Mantle Convection and the Thermal Structure of the Plates

Home , Mantle convection, Natural convection

VOL. 83, NO. B9 JOURNAL OF GEOPHYSICAL RESEARCH SEPTEMBER 10, 1978

BARRY PARSONS

Departmentof Earth and PlanetarySciences, Massachusetts Institute of Technology Cambridge,Massachusetts 02139

DAN MCKENZIE

Departmentof Geodesyand Geophysics,Madingley Rise, MadingleyRoad, CambridgeCB3 0EZ, England

Though a simplethermal model of plate creationmatches the observedbathymetry and heatflow with considerableaccuracy, it only doesso becausea constanttemperature is imposedas a lower boundary condition.A more realisticmodel is developedhere, basedon the idea that the thermal structureof the plate becomesunstable and leadsto the developmentof small-scaleconvection. Convective heat transport then suppliesthe heat flux neededto match the observationsrather than an artificial constanttemperature boundarycondition. The temperaturedependence of the rheologyis representedin a simplemanner. Below a given temperaturethe material is assumedto move rigidly, defining an upper mechanical boundarylayer. Beneaththis rigid layer, wherethe temperaturesare greater,the material is assumedto have a constantNewtonian viscosity.The part of the viscousregion where there are significantvertical temperaturegradients, immediately below the mechanicalboundary layer, forms a thermal boundary layer. As the plate cools,both the mechanicaland thermal boundarylayers increase in thickness.A local critical Rayleighnumber criterion is usedto testthe stabilityof the thermal boundarylayer. On this basis a convectiveinstability is predicted,its occurrencecoinciding with the breakdownof the linear dependenceof the depthof the oceanfloor on the squareroot of age.Though the small-scaleconvection which developsfrom this instability modifies the thermal structure, the basic observationalconstraints are shown to be satisfied.The stability criterion is further testedin two differentlaboratory experiments. These experimentsalso illustrate a possibleform for the instability, with cold densematerial breaking away from the baseof the plate and being replacedby hotter material from below.

1. INTRODUCTION plied to the plate from beneath and heat lost by conduction through it. Small-scaleconvection in the upper mantle has The effectsof the cooling of the oceanicplate as it moves been suggestedas a meansof maintaining the necessaryverti- away from a midoceanridge have receiveda good deal of cal heat transport [Richter, 1973; Richter and Parsons, 1975; attention, both theoretical and observational.Forsyth [1977] McKenzie and Weiss, 1975]. In part this suggestionstemmed recentlyreviewed the different measurementsthat have been from the difficulty encounteredin numerical experimentsin made. Variations in mean heat flow, depth, and velocity of preventingconvection in boxes whose horizontal extent was surfacewaves with age all seemto reflectchange• in thermal severaltimes that of the vertical dimensionfrom breaking up structure. Parsonsand $clater [1977] have shown that varia- into many smallercells. In a model of the thermal evolutionof tionsin meandepth and'heat flow can be adequatelydescribed the plates which includesthe possibilityof small-scalecon- by a model of the type originally discussedby McKenzie vection, it is necessaryto distinguishbetween two types of [1967]. In this model a slab of constantthickness moves away boundary layers. The mechanicalboundary layer movesas a from the ridgecrest, its bottom boundarybeing maintained at rigid spherical cap and carries the magnetic anomalies. Be- constant temperature.A thicknessof 125 km for the slab is neath the mechanicalboundary layer is a thermal boundary given by a simple inversionmethod based on the analytical layer which is also cooled by heat lost upward but within propertiesof the model [Parsonsand Sclater, 1977]. which motionsnot simply relatedto the motion of the surface The phaseand group velocitiesof Rayleigh wavesfor peri- can occur. Neither boundarylayer has sharp boundaries,and ods from 20 to 200 s are found to increasewith increasingage hencethis modelis more physicallyrealistic than that usedby of the ocean floor [Kauselet al., 1974; Weidner, 1974; Leeds, McKenzie [1967]. 1975;Forsyth, 1975, 1977].These data can be invertedto give An observationthat initially seemedto support the notion the seismicvelocity structurein different age zones [Leedset of small-scaleconvection as the principal means of vertical al., 1974; Leeds, 1975; Forsyth, 1975, 1977]. Although the heat transportin the upper mantlewas the apparentapproach resolutionis limited, the overall effectof cooling in increasing of the heat flow measurementsto a constantbackground value the seismicvelocities is clear. The top of the low-velocityzone in old ocean basins.Although on closerexamination the heat is reasonablywell defined. Despite uncertaintiesabout the flow cannot in fact discriminatebetween an approachto a exactshape of the solidus[Millhollen et al., 1974;Green and constantlevel and continuedcooling, an equivalentargument Liebermann,1976], relating the sharp decrease in seismic veloci- can still be madeon the basisof the variationof depthwith age tiesto the onsetof partial meltingprovides a further constraint [Parsonsand Sclater, 1977].The meandepth of the oceanfloor on the thermal structure.Forsyth [1977] concludedthat the behavesin a very simpleway as a function of age (Figure 1). available data were consistentWith the model of a cooling Out to an age of about 70 m.y. the depth increaseslinearly as plate, 125 km thick, in a peridotiticmantle. t •/:, where t is the age of the oceanfloor. For larger agesthan Sucha model requiresa supplyof heat, the heat flux at the this the linear relationshipbreaks down, and the depth in- baseof the plate increasingfurther away from the ridge crest. creasesless rapidly, followinga simpleexponential decay to a A balance is reachedin the older regionsbetween heat sup- constantvalue. For agesyounger than 70 m.y. the form of the Copyright (D 1978 by the American GeophysicalUnion. variation can be accountedfor by a variety of thermal models. Paper number 8B0463. 4485 0148-0227/78/098 B-0463501.00 4486 PARSONSAND MCKENZIE: CONVECTION BENEATH THE OCEANIC PLATES

