Geochemistry Volume 6 Geophysics Geosystems · lithosphere and may be the result of magma...

Plume-ridge interaction, lithospheric stresses, and the originof near-ridge volcanic lineaments

Eric Mittelstaedt and Garrett ItoSchool of Ocean and Earth Science and Technology, University of Hawai’i, 1680 East-West Road, Honolulu, Hawaii96822, USA ([email protected])

[1] In many hot spot–ridge systems, formation of off-axis volcanic chains or lineaments occurs betweenthe hot spot centers and nearby mid-ocean ridges. In some cases, such as Galapagos, Kerguelen, andpossibly the Mid-Pac Mountains and Tristan, these lineaments appear to meet in a focus zone near the hotspot and fan outward in the direction of the ridge axis. The origins of these lineaments are not well knownand do not easily fit into typical conceptual models of ridge or hot spot volcanism. It has been proposed forthe Galapagos region that such lineaments are caused by channeling of hot asthenosphere from off-axismantle plumes toward mid-ocean ridges, where enhanced volcanism initiates island formation (Morgan,1978). Alternatively, other workers suggest that these lineaments are controlled by patterns of stress in thelithosphere. We examine this latter hypothesis by considering the effects of buoyant uplift andasthenospheric shear on the base of the lithosphere induced by an expanding mantle plume. Using thinplate theory, we calculate the two-dimensional (plan view) pattern of depth-integrated stresses in a plate ofvarying thickness. Both a straight, continuous ridge and a ridge-transform-ridge system are simulated by adisplacement discontinuity, boundary element model. Ridge and transform segments have imposed normaland zero shear tractions, and we test different idealized tectonic or far-field stress conditions. Assumingthat volcanism is promoted by lithospheric tension and aligns along lineaments parallel to trajectories ofleast tensile stress, calculations reproduce the rough fan-shaped pattern of lineaments between theGalapagos Archipelago and the Galapagos Spreading Center. The focus of the fan shape is over the plumecenter when it interacts with a straight ridge, but the fan focus appears closer to a segmented ridge, offsetby a transform fault, if the ridge segments are more tensile than the far-field stress. This condition providesa plausible explanation for the apparent focus at the Galapagos north of the hot spot center. The width ofthe fan pattern along the ridge axis is predicted to increase as the plume-ridge separation increases and asthe far-field stress becomes more tensile parallel to the ridge. The observed width along the GalapagosSpreading Center is consistent with a nearly isotropic remote stress. Models predict the lithospheric tensioncaused by plumes to promote lineament formation on only young lithosphere. In support of this prediction,a compilation of data from 23 hot spots shows that lineament formation is common near hot spots withplume-ridge separations of <1250 km or on lithosphere of <25 Ma, but not on older lithosphere.

Components: 11,534 words, 10 figures, 1 table.

Keywords: Galapagos; hot spot; lithospheric stress field; plume-ridge interaction; volcanic lineaments.

Index Terms: 3035 Marine Geology and Geophysics: Midocean ridge processes; 3037 Marine Geology and Geophysics:

Oceanic hotspots and intraplate volcanism; 8164 Tectonophysics: Stresses: crust and lithosphere.

Received 6 October 2004; Revised 24 February 2005; Accepted 16 March 2005; Published 1 June 2005.

Mittelstaedt, E., and G. Ito (2005), Plume-ridge interaction, lithospheric stresses, and the origin of near-ridge volcanic

lineaments, Geochem. Geophys. Geosyst., 6, Q06002, doi:10.1029/2004GC000860.

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Published by AGU and the Geochemical Society

AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES

GeochemistryGeophysics

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Volume 6, Number 6

1 June 2005

Q06002, doi:10.1029/2004GC000860

ISSN: 1525-2027

Copyright 2005 by the American Geophysical Union 1 of 20

1. Introduction

1.1. Mantle Plume-Ridge Interaction

[2] A well studied form of magmatic processinvolves hot spot–ridge interaction. Current inter-actions between at least 21 hot spots and nearbyridges produce geophysical and geochemicalanomalies along 15–20% of the global mid-oceanridge network [Ito et al., 2003]. Geochemicalanomalies show variations in noble gas and isotoperatios and trace element concentrations [Detrick etal., 2002; Hanan et al., 1986, 2000; Ito et al., 2003;Keller et al., 2000; Sinton et al., 2003; Taylor et al.,1995]. Geophysical observations demonstrate thepresence of elevated topography, negative gravityanomalies [Ito and Lin, 1995; Nadin et al., 1995;Olson, 1990; Richards et al., 1988], and anomalouscrustal production [Ito et al., 2003; Sinton, 1992;White et al., 1992]. Together, these observations notonly reveal the importance of hot spot–ridge inter-action on the structure and composition of theoceanic lithosphere, but they also support the notionthat many of these systems involve interaction withmantle plumes.

[3] Manifestations of plume-ridge interaction arefound across the ocean basins. Asymmetric spread-ing and ridge reorientations at many hot spot–ridgesystems including Iceland, Kerguelen, the Galapa-gos, Shona and Louisville [Hardarson et al., 1997;Small, 1995; Wilson and Hey, 1995] suggestchanges in large scale plate shape and plate motion[Muller et al., 1998]. Also, formation of volcaniclineaments between off-axis hot spot centers andnearby ridges leads to the creation of new islandsand seamounts [Harpp et al., 2003; Harpp andGeist, 2002]. The origin of these lineaments is apoorly understood expression of hot spot–ridgeinteraction and their presence provides an oppor-tunity to extend our general understanding ofasthenosphere-lithosphere dynamics.

1.2. Near-Ridge Lineaments

[4] At many hot spot–ridge systems, volcaniclineaments extend from off-axis hot spots to nearbymid-ocean ridges (Figure 1). Examples includeLouisville [Lonsdale, 1988; Small, 1995; Vlastelicet al., 1998], Kerguelen [Small, 1995], Reunion[Dyment, 1998; Morgan, 1978], Tristan de Cuhna,Musicians [Kopp et al., 2003], the Galapagos[Harpp et al., 2003; Harpp and Geist, 2002;Morgan, 1978; Sinton et al., 2003], the LineIslands (Mid-Pac Mountains), and possibly theDiscovery and Shona hot spots [Small, 1995].

Morphologies range from continuous ridges atRodrigues, Hollister, and Genovesa ridges, toaligned but distinct seamounts and islands at theWolf-Darwin lineament and Tristan de Cunha.Lineaments show arcuate to nearly linear patternsthat, in the cases of Galapagos [Sinton et al., 2003],Kerguelen and possibly the Line Islands (Mid-PacMountains) and Tristan, fan out toward ridges froma focus zone near the hot spot (Figure 1). Mostlineaments of this type occur on young, weaklithosphere and may be the result of magmaexploiting the lithospheric stress pattern associatedwith plume-ridge interaction [Harpp et al., 2003;Harpp and Geist, 2002; Sinton et al., 2003].

1.3. Galapagos Lineaments

[5] Recent studies have focused on the origin andcharacteristics of off-axis lineaments found nearthe Galapagos Archipelago [Harpp et al., 2003;Harpp and Geist, 2002]. We focus on the Galapa-gos region because extensive morphological[Sinton et al., 2003], geochemical [Cullen andMcBirney, 1987; Detrick et al., 2002; Geist etal., 1986, 1999; Harpp et al., 2003; Harpp andGeist, 2002; Schilling et al., 2003; Sinton et al.,1996], and geophysical investigations [Canales etal., 1997; Feighner and Richards, 1994; Ito etal., 1997; Schubert and Hey, 1986; Werner et al.,2003; Wilson and Hey, 1995] provide better con-straints on models of lineament formation thancurrently possible at other hot spot systems.

