OHIO 

m 

UNIVERSITY 

Zonal Winds 

Between 25 and 120 KM 

Retrieved from Solar Occultation 

Spectra 

John H. Shaw 

Department of Physics 

Garrett W. van Cleef 

Graduate Research Associate 

National Aeronautics and Space Administration 

Washington, D.C. 20546 

Grant No. NAGW-1129 

Final Report 

RF Project 766291/719915 

July 1989 
ZONAL WINDS BETWEEN 25 AND 120 KM RETRIEVED 

FROM SOLAR OCCULTATION SPECTRA 

DIS S ERTATIO N 

I Presented in Partial Nfilment of the Requirements for 

the Degree Doctor of Philosophy in the Graduate 

School of The Ohio State University 

Garrett Warren van Cleef 11, B.Sc. 

The Ohio State University 

1989 

DISSERTATION COMMITTEE: 

J. H. Shaw 

C. D. Andereck 

R. D. Knight 

C. V. Heer 

Approved by 

Adviser 

Department of Physics 
ZONAL WINDS BETWEEN 25 AND 120 KM RETRIEVED 

FROM SOLAR OCCULTATION SPECTRA 

BY 

Garrett W. van Cleef 11, B.S. 

The Ohio State University, 1989 

Professor John H. Shaw, Adviser 
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ABSTRACT 

Atmospheric winds at heights between 25 and 120 km have been retrieved I 

with precisions of 5 ms-' from the Doppler shifts of atmospheric absorption 

lines measured from a satellite-borne instrument. Lines of the u3 COZ and 

u2 Ha0 rotation-vibration bands caused by gases in the instrument allowed 

the instrumental frequency scale to be absolutely calibrated so that accurate 

relative speeds could be obtained. By comparing the positions of both sets of 

instrumental lines the calibration of the frequency scale was determined to be 

stable to a precision of < 2 x 16T5,km-l during the course of each occultation. 

It was found that the instrumentdl resolution of 0.015 cm-l after apodization, 

the signal to noise ratio of about 100 and stable calibration allowed relative 

speeds to be determined to a precision of 5,ms-l or better by using small 

numbers of absorption lines between 1600 and 3200 cm-'. Absolute absorption 

line positions were simultaneously recovered to precisions of 5 x cm-' or 

better. 

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The wind speed profiles determined from four sunset occultations and 

one sunrise occultation show remarkable similarities in the magnitudes and 

directions of the zonal wind velocities as functions of height. These wind profiles 

appear to be manifestations of atmospheric tides. The periodic heating of the 

atmosphere primarily by ozone absorption and water insolation can force global 

1 
circulation of the upper atmosphere as described by Lindzen and Chapman 

(1970). The diurnal and semidiurnal Fourier components of the heating rates 

as predicted by Groves (1983) were shown to be significant in driving the tidal 

circulation. The linear or "classical" tidal theory [Lindzen, Chapman, 19701, 

however, predicts wind speeds which grow exponentially with height due to 

the exponential decrease in the hydrostatic background pressure. The growth 

was found to lead to wind speeds much larger in amplitude than revealed by 

the spectroscopic measurements. The classical theory must break down in a 

nonlinear way and the consequences of nonlinear and adiabatic instabilities 

present in such growth in amplitude is discussed. 

.. 

11 
VITA 

1983 B.S., Stevens Institute of Technology, Hoboken, 

New Jersey 

1983-1989 Department of Physics, Ohio State University, 

Columbus, Ohio 

PUBLICATIONS 

van Cleef, G. W., J. H. Shaw, C. B. Farmer, Zonal winds between 25 and 

120 km obtained from solar occultation spectra, Geophys. Res. Lett., 14, 

1266-1268, 1987. 

RESEARCH AREA 

Infrared spectroscopy and atmospheric physics. 

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PRECEDiKG PAGE BLANK NOT FILMED 
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TABLEOFCONTENTS 

ABSTRACT.. ......................... i 

ACKNOWLEDGMENTS .................... iii 

VITA .............................. iv 

LIST OF FIGURES ...................... viii 

LIST OF TABLES ....................... xiv 

INTRODUCTION .................... 1 

Mission Objectives ................. 1 

Overview of Thesis Research ............. 3 

I SPECTROSCOPIC THEORY ................ 9 

1.1 The Michelson Interferometer and ........... 9 

Fourier Transform Spectroscopy 

1.2 Measuring Doppler Shifts from ATMOS Spectra .... 21 

1.3 QuantumMechanics of Infrared Spectroscopy ..... 22 

1.4 Normal Coordinates and Vibrational Energy for .... 26 

Triatomic Molecules 

1.5 Rotational Energy Levels .............. 34 

1.6 Rotation-Vibration Spectra ............. 36 

1.7 Radiative Transfer ................. 40 

1.8 Line Shapes .................... 44 
1.9 Absorption and Emission Coefficients . . . . . . . . . 1.10 Optical Depth, Equivalent Width, and Curve of Growth . 1.11 Measurement' Precision Theory . . . . . . . . . . . . 

I1 EXPERIMENTAL METHOD . . . . . . . . . . . . . . . 

11.1 Instrumental Calibration . . . . , . . . . . . . . . 11.2 'Scale Uncertainty a(c) . . . . . . . . . . . . . . . 11.3 Solar and Ratioed Spectra . . . . . . . . . . . . . . 11.4 Estimating the Noise Level . . . . . . . . . . . . . 11.5 Estimating Speeds Along the Line of Sight . . . . . . . 11.6 Relative Speed Measurements Using v3 COa . . . . . . Lines Between 2330 and 2372 cm-' 

11.7 Extending Relative Speed Measurements and . . . . . . the Determination of 

11.8 Weighted Averages of Relative Speed Profiles . . . . . . 11.9 Extensions to Lower Tangent Heights by Using . . . . . 

H2O and N2O 

11.10 Line Positions Obtained and Calibration . . . . . . . . Standards 

11.11 Testing the Measurement Theory . . . . . . . . . . . 

11.12 Estimating the Doppler Shift due to Orbital . . . . . . Motion and the Earth's Rotation 

11.13 Density Weighted Residual Winds . . . . . . . . . . . 11.14 Conclusion . . . . . . . . . . . . . . . . . . . . . 

I11 LARGE SCALE DYNAMICS OF THE CIRCULATION OF . . THE EARTH'S ATMOSPHERE 

111.1 The Atmospheric Temperature Distribution . . . . . . 

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62 

72 

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101 

104 

109 

111 

117 

123 

126 

136 

142 

149 

167 

179 

174 

184 
111.2 Fluid Equations of Motion ............. 186 

111.3 Hydrostatic State and the Thermal Wind ....... 189 

111.4 Linear Waves on a Hydrostatic Spherical ....... 196 

111.5 F'ree Planetary Waves . The Homogeneous ....... 208 

111.6 Atmospheric Thermal Tides ............. 216 

Atmosphere 

Solutions to the Linear Wave Equations 

SUMMARY ....................... 226 

Appendix 

A Scaling the Nonlinear Equations of Motion ......... 229 

B The Atmospheric Buoyancy Frequency ............ 

D Method for Solving the Homogeneous Vertical ........ 254 

E The WKB Approximation ................. 260 

F The Upper Boundary Conditions .............. 261 

G Integrating the Vertical Structure Equation ......... 264 

H Energy Transport in Linear Wave Theory .......... 269 

J Including Radiative Damping into the ........... 273 

K Linearized Temperature Perturbation ............ 277 

241 

C Method for Determining the Eigenfunctions OE'(p) 

Vertical Structure Equation 

Linear Wave Theory 

LIST OF REFERENCES ................. 278 

vii 
LIST OF FIGURES 

Figures 

I SPECTROSCOPIC THEORY 

1.1.1 The simple Michelson interferometer ......... 1.1.2 sincx ....................... 1.1.3 Absorption line from ATMOS spectra without ..... 1.1.4 Absorption lines with different apodieation ....... 

1.4.1 Normal mode vibrations for C02 molecule ....... 

1.4.1 Normal mode vibrations for H2O molecule ....... 

1.11.1 Line saturation .................. 1.11.2 Curve of growth for Doppler and Lorentz lines ..... 

1.12.1 Parabolic fitting function .............. 1.12.2 Gaussian fitting function .............. 1.12.3 C02 absorption line taken from ATMOS spectra 

1.12.3 HzO absorption line taken trom ATMOS spectra 

.... 

.... 

I1 EXPERIMENTAL METHOD 

2.1.1-2 Instrumental lines in solar spectra ........... 2.1.3-4 6' from instrumental lines .............. 2.1.5-6 ~(6') from instrumental lines ............. 2.1.7 a' from instrumental lines for SR02 .......... 2.1.8 a' from instrumental lines for SS06 .......... 2.1.9 a' from instrumental lines for SSO9 .......... 2.1.10 a' from instrumental lines for SSII .......... 

viii 

19 

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33 

33 

61 

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69 

69 

70 

70 

82 

783 

84 

90 

90 

91 

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2.1.11 a' from instrumental lines for SS13 .......... 92 

2.1.12 a(aL) from C02 instrumental lines .......... 92 

2.1.13 a(ak) from H2O instrumental lines .......... 93 

2.1.14 A(ak) between C02 and H2O instrumental lines .... 93 

for the five occultations 

2.2.1 cL and from C02 instrumental lines ....... 98 

2.2.2 @ and b(ek) from H2O instrumental lines ....... 98 

2.2.3 Correlations C' determined from all five occultations ... 99 

2.3.1 Complete Filter-3 solar spectrum ........... 103 

2.3.2 Solar CO band heads near the u3 C02 region ..... 103 

2.4.1 Noise level in the ratioed spectra- ........... 107 

2.6.1 Pairs of instrumental and atmospheric lines ...... 114 

2.6.2 Overlapping of lines from ATMOS spectra ....... 114 

2.6.3 SR02 relative speed profile determined from COz lines . 115 

2.6.4 Sunset relative speed profiles determined from C02 lines . 115 

2.6.5 Measurement uncertainty a(uL) for Figures 2.6.3-4. ... 116 

2.7.1 Precisions of v3 CO1 line positions .......... 121 

2.7.2 Precisions of wind speeds before and after iteration ... 121 

procedure 

2.7.3 Precisions of v3 CO? lines positions .......... 122 

2.7.4 Precisions of wind speeds before and after iteration ... 122 

procedure 

2.8.1 First extended relative speed profiles for sunset .... 125 * 

occultations 

2.8.2 Precisions of first extended relative speed profiles .... 125 

2.9.1 Precision of u2 H2O line positions ........... 2.9.2 Precisions of wind speeds from v2 H2 0 lines 

129 

...... 129 

ix 
2.9.3 Second extended relative speed profiles for sunset . . . 130 

occultations 

2.9.4 Precisions of second extended relative speed profiles . . 130 

2.9.S Precisions of 2x4 H10 line positions . . . . . . . . . . 131 

2.9.6 Precisions of wind speeds from 2v2 H2O lines . . . . . 131 

2.9.7 Third extended relative speed profiles . . . . . . . . . 132 

for the sunset occultations 

2.9.8 Precisions of third extended relative speed profiles . . . 132 

2.9.9 Precisions of 2v1 N2O line positions . . . . . . . . . . 133 

2.9.10 Precisions of wind speeds from 2v1 N20 lines . . . . . 133 

2.9.11 Fourth extended relative speed profiles . . . . . . . . 134 

for the sunset occultations 

2.9.12 Fourth extended relative speed profiles . . . . . . . . 134 

for the sunrise occultation 

2.9.13 Precisions of the fourth extended relative speed profiles . 135 

2.10.1 Differences between standard line positions and retrieved 140 

line positions 

2.10.2 Differences between standard line positions and retrieved 140 

N20 line positions 

2.10.3 Peak absorptances for C02 instrumental lines for . . . . 141 

each occultation 

2.10.4 Peak absorptances for HzO instrumental lines for . . . . 141 

each occultation 

2.11.1 At for u3 C02 atmospheric lines . . . . . . . . . . . 144 

2.11.2 AP for 212 Ha0 atmospheric lines . . . . . . . . . . . 144 

2.11.3 A; for 214 H2O atmospheric lines . . . . . . . . . . 145 

2.11.4 A; for 2v1 N2O atmospheric lines . . . . . . . . . . 145 

2.11.5 ApAP for v3 C02 atmospheric lines . . . . . . . . . . 146 

2.11.6 ApAP for u2 H2O atmospheric lines . . . . . . . . . . 146 

X 
2.11.7 ApAC for 2v2 H2O atmospheric lines . . . . . . . . . 147 

2.11.8 ApAC for 2v1 N2O atmospheric lines . . . . . . . . . 147 

2.12.1 t-y planar projection of orbital positions . . . . . . . . 160 

2.12.2 2-2 planar projection of orbital positions . . . . . . . . 160 

2.12.3 Close up of x-y planar positions for the sunrise occultations 161 

2.12.4 Close up of x-y planar positions for the sunset occultations 161 

2.12.5 Close up of 2-y planar positions of orbital and tangent . 162 

points for the four sunset occultations 

2.12.6 Orbital-rotational motion with relative speed profiles . . for sunsets 

2.12.7 Orbital-rotational motion with relative speed profile . . for sunrise 

2.12.8 Uncertainty in least squares fit for orbital-rotational motion 

2.12.9 Residual speed profiles for four sunset occultations . . . 2.12.10 Residual speed profile for the sunrise occultation . . . . 2.12.11 Combined uncertainty of residual speed . . . . . . . . 2.12.12 Angle between line of sight and zonal direction . . . . . for the sunsets 

2.12.13 Angle between line of sight and zonal direction . . . . . for the sunrise 

2.13.1 Concentration c(h) for COz from ATMOS standard . . . atmosphere 

2.13.2 Concentration c(h) for H2O kom ATMOS standard . . atmosphere 

2.13.3 Concentration c(h) for N2O from ATMOS standard . . atmosphere 

2.13.4 Tangent point geometry . . . . . . . . . . . . . . . 2.13.5 Concentration fraction of COZ for various tangent heights 

2.13.6 Concentration fraction of Ha0 for various tangent heights 

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2.13.7 Concentration fraction of NzO for various tangent heights 

2.13.8 Layer thickness of CO2 for various tangent heights ... 2.13.9 Layer thickness of HzO for various tangent heights ... 2.13.10 Layer thickness of N10 for various tangent heights ... 2.13.11 Layer thicknesses near the tangent point tk ....... 2.13.12 Density weighted speed profiles for the four sunset 

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175 

176 

176 

... 177 

occultations 

2.13.13 Density weighted speed profiles for the sunrise ..... 177 

occultation 

2.13.14 Zonal average of sunset wind profiles ......... 178 

2.13.15 Standard deviations of the zonal average winds .... 178 

I11 LARGE SCALE DYNAMICS OF THE CIRCULATION OF . . 181 

THE EARTH'S ATMOSPHERE 

3.1.1 Midltitude temperature profile ............ 185 

3.3.1 Zonal mean temperatures for equinox conditions .... 194 

3.3.2 Zonal mean temperatures for solstice conditions .... 194 

3.3.3 Zonal mean winds for equinox conditions ....... 195 

3.3.4 Zonal mean winds for solstice conditions ....... 195 

3.4.1 Eigenfrequencies for d = -1 ............. 206 

3.4.2 Eigenfiequencies for 8 = 0 .............. 206 

3.4.3 Eigenfrequencies for 8 = 1 .............. 206 

3.4.4 Eigenfrequencies for d = 2 .............. 206 

3.5.1 Temperatureamplitudeof8day wave t 42 km ..... 212 

3.5.2 Geopotential height anomolies of the 5-day wave observed at 45 

km ........................ 213 

3.5.3 ATMOS standard temperature profile ......... 214 

3.5.4 Surface pressure oscillation of the 5-day wave ..... 214 

3.5.5 Computed pressure perturbation for the 5-day wave . . 215 

3.5.6 Computed zonal winds for the 5-day wave ....... 215 

xii 
3.6.1 Variation of solar heating . . . . . . . . . . . . . . 3.6.2 Observed and theoretical northerly diurnal winds . . . . 3.6.3 Diurnal heating rates . . . . . . . . . . . . . . . . 3.6.4 Amplitude of diurnal temperature perturbation . . . . 3.6.5 Amplitude of diurnal zonal wind perturbation . . . . . 3.6.6 Amplitude of diurnal meridonial wind perturbation . . . 3.6.7 Semi-diurnal heating rates . . . . . . . . . . . . . . 3.6.8 Amplitude of semi-diurnal temperature perturbation . . 3.6.9 Amplitude of semi-diurnal zonal wind perturbation . . . 3.6.10 Amplitude of semi-diurnal meridonial wind perturbation 

... 

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LIST OF TABLES 

Tables 

INTRODUCTION 

1.1 F'requency transition ranges of the ATMOS optical filters along 

with the principle molecular species that can be analyzed in these 

regions ....................... 7 

1.2 Summary of occultation spectra obtained during the Spacelab-3 

mission ....................... 8 

I SPECTROSCOPIC THEORY 

1.11.1 Form factors for parabola-parabola fit for various values of m 71 

1.11.2 Form factors for parabola-gaussian fit for various values of m 71 

1.11.3 Estimated measurement uncertainty kom parabola-gaussiaa least 

squares fit using direct sum ............. 71 

I1 EXPERIMENTAL METHOD 

2.1.1 Check for instrumental drift for the five occultations obtained 

from v3 C02 lines .................. 85 

2.1.2 Check for instrumental drift for the five occultations obtained 

from v2 H2O lines .................. 85 

2.1.3 v3 C02 standards used is primary instrumental calibration 86 

2.1.4 v2 H2O standards used in primary instrumental calibration 86 

2.1.5 Estimated global offsets a for the five occultations kom vs C02 

lines.. ...................... 87 

2.1.6 Estimated global offsets a for the five occultations from u2 H2O 

lines.. ...................... 87 

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2.1.7 Differences between global offsets a for the five occultations 87 

2.1.8 u3 C02 standards compared to observed instrumental positions 

after primary calibration . . . . . . . . . . . . . . . 88 

2.1.9 u2 Ha0 standards compared to observed instrumental positions 

after primary calibration . . . . . . . . . . . . . . . 89 

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2.2.1 c determined kom v3 C02 lines . . . . . . . . . . . . 100 

2.2.2 e determined from u2 HzO lines . . . . . . . . . . . . 100 

2.2.3 Correlations of ek between HzO and C02 regions . . . . 100 

2.4.1 Estimates of line noise from C02 instrumental lines . . . 108 

2.4.2 Estimates of line noise fiom HzO instrumental lines . . 108 

2.10.1 Offsets between various linelisting . . . . . . . . . . 141 

2.11.1 Estimates of line noise determined from atmospheric lines 148 

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INTRODUCTION 

Mission Objectives 

The Atmospheric Trace Molecule Spectroscopy experiment (ATMOS) was 

designed to obtain high resolution infrared absorption spectra from an orbiting platform in space. The instrument was managed for the National Aeronautics and Space Administration (NASA) by the Jet Propulsion Laboratory 

(JPL) under the direction of C. B. Farmer and was first flown in the spring 

of 1985 as a payload of the shuttle Spacelab-3 mission. This project has also 

become a part of NASA’s Upper Atmospheric Research program (UAR). 

The major goal of the UAR is to obtain quantitative data pertaining to the 

state of the upper atmosphere over a long period of time to determine the effects of natural and perhaps man-made changes of the atmosphere’s physical 

state. Information has been obtained from the Spacelab-3 experiment by the 

ATMOS experimental team concerning the volume mixing ratios (VMRs) 

and concentrations of a number of minor and trace gases in the atmosphere. 

Temperature and pressure variations with height were also obtained by analyzing the strengths and equivalent widths of the absorption lines and these 

measurements are related to upper atmosphere dynamics. The purpose of 

the research described here was to determine if accurate wind speeds along 

the line of sight can be obtained to high precision. This was achieved. In 

addition the influence of the diurnal and semi-diurnal Fourier components of 
2 

the heating of the lower atmosphere on the upper atmospheric circulation is 

discussed in this report. 

The ATMOS instrument is a fast Fourier-transform (FFT) interferometer 

which covers a wide spectral range from 600 to 5000 cm-'. This spectral 

range is further divided into smaller regions by the use of specialized filters 

to increase the signal to noise ratio of the processed spectra. With a scan 

time of about 2 s the spectra have an unapodized resolution of about 0.01 

cm-l, a digital frequency spacing of 0.0075 cm" and a vertical resolution of 

about 4 km. A variety of programs is available on a Prime 955 computer at 

the ATMOS data analysis center to study the spectra. 

The mission's strategy was that, by using the sun as the source of radiation, absorption spectra of about 40 molecular species in the earth's atmosphere could be observed in successive spectra during sunset and sunrise 

solar "occultations". An occultation, or more precisely a limb observation, 

is defined as when the earth's atmosphere intersects a line between the orbiter and the sun during an orbit. From these spectra the properties of the 

atmospheric constituents and the dynamical state of the atmosphere in regions near the tangent points of observation were studied. A tangent point 

is defined as the point above the earth's surface where a light ray from the 

sun to the orbiter is closest to the earth's surface. This distance above the 

earth's surface is referred to as the tangent height. A description of the optical filters used to define several spectral regions and a list of some molecules 

whose infrared transitions occur in these regions are given in Table 1 [Farmer, 

19871. 

During the course of the mission two latitude bands were sampled. Mid- 
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latitude sunsets occurred near 30' N and sunrises near 50" S. Since the orbital 

period of the platform was about 90 minutes a wide range of longitudes was 

available for study in the two latitude bands. The Spacelab-3 Mission (SL-3) 

was launched on board the shuttle Challenger on April 29, 1985 at 21:00:30 

EST and Table 2 lists the geographical locations, times, and filter region 

for each occultation. For each occultation a set of spectra was obtained 

in a period of a few minutes and hence a cross section of the atmospheric 

state could be sampled as a function of altitude for the latitude and longitude 

about the tangent points. A total of 19 such occultations accumulated almost 

1200 atmospheric spectra. In addition, more than 1000 solar spectra were 

obtained at altitudes far above those where infrared absorption by telluric 

gases is significant. 

Overview of Thesis Research 

After the completion of the mission in May 1985 the raw data were processed and soon afterwards the first transformed spectra were available for 

analysis. It was decided to study the possibility of measuring winds from 

the spectra and to estimate the accuracy which can be expected. Due to the 

relative motion of the shuttle orbiter with respect to the atmosphere as well 

as possible motions caused by atmospheric circulations the absorption line 

positions were expected to change significantly from spectrum to spectrum. 

It was determined that the relative speeds between the observer and the 

atmosphere surrounding the tangent points could be determined by measuring the Doppler shifts of groups of absorption lines and the relative speeds 
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associated with these shifts. 

To measure the Doppler shift of a moving absorber, the frequencies of 

two lines must be compared, the rest frame position and the position in the 

moving frame. If c is the speed of light, v is the relative speed of the absorber 

to the orbiter, and c is the wavenumber (in cm-') of the line in the rest frame, 

then the new wavenumber in the moving frame is shifted from the rest frame 

position by the amount 

V 

C 

Ac= -c. 

The relative speed v can be determined by this shifts in frequency by 

v=- C. 

Ac 

c 

The accuracy of v is limited by how well known the rest frame position c is 

relative to the instrument's "true" rest frame position for that transition. It 

was determined that even if the instrumental calibration is not "absolute", 

and hence line positions determined using it were not absolute, accurate wind 

speeds could, nonetheless, still be determined. This is true since u depends 

not on the Doppler shift Ac alone, but on the ratio Ac/c. Average calibration 

offsets will cancel in the determination of v and this is discussed in 511.4. This 

technique was found to require that for at least one telluric absorption band, 

a set of corresponding lines due to gases inside the instrument needed to be 

present in the spectra. The accuracy was then transferred to other spectral 

regions by a "bootstrapping" technique which utilized the wind speeds determined from bands in which instrumental lines were available to determine 

the positions of new absorption lines relative to the instrument's calibration. 

It was determined that a sufficient number of spectral lines was available 
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in the spectra (- 20-40 lines/spectrum) to obtain precisions of 5 ms" or 

better by measuring the Doppler shifts of telluric absorption lines [Shaw, 

19851 using realistic estimates of the noise level. It was also determined that 

upper limits of fluctuations in the calibration stability of the instrument were 

needed since the precisions of the wind speeds would be ultimately limited 

by these. Such uncertainties were determined to be extremely small and did 

not contribute more than 1-2 ms-' errors. 

Accurate relative speed profiles were estimated from spectra with tangent 

heights between 25 and 120 km and were found to have precisions from 3 to 

5 ms-' throughout this range. In order to obtain wind speeds relative to the 

rotating frame of the earth the orbital motion of the shuttle and the rotational 

motion of the tangent points were estimated from telemetry data provided by 

Goddard Space Flight Center. This motion was subtracted from the relative 

speeds between the orbiter and the atmosphere estimated from the Doppler 

shifts to obtain wind speeds relative to the earth's surface. Uncertainties due 

to errors in estimating the line of sight velocity components were found to 

be less than 2 or 3 ms-'. The residual wind speed profiles were found to 

indicate essentially the zonal component of the atmospheric flow. 

Similarities of the wind speeds with height were found in all four sunset 

occultations analyzed. These profiles were found to have wavelike characteristics and further investigation of this phenomenon was done. (Since only 

one sunrise occultation was available in the filter used for wind speeds such 

comparisons with other sunrise measurements are unavailable at the current 

time.) Such variations in altitude can be described by the theory of atmospheric tides [Siebert, 1961; Chapman, Lindaen, 19701. The source of the 
6 

excitations and their “wavelike” nature can be explained by the periodic absorption of solar radiation by the atmosphere, particularly by ozone between 

20 and 90 km and water vapor below 10 km altitude. Since this happens on a 

global scale the response of the atmosphere is also on this scale. The notion 

of tidal eigenfunctions and accompanying vertical structure equations were 

derived and solutions to these mathematical models were computed. There 

were similarities of the phases predicted between the theory and the ATMOS 

results. The “classical” theory of tides, however, was found to be inadequate 

in describing the magnitudes of the waves at heights above about 85 km since 

it ignores the contributions due to non-linearities and adiabatic instabilities. 

The wind speeds measured do contain information of the geometric distribution of global heating rates regarding the absorption of solar radiation in the 

atmosphere. This can be attributed to the global scale of the atmospheric 

response to tidal forcing. Further study into the non-linearities in the fluid 

dynamics as well as future measurements of winds in a manner similar to the 

ATMOS project may yield important data about the geographical distribution of the gaseous absorbers. 

a 
7 

Table 1 Frequency transition ranges of the ATMOS optical filters together 

with the principal molecular species that can be analyzed in these regions 

[Farmer, 19871. 

I Filter 1 Filter 2 Filter 3 Filter 4 

600-1200 cm-' 1100-2000 cm" 1580-3400 cm-' 3100-4700 cm-' 

co2 co2 c02 c02 

H20 H2 0 h20 h20 

03 03 

CHI CHI 

co 

n20 

"3 H202 NO HF 

"02 H02 (Nod (HCN) ("03) 

CCL2F2 NO2 (H202) 

HN04 

CC13F (NO) HDO 

CHCClF2 N2OS h2c0 

CH3C1 HN03 

CC14 HC1 

COF2 HOC1 (CH3CI) 

COClF CFI ocs 

CION02 

c10 so2 HCN 
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Table 2 Summary of occultation spectra obtained during the Spacelab-3 mission. Latitudes and Longitudes refer to both high (- 120 km) and low (- 

0 km) tangent heights. MET - Mission Elapsed Time, GMT - Greenwich 

Mea rime. 

Occ Opt MET GMT Lat Long -, (E) , 

Type Fil Start 60 km LowIHigh LowIHigh 

SROl 3 0/07:13 119/23:17 49.9148.5 S 118.0/117.0 

SSOl 

ss02 

SR02 

SS03 

SS04 

SR03 

SS05 

SS06 

SS07 

SS08 

SR04 

SR05 

SR06 

SR07 

sso9 

SSlO 

SSll 

ss12 

SS13 

1 

1 

3 

2 

4 

6 

1 

3 

2 

1 

4 

2 

1 

5 

3 

4 

3 

2 

3 

0/08:08 

0/18:52 

0/19:27 

0/21:55 

1/00:55 

1/01:34 

1/02:28 

1/07:01 

1/08:33 

1/10:05 

1/15:18 

1/16:49 

1/18:21 

1/19:53 

2/02:53 

2/04:25 

2/05:51 

2/07:27 

2/01:22 

120/00:12 

120/11:20 

120/10:54 

120/13:57 

120/17:00 

1201 17:36 

120/18:32 

120/23:07 

121/00:38 

121/07:21 

121/08:53 

12 11 10:24 

121/11:56 

121/17:26 

121/18:58 

121/20:30 

121/23:33 

121/02:10 

121122:Ol 

34.5135.9 N 

32.6134.1 N 

49.1147.5 S 

33.6131.9 S 

33.0131.3 N 

48.6146.9 S 

32.8131.1 N 

31.9130.1 N 

31.7136.0 N 

31.4129.6 N 

47.4 f45.6 S 

47.3145.5 S 

47.2145.4 S 

47.1145.1 S 

28.5126.6 N 

28.1126.2 N 

27.8125.9 N 

27.3125.3 N 

27.3/25.3 N 

276.31277.1 

115.31116.1 

294.61293.3 

70.2169.3 

24.2123.3 

202.7/201.4 

1.210.3 

292.11291.2 

269.21268.2 

246.21245.2 

356.01354.7 

332.61331.8 

310.1/308.8 

287.21285.8 

16.1115.2 

353.11352.2 

330.11329.2 

284.11283.2 

204.11283.2 
9 

CHAPTER I 

SPECTROSCOPIC THEORY 

1.1 The Michelson Interferometer and 

Transform Spectroscopy 

In the classical Michelson interferometer Figure 1.1.1 light 

Fourier 

from the source 

00 is collimated at C into a beam which is incident on a half-silvered beam 

splitter B. The reflected ray travels to a stationary mirror S and it is reflected 

back to the beam splitter. The partially transmitted ray travels to a movable 

mirror M and is reflected back to the beam splitter. The two split beams 

recombine at 0 and are sent to the detector D. Since M is movable, the two 

beams interfere both constructively and destructively as the path difference 

2 changes. 

For simplicity we consider an incident beam with frequency f. The 

wavenumber c is related to the frequency f and the speed of light c by 

1 

J 

C 

c -. 

If we assume the amplitude A of this monochromatic incident beam is split 

evenly between both arms M and S with path difference z and if we neglect 
10 

losses in intensity in the beam due to dispersion then the complex amplitude 

of the beam at the detector is 

1 AD = -A(1+ 2 eG), 

where the phase difference is 

6 = 2ncx. (1.3) -- 

The intensity at the detector D is 

(1.4) 

1 1 I(c,z) = IAD~' = #I + cosb)~' = #I + cosb)Io(c). 

Suppose that I(c) represents the spectral radiance (power/cm-') then 

total signal Io(z) received at the detector D is the sum of all contributions 

at all frequencies 

Io(z) = iw de f(1 + cos 6) I(c), 

= ; 1- dc Io(c) + ; 1= de cos 6 I(c), 

1 

-2 = - [I+I(z)]. 