2000I wherethe velocityu is taken to be constant.Other termsare L NORTHPACl fib definedin Figure2. The changein depthacross the columnis !• • Meandepth and standard d.... t,on given by •- • -- Theoreticalelevahon, platemodel 3000 -' Lineart•/z relahon d(x+ dx) - d(x) = apo

_ (po - pw) ßfo [T(x + dx, z) - T(x, z)] dz (2) where a is the volumetricexpansion coefficient, po the refer- 4000- encemantle density, and pwthe densityof seawaterand both (1) and (2) are correctto firstorder in (aT). The differencein internalenergy between material leaving and enteringthe col- 5000 T umn is thereforeproportional to the changein depth.Com- 6000 bining (1) and (2) resultsin the relation -(po-pw)C•, Od + q,= qo+ H(z)dz (3) 5 6 8 13 fo 7ooo I I 0 I 2 :5 4 5 6 7 8 9 I0 II 12 1:5 wheret = x/u, the ageof the oceanfloorß As d(t) andq•(t) are •/•GE (M.Y. B. P.) presumedknown, then the right-hand side of (3)is alsoknown Fig. 1. Plot of meandepth in the North Pacificversus the square as a functionof t. If q0 and H(z) can be neglected,then root of age. Numbers at the bottom of the figure denote selected Cenozoicand Mesozoicmagnetic anomalies [from Parsonsand Scla- equatingthe left-handside of (3) to zeroprovides a relation let, 1977]. betweenvariations of depthand surface heat flux. Lister [ 1977]

derivedsuch a relationshipfrom the t•/•' and t -•/•' dependences_ of the two observableswhich hold for small enought. Thermal modelsof a platethat predictthe observedfunctions d(t) and One is the coolingplate model described above. Another sim- q•(t) haveidentical total heat sources given by theright-hand ply allowsthe materialto loseheat at the upperboundary sideof (3). They differonly in theway that the heatsources are without restrictingthe depth to which cooling can occur partitionedbetween q0 and H(z) and,more important, in the [Davisand Lister, 1974].A third modelallows the bottom heat transportmechanism used to supplyqo. Forsyth [1978] boundaryto migrateand assumes that it isat themelting point suggestedthat the requiredheating could be providedby ra- of the material[Parker and Oldenburg, 1973]. For sufficiently dioactive sourcesdistributed in the upper 300 km of the smallages the behavior in all thesemodels is principally deter- mantle, the vertical heat transport mechanism being con- minedby coolingfrom similar initial temperature profiles that ductive.Schubert et al. [1976] usedshear stress heating pro- are uniformwith depthright up to or almostup to the upper ducedin the large-scaleflow associatedwith the platemotion. boundary.However, for greaterages the coolinghas pene- The verticalheat transport mechanism was still predominantly trateddeep enough for theinfluence of somebottom boundary conductive.Each of the above mechanismsproduces similar conditionto be felt. Heat is suppliedso that the depthto any thermal structuresand hencecan account reasonablywell for givenisotherm increases less rapidly, and accordingly,the the observations.The problemof the stabilityof a large-scale depthof theocean floor increases less rapidly. The alternatives flow with sucha thermal structurein the upperboundary layer to the plate modelcan be modifiedto allow for an input of hasnot beenconsidered. This providesa meansof discrimina- heat from below[Crough, 1975; OMenburg, 1975], but thento ting betweenthe differentmodels, since some form of small- a greateror lesserdegree they approximate that model. scaleconvection will clearlydevelop in thosecases that can be Alternative mechanisms,other than small-scaleconvection, shown to be unstable. havebeen suggested in orderto supplythe requiredheating The schemethat we attemptto justify in this paper is illus- and reproducethe observed variation of depthand heat flow tratedin Figure3. Materialmoves away from the ridgecrest, with age [Forsyth,1977; Schubert et al., 1976].Any thermal losingheat conductively with the upper boundary fixed at 0øC. modelof a platethat satisfiesthe observedvariation of surface There is no small-scaleconvection at this point, becausethe heatflow and depthwith ageis constrainedto havethe same overturningof the large-scalecirculation precludes the exis- total heatsources provided by a combinationof internalheat tenceof the cold upperconductive boundary layers of such generationand heating from below. This canbe seenby con- convectioncells. Initially, the onlyheat brought from belowis sideringthe energybalance of a columnthrough the plate, that transportedin the large-scaleflow, and coolingoccurs heighta, and width dx, observedin a stationaryreference exactlyas for a half space.The mechanicalproperties of the frame(Figure 2). We assumethat lateraltemperature varia- materialwill vary rapidlywith temperature.We representthis tionsin the plate are confinedabove the baseof the column. temperaturedependence by assumingthat below a certain Horizontal heat conductionis neglected,this being a good temperaturethe materialbehaves rigidly. Above that temper- assumptionaway from the ridgeaxis. The energybalance is ature the material is treatedas having a uniform Newtonian then viscosity.This is undoubtedlyan oversimplificationof com- plex rheologicalbehavior, but sucha two-layermodel has proveduseful in discussingstress balances in the large-scale poCou [T(x + dx, z)- T(x, z)] dz + (qu- qo)dx flow [Richter,1977; Richter and McKenzie,1978]. Also the calculationsof Schubertet al. [1976]suggest that sucha repre- = dx H(z) dz (1) sentationof therheology should be adequate for thepurposes of this paper. PARSONSAND MCKENZIE:CONVECTION BENEATH THE OCEANIC PLATES 4487

Tqu d(x)" ._--- d(x+dx) z=a

,•oatooC puT(x,z)dz •oa/ooC puT(x+dx ,z )dz

z=O qb

Fig. 2. Sketchto illustratethe energy balance for a columnof materialin theplate. The relation of theenergy balance to the changein depthd(x) acrossthe column,and to the upperand bottomheat fluxes,q• and qo,is discussedin the text. Othervariables are asfollows: T(x, z) is the temperature,p0 the density,C, the specificheat, u the platevelocity, and H(z) the radioactive heat content of the column.

