Microscopic model of rock melting beneath landslides calibrated on the mineralogical analysis of the Köfels frictionite

Abstract

Friction rocks are produced by concentrated shear and partial melting at the Earth surface. Although the commonest examples are known from earthquake-generated layers (pseudotachylytes), such rocks may also derive from the slippage of rock avalanches. While numerous studies have been dedicated to earthquake-generated pseudotachylytes from a petrological, geochemical and physical viewpoint, fewer investigations have focused on the corresponding rock avalanche rocks (also termed frictionites), especially concerning the mechanics of formation and its relationship to the mineralogical composition of the original rock. In this work, we introduce a numerical model for the melting of a crystalline micro-breccia (gouge) in the shear layer of a rock avalanche due to heat generated at the sliding surface. The motion of the landslide is calculated and the resulting frictional heat is used to compute the melting of the crystalline gouge. We constrain the model based on calibration of the Köfels frictionite in Austria, the best-known example of landslide-frictionite association. We have collected samples of frictionite and of the original rock consisting of gneiss rich in alkaline feldspar and have analysed it chemically and mineralogically. Not only should the model calculate the temperature increase to fuse the average gneissic rock, we also require that the simulated percentages of mineral species should reasonably reproduce the data. The compositional data of both the original rock and the frictionite, whose mineralogical species have different melting temperatures, latent heats and thermal conductivities, allow us to constrain the numerical simulations. A second aspect considered in this work is the particular pumiceous texture derived from growth and coalescence of bubbles as a consequence of decreasing ambient pressure. The simulations show a satisfactory agreement with data but also discrepancies that are probably due to the limitations of the numerical model and to the irregular grain shape of the rock gouge in the real data. Simulations indicate that although temperatures were higher than the melting temperature of all the species, complete melting of the original crystals was not reached because of the limited duration of the landslide flow. The model could be of interest for future quantitative investigations of landslide frictionites.

Appendices

Appendix 1: Analytical calculation of the velocity for an exponential profile of the topography

The velocity of the centre of mass in a lumped mass model in which the centre of mass of the landslide is represented by a point moving along the terrain is calculated analytically for the case of an exponential-like topography of the form y = y ₀ e ^− Λx.

Because $\frac{\mathrm{d}U}{\mathrm{d}t}=\frac{\mathrm{d}U}{\mathrm{d}l}\frac{\mathrm{d}l}{\mathrm{d}t}=\frac{U\mathrm{d}U}{\mathrm{d}l}=\frac{1}{2}\frac{\mathrm{d}{U}^2}{\mathrm{d}l}$ where dl is the path length, and using the equation of motion (1), we find

$$ \frac{1}{2}\mathrm{d}{U}^2=g\left[ \sin \beta \kern0.28em \mathrm{d}l- \tan \phi \cos \beta \kern0.28em \mathrm{d}l\kern0.28em \right] $$

(16)

from which, considering that sin β dl = dy = (dy/dx)dx and that cos β dl = dx, it follows by direct integration that the velocity as a function of the horizontal coordinate is

$$ U=\sqrt{2g\left[{y}_0\left(1-{e}^{-\varLambda x}\right)- \tan \phi x\right]}. $$

(17)

Note that after a certain distance, the term tanϕ x prevails in the argument of the square root of Eq. (17), and thus, the velocity starts to decrease. This is noticeable in Fig. 6a for the case of highest friction coefficients for x > 1300 m.

After a distance x ₁ a linear topography with inclination β ′ is assumed. The velocity for x > x ₁ can be calculated as

$$ U=\sqrt{U_1^2+2g\left( \sin \beta^{\prime }- \tan \phi \kern0.5em \cos \beta^{\prime}\right)\left(x-{x}_1\right)\;} $$

(18)

where ${U}_1=\sqrt{2g\left[{y}_0\left(1-{e}^{-\varLambda {x}_1}\right)- \tan \kern0.1em \phi \kern0.28em {x}_1\right]}$ is the velocity at the transition point x₁. The values adopted here for the Köfels landslide are y ₀ = 600 m, Λ^− 1 = 1250 m, x ₁ = 1250 m, x ₁ = 2000 m and β ′ = −12°.

