Characteristics and consequences of flow in the lower crust

JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 105, NO. B5, PAGES 11,029-11,046, MAY 10, 2000
Characteristics and consequences of flow
in the
lower
crust
Dan MCKenzie, Francis Nimmo, and James A. Jackson
Institute of Theoretical Geophysics,Bullard Laboratoriesof the Department of Earth Sciences
University of Cambridge,Cambridge,United Kingdom
P.B. Gans
Department of GeologicalSciences,University of California, Santa Barbara
E.L.
Miller
Department of Geologicaland Environmental Sciences,Stanford University, Stanford, California
Abstract. In someplaces,there is strongevidencethat the lower continentalcrust
has flowed so as to smooth out variations in crustal thicknesscausedby differential
crustal extension or shortening. In order to better understand the processes
involved, we investigatethe behavior of a fluid layer over a fluid half-spaceto see
how such a systemrespondsto the deformation of its upper and lower boundaries.
This simple system can be used to study both the decay of crustal thickness
contrastsand the behaviorof a thin lithosphericsheet. The changingresponseof
the systemto variations in density and viscositycontrastsand to different boundary
conditionsimposed on the fluid interface can easily be studied analytically. The
most important results are that variations in crustal thicknesson a wavelengthof
a few times the thicknessof the flowingchannelwill decay quickestand that large
lateral variations in crustal thicknesscausethe fluid to develop a steep front, which
may causea topographic step above it at the Earth's surface. Deformation within
the channel will be principally by simple shear. The clear association of lower
crustal flow with regionsof thickened crust and magmatic activity suggeststhat
both can reduce the viscosityof the lower crust to levels at which flow can occur.
The smoothingof crustal thicknesscontrastsleadsto differential vertical motions,
and is thus a method by which substantial tilting can occur without faulting.
This differential uplift may be responsiblefor rotating and exhumingsomeof the
detachmentfaults in metamorphiccore complexesin the Basin and Range province
of the western United States. It is also a method of causingstructural inversionin
basinsthat does not require the reactivation of normal faults as thrusts or reverse
faults.
1. Introduction
oceanic and continental lithosphere is the thicknessof
the crust. Differencesin crustalthicknessproducecorThe successof plate tectonics as a descriptionof responding
variationsin elevationand henceof gravitaoceanictectonicshas encouragedattempts to use the tional potentialenergy.Suchforcesare moreimportant
same ideas to describe continental deformation. Such in continental regions, where the crust is thick, than
effortshave been partially successful:the relative mo- they are in the oceansand tend to equalizethe crustal
tion betweenthe large aseismicparts of continentscan thickness.
be usefullydescribedby relativeangularvelocities.HowA number of observations of continental deformation
ever, the zones of distributed continental deformation suggestthat, at least in some places,the brittle deforthat are sometimes more than 2000 km across cannot be mation of the upper crust is decoupledfrom the dedescribed
by rigid motions[e.g.,Englandand Jackson, formation of the lower crust. One example, described
1989]. One obviousand important differencebetween by Gans[1987]and illustratedin Figure 1, is from the
easternBasinandRangeprovinceof the Western
United
Copyright
2000by the American
Geophysical
Union.
Paper number 1999JB900446.
0148-0227/00/1999JB900446509.00
States. Figure 1 showsthat variationsin the extension
factor/• of the upper crust, from 1.4 to 4.0, are not associatedwith correspondingchangesin topographyor
crustal thickness. Although some igneousmaterial is
11,029
/
11,030
MCKENZIE
ET AL.:
FLOW
IN THE
LOWER
CRUST
Figure 1. Geologicstrip map and generalizedcross-sectionacrossthe eastern Great Basin,
United States[from Gans,1987]. StippledareasrepresentPrecambrianto Mesozoiccarbonate
and clastic rocks. Highly extended domains are highlighted with slanted lines. Note that the
Moho is almost
fiat.
likely to have been added to the crust during extension,
suchadditionswill not have been localizedenough,nor
have had sufficient volume, to account for the lack of
Moho topography. Thus lower crustal material must
have flowed from regionswhere the amount of stretch-
Figure 1 and with why, how, and when it occurs,rather
than with the more localized flow that must occur at
the baseof tilted upper crustal blocks.
In section 2 we summarize previous models of lower
crustal flow and then investigate the behavior of our
ing is smallto thosewhereit is large[Gans,1987].
own simple model, in which a fluid layer overliesa halfLower crustal flow is also required to account for space. We then discussthe conditionsunder which flow
structures seen on deep reflection profiles, especially might be expectedto occur in nature and identify the
those across the continental
shelf around the United
features of our model that are likely to be applicable.
Kingdom[e.g.,Brewerand Smythe,1984;Kusznirand We then attempt to estimate the probable timescales,
Matthews,1988]. On almostall of theseprofilesthe viscosities,and velocitiesthat might be expected and
large normal faults that offset the layers of the upper discusshow our results may apply to geologicalprobcrust do not cut the subhorizontal
reflectors of the lower
crust or the Moho, even though large planar structures
that appear to be faults are not uncommonin the upper mantle. In these reflection profilesthe faults in the
upper crust often resemble tilted blocks or dominoes
and are typically spaced 15-20 km apart. Flow on this
length scalemust occur at the base of the blocksand
is, indeed, assumedto have done so by those who use
simplemodelsto model the vertical motionsin suchter-
lems.
2. Model
and
Previous
Work
Authors in the past have studied lower crustal flow
both analytically and numerically and have differed in
their choiceof rheology,boundary conditions,and num-
ber of layers. Kusznir and Matthews[1988], Gratton
[1989],and Kruse et al. [1991]wereinterestedin es-
rains[e.g.,Jacksonet al., 1988; Yielding,1990]. This sentially the same problem that we addresshere, with
paper is primarily concernedwith the nature of lower Gratton's emphasisprincipally on the decay of crustal
crustal flow over the -• 100 km length scaleimplied by rootsbeneathmountains.KusznirandMatthews[1988]
MCKENZIE
ET AL.: FLOW IN THE LOWER CRUST
11,031
realized the importance of crustal thicknessin reducing
the viscosityof the lower crust and obtained an expression for the timescale
on which
crustal
thickness
con-
trasts decay. Their expressionsagree with ours when
a printing error in their equation in Appendix 2 is cor-
rected (N.J. Kusznir, personalcommunication,1990).
Kruse et al. [1991]and Kusznir and Matthews[1988]
considereda power law theology but did not allow the
lowercrustalchannelwallsto deform.MCKenzie [1988]
used the nonlinear
solutions
for channel
flow obtained
half-space p2, v2
by Huppert[1982]to showthat sharptopographic
fronts Figure 2. Cartoon to illustrate the modeldescribedin
could developeven when the rheologywas viscous.Bird
the text. A fluid layer of thicknessh, viscosityt/z, and
[1991]foundthe samebehaviorwith a powerlaw the- density pz overliesa fluid half-spacewith viscosityt/2
ology,a resultconsistentwith our findings(seebelow). and density P2.
More recently,both Nakada[1994]and Royden[1996]
have investigatedthe effectsof mantle flow on lower
crustaldeformation,whilstHopperandBuck[1996]examined how a ductile
lower crust influences rift mor-
phologies.Zhong[1997]useda modelsimilarto that analyzed below, but with a Maxwell rather than a simple
viscoustheology,to argue that short-wavelengthtopography could remain uncompensatedfor extensiveperi-
ods. Zhao and Morgan[1987]arguedthat the Indian
another boundary condition on the upper or lower surface of the layer: that the horizontal velocity vanishes.
Though this condition will be referred to as "rigid," it
is a condition on the horizontal, not the vertical velocity. A boundary can be both rigid and deformable at
the same time if it consistsof a thin sheet of strong
material separating the layer from the half-space. This
would, for instance, be the appropriate boundary condition to use for a layer of fluid separated from a fluid
half-spaceby a thin plastic film.
Below we argue that creepingflow only becomesim-
crust is driving lower crustal thickeningbeneath Tibet.
The deformation they studied by finite element methods is equivalentto one of the modesdiscussedbelow.
