## 1. Introduction

The existence of an aerosol layer in the stratosphere was first described by Junge et al. (1961). The frequent large volcanic eruptions of the last several decades have also directed attention toward the stratospheric aerosol, which can be considered to be composed of a background component that is perturbed from time to time by these eruptions. Since the lifetime of the stratospheric aerosol is much longer than that of the tropospheric aerosol, the net change in radiative forcing following a large volcanic eruption is mostly due to the stratospheric aerosol.

The stratospheric aerosol depends on nucleation, heteromolecular condensation of water vapor and H_{2}SO_{4} vapor, coagulation, and sedimentation. In principle, the standard chemical transport model for trace gases can be applied to stratospheric aerosols. However, unlike trace gases, aerosols undergo a production and growth process in the stratosphere with the radii of the particles generally increasing with time. The radius of a stratospheric aerosol particle controls its sedimentation rate, and a physical description of the growth process is required in order to treat production, depletion, and transport processes properly. The size distribution is particularly important for radiative transfer calculations, since the radiative forcing can be very different for aerosols with the same mixing ratio but with different size distributions.

Theoretical work to explain the production and depletion process of stratospheric aerosols can be traced back to Junge et al. (1961). Hamill et al. (1977a,b) and Yue and Deepak (1979) analyze the nucleation, growth, and coagulation processes of the stratospheric aerosol, and Rosen et al. (1978) discuss a simple analytical model. Turco et al. (1979) and Toon et al. (1979) use a one-dimensional numerical model to simulate the production and depletion of stratospheric aerosols. In their work the nucleation, condensation, coagulation, and evolution of aerosol size are all related in a solution that shows reasonable agreement with observations.

Two-dimensional models for the stratospheric aerosol are discussed by Pitari et al. (1993) and Weisenstein et al. (1997). The physical principles embodied in the two-dimensional models are similar to those used by Turco et al. (1979). The aerosol evolution following the large Pinatubo eruption (Russell et al. 1993) is successfully simulated in the two-dimensional model (Bekki and Pyle 1994; Tie et al. 1994; Weisenstein et al. 1997).

Although the stratospheric aerosol has been studied using one- and two-dimensional numerical models, the physical relationships between nucleation, growth, and sedimentation processes are not explicitly revealed in the numerical solutions. We approach the problem through the four-dimensional continuity equation for the aerosol where the extra dimension, beyond the usual three, is related to the change of particle size and hence the physical processes of nucleation, condensation, sedimentation, and background air motion. The effects of turbulent diffusion are parameterized by partitioning the physical quantities into mean and eddy components and relating the diffusive effect of the eddies to gradients in the means. The four-dimensional continuity equation unifies transport and other physical processes for the aerosol in a single equation.

In the radius region beyond which coagulation is important, an analytical solution is obtained for the two-dimensional continuity equation (reduced from the four-dimensional continuity equation by omitting horizontal spatial variation) that demonstrates how particle nucleation, growth, and sedimentation are related through the characteristic curve of the partial differential equation. The results illustrate which factors determine the aerosol size distribution for different radius ranges.

## 2. The continuity equation for the aerosol

*n*(

*r,*

**x**,

*t*) is the size distribution of the aerosol, which is a function of the radius

*r*of the particle, the spatial location

**x**, and time

*t.*The aerosol growth rate,

*g*=

*g*

_{c}+

*g*

_{a}, due to condensation and coagulation, is discussed in the following section. The air velocity is

**V**, and

*q*(

*r,*

**x**,

*t*) is the nucleation rate that is the source for the size distribution. Integrating (1) with respect to

*r*, with boundary conditions

*n*(0,

**x**,

*t*) =

*n*(∞,

**x**,

*t*) = 0, giveswhereis the number concentration of the aerosol particle, andis the number concentration source (nucleation rate). Equation (5) is the well-known three-dimensional continuity equation for number concentration

*N*

_{0}. The mass of aerosol particles is not conservative in the four-dimensional continuity equation, since sulfuric vapor and water vapor are being continuously transformed into particle mass. Since the nucleation and growth processes compete for sulfuric vapor and water vapor simultaneously, the nucleation–growth rate will be limited by the amount of vapor available.

