Microphysical Influences on Electro-Optic Signal 

Propagation in Complex Areas 

by Arnold Tunick and Gary Carhart 

ARL-MR-0633 December 2005 

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Army Research Laboratory 

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ARL-MR-0633 December 2005 

Microphysical Influences on Electro-Optic Signal 

Propagation in Complex Areas 

Arnold Tunick and Gary Carhart 

Computational and Information Sciences Directorate, ARL 

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Microphysical Influences on Electro-Optic Signal Propagation in Complex 

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Arnold Tunick and Gary Carhart 

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14. ABSTRACT 

This paper presents a finite-difference computer model to predict the microphysical influences on electro-optic signal 

propagation in complex areas, e.g., around single and multiple building arrays and forests. The model incorporates physicsbased algorithms to account for advection, the pressure gradient, eddy diffusion, and drag forces due to vegetation (e.g., open

fields or forests). Note that mechanisms to account for heating, cooling, and moisture flux are not included at this time. Our 

main concern is to maintain a computationally efficient program code that is also flexible with regard to modifications and 

debugging. This allows us to apply the computer model to different kinds of tests. Nevertheless, we anticipate that this 

research will contribute much useful information related to laser optic signal propagation such as the impact of buildings and 

forests on future optical turbulence (Cn2) calculations. 

15. SUBJECT TERMS

computational fluid dynamics; wind flow; buildings; forests; optical turbulence; laser optics 

16. SECURITY CLASSIFICATION OF:

19a. NAME OF RESPONSIBLE PERSON

Arnold Tunick 

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Standard Form 298 (Rev. 8/98) 

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iii 

Contents 

List of Figures iv

Acknowledgment v

1. Introduction 1

2. Model description 2

3. Model results 4

4. Discussion 8

5. Summary and Conclusions 11

References 12

Distribution List 14
iv 

List of Figures 

Figure 1. Horizontal (a) and vertical (b) cross-section of the vector wind field around a single 

building. Initial wind velocities are u = 3.0 ms

–1 and v = w = 0.0 ms–1. The horizontal 

cross-section is taken at z = 5.0 m........................................................................................... 4 

Figure 2. Horizontal (a) and vertical (b) cross-section of the vector wind field around a five 

building array. The initial wind velocities are u = 3.0 ms

–1 and v = w = 0.0 ms–1. The 

horizontal cross-section is taken at y = 13.0 m and z = 5.0 m................................................. 5 

Figure 3. Vertical cross-section of (a) the vector wind field and (b) wind velocity contours (units 

ms

–1) around a uniform forest canopy. The canopy height is h = 15.0 m. The leaf area 

index is LAI = 4. ..................................................................................................................... 6 

Figure 4. Selected wind velocity profiles taken from the results shown in figure 3, i.e., the initial 

wind velocity (log-law) profile (solid line), the profile +20 m upwind (large dashed line), at 

the leading edge (dotted line), at the forest center (dash-dotted line), at the trailing edge 

(dash-double dotted line), and +20 m downwind of the forest (small dashed line)................ 7 

Figure 5. Horizontal cross-section of the vector wind field around the IOL and adjacent 

buildings. The initial wind velocities are u = w = 0.0 ms

–1 and v = 4.0 ms–1. The horizontal 

cross-section is taken at z = 6.0 m above ground level. The outlined (dashed-double dotted) 

area identifies forests along the western border of the building complex. ............................. 8 

Figure 6. Time series of (a) path-averaged optical turbulence (scintillometer) data and (b) 

microphysical characterization data collected at the ARL A_LOT Facility on 12 December 

2004....................................................................................................................................... 10 
v 

Acknowledgment 

The author gratefully acknowledges Ronald Meyers and Keith Deacon of the U.S. Army 

Research Laboratory (ARL) for offering helpful comments and discussions on this topic. Partial 

funding for this study was provided by the U.S. Army Engineer Research and Development 

Center and ARL. 
vi 

INTENTIONALLY LEFT BLANK. 
1 

1. Introduction 

The Army Research Laboratory (ARL), Atmospheric Laser Optics Testbed (A_LOT) is a unique 

experimental facility with which to measure optical turbulence (Cn2) intensity and investigate its 

effects on infrared imaging and laser optics communications (1,2). Within the A_LOT, a near 

horizontal, 2.3 km optical path extends from the top of a tall water tower to the Intelligent Optics 

Laboratory (IOL) rooftop at ARL. However, complex microphysical influences, e.g., irregular 

wind flow patterns around the IOL and the water tower and the effects from wind shears and 

temperature changes across the top of nearby forest canopies, may affect the A_LOT measured 

data and research applications. To this end, computer simulation models may provide some 

meaningful results even though all the pertinent landscape and/or canopy characterization data 

along the optical path may not yet be known or available. 

