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## APPENDIX A. FURTHER REMARKS ON THE RKE MODEL

Papers [I], [II] and [VI] invoke a number of concepts that look plausible and innocuous at first, but warrant a more detailed discussion. In this appendix, we collect issues that do not directly affect the model equations.

###

#### A.1 Viscous shear vs inelastic collisions

Bartelt and Buser distinguish between viscous shear work and inelastic collisions associated with RKE. The first notion stems from a macroscopic description of the granular material, while the second notion corresponds to a microscopic viewpoint. The kinetic theory of granular materials shows in a precise mathematical way how particle collisions give rise to a close analogue of viscosity in fluids (Jenkins and Savage, Reference Jenkins and Savage1983). As in the theory of turbulence, correlations of fluctuation velocities create a contribution to the stress tensor in a granular assembly. This implies that one cannot separate these notions from one another, as is done in [I] and [II].

#### A.2 Work done by random particle motion

Equations (I.5) and (I.6) stipulate that the forces arising from the random motion of particles average to zero because of their randomness and thus RKE cannot be converted to kinetic or potential energy. In papers [V]–[VII], this statement is tacitly revoked and *γR*
_{
K
} is effectively used as the dispersive pressure. But where precisely lies the flaw in the argument of [I]? One can trace the problem to the stipulation that random motion produces random forces that average to zero. However, forces are exerted by one system on another. If system A is a granular assembly and we disregard static electricity, A can exert a force on some other system B only if A and B are in contact with each other. System B would typically be a container wall, creating a boundary for the granular assembly. This causes an asymmetry: particles approaching the right-hand side wall have a wall-normal velocity component *v*
_{⊥} > 0, but after the collision, *v*′_{⊥} < 0. The *force* on the wall depends, not on the average of *v*
_{⊥} and *v*′_{⊥}, but on *v*
_{⊥} − *v*′_{⊥}, which necessarily is larger than 0 due to the presence of the boundary. If the particle collisions with the wall are strong and frequent enough, they will push the container wall outward and do mechanical work.

#### A.3 Energy balances

The energy balance (I.19) (or its equivalent forms (I.7) and (I.11)) looks deceptively simple and straightforward:

However, [I] emphasises that
$\dot {W}_f$
is always negative because the friction force opposes the direction of motion. *Subtracting* a negative work rate from the avalanche energy would therefore increase that energy, thus the sign of this term must be changed. We will henceforth consider the equation with corrected sign. To emphasise the conservative character of the gravitational force in contrast to the dissipative nature of friction, we will apply (I.9),
$ \hskip 0.9pt \dot{\hskip -2pt U} = - \hskip 0.9pt \dot{\hskip -0.9pt W}_g$
, in reverse and use the gravitational potential energy instead of the gravitational work rate. We thus discuss the equation

The authors apply this energy balance in the framework of a depth-averaged flow model. Moreover, the shear is assumed to be concentrated in a very thin bottom layer, i.e., one assumes the velocity profile to be uniform inside the avalanche and the slip velocity to be equal to the (depth-averaged) flow velocity. Let us therefore test (I.19') by considering a block of mass *m* sliding on a horizontal surface, with friction coefficient *μ* and initial velocity *u*
_{0}. The equation of motion is readily integrated:

and gives the kinetic energy

The time-dependent term on the right-hand side exactly equals the work done on the block by the external friction force,

as *s*(*t*) = *u*
_{0}
*t* − *μ*
*gt*
^{2}/2. Thus, we find *K*(*t*) = *K*
_{0} + *W*
_{
f
}(*t*). Since there is no gravitational work in this case, this corresponds to (I.19'), but with
$\dot {R} + {\dot \Phi } = 0$
. However, the frictional work surely has been converted into heat (
$\dot {\Phi } \gt 0$
), so why does it not show up in the balance equation? The answer is that the heat is not generated inside the sliding block but at the boundary and (I.19') lacks a term describing the heat flux across the boundary of the block. The correct form of Eqn (I.19) would therefore be

where *Q*
_{a} is the sum of the granular and thermal heat fluxes into the avalanche, integrated over the control volume surface.

