Phase 3 Methodology

Snowpack — iSnobal

Two-layer energy-balance snowpack with cold content:

$cc = c_i \cdot z_s \cdot \rho_s \cdot (T_s - T_\text{freeze})$

Melt occurs only when $cc \ge 0$; available mass flux

$M = \frac{cc}{L_f}, \quad L_f = 334{,}000\ \text{J/kg}.$

Compaction follows the Kojima viscosity law

$\eta = 3.6 \times 10^{6} \exp(0.08 (T_\text{freeze} - T_s) + 0.021 \rho_s), \quad \frac{d\rho_s}{dt} = \rho_s \frac{\sigma}{\eta}.$

Hydrodynamics — Saint-Venant 1-D

$\frac{\partial A}{\partial t} + \frac{\partial Q}{\partial x} = q_\text{lat}$

$\frac{\partial Q}{\partial t} + \frac{\partial}{\partial x}\!\left(\frac{Q^2}{A}\right) + g A \frac{\partial h}{\partial x} = -g A S_f$

Manning friction slope: $S_f = n^2 u |u| / R^{4/3}$.
Solved via HLL flux with semi-implicit friction; CFL ≤ 0.5.

2-D Shallow Water

Lax-Friedrichs flux + semi-implicit Manning friction on a PyTorch tensor grid.
GPU acceleration via CUDA available on SageMaker. Wet/dry threshold 1 mm.

Avalanche — Voellmy-Salm

Depth-averaged closure:

$\tau = \mu \rho g h \cos\theta + \frac{\rho g u^{2}}{\xi}.$

Monte Carlo over $(\mu, \xi)$ via Sobol QMC: $\mu \in [0.10, 0.25]$, $\xi \in [1000, 2500]\ \text{m/s}^2$.

Quantum Amplitude Estimation

Classical MC: $\sigma \sim 1/\sqrt{N}$.
IAE (Brassard et al. 2002): $\sigma \sim 1/N$ — quadratic speed-up.
Deployed on Qiskit Aer; classical MC fallback on Lambda runtimes.

Buckingham Π

For each solver the primary dimensionless groups (Fr, Re, Stefan, Bowen) are declared and invariance under characteristic-scale rescaling is asserted. CI enforces $\Delta\Pi < 10^{-6}$.

Risk Aggregation

$P_\text{GLOF} = w_1 P_\text{breach} + w_2 \sigma(Q_\text{surge}) + w_3 (1 - e^{-A/A_0}) + w_4 \sigma(\text{SWE})$

Weights calibrated against the historical 2010–2024 archive in Phase 8.