In a previous study, Tunick (3) reported on the characterization of complex (e.g., urban) signal 

propagation environments via physics-based, computational fluid dynamics (CFD) models. 

However, CFD program codes are (as a rule) quite computationally intensive (4,5). For 

example, CFD codes can require 1 to 8 hours or more of execution time on multiprocessor supercomputers. In addition, CFD models are generally cumbersome to modify and debug. In 

contrast, it was found that a useful mathematical representation of several key microphysical 

parameters inside and above forests could be obtained by means of a rapid, steady-state, secondorder turbulence closure model, with an embedded radiative transfer and energy budget 

algorithm to predict the heat source (6,7). However, development of this later computer model 

made it possible to generate realistic profiles for the wind speed and temperature within and 

above uniform vegetative canopies only. Hence, we desired an intermediary model, which 

would be computationally efficient and, at the same time, complete in physics to characterize the 

propagation environment around different building geometries embedded in the model grid (as 

well representation of canopy drag forces). In other words, we are interested in models that are 

still reasonably fast but have enough flexibility to apply to the types of field tests envisioned in 

future works (e.g., studies related to free-space laser optics communications from within the 

ARL A_LOT). 

In this paper, we present a finite-difference computer model to predict the three-dimensional 

microphysical influences on signal propagation in complex areas, e.g., around single and 

multiple building arrays and forests. So far, the model accounts for advection, the pressure 

gradient, and drag forces due to vegetation (e.g., open fields or forests). The model incorporates 

first-order (eddy diffusivity) turbulence closure for neutral stability (see Mellor and Yamada (8)). 

Note that mechanisms to account for heating, cooling, and moisture flux are not yet considered 

(in order to maintain computational efficiency and flexibility with regard to modifications and 

debugging). Nevertheless, our research strategy includes the development of such algorithms, 
2 

which can be implemented in future works to improve existing optical turbulence (Cn2) 

modeling and analysis around the ARL A_LOT Facility. 

2. Model description 

Following the numerical methods discussed by Meyers (9) as well as those found in several key 

texts on fluid mechanics (10-14), the simplified Navier Stokes equation (in non-conservative 

form) for the current model, neglecting Coriolis1

 forces, can be expressed as 

u u ( u) k u f

t

u r r r r r r r r

+ ⋅ ∇ = ∇ ∇ ⋅ + ∇ +

∂

∂ 2 β (1) 

where t is the independent variable time, u = u, v,w r is the wind velocity vector, β is a constant, k

is the eddy viscosity (assumed constant), and f

r

is the drag force acceleration due to vegetation. 

In equation 1, the advection term is 

( ) u

z

w

y

v

x

u u u ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂ ⋅∇ = r r , (2a) 

( ) v

z

w

y

v

x

u v u ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂ ⋅∇ = r r , (2b) 

and 

( ) w

z

w

y

v

x

u w u ⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂ ⋅∇ = r r . (2c) 

The divergence portion of the pressure gradient term is 

z

w

y

v

x

u A u

∂

∂

+

∂

∂

+

∂

∂ = ∇ ⋅ =r r , (3) 

and the Laplacian portions of the eddy diffusion term are 

2

2

2

2

2

2

2

z

u

y

u

x

u

u

∂

∂

+

∂

∂

+

∂

∂ ∇ = , (4a) 

 1An earlier survey of the literature (15) noted that other than a few authors, e.g., Shinn (16), Holland (17), and Wilson and 

Flesch (18), most exclude the effect of the Coriolis force as having negligible effect on the scales of motion considered, i.e., wind 

flow through the forest canopy layer. 
3 

2

2

2

2

2

2

2

z

v

y

v

x

v

v

∂

∂

+

∂

∂

+

∂

∂ ∇ = , (4b) 

and 

2

2

2

2

2

2

2

z

w

y

w

x

w

w

∂

∂

+

∂

∂

+

∂

∂ ∇ = . (4c) 

Finally, the vegetation drag force accelerations are 

f C Au u x d

r = , (5a) 

f C Au v y d

r = , (5b) 

and 

f z Cd A u wr = , (5c) 

where Cd is the drag coefficient (0.1-0.2), A is the leaf area density (see ref. (6)), and u

r is the 

magnitude of the total wind. 