[6] A series of approximately seven volcanicchains arrayed in a distinctly fan-shaped pattern,emanate from a focal zone just north of SantiagoIsland [Sinton et al., 2003] toward the GalapagosSpreading Center (GSC) between �92�300W and89�W (Figure 1a). The majority of these volcaniclineaments are concave toward the GSC and curveto meet the ridge at nearly right angles both to thewest and east of the large (�100 km offset)transform at 91�W [Sinton et al., 2003]. Manyresearchers have speculated as to the origin of themost prominent of these volcanic chains, the Wolf-Darwin Lineament (WDL). Originally, Morgan[1978] proposed that Wolf and Darwin Islands nearGalapagos and Rodrigues Island near Reunion areexamples of ‘‘a second type of hot spot island,’’formed by an asthenospheric channel connectingthe off-axis hot spots to the ridge axis. Enhancedvolcanism at the intersection of the channel and theridge creates seamounts and islands which aresubsequently rafted away by plate motion. Thishypothesis predicts that ages along the lineamentsshould increase away from the ridge, equal to the

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age of the underlying crust. Volcanic rocks sam-pled on the WDL, however, do not reveal thissimple age progression and lineament ages are 1–5 m.y. younger on the WDL [Sinton et al., 1996;White et al., 1993] and 10–15 m.y. younger on

Rogrigues [Bonneville et al., 1988] than Morgan[1978] predicts. The gravity analysis of Feighnerand Richards [1994] suggests that the WDL occursnear the boundary of a discontinuity in effectiveelastic plate thickness where a lithospheric fault

Figure 1. Volcanic lineaments are seen at several hot spots, including (a) the Galapagos, (b) Kerguelen, (c) LineIslands/Mid-Pacific Mountains, (d) Reunion, (e) Tristan de Cuhna, and (f) Louisville (white dashed lines and ellipseindicate possible hot spot track and current plume center locations). Predicted bathymetry from satellite altimetry[Smith and Sandwell, 1997]. Seamount ages, plume locations, and plate reconstructions from Coffin et al. [2002],Davis et al. [2002], Geli et al. [1998], Muller et al. [1997], O’Connor and Roex [1992], O’Connor and Duncan[1990], and Wessel and Kroenke [1998]. Black circles indicate the approximate center of volcanism at the labeledtimes (Ma). Seafloor isochrons show approximate mid-ocean ridge geometries at the labeled times. Plume-ridgeseparation is the distance between the circles and the ridge at the appropriate age.


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may have occurred allowing magma to reach thesurface.

[7] Influence of the 91�W transform fault onlineament formation is suggested by the system-atic decrease in lineament trend [Sinton et al.,2003] and decrease in curvature as the linea-ments approach the transform. Harpp and Geist[2002] and Harpp et al. [2003] hypothesize thatthe WDL and Genovesa ridge are consequencesof lithospheric tension associated with the 91�Wtransform fault along the GSC. The transformmodel proposed by Harpp and Geist [2002],however, predicts a lineament curvature oppositeto that observed and increasing curvature nearthe transform corner [see Gudmundsson, 1995].Alternatively, Sinton et al. [2003], suggest thatplate parallel gravitational stresses due to litho-spheric uplift from an impinging plume willproduce a radial pattern of least tension promot-ing radial dike orientations [Ernst and Buchan,1997]. This hypothesis predicts the radiatingvolcanic lineaments to be straight rather thancurved and, as noted by Sinton et al. [2003],predicts them to radiate from the center of theplume whereas the Galapagos lineaments radiatefrom an area on the northern edge of thearchipelago. Both the prediction of transform-induced stresses and plume-induced stressesindividually fail to adequately explain the forma-tion of the Galapagos lineaments. We proposethat lithospheric stresses due to the combinedeffects of the plume and segmented ridge canexplain the general lineament pattern, focuslocation, and decreasing curvature near the trans-form fault.

[8] The goal of this paper is to explore how hotspot–ridge interaction can influence the litho-spheric stress field and thus the pattern ofvolcanic lineaments. Our study includes quanti-tative tests of the hypotheses of Harpp et al.[2003], Harpp and Geist [2002] and Sinton et al.[2003]. We assume that the aforementioned hotspot–ridge systems involve buoyant astheno-sphere rising and spreading beneath the litho-sphere. In this context we refer to theseinteractions as ‘‘plume’’-ridge interactions. Wecalculate two-dimensional (2-D), plan view,depth-integrated stresses in a plate of varyingthickness subject to loads due to plume shear,plate parallel gravitational body forces andboundary tractions along both a straight ridgeand a ridge-transform-ridge system. On the basisof the model results, we address the implicationsfor near, but off-ridge volcanism for the partic-ular case of the Galapagos as well as otheroceanic hot spots.

2. Conceptual and Mathematical Model

2.1. Conceptual Model

[9] Figure 2 illustrates our conceptual model offorces involved in plume-ridge interaction. Weapproximate the lithosphere as an elastic plateand treat the zone of rifting near the ridge axisand the region of strike-slip motion along trans-form faults as an internal boundary. We refer to thisregion of nonelastic deformation along the ridge asthe plate boundary zone, PBZ (Figure 2). Thepattern of stress in the plate is a sum of all forces

Figure 2. Conceptual model. Red line along the ridge axis represents the plate boundary zone (PBZ) of magmatismand nonelastic deformation, with arrows showing sense of motion across the transform fault. Anomalously buoyant(mantle plume) asthenosphere uplifts the plate, causing plate parallel gravitational forces (qg). Large arrows showflow of buoyant asthenosphere, which introduces shear along the base of the plate qs. Rp denotes extent over whichplume flow is assumed to remain nearly radial. At radii greater than this, plume flow deviates due to plate motion.Both plume uplift and asthenospheric shear introduce tension in the plate.



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acting on the plate. These forces include tractionsalong the PBZ and the distal plate boundaries, thepull of gravity on surface topography (whichincludes the deepening with seafloor age) as wellas asthenospheric shear on the base of the plate[Forsyth and Uyeda, 1975]. Indeed the state ofstress near mid-ocean ridges varies from region toregion, ridge to ridge (J. Reinecker et al., The 2004release of the World Stress Map, available at http://www.world-stress-map.org). To make our modelsgeneral and to focus on the local effects of plume-ridge interaction, we simplify the effects of all ofthe distal forces as a uniform far-field stress andthen calculate the perturbations caused by plume-ridge interaction. First, we consider a case ofisotropic far-field stress equal in magnitude to thenormal stress (either tensile or compressive) on theridge. Ridge-normal compression could be causedby the excess morphological high at the ridge axiswhile ridge-normal tension could be caused byseafloor spreading and is evident by extensivenormal faulting in the PBZ. Second, we illustratethe effects of a nonisotropic far-field stress. Finally,we consider a PBZ stress that is more tensile thanthe far-field isotropic stress.

[10] Locally, the stress pattern near a plume-affected ridge will be perturbed by plume-inducedstresses. The plume-induced loads on the plate areexcess asthenospheric shear (qs, bold terms denotevector quantities), caused by buoyant astheno-spheric material (mantle plume) spreading radiallybeneath the lithosphere, and the pull of gravity (qg)down the slope of the anomalously upliftedlithosphere [Westaway, 1993]. The combinationof these loads creates horizontally varying litho-spheric stresses. We assume magma will penetratethe lithosphere and erupt along lineaments parallelto the direction of least tensile resultant stress, asintegrated through the thickness of the plate.

