JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 97, NO. El0, PAGES 15,967-15,976, OCTOBER 25, 1992
Pancakelike
DAN
Domes
on Venus
MCKENZIE
Institute o! Theoretical Geophysics,Department o! Earth Sciences,Cambridge, England
PETER G. FORD, FANG LIU AND GORDONH. PETTENGILL
Center •or SpaceResearch,MassachusettsInstitute o• Technology,Cambrid_qe
Theshape
ofseven
largedomes
ontheplainsofVenus,
withvolumes
between
100and1000km3,
is comparedwith that of an axisymmetricgravity current spreadingover a rigid horizontal surface. Both the altimetric profilesand the horizontal projection of the line of intersectionof domes
on the synthetic aperture radar imagesagree well with the theoreticalsimilarity solutionfor a
NewtonJanfluid but not with the shape calculated for a rigid-plastic rheology nor with that for
a static model with a strong skin. As a viscouscurrent spreads,it generatesan isotropic strain
rate tensorwhosemagnitudeis independentof radius. Sucha flow can accountfor the randomly
oriented cracks that are uniformly distributed on the surface of the domes. The stressinduced
by the flow in the plainsmateriM belowis obtainedand is probablylarge enoughto producethe
short radial cracks in the surface of the plains beyond the domes. The viscosity of the domes
can be estimated from their thermal time constantsif spreadingis possibleonly when the fluid
is hot and lies between 1014 and 1017 Pa s. Laboratory experimentsshow that such viscosities
correspondto temperatures of 610ø to 700øC in dry rhyolitic magnaas.These temperatures agree
with laboratory measurementsof the solidustemperatureof wet rhyolite. These resultsshowthat
the developmentof the domescan be understoodusingsimplefluid dynamicalideasand that the
magmasinvolvedcan be producedby wet meltingat depthsbelow10 kin, followedby eruption
and degassing.
1. INTRODUCTION
viscosityobtainedby usingHuppert's[1982]expressions
is
domes[Headet al., 1991; Pavri et al., this issue]that were
unrealisticallylarge and that the dome shapeis instead controlled by stresseswithin a thin skin of cool, highly viscous
probably formed by the extrusion of viscousmagmas from
material
The plains of Venus contain large numbers of small
central
conduits.
Similar
structures
are found
that
forms
at the surface
of the dome.
Since the
on Earth
altimeter on Magellan can measurethe crosssection of the
whererhyoliticand dacitic magmasare extruded[seeFink, dome in Rusalka Planitia to an accuracy of about 100 m,
1987],and thoseon Venusare alsolikely to be formedfrom a profile across this dome can be used to test these three
silicic magmascontainingmore than 60% SiO9•. We chose models and in this way to constrain the mechanismsthat
control the shapes of domes on Venus and the theology of
seven large domes for detailed study. The largest is in
magmas involved.
Rusa.lka Planitia, and it is close to the equator where the
We first show that the profile of the dome in Rusalka
altimeter footprint along the track is only 2 km. Since the
diameter of this dome is 34 km, its shape can be accurately Planitia agrees well with that expected from Huppert's
measuredwith the altimeter on Magellan. The other six [1982] model but not with that calculatedfrom a rigidplastic theology. A static model with constant surfacetendomesillustrated in Figure 1 are at about -30øN and are all
smaller than that in Rusalka Planitia.
One altimeter
track
sion alsodoesnot give a shapethat agreeswith the observed
crosses close to the axis of one of these domes but does not
profile. We then show that a viscousmodel can also account
provide a detailed profile becausethe footprint spacing is for the observed shape of the curve of contact between two
about 5 kin.
overlapping domes, and for a number of other features on
thesyntheticapertureradar(SAR) images.Huppert's[1982]
Several suggestionshave been made to account for the
observedshapesof silicicdomeson Earth. Huppert[1982] expressionsallow the viscositiesof the magma involved to
arguedthat they were producedby an axisymmetricgravity be estimated, and the values obtained are similar to those
current spreadingover a rigid horizontal surface and calcu- measuredin the laboratory. This agreementin turn allows
lated the profileof sucha dome when the theologyis that of a the temperature of the magma to be estimated if its compositionis that of a rhyolite(rhyoliteand graniteare extrusive
viscousfluid. If, however,the theologyof the magma is better describedby that of a Bingham fluid with a yield stress and intrusiverocksof the samecomposition).The temper[seeFink, 1987;Blake,1990],then the shapeof the domeis atures obtained in this way agree with that found in the
described
by the expression
obtainedby Nye[1952].Another laboratory for the wet solidusof such a magma. Finally, we
modelfor suchdomeswas proposedby Iverson[1987]and speculate about the geologicalprocesseswithin Venus that
by Fink and Griffiths[1990].They arguedthat the magma may generate the silicic magmas that form the domes.
2. DOME
Copyright 1992 by the American Geophysical Union.
