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Inferring nonlinear mantle rheology from the shapeof the Hawaiian swellN. Asaadi1, N. M. Ribe2 & F. Sobouti1

The convective circulation generated within the Earth’s mantle bybuoyancy forces of thermal and compositional origin is intimatelycontrolled by the rheology of the rocks that compose it. These candeform either by the diffusion of point defects (diffusion creep,with a linear relationship between strain rate and stress) or by themovement of intracrystalline dislocations (nonlinear dislocationcreep)1,2. However, there is still no reliable map showing where inthe mantle each of these mechanisms is dominant, and so it isimportant to identify regions where the operative mechanismcan be inferred directly from surface geophysical observations.Here we identify a new observable quantity—the rate of down-stream decay of the anomalous seafloor topography (swell) pro-duced by a mantle plume—which depends only on the value of theexponent in the strain rate versus stress relationship that definesthe difference between diffusion and dislocation creep. Com-parison of the Hawaiian swell topography with the predictions ofa simple fluid mechanical model shows that the swell shape ispoorly explained by diffusion creep, and requires a dislocationcreep rheology. The rheology predicted by the model is reasonablyconsistent with laboratory deformation data for both olivine3 andclinopyroxene4, suggesting that the source of Hawaiian lavas couldcontain either or both of these components.

Three distinct approaches have been used to constrain the rheolo-gical structure of the Earth’s mantle. The most versatile approach islaboratory deformation experiments wherein the relation between thestrain rate _e and the deviatoric stress t in a deforming sample is mea-sured under controlled physical and chemical conditions1,2. Suchexperiments yield rheological laws of the form _e~Dtn, where n is apower-law exponent and D depends in general on pressure, temper-ature, grain size and the chemical composition and mineralogy of thesample. Diffusion creep has n 5 1, whereas n 5 3.5 for dislocation creepof olivine5, the dominant mineral in the uppermost mantle. However,application of these laws to the Earth is subject to large uncertainties,both because several of the properties on which D depends are poorlyconstrained in the mantle and because geological strain rates are 6–8orders of magnitude lower than those in the laboratory.

A second, geodynamical approach is to infer the depth variation ofmantle viscosity from the rates of surface rebound following deglacia-tion events6, the long-wavelength components of the Earth’s nonhy-drostatic gravitational equipotential surface (geoid)7, or both together8.A robust result of this approach is that the lower mantle must be moreviscous (by at least a factor of ten) than the upper mantle. However,studies using this approach typically assume a linear rheology (n 5 1)throughout the mantle, and consequently provide little informationabout the relative importance of dislocation versus diffusion creep.

The third approach is seismological, and exploits the fact that dis-location creep (but not diffusion creep) causes the crystallographicaxes of the crystals in a deforming rock to become aligned in a non-random way9. The resulting dependence of elastic wave speed on thepropagation direction (seismic anisotropy) can be detected either by itseffect on surface waves10 or by the splitting of shear waves that it

causes11, and provides unambiguous evidence for active dislocationcreep somewhere along the path between the source and receiver.However, the precise location of the anisotropic region along this pathis difficult to constrain.

Here we show that a more direct determination of the operativedeformation mechanism is possible in a well-defined portion of theEarth’s uppermost mantle that lies beneath the Hawaiian islands. Sincethe earliest days of plate tectonic theory, the Hawaiian islands haveinspired many fundamental new ideas in global geophysics, includingthe concept of a hotspot as a fixed locus of mantle melting12,13, theassociation of hotspots with plumes rising from the lower mantle13–15,the lithospheric flexure model of isostasy16, and the recognition thathotspots can experience episodes of rapid migration17. Below we showthat Hawaii can also make a novel contribution to our understandingof mantle rheology.

The largest geophysical signature produced by the Hawaiian plumeis a broad topographic swell some 1,400 m high and 1,000 km wide(Fig. 1). According to the principle of gravitational equilibrium (iso-stasy), the downward buoyancy force on the swell (which displaces seawater) must be compensated by an upward buoyancy force providedby low-density material beneath: in this case, the anomalously hotmaterial that ascends in the Hawaiian plume and then spreads laterallyover the base of the Pacific plate.

1Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran. 2Universite Pierre et Marie Curie (Paris 6), Universite Paris-Sud, CNRS, Lab FAST,Batiment 502, Campus Universitaire,91405 Orsay, France.

