Molecular Diffusion of NO and
in the Thermosphere
4
N( S)
Justin D. Yonkera, Scott M. Baileyb, Larry J. Paxtonc
a- Department of Physics, Virginia Tech, Blacksburg, VA, [email protected]
b- Bradley Department of Electrical and Computer Engineering, Virginia Tech, Blacksburg, VA
c- Johns Hopkins University/Applied Physics Laboratory
Abstract
In this paper the effect of molecular diffusion on the minor species NO and N(4S) from
110-170 km is examined and the relevant diffusion coefficients are calculated. Below
about 110 km, eddy diffusion is prevalent in the atmosphere and the densities of the
main atmospheric components decrease with altitude in accord with the mean scale
height. In this state of 'perfect mixing', the mixing ratios are approximately constant
with altitude. Above 110 km, however, collisions are infrequent, turbulence is difficult
to sustain, and the neutral atmosphere tends to a state of diffusive equilibrium in which
the densities fall off in accord with their individual scale heights. Vertical transport in
this region is dominated by molecular diffusion and driven by gradients in density,
temperature, and pressure.
To calculate the relevant transport coefficients requires knowledge of the associated
collision integrals of various orders. Such integrals appear in the collisional term of the
Boltzmann equation and are needed to solve for the distribution function. Tabulations
of the required collision integrals have recently appeared in the literature [Capitelli et al
(2000), Wright et al (2005)] and are here used with NRLMSIS and expressions from
Hirschfelder, Curtiss, and Byrd (1954) to calculate the molecular diffusion coefficient, D,
and the thermal diffusion factor, αT, for NO and N(4S) as a function of thermospheric
altitude.
These diffusion coefficients are then incorporated in a time-dependent, 1D
photochemical model of thermospheric NO and compared with one year of equatorial
data taken by the Student Nitric Oxide Explorer (SNOE) at 11 AM LST. Because the
vertical flux driven by the pressure and concentration gradients is typically much larger
than that due to the temperature gradient, it is concluded that the thermal diffusion
factors for both NO and N(4S) can be safely ignored for 1D modelling of NO in the
thermosphere. Calculated values of the NO molecular diffusion coefficient and vertical
velocity are compared with those of previous authors.
Description of the Model
Above 110 km, transport due to eddy diffusion can be ignored. In this region the model
calculates the NO and N(4S) density by solving the continuity equation (Bailey et al, 2002):
∂ni
∂Φ i
= Pi − Li ni −
∂t
∂z
(e.1)
where i=NO or N(4S). The vertical flux due to molecular diffusion is given by (Nicolet, 1960):
 ∂ni
ni ∂T
ni 
i
Φ i = − Di 
.
 ∂z + (1 + α T ) T ∂z + H 
i 

(e.2)
The first term on the right represents transport due to concentration gradients, the second
due to thermal gradients, and the third due to the partial pressure scale height,
1
1 ∂pi mi g
=
=
.
Hi
pi ∂z k BT
P=Chemical production (mol cm-3s-1)
D=Averaged molecular diffusion coefficient (cm2s-1)
αT=Averaged thermal diffusion factor (unitless)
Z= Altitude (cm)
L=Chemical loss frequency (s-1)
H=Pressure scale height (cm)
φ=Diffusive Flux (mol cm-2s-1)
T=Temperature (K)
Density Scale Height
Another way to view equation (e.2) is to write it as follows (Chamberlain and Hunten,1987):
 1
1
Φ i = Di ni 
−
*
*
H
H iE
 i
1
1 ∂ni
=−
*
Hi
ni ∂z
i
1
mi g (1 + αT ) ∂T
=
+
*
H iE k BT
T
∂z




