Thermal erosion of felsic ground by the laminar flow of a basaltic lava, with application to the Cave Basalt, Mount St. Helens, Washington

Authors


Abstract

[1] The thermal erosion of cold felsic ground by the steady laminar flow of a hot basaltic lava is examined theoretically and experimentally. Initially, a chill layer is grown and then remelted at the base of the lava flow. A steady thermal erosion velocity is then established, which is limited by the buoyant instability of the melted ground or by the effective freezing temperature of the basaltic lava. When the theoretical analysis is applied to the longest lava tube system of the Cave Basalt on Mount St. Helens, it is found that about 100 days of flow is sufficient to produce the observed ground erosion in Little Red River, Ape and Lake Caves.

1. Introduction

[2] The thermal erosion of the underlying ground during the laminar flow of a lava has been reported at two diverse locations on Earth. The first location is on the Big Island of Hawaii, where basaltic flows about 1 m deep have been observed to erode 5–15 metres into the underlying solidified basalt over several months to create deep lava tubes (Figure 1) [Swanson, 1973; Peterson and Swanson, 1974; Kauahikaua et al., 1998]. The second location is in the summit crater of Oldoinyo Lengai, Tanzania, where carbonatite flows about 13 cm deep have been observed to erode underlying solidified carbonatite at a rate of about 2 mm/min [Dawson et al., 1990; Pinkerton et al., 1990, 1995].

Figure 1.

A photograph of a roofed lava tube, formed during the 1969–1971 Mauna Ulu eruption of Kilauea Volcano, Hawaii Volcanoes National Park. Note the ledges within the tube, indicating different levels of flows. (Photo by J.B. Judd, 21 October 1970. Reproduced from the U.S. Geological Survey Photographic Library: http://libraryphoto.cr.usgs.gov/).

[3] Motivated by these exciting field observations, Kerr [2001] conducted a theoretical and experimental investigation of thermal erosion by a laminar lava flow. He determined the final steady state thermal erosion velocity and the thicknesses (and timescales) of the associated thermal boundary layers in both the lava and the underlying ground. He also analyzed the growth and melting back of a chill layer that initially forms on the base of the lava flow. He then showed that these theoretical predictions were in excellent agreement with laboratory experiments in which hot molten wax flowed over and melted an underlying bed of solidified wax. Finally, he applied the theory to predict rates of thermal erosion consistent with the observations in Hawaii and Tanzania.

[4] In all the above work, the hot lava was assumed to be compositionally identical to the ground it was flowing over. However, in many geological situations, the lava will have a chemical composition that is markedly different to that of the underlying ground. For example, in the Cave Basalt on Mount St. Helens [Greeley and Hyde, 1972; Greeley et al., 1998], the basaltic lava flowed over and eroded a dacitic pyroclastic flow, which led to contamination of the basaltic lava [Williams et al., 2004]. The aim of this paper is to gain a dynamical understanding of this geophysically and geochemically interesting problem. I begin in section 2 by extending the theoretical analysis of Kerr [2001] to describe the thermal erosion of felsic ground by a basaltic lava. This thermal erosion is illustrated in section 3 by a laboratory experiment that examines the compositional convection produced when the erosion releases buoyant melted ground. In section 4, I summarize field observations of the Cave Basalt lava tubes and list the key physical properties of the lava and the ground. Finally, in section 5, the theory and magmatic parameters are combined to investigate the development of the main lava tube system of the Cave Basalt.

2. Theory

[5] In this section, I analyze theoretically the thermal erosion of felsic ground by the laminar flow of a basaltic lava. In this case, the freezing temperature Tf of the lava is expected to be greater than the melting temperature Tm of the ground, and the melted ground is expected to have a lower density and a much greater viscosity than the overlying basaltic lava (see Table 1 for a summary of all notation). In examining the resulting heat transfer and phase changes, I divide the temporal evolution of the lava flow into three steps: the establishment of a steady state thermal boundary layer at the base of the flow (over times of order an hour), the growth and melting back of a basaltic chill layer on the base of the flow (over times of order a week), and the final steady state thermal erosion of the felsic ground (over times of order a few months).

