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A simple model of convection in the terrestrial planetsA. C. Fowler ab

a Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, U.S.A. b MathematicalInstitute, Oxford, England

Online Publication Date: 01 February 1985

To cite this Article Fowler, A. C.(1985)'A simple model of convection in the terrestrial planets',Geophysical & Astrophysical FluidDynamics,31:3,283 — 309

To link to this Article: DOI: 10.1080/03091928508219272

URL: http://dx.doi.org/10.1080/03091928508219272

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Geophjr As1roph.y.s. Fluid Dynamics, 1985, Vol. 3 I, pp. 283-309 0309- 1929 w 3 104-OZ83 $1 X.50,O !i: 1985 Gordon and Breach, Science Publishers, Inc. aiid OPA Lid. Printed in Greal Bnlain

A Simple Model of Convection in the Terrestrial Planets

A. C. FOWLERt Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, U.S.A.

(Rrceiz:ed March 8, 1984; in,f ina/,fbri i i Airgirsi 30. I Y K 4 )

In this paper we study analytically the simplest fluid mechanical model which can mimic the convective behavior which is thought to occur in the solid mantles of the terrestrial planets. The convecting materials are polycrystalline rocks, whose creep behavior depends very strongly on temperature and probably also on pressure. As a simple model of this situation, we consider the flow of a Newtonian viscous fluid, whose viscosity depends strongly on temperature (only), and in fact has an infinite viscosity below a certain temperature, and a constant viscosity above this temperature. This model would also be directly relevant to the convection of a melt beneath its own solid phase (e.g. water below ice, though in that case there are other physical complications).

As a consequence of this assumption, there is a vigorous convection zone overlain by a stagnant lid, as also observed in analogous laboratory experiments (Nataf and Richter, 1982). The analysis is then very similar to that of Roberts (1979), but the extension to variable viscosity introduces important differences, most notably that the boundary between the lid and the convecting zone is unknown, and not horizontal. The resulting buoyancy induced stresses near this boundary are much larger than the stresses produced by buoyancy in the side-wall plumes, and mean that the dynamics of this region, and hence also the heat flux, are independent of the rest of the cell. We give a first order approximation for the Nusselt number-Rayleigh number relationship.

1. INTRODUCTION

The study of solid-state convection in the mantle of the Earth (and by extension, in othcr terrestrial planets) over the last fifteen years

?Present address: Mathematical Institute, 2 4 2 9 St. Giles’, Oxford OX1 3LB,

283 England.

Downloaded By: [University of Oxford] At: 11:58 24 June 2009

284 A. C. FOWLER

has raised a number of issues of active concern. Among these, one may mention the controversy over the depth extent of mantle convection, where mineralogical (Jeanloz and Thompson, 1983) and geochemical studies (Allegre et al., 1979; O’Nions et al., 1979; DePaolo, 1981) tend to suggest a layered convecting system, whereas post-glacial rebound studies (e.g. Peltier, 1980) show a relatively constant sub-lithospheric viscosity.

The point of relevance for the present paper (which is not concerned with layering) is that a number of authors have argued dynamically in favor of whole mantle convection, on the basis of the nature of constant viscosity convection (O’Connell, 1977; Elsasser et al., 1979; Peltier and Jarvis, 1982). Indeed, it is a commonly held opinion that constant viscosity convection is a useful, perhaps even accurate, analogue for mantle convection. This idea stems from the original studies of Turcotte and Oxburgh (1967) and McKenzie et al. (1974), and continues to play an important role (e.g. Hewitt et al., 1981; Jarvis and Peltier, 1982).

Another instance of this reliance on constant viscosity results is in the popular study of Earth’s thermal history. This has been inves- tigated by numerous authors (Sharpe and Peltier, 1978; Davies, 1980; Turcotte, 1980; McKenzie and Weiss, 1975; Schubert et al., 1979; Schubert et al., 1980), generally on the basis that mantle convection can be parameterized in the same way as laboratory convection of constant viscosity fluids.

There is a serious flaw in treating constant viscosity convection as an analogue for convection in terrestrial mantles, because experimental and theoretical studies show that the viscosity of mantle-type rocks is extremely sensitive to temperature (and probably also pressure): in particular, the Earth’s lithosphere behaves (on a large scale) essentially rigidly because of this. Furthermore, the strength of the analogy between Benioff zones and downwelling cold plumes in constant viscosity convection is confounded by the observed fact that in the convection of a material with strongly temperature- dependent viscosity, the top, cold layer is so rigid that it does not subduct at all, even in vigorous convection. This essentially straight- forward fact is ‘clearly demonstrated in experiments by Nataf and Richter (1982), and can be seen in various numerical results, where, for example, the inclusion of weak zones is necessary to encourage subduction (Kopitzke, 1979; Schmeling and Jacoby, 1981).


