Power Spectral Densities of Atmospheric

Aerosol Particle Counts

by Chatt C. Williamson, Steven C. Hill, Dennis M. Garvey,

Michael L. Larsen, and Cheryl L. Klipp

ARL-TR-5064 January 2010

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Army Research Laboratory 

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ARL-TR-5064 January 2010 

Power Spectral Densities of Atmospheric 

Aerosol Particle Counts 

Chatt C. Williamson, Steven C. Hill, Dennis M. Garvey, 

Michael L. Larsen, and Chrryl L. Klopp 

Computational and Information Sciences Directorate, ARL 

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Power Spectral Densities of Atmospheric Aerosol Particle Counts

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14. ABSTRACT

The velocity components of the atmospheric wind typically have a power spectral density (PSD) with a slope of –5/3 in an “inertial subrange” 

of frequencies, roughly from 0.01 to >10 Hz depending on the wind speed and stability. As passive scalars, atmospheric temperature and 

humidity have a relatively small direct effect on the winds, and they also typically have a –5/3 slope in this frequency range. Atmospheric 

aerosol particles are also expected to behave as passive scalars, and PSD slopes of –5/3 have been reported for some atmospheric aerosol 

measurements with a volume-based light-scattering instrument. Herein, we present aerosol measurements obtained with a single-particle 

optical counter over two four-day periods with concurrent sonic anemometer measurements of wind speed and temperature. PSDs of the 

aerosol particle counts, temperature, and winds are presented and compared in the frequency range 10–4 to 10 Hz for 5 min subintervals. As 

expected, the PSDs of the velocity components and of the temperature decrease with a slope near the predicted –5/3. But the slopes of the 

PSDs of the aerosol particle counts are near –5/3 for only a small fraction of the subintervals; generally the magnitude of the slopes is much 

smaller than –5/3 and often close to zero. The discrete nature of the particle counts results in a shot-noise floor, which limits the slopes in 

PSD that can be measured at low particle concentrations and high sample rates. Interestingly, an overall slope of –7/6 is found over a large 

frequency range when the entirety of both of the four day periods is considered.

15. SUBJECT TERMS

Power spectral density8, aerosol, Welch’s method, atmosphere

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Chatt C. Williamson

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iii

Contents 

List of Figures iv

1. Introduction 1

1.1 Overview .........................................................................................................................1

1.2 Background .....................................................................................................................2

1.2.1 Turbulent Eddies .................................................................................................2

1.2.2 Spectra of Turbulence – Models and Measurements ..........................................2

1.2.3 Spectra of Turbulence – Passive Scalars .............................................................3

1.3 Approach .........................................................................................................................5

2. Methods 5

2.1 Experimental ...................................................................................................................5

2.2 Analysis ...........................................................................................................................6

3. Results and Discussion 7

3.1 Particle Counts Measured versus Time ...........................................................................7

3.2 Power Spectral Densities of Wind, Temperature, and Aerosol for 90 h Periods ............8

3.3 Limitations on Frequency Spectra as a Function of Particle Count (Shot Noise Floor)10

3.4 Time-Dependence of the Slopes of the Power Spectral Densities of Wind, 

Temperature, and Aerosol for 5 min Intervals .......................................................................12

3.5 Slope of Aerosol Power Spectral Densities with Low and High Aerosol Concentrations12

4. General Discussion 18

5. Summary 19

6. References 20

Appendix. Effect of Dropping 15 s of Data 23

Distribution List 25 
iv

List of Figures

Figure 1a. Aerosol particle counts measured at 20 Hz using the Climet particle counter 

during the period July 3–7, 2007. .............................................................................................8

Figure 1b. Aerosol particle counts measured at 20 Hz using the Climet particle counter, July 

13–17, 2007................................................................................................................................8

Figure 2. PSDs estimated using the modified Welch method for the u component of wind, 

the temperature (Tv), and aerosols. The data for aerosols was binned into two different 

sampling frequencies before analysis. The lines with slopes of –5/3 (solid) and –7/6 

(dashed) are for comparison with the winds and the aerosols, respectively. The slope of 

the wind (u) is similar to –5/3 in the inertial subrange (10–1 to 101 Hz)....................................9

Figure 3. PSDs estimated using the modified Welch method for the aerosols for the two 

sampling periods (July 3–7 and July 13–17, 2007). The line with slope –7/6 (dashed) is 

for comparison. ........................................................................................................................10

Figure 4. PSDs estimated for a 5-min sample of measured particles (blue) and for the sample 

after 90% (green) and then 99% (red) of the particles have been randomly eliminated. .......11

Figure 5. Estimate of the slope of the PSDs calculated for the winds (blue), sonic 

temperature (green), and aerosol number concentrations (red). ..............................................12

Figure 6. Mean particle count of aerosols, TKE, and slope of aerosol PSD for each 5-min 

window of a 90 hr data collection beginning July 3, 2007. ....................................................13

Figure 7a. Aerosol particle concentrations measured at 20 Hz using the Climet OPC, July 3–

7, 2007, during a period where the concentration is lowest. The average particle count 

is about 9.21 per bin.................................................................................................................14