Rather than develop a complicated model which relies on the poorly known theologicalproperties of the
lower crust, we use the simplest model that is likely portant whenthe temperature(in K) exceedsabout 0.7
to contain the important features that govern crustal of the solidus temperature. Because the solidus temflow (seeFigure2). It consists
of a layerof fluid with perature of the upper mantle is , 1200øC, it will only
densitypz and viscosityth overlyinga fluid half-space flow when the temperature exceeds750øC. Though this
of densityP2 (> pz) and viscosityt/2. For the caseof estimate is uncertain, the creep rate of the upper manlowercrustalflow,t/2 (representing
the uppermantle)is tle is generally less than that of the lower crust when
probablymuchgreaterthan th (representing
the lower they are at the same temperature. Hence the Moho will
crust), but as we shall see,we can use this model to act as a rigid boundary to lower crustal flow. The rigid
investigateother situationsas well. Under most condi- layer beneath the Moho will, however, probably be no
tions of fluid flow the velocity field doesnot correspond more than a few tens of kilometres thick in most places.
The other uncertainty that affects the calculations
to eitherpureor simpleshear,sinceboth components
of
is whether lower crustal flow is Newtonian
or involves
the velocityare nonzero,asis the vorticity,and both the
velocity and the vorticity are functions of both x and power law creep. The transition between these regimes
z. However, when the wavelength of the disturbance depends on both temperature and shear stress and is
is large comparedwith the layer thickness,certain im- poorly constrainedby existing laboratory experiments.
portant flows occur which are good approximationsto However, the numerical experiments on power law creep
carriedout by Kruseet al. [1991]showthat the useof a
simple or pure shear.
Since both the upper and lower surface of the layer nonlinear stress-strain relationship rather than a Newcan independently move vertically, they are referred to tonian viscositydoesnot lead to qualitative differences
as deformable boundaries. When they do so, the verti- in behavior. We therefore adopt a constant, Newtonian
cal fluid velocity at the boundary must equal the rate viscosity,since this greatly simplifiesthe calculations.
at which the boundary deforms. Other boundary con- Nor do we include the flexural rigidity of the upper
ditions that must be satisfied are that the normal and
crust,unlike Kaufman and Royden[1994]. In addition
tangential stressmust be continuousand the vertical to calculatingthe velocity fieldsand responsetimes of
and horizontal velocities at the interface between the
the model we also obtain the gravity field and its relalayer and the half-spacemust be the same. Though the tion to the deformation of the upper surface.
shear stress must vanish on the upper surface of the
We emphasizethat our treatment assumesan instanlayer, there is no restriction on the horizontal velocity. taneousdisturbanceof the layer boundaries,the decay
This boundary condition will be referred to as "free" of whichwe then investigateasa functionof wavelength,
or "stressfree." It is also sometimes useful to impose densitycontrast, and viscositycontrast. At least in
11,032
MCKENZIE
ET AL.: FLOW
some places, such as the Basin and Range province
of the western United States, there is evidence that
lower crustal flow occurredalmost synchronouslywith
regional extension. In the absenceof such flow large
variationsin surfaceand Moho topographywould have
been produced if the upper and lower crusts had extended by the same amounts in the same place. We
discuss this further
2.1.
Small
IN THE LOWER
CRUST
(a)
4
3
in section 3.2.
Perturbations
and
Linear
Behavior
Becausethere are two surfacesinvolvedin the model,
there must be two responsetimes that characterizethe
system. In general, both decay modes involve movement of both interfaces.
When
the deformations
•
of the
-2
upper and lower surfaceof the layer are small compared
with its thicknessh, the equations governingthe de-
formation are linear (see Appendix A). A number of
asymptoticsolutionsto the equations,correspondingto
limiting valuesof viscositycontrastor wavelengthA, can
be obtained analytically, and we present the results as
plots of responsetime •- against wavenumberk, where
k = 2•r/A. All the calculationswe showwere carried
out with Px= 2.8 Mg m-a and P2= 3.3 Mg m-a. It is
convenientto measureall lengths in terms of the layer
thickness and all times in terms of a natural
i
i
i
-1
0
1
log(k)
(b)
0.0
rigid lowerboundary
=
-0.2
stress continuous
o
J:3 -0.4
timescale
to of the model, where
•
tO- /]l/Plgh,
-0.6
(1)
o
where /]1 and pl are the viscosity and density of the
layer. Throughout the following,we neglectflow out of
e -0.8
c
the plane.
2.1.1. A low-viscosity layer over a more vis-1.0
cous half-space. The simplest problem to solve is
0.0
0.2
0.4
0.6
0.8
1.0
1.2
when the wavelengthof the deformation is also small
horizontalvelocity
comparedwith h. The behavior of each surfaceis then
independent,and the decay times for disturbancesof Figure 3.
(a) Plot of response times w• versus
wavelengthA are easilyfound. The time •"arequiredfor wavenumberk• for a density contrast R - 1.18 and
a disturbance
of the uppersurfaceto decayto l/e of its viscositycontrastr - •,//]1 = 50. (b) Plot of velocinitial
ity versusdepth within the layer for the same density
size is then
(2) and viscositycontrastsas in Figure 3a, when k• = 0.1.
-
Depth and velocity are normalized to the thicknessof
the layer. The solid curve is the velocity when the lower
Ta= 4•r/]l/Plg•,
(3) boundary of the channel is that the stressand velocity
are continuousbetween the layer and the half-space,
wherek• - hk - 2•rh/Aand the primesshowthat the and the dashed curve is the velocity when the lower
variableshave beenscaled(AppendixA). The corre- boundary is rigid and deformable.
or
spondingtime Tbfor the lowerboundaryis
Tg-2k'(r
1)
(R-+1)
face decay faster than those on the interface between
(4) the half-spaceand the layer becausethe density con-
or
4•r(/]1+/]2)
Tb
-- (P2
--Pl)g•'
(5)
where
R -- P2/Pl,
r -- /]2//]1.
(6)
trast acrossthe top surfaceis greater. If p•, - px, the
only force driving the flow is the deformationof the upper surface. The time •"atherefore correspondsto the
isostaticresponsetime of the systemto surfaceloads.
The analytic solutionsgiven in (3) and (5) are only
valid when h >> A. The more general variation of responsetimes with wavenumberis shown in Figure 3a,
Sincepl > (p2- Pl) and/]1 < (/]t +/]2), it followsthat
Tb > •. In other words,disturbancesto the top sur- whichcontainsa plot of log•"a
• andlog•"gasfunctions
of
MCKENZIE
Longwavelength
(k' << 1)
ET AL.: FLOW
Shortwavelength
(k'>>
1)
Ta
IN THE LOWER
CRUST
11,033
if h _• 10 km, then ,k _• 200 km. Longer-and shorterwavelengthdisturbances
will decaymore slowly. The
dimensionless
wavenumber
for the mostrapidlydecaying disturbanceis not very different from that of 1.05
whichSmith[1975]foundfor the mostrapidlygrowing
disturbance
whena Newtonianlayeris strainedby pure
shear. This value increases to 1.57-1.85 when the mate-
rial is non-Newtonian
[Smith,1975].In detail,however,
$mith's problem is rather different from that consid-
eredherebecausegravityis absentandthe instabilityis
drivenby a large-scale
viscousflowwhichis not present
in our model.
The flowassociated
with the response
time wgmust
Figure 4. Sketchesof the deformation associatedwith
for removingthe crustalthicknessvariathe two responsetimes of the systemin Figure 2 at very be responsible
long and very short wavelengths.
tionsgeneratedby the differentialstretchingin Figure
1. The positionand shapeof the minimumin wg(k•),
and the valuesof Tgand k• at that minimum,depend
on the densityand viscositycontrasts,as will be seenin
logk' for r/2/rh = 50 whenthe shearstressis continuous Figures 5 and 6. The important result is that there is a
on the lower interface of the layer. The types of deforminimumin Tg(k•) and that it occurswherethe wavemation associatedwith the two responsetimes at large
lengthis a few timesthe layerthickness
(,k= 27rh/k•).
Nakada[1994]founda similarresult,althoughhe does
Figure4. When k• >> i (Figure4, right), both response
not showan increasein wgat shortwavelengths.
and small wavelengthare illustrated in the cartoonsin
times are proportionalto k•. When k• (( I (Figure 4,
left), T• is againproportionalto k•, and both the upper
When the viscosityof the layer and the half-spaceare
the samebut their densities
aredifferent(Figure5), the
and lower boundaries of the layer deform at the same
behavior
of
•
becomes
simple.
