Fowler, A. C.

English

ORA - Oxford University Research Archive

The depth of convection in a fluid with strongly temperature and pressure dependent viscosity

Oxford University Research Archive (ORA)

Oxford University Research Archive

GEOPHYSICAL RESEARCH LETTERS, VOL. 9, NO. 8, PAGES 816-819, AUGUST 1982 THE DEPTH OF CONVECTION IN A FLUID WITH STRONGLY TEMPERATURE AND PRESSURE DEPENDENT VISCOSITY A.C. Fowler Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 Abstract. We discuss the equations describing convective motion of a fluid with temperature and pressure dependent rheology. By combining observational constraints from plate tectonics with theoretical constraints deduced from the model equations and parameter values, we are led to the conclusion that shallow upper mantle con- vection is consistent with the equations of fluid dynamics. The Nature of Mantle Convection In studying models of the convective flow in the earth's mantle (for reviews, see Oxburgh and Turcotte (1978), Turcotte (1979)), geophysicists have been hampered by an unusual set of circum- stances: these are, that direct observational evidence is sketchy and requires interpretation; direct laboratory simulation appears insurmount- ably difficult (due to a lack of suitable mater- ials, and to the inability to scale all the para- meters properly); direct numerical computation of even a two-dimensional convective flow with a relevant rheology has not been obtained. These circumstances have produced a situation in which various fundamental properties of the convective flow are not well understood; for example, the depth of convection is still a contentious issue, with various authors arguing for whole-mantle or layered upper and lower mantle convection, respectively. Because theoretical and numerical studies of the convection of a strongly nonlin- early rheologic fluid are effectively lacking (in the context of the earth), there is a tendency to make deductions on the nature of the real flow from much simpler (often, for example, Newtonian) models in the hope that they are accurate enough (in some sense). Of the various complications associated with mantle convection, possibly the most obvious and perhaps the most significant is the dependence of viscosity on temperature and pressure. Commonly quoted activation energies for mantle-like mater- ials are E* = 122 kcal/mole (Goetze 1978), and with a basal temperature of (say) T b = 2000 K, one finds the dimensionless parameter which measures the strength of viscosity dependence on temperature, E*/RT b ~ 30. The pressure depend- ence is of a similar order of magnitude, and operates in the opposite direction. Even if the activation energy were much less, E*/RT b would still be large, and it is the fact that it occurs in an exponential dependence that is so crucial. Local variation of the temperature by a few hun- dred degrees Kelvin will alter the viscosity by orders of magnitude, and this can hardly but affect the flow. We suggest that the rheology Copyright 1982 by the American Geophysical Union. Paper number 2L0773. 0094-8276/82/002L-077353.00 will therefore control the flow, and thus that models which ignore the rheology must be care- fully examined to see if conclusions deduced from them will carry over. Particularly we will briefly propose an analogue of Turcotte and Oxburgh's (1967) boundary layer theory, to see if the conclusions of Elsasser, Olson and Marsh (1979) will carry across: we find they do not. (See also O'Connell and Hager 1980, for a differ- ent viewpoint.) The idea of rheology controlling the internal temperature has been developed by Tozer (1972, 1977, 1978), but we emphasize that his is essentially a global idea, whereas we shall be concerned here with a local dynamic control. The Effect of a Strongly Temperature and Pressure Dependent Rheology Let us consider the convection of a fluid whose viscosity is given by n = n0exp[ (E* + pV*)/RT], (•) where B0 is a reference viscosity, E* is the activation energy, p is the pressure, V* is the activation volume, R is the gas constant, and T is the absolute temperature. To fix ideas (only), we consider prescribed temperatures T b on the base, and T O (substantially less then T b) on the surface. E*/RT b is a dimensionless measure of how much the exponent varies through the flow. If we denote c = RTb/E* << 1, (2) then the viscosity contrast between top and bot- tom is exp[0(1/c)]. If 1/e • 30, then exp(1/e) • 1013 , which is beyond the range of any laboratory or numerical experiment so far conducted. Verti- cal temperature profiles for which T/(E*+ pV*) are constant are isoviscous: for a single phase mantle in which pressure is essentially litho- static, such isoviscous temperatures are linear functions of depth. Now suppose the maximum value of T/(E*+ pV*) within a convection cell gives a corresponding minimum viscosity •m- If the temperature else- where is much smaller than the corresponding iso- viscous profile (with • = •m), then it follows from the smallness of e that the viscosity in such regions will be exponentially large in com- parison with •m. From the point of view of the minimum viscosity material, the high viscosity material will appear effectively rigid. Note: we do not mean rigid in the sense of behaving elas- tically rather than viscously, but dynamically rigid in the sense that such high viscosity material will appear rigid on time scales rele- vant to the convection. Since an applied stress induces the largest velocity in the least viscous 816 Fowler: Depth of Convection in a Thermoviscous Fluid 817 fluid, we might expect the convective time scale to be dictated by such fluid. Effectively rigid material will then not sig- nificantly deform on time scales of interest, and it seems unlikely that it can participate sub- stantially in the circulatory motion: a little thought suggests that one plausible