Equation 1 is solved explicitly forward in time, with the advection term upwind differenced in 

space and the diffusion term central differenced in space. As an example, the equation for the ucomponent of the wind can be discretized as 

( ) ( ) ( )

[ ] ( )( )( ) ⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎥

⎦

⎤

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎢

⎣

⎡

+ + + ⋅

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Δ

− +

+

Δ

− +

+

Δ

− +

+

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Δ

− +

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Δ

− +⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Δ

− +⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

Δ

− −

= + Δ

+ − + − + −

+ −

− − −

+

n

i j k

n

i j k

n

i j k

n

d i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

i j k

n

n i j k

i j k

n

i j k

n

n i j k

i j k

n

i j k

n

n i j k

i j k

n

i j k

n

i j k

C A u v w u

z

u u u

y

u u u

x

u u u

k

x

A A

z

u u

w

y

u u

v

x

u u

u

u u t

, ,

2

, ,

2

, ,

2

, ,

2

, , 1 , , , , 1

2

, 1, , , , 1,

2

1, , , , 1, ,

1, , 1, ,

, , , , 1

, ,

, , , 1,

, ,

, , 1, ,

, ,

, ,

1

, ,

2

1

2 2 2

2 β . (6) 

The boundary conditions applied in this model are as follows: inflow (typically from the west) is 

a specified value (Dirichlet boundary condition), such as a constant or log-law velocity profile. 

Outflow (typically to the east) is constrained to have a constant flux, i.e., zero gradients 

(Neumann boundary condition). The north and south sides are also constrained to have a 

constant flux. The vertical velocity at the bottom of the model is a specified value (e.g., w = 0) 

and at the model top the vertical velocity has a zero gradient. 
4 

3. Model results 

The model was applied to four cases, i.e., a single building in the center of the grid to test wind 

flow symmetry, an array of five buildings (also to simulate wind flow symmetry), a forest 

canopy centered in the model grid to simulate leading edge flow separation, and a larger and 

more complex building array to simulate wind flow characteristics around the IOL and adjacent 

ARL structures. Grid spacing for the single and multiple building array was uniform, i.e., Δx = 

Δy = Δz = 1.0 m, where the number of grid points was 27 x 27 x 12 for the single building and 50 

x 50 x 20 for the array. Here, Δt = 0.001 s and the model was executed for 14K time-steps. For 

the forest application, the grid spacing was Δx = Δy = 2.0 m and Δz = 1.0 m, where the number of 

grid points was 60 x 60 x 30. Here, Δt = 0.001 s and the model was executed for 40K time-steps. 

For the IOL and adjacent buildings, the grid spacing was Δx = Δy = 2.5 m and Δz = 2.0 m, where 

the number of grid points was 100 x 100 x 25. Here, in contrast, Δt = 0.001 s and the model was 

executed for 20K time-steps. The computer model was executed on a 1.8 GHz desktop PC with 

approximately 512 MB RAM. Model runtimes were approximately 2 minutes for the single 

building, 20 minutes for the five building array, 50 minutes for the forest canopy, and 2 hours 

and 20 minutes for the IOL and adjacent buildings. 

Figure 1 presents horizontal and vertical cross-sections of the vector wind field around a single 

building located in the middle of the model grid. Here, the initial wind velocities are u = 

3.0 ms

–1 and v = w = 0.0 ms–1. The horizontal cross-section is taken at z = 5.0 m above ground 

level. This result is very encouraging. The wind flow patterns going around (and over the top 

of) the single building appear quite reasonable. 

 (a) (b) 

Figure 1. Horizontal (a) and vertical (b) cross-section of the vector wind field around a single building. Initial wind 

velocities are u = 3.0 ms–1 and v = w = 0.0 ms–1. The horizontal cross-section is taken at z = 5.0 m. 

Y 
5 

Figure 2 presents horizontal and vertical cross-sections of the vector wind field around a five 

building array. Again, the initial wind velocities are u = 3.0 ms

–1 and v = w = 0.0 ms–1 and the 

cross-sections are taken at y =13.0 m from the south edge and z = 5.0 m above ground level. 

These results are also quite encouraging, especially since they provide good visual confirmation 

(i.e., strong symmetry) that the program coding is error-free. 

 (a) (b) 

Figure 2. Horizontal (a) and vertical (b) cross-section of the vector wind field around a five building array. The 

initial wind velocities are u = 3.0 ms–1 and v = w = 0.0 ms–1. The horizontal cross-section is taken at 

y = 13.0 m and z = 5.0 m. 

Figure 3 presents vertical cross-sections of the wind field around a uniform forest canopy located 

in the middle of the model grid. The canopy height is h = 15.0 m. The leaf area index is LAI = 4. 

