RESEARCH ARTICLE

Tumulus development on lava flows: insightsfrom observations of active tumuli and analysisof formation models

Steven W. Anderson & Suzanne E. Smrekar &

Ellen R. Stofan

Received: 2 September 2010 /Accepted: 27 December 2011 /Published online: 27 January 2012# Springer-Verlag 2012

Abstract Here, we use observations of active flows alongwith detailed morphometric field measurements of morethan 70 tumuli on flows at Mount Etna (Italy), Kilauea,and Hualalai (US) volcanoes to constrain a previously pub-lished model that estimates the pressure needed to formtumuli. In an attempt to discover the nature and magnitudeof pressure variations within active lava flow interiors, wethen consider how tumuli differ from idealized circularplates. We incorporate observations of active tumuli andfind that they may grow asymmetrically yet produce asymmetrical tumulus and can form where the flow pathsignificantly changes direction. Bending models of clampededges provide the most reasonable head estimates for thetumuli in our study. Tumulus formation requires the propercombination of cooling and effusion rate. If cooling is tooextensive and effusion rate too low, the crust will providetoo much resistance to bending. If cooling is too limited andeffusion rates too high, crusts will not develop or haveinsufficient strength to resist fracture and subsequent break-outs. We do not find it surprising that tumuli are rarelyfound over well-established lava tubes that typically haverigid, walls/overlying crusts that exceed 2 m in thickness

and provide too much resistance to bending. Silicic flowslack tumuli because the viscosity gradients within the floware insufficient to concentrate stress in a localized area.

Keywords Lava . Inflation . Tumuli . Modeling . Viscosity

Introduction

The emplacement of basaltic lava flows is one of the mostcommon planetary surface-forming processes in the solarsystem. Assessments of lava flow surface morphology andfeatures provide clues about the processes occurring duringeruptions and may provide insight regarding the formationof flows not witnessed during emplacement. Tumuli arecircular to elongated domes, ∼1–10 m in height, found onsome basaltic lava surfaces (Fig. 1) and are the surfaceexpression of magmastatic overpressure within the flows(Walker 1991; Rossi and Gudmundsson 1996; Anderson etal. 1999, 2000). They occur in many, if not all, major maficvolcanic centers on Earth, and high resolution images fromthe HiRISE camera on Mars Reconnaissance Orbiter make itpossible to recognize tumuli on Mars as well (Keszthelyi etal. 2008; Giacomini et al. 2009). The distribution patterns oftumuli on lava flow surfaces (Glaze et al. 2005) suggest thatareas of high magmastatic overpressure vary in time andlocation within an active flow interior and result in theformation of internal pressure gradients that may be intri-cately linked to interior flow paths. These flow paths arebelieved to vary in form, from anastamosing networks ofpreferred pathways sandwiched between upper and lowercooled crusts (Walker 1991; Anderson et al. 1999; 2000;2005; Duraiswami et al. 2001) to wider sheet-like forms(Hon et al. 1994), to well-established tubes that feed surfacefeatures such as ephemeral boccas, breakouts, and tumuli

Editorial responsibility: A. Harris

S. W. Anderson (*)MAST Institute, University of Northern Colorado,1210 Ross Hall,Greeley, CO 80639, USAe-mail: steven.anderson@unco.edu

S. E. SmrekarJet Propulsion Laboratory, California Institute of Technology,Pasadena, CA 91109, USA

E. R. StofanProxemy Research,Laytonsville, MD 20882, USA

Bull Volcanol (2012) 74:931–946DOI 10.1007/s00445-012-0576-2

(Duncan et al. 2004). Understanding how overpressure isdistributed within an inflating flow both spatially and tempo-rally may provide clues regarding the mechanics of the infla-tion process and the evolution of internal lava distributionnetworks as a function of known eruption parameters (i.e.,flow rate, slope, time, underlying topography).

Here, we use observations of actively inflating tumulialong with detailed morphometric field measurements ofmore than 70 tumuli on flows at Mount Etna (Italy), Kilaueaand Hualalai (HI, USA) volcanoes to better understandtumulus formation. We use these data to evaluate a modelfor estimating the overpressure needed for tumulus forma-tion (Rossi and Gudmundsson 1996) in an attempt to dis-cover the processes occurring within active lava flowinteriors. Rossi and Gudmundsson (1996) constrained theirmodel with idealized morphometric values and estimatedthat approximately 8–40 m of magmastatic head is neededto bend the surface crust of an active lava flow into thecharacteristic whale-back shape of a tumulus. Using actualmorphometric measurements from field sites, we find thatthe pressure estimates using the Rossi and Gudmundsson(1996) model vary by several orders of magnitude over timeand space within a single lava flow.

Although the model gives reasonable values of magma-static overpressure for many of our measured tumuli, it alsoproduces some values that may be unreasonably high (ex-ceeding the topographic relief available to generate thecalculated magmastatic head) or low (<0.1 m of magma-static head, which is insufficient to overcome the tensilestrength of the crust). In this paper, we consider otherformation models that differ from the idealized circular platemodel examined by Rossi and Gudmundsson (1996) andprovide a wider range of inflation pressure predictions for a

variety of tumuli measured in the field. In particular, weexamine the effects of (1) circular vs. elliptical plates, (2)broken tumulus edges (simply supported) vs. tumuli whosecrusts are continuous with the surrounding flow (clampededges), (3) localized pressure sources beneath the plate vs.uniform loading of the plate, (4) strength of the elastic andviscoelastic crust, and total crustal thickness, (5) small-versus large-scale deflection, and (6) fracture and lifting ofthe plate vs. sustained bending. We also incorporate obser-vations of active tumuli and find that even asymmetricgrowth can produce a symmetrical tumulus and that tumulican form where the flow path significantly changes direc-tion. We show that examining a more complete range oftumuli behavior allows better prediction of overpressure andgreater insight into tumulus formation, the structure of thesurface crust, and the nature of subcrustal flow pathways.

Background—tumuli, pahoehoe flow types, internalstructure, and modeling of active lavas

Tumuli are most commonly found on basaltic pahoehoelavas, although they may also form on a’a flows (Duncanet al. 2004). They range in size from meter scale to manytens of meters in diameter, with heights that may exceed10 m. Tumuli may be roughly circular in planform, with acharacteristic whale-back shape, or they may exhibit a high-ly elongate planform. The distinction between large, elon-gate tumuli and pressure ridges is not clear, although Walker(1991) suggested that the width of a pressure ridge wouldmeasure less in planform than the sum of the lengths of itstwo flanks There is no well-established lower limit ontumulus size, although for our study we defined tumuli byhaving at least 0.5 m of uplift, a traceable outer margin, andat least one axial fracture that displayed morphology indic-ative of the tumulus-forming process as described by Walker(1991) and Anderson et al. (1999). Walker (1991) found thattumuli tend to form on shallow slopes in lava flows experi-encing a modest amount of extension. He defined threetypes of tumuli: shallow-slope tumuli, moderate-slopetumuli situated on slopes of 4° or more, and flow-lobetumuli that form near advancing flow-fronts and inflatealong the entire lobe width. Shallow-slope tumuli weredescribed as typically larger than those on moderate slopes,and as forming in clusters or trains. Walker (1991) did notfind any association between tumulus trains and tubes butsuggested that tumulus trains may form over lesser, moretransient tubes. Duraiswami et al. (2001) describe tumuli asbeing common on both hummocky and thicker sheet lobesin the Deccan volcanic province. They interpret the align-ment of tumuli to indicate that they have developed alonganastomosing tube systems, and the Deccan tumuli aresimilar in their general morphology to tumuli in Hawaii.

