Geophys. J . R. astr. SOC. (1987)89,549-577
Analysis of the cooling of a variable-viscosityfluid with
applications to the Earth
w . Roger Buck Lamont-Doherty Geological Observatory of Columbia
University, Palisades, New York 10964, USA
Accepted 1986September 24. Received 1986July 7; in original form 1985June 22
Summary. Analysis shows that the convective cooling of a fluid with a
temperature-dependent viscosity should exhibit simple behaviour which is
consistent with geophysical data on the cooling of the oceanic lithosphere
and asthenosphere. The cooling rate of convecting fluid is predicted to vary
approximately linearly with (time)-'12. This relationship has also been demonstrated for numerical solutions of the full governing equations of convection.
It is found that a similarity solution can describe the evolution of temperature
within the stagnant high-viscosity lid which forms over an actively convecting
region. The rate of thickening of the lid (or lithosphere) is directly proportional to a single parameter, A; and the overall cooling rate is also a function
of A. Expressions are derived which relate changes in the parameter (A) to the
initial average viscosity (or temperature of the fluid) and to the temperature
and pressure dependence of viscosity. With assumptions about how the
cooling of variable-viscosity fluids can be used to describe the cooling of the
oceanic lithosphere and asthenosphere, values of sea-floor depth and the
isostatic geoid height can be predicted as a function of time. Sea-floor depth
is predicted to increase linearly with
and geoid height to vary nearly
linearly with time in agreement with observations. The rate of change of the
depth or geoid height is a strong function of the initial temperature of the
convecting region. The mantle is taken to have physical properties appropriate
for olivine. A change in the mantle temperature under the ridge of 150K
causes only about a 20 per cent change in the rate of subsidence but causes
nearly a 100 per cent change in the rate of change of the geoid height with
time. The observed variations in these rates over different sections of oceanic
ridges are of about this magnitude. Simple conductive cooling would require
more than l000K variation in mantle temperature to account for the geoid
height observations. This analysis can also be used to estimate the cooling rate
of magma sills.
Key words: convection, viscosity, lithosphere, mid-ocean ridge, mantle
heterogeneity, geoid-age relation
550
W. R. Buck
Introduction
The problem of convection driven by the cooling of a fluid from above has many applications in physics and engineering (eg. Curlet 1976; Ingersoll 1966). Geophysicists are particular interested in the case when the viscosity of the fluid is a strong function of temperature. The Earth’s mantle is thought to be a temperaturedependent viscous fluid (Weertman
& Weertman 1975). The temperature dependence of the viscosity of the mantle is the
reason why the lithospheric plates behave rigidly and the hotter asthenosphere below can
flow to allow the plates to move. In a numerical study of the cooling of a temperaturedependent fluid, Buck & Parmentier (1986) find a very simple relationship between the rate
of convective cooling and time. Namely, the heat flux varies approximately linearly with
(time)-”?. This simple behaviour suggests that it may be possible to describe analytically the
rate of cooling of the fluid and the temperature profile and thickness of a stagnant highviscosity lid which forms at the top of the cooling fluid.
The cooling of the lithosphere has usually been treated as a purely conductive phenomenon (Turcotte & Oxburgh 1967), at least up to cooling ages of about 80Myr (Parsons &
Sclater 1977). Furthermore, it has been assumed that if a small scale of convection did
develop beneath cooling lithosphere then there would be a sudden and large change in the
rate of cooling and thickening of the lithosphere (Parsons & McKenzie 1978; Houseman &
McKenzie 1982). Recent work on the interaction of convection in the asthenosphere and the
lithosphere has indicated that such convection is likely to begin only a few million years
after the lithosphere begins to cool and that there is not an abrupt change in the rate of
cooling of the lithosphere (Buck 1983, 1984; Fleitout & Yuen 1984; Buck & Parmentier
1986). There are also data whch may require that convection begins under very young lithosphere. Haxby & Weissel (1986) have suggested that short wavelength (- 200km) gravity
undulations derived from SEASAT altimetry data may be due to small-scale convection
beneath oceanic lithosphere. They observe these features over ocean floor formed as recently
as 6 Myr. Numerical experiments on the effects of small-scale convection described here, and
in greater detail in Buck & Parmentier (1986), have shown that convection can produce the
observed magnitude of the SEASAT gravity signals and not lead to unreasonable rates of
subsidence of the ocean floor (Buck 1985). The subsidence rate is changed when convection
is active beneath the lithosphere, but subsidence and the thickness of the lithosphere vary
linearly with (time)”2 as they do for purely conductive cooling of a half-space. Likewise the
isostatic geoid height varies linearly with time as it does for conductive cooling. The slope of
the geoid height-time relation is strongly dependent on the viscosity parameters.
The aim of this work is to derive an approximate analytic relationship between the rate of
cooling and the physical properties which control the strength of convection in a variableviscosity fluid cooled from above. This relationship is then used to predict changes in geophysical observables due to variations in physical properties of the convecting fluid. This
approach is fusdamentally different from the standard parameterized convection treatment
which has been applied to the thermal evolution of whole planets (e.g. Sharpe & Peltier,
1978; Schubert, Stevenson & Cassen 1980; Turcotte, Cook & Willeman 1979) where the
lithospheric thickness is taken to be in equilibrium with the heat flux advected to its base.
This is not the case in the transient cooling problem described here.
The analysis is based on the assumption of a strongly temperaturedependent viscosity
and of a Nusselt number (Nu) which depends on the Rayleigh number (Ra) to a fvred power.
Care is given to the definitions of the Rayleigh number and the Nusselt number since this
problem is a transient (non-steady state) one and viscosity variations limit convection to
only part of a layer of fluid. Detailed analysis of 2-D numerical calculations (Buck &
Cooling of a variable-viscosity fluid
551
Parmentier 1986) shows that the Ra-Nu relation holds for all model parameters considered.
Next, it is shown that, to leading order, the convective heat flux through a layer of variableviscosity fluid cooled from above should have the same functional dependence on time as
the heat flux in simpler cooling problems which can be described using a similarity solution.
The numerical experiments are shown to exhibit this behaviour. Therefore, the formalism of
the simple moving boundary problem (e.g. where a phase change occurs when the temperature drops below a given value) can be applied to this problem. In this case, the interface
between the conductive lid and the convecting region, including the flow boundary layer, is
analogous t o the moving boundary. Parker & Oldenburg (1973) have also treated the cooling
of the lithosphere as a moving boundary problem, but the heat supplied to their boundary
was due to the latent heat of solidification not a convective heat flux, as in the present case.
The rate of thickening of the conductive lid and the average temperature of the lid are both
decreased by the convective heat flux. Both effects can be described by a single parameter,
A, which is related to the viscosity parameters of the system assuming the derived relations.
The numerical experiments are shown to agree with the estimated values of h for a given set
of model parameters. Since it is difficult to predict the convective heat flux at a reference set
of conditions, the expressions that are derived constitute a parameterization of the effect of
changes in viscosity or fluid temperature on the rate of cooling of the fluid.
