Lengths and hazards from channel-fed lava flows on Mauna Loa, Hawai‘i, determined from thermal and downslope modeling with FLOWGO

Abstract

Using the FLOWGO thermo-rheological model we have determined cooling-limited lengths of channel-fed (i.e. ‘a‘ā) lava flows from Mauna Loa. We set up the program to run autonomously, starting lava flows from every 4th line and sample in a 30-m spatial-resolution SRTM DEM within regions corresponding to the NE and SW rift zones and the N flank of the volcano. We consider that each model run represents an effective effusion rate, which for an actual flow coincides with it reaching 90% of its total length. We ran the model at effective effusion rates ranging from 1 to 1,000 m³ s⁻¹, and determined the cooling-limited channel length for each. Keeping in mind that most flows extend 1–2 km beyond the end of their well-developed channels and that our results are non-probabilistic in that they give all potential vent sites an equal likelihood to erupt, lava coverage results include the following: SW rift zone flows threaten almost all of Mauna Loa’s SW flanks, even at effective effusion rates as low as 50 m³ s⁻¹ (the average effective effusion rate for SW rift zone eruptions since 1843 is close to 400 m³ s⁻¹). N flank eruptions, although rare in the recent geologic record, have the potential to threaten much of the coastline S of Keauhou with effective effusion rates of 50–100 m³ s⁻¹, and the coast near Anaeho‘omalu if effective effusion rates are 400–500 m³ s⁻¹ (the 1859 ‘a‘ā flow reached this coast with an effective effusion rate of ∼400 m³ s⁻¹). If the NE rift zone continues to be active only at elevations >2,500 m, in order for a channel-fed flow to reach Hilo the effective effusion rate needs to be ≥400 m³ s⁻¹ (the 1984 flow by comparison, had an effective effusion rate of 200 m³ s⁻¹). Hilo could be threatened by NE rift zone channel-fed flows with lower effective effusion rates but only if they issue from vents at ∼2,000 m or lower. Populated areas on Mauna Loa’s SE flanks (e.g. Pāhala), could be threatened by SW rift zone eruptions with effective effusion rates of ∼100 m³ s⁻¹.

Minor modifications to FLOWGO

A few modifications were made to the original version of FLOWGO to improve its ability to model aspects of channelized flows. First, the original version of the program included a term for conductive heat loss into the flow base but inadvertently left out such heat loss to the levees. At the vent, where without independent constraint on channel dimensions we set depth and width equal, conductive heat loss to the levees is twice that to the base. Making the change to include levee heat loss has very little effect on overall flow length when compared to the Mauna Loa, Kīlauea, and Etna test cases because along most of the flow, width is considerably greater than depth. However, its inclusion makes FLOWGO more realistic.

The second minor change has to do with the treatment of yield strength. Jeffreys equation (Jeffreys 1925) as modified by Moore (1987) is the following:

$$V_c = \text{ }\left[ {\text{d}^2 \rho _{\text{lava}} g\text{sin}\,\theta /3\eta _{\text{lava}} } \right]\left\{ {1 - 3/2\left({{{YS_{\text{core}} } \mathord{\left/ {\vphantom {{YS_{\text{core}} } {\tau _{\text{base}} }}} \right. \kern-\nulldelimiterspace} {\tau _{\text{base}} }}} \right) + 1/2\left({{{YS_{\text{core}} } \mathord{\left/ {\vphantom {{YS_{\text{core}} } {\tau _{\text{base}} }}} \right. \kern-\nulldelimiterspace} {\tau _{\text{base}} }}} \right)^3 } \right\}$$

(2)

V_c=mean velocity in the channel, d=channel depth, ρ_lava=lava viscosity, g=gravity, θ= underlying slope, η_lava=lava viscosity, YS_core=lava yield strength, and τ_base=the shear stress at the base of the flow. The portion of Eq. 2 in square brackets is the Newtonian part and that in the curly brackets is the non-Newtonian part. Note that if YS_core=0, the non-Newtonian part becomes unity.

τ_base is a limiting value in the sense that as long as the flow core is more fluid than the shear stress required to deform lava at the flow base (YS_core<τ_base), lava will flow. As soon as YS_core=τ_base (due to cooling of the flow core), then the internal strength of the lava will be greater than the stress driving deformation at its base, and flow will stop (the value in the curly brackets becomes zero).

