Tracking snowpack evolution through the winter season
2026-02-27
The mountain snowpack of western North America is the water tower that supplies irrigation for California’s Central Valley, municipal water for Denver, Phoenix, and Salt Lake City, and hydropower for the entire Pacific Northwest. In an average year, the Colorado River basin snowpack holds roughly 15 km³ of water — more than the capacity of Lake Mead. The timing and magnitude of spring runoff from that snowpack determines whether reservoirs fill, whether irrigation water is available in August, and whether summer low flows in rivers remain high enough for salmon. Getting that forecast right is worth billions of dollars annually.
Snow accumulation and melt modelling traces the evolution of the snowpack from the first snowfall of autumn through peak accumulation in late winter to the eventual complete melt in spring. The key variable is Snow Water Equivalent (SWE): the depth of water that would result if the entire snowpack melted instantaneously. SWE is what the hydrologist actually cares about — not snow depth, which varies with density, but the water mass stored. This model builds two levels of melt prediction: the simple degree-day approach, which accumulates temperature above freezing and applies a melt factor, and the energy balance approach that accounts for radiation and turbulent heat exchange. Both are in operational use; understanding why the simple approach works so well, and where it fails, is as important as knowing the equations.
How much water will this snowpack release during spring melt?
Snow Water Equivalent (SWE) is the key variable:
\text{SWE} = \rho_{\text{snow}} \times d_{\text{snow}} / \rho_{\text{water}}
Where: - \rho_{\text{snow}} = snow density (kg/m³) - d_{\text{snow}} = snow depth (m) - \rho_{\text{water}} = 1000 kg/m³
Example: 1 meter of snow at density 300 kg/m³:
\text{SWE} = \frac{300 \times 1.0}{1000} = 0.30 \text{ m} = 300 \text{ mm}
This means: If all snow melted instantly → 300 mm of water runoff!
Applications: - Water supply forecasting: How much water in reservoir? - Flood prediction: Peak flow timing and magnitude - Irrigation planning: Allocate water to farmers - Hydropower: Generation scheduling - Climate monitoring: Trends in snow storage
The mathematical question: Given daily weather (precipitation, temperature), how do we model the evolution of SWE from fall through spring melt?
Change in SWE over time:
\frac{dS}{dt} = P_{\text{snow}} - M - E
Where: - S = SWE (mm or kg/m²) - P_{\text{snow}} = snowfall (mm/day) - M = melt (mm/day) - E = sublimation/evaporation (mm/day)
Accumulation season (winter): - P_{\text{snow}} > 0, M = 0 → SWE increases - Peak SWE typically late March-April (mid-latitudes)
Melt season (spring): - P_{\text{snow}} = 0, M > 0 → SWE decreases - SWE → 0 by May-June (lower elevations)
Temperature threshold:
P_{\text{snow}} = \begin{cases} P & \text{if } T_{\text{air}} \leq T_{\text{threshold}} \\ 0 & \text{if } T_{\text{air}} > T_{\text{threshold}} \end{cases}
Simple: T_{\text{threshold}} = 0°C
Refined: Transition zone (-1°C to +3°C):
f_{\text{snow}} = 1 - \frac{T_{\text{air}} - T_{\min}}{T_{\max} - T_{\min}}
Where f_{\text{snow}} = fraction falling as snow (0-1).
Mixed precipitation: Common near 0°C.
Typical mid-latitude mountain:
Oct-Nov: First snowfall, shallow snowpack
Dec-Feb: Accumulation, cold temperatures, little melt
Mar: Peak SWE (maximum water storage)
Apr: Melt onset, SWE decreases
May: Rapid melt, major runoff
Jun: Snowpack depleted (lower elevations)High elevations: Snow persists year-round (permanent snowfields).
Empirical relationship:
M = \begin{cases} a(T_{\text{air}} - T_{\text{melt}}) & \text{if } T_{\text{air}} > T_{\text{melt}} \\ 0 & \text{otherwise} \end{cases}
Where: - M = melt rate (mm/day) - a = degree-day factor (mm/(°C·day)) - T_{\text{melt}} = melt threshold (typically 0°C)
Typical values: - a = 3 mm/(°C·day) for open sites - a = 2 mm/(°C·day) for forest - a = 4-6 mm/(°C·day) for glaciers
Daily time step:
M_{\text{today}} = a \times \max(0, T_{\text{mean}} - 0)
Cumulative measure:
\text{PDD} = \sum_{t=1}^{n} \max(0, T_{\text{mean},t})
Total melt over period:
M_{\text{total}} = a \times \text{PDD}
Example: Spring with temperatures 2, 4, 6, 8°C over 4 days:
\text{PDD} = 2 + 4 + 6 + 8 = 20 \text{ °C-days}
M_{\text{total}} = 3 \times 20 = 60 \text{ mm}
Add radiation term:
M = a_T(T - T_{\text{melt}}) + a_R(1 - \alpha)Q_{\text{SW}}
Where: - a_R = radiation factor (mm·m²/(W·day)) - \alpha = albedo - Q_{\text{SW}} = incoming shortwave (W/m²)
Accounts for: Sunny warm days melt more than cloudy warm days.