*w*

_{z}is the background vertical air velocity (assumed constant), and

*w*

_{r}the aerosol settling velocity.

### a. Microphysical processes for stratospheric aerosols

*Q*

*πr*

^{2}

*β*

_{A}

*N*

_{B}

*G*

*kT*

*r** is the critical radius that the embryonic droplet requires to be activated,

*r** is very small (about 0.005

*μ*m),

*N*

_{B}is the concentration of H

_{2}O gaseous molecules,

*β*

_{A}is the rate at which sulfuric acid molecules impinge upon a critically sized embryo, and

*G*

^{*}

_{p}

_{2}SO

_{4}vapor pressure and water vapor pressure, and it is sensitive to the temperature. Recent calculations show that the maximum nucleation rate occurs at the tropopause, where temperature has its minimum (Weisenstein et al. 1997; Hamill et al. 1997). Our interest is in the physics of the evolution of the stratospheric aerosol, and we do not calculate the nucleation rate but rather adopt the H

_{2}SO

_{4}vapor profile and nucleation rate of Weisenstein et al. (1997). This vertical profile of nucleation rate is similar to that of Hamill et al. (1997), but with a smaller peak value at about 16 km. The Hamill et al. (1997) calculation is consistent with the measurement results of Brock et al. (1995), although it is emphasized that the nucleation rate is subject to considerable uncertainty.

*g*

_{c}for the stratospheric aerosol by heteromolecular condensation iswhere

*r*is the radius of the aerosol particle;

*K*

_{n}is the Knudsen number

*K*

_{n}=

*l*

_{eff}/

*r,*where

*l*

_{eff}is the mean free path for the sulfuric molecules in air;

*D*

_{m}is the molecular diffusion coefficient;

*P*

_{a}is the H

_{2}SO

_{4}vapor pressure;

*P*

^{0}

_{a}

_{2}SO

_{4};

*λ*= (1.333

*K*

_{n}+ 0.71)/(

*K*

_{n}+ 1);

*V*

*χ*is the fraction of the sulfuric molecule number in the drop with

*χ*=

*n*

_{a}/(

*n*

_{a}+

*n*

_{b}), where

*n*

_{a}and

*n*

_{b}are the number of acid (

*a*) and water (

*b*) molecules in the droplet. We use the formula of Steele and Hamill (1981) to obtain the aerosol composition, which is based on the ambient water vapor pressure and temperature. We obtain the value of

*P*

^{0}

_{a}

_{2}SO

_{4}vapor pressure

*P*

_{a}from Weisenstein et al. (1997).

*g*

_{a}is the growth rate due to coagulation, and

*K*(

*υ, u*) is the coagulation kernel between particles of volume

*υ*and

*u.*The kernel of Fuchs (1964) is commonly used. Coagulation is important when there is a high number concentration of particles. For the stratospheric aerosol this is the case for particles just after nucleation, and coagulation is a dominant factor at this stage but becomes less important as particle radius grows.

### b. Particle velocity

*ρ*is the density of the particle;

*G*is the acceleration due to gravity;

*ν*is air viscosity;

*B*′ is a correction factor obtained from measurements (Kasten 1968),

*B*

*K*

_{n}

*K*

_{n}

*e*

^{−0.87/(Kn)}

*K*

_{n}=

*l*

_{eff}/

*r*is the Knudsen number mentioned previously. Since

*l*

_{eff}∝ 1/

*N*∝ 1/

*P,*where

*N*is number concentration of air molecules and

*P*is the pressure, we write

*l*

_{eff}=

*e*

^{z/zl}

*l*

_{o}

*l*

_{o}is the mean free path at the surface and

*z*

_{l}is the scale height taken here to be

*z*

_{l}= 7.6 km. In the stratosphere

*l*

_{eff}is always much larger than the radius of the stratospheric aerosol, and the exponential term in (10) is expanded to give

*w*

_{r}

*w*

_{1}

*r*

^{2}

*w*

_{2}

*re*

^{z/zl}

*w*

_{1}= 0.0744

*ρG*/

*ν, w*

_{2}= 0.371

*ρGl*

_{0}/

*ν,*and 0.7 is used instead of 1 as the first term on the right-hand side of (10) in order to obtain a better fit. Figure 1 compares

*w*

_{r}obtained from (9) and (11) and shows that the difference is small.