Figure 7b. Aerosol particle concentrations measured at 20 Hz using the Climet OPC, July 3–

7, 2007, during a period where the concentration is highest. The average particle 

number is 62.8 per bin. ............................................................................................................15

Figure 8. The slope of the linear fit of the PSD for a fixed frequency interval (0.05 to 1.0 

Hz) and a given mean particle count........................................................................................16

Figure 9. Right hand interval bound of the frequency interval for which the linear fit of the 

PSD was closest to –5/3 slope, with a left hand bound of 0.05 Hz and a given mean 

particle count............................................................................................................................17

Figure 10. Right hand bound of the frequency interval for which the linear fit of the PSD is 

within 1% (blue), 5% (red), or 10% (green) of –5/3 slope, with a left hand bound of 0.05 

Hz and a given mean particle count........................................................................................18

Figure A-1. PSDs for aerosols July 13–17, 2007, with and without the zeros removed. The 

effect appears small..................................................................................................................23

Figure A-2. Same as for figure A-1 though limiting the display to the frequency interval in 

the region of most significant difference for the two PSDs....................................................24
1

1. Introduction

1.1 Overview

Atmospheric aerosols have many impacts on humans and the earth’s environment, as they affect 

the health of humans, other animals, and plants. Both biological particles (e.g., tuberculosis, 

plague) and non-biological particles (e.g., asbestos, polycyclic aromatic hydrocarbons (PAH) 

from combustion) have important health effects. Atmospheric aerosols also affect global 

climate, directly by absorbing and scattering radiation and indirectly by acting as cloud 

condensation nuclei (CCN). They are also important in atmospheric chemistry because many 

atmospheric reactions occur inside or on the surfaces of particles, and agglomeration of particles 

as well as adsorption of gases onto the surface of particles is common. Atmospheric aerosols can 

affect communications and imaging through the atmosphere because of their effects on the 

propagation of light and other electromagnetic radiation. Because of the importance of 

atmospheric aerosols for many areas, and the importance of measuring and understanding 

aerosols (in total, and particular types of aerosols, such as biowarfare or chemical warfare 

agents), a better understanding of the statistics of atmospheric aerosols is desirable. The 

statistics of atmospheric aerosols appears to be more complex than the statistics of atmospheric 

turbulence—defined in terms of wind velocities—because aerosols are generated in many ways;

their behavior depends strongly on size and chemical composition; and they grow, shrink, and/or 

age with time in ways that depend upon temperature, humidity, sunlight, and the vapor pressure 

of gaseous species.

The primary topics of this report are our measurements of atmospheric aerosols, counted one at a 

time, and our calculations of the power spectral density (PSD) of the aerosol particle counts over 

a large frequency range. The measurements were made at a fixed point in the atmospheric 

boundary layer, on the roof of a building, about 1 m from the flat surface of the roof. We 

achieved the very large range of frequencies (about 5 orders of magnitude) and high frequencies 

in the aerosol PSD by making single-particle measurements using an instrument that draws air 

through at 28 liters/min. One objective was to determine under what conditions and over what 

frequency range the aerosol PSDs calculated from counts of single-particles exhibited a –5/3 

slope. We also discuss and speculate about how this measured aerosol PSD might depend upon 

the time the added aerosol had to mix before it reached the detector; we also speculate on 

changes due to atmospheric chemistry and on the strength and variability of the winds. 

One difficulty in discussing atmospheric turbulence is that the term “turbulence” is used 

differently by different researchers, where possibly the main difference pertains to the regions 

where the term “turbulence” applies. For example, for some researchers the regions in which 

turbulence occurs may be defined in terms of frequency (spectral analysis), or distance from the 

ground, or height within the boundary layer. Some researchers limit their definition of 
2

“turbulence” to regions in which Kolmogorov’s assumptions for isotropic turbulence are valid 

(possible symmetries of the Navier-Stokes equations pertain; the turbulent flow is isotropic, selfsimilar, and has a non-vanishing dissipation rate per unit mass (see Frisch, 1995, Ch. 6)), i.e., to 

an “inertial subrange,” lying between the “energy input range” at the low frequency end and the 

“dissipation range” at the high end or to regions where the PSD of winds should follow a –5/3 

power law. Kundu (1990) quotes Marcel Lesieur as, “turbulence is a dangerous topic which is at 

the origin of serious fights in scientific meetings since it represents extremely different points of 

view, all of which have in common their complexity, as well as an inability to solve the problem. 

It is even difficult to agree on what exactly is the problem to be solved.” Mandlebrot (1983) 

asked, “Should the term ‘turbulence’ denote all unsmooth flows, including much of meteorology 

and oceanography? Or is it better to reserve it for a narrow class, and, if so, for which one? Each 

scholar seems to answer these questions differently.” In this report we use turbulence in a 

broader sense, and include much of meteorology and unsmooth flows.