The first term I' is the total incident intensity while the second term I(z) is 

the modulated part and is given by 

In Fourier spectroscopy the interference pattern I(=), also known as an interferogram, is measured and from it the the spectral radiance I(c) is determined. From the theory of Fourier transforms we have the relationship 
11 

where it is assumed that I(<) = 0 for c < 0. This can be inverted [Mathews, 

Walker, 19701 to give 

I(c) a lW dx cos2ncx I(x) -00 

and the pair of equations (1.7-8) are a Fourier-cosine pair. This technique of 

spectroscopy has the multiplex advantage which improves the signal to noise 

since all spectral features are simultaneously recorded in the interferograms 

[Thorne, 19881. 

.* 

In practice the maximum path difference is not infinite but is usually 

confined to a limited region say, -L < 1: < L. Then >(z) is zero outside this 

range and the spectrum (1.8) is 

I'(c) OC /" dx cos27rcx I(x). (1.9) -L 

Some spectral information is lost due to the finite path length with a subsequent degradation of the instrumental resolution. If we consider the case of 

a monochromatic input signal Io at wavenumber e' 

I($) = Io qc - e'), (1.10) 

then the output interferogram is a cosine function 

I(x) = Io cos27rc'x. (1.11) 

Since data are only available for -L < x < L the computed spectrum I'(c) 

is 

I'(c) a Io /" dx cos 2ncx cos 27rc'x 

-L 

a IOL [sinc 24~' + c)L + sinc2n(c' - c)L] (1.12) 
12 

where the sinc function is defined as 

sin y 

Y sincy = -. (1.13) 

Since the first term on the right hand side of (1.12) in negligible compared 

to the second the output interferogram for the monochromatic wave is 

1'( e) oc Io L sinc 27r( c' - c) L. (1.14) .- . . 

Figure 1.1.2 shows the first two zeros of the sinc function f1/2L. The resolution of the spectra is approximately 1/2L and corresponds to the region of 

maximum instrumental response to a spectral feature. Outside this region 

the sinusoidal variation of the instrumental response is due to the finite path 

difference. As L, + 00 the sinc function approaches a &function. and we 

retrieve all spectral information. 

Now consider the case where the radiation is described by a spectral 

density I(c) 

I(c) = de' 6(c - <')I(<'). -OD 

Then, from (1.7), the interferogram is 

OD 

I(z) a [OD de cos2ncz dc'I(c') 6(c - e'), .. 

J-OD J--00 

OD 

- 10D de' cos 27rc'z I(c'). 

(1.15) 

(1.16) 

Since data are only gathered for -L c z < L the resolved spectrum is, from 

L lL dz cos 2ncz I(z), 

dz cos 2ncz cos 27rc'z I(<'). (1.17) 
13 

We see that (1.17) can be written 

I'(c) = /= de' i(c, e') I(c'), -00 

(1.18) 

if we define the normalized instrument function 

L 

i(c,c') G lLd+ cos2nc'x cos2ncx, 

= L [sinc2n(c + c')L + sinc2n(c - c')L]. (1.19) 

It can be shown [Thorne, 19881 that this form is normalized 

(1.20) 

We see that (1.19) represents a response centered around c = fc'. This is 

true since Fourier transforms of functions in the x domain always have mirror 

images about zero in the c domain. If we define 

&(e, e') G L sinc &rL(c f e'), (1.21) 

then the instrument function (1.19) is 

i(c, e') = i+(c, e') + i-(c, e'). (1.22) 

Usually we refer to the region c > 0 so that the contribution of i+ is usually negligible for c 3 1/2L. We can then say the instrument function is 

approximately 

i(c, e') i(c - e') E %-(e, e') = 2L sinc2nL(~ - e'). (1.23) 

The convolution between two functions A(c) and B(c) is defined as 

A(c) * B(c) 1 -- de' A(c -.c')B(c) ( 1.24) -00 
I 

I 

I 

I 

1 

1 

I 

1 

I 

‘I 

I 

8 

.I 

I 

.I 

t 

I 

I 

1 

14 

so then (1.18) can be expressed 

I‘(c) = ;(e) * I(c). (1.25) 

The resolved spectrum I’(c) is the convolution of the true spectrum I(c) with 

the instrumental function i. 

The resolution 6c = 1/2L caused by the finite path length allows information from nearby wavenumbers c’ to contribute to the partially resolved 

spectrum at c which is given by (1.25). The sinc function has a “ringing” 

at wavenumbers not centered near the maximum of the instrument function 

which can contribute significant interferences between different wavenumbers in the spectra. Figure 1.1.3 shows the ringing caused by this instrument 

function about an absorption line taken from the ATMOS spectra. The deep 

central spike is the instrumental response at the spectral line center while 

the side lobes are artifacts of the finite path difference and not real spectral 

features. This ringing can be mathematically filtered out through a process 

called apodization - literally meaning “cutting of the feet” [Thorne, 19881 - 

but with a complementary loss of spectral resolution . From the property 

of Fourier transforms it can be shown that the transform of a product of 

functions is the convolution of their transforms 

We can now imagine multiplying the interferograms I(z) by an “apodizing 

function” a(%). An apodized spectrum can be retrieved from this apodized 

interferogram a(t) I(z) using (1.25) 
15 

If we define the apodized instrument function 

(1.28) 

then by using (1.25) the apodized spectrum (1.27) can be shown to be 

I'(c) = Iw dc'i,(C, c') I(C'). (1.29) -00 

The apodizing function a(z) can be freely chosen to achieve whatever mathematical filtering is desired but always at the cost of spectral resolution. There 

are several apodizing functions available to the user when computing the ATMOS spectra and they are described in detail by Norton and Beer (1976). 

Figure 1.1.4 shows the absorption line used in Figure 1.1.3 after convolution 

with three different apodization functions available on the ATMOS computer 

and described by Norton and Beer [1976] were used. These apodizing functions give more smoothing of the spectral while degrading the resolution from 

0.0125 cm-' to about 0.015 cm-l between the several functions. The apodizing functions widen the spectral features and cause the peak absorptances of 

the lines in the spectra to decrease. The apodizing function chosen in this 

study eliminated false spectral features greater than the noise level and had 

a resulting resolution of about 0.015 cm-l. This was found to be acceptable 

within the measurement requirements $1.12. 

If an interferogram I(x) is sampled at intervals Ax features at other 

wavenumbers may be superimposed on real features unless the value of Ax 

is properly chosen. This "aliasing" is due to the periodicity of the Fourier 

transform and is described by Thorne (1988). Thorne shows that the inter- 
16 

ferogram must be sampled at least every 

1 AX = - 

2C& 

if the minimum wavenumber is C- to avoid 

(1.30) 

this sort of overlapping. Special 

optical filters must also be used to prevent false lines from appearing in the 

spectra. The ATMOS filters were carefully chosen to avoid aliasing [Farmer, 

19871. 

Absolute line positions, are ultimately limited by the accuracy of the HeNe laser wavelength used to calibrate the instrument. If the recorded value 

of the path difference 2' is related to the true path difference z by 

2' = (1 + a)x (1.31) 

where la1 a 1 then the interferogram (1.7) is 

(1.32) X' I(2) = I (G) = 1: de' cos 2nc' (L) I(q'). l+a 

This shows how the interferogram is related to the path difference recorded by 

the instrument. These interferograms are then processed by (1.9). However, 

since the path difference is uncertain, we obtain a spectrum 

L 

I'(c) = lL dz' cos 2xcz' I(%) 

. (1.33) -rn -L l+a 

The instrument function is then 

io(q,c') = L [sinc2n(~ + (1 +a)c') L + sinc2x(c - (1 + a)c') L]. (1.34) 

From this result the uncertainty in the laser frequenEy causes the path lengths 

to be in error which causes features at frequency c in the convolved spectra to 
17 

1 

I 

8 

I 

I 

I 

:I 

‘I 

8 

8 

I 

8 

1 

I 

be shifted by -ac. Typical errors in the He-Ne frequency are given by c(a) - 

O( [Norton, 19893 and line positions determined from the spectra should 

be uncertain to this degree. It is shown in 511.4 that these uncertainties do 

not &ect the accuracy of the winds from Doppler shifts provided that a set 

of instrumental lines with atmospheric counterparts can be measured in the 

spectra. 

Due to the digital nature of the interferogram data and the presence of 

noise in the detector signals the location of zero path difference (ZPD) in the 

interferogram can be in error. The point of zero path difference is defined as 

that value of z for which all frequencies interfere constructively. For an ideal 

interferogram this point is taken as the origin of z. Errors in locating ZPD 

can introduce apparent asymmetries in the interferograms which may cause 

frequency dependent calibration errors in the resolved spectra I’(c). To see 

this, suppose that an estimate of ZPD is in error by an amount (. Then the 

measured interferogram (1.7) is 

(1.35) 

Using the form of this interferogram which is sampled from -L < z < L. we 

have 

00 

- LW dc’I(c‘) /” dz cos2?rcz cos27rc’(z + () 

-L 00 

- LW de‘ i(c, e’) I($’). (1.36) 

The instrument function contains the phase error and can be evaluated using 
I 

I 

'I 

't 

8 

:I 

8 

I 

II 

I 

18 

standard integration 

L 

i(c, q') = I_, dz cos 27r;z cos 27r;'(z + () 

= L [sinc 27r(c - c')L + sinc 27r(c + c')L] cos 27rc't 

N L sinc 27r(c - c')L cos 27rc't. (1.37) 

The instrument response function i(c, e') is modulated by the term cos elf. . 

For values of e'( << 1 the interferogram will peak when e' = c and should 

produce symmetric line shapes. For large frequencies where e'( - 1 the 

instrument function will not peak at c and the phase errors can produce 

significant distortions in the line shapes which may cause systematic errors 

in the positions of spectral features. 

An estimate of the magnitudes of these systematic fluctuations in line 

positions set a limit of the accuracy of the relative speed measurements. The 

sampling interval Ax of two fringes of the He-Ne laser used for calibration 

was used to obtain the interferograms [Norton, 19891. The wavelength of 

the laser is about 633 nm which corresponds to Ax 21 1.27 x lo-' cm. The 

precision of determining ZPD was estimated to be 1000 times better than this 

value [Norton, 19891 which corresponds to e(() - O(lO-') cm. For typical 

infrared transitions the distortions in the instrument function are negligible. 

With this degree of precision distortions the phase errors are negligible and 

we can therefore expect the calibration of the instrument to be extremely 

stable. 
8 

1 

1 

I 

I 

I 

1 

1 

1 

00 

B 

19 

Figure 1.1.1 Simple Michelson interferometer. The ATMOS interferometer 

has a much more complex optical arangement [Farmer, 19871, however, the 

basic interferometric principles are the same as in this simple design. 

4 i 

Figure 1.1.2 Instrumental response function sincz. The amplitude of the 

oscillation decreases as Izl increases. 
20 

8 

Figure 1.1.3 Absorption line from ATMOS spectra without apodization. Notice the sinc function behavior of the instrumental response function away 

the line center where it is a maximum. 

Figure 1.1.4 Same absorption above with apodizing functions 1, 2 and 3 

[Norton, Beer, 19761. The figures have been displaced vertically for clarity. 

Notice that the ‘ringing’ in Figure 1.1.3 has been eliminated at the expense 

of instrumental resolution. The peak absorptance of the line is also smaller. 

Apodization ‘2’ was used in this study. 

8 
I 

'I 

1 

I 

I 

I 

I 

8 

i 

21 

1.2 Measuring Doppler Shifts from ATMOS Spectra 

Due to the geometry of the experiment the measurements are confined 

to the components of atmospheric motion along the line of sight. In order 

to determine the components we need to measure to total relative motion 

between the gas near the tangent points and the orbiter and then subtract 

from these results motion due to the shuttle's orbit and the earth's rotation. 

The residual speed is then the component of wind relative to the earth's 

rotating frame along the tangent ray path. In order to obtain the relative 

speed it is useful to observe the actual positions of absorption lines in the 

ATMOS spectra and compute the Doppler shifts of their wavenumbers from 

a rest frame position. The observed position cj of a feature in the moving 

frame is related to its position {j in the rest (non-moving) frame by the 

non-relat ivis t ic Doppler shift 

. 

(1.38) 

Here v is the relative speed between the orbiter and the absorber, c is the 

speed of light, and /3 v/c which is O(lO-s) for orbital speeds. In order to 

obtain winds to 5 ms-l we need to be able to determine c to about 0.00005 

cm-'. Equivalent precisions can be obtained by using many absorption lines. 
I 

I 

-1 

1 

’I 

1 

I 

‘I 

I 

R 

I 

1 

I 

:I 

.I 

I 

1 

I 

:n 

22 

1.3 Quantum Mechanics of Infrared Spectroscopy 

The ATMOS interferometer is designed to measure the absorption of solar 

radiation in the infrared. In this section we examine the physical processes 

involved and describe the quantum mechanical basis for the spectra. 

The energy of a molecule depends on the locations and motions of its 

nuclei, center of mass, orientation, and electronic state. Let the locations 

of all the nuclei relative to the molecule’s center of mass R” be described 

by the vector R. The electronic energy of the molecule is then described by 

E(R). This energy is evaluated with stationary nuclei. However, due to the 

uncertainty principle 

(1.39) 

no real molecule can have fixed nuclei i since the more accurately their positions R’ are determined the larger is the uncertainty in their momenta 

Pi. The nuclei therefore “oscillate” about an equilibrium configuration R‘q 

in accordance with the uncertainty principle. Also, the molecule cannot be 

considered to have a fixed center of mass R” and from the uncertainty principle it has some translational momentum. The molecule also cannot have a 

fixed orientation since the uncertainty in its angular momentum would then 

be infinite. All these three types of motion must be considered as part of the 

wave function. 

ti 

Q(R’+,v,r) u(e,v,z) 2 5 

The nuclear motions can be treated within the context of the BornOppenheimer approximation [Berry, Rice, Ross, 19801. The lighter electrons 

are considered to move much more rapidly than the heavier nuclei. For every 
23 

1 

1 

I 

1 

(I 

configuration of the nuclei during their slower scale motions, the electrons 

are considered to attain an “equilibrium” state. The Hamiltonian 7? of the 

molecule can be assumed to be the sum of the nuclear kinetic energy fin and 

the energy of the electronic state 

7?=%+%. ( 1.40) 

Let the positions of all the electrons relative to the center of mass R” be 

given by the vector r. Then Born-Oppenheimer approximation states that 

the wave function +(r,R) of the molecule is separable into a nuclear and 

electronic wave function 

+(r, R) +n(R)+e(r, R). (1.41) 

The electronic part satisfies the relation 

fie+e(r, R) = JqR)+e(r, R). ( 1.42) 

By using the Schr6dinger equation 

7?+ = E$ ( 1.43) 

it can be shown [Berry, Rice, Ross, 19801 from (1.41-3) that 

The second term on the left hand side is neglected since there are only 

small interactions between electronic and nuclear motions. This is the BornOppenheimer approximation and (1.44) can then be expressed as 
I 

l 

’I 

I 

I 

I 

I 

‘I 

I 

I 

I 

I 

:I 

I 

:I 

LI 

I 

I 

.I 

24 

The quantity E(R) is the effective potential energy field caused by the electrons in which the nuclei move. 

Now assume at that the energy is separable into three parts, translation 

of the center of mass, internal vibration, and rotation of the molecule’s orientation. (Even though this is not quite accurate and there are corrections to 

this assumption, it will suffice for the following discussion). Then the nuclear 

Hamiltonian can be written as 

~ _, 

Let the total mass of the molecule be M, the individual nuclear masses be 

mi, the center of mass momentum operator be P-, the nuclear momenta 

relative to the center of mass operator in the “molecular coordinate frame” be 

Pi, the angular momentum of the molecular coordinate frame relative to the 

observer’s frame operator be L, the molecular moment of inertia be I, and the 

set of angles needed to specify the orientation of the molecular coordinate 

system in the observer’s frame. The moments of inertia are evduated by 

~y = mi(t.;l+ z:), 

i 

I, = Cmi(c: + y?). 

i 

The the Hamiltonians in (1.41) can be expressed as 

(1.47) 

(1.48) 
I 

I! 

I 

! 

I 

I 

I 

I 

I 

I 

I 

.I 

1 

1 

.I 

.I 

1 

I 

.I 

I 

25 

The wave function is 

$n = $tr-$int$rot (1.49) 

and the energy becomes 

The three eigenvalue equations for the translational, vibrational and rota- 

(1.51) 

The translational energy levels form a continuum when the 1eng.h scale of 

the motion is large. The occupation of translational energy levels is related 

to the temperature of the gas and, as we shall see, the Doppler shifts due 

to these thermal motions cause the absorption lines to have a width characteristic of their average motion. The vibrational energy levels are discrete 

and are well approximated by the harmonic oscillator model near the minima 

of their vibrational potential wells. Infrared transitions between vibrational 
26 

and rotational levels are observed in the ATMOS spectra for many atmospheric gases. The rotational levels have quantized angular momentum due 

to the azimuthal periodicity of the wave function and accompany vibrational 

t rami tions. 

1.4 Normal Coordinates and Vibrational Energy for 

Triatomic Molecules 

There were about 40 molecular species whose spectra were measured with 

the ATMOS interferometer. In this study COZ, HZ0 and NZ0 were used to 

obtain wind speeds. We will therefore concentrate on the quantum mechanics 

of vibration of triatomic molecules which describe the spectra of these three 

molecules. 

A molecule with N atoms has 3N independent nuclear coordinates in 

general. However, not all of these coordinates are required in describing the 

internal molecular energy EV+ In the absence of external fields the center 

of mass coordinates Rm and the orientation fl do not affect the vibrational 

energy. For linear molecules such as COZ and NZO, two angles (spherical 

coordinates) are needed to specify the orientation of the symmetry axis, 

while for non-linear molecules such as HZ0 three angles (Euler angles) are 

needed to specify the complete orientation. Thus there are n = 3N - 5 or 

3N - 6 vibrational degrees of freedom. For linear triatomic molecules this 
27 

corresponds to n = 4 degrees of freedom while for non-linear molecules we 

have n = 3 degrees of freedom. 

In a triatomic molecule the vibrational energy is a function of three or 

four independent generalized coordinates q,,, Q = 1, .., n [Herzberg, 19451. 

The Cartesian coordinates R; of each nucleus i are related to the generalized 

coordinates by 

R; = %+I). (1.52) 

There is a preferred set of these coordinates q in terms of which the vibrational motion can be expressed as simply as possible, the normal coordinates 

[Herzberg, 19451. In this approach we assume a well-defined equilibrium geometry about which the nuclei perform small oscillations. This equilibrium 

geometry is called a molecular frame. The nuclei are assumed to perform 

oscillations in a potential energy field V(q) E E(R(q)) about an equilibrium position given by q'q. Since this equilibrium position is defined to be a 

potential energy minimum in the context of the Born-Oppenheimer approximation, then (aV/aq,,)q.g = 0. If we define our normal coordinates to have 

a zero value at equilibrium, q'q G 0, then the potential energy function can 

expressed as 

(1.53) 

from a Taylor series expansion about the equilibrium point. For sufficiently 

small oscillations all of the extra terms in the expansion (1.53) can be dropped 

as well as V(0) since it is a constant for the equilibrium geometry. This 

harmonic approximation is only useful for the states near the bottom of the 

potential well and additional anharmonic terms need to be included for a 
I 

I 

28 

more accurate solution elsewhere [Herzberg, 1945). 

If we define the force constants [Berry, Rice, Ross, 19801 

then the potential energy in the harmonic approximation is 

The kinetic energy of the molecule is 

(1.54) 

(1.55) . 

(1.56) 

It can be shown [Fetter, Walecka, 19801 that, in terms of the normal coordinates, 

1 .. T = - ~msrl9oqu~ 

2 uu’ 

where the reduced mass matrix is defined by 

(1.57) 

(1.58) 

which becomes a real, symmetric, constant matrix for the equilibrium geometry. 

The Hamiltonian operator BVib could be obtained from (1.48) and (1.55), 

however the Schriidinger equation would be very difficult to solve due to the 

coupling &,,,,I between terms. It can be shown [Fetter, Walecka, 19801 that a 

set of normal coordinates can always be determined so that the matrices IC 

and m are diagonalized. In this case the potential energy is given by 

(1.59) 
29 

and the kinetic energy is 

where we use p,, = m,,q,,. Then the Hamiltonian operator becomes 

where 

(1.60) 

(1.61) 

(1.62) 

The vibrational Hamiltonian can be written as a sum of noninteracting harmonic oscillators and the wave function can be written as the product 

The energy is then given by the sum 

(1.64) 

where 

The operator (1.62) is the Hamiltonian of a simple quantum mechanical 

harmonic oscillator and the eigenfunctions $t,iho(qu) are well known. The 

energy eigenvalues are given by 

where 

(1.66) 

(1.67) 
30 

and 

vu = 0,1,2,---; u = l,---,n. (1.68) 

For small displacements the oscillations about the equilibrium configurations 

reduce to those of a set of independent harmonic oscillators of frequency w,,. 

Any small amplitude motion is then a linear combination of the harmonic 

oscillators. 

The determination of the types of motion corresponding to each normal 

mode can, in general, be determined from the masses of the nuclei and the 

equilibrium configuration. One method [Berry, Rice, Ross, 19801 for determining the normal mode motions is to examine the molecule in the molecular 

frame. Here the sum of the momenta of the nuclei is zero 

(1.69) 

i 

Since the coordinate system rotates with the molecule the total angular momentum must also vanish 

CLi = 0. (1.70) 

i 

Since the particles are in simple harmonic motion the momentum components 

are a quarter of a cycle ahead of the displacements 

so that (1.69) becomes 

CrniR, = 0. 

(1.71) 

(1.72) 

i 

Since each nucleus i oscillates about an equilibrium position Gq each angular 

momentum is 

Li = miRtqx& (1.73) 
31 

so that 

CmiRiqx& = 0. (1.74) 

Combining (1.72) and (1.74) a set of relative coordinate displacements can be 

determined for each type of molecular configuration. Figure 1.4.1 and 1.4.2 

show the various vibration modes for the C02 and H20 molecule respectively. 

i 

To evaluate the relative vibrational frequencies of the various modes re- . 

quires a knowledge of the potential energy surface E(R) and this is not always 

available. However the relative magnitudes of the frequencies can occasionally be established. Consider the CO2 molecule in the following example. 

This molecule has four modes, an asymmetric stretching (v3), symmetric 

stretching (VI), and. two degenerate bending modes (~2). For a given displacement, the bending modes require less stretching of the bonds and hence 

smaller changes in the bond energy than the stretching modes. Therefore 

the bending modes should have lower frequencies. Also since the C atom 

is lighter than the 0 atom, and since the asymmetric mode requires larger 

bond stretching than the symmetric mode, the asymmetric stretching mode 

should have a higher frequency. Thus the frequencies should be ordered as 

c(asymm stretch) > c(symm stretch) > <(bending). (1.75) 

This is in fact true for the C02 molecule where it is found spectroscopically that C(asymm stretch) = 2349cm", <(symm stretch) = 1388cm-', and 

<(bending) = 667cm-'. These are referred to as the us, u1, and v2 vibration 

modes respectively. 

Each vibrational mode can be excited to many energy levels. In the 

harmonic oscillator an infinite set of energy eigenvalues axe spaced at inter- 
32 

vals of tiW. In real molecules, the higher vibrational energy levels stretch 

the molecule. At farther separations the molecule breaks apart because the 

quantum mechanical binding energy of the electron cloud becomes weaker 

than nuclear repulsion [Berry, Rice, Ross, 19801. The bound energy levels 

become more closely spaced as the ibational quantum number increases since 

the average curvature of the potential well represented by the IC, well is reduced. This due to deviations of the binding energy from a pure parabolic 

potential and is shown in Figure 1.4.3. 

. 

Each mode of oscillation may have more than one quantum of vibrational 

energy. In fact, different modes of oscillation may be present simultaneously. 

The modes are classified by the number of vibrational quanta associated with 

the state [see Herzberg, 19451. As an example, the first excited asymmetric 

vibrational mode of COP is labeled the 'us' which stands for one quantum of 

the n = 3 vibrational state. Two quanta of the symmetric stretching mode 

is labeled '2~'. Combinations are designated by a sum. For the vibrational 

state with one quantum in the symmetric stretching n = 1 and two quanta 

in the bending n = 2 modes is labeled VI + 2ua. Further elaboration on 

rotational-vibrational states and nomenclature for many molecules is given 

by Herzberg (1945). 
t'3 

Figure 1.4.1 Normal mode vibrations for the COa molecule. 

Y 

Figure 1.4.2 Normal mode vibrations for the Ha0 molecule. 

33 
34 

1.5 Rotational Energy Levels 

In the case of diatomic molecules (such as CO) and linear triatomic 

molecules (such as CO2 and N2O) the rotational energy can be approximated 

reasonably well by the rigid symmetric top model 

Io 3 I, = I, # I, (1.76) 

where the z axis is defined to be the bonding axis. Then the rotational 

Hamiltonian can be expressed as 

1 L2 1 1 

210 2 I, ",,t=-+- ( --- :,> 2, (1.77) 

where L2 is the total angular momentum operator (squared) and the z component of the angular momentum operator is Lz whose eigenfunctions are 

the spherical harmonics Xm(0,4) [Berry, Rice, Ross, 19801, 

In the case of the rigid rotor the energy levels are 

(1.78) 

(1.79) 

(1.80) 

forl=0,1,2 ,... andm=-1 ,..., 1. 

If the atoms are considered as mass points then there is no z component 

of the angular momentum and the levels are 21 + 1 degenerate. However, in 

a linear molecule the contributions to I, from the electrons and the non-aero 
35 

nuclear radii are both very small, I, < Io, and the coefficient of i: in (1.80) 

is much larger than that of Lz. For example, the energy associated with 

is very large when I;, # 0 in all states except the ground states when 

it vanishes [Goody, 1964; Berry, Rice, Ross, 19801. In any case, this second 

term can be included into the electronic energy since it is a constant and the 

rotational energy can be described by 

(1.81) 

for J = 0, 1,2, e. The quantum number J is used instead of 1 in describing 

molecular angular momentum and in the case of linear molecules Ah is identified as the z component of angular momentum where A is an integer. For 

a linear molecule A = 0,1,2, -e and therefore the total angular momentum 

must be equal to or greater than A, J = A,A + l,A + 2,... . Most of the 

infrared rotational-vibrational spectra described in this study are associated 

with molecules that are in the electronic ground state which corresponds to 

A = 0. In the case of the asymmetric top where all moments of inertia are 

unequal such as the case for HZ0 there is no axis about which the angular 

momentum is conserved and the solutions cannot be described by spherical 

harmonics. The solutions still have 21 + 1 levels for each value of I, but they 

are complex and there are no degeneracies [Herzberg, 1945; Goody, 19641. 
36 

I. 6 Rotation-Vibration Spectra 

The moment of inertia Io is related to the bond lengths between atoms. 

When a molecule undergoes vibrational oscillation, the anharmonicity in the 

potential well forces the average equilibrium bond length for each vibrational 

state to increase. This increases the moment of inertia and decreases the 

energies of the rotational levels (1.81). The modified rotational energy levels 

are described by [Berry, Rice, Ross, 19801 

. , 

(1.83) 

where B, is the rotational constant for the vibration level u, a, is a constant, 

and B, = h/4~c& is the constant which characterizes the energy of the 

rigid rotator. The energy levels (1.83) describe linear molecules such as GO, 

COP and NPO. The energy levels are much more complicated for nonlinear 

molecules such as HzO and CH4 and cannot be described by (1.82-3). 

If the energy of the incoming photon is equal to the difference between 

two energy levels of a molecule, a transition between the two states may 

be induced provided the molecule can interact with field as a whole. The 

probability of a transition between two states is proportional to Ipn,,, I2 where 

JJ,,~, is the dipole between the two states n and n' 

(1.84) 
37 

The dipole operator p is usually given by er, e being the charge of the electron and r the position vector. There are additional terms to the transition 

probability involving quadrupole moments, magnetic dipole moments, and so 

on, however rates of these types of transitions are usually of an order of magnitude or more smaller than those for electric dipole transitions [Steinfeld, 

19851. For the rigid rotator it can be shown [Goody, 1964; Berry, Rice, Ross, 

19801 that the probability for a dipole transition is non-zero if AJ = fl. For 

the harmonic oscillator the dipole selection rule is Au = fl. A molecule can 

undergo transitions in which both the vibrational and rotational quantum 

numbers change according to the selection rules. I will be concerned here 

with rotation-vibration transitions in the ground state since these are the 

dominant type of transitions observed in the ATMOS spectra. 

In (1.82-3) we see that the rotation-vibration energy is proportional to 

the constant hc. We can define a new quantity which describes the rotational 

part of the energy 

(1.85) 

(1.86) 

The unit of measurement of this quantity is cm”. The differences between 

energy levels Ac described in the manner of (1.85-6) is in terms of wavenumbers c which is the number of wavelengths of radiation within a centimeter 

of length in a vacuum. If A is the wavelength in vacuum then the frequency 

f is related to it by 

(1.87) C f =x. 
38 

I 

1 

I1 

I 

For a transition of energy difference AE the corresponding frequency of radiation is 

(1.88) AE 

h f=-=cA&EcCC. 

Explicit 1 y, 

f a c = l/X. (1 39) 

Therefore frequency and wavenumbers are proportional to each other. 

In an absorption or emission process the energy of each photon involved 

must equal the difference between the energy levels. For linear molecules, 

if the lower energy level of the vibrational mode is given by the quantum 

numbers Y, J and that of the upper energy level by v', J' then the energy 

difference, in wavenumbers, is 

c = (v' - Y)C, + J'(J' + 1)Byl - J(J + 1)B" (1.90) 

where 

tiw, 

Ce = - hc (1.91) 

for the vibrational mode. For absorption processes in which Au = +l there 

are two sets of rotation lines, those with AJ = +1 and those with AJ = -1. 

AJ = 0 is prohibited in dipole transitions in the ground electronic state since 

there is no net dipole moment between the states. The set of transitions 

where AJ = +1 is called the R branch and is given by J' = J + 1. The line 

positions for this branch can be written in the form 

(1.92) 
1 

I 

I 

1 

I 

8 

I 

I 

t 

‘I 

1 

‘I 

I 

I 

‘8 

I 

IS 

‘I 

I 

39 

The case where AJ = -1 is called the P branch and is given by J’ = J - 1. 

The line positions are 

The quadratic terms involve the differences due to vibrational stretching between the rotational constants. In an infrared transition where the electronic . . . 

states remain the same for reasons explained earlier the rotational constant 

decreases with increasing the vibrational quantum number u. From (1.92) it 

can be seen that a value for J is eventually reached where the frequencies begin to decrease with increasing J. This causes a band head - such as the solar 

CO band heads observed in the ATMOS spectra $11.3 - and occurB in the 

R branch in rotation-vibration spectra since c B, [Berry, Rice, Ross, 

19801. The B, values, however, are usually very close to one another and it 

is difficult to observe band heads in spectra taken at ordinary temperatures. 

Typical values for the rotational constants is of the order of several cm” for 

atmospheric gases such as C02 and N2O. The lines are spaced far enough 

apart that lines of the same vibration band do not interfere with each other. 
40 

'I 

I 

!I 

I 

8 

I 

8 

:I 

'I 

I 

1.7 Radiative Transfer 

The energy of a molecule in the ground electronic state consists of translational, vibrational and rotational energy. The vibrational and rotational 

energy are associated with internal degrees of freedom while the translational motion is related to the center of mass coordinates. The molecules 

can transfer energy among themselves via collisional and radiative processes. 