CONVECTION IN TERRESTRIAL PLANETS 285

The purpose of this paper is therefore to initiate the analytic study of convection in a variable viscosity material, the potential applic- ation being to the mantles of terrestrial planets (e.g. Mars, the Moon, perhaps Venus), but not the Earth, insofar as we will analyze a situation in which subduction does not occur.

The (numerical) study of variable viscosity convection was init- iated quite early (Foster, 1969; Torrance and Turcotte, 1971), and experimental results were obtained by Booker (1976) and Booker and Stengel (1978), but there seems to have been little emphasis on this aspect of the problem, perhaps because of the stagnant lid effect, which suggested that constant viscosity convection was a more appropriate analogue. A notable exception is the work by Tozer (1972). Recently, however, a number of authors have considered convection in fluids with temperature-dependent viscosity. These include experiments (Nataf and Richter, 1982; Richter, 1978; Stengel et al., 1982), theoretical analysis (Fowler, 1982, 1983; Kenyon and Turcotte, 1983; Morris, 1982), and a variety of numerical comput- ations (DeBremaecker, 1977; Daly, 1980; Hsui, 1978; Jacoby and Schmeling, 1982; Parmentier and Turcotte, 1978; Cserepes, 1982; Christensen, 1981, 1982, 1983a, b).

The purpose of the present paper is to provide an analytic solution for the state of convection in a fluid with non-constant viscosity. As such, the analysis may be thought of as a generalization of Roberts’ (1979) analysis. Ideally, the appropriate rheological flow law that should be used is one in which the viscosity is a function of temperature 7: pressure p and stress z:

It is probably the dependence on temperature which is most significant, and a simple tractable resemblance to (1.1) which reflects the temperature dependence is a purely exponential law:

The analysis of this case has in fact been recently attempted (Morris and Canright, 1984; Morris, 1984; Fowler, 1984a), in the case where the exponent b is “large”; that is,


286 A. C. FOWLER

Here, & and To are prescribed basal and surface temperatures respectively. Numerical studies of both (1.1) and (1.2) when, respectively, E*/RT, and b(T,-To) are large, show that the bulk of the temperature jump across the convecting layer occurs across the stagnant lid, below which a roughly isoviscous convecting zone occurs. If the basal temperature of the lid is (approximately) given by 7;, then one can estimate the ratio

[this for the viscosity given by (1.2)]. In this paper, we will adopt the simplest possible flow law which

mimics this behavior; that is, that q is piecewise constant above and below some temperature T,:

~ = c Q , T<7;,

(1.5) y=constant, T > 7;.

This law mimics the physics exhibited by (1.2) [and thus to an extent, that of (1.1)], provided we choose 7; in keeping with (1.4), that is

This apparently simple law actually causes a good deal of complication to occur in the analysis.

It should be pointed out that when the title claims that we are studying a simple model of terrestrial mantle convection, the emphasis is on the simplest law which attempts to mimic the above physics, but will not (obviously) simulate the actual behavior. For example, one might point out that in the Earth, one has To-300, 7; - 1500, T, - 3000 (core-mantle boundary), so that (T, - To)/(& - 7;) - O( l), as opposed to (1.6). But of course, much (if not all) of the increase in temperature through the convecting zone of the Earth is due to adiabatic heating, which is not considered here. The point is to understand the flow structure and, in particular, how it differs from a constant viscosity convection.



