UNIVERSITY OF CINCINNATI
Date:______
I, ______, hereby submit this work as part of the requirements for the degree of: in:
It is entitled:
This work and its defense approved by:
Chair: ______
Theoretical Analysis of the Temperature Variations and the Krassovsky Ratio for Long Period Gravity Waves
A dissertation submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of
DOCTOR OF PHILOSOPHY (Ph.D)
in the Department of Physics of the College of Arts and Science 2008 by
Tharanga Manohari Kariyawasam
M.S., University of Cincinnati, Cincinnati, OH B.S., University of Colombo, Colombo, Sri Lanka
Committee Chair: Professor Tai-Fu Tuan
i
Abstract
Based on the assumption that they are caused by atmospheric gravity waves rather than atmospheric tides, this study aims at developing a theoretical analysis of the long period (~ 8 hour) fluctuations of both the Meinel OH band intensity and the rotational temperature. Eddy thermal conduction and eddy viscosity is included in the calculation. In addition, to account for the very long periods (~ 8 hour), Coriolis force due to earth’s rotation will also be taken into account by employing the “shallow atmosphere” approximation. The current theoretical analysis differ from the prior models in that it will include the Coriolis force and the model will deal with very long periods, and in addition the height varying background wind is also included in the discussion. Long period fluctuations in the airglow have been measured in many recent experimental observations (Taylor M.J., Gardner L.C., Pendleton W.R., Adv. Space Res., 2001). The Krassovsky ratio which determines the efficiency of producing an intensity fluctuation for a given temperature fluctuation, and also the phase difference between the intensity and temperature fluctuation will also be calculated based on the gravity wave assumption.
ii
iii Acknowledgements
First and foremost, I would like to convey my sincere gratitude to my faculty advisor,
Professor Tai-Fu Tuan, for his invaluable guidance and relaxed, thoughtful insight in
completing this work. I greatly appreciate the sincere support that he has provided me in
all my endeavors over the years. I also would like to extend my deepest gratitude to my
dissertation committee members Professor Paul Esposito, Professor Rostislav Serota,
Professor Rohana Wijewardhana and Professor Bernie Goodman for their comments,
advice and for helping me in many ways to complete my research work successfully.
I am particularly thankful to Professor Frank Pinski, Dr. Richard Gass and Professor
Nageswari Shanmugalingam for sharing their constructive ideas and for their help in my research project. I would also like to thank Dr M. J Taylor and Dr. J.R Winick for sharing their insight and for the helpful discussions. I am indebted to the faculty members at the
Physics Department for their many advice and lessons in Physics. I am grateful to Donna
Deutenberg, Elle Mengon, John Whitaker and Melody Whitlock and all other staff members at the Physics Department for their cooperation and support. I wish to thank my friends, at the Department of Physics, for the most enjoyable time and for the wonderful memories. Many thanks, to the Department of Physics and the Graduate
Student Association for the financial support.
A special thanks to my husband for his understanding, continuous encouragement and
guidance without reservation. Finally, I wish to thank my parents for their love and guidance, without which I would never have enjoyed so many opportunities.
iv TABLE OF CONTENTS
Abstract……………………………………………………………………………….. i Acknowledgement……………………………………………………………………. ii Table of Contents………………………………………………………………….…. iii
CHAPTERS
1 Introduction 1.1 Atmospheric Gravity waves 01 1.2 Airglow Emissions in the Middle Atmosphere 03 1.3 Krassovsky Ratio Method 07 1.4 Experimental observations 1.4.1 Taylor’s Observations 09 1.4.2 Oznovich’s Observations 11 1.5 Shallow Atmosphere Approximation 12
2 Mathematical Formulation 2.1 Basic Hydrodynamic Theory 14 2.2 Gravity wave Model 18 2.3 Atmospheric Model Used 23 2.4 Airglow Response 2.4.1 Eulerian continuity equation 26 2.4.2 The internal gravity wave response 28 2.4.3. Brightness Weighted Temperature Method 30
3 Hydroxyl Chemical Kinetic Model 3.1 Background 34 3.2 OH Vibrational Kinetic Model 36
4 Results & Discussions 4.1 Numerical Results and Discussions 43 4.2 Conclusion 50
5 Figures 52 6 Bibliography 91 7 Appendices 7.1 Appendix A 98 7.2 Appendix B 101 7.3 Appendix C 113
v CHAPTER 1
INTRODUCTION
1.1 Atmospheric Gravity Waves
Low altitude tropospheric sources such as large earth quakes or nuclear explosions are capable
of generating upper atmospheric modes with very long periods which are observed as
Traveling Ionospheric Disturbances (TIDs). Gravity waves are these neutral atmospheric
disturbances that are generated by various low altitude sources like earthquakes, artificial wave
sources such as nuclear explosions and high altitude sources like hurricanes and tropospheric
thunderstorms. Due to these gravity waves originated in the lower atmosphere by the weather
related disturbances and/or orographic forcing, large amounts of energy and momentum are
transported to the upper atmosphere. Neutral air is also known to propagate in the
thermosphere by solar EUV heat input within the thermosphere.
A discrete spectrum of atmospheric gravity wave modes are supported by the earth’s
atmosphere and different scientists have studied disturbances caused by various gravity waves
with periods ranging from a few minutes to several hours. In essence, the gravity waves
generated by these different sources are capable of propagating vertically and horizontally, interact non- linearly and greatly influence the constituent densities and the energy and the momentum flows of the atmosphere. Since C. O. Hines [Hines, 1960] published his universally accepted and widely used linear gravity wave theory in 1960, extensive theoretical as well as experimental studies of the gravity waves and their effects displayed in diverse atmospheric phenomena has become a major goal of atmospheric science research.
Experimentally obtained measurements of wind data using radar techniques, chemical and
rocket smoke trail releases, minor constituent concentrations taken using a variety of
instrumentation, temperature and temperature fluctuation measurements obtained using lidar
experiments, etc are studied and analyzed by scientists in atmospheric modeling to establish
density, pressure, temperature and background concentration profiles. Experimental detection
of minor constituent concentrations and temperature fluctuation measurements are conducted
using radar, lidar and satellite aided observational techniques. Images of the structure of
gravity waves are taken using all-sky imagery. Using the observational data gathered with the
aid of lidar, rockets, all-sky imagery, etc, scientists engaged in atmospheric modeling of
gravity waves attempts to explain experimental data to better understand and predict the
atmospheric effects.
2 1.2 Airglow Emissions in the Middle Atmosphere
Throughout the middle atmosphere these gravity waves are known to be a major source of
meso-scale fluctuations, especially in the mesosphere and the lower thermosphere (MLT)
region, where profound impact is observed on the temperature structure and the general
circulation. Early evidence of minor constituent motions is available in the form of auroral
displays, wave-like patterns in the radio echoes and meteor trail distortions. At MLT heights
the upward propagating atmospheric gravity waves create substantial fluctuations in the line of
sight column brightness and rotational temperature of several airglow emissions by perturbing
the concentration profiles of the reacting minor species whose reaction rates are also
temperature dependant. When observed during daytime the airglow is called dayglow and
when observed in the night it is referred to as nightglow. The airglow emissions are used
extensively to study temperature and other fluctuations in the upper atmosphere and also to
study the temporal and spatial variations of the concentration profiles of the reacting minor
species.
In studying the turbulence and the stability of the middle atmosphere it is important to factor
into consideration atmospheric dynamics and chemistry because this height region is abundant
of dynamical and photochemical interactions. The propagation of atmospheric gravity waves in the middle atmosphere affects the emissions of various wave lengths, produced due to decaying of atmospheric constituents. Atmospheric dynamics deals with these effects of
Internal Gravity Waves (IGW) on the mesospheric airglow emissions. Atmospheric chemistry
3 is important because the propagation of the IGW changes the temperature dependant rate
coefficients of chemical reactions by changing the neutral temperatures of the atmosphere and
also influences the chemically active constituent concentrations. The kinetic energy densities
and the temperature fluctuations can be calculated using the mesospheric airglow emissions.
Since the photochemistry of the middle atmosphere is affected by the propagating atmospheric
gravity waves, various scientists have used the response due to gravity wave perturbation on
middle atmosphere photochemistry as a probe in observing and analyzing gravity wave
dynamics. Theoretical calculations made by Porter et. al. [1974] on OI 6300 Å and NI 5200 Å,
1 Weinstock [1978] for O2 ( Σ) and OH emissions and Hatfield and Tuan [1981] for H, O3 and
OH emissions, Walterscheid et, al [1987], Makhlouf et. al., [1995] for OH, etc and also
experimental observations made by Krassovsky [1972]; Krassovsky et. al., [1977]; Takahashi
et. al., [1985]; Taylor et. al., [1987]; Viereck and Deehr, [1989], Taylor et. al., [1991],
Oznovich et. al. [1995], Drob [1996], Reisin and Scheer [1996], etc. have attempted to study
airglow response to gravity wave propagation.
Theoretical models developed by various scientists have included effects of Coriolis force,
molecular viscosity, thermal conductivity, horizontal background wind and ion drag. Hines
[1960] is a basic model that has assumed the atmosphere to be infinite, homogeneous and omitted from the theoretical formulation the above effects. Pitteway and Hines [1963] have
theoretically formulated the energy dissipation by molecular viscosity and the damping due to
thermal conduction. Midgley and LieMohn [1963] have included both the thermal conduction
and the molecular viscosity into the calculations. Francis [1973] has considered molecular
4 viscosity, thermal conduction and the ion drag. Klostermeyer [1972a] have incorporated the
background wind into the formulation. Hickey [1987] in his theoretical model has incorporated
Coriolis force, thermal conductivity, molecular viscosity and the ion drag. But their study
mainly focuses on the analysis of the dispersion equation.
The objective of our study is to develop a theoretical analysis of the long period (~ 8 hour)
fluctuations of both the Meinel OH band intensity and the rotational temperature based on the
assumption that they are caused by atmospheric gravity waves rather than atmospheric tides
and to theoretically formulate and analyze the Krassovsky ratio and the temperature variation
due to these very long period (~ 8 hour) gravity wave. Therefore in order to account for the
very long periods (~ 8 hour) in the fluctuation, Coriolis force due to earth’s rotation was taken into account. In addition our model includes effects of thermal conduction and molecular
viscosity in the calculation. The most important difference between our analysis and the previous treatments is the consideration of very long periods (∼ 8 hour) and the inclusion of a
complete chemical scheme. The model with the inclusion of Coriolis force and dissipation
effects has not been treated before for such long gravity wave periods.
Our kinetic chemical scheme based on Makhlouf et. al. [1995] includes the vibrational level
dependence. It includes the basic five reaction kinetic scheme for total hydroxyl number
density employed by Walterscheid et. al. [1987] and in addition it also incorporates quenching
of [OH (n)] by O2 and N2, radioactive decay equation of [OH (n)] and reactions that involve
additional sources and sinks. Their theoretical calculation is based on Hines [1960] model
5 atmosphere. The theoretical model used by Hines in “Internal Atmospheric Gravity Waves at
Ionospheric Heights” (Hines, [1960]) assumes a uniform background temperature in the
absence of background wind and also assumes the atmosphere to be infinite and unbounded.
Even though our OH airglow model closely follows the chemical scheme developed by
Makhlouf et. al., in calculating the airglow response on the Krassovsky ratio due to the
propagating IGW, we differ in one fundamental aspect. Makhlouf et. al. [1995] used the Hines
[1960] gravity wave model which provides simple analytic solutions. The differential
equations in this model govern the propagation of atmospheric gravity waves in uniform
isothermal, isotropic, adiabatic and non viscous atmosphere. Our model is formulated to allow for a more realistic atmosphere. The gravity wave model that we have used considers a non- adiabatic (includes the effects of Coriolis force due to earth’s rotation) and a viscous
atmosphere (includes dissipation effects of eddy thermal conduction and molecular viscosity)
in our calculations. Even though our model is a less convenient model requiring complex
computations to numerically solve the coupled hydrodynamic equations, this is a more
consistent and a realistic analytical model. It allows us to conduct meaningful comparisons of
our results with the experimental results obtained for mid and high latitude long period (~ 8
hours) gravity waves by various investigators.
Also our hydrodynamic theory follows Hickey et al. [1987]. A major difference between the
problems that we study and that of Hickey et. al. [1987] is the focus of the problem. Hickey et.
al. deals with the calculation and the analysis of the dispersion equation characteristics. The
6 primary focus of our study is to examine the effects of the linear atmospheric gravity wave on
the airglows and the minor constituents in the middle atmosphere by studying the Krassovsky
ratio. The OH airglow model that we have adopted takes into consideration the vibrational
dependence, the quenching effects and the OH production from other levels allowing a more
realistic observation of the gravity wave response on the vibrational level dependence. And at
the same time treats the full hydroxyl chemical kinetic model
1.3 The Krassovsky Ratio Calculation
Early photometric investigations of airglow emissions were considerably concerned about the
spatial and temporal variability induced by gravity waves of various vertical and horizontal
scale sizes. The Krassovsky ratio was proposed by V. I. Krassovsky in 1972 (Krassovsky,
[1972]. The Krassovsky ratio method provides an excellent technique in probing the adiabatic
processes in the MLT heights. By relating the relative fluctuation in column brightness ( ΔB )
to the relative fluctuation in the rotational temperature ( ΔT ), Krassovsky ratio determines the efficiency of producing a brightness fluctuation for a given temperature fluctuation. It is defined as,
⎛ΔB ' ⎞ ⎜ nn ⎟ B ' η = ⎝ 0nn ⎠ (1) n ⎛ ⎞ ⎜ΔT ⎟ ⎝ T0 ⎠
7
It measures the airglow response to a periodic dynamical perturbation such as an internal
gravity wave and has been used extensively in studying wave dynamics. The variables B0nn’B and T0 are the unperturbed column brightness between the vibrational levels n and n’ and the
steady state mean temperature. ΔB ' and ΔT are the fluctuations in brightness and ambient nn
temperature due to the propagation of gravity wave. In our calculation the unperturbed mean
values are approximated to local mean values (Viereck and Deehr, [1989]).