With certain assumptions, changes in h are related to data that depend on the thickness
and thermal structure of the oceanic lithosphere and asthenosphere: the rate of subsidence,
the heat flux, and the rate of change of the geoid height with age. The most important
geophysical application of this parameterization should be in estimating the effect of lateral
heterogeneity of asthenospheric temperatures along the ridge crest on the rate of subsidence
of the ocean floor and on the rate of change of geoid height with age.
Numerical calculation results
The rate of cooling of a fluid with viscosity which depends on temperature and pressure will
be affected by the vigour of the convection in that fluid. When a box of temperaturedependent fluid is cooled from above, a region may develop at the top of that box where
viscosities are so h g h as to preclude any significant convective flow there; this has been shown
in the laboratory experiments of Richter, Nataf & Daly (1983) and the theoretical work of
Jaupart (1981). Schematic boundaries separating the regions where advective heat flow
dominates over conduction (the convecting region), the region where it is insignificant (the
conductive lid), and the boundary layer are shown in Fig. 1.
For boxes of various widths (w> and depths (D),the 2-D Navier-Stokes equations of
energy, mass and momentum conservation in a variable viscosity, infinite Prandtl-number
fluid (Turcotte, Torrance & Hsui 1973) are studied using standard finite difference methods
(Parmentier 1975). The flow boundary conditions are free slip on the bottom and sides of
the box and fixed at the top. The viscosity (p) is taken to depend on absolute temperature
( T ) and pressure (P) through a relation appropriate for olivine (Weertman & Weertman
1975) given by:
where E is the activation energy; V is the activation volume, R is the universal gas constant,
and A . controls the average viscosity. Values of E and V are in the range of those experimentally determined for creep in olivine (Goetze 1978; Kohlstedt, Nicholas & Hornack
1980; Reddy et al. 1980; Sammis, Smith & Schubert 1981). A . is based on geophysical
estimates of shallow mantle viscosity which are discussed in Buck & Parmentier (1986).
552
W.R . Buck
Temperature
Tcr
Flow Boundary Layer
Depth
Z
-
"-rI
L
1
Figure 1. Schematic temperature profile and its relation to the regions of cooling fluid. In the conductive
lid heat is only transported by diffusion. The bottom of the conductive lid is defined to be a point of
constant temperature ( T P )at depth ZL which moves down with time. The temperature of the convecting
region (T,) also decreases with time from an initial value of T,. The surface temperature is T o . Zb is the
bottom of the flow boundary layer.
The temperature at the top of the box is fixed to be 273 K (or 0°C). Ideally, the interior
of the box would initially be set at a constant temperature, but this leads to numerical difficulties since temperature gradients are then infinite at time zero. Therefore, the initial temperature profile for all the numerical calculations is that resulting from 5 Myr of conductive
cooling of a half space initially at temperature T,. Small temperature perturbations (< 1K)
are given to the finite difference grid points to allow convection instabilities to develop.
Table 1 gives the physical parameters which are common to all the cases for the fluid in the
Table 1. Parameters set for numerical models.
Symbol
K
Q
g
P
K
CP
R
E
V
Cls
Name
Diffusivity
Thermal expansion
Acceleration of gravity
Mantle density
Conductivity
Specific heat
Gas constant
Activation energy
Activation volume
Reference viscosity at depth 150 km
Value
3.OXlO'
9.8
3500
3.2
900
8.31
Varied
Varied
Varied
Units
mZs-'
K-'
m sT2
kg
J (m-s-K)-'
J (kg - K)-'
J (mol K)-'
kcal mol"
cm3mol-'
Pa SK'
Cooling of a variable-viscosity fluid
553
box. Table 2 shows defined parameters which describe the calculation results, and Table 3
gives the parameters which are varied from case to case. Instead of tabulating the values of
A . for the cases, the value of the viscosity (p,) at a standard temperature and pressure
(1 573 K and a pressure corresponding to a depth of 150 km) is given.
Typically, in a model calculation, convection begins immediately under a stagnant highviscosity lid. The flow pattern for one calculation is shown in relation to the horizontally
averaged temperature, viscosity and advective heat flux in Fig. 2 .
The boundary layer which separates the convecting and conducting regions can be
defined in terms of the horizontal average of the vertical advective heat flux [qc(z)]
where w is the vertical velocity at a point and W is the width of the box. Below the region
where the convective heat flux is negligible, 4c varies nearly linearly with depth. Fig. 2 shows
the relation between q C ( z ) and other quantities at one time in one of the numerical cooling
calculations. This shows that when the viscosity is lowest, the stream function values are
high and the advective heat flux is significant. A straight line is fit through two points of this
curve at 0.2qcmax and 0.8qCmaxas illustrated in Fig. 3 where 4cmaxis the maximum
advective heat flux for all depths in the box. The depth at which the value of this linear
function is zero is considered to be the top of the boundary layer (23.The bottom of the
boundary layer (zb) is defined as the point where this line intersects qcmax(also in Fig. 3).
The region below z b is taken to be the convecting region and the average temperature
defined there as Tcr.
The depth of the bottom of the conductive lid (zL) is defined as the depth where the
horizontally averaged temperature is 90 per cent of the temperature in the convecting region
(Tcr- To). Although the depth of the bottom of the conducting region (zL) is not exactly
Table 2. Defined parameters.
Length scale for the flow weighted by advective heat flux
Dissipation weighted average viscosity
Nusselt number
Rayleigh number
Vertical component of velocity
Horizontally averaged advective heat flux
Initial temperature inside box
Average temperature of the convecting region
Temperature at bottom of conductive lid
Average nondimensional temperature of conductive lid
Temperature drop across convecting region
Depth of top of thermal boundary layer
Depth of bottom of thermal boundary layer
Depth of the base of conductive lid (or lithosphere)
Ratio of ito p (zb)
Slope of Nu versus temperature
Factor relating Nu and dT,/dt
Proportionality between In (Nu) and time
Parameter which defines the rate of movement of the base of the lithosphere; depends on
convective vigour
Total heat flux out of the convecting region
Subsidence
Isostatic geoid anomaly
12
14
15
17
18
19
20
22
23
Case no.
10"Pa s-')
5 .O
1.o
1.o
1.o
1.o
1.o
1.o
5 .o
1.o
(X
kS
110
110
110
102
80
110
110
110
110
(m)
kcal
E
7.5
75
7.5
0 .o
7.5
I .5
7.5
7.5
7.5
V
180
150
190
190
160
170
180
150
120
I
0.96
1.17
1.07
0.87
0.93
0.94
0.97
0.96
1.02
h
19.2
11.6
14.0
21.9
20.3
20.3
19.2
20.3
16.4
t o= 64
(5)
6 .O
4.4
4.2
6 .I
5.2
6.8
6 .O
7.6
7.8
4.6
5.3
5 .O
0 .o
4.4
45
4.6
4.6
4.8
170
88
110
160
150
180
170
180*
200
174
92
105
131
165
200
174
218
181
2
4
-5
-17
9
10
2
18
10
Percentage
difference
Table 3. The f i s t five columns give the model case numbers and the parameters which define them. The reference viscosity L ( ~is the viscosity at the initial temperature, T,, and at a pressure corresponding to 150 krn depth. It is determined by parameter A, in (1). Next, the parameter h defined by (22) for the model case.
qcmax is the value of the maximum horizontally averaged heat flux at 64 Myr model time. The values of c,, c, and c, are defined in the text and are calculated
using the model parameters given in the table. The combinations of these parameters are those that go into equation (25) and are used to get the predicted value of
d (qc max)/d ( t - ' / z )The
.
graphical value of d (qc max)/d ( t P Z )is estimated from Fig. 11 at time = 70 Myr. Case 22* was not shown in Fig. 11, but was determined
the same way.