(e.3)
(e.4)
(e.5)
Hi* is the actual density scale height, while HiE* is the density scale height for a constituent in
diffusive equilibrium. In equilibrium the vertical flux is zero and Hi*= HiE*. This describes Ar and
He well, but for NO and N(4S) chemical production keeps the density at most altitudes larger
than it would be in diffusive equilibrium. As the densities do not fall off as fast as the equilibrium
distributions, both species will exhibit a net vertical transport downward. With knowledge of the
molecular diffusion coefficent, D, and the thermal diffusion factor, αT , both of which are defined
through the collision integrals, the diffusive fluxes and velocities can be calculated.
( n ,s )
Ω
Definition of the Collision Integral,
The collision integral of order n,s at temperature, T, is defined classically by three integrals
(Capitelli et al, 2000):
(1) the deflection angle, Θ, at impact parameter, b ,and relative collision energy, E
∞
θ (b, E ) = π − 2b ∫
rc
dr
1−
2
b V (r )
−
r2
E
rc=distance of closest approach
V(r)=Intermolecular potential
(2) the collision cross section, Q(E), obtained by averaging over the impact parameter.
n
−1 ∞
∞
 E 
 [1 + (−1) ] 
n
Qn ( E ) = 2π 1 −
 ∫ db[b(1 − cos θ (b, E )]
2(n + 1)  0

(3) the cross-section averaged over relative collision energy

−
F ( n, s )
2 ( n ,s )
 k BT  ( s +1)
σ Ω (T ) =
e
E Qn ( E )dE
s+2 ∫
2(k BT ) 0
F ( n, s ) =
4(n + 1)
π ( s + 1)![2n + 1 − (−1) n ]
-F scales the collision integral to that for scattering by a hard sphere of diameter σ.
-Subscripts on Ω indicate the colliding species, superscripts the order.
Interpretation of the molecular diffusion coefficient, D
Consider a two-component (binary) gas mixture with components 1 and 2 initially
partitioned in the left and right half of the box. When the partition is removed, each gas
will have a bulk velocity opposite in sign that works to remove the concentration gradient. If
the box is maintained at constant pressure and temperature and in the absence of any other
differential accelerations acting on the components (e.g. electric fields and ions), the bulk
velocity of the component 1 relative to 2 is given by (Chapman and Cowling, p.144):
ni=number density of species i=1,2 ,
2
n= n1 + n2
n 
1 ∂n1 
C1 − C2 = −
 D1−2

D1-2=molecular diffusion coefficient for
n1n2 
n ∂r 
binary mixture of species 1 and 2
For applications to the thermosphere, consider component 1 to be the minor species
NO or N(4S) and component 2 to be N2, O2, or O so that n2=n. The flux, Φ1=n1C1, of the
minor species relative to the stationary background major species is then given by
 ∂n1 
Φ1 = − D1−2 

 ∂r 
At equilibrium Φ1=0 and the concentration gradient has been removed. In terms of the
collision integrals, D1-2 is given by (Hirschfelder et al, p. 539)
1/ 2
D1−2
3  2k BT 
=


16n  πµ 
1
1
T3
=
.
002628
(1,1)
(1,1)
2 µ (σ 12 ) 2 Ω12
p (σ 12 ) 2 Ω12
where the second term holds for the pressure, p, in dyne/cm2, and collision diameter, σ12, in Å2.
Calculated Binary Molecular Diffusion Coefficients at 1 atm
Figure 1
These show the molecular diffusion coefficients as a function of temperature
at a constant pressure of 1 atm for NO and N(4S) in binary mixtures with the
three major thermospheric gases. The coefficients for the mixtures with
atomic oxygen are largest due to the smaller relative diameter of the N-O
and N2–O collisions. For comparison, the result of Bzowski et al (1990) for
NO-N 2 is also shown.
Interpretation of the thermal diffusion factor, αT
In the absence of any other differential accelerations acting on the gas, when a temperature
gradient is applied to an initially uniform, two-component (binary) gas mixture that is
maintained at constant pressure, the bulk velocity of the component 1 relative to
component 2 is given by (Chapman and Cowling, p.144):
n2 
1 ∂n1
1 ∂T 
+ DT
C1 − C2 = −
 D1−2

n1n2 
n ∂r
T ∂r 
ni=number density of species i=1,2 , n= n1 + n2
D1-2=molecular diffusion coefficient for gas of species 1 and 2
DT= thermal diffusion coefficient
Again, consider component 1 to be the minor species NO or N(4S) and component 2 to be
N2, O2, or O so that n2=n. The diffusive flux of the minor species relative to the stationary
background major species is then given by
1 ∂T 
 ∂n1
Φ1 = n1C1 = − D1−2 
+ n2 kT