Many scientists (Hecht et. al., [1987]; Viereck and Deehr, [1989]; Swenson et. al., [1990],
Makhlouf et. al. [1995], Taylor et. al., [2001]) have made Krassovsky ratio computations involving monochromatic gravity waves for a range of gravity wave periods. In the literature very few experimental measurements involving long period gravity wave have been conducted
that can be used to theoretically calculate the Krassovsky ratio for long period gravity waves.
Drob [1996], Oznovich et. al. [1995] and Viereck and Deehr [1989] have obtained very high
latitude measurements for long period wave signatures using long polar nights, which can be
used for Krassovsky ratio calculation. Long period gravity wave observations in the mid
latitudes using optical observations are very uncommon. Only a very few scientists (Reisin and
Scheer [1996], Taylor et. al. [2001]) have conducted Krassovsky ratio derivations for the long
period gravity waves with wave periods higher than ~ 6 hours for these latitudes. Focusing on quasi monochromatic wave events obtained from 2 sites located mid-latitude at 32º S and 37º
N, Reisin and Sheer [1996] have conducted a study using a wide range of wave periods. Using
8 a high performance imaging system Taylor et. al. [2001] has recorded long period wave signatures with wave periods close to terdiurnal tide (∼ 8 hour).
Even though the most complete study by Reisin and Sheer [1996] was conducted using a wide range of wave periods for OH (6, 2) and O2 (0, 1) band data, since it focuses mainly on the semidiurnal period range (or T ≈ 12 ± 2 hours) it was not used in our calculations, as our main focus is on significantly different part of the wave spectrum with wave periods close to 8 -hr terdiurnal tide. The wave sequence of approximately 10 complete wave cycles of 8 hour period, observed by Drob [1996] was also not used in our calculation because of the unusualness associated with this particular wave train. Oznovich et. al. [1995] and Taylor et. al.
[2001] measurements are the most detailed and therefore they were used in combination with our gravity wave model for Krassovsky ratio computations.
1.4 Experimental Observations
1.4.1 Taylor’s Experimental Observations
According to Taylor et. al., [2001] the CEDAR Mesospheric Temperature Mapper (MTM - a high performance solid state imaging system) with its ability to provide stable and linear measurements is appropriate in taking seasonal long term measurements. They used the MTM to take high precision nocturnal field measurements in rotational temperature and wave
9 induced OH mesospheric intensity perturbations in the OH Meinel (6, 2) band emission. Their
study have mainly focused on IGW perturbations with very long periods (~ 8 hour) which they
have found to take place rather frequently (Taylor et. al. [1999]; Pendleton et. al. [2000])
especially in the fall and early winter seasons. During the campaign, measurements were made
on fall and the early winter months to take advantage of the long winter nights, centered on the
new moon from the locations given in Table 01.
Taylor used the high precision measurements in rotational temperature and wave induced OH
mesospheric intensity perturbations in the OH Meinel (6, 2) band emission, obtained using the
high performance MTM, to study IGW perturbations with very long periods (~ 8 hour). The seasonal measurements taken by Taylor with the MTM from about 250 days of data have shown unambiguous evidence of long period wave oscillations in rotational temperature and the gravity wave band intensity. Using subsequent investigations conducted by Krassovsky ratio method, they studied the magnitude and phase relationships between the temperature and
the induced intensity for the observed wave phenomena.
Site Location Observations Days
Bear Lake Observatory (BLO), 41.6º N, 111.6º W October 6-7, 1996 (UT 281)
Utah October 17-18, 1996 (UT 292 )
Ft Collins, Colorado 40.58º N, 105.08º W November 2-3, 1997 (UT 307)
December 23 - 24, 197 (UT 358 )
10
Eureka, Canada 79.59º N, 85.56º W December 21/22, 1993 (UT 356)
December 31/January 1, 1994 (UT 01)
Table 01: Observation sites and dates for the experimental data for Taylor et. al (2001) and
Oznovich et. al. (1995) used for our Krassovsky Ratio calculations.
1.4.2 Oznovich’s Experimental Observations
Making use of the high latitude complete darkness ( λ 80°≥ ) of the winter polar cap,
oscillations in the OH rotational temperature and the airglow brightness were observed over
Eureka, Canada (79.59N, 85.56W) under mostly (60%) clear weather conditions. These uninterrupted 24 hour, optical nightglow ground based measurements of the Meinel OH (3, 1) band, conducted using a Michelson interferometer (MI) were expected to monitor the oscillations in the MLT region with very high temporal and spatial resolution. They observed
5 and 3 complete cycles of marked ~ 8 hour oscillations in airglow brightness and rotational temperature on December 21/22, 1993 (UT day 356) and December 31/January 1, 1994 (UT day 1).
11 Site Observation Days Period (hour)
Bear Lake Observatory October 6-7, 1996 (UT 281) 9.2 ± 1.0
(BLO) October 17-18, 1996 (UT 292 ) 6.7 ± 1.5
Ft Collins, Colorado November 2-3, 1997 (UT 307) 8.6 ± 1.0
December 23 - 24, 197 (UT 358 ) 9.2 ± 1.5
Eureka, Canada December 21/22, 1993 (UT 356) 8.1 ± 1-2
Table 02: gravity wave periods observed for various observation sites by Taylor et.al [2001] and Oznovich et. al. [1995]
1.5 Shallow Atmosphere Approximation
Using the shallow atmosphere approximation the Coriolis force is taken into account.
According to Phillips [1966] shallow atmosphere approximation neglects the horizontal
component of the Earth’s rotation vector in the momentum conservation equations. This is
effectively equivalent to neglecting a force in the zonal direction which is proportional to the
vertical velocity component and a force in the vertical direction which is proportional to the
zonal velocity component.
With increasing wave periods, even though the importance of Coriolis force increases
(Volland, [1969a]); the vertical velocity of the gravity wave motion generally decreases
12 (Georges, [1968]). Therefore under shallow atmosphere approximation for gravity wave propagation, neglecting a force in the zonal direction which is proportional to the vertical velocity component would be a good approximation. However, long period gravity waves have significant zonal velocity components. Therefore neglecting a force in the vertical direction which is proportional to the zonal velocity component under the shallow atmosphere approximation would not be a good approximation for long period gravity wave propagation, theoretically. But since most long period waves are only created at high altitudes in the thermosphere and propagate predominantly in the meridianal direction to lower altitudes in practice, it is to be a good approximation still. Even though shallow atmosphere approximation has been well justified for large scale, quasi-horizontally propagating winds in the earth’s atmosphere by Phillips [1966], it has also been successfully applied to gravity wave propagation in the atmosphere by Richmond and Mitsushita, [1975].
13 CHAPTER 2
MATHEMATICAL FORMULATION
2.1 Basic Hydrodynamic Theory
The equations of fluid motion takes into consideration thermal conductivity, molecular
viscosity and the Coriolis force for the atmosphere at the airglow heights considered in the
dissertation (70-120 km). The Coriolis force due to earth’s rotation, thermal conduction and
thermal viscosity are likely to be important for long period gravity waves in the mesospheric range and therefore they are included into the calculation. With increasing altitude temperature
dependant κ 0 increase and mean density ρ decreases exponentially therefore the ratio 0 ρκ increases with height. Hence the inclusion of heat conductivity for diurnal wave propagation becomes important. For very long period (~ 8 hours) gravity waves Coriolis force due to earth’s rotation needs to be taken into account to describe the advection of earth’s momentum due to its rotation. The Coriois force in our model is taken into account under the limitations of the shallow atmosphere approximation (Phillips, [1960]). The considered mass conservation, momentum conservation, energy conservation and the ideal gas equation are as follows:
14 Dρ ρ Vdiv =+ 0)( (1) Dt
VD ρρ gradPg ρ )(2 −×Γ−−= divV σ (2) Dt
DT ∂Vi Cv ρ + VdivP )( + ∑∑σ ik κ 0 T =∇∇− 0).( (3) Dt ik ∂xk
= ρμ TP (4)
Where, T, P and ρ are the temperature, pressure and density variables respectively. Cv is the specific heat at constant volume and μ is a constant proportional to the universal gas constant.
,, kkk zyx : Wave numbers in the x, y, and z directions
,, ΔΔΔ WVU : Velocity field perturbations in the kx, ky, kz
directions
− 71.07 τ z ×= zT )(1034.3)( : Coefficient of viscosity
κ z ×= − zT )(1071.6)( 71.04 0 : Coefficient of thermal conduction
) ) ˆ Γ+Γ+Γ=Γ zyx kji : Coriolis force term
)( ˆ )( ˆ Δ+Δ+Δ= )( kWjViUV ˆ : Particle velocity field
D ∂ rr = V.∇+ : Eularian derivative. T ∂tD
⎛ ∂V ∂V 2 ⎞ ⎜ i k ⎟ ik −= τσ ⎜ + − δ ik ()Vdiv ⎟ : Viscous stress tensor ⎝ ∂xk ∂xi 3 ⎠
15 R0 : Universal gas constant
ω : Intrinsic angular wave frequency
C γ : Ratio of specific heats (γ = p ) Cv
δ ik : Kronecker delta
A rectangular coordinate system (Figure 8) has been chosen such that the x, y, and z axes point towards geographic south, east and vertically upward direction respectively. Wave numbers in the horizontal directions are taken to be real and non height varying. The constant kx and ky values assume that boundary condition to any problem would not introduce any horizontal, exponential decrease of amplitude as a requirement (Hines, [1960]). Allowing for variation of phase with height for the internal gravity waves, the height varying vertical wave number kz needs to be complex. In such a coordinate system the Earth’s angular velocity and gravitational acceleration gr are given as,
()Γ−=Γ Cos 0 ,0, ΓSinλλ 0
() (,0,0 −= gZg )
−− 15 Here ×=Γ 1029.7 s and λ0 is the geographic latitude.
Using the shallow atmosphere approximation, the Coriolis force is taken into account (Phillips,
[1996]). Because of the propagation of gravity waves in the atmosphere the dynamical
16 variables ρ P, T and V contain perturbation components which vary harmonically in the x, y,
z directions (directions of wave propagation) and on time t. According to the perturbation theory the temperature, pressure and the density fields with the propagation of the gravity wave can be written as,
0 Δ+= PPP
0 Δ+= ρρρ (5)
0 Δ+= TTT
Where P0, ρ 0 and T0 are the mean values of background pressure, density and temperature.
The perturbation amplitudes are small enough to be treated in the linear approximation. The
perturbed quantities due to the propagation of the atmospheric gravity wave are given as ΔP ,
Δρ and ΔT . In the absence of ambient horizontal winds the monochromatic plane wave
solutions of the hydrodynamic equations are considered to be of the form:
z 2 H ()ω −−− zyx zkykxkti ()0 ()Δ+= () eezPzPzP z 2 H ()ω −−− zyx zkykxkti ()0 ()Δ+= ρρρ () eezzz z ()ω −−− zkykxkti () ()Δ+= () 2 H eezTzTzT zyx 0 (6) z ()ω −−− zkykxkti Δ=Δ () 2 H eezUU zyx z ()ω −−− zkykxkti Δ=Δ () 2 H eezVV zyx z ()ω −−− zkykxkti Δ=Δ () 2 H eezWW zyx
17 zRT )( Where the scale height is defined as, zH )( = 0 . The structure of the background )( gzM
temperature is shown in figure 1 and is far from isothermal. Therefore assuming horizontal
stratification a multilayer model for the quite, atmosphere is considered to account for the
uniform isothermal atmosphere. A step size of 1 km is considered for each isothermal layer in
our multilayer model for purposes of numerical calculation. For plane wave solutions to exist,
τ κ and 0 are considered constant for each thin homogeneous slab. P0 P0
2.2 Gravity Wave Model
Theoretical models developed by various scientists have included effects of Coriolis force,
molecular viscosity, thermal conductivity, horizontal background wind and ion drag. Hines
[1960] is a basic model that has assumed the atmosphere to be infinite, homogeneous and omitted from the theoretical formulation the above effects. Pitteway and Hines [1963] have
theoretically formulated the energy dissipation by molecular viscosity and the damping due to
thermal conduction. Midgley and LieMohn [1963] have included both the thermal conduction
and the molecular viscosity into the calculations. Francis [1973] has considered molecular viscosity, thermal conduction and the ion drag. Klostermeyer [1972a] have incorporated the
background wind into the formulation. Hickey [1987] in his theoretical model has incorporated
18 Coriolis force, thermal conductivity, molecular viscosity and the ion drag. But their study
mainly focuses on the analysis of the dispersion equation.