$
R
-$I
.f
P
wl
vl
Cooling of a variable-viscosityfluid
555
i
(Pa-s)
AVERAGE
TEMP
1
I
(OK)
HEAT
FLUX
STREAM FUNCTION
-
400.
0.
WIDTH (km)
' DIFF
ADV
273.
400.
1573
(rnW/rn'l
17.
21.
-10.
40.
-100
100.
Figure 2. Equally spaced contours of the stream function between maximum and minimum values plus
four profiles of horizontally averaged quantities for a model time 10Myr after the start of numerical
cooling calculation 15. The parameters which characterize the model are given in Table 3. The contours
show that the flow does not penetrate into the region at the top of the box where the temperatures are
low and the viscosities are high. The profile of advective heat flux [nondimensionalized as given by (211
also indicates that the flow has not penetrated to the region of high viscosity at the top of the box nor
into the region at the bottom where the pressure dependence of viscosity has led to higher viscosities. The
difference between the temperature of the model case and the temperature resulting from conductive
cooling of a halfspace (COND DIFF) shows that convection has cooled a broad region while temperatures
at shallow depth are relatively higher.
0.
Qc max
Convective Heat Flux
Figure 3. Illustration of the method of estimation of the boundary layer thickness ( a Z ) and the position
of the top (zt) and bottom (zb) of the boundary layer from the variation of the convective heat flux (qc).
Since the q c is nearly a linear function of depth in the range where it changes most rapidly a straight line
can be fit through that region. The two points which are somewhat arbitrarily chosen to define that line
are the points where q c equals 20 and 80 per cent of qcmax. Varying those arbitrary values does no1
greatly alter the resulting estimate of the boundary layer position, but a consistent way of doing the
estimate must be chosen. The depth at which this line has a value of zero for q c is defined as zt and where
it equals qmax is the bottom (Zb).
556
W. R. Buck
equal to the top of the boundary layer (zt), this definition allows easy comparison of the lid
temperature structure between different cases. The non-dimensional average temperature in
the lid (TL) is defined as
Dividing the average lid temperature by 0.9 (Tcr - T o )allows T L to be used as a measure of
the shape of the average temperature profile in the lid. A value of TL of 0.5 indicates a linear
temperature profile and a value of 0.6 corresponds to the error function profile which results
from conductive cooling of a half-space. Fig. 4 shows that for the numerical cases TL
decreases from an initial value of 0.6 (due to initial conditions) to a lower value which is
relatively constant in time.
(Time) (m.y.), D=400 km
20
40
60
80
100
120
.
I
I
I
I
I
I
5
10
15
20
25
30
(Time) (m.y.), D=200 km
Figure 4. Values of the average lid temperature ( T L )defined by equation (3) are plotted against time for
the cases indicated. Table 3 gives the parameters used in each case. A value of nondimensionalized temperature of 0.60 corresponds t o the conductive solution and a value of 0.50 indicates a linear temperature
profile. Note that after an initial period of adjustment the values of T L are nearly constant. To dimensionalize the time two length scales for the depth of the box are used: 400 and 200 km. The larger depth
gives the larger times.
Cooling of a variable-viscosityfluid
557
Another feature of the numerical calculations which shows a regular behaviour in time is
zL, the depth of the base of the conductive lid. When plotted versus (time)’l2 it is nearly
linear (Fig. 5 ) , but the plots have slopes which are less than that predicted for conductive
cooling of a half-space. It is the constancy of TL in time and the linear trend versus (time)”2
for lid thickness for the numerical calculations which have induced the following analysis.
Two standard parameters are calculated through time for the convecting regions. They are
the Rayleigh number (Ra) and the Nusselt number (Nu). The physically meaningful definition of the Rayleigh number for this problem is
pgol A T1
Ra =
(4)
K P
where p (density), g (acceleration of gravity), and (Y (coefficient of thermal expansion) are,
the same for all calculations (Table 1). Because this problem is transient and viscosities vary
across the convecting region, the values of AT (temperature drop), I (length scale), and p
(average viscosity) must be defined in a self-consistent and physically reasonable way. These
values change with time in the calculations. To define the temperature difference for the
/
t/
0
ZLS x
/
2 z t
w
1.
0
( Ti
2.
3.
4.
5.
(m.yV2, D = 200 km.
Figure 5. The nondimensional depth to the bottom of the conductive lid, ZL, calculated for case 15
plotted versus (time)”’. Depth is nondimensionalized by dividing by the depth of the box, D.
558
W.R. Buck
convecting region (AT), the horizontally averaged temperature [Th(z)] is used. The temperature difference between the top and bottom of the boundary layer AT = Th(zb) - Th(z&
The top and bottom of the boundary layer are defined in terms of the vertical advective heat
flux, as illustrated in Fig. 3. The vertical advective heat flux is also used to define the
thickness of the region over which the flow is vigorous as
The viscosity to be used in the calculation of the Rayleigh number is an average weighted by
the second invariant of the strain rate tensor ( E ) , suggested by Parmentier (1978),
where A is the area of the box.
Finally, the Nusselt number is defied as the maximum horizontally averaged vertical
heat flux (qcma.J divided by the steady state conductive heat flux over the convecting region
qc max 1
NU=-.
KAT
(7)
This is a slight variation on the normal definition of the Nusselt number, which is the ratio
of the total heat flux in the presence of steady-state convection to the heat flux across the
same region when there is only conductive transport of heat (McKenzie, Roberts & Weiss
1974). The convection here is quasi steady state, that is, the heat flux is changing slowly
with time, but at a given time the convection is in equilibrium with the heat flux it is transporting. The use of the maximum convective heat flux (qcmax),instead of the total heat
flux, in the definition of the Nusselt number is justified because at the level where the convective heat flux is a maximum the conductive heat transport is negligible.
Theory predicts that for steady-state convection the relation between Ra and Nu should
be given by
Nu = a Rab,
(8)
where b is a constant between 1/5 and 1/3 (Roberts 1979) for convective flow between
fixed boundaries. Laboratory experiments by Booker (1976) and Richter et al. (1983) have
shown that this relation holds even for fluids which have a viscosity which strongly depends
on temperature for Rayleigh numbers up to lo6. Christensen (1984) has called this relation
into question for variable viscosity convection, but his numerical calculations assume
different boundary conditions (i.e. he considers the top boundary to be free to slip) and the
viscosity he used to calculate his Rayleigh number is different from that used here.
