Chapter 3, Problem 27P

Figure P3.16 shows a schematic description of the global carbon cycle (Li,). In the figure, m_A(t) represents the amount of carbon in gigatons (GtC) present in the atmosphere of earth; mV(t) the amount in vegetation; m_s(t) the amount in soil; m_SO(t) the amount in surface ocean; and m_IDO(t) the amount in intermediate and deep−ocean reservoirs. Let u_E(t) stand for the human generated CO₂ emissions (GtC/yr). From the figure, the atmospheric mass balance in the atmosphere can be expressed as:

$\frac{d{m}_A}{dt}\left(t\right)=u_E\left(t\right)-\left(k_{O1}+k_{L1}\right)m_A\left(t\right)+k_{L2}{m}_V\left(t\right)+k_{O2}{m}_{SO}\left(t\right)+k_{L4}{m}_S\left(t\right)$

where the k’s are exchange coef?cients (yr−1).

1. Write the remaining reservoir mass balances. Namely, write equations for ,
   $\frac{d{m}_{SO}\left(t\right)}{dt},\frac{d{m}_{IDO}\left(t\right)}{dt},\frac{d{m}_V\left(t\right)}{dt}$, and $\frac{d{m}_S\left(t\right)}{dt}$
2. Express the system in state-space form.

FIGURE P3.16 Global carban cycle¹³