Thomas P Ferrand
email: thomas.ferrand@uni-orleans.fr

Stefan Nielsen

Loïc Labrousse

Alexandre Schubnel

Scaling seismic fault thickness from the laboratory to the field

Pseudotachylytes originate from the solidification of frictional melt, which transiently forms and lubricates the fault plane during an earthquake. Here we observe how the pseudotachylyte thickness a scales with the relative displacement both at the laboratory and field scales, for measured slip varying from microns to meters, over six orders of magnitude. Considering all the data jointly, a bend appears in the scaling relationship when slip and thickness reach ~1 mm and 100 µm, respectively, i.e. MW > 1. This bend can be attributed to the melt thickness reaching a steady-state value due to melting dynamics under shear heating, as is suggested by the solution of a Stefan problem with a migrating boundary. Each increment of fault is heating up due to fast shearing near the rupture tip and starting cooling by thermal diffusion upon rupture. The building and sustainability of a connected melt layer depends on this energy balance. For plurimillimetric thicknesses ( > 1 mm), melt thickness growth reflects in first approximation the rate of shear heating which appears to decay in ⁄ to , likely due to melt lubrication controlled by melt + solid suspension viscosity and mobility. The pseudotachylyte thickness scales with moment M0 and magnitude MW; therefore, thickness alone may be used to estimate magnitude on fossil faults in the field in the absence of displacement markers within a reasonable error margin.

Introduction

Originally evidenced in quartz-bearing rocks [START_REF] Shand | The pseudotachylyte of Parijs (Orange free State), and its relation to 'Trap-Shotten Gneiss' and 'Flinty Crush-rock[END_REF]Philpotts, 1964) and for large earthquakes [START_REF] Mckenzie | Melting on fault planes during large earthquakes[END_REF], frictional melting of the fault surface at depth is a common phenomenon (e.g. [START_REF] Toyoshima | Pseudotachylite from the Main Zone of the Hidaka metamorphic belt, Hokkaido, northern Japan[END_REF][START_REF] Kanamori | Frictional melting during the rupture of the 1994 Bolivian earthquake[END_REF]Otsuki et al., 2003;Hirose & Shimamoto, 2005a;2005b;[START_REF] Spray | Frictional melting processes in planetary materials: From hypervelocity impact to earthquakes[END_REF][START_REF] Beeler | Earthquake source properties from pseudotachylite[END_REF][START_REF] Ferrand | Dehydration-driven stress transfer triggers intermediate-depth earthquakes[END_REF][START_REF] Aubry | Frictional heating processes and energy budget during laboratory earthquakes[END_REF]. The result of this melting and fast quench is the formation of pseudotachylytes (e.g. [START_REF] Shand | The pseudotachylyte of Parijs (Orange free State), and its relation to 'Trap-Shotten Gneiss' and 'Flinty Crush-rock[END_REF][START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF] Located both in the fault vein and injection veins, they consist of either glassy or cryptocrystalline rocks (e.g. [START_REF] Toyoshima | Pseudotachylite from the Main Zone of the Hidaka metamorphic belt, Hokkaido, northern Japan[END_REF]Lin, 1994;[START_REF] Dobson | Diffusion profiles around quartz clasts as indicators of the thermal history of pseudotachylytes[END_REF][START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF] originating from the solidification of the frictional melt [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF], along with clasts remaining from the host rock.

Pseudotachylytes have also been described as the result of landslides [START_REF] Masch | Electron microscopy study of hyalomylonites-evidence for frictional melting in landslides[END_REF] and meteoritic impacts [START_REF] Dietz | Vredefort ring structure: meteorite impact scar?[END_REF][START_REF] Wilshire | Why is earthquake prediction research not progressing faster[END_REF]

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fault itself but a significant part of the damage zone, containing millimetric and micrometric faults, can remain unaltered (e.g. [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF][START_REF] Scambelluri | Fossil intermediate-depth earthquakes in subducting slabs linked to differential stress release[END_REF][START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF].

Comparing natural pseudotachylytes to their experimental analogues is key for understanding earthquakes mechanics (e.g. Di [START_REF] Toro | Natural and experimental evidence of melt lubrication of faults during earthquakes[END_REF]. Laboratory studies have been reporting pseudotachylytes in various lithologies (Table 1;Table 2; The moment magnitude MW of an earthquake is directly related to the average displacement on the fault plane, which allows geologists to give reasonable estimates of MW based on direct field observations. However, deep lithologies often exhibit a lack of clear displacement markers. Based on field measurements on pseudotachylytes in the Outer Hebrides, Scotland, [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF] showed that, at the field scale, the relationship between and follows a power law. However, field observations of pseudotachylyte thickness are to be considered carefully, because discrete accumulations of glassy or fine-grain material,

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either in pull-aparts, dilational jogs (e.g. [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF] or tensile fractures (e.g. Di Toro et al., 2005;[START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF] have to be fully documented for a thorough average thickness estimate. Earthquake scaling may help understand slip weakening mechanisms [START_REF] Abercrombie | Can observations of earthquake scaling constrain slip weakening?[END_REF]. Large and small earthquakes present striking similarities (e.g. [START_REF] Kita | Physical mechanisms of oceanic mantle earthquakes: Comparison of natural and experimental events[END_REF][START_REF] Ide | Frequent observations of identical onsets of large and small earthquakes[END_REF][START_REF] Brodsky | The importance of studying small earthquakes[END_REF] and appear to nucleate similarly [START_REF] Ide | Frequent observations of identical onsets of large and small earthquakes[END_REF][START_REF] Abercrombie | Small and large earthquakes can have similar starts[END_REF], which supports that laboratory analogues (e.g. Mw -6, fault = 1mm; Ferrand et al., 2017) may be representative of natural processes (e.g. Mw 6+, fault > 1km; [START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF]. However, it is not clear which parameters control the evolution of small-scale ruptures into larger events. Understanding to what extent laboratory analogues of earthquakes are representative of larger-scale natural events is key for the current advances in earthquakes physics and earthquake magnitude prediction [START_REF] Kilb | The initial subevent of the 1994 Northridge, California, earthquake: Is earthquake size predictable[END_REF][START_REF] Wilshire | Why is earthquake prediction research not progressing faster[END_REF].

