Moving lithospheric plates and mantle convection

Geophys. J. R . astr. SOC.(1979) 57, 209-228

Moving lithospheric plates and mantle convection

Richard A. LUX* Department of Mechanical and Aerospace Sciences, University of Rochester, Rochester, New York 14627, USA

Geoffrey F. Davies? Department

of Geological Sciences, University of Rochester, Rochester, New York 14627, USA

John H. Thomas

Department of Mechanical and Aerospace Sciences, University of Rochester, Rochester, New York 14627, USA

Received 1978 November 29;in original form 1978 July 17.

Summary. The coupling between a rigidly moving lithospheric plate and a convecting mantle is investigated using a simple two-dimensional numerical model that incorporates a horizontally moving upper boundary, simulating the effect of a moving plate, over a fluid layer heated from below. The moving boundary strongly controls the horizontal length scale of convection cells when its velocity is greater than the free convective velocity (i.e. the velocity with which the fluid would convect under a stationary boundary). In a box of aspect ratio 4 (width/depth), a transition in flow structure occurs from several equidimensional convection cells under a slowly moving boundary to a single long convection cell under a rapidly moving boundary. The flow structure transition occurs approximately when Pe/RaW3= 0.04, where the Peclet number, Pe, measures the (prescribed) velocity of the upper boundary, and the Rayleigh number, Ra, measures the heating of the fluid layer. Near the transition, the flow tends to be unsteady; this behaviour can be well understood in terms of the instability of the thermal boundary layers, which can be characterized by a local Rayleigh number. Using conventional estimates of mantle parameters, the mantle is either near or above the transition to single-cell convection, whether upper-mantle or whole-mantle convection is assumed. The net tangential force exerted by the fluid on the upper boundary varies approximately linearly with the boundary velocity above the transition, and it is positive (driving) for boundary velocities ranging from the value at the transition to about three times the transition Now at: Department of Terrestrial Magnetism, Camegie Institution of Washington, 5241 Broad Branch Road, N.W., Washington, DC 20015, USA. t Now at: Department of Earth and Planetary Sciences, Washington University, St Louis, Missouri 631 30, USA.

210

R . A . Lux, C. F. Davies and J. H. Thomas value. Boundary velocities corresponding to zero net force are close to those expected for convection under a free-slip boundary. When scaled to wholemantle convection, the shear stress on the upper boundary is of the order of 10 bar, and above the transition to a single cell the net force on a large plate is sufficient to overcome even kilobars of frictional resistance at plate boundaries. These results indicate that mantle convection and lithospheric plate motions are likely to be strongly coupled, with the rigid moving plates strongly controlling the mantle flow structure, with single cells under the faster plates, but with the mantle convection cells possibly exerting large driving forces on the plates.

Introduction

Over the past 10 years a good kinematic description of the motions of the Earth’s tectonic plates has evolved. Although the positions and velocities of the plates are known, the mantle flow beneath them is quite unknown due to the lack of direct observations. At present there is a controversy concerning the driving mechanism of plate tectonics, but some distinct alternative models have gradually emerged. These can be characterized broadly by whether the whole mantle or only the upper mantle is involved in the flow. Central to this controversy is the interaction of the plates with the mantle. Although the lithosphere and the mantle together form a single, large convecting system, the forces generated by the mantle are often considered separately from those generated by the plates. If convection in the mantle is driving the motions of the plates, the mantle is ‘active’. If, on the other hand, the negative buoyancy of the subducting slab drives the flow, then the mantle is ‘passive’. ‘Upper-mantle’ models tend to assume the mantle is passive (e.g. Forsyth & Uyeda 1975; Richardson, Solomon & Sleep 1976; Chapple & Tullis 1977), both because of the difficulty of treating an active mantle and because of the difficulty of producing stable convection cells with large aspect ratio (width to depth) (Richter 1973a, b; Houston & De Bremaecker 1975). ‘Whole-mantle’ models allow the possibility of either an active or passive mantle (Davies 1977a, 1978; Hager & O’Connell 1978) and seem more feasible from a fluid-dynamical point of view. The variety of opinions only illustrates that deducing the dynamics of the mantle is extraordinarily complex. The eventual resolution of the controversy will depend on systematic investigations into basic fluid-dynamic phenomena. Here we concentrate on the coupling between the moving lithospheric plates and the flow beneath. It is found that the interaction of the moving plates with convection is likely to be important for the mantle and that the principal effect of the plates is to control the scale of convection cells in an active mantle. Thermal convection has been invoked to provide a driving force for plate motions and, in addition, to determine the flow structure in the mantle. As reviewed by Whitehead (1976), there are some important differences between mantle convection and laboratory convection at high Rayleigh number. Both theoretical and experimental studies of Rayleigh-Btnard convection (fluid layer heated from below) show that convection occurs in a regular pattern of nearly equidimensional cells, with the width of these cells of the order of the depth of the convecting fluid layer. The pattern of plates and plate motions on the Earth is irregular and does not resemble normal Rayleigh-Btnard convection. More important, the horizontal scale of some plates, such as the Pacific, is considerably greater than the depth of the mantle. Convection cells of large aspect ratio (width to depth) are required to achieve the mass transport associated with production and subduction of lithospheric plates. In addition, if the mantle produces a net force on an overlying plate sufficient to drive it, then not only