2. MATHEMATICAL MODEL: SCALING

We consider two-dimensional convection of a fluid whose rheology is given by (1.2) in a rectangular enclosure of height d, length ad, with stress-free walls. The dimensionless equations describing the flow (for T > ?;) are

o-'du/dt= -Vp+V2u+R0j,

dO/dt=V20, divu=Q, (2.1)

where

are Prandtl and Rayleigh numbers, respectively. Here u, p , 0 are velocity, pressure, temperature scaled respectively with u/d, r p / d 2 , and AT: specifically the temperature has been written as

T=T*+(AT)O, (2.3)

T* and AT being defined below. Introducing the streamfunction I),

where x, y are dimensionless horizontal and vertical coordinates scaled with d, and the vorticity

0 = u, - v,, (2.5)

(2.1) can be reduced to

where we have let g+co. These equations are valid in T > T , and if we prescribe a basal temperature > 7;, and a surface temperature


288 A. C. FOWLER

To<7;, then we expect the solution to have a stagnant lid T<7; adjoining the top surface y = 1, beneath which convection occurs for T < 7;. If R>> 1, we expect the convecting zone to have the typical nature of such flows (Roberts, 1979): an isothermal interior, sur- rounded by thermal boundary layers. We now choose T" and AT in order that 0 = - 1 corresponds to T = 7;) and so that 8=0 in the isothermal interior. Of course, at this stage, the isothermal temperature is unknown, but we can in principle choose j3 so that 0 = j3 > 0 corresponds to T=T,. Then fl is to be determined as part of the solution. With this choice, we define

Let us define the isotherm T = 7; (d= - l), i.e. the lid boundary, to be y = 1 - ys(x), y < 1. Then boundary conditions for the flow are the following:

on x = 0, a: 0, = * = l),, =o, (2.8)

ony=O: d=P, *=*yy=o, (2.9)

o n y = l - y s : d=-1 , $=+,=O. (2.10)

Additionally,

Vz8=0 in l-ys<y<l,O<x<u,

d = - 1 - A o n y = l , (2.11)

where A is determined from (2.7):

We choose y such that S- 1. In addition, suppose

ys' -A; (2.13)

one can have A - y , or i < < y : either may be important. If A - y , then if


CONVECTION IN TERRESTRIAL PLANETS

we assume y<<1, the solution of (2.11) is

8- - 1 - A + A( 1 -y)/y~+ O(yzA);

alternatively, if i.<<y, y < 1, then the lid base is almost flat, and

8- - 1 - A + A( 1 - y)/yB + 0(1A),

where (if 1 ~ y ) we write

Y S = yB + AS,

289

(2.14)

(2.15)

(2.16)

Thus, we have the extra thermal boundary condition on 1-7s:

1-y<<l: B,=-A/ys ony=l-ys;

I < < y < l : 8,=-A/yB ony=l-yB-LS. (2.17)

This extra boundary condition determines the location of the lid base.

We must now rescale the equations to determine the parameters 1, y and the boundary layer thicknesses. The physics of the flow is akin to that given by Roberts (1979), i.e. plumes at the side generate vorticity which drives the interior (biharmonic) flow. At the roof (base of the lid), however, there may or may not be a shear layer (Fowler, 1984b), depending on the slope-which is at present unknown. Let us define the top thermal boundary layer (at y = 1 - y s ) thickness to be 6. Then (2.17) implies

6 = y/A. (2.18)

Balance of the temperature equation implies t,b - 1/6 in this layer, and thus o - l/d3 there.

Now let us suppose that the top layer is a simple shear layer. Then matching to the core (the isothermal interior) would imply t,b - 1/d2 there. If the plumes are of thickness [XI, t,b - [x]/S2 in the plumes. If 0 is convected round the corners (Roberts, 1979), then 8$ has the same magnitude in thermal boundary layer and plume. If 8-[0] in the plume, then [0][~]/6~-1/6, i.e. [8]-6/[x] ( 5 1 ,


290 A. C. FOWLER

by assumption). Therefore (vorticity-buoyancy balance) a-NR[Q][xl-R6 in the core. However, a shear layer must also balance vorticity and buoyancy, whence w-R61 [since if y = 1 -ys - hX then 3, = 8, - ( y s ' / 6 )d , - O(/z /h)] ; but (1) in the shear layer must be >> the core value (otherwise the velocity cannot match), which requires R6A >>R6, which contradicts the assumption 25 1.

Alternatively, with no shear layer at the top, $-1/6 in the boundary layer would match to + - l / d 3 in the core, hence $ - [x]/S3 in the cold plume, hence [O] - S2/[x] there, hence o - R [ Q ] [ x ] - Rh2 gives the core vorticity. On the other hand, the boundary layer vorticity is w-R61. We can assume A 2 6 (if 1<<6, there is no solution, since the convective flow sees a flat roof, and the heat flux out cannot match the heat flux through the lid): then if ;1>>6, the vorticity in the shear layer >> that in the core; again this is inconsistent, and the resolution is that the top thermal layer i s a shear layer, but the velocity jump across it is much smaller than the ambient core value (cf. Fowler, 1984b). If 2-6, then this scaling is appropriate, and the lid is analogous to Roberts' (1979) case of a no- slip boundary.

