Appendix O — Aerosol: Mathematical Details

Implementation status

Model	DITL	PACE V3	PACE V4	Evaluation	Planned
—	x	x	x	—	—

This chapter provides the mathematical relationships used to convert between number- and volume-based aerosol size distributions and to calculate the moments, effective radius, and effective variance used by the FastMAPOL aerosol model.

The operational aerosol representation and the definitions of the Aer-1 and Aer-2 models are described in Appendix K.

O.1 Number and Volume Size Distributions

For a single log-normal aerosol mode, the volume distribution is

\[ \frac{dV(r)}{d\ln r} = \frac{V_0}{\sqrt{2\pi}\sigma_v} \exp \left[ -\frac{(\ln r-\ln r_v)^2}{2\sigma_v^2} \right], \]

and the corresponding number distribution is

\[ \frac{dN(r)}{d\ln r} = \frac{N_0}{\sqrt{2\pi}\sigma_n} \exp \left[ -\frac{(\ln r-\ln r_n)^2}{2\sigma_n^2} \right]. \]

Here, \(V_0\) and \(N_0\) are the integrated volume and number densities, \(r_v\) and \(r_n\) are the volume-mean and number-mean radii, and \(\sigma_v\) and \(\sigma_n\) are the corresponding logarithmic standard deviations.

Particle volume and number are related through

\[ dV(r) = \frac{4\pi}{3}r^3dN(r). \]

Therefore,

\[ \frac{dV(r)}{d\ln r} = \frac{4\pi}{3}r^3 \frac{dN(r)}{d\ln r}. \]

O.2 Conversion Between Number and Volume Distributions

The equivalent log-normal number and volume distributions have the same logarithmic width:

\[ \sigma_n=\sigma_v=\sigma. \]

Their characteristic radii are related by

\[ r_n = r_v\exp(-3\sigma^2). \]

The corresponding integrated number and volume densities satisfy

\[ N_0 = V_0 \frac{3}{4\pi r_v^3} \exp(4.5\sigma^2), \]

or equivalently,

\[ N_0 = V_0 \frac{3}{4\pi r_n^3} \exp(-4.5\sigma^2). \]

These relationships allow FastMAPOL’s volume-based retrieval state to be converted into the number-density representation required for calculations of particle extinction and other number-weighted quantities.

O.3 Distribution Convention

A particle size distribution may be expressed with respect to either \(r\) or \(\ln r\). The two representations are related by

\[ \frac{dN(r)}{dr} = \frac{1}{r} \frac{dN(r)}{d\ln r}, \]

and

\[ \frac{dV(r)}{dr} = \frac{1}{r} \frac{dV(r)}{d\ln r}. \]

This distinction is important when comparing size-distribution parameters and moment definitions among different references and numerical implementations.

O.4 Moments of the Number Distribution

Define the \(k\)th moment of the number size distribution as

\[ m_k = \int_0^\infty r^k \frac{dN(r)}{dr}\,dr. \]

Equivalently,

\[ m_k = \int_{-\infty}^{\infty} r^k \frac{dN(r)}{d\ln r}\,d\ln r. \]

For a log-normal number distribution,

\[ m_k = N_0r_n^k \exp \left( \frac{k^2\sigma^2}{2} \right). \]

Substituting the number-volume relationships gives an equivalent expression in terms of the volume-distribution parameters:

\[ m_k = V_0 \frac{3}{4\pi} r_v^{k-3} \exp \left[ \frac{(k-3)^2\sigma^2}{2} \right]. \]

For \(k=3\),

\[ m_3 = \frac{3V_0}{4\pi}, \]

showing the direct relationship between the third number-distribution moment and total particle volume.

O.5 Effective Radius

The effective radius is the area-weighted mean particle radius:

\[ r_{\mathrm{eff}} = \frac{ \int r^3(dN/dr)\,dr }{ \int r^2(dN/dr)\,dr } = \frac{m_3}{m_2}. \]

For a log-normal number distribution,

\[ r_{\mathrm{eff}} = r_n \exp \left( \frac{5}{2}\sigma^2 \right). \]

Using

\[ r_n=r_v\exp(-3\sigma^2), \]

this can also be written as

\[ r_{\mathrm{eff}} = r_v \exp \left( -\frac{1}{2}\sigma^2 \right). \]

Thus,

\[ r_n < r_{\mathrm{eff}} < r_v \]

for a distribution with finite width.

