Frictional melt is implied in a variety of processes such as seismic slip, ice skating, and meteorite combustion. A steady state can be reached when melt is continuously produced and extruded from the sliding interface, as shown recently in a number of laboratory rock friction experiments. A thin, low-viscosity, high-temperature melt layer is formed resulting in low shear resistance. A theoretical solution describing the coupling of shear heating, thermal diffusion, and extrusion is obtained, without imposing a priori the melt thickness. The steady state shear traction can be approximated at high slip rates by the theoretical form τss = σn1/4 (A/) under a normal stress σn, slip rate V, radius of contact area R (A is a dimensional normalizing factor and W is a characteristic rate). Although the model offers a rather simplified view of a complex process, the predictions are compatible with experimental observations. In particular, we consider laboratory simulations of seismic slip on earthquake faults. A series of high-velocity rotary shear experiments on rocks, performed for σn in the range 1–20 MPa and slip rates in the range 0.5–2 m s−1, is confronted to the theoretical model. The behavior is reasonably well reproduced, though the effect of radiation loss taking place in the experiment somewhat alters the data. The scaling of friction with σn, R, and V in the presence of melt suggests that extrapolation of laboratory measures to real Earth is a highly nonlinear, nontrivial exercise.
 The primary aim of this study is to interpret the experimental observations obtained in rock experiments done with high-velocity rotary friction apparatus (HVRF), in order to possibly extrapolate them and use them in earthquake source studies for natural faults [Hirose and Shimamoto, 2005a; Di Toro et al., 2006a, 2006b]. With this aim, we build a mathematical model that captures the essential features of frictional melt dynamics, based on thermal diffusion and inhomogeneous viscous flow. Though our approach carries some similarities with the quantitative mathematical approach of Fialko and Khazan , there are fundamental differences in the outcome essentially due to considering the convergence to the steady state, the moving boundary condition and the melt extrusion from the sliding interface.
 Since we wish to obtain a fully analytical solution that provides insight into the basic dynamics of the process, we recur to a number of assumptions. As a consequence, several features of the HVRF experiments are simplified in the theoretical model, in particular, the cylindrical geometry, the heat loss through radiation and the inhomogeneous composition of the melt. In spite of this, we find that the theoretical model captures surprisingly well the essential features observed in HVRF experiments. We note that a treatment specifically accounting for all the features of HVRF experiments is only possible through numerical modeling, and is beyond the scope of this manuscript.
 In section 2, we discuss the heat balance between the melt layer, where heat is produced by viscous shear, and the solid wall acting as a heat sink due to both thermal diffusion and latent heat absorption. We argue that a steady state is reached under a constant shear rate. Under favorable conditions of slip on a dry fault, melting temperature is readily reached. Shortly thereafter, a steady state condition is achieved, provided that rock shortening and melt extrusion are accounted for. Building along the same lines as Di Toro et al. [2006b], we show that the duration of the transient depends on slip rate, normal stress and the properties of the host rock. As a consequence, an apparent slip-weakening distance develops, which is not a constitutive parameter but depends on the dynamic loading conditions (i.e., slip rate and normal stress). The predicted transient duration is in agreement with that observed in laboratory experiments, and the apparent slip-weakening distance is in agreement with both experiments and values deduced from field studies on natural pseudotachylyte-bearing faults [Hirose and Shimamoto, 2005b; Di Toro et al., 2006b].
 In section 3, we solve the steady state problem of viscous shear heating in a melt layer of inhomogeneous viscosity. The thickness of the melt is a result of thermal equilibrium between the melt and the solid. To our knowledge, this is a totally unprecedented analytical result that may apply to a number of lubrication processes where a steady state approximation can be assumed. This allows to derive the temperature distribution inside the melt layer, showing super heating of the order of ≈200°C above melting temperature for standard faulting conditions, close to the center of the melt layer. The predicted superheating is in agreement with microstructural observations in pseudotachylytes [Di Toro and Pennacchioni, 2004].
 In section 4, we recall some classic results about the viscous force developed by the extrusion of a liquid film, and how it relates to the normal stress across a frictional interface. We also recall basic notions of hydrodynamic pressurization. This discussion allows to predict a strongly non linear relationship between friction (i.e., shear traction) τss and normal stress σn in the presence of melt.
 In section 5, we assemble the equations of sections 2, 3, and 4 to obtain the steady state friction τss as a function of the normal load σn, the sliding velocity V and the radius R of contact area. Rock and melt properties, instead (thermal diffusivity κs, capacity c, latent heat L, mass density ρ, melting temperature Tm and parameters describing the viscosity dependence with temperature) are grouped by means of dimensional analysis in one characteristic stress Γ and one characteristic velocity W.
 In section 6, we describe the experiments that were performed on gabbro, tonalite and peridotite under a variety of normal stress, slip rates and sample sizes in order to test the validity of our theoretical analysis.
 In section 7, we compare the steady state friction law obtained in section 5 to the experiments described in section 6 and argue that heat loss through radiation (a feature not present in buried, natural faults) is not negligible in the samples and introduces a visible bias in the low stress range. Experimental results for three rock types (gabbro, tonalite and peridotite) under a variety of normal stress, slip rates and sample sizes are compared to the theory. Finally, the relevance of the model for ice sliding is discussed in the light of experimental results in the literature.
 Throughout this manuscript the terms “friction”, “shear traction” and “shear resistance” all refer to the traction across a sliding interface; friction is intended not only in the strict connotation of rubbing across a dry, solid/solid contact, but also that of sliding shear resistance of a viscous layer.
2. Melting and the Heat Balance
 Before we proceed to our analysis, we briefly illustrate here the geometry of the problem. A schematic cartoon of the shear/shortening mechanism analyzed is presented in Figure 1. Two types of motion are present: shear rate V (relative motion of the rock compartments parallel to the frictional interface) and shortening rate v (convergence of the two rock compartments). The middle grey area represents melt due to frictional heating.
 It is important to note that continued melting during the frictional process wears out the rock volume from the sliding blocks, producing a migration of the solid/melt boundary. This class of problem involving a phase transition is frequently referred to as the Stefan problem [Carslaw and Jaeger, 1959]. In this particular case, we point out that while new melt mass is driven into the interface by continued rock melting, part of the melt is extruded laterally from the interface (Figure 1, t3, grey arrow noted ext). If steady state is reached, extrusion and melting compensate each other resulting in a constant thickness ω. Such an extrusion process applies to frictional melt in many circumstances, including some observed natural faults (see section 2.2).
 A symbolic box B is added in Figure 1 (t3, overprinted lighter rectangle), representing a control volume for the analysis of mass and heat flow. In our choice of referential for the Stefan problem solution, we opted for an Eulerian reference frame: The control volume in Figure 1 is attached to the migrating solid/melt boundary (dotted vertical line), and the mass particles flow through it.
 In Figure 1 (t3) the black arrows represent the mass flow through the control volume and the top arrow (ext) represents extrusion of melt. The vertical lines within box B represent temperature isovalues (temperature grows from right to left in B). At the steady state, the isotherms remain fixed while the mass flows across them and through the control volume.
 The heat source in our problem is friction (and viscous shear) across the sliding interface. The rate of heat production is = τV (where τ and V are shear traction and slip rate across the fault, respectively). The product τV will appear ubiquitously in the context of the steady state analysis, so we will first solve the problem as a function of τV, and obtain the single value of traction τ subsequently. Heat transfers may happen by diffusion and advection, and two heat sinks are provided by melting (latent heat) and, under experimental conditions, radiation loss.
 We consider the solid walls on each side of the sliding interface as two infinite half-spaces; this is acceptable provided that the duration of the studied process is small with respect to time of diffusion across the model, be it the Earth crust or the laboratory sample. In the latter case, the approximation holds within certain limits to be discussed.
