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Initial growth of new particles¶

Case categorization¶

After calculating the nucleation rate \(J^*\) and the properties of new particles (nuclei) using physics-based or empirical parameterizations (see, e.g., this page on binary nucleation and this page on nucleation in the PBL), the dry volume of a new particle, \(V_{d,nuc}\) (unit: \(\rm m^3\)), and the dry diameter \(D_{d,nuc}\) (unit: m) are diagnosed as follows.

\[ \begin{align} V_{d,nuc} &= \frac{n_{H_2SO_4} M_{w,SO_4}} {N_{\mathrm{A}} \rho_{H_2SO_4}} \,, \\ D_{d,nuc} &= \left( \frac{6 V_{d,nuc}}{\pi} \right)^{\frac{1}{3}} \,. \end{align} \]

This dry diameter is compared with the predefined lower and upper bounds of particle diameter (unit: m) of the smallest lognormal mode. In MAM4, these are the smallest and largest diameters of the Aitken mode given by

\[ \begin{align} D_{p,lo} &= \exp \left[ 0.67 \ln (8.7 \cdot 10^{-9}) + 0.33 \ln (26 \cdot 10^{-9}) \right] \,, \\ %% about 12.5 nm D_{p,hi} &= 52 \cdot 10^{-9} \,. \end{align} \]

If \(D_{d,nuc} > D_{p,lo}\), then the new particles are sufficiently large to be described by the lognormal modes. In this case, MAM will let \(J_{nuc} = J^*\) and return to the host model the tendencies given at the end of the section on notation and assumptions.
More often, we will find that \(D_{d,nuc} \leq D_{p,lo}\). The initial growth of the newly nucleated small particles is parameterized using a revised version of the method described in Kerminen and Kulmala (2002).

Note

Worth revisiting whether \(D_{p,lo}\)=8.7 nm makes more sense.

Assumptions for the initial growth¶

The parameterization of Kerminen and Kulmala (2002) use the following simplifying assumptions.

The only important sink for the newly formed nuclei is their coagulation with the pre-existing particles. That is, the self-coagulation of fresh nuclei is ignored.
The only important source of mass for the growth of nuclei is the condensation of chemical gases, and the condensation rate is assumed to be constant within one time step.
The population of pre-existing larger particles and the concentration of condensible vapors remain unchanged within one time step.
The condensable vapor species responsible for nuclei growth are non-volatile.

The apparent nucleation rate¶

The growth of fresh nuclei due to the condensation of sulfuric acid gas and the decrease of nuclei number concentration due to coagulation with pre-existing particles are taken into account to estimate the so-called apparent nucleation rate, i.e., the source term for the Aitken-mode particle number concentration:

\[ J_{nuc} = J^* \exp \left( \frac{\eta}{D_{nuc,fin}} - \frac{\eta}{D_{nuc,ini}} \right) \]

where \(D_{nuc,ini}\) and \(D_{nuc,fin}\) are the wet diameters (unit: nm) before and after nuclei growth,

\(D_{nuc,ini} = \max (2r^*, 1)\)

\(D_{nuc,fin} = 10^9 D_{p,lo} f_{v}^{\frac 13}\),

where \(f_{v}\) is the ratio of wet volume to dry volume of a particle estimated using a simple Kohler approximation for \(\rm NH_{4}HSO_{4}\) (This approximation follows the Kohler theory used for the aerosol water uptake in MAM but neglects the surface curvature effect. The composition is assumed as pure ammonium bisulfate with hygroscopicity = 0.56),

\[ f_{v} = 1 - \frac{0.56}{\ln (\min(\max(\rm RH,0.1),0.95))} \]

The quantity \(\eta\) in the expression of the apparent nucleation rate is given by

\[ \begin{align} \eta &= \frac{\gamma CS^\prime}{GR} \,. \\ \end{align} \]

with

\[ \begin{align} \gamma &= 0.23 D_{nuc,ini}^{0.2} (\frac{D_{nuc,fin}}{3})^{0.075} (\frac{\rho_{nuc}}{1000})^{-0.33} (\frac{T}{293})^{-0.75} \,. \label{eq:gamma} \\ \end{align} \]

Note that this expression of \(\gamma\) has omitted one factor in Eq. (22) of Kerminen and Kulmala (2002) that involves the number mean diameter of the pre-existing particle population. That factor is assumed to be close to 1.

\(\rho_{nuc}\) is the density of nuclei (unit: kg per \(\rm m^3\) of nuclei) after considering the aerosol water uptake by sulfate,

\[ \rho_{nuc} = \frac{\rho_{H_2SO_4}}{f_{v}} \,.\]

\(GR\) is the nuclei growth rate (unit: \(\rm m~s^{-1}\)) estimated as

\[ \begin{align} GR &= \frac{3.0 \cdot 10^{-9} v_{H_2SO_4} M_{w,H_2SO_4} c_{n,H_2SO_4}} {\rho_{nuc}} \,. \\ \end{align} \]

\(v_{H_2SO_4}\) is the approximated mean molecular speed of H\(_2\)SO\(_4\) gas (unit: \(\rm m~s^{-1}\)),

\[ \begin{align} v_{H_2SO_4} &= 14.7 \sqrt{T} \,, \\ \end{align} \]

\(CS^\prime\) in the expression of \(\gamma\) is referred to as the condensation sink in Kerminen and Kulmala (2002). The calculation of \(CS^\prime\) in MAM uses the H\(_2\)SO\(_4\) condensation coefficient (\(C_{sum,H_2SO_4}\) unit: \(s^{-1}\)) from the gas-aerosol exchange parameterization, and is different from Kerminen and Kulmala (2002):

\[ CS^\prime = \frac{C_{sum,H_2SO_4}} {4 \pi \mathbb{D}_{g,H_2SO_4} \alpha_{H_2SO_4}} \,, \label{eq:cs_prime} \\ \]

\(\mathbb{D}_{g,H_2SO_4}\) is the approximated gas diffusivity of H\(_2\)SO\(_4\) gas (unit: \(\rm m^2~s^{-1}\)),

\[ \mathbb{D}_{g,H_2SO_4} = \frac{6.7037 \cdot 10^{-6} T^{0.75}} {c_{air}} \,. \\ \]