The definition of the terms given here that go into the Raleigh number do not scale
simply with the temperature and pressure dependence of viscosity. However, the change in
Nu for a given change in the average viscosity or the temperature of the fluid can be predicted through the Ra-Nu relation, once the value of the Nusselt number at a reference
temperature is prescribed. Thus, this is a parameterization of the change of heat flux of a
convecting system given a reference state. Eventually, the physics of variable-viscosity flow
Cooling of a variable-viscosity fluid
559
may be sufficiently well understood that the effects of changes in E and V on the Rayleigh
number can be more rigorously quantified. Then for a given set of viscosity parameters, a
Rayleigh number and a Nusselt number can be predicted for a reference state.
Fig. 6 shows the relation between log(Nu) and log(Ra) for several cases. For a range of
Rayleigh numbers for each-case considered there is a linear relation between these quantities.
The slope of this plot is generally between 0.25 and 0.30 [= b in equation (S)]. The points
which do not lie on linear trends are all for times early in the calculation, when the average
lid temperature (TL) was varying. For the points which do not fit, the Nusselt number is
lower than expected from the linear relationship. The times when TL is constant and
equation (7) holds, which shall be referred to as the interval of ‘transient equilibrium’ of the
lid with the convecting region, shall be concentrated upon.
The heat flux out of the convecting region affects the rate of cooling of the lid and the
value of TL.The Nusselt number describes that heat flux, and it shall be investigated how it
varies with time during cooling. Since the Nusselt number depends on the Rayleigh number
[via (8)], it is necessary to find the parameters that are causing the variation of the Rayleigh
number with time. The average viscosity changes by a large amount through a calculation,
1.2
1.1
n
z‘
W
1
Case 14 A
Case 18 rn
Case20 0
0
t
Q
lmO
I
Case
/ ‘ /1
0.9
Slope = .30
/?
H0X-=
0.7
0.6
”
890’a’
a
Case 14 + Case 20
Slope = .25
l0
3.2
3.6
4.0
4.4
4.8
5.2
Log ( R 4
Figure 6. A plot of log(Nu) versus log(Ra) where the Nusselt number (Nu) is given by (7) and the
Rayleigh number (Ra) is given by equations (4)-(6) for several cases. All of the cases can be fit with
straight lines with slopes, which equals b in equation (S), between about 0.2 and 0.3. The offset of the
points for case 18 relative to the other cases indicate that Nu and Ra may not be perfectly defined for all
different possible viscosity parameters, but given the large variation in parameters used in the calculations
the correspondence of the results is very good.
W. R. Buck
560
while the other parameters that define the Rayleigh number do not. The area where the
viscosity is minimum is the area where the strain rate is greatest and, therefore, 6 is highest,
as illustrated in Fig. 6. Therefore, the average viscosity (p) is weighted most heavily toward
the minimum viscosity. The minimum viscosity occurs at the base of the thermal boundary
layer (zb) because this is the shallowest depth at which the nearly isothermal convecting
region occurs. The viscosities deeper than zb are higher if the pressure dependence of
viscosity [V in (l)] is greater than zero. When V = 0 the convecting region is nearly
isoviscous, except in the narrow boundary layer. Fig. 7 shows a plot of the ratio of the
average viscosity (p) defined by ( 5 ) versus the viscosity at the base of the boundary layer
[p(zb)] for two numerical cooling calculations through time. In case 20, v = 7.5 cm3mol-'
and in case 17, V =0. In both cases p was linearly related to p(zb), except at the start of the
calculation, but the ratio of the relation is different. The ratio was nearly 1.O for case 17 (no
pressure dependence on viscosity) and about 5.0 for V = 7.5 cm3mol-' in case 20.
In all the numerical cases, the ratio is nearly constant for a given model. Thus throughout
a calculation, the effective average viscosity
can be related to 1-1 (zb) as
(n)
p =c1I-1 (zb)
(9)
9
where c1 is a constant that depends on V. Three factors affect the value of the minimum
viscosity [p(zb)]. One is the average viscosity determined by A,, in (1). The other two factors
61
I
I
1
I
0
@
O
I
h
0
0
Case 20
0 Average = 5.40
Std. Dev.= 0.16
II
W
N
v
0
v)
a
.+
.cn
3-
Case 17
El Average = 0.82
Std. Dev.= 0.08
.-0
>
cc
0
0
.+
cd
CT
2-
r
--la- -a- ,T
1-
1
0
&
b
I
I
I
I
20
40
60
80
9 -L
*-
1
100
Time (m.y.)
Figure 7. The time variation of the ratio of the average viscosity (d,
as defined by ( 6 ) and the horizontally averaged viscosity at t h e base of the boundary layer [c((zb)] is shown for the two cases which
differ in activation volume. For case 17, the activation volume is effectively zero while for case 20 it is
7.5 cm3mol-'. Over the time of the calculations the ratio is nearly constant.
Cooling of a variable-viscosity fluid
561
are more interesting because they change with time. One is the average temperature of the
convecting region (Tcr) which affects viscosity in proportion to the magnitude of the activation energy ( E ) in (1). The other is the depth to the bottom of the thermal boundary layer
(zb), since pressure (P)in (1) depends on depth. The effect of variations in the average temperature (Tcr) and in the depth (zb) on the Nusselt number will be considered. Furthermore,
it will be shown that both effects are consistent with the Nusselt number being proportional
over a significant range of time.
to P 2
Theoretical treatment of the cooling problem
If only the temperature dependence of the viscosity is considered, then (1) reduces to
Combining (10) with (4) and (8) gives a relation between the Nusselt number and
temperature
where
).
A , = a ( CrpgATl3
KClAO
Fig. 8 shows that the relation between ,,eat flux and temperature is nearly linear. Fig. 9
shows that the relation between the heat flux and the temperature of the convecting region
I
1
I
I
I
I
I
1260
1280
1300
Theoretical Heat
Transporting
12
11
c
101
1200
1220
1240
(Tc,-To, ( O K )
Figure 8. The dashed line is a plot of the Nusselt number (Nu) versus temperature (TI given by (11). Here
E = 100 kcal mol-', b = 0.25 and Tref= 1523 K and Nu (T,,f) = 10. There is a change in Tof 100 K for this
plot. The solid line is the best fitting straight line approximation to this relation over the same range.
W. R. Buck
562
(Tc,- To)
1220
1240
(OK)
1260
1280
1300
14
12
10
NU
8
6
4
2
.94
.96
.98
1.o
Figure 9. The variation of the Nusselt number (Nu) with changing temperature of the convecting region
(Tcr) is shown for two of the numerical cases considered here by plotting Nu against (Tcr - To)/(Tm- To).
The trends are approximately linear; however, the change in Nu is greater than is predicted solely as a
consequence of cooling a temperature-dependent fluid. The pressure dependence of viscosity also
contributes as discussed in the text.
is also linear for the numerical calculations. The temperature of the convecting region (Tcr)
is simply related to the heat flux out of the region or Nu by
where c2 =KAT/(l2pc,). Differentiating (1 1) with T = T,, and combining with (12) gives
where
Cooling of a variable-viscosityjluid
/
I
I
I
I
I
0.12
0.14
0.16
0.18
563
i
I
(m.y.)-’I2
Figure 10. This shows the dependence of Nu ( t ) for (14) and the linear approximation to that relation
given by (15) on ( t ) - l I 2 . The two plots were made to coincide at time f = 50 Myr, which corresponds to
(t)-”*= 0.141. The time range for the plots is 50 Myr.
where Gef is a reference temperature. Integrating equation (13), assuming t = t o when
Nu (0 = Nu(Tref),gives
Nu(r) = Nu(to) exp [-c2c3(t - t o ) ] .
