Modeling lava lake heat loss, rheology, and convection



[1] Measurements at Erta Ale's lava lake and theoretical equations for lake rheology, density driven convection and thermally-driven plume ascent allow the constraint of lake dynamics. Cooling and crystallization expected from surface heat losses imply a viscosity increase from 150 Pa s to 300–1800 Pa s for cooled surface layers. Convection is expected to proceed vigorously under low viscosity conditions driving rapid (0.1–0.4 m s−1) surface motions and sluggishly under moderate-to-high viscosity conditions to drive slower motions (<0.08 m s−1). Convection is likely driven by small (∼6 kg m−3) density differences, where surface cooling can influence lake rheology and explain variable rates of surface convective motion.

1. Introduction

[2] Convection allows a lava lake to remain active. Buoyant magma ascending into the lake replaces cooled magma leaving the lake, allowing the lake to remain fluid while maintaining a small and stable (non-overflowing) volume [Francis et al., 1993]. This implies circulation between the lake and a deeper system to which it is connected via an open conduit, as well as within the lake itself. Such circulation was noted within Kilauea's Halemaumau lake during 1911–17 by Perret [1913] and Jaggar [1917]. Convection within Kilauea's lakes result in plate-tectonic-like surface motions [Duffield, 1972]. Such motions involve smooth and steady movement of crust away from upwelling zones towards sinks where crust is consumed [Duffield, 1972]. Rates of motion and vigor of convection can, however, vary over minutes-to-hours [e.g., Flynn et al., 1993]. Thermal data collected at Erta Ale (Ethiopia) during 2002 [Harris et al., 2005] reveal phases of vigorous and sluggish convection characterized by high (0.1–0.4 m s−1) and low (<0.08 m s−1) surface velocities, respectively (Table 1). I use these data to constrain lake power loss, rheological and convection regimes. My aim is to assess the lake rheology and dynamics during vigorous and sluggish phases, assuming that surface velocity is related to thin crusts forming on the exposed upper edge of a convecting layer [Harris et al., 2005].

Table 1. Lake Surface Velocity and Temperature (Tsurf) Measured During Vigorous and Sluggish Convection at Erta Ale, With Heat Flux, Cooling and Crystallization Calculated Using M of 650 and 350 kg s−1 During Vigorous and Sluggish Convection, Respectivelya
 VigorousSluggish VigorousSluggish
  • a

    Measurements are from Harris et al. [2005]. Resulting lake temperatures, crystallinities and viscosities are given on the right. ηm(T) is calculated using equation (2). This is used in equation (3) to give mixture viscosity for each temperature given no new crystal growth, i.e., a crystal content of 25% [η(ϕ = 25)]. Equation (3) is next used to calculate mixture viscosities for each crystallinity [η(ϕ)] given no cooling (i.e., T = 1190°C). In calculating η(ϕ), ϕ is obtained from phenocrysts (25%) plus new crystals grown for each case.

Velocity, m s−1  Lake Temp, °C  
   Max0.380.08   Max11721178
   Min0.040.01   Min11401128
   Mean0.120.03   Mean11601172
Tsurf, °C  Crystallinity, %  
   Max650560   Max4145
   Min440280   Min3129
   Mean540340   Mean3531
Qrad, MW  ηm(T), Pa s  
   Max37.424.8   Max6878
   Min13.34.8   Min4744
   Mean22.57.3   Mean5447
Cooling, °C  η(ϕ = 25), Pa s  
   Max5062   Max262300
   Min1812   Min181170
   Mean3018   Mean208182
Crystals grown, %  η(ϕ), Pa s  
   Max1620   Max7311301
   Min64   Min236201
   Mean106   Mean343238

