Surface energy balance — how much solar energy stays at the ground?
2026-02-26
You should know
That energy is conserved, sunlight can be reflected or absorbed, and warmer objects tend to emit more heat.
You will learn
How to track the main radiation streams at the surface, what albedo does, and how net radiation controls surface heating.
Why this matters
Radiation is the energy source behind snowmelt, evaporation, warming soils, and plant growth. Many environmental models start here.
If this gets hard, focus on…
Just keep asking two questions: what energy comes in, and what energy leaves? Net radiation is the difference between those two totals.
The Greenland ice sheet has a surface albedo of around 0.80 — it reflects 80% of incoming solar radiation straight back to space. Fresh snow in the Canadian Rockies reflects 0.85 or more. A conifer forest might reflect 0.10. Dark ocean water on a calm day, as little as 0.05. These differences seem like surface detail, but they have continental-scale consequences: the ice sheet’s high albedo is part of why it persists, while the boreal forest’s low albedo actively warms the surrounding climate by absorbing radiation that snow-covered tundra would have reflected. The 2012 record melt of Arctic sea ice exposed dark ocean that absorbed rather than reflected summer solar energy — a feedback that is still accelerating warming.
Net radiation is the difference between all the radiation arriving at a surface and all the radiation leaving it. It is the fundamental quantity that drives surface heating, evaporation, snowmelt, and photosynthesis. Calculating it requires tracking four separate radiation streams: incoming shortwave from the sun, reflected shortwave, incoming longwave from the atmosphere, and outgoing longwave emitted by the surface itself. The Stefan-Boltzmann law connects temperature to outgoing longwave emission; albedo controls reflected shortwave; and the two together determine how much energy is available to drive all the physical and biological processes at the surface.
How much solar energy actually warms the ground?
Sunlight arrives at Earth’s surface. Some bounces off (reflection). Some is absorbed and heats the surface. The surface also radiates energy back to the atmosphere as infrared heat.
The net radiation is the difference between energy absorbed and energy lost. This net radiation drives all surface processes: - Heating the soil and air - Evaporating water - Melting snow - Photosynthesis
The mathematical question: How do we calculate net radiation from incoming solar radiation, surface properties, and temperature?
At Earth’s surface, four radiation streams matter:
Incoming shortwave (S_{\downarrow}): Direct and diffuse solar radiation from the sun (visible and near-infrared, 0.3–3 μm wavelength)
Reflected shortwave (S_{\uparrow}): Solar radiation bounced back by the surface (same wavelengths)
Incoming longwave (L_{\downarrow}): Infrared radiation emitted downward by the atmosphere (thermal radiation, 3–100 μm wavelength)
Outgoing longwave (L_{\uparrow}): Infrared radiation emitted upward by the surface (thermal radiation)
Net radiation:
R_n = (S_{\downarrow} - S_{\uparrow}) + (L_{\downarrow} - L_{\uparrow})
Or equivalently:
R_n = \text{Net shortwave} + \text{Net longwave}
Shortwave (solar): - Originates from the sun (temperature ~5,800 K) - Peak wavelength ~0.5 μm (green visible light) - Intensity varies with time of day, season, cloudiness - Measured in W/m² (watts per square meter)
Longwave (thermal infrared): - Originates from Earth’s surface and atmosphere (temperature ~200–300 K) - Peak wavelength ~10 μm (thermal infrared, invisible to the eye) - Present day and night (doesn’t require sunlight) - Measured in W/m²
Why separate them? Different physics. Shortwave is controlled by albedo (surface color). Longwave is controlled by temperature (Stefan-Boltzmann law).
Albedo (\alpha) is the fraction of incoming solar radiation that is reflected:
\alpha = \frac{S_{\uparrow}}{S_{\downarrow}}
Albedo ranges from 0 (perfectly absorbing, black) to 1 (perfectly reflecting, mirror).