T ∂r 
 ∂r
with
DT
kT =
D1−2
where kT is called the thermal diffusion ratio. Equilibrium is obtained when Φ1=0. The result:
1 ∂n1
1− 2 1 ∂T
= αT
n1 ∂r
T ∂r
with
kT = α
1− 2
T
n1n2
.
2
n
In the presence of a temperature gradient a steady state is reached in which the thermal
gradient is proportional to the density gradient with αT the coefficient of proportionality.
In the second-order Chapman-Cowling approximation the thermal diffusion factor, αT1−,2
is defined in terms of the thermal conductivity, λ, by (Hirschfelder et al, p. 534):
α
1− 2
T
x1 x2 S (1) x1 − S ( 2 ) x2
= (6C − 5)
λ12
X λ + Yλ
λ1 = 1989.1 ⋅10−7
1
T
( 2, 2 )
m1 (σ 1 ) 2 Ω11
λ2 = 1989.1 ⋅10−7
1
T
m2 (σ 2 ) 2 Ω(222, 2 )
S
m + m2 λ1−2 15  m2 − m1 

 −1
= 1
−
2m1 λ1 4 A12*  2m1 
S ( 2)
m + m1 λ1−2 15  m1 − m2 

 −1
= 2
−
2m2 λ2 4 A12*  2m2 
Xλ =
Yλ =
x12
λ1
2
1
x
λ1
+
U
2 x1 x2
(1)
λ1−2
+
+
λ1−2
U
(1)
U
( 2)
2
4 * 1  12 *
 m1 (m1 − m2 )
= A12 −  B12 + 1
+
15
12  5
2m1m2
 m2
2
4 * 1  12 *
 m2 (m2 − m1 )
+
= A12 −  B12 + 1
15
12  5
2m1m2
 m1
U (Y ) =
x22
λ1
2 x1 x2
=mole ratio of species 1
x2 =mole ratio of species 2
λ1 =thermal conductivity for pure gas of species 1
λ2 =thermal conductivity for pure gas of species 2
λ12 =thermal conductivity for mixture of species 1 and 2
µ =reduced mass
x1
1
T
( 2, 2 )
2 µ (σ 12 ) 2 Ω12
λ1−2 = 1989.1 ⋅10−7
(1)
*
12
U +
Y
x22
λ1
U
( 2)
A12* =
2
5  12
4 * (m1 + m2 ) 2 (λ1−2 ) 2 1  12 *
m
 (m − m2 )
A12
−  B12 + 1 1 −  B12* − 5  1
15
4m1m2
λ1λ2 12  5
 m1m2
 m2 32  5
( 2, 2 )
(1, 2 )
(1, 3)
(1, 2 )
Ω12
− 4Ω12
(5Ω12
) * Ω12
*
=
=
,
,
B
C
12
12
(1,1)
(1,1)
(1,1)
Ω12
Ω12
Ω12
These equations are used in the following to calculate α in binary mixtures
where 1=NO or N(4S) and 2=N2, O2, or O.
1− 2
T
Thermal Diffusion Factors for Atomic
Oxygen-Nitrogen Mixtures
Figure 2 *
Figure 3 *
Figure 2 shows α calculated at varying O and N densities and temperatures. It compares
excellently with Figure 10 of Levin et al (1990) (not shown) and so demonstrates the
robustness of the code. Figure 3 shows the sign convention which is that the flux due
to the temperature gradient is such that the lighter component tends to the hotter region.
O− N
T
*-Both titles on figure 2 and 3 should read “Thermal Diffusion Factor” rather than “Ratio”.
Figure 4
Thermal Diffusion Factors
Figure 5
Both plots show α for mixtures where, as appropriate to the thermosphere, the ratio of the
density of the minor species N(4S) and NO to that of the major species tends to zero. The
Capitelli plots show the expected classical behavior as T->0. The magnitudes are consistent
with the interpretation that the thermal diffusion factor is proportional to the mass
difference of the colliding species while the signs again show that the heavier component
in the mixture tends toward the colder region.
1− 2
T
Collision Integrals Used in Calculations
Two comprehensive compilations of collision integrals for thermospheric species have recently
appeared. In the following figures, results labelled ‘Capitelli’, use the coliision intergrals of
Capitelli et al (2000). Results labelled ‘WSLP’, ‘Wright (2005)’, or those that are
unlabelled use the collision integrals of Wright (2005), except in the following cases:
Table 1
Collision Integral (all orders of n,s)
Reference
ΩN-N
ΩN-O
ΩO-O
Levin, Partridge, Stallcop (1990)
ΩN-N2
Stallcop, Partridge, Levin (2001)
ΩN2-N2
Stallcop, Partridge, Levin (2000)
Although the results obtained by the Capitelli group are good compared to the experimental