The objective of our study is to develop a theoretical analysis of the long period (~ 8 hour)
fluctuations of both the Meinel OH band intensity and the rotational temperature based on the
assumption that they are caused by atmospheric gravity waves rather than atmospheric tides
and to theoretically formulate and analyze the Krassovsky ratio and the temperature variation
due to these very long period (~ 8 hour) gravity waves to compare with experimentally
observed results. Therefore in order to account for the very long periods (~ 8 hours) in the
fluctuation, Coriolis force due to earth’s rotation was taken into account. In addition, our model
includes effects of thermal conduction and molecular viscosity in to the calculation. With
increasing altitude temperature dependant thermal conductivity coefficient κ 0 increase and
mean density ρ 0 decreases exponentially therefore the ratio ρκ 00 increases with height.
Hence the inclusion of heat conductivity for diurnal wave propagation becomes important. The most important difference between our analysis and the previous treatments is the consideration of very long periods (∼ 8 hour). The model with the inclusion of Coriolis force
and dissipation effects has not been treated before for such long gravity wave periods.
In three dimensions, we consider a horizontally stratified, non-adiabatic (with the inclusion of
thermal conductivity and the molecular viscosity) and an anisotropic (with the inclusion of
Coriolis force) atmosphere, in which the ambient photochemical system is perturbed by the
monochromatic IGW. The above model is used to calculate the temperature, pressure, vertical
19 and horizontal components of the particle velocity perturbations due to the propagation of the
IGW. The theoretical models developed for the atmosphere by various scientists (Francis
[1973a, b], Klostermeyer [1972a], Volland [1969], Hickey [1987]) have used the multilayer
model. Some of the theories used by other scientists for the atmosphere include the ray theory,
the WKB (Wentzel-Kramers-Brillouin) method and the optical potential model.
The thermodynamic quantities are substituted into the hydrodynamic equations. Due to the propagation of gravity waves in the atmosphere the dynamical variables ρ ,, TP and V contain perturbation components which vary harmonically in the x, y and z directions (directions of wave propagation) and on time t. We neglected terms of second order and higher in our equations. If the perturbations are due to propagating plane waves, then in the absence of ambient wind the obtained linearized coupled first order hydrodynamic equations take the following form,
The mass conservation equation;
ρ TAAPAWAVAUA =Δ+Δ+Δ+Δ+Δ+Δ 0)()()()()()( 11 12 13 14 15 16 (7)
) The ) ),, kji components of the momentum conservation equation are,
ρ TAAPAWAVAUA =Δ+Δ+Δ+Δ+Δ+Δ 0)()()()()()( 21 22 23 24 25 26 (8)
ρ TAAPAWAVAUA =Δ+Δ+Δ+Δ+Δ+Δ 0)()()()()()( 31 32 33 34 35 36 (9)
20 ρ TAAPAWAVAUA =Δ+Δ+Δ+Δ+Δ+Δ 0)()()()()()( 41 42 43 44 45 46 (10)
Energy conservation equation;
ρ TAAPAWAVAUA =Δ+Δ+Δ+Δ+Δ+Δ 0)()()()()()( 51 52 53 54 55 56 (11)
The coefficients for the above equations are given in Appendix A. And assuming an atmosphere in which molecules are all identical, interacting only by collisions and moving in straight lines between collisions the ideal gas equation can also be expressed as follows;
ΔP ΔT Δρ = + (12) P Τ ρ 000
Where
iz P ΔΔ ρ (ω zyx zkykxkti −−−− ) ΔΔ TV ,,, ∝ e 2H P ρ00
From the above equations (6)-(11) a set of dimensionless coupled algebraic equations can be derived by eliminating temperature perturbation as follows,
21 ΔUk ω s m κ − iα − 11 h0 1 ΔVk ω s s κ 0 νR − ho γ −1 ho ΔWk ω 3 sR 2 −+ ηβηη − − ακη )3( sisicsm 0 =0 Δ PP η + 3 mRicsm 2 −+ − ακηβηη )3( mim 0 0 + + 4)2()2( Rimis −− − iακβηηακηακη Δ ρρ 0
(13)
Where,
k x km 222 2 1 s = m = h0 += kkk yx iR ακκ +−= 1 α = kh0 kh0 h0 Hk
2 2 ⎛ 11 ⎞ ω iωμ λ kTi h00 κ ⎜k z += ⎟ β = 2 η = ν = kh0 ⎝ 2H ⎠ gHk h0 3P0 ωP0
fω c = 2 gHkh0
Since we included dissipation, waves generated by low atmospheric sources and propagating to higher attitudes are usually not reflected and detected at earth’s surface. Therefore the height and the exact form of upper boundary become unimportant and all upper boundary conditions which satisfy the fact, the upper boundary considered doesn’t act as a source could be used to derive the gravity wave modes (Francis, [1973]). Therefore we solved the full set of 6 coupled algebraic equations generated by the above linearized equations subject to the rigid surface boundary condition at z = 0 and radiation condition at infinity
22 The perturbations due to the long period gravity wave in the horizontal velocity
components (, ΔΔ VU ), vertical velocity component ΔW and the background temperature ΔT can be calculated using the above uniform isothermal gravity wave model. To justify our numerical method we considered smaller step sizes of 0.1km and 0.01km and checked the stability of the solutions obtained. The obtained solutions to the full set of coupled equations was then used to calculate the Krassovsky ratio and the phase difference between the brightness and temperature fluctuation based on the long period (∼ 8 hour) gravity wave
assumed.
2.3 Atmospheric Model Used
The 1972 COSPAR profiles and Jacchia Reference Atmosphere will be used for
temperature, mean molecular mass, gravitational constant and the ratio of specific heats.
Experimentally obtained COLORADO atmospheric profiles with latitude and longitude 45ºN
and 330ºW was used to obtain O, H, HO2 and O3 equilibrium profiles. Figure 1 shows the
atmospheric temperature profile used. Figure 2 shows the mean density variation with height
obtained from the COSPAR profiles with the mean density variation plotted on a logarithmic
scale in the inset. Figure 5 shows the major species number density profiles of [N2] and [O2].
Figure 6 shows the atomic oxygen, atomic hydrogen, ozone and perhydroxyl radical minor
species number densities obtained from the COLORADO atmospheric profiles. Figure 7 shows
23 the equilibrium hydroxyl number density values for the different vibrational levels plotted also
using the COLORADO atmospheric profiles.
The coefficient of thermal conductivity and the coefficient of viscosity are taken from
+ Dalgarno [1962] and it is assumed that O2 and atomic oxygen (O ) are the major neutral and
ionic constituents. In order to obtain boundary conditions the horizontally stratified
atmosphere is approximated by a model atmosphere consisting of a number of thin isothermal, homogeneous slabs of approximately 1km in thickness, in each of which the parameters T0,
τ κ 0 M, , and g are considered constant. The vertical wave number kz varies from layer to P0 P0 layer.
Waves with horizontal phase velocities Vph = /70,50,30 sm were used (with an arbitrary
wave propagation direction of β0 = 45° with the x direction - geographic south) for the
calculations for all locations considered. The corresponding horizontal wave length values vary
from 723.6 to 2318.4 km (See Table 03 below). The geographic latitudes used in the
calculations were λ0 41,6.41 80°°°= NandNN for Bear Lake Observatory (UT), Ft Collins
(CO) and Eureka (Canada) respectively. Since diatomic gases N2 and O2 predominate below
200km altitude range the ratio of specific heats γ = 4.1 value was used for our calculation.
Even though eddy viscosity dominates for heights below turbopause (105km), for our MLT
height range (70-120km) we used the coefficient of viscosity given by Dalgarno [1962].
24
Location UT Day Period (hr) -Horizontal wave length (km)
Vph = 30 m/s Vph = 50 m/s Vph = 70 m/s
Bear Lake 281 9.2 ± 1.0 993.6 1656.0 2318.4
Observatory, 292 6.7 ± 1.5 723.6 1206.0 1688.4
UT
Ft. Collins, 307 8.6± 1.0 928.8 1548.0 2167.2
CO 358 9.2 ± 1.5 993.6 1656.0 2318.4
Eureka, 01 8.1± 1.2 874.8 1458.0 2041.2
Canada 356 8.1± 1.2 874.8 1458.0 2041.2
Table 03: Horizontal wave lengths used for the calculation of Krassovsky ratio for different locations for the considered horizontal phase velocities
25 2.4 Airglow Response
2.4.1 Eulerian Continuity Equation
The vibrational level dependant Eulerian continuity equation which describes the number density for [OH (n)], n = 0... 9 and the minor species take the following form,
∂N r n −−= ()VNdivNLQ (14) ∂t nnn n
Where, n ( = 0-9) vibrational level
Nn vibrational level dependant species ([OH (n = 0-9)]) concentration
Qn production rate for the vibrational level n
Ln loss rate for the vibrational level n
V Gravity wave velocity field ( ,, ΔΔΔ= WVUV ) in Cartesian coordinates
Using the rate constants, the vibrational level dependant photochemical production and loss rates are calculated for the full set of reactions. The vibrational level dependant volume emission rates are calculated using the equation
' ' NAI (15) nn = nn n
26
Where Ann´ are the Einstein coefficients and Nn is the population in vibrational level n. The
following equation (16) which describes the photochemical equilibrium in the absence of
internal gravity wave with no horizontal background wind in the unperturbed steady state can
be obtained from the above continuity equation (equation (14)):
0 −= NLQ nnn 000 (16)
Under the photochemical equilibrium, the steady state OH (n = 1-9) concentrations, that can be used to calculate the unperturbed profiles, using our OH vibrational kinetic model and the corresponding reaction rates takes the form:
Qn0 n0 = []()nOHN 0 = (17) Ln0
The calculated unperturbed level profiles using the complete set of reactions assuming photochemical equilibrium is given in figure 7. The vibrational level dependant steady state volume emission rate can then be calculated using,
nn ′0 = nn ′ [()nOHAI ]0 (18)
27 Where Ann’ are the Einstein coefficients and the [OH (n)]0 are the unperturbed vibrational level
dependant OH densities. The ground observed total airglow intensity can be calculated by
integrating over an emission band.