1.2 Background

1.2.1 Turbulent Eddies

Turbulence is often envisioned as being composed of eddies of many sizes. Richardson (Kundu, 

1990, ch. 14.2) suggested that energy is added to the atmosphere at large scales—or equivalently, 

into large-scale eddies—and then cascades downward into smaller eddies, and is eventually 

dissipated by viscous motion. That the eddy concept includes rotation is enhanced by 

Richardson also referring to eddies as “whorls.” Although Webster’s Dictionary defines an 

“eddy” as “a current of air, water, or other fluid moving against the main current and having a 

circular motion,” and although in everyday usage, “rotation” seems to be a fundamental aspect of 

an eddy, some researchers appear to envision turbulence in terms of random variations, with no 

initial concept of rotation. Here we will largely go with Pope’s definition (Pope, 1998): “An 

‘eddy’ eludes precise definition, but is conceived to be a turbulent motion, localized within a 

region of size l, that is at least moderately coherent over this region.”

1.2.2 Spectra of Turbulence – Models and Measurements

The power spectrum of the wind velocities in turbulence was shown theoretically by

Kolmogorov (Frisch, 95) to follow a –5/3 power law in an inertial subrange when several 

conditions apply. He assumed a homogenous, stationary, isotropic cascade process where there 

is no energy loss until dissipation occurs at the highest frequencies, and so the rate of energy 

flow from lower frequencies is a constant. There was no assumption of vortices or eddies.

Onsager (1945; 1949) began from an assumption of vortices that could interact (similar to 

Richardson’s initial hypotheses) and arrived at a –5/3 frequency dependence. He also showed 

that for the 2-D case, at least, energy added to small eddies could transfer to larger eddies to take 

the system in the direction of the –5/3 sloped PSD. 
3

The PSD of the winds in the atmospheric boundary layer, particularly the surface layer, 

calculated from measured wind velocity components has in fact been observed to follow a –5/3 

power law over a range of frequencies, which may extend from about 0.01 to >10 Hz (Frisch, 

1995; Kaimal, et al; 1972; Kaimal, 1973). The extent of this range is smaller for the vertical 

velocity component compared to the longitudinal and transverse components, i.e., the “cutoff” on 

the low frequency end occurs at a higher frequency or shorter wavelength. 

1.2.3 Spectra of Turbulence – Passive Scalars

The –5/3 power law has also been suggested to apply to “passive scalars,” such as temperature 

and humidity (Batchelor, 1953). It is argued that passive scalars should follow the turbulence 

structure of the winds. Warhaft (2000), however, reporting on results of wind tunnel studies, is 

less supportive of the view that the PSDs of passive scalars necessarily follow a –5/3 slope. A 

passive scalar should have no more than a very small effect on the motion of the suspending gas, 

and should largely follow the flow of the gas. Aerosols in the atmosphere typically have only a 

tiny mass compared to that of the air in which they are suspended. Also, most aerosols have 

small settling velocities—e.g., the settling velocity in still air for a 1 mm water droplet is 

35 mm/s. Blackadar (1997) states: “Most properties other than momentum and heat are passive 

in the sense that their distribution plays no part in the turbulent mechanisms that spread them. 

Included in the broad list of such properties are most contaminants ....”, e.g., aerosols. What are 

the conditions, then, for which aerosols behave as passive scalars, and over what frequency 

ranges do spectra of their concentrations exhibit a –5/3 slope?

Porch and Gillette (1977) measured a –5/3 PSD for atmospheric aerosols under conditions where 

there were high levels of wind-blown dust, and it is not difficult to imagine that the addition of 

aerosols might go according to the variance of the winds that are blowing the aerosols into the 

air. This is the only reported measurement of a –5/3 slope for atmospheric aerosols that we have 

found in the open literature. In this case, an eyeball alignment fits the measurement to a –5/3 

slope over a frequency range of 0.1 to 5 Hz. In measurements of plumes of obscurant smokes 

with optical particle counters, Garvey et al. (1982) found slopes of the PSDs to be about –5/3 

over a limited range (no more than 0.03 to 0.2 Hz). In these tests, the particle counters covered 

nominal size ranges of radii from 0.12 to 0.42 μm, 0.25 to 1.0 μm, and 0.5 to 10 μm. There was 

no obvious size dependence of the measured spectra, though the number concentration in a given 

size interval had to be high enough to reflect the aerosol concentration at a statistically 

significant level. PSDs with slopes of –5/3 over at least part of the frequency range from 0.1 to 

5 Hz were also found in later measurements of artificial fogs with a lidar. In these measurements 

and those of Porch and Gillette, where the sampling rates were large enough to extend the PSDs 

to frequencies greater than 5 Hz, the slopes tended to flatten out at frequencies above that value. 

A –5/3 slope in the PSD has been observed for atmospheric aerosol backscatter using lidar 

measurements (Radkevich, et al., 2007). In backscattering measurements of this type—although 

aerosols of all sizes contribute to the signal—(i) the scattering efficiencies for particles much 
4

smaller than the wavelength decrease rapidly as the size decreases; (ii) the scattering efficiencies 

for particles greater than about a wavelength are around two, and so the scattering cross-sections 

increase somewhat proportional to cross sectional area; and (iii) for a given particle crosssection, the backscattering by spherical aerosols is typically several times larger than for nonspheres with equivalent cross-sections. Because aerosols may be hygroscopic, the scattering 

from aerosols may increase rapidly as humidity increases, in ways that differ from changes in 

particle numbers in certain size ranges. The differences between the effects of humidity on 

backscattering and on particle numbers will depend upon the initial particle-size distribution, the 

thresholds chosen for counting particles, and on the wavelengths at which scattering is measured. 