state will be that 'cold', 'rigid' material will exist in one or more quasi-stagnant layers throughout the cell, their location depending on the relation of the isoviscous temperature profiles to the in- terior (adiabatic) temperature. Particularly, vigorous convection (high Rayleigh number) will induce a top cold thermal boundary layer, which will be 'quasi-rigid' and preferentially stag- nant; this rigid boundary layer may be defined as the lithosphere. The location of other such rigid regions depends on the precise rheology, and is discussed further below. In the real earth, of course, such stagnant plates do not exist, rather the tectonic plates are active, moving relative to each other at speeds of •< 10 cm y-1. The reason for such activity is the nature of the motion at oceanic trenches, where the surface plates are subducted downwards into the upper mantle. The discussion above, however, raises the question of how such cold, rigid slabs are able to be subducted a__t all: various studies (Turcotte et al 1977, Turcotte et al 1978) have shown convincingly that subduction may be understood on the basis of non- hydrodynamic phenomena such as partial melting and elastic yielding, but the problem of initia- tion remains puzzling. For a thermoviscous fluid, the growing insta- bility of the cold layer as it thickens is over- compensated by the increasing effective viscos- ity, and purely buoyant effects will not subduct the lithosphere (Yuen, Peltier and Schubert 1981) y=l is the top surface, and y=0 is the base. We define the basal viscosity •m by •m = •0exp [ (E* + 00gdV*)/RT b ], (5) where E* is activation energy, V* is activation volume, R is the gas constant; it then follows from (1) that n = nmeXp[{(l+•) (1-0)-•y}/e0], (6) where s = RTb/E*, • = 00gdV*/E*. (7) With values O0 ~ 3 5 gm cm -3 ~ d ~ , g 10ms -2 ß , 700 km, V* ~ 11 cm 3 mole -1 , E* ~ 122 kcal mole -1 T b ~ 2000 K, R ~ 8 3 J mole -1 K -1 ß , we have 1/30, • ~ 1/2. (8) These parameters are for shallow mantle convec- tion, but for illustration only, to show that • << 1, • ~ 1. In terms of 0, we have proposed that a basal boundary layer will have a temperature jump A8 ~ O(•). (6) then implies that • varies by <• O(1) within the layer, and so the isoviscous temperature 8rhe which one would expect in the cell interior is given by 0 = Srh e = 1- •y/(l+•) + O(e). (9) We reiterate this point: if (1+•)(1-8) - •y becomes positive and 0(1) in the cell interior, i.e. 0rhe- 8 ~ O(1), then •/•m ~ exp[O(1/e)], and such material is effectively rigid (corres- ponding considerations apply if 8 - 8rhe ~ O(1)). - non-hydrodynamic processes must be relevant. Therefore we expect that convection in a fluid On the other hand, the decrease in viscosity in a with such a rheology will constrain the interior basal thermal layer will greatly enhance insta- bi!ity. Calculations of Schubert et al (1969) can be adapted (Fowler, 1981) to show that a temperature difference across the layer greater than O(eT b) will be catastrophically unstable, and hence in a steady state such a difference will not occur. In effect, the basal thermal boundary layer vanishes, and the temperature and viscosity are (approximately) continuous at the base. This is really just another instance of the isoviscous nature of the flow. Constraints on the Viscosity temperature profile to be approximately isovis- cous. This is in line with models of the earth's viscosity which are consistent with isostatic rebound data (Cathles 1975, ?eltier 1980). How- ever, the 'isoviscous' nature of the flow is paradoxically due to the strongly temperature- dependent rheology, and it remains to be seen what other effects the theology will have. The above discussion raises an immediate prob- lem. If the rheology chooses the temperature, what does the temperature equation do? There does not seem to be any possible way in which it does not also choose the temperature; particu- larly, for a vigorously convecting fluid at high Consider the convection of a fluid in a layer Rayleigh number, the appropriately scaled temper- of depth d. We nondimensionalise the temperature ature equation will be of singular perturbation T and depth y by putting type, and in the absence of viscous or radiogenic T = Tb0, y = dy*, (3) where T b is the basal temperature. If O 0 is the ambient density (with the Boussinesq approxima- tion), g is the constant acceleration due to gravity, then the (lithostatic) pressure is given may then write by 0 • Oad = exp[-Dy], p = 00gd(1-y) , (4) where source terms, we can expect that thermal boundary layers will occur, but that elsewhere (i.e., in the interior) the entropy will be nearly cons- tant, which in the present context implies an internal adiabatic temperature. In view of the small temperature jump across the basal layer, we (10) where henceforward we drop the asterisk on y*, D : •gd/c (11) P 818 Fowler: Depth of Convection in a Thermoviscous Fluid is the dissipation number, e is the thermal expansion coefficient, Cp is the specific heat. For shallow mantle convection, D ~ 0.2; for whole mantle convection, D ~ 0.86. We now have two proposed temperatures for the sublithospheric flow; furthermore, these are formally distinct (one is exponential with depth, the other lin- ear), and therefore not in principle reconcil- able. This is a puzzle, and apparently the solution is the following. We do not require Bad to equal 8rhe exactly (O(s) difference is certainly toler- ated): what we formally require is that I•ad- •rhel << 1, and this can be satisfied provided the parameters D and B are appropriately prescribed. Specifically, we require formally that D << 1, so that Bad • ! - Dy + «D2y 2 + ..., and then also that D- B/(i+B) << 1, so that 1Bad - 8rhel T O[ID - B/(i+B) I, D2]. This requires (if Bad- 8rhel ~ O(½)) that D - •/(1+]•) ~ D 2 ~ 0(s). (12) It is evident that shallow mantle convection