Originally, FLOWGO determined τ_base from the following:

$$\tau _{\text{base}} = \rho _{\text{lava}} gd\,\text{sin}\,\theta$$

(3)

Although post-eruption yield strength is commonly determined from measuring the dimensions of a flow that has come to rest on a given slope (e.g. Hulme 1974), there is no physical reason why during an eruption the yield strength of flowing lava should be a function of the underlying slope. Additionally, inspection of Eq. 3 shows that as the underlying slope approaches zero, τ_base approaches zero. This causes the non-Newtonian portion of Eq. 2 to become negative and a modeled flow would stop purely due to the underlying slope, even if it has cooled only slightly.

There is another method to calculate τ_base that is temperature and crystallinity dependent and slope independent (Dragoni 1989; Pinkerton and Stevenson 1992):

$$\tau _{\text{base}} = \left\{ {6500\phi _{\text{base}}^{2.85} } \right\} + b\left\{ {\text{exp}\left[ {c\left({T_{\text{erupt}} - T_{\text{base}} } \right) - 1} \right]} \right\}$$

(4)

Here, φ_base=the mass fraction of crystals in the flow base, b and c are constants that have values of 10⁻² Pa and 0.08 K⁻¹, respectively (Dragoni 1989), T_erupt=the eruption temperature, and T_base=the temperature of the flow base. Because the temperature difference term (T_erupt–T_base) is on the order of 400°C, the exponential term becomes very large. YS_core is derived from a similar formula (Dragoni 1989; Pinkerton and Stevenson 1992):

$$YS_{\text{core}} = \left\{ {6500\phi _{\text{core}}^{2.85} } \right\} + b\left\{ {\text{exp}\left[ {c\left({T_{\text{erupt}} - T_{\text{core}} } \right) - 1} \right]} \right\}$$

(5)

Here φ_core=the mass fraction of crystals in the flow core (which is temperature-dependent and calculated by FLOWGO at each increment downflow). Importantly, the temperature difference (T_erupt−T_core) is zero at the vent and increases only slightly down flow, meaning that the exponential term here is considerably less than it is for τ_base (Eq. 4 ). Because of this difference in the exponential terms, the quantity YS_core/τ_base in Eq. 2 is always extremely small and the resulting flow is essentially Newtonian.

Additional consideration of a channelized flow indicates that it isn‘t the shear stress at the base of the flow that is critical. Instead, it is the shear stress at the base of the channel. Eq. 4 can therefore be modified to:

$$ \tau _{{{\text{base - of - channel}}}} = {\left\{ {6500\phi ^{{2.85}}_{{{\text{base - of - channel}}}} } \right\}} + b{\left\{ {{\text{exp}}{\left[ {c{\left( {T_{{{\text{erupt}}}} - T_{{{\text{base - of - channel}}}} } \right)} - 1} \right]}} \right\}} $$

(6)

The difference between the eruption temperature and that at the base of the channel (T_erupt−T_{base-of-channel}) is much closer to that between the eruption temperature and the flow core temperature (T_erupt−T_core) and therefore so are the exponents associated with these terms. The replacement of YS_core/τ_base (Eq. 2 ) with YS_core/τ_{base-of-channel} (Eq. 1) means that the yield strength portion of the velocity equation is no longer negligible. Additionally, the yield strength portion of the velocity equation is now dependent on temperature and crystallinity (the mass fraction of crystals at the base of the channel, ϕ_{base-of-channel}, is calculated by FLOWGO) and independent of the underlying slope.

In the original FLOWGO model, T_hot, the temperature of lava exposed in cracks in the surface crust, was set to always be T_core−140°C, based on field measurements at Etna and Kīlauea (e.g. Calvari et al. 1994; Flynn and Mouginis-Mark 1994). However, it seems unlikely that a constant difference in temperature would be maintained as T_core decreases downflow. Instead, the difference between T_core and T_hot (as well as that between T_core and T_{base-of-channel}) should decrease as the flow cools. At the vent, the eruption temperature T_erupt=1140°C, the solidus temperature T_solid=980°C, T_core=T_erupt, T_hot=T_core−140°C, and we consider that T_{base-of-channel}=T_hot. The starting conditions therefore are that T_hot and T_{base-of-channel} have a value of T_core−0.875(T_core−T_solid). The final minor FLOWGO change was to maintain this relative temperature spacing. This has the added benefit of preventing both T_hot and T_{base-of-channel} from reaching values less than T_solid.