Are mantle plumes adiabatic?

Ctirad Matyska a;*, David A. Yuen b

a Department of Geophysics, Faculty of Mathematics and Physics, Charles University, V Holes­ovic­kach 2, 180 00 Prague 8,Czech Republic

b Department of Geology and Geophysics and Minnesota Supercomputer Institute, University of Minnesota, Minneapolis,MN 55415-1227, USA

Received 13 March 2001; accepted 30 April 2001

Abstract

The issue concerning the state of adiabaticity of mantle plumes has been examined in a cartesian two-dimensional boxwith an aspect-ratio of six. We have investigated in the quasi steady-state regime high Rayleigh number convection withboth depth-dependent viscosity and thermal expansivity for both the Boussinesq and the extended Boussinesqapproximations. We have also assessed the influence of various forms of thermal conductivity and internal heating. Wehave generalized the classical Bullen's parameter equation from one dimension to multidimensions. For assessing thelocal state of adiabaticity inside plumes and in their surroundings, we have extracted from the local geotherms and thelocal thermodynamic properties the corresponding Bullen's parameter profiles and the two-dimensional mapsportraying the state of adiabaticity in the mantle. Histograms characterizing the frequency of adiabaticity are alsoemployed for quantification purposes. In general, superadiabatic thermal gradients are found inside the thick plumelimbs and sometimes along the central part of the plume. The centers of plume heads are subadiabatic or nearlyadiabatic, but the edges of the plume heads are strongly subadiabatic. Alternating strips of subadiabaticity andadiabaticity are found in the downwellings. The ambient mantle outside the plumes is generally adiabatic and issometimes perforated with islands of marked deviations from adiabaticity. ß 2001 Elsevier Science B.V. All rightsreserved.

Keywords: numerical models; geothermal gradient; mantle; convection

1. Introduction

There are many good reasons for examiningboth the local and global degree of adiabaticityin the mantle. Foremost are: (1) the e¤ciency ofheat transport in the mantle as it has been com-mon in thermodynamic considerations that heat

conduction is negligible outside mantle boundarylayers because of the vigorous nature of convec-tion, and thus one long-standing assumption ingeophysics has been the adiabatic state of themantle ; the related question is (2) the extrapola-tion of equation of state parameters in the lowermantle [1,2] by means of the adiabatic gradient oftemperature and the other problem is (3) under-standing the thermodynamic state under the litho-sphere and subducting slabs, especially in the re-gions where the lithosphere interacts with hotplume heads and where diapirs may rise through

0012-821X / 01 / $ ^ see front matter ß 2001 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 0 1 ) 0 0 3 6 1 - 2

* Corresponding author. Tel. : +420-2-21912538;Fax: +420-2-21912555; E-mail : ctirad.matyska@m¡.cuni.cz

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the wedge region [3]. These problems can bestudied by numerical models of thermal con-vection, which yield basic physics of heat trans-port in a moving continuum, such as the Earth'smantle, although in the real Earth the situationcan be complicated by the presence of phasetransitions, hydrous phases, complex rheologicalzones and chemical strati¢cation. Matyska andYuen [4] have already shown that the departuresfrom a globally averaged adiabatic temperaturegradient can be assessed quantitatively fromcalculating Bullen's parameter [5] from the aver-aged temperature pro¢les derived from mantleconvection calculations. Bunge et al. [6] have re-peated these ¢ndings with a three-dimensionalspherical model and found good agreement tothe two-dimensional results of Matyska andYuen [4].

