Exponential decay through space — modelling vertical gradients in forest light
2026-02-26
Walk into an old-growth Douglas fir forest in British Columbia and the light drops immediately — not gradually but in steps, as you pass beneath successive tiers of canopy. Instruments record what the eye perceives: PAR (photosynthetically active radiation) that might be 1,500 µmol/m²/s in full sun drops to 600 at the first canopy layer, 150 beneath the next, and sometimes below 10 at the forest floor. That is less than 1% of full sunlight — enough for shade-tolerant ferns and mosses but insufficient for most tree seedlings. The vertical light gradient is not incidental to how a forest works; it is the primary structure that determines species composition, carbon allocation, and the competitive dynamics between trees.
The mathematics describing this gradient is Beer’s Law — an exponential attenuation equation that appears in an remarkable number of contexts: light through water, X-rays through tissue, radar through soil, sound through atmosphere. In each case, an absorbing medium removes a constant fraction of the signal passing through each layer. The cumulative effect is exponential decay. Deriving Beer’s Law from first principles, and understanding how leaf area index controls the shape of the light profile, is the foundation for canopy radiative transfer models used in everything from carbon cycle accounting to satellite image interpretation.
How does light intensity change as you descend through a forest canopy?
Stand at the top of a rainforest and measure light intensity: 100% of incoming solar radiation. Drop down 10 meters into the canopy: maybe 60% remains. Another 10 meters: 35%. At the forest floor: perhaps 2%.
Why does this happen? Each leaf intercepts and absorbs some fraction of the light hitting it. The cumulative effect of many layers creates an exponential decline.
The mathematical question: Can we predict light intensity at any height, given the density of leaves above?
Imagine sunlight as a stream of photons raining down from above. As the stream passes through each infinitesimal layer of leaves: - Some photons are absorbed (photosynthesis, heat) - Some are scattered (reflected, transmitted through gaps) - The remainder continues downward
The rate of loss is proportional to: 1. The light intensity at that height (more light → more absorption) 2. The leaf density at that height (more leaves → more interception)
This is exactly the structure of exponential decay — but in the vertical dimension rather than time.
Leaf Area Index is the total one-sided leaf area per unit ground area.
Units: \text{m}^2 (leaf) / \text{m}^2 (ground) = dimensionless
Example: An LAI of 5 means there are 5 square meters of leaf area above every square meter of ground.
For a canopy of height H with uniform leaf density:
\text{LAI} = \text{leaf area density} \times H
Let: - I(z) = light intensity at height z (units: W/m² or μmol photons/m²/s) - z = height above ground (m) - k = extinction coefficient (dimensionless) - L(z) = cumulative leaf area index above height z
The rate of light loss with decreasing height:
\frac{dI}{dz} = -k \cdot I(z)
Negative sign: Light decreases as z decreases (moving down through the canopy).
This is identical in form to radioactive decay — but the independent variable is height, not time.
Separation of variables:
\frac{1}{I} \frac{dI}{dz} = -k
Integrate:
\int \frac{1}{I} dI = \int -k \, dz
\ln I = -kz + C
Exponentiate:
I(z) = e^{-kz + C} = e^C \cdot e^{-kz}
At the top of the canopy (z = H), let I(H) = I_0 (incoming radiation):
I_0 = e^C \cdot e^{-kH}
e^C = I_0 e^{kH}
Substitute back:
I(z) = I_0 e^{kH} \cdot e^{-kz} = I_0 e^{-k(z - H)}
Or, measuring from the top of the canopy with z' = H - z (depth into canopy):
I(z') = I_0 e^{-kz'}
This is Beer’s Law (also called the Beer-Lambert Law or Beer-Bouguer Law).
If leaf area density is uniform, the cumulative LAI from the top to depth z' is:
L = \text{LAI}_{\text{total}} \times \frac{z'}{H}
Then:
I = I_0 e^{-kL}
This form is more commonly used in ecology: light depends on cumulative leaf area, not just height.
Problem: A forest canopy has total LAI = 6, extinction coefficient k = 0.5, and incoming light I_0 = 1000 μmol/m²/s.
(a) Light at forest floor
I = I_0 e^{-kL} = 1000 \cdot e^{-0.5 \times 6} = 1000 \cdot e^{-3}
e^{-3} \approx 0.0498
I \approx 1000 \times 0.0498 = 49.8 \text{ μmol/m²/s}
About 5% of incoming light reaches the forest floor.