PROFILES
In plan view the sevendomes are almost perfectly circular and have radii that are large comparedwith their heights
Paper number 92JE01349.
(Figure 2). Their shapesuggeststhat they were formed
0148-0227]92]92 JE-01349505.00
by eruption from central conduits whose maximum width
15,967
15,968
MCKENZIE
11.2
I
ET AL.: PANCAKELIKE
11.6
I
I
•,
•
.":-•'v=",.,
....
...,, '"".
.
• • .
. ••-•
....
:•c...
........ '•
•.' ,., •
.-•'
•.•'.-','•'
- ......
,. ..?.,
......
12.0
I
12.4
I
• •
'-?
DOMES ON VENUS
,/
-29.4
•.:.•'i--:...: ...•---:'•.'
.
.'-•,,.,
' •'
"-%•'
"•'••
.,•,.'..:...--,,,',,
,.•• ' •',•.,•
-29.8
........
•'::-..:.,
• .....
-?:::0•-
'•"•'•:
•::,,,;.
•::,,.:,•:•
,•,"•
.. •,•.•..•..• ..
^f,4(c)
^f,4(b)
4(a)
Fig. 1. A SAR imageof the domeseastof Alpha. The pairs of domesthat are shownin plan view in Figure 4 are
marked by labelled arrows.
was small compared with the present radii of the domes.
Their circularity requires the surface onto which they were
erupted to have been planar and horizontal. We examine
three models that have been proposed to account for the
shape of similar domes on Earth. In two of these the rate of
releaseof gravitation energyby the spreadingof the domeis
restricted by internal stresses.These are viscousstressesin
3(a)
I
4(a) A
II
,
Huppert's[1982] model and stressescausingplastic failure
in Nye's [1952]rigid-plasticmodel. In the third model that
we consider, the spreading tendency is resisted by constant
4(a) B
tension
,
in a thin
skin.
tail by Huppert[1982],who showsthat the heighth of the
4(b) A
surface of a dome is related
'
=
t
I
to the radial
distance
r from its
center by
4(b) B
ß
surface
The spreadingof an axisymmetric viscousdrop of fluid
over a rigid horizontal surfacehas been studied in somede-
-
where ho is the height at r = 0 and r0 is the radius of the
, dome. This expressionis only valid when r0 ),) h0 and does
not apply near the edge of the dome where r0 -r • h0.
I
Equation (1) appliesonly when the volumeof magmain
the dome is constant
4(c) A
and will therefore
not describe
dome
growth. However, Huppert found experimentally that the
' shapedescribedby equation (1) was rapidly established
when fluid ceasedto be supplied to the dome and also argued on theoretical grounds that any initial shape would
asymptotically approach this similarity solution. Equation
4(c) B
(1) shouldthereforeprovidea good approximationto the
0
10
20
30
•
40
shapewhenthe durationof the eruptionre is shortcom-
Fig. 2. Sections without vertical exaggeration through the cen-
pared with the thermal time constant r• of the dome. We
will for simplicity assume that r• ),) r•, but there is at
present no evidence that this assumptionis correct for Venu-
ters of the domes
sian domes.
km
in Table
1.
MCKENZIE
ET AL.: PANCAKELIKE
Figures 3a-3e show two profilesthat were chosenbecause
they pass close to the center of a large dome in Rusalka
DOMES ON VENUS
15,969
with sucha theologywasobtainedby Nye [1952]. Usingthe
same notation as before, his expression is
Planitia (Figure 3e). Sucha restrictionis important,both
becauseDoppler sharpening of the radar pulse returned to
the altimeter can only reduce the size of the footprint in the
direction that is along the track and because the shape of
the center of the dome is most strongly affected by its theology. The size of the footprint in the across-trackdirection
cannot be improved by Doppler sharpeningand is never better than 15-30 km.
Profiles
must therefore
cross close to the
center of the dome, where the approximation that the shape
changesonly slowly in the cross-trackdirection is likely to
be satisfied,if the shape of a dome is to be accurately measured at intervals that are small compared to 15 km. The
and has a discontinuityin dh/dr at r = 0. Sincethe shear
stressin the plastic region is independentof strain rate, the
dome shape calculated for such a theology is independent of
the eruptionrate. Blake[1990]hascarriedout a numberof
experiments with a suspensionof clay in water that has a
theologysimilar to that of a rigid-plasticmateriM and found
that Nye's expressionprovides a good fit to his experiments
(noticethat Blake usesthe samesymbolR to denoteboth
r and r0- r). However,as Figures3b and 3d show,Nye's
domein Figures3a-3eis closeto the equatorof Venus(Table expressiondoes not fit the altimeter observations.The rms
1), wherethe spacecraftis closeto the planet and therefore misfit is 121 m for the profile from orbit 3067 and 165 m from
its horizontal velocity is large. For both these reasonsthe
spacing between the Doppler sharpenedfootprints of the altimeter is only about 2 km and shows how effectively this
techniquecan increasethe spatial resolutionof the altimeter
[Fordand Pettengill,this issue].Huppert's[1982]modelwas
also used to fit the profile across the dome east of Alpha
shownin Figure 3f at a latitude of about -30øN. At this
latitude the footprint spacingincreasesto about 5 km becauseMagellan is farther from the planet and is travelling
more slowly. This profile therefore providesa lessseveretest
of any theory of dome formation than do those across the
dome in Rusalka
Planitia.