Local features, in particular plumes, are com-monly held to be adiabatic from investigations onstarting plumes in laboratory studies [7^9], or inanalytical or numerical models [10,11], especiallyin strongly time-dependent £ow situations, suchas in the aftermath of a £ushing event in convec-tion with phase transitions, e.g. [12,13]. By usingthe di¡erence between the actual temperature gra-dient and the adiabatic gradient, Leitch and Yuen[14] found extensive regions of subadiabaticityoutside of the cold downwellings in two-dimen-sional spherical-shell compressible convection ata relatively low Rayleigh number of 3U105. Stein-bach and Yuen [15] have found with Lagrangianpassive tracers that the local temperature ¢eldin the vertical £ow might depart signi¢cantlyfrom the adiabatic state. In this study we willgeneralize the analytical expression for Bullen'sparameter to three dimensions. This extensionof Bullen's equation to multidimensions will allowus to conduct a local interrogation of the adiaba-ticity of plumes in a more accurate fashion thanwas done in [14], where di¡erences between twosimilar numbers were involved, and the Lagran-gian technique of Steinbach and Yuen [15], inwhich few tracers were employed and the time-histories were di¤cult to monitor. We will alsodisplay the local evaluations of mantle adiabatic-ity.

2. Model description

It is well expected from physical considerationsthat adiabaticity would be mostly likely to prevailunder vigorous convection. Hence we have fo-cused on this regime rather than on low Rayleighnumber (Ra) convection in order to ¢nd patchesof non-adiabaticity, which may still remain. As in[4], we have focused on two-dimensional modelsin order to have better spatial resolution for re-solving the thin features in high Rayleigh numberconvection, which we have taken to have Ra ex-ceeding 106. Thus we have employed a two-di-mensional rectangular model with an aspect-ratioof six. A large aspect-ratio is employed here be-cause of the depth-dependent properties which in-duce long-wavelength circulation [16]. The mo-mentum and energy equations for an extendedBoussinesq £uid without inertial terms weresolved for impermeable and free-slip boundaryconditions. The equations take the form:

9Wv � 0 �1�

9W�j �z��9v� �9vT ��3RasK �z�Tez39p � 0 �2�

DTDt� 9W�k9T�3vW9T � R�DK �z��T0 � T�vz�

DRas

j �z��9v� �9vT� : 9v �3�

where v = (vx, vz), T and T0 are, respectively, di-mensionless velocity, temperature (normalized bythe temperature drop vT over the mantle) andsurface temperature, p is the dimensionless dy-namical pressure, ez is the unit vector pointingdownward, t is the dimensionless time, j(z) isthe depth-dependent dimensionless dynamic vis-cosity normalized by a reference dimensional val-ue js, K(z) is the depth-dependent dimensionlessthermal expansivity normalized by Ks, R is thedimensionless internal heating and k is the di-mensionless thermal conductivity normalized byks. The symbol `W' represents the scalar productof vectors, the symbol ` :' represents the totalscalar product of two tensors and `T ' denotes

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the transposition. The Rayleigh number Ras =b2

s cpKsvTgd3/ jsks, (bs is a reference density, cp

is the speci¢c heat under a constant pressure, gis the gravity acceleration, and d is the mantlethickness) was ¢xed to Ras = 107, which is therepresentative value of the surface Rayleigh num-ber in whole mantle convection, and the dissipa-tion number D =Ksgd/cp was chosen equal to 0.5[16]. We can recover the classical Boussinesq ap-proximation by setting D = 0.

In order to satisfy Eq. 1, we reformulated theequations by means of the stream function i,which is de¢ned by the relations vx = Di/Dz andvz =3Di/Dx. The spatial derivatives in Eqs. 2and 3 were performed by an eighth-order ¢nitedi¡erence scheme [17] employing 768U128 gridpoints for the heat Eq. 3 and a coarser gridwith 192U32 grid points to discretize the momen-tum Eq. 2. The time-stepping has been carried outwith a second-order explicit Runge^Kuttascheme.