2.4.2 The Internal Gravity Wave Response
The internal gravity wave response was investigated using the following method. To inspect
the OH airglow response due to the propagating IGW, the continuity equation is linearized to
the first order (X = X0 + X) about their steady state values (X0). And it takes the following
format:
Δ∂ N ∂ ∂ ∂ ∂ n −Δ−Δ−Δ= NLNNLQ ()−Δ NU ()−Δ NV ()Δ−Δ WW ()N ∂t 0 0 nnnnnn 0 ∂x n0 ∂y n0 ∂z ∂z n0
(19)
In our model the fluid parameter perturbations are assumed to be due to monochromatic plane wave propagating in an arbitrary direction and (in the absence of horizontal background wind)
⎡ ⎛ zi ⎞⎤ ⎜ ⎟ it is assumed to take the formexp ⎢ ⎜ω zyx zkykxkti −−−− ⎟⎥ . The equilibrium state of ⎣ ⎝ 2 H ⎠⎦
the photochemical system is assumed to be perturbed by this propagating gravity wave. The
airglow response is linearized to the first order and all terms which are second order and higher
are neglected because the gravity wave amplitude is taken to be too small to be considered. In
28 horizontal homogeneity assuming arbitrary plane waves of the above format, the linearized first order equations take the following format,
ΔNn ΔQn ⎡ ⎛ i ⎞ 1 ∂Nn0 ⎤ ()n0 + iL ω = xn y ⎢iVikUikL ⎜ ++Δ+Δ+Δ− kz ⎟ − ⎥ ΔW (20) Nn0 Nn0 ⎣ ⎝ 2H ⎠ Nn0 ∂z ⎦
Where
Nn0 vibrational level dependant species background concentration
ΔQn production rate fluctuation for the vibrational level n
ΔLn loss rate fluctuation for the vibrational level n
ΔNn vibrational level dependant fluctuation in the species ([OH (n = 0-9)], [H],
[HO2], [O] and [O3]) concentration by the propagating gravity wave
A total of 14 coupled algebraic equations are derived by applying the above linearized
continuity equation for each vibrational level (n = 0 – 9) taken as individual, chemically active
species and the other molecular species H, HO2, O and O3. [ΔOH (n=0-9)], [ΔH], [ΔHO2],
[ΔO] and [ΔO3] are the corresponding density perturbations obtained by solving the 14 coupled
equations generated by the propagating gravity wave. The obtained IGW perturbed, molecular
species profile figures using the complete set of reactions are given. The vibrational level dependant hydroxyl profiles that are generated as a result of the propagating long period atmospheric gravity wave are given the figures (28 - 37). These obtained hydroxyl density
29 perturbations are used to calculate the vibrational level dependant, volume emission rates. The
volume emission rate perturbed to the first order is obtained as follows:
' ' III ' nn nn 0 Δ+= nn (21) ' ' nOHAI nn 0 nn Δ+= [()]
2.4.3 Brightness Weighted Temperature Method
In our Krassovsky ratio calculation, for the uniform isothermal atmosphere with 1 km thick
layers (for the considered multilayer model), the temperatures used from the 1972 COSPAR profiles are the mean local temperatures. In an effort to employ a more realistic model for our
calculation, like many theoreticians and modelers have done, we have adopted the brightness
weighted temperature method (Walterscheid et. al., [1987], Schubert et. al., [1991], Makhlouf
et. al., [1995]) associated with a column integrated measurement to determine the fractional
temperature variation and the steady state effective temperature to calculate the Krassovsky
ratio. We also use the plane parallel approximation. It assumes that in the horizontal direction
medium stretches to infinity and therefore all optical properties are considered to be
independent of horizontal position in the homogeneous plane parallel slab. Under the plane parallel approximation the brightness weighted temperature is defined as:
30 ' ()(),,,, dlzyxTzyxI ∫ nn T n =〉〈 (22) B ' nn
Where
B ' = Vibrational level dependant brightness nn
Vibrational level dependant volume emission rate, and I nn' =
=secθ ddl θ (23)
The integration is conducted along the line of sight (LOS) and θ is the zenith angle to the line
of sight. Linearizing to the first order, the brightness weighted temperature takes the form:
n n0 TTT 〉〈Δ+〉〈=〉〈 n (24)
Where, the steady state effective BWT for the vibrational level n and its variation due to the propagation of IGW is defined as follows:
' ()0 ()dlzTzI T = ∫ nn 0 (25) n0 Bnn 0'
31 ' ( )Δ ( ,, )dlzyxTzI ∫ nn 0 −+=〉〈Δ TTTT 321 nnnn with T1n = Bnn' 0
Δ ' ()(),, dlzTzyxI ∫ nn 0 T2n = (26) Bnn' 0
ΔB ' ' () ()dlzTzI nn ∫ nn 0 0 T3n = × Bnn' 0 Bnn' 0
0 zT )( is the mean local temperature at height level z. The brightness for a transition from vibrational level n to n’ (Bnn’) measured by an airglow instrument linearized to the first order is defined as follows:
' ' Δ+= BBB ' with ' = ' ( ,, )dlzyxIB nn nn 0 nn nn ∫ nn
' = ' ( )dlzIB (27) nn 0 ∫ nn 0
' Δ=Δ ' ( ,, )dlzyxIB nn ∫ nn
The corresponding column integrating airglow brightness, its fluctuation due to the propagating gravity wave, brightness weighted temperature, and its fluctuation for the Meinel
(6,2) band values hence take the following form and were used for our Krassovsky ratio calculation for the Meinel hydroxyl (6,2) emission
= dlzIB (28) 062 ∫ 062 ()
Δ=Δ ,, dlzyxIB (29) 62 ∫ 62 ()
32 ()0062 ()dlzTzI T = ∫ (30) 06 B062
062 ()Δ (,, )dlzyxTzI Δ 62 (),, 0 ( )dlzTzyxI ΔB T =Δ ∫ + ∫ − T 62 (31) 6 60 B062 B062 B062
We then used the vibrational level dependant volume emission rate, the atmospheric steady state effective temperature, the fluctuation in the vibrational level dependant volume emission rate and the fluctuation in the steady state temperature due to the IGW propagation for the OH
Meinel (6, 2) band data to calculate the Krassovsky ratio values.
33 CHAPTER 4
HYDROXYL CHEMICAL KINETIC MODEL
3.1 Background
Waves generated by weather related disturbances such as thunderstorms, tropospheric
convection and/or many other different sources such as volcanic eruptions and earthquakes in
the lower atmosphere has the ability to move large amount of energy into the upper atmosphere
and thereby impacting the general circulation and the temperature structure of the Mesosphere
and Lower Thermosphere (MLT) region . Middle atmosphere studies should take into
consideration both the dynamics and the chemistry of this photo-chemically and dynamically
active region. The propagation of the IGW influences the photochemistry by changing the
concentration of the chemically active constituents and also by changing the local temperatures
and thus the temperature dependant rate coefficients of chemical reactions.
Evidence of temporal and spatial structures has been observed in hydroxyl airglow and such
structures have been related to the passage of atmospheric internal gravity waves (IGW) [Hines
1960]. The study of internal gravity waves and their role in the dynamics of the middle
atmosphere therefore can be done by observing the OH emission layer. Observations
34 conducted by Krassovsky, [1972]; Takahashi et al., [1985]; Viereck and Deehr, [1989], Taylor et al., [1991]; Oznovich et. al. [1995], Drob [1996], Reisin and Scheer [1996], etc have witnessed spatial and temporal structures in hydroxyl airglow emissions. These early photometric investigations of airglow emissions were considerably concerned about the spatial and temporal variability induced by gravity waves of various vertical and horizontal scale sizes. These structures are known to be consequential to the passing of IGWs. Therefore the role of IGW in the middle atmospheric dynamics can be studied using the emission layer observations.
By coupling a five reaction photo-chemical model from a set of reactions for the minor constituents OH, H, O3, O and HO2 and background major gas M (O2+N2) Walterscheid et. al.
[1987] formulated an Eulerian photochemical-dynamical model for the hydroxyl airglow.
Quenching effects and the background wind were not taken into account in this dynamical model for the isothermal atmosphere. Since they only considered a single height level for their
Krassovsky ratio calculation and integration over the entire hydroxyl layer was not conducted,
the computed Krassovsky ratio results are more applicable to single atmospheric height level,
that is for either gravity waves with large (when compared to emission layer thickness) wave
lengths or for a thin emitting layer. The calculation was broadened to a finite thickness
emitting layer by Schubert et. al. [1988], using Brightness Weighted Temperature Model
(BWT) to obtain Krassovsky ratio applicable to a column-integrated measurement. Molecular
viscosity and the eddy diffusion effects were added to the calculation by Schubert et. al.
[1991]. Tarasick and Shepherd [1992b] in their calculation included quenching of single
35 vibrational levels but with the introduction of simplifying approximations they avoided solving
the full set of photochemical-dynamical equations.
3.2 OH Vibrational Kinetic Model
Our Hydroxyl vibrational kinetic model with vibrational level dependence is based on
Makhlouf et al. [1995]. In linearizing the coupled continuity equations to examine the reaction of the major and minor photochemical constituents to the propagating atmospheric gravity wave, we used a method similar to Walterscheid et. al. [1987]. But in addition to the five species studied by Walterscheid [1987] in some equations we also included vibrational level dependent loss effects using the nine lowest vibrationally excited states of hydroxyl. Hence we have used a model which has employed an expanded, complete reaction set for production and
loss of OH that includes production and loss of OH vibration, chemical quenching of hydroxyl
by atomic oxygen, collisional quenching by the molecular species O2 and N2 and a few minor
but non-negligible reactions from Winick [1983]. The considerable effect chemical quenching
has, on the integrated column radiance and the hydroxyl emission profile in the steady state is
well recognized. We have also separately computed the photon emission rates for each
vibrational transition rather than taking them to be equal to the production rate of the
vibrationally excited OH. The reactions used for our model are as follows:
36 k1n (R1) OH 3 ⎯+ ⎯→ )( + OnOH 2 n = 6-9
k 2 n (R2) HOO 2 ⎯+ ⎯→ )( + OnOH 2 n = 0-3
k3 (R3) 2 ⎯++ ⎯→ 3 + MOMOO
k4 (R4) 2 MOH ⎯++ ⎯→ 2 + MHO
k5 (R5) 3 HOO 2 ⎯+ ⎯→ += 2)0( OnOH 2
k6 (R6) )0( OnOH 3 ⎯+= ⎯→ + OHO 22
k7 (R7) )0( HOnOH 2 ⎯+= ⎯→ 2 + OOH 2
k8n (R8) )( ⎯+ ⎯→ + OHOnOH 2 n = 0-9
,kk ,99 Rnn (R9) )( OnOH 2 ←+ ⎯→⎯⎯ )1( +− OnOH 2 n = 1-9
,kk ,1010 Rnn (R10) )( NnOH 2 ←+ ⎯→⎯⎯ )1( +− NnOH 2 n = 1-9
(R11) nOH )( ⎯⎯→A , ⎯−Nnn )'( +− hnnnOH n = 1-9, n´ = 1-6, n-n´ ≥ 0
And the collisional excitation, reverse reactions are governed by the following expressions,
kk 9,9 nRn −= − nn −1 BTkEE ]/)(exp[ (32)
Rn = kk 10,10 n −− nn −1 BTkEE ]/)(exp[ (33)
Where En is the energy of vibrational level n, kB is the Boltzmann constant and T is
background temperature in Kelvin. The vibrational level dependent rate coefficients 1 ...... ,, kk 10
37 are listed in Table (4). From TurnBull and Lowe [1989] band averaged Einstein coefficients
A , −nnn ' (Table 5) are obtained. The coefficients of reverse reactions k9n,R and k10n,R which result
in excitation by collision are calculated using the above equations (32) and (33) where kB and
En are Boltzmann constant and the energy of the vibrational level n.
Rate Value Comments and References Constant -10 b k1n 1.4 × 10 exp(-470/T) b(n) b(n) is the branching ratio for reaction (R1) ,
b(9) = 0.48, b(8) = 0.27, b(7) = 0.17, b(6) = 0.08, b(n =1-5) = 0
-11 c k2n 3.0 × 10 d(n) d(n) is the branching ratio for reaction (R2) ,
d(0)=0.52, d(1)=0.34, d(2)=0.13, d(3)=0.01, d(ν=4-9)=0
-34 2.3 k3 6.0 × 10 (300/T) three-body O3 production
-32 1.6 k4 5.7 × 10 (300/T) three-body HO2 production
-14 k5 1.1 × 10 production of OH(0) only
-12 k6 1.6 × 10 exp(-940/T) chemical loss of OH(0) only
-11 k7 4.8 × 10 chemical loss of OH(0) only
-11 k8 a8(n) × 10 a8(0)=4.9, a8 (1)=10.5, a8 (2-9)=25
38
-13 k9 a9(n) × 10 a9(1) = 1.3, a9(2) = 2.7, a9(3) = 5.2,
a9(4) = 8.8, a9(5) =17.0, a9(6) = 30.0,
a9(7) = 54.0, a9(8) = 98.0, a9(9)=170.0
-14 k10n a10(n) × 10 a10(1) = 0.58, a10(2) = 1.0, a10(3) = 1.7,
a10(4) = 3.0,a10(5) = 5.2, a10(6) = 9.1,
a10(7) = 16.0, a10(8) = 27.0, a10(9) = 48.0 Table 4: Rate constants for OH (n) Production and Loss Processes a Reaction rate constant units are s-1 for a uni-molecular reaction, cm3s-1 for a two-body reaction and cm6s-1 for a three-body reaction. b Klenerman and Smith [1987]. c Kaye [1998]
A n = 1 2 3 4 5 6 7 8 9
n-n´
0 22.74 15.42 2.032 0.299 .051 0.010
1 30.43 40.33 7.191 1.315 0.274 0.063
2 28.12 69.77 15.88 3.479 0.847 0.230
3 20.30 99.42 27.94 7.165 2.007 0.620
4 11.05 125.6 42.91 12.68 4.053
5 4.00 145.1 59.98 19.94
6 2.34 154.3 78.64
7 8.60 148.9
8 23.72
39 ' ' ' 0,61,91 , −nnn ( nnnnA ≥−−=−= )
Table 5: Einstein coefficients for OH(n) radiative decay reaction (R11)
The fourteen species representing [OH(n = 0-9)], [H], [HO2], [O] and [O3] are coupled in the
kinetic scheme considered. In addition to the basic five equations of Walterscheid at. al.
[1987], we have also considered an extended reaction set in order to account for the vibrational
level dependence of loss effects and other non-negligible effects. The five basic reactions of
Walterscheid et. al. [1987] for minor species constituents OH, O, O3, H and HO2 are as follows: (R1) the Ozone process which is the primary reaction for excited OH (n = 6-9
)production (since Bates and Nicolet [1950])) where the branching ratios for n = 6-9 are taken from Klenerman and Smith [1987], the perhydroxyl process (R2) which is likely the secondary reaction for OH (n = 0-3), (even though all levels for n ≤ 6 are energetically accessible in the reaction ) production and also the primary loss mechanism for HO2. In comparison to its ability to produce vibrationally excited OH during day time (LeTexier et. al., [1987]) the night time OH production is of minor importance (Kaye, [1988]) for this reaction. (R3) and (R4) are the three body O3 production and HO2 production reactions. (R8) is the reaction for the sudden
death chemical quenching for the loss of OH molecules by atomic oxygen. The reaction rates
are somewhat controversial for this reaction. The rate constant for n = 0 was adopted from
Hampson and Garvin [1977] and n = 1 from Spencer and Glass [1977]. No experimental values
are available for n = 2 – 9, Sivjee and Hamwey [1987] however have proposed a value of
4.0×10-10 cm3s-1 which is greater than the gas kinetic collisional limit (2.5×10-10 cm3s-1) in the
hydroxyl airglow layer, therefore we have adopted the gas kinetic collisional limit (every
40 collision with atomic oxygen destroys an OH(n) molecule) as the rate coefficient for these
vibrational levels. In fact this rate (2.5×10-10 cm3s-1 for n = 2-9) is in agreement with the rocket measurements of Baker, [1978].