## 3. The equation for the steady state

Equations (13)–(14) apply over the complete radius range including that part of the range where the complicated effect of coagulation in the growth term *g* = *g*_{c} + *g*_{a} is important. In this complex situation analytic solutions to (14) are not available.

*r*< 0.02

*μ*m (Hamill et al. 1977b). If we restrict attention to that part of the radius range

*r*> 0.02

*μ*m that is dominated by condensation, then

*g*≈

*g*

_{c}. The growth term for condensation can be written aswhere

*α*=

*V*

*D*

_{m}(

*P*

_{a}−

*P*

^{0}

_{a}

*kTχ,*and

*β*=

*λl*

_{eff}. The correction factor

*λ*is close to unity, and in the stratosphere,

*l*

_{eff}is one to two orders of magnitude larger than the aerosol radius. We may therefore neglect

*r*in (12). There is a factor of

*l*

_{eff}in

*D*

_{m}(Kennard 1938), which cancels with that in

*β.*We use

*r*= 0.45 (

*μ*m) in the correction factor, which causes error of about <5%.

*ŵ*

_{z}=

*w*

_{z}/(

*g*

_{c}

*z*

_{l}),

*ŵ*

_{1}=

*w*

_{1}(

*g*

_{c}

*z*

_{l}), and

*ŵ*

_{2}=

*w*

_{2}/(

*g*

_{c}

*z*

_{l}) related, respectively, to the scaled vertical motion and the two terms in the particle fall velocity.

### a. The characteristic curve relating particle position and radius

*ŵ*are constant as

*e*

^{−[z/zl+ζ(r)]}

*η*

*r*

*C,*

*η*(

*r*) =

*ŵ*

_{2}∫

^{r}

*r*′

*e*

^{−ζ(r′)}

*dr*′,

*ζ*(

*r*) = −

*ŵ*

_{z}

*r*+ ⅓

*ŵ*

_{1}

*r*

^{3}, and

*C*is a constant of integration. For an initial radius and position (

*r*

_{m},

*z*

_{m}) for the particle,

*C*= exp{−[

*z*

_{m}/

*z*

_{l}+

*ζ*(

*r*

_{m})]} −

*η*(

*r*

_{m}), and

*e*

^{−[z/zl+ζ(r)]}

*e*

^{−[zm/zl+ζ(rm)]}

*η*

*r*

*η*

*r*

_{m}

*η*

*r, z*) as the particle grows by diffusion while falling through the atmosphere.

### b. Number density

*e*

^{z/zl}

*D*is another constant of integration.

*H*as a function height for a given radius

*r*

_{m},

*H*

*z*

_{m}

*n*

*r*

_{m}

*z*

_{m}

*r, z*) and (

*r*

_{m},

*z*

_{m}) are two points on the characteristic curve.

### c. Application to the stratospheric aerosol

*r*>

*r*

_{m}= 0.02

*μ*m (Hamill et al. 1977b). As this radius range is beyond the critical activation radius for the aerosol, the source

*q*is zero and we finally arrive at solutions of the form

*e*

^{−[z/zl+ζ(r)]}−

*e*

^{−[zm/zl+ζ(rm)]}= Δ

*η*

*n*(

*r, z*) =

*e*

^{−[z/zl+ζ(r)]}

*e*

^{[zm/zl+ζ(rm)]}

*H*(

*z*

_{m}) for

*r*≥

*r*

_{m}.