2.2. Mathematical Method

[11] The lateral extent of the lithosphere is largecompared to its thickness; therefore we use thinshell theory to develop our 2-D governing equa-tions of an elastic plate. To describe stresses in theplate, we use nonlithostatic, depth-integratedstresses, or stress resultants, Nij (see Appendix A),

Nij ¼Z h

0

sijdx3 ð1Þ

(i, j = 1, 2 for the lateral directions), where sij arestresses throughout the lithosphere, h is lithospherethickness, and x3 is the vertical axis corresponding

to depth in our model where x3 = 0 is the surface ofthe plate (Figure 2).

[12] Strictly, the reference state of the lithosphere isa curved ‘‘shell’’ due to the deepening of mantleisotherms with the square root of distance from theridge, however, scaling arguments show that a flatplate provides a good approximation for our prob-lem (see Appendix A). For a flat plate of uniformthickness, equilibrium is solved by a balance oflaterally varying stress resultants, Nij, and plume-induced loads, which behave mathematically asbody forces (q = qs + qg)

@Nij

@xiþ qj ¼ 0 ð2Þ

(summation over i is implied). The constitutiverelation of stress resultants to strain along the mid-plane of the plate (eij) is given by Novozhilov[1959]

N11

N22

N12

26666664

37777775

¼

Eh 1� v2ð Þ�1Ehv 1� v2ð Þ�1

0

Ehv 1� v2ð Þ�1Eh 1� v2ð Þ�1

0

0 0 Eh 1þ vð Þ�1

26666664

37777775�

e11

e22

e12

26666664

37777775;

ð3Þ

where E is Young’s modulus and n is Poisson’sratio (see Table 1 for values used). The compat-ibility relation is [Novozhilov, 1959]

1� nð Þ @2N11

@x22þ @2N22

@x21

� � n

@2N22

@x22þ @2N11

@x21

� ¼ 2

@2N12

@x1@x2

ð4Þ

and when combined with (2) yields

r2 N11 þ N22ð Þ ¼ � 1

1� vð Þ@q1@x1

þ @q2@x2

� : ð5Þ

[13] Lithospheric cooling increases plate thickness,h, perpendicular to the ridge and ridge segmenta-tion introduces age discontinuities across the frac-ture zones. For laterally varying elastic parameters(Eh(x1, x2)) the equations of equilibrium and com-patibility can be expressed as (see Appendix B)

@sij@xi

þ bj ¼ 0 ð6Þ



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and

1� nð Þ @2s11@x22

þ @2s22@x21

� � n

@2s22@x22

þ @2s11@x21

� ¼ 2

@2s12@x1@x2

;

ð7Þ

respectively. Combining (6) and (7) yields thesingle governing equation

r2 s11 þ s22ð Þ ¼ � 1

1� vð Þ@b1@x1

þ @b2@x2

� : ð8Þ

In (6)–(8), sij (= Nij/h) are depth-averaged stressesand bi are the combined loads due to uplift andplume shear, q, as well as ‘‘fictitious’’ body forceterms arising due to lateral changes in h (seeAppendix B)

bi ¼qi

hþ @h

@xj

sijh

i6¼j

þ @h

@xi

siih: ð9Þ

The form of equations (6)–(8) is identical to theequations of plane stress and allows solutionsto be found using an Airy stress functionformulation.

[14] At straight ridge axes, lithospheric thickness isassumed to vary only perpendicular to the ridgeand as such, the fictitious body forces will onlyinclude terms with gradients of h in the ridgenormal direction (x1). Near a transform, gradients

in thickness parallel and perpendicular to the ridgeaxis will exist. Though (8) and (9) are formulatedin terms of mean stress, sij, we will show results interms of depth-integrated stresses, or stress resul-tants, Nij.

2.3. Plume Forces, Lithospheric Strength,and Plate Boundary Conditions

[15] In our model, loads are introduced by thebuoyant plume asthenosphere and along the ridgePBZ. These loads generate the local stress pattern.The loads introduced by the plume on the litho-sphere, q, include plume shear, qs, and the plate-parallel pull of gravity due to plate uplift, qg. Inorder to calculate q, we approximate the plumespreading beneath the plate as an axisymmetricviscous gravity current.

[16] The flow of plume material near a ridge iscontrolled by plate-driven corner flow, gravity-driven plume expansion and flow along the slopeof the lithosphere [Ribe, 1996]. For simplicity,we restrict consideration to the axisymmetricbuoyant ‘‘self spreading’’ term of the equationdescribed by Ribe [1996]. Thus the radius of ourplume is not the full extent of the plume beneaththe lithosphere or along the ridge axis, but theregion where axisymmetric flow dominates(Figure 2). Outside of this radial zone, plume

Table 1. Model Parameters

Parameter Description Value Units

bi total loads on the plate - NE Young’s Modulus 7 � 109 Pa

eij strain - -

G shear modulus 3 � 1010 Pah plate thickness - mhp plume flow thickness - m

k thermal diffusivity 1 � 10�6 m2/s

L 1/2 width of lineament extent along ridge axis - mNij stress resultants - GN/mqs load due to plume shear - Nqg load due to gravitational body forces - Nq total load on the plate due to the plume - Nr radial distance from plume center - mRp radius of radially dominated plume flow 3.5 � 105 mu velocity of plume spreading - m/sxp plume-ridge separation distance - mx1, x2, x3 coordinate directions - -m viscosity 1018–1019 Pa sDr density deficit of plume material 20 kg/m3

sijaverage stress - Pa

n Poisson’s ratio 0.25 -DNf (Nf11 � Nf22), far-field differential stress - GN/mDNp (Np11 � Np22), plume differential stress - GN/m



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flow departs from axisymmetry, but the flow isslower than in the radial zone and is not con-sidered here.

[17] To compute qs we must determine the verticalvelocity gradient of plume-flow beneath the litho-sphere. Conservation of momentum for a thin layerof fluid yields the solution for radial flow, u, as afunction of radial distance from the center of theflow, r, and depth below the lithosphere (x3 � h)[Huppert, 1982]

u r; x3; tð Þ ¼ � 1

2

Drgm

@hp@r

x3 � hð Þ 2hp � x3 � hð Þ� �

; ð10Þ

where g is the acceleration due to gravity, Dr isthe density contrast between the plume andsurrounding mantle, hp is the thickness of theplume flow and m is dynamic viscosity (Figure 3).

The shear along the base of the plate is found tobe

qs ¼ m@u

@x3

x3�hð Þ¼0

¼ � 1

2Drg

@hp@r

2hp ð11Þ

and is proportional to (hp2/2) (Figure 3). Huppert

[1982] shows that the equations describingconservation of mass and momentum in a thinlayer of fluid can be formulated into a single,second order, ordinary differential equationdescribing the shape of the flow, hp(r) [Huppert,1982, equation (2.25)]. We solve this equationfor hp(r) and @hp/@r (in (10) and (11)) using asecond-order Runga-Kutta method.

[18] Anomalously buoyant plume material beneaththe lithosphere will also produce topographic up-lift, the slope of which causes plate parallel grav-itational forces. If the radius of the plume is largecompared to the flexural wavelength, the height ofisostatic topography, ht, depends on the product ofthe plume thickness, Dr, and the difference be-tween the mantle density, rm, and the ocean den-sity, rw:

ht ¼Dr

rm � rwð Þ hp: ð12Þ

The plate-parallel gravitational body force due toplate uplift (qg) is proportional to ht and the slopeof the lithosphere, @ht/@r,

qg ¼@ht@r

rl � rwð Þgh; ð13Þ

where rl is lithospheric density. The total plumeforce is q = qs + qg (Figure 3).

