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Let $X$ and $Y$ be horizontal coordinates in a fixed Cartesian coordinate system and $Z(X,Y)$ denote a mountain profile parameterized in $X$ and $Y$. The independent variables $x$ and $y$ denote the arclength along the surface topography, the $z$-coordinate is perpendicular to the profile (see Fig. above). The coordinates $x$, $y$, and $z$ define the surface induced coordinate system. Its orientation varies with the position on the surface, such that the vector of gravitational acceleration $\bf g\,=\,\left(g_x,g_y,g_z\right)$ has three non-zero components in general, in each case functions of $x$ and $y$. Time $t$ completes the set of independent variables for the system.

$$

\partial_t H + \partial_x \left ( H U_x \right ) + \partial_y \left ( H U_y \right ) = \dot Q(x, y, t) ,

$$

The topography, *Z(X, Y)*, is given in a Cartesian framework, *X* and *Y* being the horizontal coordinates. The surface induces a local coordinate system, *x, y, z*. It is discretized such that its projection onto the X-Y plane results in a structured mesh, see picture above.

We then solve the following system of differential equations for the avalanche flow height, *H(x,y,t)* and velocities, *Ux (x,y,t)* and *Uy (x,y,t)*, at time *t*. From first principles of mass and momentum conservation, the fundamental balance laws are derived:

\(\partial_t H + \partial_x \left ( H U_x \right ) + \partial_y \left ( H U_y \right ) = \dot Q(x, y, t)\)

\(\partial_t \left(H U_x \right) + \partial_x \left( c_x \,H U_x^2 + g_z k_{a/p} \frac{H^2}{2} \right) + \partial_y \left( H U_x U_y \right) = S_{gx} - S_{fx}\)

\(\partial_t \left(H U_y \right) + \partial_x \left( H U_x U_y \right) + \partial_y \left( c_y H U_y^2 + g_z k_{a/p} \frac{H^2}{2} \right) = S_{gy} - S_{fy}\)

\(\dot{Q}(x,y,t)\) in the mass-balance equation denotes the mass production source term, referred to as the snow entrainment rate or the snow deposition rate. The field variables of interest are the avalanche flow height, *H(x, y, t )*, and the mean avalanche velocity, *U(x, y, t )*. The magnitude and direction of the flow velocity are given by $U=\sqrt{U_x^2 + U_y^2}$\(\) and the unit vector ${n}_U:=\,\frac{1}{U} \;( U_x , U_y )^{T}$\(\), respectively. The right-hand sides of the momentum equations and sum to the effective acceleration of the avalanche. The terms

\(S_{gx} = g_x H \qquad \text{and} \qquad S_{gy} = g_y H\)

define the gravitational accelerations in the *x* and *y* directions, respectively. The acceleration normal to the avalanche slope is given by *g _{z}*. The friction \(S_f\,=\,\left( S_{f,x}, S_{f,y}\,\right)^T\) in the Voellmy-Salm (VS) model is given by

\(\)$$ S_{fx} = {n}_{U_x} \, \left [ \mu g_z H + \frac{g {U}^2}{\xi} \right ] \qquad \text{and} \qquad S_{fy} = {n}_{U_y} \, \left [ \mu g_z H + \frac{g {U}^2}{\xi} \right ].$$

The VS approach splits the total basal friction into a velocity independent dry-Coulomb term which is proportional to the normal stress at the flow bottom (friction coefficient *μ*) and a velocity dependent “viscous” or “turbulent” friction (friction coefficient *ξ*) (Salm, 1993). The division of the total basal friction into velocity independent and dependent parts allows the modeling of avalanche behaviour when the avalanche is flowing with a high velocity in the acceleration zone and close to stopping in the runout zone.