(14)
Expanding t around t oin termsof t-”’ gives
Nu ( t ) = Nu ( t o )[ 1 t 2cZc3(f’” - f;’”)
$‘I.
To show that (15) is a good approximation to (14) for the range of parameters considered
here, values of Nu(t)/Nu (to)were calculated using both formulas. Using E = 100 kcal mol-I,
pc,= 3 x
Jm-3K, Nu(to) = 20, 1 = 200km, A T = 100K, K = 3.2 J ( m s K)-’ and
Tref= 1550K, it is found that the average error due to the approximation was less than 2 per
cent over a period of 50Myr with t o centred on 50Myr (see Fig. 10). Therefore, for a
cooling convecting fluid with temperature dependent viscosity given by (l), the Nusselt
number, or the convective heat flux, can be shown to vary, to first order, with f ’ j 2 .
A similar analysis can be applied to the variation of the Nusselt number with the depth to
the isothermal convecting region or the pressure there, which is linearly related to depth.
However, a further assumption must be made, namely that zb varies as t”’, which is the case
for the numerical results. I t will then be shown that this causes the Nusselt number to vary
as t-”’.
Holding temperature constant so that viscosity only varies with Zb, (I) can be rewritten
where v‘ = Vpg and A2 = A oexp (E/RT,,f). Since the Nusselt number (Nu) depends on the
W.R . Buck
564
Time (M.Y)
(a)
100 80 70 60
24 -
-
1
1
1
1
50
40
30
I
1
I
-
A
-
20 -
w
E
-2
E
=
16
X
cs!
12
o C A S E 14
v
CASE 17
0 CASE 18
0
A C A S E 20
*
I
I
.I0
I
I
.I2
I
I
.I4
I
I
.I6
I
I
I
.I8
Figure 11. The variation of q c m a x versus (time)-"* for the numerical calculations. qcmax is directly proportional to the Nusselt number (Nu). The relation is essentially linear in the time interval when T is
relatively constant for the model calculations. Table 3 gives the parameters which define the model cases.
(a) shows four cases which have the same box geometry, and initial conditions, but vary in the parameters
which control the viscosity. (b) shows that the cases have the same viscosity parameters, but differ in the
box geometry or the initial conditions.
average viscosity through (Z), ( 7 ) and ( 8 ) , it will depend on zb as
Nu (zb) = A (A2)-*exp
(S)
with A l defined in (1 1). The boundary layer thickness changes little during the numerical
calculations compared to the thickness to the conductive lid (zL), and it is small compared
with zL,so zb is approximately equal to zL.zLis assumed to be equal to 2 h (Kf)'12, where
is a parameter which will be discussed in the next section and K is the thermal diffusivity.
Thus by changing (17) the Nusselt number as a function of time is
565
I00 80 70 60
I
I
I
I
50
40
30
I
I
t
-
-
-
-
20 -
-
-
-
16-
-
X
-
-
5
12-
24
N
E
\
20
W
0 CASE 12
-
V
CASE 15
0 C A S E 19
8-
A C A S E 23
I
I
.I0
because we have replaced
zb
I
I
.I2
I
I
.I4
I
I
.I6
I
I
-
-
I
.I8
with 2h(~t)’”.Expanding (18) around to in terms oft-’” gives
Nu (t) = Nu (to) [ 1 t ( f ’ 1 2 - t;’”) c 4 t 0 ] .
(19)
As with the variation of Nusselt number with temperature, the variations due to changes in
are approximately linear with t-‘/2.
It will now be shown that when Nusselt number varies linearly as f ’ ” , as it approximately does for this problem, then the cooling of the conductive lid can be described by a
similarity solution. The treatment of this solution follows Carslaw & Jaeger (1959), and the
problem is similar to a moving boundary problem where there is a change of phase, which
was first treated by Stefan (1891). The difference between the base of the boundary layer
and the base of the lid is taken to be neghgible. The heat flux there is the maximum convective heat flux
zb
where K is the thermal conductivity. The position of zL is assumed to move with time at a
rate given by
zb =
2h ( K t ) ” 2 ,
where
(21)
X is a constant which depends on the Nusselt number of the system at a given time.
566
W.R. Buck
The temperatures in the lid must satisfy the conductive heat transport equation. Such a
solution is
~ ( z t, ) = B
erf [ z / 2 ( ~ t ) "+~T,,.]
(22)
At depth z b the temperature is assumed to be T,, + To(see Fig. 1) so the constant B is given
by T,,/erf(X), where T,, is the average temperature of the convecting region. Since T,,
varied slowly in the numerical cases considered here, T,, is taken t o be constant when considering the temperature structure of the lid. If there were no convection, just conduction at
all depths, then X would be infinite.
Combining (20), (21) and (22), we find that the convective heat flux must vary linearly
with t-'I2 for (21) to hold. This gives
Next, we can show that, for this problem, the average temperature of the lid is constant
in time and depends on A. The lid similarity temperature (TL)is defined as the average temperature down to the bottom of the lid divided by the temperature there (T,., + To). Integrating (22) down to z b and dividing by T,, + Toyields
Thus TL is a function only of X and does not vary with time. When Tp- To= 0.9(T,,- To)
then TL is analogous to the average lid temperature calculated for the numerical results
which was shown to be nearly constant in time for the numerical calculations. Using this
equation, a value of X can be calculated for a given value of TL. For the numerical cases the
integration was done only down to the point where the temperature equalled 0.9 Tcr.Thus,
the value of X obtained using (24) must be corrected to reflect the value of X which determines the rate of motion of the point where the temperature is T,,. The results of doing this
are shown as values of h for the numerical cases in Table 3 .
Comparison between theory and numerical results
The combined effect of variations in T,, and t b on the convective heat flux is simple to get
by differentiating and adding (1 5) and (19). The Nusselt number (Nu) is then linearly related
to the convective heat flux (4,
via (7) to give
Fig. 1 1 shows plots of measured values of qcmaxversus f ' 1 2 for the numerical cases, and it is
clear that the trend is linear in the region of 'transient equilibrium'. The behaviour is similar
to that predicted for the Nusselt number (see Fig. lo).
To compare the predictions of this theory against the results of the numerical calculations,
we must calculate the parameters cz, c3 and c4 and be able to relate them to features of
results of the model. Table 3 gives the values of the slopes from the plots in Fig. 1 1 along
with the predicted values of the slopes based on the calculated values of c2, c3 and c4. The
value of X for each calculation is estimated by calculating TL and relating it to h by (24). In
the estimation of these parameters, no effort was made to adjust for the differences in the
depth extent of the efficient cooling by convection (Az), which would affect the value of
Cooling of a Variable-viscosityfluid
567
cz, nor was the value of to allowed to vary from case to case. Even so only for run 17 and
case 22 was the predicted slope more than 1 0 per cent different than the calculated values.