Slip on a plane lubricated with a silicate melt depends on the effective thickness and continuity of the melt layer, and on the viscosity of the melt. In nature, however, faults are not planes. Initial fault geometry is controlled by pre-seismic creep and gouge evolution (e.g. Bayart et al., 2018) and significantly impacts fault rheology (e.g. [START_REF] Johnson | Fault geometry and slip distribution of the 1999 Chi-Chi, Taiwan earthquake imaged from inversion of GPS data[END_REF][START_REF] Nielsen | Scaling in natural and laboratory earthquakes[END_REF][START_REF] Romanet | Fast and slow slip events emerge due to fault geometrical complexity[END_REF], as it may define the rock volume to be melted. Slip lubrication leads to an asperity-scale dynamic stress drop much higher than seismological values (e.g. Passelègue et al., 2016a). Geometrical heterogeneities should control the cumulative melt volume. Strong-motion studies show that the rise time, i.e. required for slip to occur, is significantly shorter than the overall rupture duration [START_REF] Beroza | Short slip duration in dynamic rupture in the presence of heterogeneous fault properties[END_REF], which supports a "pulse" rupture type [START_REF] Heaton | Evidence for and implications of self-healing pulses of slip in earthquake rupture[END_REF].

Here, we compile a number of laboratory and field observations of pseudotachylyte thicknesses versus their associated displacements (Table 1; Table 2; Fig. 1; Fig. 2), for slip values ranging from < 10 µm to > 1m. Based on energy considerations, we investigate the possible scaling of pseudotachylyte thickness with displacement and moment magnitude. This may help understand why some small ruptures propagate into

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Field and laboratory observations

As illustrated in Fig. 1, several works have reported fault thickness and relative displacement since the observations of [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF]. We compile documented and data (Fig. 2) from both natural and experimental earthquakes, deriving from various geodynamic contexts (e.g. mantle vs. crust) or experimental techniques (e.g. stick-slip, rotary shear and D-DIA; Table 1). Friction -until melting temperature is reached -and subsequent viscous shear in the molten layer provide a heat source, which is counteracted by the latent heat of melting , the thermal diffusion into the country rock, depending on its heat capacity , and advection with the loss of melt toward lateral tensile cracks or zones of dilation (i.e. jogs or stepovers). In the following, we note the average melt volume per unit fault surface (including the injection veins, e.g. Di [START_REF] Toro | Fault lubrication during earthquakes[END_REF]) and a the residual thickness after melt injects in the cracks of the damage zone. If such injection is more frequent for large pseudotachylytes ( > 1 mm) than for small pseudotachylytes ( < 1 mm), then we may see a discrepancy where < for the former and ~ for the latter (Fig. 1a).

Data compilation

In Fig. 2 the log-log plot shows different log-linear trends and a break in slope ( ≈ 1 mm, ≈ 100 µm) suggesting a change in thermal or hydraulic regime of the dynamic system. Except for [START_REF] Barker | Pseudotachylyte-generating faults in central Otago, New Zealand[END_REF]

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This article is protected by copyright. All rights reserved. and Di [START_REF] Toro | Natural and experimental evidence of melt lubrication of faults during earthquakes[END_REF], large-scale data, i.e. > 1 mm and > 100 µm, show a relatively close fit to the power law described by [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF]:

= β • (1)
where β = 43600 m -1 . On the other hand, for < 1 mm and < 100 µm, the data deviate from the loglinear trend of eq. ( 1). Note that the discrepancy between and a is not sufficient to account for the bend at a~1 mm. For example, if the scaling observed at the sub-metric scale was valid at a larger scale, the Balmuccia HP pseudotachylyte (1.6 < < 1.9 m; ≈ 5 mm; [START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF] should have an effective thickness ≈ 1 m. There is no trace of such abundance of melt at the outcrop scale; in addition, according to first-order calculations the magnitude associated to ≈ 1 m would be larger than 9.

The data from [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF] and most data from [START_REF] Barker | Pseudotachylyte-generating faults in central Otago, New Zealand[END_REF] Regarding other experimental studies, the observations correspond to displacement as long as the samples are cut perpendicular to the fault and the uncertainty mostly relies on the "single-jerk" assumption, especially for the stick-slip design (section 4.1). Based on a limited number of FIB sections, D-DIA studies consistently report values of some tens of nanometers. Values as small as 20 nm are associated to local larger amorphous pockets along the fault, up to 200 nm thick [START_REF] Ferrand | Dehydration-driven stress transfer triggers intermediate-depth earthquakes[END_REF].

Scaling of pseudotachylyte thickness under the adiabatic assumption

Considering the rock volume enduring frictional heating as a closed system within a very short timescale, the adiabatic assumption can be considered [START_REF] Sibson | The habitat of fault-generated pseudotachylyte: Presence vs. absence of friction-melt[END_REF][START_REF] Rempel | Thermal pressurization and onset of melting in fault zones[END_REF]. For each increment of rock on the rupture path, the duration of shear is extremely short [START_REF] Heaton | Evidence for and implications of self-healing pulses of slip in earthquake rupture[END_REF]; Beroza

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& Mikumo, 1996), i.e. not the full slip timescale but the timescale of the initial shear and rupture. In other words, the local onset of efficient melt lubrication, which favors slip, actually corresponds to the local onset of cooling. Such short timescales also imply undrained conditions likely applying to the dynamic melt layer, at least for small events ( < 1 mm).

Assuming a Newtonian melt, negligible fracture energy and negligible viscoelastic effects (section 4.2), the total shear heating results from the time integral of shear power [START_REF] Nielsen | Frictional melt and seismic slip[END_REF]):

= !′# • $ !′# • %!′ = & # • % %! & • %!′ = ' • & # • % ′ = ¯• (2)
with ¯ the average shear stress during the slip at a velocity $. The heat can be partitioned into specific heat, latent heat for phase transitions (e.g. melt) and other heat sinks (e.g. diffusion, melt extraction).

Considering an adiabatic pseudotachylyte formation [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF]), all the heat ) generated in the process zone is dissipated within the molten layer on the rupture timescale and is directly related to its thickness ) :

) = • = * • • ∆, + # • ) (3)
with the latent heat of fusion, * the density, . the heat capacity and ∆, = , / -, 1 # the process-zone heating during the dynamic rupture propagation (, / : melting point; , 1 : initial host-rock temperature). This implies that the thickness of the molten layer is a linear function of as follows:

) = 2 • (4) with 2 = 3* • • ∆, + #4 ⁄
. As shown on Fig. 2, ) strongly depends on . Assuming that = 0.3

MJ•K -1 , * = 3000 kg•m -3 , . = 1 kJ•kg -1 •K -1 and ∆, = 1000 K then 2 ≈ 3.10 -3 for = 10 MPa.
The adiabaticity is satisfied if is larger than the heat diffusion distance within the duration of the slip ( ≫ √9 • : • !). However, is dynamically growing during the slip, and the ratio ! ⁄ is not necessarily constant throughout the spectrum of observed vein sizes.