Moving lithospheric plates

21 1

must the magnitude of the basal shear stress on a plate be sufficiently large, but also there must not be neighbouring convection cells under the plate that rotate in an opposite sense and cancel each other. In other words, the flow structure must be organized on a horizontal scale comparable to that of the plate. Whether the mantle is active or passive, the most efficient way of achieving this, of course, is to have just one cell associated with each plate. Such convection cells will not be equidimensional since the plate size (length 104km) is much greater than the active fluid depth, even with whole-mantle convection (depth 3000 km). Thus, with whole-mantle convection, aspect ratios of 3 or 4 are required, while with upper-mantle convection, aspect ratios up to 20 or 30 are required. The long, rigid lithospheric plates are our primary observation of mantle convection and yet are the feature that is least well incorporated into convection theory. In this study their effect is simulated by a rigid upper boundary that may move in its own plane. Convection cells of large aspect ratio are rarely observed in more conventional fluids. Some results of a numerical model by Richter (1973a, b) suggest that a moving boundary is important in this regard. His model includes moving upper and side boundaries on a box of aspect ratio 6 and rather complicated thermal boundary conditions. As discussed by Richter, the primary effect of the moving boundary is to alter the flow structure in the direction parallel to the boundary motion, to produce a single long convection cell. If the boundary is moved slowly, compared with the free convective velocity, the flow structure is characteristic of Rayleigh-=nard convection. The cells are about as wide as the fluid is deep. If the boundary is moved near to or faster than the free convective velocity, the flow structure is characteristic of a boundary-driven flow and the cells become elongated in the direction of the moving boundary. This effect, along with several others, is demonstrated in the results of the present model, and is systematically investigated. The interaction of a moving boundary with convection has also been studied numerically by Torrance & Turcotte (1971), Torrance eC al. (1972) and Parmentier & Turcotte (1978), and experimentally by Richter & Parsons (1975). All of these numerical models include other effects besides the moving boundary, such as variable viscosity or a subducting slab, and only a few cases were run for each model, so that it is difficult to reach any general conclusions. The model developed in the next section includes only the action of the moving boundary on convection, in order to isolate its effect. Numerical models are generally restricted to two dimensions by finite computing resources, and our study is no exception. The artificially enhancqd stability of twodimensional convection, due to the exclusion of three-dimensional modes of instability, is a weakness of most numerical studies, and yet, as demonstrated by McKenzie, Roberts & Weiss (1974), considerable understanding of basic fluid dynamics has resulted from twodimensional models. Except for the third dimension, the model detailed below is similar to the configuration studied experimentally by Richter & Parsons (1975). Richter & Parsons show that flow in the third dimension, in the form of rolls with axes parallel to the boundary velocity, is important in their experiment, but in the discussion to follow we argue that this flow is not likely to be a major factor in the dynamics of the mantle. Even if such three-dimensional flow is important for the mantle, the results presented here still demonstrate the basic effect of a moving upper boundary on the horizontal length scale of convection.

Flow model and governing equations Mantle convection has been modelled numerically in a number of studies, and it has been shown that many important features can be investigated with relatively simple models

212

R . A . Lux, G. F. Duvies and J. H. Thomas y: I

T:O

___

+=O a = u b dL,o dX

] - ;%=o

w +: =0O

w

y = 0 --I

T=I

I

x=o

+=O

wzO

,

I

I

,,L d

Figure 1. Model geometry and boundary conditions. The fluid-filled box is heated from below and the top boundary moves with velocity u b towards the right. The box has length 4d throughout this study.

assuming a viscous fluid in the limit of large Prandtl number, in which case inertial effects are unimportant (see, e.g. Richter 1973a; McKenzie et al. 1974). Most such models have assumed that only the upper mantle (700 km depth) is involved in convection, but it has been argued that the geophysical evidence for this is not compelling, and that whole-mantle (3000 km depth) convection should also be considered (O’Connell 1977; Davies 1977a; Elsasser, Olson & Marsh 1979). The model described below allows the possibility of either upper-mantle or whole-mantle convection. The model, shown in Fig. 1, consists of a two-dimensional rectangular box filled with a Newtonian Boussinesq fluid of constant material properties that is heated from below. The length of this box is chosen to be four times its depth primarily because longer boxes, more representative of upper mantle geometry, require prohibitive amounts of computer time. The Boussinesq approximation allows density variations to occur in the gravity force term of the momentum equation. In all other respects the fluid is incompressible. The top and bottom boundaries are maintained at constant temperature and the side walls are insulating. The side walls and bottom boundary are free-slip, and the top boundary is no-slip and may move at a constant prescribed velocity from left to right. Except for the moving no-slip top boundary, this model is quite similar to the one used by McKenzie et al. (1974). The equations governing convection in the mantle have been discussed at length elsewhere (Peltier 1972; Turcotte & Oxburgh 1972; Richter 1973a; McKenzie et ul. 1974), so only a brief outline will be presented here. It is important to note that the governing equations, model geometry and boundary conditions used here are the simplest possible. In particular, the two-dimensional rectangular geometry is a poor approximation to the geometry of the mantle. The purpose of this investigation is to understand the basic interaction of convection and a moving boundary, and on this ground such a simple model is not only justified but desirable. The relative importance of forces generated by a tangentially moving boundary and thermally driven buoyancy forces can be approximately determined from the way in which the governing equations scale into non-dimensional form. The equations governing convection can be non-dirnensionalized in different ways, depending on whether the flow is primarily free or forced convection. Fluid motions driven by gravity alone are usually called free convection, in contrast to forced convection driven by surface forces. The non-dimensionalization follows that of Ostrach (1953), Peltier (1972) and McKenzie et al. (1974). If we let primes denote dimensional variables, we have