Two valuesof r0 were first estimated by measuringthe
locationof the domecenteron the digital SAR image and of
the two points whereeachaltimeter track crossedthe margin
of the dome. The values of r0 and h0 were then found by
orbit 1277. Therefore the ability of a profile to discriminate
betweenthe two theologicalmodelsis greater when the track
passes
closeto the centerof a dome(Figure3e).
The third model that has been proposed to account for
the shape of silicic domes depends on the strength of a thin
skin of cool material at the surfaceof the dome [Iverson,
1987;Fink and Griffiths,1990]. These authorsarguedthat
the magma viscositycalculatedfrom Huppert's[1982]expressions
[seeHuppertet al., 1982]was muchgreaterthan
that observedin the laboratory, and therefore that someprocessother than viscousspreading must be involved. How-
ever, Webband Dingwell[1990]haverecentlymeasuredviscosities
of 10•4 Pa s in dry rhyolitemeltsat a temperature
of about 700øC, and therefore this objection to Huppert et
al.'s proposalsno longerapplies(seebelow). It is nonethe-
lessof interest to discoverwhether a thin strong skin under
fittingequation(1) to the altimeterprofilesby leastsquares, constant tension can account for the observed shapes. To
with the requirement that r0 lay between the two estimates do so, we used the differential equations given by Iverson
obtained from the SAR image. Once r0 and h0 are known [1987]that are satisfiedby any radial profileand retain his
the volume Q of a dome is easily obtained by integration of notation
equation(1)
37r
q = q-h.0,0
and is listed
in Table
1. The
(2)
rms misfits
If the reflection
is from
a smooth
horizontal
surface,the error is governedby the pattern of phasemodulation of the radar and is no more than a few meters.
How-
ever, if the surfaceis rough, the shape of the returned pulse
is controlled by energy scattered from regions that are not
directly beneath the track of the spacecraft. Doppler sharpening can remove returns from regionsahead or behind the
spacecraft, but the returned pulse is still broadened by scattering from regionsat right anglesto the spacecraftvelocity
vector. This effect controls the accuracy with which the
arrival time of the reflected pulse can be measured. The
surfaceof the domesis rough and strongly affectsthe shape
of the returned pulse. Therefore the likely error in height
along the profilesin Figure 3 is probably as great as 100 m,
and the differences
between
the observed and the calculated
shapesin Figure 3a and 3c are unlikely to be significant.
An alternative model for the theology of a silicic magma
is that of a rigid-plastic material, and the shape of a dome
_"(a)
rt
&--'
dr'= •1 "- rl2
are 51 m for or-
bit 3067, and 90 m for 1277. It is not straightforward to
compare these misfits with estimates of the likely errors in
the heights measured by the altimeter, becausethese are
strongly dependenton the reflection propertiesof the Venusjan surface.
=
dr t
2
where
,
r =
T
,
•/at/pg'
z =
z
V/at/pg
'
r/=sin•
a is the shear stress within the skin of thickness t, p is the
melt density, g is the acceleration due to gravity, r is the
radial and z is the vertical coordinate, and • is the angle
betweenthe tangent to the surfaceand the horizontal. Following Iverson, z was taken to increasedownwards,which is
the oppositeconventionto that usedby Huppert. Equations
(4) and (5) wereusedwhenI•1 < •/4. When this condition
was not satisfied,
dr'
--
I•
dz' V/1_ it2
V/1- it2
dz'
were used instead, where
COS•)
(6)
(7)
15,970
MCKENZIE
ET AL.: PANCAKELIKE
DOMES ON VENUS
Orbit #3067
Orbit #1277
(c)
(a)
,
,
,
Huppert'sviscoustheory
Huppert'sviscoustheory
ß
1.0
1.0
0.5
0.5
ß
-0.5
i
i
-10
-0.5
i
0
ß
ß
ß
i
-10
10
i
i
0
10
Distance from center of dome
km
Distance from center of dome
ß
ß
ß
2O
km
(d)
(b)
........
Nye
E
Iverson
....
1.0
1.0
0.5
0.5
-0.5
'
'
'
-10
0
10
Distance from center of dome
-0.5
km
........