175° W 170° W 165° W 160° W 155° W 150° W

15° N

20° N

25° N

30° N

300 600 900 1,200m

0

Figure 1 | Residual topography of the Hawaiian swell. This is defined as theobserved seafloor topography29 minus the reference topography predicted by athermal model for cooling oceanic lithosphere30. Negative values of residualtopography appear as dark blue, and values exceeding 1,300 m as red-brown.The white line is a segment of a great circle that fits the central axis of the islandchain between longitudes 159–173uW. The green dot shows the currentlocation of the centre of the Hawaiian hotspot (Kilauea volcano), and the whitedot is its projection onto the central axis. Our analysis considers only the mainpart of the swell southeast of the green line.

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Focusing now on the main part of the swell east of 173uW longitude,we note first that its width is smaller in the middle (around 163uW)than to either side, reflecting a past episode of reduced (by about 20%)upward flux of material in the Hawaiian plume18. However, the parts ofthe swell to the left of (older than) and the right of (younger than) theconstriction have nearly the same amplitude and width. This crucialobservation motivates the following physical model.

We consider the Hawaiian plume as a fixed source of buoyant fluidbeneath a plate moving at constant speed19,20 (Fig. 2a). The fluidspreads laterally over the plate’s base to form a broad plume head witha thickness that is small compared to its lateral dimensions. Becausethe flow within it is dominantly simple shear on horizontal planes, theviscosity g~t=_e corresponding to the rheology _e~Dtn is:

g~D{

1n _e

1{nn ð1Þ

where _e~LuLz

��������, u is the horizontal velocity and z is the vertical coord-

inate.The thickness of a plume head with the rheology of equation (1) is

governed by a partial differential equation that can be derived using the‘lubrication’ theory of slow viscous flow in thin layers (Methods). Itcontains three parameters: the plate speed U, the volumetric supplyrate Q of the buoyant material, and the parameter s 5 D(gdr)n/[(n 1 2)(n 1 3)] that characterizes its rate of lateral spreading, whereg is the gravitational acceleration and 2dr is the density deficit of the

plume relative to the ambient mantle. A scaling analysis of the equa-tion shows that the width of the plume head is proportional to:

L0~sQ2nz1

U2nz2

� � 13nz1

ð2Þ

Conservation of the downstream volume flux then implies that theplume head’s thickness is proportional to h0 5 Q/(UL0). Finally, iso-stasy implies that the amplitude of the topography is proportional to:

S0~Qdr

UL0 r0{rwð Þ ð3Þ

where r0 5 3,400 kg m23 is the density of the ambient mantle andrw 5 1,000 kg m23 that of sea water.

Figures 2b and c show the steady-state numerical solutions of thelubrication equation for diffusion creep (n 5 1) and dislocation creep(n 5 3.5) rheologies. Gravitational spreading is less efficient for aplume head with a dislocation creep rheology, which decays andwidens more slowly downstream because the viscosity increases fromthe centre (where _e is largest) towards the edges (where _e is smaller).The influence of the rheology is revealed by a similarity solution of thelubrication equation that is valid at distances x far downstream fromthe hotspot (Methods). It predicts that the swell topography along theaxis y 5 0 should decay downstream as:

S!S0x{x0

L0

� �{1

3nz2ð4Þ

where x0 is the virtual position of the hotspot. Accordingly, S /(x 2 x0)20.2 for diffusion creep and S / (x 2 x0)20.08 for dislocationcreep with n 5 3.5. The rate of downstream decay of the swell is there-fore a sensitive indicator of the rheology of the buoyant material thatcompensates it.

To compare the predictions of the lubrication model with the obser-vations, we first exclude the parts of the residual topography that areunrelated to the swell (Methods). Using a least-squares procedure, wedetermine the values of L0 and S0 for which the numerical solutions ofFigs 2b and c best fit the non-excluded portion (1.73 106 km2) of theresidual topography. These solutions are shown in Fig. 3a (for diffusioncreep) and Fig. 3b (for dislocation creep), together with two contours(0 m and 700 m) of the residual topography. The diffusion creep solutiondoes not match well the shape of the swell. It decays too rapidly down-stream, and is too narrow near the hotspot and too wide farther down-stream. In contrast, the dislocation creep solution matches much betterthe slow downstream decay of the swell and the shape of its boundary.