d tt=-tt, UO

T’ = TAT + To,


213

where the reference velocity Uo (discussed below) is characteristic of flow velocities. The length scale d is the depth of the fluid layer, the time-scale d/Uois a turnover time, AT is the temperature difference maintained across the fluid layer and To is a reference temperature. Velocity components u and u are in the horizontal and vertical directions respectively. It is useful to define a stream function $, so that the incompressibility condition is satisfied and pressure may be eliminated. The appropriate stream function formulation of the momentum equation is

The energy equation is simply

DT

1

Dt

Pe

----

V’T.

The non-dimensional numbers are defined as: Pe = Peclet number = Uod/K, Ra = Rayleigh number = g a d 3 A T / ~ u , where g is the acceleration of gravity, K is the thermal diffusivity, CY is the coefficient of thermal expansion and v is the kinematic viscosity. Values of these parameters appropriate to the mantle are given in Table 1. In forced convection, a velocity typical of the externally driven flow, such as the boundary velocity, ub, can be chosen as the velocity scale Uo. In the stream function equation (1) we see that the importance of buoyancy is determined by the ratio Ra/Pe and that, in the absence of boundary motion, horizontal variations in temperature drive the flow. For free convection the choice of a characteristic velocity Uois not obvious. Since a freely convecting fluid is driven solely by the horizontal temperature gradient aT/ax, we may choose the characteristic velocity scale Uo=g&d2AT/usuch that the driving term (Ra/Pe) (aT/ax)is of order 1. Using this scaling the momentum and energy equations become

aT V4$ =-

(3)

ax

Table 1. Assumed values of mantle properties (see McKenzie & Weiss 1975). The range of values of Pe and Ra correspond to the range of observed plate velocities and the uncertainty in mantle viscosity.

d

Pe = U , , ~ / K Ra = a g A T d ’ / ~ v

Upper mantle

Whole mantle

i x 10’cm 200 -2000 105’’

3 x 108cm 103-104 106 1

*

R. A . Lux, G. F. Davies and J. H. Thomas

214 and DT

Dt

-

1

Ra

V2T.

(4)

As a result of this scaling we see that the Rayleigh number is equivalent to a Peclet number for free convection (Turner 1973, p. 209). The velocity scale Uo=gad2AT/vis chosen for this study. These two methods of non-dimensionalization give insight into the problem of combined forced and free convection. They show that the ratio of driven velocity to free convective velocity, Ud/Uf = Ub/(agdzAT/v)= Pe/Ra, will determine the relative importance of forced and free convection. Note that Pe/Ra is a ratio of velocity scales, and not of actual fluid velocities. In practice we find that measured free-convective velocities are approximately Uf, suggesting that Ra be replaced by Ra/Rc, where Rc is the critical Rayleigh number. Thus, for Pe/(Ra/Rc) s 1 forced convection will dominate, and for Pe/(Ra/Rc) Q I free convection will dominate. A better estimate of the free-convective velocity can be made by considering the simple boundary-layer arguments of McKenzie e l al. (1974) and McKenzie & Weiss (1975). We find that Ud/Uf= PeRa-2’3(Z/d)1’3, where 1 is the length of the box. This result is basically in agreement with the crude scaling above, only now the aspect ratio ( l / d )appears. The Ra-’I3 dependence seems to give a better fit to the results of our numerical calculations than Ra-’. The non-dimensional grouping PeRa-2’3 (Z/d)”3 shows how the relative effect of the moving boundary scales with the three important parameters, boundary speed (Pe), heating (Ra) and aspect ratio (l/d). Increasing Ra tends to decrease the effect of the moving boundary, while increasing Pe or l/d tends to increase the effect.

Numerical methods The numerical methods we have used are well established and easy to apply. If we introduce the vorticity w (defined below), then the fourth-order stream function equation (3) can be separated into a pair of second-order Poisson equations,

for which efficient matrix reduction methods are available (Sweet 1974; Swarztrauber & Sweet 1975). For prescribed stress boundary conditions, the Poisson equations can simply be solved successively as done by McKenzie et al. (1974). A complication arises for a prescribed boundary velocity because the boundary conditions on vorticity are then not explicit (Richter 1973a). On the boundary the vorticity is proportional to the shear stress and, since the shear stress is not known until the flow field is found, we lack a boundary condition on w . The Poisson equation for the stream function has an over-prescribed boundary condition, since both J, and aJ,/ay are given. This dilemma is resolved by using the condition on J, for the stream function equation and the condition on aJ,/ay in a Taylor series expansion for w at the boundary for the vorticity equation. To ensure numerical stability with this boundary condition, it is necessary to iterate between the equations; an efficient way of optimizing this iteration has been given by Ehrlich & Gupta (1975). The iteration technique for the coupled Poisson equations was tested against analytical solutions of the biharmonic equation given by Davies (1977b) and shown to be very accurate. The solution of the convection problem described here evolves with time, so


