## 1. Introduction

[2] Many studies to date have described numerical techniques for solving the differential coagulation equation [e.g., *Lushnikov*, 1975; *Turco et al.*, 1979; *Suck and Brock*, 1979; *Gelbard and Seinfeld*, 1980; *Tsang and Brock*, 1982; *Seigneur*, 1982; *Friedlander*, 1983; *Warren and Seinfeld*, 1985; *Tzivion et al.*, 1987; *Toon et al.*, 1988; *Strom et al.*, 1992; *Kim and Seinfeld*, 1992; *Jacobson et al.*, 1994; *Kostoglou and Karabelas*, 1994; *Jacobson and Turco*, 1995; *Binkowski and Shankar*, 1995; *Kumar and Ramkrishna*, 1996; *Fassi-Fihri et al.*, 1997; *Russell and Seinfeld*, 1998; *Trautmann and Wanner*, 1999; *Fernandez-Diaz et al.*, 2000; *Bott*, 2000; *Jeong and Choi*, 2001; *Sandu*, 2002]. Many studies have also examined solutions to condensational growth equations [e.g., *Mordy*, 1959; *Middleton and Brock*, 1976; *Gelbard and Seinfeld*, 1980; *Varoglu and Finn*, 1980; *Smolarkiewicz*, 1983; *Friedlander*, 1983; *Tsang and Brock*, 1986; *Whitby*, 1985; *Brock et al.*, 1986; *Tsang and Korgaonkar*, 1987; *Toon et al.*, 1988; *Rao and McMurry*, 1989; *Tsang and Huang*, 1990; *Gelbard*, 1990; *Pilinis*, 1990; *Kim and Seinfeld*, 1990, 1992; *Jacobson and Turco*, 1995; *Lister et al.*, 1995; *Binkowski and Shankar*, 1995; *Jacobson*, 1997c; *Kleeman et al.*, 1997; *Gelbard et al.*, 1998; *Chock and Winkler*, 2000; *Nguyen and Dabdub*, 2001; *Sandu and Borden*, 2001]. A third group of studies has examined solutions to equations for nonequilibrium dissolutional growth plus equilibrium reversible chemistry of electrolytes [*Jacobson et al.*, 1996; *Meng and Seinfeld*, 1996; *Jacobson*, 1997a, 1997b, 1997c; *Meng et al.*, 1998; *Sun and Wexler*, 1998a, 1998b; *Song and Carmichael*, 1999; *von Salzen and Schlünzen*, 1999; *Capaldo et al.*, 2000; *Pilinis et al.*, 2000].

[3] In most studies to date, one aerosol size distribution has been considered. In some studies, more than one distribution has been considered [e.g., *Toon et al.*, 1988; *Jacobson et al.*, 1994; *Fassi-Fihri et al.*, 1997; *Kleeman et al.*, 1997; *Russell and Seinfeld*, 1998; *Jacobson*, 2001]. One reason to treat multiple distributions and interactions among them is that real particles exist in a variety of mixing states [e.g., *Andreae et al.*, 1986; *Levin et al.*, 1996; *Murphy et al.*, 1998; *Pósfai et al.*, 1999; *Silva et al.*, 2000; *Guazzotti et al.*, 2001; *Okada and Hitzenberger*, 2001; *Naoe and Okada*, 2001]. Particles near their sources are generally externally mixed, whereas those far from their sources are often partially or completely internally mixed. When only one size distribution is treated in a model, all material is considered internally mixed at the emission source. Such a representation can distort predicted chemical compositions [e.g., *Kleeman et al.*, 1997]. When multiple distributions without interactions among them are treated, all distributions are treated as externally mixed. This is unrealistic away from source regions. When multiple distributions with interactions among them are treated, some distributions can be treated as externally mixed whereas others can be treated as partially or completely internally mixed. The radiative effects of externally and internally mixed particles differ. For example, the modeled global direct radiative forcing of black carbon (BC) may be about a factor of two higher when BC is treated as an internally mixed core surrounded by a shell compared with when it is treated as externally mixed [*Jacobson*, 2000]. Simulations of the global scale mixing state and resulting direct forcing of BC (which used the numerical techniques and size distributions treated here) suggest that both may fall between those of an external and internal mixture but closer to those of an internal mixture [*Jacobson*, 2001].

[4] In this paper, techniques for solving (1) coagulation, (2) nucleation/condensational growth simultaneously, and (3) coupled nonequilibrium dissolutional growth/equilibrium reversible chemistry of electrolytes, all between the gas phase and multiple aerosol size distributions, are given. All techniques conserve moles or volume (including between the gas and aerosol phases), all techniques except the reversible chemistry technique are noniterative, and all techniques are unconditionally stable (where a stable solution is defined as one in which the difference between the numerical and exact solution is bounded regardless of the time step or integration time [e.g., *Celia and Gray*, 1992, p. 216]). The techniques make use of a consistent treatment of particle number concentration, mole concentration, volume concentration, solution density, nonsolution density, and refractive index.

[5] In the following sections, the numerical techniques are described. The techniques are then analyzed in box model simulations to explore physical processes, chemical processes, and interactions affecting aerosol particle evolution. In one case, results are compared when a different method of treating the aerosol size distribution is considered. Equations for the calculation of particle real and imaginary refractive from compositions in multiple distributions are also given.