However,wgstill has
velocity. The correspondingflow is that of the isostatic
a minimumbecausethe flow requiredto removelong-
responseof a half-spacewith viscosityr/2. The response
wavelengthvariations is restricted to the layer. The
time is controlledby flow within the half-space,and the
minimumin Tg(k•) is rathershallower
than in Figure3a
only flow that occurs within the layer is that associand
has
shifted
position,
to
a
smaller
valueof wg,anda
ated with bending. The flow associatedwith the other
response
time Tgis of moregeological
interest.It cor-
largervalueof k•. Notethat • is alwayslessthan•
for a givenvalueof k• in both Figures3 and 5. At long
respondsto the decay of compensatedcrustal thickness
wavelengths(i.e., k• • 1), isostaticcompensation
is
variations by the lateral flow of the crust.
alwaysattained more quickly than the decayof crustal
The horizontal velocity profile in Figure 3b shows thickness contrasts.
that the flow is concentratedin the layer. Becausethe
2.1.2. A viscous layer over an inviscid fluid.
viscosityof the half-spaceis so much greater than that
A very different type of behavior occurswhen r/x >>
of the layer, the velocity at the interface is small and
the flow is similar to that with a rigid deformablelower
boundary, shown with a dashed line in Figure 3b. It
3
is this modethat Zhao and Morgan[1987]analyzedby
finite element methods in their study of Tibetan deformation. Since k• (( 1, the wavelength of the crustal
thicknessvariations is large comparedwith the layer
thicknessand the horizontal velocity is large compared
with
the vertical.
Hence the deformation
within
the
layer is principally by simpleshear,with only a small
componentof pure shear, and the viscousresistanceis
large. If the amplitudeof the deformationis kept constant as k• decreases,the gravitationalforcedriving the
flow also remains constant, but the distance that the
material in the layer must be transported increasesas
the wavelengthof the disturbanceincreases.It is for
this reasonthat Tgincreases
so rapidly as the wavelengthincreases.The asymptoticanalysishasnot been
-2
I
I
I
-1
0
1
log(k)
doneforthiscase,butFigure3asuggests
thatwg•ck•-2 Figure 5. Plot of responsetimes versuswavenumber,
whenk• (• i and r/2 • rh. As Figure3 shows,wg(k
•)
as in Figure 3a, but in this casewith no viscositycon-
has a minimum value when k • _• 0.3 or ,k _• 20h: thus,
trast (r = r/2/rh = 1).
11,034
MCKENZIE
ET AL'
FLOW IN THE LOWER CRUST
When k• • 1, • • h, and the upper and lowerboundSolomonet al. [1982],whouseda viscouslayeroverly- aries of the layer move independently. The behavior of
ing an inviscidhalf-spaceof different density to study r• when k• (( I is alsothe sameas beforeand correthe recoveryof impact basins. Figure 6a showsthe be- spondsto isostatic recovery with a time constant conhaviorof ra•(k•) and r•(k •) whenr/2/r/1= 0.004, and trolled by the propertiesof the half-space. The behavanalytic expressionsfor r = 0, k• (( I are obtained ior of r• ask• -• 0 is, however,
quitedifferentfromthat
in Appendix A. The numbersin Figure 6a against the in Figures 3 and 5 becausethe crustal thicknesscan
changeby deformingthe layer by pure shear. The vesegmentsof the curvesshowthe value of n where
locity profileis illustrated in Figure 6b as the solidline
r' cr k'".
(7) and showsthat the velocity gradientswithin the layer
are negligible.That the stressfree boundary condition
(a)
on the lower boundary of the layer is indeed respon-
02. This problemis closelyrelated to that studiedby
siblefor this behavioris shownby calculating
r• with
• • e rigid
lower
boundary
a rigid instead of a free boundary, shownby the line
' • ß• -2
4
'.. ßß •
with shortdashesin Figures6a and 6b. The time r•
stress
continuous
is then proportionalto k•-2, as the asymptoticanalysisin AppendixA (equation(A23)) alsoshows.Thus
viscousdissipationrate is much smaller and the relaxation time much faster when the layer deformsby pure
shear, rather than by simple shear. This result is also
clea.
r from Figure 6a, which showsthat, for a value of
/4
k• = 10-2, r• -• 105forsimple
shear(dashed
curve)but
r• - 10•'5 forpureshear(solidcurve).
Because
r• is constantwhenk' • 1 andthe lower
1
•
-2
I
I
I
-1
0
1
log(k)
boundary is free, all perturbationsdecay at the same
rate. Therefore the shape of the deformation does not
change.EnglandandMCKenzie [1982]foundthe same
(b)
0.0
----'•
rig,;
lower
'bound;•ry
J I
behavior for a deforming thin sheet with Newtonian
viscosity,whereboth upper and lowerboundarieswere
free, and were for this reasonunable to producesteep
surfacegradients.A non-Newtonianrheology,however,
did producesuchgradients.Thereforewhetherthe flow
is Newtonian or non-Newtonian has an important in-
i-0.8
0'6
Figure 6. (a) Plot of responsetimesversuswavenumber, as in Figure 3a, but with the viscosity contrast
r - r/2/r/• - 0.004. The behavioris similar to that of
a viscouslayer overlyingan inviscidfluid. Two curves
are shownfor r•: the solidcurveshowsthe behavior
-1.0
_
0.0
0.2
0.4
0.6
0.8
1.0
1.2
horizontalvelocity
(c)
when the stressis continuouson the lower boundaryof
the layer and correspondsto the conditionsusedin thin
sheetanalysisof lithospheredeformation[e.g.,England
and MCKenzie, 1982];the dashedcurveis for a rigid
and deformablelower boundary to the layer. The numbers adjacent to the curves are the slopesin log-log
space(seeequation (7)). (b) Plot of velocityversus
depth within the layer for the same density and viscosity contrastsas in Figure 6a, when k• = 0.01. Depth
and velocityare normalizedto the thicknessof the layer.
The solid curveis the velocity when the lower boundary
Za
E
condition
Zb
on the channel
is that
the stress and veloc-
ity are continuousbetween the layer and the half-space
(the thin sheetapproximation),and the dashedcurve
is the velocity when the lower boundary is rigid and de-
formable.(c) Plot of admittanceZ versuswavenumber
for the viscosityand density contrastsfor the continuous
stresscurvesin Figure 6a. The curve Za corresponds
to
-2
-1
0
log(k)
1
the deformationassociated
with response
time ra• , and
Zbwith r•.
MCKENZIE
ET AL.:
FLOW
IN THE
LOWER
CRUST
11,035
fluenceon the solutionin this limiting case. The same and (B14)). It is straightforward
to showthat a layer
doesnot appear to be true for channelflow in the lower of fluid spreadingover a rigid but deformablesurface
crust [Gratton,1989; Kruse et al., 1991];we demon- satisfies
the sameequationthat Huppert[1982]solved
strate later that for larger perturbations even Newto- (AppendixB). Furthermore,only the numericalconnian fluidscan developsteepsurfacegradients.
stant changesif a rigid but deformableboundaryconThe complicatedbehaviorof Ta
• betweenk'- 1 and dition is imposedon the lowersurfaceof the layer. It is
k' - 0.1 can also be understoodwith the help of the
the rigidity rather than the deformabilitythat controls
asymptotic expressionsin Appendix A. When the vis- the behavior.As longas at leastoneboundaryis rigid
cosity of the half-spaceis zero, the isostaticresponse (i.e.,thehorizontal
velocitycomponent
at the boundary
is controlledby the viscosity of the layer and the time is zero),the deformation
of the layerwill be by simple
constantis proportional
to k'4 (equations
(A20) and shear and a flow front will develop. Whether or not
(A22)). This behavioris clearlyshownin Figure 6a. the boundariesare deformablechangesthe numerical
However,as the wavelengthincreases,the resistanceof
the layerto deformationrapidly becomesunimportant,
and the isostaticresponseis governedby the properties
of the half-space.