The equation of motion implies
$\dot {K} + \hskip 2pt \dot{\hskip -2pt U} = {\dot W}_f$
; thus we get
$ \hskip 0.9pt \dot{\hskip -0.9pt R} + {\dot \Phi } = Q_{{\rm a}}$
, but we do not know the value of *Q*
_{a}. We can obtain some qualitative insight if we consider an (infinitesimally) thin control volume along the interface, in which all the shear is concentrated. Friction converts kinetic energy of the sliding block into heat and RKE inside the shear layer at a rate
$- {\dot W}_f \gt 0$
. With its infinitesimal volume, the shear layer has, however, only a vanishingly small capacity for storing this energy. This means that the total heat and RKE flux out of the shear layer into the avalanche, *Q*
_{
a
}, and the snow cover, *Q*
_{
s
}, must equal
$- \hskip 0.9pt \dot{\hskip -0.9pt W}_f$
. Clearly, *Q*
_{
s
} > 0 in a snow avalanche so that
$0 \lt Q_{{\rm a}} \lt - \hskip 0.9pt \dot{\hskip -0.9pt W}_f = \hskip 0.9pt \dot{\hskip -0.9pt R} + {\dot \Phi }$
. Comparing with (I.17),

one sees that Bartelt and Buser assume Q_{s} ≈ 0,
$ \hskip 0.9pt \dot{\hskip -0.9pt R} = \alpha Q \, -\beta R$
, and
$\dot {\Phi } = (1\,-\,\alpha ) Q + \beta R$
. With these assumptions, the balance Eqn (I.19') reduces to

In contrast, the original Eqn (I.19') (after correcting the sign error) would imply that the snow cover absorbs all the frictional work. The preceding analysis also applies to a flow with internal shear with a few modifications.

In Section 7 of [VI], Buser and Bartelt attempt to show that their equation system conserves energy. To this end, they split the (non-random) kinetic energy into two components, defined as
$K^{xy} \equiv \overline {\rho {\bi u}^2}/2$
and
$K^z \equiv \overline {\rho w^2}/2$
, and state the following balance equations:

Among multiple issues connected with (VI.41), we mention the following: (i) The kinetic energy balances should be derived directly from the momentum-balance equations. In doing so, Eqn (VI.40) would receive a contribution from the (slope-parallel) gradient of normal stresses, and Eqn (VI.41) would be supplemented by a contribution due to dispersive pressure. (ii) The model is not fully closed in the sense that evaluating
$\dot {W}_f^z = \int _0^h {\bf \nabla}\,\cdot \,{\bi S}(z)\, {\rm d} z$
would require constitutive expressions for the shear stresses **
***S*
(*z*) across the flow depth. The Voellmy-type bed-friction law provides only the bed shear stress, **
***S*
_{
b
}. (iii) Equation (VI.41) should contain either the rate of change of potential energy, D_{
t
}(*R*
_{
V
}
*h*), or the work rate of gravity,
$- \hskip 0.9pt \dot{\hskip -0.9pt W}_g^z$
, but not both. Gravity being a conservative force, the change of potential energy is the opposite of the work done by gravity, thus
$ {\rm D}_t(R_V h) = + \hskip 0.9pt \dot{\hskip -0.9pt W}_g^z$
with the sign convention of Eqn (VI.5). When this is taken into account, (VI.41) degenerates to
$\dot {K}_z = \,-\, \hskip 0.9pt \dot{\hskip -0.9pt W}_f^z$
. As mentioned above, this relation lacks the main term, namely the contribution from the dispersive pressure gradient.

## APPENDIX B. COMPARISON WITH THE GENERAL BALANCE EQUATIONS FOR MASS, MOMENTUM AND FLUCTUATION ENERGY

Further insight into the significance of the constitutive assumptions in the density-changing RKE model can be obtained by comparing it with the general depth-averaged balance equations for mass, momentum and fluctuation energy, of which it has to be a special instance if it is to be consistent. For simplicity, consider flow down a straight, rigid incline at an angle *θ* to the horizontal. We take *x* in the flow direction, *y* horizontal in the sliding plane and z normal to the incline, with origin at the base and positive upward, *x*
_{
α
} = (*x*, *y*, *z*)^{T}, *ρ* the average mass density of the grains, and *u*
_{
α
} = (*u*, *v*, *w*)^{T} the components of the ensemble-averaged grain velocity. We will first state the equations for a general 3-D flow and then discard the variations along the *x*- and *y*-directions to make the comparison simpler.

From the general principles of fluid mechanics, the mass-balance equation must take the local form

(We use tensor notation here to emphasise that these equations are 3-D and switch to vector notation after depth-averaging. Summation over repeated indices is understood.) Take *σ*
_{
αβ
} to be the components of particle stress and *f*
_{
α
} = *g*(sin*θ*, 0, − cos*θ*)^{T} the components of external force per unit mass, with *g* the gravitational acceleration. Then the local balance of linear momentum is

With
${\cal K} \equiv (1/2)\rho u_{\alpha} u_{\alpha} $
, the balance of mechanical energy reads