is reasonably consistent with (12), whereas whole mantle convection is definitely not. How can 'given' parameters be prescribed? Examining D and •, we see that each is proportional to depth d, and therefore (12) can always be satisfied provided the depth is sufficiently small. This implies, since the depth is an externally pre- scribed constant, that the depth of vigorous convection (limited by (12)) may be smaller than the layer depth. From previous discussion, we might expect the remainder of the layer to con- sist of more viscous, quasi-stagnant fluid. Such behavior is suggested by various numerical compu- tations (for example Torrance and Turcotte (1971)). The constraint (12) suggests that even if one attempted to achieve whole mantle convection, the natural convecting depth would be shallow, and the lower mantle would behave effectively stag- nantly, or convect separately. A more detailed scaling analysis (Fowler, 1981) suggests that the upper mantle convection will admit an alge- braic variation in viscosity (of perhaps 2 to 4 orders of magnitude), and this is in line with a numerical simulation of Parmentier and Turcotte (1978). The constraints (12) are replaced by D << 1, • - (i+•)D = m½ln(D/½), (13) where for a stress free base, m • 2, for a no slip base m m 5. It is perhaps revealing that with B ~ 0.5, D ~ 0.2, ½ ~ 1/30, we find m m 3.35 from (13), whereas for whole mantle con- vection with D ~ 0.86, B ~ 2, (13) would imply m < O. We should emphasize that it is the orders of magnitude which are important here, not the precise quantitites. The present discuss. ion is therefore fairly robust under variation of mater- ial properties such as activation energy, etc. Discussion What we have attempted to show is that purely mechanical processes will have the effect of pre- ferentially selecting a shallow depth of vigorous convection. Such a conclusion appears to receive observational support from the compressive nature of deep earthquakes (Isacks and Molnar 1971), and the fact that such earthquakes (identified with the integrity of slabs st such a depth) do not occur below 700 kilometre depth. In addition, a seismic discontinuity at ~ 670 km may be due to a phase change and a chemical discontinuity, either of which could act as a barrier to convection. Also, the different nature of ocean island ba- salts (due to hot spots) and mid-ocean ridge ba- salts circumstantially suggests a chemically layered mantle (Anderson 1979, 1980, 1981). The apparent coincidence between a chemical discon- tinuity at ~ 700 km acting as a barrier to con- vection, and the apparently mechanically pre- ferred depth is readily explained in terms of subducting slabs which are able to maintain the chemical layering at depth. For example, if a layer is too deep, slabs preferentially sink to the natural convective depth, thus providing a natural fluid dynamic mechanism whereby a chem- ically layered mantle could occur. References Anderson, D.L., Chemical stratification of the mantle, J. Geophy•. Res. 84, 6297-6298, 1979. Anderson, D.L., Early evolution of the mantle, Episodes No. 3, 3-7, 1980. Anderson, D.L., The chemical composition and evolution of the mantle, preprint, 1981. Cathles, L.M., III, The viscosity of the earth's mantle, Princeton University Press, Princeton, N.J., 1975. Elsasser, W.H., P. Olson, and B.D. Marsh, The depth of mantle convection, J. Geophys. Res. 84, 147-155, 1979. Fowler, A.C., Implications of scaling and non- dimensionalisation for the convection of the earth's mantle, preprint, 1981. Goetze, C., The mechanics of creep in olivine, Phil. Trans. Roy. Soc. A 288, 99-119, 1978. Isacks, B., and P. Molnar, Distribution of stresses in the descending lithosphere from a global survey of focal-mechanism solutions of mantle earthquakes, Rev. GeophYS. Space Phys. •, 103-174, 1971. O'Connell, R.J., and B.H. Hager, On the thermal state of the earth, in Physics of the Earth's Interior, 270-317, Soc. Italiana di Fisica, Bologna, Italy, 1980. Oxburgh, E.R., and D.L. Turcotte, Mechanisms of continental drift, Rep. Prog. Phys. 41, 1249-1312, 1978. Parmentier, E.M., and D.L. Turcotte, Two-dimen- sional mantle flow beneath a rigid accreting lithosphere, Phys. Earth Planet. Int. 17, 281-289, 1978. Peltier, W.R., Mantle convection and viscosity, in Physics of the Earth's Interior, 361-431, Soc. Italiana di Fisica, Bologna, Italy, 1980. Schubert, G., D.L. Turcotte, and E.R. Oxburgh, Stability of planetary interiors, Geophys. J. R. astr. Soc. 18, 441-460, 1969. Torrance, K.E., and D.L. Turcotte, Thermal con- vection with large viscosity variations, J. Fluid Mech., 4__7, 113-125, 1971. Tozer, D.C., The present thermal state of the terrestrial planets, Phys. Earth Planet. Int. •, 182-197, 1972. Tozer, D.C., The thermal state and evolution of Fowler: Depth of Convection in a Thermoviscous Fluid 819 the earth and terrestrial planets, Sci. Progr. 64, 1-28, 1977. Tozer, D.C., Terrestrial planet evolution and the observational consequences of their formation, in The Origin of the Solar System, ed. S.F. Dermott, Wiley, New York, N.Y., 433-462, 1978. Turcotte, D.L., Convection, Revs. Geophys. Space Phys. 17, 1090-1098, 1979. Turcotte, D.L., W.F. Haxby, and J.R. Ockendon, Lithospheric instabilities, in Island Arcs • Deep Sea Trenches• and Back Arc Basins, ed. M. Talwani and W.C. Pitman III, 63-69, A.G.U., Washington, D.C., 1977. Turcotte, D.L., D.C. McAdoo, and J.G. Caldwell, An elastic-perfectly plastic analysis of the bending of the lithosphere at a trench, Tectonophys. 47, 193-205, 1978. Turcotte, D.L., and E.R. Oxburgh, Finite ampli- tude convective cells and continental drift, J. Fluid Mech. 28, 29-42, 1967. Yuen, D.A., W.R. Peltier and G. Schubert, On the existence of a second scale of convection in the upper mantle, Geophys. J. R. Astro. Soc. 65, 171-190, 1981. (Received September 14, 1981; accepted May 14, 1982.) 