Molecules can lose or gain translational energy due to elastic collisions with 

other molecules. The internal and translational energy can also be exchanged 

in inelastic collisions. When a gas is subjected to radiation of frequencies 

comparable to differences in energy levels of its molecules energy can be absorbed and emitted. The.populations of the various energy levels is therefore 

related to the balance between collisional and radiative processes. 

In the lower atmosphere below 70 km the collision rates are larger than the 

radiative rates due to the higher molecular density [Gao, 19881. The populations of the the rotational-vibrational levels me then dominated by molecular collisions and are described by the Boltemann distribution [Thorne, 

19881. A quantity called the temperature T can be defined which specifies 

the mean kinetic energy tkT of the colliding particles and the particles follow 

a Maxwellian velocity distribution. This is called local thermodynamic equilibrium (LTE) since the molecules cannot be considered to be in complete 

thermodynamic equilibrium due to radiative processes between themselves, 

the rest of the atmosphere, space and solar radiation. The molecules can be 

considered to be in local thermodynamic equilibrium with the total radiation 
I 

8 

'1 

I 

I 

'I 

8 

I 

I 

I 

41 

of the atmosphere which is described by the Planck blackbody distribution 

at the kinetic temperature temperature T [Thorne, 19881. In the following 

derivations we assume that the condition of LTE is adequate and this was 

found to be approximately true by Gao (1988) for the u3 rotation-vibration 

transitions of COO at altitudes between 70 and 130 km. 

SPONTANEOUS EMISSION 

Spontaneous emission at a frequency fo occurs naturally from an excited 

state 8 to a lower energy state 8' . It is induced by zero point oscillations in 

the radiation field [Cohen-Tannoudji, 19771. The rate of this process A,,,, 

is related to the electric dipole matrix element p,,, = (glp18') between the 

states by 

(1.94) 

[Thorne, 19881. The mean lifetime between the states is then given by 

STIMULATED EMISSION AND ABSORPTION 

Besides the spontaneous emission of radiation from an excited state to 

a lower energy state there can also induced transitions in the presence of 

external fields. Suppose a lower energy state has the energy hf,, and the 

upper state has the energy hf,. An electromagnetic wave with frequency 
42 

fo = f, - f,~ can be described by an electric field 

Each element of the sum (1.96) has a component that oscillates with the 

frequency of each state. It is these resonances with the dipole moment of 

each state that eventually induces transitions between the two states [Aller, 

19631. The photon of frequency fo can drive a transition between the two 

states, with an efficiency which depends on the strength of the dipole between 

the states. The electric field can induce both absorption and emission of 

a photon and the rates of both of these processes axe proportional to the 

average photon intensity 1 near the transition frequency YO [Thorne, 19881 

.. 

(1.97) 

(1.98) 

respectively. The coefficients A,,,,, B,,,, and B,,,, are commonly referred 

to as Einstein coeficienb and are used to describe the rates of the various 

emission and absorption processes. 

RADIATIVE BALANCE 

If n,, and n, are the density of molecules in the lower and upper state 

respectively then the relationship between the absorption and emission processes in radiative equilibrium is given by the relation 

- - n,, B,,,, I = n, A,,,, + n, B,,,, I (1.99) 
8 

I 

I 

1 

‘I 

$1 

I 

I 

1. 

8 

1 

‘I 

I 

I 

I 

R 

8 

I 

I 

43 

since in equilibrium the rate of photons emitted is equal to the rate of photons 

being absorbed. Then the photon intensity can be solved for 

(1.100) 

The ratio of the populations n:/n, is undetermined. In local thermodynamic 

equilibrium at temperature T the ratio of the populations in each state is . 

given by Boltzmann’s equation [Steinfeld, 1985; Thorne, 19881 

where gi axe the degeneracies of each state. Then 

However from Planck’s Law [AUer, 19631 

(1.101) 

(1.102) 

(1.103) 

for a blackbody radiator at the temperature T. Since If varies slowly over 

the width Au of the line, I’ can be replaced with Ifo 

(1.104) 

Since this must hold for all T it can be shown that the Einstein coefficients 

are then related to each other by 

- A,+,:/B,+,: - 2h fi/ca 

ehfoIk* - 1 (g,: B,t+,/g, B,,,:) ehfolkT - 1 ’ 

(1.105) 
1 

t 

I 

I 

‘1 

I 

I 

I 

1 

1 

‘I 

.I 

1 

I 

8 

I 

‘I 

a 

44 

and depend only on the electric dipole matrix element and the energy difference between the states. 

The case where LTE is not valid is discussed by Gao (1988) and Thorne 

(1988). At altitudes in the atmosphere high enough where molecular collisions are less frequent than radiative processes, the populations of rotationalvibrational energy levels are dominated by absorption and emission processes. 

The kinetic temperature T is then not adequate to describe the populations 

of the states. The lower atmosphere is the main source of photons present 

for absorption and the blackbody radiation distribution which describes them 

may not correspond to the local kinetic temperature of the gas [Gao, 19881. 

The atmosphere is then considered to be in a state of non-LTE (NLTE). 

The grand effect of NLTE is to alter the populations of the various occupied 

states. It was found by Gao (1988) that between 70 and 130 km the deviations of the atmosphere from LTE do not affect the populations of rotational 

levels significantly from the Boltzmann distribution at a kinetic temperature corresponding to the temperature of the lower atmosphere. Although 

LTE/NLTE is important for determining the populations of various states, 

it is not important for obtaining wind speeds $811. 

1.8 Line Shapes 

The shapes of spectral lines depends on the molecules and their physical 

conditions. In this section we examine some of the most important physical 
45 

processes which describe spectral line shapes. Due to the uncertainty principle the finite life times of excited molecular states causes the lines to have a 

natural width (in photon energy) and shape. This width is a characteristic 

fundamental to the molecule and depends only on the frequency of the transition and the dipole matrix elements between the states (1.94). The thermal 

motions of the molecules also causes the frequencies of radiation involved in 

the absorption/emission processes to be Doppler shifted from the natural line 

shape as described by (1.1). The distribution of velocities and hence Doppler 

shifts are governed by the Maxwell velocity distribution. Finally, a molecule 

may also undergo collisions with other molecules in the gas. A molecule can 

interact with the radiation field during the period of time between collisions. 

A sum of all frequencies over that period of time can appear to the molecule 

as the correct driving frequency due to the properties of the Fourier transform. Frequencies well away from the transition frequency may drive the 

transition. All of these mechanisms as well as combinations of them can be 

described by a line profile 4(c) which describes the normalized distribution 

of the photon energies which induce transitions. 

. 

The width A E of an energy level 8 is implied by the uncertainty relation 

ti AE.7, 2 - 2 (1.107) 

where r, is the mean lifetime of the state. If the lower state has an infinite 

lifetime, which is the case for transitions to the ground state, then width of 

the transition, in wavenumbers (1.85), is 

1 

(1.108) 
1 

I 

I 46 

I 

I 

I 

1 

I 

I 

I 

1 

I 

.I 

1 

I 

8 

8 

I 

I 

In general, each transition has a natural lifetime given by 

7,-1 = A,,,, 

0' 

(1.109) 

where s' denotes all lower energy states and A,,,, is given by (1.94), the 

coefficient of the wave function of the state s is then of the form 

since the square modulus of the wave function represents the probability of 

finding the state and is the probability density of a decay between 

times t and t + dt. Consider a transition at frequency fo. The decay causes 

the emission of photons about the frequency fo. The electric field E due to 

this transition caused by emission decays as [Thorne, 19881 

where wo = 27rfo = 27rcco. The strength of the electric field decays with 

increasing time and has an oscillating part due to the transition frequency. 

This electric field has a Fourier transform 

00 

E(w) = (274-1 1- dt E(t)e'"'. 

Since E(t) is only defined for t 2 0, 

1 

27 

-1 

~(w) a [- - i(w - wO)] . 

(1.112) 

(1.113) 

The line intensity is proportional to the square of the electric field [Jackson, 

19751 I, = IE(w)12 and it can be shown by integration that 

1 Am 

x (C - CO)' + (Am)' I(C) = Io - (1.114) 
47 

where Am = (47rcr,)-l is the natural line width in wavenumbers which is 

in agreement with the width predicted by the uncertainty principle (1.108). 

The total line intensity integrated over all wavenumbers is defined as 

The line profile for natural broadening is 

(1.115) 

(1.116) 

and this is normalized as Jym t#~(c) de = 1. 

As an example, for an infrared transition to the ground state r, = AZ-.~. 

Since fo = c co then (1.94) becomes 

- 

16x3$ 

IC121 I' 3Eo h (1.117) 

For a typical u3 COS transition we can choose, for exampleoul-m, co = 2330 

cm-' and p12 - Cm as typical values. Then A3+1 - 3.6 x lo4 Hz. 

This gives Am - 7 x 10-ecm-l as the natural width which indicates that 

these lines are very narrow, about six to seven orders of magnitude smaller 

than the resolution of the ATMOS interferometer. 

The thermal motions of the molecules in a gas cause the frequencies of 

the absorbed or emitted photons to be Doppler shifted 

21 

=- e; - eo 

CO C 

(1.118) 

where e; is the wavenumber in the observer's frame, co is the frequency in the 

molecule's rest frame and 'v is the relative speed between the molecule and 

the observer along the line of sight. The fraction of molecules moving with 
1 

I 

I 

1 

'I 

.I 

I 

I 

I 

I 

I 

'I 

;I 

I 

I 

I 

48 

speed v and v + dv along the line of sight is given by a normalized Maxwell 

distribution [Thorne, 19881 

(1.119) 

(1.120) 

From this result the Doppler line profile is given by 

(1.121) 

The Doppler width is 

ACD=CO E* - (1.122) 

As an example, for a temperature of 300 K, co = 2300 cm'l, and a COZ mass 

of 44 gm/mole, Am = 2.6 x 10-3cm'1. Note that the Doppler broadening is 

much larger than natural broadening 

ACD >> ACN. (1.123) 

in the infrared regions but is still about 5 times smaller than the resolution 

of the ATMOS interferometer. 

A molecule can also have collisions with other molecules. Let the mean 

time between'collisions be TC. For simplicity consider a gas with a density 

such that the time between collisions is much greater that the interaction 

time q, the time for the collision processes. If the time between collisions 

is t' the radiation affecting the molecule will subject to random phase interruptions caused by collisions [Rybicki, Lightman, 19791. Specifically, between 

collisions, the component of the electric field which affects the molecule is 

~'(t) oc e-iwo' (1.124) 

I 

I 
I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

I 

49 

for 0 5 t 5 t' where again wo = 27rf0 = 27rcco. By using (1.112) the Fourier 

transform of this field is 

sin [(w - wo)t*/2] 

(w - wo)/2 E'(w, t') a exp [i(w - wo)t'/2] 

Again since I'(w, t') = I€'(w, t*)l2 then 

(1.125) 

(1.126) 

The probability dP(t') that the molecule has a collision between time t' and 

t' + dt' is 

dP(t') = ~~'e"''l"dt*. (1.127) 

The final intensity distribution is given by the average of the individual intensity distributions (1.126) over the probability function (1.127) 

*w 

I(w) = T;' I'(w,t')e-f'/Tcdt'. (1.128) 

This integral can integrated by standard methods to be 

1 ACC I(<) = Io - (c - coy + (&cy (1.129) 

where the collisional width, also cded the Lorentz width, is given by 

ACC = (~TCTC)-~ (1.130) 

and Io is the total line intensity (1.115). The profile for pressure broadening 

is also a Lorentzian like natural broadening 

(1.131) 
50 

The mean time between collisions can be estimated by 

TC = (mean free path)/(thermal speed) = - (1.132) 

where n is the density of molecules, Z is an average collision cross section, 

m is the molecular mass. From (1.130) 

I n 

(1.133) 

by using the ideal gas equation P = nkT. Since atmospheric densities decrease with height exponentially we expect that collision broadening to be 

important primarily near the surface. As height increases the collisional 

frequencies decrease in a like manner and we expect that the lines are governed by the Doppler shape. Above about 50 km the primary line shapes 

are Doppler with small Lorentz contributions. Below about 20 km pressure 

broadening is important. 

In the regions between 20 and 60 km altitudes the atmospheric absorption lines can show both Doppler and Lorentz (pressure) broadening. The 

observed line shape is actually a set of Lorentz profiles averaged over the 

Maxwell velocity distribution [Rybicki, Lightman, 19791. The line shape is a 

convolution of the Doppler and Lorentz profiles and is called the Voigt profile 

b(c) = h(c) * 4L(c) = /w de' 4D(c' - c) &(e'). (1.134) -00 

This profile is normalized since the Doppler and Lorentz profiles are normalized and this can be seen by integration 

= 1. (1.135) 
t 

ff 

1 

4 

1 

i 

I 

1 

1 

I 

51 

The Voigt line shape is governed by two widths, the Doppler width Am and 

Lorentz width Acc. When pressure broadening becomes dominant Acc > 

Am and the line shape becomes a pure Lorentzian. In the upper atmosphere where pressure broadening is negligible compared to Doppler broadening Am >> As the line center has a Doppler shape while the ‘wings’ - the 

absorption outside of the Doppler width - have the Lorentz shape. 
52 

I 

1 

I 

1 

1 

1.9 Absorption and Emission Coefficients 

The ATMOS spectra show the absorption of the atmospheric gases dong 

the line of sight between the observer and the sun. The ray path passes 

through all altitudes above the tangent point of observation. In this section . __ . 

we discuss how the transmitted intensity depends on the amount of absorber 

along the line of sight. We use this information to examine how important 

the contributions of the absorber in regions far away from the tangent point 

contribute the the observed line positions in 511.13. 

The line shapes discussed in the previous section apply to large ensembles 

of molecules undergoing radiative and collisional transitions. Large amounts 

of gas along the line of sight between the radiation source and the instrument 

enable the molecules to absorb significant amounts of the incoming radiation 

due to cumulative effects of absorption. The absorption coefficient a, is 

defined to describe the loss of beam intensity d1, at wavenumber c as it 

travels over a distance ds of the medium by 

(1.136) 

A spontaneous emission coefficient j, is also defined as the power emitted per 

unit volume. This can be assumed to be isotropically distributed since there 

is no preferred direction of radiation in an emission. If an incident beam 

with parallel rays such as solar rays in the ATMOS experiment is used, then 

the spontaneous emission term j, can be neglected since most of the emitted 

radiation goes out of the field of view. 
1 

1 

I 

I 

I 

f 

1 

I 

I 

b 

1 

E 

I 

I 

E 

I 

I 

I 

a 

53 

Let s be the upper energy state of a transition, s‘ the lower energy state 

and n the number of active molecules per unit volume. Then hc c n,A,,,id(C) 

is the power emitted by spontaneous emission per unit volume per unit 

wavenumber where hcc is the photon energy and n,A,+,t+(c) is the rate 

of photons emitted per unit wavenumber per unit volume. This radiation 

caused by this term is isotropically distributed and can be neglected relative ~ __. 

to the beam absorption. 

The power absorbed per unit wavenumber per unit volume is the difference between absorption and stimulated emission since the stimulated emission will be polarized with respect to the incoming radiation [Steinfeld, 19761 

and can be thought of as “negative absorption” 

Since the intensity IC is the power per unit wavenumber per unit area the 

power absorbed per unit wavenumber per unit volume is the change in intensity per unit length along the beam 

-- - hc c (n,l B,l,, - n,B,,,~). ds (1.137) dI, 

Then the absorption coefficient can be determined from (1.136) 

If the system is in thermodynamic equilibrium with itself, which is the case 

with of an atmosphere in LTE and will be assumed to be approximately true 

for NLTE [Gao, 19881 then from (1.101) 

(1.139) 
I 

1 

I 

1E 

t 

I 

2 

I 

1 

1 

'1 

E 

11 

I 

'I 

1 

'1 

8 

:I 

54 

By using this in (1.138) and using the Einstein relation (1.106) 

a, = hccn,:B,:,,(1 - e-hcc/CT)c$(c). (1.140) 

The transmitted intensity neglecting the spontaneous emission term is 

the solution of (1.136) 

- = -a& ds (1.141) 

which can be solved 

I, = I,(O)e-J5+& (1.142) 

where L is the total path length the radiation travels through the absorber 

and I,(O) is the radiation intensity in the absence of any absorption. 

It was shown 51.8 that the line widths are much smaller than the central 

wavenumber $0. Also, at infrared wavenumbers and at typical atmospheric 

temperatures the exponential factor in (1.140) is much less than unity and 

the absorption coefficient can be well approximated by 

Then the transmitted intensity (1.142) can be written as 

I, = I,(0)e-k~u. (1.144) 

The number of molecules per unit area in the lower state is defined by 

L 

u = 1 dsn,: (1.145) 

where it is should be recalled that n,: is the density of molecules in the lower 

state. The absorption cross section per molecule le, is given by 

le, = Sc$(c). (1.146) 
I 

1 

1 

P 

11 

I 

I 

I 

1 

and the line strength is defined as 

S = hcCoB,,,,. 

55 

(1.147) 

The line strength is also equal to the integrated ausorption coefficient kc since 

the line profile is normalized and this can be shown by integrating (1.146) 

1 

It 

a 

I 

'I 

t 

D 

8 

1 

. 

(1.148) 
I 

1 

1 

'I 

8 

II 

1 

t 

1 

56 

1.10 Optical Depth, Equivalent Widths, and Curve 

of Growth 

The optical depth r, at a given wavenumber c is related to the mean free 

path of a photon [Thorne, 19881. We define 

r, = k,u (1.149) 

so that (1.144) is 

Ic(rc) = I,(O) e-.''. (1.150) 

For r, < 1 the medium is "optically thin" while for r, >> 1 it is "optically 

thick". For the case of optically thin layers we can expand (1.150) 

I&,) = I,(O) (1 - 7, + .-). (1.151) 

In this case the transmitted intensity I, decreases linearly with an increase in 

the amount of absorber u. As the amount of absorber u increases this linear 

behavior no longer holds true and saturation is reached. The absorption line 

initially saturates at the line center and then outwards towards the "wings". 

Figure 1.10.1 shows this saturation as the amount of absorber increases. 

The total loss of incident radiation due to absorption by the line is 

dc [I,(O) - I,(r,)] = Power Loss/Area. (1.152) 

The ratio of this quantity to the incident intensity per unit wavenumber 

defines the equivalent width 

(1.153) 
57 

From (1.150) we have 

(1.154) 

The equivalent width is independent of the instrumental resolution [Thorne, 

19881. Although a decrease in resolution will cause the observed line to 

"spread out" over a greater spectral region, the equivalent width remains 

the same as the 'true' line. Thus measurements of the equivalent widths - . 

of absorption lines can be used to obtain information regarding the total 

absorption of the line and properties associated with it such as temperature, 

pressure, and absorption cross sections u [Gao, 19881. If the instrumental 

resolution becomes larger and hence the resolved width of the line at, say, 

half maximum absorption increases, the peak absorptances must decrease 

since the equivalent width with remains the same. 

A typical characteristic of an absorption feature is its width Ac as described in $1.9. This is characteristic of Doppler and Lorentz lines, as well as 

the convolution of the two which is called a Voigt line. If we change variables 

c - eo z=- 

AC 

(1.154) can be written as 

For the Doppler line (1.121) 

For the Lorentzian (1.131) 

(1.155) 

(1.156) 

(1.157) 

(1.158) 
8 

i 1 

58 

We see that in both cases the kaction of the equivalent width (EW) to the 

line width W/Ac is a function of Su/Ac. If we define 

su 

AC 

a=- 

then (1.157-8) can be written 

r,(Doppler) = 

11 II r,(Lorentz) = -- 

7r 1 + 23 a* 

(1.159) 

(1.160) -- 

(1.161) 

For the two cases I computed the integrals (1.150) as a function of a and 

are plotted in Figure 1.10.2. Both curves have the same slope in the reeon 

where a < 1. This linear region is predicted by (1.156). For small r, we have 

w 

AC 

- N Le dz 7,. (1.162) 

For the Doppler and Lorentz lines we have 

The growth rate is the same for both type of lines in the linear regions. The 

equivalent width is then directly proportional to the amount of absorber. The 

linear growth for both Doppler and Lorentz lines continue until log Su/Ac - 

1/2 which can be seen in Figure 1.10.2. 

As the amount of absorber continues to increase the equivalent width of 

a Doppler line WD approaches almost ten times the line width Am and increasing the amount of absorber u does not increase the equivalent width significantly beyond this which can be seen in Figure 1.10.2. Since the Doppler 
59 

width Am depends primarily on the temperature T and not the pressure the 

equivalent width WD will not change significantly, with height. 

For a pressure broadened line, the curve in Figure 1.10.2 shows that the 

equivalent width of a Lorents line Wc increases indefinitely relative to the line 

width Acc. Since the line width depends directly on the pressure P (1.133), 

as the altitude decreases and the amount of absorber along the line of sight 

remains significant, the increasing pressure causes the equivalent width Wc 

of the lines to grow without limit. In the case of strong absorption lines such 

as the u3 COa band, large spectral regions can become opaque due to strong 

pressure broadened saturation. 

. 

The increasing equivalent width limits the altitude ranges where pressure 

broadened lines may be useful for measuring wind speeds since the lines as 

observed by the instrument must not be distorted $1.12. If the equivalent 

width of the line is much smaller than the instrumental resolution, the line 

will yield approximately the same shape as an infinitesimally thin absorption 

line with the same equivalent width. “Flattening” of the line shape near the 

line center due to saturation can not be seen in the spectra and does not 

significantly effect the resolved line shape provided that Am < resolution. 

If the equivalent width becomes significant with respect to the instrumental 

resolution we expect that spectral features will become broadened. As shown 

in the next section, increasing the width of a feature in the processed spectra 

can significantly degrade the precision of determining line positions. Since 

the equivalent width of a Doppler line WD is limited to about ten times 

the Doppler width A~D, and since the Doppler width is about 100 times 

smaller than the instrumental resolution, Doppler lines can be used over 
1 

1 

I 

I 

1 

I 

60 

altitude ranges spanning tens of kilometers for obtaining wind speeds without ' 

having to be concerned with significant increases in the Doppler equivalent 

width. On the other hand, the equivalent width of pressure broadened lines 

increases rapidly with increasing pressure and absorber amount u. Pressure 

broadened lines therefore have smaller ranges over which they can be useful 

for measuring Doppler shifts and lines used for measuring relative speeds 

need to be carefully chosen to avoid the effects of pressure broadening in the 

resolved spectra. 
61 

e 

- 

- - 

- - " " I' I' 

Figure 1.10.2 Curve of growth for Doppler and Lorentz lines computed from 

(1.156). The equivalent width W, of the Doppler lines becomes almost 10 

times its Doppler width Am as the amount of absorber u increases. The 

equivalent width of the Lorene lines WC increases with u without limit relative to its collision broadened width Acc. 
62 

I. 11 Measurement Precision Theory 

The purpose of this section is to examine how the measurement precision 

of the wavenumber of an absorption line is affected by the resolved shape 

of the line with a noisy background and digital frequency spacing. In the 

ATMOS spectra the noise level is about u(N) = 1% of the background 

signal and the wavenumber spacing is 0.0075 cm-’. For an absorption line 

with a typical resolved width of .015 cm-’ at half maximum, there are only 

3 points to use to estimate the line position. The finite resolution of the 

spectra causes the lines to be “spread out” and the data used to obtain 

line positions correspond to the true spectra convolved with the (apodized) 

instrument function. Therefore the ‘line shapes’ present in the spectra are 

degraded by the instrumental resolution. In this section the ‘width’ of a line 

is the apparent or geometric width as it appears in the processed spectra. 

This instrumental width is limited by the resolution of the spectra. 

Provided pressure saturation is not present, we can approximate the geometric shape of the a line in the convolved spectra by a parabola of the 

form 

(1.165) 

for -w/2 5 x 5 w/2 where z = 0 corresponds to the line center, Ap is the 

geometric peak absorptance of the line and w the geometric width at half 

maximum. Figure 1.11.1 shows an example of this function. In order to 

test our ideas, we allow this function to describe the shape of the absorption 

line as it would be measured in the absence of noise. We can model the 
63 

actual observed data with this “clean” signal and a random distribution 

of noise added to each point. A random noise distribution N(z) with a 

standard uncertainty u(N) is added to this “noiseless” signal g(z) to produce 

an “observed signal” g( 2) + N( 2). 

Having obtained the data for this line suppose, for simplicity, that a trial 

function of the form 

f(z) = (2Ap/WZ)(Z - Y)’ (1.166) 

is to be fit to the observed signal g(z) + N(z) by the least squares method. 

We wish to determine the best fit and hence the line position from this fit. 

We sample the signal g(z) + N(z) at the n points zj. The measured line 

center yo is the value of y which minimizes the least squares sum 

(1.167) 

The least squares sum M can be approximated by an integration over the 

range of sampled points 

The second term on the right hand side is essentially zero since N(z) is a 

random function with a mean of zero. The last term can be ignored since it 

is a constant which only depends on the noise distribution c(N). This gives 

MN (1.169) 
D 

1 

8 

I 

8 

I 

I 

I 

1 

I 

I 

I 

,t 

I 

I 

I 

,I 

:I 

,a 

64 

The value yo is chosen which minimizes M by the condition 

q = 0. (1.170) 

At the stationary point yo the precision is given from elementary statistics 

[Bevington, 19691 by 

For our model functions 

so that 

2nAi 1 

M= (y4 + py). az( N)w4 

The minimum occurs at yo = 0 which gives 

(1.171) 

(1.172) 

(1.173) 

(1.174) 

This relates the measurement precision to the geometric line width w, the 

noise-to-signal ratio c(N), the number of points sampled n, and the geometric 

peak absorptance of the line Ap. 

The result for yo) which comes from the integral approximation can be 

tested against a more direct calculation to estimate the uncertainties from 

the least squares fit. Once again introduce the least squares sum 

(1.175) 

where this is related to the previous example by n = 2m + 1 and zj = 

(j/m)(w/2) for -m 5 j 5.m. Using the same functional forms f(z) = 
65 

(2AP/W2)(2 - Y)' and g(z) = (2AP/w2)z2, the least squares sum can be 

expressed, after discarding terms involving the random function N(z) for 

the same reasons given above, as 

The sum (1.177) can be shown to be 

(12m2 6m)y' + (4m2 + 6m + 2)w2y2 4 M= (1.178) 3m u2( N)w* ' 

Once again the measurement uncertainty can be estimated from (1.165) with 

yo = 0 which gives 

(1.179) 

The result of this calculation shows the same dependence on the parameters u(N), w, and Ap as the result from the integral approximation except for 

a multiplicative form factor f' which only depends on the number of points 

n 

(1.180) 

* 

Table 1.11.1 shows how these form factors are related for various values of nz. 

The values listed show that the integral approximation is higher by as much 

as 20% for small n than the direct sum calculation with better agreement as 

n increases. 

This method of determining line positions is similar to the procedure 

which is followed by the line-finder program [Norton, 19861 which determines 
66 

the line positions from the ATMOS spectra. The model for the absorption line shapes is somewhat simplified and it is of interest to see how this 

parabola-parabola least squares fit compares to a fit of a parabola to a gaussian absorption. This pair of functions is chosen since the true absorption 

lines ‘taper off’ at wavenumbers away from the line center similar to a gaussian shape. Figure 1.11.2 shows such a sample gaussian described by 

(1.181) 

Again A, represents the geometric peak absorptance and w is the geometric 

width at half maximum. 

Although the gaussian line shape is chosen rather arbitrarily the dependence of the measurement precision to the geometric width w, the peak 

absorptance A, and the noise amplitude Q(N) remains the same as for the 

parabola-parabola fit. To see this, we fit 

to g(z) for -w/2 5 ?: 5 w/2 by both the integration approximation and 

the direct sum methods. To solve for the measurement uncertainty we again 

followed the procedure above (1.169-71). The rather involved calculations 

were performed using the symbolic manipulation program MACSYMA at the 

VAX-8600 facility in the Physics Department at the Ohio State University. 

It was found that, by closely following the procedure (1,169-?l), the result 

using the integration approximation is 
’\ 1 

’ I 

1 

1 

67 

(1.183) 

Once again the result is a form factor multiplied by the product q(N)wA;’. 

For the direct sum calculation the resulting measurement uncertainty following a procedure identical to (1.175-79)) also evaluated by MACSYMA, is 

given by 

(1.184) 3 -l 

(1.185) 

The form factors are shown in Table 1.11.2. Again the integration approximation introduces larger uncertainties than the direct sum method, however, even more interesting is that the values for the parabola-parabola and 

parabola-gaussian direct sum methods give approximately the same answer. 

The former case has slightly smaller form factors. These results suggests 

that the measurement precision may not be very sensitive to the details of 

the geometric line shapes which can be seen by comparison of the direct sum 

methods which are more accurate. 

The calculations above suggest that the measurement precision is proportional to the noise level a(N), the geometric line width w, and inversely 

proportional to the peak absorptance A,,. The signal to noise ratio (SNR) 

in the ATMOS spectra is approximately 100 in the u3 COZ region near 2350 

cm-l which corresponds to r(N) = 0.01. For the apodized spectra the width 

at half maximum peak absorptance is about 0.015 cm”, which can be attributed to the instrumental resolution and can be seen from a few sample 
68 

line shown in Figures 1.11.3-4. (The ATMOS spectra is digitally processed 

at intervals of 0.0075 cm-’ and due to a “smoothing” procedure written by 

Norton the drawings of the absorption lines appear as curves.) The width of 

the line is approximately twice the digital spacing which gives n = 3. Table 

1.11.3 shows estimated measurement uncertainties using these parameters 

for lines with various values of Ap. .. 

From this information we can estimate the precisions by which Doppler 

shifts can be measured. From 1.2 the ancertainty in measuring relative speeds 

v can be determined 

U(V) = ~(Ac)/c C. (1.186) 

The measurement error per line is dependent both on the precision by which 

line positions can be obtained U(C) and -the central wavenumber c. The 

measurement precision Q( e) is frequency independent. Therefore, for lower 

wavenumber regions the error in measuring the relative speed a(v) increases. 

Using the data presented in Table 1.11.3 we see that, as an example, for lines 

with a peak absorptance of about 30 % and central wavenumber c -2000 

cm-’ the precision of measuring relative speeds is about 25 ms-l per line. 

Therefore a set of 25 lines can bring the mean precision to 25 ms-’/fi =5 

ms”. This measurement precision theory predicts that precisions of obtaining 5 ms-’ are within the capabilities of the interferometer. Using higher 

frequency regions and stronger lines it is feasible to improve these statistics. 

However, due to the broadening effects discussed previously which may limit 

the altitude ranges for using atmospheric lines to obtain Doppler shifts, a 

wind speed measurements with precisions of 5 ms-’ was determined to be 

within the capabilities of the ATMOS instrument. 
W 

69 

Figure 1.11.1 Function g(z) = (2AJW')z' and fitting function f(z) = 

(24/Wa)(z - y)' to describe line profiles. 