Nye
'
'
'
-10
0
10
Distance from center of dome
Iverson
km
Fig. 3. (a) Altimeter profile acrossthe domein RusalkaPlanitia from orbit 1277, shownin Figure 3c, from
footprint 627 on the left to 644 on the right, plotted as a functionof the distancealongthe track, with the origin
at the point of closestapproachto the centerof the dome(Table 1). The heavyline showsthe best fit obtained
usingequation(1) for a newtonJan
viscous
rheology
andtheparameters
in Table1. (b) The sameprofileasFigure
34, with the bestfitting curvefor a rigid-plasticrheologyfrom equation(3), shownby the heavydashedline. The
dotted line showsa skin model with a similar aspectratio to that observed,obtained by integrating equations
(4)-(7) with h' = 3 x 10-s. (c) and (d) Asfor Figures34 and3b,but for orbit3067,fromfootprint670on the left
to 688 on the right. Two points at the level of the plains plot inside the calculated profile of the dome because
the echofrom the plains was so muchstrongerthan that from the edgeof the dome that the time of arrival of the
latter couldnot be measured.(e) Groundtrack of the altimetric profilesin Figures34-3d. The continuouscircle
showsthe radiusof the domeobtainedfrom fitting orbit 1277,the dashedcirclethat from 3067. (f) A profilefrom
orbit 576 from footprint 1454 on the left to 1461 on the right, acrossthe westernof the two most easterlydomes
eastof Alpha (seeFigure44 for the altimetertrack). The heavyline showsthe bestfitting modelwith NewtonJan
viscosity.
Integration was started with •/= 0 and z' = h', where
/,'=
/'
y/at/pg
that the aspect ratio, defined to be the ratio of the radius
(S)
to the heightof the dome,was~ I unlessh' wasverysmall.
The observedaspect ratio of the dome in Figures 3b and
3d is about 10 and could only be reproduced with a value
and h is the height of the origin of the coordinate system
of h' of between10-s and 10-9 . The top of the domeis
abovethe centerof the dome, and first usedequations(4)
and (5), changingto (6) and (7) when• • •r/a, then back
to (4) and (5) when I•bl _< •r/4. Integrationwas stopped
whenI•bl• 0.01. A schemesimilarto that of Iverson'swas
used,except that all four equationswere integrated usinga
then very fiat. Such a result is to be expected from the
behavior of drops of mercury, whoseshape is controlledby
a constant surface tension, and therefore satisfiesequations
fourth-order Runge-Kutta integration, and 0.5 was replaced
by 1.0 in two placesin line 1920 of his code.
(4)-(7). Whensuchdropsaresmall,they areapproximately
spherical,but the top of large aspect ratio pool of mercury
is sufficiently fiat to act as mirror that reflects with little
distortion.
The valueof h' in equation(8) is the dimensionless Figures 3b and 3d showthat the profile of a dome whose
height above the center of the dome at which the pressure shape is maintained by a constant skin tension does not
within the fluid of the dome would be zero. Integration agree well with that observed. It is of course possibleto
of the equationsstarting with variousvaluesfor h• showed maintain a dome of any shape by suitable forces within a
MCKENZIE
ET AL.: PANCAKELIKE
DOMES ON VENUS
15,971
Figure 3] showsthe best fit to one of the domes illustrated in Figure 4a, with arms misfit of 11 m. Though the
fit is good, the 5-km spacing between the altimetric foot-
(e)
printsis too largeto providea goodtest of Huppert's[1982]
model. An attempt was also made to use a profile from orbit 571, acrossa large dome in Tinatin Planitia at 12.10øN,
7.63øE. Unfortunately,both the dome and the surrounding
plainshave been deformedand tilted sinceit was emplaced.
It is not obvious how such effectscan be removed, and unlessthey are, fitting the calculated profile to that observed
is meaningless.
These three comparisonsbetween the calculated and observed shapesof the Venusian domes show that the viscous
N
ß
ß
ß
0
10
km
model of Happert [1982] best fits the altimetric profiles.
20
The other two models are therefore
Orbit #576 across dome E. of Alpha
(0
not further
discussed.
Though the dynamics of the spreadingappear to be controlled by viscousforces, rather than by the properties of
the cold skin, such a skin must be present.
0.5
3. DOME INTEP, SECTIONS AND SURFACE FEATURES
The altimeter profiles discussedin the last section provide the most direct
test of the various
models
that
have
beenproposedto accountfor the shapeof the magma domes.
However, the SAR imagescan also be usedfor this purpose,
because the position of the line that marks the junction
between two overlapping domes depends on their shapes.