Table 1. Physical Parameters
SymbolUnitsDescription
cJ kg−1 °C−1specific heat of the lava (including latent heat release by crystals)
cb, cgJ kg−1 °C−1specific heats of basalt and the ground
dmdepth of the lava flow
gm s−2acceleration due to gravity
hlmthermal boundary layer thickness at the base of the lava flow
hmmthickness of the layer of melted ground
hfmmaximum thickness of the chill layer
hgmlength scale of the temperature profile in the ground
k, kgW m−1 °C−1thermal conductivities of the lava and the ground
tfstime of the maximum thickness of the chill layer
tflowstotal flow time
tgstime to grow the temperature profile in the ground
tlstime to grow the thermal boundary layer at the base of the lava flow
tmstime at which the ground underlying the chill layer begins to melt
trstime for complete remelting of the chill layer
u(z)m s−1velocity of the lava flow
qm2 s−1lava flow rate per unit width
wmwidth of the lava flow
x, zmdownstream and cross-stream coordinates
xcmdownstream distance at which Ti = Tf
C1, C2, C3, C4 constants evaluated in Figures 4 and 16 of Huppert [1989]
F contamination of the flowing basaltic lava
HW m−2heat flux to the base of the flowing lava
HmW m−2heat flux through the layer of melted ground
LJ kg−1latent heat released by the lava during chill formation
LbJ kg−1latent heat of the crystallizing phases in the basalt
LgJ kg−1latent heat of the ground
P constant taken from Kerr [1994]
Pe Péclet number
Re Reynolds number
T°Ctemperature
Tf°Cfreezing temperature of the lava
Tg, Tl°Cinitial temperatures of the ground and the lava
Ti°Ctemperature at the top of the layer of melted ground
Tm°Cmelting temperature of the ground
Um s−1surface velocity of the lava flow
Vm s−1steady state thermal erosion velocity
X, Xc crystal fraction, critical crystal fraction
α°slope of the ground
δ parameter defined in equation (12)
κ, κgm2 s−1thermal diffusivities of the lava and the ground
λmwavelength of the buoyant plumes of melted ground
μ, μmPa sviscosities of the lava and the melted ground
νm2 s−1kinematic viscosity of the lava
ρ, ρg, ρmkg m−3densities of the lava, the ground and the melted ground
equation image, equation imageg, equation imagem Stefan numbers defined in equations (18), (11), (19)

2.1. Thermal Boundary Layer

[6] Consider the steady flow shown in Figure 2, where the lava has a density ρ, viscosity μ and a uniform initial temperature Tl (at x = 0). For simplicity, the flow is taken to be two-dimensional, which is reasonable if the lava channel width w is much greater than the flow depth d [Tallarico and Dragoni, 1999]. Laminar flow down a slope α has a parabolic velocity profile

equation image

where the surface velocity

equation image

and g is the acceleration due to gravity. The flow rate per unit width is

equation image

The flow is characterized by a Reynolds number

equation image

where the kinematic viscosity ν = μ/ρ. The assumption of laminar flow is valid provided that the Reynolds number is less than about 500 [Hulme, 1973], which is typically the case for modern day terrestrial lavas [see Kerr, 2001].

Figure 2.

Schematic diagram of a laminar lava flow. The lava has a density ρ, a viscosity μ, and a thermal diffusivity κ. The two-dimensional flow has a depth d and is on a slope of angle α. The flow has a parabolic velocity profile that varies from zero in contact with the ground to a maximum velocity U at the upper surface.

[7] Since the thermal diffusivity κ of the lava is much smaller than its kinematic viscosity ν, the Péclet number of the flow

equation image

is generally very large [Kerr, 2001]. As a result, heat transfer from the lava to the ground forms a thin thermal boundary layer in which the velocity gradient can be taken to be linear. If alongstream diffusion of heat is neglected and the temperature at the base of the flow is taken to be Tf, then the advection-diffusion equation for the steady temperature field can be solved to find the heat flux to the base of the lava flow

equation image

where k = κρc and c are the thermal conductivity and heat capacity of the lava, and the constant Γ(equation image) = 0.8929795… [Kerr, 2001]. From equation (6), the thickness of the steady state thermal boundary layer can be defined as

equation image

while the timescale to grow this thermal boundary layer can be estimated as

equation image

In a typical basaltic lava tube, hl is much less than d and tl is short compared to the duration of the flow (e.g., for the Hawaiian lava tubes described by Kauahikaua et al. [1998], Kerr [2001] showed that hl ranges from about 1 to 5 cm over downstream distances x of 10 to 1000 m, while tl ranges from about 0.3 to 6 hours). As a result, the heat flux H be taken as steady when examining freezing or melting at the base of the lava flow over much longer timescales (see sections 2.2 and 2.3).