Fig. 1 Bend in this active pahoehoe flow was the site of the growth ofa tumulus. Photo taken at 6:54PM local time on 10/2/05. Black linerepresents the approximate position of a line marking the midpointbetween the two lobe margins

932 Bull Volcanol (2012) 74:931–946

Tumuli are not found on all inflated pahoehoe surfaces.Self et al. (1998) defined two types of inflated pahoehoeflows: sheet flows and hummocky flows. Sheet flows arecomposed of lobes lacking tumuli that form from relativelycontinuous, rapid emplacement over shallowly sloping,smooth surfaces (Self et al. 1998). Hummocky flows havemany discrete tumuli and form on rougher surfaces onsteeper slopes, with relatively slow, discontinuous emplace-ment (Self et al. 1998). Self et al. (1998) suggested thattumuli form over depressions, indicating that the distribu-tion of tumuli should reflect the pre-existing topography.

Rossi and Gudmundsson (1996) studied tumuli on theThrainsskjoldur flow field in Iceland, defining three types oftumuli based on their distance from the vent: lava-coatedtumuli, upper slope tumuli, and flow lobe tumuli. Flow lobetumuli typically do not have the lava outpourings commonon lava-coated and upper slope tumuli and form when smalllava lobes, or tongues, inflate and form the characteristicwhale-back form. Because they are inflated flow lobes,flow lobe tumuli tend to have large aspect ratios(length/width) compared with the other two types. Rossiand Gudmundsson (1996) calculated magma overpres-sures needed to lift the surface crust to form a tumulus,suggesting that tumuli can be used to study the varia-tions in overpressure within the inflating flow. This workhas been applied to flows at Hawaii (Anderson et al. 1999),Mt. Etna, Sicily (Duncan et al. 2004), and other flows inIceland (Mattsson and Hoskuldsson 2005).

Mapping by Duncan et al. (2004) showed that largetumuli may serve as a source of material and pressure forsatellite tumuli. They mapped a section of an a’a flow with alarge tumulus that served as the source for flows withadditional nearby tumuli, creating a large pahoehoe tumuluscomplex within a larger a’a flow. They also noticed that notall tumuli were located directly over underlying tubes, sug-gesting that a subsurface distributary network of tube-likepathways must exist independent of the tube, similar to thatdescribed by Walker (1991), Anderson et al. (1999) andDuraiswami et al. (2001) for pahoehoe flows.

Relating surface morphology to subcrustal flow structureis difficult, given the lack of direct observations of an activelava flow interior. Specifically, the processes capable ofgenerating magmatic overpressure for tumulus growth in anetwork of thermally and mechanically preferred pathwayswithin an inflating lava flow are not well understood. Paststudies have not shown a strong correlation between thelocation of major lava tubes and tumuli (Walker 1991;Byrnes and Crown 2001). Byrnes and Crown (2001) triedto relate the surface morphology of flows at the Mauna Uluflow field to the tube system mapped by Holcomb (1976).They concluded that the units they mapped, which includedan inflated unit, did not correlate to major tubes and, there-fore, may be related to a smaller scale subcrustal distributary

network. Similarly, Anderson et al. (1999, 2005) proposedthat a network of thermally preferred pathways, or viscousfingers (Saffman and Taylor 1958), exist under the crust ofan inflating flow that could concentrate pressure and giverise to inflation features such as tumuli or lava rises onhummocky flows. Viscous fingering has been cited as animportant process in a number of volcanic settings, includ-ing fissure eruptions (Wylie et al. 1999a, b; Helfrich 1995),mantle plume generation (Weeraratne et al. 2003), inflatinglava flows (Anderson et al. 1999, 2000, 2005; Duraiswamiet al. 2001; Schaefer and Kattenhorn 2004; Houle et al.2009) and perhaps for intrusive bodies (Hallot et al. 1996,and the fingered sheet intrusions of Pollard et al. 1975).Basaltic lava flows typically display rheologies that are nearNewtonian for material above ∼1125°C and Bingham forcooler temperatures (e.g., Shaw 1969; Pinkerton and Sparks1978), and Saffman–Taylor fingers are common in bothNewtonian and yield-strength fluids (Coussot 1999).

The surface morphology of the cleft-forming fracturesfound dissecting most tumuli capture the kinematics oftumulus emplacement (Walker 1991; Anderson et al. 1999,2000). Anderson et al. (1999) described a characteristicthree-zoned fracture morphology consisting of an upperelastic crust that fractures in a crudely columnar fashion.The columnar aspect of the fracturing requires plane strainconditions that are accounted for by cooling alone (Ryanand Sammis 1981) and, therefore, represents elastic crustformation prior to inflation. A middle, planar-fracture zonerequires a change in the stress state and is best described as atensional regime created during inflation and bending ofhotter, more viscoelastic crust directly beneath the cooler,elastic, columnar-fractured crust. Anderson et al. (1999)proposed that this region represents an area of transitionbetween the elastic crust forming overlying columnar frac-tures and the molten viscous lava in the tumulus interior. Alower zone consisting of alternating bands of dark graymaterial interspersed with reddish material representingelastic fracturing followed by ductile deformation lavarequires a stress state that changes in intensity, first allowingthe cooling front to proceed ahead of the crack tip duringlow stress periods, followed by a high stress period thatforces the crack tip to penetrate completely through theelastic zone into hot ductile material. These changes in stressintensity, or stress “pulses,” are consistent with the move-ment of viscous fingers in the interior that change the focusof inflation within the flow (Anderson et al. 2005). Thepresence of viscous fingers allows a flow to grow witha relatively constant overall flow rate, yet produces acrusted extrusion that inflates unevenly as the volume isdirected to discrete parts of the flow, allowing for theformation of tumuli and hummocky surface topographyand other inflated features such as lava rise pits andlava rises.

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To a first approximation, the overpressure within a lavaflow available to drive the inflation process may bethought of as that produced by a column of magma andis thus a product of the column height (or magmastatichead), lava density, and gravity. Rossi and Gudmundsson(1996) proposed a model of a bending elastic plate toestimate the average inflation pressure needed to createtumuli. Specifically, they used a circular plate with simplysupported edges and assumed small deflections. Usingidealized parameters for thickness of the viscoelasticcrustal thickness, height, and diameter of tumuli, theyestimate a magmastatic pressure of 0.2–1 MPa. For adensity of 2,500 kg/m3, this requires a magmastatic headof 8–40 m, which was less than the >100 m of relief intheir study area.