215

the boundary condition on w must be found at each time step by the iteration method. This procedure is time consuming, so to increase the rate of convergence of this iteratior? a boundary condition on w accurate to first order (Taylor series) is used. The energy equation (4) is solved using an alternating-direction implicit (ADI) method similar to that described by Houston & De Bremaecker (1974). The AD1 method combined with block cyclic reduction was shown to be both efficient and accurate for solving the incompressible driven cavity problem by Smith & Kidd (1975). Upwind differencing is used for stability of the convective terms, even though it is less accurate than central differencing (de Vahl Davis & Mallinson 1976). To ensure accuracy and stability, time steps are always taken to obey the Courant condition (uAt/Ax) G 1 (Roache 1972). The Nusselt number, Nu, defined as

has proved to be a good diagnostic. Nu measures the ratio of total heat flux to the flux which would have been carried by conduction alone. Since the side walls are insulating, the horizontally averaged Nusselt number should be constant with height, and the extent to which it is gives a rough measure of both accuracy and convergence. The change with time of the Nusselt number is used to estimate when a flow has reached steady state, as suggested by Plows (1968). In general, a steady state solution is found, but for some values of Pe and Ra, near the transition in flow structure from a single cell to multiple cells, the solution is inherently time-dependent. This time-dependent flow is consistent with the instability of the thermal boundary layer discussed further below. Considerable confidence in our numerical technique has been gained through comparisons with the numerical results of McKenzie et al. (1 974), showing almost identical streamline, vorticity and isotherm patterns. Grid spacings of A x = Ay = 1/32 provide reasonable accuracy except for runs of Ra = lo6 where the boundary layer is quite thin compared to the mesh size. Many grid points are required for the flow calculations with large aspect ratio, and typical solutions (such as those shown in Figs 2, 3 and 4) require several hours of CDC 6600 computer time.

Results FLOW STRUCTURE

Our computations show two general types of flow; (1) a multi-cell, Rayleigh-Benard flow consisting of three cells for low boundary speeds and (2) a larger aspect-ratio flow consisting of one cell for high boundary speeds. As either the Peclet number (range 0-5000) or the Rayleigh number (range 5000-106) is varied, a transition between these two flows is observed. This transition is not sharp and undergoes hysteresis, depending on the side from which the transition is approached. The streamlines and isotherms for a sequence of flows illustrating this are shown in Figs 2 , 3 and 4. The sequence in Fig. 2(a-e), for increasing Pe, is for Ra = lo4.The abbreviated sequence in Fig. 3(a, b), also for increasing Pe, is for Ra = 10’. It can be seen from Figs 2(c) and 3(b), which both have Pe = 20, that the effect of the moving boundary is more pronounced at lower Ra. The sequence in Fig 4(a-e) is for R a = lo5 and starts at high Pe with a single long cell, which becomes unstable and tends to break into smaller cells as Pe is reduced through successively lower values. Note that the initial conditions for the flows shown in Figs 2, 3 and 4 are taken to be the previous flows in the sequence. For example, the flow

216

R. A . Lux, G. F. Davies and J. H. Thomas

Figure 2. Steady-state isotherms (above) and streamlines (below) for Ra = lo4. As Pe is increased, this sequence shows the transition from multi-cell to single-cell flow structure. In all figures the isotherms are contoured evenly between 0 and 1. The streamlines are contoured evenly between 0 and Q m i n (a negative for clockwise circulation) or Q m a x (Q positive for counterclockwise circulation) for each cell. The single-cell flows have nine contour lines per cell and the multi-cell flows have four per cell. (a) Qmin = - 1.38 X (b) Pe = 8, Qma, = Normal Rayleigh-%nard pattern, Pe = 0, Qmax = 1.38 X Q m i n = - 1.59 X lo-’. (d) Pe = 30, = - 1.50 X l o + . (c) Pe = 20, Qmax = 9.53 X 1.27 x lo-), = - 1.93 X (e) Pe = 50, = - 1.66 X


217

(e ) Figure 2 continued

in Fig. 2(a) is the initial condition for the flow in Fig. 2(b). Likewise, the flow in Fig. 2(b) is the initial condition for the flow in Fig. 2(c), and so forth. The flows resulting from many numerical runs may be categorized as‘being either multicell or single-cell, with the two types of flow occupying different regions of the Pe-Ra plane as shown in Fig. 5. In between these two types lies a transition region that marks a change in cell structure. The dependence of flow structure on Peclet number and Rayleigh number is roughly delineated by the dashed line in Fig. 5 corresponding to Pe = 0.04 RaZ3. Although it is only an approximate guide, above this line the moving boundary predominates and the flow is primarily single-cell while below it thermal effects are more important and the flow is primarily multi-cell. This supports our scale analysis and simple boundary layer theory noted above. The transition region is discussed in more detail in a later section. Fig. 5 shows that the transition in flow structure occurs for parameters representative of both upper-mantle and whole-mantle convection. This suggests that the coupling between the mantle and the plates is likely to be important whatever the depth of mantle convection. The boxes in Fig. 5 outline the range of estimates of Pe and Ra for the upper mantle and whole mantle, as given in Table 1. The uncertainty of the parameters used to define Pe and Ra make the distinction between the upper and whole mantle rather tentative. In general, however, the upper mantle has a lower Pe and Ra than the whole mantle and this is reflected in their locations in Fig. 5. Note also that the flow transition has been located only for an