As coolingproceeds, the spacingof the isothermsincreases. convecting region and supplied below it from the lower The mechanicalboundary layer lies betweenthe surfaceand mantle. The upper thermal boundary layer of the small-scale the isothermchosen to representthe transition between rigid convectionoccupies the region immediately below the me- and viscous behavior. This isotherm and another chosen iso- chanicalboundary layer, hence the namefor thisregion given thermare usedto providea measureof thetemperature gradi- above. ent withinthe viscousregion immediately below the mechani- The small-scaleconvection maintains a mean temperature cal boundarylayer. We call the region betweenthese two structure in the upper mantle that is essentiallyadiabatic, isothermsthe thermalboundary layer. The thicknessesof the apart from its upperboundary layer. As the platesmove apart, two boundarylayers will increasewith age as shown in Figure material can ascendadiabatically with little distortion of the 3. If at an ageof 70 m.y.the thickness of thethermal boundary isothermsin the uppermantle. This is distinctlydifferent from layerhas increased sufficiently that it becomesunstable on the a convectivemodel in whichthe platesform the upperbound- basisof a local Rayleighnumber criterion for the boundary ary layer of a singleconvection cell [Turcotteand Oxburgh, layer,then cold materialwill breakaway from the bottomof 1967]. In that casethere is a large temperaturecontrast be- the lithosphere, and mantle material at the assumed basal tween the ascendingmaterial and the adiabaticcore of the cell, temperaturewill replaceit. As coolingproceeds further, the which providesa major componentin driving the cell. small-scaleconvection increases in amplitudeuntil a balanceis Oncethe instabilityoccurs, the overturningproduced by the reachedbetween the heat conductedthrough the plate and onset of small-scaleconvection effectively keeps the mean heattransported vertically by the small-scaleconvection. This, temperatureconstant at the depthto whichcooling had pene- in turn, dependson the heat sourcesdistributed within the trated at that time. This depththen effectivelydefines the plate

0 50 I00 150 -- Age,m.y.B.P. i i i i

•:.:•:...:•:•....::•...•::•••.•:•.•....•..:••..:•:••:•••:..:•.•.•======.... ::::::::::::::::::::::::::::::::::::::::::::• •i• • • •:• Thermalbarbarylayer Onset of the instability [•/iscousI Thermalstructure ofplate and small-scale convectionapproach equilibrium

Fig. 3. Schematicdiagram showing the divisionof the plate into rigid and viscousregions and the occurrenceof an instability in the bottom viscouspart of the plate. 4488 PARSONSAND MCKENZIE: CONVECTIONBENEATH THE OCEANICPLATES thicknessdetermined from fits to the variationsof depth and K = 8 X 10-7 m' s-1; heat flow versusage and is the sum of the thicknessesof the Cj, = 1.17 X 10a J kg-1 øC-1; mechanicaland t.hermal boundary layers.Both are included a = 3 X 10-5 øC-1; becausesignificant mean temperaturedifferences relative to Rac = 10•. the temperatureprofile underthe ridgecrest occur above this depth, and the variation of depth versusage is determined The value of 4.25 eV for the activationenergy is at the lower principallyby suchtemperature differences. end of the range of experimentallydetermined values. Post The basic argument that we use below dependson the [1977] obtained4.06 eV for the high-temperaturecreep of Mt. existenceof a local critical Rayleigh number criterion to pre- Burnett dunite in a hydrous environment. Higher values dict instabilityin the thermalboundary layer. Although such a around 5.4 eV are found for dry olivine [Kohlstedtand Goetze, criterion is intuitively reasonable,it cannot be rigorouslyjusti- 1974].There is a good deal of uncertaintyassociated with the fied. Therefore after the use of this criterion in section 2 in preexponentialcoefficient in (5), in particular for the grain connectionwith the hypothesisdescribed above, its appli- size. In the upper 100 km the pressureeffect is small,and the cation in a variety of other physicalsituations is also consid- temperaturevariation alone is plotted in Figure 4. Decreasing ered. In all cases,agreement is obtainedwith either numerical the temperaturefrom 1300ø to 900øCincreases the viscosity by or experimentalresults. This providesconfidence that the pre- slightlymore than 4 ordersof magnitude.We havepicked the dictions based on a local critical Rayleigh number criterion isotherm will be at least qualitatively correct. T' = 0.75T1 (6) 2. THE INITIATION OF SMALL-SCALE CONVECTION Figure 3 illustratesthe schemethat is to be tested.Material or T' = 975øC in this case,to representthe transition from is assumedto rise adiabaticallyat the ridge crest.Estimates of quasi-rigidto viscousbehavior. As material movesaway from the temperaturebeneath spreading centers have beenmade in the ridge crestand coolingproceeds, the temperatureprofile is a number of different ways and give reasonableagreement. In initially given by order to form olivine tholeiitesby partial melting of a model pyrolite mantle, temperaturesbetween 1300 ø and 1400øCfor T = T1 err {z/[2(gt)l/']} (7) the depth range 30-60 km are required [Wyllie, 1971; Green, 1972]. Application of plagioclasegeothermometry to ocean where z is the depth, t the time, and • the thermal diffusivity. ridge basaltsgives temperatures ranging from 1200ø to 1400øC From (6) and (7) the thicknessof the mechanicalboundary [Scheidegger,1973; Frey et al., 1974]. Temperaturesaround layer thereforegrows as 1350øC were obtained independentlyof the other parameters A = 1.63(gt)1/' (8) of the plate model by fits to the empirical depth-agecurve [Parsonsand Sclater, 1977]. For the calculations below, a In order to provide a measureof the temperaturegradient in uniform temperature the viscousregion we follow another isothermchosen to be T1 = 1300øC (4) T" = 0.97T1 (9) is assumed,the small adiabatic gradient being neglected. The decisionto subdividethe coolingplate into two regions, one quasi-rigidand the other uniformly viscous,is basedon 3O the very rapid variation of viscositywith temperature.A number of different mechanismsfor creep in solidsexist [Stocker and Ashby, 1973]. Dislocation creep and diffusion creep, in particular the former, are appropriateto the range of stresses and temperaturesin the uppermantle. These mechanisms have an exponentialdependence on temperaturearising from the T T' form of the diffusioncoefficient, both processesdepending on the diffusion of oxygen ions within the lattice. As an example, E • •o the viscosityfor Nabarro-Herring creep is given by