Figure 1.11.2 Function g(z) = Ap(l - e-4h2(s/w)a) to describe gaussian line 

profile. 
'I 

'I 

i 

1 

:I 

1 

I 

'I 

I 

I 

g!!...ee!e.-.7 r .................. ' 1---r 

- 

6' A ................. 1 ................................ d.. ..... I -_: 

2PI.m I.rB0 pY.0 

Figure 1.11.3 A CO, absorption line taken from ATMOS spectra. 

I 'T"--'1- --'I-- ..._ -. 

- 

................. f: - I L I 1 .--..._- J 

1CIL.O I.rB0 lac% 

Figure 1.11.4 An Ha0 absorption line taken from ATMOS spectra. 

70 
I 

I 

I 

I 

'l 

E 

I 

'I 

I 

'I 

I 

'I 

I 

I 

I 

I 

'I 

Table 1.11.1 Form factors for parabola-parabola least squares fit for various 

values of m. 

m n = 2m+ 1 Integral Direct Sum 

1 3 0.50000 0.35355 

2 5 0.38723 0.31623 

3 7 0.32733 0.28347 

4 9 0.28868 0.25820 

5 11 0.26112 0.23837 

Table 1.11.2 Form factors for parabola-gaussiaa least squares fit for various 

values of m. - - 

m n = 2m+ 1 Integral Direct sum 

1 3 0.62051 0.35355 

2 5 0.48065 0.32063 

3 7 0.40622 0.28924 

4 9 0.35825 0.26440 

5 11 0.32405 0.24464 

Table 1.11.3 Estimated measurement uncertainty from parabola-gaussiaa 

least squares fit using direct sum. 

0.2 

0.3 

0.4 

2.65 

1.77 

1.33 

0.6 

0.7 

0.8 

0.88 

0.76 

0.66 
CHAPTER I1 

EXPERIMENTAL METHOD 

11.1 Instrumental Calibration 

72 

CALIBRATION MODEL 

To get useful wind speed measurements we must be able to measure 

Doppler shifts precisely for each spectrum and must be able to intercompare 

the spectra. The comparison of relative speeds determined from a series of 

spectra is limited by the calibration stability of the interferometer. As well 

as noise in the spectra, there is the possibility that the calibration contains 

random offsets due to errors in locating ZPD in each interferogram 51.1. 

An estimate of the magnitude of this scde uncertainty provides information 

about the precision of the ZPD locator program as well as setting limits of 

the accuracy of relative speed measurements. Sets of spectra may also have 

slowly changing calibration offsets caused by variations in the environment 

of the interferometer as well as “man-made” calibration offsets introduced 

b 
I 

I 

b 

I 

1 

t 

1 

I 

'1 

1 

t 

1 

9 

,I 

I 

c 

I 

1 

e 

73 

during the processing stage [Norton, 19861. In order to obtain accurate measurements of Doppler shifts it is necessary to calibrate the spectra accurately. 

Of the four spectral filters used with the interferometer only Filter 3, 

which covers the region 1600-3200 cm-', provided spectra with lines due to 

gases trapped or around the instrument by which the calibration stability 

could be analyzed. The strongest lines of the u3 COZ and ua Ha0 vibration 

bands from 2300 to 2380 cm-' and 1600 to 2000 cm" respectively appeared 

in spectra taken at tangent heights far above the atmosphere. These lines 

appeared to be related to the environment of the instrument and were due to 

residual gas trapped in the instrument as well as gas floating in space with 

the orbiter. These lines were called instrumental lines and it was found that 

they provided a tool for directly measuring the behavior of the calibration 

system. Figures 2.1.1-2 show some of these absorption lines as they appeared 

in the ATMOS spectra. In Figure 2.1.1 the u3 COa band is shown as well as 

the t) = 0 + 1 band head of solar CO. In Figure 2.1.2 some strong lines of 

Va Ha0 band are shown together with other weak lines of solar CO. 

In order to test the accuracy of the instrumental calibration the measured 

positions from a set of instrumental lines cj" of each line j in each spectrum 

k were assumed to be related to accepted standard positions ij by the model 

Here, a is an offset which is used to model any wavenumber errors in the 

calibration of the interferometer. This global ofaet is assumed to vary slowly 

over large wavenumber regions of the spectra and we will consider it be relatively constant over regions of several hundred cm-'. The source of this kind 
74 

'1 

I 

of offset is assumed to be related to environmental effects or arbitrarily introduced data processing artifacts [Norton, 19861. The calibration noise due to 

phase errors in the processing stage is given by 8, a frequency independent 

scale error for the spectrum k, and e:, the measurement error of the line j 

in the spectrum k. It was shown in 51.1 that the calibration uncertainties 

due to phase errors should be negligible but they are still included here. It is 

assumed that all of these errors are independent of each other. The position 

cf of each spectral line j in a given spectrum k is measured by a line-finder 

program developed by Norton (1986) which estimates the positions of absorption lines from the spectral data and its precision of O(lO-'cm-') is 

discussed in $1.11. The measurement uncertainty a(ej) and scale uncertainty 

a(€) both contribute to the position uncertainty a(cj) and it is necessary to 

estimate the magnitudes of each of these. 

DRIFT PARAMETER 6 

Before proceeding along these lines it is important to check the calibration 

stability in each set of spectra for each occultation. A reference spectrum r 

was arbitrarily chosen, in this case at the highest tangent height, and a check 

for a calibration drift factor 6 was made. It is assumed that there exists the 

possibility that 

c; = (1 + 6)c; (2.2) 

where k are the other spectra in the set and b may change between spectra in 

an occultation, a sort of calibration "drift" factor. Stability will be inferred 

if there are no statistically significant deviations away from b = 0. The 
1 

I 75 

1 

measured positions of each of the CO1 and H10 lines j in each spectrum le 

were compared to their corresponding positions in the reference spectrum r 

and a value for 6; determined by 

(2.3) +-- ci” 1. 

c; 

Jk (6;)j (2.4) 

An average value 

and the uncertainty ~(6~) for each spectrum was determined. The brackets 

stand for an average over the subscript j - i.e. all of the lines used in spectrum 

IC. 

Estimated values of 6k for the five occultations are shown in Figures 

2.1.3-4 for CO2 and H10 instrumental lines respectively. The figures show 

the distribution of the drift factor of each spectrum as a function of tangent 

height. The values for 6 have a scatter of about 5~10-~ and about 8~10’~ 

for the COS and HpO lines respectively. There does not appear to be any 

‘trend’ to the data and no apparent calibration drift. Figures 2.1.5-6 show 

the respective uncertainties ~(6~). The uncertainties are of the order as the 

scatter of the 6. This indicates that the scatter of the drift factors 6 are due 

to noise in the data and are not caused by any real drift. Tables 2.1.1-2 list 

the average over all spectra k and the standard error ~(6) for CO1 and 

H2O lines respectively for the five occultations. From these results there is 

no evidence of a measureable drift of the kequency calibration during an 

occultation since the average values of 6 found are of the same order as 
76 

I 

a 

the measurement uncertainty in all of the spectra k. This indicates that 

the calibration does not "drift" between spectra during an occultation. The 

fractional change of a line position predicted in Table 1.11.3 is of the same 

order as these fluctuations. As an example, a line with a peak absorptance 

of 40 % at a frequency of 2330 cm" can be predicted to have a fractional 

measurement error of 1.33 x 10-'/2330 = 5 x lo-'. This corresponds to wind 

speed errors of about 15 ms-' per line. 

GLOBAL CALIBRATION OFFSET a 

It is useful to compare the instrumental line positions to a set of accepted 

standards in order to estimate a for each occultation. Assume that such a 

set of standards ij exist. Sets of COZ and HoO standards recommended by 

Brown (1986) were used and are listed in Tables 2.1.3-4 respectively. From 

(2.1) we can write 

a = cj"/$ - 1 + error. (2.6) 

For each line j in each spectrum k a value 

is computed. The average value over all lines 

ak = 

was determined for each spectrum and is displayed in Figures 2.1.7-11 for all 

five occultations using COZ and HZO instrumental lines. The data show a 

variation from spectrum to spectrum of about 5 x lo-'. Notice that there 
77 

does seem to be a significant offset on the average between the values of a 

computed for the COZ and Ha0 lines. Figures 2.1.12-13 show b(ak) for the 

COZ and H20 lines respectively. The uncertainties for the COO lines are 

about 2-3~10-~ which is of the order of the average fluctuations of ab. For 

Hz0 the values for are about 4x10" and they are significantly larger 

than the COa values but are also consistent with the average fluctuations of 

ak. This information indicates that the fluctuations of measured values of 

ak are due noise in the spectra rather than calibration fluctuations. The null 

result above for the calibration drift factor 6 also confirms this. 

Since ak was found to vary randomly due to instrumental noise we determined the mean value for a in each occultation 

a = 

The mean values a and mean standard error a(a) are given in Table 2.1.5-6 

for the five occultations as computed from the u3 COZ and u2 HZ0 bands 

respectively. Table 2.1.7 lists the differences between the two sets and it is 

evident, since the offset is larger than the mean errors, that the difference 

in their offsets Aa = ace, - aHIO are statistically significant. Figure 2.1.14 

shows the differences Aak for all occultations. The general trend of the 

difference of the offsets Aak appears to be between about 3-5~10-~. The 

average difference between the instrumental global offsets a is +1.38&0.41 x 

This indicates that the discrepancy between the positions determined 

from the instrumental lines and the standards is significant. Since, according 

to Brown (1986), the C02 standards are "good" (accurate to lo-' cm-l) the 

HZ0 standard frequencies may be too low by this factor. This also suggests 
'R 

'I 

1. 

:I 

I 

I 

1 

18 

1 

78 

that our assumption that the "instrumental" lines are at rest with respect to 

the interferometer may not be correct. This question of whether or not the 

"instrumental" lines are at rest is examined in $11.10. 

Comparing the differences in Aa of the instrumental lines to the standard 

positions is like 'comparing apples and oranges'. Either set of standards may 

be 'correct' but there is no way of determining it from the data. However, 

we do have insight into the magnitudes of the discrepancies. It is shown 

in $11.5 that any ubsolute calibration offsets do not affect the accuracy in 

which Doppler shifts can be determined, provided they are consistent from 

spectrum to spectrum and that a set of instrumental 'absolute' standards are 

available, such as the instrumental COa and H10 lines. 

The systematic relative velocity Av shift caused by the global offset a 

for the two sets of instrumental lines is listed in Tables 2.1.5-6. The velocity 

shift is given by Av = a c where c = 299 792 458 ms" is the speed of 

light. The mean errors in the relative velocity shift a(Av) = ca(a) are 

between 1 and 3 ms-'. The mean values Av are an order of magnitude or 

more larger than these uncertainties and therefore there were statistically 

significant calibration offsets between occultations. The sunset occultation 

SSO9 was found to have a particularly large offset of about 80 ms-l compared 

to the other occultations. These offsets were probably introduced during 

the processing of the spectra and are artificial in origin [Norton, 19861 - 

specifically, the calibration scales were multiplied by certain factors during 

testing of the spectra. It is evident that the wavenumber calibration of 

the spectra need corrections, to agree with calibration standards as well as 

consistency between occultations, for wind speeds to be measured. When 
‘I 

I 

I 

:I 

1 

,I 

8 

‘I 

,I 

t 

79 

instrumental lines are present in the spectra this calibration can be achieved 

since the instrumental lines mark the rest frame positions. 

The frequency scales of all subsequent spectra were shifted by the correction factor 1 - Q to compensate for these offsets for each occultation. The 

positions of the instrumental u3 CO2 lines were thus forced to agree with the 

standards. We can say that the spectra were calibrated with respect to the 

CO2 standards. The CO2 correction factors used are given in Table 2.1.5. As 

discussed above, the absolute calibration of the instrument is only as good 

as the u3 line standards themselves. We accepted these standards as the 

‘best adable’ and calibrated the instrument relative to these. However, it 

is shown in 511.4 that the accuracy of the relative speeds is independent of 

the set of standards used provided that instrumental lines are available in 

the spectra for cslibration. 

After these correction factors were applied to the frequency calibration 

it was then possible to make new estimates of the CO2 and HZO standards 

from the instrumental lines. All of the instrumental lines from n = 125 

spectra from the five occultations were used. Tables 2.1.8-9 list the new 

positions from the ATMOS instrumental lines as compared to Brown’s values, 

A$ = Cmt - mrorn for both regions. After calibration, ~H~O was determined 

and the instrumental positions of the H20 lines differ from Brown’s standards 

by a factor a = -1.25 f 0.36 x which agrees with the estimate given 

previously. This can be seen by the multitude of minus signs in the differences 

column in Table 2.1.9. The average global offset for the COa lines vanishes 

as required. 

Also listed in Tables 2.1.8-9 are the precisions of the new positions. It 
80 

is worth noting that Brown's standards are known to lo-' cm-' accuracy 

while the precisions of the ATMOS values are almost an order of magnitude 

smaller than this. This suggests that there is an improvement of a factor of 

almost five in the line position precisions, particularly in the vs CO1 lines. 

However, most of the differences between the instrumental lines and Brown's 

standards are within lo" cm-l. Hence, this agreement suggests that the 

ATMOS interferometer can be used for determining precise relative positions 

for other lines based on Brown's va CO1 values. 

From this analysis we found that the instrumental frequency calibration 

remains stable during the course of each occultation. Provided that scale 

errors do not exist, and we show in the next section that this appears to be 

true, we can rely on the frequency calibration of the interferometer to measure Doppler shifts and positions of lines relative to the overall calibration 

of the instrument. Measurement of the fluctuations in ak and Sk also hint 

that the measurement precision theory may be reliable and a more quantitative treatment is given in 511.11. After calibrating the spectra is was found 

that the measured positions of the instrumental COa and H10 lines were 

significantly improved over the precisions of the standards. Typical precision 

of 2-3~10-~ cm-' were found. This indicates that the differences between 

the values determined from ATMOS and the standard positions provided by 

Brown are statistically significaat. 

The absolute error in estimating Doppler shifts can now be estimated 

since instrumental lines were available in the spectra for calibration. The 

rest frame positions i of these lines were determined from the spectra to 

precisions of about 3 x cm". For 20 of these lines and using an average 

c- a 
81 

frequency of about 2350 cm-' the absolute uncertainty in relative velocity v 

is determined by 

(2.10) 3 x 10-5 c < 1 ma-'. 2350 absolute uncertainty = 

The instrumental lines allow us to calibrate the instrument to obtain absolute 

Doppler shifts to < 1 ms'l provided the number of spectra n and the number 

of lines N are large enough so that the factor a is large enough to reduce 

the mean uncertainties. 

1 

1 
o, ...................................................... , ....................................... ......, ......... ........, ................ 82 0. 

-. Ir %a,, 

ai 8i 

231 o .m llMI0 as 

._ - ................ L ................. L ................. : ................. 1 ................. 1 ..-...-......- --..- 

Figure 2.1.1 Instrumental u3 COa abrorption lines and 0 CO u = 0 + 1 band 

head observed in solar spectra. The rg COa instrumental absorption lines 

are used for calibration. The 0 CO lines tend to interfere with atmospheric 

and instrumental lines causing unwanted line distortions. 

ai 8; 

0' ................ 1 ................. L ................. i ................. 1 ................. 1 ..............._ i ...... 

Figure 2.1.2 Instrumental va Ha0 absorption lines intermixed 0 CO lines 

observed in solar spectra. The stronger lines as well aa some weaker lines 

correspond to the instrumental lines. The water molecule is not linear and 

its spectra does not display evenly spaced rotational levels like COa. Water 

lines must be carefully checked for possible interferences with 0 CO lines. 
I 

I 

I 

II 

I 

I 

b 

2 

(3 

0 

b 

I 

9 

I 

I I I I I I I 

I I I I I I I 

I I I I I I I 

I I I .I 1.1 I 

ti0.m 10.0 ;n.a lao ha 

TWlT YIWT IMI 

83 

Figure 2.1.3 The test for calibration drift a* using ys COZ instrumental lines 

for the five occultations. The data points appear to be randomly scattered 

about the origin with a variance of about 5 x lo-* and this is due to measurement error associated with locating line podtions. 

Figure 2.1.4 The test for alibration drift 6' using va H10 instrumental lines for 

the five occultations. The data points appear to be randomly scattered about 

the origin with a variance of about 8 x lo-" and this is due to measurement 

error associated with locating line positions. 
84 

r - - - -I- - - - '1"" 1---- I I I I I I 

I I I I I 

Figure 2.1.5 Measurement errors ~(6~) of the 6k values in Fig. 2.1.3 using y 

COS instrumental linea for the five occultations. The scatter per measurement is of the same order of magnitude as the scatter of dues in 2.1.3. The 

scatter of points in 2.1.3 are due to random errosr in the line positions. 

I I I I 1 I I: I I I I I 

Figure 2.1.6 Measurement errors ~(6~) of the Jk values in Fig. 2.1.4 using 

HZ0 instrumental lines for the five occultations. The scatter per measurement is of the same order of magnitude as the ecatter of dues in 2.1.4. The 

scatter of points in 2.1.3 are due to random errors in the line positions. 
I 

I 

85 

1 

I 

I 

1 

I 

I 

I 

I 

I 

1 

I 

,I 

Table 2.1.1 Check for instrumental drift for the five occultations obtained 

from u3 COz lines. 

Occultation 6( x lo-') cr(6)( x lo-') Ref. Spectrum 

SR02 -0.72 1.96 A50430113052 

SS06 -0.62 2.15 A50430230523 

sso9 -0.88 2.30 A50501172502 

SSll +0.58 3.01 A50501202822 

SS13 -0.29 3.11 A50501233132 

Table 2.1.2 Check for instrumental drift for the five occultations obtained 

from uz HzO lines. 

Occultation 6( x lo-') u(6)( x lo-') Ref. Spectrum 

SR02 -4.70 3.55 A50430113052 

SS06 +0.80 3.95 A50430230523 

sso9 -1.02 4.12 A50501 172502 

SSll +1.20 3.57 A50501202822 

SS13 +1.14 3.94 A50501233132 
I 

'8 

I 

'I 

I 

,I 

8 

'I 

,I 

I 

86 

Table 2.1.3 u3 C02 standards given by Brown (1986) used in primary instrumental calibration. 

No. Position (cm-') Rot 

1 2330.5571 P22 

2 2334.1563 P18 

3 2335.9140 P16 

4 2337.6580 P 14 

5 2339.3722 P12 

6 2341.0620 P 10 

7 2342.72 72 P8 

8 2345.9841 P4 

9 2354.4333 R6 

10 2355.8894 R8 

11 2357.3207 R10 

12 2358.7272 R12 

13 2360.1089 R14 

14 2361.4659 R16 

15 2362.7980 R18 

No. Position (cm-') Rot 

16 2364.1053 R20 

17 

18 

19 

20 

21 

22 

23 

24 

25 

26 

27 

28 

29 

2365.3876 

2366.6451 

2369.0853 

2370.2679 

2371.4255 

2372.5581 

2373.6656 

2374.748 1 

2375.8055 

2376.8377 

2377.8448 

2378.8268 

2379.7836 

R22 

R24 

R28 

R30 

R32 

R34 

R36 

R38 

R40 

R42 

R44 

R46 

R48 

Table 2.1.4 u2 H2O standards given by Brown (1986) used in primary 

instrumental calibration. 

No. Position (cm-') Vib-Rot 

1 1616.7116 110 101 

2 1627.8275 202 111 

3 1647.4041 422 413 

4 1669.1683 413 322 

5 1669.3929 313 202 

6 1688.3785 725 716 

7 1704.4534 432 423 

8 1714.0337 625 616 

9 1715.1551 606 515 

10 1718.6117 322 211 

11 1730.0550 615 524 

No. Position (cm-') Vib-Rot 

12 1756.8188 716 625 

13 1771.2875 331 220 

14 1775.634 1 726 615 

15 1790.9518 827 716 

16 1799.6156 431 322 

17 1802.4797 918 827 

18 1825.2016 634 523 

19 1844.1806 441 330 

20 1867.8527 542 431 

21 1889.5694 643 532 

22 1942.516 1 652 541 
87 

Table 2.1.5 Estimated global offsets a for the five occultations from u3 COa 

lines. Lines 1-20 were used for this analysis. 

Occultation a( x lo-') 8( a)( x lo-') Au (m/s) 8( Au) (m/s) 

SR02 +2.66 0.39 +7.97 1.17 

SS06 -0.55 0.48 -1.65 1.42 

sso9 -27.30 0.43 -81.84 1.29 

SSll -7.13 0.63 -21.38 1.89 

SS13 -2.96 0.52 -8.87 1.56 

Table 2.1.6 Estimated global offsets a for the five occultations from ua Ha0 

lines. 

Occultation a( xlO-') a(a)( xlO-') Av (m/s) b(Av) (m/s) 

SR02 -0.53 0.74 -1.59 2.22 

SS06 -2.88 0.88 -8.63 2.64 

sso9 -26.32 0.78 -78.91 2.34 

SSll -8.56 0.77 -25.66 2.31 

SS13 -3.87 0.67 -11.60 2.01 

Table 2.1.7 Differences between global offsets a for the five occultations. 

Occultation Aa( x lo-') a(Aa)( x lo-*)- 

SR02 +3.19 0.84 

SS06 +2.33 1.00 

sso9 -0.98 0.89 

SSll +1.43 0.99 

SS13 +0.91 0.85 
I 

t 

1 

I 

I 

I 

I 

I 

1 

I 

8 

I 

'I 

1 

I 

8 

'I 

,8 

,s 

88 

Table 2.1.8 y COZ standards given by Brown (1986) as compared to observed instrumental positions after primary calibration for II = 125 spectra. 

All frequencies in cm-' where applicable. 

No. c~rorn %ut A~ (xio-4) a(Cmt) (~10-4) 

1 2330.5571 2330.55709 -0.1 0.20 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

2334.1563 

2335.9140 

2337.6580 

2339.3722 

2341.0620 

2342.7272 

2345.9841 

2354.4333 

2355.8894 

2357.3207 

2358.7272 

2360.1089 

2361.4659 

2362.7980 

2364.1053 

2365.3876 

2366.6451 

2369.0853 

2370.2679 

2371.4255 

2334.15633 

2335.91938 

2337.65799 

2339.37219 

2341.06197 

2342.72719 

2345.98410 

2354.43342 

2355.88941 

2357.32067 

2358.72723 

2360.10892 

2361.46586 

2362.79803 

2364.10525 

2365.38761 

2366.64511 

2369.08528 

23 70.26 79 1 

2371.42548 

+0.3 

-0.2 

-0.1 

-0.3 

-0.3 

-0.1 

0.0 

+1.2 

+1.0 

-0.3 

+0.3 

+0.2 

-0.4 

+0.3 

-0.5 

+0.1 

+0.1 

-0.2 

+0.1 

-0.2 

0.19 

0.18 

0.20 

0.21 

0.21 

0.24 

0.34 

0.26 

0.22 

0.22 

0.19 

0.18 

0.17 

0.19 

0.20 

0.20 

0.20 

0.25 

0.29 

0.31 
89 

Table 2.1.9 v2 H20 standards given by Brown (1986) as compared to observed instrumental positions after primary calibration for n = 125 spectra. 

All frequencies in cm'l where applicable. 

No. cl~rorn chat AC (~10-4) (~10-4) 

1 1616.7116 1616.71148 -1.2 0.20 

2 1627.8275 1627.82747 -0.3 0.26 

3 1647.4041 1647.40405 -0.5 0.24 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

1669.1683 

1669.3929 

1688.3785 

1704.4534 

1714.0337 

1715.1551 

1718.6117 

1730.0550 

1756.8 188 

1771.2875 

1775.6341 

1790.9518 

1799.6156 

1802.4797 

1825.2016 

1844.1806 

1867.8527 

1889.5694 

1942.5161 

1669.16834 

1669.39287 

1688.37845 

1704.45341 

1714.03360 

1715.15508 

1718.61159 

1730.05499 

1756.81878 

1771.28742 

1775.63409 

1790.95184 

1799.6 1557 

1802.47976 

1825.20154 

1844.18061 

1867.85279 

1889.56936 

1942.51614 

+0.4 

-0.3 

-0.5 

+0.1 

-1.0 

-0.2 

-1.1 

-0.1 

-0.2 

-0.8 

-0.1 

+0.4 

-0.3 

+0.6 

-0.6 

+0.1 

+0.9 

+0.4 

-0.4 

0.34 

0.21 

0.38 

0.21 

0.30 

0.21 

0.22 

0.34 

0.24 

0.25 

0.42 

0.31 

0.30 

0.44 

0.23 

0.18 

0.33 

0.27 

0.36 
n 

c 

I P e+ 

8 

I I I I t 

I I I I I 

90 

I I I I I I 

r--- 

I 

I 

I 

I 

I 

I 

I 

I 

L-,, 

.---- 

Figure 2.1.7 ark determined from COS (did line) and H10 (dotted line) 

instrumental lines for occultation SR02. The ak values for CO1 are generally 

greater than the Ha0 values. 

1 1 I I L 

I I I I bI I I I 

tn 

.. 

.. I jiI .. I Ii . I I I I 

I I I !I I I I 

I 22 1 I I I I I 

CW 

1 1w.00 I'ZO.~, ti0.w ib0.m ia1.00 LOOQ a0.m LO.W TRNGENT EIWT la) 

Figure 2.1.8 ark determined from COO (solid line) and €I10 (dotted line) 

instrumental lines for occultation SS06. The a* values for CO1 are generally 

greater than the Ha0 values. 
91 

I I I 1 L 

I I I I 

I I I I I 

c- h t 

Figure 2.1.9 ak determined from COa (solid line) and Ha0 (dotted line) 

instrumental lines for occultation SSO9. The ak values for COa are generally 

greater than the HZO values. 

I I I I I I 

I I I I I I 

SJ I I I I I I I 

rn 

I 1m.m 120.00 140.00 lw.m l(o.00 2b.m zil3.w z'co.oo 

TRNGENT t~m (mi 

Figure 2.1.10 ah determined from COa (solid line) and Ha0 (dotted line) 

instrumental.lines for occultation SS11. The ah values for COa are generally 

greater than the Ha0 values. 
92 

I 

I 

'1 

'1 

I 

I 

'I 

III 

i II I I :.: I I 1 

I I I I I I I 

TRNOENT l€LIMT (K)o 

I I 11, I I 

9- mr 100.00 1a.m Iro.00 IW.00 li0.W 2kl.al h.m i0.w 

Figure 2.1.11 ah determined from COS (did line) and HZO (dotted line) 

instrumental lines for occultation SR13. The d values for COa are generally 

greater than the HZ0 values. 

Figure 2.1.12 @(ak) from CO? instrumental lines for the five occultations. 

The mean uncertainties are typically about 2 x lo-" which is typically the 

magnitude of the fluctuations of the ah in Figs. 2.1.9-13 for COZ. 
I 

I 

I 

I 

I 

I 

I 

I 

I 

'I 

.I 

'I 

11 

I 

'I 

1 

D 

:I 

.I 

93 

I I I I 

Figure 2.1.13 *(ak) from HZO instrumental lines for the five occultations. 

The mean uncertainties are typically about 4 x lo-" which is typically the 

magnitude of the fluctuations of the ak in Figs. 2.1.9-13 for H10. 

TANGENT NIWT (KN) 

Figure 2.1.14 A(ak) between COS and HSO instrumental lines for the five 

occultations. The large fluctuations are due to measurement errors, however, the average of the differences tends to be larger than zero. The mean 

difference is +1.38 f 0.41 x for all occultations. 
I 

I 

1 

'I 

'I 

I 

94 

11.2 Scde Uncertainty +) 

In the previous section we have calibrated the spectra using about 30 u3 

instrumental lines of COZ. Global calibration errors are not the only source 

of errors in the measurement of line positions. The scale uncertainty Q(E) sets 

the ultimate limit to the precision by which wind speeds may be obtained. 

To estimate this scale uncertainty, consider (2.1) after correcting all spectra 

by the calibration factor 1 - a $11.1. We have 

= ij + e$ + 2. 

which can be rearranged 

2 -k = - cj - ej. 

(2.11) 

(2.12) 

Averaging this over all lines j in each spectrum k gives 

(2.13) 2 k = - ij)j - (ej)j. 

For a sufficiently large sample size we expect that measurement errors due 

to noise vanish 

(e$)j -+ 0. (2.14) 

If we impose this condition then (2.12) becomes 

The COZ and HZ0 line positions determined from the ATMOS spectra 

in Tables 2.1.8-9 were used for the standard positions <j for both regions. In 
I 

I 

E 

u 

8 

1 

C 

I 

1 

95 

Figures 2.2.1-2 values for 8 from 125 spectra are plotted for both spectral 

regions respectively. The estimated values of the scale errors 8 appear to 

scatter between 5 and ~OX~O-~ cm". Also plotted is the mean uncertainties in measuring the scale error a(@). We see that the error in the # is 

approximately the same as their scatter between spectra. Values of the scale 

error can be determined by defining the average of the scale errors for each 

occultation 

€E (8)& (2.16) 

for each occultation. Tables 2.2.1-2 list e and a(€) for the C02 and Ha0 

lines respectively for each of the five occultations. It is evident that le1 5 

a(€) in both regions. This indicates that the scale uncertainty is less than 

5 x lO-'cm-'. It should also be noted that a(€) N a(#). This indicates 

that the measured values 8 may in fact be due to noise instead of being 

due to a true scale uncertainty. More precisely stated, the approximation 

(e!). may not vanish in each spectrum but may have a random fluctuation 

due to the finite sample size of about 20 lines. To test this we examined the 

correlations between the estimates for 8 in each spectrum k for the H20 and 

COa regions. The correlations 

3 

Ck = 4,040, (2.17) 

were determined in each spectrum and displayed in Figure 2.2.3 for all the 

spectra 1. The correlations appear to be evenly distributed about zero. This 

suggests that there a correlation does not exist. The total correlation 

(2.18) 
96 

'b 

1 

'I 

.c 

1 

8 

was then determined and values for C and a(C) are listed in Table 2.2.3. 

The correlation overall correlation C is of 0(3 x lo-'") cm" and and the 

mean uncertainties a(C) is also 0(3 x 10"") cm-'. The upper limit of 

the correlations is therefore determined by the uncertainties b(C). Since the 

correlation is determined from the positions of two lines, the upper limit of the 

scale uncertainties is approximately determined by S(C) N 28'(~). Using the 

values of a(C) in Table 2.2.3 this gives an upper limit of the scale fluctuations 

of about 1.3 x cm-'. This means that the frequencies obtained by the 

interferometer are good to at least cm-'. The results of the correlation 

analysis do not show any significant evidence of a calibration uncertainty. 

Since the correlations effectively vanish the values of h plotted in Figures 

2.2.1-2 are due to measurement errors ef and not a true scale uncertainty. 

As previously stated, due to the number of lines sampled, N = 20, there may 

still be significant fluctuations in the average of the measurement errors of 

all the line positions obtained in spectrum k 

ek = (ej)j k 

of the order 

o(ek) = N-'/'a(e). 

(2.19) 

(2.20) 

Using typical values for a(e) listed in Table 1.12.3, for example 2.65 x 

lO-'crn-' for peak absorptances of about 20% and using N = 20 we get 

an estimation for cr(ek) = 5.9 x lO"cm-'. This agrees with the values estimated for a($) in Tables 2.2.1-2. 

This suggests there is no evidence for the presence of a scale uncertainty. 