2.2. Growth and Melting Back of the Chill Layer

[8] After the lava makes contact with cold ground at an initial temperature Tg, the heat flux into the ground is initially much greater than the initial heat flux from the lava to the floor (because TfTg is much greater than TlTf). This initial difference in heat fluxes results in the growth of a chill layer at the base of the flow [Huppert, 1989; Bruce and Huppert, 1989, 1990; Petford et al., 1994; Kerr, 2001]. With time, the conductive heat flux into the ground steadily decreases until it becomes less than the steady state heat flux H from the lava to the base of the flow, which leads to melting back of the chill layer. From the work of Huppert [1989], it can be estimated that the basal chill layer has a maximum thickness

equation image

and is grown in a time

equation image

where the constants C1 and C2 are given as functions of a ground Stefan number

equation image

in Figure 4 of Huppert [1989], L is the latent heat of the lava, and kg, ρg, cg, and κg = kg/ρgcg are respectively the thermal conductivity, density, heat capacity, and thermal diffusivity of the ground. After the time tf, melting back of the basal chill layer depends on equation imageg and a second parameter

equation image

Complete remelting of the chill takes a time

equation image

but the underlying felsic ground reaches its melting temperature Tm and begins to melt at an earlier time

equation image

where the constants C3 and C4 are given as functions of equation imageg and δ in Figure 16 of Huppert [1989].

2.3. Steady State Thermal Erosion

[9] After the meltback time tr, the lava begins to erode the underlying ground at a steady velocity V. In determining this erosion velocity V for the turbulent flow of a komatiite lava, Huppert [1989, p. 307] made the assumption that the melted ground would be instantaneously ablated and mixed with the overlying flow. However, in the current case of the laminar flow of a basaltic lava, a layer of melted ground will exist between the basalt and the felsic ground, as shown in Figure 3.

Figure 3.

Schematic diagram of the eventual steady state temperature field near the base of the lava flow. The lava has an initial temperature Tl, while the ground has an initial temperature Tg and a melting temperature Tm. Between the lava and the ground lies a layer of melted ground of thickness hm. The temperature at the top of this melted layer is Ti. Also shown are the thermal erosion velocity V and the thermal boundary layer thickness hl.

[10] The thickness hm of this buoyant layer will be controlled by one of two physical constraints, which result in two distinct thermal erosion regimes that dominate at different downstream distances x (see Table 2). The first thermal erosion regime results from the constraint that the melted ground cannot rise unless its thickness is large enough that the timescale of the gravitational instabilities that drain the layer is comparable to the time to grow the melt layer [Lister and Kerr, 1989; Kerr and Lister, 1992], which results in a melt thickness given by

equation image

where the melted ground has a viscosity μm and a density ρm (which is taken to be equal to ρg), and the constant P = 6.222… when μmμ [Kerr, 1994]. Since μmμ, the buoyant melted ground is expected to rise in sheets [see Jellinek et al., 1999], with a wavelength λ ≈ 0.94 πhm [Kerr, 1994], which are expected to be oriented parallel to the flow direction of the lava [see Hart, 1971].

Table 2. Steady State Thermal Erosion Regimes
0 ≤ xxcxxc
TlTiTfTi = Tf
hm given by equation (15)hm given by equation (22)

[11] In the second thermal erosion regime (see Table 2), the temperature Ti at the top of the melt layer is constrained to be equal to the freezing temperature Tf of the lava. This regime occurs because, if Ti ever falls below Tf, then freezing will occur at the base of the basalt lava that will prevent the buoyant melted ground from rising, which in turn will force the thickness hm of melted ground to increase until Ti is equal to Tf (at which point, the basal freezing will be removed and the melted ground will be able to rise).

[12] When the downstream distance x is sufficiently large, the steady state thermal erosion is in the second regime where Ti = Tf (see Table 2), and the erosion velocity V is given by interfacial heat flux condition at the base of the flowing basaltic lava:

equation image

where Lg is the latent heat of the ground. Combining equations (6) and (16) then gives

equation image

where

equation image

is a Stefan number that is typically large [Kerr, 2001]. Similarly, another Stefan number that characterizes heat transfer in the melt layer

equation image

is also typically large, which enables the heat flux Hm through the melt layer to be accurately estimated as

equation image

When equation (20) is combined with the interfacial heat flux condition at the base of the melt layer

equation image

the melt layer thickness can be estimated as

equation image

where hg = κg/V is the length scale over which the temperature exponentially decreases in the ground underlying the melt layer and tghg2g = κg/V2 is the timescale required to establish this steady temperature profile [Kerr, 2001].

[13] If the downstream distance x is decreased, then the heat flux H and melting velocity V increase (see equations (6) and (17)). This in turn decreases the melt thickness hm given by equation (22), until it becomes equal to the thickness given by equation (15) at a critical downstream distance

equation image

where the erosion velocity is given by

equation image

[14] For downstream distances less than xc, the thermal erosion is in the first regime described in Table 2, and there are 3 unknown quantities to determine: the thermal erosion velocity V, the melt thickness hm, and the temperature Ti at the top of the melted ground layer (which lies somewhere between Tf and Tl). These quantities are linked by 3 coupled equations: the melt thickness equation (15), the interfacial heat flux condition at the top of the melt layer

equation image

and the interfacial heat flux condition at the base of the melt layer

equation image

These equations can be solved by substituting equation (15) into equation (26) to eliminate hm, leaving an expression for Ti that can then be substituted into equation (25) to yield an equation for V.

