GEOPHYSICAL
RESEARCH LETTERS, VOL 21, NO. 12, PAGES 1177-1180, JUNE 15, 1994
A theoreticalmodelof coolingviscousgravity currentswith
temperature-dependent
viscosity
David
Bercovici
Department of Geology & Geophysics,University of Hawaii, Honolulu
lens-shapedsimilarity profilesof the constant viscosity
case. In particular, fluid near the perimeter of the flow
is cooler and more viscousthan at the flow center, and
currents(e.g., lava flowsand mantleplumeheads)are thus can act as a plug causingthe interior of the flow to
composedof coolingfluid with temperature-dependent inflate and/or changeshaperadically. In this paper we
viscosity. An axisymmetric gravity current theory ac- examine a basic non]inear axisymmetric gravity current
counting for these thermo-viscouseffectsis thus pre- theory which accountsfor these thermo-viscouseffects.
sentedand exploredhere. Unlike isoviscousgravity cur-
Abstract. Gravity current theory has applicationsto
any geophysicalphenomenainvolving the spreadingof
fluid on a horizontal interface. Many geologicalgravity
rents[Huppert,1982],coolingvariable-viscosity
currents Theory
do not conserveshape and can undergoa somewhatexThe theoretical model describesa cylindrically axotic evolution. For large viscosity contrasts between
isymmetric,
radially flowing, hot fluid mass cooling
cold and hot fluid, a constant volume, initially hot,
through
two
horizontal
boundaries,at leastoneof which
domed current collapsesrapidly into a fiat plateau with
is
deformable.
The
fluid
viscosityis
a steep edge. Gravity currents ejected at a constant
volume flux from a central conduit also spread with
n0)a flattened plateau shape. Continuously fed currents
that have a large hot initial volume develop an outwhere r/h and r/, are the viscositiesof the hottest and
r/n
+(r/•- r/•)0
wardly propagatinginterior plateau. Regardlessof initial state, continuously-fed,variable-viscositycurrents
growprimarily by thickening;this contrastssignficantly
from isoviscouscurrents which grow almost entirely by
spreading.
coldestfluid, respectively(thus r/h _<r/c) and 0 is the
dimensionless
temperature. Althoughthis rheologyis
simplified,the inversedependenceof viscosityon temperature captures the essential behavior of viscousflu-
ids; in particular, viscosityvariationsare largestwhere
the mean temperatureis coldest.For simplicity,we impose the same isothermal condition of 0 -
Introduction
(1)
0 at both
upper and lowerboundaries(heightsz- 0 and z- H,
where H is the current'sthickness);elsewherein the
Viscousgravity currents- i.e., fluid massesspreading
fluid 0 < 0 _<1. Thus, to first order
under their own weight- are widely studiedphenomena
z
z
with applicationsto a large number of geologicalprob-
0- 6©•(1-•)
lems,from lava flowson Earth [e.g., Crisp and Baloga,
1990;Fink and Griffiths,1992]andVenus[McKenzieet
© - • f• Odz.Higher
order
contributions
to
al., 1992],to buoyantdiapirsor plume-heads
spreading where
the temperaturemay occur,especiallynear the centerof
beneatha rigid surface[Olson,1990]. Symmetrically
the current,but (2) mustalwaysrepresent
the dominant
spreadingisoviscousgravity currents are well described
by analyticalsimilaritytheory[Huppert,1982];i.e., they
contribution.
Radial motion of the current is described
conservetheir basic shape as they spread. For many by the axisymmetricStokesequationfor shallow-layer
approxgeologicalgravity currents, however, the fluid is often flow with the hydrostaticand long-wavelength
imations
(and
thus
vertical
variations
in
viscous
stress
coolingfrom an initially hot state, or being fed by a
provide
the
dominant
resistive
force):
sourceof hot fluid. Examples of this are lava flows and
mantle plume heads. Moreover, many geologicalflu(3)
ids have temperature-dependentviscositieswherein the
fluid becomesweaker as its temperature increases.The
combinedeffectsof coolingand temperature-dependentwhereg is gravity,vr is radial velocityand Ap is the abviscositylead to significantdeviationsfrom the basic solutedensitycontrastacrossthe deformingboundary.
BecauseAp is treatedas a constant,the gravitycurrent
is assumedto havean intrinsicdensitycontrastwith its
surroundings
whichis muchlargertha.nthermal density
Copyright1994by the AmericanGeophysical
Union.