In addition, a generic leaf area density profile is assumed (see ref. 6,7). Although somewhat 

difficult to visualize using small vector arrows, the wind field (shown in figure 3a) does separate 

(diverge) at the forest edge. Part of the flow goes downward into the trunk space of the forest, 

below the layer of leaves and branches, and part of the flow goes upward through the canopy top 

and above. While streamlines may be generated in future works to better visualize these model 

data, this result is highly consistent with previously published works (16, 19, and 20). Moreover, 

figure 3b shows clearly that the magnitude of the winds decrease with distance into the forest 

(due to cumulative drag effects), especially in the upper half of the forest stand. In addition, 

secondary, low level wind speed maxima develop through the forest leading edge. 

Y 
6 

 (a) (b) 

Figure 3. Vertical cross-section of (a) the vector wind field and (b) wind velocity contours (units ms–1) around a 

uniform forest canopy. The canopy height is h = 15.0 m. The leaf area index is LAI = 4. 

Figure 4 presents selected wind velocity profiles taken from the model data shown in figure 3, 

i.e., the initial wind velocity (log-law) profile, the profile +20 m upwind, the profile at the 

leading edge, the profile at the forest center, the profile at the trailing edge, and the profile +20 m 

downwind of the forest. The wind speed profile in the open fields is logarithmically increasing 

with height above the roughness plane. Inside the forest, the winds are shown to decrease 

rapidly as momentum becomes depleted through the layers of leaves and branches (due to 

increased drag). In addition, for the profile in the forest center, the model produces a secondary 

wind speed maximum at a height of about z = 4.0 m. Shinn (16) and Shaw (21) discuss such 

low-level wind maxima in detail. Note that the model results shown here will be sensitive to 

variations in the assumed leaf area index (LAI) and leaf area density distribution in the canopy 

(6,7). 
7 

Figure 4. Selected wind velocity profiles taken from the results shown in figure 3, i.e., the initial wind velocity (loglaw) profile (solid line), the profile +20 m upwind (large dashed line), at the leading edge (dotted line), 

at the forest center (dash-dotted line), at the trailing edge (dash-double dotted line), and +20 m 

downwind of the forest (small dashed line). 

Figure 5 presents a horizontal cross-section of the vector wind field around the IOL and adjacent 

buildings. Here, the initial wind velocities are u = w = 0.0 ms

–1 and v = 4.0 ms–1. The horizontal 

cross-section is taken at z = 6.0 m above ground level. The (dashed) outlined area identifies 

forests along the western border of the building complex, for which a constant drag force with 

height (up to the canopy top, h = 18.0 m) is assumed. More complete canopy characterization 

data, like those implemented in the previous example, may be incorporated in future works. 

Nevertheless, this model result is very appealing. It demonstrates that we have completed 

several successful steps towards obtaining a useful mathematical representation of the wind flow 

around the IOL and adjacent buildings. Naturally, these results (as well as those shown above) 

will need to be confirmed by measurements. Yet, if determined to be realistic, this kind of data 

will be very important for electro-optic and acoustic characterization and prediction. 
8 

Figure 5. Horizontal cross-section of the vector wind field around the IOL and adjacent buildings. The 

initial wind velocities are u = w = 0.0 ms–1 and v = 4.0 ms–1. The horizontal cross-section is taken 

at z = 6.0 m above ground level. The outlined (dashed-double dotted) area identifies forests along 

the western border of the building complex. 

4. Discussion 

Note that the wind flow model presented in this report is still a program code in development, 

even though it already exhibits some powerful capabilities. Additional algorithms and/or 

alternate numerical schemes may be implemented in future works to improve the model. For 

example, temperature and moisture prediction may be investigated via the following expressions, 

i.e., 

θ θ

θ θ θ

κ

θ θ θ θ S

x y z z

w

y

v

x

u

t

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂ = − ∂∂

2

2

2

2

2

2

, (7) 
9 

and 

q q S

z

q

y

q

x

q

z

wq

y

vq

x

uq

t

q +⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂

+⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

+

∂

∂

+

∂

∂ = − ∂∂

2

2

2

2

2

2

κ , (8) 

where θ is air temperature (in units oK), q is specific humidity (in units kg kg-1), κθ and κq are the 

eddy diffusivities for heat and water vapor, respectively (in units m2s

-1), and Sθ and S q are the 

source terms for heat and moisture. The quantity Sθ for forests can be determined via the 

radiation and energy budget equations presented in (7). Alternately, a simpler form for the heat 

source was shown by Shaw and Schumann (22), i.e., 

( )

dz

dQ z Sθ = , (9) 

where Q(z) = Q(h)exp(–αF). Here, Q is the kinematic heat flux, h is the height of the canopy 

top, α = 0.6 is an extinction coefficient, and = ∫

h

z

F Adz is the non-dimensional cumulative leaf 

area index. Naturally, for open fields, buildings, and/or pavement, other suitable heat source, 

moisture source, and energy budget relations can be applied. 