Observations of active tumuli and field data

On October 3, 2005, we observed the growth of twotumuli on the coastal plain of Kilauea volcano, Hawaiiin a small (∼100×100 m) flow field situated on a slopeof <2° that was fed by a small channel that remainedopen during the growth of these features. One tumulusgrew during the overnight hours of October 2 and 3 toa height of approximately 1.5 m and was approximately5 m in width. The tumulus was too hot to allow directmeasurements of morphometry, but images shot at adistance of approximately 10 m show relatively thincrusts of <5 cm for the crudely columnar upper crustand approximately 10 cm for the planar zone. The finalshape was fairly symmetrical about the central cleftfracture. We observed that both the axial and circumfer-ential fractures at the edges of the uplifted platesformed early in the inflation process. Detailed GPSmeasurements show that this tumulus grew directlyabove a ∼90o bend in a flow lobe that was visible lessthan 24 h earlier (Fig. 1). No active lava was observedemanating from this tumulus, although a pahoehoetongue was clearly evident at the base of the axialcrack. No additional growth was observed after ourinitial observation, constraining the emplacement timeto a maximum of 22 h.

A second tumulus was observed several meters away andwas in the process of growing when we first observed itduring the afternoon of October 3. When first observed at11:39AM local time, only one side of the tumulus had liftedsubstantially (to a height of approximately 2 m) above thesurrounding active flow. This tumulus also displayed crustalthicknesses that were similar to the first tumulus describedhere, and active pahoehoe lava was observed flowing from acrack near the edge of the tumulus, approximately 1–1.5 mabove the surrounding flow (Fig. 2). Incandescent material

was observed at the base of the axial crack but did notextrude. By 2:14PM local time, the other side of the tumulushad lifted to approximately the same height as the first,forming a more symmetrical tumulus (Fig. 2). No additionalsignificant growth of the tumulus was noted after this ob-servation, showing that some tumuli may inflate over rela-tively short (several hours) time periods and that a tumuluswith a relatively symmetrical final shape may not inflateevenly.

In February 2007, we studied an actively inflating100×200 m area on the coastal plain of Kilauea andfound two tumuli that formed over a 2-day period. Ter-restrial Light Detection and Ranging (LiDAR) data wereobtained for the pre- and post-eruption surfaces (fordetails of the LiDAR field methodology, see Andersonet al. 2008). We compared the locations of the tumuli

Fig. 2 Asymmetric growth of a tumulus. Top photo taken at 4:39PMlocal time on 10/3/05. Bottom photo was taken 1 h and 35 min later.Note the asymmetric growth of the tumulus, as well as the height of thebreakout in left photo ∼1.5 m above the surrounding flow surface. Theheight of the tumulus is approximately 3 m above the surrounding flowsurface

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with the pre-eruption surface and found that one tumulusformed over a ∼2×2×1 m deep low area in the pre-eruption topography, whereas the other tumuli formedin one of the highest and flattest parts of the field area.Both tumuli were <10 m in length and width and <4 min height. We did not observe any active outpouringsduring emplacement but cannot rule out their occurrenceduring times when we were not present. Extreme heatfrom the cooling flows prevented close inspection todetermine if outpourings had occurred.

We obtained data also from the following field sites: the1801 lava flow from Hualalai volcano (HI) (Kauahikaua etal. 2002), flows from the 1983-present eruption of Kilaueavolcano (HI), the 1983 and 1792 flows from Mount Etna(Italy), and two other Etnean flows near Bronte and Maletto(Italy) emplaced during the past 500 years that show theiroriginal flow surface structures. We also mapped a sheetflow lobe at Kalapana (HI) emplaced during the 1983–present eruptions.

We used real-time differential Trimble ProXR GPS(dGPS) to map the height, shape, area, and dilated sur-face fractures at 20 inflated structures in Hawaii (tumuli,lava rises, and sheets) where the estimated effusion ratesfor eruptions producing these flows ranged from ∼1 to10 m3/s (Heliker and Mattox 2003; Rowland and Walker1990). Post-processing with base-station corrections pro-vided data with horizontal and vertical accuracies ofapproximately 0.3 m. At Mount Etna, we also measuredfour tumuli on the 1983 a’a flow (8 m3/s; Duncan et al.2004 and references therein), 23 tumuli on the 1792 flow(∼2.5 m3/s (Romano and Sturiale 1982), four tumuli onthe 1651 flow near Bronte (Romano and Sturiale 1982),and 21 tumuli on the prehistoric Balze Soprano flownear Maletto (Romano and Sturiale 1982; Guest et al.2011) (Table 1). Tumuli were located on local slopes,determined in the field using dGPS, ranging from 1.1°to 5.6°. We estimated the flow field slope at each sitefrom the highest and lowest point measured during ourdGPS surveying, except at the 1801 Hualalai site wherewe measured two features separated by a only fewmeters. This was not an adequate distance to providea reasonable slope estimate, so we calculated the 1801Hualalai slope between our features and the nearby(∼1 km) coastline over which a relatively constant gradeexists.

At each tumulus, we measured the thickness of theelastic crust present during formation to the nearestcentimeter, which we believe is analogous to the crude-ly columnar zone described by Anderson et al. (1999).We also measured the thickness of the viscoelastic crustto the nearest centimeter, which we believe is analogousto the middle, “planar” fracture zone discussed inAnderson et al. (1999). Figure 3 shows the differences

in dimensions for tumuli in Hawaii and at Mount Etnathat may relate to formation history, with tumuli at Etnadisplaying significantly thicker crusts. Many tumuli inour study displayed lava outpourings in the centralfracture, although we did not systematically note thisfor each tumulus, and we did not observe any tumulithat were extensively lava coated.

Modeling

Rossi and Gudmundsson (1996) proposed a model of abending elastic plate to estimate the average inflation pres-sure needed to create tumuli. Specifically, they used a cir-cular plate with simply supported edges and assumed smalldeflections. Using idealized parameters for the thickness ofthe viscoelastic crustal thickness, height, and diameter oftumuli, they estimate a magmastatic pressure of 0.2–1 MPa.For a density of 2,500 kg/m3, this requires a magmastatichead of 8–40 m. However, some field settings may notprovide sufficient relief to produce head values above5 m (see Duncan et al. 2004), and our observations ofsmall lava outpourings on actively inflating tumulioccurring only 1–2 m about the flow surface and smalloutflows on older tumuli confined to the central fracturesuggest that, in some cases, magmastatic head valuesduring emplacement may be up to an order of magni-tude less than the values originally suggested by Rossiand Gudmundsson (1996). The pressure must be greatenough to locally lift and bend the crust above the adjacentflow surface, but in some cases, it may not have to exceed theheight of the tumulus. For those tumuli that are extensive-ly lava coated (Rossi and Gudmundsson, 1996) or per-haps not lava coated at all [where the base of the entirebending crust stays under compression during formation(Gudmundsson 1999; Andrew and Gudmundsson2007)], magmastatic head values that exceed the tumu-lus height may be more appropriate. Therefore, modifi-cations are needed in order to make the modelingapplicable to tumuli that display field relations thatvary from the idealize tumulus used by Rossi andGudmundsson (1996).

