JOURNAL OF GEOPHYSICALRESEARCH,VOL. 91, NO. B2, PAGES1961-1974,FEBRUARY 10, 1986
ConvectionBeneathYoung OceanicLithosphere'
Implicationsfor Thermal Structureand Gravity
W. ROGER BUCK 1
Department
ofEarth,Atmospheric,
andPlanetary
Sciences,
Massachusetts
Instituteof Technology,
Cambridge
E. M.
P ARMENTIER
Departmentof GeologicalSciences,
BrownUniversity,Providence,
RhodeIsland
Small-scale
convection
underthe oceanic
lithosphere
whichbeginsin thefirst5 m.y.of coolingcan
producea gravity signalwith the amplitudeand wavelengthobservedfor large areasof the central
Pacific and southern Indian oceansusing Seasat altimeter data. The trend of the observed anomalies is
parallelto the directionof platemotionas mightbe expectedif they wereproducedby small-scale
convection.
Modelspredictthat the wavelengths
of gravityanomalies
increase
morerapidlywith age
thanis observed.
The persistence
of shortrelativelyuniformwavelength
anomalies
(< 200km) to crustal
agesof 50 Ma mayindicatethat theywereproduced
whenthelithosphere
wasveryyoungandthin and
were"frozenin" ascoolingthickened
the elasticlithosphere.
Small-scale
convection
whichbeginsunder
very younglithospheredoesnot violate other geophysicaldata suchas the rate of seafloorsubsidence
and variationsof geoidheight with age. After convectionhas begun,the subsidence
due to thermal
contractionwithin the lithospherevarieslinearly with age, in the absenceof mantle heat sources,
althoughthe rate of changeof thesequantitiesis affectedby convection.Much of the variationof the
geoidheightacrossfracturezonescan be fit by a modelwhichincludessmall-scaleconvection.
INTRODUCTION
effect of shearingon the form of thermal convectiveinstabilities,
including early experimentalwork by Graham [1933]
Convection beneath the oceanicplates on a scale smaller
than the horizontal dimensionsof the lithosphericplateshas and theoreticalstabilitystudiesby Ingersoll['1966] and Gage
been suggestedto explain several geophysicalobservables. and Reid ['1968].Richter ['1973] showedthat finite amplitude
This providesone suggestedexplanationfor the deviation of convective motions in an infinite Prandtl number fluid could
seafloorsubsidence
with age from that predictedby simple be reoriented by shearing and suggestedthat large-scale
conductivecooling of the oceaniclithosphere[Parsonsand mantle flow associatedwith plate motions could control the
McKenzie, 1978]. More recently, in their analysis of Seasat form of small-scaleconvectionbeneatha plate. This was coraltimeter data, Haxby and Weissel[1986] have noted linear roborated by the laboratory experimentsof Richter and Pargravity anomalies which trend in the direction of plate sons[1975] and Cutlet [1976].
Theoretical studies of the stability of the top thermal
motion. They have suggestedthat these featuresmay be the
boundary layer of the large-scalemantle flow have also been
result of small-scale convection. Based on theoretical considerations, Richter [1973] predicted that small-scale convection carried out [Parsons and McKenzie, 1978; Jaupart, 1981;
should take the form of two-dimensional
rolls with axes ori-
ented in the direction of plate motion, thus providing an explanation for the form of the observedgravity anomalies.In
this paper we describenumerical calculationsaimed at understandingsmall-scaleflow which may occur under the oceanic
plates. The purpose of this work is to investigatewhether
modelswhich are consistentwith subsidence-age
data for the
oceansand other geophysicaldata can producethe observed
gravity features.
We first review previous work on small-scale convection
and then discussthe formulation of approximate models of
convectionand the calculationof severalgeophysicalobservablespredictedby the models.A rangeof modelsis considered
basedon laboratory measurements
of physicalpropertiesof
mantle mineralsand estimatesof mantle viscosity.The predictions of the modelsare comparedwith data for subsidenceof
the oceanfloor and gravityfor the oceans.
A number of investigationshave been carried out on the
Yuen et al., 1981; Yuen and Fleitout, 1984]. Parsons and Mc-
Kenzie [1978] treated a mantle of uniform viscositybelow a
fixed boundary and found that a thermal boundary layer
could go unstableafter 70 m.y. of coolingif its viscositywere
• 1021Pa s. Yuenet al. [1981] considered
a viscositystructure resultingonly from temperature-dependent
viscosity.For
viscositieswhich depend only on temperatureand which are
consistentwith postglacial rebound estimatesof whole mantle
viscosity,they concludethat no instabilitiesdevelopin a cooling boundary layer for a time equal to the age of the oldest
oceanicplates (200 Ma). Jaupart and Parsons[1985] studied
the linear stability problem for a depth-dependentviscosity
structure and concluded that for the base of the oceanic litho-
sphereto go unstableafter 70 m.y. of conductivecooling re-
quiredaverageviscosities
thereof the orderof 1021Pa s. They
also noted that the ratio
of the maximum
to the minimum
viscosityin the convectingregion was at most about a factor
of 10. Yuenand Fleitout [1984] concludedthat viscositywhich
dependson pressureas well as temperature is required to
allow boundarylayer averageviscosities
to be low enoughfor
•Now at Lamont-Doherty
Geological
Observatory
of Columbia
small-scale
convection
to
occur
under
the
oceanplates(i.e., a
University, Palisades,New York.
low-viscosity zone) and still match other constraints on
Copyright1986by the AmericanGeophysicalUnion.
mantle viscosity.Our first finite amplitude calculations[Buck,
1983] led to the same conclusion.
Paper number 5B5483.
0148-0227/86/005B-5483$05.00
Two previous studieswhich have consideredthe time evolu1961
1962
BUCK AND PARMENTIER: CONVECTION BENEATH OCEANIC LITHOSPHERE
Fig. 1. Small-scaleconvection beneath the oceanic lithosphere having the form of rolls oriented in the direction of
plate motion. Flow in a vertical plane parallel to the ridge (spreadingcenter)axis is calculatedfrom a two-dimensional,
transientconvectivecoolingmodel with boundaryconditionsshownin Figure 2. The streamlinesshownare from Figure
3b at a time (lithosphereage)of 9 Ma.
tion of convection are similar in formulation to the present
work [Houseman and McKenzie, 1982; Fleitout and Yuen,
1984]. Both studies are concerned with the possibility that
small-scaleconvection can explain a decreasein the rate of
seafloor subsidenceafter an age of about 70 Ma. The formulation
of
Houseman
and
McKenzie
cannot
allow
for
the
motion of the boundary between the lithosphere and the convecting region below becausethey treat a convectingregion
with constant viscosityand a rigid lithosphere.In their model
the boundary layer could not go unstable until cooling had
penetrated through the rigid lithosphere. Fleitout and Yuen
[1984] considereda temperature- and pressure-dependentviscosity, thus avoiding the need to define a lithosphereof constant thickness.However, the wavelength and depth of penetration of the flow were prescribed,and little variation of the
viscosity parameters was considered. Because these studies
were concerned with the approach of the lithosphere to a
thickness in equilibrium with the background mantle heat
flux, convection was driven by both heating from below and
heating from within.
