JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 99, NO. B5, PAGES 9543-9559, MAY 10, 1994
Diapiric ascentof magmasthroughpowerlaw crustand mantle
RobertoFerrezWeinberg
• andYuri Podladchikov
2
TheHansRambergTectonicLaboratory,
Instituteof EarthSciences,
UppsalaUniversity,Uppsala,Sweden
Abstract. Therehasneverbeena convincing
explanation
of thewayin whichdiapirsof molten
granitecaneffectively
risethrough
mantleandcrust.We argueherethatthisismainlybecause
thecountry
rockshavepreviously
beenassumed
tobeNewtonian,
andwe showthatgranitoid
diapirsrisingthroughthermallygradedpowerlaw crustmayindeedriseto shallowcrustallevels
whilestillmolten.Theascentvelocityof diapirsis calculated
throughan equation
withtheform
of theHadamard-Rybczynski
equation
for theriseof spheres
through
Newtonian
ambientfluids.
Thiswell-known
equation
iscorrected
by factors
dependent
onthepowerlawexponent
n of the
ambientfluid andtheviscositycontrast
betweenthedropandtheambientfluid. Thesecorrection
factors
werederived
fromresults
reported
in thefluidmechanical
andchemical
engineering
literature
for theascentof Newtonian
dropsthrough
powerlawfluids.Theequation
allows
calculation
of theascent
ratesof diapirsby directapplication
of rheological
parameters
of rocks.
Thevelocityequation
isnumerically
integrated
fortheascent
of diapirsthrough
a lithosphere
in
whichthetemperature
increases
withdepth.Thedepthof solidification
of thediapiris
systematically
studied
asa function
of thegeothermal
gradient,
buoyancy
of thebody,solidus
temperature
of themagma,andrheologicalparameters
of thewall rock. The resultsshowthat
whenthewallrockbehaves
asa powerlaw fluid,thediapir'sascentrateincreases,
withouta
similar
increase
intherateofheat
loss.Inthisway,diapirs
rising
at10to102 m/yrcanascend
intothemiddleor uppercrustbeforesolidification.Strainratesoftening
ratherthanthermal
softening
isthemechanism
thatallowsdiapirism
tooccurat suchrates.Thethermal
energyof
thediapiris usedto softenthecountry
rockonlyat latestages
of ascent.Thetransport
of
magmas
through
thelowercrustandmantleasdiapirsis showntobeaseffectiveasmagmatic
ascentthroughfractures.
Introduction
1990; Kerr and Lister, 1988; ROnnlund,1989; Lister and
Kerr, 1989;PolyakovandPodladchikov,
1990;Weinberg,
Silicic volcanismdemonstrates
that granitemagmas
1992].
aroundgranitoid
commonlyreachshallowcrustallevels. Many authors Severalworkershaveusedthestructures
challengediapirismasa feasibleascentmechanismbecause plutonsto concludethat the viscosityof the plutonsmust
it hasneverbeenconvincingly
demonstrated
thatmagmatic havebeensufficientlysimilarto their wall rocksduring
diapirscan reachthe shallowcrustin geologically their Stokes-likeascentandthat the granitoidsmusthave
been highly crystallized [Berger and Pitcher, 1970;
Ramberg, 1970; Soula, 1982; Bateman, 1984]. Another
Mawer, 1992]. Theascentof a diapirmaybe closely
approximated
by theslowtranslation
(negligible
inertia)of groupof workersfollow Grout [1932] and suggestthat
reasonabletimes[e.g.,Patersonet al., 1991; Clemensand
a viscous
dropthrough
a viscous
fluid [e.g.,Marsh,1982;
deformation of Newtonian wall rock is confined to narrow
Schmeling
et al., 1988;Cruden,1988,1990;Weinberg, aureoleswhich are thermallysoftenedby heat from the
1992]. The flow of an infiniteNewtonian
fluidpasta aliapit (hot-Stokesmodels [Marsh, 1982; Morris, 1982;
sphere(Stokes flow; Figure 1) is a well-documented Ribe, 1983; Daly and Raefsky, 1985; Mahon et al.,
problemin fluid dynamics[e.g., Batchelor, 1967], and 1988]). Accordingto calculationsby Marsh [1982] a
therefore
moststudies
dealingwiththeascentof diapirs sphere3 km in radiuswouldbe ableto riseonlyhalfway
simplifythewallrockto a Newtonian
fluidanddisregard through a Newtonian lithospherebefore solidifying.
bodiesfollowingthe samepath, while it is
theeffectsof temperature
[Biotand Odd,1965;Whitehead Subsequent
andLuther,1975;Marsh,1979;MarshandKantha,1978; still warm, could reach higher levels. The effect on
Ramberg,1981;Schmeling
et al., 1988;Cruden,1988, diapiricascentof increasingviscosityof the ambientfluid
hasbeenstudiedrecentlyby Kukowskiand Neugebauer
1Nowat Research
Schoolof EarthSciences,
Australian [1990].
However,it is well knownfrom laboratoryexperiments
National University, Canberra.
2permanently
at Institute
of Experimental
Mineralogy,that at naturalstrainrates,pressures,and temperatures,
RussianAcademyof Sciences,Chernogolovka.
mantleandcrustalrocksarelikely to behaveaspowerlaw
fluidsaccordingto
Copyright 1994 by the AmericanGeophysicalUnion.
• = A e-E/RTc•nforuniaxial
stress
(1)
(seenotationlist and Kirby [1983], Paterson[1987], and
WillcsandCarter[1990,andreferences
therein]).Whenn =
Paper number 93JB03461
0148-0227/94/93 JB03461 $05.00
9543
9544
WEINBERG AND PODLADCHIKOV: DIAPIRIC ASCENT OF MAGMAS
The equationscontroling diapiric ascentare integrated
numericallyto calculatethe depthand time of solidification
of magmaticdiapirsrising throughlithospherewith defined
geothermalgradients. Several parametersare found to
control the ascent. The diapir's velocity, temperature,
viscosity,and Peclet and Nusseltnumbersare calculated,
togetherwith the effective viscosityof the surrounding
mediumand the thicknessof the diapir'sthermalboundary
layer. The calculations take account of the effects of
thermal softeningof the wall rock by includingthe drag
correction derived by Daly and Raefsky [1985]. We
illustrate the approachby applying our equationsto two
examples: (1) a diapir with the well-constrained
characteristics of the Tara grantdiorite in Australia,
ascendingthrougha crustwith the propertiesof Westerly
granite [Miller et al., 1988], and (2) the rise of a 10-km
radiusmantle diapir from a subductingslab 100 km deep.
Figure 1. Streamlinesin and arounda spheretranslating This diapir risesto the baseof a 40-km-thick crustwhere
throughNewtonianfluid (a) viscoussphere(b) solidsphere it triggersthe formation of a crustal diapir that ascends
(Stokescase;redrawnfrom Schmelinget al. [1988]). The througha layeredcrust.
The results indicate that magmatic diapirs may rise
streamlines shown will concentrate towards a sphere
throughthe mantle or lower crustone orderof magnitude
translatingthroughpower law fluids.
fasterthanpredictedby earlier studies.In effect,diapirism
throughpower law surroundings
can accountfor magmas
1, strain rate varies linearly with stress, the constant reaching shallow crustal levels. In contrastto previous
viscosityis independentof the strainrate, and the fluid is results,we showhere that diapirscan rise from the Moho
said to be Newtonian. When n is larger than 1, as in most to shallow
crustin timespans
of only104to 105years.
rocks, the relationshipbetween strain rate and stressis Alternatively,melts can rise diapiricallyfrom the melting
nonlinear, and the effective viscosity of the material zone
of subducting
plates
tohighcrustal
levels
in 105to
decreaseswith increasingstrainratesor stress. Although 106years. This approximates
the delaybetweenthe
extrapolatingthe resultsof laboratoryexperimentson rock initiation of subductionand the start of arc volcanicity
theology to geological systems is difficult (see the [Marsh, 1982].
thoroughdiscussionby Paterson [1987]), rock theology
determinedby thesestudiesprobablybetterapproximates
the actual behavior than does the simplification to
Newtonian
behavior.