As a starting point, we use our field data with the Rossiand Gudmundsson (1996) to calculate the pressure neededto form each tumulus. As will be shown, the model providesreasonable values of magmastatic head (enough to exceedthe strength of the crust but does not exceed the topographicrelief available to drive the flow downhill) for many, but notall, tumuli. In some cases, the model predicts magmastaticpressures that are not enough to deform the crust and, inother cases, predicts very high values that cannot be gener-ated topographically. We therefore investigate a number of

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Table 1 Tumuli data

Flow Tumulus number Length (m) Width (m) Elastic crust (m) Viscoeastic crust (m) Wmax Magmastatic head (m)

Etna 1792 2 31 17 0.03 0.24 3 0.0045

Etna 1792 3 23 13 0.32 0.42 6 32.31

Etna 1792 6 11 7 0.07 0.23 2.5 1.67

Etna 1792 8 101 42 0.65 0.6 9 3.72

Etna 1792 15 28 14 0.07 0.24 3 0.12

Etna 1792 16 40 17 0.14 0.9 4 0.61

Etna 1792 17 68 17 0.1 0.4 5 0.28

Etna 1792 18 15 15 0.36 0.9 3.5 15.14

Etna 1792 28 75 16 0.08 0.65 5.75 0.21

Etna 1792 21c 29 17 0.22 0.26 6.5 3.88

Etna 1792 5 31 9 0.2 0.4 4 22.89

Etna 1792 7 11 7 0.64 0.15 2.5 1281.26

Etna 1792 9 63 18 0.06 0.4 7 0.067

Etna 1792 10 33 18 0.21 0.32 4.5 1.86

Etna 1792 11 136 50 0.09 0.47 3.5 0.0019

Etna 1792 12 37 18 0.06 0.22 6.5 0.062

Etna 1792 13 33 10 0.02 0.2 3.8 0.014

Etna 1792 14 34 18 0.27 0.35 6.5 5.72

Etna 1792 21 51 12 0.02 0.6 3.75 0.0067

Etna 1792 22 24 13 0.1 0.55 2.75 0.45

Etna 1792 23 17 8 0.02 0.26 3 0.027

Etna 1792 24 24 15 0.02 0.4 4.5 0.0033

Etna 1792 27 29 14 0.22 0.78 4.75 6.18

Etna Bronte 1 17.5 9 0.138 0.094 3 5.64

Etna Bronte 2 10 8 0.01 0.03 4 0.0045

Etna Bronte 3 25 18 0.132 0.39 4 0.41

Etna Bronte 4 30 15 0.01 0.55 4 0.00037

Etna 1983 3 35 18 0.068 0.41 8 0.11

Etna 1983 3d 23 18 0.096 0.442 3 0.11

Etna 1983 3c 10 6 0.138 0.094 3 28.55

Etna 1983 3a 35 18 0.57 0.094 8 66.24

Etna Maletto 7 29 17 0.56 0.55 2.25 0.00075

Etna Maletto 10 21 18 0.26 0.27 3.75 22.20

Etna Maletto 11c1 47 19 0.26 0.49 4.75 0.11

Etna Maletto 11c2 47 19 0.5 0.17 4.75 1.14

Etna Maletto 12 27 15 0.32 0.28 3 2.94

Etna Maletto 13 37 17 0.33 0.36 4.5 3.00

Etna Maletto 15 29 22 0.18 0.22 3.25 21.38

Etna Maletto 18 28 17 0.45 0.4 3.25 9.11

Etna Maletto 22 32 17 0.3 0.52 3.75 5.69

Etna Maletto 23 40 27 0.55 0.6 3.75 5.51

Etna Maletto 24 29 22 0.3 0.53 4 2.16

Etna Maletto 26 73 20 0.36 0.35 3.25 4.44

Etna Maletto 27 143 73 0.19 0.45 6.5 0.0073

Etna Maletto 28 137 39 0.2 0.45 3.5 0.056

Etna Maletto 32 64 37 0.11 0.4 3.25 0.010

Etna Maletto 33 74 41 0.37 0.32 3.5 0.29

Etna Maletto 34 26 19 0.31 0.4 2.5 2.68

Etna Maletto 35 44 23 0.16 0.6 3 0.20

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model modifications, each of which carries a set of assump-tions listed here:

1. Small-deflection elastic bending of a circular plate(Rossi and Gudmundsson 1996)

(a) Simply supported plate edges(b) Uniform overpressure is applied to the plate(c) Plate is free to rotate at the edges(d) Plate is circular(e) Pressure is continuously applied(f) Depth of pressure source has no effect(g) Ratio of maximum deflection to crustal thickness

<1 (small deflection)(h) Pressure source is as wide as the tumulus

2. Small-deflection elastic bending of a circular plate—clamped edges (Ugural 1981)

(a) Clamped plate edges (requires additional stress tobend edge of plate)

(b–h) Same as for number 13. Small-deflection elastic bending of an elliptical plate

(b) Plate is elliptical (Ugural 1981)(a), (c–h) Same as for number 1

4. Small-deflection elastic bending of a circular plate—narrow pressure source (Ugural 1981)

(h) Pressure source is at the center of the plate(a–g) Same as for number 1

5. Piston model

(a) Bending is only applicable until the edges of thetumulus fracture sufficiently to allow the plate torise simply through magmastatic uplift.

(b) The pressure need not be uniform or continuous(c) The plate may be any shape(d) The final height is determined by the magmastatic

pressure at the time the lava attains sufficientstrength through cooling to support the lifted piston.

6. Large-deflection elastic bending of a circular plate

(g) Ratio of maximum deflection to crustal thicknessexceeds 1 (large deflection)

(a–f), (h) same as for number 1

Several aspects of emplacement are not accounted for in anyof the models. For our modeling, we assume that the platesmaintain a constant thickness and strength during emplacement,though this is unlikely given our observations showing thattumuli form over a period of a day or two. During this time,additional cooling and fracturingmay change the plate thicknessand strength. In addition, we assume a constant application ofpressure, except for number 5, though this seems unlikely for a

Table 1 (continued)

Flow Tumulus number Length (m) Width (m) Elastic crust (m) Viscoeastic crust (m) Wmax Magmastatic head (m)