The above analysis can be refined by accounting for the variation ofthe Hawaiian plume flux Q during the past 20 million years. This gaverise to an along-chain variation of the swell’s cross-sectional area A(x),which has a minimum around 163uW longitude (Fig. 1). Figure 3 showsthe solutions of the time-dependent lubrication equation for n 5 1 andn 5 3.5 that provide the best fit to the distribution A(x) determinedfrom the residual topography data18 (Methods). Again, only the solu-tion with n 5 3.5 provides an acceptable fit to the residual topography.

We now consider the depth of the low-density material compens-ating the swell, which is not immediately obvious. An estimate isprovided by the geoid–topography ratio (GTR), the ratio of the anom-alous height of the gravitational equipotential surface to the topo-graphy anomaly. In the limit of a very broad swell, the GTR (inmetres of geoid per kilometre of topography) is approximately one-tenth the average depth d (in kilometres) of the low-density compens-ating material. Early estimates of GTR < 4–5 m km21 for theHawaiian swell21 suggested d < 40–50 km, above the base of normaloceanic lithosphere of the same age (about 90 million years). However,those estimates were too low, owing to incomplete removal of the effectof the volcanic islands, which are more shallowly compensated (by theplate flexure mechanism) than the swell itself. Refined estimates thatcorrect for this bias22 yield GTR < 7–8 m km21, indicating that the

0.40.5

0

1

0.40.5

0.6

01234

x/L0

a

b

c

n = 1

n = 3.5 1

0

y

xz

L

U

h

Q

y/L0

y/L0

ρ0 – δρ

ρ0

Figure 2 | Lubrication-theory model for the Hawaiian swell. a, Modelgeometry (oblique cut-away view). Buoyant fluid with a density deficit 2drrelative to the ambient mantle and a diffusion creep (n 5 1) or dislocation creep(n 5 3.5) rheology is released at a volumetric rate Q beneath a plate moving atspeed U. The fluid spreads laterally to form a thin plume head of thickness h andhalf-width L xð Þ?h. b, Steady-state plume head thickness h(x, y) predicted bythe model for n 5 1 (portion y $ 0 only). The contour interval is 0.1Q/(UL0),where L0 is defined by equation (2). The heavy black line shows the widthpredicted by the analytical similarity solution (Methods). c, Same as b, but forn 5 3.5.

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low-density material is at depths near or slightly above the base ofnormal lithosphere. This result, together with the well-defined lateralboundaries of the swell, implies that the region in which our inferenceof dislocation creep deformation applies is well constrained in all threedimensions. Our result is also consistent with geodynamical modellingof the apparent thermal age of Pacific lithosphere inferred from seis-mology, which suggests that dislocation creep is the dominantdeformation mechanism at depths ,410 km throughout the Pacificupper mantle23.

We now use the numerical solution of Fig. 3d to estimate the valuesof the Hawaiian plume buoyancy flux B ; Qdr and the rheologicalprefactor D in equation (1) (Methods). The best-fitting values of L0 andS0 for that solution imply B 5 5,610 kg s21, within the range (2,800–8,700 kg s21) of previous estimates20,24,25. The values also implyD 5 (8.7–19.8) 3 10238 kg m21 s212/7, which is more easily under-stood by translating it into the minimum viscosity gmin within theplume head. For the solution of Fig. 3d and a realistic range of valuesof dr (Methods), the maximum strain rate within the plume head is_emax 5 (3.1–3.7) 3 10213 s21. Equation (1) then implies gmin 5 (2.3–3.3) 3 1019 Pa s.

The above viscosity estimate can be compared with laboratory-based rheological laws for dry26,27 olivine aggregates as a function oftemperature, pressure and strain rate3 (Methods). For strain rates(3.1–3.7) 3 10213 s21 and representative values of temperature andpressure in the Hawaiian plume head, the experimental rheologicallaw predicts a viscosity gexp 5 (3.2–15) 3 1018 Pa s, a factor 1.6–10.3lower than gmin.