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Looking at the entire dataset, the adiabatic scaling reproduces the observations pretty well (Fig. 2). This is particularly true at small scale ( < 1 mm), because of small associated time scales (10 to 1000 µs only), during which sliding is likely to occur under quasi-adiabatic and undrained conditions.

On the contrary, at a larger scale ( > 10 mm), melt can advect and temperature diffuse, to such an extent that the effective thickness of the silicate melt layer increases less with sliding (Passelègue et al., 2016b). Effects of both temperature diffusion and melt advection should depend on time and space scales, respectively via thermal diffusivity and melt viscosity. Notably, the concept of "seismic suction pump" was proposed to explain near-fault drops of fluid pressure during rupture propagation [START_REF] Sibson | Earthquake rupturing as a mineralizing agent in hydrothermal systems[END_REF] and has recently been confirmed in laboratory experiments producing stick-slip events systematically associated with near-instantaneous drops in fluid pressure [START_REF] Brantut | Dilatancy-induced fluid pressure drop during dynamic rupture: Direct experimental evidence and consequences for earthquake dynamics[END_REF]. This effect has been extended to the mobility of the rupture-induced transient melt observed at high pressure [START_REF] Ferrand | Reproduction expérimentale d'analogues de séismes mantelliques par déshydratation de l'antigorite et comparaison à des pseudotachylites naturelles[END_REF][START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF]. The link observed by [START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF] between and could be the result of a combination of the dynamic rupture process, as observed in the laboratory (e.g. Passelègue et al., 2016a) and of melt sucking in tension cracks [START_REF] Ferrand | Reproduction expérimentale d'analogues de séismes mantelliques par déshydratation de l'antigorite et comparaison à des pseudotachylites naturelles[END_REF]. Thus, the deviation in the pseudotachylyte scaling law could be interpreted as due to both scale-dependent heat conduction and melt pumping effects. Therefore, in section 3 we remove the adiabatic assumption.

Pseudotachylyte formation with heat diffusion

The bend observed in the scaling law can be attributed to the effect of melting dynamics under shear heating at large scales. At first order, the dataset seems to follow the adiabatic trend (eq. (4)) at small scales (D < 1 mm) and Sibson's empirical law (eq. (1)) at larger scales (Fig. 2). Separating the dataset between low confining pressures (< 0.5 GPa; Fig. 3) from high confining pressures (> 0.5 GPa; Fig. 4), the intersection/transition from eq. (4) and eq. (1) appears less well defined, for several reasons discussed

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To better understand the apparent discrepancy between small-scale and large-scale observations, it is

Stefan problem: heat balance at the advancing melt front

* • • ; = : • * • . • < =, => ? + - =, => ? - A (5)
The +/-signs refer to the heat flow at the melting front toward the rock and toward the melt, respectively, and ; is the advancement velocity of the melting boundary, i.e. the boundary between melt and country rock.

For simplicity we assume that density and diffusivity are the same in either melt or solid phases, and constant. It can be shown (Nielsen et al. 2010) that:

=, => ? - ≈ !# • $ !# 2 • : • * • . ( 6
)
resulting in:

* • • ; = : • * • . • =, => ? + + 1 2 • !# • $ !# (7)

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A nearly steady-state situation can be achieved after an initial transient, provided that • $ changes gradually. In such case

CD CE F + = G H
• ∆, [START_REF] Nielsen | Frictional melt and seismic slip[END_REF] and eq. ( 7) reduces to:

; !# = !# • $ !# 2 • * • + • ∆,# (8)
Importantly, time integration of eq. ( 8) yields eq. ( 4), indicating that the steady-state replicates the adiabatic situation. Indeed, the heat loss through diffusion is retrieved when the melting boundary advances, thus resulting in no net loss in steady-state. However, an initial transient can be predicted based on dimensional considerations and simple analytical solutions of the diffusion equation (Nielsen et al., 2008, eq. 24) where no steady-state has yet been reached:

I = 8 • $ • K * • + • ∆,# L (9)
Substituting = 500 MPa, $ = 1 m/s, ∆, = 1000 K and peridotite parameters in eq. ( 9) we obtain I = 0.59 mm, in agreement with I ≈ 0.1-1 mm in Fig. 4.

Modelling results

For sake of clarity, solutions of the Stefan problem are given and compared to observational data separating low (P < 0.5 GPa) and high-pressure (P > 0.5 GPa) events, respectively in Fig. 3 and Fig. 4. No steady-state is assumed in these models, where we use eq. ( 7) to obtain the velocity ;, and a numerical solution of the diffusion equation to obtain the term A time-dependent heat source !# • $ !# is imposed and the resulting !# is compared to the observed pseudotachylyte thickness # (gray curves, Fig. 3 and Fig. 4) under the assumption that

# = $ • !#; $ = 1 m • s .
We use values of , *, . and : which are broadly relevant for the variety of cases discussed here, but more closely related to peridotite.

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We test the model using a trial heat which remains relatively high and approximately constant during a very short initial slip stage where • $ = Q , subsequently switching to of the form In all three cases the initial thickness is relatively low, in agreement with D-DIA datapoints, but rapidly increases because the melting front is still in the initial transient stage. At about ≈ 0.1 -1 mm ( ≈ 10 _ m) a quasi-steady-state is reached where thickness increases as ∝ √ in agreement with the imposed power which decays in ≈ 1 √ ⁄ for curves [START_REF] Abercrombie | Can observations of earthquake scaling constrain slip weakening?[END_REF] and [START_REF] Abercrombie | Small and large earthquakes can have similar starts[END_REF], therefore closely matching the steadystate/adiabatic approximation of eq. ( 6). For case [START_REF] Andersen | Fossil earthquakes recorded by pseudotachylytes in mantle peridotite from the Alpine subduction complex of Corsica[END_REF] the thickness increases as ∝ again in agreement with the steady-state approximation of eq. ( 6) but assuming that the power is roughly constant throughout the slip.

