3.1 Basic definitions of $h_{stop}$ , $h_{start}$ and the empirical flow rule

Pouliquen (Reference Pouliquen1999a ) performed experiments with spherical glass beads on rough beds and measured the deposit thickness $h_{stop}(\unicode[STIX]{x1D701})$ left by a steady-uniform flow as a function of the inclination angle $\unicode[STIX]{x1D701}$ . The inverse function of this deposit thickness $\unicode[STIX]{x1D701}_{stop}(h)$ was found to have the following empirical fit, given here in Pouliquen & Forterre’s (Reference Pouliquen and Forterre2002) reciprocal form,

(3.1) $$\begin{eqnarray}\displaystyle \tan \unicode[STIX]{x1D701}_{stop}=\tan \unicode[STIX]{x1D701}_{1}+{\displaystyle \frac{\tan \unicode[STIX]{x1D701}_{2}-\tan \unicode[STIX]{x1D701}_{1}}{1+h/\mathscr{L}}}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D701}_{1}$ is the slope inclination below which a layer of any thickness comes to rest and $\unicode[STIX]{x1D701}_{2}$ is the angle above which it is not possible for a uniform layer of grains to remain static on the slope. The parameter $\mathscr{L}$ (having the dimensions of a length) is the characteristic flow depth over which a transition between the angles $\unicode[STIX]{x1D701}_{1}$ and $\unicode[STIX]{x1D701}_{2}$ occurs and, as such, it is dependent on the properties of the grains and on the bed roughness. It follows that for $\unicode[STIX]{x1D701}\in (\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2})$ the deposit layer thickness may be written as

(3.2) $$\begin{eqnarray}\displaystyle h_{stop}(\unicode[STIX]{x1D701})=\mathscr{L}\left({\displaystyle \frac{\tan \unicode[STIX]{x1D701}_{2}-\tan \unicode[STIX]{x1D701}_{1}}{\tan \unicode[STIX]{x1D701}-\tan \unicode[STIX]{x1D701}_{1}}}-1\right). & & \displaystyle\end{eqnarray}$$

Experimental measurements of the flow thickness $h$ and the depth-averaged velocity $\bar{u}$ have been found to collapse (Pouliquen Reference Pouliquen1999a ; Forterre & Pouliquen Reference Forterre and Pouliquen2003; Edwards et al. Reference Edwards, Viroulet, Kokelaar and Gray2017) for an empirical flow rule of the form

(3.3) $$\begin{eqnarray}\displaystyle Fr={\displaystyle \frac{\unicode[STIX]{x1D6FD}h}{h_{stop}(\unicode[STIX]{x1D701})}}-\unicode[STIX]{x1D6E4}, & & \displaystyle\end{eqnarray}$$

which relates the Froude number,

(3.4) $$\begin{eqnarray}\displaystyle Fr={\displaystyle \frac{|\bar{u}|}{\sqrt{gh\cos \unicode[STIX]{x1D701}}}}, & & \displaystyle\end{eqnarray}$$

to the flow height and the deposit depth $h_{stop}$ . The constant of proportionality $\unicode[STIX]{x1D6FD}$ and the offset $\unicode[STIX]{x1D6E4}$ have been measured as $\unicode[STIX]{x1D6FD}=0.136/\sqrt{\cos \unicode[STIX]{x1D701}}$ with $\unicode[STIX]{x1D6E4}=0$ for flows of spherical glass beads on a rough bed of the same material (Pouliquen Reference Pouliquen1999a ), $\unicode[STIX]{x1D6FD}=0.65/\sqrt{\cos \unicode[STIX]{x1D701}}$ with $\unicode[STIX]{x1D6E4}=0.77/\sqrt{\cos \unicode[STIX]{x1D701}}$ for flows of sand on a rough bed of the same material (Forterre & Pouliquen Reference Forterre and Pouliquen2003) and $\unicode[STIX]{x1D6FD}=0.63$ with $\unicode[STIX]{x1D6E4}=0.40$ for flows of carborundum on a rough bed of spherical glass beads (Edwards et al. Reference Edwards, Viroulet, Kokelaar and Gray2017). Note that the alternative definition of the Froude number $\hat{Fr}=|\bar{u}|/\sqrt{gh}$ by Pouliquen (Reference Pouliquen1999a ) and Forterre & Pouliquen (Reference Forterre and Pouliquen2003) gives rise to the apparent angle dependence in $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6E4}$ , but this is a small correction and they should be thought of as constants. Using (3.3) to substitute for $h_{stop}(\unicode[STIX]{x1D701})$ in (3.2) and the fact that $\unicode[STIX]{x1D707}=\tan \unicode[STIX]{x1D701}$ for steady uniform flows, Pouliquen (Reference Pouliquen1999a ) determined an empirical dynamic friction law for flows on rough beds provided that the Froude number $Fr>\unicode[STIX]{x1D6FD}$ , which is the value that they assumed was the minimum steady uniform Froude number, i.e. $\unicode[STIX]{x1D6FD}_{\ast }=\unicode[STIX]{x1D6FD}$ .

Pouliquen & Forterre (Reference Pouliquen and Forterre2002) combined this dynamic rough bed friction law with Daerr & Douady’s (Reference Daerr and Douady1999) concept that once the grains were stopped (i.e. $Fr=0$ ) flow could not spontaneously start again until the inclination angle was increased past a critical value, $\unicode[STIX]{x1D701}_{start}(h)$ . This was similarly found to have an empirical fit of the form

(3.5) $$\begin{eqnarray}\displaystyle \tan \unicode[STIX]{x1D701}_{start}=\tan \unicode[STIX]{x1D701}_{3}+{\displaystyle \frac{\tan \unicode[STIX]{x1D701}_{2}-\tan \unicode[STIX]{x1D701}_{1}}{1+h/\mathscr{L}}}, & & \displaystyle\end{eqnarray}$$

where the third critical angle, $\unicode[STIX]{x1D701}_{3}$ , is the minimum angle at which it is possible for a layer of any thickness to accelerate from stationary and is the asymptote for large $h$ of the curve $\unicode[STIX]{x1D701}_{start}(h)$ . The inverse function of this curve gives the thickness of a flow that is initiated at the critical slope angle as

(3.6) $$\begin{eqnarray}\displaystyle h_{start}(\unicode[STIX]{x1D701})=\mathscr{L}\left({\displaystyle \frac{\tan \unicode[STIX]{x1D701}_{2}-\tan \unicode[STIX]{x1D701}_{1}}{\tan \unicode[STIX]{x1D701}-\tan \unicode[STIX]{x1D701}_{3}}}-1\right). & & \displaystyle\end{eqnarray}$$

This slope angle-dependent critical flow thickness is infinite at $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D701}_{3}$ and is undefined for $\unicode[STIX]{x1D701}<\unicode[STIX]{x1D701}_{3}$ . For angles $\unicode[STIX]{x1D701}\leqslant \unicode[STIX]{x1D701}_{3}$ a static uniform layer of any thickness will therefore remain stationary unless it is disturbed.

In the intermediate regime $Fr\in [0,\unicode[STIX]{x1D6FD}]$ there was no further experimental information and Pouliquen & Forterre (Reference Pouliquen and Forterre2002) suggested a Froude number dependent power law interpolation between the minimum dynamic and maximum static friction. Since the maximum static friction was higher than the minimum dynamic friction, the intermediate friction was a monotonically decreasing function of $Fr$ . Pouliquen & Forterre’s (Reference Pouliquen and Forterre2002) definition of a multivalued static friction, monotonically decreasing intermediate friction and monotonically increasing dynamic friction, as shown in figure 1(b), provide the essential ingredients for frictional hysteresis.

3.2 The transition thickness $h_{\ast }$ and Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction law

One assumption underlying Pouliquen & Forterre’s (Reference Pouliquen and Forterre2002) friction law is that the slowest steady-uniform flow, with Froude number $\unicode[STIX]{x1D6FD}$ , has the same thickness as the resulting deposit $h_{stop}$ . However, as shown in the erosion–deposition wave simulations of Edwards & Gray (Reference Edwards and Gray2015) (see their figure 15) in which $\unicode[STIX]{x1D6FD}_{\ast }=\unicode[STIX]{x1D6FD}$ , the resulting flow actually comes to rest to form a deposit that is thinner than $h_{stop}$ . As a consequence of this, and other deficiencies that are described in greater detail in § 6, Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) introduced the idea that the minimum thickness for a steady-uniform flow $h_{\ast }$ was greater than $h_{stop}$ and that it occurred at Froude number $\unicode[STIX]{x1D6FD}_{\ast }>\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ . This is consistent with Pouliquen’s (Reference Pouliquen1999a ) empirical flow rule, since all the steady-uniform flows are measured for $h>h_{stop}$ and $Fr>\unicode[STIX]{x1D6FD}$ . Both Forterre & Pouliquen (Reference Forterre and Pouliquen2003) and Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) also only observed steady-uniform flows that had Froude numbers significantly greater than $\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ in their experiments with angular sand and carborundum particles.

Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) assumed that $h_{\ast }(\unicode[STIX]{x1D701})$ occurred at a linear combination of $h_{stop}(\unicode[STIX]{x1D701})$ and $h_{start}(\unicode[STIX]{x1D701})$ and, hence, by the flow rule (3.3) the transition Froude number $\unicode[STIX]{x1D6FD}_{\ast }$ varied with the inclination angle. This approach allowed Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) to model the formation of eroded troughs, lateral levees and elevated channels for mono-disperse flows of carborundum on an erodible bed. While this was a reasonable first approximation, one consequence of this definition, which was not immediately apparent at the time, was that if $\unicode[STIX]{x1D701}<\unicode[STIX]{x1D701}_{3}$ then $h_{start}$ was undefined and hence the transition thickness $h_{\ast }$ and $\unicode[STIX]{x1D6FD}_{\ast }$ were also undefined. Furthermore, variation in $\unicode[STIX]{x1D6FD}_{\ast }$ with inclination leads to deposit layers that are significantly thicker than $h_{stop}$ being produced for shallow slope angles, as will be shown in § 6. Predictions of deposit layer depth are, however, still in good agreement with $h_{stop}$ on steeper inclinations, which explains why Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) were able to qualitatively match simulations with experiments in the small range of slope angles $34.0^{\circ }\leqslant \unicode[STIX]{x1D701}\leqslant 35.2^{\circ }$ that they investigated. This paper therefore introduces a relatively minor, but important, modification that the transition is instead assumed to occur at a fixed Froude number $\unicode[STIX]{x1D6FD}_{\ast }$ for all slope angles, which means that the friction law is now valid for all inclinations. The flow rule (3.3) then implies that the minimum steady-uniform flow thickness $h_{\ast }(\unicode[STIX]{x1D701})$ is a constant multiple of $h_{stop}(\unicode[STIX]{x1D701})$

(3.7) $$\begin{eqnarray}\displaystyle h_{\ast }(\unicode[STIX]{x1D701})=\left({\displaystyle \frac{\unicode[STIX]{x1D6FD}_{\ast }+\unicode[STIX]{x1D6E4}}{\unicode[STIX]{x1D6FD}}}\right)h_{stop}(\unicode[STIX]{x1D701}), & & \displaystyle\end{eqnarray}$$

which has the advantage that it is defined for all slope angles in the range $\unicode[STIX]{x1D701}\in [\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2}]$ . The variation of $h_{\ast }(\unicode[STIX]{x1D701})$ with the inclination is shown in figure 1(a). To ensure that $h_{\ast }(\unicode[STIX]{x1D701})>h_{stop}(\unicode[STIX]{x1D701})$ , the transition Froude number $\unicode[STIX]{x1D6FD}_{\ast }>\max (\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4},0)$ . This condition ensures that $\unicode[STIX]{x1D6FD}_{\ast }$ is strictly positive, and hence that the frictional transitions are well defined, even if $\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}<0$ as in the case of sand. A further constraint on the transition thickness is that $h_{\ast }(\unicode[STIX]{x1D701})<h_{start}(\unicode[STIX]{x1D701})$ for all inclinations, which using (3.7) implies that

(3.8) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{\ast }<\unicode[STIX]{x1D6FD}{\displaystyle \frac{h_{start}(\unicode[STIX]{x1D701})}{h_{stop}(\unicode[STIX]{x1D701})}}-\unicode[STIX]{x1D6E4}. & & \displaystyle\end{eqnarray}$$

The right-hand side of (3.8) has a minimum at $\unicode[STIX]{x1D701}=\tan ^{-1}((\unicode[STIX]{x1D707}_{2}+\unicode[STIX]{x1D707}_{3})/2)$ , where $\unicode[STIX]{x1D707}_{i}=\tan \unicode[STIX]{x1D701}_{i}$ for $i=1,2,3$ . It follows that the transition thickness $h_{\ast }<h_{start}$ for all angles provided

(3.9) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{\ast }\leqslant \unicode[STIX]{x1D6FD}_{\ast }^{max}=\unicode[STIX]{x1D6FD}\left({\displaystyle \frac{\unicode[STIX]{x1D707}_{2}+\unicode[STIX]{x1D707}_{3}-2\unicode[STIX]{x1D707}_{1}}{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{3}}}\right)^{2}-\unicode[STIX]{x1D6E4}. & & \displaystyle\end{eqnarray}$$

This has a value of $\unicode[STIX]{x1D6FD}_{\ast }^{max}=0.47$ for the material properties in table 1, which is equal (to two decimal places) to the value of $\unicode[STIX]{x1D6FD}_{\ast }$ that was chosen by Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) to match deposit layer depths $h_{deposit}(35.2^{\circ })\approx h_{stop}(35.2^{\circ })$ for carborundum particles on a rough bed of glass beads. In fact, in § 4 it is explicitly shown, by performing direct numerical computations, that a single value of $\unicode[STIX]{x1D6FD}_{\ast }=0.466$ gives the correct experimental deposit layer depth $h_{stop}(\unicode[STIX]{x1D701})$ for all inclination angles. It is also in approximate agreement with the minimum observed Froude number for steady-uniform flows in Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) flow rule experiments. However, for other materials and slope roughnesses $\unicode[STIX]{x1D6FD}_{\ast }$ may not be equal to $\unicode[STIX]{x1D6FD}_{\ast }^{max}$ and should rather be inferred directly from the flow rule experiments. As such, $\unicode[STIX]{x1D6FD}_{\ast }$ may be considered as another intrinsic rheological property, whereas $h_{stop}$ (and therefore $h_{\ast }$ too) is an emerging property.

3.3 The modified friction law

The modified friction coefficient in the dynamic $(Fr\geqslant \unicode[STIX]{x1D6FD}_{\ast })$ , intermediate $(0<Fr\leqslant \unicode[STIX]{x1D6FD}_{\ast })$ and static $(Fr=0)$ regimes with a constant $\unicode[STIX]{x1D6FD}_{\ast }$ is therefore

(3.10) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}=\unicode[STIX]{x1D707}_{1}+{\displaystyle \frac{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}}{1+h\unicode[STIX]{x1D6FD}/(\mathscr{L}(Fr+\unicode[STIX]{x1D6E4}))}}, & Fr\geqslant \unicode[STIX]{x1D6FD}_{\ast }, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}=\left({\displaystyle \frac{Fr}{\unicode[STIX]{x1D6FD}_{\ast }}}\right)^{\unicode[STIX]{x1D705}}\left(\unicode[STIX]{x1D707}_{1}+{\displaystyle \frac{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}}{1+h\unicode[STIX]{x1D6FD}/(\mathscr{L}(\unicode[STIX]{x1D6FD}_{\ast }+\unicode[STIX]{x1D6E4}))}}-\unicode[STIX]{x1D707}_{3}-{\displaystyle \frac{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}}{1+h/\mathscr{L}}}\right) & & \displaystyle \nonumber\\ \displaystyle \qquad +\,\unicode[STIX]{x1D707}_{3}+{\displaystyle \frac{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}}{1+h/\mathscr{L}}}, & 0<Fr\leqslant \unicode[STIX]{x1D6FD}_{\ast }, & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}=\text{min}\left(\unicode[STIX]{x1D707}_{3}+{\displaystyle \frac{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}}{1+h/\mathscr{L}}},|\text{tan}\,\unicode[STIX]{x1D701}\boldsymbol{i}-\unicode[STIX]{x1D735}h|\right), & Fr=0, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D705}$ is the interpolation power in the intermediate regime. Values of the friction angles $\unicode[STIX]{x1D701}_{i}=\tan ^{-1}\unicode[STIX]{x1D707}_{i}$ , $i=1,2,3$ and the flow rule parameters $\unicode[STIX]{x1D6FD}$ and $\mathscr{L}$ measured by Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) together with $\unicode[STIX]{x1D705}$ are given in table 1. This law reduces to the Pouliquen & Forterre (Reference Pouliquen and Forterre2002) friction coefficient for glass beads when $\unicode[STIX]{x1D6FD}_{\ast }=\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6E4}=0$ . The key difference to Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) law is that $\unicode[STIX]{x1D6FD}_{\ast }$ is now constant, rather than being a function of the slope angle, i.e. $\unicode[STIX]{x1D6FD}_{\ast }(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D6FD}h_{\ast }(\unicode[STIX]{x1D701})/h_{stop}(\unicode[STIX]{x1D701})-\unicode[STIX]{x1D6E4}$ , where  $h_{\ast }(\unicode[STIX]{x1D701})=(1-a)h_{stop}(\unicode[STIX]{x1D701})+ah_{start}(\unicode[STIX]{x1D701})$ and $a$ is a constant. This immediately implies that the friction law (3.10)–(3.12) is valid for all inclinations, not just where both $h_{stop}$ and $h_{start}$ exist, i.e. the range $[\unicode[STIX]{x1D701}_{3},\unicode[STIX]{x1D701}_{2}]$ . This small change therefore represents a significant extension of the range of validity of the theory.