The behaviorof the ratio of the gravity anomalyto
the surfacedeformation,known as the admittance, Z,
as a functionof k' for the two modesof decayis illustrated in Figure 6c. The isostatic responseshows
a largeadmittanceZa (dashedcurve)that varieslittle
with wavelength.Hencethere is a largeamountof gravitational potential energyassociatedwith this deformation that is available to drive the flow. As would be ex-
pected, the gravity anomaly associatedwith crustal flow
is small (solid curve) becausethe crust is isostatically
compensated. Hence the small value of Zb calculated for
•-•agrees
with the absence
of largegravityanomalies
associatedwith long-wavelengthcompensatedvariations
in crustal thickness.
This behavior
is unlike that found
by Zhong[1997],whoseMaxwellrheologicalmodelcontains some modes with long time constantsthat also
have large valuesof Z. This differencesuggeststhat it
is the elastic forces in his model that cause this difference.
2.2. Larger Perturbations:
and
Channel
Nonlinear
Behavior
Flow
factorsbut not the exponentof k'. Furthermore,if the
viscosityof the lower crust is much lessthan that of the
upper mantle, the style of flow in the lower crust is little
affectedby extension
of the uppermantle(Figure3b).
Similarly,it makeslittle differencewhetherit is the upper mantle or the upper crust (or both) that is rigid,
sincethe rigidity of either boundary will force channel
flowto occur. If, however,both upper and lowerboundariesare stressfree, Huppert's analysisdoesnot apply
becausepure sheardeformationis then allowed,and the
form of the solutionsis completelychanged.
Huppert[1982]also remarksthat almost all initial
shapeswill approach the similarity solution with increasingtime. Figure 7 showsthe evolution of three
differentinitial liquid shapes,obtainedby solvingHuppert's equation 2.9 with an implicit finite difference
scheme. The thin line is the shape predicted by the
similarity solution, and the dotted line is an initially
triangular distribution having the sametotal area. Althoughthis latter shapedoesnot possess
a sharpfront
initially, it approachesthe similarity solutionovertime.
The bold line is the evolutionof a shapewith a smaller
initial lateral thicknesscontrastand showsthat a large
contrastis requiredto causethe sharpfrontsto develop.
The linear theory in Appendix A givessomeindication of how such fronts arise when one or both bound-
ariesare rigid. Under thesecircumstances
(A23) shows
The discussionabove is entirely concernedwith the
behavior of perturbations to the surface or the interface the amplitude a of which is small comparedwith
h. Thoughthe value of a/h at whichthe approximation ceases to be useful cannot be estimated
without
numericalexperiments,it is only likely to be accurateif
a/h < 0.1. If this conditionis not satisfied,the flow
depends nonlinearly on the amplitude of the disturbanceand numericaltechniquesare generallyrequired
to solvethe flowproblem.However,as Huppert[1982]
showed,progresscan still be made analytically if certain conditionsare satisfied. He discussedthe problem
of magmaspreadingover a rigid undeformablesurface
and obtained a similarity solution that was valid when
the horizontalextent of the magmawaslargecompared
with its thickness. Becausethe shape of the layer is
givenby a similarity solution,it doesnot changewith
that •-bc• h-3. Hencedisturbances
with a particular
wavelengthdecayvery muchfaster on a thick layer than
on a thin one. This effectallowsthe liquid in Figure 7
to flow rapidly toward the front, eventhoughthe slopes
of the upper and lower interfacesare small. Only as it
approaches
the front doesits velocitydecrease,causing
the liquid to pile up and in this way maintain the front.
It is also important that the shape is independent of
the viscosityof the spreadinglayer: only the rate of
evolutionof the shapeis controlledby the viscosity.
2.3. Summary
This discussionshowsthat many important features
of the responseof the crust and upper mantle to variations in crustal thicknesscan be understoodusing simple analytical expressions.An important result is that
variationsin lowercrustalthicknesswhosewavelengthis
tirneif the x axisis suitablyscaled(seeequations(B13) a few timesthe thicknessof the layer will decaythe most
11,036
MCKENZIE ET AL.' FLOW IN THE LOWER CRUST
(b)
(a)
4
0.64 ka
0 ka
E
'"
2
%
%
.
similarity .....
i
-1
thin •i
thick
I
i
i
i
(d)
(C) 4
6.4 ka
E
,,.
64.3 ka
2
I
0
10
I
20
I
30
i
40
0
i
10
20
i
30
40
Distance, km
Distance, km
Figure 7. (a) Initial topographicprofilesfor two models.In all casesthe kinematicviscosityv
is 10TMm2 s-1 andthe acceleration
dueto gravityg is 10 m s-2. The boldlineis a triangular
profilesuperimposed
on a layerof thickness
of 2 km, andthe dottedlineis a profileimposedon a
layerwhosethickness
is 100m. The finelinesin Figures7b-7dshowthe similaritysolutionfrom
Huppert's
[1982]equation2.12havingthe sametotal areaasthetriangularprofile.The evolution
of initially triangularprofilesis calculatedfrom implicitfinite differencesolutionto Huppert's
[1982]equation2.9, andthe similaritysolutionprofileis givenby hisequation2.12. Shapeof
fluids(b) after640years,(c) after6.4x 103 years,and(d) after64.3x 103 years.Notethat a
sharpstephasdeveloped
onlyfor the dottedline, whichcloselyresembles
the similaritysolution.
A largethicknesscontrastis neededfor this effectto occur.
rapidly. This wavelengthdependencewas also noticed 3. Geological Consequences of Flow
by Kusznir and Matthews[1988],Kruse et al. [1991], in the Lower
and Nakada[1994].We foundthat the decayin crustal
thicknesscontrastsoccursmore slowly than the attainment of isostatic equilibrium. This result is likely to be
general.
Another important result is that any local region of
low-viscosity,lower crustal material at the baseof thick
crust will spreadoutward. If there are large lateral variations in the thicknessof this low-viscosityregion and
if it is boundedeither aboveor below(or both) by highviscositymaterial, the fluid will developa sharp front
as it flows.
Crust
In this section we examine some of the consequences
of lower crustal flow and their significancefor various
geologicalproblems. It is not our intention to use the
simplifiedand idealized treatment of flow in section2 to
attempt a detailed explanation of the geologyin particular localities but rather to point out how lower crustal
flow might influencethe developmentand geometryof
somegeologicalphenomena.
It is generallyacceptedthat the viscosityof the lower
crust is lower than that of both the mantle and the up-
MCKENZIE
ET AL.'
FLOW
IN THE
LOWER
CRUST
11,037
per crust [Meissnerand Kusznir, 1987; Strehlauand
Meissner,1987],and thereforer • 1. When the lower
crust flowsso as to equalizecrustal thickness,the relevant problemis channelflow with a rigid deformable
surfacebelowand either a rigid or free deformablesurfaceabove.The flowis likely to be principallyby simple
shear,and Huppert's[1982]equationsapply.The channel that flows in the lower crust is unlikely to exceed
• 30 km in thickness, and deforms so as to smooth
Mohotopographyon a scaleof 100-200km (Figure 1),
sothat k/ _• I for the problemsof interesthere. The two
responsetimes that concernus are likely to behavein a
similarwayto thosein Figure3a whenk/ _• 1. The time
•-,•isthenalways
muchsmaller
than•-g,soisostatic
equilibrium is quicklyestablished,and the negativeslopeof
d•"g[dk
• at longerwavelengths
will leadto the develop-
I >35km• 30-35km
mentof fronts. Noticethat this is true because•h
The actual value of the mantle viscosityis unimportant.
3.1 What
Determines
Whether
Flow
Occurs?
Lower crustal flow is expected to occur even when
the upper mantle and crust can deform by brittle failure. Creepingflow at geologicallyimportant rates can
occurwhen the absolutetemperature of a solidexceeds
about 0.7 Ts, whereTs is the solidustemperaturein K
[Stockerand Ashby[1973].The solidustemperatureof
the lower crust is 700ø- 900øC, so flow shouldoccur
wherethe temperatureexceeds400ø- 500øC. Hence,if
the crustalthicknessis 30 km and the temperaturegra-
dientis 20øCkm-1, the bottom5-10 km of the crust
will be able to flow. These temperature estimatesare
W
100
•
0
40 •
200
I
ß
I
300
I
400km
I
I
••
Moho
Moho
CENTRAL GRABEN
halfgn}ben
W
50
100./] I k 150km
E
'-"'- •-"•"' -.• '"' •"_" •__.•..•-•.•1 o
0t I, •r•T'11
10
,.•,. MOnO
w
!:
40
MOIST
considerably
lowerthan thoseof Kruse et al. [1991],
Figure 8. Map of estimated Moho depth around the
who usevaluesof the activationenergyfor creepdetermined in the laboratory to estimate that lower crustal
flow will only occurovergeologicaltimescaleswhenthe
temperature exceeds700ø- 1000øC. Both temperature estimatesare lessthan the temperature of basaltic
magma, which is likely to be one of the main causesof
increasesin lower crustal temperature that lead to flow
(seebelow).