 If left to evolve for a few seconds, HVRF experiments where melt is formed all lead to a stationary state (see section 6), suggesting an energy balance that we characterize as a first step of our analysis. Thermal diffusion in a half-space may reach a stationary state provided that a moving boundary condition is introduced. This condition is met in the case of frictional melt, because the solid wall is being etched out as solid mass from the edge is turned into melt. The solid/melt interface propagates into the rock at a velocity ν, acting as a moving boundary.
 If we anchor our referential to the solid/melt interface (Figures 1 and 2) , neglect gradients parallel to the interface, and consider diffusion into the solid equation (1) reduces to
valid only on the solid side of the boundary (the solution within the melt incorporates a heat source term and is treated in section 3). In (2)ν = −vz is the shortening rate, i.e., the velocity of melt/solid boundary advancement, while
is the distance from the solid/melt boundary, z is the distance form the center of the melt and ω is the half thickness of the melt layer (see Figure 2).
 As outlined in Figure 2, after onset of melting our boundary conditions are of fixed temperature T = Tm (melting temperature) at the melting front z = ω; at infinity T = Ti (initial temperature in the rock). If melt produced by the heating is removed from the interface at the same rate as it is produced, we can consider a steady state solution with ν = const. for the temperature inside the solid, by replacing ∂T/∂t = 0 into (2) and immediately obtain the solution
Equation (4) indicates the formation of a thermal boundary layer of finite, constant thickness b = κ/ν in the solid block, inversely proportional to v. As the thickness of the thermal boundary layer is constant at steady state, the amount of heat stored in the solid walls does not increase with time, which equates to an adiabatic condition under steady state. Removal of melt occurs by extrusion in both natural faults (where most of the melt is injected into veins in the sidewalls [Di Toro et al., 2005]) and laboratory samples (simply ejected from the border of the cylindrical sample [Di Toro et al., 2006a]). We point out that a natural rock is usually constituted by different minerals whose melting temperatures may vary of a few hundred °C [Spray, 1992]. Tectonic pseudotachylytes (i.e., natural solidified friction melts produced during seismic slip) consist of clasts of minerals having a high melting temperature (not assimilated by the melt) suspended in a glassy-like matrix. As a consequence, a model assuming a single melting temperature for the whole mineral assemblage is a simplification. On the other hand, the boundary between melt and solid, in most examples of natural faults and laboratory samples, appears well defined, so that the assumption of an average temperature for melting is an acceptable approximation.
2.2. Steady State Heat Balance
 The shortening rate ν at the steady state can be predicted by writing the balance of heat across a control volume around the solid/melt boundary under constant heat rate production = τV.
 The mass flow is represented by black arrows that enter and exit the symbolic control volume, box B in Figure 1 (t3). A particle inside the solid moving toward the interface (1) enters the control volume at a temperature Ti, increases in temperature up to Tm as it approaches the interface ω, (2) absorbs a latent heat during melting (crossing the interface at ω toward the melt) (hence the particle approaches the center of the melt: As discussed in section 3, owing to inhomogeneous viscous shear the melt may undergo an additional temperature increase before extrusion, on average ΔT), and (3) finally, the particle flows along the interface thus exiting from the control box B, due to the net flow generated by extrusion (ext.).
 Let us now quantify the heat transfers due to the above mass flows 1–3. If ν is the shortening rate of the solid block, melting rate of solid mass per unit area of fault is νρ, where mass density is ρ. Rate of melt production is (where the breve denotes parameter values inside the melt). Mass conservation yields
Furthermore, under the steady state, the melt layer remains of equal thickness [Hirose and Shimamoto, 2005a] so that the net flow rate out of the control volume in flow 3 (extrusion rate ) should be equal to the rate of melt production:
 Heat entering the box with mass flow 1, per unit area and time, is
while heat loss due to latency (b) is
where L is latent heat. Assuming that a proportion of solid clasts ϕ survive inside the melt, then the value for latent heat is substituted by an effective value L → L (1 − ϕ). Heat leaving the box with mass flow 3 is
(Note that slight temperature variations of capacity and density within each phase are neglected.)
 Finally, the heat source in the problem is the shear work; balancing out all terms (7)–(9) against heat production rate τV, and letting τss be the steady state shear stress, we obtain the equality:
(the factor of two is introduced to account for the presence of a boundary on each side of the melt layer). Equation (10) is the precise steady state balance accounting for finite superheating ΔT above melting temperature, while (11) is an approximation which has been used elsewhere [Sibson, 1975; Di Toro et al., 2006b] for geological estimates of friction on exhumed faults. It is now clear that such estimate implicitly assumes that ΔT is negligible and that c ≈ . Note that heat loss through radiation or circulation/sublimation of fluids (other than melt) is not considered here. A simplified form to account for radiation loss is proposed below (section 6), and used in some of the comparisons with theory.
2.3. Shortening Rate and Latent Heat
 We shortly give here the general expression for the rate of migration of the moving boundary (shortening rate), in a Stefan problem with latent heat L.
 According to Fourier's law, heat transfer is proportional to temperature gradient, hence heat entering the boundary at z = ω+ (ξ = 0+) is
while the heat transfer away from the melt and into the boundary, can be expressed as
and the difference between heat entering and leaving the melt/solid boundary goes into latent heat:
Note that this evaluation is valid both at steady state or not; however, we shall use it below (section 3) in order to derive a boundary condition at the steady state.
2.4. Boundary Condition on Temperature Gradient
 According to the steady state solution (4), we derive the temperature gradient at ω toward the solid:
Substituting (15) into (14) and then replacing ρν according to (10), we may compute the temperature gradient in the melt at the solid/melt boundary:
where the second term in parenthesis may be neglected in the light of typical magnitudes of the parameters (typical magnitude orders in fault and rock experiments are ΔT ∼ 102 K, Tm ∼ 1400 K, Ti = 273 K, L ∼ 3 105 J kg−1, c ∼ 103 J kg−1 K−1, ∼ c, τV ∼ 107 Jm−2 s−1. For example, see parameters of gabbro, tonalite, and water in Tables 2 and 3 for solid or liquid state, respectively). The above condition on the temperature gradient can be supplemented by the requirement that the solid/melt boundary is at melting temperature:
The boundary conditions (16) and (17) are used in section 2 for the derivation of the temperature profile within the melt layer.
Indicative values used in this study, unless otherwise specified. Values are estimated from lab measures on the rock samples and from their mineral composition. Parameters κ, ρ, c assume an average temperature between melting (Tm) and initial (Ti) temperature (extrapolated from Holland and Powell ). Temperature variations of the constitutive parameters listed here are neglected.
Indicative values used in this study, unless otherwise specified. Variations of , , with temperature are neglected.
0.344 × 10−6
0.72 × 10−6
0.15 × 10−6
2.5. Duration of the Transient
 One key assumption of the forthcoming transient analysis is that the melt layer is thin, in the sense that the time for heat diffusion across the melt thickness is short with respect to the total duration of the sliding. In other words, (2ω)2/κ ≪ tp where tp is the duration of the process, 2ω the melt thickness and κ the thermal diffusivity in the melt. As a consequence, heat diffusion through the melt is for all practical purposes instantaneous with respect to the slower heat diffusion through the sidewalls of the fault or the solid sample of dimension H ≫ ω. This applies to the frictional melt experiments where ω < 0.15 mm, and also to several examples of documented faults with pseudotchylyte [Dis Toro et al., 2005].
 One main consequence is that the time delays in the evolution of the sliding interface are essentially controlled by the slowness of heat diffusion in the rock solid walls, which act as buffers in the heat balance problem. The thermal evolution of the melt layer constantly and swiftly adapts to the slower evolution of the solid walls, in particular, their efficiency as heat sinks.