variations. The vertically averagedvr is
o(rloz]h
0----APg'•-r
+•z
Paper number94GL01124
1•Hv,.dz-- ApgH
•(r•+ •'0)OH(4)
U- •
0094-8534/94/94GL-01124503.00
1177
1178
D. BERCOVICI:
The dimensionless
constant
COOLING
VISCOUS
C is either 3 if one horizon-
tal boundaryis free-slipand the other no-slip(as appropriatefor surfacegravity currents,e.g., lava flows),
or 12 if both boundariesareno-slip(appropriatefor low
viscositydiapirsin a high-viscosity
matrix). Similarly,
GRAVITY
CURRENTS
which permits nearly isothermal initial conditions in
the current but a smooth temperature transition at its
edge. As predicted by similarity theory, the isoviscous
(y = 0) currentmaintainsthe shapeprescribed
by (9) as
it spreads(Figure 1). In contrast,the variable-viscosity
r/'/(r/c- r/n)is • if oneboundary
isfree-slip
theother (• - 1000)gravity currentdoesnot conserveshape,bea if both boundariesare no-slip. Conserra- comingvery fiat and steep-sided(bottom frame, Figure
no-slip,or õ
1). The steepeningof the flow front allowsthe lowtion of mass requires
viscosity material in the center of the gravity current
on
to push out the cold viscousmaterial near the edge of
at + r
=o
(5)
0©
Since most of this current
has low viscos-
ity, it collapsesvery rapidly, spreading approximately
and temperature transport is describedby
O©
the current.
100 times faster than in the • -
0 case. Because of the
rapid spreading, the nearly isothermal core is retained
12•
at + U0-7= - H
(6)
•vheren is thermaldiffusivity.We substitute(4) into(5)
and (6); afternondimensionalizing
H by Ho (a reference
thickness)
r byß/•xpg/•
• t by•12• and
V12Cnr•, we obtain
at = r Or
•
O©
OHO©
at = ( + )H_•
n
(7)
(8)
and stretched
Continuously
outward
with
the current.
Fed Currents
The thicknessof a gravity current fed by a constant
volume flux at its center is singular at r - 0. We therefore prescribe an inner vertical boundary at r- ri • 0
(whichrepresentsthe outerboundaryof the supplyconduit). The dimensional
volruneflux throughthisboundary is a constant Q, and thus
r(1+ uO)H
3O.H
Or= -1 at r- ri
(11)
where u - r/'/r/n is the only free parameter[seealso
Bercovici,1992]. Theseequationsare solvednumerically via basic finite differenceswith appropriate Courant
conditions on the time step.
Numerical
Experiments
In this sectionwe examinegravity currentswith 1)
constantvolumeand 2) continuousmasssupplyfrom a
central conduit. The first caseis associatedwith spread-
ing isolateddiapirs,the secondto continuouslyfed lava
flows and plume heads. In both caseswe considertwo
differentviscositycontrasts,y - 0 (constantviscosity)
and y- 1000(representative
of silicatematerials).
Constant
Volume
Currents
For constant volume gravity currents, we apply the
boundaryconditionthat 0___H
_ 0o
_
0 at rOr
Or
--
0; the
boundary condition at the outer radius is arbitrary since
we do not allow the gravity current to spreadthat far.
Our initial
condition
for H is
1. Perspectiveviewsof dimensionless
thickness
H(r,
0)-{(1- r'lr•v
)•/•for
r_•rN (9)HFigure
for constant volume gravity currents with different
0
for r > rN
viscosity contrasts y. Profiles of temperature © across
conditionhas the shapeof the similaritysolution[Hup- the center of the current are shown on the right edge
pert, 1982]for an isoviscous
current. In this case,the of each frame; the top profile has a maximum value of
i and subsequent profiles use the same vertical scale.
thicknessscale Ho is simply the starting thicknessof
Temperature for the y - 0 case is scaled up by a factor
the gravity current. The initial conditionfor t3 usesa of 100 and hence shown as a dashed curve. The top
super-Gaussianshape
framedisplaysthe initial conditions(from (9) and (10)
where rN is the initial radius of the current's edge;this
O(r,0)- e-(•/•)•ø
(10)
with rs - 2.5). Dimensionless
time t, maximumH
(rear left cornerscale)and • are indicated.
D. BERCOVICI:
COOLING
VISCOUS
GRAVITY
CURRENTS
1179
past this radius. As the edge of the current movesoutside the hot low viscosity zone it becomesmore rounded.
2•aps . Wealsorequire
that© - 1 at r - ri, i.e., In the casewith a large hot starting volume, the inwe assumefluid is injected at the maximum tempera- jected volume flux is at first accomodated by the iniwhere we have defined the reference thickness as Ho =
(C'rl,:Q)
1/4
ture.
The initial
H(r,O)-
condition
on H is
r
{(T•uløg(rNI
0
tial spreadingof the current(Figure2). However,most
of the current's heat diffusesaway fairly rapidly, leaving a small hot core at its center. This has no effect
forri _•r _•rN
on the
for r > rN
(12)
which correspondsto a gravity current at temperature
© = I flowing with constantvolume flux (i.e., with
(11) appliedto the wholefluid). The initial condition
on © is prescribedby (10). We considertwo initial
conditions:1) wherethereis essentially
no pre-existing
gravitycurrent(corresponding
to directejectionof fluid
y -
0 current
which
is therefore
not shown.