As we showedabove,providedthe upper mantle does
not flow horizontallyin responseto lower crustalflow,
the behavior of the upper crust has a minor influence
sectionsto show the refraction profile of Barton and
British Isles [after Meissneret al., 1986] and cross
Wood[1984]acrossthe Central Grabenof the North
Sea[adaptedfrom Barton and Wood,1984],and along
the lines of the MOIST seismicreflection profile north
of Scotland[adaptedfrom Brewerand Smythe,1984].
The Central Graben sectionis alongthe line CG on the
map and is shownwith sedimentarycoveras solid area.
The MOIST sectionis a migrated line drawing along
the line marked M on the map.
(compareequations(B9) and (Bll)). Thesearguments phy hasbeenpreservedfor many millionsof years. The
Alps and Appalachiansstill have deep roots and dewherethe crustalthickness
exceeds
20 or 25 km [seealso pressedMohos[e.g.,Bols and ECORS, 1990;Jameset
Kusznirand Matthews,1988]. Henceit shouldlargely al., 1968];the northernand centralNorth Sea (Figbe restrictedto continents,though it may also be im- ure 8) and the AegeanSea,both extensionalprovinces,
portant in oceanic regions such as Iceland, where the still haveelevatedMohos[Bartonand Wood,1984;Hok
crust is thick and the heat flow is high.
liger and Klemperer,1989;Makris and Stobbe,1984].
Any contrastin crustal thicknessgeneratesbuoyancy Even amongrifted continentalmargins,wherethe difshowthat the lower crustal flow is only likely to occur
forcesthat act so as to try and make the crustal thicknessuniform. However,theseforcesare not alwayssuccessfulin removing thicknesscontrasts, and there are
ference between oceanic and continental
crustal thick-
nessmightbe expectedto producesubstantialbuoyancy
forces[e.g.,Bott andDean,1972],thereare some,such
many placeson the continentswhere Moho topogra- asthe Armorican-Biscaymargin, that showa steadyde-
1,038
MCKENZIE
crease in the thickness of continental
ET
AL'
FLOW
IN THE
LOWER
CRUST
crust toward the
Figure 3a, which showsthe wavelengththat relaxes
oceanovera distanceof 100-200km [e.g.,Ginzburget fastest. If the channelis thin, this wavelengthwill be
al., 1985].Thereforethereareplaceswherethestrength relativelyshort (k• och/•): perhapsenoughto smooth
of either the lower crust or the uppermostmantle is suf- the Moho beneathtilted blocksin the upper crust,but
ficientto resistthe buoyancyforcesandmaintaina long- insufficient
to smooththe longer-wavelength
variations
wavelengthtopography on the Moho. However, in all in crustal thickness between stretched and unstretched
theseexamples,there is no sign of shorter-wavelengthupper crustal terrains. As the channel thickness inMoho topographymimicking the sawtoothtopography creases,the time requiredfor this longer-wavelength
of tilted blocksin the upper crust, and so someflow has flowwill decrease
dramatically.In particular,overthickoccurred,at least at these shorter wavelengths.
enedcrust that is heated, for example,by basalticunElsewhere,there are placeswhere a strong casemay derplating, could have a low-viscositycrustal channel
be made that flow in the lower crust has reduced or re- that is thick enoughto floweasilyandto compensate
for
moved long-wavelengthtopography on the Moho. Sev- uppercrustalstrain variationson a very shorttimescale.
eral authors have suggestedthis effect in parts of the
The association
of lowercrustalflowwith magmatism
Basin and Range provinceof the westernUnited States is particularlystrikingin the Basinand Rangeprovince.
[e.g.,Gans,1987;BlockandRoyden,1990;Kruse et al., Gans[1987]and Ganset al. [1989]pointedout the close
1991;Bird, 1991]. The interiorof the Tibetan Plateau association
in spaceandtime of extensionalfaultingand
is also extremely fiat, suggestingthat flow may have volcanismin many parts of the Basin and Rangeand
occurredover hundredsof kilometers[Fieldinget al., arguedthat the influx of mantle derivedbasalt may
1994]
have weakenedthe crust sufficientlyto causeit to colLower crustal flow is also likely to have occurredim- lapse. In particular, the areas where the upper crust
mediately north of Scotland, where deep seismicreflec- has extended the most and where lateral flow of lower
tion profilesshowfault-boundedhalf graben, that must crustal material from beneath adjacent less extended
represent a significant amount of upper crustal exten- uppercrustalareasmust haveoccurred(Figure 1) are
sion, almost at the sea bed (Figure 8) [e.g., Brewer all areasof voluminousvolcanismimmediatelyprior to
and Smythe,1984]. There is, however,no evidence and synchronous
with the extension. This magmatic
of postrift thermal subsidence,and differencesin the activity may have reducedthe lower crustal viscosity
Moho depth beneath this region and beneath the adja- by introducingheat (seesection3.2). The association
cent Scottish mainland, which has undergoneinsignifi- of lower crustal flow with igneousactivity is not surcantstretchingin the uppercrust[e.g.,Meissneret al., prising. Whether, and how much, basalticmagma is
1986],are small. Theseobservations
suggestthat the generatedby adiabaticdecompression
duringextension
lower crust may have flowed from under Scotland to- dependson the amount of stretching,the potential temwardthe offshore
region[seealsoKusznirandMatthews, perature of the asthenosphere,and the initial thickness
1988].Alternatively,the rifting may havebeenaccom- of the lithosphere
[MCKenzieandBickle,1988].Most
paniedby magmatism[White and Lovell,1997]which intracontinental sedimentary basinshave stretched less
reduced the difference in crustal thickness.
than a factor (/3) of two and will not havegeneratedsufIn these examples, there are some obvious patterns.
The lower crust has flowed during extension in places
where earlier shortening had probably thickened the
crust to -• 40 km, such as in the Basin and Range
ficient volumesof partial melt to affect the total crustal
thickness. Where asthenospheretemperatures are elevated or whereextensionfactorsare extreme,it is easy
to generatea few kilometersof partial melt, which, as
province[Coney and Harms, 1984] and in Scotland we demonstratebelow (section3.2), is sufficientto re[Watson,1984;McClay et al., 1986]. In contrast,the duce the viscosityof the lower crust enoughto cause
best examplesof extension creating significant varia- flow. Thus there are circumstances
in which we might
tions in crustal thickness over relatively short distances expectigneousactivity to accompanyextension,though
areplaces,suchasthe SaltonTroughin California[Fuis the adiabatic decompressionresulting from the extenet al., 1984]or the North Sea[Bartonand Wood,1984], sionmay not be the mechanismby which the magmas
where either the crust was never thick, or earlier ex- are produced. In the Basin and Rangein particular, the
tensionand/or erosionhad already uniformlythinned onsetof magmatic activity appearsto predate extension
it to _<• 30 km. These observationssuggestthat only at the surface[Ganset al., 1989].
crust thicker
than
-• 30 km has a weak lower crustal
channelthat is thick enoughfor flow to occurover long
distancesin a reasonable geologicaltime and in this
way to remove Moho topography. We would clearly
expect this relationship from the behavior shown in
We conclude that both initial
crustal thickness and
the presenceof magmatic activity play an important
role in encouraginglong-wavelengthflow in the lower
crust. There are placeswhere the crust was thick prior
to extension and yet Moho topography remains. One
MCKENZIE
ET AL.: FLOW
example is the northern Aegean Sea, where deep offshore basins that are bounded by large normal faults
that have elevatedthe Moho beneath them [Makris
and Stobbe,1984; Mercier et al., 1989; Taymaz et al.,
1991]. Most of this extensionprobablyoccurredin the
last 5 Myr and was accompaniedby only minor igneousactivity [Bellonet al., 1979].Anotherexampleis
the Caledonian-Appalachianorogen. The Appalachians
IN THE
LOWER
CRUST
11,039
be short, of the order of 1-10 Myr, and that significant flow can begin during the extension. If it does,
then there may never be much differencein elevation
between relatively stretched and unstretchedareas of
the upper crust, nor much topography on the Moho,
thoughpresumablysomewould havebeen necessaryto
drive the flow.