[15] Finally, I note that in the limit that x → 0, hl → 0 and TiTl (see Table 2). The heat flux through the melt layer is then given by

equation image

which, when combined with equations (15) and (21), predicts a erosion velocity at the start of the flow of

equation image

and a melt thickness of

equation image

3. Experiment

[16] In section 2.3, I explained that thermal erosion of a felsic substrate will produce a layer of buoyant melted ground that is gravitationally unstable (see Figure 3). To explore the nature of the resulting compositional convection, I conducted a simple analogue laboratory experiment, which is able to simulate the first thermal erosion regime given in Table 2.

[17] The experiment, which is sketched in Figure 4, used a channel that was 120 cm long and 4.8 cm wide. The channel had sidewalls and removable end walls made of Perspex, and an aluminum base that sat on two copper heat exchangers. The ground material was a water-soluble, polyethylene glycol wax (PEG 600) that was dyed a blue color (using a powdered dye). The wax had a melting temperature Tm of about 20°C. The molten wax was first poured into the channel as it lay horizontally, and then solidified by pumping cold coolant through the underlying heat exchangers. The temperature of the final solidified wax block was measured to be about 10°C.

Figure 4.

Schematic diagram of the apparatus used in the thermal erosion experiments.

[18] The experiment was started by removing the end walls of the channel and tilting the channel to an angle of 3.5°. A 60 wt. % aqueous sucrose solution, at a temperature Tl = 33.5°C, was then allowed to flow under gravity from an overlying reservoir through a tube onto a 20 cm long piece of polystyrene foam insulation at the upper end of the channel, from where it flowed downstream over the cold solidified bed of wax, whose thickness (2.75 cm) was initially equal to that of the polystyrene foam (see Figure 4). The experiment was video recorded from above, and photographed from above and from the side.

[19] The experimental parameters are summarized in Table 3. The volume flow rate of the sucrose solution was 19.2 ml/s. The flow in the channel was observed to be laminar, which is consistent with its Reynolds number of 17.8. The Péclet number of the flow was very large (about 4000).

Table 3. Parameter Values of the Experiment
SymbolUnitsValue
dmm3.6
gm/s29.8
wcm4.8
qcm2/s4.0
Pe 4000
Re 17.8
Tf°C−12.5
Tl°C33.5
Tm°C20
Tg°C10
Um/s0.17
α°3.5
κmm2/s0.10
μPa s0.0287
μmPa s0.178
νmm2/s22.4
ρkg/m31281
ρgkg/m31210
ρmkg/m31126

[20] In the experiment the flow of the hot sucrose solution caused the underlying wax block to be thermally eroded, at a rate that decreased with distance downstream from the polystyrene foam (Figure 5). The erosion released buoyant plumes of blue melted wax which were swept downstream as they rose into the clear sucrose solution, producing lines oriented parallel to the flow direction (Figure 6). The plumes first reached the surface of the flow about 1.5 cm downstream of the polystyrene foam (see Figure 5).

Figure 5.

Side view of first 15 cm of the wax block, after 15 min 23 s of the thermal erosion experiment. The thermal erosion has created a “lava fall” at edge of the polystryene foam insulation, below which the erosion decreases with distance downstream. Below the lava fall, buoyant plumes of blue melted wax are swept downstream as they rise into the overlying clear sucrose solution.

Figure 6.

Oblique view of the last 40 cm of the channel (after 6 min 4 s), with the end of the channel at the bottom of the photograph. The buoyant plumes of blue melted wax are swept downstream as they rise, producing lines oriented parallel to the flow direction. (The blue strip on the right is just a reflection of the flow in the sidewall of the channel.)

[21] This experiment successfully demonstrates the compositional convection that results when the melted ground is less dense but more viscous than the overlying hot fluid. In this experiment, the compositional convection controls the rate of thermal erosion at all distances along the channel (i.e. the thermal erosion is always in the first regime described in Table 2), since the freezing temperature of the eroding sucrose solution (Tf = −12.5°C) is below the melting temperature Tm of the wax ground (which ensures that the second regime in Table 2 cannot arise).