In order to account for the vibrational level dependence, in addition to the above we have also added the equations (R9) – (R11) and as additional sources and sinks equations (R5) – (R7).
The rate coefficients for equations (R9) and (R10) which describes collisional quenching of
hydroxyl by O2 and N2, are uncertain, the experimental measurements are incomplete and there are only a only a few theoretical guides available. SSH theory (Schwartz, Slawsky, Herzfeld) for vibrational relaxation states that single quantum transition is the most probable method by which quenching by molecular species occurs specially for lower vibrational levels. For higher vibrational levels application of SSH theory is questionable (Dodd et. al., [1991]), Therefore for vibrational states n = 1 - 6 the rate coefficients are adopted from Dodd et. al., [1991], for n
= 7 - 8 the rate coefficients are adopted from Knutsen and Copeland [1993] and for n = 9 the rate coefficients are adopted from Chalamala and Copeland [1993]. The rate coefficients for quenching by N2 are a factor of 20-35 lower than the coefficients for quenching by O2 and therefore the reaction rates for quenching by N2 has a much less significance. The quenching
rates however increase almost as much between the vibrational states n = 1 and n = 9. Showing
itself as an attenuation at the bottom side of the profile and a rising at the peak height levels.
The effect of quenching by both O2 and N2 on the OH Meinel band volume emission profiles
will be felt most strongly at lower altitudes.
41 As a result of large vibrating molecule anharmonicity, in addition to the single transitions,
multi-quantum transitions are also favored for reaction (R11) which deals with radiative decay
for various vibrational levels. The largest rate is the first overtone (Δn = 2) which occurs for
n > 2 but there are higher overtones (Δn ≥ 3 ) which may even be greater than the fundamental.
The total radiative loss rate changes roughly with the vibrational level, for example at 200 K temperature it increases from 22.74 s-1 (n = 1) up to 275.9 s-1 (n = 9) (Turnbull and Lowe,
[1989]). The reactions (R5) – (R7) are only small contributors. But they are also included as
additional sources and sinks in an attempt to complete the reaction set and to bring the sum
[OH (n)] calculated using kinetic model closer to the 1-D diurnal model.
In our hydroxyl kinetic model calculation, the O2 and N2 major species number densities are
assumed to be affected by gravity wave dynamics only, and are assumed to change only
negligibly as a result of photochemistry.
42 CHAPTER 4
NUMERICAL RESULTS & DISCUSSIONS
4.1 Numerical Results and Analysis
In our study we developed a theoretical analysis of the long period (~ 8 hour) fluctuations of
the Meinel OH band intensity and the rotational temperature, to theoretically formulate and
analyze the Krassovsky ratio and the temperature variation due to very long period gravity
waves. Our model was constructed to allow for a realistic atmosphere. Therefore the gravity
wave model that we used includes the effects of Coriolis force due to earth’s rotation to
account for the non-adiabatic nature of atmosphere and the dissipation effects of thermal
conduction and molecular viscosity to allow for a viscous atmosphere in our calculations. Our
chemical kinetic scheme for OH Meinel (6, 2) airglow is a complete chemical scheme and it
also includes the vibrational level dependence in a number of reactions. The atmospheric
models that we used include 1972 COSPAR profiles, Jacchia Reference Atmosphere and the
experimentally obtained COLORADO atmospheric profiles.
In this study of Krassovsky ratio, the method of approach was as follows. The non-
dimensionalized full set of coupled equations obtained using our uniform isothermal gravity
43 wave model was numerically solved to find the perturbations due to the IGW in the horizontal velocity components (, ΔΔ VU ), vertical velocity component ΔW and temperature
perturbation ΔT . Our numerical analysis was conducted using data obtained from three mid
and high latitude locations, namely Bear Lake Observatory (UT), Ft Collins (CO) and Eureka
(Canada). Using data for each station for different phase velocities, the gravity wave response
on temperature, horizontal velocity components and the vertical velocity component was deduced. The perturbation data was then used to investigate the internal gravity wave reaction on the vibrational level dependant hydroxyl number densities and the associated minor species concentrations employing the following scheme.
Using the linearized continuity equation, 14 coupled algebraic equations are derived and
numerically solved with each vibrational level of [OH (n = 0 – 9)] and the other molecular
species H, HO2, O and O3 taken as individual, chemically active species. Our study focuses on
monochromatic wave events with significantly long (~ 8 hr) observed periodicities. Our
analyses spanned a wide range of phase velocities. For two of the locations, specifically for
Bear Lake Observatory (BLO - UT day 281) and for Ft Collins (UT day 307) this computation
is conducted for the following five horizontal phase velocity values Vph = 10 m/sec, 30 m/sec,
40 m/sec, 50 m/sec, 70 m/sec and 90 m/sec. The objective was to analyze the long period IGW
response on the different vibrational levels of OH Meinel (6, 2) emissions and on the minor
species concentrations. In calculating the Krassovsky ratio for the mid and high latitude
locations considered for the days specified, three horizontal phase velocities are used, the
chosen values are Vph = 30 m/sec, 50 m/sec and 70 m/sec. The obtained perturbations were
44 then used for computing vibrational level dependant volume emission rates, the fluctuations in
the vibrational level dependant volume emission rate, the atmospheric steady state effective temperature and the fluctuation in the steady state temperature for different horizontal phase velocities for all locations considered, to calculate the Krassovsky ratio.
With the inclusion of our long period gravity wave model, we observe large temperature
perturbations (see Figures 9, 10). In high latitudes solar driven higher order modes are capable
of producing high amplitudes of the MLT temperature and the airglow brightness. Almost all
of the considered observations are conducted in the late fall and winter months to take
advantage of long nocturnal measurements. But it is not possible to attribute the origin of these
observed higher amplitudes of the rotational temperature and Meinel (6, 2) OH airglow
brightness to global solar forcing, due to the fact that higher order modes are not strongly
excited by solar forcing and due to the absence of solar forcing in the winter months. Therefore
any such higher order mode explanations can be excluded. Also according to predictions of
Forbes [1982 a, b], in the high latitude Polar Regions, the amplitudes of the MLT temperature
and the airglow brightness due to the atmospheric tidal oscillations are very small at the winter
solstice. Therefore we can conclude that these high temperature perturbations are a response to the propagating long period IGW.
Next we wish to compare the Krassovsky ratio results that we obtained for our gravity wave
model against experimental observations considered. The calculated Krassovsky ratio
45 ⎛Δ Bnn ' ⎞ η = ηnn iφn ][exp|| is a complex quantity. The phase relation between the ⎜ ⎟ and B ' ⎝ 0nn ⎠
⎛ ⎞ the ⎜ΔT ⎟ was also calculated to compare our results with the results of Taylor et. al. [2001] ⎝ T0 ⎠
and Oznovich et. al. [1995]. The calculated values for the Krassovsky ratio, seems responsive
to the horizontal phase velocity. Therefore we investigated the effects of phase velocities, by
considering a range of values for the Vph for the upward wave energy propagation. In
calculating the Krassovsky ratio for the mid and high latitude locations considered for the days
specified, three horizontal phase velocities are used, the chosen values are Vph = 30 m/sec, 50
m/sec and 70 m/sec. An inventory of Krassovsky ratio values that we attained through our
numerical analysis is given in the table below.
Location UT Day Period Vph (m/s) Magnitude ηn Phase φn
Bear Lake 281 9.2 30 9.68 -0.84º Observatory, 50 10.12 -0.02 º UT 70 6.00 -1.64 º
Bear Lake 292 6.7 30 0.80 -0.05 º Observatory, 50 1.46 1.75 º UT 70 5.27 2.93 º
Ft. Collins, 307 8.6 30 1.58 9.75 º CO 50 2.13 13.08 º 70 1.17 -75.51 º
46
Ft. Collins, 358 9.2 30 0.52 88.31 º CO 50 0.81 -83.17 º 70 1.42 -72.18 º
Eureka, 356 8.1 30 10.16 -0.01 Canada 50 9.07 0.15 70 10.06 0.02 Table 06: The Krassovsky ratio values for BLO UT days 281 and 292, Ft Collins UT days 307
& 358 and Eureka UT day 356 obtained from our theoretical model. The phase relation between relative perturbation in brightness and relative temperature perturbation is also calculated.
Taylor et. al. [2001] recorded two long period wave oscillations, showing well defined sinusoidal wave patterns at Bear Lake Observatory (BLO), Utah. For the wave observed on the night of October 6-7, 1996 (UT day 281) period of oscillation was 9.2 hrs and for the wave observed on the night of October 17-18, 1996 (UT day 292) the period was 6.7 hrs. The calculated magnitude for the Krassovsky ratio for the observed wave pattern for UT day 281 was η ±= 110|| and the phase shift between the rotational temperature and the intensity perturbations was φ 1131 °±°−= . The corresponding values for the Krassovsky ratio magnitude and the phase shift for the long period oscillation with a period of 6.7 ± 1.5, on the night of October 17 – 18, 1996 was η = ± 0.36.7|| and φ = − °± 2666 ° . The η magnitudes that we obtained with our model, for both observation days for this site yields high numerical values, similar to Taylor et. al. [2001]. For UT day 281 data, the measured Krassovsky ratio
47 values agrees within the error bars with our model for Vph = 30 and 50 m/s values. For UT day
292 data our model agrees within the error bars with the measured Krassovsky ratio values for
Vph = 70 m/s.
Similar computations done on the measurements from at Ft. Collins, CO on the nights of 2-3
November 1997 (UT day 307) and 23-24 December 1997 (UT day 358) have yielded values
for the Krassovsky ratio magnitude and the phase shift as η = ± 7.19.3|| , φ °−= ± 4194 ° and η ±= 12.2|| , φ 311 °±°−= . The corresponding wave periods measured for the above
oscillations were 8.6 ± 1.0 hr and 9.2 ± 1.5 hr. The η magnitudes that we obtained with our
model, for both observations for this site yields low numerical values. The Low magnitude
values our model attained for this particular location is in agreement with the results of Taylor
et. al. [2001]. For UT days 307 and 358 data, the measured Krassovsky ratio values agrees within the error bars with our model for Vph = 50 m/s and Vph = 70 m/s respectively (See Table
06).
Oznovich et. al. [1995] had observed 5 and 3 complete cycles of marked ~ 8 hr oscillations in
airglow brightness and rotational temperature on December 21/22, 1993 (UT day 356) and
December 31/January 1, 1994 (UT day 01). Their December 21 observation with measured
period T = 8.1 ± 1-2 hr yielded Krassovsky ratio value of η = ± 7.18.8|| with phase valueφ 9107 °±°−= . The results from our calculations for UT day 356 are in very good
agreement with this experimental measurement (See Table 06). In particular both, our data and
48 the results obtained by Oznovich et. al. [1995] has obtained high magnitudes for the
Krassovsky ratio.
We next examine the phase relationships for our calculated results. According to the
computations conducted by Sivjee and Walterscheid [1994] the OH airglow brightness
modulations lead the temperature modulations for non-migrating tides. And for gravity wave
induced perturbations it is the temperature modulation that leads airglow brightness. As the
above results indicates obtained values for their mid-latitude phase shift calculations were
predominantly negative in sign indicating that the induced intensity perturbation waves lag the
temperature perturbation waves. In comparison, according to our results the negative phase of
η obtained for all three horizontal phase velocities for BLO UT day 281 is in full agreement
with Taylor et. al [2001] but for BLO UT day 292 we obtained negative phase shift for Vph =
30 m/s only. However the phase values that we obtained are lower than the values obtained by
Taylor et. al. [2001]. For Ft. Collins (CO) we obtained higher phase values for both the considered days. We obtained high negative phase values for the horizontal phase velocities
Vph = 70 m/s and Vph = 50 and 70 m/s for UT day 307 and UT day 358 respectively. Therefore
we can conclude that the negative phase values we invariably obtained for the Krassovsky ratio
for the different mid latitude locations agrees with the gravity wave approximation. As a result
for the mid latitude data the observed perturbations cannot be a result of a tidal wave by origin.
Due to the large horizontal wave lengths observed (λx > 3400km) , Oznovich et. al. has
attributed the ~ 8 hr oscillations to an inertio-gravity wave rather than an internal gravity wave
49 taking into account the influence of the earth’s rotation. And the other likely explanation they have provided for the observed ~ 8 hr oscillations was a non-migrating tide. For Eureka
(Canada) we obtained low φ values for UT day 356. Here we obtained negative phase value for the horizontal phase velocity for Vph = 30 m/s. Therefore from our theoretical analysis for the long period oscillations considered, even though some indicate small φ values, invariably the phase shift was negative. Indicating that the, temperature perturbations led the airglow brightness perturbations. Therefore the negative phase of the Krassovsky ratio that we obtained confirms that the ∼ 8 hour oscillation measurements taken on the December 21, 1993 cannot be attributed to a the presence of a non-migrating tide even though zonally symmetric tide is expected to be significant at high latitudes. In an attempt to address the conflicting phase relationships we look at the gravity wave linearization process that we employed. We can conclude that the first order linearization that we employed using the perturbation theory may have introduced unanticipated limitations and therefore it may be worthwhile to investigate the problem without introducing the first order linear approximations.