An alternative mineralogical model for the Hawaiian plume positsthat a substantial portion of the source region of Hawaiian lavas isentirely free of olivine, being dominated instead by clinopyroxene28.It is therefore of interest to compare the viscosity inference from ourmodel with the predictions of laboratory deformation experiments on

natural clinopyroxene aggregates4, for which n 5 4.7. The solutions ofthe lubrication equation with n 5 4.7 that best fit the Hawaiian residualtopography are shown in Supplementary Fig. 3. The maximum strainrate is _emax~ 6:3{7:6ð Þ| 10{13 s{1, which corresponds to a viscositygmin 5 (1.0–1.5) 3 1019 Pa s. For the same values of temperature andpressure as previously, the experimental rheological law for clinopyr-oxene (Methods) predicts a viscosity gexp 5 (3.4–7.5) 3 1018 Pa s, afactor of 1.3–4.4 lower than gmin. Our model predictions thus agreeslightly better with the laboratory data for clinopyroxene than withthose for olivine. However, the error bars are large because the applic-ability of laboratory-based rheological laws to mantle conditions isdifficult to demonstrate. We conclude that our inferred rheology isprobably consistent with a source comprising either olivine-rich orclinopyroxene-rich components, or both.

METHODS SUMMARYThe lubrication model with constant s was validated against a three-dimensionsalnumerical code for convection in a fluid with viscosity dependent on temperature,pressure and strain rate (Supplementary Figs 1 and 2). The lubrication equation wassolved numerically using an explicit finite-difference method. The scales L0 and S0

that yield the best fit of a given dimensionless solution of the lubrication equation tothe residual topography data were determined by maximizing the variance reduction.For non-steady-state solutions, the time-varying flux Q(t) was also adjusted iterativelyto obtain the best possible fit between the cross-sectional area distribution A(x) of theswell predicted by the lubrication solution and that estimated from the observations.

Full Methods and any associated references are available in the online version ofthe paper at www.nature.com/nature.

Received 14 September 2010; accepted 7 March 2011.

Published online 11 May 2011.

1. Poirier, J.-P. Creep of Crystals (Cambridge University Press, 1985).2. Karato, S.-I. Deformation of Earth Materials (Cambridge University Press, 2008).

300

600

900

1,200

m

0

165° W

160° W

155° W

150° W

20° N

25° N

30° N n = 1, constant flux n = 3.5, constant flux

n = 3.5, variable flux

0.8

0.9

1.0

A0.4

0.6

0.8

1.0Q

0.8

0.9

1.0

0.4

0.6

0.8

1.0

n = 1, variable flux

a b

c d

Q

A

Figure 3 | Comparison of the lubrication model predictions with theresidual topography of the Hawaiian swell. Data are shown as obliqueMercator projections. The solid lines are smoothed contours of the residualtopography for 0 m (white) and 700 m (black). The green curve bounds theareas excluded from the analysis (Methods). Colours show the solutions of thelubrication equation for diffusion creep (n 5 1; a and c), dislocation creep(n 5 3.5; b and d), constant plume flux (a and b) and variable plume flux (c and

d) that best fit the residual topography (Methods). The 700 m contour of eachsolution is shown by a dotted black line (compare with the solid black line). Theflux Q(t 5 2x/U) and the swell’s cross-sectional area A(x) for the two variable-flux cases are also shown below, each normalized to its value at the left of thefigure. The diamond symbols are estimates (similarly normalized) of A(x)obtained from the residual topography18.

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3. Keefner, J. W., Mackwell, S. J., Kohlstedt, D. L. & Heidelbach, F. Dependence of thecreep of dunite on oxygen fugacity: implications for viscosity variations in Earth’smantle. J. Geophys. Res. doi:10.1029/2010JB00748 (in the press).

4. Bystricky, M. & Mackwell, S. Creep of dry clinopyroxene aggregates. J. Geophys.Res. 106, 13443–13454 (2001).

5. Bai, Q., Mackwell, S. & Kohlstedt, D. L. High-temperature creep of olivine singlecrystals. 1. Mechanical results for buffered samples. J. Geophys. Res. 96,2441–2463 (1991).

6. Peltier, W. R. Glacial isostatic adjustment. 2: Inverse problem. Geophys. J. R. Astron.Soc. 46, 669–705 (1976).

7. Hager, B. H., Clayton, R. W., Richards, M. A., Comer, R. P. & Dziewonski, A. M. Lowermantle heterogeneity, dynamic topography and the geoid. Nature 313, 541–545(1985).