• $ = Q • R ' S ' (Q constant in J•m -2 •s -1 )

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Once the steady-state thickness is reached ( > I ), we retrieve the trend compatible with Sibson's observation such that ``= A • / with A = V1/β ≈ 5•10 -3 m 1/2 , in agreement with stress decay ̅ ∝ 1/√ .

As mentioned above, [START_REF] Barker | Pseudotachylyte-generating faults in central Otago, New Zealand[END_REF]

Discussion

The results presented in section 3.2, if carefully considered in light of limitations of both the dataset (section 4.1) and the model (section 4.2), might yield a tool to estimate the magnitude of an earthquake by simply looking at the thickness of the resulting pseudotachylyte (section 4.3).

Limitations of the dataset

Displacement-thickness or offset-thickness datasets are presented along with associated uncertainties (section 2; Fig. 2). Here we detail, separately, the factors likely to control displacement and thickness.

Displacement estimates depend on the experimental design: for the D-DIA experiments, several lines of evidence suggest that each transformation-induced fault records a single seismic event (Schubnel et

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experimental fault segment is not confined, which induces fast extrusion of the melt out of the fault plane.

Finally, rotary-shear experiments do not reproduce dynamic ruptures but slip generated artificially by a motor, although it provides the dynamic molten layer extrusion (section 2.1). For normal stress larger than 20 MPa and abrupt accelerations to target slip rates of 3 m.s -1 , rotary-shear data are well fitted by the model presented in section 3 (Fig. 2; Niemeijer et al., 2011).

Limitations in the physical model

The occurrence and efficiency of melt lubrication depend on the melting point and melt viscosity, which are not explicitly taken into account in our model (section 3). However, these key factors are implicitly contained within the characteristic time ! c and associated distance c (onset of weakening). Studies reporting pseudotachylyte thickness and displacement data investigate various lithologies and contexts (Fig. 1, Fig. 2, Table 1), but the same physics should apply. In the same conditions of pressure, temperature, stress and strain rate, similar materials rupture in the same way. Hereafter we discuss the influence of rock composition, pressure, temperature and strain rate on the melting point and melt viscosity.

Melt viscosity is linked to rock chemistry and temperature (e.g. [START_REF] Taniguchi | Entropy dependence of viscosity and the glass-transition temperature of melts in the system diopside-anorthite[END_REF][START_REF] Spray | Viscosity determinations of some frictionally generated silicate melts: Implications for fault zone rheology at high strain rates[END_REF][START_REF] Suzuki | Viscosity of albite melt at high pressure and high temperature[END_REF]. Between 1500 and 2000°C at 1 atm, ultramafic melts are at least one order of magnitude less viscous than crustal melts with higher silica contents [START_REF] Holtz | The effects of silica and water on the viscosity of hydrous quartzofeldspathic melts[END_REF][START_REF] Karki | First principles viscosity and derived models for MgO-SiO2 melt system at high temperature[END_REF] 1991). The viscosity of dacite or albite melts drops by one order of magnitude when the pressure increases by 2 GPa [START_REF] Tinker | High-pressure viscometry of polymerized silicate melts and limitations of the Eyring equation[END_REF]. These factors contribute to a drastic reduction of the melt viscosity at depth.

Peridotitic melts may have a viscosity as low as 1 Pa.s at ambient pressure [START_REF] Toplis | Shear viscosities of CaO-Al2O3-SiO2 and MgO-Al2O3-SiO2 liquids: Implications for the structural role of aluminium and the degree of polymerisation of synthetic and natural aluminosilicate melts[END_REF], which

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drops to ~ 10 -1 Pa.s between 1 and 3 GPa [START_REF] Suzuki | Viscosity of albite melt at high pressure and high temperature[END_REF]Liebske et al., 2005), equalling diopside viscosity [START_REF] Taniguchi | Entropy dependence of viscosity and the glass-transition temperature of melts in the system diopside-anorthite[END_REF], whereas higher SiO2 contents (e.g. andesite composition) exhibit viscosities of ~10 Pa.s [START_REF] Vetere | Viscosity of andesitic melts -new experimental data and a revised calculation model[END_REF].

At high slip rates, displacement is accommodated over finite distances with extremely high strain rates, i.e. far above the conditions for viscoelasticity. Whether a melt behaves as a solid or a liquid depends on its relaxation time [START_REF] Dingwell | Structural relaxation in silicate melts and non-Newtonian melt rheology in geologic processes[END_REF][START_REF] Hayward | Rheological controls on asperity weakening during earthquake slip[END_REF]. Most geological materials have non-Newtonian viscosities, especially strain-rate dependent. For seismic slip and at temperatures above the melting temperature, this could result in viscosities being many orders of magnitude less than standard laboratory measurements [START_REF] Webb | Non-Newtonian rheology of igneous melts at high stresses and strain rates: Experimental results for rhyolite, andesite, basalt, and nephelinite[END_REF]). In addition, a moderate solid fraction may increase the viscosity of the solid+melt suspension by one order of magnitude [START_REF] Costa | Viscosity of high crystal content melts: dependence on solid fraction[END_REF][START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF].

The solid fraction is likely to be high in rocks made of minerals with very different melting point (e.g. [START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF]. Inasmuch as melt viscosity is sufficiently low, the melt flux toward these cracks may also represent an important heat sink.

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The viscosity in the fault vein is expected to increase due to both extrusion (fault thinning) and enhanced quench (solidification). The suction effect dynamically migrates as long as there is still enough lubricant to propagate the rupture. Alternative lubrication mechanisms, such as thermal pressurization [START_REF] Rempel | Thermal pressurization and onset of melting in fault zones[END_REF][START_REF] Viesca | Ubiquitous weakening of faults due to thermal pressurization[END_REF] could play a significant role if melting conditions are not fulfilled.

As mentioned in section 3, the study of [START_REF] Barker | Pseudotachylyte-generating faults in central Otago, New Zealand[END_REF] reported a pseudotachylytes population with and consistent with adiabatic formation (section 2.3, Fig. 2), indicating higher values of average shear stress > 100 MPa. According to eq. ( 3), this could also be due to a lower *, a lower or a lower ∆, for the scaling law proposed in this study. The fault zone is ≈ 1 km thick and contains ≈ 4 % of pseudotachylyte veining. Large volumes of pseudotachylytes formed in anhydrous felsic granulites, but in significantly smaller volumes in underlying amphibolite-facies rocks [START_REF] Camacho | Large volumes of anhydrous pseudotachylyte in the Woodroffe Thrust, eastern Musgrave Ranges, Australia[END_REF]. The high Si content and the anhydrous conditions probably maintained a high melt viscosity, which means that a larger melt layer is required for a given amount of slip to account for the same stress drop. In addition, the zone is characterized by numerous seismic events, some of which potentially affecting the same fault segment, consistently with repeating earthquakes inferred by seismologists (Uchida & Bürgmann, 2019).

