Root Growth in Soil (2D rhizotron)

What the model does

A plant root system is simulated in a 24 x 32 cm vertical slice of soil (a rhizotron). Two fields live on a 48 x 64 grid: soil water content θ and a dissolved nutrient amount N. Roots are a set of growing tips; every tip leaves a trail of segments, and every cm of root takes up water and nutrient from the cell it sits in. Day 0 starts with one primary axis and four seminal roots.

Water and nutrient transport

Both fields obey a diffusion equation solved with an explicit conservative finite-volume scheme: ∂θ/∂t = ∇·(D(θ)∇θ) − kw·L·(θ−θwp) with the soil-water diffusivity D(θ) = D0(θ/θs)3 (a simple power law that makes dry soil conduct poorly; D₀ = 10, 30 or 90 cm²/day for clay, loam, sand), and ∂N/∂t = Dn∇²N − Imax·L·N/(Kn+N). The sink terms use the root length L inside each cell. The surface row relaxes toward the irrigation water content, the bottom row drains freely, and the sides are closed. The time step 0.002 day satisfies dt ≤ dx²/(4 Dmax).

Root growth rules

A tip elongates at v = vmax(order) · fw(θ) · (0.25 + 0.75 fn(N)) · (1 − compaction·depth factor) with fw = clamp((θ−θwp)/(θopt−θwp)) and fn = N/(Kn+N). Its direction angle follows dα/dt = −g sin(α−αset) + h·turn(∇θ) + c·turn(∇N) + noise: gravitropism pulls axes down and laterals toward their set angle, hydrotropism and chemotropism turn the tip toward wetter or richer soil. Laterals appear every s/(1+p·Nlocal) cm of tip growth, so nutrient-rich soil is more densely branched (root proliferation). Lateral order 1 to 3 have shorter maximum lengths and lower growth rates.

Carbon economy

Photosynthesis 0.6 · light · LA(S) with leaf area LA = 6 S0.66/(1+S/30) is split by the allocation slider: the root share pays cost carbon per cm of new root (growth slows by a common factor when the pool is empty), the shoot share builds shoot mass S in proportion to nutrient sufficiency. This is the feedback behind the classic root-to-shoot balance: starve the plant of nutrient and the shoot stops growing; starve it of carbon and the roots stall.

Independent numerical checks (run in the page)

(1) An isolated axis in fixed conditions must reach exactly v·f·t cm. (2) The live nutrient budget: nutrient left in the soil plus nutrient taken up equals the initial amount plus fertigation and any pulses you added (conservation of mass, to rounding error). (3) A point source diffusing in the grid has variance 2 D t and conserved mass.

Honest simplifications

The slice is two-dimensional; real root systems are three-dimensional and contain hairs, mycorrhizae, exudates and microbes that are not modelled. Water flow ignores gravity (only diffusion and bottom drainage) and hysteresis; the nutrient is a single mobile ion without sorption; tropism strengths are tuned to be visible, not measured for a species. The numbers teach how the controls trade off and are not a crop forecast. Educational model.