived from r/(T, z) based on the "duro" rheology, where (T) is very much above 0.5 in the interior [Fleitout and Yuen, 1984a]. By con- trast, the mean field average temperature for r/(T, z) is higher than the two-dimensional results for small viscosity contrasts, and we can observe that the inclusion of pressure dependence in the viscosity law enlarges the range of Ar/where this state- ment is valid. On the other hand, for higher values of Ar/ the reversed situation takes place.

4.3. Comparison of the Dependences on the Wave Number

In order to study the effects of different cell configurations, we now examine how the solutions depend on the wave number k, which is inversely proportional to the cell aspect ratio a. This relationship is given by k = rc/a.

For the purpose of understanding better the properties of heat transfer it is of importance to determine the optimal wavenumber that maximizes the Nusselt number. For con-

stant viscosity and stress-free convection this optimal wave number is around rc [Olson and Cotcos, 1980; Hansen and Ebel, 1984; Quareni and Yuen, 1984]. For low Rayleigh num- bers we find in the bottom panel of Figure 5 that both the mean field and two-dimensional results predict that maximum heat transport is achieved at long wavelength. This also means

/ II i i ø"l_ 4 '

0.4 1000 • I

/ i r 1

,,, 11,,o o . ,, i ! , , , , , ,

-2000 -1000 0 1000 2000 0 0.2 0.4 0.6 0.8 1

Fig. 2. Mean horizontal velocity •2 and temperature •' as function of depth for a temperature-dependent rheology with two different viscosity contrasts Aq = 250 and 1000 and a Rayleigh number Ra o = 5 x 10 '•. Solid curves represent the mean field solutions' dashed curves denote the two-dimensional ones. An aspect ratio of unity has been assumed.

12,638 QUARENI ET AL.' HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY

o

0.2

z 0.6-

0.8-

1.0 0

A r•-- 500, e c = 1000, Ra = 50,000

-

-

I I I I 1%,,..1 •"4.....3 I

0.6 0.8 1.0 -400 -200 0 200 400 10 '2 10 '1 1 10

Fig. 3. Mean temperature T, horizontal velocity if, and viscosity r/ as function of depth for a temperature- and pressure-dependent rheology with a total viscosity contrast At/= 500 and exp (C) = 1000. The Rayleigh number Ra o is set equal to 5 x 104. Solid curves represent the mean field solutions' dashed curves denote the two-dimensional ones. The aspect ratio is the same as in Figure 2.

that at low Rayleigh numbers the upper viscous boundary layer is much more active in a circulation with long cells [Daly, 1980a]. In this situation the optimal wavelength derived from mean field calculations is about 70% larger than that derived from two-dimensional calculations. However, for

higher Rayleigh numbers the optimal wave number increases, as shown in the top two panels of Figure 5. For Rao = 10 '•, corresponding to RaT of 0(106), we find that the optimal wave number has now shifted back to the range associated with constant viscosity. Although the Nusselt numbers are greater and the deviations between the two methods become bigger with greater vigor of convection, the characteristic of Nu versus k is revealed quite well with the mean field approach. One would not, of course, expect that the mean field equations could capture the bifurcation of the Nusselt number shown in

0.9 I i I I I I I I

o.8 - Rao-10,000 . 0.7

0.6 -

0.5 I I I I I I I I'

0.6 Raø=10'000 0.5

0.4

0.5

0.4

0.3

1 4 16 64 9-50 1000 4000 16,000

Fig. 4. Globally averaged temperature (T) plotted as function of the total viscosity contrast for temperature-dependent rheology (top panel) with Ra o = 104 and for temperature- and pressure-dependent rheology (bottom panels) with Rao = 10 '• and exp (C) = 64 and with Ra o = 5 x 10 '• ane exp (C)= 1000. Solid curves represent the mean field solutions; dashed curves denote the two-dimensional ones. Same aspect ratio is used as in Figure 3.

the top panel. This bifurcation is due to a change of the flow pattern [Christensen, 1984b]. The greater discrepancy found at higher Rayleigh numbers is due to the presence of well- developed plumes which induce strong lateral heterogeneities, thus rendering the mean field assumption less valid.