In the Basin and Range province, upper crustal ex-
[Gans,1987,
havea crustalroot [Jameset al., 1968]that did not flow tensionand magmatismare synchronous
away as a result of the adjacent extensionwhen the At- Gans et al., 1989], and there is evidencethat lower
lantic Ocean opened. Yet the Scottishpart of the same
orogenno longer has a Moho significantlydeeperthan
its adjacent stretchedoffshoreregion. It may be significant that the Scottish part alone was affected by the
Tertiary igneousactivity that accompaniedthe opening
of the North Atlantic (seesection3.2). At the other extreme, there are placeswhere extensionaccompaniedby
igneousactivity has nonethelessproducedan elevated
Moho. An example is the Salton Sea region in Califor-
crustal flow was also synchronouswith the extension.
Had the flow not occurredduring extension,the highly
extendedcorridorsof upper crust, where/• > 2, would
have been topographicdepressions1-3 km deep and
would have thick accumulationsof sedimentary and vol-
canic rocks. Although thick sectionsof suchrocksare
preserved
locallywithin the half grabenof the extended
corridors,there is no evidencethat the corridorsthemselveswereregionaldepocenters.The mostcompelling
nia [Fuiset al., 1984],thoughherethe crusthad not evidencethat they were not comesfrom the distribution of regionallyextensiveignimbrite sheets. Both
been thickened prior to extension.
the KalamazooTuff in east central Nevada [Gans et
al., 1989]and the PeachSpringsTuff alongthe Col-
3.2 Scale and Timing of Flow
oradoRiver extensionalcorridor[Youngand Brennan,
An important result of the analysis in section 2 is 1974]wereeruptedwithinor adjacentto areasthat were
that the relaxation
of crustal thickness contrasts ocundergoing
localizedlargemagnitudeextensionat the
curs on a timescalethat is very wavelengthdependent. time they wereerupted.Theseignimbritesformedthin,
The lengthscalethat relaxesfastestcorresponds
to .-- 10 regionallyextensive
sheetsandweredeposited
on both
times the thicknessof the flowing layer, or in the region highly extendedand little extendedground. For exof 100-200 km for a channel 10-20 km thick. Substanample,the KalamazooTuff, which was eruptedfrom
tially longer and shorter wavelengthsrelax much more near the center of the extensional corridor in the eastslowly(Figure3a). This effectdependson the shapeof ern part of centralNevada,is foundas far east as the
the minimumin •-•(k•) and henceon the viscositycon- ConfusionRangeand as far west as the Butte Mountrast, whichis not well known in nature. Thus the min- tains (seeFigure1 and Ganset al. [1989]),both of
imumin •-•occursat/k •_ 20h(Figure3a) for a viscosity which have undergonelittle extension. Similarly, the
contrast of r = 50. If, however,r = 10, the minimum in PeachSpringsTuff is found from the central Mojave
•-• occursat A _• 10h. If heatfromigneousintrusions
is Desertto the ColoradoPlateautransitionzone[Glazner
important in lowering the lower crustal viscosity,only
thosewavelengthswhosetime constantis shorter than
the thermal
time
constant
will be able to relax.
We
shouldtherefore look for geologicalevidenceof flow on
the scale of -• 100-
200 km. This is the scale on which
et al., 1986]and crosses
severaldomainsof highlyextended and relatively unextendedupper crust. These
sheetswereemplacedby flow over the groundsurface
and are typicallyonly tens of metersthick. They are
not significantly
thickerwithin the extensionalcorri-
flow has occurred in Figure 1.
dorsthan outside. As ignimbritestypically pondwithin
The geologicaland thermalhistoryof an extendedre- topographic
depressions,
there cannothavebeenany
gion is greatly influencedby whether the lower crustal largelong-wavelength
elevationdifferences
betweenexflow was contemporary with, or later than, the exten-
tended and unextended areas. Extension at the latitude
sion. If we take the minimumvalueof •-• = 316 from of the WhippleMountainsoccurredprincipallybetween
Figure3a, and a valueof •ll/Pl = 1017m2 s-l, then 20and17Ma [Davis,1988;NielsonandBetatan,1990],
equation(A14) givesa minimumresponse
time % of 7
Myr for a lower crustal layer 15 km thick. This esti-
so for the 18.5 Ma Peach SpringsTuff [Nielsonet al.,
1990]to haveflowedacrossa regionwith little variation
mate is proportional to the viscosity,which is uncertain by at least an order of magnitude, and also dependson the viscositycontrast,r. Thus if r - 10, the
minimum responsetime •-bis only 3 Myr. Nonetheless,
in elevation, lower crustal flow must have occurredon a
timescaleof • I Myr. This is consistentwith our assertion that significantrelaxationof long-wavelength
topo-
these calculations indicate that the response time can
a time scale of only a few million years.
graphicand Moho elevationcontrastscan occurwithin
11,040
MCKENZIE ET AL.' FLOW IN THE LOWER CRUST
If flowwereto occurlongafter extensionandthe associatedthermalsubsidence
had ceased,
it wouldproduce
uplift of the synriftand postriftsediments.Suchuplift mayraisethe sediments
abovesealevel,wherethey
couldbeeroded.KaufmanandRoyden[1994]havesug-
Detachment
gestedthat postrift uplift observedin the Halloran Hills
area in eastern California is due to later lower crustal
Anotherwayin whichpostextensional
flowmightbe Figure 9. Sketchto showhow the spreadingof a fluid
recognizedis by hydrocarbon sourcerocks that are too
layerin the lowercrustcan producethe propagationof
mature for their present depth of burial. Postexten- a detachment. Rotation of the lower plate occursas the
sional flow provides a mechanismfor structural inver- fluid front passesbelow. The thicknessof the fluid layer
must exceedthat of the upper plate if this mechanism
sion that does not involve the reactivation of old normal
to operate. The geometryin the regionof the front
faults as reversefaults, though this latter mechanism is
is not correct in detail.
certainlydoesoccurin places[StoneIcy,1982;Roberts,
1989;de Graciansky
et al., 1989;Letouzey,
1990].The
regionof extensionalhalf grabenimagedby the MOIST
seismicreflectionline north of Scotland(seeFigure 8 lower crustal flow could keep pace with upper crustal
and Brewer and Smythe,[1984])may have beenup- stretchingand preventthe formationof significantlong
lifted by lowercrustalflow. In this regionthe Moho is wavelengthMoho and surfacetopography. In the secapproximately the same depth as it is under Scotland. ondcase,magmatismand flow may followthe inception
There is no apparent postrift subsidenceabovethe half of stretching by a period that dependson the initial
graben,whoseuppermostsynrift infill of Permo-Triassic mantle temperature,the amount of stretching,and the
sediments
is foundalmostat the surface[Kirton and strain rate if the magmatismis related to the extension
Hitchen,1987].Clearly,lowercrustalflowhasoccurred itself[MCKenzieand Bickle,1988]or by an arbitrary
here and has removed any crustal thicknesscontrast periodif it is unrelatedto extension.In the latter case,
betweenScotlandand offshore. However,the Permo- deep basinsmay form during stretchingand then be
Triassic
sediments
arecontinental
or lacustrine
[Ziegler, uplifted by the flow.
