In the figure on the left — the Grow-Decay diagram — carbon first enters the system as CO\(_2\) through Net Primary Productivity in the tree and moss pools. It is then transferred by vegetation turnover to the litter and eventually soil pools as biomass. During this process, carbon will also leave each pool — including the pyrogenic carbon pool from burn events — through respiration as CO\(_2\).

The figure on the right — the Burn diagram — displays the carbon transfers during a burn event. During a burn event, carbon will leave each of the pools and go into the atmosphere as CO\(_2\). However, there will still be pyrogenic carbon remaining in the biomass that was not combusted completely.

As burn events continue to occur, the pyrogenic carbon continues to build up in the soil in pools called burn cohorts.

The dimensional equations are:

Tree: \(\frac{dC_T}{dt} = U(t) a_{T} - k_{T} C_{T}\)

Moss: \(\frac{dC_m}{dt} = U(t) a_{m} - k_{m} C_{m}\)

Litter: \(\frac{dC_l}{dt} = f_{Tl} k_{T} C_{T} + f_{ml} k_{m} C_{m} - k_{l} C_{l}\)

Soil: \(\frac{dC_s}{dt} = f_{Tl} k_{T} C_{T} + f_{ms} k_{m} C_{m} + f_{ls} k_{l} C_{l} - k_{s} C_{s}\)

Burn: \(\frac{dC_{bi}}{dt} = -k_{s} b_{s}^{i} C_{bi}\)

The dimensional equations were and normalized to the natural primary production over the decay rate of the soil (carbon area density/time (\(\frac{g-C}{m^2 \ yr}\))), with the following assumptions:

\(\bar{t} = t k_{s}\) and \(C_{x} = {\frac{u(t)}{k_s}\bar{C}_{x}}\)

This resulted in the following nondimensional equations:

Tree: \(\frac{d\bar{C}_T}{d\bar{t}} = a_{T} - \frac{k_{T}}{k_{s}} \bar{C}_T\)

Moss: \(\frac{d\bar{C}_{m}}{d\bar{t}} = a_{m} - \frac{k_{m}}{k_{s}} d\bar{C}_{m}\)

Litter: \(\frac{d\bar{C}_l}{d\bar{t}} = \frac{k_T}{k_s} f_{Tl} C_{T} + \frac{k_{m}}{k_{s}}f_{ml} \bar{C}_{m} - \frac{k_{l}}{k_{s}} \bar{C}_l\)

Soil: \(\frac{d\bar{C}_s}{d\bar{t}} = \frac{k_T}{k_s} f_{Tl} \bar{C}_T + \frac{k_{m}}{k_{s}} f_{ms} d\bar{C}_m + \frac{k_l}{k_s} f_{ls} d\bar{C}_l - d\bar{C}_s\)

Burn: \(\frac{d\bar{C}_{bi}}{d\bar{t}} = -b_{s}^{i} \bar{C}_{bi}\)

When assuming steady state for both dimensional (\(\frac{dC_x}{dt} = 0\)) and nondimensional equations (\(\frac{d\bar{C}_x}{d\bar{t}} = 0\)), the analytical solution for each state variables are as follows:

Dimensional | Nondimensional |
---|---|

\(C_T = \frac{a_T}{k_T}u(t)\) | \(\bar{C}_T = \frac{k_s}{k_T}a_T\) |

\(C_m = \frac{a_m}{k_m}u(t)\) | \(\bar{C}_m = \frac{k_s}{k_m}a_m\) |

\(C_l = \frac{u(t)}{k_l}(f_{Tl}a_T + f_{ml}a_m)\) | \(\bar{C}_l = \frac{k_s}{k_l}(f_{Tl}a_T + f_{ml}a_m)\) |

\(C_s = \frac{u(t)}{k_s}(f_{Ts}a_T + f_{ms}a_m + f_{ls}f_{Tl}a_T + f_{ls}f_{ml}a_m)\) | \(\bar{C}_s = f_{Ts}a_T + f_{ms}a_m + f_{ls}f_{Tl}a_T + f_{ls}f_{ml}a_m\) |

\(C_{bi} = 0\) | \(\bar{C}_{bi} = 0\) |

When assuming steady state during decay, there is no input into the burn pool, and therefore its state variable is equal to zero.

This model is currently under developement by the Todd-Brown Lab at the University of Florida under an MIT License, manuscript in prep. Please contact ktoddbrown <@at@> ufl.edu for further details.

The Grow-Burn-Decay model is currently under developement by the Todd-Brown Lab at the University of Florida under an MIT License, manuscript in prep.
Please contact ktoddbrown <@at@> ufl.edu for further details.