In the measurements of Radkevich et al., (2007) the smallest sizes resolved are on the order of 

3 m.

Single-particle statistics of atmospheric aerosol have been studied only to a limited extent. 

Following similar work with cloud droplets (e.g., Kostinski and Shaw 2001; Shaw et al. 2002) 

and precipitation (e.g., Kostinski and Jameson 1997; Larsen et al. 2005), most microphysical 

studies of aerosol particle number fluctuations have been conducted in the spatial domain rather 

than the PSD frequency domain (see, e.g., Preining 1983; Larsen et al. 2003; Larsen 2007). The 

discovery of non-trivial statistical spatial structure, however, suggests that the PSD could have 

non-trivial organized structure.

The statistics of atmospheric aerosols appears to be more complex than the statistics of 

atmospheric turbulence, defined in terms of wind velocities, for reasons such as the following. 

(a) Atmospheric aerosols in some size ranges and some situations probably act as passive 

scalars in the turbulent winds, and so it can be argued that they should have similarities in 

their statistical behavior, although the physical arguments for this may vary with the 

sources of aerosols.

(b) The “aerosol concentration” may appear to be a continuously varying function of space and 

time when sample volumes are sufficiently large, but it is not continuous when the average 

number of particles in the sample volume is small or is less than one. The arrival times for 

such particles have been reported to deviate significantly from a Poisson distribution, at 

least for some cases (Larsen et al. 2003; Larsen 2007).

(c) Aerosols diffuse much more slowly than gases (e.g., water vapor) or temperature (heat), 

and so mixing them should occur more slowly, especially in ranges where diffusive mixing 

is significant compared to convective mixing.

(d) Atmospheric aerosols of different sizes may behave differently, largely because of 

gravitational settling rates and inertia. The largest particles may not follow the winds as 

well as typical “passive scalars” because their inertia is large, and their settling velocities 

may be much more significant.
5

(e) Atmospheric aerosols may behave differently depending upon their composition and 

properties that depend upon composition. For example, ammonium sulfate particles 

increase in size as humidity increases. Particles differ in how well they may act as CCN. 

Those that are better as CCN may be washed out more rapidly.

(f) Sources of aerosols (especially of non-water-droplet aerosols) may be more localized than 

are the sources of winds, in which the very large eddies have the most energy in power 

spectra. Example sources of aerosols are wind-blown dust from soil or other surfaces, such 

as the leaves of plants, or from plant debris; particles that are generated photochemically 

from vapor-phase terpenes emitted by plants; and combustion-generated particles from 

vehicular traffic, biomass burning, etc. The combustion emissions include particles such as 

fly ash or black carbon, which are emitted directly, but also low volatility molecules such 

as PAH, which are emitted as gasses at high temperature, but then condense onto the 

surface of other aerosols.

(g) “Aging” of atmospheric aerosols is known to be important in modifying aerosols and their 

properties (Rudich). For example, organic compounds in aerosols may be oxidized by 

ozone and some of the products may be volatile, or they may be more hydrophilic and 

more efficient as CCN. Or, biological particles emitted into the atmosphere may become 

nonviable with time because of chemical reactions, which may be photochemical (e.g., 

formation of thymidine dimers in DNA) and/or may be with oxidizing molecules in the 

atmosphere (e.g., reactions of ozone with surface sugars on bacteria). 

1.3 Approach

Our primary objective in this report is to measure the aerosol concentrations, winds, and sonic 

temperature, and from these to calculate, in the inertial subrange, the slope of the PSDs of 

aerosol concentration and sonic temperature, as well as the fluctuations of the winds. We find 

that the calculated slope of the PSD of aerosols can be far different from the –5/3 slope predicted 

in the inertial subrange even when the slopes of the PSDs of the winds and temperatures behave 

as predicted. We also find that the shot noise floor associated with sampling individual particles, 

with relatively small particle counts per bin, can limit the highest frequencies at which a –5/3 

slope is observed.

2. Methods

2.1 Experimental

The vector wind components (u, v, w) and sonic temperature were measured using an RM 

Young Model 81000 sonic anemometer. The operational frequency of the anemometer is 

approximately 160 Hz with a maximum sampling frequency of 32 Hz. For this experiment, the 

anemometer was operated using a 20 Hz sampling frequency. The output data from the 
6

anemometer is provided in ASCII text through an RS232 serial port. This output data was 

archived using a custom data acquisition system.

Particles were measured using a Climet Optical Particle Counter (OPC), (Climet, CI-8060). The 

particle scattering voltage versus time (as measured by a PMT inside the Climet) was recorded 

using a National Instruments data acquisition board run with LabVIEW. The PMT voltages, 

with time stamps, were subsequently processed to pick out peak heights and arrival times. 

