Snow Accumulation and Melt modelling
modelling Level 3

Snow Accumulation and Melt modelling

How much water is stored in the snowpack? When will melt begin? What's the peak runoff timing? Snow accumulation and melt models track snowpack mass balance through the season—accounting for precipitation, sublimation, and melt. This model derives degree-day models, implements energy balance approaches, and introduces Snow Water Equivalent (SWE) as the key hydrological variable.

Prerequisites: mass balance, degree day integration, temporal modelling, water equivalent

Updated 13 min read

1. The Question

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?

2. The Conceptual Model

Mass Balance Equation

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)

Rain vs. Snow Partitioning

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.

Seasonal Cycle

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).

3. Building the Mathematical Model

Degree-Day Melt Model

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:

\[M_{\text{today}} = a \times \max(0, T_{\text{mean}} - 0)\]

Positive Degree Days (PDD)

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:

\[\text{PDD} = 2 + 4 + 6 + 8 = 20 \text{ °C-days}\] \[M_{\text{total}} = 3 \times 20 = 60 \text{ mm}\]

Enhanced Degree-Day Model

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.

Energy Balance Approach

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

Snow Density Evolution

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

4. Worked Example by Hand

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

Solution

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.

5. Computational Implementation

Below is an interactive snow accumulation/melt simulator.

Peak SWE: -- mm

Peak date: --

Snow-free date: --

Total melt: -- mm

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!

6. Interpretation

Water Supply Forecasting

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.

Flood Forecasting

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.

Climate Change Impacts

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

Drought Detection

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

7. What Could Go Wrong?

Temperature-Only Models

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.

Rain-Snow Threshold

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.

Sublimation Ignored

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).

Spatial Variability

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.

8. Extension: Utah Energy Balance (UEB) Model

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.

9. Math Refresher: Integration & Accumulation

Definite Integral

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.

Numerical Integration

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

\[M_{\text{total}} = 0 + 0 + 5 + 10 + 15 + 20 = 50 \text{ mm}\]

Summary

  • Snow Water Equivalent (SWE) quantifies water stored in snowpack
  • Mass balance: SWE change = snowfall - melt - sublimation
  • Degree-day models: Empirical melt based on temperature ($M = a(T - T_{melt})$)
  • Typical DDF: 2-6 mm/(°C·day) depending on exposure and cover
  • Seasonal cycle: Accumulation (Oct-Mar), peak SWE (Apr), melt (Apr-Jun)
  • Rain-snow threshold: Temperature determines precipitation phase (~0°C)
  • Applications: Water supply forecasting, flood prediction, drought monitoring
  • Climate change: Earlier melt, lower peak SWE, more rain vs. snow
  • Advanced models: Energy balance (UEB, SNOBAL) for detailed physics
  • Spatial variation: Elevation, aspect, canopy coverage affect accumulation and melt
  • Foundation for understanding mountain water resources and seasonal streamflow

References