In any event, we choose scales so that vorticity balances buoyancy in all thermal layers (and plumes). It is useful to define the (cold) plume thickness [x] = 6,. Then $ - 116: in the core, so co - R6:/6 at the edge of the plume: thus

6: = 6/R. (2.19)

We define

(2.20)

dropping overbars, (2.6) can thus be written

In the two plumes and lower boundary layer, 8 advects round as it diffuses. The layers are of the same thickness ( 6 J , and so %-6,/6 in



all of them. Therefore we define the basal temperature by

B= 6- lhcf l , PWO(1). (2.22)

(2.18) and (2.19) are two relations for the three unknowns d,, 6 and y. A third comes from balancing vorticity and buoyancy in the thermal layer at the top. From (2.21), $ N 6:/6 ~ o d ' , o N 62(6/62)(i/d) nd2/6:, thus o N 6:/h3 - i62/62, that is

6," = /Id5, (2.23)

or, using (2.19)

iRd4 = 1. (2.24)

If 1 is known, this gives the necessary third relation. As discussed, we can assume 65lzy. The two end-members of this set are

6 = R - 1 / 5 , y=AR- ' l5 , g c = R - 3 / 1 0 ; (2.25)

and

Notice that, from (2.23), p - 6,/6 << 1. There is apparently nothing to rule out either parameterization above, nor indeed any intermediate one: 6<<A<<y; a further argument is necessary to eliminate these, but both (2.25) and (2.26) give recipes for solving the resulting equations. This is done in the following two sections.

3. ANALYSIS (i): A = y

We choose

in accordance with (2.26). Away from the boundaries, ddldt-0, and


292 A. C. FOWLER

(in a steady state) I9 is constant. By choice of B, then

(despite the fact that S/S; >> 1). The boundary conditions on the core flow will be

* = * y = 0, y = 1 - y s ;

* = * y y = 0, y=o;

*=o,$,li/xx= -c,, x=o (3.3)

where CI and C, will be the integrated buoyancy in the plumes. Note $ y = O (to leading order) on y=l-ys, despite the fact that there is a shear layer there (cf. Fowler, 1984b).

In the plumes, we rescale thus:

Denoting boundary layer variables by capitals, the leading order equations are then (for example near x = 0)

from the energy equation we find

m

0 0 7 OdY = C, = J Ov&X = wpup, (3.6)

where wp,vp are the vorticity and velocity experienced by the core flow at the plume. Hence one obtains (3.3), [and similarly (3.3),] directly.


CONVECTION IN TERRESTRIAL PLANETS

In the basal thermal boundary layer

y - 6 0 *-do e-6,/6,

and so

293

(3.7)

We choose B=@(x,O) so that the heat flux in at the base equals that out the top.

The upper thermal boundary layer is also a shear layer: we put

the leading order equations for the rescaled variables are then

(3.10)

with boundary conditions

Y = O e= -i,ey=i/s,yr=yly=o;

Y -+a: e+o,R+o. (3.1 1)

The only comment is on the condition on R at 00: we have o-6 f /d3 -All2>> 1 , from (2.26). Therefore the core vorticity is O ( K 1 I 2 ) relative to R, and (to leading order) R-0. Similarly $-6f/6<<1, thus the core flow sees a shear velocity -6f/62-[A312R-1]1/5; this is << 1 provided R>>A3I2: in fact, we need R Z A 4 for 75 1. We actually require y<< 1 (so O Y = l/s on Y =O- otherwise one has to solve the full lid conduction problem), and also y >> 6, so there is no 8, term in (3.10). Thus, strictly, R >> A4 >> 1.

Initial conditions for (3.10) are of similarity type, arising from matching to the thermal structure in the corner. It is because of the fact that this layer is a shear layer, that there is a similarity solution


294 A. C . FOWLER

to (3.10): the boundary layer chooses its own scales, and is (to leading order) unconcerned with the rest of the flow.

We define the similarity variable

4 = Y / S ( X ) ,

and put

H=g(t), Q=ss’lz((), ’Y=s3s‘f(t).

Then [after integrating (3.10), once]

h’= -g, f ” = - h, g” + S [ S % ‘ ] ’ f s ’ = 0,


h( a) = 0, f ( 0 ) = f ’ ( 0 ) = 0,

g(0) = - 1, g‘(0) = 1, g(G3) =o.