*p*=

*e*

^{−z/zl}

*h*(

*p*

_{m}) =

*n*(

*r*

_{m},

*z*

_{m}) =

*H*(

*z*

_{m}), to give

*pe*

^{−ζ(r)}

*p*

_{m}

*e*

^{−ζ(rm)}

*η*

*p*

_{m}

*e*

^{−ζ(rm)}

*p*−

*e*

^{ζ(r)}Δ

*η*)

*e*

^{−ζ(r)}= (

*p*− Δ

*p*)

*e*

^{−ζ(r)}. According to (27), the size distribution

*n*(

*r, p*) is obtained from the reference profile

*h*(

*p**) at the pressure

*p** = (

*p*− Δ

*p*)

*e*

^{−Δζ}scaled by the factor

*p*/(

*p*− Δ

*p*). Equation (27) demonstrates the growth of the particles as they fall through the atmosphere from pressure

*p** to

*p.*Sinceand Δ

*p*=

*e*

^{ζ(r)}Δ

*η*are positive, the scaling factor

*p*/(

*p*− Δ

*p*) is greater than unity implying the growth of size distribution as the particles fall through the atmosphere. The size distribution at

*p*is determined by the reference profile at

*p**, where

*p*=

*p**

*e*

^{Δζ}+ Δ

*p*; and generally

*p** <

*p*so the particles fall from some higher level. Since Δ

*ζ*= −

*ŵ*

_{z}(

*r*−

*r*

_{m}) + ⅓

*ŵ*

_{1}(

*r*

^{3}−

*r*

^{3}

_{m}

*p** >

*p*if the vertical velocity

*ŵ*

_{z}is positive and sufficiently large, thereby allowing the particles to be swept upward. Taken together, (25) and (26) explicitly relate the position, radius, and size distribution of the aerosol.

### d. The effects of diffusion

*w*

*w*

_{z}−

*w*

_{r}. The means represent resolved values of the variables; and the covariances, the effects of unresolved or subgrid scales on the resolved scales. These are parameterized in terms of resolved quantities in the usual way through mean field theory aswhere

*D*=

*D*

_{m}+

*D*

_{e}is the sum of the molecular diffusivity and eddy diffusivity. The molecular diffusion coefficient has been used in the preceding section (Kennard 1938). We take the vertical eddy diffusion coefficient from the results of Garcia and Solomon (1983) for the stratosphere and adjust eddy diffusion coefficients in the troposphere following Dickenson and Chang (Turco et al. 1979). The covariance

*g*′

*n*′

*w*

_{e}=

*w*

_{z}− (

*D*/

*n*

*z*)

*n*

*D*/

*n*

*z*)

*n*

*w*

_{e}=

*w*

_{z}+ |(

*D*/

*n*

*z*)

*n*

## 4. Results

The discussion in the previous section indicates how the size distribution depends on condensation, particle settling velocity, diffusion, and background air velocity. Here we evaluate and discuss solutions for the characteristic curve and for number density.

### a. The characteristic curve

The analytic solution for the characteristic curve (20) is an approximation based on a constant average value of *g*_{c}. A more detailed approach treats *g*_{c} as a function of height, by including, for example, the change in the weight percentage of liquid H_{2}SO_{4} in the aerosol particle and the sulfuric vapor pressure with height. Under these conditions (20) is used in calculation of the characteristic curve by applying it piecewise in the form exp{−[*z*_{i+1}/*z*_{l} + *ζ*(*r*_{i+1})] − exp{−[*z*_{i}/*z*_{l} + *ζ*(*r*_{i})]} = [*η*(*r*_{i+1}) − *η*(*r*_{i})], where (*r*_{i}, *z*_{i}) and (*r*_{i+1}, *z*_{i+1}) are neighboring points along the characteristic curve, and where *g*_{c} varies from point to point.