results for DO-O2 and DN-N2 at 300 K., their focus was on aerobraking temperatures which far
exceed those typically found in the thermosphere. Due to this, and because the differences are
small in many cases, in much of what follows only the results obtained from the WSLP
tabulations are illustrated. Also, the Levin, Partridge, and Stallcop group employ quantum
methods, rather than classical, to calculate the collision cross sections as a function of the phase
shift and angular momentum quantum number, instead of the classical impact parameter and
deflection angle. Errors range from 5% for atom-atom interactions to 25% for
molecule-molecule interactions.
Thermospheric Molecular Diffusion Coefficients
Figure 6
These figures show the binary molecular diffusion coefficients as a function of altitude using
NRLMSIS temperatures. The plot labelled “Average” is obtained using the following equation
from Banks and Kockarts (1973b, p.34) for the diffusion coefficient in a multicomponent mixture:
nj
1
=∑
.
Di j ≠i nDi − j
Between 150-160 km, the calculated DNO varies from 6e8 to 1e9 cm2/s which is significantly
higher than the values of 2e8 to 4e8 cm2/s cited in Banks and Kockarts (1973a, p.350).
Thermospheric Thermal Diffusion Factors
Figure 7*
The general trend of the magnitudes with increasing altitude is explained by two factors: the increasing frequency of
collisions with atomic oxygen and the decreasing frequency of collisions with O2 . For N(4S), the average mass difference
is smaller, and so the magnitude of αT decreases. The opposite is true for NO. The plots are calculated using NRLMSIS
temperatures and the same type of averaging as in Figure 6. The validity of this method is called into question by
Haugen et al (2005) who states ‘multicomponent thermal diffusion coefficients may not be estimated from binary data’
and illustrates the proper method. While this is true for typical fluids, the diluteness of the thermosphere, and the
relative smallness of the thermal diffusive flux, suggests that the error incurred with this averaging scheme is minor.
*-The title on Figure 7 should read “Thermal Diffusion Factor” rather than “Ratio”.
Comparison with Observations
Figure 8
Shown is the average of one year of zonally-averaged SNOE data and the average NO density
profiles for one year calculated using equation (e.1) and the diffusion coefficients shown in
Figures 6 and 7. The average model/data discrepancy at all altitudes above 110 km is 16%
when using the Capitelli collision integrals and 14% when using the WSLP integrals. Inclusion
of the thermal diffusion factors improves the model/data discrepancy by .07% for both sets.
For reasons shown in the remaining figures, the thermal diffusion factor may be safely
neglected in photochemical/transport models of NO in the thermosphere.
Calculated N(4S) and NO Densities
and Density Gradients
Figure 9
Figure 10
These figures show that including the thermal diffusion factor for N(4S) very slightly affects the
modelled density and density gradient, though the similar effect is not discernible for
NO . From equation (e.2), the diffusive flux resulting from the concentration gradient is
obtained when the bottom figures are multiplied by the corresponding molecular diffusion
coefficients (see Figure 6).
Diffusive Fluxes and Vertical Velocity of NO
Figure 12
Because the NO density decreases with altitude (Figure 10), diffusion due to the concentration gradient