(3.14) collapses to an equation for f(0,

f ” + / I f f ’“ = 0,

where

A = s[s3s’]’ =constant.

We require

f ” ( a) [ =,f”’( m)] = 0, f ( 0 ) = f ’ (0) = 0,

f”‘(0) = - 1, . f ‘ ” (O) = 1.

To solve this, we first solve the problem

f” + 2771’ = 0,

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)



for a given A", by shooting with values y"(0) and 7"(0). Then a solution of (3.16) is given by

f = b.7(~{) , A = c4Ab = 1/C3, (3.20)

where

c = 1/7'"(0). (3.21)

The computed value of A is 11.74. We now solve (3.17) for s. Two integrations yield

0 (3.22)

where k is constant (we have assumed s = 0 at x=0, consistent with the similarity initial conditions). To compute k, we use the integral form of the temperature Eq. (3.10). This yields

thus,

where N is the Nusselt number. We have to be careful in interpret- ing (3.23), because s; f3dY includes both the contribution from 8- 1 computed above, and also the "ghost" of the plumes. In the upwelling plume Q-6,/4, $-dC; at the top left corner, this small temperature excess is advected round into a region Y - Y-6/6, (since it lies on the fringe of the shear layer); thus, with $ - 6,, 1 - y - ys- 6/6,, the temperature equation for @-6,/6 is (with appropriate coordinates)

YYex -Y',Oy - O(Sf/S') << 1. (3.24)

Thus to leading order, 0 is advected across the thermal layer, and


296

thus

A. C . FOWLER

[ 1 @ d Y ] . 0 -0; (3.25)

from this it follows that [SgBdY] ; in (3.23) is approximately given by 1; OdY l a , where 8 is given by (3.13). Hence, we require

This gives, using (3.14), (3.15) and (3.17),

= A ' { [ k ++As( a) 3 ] - k }

= A ' s3( a) s'( a) = A ~ [ k + ? jAs (~ )~] ' / ' . (3.27)

Thus, we require

k=O.

(3.22) then simplifies to

s = EX215 , (3.29)

where

c" = (25A/6)'I5 M 2.177. (3.30)

The heat flux decreases as x - ' . ~ , and the Nusselt number is

a N-(Ad)-' S s- ldx = 0 . 7 6 6 ~ ~ / ~ ( R A ) ' / ~ , (3.31)

0



where the coefficient is given by 5 / 3 L This definition of the Nusselt number is u times that defined by Roberts (1979).

So far as it goes, this solution is satisfactory. In practice, one has to match the core flow to regions in the corners where the stress is large. The fact that s’-+oo at the top left corner is a complication of this problem. One can expect that the relaxation of the corner flow singularity might additionally relieve the singularity in s: for otherwise, there would be a singularity associated with the infinite slope of s at x= O as well as that due to the plume vorticity. Thus in reality, one would hope that the slope in s would become “large”, but not infinite. An analysis to pursue this point will not be undertaken here.

4. ANALYSIS (ii): A=d

In this case, we have

and require R 2 A5 for y = 6A 5 1. The analysis is now largely similar to Roberts’ (1979) fixed-surface case. In the core, 8=0 and t,b satisfies

v4* = 0, (4.2)

with (at leading order)

*=o, *,=o ony=O and 1-yB,

* = 0, *&xx = c, on x=u,

where C, and C, are to be determined (from the plume solutions). In the plumes (and the basal thermal boundary layer),

and thus (at leading order) the plume equations are (as before)

Y = v,(y)X, 7 @dY = C, = v p o p , (4.5) 0

G.A.F.D.- E


298 A. C . FOWLER

for the left-hand plume, and similarly for the right-hand plume. The temperature equation in the basal layer is similar. If we define (cf. Roberts, 1979) s as arc length along the boundary in the direction of flow (downwards at x=u), and U as the core velocity at the boundary in the direction of s, then the temperature equation in plumes and basal layer can be written as a diffusion equation, using Von Mises' transformation:

(except in the top thermal layer). We define

T = 0 at x = a, y = 1 - yB (top right);

T = TI

T = T2 at x=O,y=O(bottorn left);

at x = a, y = 0 (bottom right); (4.7)

T = T3 at x = 0, y = 1 - yB (top left).