The estimate of the scale uncertainty in $1.1 predicted by the magnitude 
'I 

'I 

97 

of the phase errors is several orders of magnitude smaller that the estimates 

a(&). From here on it is assumed that e" G 0 and that all fluctuations are due 

to measurement uncertainty. This result simplifies the analysis significantly. 

The scale uncertainty appears to be negligible and the relative accuracy of 

the wind speed measurements is not limited by this effect. 
98 

I 1. *I . ' I I I 

, 8 I I I I 

I I I I I I 

I I I I I I I 

I I I 1 I I I 

I I I I I 1 I 

I I I I 

Figure 2.2.1 @ for the five occultations determined from C02 lines (points) 

and a(@) (lines). The scatter of the data about zero is of the same order 

of magnitude of the mean uncertainties. This indicates that the scatter in 

points is probably due to noise instead of a true scale fluctuation. 
I 

IC 

I 

' 

1 

8 

E 

,I 

,I 

t, 

99 

Figure 2.2.3 Correlation Ck of the C02 and Ha0 8 determined from all 

five occultations. The scatter is within 0.5~10'~ cm'l. The scatter of this 

correlation about zero means that the scale uncertainty m)is less than 2 x 

cm-'. 
I 

s 

t 

I 

1E 

1 

I 

E 

I 

Table 2.2.1 e determined from u3 COZ lines. 

Occultation e( x 10-6cm-’) ij(~)( x 10-6cm-1) 

SR02 -0.083 4.533 

SS06 -0.114 4.920 

sso9 -0.090 5.269 

SSll -0.095 6.767 

SS13 -0.094 7.184 

Table 2.2.2 e from ul HZ0 lines. 

Occultation e( x 10-6cm”) B(E)( x 10-6cm-1) 

SR02 -3.356 6.521 

SS06 -1.766 6.820 

sso9 +3.964 7.096 

SSll -0.258 6.125 

SS13 +0.599 6.721 

Table 2.2.3 Correlations of 8 between HZO and COZ regions. 

Occultation C( x 10-lOcm”) B( C)( x 10-lOcm”) 

SR02 +0.838 6.507 

SS06 -2.168 6.074 

sso9 -12.62 7.267 

SSll +7.728 5.809 

SS13 -6.488 8.475 

all -3.362 3.346 

100 
101 

11.3 Solar and Ratioed Spectra 

Figure 2.3.1 shows a condensed full Filter-3 solar spectrum collected at 

tangent heights above 120 km where there is not any significant atmospheric 

absorption. The spectral features are condensed due to the large range of 

wavenumbers 1600 < c < 3200 cm-’ shown in a small space on the page. 

Many spectral features are evident, particularly the solar ‘0’ CO fundamental 

(Av = 1)’ - e.g. u = 0 -+ u = 1, u = 1 + u = 2, etc. - rotationvibration bands with wavenumbers less than 2330 cm-’. Figure 2.3.2 shows 

a ‘magnification’ of the 0 CO band heads near 2300 cm-’ along with the 

v3 Cop instrumental lines. Instrumental water lines protrude below 2000 

cm-’. The lines in the spectrum above the u3 COa region c > 2380 cm” 

are other solar lines. The 0 CO bands can distort other spectral features 

which may be of interest. The instrumental COa lines cannot be used for 

cdbration in the region where the u3 COa lines overlap with the 0 CO lines 

since interferences distort the line shapes. Distortions can cause systematic 

errors in determining their positions c. The lines used in this report were 

carefully selected to avoid these problems. 

The spectral regions where the absorption bands used to obtain wind 

speeds in the atmosphere are located in Figure 2.3.1. All of the lines in 

the solar spectra appear to be superimposed on a ‘background signal’ S. 

The background signal appears to vary with wavenumber c. This happens 

because the detector sensitivity and the filter transmittance vary with the 

wavenumber (Shaw, 19891. The background signal as well as the instrumental 
1E 

102 

‘I and solar lines superimposed on it serve aa the input radiation intensity I,(O) 

described in $1.9 (1.144). The solar features are present in all of the ATMOS 

spectra, with or without atmospheric absorption. 

The positions and shapes of the solar features and instrumental lines remained essentially unchanged during each occultation. It was desirable to 

‘factor out’ these lines since they distort atmospheric features. The solar features serve as the background I,(O) in (1.144) for the atmospheric absorption 

exp[-k,u] as described in $1.9. The signal I,(O) was estimated by averaging 

all of the solar spectra taken above the atmosphere for each occultation [Norton, 19861. This produced solar spectra I,(O) with a very high SNR - 500:l 

[Norton, 19861. The atmospheric absorption was then estimated by taking 

the ratio of the spectra containing atmospheric features I, to the solar average I,(O). These were termed ‘ratioed’ spectra and from (1.144) they are 

defined to be t 

(2.21) 

The ratioed spectra made by Norton (1986) were used for the majority of 

the atmospheric studies conducted by the ATMOS team. 
q.""'."........ v .................................... 1' ............................................ .'...... ~ ......"...."..,... .............. 0. 

103 

0 -.... ............ ................. L ................................... 1 ................. 1 ................... ._.-._-_.. :" ....._..... 'ma -la PQ, 

Figure 2.3.1 A complete Filter-3 solar spectrum. Instrumental lines of COS 

and H90 are superimposed on background solar spectra. Solar CO lines are 

present below - 2300 cm" and the variable output of the detector is evident. 

.. .................................... Or""'-'""" 1' ......................................................................... I"" ............ @ co 9: 

-. 

.............. ..L .............. I ............................ .I ............... 1 .................. ! ..................................... 

2220 .m im~ro 2380.0 

Figure 2.3.2 Av = 1 0 CO band heads and instrumental u3 CO, lines observed in the solar spectra. This covers a slighly large wavenumber range 

than Figure 2.1.1. 
104 

11.4 Estimating the Noise Level 

The interferograms included noise in the instrumental signals caused by 

fluctuations of the light source, photon noise caused by random arrival of 

photons from the source, thermal fluctuations in the detector, and external 

vibrations of the instrument [Norton, 1986; Thorne, 19881. The noise in the 

interferograms I(z) is transferred into noise in the spectra I'(c) through the 

transform (1.9). In $1.11 it was shown how this noise affects the precision by 

which the positions of the absorption lines can be measured. 

The ratioing process does not affect the signal to noise ratio SNR, of 

the spectra significantly. In the non-ratioed spectra the noise h/, in the 

signal I,(O) has random fluctuations a(h/,). The ratioing process produces 

a constant background signal of unity by definition (2.20). The noise N, in 

the ratioed spectra R, then has fluctuations 

(2.22) 

In regions where solar absorption features are not present the background 

signal S, and solar signal I,(O) are the same. The SNR of the ratioed spectra is dependent on the background signal in the denominator of (2.21). In 

regions where solar absorption features are strong the denominator of (2.21) 

may become large and the noise increases. To get the best results in measuring line positions in the ratioed spectra R, it is desirable to avoid spectral 

regions where the noise level caused by solar features is high. An estimate of 

the SNR, of the ratioed spectra & can be determined by examining ratioed 
I 

105 

spectra above tangent heights where atmospheric absorption is significant. 

Figure 2.4.1 shows one such spectra. We see that between 1600 and 2000 

cm-', for example, the SNR is lower than near 2400 cm". By visually comparing the wavenumber variations in fluctuations of the ratioed spectrum in 

Figure 2.4.1 to the wavenumber variations of the background signal in the 

unratioed solar spectrum in Figure 2.3.1 the correlation between the two is 

evident. This is predicted by (2.21). 

Since scale errors 8 are negligible, measurement uncertainties of the line 

-j in the spectrum k can be determined from (2.10) 

e; = c; - 6. 

By using these values, from the theory 81.11 

(2.23) 

(2.24) 

where c(N) = l/SNR, is the noise to signal ratio and is described by (2.21) 

in the ratioed spectra &. By measuring the positions of instrumental lines 

of HZ0 and CO1 in the unratioed spectra we can estimate the signal to noise 

ratios for both regions in the ratioed spectra by using (2.23). We expect the 

relation (2.21) to hold true. From examination of Figure 2.3.1 the SNR of 

the H10 region should be almost a factor of two smaller than the COS region. 

The Line noise will be defined here to be fnc(N)w, fn being the form 

factor, u(N) = l/SNR the signal fluctuations and w the geometric width 

of the line as described in $1.11. For a given spectral region the signal fluctuations u(N) are approximately a constant and since fn and w are the 

same for similar lines, the line noise defined in (2.23) is approximately a constant. The line noise over all lines -j in each region were determined using 

hk 

Apjc(ej) c((%jej)h) --L fnu(N)w, 
106 

(2.23) and are listed in Tables 2.4.1-2 for each occultation individually and 

all collectively for the CO1 and H10 instrumental lines respectively. The 

line noise for the COa region is almost a factor of two larger than that the 

H10 regions in agreement with (2.21). For the CO1 instrumental lines the 

line noise of about 0.47~10-~ cm" is in close agreement with the estimate 

(2.23) predicted with rough estimates of the line parameters fn = 0.35355, 

c(N) = 0.01, and W = 0.015 cm-' of 0.54 x 10'' cm". It appears that the 

measurement uncertainty theory can be used to describe the precision of the 

line-finder program quite accurately. 
o,. . ... . . . . . ... . . . . .,.. . . . . . . . . . . . . . . . .. .. . . . . . . . . .. . . . .. ,..... . . . . . . . . . . . . . . . . . . . . . . . . ... .. .. . . . ....... .......,..... . , .. __... ..,......... ., .... .. 0 

107 

s 

'I 

I 

I 

I1 

I 

I 

. . .-. . . 

r. 

8; 

9: 

1wo.m UFMH) 32oo.a 

0: 

.__....___._....__ L _................ L ...,. .. l..... ._......., 1 .................. ! .................. ! .................. 

Figure 2.4.1 The noise level in the spectra after the ratioing process. This 

spectrum was taken at a tangent height where atmospheric absorption is 

absent. This represents the background noise level of the spectra. Note the 

variations in the spread of the noise with wavenumber. 
Table 2.4.1 Estimates of noise level from CO1 instrumental lines. 

Occultation mean ( x 10-~~~--1) 

SR02 4.94 

SS06 4.55 

sso9 4.42 

SSll 4.67 

SS13 4.79 

d 4.67 

Table 2.4.2 Estimates of noise level from Ha0 instrumental lines. 

Occultation mean (x iO-6cm-i) 

SR02 8.30 

SS06 8.01 

sso9 7.79 

SSll 8.10 

SS13 8.09 

all 8.10 

108 
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11.5 Estimating Relative Speeds Along the Line of 

Sight 

The method of estimating individual line positions and the calibration 

stability of better than 2 x cm-' determined in $11.3 imply that wind 

speeds can be obtained to a precision of 5 ms-' or better. It was also pointed 

out by Shaw (1985) that these kinds of precisions could be achieved using 

Filter 3 spectra provided that the SNR was large enough. From the data in 

Table 1.12.3 we can estimate the sample size needed to attain the precision 

of 5 ms-' or better. From (1.186) the uncertainty per line j of the velocity 

is given by 

C 

u(uj) = u( cj) (2.25) 

From the measurement theory (1.180) the uncertainty is proportional to A;'. 

For a line with a peak absorptance Ap li 20% and a position of tj N 2350cm" 

we see that ~(vj) li 35 ms-'. Using N lines the precision of the relative 

velocity is then 35 ms-l/fi so we need about 50 lines to attain 5 ms-'. 

However, for stronger lines, using a peak absorptance 4 N 70% for example, 

this sample size can be considerably smaller. By using a value of u(cj) N 

0.76 x lO-*cm-' we find that c(uj) N 10 ms-' which requires a sample 

of size of only 4 lines to attain the desired precision. In practice the peak 

absorptances of the lines used are primarily in the 20-70% range and a sample 

size between 20 and 30 lines is usually available. Therefore we can predict 

that relative speeds can be determined to at least 5 ms" and most likely 

ci 
110 

'I 

better using single bands of atmospheric lines. 

The accuracy by which line Doppler shifts can be determined is independent of any wavenumber dependent calibration errors, provided that line 

positions in the rest frame can be measured by the instrument. To see this, 

the wavenumbers measured by the instrument cmt can in principal be shifted 

from the "true" wavenumber ct- by a small amount and this offset can be 

represented by 

cmt = (1 + a) ctme (2.26) 

where a is may be regarded as constant over small spectral regions. Note 

that this a is not the same factor defined in $11.1. If the "true" rest frame 

position is given by it, then the corresponding position as measured by the 

instrument will be at 6-t = (1 + a) &me. For Doppler shifted lines then 

cit = (1 + +me = (1 +a) (1 + ;) hm 

(2.27) 

which is independent of the shift a. This is because the form l+a is the same 

for both the rest frame and moving frame positions. This result indicates 

that accurate relative speed measurements can be made provided that the 

rest frame positions +jt are known and that the calibration of the ATMOS 

interferometer remains stable throughout the course of observation. 

There are typically about 30 lines of the u3 CO2 band which are visible in 

the spectra and can used to measure Doppler shifts. Relative speeds can be 

obtained to 5 ms-' with these lines provided that the lines are not too weak 

or saturated. From (2.24) we see that the uncertainty of measuring Doppler 

shifts is wavenumber dependent. As an example, for the y COO band neax 
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667 cm-l, many more lines are needed. For a wavenumber of about 1/4 of 

the u3 lines the uncertainty u(vj) will be about 4 times as large. We would 

need 16 times as many lines. From a practical point of view, relative speed 

measurements should be confined to the higher wavenumber regions of the 

spectra. 

11.6 Relative Speed Measurements Using v3 CO2 

Lines Between 2330 and 2372 cm-l 

The ATMOS instrument was found to have a stable frequency calibration as well as an excellent SNR of about 1OO:l. This enables precise relative 

speeds to be obtained from a small number of lines in the mid-infrared regions [Shaw, 19851. Below about 120 km atmospheric absorption lines of the 

u3 COZ absorption band become visible in the spectra between 2330 and 2370 

cm”. Figure 2.6.1 shows several of these lines along with their instrumental 

counterparts taken from an unratioed spectrum for illustration. These atmospheric lines were used to measure relative speeds from tangent heights of 

about 120 km to about 80 km where overlapping with other CO1 bands and 

line saturations occur. A typical overlap of CO1 lines from different bands 

taken in successive spectra is shown in Figure 2.6.2. As the tangent height 

decreased the lines begin to interfere with each other. The spectra are displaced vertically for clarity. The u3 lines used in this study were chosen to 
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112 

be those lines which displayed minimal interferences. These were considered 

to be "good" lines. Another criterion used was that the peak absorptances 

range from 10-70% because resolved line weaker than these became distorted 

by the background noise while stronger lines become saturated. This caused 

the curvature of the line near the line center to become flat which increased 

the measurement error (the geometrical or spectral width w of the line increases dramatically). 

With the above criteria for choosing absorption lines, the instrumental 

lines given in Table 2.1.8 were used as the standards ij in (2.26). The relative 

speed for v; for each spectrum k and line j was measured using (1.1) 

k 

v;= (?-I) c. 

From this the relative speed vk for each spectrum k was determined 

vk = (Vj")j 

(2.28) 

(2.29) 

and its corresponding mean deviation i+(vk) determined. This was performed 

for each of the occultations and the resulting relative speed profiles were 

determined. Figures 2.6.3-4 show the relative speed versus ray tangent height 

using the v3 Cot atmospheric lines for the sunrise and sunset occultations 

respectively. The relative speed profiles are typically of the order of 6000 

ms-'. This is due to the orbital motion of the satellite and and this motion 

is estimated and subtraced from the profiles to obtain zond'wind in the 

atmosphere around the tangent point 511.12. 

The relative speed profiles show a variability with tangent height. The 

'wavelike' structure of the relative speed profiles is immediately recognized, 
113 

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particularly in Figure 2.6.4 where the four sunset profiles demonstrate a 

remarkable similarity with regards to altitudes of peaks and valleys. Large 

shears, especially between 105 and 120 km aze apparent. The sunrise profile 

in Figure 2.6.3 demonstrates similar features. Figure 2.6.5 displays the mean 

error 8(vk) for all of these occultations. Note that b(vk) increases at large 

heights because the lines become very weak. At the lowest tangent heights 

the increase in b(vk) is due to overlapping of lines and saturation. The curves 

all follow the same general trend and, as proposed above, precisions of 5 ms” 

are indeed found. Between 80 and 110 km these lines achieved a maximum 

precision of -2.5 ms-l. Since the calibration of the instrument is stable 

these precisions indicate that the features observed are real. Also evident in 

Figure 2.6.4 axe relative differences of over 100 ms” between SS06 and the 

rest of the sunset profiles. The relative speed profiles for SSOS, SS11, and 

SS13 showed a trend of increasing relative speed with respect to the SS06 

profile as the mission progressed. This is due to the changing geometrical 

orbit of the instrument and its component of velocity along the line of sight. 
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a I 

1 t I 1 I 

2362 2363 - 236* 2x5 2366 

Im-7 

Figure 2.6.1 Pairs of instrumental and atmospheric lines. The atmospheric 

lines are red-shifted (towards the left) from the instrumental lines as the 

tangent height decreases kom 112 (a) to 90 km (c) in a sunset occultation. 

The spectra are displaced for clarity. Note the increasing peak absorptiona 

of the atmospheric lines as the tangent heights decreases. 

Figure 2.6.2 Overlapping of lines from ATMOS spectra. Notice how the line 

shapes become distorted by other spectral features as the tangent height 

decreases. Lines must be examined for interferences before including them 

in wind measurements. The spectra have been slightly displaced vertically 

for clazity. 
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Figure 2.6.3 Relative speed profile determined using u3 CO1 atmospheric 

lines for the sunrise occultation. There appear to be rather large shifts in 

the relative speeds near 80 and 105 km. 

Figure 2.6.4 Relative speed profile determined using y CO1 atmospheric 

lines for the sunset occultations. The four occultations (6) SS06 - (9) SSO9 

- (11) SSll - (13) SS13 are displaced due to changes in the orbital motion 

511.12. The appear to have similar relative speed shears near 80 and 105 km 

as well as ‘minima’ near 95 km. These are due to variations in the upper 

atmospheric wind speeds. 
116 

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a 

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%!on :dm i.a I.! ba A 

pREcIsIm L WYCI 

Figure 2.6.5 Measurement uncertainty a(d) for all occultations using COa 

atmospheric lines. A precision of better than 5 ms" is achieved between 80 

and 110 km. The increase outside of this region of high precision is due to the 

weakness of lines above 110 km and line saturation and interferences below 

80 km. 
117 

11.7 Extending Relative Speed Measurements and 

the Determination of knst 

We now proceed to extend these measurements to lower altitudes using absorption lines in which no instrumental counterparts are available for 

determining the rest frame positions. The profiles in Figures 2.6.3-4 were 

determined by directly comparing the instrumental COa lines to their atmospheric counterparts and thus directly measuring the Doppler shifts. In 

general) instrumental lines are not available for direct comparison at lower 

tangent heights. Fortunately there is a way of getting around this problem. 

We know that the calibration of the instrument is stable, we have determined 

relative speeds using the calibrated instrumental lines, and there are more 

lines entering into the spectra as the tangent height decreases. We must use 

these new atmospheric lines that are entering into the spectra, however we 

need to know tkt for each of these lines. 

Consider a set of new atmospheric lines which have entered the spectra. 

Their positions are Doppler shifted by (1.1). It should now be understood 

that we itre referring our wavenumbers to the calibration system of the instrument and are no longer referring to ‘absolute standards’ which were used 

to calibrate the vs COa region. As shown in the previous section (2.26) 

absolute standards are no longer needed to measure absolute relative wind 

speeds. Let us suppose that the relative speeds vC for a set of spectra k have 

been determined to a reasonable degree of precision (5 m/s or better). Then 

we can invert (2.26) to obtain an estimate of the rest frame positions of each 
118 

of the new lines which have appeared in the spectra for which vk had been 

estimated 

(2.30) 

from the observed atmospheric line position ~i for each new line j in each 

spectrum k. We can estimate the rest frame position from the average 

ij = <i:)b (2.31) 

as well as its mean error a(ij). Having determined these rest frame positions 

the new lines may be used to extend the wind below the current minimum 

tangent height. 

One problem with this technique as it is presently stated is that the line 

positions tj determined are only as precise as the relative speed profiles uk 

from which they were obtained. On the other hand, if the rest frame positions 

have a degrading precision then the relative speeds themselves will become 

less precise. An iterative scheme proved to be useful in dealing with this 

problem. Specifically, an initial set of relative speeds vk,(0) was determined 

from an initial set of rest frame positions Q") over an altitude range for all 

of the occultations. nom these initial relative speeds u'*(O) an updated List 

of line positions was determined 

Then an updated relative speed profile was determined 

(2.32) 

(2.33) 

(2.34) 

(2.35) 
119 

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This procedure was then repeated until a desired level of convergence in the 

mean uncertainties of the relative speeds b(vk) and the line positions a(ij) 

were both obtained. I chose value of 0.1 ms" and cm-' respectively. 

The relative speed profiles shown in Figures 2.6.3-4 were recomputed 

along with the rest frame positions of the atmospheric lines by using the 

above procedure for the lines specified in Table 2.1.8. The precisions in lines 

positions a(ij) were found to converge to less than cm" in usually 

three or four iterations with a simultaneous convergence of s(vk) less than 

0.1 ms-'. Figure 2.7.1 shows the precisions of the positions determined from 

measuring instrumental lines (bold line) and those positions computed from 

the atmospheric lines using the iteration procedure (points). The precisions 

determined using the atmospheric lines are more uniformly distributed with 

a value near 0.28 x lo'* cm" than the positions determined from the instrumental lines. The apparent degradation of the precision is actudy due to 

the smaller sample size of atmospheric spectra used (about 60) compared to 

the number of high altitude spectra used (about 125). However, the results 

do show that by using the velocity profiles to determine line positions, we can 

obtain satisfactory results, particularly in the case where only a few atmospheric lines are originally present for initial line position estimates. Figure 

2.7.2 shows the precisions obtained using instrumental lines (bold lines) and 

atmospheric lines (points). The slight improvement below 100 km is not 

significant. 

At wavenumbers between 2372 and 2380 cm" eight u3 COa lines become 

reasonably strong (Ap > 15%) at about 100 km and are useful for extending 

the relative speed profiles. Their positions 9"' were estimated from (2.27- 
120 

8) using the relative speed profiles in Figures 2.6.3-4 above 80 km. The 

line positions were then used to extend the relative speed profiles to lower 

tangent heights. The iteration procedure was used for the lines between 

about 70 and 100 km. Figure 2.7.3 shows the improvements in the lines 

precisions between the first estimate (solid line) and the converged result 

(points). An improvement of about 5 x 10-6cm-1, although small, is evident. 

The precisions in the relative speeds are shown in Figure 2.7.4 and there does 

not appear to be a significant difference before and after the iteration. Since 

the range of altitudes where these CO2 lines appear overlap the relative 

speeds in Figures 2.6.3-4 is large (30 km) the sample size of lines for the new 

rest positions is large enough to ensure fairly precise initial estimates ('I of 

about 3~lO-~ cm-'. This is not the case in general, however. 

The above results show that the initial relative speed profiles can be used 

to determine the unknown rest frame positions of atmospheric lines. In turn, 

these rest kame positions can then be used to determine relative speeds at 

extended tangent height ranges. Since the line positions and relative speeds 

are determined simultaneously, the precisions of the initial relative speed 

profiles are maintained. This indicates that by using the calibration stability 

of the instrument, relative speeds with precisions and accuracies similar to 

those determined by using the positions measured from the instrumental lines 

can be determined from other atmospheric lines. We used this procedure to 

extend relative speed profiles to 25 km by using N10, CO1 and H20 lines 

$11.9. 
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Figure 2.7.1 Precisions of v3 CO2 rest frame positions determined from atmospheric lines (6 < 2372 cm-') before (line) and after (points) iteration 

procedure. The improvement in line positiona ir alight. * 

I I 1 8 I I 

1 I I I I I 

Iw 

m c4D I$.m :- 2040 a40 ir Pa€,;% IH:YC, ' 

Figure 2.7.2 Precisions of relative speeds determined from y COa atmospheric lines (6 < 2372 cm-l) before (line) and after (points) iteration procedure. There is only a slight improvement in precisions from using the 

measured instrumental line positions. 
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122 

Figure 2.7.3 Precisions of u3 COZ ;est frame positions determined from atmospheric lines (; > 2372 cm-l) before (line) and after (pohts) iteration 

procedure. There are improvements of up to 1 x l'O-6 cm" in the line 

positions. L 

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Figure 2.7.4 Precisions of relative speeds determined from u3 CO-, atmospheric lines (t > 2372 cm-l) before (line) and after (points) iteration procedure. The improvement in the precisions is very small. Note that the 

precision of 5 ms-' has been maintained to 70 km usingthese lines. The 

precision degreades above 90 km due to the weaknesses of the lines. 
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11.8 Weighted Averages of Relative Speed Profiles 

We have obtained relative speed profiles from different sets of lines and we 

wish to combine the results to obtain the best profile. The speed profiles can 

be combined by calculating a weighted average. This takes into account the 

varying precisions of the relative speeds with height. Let us assume we have 

several profiles I for each spectrum k given by uf with a mean uncertainty 

a(vf). Then the weighted average is 

uk = cw:.: 

1 

where the weights w: are given by [Bevington, 19691 

The weighted uncertainty is then determined by [Bevington, 19691 

1 1 -- 

8Z("k) - bl(v:)' 

(2.36) 

(2.37) 

(2.38) 

Profiles combined in this method are called "extended" or "merged" relative speed profiles and are then used to determine initial estimates of line 

positions for further extension to lower tangent heights. Figures 2.8.1 and 

2.8.2 show the extended relative speeds uk (for sunsets) and their weighted 

precisions 8(vk) determined using this method. It can be seen in Figure 

2.8.2 that a 5 ms-' precision or better is maintained to about 70 km. In the 

region of overlap between the two sets of y COZ lines between 80 and 100 

km described in the previous two sections we have approximately 30 total 
124 

lines. The average peak absorptance is about 50% over this range. Using 

a noise level of fno(N) w = 5 x cm-' from Table 2.4.1 the predicted 

measurement uncertainty per line from (1.180) is 

5 x 1O-'cm" 

.SO uncertainty = = 10-~~~-'. (2.39) 

For a central frequency of about 2350 cm-' for 30 CO1 lines the predicted 

uncertainty in the relative speeds is 

10-4 

2350 a a(vk) = -- 21 2.5 ms-'. 

This is consistent with the observed precisions shown in $11.7. 

(2.40) 
q 1 1 

I 8. 

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-1 i 

_?I I 

125 

R 

Figure 2.8.1 First extended relative speed profiles vk for the four sunset occultations. The speed profiles using v3 COZ absorption lines which did not 

have instrumental counterparts have been combined with the sunset speed 

profiles in Figure 2.6.4. Notation is the same as in Figure 2.6.4. 

C 

7 

Figure 2.8.2 Precisions of first extended relative speed profiles 8(d) for ad 

occultations. The precision of 5 ms-l is maintained from 70 to 110 km. 
'I 

126 

11.9 Extensions to Lower Tangent Heights by Using 

H20 and N20 

In the spirit of the last two sections the profiles were extended by using 

three more bands. Between 40 and 80 km the ua Ha0 atmospheric counterparts to the Ha0 instrumental lines given in Table 2.1.9 were used for an 

extension. Between 30 and 65 km lines of the 2ua Ha0 band near 3150 cm" 

was used and finally between 20 and 40 km lines of the 2u1 Ng0 near 2550 

cm" was used to complete the profiles. 

The iterative technique was chosen to estimate the positions of the ua 

water lines instead of relying on the instrumental positions as was done for 

the u3 Cot lines for a reason which will made clear shortly. In fact, we 

would like to compare the positions retrieved for both sets of lines and this is 

done in the following section. The initial positions Cy) were determined from 

the spectra between 50 and 80 km and the relative speed profiles in Figure 

2.8.1. The precisions a(@') are shown by the bold line in Figure 2.9.1. The 

iterative scheme was then carried out and the precisions of the line positions 

are given by the points in Figure 2.9.1. There is an improvement in the 

mean uncertainties to better than 8 x cm-'. Figure 2.9.2 shows the 

initial (connected points) mean uncertainties for the estimated relative speeds 

between 40 and 80 km using these lines and the improved mean uncertainties 

(points). Between 45 and 60 km there is an improvement of up to 3 or 4 

ms-'. The resultant profiles were then merged with those given in Figure 

2.9.1 and these are shown in Figures 2.9.3. The weighted mean uncertainties 
127 

are shown in Figure 2.9.4. The mean uncertainty a(d) is 5 m/s or better 

between 50 and 110 km. The relative speeds appear to "flatten out" below 

75 km and do not appear to show the rapid variations with height which are 

noticeable above this level. 

The extension method at this point is fairly straightforward. The next set 

of lines which were found to be useful is the 2uZ Ha0 vibration band which 

is approximately at twice the frequency as its ua counterpart. However, the 

sample of nine lines chosen in the overlap regions between 30 and 65 km 

is much smaller than the v2 sample size of more than 20. This does not 

cause significant problems since the higher frequency of these lines helps the 

statistics (2.25). Figure 2.9.5 shows the initial estimates and final estimates 

in the line position precisions (line and points respectively). In every cwe 

there is a marked improvement. Figure 2.9.6 shows the initial and final mean 

uncertainties in the relative speeds and below 40 km there is an improvement 

of several ms-l for most occultations. Finally the resultant relative speed 

profiles were merged with Figure 2.9.3 and are shown in Figure 2.9.7. The 

mean uncertainties are shown in Figure 2.9.8. Precisions are now about 5 

m/s in most cases from about 35 to 110 km. 

The final set of lines which were found to be useful is the 214 NZ0 vibration band. There were 61 lines available in the overlap regions between 30 and 

40 km. Even though lines were relatively weak in these regions (A, E 10%) 

the large sample size improved the statistics. Figure 2.9.9 shows the initid 

estimates and final estimates in the line position precisions (line and points 

respectively). In every case there is again a significant improvement to less 

than lo-' cm-'. Figure 2.9.10 shows the initial and final mean uncertainties 
128 

in the relative speeds and below 30 km there is an improvement of several 

ms-l for most occultations. Finally the resultant relative speed profiles were 

merged with Figure 2.9.8 and are shown in Figure 2.9.11-12 for the sunsets 

and sunrise respectively. The mean uncertainties are shown in Figure 2.9.13. 

Precisions are now between 2 and 5 ms" in most cases from about 25 to 110 

km. 

From these results we see that relative speeds precisions of better than 5 

ms-' can be obtained by measuring rest frame positions from the atmospheric 

lines. The calibration stability of the instrument is adequate to maintain 

these precisions throughout the middle atmosphere. The accuracy of the 

profiles is determined by using the C02 instrumental lines to calibrate the 

wavenumber scale. The iteration method of extending the relative speeds to 

lower tangent heights and using new atmospheric bands as they appear in 

the spectra does not degrade the precisions of obtaining line positions and 

speeds. 
129 

8 

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I I I I 1 I I 

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Figure 2.9.1 Precisions of y Ha0 rest frame positions determined from atmospheric lines before (line) and after (points) iteration procedure. There is 

a noticable improvement in the line pwitions of up to lo-' cm". 