1982],sothe lackof postriftsediments
may be because Figure 9 showsa sketch of the first case, in which
the Permo-Triassic
(or older)extensionoccurredabove the flowinglower crustal layer is simultaneouswith exsea level in thickened crust that subsided at the end of tension. As we arguebelow (section3.3), there is evtheCaledonian
orogeny
[McClayet al., 1986].Flowmay idencethat, at least in the Basin and Range, this exhave occurredat this time, as the Devonian extension tensiontakes the form of a throughgoingfault, which
wascertainlyaccompanied
by intrusiveigneousactivity in our model is progressivelyrotated and exhumed as
[Ziegler,1982],ratherthanin the muchlaterPaleocene-the front propagates.If Figure 9 describeshow detachEoceneigneousepisode.Suchigneousactivity compli- mentsare uplifted, the thicknessof the flowinglayer
catesinterpretation of the data, sincemelt will flow into may be estimated from the amount of structural relief
regionswhere thinning was greatest and thus reduce seenas the front passes.If the detachment is to reach
crustalthicknesscontrastsin a mannervery similarto the Earth's surface,the thicknessof the layerof viscous
lowercrustalflow. The regionaluplift of Scotlandis al- lowercrust which is flowingmust exceedthe thickness
mostcertainlydue to underplating[White and Lovell, of the upperplate. Use of equation(B15) then allows
1997]. The data are thus equivocal,but the contrast an estimateof the lowercrustalviscosity.If mylonites
is strikingbetweenthe MOIST area north of Scotland, are uplifted from depths of -• 10 km over distancesof
areais 3 x l0 s m•', so
whichwasaffectedby the early Tertiary basalticvolcan- -• 30 km, thenthe cross-sectional
ism and has alsoexperiencedlower crustal flow, and the
Central and Viking Graben east of Scotland,wherethe
Permo-Triassicsedimentsare typically between3 and 5
km below the seabed, and the Moho remains elevated
(Figure8).
thatq - 5 x 107m•',g' - 10.9g(fromequation
(Bll)),
and x - 3 x 104 m. Then
k - 6 x 10TM/r
h mmyr- [
(8)
The speedof propagationof suchfronts is probably
in the regionof 10 mm yr-[ ratherthan i mm yr-[
case is a situation in which extension is initiated over a
or i m yr-[. This argument
suggests
that /]1 ( 6 x.
rising mantle plume or over a subductionzone, where 10x9Pa s. Similarvalueshavebeenderivedby Kruse
igneousactivity and perhapsalsocrustalthickeninghad et al. [1991]andKaufmanandRoyden[1994].
We can thus envisagetwo extreme cases. The first
already reduced the viscosity of the lower crust. Here
Such low viscositiesmay well require a local heat
MCKENZIE ET AL.: FLOW IN THE LOWER CRUST
11,041
source,suchas an igneousintrusion. The amountof temporary with the faulting. In most of the core complexes of the western United States, there is a strong
easilyestimated.Let us assumethe lowercrustto be likelihood that this is the case, as we have already dismelt needed to soften the lower crust substantially is
at a temperatureof 400øCand the basalticintrusionto
be at 1200øC: an excessof 800øC. If 3 km of basalt is
intruded,this will raisethe temperatureof •0 9 km of
lowercrustby 300øC.Thisis enoughto induceflow(see
section1) and is compatiblewith the amountsof melt
that canbe producedby stretching[seeMCKenzie and
Bickle,1988],particularlyif/? > 2 or if the upperman-
cussed.
It is, however,difficultto understandhow the processesdiscussed
abovecan bring the upper surfaceof
the lower crustal channel to the Earth's surface. The
top of the channelis formedby rocksthat wereoriginallyat midcrustaldepths.Mostof the shearing
occurs
bysimpleshearwherethechannel
thicknesss
isuniform
tle is hotter than normal. If the reduction in viscosity is
and little vertical motion is occurring.Futhermore,this
dueto magmaticaddition,the timescaleoverwhichthe
temperatureand viscosityof a layerof thickness10 km
shearingmust occurbeneaththe upliftedplate, since
the flowis drivenby gravity.The observed
structureof
corecomplexes
is rather different:the shearingoccurs
in front of the uplift, and the zoneinvolvedis then uplifted. The metamorphicfaciesof the deformedrocks
variesfrom lower greenschist
faciesin the upper mylonitesto uppergreenschist
or amphibolite
faciesat the
deepest
levelsthat areexposed
[e.g.,Listerand$noke,
1984;Davis,1983;Davis et al., 1986]. All thesefea-
would return to their normal values can be estimated
from the usual thermal time constant rth:
=a
(9)
wherea is the thicknessof the layer and n is the thermal
diffusivity,to be •0 0.5 Myr.
3.3 Detachment Faulting and Tilting in
Metamorphic Core Complexes
tures are consistentwith the model sketchedin Figure
9, wherethe detachment
is lockedontothe propagating
front of the lower crustal material but the flowing lower
Over the last two decades,there has been much debate about the origin of the metamorphiccore com- crust doesnot form the lower plate of the detachment
plexesof the western United States. In these com- itself.
plexes,relativelyunmetamorphosed
but highlyfaulted
cover rocks of the upper plate are juxtaposed against 4. Conclusions
metamorphosedand penetratively deformedunderlyWe have used simple analytic expressionsto invesingrocksof the lowerplatealongshallow-dipping
faults
knownasdetachment
faults[Crittendenet al., 1980].
Lower crustal flow providesa mechanismfor rotat-
tigatethe characteristics
of flowin a channelthat deforms so as to smooth out variations in crustal thick-
ing the detachmentfaults in core complexes
to shal- ness.These expressionsshowhow variationsin density
lower dips that does not involve movementon later and viscosityaffect the length scaleand timescaleon
generations
of steepfaults. This has beenrecognized which flow occurs,and also how the flow is influenced
by GansandMiller [1983],Buck[1988],Wernickeand by the boundaryconditionson the top and bottomof
Axen[1988],Spencer
andReynolds
[19901
andBlockand the channel. The most important results are that the
Royden[1990],amongothers. By removingmaterial timescaleof the flow is very dependenton wavelength,
so variations in crustal thicknesson a length scale of
ing it beneathregionsof extendedupper crust, large •0 10 times the channel thickness will decay the most
from beneath unextended horst blocks and reposition-
rapidly. As the body of low-viscosity
material flows
awayfromregionsof thickcrust,it doessoasa frontbeuppercrustalrocks.Buck[1988]showshowthe upper lowa topographicstepin the Earth'ssurface.Underreparts of faults can rotate to shallowerdips than their alistic boundaryconditionsthe deformationwithin the
lowerparts, thus acquiringa convex-upcurvature. He channelwill be mainly simple shear. Such a front can
suggests
that the flatter, upperpartsof the faultsbe- drive a detachment fault in front of it, in the same way
relative vertical motions can occur over a scale of a few
tens of kilometers,which in effect, producestilting of
comeinactive and that new faults form as steepersplays
as marbles on a carpet can be rolled by moving one's
offthe deeperpart of the fault, cuttingthroughthe former hangingwall. In this way flat, inactivefaults can
be exposedat the surface,thoughwhen they were active, they movedin a steeperdip rangecompatiblewith
seismological
observations
of activenormalfaultstoday
hand across the floor underneath.
We also examined the conditions under which lower
crustalflow is likely to occurover wavelengthsof 100200km in a geologically
reasonable
time. The strongassociation of lower crustal flow with areas of both thick-
[Jackson
andWhite,1989].Essential
to thismechanismenedcrust and basalticmagmatismsuggeststhat both
arebotha verysmalleffectiveelasticthickness
andflow effectsare important in reducingthe lowercrustalviscosityto a levelat whichflowis likely.Thereis evidence
in the lower crust. Furthermore, the flow must be con-
11,042
MCKENZIE ET AL.' FLOW IN THE LOWER CRUST
from parts of the Basin and Range provincethat the If • is expanded in a Fourier series
lowercrustalflow accompanied
the extensionof the upß = • sinkx,
per crust,sopreventingthe development
of topographic
contrasts between extended and unextended areas at
the Earth's surface. The timescale on which such flow
occurs puts constraints on the likely viscosity of the
lower crustal
channel.
(AS)
then
0 - (A+ Bz)e•z + (C+ Oz)e-•z,
(A6)
whereA, B, C, and D are constants.
Boundaryconditionsare imposedon u, w and the
shear
stresser,• using
extendedareasof the upper crust and lower unextended
If the lowercrust flows,it will elevatethe depressed,
regionssuch as horst blocks. Becausethe channel that
flowscan be in the region of 10-30 km thick, it can
produce relative vertical displacementsof severalkilo-
u=
-[kA + (1+ kz)B]e• sinkx
+[kC- (1- kz)D]
e-kzsin
kx
meters in amplitude over distances of a few tens of kilo-
w - kOcoskx
(A7)
(A8)
meters. Becauseit providesa mechanismfor tilting
that doesnot require faulting, it may help resolvethe
-2, [k2C- k(1- kz)DJe-• sinkx,
discrepancybetweenthe very low angledips of detachment faults in metamorphiccorecomplexesand the sub- where0 is the dynamicviscosity.
stantiallysteeperdipsseentoday in largenormalfaults
The kinematicboundaryconditionrequiresthe vethat move in earthquakes. Lower crustal flow is also a locity on any boundaryto equal the rate at which the
mechanismby which extendedbasinscan later be up- boundary
deforms.If any suchsurfaceis described
by
lifted, particularly if the flow occurred after extension d + {, where d is constant,then
axz
- -2,[k2A
+k(1
+kz)B
1e• sin
kx (A9)
had ceased. This mechanism is an alternative
to struc-
tural inversion by the reactivation of old normal faults
as reversefaults, which is also known to occur.