8. Mitrovica, J. X. & Forte, A. M. A new inference of mantle viscosity based upon jointinversion of convection and glacial isostatic adjustment data. Earth Planet. Sci. Lett.225, 177–189 (2004).

9. Karato, S. Jung. H., Katayama, I. & Skemer, P. Geodynamic significance of seismicanisotropy of the upper mantle: new insights from laboratory studies. Annu. Rev.Earth Planet. Sci. 36, 59–95 (2008).

10. Montagner, J. P. & Tanimoto, T. Global upper mantle tomography of seismicvelocities and anisotropies. J. Geophys. Res. 96, 20337–20351 (1991).

11. Long, M. D. & Silver, P. G. Shear wave splitting and mantle anisotropy:measurements, interpretations, and new directions. Surv. Geophys. 30, 407–461(2009).

12. Wilson, J. T. A possible origin of the Hawaiian islands. Can. J. Phys. 41, 863–870(1963).

13. Morgan, W. J. Convection plumes in the lower mantle. Nature 230, 42–43 (1971).14. Montelli, R. et al. Finite-frequency tomography reveals a variety of plumes in the

mantle. Science 303, 338–343 (2004).15. Wolfe, C. J. et al. Mantle shear-wave velocity structure beneath the Hawaiian hot

spot. Science 326, 1388–1390 (2009).16. Watts, A. B. & Cochran, J. R. Gravity anomalies and flexure of the lithosphere along

the Hawaiian-Emperor Seamount Chain. Geophys. J. R. Astron. Soc. 38, 119–141(1974).

17. Tarduno, J. A. et al. The Emperor seamounts: southward motion of the Hawaiianhotspot plume in Earth’s mantle. Science 301, 1064–1069 (2003).

18. Davies, G. F. Temporal variation of the Hawaiian plume flux. Earth Planet. Sci. Lett.113, 277–286 (1992).

19. Olson, P. in Magma Transport and Storage (ed. Ryan, M.) 33–51 (John Wiley, 1990).20. Ribe, N. M. & Christensen, U. Three-dimensional modelling of plume-lithosphere

interaction. J. Geophys. Res. 99, 669–682 (1994).

21. Marks, K. M. & Sandwell, D. T. Analysis of geoid height versus topography foroceanic plateausandswellsusing nonbiased linear regression. J.Geophys.Res.96,8045–8055 (1991).

22. Cserepes, L., Christensen, U. & Ribe, N. M. Geoid height versus topography for aplume model of the Hawaiian swell. Earth Planet. Sci. Lett. 178, 29–38 (2000).

23. van Hunen, J., Zhong, S., Shapiro, N. M. & Ritzwoller, M. H. New evidence fordislocation creep from 3-D geodynamic modeling of the Pacific upper mantlestructure. Earth Planet. Sci. Lett. 238, 146–155 (2005).

24. Sleep, N. H. Hotspots and mantle plumes: some phenomenology. J. Geophys. Res.95, 6715–6736 (1990).

25. Ribe, N. M. & Christensen, U. The dynamical origin of Hawaiian volcanism. EarthPlanet. Sci. Lett. 171, 517–531 (1999).

26. Hirth, G. & Kohlstedt, D. L. Water in the oceanic upper mantle: implications forrheology,melt extraction and the evolution of the lithosphere.Earth Planet. Sci. Lett.144, 93–108 (1996).

27. Karato, S.-I. Insights into the nature of plume-asthenosphere interaction fromcentralPacific geophysicalanomalies.EarthPlanet. Sci. Lett.274,234–240 (2008).

28. Sobolev, A. V., Hofmann, A. W., Sobolev, S. V. & Nikogosian, I. K. An olivine-freemantle source of Hawaiian shield basalts. Nature 434, 590–597 (2005).

29. Smith, W. H. F. & Sandwell, D. T. Global seafloor topography from satellite altimetryand ship depth soundings. Science 277, 1956–1962 (1997).

30. Stein, C. & Stein, S. A model for the global variation in oceanic depth and heat flowwith lithospheric age. Nature 359, 123–129 (1992).

Supplementary Information is linked to the online version of the paper atwww.nature.com/nature.