It is important to note that in the static regime (3.12) the friction can take a maximum value of $\unicode[STIX]{x1D707}_{start}=\unicode[STIX]{x1D707}_{3}+(\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1})/(1+h/\mathscr{L})$ , but any state in which the grains can be held static by a lesser friction is also permissible, as shown in figure 1(b). A general two-dimensional static momentum balance implies that the friction must precisely match the gravitational force pulling the grains downslope as well as any pressure gradients that may be driving them in other directions. As a result $\unicode[STIX]{x1D707}=|\text{tan}\,\unicode[STIX]{x1D701}\boldsymbol{i}-\unicode[STIX]{x1D735}h|$ , where $\boldsymbol{i}$ is the unit vector in the downslope direction and $\unicode[STIX]{x1D735}h$ is the gradient of the thickness. This force also needs to be oriented to oppose the net forces in the static momentum balance.

3.4 Motion of a uniform depth layer

The properties of the friction law (3.10)–(3.12) can be neatly summarized by considering its effect in the case of a uniform flow of constant thickness $h$ . For this situation the momentum balance (2.2) reduces to

(3.13) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}\bar{u}}{\text{d}t}}=gS\cos \unicode[STIX]{x1D701}, & & \displaystyle\end{eqnarray}$$

where the non-dimensional net acceleration $S=\tan \unicode[STIX]{x1D701}-\unicode[STIX]{x1D707}$ determines whether the flow accelerates ( $S>0$ ), decelerates ( $S<0$ ) or moves at constant speed $S=0$ . This can be visualized by plotting the zero contour of the non-dimensional net acceleration $S$ as a function of the Froude number $Fr$ and the thickness $h$ as shown in figure 4. In the blue shaded region $S$ is positive and the flow is accelerated, which is indicated by arrows pointing to the right, while in the white region the flow is decelerated and the arrows point to the left. On the thick black line between the two domains $S$ is equal to zero and the flow will move at constant speed. It follows that all flows whose thickness is below $h_{\ast }$ are decelerated towards the static state, while flows whose thickness is above $h_{start}$ are either accelerated, or decelerated, towards steady-uniform flow. In the hysteretic regime between $h_{\ast }$ and $h_{start}$ initially static layers of grains are stable to small perturbations, because of the decelerative region immediately adjacent to the $h$ axis. The interpolation parameter $\unicode[STIX]{x1D705}$ , between the static and dynamic friction regimes, controls the necessary magnitude of finite sized disturbances (represented by an instantaneous Froude number) that are needed to destabilize a static layer of grains. Pouliquen & Forterre (Reference Pouliquen and Forterre2002) chose a value of $\unicode[STIX]{x1D705}=10^{-3}$ , which led to unphysically high sensitivity and under-resolution of the friction law (discussed in § 6.1). The stability of static layers to finite perturbations is better described by order-unity values of $\unicode[STIX]{x1D705}$ that give a roughly linearly decreasing zero contour in figure 4 between $Fr=0$ and $Fr=\unicode[STIX]{x1D6FD}_{\ast }$ , which forms an upper boundary to an approximately triangular decelerating region between $h_{\ast }$ and $h_{start}$ . Since this region is much wider at the bottom than at the top it is much easier to destabilize material whose thickness is closer to $h_{start}$ than $h_{\ast }$ . Once material in this region has been mobilized, however, it will accelerate/decelerate towards steady-uniform flow. A contour plot of the non-dimensional net acceleration $S$ , such as in figure 4, therefore provides a useful way of visualizing the hysteretic behaviour of the friction law.

Figure 4. The zero contour of the non-dimensional net acceleration $S=\tan \unicode[STIX]{x1D701}-\unicode[STIX]{x1D707}$ (solid black line) as a function of the Froude number and the flow thickness for the material properties of carborundum. Flows accelerate when $S>0$ (blue region and right arrows) and decelerate for $S<0$ (white region and left arrows). The solid vertical black line at $Fr=0$ shows the multivaluedness of the friction coefficient for stationary material. The region of positive net acceleration reaches $h=h_{\ast }(\unicode[STIX]{x1D701})$ (solid orange line) at $Fr=\unicode[STIX]{x1D6FD}_{\ast }$ (dashed black line), where $h_{\ast }(\unicode[STIX]{x1D701})$ is a constant multiple (3.7) of $h_{stop}(\unicode[STIX]{x1D701})$ (solid red line), and approaches $h=h_{start}(\unicode[STIX]{x1D701})$ (solid green line) for small Froude numbers. Note the interpolation parameter $\unicode[STIX]{x1D705}=1$ produces an approximately linearly decreasing $S=0$ contour between $Fr=0$ and $\unicode[STIX]{x1D6FD}_{\ast }$ . The material properties are given in table 1 and the slope angle-dependent properties in table 2.

Note that the qualitative picture in figure 4 is the same for Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction coefficient, in the reduced region of parameter space $\unicode[STIX]{x1D701}\in (\unicode[STIX]{x1D701}_{3},\unicode[STIX]{x1D701}_{2})$ for which their law is valid, although it will be shown in § 6 to produce deposit layers that are much thicker than $h_{stop}$ on shallow inclinations.

4.1 Numerical method

High-resolution shock capturing numerical methods (e.g. Nessyahu & Tadmor Reference Nessyahu and Tadmor1990) are required to solve the standard depth-averaged avalanche equations (e.g. Gray et al. Reference Gray, Tai and Noelle2003), which are a system of hyperbolic equations. The inclusion of the depth-averaged $\unicode[STIX]{x1D707}(I)$ -rheology (Gray & Edwards Reference Gray and Edwards2014; Baker et al. Reference Baker, Barker and Gray2016a ) changes this system into a set of (advection dominated) convection–diffusion equations. Therefore the closely related semi-discrete high-resolution non-oscillatory central schemes of Kurganov & Tadmor (Reference Kurganov and Tadmor2000) are used to solve the equations here. The time stepping is handled with a second-order Runge–Kutta method limited by upper bounds of both a Courant–Friedrichs–Lewy (CFL) number of 0.125 and a maximum step size of $10^{-4}$  s (as used by Edwards & Gray Reference Edwards and Gray2015; Edwards et al. Reference Edwards, Viroulet, Kokelaar and Gray2017), which helps with the extra constraints on the stable time step beyond the CFL condition due to the source and, in particular, viscous terms. The one-dimensional numerical domain spans $x\in [0,2]$  m and is discretized linearly over 2000 grid points. At the upstream boundary, $x=0$ , there is no advective or diffusive flux, which is implemented by setting $m=0$ and $\text{d}m/\text{d}x=0$ there. There is free outflow at the downstream boundary $x=2$  m, which is imposed via a linear extrapolation of the values of $h$ and $m$ from the final two columns of interior cells. In order to solve the system, the depth-averaged equations (2.1)–(2.3) are written in conservative vector form as

(4.1) $$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{w}}{\unicode[STIX]{x2202}t}}+{\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{f}(\boldsymbol{w})}{\unicode[STIX]{x2202}x}}=\boldsymbol{S}(\boldsymbol{w})+{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}}\left(\boldsymbol{Q}\left(\boldsymbol{w},{\displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{w}}{\unicode[STIX]{x2202}x}}\right)\right), & & \displaystyle\end{eqnarray}$$

where $\boldsymbol{w}=(h,m)^{\text{T}}$ is the vector of conserved variables $h$ and $m=h\bar{u}$ . The resulting convection flux function $\boldsymbol{f}$ , source term vector $\boldsymbol{S}$ and diffusive flux function $\boldsymbol{Q}$ are

(4.2a-c ) $$\begin{eqnarray}\displaystyle \boldsymbol{f}=\left(\begin{array}{@{}c@{}}m\\ {\displaystyle \frac{m^{2}}{h}}+{\displaystyle \frac{h^{2}}{2}}g\cos \unicode[STIX]{x1D701}\end{array}\right),\quad \boldsymbol{S}=hg\cos \unicode[STIX]{x1D701}\left(\begin{array}{@{}c@{}}0\\ \tan \unicode[STIX]{x1D701}-\unicode[STIX]{x1D707}\end{array}\right),\quad \boldsymbol{Q}=\unicode[STIX]{x1D708}h^{1/2}\left(\begin{array}{@{}c@{}}0\\ {\displaystyle \frac{\unicode[STIX]{x2202}m}{\unicode[STIX]{x2202}x}}-{\displaystyle \frac{m}{h}}{\displaystyle \frac{\unicode[STIX]{x2202}h}{\unicode[STIX]{x2202}x}}\end{array}\right), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

respectively. The friction $\unicode[STIX]{x1D707}$ in the source term vector $\boldsymbol{S}$ is given by (3.10)–(3.12) with the material parameter values given in table 1.