One of the fundamental ¢ndings in thermalconvection dynamics is that increase of viscosityand decrease of thermal expansivity with depthcan produce huge slowly evolving plumes inthe lower mantle [16]. As the aim of this studyis to deal with the plume geotherms, this is thereason why we have followed this study and choseagain the thermal expansivity pro¢le (see also[18]) :

K �z� � 8�2� z�3 �4�

where znG0,1f is the dimensionless depth. Simi-larly, the viscosity increase with depth was mod-elled to be:

j �z� � exp�4z� �5�

In general, the thermal conductivity varies withtemperature and pressure because the phononpart of the conductivity [19] decreases with tem-perature and increases with pressure (depth),whereas its radiative part strongly increases withtemperature. To test the in£uence of variable ther-mal conductivity on the mantle plume thermalstructure, we have followed [19] and [4], used

the model :

k�z;T� � �1:0� 2:5z� T0

T0 � T

� �0:3

exp�30:447T��

1:03�T0 � T�3 �6�

and compared the results with those obtained forthe constant thermal conductivity k = 1.

The Bullen parameter R is a dimensionless ratiobetween the depth-gradient of density and theadiabatic density depth-gradient. It was intro-duced when only depth-dependent seismic Earthmodels were available [5] simply by:

R �z0� � x �z0�b �z0�g�z0�

dbdz0�z0� �7�

where zP= zd is dimensional depth andxx(zP) = vp

2(zP)343v

2s (zP) is the seismic parameter.

Therefore, the Bullen parameter was consideredas the quantity assessing horizontally averageddeviations from adiabaticity.

Taking into account only deviations of densityfrom adiabatic density pro¢les due to thermal ex-pansion and going back to dimensionless coordi-nates, the Bullen parameter corresponding to thethermal ¢eld T(x,z) is (compare [4]) :

R �x; z� � 13K sK �z�x �z�vT

gd

DTDz�x; z�3DK �z��T0 � T�x; z��

� ��8�

We have used the surface value of thermal ex-pansivity Ks = 3.1035 K31, vT = 3725 K, g = 10 ms32, d = 2890 km and depth-dependent xx(z) corre-sponding to the PREM value [20] in the mantle.We would like to emphasize here that the Bullenparameter R de¢ned above is a function of boththe vertical and horizontal coordinates. Eq. 8 canalso be extended to three-dimensional situationsand R can be a function of all three spatial vari-ables. In other words, the Bullen parameter char-acterizes the state of the convection model at aparticular location and time instant. The valueof 1 denotes perfect adiabaticity of a vertical geo-

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therm. Its magnitude higher than 1 correspondsto a local subadiabatic state whereas its magni-tude lower than 1 marks a local superadiabaticstate. The expected resolution from seismic datamay only allow detection of deviations in R from1.0 up to 2 or 3% [1].

3. Results

In Fig. 1 we show for Ras = 107 ¢ve snapshotsof the temperature ¢eld for the depth-dependentthermal expansivity (Eq. 4) and the depth-depen-dent viscosity (Eq. 5) The dominant plumes ap-pearing in each panel are found in a quasi-equi-librium state after a long-time integration. Wehave studied the following physical systems, wheredi¡erent approximations for the thermal conduc-tivity, internal heating and dissipation were takeninto account (from the top to the bottom):

1. Extended Boussinesq model with D = 0.5, var-iable thermal conductivity k = k(z,T), see Eq. 6,and no internal heating.

2. Extended Boussinesq model with D = 0.5, con-stant thermal conductivity k = 1 and no inter-nal heating.

3. Classical Boussinesq case obtained by puttingD = 0 with constant thermal conductivity k = 1and no internal heating.

4. Extended Boussinesq model with D = 0.5, var-iable thermal conductivity k = k(z,T), see Eq. 6,and the internal heating R = 12.