Reflected shortwave:
S_{\uparrow} = \alpha \cdot S_{\downarrow}
Net shortwave (absorbed):
S_{\text{net}} = S_{\downarrow} - S_{\uparrow} = S_{\downarrow}(1 - \alpha)
Examples of albedo:
| Surface | Albedo (\alpha) |
|---|---|
| Fresh snow | 0.80–0.95 |
| Old snow | 0.40–0.70 |
| Sea ice | 0.50–0.70 |
| Desert sand | 0.30–0.40 |
| Grassland | 0.15–0.25 |
| Deciduous forest | 0.15–0.20 |
| Conifer forest | 0.05–0.15 |
| Ocean | 0.06–0.10 |
| Asphalt | 0.05–0.10 |
Key insight: Dark surfaces (forests, oceans) absorb most solar radiation. Light surfaces (snow, ice) reflect most.
The Stefan-Boltzmann Law relates temperature to thermal radiation emitted by a surface:
E = \sigma T^4
Where: - E is emitted radiation (W/m²) - \sigma = 5.67 \times 10^{-8} W/m²/K⁴ (Stefan-Boltzmann constant) - T is absolute temperature (Kelvin)
Example: A surface at 15°C = 288 K emits:
E = 5.67 \times 10^{-8} \times (288)^4 = 390 \text{ W/m}^2
This is a LOT of energy — comparable to incoming solar radiation!
Real surfaces are not perfect “blackbodies.” They emit less than the theoretical maximum.
Emissivity (\varepsilon) is the ratio of actual emission to blackbody emission (0 ≤ ε ≤ 1).
Actual emission:
L_{\uparrow} = \varepsilon \sigma T_s^4
Where T_s is the surface temperature.
Typical emissivities: - Water: ε ≈ 0.95–0.98 - Vegetation: ε ≈ 0.95–0.99 - Soil: ε ≈ 0.90–0.95 - Snow: ε ≈ 0.98–0.99
Most natural surfaces have high emissivity (ε > 0.9) in the thermal infrared.
The atmosphere emits downward infrared radiation. This depends on: - Air temperature (warmer air emits more) - Water vapor content (water vapor is a strong greenhouse gas) - Cloud cover (clouds emit strongly; cloudy nights are warmer)
Simple approximation (clear sky):
L_{\downarrow} = \varepsilon_a \sigma T_a^4
Where: - \varepsilon_a is the effective emissivity of the atmosphere (typically 0.7–0.9) - T_a is air temperature (K)
With clouds: L_{\downarrow} increases, approaching \sigma T_a^4 for complete overcast.
Combining all four streams:
R_n = S_{\downarrow}(1 - \alpha) + L_{\downarrow} - \varepsilon \sigma T_s^4
Interpretation: - First term: absorbed solar radiation - Second term: atmospheric infrared warming - Third term: surface infrared cooling
Day vs. Night: - Daytime: S_{\downarrow} large → R_n typically positive (surface gains energy) - Nighttime: S_{\downarrow} = 0 → R_n = L_{\downarrow} - L_{\uparrow} typically negative (surface loses energy)
Problem: A grassland surface has: - Albedo: \alpha = 0.20 - Emissivity: \varepsilon = 0.96 - Surface temperature: T_s = 25°\text{C} = 298 K - Air temperature: T_a = 20°\text{C} = 293 K - Atmospheric emissivity: \varepsilon_a = 0.80 (clear sky) - Incoming solar radiation: S_{\downarrow} = 800 W/m² (midday sun)
Calculate the net radiation.