The purpose of this study is to calculate the effect of smallscaleconvectionon the cooling of oceaniclithospherewhich is
less than 70 Ma in age. We consider viscositieswhich are in
accord with laboratory and geophysical estimates for the
mantle. A wide range of viscosity parameters were studied in
order to estimate the sensitivity of the convecting system and
to identify the values which can match geophysicaldata on
oceanic lithosphere.In our formulation the boundary layers
can go unstable and convectionbegin at a time which is determined by the viscosityparameters.In our problem the litho-
MODEL
FORMULATION
Small-scaleconvectionin the form of rolls with axesparallel
to the direction of plate motion is illustrated in Figure 1. As in
previous studies[Housemanand McKenzie, 1982; Fleitout and
Yuen, 1984], we simplify the three-dimensional problem to
consideronly two-dimensional flow in a vertical plane parallel
to a ridge crest. In doing so, we ignore the effect of vertical
gradients in horizontal velocity perpendicular to the ridge and
both the thermal and mechanical coupling between vertical
planes parallel to the ridge. As a result of this assumption, the
effect of vertical shearing by plate motion on the thermal and
mechanical structure is also ignored. These approximations
reduce the problem to one of time-dependent two-dimensional
convection. The plane of the calculation is considered to move
with the plate, so model time correspondsto the age of oceanic lithosphere and is proportional to distance from the ridge
crest.
The depth of penetration of the small-scale cells into the
mantle must depend in part on the structure of the large-scale
flow. Here, we consider no penetration deeper than 400 km
since we are mainly concernedwith the effects of small-scale
convection soon after it has begun, when the cells are of small
vertical scale. For greater penetration depths, the interaction
of the large- and small-scaleflow will almost certainly be more
complicatedthan we assumehere. Also the gravity anomalies
describedby Haxby and Weissel [1986] generally have wavelengths of about 200 km. The depth of penetration of convection cells should be of the same order as the wavelength of the
gravity anomalies they produce, as will be seen in the model
results.
sphereand the convectingregion are allowedto interact,and
With these assumptions and simplifications our problem
the thicknessof the lithospherechangeswith time. We consid- reduces to studying thermal convection in a box of variable
er the viscosity to be temperature- and pressure-dependent, viscosity fluid driven by cooling from above. This box is
but we do not constrain the wavelength and depth of penetration of the convection
in some of the calculations.
In the first
40-70 m.y. of lithospheric cooling, heat sourcesin the mantle
should have a small effect on the rate of cooling. Therefore
heat sources are not included
in these models.
shown in Figure 2. We define a region of calculation (or box)
to be of width L and depth D. In that region we solve the
two-dimensional Navier-Stokes equations for mass, momentum, and energy conservation [Batchelor, 1967-1. They are
modified for flow in the earth's mantle by dropping inertia
BUCK AND PARMENTIER.' CONVECTION BENEATH OCEANIC LITHOSPHERE
TABLE 1. ParametersUsed for Nondimensionalizingthe
GoverningEquationsand Calculatingthe Model Results
terms and terms that depend on material compressibility[Tur-
cotteet al., 1973]. The valuesof the physicalparameterswhich
were usedare givenin Table 1.
The governingequationswere solvedusing finite difference
approximations with centered differences for the diffusion
terms and upwind differencesfor the advection terms. Forward time steppingwas usedfor the time derivatives.We used
variable spacing of grid points in a difference scheme developed by Parmentier[1975]. This allowed higher resolutionin
the regionsof the largest gradientsof viscosityand flow, without an excessivenumber of points overall. In the region of
highest resolution the grid spacing is uniform, so formal
second-order accuracy in the centered difference approximations is preserved [Roache, 1982]. The grid point positions
are shown in Figures 3 and 4 as marks around the boxes.
Resolution of the solutions on the grids used here was established in two ways. First, the numerical experiments were
done on successively
refined grids until the same results were
achievedon two different grids. Second,the heat flux out of
the grid was comparedto the averagerate of changeof temperature within the box to ensure conservation of energy.
Cases 12 and 20, described later and in Table 2, were identical
except that the number of grid points in each direction was
greater by a factor of 1.5 for case20. The maximum difference
in averagetemperature (Figure 5) was lessthan 5%.
Since olivine
is considered
to be the dominant
mineral
in
the upper mantle [Ringwood, 1975], we adopt a relation for
the dynamic viscosity /• [Weertman and Weertman, 1975]
given by
/•(T, P) = A exp ((E + P V)/R T)
(1)
where E is the activation energy, V is the activation volume, A
is a constant varied to adjust the average viscosity, and R is
the universal gas constant.The value of the activation energy,
which controls the temperature dependenceof the viscosity,is
estimated from laboratory data. Geotze [1978] summarizes
measurementsof creep in olivine giving 520 _+20 kJ/mol as a
value for E. Based on dislocation recovery during static annealing,Kohlstedtet al. [1980] find that E = 300 + 20 kJ/mol.
We generally use values intermediate to these. The temperature fields displayedlater do not include an adiabatic gradient, but the temperaturesused to calculate the viscosityfrom
X,(•
T = To , u = u/ = O
CONDUCTIVE LID
Z,•(/
aT
ax
aT
APPROXIMATELY
•'7-
uT = 0
ISOTHERMAL
CONVECTING
a.___.•
= 0
ax
Symbol
REGION
T = T,,
aT
az
Units
diffusivity
lengthscale
10-6
4.0 x 105
m:/s
m
AT
temperature
1300
øC
3.0 x 10-5
1/øC
scale
•
thermal expansion
coefficient
/•
g
viscosity
accelerationof
gravity
1.0 x 1021
9.8
Pa s
m/s:
p,•
Pw
mantledensity
water density
3500
1000
kg/m3
kg/m3
K
conductivity
3.2
J/m s øC
cv
specificheat
900
J/kg øC
equation (1) did include a contribution due to an adiabatic
gradient of 0.3øC/km. Sammiset al. [1981] show that estimates of the activation volume based on both experimental
and theoretical methods give a range for olivine of 10-20
x 10-5 m3/mol. The activation volume, which controls the
pressure dependenceof the viscosity, is critical to reconciling
different estimates of mantle viscosity based on geophysical
observations.