2. Heat Loss and Convection at Stagnant and Active Lakes

[3] Stagnant (steadily freezing) lakes are comprised of bodies of lava that have become trapped within topographic depressions. Examples include the Hawaiian lava lakes forming within Kilauea Iki crater in 1959, Alae in 1963 and Makaopuhi in 1965. In contrast, an active lake is continually fed by fresh magma so that the lake volume is constantly replenished. This prevents freezing and allows the lake to remain molten. In stagnant cases a thick, stable roof forms below which a steadily cooling and crystallizing core convects until such time as the lake freezes solid. Such bodies thus cool in place to form lakes with thick, stationary crusts of solidified lava that form rigid lids or roofs on the convecting core [e.g., Wright and Okamura, 1977; Peck, 1978]. The lid, being a thick and solid roof, is decoupled from the fluid, convecting core and hence provides no insights into internal motions. Such scenarios have been modeled by Carrigan [1987], Worster et al. [1993], and Jellinek and Kerr [2001], where heat is lost by convection within the core and conduction across the lid. This scenario is different to that at an active lake where continuous surface renewal means that stagnant lids do not develop. Instead the surface is a thin, mobile, skin that rides upon an underlying molten layer and from which heat is lost mostly by radiation [Harris et al., 1999]. Thus, zones of surface motion form that are characterized by movement from up-welling regions, where crust forms, to sinks where crust is consumed [Duffield, 1972]. Crust thickens with the square root of time (t) as the surface cools [Turcotte and Schubert, 2002], so that crust thickens with distance from the source [Harris et al., 2005]. Crusts were observed to thicken with √t at Alae and Makaopuhi lakes [Peck et al., 1966, Wright and Okamura, 1977].

3. Crust Ages and Temperatures at Erta Ale

[4] Given the minimum recorded surface velocity at Erta Ale (0.01 m s−1) and maximum travel distance (40 m, the typical width of circulating zones [Harris et al., 2005]), the surface (upon reaching the sink) will be up to 1.1 h old. This surface crust will cool by radiation, behaving similarly to newly exposed pahoehoe where, although rapid initial cooling will form a glassy coating, the surface layer does not cool below 800°C until 10–12 minutes after formation [Hon et al., 1994]. The surface crust is composed of a ductile, visco-elastic, layer whose upper and lower contacts are defined by the 800 and 1070°C isotherms. Following Hon et al. [1994], the brittle layer (i.e., the layer between the 800°C isotherm and the surface) will have a thickness of 2.5 cm after 1.1 h, and that of the 1070°C isotherm (the base of the visco-elastic layer) will be 8 cm. Surface temperature will exponentially decay with time, with the empirically-derived model of Hon et al. [1994] being in agreement with theoretical models [Harris et al., 2007] and predicting a surface temperature of 300°C after 1.1 h. This is consistent with minimum surface temperatures recorded at Erta Ale (Table 1). The lake surface will thus be comprised of a thin hot crust behaving as a visco-elastic skin on the underlying molten layers; thickening away from upwelling zones and being consumed at sinks. The skin is thus intimately coupled to the molten underlying layers, so that its movement, temperature and age record the motion, velocity and heat flux of those layers.

4. Heat Loss, Cooling and Crystallization

[5] Measurements at Erta Ale during 2002 gave mean surface temperatures (Tsurf) of 340 and 540°C during sluggish and vigorous convection, respectively (Table 1). Given a lake area (A) of 910 m2 [Oppenheimer et al., 2004] these convert to radiative power losses (Qrad = AσTsurf4, σ being the Stefan-Boltzmann constant) of 7 and 23 MW during the two convection modes, respectively (Table 1) and compares with 10–25 MW obtained using thermal image data in 2003 by Oppenheimer et al. [2004]. Following Francis et al. [1993] cooling (ΔT) or crystallization experienced by lake lava can now be estimated:

