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WILLIAMS ET AL.: Thermomechanical erosion of lava at the Katinniq Ni-Cu-(PGE) deposit
APPENDIX
Low-Viscosity Lava Emplacement and Erosion Model
Flow over Dry, Consolidated Substrates
The following is our base model for low-viscosity, turbulent flow emplacement over anhydrous, consolidated substrates, as described in Williams et al. (1998, 1999, 2001). Our model quantifies the evolution of the turbulent lava flow subject to steady-state erosion, since the transient growth and meltback of a chilled margin only lasts for a time of order 30 s (Huppert 1989).
Given a set of initial conditions (i.e., lava and substrate major oxide compositions; lava eruption, liquidus, and solidus temperatures; lava flow thickness, ground slope, ground melting temperature, and ambient temperature), our model calculates the initial values of important temperature- and composition-dependent thermal/rheological properties of the lava and substrate (i.e., density, viscosity, specific heat, thermal conductivity, heat of fusion) using algorithms from experimental petrology (Table 1). Auxiliary equations are then used to calculate additional lava properties, including crystallinity (X):
(A1)
bulk viscosity (mb):
(A2)
when X < 0.3 and
(A3)
when X ≥ 0.3 (Pinkerton and Stevenson 1992); flow velocity (u), friction coefficient (l), and Reynolds number (Re, for describing flow and heat transfer in wide, non-circular full lava tubes (i.e., width w > d, depth): e.g., Holman 1990) (solved iteratively):
(A4)
(A5)
(A6)
Thermal conductivity (kl, temperature-dependent)[1]:
(A7)
Prandtl number (Pr):
(A8)
forced convective heat transfer coefficient (hT):
(A9)
lava erosion rate (um): (A10)
and degree of contamination of the lava by assimilated substrate (S(x)):
, (A11)
Lava physical properties (density, specific heat, flow thickness, flow velocity, and heat transfer coefficient) are used to determine the heat loss from the model flow in a numerical solution of a first-order ordinary differential equation, which gives lava temperature as a function of distance downstream (Huppert and Sparks 1985):
(A12)
(A13)
In Equation (A12), we assume that heat loss occurs by forced convection within the turbulent lava to the top and base of the flow, and from thermal erosion that removes the underlying substrate. Heat is gained back from release of latent heat of crystallization.
The steady-state crustal thickness (hcs) at any given distance is calculated assuming a balance between the forced convective heat loss to the top of the lava flow (maintained at the lava solidus, Tsol), the conductive heat loss through the growing crust (base at Tsol, top at the contact temperature with sea water, Tcs), and the natural convective heat loss from the top of the crust to cold sea water:
(A14)
in which Tcs is given by:
(A15)
Since the thermally eroded substrate (S) will be completely assimilated into the turbulently flowing lava (Jellinek and Kerr 1999), we calculate the compositional change in the liquid lava at each model distance increment using the following mass balance expression:
(A16)
The compositional change in the liquid lava due to the crystallization of olivine (X) can also be determined using the following mass balance expression:
(A17)
Mold in Eq. (A17) is Mnew from Eq. (A16). Equation (A17) is used in conjunction with partition coefficient and stoichiometric algorithms to calculate olivine composition at each model increment. Olivine-liquid partition coefficients for Fe2+, Mg, and Ca are from Beattie et al. (1991, 1993); coefficients for Ti and Al are from Kennedy et al. (1993). Equation (A17) results in a new liquid lava composition that is used to recalculate the temperature- and composition-dependent thermal, rheological, and fluid dynamic properties of the lava at each model increment. Thus, the physical and geochemical evolution of the lava flow with distance is simulated.