218

R . A . Lux, G. F. Davies and J. H. Thomas

(b) Figure 3. Similar to Fig. 2 but with Ra = 10'. (a) Pe = 0, J'max = 4.43 X (b) Pe = 20, =4.41 X ni,,* = -4.73 X

=

- 4.43

X

aspect ratio of 4. Higher aspect-ratio boxes (corresponding to upper mantle flow) would presumably undergo the transition at higher values of Pe, as discussed later. BOUNDARY STRESS

In addition to observing the flow structure, it is possible to measure the net force exerted on the moving plate by the fluid. By integrating the shear stress along the top boundary, we obtain the net tangential force per unit width on the plate. The variation of the net force on the plate with Peclet number is shown in Fig. 6 for several values of Rayleigh number. The net force per unit width has been made dimensional by using a viscosity p of 1 OZ2poise, a diffusivity K of 10-'cm2s-' and a depth d of 108cm. An interesting result is found for flows of one cell. Near the transition region the net force changes sign. The force is positive (flow driving the plate) for slower boundary speeds and negative (plate driving the flow) for higher speeds, and seems to be a linear function of plate velocity. The magnitude of the net force increases with Rayleigh number, and the straight lines fitted to the data are all approximately parallel. The net force is a function of flow structure. All the points in Fig. 6 correspond to singlecell flows; although the straight lines can be extrapolated in the direction of increasing Peclet number, they cannot be extended to zero Peclet number because the flow changes to

Moving lithosphen'c plates

219

Figure 4. This sequence at Ra = 10' shows the development of time-dependent flow as Pe is decreased. The flows in (c), (d) and (e) are inherently time dependent. (a) Pe = 500, $,,,in = - 1.21 X (b) (d) Pe = 100, Qmin= - 7.48 X (c) Pe = 200, = 8.09 X Pe = 400, Qmin = - 1.03 X 10". (e) Pe = 5 0 , $ m i n = 7.64 X -

-

220

R . A . Lux, G.F. Davies and J. H. l30mas /

(e) Figure 4 continued

4-

w.m.

0

u.m. w

a

3-

0 0 -

n n

0-

3

I

I

I

I

4

5

6

7

log Ra

Figure 5. The flow structure as a function of Pe and Ra. The flows are identified as being either steady single-cell 0 , steady multi-cell A , time-dependent single-cell , or time-dependent multi-cell 4. The dashed diagonal line Pe = 0.04 RaZi3approximately indicates the transition between single-cell and multi-cell flows. Above this line the flow structure is primarily single-cell and below it is primarily multi-cell. The solid diagonal line Pe = 0.2 Ra0.6indicates zero net tangential force on the moving boundary. Above this line the moving boundary drives the fluid. Below this line but above the dashed flow transition line, single-cell convection drives the boundary motion. The boxes (u.m.1 and (w.m.) indicate the portions of the Pe-Ra plane corresponding to the upper mantle and whole mantle respectively.

:c 2.


22 1

Figure 6. Net tangential force per unit width applied to the moving boundary by single-cell convection as a function of Pe. The straight lines are fitted to points having the same Ra as indicated. For a positive net force the fluid is driving the boundary, for a negative net force the boundary is driving the fluid. As in Fig. 5 , c denotes steady single-cell flow and denotes time-dependent single-cell flow.

multi-cell structure here and the net force drops by approximately two-thirds, corresponding to two cells driving and one cell opposing the boundary motion. The points of zero net force are well located by interpolation in Fig. 6 and are used to define the solid diagonal line plotted in Fig. 5. This line of zero net force is well defined, unlike the dashed line indicating the flow transition, and is given by Pe = 0.2 RaoB6,in rough agreement with the functional relation Pe a Ra2’3 predicted by the boundary layer theory. If the mantle parameters plot in the upper parts of the boxes in Fig. 5, then of course some other driving force must act to maintain the boundary (i.e. plate) motions. However, it is important to note that there is a region between the dashed (flow transition) line and the solid (zero net force) line in Fig. 5 where the flow consists of a single long convection cell that still provides a net driving force on the moving boundary. An average basal shear stress may be calculated by dividing the net force by the length of the box (4 x iO*cm). Between the dashed and solid diagonal lines of Fig. 5, this average stress is of the order of 10 bar. The magnitude of this stress suggests that a mantle cell of aspect ratio 4 is capable of overcoming frictional resistance at plate boundaries even if boundary frictional stresses are of the order of kilobars (Hanks 1977; Davies 1978). In this case it seems that an active mantle is capable of driving plate motions. More calculations are necessary to determine whether flows of aspect ratio larger than 4 are capable of driving the moving boundary. However, both the boundary layer theory and our limited experience with flows in boxes with l/d = 5 indicate that an increase in aspect ratio requires an increase in boundary velocity in order to maintain stability. This is consistent with suggestions that upper-mantle models might require active plates, while whole-mantle models allow passive plates. In the case of the upper mantle, the fluid may not be able to drive extremely high aspect-ratio cells (l/d = 2 0 ) with high enough velocity to ensure stability. TRANSITION FLOWS

Flows in the transition region (Fig. 4(c, d, e)) are unsteady and are characterized by the formation of thermal ‘blobs’ that begin to separate from the thermal boundary layer. Figs 7

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R . A . Lux, G. F. Davies and J. H. Thomas

(C)