Figure 2 shows the characteristic curves for initial radius *r*_{m} = 0.02 *μ*m and position values *z*_{m} = (20, 25, 30, 35) km for three cases of background vertical air motion, namely, *w*_{z} = (0, 10^{−4}, −10^{−4}) m s^{−1}. The characteristic curves in Fig. 2 trace out the particle trajectories in (*r, z*) space. For the case of no background vertical air velocity (Fig. 2a), the particles grow as they descend. Particles that start their journey at higher altitudes have time to grow to greater radius before passing out of the region. The curvature of the trajectories reflects the varying fall speed of the particles as a function of radius and, to a lesser extent, variations in the condensation growth rate. The curves are steepest for larger particles, since the fall speed increases rapidly with the increase in radius (Fig. 1). For upward background air velocity (Fig. 2b), small particles in the lower part of the region may have fall speeds that are less than the upward air velocity and so are swept upward initially until they grow heavy enough so that they begin to fall. Small particles in the upper part of the region are less likely to be swept upward since their settling velocity is large due to the lesser air density there. Particles that begin with the same radius *r*_{m} at different heights in the atmosphere follow trajectories that tend to produce similar radii at the bottom of the region. Downward air velocity (Fig. 2c) accelerates the particles’ fall so they have less time to grow. The particles reach the bottom of the region with smaller radii than for the case of no or upward vertical motion. As discussed in section 3d, the effect of diffusion is expected to be similar to that of vertical upward motion.

### b. Reference profiles

The coagulation process is important in the range *r** < *r* < 0.02 *μ*m, where *r** is the activation radius for the aerosol. According to Weisenstein et al. (1997), the size distribution is neither a lognormal nor a gamma distribution. This size range contributes very little to the net radiative forcing due to the aerosols, however. Turco et al. (1979) calculate the size distribution in two size ranges; the range *r* < 0.02 *μ*m includes a source term while the range *r* > 0.02 *μ*m does not. The size distribution from the first calculation provides the boundary condition for the second calculation. This method is also used in the recent two-dimensional model of Pitari et al. (1993). We follow this general approach to calculate the reference profile *n*(*r*_{m}, *z*_{m}) = *H*(*z*_{m}), which is used in the range *r* > 0.02 *μ*m, where the nucleation source and growth by coagulation are no longer important. The reference profile of size distribution at *n*(0.02 *μ*m, *z*_{m}) may be also obtained from observations.

*r*< 0.02

*μ*m the source function is written as

*q*(

*r, z*) =

*Q*(

*z*)

*δ*(

*r*−

*r**), where

*Q*(

*z*) is the nucleation rate profile. In the numerical calculation, the

*δ*function is approximated aswhere Δ

*r*is the bin interval. We then solve (30) numerically to obtain

*n*(

*r*

_{m},

*z*

_{m}). Here,

*g*

*w*

_{r}are evaluated using (2) and (9) without further approximation. Tropical temperature and water vapor profiles are used. The H

_{2}SO

_{4}vapor and nucleation rate profiles are the same as those of Weisenstein et al. (1997). The saturation sulfuric acid vapor value are calculated based on Gmitro and Vermeulen (1964).

As usual, the radius range is divided into a number of bins. These bins are spaced geometrically with *r*_{i+1} = *τr*_{i} = *τ*^{i+1}*r**, and we use *τ* = 2. The grid interval for *z* is 100 m, and the vertical domain is from the surface to 38 km, close to that of Weisenstein et al. (1997). The numerical treatment of coagulation follows Yue and Deepak (1979).

Figure 3 shows the vertical profile of *n*(*r*_{m}, *z*_{m}) = *H*(*z*_{m}) for *r* = *r*_{m} = 0.02 *μ*m. This serves as the boundary condition for the second calculation. The result is not sensitive to resolution in *r.* Figure 3 also shows the results for background vertical velocities of *w*_{z} = ±10^{−4} m s^{−1} and indicates that the size distribution at *r*_{m} = 0.02 *μ*m is sensitive to vertical velocity in the lower stratosphere.

### c. Equilibrium solutions

*H*(

*z*

_{m}) in Fig. 3, and with

*q*= 0, (23) becomes

*n*

*r, z*

*e*

^{−[z/zl+ζ(r)]}

*e*

^{[zm/zl+ζ(rm)]}

*H*

*z*

_{m}

*w*

_{z}= 0 in the absence of diffusion. The size distribution is plotted as

*dN*/log

*r*=

*nr*for

*N*=

^{r}

_{0}

*n dr.*The size distributions of Fig. 4a show the characteristic increase with

*r*to a maximum followed by a rapid decline. Compared to the results of Weisenstein et al. (1997) (their Fig. 9a), the curves in Fig. 4a decline too rapidly for

*r*> 0.1

*μ*m, especially for larger values of

*r,*at 18 and 25 km. The size distributions do not attain a radius of 1

*μ*m, which is the expected maximum value for stratospheric aerosols. In conjunction with the lack of particles in the larger size ranges, there is an excess of particles in the smaller size ranges. In the absence of diffusion, the particles apparently sweep through the region too rapidly to grow to the expected sizes.