Increasing the pressure dependence can also cause a shift of the optimal wave number for heat transfer toward smaller

Nu

20 10.5

19

18

17

16

12

11

10

/ Ra o = 10,000 e c =1

/•7 = 1000 I I I I I I I

- 10

- 9.5

-9

- 8.5

-8

- 7.5

-7

- 6.5,

-6

I I I I I I I

7 - 5.5

6 -5

5- / "• -.• - 4.5

0 I 2 3 4 5 6 7 8

k

Fig. 5. Nusselt number Nu versus wave number k from mean field (solid curves) and two-dimensional (dashed) calculations for temperature-dependent viscosity with At/= 1000. Three cases have been examined with different Rayleigh numbers Ra o = 1000, 3000, and 10,000, from bottom to top.

QUARENI ET AL.' HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY 12,639

9 I i i i i

5 -

8-

Nu 7-

6-

5-

7 -

/x7 =25o .,o=2OOO i i i i i i i

c e =64

i i i i i i i

c e = 1000

6.5

-6

- 5.5

-5

- 4.5

5.5

4.5

- 5.5

-5

- 4.5

-4

1 2 3 4 5 6 7 8

Fig. 6. Nusselt number Nu versus wave number k from mean field (solid curves) and two-dimensional (dashed) calculations for temperature-dependent (top panel) and temperature- and pressure- dependent (bottom two panels) viscosity with At/= 250. The amount of pressure dependence in the top two panels is such that exp (C) = 64 and 1000, respectively.

larger values, as it is shown in Figure 6 for Rao = 2000 and r/(T, z) with a total viscosity contrast of Ar/- 250. In the comparison with the two-dimensional curves, we can again observe that the trends are well preserved by the mean field approximation and that for this situation of relatively low Rayleigh number the quantitative agreement is better than 40%.

The existence of the maximum in the Nu versus k curve is

related to two competitive mechanisms, the mobility of the lid at the top of the convective circulation and the dependence of the vertical convective heat transport on the surface area of the cell. As noted already, for longer cells the top lid becomes more mobile, thus allowing for greater amounts of heat trans- fer. This behavior in displayed in Figure 7, where the horizon- tally averaged surface velocity (%), where (Us) = t2(z = 0) in the mean field formulation, is plotted as a function of the wave number for both r/(T) and r/(T, z). It should be obvious from Figure 7a that the ratio between the surface velocities of long- and short-wavelength cells increases with the vigor of convec- tion. Cells associated with whole mantle convection, e.g., the Pacific plate, typically have aspect ratios greater than two. Thus dimensionless surface velocities, characteristic of plate tectonic velocities, between 0(102 ) and 0(103), can be attained even for large viscosity variations. Second, we notice that for long cells the mean field approximation predicts higher hori-

zontal velocity at the surface than the two-dimensional ap- proach, while for smaller aspect ratios the opposite tendency occurs. Finally, this same phenomenon is found for temperature- and pressure-dependent viscosity (cfi Figures 7b and 7c), although the surface velocities are smaller for the same Rao (Rao-- 10 3) than those derived from temperature- dependent viscosity.

4.4. Comparison of Heat Transfer Characteristics

The properties of steady state heat transfer which are of vital interest to the study of the earth's thermal history are compared in this subsection. As the viscosity drop At/can be specified beforehand, it is physically more intuitive in studying a cooling convective system such as the earth, to use Rat [Nataf and Richter, 1982; Richter et al., 1983; Christensen, 1984b], which is only known "a posteriori." This is based on the averaged temperature (T) of the convective system. For this reason, the behavior of the Nusselt number should be studied as a function of Rat rather the Rao for parametrized convection models. For a given Rao, we begin from a case with constant viscosity for which (T) = 0.5; we then increase the viscosity contrast by factors of 4 to trace out the solid curves for mean field shown in Figures 8 and 9. In the pres- ence of variable viscosity the average temperature increases, thus making Rar greater than Rao by as much as a factor of 100 or more. Initially, the Nusselt number increases, then it reaches a flat region, which corresponds to a regime where the interior circulation becomes decoupled from the moving lid, as has first been noted by Christensen [1984b, 1985a, b] for the "molle" rheology. Even though there are rather substantial differences in Nu, the two curves definitely have similar trends. The logarithmic slope/• decreases with increasing Rat, due to the greater viscosity contrast. We find that /• predicted by mean field is greater than that derived from the two- dimensional results. This discrepancy is greater for larger Rao. Hence this decrease of/• with Rat can be documented with the mean field method. This result, shown by both mean field and two-dimensional calculations, may have important impli- cations [Christensen 1984b, 1985a, b] for thermal history cal- culations by parameterized convection which usually assumes /• to be around 0.3 [e.g., Sharpe and Peltier, 1978; Schubert et al., 1979], as suggested by constant viscosity convection.