To determine X in terms of the coefficients of viscosity A , E and V, (23) can be differentiated and combined with (25), (7) and (8) t o give
where R$ef is the critical Rayleigh number defined by (4) with p = k e f , and the heat flux
qcmaxis taken to be zero when p ( t o ) equals pref.Instead of considering variations in the
pre-exponential in the viscosity relation, A o , this expression has been written using the
average viscosity, ,ci, which is directly proportional to A o . This relation shows that X does
not depend on time for this theoretical development, but only on the physical parameters
of the system. The most important parameter which affects X is the average viscosity,
p. This is particularly interesting because the average viscosity of the asthenosphere may vary
from place to place, as will be discussed later. In Fig. 12, the variation of X as a function of
the log of the average viscosity is plotted. This is shown to be a nearly linear relation with a
change of about 0.2 in X for a change of a factor of 10 in the average viscosity, for b (the
exponent in Ra-Nu relation) equal to 0.25. Since the average viscosity is taken to be an
exponential function of temperature, then X is essentially linear with changes in temperature
1.8
1.6
1.4
A
1.2
1.o
0.8
0.6
0
1
2
3
4
Log(Pref/p)
Figure 12. The variation of h as a function of log [/+ef/(P)] as given by (26) assuming b = 0.25 and that
a = 1.0 when p = 0.09 M.This assumes that a l l the variation in A is due to differencesin the average initial
viscosity. Taking the cause of the viscosity variations to be differences in the initial temperature of the
convecting region (T,) gives the axis at the top of the figure. The difference in T, from some reference
value, T, ref, which would cause the viscosity variations on the lower axis is shown. The activation energy
is taken t o be 1 0 0 k cal m o l ~and
’ T, ref, is 1200°C.
568
W.R. Buck
around a reference value ( T , ref). For E = 100 kcal mol-' values of (T, - T, ref) are shown
in Fig. 12 which correspond to the variations in average viscosity. Between cases 12 and 14,
there is a change in the average viscosity of a factor of five, which would correspond to a
difference in temperature of about 80°C for the value of E used. Equation (26) or Fig. 12
predicts that the difference in h for the two cases should be about 0.15, in excellent
agreement with the value derived from the numerical experiments of 0.13.
The change of h due to a change of the temperature dependence of viscosity E, is as
simple to estimate as that due to changes in the average viscosity. This is because changing
E causes a change in AT, the temperature difference across the convecting region. It is found
that AT is inversely proportional to E. T h s can be understood as a consequence of the
viscosity difference across the convecting region being roughly constant. This constancy is
found not only in these calculations but is also noted by Jaupart (1981), who studied the
linear stabdity of fluid with depth-dependent viscosity. Since E and AT are inversely proportional, the term c2c3 does not change for a change in E. Therefore only the change in AT in
(26) will affect A. Since AT also enters into the Rayleigh number [see (4)],the change in the
right-hand side of (26) due to a change in AT should be equal to the ratio of the old value of
E over the changed value of E, all raised to the (1 + b) power. Case 18 had the same viscosity
parameters as case 14 except that E was 80kcalmol-' as in case 14. For b = 0.25 the change
in the right-hand side of(26) due to thischange inEisaboutafactorof 1.5 [=(110/80)1.25].
This is about the same change as would be predicted for a decrease in the average viscosity of
a factor of five, which is the change between case 12 and cases 14 and 18. Therefore, it is
not surprising that the values of X calculated for cases 18 and 12 are almost equal.
It should be noted that the linear relation between qcmax and f - l i 2 shown for the
numerical cases generally is not valid in the early times of model calculations. However, for
case 15, which was the only case where a wavelength of flow was not imposed (see Buck &
Parmentier 1986) the relation is valid at the earliest times (Fig. lo). Since this less constrained system is more likely to reflect the behaviour of a natural system, then the assumption of a linear relation between heat flux out of the convecting region and f-'",
which is
required for (26) to hold, is valid to earIier cooling times. For time r = 0 the approximation
clearly breaks down because the heat flux out of the convecting region cannot be infinite.
A parameterization of the effects of non-Newtonian viscosity was not explicitly considered. The stresses depend on the wavelength and Rayleigh number of the convective flow.
For the small wavelengths and moderate Rayleigh numbers used in my model calculation,
the effect of stress should not be large if it is appropriate to use a cut-off stress of about
10 barsbelowwhichthe rheology is stress independent, as is done in Fleitout & Yuen (1984).
Application to geophysical problems
In the preceding sections a mathematical description of the temperatures in a cooling lid
over a variable viscosity convecting region was developed. The dependence of the only parameter (A) needed to describe the system on average viscosity, on the temperature dependence of viscosity (E), and on the pressure dependence of viscosity ( V ) was shown. The
variation of the geophysical observables of lithospheric subsidence, local isostatic geoid
height variations, and heat flow will be discussed in terms of their variation with A. Also, the
use of the relations described here to estimate the rate of cooling of convecting magmatic
intrusions will be briefly discussed.
A word should be said about the assumptions used to relate the simple 1-D mathematical
model presented here to the case of the 3-D mantle of the Earth. As has been shown, the I-D
model matches the horizontally averaged temperature of the 2-D numerical calculations
Cooling of a variable-viscosityfluid
569
quite well. Thus, this theory should be good when there is no convection other than that
driven by cooling from above. Even if that flow is 3-Din nature, this should be true since the
same kind of Ra-Nu relations [see (8)] should hold for a simple 3-D flow driven by cooling
from above, as laboratory experiments show (Booker 1976). However, there are special
problems in applying these results to the cooling of the oceanic lithosphere. First, there is a
large scale of flow associated with the motion of the lithospheric plates. This flow should be
perpendicular to the flow calculated in the numerical runs considered here (Richter 1973;
Richter & Parsons 1975), as illustrated in Fig. 13. The present results are applicable to the
cooling of the oceanic lithosphere only to the extent that there are no vertical gradients of
velocity in the direction of the large-scale flow. The depth range for which this is a good
assumption depends on the form of the large-scale flow.
(b)
0
-h
E
Y
;r
v
I
ICONC
t DlFF
ta
w
n
!(OK)
I
I
0.
ASTHENOSPHERE
I
1
I
I
I
I
400
2
I
I
2c
Figure 13. (a) Geometric relation envisioned between smallscale convection and the largescale flow
under a ridge crest. (b) Shows the average temperatures for case 20 at model time 40 Myr and approximately how the model temperature profile is taken to relate to the oceanic lithosphere and asthenosphere.
S denotes water depth or subsidence.
570
W.R. Buck
The second assumption made in applying these results to the cooling of the oceanic lithosphere and asthenosphere is that cooling at all depths leads to subsidence of the surface.
Much work has been done on the relation between temperature variations in a convecting
fluid and surface deformation and geoid variations (McKenzie 1977; Parsons & Daly 1983).
However, one must be able to specify the aspect ratio of the large-scale mantle flow, its
depth extent, and its viscosity structure in order to relate the temperature variations caused
by small-scale convection to the surface deformation and geoid. Therefore, I make the
simple assumption that cooling at all depths is weighted the same.
