# Mathematical Model

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 gz. 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.