=
•
dt
(A10)
,
Ox
If we assumethat the decayis exponentialwith a time
Appendix A: Linear Theory
constant
r (whichis not necessarily
real),then
In general, any disturbanceto either the top or the
bottom of the layer generatesa flow that movesboth.
• - •0 exp(-t/r).
(All)
The equations are linear and therefore have solutions Continuityof normalstresson z - h requiresa• to be
of the form exp(at)f(x, z), wherea may be complex. continuous,where
The interfaceconditionsare that both componentsof
Ow
'the velocityand the normal and tangentialstressmust
o'zz
-- pgs
e- P1+ 2rlOz'
(A12)
be continuousand that the rate at which any interface
movesmust equal the normal velocity. This kinematic
whereP1 is the pressureperturbationdue to the flow.
boundaryconditionis conveniently
combinedwith the
Wecannowuse(A10)to eliminate
• from(A12),and
normal stresscondition. The problem then reducesto
expressP• and w in terms of 0:
findingthe roots of a functionthat is quadraticin •
(seebelow). The solutionsare found to be real for all
values of the horizontal wavenumber k.
a•- (krpgO
+3kq
dOridaO
)sin
kx (A13)
dz
The creepingflow of an incompressible
viscousfluid
in the absence
of buoyancyforcesis governed
by
k dz a
and p is the density of the fluid. If h is the thickness
of the layer,(A?)-(Ala) may be madedimensionless
by
V.v-O
(A1) substituting
and by the curl of Stokes'sequation
(x,z)-n(x',z'),
V•w-0,
r-•r
plgh '
t-
/n
(A14)
(A2)
wherepl is the densityand 0• the viscosityof the layer.
The modelthen requiresA, B, C, and D to be caldimensional
in the x,z plane,v (= (u,0,w)) can be culatedfor the layerand the half-space,z' < -1. Since
the velocitymusttend to zeroas z -• -oe, C and D
describedby a stream function •:
wherew - V x v is the vorticity.
If the flow is two-
for the lowerlayermust be zero. On the uppersur-
v-
-T;,0,
.
(A3)
Thevelocity
v thensatisfies
(A1) and(A2) if
V4•I
• --0.
(A4)
face of the layer, z' - 0, O'xz
' and O'zzare required to
be zero. The last four conditions
are providedby the
requirementthat u• and • should be continuous on
z• - -1 , and either by continuityof axz
' and u' or by
u• - 0 for both the layerand the half-space.The first
of these casesrequiresthat
MCKENZIE
r-
2k
ET AL.'
LOWER
r + 2k
1
k
(r- 1)(1- k)e-•
-[r(R-1)-2k(r-1)le
-•
IN THE
0
k
(r - 1)ke-k
[r(R-1)-2k(r-1)le
FLOW
=0
-•
CRUST
11,043
0
-1
-(r + 1)ke•
[r(R-1)-2k(r+l)]e
(r + 1)(1+ k)e•
-[r(R - 1) - 2k(r + 1)]ek
•
(A15)
The secondis the rigid boundary condition and requires
that
r - 2k
0
r + 2k
k
1
k
ke-•
[r(R-1)-2k(r-1)]e -•
(1 - k)e-•
-[r(R-1)-2k(r-1)]e
= 0
0
-1
-ke •
-k [r(R-1)-2k(r+l)]e
(1 + k)e•
• -[r(R-1)-2k(r+l)]e
•
(A16)
where the primes have been omitted and
r/2- rr/x, p2 - Rpx
where the subscripts 1 and 2 refer to the layer and
half-space,respectively.Clearly, (A15) and (A16) are
quadratic in r, and therefore there are two decay times
for the model. When k' >> 1, the equations are easily solved because the movement of the upper surface, z' = 0, does not affect the shape of the interface
z• = -1, and vice versa. Hence the decay time for
disturbancesof the upper surface can be found by requiring C and D to be zero in the layer and ß - 0 in
the half-space. There is then only one value of r • that
satisfiesthe equations
or
r• - 2qxk/pxg - 4•rqx/pxgA.
(A18)
Similarly,setting A - B - 0 in the layer gives
rg- 2k'(r + 1)/(R- 1)
2k(qx+ •2)
=
_ px)g
4•r(qx+ •2)
- px)gX'
r• - 4rlxhak4/3p2g
- 64•r4rlxha/3p2gA4,
(A22)
r• - 3R/k'2(R- 1),
3•xp2
3•xp2•2
n - px(p2
- px)ghak
2= x2p(p2
- px)gha'
(A2a)
The generalexpressions(A15) and (A16) were solved
by iteration, starting with the largest value of k• and
using(A18) and (A19) as trial valuesfor r} and
The resulting values were used as the trial values for
the next smaller value of k •.
A usefulquantity known as the admittance Z can be
calculated directly from the geophysicalobservationsis
the ratio of the gravity anomaly to the surface deformation at a particular wavenumber. It is given by
Z(k')- 3gpxI + (R-1)• -
(A24)
'
where a is the radius of the Earth, Pe the mean density
of the Earth, and •x, •2 are the deformationof the upper
and lower surfacesof the layer, respectively,obtained
or
n-
r•a- 4k'4/3R,
(A17)
(A19) from (A10). The curvesin Figure 6c are calculated
Equations(A18) and (A19) are the solutionsto both
(A15) and (A16) whenk• -4 oc. The time • is only
positive when p• • px. When p• • px, the model
is gravitationallyunstable,and any disturbancegrows
exponentially. A finite amplitude model of such an instability, with nonlinearrheologyis discussedby Bott
usingPx- 2.8Mg m-•
]
Pe- 5.5Mg m-•
ß
Appendix B' Nonlinear Theory
The expressions
givenby Huppert[1982]are easily
adaptedto the caseof a lighter liquid spreadingover a
denserlayer, with u = 0 on the interface. If the upper
surfaceof the layer is h(x) and its thicknessis a(x),
[1999].The otherlimitingcase,k• • 1, is harderto calthen if the shearstressvanisheson z = h(x),
culate. Two casesare of interest. When q• - 0, (A15)
gives
T• -- kt4/3n,
ra -- qxhakq/3p2g
- 16x4q•ha/3p2gA
4,
(A20)
rg- 4R/(R- 1), n - 4,xP2/Px(P2
- p•)gh. (A21)
-o.
If the horizontal velocity is zero on the lower boundary
of the layer,
= 0.
If u is zero on the interface,the correspondingexpres- The distance of the lower boundary of the spreading layer below the surfaceof the liquid half-spaceis
sionsfrom (A16) are
11,044
MCKENZIE
ET AL.' FLOW IN THE LOWER CRUST
a(x) - h(x). Sincethe layer is thin, the viscouscontribution to the normal stresscan be neglected,and the
boundaryconditionon azz at the baseof the layer re-
where
/ 3T]l
)1/5
Pl g'q3t
•N- 1.411...
quires that the pressure in the two fluids should be the
same. Therefore
The velocityVF with which the front travelscan be
obtainedfrom (B14)'
gPIa - gp•.(a - h)
or
dy -- 0 -a
(B14)
dt +
x
--
dx.
t
Hence
whereR-
p•./pI > 1. Huppert[1982]showsthat the
VF---(OY/Ot)x
= Plg'qSY•
(B15)
(OY/OX)t 15r]1
x4 '
balancebetweenthe pressuregradient and the viscous
forcesrequiresthat
0h
r/i 0•'u
=
Ox Pig 0z•"
(B4)
Acknowledgments.
This work was supportedby Natural Environment Research Council and the Royal Society.
Earth
Sciences contribution
5856.
Integration,subjectto (B1) and (B2), then gives
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