 The efficiency of heat absorption by the solid depends on the thermal profile in the solid itself. It is initially a very efficient buffer because the temperature gradient is steep, but as heat diffuses into the solid the gradient drops to its steady state limit (15), so that more heat remains trapped in the melt layer, inducing temperature increase and more effective weakening.
 We may consider two distinct phases in the transient, one before the occurrence of pervasive melt (0 < t < tm) and one after (tm < t < tc). Pervasive melt is defined in opposition to flash melting, the formation of isolated melt drops that form prematurely at points of asperity contacts [Rice, 1999; Hirose and Shimamoto, 2005a]. After pervasive melting, the interface evolves to an equilibrium condition (steady state) within a characteristic time interval tc. As a consequence, the total transient duration from the onset of slip is tc + tm. We anticipate that assuming a dry environment, realistic faulting conditions generally yield to tm ≪ tc (see section 6).
 The detail of complex frictional processes taking place before pervasive melting [Hirose and Shimamoto, 2005a] surpasses the aim of this study. However, during such phase the temperature evolution at the interface is readily modeled by solving the diffusion equation with an imposed heat flow as a boundary condition. Assuming a roughly constant heat rate production τV, equates to fixing the temperature gradient at the boundary:
which, equated to the melting temperature Tm, yields the time tm for pervasive melt at the interface:
 As for the value of τV during transient, we may use an estimate of significant average, under a few simplifying assumptions. In the experiments slip rate is actually V = Vss, because the velocity is imposed by the machine. The average stress can be estimated from the experimental trends. Since = (1/tc) τ(t) dt, and observing that the friction drops to very low values at high rates is roughly exponential in time, we can adopt as a rule of thumb ≈ μσn (1 − 1/e) where μ ≈ 0.8 is the solid friction coefficient.
 After melting time tm, the condition changes to that of a moving boundary where temperature T = Tm is imposed. By using the thermal distribution at time t = tm as an initial condition, the subsequent evolution toward steady state may be obtained by solving equation (2) stepwise in time, with shortening rate ν(t) estimated within each interval according to (10). We do not develop the complete transient solution here; instead we estimate an order of magnitude for the duration of transient and equivalent slip-weakening distance, based on a crude approximation.
 A moving source solution proposed by Carslaw and Jaeger  may be used to solve (2) by taking the Laplace transform. However, it assumes that at the time of melting onset, temperature variations in the solid are negligible; moreover, it assumes that shortening rate ν is constant from the beginning of the transient. Examples of numerical simulations by Landau  and laboratory experiments performed on rocks [Hirose and Shimamoto, 2005a] show that v actually differs significantly from its steady state value only in the first portion of the transient, but less than a factor of 2 over most of the transient duration. With the caution required by the above dissimilarities, we may adapt the moving source solution to our case as
given that lima→∞erfc(a) = 0 and lima→ ∞erfc(−a) = 2. It is clear that at small ξ (z → ω+), the convergence of (21) to the steady state value is controlled by thes characteristic time tc = 2 κ/ν2 appearing in the erfc functions. As a consequence, assuming that v during the transient is comparable to its steady state value as in (10), and assuming that heating of the solid previous to melting time is negligible, the convergence time should scale approximately (τV)−2.
 Using tc + tm ≈ tc, an apparent slip weakening distance δc may be obtained by multiplying the characteristic time with slip rate V:
which is a clear violation of the concept of a critical slip distance as fixed constitutive parameter. Weakening time and apparent weakening distance are both dynamically controlled by slip conditions and cannot be derived a priori solely from the structure of the interface or the properties of the rock. Accessorily, we may now compare (23) and (20) to confirm that tc is dominant with respect to tm.
 For a detailed solution of the transient, however, use of some numerical algorithm is required (see, for example, Landau  assuming constant heat production rate or Fialko and Khazan ). Graphics representing the temperature profile evolving in time until it reaches the steady state, according to solutions (18) and (21) are given by Di Toro et al. [2006b].
3. Solution of the Shear Heating Problem
 In this section we obtain the profiles of temperature and viscosity across the melt layer, and a relation between melt thickness h, slip rate and shear traction τss, at the steady state.
 We start by the general heat diffusion problem as described by equation (1), and eliminate all the terms that vanish or may be considered reasonably small. The steady state implies that ∂T/∂t = 0. Since the melt layer is thin, we surmise that all derivatives parallel to the sliding interface may be neglected given the small aspect ratio. Indeed, a priori values R ≈ 10−2 m and h ≈ 10−4 m as fault parallel and fault perpendicular dimensions of the melt layer between two extrusion heads (in agreement with typical values observed both in HVRF experiments and natural faults) yield
As a consequence, the diffusion equation within the melt reduces to
In addition, a priori estimates show that within the melt, the heat source density (second term) is relatively dominant against the advection (third term), so that ν ∂zT may be discarded. Indeed, an estimate for the second term is roughly τssV/2 ρ c ω. An upper bound estimate for the third term is provided by replacing the temperature gradient by its boundary expression (16), yielding ν τssV/κ ρ c. The ratio between the two yields an upper bound for the dimensionless Peclet number νω/κ of about 0.1 (using typical values ω ≈ 0.510−4 m, κ ≈ 10−6 and ν ≈ 10−3 m s−1). Assimilating shear to a laminar flow under variable viscosity η,
Finally, we can write
With no loss of generality, we may immediately predict a general feature of the solution by introducing the variable change ψ = zτss in (28) to obtain
illustrating that the general temperature profile is the same for any τss value, except for a contraction 1/τss along the z axis. The temperature solution thus exhibits self-similarity in ψ = zτ. As a consequence, the melt thickness 2ω will be inversely proportional to τss, in the same way as in the simplified constitutive equation for a constant viscosity, τss = ηV/(2ω).
 Rock melt viscosity dependence on temperature is often described in terms of the Arrhenious law
where T is in kelvin. In order to find the general solution of the differential equation (28), a regularized, integrable form of viscosity η(T) should be used. The term B/T may be replaced by it series expansion B/Tm − B (T − Tm)/Tm2 in the vicinity of a reference temperature T ≈ Tm (in our case the melting temperature) to obtain the well known Nahme's approximation [Costa and Macedonio, 2002] which, after some algebra, can be written as
Note that we neglect possible deviations of the fluid from newtonian behavior. Values of ηc and Tc may be estimated based on composition, clast content and a well known empirical viscosity law [Shaw, 1972; Spray, 1993]. Note that this only leads to a very rough estimate, given that the result is highly dependent on clast concentration and melt composition; furthermore, it assumes that the melt behaves as a newtonian fluid (viscosity independent of shear rate) which is probably not the case here. As a consequence, a priori estimates of ηc, Tc are only indicative for the time being, and shall require a substantial analysis of experimental and natural samples which has just started. Indeed, the chemical composition is different from the original rock due to selective melting [Spray, 1992] and does vary dynamically during the experiment [Hirose and Shimamoto, 2005a]. For instance, unpublished electron microprobe analysis indicate that the glass composition (i.e., the matrix) of the artificial pseudotachylyte produced in high-velocity rock friction experiments conducted on tonalite is far more basic (52% SiO2) than that of the starting tonalite (66% SiO2), though SiO2 content is only one of the chemicals that affect melt viscosity (the other being alkalis, H2O content, etc. [Philpotts, 1990]). In the case of experiments conducted on gabbro, the difference in composition between glass an gabbro is less marked at the steady state [Hirose and Shimamoto, 2005a].
