Odd oxygen ($O_x=\mathrm{O}+{\mathrm{O}}_3$)

A chemical-family bookkeeping variable in a simple stratospheric model. Interconversion of atomic oxygen and ozone preserves this count, while $\mathrm{O}+{\mathrm{O}}_3\to 2{\mathrm{O}}_2$ removes two units. The definition must be extended consistently when including additional reservoirs or excited atomic states.

A basic oxygen–ozone mechanism combining oxygen photodissociation, three-body ozone formation, ozone photodissociation and $\mathrm{O}+{\mathrm{O}}_3\to 2{\mathrm{O}}_2$. The last reaction removes two units of odd oxygen; a catalytic cycle can add another loss pathway.

With photolysis switched off, constant molecular oxygen and collider reservoirs give $\dot{x}=-x(a+by)$ and $\dot{y}=x(a-by)$ for atomic oxygen $x$, ozone $y$, $a=k_1[{\mathrm{O}}_2][M]$ and $b=k_3$. Dividing the equations and integrating givesFor $y_0<a/b$, ozone grows toward a finite-reservoir endpoint below $a/b$. Its value at sunrise depends on the dark-interval duration; $a/b$ is an approximate quasi-steady balance while atomic oxygen remains available.

Under fast Chapman cycle partitioning, an odd oxygen family abundance $X$ can satisfy $\dot{X}=P-k{X}^2$. At equilibrium $P=k{X}_e^2$, its turnover lifetime is $X_e/P=1/(k{X}_e)$, while linearizing gives $\delta \dot{X}=-2k{X}_e\delta X$ and relaxation time $1/(2k{X}_e)$. A chemical lifetime is therefore not automatically an eigenmode relaxation time. The often-used expression $1/(2k_3[\mathrm{O}])$ also assumes ozone dominates the family abundance.

Suppose a reduced steady balance is $P=A{X}^2+BX$, where $X=[{\mathrm{O}}_3]$, and hold $P$ and all specified background concentrations fixed. With $u=1/T$, $d\ln A/du=-a$ and $d\ln B/du=-b$, implicit differentiation gives $d\ln X/du=(aA{X}^2+bBX)/(2A{X}^2+BX)$. The quadratic loss term includes temperature-dependent O/O3 partitioning as well as the direct reaction coefficient. Holding the loss ratio fixed over a finite change is an additional approximation; solving the new quadratic provides a check.