[19] In order to understand the response of thelithosphere to the imposed forces, we calculatedepth-integrated tensile yield strength (Figure 4).As commonly done, the lithosphere is assumed tobehave in a brittle fashion at shallow depths andtransition to ductile behavior with depth [Buck andPoliakov, 1998; Chen and Morgan, 1990; Shawand Lin, 1996]. We assume a two-layer lithospherewith a top layer of basaltic crust having a lowerductile strength than the underlying peridotitemantle [e.g., Shaw and Lin, 1996]. Thickening ofthe lithosphere is controlled by a half-space coolingmodel [e.g., Parsons and Sclater, 1977]. A coolingplate strengthens with age, but like Small [1995],we predict the integrated strength to increase grad-ually at young plate ages (<�1.8 m.y. for a crustalthickness of 10 km), when the crust is supplyingmost of the strength, and to increase more rapidly

Figure 3. (top) Cross-sectional view of the modelplume flow beneath the lithosphere (here sketched withconstant thickness). Velocity with depth in the plume iszero immediately beneath the lithosphere and increasesto tens of cm/yr at the base of the plume layer. (bottom)On average the plume shear force (qs, gray) isapproximately 40% of the total force (q, black) exertedon the lithosphere due to plume shear and uplift. Largeforces at r = 0 and r = Rp are consequences of theoreticalsingularities in the plume solution and are calculated atthese locations using an approximate solution [Huppert,1982].



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at greater plate ages when the mantle begins tosupply strength (Figure 4). Another interestingprediction is that thicker crust will produce aweaker lithosphere which, in turn, may influencethe maximum plume-ridge separation distancewhere lineament formation is possible. When in-cluding the transform fault in our models, thestrength variations due to the age discontinuityacross the transform and fracture zone are presentfor the entire length of the model.

[20] The boundary conditions include imposedstress resultants in the far field and along the ridgeaxis. The imposed far-field (resultant) stresses maybe tensile or compressive and are assumed to becaused by all of the loads on the plate, except bythe local forces associated with plume-ridge inter-action. The local ridge axis is subject to imposednormal tractions equal to the value of the far-fieldstress perpendicular to the ridge. We assume thatextension and accretion in the PBZ prevent anysubstantial shear stress and therefore impose a zeroshear stress condition at the ridge axis. For asegmented ridge offset by a transform fault, thenormal and zero shear stress conditions are identi-cal along both the ridge and transform segments.Although shear may be present along the transform

at the Galapagos, it does not significantly affect ourmodel solutions.

2.4. Numerical Implementation

[21] Stress resultants are calculated throughout theplate by solving equation (8) with the right handside approximated by point loads. The well knownproblem of calculating stress in an infinite platedue to a directed point force is known as Kelvin’sproblem [Barber, 2002; Crouch and Starfield,1983; Timoshenko and Goodier, 1934]. We use aplane stress solution of Kelvin’s problem to calcu-late the effect of plume shear and uplift at anygiven point in the model space. By summingKelvin solutions due to the appropriate forces ateach location, we simulate the effect of the wholeplume on the lithosphere.

[22] The boundary conditions along the PBZ areimplemented using a displacement discontinuity,boundary element model [Crouch and Starfield,1983]. The boundary elements method is used foreffective modeling of crack-like behavior of theridge. We begin with an idealized case of a straightridge axis and extend the ridge well beyond theregion of interest (i.e., 10 times the plume radius inboth directions) to minimize the influence of theridge ends on the solutions.

[23] With the Kelvin point forces (imposing theloads q) and the ridge boundary condition applied,we solve equation (8) using an Airy stress functionapproach to yield a first approximation to Nij and biwith the ‘‘fictitious’’ body forces initially set tozero. Achieving the final solution requires severaliterations over the whole domain to accuratelysolve for the ‘‘fictitious’’ body forces due to thevarying plate thickness (equation (9) andAppendix B). With each successive iteration, thedifference between the previous and new stressfields diminishes indicating convergence towardthe solution. A solution is assumed to besufficiently accurate when the difference betweensuccessive iterations is less than 0.1% of themaximum stress in the plate.

3. Results

3.1. Straight Ridge

[24] To provide an intuitive understanding of ourresults we show a series of solutions with thedifferent plate loads sequentially added to themodels (Figure 5). For each model, we set a plumevolume flux of 90 m3/s [Ribe and Delattre, 1998],

Figure 4. (top) Lithospheric strength envelopes at 10and 150 km from the ridge axis demonstrate the strengthof the lithosphere with depth for a 7 km thick crust.(bottom) Integrated strength of the lithosphere increaseswith thermal age t of the lithosphere outside of the PBZ.In young lithosphere the ductile strength dominates thelower crust, while in older lithosphere, brittle behavior ismost important. Left side of diagram (t � t0 = 0)corresponds to the edge of the PBZ, assumed to havecooled as an infinite half-space to a thermal age of t0 =0.5 m.y.



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a seafloor spreading rate of 30 km/myr, a radius ofaxisymmetric plume flow of 350 km, and a plumedensity deficit appropriate for a temperature excessfrom local mantle of 200 K. Complete modelparameters and variables are described in Table 1.First, we consider a plate with uniform thickness,no ridge axis, zero far-field stress and subject onlyto the shear due to radial plume expansion. Stressresultants due to plume shear alone show a radialpattern of least tensile stress resultant trajectorieswith most tensile magnitudes being proportional to(hp

2/2) and having a maximum at the plume center(Figure 5a). Trajectories of least tensile stressresultants may control the preferred paths of lateralmagma propagation through the lithosphere (e.g.,dikes will tend to open in the direction of maxi-mum tension) while stress resultant magnitudes arelikely to influence the ability of magma to pene-trate the lithosphere. Plume shear induces �40% ofthe imposed stress due to plume-lithosphere inter-

action (Figure 3). Next we add the effects oftopographic uplift (Figure 5b). Plate parallel grav-itational body forces due to dynamic topographyintroduce the remaining �60% of the stress resul-tant magnitude. The magnitude of plate parallelgravitational forces is proportional to lithosphericthickness and height of uplift (equation (13)). Thethinning of the lithosphere toward the ridge axistherefore produces a nonaxisymmetric pattern ofstress resultants with deviation from the radialpattern greatest where gradients in lithosphericthickness are largest.

[25] Adding the effects of variations in plate thick-ness causes the orientations of least tensile stressresultants to rotate toward the gradient in platethickness (i.e., toward a ridge axis; Figure 5c) andincreases the magnitude of the most tensile stressby �10%. Finally, introduction of an isotropic far-field tension and a ridge boundary, equal to the

Figure 5. Calculated solutions are due to several components of plume-ridge interaction, shown with different plateloads sequentially added to the model. (a) The axisymmetric shear due to plume flow qs, (b) plus the gravitationalbody forces due to dynamic topography qg, (c) plus the effects of varying plate thickness (see equation (9)), (d) plusthe ridge boundary condition. Contours are most tensile stress resultant (GN/m), and ticks are stress trajectories of theleast tensile stress resultant. The plume center is denoted by a black dot. Dotted white lines denote the future locationof the ridge, while the solid white line in Figure 5d denotes the implemented ridge boundary.



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far-field condition, increases tension uniformlythroughout the plate. The zero shear stress (resul-tant) condition on the ridge tends to orient leasttensile trajectories perpendicular to the ridge. In thearea between the plume and the ridge, the stressresultant trajectories form a fan shape with a centerslightly ridgeward of the plume and curve to meetthe ridge at nearly right angles (Figure 5d).