4. The Cave Basalt, Mount St. Helens

4.1. The Lava Tubes

[22] The Cave Basalt is a 0.24 km3 pahoehoe basalt flow erupted on the southwestern flank of Mount St. Helens about 1900 years ago [Greeley and Hyde, 1972; Greeley et al., 1998; Williams et al., 2004]. The flow was emplaced in a stream valley cut into dacitic block and ash flow and lahar deposits. The flow field contains numerous lava tubes [see Greeley and Hyde, 1972, Table 1]. The longest tube system discovered is about 9 km long (see Figure 7), and includes segments named Little Red River Cave, Ape Cave (Figure 8), Lake Cave, and Ole's Cave.

Figure 7.

A map of the Cave Basalt, Washington State, USA, based on new geologic mapping by Michael A. Clynne (USGS), showing the main tube system (courtesy of Dr David Williams, Arizona State University).

Figure 8.

A photograph of Ape Cave, which lies at the southern end of the Mount St. Helens National Volcanic Monument. Ape Cave is one of the longest continuous lava tubes in the continental United States, with a passageway length of about 4 km. (Reproduced from the US Forest Service website: http://www.fs.fed.us/gpnf/recreation/ape-cave/index.shtml, which provides information for visitors on hiking along the length of the cave.)

[23] In the Little Red River Cave, Greeley et al. [1998] report that the ground has been eroded “from 5 to 15.5 m over an average preflow slope of 4.5°, with the greatest erosion occurring at the upflow part of the tube where the slope was ∼10°”. Similarly, they report erosion of 2 to 6 m in Ape Cave over an average slope of 3.3°, and erosion of 1 to 4 m in Lake Cave over an average slope of 2.6°.

[24] The tube widths w reported by Williams et al. [2004] were 3.8 m in Little Red River Cave, ∼3 m in Upper Ape Cave, and ∼5 m in Lower Lake Cave. From observations of the heights of the lowest lava strandlines above the tube floors in Little Red River and Ape Caves, Williams et al. [2004] estimate that the lava flow depths were in the range of 0.8 to 1.2 m. Here I will assume the flow depth d in the Little Red River Cave was 1.0 m where the slope was 4.5°. If I then conserve the volumetric flow rate qw throughout the tube system, I find that the flow depth was 0.77 m at the start of the Little Red River Cave (where α = 10°), 1.2 m in Ape Cave (where α = 3.3° and w = 3 m), and 1.1 m in Lake Cave (where α = 2.6° and w = 5 m).

4.2. The Basalt Lava

[25] The lava flow was a high-Al basalt, with a composition of about 50 wt.% SiO2 and 17 wt.% Al2O3 [Greeley and Hyde, 1972]. Williams et al. [2004] report that the lava had an eruption temperature Tl of about 1160°C, a liquidus temperature of 1205°C, a solidus temperature of 1050°C, a density ρ of 2510 kg/m3, and a melt viscosity of 63 Pa s. Since the erupted lava contained a volume fraction X ≈ 0.3 of olivine and euhedral plagioclase phenocrysts [Williams et al., 2004], I use Marsh [1981] to infer that the lava had a bulk viscosity that was larger than that without crystals by a factor of (1 − X/Xc)2.5, where the critical crystallinity Xc = 0.6, which gives a value of μ = 356 Pa s. This bulk viscosity predicts a volumetric flow rate qw of 6.9 m3/s, which would have erupted the estimated Cave Basalt flow volume of 0.24 km3 [Williams et al., 2004, p.179] in a time of ∼400 days.

[26] The thermal conductivity of the basalt is taken to be k = 1.1 W/m/K [e.g., Murase and McBirney, 1973; Touloukian et al., 1989; Büttner et al., 1998]. Given the heat capacity of basalt cb = 1560 J/kg/K and the latent heat of the crystallizing phases Lb = 500 kJ/kg (given in Williams et al. [2004, Table 3]), the change in crystal fraction with temperature (dX/dT ≈ −1/155 K−1) implies that the lava has an effective heat capacity

equation image

of about 4800 J/kg/K. The effective thermal diffusivity of the lava is then about κ = 0.9 × 10−7 m2/s. Finally, I note that the critical crystallinity Xc = 0.6 corresponds to an effective freezing temperature Tf of 1112°C, and an effective latent heat release of L = 0.4Lb = 200 kJ/kg during the initial growth of the chill layer at the base of the lava flow.