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.--- J-A 
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Figure 2.9.3 Second extended relative speed profiles uk for the sunset occultations. This is obtained by combining the speed profiles determined with 

the atmospheric u:, Ha0 lines with the profiles determined using the ys COa 

atmospheric lines. Rapid wind vaxiations do not appear to occur below 70 

km. The separation of the profilea due to changes in the orbital speed is 

evident. 9, 

8 I I 

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Figure 2.9.4 Precisions of second extended relative speed profiles 8(uh) described above for all occultations. A precision of 5 ms-' is maintained above 

50 km. 
131 

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Figure 2.9.5 Precisions of 2v2 HZO rest kame positions determined from 

atmospheric lines before (line) and after (points) iteration procedure. Precisions are improved from initial estimater by up to O.SxlO-' cm". 

I 

Figure 2.9.6 Precisions of relative speeds determined from 2u2 HS0 atmospheric lines before (lines) and after (points) iteration procedure. Precision 

of about 5 ms" are maintained betyween 35 and 50 km. Increases above 

that are due to the weaknesses of the lines. Relative speed precisions do not 

improve signifgicantly before and after the iteration procedure. 
132 

Figure 2.9.7 Third extended relative speed profiles uk for the sunset occultations. This is obtained by combining the speed profiles determined with 

the atmospheric 2va Ha0 lines with the profiles determined using the y CO2 

and va Ha0 atmospheric lines. 

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I I 1 I 

I 1 I 

I I I mr ma a I 

&iI,ra;'?WWe'P - 

Figure 2.9.8 Precisions of third extended relative speed profiles a(vk) for all 

occultations which corresponds to the speed profiles above. A precision of 5 

ms" is mainted between about 35 and 110 km. 
'8 

f 

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04 1 1 cvr I I 

M20.00 234.w 23w.w 25m-00 ma 

FREOUENCY fCH-1) 

Figure 2.9.9 Precisions of 2v1 NZ0 rest frame positions determined from 

atmospheric lines before (line) and after (points) iteration procedure. Xmprovements in precision of more than 1 x 10" cm" between the initial and 

final estimates is evident, particularly near the band origin near 2560 cm" 

and the high J lines. 

I I 1 %!OO ka, Ib.00 lhl io40 a% sa.00 I 3s 4 

PRECISION f WSECI 

Figure 2.9.10 Precisions of relative speeds determined from 2y NZ0 atmospheric lines before (lines) and after (points) iteration procedure. A precision 

of 5 ms" is maintained between about 5 and 38 km. Below 25 km line saturation due to pressure broadening degrades the precisions. 
134 

s 

Figure 2.9.11 Fourth extended relative rpeed profiles vk for the sunset occultations obtained from combining speed profiles determined using the 2ul 

N10 absorption lines to the profiles using the us COa, ua H)a0 and 2ua Ha0 

atmosphwewric absorption lines. The gentle variation in relative speed with 

height below 60 km is evident as well as rapid variations with height above 

70 km. The similarities in the variations with height in the profiles is evi- 

: ): : I I 

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Cvr 

dent. The separations between occultations ir due to changes in the orbital 

geometry $11.12. - I I 

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15.0 I:,.so zl i.00 

PRECKSKm (WSLC 1 

Figure 2.9.13 Precisions of the fourth extended relative speed profiles a(d) 

for all occultations described in Figures 2.9.11-2. A precision of 5 ms" or 

better is maintained between 25 and 110 km for all occultations. 
136 

11.10 Line Positions Obtained and Calibration 

Standards 

COMPAFUNG RETRIEVED AND STANDARD LINE' POSITIONS 

It has been shown that the line positions determined from the wind retrieval method are precise to O(10" cm-') and accurate to 10' cm-'. These 

were compared to a set of standard for the Cop, HpO and Np0 line positions 

compiled in the ATMOS line listing by Brown (1986). The differences between retrieved values and the standards recommended by Brown &,- 

or in the ATMOS line list <*~~os are shown in Figure 2.10.1. For the up Hp0 

lines there does appear to be a difference between %he positions determined 

from the atmospheric lines and the ATMOS standards positions of about 

lo-' cm-' for most of the lines. The other bands appear to be distributed 

more or less about zero. Some of the NZO lines have significant shifts of up 

to 5 x lo-' cm-l. Since the precisions of O(lO-' crn-l) we obtain are of an 

order of magnitude better than this the ATMOS instrument can be useful 

for improving rotation-vibration line wavenumbers and B values $1.6. For 

example, the 2up Hp0 lines were only initially known to about cm-l 

accuracy [Brown, 19861 and the precisions we obtain of 5 x 10' cm-l is 

almost two orders of magnitude better than this. The low precision of the 

ATMOS line list for these lines is reflected in the large overall variation in 

the differences for these lines near 3200 cm-'. 
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We can estimate the offset a,b between separate line lists ‘a’ and ‘b’ 

(2.41) 

where ‘a’ and ‘b’ are two different sets of line positions for the same set of 

absorption lines. Table 2.10.1 gives values of a& between various linelists. 

The line positions determined from the interferometer are compared to accepted standards. For the Cop and up HpO lines comparisons are also made 

between the instrumental lines and the positions determined from the atmospheric lines. 

We see from the data that the Cop line positions agree within the uncertainty of about 3 x cm-’ for the standards and positions determined 

using ATMOS. This is to be expected since we forced the instrumental line 

positions to agree with the siandards during the wavenumber cslibration. 

Similarly, positions determined using the atmospheric lines agree with the 

positions of the instrumental lines since these positions were derived from 

Doppler shifts, which in turn were derived initially from the instrumental 

positions. The measured positions of the 2up HpO lines were also found to 

agree with the ATMOS linelisting standards. However, since they are only 

accurate to - cm-’, this indicates only that we are obtaining at least 

that order of accuracy. The average positions of the 2vl Np0 lines were found 

to agree with the ATMOS linelisting and a comparison between the positions 

determined using the atmospheric Lines and the standards is given in Figure 

2.10.2. 

THE H10 LINE POSITION PROBLEM 
138 

In the case of the up HpO lines curious discrepancies are apparent. First 

of d, we see that, from comparing the instrumental lines to the standards 

provided by Brown, the difference may be significant but is of the order of the 

measurement erros. Secondly, and most importantly, there is a significant 

discrepancy between the positions determined from the atmospheric HZ0 

lines and the instrumental HpO lines. From the data in Table 2.10.1 we see 

that this shift corresponds to a Doppler shift of almost +15 ms-' between 

the standards where 

+it < iatm- (2.42) 

Due to this discrepancy and the marginal agreement with Brown's standards 

there is thirdly a significant offset between the atmospheric standards and 

Brown's standards. This discrepancy poses and interesting problem. If we 

calibrated the instrument with respect to the water lines instead of the carbon dioxide lines then the positions of the water lines determined from the 

atmospheric lines would be in agreement with the instrumental lines, aince 

these positions would have been used to determine the Doppler shifts. This 

would cause a shift of -15 ms-l in the recovered speed profiles. This, when 

determining the positions of the COP lines from the atmospheric lines would 

cause them to be shifted -15 ms-' from their instrumental counterpszts 

+it > iatm. (2.43) 

Which calibration is the right one to use? Since the offset is significant 

it implies one of the two components of the gas, namely the Cop or HZO, 

is moving with respect to the instrument. In the first case when we have 

calibrated relative to the Cop standards we find that the HpO instrumental 
139 

frequencies are lower than those determined from the Ha0 atmospheric lines 

- or synonomously with the COO calibration via Doppler shifts. If we accept 

the accuracy of the positions determined from the Ha0 atmospheric lines 

then the negative shift in the HO0 instrumental lines indicates that they 

are moving away from the instrument. On the other hand, if we calibrate 

relative to the Hz0 standards the COO instrumental lines will have a positive 

shift relative to the COO atmospheric line positions. This means that the 

carbon dioxide lines are moving toward the instrument. The first implies a 

source of Ha0 diffusing into space at a bulk velocity of 15 ms-l. The second 

implies COO sources converging on the instrument. There is no reasonable 

explanation for this latter kind of source. Therefore we should accept the 

former. This outgassing, or leakage of a trace amount of HOO vapor from the 

shuttle itself, is the diffusion of a trace cloud of water vapor into space. The 

source of this outgassing may be due to water vapor desorbing from the outer 

shell of the shuttle itself. Shaw (1988) pointed out that this may be due to 

the high level of humidity present at the launch site. This also suggests that 

Hz0 absorption lines may not be useful for absolute line position calibration. 
I 

'l 

1 

1 

1 

I 

I 

I 

I 

I 

140 

I I I 1 I I 

Figure 2.10.1 Differences between standard line positions (Brown's or ATMOS) and retrieved line positions for u3 C02, u2 H20, 2uz HzO and 24 NzO 

from atmospheric lines. 

Figure 2.10.2 Differences between standard line positions (Brown's or 

ATMOS) and retrieved line positions for the 24 NzO from atmospheric lines. 

This is the same data shown in Figure 2.10.1 but magnified for clarity. Notice 

that the differences in the position appear to systematically vary between the 

band center and the high J lines. These improvements in positions may be 

significant. 
141 

0 

s 

0- 

w 

:la t"' 

K 

0 

v) ah 

Yo 

0 

t 

- M:M:&t: 1 :a 

1: P85:H//w1 :E!/ 4 - - POf/HM/W2/I.SS09 

91 

6 

t l 
I 

I 

I 

I 

Table 2.10.1 Offsets between various line lists ‘a’ and ‘b’. A - Positions determined from atmospheric lines. I - Positions determined from instrumental 

lines. LRB - Positions recommended by Brown (1986) for instrumental calibration. ATMOS - Positions srchived in the ATMOS line listing. 

Band Linelist ‘a’ Linefist ‘b’ a& (X a(a&) (Xlo-’) 

v3 co2 A LRB -0.08 0.49 

I LRB $0.04 0.33 

A I -0.12 0.50 

2~1 N20 A ATMOS -0.12 0.65 

2~2 HzO A ATMOS -2.83 3.95 

Y H20 I LRB -0.90 0.71 t 

A I +4.63 0.90 t 

A LRB +3.73 0.61 t 
1 

11 

I 

I 

I 

I 

I 

I 

I 

I 

1 

I 

1 

I 

I 

I 

.I 

:I 

142 

11.11 Testing the Measurement Theory 

In $1.11 a theory was developed to predict the uncertainties in the measurement of line positions attainable using the ATMOS spectra. It was shown 

that the measurement uncertainties behave as 

(2.44) 

where f is a form factor which depends on the geometrical shapes of the 

lines and the sampling process, w is the width of the line at half maximum, 

Ap is the peak absorptance and aw) is the background noise level. The 

atmospheric lines were used to test this theory. 

After the relative speeds uL were determined with the methods discussed 

previously, each atmospheric line position rjk was shifted to a rest frame 

position tf using 

(2.45) 

Then the difference between this value and the rest frame position estimated 

by (2.29) was calculated for each line 

(2.46) 

These values were then plotted versus Aij. If the measurement theory is a 

realistic one then we would expect the spread of the distribution of A$ to 

have an envelope width proportional to A;'. Figures 2.11.1-4 show this kind 

of behavior for the four bands used in the analysis - u3 COa, HaO, 24 

H,O, and 2v1 Na0 respectively. Indeed, the uncertainty of A$ does decrease 
143 

I 

I 

I 

I 

I 

I 

I 

I 

I 

$1 

I 

‘I 

with increasing Ap. Since f, Q(N) and W are approximately constant for 

each band we use (2.43) 

If we plot the distribution 4j A$; versus A:j we would expect to see a 

distribution whose width is independent of the peak absorptance. Figures 

2.11.5-8 show these distributions for the four atmospheric bands used in the 

analysis. Table 2.11.1 list the distribution deviation a(Av4) for each of the 

bands. We see that the COa lines have the smallest line noise $11.4 followed 

by the N20 band and then the HZ0 bands. This is in agreement with the 

ordering of the total background signal discussed in $11.4. We can also see 

upon comparing the u3 CO2 and u2 H20 noise levels computed from the 

atmospheric lines to those computed from the instrumental lines that the 

atmospheric values are higher. This may be caused by an increase in the 

line noise due to the ratio process. The ratio of the noise levels determined 

from the atmospheric v3 CO2 lines to the u2 Ha0 lines is about 1.6 from 

Table 2.11.1. A similar ratio determined from the values in Tables 2.3.1-2 is 

about 1.7. These two values are approximately the same which again suggests 

that the measurement precision theory of 51.11 can be used to describe the 

measurement uncertainties of line positions from the spectra. 
I 

I 

I 

I 

1 

1 

I 

i 

I 

I 

I 

I 

I 

I 

I 

i 

.I 

I 

.I 

144 

Figure 2.11.1 Ai for the u3 COa atmospheric lines as a function of the measured peak absorptance Ap. The spread of the data is larger for small peak 

absorptances and becomes smaller as the peak rbsorptance increases. 

Figure 2.11.2 Ai for the ua Ha0 atmosplieb’c lines as a function of the measured peak absorptance Ap. The spread of the data is larger for small peak 

absorptances and becomes smaller as the peak absorptance increases. 
1 

IB 

I1 

I 

1 

1 

I 

I 

1 

I 

I 

)I 

0 I 

1 

I 

U 

I 

I 

.I 

p :I--- --- 

I I I I 

I. I I I 

145 

I* I I I I I I 

I I I I I I I 

1.1 I I I I I n, 

ea d-m %L l3iahl&* 

Figure 2.11.3 A< for the 2u2 Ha0 atmospheric lines as a function of the 

measured peak absorptance A,. The spread of the data is larger for small 

peak absorptances and becomes smaller as the peak absorptance increases. 

I I 1 1 $- I I I I 

I I I I 13 I I I 1 

Figure 2.11.4 Ai for the 2vI N20 atmoipheric lines as a function of the 

measured peak absorptance Ap. The spread of the data is larger for amall 

peak absorptances and becomes smaller as the peak absorptance increases. 
I 

I 

'I 

1 

1 

li 

I 

I 

I 

1 

:I 

.I 

I 

1 

1 

I 

:I 

.i 

E +Q 

146 

I I 

I I I I 

I I I I -' -4 

:: : I I I I :: !: . X 

L 

fb 0 

U 

Q 

I I I I I I LI 

I I I I I I I 

I I I I 

Figure 2.11.5 ApAi for the u3 COz atmospheric lines as a function of the 

measured peak absorptance Ap. The spread of the data appesrs to remain 

constant as a function of the peak absorptance in agreement with the measurement precision theory. 

b 

I 

4 

1 I 1 , I I I I :: %mi I I I I I* 

Figure 2.11.6 &Ai for the uz Hz0 atmospheric lines. The spread of the data 

appears to remain constant as a function of the peak absorptance. 
1 

I 

1 

I 

1 

E 

i 

I 

I 

147 

Figure 2.11.7 ApAi for the 2va H20 atmospheric lines. The spread of the 

data appears to remain constant -ts a function of the peak absorptance. 

I I I I I. I I: I I I I I I 

Figure 2.11.8 ApAi for the 2vI NZ0 atmospheric lines. The spread of the 

data appears to remain constant as a function of the peak absorptance. 
14B 

Table 2.11.1 Estimates of he noise determined from atmospheric lines. 

Band mean (xlO-'cm'') 

Y COa 6.1 

u2 H20 9.8 

2~2 Ha0 9.5 

2~1NzO 7.5 
'a 

E 

IU 

1 

I 

c 

I 

149 

11.12 Estimating the Doppler Shift due to Orbitd 

Motion and the Earth's Rotation 

THE EPHEMERAL DATA 

The relative speed profiles presented in Figures 2.9.11-12 contain three 

independent Doppler shifts, the projection of the orbital velocity of the shuttle along the line of sight, the rotational speed of the earth along the line of 

sight, and the atmospheric winds along the line of sight. We wish to compute 

the effects due to the first two and then to subtract these from the relative 

speeds profiles to obtain residual wind speed profiles along the line of sight. 

The positions of the orbiter with respect to the earth were provided by Goddard Space Flight Center and were tabulated in terms of geocentric distance 

T, longitude 4, and latitude 8. Positions were evaluated at times t at six 

second intervals for each occultation. The orbital data are given in terms of 

the coordinate system of the earth rotating with the angular speed 

27r n=- 

T, (2.48) 

where T, is the sidereal day of 23h 56m. In order to examine the data in 

terms of a stationary frame it was necessary to translate the longitudes appropriately. This stationary frame can be arbitrarily chosen for the purposes 

of computing the relative motions of the orbiter with respect to the tangent 

point. I chose an initial time to = 120d00 : 00 : 00 GMT. The sampled times 

ti for each point i were converted to an elapsed time 

Ti = ti - to. (2.49) 
150 

The longitudes $i were also translated to the reference frame at to by 

4; = 4i + a~i (2.50) 

so that a longitude $i given at the elapsed time 7; corresponds to the longitude 

4: at the time to. The positions are given in terms of this geocentric stationary 

coordinate system. It is useful to transform this spherical coordinate system 

into a Cartesian system via 

Xi = Ti COS COS 4:, 

yi = ~i cos& sin$:, 

zi = t'i sin&. 

(2.51) 

(2.52) 

(2.53) 

These positions were computed for all of the occultations and are presented in 

Figures 2.12.1-2 for 2-y and x-z planar projections respectively. The figures 

show that the part of the orbit tabulated in the Goddard ephemeris closely 

follow a circular orbit, however there are notable changes in the orbit. Figures 

2.12.3-4 are close up diagrams of the 2-y planar projections for the sunset 

and sunrise occultations respectively, There is evidently a regular drift in 

the orbit. This drift is responsible for the shifts in the relative speeds for the 

sunset occultations shown in Figure 2.9.11. The gap between the profiles for 

SS06 and SSO9 in Figure 2.9.11, for example, is related the change in position 

of the orbit between these two occultations which is evident in Figure 2.12.4. 

The ephemeris data also give positions of the tangent points in the same 

e 

terms as for the orbital data. Again the longitudes 4;- are translated by 

(2.54) 
I 

I 

I 

8 

li 

0 

1 

I 

I 

151 

and the Cartesian coordinates are given by 

The 2-y planar projections of the tangent points as well as orbital data for 

the four sunset occultations is shown in Figure 2.12.5. The orbital drift 

is evident. It is useful to note that the ephemeris data are collected at 

time intervals At - 6s. The occultation takes about 5 min which gives us 

n =- 300s / 6s = 50 data points to use for the analysis below. 

LEAST SQUARES METHOD 

To get the Doppler shift from the positions of the orbiter and tangent 

points in terms of the Cartesian coordinate system we need to compute the 

rates of change of these positions. We wish to determine v = dr/dr where 

r = r(r) is a function to be determined from the data ri, ri. It was found that 

errors in the positions of the orbiter and tangent point caused uncertainties 

in the velocity of over 20 ms" between points using simple finite differences 

for the time derivative of the positions. A least squares method was used to 

determine the smooth trajectories of the orbiter and the tangent point as a 

function of elapsed time r. A polynomial of order m was fit to the observed 

data of the form 
152 

€= 

These are three independent sets of equations. If we consider the set of 

equations for the xi and that there are n independent data points, we can 

r 1 

€1 

4 

Q 

write 

where 

X= 

x=Te+e 

I: : : 

and the column of model errors is given by 

9 €= 

€0 

€1 

€2 

€m 

(2.61) 

(2.62) 

(2.63) 

In order to get the best set of parameters which best fit the measured 

positions xi in a least squares sense we need to find the least squares estimator 

e* for the coefficients that satisfies 

6 - Te*)T (x - T(*)] = 0. (2.64) 

Here, (x - T(*)= (x - T€*) is the residual sum of squares. By differentiating 

(2.63) it can be shown that 

(T~T)~* = T~~. (2.65) 
153 

So solving for e*, 

e* = (T=T)-~T=x. (2.66) 

The matrix TTT is an (m + 1) x (m + 1) symmetric matrix. 

The assumptions for the least squares method are that the errors y arise 

from distributions which have a mean of zero 

E(€) = 0 (2.67) 

and have a common unknown variance variance u', but zero covariance 

E(€€=) = 21. (2.68) 

Under these conditions e* is an unbiased estimator for 

E(x) = T( and it can be shown that 

[Myers, 19861 since 

E(€*) = €- (2.69) 

The estimator (* also has the minimum variance [Koerts, Abrahamse, 19691. 

The parameters ai* have been estimated by (2.65) and we wish to estimate the errors E*. By using (2.60) 

where 

N = I - T(TTT)-lTT. (2.71) 

The n x n matrix N has the following properties (Graybill, 19761 

N = N~, (2.72) 

NT = 0, (2.73) 
154 

N = NN = N=N = NN*, 

trace N = n - (m + 1) = n - m - 1. 

(2.74) 

(2.75) 

The last property can be proven since if A and B are square matrices of 

the same order then trace(A + B) = trace A + trace B and trace (AB) = 

trace (BA). Then 

trace N = trace [I - T(TTT)-'TT] 

= trace I - trace [T(TTT)-'TT] 

= trace I - trace [(TTT)-'(TTT)] 

- n-(m+l) (2.76) 

since T*T is an (m + 1) x (m + 1) matrix. 

We can now find an estimate 2 for the unknown variance a' of the errors. 

One way is to estimate it by the mean square of the residuals 

dl - n-l E +T E - n-'(Nx)TNx=n-'xTNTNx =n."xTNx. (2.77) 

This estimate is biased [Myers, 19861 since E(s2) # u2 

E(u2) = n-'E[x*Nx] = n"E[zN~l (2.78) 

using (2.60) and (2.72). At this point we make use of a theorem [Graybill, 

19761 on the expected value of a quadratic form characterized by a matrix 

Q. Given a random vector E with a mean E(€) = p and E(d) = a21 then 

E(E~QE) = a' trace Q + pTQp. (2.79) 

In (2.77) we identify p + 0 and Q + N so that 

E(8') = u2n-* trace N = (n - m - l)n-'a2. (2.80) 
155 

We see then that even though g2 is a biased estimate the quantity u*' E 

A/(n - m - 1) is an unbiased estimate. Therefore 

b*'= (n - m - I)-' [X - T(*IT [X - "('1 (2.81) 

is an unbiased estimate of the variance. Now the variance-covariance matrix 

X for (* can be evaluated by computing X G E{(( - e*)(( - e*)=} and it 

has been shown [Myers, 19861 that 

x E E{(< - (*)(( - (*)*} = U*~(T~T)--~. (2.82) 

Having estimated the coefficients fj we wish determine the velocity vectors from the least squares fit. We estimate the position z(r) by the function 

z'(7) = Cfp. (2.83) 

m 

If we define the vector 

then the variance is given by [Myers, 19861 

fP(Z*) = PTXP. 

The velocity can be estimated from (2.79) by 

(2.84) 

(2.85) 

(2.86) 
I 

8 

1 156 

If we define the vector 

dr +=- 

dr 

then the variance is estimated by 

(2.87) 

ByV:) = +=xi. (2.88) 

The least squares method is similarly applied to the y and z variables as well 

for both the orbiter and tangent point. 

CORRECTION FOR THE EARTH'S ROTATION 

FinaJly we account for the rotational motion of the earth. At the equator 

the tangential velocity of the earth rotation is 467 ms'l and causes significant 

Doppler shifts. The velocity vector of the rotation at the tangent point rtw 

(2.89) 

where n is the angular velocity of the earth's rotation. When the vector 

functions r+t-(r) and r*Orb(r) have been respectively computed and their 

respective velocities Vt=g and v*Orb(7) determined the relative speed along 

the line of sight can then be estimated. The total contribution to the relative 

speed profiles caused by orbital motion and the earth's rotation is given by 

The unit vector between the orbiter and the tangent is calculated by 

where r*orb-tq is the distance between the orbiter and the tangent point. 

Finally the velocity along the line of sight is given by 

Av* = Av* - ii. (2.92) 
157 

The quantity Av' must be subtracted from the relative speed profiles determined in $11.9 to obtain residual wind speeds along the line of sight. The 

variance of Av' can be estimated by expanding (2.91) 

Av' = Av: ha + Avy' A,, + Aut ii,. (2.93) 

Since the error in ii is negligible and the Aut are uncorrelated then 

(2.94) 

where the variances in the components are determined by (2.87). 

These calculations were carried out for the ephemeris data described in 

Figures 2.12.1-2 for data corresponding to tangent heights from -40 to 300 

km. In each occultation the elapsed time r wm determined relative to the 

time when the tangent height was nearest 30 km. This allowed us to obtain 

the best fit for the orbital/rotational speeds with tangent heights between 30 

and 110 km since this altitude range represents the region where the 'origin' 

of the polynomial fit is located and the fit is better in this region than at the 

ends near 300 km and -40 km. In Figure 2.12.6-7 the relative speed profiles 

511.9 together with their orbital/rotational motion contributions are plotted 

for the sunset and sunrise occultations respectively. The slopes of the orbital/rotational motion curves follow the trends of the relative speed profiles 

determined from the atmospheric lines. Figure 2.12.8 shows the uncertainty 

~(Av') from the least squares analysis using (2.90) for all five occultations. 

The uncertainties in the orbital/rotational speeds were about 3 ms'l near 

60 km tangent height. SSll and SS13 have precisions by as much as 5 ms" 

worse than the other three occultations above 120 km since there are not as 
158 

many ephemeris data available for these two occultations. However, these 

precisions are still better than 5 ms” below 100 km. 

PRECISION OF RESIDUAL SPEED PROFILES 

The residual speed along the line of sight, which is the wind motion, 

is computed by subtracting the orbital-rotational motion from the relative 

speed profiles. The residual speeds for the four sunset and the sunrise occultations are plotted in Figures 2.12.9-10 respectively. We immediately see the 

striking similarity between the sunset profiles. The measurement precision 

is determined from the precision displayed in Figures 2.12.8 and 2.9.13 

a’(residual) = B’(re1ative) + B’(orbit - rotation). (2.95) 

The precisions for the residual speed profiles are shown in Figure 2.12.11 

This find result shows a total measurement uncertainty of about 5 m/s between 25 and 110 km for the five occultations. Since the amount of absorber 

of COZ above 110 km rapidly falls off and the orbital-rotational uncertainties becomes large with increasing tangent height the uncertainties in the 

residual wind speeds becomes large. Below 20 km the uncertainties becomes 

large due to the saturations of N20 lines as well as orbital-rotation uncertainties. Between these regions we have obtained the desired total measurement 

- 

precision of 5 ms-’. 

THE LINE OF SIGHT ORIENTATION 

The orientation of the line of sight relative to the earth’s surface was also 

determined. This calculation was performed by measuring the angle between 

the vector ii and the east-west unit vector 6 using standard trigonometry. 

In Figures 2.12.12-13 this angle is plotted as a function of tangent height 
159 

for the sunset and sunrise occultations respectively. From these data we can 

see the line of sight is inclined to the zonal direction by about 17-18' and 

21-23' for the sunsets and sunrise respectively. Therefore we can conclude 

that residual speeds measured are very nearly the zonal component of the 

wind speeds. In Figure 2.12.9 a positive residual speed is eastward for the 

sunsets while a negative residual speed in Figure 2.12.10 is eastward for 

the sunset. We see that in both figures an eastward prevailing wind of 25 

and 50 m/sec between tangent heights of 20 and 70 km are present in the 

sunsets and sunrise respectively. Above 80 km rapid variations of the zonal 

wind with height are present. The similarity of zonal wind speed with height 

between the various sunset wind speed profiles in Figure 2.12.9 is particularly 

interesting and various possible causes of the observed wind speed features 

is discussed in §§III. 

The Doppler shifts introduced by the orbital motion of the instrument 

as well as the rotation of the earth have been estimated to 5 ms-' or better 

between tangent heights of 20 and 120 km from the ephemeris data. The 

ephemeris data can be used to determine absolute zonal wind speeds from the 

relative speed profiles determined in 511.9 without a significant degradation 

of the total precision from 5 ms" over the tangent height range. 
160 

d 4 

Figure 2.12.1 2 - y planar projection of orbitd positions computed using the 

ephemeris data showing both sunset (SS) and sunrise (SR) positions for all 

of the occultations. The orientation of the coordinate system was arbitrarily 

chosen, however, the circular shape of the orbit can be seem. The identification of points is not important in this illustration. There are variations in 

the orbital path. 

il 

'La 2s- -iza i2.m &a *.a ha 

Figure 2.12.2 Same as in Figure 2.12.2 except for the z - z planar projection. 

x lI)o .Id 
161 

Figure 2.12.3 Close up t - y planar projection of orbital positions for the 

sunrise occultations. There is a noticable changing of the orbit between 

occultation (1) ,,SROl through (7) SR07. 

'I I 

Q La ij '1wm 1lRa h0.m *a a 

x ral*.lo' - dn 

Figure 2.12.4 Close up t - y planar projection of orbital positions for the 

sunset occultations. There is a noticable changing of the orbit between occultations (1) SSOl through (13) SS13. This changing of the orbital geometry 

causes the shifts between the relative speed profiles of up to 200 ms'l in 

Figure 2.9.11. 
8 

162 

r; 

Figure 2.12.5 Close up z-y planar projection of orbital and. tangent point 

positions for the four sunset occultations - (6) SS06 - (9) SSO9 - (11) SSll - 

(13) SS13. The progression of points as time increases during the occultation 

is indicated. This changing of the line of sight geometry causes the shifts 

between the relative speed profiles of up to 200 ms" in Figure 2.9.11. 
163 

Figure 2.12.6 Orbital-rotational motion plotted with relative speed profiles 

for the sunset occultations - (6) SS06 - (9) SSO9 - (11) SSll - (13) SS13. The 

points on the approximately 'straight' curve represent relative speeds due 

to the orbital-rotational motion. The separations of the orbital-rotational 

motion are in agreement with the separation of speed profiles obtained by 

measuring the Doppler shifts of atmospheric lines. The rapid variations above 

70 km are due to 

Figure 2.12.7 Same as above except for the sunrise occultation - (2) SR02. 
Figure 

all five 

20 and 

164 

2.12.8 Uncertainty in least squares fit for orbital-rotational motion for 

occultations. The precision of 5 ms-l or better is maintained between 

100 km for_ all occultations. 

Figure 2.12.9 Residual speed profiles for the sunset occultations 

(9) SSO9 - (11) SSll - (13) SS13. 

- (6) SS06 - 
Figure 2.12.10 

Figure 2.12.11 

165 

Residual speed profile for the sunrise occultation n - (2) SRO2. 

Corn - all five occultations. A precision of about 5-6 ms-' is maintained between 25 and 110 

km. 
166 

. 

-. . . -. ...... . ... . . . ... 

1-0 14- 15- IS- ISw 15*11 1.m. 

mot /em8 

Figure 2.13.2 Concentration c(h) for Ha0 from ATMOS rtandard atme 

sphere. 