• 0.0
O
Equation(1) allowsthe projectionof this line onto a horizontal plane to be calculated analytically from the radii r•
and r• of the two domes, their central heights h• and h•,
and z] the separation of their centers. If the origin of the
coordinate system is taken to be at at the center of dome 1,
and the +x direction to be along the line joining the centers of the two domes, then the horizontal projection of the
Huppert's viscoustheory
i
-0.5
i
-10
i
0
10
Distance from center of dome
km
line of intersection
Fig. 3. (continued)
between
the two domes forms an arc of
a circle whose center is at
o)
thin skin. However, the shape is only uniquely defined by
the value of h' if the tensional force within
(9)
with radius R given by
the skin is re-
quiredto be constant.Iverson[1987]imposedthiscondition
but did not explain the physical processby which this state
is maintained. His model assumesthat it is the strength
of the skin, rather than the fluid mechanics of melt transport, that controls the shape of the dome. However, it is
unclear how such a dome could grow, sinceits volume could
only changeif the skin were to break. Once it did so, the
magma would rapidly flow through the break and produce
• (x,)2
r•r2 +1) --•1
R2=•-c•a
(1
(10,
where
(•3
1
The
intersection
is a straightline when/• -- 0, or r]/r2 -3
3
h]/h2. The valuesof hi and h2 were obtainedfrom the alti-
the type of compositeflowseenin Fink and Griffiths'[1990] metric observations, and rl, r2 and x• were calculated from
photographs. No such structures are visible on the domes
on Venus.
TABLE
Figure
Latitude,
deg
Longitude,
deg
1.
the positionsof three pointson the edgeof eachdome. Mercator projectionsof three pairs of domesin Figure 4 show
Domes
Radius,
Height,
Volume,
km
km
kills
Rusalka
1277
3a
-2.867
150.890
17.2
1.48
1040
Rusalka
3067
3c
-2.867
150.890
15.8
1.40
82O
12.305
13.4
0.44
186
12.305
12.7
0.55
209
12.428
12.3
0.67
237
11.854
13.7
0.97
428
11.999
13.3
1.26
514
11.432
12.9
0.45
176
11.223
13.1
1.05
457
Alpha
Alpha
Alpha
Alpha
Alpha
Alpha
Alpha
E
E
E
C
C
W
W
3]
4a, A
4a, B
4b, A
4b, B
4c, A
4c, B
-29.726
-29.726
-29.711
-29.632
-29.781
-29.747
-29.736
•The temperatureestimatedfrom equation(24).
Pas
1.0
8.9
3.7
1.3
3.6
1.9
7.8
4.5
3.1
x
x
x
x
x
x
x
X
X
øC
10 ]?
10 •s
1014
10 •5
10 •5
10 •s
1016
10 •4
1016
years
610
7400
610
6600
700
65O
670
970
66O
1400
63O
3000
610
5100
690
65O
620
3500
15,972
MCKENZIE
ET AL.' PANCAKELIKE
DOMES ON VENUS
azimuth and the angle between the line joining the spacecraft to the feature and the vertical, measuredon the surface
of Venus. The heavy dashedlines in Figure 4 show the posi-
(a)
lO
tion of the intersection
curves after this correction
has been
applied, and they agree well with those observedin all three
cases. Such agreementis surprising,since the flow during
the emplacement of the seconddome in each pair must be
influencedby the presenceof the first. Indeed, the original
aim of the calculation of the shape of the line of intersection was to discoverwhich of the two overlappingdomeswas
emplacedfirst. The agreementbetween the calculated and
observedlines suggeststhat the presenceof the first dome
has only a minor influenceon the flow that emplacesthe second and that it is therefore not easy to discoverwhich is the
older dome of each pair from their shapes alone. An experimental test of these results can easily be carried out in the
laboratory, and such a program has recently been started
E
-lO
I
(b)
lO
by H. E. Huppert (personalcommunication,
1992). Preliminary resultssuggestthat the shapeof viscousspreading
domes is indeed rather little affected by obstaclesin their
path.
l:::o
The good agreement between the calculated and observedshapessuggeststhat the velocity field calculatedby
Huppert [1982]can be usedto study the evolutionof the
A
-lO
domes.He shows(equation2.19) that the radial velocityu
is
i
-10
i
i
0
10
i
u(r,z
t)= Pg
Ohz(2hz)
'
2rt Or
i
20
30
(11)
wherep (_•2.4Mg m-a) isthedensity
andr/theviscosity
of
thedomemagma,andg (= 8.87m s-a) is theacceleration
(c)
due to gravity. The radial velocity at the surfaceof a dome
lO
is therefore
3r
l:::o
where t is the time sincethe eraplacementof the dome, and
the three componentsof the horizontal strain rate tensor •
o
are
cgu
A
-lO
-1
0
i
i
0
10
i
i
20
30
krn
Fig. 4. (a) The heavy lines showthe Mercator projectionof the
outline of the eastern pair of domes in Figure 1. The centers,
shown by the larger open dots, are 10.91 km apart, and the x
axis
has an azimuth
of 80.46 ø .