 In addition, it is possible to infer adequate values of ηc, Tc directly from the experimental measures of friction (see section 7). Auspiciously, the empirical estimate and the data fit yield compatible results, such that ηc, Tc fall within the same ranges. With the viscosity defined as (31), the differential diffusion equation becomes
Given the symmetric boundary conditions on both sides of the melt layer, we seek for a solution that is symmetric with respect to the melt center, with no discontinuity either in T or ∂T/∂z. This lets presume a solution including some trigonometric function where the spatial period is linked to the melt thickness and the boundary conditions, and incorporating a logarithm as the inverse of the viscosity exponential. After some algebra, the application of boundary conditions (16) and (17), and the requirement of symmetry in z = 0, we obtain (see details in Appendix B) the steady state temperature distribution for a given τssV product:
where the characteristic velocity W has been defined as
Imposing the boundary condition (17) to the temperature solution (34) yields a relationship between melt thickness ω, shear stress τss and slip rate V:
Note that by arbitrarily fixing ω, the above equation may yield a first constitutive relation for the steady state value of τss. However, it is not very useful by itself because the melt thickness ω is variable and yet unknown: In order to define ω, we introduce the normal stress dependence in section 4.
 On combining (34) and (31) we obtain the viscosity profile
and the shear rate profile can now be obtained according to (27). At slip rates V > W, the approximation
can be made. The above functions decrease monotonously with V, except for the lowermost slip rates below or close to W.
 Solutions (34) and (36) confirm that variations in traction τss only affect the melt thickness. Indeed, τss in equation (34) will contract or expand the temperature profile along the z axis, but the shape of T(z) will be the same upon rescaling. To exemplify this, we may introduce the dimensionless distance z′ = z/p, where p = ηcW/τss, into the temperature profile, and note that a variation of τss induces a change in the z′ scale. This is a consequence of the self-similarity already discussed above, in relation to equation (28). The boundaries of the melt (z = ±ω) where the condition T = Tm has to be met, should expand or contract accordingly, as defined in equation (36). While variations in V will affect the melt thickness too, they will also have a more complex effect on the shape of the temperature profile.
 Furthermore, we may define an equivalent shear viscosity ηs for shear across the melt such that
where V/2ω is the average strain rate across the melt. Since the slip rate V is due to the integral of shear rate across the melt, we have
which yields to the definition of ηs as
However, using the value of ω according to (36) and inserting it into equation (39), we also obtain
and we remark that ηs only depends on slip rate V. A check for consistency is readily done by inserting the temperature profile (34) into the viscosity law (31), and integrating it across the thickness ω: the same expression as (42) is obtained.
Figure 3 represent a typical solution of the temperature profile inside the melt, and the exponential decay of temperature outside the melt and into the solid wall. Figure 4 represents temperature, viscosity, shear rate and displacement rate inside the melt, for two different slip rates, assuming typical Earth fault parameters.
 We remark that increases in either V or τss result in thinning of the melt layer, but while variations in τss do not affect the shape of temperature and viscosity profiles, variations in V do alter them. Indeed, increased slip rates reinforce inhomogeneity, inducing a concentration of shear toward the middle of the melt layer, with higher peak temperatures. Instead, slower slip rates produce rather even shear distributions, more akin to shear within a layer of homogeneous viscosity.
 As a consequence, faults that slip faster should have a higher temperature peak in the middle of the melt. Microstructural analysis of pseudotachylyte bearing faults may allow estimates of the maximum temperature that has been reached in the fossil melt layer (for example, by identifying which minerals are still present as clasts). Noteworthy, some natural pseudotachylytes are depleted in clast content in the inner part of the vein: This is consistent with the expectations from our theoretical analysis [Di Toro and Pennacchioni, 2004, Figure 6]. Thus in principle the coseismic slip rate of fossil faults could be retrieved from clast size distribution across the pseudotachylyte thickness.
 Interestingly, equation (36) relates the melt thickness directly to slip rate V and shear traction τss, independently on the geometry and on the normal stress. This is confusing at first sight since, clearly, normal stress does have a strong effect on melt thickness. However, such an effect is accounted for indirectly in (36), through the shear stress term τss (a normal stress increase will also be accompanied by a shear stress increase (see section 5) corresponding to a reduction of thickness).
 An advantage of (36) is that it could allow a direct estimate of the two parameters governing the melt viscosity (ηc, Tc). Measuring stress under two different velocities, for example, yields a system of two equations and two unknowns, provided that the melt thickness can be measured with sufficient precision during the sliding. Unfortunately, the thickness, for the time being, can only be measured a posteriori, by analyzing a thin section of the sample. Since the melt thickness is most probably different at the end of the experiment than during the sliding phase, owing to continued extrusion for an unknown lapse of time, it was so far impossible to use this method for a reliable parameterization of the viscosity law.
4. Effect of Normal Stress
4.1. Extrusion Dynamics
 The next step of our theoretical analysis is to introduce normal stress dependence. As illustrated by the solution of shear heating problem in section 3, no dependence of normal stress has been introduced so far, as viscosity of the melt only depends on slip rate. Thus the effect of normal stress will essentially be introduced by its coupling to melt thickness 2ω.
 In the cylindrical experimental samples, the expulsion of melt from the sample edges can be modeled as the extrusion of a viscous fluid squeezed out from a disk of radius R. For less trivial geometries (e.g., natural faults) the problem can be reduced to similar terms by arguing that R is the average flow length of the melt before it is evacuated through an injection vein. At the steady state, the thickness of the melt layer is constant, so that melt is produced at the same rate at which it is expelled from the sliding interface: We may equate the rate of melt production ρνπR2 to the rate of melt expelled at the edges, r 2πRh, where r is the radial flow velocity averaged over the melt thickness h at the edge r = R. The value of r itself may be obtained by solving the problem of a laminar viscous flow driven by the difference of pressure gradient due to the compression of the melt layer. The resistance that a fluid film offers to compression is known as viscous force; there exists well-known solutions for cylindrical symmetry under the assumption of a constant, equivalent extrusion viscosity ηe within the melt [Lee and Ladd, 2002]:
for squeezing rate ν, normal stress against the rigid disk σn. It is noteworthy that the above form, developed in the cylindrical case, is generally valid for other geometries (for example a two-dimensional channel flow) with the exception that the dimensionless geometrical factor C will change. For a solid cylinder, C = 3/16. In case of a hollow cylinder, the situation encountered in the HVRF experiments, the factor C accounts for the presence of an inner radius Ri and an outer radius R, and the geometrical factor becomes [Fialko and Khazan, 2005]
Typical values of α in HVRF experiments on hollow samples, yield 0.012 < C < 0.046. On solid cylindrical samples, C ≈ 3/16. For a 2D symmetry instead of cylindrical, where R is half the distance between two extrusion lines, it can be shown that the geometrical factor becomes C = 1/2. For earthquake faults, we may consider that the sliding area is constituted by a number of intact patches surrounded by extrusion veins, assume that each patch is roughly circular and adopt C ≈ 3/16. On the other hand, we may assume a 2D symmetry (fault bands of width R separated by parallel extrusion veins) and use C = 1/2. A combination of both models implies an intermediate value 3/16 < C < 1/2. As for the value of R on natural faults, field observations (e.g., the Gole Larghe fault system [Di Toro et al., 2005]) suggest that it be of the order of decimeters to meters.