Etna Maletto 37 130 60 0.42 0.43 3.25 0.08

Etna Maletto 38c1 110 45 0.45 0.58 4 0.41

Etna Maletto 38c2 110 45 0.14 0.52 4 0.012

Hawaii 1994 1 31.92 17.367 0.11 0.192 2.829 0.19

Hawaii 1994 2 24.78 7.34 0.053 0.127 1.394 0.33

Hawaii 1994 3 18.09 6.767 0.145 0.185 2.643 18.03

Hawaii 1994 4 7.64 4.417 0.163 0.123 4.153 221.8

Hawaii 1994 5 8.75 2.947 0.01 0.0683 0.816 0.05

Hawaii 1994 6 35.52 17.45 0.091 0.151 0.526 0.02

Hawaii 1994 7 28.43 22.53 0.181 0.281 5.3 0.57

Hawaii 1994 8 18.55 16.673 0.131 0.13 3.091 0.42

Hawaii 1994 9 5.11 2.937 0.012 0.218 1.757 0.19

Hawaii 1994 10 6.69 2.877 0.0617 0.138 0.847 13.63

Hawaii 1994 11 12.36 4.883 0.00667 0.086 0.845 0.0020

Hawaii 1994 12 21.74 5.837 0.0922 0.165 1.491 4.72

Hawaii 1994 13 24.32 6.733 0.075 0.167 1.819 1.75

Hawaii 1994 14 12.53 5.837 0.0138 0.0888 1.003 0.010

Hawaii 1994 15 7.25 3.36 0.0527 0.0683 1.089 5.87

Hawaii 1994 16 10.7 4.563 0.016 0.2 3.009 0.13

Hawaii 1801 1 108.77 32.33 0.325 0.164 4.153 0.61

Hawaii 1801 2 30.38 8.92 0.184 0.195 1.455 6.72

Hawaii 1801 3 19.23 10.1233 0.239 0.208 2.14 13.05

Hawaii 1801 4 20.42 10.84 0.161 0.217 2.61 3.70

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number of reasons. First, the lowermost zones of central cleftfractures in tumuli typically show evidence of varying stressintensities that may be related to variations in the subcrustalmovement of lava (Anderson et al. 1999). Furthermore, the

stress on a flow crust produced by the injection a unit volumeof lava will steadily decline as the flow grows—the crust on asmall flow must extend more per unit area than on a larger flowto accommodate every cubic meter of new lava (Anderson et al.

Fig. 3 Measured morphometric parameters for tumuli in this study

938 Bull Volcanol (2012) 74:931–946

2005). Moreover, the degree to which the flow grows horizon-tally will affect the rate of inflation and any obstacles or impedi-ments to lengthening and widening the flow will affectoverpressure within the extrusion. Finally, any change in theoverall flux into an inflating flow from the source region maychange the overpressure within the flow.

Rossi and Gudmundsson (1996) modeled tumuli growth assmall-deflection elastic bending of a circular plate, and thisserves as our starting point as we attempt to account for the fullrange of morphometries displayed by the tumuli in our fieldareas and look for modifications that may provide for lowerestimates of magmastatic head that may be more consistent withour observations of active and older tumuli in this study and forthose tumuli in settings where the topography is incapable ofgenerating >10m of magmastatic head. The bending moment ofa plate is described by the flexural rigidity (D) of an elastic crustover the radius of curvature. For a tumulus, the radius of curva-ture is assumed to be the height of the tumulus. We obtain D by

D ¼ Eh3

12 1� v2ð Þ ð1Þ

where E is the Young's modulus, v is Poisson's ratio, and h is theviscoelastic crustal thickness.We use values of 5GPa and 0.3 forE and v, respectively (Gudmundsson 1983; Rossi andGudmundsson 1996). The specific bendingmodel used byRossiandGudmundsson (1996) assumes simply supported plate edgesand that a uniformmagmatic overpressure is applied to the plate.A simply supported plate is free to rotate at the edges. For thismodel, themagmastatic overpressure (pm) needed to bend a platewith radius a to height of wmax tumulus is given by

pm ¼ 64Dwmax

a4ð1þ vÞð5þ vÞ ð2Þ

Assuming that the magmastatic overpressure is caused bythe downhill flow of the lava from its source vent, theequivalent magmastatic head, Δz, is

Δz ¼ pmgρm

ð3Þ

where g is the acceleration of gravity, and ρm is the densityof basaltic magma (2,500 kg/m3, Rossi and Gudmundsson1996). Substituting Eq. 2 into Eq. 3 and gives a magmastatichead of

Δz ¼ 64Dwmax

gρma4ð1þ vÞð5þ vÞ ð4Þ

The predicted magmastatic head for measured tumuli onHawaii and Etna using this model are given in Table 1 usingfield-measured thicknesses of the elastic crust to calculate Drather than the idealized elastic thickness used by Rossi andGudmundsson (1996). However, since the elastic crusts mea-sured at our tumuli (the crudely columnar zone of Anderson et

al. (1999)) are fractured prior to tumulus formation, it isdifficult to assess to what extent the elasticity is affected bythe presence of these joints, although we note that similarvalues of magmastatic head are obtained if we use the mea-sured viscoelastic crustal thicknesses (planar zone of Andersonet al. (1999) instead of the elastic crustal thickness. The curvedshape of the sides ofmost tumuli makes it clear that one or bothof these layers is experiencing bending.

Many of the predicted values using field-measured elasticcrust thicknesses are unreasonably high or low (values muchsmaller or larger than the height of the tumulus, or values manytimes greater than the relief available to drive the flow down-hill). In addition, since little, if any, lava flows out of activetumuli, and since many of our tumuli displayed some limitedlava coatings, we assume that the predicted elevation head formost of our tumuli may be closer to the range of the tumuliheights, or 1–10 m. We note here that the flanks of the tumulimay not exhibit outpourings even though pressurized lava mayexist just beneath the bending plate (as predicted by Andrewand Gudmundsson 2007). However, tumuli in the field may notform in an idealized manner and outpourings of material dooccur. For the active and older tumuli in our study, these out-pourings are confined to the lower flanks and central fracturevalleys, allowing for the possibility that magmastic head valuesapproximately equal to the tumulus height may be all that isnecessary for tumulus formation.

Most of the tumuli that have very small values of predictedmagmastatic head are those with the largest widths (see Table 1),as expected given the a−4 term in Eq. 4. Some of the widerfeatures may be better classified as lava rises in terms of theirgenesis, although the broad features measured here all display thecharacteristic cleft fracture and are therefore morphologicallyconsistent with tumuli. We discuss possible differences in theformation mechanism for these features below. Most of the nar-rower tumuli (widths <25 m) have unreasonably high predictedmagmastatic heads, with the highest predicted values in excess of1 km, as the model’s a−4 term strongly controls head estimates.

For basic models of plate bending, the approach taken byRossi and Gudmundsson (1996) predicts greater uplifts for agiven pressure than the other models we investigated. In thissection, we explore a range of simplemodels that are applicableto many tumuli but require a larger magmastatic head toproduce a given tumulus height (Fig. 4). Equations for differentplate configurations are taken from Ugural (1981). Our objec-tive is to determine if different models can bring more of thepredicted elevation heads into a reasonable range. The goal ofthe modeling is not to perfectly predict the magmastatic pres-sure needed to form tumuli. Instead, understanding the modelmodifications needed to predict reasonable pressure estimatesallows us to better understand processes that govern tumuligrowth under various conditions.