Previous
Studies
Somerecentstudiestry to assessthe effectsof powerlaw
wall rocks on diapiric rise rate [Morris, 1982; Mahon et
The slow translationof spheresthroughfluids has been
al., 1988;Miller et al., 1988; England, 1992]. Mahon et thoroughlystudied. Hadamard [1911] and Rybczynski
al. [1988] found low ascent velocities for diapirs rising [1911] extended to viscous spherical drops, Stoke's
through wall rock of olivine theology at crustal equationfor the velocity of a solid spherefalling through
temperatures. Miller et al. [1988] use the theology of an infinite Newtonian fluid:
Westerly graniteto calculatethe ascentrate of diapirsand
to estimatethe strain rates, with which they determinethe
(2)
effective viscosityof the wall rock, on the calculationsby
Mahon et al. [1988]. The influenceof powerlaw behavior
of the wall rock on diapirism has been hindered by the
absencein the literatureof a simpleequationfor therise of (see notationlist for symbols). Subsequentworks have
a Newtonian drop through power law fluids. Another treatedthe translationof spheresthroughfluids of other
hinderance is the lack of a rule for the definition
of
rheologies. Rather than the simpleuniaxial stressin (1),
effective viscosityof media with strongspatialvariationin the main interesthereis the translationof spheresthrough
1Apg
h)
V=5
Itr2(It
[,It
++3gsP
5gsphJ
strain rate.
power law fluids which, for two- or three-dimensional
mustbe describedby the more generalequation
This paperpresentsan equationfor the final velocity of stresses,
viscousbuoyant dropsrising throughpower law ambient [Turcotteand Schubert,1982; Wilksand Carter, 1990]
fluids. This equation is then applied to the problem of
(3)
magmaticdiapirsrising throughthe lithosphere.We start
O'ij:K• m-1•
by reviewing the fluid mechanical and chemical
m=1./n,
and• isthesecond
invariant
ofthestrain
engineeringliteratureon the rise of dropsthroughpower where
rate
tensor
eij:
law fluids and rewriting thosesolutionsin the form of the
well-known Hadamard-Rybczynski
equation(2). We then
extrapolatethe known solutionsto highervaluesof n that
are of interest in geology, and we show how the
rheologicalparametersof rockscanbe usedin the equation.
WEINBERG AND PODLADCHIKOV:
DIAPIRIC ASCENT OF MAGMAS
The velocity equationfor dropsrising throughpower
law fluidscanbe writtenas (seeAppendixA for derivation)
2 (APR)n rn+l
V = 9•
KnXn
.
(5)
9545
X(S)
=(g;l'5gsph•
(6)
+ gsph J
For the limiting casesof a solid spherefalling through
power law fluids, and for viscousdropsfalling through
Newtonianfluid, the dependency
of X canbe extrapolated
in a self-consistent way to all values of n and S =
Mhatre and Kintner [1959] foundexperimentallythatthe
by
terminal velocity of viscousdropsfalling throughpower [tsph/Yteff
law fluid is predictedby the apparentNewtonianviscosity
of the power law fluid. Crochetet al. [1984] summarizes
severalapproachesdescribedin the literatureto determine
3 }.teff
1
the valuesof X for solid spheresas a functionof n. More
recentwork on solidsphereswascarriedout by Dazhi and wheregeff is the effectiveviscosityof theambientpower
Tanner [1985] andKawaseand Moo-Young[1986]. The law fluid
formeralsostudiedhow walls of differentgeometriesaff•t
the velocity of spheres. The studyof motion of inviscid
K n 6n- 1
(GMGgeff
[teff
+
1.5[tsph)
V=1Aogr2
(X•o
+[tsph
• (7)
=
bubbles in •ower law fluids started with Astarita and
Apuzzo [1965], who used semiquantitativeargumentsto
find the expressionfor the drag coefficient. Nakano and
Tien [1968] foundanalyticallytheupperboundof the drag
coefficient of a Newtonian spherical drop of variable
viscositymoving througha power law fluid of n valuesup
to 1.667 (m=0.6). Bhavaraju et al. [1978a, b] found that
the ratio betweenthe velocity of a swarmof bubblesand a
single bubble increasesas n increases. This result was
extendedto high n valuesby Chhabra [1988], who found
that for n > 2, swarmsof bubblesmay rise fasterthan a
singlebubble,and for n=3.33, bubble swarmsmay rise
twice as fast as a singlebubble.
The transfer of the heat or mass of spheres to
surroundingpower law fluids was studiedby Hirose and
Moo-Young [1969] and Kawase and Moo-Young[1986,
andreferencestherein]. We useherethe resultsobtainedby
HiroseandMoo-Young[1969] to calculatetheheattransfer
from the sphereto the ambient fluid. The efficiency of
heat (or mass)transferis usuallydescribedas a dependency
of the Nusselt number Nu on the Peclet number Pe.
Since
neitherof theseparametersdependon the theologiesof the
fluids, they retain the samedefinition for both power law
and Newtonian fluids (see notation list). Although the
definitionof Nu and Pe neednot be changedfor power law
fluids, the relationshipbetweenNu and Pe doesneed to be
corrected by a factor that depends on the only
dimensionlessparameterin the theological law, namely,
thepowerlaw exponentn (seeequations(19) and(20)).
Rewriting Known X Values to Fit the
Hadarnard-Rybczynski Equation
(8)
geff (Apgr)n_
1
and the parametersXsol,M, and G are functionsof m=1In
Xsol= 1.3 (1-tn2)+ m
M = 0.76+0.24m
G = 2.39 - 5.15tn + 3.77tn2.
(9)
Whereas Xsol correctsthe velocity for a solid sphere,a
combinationof Xsol, M and G is necessaryto correctthe
velocity of a viscousdrop. M and G were obtainedby
recalculating
the dataof Nakanoand Tien [1968] in orderto
fit
(7) (Figure 2).
Values resulting from the
multiplication of Xsol and M equal 3/2 of the correction
factorfor theinviscidsphereof Nakanoand Tien[1968] (Y
in their Figure 1).
Effective viscosityis usually defined by assumingsome
value of the strain rate. It is impossibleto make suchan
assumptionin the casestudiedhere. Instead,it is possible
to calculate the characteristic stress o=Apgr, for the
uniaxial condition (1)
o
geff-
•
o
=
Kn
Kn
-
Ae-E/RT
on on-1- (Apgr)
n-1
The constant6n-1 that appearsin (8) resultsfrom our
particular geometry and gives a quantitatitvemeaning to
our definition of effective viscosity. If we consider a
buoyantdrop and substitutethe ambientpower law fluid
with a Newtonianfluid with viscosityequalto the effective
viscositycalculatedin (8), the ascentvelocity will remain
the same [Mhatre and Kintner, 1959]. This definition of
effective viscosity also predictsthat the transitionof the
drop velocity from the solid to the inviscidregime occurs
at viscositycontrastsS--l, similar to Newtonianfluids (for
The velocity of a viscous sphererising through power which S is simply the ratio of Newtonianviscosities).
law fluids may be calculated from (5) if the correction
Equations(5) and (7) are essentiallysimilar; they differ
factorX is known. The valueof X for solidspheres(called only in the definition of the effective viscosityand in the
X$ol here), is a known function of n. Equation (9) and expansion of X into two terms: Xsol and the viscosityFigure 2 showthe polynomialfitting of the valuesof X$o1 dependent term. Equation (7) reduces to Hadamardgiven by Crochet et al. [1984]. From the Hadamard- Rybczynski'sequation when n=l, and it fits the existing
Rybczynskiequation,the dependency
of X on the viscosity data well (Figure 2). There are two advantagesof our
ratio for Newtonian fluids (n = 1), follows from (2)
definition of geff: first, there is no need to fix the strain
Introduction
9546
WEINBERG AND PODIAd2X2HIKOV' DIAPIRIC ASCENT OF MAGMAS
1
1.6
- Ae-E/RT
3(n+1)/2
(10)
1.4
wherethe valuesof A in Pa-n s-1,E in J/mol,andn can
be takendirectlyfrom thegeologicalliterature[e.g.,Kirby,
1.2
1983].