From asymptotic analysis, Morris and Canright F1984] have derived an expression for r/(T) which gives a very concise re- lationship for the quantities involving Nu, At/, and Rat. Fol- lowing this hint, we find that the curves with different Rao in Figures 8 and 9 can really be collapsed into universal curves according to empirical relations given by Nu = ax(Rara2/Arla3), where ax, and a2, and a 3 are free parameters obtained from least squares fit. These results from mean field calculations are summarized in Figure 10 for both rheologies. Our finding that an universal curve exists in the case of r/(T, z) holds the hope that an asymptotic relation for temperature- and pressure- dependent rheology, similar to that derived by Morris and Canright for r/(T), may be found in the future.

4.5. Comparison of the Averaged Surface Velocity With Asymptotic Theory

Another fluid dynamical quantity which the mean field method can predict correctly is the variation of the averaged horizontal velocity at the surface with At/. By means of matched asymptotics, Fowler [1984, 1985] has derived several useful expressions describing the behavior of high Rayleigh number convection in strongly variable viscosity fluids for the

12,640 QUARENI ET AL.: HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY

1200- A•=10 a e c=1 4000 • • A•=10 a '

f ,1 C;lO _ •ooo • 1 •søø •u•)•o • - 900 Rao= 105 13000 I \\ Ra =105 8OO

•2500 • - •oo o •- •-

600 • 2000 (Us / , , , , , - q1500 60

- ooo Us)o 200 • h •c [ 03

0 0 0 I I I I i i 0 i 2 3 4 5 6 0 i 2 3 4 5 6 7

k k

Fig. V. Averaged surface horizontal vdocity •u• versus wave number k from mean field (solid curves) and two- dimensional (dashed curves) results for temperature-dependent (Figure 7a) and temperature- and pressure-dependent viscosity (Figures 7b and 7c) with a total viscosity contrast • = 103. (a) Two Rayleigh numbers Ra o = 103 and 10 • have been considered for both mean field and two-dimensional approaches and the curve for Ra o • 10 s is plotted for mean- field only. (b) and (c) The Rayleigh number is equal to 103 and the pressure dependence parameter exp (C) is equal to 103 and 64, r•sp•ctivdy.

"molle" rheology. In particular, the mean surface velocity (Us) is given by [Fowler, 1984]

a3/SRao4/5(ln At/) 2 (Us) • 0.02 Ar/•/5 (22)

It is important to appreciate the fact that the above asymp- totic expression is supposed to be valid for large values of Rao and At/. One sees readily that this function for (Us) contains a maximum with respect to the viscosity contrast. This maxi- mum is predicted to occur at At/,-, 2 x 10 ½. Physically, we may interpret (22) as having two contributors, one with the logarithmic dependence of At/, the other with the dependence on At/with an exponent of -1/5. The first factor, arising from an increase of longitudinal stress, tends to increase the surface velocity and dominates for small values of At/. The second one represents the effect of decoupling between the lid and the underlying flow and prevails for large Ar/. We show in Figure 11 the surface velocity (Us) as a function of Ar/ for the mean field calculations. At low values of Rao, where the asymptotic theory would not apply strictly, the maximum is attained for Ar/between 10 3 and 10 ½. However, as Rao is increased beyond 5 x 10 '•, the maximum is shifted to 0(10 •) in good agreement with Fowler's asymptotic result.

5. COMPARISON OF TIME-DEPENDENT CALCULATIONS

Time-dependent calculations are much costlier than steady state computations. Thus, the question naturally arises as to whether the mean field method can capture some of the details of thermal evolution obtained by integrating the full set of two-dimensional equations. From the work on finite Prandtl number convection [Toornre et al., 1982] it has commonly been thought that the mean field approximation is inadequate

to describe time-dependent phenomena encountered in con- vection. To be sure, we would not recommend that the mean field equations be employed in studying problems involving hydrodynamic transitions leading to turbulence. It has been

tl I I I I - 1 00 2501000 200•0 4000 50 Rao = 200,000 64 z:••....• . 4 16•...•._.___,,_.• 2000