 Since the viscosity across the melt is inhomogeneous, we solve the Navier Stokes equation accordingly and then derive the expression for an equivalent homogeneous viscosity ηe, resulting in the same net radial extrusion, in order to use the simpler equation (43). According to the flow solution under cylindrical symmetry, neglecting centrifugal force and radial variation of viscosity yields the radial flow rate vr(r, z) inside the melt:
 It follows that the net flow rate across a section of the melt layer of thickness 2ω is
where P is the fluid pressurse in the melt. In the case of a homogeneous viscosity ηe, we may write
 By equating (46) and (47), we can express the equivalent extrusion viscosity ηe as
while we recall that the equivalent shear viscosity ηs appearing in equation (41) was expressed as
 In principle, since ηe ≠ ηs, we cannot use a single equivalent viscosity for both shear and extrusion dynamics. However, through appropriate integration of expressions (48) and (41) according to the viscosity profile obtained in (37), it can be shown that 1 < ηe/ηs < 1/3, depending on the ratio V/W. For small V, the viscosity profile is homogeneous and both expressions are equal. At large V, the profile is highly inhomogeneous but the maximum difference between the two expressions is a factor of 3. In particular, we shall make the assumption that V>>W and use the following limit for further developments:
Equation (43) assumes that all the normal stress across the interface is supported by the viscous extrusion force. However, other factors may affect the coupling of h and σn and complicate the relationship described in (43). The contribution of hydrodynamic pressure [Brodsky and Kanamori, 2001] at high slip rates could be accounted for by adding a term to σn resulting in
where H is a topographic variation of the interface boundary, and λ its wavelength. The first term on the right-hand side of (52) is not included in the analysis of Brodsky and Kanamori  because the extrusion process was not considered.
which allows to evaluate whether the hydrodynamic push dominates in the effective support of normal load (χ ≪ 1) or whether the viscous force dominates (χ ≫ 1).
 Using tentative values of H and λ, suggested by preliminary inspection of experimental samples [Hirose and Shimamoto, 2003] and thin sections of natural faults, shows that the two terms may be of comparable magnitude at the lower range of shortening rates v, while viscous force may dominate in the higher range of v. Shortening rate will increase with normal stress, so that viscous force may be dominant under large normal loads (several kilometers in depth in the Earth Crust). However, the outcome essentially depends on actual values of H and λ and how they vary as a function of the experimental conditions, so that for the moment a reliable statement on the value of χ is premature.
4.3. Hertzian Contact and Other Effects
 Further effects on σn should be included: (1) the normal stress may be partially supported by the center of the rotating samples, in the case of full core cylindrical samples, where an almost solid-solid contact is achieved due to the thinning of melt layer. (2) A bias may be due to solid clasts and sample surface roughness that are about the same dimension as the average melt thickness, thus providing shouldering and solid support across the interface. (3) The formation of bubbles within the melt is observed in the thin sections of experimental samples (see Hirose and Shimamoto [2005a] and SEM investigations of experimental samples used in section 7 of this study), under relatively low normal stress conditions; one consequence is that the volume increase produces overpressure, which also provides partial support of normal load. The contribution of effects 1, 2, and 3 is very complex, but we may add a generic, empirical term to the normal stress expression to account for effects other than the viscous extrusion force.
5. Theoretical Traction Under the Steady State
 On the basis of the sections 2–4, we now assemble a system of coupled equations where the variables are melt half thickness ω, traction τss, equivalent viscosity σe, normal stress σn, shortening rate v and slip rate V. Variations of viscosity and temperature across the melt layer are implicitly accounted for, by use of solutions (36)–(42) obtained in section 3. The most interesting question, both from the physical and the geological point of view, is how the value of friction (or shear traction) responds to given conditions of normal stress and slip rate. Accordingly, we solve the system in order to predict τss under the steady state.
 In its most straightforward combination, the shear traction can be derived from equations (11), (36), (43), and (51) yielding
where the parameters have been arranged, for clarity, in the following dimensional groups:
with dimensions of stress and velocity, respectively (ηcW/R also has stress dimensions); the dimensionless geometrical factor C is defined in equation (44). We note here that W can be interpreted as a critical slip velocity where a crossover takes place in friction from a velocity-hardening behavior (quasi-viscous friction) to a velocity-weakening behavior (unstable regime). Indeed, in (55), steady state friction τss increases with slip rate V for V < W but decreases with V for V > W.
 Frictional melt is unlikely to occur on earthquake faults at low slip rates; hence the following limit obtained by a Taylor expansion at high slip rates, may be used as a good approximation for V > W:
where A conveniently groups most of the rock, melt parameters and geometrical factor C(44), in a single term with units m1/2Pa3/4 such that
As the log term in (57) evolves slowly at V ≫ W, the velocity dependence of traction mimics V−1/2 in a given velocity interval, i.e., the traction exhibits rate weakening of order 1/2.
6. Experimental Results
 In order to specifically test the dependence on normal stress, slip rate, radius and geometry appearing in equation (57), we performed a new series of high-velocity friction experiments on India Gabbro rock samples. Incidentally, the experimental data also allows to illustrate the shortening rate dependence demonstrated in equation (11), the contraction of the transient duration determined in equation (23) and to outline the effect due to heat loss by radiation through the sample edges, to be discussed below. We selected India Gabbro because of the profitable previous expertise on this rock type [e.g., Tsutsumi and Shimamoto, 1997; Hirose and Shimamoto, 2003, 2005a, 2005b]. In addition, thermal fracturing is less severe in gabbro than in more acid, quartz-bearing rocks [Ohtomo and Shimamoto, 1994], thus facilitating the experiments. Finally, the relatively small grain size of India Gabbro yields better data than other rock types (e.g., the stress curves are smoother). The experimental axial load was varied from 0.25 to 15.1 MPa, the slip rate from 0.94 to 2.58 m s−1 and a variety of sample radii were used (full cylindrical samples of 19.5 mm, 11.2 and 10.9 mm and hollow-shaped cylinders with external/internal radius of 19.5/13.3 mm and 19.5/7.7 mm). All experiments were conducted in the high-velocity rotary friction (HVRF) at Kyoto University, described in [Shimamoto and Tsutsumi, 1994; Hirose and Shimamoto, 2005a]. The new results are complemented with data from previous experiments on tonalite [Di Toro et al., 2006a], gabbro [Hirose and Shimamoto, 2005a], peridotite and monzodiorite (T. Hirose, unpublished, 2006). For a summary of the different experimental set up and main experimental results, see Table 4.
Table 4. Experimental Data Used in the Figures 6–9
 A typical experiment is displayed in Figure 5a, conducted on a hollow-shaped sample (19.5/13.3 mm) under constant, relatively low normal stress (0.75 MPa). Given the low normal stress applied, shortening was slow and this allowed to reach a total slip of 280 m. Note that typical slip for experiments under larger loads (5–15 MPa) are much shorter, as illustrated in Figure 5b. The large displacement allows to investigate in detail the dependence of shear stress with, for instance, slip and slip rate. The shear stress evolution in Figure 5a is rather convoluted, going through several peaks before it finally reaches a minimum at the steady state.
 On the other hand, an increase in normal stress both simplifies and contracts the duration of weakening, as illustrated in Figure 5b: in this case, friction immediately starts to weaken in a single, short episode directly yielding to the steady state. The evolution of the fault products (presence of debris, melt composition, clast content, etc.) and shear stress with slip was investigated in detail in the case of gabbro experiments [Hirose and Shimamoto, 2005a]. In particular, a continuous film of melt decorates the sliding surface starting from the second peak friction. In all the experiments performed, an increase in slip rate corresponds to a lower steady state friction and an increased shortening rate v. An increment in normal stress σn amplifies both friction and shortening rate.
 By changing the sample size and shape (solid to hollow), and thus by changing the melt escaping distance R − Ri, we verified that under a given slip rate and normal stress, the steady state friction drops with increasing distance, that is, lubrication is more effective if melt is not allowed to escape from the sliding surface.