v = 10moDokbTR'exp (E+pV) kbT (5) 15 where v is the kinematic viscosity,k, Boltzmann'sconstant, R the mean grain radius, Do the referenceconstant in the diffusion coefficient,mo the massof an oxygenion, E an activation energy, V an activation volume, T the absolutetemperature, •o[i i , i I I i I I andp the pressure.The followingtabulation gives the valuesof 500 I000 1500 physicalparameters that have beenused in this paper: T,ø½

0o = 3.33 Mg m-3; Fig. 4. Plot of viscosityversus temperature for the creeplaw given g = l0 m s-1' by (5) and physicalparameters listed in the tabulationin the text. Any R = 0.5 mm; other creeplaw will displaythe samestrong temperature dependence. Do= 5X 10-4m's-•; The arrow labeledT' denotesthe temperaturechosen to representthe divisionbetween rigid and viscousbehavior. T• is the mantletemper- E = 4.25 eV; ature below the plate. The dashedline showsthe value of viscosity v0 = 2 X 1017m' s-1; requiredfor the instability to occur after 70 m.y. PARSONS AND MCKENZm: CONVECTION BENEATH THE OCEANIC PLATES 4489

•,(o½) Rac the critical Rayleighnumber. The valuesthat we haveused are given in the precedingtabulation. Substitutingsuitable o0 $oo 6oo valuesfor the physicalparameters in (12) wouldgive the age, to, at which an instability would occur. Becauseof uncer- taintiesin the valuesfor the viscosity,rather than do this, we assertthat t• = 70 m.y. as suggestedby the variation of depth with age and ask what value of v is neededfor (12) to hold. Depth (kin) Using the valuesin the tabulation,a valuefor the viscosityof 2.33 X 10•e ma s-• (13) is obtained.This is not unreasonable,considering the values

700 .... plotted in Figure 4 and that (13) representsan averageviscos- ity acrossthe boundarylayer. values for the meanmantle viscosity of 2 X l0 •7 m• s-• are obtained from studies of postglacialuplift [Cathies, 1971; Peltlet and Andrews,1976]. T ¾(øC) Richter and McKenzie [1978] calculatedvalues ranging from o $oo 600 2 X l0 TMm • s-• to 2 X l0 •7 m• s-• by consideringthe balance o betweenviscous stresses and the buoyancyforces driving the motion of subduetedplates. In their modelsthey alsorequired a low-viscositylayer to decouplethe plate motion from the mantle beneath;values in this layer as low as 10" m• s-• were Depth suggested.Hence we seethat the value given here falls within B (km) the range of probable values obtained by various different methods.

3. THE EFFECT OF SMALL-SCALE CONVECTION

700 ON THE THERMAL STRUCTURE Fig. 5. Isothermsand horizontallyaveraged temperature structure The model used above to determine where the thermal for steadystate convectionin squareboxes for (a) 30 mW m-2 from boundary layer becomesunstable uses the thermal structure belowand 40.7 nW m-8 from within and(b) 83.6 nW m -8 from within. Both have the same mean heat flux at the surface and are calculated given when conduction is the only means of vertical heat for a 24 X 24 mesh.Isotherms are contouredat intervalsof 60øC [after transport.The developmentof the boundarylayer instability McKenzie et al., 1974]. leads to convectiveheat transport, and therefore the mean temperaturestructure can only be obtained by solving non- linear equations.We shall not attempt this here. Instead we The isothermsT' and T" define a thermal boundary layer can usethe sameboundary layer stabilitycriterion to construct acrosswhich there is a temperaturedrop a simplemodel representingsteady state convection. The ana- AT = T" - 7" (10) lytical relationshipsobtained in this way can be used to test whetherthe depthand heatflux for infinitelyold seafloor, here The thicknessof the thermalboundary layer growsas calledthe equilibriumstate, agree with the extrapolationsob- 6 = 1.43(gt) zn (11) tained by fitting a plate model to the observations. The startingpoint is a schemesuggested by Howard [1966] The thermal boundarylayer is assumedto becomeunstable as being equivalentto turbulent Rayleigh-Benardconvection when a locally defined Rayleigh number exceedsa critical in some average sense.This is adapted here to the case of value. In other words,the local stabilitycriterion is givenby convectiondriven by internal heat sourcesor a combinationof internal heat sources and fixed heat flux from below. This gaA T68/•v = Rs, (12:) modeof heatingis more appropriateto the uppermantle than where g is the gravitational acceleration,a the volumetric the uniform temperaturedifference used in Rayleigh-Benard thermalexpansion coefficient, •, the kinematicviscosity, and convection[McKenzie et al., 1974]. The steadystate in such