For •/ - 1000, a narrow fiat plateau begins to inflate
within the hot low-viscosity region because a larger
hydrostatic head is required to push out the newly
cooled high-viscosityfluid. The plateau slowly propagates out toward the edge of the gravity current, and
as its front passesinto the cold, high-viscosityregion
it becomes rounder, similar in shape to the isoviscous
againsta surface);and 2) wherethere is a largehot current. During this time, the plateau quadruples in
initial current(associatedwith a startingplumehead thickness. Moreover, the edgeof the gravity current reimpingingon a surfacewhilebeingtrailedby a conduit). mains completelystatic until the plateau front reaches
it. Although the choiceof initial volume size is somesentiallymaintainsthe shapeprescribedby (12) as it what arbitrary, these results are fairly robust. A thicker
propagates(Figure 2). (Becauseof the inner vertical hot initial gravity current is, at first, too thick to be
boundary, however, this case is not identical to Hup- supported by the volume flux and collapsesrapidly to
pert's[1982]sinfilaritysolutionfor the isoviscous
gravity a thinner current (similar to the constantvolume curThe current
with
no initial
volume and • =
0 es-
current.) During the simulation,the currentincreases rent, Figure 1) after whichits heat diffusesaway and
the interior plateau forms as in the above case. If the
In contrast,the gravity current with • - 1000 prop- initial current is thinner the formation of the plateau is
agatesoutwardas a fiat, steep-sided
plateau(Figure simply accelerated.
in thickness less than 80% and most of this before t = 1.
2), clearlynot conserving
the shapeprescribed
by (12). Small-Slope Caveat
This current increasesin thicknessby over 400• and
The gravity currents presented here develop steep
propagatesfaster than the isoviscous
current. The high
temperature regionof the current, however,doesnot ex- edgesor plateau fronts, especiallyfor large •. As noted
tend beyond r • 2 even as the current's edgepropagates by Hupper![1982],gravity currenttheoriesare derived
v=O
v= 10 3
c;1t=O .• '•":
•'h%•::.•......::
..... ' t=O...:v2:•"'":'""•
...........
:•'":
':"':':'"':"
' '"':"':>'"'
'•
..:.
.......
ßii.._.•111:
...•'
•:•:;:•
'%'
..........
•.:;..__-;•:.:•:•'"::"
'•]:;:i;
''......•.•i:!•._.•i••:•
...::...:''""'
':'?'"':":'::'•:"...%:.::•
:"':•'--'f••i•
-•ili
" .........
•;'•;:}:.::.'_-:.:::
.!i :.:
• ,.½
•'";
'•
':•:ii,;.;;•:.,
:,•_•:.•:•::..::,.:
•.........................
..•:.
................
•-'"'::'
::"
............
•.•:•,?......:....::
:.:.. .............
Figure 2. Perspective
viewsof the thickness
H (with profilesof O on the right edge)of continuously-fed
gravity
currents for different times t and viscosity contrasts y. Left and middle columnsshow currents with essentially
no starting volume, i.e. ri -- i and rs - 1.1. The right columnshowsa current with a large, hot initial volume,
i.e., rs = 8.8. Initial conditionsare specifiedby (12) and (10). MaximumE) is I for all frames. Starting and
ending maximum H are indicated by the rear left corner scales.
1180
D. BERCOVICI:
COOLING
VISCOUS
with the small-slope assumption; i.e., motion within the
current is modelledwith channelflow. This assumption
is far from correct at the edge of the gravity current.
Historically,however,thesetheorieshave provensupris.ingly successfi•lin predicting the steep fronts in laboratory generatedgravity currents[e.g., Huppert, 1982;
Didden and Maxworthy, 1982], even for variable rhe-
ology [e.g., œiu and Mei, 1989]. Thus the channelflow model appearsto captureenoughof the important
physicsto at least partially describethe flow front.
GRAVITY
CURRENTS
geststhat gravity currentswith temperature-dependent
viscositygrow by thickening,then it also impliesthat
the availableplume-headuplift is significantlygreater
than that of a constant-viscosityplume head and may
provideenoughbuoyantstressto generatefeaturessuch
as swells and coronae.
Acknowledgments.
The author is gratetiff to Stuart
Weinstein, Stephen Self, Peter Olson and John A. White-
head for useful discussionsand/or reviews. This work is
supported by NSF grant EAR-9303402.
Discussion and Applications
In this paper we have briefly examined a theoretical model for coolinggravity currentswith variable vis-
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primarily by thickening, as opposedto isoviscouscurthermal components, J. Geophys. Res., 95, 1255-1270,
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1990
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[e.g., Olson,1990]. Plumesin the Earth, however,are
primarily thermally buoyant, whereas the theory here is
more applicable to plume heads with a predominantly
chemical density heterogeneity. Moreover, plate motions on Earth tend to draw plume heads into ellip-
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and eruption conditionsfrom Magellan data, J. Geophys.
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sibly result from mantle plumes, also maintain plateau Schoolof Ocean & Earth Science& Technology,University
shapes[Squyreset al., 1992]similarto thosesuggested of Hawaii, 2525 Correa Road, Honolulu, HI 96822
by our gravity currenttheory [seealsoBercoviciand
February16, 1994;accepted
March25, 1994.)
Lin 1993]. Finally,sincethe theoryof this papersug- (received