Figure 7. The formation of blobs and time-dependent flow. A sequence, increasing in time, is shown for flowshaving Pe = 200 and Ra = 10’. (a) Non-dimensional time t = 1800, $,,,in = - 9.7 X (b) t = = - 7.91 X (d) t = 5400, 3000, $,,,in = - 7.66 X (c) t = 3600, $,,,in = - 7.97 X (f) t = 7200, $,,,in = - 7.63 X (e) t = 6300, qmin = -8.97 x


223

(f)

Figure 7 continued

and 8, which show time sequences at constant Pe and Ra, illustrate the formation of these blobs and also show the time-dependent behaviour of the flow. I f a large aspect-ratio cell is slowed down, the thermal boundary layers on the upper and lower boundaries will thicken. A similarity solution shows that the boundary layer thickness F behaves as 6 a: ( K X ‘ / U ’ ) ” * , where u ‘ is the horizontal velocity in the boundary layer and x’ is the horizontal distance from the end wall in the direction of flow. The thickness of the boundary layer increases along its length and is thinner for faster velocities. The thermal boundary layers transport

224

/ . 1

Figure 8. The Same as in Fig. 7 only Pe = 500 and Ra = lo6. (a) t = 2100, $min = - 2.7 x (c) t = 4200, $min = - 3.56 X lo-". (d) t = 6000, $max = 2.87 X r = 3300, $min = - 2.58 X $min= - 3.3 x

(b)


225

heat vertically by conduction; thus the convective instability of a thermal boundary layer may be understood in terms of a local Rayleigh number R B L for the boundary layer (Howard 1966; Busse 1967; Foster 1971; McKenzie et al. 1974; McKenzie & Weiss 1975; Parsons & McKenzie 1978). If the boundary layer thickness and velocity are used to define a local Rayleigh number, we find R B L = RaPe-3/2(x'/d)3/2.Thus the thermal boundary layer is stabilized by large velocities (large Pe) and destabilized by increased heating (large Ra). Note that R B L is a function of aspect ratio as well, and the boundary layer is less stable with increasing length. The thermal blobs grow to larger size and occur more frequently along the bottom boundary layer. The bottom boundary is free-slip and has a lower critical Rayleigh number than the no-slip top boundary; consequently, it is less stable. This simple notion of boundary layer instability explains qualitatively the formation of blobs and the mechanism for the transition in flow structure. The flow in the transition region is time dependent because the boundary layer is unstable. When a steady, large aspect-ratio flow is slowed down, the blobs grow, detach from the boundary layer and entrain more fluid as they rise (sink). Before they can reach the upper (lower) boundary, the main circulation of the convection cell sweeps them to the ends of the box. The singlecell flow is likely to change to multi-cell flow if the blobs grow faster than they are swept away. If a cell of large aspect ratio is to remain stable, the boundary velocity must be sufficient to counteract the destabilizing influence of heating and the large cell width. Discussion

The model employed in this study incorporates one distinctive feature of the lithosphere; its horizontal velocity is independent of position. This feature results from the strong temperature dependence of mantle rheology, which, when combined with the cool upper thermal boundary layer, produces a mechanically strong layer which tends to move rigidly. A second consequence of the temperature dependence of the rheology, closely related to the first, is that the cool upper thermal boundary layer tends to be frozen into the lithosphere and thereby stabilized against downwelling. Although this second feature has been neglected here, as in many other studies, its effect will tend to justify the neglect of three-dimensional flow in the present model. Richter & Parsons (1975) performed laboratory experiments closely analogous to the present numerical experiments except that three-dimensional flow was allowed. In their experiments, a fluid layer was heated from below and a moving upper boundary was achieved by dragging a thin mylar sheet acrbss the top of the fluid. The effect of moving the boundary rapidly was to produce convective rolls with axes parallel to the boundary velocity and widths comparable to the fluid depth. In other words, the moving boundary extended the horizontal scale of the cells in the direction of the boundary motion, but not transverse to it. The present numerical model demonstrates the extension of the horizontal scale in the direction of boundary motion, but the longitudinal rolls do not occur because three dimensional motion is precluded. Clearly, the presence of longitudinal rolls would be an important factor, since the heat which they transport would not be available to power the larger scale shear flow and boundary motion. It is not obvious, however, that the model of Richter & Parsons (1975) is a good approximation to the Earth, since it includes only one of the two important features of the lithosphere: the mylar sheet simulates the rigid motion of the lithosphere, but not the tendency of the lithosphere to trap the upper thermal boundary layer. Thus, in the experiment the upper boundary layer is free to develop instabilities in the third dimension which grow into the longitudinal rolls, whereas in the Earth this tendency will be strongly inhibited and the rolls may develop only weakly or not at all.