T T

t-O

t-t o

Fig. 6. Illustrationof a simplifiedmodel for convectionat highRayleigh numbers driven by internalheating. 4490 PARSONSAND MCKENZIE: CONVECTION BENEATH THE OCEANIC PLATES convectionproduces a mean temperatureprofile which is es- Averagingthe temperatureprofile (equation (14)) overthe sentially isothermalfrom the bottom of the layer up to a thin interval (0, t•) givesthe mean temperaturestructure of the boundarylayer acrosswhich the temperaturedrops rapidly to boundary layer: match the zero temperatureboundary condition on the upper surface(Figure 5). A boundary lay6r at the bottom of the T = T• [erf(•)+ (2•/'/1'1/9') .exp (-•'•)- 2•'• erfc (•)] (21) convectinglayer only developssignificantly as the proportion where//= z/2(tctc)1/•'. Despite the factthat the abovemodel is of the total heating provided by heat flux from below in- very crude, the reasonablevalue obtained for Rac and the creases. correctform of the relationship(equation (18)) suggestthat Following Howard [1966], we assumethat this can be mod- reasoningbased on the localstability of the boundarylayer eled in a time-averagedsense by starting with a completely gives sensibleresults. isothermal layer, temperature T = Ta, and at time t = 0, In the equilibriumstate the mean thermalgradient in the imposing an upper boundary condRion T = 0 (Figure 6). As mechanicalboundary layer is linear,and its thickness •4' (Fig- before, an upper thermal boundary layer developsby con- ure 7) dependson the heat flux F throughits base: ductivecooling, the temperaturebeing given by A'= [k(T1- T•)]/F (22) T = Ta erf {z/[2(Kt)l/2]} (14) Parsonsand Sclater [ 1977]estimated the heatflux throughthe This boundary layer becomesunstable at a time t = to,when a baseof the platein the equilibriumstate to be 34 mW m-", so local Rayleigh number definedfor the boundarylayer reaches that A' = 91 km. The lengthscale • of thethermal boundary some critical value. The local stability criterion is given by layercan be calculated from (15). Valuesofv = 2.33X 1016m" ga T, ba/Kv = Rac (15) s-1, T• = 325øC,Ra• = 10•, and otherwiseas in thetabulation, Wereused. We find that • = 58 km and,substituting this value wherethe boundary layer thicknessat this time is taken to be in (17), wefind a valueof F = 40 mW m-" for theequilibrium • = 2(•tc) 1/• (16) heatflux. Considering the uncertainty in theequilibrium value of the heatflux andthe crudenessof the model,this is quite that is the thermal diffusion length scaleat t = tc. When the reasonableagreement. boundary layer becomesunstable, the fluid in the boundary In the equilibriumstate, (21) givesthe temperaturestructure layer is replacedin a time short in comparisonto tc [Howard, in the thermalboundary layer. If depthsare measureddown- 1966] by fluid at the temperatureTa. Hence the initial stateis ward from the top of the mechanicalboundary layer, the restored,and the processrepeats itself (Figure 6). The average temperaturein both the boundarylayers in the equilibrium heat flux over the interval (0, tc) is state is given by

2kTa 4kTa r(z) = [(r, - 0 < z < a' F = (•rKtc)l/O.-',/1.1/2(• (17) T(z) = Ti + T•[(2/•r1/") • exp(-•") (23) with k (= pCp•) the thermal conductivity,p beingthe density -(1 + 2•")erfc(//)] z > A' and Cp the specificheat. In the caseof convectiondriven by internal heating, it is F rather than T,• which is specified,and where• = (z - A')/b in thiscase. The elevationof the ridge T,• adjusts until a balance is obtained with the internal heat crestrelative to infinitelyold seafloor is givenby sources.From (15) and (17) a relation betweenF and T,• can be obtained: eac- COo- ow) [T1- T(z)]dz (24) if variationsin depthof theocean floor are isostatically com- lOgloTa= 0.75lOglo F+ 0.25 lOglo 64paCpaKag a (18) pensated.Parsons and Sclater[1977] estimated a ridgecrest McKenzie et al. [1974] obtained a similar relation based on elevationof 3900m relativeto an asymptoticdepth of 6400m. their numerical calculations: Substitutionof (23) in (24) resultsin a valueof enc= 3470m, in good agreementwith the observations,since most of the logloT• - 0.76 lOgloF + 3.58 (19) difference from the observedvalue is due to the values of a and when Ta is given in degreesCelsius and F is measuredin watts T1 adopted here for simplicity,which differ from thoseesti- per square meter. This relationship was actually derived for matedby Parsonsand Sclater [1977]. As well assatisfying the the maximum temperaturein the cell, but as can be seenfrom equilibriumconstraints reasonably well, the convectivemodel the isothermsand mean temperaturestructure shown in Fig- automaticallygives a linearrelationship between depth and t 1/" ure 5b, the relationshipshould hold for the temperaturejump out to 70 m.y. and the breakdownof the relationshipat this acrossthe boundary layer in the mean temperaturestructure. age.We havenot attemptedto solvethe nonlinearequations The simpletime-averaged model basedon a local critical Ray- describingthe developmentof the instabilityand so cannot leigh number criterion gives the correct form of the relation. testthe behaviorof the convectivemodel between 70 m.y.and The value of Rac that is to be applied to the stability of the the equilibrium state. boundary layer can be obtained by equatingthe constantsin For a platemodel the properties of themodel are principally (18) and (19). Using the values of the physical parameters determinedby the plate thickness.In the caseof the convective given by McKenzie et al. [1974], we find that model, however,the thermalboundary layer has a thickness andtemperature drop across it whichdepend not onlyon the Rao = 473 (20) heatflux but alsoon the viscosityvalue. This dependencecan be seenin (18). As an illustration,we canask what viscosity The critical Rayleighnumber for free-freeboundaries is Ra• = valueis needed,given Ta = 325øCto satisfythe observedheat 658, with constant temperature specified on the bottom flux of 34 mW m-•. Using(15) and (17), a viscosityvalue boundary, or Ra• = 385, with constant flux on the bottom boundary. v = 3.9 X 1016m •' s-1 (25) PARSONSAND MCKENZIE: CONVECTIONBENEATH THE OCEANICPLATES 4491