Mathematical Institute Eprints Archive

GEOPHYSICAL RESEARCH LETTERS, VOL. 9, NO. 8, PAGES 816-819, AUGUST 1982 
THE DEPTH OF CONVECTION IN A FLUID WITH STRONGLY TEMPERATURE AND 
PRESSURE DEPENDENT VISCOSITY 
A.C. Fowler 
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 
Abstract. We discuss the equations describing 
convective motion of a fluid with temperature 
and pressure dependent rheology. By combining 
observational constraints from plate tectonics 
with theoretical constraints deduced from the 
model equations and parameter values, we are led 
to the conclusion that shallow upper mantle con- 
vection is consistent with the equations of fluid 
dynamics. 
The Nature of Mantle Convection 
In studying models of the convective flow in 
the earth's mantle (for reviews, see Oxburgh and 
Turcotte (1978), Turcotte (1979)), geophysicists 
have been hampered by an unusual set of circum- 
stances: these are, that direct observational 
evidence is sketchy and requires interpretation; 
direct laboratory simulation appears insurmount- 
ably difficult (due to a lack of suitable mater- 
ials, and to the inability to scale all the para- 
meters properly); direct numerical computation of 
even a two-dimensional convective flow with a 
relevant rheology has not been obtained. These 
circumstances have produced a situation in which 
various fundamental properties of the convective 
flow are not well understood; for example, the 
depth of convection is still a contentious issue, 
with various authors arguing for whole-mantle or 
layered upper and lower mantle convection, 
respectively. Because theoretical and numerical 
studies of the convection of a strongly nonlin- 
early rheologic fluid are effectively lacking (in 
the context of the earth), there is a tendency to 
make deductions on the nature of the real flow 
from much simpler (often, for example, Newtonian) 
models in the hope that they are accurate enough 
(in some sense). 
Of the various complications associated with 
mantle convection, possibly the most obvious and 
perhaps the most significant is the dependence of 
viscosity on temperature and pressure. Commonly 
quoted activation energies for mantle-like mater- 
ials are E* = 122 kcal/mole (Goetze 1978), and 
with a basal temperature of (say) T b = 2000 K, 
one finds the dimensionless parameter which 
measures the strength of viscosity dependence on 
temperature, E*/RT b ~ 30. The pressure depend- 
ence is of a similar order of magnitude, and 
operates in the opposite direction. Even if the 
activation energy were much less, E*/RT b would 
still be large, and it is the fact that it occurs 
in an exponential dependence that is so crucial. 
Local variation of the temperature by a few hun- 
dred degrees Kelvin will alter the viscosity by 
orders of magnitude, and this can hardly but 
affect the flow. We suggest that the rheology 
Copyright 1982 by the American Geophysical Union. 
Paper number 2L0773. 
0094-8276/82/002L-077353.00 
will therefore control the flow, and thus that 
models which ignore the rheology must be care- 
fully examined to see if conclusions deduced from 
them will carry over. Particularly we will 
briefly propose an analogue of Turcotte and 
Oxburgh's (1967) boundary layer theory, to see if 
the conclusions of Elsasser, Olson and Marsh 
(1979) will carry across: we find they do not. 
(See also O'Connell and Hager 1980, for a differ- 
ent viewpoint.) The idea of rheology controlling 
the internal temperature has been developed by 
Tozer (1972, 1977, 1978), but we emphasize that 
his is essentially a global idea, whereas we 
shall be concerned here with a local dynamic 
control. 
The Effect of a Strongly Temperature and 
Pressure Dependent Rheology 
Let us consider the convection of a fluid 
whose viscosity is given by 
n = n0exp[ (E* + pV*)/RT], (•) 
where B0 is a reference viscosity, E* is the 
activation energy, p is the pressure, V* is the 
activation volume, R is the gas constant, and T 
is the absolute temperature. To fix ideas 
(only), we consider prescribed temperatures T b on 
the base, and T O (substantially less then T b) on 
the surface. E*/RT b is a dimensionless measure 
of how much the exponent varies through the flow. 
If we denote 
c = RTb/E* << 1, (2) 
then the viscosity contrast between top and bot- 
tom is exp[0(1/c)]. If 1/e • 30, then exp(1/e) • 
1013 , which is beyond the range of any laboratory 
or numerical experiment so far conducted. Verti- 