4.2 Conclusion
For both observation days for BLO, the Krassovsky ratio magnitudes that we obtained with our model produced high numerical values, consistent with results of Taylor et. al. [2001]. For UT days 281 and 292 data, our model agrees within the error bars with the measured Krassovsky ratio values of Taylor et. Al [2001] for Vph = 30 and 50 and Vph = 70 respectively. In
50 agreement with the results of Taylor et. al. [2001], the η magnitudes that we obtained with our
model for both observations for Ft Collins site yields low numerical values. And for UT days
307 and 358 data, the measured η values agrees within the error bars with our model for Vph
= 50 m/s and Vph = 70 m/s respectively. Also for Oznovich [1995] data, in comparison the results from our calculations for UT day 356 are in very good agreement with this experimental
measurement. In particular we obtained high η in agreement with the results obtained by
Oznovich et. al. [1995].
The negative Krassovsky ratio phase values that we consistently obtained from our theoretical
analysis for the different mid latitude locations for different horizontal phase velocities agrees
with the gravity wave approximation. Therefore with our knowledge of the tidal phase
relationship, we can conclude that for the mid latitude data, the observed perturbations cannot
be a result of a tidal wave by origin. For the high latitude location (Eureka, Canada) our
calculations for the long period oscillations considered, invariably produced negative phase
shifts even though some φ obtained values were small. Which indicates that the, temperature
perturbations led the airglow brightness perturbations. Therefore we can conclude that the
negative phase of the Krassovsky ratio that we obtained confirms that the ∼ 8 hour oscillation
measurements taken on the December 21, 1993 cannot be attributed to a the presence of a non-
migrating tide. For many of the wave perturbations we discussed here the apparent conflicting
phase relationships promote further investigation of the origin of these waves.
51
CHAPTER 5
FIGURES
52 200
160
120 Thermosphere
Mesopause
Altitude (km) 80
Mesosphere
Stratopause 40 Stratosphere Tropopause Troposphere 0 0 200 400 600 800 1000 1200 1400 Temperature (K)
Figure 1: Atmospheric temperature profile
53 120 Mean Density variation with Height 115 on a Logarithmic Scale for z = 1 - 150 km 150
Altitude (km) 110 120
90 105 (km) Altitude
60 100 30
95 0 1.0E-09 1.0E-07 1.0E-05 1.0E-03 1.0E-01 1.0E+01 90 -3 Log ρ0 (kg/m )
85
80
75
70 0.0E+00 1.0E-05 2.0E-05 3.0E-05 4.0E-05 5.0E-05 6.0E-05 7.0E-05 8.0E-05 9.0E-05 -3 ρ0 (kg/m )
Figure 2: Mean density variation with height and in the inset the mean density variation is plotted on a logarithmic scale
54 160
140
120
100 160 140
Altitude (km) Altitude 80 120 100 60 80 60 Altitude (km) Altitude 40 40 20 0 20 1.0E-10 1.0E-08 1.0E-06 1.0E-04 1.0E-02 1.0E+00
Log10 (τ/P0)
0 0.0E+00 1.0E-02 2.0E-02 3.0E-02 4.0E-02 5.0E-02 6.0E-02 7.0E-02
-1 2 τ/P0 (kg ms K)
Figure 3: Altitude variation of Molecular Viscosity coefficient over mean pressure (τ/P0)
55 160
140
120
100
160 140 80 120 Altitude (km) Altitude 100 60 80 60 Altitude (km) Altitude 40 40 20 0 20 1.0E-07 1.0E-05 1.0E-03 1.0E-01 1.0E+01 1.0E+03
Log10 (κ0/P0)
0 0.00E+00 2.00E+01 4.00E+01 6.00E+01 8.00E+01 1.00E+02 1.20E+02 1.40E+02 -1 2 κ0/P0 (kg ms K)
Figure 4: Altitude variation of Thermal Conductivity coefficient over mean pressure (κ0/P0)
56 120
115 Oxygen Nitrogen
110
105
100
95
Altitude (km) Altitude 90
85
80
75
70 1.E+10 1.E+11 1.E+12 1.E+13 1.E+14 1.E+15 1.E+16 Number Density (cm-3)
Figure 5: Major species number density profiles of [N2] and [O2]
57 120
110
100
Altidude (km) 90
O 80 H OZONE HO2
70 1.E-04 1.E-02 1.E+00 1.E+02 1.E+04 1.E+06 1.E+08 1.E+10 1.E+12
Number Density (cm-3)
`Figure 6: Unperturbed minor Species number density profiles of atomic oxygen, atomic hydrogen, ozone and HO2 obtained from COLORADO atmospheric profiles
58 110
100
90 ν = 9
Altitude (km) ν = ν = 80
70 0.0E+00 2.0E+03 4.0E+03 6.0E+03 8.0E+03 1.0E+04 1.2E+04
OH Number Density (cm-3)
Figure 7: Equilibrium photochemical altitude [OH(n)] profiles for n = 1-9 generated using the complete hydroxyl vibrational model
59
Vertically Upward (Z)
East (Y)
β0
k South (X) h
Figure 8: The chosen rectangular coordinate system with the x, y, and z axes pointing towards geographic south, east and vertically upwards, respectively. The horizontal wave propagation direction is assumed to be at an arbitrary direction β0 with the geographic south.
60
120
Altitude (km) 110
100
90
80
70 -5.0E+07 0.0E+00 5.0E+07 1.0E+08 1.5E+08 2.0E+08
Unperturbed Profile [H] (cm-3)
Figure 9: The unperturbed atomic hydrogen number density ([H]) from COLORADO atmospheric profiles
61 T = 9.2 hr Location: Bear Lake Observatory Day: UT 281
120
115
110
105
100
95
Altitude (km) 90 Vph 10 85 Vph 30 Vph 40 Vph 50 80 Vph 70 Vph 90 75
70 -2.0E+10 0.0E+00 2.0E+10 4.0E+10 6.0E+10 8.0E+10 1.0E+11 1.2E+11 Δ[H] (cm-3)
Figure 10: Gravity wave response wave amplitude of atomic hydrogen number density (Δ[H]) for different phase velocities for Bear Lake Observatory location (UT day 281)
62 T = 9.2 hr Location: Bear Lake Observatory Day: UT 281
120 Vph 10 115 Vph 30 Vph 40 110 Vph 50 Vph 70 Vph 90 105 Unperturbed [H] profile
100
95
Altitude (km) Altitude 90
85
80
75
70 1.0E+00 1.0E+01 1.0E+02 1.0E+03 1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08 1.0E+09 1.0E+10 1.0E+11 1.0E+12 Log Δ[H] (cm-3)
Figure 11: The unperturbed atomic H number density ([H]) and the gravity wave response wave amplitude of H number density (Δ[H]) for different phase velocities for Bear Lake Observatory location (UT day 281) plotted on a logarithmic scale for ease of comparison
63
120
Altitude (km) 110
100
90
80
70 -5.0E+05 0.0E+00 5.0E+05 1.0E+06 1.5E+06 2.0E+06 2.5E+06 3.0E+06 3.5E+06 4.0E+06
-3 Unperturbed Profile [H02] (cm )
Figure 12: The unperturbed perhydroxyl number density ([HO2]) from COLORADO atmospheric profiles
64 T = 9.2 hr Location: Bear Lake Observatory Day: 281 120 Vph 10 Vph 30 Vph 40 Vph 50 110 Altitude (km) Vph 70 Vph 90
100
90
80
70 -2.0E+10 -1.5E+10 -1.0E+10 -5.0E+09 0.0E+00 5.0E+09 1.0E+10 1.5E+10 2.0E+10 2.5E+10
−3 Δ[ΗΟ2] (cm )
Figure 13: Gravity wave response wave amplitude of perhydroxyl radical, number density (Δ[HO2]) for different phase velocities for Bear Lake Observatory location (UT day 281)
65
T = 9.2 hr Location: Bear Lake Observatory Day: UT 281 120 Vph 10 115 Vph 30 Vph 40 110 Vph 50 Vph 70 105 Vph 90 Unperturbed [HO2] profile 100
95
Altitude (km) 90
85
80
75
70 1.0E-04 1.0E-02 1.0E+00 1.0E+02 1.0E+04 1.0E+06 1.0E+08 1.0E+10 1.0E+12
-3 Log [HO2] (cm )
Figure 14: The unperturbed atomic H number density ([H]) and the gravity wave response wave amplitude of H number density (Δ[H]) for different phase velocities for Bear Lake Observatory location (UT day 281) plotted on a logarithmic scale for ease of comparison
66
120
110 Altitude (km)
100
90
80
70 -1E+11 0 1E+11 2E+11 3E+11 4E+11 5E+11
Unperturbed Profile [O] (cm-3)
Figure 15: The unperturbed atomic oxygen number density ([O]) from COLORADO atmospheric profiles
67
T = 9.2 hrs Location: Bear Lake Observatory Day: UT 281
120
115
110
105
100
95
Altitude (km) 90
85 Vph 30 80 Vph 40 Vph 50 75 Vph 70 Vph 90 70 -2.0E+10 0.0E+00 2.0E+10 4.0E+10 6.0E+10 8.0E+10 1.0E+11 1.2E+11 Δ[Ο] (cm−3)
Figure 16: Gravity wave response wave amplitude of atomic oxygen number density (Δ[O]) for different phase velocities for Bear Lake Observatory location (UT day 281)
68 T = 9.2 hrs Location: Bear Lake Observatory Day: UT 281
120
115 Vph 30 Vph 40
Altitude (km) Altitude 110 Vph 50 Vph 70 105 Vph 90 Unperturbed [O] profile 100
95
90
85
80
75
70 1.0E-01 1.0E+0 1.0E+0 1.0E+0 1.0E+0 1.0E+0 1.0E+0 1.0E+0 1.0E+0 1.0E+0 1.0E+0 1.0E+1 1.0E+1 1.0E+1 0 1 2 3 4 5 6 7 8 9 0 1 2 Log Δ[Ο] (cm−3)
Figure 17: The unperturbed atomic H number density ([H]) and the gravity wave response wave amplitude of H number density (Δ[H]) for different phase velocities for Bear Lake Observatory location (UT day 281) plotted on a logarithmic scale for ease of comparison
69
120
115
110
105
100
95 Altitude (km) Altitude 90
85
80
75
70 0.0E+00 1.0E+08 2.0E+08 3.0E+08 4.0E+08 5.0E+08 6.0E+08 7.0E+08 -3 Unperturbed Profile [O3] (cm )
Figure 18: The unperturbed ozone number density ([O3]) from COLORADO atmospheric profiles
70 T = 9.2 hrs Location: Bear Lake Observatory Day: UT 281 120 Vph 10 115 Vph 30 Vph 40
Altitude (km) Altitude 110 Vph 50 Vph 70 105 Vph 90
100
95
90
85
80
75
70 -1.0E+11 0.0E+00 1.0E+11 2.0E+11 3.0E+11 4.0E+11 5.0E+11 6.0E+11 7.0E+11 -3 Δ[O3] (cm )
Figure 19: Gravity wave response wave amplitude of ozone number density (Δ[O3]) for different phase velocities for Bear Lake Observatory location (UT day 281)
71
T = 9.2 hrs Location: Bear Lake Observatory Day: UT 281 120
115
110 Altitude (km) 105
100
95
Vph 10 90 Vph 30 Vph 40 85 Vph 50 Vph 70 80 Vph 90 Unperturbed O3 profile 75
70 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04 1.E+05 1.E+06 1.E+07 1.E+08 1.E+09 1.E+10 1.E+11 1.E+12 -3 Log [O3] (cm )
Figure 20: The unperturbed atomic H number density ([O3]) and the gravity wave response wave amplitude of H number density (Δ[H]) for different phase velocities for Bear Lake Observatory location (UT day 281) plotted on a logarithmic scale for ease of comparison
72 Vph = 50 m/s
120
115 Altitude (km) 110
105
100
95
90
85 BLO Day 281
80 BLO Day 292 Eureka Days 01 & 356
75 Ft Collins Day 307 Ft Collins Day 358 70 -5.0E+09 0.0E+00 5.0E+09 1.0E+10 1.5E+10 2.0E+10 2.5E+10 3.0E+10 3.5E+10 Δ[H] (cm-3)
Figure 21: Gravity wave response of atomic hydrogen number density (Δ[H]) for different locations for the horizontal phase velocity Vph = 50 m/s
73 Vph = 50 m/s 120 BLO Day 281 BLO Day 292 115 Eureka Days 01 & 356 Ft Collins Day 307 110 Ft Collins Day 358
105
100
95
Altitude (km) 90
85
80
75
70 -5.0E+08 0.0E+00 5.0E+08 1.0E+09 1.5E+09 2.0E+09 2.5E+09 -3 Δ[HO2] (cm )
Figure 22: Gravity wave response of perhydroxyl number density (Δ[HO2]) for different locations for the horizontal phase velocity Vph = 50 m/s
74 Vph = 50 m/s
120
115
Altitude (km) Altitude 110
105
100
95
90
85 BLO Day 281 80 BLO Day 292 Eureka Days 01 & 356 75 Ft Collins Day 307 Ft Collins Day 358 70 -1E+10 0 1E+10 2E+10 3E+10 4E+10 5E+10 6E+10 7E+10 8E+10 9E+10 Δ[O] (cm-3)
Figure 23: Gravity wave response of atomic oxygen number density (Δ[O]) for different locations for the horizontal phase velocity Vph = 50 m/s
75 Vph = 50 m/s
120
115
110
105
100
95
Altitude (km) 90
85 BLO Day 281 BLO Day 292
80 Eureka Days 01 & 356 Ft Collins Day 307 75 Ft Collins Day 358
70 -5.0E+10 0.0E+00 5.0E+10 1.0E+11 1.5E+11 2.0E+11 2.5E+11 3.0E+11 3.5E+11 -3 Δ[O3] (cm )
Figure 24: Gravity wave response of ozone number density (Δ[O3]) for different locations for the horizontal phase velocity Vph = 50 m/s
76 S0
S = 0
μ = 1 z z μ > 0 y dz θ´ ds θ´ θ Reference Level 0 μ = 0 ϕ0
x μ < 0 μ = -1 ds = dz / μ with μ = cos θ´
Figure 25: Plane parallel geometry illustration. The solar radiation in incident from (θ0,ϕ0). For upwelling radiation m>0 for down-welling radiation m<0, for zenith (m=1) for nadir m = -1 and for horizontal direction m = 0.