equation image

M, f, Λ and cp being lava mass flux, mass fraction of crystallization within the lake, heat of crystallization (3.5 × 105 J kg−1) and lava specific heat capacity (1150 J kg−1 K−1). This allows power and mass flux to be plotted as a function of ΔT (Figure 1a) or crystallization (Figure 1b). Using M of 350–650 kg s−1 (calculated from SO2 data by Oppenheimer et al. [2004]), equation (1) can be solved with the assumption of no crystallization, i.e., ΔT = Qrad/Mcp. This gives a maximum bounds on ΔT of 60 and 50°C during sluggish and vigorous periods, respectively (Table 1 and Figure 1a), and compares with cooling by 40°C measured by Wright and Okamura [1977] in the upper 10 m of the lake active within Kilauea's Makaopuhi crater. Solving equation (1) for no cooling (f = Qrad/MΛ) gives maximum crystallization of 20 and 16% during vigorous and sluggish periods (Table 1 and Figure 1b). Assuming a maximum lake temperature of 1190°C (maximum obtained by Burgi et al. [2002]) cooling necessary to balance heat losses could reduce lake temperatures to 1130°C (Table 1) or, given an initial crystal content of 25% (phenocryst content of Barberi et al. [1973]), crystallinities could increase to 45% (Table 1).

Figure 1.

(a) Lava lake cooling or (b) crystallization required to balance heat losses for mass fluxes of 1–108 kg/s. Solid points locate Table 1 results and gray zone indicates the Erta Ale result field. (c) Variation in mixture viscosity with temperature for a range of crystallinities (dashed lines give relationships for crystallinities of 25, 35 and 45%). Open points give temperature dependent viscosities calculated for a crystal content of 25%; solid points give the crystal dependent viscosities calculated for T = 1190°C (Table 1). Trends (1) and (2) mark viscosity paths for decreasing temperature and increasing crystallinity, and gray zone indicates the Erta Ale result field. (d) Relationship between thermal power and plume vertical velocity calculated using equation (6) for viscosities of 1–105 Pa s (dashed lines give the relationships for viscosities of 150 and 1500 Pa s). Solid points show velocity and power combinations given in Table 1. These indicate that vigorous convection (high velocity and power flux: light gray zone) would be associated with viscosities of ∼102 Pa s and sluggish convection (low velocity and power flux: dark gray zone) would be associated with 103–104 Pa s. Movement between the two end members requires cooling and crystallization (as indicated by arrow), so that viscosity increases by up to two orders of magnitude, as expected from heat loss, cooling and viscosity calculations based on the field measurements of lake surface temperature, as we move between the two cases.

5. Temperature and Crystal Dependent Viscosity

[6] Cooling and crystallization will increase lava viscosity (Figure 1c). Following Shaw [1972], melt viscosity (ηm) can be calculated as a function of composition and temperature:

equation image

here s is the slope for the viscosity-temperature relationship, and cT and cη are temperature and viscosity dependent constants with values of 1.5 and −6.4. Using the composition of two Erta Ale lake samples given by Barberi et al. [1973] a slope of 2.3 is appropriate and gives a melt viscosity at 1190°C of 40 Pa s, increasing to 80 Pa s at 1130°C. Giordano et al. [2006] provide a non-Arrhenian model for anhydrous metaluminous melts which, for Erta Ale's composition, predicts similar but higher melt viscosities (80 Pa s at 1190°C, 150 Pa s at 1130°C). Some of this difference is due to the deploymerizing effect of water considered by Shaw [1972]. Applying Shaw [1972] under anhydrous conditions gives 50 Pa s at 1190°C and 100 Pa s at 1130°C, reducing the difference between the two models to 30–50 Pa s.

[7] The effect of crystals on mixture (melt plus crystals) viscosity [η(ϕ)] can now be assessed using the Einstein-Roscoe relationship [Marsh, 1981]:

equation image

where ϕ is crystallinity and ϕmax is the maximum crystal content that can be attained (∼60%) [Pinkerton and Stevenson, 1992]. Given a 25% phenocryst content, the 40 Pa s melt viscosity yields a minimum mixture viscosity of 150 Pa s for lava at 1190°C. While cooling by 60°C may increase the mixture viscosity to 300 Pa s, crystallization has the potential to push viscosities up to 1300 Pa s (Table 1) or more (Figure 1b). I thus define two rheological regimes for lake lava: (1) low viscosity, for newly arrived lava that has not yet undergone cooling and crystallization, and (2) moderate-to-high viscosity, for lava that has undergone cooling and crystallization (Table 2). Maximum viscosities of ∼1800 Pa s are expected for lava that has crystallized by 45% and cooled by 35°C (Table 2).