The velocity of sphericaldiapiric bodiesrising through
anypowerlaw rockmaybe calculated
by (7), (8), and(10).
The next sectiondiscussesthe influenceof ellipticity of
the drop on the rising velocityas a functionof the power
law exponentn. Following that we describethe computer
codeusedto explorethe effectsof powerlaw rheologyand
lithospherical
temperature
gradientson diapiricsystems.
1.0
0.8
0.6
0.4
0.2
0
I
I
I
I
I
0.2
0.4
0.6
0.8
1.0
Velocity Versus E!!ipticity
1.2
m=l/n
The correction factors derived above are based on results
sphere
ß Xsol= 1.31+ m- 1.29m2
R= 0.9992
ß
R = 0.9910
M = 0.76 + 0.23 m
ß G=2.39-5.15m+3.77m 2 R=0.9981
Figure 2. The correctionfactorsfor spheres:Xsol from
Crochet et al. [1984] and G and M obtainedby the fit of
thedataof Nakanoand Tien [1968]by equation(7).
rate in order to estimate •teff, and second,the velocity
equationis similar to Hadamard-Rybczynski's
equation.
Since Xsol, G, and M always approximateunity, rough
estimationsof the velocityof naturaldiapirsmay disregard
these factors and simply use Hadamard-Rybczynski's
equationtogetherwith (8).
Procedure for Extrapolation
relatedto sphericalbubblesanddrops. Althoughthiswork
and all previousworks that attemptedto integratethe rise
of diapirs through the lithosphereassumethe spherical
diapirs [e.g., Marsh, 1982; Miller et al., 1988; Mahon et
al., 1988], diapirs may deform as they rise due to
heterogeneities
of the wall rock properties,interactionwith
other rising diapirs,asymmetricalthermalaureolearound
the diapir and, most importantly,the upward increasein
viscosityof the wall rock due to temperaturedecrease.The
latter effect causes diapirs to expand laterally on the
horizontalplaneand to shortenvertically(flat-lyingellipse
in Figure 3). In an earlierpaperWeinberg[1993] studied
the velocity of two-dimensional(2D) solidellipsesrising
throughpower law fluids and showedthat the velocity of
flat-lying ellipses decreasesas n increases,whereasthe
velocity of ellipses rising parallel to their long axes
(upright ellipses), increases. The velocity changewith
ellipticity is small for aspectratios up to 2, but increases
with
The values of X for inviscid spheresknown from the
n of the ambient
fluid.
The
ratio
between
the
velocity of a solid circle and a flat-lying solid ellipse of
literatureextendonly up to n=1.667(m=0.6 [Nakanoand suchan aspectratio is 1.5, for n=3. Although these2D
Tien, 1968]). In order to extend these to geologically resultsareonlyqualitativelyvalid for the 3D case,theyare
interesting values of higher n, we extrapolated the likely to be an upper bound for an oblate flat-lying
correctionvaluesby the samefunctionaldependence(9). ellipsoid.
Errors due to extrapolationare expectedto be only minor
Figure 3 extendsthe work of Weinberg [1993] to low
due to the shapeof the equationand to the smallvariations viscosity flat-lying ellipses and to higher n values,
in their values.
Further research in this area and the
simulatinginviscidmagmaticdiapirsflatteneddue to the
derivation of values for these factors to higher n could rise through a lithosphere with upward increase in
easilycorrectour extrapolation.
viscosity. The velocities were calculated numerically
using a finite differencecomputercode developedby H.
Schmelingand describedin Weinberg and Schmeling
[1992]. Low-viscosity ellipses show an enhanced
From Rheological Parameters of Rocks
differencebetweenthe velocity of circles and ellipsesas
to the Velocity Equation
comparedto similar resultsfor solid ellipses. However,
the velocity decreaseis still as small as a factorof 2 when
n=3 for an ellipse aspectratio of 2. The influence on
The geological literature contains several laboratory velocity of a weak spheredeformation(aspectratiosclose
studiesof the rheologiesof rocks [e.g., Kirby, 1983; to one) can then be assumedto be negligible in our
Paterson, 1987; Wilks and Carter, 1990] as well as calculations. In the mantle and in the deep crust the
rheologiesof magmasandmelts of differentcompositions isoviscouslines causedby the geothermalgradienthave
at diversetemperatures,pressuresand crystaland bubble largeverticalspacingas comparedto the radii of common
contents[McBirneyandMurase,1984;Speraet al., 1988]. diapirs. Thus in theseregionsdiapirsmay be assumedto
For simplicity, we assumehere that all magmas are be spherical.However,as diapirsrise to shallowercrustal
Newtonian fluids, and we calculate the value K n in (7) levels the vertical distance of isoviscous lines decreases and
lateralexpansionand verticalflatteningof diapirsbecome
throughthe equation
WEINBERG AND PODLADCHIKOV:
DIAPIRIC ASCENT OF MAGMAS
9547
t*=H2/k
4.5
.
3.5
O
inviscidellipsea/b=2
ß
solidellipsea/b=2
A
inviscidellipsea/b=1.5
V
inviscidellipsea/b=1.2
.
0
-
L*=
.
H
p. = Ap
T*=E/R
2.5
The remainingsix variables(geothermalgradientdT/dH,
Tsol,g, r, K n, and n) were used to form six independent
O
dimensionlessnumbers that control the final depth'of
crystallizationof thediapir:
1.5
0
1
2
3
4
5
6
n
the compositionalRayleighnmnberRa modifiedhere for
powerlaw fluid
Ra = (Apœr)
nr H
Figure 3. The influence of ellipticity on the velocity of
two-dimensional
objectsrisingthroughpowerlaw fluidsof
differentpower-lawexponentsn. The ellipsesrise parallel
to the short axis as shown in the insert.
k K n 6 n-1
where
gn =
The velocities are
3n+1/2 A
given as the ratio of the velocityof circles(Vc) to that of
geothermalgradientgt
the ellipse (Ve). "Inviscidellipses"correspondto ellipses thedimensionless
3 ordersof magnitudelessviscousthan the ambientfluid,
•T HR
and threedifferent ellipticities are shownfor n=5. "Solid
gt=•H
E
ellipses"correspondto ellipses3 ordersof magnitudemore
viscousthan the ambient fluid [from Weinberg, 1993].
All calculations were carried out for ellipses of equal the effective viscosityof the sphere
buoyancyand box width of 5a (a is the horizontalaxis of
[tsp
hk
the ellipse). Solid ellipsesare lessinfluencedby the shape
than inviscid ones, and both show that increasing n the
influenceof the shapeon the velocityincreases.
ApgrH•
the solidustemperature
T'sol- TsolR/E
more important. At theseshallowlevels our velocity
and the radius r'=r/H, and r•
estimates,
calculatedbelowfor wall rocksof n = 3, may be
overestimatedby a factor of 2 if their horizontal radius
becomestwice that of the vertical radiusbefore the diapir Table 1. ParametersControllingthe Ascentof the Tara
Granodiorite
solidifies. However, a decelerationduring late stagesof
emplacementwill have very little influenceon the depthof
Tara
solidificationof the diapir (seeFigure 5).