,..- •" 16 10 5 i I I I I

I I I I* I

100 - 16,ooo 24,000- 50 Rao=50'000 lOOO 4o0•.•,••_• _

Nu 4 16 64• '•"•'• 10 • •-- -- •-4'* - 2'• - -• 0•0-• - --'•000 • 4 16 - 5 I A•r•=l I I I I -

i

I I • 1000 • 4000

/ 64 25••.•.•.•• ' '16,000 16

F •.•• • ..... ' ......... 16,000 101 ...- •-•-. '- ;•o'-'•o•- 4000 5 F •'''• 16

ZXl,,• = 1 Rao =10,000 1 i i I I

104 10 5 10 6 107 10 8 Ra T

Fig. 8. The Nussclt number Nu versus the Rayleigh number based on the interior temperature Rar (cf. equation (21)) for temperature-dependent viscosity for mean field (solid curves) and two-dimensional (dashed) results. In each panel the Rayleigh number based on the top viscosity Ra o (cf. equation (3a)) is fixed (from bottom to top, Ra o --- 104, $ x 104 and 2 x 105). The aspect ratio is 1.

QUARENI ET AL.: HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY 12,641

1o

I i

lO

ß ' 250 10•_•.......2000 16

•'"" 4 Aq--•_

Rao= 50,000 e c =64

I I

1000 _2000

64 257.•• - 16•..7- •'• .... _ ...... 1000

1 Rao= 10,000

e c =64

104 10 5 10 6 10 7 Ra

Fig. 9. The Nusselt number Nu versus the Rayleigh number based on the interior temperature RaT (cf. equation (21)) for temperature- and pressure-dependent viscosity with exp (C)= 64 for mean field (solid curves) and two-dimensional (dashed) results. In each panel the Rayleigh number based on the top viscosity Ra o (cf. equation (3a)) is fixed (from bottom to top, Ra o = 104 and 5 x 104). The aspect ratio is the same as in Figure 9.

shown by Quareni and Yuen [1984] that one can obtain an agreement to within 10% between the transient solutions of mean field and two-dimensional results for constant viscosity. Dynamical features depicting the transition from double to single layer convection [Boss and Sacks, 1984] can also be captured by the mean field method, as shown by Quareni and Yuen [1984], who found good agreement in the duration of double-layer convection. Fleitout and Yuen [1984b] have found that for the "duro" rheology the evolution of the surface features can be computed to better than 5% accuracy for seafloor ages greater than 75 Ma.

In Figure 12 we show the comparison for time-dependent convection in an uniformly internally heated fluid with a Ray- leigh number (equation (3b)) Rao = 1.5 x l0 s. This vigor of convection is characteristic of circulation restricted to the

upper mantle. The "molle" viscosity is taken to be dependent only upon temperature with ,4 = 90; an aspect ratio of 2 has been imposed for the cell size. The dimensionless inverse decay time for the internal heating is 2 = 10. The initial conditions are taken from steady state profiles associated with a constant internally heated system, i.e., H = 1. Small perturbations in 0 have been imposed for the mean field, while the two- dimensional initial profiles are still not strictly convergent by the steady state algorithm. We then show the two different results (solid curves indicating mean field, dashed curve de- noting two-dimensional solutions) of the averaged surface heat flux Qsur and the globally averaged temperature (T) as func- tions of time up to t = 0.27, corresponding to a time of 3 billion years based on a scaling of upper mantle convection. We have normalized the surface heat flow with respect to the steady state value for constant internal heating.

Small enough time steps, At = 0(10-s) have been employed at the initial stages of the mean field calculations to insure that the initial oscillations in the surface heat flux are not due

to numerical artefacts. We have also monitored the energy budget at each time step and found agreement to better than

Nu

10 4 10 5 10 G

Ro T/A'r} 0.2G

Nu IO

-i [ t i Illi[ i [ I I lll[ I I I I i till I .• : ß ROo=10 4 .; eC=64 I -J -,• ,, =5xlO?; ,, =64_l -.!

o ,, =5xl04; "=1031 oY --[

1 t ,• o._!

_1 I I I I1•1 I I I I IIt•l • I I I II1•1 "1 10 4 10 5 10 6

ROT/A•0.12õ

Fig. 10. Universal curves of the Nusselt number versus RaT and Ar/collapsed following the asymptotic scaling of Morris and Canright [1984]. The results are taken from mean field calculations. The sym- bols for the various Rao are given. The top panel contains results for r/(T), while the bottom contains the results for r/(T, z).