The heat flux out of the surface of the lithosphere and the subsidence due to thermal
contraction scale simply with h. The surface heat flux is derived by differentiating the temperature profile [see ( 2 2 ) ] at z = 0. This gives a surface heat flux which is the same as
that which would be calculated for conduction alone, increased by the value l/erf A.
Fig. 13(b) illustrates how model temperature changes are related to subsidence. Since subsidence (S)should reflect thermal contraction due to the integrated heat flux out of the
surface, it too will be equal to the conductive value times l/erf A. Therefore the predicted
subsidence will vary linearly with the square root of age for this model, but the slope of
subsidence versus (time)”2 of the seafloor will depend on the vigour of convection:
If h scales with initial temperature of the mantle under the ridge then we can predict the
change in the rate of subsidence for a given change in temperature. This is plotted in Fig. 14
which shows that the change in the rate of subsidence is only about 30 per cent for a change
in mantle temperature of 200°C. The rate of subsidence is known to vary along different
sections of ridge crest and systematic studies of subsidence are presently underway (Hayes
1986; Cochran 1986). The variations in surface heat flux should be of the same magnitude,
Tm-Tmref( O K )
50
100
200
300
I
0’71
2
-
0.6
-2E=+
0.5
0.1
O’*
Derivative of Subsidence
with (time)’”
A
Convection
1
1.8
1.6
1.4
1.2
1 .o
0.8
Figure 14. The derivative of subsidence (S) with respect to
versus h as predicted by (27). Also
shown on the top axis is the difference in initial mantle temperature (T,) which would give rise to the
indicated variations in h as discussed in the text for E = 100 kcalmol-’. The solid line shows the effect of
variations in h and in (27). The dashed tine is the variation in the rate of subsidence for a conductive
model which depends only on Tm not on A. Here K = 7.0 ~ l O - ’ r n ~ s -T,
~ , - To = 1300 K is taken and
other constants as defined in Table 1 .
Cooling of a variable-viscosityfluid
571
but would be difficult to resolve in the data due to the uncertainty of the measurements.
The isostatic geoid height is very sensitive to the distribution of temperatures with depth and
therefore to the effects of small-scale convection. I will show that convection under a
cooling lithosphere will lead to isostatic geoid anomalies which decrease approximately
linearly with age but that the predicted slope of the geoid height-age relation will depend
strongly on the convective vigour. Before a relation between geoid height and h can be
derived, it must be shown how the depth range of convection changes with time.
In the numerical experiments described in Buck & Parmentier (1986), the depth range of
convection is seen to increase with time. The linear gravity features observed over several
areas of the oceanic plates may show little increase in wavelength with crustal age (Haxby &
Weissel 1986), however, this may reflect the interaction of convective stresses with the
growing elastic lithosphere. If the aspect ratio of the convection cells remains close to unity
as they grow, then the wavelength will be about twice the depth range for convection. Since
the dependence of viscosity on temperature limits the temperature drop across the convecting region, it is reasonable that the pressure dependence of viscosity controls the depth range
of convection (2). The simplest assumption that can be used to derive an expression for 1 is
that a constant ratio between the viscosities at the top and bottom of the convecting region
must be maintained. Using equation (1) a given drop in temperature at the top of the region
(T, - Tc,) will be related to 1 by
where P is an average value of the pressure in the region. Next, the total amount of heat
available for cooling of the convecting region must be estimated. The total amount of heat
removed from the convecting region at any time (Qcr) is given by the integral of (21) minus
the heat of cooling material added to the lid
Qc, ( t ) can be related to the change in temperature of the convecting region and the thickness
of the convecting region ( I ) through (21)
Since Q,, depends on t'" then I will depend on ? I 4 . The rate of change of I is a strong
function of the pressure dependence of viscosity, as it is proportional to VG'/'. Fig. 15
shows 1 as a function of time for two values of V', along with the convective wavelength
assuming the convection cells have a unit aspect ratio. The size of the convection cells varies
slowly, after an initial period of rapid growth. Thus, the assumption made earlier that the
depth range of convective cooling (I) was constant in time is consistent with (30). The
numerical calculations done so far do not allow comparison with the prediction of cell size
versus time, since the box size and boundary conditions limited the growth of the cells,
Appropriate calculations are now being carried out to check these predictions.
The isostatic geoid height depends on the dipole moment of the density distribution
(Haxby & Turcotte 1978) and thus on the shape of the temperature profile in the lid and in
the convecting region. The contribution to the isostatic geoid height from the conductive lid
is achieved by integration of the dipole moment of theetemperature distribution given by
( 2 3 ) down to depth ZL. The contribution due to the assumed drop of temperature, the
572
I
1
I
I
Figure 15. The variation in length scale of active convection ( I ) versus time given by equation 30 for two
values of the activation volume V assuming A = 0.9 in both cases. Also shown is the wavelength for the
convection assuming a unit aspect ratio for the cells. Constants as defined in Fig. (14).
convecting region is then added to this t o give
Time (rn.y.),
0.200 km
-1.0
II
0
-15.0E
v
I
20.0-
0
50
75
100
Tirne(rn.y.), 0.400 km
Figure 16. Plots of geoid height (If) versus time are shown for conductive cooling of a halfspace, the
25
results of numerical model case 15 and the analytic model given by (31) for A = 1.0. Two length scales
(or box depths, D ) are used to dimensionalize the model results as discussed in Fig. 4. The analytic prediction is nearly linear with a slope almost twice that for conduction and also nearly matches the results of
the numerical model which had the same value of A . Values for K , of and T, as in Fig. (14).
Cooling of a Variable-viscosityfluid
573
where second order terms in the integration have been neglected. The predicted geoid height
is very close to being linear with time, as shown in Fig. 16. Fig. 17 shows how strongly the
slope of geoid height-age (dH/dt) depends on X and therefore on the initial temperature of
the convecting region.
A very different application of the parameterization derived here is to the cooling of
magma bodies from above, which has been described by Carrigan (1984). He considers
magma bodies that are insulated on their sides and that, because of temperature dependent
viscosity of the magma, have a conductive lid over a convecting region. In his numerical
calculation, he considers Rayleigh numbers in the same range as the numerical cases already
described from Buck & Parmentier (1985), although the length scales involved are much
smaller. The lower viscosity of magma offsets this. Carrigan (1984) develops a parameterization in terms of a set of ordinary differential equations which must be solved numerically.
He is mainly concerned with the average temperature of the convecting region. In his treatment, he does not have to make the simplifying approximation that the depth extent of the
convecting region is constant in time. That approximation must be made to derive the
analytic parameterization described here. The advantage of this treatment is that someone
who wishes to use the parameterization does not have to program the differential equations
on a computer, but merely has to evaluate the expressions derived here.
When only the temperature dependence of viscosity is considered in ( l ) , the temperature
of the convecting region is approximated by:
c3
where T, is the initial temperature, Nu(t,) is the initial Nusselt number for the system, and
c2 and cg are as defined before. This was derived by combining (1 2) and (1 4) and integrating
taking To= 0. Equation (32) predicts the exponential decrease in the interior temperatures
w h c h were reported by Carrigan (1984). A direct comparison with h s results is difficult
because, in the two cases he presents, he changes the initial conditions of temperature, the
03-
Derivative of Geoid Height
wlth Time
x
E
-5.