Temperature,øC vection producechanges in the refractiveindex. For instance, o 5oo I000 1500 cold downwellingfluid has a higher refractiveindex than sur- ß " ' ' ' I ' ' ' ' I' • • • ,• I • • • ' rounding fluid and causesconvergence of the light beam. A translucentscreen interrupts the beam after it passesthrough the fluid. The photographshown in Figure 8 representsa view '"".. -r-I II .""'"'"'••...... / T,•I from above the convectinglayer, bright regionsmarking cold '",., I 50 downwellingfluid and dark regions hot upwelling fluid. The - '...... • I• Plate working fluid was a Dow Corning 200 silicone oil with a '-... • thickness Prandtl number (= ,/K) of 8600, the large Prandtl number limit being appropriate for mantle convection.Values of the

E physicalparameters for the oil are po = 0.971 Mg m -a, •, = ...... _ 10-a m•' s-•, a = 9.6 X 10-4 øC-•, and • = 1.16 X 10-7 \ i - \ i m•' s-•. The viscosityis, to a good approximation, indepen- dent of temperature. _ ofsmall-scale convection Convection producedby cooling from above has been con- ' sideredin connectionwith the coolingof the surfacesof oceans 150 - and lakes.Spangenberg and Rowland[ 1961] and Foster [ 1965b] observedconvection induced in water by evaporative cooling at its upper surface.In the former casea Rayleigh number of 1183 was calculated from the temperature structure in the upper boundary layer at the onset of convection.The experi- l mentshere were performedto test further the applicabilityof a Fig. 7. The solid curveis the equilibriumthermal structureof the local critical Rayleigh number criterion for the upper bound- plate and showsthe part played by the thermal boundary layer of ary layer and to provide an illustration of the form of con- small-scale convection. The mean convective thermal structure differs vection that might be expected. from a linear profile expectedfor the plate model at the baseof the thermal boundarylayer. The dottedcurve is the temperaturestructure Two experimentswere performedfor layer depthsd of 5 and given by (7) for an age of 70 m.y. immediatelybefore the onsetof 12 cm. Both boundarieswere maintained at the sametemper- convection.Parameter A' is the thicknessof the mechanicalboundary ature (40øC) for a time greaterthan the thermal time constant layer in equilibrium,and 6 is the thermallength scale governing the d•'/• of the layer.This producesan initiallyisothermal layer of time-averagedtemperature structure in the boundary layer. Also fluid. At t = 0 the heating of the bottom boundary was shownis the plate thicknessobtained by fits to depth and heat flow data. stopped, and the temperature of the upper boundary was rapidly reduced. The bottom boundary temperature varies very slowly subsequentlyin comparisonto that of the upper is obtained,slightly greater than the value in (13). Figure 7 boundary,so that essentially,the fluid was being cooledfrom showsthe mean temperaturefor the convectivemodel in this abovewith zero heat flux from below. The instabilityoccurs in case.The temperaturestructure differs significantly from the the cold upper boundary layer, cold fluid breaking away and lineargradient expected for the platemodel only at the baseof forming concentrated downwellings. The boundary layer the thermal boundary layer. It is encouragingthat both the thicknessat this point, t = to, is estimatedfrom 6 = 2(gt•)x/•', onset of small-scaleconvection and the asymptoticbehavior the thermal diffusion length scale.The value of the boundary can be matchedwith viscosityvalues lying within the rangeof layer thickness,b • 1.9 cm, was alwaysmuch smaller than the other estimates. depth of the fluid layer. A value for the temperaturedrop It should be noted that the definition of the thickness of the acrossthe boundarylayer was obtainedfrom the mean differ- thermal boundary layer used in this sectiondiffers slightly ence betweenthe upper boundary temperatureand the initial fromthat usedin the previoussection. In the equilibriumstate temperatureof the fluid over the interval (0, t•). From (12), the temperatureprofile' in the mechanicalboundary layer is values of Ra• • 2.1 X 103 were obtained. This provides an essentiallyfixed, and only the turning over to fluidbeneath it is upper limit to the critical Rayleigh number for the boundary considered.Before the instabilityoccurs, the temperatureis layer, since the instability must develop to a certain extent alsovarying in the mechanicalboundary layer, and hencethe before it can be observed with this visualization method. The gradientsin thethermal boundary layer will be different.It is local stability criterion applied to the upper boundary layer clear,however, that in all casesthe boundarylayer thickness is therefore provides a reasonableguide to the onset of con- proportionalto (Ktc)•/•'. vection in the fluid.

4. BOUNDARY LAYER INSTABILITIES The form of the convection always consistedof localized IN LABORATORY EXPERIMENTS downwellings.The upwellingoccurs over a diffusearea, since no localizeddark upwelling regionsare observed.An example a. TransientCooling From A bore of the developedform of the convectionis given in Figure 8. The experimentsdescribed here were performed by usinga The horizontal length scale of the convectivepattern was convectiontank basedon the designof Chenand Whitehead observed to decreaseas the convection developed after the [ 1968],which is fullydescribed elsewhere [Richter and Parsons, initial instability. Downwellingsare seento developwith time 1975; Whiteheadand Parsons, 1978]. A layer of fluid is in different parts of the tank. If attention is concentratedon a boundedabove and belowby horizontalglass surfaces which small area, the mean effect of new downwellingsdeveloping are maintained as isothermalboundaries to a good approxi- and moving acrossthis area can be seento be equivalentto the mation.These are separatedby machinedspacers which con- time-averagingprocess considered above. These results are trol the depthof the layer.Convection is observedby passinga consistent with the observations of Whitehead and Chen slightlydivergent beam of light throughthe layer of fluid. [1970], Spangenbergand Rowland[1961], and Foster [1965b]. Horizontal temperaturedifferences associated with the con- Convectiondominated by localizeddownwellings and diffuse 4492 PARSONSAND MCKENZIE: CONVECTIONBENEATH THE OCEANIC PLATES

ß :.:

Fig. 8. Exampleof the form of convectionproduced by cooling from above. Note the predominanceof localizedcold downwellings(bright regions).