(

Implication: Moment magnitude vs pseudotachylyte thickness

Keeping in mind the limitations detailed above, we can try a tentative scaling of the pseudotachylyte thickness with the earthquake magnitude. The stress drop ∆ during an earthquake is controlled by the fault geometry such that [START_REF] Kanamori | The Energy Release in Great Earthquakes[END_REF]:

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∆ = • X • d ⁄ (10
) with d is the fault length and X is the shear modulus, e.g. ∼ 8. 10 Pa for olivine, and is a geometrical constant ≈1. The moment magnitude f is defined as:

f = X • • d (11)
Combining (10) and (11), the displacement can be expressed as a function of f as follows:

= < ∆ A ] ⁄ • f ] ⁄ X (12)
As a consequence, for an adiabatic process, gh1gi varies as Vf j :

)kl)m = < ∆ A ] ⁄ • f ] ⁄ X • 2 (13)
with 2 = 3* • • ∆, + #4 ⁄
, the thickness )kl)m can be expressed as a function of ∆Y, and f :

)kl)m = < ∆ A ] ⁄ • f ] ⁄ • * • • ∆, + # • X (14)
Considering large-scale steady-state, `` varies as Vf n :

``= < ∆ A ] ⁄ • f W ⁄ X ⁄ • 2 (15)
To summarize this subsection, )kl)m and `` vary as Vf j and Vf n respectively. In other words, the pseudotachylyte thickness saturates for large earthquakes (Fig. 5) and cannot be much larger than 5 cm, which corresponds to a Mw 9, i.e. tens of meters of slip. (f o 9). Hence, larger pseudotachylyte veins found on the field are likely to represent "injectites" (tension cracks), local pull-apart structures or the result of successive events.

We define the crossover moment f c as the intersection between the calculations considering adiabatic and non-adiabatic processes, that can be written:

f c = < ∆ A • K * • • ∆, + # • X L W • 2 ] (16)
Keeping the same assumptions as for eq. ( 9), the crossover seismic moment f c would be around 5.10 13 to 10 14 , i.e. f o ≈ 3, mostly depending on and ∆ (Fig. 6). When f ≤ f c measuring could provide a reasonable estimate of f o , but for larger events efficient melt lubrication would induce

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significant uncertainties, with fault veins thicknesses of some millimeters potentially corresponding to magnitudes ranging from 2 to 7. This implies that field observations of pseudotachylyte thickness should be used carefully in the establishment of energy balances. The total melt volume produced per surface unit appears much more adequate, which requires a thorough investigation at the outcrop scale to estimate the melt amount intruded within the damage zone. In the field, as large pseudotachylyte exposures are necessarily truncated for large events, it is only possible to give minimum estimates of either or f o . Nonetheless, carefully considering all the parameters detailed in this study, notably compositional considerations and P-T conditions, some estimate could be made. Furthermore, in the absence of displacement marker at a large pseudotachylyte fault vein ( > 1 mm), studying the damage zone could give some idea of the size of the main rupture surface. Our model could be used to upscale small-scale observations for either adiabatic or diabatic ruptures.

Implication: energy balance

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Conclusions

Pseudotachylyte thickness scales with slip over six orders of magnitude for measured slip varying from microns to meters, which can therefore be cautiously estimated in the field even in the absence of displacement markers. As a consequence, the thickness scales with moment M0 and magnitude Mw. Network of ultramylonite-like veins highlighted in (f) ( = 2 cm; = 120 µm). See the associated Table 1 for details and additional references.

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that is not considered here (section 2.1). Gray sidebars recall the limits of Light Microscopy (LM) and Scanning (SEM) and Transmission Electron Microscopy (TEM). Calculations provided in section 2.2. The density is considered to be equal to 2900 kg.m -3 , and its variability does not have much influence on the results). Uncertainties about the dynamic temperature rise r, are considered. The range of average shear stress is consistent with typical seismological estimates. See Table 1 for information about the datasets.

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  ; Ueda et al., 2008; Obata & Karato, 1995; Andersen & Austrheim, 2006; Andersen et al., 2014; Deseta et al., 2014a, 2014b).

  , which are not the topic of the present study. An alternative mechanism has been proposed to explain the formation of pseudotachylytes through intense comminution leading to slow mechanical amorphization (e.g. Wenk, 1978; Pec et al., 2012; Hayward et al., 2016; Marti et al., 2020), which is unlikely when either asymmetrical damage/injection (e.g. Griffith et al., 2012; Ferré et al., 2015; Thomas et al., 2017; Petley-Ragan et al. 2019) or fractionated crystallization (e.g. Warr & van der Pluijm, 2005; Ferrand et al., 2018) are observed. Alteration or retrograde metamorphism during the exhumation from great depth (up to >80 km) often affects the preservation of pseudotachylyte (e.g. Kirkpatrick et al., 2009; Kirkpatrick & Rowe, 2013; Phillips et al., 2019). Nevertheless, pseudotachylyte remnants have been observed in various places around the world, especially within lower-crust and upper-mantle outcrops (e.g. Sibson et al., 1975; Obata & Karato, 1995; Barker, 2005; Nielsen et al., 2010b; Di Toro et al., 2006; Andersen et al., 2014; Scambelluri et al., 2017). Rheological transitions and fluid-rock interactions tend to entirely transform the mineralogy on the

Fig. 1 ;

 1 Fig. 2), during rotary-shear experiments (e.g. Spray, 1987; Di Toro et al., 2006; Hirose & Shimamoto, 2005a; Del Gaudio et al., 2009; Niemeijer et al., 2011), stick-slip experiments (e.g. Passelègue, 2014; Passelègue et al., 2016a; Brantut et al., 2016; Hayward et al., 2017; Lockner et al., 2017) and high-pressure experiments (Incel et al., 2017; Ferrand et al., 2017). Rotary-shear experiments have been able to reproduce "seismic" slip velocities (i.e. ≥ 1 m•s -1 ) and metric displacements characteristic of earthquakes of MW > 4 (i.e. > 2 cm), providing data that scale with natural earthquakes (Nielsen et al., 2010b; Hirose & Shimamoto, 2005b; Di Toro et al., 2011). Rotaryshear experiments shed new insights on the weakening mechanisms but explain only the frictional part of the seismic process, i.e. they do not include dynamic propagation, and are undertaken at normal stresses that are far lower than would be expected at earthquake hypocentres. Stick-slip experiments encompass both the seismic rupture nucleation and propagation stages (Passelègue et al., 2016a; 2016b; Aubry et al., 2018; Marty et al., 2019) but remain limited by the millimetric to pluricentimetric size of the sample, so that the laboratory magnitudes rarely exceed MW -3 (Passelègue et al., 2016a). In consequence, connecting laboratory results with seismological and field observations remains difficult and controversial.