(b) Depth where I = 0.1 I_0
$0.1 I_0 = I_0 e^{-kL}$
$0.1 = e^{-0.5L}$
Take natural log:
\ln(0.1) = -0.5L
-2.303 = -0.5L
L = \frac{2.303}{0.5} = 4.61
Light drops to 10% at cumulative LAI of 4.6 (about 77% of the way through the canopy).
(c) Fraction penetrating
From part (a): I_{\text{floor}} / I_0 = 0.0498 \approx 5\%
5% penetrates to the ground.
Below is an interactive model showing vertical light profiles.
<label>
Total LAI:
<input type="range" id="lai-slider" min="1" max="10" step="0.5" value="6">
<span id="lai-value">6.0</span>
</label>
<label>
Extinction coefficient (k):
<input type="range" id="k-slider" min="0.2" max="1.0" step="0.05" value="0.5">
<span id="k-value">0.50</span>
</label>
<label>
Incoming light (I₀):
<input type="range" id="i0-slider" min="500" max="2000" step="100" value="1000">
<span id="i0-value">1000</span> μmol/m²/s
</label><p id="floor-light"></p>
<p><strong>Biological context:</strong> Shade-tolerant species can photosynthesize at 1-2% of full sun. Sun-adapted species typically need >10%.</p>Try this: - Increase LAI to 10 (dense tropical forest). Floor light drops below 1%. - Decrease k to 0.3 (sparse leaves, less absorption per unit LAI). More light penetrates. - Increase k to 0.8 (dense clumped leaves). Light drops off faster. - Set LAI = 2 (savanna woodland). Significant light reaches the understory.
The extinction coefficient k depends on: 1. Leaf angle distribution — horizontal leaves (k \approx 1) block more light than vertical leaves (k \approx 0.3) 2. Leaf optical properties — dark leaves absorb more than shiny leaves 3. Canopy clumping — clumped foliage increases k (gaps let light through, but dense clumps block heavily)
Typical values: - Grasslands, crops: k \approx 0.4–$0.5$ - Temperate deciduous forests: k \approx 0.5–$0.6$ - Tropical rainforests: k \approx 0.6–$0.8$ - Conifer forests: k \approx 0.4–$0.5$ (needle geometry)
Plants adapted to deep shade (ferns, mosses, shade-tolerant tree seedlings) can photosynthesize efficiently at 1–5% of full sun. They have: - High chlorophyll concentration - Larger, thinner leaves (more light capture per unit mass) - Lower light compensation point
This allows stratification: tall sun-adapted trees above, shade-tolerant species below.
In temperate deciduous forests, LAI varies dramatically: - Summer: LAI = 5–6, floor light 2–5% - Spring (pre-leaf-out): LAI = 0, floor light 100%
This creates a spring ephemeral window: wildflowers bloom before the canopy closes.
Real canopies are clumped — leaves cluster around branches, creating gaps. Beer’s Law assumes a random, homogeneous distribution.
Correction: Use an effective LAI or apply a clumping factor to k.
Beer’s Law treats light as purely absorbed. In reality: - Some light is scattered by leaves (diffuse radiation) - Some is reflected from leaf surfaces - Some is transmitted through leaves (especially green light)
A more complete model tracks direct vs. diffuse radiation separately.
The derivation assumes vertical light rays. In reality, the sun’s angle changes throughout the day.
When sunlight is oblique, the effective path length through the canopy increases, so light attenuation is stronger.
Correction: Replace k with k / \cos(\theta), where \theta is the solar zenith angle.
LAI is cumulative (total leaf area above you). Leaf area density is LAI per unit height (m² leaf / m² ground / m height).
Beer’s Law applies to any direction where absorbing material is distributed.
Examples: - Atmosphere: Pressure decreases exponentially with altitude - Ocean: Light attenuates exponentially with depth (coastal waters: ~10 m penetration; open ocean: ~100 m) - Soil: Water infiltration follows exponential profiles in some cases - Beer (yes, the beverage): Light absorption through a liquid — the original application by August Beer (1852)
The mathematical structure is always the same:
\frac{dI}{dx} = -\alpha I
where \alpha is the attenuation coefficient and x is the distance through the medium.
In Model 3, we saw:
\frac{dN}{dt} = rN \quad \Rightarrow \quad N(t) = N_0 e^{rt}
Here, the independent variable was time.
In this model:
\frac{dI}{dz} = -kI \quad \Rightarrow \quad I(z) = I_0 e^{-kz}
The independent variable is height (or depth).
The mathematics is identical. Differential equations don’t care whether you’re moving through time or space. The structure of the equation determines the solution.
This insight — that spatial and temporal processes can obey the same equations — is foundational to mathematical physics and physical geography.