A 13 km tidewater valley glacier gains snow above the equilibrium line and loses ice below it. Warm the climate and watch the melt, the retreat, the calving and the sea-level contribution.
The science
Temperature and degree days. Air temperature at elevation z follows T = T₀ + ΔT − Γ z + A sin(2πt) with Γ = 6.5 K/km, A = 9 K and T₀ = 3 °C at sea level. Melt is proportional to positive degree days: PDD = ∫ max(T, 0) dt (closed form for the sinusoid), melt = DDF · PDD with DDF = 4 mm w.e./(K day) for snow and 7 mm for ice. Precipitation P = P₀(1 + 0.0006 z) falls as snow while T < 1 °C.
Mass balance and ELA. The annual balance b(z) = snow − melt (converted to ice equivalent) is positive above the equilibrium line altitude (ELA) and negative below it. Warming raises the ELA, shrinking the accumulation area.
Ice flow. Shallow-ice approximation with Glen's law (n = 3, A = 2.4×10⁻²⁴ Pa⁻³ s⁻¹): q = −(2A/(n+2)) (ρ g)ⁿ H^(n+2) |∂s/∂x|^(n−1) ∂s/∂x and ∂H/∂t = −∂q/∂x + b on 170 cells of 100 m, explicit with a diffusion-limited time step.
Calving. A marine front cell calves when its thickness is below the flotation thickness plus a margin: H < (ρw/ρi) d + hab. Calved volume is tracked separately from melt.
Sea level. Mass lost from land ice raises sea level by Δh = ΔV ρi / (ρw Aocean) with Aocean = 3.61×10¹⁴ m², i.e. about 362 Gt of ice per millimetre. Ice already floating does not add.
Simplifications. 1-D flowline with a fixed 2 km width, no basal sliding, no ice-ocean melting, no seasonal snow-pack memory, calving rule is a simple height-above-buoyancy test, and the surface drawn in 3D is a lens-shaped cross-section of the 1-D thickness. The parameters are illustrative round numbers, not a model of any real glacier or a projection. Models built with Blender, scene by Three.js.