# Mathematical Model

Avalanche flow is characterized by unsteady and non-uniform motion with varying height and velocity. The field variables of interest are therefore the avalanche flow height $H(x,y,t)$ and the mean avalanche velocity ${U}(x,y,t):=(U_x(x,y,t),U_y(x,y,t))^{T}$. The magnitude and direction of the flow are given by $$\left\lVert{U}\right\rVert=\sqrt{U_x^2+U_y^2}$$ and the unit vector $${n}_U:=\frac{1}{\left\lVert{U}\right\rVert} \;( U_x , U_y )^{T}$$ respectively (see Figure on the right).

From first principles of mass and momentum conservation the fundamental balance laws are derived. Depth-averaging of these results into a set of partial differential equations for $H$ and ${U}$. Snow avalanches exhibit a shallow flow geometry, meaning that the shallowness parameter defined by the characteristic flow height ($h \approx 1.0 m$) and characteristic length ($l \approx 200 m$) $$\epsilon:=\frac{\text{h}}{\text{l}}$$ is small, $\epsilon << 1$. It is this geometrical property which justifies a model formulation in terms of depth-averaged field variables. We find the following mass balance in terms of the height

$$\partial_t H + \nabla \cdot \left( H {U} \right) = \dot Q$$

where $\dot{Q}(x,y,t)$ denotes the mass production source term, refered to as the snow entrainment rate ($\dot Q > 0$) or the snow deposition rate ($\dot Q < 0$). The componentwise depth-averaged momentum balance is given by

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

$$\text{and}$$

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

The right hand side terms add up to an effective acceleration. More explicitely $S_{gx} = g_x H$ and $S_{gy} = g_y H$

denote the driving, gravitational accelerations in the $x$ and $y$ directions, respectively. $S_{fx}$ and $S_{fy}$ represent the friction, which are given by

$$S_{fx} = \mu g_z H + {n}_{U_x} \,\frac{g \left\lVert{U}\right\rVert^2}{\xi}$$

$$\text{and}$$

$$S_{fy} = \mu g_z H + {n}_{U_y} \,\frac{g \left\lVert{U}\right\rVert^2}{\xi}$$

The VS approach splits the total friction into a velocity independent dry-Coulomb term which is proportional to the normal stress at the flow bottom (friction coefficient $\mu$) and a velocity dependent “viscous” or “turbulent” friction (friction coefficient $\xi$). The division of the total friction into velocity independent and dependent parts allows the modelling of avalanche behaviour when the avalanche is flowing with high velocity in the acceleration zone and close to stopping in the runout zone. The VS approach therefore allows the modelling of avalanche behaviour from initiation to runout.