Figure P3.16 shows a schematic description of the global

Chapter 3, Problem 27

(choose chapter or problem)

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 \((\mathrm{GtC})\) present in the atmosphere of earth; \(m_V(t)\) the amount in vegetation; \(m_s(t)\) the amount in soil; \(m_{S O}(t)\) the amount in surface ocean; and \(m_{I D O}(t)\) the amount in intermediate and deep ocean reservoirs. Let \(u_E(t)\) stand for the human generated \(\mathrm{CO}_2\) emissions (GtC/yr). From the figure, the atmospheric mass balance in the atmosphere can be expressed as:

\(\begin{aligned} \frac{d m_A}{d t}(t)= & u_E(t)-\left(k_{O 1}+k_{L 1}\right) m_A(t)+k_{L 2} m_V(t) \\  & +k_{O 2} m_{S O}(t)+k_{L 4} m_S(t) \end{aligned}\)

where the k's are exchange coefficients \(\left(\mathrm{yr}^{-1}\right)\).

(a) Write the remaining reservoir mass balances. Namely, write equations for \(\frac{d m_{S O}(t)}{d t}, \frac{d m_{I D O}(t)}{d t}, \frac{d m_V(t)}{d t}\), and \(\frac{d m_S(t)}{d t}\)

(b) Express the system in state-space form.