Flow over Unconsolidated, Water-rich Substrates
When the substrate under a turbulent komatiite flow in unconsolidated and water-rich (i.e., a sediment), under certain circumstances disaggregation will occur due to the vaporization of intergranular water. When intergranular water in the substrate is heated by the overlying komatiite, it will rise in temperature to the boiling point (Tvap~316˚C at a pressure of 100 bars), at which point the water vaporizes. Because water expands as it changes phase from liquid to vapor, this expansion may fragment and/or fluidize the unconsolidated substrate, and could enable mixing with the lava (i.e., thermo-mechanical erosion) before subsequent melting in the lava. The energy conservation equation to describe this situation of thermo-mechanical erosion is:
(A18)
In this equation, Edg is the energy required to disaggregate the ground (i.e., to heat it up to Tvap and then vaporize the water):
(A19)
and Ehg is the energy required to heat the disaggregated and mechanically eroded ground up to the lava temperature:
(A20)
where fw is the volume fraction of water in the substrate = 0.25, Lvap is the heat of vaporization of the seawater (J/kg) and cvap is the specific heat of the water vapour (J/kg ˚C). In the heat transfer coefficient, mg is now the viscosity of the fluidized bed, ~1 Pa s (Davidson et al. 1977). The erosion rate um is given by:
(A21)
This erosion rate then determines the upflow velocity uvap of steam produced by vaporization of intergranular sea water:
(A22)
For fluidization of an unconsolidated substrate (Davidson and Harrison 1963), the steam upflow velocity must be greater than about 1/70 of the settling velocity (us) of particles in the water vapor, which depends on the particle density (rp ~2600 kg/m3 for sediment), the density of steam at depth (rvap ~55 kg/m3), the viscosity of the steam (mvap ~0.00002 Pa s) and the particle diameter (d). Our model predicts that typical steam upflow velocities are ~0.4-0.2 cm/s, which suggests that particles up to the size of very fine sand (<0.1 mm) could have been fluidized in this manner.
List of Symbols:
cb lava bulk specific heat, J kg-1 K-1.
cg substrate specific heat, J kg-1 K-1.
cl lava liquid specific heat, J kg-1 K-1.
cvap water vapor specific heat, J kg-1 K-1.
cw sea water specific heat, J kg-1 K-1.
d particle diameter, cm.
Edg energy to disaggregate ground, J m-3.
Ehg energy to heat ground, J m-3.
Emg energy required to melt substrate, J m-3.
fL fraction of heat of fusion to partially melt substrate. Typically = 0.4.
fw fraction of water in substrate.
g sea water convection parameter, -.
g gravitational acceleration, m s-2.
h lava flow thickness, m.
hcs steady-state crustal thickness, m.
hT lava convective heat transfer coefficient, J m-2 s-1 K-1.
kc lava crust thermal conductivity, J m-1 s-1 K-1.
keff lava effective thermal conductivity, J m-1 s-1 K-1.
kl lava thermal conductivity, J m-1 s-1 K-1.
kw sea water thermal diffusivity, m2 s-1.
Lg substrate heat of fusion, J kg-1.
Ll lava heat of fusion/crystallization, J kg-1.
Lvap sea water heat of vaporization, J kg-1.
Mnew mass (major oxide composition) of lava at current model timestep
Mold mass (major oxide composition) of lava at previous model timestep
Molv mass (major oxide composition) of olivine at current model timestep
l lava friction coefficient, -.
mb lava bulk viscosity, Pa s.
mg substrate melt dynamic viscosity, Pa s.
ml lava liquid dynamic viscosity, Pa s.
mvap steam dynamic viscosity, Pa s.
mw sea water dynamic viscosity, Pa s.
y ground slope, degrees.
Pr lava Prandtl number, -.
rb lava bulk density, kg m-3.
rc lava crust density, kg m-3.
rg substrate density, kg m-3.
rl lava liquid density, kg m-3.
rvap water vapor density, kg m-3.
rw sea water density at ambient temperature, kg m-3.
Q0 initial 2-D flow rate, m2 s-1.
Q(x) flow rate, m2 s-1.
Re lava Reynolds number, -.
S(x) lava degree of contamination by substrate, -.
DS change in lava contamination per model timestep, -.
t time since flow began, s.
T lava temperature, ˚C.
Ta ambient temperature of the environment, ˚C.
Tcs steady-state crustal surface temperature, ˚C.
TLiq lava liquidus temperature, ˚C.
Tmg substrate melting temperature, ˚C.
TSol lava solidus temperature, ˚C.
Tvap sea water vaporization temperature, ˚C.
u lava flow velocity, m s-1.
um erosion rate of the substrate, m s-1.
uvap steam upflow velocity, m s-1.
x distance from source vent, m.
X volume fraction of crystals, -.
DX change in lava crystallinity per model timestep, -.
x'(T) rate of change of crystal fraction with temperature.
[1] We use Equation (A7) because the very low thermal conductivities at high temperatures found by Snyder et al. (1994) may be flawed (Shore 1995).