## The Grow-Burn-Decay model v 0.2.2

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.

## Equations

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

### Steady state solutions

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.

# Release notes

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.

## Initial carbon pools

Initial carbon stock in each pool [g-C m-2 yr-1]

## Grow

### Net Primary Productivity

NPP of trees and moss after the burn [min] and in mature ecosystem [max] (g-C m-2 yr-1)

### Vegetation Turnover Time:

The inverse of the vegitation death rate [yr].

### Recovery Year

Time it takes for the NPP to reach maturiture rate after burn event (yr)

## Burn

### Effectivness of reburn

Change in lost carbon per burn event (X^n*orginal_burn). <1: less carbon is burned with each burn event. >1: carbon is more vulnerable to burn losses.

### Response to fire

Fraction of original carbon stock that remains in the origianl pool [remain] or transferred to pyrogenic carbon pool [remain + pyro] [g-C/g-C]. The remaining carbon is burned off as CO2. Note that soil is always entirely transfered to the pyrogenic carbon pool.

## Decay

### Effectiveness of decay after burn

Change in decay rate per burn event (X ^ number_burns * orginal_decay). 1: no effect of burns on orginal decay rate. 0: burned carbon does not decay

### Decay turnover time

Turnover time for the decaying carbon pools [litter, soil] or one over the decay rate (yr).

### Carbon Transfer

Fraction of carbon from decaying pools transferred to other pools (g-C/g-C)

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.