7. Comparison of Data and Theoretical Curves
 There are several significant differences between real Earth faults and a laboratory HVRF experiment. In particular, HVRF experiments have lower normal stress, smaller size, cylindrical geometry and heat loss through radiation, features not found in real Earth faults. These differences should be carefully taken into account when extrapolating laboratory data to real faults. This also suggests that slightly different models should be used to describe either natural faults or experimental faults. In a sense, the physics of frictional melting in the laboratory experiments is more complex than that one may expect on natural faults, because additional factors participate in the problem due to finite size and cylindrical geometry in HVRF.
 As a consequence, the fit of HVRF experimental data based on a theoretical model should include, ideally, nontrivial effects due to effects of radiated heat, sample geometry, finite size and velocity variations across the radius of the sample. A complete mathematical description can only be attempted with numerical simulation tools, a task that we leave for future studies. In our present approach, we rather adopt the analytical model developed in sections 2–5, arguing that it can be extended simply but efficiently, to include the gross features of the experimental conditions. The essential divergence can be summed up in two points: the cross-radius variability in the rotary experiments and the heat loss through radiation at the border of the samples.
 The radial variations across the rotating sample, may be accounted for by using effective values that correspond to a balanced mean across the radius. This approach was proposed by Shimamoto and Tsutsumi  and Hirose and Shimamoto [2005a] in order to quantify the average friction and slip rate across annular or cylindrical samples. It can be shown that the averaging approach is correct provided that the variable has a linear dependence with radius (Appendix C). In other cases the average is less accurate, but the error introduced is usually not large, since the dominant area is always represented by the peripheral annuli of the sample, while the central region of the sample is not representative. As shown in Appendix C, for the case of frictional melt at the steady state, a constitutive relation predicted by (57) yields identical behavior for averaged and actual values, except for a small constant factor. In addition, experiments performed with hollow samples (annular shape) where minimum and maximum radii are similar, show no fundamental difference with those performed on full cylinders [see Di Toro et al., 2006a, supplementary material]. These observations are comforting and confirm that the use of radial average is reasonable and meaningful in frictional melt problems.
7.1. Effect of Radiation Loss
 The radiation effect has a clear signature in the experimental data, it can be quantified and added to the heat balance. Indeed if we plot the shortening rate versus heat rate τssV, we expect a linear dependence that goes through the origin as a consequence of balance (10), in the absence of radiation or additional heat losses. The experimental data appear to be aligned indeed (Figure 6), but an overall offset indicates that an additional heat sink is present.
 In order to quantify the radiation, we start by remarking that according to (4), the thermal boundary layer of the sample which is subject to significant heating, shrinks in proportion to the increase of shortening rate ν. Thus the perimeter of the cylindrical sample is heated on a portion s of its surface s ≈ 2πR/(ν/κ). However, the maximum temperature is bounded by the melting temperature Tm at the interface. By matter of consequence, the overall radiation emitted from the sample perimeter will be inversely proportional to ν. Thus radiation can be essentially described as a factor ϕ/ν, where ϕ is a constant incorporating all information on values such as sample radius, radiation coefficient and bounding temperatures that remain unchanged even when load, slip rate and stress are altered. Incorporating the radiation factor into the heat balance (11), we obtain
Solving for ν yields the expression
where we recall that the notation
with stress dimensions has been introduced for clarity. The above theoretical expression for ν is represented together with data points for a variety of experiments in Figure 6. It is interesting to note that when heat rate τssV becomes small with respect to radiation ϕ, there is a limit where no real solution of (60) exist. This shows that no stable melt dynamics is possible when insufficient heat is produced for both melting and radiation. On the other hand, when heat production τssV becomes large, (60) tends to recover the radiationless form (10). Thus in the experimental data, the effect of radiation should be dominant at small τssV but negligible at large τssV. There appears to be a gap in the data at low values of τssV, possibly corresponding to the absence of solution at low heat rates in the presence of radiation loss.
 The ϕ factor quantifying radiation, can be estimated empirically by fitting the experimental data with (60) as illustrated in Figure 6. Regarding the series of experiments in object, we obtain a reasonable fit for values 10 < ϕ < 20 J m−1 s−2 (the curve is plotted for ϕ = 20 J m−1 s−2). The ϕ value depends on sample dimensions and composition, so that it has no general significance, but the above value should not vary widely for rock experiments with a similar setup. The equivalent heat loss per square meter of sliding surface, assuming a shortening rate of the order of ν ≈ 1 mm s−1, is about 2 104 J m−2 s−1. In the presence of radiative loss the behavior is more complex, especially at low normal stress, as illustrated in Figure 7 by both the data and the theoretical curve based on equation (60). In particular, as mentioned above radiation loss introduces a condition of minimum heat production rate, below which there exists no stable melting solution. As a consequence, instability and leaping is likely to occur at low normal stress, generating a stick-slip-like behavior indeed observed in some HVRF experiments (T. Hirose, unpublished data). Under larger normal stress, radiation loss does not grow while shear traction and melt production increase, so that the effect of radiation gradually vanishes in the heat balance and the behavior tends toward (57).
 Ideally, all the experimental results obtained under variable conditions of velocity, size and normal stress, and for various rock compositions, could be collapsed on a single curve with the adequate choice of dimensionless parameters. For example, according to equation (57), a general representation of shear versus normal stress could be obtained by using the normalization:
independently on slip rate, material composition type or sample size, while the velocity dependence could be illustrated by
independently of material composition, sample size or normal stress, and similarly for size dependence.
 Unfortunately, though radiation loss should play no role on natural faults, it may slightly alter some experimentally estimated frictional values, in particular, at low normal stress (at high loads the heat production τV is much greater but the radiation ϕ remains bounded as explained above, thus becoming negligible). Given the variety of experimental conditions, and the nonlinear effect of radiation expressed in equation (60), the use of a general normalization system of the variables is not straightforward.
 As a consequence, in order to illustrate the dependence of shear stress on either velocity, normal stress, composition or sample size, we separated the experimental results in different subsets. Within each subset, only one of the above parameters varies from one experiment to the other, while the other parameters are kept similar.
7.2. Normal Stress Dependence and Composition Effect
 For example, in order to illustrate the normal stress dependence of shear traction, Figure 7 shows the behavior of rocks within three different composition sets: data set a, gabbro-monozodiorite; data set b, peridotite; and data set c, tonalite.
 Data set a groups experiments done on full cylindrical samples of radius R ≈ 11 mm of gabbro and monzodiorite rocks (those are very similar rocks in mineral assemblage and chemical composition), under effective slip rates V ≈ 1.14 m/s, while normal stress conditions span a wide interval 0.8–20 MPa (Figure 7, triangles and circles). We needed to apply a renormalization on a few data points used because the corresponding experiments used a slightly different sample radius R′ ≠ 11 mm and/or slip rate V′ ≠ 1.14 m s−1. In order to eliminate spurious variations due to the different radii and velocities (and obtain a homogeneous data set where the only variable is normal stress), we normalized the measured shear stress by and . The normalizing correction is based on equation (57), which predicts an inverse square root dependence on velocity and radius.
 Data sets b and c include experiments performed under similar conditions of normal stress, slip rate, and sample size, but on rocks of quite different composition: tonalite samples are relatively richer in silica (Figure 7, stars), while peridotite samples are depleted in silica (Figure 7, diamonds). Since the viscosity increases dramatically with silica content, a difference of a few orders of magnitude would be expected provided that melt thickness and temperature remained the same for all experiments. However, as seen from Figure 7, the difference is small (at most a factor of 3).