4.3. The Felsic Ground

[27] The dacite tuff had an initial temperature Tg of about 20°C, a solidus temperature of 950°C, a liquidus temperature of 1144°C, a heat capacity cg = 1480 J/kg/K, and a latent heat Lg = 200 kJ/kg [see Williams et al., 2004, Table 3]. The ground is assumed to have a density ρg of 2100 kg/m3 [e.g., Kueppers et al., 2005; Platz et al., 2007], which corresponds to a porosity of 0.22 if the rock density is taken to be 2700 kg/m3. I estimate that the rock had a thermal conductivity of 1.8 W/m/K [based on measurements by Murase and McBirney, 1973 for a Mount Hood Andesite with 60 wt. % SiO2], which corresponds to the ground having a bulk thermal conductivity kg of 1.26 W/m/K (using equation (5') of Horai [1991] with a porosity of 0.22) and a thermal diffusivity of κg = 4.1 × 10−7 m2/s. The effective melting temperature of ground is taken to be halfway between its liquidus and solidus temperatures (i.e. at Tm = 1047°C), where the ground will consist of 39% melt, 39% crystals and 22% pore space. The melted ground layer is assumed to have a density ρm of 2100 kg/m3 and a viscosity μm of about 2 × 104 Pa s [Williams et al., 2004].

5. Thermal Erosion

5.1. Theoretical Predictions

[28] Using the theoretical analysis in section 2, together with the physical parameters of the Cave Basalt flow outlined in section 4 (see Table 4), the evolution of this lava tube system can now be quantitatively examined. In particular, I aim to determine whether the formation of these caves can be explained as being due to thermal erosion by the flowing basaltic lava.

Table 4. Basalt and Ground Parameters
SymbolUnitsValue
cJ kg−1 °C−14800
cgJ kg−1 °C−11480
gm s−29.8
kW m−1 °C−11.1
kgW m−1 °C−11.26
C1 0.35
C2 0.78
C3 3.7
C4 3.1
LJ kg−1200
LgJ kg−1200
Tf°C1112
Tg°C20
Tl°C1160
Tm°C1047
Xc 0.6
δ 0.94
κm2 s−10.9 × 10−7
κgm2 s−14.1 × 10−7
μPa s356
μmPa s2 × 104
vm2 s−10.142
ρkg m−32510
ρg, ρmkg m−32100
equation image 6.5
equation imageg 0.15
equation imagem 17.9

[29] The theoretical predictions are summarized in Table 5. The surface velocity U ranged from 3.5 m s−1 at the steep, upflow end of Little Red River Cave down to 1.9 m s−1 in Lake Cave. The Reynolds number Re ranged from 16 in Ape Cave to 10 in Lake Cave, which demonstrates that lava flow in the tube system was always in the laminar regime.

Table 5. Thermal Erosion Predictions for Little Red River Cave, Ape Cave, and Lake Cave
ParameterUnitsUpflow End, Little Red River CaveLittle Red River CaveApe CaveLake Cave
α°104.53.32.6
dm0.771.01.21.1
wm3.83.8∼3∼5
Um s−13.52.72.91.9
qm2 s−11.81.82.31.4
Re 13131610
Pe 2.0 × 1072.0 × 1072.5 × 1071.5 × 107
xcm2.51.51.30.9
xm2.5–1010–100010–100010–1000
hlcm0.5–0.91.0–4.81.1–5.01.2–5.5
tlhours0.09–0.220.32–6.90.35–7.50.43–9.3
hfcm4.9–7.89.4–439.8–4510.9–51
tfdays0.11–0.280.39–8.50.43–9.20.53–11.5
tmdays0.4–1.11.6–341.7–372.1–46
trdays0.5–1.31.9–402.0–442.5–54
Vcm d−122.3–14.111.8–2.511.3–2.410.1–2.2
hmcm0.9–1.41.7–7.81.8–8.12.0–9.1
hgm0.16–0.250.30–1.40.31–1.50.35–1.6
tgdays0.7–1.82.6–552.8–603.5–75

[30] The Péclet number was very large (Pe > 107), which resulted in the thermal boundary layer at the base of the lava flow having a thickness hl that was always small compared to the lava depth at typical downstream distances x. Although hl is proportional to x1/3 (see equation (7)), the growth of this thermal boundary layer would have been regularly interrupted at lava falls [Kerr, 2001]. Lava falls are commonly observed in Hawaiian lava tubes [e.g., Peterson and Swanson, 1974; Allred and Allred, 1997; Kauahikaua et al., 1998; Greeley et al., 1998; Porter, 2000], where they cause significant disruption to the laminar flow that often brings interior lava to the surface. Lava falls were also present in the Cave Basalt lava tubes (e.g., in Little Red River Cave [see Greeley and Hyde, 1972, Figure 5], in Ape Cave (see Figure 9, and Williams et al. [2004, p.178]) and in Beaver Cave [see Greeley and Hyde, 1972, Figure 11]). Since Kauahikaua et al. [1998] report “as many as six or more” lava falls in the 11.3 km length of recent Hawaiian lava tubes, I assume that a similar frequency of lava falls in the Cave Basalt tubes imposed a upper limit on x of about 1 km. Over distances x of 10 to 1000 m, hl ranged from 1 to 6 cm, while the predicted time tl to grow the boundary layer ranged from only 0.3 to 9 hours (Table 5).