Calibration of this system was accomplished using polystyrene latex spheres aerosolized with a 

nebulizer followed by a drying tube (Royco). Using this calibration, we selected a threshold of 

PMT voltage that corresponded to 0.85 μm diameter PSL spheres (purchased from Duke 

Scientific). The particle counts were subsequently collected into 50 ms bins (20 Hz sampling) or 

10 ms bins (100 Hz sampling). 

The algorithm used to find the particle arrival times and peak heights employs a Schmitt trigger. 

Typically, when a particle is detected, the PMT voltage is high for around 5 to 15 μs. The 

background voltage is assumed to have a noise that is small compared to the threshold voltage 

chosen. However, we have observed (as others have previously) that the “background” voltage 

of the OPC occasionally increases slowly (with rise times on orders of 10s to 100s of μs) to 

voltages that may even exceed the threshold voltage. One possible reason for the slow increase 

in “background” may be that a particle is in a position where it moves slowly, but still scatters 

light that gets to the PMT. Another possible reason is that a very high concentration of (possibly 

small) particles entered the OPC. Cigarette smoke could possibly cause such an effect. This 

floating background could distort the particle statistics, for example, by appearing as extra 

particles with sizes above threshold, in cases where the “background” has floated to a significant 

fraction of threshold. If we do further work with the Climet OPC, we plan to develop more 

sophisticated algorithms to keep track of the “background” and count particles that have peak 

scattering that is greater than the sum of the background and the threshold chosen. Since—as 

best we can tell—these floating backgrounds are a relatively rare occurrence, we think that their 

effect on the statistics presented here is small. The discussion in the appendix on the effects of 

zeros in the particle counts adds to our impression that the effects of rare changes in particle 

counts are not so likely to cause large changes in the statistics. 

The measurement site was the roof of Bldg 202 at the U.S. Army Research Laboratory (ARL)

Adelphi Laboratory Center. The measurements were made for two periods: July 3–7 and July 

13–17, 2007.

2.2 Analysis

If the estimations of the PSDs are calculated directly from the data for aerosols, but also for 

winds and sonic temperatures, there are high frequency oscillations in the PSDs that are so large 

that it is difficult to discern where the underlying signal is. As described by Manolakis et al., 

(2000), in a section on estimating power spectra of stationary random signals, there are two main 

approaches to reducing the variance and smoothing the PSDs. These two approaches should 
7

give similar results: (1) average the PSDs over frequency (e.g., by taking a running average), 

and (2) divide the input data into smaller ranges, compute PSD estimates for each of these, and 

then average the PSD estimates (the Welch or Welch-Bartlett method is an example of this 

approach). We calculated PSDs using both approaches (1) and (2), but then settled on approach 

(2) for the analysis shown here. 

The PSDs for the aerosols, wind vector components, and sonic temperature were estimated using 

a modified Welch method. The modified Welch method computes an estimated PSD for a 

number of subwindows of the data being analyzed. Each subwindow is created using standard 

numerical filtering techniques such as a Hamming or Hanning window. The estimated spectral 

densities of each of these subwindows are then averaged to give the final estimated PSD. This 

technique tends to produce “smoother” spectra than does using a periodogram to estimate the 

spectral density. We employed the MATLAB routine pwelch, which implements the previously 

described method. By default, pwelch applies a Hamming window to eight overlapping 

segments of the input data. A second method used to provide an even smoother estimate of the 

PSD was implemented. This method also relied upon pwelch. The method makes successive 

calls to pwelch, using a smaller window size with each successive call. The lowest valid decade 

of frequency spectra are then saved. This procedure is repeated until the algorithm has covered 

all the viable frequency ranges. 

The LabVIEW programs were written in such a way that there was a need to have some 

downtime to write the data to the hard drive. This time amounted to 1.5 s (30 samples of zero in

the 20 Hz data) every 5 min, and two samples of unusable data (at 20 Hz) every 15 s. The 

appendix illustrates PSDs estimated using the Welch method for aerosols of July 13–17, 2007, 

with and without the zeros removed, and find that the effects on the PSD of removing these 

missing data points are small. 

3. Results and Discussion

3.1 Particle Counts Measured versus Time

The particle counts measured at Adelphi, MD, are shown in figure 1 for two sampling periods: 

July 3–7 and July 13–17, 2007. The peak particle count per 0.05 s bin was a little over 500. The 

average particles per 0.05 s bin was 33.4 and 29.4 for the two sampling periods, respectively.
8

Figure 1a. Aerosol particle counts measured at 20 Hz using the Climet particle counter during the 

period July 3–7, 2007. 

Figure 1b. Aerosol particle counts measured at 20 Hz using the Climet particle counter, July 13–17, 2007.

3.2 Power Spectral Densities of Wind, Temperature, and Aerosol for 90 h Periods

The PSDs of the winds (u, v, w), temperature and aerosol concentration, all estimated using a 

modified Welch method, are illustrated in figure 2, which shows the data taken July 13–17, 

2007. The curves have been normalized to have the same value at 0.2 Hz. The aerosol PSD is 
9

calculated for the same data using two different sampling rates (20 Hz and 100 Hz). The inertial 

subrange, as defined by frequencies where the PSD of the winds decrease with a slope of –5/3,

appears to be in the range 0.1 Hz to about 10 Hz. Although the slope of the temperature is close 

to –5/3 in this range (10–1 to 101 Hz), the slope of the PSD of the aerosols is closer to –7/6. The 

next section discusses how a shot-noise floor might lead to flattened slopes of PSDs of aerosols.