Figure 5. The flow thickness $h$ and downslope velocity $u(x,z,t)$ at a sequence of time steps (a–d) for a slope inclined at $\unicode[STIX]{x1D701}=36.3^{\circ }$ and with an initially uniform static layer of initial thickness $h(x,0)=2.4$  mm, which is $0.5$  mm greater than $h_{start}(36.3^{\circ })$ (solid green line). The filled region shows the thickness and the contour scale within it denotes the velocity, which is reconstructed from the depth-averaged downslope velocity $\bar{u}(x,t)$ by assuming an exponential profile (4.3) through the avalanche depth. There is no further inflow at $x=0$ , but there is free outflow at the downstream boundary. A travelling wave at the rear of the steady-uniform flow region (dash-dotted black line) passes through the material at a constant wavespeed that is greater than the surface velocity. This is followed by a slightly slower travelling deposition wave (between the dashed black line and the dotted black line) that connects the static deposit and the transition thickness $h_{\ast }$ (solid orange line). These waves catch up with surface particles, which are shown with light blue circular markers, and they are deposited on the surface of the final deposit layer that has a thickness of approximately $h_{stop}$ (solid red line). The material properties are given in table 1 and there is a movie of the simulation in the online supplementary material (movie 1).

4.2 Simulating the formation of an $h_{stop}$ layer

A numerical simulation of the formation of an $h_{stop}$ layer on a slope of $36.3^{\circ }$ and with a static initial layer thickness $h(x,0)=h_{start}(36.3^{\circ })+0.5$  mm $=2.4$  mm, which matches the experiment of § 1, is shown in figure 5. When the simulation commences this layer accelerates towards steady-uniform flow and at its trailing edge a non-uniform flow develops. The two-dimensional downslope velocity field is reconstructed from the depth-averaged velocity $\bar{u}(x,t)$ by assuming an exponential velocity profile (Wiederseiner et al. Reference Wiederseiner, Andreini, Epely-Chauvin, Moser, Monnereau, Gray and Ancey2011; Kamrin & Henann Reference Kamrin and Henann2015),

(4.3) $$\begin{eqnarray}\displaystyle u(x,z,t)={\displaystyle \frac{\unicode[STIX]{x1D706}\bar{u}(x,t)}{\exp (\unicode[STIX]{x1D706})-1}}\exp \left(\unicode[STIX]{x1D706}{\displaystyle \frac{z}{h}}\right), & & \displaystyle\end{eqnarray}$$

where the constant $\unicode[STIX]{x1D706}=1.4$ is chosen to match surface velocities. As the flow accelerates, a small pile of material is formed near the inflow due to frictional hysteresis, but by 40–50 cm downstream a uniform deposit is left behind, whose thickness $h_{deposit}(36.3^{\circ })\approx h_{stop}(36.3^{\circ })$ . Indeed, Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) chose the transition Froude number $\unicode[STIX]{x1D6FD}_{\ast }=0.466$ precisely because it produced the correct deposit layer thickness in the range of inclinations that they studied.

The thick steady-uniform flow and the thinner static deposit are connected by two travelling waves, as shown by the diagonal lines in figure 5. The first travelling wave marks the rear of the region of steady-uniform flow and moves slightly quicker than the second wave, which rapidly deposits material and connects the minimum steady-uniform flow thickness $h_{\ast }$ with the final deposit. In between the two waves is a slowly expanding region, within which the grains slow down and the flow thins from the steady-uniform value to $h_{\ast }$ . The speed of the second wave is calculated to be $0.22$  m s $^{-1}$ , which is in good agreement with the experimentally measured value of $0.24$  m s $^{-1}$ in § 1, for the same steady-uniform flow thickness and slope angle. Both travelling waves actually move faster than any of the grains, so a typical surface particle (shown in figure 5 by light blue markers) will initially accelerate and then travel downslope at constant speed, until it is caught up first by the steady-uniform flow wave and then by the rapid deposition wave. As the first wave passes by, the flow thins and the particles slow down, before the second travelling wave quickly brings the grains to rest on the surface of the $h_{stop}$ deposit. An animation of the process is available in the online supplementary material (https://doi.org/10.1017/jfm.2019.517).

Figure 6. Space–time $(x,t)$ plot showing the surface velocity $u(x,h,t)$ for a flow at $\unicode[STIX]{x1D701}=36.3^{\circ }$ and with an initial stationary layer of thickness $h(x,0)=h_{start}(36.3^{\circ })+0.5~\text{mm}=2.4$  mm. Individual surface particles are tracked (light blue lines) to visualize the particle trajectories through the travelling wave at the back of the steady-uniform region (dash-dotted black line), the slowly expanding region and the subsequent travelling deposition wave (between the black dotted and dashed lines). The vertical lines indicate stationary material and diagonal lines represent moving grains. The first travelling wave moves slightly faster than the second travelling wave, which rapidly deposits the grains once they have been slowed down in the expanding region (between the dotted and dash-dotted black lines). The material properties are given in table 1.

The downslope surface velocity can also be visualized as a space–time plot as shown in figure 6. Since the surface velocity contours are separated at the friction transition thickness into regions of two slightly different gradients, this gives further support to the observation that the deposition process occurs by way of two travelling waves that move downslope at slightly different constant speeds and are connected by a slowly expanding region. A series of surface particle trajectories are also plotted on top of the contours in figure 6 and these all have the same shape. In the region of steady-uniform flow they propagate downslope with a constant surface velocity of $0.22$  m s $^{-1}$ , but as the deposition wave approaches with a slightly faster speed they slow down and stop to form vertical parallel lines. This agrees well with the experimental observations from the space–time plot in figure 2.

Setting $\unicode[STIX]{x1D712}=1$ in the momentum balance (2.2), which is equivalent to an exponential velocity profile (4.3) with $\unicode[STIX]{x1D706}\rightarrow 0$ , results in a surface velocity of $0.12$  m s $^{-1}$ that is equal to the depth-averaged steady-uniform flow velocity. Choosing $\unicode[STIX]{x1D712}\approx 1.16$ produces a velocity reconstruction that is in better agreement with the experimentally measured surface velocity.

4.3 Deposit layer depths at different inclination angles

Figure 7. A series of numerical simulations showing the evolving flow thickness starting from $h(x,0)=h_{\ast }(\unicode[STIX]{x1D701})$ (dashed black line) at slope angles (a) $\unicode[STIX]{x1D701}=31.9^{\circ }$ , (b) $\unicode[STIX]{x1D701}=34.1^{\circ }$ , (c)  $\unicode[STIX]{x1D701}=36.3^{\circ }$ , (d) $\unicode[STIX]{x1D701}=38.5^{\circ }$ and (e) $\unicode[STIX]{x1D701}=40.7^{\circ }$ . In each panel the flow thickness is shown at one second intervals by light blue lines and the blue shading increases in intensity with increasing time. The initial layer depths are equal to the thickness of the friction law transition $h_{\ast }$ (solid orange lines), which are less than $h_{start}(\unicode[STIX]{x1D701})$ (solid green lines). An initial momentum $m(x,0)=h_{\ast }\bar{u}_{\ast }$ is therefore imposed in order to start the flow, where $\bar{u}_{\ast }=\bar{u}_{\infty }$ is given by the steady-uniform flow velocity relation (4.4). As the deposition wave travels through the system all the flows eventually leave a deposit that is close to $h_{stop}(\unicode[STIX]{x1D701})$ (solid red lines). The material properties are given in table 1 and the slope angle-dependent properties in table 2. Movies of the full simulations are available in the online supplementary material (movies 2–6).