Granodiorite
The calculationscarried out below could be improved
and correspondmore closely to nature if both the shape
evolution of the diapir and the correction of diapir's r(km)
3.0
velocity as a function of its shapewere known. However,
on the basis of the discussion above, we believe that the
H(km)
assumptionof a spherical shape will not considerably Tinitmagma(øC)
change our results. We will show that a much more
important source of error in the calculations is the TsoI (øC)
relatively high error bar in the determinationof the flow k (m2/s)
law parametersof rocks.
Ap (kg/m3)
Computer Code
A computercodewas developedin order to integratethe
velocityequationfor theascentof spherical(or cylindrical)
diapirs through the lithosphere. The ascentrate of the
diapir consideredhere is controlledby 10 variables. Four
of these were chosen to be the characteristic dimensions
25.0
770230
650
7x10'7
550
Geothermal gradient(øC/km)
30
E (kJ/mol)
14 2
A (MPa'n/s)
2x10'4
n
2.0
Paramaters are fromMiller eta/. [1988]. The values of E, A,
and n are from Westerly granite [Hansen and Carter, 1982]
9548
WEINBERG AND PODIaM2K2HIKOV:DIAPIRIC ASCENT OF MAGMAS
illustratethe useof the program,and to give an idea of the
dependence of solidification depth on the several
parameters,resultsfor relatively well-known geological
systemsare presentedbelow.
The code considerstile effect of heat softening of the
wall rock on the velocityand calculatesthe coolingof the
diapirat eachascentstep(a detaileddescription
of thecode
and the equationsusedis given in AppendixB). We
studied systematic variations in five of the six
dimensionlessparametersthat control ascentand depth of
Application to Selected Geological Systems
The
Tara
solidification
ofthediapir.Wedidnotstudy
g'sph
because
As a first example, we simulatethe ascentof the Tara
granodiorite through the Cootlantra granodiorite in
the diapir was assumedto have a viscositywell below that
of the wall rock, behavingas an invis½idbody. Resultsfit
by the leastsquaresmethodarepresented
in AppendixC in
the form of an equationthat allowscalculationof the depth
of solidification as a function of all five parameters. To
a)
Granodiorite
southeasternAustralia [Miller et al., 1988]. We chosethis
diapir becausethe parameterscontrolling its ascentare
relativelywell constrained(Table 1). Figures4a-4c show
H.eff (Pa. s)
1015
1016
14
1017
I
1019
10la
i
i
14.8 km
16
37600y •
•_
'..,.
. •
ß....
..,,,,,,
J
• •,';'7=
2.0
18
V no heat softening
........... V heatsofteningfor7 > 2
Peff
24
260.01
•
•1
0.1
i
100
10
Velocity (m/y)
b)
•T (rn)
0
400
800
1200
1600
2000
14
16
T no heat softening
18
T heat softening
(5I- noheatsoftening
5-I- heatsoftening
•
22
24
26
700
750
800
850
900
950
1000
1050
m(K)
Figure 4. (a) Velocity and effective viscosityof the wall rock [left for spheresrising with and
without thermalsofteningof the ambientrock and constantdensitycontrast. Calculationof [left is
basedon (8) usingthe regional temperatureat the diapir'sdepthand ignoringsofteningdue to heat
releasedby the diapir. Thermal softeningof the wall rock is included in the calculation of drag
reduction.
(b)Regional
temperature
(T),temperature
of thesphere
(Tsph),
thickness
ofthemetamorphic
aureole(fit), andaureoletemperature
(Tmet)asa functionof depth.The viscosityof thesphereincreases
as it rises,but is alwayssufficientlylower thanthat of its wall rocksfor it to be consideredinviscid.(c)
Pe and Dr.
WEINBERG AND PODLA[X2HIKOV:
DIAPIRIC ASCENT OF MAGMAS
9549
Drag reduction(Dr)
c)
0.3
14
0.4
I
0.5
'•1
0.6
0.7
I
I
0.8
I
0.9
1.0
I
I
1.2
16
18
•
Pe no heat softening
Pe heat softening
Drag reduction(Dr)
20
22
24
26
1
10
1O0
1000
104
Log Pe
Fig. 4. (continued)
the results of calculations for two cases, one in which the
maximum
rise rate is correctedfor thermalsofteningof the wall rock
and one without suchcorrections.Both diapirsrise from a
startingdepth of 25 km at 28 m/yr but rapidly decelerate
following the sameevolutionarypath (Figure4a-4c) until
7=2. At this point, the code startsto calculate the drag
reduction(Dr) for the thermallysoftenedcase,causingthe
abruptaccelerationat a depthof 17.5 km (Figure4c). This
unnatural jump arises only because of the absence of
solutionsfor Dr when ¾<2. In naturethe velocitiesof the
two casesare expectedto gradually diverge, so that the
thermally-softenedcase finally crystallizes at a slightly
more shallow depth than predicted here. The depth of
The unrealistically high Pe just before solidification
(Figure4c) resultsfrom theconstantdensitydifferenceAp
usedthroughoutthe calculation.In nature,Ap is expected
to decrease as the cooling magma crystallizes. To
illustratethis, we calculatedthe ascentof the samediapir
solidification
of bothdiapirs(i.e., whenTsph= Tsol=
650øC) is around 15 km, and the thermally softenedcase
rises only 300 m further than that without thermal
softening(Figure4a).
The diapir rises significantly more slowly at higher
crustallevels,as I•eff increases,with decreasingT (Figure
4a). Two thirds of the total ascenttime is spentin rising
the last 3 km before solidification.The temperatureprofile
of thecrustanddiapir(T andTsph,
respectively),
andthe
thickness
andtemperatures
of themetamorphic
aureole(fit
and Tmet, respectively) are summarizedin Figure 4b.
During the first 5000 years the diapir cools 25øC as it
ascends7.5 km through a crust cooling accordingto the
geothermalgradient (in degreesCelsius per kilometer).
The maximumtemperaturedifferenceis 230øC at a depth
of 17 km.
From then on, the diapir cools at
approximatelythe samerate as the geothermalgradientbut
is 210øC warmer than its wall rocks, and it eventually
solidifies at a depth of 15 km. The diapir rises so fast
throughrocks so hot in the first 7-8 km of ascentthat it
losesvery little heat. As the diapir slowsat highercrustal
levels, the cooling rate per risen meter increasesand the
thermal disturbance.
butwithalinear
decrease
inApfrom550kg/m
3 tozeroat
solidification(Figure 5). ComparisonbetweenFigures4a
and 5 showsthat lower Pe leadsto solidificationoccurring
500 m deeperafter slowerascentandthatthermalsoftening
becomesmore importantat final emplacement.
Systematic study of the way in which various
parameterscontrolthe depthof solidificationof a Tara-like
granodioritewas carriedout by varyingone parameterand
keeping all others constantas in Table 1 (Figure 6).
Changesin the two valuescontrollingthe buoyancyof the
diapir have little influence(Figure6a) bat, as is clear from
(7), the radius r is more important than Ap. The
logarithmicshapeof the dependencies
indicatesincreased
controlof buoyancyon the solidificationdepth. Initial and
solidustemperatureof the magmaalsohavelittle influence
on the solidificationdepth(Figure6b).