Figure 2. PSDs estimated using the modified Welch method for the u component of wind, the temperature 

(Tv), and aerosols. The data for aerosols was binned into two different sampling frequencies 

before analysis. The lines with slopes of –5/3 (solid) and –7/6 (dashed) are for comparison with 

the winds and the aerosols, respectively. The slope of the wind (u) is similar to –5/3 in the inertial 

subrange (10–1 to 101 Hz). 

The PSDs for the aerosols for the two sampling periods (July 3–7 and July 13–17, 2007) are 

illustrated in figure 3, without u and Tv to allow easier comparison. Both have a slope similar to 

–7/6 in the range of about 0.0005 to 0.3 Hz. Below about 0.0005 there is an increase in the 

power in the July 3, 2007 data. 
10

Figure 3. PSDs estimated using the modified Welch method for the aerosols for the two 

sampling periods (July 3–7 and July 13–17, 2007). The line with slope –7/6 

(dashed) is for comparison. 

3.3 Limitations on Frequency Spectra as a Function of Particle Count (Shot Noise Floor)

At sufficiently low sample volumes and/or high sample rates, the number of aerosol particles (of 

whatever type or size range chosen for study) in a sample may become shot noise limited. Other 

passive scalars such as temperature, humidity, or concentrations of gases, are typically measured 

in terms of real numbers. They also have an upper frequency above which the power spectrum is 

resolution limited.

The PSD of a single delta function (or single particle at one instant of time) is flat. Also, the 

PSD of a sampled Poisson distribution with average number of N particles per sample is flat. It 

has power at the highest frequencies measured. For a Poisson distribution, the variance-to-mean 

ratio is 1.0, so if the average particle count per binned counting interval for the OPC is 100, the 

square root of the variance (the standard deviation) would typically be about 10. 

To investigate the relations between particle counts and whether or not a –5/3 slope could be 

observed, we picked a sample window that did have a –5/3 slope, and then randomly omitted 

different fractions of the particles. The results are shown in figure 4.
11

Figure 4. PSDs estimated for a 5-min sample of 

measured particles (blue) and for the sample 

after 90% (green) and then 99% (red) of 

the particles have been randomly eliminated. 

One observation is that as the particle number decreases, the –5/3 slope is maintained but only in 

a smaller frequency interval, one that includes the lowest frequencies. The frequency at which 

the line at –5/3 bends to a slope near zero (i.e., the bend frequency”) moves toward lower 

frequencies as mean particle count decreases.

Also, as the particle count decreases by a factor of 10, the power in the low frequency region 

(where the slope is near –5/3) decreases by a factor of 102

, as expected for cases with 

constructive interference between the Fourier components. On the other hand, the power in the 

high frequencies (where the slope is approximately zero) drops by a single decade, which is 

consistent with a Poisson distribution where the variance to mean ratio is one. That is consistent 

with the Fourier components having random phase relations with each other. These results have 

a similarity to mode-locking of a laser. When the laser is not locked, the fields of the modes 

have random phase relations to other modes. The energy output is constant in time and increases 

linearly with the number of modes that have a given energy. When the laser is mode-locked, the 

fields of the modes are all in phase at specific times, the fields add so that the total field at these 

specific times is N times larger than any one mode, and the power output, proportional to the 

field, squared, is N2 times as large. 

One can imagine an appropriate averaging of curves like those in figure 4, generated with many 

different numbers of particles where the different curves have different “bend frequencies,” to 

obtain slopes that may be near linear, but with slopes much flatter than –5/3—e.g., with a slope 

of –7/6, as in figure 3.
12

3.4 Time-Dependence of the Slopes of the Power Spectral Densities of Wind, 

Temperature, and Aerosol for 5 min Intervals 

The observation of –7/6 slope for the aerosols raises questions about possible effects of shot 

noise (discussed in the next section) and the time variation of the slopes. Figure 5 shows the 

slope of the PSD of the wind u component, sonic temperature, and aerosol concentration for each 

5 min interval of the 90 hr sampling period beginning July 3, 2007. Each slope was calculated 

for a 5 min interval, and the slope was determined by the computer for the PSD between 0.05 

and 1 Hz. Each of these is shown versus time for the first 90 h of the measurement time. 

Although the slopes of the PSDs of the wind and Tv are consistently near –5/3 (within about 

10%), the slopes of the PSDs of the aerosols vary from about –3/2 to around 0. 

Figure 5. Estimate of the slope of the PSDs calculated for the winds (blue), 

sonic temperature (green), and aerosol number concentrations (red). 

3.5 Slope of Aerosol Power Spectral Densities with Low and High Aerosol Concentrations 

Figure 6 shows the mean aerosol particle count per bin, the TKE, and the slope of the aerosol 

PSD for each 5-min window of a 90 hr data collection beginning July 3, 2007. The times when 

the slopes of the aerosol PSDs are relatively large tend to occur when the particle numbers are 

larger. Correlations between the TKE and the slope of the aerosol PSD are not obvious. 