The granular temperature, *T*, is defined as one-third of the mean square of the particle velocity fluctuations and thus relates to *R*
_{
K
} as (3/2)*T* ≡ *R*
_{
K
}. Let *q*
_{
α
} be the components of the flux of fluctuation energy, and *Γ* the rate of collisional dissipation per unit volume. *R*
_{K} then has to obey the following advection–diffusion–dissipation equation:

Next, we average over z between the bed at *z* = 0 and the surface at *z* = *h*(**
***x*
, *t*). We use the notation **
***u*
≡ (*u*, *v*)^{T}, *u*
_{
α
} ≡ (**u**, *w*)^{T}, ∂_{
α
} ≡ (**∇**, ∂_{
z
})^{T}. The 3-D stress tensor decomposes into the 2-D tensor *σ*
_{
ab
} ≡ **
***σ*
, the 2-D vector *σ*
_{
az
} = *σ*
_{
za
} ≡ **
***S*
, and the 2-D scalar *σ*
_{
zz
}, with *a*, *b*∈{*x*, *y*}. **
***S*
_{b} is the bed shear stress. For any field *ψ*(**
***x*
, *z*, *t*), the depth average can be written as
$h \bar {\psi }({\bi x},\,t) \equiv \int _0^{h({\bi x},\,t)} \psi ({\bi x},\,z,\,t) {\rm d} z$
. Leibniz's rule, e.g.,
$\partial _t \int _0^h \psi ({\bi x},\,z,\,t) {\rm d} z = \int _0^h \partial _t \psi ({\bi x},\,z,\,t) {\rm d} z + \psi ({\bi x},\,h,\,t) \partial _t h({\bi x},\,t)$
, and the kinematic boundary condition,

are repeatedly used together with the boundary conditions *w*(**
***x*
, 0, *t*) = 0 and *σ*
_{
αβ
}(**
***x*
, *h*, *t*) = 0. For simplicity, we assume the bed to be non-erodible and the density to be constant with depth. In this case, the height *h* is a useful variable. When the density varies strongly and the upper edge may not be well defined, it is better to work with mass hold-up
$m = h\bar {\rho }$
and the centre-of-mass,
$Z = \overline {z\rho }/\overline {\rho }$
. We can rewrite the kinematic boundary condition as a volume-balance equation. Thus, the system is governed by five balance equations for, respectively, the volume, the mass, the linear momenta in the flow plane and normal to the bed, and the fluctuation energy:

In addition, one must specify expressions relating the average of products of fields to the product of averages, constitutive equations relating the stresses
${\bar {\bi \sigma }}$
, **
***S*
, *σ*
_{
zz
} to the fields *h*,
$\bar {\rho }$
,
$\hskip 0.9pt\bar {{\hskip -0.9pt{\bi u}}}$
,
$\bar {w}$
and
$\hskip 0.9pt\bar {\hskip -0.9ptR}_K$
, and boundary conditions for the fields and stresses. Within the stated framework, these equations are general. Note that the height equation (B8), mass balance (B9) and the *z*-momentum balance (B11) need to be used jointly to determine the flow depth and the density. Simplifying assumptions are needed to close the equations and to make them tractable. However, any approximations have to be compatible with the general structure of Eqns (B8)–(B12). Equation (B8) can be thought of in at least three different ways: Firstly, as we have written it here, as a volume-balance equation; secondly as a kinematic equation
$\partial _t h + \overline {{\bi u}}\,\cdot \,{\bf \nabla} h = w_h$
; and thirdly as an equation for the centre-of-mass *h*/2. This last interpretation is the most general and most useful since it corresponds to the gravitational potential energy and is well defined for any density distribution including when there is no well-defined upper surface.

Now we may compare these equations with the extended RKE model of [V] and [VI]. The indices Φ and Σ refer to the dense flow and the snow cover, respectively. One readily identifies *M* with
$h\bar {\rho }$
. First, we focus on the equations for *M*
_{Φ}, *M*
_{Φ}
*u*
_{Φ}, *M*
_{Φ}
*v*
_{Φ} and *Rh*
_{Φ}; we will discuss the equations for *h*
_{Φ}, *M*
_{Φ}
*w*
_{Φ} and *N*
_{
K
} afterwards. The left-hand sides of Eqns (B8)–(B12) agree with Eqns (VI.30) and (VI.37)–(VI.39) if one approximates the depth-averages of products of fields by the products of the depth-averaged fields, assuming uniform profiles. The source terms proposed in [VI] are summarised by the first four rows of Eqn (VI.39):