(Source: Wilford Zdunkowski et.al. [2007])
77
120
BLODay281 Vph30 BLODay281 Vph40 110
BLODay281 Vph50 Altitude (km) BLODay281 Vph70 BLODay281 Vph90 EurekaDays01&356 Vph30 100 EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 FtCollinsDay358 Vph30 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70 90
80
70 -9.0E+09 -7.0E+09 -5.0E+09 -3.0E+09 -1.0E+09 1.0E+09 Number Density (cm-3)
Figure 26: OH Meinel (6,2) number density gravity wave response for the vibrational level (n = 0) for different locations considered for various horizontal phase velocities
78 120.00 BLODay281 Vph30 BLODay281 Vph40 BLODay281 Vph50 BLODay281 Vph70 BLODay281 Vph90 110.00 Altitude (km) EurekaDays01&356 Vph30 EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 FtCollinsDay358 Vph30 100.00 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70
90.00
80.00
70.00 -8.0E+05 -7.0E+05 -6.0E+05 -5.0E+05 -4.0E+05 -3.0E+05 -2.0E+05 -1.0E+05 0.0E+00 1.0E+05 Number Density (cm-3)
Figure 27: OH Meinel (6,2) number density gravity wave response for the vibrational level (n = 1) for different locations considered for various horizontal phase velocities
79 120 BLODay281 Vph 30 BLODay281 Vph 40 BLODay281 Vph 50 BLODay281 Vph 70 BLODay281 Vph 90 110 Altitude (km) Altitude EurekaDays01&356 Vph30 EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 FtCollinsDay358 Vph30 100 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70
90
80
70 -1.1E+05 -9.0E+04 -7.0E+04 -5.0E+04 -3.0E+04 -1.0E+04 1.0E+04 Number Density (cm-3)
Figure 28: OH Meinel (6,2) number density gravity wave response for the vibrational level (n = 2) for different locations considered for various horizontal phase velocities
80 120 BLODay281 Vph 30 BLODay281 Vph 40 BLODay281 Vph 50 BLODay281 Vph 70 BLODay281 Vph 90 110
EurekaDays01&356 Vph30 Altitude (km) EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 FtCollinsDay358 Vph30 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70 100
90
80
70 -4500 -3500 -2500 -1500 -500 500 1500 Number Density (cm-3) Figure 29: OH Meinel (6,2) number density gravity wave response for the vibrational level (n = 3) for different locations considered for various horizontal phase velocities
81 120 BLODay281 Vph 30 BLODay281 Vph 40 BLODay281 Vph 50 BLODay281 Vph 70 110
Altitude (km) BLODay281 Vph 90 EurekaDays01&356 Vph30 EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 100 FtCollinsDay358 Vph30 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70
90
80
70 -300 -200 -100 0 100 200 300 400 500 Number Density (cm-3)
Figure 30: OH Meinel (6, 2) number density gravity wave response for the vibrational level (n = 4) for different locations considered for various horizontal phase velocities
82 120 BLODay281 Vph 30 BLODay281 Vph 40 BLODay281 Vph 50 BLODay281 Vph 70
110 Altitude (km) BLODay281 Vph 90 EurekaDays01&356 Vph30 EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 100 FtCollinsDay358 Vph30 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70
90
80
70 -150 -100 -50 0 50 100 150 200 250 300 Number Density (cm-3)
Figure 31: OH Meinel (6, 2) number density gravity wave response for the vibrational level (n = 5) for different locations considered for various horizontal phase velocities
83 120 BLODay281 Vph 30 BLODay281 Vph 40 BLODay281 Vph 50 BLODay281 Vph 70 110 BLODay281 Vph 90 Altitude (km) Altitude EurekaDays01&356 Vph30 EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 FtCollinsDay358 Vph30 100 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70
90
80
70 -90 -50 -10 30 70 110 150 190 Number Density (cm-3)
Figure 32: OH Meinel (6, 2) number density gravity wave response for the vibrational level (n = 6) for different locations considered for various horizontal phase velocities
84 120 BLODay281 Vph 30 BLODay281 Vph 40 BLODay281 Vph 50 BLODay281 Vph 70 110
Altitude (km) Altitude BLODay281 Vph 90 EurekaDays01&356 Vph30 EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 100 FtCollinsDay358 Vph30 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70
90
80
70 -40 -20 0 20 40 60 Number Density (cm-3)
Figure 33: OH Meinel (6, 2) number density gravity wave response for the vibrational level (n = 7) for different locations considered for various horizontal phase velocities
85 120 BLODay281 Vph 30 BLODay281 Vph 40 BLODay281 Vph 50 110 BLODay281 Vph 70 BLODay281 Vph 90 Altitude (km) EurekaDays01&356 Vph30 EurekaDays01&356 Vph50 100 EurekaDays01&356 Vph70 FtCollinsDay358 Vph30 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70 90
80
70 -5.0E-15 0.0E+00 5.0E-15 1.0E-14 1.5E-14 Number Density (cm-3)
Figure 34: OH Meinel (6, 2) number density gravity wave response for the vibrational level (n = 8) for different locations considered for various horizontal phase velocities
86 120 BLODay281 Vph 30 BLODay281 Vph 40 BLODay281 Vph 50 BLODay281 Vph 70 110 BLODay281 Vph 90 Altitude (km) EurekaDays01&356 Vph30 EurekaDays01&356 Vph50 EurekaDays01&356 Vph70 FtCollinsDay358 Vph30 100 FtCollinsDay358 Vph50 FtCollinsDay358 Vph70
90
80
70 -30 -20 -10 0 10 20 30 40 50 Number Density (cm-3)
Figure 35: OH Meinel (6,2) number density gravity wave response for the vibrational level (n = 9) for different locations considered for various horizontal phase velocities
87 120
115 OH (n = 2) OH (n = 3) 110
105
100
95
90
85
80
75
70 -3.0E+04 -2.5E+04 -2.0E+04 -1.5E+04 -1.0E+04 -5.0E+03 0.0E+00 5.0E+03
Figure 36: hydroxyl number density for the vibrational level n = 2 and n = 3 plotted for Bear Lake Observatory location for UT day 281 for the 50 m/s horizontal phase velocity
88 14
12 Location = Bear Lake Obsevatory Period = 9.2 hr Day = UT 281
10
8
Magnitude 6 η
4
2
Location = Ft Collins Period = 8.6 hr Day = UT 307 0 0 102030405060708090100
Phase Velocity (m/s)
Figure 37: Krassovsky Ratio magnitude variation with phase velocity for Bear Lake Observatory (day 281) & Ft Collins (UT day 307)
89 12
10
8 η
6 Magnitude BLO, UT Day 281 4 BLO, UT Day 292 Ft Colloins, CO Day 307 Ft Collins, CO Day 358 Eureka, Canada Day 356 2
0 20 30 40 50 60 70 80 Horizontal Phase Velocity (m/s)
Figure 38: Summery plot of Krassovsky Ratio magnitude (|η|) variation with horizontal phase velocity for the high and mid latitude locations considered
90 CHAPTER 6
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97 Appendix A
The coefficient values obtained for the linearized coupled first order hydrodynamic equations are as follows:
The coefficients for the mass conservation equation are,
11 = kA x ρ0
12 = kA y ρ 0
⎛ ∂ρ 0 ⎛ i ⎞⎞ 13 =⎜iA ρ 0 ⎜ ++ k z ⎟⎟ ⎝ ∂z ⎝ 2H ⎠⎠
A14 =0
A15 −= ω
A16 =0
The coefficients for the i) component of the momentum conservation equation are,
⎛ 2 2 ⎞ ⎛ 4k 2 i ⎛ i ⎞ ⎛ i ⎞ ⎞ ⎜iA τωρ ⎜ x k ⎜ +−++= k ⎟ ⎜ ++ k ⎟ ⎟⎟ 21 ⎜ 0 ⎜ y z z ⎟⎟ ⎝ ⎝ 3 H ⎝ 2H ⎠ ⎝ 2H ⎠ ⎠⎠ ⎛ τ ⎞ A22 ⎜ 2ρ 0 z +Γ−= kk yx ⎟ ⎝ 3 ⎠ ⎛ ki ττ k ⎛ i ⎞ ⎞ ⎜ xx ⎟ A23 ⎜2ρ 0 y +−Γ= ⎜ + k z ⎟ ⎟ ⎝ H ⎝ 23 H ⎠ z ⎠
24 −= ikA x
A25 =0
A26 =0
The coefficients for the )j component of the momentum conservation equation are,
98
⎛ τ ⎞ A31 ⎜2ρ0 +Γ= kk yxz ⎟ ⎝ 3 ⎠ ⎛ 2 ⎞ ⎛ 2 4 2 i ⎛ i ⎞ ⎛ i ⎞ ⎞ ⎜ τωρ ⎜ kkiA ⎜ +−++= k ⎟ ⎜ ++ k ⎟ ⎟⎟ 32 ⎜ 0 ⎜ x y z z ⎟⎟ ⎝ ⎝ 3 H ⎝ 2H ⎠ ⎝ 2H ⎠ ⎠⎠ ⎛ ki ττ k i ⎞ ⎜ yy ⎛ ⎞⎟ A33 ⎜ 2ρ0 x ⎜ ++−Γ−= k z ⎟⎟ ⎝ H ⎝ 23 H ⎠⎠
34 −= ikA y
A35 =0
A36 =0
) The coefficients for the k component of the momentum conservation equation are,
⎛ 2 ki ττ k xx ⎛ i ⎞⎞ A41 ⎜ 2ρ0 y ++Γ−= ⎜ + k z ⎟⎟ ⎝ H ⎝ 233 H ⎠⎠ ⎛ 2 ki ττ k i ⎞ ⎜ yy ⎛ ⎞⎟ A42 ⎜2ρ0 x ++Γ= ⎜ + k z ⎟⎟ ⎝ H ⎝ 233 H ⎠⎠ ⎛ 2 ⎞ ⎛ 22 4i ⎛ i ⎞ 4 ⎛ i ⎞ ⎞ ⎜ τρ ⎜ kkiA −++Ω= ⎜ + k ⎟ ⎜ ++ k ⎟ ⎟⎟ 43 ⎜ 0 ⎜ yx z z ⎟⎟ ⎝ ⎝ H ⎝ 23 H ⎠ ⎝ 23 H ⎠ ⎠⎠ ⎛ ⎛ i ⎞ 1 ⎞ 44 ⎜iA ⎜ +−= k z ⎟ + ⎟ ⎝ ⎝ 2H ⎠ H ⎠
45 = gA
A46 =0