Figure 9.

A sketch of the upper part of Ape Cave, showing the position of a 8-ft (2.5 m) high lava fall. (Reproduced from the U.S. Forest Service Website: http://www.fs.fed.us/gpnf/recreation/ape-cave/index.shtml.)

[31] As explained in section 2.2, the initial lava flow was accompanied by the growth and subsequent melting back of a chill layer on the floor of the lava channel. For the parameter values listed in Table 4, equation imageg = 0.15 and δ = 0.94, which enables the evaluation of C1 = 0.35, C2 = 0.78, C3 = 3.7 and C4 = 3.1 using Figures 4 and 16 of Huppert [1989]. Over distances x of 10 to 1000 m, the chill layer had a maximum thickness hf of 9 to 51 cm (Figure 10a) and was grown in a time tf of 0.4 to 12 days. The chill was then remelted in a time tr of 1.9 to 54 days (Figure 10b), but the underlying ground reached the effective melting temperature Tm at a slightly earlier time tm of 1.6 to 46 days (Table 5).

Figure 10.

(a) The maximum thickness hf of the initial chill layer at the base of the lava flow and (b) the time tr to grow and remelt it as a function of downstream distance x for Little Red River Cave (red solid line), Ape Cave (black dot-dashed line), and Lake Cave (blue dashed line). In Little Red River Cave, the ground slope is 10° at its upflow end (where x is taken to be 2.5–10 m) and 4.5° elsewhere (where x is taken to be 10–1000 m).

[32] After removal of the chill layer, the ground thermally eroded at a constant velocity (see section 2.3). For the parameter values listed in Table 4, the values of the Stefan numbers are equation image = 6.5 and equation imagem = 17.9. The critical downstream distance xc defined by equation (23) is found to be only 1–2 m (Table 5). For downstream distances greater than xc, thermal erosion was controlled by the constraint that Ti = Tf (see Table 2), and the erosion velocity V is given by equation (V). Over distances x of 10 to 1000 m, the erosion velocity ranged from 12 to 2.2 cm/d in the 3 caves (Figure 11a). The thickness hm of the melted ground layer ranged from 1.7 to 9.1 cm (Figure 11b), while the thickness hg of the underlying thermal boundary layer in the ground ranged from 0.3 to 1.6 m (Figure 11c). I also note that, although the thin layer of viscous melted ground was able to creep downstream, its surface velocity (m sin α hm2/2μm) was negligible (i.e. it is predicted to be a factor of 10−6 to 10−4 smaller than the lava surface velocity U).

Figure 11.

(a) The steady state thermal erosion velocity V, (b) the thickness hm of the buoyant layer of melted ground, and (c) the length scale hg of the temperature profile in the ground as a function of downstream distance x for Little Red River Cave (red solid line), Ape Cave (black dot-dashed line), and Lake Cave (blue dashed line). In Little Red River Cave, the ground slope is 10° at its upflow end (where x is taken to be 2.5–10 m) and 4.5° elsewhere (where x is taken to be 10–1000 m).

[33] To quantify the thermal erosion at the steep, upflow end of Little Red River Cave (shown in Figure 4 of Greeley et al. [1998]), I take a downslope distance x of only 2.5 to 10 m. Here the basal chill layer was thinner (hf = 5 to 8 cm; see Figure 10a) and was rapidly remelted (tr = 0.5 to 1.3 days; see Figure 10b), and the steady state erosion velocity V was greater, ranging from 22 to 14 cm/d (Figure 11a). This high erosion velocity also resulted in a thinner melted ground layer (hm ranged from 0.9 to 1.4 cm; see Figure 11b) and a thinner thermal boundary layer in the ground (hg ranged from 0.16 to 0.25 m; see Figure 11c).

[34] From Figures 1011, it is clear that the dominant factor affecting the thermal erosion in the caves is the downstream distance x from a lava fall, rather than the ground slope α (as suggested by Greeley et al. [1998, p. 27, 334]) or the tube width w.

5.2. Discussion

[35] The predicted erosion velocities for the Cave Basalt tubes listed in Table 5 are found to be comparable to the rates of thermal erosion measured in Hawaiian lava tubes. For example, Kauahikaua et al. [1998] observed an erosion rate of 10 cm/d for 6 to 7 weeks at the 450W skylight in 1994, and an erosion rate of 4.1 cm/d (i.e., ∼5 m in 4 months) at the 2010-foot skylight in 1993. Similarly, Swanson [1973] reported the erosion of as much as 15 m in about 8 months (i.e. at about 6 cm/d) in similar lava tubes.