An averagemantle viscosityof about 102• Pa s is required
by postglacial rebound [Cathies, 1975; Peltier and Andrews,
1976]. Several geophysical observations require much lower
viscositiesat shallow depths in the mantle under the oceanic
lithosphere and under tectonically active regions of the continents. Passey [1981] has analyzed the rebound of dried lakes
in Utah and infers shallow mantle viscosities lower than 1019
Pa s. Richter and McKenzie [1978] and Weins and Stein
[1985] require asthenosphericviscositiesbeneath the oceans
in the rangeof 1018-1019Pa s basedon the distributionof
stressesin the oceanic plates. Viscosity must increase with
pressureand therefore depth to reconcile low viscositiesat
shallow depths and higher average viscositiesfor the mantle.
Figures 3 and 4 show viscosity calculated with equation (1)
plotted versus depth for the viscosity parameters given in
Table
2.
At the top of a cooling variable viscosityfluid, temperatures
are low and temperature gradients are high. Therefore viscosity, given by equation (1), in the top of the box can be so
large that flow is negligible in that region. In this lid, which is
analogous to the thermal lithosphere, heat transfer takes place
exclusively by conduction. Below this region convection is the
dominant mechanism of heat transport. Since we are considercondition, both the lid thicknessand the vigor of the convection in the interior will change with time. It is the interaction
of the cooling, thickening lid with the underlying convecting
region which is of interest. The convection is driven by the
temperature gradients at the base of the lid, and in turn the
rate of thickening of the lid (or lithosphere)is affectedby the
convection.
./T = 0
a.-,q= 0 ,
Value
•c
L
Boundaries
az
Name
ing the transient cooling of a fluid and not a steady state
BOUNDARY LAYER
•T: 0
1963
•
= 0
Fig. 2. The boundary conditions for the numerical experiments
describedin this paper. The conductive lid is the region where the
advectiveheat flux is negligiblecomparedto the conductiveheat flux.
The convectiveboundary layer is defined in the text as are the boundary conditions.
on all sides of the box are taken
to be shear
stress-free.However, it is computationally more efficient to
place a no-slip (fixed) boundary at the depth in the lid where
viscosity is three orders of magnitude above the minimum
viscosityin the box. Becausethe viscosityis so high in the
cold lid, there is effectivelyno flow there. Calculations with the
boundary at shallower depth in the lithosphere, where the
viscosity is higher, give the same results but require more
computer time. The thermal boundary conditions are fixed
temperature (0øC) at the top and insulating on the sides and
1964
BUCK AND PARMENTIER'CONVECTION BENEATHOCEANIC LITHOSPHERE
z
o
(•4)
o
Hœd30
Z
0
0
(w•)
•
•'
n
--
•
Hñd30
o
••'
ov
mira
n1.1.1
d
1.1.1 o
I--.
v
c;
(w•i)
H/riB0
ci
o
6
(w4) Hñd30
o
o
BUCK AND PARMENTIER:CONVECTIONBENEATHOCEANIC LITHOSPHERE
rr
d
1965
E
o_
i'
I-d
(w•i)
t-
O
Z
rr
d
(w•)
hJ. d3C]
o
hJ_d3a
o
d
I--
•
x
z
t-
O
Z
i,
d
(w•)
hJ. d3C]
o
(W•l) HJ. d3a
o
1966
BUCK AND PARMENTIER' CONVECTION BENEATH OCEANIC LITHOSPHERE
20.
calculation to another, as listed in Table 2. A random initial
GRAVITY
temperatureperturbation was used in only one of the models.
This is ca•e 15 which was also carried out in the widest box.
(mgals)
This model is designedto examinechangesin the depth of
penetrationand wavelengthof the convectioncellswith time.
-20.
AV
I TEMP
COND
TEMP
)
400.
,
273.
,
1573.
200.
-2OO.
STREAM
Q(z)
=•
FUNCTION
adV
HEAT
400
'
WIDTH
17.
21.
w(x,
z)T(x,
z)dx
(2)
where w is the vertical component of the velocity. The convecting region is consideredto be all the area at greater depth
where the temperature is nearly uniform. In the conductivelid,
(Pa-s)
120.
below a deviatoric stress of 10 bars the
the level of maximum horizontally averaged advective heat
flux Q:
I
VI
0.
their formulation,
viscosity is Newtonian. The deviatoric stressesin our calculations are generally less than this value becausethe convective
wavelengthsare small.
The box can be divided into three regions based on the
mode of heat transfer: a conductive lid, a convective thermal
boundary layer, and a nearly isothermal convecting region
(see Figure 2). The top of the convecting region is defined as
I-
(øK)
In the other model casesonly one convection cell is induced
by a periodic temperature perturbation. These smaller, simpler
casesare used to study the effect of varying parameters and to
examine the effectsof different convectivewavelengths.
Calculations with non-Newtonian viscosityhave been carried out but are not discussedin detail here. Using nearly the
same parameters for stressdependenceof viscosity as Fleitout
and Yuen [1984], we found no effect on our calculations. In
-1.
the advective heat flux is effectively zero. This region extends
to a depth where the average temperature exceeds90% of that
in the convectingregion. Thus we define the base of the lid ZL
to correspond to this temperature. The average temperature in
the conductive lid TL is defined as
T•= •
FLUX
(3)
T(x,z)dxdz
Figure 5 showsvaluesof T• for a number of the calculations.
We considerthe lithosphereto be analogousto the conduc-
4.
(km)
Fig. 4. The samequantitiesas in Figure 3 for case20 at a time 56
m.y. into the calculation.The isothermscorrespondto temperatures
tive lid and calculate
the seafloor
subsidence
due to vertical
thermal contraction and isostaticequilibrium. In terms of T•,
the subsidenceis given by
of 1160ø, 1185ø, 1210ø, and 1235øC in order from top to bottom.
bottom. Both thermal and stressboundary conditions on the
vertical sidesare equivalentto a reflectioncondition.
A horizontally uniform initial temperature resultingfrom 5
m.y. of conductivecooling with an initial box temperatureTm
of 1300øCis adopted.Convectionmay begin earlier than this
for some of the viscosity structures we examine, but temper-
ature and viscositygradientsare so large at smallertimesthat
they are difficult to resolve even on relatively fine grids. To
induce convective motion, two types of initial temperature
perturbationsare superimposed
on the horizontallyuniform
temperatureprofile.In the first, a randomperturbationof less
than 1øC was introduced at each grid point. In the second,a
periodictemperatureperturbationwith a wavelengthequalto
twice the width of the box and with a IøC amplitudewas used
to inducethe growth of only one convectivewavelength.
