Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2022 / 5 / c / Solution

Define the two concentration differences $u=C_a-C_b$ and $v=C_b-C_d$. For zero emissions and $R_a=R_b=R$, their ordinary differential equation isWriting the positive decay rates as ${\mu }_i=1/{\tau }_i$, their characteristic equation isConsequentlyFor $r\gg s$, the first normal mode rapidly equalizes the fast reservoirs, giving ${\mu }_1\simeq 2r/R$. The slower evolution treats them as a combined capacity $2R$ exchanging with $R_d$, giving ${\mu }_2\simeq s[1/(2R)+1/R_d]$. ThereforeThe timescale ratio is ${\tau }_2/{\tau }_1\simeq 4r/s\gg 1$ in the joint limit. Since the conserved normal mode has ${\tau }_3=\infty$, the hierarchy is ${\tau }_1\ll {\tau }_2\ll {\tau }_3$.