O.6 Effective Variance

The effective variance is defined as

\[ \nu_{\mathrm{eff}} = \frac{ \int (r-r_{\mathrm{eff}})^2 r^2(dN/dr)\,dr }{ r_{\mathrm{eff}}^2 \int r^2(dN/dr)\,dr }. \]

Using the distribution moments,

\[ \nu_{\mathrm{eff}} = \frac{m_4m_2}{m_3^2}-1. \]

For a log-normal distribution,

\[ \nu_{\mathrm{eff}} = \exp(\sigma^2)-1. \]

For sufficiently narrow distributions,

\[ \nu_{\mathrm{eff}} \approx \sigma^2, \]

and

\[ r_{\mathrm{eff}} \approx r_v. \]

O.7 Moments of the Volume Distribution

Because FastMAPOL uses volume density in the retrieval state vector, the same quantities can be expressed directly using volume-distribution moments.

Define

\[ m'_k = \int r^k \frac{dV(r)}{d\ln r} \,d\ln r. \]

The effective radius is then

\[ r_{\mathrm{eff}} = \frac{m'_0}{m'_{-1}}, \]

and the effective variance is

\[ \nu_{\mathrm{eff}} = \frac{ m'_{-1}m'_1 }{ (m'_0)^2 } -1. \]

For a log-normal volume distribution,

\[ m'_k = V_0 r_v^k \exp \left( \frac{k^2\sigma^2}{2} \right). \]

These expressions are mathematically equivalent to those derived from the number distribution.

O.8 Multimode Aerosol Distributions

For a multimode aerosol,

\[ \frac{dV(r)}{d\ln r} = \sum_i \frac{V_{0,i}}{\sqrt{2\pi}\sigma_i} \exp \left[ -\frac{(\ln r-\ln r_{v,i})^2} {2\sigma_i^2} \right]. \]

The corresponding total moment is obtained by summing the moments of the individual modes:

\[ m_k = \sum_i m_{k,i}. \]

Therefore,

\[ r_{\mathrm{eff}} = \frac{ \sum_i m_{3,i} }{ \sum_i m_{2,i} }, \]

and

\[ \nu_{\mathrm{eff}} = \frac{ \left(\sum_i m_{4,i}\right) \left(\sum_i m_{2,i}\right) }{ \left(\sum_i m_{3,i}\right)^2 } -1. \]

Using volume-distribution moments, the equivalent expressions are

\[ r_{\mathrm{eff}} = \frac{ \sum_i m'_{0,i} }{ \sum_i m'_{-1,i} }, \]

and

\[ \nu_{\mathrm{eff}} = \frac{ \left(\sum_i m'_{1,i}\right) \left(\sum_i m'_{-1,i}\right) }{ \left(\sum_i m'_{0,i}\right)^2 } -1. \]

The same expressions can be applied to the fine or coarse aerosol distributions by restricting the summation to the corresponding modes.

O.9 Relationship Between Volume and Optical Weighting

The distinction between volume and number weighting is important when interpreting multimode aerosol properties. For the same total particle volume, smaller particles correspond to a substantially larger number of particles than larger particles.

For two aerosol modes, the extinction contribution can be expressed as

\[ \tau = C_{\mathrm{ext},f}N_f + C_{\mathrm{ext},c}N_c. \]

Using volume density,

\[ \tau = C_{\mathrm{ext},f} \frac{3V_f}{4\pi r_{v,f}^3} \exp(4.5\sigma_f^2) + C_{\mathrm{ext},c} \frac{3V_c}{4\pi r_{v,c}^3} \exp(4.5\sigma_c^2). \]

Thus, the relative optical contribution of two modes depends not only on their volume fractions but also on particle size and extinction efficiency.

This distinction explains why the fine-volume fraction and fine-mode optical-depth fraction are not equivalent and why both quantities provide useful but different information about the retrieved aerosol distribution.

O.10 Summary

The mathematical formulation presented here provides the connection between the volume-based aerosol representation used in the FastMAPOL state vector and the number-, area-, and optically weighted quantities used to characterize aerosol properties. The same moment framework applies to individual log-normal modes, multimode distributions, and fine/coarse aerosol partitions.