In some cases of tumulus formation, a model in which theedges of the plate are clamped, rather than simply supported,

Bull Volcanol (2012) 74:931–946 939

better describes the process of formation. In this model,additional pressure is required to bend the plate near its edges.This may be applicable for flow lobe tumuli where the edgesof the flow lobe must bend without significant fracturing thatwould breach the pressure source and cause breakouts. Themagmastatic pressure required for bending a circular platewith clamped edges to a height wmax is given by

pm ¼ 64Dwmax

a4ð5Þ

For a Poisson’s ratio of 0.3, the pressure to bend both thetumulus along its axis and at its edges to a obtain a given

height is approximately a factor of 4 greater than if the edgesare assumed to rotate freely, which translates most of thevalues of predicted magmastic head that are <1 m using amodel with simply supported edges to above 1 m using theclamped edge model.

Predictions of excessively large magmastatic heads arereduced to more reasonable values for many tumuli using anelliptical rather than circular plate. Most tumuli are ellipticalin shape. Of the 72 tumuli in this study, 34 have a length towidth ratio >2 and 63 have a ratio >1.5. For an ellipticalplate with clamped edges, the predicted magmastatic head isgiven by

Fig. 4 Comparison ofelevation head required fordifferent bending models as afunction of width for allmeasured tumuli in Hawaii.Lower dashed line marks 1 m ofmagmastic head, and the upperdashed line represents 10 m ofhead, the boundaries we assumeare reasonable for the majorityof tumuli in our field settings.Larger head values may bereasonable for tumuli fromother volcanoes. Red symbolsindicate Etnean tumuli, whereasblue symbols representHawaiian tumuli

940 Bull Volcanol (2012) 74:931–946

pm ¼ wmax8D3

b4þ 2

a2b2þ 3

a4

� �ð6Þ

where b is the minor axis, and a is the major axis. For agiven width, bending an elliptical plate requires less pres-sure per unit area than bending a circular plate. Thus, for a02b, the predicted magmastatic pressure is a slightly less thanhalf (29.5/64) that predicted for the case when a0b. For themaximum length/width ratio observed in this data set, 4.7,the predicted magmastatic pressure for an elliptical model is∼0.4 that of a circular plate, assuming clamped edges forboth. This model does not preferentially provide a better fit(one with a magmastatic head between 1 and 10 m) for moreelliptical tumuli. However, it does slightly reduce the prob-lem of very large values of magmastatic heads for some ofour measured tumuli.

If the pressure source is narrower than the actual platethat is lifted, the pressure source has to be considerablyhigher than if it covers the entire width of the plate. Themagmastatic pressure for a circular plate with simply sup-ported edges and a pressure source at the center of the platesof radius c, where c<a is given by (Ugural 1981):

pm ¼ wmax64D

a4þ 8D

a48þ 8

3c4a

c

� �4� 2

a

c

� �2� 2

a

c

� �4ln

a

c

� �� �� �

ð7ÞFor example, if the pressure source is half the width of

the plate that is lifted, the term in curved brackets will be∼44, making the pressure required to produce uplift appre-ciably higher.

Examining a wider range of bending models shows thatthe magmastatic pressure can easily be increased to reason-able levels by incorporating a larger range of tumuli geom-etries and edge conditions, or reduced by accounting forshape. However, these models still predict an unreasonablylarge magmastatic head (>103–104 m) for some tumuli inour study (Fig. 4). For these tumuli, equations of bendingmay not be applicable at all stages of formation. The prom-inent fractures that define tumuli indicate that the uplift ofthe plate causes bending stresses in excess of the strength ofthe plate. Once the tumulus crust has fractured through theelastic and viscoelastic crust, the internal pressure of theflow may only act to lift the plates, without the need toinvest additional stress for further bending, if the tumuluscrust is able to detach from the flow along the edges. In thisstage of development, the tumulus height is equivalent to themagmastatic head. We demonstrate this “piston model” byshowing that the measured crust is not thick enough tosupport the maximum bending stress predicted as a functionof the measured height and width. Under most assumptionsfor reasonable geometry and tensile strength, the maximumstrength of fractured basalt is estimated to be between 1 and6 MPa (Rossi and Gudmundsson 1996). The maximum

bending stress of the plate is a function of the assumedgeometry and boundary conditions, as in the models de-scribed above. For a circular plate with simply supportededges, the maximum bending stress is given by

σmax ¼ 3ð3þ vÞ8

pma

t

� �2ð8Þ

Substituting in D and pm from Eqs. 1 and 2 above,respectively, we can write the maximum stress as a functionof the height, width, flexural rigidity, and crustal thicknessof the tumulus:

σmax ¼ 2ð3þ vÞð5þ vÞ

ð1þ vÞð1� v2Þ

Etwmax

a2ð9Þ

If we assume the larger maximum stress value of 6 MPa(Rossi and Gudmundsson 1996), we can solve for the crust-al thickness at which the strength of basalt will be exceeded:

t ¼ ð5þ vÞð1� v2Þ2ð3þ vÞð1þ vÞ

6x106a2

Ewmaxð10Þ

This predicted value of crustal thickness at failure isshown relative to the measured elastic and viscoelasticcrustal thickness (Fig. 5). Figure 5 shows that the measuredcrustal thicknesses for both the elastic or viscoelastic crustsare far larger than the crustal thickness at which failure ispredicted. Similarly, we can use measured values of crustalthickness and solve for the height at which failure is pre-dicted (maximum tensile stress reaches 6 MPa). Again, wefind that the measured values of height are much larger(with two exceptions) than those needed for failure. Thesecomparisons show that failure of the crust, forming the axialcrack and perhaps additional cracks, happens well before atumulus reaches its final height or crustal thickness, andtherefore, the bending equations may only be applicableduring the early stages of formation. It follows that tumuliforming in this manner may have morphologies that differ,especially along their perimeters, from those that formedsolely from bending of the crust.

A further indication that bending equations have limitedapplicability beyond the early stages of inflation is that thesimple models described are valid only for small deflectionsand become inaccurate when the ratio of maximum deflectionto crustal thickness exceeds 1 (Ugural 1981). This is the valueat which large deflection models, which include the effects ofmembrane stresses, are needed to fully describe the deforma-tion. This ratio is >3 for all the tumuli in this study, whether aelastic or a viscoelastic crustal thickness is used. Using theformulation for small deflections results in a serious over-estimate of the pressure required to form tumuli, as the pres-sure is a quadratic rather than linear function for large deflec-tions. If we were to use large deflection bending models tocalculate the implied magmastatic head, the values increase by

Bull Volcanol (2012) 74:931–946 941

an order of magnitude or more. If we assume a circular platewith simply supported edges, for a ratio of the deflection to thecrustal thickness of ∼1.1, the pressure for large deflectionbending needed to achieve a given uplift is twice that for smallbending due to the additional effect of membrane stresses. Theincrease in the amount of pressure needed to create bending isan exponential function of the ratio of deflection to crustalthickness and increases very rapidly for ratios above 1.1.Thus, large-deflection models may be needed to estimateformation pressures of tumuli that form in flows with thick,strong crusts that resist fracture and bend through a proportionof the emplacement process.