Acknowledgements We thank A. Davaille, C. Herzberg, S.-I. Karato and D. Kohlstedt fordiscussions and advice. Thiswork was supported by theFrenchembassy inTehran andby the SEDIT programme of INSU and the ANR (grant PTECTO) in France.

Author Contributions N.A. derived the lubrication equation, determined the similaritysolution and the full numerical solutions of that equation, and analysed the topographydata. N.M.R. proposed the idea for the study, determined the three-dimensionalnumerical solutions with temperature-dependent rheology, and wrote the manuscript.F.S. co-directed the parts of the work done in Zanjan. All authors discussed the resultsand commented on the manuscript.

Author Information Reprints and permissions information is available atwww.nature.com/reprints. The authors declare no competing financial interests.Readers are welcome to comment on the online version of this article atwww.nature.com/nature. Correspondence and requests for materials should beaddressed to N.A. (n_asaadi@iasbs.ac.ir) or N.M.R. (ribe@fast.u-psud.fr).

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METHODSDerivation and solution of the lubrication equation. In the following, the sub-scripts a and b take on the values 1 or 2, corresponding to vector components inthe x- and y-directions, respectively.

Because the plume head is thin, the pressure inside it is nearly hydrostatic. Itsvalue relative to the pressure outside is:

p~gdr h{zð Þ ð5Þ

where g is the gravitational acceleration and 2dr is the density deficit of the plumerelative to the ambient mantle. The horizontal flow in the plume head is aPoiseuille flow driven by the lateral gradient of the pressure. In the lubricationapproximation, the horizontal velocity components ua satisfy:

Lz gLzuað Þ~Lap:gdrLah ð6Þ

where the viscosity g is given by equation (1). The solution of equation (6) forwhich the slip ua 2 Ud1a at the plume head’s upper surface z 5 0 and the sheartraction ghzua at its lower surface z 5 h both vanish is:

ua~Ud1a{D

nz1gdrð Þn +hj jn{1 hnz1{ h{zð Þnz1� �

Lah ð7Þ

where d1a is the Kronecker delta symbol.Conservation of mass over the whole thickness of the plume head requires:

LthzLb

ðh

0ubdz~

Qpa2

exp {r2�

a2�

ð8Þ

where the source of buoyant fluid has been represented by a radially symmetricGaussian distribution of width a with r2 5 x2 1 y2, and summation over therepeated subscript b is understood. Substitution of equation (7) into equation(8) finally yields:

LthzULxh~+: s +hj jn{1+ hnz3� � �

zQ

pa2exp {r2

�a2

� ð9Þ

where s 5 D(gdr)n/[(n 1 2)(n 1 3)].Equation (9) was prepared for numerical solution by rewriting it in terms of the

dimensionless variables hUL0/Q, x/L0, y/L0 and tU/L0. Solutions were obtainedusing an explicit finite-difference method. The gravitational spreading term (firstterm on the right-hand side) was discretized using a centred difference schemewith fourth-order accuracy. The advection term Uhxh was discretized usingsecond-order accurate upwind differencing. Time stepping was performed expli-citly using a second-order accurate midpoint method.

Far downstream from the hotspot, the source term in equation (9) is negligible,and gravitational spreading is primarily in the y direction. The steady-state form ofequation (9) then takes the form:

ULxh~s nz3ð ÞLy hnz2 Lyh�� ��n{1Lyh

�ð10Þ

where s has been assumed constant. Conservation of the downstream volume fluxrequires:

UðL

{Lh dy~Q ð11Þ

where L(x) is the half-width of the plume head. Equations (10) and (11) admit asimilarity solution of the form:

h~Q

UL xð ÞH fð Þ ð12Þ

where f~y

L xð Þ .The appropriate boundary conditions are:

0~L x0ð Þ~H’ 0ð Þ~H 1ð Þ ð13Þ

where x0 is the virtual position of the hotspot as seen by the solution far down-stream and the prime indicates d/df. Substitution of equation (12) into equations(10) and (11) yields ordinary differential equations for L(x) and H(f) that can besolved subject to the conditions (13) to yield:

H fð Þ~c1 1{ fj j1zn

n

� � n2nz1

ð14Þ

and

L~c2L0x{x0

L0

� � 13nz2

ð15Þ

where

c1~C 5n2z5nz1

nz1ð Þ 2nz1ð Þ

�

2C 2nz1nz1

�C 3nz1

2nz1

� ð16Þ

c2~ nz3ð Þ 3nz2ð Þ nz12nz1

� �n

c2nz11

� 13nz2

ð17Þ

and C is the Gamma function. For n 5 3.5, c1 < 0.6775 and c2 < 0.9432. Thevirtual hotspot position x0 is determined by least-squares fitting of equation(15) to the full numerical solution of equation (9), yielding x0/L0 5 0.753 forn 5 1 and 0.272 for n 5 3.5.Validation of the lubrication model. Although the gravitational spreading para-meter s depends on temperature and pressure in general, we treat it as constant inthis study. Here we justify this assumption using a three-dimensional numericalcode20,31 for convection in a fluid whose viscosity g depends on temperature,pressure and strain rate according to:

g~gminz g{1maxzg{1

1

� {1, ð18Þ

g1~g0_e1

_e0

� �1{nn

expEzpV

nRT

� �, ð19Þ

_e1~ I2zI2min

� 1=2 ð20Þ

where I~ 2eijeij� 1=2

is the second invariant of the strain rate tensor eij. In equa-tions (18) to (20), g0 5 1021 Pa s is a reference viscosity, and gmin 5 0.001g0 andgmax 5 500g0 are the minimum and maximum allowable viscosities, respectively._e0~2|10{15 s{1 is a reference strain rate, and Imin 5 10215 s21 is the minimumallowable value of I. Finally, T is the absolute temperature, p is the pressure, E is theactivation energy, V is the activation volume, and R is the universal gas constant.

The equations of conservation of mass, momentum and energy are solved usinga hybrid spectral (in the two horizontal directions) and finite-difference (in thevertical direction) method in which the coupling of different spectral modes bylateral viscosity variations is calculated iteratively20,31. A thermal plume is gener-ated by imposing a hot temperature anomaly DTmaxexp(2r2/a2) on the bottom ofthe model box (depth d 5 400 km). The thermal plume interacts with the shearflow generated by motion of the upper surface of the model box at a constantvelocity U 5 2.7 3 1029 m s21 (8.6 cm yr21). Effects of melting-induced deple-tion25 are neglected for simplicity.

Supplementary Fig. 1a shows the steady-state temperature field for a plume withbuoyancy flux B 5 2,340 kg s21 and DTmax 5 200 K in a fluid with diffusion creep(n 5 1) rheology with E 5 400 kJ mol21 and V 5 6.1 3 1026 m3 mol21. To com-pare this three-dimensional solution quantitatively with the lubrication model, weuse the former to calculate the isostatic topography:

Siso~r0a

r0{rw

ðd

0dT dz ð21Þ

where dT(x, y, z) is the temperature anomaly (temperature with the plume presentless the temperature in a second solution withDTmax 5 0) and a 5 3.5 3 1025 K21

is the coefficient of thermal expansion. Supplementary Fig. 1b shows Siso on thecentral axis y 5 0 as a function of x 2 x0 with x0 5 163 km (black line), togetherwith the power-law decay of equation (4) with n 5 1 that best fits it (red line). Thedownstream decay of the topography agrees closely with the prediction Siso /(x 2 x0)20.2 of the lubrication model.

Supplementary Fig. 2 is for a plume with the same values of B,DTmax, E and V asin Supplementary Fig. 1, but with a dislocation creep rheology (n 5 3.5). Again, thetopography decays downstream in accordance with the lubrication predictionSiso / (x 2 x0)20.08.

Supplementary Figs 1 and 2 show that the lubrication model accurately repro-duces the dynamics of the more complicated three-dimensional code for bothdiffusion creep (n 5 1) and dislocation creep (n 5 3.5) rheologies, even in thepresence of strongly temperature-dependent viscosity. The physical reason forthis is that the largest temperature variations in the three-dimensional code areconfined near the upwelling plume conduit, so that lateral variations of temper-ature throughout most of the plume head are relatively small (Supplementary Figs1a and 2a). A lubrication model with constant s can therefore be used instead ofthe full three-dimensional code, which, because of its high computational cost andmultiplicity of parameters, would render intractable the task of finding the modelwith the optimal fit to the residual topography data.Comparison of the model predictions with the residual topography. The firststep is to exclude those portions of the residual topography that are unrelated tothe swell, which include the Hawaiian islands, the surrounding moat produced bylithospheric flexure, and the nearby Mid-Pacific mountains. These areas areenclosed by the green lines in Fig. 3. We also excluded all topography outsidethe range 0–1,300 m.