upwelling regions is typical of numerical results for internal ically in Figure 9, by a combinationof a uniform heat source heating (Figure 5b). This contrasts with Rayleigh-Bernard over the top surfaceat one end of the tank and an isothermal convection,where upwelling and downwellingfluid occursym- heat sink over the other end of the tank. The source and sink metrically.The instabilityin the bottom part of the plate that extendthe whole width of the tank. The production of a large- was discussedabove might be expectedto take the form of scale flow in this way is to be expected,but what was unex- localizeddownwellings falling from the baseof the plate. pectedwas the behavior in planestransverse to the main flow (Figure 10). As the large-scaleflow entered the region under b. ObservedSmall-Scale Instability the isothermalheat sink, coolingproduced an upper boundary in a Large-Scale Flow layer which eventuallybecame unstable,producing convective Recently, Curlet [1976] conducted an experiment that has rolls aligned along the direction of the large-scaleflow. certain features in common with the suggestedscheme for an Curlet [1976] calculated a local Rayleigh number for this instability at the baseof the plate. Convectionin glycerinewas upper boundary layer and showedthat the onsetof convective studied in a plexiglassenclosure, with a horizontal crosssec- rolls occurred when Ra• '-• 2 X 10s, in agreementwith the tion of 24 inchesX 20 inchesand a depth of 5 inches,in order experiment in subsectiona. For the purposesof comparison to understandpossible convective processes that might occur with possiblemantle convection,various features of this exper- in glassfurnaces. Neutrally buoyant anthraceneparticles were iment should be noted. First, the horizontal scale of the small- mixed in with the glycerine. The flow in longitudinal and scale convectioncells is much lessthan the depth of the tank. transversevertical sectionsthrough the tank could be visual- For the particular combinationof boundary conditionsin this ized by illuminating the particleswith a planar light sourceand experiment, where a small percentageof the heat lossoccurs at taking streak (long exposure)photographs. A large-scaleflow the side boundaries, one observesa mean temperature struc- along one length of the tank was forced, as shown schemat- ture under the sink which is potentially unstablein the upper PARSONSAND MCKENZIE: CONVECTION BENEATH THE OCEANIC PLATES 4493 boundary layer but stabilizing over a large part of the region equally to their models,and it appearspossible that in both below that. Such a temperaturestructure tends to prevent the casesan instability will occur leading to the developmentof penetration of the downwellingfluid into the fluid below small-scale convection. We conclude therefore that the mode [Whiteheadand Chen, 1970].Second, the longitudinalalign- of vertical heat transport beneaththe platescannot be solely ment of the small-scale convection cells is a result of the inter- conductive.Though the boundarylayer theory usedabove is actionwith the large-scaleflow [Richterand Parsons,1975], a crude, the resultssuggest that it shouldnot be difficult with a shear flow tending to suppresstransverse rolls. more exact theory to obtain a consistentconvective system. We haveconsidered only one boundarylayer instability.Other 5. DISCUSSION AND CONCLUSIONS instabilities in the total circulation related to plate motions A local Rayleigh number criterion used to test the stability may be possible.These can only augmentthe argumentsfor of a boundary layer has been applied to the evolving thermal convectionon a different horizontal length scalebeneath the structureof the plates. It was demonstratedthat it is likely that plates.The occurrenceof a convectiveinstability and develop- an instability in the bottom viscousregion of the plate could ment of small-scaleconvection provide a consistentway of lead to the developmentof small-scaleconvection. The same modelingthe observedbehavior, which is alsoconsistent with local Rayleigh number criterion applied to boundary layers the results of laboratory and numerical convection experi- has been shown to provide satisfactoryresults in a number of ments. As the developmentof small-scaleconvection can ac- examples.The studiesof Foster [1965a], Howard [1966], and count for the observationsby itself, it is not necessaryto Whiteheadand Chen [1970] give someindication of why this is supposethat additional effectssuch as shearstress heating so. A cooling boundary layer above a semi-infinite region is contribute significantlyto the thermal structure. alwaysunstable. For a regionof finite depththere will be some The depthto whichcooling can penetrateinto the mantleis small critical Rayleigh number for the region as a whole to essentiallythe sum of the thicknessesof the mechanicaland become unstable. However, the stability test is applied to a thermal boundarylayers. Figure 7 showsthe relation of these basicstate that is time dependent.Any disturbancesmust grow to the plate thicknessobtained by fits to depthversus age and fast enoughnot to be swampedby changesin this basicstate. heat flow versusage observations.Inversion of observedsur- The disturbancefor the layer as a whole will have the largest face-wavevelocities gives the depth to the low-velocityzone wavelengthand low growth rates.As the coolingproceeds, the [Forsyth,1977], and this is assumedto denotethe onset of boundarylayer itself becomesunstable, and this disturbance partialmelting. In the pastthis depth was sometimes identified has a fast enough growth rate to dominate the time depen- with the thicknessof the plate usedin thermal models.Figure denceof the basic state. Hencethe stability of the whole region 11 shows two estimates of the solidus of peridotite in the appearsto be governedby the stabilityof the boundarylayer. presenceof water and the depthand temperatureobtained for In principle, the stability of any two-dimensionalflow put the bottom of the plate modelthat fits the depthand heatflow forward as the large-scalecirculation associatedwith plate versusage data. It is clearthat the onsetof partial meltingwill motions could be tested exactly along the lines used by Busse occur at shallower depths than the plate thickness,so that [1967] for the stability of convectiverolls. In practice, how- estimates obtained from surface waves should differ from that ever, this is a difficult procedure. In any case, little is known obtained from thermal models. Figures 7 and 11 show that about the circulation as a whole, solutions of the equations partial meltingcan occur within the thermal boundarylayer, governing the behavior, or the theology of the material in- and a questionthat must be answeredis whetherthe partial volved. The part that seemsto be best understood is the meltingeffects the theologicalbehavior and hencethe thermal thermal structureof the plates,the boundarylayer of the large- structure. scaleflow. The abovecriterion providesa simple,albeit crude, Suchexperiments as are available[Auten et al., 1974;Auten method of testingfor stability.Forsyth [1977] and Schubertet and Gordon, 1975; Arzi, 1974] suggestthat the presenceof al. [1976] have suggestedalternative explanations for the flat- small amountsof melt do not significantlyaffect the viscosity. tening in the depth versus age curve as discussedin the in- In order to reducethe viscositydrastically it is not just suf- troduction, the mode of vertical heat transport being con- ficient that the melt wet the grain boundaries,forming a conductive in both cases. However, in neither case was the nected network of melt. In fact, in the derivation of the effec- stabilityconsidered. The thermalstructures of thesemodels, as tive viscosityproduced by diffusioncreep [Herring, 1950; forthe slab model, are closely, similar. This must always be so McKenzie,1968] the shear stress on the grain boundaries is in the age rangewhere the t•/2 dependenceof oceandepths assumedto vanish.The amountof meltingmust be enough holds. Hence the above argumentsabout stability apply that the probabilityof solid-solidgrain bondingis reduced,