The
fine
dashed
3
u
3
&•= O-'-•=
16•' &•=0' &•-- r -- 16t
B
line
shows
the
projection onto the horizontal plane of the line of intersection of
the two domes, that with heavy dashes shows the intersection
line after the slant range correction has been applied, using an
azimuth of 84.74ø, measured clockwise from N, and an inclination
of 30.3ø for the illumination, both measured at the domes. The
solid dots are points on the intersection measured from the SAR
images. The arrow labeled N shows the northward pointing vec-
tor. (b) As for Figure 4a, but for the two centerdomesin Figure
1. The dome centers are 18.66 km apart, and the x axis has an
(13)
where • is now the azimuthal angle. Since the radial and
azimuthal strain rates are equal and &• is zero, the surface
of the dome undergoesuniform extensionat the samerate in
all directions. Furthermore, the strain rate is independent
of the radius. The SAR images of the domes in Figure 1
showthat someof their central regionscontain a number of
irregular cracksthat do not have any preferred orientation.
They are also uniformly distributed over the central region
of the domes. The strain rate describedby equation(13)
can produce such cracks in the cooler crust of the dome as
the interior
continues
to flow.
Because the horizontal
strain
rate tensor is isotropic and independent of radius, the cracks
should not show any preferred orientation, nor should their
density vary with radius. The cracks form within a thin
cold layer of high viscosityat the surfaceof the dome whose
azimuth of 140.68ø. (c) As for Figure 4a, but for the two western influenceon the flow is ignoredin Huppert's[1982]calcudomes in Figure 1. The dome centers are 20.03 km apart, and
lations. That such an approximation is valid is suggested
the x axis has an azimuth
of 97.26 ø.
by the good agreementbetweenthe predicted and observed
shapes. There is also some evidence that the formation of
such cracksis influenced by the viscosityand temperature of
the shape of the calculated intersect lines. However, they the dome-formingmagma. Of the domeseast of Alpha, the
cannot be directly compared with the SAR image, because three thickest domes, whose viscosityis highest and temthe apparent position of a feature on the surface depends perature is lowest, are the two illustrated in Figure 4b and
on the height of the feature as well as on the viewing angle dome B in Figure 4c. These are the three with the most
from the spacecraft. This correction depends both on the obviouscracksin Figure 1.
MCKENZIE ET AL.' PANCAKELIKE DOMES ON VENUS
The SAR image also shows that radial striations are
present near the margins of the domes. Though the ap-
15,973
tanceof the material over whichthe domesare spreadingto
the two modes of failure.
If it is similar to that where Ven-
proximationsmadein derivingequation(1) fail in this re- era 13 and 14 landed, its yield strength is lessthan 25 MPa
gion,Huppert's[1982]expressions
for the velocityshowthat [Surkovet al., 1984],andit is likelyto fail in the mannerobd• >> dzz and •z• = 0. Hencevertical cylindricalsurfaces served.The fracture geometryvisibleon Venusjust beyond
undergo almost uniaxial extension in the azimuthal direction. Such a strain rate acting on a surfaceinclined to the
horizontal could produce radial tension fractures like those
the edge of the domesdiffers from that observedat Mount
observed.
Mount St. Helens,extensivethrusting occurredin associa-
St. Helens[Chadwick
et al., 1988]in severalways.On Venus
the fractures are short and radial with no visible thrusts. At
FigureI showsthat the plainsjust beyondthe outeredge tion with the radial fractures,and Chadwicket al. [1988]
of severalof the domes are cut by small short radial extensional fractures spaced at intervals of 2-3 km around the
perimetersof the domes. No associatedthrusting is visible.
These fracturesprobably result from stressesinducedby the
argued that both were producedby stresseson the walls of
the vertical conduit that suppliedthe dome with magma.
A similar originfor thoseon Venusis unlikely' they extend
only a few kilometersbeyondthe edgeof the domes,a dis-
spreadingof the domesoverthe plains.The force/unitarea tance that is small compared with the dome radius. The
fi that actson a surfacewhosenormalis nj is
spacingbetweenadjacentfracturesis also small compared
with the domeradius.It is morelikely that the spacingbef i = o'ijnj
tween the fractures is controlled by the thicknessof a weak
The shear stressesa• and a•
equation(11)
are easily calculatedfrom
. (au av) ,
----
o'z4,= 0
(14)
Huppert's[1982]equation(2.23) gives
(19)
where r/is the viscosityand
av
•zz>>a--•
The viscosityof the magma is an important parameter
that can also be estimated from the shape of the domes.
(1•
pgQ3
t],l
)-i/sro =•v
where v is the vertical velocity. Since
au
layer near the surface.