The boundary conditions for 0 are

on Y=O: Oy=O, O<T<T,,T,<T<T3,

This temperature distribution is convected into the fringe of the thermal boundary layer, where it is advected across without change. The accumulated coolth of the main thermal layer is released at x = a , essentially as a delta function to the 0 distribution. Hence, for T > T3, the initial condition at T3 + is

where OdY l a is computed from the top thermal boundary layer.



One then continues to solve (4.7) and (4.8) with T-T-T,, etc. From this solution one obtains C, and Cr (hence the interior flow, hence TI, T,, T3).

In the thermal boundary layer [near the lid base at y = 1 -yB- 6S(x)], we put

y=l-yB-GI:

the leading order equations are

Y Y O X - Y J , = @,Y,


on Y=S(x): @= -1,

lj-62, 0-1; (4.10)

Y y y = - O, wYy = Ox, (4.11)

e,=i/B, Y=Y,=o,

as Y-co: @+O, o+o0(x), (4.12)

where o0 is -ljyyJy=l-yB, i.e. the vorticity on 1 -yB as determined by the core solution. These are slightly different to Roberts' (1979) equations, which lack the term 0, in (4.11)3, apparently due to an oversight. The appropriate initial conditions for (4.12) are similarity ones, but unfortunately no similarity solution to (4.11) and (4.12) is possible, for a general o0(x) . Worse, the determination of o0 involves a complicated coupled problem: that is, o0 is determined in terms of Cl and C,; these are given by

m 0 0 1

C,=J@(y,O+)d'J', C,=C-C,, C = j O d Y l , (4.13) 0 0 a

where 0 is determined from (4.6) and (4.7), C= C, + C, because of (3.23), and C = st OdY l a is the top thermal boundary layer negative buoyancy. Thus, plume, core and upper layer are all coupled together, and a sequence of numerical solutions is necessary to find C (which then determines the Nusselt number as N = C/6A). Thus, we should compute the core solutions as functions of C, and C,, to find

wO(x) =oO(X; C,, Cr), = Tl(Cl, Cr), (4.14)

T2 = T,(Cl, Cr), T3 = 7"(c1, cr);


300 A. C. FOWLER

solution of (4.6) and (4.7) together with periodicity (period T3) yields a('€', T; T,, T,, T3, p), where also p can be determined as p(C, TI, T,, T3) (so that the heat flux in at the base equals that out at the top). Hence we use (4.13) to compute

and also (4.11) and (4.12) to find S(x) and

c = C[OO]. (4.16)

Then (4.14), (4.15) and (4.16) give seven independent relations to determine the seven unknowns T,, T,, T3, C,, C,, C and wo(x).

However, the crux of the problem is the solution of the boundary layer Eqs. (4.11) and (4.12). One can plausibly doubt that these equations have a solution at all, for various reasons. One suggestion of this is the fact (easily shown) that if 6, in (4.11)3 is replaced by zero, there is no solution, since in that case S(x) is uncoupled from the problem. However, the most cogent reason arises from examining the initial conditions for (4.11). These are determined from matching the solutions (if they exist) to (4.11) to a further rtgime near the corner, in which x-6 , l - ~ B - y - 6 ~ ' ~ , $-d5I2, 0 - 6 ~ ~ ~ ~ [the reason for these scalings is considered in further detail by Fowler (1984a)l. By solving for this corner region, one finds that as x+O in the boundary layer, OpNx-1/2 on Y = S , which is evidently inconsistent with (4.12). The inference is that the initial conditions for (4.11) are inconsistent with the boundary conditions (4.12), and thus that this scaling is implausible. Since we already have a solution of the problem in Section 3, this is not a catastrophe.

Thus, on the basis of the asymptotic matching involved, we suggest that the scaling of this section may be inappropriate for a solution to the complete equations. It is in fact possible that the Eqs. (4.1 1) with boundary conditions (4.12) have no solution, which would certainly rule out their relevance. Until some better found- ation for the existence of solutions to (4.11) and (4.12) is put forward, it is perhaps irrelevant to attempt a numerical solution of the coupled equations described earlier.