1% among the rate at which the interior temperature changes, the heat transported away and the heat produced by volu- metric heating. As we have already observed in the steady state cases, the mean field interior temperature is lower than the two-dimensional one as is also the surface heat flux. The

first part of the mean field solution is characterized by period- ic upwellings and downwellings which have their origins in buoyant instabilities within the bottom and top boundary

o I I I I I 1 10 102 103 104 105 106

Fig. 11. Averaged horizontal surface velocity (Us), from mean field solutions, is plotted as a function of the viscosity contrast At/ in case of a temperature-dependent rheology. Four Rayleigh numbers Rao have been considered, Rao- 10 3, 10 4, 5 x 10 4, 2 x lO s. The aspect ratio is kept the same as in Figure 10.

12,642 QUARENI ET AL.' HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY

1.5 1 [ [ [ [ [ 1.0. , • -

0.5 ;i'"'-.. -

(T) I ' , , , 0.05 ' - ' - - - ' - -• 0

o o.1 0.2

t

Fig. 12. The normalized surface heat flux Qsur and the globally averaged temperature (T) are plotted as functions of time for time- dependent convection in an uniformly internally heated fluid with a Rayleigh number (equation (3b)) Ra o = 1.5 x 10 •. The dimensionless inverse decay time for the internal heating is 2 = 10. The viscosity is taken to be temperature-dependent, with A - 90; an aspect ratio of 2 has been imposed for the cell size. Solid curves represent the mean field solutions; dashed curves denote the two-dimensional ones. In order to simulate the bifurcation of the two-dimensional solution into two cells at t = 0.1, the aspect ratio has been halved in the mean field computation (dotted curve).

layers. These finite amplitude perturbations induce global flows which cause oscillations in the surface heat flux. It is

somewhat surprising that the two-dimensional boundary layer instabilities can also be picked up by the mean field formu- lation. Due to the nature of the mean field approach, the only possible degree of freedom in the motion is restricted to the vertical dimension. Hence these oscillatory perturbations are very much larger than the boundary layer instabilities found by the two-dimensional calculations, which can be swept about the global circulation.

After t = 0.1, about 1 billion years, the two-dimensional convective pattern bifurcates into two cells. In the mean field scheme we try to simulate this phenomenon by halving the aspect ratio at t = 0.1, but it does not help to narrow the gap existing between the curves obtained by the two approaches. The solution obtained with the mean field approximation having the aspect ratio equal to 1 at t = 0.1 is represented by a dotted line in Figure 12 for the surface heat flux. There is no visible difference in the average temperature profile.

We have also checked in the case of the mean field calcula-

tions whether the observed oscillations in Figure 12 may be caused by the second-order discontinuity in the heating func- tion imposed at t = 0. To ascertain whether the periodic surges have indeed dynamical origins in the boundary layers, we have also carried out a calculation with constant heating, using the same initial profiles as for the time-dependent heat- ing situation. We'find (Figure 13) that boundary layer instabil- ities also occurred in the case of constant heating (dashed curve). The oscillations of the two cases remained in phase until about t = 0.03. Thereafter the two solutions departed

from each other, with the constant-heating case finally settling down to a steady state after t = 0.08 and the time-dependent case cooling with a longer time scale.

The efficiency of convective readjustment in the presence of heating in the interior and coming from below can be mea- sured by means of the flux ratio F(t) [Daly, 1980b] defined as

Q•(t) F(t) - (23)

F(t) + H(t)

where F(t) is the heat flux from below and H(t) is the rate of internal heating. A value of unity for F indicates that there is a perfect balance between the heat flowing out of the convecting region and the heat produced within. An important quantity which parametrized convection predicts is the ratio of radio- genic heat production over the heat loss. This quantity is called the Urey ratio [Christensen, 1985b] Ur. It is related to F by Ur--1/F. From the results reported in Figure 12 we may construct the corresponding curves for F. The compari- son of the temporal variation in F is shown in Figure 14. We observe that there are large oscillations in the beginning, which are related to the boundary layer instabilities discussed above. The dotted line represents the case where an aspect ratio of one has been employed in the mean field calculations after t = 0.1. There is one marked difference in F. For t less

than 0.025, about 250 m.y., F predicted by the mean field drops at times below unity during periods where there is rapid surface cooling due to the detachment of a cold blob. One can observe this in the oscillations of Qsur in Figure 12 which represent the alternating episodes of hot mass impinging the top boundary layer or of cold mass breaking off from the top. This tendency reflects a definite drawback of the mean field theory in monitoring the initial stages of thermal evolution in that large thermal disequilibrium is predicted. From t = 0.04 on, the agreement in F becomes closer to better than 20%. For larger times, past t - 0.15, the discrepancy becomes larger and widens asymptotically.