A list of the model parameterswhich are common to all the
numerical experimentsis given in Table 1. The averageviscosity (expressedas a referenceviscosityand controlledby
parameterA), the activationenergyE, the activationvolume
V, the width L, and depth D of the box are varied from one
Both subsidence
and Z• are plotted as a functionof t t/2 in
Figure 6 for two of the cases.Since the averagetemperature of
TABLE
2.
#ref,
Parameters
E,
Which
V,
Define
L,
Case 10asPas kJ/mol cm3/mol km
12
14
15
17
18
19
the Numerical
Number of
Grid
Points
D,
km
400
400
400
400
400
400
Cases
1.0
5.0
1.0
0.5
5.0
1.0
420
420
420
420
300
420
10.0
10.0
10.0
2.5
10.0
10.0
120
120
400
120
120
120
0.80
0.93
0.87
0.74
0.78
0.79
2,320
2,320
10,201
2,320
2,320
4,400
20
1.0
420
10.0
120 •00 0.81
4,400
21
22
0.5
1.0
420
420
2.5
10.0
120
60
400
400
-"
0.80
4,400
4,400
23
1.0
420
10.0
120
300
0.84
3,560
The referenceviscosityttref, the value of viscosityat the start of a
calculation at 150 km depth in the model box, definesthe value of A
in equation (1). The other parametersare describedin the text.
Buck: AND PARMENTIER' CONVECTION BENEATH OCEANIC LITHOSPHERE
1967
our modelsthe value of/• will dependon the vigor of convection beneath the lid. A relationshipbetween/• and the viscosityas well as other model parametersare givenin a related
(a)
.6O
paper [Buck, 1986]. S(t) shoulddependon ,•TmOCt)
•/•. The
parameter/• is found to be proportionalto the value of TL.
Values of/• determinedfrom the presentmodelsare given in
Table
.58
.56
2.
An alternative
%',
_
•,",._',...
...........................
xxx,,
'-••
•__•___
_
L -,
estimate
of the subsidence
is based on the
change in the average temperature of the upper part of the
box to a prescribed,constant depth of compensation [Jarvis
and Peltier, 1982; Houseman and McKenzie, 1982; Fleitout and
Yuen, 1984]. The material above the depth of compensationis
assumedto be in isostatic equilibrium. The subsidencecalcu-
lated usinga depth of compensationof 150 km is effectively
the same as that calculatedfrom equation (4) becausethe
changeof temperaturewith time below the boundarylayer is
small comparedto that in the conductivelid. The depth of
compensation chosen affects the subsidencethat is calculated
.54
_
-
I
i
I
I
I
I
20
40
60
80
I00
120
TIME
(Ma)
from our model results.If the depth of compensationwere
taken to be at the bottom of the box, convectivecooling
would resultin fastersubsidence
than conductivecooling.
The isostaticgeoid anomaly [Haxby and Turcotte, 1978] is
given by
-27•G
(b)
H(t) 6O
+
[r•-
r•(z)]z
(6)
where T•(z) is the horizontally averaged temperature at a
depth z. This expressionis valid only if the density variation
producing the anomaly is isostaticallycompensatedand small
in vertical compared to horizontal dimensions.Thus it should
be valid as long as variations in temperature and therefore
density, below the lithosphereare small as should be the case
for the geoid offset acrossfracture zones.
The gravity anomaly at the top of the box is also calculated
from our models.Three componentscontribute to the anomaly. One is due to temperatureand thereforedensityvariations
.58
.56
in the box. A secondis due to the deformationof the top
surface of the box as a result of convectivestresses.Third,
vertical normal stressvariations due to horizontal temper-
54
ature differences within the conductive lid contribute
20
40
60
TIME
80
I00
I
0
(Ma)
Fig. 5. Values of the averagelid temperatureTL definedby equation (3), then normalizedby the temperatureat the baseof the lid as a
function of time for the casesindicated. Table 2 gives the parameters
usedin each case.The casesdisplayedin Figure 5b all have the same
viscosity parameters but different box widths W, and case 15 had
different initial conditions
to the
deformationof the top boundaryof the box. The first component of the anomaly is calculatedby numericallyintegrating
the followingexpressionfor the verticalcomponentof gravity
GT due to distributedtwo-dimensionaldensityanomalies:
GT(X')
= 2GPm•
[Th(z
) -- T(X,
z)](x--x')
Df+3L
Z2+ z2
dx dz
.J-2L
(7)
than the others. A value of nondimensiona-
lized temperature of 0.60 correspondsto the conductivesolution and
a value of 0.50 correspondsto a linear temperatureprofile.Note that
for timesgreaterthan 50 Ma the valuesof TLare relativelyconstant.
where G is the gravitational constant and the other values are
as defined above. The temperature structure outside the box is
the lid changesby less than 10%, the subsidenceoccurs primarily as a result of increasingZL. Convection changesthe
slope of the curves, but during most of the calculation they
remain linear on such a plot. This dependenceof ZL on time
effects.
can be written
as
Z,•(t)= 2/4•ct)
•/2
(5)
where •cis the thermal diffusivity.For purely conductivecool-
ing,/• is givenby err- x(0.9)[CarslawandJaeger,1959].For
assumedto be horizontally periodic with wavelength 2L. The
range of integration is over 2.5 wavelengthsto avoid any edge
To determine the component of the gravity anomaly due to
the flow, we must calculate the normal stress azz on the
bottom boundary of the conductive lid. The normal stressat
any boundary point is calculated using equations given by
McKenzie [1977] and Parmentier and Turcotte [1978]. The
stress at the surface of the box must include
the effect of
temperature and therefore vertical normal stressvariations in
the conductivelid (at). Neglecting shear stresseson vertical
1968
BUCK AND PARMENTIER' CONVECTION BENEATH OCEANIC LITHOSPHERE
CASE
(a)
CASE
15
20
O.
CONDUCTIVE
-0.25
(b)
10.
0.3
-0.25
0.3
j/
ZL= A2•/Kt
/
/
O.
i
i
i
i
i
i I I I
O.
O.
L
I
I
I
I
• i • !
10.
10.
(TIME) 4/2 (Ma) •/2
(TIME) 4/2 (Ma) 4/2
Fig. 6. The valuesof two parameters
calculated
for cases15and20 plottedversust•/2.(a) The nondimensional
value
(T•.- T=)Z•.usedto calculatethe subsidence
duc thermalcontractionwithinthe lithosphere,
as described
in the text.(b)
Z•., the nondimensional
depth to the bottomof the conductivelid.
planeswithin the lid, the total normal stressat the surface(as)
is the sum of a:: and at. The stressat the surfaceis adjusted
so that the averageis zero. The gravitational effect of these
stressesin our model is determined by the resultingelevation
E(x) of the surface.To determinethis, we must assumea flexural rigidity D of the elasticlithosphere.If we assumeD to be
zero, resulting in a pointwise isostatic response,hydrostatic
stresses due to elevation
(azz),and GL, which is the component due to the stressesproducedby the temperaturevariations in the lithosphere(at).