  important to come back to the definition of the seismic rupture, which is the dynamic migration of a mechanical instability (e.g. Ohnaka, 2003; Passelègue et al., 2016a), thus implying a migration boundary of the melt front. The full problem of melting coeval with heat diffusion requires solving a Stefan problem, where the heat balance is defined at the advancing melt front (Carslaw & Jaeger, 1959; Nielsen et al., 2008). In section 3.1, we develop the model, whose results in section 3.2 show that the key is in the first moments of the sliding and that Sibson's empirical law differs from the adiabatic model (section 2.2) for large events in case of inefficient lubrication.

Following

  Nielsen et al. (2010b), a Crantz-Nicolson finite difference scheme is used to numerically solve the 1-D diffusion equation with an advection term, coupled to the boundary condition:

(

  Nielsen et al., 2010b). The melt volume per unit fault surface is thereafter obtained by !# = P ; !′#%!′ .

  in agreement with high-velocity weakening observed in rotary-shear experiments[START_REF] Nielsen | Scaling in natural and laboratory earthquakes[END_REF] and with the fit initially proposed by[START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF]. To further support such choice for • $, we note that high-velocity friction can be fitted(Nielsen et al. 2016) by ∝ 1 √ ⁄ , resulting in ̅ • ∝ √ . For slip velocity during experiments and earthquakes, we may assume an indicative slip history with an initial acceleration U of slip such that $ = U • ! = √2 • • U up to a critical distance I , followed by a modest variation such that $ ≈ Const ≈ V2 • U • I , resulting in an initial heat power • $ ∝ √2U ∝ Q , followed by • $ ∝ R ' S ' . The gray curves in Fig. 3 and Fig. 4 are obtained from modelling with Q = 500 • 10 W J•m -2 •s -1 (as an indication, this value is compatible with $ = 1 m•s -1 and = X • Y Z = 0.5 × 1 GPa, and it is assumed that the volume loss by injection is relatively limied therefore ≈ ). Three different values were modelled for the onset of weakening: [1] ! I = 7.55 • 10 W s (i.e. I = 2.88•10 -5 m; corresponding to the onset of melting at , = , / ), [2] ! I = 10 ] s and [3] ! I = ∞ (the latter case results in a constant power Q throughout the slip; arguably, this may account for absence of weakening due to high viscosity of colder or more felsic melt, to match the datapoints in the higher part of the diagram).

  The thickness measured in the field may vary over more than one order of magnitude along strike, either due to dilational and contractional jogs or because of melt flux into the damage zone (e.g.[START_REF] Sibson | Generation of pseudotachylyte by ancient seismic faulting[END_REF][START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF]. Field studies show that pseudotachylytes might be present as discontinuous patches along a fault, with melting only along contractional jogs[START_REF] Griffith | Static stress drop associated with brittle slip events on exhumed faults[END_REF] Kirkpatrick & Shipton, 2009). Experimental data also show significant variation of the pseudotachylyte thickness (e.g.[START_REF] Hayward | Melt welding and its role in fault reactivation and localization of fracture damage in seismically active faults[END_REF]. For stick-slip experiments, only cumulative average values are available. For D-DIA experiments, TEM on FIB sections is required to measure the fault thickness. In rotary-shear experiments, the sample is only meant to be an equivalent of a limited fraction of a fault, while other techniques may encompass complete fractures, i.e. ruptures area smaller than the sample size. In addition, the

  lower melting point). The data of Di[START_REF] Toro | Natural and experimental evidence of melt lubrication of faults during earthquakes[END_REF] are also consistent with the adiabatic assumption, which suggests that quartzo-feldspathic rocks would not achieve efficient weakening after the onset of melt-assisted lubrication. Feldspar has a lower melting point (~1500°C;[START_REF] Taniguchi | Entropy dependence of viscosity and the glass-transition temperature of melts in the system diopside-anorthite[END_REF][START_REF] Suzuki | Viscosity of albite melt at high pressure and high temperature[END_REF] than the other minerals constituting the rocks of the dataset, but the high Si/Mg ratio maintains a relatively high melt viscosity at low pressure (< 0.5 GPa;[START_REF] Suzuki | Viscosity of albite melt at high pressure and high temperature[END_REF].The Woodroffe Thrust fault zone (Australia, e.g.[START_REF] Camacho | Large volumes of anhydrous pseudotachylyte in the Woodroffe Thrust, eastern Musgrave Ranges, Australia[END_REF] appears as a counterexample

A

  recent study showed that the energy dissipated by fracturation during the propagation of the seismic rupture depends on the size of the asperities of the fault surface, following a power law (Passelègue et al., 2016b). The nucleation length directly depends on the fracture energy, which means that it depends on the asperity size (Ohnaka, 2003; Passelègue et al., 2016b). The energy balance of fault slip would suggest that fracture energy G dissipates in fault weakening due to coseismic processes (including melting) that would reduce the heating. Seismological inferences of G do show that in itself it is an increasing function of slip (e.g. Rice, 2006; Viesca & Garagash, 2015).Interestingly, the scaling relationship of G with D also shows a change of slope at D ~ 1 mm, varying as D 2 for D < 1 mm and as D 2/3 for D > 1 mm[START_REF] Viesca | Ubiquitous weakening of faults due to thermal pressurization[END_REF]. Furthermore, D varies as a 2 for D > 1 mm, thus G varies as a 4/3 ; and for D < 1 mm D may actually vary as a 2/3 , which would mean that G varies as a 4/3 as well. In other words, the pseudotachylyte thickness would scale with the fracture energy with no impact of scale.