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The tendency of the lithosphere to stabilize the upper thermal boundary layer has been demonstrated in several studies. McFadden & Smylie (1968), Houston & De Bremaecker (1975) and Richter & Daly (1978) have studied convection in fluids with depth-dependent viscosity in which the viscosity at the top of the fluid is much greater than in the interior. The effect is to increase the horizontal scale of the convection cells; evidently fluid in the upper thermal boundary layer must cool for a longer time and develop larger buoyancy forces in order to overcome the resistance of the more viscous layer. In the Earth, this effect will operate both parallel to and transverse to the lithospheric velocity, so that it will tend to suppress the longitudinal rolls and enhance the extension of the horizontal scale in the direction of plate motion. The development of longitudinal rolls will depend on the extent to which the thermal boundary layer extends below the lithosphere into the asthenosphere. Parsons & McKenzie (1978) have investigated a model in which this condition is explicitly assumed to exist, but have not directly addressed the question of whether it exists in the Earth. We conclude from these considerations that the Earth's behaviour is somewhere intermediate between the strictly two-dimensional flow of our numerical model and the fully three-dimensional flow of the laboratory model of Richter & Parsons (1975); it is probably closer to the two-dimensional model. In any case, the two-dimensional model must be relevant to the large-scale part of the flow associated with plate motions. Clearly, these questions deserve further investigation. Another question which must be considered is that of time-scales for the approach to equilibrium in comparison to time-scales in the Earth. In order to conduct our numerical investigation systematically, it was necessary to seek steady states of convection (although these are not always achievable). Richter & Parsons (1975) pointed out that convective systems can approach steady state very slowly, and when scaled to the mantle the time-scale might approach or exceed the age of the Earth. The present calculations were not intended to simulate the unsteady flow with any accuracy, but some rough estimates indicate that, when scaled to the mantle, the time-scales are indeed quite long. For example, the nondimensional time elapsed between Fig. 2(c) and (d) was 3000, which corresponds (using the parameters of Table 1) to about 4.7 x 109yr (upper mantle) or 8.6 x 10"yr (whole mantle). Likewise, the time interval between the frames in Fig. 7 corresponds to about 10syr (upper mantle) or 109yr (whole mantle). Although the time-scales of Figs 2 4 are presumably some measure of the time it takes for a plate to influence the flow structure, they are not directly applicable to the Earth, since the sequences in these figures are not expected' to simulate any sequence in the Earth. An important feature of the Earth is that the plqtes grow and shrink on time-scales of about 10Syr because of the relative migration of ridges and trenches, as discussed by Garfunkel (1975) and Davies (1977a). Convection cells will tend to grow and shrink with the plates, so that it may be possible to develop a large cell under a large plate in about 10'yr. On the other hand, the relatively rapid changes in plate size mean that steady state flow will never be closely approached. Some well-chosen simulations might illuminate this process. The time-scale question is also relevant to the development of longitudinal rolls. Richter & Parsons (1975) found that the time required for the development of rolls, scaled to the upper mantle, is about lo7 to 108yr, depending on the plate velocity. Their results indicate that the time-scale for the development of whole-mantle rolls would be an order of magnitude larger and thus that whole-mantle rolls are unlikely. Another way to reach this conclusion is to note that rolls develop in a system with two distinctly different length scales, such as the width and depth of a high aspect-ratio box. With whole-mantle convection, these length scales would differ by a factor of 4 at most; there would not be room for ioiigitudinal rolls (of whole-mantle scale) to develop significantly. Thus the use of


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two-dimensional flow models seems to be even more strongly justified for whole-mantle convection. Conclusions We have shown that two-dimensional convective cells of large aspect ratio can occur in a fluid with a horizontally moving rigid upper boundary. The conditions under which these occur depend on three parameters: (1) the Peclet number, which measures flow speed, (2) the Rayleigh number, which measures heating and (3) the aspect ratio. Results of computations over a wide range of Peclet and Rayleigh numbers suggest that the coupling between a moving lithospheric plate and the mantle is very important in determining the convective flow structure in the mantle. Cells of aspect ratio 4 are found to be stable for likely mantle conditions. It was found that these large cells may either drive or resist the boundary motion which stabilizes them and that they can generate driving stresses which, when scaled to whole-mantle cells under large plates, would be sufficient to overcome any likely resistance at plate boundaries. The model used here is very idealized, since the goal was to investigate the phenomenon rather than to simulate the Earth in detail, but these conclusions are likely to apply to more complex models and to the Earth. The same phenomenon can be seen in the more complex models of Richter (1973a), Houston & De Bremaecker (1975), De Bremaecker (1977) and Parmentier & Turcotte (1978). Allowing flow in the third dimension might affect the results, but it is argued that the lithosphere stabilizes the thermal boundary layer and suppresses such flow in the mantle. Although plate sizes and velocities change on timescales comparable to the time required for plate motion to affect mantle flow, the effect should still be substantial, and in fact a growing plate may provide a mechanism for rapidly generating a cell of large aspect ratio. It is not expected that different boundary conditions (such as a no-slip lower boundary), spherical geometry or internal heating will substantially change these conclusions. Results with internal heating and with different moving-boundary configurations will be reported later. Acknowledgments We are grateful to the University of Rochester’s Laboratory for Laser Energetics for generously providing computer time and to Roland Sweet of the National Center for Atmospheric Research for providing several subroutines. This research was supported by National Science Foundation grant EAR 77-14672. References Busse, F . H., 1967. On the stability of two-dimensional convection in a layer heated from below, J. Math. Phys., 46,140-150. Chapple, W. M . & Tullis, T. E., 1977. Evaluation of the forces that drive the plates, J . geophys. Res., 82, 1967-1984. Davies, G. F., 1977a. Whole-mantle convection and plate tectonics, Geophys. J . R. asfr. Soc., 49, 459486. Davies, G. F., 1977b. Viscous mantle flow under moving lithospheric plates and under subduction zones, Geophys. J. R . astr. Soc., 49,557-563. Davies, G. F., 1978. The roles of boundary friction, basal shear stress and deep mantle convection in plate tectonics, Geophys. Res. Lett., 5 , 161-164. De Bremaecker, J. Cl., 1977. Convection in the earth’s mantle, Tectoriophys., 41, 195-208. de Vahl Davis, G . & Mallinson, G. D., 1976. An evaluation of upwind and central difference approximations by a study of recirculating flow, J . Compt. Fluids,4, 29-43.

