Linear three-reservoir carbon model ($D\dot{\mathbf{C}}=E{\mathbf{e}}_a-L\mathbf{C}$)

A capacity-weighted exchange model for atmospheric, fast surface and deep ocean carbon dioxide anomalies. Positive diagonal capacities form $D$, while $L$ is the weighted path-graph Laplacian matrix with exchange conductances $r,s$. Then ${\mathbf{C}}^{T}L\mathbf{C}=r(C_a-C_b{)}^2+s(C_b-C_d{)}^2\ge 0$. The homogeneous generator $-D^{-1}L$ is similar to a symmetric negative-semidefinite matrix, with two decaying normal modes and one zero normal mode for equal concentrations. Summing all budgets proves that the capacity-weighted total is conserved after emissions stop. The long-time concentration from total input $M$ is $M/(R_a+R_b+R_d)$, rather than zero.