  Experimental and natural faults show striking similarities. Regardless of design, the scaling law follows the same trend. The bend observed in the scaling law is interpreted as the signature of pseudotachylyte thickness saturation for large Mw due to scale-dependent efficiency of melt-assisted lubrication and possibly to variations in viscosity and mobility of the pseudotachylyte met suspension infill. The thickness of the molten layer cannot be much larger than 5 cm, which corresponds to a Mw 9, i.e. tens of meters of slip. Hence, larger reported pseudotachylyte veins most likely correspond to injection veins or local dilational jogs along longer and thinner faults.

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Displacement as a function of fault thickness: datasets and adiabatic assumption. Observations

Figure 3 :

 3 Figure 3: Scaling seismic fault thickness with displacement as a Stefan problem (low pressures). Gray

Figure 4 :

 4 Figure 4: Scaling seismic fault thickness with displacement as a Stefan problem (high pressures). Gray

Figure 5 :

 5 Figure 5: Fault thickness and direct magnitude estimates. a: pseudotachylyte thickness (data) and dynamic molten zone width (calculations) vs. earthquake magnitude fw , with fw = 2 3 ⁄ log f -6.07 (section 4.3) and f from eq. (12). Warm colors correspond to the adiabatic process (eq. 15) whereas the blue color is used for the diabatic process (Sibson's law), which actually corresponds

Figure 6 :

 6 Figure 6: Crossover moment f I and moment magnitude f I . f Iw as function of and ∆Y showing limited influence of ∆,. The gray box indicates the range of values consistent with observation, as discussed in section 3.3 of the main manuscript.

  Both field observations (e.g. Sibson, 1975; Ueda et al., 2008; Obata & Karato, 1995; Andersen & Austrheim, 2006; Nielsen et al., 2010a; 2010b; Andersen et al., 2014; Scambelluri et al., 2017; Ferrand et al., 2018) and laboratory analogues (e.g. Passelègue et al., 2016a; Incel et al., 2017; Brantut et al., 2016;

	Ferrand et al., 2017; Aubry et al., 2018) have revealed that the displacement is closely related to the
	thickness of fault veins.

  Fig. 2 with the associated displacement estimate of ≈ 3 m. Rotary-shear experiments constitute the most reproducible experimental dataset of seismic slip (e.g. Hirose & Shimamoto, 2005a; 2005b; Di Toro et al., 2006; Violay et al., 2015) and is for now the only technique reaching slip value typical

	fault network (e.g. Nielsen et al., 2010b) or taking into account the internal consistency of the dataset (e.g.
	Barker, 2005). 3D photogrammetry for instance, will definitely allow future studies on pseudotachylytes
	are apparent offset only, since displacement markers were not systematically identified on an exposed surface parallel to the slip direction. Theoretically, an offset corresponds to a true displacement = [ ; ∝[. The large datasets of Nielsen et al. (2010b) and Di Toro et al. (2006) correspond to true displacement data, i.e. taking into account the geometry of the fault and the observed finite offset. Regarding the fault network from Scambelluri et al. (2017), the metagabbro sample was cut perpendicular to shear, which means that the observed offsets are true displacements with a negligible error. Regarding the HP fault described by Ferrand et al. (2018), the average apparent offset is ≈ 1.2 m for the main slip surface, which corresponds to a displacement of 1.75 ± 0.15 m, i.e. ≈45% larger than the measured offset. Considering that the rest of the dataset (red crosses, Fig. 2) follows the same offset-displacement relationship, a small uncertainty is taken into account (very small in log-log). For other field studies (Obata & Karato, 1995; Andersen et al., 2014), only few displacement values are provided, which we associate with uncertainties of + 50%. Only a few studies (Di Toro et al., 2005; 2006; Pittarello et al., 2008; Ferrand et al., 2018) report the average thickness of pseudotachylytes measured over several meters of fault length. For the other studies, estimates can still be cautiously considered, either regarding the large number of data collected on the to reach a sounder statistical approach, including the determination of the actual volume and 3D distribution of pseudotachylyte networks. Regarding the study of Obata & Karato (1995), only the minimum and maximum values of a range of observed thicknesses is provided, which we tentatively Accepted Article associate in
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of natural earthquakes with Mw > 4 (> 2 cm). In particular, this technique evidences dynamic friction values approaching zero during melt lubrication (Di

[START_REF] Toro | Friction falls towards zero in quartz rock as slip velocity approaches seismic rates[END_REF]

. However, in rotary-shear experiments, the slip surface is not confined within a larger sample (D-DIA experiments for instance) or a jacket (stick-slip experiments). Melt is therefore easily extruded, leading to much smaller final melt thickness (a*; Fig.

2

; Table

2

). Nonetheless, some publications provide a record of the axial shortening during rotary-shear experiments (Di

[START_REF] Toro | Natural and experimental evidence of melt lubrication of faults during earthquakes[END_REF] Niemeijer et al., 2011)

, which actually relates to the cumulative melt (+ clasts) thickness (w*; Fig.

2

; Table

2

).

  and Di[START_REF] Toro | Natural and experimental evidence of melt lubrication of faults during earthquakes[END_REF] reported large-scale pseudotachylytes that do not follow Sibson's law, but rather a linear relationship between and , suggesting an adiabatic process and inefficient melt lubrication. The data of[START_REF] Barker | Pseudotachylyte-generating faults in central Otago, New Zealand[END_REF] indicate > 100 MPa when the data of Di[START_REF] Toro | Natural and experimental evidence of melt lubrication of faults during earthquakes[END_REF] would indicate ≈ 10 MPa (Fig.2). Contrary to all other studies focusing on

isotropic rocks,

[START_REF] Barker | Pseudotachylyte-generating faults in central Otago, New Zealand[END_REF]

describes pseudotachylytes in a quartzo-feldspathic schist, suggesting that the bend is host rock-dependent. The parameters controlling melting point and magma viscosity, and how the latter influence our modelling results, are discussed in section 4.2. In any case, according to our modelling results, Sibson's empirical law actually corresponds to efficient melt lubrication case, while scaling relations closer to the adiabatic (linear) trend correspond to events where weakening was not significant.