In summary, models that incorporate clamped edges andpressures sources that are narrower than the width increasethe pressure requirements for tumulus formation. Models forelliptical plates, broken or detached edges, and large-deflection bending require less magmatic overpressure thanin the original model of Rossi and Gudmundsson (1996).These model modifications provide us with the quantitativetools for evaluating tumuli that depart from the idealizedconfiguration of a small-deflection bending of a circularplate that is simply supported along its edges.

Discussion

One of the primary goals of this work is to better understandthe subcrustal flow structure and sources of localized

magmatic overpressure of an inflating lava flow through acombination of observations, field measurements, and mod-el modifications. The observations of tumuli forming athorizontal or vertical bends in a flow pathway are consistentwith previous studies that suggest that there is a clear linkbetween tumuli and the nature of the subcrustal flow path-ways (Walker 1991; Anderson et al. 1999; Duncan et al.2004; Glaze et al. 2005). They are also consistent withmodels of pipe flow that show higher pressures upstreamfrom bends in flow path and higher flow velocities along theoutside margins of a pipe bend when carrying pressurizedfluids (Berger et al. 1983 and references therein). For thosetumuli that form where there are no clear vertical or hori-zontal bends, it is conceivable that restrictions in pathwaydiameter, changes in the cross sectional shape of the path-way, or increased cooling and crystallization within a tubecould also result in large pressure gradients that may giverise to localized inflation. Therefore, changes in the dimen-sions of, or viscosity gradient within, the pathway couldarise in response to the underlying topography (resulting inchanges of flow direction and constrictions) or emplacementhistory (such as changes in extrusion rate and cooling) andform tumuli. In all cases, however, a preexisting flow path isneeded in order to concentrate and increase the pressurelocally to form tumuli.

The tumulus depicted in Fig. 2 formed along a sharp(∼90°) bend in a flow lobe. However, the actual upliftbecame apparent approximately 24 h after the formation of

Fig. 5 Predicted values ofcrustal thickness at failure isshown relative to the measuredelastic and viscoelastic crustalthickness. For nearly everytumulus in this study, themeasured crust is larger thanthat predicted for failure. Thissuggests that crustal failure willoccur well before a tumulusreaches its final height orcrustal thickness, and therefore,the bending equations may onlybe applicable during the earlystages of formation for sometumuli

942 Bull Volcanol (2012) 74:931–946

the lobe, and by this time, the entire area within 50 m of thisbend had been resurfaced with new flows. If we had notobserved the area a day earlier, we would not have knownthat the tumulus was related to a particular lobe, whichshould serve as a warning to those who are attempting todelineate between flow lobe tumuli and shallow slopetumuli in the field. For this particular tumulus, it is not clearwhether the edges of the original flow lobe and pahoehoetoes remained intact and later inflated or whether thesemargins were partially “resorbed” as suggested by Self etal. (1998) and described by Burkhard (2003) as new flowsquickly advanced over the original flow lobe margins,allowing the flow path to act as a hot, viscous finger uncon-fined by the original flow margins.

Envisioning the inflated flow interior as a network ofintertwined viscous fingers provides a conceptual frame-work for integrating our observations into a discussion ofthe kinematics of tumulus formation. Viscous fingers maybe best thought of as flow paths with compliant walls thatrespond to changes in subcrustal flow conditions. In orderfor localized inflation to occur, it must be easier for thecompliant pathway to extend vertically and bend the over-lying crust rather than extend horizontally. It seems reason-able that most viscous fingers will extend horizontallyagainst viscous material rather than vertically (which wouldrequire bending and or breaking an elastic or viscoelasticcrust), which suggests that tumulus growth requires a spe-cial set of conditions where horizontal accommodation ofviscous finger movement is inhibited. Pre-eruption topo-graphic obstacles could provide resistance to horizontalmovement within a viscous flow interior, which may ex-plain the location of the 2007 tumulus that formed above a1-m deep depression in the pre-eruption topographic sur-face. For tumuli that do not form over any obvious topo-graphic obstacle or bend in a flow path, we suggest thatvariations in the properties of the lava in the flow interiorform some resistance to horizontal movement. When pahoe-hoe flows grow at their fronts through the formation of toesand small breakouts/channels, the outer crust or channelwalls cool relatively quickly. As the flow continues togrow, the cooled crusts of these small channel wallsor toes are either buried by new material or partiallyresorbed by the flow as inflation breaks down the smallsepta between these features (Self et al. 1998). Howev-er, the physical properties of these once-cooled featuresshould provide some strength and resistance within theflow and inhibit some horizontal movement within themostly liquid interior of an inflating flow. Therefore,based on the locations of the tumuli observed duringemplacement, we envision that the interior of mostinflated flows are most likely networks of thermallyand mechanically preferred pathways with compliantwalls that respond in size and shape to changes in flow

dynamics, separated by stronger yet still molted materi-als representing resorbed septa between once-cooledsurface features.

In sheet flows (Hon et al. 1994), the zone of verticalgrowth is spread more evenly across the flow surface,whereas tumuli grow at discrete locations. This suggeststhat the plumbing systems within hummocky flows thatdisplay tumuli are different from those in sheet flows. Sev-eral authors (Anderson et al. 1999, 2000, 2005; Duraiswamiet al. 2001; Schaefer and Kattenhorn 2004) have suggestedthat the development of small viscous fingers (consistentwith the transient tubes of Walker 1991) in the interior,which localize flow and impart stress on the overlying crust,is consistent with the development of the characteristichummocky appearance of hummocky flows and their asso-ciated fractures, whereas broader, less localized thermallyand mechanically preferred pathways may exist within sheetflows. The lack of association between tumuli and majorlava tubes (Walker 1991; Byrnes and Crown 2001) alongwith the random distribution pattern of tumuli on someHawaiian and Icelandic flows (Glaze et al. 2005) is alsoconsistent with a model of an inflating flow where viscousfingers localize stress for periods of time and then move inresponse to changing flow conditions.

Our goal in finding model modifications that lead toreasonable magmastatic head values for our tumuli is tobetter identify and quantify the processes at work in devel-oping the hummocky surfaces that typify long-durationinflating lava flows. A closer look at our field data, andtrends in the modeling results, provides several insightsregarding the emplacement of these flows. Tumulus forma-tion requires the proper combination of cooling and effusionrate. If cooling is too extensive and effusion rate too low, thecrust will provide too much resistance to bending for thelimited flow rate to overcome. If cooling is too limited andeffusion rates too high, crusts will not develop or have insuf-ficient strength to resist fracture and subsequent breakouts.Tumuli therefore require a crust-dominated flow (Kilburn1993) where vertical growth is either preferred over, or is ableto occur with, lateral growth of the flow. Therefore, environ-ments that retard the downslope flow of crust-dominatedextrusions, such as low local slopes, should favor verticalgrowth and the development of tumuli.