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Ehrlich, L. W. & Gupta, M. M., 1975. Some difference schemes for the biharmonic equation, SIAM J. Numer. Anal., 12, 773-790. Elsasser, W. M., Olson, P. & Marsh, B. D., 1979. The depth of mantle convection, J. geophys. Res., 84, 147 -1 55. Forsyth, D. & Uyeda, S., 1975. On the relative importance of the driving forces of plate motion, Geophys. J . R . astr. SOC.,43, 163-200. Foster, T. D., 1971. Intermittent convection, Geophys. Fluid Dyn., 2, 201-217. Garfunkel, Z., 1975. Growth, shrinking, and long-term evolution of plates and their implications for the flow pattern of the mantle, J . geophys. Res., 80,4425-4432. Hager, B. H. & O’Connell, R. J., 1978. Subduction zone dip angles and flow driven by plate motion, Tectonophys., 50, 111-133. Hanks, T. C., 1977. Earthquake stress drops, ambient tectonic stresses and stresses that drive plate motions,Atre Appl. Geophys., 115,441-458. Houston, M . H., Jr & De Bremaecker, J . CI., 1974. AD1 solution of free convection in a variable viscosity fluid, J. Comp. Phys., 16,221-239. Houston, M. H., Jr & De Bremaecker, J. Cl., 1975. Numerical models of convection in the upper mantle, J. geophys. Res., 80,742-751. Howard, L. N., 1966. Convection at high Rayleigh number, in Proceedings o f t h e Eleventh International Congress of Applied Mechanics, Munich (Germany), ed. Gortler, H., Springer-Verlag, Berlin. McFadden, C. P. & Smylie, D. E., 1968. Effect of a region of low viscosity on thermal convection in the mantle, Nature, 220,468- 469. McKenzie, D. P., Roberts, J. M. & Weiss, N. O., 1974. Convection in the Earth’s mantle: towards a numerical solution, J. Fluid Mech., 62,465-538. McKenzie, D. P. & Weiss, N. O., 1975. Speculations on the thermal and tectonic history of the Earth, Geophys. J . R . astr. Soc., 42,131-174. O’Connell, R. J., 1977. On the scale of mantle convection, Tectonophys , 38, 119-136. Ostrach, S., 1953. New aspects of natural-convection heat transfer, Trans. ASME, 75, 1287-1290. Parmentier, E. M . & Turcotte, D. L., 1978. Two-dimensional mantle flow beneath a rigid accreting lithosphere, Phys. Earth planet. Int., 17, 281-289. Parsons, B. & McKenzie, D., 1978. Mantle convection and the thermal structure of the plates, J. geophys. Res., 83,4485-4496. Peltier, W. R., 1972. Penetrative convection in the planetary mantle, Geophys. P7uid Dyn., 5,47-88. Plows, W. H ., 1968. Some numerical results for two-dimensional steady laminar BBnard convection, Phys. Fluids, 11,1593-1599. Richardson, R. M., Solomon, S. C. & Sleep, N. H., 1976. Intraplate stress as an indicator of plate tectonic driving forces, J geophys. Res., 81, 1847-1856. Richter, F. M., 1973a. Dynamical models for sea floor spreading, Rev. Geophys. Space Phys., 11, 223287. Richter, F . M., 1973b. Convection and the largescale circulation of the mantle, J. geophys. Res., 78, 8735-8745. Richter, F. M. & Daly, S. F., 1978. Convection models having a multiplicity of large horizontal scales, J . geophys. Res., 83,4951 -4956. Richter, F. M. & Parsons, B., 1975. On the interaction of two scales of convection in the mantle, J. geophys. Res., 80,2529-2541. Roache, P. J., 1972. Computational Fluid Dynamics, Hermosa Publishers, Albuquerque, New Mexico. Smith, R. E., Jr & Kidd, A., 1975. In Numerical Studies of Incompressible Viscous Flow in a Driven Cavity, pp. 61-82, Staff Langley Research Center, NASA SP-378. Swarztrauber, P. N. & Sweet, R., 1975. Efficient FORTRA N subprograms for the solution of elliptic partial differential equations, Report No. NCAR-TNIIA-109, National Center for Atmospheric Research, Boulder, Colorado. Sweet, R . A., 1974. A generalized cyclic reduction algorithm, SIAM J. Numer. Anal., 11, 506-520. Torrance, K., Davis, R., Eike, K., Gill, P., Gutman, D., Hsui, A., Lyons, S. & Zien, H., 1972. Cavity flows driven by buoyancy and shear,J. FluidMech., 51,221-231. Torrance, K . E. & Turcotte, D. L., 1971. Structure of convection cells in the mantle, J. geophys. Res., 76,1154-1161. Turcotte, D. L. & Oxburgh, E. R., 1972. Mantle convection and the new global tectonics, Ann. Rev. FluidMech., 4, 33-68. Turner, J . S., 1973. Buoyancy Effects in Fluids, Cambridge University Press. Whitehead, J. A., Jr, 1976. Convection models: laboratory versus mantle, Tectonophys., 35, 21 5-228.