  estimate at the scale of the saw-cut after seismic events (Passelègue et al., 2016a;[START_REF] Brantut | Dynamic weakening and amorphization in serpentinite during laboratory earthquakes[END_REF][START_REF] Aubry | Frictional heating processes and energy budget during laboratory earthquakes[END_REF]. This could actually explain why displacements are up to one order of magnitude larger in stick-slip experiments compared to faults with the same thickness in the D-DIA (Fig.2), even though

	differences in machine stiffness may also be at play. Considering multiple pseudotachylyte generation on
	the same fault plane, a significant increase of / would be observed. However, publications highlight the
	high strength of faults welded by pseudotachylyte (Di Toro & Pennacchioni, 2005; Mitchell et al., 2016;
	Proctor & Lockner, 2016; Hayward & Cox, 2017). Thus, new ruptures are not expected to affect preexisting
	pseudotachylytes except for stick-slip experiments or in case of alteration. In nature, pseudotachylytes
	may also form from shear zones exhibiting a significant displacement before rupture nucleation (e.g.

al., 2013; Ferrand et al., 2017; Incel et al., 2017), which is uncertain for stick-slip experiments that give an

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  . At 1800°C, a melt of MgO or Mg2SiO4 composition exhibits a viscosity six orders of magnitude lower than SiO2 (Karki et al., 2013). Melt viscosity is primarily governed by SiO2 tetrahedra polymerization, strongly limited by cation diversity (e.g. Whittington et al., 2000; Le Losq et al. 2013; Toplis & Dingwell, 2004).

	The viscosity of silicate melts decreases with pressure (e.g. Gupta, 1987; Xue et al., 1991; Bottinga &
	Richet, 1995; Suzuki et al., 2002; Liebske et al., 2005) due to a change in silicon coordination (Xue et al.,

  [START_REF] Brantut | Dynamic weakening and amorphization in serpentinite during laboratory earthquakes[END_REF]. In small amounts, water lowers the melting point of peridotite by ~100 °C, inducing olivine melting around 1300 °C (San Carlos, 0.32 wt.% H2O;[START_REF] Gaetani | The influence of water on melting of mantle peridotite[END_REF]. Water also drastically lowers melt viscosity[START_REF] Kushiro | Viscosities of basalt and andesite melts at high pressures[END_REF][START_REF] Richet | Water and the viscosity of andesite melts[END_REF][START_REF] Vetere | Viscosity of andesitic melts -new experimental data and a revised calculation model[END_REF]. As a consequence, tension cracks forming in the dilatant quadrant as rupture propagates (e.g. Okubo et al., 2019) generate transient depressions able to suck the melt out of the fault core

tonalite; Di

[START_REF] Toro | Natural and experimental evidence of melt lubrication of faults during earthquakes[END_REF]

, whereas peridotites likely endure congruent melting

[START_REF] Ferrand | Energy balance from a mantle pseudotachylyte, Balmuccia, Italy[END_REF]

. Furthermore, exothermic crystallization

[START_REF] Burbank | A source of heat-energy in crystallization of granodiorite magma, and some related problems of volcanism[END_REF][START_REF] Blundy | Magma heating by decompression-driven crystallization beneath andesite volcanoes[END_REF]

) may inhibit the viscosity decay.

Water, either aqueous or structurally bound, is key to activating weakening processes (e.g.

Mei & Kohlstedt, 2000a; 2000b;[START_REF] Violay | Effect of water on the frictional behavior of cohesive rocks during earthquakes[END_REF]

. Dehydration reactions induce dynamic weakening that favors strain localization and seismic sliding (e.g.

[START_REF] Brantut | High-velocity frictional properties of a clay-bearing fault gouge and implications for earthquake mechanics[END_REF]

. Conversely, during sliding-induced antigorite dehydration, a pressurized low-viscosity silicate melt leads to extremely efficient lubrication (

Table

  All the faults are reported as pseudotachylyte, except the ones ofSchubnel et

	al. (2013; "transformational faulting" during olivine-spinel transition). Data from Scambelluri et al. (2017)
	are from the fault network shown in Fig. 1. The datasets of Barker (2005) and Sibson (1975) are minimum
	estimates of , which could be one or two orders of magnitude larger (see discussion in section 2.1).

). Field-scale data are compared to micrometer-scale and nanometerscale observations. A distinction is made between the pseudotachylyte thickness (fossilized trace of the seismic rupture, i.e. observational data) and the dynamic thickness (volume of melt produced by surface unit, i.e. simple adiabatic model). Values of and should slightly differ considering off fault damage and melt intrusion (section 2.

2).

Rotary-shear experiments provide two distinct datasets: sample axial shortening w* (cumulative melt thickness) and the final pseudotachylyte thickness a*

(Hirose & Shimamoto, 2005;[START_REF] Del Gaudio | Frictional melting of peridotite and seismic slip[END_REF]

Table 1 -Dataset collected from various experimental and field works and presented in Fig. 2.

 1

	Tables
	a	D	Material	Reference
			D-DIA experiments
	200 nm 20 nm	10 µm 10 µm	Olivine-antigorite aggregates	Ferrand et al. (2017)
	10 nm	8 µm
	50 nm	8 µm	Lawsonite blueschist	Incel et al. (2017)
	30 nm	6 µm
	100 nm 150 nm	10 µm 30 µm	Ge-olivine	Schubnel et al. (2013)
			Stick-slip experiments
	1.5 µm	250 µm		Passelègue et al. (2016a)
	8 µm 7 µm 7 µm	1.2 mm 1.2 mm 4.2 mm	Westerly granite	Aubry et al. (2018, supp. Mat.) Lockner et al. (2017)
	2.5 µm	500 µm	Alpine Corsica antigorite	Brantut et al. (2016)
	2-4 µm	1.35 mm
	2-4 µm 3-4 µm	1.09 mm 2.28 mm	Fontainebleau sandstone, > 99% quartz	Hayward & Cox (2017)
	50-80 µm	2.28 mm
			Field work
	50 µm	6 mm
	20 µm	2 mm
	70 µm 100 µm 100 µm	1 mm 1.6 mm 10 mm	Lanzo peridotite, Italian Alps	Scambelluri et al. (2017)
	20 µm	500 µm
	60 µm	1 mm
	4 mm	1 m
	3 mm	20 cm
	12 µm	2 cm
	5 mm	1.7 m
	100 µm 300 µm 1 mm	6 mm 2 cm 5 cm	Balmuccia peridotite, Italian Alps	Ferrand et al. (2018)
	10 µm	100 µm
	5 µm	200 µm
	6 µm	300 µm
	20 µm	400 µm

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Acknowledgments

We thank Sarah Incel for providing the SEM image of the dehydrated lawsonite blueschist. This research is supported by the International Research Promotion Office of the Earthquake Research Institute and the European Research Council, grant no. 681346-REALISM. All datasets used in this paper have already been used by previous studies and are fully presented in the tables.

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