5. DISCUSSION

In our analysis, we have found at least two possible flow regimes in (Sections 3 and 4) with different structures, notably a “flat” versus “sloping” lid base. However, we have suggested that the flat base scaling (Section 4) may be inconsistent, so that the solution in Section 3 may be the only viable one. The major point still requiring discussion is the possibility of an intermediate regime 6<<A<<y. In this case, analysis can be performed analogously to Sections 3 and 4, but the prescription of I is unavailable. The parameters A, 6, y and 6, are related by

but no final relation naturally appears. To obtain such a relation, we do the “lid-stripping” discussed by Nataf and Richter (1982). Thus, when 6<<I<<y, the temperature in the lid is given by (2.19, and

Thus, the Nusselt number (as defined here) is

AN=af(R,A,a)=a/6+0(A). (5.3)

Now suppose we change To by dT,, and the depth d by 6 d , in such a way that -6To/6dxANAT/d. Then the solution for d + 6 d is the same as that for d, except that the lid is 6d thicker. Particularly, the “Nusselt” number AN is the same. This gives a differential relation for f ( R , A, a), which is incidentally for 6 $2 < y , thus including the results of Section 4.

Notice A = ( T - To)/AT from (2.7) and (2.12), thus

furthermore,


302 A. C. FOWLER

6(AT)=O, and

(6R)lR = 3 ( 6 d ) / d .

The lid-stripping prescription above states that [using (5.4)]

6 f = O if 6A= f 6 d l d ; (5.7)

since 6,f=fK6R+jAdA+S,6a, use of ( 5 3 , (5.6) and (5.7) leads to the partial differential equation for f :

The general solution of this first order equation is

f = f o [aeAif , Re - 3A2/f 1, (5.9)

for some as yet arbitrary function fo. Notice that, since A N = a f , we require A/ f < 1 in (5.9).

To calibrate fo, we use the results of Section 4 (assuming these are relevant: if, as we have suggested, they are not, then this would immediately suggest that no solution for Accy is possible). The Nusselt-Rayleigh number relation in Section 4 would be of the form

AN = af- C(U) R”’; (5.10)

strictly, this derivation assumes y < < l [since if y - I , then the core flow will depend on y as well, and hence C=C(a,y)]. In the case y c 1 , identification of (5.9) and (5.10) shows that the flux law for 7 5 1 , 6 % A < y , is of the form

f - a ~ 1 ~ ( ~ ~ A 1 . f ) - 8Ais.f (5.11)

which implicitly defines N . It immediately follows from (5.11), since A/ f 5 1, that 6 - 11 f - R1/’, and hence A - 6 automatically. Provided the boundary layer scaling in Section 4 is the only one of its type possible, this shows that the intermediate range 6<<A<<y is untenable.

The relation (5.1 1) then provides the parameterisation for N when



R - A 5 , while (5.10) is valid for R>>A5. One can see what happens for 1 << R << A5 by examining (5.1 1) as A/ f increases. For simplicity, we take the expected estimate of C as a+O (cf. Roberts, 1979):

C(a) = Ca3i5, (5.12)

which should suffice for rough discussion. Then

1.e.

4 e p 4 = kA/RIi5, (5.14)

where

4 = A / f -y, k=a2I5/C. (5.15)

For A<<R'l5, 4 is small [the large solution of (5.14) is inadmissible since then ?>>>I. As A/R'is increases to e- ' /k , C#J increases to U(1). For A/R'I5 above this critical value, there is no solution to (5.14); it seems reasonable that this may correspond to the non-existence of a core flow at too large aspect ratios [a/(l-y)22, Roberts (1979)], and that in this case the single cell would bifurcate to two or more cells at some O(1) value of Cp, thus reducing a, and hence reducing k, and increasing the permissible LI/R"~ ( e P 1 / k ) . It should be emphasized that the above discussion will apply if the equations in Section 4 have a solution, but that our contention is that this is probably unlikely, so that Section 3 describes the appropriate solution.

6. CONCLUSIONS

We have presented an asymptotic analysis for the convective flow of a fluid whose viscosity is infinite for T < IT;, constant for T > T . The flow depends on the parameters


304

where the aspect ratio

and To are prescribed basal and surface temperatures, and

a = l /d (length/depth). (6.3)

For R >> 1, we find two possible boundary layer structures for the flow, differing primarily in the thickness and shape of the cold, rigid lid on the flow. These are described in terms of a parameterization in terms of the Nusselt number,

For case (i), in Section 3, we find

with a relative error of O(A3’20R-11’0 ) ( - B ) . For this case, we require R 2 A 4 (so that yS l), and A>> 1 (so there is a shear layer at the top). For the particular formula (6.5), which uses also the assumption y<<l, we in fact require R>>A4. For this solution, the lid base is given by

although this result breaks down near x=O. Thus the heat flux decreases as x - O - ~ .