With the erosion rate set to 0, this becomes in our notation

[V] and [VI] model the term
${\bf \nabla}\,\cdot \,(h\bar {{\bi \sigma }})$
on the left-hand side of (B8) as
$(1/2){\bf \nabla}(\bar {\rho } h^2 g_z)$
and neglect shear stresses in vertical planes (as virtually all quasi-3-D avalanche models do). The RKE-modified Voellmy friction law is used to model the bed shear stress **
***S*
_{b} – however, now with *R*
_{
V
} instead of *R*
_{
K
} determining the decrease of the friction parameters (cf. Eqns (VI.35) and (VI.36)). The slope-parallel diffusive flux of RKE is neglected (**
***q*
≈ 0), as mentioned earlier. In (B11), the dissipation *Γ* is assumed proportional to *R*
_{
K
} + *R*
_{
V
} = *R*
_{
K
} − *M*(*k* − *k*
_{0})*g*
_{
z
} rather than to *R*
_{
K
}
^{3/2} as suggested by kinetic theory. Neither the different exponent nor the appearance of *R*
_{
V
} in *Γ* is incompatible with the general framework because the latter does not specify the form of *Γ*, but it is a clear departure from kinetic theory.

To the extent that Bartelt and co-workers assume the shear layer to be infinitesimally thin, the supply of RKE could be described by setting the boundary flux term
$q_z \vert_0 = \alpha {\bi S}_{{\rm b}} \cdot \hskip 0.9pt \bar {{\hskip -0.9pt{\bi u}}}$
. But it appears more natural to regard *α*
**
***S*
_{b} · **
***u*
|_{0} as the limit of
$h {\overline {\bi S}\,\cdot \,\partial _z {\bi u}}$
when the thickness of the shear layer, *δ*
_{s}, tends to zero: in the shear layer, the shear stress is approximately equal to **
***S*
_{b}, and the shear rate is
$\partial _z {\bi u} \,\approx \hskip 0.9pt\bar {{\hskip -0.9pt{\bi u}}}/\delta _s$
. Integration over *z* from 0 to *h* then gives
${\bi S}_b \cdot \, (\bar {{\hskip -0.9pt{\bi u}}}/\delta _s) \delta _s = {\bi S}_b \,\cdot \hskip 0.9pt \bar {{\hskip -0.9pt{\bi u}}}$
. This would, however, impose *α* = 1.

The extended RKE model of [V] and [VI] thus neglects all terms on the second line of Eqn (B12) except the third. The first and fourth terms describe RKE generation due to shear along the vertical planes. According to the standard scaling arguments for shallow flows based on the aspect ratio *ε* ≪ 1,
$\hskip 0.9pt\bar {{\hskip -0.9pt{\bi u}}}$
and ∂_{
z
} are *O*(1) while *h*, *w* and **∇** are *O*(*ε*). Thus, only the third term,
$h \overline {{\bi S}\,\cdot \,\partial _z {{\bi u}}}$
, is *O*(*ε*) while the others are *O*(*ε*
^{2}) or *O*(*ε*
^{3}) and would be negligible. However, the second term,
$h\overline {\sigma _{zz}\partial _z w}$
, is special in that it is present even if the flow does not deform in the tangential directions of the incline, but changes density. Moreover, it describes how RKE is transformed into potential (i.e., non-random) kinetic energy as the flow expands in the bed-normal direction. This term thus implements the feed-back mechanism governing density changes and must not be discarded. A more detailed scale analysis of Eqn (B11) would need to introduce different timescales and is beyond the scope of this paper, but will be invaluable in the construction of an improved, consistent model.

Finally, comparing the last three equations in the system (VI.30), (VI.37)–(VI.39),

with Eqn (B11) is not straightforward because the extended RKE model here departs from the canonical approach based on the fundamental balance equations. We first note that, if one assumes uniform density and velocity profiles, one may insert Eqn (B9) into Eqn (B11) to obtain

Using the equation for *M*
_{Φ} with
$\dot {M}_{\Sigma \rightarrow \Phi }$
, which is identical to Eqn (B9), the same procedure can be applied to Eqn (B15) and yields (in our notation)

The neglected term is *O*(*ε*), thus smaller than each of the other two terms on the right-hand side of Eqn (B17), but of the same order as D_{
t
}
*w* and the sum of the *O*(1) terms. Equation (B14) appears to contain a misprint – *w*
_{Φ} is the centre-of-mass velocity, thus there should be a factor 2 on the right-hand side. Even so, this equation is in conflict with the kinematic boundary condition (B7) because **∇** · **u**
_{Φ} ≢ 0. In fact, tracing its derivation in [VI], one sees that it *should* be the kinematic boundary condition rather than a dynamical equation. We have discussed the reasons why Eqn (B16) is not valid in Section 3; comparing it with Eqn (B11), it is apparent that it needs to be replaced by a proper constitutive law for *σ*
_{
zz
} or, equivalently, the pressure.