It is to be emphasized that the time-dependent case exam- ined above is not exactly easy, since there have been boundary layer instabilities prevailing during the initial stages, which would correspond to the first several hundred million years of the thermal history. However, the consensus from this limited investigation is that for geophysical purposes the mean field

2.0

--- H(t):H o exp(-IOt)

• I •, -- H(t)=H o 11

n I I

] I I o o.oz 0.04 o. o6 o.os

t

Fig. 13. Comparison of mean field surface heat fluxes for constant and time-dependent heatings. The theological and physical parame- ters are taken from Figure 12. The inita] profiles for T and • are identical for the two heating configurations. Solid and dashed curves denote time-dependent and constant heatings, respectively. The e- folding time for time-dependent heating is indicated.

QUARENI ET AL.: HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY 12,643

0 0.1 0.2 0.3 t

Fig. 14. F (cf. equation (23)), the inverse Urey number, versus time from the solution shown in Figure 12. Solid curve represents the mean field solutions; dashed curve denotes the two-dimensional ones. The dotted curve refers to the mean field solution obtained by halving the aspect ratio at t = 0.1 (see Figure 12).

formulation is able to pick up some of the physics present in two-dimensional convection, as it has been shown able to re- produce in a correct way the general evolutionary behavior of the interior temperature, surface heat flux, and the Urey ratio.

6. CONCLUSIONS

Our understanding of convective circulation in variable vis- cosity fluids is currently in a state of rapid evolution. There can be no doubt at this point that understanding the physics of mantle convection is not complete without incorporating the effects of variable viscosity. For instance, there still remain open questions, such as the deviation from an adiabatic profile due to viscous dissipation and internal heating, which are per- tinent to convection in anelastic fluids [Jarvis and McKenzie, 1980] with variable viscosity. At the same time, it is also extremely important to assess the limitations of this method for simpler systems, such as variable viscosity convection for a Bousinnesq fluid in order that one may have better guidelines for using the mean field equations in the future.

Our results yield the following conclusions: 1. For small Rayleigh numbers, mean field seems able to

approximate two-dimensional solutions satisfactorily. This be- havior is to be expected, as the mean field formulation be- comes exact for Rayleigh numbers close to the critical.

2. For higher Rayleigh numbers the differences become bigger due to the interior thermal heterogeneities induced by the descending cold boundary layer. However, the trends for the Nusselt number as a function of the wave number and of

the interior Rayleigh number are still predictable. 3. Mean field interior temperatures are greater than the

two-dimensional predictions. The steady state Nusselt num- bers from mean field are always greater than those from the full set of equations; the power law index fl is also greater. The decrease of fl with Ra T can also be found in the mean field formulation. There is little difference in the efficacy of mean field theory as applied to r/(T) and r/(T, z), with temperature- dependent viscosity perhaps holding a slight edge.

4. Results for time-dependent calculations show that the evolution of the averaged interior temperature and surface heat flow show good agreement between mean field and two- dimensional formulation. However, we have found that there is a tendency for very large thermal instabilities to appear in the mean field equations during the first billion years. This

phenomenon is caused by the geometrical constraints imposed by the very nature of the mean field approximation. However, it is still possible to correlate some of the mean field unstable events with boundary layer instabilities found in the two- dimensional calculations. The Urey number exhibits large dif- ferences during the spin-up stages. But the same long-term secular trend after 0(1 billion years) can be found by both method. The overall time scale, F/(dF/dt), can be found by both methods past the initial transient period.

In sum, we have compared here the results for the "molle" rheology, which is useful for studying the thermal evolution of planets in which the surface is mobile. We note, on the other hand, that the "duro" rheology is relevant for studying con- vection beneath immobile plates, such as old continental cra- tons, or in other planetary bodies in which plate tectonics are not visibly operative. Because of its many potential appli- cations, we believe that a similar type of comparison for the "duro" rheology is also in order.