The main parameterscontrolingthe solidificationdepth
are A, E, and n (Figures0c and 6(1). All threeparameters
werevariedhereby theuncertainties
of thevaluesobtained
in laboratoryexperiments(for example,seetherheologyof
olivine of Kirby [1983]). Increasingn haslittle effect but
leads to shallower emplacement for the Tara-like
granodiorite. The most importantcontrolson the depthof
solidificationstudiedhere are A and E. A likely spanfor
the preexponentialparameterA causeschangesin the final
plutondepthof nearly 5 kin, whereasthe activationenergy
E changesit by nearly 16 km. At one extreme of E, the
diapir hardly leaves its source; at the other extreme the
diapir almost reaches the surface. Miller et al. [1988]
metamorphic
aureolewidensconsiderably
(fit,Figure4b). estimatedthe solidificationdepthof the Tara granodiorite
above,thisdepthcan easily
At this stage, Pe decreases below 100 and thermal to 10-12 km. As demonstrated
softeningbecomesmore and more important(decreasein be obtained by adjusting slightly the values of the
Dr, Figure 4c). Eventually,the diapir solidifiesand all its theological properties of the wall rock (the Westerly
remaining heat is conductedto the wall rock, causing granite).
9550
WEINBERG AND PODLADCHIKOV:
DIAPIRIC ASCENT OF MAGMAS
!.tef
f (Pa. s)
10 •
14
10 •?
I
10 •
I
10 •
I
1020
I
16.1 km 35600y
16-
....................
...............
. --- _ __
16.4km
44100y-'-• "•'•'-,,,....•
V no heat softening
......... V softeningfor 3'->2
16-
x•..
%fi no heat softening
tZeff heat softening
----
•',.
22
24-
I
26
I
10-5
I
I
0.001
I
I
0.1
10
Velocity (m/y)
Figure 5. V and •teff for spheresrisingwith and withoutthermalsofteningof the ambientrock where
Apdecreases
linearly
from550kg/m
3 tozerojustbefore
solidification.
Diapiric
Ascent From a Subducting Slab
This calculation is a simplified model of the rise of a
batch of basic magma from a subductingslab. The batch
rises through the overlying mantle to underplatelight
continental
crust below
the Moho.
The arrival
of the
mafic magmaat the Moho meltsa smallersilicic diapir
which ascends through the continental crust. The
a)
parameters
usedin the calculations
are listedin Table 2. Iu
the first 60 km we use the geothermalgradient of the
mantle and the theologyof olivine [Kirby, 1983]. The
continental
crust is 40 km thick and divided
into two
layers: a lower 15 km formed by Adirondacksgranulite
(rheologygivenby Wilksand Carter [1990]) andan upper
25 km of Westerlygranite(theologygivenin Hansenand
Carter [1982]).
Ap(kg/m
3)
300
250
13.0
350
I
400
450
I
I
I
500
550
I
I
600
650
Ap Depth = 2244 (12.9 -In
r
Depth = 5740 (10.6 -In r)
14.0
15.0
16.0
•
2000
I
2500
.
I
3000
I
3500
!
4000
4500
r (m)
Figure6. Systematic
variation
of parameters
controlling
thedepthof emplacement
of a Tara-like
granodiorite:(a) r andAp, (b) initialandsoliduste•nperature
(TinitandT$ol,respectively),
(c) E andA,
and(d) n. Eachparameterwasvariedindividuallywhile theotherswerekeptconstant(seeTable 1).
WEINBERG AND PODLADCHIKOV:
DIAPIRIC ASCENT OF MAGMAS
9551
Tinit(øC)
b)
760
780
800
i
I
14.0
820
840
I
I
I
660
I
680
14.5
•.15.0
15.5
-
Tinit
Depth= 6570 (8.9- In Tinit)
Tsol
Depth= 26598 log (Tsol) - 59948
16.0
I
620
6OO
I
640
700
T sol(øC)
c)
E (1OsJ/mol)
0.8
6.0
1 .o
1.2
I
I
1.4
1.6
1.8
I
I
I
x
2.2
I
Depth
=8.18x103
A'0'0709
Depth
= 1.962
x 104(InE- 11.1•'
8.0
x
2.0
E
10.0
x
x
\
12.0
\
14.0
16.0
18.0
20.0
I
22.0
10 's
I
lO-3
lO-•
10 '2
LogA (MPa-nS'1)
Fig. 6. (continued)
In this calculation we have not built in any pause by this layer to initiate its ascent as a Rayleigh-Taylor
betweenthe arrival of the mantle diapir at the Moho and instability is unknown and could be similar to the total
the departureof the crustalsilicicdiapir; instead,the diapir ascent time.
The results for this system (Figure 7) show a diapir
is assumedto rise straight through the Moho despite its
changein chemistry.The appropriate
pausecouldbe added rising through60 km of mantle in 0.18 m.y. (Figure 7a)
to the total ascent time if its duration were known.
and reachingthe baseof the crust 167 K warmer than the
Although Hupp_ertand Sparks [1988] calculatedthat it surroundingtemperature(Figure 7b). It then melts a
wouldtake102-103yearsfor a basicmagmato melta volume of acidic magma that rises as a sphereof 4 km
thatrisesin 0.17
layer at the bottomof the crust,the incubationtime taken radius(meltingprocessis not considered)
9552
WEINBERG AND PODLADCHIKOV:
DIAPIRIC ASCENT OF MAGMAS
d)
14.4
Depth =
' '
14.5
14.6
14.7
15.8 -
15.9
-
15.0
-
15.1
1.4
I
1.6
I
1.8
I
2.0
I
2.2
I
2.4
2.6
Fig. 6. (continued)
m.y. to a depth of 15.8 km, where it solidifies. Two
jumps in the velocity can be seenin Figure 7a. The first
correspondsto the change from mantle to lower crustal
diapir at a depthof 40 km, and the secondcorresponds
to
the changein rheologyfrom granuliteto Westerly granite
at a depth of 25 km. Despite the decreasein density
contrastthe sphererises through the granite 1000 times
faster than through the granulite at the transition depth.
This is becauseeffective viscosity of the granite is low
comparedto the granuliteat the sametemperature(Figure
7a). A smallerjump also occursin eachrock type, every
time ¾becomes> 2.0, and Dr begins to affect the ascent
velocity(Figure 7c).
Discussion
All theresultspresentedhererefer to diapirsrisingin the
absence of external tectonic stresses. Such stresses could
considerablyaffect our conclusionsand changethe ascent
velocity,depthof solidification,and shapeof thebody.
Table 2. Parameters
Controllingthe Ascentof a MantleDiapirThat Melts a ContinentalCrustalDiapir
Mantle Diapir
Lower Crust
(AdirondacksGranulite)
Upper Crust
(Westerly Granite)
r(km)
10.0
5.0
5.0
H(km)
100.0
40.0
25.0
Tinitmagma(øC)
1302
1001
Tsolmagma(øC)
850
650
650
k (m2/s)
10'6
10'6
10'6
Ap (kg/m3)
500
500
400
Geothermalgradient(øC/kin)
5
25
25
E (kJ/mol)
533
243
142
A (MPa'n/s)
6.34x104
8x10'3
2x10'4
n
3.5
3.1
2.0
Values of E, A, and n for the mantle are averagesfor olivine from Kirby [1983], Adirondacksgranulite from
Wilks and Carter [ 1990] simulatesthe lower crust and Westerly granitefrom Hansen and Carter [1982] the
upper crust.
WEINBERG
a)
AND PODLADCHIKOV:
DIAPIRIC
ASCENT
1016
0
1017
9553
Profile
Izeff (Pa. s)
1015
OF MAGMAS
1018
1019
1020
I
15800r•
Westerly
Ap = 400 kg/m3
3.56 10'"y
r=
5km
20
Adirondacks
¾= 2.0
Ap = 500 kg/m3
r=5km
40
Moho
j J
Mantle
(olivinerheoiogy)
Ap = 500 kg/m3
r= 10kin
80
100
0.01
1
0.1
lO
100
IOO0
Velocity(m/y)
---- --
Peri
,
Velocity
b) o
Westerly
granite
20
Adirondacks
granulite
40
Moho
Mantle
80
100
600
800
1000
1200
1400
1600
T (K)
Figure 7. A diapir risesfrom a subductingslab througha mantle with olivine theology. At its
arrivalat the Moho, themantlediapirmeltsa smallercrustaldiapirwhichthenrisesthrougha layered
crustformedby Adirondacksgranuliteat the baseandby Westerlygraniteat the top. (a) Velocity and
geffasa function
of depth,(b) Tsph,TmandTrue
t, (c) dragreduction
Dr when¾> 2; for smaller
7
values, Dr=-l.