Figure 7 shows the PSDs (estimated using the modified Welch method) of the aerosol particle 

concentrations measured at 20 Hz for the time periods illustrated in figure 4, in the July 3–7, 

2007 data, where (a) is for the period where the concentration is low, and (b) is for the period 

from where the concentration is high. As expected from figure 4, the slope of the aerosol PSD 
13

remains near –5/3 over a much larger frequency range when the aerosol particle counts are high,

as compared to when the aerosol particle counts are low. 

Figure 6. Mean particle count of aerosols, TKE, and slope of aerosol PSD for each 5-min window of 

a 90 hr data collection beginning July 3, 2007. 

NOTE: Top: Average aerosol number per 50 ms sample period in the 5-min window. Middle: TKE for each 

window. Bottom: slope of PSDs (estimated using the modified Welch method) of the aerosol particle 

concentrations measured at 20 Hz.
14

Figure 7a. Aerosol particle concentrations measured at 20 Hz using 

the Climet OPC, July 3–7, 2007, during a period where 

the concentration is lowest. The average particle count 

is about 9.21 per bin.
15

Figure 7b. Aerosol particle concentrations measured at 20 Hz 

using the Climet OPC, July 3–7, 2007, during a

period where the concentration is highest. The 

average particle number is 62.8 per bin.

The mean particle count per bin required to observe an aerosol-PSD slope of a given magnitude 

is explored further in figure 8. Here, when the average particle count per bin is greater than 100 

(for 6000 bins at 20 Hz), the slope can be within a few percent of –5/3. On the other hand, when 

the average particle count per bin is less than about 3, the slope is less negative than –0.5. 
16

Figure 8. The slope of the linear fit of the PSD for a fixed frequency 

interval (0.05 to 1.0 Hz) and a given mean particle count.

Figures 9 and 10 investigate the size of the frequency interval, for which the linear fit of the PSD 

approximates a –5/3 slope. Figure 9 indicates the right-hand bound of the interval, for which the 

fit is closest to a –5/3 slope plotted against the mean particle count per bin. Figure 10 considers 

the largest intervals in which the slope of the linear fit is within 1%, 5%, or 10% of –5/3.
17

Figure 9. Right hand interval bound of the frequency interval 

for which the linear fit of the PSD was closest to –5/3 

slope, with a left hand bound of 0.05 Hz and a given 

mean particle count.
18

Figure 10. Right hand bound of the frequency interval for which the linear 

fit of the PSD is within 1% (blue), 5% (red), or 10% (green) 

of –5/3 slope, with a left hand bound of 0.05 Hz and a given 

mean particle count.

4. General Discussion

The work presented here shows, to our knowledge, the first calculations of the PSDs of aerosol 

particle counts in the atmosphere. Previous PSDs of “aerosols” reported in the atmosphere were 

PSDs of aggregated scattering from all the particles in a volume (defined either by the scattering 

volume of the nephelometer (Porch and Gillette, 1977), or by the resolution of the lidar system

(Radkevich, 2007), which contains many particles).

The aerosol particle count power spectrum may have more energy at higher frequencies, relative

to the turbulence fluctuations of wind velocity or passive tracers, such as temperature or 

concentrations of added gases, especially if the air mass has had a long time to be away from its 

main sources of aerosols. If the air mass sample is many hours from the time the majority of the 

aerosols were injected, or many miles from the main sources of the aerosols in it, and, as this air 

mass moved, turbulence mixed and reduced the relative fluctuations in aerosol concentration, 
19

then maybe the variations in aerosol concentration leftover from the turbulent mixing are still 

there, but are small relative to the Poisson variations.

Particles in an air mass might be distributed relatively homogeneously, if, for example, a large 

fraction of the aerosol in the air mass came from the Pacific Ocean or from photochemical 

generation of particles from gaseous OC, and the gaseous OC and the sunlight were relatively 

homogeneous. Consider an air mass in which turbulent mixing has occurred over a relatively 

long time after the sources have mostly stopped adding aerosol to the mass. For example, the 

source that was added was from a well-mixed tank of 0.5 μm PSL mixed with SF6. The SF6 and 

aerosols in it are then well-mixed by turbulent airflows and diffusion.

The measurable aerosol concentration depends upon the instrument. The smallest turbulent 

eddies studied in the atmosphere have sizes on the order of a few millimeters. (Stull, 1988, p.54). 

5. Summary

Aerosol particle counts, winds, and temperature were measured on a rooftop in Adelphi for two 

90-hr windows in July 2007. The particle counts were binned to 20 Hz and to 100 Hz. PSDs 

were calculated using modified Welch methods.

The slope of the aerosol PSD was observed to be about –7/6 over 4 orders of magnitude, for each 

4-day period. However, the slopes of the aerosol PSDs for each of the 5-min windows in the 

90 hr periods were highly variable. Most of the slopes had magnitudes less than 0.5, and a large 

majority of the slopes had magnitude less than 1. Slopes near –3/2 were very intermittent. 