Source J--T Sink UnJformheatin,• T IsothermalT

Large-scalecirculation

Fig. 9. Sketchof a longitudinalsection of the experimentperformed by Curlet [1976]. The dashedlines marked A and B denotetransverse sections for which the flow is illustratedin the photographsof Figures 10a and 10b,respectively. 4494 PARSONSAND MCKENZIE: CONVECTIONBENEATH THE OCEANIC PLATES

Fig. 10. (a) Photographshowing the flow in a transversesection under the source region. The smoothcurved streaks on the bottomof the photographrepresent flow drivenby heatlosses at the sideboundaries which is weakin comparisonto the principallongitudinal large-scale flow. (b) Photographillustrating the developmentof small-scalerolls underthe cooling region.These occur when a localRayleigh number, calculated for the upperboundary layer that formsin the sinkregion, exceedsa critical value [from Curlet, 1976]. PARSONSAND MCKENZIE: CONVECTION BENEATH THE OCEANIC PLATES 4495

mechanismfor isostaticcompensation. The exact wavelength of the small-scaleconvection seriously affects attempts to de- 4O tect it. The theoreticalstudies of Foster [ 1965a]and Whitehead 40 ; and Chen [1970] suggestthat the wavelengthcould be inde- pendentof the depth, beingdetermined solely by the amount

3O 30 -- of heating. The wavelengthwould also be affected by the viscositystructure in the upper mantle, any increasein viscosity with depth tendingto reducethe penetrationand hencethe wavelengthof the convection.If the wavelengthis too small =20 2O (<400 kin), the masking effect of the overlyingelastic plate will make it difficult to detect by trying to observeassociated depth and gravity anomalies [McKenzie and Weiss, 1975]. I0 IO Without a direct demonstrationof small-scaleconvection, dis- criminatingbetween thermal modelsof the plateswill depend solelyon theoreticalarguments such as those given above.It is ] i I i possiblebut unlikelythat a full nonlinearstudy of the thermal 800 1200 1600 800 1200 1600 boundarylayer instabilitycombined with the depth-agecurves Temperature(ø C) will provide information about any convectivemotions in- volved. Fig. 11. The temperaturesand depth,with estimatederrors, deter- minedfor the baseof the plate [from Parsonsand Sclater, 1977],com- Acknowledgments. This researchwas started while both authors paredto two proposedmelting curves for peridotitein the presenceof were participants in the GeophysicalFluid Dynamics Program at small amounts of water. Curve A is from Millhollen et al. [1974], Woods Hole Oceanographic Institution in 1975. We would like to and curve B is from Green [1973]. thank Willem Malkus for encouragementthen and since.Nigel Cutlet kindly gaveus permissionto reproducethe photographsin Figure9. and any interlockingof grainsis small;otherwise, the viscosity We are grateful to Jack Whitehead for experimentalfacilities and to is still primarily determined by transport within the solid Bob Frazel for assistancein buildingapparatus and with photography. The work was principally supportedby the Office of Naval Research grains.Although it is difficult to be specificwith suchmeager under grant N0014-75-C-0291. Research on mantle convection at experimentalevidence, it is likely that the viscosity is not Cambridge, England, is supportedby a grant from the Natural Envi- significantly reduced until the fraction of melt exceedsa criti- ronmental Research Council. cal amount (•20%; Arzi [1974]). The small amounts of melt that can be expectedin the low-velocityzone, consideringthe REFERENCES relation of the temperaturesto the wet and dry melting curves Arzi,A. A., Partialmelting in rocks:Rheology, kinetics and water [Wyllie, 1971], should not substantiallyaffect the viscosity. diffusion,Ph.D. thesis,Harvard Univ., Cambridge,1974. Any flow in this region will be undisturbedby the presenceof Auten,T. A., andR. B. Gordon,Compresslye creep rates of partially partial melt. This is consistentwith the observationof Forsyth melted AI-Ga alloys, Met. Trans., 6a, 584-586, 1975. [ 1977]that changesin seismicvelocity structure clearly showed Auten,T. A., R. B. Gordon,and R. L. Stocker,Q andmantle creep, Nature, 250, 317-318, 1974. the effectsof cooling within the low-velocityzone. Busse,F. H., On thestability of two-dimensionalconvection in a layer In a similar way the effectiveelastic plate thicknessobtained heatedfrom below,J. Math. Phys.,46, 140-150, 1967. from modelsrepresenting the flexural effectsof excesscrustal Cathies,L. M., The viscosityof the earth'smantle, Ph.D. thesis, loading[Watts and Cochran,1974; McKenzie and Bowin, 1976] Princeton Univ., Princeton, N.J., 1971. Chen, M. M., and J. A. Whitehead, Evolution of two-dimensional is obviouslydistinct from either the depth to the low-velocity periodicRayleigh convection cells of arbitrarywavenumbers, J. zone or the thermally definedplate thickness.The measures Fluid Mech., 31, 1-15, 1968. provided by these different observationaltechniques are all Crough,S. 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