(15)
substitution gives
Ou 2gphoa
r
=
Oz
3,1r•o
h
(16)
Equation(19) can be usedto estimater/if t is known.The
viscosityof the magma dependsexponentially on tempera-
ture [WebbandDingwell,1990],andthereforespreadingwill
and hence
effectivelyceasewhen the magma cools. The time scale rc
for such cooling is
Anestimate
of themagnitude
of Ou/Ozandat, canbe
obt•ned by substituting
(20)
Tc•
where•(_• 10-• m2 s-1) is the thermaldiffusivity.The
valuesfor ho in Table 1 give characteristicthermal times of
r ~ ro, h ~ ho
700 to 7000years.Settingt = r• in equation(19) gives
to give
Ou gph2o
ho
-~ •,rlro arz ~ gpho-Oz
ro
(18)
Q•
r/=gph•o
3•r2
• •0
(21)
Sincethe normalstressazz is gpho, azz]½rrz~ )t, where Equation(21) wasusedto obtainthe estimates
of r/in Table
)t = ro/ho is the aspect ratio of the dome. Substitution 1. This rather crude theory is usefulas a guide to the likely
of h0 = 0.5 km and r0 = 13 km givesaz• ~ 11 MPa and
a• ~ 0.4 MPa. The stressfield imposedby the spreading
domeson the plainsmateriMis independent
of the viscosity,
is in the radial direction,and increasesmonotonicallywith
radius.ThoughHuppert's[1982]expressions
givean infinite
order of magnitude of the viscosity.
4. COMPARISON wrrH LABOIL¾1'ORYEXPERIMENTS
It is importantto discover
if the propertiesof the magma
that forms the domes on Venus are consistent with labora-
valuefor arz at the edgeof the dome,they are not valid when
ro-r < ho. Nonetheless,
the real radial stressexertedby the
domenearits marginmust be large. Sucha stressproduces
an extensionalstresswithin the plainsbeneaththe domes,
tory determinationsof magmaproperties.The rheological
experimentsthat are mostsuitablefor this purposeare those
of Webband Dingwell[1990],whomeasuredthe viscosityof
fibersformedfrom silicatemeltswith a varietyof composiand a radial compression
beyondtheir margins,wherethe tions. They were particularly interestedin whether a visfracturesare visible. Failure in extensioncan only occuron cousrheologycoulddescribethe creepbehaviourof the melt.
planesthat are normalto the leastprincipalstress,whereas Their measurementson rhyolite were more extensivethan
failurein shearoccurson planesat approximately45ø to the thoseon othercompositions
and will be usedfor comparison
greatestand least principal stresses.Beyond the dome mar- with viscosity
estimatesfromVenus.Thoughthereis no digins the planes on which extensional failure can occur are
rect evidencethat the dome-forming
magmaon Venusis a
radial and vertical,whereasthoseon whichthrustingmay rhyolite,basalticmagmascrystallizeat temperatureswhere
take place are tangentialto the dome margin, and dip at their viscosityis very much lower than thoselisted in Table
about45ø. Whetherfailurewill occurdependson the resis- 1.
15,974
MCKENZIE
ET AL.: PANCAKELIKE
DOMES ON VENUS
ßooooo
Webband Dingwell[1990]found that the strain rate of
Estimated
temperature
range
ofdome
magmas
rhyolitic magmas varied linearly with stress provided the
strain rate d was less than dc, where
•c ~
10-3
Granite solidus
750
i
i
(22)
where
K•oisthesteady
statebulkmodulus
(• 25GPa)and oO71210
r/is the viscosityas d -• O. The strain rate can be estimated
•
from
equation
(18)
and
the
values
ofpand
gused
previously
•
Ou
0•
c9--•
~3x1 /r/
•_650
(23) •o
whereas
equation(22) gives• ~ 3 x 107/,/. Therefore,
the
calculatedstrainrate is sufficiently
low for the theologyto
be described by a Newtonian viscosity, except close to the
600
0
marginof the domeswhereHuppert's[1982]approximations
are anyway not valid. This conclusion is consistent with
the argument in section 2, which Mso suggestedthat the
relationship between stressand strain rate was linear.
1
Pressure
2
3
GPa
Fig. 5. The temperature range estimated from the viscosity in
Table 1, compared with the solidus for wet granite from Table 2
of Merrill et aL [1970].
Webband Dingwell's[1990]resultscan Msobe usedto
estimate the temperature of the magma from its viscosityif
it is of rhyolitic composition. The viscositiesthey measured
its viscosity. Despite these reservations,the estimate from
rangedfrom about 109to 10TMPa s, and the temperature equation(24) is probablya usefullowerboundon the likely
variation was described by
log•0,/= 2.298x 104/(T+ 273)- 9.15
viscosity and can be used to discover how long the domes
(24)
can retain their shape. Equations(1) and (11) give V, the
horizontal velocity at the outer edge of the dome
where T is the temperature in degrees Celsius. Equation
V= "g•
3r/To
(24) wasusedto calculatethe magmatemperaturesin the
last column of Table 1, which range from 610ø to 700øC.