cal temperature profiles for which T/(E*+ pV*) are 
constant are isoviscous: for a single phase 
mantle in which pressure is essentially litho- 
static, such isoviscous temperatures are linear 
functions of depth. 
Now suppose the maximum value of T/(E*+ pV*) 
within a convection cell gives a corresponding 
minimum viscosity •m- If the temperature else- 
where is much smaller than the corresponding iso- 
viscous profile (with • = •m), then it follows 
from the smallness of e that the viscosity in 
such regions will be exponentially large in com- 
parison with •m. From the point of view of the 
minimum viscosity material, the high viscosity 
material will appear effectively rigid. Note: we 
do not mean rigid in the sense of behaving elas- 
tically rather than viscously, but dynamically 
rigid in the sense that such high viscosity 
material will appear rigid on time scales rele- 
vant to the convection. Since an applied stress 
induces the largest velocity in the least viscous 
816 
Fowler: Depth of Convection in a Thermoviscous Fluid 817 
fluid, we might expect the convective time scale 
to be dictated by such fluid. 
Effectively rigid material will then not sig- 
nificantly deform on time scales of interest, and 
it seems unlikely that it can participate sub- 
stantially in the circulatory motion: a little 
thought suggests that one plausible state will be 
that 'cold', 'rigid' material will exist in one 
or more quasi-stagnant layers throughout the 
cell, their location depending on the relation of 
the isoviscous temperature profiles to the in- 
terior (adiabatic) temperature. Particularly, 
vigorous convection (high Rayleigh number) will 
induce a top cold thermal boundary layer, which 
will be 'quasi-rigid' and preferentially stag- 
nant; this rigid boundary layer may be defined as 
the lithosphere. The location of other such 
rigid regions depends on the precise rheology, 
and is discussed further below. 
In the real earth, of course, such stagnant 
plates do not exist, rather the tectonic plates 
are active, moving relative to each other at 
speeds of •< 10 cm y-1. The reason for such 
activity is the nature of the motion at oceanic 
trenches, where the surface plates are subducted 
downwards into the upper mantle. The discussion 
above, however, raises the question of how such 
cold, rigid slabs are able to be subducted a__t 
all: various studies (Turcotte et al 1977, 
Turcotte et al 1978) have shown convincingly that 
subduction may be understood on the basis of non- 
hydrodynamic phenomena such as partial melting 
and elastic yielding, but the problem of initia- 
tion remains puzzling. 
For a thermoviscous fluid, the growing insta- 
bility of the cold layer as it thickens is over- 
compensated by the increasing effective viscos- 
ity, and purely buoyant effects will not subduct 
the lithosphere (Yuen, Peltier and Schubert 1981) 
y=l is the top surface, and y=0 is the base. We 
define the basal viscosity •m by 
•m = •0exp [ (E* + 00gdV*)/RT b ], (5) 
where E* is activation energy, V* is activation 
volume, R is the gas constant; it then follows 
from (1) that 
n = nmeXp[{(l+•) (1-0)-•y}/e0], (6) 
where 
s = RTb/E*, • = 00gdV*/E*. (7) 
With values O0 ~ 3 5 gm cm -3 ~ d ~ , g 10ms -2 ß , 
700 km, V* ~ 11 cm 3 mole -1 , E* ~ 122 kcal mole -1 
T b ~ 2000 K, R ~ 8 3 J mole -1 K -1 ß , we have 
1/30, • ~ 1/2. (8) 
These parameters are for shallow mantle convec- 
tion, but for illustration only, to show that 
• << 1, • ~ 1. 
In terms of 0, we have proposed that a basal 
boundary layer will have a temperature jump 
A8 ~ O(•). (6) then implies that • varies by 
<• O(1) within the layer, and so the isoviscous 
temperature 8rhe which one would expect in the 
cell interior is given by 
0 = Srh e = 1- •y/(l+•) + O(e). (9) 
We reiterate this point: if (1+•)(1-8) - •y 
becomes positive and 0(1) in the cell interior, 
i.e. 0rhe- 8 ~ O(1), then •/•m ~ exp[O(1/e)], 
and such material is effectively rigid (corres- 
ponding considerations apply if 8 - 8rhe ~ O(1)). 
- non-hydrodynamic processes must be relevant. Therefore we expect that convection in a fluid 
On the other hand, the decrease in viscosity in a with such a rheology will constrain the interior 
basal thermal layer will greatly enhance insta- 
bi!ity. Calculations of Schubert et al (1969) 
can be adapted (Fowler, 1981) to show that a 
temperature difference across the layer greater 
than O(eT b) will be catastrophically unstable, 