Finally, the average heat flux out of the top of the box
(Qs(t))is given by the product of the average temperature
gradientat z = O.and the conductivityK.
Os(t)
= KZ1fo
•dT
(x,o)
1:
dx
of the surface must match the normal
stressat each point, giving
•(x) =
(•-
(8)
where g is the acceleration of gravity. If D is nonzero, the
elevationwill be reducedby an amount which dependson the
wavenumber k of each Fourier component of the stressdistribution. For an elastic layer thicknessof 10 km and using
valuesfor elasticparametersfrom Watts and Steckler[1980],
the flexuralrigidityD -- 10•-3kg m•-/s•-.For thisflexuralrigidity, a stressdistribution with a wavelengthlessthan 200 km
producesalmostno surfaceelevationaccordingto the relation
for the damping effect of a thin elastic layer [McKenzie and
(9)
where dT/dz is estimated using a centered finite difference
approximation.
RESULTS
The results of a calculation (case 15) within a box repre-
sentinga 400 x 400 km region of the mantle are illustratedin
Figure 3. Figure 3 showsseveralquantitieswhich describethe
flow at four times. The temperature contours give an idea of
the rate of movement of cold sinking and hot rising material.
Advective
heat transfer
dominates
the conductive
transfer
in
the region where isothermsare distortedfrom horizontal. The
streamlines show the number of convection cells at a given
time and the depth of penetration of the flow. The cells are
Bowin, 1976].
seen to grow larger during the early part of the calculation.
The gravity anomaly at a point above the surfacedue to an The initial wavelength of the flow is controlled by the thickelevation anomaly is calculatedassumingthat the extra mass nessof the thermal boundary layer which first becomesunstadue to the surface elevation can be considered an infinite sheet
ble. Jaupart [1981] points out that the fastestgrowing waveat a depth of 4 km below the sea surface.This is a good length of the instability for a boundary layer in which visapproximation for featureslike those discussedhere with a cositydecreasesexponentiallywith depth shouldbe betweenn
wavelengthgreater than severaltens of kilometers.Then the and 2n times the boundary layer thickness.The boundary
total gravity anomaly causedby elevation is given by the sum layer definedhere is the region where both the advectiveand
of Go,which is the part of the signalproducedby flow stresses conductiveheat flux vary rapidly with depth. After 2 m.y., the
BUCK AND PARMENTIER: CONVECTION BENEATH OCEANIC LITHOSPHERE
wavelengthof the flow in Figure 3 is about 60 km. This is
consistentwith an initial boundary layer thicknessof about 10
km. In just another 2 m.y. the wavelengthof the flow increases
to nearly 120 km. The growth of the cellsis rapid early in the
calculation then later slows and finally stops when the box is
filled. The slowing of the growth of the cells dependson the
pressuredependenceof the viscositysincethis causesthe viscosity to increasewith depth. In a model where viscositydid
not depend on pressure,the cellsfilled the box more rapidly
than for any of the other cases.
The plots of the advective heat flux, shown for different
times in Figure 3, exhibit some interesting features. For a
model time of 2 m.y. the convectionis just starting to develop
and very little heat is being transported by the flow. At the
next two times, 4 and 5 m.y. into the calculation, the plots of
(a)
1969
o
-
_
I
12-
•
'•. _'•"",,a,
• _
16
advective heat flux have an extra local maxima due to a large
amount of cold material from the original unstable boundary
18
layer moving down. The profilesof the advectedheat flux for
the rest of the calculation
look
more
like that for case 20
shown in Figure 4. There the advectiveheat flux is a maximum at the base of the boundary layer and decreasesmonotonically with depth.
The horizontally averagedtemperature profiles in Figure 3
show gradients in the conductivelid but relatively uniform
temperaturebelowthe boundarylayer.The differencebetween
the horizontally averagedtemperatureat a given depth and
the temperaturewhich would result from purely conductive
cooling at the same time is also shown. In the convecting
region the temperaturesare lower than they would be in the
absenceof convection,while in the conductive lid the temperatures are higher than they would be for purely conductive
20
two cellsshouldvary approximatelyas x-•/2. Thereforethe
smaller the cell, the higher the horizontally averaged value of
the heat flux across the boundary layer. After about 20 m.y.
the value of T• becomesremarkably constant.
The isostaticgeoid anomaly H(t) for case 15 as a function of
time, calculated using equation (6), is shown in Figure 7 along
with resultsfor severalother cases.The slope of thesecurvesis
60
TIME
80
I00
(Ma)
(b)
-
CONDUCTIVE
.25
\
•.
.20
['•.\\••
cooling.
The average temperaturein the lid normalized by the temperature at the base of the lid (TL), shown in Figure 5, is a
measureof the temperatureprofile in the conductivelid. A
value of TL of 0.6 correspondsto purely conductivecooling
and a value of 0.5 resultsfrom a linear temperature gradient
reflectingsteady state heat conduction.High heat flux from
the convectingregion resultsin a thinner lid in which temperaturesmore closelyapproximatethe steadystate distribution.
For case15, TLdecreases
from the value for purely conductive
coolingfaster than for the other caseswhere only one convective wavelengthis presentin the box. The small cells,present
early in the run for case 15, are more efficientin transferring
heat out of the convectingregion than are longer-wavelength
cells. The local heat flux acrossthe boundary layer at a given
horizontal distance x from the center of upwelling between
40
E
"o .lo
.o5
_
I
I
20
I
I
40
TIME
,
I
6O
,
I
8O
IO0
(Ma)
Fig. 7. (a) The variation of the isostatic geoid height H given by
equation (6) with time for several of the casesincluding purely conductive cooling. (b) The values of the model geoid height slope (dH/dt)
versus time. The slope can be related to a geoid-age slope derived
from the observedgeoid offset acrossfracture zones [Cazenave, 1984].
the lid lag the changein cell size sincetime is required for the
lateral
differences in advective
heat flux to be conducted
into
proportionalto )•2•T,,•cas shownby Buck[1986]. The slopeis the lid. The componentswhich make up the total model gravmore appropriatefor comparingwith the data for the oceanic ity anomaly (G•, G•, and Gr) are shown in Figure 8 at one
lithosphere.The time derivative of model geoid heightsis also
shown in Figure 7 along with the geoid-ageslope [Cazenave,
1984] derived from the geoid height offset acrossoceanicfrac-
time for case 15. Clearly, most of the total anomaly arises due
to the combination of stressesat the base of the lithosphere
(G•) and vertical normal stressvariations through the lithoture zones.
sphere(G•), both of which will be reducedin magnitudeby the
The total gravity anomaly associatedwith the small-scale flexural rigidity of the elastic lithosphere. Admittance studies
convective rolls is shown in Figure 3, assuming no flexural have not yet been done for the region of the Central Pacific
damping of the signal. The amplitude of the anomalies in- studied by Haxby and Weissel [1986], but as shown in Figure
creaseswith time, especiallyafter the cells ceaseto grow very 11, there is a positive correlation betweengravity and topograpidly. This is becausethe effect of temperaturevariations in raphy as would be predictedby this model.