Net shortwave:
S_{\text{net}} = S_{\downarrow}(1 - \alpha) = 800 \times (1 - 0.20) = 800 \times 0.80 = 640 \text{ W/m}^2
Incoming longwave:
L_{\downarrow} = \varepsilon_a \sigma T_a^4 = 0.80 \times 5.67 \times 10^{-8} \times (293)^4
= 0.80 \times 5.67 \times 10^{-8} \times 7.357 \times 10^9 = 334 \text{ W/m}^2
Outgoing longwave:
L_{\uparrow} = \varepsilon \sigma T_s^4 = 0.96 \times 5.67 \times 10^{-8} \times (298)^4
= 0.96 \times 5.67 \times 10^{-8} \times 7.880 \times 10^9 = 429 \text{ W/m}^2
Net longwave:
L_{\text{net}} = L_{\downarrow} - L_{\uparrow} = 334 - 429 = -95 \text{ W/m}^2
(Negative because surface radiates more than it receives from atmosphere)
Total net radiation:
R_n = S_{\text{net}} + L_{\text{net}} = 640 + (-95) = 545 \text{ W/m}^2
Interpretation: The surface gains 545 W/m² during midday. This energy heats the ground, the air, and drives evaporation.
Below is an interactive radiation balance calculator.
<label>
Surface type:
<select id="surface-type">
<option value="snow">Fresh Snow (α=0.85)</option>
<option value="sand">Desert Sand (α=0.35)</option>
<option value="grass" selected>Grassland (α=0.20)</option>
<option value="forest">Conifer Forest (α=0.10)</option>
<option value="ocean">Ocean (α=0.08)</option>
</select>
</label>
<label>
Solar radiation (W/m²):
<input type="range" id="solar-slider" min="0" max="1200" step="50" value="800">
<span id="solar-value">800</span> W/m²
</label>
<label>
Surface temperature (°C):
<input type="range" id="surf-temp-slider" min="-20" max="40" step="1" value="25">
<span id="surf-temp-value">25</span> °C
</label>
<label>
Air temperature (°C):
<input type="range" id="air-temp-slider" min="-20" max="40" step="1" value="20">
<span id="air-temp-value">20</span> °C
</label>
<label>
Cloud cover:
<input type="range" id="cloud-slider" min="0" max="1" step="0.1" value="0">
<span id="cloud-value">0.0</span> (0=clear, 1=overcast)
</label><h4>Energy Balance:</h4>
<table>
<tr>
<td><strong>S↓ (incoming solar):</strong></td>
<td><span id="s-down"></span> W/m²</td>
</tr>
<tr>
<td><strong>S↑ (reflected):</strong></td>
<td><span id="s-up"></span> W/m²</td>
</tr>
<tr style="border-top: 1px solid #ddd;">
<td><strong>Net shortwave:</strong></td>
<td><span id="s-net"></span> W/m²</td>
</tr>
<tr>
<td><strong>L↓ (atmospheric IR):</strong></td>
<td><span id="l-down"></span> W/m²</td>
</tr>
<tr>
<td><strong>L↑ (surface IR):</strong></td>
<td><span id="l-up"></span> W/m²</td>
</tr>
<tr style="border-top: 1px solid #ddd;">
<td><strong>Net longwave:</strong></td>
<td><span id="l-net"></span> W/m²</td>
</tr>
<tr style="border-top: 2px solid #333;">
<td><strong>Net radiation (Rn):</strong></td>
<td><span id="r-net" style="font-weight: bold;"></span> W/m²</td>
</tr>
</table>Try this: - Switch to snow: Albedo jumps to 0.85 → most solar radiation reflects → low net shortwave - Set solar to 0 (nighttime): Net radiation goes negative (surface cools by IR emission) - Add clouds: Incoming longwave increases → net radiation less negative at night - Ocean vs. forest: Dark surfaces absorb more solar → higher net radiation
Key insight: Albedo dominates daytime energy balance. Temperature dominates nighttime balance.
During the day, S_{\downarrow} is large. For most surfaces:
R_n \approx S_{\downarrow}(1 - \alpha) + (L_{\downarrow} - L_{\uparrow})
Net radiation is positive → surface warms.
Example: Grassland at solar noon (S_{\downarrow} = 1000 W/m², \alpha = 0.20):
S_{\text{net}} = 1000 \times 0.80 = 800 \text{ W/m}^2
Even with longwave cooling (−100 W/m²), net radiation is ~700 W/m².
At night, S_{\downarrow} = 0. Net radiation simplifies to:
R_n = L_{\downarrow} - L_{\uparrow}
Typically negative (surface cools).