Figure 2.13.3 Concentration c(A) for NaO from ATMOS rtandard atmosphere. 
166 

Figure 2.12.12 Angle between the line of sight and the zonal direction for the 

sunset occultations. The angle is approximately 17O.5 between the line of 

sight and the zonal direction indicating that the wind speeds are essentially 

the zonal component. $7 

t----r---- 

4-00 i190 ri.00 h.00 A40 25. 

f3- (DEOIKESl 

Figure 2.12.13 Angle between the line of sight and the zonal direction for the 

sunrise occultation. The angle is approximately 22' between the line of sight 

and the zonal direction indicating that the wind speeds are essentially the 

zonal component. 
167 

I 

I 

I 

1 

I 

I 

I 

'I 

11.13 Density Weighted Residual Winds 

The wind speeds determined in the previous section were computed under the assumption that the observations were made solely on gases at the 

tangent points. However, the line of sight penetrates all levels above the 

tangent height. We have seen that there are wind variations with height. If 

these variations are rapid enough, which may be the case for the zonal wind 

profiles determined in the previous section, they may cause a 'smearing' of 

the results and degrade the vertical resolution of the profiles. We wish to 

examine how this affects the vertical resolution of the results. 

To begin with we can examine the number density of each gaseous component used to obtain wind speeds, namely CO', H'O, and N'O. Figures 

2.13.1-3 show the concentrations as a function of height c(h) as given in the 

ATMOS data base. We wish to determine the total fraction of absorber 

present as a function of distance about the tangent point. Figure 2.13.4 

shows the typical geometry of a tangent ray. Refraction in the atmosphere is 

not taken into consideration since this model is only assumed to be a first approximation to begin with. The parameter 1: is the distance from the tangdnt 

point along the line of sight. Given the tangent height t and the distance 1: 

the altitude above the surface h,(z) is given by geometry 

ht(2) = TO -k t)' -k 2' - To. (2.96) 

Let the total path length through the atmosphere for a given tangent height 
168 

t be represented by 

where is "top" to the atmosphere and can usually be picked to be above 

the region where the wind measurements were carried out. This is a reasonable assumption since the amount of infrared absorber decreases rapidly as a 

function of height. We can then determine the number of molecules per unit 

area N: along the line of sight 

(2.98) 

Having determined this we want to determinebthe total fraction f:(z) of 

absorber within a distance z to tangent point dong the line of sight 

(2.99) 

The integrated fraction was determined for the concentration profiles 

given in Figures 2.13.1-3. A maximum height of 150 km was assumed for 

hmm. Figures 2.13.5-8 show the total fraction (2.98) as a function of z for 

various values of the tangent height t. It is found that 90% of each species is 

within 400 km of the tangent point, the remainder being distributed about 

the ray. For N10 the 90% value threshhold is about 250 km. 

It is of even more importance to estimate the'range of altitudes which 

contain certain fractions of the total absorber along the line of sight. Figures 

2.13.5-8 were transformed using (2.95) to examine how the integrated fraction 

is related the h:(z) - t. For a given fraction of the total absorber along the 

line of sight a value of a "layer thickness" ht(z) - t can be determined. The 
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layer thickness for the three components are plotted in Figures 2.13.8-10 for 

various tangent heights t. We see that for COa over the range of tangent 

heights where the atmospheric v3 lines are useful that 90% of the absorber 

is contained in the first 10 km or less of the above the tangent point. Since 

the spectra are sampled at approximately 4 km intervals, then 70 and 80% 

of the absorber is contained between a tangent height and the next highest 

sampling height. Between a tangent point and two sampling spectra more 

than 90% of the absorber is present. We therefore expect that the vertical 

resolution of the wind speed at a given tangent height is being degraded 

primarily by the next highest level. 

Figure 2.13.11 is a diagram which defines a “layers” model. Basically, we 

can characterize the atmosphere for spectrum k at the tangent height tk by 

the gas lying within fR/2 of the tangent height where R is the sampling 

resolution of approximately 4 km. We see from the diagram that the layer 

thickness for the tangent spectrum k is R/2 or about 2 km. The layer thickness for the next level k + 1 is the full resolution R or about 4 km. From 

Figures 2.13.8-10 we see that approximately 60% of each absorber is contained within the first 2 km. This corresponds to the first dotted line above 

tk in Figure 2.13.11. In the next layer between 2 and 6 km an additional 20 

to 30 9% of the absorber is present. In the next layer contains about 10% of 

the absorber, and so on. So we can assume, to a first approximation, that the 

winds measured from Doppler shifts in spectrum k are a weighted average 

of the “true” winds, the weights being the percentage of absorber each layer 

[Shaw, 19851. 

The above argument suggests the following approximate weighting scheme. 
170 

' 1 

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Let each measure wind speed at the layer k be given by vk and let the "true" 

speed be given by v'~. Then by the above arguments we have that 

vk 1~' 0.6~'~ + 0.2~'"' + 0.1~'~'~ + (2.100) 

which is a set of equations for all the spectra k and this can be inverted. Figures 2.13.12-13 show the wind profiles predicted by this method. Comparing 

these figures to Figures 2.9.9-10 show that in the vicinity of large wind variations that the weighted profiles have an enhanced vertical resolution. The 

peaks are shifted slightly in altitude and the magnitudes are larger by up to 

50 m/s near 105 km. 

The layers model can be used to improve the vertical resolution since the 

ATMOS instrument has a limited sampling resolution of about 4 krn between 

spectra. The interferences of lines in different layers in the spectra is caused 

by the rate by which the density changes with height and the curvature of 

the earth and is not related to the vertical sampling interval of about 4 km of 

the instrument. If the vertical sampling interval can be decreased in future 

missions it may be possible, however, to determine the contribution of each 

layer to the observed wind profiles. 

Positive speeds in Figures 2.13.12-3 correspond to an eastward motion 

in the northern latitudes and a westward motion in the southern latitudes 

respectively. Figure 2.13.14 shows a zonal average from the four sunset occultations and Figure 2.13.15 shows the standard deviations about these mean 

values. Between 25 and 65 km there was an eastward wind of about 25 

ms-' which remained steady within the precision of 5 ms-' over the period 

of observation and range of longitudes as indicated by the small standard 

I 
171 

deviations in Figure 2.13.15. The consistency in the shapes of these profiles 

also confirms the stability of the instrument calibration. The wind motion 

reverses at 65 km and above 80 km. All of the sunset occultations show maxima near 85 and 105 km. The variations in these maximum speeds appear 

to be real. However, if the wind sheets are thin, the variations may be due 

in part to a lack of vertical resolution. These variations in maximum speeds 

may also be due to tides [Chapman and Lindzen, 19701 or breaking gravity 

waves [Fritts, 19841 (see @I11 for the definitions of these terms). The minima 

near 90 km correspond to an eastward wind of about 25 ms’l. 

The sunrise occultation in Figure 2.13.13 shows a single eastward maximum of about 100 ms” near 105 km. From 80 to 95 km there is a westward 

motion of about 50 ms-’ which is opposite to the direction observed in the 

northern hemisphere. A wind shear of about 20 ms-lkrn-l occurs between 

84 and 78 km. From 30 to 75 km the eastward motion of about 50 me” is 

nearly twice that observed in the northern hemisphere. 

Because the observations were made over a 24 hour period we can conclude that there were vertically stratified zonal winds at sunset between 25 

and 120 km altitudes near 30”N over a wide range of longitudes. The single 

occultation available for the southern hemisphere also shows winds with magnitudes, directions, and shears which have similarities to the northern hemisphere profiles. The measurements were made near the vernal equinox and 

since the asymmetries of the mesospheric circulation between hemispheres 

may be dependent on the seasons it would be useful to have more measurements at other latitudes and seasons [Labitzke, 19851. 
8 

0 

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1 

1 

Figure 2.13.1 

sphere. 

10- I=- 1UY IU I.UU 

-1 

Concentration c( h) for 

172 

iaii labia iaria 

C,02 from ATMOS standard atmo- 
173 

++ 

-7 

0 

Figure 2.13.4 Tangent point geometry of the line of sight. 

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Figure 2.13.5 Concentration fraction as a function of 2 for COa computed 

using data in Figure 2.13.1 for several tangent heights; (a) 60 km, (b) 70 km, 

(c) 80 km, (d) 90 km, (e) 100 km. 
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1 

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t 

t 

1 

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Figure 2.13.6 Concentration fraction as a function of z for HzO at various 

tangent heights; (a) 30 km, (b) 40 km, (c) 50 km, (d) 60 km, (e) 70 km, (f) 

80 km. 

Figure 2.13.7 Concentration fraction as a function of z for N2O at various 

tangent heights; (a) 20 km, (b) 25 km, (c) 30 km. 
175 

--1---- 1---- r - - - -1- - - - 1---- r---- I I I I I I 

I I I 1 I 

I 1 I I I 

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Figure 2.13.8 Concentration fraction M a function of layer thickness for COa 

at various tangent heights; (a) 60 km, (b) 70 km, (c) 80 km, (d) 90 km, (e) 

100 km. 

Figure 2.13.9 Concentration fraction as a function of layer thickness for HtO 

at various tangent heights; (a) 30 km, (b) 40 km, (c) 50 km, (a) 60 km, (e) 

70 km, (f) 80 km. 
176 

Figure 2.13.10 Concentration kaction as a function of layer thickness for N10 

at various tangent heights; (a) 20 km, (b) 25 km, (c) 30 km. 

t k*' 

Figure 2.13.11 Illustration of layer thicknesses near the tangent point tk. 

c- 3 
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Figure 2.13.12 Density weighted zonal wind speed profiles for the sunset 

occultations. Positive speeds refer to an eastward flow; (6) SS06, (9) SSO9, 

(11) SS11, (13) SS13. 

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D.W 

Figure 2.13.13 Density weighted zonal wind speed profiles for the sunrise 

occultation SR02. Negative speeds refer to an eastward flow. 

2. . 
178 

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awnm 

Wnd sp.d (m/au) 

Figure 2.13.14 Zonal average of the wind speeds for the four sunset occultations near 30" N. A positive rpeed ir easiward [van Cleef, Shaw, Farmer, 

19871. 

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0 m304050 

slandord 0.Vacim (mine) 

Figure 2.13.15 Standard deviations of the zonal average winds shown in Figure 2.13.14 [van Cleef, Shaw, Farmer, 19871. 
179 

11.14 Conclusion 

Precisions of about 3 x lo-' cm" for the assumed rest frame positions 

of the u3 COO instrumental lines were sufficient to allow for stable frequency 

calibration and wind speeds to be accurate to better than 1 ms-'. The calibration scale was found to be stable to better than 2~lO'~ cm-' and there 

were no indications of random calibration offsets between spectra. Thus 

Doppler shifts can be measured to accuracies of better than 5 ms". Analysis of the orbital ephemeris data indicated that relative speeds between the 

orbiter and the earth with accuracies of 5 ms" or better could be obtained. 

We therefore successfully obtained zonal winds to precisions of 5 ms" between tangent heights of 25 and 120 km. 

The relative positions of the lines used to obtain wind speeds were obtained to precisions of 5 x cm-' or better. There appeared to be significant improvements in the precisions of these line positions as compared 

to other sources [Brown, 19861. The line positions were compared to other 

estimates of rest frame positions and were found to agree with them within 

the known accuracies of about lo-' cm-l cm-l for the 24 H20 lines). 

The positions therefore agreed with the uncertainties of the standards. The 

accuracies of the rest frame positions, however, had no effect on the accuracy 

by which Doppler shifts could be measured since instrumental lines of CO1 

were available for absolute calibration. 

The zonal wind speeds obtained by this method showed significant variations with height of up to the order of 100 ms-'. The similarity in the sunset 
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wind profiles, taken at different longitudes but at similar latitudes and local 

times, indicate that the winds may be directly related to the atmospheric dynamics at sunset. Since there was only one sunrise occultation was available 

in the Filter 3 spectra similar conclusions cannot be made at this time. 
181 

CHAPTER I11 

LARGE SCALE DYNAMICS OF THE 

CIRCULATION OF THE EARTH’S 

ATMOSPHERE 

In this chapter we examine several types of activity which are characteristic of the large scale circulation of the middle and upper atmosphere. This 

discussion is motivated to understand the ‘oscillating’ nature of the ATMOS 

wind profiles obtained from the spectral data. The different classifications of 

motion discussed in this chapter can be attributed to some of the observed 

features of the zonal winds shown in Figures 2.13-15. It was found that atmospheric tides, planetary scale oscillations produced by the daily or diurnal 

ultraviolet absorption and subsequent heating of the atmosphere by boundary layer water vapor and stratospheric ozone, are significantly large enough 

to produce the kind of wind speeds measured during the mission, particularly above stratospheric heights. In the stratosphere the wind motion is 

dominated by the thermal wind which can produce zonal wind motions of 

up to several tens of ms-’ and these appear in the ATMOS wind profiles at 

approximately the correct magnitude predicted [Salby, 19811 of several tens 

of ms-’. 
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182 

This chapter presents the linear theory of global scale atmospheric disturbances. The linearized theory predicts that the amplitudes of these disturbances, in terms of velocity, pressure apd temperature perturbations from 

the mean state, will grow with height proportional to pa where pb is the 

mean background density. The exponential increase with height is due to energy transporting through the exponentially decreasing background density 

imposed by this linear theory [Kato, 19801. The linear theory presented in 

this chapter includes radiative damping or infrared cooling [Andrews, Holton, 

Leovy, 19871 primarily due to the radiative properties of the 15 pm ua CO:, 

vibration band [Dickinson, 19841 but neglects dissipative mechanisms such as 

the kinematic viscosity which varies as [Andrews, Holton, Leovy, 1987; 

Pedlosky, 19871. 

-111 

-112 

In the previous chapter we showed how zonal wind speeds between 25 and 

120 km were obtained with a precision of 5 ms” from the ATMOS spectra. 

The results indicated that there were variations of about 100 ms-’ in the wind 

speed over changes of only a few kilometers in the tangent height. The profiles 

in Figures 2.13.12-13 show that the amplitudes of the wind variations become 

larger with increasing height. The variations appear to begin at about 70 km 

with an amplitude of about 20 ms-l and increase to almost 150 ms’l at about 

110 km. The sunset measurements occurred at a variety of longitudes at the 

same latitude and the wind structure appeared to be preserved betwcen the 

occultations. This regularity of the wind patterns suggests that they axe 

caused by global scale dynamics. Wind measurements by other researchers 

[Reed, 1969; Manson, Meek, 1986; Wallace, Tadd, 1974; Muraoka, Sugiyama, 

kawahira, 1988 - for example] indicate that the behavior observed by ATMOS 
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183 

is typical of mesospheric and lower thermospheric dynamics. 

The atmosphere heats and cools on a regular daily basis and this forces the 

upper atmospheric disturbances are periodic in nature. The classical or finear theory of atmospheric tides first described by Siebert (1961) and later by 

Chapman and Lindzen (1970) and Kat0 (1980) predicts atmospheric disturbances of pressure, density, temperature and wind due to atmospheric heating which propagate to thermospheric heights. The exponentially decreasing 

density of the atmosphere with height allows the amplitudes of pressure, 

density, temperature and velocity perturbations waves to grow with height 

[Etts, 19841. The classical theory of tides [Siebert, 1961; Chapman and 

Lindzen, 19701 predicts unlimited growth in wave amplitudes with height. 

This is due to the linearity of the theory. However, instabilities from the 

nonlinear terms in the fluid dynamical equations equations cause the waves 

to break which can lead to turbulence. Adiabatic instabilities due to large 

local potential temperature perturbations also may contribute to the breakdown of the tidal waves [Lindsen ,19811. It is desirable to see if we can 

propose a theory which can ‘explain’ the nature of the wind speeds obtained 

by ATMOS. 

In order to test the linear theory a model of atmospheric heating rates 

proposed by Groves (1982) was used as a periodic heating term and the 

response of the atmosphere is examined. The linear theory predicts that 

the leading diurnal tidal mode may be responsible for the regular structure 

observed in the ATMOS results. However, since the linear theory does not 

‘break’, this response is overestimated. 
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111.1 The Atmospheric Temperature Distribution 

The earth's atmosphere has an average altitude dependent temperature 

distribution similar to the one given in Figure 3.1.1 [from Andrews, Holton, 

Leovy, 19871. Near the surface of the earth convection and adiabatic expansion of the atmosphere cause the temperature of the atmosphere to decrease 

with height [Kato, 19801. The temperature minimum near 12 km is caused 

primarily by infrared emission of H2O vapor. The region of the temperature 

decrease is known as the troposphere and the region where the minimum 

occurs is known as the tropopause. Above the tropopause the atmospheric 

temperature rises to a maximum near 50 km. The region between about 10 

and 50 km where the temperature rises is known as the stratosphere and this 

is due mainly to the absorption of solar ultraviolet radiation by 03 [Andrews, 

Holton, Leovy, 19871. The maximum near 50 km is called the stratopause 

and occurs because the 03 concentration decreases rapidly above this level 

and reduces the solar energy absorption. Above the stratosphere the temperature decreases again primarily due to infrared radiative cooling of COa 

[Kato, 19801. The temperature continues to decrease to a minimum near 85 

km where the temperature again again begins to increase due to radiative solar heating by dissociation of molecular oxygen and by ionization of 0 [Kato, 

19801. The region between 50 and 85 km where the temperature decreases 

is known as the mesosphere and the region of the temperature minimum at 

about 85 km is known as the mesopause. The region above 85 km where the 

temperature increases is known as the thermosphere. Above the mesopause 

the amount of ionization of atomic species increases with height and the 
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185 

atmosphere can conduct electricity and interact with magnetic fields. This 

region is known as the ionosphere and its electromagnetic interactions play 

a dominant role in the atmosphere above 150 km [Volland, Myers, 19771. 

0.02 

Figure 3.1.1 Midlatitude temperature profile based on U.S. Standard Atmosphere (1976) [from Andrews, Holton, Leovy, 19871. 
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111.2 Fluid Equations of Motion 

The atmosphere's circulation is governed by the Navier-Stokes equation 

of fluid dynamics which can be expressed, as seen in a rotating coordinate 

frame, by [Pedlosky, 19871 

Du 

P [ +2a x u] = -VP+pV@ +7 

where u is the velocity as observed in the rotating frame, the mass density 

of the fluid is defined by p, the pressure is given by P, the constant angular 

velocity of the earth's rotation is 52 = 7.272 x the potentia3 of 

conservative forces is a, and other non-conservative forces such as friction 

are represented by the term 3. The factor 2fl x u, the Coriofis acceleration, 

comes about from the coordinate transformation to the rotating frame of the 

earth. The total time derivative D/Dt is given by 

DB 

-- - -+u*v 

Dt - 6t 

and represents the total time rate of change of any property following a fluid 

element. The conservation of mass is governed by the continuity equation 

and this is commonly expressed as [Pedlosky, 19871. 

BP DP 

- + V*(pu) = - +p v. u = 0. 

at Dt (3.3) 
187 

For the atmosphere, the ideal gas law is used for the equation of state. 

This is given by the relation 

P P=-RT M (3.4) 

where R is the gas constant, Tis the kinetic temperature and M is the average 

molecular weight. The energy balance of the atmosphere is described by the 

thermodynamic equation [Pedlosky, 19871 

- 

D9 MdN = -- 

Dt CpT 

where the potential temperature 9 is defined as 

RICP 

JET($) . 

(3.5) 

(3.6) 

Here Po is a constant reference pressure at temperature To and 31 is the 

heating rate per unit mass. Since Cp = R + Cv where CV and Cp are 

the specific heats at constant volume and pressure respectively, then n 

R/Cp = 1 - 7-l where 7 = Cp/Cv = 1.4 is the ratio of specific heats 

[Pedlosky, 19871. The five relations given in (3.1, 3, 5) and the equation of 

state (3.4) are used to describe atmospheric dynamics at all time and length 

scales. 

The equations of motions (3.1, 3, 5) can be transformed into spherical 

coordinates representing the height z above the earth with radius a, latitude 

8 and longitude 4. We denote the eastward or zonal component of the velocity 

by u, the northward or meridonial component of the velocity by u, and the 

upward or vertical component of velocity by w. The zonal, meridonid and 

vertical momentum equations are [Pedlosky, 19871 
I 188 

Du wu vu 

Dt a a 

- + - - - tan d - 252 sin d v + 252 COS d w 

(3.7) 

1 ap F+ 

=- -+---, 

- + - + -tan d + 252 sin d u = --- + -9 Dt a a Pa ad P 

pacosd 84 p 

(3-8) 

Dv wv u' 1 ap 3# 

(3.9) 

respectively where the gravitational acceleration is given in (3.1) by V9 = 

-gi 'v -9.8 ms-' i. In this coordinate system the continuity equation (3.3) 

is 

aw 2w 1 

These equations (3.7-10) along with 

(3.10) ( z(v 8 cos e) + 5)) = 0. 

84 

the thermodynamical relationship (3.5) 

D3 

-- --- 37i 

Dt Cp T (3.11) 

and the spherical coordinate representation of the time derivative operator 

(3.2) 

(3.12) 

are used to describe the dynamics in the spherical geometry of the earth. 
189 

111.3 Hydrostatic State and the Thermal Wind 

The simplest static solution to the dynamical equations (3.7-11) is the 

time independent hydrostatic case where the atmospheric velocity vanishes 

everywhere u = 0. This state can only occur if there are no horizontal temperature gradients. Assuming that this state represents the background is 

what is known as the hydrostatic approximation. If we assume that nonconservative forces vanish then from (3.7-11) the conditions for this solution 

are 

(3.13) 

(3.14) 

(3.15) 

'H = 0. (3.16) 

The hydrostatic pressure field (denoted by the 'h' subscript) is independent 

of longitude 4 and latitude 8. The height dependence of the hydrostatic with 

height z varies according to (3.15). By this relationship and the equation of 

state (3.4) the hydrostatic density and temperature (sirnilax to Figure 3.1.1) 

must also only depend on the height z and the heating 31 must also vanish 

for the hydrostatic state to be realized. 

The hydrostatic case is a highly idealized approximation to the true atmospheric state. Figure 3.3.1-2 show the zonal mean temperatures for altitudes of up to approximately 160 km respectively during equinox and solstice 

conditions [from Salby, 19811. The average over latitudes is similar to the 

temperature profile in Figure 3.1.1 and there are variations between seasons 
190 

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as well. From these data we see there are significant deviations of up to 

10% in the temperature at a given height between different latitudes for each 

season and that the zonally averaged temperature background Tb is not independent of latitude 8. This temperature variation violates the conditions 

for the hydrostatic state which implies that the velocity field will not vanish 

and the horizontal temperature gradients induce a zonal velocity field. This 

is called the thermal wind and by assuming this to be the background state 

is known as the thermal wind approximation. 

The simplest case for the thermal wind is a steady state (a/& = 0) 

zonally symmetric solution (8/84 = 0) from the temperature data. The 

zonal mean wind can be derived from the latitudinal and vertical temperature gradients and assumes that the vertical and meridonial velocities vanish 

[Andrew, Holton, Leovy, 19871 u = w = 0. In the absence of dissipation, the 

dynamicd equations become (3.7-11) 

(3.17) 

(3.18) 

(3.19) 

(3.20) 

?i = 0. (3.21) 

These equations show that the fields are zonally symmetric and exist in the 

absence of heating. A differential equation which can describe the zonal 

velocity ub in terms of the background temperature Tb can be derived kom 

these equations. By combining (3.18-9) and using the equation of state it can 
191 

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be shown [Kato, 19801 that to a first (linearized) approximation the zonal 

mean winds ub can be derived from the zonal mean temperature Tb by the 

thermal wind equation 

(3.22) 

The differential equation which describes the relationship between the pressure and density can be determined by combining (3.18-9) 

afi coteah - + -- = -fig. 6% a a9 (3.23) 

The mean zonal wind determined from the mean temperature profiles 

in Figures 3.3.1-2 are given in 3.3.3-4 for equinox and solstice conditions 

respectively [after Salby, 19811. Measurements of zonal mean temperatures 

are routinely conducted during the va.rious month’s of the year [see Barnett, 

Carney, 19851. The zonal mean winds change as a function of height and 

latitude with the seasons. An eastward jet of about 50 ms-l develops in the 

southern hemisphere during the equinox near 50’ S around 9 scale heights 

or 60 km and continues to increase to about 80 ms-l during the southern 

winter. The eastward maximum zonal wind of about 70 ms-’ between 40 

and 80 km in sunrise zonal wind profile SR02 in Figure 2.13.13 determined 

from the ATMOS experiment can be explained by this jet near 50’ S since 

the mission was flown about one month after the vernal equinox. The sunset 

wind speeds SS06, SSO9, SSll and SS13 in Figure 2.13.12 show an eastward 

flow of about 20 ms-l near 50 km. This is about 10 ms” less than the value 

predicted for the equinox conditions of about 30 ms-’. However, between 

the equinox and summer solstice the zonal wind speed will decrease with 
192 

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time and changes direction which is evident by comparing the zonal winds 

in Figures 3.3.3-4 at 30' N. 

The thermal wind due to latitudinal gradients of the zonal mean temperature can explain the eastward wind below 70 km which was observed 

observed in the ATMOS wind data. The consistency the zonal winds features of the sunset wind profiles in Figure 2.13.12 can be explained by the 

zonal symmetry of the thermal winds. Thermal wind features below 70 km 

can be expected in future ATMOS missions. Seasonal as well as latitudinal 

variability in the zonal winds should be expected [Barnett, Corney, 19851. 

The ATMOS wind profiles confirm the steadiness of the zonal wind over a 

wide range of longitudes. The deviations of the averages of the four sunset 

profiles in Figure 2.13.15 are between 5 and 10 ms-l below 60 km. This observation may be useful for calibrating spectra in which instrumental lines are 

not available for calibration 811.1 since bith observation and theory indicate 

that the zonal mean winds are very steady and predictable. By assuming 

that the thermal wind is steady, it may be useful in future missions to adjust the wavenumber calibration of the spectra to obtain the best mutual 

correlations with wind speeds below 60 km. 

The temperature profiles Ta in Figures 3.3.1-2 and the corresponding 

zonal winds q, in Figures 3.3.3-4 show that the atmospheric state is highly 

non-uniform with latitude and height. Above 13 scale heights or about 85 km 

the thermal wind vanishes since the zonal mean temperature in thermosphere 

effectively becomes independent of latitude [Salby, 19811 which can be seen 

in Figures 3.3.1-2. However, the ATMOS measurements indicate that there 

exist winds with zonal speeds of over 100 ms-' and large wind shears. This 
193 

cannot be explained by the thermal wind. We can show that time dependent 

global waves play an important role in the circulation of the middle and upper atmosphere. From the perspective of modeling atmospheric circulation, 

the mean zonal-wind and temperature profiles can serve as a background 

upon which time dependent fluctuations occur [Lindzen, Hong, 1974; Waltersheid, Venkateswaran, 19791. Rossby waves, which are homogeneous free 

atmospheric oscillations having periods on the order of days, can interact 

with the zonal-mean circulation and modify the zonal-mean winds predicted 

by the thermal wind equation (3.22) [Dickinson, 19691. Atmospheric tides 

[Chapman and Lindzen, 19701, forced by the periodic heating and cooling of 

atmosphere by absorption of ultraviolet solar radiation by ozone and water 

vapor, have periods of a day and fractions thereof and interact with Rossby 

waves and the zonal mean flow to produce net accelerations as well [Teitelbaum, Vial, 1981; Zurek, 19861. The mean zonal winds conversely affect 

the structure of the planetary waves and tides [see Lindzen and Hong, 1974; 

Waltersheid, Venkateswaran, 19761 and these types of atmospheric activity 

may be highly interdependent. 

This section has shown that the eastward winds of about 30 m8-l and 

70 ms-l in the northern and southern hemisphere ATMOS wind profiles 

below about 70 km can be described by the regular thermal wind associated 

with latitudinal temperature gradients of the lower and middle atmosphere. 

However, these represent are slowly varying with a time scale of the order of 

the seasons. Other time dependent effects also occur the wave like character 

of the ATMOS wind profiles $11.13 may be caused by other dynamics. These 

are discussed in the next few sections. 
194 

lo) TEMPERATURE 1.I) 

Figures 3.3.1 Zonal mean temperaturea for equinox conditions [-om Sa 

19811. 

'Y9 

Figures 3.3.2 Zonal mean temperatures for winter soltice conditions [fiom 

Salby, 19811. Summer soltice conditions are approximately the same with by 

interchanging N and S. 
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LATITUDE 

Figures 3.3.3 Zonal mean winds for equinox conditions. Solid contours are 

positive (eastward) speeds while dashed contours are negative (westward) 

speeds [from Salby, 19811. 

=Ibl 

7 

ZONAL VELOCITY (nlul 

'I , I 1 I 

I6 "t I 

LATITUDE 1-l 

Figures 3.3.4 Zonal mean winds for winter soltice conditions. Solid contours 

are positive (eastward) speeds while dashed contours are negative (westward) 

speeds [from Salby, 19811. Summer soltice conditions are approximately the 

same with by interchanging N and S. 

i I 
196 

111.4 Linear Waves on a Hydrostatic Spherical 

Atmosphere 

We have seen that winds occur in the upper atmosphere. Indeed, the 

atmosphere can support a variety of free and forced wave motions. In the next 

few sections we briefly review the linear wave theory of global scale dynamics. 

The waves which have meteorological significance range from acoustic gravity 

waves [Kato, 19801 which are free oscillations with periods of several hours or 

less due to stratification of the background temperature, atmospheric tides 

[Chapman and Lindzen; 19701 with periods of a day or less and planetary 

or Rossby waves with periods of several days or more [Andrews, Holton, 

Leovy, 19871. Of these three types of waves the latter two, atmospheric 

tides and Rossby waves, are significantly affected by the variation of Coriolis 

acceleration with latitude 6. I will not examine acoustic gravity waves in 

this report due their limited horizontal length scale [Gossard, Hooke, 1975; 

Kato, 19801. However, they can be important in decting the zonal mean 

circulations due to wave breaking [F'ritts, 19861 and subsequent transfer of 

momentum due to turbulent eddy diffusion [Andrews, Holton, Leovy, 19871. 

We will examine, therefore, waves which are of the greatest importance for 

the large scale circulation of the middle atmosphere since it is not possible 

to examine all waves in detail. 

We shall only examine waves in an atmospheric background state which 

is vertically stratified without horizontal temperature gradients. This corresponds to the hydrostatic approximation. It was mentioned that the actual atmosphere is not in a hydrostatic state and possesses a mean thermal 
197 

wind. The presence of background winds can affects the way atmospheric 

waves propagate through the atmosphere [Lindzen, Hong, 1974; Waltersheid, 

Venkateswaran, 19791. Although background winds tend to complicate linear wave calculations, qualitatively the results are similar to those obtained 

using the more tractable hydrostatic background approximation and can be 

found in the literature [Lindzen, Hong, 1974; Waltersheid, Venkateswaran, 

19791. 

The system of linear equations describing fluid motions on a sphere was 

originated by Laplace (1799, 1825) and is useful for understanding atmospheric tides and global modes (Rossby waves). The purpose of the following 

discussion is to examine some of the most important solutions. 

The nonlinear equations which describe global scale dynamics are described in Appendix A. The final equations (A.61-7) are presented in dimensionless form and represent the behavior of the system in terms of the length 

and time scales for global scale disturbances. The scaling of the variables 

shows which terms in the dynamical equations are of primary and secondary 

importance. These derivations, although extensive, are the underlying basis 

for the linear wave theory of disturbances on a vertically stratified rotating 

spherical atmosphere. The description of linear wave theory which follows is 

based on the scaling derivations. 