[26] Ultimately, the extent and shape of thetrajectory pattern is produced by the combinationof the above forces. When the differentialstresses caused by the far-field and ridge con-ditions are small, the stress pattern is dominatedby the axisymmetric plume forces and displays afan shape. Such conditions are maintained whilethe differential far-field stress is small or zero,the plume-ridge separation is relatively large(xp/Rp > 0.5), and when the normal stress onthe ridge is equal to the far-field stress (Figure 5).Making the ridge and far-field conditions more

tensile or compressive does not change thepattern of stress trajectories, but simply addsuniformly to the stress field.

[27] Variations in the pattern of least tensile stresstrajectories with xp and with the imposed differen-tial far-field stress resultants (DNf = Nf11 � Nf22;see Figure 6) can be characterized by predictions ofL, the half-width along the ridge axis that linea-ments will intersect the ridge axis at an anglegreater than 45� (Figure 6, inset). Values of Lnormalized by Rp increase with normalized sepa-ration (xp/Rp). At small separations (xp/Rp < �0.4)the increase in width is rapid, at intermediateseparations (�0.4 < xp/Rp < �0.6) the increase ismore gradual, and for larger separations (xp/Rp >�0.6) width increases again rapidly with separa-tion. Increasing the relative far-field tension per-pendicular to the ridge (DNf) roughly preserves thedependence on normalized separation, but shiftsthe curve to smaller normalized widths (crosses,

Figure 6. Model predictions of normalized lineament half-width, L/Rp, increase with normalized plume ridgeseparation, xp/Rp, and differential far-field stress resultant (DNf = Nf11 � Nf22). The predicted pattern of least tensilestress trajectories forms a fan-shaped pattern for xp/Rp greater than 0.5. For smaller plume-ridge separations thetrajectories maintain ridge perpendicular orientations between the ridge and plume center. Solid curves are best fitcubic functions, which are shown to emphasize the general trends of the model predictions.



10 of 20

Figure 6). Decreasing DNf to negative valuesreduces the normalized widths (asterisks, Figure 6).

[28] The dependence of normalized width on thedifferential far-field stress resultant is better illus-trated in Figure 7. Here we show L/Rp at xp/Rp =0.72 and vary DNf normalized by the differentialstress resultants due to plume-induced forces in theabsence of the ridge and far-field conditions (taken30 km south of the ridge axis), DNp. For increasingvalues of DNf/jDNpj, L/Rp decreases until all leasttensile trajectories are ridge parallel at DNf/jDNpj =1.1. This decrease in L/Rp with differential far-fieldstress is well modeled by a quadratic relationshipover the majority of far-field conditions. Withnegative values of DNf/jDNpj, all trajectories even-tually become axis-perpendicular, causing L toincrease rapidly with decreasing DNf/jDNpj anddeviate from the quadratic best-fit line (Figure 7,leftmost points).

3.2. Segmented Ridge

[29] Finally, we test the importance of a transformfault. Transform faults have significant effects onthe near-ridge lithospheric stress field [Behn et al.,

2002; Pollard and Aydin, 1984] and the 91�Wtransform fault along the GSC is an example oflarge scale segmentation that may affect the linea-ment pattern. We model a transform fault 125 kmin length located north of the plume center. Thelengths of the two ridge segments are limited to�400 km which is the approximate distance of thenext set of transform faults of the GSC to the eastand west. Lithospheric thickness is discontinuousacross the transform fault throughout the modeldomain (i.e., we assume the fracture zone extendsacross the whole model).

[30] First we look at the solutions for a ridge withnormal tension equal to the isotropic far-fieldstress. Stress resultants due solely to the far-fieldand ridge boundary conditions are isotropic with-out the plume forces (Figure 8a). With forces dueto a plume of viscosity 1018 Pa s, a focus zone ofleast tensile stress trajectories is seen south of thelower transform corner and ridgeward of the plumecenter, but is poorly defined (Figure 8c). Trajectoryorientations deviate from the general fan-shapedpattern in the inside corner of the transform. Withthe larger plume viscosity of 1019 Pa s, a focus isapparent near the plume center, while the insidecorner region again displays trajectories that devi-ate from the fan-shaped pattern (Figure 8d).

[31] Next we examine solutions where the normaltension along the ridge is greater than the far-fieldstress by 100 GPa m. As in the previous case, wefirst examine the effect of the far-field and ridgeconditions alone. Trajectories radiate from a distinctfocus zone just south of the lower ridge-transformintersection and curve back toward the ridge axis.Near the left (west) side of the transform, the strikeof the stress resultant trajectories is nearly parallelto the transform fault and rotates clockwise withdistance from the transform (Figure 8b). Stressresultant magnitudes are highly tensile close tothe transform, but become compressive south ofthe transform tip. Inclusion of a plume producesstress resultant magnitudes that are more tensilethroughout the plate, reduces the compressive stresssouth of the transform tip, and produces a generalfan-shaped pattern of least tensile stress trajectories(Figures 8d and 8f). With a plume viscosity of1018 Pa s, the apparent trajectory focus zone remainsidentical to that without plume forces, but the stresstrajectories to the left of the transform rotate coun-terclockwise with increasing distance from thetransform; in the opposite sense of the trajectorieswithout the plume forces (Figure 8d). With a largerplume viscosity of 1019 Pa s, the apparent focus

Figure 7. Model predictions of lineament half-width(see Figure 6), L, versus the ratio of the differentialremote stress resultant (DNf = Nf11 � Nf22) and themagnitude of the differential plume stress resultant,DNp, 30 km south of the ridge axis between the plumecenter and ridge in a calculation with Nf11 = Nf22 = 0.The width of the fan-shaped region diminishes as DNf/jDNpj increases. For a plume ridge separation distance,xp/Rp, of 0.7, most of the model predictions are wellapproximated by a quadratic fit, but a cubic fit isnecessary to fit the total range of predictions. Usingmeasurements of the Galapagos lineaments, the stressregime near the GSC falls within the gray box (see textfor details). The black star, red dot, and blue crosscorrespond to symbols in Figure 6.



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shifts to a location close to the plume center withsubtle changes in trajectory orientation near thetransform (Figure 8f). Overall, the inclusion of theridge offset maintains the general fan-shaped patternof least tensile stress trajectories, but deviates fromthe solution due to a straight ridge near the transformfault where a focus zone is observed.

[32] To assess where volcanic lineaments are mostlikely to form we divide the predicted stress resul-tants by the lithospheric yield strength. For bothcases with and without a transform, the fraction ofthe lithospheric yield strength is largest between themodel plume center and the ridge axis, where theplate is youngest. To the south of the model plumecenter the fraction is relatively low (Figures 9b and

9c). Model results therefore show that lineamentformation is most likely to occur between the plumeand ridge axis in a fan-shaped pattern.

4. Discussion

4.1. Galapagos Lineament Pattern

[33] It is well known that dikes and fractures tendto propagate along trajectories of least tensile stress[Ernst and Buchan, 1997; Glen and Ponce, 2002;Muller et al., 2001; Muller and Pollard, 1977;Ode, 1957; Pollard and Aydin, 1988]. Islandsalong the lineaments to the west of the southwardprojection of the 91�W transform fault, displaynorth-northwest trending fractures, whereas east-

Figure 8. (a) Ridge-transform boundary stresses equal to the isotropic far-field stresses produce an isotropic stressfield. (b) Least tensile stress resultant trajectories and magnitudes of most tensile stress resultant due to a segmentedridge show many of the characteristics of the Galapagos lineaments when the ridge is more tensile than the far fieldby 100 GPa m but do not explain the orientation of the Wolf-Darwin lineament. (c and d) Added effects of a plumewith viscosity of 1018 Pa s show features consistent with the Galapagos lineaments when the ridge is more tensilethan the far field, but deviate near the inside transform corner when the ridge and far-field stresses are equal. (e and f)A plume with a viscosity of 1019 Pa s is more consistent with other hot spots, such as Kerguelen. White lines denotethe location of the ridge axis, and black dots at the origin denote the model plume center.