5. Extended Boussinesq model with D = 0.5, con-stant thermal conductivity k = 1 and the inter-nal heating R = 12.

The magnitude of internal heating R = 12 wouldcorrespond to to the chondritic value of radiogen-ic heating from the dimensional analysis em-ployed (e.g. [14]). We note that the temperature¢eld is not in a steady-state, strictly speaking, inany of these ¢ve cases, but the evolution of thehuge plumes takes place very slowly as comparedto the strongly time-dependent behavior of thesmall cold downwellings (see also recent three-di-mensional simulations [21], which revealed similartendencies but in three dimensions). Thus, it was

relatively easy to choose the appropriate snap-shots with well-developed and very stable plumesfor obtaining representative interior plume tem-perature distributions and also in the ambientmantle. We have taken this tack of investigatingthoroughly the adiabaticity of a well-de¢ned quasisteady-state situation before venturing into tran-sient situations associated with surface boundaryconditions (e.g. [22]), which in fact will producemany non-adiabatic situations. The other conse-quence of increasing viscosity and decreasing ther-mal expansivity with depth is the relatively lowmantle temperature, which results in creation ofthick superadiabatic boundary layer at the bot-tom. The changes of the thermal conductivity ac-cording to Eq. 6 can slightly increase the averagetemperature but they are not able to compensatefor the cooling in£uence of the interior temper-ature due to the viscosity increase and decreasein thermal expansivity.

The corresponding two-dimensional distribu-tions of the local Bullen parameter computedfrom the two-dimensional temperature ¢elds inFig. 1 (see Eq. 8) are shown in Fig. 2. Thesepanels represent two-dimensional maps, showingwith high precision the regions of adiabaticity towithin 5% (yellow color), subadiabaticity (orangeand red) and superadiabaticity (blue and darkblue). The highest superadiabatic temperaturegradient, as shown by the vivid blue color, lieswithin the boundary layers, which is not surpris-ing, but what is interesting are the dramaticchanges of the vertical temperature gradient asso-ciated with falling cold blobs in the upper mantle.Their bottom and top parts have respectivelystrongly superadiabatic and subadiabatic gra-dients. The top part of the blob would correspondto the top part of a subducting slab, which meansthat the mantle above a subducting slab would bestrongly subadiabatic. Very interesting results areobtained for the plumes. In all cases the hotplume heads produce horizontally far-reachingzones under the top boundary layer with stronglysubadiabatic gradient of temperature but the cen-ters of the plume heads are nearly adiabatic. Onthe other hand, the plume limbs are slightly super-adiabatic but internal heating can decrease themagnitude of the vertical temperature gradient

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both in the plume limbs and in the bottomboundary layer. The ambient mantle surroundingthe plumes is adiabatic and not so around thecold downwellings.

The detailed statistics of the distribution of theBullen parameter is in Fig. 3. The shaded histo-grams were obtained from the distributions of Rover the whole computational box, thin line histo-

Fig. 1. Five snapshots of the dimensionless temperature ¢eld T for the surface Rayleigh number Ras = 107, two di¡erent magni-tudes of the dissipation number D and the internal heating R, and two di¡erent choices of the thermal conductivity k. The upperpanel: D = 0.5, k is variable (see Eq. 6) and R = 0; the second panel: D = 0.5, k = 1 and R = 0; the middle panel: D = 0, k = 1 andR = 0; the fourth panel: D = 0.5, k is variable and R = 12; the bottom panel: D = 0.4, k = 1 and R = 12. In all cases the viscosity jincreases with depth (see Eq. 5) and the thermal expansivity K decreases with depth (see Eq. 4).

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grams are for squares with the unit width sur-rounding the plumes and thick line histogramscorrespond to rectangles with the width of 0.25surrounding the plume centerlines. If the wholecomputational box is taken into account, thestate, when the Bullen parameter deviates by less

than 5% from the adiabatic value, is dominatingand its frequency is slightly more than 50%. How-ever, the statistics of R within the plume is di¡er-ent: if there is no internal heating (R = 0), theadiabatic state is suppressed and the frequencyof slightly superadiabatic state (R between 0.85

Fig. 2. Five snapshots of the local Bullen parameter R computed from the temperature ¢elds in Fig. 1 by means of Eq. 8. Thecontour interval is 0.1. The yellow color portrays the `adiabatic' regions, where the local Bullen parameter R is inside the interval1 þ 0.05. The regions with superadiabatic local vertical gradient of temperature are blue and those with subadiabatic local verticalgradient of temperature are orange^red.