Clear night (low L_{\downarrow}): Strong cooling → frost possible
Cloudy night (high L_{\downarrow}): Weak cooling → warmer
This is why frost forms on clear nights! Clouds act like a blanket, re-radiating surface heat back down.
Snow-albedo feedback is a major climate amplifier:
Arctic amplification: Polar regions warm faster than tropics partly due to this feedback.
Earth’s average albedo: \alpha \approx 0.30 (clouds 0.50, oceans 0.08, land ~0.15 average)
Incoming solar (top of atmosphere): ~1361 W/m²
After reflection: $1361 = 953$ W/m² absorbed
This absorbed energy must be balanced by outgoing longwave radiation to maintain stable temperature.
Greenhouse effect: Atmosphere absorbs some L_{\uparrow}, emits L_{\downarrow} back → surface warmer than it would be without an atmosphere.
Albedo (\alpha) applies to shortwave (solar) radiation (0.3–3 μm).
Emissivity (\varepsilon) applies to longwave (thermal IR) radiation (3–100 μm).
A surface can have low albedo (dark, absorbs solar) but high emissivity (efficient IR radiator).
Example: Vegetation is dark (α ~ 0.15) but radiates efficiently (ε ~ 0.97).
Real albedo varies with wavelength. Snow reflects visible light (appears white) but absorbs some near-infrared.
Broadband albedo (used here) is the average over all solar wavelengths. More sophisticated models use spectral albedo (albedo as a function of wavelength).
Albedo changes with: - Sun angle: Ocean albedo increases when sun is low (more glint reflection) - Moisture: Wet soil is darker than dry soil - Snow age: Fresh snow (α = 0.9) → old snow (α = 0.5) as it compacts and collects dirt - Vegetation state: Green leaves (α ~ 0.20) → dry leaves (α ~ 0.30)
Our model assumes a flat, horizontal surface. Real terrain has: - Slopes: Affect solar angle (Model 6 covered this) - Shading: Mountains cast shadows - Multiple reflections: Light bounces between surfaces (canyon walls, forest canopy)
Temperature must be in Kelvin for Stefan-Boltzmann:
L_{\uparrow} = \varepsilon \sigma T^4
If you use Celsius, you’ll get nonsense results!
Conversion: T_K = T_C + 273.15
Net radiation doesn’t all go into warming the ground. It’s partitioned among:
R_n = H + LE + G + S
Where: - H = Sensible heat flux (heating the air) - LE = Latent heat flux (evaporating water; E is evaporation rate, L is latent heat) - G = Ground heat flux (heating the soil) - S = Storage (heating biomass, water bodies)
Next model will explore how R_n is partitioned between sensible and latent heat — the foundation of evapotranspiration modelling.
The Stefan-Boltzmann law comes from Planck’s law integrated over all wavelengths.
Physical origin: Hotter objects emit: - More total energy (T⁴ dependence) - At shorter wavelengths (peak shifts to blue for hot objects, red for cool)
Wien’s displacement law: Peak wavelength \lambda_{\text{max}} \propto 1/T
Sun (T ≈ 5800 K): peaks at 0.5 μm (green visible light)
Earth (T ≈ 288 K): peaks at 10 μm (thermal infrared)
$288^4 = 288 \times 288 \times 288 \times 288$
Easier method:
$288^4 = (288^2)^2 = (82944)^2 = 6,879,707,136$
Then multiply by \sigma = 5.67 \times 10^{-8}:
E = 5.67 \times 10^{-8} \times 6.88 \times 10^9 = 390 \text{ W/m}^2
For a small change \Delta T around temperature T:
\Delta E \approx 4\sigma T^3 \Delta T
Example: At T = 288 K, a 1 K increase:
\Delta E = 4 \times 5.67 \times 10^{-8} \times (288)^3 \times 1 = 5.4 \text{ W/m}^2
Surface emission increases by 5.4 W/m² per degree warming.