 This illustrates that the frictional melt dynamics process introduces a feedback effect between thickness, temperature and viscosity of the melt, resulting in a quite similar, weak effective shear traction at the interface independently on the rock type. This feature has been noted elsewhere based either on experimental [Di Toro et al., 2006b] or numerical studies [Sirono et al., 2006]. In addition, the melt composition is fairly different from the original rock: While several silica-rich minerals do not reach melting temperature (e.g., quartz, melting point 1713°C at 0.1 MPa [Spray, 1992]), the iron and magnesium rich minerals (e.g., biotite, melting point 700°C at 0.1 MPa [Spray, 1992]) melt at lower temperatures producing a less viscous, silica-depleted melt ([Shand, 1916; Spray, 1992]). On the other hand, this is partly compensated by the fact that solid clasts remain in suspension increasing the apparent melt viscosity. Given the above considerations on the chemical complexity of dynamic melting process, it is obvious that considerations based on host rock composition alone do not lead to a reliable viscosity prediction.
 The nonlinear effect of normal stress in the presence of melt lubrication is clearly visible in Figure 7; even in the presence of radiation loss, the trend is compatible with a σ1/4 dependence (as in equation (57)).
 However, according to solution (55), the shear traction versus normal stress curve should go through the origin. This is the case also for a solution which includes the effect of radiation loss (60) on the resulting shear resistance curves (not shown here). On inspecting the normal stress dependence of the data as plotted in Figure 7, though, a slightly better match is obtained by adding a finite normal stress value σo to the viscous push in equation (55):
(where for convenience, f( ) represents the complex rate dependence of equation (55), and A is the parameter group defined in equation (58)).
 The normal stress effect is illustrated in Figure 7, which was obtained by selecting a subset of experiments on gabbro with identical geometry, and identical radius R and slip rate V apart from negligible variations. Thus with the values of R, V imposed, terms Γ and σo could be adjusted in equation (64) in order to obtain the theoretical curve yielding the best data fit. The same operation was repeated for peridotite and tonalite, which lay on different curves because of different chemical composition of the melt. Values of σo ≈ 1.2, 1.5 and 4 MPa are found to suitably match the data for gabbro, tonalite and peridotite experiments, respectively. As for the right-hand factor (Af(V)/) in (64) the three curves are obtained for values of 3.75 × 104, 6.9 × 104 and 3 × 104 Pa3/4, respectively.
 Several hypothesis can be outlined to explain this σo effect. An attractive explanation is the presence of an elastohydrodynamic effect [Brodsky and Kanamori, 2001], but a confirmation that such process is effective will rely on accurate studies of the actual melt/solid interface roughness on both natural faults and laboratory experimental samples in the presence of melt, yet to be achieved. In alternative, we may assume a partial support of the load by contacting asperities on the irregular sample surfaces (see section 4.3), or the pressurization due to bubble formation and degassing of melt under low normal stress. A quantitative model including such effects requires further research, including extensive analysis of thin sections of experimental and natural fault samples.
7.3. Velocity Dependence
 In order to illustrate the effect of slip rate, data from experiments performed on gabbro under similar conditions of normal stress and sample size were gathered. Two different subsets were obtained, one at normal stress of 0.8 MPa and one at 1.5 MPa, covering a range of slip rates between 0.94 and 2.57 m s−1. The two subsets were normalized in order to align on a single curve in spite of the difference in normal stress, and then represented in Figure 8. The grey curves are obtained using the slip rate V dependence
in an attempt to obtain a suitable value of W (as defined in equation (35)). With no restriction on the overall amplitude, it was found that values within the interval W = 0.037–0.14 m s−1 all adequately fit the data. We note that this range is in agreement with velocity weakening estimated through inversion of seismic source data [Nielsen and Olsen, 2000]. The uncertainty on W obtained in Figure 8 is almost a factor of 4, given the current experimental data. Indeed the data points fall in the higher velocity range where the tail of the distribution is not critically dependent on W. This lack of resolution may be reduced if either (1) experiments at lower slip rates could be performed (but this involves technical challenges related to the rigidity of the machine) or (2) the dynamic thickness of the melt layer could be measured in situ during active slip, and used with equation (36) (but the dynamic measure of ω is another technical challenge).
 In spite of the lack of resolution on W, the experimental results are compatible with the theoretically trend predicted, as illustrated by the agreement between the velocity-weakening curves and the data.
 Note that the parameter fitting of Figure 7 with equation (64), implicitly uses the constitutive parameters of rock and melt, which are all known except for ηc, Tc which govern the melt viscosity. Similarly, the above analysis of the velocity dependence and fit of Figure 8 with equation (65) allowed to constrain the value of characteristic velocity W which also contains ηc, Tc, though only within a wide uncertainty domain. Thus combining the values obtained for gabbro by fit of both normal stress and velocity dependence, allows to define a domain of possible parameters ηc, Tc governing the temperature dependence of viscosity under steady state conditions. In particular, we note that by selecting the upper range of W = 0.14, we obtain tentative values of Tc ≈ 8°C, ηc ≈ 4.5 103 Pa though the significance of such a result is reduced by the lack of resolution on W (we stress that such values are purely indicative, and that they are sensibly different from alternative estimates, for example, Tc ≈ 74°C, ηc ≈ 15 103 Pa obtained through an empirical viscosity prediction based on the composition of the solid rock [Shaw, 1972] corrected for the presence of 20% of clasts [Spray, 1993]).
7.4. Sample Size Dependence
 Finally, let us comment on the role of sample size R. This part is less trivial because (1) as the sample preparation is performed using standard machine drills, there is a small number of different sizes available, (2) the limited machine power and the fragility of smaller samples imposes an upper and a lower bound on the sample size, respectively (as a consequence, there is a very small number of data points in the subset used to characterize size dependence), and (3) the effect of radiation loss varies with sample geometry, because of large variations in the ratio of frictional surface to air-exposed surface. This introduces variations on τ which are, at the time being, very difficult to remove from the data when the sample geometry changes. Indeed, the quantification proposed in equation (60) assumes a constant geometry of the system, represented by the ϕ term. We argue that radiation loss is in proportion to the outer and inner surface of the cylindrical tube, sr ≈ 2 b π (R + Ri) if b is the overheated region of the sample. On the other hand, the production of heat by friction is proportional to the sliding contact surface, ss = π (R2 − Ri2). As a consequence, the relative incidence of radiation loss on the traction will vary nonlinearly, as (R + Ri)/(R2 − Ri2), through ϕ term of equation (60).
 While the viscous extrusion dynamics for a full core, cylindrical sample only depends on one dimension (sample radius R), for many hollow samples the extrusion takes place both at the inner radius Ri and the outer radius R, so that both R and the geometrical correction factor C(α) based on α = Ri/R need to be accounted for (see equation (44)) as proposed by Fialko and Khazan . As a consequence, for hollow samples the size and geometry should affect shear resistance τ as R−1/2C−1/4, according to equations (55) and (44), provided that radiation loss is negligible, which, of course, is not the case in most of the lower load experiments (as illustrated in Figure 6). In the case where radiation is significant, the effect of sample geometry affects the traction both through its role in viscous extrusion and its role in radiation loss. It follows that the signature of sample size is thus far from clear in the present data set. For simplicity, a subset of samples for experiments done under similar conditions but with different radii were selected for Figure 9, where traction is shown as a function of the quadratic difference between outer and inner radii (Re = ): Weakening is indeed associated to an increase in sample dimensions.
8. Sliding on Ice
 Slip during one earthquake is just a small fraction of the total fault length, as a rule of thumb the ratio is about 10−4. The opposite situation occurs for ice sliding: The sliding object, usually small (skate, ice block, ski, etc.), is displaced of several times its own length, constantly migrating on fresh ice surface during its movement. This is true in most current situations that we may witness in everyday's life, which exclude the cases of extremely large objects or ice quakes inside glaciers.