Time and Velocity Constraints
significant. The upper time limit is in accord with the
solidificationtime of the diapirsmodeledhere. Marsh
[1982] found that in the subductionzone beneathScotia,
The velocity and time taken for diapiric ascentderived the responsetime of volcanism to a young subducting
here fit the few geologicalconstraints
availablereasonably plate is 1 m.y. He arguedthat the viscosityof the wall
well. Mahonet al. [1988] arguethatgranitoiddiapirshave rocks
in contact
withthediapirneeds
tobe1017to 1018
toascend
inmorethan104years
andlessthan105years. Pa s for diapirismto occurwithin reasonabletimes. Our
Thisisbecause
it takesmorethan104yearsforthermal calculationsdid not considerthetime to producebatchesof
softening
of thewallrocktobeeffective,
andlessthan105 crustalmagmafrom the mantlemagma. Nonetheless,the
years for large diapirs to solidify. This work showsthat
strain rate is so much more important than thermal
diapirs took a very reasonable0.35 m.y. to rise from
sourceto 15 km from the surface.The effectiveviscosity
softening
astorender
thelowerlimitof 104yearshardly of the wall rock approximatesthe requirementsof Marsh's
9554
WEINBERG AND PODLADCHIKOV: DIAPIRIC ASCENT OF MAGMAS
20
40
80
ß
lOO
0.2
•
,
,
i
0.3
,
,
,
•
i
,
,
,
,
0.4
i
•
0.5
ß
,
,
i
•
,
,
ß
0.6
i
•
0.7
•
,
,
0.8
Drag reduction(Dr)
Fig. 7. (continued)
calculations (Figure 7a). A diapir formed near the
subductinglithospheremust rise faster than 3cm/yr in
order not to be draggeddeeperinto the mantle with the
descending
plate [Marsh 1979]. The mantlediapirhasan
initial velocityof 190 m/yr, for the parameters
in Table 2
(Figure 7), and only diapirshaving a radiussmallerthan
1.4 km would be draggeddownwardby the subducting
plate. To form suchlarge magmabodies,shearingalong
the top of the subductingplate (and in the overlying
mantlewedge)andconsequent
disruptionof magmabatches
into small bodieshave to be slower than the processesof
meltingandaccumulationof magmas.
The depth of solidificationof a diapir may differ from
that calculatedabove if (1) the theologicalparametersare
slightly different, (2) two or more diapirsfollow the same
path (see Marsh [1982] for hot-Stokesmodels), (3) a
swarmof diapirsrise simultaneously
(fasterascentrates
[see Chhabra, 1988]), and/or (4) a rigid top boundaryis
Bateman, 1984; Paterson et al., 1991; Clemens and
Mawer, 1992]. The low ascentratesof diapirsin previous
calculationsmakethe ascentof magmasthroughfracturesa
clearlymoreeffectivetransportmechanism.Here we will
discuss some difficulties related to fracturing due. to
buoyancyforcesandcomparetherelativeeffectiveness
of
magmatransportby the two mechanisms.
Beris et al. [1985] showed that for a sphere to rise
present.
2000 MPa
Thermal
throughBinghamfluids,the dimensionless
groupYg,
corresponding
to theratio of theyield stressto theexternal
forceactingon thesphere,hasto be smallerthan0.143:
2x¾•r
2 _ 3 xy <0.143
Yg- 4•:Apr3g
2 Apgr
3
whereXyis theyieldstress
for theambient
rocks.For
normalstresses
on = pgH (lithostatic
pressure)
of 200 -
Xy = 50+ 0.60n
Aureole
In order for diapiric ascentto occur at practical rates,
Marsh [1982] invoked in his hot-Stokesmodelswarming
of the countryrocksto theirsolidustemperature.Paterson
et al. [ 1991] arguedagainstdiapirismbasedon the lack of
extensivemigmatitesaroundmost upperand mid crustal
plutons. However, we have shown here that strain rate
softening of the wall rock is an effective substitutefor
thermal softening. There is thereforeno need to expect
wide aureolesof migmatitesarounddiapirs.
(11)
(12)
whereXyistheyieldstress
[Byerlee,
1978]. Substituting
(11) in (12) yields
Apgr>
50 + 0.6 pœH
0.0953
Thus, in order for the diapir to move at all, its buoyancy
stresshas to be approximately6 times larger than the
normal stressmeasuredat the centerof thebody. It is clear
that the diapirwill not movethrougha plasticfluid if its
buoyancydoesnot exceedthe yield strength. Extremely
large magma bodies,high pore pressure,proximity to a
Magma Transport Through Fractures
free surface,externalregionalstresses,
or concentration
of
theseregionalstresses
at the tip of a dike mustbe invoked
Most recentchallengeson the diapiric ascentof magmas in orderfor the risethroughfracturingto be possible.
Clemensand Mawer [1992] offeredtransportof magma
have been basedmainly on the absenceof a soundtheory
that explains how diapirs can ascend through stiff throughself-propagatingfracturesas an alternativeto
lithosphericrocks in geologicallymasonabletimes [e.g., diapirism. However, a dike cools much faster than a
WEINBERG
AND PODLADCHIKOV:
sphereof the same volume. Marsh and Kantha [1978]
concludedthat for a given volumeof magmato reachthe
samedepth,it musttravel104timesfasterin a dikethan
in a sphericaldiapir. Our resultssuggestthat diapiric
ascentratesof 10-50 m/yr may be commonin the lower
crust. Such velocities are 1 order of magnitudemore
efficientin transporting
magma
at 3x104m/yralonga 1km-long and 3-m-thick dike [from Clemensand Mawer,
1992, p. 349]. However,as the diapirrisesanddecelerates,
fracturing becomes easier due to decreasedlithostatic
pressure(On) and may at a certain critical point become
more efficient in transportingmagmathan diapirism. It is
possiblethat at a certain depth the systemchangesfrom
diapirismto fracturing[Rubin,1993].
Another argument used against diapirs of buoyant
magma [e.g., Clemensand Mawer, 1992] is the natureof
the structuresin experimentscarried out by Ramberg
[ 1981], where low-viscositymodel magmasrose through
high-yiscosityoverburden. However, it is known from
DIAPIRIC
ASCENT
OF MAGMAS
9555
and M calculated from known results for drops were
extrapolatedto high n valuesandthenappliedto geological
systems.The most importantadvantagesof our equations
(7) and (8) are that with our definition of the effective
viscosity (8), the velocity equation becomessimilar to
Hadamard-Rybczynski's
equation; the known valuesof the
theology of rocks can be directly applied to calculate the
effective viscosity of the wall rock (].teff); it is not
necessaryto assumea strainrate in orderto calculategeff,
insteadthe buoyancystressof the diapir is used; and since
the correction factors are always close to unity, the
velocity may be approximatedby calculatinggeff and
usingHadamard-Rybczynski's
equation.Any errorsarising
from the extrapolationof the correctionfactorsto high n
may be remedied when these factors are eventually
determinedfor high n values. More importantsourcesof
error are the uncertaintiesconcerningthe theologiesof
rocks [Paterson,
1987].