Figure 8. A series of numerical simulations showing the evolving flow thickness starting from $h(x,0)=h_{start}(\unicode[STIX]{x1D701})+0.1~\text{mm}$ (dashed black line) at slope angles (a) $\unicode[STIX]{x1D701}=33.0^{\circ }$ , (b) $\unicode[STIX]{x1D701}=35.2^{\circ }$ , (c) $\unicode[STIX]{x1D701}=37.4^{\circ }$ , (d) $\unicode[STIX]{x1D701}=39.6^{\circ }$ and (e) $\unicode[STIX]{x1D701}=41.8^{\circ }$ . In each panel the flow thickness is shown at one second intervals by light blue lines and the blue shading increases in intensity with increasing time. All of the initially static layers have a thickness $h>h_{start}(\unicode[STIX]{x1D701})$ (solid green lines) and so they gain momentum. As the deposition wave travels through the system all the flows eventually leave a deposit that is close to $h_{stop}(\unicode[STIX]{x1D701})$ (solid red lines). The friction law has a transition thickness $h_{\ast }(\unicode[STIX]{x1D701})$ (solid orange lines). The material properties are given in table 1 and the slope angle-dependent properties in table 2. Movies of the full simulations are available in the online supplementary material (movies 7–11).

The friction law (3.10)–(3.12) does not just produce the correct deposit depth at one inclination angle and static layer thickness, but it does so for various initial conditions in the entire range of angles at which $h_{stop}$ layers form with carborundum particles. Figure 7 shows a series of simulations at inclinations ranging from $31.9^{\circ }$ to $40.7^{\circ }$ that start with a static layer of thickness $h(x,0)=h_{\ast }(\unicode[STIX]{x1D701})$ . Since the initial thicknesses are less than $h_{start}$ , an initial momentum $m(x,0)=h(x,0)\bar{u}(x,0)$ must be imposed to initiate a flow. The initial velocity is set to $\bar{u}(x,0)=\bar{u}_{\infty }$ , where in general, for a flow of uniform thickness $h(x,0)=h_{\infty }$ , the dynamic friction regime (3.10) implies that the steady uniform velocity is

(4.4) $$\begin{eqnarray}\displaystyle \bar{u}_{\infty }=\left({\displaystyle \frac{\unicode[STIX]{x1D6FD}h_{\infty }}{h_{stop}}}-\unicode[STIX]{x1D6E4}\right)\sqrt{gh_{\infty }\cos \unicode[STIX]{x1D701}}. & & \displaystyle\end{eqnarray}$$

Each plot in figure 7 shows a time sequence of the free surface at one second intervals, so the complete evolution of the formation of the deposit can be visualized by a shading that increases in intensity with increasing time. All the simulations show the rapid development of a travelling wave that directly connects the friction transition at $h=h_{\ast }$ to a deposit depth that is very close to $h_{stop}$ , as shown in figure 7.

Figure 8 shows a series of simulations at inclinations ranging from $33.0$ to $41.8^{\circ }$ that start with a static layer $h(x,0)=h_{start}(\unicode[STIX]{x1D701})+0.1$  mm, which therefore flows spontaneously. All of these flows leave a uniform thickness deposit by about 30–50 cm downstream that is very close to the correct experimental $h_{stop}$ value, as shown in figure 9. Indeed, the mean deviation of $h_{deposit}$ from $h_{stop}$ for all simulations is less than $3\,\%$ , whilst the maximum is less than $5\,\%$ . At low inclination angles (figure 8 b) there are clearly two travelling waves that are separated by an expanding region, but as the inclination increases (figure 8 c–e) the first and second waves move at almost the same speed, so the deposition wave as a whole looks like the back of the erosion–deposition waves observed by Edwards & Gray (Reference Edwards and Gray2015). It is also evident from figure 8 that the time taken to reach a steady travelling wave state, as well as the speed of the wave, both decrease with increasing slope angle. The travelling waves will be studied in more detail in § 5.

Figure 9. The deposit thickness (orange squares) left by simulations starting from initial thickness $h(x,0)=h_{\ast }(\unicode[STIX]{x1D701})$ (solid orange line), as shown in figure 7, and the deposit thickness (green squares) left by simulations starting from $h(x,0)=h_{start}(\unicode[STIX]{x1D701})$ (solid green line), as shown in figure 8. All the deposit thicknesses are in good agreement with $h_{stop}(\unicode[STIX]{x1D701})$ (solid red line). Simulations on a slope inclined at $\unicode[STIX]{x1D701}=36.3^{\circ }$ with various initial thicknesses (black circles) greater than $h_{start}(36.3^{\circ })$ also all collapse (downward arrow) to leave deposits (black squares) close $h_{stop}(36.3^{\circ })$ and are in good agreement with one another. An example of one of these simulations, for $h(x,0)=h_{start}(36.3^{\circ })+0.5$  mm, is shown in figure 5. The material properties are given in table 1 and the various slope angle-dependent properties for the initially uniform layer simulations in table 2.

An investigation into how the deposit depth is affected by changing the thickness of the initial layers is also crucial to our understanding of the deposition process. It has already been shown that the numerical deposits closely agree with the experimental value of $h_{stop}$ for both $h(x,0)=h_{\ast }(\unicode[STIX]{x1D701})$ and $h(x,0)=h_{start}(\unicode[STIX]{x1D701})+0.1$  mm for the whole range of valid inclinations. Figure 9 also shows the results of a series of simulations with different initial layer thicknesses on a fixed slope inclination of $\unicode[STIX]{x1D701}=36.3^{\circ }$ . All of these simulations, of which figure 5 is one example, produce a deposit depth that is within $3\,\%$ of the correct experimental value for $h_{stop}(36.3^{\circ })$ . This strongly suggests that, at least for a system of carborundum particles on a bed of glass beads, a universal $h_{stop}$ layer thickness develops due to the combination of two travelling waves that move at slightly different speeds and are connected by a slowly expanding region between them. The empirical deposit depth $h_{stop}$ is therefore insensitive to the initial conditions, since thick initial layers are able to connect to the top of a travelling deposition wave at $h_{\ast }$ , which then leaves the correct deposit thickness. Such solutions may not always exist for other granular materials, hence the form of the travelling deposition waves suggest that, in general, the most robust method of recovering the experimental deposit depth is by way of a steady-uniform flow at the minimum possible thickness $h_{\ast }$ .

Changing the shape factor from $\unicode[STIX]{x1D712}=1$ , as used here, to $\unicode[STIX]{x1D712}=1.16$ that results from the exponential velocity profile (4.3) with $\unicode[STIX]{x1D706}=1.4$ , has little effect on the deposit depth, despite changing the reconstructed surface velocity. For typical parameters, the thickness of the layer that is deposited is still within $2\,\%$ of the experimental value of $h_{stop}$ when calculated with $\unicode[STIX]{x1D712}=1.16$ .

4.4 Simulation of the experimental procedure to measure $h_{stop}$ and $h_{start}$

The experimental procedure for determining the $h_{stop}(\unicode[STIX]{x1D701})$ and $h_{start}(\unicode[STIX]{x1D701})$ functions, which was discussed in the introduction and sketched in figure 1(a), is now simulated numerically to further test the modified friction law proposed here. The simulation starts at a shallow angle $\unicode[STIX]{x1D701}=32.5^{\circ }$ . This is below $\unicode[STIX]{x1D701}_{3}$ , where $h_{start}$ is undefined, and where there is no thickness above which a static layer can spontaneously start to flow downslope. However, with the definition of $h_{\ast }$ as a multiple of $h_{stop}$ proposed in (3.7) it is still possible to have steady-uniform flows in the range $[\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{3}]$ provided there is a sufficient initial perturbation. The simulation is therefore started with a layer of thickness $h(x,0)=h_{\ast }(32.5^{\circ })=8.1$  mm that is given the corresponding minimum steady-uniform flow velocity that is possible at that inclination, according to (4.4) with $h_{\infty }=h(x,0)$ . It is important to note that the modified friction law (3.10)–(3.12) with constant $\unicode[STIX]{x1D6FD}_{\ast }$ is crucial, since it allows $h_{\ast }$ to be defined for all slope angles $\unicode[STIX]{x1D701}\in [\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2}]$ . In the case of Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction law with non-constant $\unicode[STIX]{x1D6FD}_{\ast }$ , the minimum steady-uniform flow thickness $h_{\ast }(\unicode[STIX]{x1D701})$ is not defined for $\unicode[STIX]{x1D701}\in [\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{3}]$ , since $h_{start}$ is undefined, and the model therefore fails to predict the initial collapse.