is positive (upward). Yet the flux due to the equilibrium density scale height, (the sum of the pressure
and thermal fluxes), dominates this contribution at all altitudes. Between 150-160 km the calculated NO
vertical velocity, obtained by dividing the total flux by the density, ranges from -100 to -130 cm/s
(downward) which compares well with the Banks and Kocharts (1973a) value of -140 cm/s.
Diffusive Fluxes and Vertical Velocity of N(4S)
Figure 13
Many of the features of these plots are similar to those of Figure 12, with the exception that for N(4S),
unlike NO, the flux due to the concentration gradient is negative. As shown in Figure 9, the N(4S)
density increases with altitude and is thus extremely far from diffusive equilibrium. The N(4S) total
diffusive flux is an order of magnitude larger than that of NO at the higher altitudes, and the vertical
velocity much faster.
Conclusion
Using recently published tabulations of collision integrals, the molecular diffusion coefficients
and thermal diffusion factors for NO and N(4S) were calculated for thermospheric conditions
between 110-170 km. Eddy diffusion was assumed to be nonexistent at these altitudes and the
resulting flux term in the continuity equation was determined solely by molecular diffusion.
Deviations of the actual densities from the equilibrium density scale heights resulted in a net
downward transport of both species at all altitudes.
The molecular diffusion coefficient and vertical velocity for NO from 150-160 km were computed
for the 1999 vernal equinox and compared with those given in Banks and Kocharts (1973a). Our
computed velocity ( -100 to -130 cm/s) compared favorably with theirs (-140 cm/s) while our
molecular diffusion coefficient (6e8 to 1e9 cm2/s) was about three times higher than theirs
(2e8 to 4e8 cm2/s).
Upon comparison with one year of SNOE data, the average model/data discrepancy for all
altitudes was 14% when using the collision integrals of Wright et al (2005) and 16% with those
of Capitelli et al (2000). Because of the dominant contribution to the diffusive flux from the
pressure scale height, inclusion of the thermal diffusion factor in the model changed the
results by less than .1 %. This implies that 1D modelling of thermospheric NO can be accurately
accomplished without inclusion of the thermal diffusion factor. The same statement may not
hold, however, for a 2D model that includes latitudinal temperature gradients. In this case,
transport due to the pressure scale height would vanish, and the contribution of thermal
diffusion to the horizontal diffusive flux would be much more significant.
Updated NO Chemical Reaction Rates
Table 2
Reaction
N ( 2D) + O2 → NO + O
N ( 4S ) + NO → N 2 + O
N ( 2D) + NO → N 2 + O
Rate Coefficient
6.2e(−12)(T 300)
4.35e(−12) exp(130.68 T ) ⋅ T .28
Source
Duff et al (2003)
Gamallo et al (2006)
Herron (1999)
6e(−11)
N ( 2D) + O → N ( 4S ) + O
1.65e(−12) exp(− 260 T )
Herron(1999), Fell and
Steinfeld(1990)
N ( 4S ) + O2 → NO + O
1.5e(−11) exp(− 3600 T )
JPL (2003)
N 2+ + e → 2 N ( 2P,2D,4S )
2.2e(−7)(Te 300)
NO + + e → N ( 2D,4S ) + O
3.5e(−7)(Te 300)
N 2 ( A, v > 2) + O
→ N ( D, S ) + NO
2
4
−.39
BR = (0.10,1.42,0.48)
−.69
BR = (0.85,0.15)
1.7e(−11)(T 300)
BR = (0.5,0.5)
.5
Sheehan and St.
Maurice (2004)
Sheehan and St.
Maurice (2004)
Thomas and Kaufman
(1997)
Except for the above, all rate coefficients used in the model are those of Bailey et al (2002).
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