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Once a numerical solution of equation (9) was found for a given value of n, weused a least-squares procedure (maximization of variance reduction) to determinethe values Lbf

0 and Sbf0 of the length scale L0 and the topography scale S0 such that the

re-dimensionalized solution best fits the non-excluded residual topography. For asteady-state solution with constant Q, the determination is direct. For non-steady-state solutions in which Q is a function of time, we determined Lbf

0 and Sbf0 iteratively

by adjusting the shape of the function Q(t) to obtain the best possible agreementbetween the cross-sectional area distribution A(x) of the swell topography predictedby the numerical model and that estimated directly from the residual topographydata18. An iterative procedure is required because the volume flux Q(t) of theHawaiian plume varies on a timescale that is comparable to the fluid-mechanicaladjustment time L0/U, which implies that the cross-sectional area of the swell at agiven distance x from the hotspot is not simply proportional to the plume flux at theearlier time t 5 2x/U when the cross-section was directly over the hotspot. For thenon-steady lubrication solution of Fig. 3d, Lbf

0 ~630 km and Sbf0 ~1,375 m.

Given Lbf0 and Sbf

0 , the buoyancy flux B and the rheological prefactor D of thebest-fitting lubrication solution were found by solving the two simultaneous equa-tions L0~Lbf

0 and S0~Sbf0 , where L0 and S0 are defined by equations (2) and (3)

respectively. The results are:

B~Sbf0 Lbf

0 r0{rwð ÞU ð22Þ

D~2znð Þ 3znð Þ Lbf

0

�g

� ndr1znU

Sbf0 r0{rwð Þ

� �1z2n ð23Þ

To evaluate equations (22) and (23) we assumed U 5 2.7 3 1029 m s21 anddr 5 r0aDT, where a 5 3.5 3 1025 K21 and DT 5 125–150 K is a typical excess

temperature in the plume head far from the hotspot, based on a maximum tem-perature contrast (250–300 K) of the upwelling material beneath the hotspot32.Comparison of the model predictions with laboratory experiments.Laboratory deformation experiments in the dislocation creep regime typicallyyield rheological laws of the form:

_e~Af mtn exp {EzpV

RT

� ð24Þ

where A is a constant and f is the oxygen fugacity. To evaluate equation (24)within the Hawaiian plume head downstream from the hotspot, we assumep 5 3.3 GPa (corresponding to a depth <100 km) and T 5 1,748–1,773 K, corres-ponding to an excess temperature DT 5 125–150 K relative to an ambientmantle temperature 1,350uC. For dry olivine aggregates, m 5 0.2, n 5 3.59,A~1:15|10{19 s{1 Pa{n{m, E 5 449 kJ mol21 (ref. 3), and V 5 1.7–2.8 3 1025

m3 mol21 (refs 33 and 34). In addition, we assume f [ 10{6:2,10{2:1½ � Pa (ref. 3). Fordry natural clinopyroxene aggregates, m 5 0, n 5 4.7,A~4:0|1019 s{1 Pa{n, andE 1 pV 5 760 kJ mol21 for an average confining pressure P 5 425 MPa (ref. 4).Because V is poorly known for clinopyroxene, we assume the same range of valuesas for olivine.

31. Christensen, U. & Harder, H. 3-D convection with variable viscosity. Geophys. J. Int.104, 213–226 (1991).

32. Herzberg, C. & Asimow, P. D. Petrology of some oceanic island basalts:PRIMELT2.XLS software for primary magma calculation. Geochem. Geophys.Geosyst. 9, Q09001 (2008).

33. Kirby, S. H. Rheology of the lithosphere. Rev. Geophys. 21, 1458–1487 (1983).34. Borch, R. S. & Green, H. W. II. Deformation of peridotite at high pressure in a new

molten salt cell: comparison of traditional and homologous temperaturetreatments. Phys. Earth Planet. Inter. 55, 269–276 (1989).

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