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west trending fractures are found to the east[Cullen and McBirney, 1987; Geist et al., 1986;Harpp et al., 2003; Harpp and Geist, 2002]. Thusthe fractures roughly parallel the lineaments, con-sistent with our notion that volcanic lineamentswill tend to form along trajectories of least princi-ple tension in the whole lithosphere.

[34] The ridge-transform-ridge configuration of themodel shown in Figures 8d and 9b is successful inreproducing the gross pattern of lineaments in theGalapagos region. This model successfully predictslineaments to focus south of the southern trans-form-ridge intersection. Because predicted magni-tudes of integrated stresses are appreciablefractions of the integrated lithospheric strengthonly on the youngest lithosphere, models alsopredict formation of lineaments between the pre-sumed plume center (Fenandina Island) and theGSC. While both the straight and segmented ridgemodels produce the above general characteristics,the segmented ridge does slightly better becausethe transform fault causes the apparent focus zoneto be north of the plume center more consistentwith observations (Figures 8, 9b, and 9c). Withoutthe transform, or for large plume viscosities, thefocus center is nearly at the plume center, similar tomodels that only include plume shear and uplift(Figures 5a and 5b). Models including the trans-form offset thus provide a solution to the paradoxof the lineament focus being offset from thepresumed plume center.

[35] The transform fault at 91�W was previouslysuggested to be the primary control on lineamentpatterns at the Galapagos [Harpp et al., 2003;Harpp and Geist, 2002]. Our models support asignificant contribution from the transform, butmodels also indicate that the plume effects areimportant. For example, the model with only theridge-transform effects predicts least tensile stresstrajectories that are nearly perpendicular to theWolf-Darwin lineament and other lineaments farfrom the transform fault (Figure 8b). In addition,the transform effects cause relatively compressivedeviatoric stress resultants south of the lowertransform corner which should inhibit lineamentformation. The added plume effects produce tra-jectories consistent with the Wolf-Darwin linea-ment and tensile stress resultant magnitudes nearthe southern transform tip. Overall, the lineamentpattern is better modeled by the case of a ridge thatis more tensile than the far-field stress. The focusof trajectories is better delineated and trajectoriesclose to the transform closely resemble the linea-

Figure 9. (A) Map of the Galapagos Spreading Center(denoted by subhorizontal black lines), the GalapagosArchipelago (black circle enclosing Fernandina Island,location of most recent volcanism), and the volcaniclineaments (black dashed lines), which appear to fan outfrom a focus at the white circle. Contours of fraction oflithospheric yield strength (most tensile stress resultantdivided by yield strength) for (b) a model with two ridgesegments separated by a transform fault and (c) a modelwith a single, straight ridge axis. Ridges are more tensilethan the imposed isotropic far-field stress by 100 GPa m.Ticks mark trajectories of least tension. Magmapenetration via dikes will tend to align with thesetrajectories. Models predict trajectories to fan northwardaway from an apparent center within the white circles.Black dots show the center of the model mantle plume.



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ment orientations. We conclude that the plumeeffects are critical to the fan-like pattern of Gal-apagos lineaments far from the transform, while thetransform fault effects are important to lineamentorientations close to the transform and to locatingthe focus of the fan ridgeward of the plume center.

[36] On the basis of gravity modeling, Feighnerand Richards [1994] predict a discontinuity in theeffective elastic thickness of the Nazca plate nearthe Galapagos Archipelago, with lithosphere to thewest and south of the Wolf-Darwin lineament morerigid than the lithosphere to the north and east.Lithospheric strength in our models varies bothnorth-south due to plate cooling and east-westacross the age discontinuity due to the transformfault. This roughly approximates the proposedstrength variations of Feighner and Richards[1994], only significantly differing between thetransform and the WDL. Tests on the importanceof this east-west discontinuity to the overall patternof stress resultants in our models indicate littlechange in trajectory orientations or stress resultantmagnitudes and suggest that inclusion of the morecomplex strength variations of Feighner andRichards [1994] will not significantly alter thesolutions.

[37] This said, it is important to note that wherethe stress resultant magnitudes are equal to orexceed the integrated lithospheric yield strength,the lithosphere will deform anelastically and itsbehavior is not represented by our elastic rheol-ogy. This applies especially to the inside cornerregions of the transform fault model where stressconcentrations due to the crack tips mayexceed the yield strength of young lithosphere(Figure 9b). We therefore consider our purelyelastic models as representing the minimal levelof sophistication needed to address the posedproblem.

[38] Regarding the source of magma feeding thelineaments, geochemical analysis of lavas from theWolf-Darwin lineament shows increasing plumechemical influence toward the ridge (e.g., 87Sr/86Srfrom 0.7026–0.7034) [Harpp and Geist, 2002].These chemical variations could be causedby changes in melt composition along an astheno-spheric plume channel oriented along the pseudo-faul t near Wolf Island or by tapping ofplume-contaminated mantle dispersed throughoutthe region and already processed bymelting beneaththe GSC [Harpp and Geist, 2002]. Our modelsrequire a melt source beneath the lineaments andsuggest that the lithospheric stress field controls

where and how this melt erupts, but do not requireasthenospheric channels to form the lineaments.

[39] We propose a model where the lineaments formin regions of high integrated tension that will pro-motemagma to crack its way through the lithosphereand erupt to initiate a volcanic lineament. Theprocess of diking and volcano loading decreasesthe integrated tension in the underlying lithosphereand leads to tensile stress concentrations near theends of the lineament. This will promote new vol-canism and lengthening of the lineament in amannermuch like a giant crack in the lithosphere. Volcanisminitiates wheremagma supply is high and tension is alarge fraction of the lithospheric yield strength nearthe ridge axis. Volcanism subsequently propagatesaway from the ridge roughly according to the pre-existing lithospheric stress field. Indeed, alongWolf-Darwin, ages are seen to decrease away fromthe ridge [Sinton et al., 1996] suggesting that volca-nism initiated near the ridge axis and propagatedsoutheast. Volcanic initiation near the ridge axissupports the assertion that regions of high litho-spheric yield fraction will be most penetrable bymagma. Age dating of the other Galapagos linea-ments will provide a further test of this hypothesis.

4.2. Islands of the Galapagos Archipelago

[40] Islands of the Galapagos Archipelago alsoshow aligned fractures [Darwin, 1860] and thevolcanoes themselves are aligned along rectilinear‘‘Darwinian trends’’ most readily exhibited by the‘‘J’’-shaped Isabela island [McBirney and Williams,1969]. The stresses we have modeled could haveinfluenced the formation of these islands. A similarpattern is displayed by least tensile stress trajecto-ries near the plume center in our straight ridgemodels and slightly ridgeward of the plume centerin models of a segmented ridge with a largeviscosity plume (Figures 8e, 8f, and 9c). However,the main islands are larger than the volcanic linea-ments and will introduce much larger bendingstresses which may have been more importantin controlling where adjacent volcanoes form[Hieronymus and Bercovici, 2001; ten Brink,1991]. Future studies that incorporate bending intomodels of plume-lithosphere interaction are neededto better explore the formation of the main Gal-apagos islands.

