Jun 15, 1994 - beneath a rigid surface [Olson, 1990]. Symmetrically spreading ..... Gravity current theory also has applications to man- tle plume heads ...
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 _ 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-
References
Bercovici, D., Wave dynamics in mantle plume heads and hotspot swells, Geophys. Res. Lett., 19, 1791-1794, 1992. ume gravity current, an increase in viscosity contrast Bercovici, D. and J. Lin, Mantle plume-head waves: A posv enhancessteepening of the current's edge and flatsible mechanismfor the formation of the outer-ridgesof tening of its center. Continuouslyfed gravity currents Venus coronae, Eos Trans. AGU, 7d, 189, 1993. with large v also spread as fiat plateaux, or form in- Cas, R.A.F. and J.V. Wright, l•2>lcanicSuccessions:Modern and A,cient, Allen & Unwin Publishers, London, Ch. 4, terior plateaux which propagate to the currents' edge.
cosity. The model suggeststhat for a copsrantvol-
1987.
Furthermore, these gravity currents grow in volume Crisp, J. and S. Baloga, A model for lava flow with two primarily by thickening, as opposedto isoviscouscurthermal components, J. Geophys. Res., 95, 1255-1270,
rentswhichgrow•dmostentirelyby spreading[ttuppert, 1990 1982]. Didden, N. and T. Maxworthy, The viscousspreading of plane and axisymmetric gravity currents, J. Fluid Mech., The theory presented here offers a simple model for 121, 27-42, 1982. applicationsto flow of silicatesin geolicalsettings.The Fink, J.H. and R.W. Griffiths, A laboratory analogstudy of fiat plateau shape of variable-viscositycurrentsis sugthe surface morphology of lava flows extruded from point gestive of lava depositswhich lack a dome shape, e.g., and line sources, J. Volcanol. Geotherm. Res, 5J, 19-32, mesa lavas [Cas and Wright, 1987], tortas [S. Self, 1992. pets. comm., 1994] and possiblyeven lava domeson ttuppert, H.E., The propagation of two-dimensional and axVenus[Pavri et al., 1992;McKenzie el,al., 1992]. The isymmetric viscousgravity currents over a rigid horizontal surface, J. Fluid. Mech., 121, 43-58, 1982 growth and propagat.ion of interior plateaux within continuously fed gravity currents is indicative of overflow Lin, K.F. and C.C. Mei, Slow spreadingof a sheetof Bingham fluid on an inclined plane, J. Fluid Mech., 207, 505behavior in lava flows, though in that instance the 529, 1989.
plateaux wouldessentiallyundergowave-breakingwhich McKenzie, D., P.G. Ford, F. Liu, and G.H. Pettengill, Pancannot be predicted by this theory. It, should be noted cakelike domes on Venus, J. Geophys. Res., 97, 15,967that trnly accurate models of lava flows probably re15,976, 1992.
quire incorporationof freezingcrust [e.g, Crisp and Olson, P., Hot spots, swells and mantle plumes, in Magma Transport and Storage, M.P. Ryan, ed., John Wiley & Baloga,1990].
Sons, 33-51, 1990. Gravity current theory also has applicationsto manPavri, B., J.W. Head, K.B. Klose, and L. Wilson, Steeptle plume heads spreadingbeneath oceaniclithosphere
[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-
sided domes on Venus: Characteristics,geologicsetting, and eruption conditionsfrom Magellan data, J. Geophys. Res., 97, 13,445-13,478, 1992.
Squyres, S.W., D.M. Janes, G. Baer, D.L. Bindschadle•,G. Schubert,V.L. Scharpton,and E.R. Stofan, The morphology and evolution of coronaeon Venus, J. Geophys.Res.,
tical structures[Olson, 1990]. Nevertheless,Wessel's 97, 13,611-13,634, 1992. \¾essel, P., Observational constraints on models of the [1993] study of the Hawaiian swell suggestsa signficanfly flattened plateau-like shape for a cross-section
of the swellperpendicularto the islandchain(after volcanic construction
Hawaiian hot spot s•vell, J. Geophys. Res., 98, 16,09516,104, 1993.
and associated flexure are accounted
for); this is reminiscentof the gravity currentshapes implied in this paper. Coronae on Venus, which posD. Bercovici, Department.of Geology & Geophysics, 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