[36] When the growth and melting back of the initial basal chill layer is allowed for, I find that the main Cave Basalt tube system is able to be formed in a flow time tflow of about 100 days (which is comfortably less than the eruption time of ∼400 days estimated in section 4.2). In this flow time, the total predicted thermal erosion V (tflowtr) is 9.8 to 1.0 m in Lake Cave, 11.1 to 1.4 m in Ape Cave, 11.6 to 1.5 m in Little Red River Cave, and 22.2 to 13.9 m at the steep, upflow end of Little Red River Cave (see Figure 12), which is reasonably consistent with the erosion observations of Greeley et al. [1998] summarized in section 4.1. I conclude that thermal erosion is able to explain the formation of the Cave Basalt lava tubes, as well as the active erosion measured in Hawaii and Tanzania [Kerr, 2001]. Hence there is no need to appeal to mechanical erosion [e.g., Williams et al., 2004] to understand these eroding lava tubes, although I have previously noted [Kerr, 2001, p. 26,461] that the presence of debris in the tube (e.g., from the collapse of walls and roof [Kauahikaua et al., 1998]) may increase the rate of thermal erosion by disrupting the thermal boundary layer at the base of the lava.

Figure 12.

Predicted thermal erosion after 100 days of flow as a function of downstream distance x for Little Red River Cave (red solid line), Ape Cave (black dot-dashed line), and Lake Cave (blue dashed line). The curvature seen at large x is due to the time tr to remove the initial chill layer becoming a significant fraction of the flow time tflow.

[37] I also note that the temperature in the melted ground layer is expected to range from Tm up to Tf, which means that this layer ranges from 39% melt, 39% crystals and 22% pore space at its base up to 65% melt, 13% crystals and 22% pore space at its contact with the overlying basaltic lava. As a result, the rising melted ground entering the overlying basalt will inject both melt and xenocrysts (or partially melted xenoliths), in agreement with the observations of Greeley et al. [1998] and Williams et al. [2004]. Underneath the melted ground layer lies a region of heated ground in which the temperature decreases exponentially from Tm to Tg with the length scale hg (see Figure 11c). This thermal boundary layer in the ground provides an explanation of country rocks underlying the tubes being “baked a brilliant red” [Greeley and Hyde, 1972, p. 2405; Williams et al., 2004, Figure 2c].

[38] Finally, if the lava is homogenized at a lava fall after a distance l, the contamination of the basaltic lava due to steady state thermal erosion can be quantified. Using equation (17), the volume fraction of felsic ground F incorporated into the flowing lava is given by

equation image

For the caves in Table 5, and for l = 1 km, F ranges from 1.8 × 10−4 in Ape Cave to 2.8 × 10−4. After the whole 9 km of this tube system, the cumulative contamination due to thermal erosion is only about 0.2%. This contrasts with the much larger lava contaminations (∼1–20%) that can be produced by the turbulent flow of high-temperature, low-viscosity Archean komatiites [Huppert and Sparks, 1985; Williams et al., 1998, 2001].

6. Conclusions

[39] In this paper, I have analyzed the thermal erosion of cold felsic ground by the laminar flow of a hot basaltic lava. Initially, a chill layer is grown and then remelted at the base of the lava flow. A steady rate of thermal erosion is then attained, which is driven by forced convective heat transfer from the lava to the ground, together with natural compositional convection as the melted felsic ground rises into the denser overlying basaltic lava. If the distance x downstream (from the lava source, or from a lava fall) is less than a critical value xc, the erosion velocity is limited by the buoyant instability of the melted ground (the first regime in Table 2). If the downstream distance x is greater than xc, the erosion velocity is limited by the effective freezing temperature of the basaltic lava (the second regime in Table 2), and is found to decrease as x−1/3 (see equation (17)).

[40] When the theoretical analysis is applied to the longest tube system of the Cave Basalt on Mount St. Helens, the thermal erosion velocities predicted are similar to those observed in Hawaiian lava tubes. I conclude that about 100 days of lava flow is able to explain the ground erosion reported by Greeley et al. [1998] in Little Red River Cave, Ape Cave, and Lake Cave.

Acknowledgments

[41] We thank Tony Beasley and Jesse Robertson for their assistance in conducting the laboratory experiment presented in section 3 and Stewart Turner, David Williams, and the reviewers for their helpful comments. I gratefully acknowledge financial support from ARC grant DP0772770.

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