Soil Regeneration
Quantitative soil science: from microbial kinetics and carbon pool dynamics to mycorrhizal transport models and a fully interactive RothC carbon-accumulation simulation.
1 The Living Soil
A single gram of healthy topsoil harbours more microorganisms than there are people on Earth. These communities drive decomposition, nutrient cycling, and the formation of stable soil organic matter (SOM). Understanding their kinetics is the first step toward managing soil carbon.
Michaelis-Menten Decomposition Kinetics
Microbial uptake of dissolved organic carbon (DOC) follows saturating kinetics. The rate of substrate consumption by a microbial biomass pool B is:
where Vmax is the maximum specific uptake rate (mg C g-1 B h-1), S is substrate concentration, and Km is the half-saturation constant. At low S the rate is first-order; at high S it saturates.
Carbon-Use Efficiency (CUE)
CUE determines how much assimilated carbon is incorporated into biomass versus respired as CO2:
Typical CUE values range from 0.3 to 0.6. Higher CUE means more carbon retained in the soil food web. CUE declines with temperature, substrate recalcitrance, and nutrient limitation.
Soil Food Web
2 Carbon Cycling & Sequestration
Soil carbon models partition organic matter into pools with distinct turnover times. The RothC model (Rothamsted Carbon) uses five pools; the Century model uses three primary pools. Understanding pool dynamics is essential for predicting long-term sequestration potential.
RothC Pool Dynamics
The RothC model partitions soil organic carbon into Decomposable Plant Material (DPM), Resistant Plant Material (RPM), microbial Biomass (BIO), Humified organic matter (HUM), and Inert Organic Matter (IOM). Each pool decays at a characteristic rate modified by temperature, moisture, and soil cover:
where Ii is plant/manure input to pool i, ki is the base decomposition rate, a, b, c are temperature, moisture, and cover modifiers, and alphaji is the transfer fraction from pool j to i.
Century Model
The Century model uses Active, Slow, and Passive pools with turnover times of 1-5 years, 20-50 years, and 200-1 500 years respectively. Decomposition products are partitioned between pools and CO2 according to clay content and lignin:N ratios.
MAOM: Langmuir Sorption Isotherm
Mineral-Associated Organic Matter (MAOM) forms when dissolved organic compounds bind to mineral surfaces. The sorption equilibrium follows a Langmuir isotherm:
where Q is the amount sorbed (mg C g-1 mineral), Qmax is the maximum sorption capacity, KL is the Langmuir constant, and Ceq is the equilibrium DOC concentration. MAOM is a key long-term carbon stabilisation mechanism in fine-textured soils.
Carbon Pool Flow (RothC)
3 Mycorrhizal Networks
Over 90% of land plants form symbiotic associations with mycorrhizal fungi. Arbuscular mycorrhizal fungi (AMF) extend the effective root surface area by orders of magnitude, accessing phosphorus and micronutrients beyond the depletion zone. In exchange, the plant allocates 10-20% of its photosynthate to the fungal partner.
Nye-Tinker-Barber Nutrient Uptake Model
The Nye-Tinker-Barber model describes radial diffusion of a nutrient towards a cylindrical root (or hypha) of radius a:
where De is the effective diffusion coefficient in soil, r is the radial distance from the root axis, and v0 is the water-uptake velocity at the root surface. The boundary condition at r = a is a Michaelis-Menten uptake flux.
Bidirectional Mutualism Model
The reciprocal exchange between plant and mycorrhiza can be modelled as a coupled system where each partner's investment depends on the marginal return:
where Cp and Cf are plant and fungal carbon, alpha is the allocation fraction to the fungus, beta is the phosphorus-mediated growth benefit, and rp, rf are respiration rates.
Mycorrhizal Network
4 Soil Physics & Water Dynamics
Soil structure -- the arrangement of mineral particles, organic matter, and pore spaces -- governs water retention, aeration, root penetration, and microbial habitat. Aggregate stability is a key indicator of soil health and carbon protection.
Van Genuchten-Mualem Water Retention
The Van Genuchten model relates volumetric water content to soil matric potential:
where h is the matric head (cm), thetas and thetar are saturated and residual water contents, alpha is the inverse air-entry pressure (cm-1), and n is a pore-size distribution parameter. The Mualem extension predicts hydraulic conductivity from the same parameters.
Mean Weight Diameter (MWD)
Aggregate stability is quantified by wet-sieving. The Mean Weight Diameter measures the average aggregate size weighted by mass:
where x̄i is the mean diameter of each sieve fraction and wi is the proportion of total sample mass retained on that sieve. MWD typically ranges from 0.5 mm (degraded) to >3 mm (well-aggregated). Higher SOM, fungal hyphae, and root exudates increase aggregate stability.
Green-Ampt Infiltration Model
The Green-Ampt model approximates infiltration into a soil with a sharp wetting front:
where f(t) is the infiltration rate (cm h-1), Ks is the saturated hydraulic conductivity, psi is the wetting front suction head, and F(t) is the cumulative infiltration. Compacted or tilled soils have lower Ks and infiltrate less.
Aggregate Hierarchy
5 Carbon Accumulation Simulation
This interactive simulation implements a simplified RothC model with four carbon pools (DPM, RPM, BIO, HUM). Adjust the sliders to explore how management practices -- plant input, compost application, reduced tillage, and soil texture -- affect long-term soil carbon accumulation. Use Compare to overlay multiple scenarios.
Numerical Integration (RK4 Scheme)
Pool dynamics are integrated using the Runge-Kutta 4th-order method for stability:
where each ki evaluates the derivative at progressively refined mid-step points, giving fourth-order accuracy in the time step $\Delta t$.
Simulation Parameters
Ready. Adjust parameters and click Run.