Overall, we expect that the current research (when completed) will be quite useful to improve 

optical turbulence intensity (Cn2) prediction and analysis around the ARL A_LOT Facility. Our 

work will also contribute towards improved estimates of other key signal propagation 

parameters, such as the variance in angle-of-arrival fluctuation and the log-intensity variance of 

transmitted electromagnetic signals. As an example, for propagation across distances less than 

or equal to approximately 4 km, Beland (23) gives an expression for the variance in angle-ofarrival fluctuation for spherical waves as, 

( ) dz

L

z D C z

L

A n

3

5

3

1

0

2 2 2.914 ⎟

⎠

⎞ ⎜

⎝

⎛ = ∫ − σ , (10) 

for ( ) 2

1

D >> λ L , where D (≈ 0.090 m, possibly) is aperture diameter, λ (≈ 0.94 μm, possibly) is 

wavelength, L is optical path length, and z is the vertical coordinate. Analogous expressions 

have been presented for the log-intensity (or log-amplitude) variance of transmitted 

electromagnetic signals, so that the measure of the path-averaged Cn2 due to scintillation (i.e., 

temporal fluctuations) for a spherical wave can be expressed as, 

( ) ( ) L z dz

L

z k C z

L

x n 6

5 6

5

6

7

0

2 2 0.56 ⎟ − ⎠

⎞ ⎜

⎝

⎛

σ = ∫ (11) 

where k is the wave propagation constant ( ) λ

2π . Thus, if improved microphysical 

characterization models will provide better estimates of Cn2 along more complex optical lines-
10 

of-site, then better estimates of beam displacement, 2

A σ , and intensity fluctuations, 2

σ x , are 

the kinds improved work product that may be useful in sensor performance (and/or sensor data) 

analysis. Note that for model evaluation purposes, the ARL A_LOT can provide time series of 

optical turbulence (scintillometer) and other pertinent data, e.g., like those shown in figure 6. 

 (a) (b) 

Figure 6. Time series of (a) path-averaged optical turbulence (scintillometer) data and (b) microphysical 

characterization data collected at the ARL A_LOT Facility on 12 December 2004.

Finally, a useful expression for the refractive index structure parameter(24-26) is, 

⎟

⎠

⎞ ⎜

⎝

⎛

∂

∂

z

K n

C (z)= b

2

1/3

2 h n ε

, (12) 

where b is a constant, Kh is the turbulent exchange coefficient for heat, ε is the energy dissipation 

rate, and ∂n/∂z is the vertical gradient of the index of refraction (n). In equation 12, the vertical 

gradient of the index of refraction for visible and near-infrared wavelengths (0.36 to 3 μm) can 

be expressed as a function of the partial derivatives for potential temperature (∂θ/∂z) and 

moisture (specific humidity) (∂q/∂z), i.e., 

z

q

x 10

T

P

+1.61( M ( )- M ( ))

z

x 10

T

Pq - 1.61( M ( )- M ( ))

T

P = - M ( )

z

n

-6

2 1

-6

2 2 1 2 1 v

∂

∂

∂

∂

⎟

⎟

⎠

⎞

⎜

⎜

⎝

⎛

∂

∂

λ λ

θ λ λ λ

 (13) 
11 

where the dispersion formulas (as functions of wavelength–1, i.e., σ = λ

–1) are, 

σ σ

λ 2 2 1 38.9 -

45.473

+

130 -

6839.397

M ( )=23.7134+ , (14) 

and 

λ σ σ σ

2 4 6 M 2 ( )= 64.8731+0.58058 -0.007115 +0.0008851 . (15) 

5. Summary and Conclusions 

This report presented a new finite-difference computer model to predict the surface layer wind 

flow around single and multiple building arrays and forest canopies. The model accounts for 

advection, the pressure gradient, diffusion, and drag forces due to vegetation. The model code is 

computationally efficient and extremely flexible with regard to modifications and debugging. 

Our initial model results are quite encouraging. Hence, we have successfully demonstrated new 

capabilities towards obtaining a useful mathematical representation of microphysical 

characterization data around the IOL and adjacent buildings. In conclusion, we expect that these 

kinds of simulation codes will be an important vehicle for investigating optical turbulence intensity 

(Cn2) and related laser-optics propagation effects in and around complex environments. 
12 

References 

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13 

14. Ferziger, J. H.; Perić, M. Computational Methods for Fluid Dynamics; 3rd Edition, 

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14 

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