1970
BUCK AND PARMENTIER:CONVECTIONBENEATHOCEANIC LITHOSPHERE
GRAVITY
i
i
COMPONENTS
i
i
i
I
i
assumedplate velocity of 4 cm/yr. Some of the profilesusedto
constructthis figure are shown in Figure 3. No flexural damping was included.As seenbefore,the wavelengthsof the signal
increase with time, and the amplitude also increasessome-
!
what.
I
I
,
, I
Numerical calculations with the same boundary conditions
as for the large box calculation (case 15) but with a periodic
initial temperatureperturbation were carried out for a number
I
of cases which are listed in Table 2. To illustrate
20mgals
!
!
I
I
I
,I
I
i
i
i
1
I
I
I
I
!
i
i
I
I
tion of convection
GT
I
i
I
I
Go+ GL+ GT
O.
400.
WIDTH
(km)
Fig. 8. The components of the model gravity signal described in
the text for case 15 at a time 15 m.y. into the calculation. The component due to deviatoric stressand pressure variations at the base of the
conductive lid is G•, that due to vertical normal stressvariations in
the lid is Gt•, and that due to density variations throughout the box is
Gr. No flexural damping of G• and GL was included.
The magnitude of the maximum difference in peak to
trough amplitude for the three components of the gravity
signal are shown as a function of time for case 15 in Figure 9.
Just as for the isostatic geoid anomaly the gravity anomaly
changesmost rapidly soon after the calculation is begun. The
component of the signal due to the flow-inducedstressesG•
grows very quickly at first but later maintains a nearly constant value. The componentresultingfrom lithospherictemperature variations G•. grows more slowly but continues to
grow through most of the calculation.This is partly due to the
increasing wavelength of the flow with time, which leads to a
larger contrast in the local heat flux from the convection cell
into the conductive lid. It also continues to increase with time
after the cell width has become constant because as the lid
thickens, the temperature variations extend over a greater
depth. The magnitude of the signal arising from density contrasts throughout the box (GT) is an almost constant fraction
of GL. The amplitude of GT is much smaller than that of GL
and is opposite in sign from G• and G•. The trend of G•
parallels GL becausemost of that signal originateswithin the
conductive
lid.
A contour plot of the total model gravity signalis shownin
Figure 10 for case 15. Distance is scaledwith time through an
these one-cell
calculations,the same quantities which were shown in Figure
3 for the large box calculation are shown in Figure 4 for case
20, which has the same viscosity parameters as case 15. Only
one time is shown. The single convection cell starts out penetrating only part way through the depth of the box and goes
through the stageof cell growth noted for case 15. Here there
is no increasein the width of the cell; only its depth extent
increases.For case 17, in which there was no pressuredependenceof viscosityand so no viscosityincreasewith depth, cells
rapidly filled the box, and slow downward penetration of the
cell was not observed.This shows that the viscosityincrease
with depth is important in controlling the depth of penetracells and therefore
the rate of increase of
the convectivewavelength.
In cases 17 and 19 the initial single cell broke down into
two cells. Case 19 has the same viscosityparametersas case
20, but the width of the convectingregion is twice as great (see
Table 2). Case 17 has the same box size as case 20. Cell
breakdown occurred because the initially preferred wavelengthof instabilitywas smallerthan the box width.
Several general relations between the geophysicalobservables calculated for this set of models should be pointed out.
One of the most obvious is that the rate of change of the
temperature structure of the conductive lid varies inversely
with the average viscosity in the boundary layer and also
dependson the activation energy E. The convectiveheat flux
controls the variation of the average lid temperature TL with
time (Figure 5). For case 14, which has an average viscosity5
times that of case20, T•. decreasesmore slowly from the conductive value. This sameslow changeis clearly seenin the rate
of decreasein dH/dt shown in Figure 7 and in the slow increasein the amplitudes of the gravity componentsin Figure
9. For case 18, which has nearly the same initial viscosity in
the region below the boundary layer but a lower activation
energy E, growth is faster than for case 14. The decreased
temperature dependenceof viscosity for case 18 results in a
larger heat flux to the baseof the conductivelid.
As noted before, the size of the convection cells also affects
the rate of heat transfer from the convecting region to the
conductinglid. Case 22, which has a box half the width of case
20 but with all other parameters the same, showed a much
faster decreasein T•.. The average advective heat flux for the
smallerwidthboxwasgreaterby a factorof about(2)x/2when
the viscositieswere the same in the boundary layer region.
The averagelid temperature T•. and the slope of the plot of
the lid thickness versus t •/2 remains constant for most of the
calculationsafter about 30 m.y. of model time. This is a consequenceof the negativefeedbackor self regulationof the convecting system [Tozer, 1965]. The higher the advective heat
flux, the more quickly the convectingregion cools. Cooling
causesthe viscosityto go up and the heat flux then goesdown.
When T•. is nearly constant, the deviation of T• from the
conductive value (see Figure 5) varies inversely with the
average viscosity.Since the rate of advective heat transfer is
controlled by the average viscosity in the convecting region,
the lower the viscosity,the higher the heat flux.
BUCK AND PARMENTIER' CONVECTION BENEATH OCEANIC LITHOSPHERE
[
I
i
I
i
1971
i
34
CASE 15
/
3O
/
I
/
26
/
CASE
/
22
I
I
I
I
I
26
20
22
/
/
//
18
E
18
I--
14
/
14
/
/
/
10
i
i
I
I
i
I
20
40
60
80
100
12o
lO
n-
6
G
///
20
TIME (Ma)
(a)
>.
(b)
•
40
T
i
i
i
60
80
100
I
120
TIME (M a)
Fig. 9. The maximum peak-to-troughamplitude of the three componentsof the gravity signal,definedin the text, as
they vary with time for two of the casesconsidered.G• and GL are of the samesign,while Gr is oppositein sign.In case15
the magnitudeof the signalsrelatedto temperaturevariations,Gr and GL,increasesteadilywith time as the wavelengthof
the cellsgrow. For case20 the cell width is fixed, and the magnitudeof thesesignalsdo not increaseas rapidly.