Since there is no further mass supply at the inflow, a deposition wave forms at the trailing edge of the steady-uniform flow and a deposit that is close to $h_{stop}(32.5^{\circ })$ is left on the chute. This static numerical deposit is then used as the initial condition for the next flow. To induce it, the chute inclination is increased progressively in $0.1^{\circ }$ increments until the entire layer, ignoring small movements in the thicker transient regions, gains momentum and collapses again to leave an even thinner deposit at a steeper slope angle. The process is repeated for the collapse of an entire static layer at five different inclination angles to produce a staircase of numerically predicted $h_{stop}$ and $h_{start}$ values, shown in figure 10. The angle at which the layer collapses and the theoretical value of $h_{start}(\unicode[STIX]{x1D701})$ are in good agreement for each collapse. Furthermore, the mean deposit thickness $h_{deposit}$ , taken between 1 m and 2 m downslope at the end of the simulations when all of the material has come to rest, is also shown in figure 10 to be approximately equal to $h_{stop}(\unicode[STIX]{x1D701})$ for each of the slope angles. This is the first time that anyone has successfully been able to model this complex experimental procedure and it is remarkable that the modified friction law, with an intermediate/dynamic transition at a fixed value of $\unicode[STIX]{x1D6FD}_{\ast }$ , is able to predict it so accurately. In § 6 it will be shown that all existing non-monotonic friction laws are unable to predict this behaviour, because either they (i) produce the incorrect deposit layer depth or (ii) are unable to hold grains static on the chute as the inclination is increased towards $\unicode[STIX]{x1D701}_{start}$ .

Figure 10. Numerical simulation of the experimental procedure to determine the $h_{stop}$ (black squares) and $h_{start}$ (black circles) curves for the material properties of carborundum. The first simulation requires perturbing a uniform layer of thickness $h_{\ast }(\unicode[STIX]{x1D701})$ (solid orange line), since no layer is thick enough to flow spontaneously for shallow inclinations $\unicode[STIX]{x1D701}<\unicode[STIX]{x1D701}_{3}$ . Imposing a steady-uniform flow causes this entire layer to collapse and leave a thinner deposit (leftmost downward arrow). The slope is then progressively inclined in $0.1^{\circ }$ increments until this static layer spontaneously begins to flow (rightward arrows). The inclination angle at which the collapse occurs is found to be in good agreement with $h_{start}(\unicode[STIX]{x1D701})$ (solid green line). Each successive flow leaves an increasingly thinner deposit (downward arrows), whose thickness is in good agreement with $h_{stop}(\unicode[STIX]{x1D701})$ (solid red line). The material properties are given in table 1 and the various slope angle-dependent properties for the initially uniform layer simulations in table 2.

6.1 Pouliquen & Forterre’s (Reference Pouliquen and Forterre2002) friction coefficient for glass beads

The non-monotonic friction law (3.10)–(3.12) reduces to the extended law of Pouliquen & Forterre (Reference Pouliquen and Forterre2002) for flows of spherical glass beads when the constant transition Froude number $\unicode[STIX]{x1D6FD}_{\ast }=\unicode[STIX]{x1D6FD}=0.136/\sqrt{\cos \unicode[STIX]{x1D701}}$ and the offset $\unicode[STIX]{x1D6E4}=0$ . This implies that the minimum steady-uniform Froude number is equal to $\unicode[STIX]{x1D6FD}$ , which is lower than that observed by Pouliquen (Reference Pouliquen1999a ) for most systems, and results in a minimum steady-uniform flow thickness $h_{\ast }=h_{stop}$ . The non-dimensional net acceleration $S(Fr,h)$ , with the material parameters for glass beads (table 3 a), is shown in figure 13 to display some notable differences to the modified friction law proposed here. Most significantly, Pouliquen & Forterre (Reference Pouliquen and Forterre2002) chose the power $\unicode[STIX]{x1D705}=10^{-3}$ in the interpolation between the static and dynamic regimes. This extremely small value of $\unicode[STIX]{x1D705}$ was chosen to give a rapid decrease of the friction coefficient concentrated near to $Fr=0$ and then an almost constant friction until the dynamic regime took over. In practice, however, the value of $\unicode[STIX]{x1D705}$ is so small that the friction law starts at the maximum static friction at $Fr=0$ , but at the nearest non-zero positive number that can be represented by machine precision (which is around $10^{-324}$ ) immediately drops down to a low value of $\unicode[STIX]{x1D707}$ as shown in figure 13(b). As a result the triangular decelerating region that should lie adjacent to the $h$ -axis between $h\in [h_{stop},h_{start}]$ is partially lost in finite precision calculations. This may explain why Pouliquen & Forterre (Reference Pouliquen and Forterre2002) found that their simulations of a finite mass release on an erodible layer ran out considerably further than in experiments.

Figure 13. Zero contours of the non-dimensional net acceleration $S=\tan \unicode[STIX]{x1D701}-\unicode[STIX]{x1D707}$ (solid black lines) with the material properties for glass beads (table 3 a) against the Froude number and flow thickness on (a) a linear and (b) a semi-log scale. In the blue shaded region (with right pointing arrows) $S>0$ and flows accelerate, while in the white region (with left pointing arrows) $S<0$ and flows decelerate. The solid vertical black line at $Fr=0$ in panel (a) shows the multivalued friction coefficient for stationary material. The transition between the intermediate and dynamic friction regimes occurs at $Fr=\unicode[STIX]{x1D6FD}$ (dashed lines), whilst the $h_{stop}$ and $h_{start}$ thicknesses are shown by the red and green lines, respectively. Note that in panel (b) the range of Froude numbers extends down to the minimum representable non-zero positive number on a typical computer.

The high sensitivity of Pouliquen & Forterre’s (Reference Pouliquen and Forterre2002) friction law is also unphysical, since it is known from their own experiments that in practice the deposit layers can be inclined up to $\unicode[STIX]{x1D701}_{start}$ (which may be several degrees higher than $\unicode[STIX]{x1D701}_{stop}$ ) before a flow is triggered. Static layers are therefore relatively stable to small disturbances. A larger value of $\unicode[STIX]{x1D705}$ may therefore be chosen to give a greater stability to layers in the intermediate frictional regime, which is why Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) set $\unicode[STIX]{x1D705}=1$ . Whilst this overcomes the problem of hysteresis in stationary layers, the non-zero offset $\unicode[STIX]{x1D6E4}$ in the empirical flow rule for more angular granular materials such as sand and carborundum causes further problems.

Table 3. Material properties for (a) glass beads on a bed of the same material, measured by Pouliquen & Forterre (Reference Pouliquen and Forterre2002) and (b) sand on a bed of the same material, measured by Forterre & Pouliquen (Reference Forterre and Pouliquen2003).

6.2 A friction law based on the empirical flow rule for sand

In an investigation into the long-surface-wave instability of granular materials, Forterre & Pouliquen (Reference Forterre and Pouliquen2003) performed experiments using a flow of 800  $\unicode[STIX]{x03BC}\text{m}$ diameter sand on a rough bed of the same particles. They measured a constant of proportionality $\unicode[STIX]{x1D6FD}=0.65/\sqrt{\cos \unicode[STIX]{x1D701}}$ an offset $\unicode[STIX]{x1D6E4}=0.77/\sqrt{\cos \unicode[STIX]{x1D701}}$ in the empirical flow rule. The friction angles and flow rule parameters measured by Forterre & Pouliquen (Reference Forterre and Pouliquen2003) for sand, along with an estimate $\unicode[STIX]{x1D701}_{3}=\unicode[STIX]{x1D701}_{1}+3^{\circ }$ are given in table 3(b). The offset $\unicode[STIX]{x1D6E4}$ in the flow rule alters the dynamic friction coefficient and the transition point to the intermediate regime, which now occurs at a Froude number $\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ . This would suggest, by following the same argument as Pouliquen & Forterre (Reference Pouliquen and Forterre2002), that the friction coefficient might naively be written as

Figure 14. Zero contours of the non-dimensional net acceleration $S=\tan \unicode[STIX]{x1D701}-\unicode[STIX]{x1D707}$ (solid black lines) against the Froude number and flow thickness with the material properties of (a) sand (table 3 b), for which $\unicode[STIX]{x1D6FD}<\unicode[STIX]{x1D6E4}$ , and (b) carborundum (table 1), for which $\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}>0$ . Flows accelerate when $S>0$ (blue region and right arrows) and decelerate for $S<0$ (white region and left arrows). The solid vertical black line at $Fr=0$ in panel (b) shows the multivaluedness of the friction coefficient for stationary material. The transition between dynamic and intermediate regimes occurs at $Fr=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ , which is negative for sand, implying that all flows are dynamic and even initially static ( $Fr=0$ ) flows of thickness $h>h_{steady}(0)>h_{stop}(\unicode[STIX]{x1D701})$ (solid red lines) will accelerate, including those with thickness $h<h_{start}(\unicode[STIX]{x1D701})$ (solid green lines). For carborundum with $\unicode[STIX]{x1D6FD}>\unicode[STIX]{x1D6E4}$ the transition Froude number $Fr=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ (dashed black line) is positive so the friction regimes are all well defined. The slope angle-dependent properties for (a) sand and (b) carborundum are given in tables 4(b) and 2 respectively.