Acknowledgments. We are very much indebted to Andrew Fowler and Peter M. Olson for stimulating discussions and to Manfred Koch for checking some of the two-dimensional calculations with the finite element code NACHOS. We thank Derick Balsiger for generously contributing his time to this project. Francesca Quareni wishes to thank Enzo V. Boschi of the Dipartimento di Fisica, Settore Ge- ofisica, Universitfi of Bologna, for continuous encouragment during her research. This research has been supported by National Science Foundation grants EAR-8214094 and EAR-8306738 and Petroleum Research Foundation grant 13550-G2.

REFERENCES

Boss, A. P., and I. S. Sacks, Time-dependent models of single- and double-layer convection, Nature, 308, 533-535, 1984.

Christensen, U. R., Convection with pressure- and temperature- dependent non-Newtonian rheology, Geophys. d. R. Astron. Soc., 77, 343-384, 1984a.

Christensen, U. R., Heat transfer by variable viscosity convection and implications for the earth's thermal evolution, Phys. Earth Planet. Inter., 35, 264-282, 1984b.

Christensen, U. R., Heat transfer by variable viscosity convection, II, Pressure influence, non-Newtonian rheology and decaying heat sources, Phys. Earth Planet. Inter., 37, 183-205, 1985a.

Christensen, U. R., Thermal evolution models for the earth, J. Geo- phys. Res., 90, 2995-3007, 1985b.

Daly, S. F., The vagaries of variable viscosity convection, Geophys. Res. Lett., 7, 841-844, 1980a.

Daly, S. F., Convection with decaying heat sources: Constant vis- cosity, Geophys. J. R. Astron. Soc., 61,519-547, 1980b.

Fleitout, L. M., and D. A. Yuen, Steady state secondary convection beneath lithospheric plates with temperature- and pressure- dependent viscosity, d. Geophys. Res., 89, 7653-7670, 1984a.

Fleitout, L. M., and D. A. Yuen, Secondary convection and the growth of the oceanic lithosphere, Phys. Earth Planet. Inter., 36, 181-212, 1984b.

Fowler, A. C., Mantle and magma dynamics, in Woods Hole Summer School Notes in Geophysical Fluid Dynamics, pp. 84-86, Woods Hole Oceanographic Institution, Woods Hole, Mass., 1984.

Fowler, A. C., Fast thermoviscous convection, Stud. Appl. Math., 72, 189-219, 1985.

Gartling, D. K., NACHOS: A finite-element computer program for incompressible flow problems, parts I and II, Theory and User's Manual, Sandia Lab. Rep. 77-1333, 77-1334, Sandia Natl. Lab., Albuquerque, N.M., 1978.

Gear, C. W., Numerical Initial Value Problems in Ordinary Differential Equations, Prentice-Hall, Englewood Cliffs, N.J., 1971.

Gough, D. O., E. A. Spiegel, and J. Toomre, Modal equations for cellular convection, J. Fluid Mech., 68, 695-719, 1975.

Hansen, U., and A. Ebel, Experiments with a numerical model related to mantle convection: Boundary-layer behavior of small and large scale flows, Phys. Earth Planet. Inter., 36, 374-390, 1984.

Herrring, J. R., Investigation of problems in thermal convection, d. Atmos. Sci., 20, 325-338, 1963.

Hindmarsh, A. C., GEAR: Ordinary differential equations syste,q

12,644 QUARENI ET AL.: HIGH RAYLEIGH NUMBER CONVECTION WITH VARIABLE VISCOSITY

solver, Rep. UCID-30001, Lawrence Livermore Lab., Livermore, Califi, 1974.

Jarvis, G. T., and D. P. McKenzie, Convection in a compressible fluid with infinite Prandtl number, J. Fluid Mech., 96, 515-583, 1980.

Latour, J., E. A. Spiegel, and J. Toomre, Stellar convection theory, I, The anelastic modal equations, Astrophys. J., 207, 233-243, 1976.

McKenzie, D. P., Surface deformation, gravity anomalies and convec- tion, Geophys. J. R. Astron. $oc., 48, 211-238, 1977.