02-
E
-
-
I
U
.
01
0.0
_ - / - - - - -
-
Conduction
1.8
1.6
1.4
1.2
1.0
0.8
A
Figure 17. The derivative of isostatic geoid height ( H ) with respect to time as a function of h given by
(31). As in Fig. (14)the solid line represents changes in the slope due only to changes in A and T,. The
dashed line shows the effect of changes in Tm on the rate of change of H for a partly conductive model.
Constants same as for Fig. (14).
574
W.R.Buck
initial average viscosity, and the temperature dependence of viscosity. However, if the
difference in initial conditions is a small effect compared to the temperature dependence of
viscosity E, then (32) can be used to compare hls two cases. As noted before, the term c2c3
does not change for a change in E. Only the term Nu(t,)/c3 is important, and that term only
depends on 1/E. The difference between his cases is that he takes E to be twice as large in
the second case as in the first. He observes about one half the rate of change of temperature
for the second case relative to the first. This is consistent with the prediction of (32).
Discussion
The linear gravity features observed over regions of the oceans (Haxby & Weissel 1986) may
be a product of small-scale convection. Small-scale convection would produce change in the
horizontally averaged temperature structure of the lithosphere and asthenosphere. The
equations developed here can be used to estimate the observables that depend on the average
temperature structure of the oceanic lithosphere and asthenosphere. The key question is
whether there are any data which are better fit by the present model than by the standard
half-space cooling model for the oceanic lithosphere. The average values of the surface heat
flux, the subsidence rate, and the rate of change of the geoid height with age for the ocean
basins (e.g. Parsons & Sclater 1977) can be fit equally well by either model. This is due to
the uncertainty in the physical parameters which control the model predictions. However,
the variations in the data from region to region, especially the data on rate of change of
geoid height with age, appear to be more consistent with the present model.
Andrews, Haxby & Buck (1986) find variations in the geoid height-age slope of about a
factor of two. They consider areas where the lithosphere is younger than about 50Myr to
avoid long wavelength contamination of the signal. These variations in slope may be a result
of variations in the regional temperature of the asthenosphere. Seismic velocity models of
the upper mantle (Woodhouse & Dziewonski 1984; Nakanishi & Anderson 1982) are
consistent with variations of the average temperature of the upper mantle of about 200 K,
assuming a value of the derivative of shear wave velocity with temperature of 3.4 x 10-4km
(sK)-' taken from Simmons & Wang (1971). For the half-space cooling model the slope of
the geoid height-age relation is directly proportional to the temperature difference between
the surface and the asthenosphere (Haxby & Turcotte 1978). Thus, a 200K variation in
temperature difference should give rise to less than 20 per cent of variation in the geoid
height-age slope. On the other hand, a model which includes convective cooling of the
asthenosphere gives much larger variations in geoid height-age slopes for the same temperature variations (see Fig. 17). This is primarily a result of the strong temperature dependence
assumed for the viscosity of the Earth's mantle.
Fig. 12 shows how a relatively small change in initial temperature of the convecting
region causes a large change in the parameter h when the activation energy is taken to be
100 kcalmol-I. The effect of this change in h on the predicted geoid heigh-age relation is
shown in Fig. 17. This figure shows that a change in initial asthenospheric temperature of
about ISOK would produce an increase in the geoid height-age relation of about a factor of
two. The subsidence and the surface heat flux should not be a good way to distinguish
between the models. The slightly larger predicted variations in these parameters as a function
of initial temperature for this model, compared with the half-space model, is probably not
sufficient to be clearly resolved by the data. For example, the observed variations in the rate
of subsidence are of order 30 per cent (Hayes 1986); this could be explained by either
model. The reason the geoid height may be used to discriminate between the models is that
Cooling of a variable-viscosity jluid
575
it is very sensitive to the density (temperature) distribution with depth. Convection causes
much more cooling at large depth (in the asthenosphere) than does conductive cooling.
The effect of assuming different values for the temperature and pressure dependence of
viscosity on model predictions can be estimated using the derived equations. A very different
value of the activation energy, E , would have a large effect on the model predictions for a
given change in the initial temperature. If E is taken to be much larger than 100 kcalmol-',
the temperature change required to produce a given change in h will be reduced. The reverse
is true if E is smaller. If the pressure dependence of viscosity, V , is changed it will have a
fairly small effect on the variation of h for a given change in initial temperature. However,
when the ratio E/V is changed greatly there will be a large effect on the predicted depth
extent of the flow [see (30)] and therefore also on the geoid height. A much larger value of
V would lead to a thinner convecting region and to smaller changes in the predicted geoid
height-age relation for a given change in the initial asthenosphere temperature.
Conclusions
Expressions are derived which give some theoretical understanding of the process of the
cooling of a variable-viscosity fluid. Such a cooling process, where the lower viscosity
portion of the fluid is driven to convect by the cooling from above, may have geophysical
applications. The cooling fluid is separated into two regions which behave differently: the lid
where heat is transported completely by conduction and a region where the dominant mode
of heat transfer is by convection. The two regions are coupled because there must be a
balance between the heat flux out of the convecting region and that which is conductively
transported at the base of the lid. The rate of convective transport of heat controls the rate
of thickening of the lid as the system cools. Because the rate of convective heat transfer
behaves in a simple way, the conductive lid thickness increases linearly with
and
its temperature structure can be described by a similar solution.
A simple mathematical description is given for the regular features seen in numerical
experiments on cooling fluids with different viscosity parameters. These parameters include
the temperature and pressure dependence of viscosity. Several approximations are used to
show that simple equations can describe the lid temperature with time for this problem.
Only one parameter (A) is needed to describe the cooling of the lid, and it is explicitly
shown how to relate the viscosity and other parameters of the convecting system to that
parameter. The theory developed here is shown to be consistent with the numerical results.
With certain assumptions, it is claimed that this simple system could describe the cooling
of the oceanic lithosphere and asthenosphere. The average subsidence and surface heat flow
are related t o h, while the rate of change of the convective wavelength and the isostatic geoid
height depend on the viscosity parameters as well as h.
The parameterization derived here allows one t o estimate the effect of changes in model
parameters (mainly the initial temperature of the convecting region) on the geophysically
relevant results of the models without doing costly computer models. Temperatures in
different areas of the asthenosphere may be different, and this is commonly assumed to be
the cause of seismic velocity variations between different areas. Temperature variations will
cause variations in the asthenospheric viscosity and in the rate of change of both subsidence
and geoid height. The prediction of this model is that for changes in the average asthenospheric temperature of about 200 K there will be a relatively small change in the rate of subsidence (- 30 per cent) while the geoid height-age relationship could change by 100 per
cent. This is about the magnitude of variations observed (Andrews et al. 1986).
576
W.R.Buck
Acknowledgments
Discussion with Marc Parmentier, Barry Parsons, Jean Andrews and Bill Haxby greatly
improved this work. Thanks also for the suggestions of two anonymous reviewers. Support
provided by an Exxon Graduate Teaching Fellowship while starting this work at MIT and a
Larnont Post-Doctoral Fellowship at Columbia. Lamont contribution no. 4100.
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