(2•)
Substitutionof h0 • 1 km, r0 • 10 km givesV • 1 m/m.y.
peratureswouldbe about 40øClower [Webband Dingwell, Such a velocity is sufficiently small to allow the domes to
1990]. It is unexpectedand important that thesetempera- persist for times that are comparable to that of the age of
If the melt composition was that of an andesire,these tem-
ture estimates are similar to the solidus temperature of wet
rhyolite. Figure 5 comparesthe temperature range from
Table 1 with the wet granite soliduslisted by Merrill et al.
the planet. Such a conclusion is also consistent with the
presence of fault offsets that cross the domes. The north-
westernof the two central domeseast of Alpha Regioin Fig-
[1970]and showsthat wet meltingof suchmateriMbetween ure 1 is cut by a fault that crossesthe plains and displaces
pressuresof 0.2 and 2.5 GPa can producemagmasof suitable
temperature to accountfor the Magellan observations.Such
magmaswill losemost of their water when they are erupted
onto the surface of Venus, and their viscosity will then in-
the edge of the dome downwardsand northwestwards. The
fault can only produce a sharp fault scarp if the dome is no
longer spreadingat a significantrate. Though the velocity
of water exsolvedis only about 4%, and detailed calculations
show that adiabatic expansion of the exsolvedgas produces
a temperature changeof lessthan 10øC in the magma. Such
degassingcould account for the small crater that is visible
at the center of each dome in Figure 1. Though Webb and
for very long periods.
estimatedfrom equation(25) is likely to be too large,it is
creaseby a factorof asmuchas 108. However,the fraction nonethelesssufficiently smM1to Mlow fault scarpsto persist
5. DISCUSSION
The simple fluid dynamical model of a viscous fluid
Din#well's[1990]experimentsshowthat magmasof other spreadingover a rigid surface can account for most of the
compositions can also satisfy the viscosity estimates, the
temperatures required are lower than those for a rhyolitic
melt, whereas the wet solidus temperatures of other compositions are either the same as that for granite or are higher.
Only if the melt composition is that of a granite do the
temperatures estimated from the viscositiesagree with the
solidus temperature.
Webband Din#well's[1990]measurements
can alsobe
used to estimate the viscosity of the dome materiM at the
surface temperature of Venus of about 450øC. Substitution
intoequation
(24) gives,/• 4 x 10•'•'Pas. Thisestimate
is
featuresseenon the SAR imagesand altimeter profilesof the
pancakelikedomes on Venus in some detail. These observations provide better constraintson the theological properties
of viscousmagmas undergoingflow on a large scale than do
any yet obtained from the Earth. The viscosity estimates
in Table 1 are a weighted average of the effective viscosity
that governsthe evolution of the domes. In practice, the
viscosityof the magma must vary strongly with depth and
time. Becauseof the successof the simple constant viscosity
model, a study of axisymmetric spreadingwithin a cooling
dome whoseviscosity varies with temperature is well worth
while. Such laboratory experiments have been carried out
uncertain, becauseit requires a much greater extrapolation
of Webb and Dingwell's experimental results than do the by M. V. Stasiuk(personalcommunication,
1992)and show
temperature estimatesin Table 1. The magma is also likely that the change in the profile of the dome produced by a
to crystallize as it cools, an effect that will greatly increase temperature-dependent viscosity is not large. Relative to
MCKENZIE
ET AL.: PANCAKELIKE
the constant viscosity case, temperature dependence causes
the dome to becomeslightly flatter in the middle and steeper
at the edges. Laboratory experimentsof Webband Dingwell
[1990]allow the magma temperatureto be estimatedfrom
the viscosity.If the magmasare rhyolitic in composition,the
temperature estimates are similar to thoseof the wet solidus
at depths of 10 km or more beneath the surface. Though
there are no direct measurementsof the composition of the
dome-forming magma, the observationsare all compatible
with it being that of a rhyolite. The viscosity of magmas
with less SiO• than a rhyolite have too low a viscosity at
their solidus temperatures to produce the observed shapes.
The SAR and altimeter observationsprovide no information about the geological processesthat produced the
dome magmas, and the domes are not obviously related
to any nearby structures. Table 1 shows that the magma
volumesinvolvedare between102 and 103 km3. Comparably sized explosiveeruptions of silicic magma on Earth
occur relatively frequently and were produced by Toba
DOMES ON VENUS
15,975
involved may become clearer when the evolution of Venus is
better
understood.
Acknowledgments. We wish to thank J. Brodholt, J. Fink,
H. Huppert, M. Stasiuk, S. Webb, and C. Wilson for their help,
and the Natural Enviromuent Research Council and the Royal Society for support. The paper was written while D. MCKenzie was
at Scripps Institution of Oceanography, supported by the Cecil and Ida Green Foundation and by the Scripps Institution of
Oceanography. Earth Sciences Contribution 2240.
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15,976
MCKENZIE ET AL.: PANCAKELIKE
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P. G. Ford, F. Liu, and G. H. Pettengill, Center for Space
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Cambridge, MA 02139.
DOMES ON VENUS
D. MCKenzie, Bullard Laboratories, Madingley Road, Cambridge CB30EZ, England.
(ReceivedSeptember30, 1991;
revised June 11, 1992;
acceptedJune 11, 1992.)