The modal method

Size distribution functions

Using the terminology in Raabe (1971), the linear form of a probability density function (PDF, unit: fraction per unit size interval) for the log-normal particle size distribution of mode \(i\) can be written as

\[ n_{norm,i}(D_{p}) = \frac{1}{\sqrt{2\pi} D_{p} \ln \sigma_{g,i}} \ \exp \left [ -\frac{(\ln D_{p} - \ln D_{gn,i})^2} {2\ln^2 \sigma_{g,i}} \right] \]

where \(D_{p}\) is the particle diameter (unit: m).

The corresponding PDF in logarithmic form (unit: fraction per unit interval of the logarithm of size) is

\[ n_{norm,i}(\ln D_{p}) = \frac{1}{\sqrt{2\pi}\ln \sigma_{g,i}} \ \exp \left [ -\frac{(\ln D_{p} - \ln D_{gn,i})^2} {2\ln^2 \sigma_{g,i}} \right] \,. \]

\(D_{gn,i}\) and \(\sigma_{g,i}\) are the geometric mean and geometric standard deviation of \(D_p\), respectively. \(n_{norm,i}(\ln D_{p})\) is a normal distribution of \(\ln D_{p}\) with a mean equal to \(\ln D_{gn,i}\) and a standard deviation equal to \(\ln\sigma_{g,i}\). The two expressions above are referred to as the normalized size distribution functions in MAM.

Multiplying the normalized size distribution functions by \(N_i\), the number of all particles in mode \(i\) per m\(^3\) of air, we obtain the following number distribution functions (cf., e.g. Appendix C in Whitby, 1991).

\[ n_{i}(D_{p}) = \frac{N_{i}}{\sqrt{2\pi} D_{p} \ln \sigma_{g,i}} \ \exp \left [ - \frac{(\ln D_{p} - \ln D_{gn,i})^2} {2\ln^2\sigma_{g,i}} \right ] \]

and

\[ n_{i}(\ln D_{p}) = \frac{N_{i}}{\sqrt{2\pi}\ln \sigma_{g,i}} \ \exp \left [ - \frac{(\ln D_{p} - \ln D_{gn,i})^2} {2\ln^2\sigma_{g,i}} \right ] \]

To calculate integrals over the size distribution, we note that

\[ n_{i}(D_{p}) = n_{i}(\ln D_{p})/D_{p} \,, \]

and

\[ n_{i}(D_{p}) d D_{p} = n_{i}(\ln D_{p}) d \ln D_{p} \,. \]

The choices of size range and width for the various modes in MAM are based on measurements of tropospheric aerosols (see, e.g., Easter et al., 2004 and references therein). The parameters used in MAM4 are shown in Table 1.

Table 1. Parameters of the log-normal modes in MAM4

Mode	Lower bound of \(D_{gn,d,i}\)	Upper bound of \(D_{gn,d,i}\)	\(\sigma_{g,i}\)
Aitken	0.0087	0.052	1.6
Accumulation	0.0535	0.44	1.8
Coarse	1.0	4.0	1.8
Primary carbon	0.01	0.1	1.6

Moments of the size distribution

The \(k\)-th integral moment of a generic distribution function \(n_{i}(D_{p})\) is defined as

\[ M_{k,i} = \int_0^{\infty}D_p^k\,\,n_i(D_p) dD_p \,. \]

\(k\) is referred to as the order of the moment. For the special case of log-normal distribution, the equation above can be written as

\[ M_{k,i} = N_{i} D_{gn,i}^{k} \exp \left ( \frac{k^2}{2} \ln^2 \sigma_{g,i} \right ) \tag{1} \]

The derivation can be found in Appendix C of Whitby (1991).

For the purposes of describing microphysical processes in MAM, we consider the \(0^{th}\) to \(3^{rd}\) moments:

\[ M_{0,i} = \int_{0}^{\infty} n_{i} (D_{p}) d D_{p} \equiv N_{i} \tag{2} \]

\[ M_{1,i} = \int_{0}^{\infty} D_{p} \,n_{i} (D_{p}) dD_{p} \equiv N_{i} \overline{D}_{i} \tag{3} \]

\[ M_{2,i} = \int_{0}^{\infty} D_{p}^2 \,n_{i} (D_{p}) d D_{p} \equiv \frac{N_{i} \overline{A}_{i}}{\pi} \tag{4} \]

\[ M_{3,i} = \int_{0}^{\infty} D_{p}^3 \,n_{i} (D_{p}) d D_{p} \equiv \frac{6}{\pi}N_{i}\overline{V}_{i} \tag{5} \]

\(\overline{D}_{i}\), \(\overline{A}_{i}\), and \(\overline{V}_{i}\) are the arithmetic mean diameter, surface area, and volume, respectively, of particles in mode \(i\).

Diagnosing the mean diameter

MAM's normalized size distribution functions have one parameter per mode, \(D_{gn,i}\), that will evolve with time and vary with geographical location. (Recall that the geometric standard deviation is fixed, see basic assumption no. 4 in the overview section.)

Since MAM predicts the mass and number concentrations of each aerosol mode, Eqs. (1) and (5) can be used to diagnose \(D_{gn,i}\). Let us assume the mass concentration of species \(L\) in mode \(i\) is \(q_{m,L,i}\) (unit: kmol \(L\) per kmol air), the number concentration of mode \(i\) is \(q_{n,i}\) (unit: number per kmol air), and the air concentration is \(c_{air}\) (unit: kmol air per m\(^3\)).

Equation (1) gives

\[ M_{3,i} = N_{i} D_{gn,i}^{3} \exp \left ( \frac{9}{2} \ln^2 \sigma_{g,i} \right ) = q_{n,i}c_{air} D_{gn,i}^{3} \exp \left ( \frac{9}{2} \ln^2 \sigma_{g,i} \right )\,. \]

Equation (5), together with the assumption of additive particle volume, gives

\[ M_{3,i} = \frac{6}{\pi}N_{i}\sum_{L}\overline{V}_{L} \]

\[= \frac{6}{\pi}c_{air} \sum_{L}\frac{q_{m,L,i}M_{w,L}}{\rho_{L}} = \frac{6}{\pi}\frac{\rho_{air}}{M_{w,air}} \sum_{L}\frac{q_{m,L,i}M_{w,L}}{\rho_{L}} \]

where \(\rho_{L}\) and \(M_{w,L}\) are the density (unit: kg m\(^{-3}\)) and molecular weight (unit: kg kmol\(^{-1}\)) of species \(L\), respectively.

Therefore, depending on whether the particle number concentration is expressed using \(q_{n,i}\) or \(N_{i}\), we have either

\[ D_{gn,i} = \left[ \dfrac{6}{q_{n,i}\pi\exp\left( \frac{9}{2} \ln^2 \sigma_{g,i} \right)} \sum_{L}\frac{q_{m,L,i}M_{w,L}}{\rho_{L}} \right]^{1/3} \label{eq:dgn__num_conc_per_kmol_air} \]

or

\[ D_{gn,i} = \left[ \dfrac{6}{N_{i}\pi\exp\left( \frac{9}{2} \ln^2 \sigma_{g,i} \right)} \frac{\rho_{air}}{M_{w,air}} \sum_{L}\frac{q_{m,L,i}M_{w,L}}{\rho_{L}} \right]^{1/3}. \label{eq:dgn__num_conc_per_m3_air} \]

To diagnose the geometric mean diameter of the dry particles, \(L\) should include only the dry compositions of mode \(i\); to diagnose the geometric mean diameter of the wet particles, \(L\) should also include aerosol water.