4.3. Constraints on the Tectonic StressField

[41] The differential stress (DN = N11 � N22)induced by the combination of plume forces, and



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the far-field and ridge boundary conditions controlsthe orientations of the least-tensile stress trajecto-ries and thus the width of the observed fan-shapedpattern. One quantity that characterizes the shapeof the fanning lineaments is the half-width, L,along the ridge axis. Our models show thatplume-ridge separation, xp and the differentialstress in the far field strongly influence the valueof L and may indicate one reason why patterns oflineaments differ between hot spot localities suchas Kerguelen, Tristan de Cuhna and the Galapagos.

[42] With knowledge of L, Rp, and xp from theGalapagos we can use the predictions of Figure 7to place first-order constraints on the far-field stressfield in the near-GSC lithosphere. Taking measuredvalues for xp = 260 km [Ribe, 1996], L = 255 km,and a Rp = 350 km based on the location of adistinct decrease in chemical plume influence,crustal thickness and plume-driven melt supplyseen at approximately 93�W [Cushman et al.,2004; Detrick et al., 2002], we predict the ratio ofdifferential far-field stress to differential plumestress (DNf/jDNpj) to fall between �0.2 and 0.1(Figure 7, gray box). Our models predict a nearlyisotropic far-field stress in the near-GSC litho-sphere. This estimate provides an independent pre-diction of the stress field around the Galapagos andprovides an additional constraint on lithosphericstress field models.

4.4. Global Lineament Formation

[43] Although the majority of this study focuses onthe Galapagos lineaments, volcanic lineaments ap-pear to be common manifestations of hot spot–ridge interaction. We have identified at least 12separate hot spots currently or previously locatednear to mid-ocean ridges, showing volcanic linea-ments similar to those at the Galapagos. Theyinclude Azores, Cobb, Discovery, Foundation, Ker-guelen, Line Islands, Louisville, Musicians,Reunion, Sala y Gomez, St. Paul (Amsterdam),and Tristan de Cuhna hot spots. All of the abovehot spot–ridge systems share the common charac-teristic of having a ridge axis near the inferredlocation of the hot spot based on available datesof volcanism and seafloor magnetic lineations.

[44] Our model predicts plumes to introduce inte-grated lithospheric stresses of appreciable fractionsof lithospheric yield strength only in young litho-sphere. To evaluate the off-axis extent of lineamentformation we estimate the age of the plate onwhich the lineaments were emplaced, lineamentlengths, and the relative location of the hot spot at

the time of lineament emplacement. Ages ofvolcanism along the hot spot track and lineamentsare compiled from the literature (see Figure 10 forreferences) and crustal ages are determined fromseafloor isochrons [Muller et al., 1997]. Wemeasure the maximum length of lineaments ateach hot spot by creating regional bathymetrymaps and digitally measuring the distance alongeach lineament.

[45] Figure 10 shows the relationships betweenmaximum lineament length, plate age and plume-ridge separation for 23 oceanic hot spots with errorbars in plume-ridge separation and plate age fromthe error contours of Muller et al. [1997] and errorsin lineament length measurements estimated at±20 km. Hot spots that do not show evidence forlineaments are given a lineament length of zero.Lineaments, on average, reach lengths of about75% of the plume-ridge separation distance at thetime of emplacement. Approximately linear rela-tionships between lineament length and plate age,and lineament length and plume-ridge separationare seen until a critical distance or seafloor age isreached (Figure 10). Lineaments are not seenbeyond 1250 km or in lithosphere older than�25 Ma. This cutoff is likely due to the presenceof thick, strong lithosphere resisting plumestresses, the lack of interaction between the ridgeand plume at separation >1250 km, or both.Schilling [1991] notes a cutoff in geochemicaltracers and bathymetric highs with plume-ridgeseparations of �1600 km, slightly greater thanthe maximum distance found in our collection ofhot spots with lineaments. In addition, lineamentsdo not appear to form near ridge-centered plumes.For example, at Iceland volcanic lineaments areprimarily focused along 3 major rift axes, but arenot present off-axis. This localization of volcanismmay be a consequence of efficient suction ofmagma to these rift zones, or, alternatively, the riftzones themselves may be an expression of aplume-induced stress field similar to that whichcauses off-axis lineament formation. Not only doesFigure 10 demonstrate that lineaments form pref-erentially within certain plate ages and separationdistances, it demonstrates the commonality oflineament formation at near ridge hot spots.

5. Conclusions

[46] In this study we use a 2-D model of plume-ridge interaction to examine factors contributing tothe formation of lineaments seen at the Galapagosand other oceanic hot spots. We present a new



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method of simulating patterns of stress caused bythe local factors of plume shear, gravitational bodyforces due to uplifted topography, ridge boundaryconditions, and variations in plate thickness.

[47] Least tensile stress trajectories due to a plumeinteracting with a linear ridge axis form a fan-likepattern between the plume and ridge. The trajec-tories tend to curve to meet the ridge axis atapproximately right angles if the shear stress onthe ridge is zero. While the general fan-shapedpattern is preserved beyond plume-ridge separa-tions over �0.5 times the radius of the dominantlyaxisymmetric plume flow (Figure 6, where xp > L),the pattern deviates to the east and west for smallerplume-ridge separations, but maintains a ridge-perpendicular orientation between the ridge andplume center. The addition of a transform faultcreates a more complex pattern with least tensilestress trajectories again curving to meet the ridge at

right angles, but with the focal point of the fanpattern further ridge-ward than the correspondinglinear ridge model (Figure 9).

[48] Increasing the far-field tension perpendicularto the ridge relative to that parallel to the ridgecauses the width of the fan at the ridge axis todecrease in an approximately quadratic function ofDNf. Examination of this relationship in context ofGalapagos lineaments suggests that the differentialfar-field stress resultant is nearly isotropic in theregion of the GSC.

[49] Our model stress resultant trajectories closelyresemble the pattern of lineaments seen near theGalapagos when the ridge is segmented and moretensile that the far-field stress. Results reconcile thedifference in plume center location from that oflineament focus and explain the decreasing trendand curvature of lineaments as the transform is

Figure 10. Observations of 23 oceanic hot spots show distinct patterns in lineament formation. (a) This plot of plateage at the time of lineament emplacement versus lineament length demonstrates a region of lineament formationbetween plate ages of 0.2 and 25 Ma (hot spots without lineaments are associated with a lineament length of zero).(b) Comparison of plume-ridge separation and lineament length also shows a restricted region of formation between100 and 1250 km. Both trends show an approximately linear (black dotted best fit line) increase in lineament lengthin the region of formation (gray boxes). Plate ages and errors in plate age from Muller et al. [1997]. Where error barsare not visible, the symbol is larger than the error. Hot spots are labeled as follows: A, Amsterdam; B, Bowie; Bm,Bermuda; C, Cobb; CV, Cape Verde; Cy, Canary; D, Discovery; F, Foundation; Fn, Fernando; G, Galapagos; Hw,Hawaii, I, Iceland; K, Kerguelen; L, Line Islands; Lou, Louisville; Md, Madeira; Mq, Marquesas; P, Pitcairn; R,Reunion; S, Sala y Gomez; Sc, Society; T, Tristan; Td, Trinadade. Lineament ages and hot spot tracks are compiledfrom the literature [Bonneville et al., 1988; Davis et al., 2002; Desonie and Duncan, 1990; Douglass et al., 1999;Duncan, 1984; Geldmacher and Hoernle, 2000; Geli et al., 1998; Hekinian et al., 1999; O’Connor and Duncan,1990; O’Connor and Roex, 1992; O’Connor et al., 1995; Small, 1995; Sonne, 1990; Vlastelic et al., 1998; Weis et al.,2002].



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approached. We propose that the Galapagos linea-ments form near the ridge axis where tension is alarge fraction of the lithospheric strength and thenpropagate southward roughly following the pre-dicted stress trajectories. Volcanism is probably fedby a widespread melt supply from the underlyingasthenosphere.

