## Vertical Variation

The vertical distribution of aerosol mass concentration typically shows an exponential decrease with altitude up to a height Hp and a rather constant profile above that altitude (Gras 1991). The aerosol mass concentration as a function of height can then be expressed as

where M(0) is the surface concentration and Hp the scale height. Jaenicke (1993) proposed values of Hp equal to 900 m for the marine, 730 m for the remote continental, 2000 m for the desert, and 30,000 m for the polar aerosol types. The corresponding vertical aerosol mass concentration profiles are shown in Figure 8.29.

The aerosol number concentration may increase or decrease exponentially with altitude and one suggestion of a form of the profile is (Jaenicke 1993)

Nb_Y

where

Mass Concentration, fig m"3

FIGURE 8.29 Representative vertical distribution of aerosol mass concentration (Jaenicke 1993).

and Nb is the number concentration of the background aerosol aloft. For marine aerosol H'p varies from — 290 to 440 m. Note that if H'p is negative n = -1, and (8.56) can be rewritten as

Because in this case N(0) <C NB, the equation has the correct limiting behavior both for z —> 0 and z —► oo.

These vertical profiles are rough representations of long-term averages. Significant variability is observed in aerosol concentrations in anthropogenic plumes, areas influenced by local sources, or during nucleation events in the free troposphere.

### PROBLEMS

8.1a Given the following data on the number of aerosol particles in the size ranges listed, tabulate and plot the normalized size distributions h^{Dp) = n^(Dp)/N, and h°N(logDp) = n^(logDp)/Nt as discrete histograms.

 pm pm in Inter 0-0.2 0.1 10 0.2-0.4 0.3 80 0.4-0.6 0.5 132 0.6-0.8 0.7 142 0.8-1.0 0.9 138 1.0-1.2 1.1 112 1.2-1.4 1.3 75 1.4-1.6 1.5 65 1.6-1.8 1.7 52 1.8-2.1 1.95 65 2.1-2.7 2.4 62 2.7-3.6 3.15 32 3.6-5.1 4.35 35

8.2a For the data given in Problem 8.1, plot the surface area and volume distributions ns(pp), (logDp), nv(Dp) and n°v{\ogDp) in both nonnormalized and normalized form as discrete histograms.

8.3a You are given an aerosol size distribution function such that nM{m)dm = aerosol mass per cm3 of air contained in particles having masses in the range m to m + dm. It is desired to convert that distribution function to a mass distribution based on log Dp. Show that n°M(logDp) = 6.9 mnM(m)

8.4b Show that the variance of the size distribution of a lognormally distributed aerosol is

8.5a Starting with semilogarithmic graph paper, construct a log-probability coordinate axis and show that a lognormal distribution plots as a straight line on these coordinates.

8.6a The data given below were obtained for a lognormally distributed aerosol size distribution:

 Size Interval, Hm Geometric Mean of Size Interval, (tin Number of Particles in Interval" 0.1-0.2 0.1414 50 0.2-0.4 0.2828 460 0.4-0.7 0.5292 1055 0.7-1.0 0.8367 980 1.0-2.0 1.414 1705 2.0-4.0 2.828 680 4.0-7.0 5.292 102 7.0-10 8.367 10 10-20 14.14 2

"Assume that the particles are spheres with density pp = 1.5 g cm 3.

"Assume that the particles are spheres with density pp = 1.5 g cm 3.

a. Complete this table by computing the following quantities: ANj/ADpi, ATV,/ NADpi, ASi/ADpi, ASi/SADpi, AMt/ADpl, AM-JMAD,», AN-JAlogD,,,, ANi/NA log Z)pi, ASi/AlogDph ASt/SA\ogDpl, and AM,-/A log Dpi, AM,/ M A log Dpi, where M = particle mass.

b. Plot ANi/ A log Dpi, ASj/A log Dpi, and AM,/ A log Dpi as histograms.

c. Determine the geometric mean diameter and geometric standard deviation of the lognormal distribution to which these data adhere and plot the continuous distributions on the three plots from part (b).

8.7b For a lognormally distributed aerosol different mean diameters can be defined by

where v is a parameter that defines the particular mean diameter of interest. Show that

Plot a normalized lognormal particle size distribution over a range of Dp from 0 to 7 pm with Dpg = 1.0 pm and ug = 2.0 and identify each of the above diameters on the plot. Hint: You may find this integral of use:

8.8b The modified gamma distribution (Deirmendjian 1969) has been proposed as another function that approximates ambient aerosol size distributions, n^{Dp) = ADp exp (-BD'p), where A, b, B, and c are all positive constants.

a. Plot this size distribution for the following combinations of its parameters: A = 100, b = 3, B = 2 and c = 1,2,3

A = 100, b = 2, c = 2 and B = 1.5,2,2.5 A = 100, c = 2, B = 2 and b = 3,4,5

b. Calculate the diameter Dm at which the distribution function reaches a maximum as a function of the distribution parameters.

c. Calculate the total aerosol number concentration as a function of the distribution parameters.

d. Using the results of parts (a)-(c), discuss the effect of the four parameters on the shape of the distribution function.

e. Discuss the strengths and weaknesses of this function for the fitting of atmospheric aerosol size distributions.

8.9b Assume that an aerosol has a lognormal distribution with Dpg = 5.5 pm and

Diameter v

Mode (most frequent value) Geometric mean or median Number (arithmetic) mean Surface area mean Mass mean Surface area median Volume median

'g a. Plot the number and volume distributions of this aerosol on log-probability paper.

b. It is desired to represent this aerosol by a distribution of the form

where Fv{Dp) is the fraction of the total aerosol volume in particles of diameter less than Dp. Determine the values of the constants c and b needed to match this distribution to the given aerosol.

8.10b Given the following size frequency for a dust:

 7-17.5 10 17.5-21 10 21-25 10 25-28 10 28-30 10 30-33 10 33-36 10 36-41 10 41^19 10 49-70 10

a. Plot the cumulative frequency distributions (in %) of the number, surface area, and mass on linear graph paper assuming all particles are spheres with pp = 1.6 g cm"3.

b. Is this a lognormally distributed dust?

8.11b The following particle size distribution data are available for an aerosol:

 % by Volume DP Less than 9.8 3.2 13.8 10.0 19.6 26.7 27.7 46.8 39.1 72.0 55.3 87.5

a. What are the volume median diameter and geometric standard deviation of the volume distribution of this aerosol?

b. What is the surface area median diameter?

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