The application of a bending model with clamped edgesthat provides more reasonable estimates for many tumulisuggests that the surface crust of the flow is indeed bendingat the edge of a tumulus even though it contains manycrudely columnar thermal fractures that form in responseto a plane-strain tensional stress regime (Ryan and Sammis1981). This implies that these fractures do not completelyextend through the “mechanical crust” of the flow thatprovides some elastic strength to the flow surface (or elsethe crust would simply lift up in a piston fashion). Either a

Bull Volcanol (2012) 74:931–946 943

deeper viscoelastic crust is present that has not yet fracturedin response to thermal stress or the tensional fractures thatformed in response to cooling close during flexure andprovided strength under a compressive regime as the edgesof the plate flex to accommodate bending, or both. It isimportant to note that these small-deflection models mayonly be valid during the early stages of tumuli formation.

The tumuli dimensions, including crustal thicknesses, atHawaii and Etna (see Fig. 3) indicate differences in theformation history, with tumuli at Etna clearly cooling overlonger time periods. A thick elastic (crudely columnar) crustrequires a longer period of cooling prior to the onset ofbending than a thin crust and suggests that tumulus growthon Etna is slower to initiate. One possible explanation is thatthe steeper slopes on Etna provide more downslope impetusfor flow growth, therefore delaying or retarding the verticalgrowth component of these flows. In addition, viscosity mayplay a role in how the flow responds to the stress from theinjection of new lava into a flow interior.

A large proportion of tumuli measured at Etna and inHawaii have lengths and widths in the 20–40 m range(Fig. 3 and Table 1). Tumuli with smaller lengths and widthsrequire proportionally more pressure to obtain the bendingrequired to reach a given height. It follows that pressureswell in excess of the tumulus height will be required if athick, strong crust is also present. We therefore suggest thatsmall, tall tumuli will most likely form during the earlystages of emplacement of a flow when crusts are still thin.

Rossi and Gudmundsson (1996) and Duncan et al. (2004)both describe tumuli that are lava coated, and we observedseveral tumuli with small flows both on the flanks and withinthe axial fracture. We also observed an active breakout on atumulus flank that was issuing approximately 1–2m above thesurrounding flow surface on the flank of the asymmetricallygrowing tumulus described above. This illustrates that tumulimay not remain perfectly sealed throughout emplacement,although these small breakouts may not relieve enough1pressure to stop tumulus formation. The observation of asmall flank flow during the mid-stages of growth of theasymmetrical tumulus (see Fig. 2) shows that these breakoutsare not confined to the later stages of tumulus growth.

Given the constraints of the bending models, we do notfind it surprising that tumuli are rarely associated with well-established lava tubes. Long-lived lava tubes typically haverigid, walls/overlying crusts that may exceed 2 m in thick-ness and provide too much resistance to bending. Breakoutsare commonly observed occurring above long-lived tubes(e.g., Mattox et al. 1993; Kauahikaua et al. 1996; 1998;Calvari and Pinkerton 1998) where lava from within a tubetravels along fractures to the surface when flow is restricteddown-tube, when flow rate changes alter pressure conditionswithin the flow, or when changes in tube temperature alterthe thermal stress field around the tube (Dragoni and

Piombo 2009). Bending models suggest that tumulus forma-tion is favored when crusts are thick enough to remain intactand seal pressure from a feeding pathway below, but not sothick that they will not flex under the 1–5 m of minimum-required magmastatic head. Where tumuli are seen associatedwith tubes (i.e., the focal tumuli of Duncan et al. 2004), theytend to serve as sites of sustained flow activity.

One of our original goals was to obtain precise morpho-metric measurements of tumuli on a particular flow, usequantitative models to calculate the needed formation pres-sure, and determine howmagmatic overpressure varies in bothspace and time during the emplacement of an inflated lavaflow. That goal still eludes us because we are not able todefinitively constrain some parameters such as the degree towhich edges are clamped and the widths of the pressuresources in the field and whether these constraints remainconstant over time within an active field. In addition, manyof our initial assumptions may be incompatible with observa-tions made during the formation of some tumuli. Bendingmodels that allow us to consider elliptical plates and large-deflection bending provide us with better estimates for manytumuli and provide insight regarding the processes at work inthe crust and flow interior, but usingmorphometric parametersto obtain precise estimates of the pressure in the flow interior,and thus additional insights regarding the nature of sub-crustalflow pathways, remains an unrealized goal.

Why are tumuli only found on low-viscosity flows andnot their more silicic counterparts? Viscosity of the lava mayalso play a role in the concentration of stress and formationof tumuli. In order for localized inflation to occur, pressuregradients must exist in the flow resulting in magmatic over-pressures. If we assume the flow model described earlierwhere tumuli require a flow interior consisting of viscousfingers separated by stronger material that may have origi-nally been cooled septa between flow surface features, moreviscous material would decrease the viscosity contrast be-tween material in any thermally and mechanically preferredpathways and the stronger material between the pathways andcreate a more homogenous flow interior with respect to how itresponds to stress. More viscous flows may simply lack theviscosity contrasts present in basalt flows because these flowshave a higher starting viscosity and do not typically form toesand breakouts at the flow front (that allow septa betweenfeatures to later be partially reabsorbed by the flow).

Summary

& Observations of active tumuli and morphometric meas-urements of older tumuli allow us to evaluate a range ofbending models that provide insights regarding the na-ture of an active basaltic lava flow crust and sub-crustalflow structure.

944 Bull Volcanol (2012) 74:931–946

& Tumulus formation requires a sufficient cooling andeffusion rate to produce a crust that is strong enough tobend and resist breakouts. If cooling is too extensive andeffusion rate too low, the crust will provide too muchresistance to bending for the limited stress accumulatedduring inflation.

& Bending models that incorporate clamped edges andnarrow pressure sources increase the pressure require-ments for tumulus formation. Models accounting forelliptical plates, broken or detached edges, and large-deflection bending require less magmatic overpressurethan in the Rossi and Gudmundsson (1996) model.

& In order for localized inflation to occur and producetumuli, it must be easiest for a sub-crustal pathway toextend vertically and bend the overlying crust, suggest-ing that tumulus formation requires a special set ofconditions where horizontal movement is inhibited.

& Magmatic overpressure capable of producing crustalbending at specific sites can originate from bends orconstrictions along, or changes in viscosity gradientswithin, an internal flow pathway. These pathways mayhave compliant walls that respond to changes in sub-crustal flow conditions.

& Elastic crustal thicknesses at Etna generally exceedthose in Hawaii, indicating that Etna flows cool longerbefore tumuli begin growing. We suggest that the steep-er slopes on Etna favor downslope flow growth, there-fore delaying or retarding the vertical growth componentof these flows.

Acknowledgments The authors wish to acknowledge John Guest,Angus Duncan, Jeff Byrnes, David Crown, and Mike Ramsey fordiscussions in the field that helped us develop the ideas presented here.Anderson acknowledges support from grant NNG05GL55G from theNASA Mars Fundamental Research Program. We also thank AgustGudmundsson and an anonymous reviewer for their extremely thor-ough and thoughtful critiques that helped us focus and clarify thiswork.

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