The subsidenceS(t) is nearly linearly dependent on •., and
the isostaticgeoidheightH(t) is proportionalto •.2.It is therefore important to consider the effect on ). of variations in the
model parameters.Case 14, with the highestvalue of reference
viscosity(Table 2) of all the cases,has the highestvalue of •.. It
follows that this case also has the highest value of TL, the
highestrate of subsidence,
and the largest averagevalue of
dH/dt. Decreasingthe temperaturedependenceof viscosity,by
lowering the activation energy E as in case 18, decreases4.
The average viscosityin the isothermal region is nearly the
same at the start of the calculation
for both case 14 and 18.
However, for case 18 with a smaller E, the viscosity does not
increaseas rapidly as the convectingregion cools. Therefore
the advectiveheat flux doesnot decreaseas rapidly as for case
14.
Figure 9. One apparent result is that the component due to
flow stresses(Go) is fairly constant in amplitude after an early
period of change. The early rate of change of this signal is
greater for the caseswith lower viscosity in the convecting
region. The magnitude of the constant level of Go does not
vary much with average viscosity in the boundary layer
region, but it is greater with a larger wavelengthof the flow
and with a smaller value of the activation energy E. The component of the gravity anomaly which dependson the stresses
producedby temperaturevariationsin the conductivelid GL
increasescontinuouslywith time for all the cases.The rate of
increaseis greaterfor the longer wavelengthcases.Finally, the
part of the gravity signal arising from density differences
within the lid and convectingregion Gr tends to parallel G•
but is generallylower in amplitude and oppositein sign.
Lowering the pressuredependenceof viscosity by reducing
the activation
volume
V has much
the same effect on •. as
lowering the temperaturedependenceof the viscosity.As the
depth to the baseof the boundary layer increaseswith time of
cooling,the viscositythere will be increasingbecausepressure
is proportional to depth. The viscosityin the boundary layer
controls
activation
the advective
volume
heat flux. Thus for case 17 where the
V is small
the heat
flux
decreases
at a
slower rate than it would if V were larger. Like increasing V,
reducing the depth extent of the convective cooling increases
the value of 4. This is shown by case23 for which the depth of
the box is 3/4 of that for the other cases.
The model parameterscontrol the time variation of the
gravity signal produced by convectionin a way that does not
scalesimply with the parameter ).. The amplitude of the components(Go, GL, and Gr) is shown for severalof the models in
DISCUSSION
The amplitudes and wavelengths of the gravity signals
shown in Figure 3 for case 15 are in the range reported by
Haxby and Weissel [1986] in their analysis of gravity features
derived from Seasat altimetry data for the central east Pacific.
The amplitude of the total gravity anomalies for all the small
box calculations were also in this range for at least part of the
time duration of the calculations. Figure 11 shows profiles of
gravity and bathymetry perpendicular to the trend of linear
gravity anomalies observed by Haxby and Weissel. The observed anomalies have a wavelength of 150-250 km and a
peak-to-troughamplitudeof 8-20 x 10-5 (m/s2) over ocean
floor older than 5 Ma. The highs and lows of these features
make linear trendsin the direction of plate motion.
Two important features of the calculated model gravity
1972
BUCK AND PARMENTIER' CONVECTION BENEATH OCEANIC LITHOSPHERE
TIME (M a )
5
I0
15
20
25
-6
-2
..................
..........................................:::.•::.............................
........
i'"'"'"":::
:•'!:•":
................ :•
--
200
................................... :.'-'.•
.......
.":-':::::C':':':':'i':':':':':':':':
........
........... '.'.'.'.'.'.'.'.'.':':':i:'::...'.•
•:
----'-'•' ""'•:':'"':':':':':::::i:i:i:i:!:i:i:!:!:i:i:i:i:i:!:!:!:i.::•:
-2_
:2•.7.•:.c.,.:
-6
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=========================================
_$) •_•
-.-..••
ß ::.............:....:.:.......::
:::.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:..:.....-..•.c,.,....,.•
....
'.....
"'"""•':':':•':"•:•:•'_."!i:.•.
•:'.":':'.'.-.."•::_•
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o================================
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%%
.•. ,-, -2
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400
-4 •
- 4 -,._ -•___-'"-'-'
--'-"-
,,..i;•:;.;;,%"-----..••
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".'....:'....:•
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ß......... .'.-......... •
.....
•'.'.'.'.'.'
_
,
-4 - '"
,
400
600
DISTANCE
-.
,
: •
800
I000
( km )
Fig. 10. Contour map of the total model gravity for case15 for model timesrangingfrom 4 to 25 m.y. after the start of
the calculation. Regionsof positive gravity anomaly are shaded.Time is related to distanceby assuminga plate velocity of
4 x 10-2 m/yr. The contourintervalis 2 x 10-5 m/s2 (mGal).Flexuraldampingof the signalswasnot included.Note that
the wavelengthincreasesrapidly with time along with a moderateincreasein the amplitude.
anomalies do not match the data. First, the increase in the
wavelength of the anomalies with age observedby Haxby and
Weisselis lessthan that predictedby the resultsof case15 (see
Figure 10). Second,when the effect of flexural damping due to
the elastic lithosphere is included in the calculation of the
model gravity signals, their amplitude for wavelengthsless
than 250 km become smaller than the observed signals.The
elastic lithosphere will damp signals as a function of their
wavelength. To illustrate this, we convolve a filter defined by
McKenzie and Bowin [1976] with the model anomaly compo-
nents G,and G•. to approximate the effect of the elasticlithosphere.Filtered model anomalies for different assumedvalues
of the lithospherethicknessat one time for case 15 are shown
in Figure 12. Using the elasticlithospherethicknessas a function of age, estimated by Watts and Steckler [1980], signals
with wavelengthsless than 250 km will be damped by more
than 90% for lithosphereagesgreater than 15 m.y. Only when
the small-scaleconvectivewavelengthsare greater than 400500 km will the effect of the elasticlithospherein damping the
signalsof G,and G•. becomesmall for all lithosphereages.
SEASAT
MGALS
GRAVITY
I lO
METERS
500
TOPOGRAPHY
SCALE
•
'
•bo
'
/600
Fig. 11. Seasat-derivedgravity anomalies,filtered shipboardgravity anomalies,and seafloortopography perpendicular to the trend of linear gravity highs and lows observedin Seasatdata by Haxby and Weissel[1986]. These profilesare
centeredon 0øN latitude and 145øW longitude and are oriented roughly N-S. The shipboard gravity and topography has
been filtered to remove signals with frequencies too high to be represented in Seasat data. Vertical arrows show the
location
of fracture
zones.