(6.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{Sand}=\unicode[STIX]{x1D707}_{1}+{\displaystyle \frac{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}}{1+h\unicode[STIX]{x1D6FD}/(\mathscr{L}(Fr+\unicode[STIX]{x1D6E4}))}}, & Fr\geqslant \unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}, & \displaystyle\end{eqnarray}$$

(6.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{Sand}=\left({\displaystyle \frac{Fr}{\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}}}\right)^{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D707}_{1}-\unicode[STIX]{x1D707}_{3})+\unicode[STIX]{x1D707}_{3}+{\displaystyle \frac{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}}{1+h/\mathscr{L}}}, & 0<Fr\leqslant \unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}, & \displaystyle\end{eqnarray}$$

(6.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D707}_{Sand}=\text{min}\left(\unicode[STIX]{x1D707}_{3}+{\displaystyle \frac{\unicode[STIX]{x1D707}_{2}-\unicode[STIX]{x1D707}_{1}}{1+h/\mathscr{L}}},|\text{tan}\,\unicode[STIX]{x1D701}\boldsymbol{i}-\unicode[STIX]{x1D735}h|\right), & Fr=0. & \displaystyle\end{eqnarray}$$

With the parameters for sand, $\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}<0$ , and hence all flows are always in the dynamic regime. This is shown in figure 14(a), by plotting the zero contour of the non-dimensional net acceleration $S=\tan \unicode[STIX]{x1D701}-\unicode[STIX]{x1D707}$ with the material properties for sand. Since only the dynamic regime is active, an explicit formula for the zero contour $h_{steady}$ can be derived by equating (2.3) to zero for the dynamic friction coefficient (6.1) and simplifying with (3.2) to give

(6.4) $$\begin{eqnarray}\displaystyle h_{steady}(Fr)={\displaystyle \frac{(Fr+\unicode[STIX]{x1D6E4})}{\unicode[STIX]{x1D6FD}}}h_{stop}(\unicode[STIX]{x1D701}), & & \displaystyle\end{eqnarray}$$

which is linear in the Froude number. For $h>h_{steady}$ all flows are accelerative and for $h<h_{steady}$ all flows are decelerative. As a result, all constant thickness flows will either accelerate towards steady-uniform flow or decelerate towards zero velocity. However, the friction in the static state is not properly defined, so there is an issue here too. All of these problems arise directly from $\unicode[STIX]{x1D6E4}$ being greater than $\unicode[STIX]{x1D6FD}$ for the parameters for sand, which is a different problem to that of Pouliquen & Forterre’s (Reference Pouliquen and Forterre2002) law with $\unicode[STIX]{x1D705}=10^{-3}$ .

Returning to the material properties for carborundum (given in table 1), for which $\unicode[STIX]{x1D6FD}>\unicode[STIX]{x1D6E4}$ , the dynamic, intermediate and static regimes are still properly defined in (6.1)–(6.3) if $\unicode[STIX]{x1D6FD}_{\ast }=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ . The $S=0$ contour for the carborundum parameters is plotted against Froude number and flow thickness in figure 14(b). It has many of the desired properties for a hysteretic friction law, with (i) acceleration for $h>h_{start}$ , (ii) deceleration below $h_{stop}$ and (iii) a triangular region of deceleration between $[h_{stop},h_{start}]$ adjacent to the $h$ -axis. As $h$ approaches $h_{start}$ from below this triangular region becomes progressively smaller, implying that static grains are more likely to mobilized by a disturbance. However, Edwards & Gray (Reference Edwards and Gray2015) and Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) found that numerical simulations of a depositional steady-uniform flow using the material properties of carborundum and a transition Froude number $\unicode[STIX]{x1D6FD}_{\ast }=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ did not leave a deposit layer with the correct thickness, i.e. $h_{deposit}(\unicode[STIX]{x1D701})\neq h_{stop}(\unicode[STIX]{x1D701})$ , as will be demonstrated in § 6.4.

6.3 Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction law for angular materials

The non-monotonic friction law with constant $\unicode[STIX]{x1D6FD}_{\ast }$ presented here (3.10)–(3.12) reverts back to Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) original formulation if the transition thickness is assumed to lie between the $h_{stop}$ and $h_{start}$ curves according to $h_{\ast }(\unicode[STIX]{x1D701})=(1-a)h_{stop}(\unicode[STIX]{x1D701})+ah_{start}(\unicode[STIX]{x1D701})$ . The flow rule (3.3) then implies that the corresponding transition Froude number varies with the inclination angle like

(6.5) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FD}_{\ast }(\unicode[STIX]{x1D701})=\unicode[STIX]{x1D6FD}\left(1-a+a{\displaystyle \frac{h_{start}(\unicode[STIX]{x1D701})}{h_{stop}(\unicode[STIX]{x1D701})}}\right)-\unicode[STIX]{x1D6E4}, & & \displaystyle\end{eqnarray}$$

Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) chose $a=0.5$ to give $\unicode[STIX]{x1D6FD}_{\ast }(35.2^{\circ })=0.473$ in order to produce the correct deposit layer depth at that angle. It is no surprise that this approach was successful in the small range of angles $34.0^{\circ }\leqslant \unicode[STIX]{x1D701}\leqslant 35.2^{\circ }$ that they investigated, since it is close to the constant value of $\unicode[STIX]{x1D6FD}_{\ast }=0.466$ imposed here, which results in deposit layers close to $h_{stop}$ for all permitted inclinations $\unicode[STIX]{x1D701}\in (\unicode[STIX]{x1D701}_{1},\unicode[STIX]{x1D701}_{2})$ . However, a slope angle-dependent transition Froude number (6.5) results in deposit layers that are increasingly thicker than $h_{stop}$ as the inclination is reduced towards $\unicode[STIX]{x1D701}_{3}$ , as will be demonstrated in § 6.4. Furthermore, the transition thickness $h_{\ast }(\unicode[STIX]{x1D701})$ and hence $\unicode[STIX]{x1D6FD}_{\ast }(\unicode[STIX]{x1D701})$ become undefined for $\unicode[STIX]{x1D701}<\unicode[STIX]{x1D701}_{3}$ in this formulation, since $h_{start}$ is undefined there. Zero contours of the non-dimensional net acceleration $S$ do appear qualitatively the same for Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction law as they did for the modified version, shown in figure 4. However, that only holds if $\unicode[STIX]{x1D701}\in (\unicode[STIX]{x1D701}_{3},\unicode[STIX]{x1D701}_{2})$ and not for extremely shallow slope angles $\unicode[STIX]{x1D701}<\unicode[STIX]{x1D701}_{3}$ , which is a crucial flaw that was not immediately apparent.

Table 4. Slope angle-dependent critical layer thicknesses and coefficients $\unicode[STIX]{x1D708}(\unicode[STIX]{x1D701})$ in the depth-averaged viscosity $\unicode[STIX]{x1D708}h^{1/2}/2$ for flows of (a) glass beads and (b) sand on a bed of the same material. The material properties for each are given in the respective rows of table 3.

6.4 Numerical simulations with an initial stationary layer thicker than $h_{start}(\unicode[STIX]{x1D701})$

Numerical simulations are first carried out for an initially static layer of constant thickness $h(x,0)=h_{start}(\unicode[STIX]{x1D701})+0.1$  mm, with no further inflow at $x=0$ . With these initial and boundary conditions the constant thickness section should accelerate towards a steady-uniform flow and a deposition wave should develop at its trailing edge to leave a deposit of thickness $h_{stop}$ , as was the case in the simulations shown in figure 5 for carborundum with the friction law (3.10)–(3.12). Five different cases are considered here, that are shown in figure 15. In each case the constant thickness sections do accelerate towards a steady-uniform flow and a static deposit is left on the chute.