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and 0.95) is approximately the same; in the caseswith internal heating (R = 12) the frequency of theadiabatic state becomes dominant but the fre-quency of the slightly superadiabatic state reachesthe same levels as in the cases with no internalheating. We emphasize here that 60% is aboutthe maximum of the frequency occurring within

the plume and not 90^100% as would be expectedin traditional kind of arguments drawn from thesupposed vigor of high Rayleigh number convec-tion [23].

To show the situation inside the plume in de-tail, the geotherms, i.e. the temperature pro¢les,along the centerlines of the plumes are in the

Fig. 3. Histograms showing the frequency of the Bullen parameter R in Fig. 2. The shadow histograms show the distribution ofR inside the whole cartesian computational box G0,6fUG0,1f, the thin line histograms yield the distribution of R in the squaresGpc30.5, pc+0.5fUG0,1f, where the symbol pc means the horizontal coordinate of the plume centerlines, and the thick line histo-grams delimit it inside the rectangle Gpc30.125, pc+0.125fUG0,1f.

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upper panel of Fig. 4. The pro¢les obtained forextended Boussinesq approximation with variablethermal conductivity are shown. In the case withno internal heating the relatively high temperatureT = 0.5 is reached immediately below the upperboundary layer. The internal heating resulted inhigher temperatures and thus the temperature ofplumeheads was even T = 0.7. In any of the casesstudied, there is no boundary layer at the bottomof the plumes and thus the transfer of heatthrough the boundary below the plume feet isnegligible. Under the presence of internal heatingthe plume root was even overheated due to vis-

cous dissipation and the temperature in the coreof the plume feet was a little bit higher than thebottom boundary value T = 1. Note here that oth-er mechanisms, such as Ohmic dissipation [24,25],can also contribute to overheating at the CMB.The bottom panel of Fig. 4 shows the Bullen pa-rameter pro¢les, which were derived from the geo-therms in the upper panel. Below the extremelysuperadiabatic upper boundary layer, the sameslightly subadiabatic part of the geotherms inthe centers of the plumeheads is reached in theboth cases. The plume limbs are then character-ized by deviations towards a superadiabatic state.This result is in agreement with the fact that thereis a signi¢cant horizontal heat £ow from theplume limbs to the outside due to a relativelyhigh temperature di¡erence between the plumeand the rest of the mantle. Therefore, the uprisinghot material is not cooled by the adiabatic expan-sion only but there is also a conductive contribu-tion.

As we already noted, the temperature ¢eldsstudied are not stationary in character and thequestion arises as to the in£uence from temporalvariations in the Bullen parameter. In this paperwe have focused our attention on the plumes andfor this reason in Fig. 5 we show a sequence ofsnapshots dealing with the dynamics of the pul-sating plume. We can see that newly created in-stabilities in the lower boundary layer are stronglyattracted to the already existing plume by thethermal attractor mechanism [26] which ¢nallyresults in small-scale £uctuations of the £ow in-side the plume. The corresponding maps display-ing the local Bullen parameter are in Fig. 6. Thepulsations of £ow in the plume does not substan-tially in£uence the magnitude of the Bullen pa-rameter at the head or at the base of the plume;the most remarkable changes are caused by thetilt of the plume limb. This geometrical e¡ectcan be important in the mantle because of thelarge-scale mantle circulation which can bendthe plumes rather easily at high Rayleigh number.

4. Concluding remarks

In this study we have provided e¤cient means

Fig. 4. The dimensionless temperature pro¢les and the corre-sponding Bullen parameter pro¢les obtained along the cen-terlines of the plumes in the extended Boussinesq case(D = 0.5) with variable thermal conductivity (Eq. 6). The sol-id line corresponds to the case with no internal heating (theplume in the right part of the upper panel in Fig. 1 was em-ployed) and the dashed line is for the internal heatingR = 12.