Appendix 2: Calculation of frictional heat and temperature increase for constant slope topography

Although the calculation of frictional heat production and temperature underneath a landslide requires a numerical computation, it is useful to provide an analytical solution based on simplified assumptions. In the following, the slab slides at constant slope angle β and latent heat effects due to the re-crystallization of the mineral phases are neglected. The temperature inside the computational cell of thickness δ corresponding to the shear layer underneath the slab thus varies with Eq. (4) without the latent heat term

$$ \frac{\mathrm{d}{T}_{\mathrm{S}}}{\mathrm{d}t}=\frac{J}{\left\langle \rho C\right\rangle \delta }-\frac{2\left({T}_{\mathrm{S}}-{T}_{\mathrm{EXT}}\right)}{Y\delta}\left\langle \frac{\chi }{\rho C}\right\rangle $$

(19)

where the first and last terms on the right-hand side are the energy generated by friction and the energy diffused out of the cell, respectively.

Because

$$ J= egUH\rho \tan \phi \cos \beta . $$

(20)

and considering that the velocity along slope increases in time as

$$ U=g\left( \sin \beta - \tan \phi \kern0.2em \cos \beta \right)t $$

(21)

we find the differential equation

$$ \frac{\mathrm{d}{T}_{\mathrm{S}}}{\mathrm{d}t}=\lambda\;t-\varOmega\;{T}_{\mathrm{S}}+\varOmega \kern0.1em {T}_{\mathrm{EXT}} $$

(22)

where

$$ \begin{array}{l}\varOmega =\frac{2}{Y\delta}\frac{\chi }{\rho\;C}\hfill \\ {}\lambda =\frac{e\;{g}^2\;H\; \tan \phi \kern0.24em \cos \beta \kern0.1em }{\delta C}\left( \sin \beta - \tan \phi \kern0.24em \cos \beta \right).\hfill \end{array} $$

(23)

The solution of Eq. (22) is

$$ {T}_{\mathrm{S}}=\frac{\lambda }{\varOmega^2}\left({e}^{-\varOmega\;t}-1\right)+{T}_{\mathrm{EXT}}+\frac{\lambda\;t}{\varOmega } $$

(24)

where the initial temperature is equal to the external temperature, T _S(0) = T _EXT.

For short times (t ≪ Ω^− 1), the solution Eq. (24) reduces to

$$ {T}_{\mathrm{S}}={T}_{\mathrm{EXT}}+\frac{1}{2}\lambda\;{t}^2-\frac{1}{6}\lambda\;\varOmega\;{t}^3 $$

(25)

Considering that Ω^− 1 ≈ 100–300 s, the limit (25) will be valid during a long part of the initial landslide flow. Keeping only the first two terms of Eq. (25) and using the expression for the velocity (21) gives also

$$ {T}_S = {T}_{EXT}+\frac{1}{2}\left[\frac{e \tan \phi \kern0.2em \cos \beta }{ \sin \beta - \tan \phi \kern0.2em \cos \beta}\right]\frac{H{U}^2}{C\delta } $$

(26)

which compares with the estimate given by Sørensen and Bauer (2003), who for simplicity set the square bracket in (26) to 1. Note that the first two terms in Eq. (25) do not contain the effect of heat diffusion out of the cell, as only the third does. Moreover, if δ is assumed to increase with the diffusing heat front rather than being constant, the temperature would increase as t ^3/2 (De Blasio 2007). For completeness, we provide also the opposite limit for the temperature (t ≫ Ω^− 1) in which

$$ {T}_{\mathrm{S}}=\frac{\varLambda }{\varOmega}\left(t-\frac{1}{\varOmega}\right)+{T}_{\mathrm{EXT}}. $$

(27)

Substitution of time as a function of the path L (i.e. the distance covered along the terrain) in Eq. (24) $t=\sqrt{2L/\left[\mathrm{g}\left( \sin \beta - \tan \phi \kern0.2em \cos \beta \right)\right]}$ gives also the temperature as a function of L.