For glass beads with $\unicode[STIX]{x1D705}=10^{-3}$ (figure 15 a), the flow leaves a deposit layer $h_{deposit}(23.0^{\circ })\approx h_{stop}(23.0^{\circ })$ , which is the physically correct thickness that the empirical law is derived from. However, this is simply because the extremely small value of $\unicode[STIX]{x1D705}$ means that the friction law cannot bring to rest any material that is thicker than $h_{stop}$ . Setting $\unicode[STIX]{x1D705}=1$ (figure 15 b), which is required to capture the hysteretic behaviour, deposits a layer of physically incorrect thickness $h_{deposit}(23.0^{\circ })>h_{stop}(23.0^{\circ })$ . In the case of sand (figure 15 c) a deposit layer $h_{deposit}(35.0^{\circ })\approx h_{steady}(0)$ given by (6.4) is produced. This is because the friction law is always in the dynamic regime. Since $h_{deposit}(35.0^{\circ })\neq h_{stop}(35.0^{\circ })$ the friction law for sand is also physically incorrect and, moreover, the deposit is not completely stationary. The simulation with the parameters for carborundum (figure 15 d) and the friction law (6.1)–(6.3) also leaves an incorrect deposit thickness $h_{deposit}(35.2^{\circ })<h_{stop}(35.2^{\circ })$ , which was one of the main motivations for Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017) to introduce a frictional transition at a greater Froude number. Finally, Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction law, with a slope angle-dependent transition Froude number $\unicode[STIX]{x1D6FD}_{\ast }(33.0^{\circ })=2.05$ , is shown in figure 15(e) to produce a deposit layer that is approximately $50\,\%$ deeper than $h_{stop}(33.0^{\circ })$ . In fact, simulations with this friction law on a wide range of slope angles reveal that the deposit depths become increasingly thicker than $h_{stop}$ when the slope angle is reduced, as shown in figure 16. Furthermore, simulations are not possible at all for angles $\unicode[STIX]{x1D701}\leqslant \unicode[STIX]{x1D701}_{3}$ since $h_{start}$ and therefore $h_{\ast }$ become undefined there.

Figure 15. Flow thickness $h$ is shown at one second intervals by light blue lines and the blue shading increases in intensity with increasing time for numerical simulations using the material properties of (a) glass beads with $\unicode[STIX]{x1D705}=10^{-3}$ , (b) glass beads with $\unicode[STIX]{x1D705}=1$ , (c) sand, (d) carborundum with $\unicode[STIX]{x1D6FD}_{\ast }=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ and (e) carborundum with $\unicode[STIX]{x1D6FD}_{\ast }(\unicode[STIX]{x1D701})=2.05>\unicode[STIX]{x1D6FD}$ , i.e. Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction law that has a transition thickness $h_{\ast }(\unicode[STIX]{x1D701})=(h_{stop}+h_{start})/2$ (solid orange line). The initial conditions (dashed black lines) are stationary layers of thickness $h(x,0)=h_{start}(\unicode[STIX]{x1D701})+0.1$  mm. All of the initial layers are of thickness $h>h_{start}(\unicode[STIX]{x1D701})$ (solid green lines) and should gain momentum before leaving a deposit of thickness $h_{stop}(\unicode[STIX]{x1D701})$ (solid red lines). The material properties are given in tables 1 and 3, whilst the slope angle-dependent properties in tables 2 and  4. Movies of the full simulations are available in the online supplementary material (movies 12–16).

Figure 16. For the friction law of Edwards et al. (Reference Edwards, Viroulet, Kokelaar and Gray2017), the deposit thickness (orange squares) left by simulations with $h(x,t)=h_{\ast }(\unicode[STIX]{x1D701})$ (solid orange line) and the deposit thickness (green squares) left by simulations with $h(x,t)=h_{start}(\unicode[STIX]{x1D701})$ (solid green line), e.g. that of figure 15(e), are shown to deviate increasingly from $h_{stop}(\unicode[STIX]{x1D701})$ (solid red line) as the slope angle is reduced. Furthermore, for inclinations $\unicode[STIX]{x1D701}\leqslant \unicode[STIX]{x1D701}_{3}$ (shaded area) the friction law is undefined and simulations are not possible at all in this region, whereas they are now permitted by the modified friction law presented in § 3.3. The material properties are given in table 1, except for $\unicode[STIX]{x1D6FD}_{\ast }(\unicode[STIX]{x1D701})$ which varies with slope angle (equation 3.9 of Edwards et al. Reference Edwards, Viroulet, Kokelaar and Gray2017).

It is therefore only the modified friction law (3.10)–(3.12) with constant $\unicode[STIX]{x1D6FD}_{\ast }=0.466$ that is able to accurately match $h_{stop}$ whilst also capturing the hysteretic behaviour, as has been demonstrated in § 4. Indeed, the modified friction law does not just produce the correct experimental $h_{stop}$ deposit thickness, but it does so over the complete range of inclinations where deposit layers are expected to form. This includes, in particular, angles $\unicode[STIX]{x1D701}_{1}<\unicode[STIX]{x1D701}\leqslant \unicode[STIX]{x1D701}_{3}$ , which are not permissible for Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) original formulation.

Figure 17. Flow thickness $h$ is shown at one second intervals by light blue lines and the blue shading increases in intensity with increasing time for numerical simulations using the material properties of (a) glass beads with $\unicode[STIX]{x1D705}=10^{-3}$ , (b) glass beads with $\unicode[STIX]{x1D705}=1$ , (c) sand, (d) carborundum with $\unicode[STIX]{x1D6FD}_{\ast }=\unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D6E4}$ and (e) carborundum with $\unicode[STIX]{x1D6FD}_{\ast }(\unicode[STIX]{x1D701})=2.05>\unicode[STIX]{x1D6FD}$ , i.e. Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction law that has a transition thickness $h_{\ast }(\unicode[STIX]{x1D701})=(h_{stop}+h_{start})/2$ (solid orange line). The initial conditions (dashed black lines) are stationary layers of thickness $h(x,0)=h_{start}(\unicode[STIX]{x1D701})-0.1$  mm. All of the initial layers are of thickness $h<h_{start}(\unicode[STIX]{x1D701})$ (solid green lines) and should remain static. The solid red line denotes $h_{stop}$ . The material properties are given in tables 1 and 3, whilst the slope angle-dependent properties in tables 2 and 4. Movies of the full simulations are available in the online supplementary material (movies 17–21).

6.5 Numerical simulations with an initial stationary layer thinner than $h_{start}(\unicode[STIX]{x1D701})$

As a final and perhaps most basic check on the friction law, an initial layer of thickness $h(x,0)=h_{start}(\unicode[STIX]{x1D701})-0.1$  mm is now considered. Since the static layer is thinner than $h_{start}$ is should remain static. The results are plotted in figure 17. For the case of glass beads with $\unicode[STIX]{x1D705}=10^{-3}$ (in figure 17 a) any disturbance will result in the friction jumping from the static value at $Fr=0$ to a much lower value at the next non-zero number that can be represented on the computer as shown in § 6.1 and figure 13. As a result, the static layer flows down the chute and leaves a deposit close to $h_{stop}$ , instead of remaining stationary. Choosing $\unicode[STIX]{x1D705}=1$ (figure 17 b) is sufficient to keep the material at rest. The results for sand (figure 17 c), which has an offset $\unicode[STIX]{x1D6E4}>\unicode[STIX]{x1D6FD}$ , also highlights a problem. This time the layer flows down the plane because the dynamic regime is always active, as shown in figure 14(a). When $\unicode[STIX]{x1D6E4}<\unicode[STIX]{x1D6FD}$ for carborundum, however, the intermediate and static regimes exist in the friction law (6.1)–(6.3) and the initial layer remains static as shown in figure 17(d). Finally, Edwards et al.’s (Reference Edwards, Viroulet, Kokelaar and Gray2017) friction law (figure 17 e) also ensures that flows of thickness $h<h_{start}(\unicode[STIX]{x1D701})$ are not accelerated and therefore remain stationary in simulations, as required.

In all of the cases where the friction law correctly captures the initially static layer (figure 17 b,d,e) the simulations in § 6.4 produce incorrect deposit layer depths (figure 15 b,d,e). Hence only the modified frictional law (3.10)–(3.12) is able to capture the hysteretic behaviour and leave deposits that are approximately equal to the experimental value of $h_{stop}$ for the entire range of slope angles.