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Time integration for the condensation of H\(_2\)SO\(_4\)¶

The time evolution equations presented in an earlier section describe only the condensation process. Experiences have shown that in global climate models using time steps on the order of minutes to hours, a sequential splitting between the sources and sinks of sulfric acid gas can lead to large numerical errors in the simulated tracer concentrations and process rates (Wan et al., 2013; Wang et al., 2019). Therefore, the MAM4 code embedded in EAMv1 and v2 employes a tighter numerical coupling between the production and condensation of H\(_2\)SO\(_4\) gas. The ODEs being solved in the H\(_2\)SO\(_4\) condensation parameterization are

\[ \begin{align} \frac{dq_{m,SO_4,i}}{dt} &= q_{v,H_2SO_4} C_{H_2SO_4,i}, \quad i = 1,...,I \,, \\ \frac{dq_{v,H_2SO_4}}{dt} &= \mathcal{P_{\rm chem}} - q_{v,H_2SO_4} \sum_{i=1}^{I} C_{H_2SO_4,i} \,, \\ % \frac{dq_{m,SP,i}}{dt} &= 0, \quad \text{where}\,\, SP \neq SO_4, \,\, i = 1,...,I, \\ \frac{dq_{n,i}}{dt} & = 0, \quad i = 1,...,I \end{align} \]

These equations are identical to those presentd in the earlier section except for the addtion of the gas-phase chemical production term \(\mathcal{P_{\rm chem}}\) to the second equation.

Notation¶

Let us use \(\Delta t\) to denote the overall time step size used by the aerosol chemistry and microphysics parameterizations. In other words, each parameterization is calculated for a time period of \(\Delta t\) before exchanging information with other parameterizations. (Individual parameterizations might use smaller time steps implemented as sub-steps relative to \(\Delta t\).)

MAM4 calculates gas-phase chemistry before aerosol microphysics. We use the following notation to denote the mixing ratio of H\(_2\)SO\(_4\) at different locations of the time integration loop:

\(q_{v,H_2SO_4}^n\): at time \(t = n\Delta t\) before gas-phase chemistry,
\(q_{v,H_2SO_4}^*\): after gas-phase chemistry,
\(q_{v,H_2SO_4}^{n+1}\): after H\(_2\)SO\(_4\) condensation.

The time-step mean chemical production rate is defined as

\[ \mathcal{P}_{\rm chem}^* = \dfrac{q_{v,H_2SO_4}^*-q_{v,H_2SO_4}^n}{\Delta t} \]

Linearized equations¶

The mass transfer coefficients \(C_{H_2SO_4,i} (i = 1,...,I)\) depends on aerosol particle size and hence will evolve as condensation happens. MAM4's numerical algorithm ignores this evolution within each time step and uses instead the coefficients evaluated at time \(t = n\Delta t\), denoted as \(C^n_{H_2SO_4,i}\) below. This approximation, combined with the use of time-step mean \(\mathcal{P}_{\rm chem}^*\), gives a linearized version of the ODE for the sulfric acid gas:

\[ \frac{dq_{v,H_2SO_4}}{dt} = \mathcal{P^*_{\rm chem}} - q_{v,H_2SO_4} \sum_{i=1}^{I} C^n_{H_2SO_4,i} \]

Numerical solution for sulfric acid gas¶

Assuming a positive chemical production rate, the numerical solution of the linearized sulfric acid gas equation is calculated as follows:

When the mass transfer coefficient summed over all modes has an appreciable magnitude, i.e.,

\[C^n_{H_2SO_4,total} \equiv \sum_{i=1}^{I} C^n_{H_2SO_4,i} > \dfrac{10^{-3}}{\Delta t }\]

use the analytical solution of the linearized equation:

\[ q_{v,H_2SO_4}^{n+1} = \left( q_{v,H_2SO_4}^{n}- \frac{\mathcal{P}^*_{\rm chem}} {C^n_{H_2SO_4,total}} \right) \exp\left( - \Delta t\,C^n_{H_2SO_4,total} \right) + \frac{\mathcal{P}^*_{\rm chem}} {C^n_{H_2SO_4,total}}\,. \]

For smaller but still non-zero mass transfer coefficients, i.e., \(\Delta t\,C^n_{H_2SO_4,total} \in [10^{-20}, 10^{-3}]\), approximate the exponentional function in the analytical solution above using the Taylor series to avoid large floating-point errors caused by the small denominators. This gives

\[ \begin{align} q_{v,H_2SO_4}^{n+1} &= q_{v,H_2SO_4}^{n} \left[ 1 - \Delta t\,C^n_{H_2SO_4,total} + \frac{\left(\Delta t\,C^n_{H_2SO_4,total}\right)^2}{2} \right] \\ &+ \Delta t\,\mathcal{P}^*_{\rm chem} \left[ 1 - \frac{\Delta t\,C^n_{H_2SO_4,total}}{2} + \frac{\left(\Delta t\,C^n_{H_2SO_4,total}\right)^2}{6} \right] \,.\\ \end{align} \]

When the mass transfer coefficient is smaller, i.e., \(\Delta t\,C^n_{H_2SO_4,total} < 10^{-20}\) , condensation is neglected.

\[ q_{v,H_2SO_4}^{n+1} = q_{v,H_2SO_4}^n + \Delta t\, \mathcal{P}^*_{\rm chem} \]

Numerical solution for sulfate¶

The condensed amount of sulfric acid gas is

\[ \Delta q_{v,H_2SO_4,{\rm cond}} = q_{v,H_2SO_4}^n + \Delta t\, \mathcal{P}^*_{\rm chem} - q_{v,H_2SO_4}^{n+1} \]

This amount is partitioned to different aersol modes and the updated sulfate mass mixing ratios are

\[ q_{m,H_2SO_4,i}^{n+1} = q_{m,H_2SO_4,i}^{n} + \Delta q_{v,H_2SO_4,{\rm cond}}\, \dfrac{C^n_{H_2SO_4,i}}{C^n_{H_2SO_4,total}} \]