 As defined in equation (23), the time to reach the steady state for melt dynamics is
after what a thermal boundary layer of thickness zb = κ/v is formed (see equation (4)). For a slider of length l, the duration of sliding at a given point of the ice surface is ts = l/V. As a consequence, if ts<tc the steady state in the ice is not reached: A pervasive melt layer is not formed, while the contact is only taking place at small asperities of dimension b ≪ l. In this case an additional parameter, the indentation hardness which is strongly temperature-dependent, should be introduced in order to account for the number of asperity contacts per unit area b/l. This is a possible model for the transient, initial phase of the rock friction experiments, where flash or local melting is observed.
 In such case, as indicated by Oksanen and Keikonen , the amount of heat conducted into the solid ice, at one asperity contact, in the time interval ts = b/V amounts to
while the amount of heat stored in an ice layer of thickness zb is
Equating both yields the thickness of thermal boundary into the solid ice:
as opposed to large objects with a longer contact time, obtained in our steady state treatment:
 It follows that the resulting dynamics will be quite different in either case. Note also that in the case where objects of different composition (metal, wood, Teflon,..) are sliding on ice, the symmetry used in our study will be lost, yet another difference in the problem. For the case of small objects, fast sliding, initial temperatures close to the melting point, and low normal stress the situation ts < tc may occur. The system is far from the steady state case developed in this study; instead we refer to the treatment proposed by Oksanen and Keikonen . The experimental measures they propose show a surprising variety of different trends with slip rate and normal stress, few of which are reminiscent of the ones presented in this study.
 The above considerations on the example of ice, show that the basic energy balance involving shear heating, melt, and heat transport through diffusion and advection, provides a good starting point for the analysis of a variety of situations where frictional melt dynamics is involved. However, the resulting constitutive properties such as friction dependence on velocity, normal stress and size cannot be described with a single, universal law, but may have different behavior domains depending essentially on the geometry and the scaling of the problem considered. The theoretical form proposed in equations (55)–(57) applies to the cases of large sliding objects (l ≫ Vt) and slip durations t allowing to reach a steady state in the presence of melt extrusion, conditions which do apply to seismic slip on some documented natural faults [Sibson, 1975; Di Toro et al., 2006b]. Equation (60) illustrates the additional effect of radiation loss, a condition which should be accounted for laboratory experiments where samples are partly exposed to the open air.
 Steady state friction in the presence of melt is described by a system of coupled equations involving viscous shear, temperature-dependent viscosity, thermal balance of shear heating, diffusion, latent heat and extrusion of melt under an applied normal stress. The value of the sliding friction, the final thickness of the melt layer, its viscosity and its temperature profiles are a result of the model, under a given slip rate and imposed normal stress. The predicted friction depends nonlinearly on normal stress, slip rate and also on the runaway distance of the melt before extrusion (i.e., radius of the sample of the frictional contact area).
 Since melt dynamics may have important implications for earthquake faults, we present a series of data from rock friction experiments, performed under conditions close to those of seismic fault slip in the presence of melt. The theoretical model prediction are compared to laboratory measures of steady state friction. The first remarkable point in the experimental data is the evidence of radiation loss, showing a clear signature when comparing the amount of heat production to the rate of melt production. After adding a radiation loss term in the thermal balance, the theoretical model reproduces the main aspects of the data. In particular, the frictional dependence on normal stress, sample size and slip rate observed in the data are compatible with the theoretical prediction.
 A general implication of this study, is that friction (at least in the presence of melt) depends nonlinearly on several parameters, so that extrapolation of laboratory measures is not trivial.
 For example, friction is usually assumed to increase linearly with normal stress, an assumption that does not take into account the lubrication process such as melting. Ideal lubrication with a lubricant layer of fixed thickness should yield no dependence on normal stress. However, in the complex situation of frictional melt in the presence of extrusion (a situation that also applies to exhumed seismic sources, where most of the melt is injected in the wall rock) the dependence of friction on normal stress appears to be in σn1/4.
 In addition, velocity weakening (weaker friction with increasing slip rate) is predicted and observed in the experiments; the resulting apparent friction is also strongly dependent on sample size, a feature which to our knowledge, has never been considered or conjectured before. As a consequence, transposition of laboratory results on samples of a few centimeters to the scale of meters or even kilometers on natural faults should be done with some caution.
 Another factor which should be accounted for, because it may alter the frictional behavior, is the presence of radiation loss from the samples exposed to air, void or other environmental fluids in the laboratory experiments. The radiation loss effect observed in lab experiments, may even be considered as an analogy for the presence of cooling fluids on faults in the Earth crust.
 Weak dynamic friction due to melting is observed in lab experiments, estimated from fossil faults, and predicted by the theoretical model developed here. It obviously has important implications for earthquake dynamics, such as large dynamic stress drops and abatement of heat production on faults. Though motivating for the present study, the wide range of seismological consequences is not developed here, but deserves to be discussed at length in future dedicated studies.
 Further development of this work may go in several different directions. The first could consist in using simpler, analog materials to do further experimental tests of frictional melt behavior under a variety of conditions. The opposite, longer, more complex route is to expand the model by accounting for the chemical complexity of natural rocks constituted by a mix of different minerals, which affects the viscosity, and the presence of solid clasts in variable proportion within the melt. The latter approach should probably need complex numerical tools rather than analytical solutions, in combination with a systematic microstructural analysis of experimental and natural fault rock samples.
 Finally, and more importantly for seismological studies, it would be interesting to obtain a generalization of rate-and-state friction laws nowadays implemented in earthquake fault models; such a generalization should allow to include effects of melt, fluid pressurization, hydrodynamic lubrication and other mechanisms into the friction constitutive relation, not only for the steady state condition under constant slip rate, but for the weakening transient and possibly for the whole cycle of seismic rupture.
Appendix A:: Solution of Diffusion With Imposed Heat Flow
 The well-known solution of the diffusion equation [Carslaw and Jaeger, 1959] for a heat flow Φ imposed at the border of a half-space is obtained by integrating
which, for a fixed flow Φ = τV/2, yields the temperature profile:
Appendix B:: Solution of Temperature Profile Inside the Melt
 The differential equation
has a general solution of the form
The C2 constant is merely a shift along the z axis; by symmetry with respect to the center of the melt (Figure 2), it has to be C2 = 0. We may now apply two boundary conditions, in order to obtain C1 and h. The first condition is (17), i.e., T = Tm at the boundary z = w,
which yields a first equation relating ω and the constant C1:
Taking the gradient of T(z) and applying condition (16) at z = ω−, we obtain
which yields a second equation for ω, C1:
Finally, we can equate the two above expressions (B4) and (B6) in order to constrain the value of the constant C1:
(in units of °K2 m−2). With C1 we now have the solution in closed form for both temperature profile in the melt (as in (34)) and melt thickness 2ω (as in (36)), for a given heat rate = τV.
Appendix C:: Significance of Average Velocity and Stress
 The average of a variable α(r) across a disk of radius R,
Along the same lines an average stress can be defined. However, the actual velocity dependence of stress is not trivial to retrieve. Given the radial dependence of slip rate in the rotary experiments, the equivalence () = τ() is accurate only for particular cases (τ being the actual stress and the measured average). The equivalence is trivial in the case that stress is independent of slip rate, and it holds also in the case where τ has a linear dependence on radius. For example, in the case of viscous shear τ(r) = ηr/h, if we assume that viscosity η and thickness h = 2ω are independent of r we obtain
Finally, according to (57), theory predicts that steady state shear stress essentially depends on the inverse square root of slip rate, as τ = a/. As a consequence,
Expressions (C4) and (C5) are identical for all practical purposes, albeit the slightly different factor (about 10% between 4/3 and ). These results are comforting and suggest that for the steady state of frictional melt, it is reasonable to define and to use equivalent averaged values for stress and velocity.
 Thanks to Emily Brodsky for her constructive review which improved the clarity of the manuscript.