We have shown that the
theological parametersA and E are the main variables
controllingthedepthat whicha diapirsolidifies.Improved
knowledgeof theseparameters
will considerably
deepenour
rock will behave like inviscid bubbles, and further decrease understandingof crustaldiapirs and allow more accurate
in viscosity will not alter the behavior. The structures predictionsof their development.
observedby Ramberg[1981, pp. 333-340] resultnot from
The high velocities calculated here result from our
the low viscosity of the model magmas, but from their definition of Irefl. Diapirs are able to rise fasterthrough
enormousvolumes. In Ramberg'sexperiment M, the power law fluids than throughNewtonian fluids, and such
"magmabody"had the volumeof a spherewith a diameter fast rise is not accompaniedby large heat loss (the main
equivalent to two thirds of the total thicknessof his model problem with the hot-Stokesmodels). Thus diapirs are
"crust".The resultingfracturingof his modelcrustand the emplacedat much higher levels than earlier predicted,and
disruptionof the initial "magmabody", on the one hand their thermal energy easeslate stagesof ascent. Thermal
provesthe above statementthat diapirs cannotovercome softening of the wall rocks has never been considered
the moving criterion for spheresin Bingham fluids (11), adequate to account for granite plutons reaching high
Earlier
workers
had difficulties
in
and on the other hand showsthat fracturingis preferable crustal levels.
only for such large magma bodies and close to a free accountingfor the way in which hot-Stokesdiapirs rise
surface. Ramberg'smodelcannotbe directlyextrapolated more than a few radii before solidifying. We have shown
for smallerbodiesdeepin the crustor lithosphere.
that thermal softening has little effect on the depth of
solidificationof a diapir and that strainrate softeningof
Hadamard-Rybczynski
ecluation
(2) andconfirmed
by (7)
that diapirsmore than 10z times lessviscousthan the wall
Comparison
Law
Between Hot-Stokes
and Power
Models
Thermal softening of the ambient rocks (hot-Stokes
models)has surprisinglylittle influencein the rate of rise
of a normal-sizeddiapir. Thermalsofteningis negligible
duringthe initial stagesof ascentbecausethe temperature
contrastbetweendiapirandcountryrocksis usuallysmall
power law rocks is a far more effective alternative,
allowing magmatic diapirs to ascendsufficiently fast to
reachdepthsof 10-15 km.
In summary,insteadof assuminga strainrate in advance
as has been usualin the past, we usethe buoyancystress
of the sphereand the theological parametersof rocks to
calculate a more realistic effective viscosity of the wall
rock. Diapirs may thus transportlarge quantifiesof melt
efficiently through power law mantle and crust without
needingto heattheir wall rocksto solidustemperatures.
(low ¾values). By contrast,strainrate softeningallows
the diapir to rise so fast that it loosesvery little heatwhile
ascending.As the warm diapirreachesthe coolercrust,¾
increases
and viscousdragstartsto decreasedueto thermal
softening. However,this occursonly when the diapir is
closeto its final solidification
level,andits thermalenergy Appendix A: Velocity Equation for Settling
is rapidlylostdueto theslowascentrates. The diapirmay Spheres Through Power Law Fluids
still be buoyant when it solidifies, and the high
temperaturedifferencebetweenit and its wall rock, and its
A convenientway to study the rate of translationof a
low Pe, may considerably reduce drag so that the
drop
throughpower law fluids is to measurehow much it
crystallinediapir risesconsiderablyfasterthan expected
deviates from the expectedrate of translationthrough
withoutthermalsofteningeffects.
Newtonianfluids. This is usuallydoneby measuringthe
differencein the dragforce(D) exertedon the sphereby its
Conclusions
viscoussurroundings,in termsof the dimensionlessdrag
coefficient(CD)
This work hasexpressed
the velocityof sphericaldrops
rising throughpower law fluids in the form of Hadamard-
Rybczynski'sequation. The correctionfactorsXsol, G,
8D
CD pV2•(2r)
2.
(A1)
9556
WEINBERG
AND PODLAI•HIKOV:
The Hadamard-Rybczynski
equationshowsthat the drag
coefficientCD for a viscousdropmovingslowlythrougha
Newtonian
fluid is
CD= •ee
It
!lsph
24(!1
++1.5
Ilsph)
(A2)
DIAPIRIC ASCENT OF MAGMAS
center)of the body. After calculatinggeff, the program
then calculates the velocity of the sphere, its Peclet
number(Pe) and the distancein front of the spherewhere
the temperaturedecaysto 1/e of the temperatureat the
sphere'ssurface(fit as definedby Daly and Raefsky
[1985]). The dimensionlessrelationshipsused by the
program are
where
[t'eff =
Re =
2rV
.
Ra e(-1/T')
(A3)
For power law fluids, CD and Re must be redefinedas
[Crochet et al., 1984]
3 [•Xsol(GMg'ef
f+[t's.P5hg
1 'sphi
V'= Rae(-l/7')(
g'effG
+
•
Pe = V'r'.
(B1)
The Nusseltnumber(Nu) for an invisciddiapir(very small
S) is calculatedby using the equationfor a Newtonian
Re= 2mrmv2-mp
K
(A5)
where X is a correctionfactor that dependson n and the
viscosityratio betweenthe dropandthe ambientfluid. For
solid spheresin Newtonian ambient fluids, X=I, and for a
drop of any viscosity(from (A2))
S
•
1+
whereS=•tsph/g.X also describes
the influenceof
nonlinearityfor the more generalcase(n or rn½ 1).
Severalworkershavedetermined
thedragcoefficientCD
for spherestranslatingthroughpower law fluids [e.g.,
Crochetet al., 1984]. OnceCD is known,the velocity
equationof a spherecanbe writtenby settingthedragforce
equalto thebuoyancyforce
(A6)
and substituting D in (A1) and then in (A4), with Re
definedas in (A5). Reorganizing,
we obtaintheequation
for the velocityof a buoyantsphericaldroprisingthrough
a powerlaw fluid
V=2
(Al)g)nr
n+l
9n Knxn
25], correctedby thefactorYM for powerlaw fluidsgiven
by HiroseandMoo-Young[1969]
Nu = 0.795+0.459
YMPe 1/2
where
(B3)
YM
=(1-4m(m-1))l/2
(2m-l)
1.5S
I+S
D = •4 •r3Apg
ambientfluid givenby Daly andRaefsky[1985,equation
(A7)
(A7) reducesto Hadamard-Rybczynski
equation(2) when
Appendix B' The Computer Code "Rise"
The computer code "Rise" starts by reading the six
dimensionless
parametersthat controldiapiricascentand
calculates
theeffectiveviscosityof thewall rockgeffgiven
thebuoyancy
stress
dueto thediapirandthetemperature
of
the ambientrock and diapir (assumedinitially to be the
same).The programassumes
a uniformtemperature
inside
the spherewhen it calculatesthe velocityof the top (or
We chosenot to usethe wholeequationfor Nu givenin
Hirose and Moo-Young[1969] because,when usedfor a
Newtonian ambient fluid, their solution is not in
accordance with that established in Levich [1962] and
correctedin Daly andRaefsky[1985].
The velocityin (B1) doesnot take accountof the drag
reductiondueto softeningof thewall rockscausedby heat
releasedfrom the risingdiapir. Thustheprogramcorrects
the velocity in (B1) and the Nu in (B2) for the drag
reduction,accordingto the resultsobtainedby Daly and
Raefsky [1985]. Althoughtheir solutionwas derivedfor
ambient Newtonian fluids, we believe it to be a good
approximationeven for power law wall rocks. This is
becausethelengthscaleof thermalsoftening(fractionof a
radius)is muchsmallerthanthe lengthscaleof strainrate
softening(a few radii). No dragreductionis appliedin the
firststep,sincethetemperature
contrast
betweenthediapir
and its wall rock is likely to be low duringinitial ascent.