BUCK AND PARMENTIER: CONVECTION BENEATH OCEANIC LITHOSPHERE
FILTERED
CASE 15
GRAVITY
1973
is that the low observed values of dH(t)/dt may be related to
convectioninducedby differencesin lithospherethicknessand
horizontal temperature gradients across fracture zones. This
may cause faster homogenization of the asthenospheretemperatures and lithosphere thicknessesin the vicinity of the
TIME 10 M0
fracture
zone.
Thesurface
heatfluxQs(t)shouldvarylike(T,,/,•)(K%/t)
•/2.
Uncertainties in our knowledge of the conductivity K, the
specificheat % [Schatzand Simmons,1972; Goranson,1942]
and the measured values of heat flux are sufficiently large that
data on oceanic heat flow can be matched by a variety of
models including those presented here.
The comparison of the model results to data on the subsidence of the ocean basins is of great interest. For small-scale
convection to be associated with the gravity and geoid features just discussed,it must develop in the first few million
years after the lithosphere starts to cool. This means the onset
of small-scaleconvection cannot produce the change in slope
of the subsidence-agerelation at about 70 Ma, as suggestedby
Parsons and McKenzie [1978] and Houseman and McKenzie
[1982]. A number of alternative explanations for this feature
of the subsidencedata have been given [Forsyth, 1975; Schubert et al., 1976; Parmentier and Turcotte, 1977; Heestand and
Crough, 1981; Jarvis and Peltier, 1982; Fleitout and Yuen,
1984], all involving a heat flux from the mantle brought to the
10
regals
I
4. km
baseof the lithosphereby either convectionor conduction.
Our main interest is the early evolution of the oceaniclithosphere where the effect of a heat flux from deeper in the
mantle should be negligible. We have shown that the rate of
12. km
O.
subsidence
400.
Fig. 12. The effect on the total model gravity signal (G. + Gr.
+ Gr) of flexural damping due to elastic lithospheres of different
thicknessesfor a time of 10 m.y. into the calculation of case 15. The
assumedthicknessof the elasticlithosphereH is given at the top of
eachplot.
The elastic lithosphere acts to support topography in the
same way that it suppressesthe topographic expressionof
convectivestresses.This may explain how small-scaleconvection can result in the observed pattern of short wavelength
gravity anomalies.Topography produced by convectionwhen
the elastic lithosphere is thin can be "frozen" into the lithosphereas its thicknessand therefore flexural rigidity increase
with age. This topography and the associatedgravity anomalies should not change greatly even as the convectivepattern
beneath the lithosphere changes. These gravity anomalies
would have a linear trend in the direction of plate motion as
do those
observed.
The
stresses due to convection
due
to vertical
thermal
contraction
in the litho-
sphereshoulddependon ,•o•rm(Kt)
•/2 for coolingof the mantle
WIDTH (kin)
must
be
well developedbefore the flexural rigidity of the lithosphere is
large enough to suppresstheir topographic expression.From
the model results we estimate that this requires asthenosphere
viscositiesless than about 10•8 Pa s under young oceanic
in the absenceof heat sources.The average subsidenceof the
North
Atlantic
and
the North
Pacific
ocean
basins
as esti-
mated by Parsons and Sclater [1977] can be fit by a model
with viscositieslow enough to produce the gravity signals
discussedabove. Since the cooling of the asthenospheremay
affect the subsidence, further work is being done to relate
predictions of this model to data.
CONCLUSIONS
These calculations
have shown
that small-scale
convection
can produce the magnitude of the short-wavelengthgravity
anomalies
observed
for at least one area of the oceanic litho-
sphere and can also be consistent with seafloor subsidence
data. Subsidencethat is linear with t •/2 is reproducedby
model results,but the rate of subsidencedependson the vigor
of the small-scale
convection.
Oceanic
heat flow can also be fit
with a model which includes small-scale convection. The geoid
height offsets acrossfracture zones is more nearly matched by
our results than by a model that includes only conductive
cooling. The effect of convection due to differences in lithosphere thicknessacrossfracture zones may explain the geoid
data more completely.
lithosphere. Such values are not inconsistent with an esti-
If small-scale convection can explain observed short-
dependenceof viscosityis taken into account.
The calculated derivative of the model isostatic geoid height
with time (Figure 7) reproduces the early trend in the data on
geoid height offset across fracture zones. However, to match
the magnitude of the change in dH(t)/dt reported by Cazenave
[1984] requires a lower viscosity than in any of the models
consideredhere. Since the value of dH(t)/dt should scale with
required. First, convection must begin in the first few million
years after formation of the lithosphere at a mid-ocean ridge.
In this case, viscositiesbeneath the lithosphere must have a
minima around 10TMPa s. Second,topography,which gives
22tc•ZTm,
it is possibleto find a combinationof theseparame-
models
ters which match both the average rate of subsidenceand the
rate of changeof the isostaticgeoid height. Another possibility
small-scaleconvection beneath the lithosphere increase with
time. This would result in a much larger increasein the wave-
mated averagemantle value of 102• Pa s when the pressure wavelength gravity anomalies in the oceans, two things are
rise to the observed gravity anomalies, must be produced by
the stresses associated
with convection
when the elastic litho-
sphere was thin enough that it could be easily deformed. Our
also
show
that
the vertical
and
horizontal
scales of
1974
BUCK AND PARMENTIER: CONVECTION BENEATH OCEANIC LITHOSPHERE
length of the gravity anomalies than is observed.Topography
must be supported by the strength of the elasticlithosphereas
it cools and thickens, and to preserve a relatively constant
wavelength, it must be "frozen in."
Acknowledoments. Thanks to Nail Toksoz for support in doing
this work. Discussions with Barry parsons and the reviews of Luce
Fleitout and Ulrich Christensen improved the manuscript. W.R.B.
was supported on an Exxon Teaching Fellowship and a Lamont
postdoctoral fellowship. E.M.P. was partially supported by NASA
grant NAG 5-50 and National ScienceFoundation grant OCE-8208634.
REFERENCES
Batchelor, G. K., An Introduction to Fluid Dynamics,Cambridge University Press,New York, 1967.
Buck, W. R., Small scale convection and the evolution of the lithosphere(abstract),Eos Trans. AGU, 64, 309, 1983.
Buck, W. R., Parameterization of the effects of convectionin a variable viscosityfluid with application to the earth, Geophys.d. R.
Astron. Soc.,in press,1986.
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E. M. Parmentier,Department of GeologicalSciences,Brown University, Providence, RI 02912.
(Received February 15, 1985;
revisedOctober 29, 1985;
acceptedOctober 31, 1985.)