and hence in a steady state such a difference 
will not occur. In effect, the basal thermal 
boundary layer vanishes, and the temperature and 
viscosity are (approximately) continuous at the 
base. This is really just another instance of 
the isoviscous nature of the flow. 
Constraints on the Viscosity 
temperature profile to be approximately isovis- 
cous. This is in line with models of the earth's 
viscosity which are consistent with isostatic 
rebound data (Cathles 1975, ?eltier 1980). How- 
ever, the 'isoviscous' nature of the flow is 
paradoxically due to the strongly temperature- 
dependent rheology, and it remains to be seen 
what other effects the theology will have. 
The above discussion raises an immediate prob- 
lem. If the rheology chooses the temperature, 
what does the temperature equation do? There 
does not seem to be any possible way in which it 
does not also choose the temperature; particu- 
larly, for a vigorously convecting fluid at high 
Consider the convection of a fluid in a layer Rayleigh number, the appropriately scaled temper- 
of depth d. We nondimensionalise the temperature ature equation will be of singular perturbation 
T and depth y by putting type, and in the absence of viscous or radiogenic 
T = Tb0, y = dy*, (3) 
where T b is the basal temperature. If O 0 is the 
ambient density (with the Boussinesq approxima- 
tion), g is the constant acceleration due to 
gravity, then the (lithostatic) pressure is given may then write 
by 
0 • Oad = exp[-Dy], 
p = 00gd(1-y) , (4) 
where 
source terms, we can expect that thermal boundary 
layers will occur, but that elsewhere (i.e., in 
the interior) the entropy will be nearly cons- 
tant, which in the present context implies an 
internal adiabatic temperature. In view of the 
small temperature jump across the basal layer, we 
(10) 
where henceforward we drop the asterisk on y*, D : •gd/c (11) 
P 
818 Fowler: Depth of Convection in a Thermoviscous Fluid 
is the dissipation number, e is the thermal 
expansion coefficient, Cp is the specific heat. 
For shallow mantle convection, D ~ 0.2; for 
whole mantle convection, D ~ 0.86. We now have 
two proposed temperatures for the sublithospheric 
flow; furthermore, these are formally distinct 
(one is exponential with depth, the other lin- 
ear), and therefore not in principle reconcil- 
able. 
This is a puzzle, and apparently the solution 
is the following. We do not require Bad to equal 
8rhe exactly (O(s) difference is certainly toler- 
ated): what we formally require is that 
I•ad- •rhel << 1, and this can be satisfied 
provided the parameters D and B are appropriately 
prescribed. Specifically, we require formally 
that D << 1, so that Bad • ! - Dy + «D2y 2 + ..., 
and then also that D- B/(i+B) << 1, so that 
1Bad - 8rhel T O[ID - B/(i+B) I, D2]. This requires (if Bad- 8rhel ~ O(½)) that 
D - •/(1+]•) ~ D 2 ~ 0(s). (12) 
It is evident that shallow mantle convection is 
reasonably consistent with (12), whereas whole 
mantle convection is definitely not. How can 
'given' parameters be prescribed? Examining D 
and •, we see that each is proportional to depth 
d, and therefore (12) can always be satisfied 
provided the depth is sufficiently small. This 
implies, since the depth is an externally pre- 
scribed constant, that the depth of vigorous 
convection (limited by (12)) may be smaller than 
the layer depth. From previous discussion, we 
might expect the remainder of the layer to con- 
sist of more viscous, quasi-stagnant fluid. Such 
behavior is suggested by various numerical compu- 
tations (for example Torrance and Turcotte 
(1971)). 
The constraint (12) suggests that even if one 
attempted to achieve whole mantle convection, the 
natural convecting depth would be shallow, and 
the lower mantle would behave effectively stag- 
nantly, or convect separately. A more detailed 
scaling analysis (Fowler, 1981) suggests that 
the upper mantle convection will admit an alge- 
braic variation in viscosity (of perhaps 2 to 4 
orders of magnitude), and this is in line with a 
numerical simulation of Parmentier and Turcotte 
(1978). The constraints (12) are replaced by 
D << 1, • - (i+•)D = m½ln(D/½), (13) 
where for a stress free base, m • 2, for a no 
slip base m m 5. It is perhaps revealing that 
with B ~ 0.5, D ~ 0.2, ½ ~ 1/30, we find 
m m 3.35 from (13), whereas for whole mantle con- 
vection with D ~ 0.86, B ~ 2, (13) would imply 
m < O. We should emphasize that it is the orders 