Table 2. Thermal And Rheological Conditions Calculated for High, Moderate and Low Viscosity Regimes at Erta Ale Lava Lakea
Viscosity RegimeQ, MWT, °CCrystals, %η, Pa sPr (×104)U m s−1ReRa
  • a

    Rayleigh Number (Ra) is the product of the Grashof (Gr) and Prandtl Number (Pr). Following Holman [1992], Grashof Number can be obtained from Gr = {Nu/[0.508Pr1/2 (0.952 + Pr)−1/4]}4, in which Nu is the Nusselt Number: Nu = 0.3387 Re1/2 Pr1/3 + [1 + (0.0468/Pr)2/3]1/4 in cases where Re < 5 × 105 and Re Pr > 100.

Low25119004020.301562 × 109
Low2511902515060.17242 × 108
Moderate251130–118025170–3007–100.16–0.128–207–20 × 107
Moderate101130–118025170–3007–100.10–0.086–123–7 × 107
High10119030–45700–130030–500.05–0.040.6–1.53–7 × 106
High51155451820700.010.11 × 105

6. Upwelling Zones: Density, Dimensions, and Flux

[8] I assume that upwelling zones at the lake surface relate to the arrival of ascending, buoyant spheres of low viscosity lava, and that their ascent velocity is of a similar magnitude to lateral velocities observed at the surface near these zones. Lateral velocities will likely decline as lava spreads away from the up-welling zone, and may thus represent minimum bounds, especially if surface velocity is measured at some distance from an upwelling zone feeding radial flow. Given the length scale of Erta Ale's upwelling zones (L = 1–8 m), a mean velocity (U) of 0.12 ± 0.07 m s−1 during vigorous convection (Table 1), a fluid density (ρ) of 2600 kg m−3 and a magma viscosity of 150 Pa s (Table 2), the Reynolds number is <30 (Re = ρUL/υ, υ being kinematic viscosity = η/ρ). Given that turbulent plumes are not observed until Re > ∼200 [Snyder, 1981], I assume that flow within the lake is laminar. This assumption allows calculation of the ascent velocity of buoyant spheres of low viscosity lava from [Turcotte and Schubert, 2002]:

equation image

r, g, μs and Δρp-s being sphere radius, gravity, viscosity of the surrounding fluid and density contrast between the sphere and surrounding fluid. Density difference Δρp-s can also be written:

equation image

ρs is the density of the surrounding fluid at temperature Ts, Tp is the temperature of the fluid comprising the sphere, and α is the thermal expansion coefficient (4 × 10−5 K−1) [Turcotte and Schubert, 2002]. Temperature differences between ascending fluid at 1190°C and surrounding fluid at 1130°C (Table 1) yield Δρp-s of −6 kg m−3. Given a mean velocity of 0.12 ± 0.07 m s−1 during vigorous convection (Table 1), and arrival of low viscosity lava at 150 Pa s, equation (4) gives a sphere radius of 0.6–1.2 m. These sphere radii and ascent velocities convert to volume fluxes (πr2U) of 0.1–0.8 m3 s−1, which compares with 0.13–0.25 m3 s−1 obtained from degassing rates [Oppenheimer et al., 2004] and 0.2 m3 s−1 obtained from thermal data [Burgi et al., 2002]. Buoyant ascent thus requires small (6 kg m−3 or 0.2%) density differences, with the onset of vigorous surface activity being consistent with arrival of buoyant lava to generate upwelling zones a few meters across.