*n*(

*r, z*) is contoured in Fig. 5a, where all the physical quantities are the same as those for Fig. 4a. Figure 5a indicates how the particles from higher levels increase their size as they fall to lower levels and as they are joined by new particles from the small radius boundary. Figures 5b,c give the corresponding aerosol size distributions when background vertical velocities

*w*

_{z}= ±10

^{−4}m s

^{−1}are included. The expected effect of background vertical motion is inferred from (32) asThe first term shows that the size distribution will broaden for upward motions and narrow for downward motions, and this is generally the case in Figs. 5b,c. When background air motion is upward, the particles take a longer time to traverse the region and grow to larger radius, and the size distribution is broadened. Similarly, when the background air motion is downward, the size distribution is narrowed. The second term in (33) depends on the change in the boundary profile

*H*with background vertical motion and can become important as seen in lower regions of Fig. 5b. According to Fig. 3,

*δH*/

*δŵ*

_{z}is negative in this region, and this acts to decrease the value of

*n.*The smaller number densities in Fig. 3 are swept upward into the region by this background velocity and produce a bimodal distribution.

### d. The effect of diffusion

Figure 4b displays the results of the numerical solution of (30), including diffusive effects, for *r* > *r*_{m}, *q**w*_{z} = 0, and with the other parameters the same as for Fig. 3. A finer resolution *τ* ≡ [bu1002]*r* > *r*_{m}, the size distribution is not very different if coagulation is included. Slight differences are noticeable at 18 km, where the particle number concentration is comparatively high. Therefore, at least for these calculations, coagulation can be ignored at the second stage of calculation. In these calculations the size distribution of Fig. 4b, which includes the effects of turbulent diffusion, is much broader than that of Fig. 4a, which does not. The results of Fig. 4b are much closer to the results of Weisenstein et al. (1997) (their Fig. 9a; model A).

The corresponding size distributions including diffusive effects are shown in Fig. 6. According to the discussion of section 3d, the expected effect of diffusion is generally to broaden the size distribution. Figure 6 shows how the maximum of *n* has been eroded but the distribution of *n* broadened, especially for the larger values of *r,* by including diffusion. A comparison of the size distribution of Fig. 5a with that of Fig. 6a shows that the number of particles with a given radius generally decreases at levels as the particles are diffused from lower to higher heights. The particles have more time to grow and attain larger radii and so increase the number of particles with large *r.* The net effect is to reduce the number of smaller particles and to increase the number of larger particles.

Figures 6b,c give numerical results for the calculation of size distribution when background vertical velocities of *w*_{z} = ±10^{−4} m s^{−1} are present in addition to diffusion. These results may be compared to those of Figs. 5b,c, where diffusion is absent. In both cases, upward motion broadens the size distribution while downward motion narrows it slightly. Both upward motion and diffusion (Figs. 5b and 6b) act to reduce the size distribution for smaller particles in the lower regions since both effects act to transfer particles upward. Since upward stratospheric air motion generally occurs in the tropical region, while downward stratospheric air motion occurs at higher latitudes, this would tend to support a broader size distribution of the stratospheric background aerosol in the Tropics.

Upward background air motion can result in a bimodal number distribution at a given height where the number density decreases with radius and then increases again before decreasing once more [as indicated by the curve of *n*(*r, z*) = 1 cm^{−3} *μ*m^{−1}, e.g. ]. For no or downward air motion, the number density is unimodal and decreases with the radius.