The frequency range in which a –5/3 slope can be observed depends on the mean particle count 

per bin. The shot noise floor appears to dominate the PSDs when the observed particle counts 

were below roughly 30 particles per bin for time frames that have 6000 samples or 180,000 total 

particles.

Given the shot noise floor limitation of roughly 30 particles per bin at 20 Hz, we would like in 

future work to count as many aerosol particles as possible. Since the counting algorithm in this 

study was limited to aerosol particles that were 0.85 μm and greater, a first step to address this 

limitation is to modify the algorithm to include in the count particles which are smaller. Such a 

modification to the algorithm would not necessarily be trivial using our CLIMET CI-8060,

because the floating “background” issue discussed in section 2.1 would probably become more 

important. It may be easier to switch to a light scattering instrument that did not have any 

significant floating “background”.
20

6. References

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Eyink, G. L.; Sreenivasan, K. R. Onsager and the Theory of Hydrodynamic Turbulence. Rev. 

Mod. Phys. 2006, 78, 87–135.

Frisch, U. Turbulence: the Legacy of A N Kolmogorov, (Cambridge: 1995).

Garvey, D. M.; Walters, D. L.; Fernandez, G.; Pinnick, R. G. 1982: Experimental Verification 

of the Kolmogorov Power Law from Aerosol Concentration Fluctuations. Proceeding, 

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Jaenicke, R. Is Atmospheric Aerosol an Aerosol, Faraday Discussions, 2007.

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Turbulence. Quart. J. Roy. Meteorol. Soc. 1972, 98, 563–589.

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Larsen, M. L.; Kostinski, A. B.; Tokay, A. Observation and Analysis of Steady Rain. Journal of 

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21

Larsen, M. L. Spatial Distributions of Aerosol Particles. Investigation of the Poisson 

assumption. Journal of Aerosol Science 2007, 38, 807–822. 

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23

Appendix. Effect of Dropping 15 s of Data

Due to the way the aerosol particle data were collected in order to investigate longer time periods 

than 5 min, the data needED to be concatenated. Two methods for this concatenation are time 

preserving zero padding and strict concatenation (ignore the 1.5 s of missing data). As can be 

seen in figures A-1 and A-2, there is not a significant structural difference in the PSD estimate 

between the two methods. However, in order to compare with the continuous data provided by 

the sonic anemometer, it is preferred to adopt the time preserving method.

Figure A-1. PSDs for aerosols July 13–17, 2007, with and without the zeros removed. The effect appears 

small.
24

Figure A-2. Same as for figure A-1 though limiting the display to the frequency interval in the region 

of most significant difference for the two PSDs.
25

NO. OF

COPIES ORGANIZATION

1 ADMNSTR

ELECT DEFNS TECHL INFO CTR

ATTN DTIC OCP

8725 JOHN J KINGMAN RD

STE 0944

FT BELVOIR VA 22060-6218

1 DARPA

ATTN IXO S WELBY

3701 N FAIRFAX DR

ARLINGTON VA 22203-1714

1 CD OFC OF THE SECY OF DEFNS

ATTN ODDRE (R&AT)

THE PENTAGON

WASHINGTON DC 20301-3080

1 US ARMY RSRCH DEV AND

ENGRG CMND

ARMAMENT RSRCH DEV &

ENGRG CTR ARMAMENT ENGRG

& TECHNLGY CTR

ATTN AMSRD AAR AEF T 

J MATTS

BLDG 305

ABERDEEN PROVING GROUND

MD 21005-5001

1 PM TIMS, PROFILER (MMS-P)

AN/TMQ-52

ATTN B GRIFFIES 

BUILDING 563

FT MONMOUTH NJ 07703

1 US ARMY INFO SYS ENGRG

CMND

ATTN AMSEL IE TD A Rivera

FT HUACHUCA AZ 85613-5300

1 COMMANDER

US ARMY RDECOM

ATTN AMSRD AMR 

W C MCCORKLE

5400 FOWLER RD

REDSTONE ARSENAL AL

35898-5000

1 US GOVERNMENT PRINT OFF

DEPOSITORY RECEIVING

SECTION

ATTN MAIL STOP IDAD J TATE

732 NORTH CAPITOL ST NW

WASHINGTON DC 20402

NO. OF

COPIES ORGANIZATION

1 US ARMY RSRCH LAB

ATTN RDRL CIM G 

T LANDFRIED

BLDG 4600

ABERDEEN PROVING GROUND

MD 21005-5066

1 DIRECTOR

US ARMY RSRCH LAB

ATTN RDRL ROE V W D BACH

PO BOX 12211

RESEARCH TRIANGLE PARK NC

27709

16 US ARMY RSRCH LAB

ATTN IMNE ALC HRR MAIL &

RECORDS MGMT

ATTN RDRL CIE D 

C WILLIAMSON (10 COPIES)

ATTN RDRL CIE D D GARVEY

ATTN RDRL CIE D C KLIPP

ATTN RDRL CIE S S HILL

ATTN RDRL CIM L TECHL LIB

ATTN RDRL CIM P TECHL PUB 

ADELPHI MD 20783-1197

TOTAL: 26 (24 HCS, 1 ELECT, 1 CD)
26

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