Same first-order Hénin–Dupuis soil-carbon kinetics as the 3D field version, but rendered as two 2D-native diagrams instead of a grid of columns: a stock-and-flow tank (left) showing carbon flowing in and out of the SOC pool, and a phase-plane plot (right) showing the rate dC/dt as a function of the stock C itself — the actual state space of the ODE.
dC/dt = I − k·C
C_eq = I / k (steady-state SOC, where the phase line crosses zero)
I is the annual carbon input (baseline residue + cover-crop input, Mg C/ha/yr) and k is the fractional decomposition rate, accelerated by tillage:
k = k0 · (1 + β·tillage)
k0 ≈ 0.020 /yr, β ≈ 2.5
- Tank diagram — the fill level is C; the inflow pipe width is proportional to I; the outflow pipe width is proportional to k·C, so the tank visibly stops draining once the two flows balance.
- Phase-plane plot — dC/dt = I − k·C is a straight line in (C, dC/dt) space with slope −k. The system's state is a single dot sliding along that line; it always slides toward the C-axis crossing (C_eq), which is exactly what makes C_eq a stable equilibrium regardless of starting stock.
- A tillage pass (when tillage > 0.05) sharply steepens the line's slope for an instant, jerking the operating point off the old line onto the new, steeper one — visually explaining why switching to intensive tillage causes SOC to fall even with unchanged cover-crop input.
Between slider changes, the stock is advanced with the exact analytic solution C(t+dt) = C_eq + (C(0) − C_eq)·e^(−k·dt) rather than an explicit Euler step, so the trajectory is numerically exact for any playback speed.