McKenzie, D. P., and N. O. Weiss, Speculations on the thermal and tectonic history of the earth, Geophys. J. R. Astron. Soc., 42, 131- 174, 1975.

Morris, S., and D. Canright, A boundary-layer analysis of Benard convection in a fluid of strongly temperature-dependent viscosity, Phys. Earth Planet. Inter., 36, 355-373, 1984.

Natal, H. C., and F. M. Richter, Convection experiments in fluids with highly temperature-dependent viscosity and the thermal evolu- tion of the planets, Phys. Earth Planet. Inter., 29, 320-329, 1982.

Olson, P. L., Mantle convection with spherical effects, J. Geophys. Res., 86, 4881-4890, 1981.

Olson, P. L., and G. M. Corcos, A boundary-layer model for mantle convection with surface plates, Geophys. J. R. Astron. Soc., 89, 425- 436, 1980.

Pereyra, V., PASVA3: An adaptive finite difference FORTRAN pro- gram for first order nonlinear ordinary boundary value problems, Lect. Notes Cornput. $ci., 76, 67-88, 1978.

Petzold, L. R., Differential/algebraic equations are not ODE's, SIAM J. $ci. Stat. Cornput., 3, 367-384, 1982.

Quareni, F., and D. A. Yuen, Time-dependent solutions of mean-field equations with applications for mantle convection. Phys. Earth Planet. Inter., 36, 337-353, 1984.

Richter, F. M., H. C. Natal, and S. F. Daly, Heat transfer and hori- zontally averaged temperature of convection with large viscosity variations, J. Fluid Mech., 129, 173-192, 1983.

Roberts, P. H., On nonlinear B6nard convection, in Non-Equilibrium Thermodynamics, Variational Techniques and Stability, edited by R. Donnelly, R. Hermann, and I. Prigogine, pp. 125-162, University of Chicago Press, Chicago, Ill., 1965.

Schubert, G., and C. A. Anderson, Finite-element calculations of very high Rayleigh number thermal convection, Geophys. J. R. Astron. $oc., 80, 575-601, 1985.

Schubert, G., P. Cassen, and R. E. Young, Subsolidus convective cooling histories of terrestrial planets, Icarus, 38, 192-211, 1979.

Sewell, G., TWODEPEP, a small general purpose finite-element pro- gram, IMSL Tech. Rep. 8102, IMSL Inc., Houston, Tex., 1981.

Sewell, G., IMSL software for differential equations in one space variable, IMSL Tech. Rep. 8202, IMSL Inc., Houston, Tex., 1982.

Sharpe, H. N., and W. R. Peltier, Parametrized mantle convection and the earth thermal history, Geophys. Res. Lett., 5, 737-740, 1978.

Spohn, T., and G. Schubert, Modes of mantle convection and the removal of heat from the earth's interior, J. Geophys. Res., 87, 4682-4696, 1982.

Toomre, J., D. O. Gough, and E. A. Spiegel, Numerical solutions of single-mode convection equations, J. Fluid Mech., 79, 1-31, 1977.

Toomre, J., D. O. Gough, and E. A. Spiegel, Time-dependent solu- tions of multimode convection equations, J. Fluid Mech., 125, 99- 122, 1982.

Tozer, D.C., Towards a theory of thermal convection in the mantle, in The Earth's Mantle, edited by T. F. Gaskell, pp. 325-347, Aca- demic, Orlando, Fla., 1967.

Turcotte, D. L., and E. R. Oxburgh, Finite amplitude convection cells and continental drift, J. Fluid Mech., 28, 29-42, 1967.

Yuen, D. A., and L. Fleitout, Lithospheric thinning by small-scale convective destabilization, Nature, 313, 125-128, 1985.

U. R. Christensen, Max-Plank-Institut ffir Chemie, Abteilung Ge- ochemie, 23 Saarstrasse, D-6500 Mainz, Federal Republic of Ger- many.

F. Quareni, Dipartimento di Fisica, Settore Geofisica, Universit• di Bologna, 8, viale Berti Pichat, 40127 Bologna, Italy.

G. Sewell, Department of Mathematical Sciences, University of Texas at El Paso, E1 Paso, TX 79968.

D. A. Yuen, Department of Geology and Geophysics, 310 Pillsbury Dr., S.E., University of Minnesota, Minneapolis, MN 55455.

(Received January 4, 1985; revised June 28, 1985;

accepted July 11, 1985.)