### e. Comparison with observations

Figure 7 shows aerosol concentrations for *r* > 0.15, 0.25, 0.5 *μ*m for the three background vertical velocities. They show reasonable agreement with the observations of Deshler et al. (1993, their Fig. 1) particularly for the case of downward background air motion, which would be consistent with the expected downward air motion at Laramie, Wyoming (41°N). Water vapor and sulfuric vapor profiles used in the calculation may differ from those of the observational conditions. In the calculation, the aerosol concentration decreases rapidly above 34–35 km due to evaporation (*P*_{a} − *P*^{0}_{a}

Both analytical and numerical solutions show that upward vertical air motion broadens the aerosol size distribution and that downward vertical air motion narrows it. One implication is that the size distribution of the stratospheric aerosol in the Tropics should be wider than the distribution in the subtropical region, and this is consistent with the observations by Grainger et al. (1995) and Hervig and Deshler (1998).

## 5. Summary and discussion

The intent of this study is 1) to derive the governing four-dimensional continuity equation for the growth of the stratosphere aerosol as a basis for representing the physical processes that govern the aerosol in a unified way; 2) to obtain approximate analytic solutions to the equation to illustrate the interplay of the physical processes involved; 3) to obtain numerical solutions to the equation; and 4) to compare the numerical results to the analytic results and to similar numerical results already obtained (e.g., Turco et al. 1979; Pitari et al. 1993;Weisenstein et al. 1997). For the radius range in which coagulation is unimportant, the analytical solution of the two-dimensional continuity equation illustrates how the growth rate, settling distance, and background air motion determine the equilibrium size distribution. The particle growth through condensation and motion through sedimentation are related to the characteristic curves of the two-dimensional continuity equation where the size distribution at a particular radius and height (*r, z*) is related to the size distribution at smaller radius and at a higher height.

The size distribution obtained in this way is, however, too narrow in comparison with the observations, which indicates that the particles fall too quickly and leave the stratosphere before they have a chance to grow to the expected size. The inclusion of diffusive effects broadens the size distribution and brings it into reasonable agreement with observations. The particles are diffused upward and grow to larger sizes. In this regard diffusion acts in the same manner as an upward background air velocity. Diffusion also affects the characteristic curve relating particle radius and position.

The turbulence decomposition of the terms in the four-dimensional continuity equation gives, at least formally, the correlation term *g*′*n*′*g* = *g*_{a} + *g*_{c} ≈ *g*_{c} as approximately constant for the radius range where condensation is the dominant process so that *g*′ would be zero there. However, such a term could be nonzero for smaller particle sizes and act as a further “internal diffusion” term in the equation for the stratospheric aerosol and perhaps also for cloud droplets.

We would like to thank Drs. Hamill and Weisenstein for providing us the profiles of nucleation rate and H_{2}SO_{4} vapor pressure. We are grateful to Drs. K. Abella and M. Holzer and two anonymous reviewers for their helpful comments.

## REFERENCES

Bekki, S., and J. A. Pyle, 1994: A two-dimensional modeling study of the volcanic eruption of Mount Pinatubo.

*J. Geophys. Res.,***99,**18 861–18 869.Brock, C. A., P. Hamill, J. C. Wilson, H. H. Jonsson, and K. R. Chan, 1995: Particle formation in the upper tropical troposphere: A source of nuclei for the stratospheric aerosol.

*Science,***270,**1650–1653.Deshler, T., B. J. Johnson, and W. R. Rozier, 1993: Balloonborne measurements of Pinatubo aerosol during 1991 and 1992 at 41°N: Vertical profiles, size distribution, and volatility.

*Geophys. Res. Lett.,***20,**1435–1438.Fuchs, N. A., 1964:

*The Mechanics of Aerosols.*Pergamon Press, 408 pp.Garcia, R., and S. Solomon, 1983: A numerical model of the zonally averaged dynamical and chemical structure of the middle atmosphere.

*J. Geophys. Res.,***88,**1379–1400.Gmitro, J. I., and T. Vermeulen, 1964: Vapor–liquid equilibria for aqueous sulfuric acid.

*AIChE J.,***10,**740–746.Grainger, R. G., A. Lambert, C. D. Rodgers, F. W. Taylor, and T. Deshler, 1995: Stratospheric aerosol effective radius, surface area, and volume estimated from infrared measurements.