For case (ii), in Section 4, we would obtain

where C(a) is a function of the aspect ratio. The parameterization (6.7) would be valid for R>>A5, and could be extended to R 2 A 5 by solving a “lid-stripping” partial differential equation; the result is (5.1 l), which can be written as



which is identical to (6.7) for R >>A5, but alters the proportionality as A/R’15 increases. It should be re-emphasized that this second “solution” probably does not exist, since the initial conditions for the boundary layer Eqs. (4.11) are not consistent with the “flat base” boundary conditions (4.12). This question is worthy of further study. One should also mention the apparent non-uniqueness of the flow found by Nataf and Richter (1982), which, however, they ascribe to a bifurcation in the spatial structure of the convection.

Two points can now be made. The first is that the prediction (6.5) is an asymptotic one in the limit R+cc (and A+m, such that

(actually there will be other relative errors as well). Now, it is obvious that R must be exceedingly large (say, >lolo) before we should realistically expect log-log plots of N versus R to follow a one-fifth slope law. For example, if one plots N = R’I5 (1 + aR ~ ‘/lo)

for values of R up to lo7 (as is commonly done in experiments, both laboratory and numerical), the best fit straight line will have a slope different to 0.2. One can ascribe such marked differences to the non- attainment in experiments of the quantitative asymptotic rtgime. This, of course, does not render the asymptotic result useless, since the asymptotic flow structure (the appearance of boundary layers) can be realized for R 2 1 0 5 . Of course, experimental comparison at such high Rayleigh numbers is precluded, since the flow is then turbulent.

The second comment is that one might expect, intuitively, that the heat flux through the lid would be essentially independent of the surface temperature when A is large, in the sense implied by Nataf and Richter (1982): that is, a solution for one value of To can be converted to one at a different To by simply adding a conductive slab to the roof. This lid-stripping would be valid for the scaling in Section 4 (and was used in Section 5), but it is not appropriate for the solution in Section 3, since the heat flux varies with distance from the upwelling.

It should be pointed out that a more usual definition of the Nusselt number is N , (Roberts, 1979):

R>>A4). For any finite R, the relative error is O(A3110R-1110 1

N = aN,. (6.9)

Further, the usual definition of the Rayleigh number uses T,-To

G,A,F.D.- F


306 A. C. FOWLER

instead of AT. Using (2.7)2 and (2.12), we find

AT= (G - To)/’( 1 + p + A). (6.10)

Since p<<l, this implies that the regular definition of the Rayleigh number

is approximately related to R by

R,=(l+A)R. (6.12)

For large A, we then have (6.5) and (6.7) in the form

These two parameterizations are effectively the same, but they arise through rather different flow structures. The first has a variable heat flux at the surface, whereas the second would have an almost constant heat flux, if it existed. The analysis is closely related to that of Roberts (1979), but with certain (non-trivial) differences. Most obviously, the parameterization (6.13), depends on two essentially different parameters. Further, the free boundary nature of the lower boundary (and its non-horizontality) causes high stresses in a thermal layer below the lid, which causes a weak shear layer to exist there which uncouples from the rest of the flow.

The viscosity law used (a piecewise continuous one) is clearly a vast over-simplification of the more realistic temperature dependent one, and thus, for example, one cannot immediately use the results here in studies of thermal histories. The importance of the results here are that the results of more realistic rheologies can be mimicked by identifying the artificial parameter A with the dimensionless activ- ation energy E*/R7;, as explained in Section 1, rather than with the actual temperature ratio, since adiabatic and other effects will affect the isothermality of the convecting flow. With A held constant, one could then construct a “zero-th order” thermal history calculation.



The present calculation leads the way towards more complicated theoretical analysis of more complicated rheologies. However, one immediate conclusion that one can suggest is that the (dimensionless) lid thickness is O(ARi1I5). Laboratory and numerical calculations (Nataf and Richter, 1982; Christensen, 1984) tend to have thin lids. However (identifying A with the dimensionless temperature dependence) planetary mantles have much higher (equivalent) values of A, so that stagnant lithospheres could be much thicker. For example, (6.6) would suggest that if A-40 (a typical terrestrial estimate) and R, N lo8, most of the the mantle would be stagnant. Calculations of the type presented here may thus be of use in estimating the internal state of convecting terrestrial mantles.

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