From Model 42, net energy determines melt:
M = \frac{Q_{\text{net}}}{L_f \rho_w}
Requires: - Solar radiation - Longwave radiation - Air temperature - Wind speed - Humidity
Advantages: Physical basis, accurate
Disadvantages: Data intensive, complex
Operational use: SNOTEL sites, research basins
Fresh snow: \rho \approx 50-100 kg/m³
Aged snow: \rho \approx 200-400 kg/m³
Late season: \rho \approx 400-500 kg/m³
Compaction over time:
\frac{d\rho}{dt} = c_1 \rho + c_2
Where c_1, c_2 are empirical constants.
Temperature-dependent:
\rho(t) = \rho_0 + (\rho_{\max} - \rho_0)(1 - e^{-kt})
Typical: \rho_{\max} = 500 kg/m³, k = 0.01 day⁻¹
Problem: Model SWE evolution for November-May given daily temperature and precipitation.
Data (simplified monthly averages):
| Month | Precip (mm) | Temp (°C) | Days |
|---|---|---|---|
| Nov | 50 | -2 | 30 |
| Dec | 80 | -5 | 31 |
| Jan | 100 | -8 | 31 |
| Feb | 90 | -6 | 28 |
| Mar | 70 | -1 | 31 |
| Apr | 40 | +4 | 30 |
| May | 30 | +10 | 31 |
Assumptions: - All precipitation falls as snow when T < 0°C - Degree-day factor: a = 3 mm/(°C·day) - No sublimation
November: - T_{\text{mean}} = -2°C < 0 → No melt - Snowfall: 50 mm SWE - SWE = 0 + 50 = 50 mm
December: - T_{\text{mean}} = -5°C → No melt - Snowfall: 80 mm - SWE = 50 + 80 = 130 mm
January: - T_{\text{mean}} = -8°C → No melt - Snowfall: 100 mm - SWE = 130 + 100 = 230 mm
February: - T_{\text{mean}} = -6°C → No melt - Snowfall: 90 mm - SWE = 230 + 90 = 320 mm
March: - T_{\text{mean}} = -1°C → No melt - Snowfall: 70 mm - SWE = 320 + 70 = 390 mm ← Peak SWE
April: - T_{\text{mean}} = +4°C → Melt! - Melt: M = 3 \times 4 \times 30 = 360 mm - Precip: 40 mm (rain, T > 0) - SWE = 390 - 360 = 30 mm
May: - T_{\text{mean}} = +10°C → Strong melt - Melt: M = 3 \times 10 \times 31 = 930 mm - Available SWE: 30 mm - SWE depleted: 30 mm melts, 900 mm “potential melt” unused - SWE = 0 mm (snow gone by mid-May)
Summary: - Peak SWE: 390 mm (end of March) - Melt onset: April - Snow-free: Mid-May - Total melt: 390 mm water released
Runoff timing: Major pulse in April-May as snowpack melts.
Below is an interactive snow accumulation/melt simulator.
<label>
Degree-day factor (mm/°C/day):
<input type="range" id="ddf" min="1" max="6" step="0.5" value="3">
<span id="ddf-val">3.0</span>
</label>
<label>
Elevation (m):
<input type="range" id="elevation" min="1000" max="3000" step="200" value="2000">
<span id="elev-val">2000</span>
</label>
<label>
Winter precipitation (%):
<input type="range" id="winter-precip" min="50" max="150" step="10" value="100">
<span id="precip-val">100</span>%
</label>
<label>
Show components:
<input type="checkbox" id="show-precip" checked> Precipitation
<input type="checkbox" id="show-melt" checked> Melt
</label><canvas id="snow-sim-canvas" width="700" height="400" style="border: 1px solid #ddd;"></canvas><p><strong>Peak SWE:</strong> <span id="peak-swe">--</span> mm</p>
<p><strong>Peak date:</strong> <span id="peak-date">--</span></p>
<p><strong>Snow-free date:</strong> <span id="snow-free">--</span></p>
<p><strong>Total melt:</strong> <span id="total-melt">--</span> mm</p>Try this: - Higher elevation: Colder temps, later melt, more peak SWE - Lower elevation: Earlier melt, less accumulation - Higher DDF: Faster melt (more sensitive to temperature) - More precipitation: Higher peak SWE, more water stored - Blue area: SWE accumulation through season - Light blue bars: Snowfall events - Orange bars (downward): Daily melt - Red dashed line: Temperature (when > 0°C, melt occurs) - Notice: Peak SWE in March/April, rapid melt in May!