*J. Geophys. Res.,***100,**16 507–16 518.Hamill, P., C. S. King, and R. D. Cadle, 1977a: The nucleation of H2SO4–H2O solution aerosol particles in the stratosphere.

*J. Atmos. Sci.,***34,**150–162.——, O. B. Toon, and C. S. King, 1977b: Microscopical processes affecting stratospheric aerosol particles.

*J. Atmos. Sci.,***34,**1104–1119.——, E. J. Jensen, P. B. Russell, and J. J. Mauman, 1997: The life cycle of stratospheric aerosol properties.

*Bull. Amer. Meteor. Soc.,***78,**1–6.Hervig, M. E., and T. Deshler, 1998: Stratospheric aerosol surface area and volume inferred from HALOE, CLAES, and ILAS measurements.

*J. Geophys. Res.,***103,**25 345–25 352.Hildebrand, F. B., 1966:

*Advanced Calculus for Applications.*Prentice-Hall, 646 pp.Junge, C. E., C. W. Chagnon, and J. E. Manson, 1961: Stratospheric aerosols.

*J. Meteor.,***18,**81–108.Kasten, F., 1968: Falling speed of aerosol particles.

*J. Appl. Meteor.,***7,**944–947.Kennard, K. H., 1938:

*Kinetic Theory of Gases.*McGraw-Hill, 483 pp.Pitari, G., V. Rizi, L. Ricciardulli, and G. Visconti, 1993: High-speed civil transport impact: Role of sulfate, nitric acid trihydrate, and ice aerosols studied with a two-dimensional model including aerosol physics.

*J. Geophys. Res.,***98,**23 141–23 164.Rosen, J. M., D. J. Hofman, and P. Singh, 1978: A steady-state stratospheric aerosol model.

*J. Atmos. Sci.,***35,**1304–1313.Russell, P. B., and Coauthors, 1993: Pinatubo and pre-Pinatubo optical depth spectra: Mauna Loa measurements comparisons, inferred particle size distribution, radiative effects, and relationships to lidar data.

*J. Geophys. Res.,***98,**22 969–22 985.Steele, H. M., and P. Hamill, 1981: Effects of temperature and humidity on the growth and optical properties of sulfuric acid–water droplets in the stratosphere.

*J. Aerosol Sci.,***12,**517–528.Tie, X., G. Brasseur, and C. Granier, 1994: Two-dimensional simulation of Pinatubo aerosol and its effect on stratospheric ozone.

*J. Geophys. Res.,***99,**22 545–22 562.Toon, O. B., R. P. Turco, P. Hamill, C. S. Kiang, and R. C. Whitten, 1979: A one-dimensional model describing aerosol formation and evolution in the stratosphere: II. Sensitivity studies and comparison with observations.

*J. Atmos. Sci.,***36,**718–736.Turco, R. P., P. Hamill, O. B. Toon, R. C. Whitten, and C. S. Kiang, 1979: A one-dimensional model describing aerosol formation and evolution in the stratosphere: I. Physical processes and mathematical analogs.

*J. Atmos. Sci.,***36,**699–717.Weisenstein, D. K., G. K. Yue, M. K. W. Ko, N. D. Sze, J. M. Rodriguez, and C. J. Scott, 1997: A two-dimensional model of sulfur species and aerosols.

*J. Geophys. Res.,***102,**13 019–13 035.Yue, G. K., and A. Deepak, 1979: Modeling of coagulation–sedimentation effects on transmission of visible IR laser beams in aerosol media.

*Appl. Opt.,***18,**3918–3925.

# APPENDIX

## Derivation of the Four-Dimensional Continuity Equation

*δ*

*δx δy δz.*The number of particles in this elemental volume, in the radius range

*δr,*is

*δN*

*n δr δx δy δz,*

*n*(

*r,*

**x**,

*t*) is the size distribution. Differentiating with respect to time (represented by a dot) giveswhere

*q*is the source term. This becomeswhere (

*ẋ, ẏ, ż, ṙ*) = (

*u, υ, w, g*). Using the total derivativethe four-dimensional continuity equation for

*n*is written as