The coefficients for the energy conservation equation are,
99 51 −= x PikA 0
52 −= y PikA 0 ⎛ i ⎞ 53 iPA 0 ⎜ +−= k z ⎟ ⎝ 2H ⎠
A54 =0
A55 = 0 ⎛ 2 ⎞ ⎛ 22 i ⎛ i ⎞ ⎛ i ⎞ ⎞ i μωρ = ⎜ κωρ ⎜ kkCiA ⎜ +−++ k ⎟ ⎜ ++ k ⎟ ⎟ + 0 ⎟ 56 ⎜ 0 v 0 ⎜ yx z z ⎟ ⎟ ⎝ ⎝ H ⎝ 2H ⎠ ⎝ 2H ⎠ ⎠ γ −1 ⎠
100 Appendix B
In the absence of horizontal background wind using the kinetic photo chemical model we obtained 14 coupled algebraic equations which were non-dimensionalized and numerically solved to find the molecular species altitude dependant profiles. The coefficients of the 14 coupled equations take the following form,
[]+= [ ]0,827361,1 []++ R [ ]+ R [ 21,1021,9 ]+ iNkOkOkHOkOkP ω
[] []−−−= ANkOkP 0,121,1021,92,1
3,1 −= AP 0,2
4,1 −= AP 0,3
5,1 −= AP 0,4
6,1 −= AP 0,5
7,1 −= AP 0,6
P 8,1 = 0
P 9,1 = 0
P 10,1 = 0
= 611,1 − HOkOHkP 25 ][)]0([
= 0,812,1 − HOkOHkP 20,2 ][)]0([
P 13,1 =0
101 = 714,1 0,2 −− OkOkOHkP 30,5 ][][)]0([
⎧ ⎛ i ⎞ ∂ ⎫ 1 = x[OHikQ )]0( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]0([)]0([ 0 ⎜ + kz ⎟ − []OH )0( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
R []−−= R [NkOkP 21,1021,91,2 ]
1,82,2 [] [ ] [ ]+++= + R [ ]+ R [ 22,1022,90,121,1021,9 ]+ iNkOkANkOkOkP ω
[] []−−−= ANkOkP 1,222,1022,93,2
−= AP 1,34,2
−= AP 1,45,2
−= AP 1,56,2
−= AP 1,67,2
−= AP 1,78,2
P 9,2 = 0
P 10,2 = 0
P 11,2 =0
= 1,812,2 − HOkOHkP 21,2 ][)]1([
P 13,2 =0
14,2 −= 1,2 OkP ][
⎧ ⎛ i ⎞ ∂ ⎫ 2 = x[)]1( 0 +Δ y 0 +Δ ⎨ OHiVOHikUOHikQ )]1([)]1([ 0 ⎜ + kz ⎟ − []OH )1( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
102 P 1,3 = 0
R []−−= R [NkOkP 22,1022,92,3 ]
2,83,3 [] [ ] [ ]+++= + + R [ ]+ R [ 23,1023,90,21,222,1022,9 ]+ iNkOkAANkOkOkP ω
[] []−−−= ANkOkP 2,323,1023,94,3
5,3 −= AP 2,4
−= AP 2,56,3
7,3 −= AP 2,6
−= AP 2,78,3
9,3 −= AP 2,8
P 10,3 = 0
P 11,3 =0
= 2,812,3 − HOkOHkP 22,2 ][)]2([
P 13,3 =0
= 2214,3 OkP ][
⎧ ⎛ i ⎞ ∂ ⎫ 3 = x[OHikQ )]2( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]2([)]2([ 0 ⎜ + kz ⎟ − []OH )2( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
P 1,4 = 0
P 2,4 = 0
R []+−= R [NkOkP 23,1023,93,4 ]
103 3,84,4 [] [ ] [ ]+++= + + + R [ ]+ R [ 24,1024,90,31,32,323,1022,9 ]+ iNkOkAAANkOkOkP ω
[] []−−−= ANkOkP 3,424,1024,95,4
6,4 −= AP 3,5
7,4 −= AP 3,6
8,4 −= AP 3,7
−= AP 3,89,4
10,4 −= AP 3,9
P 11,4 = 0
= 3,812,4 − HOkOHkP 23,2 ][)]3([
P 13,4 =0
14,4 −= 23 OkP ][
⎧ ⎛ i ⎞ ∂ ⎫ 4 = x[OHikQ )]3( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]3([)]3([ 0 ⎜ + kz ⎟ − []OH )3( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
P 1,5 = 0
P 2,5 = 0
P 3,5 = 0
R []−−= R [NkOkP 24,1024,94,5 ]
4,85,5 [] [ ] []+++= + + + + R [ ]+ R [ 25,1025,90,41,42,43,424,1024,9 ]+ iNkOkAAAANkOkOkP ω
[] []−−−= ANkOkP 4,525,1025,96,5
7,5 −= AP 4,6
104 8,5 −= AP 4,7
−= AP 4,89,5
10,5 −= AP 4,9
P 11,5 = 0
= 4,812,5 OHkP )]4([
P 13,5 =0
P 14,5 =0
⎧ ⎛ i ⎞ ∂ ⎫ 5 = x[OHikQ )]4( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]4([)]4([ 0 ⎜ + kz ⎟ − []OH )4( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
P 1,6 = 0
P 2,6 = 0
P 3,6 = 0
P 4,6 = 0
R []−−= R [NkOkP 25,1025,95,6 ]
5,86,6 [] [ ] []+++= + + + + + R [ ][]+ R 26,1026,90,51,52,53,54,525,1025,9 + iNkOkAAAAANkOkOkP ω
[] []−−−= ANkOkP 5,626,1026,97,6
8,6 −= AP 5,7
−= AP 5,89,6
10,6 −= AP 5,9
P 11,6 = 0
105 = 5,812,6 OHkP )]5([
⎧ ⎛ i ⎞ ∂ ⎫ 6 = x[OHikQ )]5( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]3([)]5([ 0 ⎜ + kz ⎟ − []OH )5( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
P 1,7 = 0
P 2,7 = 0
P 3,7 = 0
P 4,7 = 0
P 5,7 = 0
R []+−= R [NkOkP 26,1026,96,7 ]
6,87,7 [] [] []+++= + + + + + + R []+ R []27,1027,90,61,62,63,64,65,626,1026,9 + iNkOkAAAAAANkOkOkP ω
[] []−−−= ANkOkP 6,727,1027,98,7
−= AP 6,89,7
10,7 −= AP 6,9
11,7 −= 16 HkP ][
= 6,812,7 OHkP )]6([
13,7 −= OkP 316 ][
P 14,7 =0
⎧ ⎛ i ⎞ ∂ ⎫ 7 = x[[OHikQ )]6( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]6([)]6([ 0 ⎜ + kz ⎟ − []OH )6( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
106 P 1,8 = 0
P 2,8 = 0
P 3,8 = 0
P 4,8 = 0
P 5,8 = 0
P 6,8 = 0
R []−−= R [NkOkP 27,1027,97,8 ]
7,88,8 [] [] []+++= + + + + + + R []++ R [ 28,1028,90,31,72,73,74,75,76,727,1027,9 ]+ iNkOkAAAAAAANkOkOkP ω
[] []−−−= ANkOkP 7,828,1028,99,8
10,8 −= AP 7,9
11,8 −= 17 HkP ][
= 7,812,8 OHkP )]7([
13,8 −= OkP 317 ][
P 14,8 =0
⎧ ⎛ i ⎞ ∂ ⎫ 8 = x[OHikQ )]7( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]7([)]7([ 0 ⎜ + kz ⎟ − []OH )7( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
P 1,9 = 0
P 2,9 = 0
P 3,9 = 0
P 4,9 = 0
107 P 5,9 = 0
P 6,9 = 0
P 7,9 = 0
R []+−= R [NkOkP 28,1028,98,9 ]
8,89,9 [] [] []+++= + + + + + + R []+ R []29,1029,92,83,84,85,86,87,828,1028,9 + iNkOkAAAAAANkOkOkP ω
10,9 [] []−−−= ANkOkP 8,929,1029,9
11,9 −= 18 HkP ][
= 8,812,9 OHkP )]8([
13,9 −= OkP 318 ][
P 14,9 =0
⎧ ⎛ i ⎞ ∂ ⎫ 9 = x[OHikQ )]8( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]8([)]8([ 0 ⎜ + kz ⎟ − []OH )8( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
P 1,10 = 0
P 2,10 = 0
P 3,10 = 0
P 4,10 = 0
P 5,10 = 0
P 6,10 = 0
P 7,10 = 0
P 8,10 = 0
108 R []+−= R [NkOkP 29,1029,99,10 ]
9,810,10 [] [ ] [ ]+++= + + + + + 3,94,95,96,97,98,929,1029,9 + iAAAAAANkOkOkP ω 11,10 −= 9,1 []HkP
= 8912,10 OHkP )]9([
13,10 −= OkP 39,1 ][
P 14,10 =0
⎧ ⎛ i ⎞ ∂ ⎫ 10 = x[OHikQ )]9( 0 +Δ y 0 +Δ ⎨ OHiVOHikU )]9([)]9([ 0 ⎜ + kz ⎟ − []OH )9( 0 ⎬ ΔW ⎩ ⎝ 2H ⎠ ∂z ⎭
= []OkP 361,11
P 2,11 = 0
P 3,11 = 0
= { + 6254,11 ()]0([][ ++ + + 9,18,17,16,1 ][) + iHkkkkOHkHOkP ω}
P 5,11 = 0
P 6,11 = 0
P 7,11 = 0
P 8,11 = 0
P 9,11 = 0
P 10,11 = 0
P 11,11 = 0
12,11 −= [][23 MOkP ]
109 ( +++= )[]OkkkkP 39,18,17,16,113,11
= []OkP 3514,11
⎧ ⎛ i ⎞ ∂ ⎫ 11 x[3 y 3 +Δ+Δ= ⎨ OiVOikUOikQ 3 ][][] ⎜ + kz ⎟ − []3 ⎬ ΔWO ⎩ ⎝ 2H ⎠ ∂z ⎭
= 0,81,12 []OkP
= 1,82,12 []OkP
= 2,83,12 []OkP
= 3,84,12 []OkP
= 4,85,12 []OkP
= 5,86,12 []OkP
= 6,87,12 []OkP
= 7,88,12 []OkP
= 8,89,12 []OkP
= 9,810,12 []OkP
P 11,12 = 0
= 0,812,12 [][]+ 1,8 )1()0( + 2,8 [ )2( ]+ 3,8 [ )3( ]+ 4,8 [ )4( ][+ 5,8 )5( ]+ 6,8 [OHkOHkOHkOHkOHkOHkOHkP )6( ]
+ 7,8 [][][])7( + 8,8 )8( + 9,8 )9( ()++++ [][][]+ 3323,22,21,20,2 + iMOkHOkkkkOHkOHkOHk ω
P 13,12 = 0
( +++= 3,22,21,20,214,12 )[]OkkkkP
⎧ ⎛ i ⎞ ∂ ⎫ 12 x y +Δ+Δ= ⎨ OiVOikUOikQ ][][][ ⎜ + kz ⎟ − []⎬ ΔWO ⎩ ⎝ 2H ⎠ ∂z ⎭
110
−= 0,81,13 []OkP
−= 1,82,13 []OkP
−= 2,83,13 []OkP
−= 3,84,13 []OkP
−= 4,85,13 []OkP
−= 5,86,13 []OkP
−= 6,87,13 []OkP
−= 7,88,13 []OkP
−= 8,89,13 []OkP
10,13 −= 9,8 []OkP
11,13 {}()+++= 19181716 []HkkkkP
12,13 −= 0,8 [][]− 1,8 )1()0( − 2,8 [ ]− 3,8 [ )3()2( ]− 4,8 [ )4( ][− 5,8 )5( ]− 6,8 [OHkOHkOHkOHkOHkOHkOHkP )6( ]
− 7,8 [][][]− 8,8 )8()7( − 9,8 OHkOHkOHk )9(
= [][]2413,13 ()++++ [ 319181716 ]+ iOkkkkMOkP ω
P 14,13 = 0
⎧ ⎛ i ⎞ ∂ ⎫ 13 x y ][][ ⎨iVHikUHikQ ⎜ ++Δ+Δ= z ⎟ Hk ][ − []⎬ ΔWH ⎩ ⎝ 2H ⎠ ∂z ⎭
[][−= OkHOkP 36271,14 ]
P 2,14 = 0
111 P 3,14 = 0
P 4,14 = 0
P 5,14 = 0
P 6,14 = 0
P 7,14 = 0
P 8,14 = 0
P 9,14 = 0
P 10,14 = 0
= ()[]− 62511,14 []OHkHOkP (0)
12,14 {}()+++= 23222120 []HOkkkkP 2
13,14 −= [][24 MOkP ]
14,13 ()23222120 []++++= [ ]+ 735 [ (0)]+ iOHkOkOkkkkP ω
P 14,13 = 0
⎧ ⎛ i ⎞ ∂ ⎫ 14 = x[2 +Δ y 2 ][] ⎨iVHOikUHOikQ ⎜ ++Δ z ⎟ HOk 2 ][ − []2 ⎬ ΔWHO ⎩ ⎝ 2H ⎠ ∂z ⎭
112 Appendix C
The obtained expressions for the column integrating airglow brightness and its fluctuation due to the propagating IGW for the OH Meinel (6, 2) airglow are as follows,
Δ=Δ ),,)](6([ dlzyxOHAB 62 ∫ 62
= )()]6([ dlzOHAB 062 ∫ 62 0
Using the brightness weighted temperature method we also obtained, the effective steady state brightness-weighted temperature and its fluctuation,
)()()6( dlzTzOHA ∫ 62 []00 0 TT == 06 )()6( dlzOHA ∫ 62 []0
62 []()6 0 ()Δ (,, )dlzyxTzOHA 62 Δ[ ( )]( ,,6 ) 0 ( ) dlzTzyxOHA T =Δ ∫ + ∫ 6 dlzOHA 6 dlzOHA ∫ 62 []()0 () ∫ 62 []()0 ()
62 []()6 ()00 ()dlzTzOHA 62 Δ[]()(),,6 dlzyxOHA − ∫ × ∫ 6 dlzOHA 6 dlzOHA ∫ 62 []()0 () ∫ 62 []()0 ()
113