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of quantifying the local magnitude of the Bullenparameter inside mantle structures. We have con-centrated, in particular, our e¡ort on hot upwel-lings. This Eulerian approach of using the multi-dimensional generalization of Bullen's parameter,which is based on the local temperature ¢eldand thermodynamic parameters, is a great im-

provement over previous e¡ort in quantifyingthe adiabaticity of plumes using Lagrangian trac-ers [15] and is more e¤cient than using the for-mula for calculating R, which involves both thedensity strati¢cation and bulk modulus variations[6].

Our new approach is very similar to what is

Fig. 5. Five snapshots of temperature in ¢ve subsequent times for the extended Boussinesq case (D = 0.5) with variable thermalconductivity (Eq. 6) and the internal heating R = 12.

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commonly done in physical oceanography in theuse of the local Brunt-Va«isa«la« frequency, whichcan be expressed by means of the local Bullenparameter, as a mean of delineating the locationof the thermocline (e.g. [27,28]). To be sure,

oceanography has been more familiar with usingmultidimensional background states more oftenthan in solid-earth geophysics because of theavailability of much denser networks of sensinginstruments [29]. But in the near future solid-earth

Fig. 6. The snapshots of the local Bullen parameter R corresponding to the snapshots of the temperature ¢eld in Fig. 5. The con-tour interval is 0.1. The yellow color portrays the `adiabatic' regions, where the local Bullen parameter R is inside the interval1 þ 0.05. The regions with superadiabatic local vertical gradient of temperature are blue and those with subadiabatic local verticalgradient of temperature are orange^red.

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geophysics is entering an era of higher densityseismic networks involving both permanent andportable instruments. These denser networksshould also alter our mindset concerning usingonly globally averaged quantities, such as theclassical Bullen's parameter pro¢le. Rather weshould be considering more regionally averagedquantities, such as histograms as a local statisticaltool and also the local Bullen's parameter pro¢leas a mean of quantifying thermodynamic stateson a regional basis. The recent work by Ishiiand Tromp [30] has revealed some interesting re-gional density pro¢les, such as found in the lowermantle beneath Hawaii and also Africa, wheresome evidence for a chemical nature of these up-wellings was suggested. With these thoughts inmind, we may now view more perspicaciouslyour results on plumes obtained by applying themultidimensional version of Bullen's equation.

We have found the following results concerningthe nature of adiabaticity of plumes:

b The interior of plume limbs is slightly super-adiabatic due to the lateral heat conductioninto the ambient mantle.

b The base of the plumes can be slightly subadia-batic because of overheating due to viscous dis-sipation [31].

b The plumeheads are almost adiabatic or slightlysubadiabatic in the centers. However, they arestrongly subadiabatic at the edges. This canhave strong implications on the usage of thepotential temperature in thermodynamic con-siderations [32] and geochemical interpretationof magmas issuing from the edge of plume-heads [33,34].

b The ambient mantle surrounding the plumes isadiabatic.

This present work on plumes can be extendedto downwellings, interior of convection zones,remnant plumeheads or slabs, interaction of hotupwellings and/or cold downwellings with phasetransition zones, diapiric heads etc. The multidi-mensional (local) Bullen's parameter can thusserve as a new exciting tool for categorizing thelocal thermodynamic situation in mantle convec-tion.

Acknowledgements

We thank Brian Kennett, S. Karato and D.Yamazaki for discussions, M. Rabinowicz andan anonymous reviewer for constructive com-ments and J. Vel|msky for technical help. Thisresearch has been supported by the ResearchProject DG MSM 113200004, the Charles Univer-sity Grant 238/2001/B-GEO/MFF, the NATOGrant EST/CLG 977 093 and the GeosciencesProgram of the Department of Energy.[AC]

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