Key insight: Snowpack is a natural reservoir—stores winter precipitation, releases during spring melt when plants need water!
April 1 SWE predicts summer streamflow:
Q_{\text{summer}} = \alpha \times \text{SWE}_{\text{Apr1}} + \beta \times P_{\text{Apr-Jul}}
Where: - Q = seasonal flow volume - \alpha = coefficient (~0.6-0.8) - P = spring/summer precipitation
Regression model calibrated to basin.
Example: Sierra Nevada - SWE = 800 mm on April 1 - Predicted runoff: ~500 mm (60-70% efficiency)
Water allocation based on this forecast.
Peak flow timing:
Snowmelt contribution:
Q_{\text{melt}}(t) = \frac{M(t) \times A}{86400}
Where: - Q = discharge (m³/s) - M = melt rate (mm/day) - A = basin area (m²) - 86400 = seconds/day
Combined: Rain + snowmelt = compounded runoff.
Example: 100 km² basin, 50 mm/day melt:
Q = \frac{50 \times 10^{-3} \times 100 \times 10^6}{86400} = 57.9 \text{ m}^3\text{/s}
Plus rain: Could exceed channel capacity → flooding.
Observed trends: - Earlier snowmelt (1-2 weeks) - Lower peak SWE (20-30% reduction in some regions) - Shorter snow-covered season - More rain vs. snow (elevation threshold rising)
Implications: - Earlier peak flows (March vs. April) - Less summer baseflow (less storage) - Increased winter flooding - Reduced water availability for irrigation
SWE anomaly:
A = \frac{\text{SWE}_{\text{current}} - \text{SWE}_{\text{climatology}}}{\sigma_{\text{climatology}}}
Drought: A < -1.5 (SWE far below normal)
Example: 2015 California drought - April 1 SWE: 5% of normal (95% deficit!) - Severe water shortages
Degree-day limitation: Ignores radiation, wind, humidity.
Fails when: - Clear cold nights: Strong radiation cooling, no melt despite positive Tmean - Cloudy warm days: Less solar, slower melt than predicted - Shaded vs. sunny slopes: Same temperature, different melt
Solution: Enhanced degree-day with radiation term, or full energy balance.
Fixed 0°C threshold misses: - Mixed precipitation zone (-1 to +3°C) - Humidity effects (wet bulb vs. dry bulb temp) - Elevation gradients in precipitation phase
Better: Logistic transition function or wet-bulb temperature.
Dry, windy conditions: Significant mass loss without melt.
Example: High plains, winter Chinook winds - Sublimation: 1-2 mm/day - Over season: 50-100 mm loss
Should include: Sublimation in mass balance (especially arid regions).
Point model vs. distributed:
Within-basin variation: - North vs. south slopes (radiation) - Elevation bands (temperature) - Forest vs. clearing (radiation, wind)
Solution: Divide basin into elevation zones or grid cells, model each separately.
Physically-based multi-layer model:
Layers: 1. Surface layer (energy exchange) 2. Upper layer (recent snow) 3. Lower layer (old snow)
Each layer tracks: - Temperature - Density - Liquid water content - Ice content
Energy balance at surface: - Shortwave radiation (with penetration) - Longwave radiation - Sensible heat - Latent heat - Ground heat - Advected heat (rain)
Internal processes: - Percolation (liquid water movement) - Refreezing (cold content) - Compaction (density change)
Output: - SWE evolution - Runoff timing and magnitude - Internal energy state
Used: Research, detailed forecasting, calibrated basins.
Accumulation over time:
S(t) = S_0 + \int_{t_0}^{t} \left(\frac{dS}{dt}\right) dt
For SWE:
S(t) = S_0 + \int_{t_0}^{t} (P_{\text{snow}} - M - E) \, dt
Discrete (daily time steps):
S_{t+1} = S_t + (P_{\text{snow},t} - M_t - E_t) \Delta t
Where \Delta t = 1 day.
Trapezoidal rule:
\int_a^b f(x) \, dx \approx \sum_{i=1}^{n} \frac{f(x_{i-1}) + f(x_i)}{2} \Delta x
For total melt:
M_{\text{total}} = \sum_{i=1}^{n} M_i \times 1 \text{ day}
Example: Daily melt: 0, 0, 5, 10, 15, 20 mm
M_{\text{total}} = 0 + 0 + 5 + 10 + 15 + 20 = 50 \text{ mm}