of magnitude which are important here, not the 
precise quantitites. The present discuss. ion is 
therefore fairly robust under variation of mater- 
ial properties such as activation energy, etc. 
Discussion 
What we have attempted to show is that purely 
mechanical processes will have the effect of pre- 
ferentially selecting a shallow depth of vigorous 
convection. Such a conclusion appears to receive 
observational support from the compressive nature 
of deep earthquakes (Isacks and Molnar 1971), and 
the fact that such earthquakes (identified with 
the integrity of slabs st such a depth) do not 
occur below 700 kilometre depth. In addition, a 
seismic discontinuity at ~ 670 km may be due to a 
phase change and a chemical discontinuity, either 
of which could act as a barrier to convection. 
Also, the different nature of ocean island ba- 
salts (due to hot spots) and mid-ocean ridge ba- 
salts circumstantially suggests a chemically 
layered mantle (Anderson 1979, 1980, 1981). The 
apparent coincidence between a chemical discon- 
tinuity at ~ 700 km acting as a barrier to con- 
vection, and the apparently mechanically pre- 
ferred depth is readily explained in terms of 
subducting slabs which are able to maintain the 
chemical layering at depth. For example, if a 
layer is too deep, slabs preferentially sink to 
the natural convective depth, thus providing a 
natural fluid dynamic mechanism whereby a chem- 
ically layered mantle could occur. 
References 
Anderson, D.L., Chemical stratification of the 
mantle, J. Geophy•. Res. 84, 6297-6298, 1979. 
Anderson, D.L., Early evolution of the mantle, 
Episodes No. 3, 3-7, 1980. 
Anderson, D.L., The chemical composition and 
evolution of the mantle, preprint, 1981. 
Cathles, L.M., III, The viscosity of the earth's 
mantle, Princeton University Press, Princeton, 
N.J., 1975. 
Elsasser, W.H., P. Olson, and B.D. Marsh, The 
depth of mantle convection, J. Geophys. Res. 
84, 147-155, 1979. 
Fowler, A.C., Implications of scaling and non- 
dimensionalisation for the convection of the 
earth's mantle, preprint, 1981. 
Goetze, C., The mechanics of creep in olivine, 
Phil. Trans. Roy. Soc. A 288, 99-119, 1978. 
Isacks, B., and P. Molnar, Distribution of 
stresses in the descending lithosphere from a 
global survey of focal-mechanism solutions of 
mantle earthquakes, Rev. GeophYS. Space Phys. 
•, 103-174, 1971. 
O'Connell, R.J., and B.H. Hager, On the thermal 
state of the earth, in Physics of the Earth's 
Interior, 270-317, Soc. Italiana di Fisica, 
Bologna, Italy, 1980. 
Oxburgh, E.R., and D.L. Turcotte, Mechanisms of 
continental drift, Rep. Prog. Phys. 41, 
1249-1312, 1978. 
Parmentier, E.M., and D.L. Turcotte, Two-dimen- 
sional mantle flow beneath a rigid accreting 
lithosphere, Phys. Earth Planet. Int. 17, 
281-289, 1978. 
Peltier, W.R., Mantle convection and viscosity, 
in Physics of the Earth's Interior, 361-431, 
Soc. Italiana di Fisica, Bologna, Italy, 1980. 
Schubert, G., D.L. Turcotte, and E.R. Oxburgh, 
Stability of planetary interiors, Geophys. J. 
R. astr. Soc. 18, 441-460, 1969. 
Torrance, K.E., and D.L. Turcotte, Thermal con- 
vection with large viscosity variations, J. 
Fluid Mech., 4__7, 113-125, 1971. 
Tozer, D.C., The present thermal state of the 
terrestrial planets, Phys. Earth Planet. Int. 
•, 182-197, 1972. 
Tozer, D.C., The thermal state and evolution of 
Fowler: Depth of Convection in a Thermoviscous Fluid 819 
the earth and terrestrial planets, Sci. Progr. 
64, 1-28, 1977. 
Tozer, D.C., Terrestrial planet evolution and the 
observational consequences of their formation, 
in The Origin of the Solar System, ed. S.F. 
Dermott, Wiley, New York, N.Y., 433-462, 1978. 
Turcotte, D.L., Convection, Revs. Geophys. Space 
Phys. 17, 1090-1098, 1979. 
Turcotte, D.L., W.F. Haxby, and J.R. Ockendon, 
Lithospheric instabilities, in Island Arcs • 
Deep Sea Trenches• and Back Arc Basins, ed. 
M. Talwani and W.C. Pitman III, 63-69, A.G.U., 
Washington, D.C., 1977. 
Turcotte, D.L., D.C. McAdoo, and J.G. Caldwell, 
An elastic-perfectly plastic analysis of the 
bending of the lithosphere at a trench, 
Tectonophys. 47, 193-205, 1978. 
Turcotte, D.L., and E.R. Oxburgh, Finite ampli- 
tude convective cells and continental drift, 
J. Fluid Mech. 28, 29-42, 1967. 
Yuen, D.A., W.R. Peltier and G. Schubert, On the 
existence of a second scale of convection in 
the upper mantle, Geophys. J. R. Astro. Soc. 
65, 171-190, 1981. 
(Received September 14, 1981; 
accepted May 14, 1982.) 