[9] An alternative hypothesis is that, rather than discrete spheres, the flux of hot, buoyant magma into the lake is continuous. Such a situation gives rise to a laminar plume, which can be modeled following the treatment of Kaminski and Jaupart [2003]. In this case ascent velocity of a steady starting plume (for laminar flow with high Prandtl Numbers, Pr) can be assessed in terms of thermal power (Q) and kinematic viscosity:

equation image

in which g, α, ρ, and Cp are gravity, thermal expansion coefficient, density, and specific heat capacity, respectively. Parameter ɛ is the root of ɛ4lnɛ−2 = Pr−1, where Pr = υ/k, k being thermal diffusivity (10−6 m2 s−1). This allows U to be plotted as a function of Q and η (Figure 1d). For low viscosity conditions (Table 2), I obtain 0.17–0.3 m s−1. Increased viscosity causes calculated U to decline to 0.01–0.05 m s−1 (Figure 1d and Table 2). These results compare well with surface velocities measured during vigorous (0.1–0.4 m s−1) and sluggish (0.01–0.08 m s−1) convection, respectively (Table 1). In all cases, results are consistent with (1) convection (which requires the Rayleigh number (Ra) to exceed a critical value of ∼2000 [Carrigan, 1987]) and (2) laminar conditions, i.e., Re < 2000 (as shown by the Table 2 results). It therefore appears that the calculated viscosities and heat losses allow model-based simulations of the changing convection dynamics that fit the measured values. These good fits indicate that changing rates of surface motion may be explained by ascent and surface arrival of buoyant spheres or plumes of differing viscosity.

7. Qualitative Model

[10] Application of treatments designed for mantle plume dynamics [e.g., Kaminski and Jaupart, 2003] fit Erta Ale's dynamics well, indicating that lake dynamics are analogous to those expected for mantle plumes. On this basis a qualitative model for the possible lake dynamics is given in Figure 2. In this model, surface arrival of hot, crystal-poor (low viscosity) lava generates vigorous convection with high surface velocities (Figure 1d). Radiative heat loss from the lake surface leads to cooling and/or crystallization. This increases the viscosity and density of the surface layer causing its velocity to decrease. Following Bergantz and Ni [1999], increased crystallinity within such a roof layer may cause dripping of negatively buoyant crystal-melt material. As sketched in Figure 2, descent of negatively buoyant material may cause mixing, displacement and instabilities within the lake [e.g., Schaeffer and Manga, 2001]. The most crystalline (densest) lava sinks to the lake floor, gathering as zones of cooled lava (epimagma) as was observed at Halemaumau [Jaggar, 1917; Macdonald, 1972]. Renewed ascent of buoyant material punches through stagnating surface layers to generate new phases of vigorous surface activity. Hot, crystal-poor lava must enter the lake to replenish the volume; necessary if the lake is not to rapidly solidify. Ultimately, to conserve mass and prevent lake overflow, cooled, crystal-rich (high viscosity) lava must exit the lake at roughly the same rate as hot, crystal-poor (low viscosity) lava enters it.

Figure 2.

A model to explain variable lava lake convective regimes. Low viscosity (high temperature, crystal-poor) lava at ηL enters the lake. Material at ηL ascends to the surface where it is associated with fast surface motions. Radiative heat-loss drives cooling and crystallization, causing the viscosity and density of the surface layer to increase and the velocity at which the layer circulates to decrease. Eventually negative buoyancy causes dripping/sinking of the highest viscosity material at ηh which descends to gather at the lake floor. Displacement of moderate viscosity material by the descending, crystal-rich plume, causes ascent of moderate (ηm) and low-to-moderate (ηL-m) viscosity lava.

8. Conclusion

[11] Convection, and related surface motions, requires density differences between the surface and interior lava. Variations in the rates of the motions imply changing density and rheological conditions within convecting surface layers. I find that, while convection requires extremely small density differences, variations in the rate of convection, as expressed by surface motions, can easily be driven by cooling induced by lake surface heat losses. Transitions in the vigor of lava lake convection may thus be explained in terms of surface cooling and crystallization to increase the density and viscosity of convecting layers.


[12] This work benefited immensely from thorough reviews and insightful advice of Fred Witham, as well as reviews by Jonathan Dehn and an anonymous reviewer.