The programcalculates¾for everysubsequent
step; T is
definedas log10 of the total viscosityvariationdue to the
temperaturegradientfrom the surfaceof the diapir to the
temperatureof the ambient fluid at infinity [Daly and
Raefsky,1985]
log10e-1/(T'sph-T')
.
(B4)
Daly andRaefsky'ssolutionis limitedto valuesof ¾> 2,
so for ¾greaterthanthis valuethe programcalculatesthe
parameter
A andthedragreduction,
Dr, using
A = œNu
0.62A
Dr = 0.60+ A
(B5)
WEI•ERG
AND. PODLADCHIKOV:
where
DIAPIRIC
Nu=0.795+0.459
YM(Peœ-0.2)1/2
anddividesthevelocityfrom (7) by Dr. Dragreductionis
importantonly whenPe is betweenI and 100 [Daly and
Raefsky, 1985], and it will be shownlater that these
values occur only when the diapir is close to its final
emplacementlevel.
At every step, cooling of the sphere and its new
temperatureis calculatedaccordingto [from Marsh and
Kantha, 1978]
complexcrystallizationprocessesin a natural diapir.
When the temperatureof the rising body achieves1.1
T'sol, its viscositybeginsto increaseand it eventually
rises10 ordersof magnitude
justbeforetotalcrystallization
[Cruden, 1990]. If the diapir loses its buoyancyby
crystallizing (when T's=T'sol), the program stops
calculating.
correctionfactor function of n, defined in (9)
H
depthof the top of the diapir
K
o/•;1Inin Pasl/n, uniaxialcase
k
thermal diffusivity
L*
length scalingparameter
in
M
powerlaw exponent
Nu
Nusselt
number,
q r / )•(Tsph-T)
l-'e
Peclet number
heat flux out of the hot sphere
q
R
gasconstant,8.314 J/mol K
l"
dimensionlessradius(r'--r/H)
sphereradius
Re
Reynoldsnumberdefinedin (A3) and (A5)
Ra
compositional
Rayleighnumber,(Ap g r)n r H/
Knk6n-1
S
time stepin numericalcalculations
temperature
decayof thespherepertimestep(dt)
T
temperature(withoutany subscriptcorresponds
to the wall rock or ambient fluid)
V
The resultsof calculationsfor the depthof solidification
H of a diapir as a function of the five dimensionless
parametershave beenfit by the leastsquaresmethod. The
following equation fits the numerical results with a
maximum relative error of 5.6% (calculated by the
difference
of predicted
andcalculated
valuesdividedby the
+ 0.0573 r -0.0253 r gt - 0.0336 r Ra + 0.0137 r Ra
gt - 0.0226 r T'sol + 0.0050 r T'sol gt + 0.0110 r
T'sol Ra - 0.0024 r Tsol Ra 'gt - 0.0226 r n +
o.0111r n gt + 0.0090r n Ra-0.0045 r n Ra gt +
0.0115 r n T'sol- 0.0042r n T'so1gt- 0.0059r n
Tøsol
Ra + 0.0022 r n T'solRa gt
[tsph/ geff
time
dt
Appendix C' Solidification Depth of a Diapir
H = 0.6434 - 0.1444gt - 0.2279Ra + 0.0464Ra.gt +
0.1316T'sol - 0.0404 T'sol gt -0.0632T'sol Ra +
0.0192T'solRa gt + 0.0455n - 0.0218n gt - 0.0074
n Ra + 0.0049 n Ra gt- 0.0271 n T'sol + 0.0118 n
T'solgt + 0.0142n T'so1 Ra -0.0062 n T'solRa gt
1In
correctionfactor functionof n, defined in (9)
dT
sum of both)
9557
G
dT= 3Nu(T'sph-T3
dt
ra
forsphere. (B6)
The programassumes
uniformtemperature
ratherthanthe
OF MAGMAS
dimensionless
geothermalgradient(dT/dxHR/E)
gt
e=¾310-Y
ASCENT
dimensionlesssolidustemperature
sphere'sor diapir'svelocity
width of the box used in numerical
calculations
dimensionlesswidth (w/r)
correction factor
X(n)
correction
factorfunction
of n andgsph/geff
Yg
the ratio of the yield stressto the external force
actingon thesphere,definedin (11)
YM
correctionfor Nu of power law fluids, definedin
(B3)
Greek symbols
log10 of the total viscosityvariationin (B4)
[Daly andRaefsky,1985]
second
invariant
ofthestrain
rate
tensor
•j
Notation
definedin (B5) fromDaly andRaefsky[1985]
thicknessof themetamorphicaureole
A
preexponential
parameter,
Pa-ns-1
CD
dimensionless
dragcoefficient
Dr
dragreduction
caused
by thermalsoftening
E
activationenergy,J/mol
g
acceleration
dueto gravity
uniaxial
dij
strain rate
strain rate tensor
heat conductivity
viscosityof the Newtonianambientfluid
geff
effective viscosityof the ambient fluid defined
9558
WEINBERG AND PODLADCHIKOV: DIAPIRIC ASCENT OF MAGMAS
by
(8)
•'sph
dimensionless
viscosity
of thesphere,
g'sph=
Ap
Byerlee, J., Friction of rocks, Pure Appl. Geophys.,116, 615-
(•tsph
k)/(Ap
grH2)
density
difference
(p-Psph)
1978.
Gulf, Houston, Tex., 1988.
uniaxial stress
stress
626,
Chhabra, R. P., Hydrodynamics of bubbles and drops in
rheologically complex liquids, In Encyclopedia of Fluid
Mechanics, edited by N. P. Cheremisinoff, pp. 253-286,
Clemens, J. D., and C. K. Mawer, Granitic magma transportby
fracture propagation, Tectonophysics, 204, 339-360,
tensor
1992.
lithostaticpressure
on = pgH
Crochet, M. J., A. R. Davies, and K. Walters, Numerical
yield stress
Simulation of Non-Newtonian Flow, Rheol. Ser., Vol. 1,
352 pp., Elsevier, New York, 1984.
Cruden, A. R., Deformation arounda rising diapir modeled by
creeping flow past a sphere, Tectonics, 7, 1091-1101,
Subscripts
1988.
crit
critical
eft
effective
Cruden, A. R., Flow and fabric developmentduring the diapiric
rise of magma, J. Geol., 98, 681-698, 1990.
Daly, S. F., and A. Raefsky, On the penetrationof a hot diapir
through a strongly temperature-dependent viscosity
medium, Geophys.J. R. Astron. Soc., 83, 657-681, 1985.
Dazhi, G., and R. I. Tanner, The drag on a spherein a power-
init
initial
m
ambient fluid
law fluid, J. Non Newtoninn Fluid Mech., 17, 1-12, 1985.
England, P., The genesis, ascent, and emplacement of the
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met
metamorphicaureole
(n)
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S
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sol
solid or soliduswhen usedwith temperature
1992.
sph
Grout, F. F., Petrography and Petrology, 522 pp., McGraw
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Hadamard, J., Mouvement permanent lent d'une sphereliquide
et visqueusedans un liquid visqueux,C. R. Acacl.Sci., 152,
sphere
1735-1738,
Acknowledgments.
We would like to acknowledge our
indebtedness
to Harro Schmeling,who designedthe computercode
used to calculate the ascentvelocity of cylinders through power
law crust, and also to thank him for his continual supportin the
running of the program necessaryto completethis work. We
would also like to thank Harro Schmeling, Christopher Talbot,
and Anders Wikstr6m for inspiring discussionand ideas. The
assistanceof Sharon Ford for language editing and Christina
Wernstr/Smfor the figuresis gratefullyacknowledged.
1911.
Hansen,F. D., and N. L. Carter, Creep of selectedcrustalrocks
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