An elastic-perfectly plastic analysis of the bending of the lithosphere at a trench,

Chemical stratification of the mantle,

Distribution of stresses in the descending lithosphere from a global survey of focal-mechanism solutions of mantle earthquakes,

Early evolution of the mantle,

Finite amplitude convective cells and continental drift,

III, The viscosity of the earth's mantle,

Implications of scaling and nondimensionalisation for the convection of the earth's mantle, preprint,

Mantle convection and viscosity,

Mechanisms of continental drift,

On the existence of a second scale of convection in the upper mantle,

On the thermal state of the earth,

Terrestrial planet evolution and the observational consequences of their formation,

The chemical composition and evolution of the mantle, preprint,

The depth of mantle convection,

The mechanics of creep in olivine,

The present thermal state of the terrestrial planets,

The thermal state and evolution of Fowler: Depth of Convection in a Thermoviscous Fluid 819 the earth and terrestrial planets,

Thermal convection with large viscosity variations,

Two-dimensional mantle flow beneath a rigid accreting lithosphere,

https://ora.ox.ac.uk/objects/uuid:71e81fc7-fd2e-41ff-b0d6-cbc9732887da/files/m43f2656512fe6891a6b35e4f267d517d

The depth of convection in a fluid with strongly temperature and pressure dependent viscosity

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