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Time evolution equations for the condensation of sulfuric acid gas

Sulfuric acid is assumed to be nonvolatile in the tropospheric version of MAM. In other words, \(q_{v,H_2SO_4,sat}=0\) at tropospheric temperatures, and \({\rm H_2SO_4}\) will always condense when it exits. (Note that there is a special treatment for the stratospheric sulfate from volcanic eruptions, which includes a non-zero saturation vapor mixing ratio).

Considering only the effect of \({\rm H_2SO_4}\) condensation, the time evolution equations of relavant prognostic variables in MAM read

\[ \begin{align} \frac{dq_{m,SO_4,i}}{dt} &= q_{v,H_2SO_4} C_{H_2SO_4,i}, \quad i = 1,...,I \,, \label{eqn:ddt_so4} \\ \frac{dq_{v,H_2SO_4}}{dt} &= - q_{v,H_2SO_4} \sum_{i=1}^{I} C_{H_2SO_4,i} \,, \label{eqn:ddt_h2so4_gas} \\ % \frac{dq_{m,SP,i}}{dt} &= 0, \quad \text{where}\,\, SP \neq SO_4, \,\, i = 1,...,I, \label{eqn:ddt_other_mass}\\ \frac{dq_{n,i}}{dt} & = 0, \quad i = 1,...,I \label{eqn:ddt_number} \end{align} \]

Here \(I\) is the number of modes of interstitial aerosol. The expression of \(C_{H_2SO_4,i}\) can be found on a different page.