Past exam of the physics of atmospheres and oceans course of the University of Oxford / 2016 / 4 / Solution

Third body and steady partition. The third molecule removes excess association energy and momentum, stabilizing newly formed ozone rather than being consumed in the net reaction. Its decreasing abundance with altitude suppresses three-body ozone formation strongly. Write $x=[\mathrm{O}]$, $y=[{\mathrm{O}}_3]$, $a=k_1{n}_2{n}_M$, $j=J_3$ and $b=k_3$, with $n_2,n_M$ held as reservoirs. The kinetic equations areThe fast association/photolysis subsystem conserves odd oxygen, $x+y$, and has eigenvalues $0,-(a+j)$. Its partition-adjustment time is thereforeThis neglects the slower production and mutual-destruction terms while finding that fast mode. On its equilibrium partition, $ax\simeq jy$, and total odd-oxygen equilibrium gives $bxy=J_2{n}_2$. HenceWith fixed oxygen mole fraction and temperature, the atomic-oxygen mixing ratio scales as $x/n_M\propto n_M^{-3/2}$ in this approximation, and thus increases upward as pressure decreases.

Lifetime versus relaxation. The distinction is formalized by Chapman family lifetime versus relaxation time. In the ozone-dominated regime $y\gg x$, the family abundance is approximately $y$, and its gross chemical lifetime isThis recovers the printed ${\tau }_S$ as a family turnover time under its additional partition assumption. It is not generally an e-folding relaxation time. With rapid partitioning, the family equation is $\dot{X}=P-k{X}^2$. Linearizing about equilibrium gives relaxation rate $2k{X}_e$ but turnover rate $k{X}_e$, a factor of two difference. More generally the approximate family lifetime is $(x+y)/(2bxy)$, and its relaxation time is half that quantity. Neither can be replaced by $1/(2bx)$ when atomic oxygen dominates.

For a direct check using the complete two-species equations, the equilibrium Jacobian matrix isThe numerical high-altitude example below gives eigenmode times about $321\mathrm{s}$ and $4.0\times 10^5\mathrm{s}$, whereas the printed ${\tau }_S$ is only $5.15\times 10^3\mathrm{s}$. Thus interpreting both printed expressions as universal eigenmode time constants would be false, even though the fast partition and the ozone-dominated turnover approximations are useful.

Setting $x=y$ in the fast-partition approximation gives $a=j$, henceRetaining mutual destruction gives $a=j+bx$ and changes this estimate by less than one percent. The formal high-altitude limit of ${\tau }_F$ is $1/J_3\simeq 333\mathrm{s}$: ozone photolysis, unlike three-body association, does not slow merely because collider density decreases. This is a partition timescale, not the lifetime of all odd oxygen. Extrapolating the fixed-reservoir approximation indefinitely upward would eventually cease to be self-consistent.

Day and night. At $p=0.01\mathrm{hPa}=1\mathrm{Pa}$ and $230\mathrm{K}$, $n_M=p/(k_{B}T)=3.15\times 10^{20}{\mathrm{m}}^{-3}$ and $n_2=0.2n_M$. The exact equilibrium is conveniently obtained fromThe physical root givesTheir ratio is $156$, consistent with the requested approximate factor $150$; the simpler fast-partition estimate gives approximately $151$. The printed fast time is $331\mathrm{s}$ at this pressure.

For nighttime Chapman ozone, after sunset set both photolysis rates to zero. Ozone initially grows by converting atomic oxygen, but simply putting every initial oxygen atom into one ozone molecule neglects the simultaneous $\mathrm{O}+{\mathrm{O}}_3$ sink. The night equations areWhile sufficient atomic oxygen remains, the ozone balance tends towardThis is a useful long-night sunrise estimate, roughly thirty times the sunset ozone abundance, not an exact result for an unspecified night duration. The precise phase-plane relation isTogether with the night equations it supplies $y(t_{\mathrm{night}})$ for any supplied duration. For illustration only, a twelve-hour dark interval gives ${\chi }_{O_3}\simeq 5.8\times 10^{-5}$; no twelve-hour duration is stated by the paper. Even the infinite-dark-time endpoint is slightly below $a/b$, because the finite initial atomic reservoir is consumed. This conditional estimate avoids inventing a unique sunset-to-sunrise duration.