[50] Measurements of 23 separate oceanic hotspots show a nearly linear increase in volcaniclineament length with increasing age of the plateat the hot spot and increasing plume-ridge separa-tion distance. This linear relationship holds untilplate ages of �25 Ma and plume-ridge separationdistances in excess of �1250 km where lineamentformation is no longer observed. We conclude thatbeyond 25 Ma, plate thicknesses are too large forplume stresses to enhance the ability of magma topenetrate the plate. Also, plume-ridge distancesgreater than 1250 km probably prevent significantasthenospheric interaction of the plume and ridge.

Appendix A

[51] Because the lithosphere is thickening approx-imately with the square root of distance from theridge, it accretes in a curved form. As the lateralextent is much larger than its thickness, the litho-sphere can be approximated as a thin shell. Theshape of a thin shell may be described in terms oftwo orthogonal coordinates, a and b, which ingeneral are curvilinear [Gould, 1988]. Infinitesimalchanges in a and b are related to changes in arclength along the mid plane of the shell, dsa and dsb,according to

dsa ¼ A d a;

dsb ¼ B d b;ðA1Þ

where A and B are known as Lame parameters and,for the lithosphere, we define da = dx1 and db =dx2 (Figure 2). Using this convention, B = 1 (nocurvature parallel to the ridge) and A is found to be

A ¼ 1þ dx3

dx1

�2 !1=2

; ðA2Þ

where dx3/dx1 describes the slope of the shell mid-plane. Using this curvilinear geometry, forceequilibrium can be described in three orthogonaldirections [Gould, 1988]

BNað Þ;a þ ANba� �

;b þ A;b Nab � B;a Nb� �

þ QaAB

Ra

þ qaAB ¼ 0; ðA3Þ

BNab� �

;a þ ANb� �

;b þ B;b Nba � A;a Na� �

þ QbAB

Rb

þ qbAB ¼ 0; ðA4Þ

BQað Þ;a þ AQb� �

;b� �

� NaAB

Ra� Nb

AB

Rbþ qnAB ¼ 0; ðA5Þ

where partial derivatives are denoted by commasfollowed by coordinate directions (to be completewe show the terms involving derivatives of B eventhough they are zero). Ri is the radius of curvaturein the i direction, Qi is depth-integrated transverseshear, and qi are the loads on the plate. Theseequilibrium equations are coupled through thetransverse shear term, Qi, which also acts to couplethem with the equations of angular momentum.

[52] To simplify equations (A3)–(A5), we deter-mine the appropriate values of the Lame parameterA and the radii of curvature of the unstressedlithosphere. Assuming the slope of the mid-plane(dx3/dx1, equation (A2)) is parallel to the surface ofthe lithosphere, radius of curvature may be de-scribed by seafloor deepening due to plate coolingaway from the ridge [Stein and Stein, 1992]

z ¼ 2:60þ cffiffiffiffiffix1

p; ðA6Þ

where c = 0:365ffiffiu

p , u is the seafloor spreading rate, andx1 is the distance from the ridge. Thus equation(A2) becomes

A ¼ 1þ c2

4x1

� � 1=2ð Þ

: ðA7Þ

For u = 30 km/myr and x1 = 10–100 km off axis,(c2/4x1) is of order 10

�4 to 10�5 so A is very closeto 1.

[53] The radius of curvature in the ridge paralleldirection, Rb, is infinite because we assume allcurvature is parallel to plate spreading and radiusof curvature due to uplift is large. The radius ofcurvature perpendicular to the ridge axis is

Ra ¼1þ dx3

dx1

�2" # 3=2ð Þ

d2x3

dx21

: ðA8Þ

Again using u = 30 km/myr and x1 = 10 km, theradius of curvature near the ridge (0.33 m.y. oldcrust) is found to be Ra > �1900 km. We expectthe stress resultant terms in (A3)–(A5) to be oforder Nij/Rp, where Rp � 102 km is the character-



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istic radial extent that plume forces, qi, act on theplate. Since Ra � Rp, the terms involving Ra

�1

(and Rb�1) in (A3)–(A5) are negligible and the

resulting linear momentum equations have a formsimilar to those of plane stress (with A = B = 1)

Nað Þ;a þ Nba� �

;b� �

þ qa ¼ 0; ðA9Þ

Nab� �

;a þ Nb� �

;b� �

þ qb ¼ 0: ðA10Þ

We use these equations of equilibrium to formulateour model of plume-ridge interaction.

Appendix B

[54] To derive the governing equation for a platewith thickness, h, that varies laterally, the consti-tutive equations (equation (3)) are substituted intothe equilibrium (equation (2)) equations to yield

Eh

1� v2ð Þ@e11@x1

þ v@e22@x1

� þ Eh

1þ vð Þ@e12@x2

¼ ��q1 þ

@h

@x1

N11

h

þ @h

@x2

N12

h

¼ b1; ðB1Þ

Eh

1� v2ð Þ@e22@x2

þ v@e11@x2

� þ Eh

1þ vð Þ@e12@x1

¼ ��q2 þ

@h

@x2

N22

h

þ @h

@x1

N12

h

¼ b2; ðB2Þ

and compatibility (equation (4)) equation to yield

1� nð Þ@2 N11

h@x22

þ@2 N22

h@x21

264

375� n

@2 N22

h@x22

þ@2 N11

h@x21

264

375 ¼ 2

@2 N12

h@x1@x2

:

ðB3Þ

Terms with gradients in thickness are placed on theright hand side (r.h.s.) of equations (B1) and (B2)and are grouped as ‘‘fictitious’’ body forces intonew variables, b1 and b2, which also include theeffects of plume shear and uplift (q1 and q2; seeequation (9)). Formulation of the governingequation is achieved by combining the compat-ibility and equilibrium equations as is commonlydone in elasticity theory. To facilitate this,derivatives of equations (B1) and (B2) are takenin the 1 and 2 directions, respectively, and theresulting equations are summed:

�@2 N11

h@x21

�@2 N22

h@x22

�@b1

h@x1

þ@b2

h@x2

0B@

1CA ¼ 2

@2 N12

h@x1@x2

: ðB4Þ

Now we may solve for the general governingequation for an elastic plate of varying thickness byequating the r.h.s. of (B4) and (B3) and simplifying

r2 s11 þ s22ð Þ ¼ � 1

1� vð Þ@b1@x1

þ @b2@x2

� : ðB5Þ

The governing equation relates the change indepth-averaged stresses (sij = Nij/h) in a plate ofvarying thickness to the imposed loads (qi), subjectto specified (ridge axis) boundary conditions. Theform of (B5) is identical in form to the governingequation for plane stress and allows the solution tobe determined through the use of an Airy stressfunction method. Because the final value of the‘‘fictitious’’ body force terms depend on the stressresultants, the equations are solved iteratively.

Acknowledgments

[55] The authors would like to thank S. Martel for providing

the 2-D boundary element code used in portions of this study.

We would also like to thank N. Ribe for discussions about the

method used and A. Delorey, M. Behn, A. Oakley, and

S. Martel for helpful comments on previous versions of this

manuscript. Thoughtful reviews were provided by D. Graham,

M. Richards, and R. Buck which helped strengthen the man-

uscript. Maps were made using GMT version 3.4.2 by P.Wessel

and W. F. Smith. Mittelstaedt and Ito were funded by NSF

grants OCE03-7051 and OCE03-51234 and Ito’s start-up

money from SOEST. The computer cluster used for the

computations was funded with NSF grant OCE01-36793. This

is SOEST contribution 6545.

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