Linear atmospheric wave theory is derived from the equations of motion ' 

(A.61-7) by ignoring the nonlinear terms to a first approximation. The linear balance equations can then be written. The continuity equation (A.62) 
198 

i becomes 

(3.24) 

The horizontal momentum equations (A.63-4) axe 

the vertical momentum equation (A.65) is 

p=(A-g)P-A-, 6P 

82 

the thermodynamical relation (A.66) is 

as 

at 

EF- +h/l w = 31, 

(3.25) 

(3.26) 

(3.27) 

(3.28) 

and the relationship between the potential temperature, pressure and density 

perturbations (A.67) is 

(3.29) 

The derivation of these equations is described in Appendix A. Basically, in 

dimensionless format, the zonal, meridonial and vertical velocity components 

are represented by ut, v‘, and w respectively while the pressure, density and 

potential temperature perturbations about the hydrostatic state are represented by P, p and 9 respectively. The vertical coordinate 1: defined in 

(A.54) represents the number of atmospheric e-folding scale heights above 

1 

7 

t9= -AP-p. 

the surface, and a latitude coordinate has been introduced p = sine. The 

hydrostatic temperature distribution is contained in the function A (A.22) 
c 

199 

I 

I 

and the dimensionless local bouyancy frequency Af (A.49; see Appendix B). 

Time dependent external heating is represented by the dimensionless function 3c (A.52). This linearized approximation is used to describe atmospheric 

dynamics provided that the (dimensionless) variables have magnitudes much 

smaller than unity. 

The standard procedure [Siebert, 1961; Chapman, Lindzen, 1970; Salby, 

1979; Kato, 1980; Andrews, Holton, Leovy, 19871 for determining the solutions to the set of equations (3.24-9) is to pose the form 

I 

x exp[i(ct + s4)I). (3.30) 

The zonal wave number s must be an integral value due to the periodicity of 

the azimuthal coordinate 4. The time t is the dimensionless representation 

described in the Appendix A, the rotational period of the earth being 2n in 

this representation. The (dimensionless) frequency number Q is continuous 

in general. 

We define the number f E QE = u/2 for convenience. The periodic 

horizontal momentum equations (3.25-6) are then 

ifu'-pv' = -isP, (3.31) 

6P 

ifv'+ p u' = - (1 - Pa) --, (3.32) 

By using these equations the horizontal velocity field (u',v') can be determined in terms of the pressure P 

(3.33) 
200 

(3.34) 

It is important to note that for f 5 1 which corresponds to atmospheric 

tidal distrubances [Chapman, Lindzen, 19701 there are two latitudes where 

p = kf and this causes the denominators in (3.33) and (3.34) to vanish. The 

latitudes, which depend on the frequency of the wavesthrough f, where this 

occurs are called the critical latitudes. Brilluon (1932) first showed that the 

numerators in (3.33-4) also vanish at these latitudes and they do not impose 

any theoretical difficulties. For the diurnal tide 5111.6 the critical latitude is 

at 30" N and S. 

Following Siebert (1961) we define the dimensionless divergence 

aw awl is ' X~V.U=A-+-+ - U. az ap 1-Pi (3.35) 

Since ut and w' depend on the pressure P the horizontal part Xh of the 

divergence (3.35) is 

aut is Xh f - + - ut = if CP ap i-pl (3.36) 

where the operator C is defined as 

by substituting (3.33-4) into (3.36). The relationship (3.35) can be written 

as 

aw 

az X - A-= ifLP. (3.38) 

The remaining three relations, namely the vertical momentum (3.27), continuity (3.24), and thermodynamical (3.28) equations become 

(3.39) 
WP- (A - 2) w +x = 0, 

20 1 

(3.40) 

and 

ifFS+APw = IH (3.41) 

respectively. From (3.29) we can solve for the density p and substitute into 

(3.39-40) to obtain the vertical momentum and continuity equations 

ap 

ax ' 9 = -PP + A- 

-ifFAP 1 - ifFS - (A- 2) w+X=O. 

7 

(3.42) 

(3.43) 

The continuity equation (3.43) and thermodynamic equation (3.41) can be 

added to yield 

(3.44) 7 

A if FP = w - - (X - IH) . 

This equation can be differentiated with respect to z (A.54) to give 

The vertical momentum equation (3.42) can be rearranged 

ap 

ax A-=S+h/lP. 

(3.45) 

(3.46) 

We now solve for 9 in (3.41) and substitute this and (3.44) into (3.46) to 

show 

This result can then be substituted into (3.45) to give 

aw ax 

ax ax 

A- = 7 [- - 

(3.47) 

(3.48) 
202 

II 

1 

I 

1 

1 

Differentiating this result with respect to z gives 

(3.49) A-A-=7A[-- a aw 

ax ax 

If we now differentiate (3.38) with respect to z 

ax 6 aw 8P A- - A-A- = ifLA-. ax 6z 8z ax (3.50) 

We now substitute (3.47) and (3.49) into this equation to obtain the following 

differential equation for the dimensionless divergence X 

- . (3.51) alx - - ax - (= 82% - $) = Fb'L 1 [flX - (A - 2) 'HI 

az2 az 

This equation can be further simplified by introducing an auxiliary function 

y, which we will also term the vertical structure function, as 

x ye=/= 4- 'H (3.52) 

Equation (3.51) then transforms to 

(3.53) 

This equation can be solved by using the method of separation of variables 

as described by Siebert (1961) and Chapman and Lindzen (1970). Consider 

a set of functions {Ofi'(p)} for n = 0,1, ... which is assumed to be complete 

over the interval -1 5 p 5 1 for each chosen pair u,a. Then the vertical 

structure function y and heating 'H can be expanded on this complete set by 

(3.54) 
203 

By calling the separation constant -Xr for each n component the following 

pair of equations can be derived 

Lor + or = 0, (3.56) 

(3.57) 

The eigenvalue problem for the operator L in (3.56) is known as Laplace’s 

Tidal Equation [Longuett-Higgins, 1968; Chapman and Lindzen, 19701 since 

it was originally derived by Laplace (1799, 1825) to describe the free oscillations of an ocean. The eigenfunctions Or, commonly called Eough functions (after Hough, 1898), form a complete orthogonal set for each set u, 

8. A method for computing the eigenfunctions and eigenvalues described 

in Appendix C is based on a method similar to one described by LonguettHiggins (1968). The orthogonality of the Hough functions is also proved. 

The equation (3.57) is called the vertical structure equation [Siebert, 19611. 

The solution for yn is determined by the eigenvalue Xr and the heating term 

Wr together with appropriate boundary conditions. 

The eigenvalues of (3.56) are related to the frequency of the mode through 

Q and, for each azimuthal mode number 8, a relationship between the eigenvalues X and u can be determined. This relationship for the values of 

-1 5 8 5 +2, as computed by Longuett-Higgins (1968), is shown in Figures 

3.4.1-4. The modes where 8 > 0 (8 < 0) are called westward (eastward) 

migrating modes since the longitude 4 of constant phase decrease (increase) 

as time t increases. Negative values of the eigenvalue X exist only for modes 

with u c 1 [Longuett-Higgins, 1968; Andrews, Leovy, Holton, 19871. 
8 

8 

I 

! 

:a 

204 

Once the vertical structure equation (3.57) has been solved for each eigenmode n we can determine the physical fields. In terms of the Hough function 

expansion, equation (3.38) can be written 

dwn X, -A- = -ifArP, dx 

and from equation (3.48) 

(3.58) 

(3.59) 

By using (3.52) these can be combined to give the pressure in terms of the 

vertical structure function yn 

With this result and equation (3.44) the vertical velocity becomes 

Combining (3.60) with (3.59) the density is 

X 

The velocity fie ds u’ and V’ are determined from equations (3.33-4 

(3.60) 

(3.61) 

(3.62) 

(3.63) 

(3.64) 
II 

205 

where 

I' 

IS 

I 

.1 

I 

(3.65) 

(3.66) 

The vertical components of the physical fields (3.60-4) along with the 

Hough eigenmodes 0, and the horizontal velocity modes UA and V,l describe 

the time and space solutions of the linear wave equations for fluid motion on 

a spherical atmosphere. 

The linear theory serves as tool by which periodic atmospheric disturbances may be studied. In the sections that follow we will show that both 

homogeneous solutions and nonhomogenous solutions of the vertical structure equation (3.57) exist and that planetary waves and atmospheric tides 

are representative of these kinds of solutions respectively. It will be shown 

that the homogeneous solutions, the planetary waves, are not likely to be 

responsible for the upper atmospheric winds observed using the ATMOS instrument. However, the nonhomogeneous solutions give rise to perturbations 

from the mean state large enough to account for the global extent and the 

magnitudes of the winds observed above 70 km by ATMOS. 

i 
206 

, 

I 

Figure 3.4.1 Eigenfrequenciea of eastward traveling modes when 8 = -1 

[adapted from Longuet-Higgins, 19681. The frequency number u (the number 

of oscillations per day) is plotted against the eigendue A for various modes. 

i 

! ' %'* i 

I 

0- 

04 

Figure 3.4.2 Eigenfrequendes u/2 of stationary modes when 8 = 0 [adapted 

from Longuet-Eggins, 19681. The frequency number u (the number of oscillations per day) is plotted against the eigenvalue A for ~OUS modes. 

b 

I 
1 

1 

I 

1 

I 

I 

I 

8 

.I 

8 

I 

8 

.I 

I 

i8 

8 

oa 0 

207 

E fi 

Figure 3.4.3 Eigenfrequencies u/2 of westward traveling modes when n = 1 

[adapted from Longuet-Higgins, 19681. The frequency number u (the number 

of oscillations per day) is plotted against the eigenvalue X for various modes. 

The horizontal line at u = 2 intercepts the dispersion curves which gives 

the diurnal eigenvalues An. The Rossby 5-day wave @ and 16-day wave @ 

are marked on the intersection of the dispersion curves with the atmwpherc 

eigenvalue At) = 8.88 giving the dimensionless frequcnau c of orcillation. 

.(A Figure 3.4.4 Eigenfrequencies u/2 of westward traveling modes when n = 2 

[adapted from Longuet-Higgins, 19681. The frequency number u (the number 

of oscillations per day) is plotted against the eigenvalue X for various modes. 

The horizontal line at u = 2 intercepts the dispersion curves which gives the 

semi-diurnal eigenvalues An. The leading Hough mode is marked @. 
208 

$8 

1 

8 

8 

111.5 Free Planetary Waves - The 

Solutions to the Linear Wave 

Homogeneous 

Equations 

The linear wave theory described in the previous section is used to describe both free and forced oscillations of the atmosphere. One set of solutions 

to the linear wave equations (3.24-9) is the homogeneous solution in which 

the atmosphere undergoes free oscillations. These kinds of waves, also known 

as free traveling planetary waves or Rossby waves[Andrews, Leovy, Holton, 

19871, can exist throughout the lower and middle atmosphere in the absence 

of external forcing and have been well-documented. They appear in surface 

pressure data [Madden, 19781, upper tropospheric radio-sonde data [Madden, Labitzke, 1979; Venne, Stanford, 19791 and in stratospheric satellite 

data [Rodgers, 1976; Hartmann, 19761. 

The most prominent of all of the atmospheric free oscillations is the westward traveling 5-day period wave with zonal wavenumber a = 1. Other 

traveling modes have also been observed, among them are the a = 1,16-day 

wave [Madden, 1978; Madden, Labitzke, 1981; Madden, 19831 and the a = 3, 

2-day wave [Salby, 1981; Rodgers, Prata, 19811. Figure 3.5.1 shows the latitude variations of the temperature perturbations from the mean state for the 

5-day component determined by Rodgers (1976). There are amplitude maxima of about 0.5 K near 50° N/S. Figure 3.5.2 shows a five day component 

of the pressure variation as observed in the stratosphere over the surface of 

the earth [Hirota, Hirooka, 19841. The patterns show the d = 1 character 

of the waves with westward traveling surfaces of constant phase. The wave 

structure is symmetric about the equator and has amplitude maxima near 

I 
209 

8 

1 

I 

R 

I 

1 

1 

8 

I 

I 

~ff 

I 

*I 

I 

50" N/S latitude in agreement with Rodgers (1976). 

The existence of a wave with a period of 5 days and zonal wavenumber 

a = 1 in the earth's atmosphere is in agreement with the theory of free 

planetary modes [Salby, 19791 and is a homogeneous solution of the linear 

wave theory 5111.4. To determine the nature of the solutions, we solve (3.57) 

in the absence of heating 3.1 = 0 [Salby, 19791 

after defining the auxiliary function 

Ar 

FAa' WG- 

(3.67) 

(3.68) 

The homogeneous equation (3.67) is considered to be an eigenvalue problem 

for the parameter A [Salby, 1979; Andrews, Leovy, Holton, 19871 subject to 

appropriate boundary conditions. The value(s) of A depend8 on the vertical temperature structure through the function w. The frequency number u 

which corresponds to the eigenvalue A for each given a are then computed 

from the dispersion relations portrayed in Figures 3.4.1-7. This method provides the frequency for the free oscillations. The latitudinal Hough functions 

@r are then determined using the method in Appendix C. In Figure 3.5.1 the 

Hough function for the 5-day wave is plotted along with the observed temperature fluctuation amplitude [Rodgers, 19761 and the similarity between 

the peaks near 50' N/S and the minimum at the equator is striking. 

A single value of eigenvalue A for free oscillations of the atmosphere of 

about 8.88 was determined shown to exist for free atmospheric oscillations 

from the ATMOS standard temperature profile in Figure 3.5.3 by the method 
210 

described in Appendix D. The corresponding frequencies of oscillation can be 

determined from the graphs of the dispersion relations (see Figures 3.4.1-7). 

The value of the Lamb mode eigenvalue A = 8.88 is plotted on the 8 = 1 

dispersion relation in Figure 3.4.5. The point marked 0 corresponds to the 

eigenfrequency of u = 0.099 which corresponds to a period of about 5.1 days. 

Since this wave is solution to the homogeneous problem it can propogate 

without the need for external forcing. The 5-day wave is believed to be 

excited by random disturbances in the atmosphere and fluctuations of mean 

winds [Andrews, Leovy, Holton, 1987) and can persist for weeks at a time 

[Salby, 1979). The 5-day mode as well as the other homogeneous planetary 

waves are 'simply the result of the hydrostatic distribution of mass in the 

atmosphere, the earth's rotation, and spherical geometry' [Salby, 19811. 

We can study how free traveling planetary waves can influence observed 

wind speeds. The vertical structure equation (3.67) is homogeneous for free 

oscillations and its solutions are unique to a multiplicrrtive constant. This 

multiplicative constant can be determined by adjusting the amplitudes of the 

surface pressure oscillations predicted by the theory to fit observed amplitudes. An analysis by Burpee (1976) indicates that the component of the 

surface pressure oscillation identified with the 5 day wave is slightly less than 

1 mb near 10-20" N latitude and some of the surface pressure data in that 

study is shown in Figure 3.5.4. The five day oscillating component is evident in the figure. Using these observations, the amplitude of the computed 

pressure fluctuation from the linear wave theory 5111.4 was adjusted to the 

experimental values to have an amplitude of 1 mb at 15" N. The fractional 

deviation of the pressure from the hydrostatic value (given by P'/% = eFAP 
8 

8 

I 

8 

I 

1 

8 

I 

I 

t 

211 

in Appendix A) as function of the number of scale heights ?: and p (= sin 8, 

8 =latitude ) is shown in Figure 3.5.5. The amplitude P'/% remains below 

10 % until about t = 17 which corresponds to about 120 km altitude. The 

corresponding zonal wind speed u is shown in Figure 3.5.6. The zonal winds 

are less than 1 ms-l below z = 17 or 120 km altitude at all latitudes for the 

5- day wave. 

We therefore do not expect planetary waves to significantly influence the 

zonal wind speeds at mesospheric heights and below and conclude that this 

kind of atmospheric disturbance is not likely to be responsible for the large 

variations in winds measured during the ATMOS mission. We must therefore 

look for other effects. In the next section we show that the nonhomogenous 

solutions to the linear wave theory 5111.4 which are driven by solar heating 

of the atmosphere can indeed influence the circulation of the atmosphere in 

a significant way and is likely to be responsible for the wave-like features 

observed consistently with the ATMOS instrument. 
I 

I 

8 

I 

I 

x 

212 

r I I I 1 

LATI 1 UE 

Figuxe 3.5.1 Solid curve: temperature amplitude of the 5-day wave as a function of latitude at about 42 km altitude for November 1973, as measured by 

the Selective Chopper Radiometer on the Nimbus 5 satellite [after Rodgem, 

19761. Broken curve: latitudinal structure of the Hough function corresponding to the 5-day Rossby mode [from Andrews, Holton, Leovy, 19871. 
1 

I 

I 

8 

I: 

1 

8 

8 

1 

-i 

213 

IO 80 IO 

I ‘0 40 I ‘0 40 

zo 0 :o 

- -.o -40 - -.o 

I 0 -1 

IO 

> 

- - . 0 

41 

-80 -10 -80 -10 

0 

LOMC I 1YOl * IO 8C 80 eo 

I ‘0 40 I ‘0 40 

zo - 0 :o - 0 

*go - -ro -a3 

-10 -IC -IO 

0 -110 -93 0 90 110 

- -.o 

-10 

I 

LO*CIlUOi LOMCI tuoc 

Figure 3.5.2 The 5-day wave at 1 mb or about 45 km as observed by the 

Stratospheric Sounding Unit on the TIROS-N satellite, for 6 successive days 

in August-September 1980 [Hirota, Hirooka, 19841. The wave-number 1 

Fourier component of the geopotential height anomaly (which can be directly 

related to pressure - see Houghton, 1987). The westward-travelling pattern 

except south of 50” S. Shaded areas denote negative anomalies; contour levels 

are 20 m [from Andrews, Holton, Leovy, 19871. 
0 

214 

T 2&) 

600 

Figure 3.5.3 The ATMOS standard temperature profile used for the hydrostatic temperature background in the tidal calculations [kom Norton, 19861. 

- : !.: 

.. ... ,.. .. 

. ,.. .. .. 

:;: 

.. , .. , 

Figure 3.5.4 Time series of surface pressure averaged for groups of stations 

in 5" longitude bands from 10"-20" N in the Caribbean [from Burpee, 19761. 

I 
1 

1 

I 

1 

8 

1 

1 

I 

8 

1 

'I 

I 

.:! 

8 

'1 

I 

!I 

1 

:I 

I 

215 

Figure 3.5.5 Fractional deviation of the pressure from the hydrostatic background pressure Fi, as a function of number of scale heights t. Notice that 

the deviation remains below 10 % until z - 14 which corresponds to about 

120 km. This type of oscillation does generate significant winds until great 

heights are reached (see Figure 3.5.6). Notice the two peaks at 50' N/S which 

agrees with the data presented in Figures 3.5.1-2. 

Figure 3.5.6 Computed zonal wind speeds as a fdction of number of rcale 

heights t for the 5-day Rossby wave. Notice that the deviation remains below 

5 ms-' until z - 18 which corresponds to about 140 km. 
216 

111.6 Atmospheric Thermal Tides 

I 

I 

1 

1 

8 

1 

I 

‘,I . 

In this section we show that forced tidal oscillations, due to the periodic heating and cooling of the atmosphere, can give rise to large perturbations of the pressure, density, temperature and velocity about the mean 

background state. Atmospheric thermal tides are forced global-scale daily 

oscillations. They are driven by the diurnally varying heating of the earth’s 

atmosphere due to ultraviolet radiation absorption by water and ozone [Andrews, Leovy, Holton, 19871 between the surface and up to about 75 km 

[Chapman, Lindzen, 1970; Groves, 1982a-b]. 

The energy absorbed by the atmosphere is transformed into mechanical 

energy which is transported to great altitudes [Chapman and Lindzen, 19871. 

Energy is also radiated away by infrared cooling (also known as radiative 

damping) primarily by the 15 pm band of CO,, [Dickinson, 19841 which 

tends to restore the atmosphere to its undisturbed background temperature 

distribution [Chapman and Lindzen, 1970; Kato, 19801. The first mechanism 

is usually incorporated into the boundary conditions of the vertical structure 

equation (3.57) described in Appendix D. Radiative damping is contained in 

the heating function 7i and its inclusion in the linear wave theory 5111.4 is 

described in Appendix J. 

The diurnal heating of the atmosphere with respect to local time can be 

represented by a distribution similar to Figure 3.6.1. The heating vanishes 

on the ‘dark’ side of the planet while it has a maximum near local noon 

and decreases to zero at local sunrise and sunset. A Fourier analysis of 

this distribution yields a steady component, a diurnal component (with a 
217 

24-hour period), a semidiurnal component (with a 12-hour period), and so 

on. In order to estimate the atmospheric response a knowledge of the Fourier 

components of the heating is necessary. The phase of the heating ‘follows’ the 

sun [Seibert, 1961; Chapman and Lindzen, 1970; Andrews, Leovy, Holton, 

19871 and it is easily verified that the frequency number u and the zonal 

number 8 in (3.30) are equal in this case. It can be verified that the phase 

of the heating and the linear response travels westward and hence the term 

migrating tides is used to describe this class of linear waves. The component 

Q = 8 = 1 is called the diurnal tide and the component u = J = 2 is called the 

semi-diurnal tide. The dispersion relation for the eigenfunctions of Laplace’s 

tidal equation are plotted for the diurnal and semi-diurnal modes in Figures 

3.6.5- 6 respectively. 

For each tidal mode described by Q, a complete set of Hough eigenfunctions 0, and eigenvalues A, can be determined by the method in Appendix 

C. The corresponding vertical structure equation (3.57) can be solved and 

the local time, latitude and height dependence of the tidal modes can be 

determined $111.4 provided the Hough components of the heating term ‘H are 

known. 

In Figure 3.6.2 is shown theoretical northerly wind speeds [Lindaen, 19671 

for a hydrostatic background model and rocket measurements [Reed, 19691 

for the diurnal component. The amplitudes of these two sets of data are 

similar, however, the hours of maximum amplitude vary with altitude. The 

discrepancies between the phases with altitude are probably due to the neglect of mean winds, errors in the heating and perhaps atmospheric cooling 

in the theoretical model as pointed out by Lindzen and Hong’s calculations 
218 

(1974) for the semi-diurnal components. In the lower atmosphere (below 60 

km) the linear tidal theory can be used to describe wind speeds since the 

amplitudes of the waves are small (horizontal speeds < 10-20 ms-l) and the 

linearization of (A.61-7) is a good approximation. 

To further understand the tidal phenomena, we can calculate the solution 

to the linear theory using a model heating function and temperature profile. 

The ATMOS standard temperature profile in Figure 3.5.3 was used to model 

the hydrostatic background. The heating model given by Groves (1982ab) was chosen since it includes results for diurnal and semi-diurnal heating 

components of both water vapor in the boundary layer and stratospheric 

ozone. The forcing model data describe the vertical structure of the heating 

'H, for each Hough component (see 3.55). The model also describes the 

heating components as function of the season. Since the ATMOS mission 

was flown near the vernal equinox the heating profiles corresponding to this 

time of the year was used. Figures 3.6.3 and 3.6.7 show the latitude and 

height dependences using (3.55) for the diurnal and semi-diurnal components 

respectively as given by the model [Groves, 1982-a-b]. 

The solutions to the linear theory $111.4 was applied to the diurnal and 

semi-diurnal modes using the heating rates by Groves (1982a-b). The calculated temerature and horizontal velocity components in Figures 3.6.46 and 

3.6.8-10 for the diurnal and semidiurnal components respectively. These results show the exponential increase of the amplitudes of the physical fields 

with scale height 1: as described by (3.60-6). From these figures we see 

that above 1: = 10 or about 70 km the upper atmosphere is extremely active mechanically. Dissipative mechanisms such as molecular viscosity in the 
219 

I 

I 

I 

I 

I 

I 

I 

I 

I 

1 

'I 

I 

:I 

I 

I 

I 

I 

.I 

-1 

lower thermosphere and infrared cooling by COa [Dickinson, 19841 in the 

stratosphere and mesosphere along with non-linear wave breaking near the 

mesopause [F'ritts, 19841 and subsequent generation of turbulence play'a role 

in damping the amplitudes of these waves. Radiative damping using the 

Newtonian cooling approximation [Chapman and Lindzen, 1970; Andrews, 

Holton, Leovy, 19841 can be added to the linear wave theory 5111.4 with 

little added complexity to the integration procedure (see Appendix J) and 

was used in my calculations. Lindzen and Hong (1974) have modeled mean 

winds as well as molecular viscosity and radiative damping effects in their 

calculations of semi-diurnal tidal perturbations and have found that velocity 

amplitudes reach a maximum value of about 100 ms-l about 150 km. 

Although the inclusion of radiative damping into the lineaz wave theory 

dampens the amplitudes by about 20 % [Chapman and Lindzen, 19701, it is 

not sufficient to prevent the growth of the waves with height. Other mechanisms become important. Large amplitudes of the potential temperature 9 

near 70 km [Andrews, Holton, Leovy, 19871 can cause local adiabatic instabilities since the static stability Na becomes negative and the local buoyancy 

frequency N becomes imaginary (see Appendix B). The atmosphere can then 

no longer support the wave and local turbulence and subsequent cascade of 

energy to smaller length and time scales occurs [Lindzen, 19811. At about the 

same altitude where adiabatic instabilities set in, non-linearities also become 

significant [Andrews, Holton, Leovy, 19871. When the horizontal velocity 

become of the order greater than 0.1 (in the dimensionless representation of 

Appendix A) or about 50 ms-l then advective instabilities due to the u - Vu 

terms in the fluid equation 5111.2 at wind shears become important [Wein- 
‘I 220 

stock, 19761. The linearized approximation $111.4 which was used to compute 

the atmospheric response to the diurnal heating of the atmosphere is then 

no longer adequate for describing the dynamics. 

From the results we conclude the linear theory calculations of the tidal 

response to diurnal atmospheric heating suggest that the zonal winds above 

70 km observed during the ATMOS mission are caused primarily by the tidal 

waves propagating into the thermosphere. The waves in this region break 

due to non-linearities and static instabilities generated by tidal perturbations 

[Lindaen, 1981; Andrews, Holton, Leovy]. Non-linearities in the theory become important and make the problem complex and hence difficult to treat 

theoretically. I have currently reached this point in my study of atmospheric 

tides. My study of the ATMOS wind profiles is by no means completed and a 

more general method for incorporating the non-linear and adiabatic breaking 

of the tides needs to be developed. 
221 

LOCAL TIME (hours). 

Figure 3.6.1 Variation of solar heating at a point in midlatitudes (heavy 

curve). The horizontal line represents the diurnal or zonal average of the 

heating. This is assumed to be balanced by a eonal-mean infrared cooling so 

that the net heating vanishes. The term H im represeated by the departure of 

the heavy curve from the dashed me [from Andrcwr, Holton, Leovy, 19871. 

- OBSERVED 

THEORE TIC ALL 

5 IO 0 IO 

AYPLITUDE fm B-') . WAS Oyyr of moa.) 

Figure 3.6.2 Observed northerly diurnal wind rpeeds in the stratosphere determined from rockets launched at Ascension Island (8"s) and theoretical 

calculations by Lindzen (1967) based on classical tidal theory [Reed, 19691. 
I 

I 

1 

E 

i 

~ 

I 

E 

P 

1 

f 

E 

I 

'1 

Y 

k 

222 

Figure 3.6.3 Diurnal heating rates per unit mass 31 as a function of scale 

heights z and latitude 8 used as the driving term in the tidal model for 

equinox conditions [adapted from Groves, 1982a-b]. There is heating due of 

water vapor near the surface 2 = 0 and ozone in the stratosDhere 4 < z < 12. 

Figure 3.6.4 Amplitude of the diurnal temperature perturbation determined 

using the linesr wave theory as a function of scale heights z and latitude 

8. The exponentially increasing amplitude of the temperature perturbation 

with scale height z is evident. 
I 

II 

i 

$8 

4 

I 

i 

1 

d 

223 

Figure 3.6.5 Amplitude of the diurnal zonal velocity perturbation determined 

using classical the linear wave theory as a function of scale heights 2 and latitude 6. This theory predicts that zonal winds can grow to great amplitudes 

> 100 ms" or more at mesospheric heights and propogate into the upper 

atmosphere. The great amplitudes determined here do not occur in the atmosphere, however, since nonlinear and adiabatic instabilities break the wave 

and prohibit its 

lag, v 

Figure 3.6.6 Amplitude of the diurnal meridonial velocity perturbation determined using classical the linear wave theory M a function of scale heights 

z and latitude 8. 
224 

Figure 3.6.7 Semi-diurnal heating rates per unit mass 3.1 as a function of 

scale heights z and latitude B used as the driving term in the tidal model for 

equinox conditions [adapted kom Groves, 1982a-bI. There is heating due of 

water vapor near the surface z = 0 and ozone in the stratosphere 4 < z < 12. 

Figure 3.6.8 Amplitude of the Semi-diurnal temperature perturbation determined using the linear wave theory as a function of scale heights 2 and 

latitude 8. The exponentially increasing amplitude of the temperature pcrturbation with scale height z is evident. 
225 

Figure 3.6.9 Amplitude of the semi-diurnal zonal velocity perturbation determined using classical the linear wave theory as a function of scale heights 1: 

and latitude 8. The semi-diurnal winds become large > 100 ms-l at greater 

altitudes than the diurnal winds, however, the exponential growth of the 

wave will eventually force it to break and prohibits its amplitude growth 

with scale height 2. 

Figure 3.6.10 Amplitude of the semi-diurnal meridonial velocity perturbation 

determined using classical the linear wave theory as a function of scale heights 

1: and latitude 8. 
U 

I 

E 

c 

1 

I 

I 

d 

i 

i 

t 

J 

I 

B 

a 

I 

I 

1 

i 

226 

SUMMARY 

I have presented many of the aspects of spectroscopy which were important for obtaining wind speeds in the upper earth's atmosphere. The primary 

objective was to determine whether or not such measurements were indeed 

possible considering the technical limitations of the ATMOS interferometer. 

These have been discussed in some detail, primarily as an instructive guide 

for future research with ATMOS or other similar instruments. 

During the course of the research I have determined that the instrument 

is adequate for obtaining accurate wind speeds to precisions of 5 m/s. The 

technique described, however, is somewhat limited in its ability to obtain 

as much wind data as possible during the course of what may be a typical 

ATMOS mission due to other priorities. The lack of instrumental lines available for calibration in all but Filter 3 means that only a fraction of the total 

number of occultations are available to obtain wind data. 

The calibration analysis indicated that the instrument was stable to better than 2 x lo-' cm-l from spectrum to -spectrum and does not reveal 

significant fluctuations. This stability indicated that the instrument wa8 

providing consistent frequencies. Comparison between positions determined 

from atmospheric lines via the instrumental calibration indicated that the 

ATMOS data can be used to improveme accepted frequency standards, particularly those line positions where the accuracy and/or precision are not well 

known. The atmosphere itself provides a natural laboratory for spectroscopic 

measurements of trace gases, many species which are difficult to observe on 
227 

earth laboratories. 

The instrumental lines which were present in the spectra indicated that 

the amount of COP present tended to decrease in time. This information was 

consistent with the theory of outgassing by the instrument. The equivalent 

width of the HzO lines appeared to remain relatively constantly throughout 

the mission. This information in addition to a Doppler shift of about +15 

ms-l away from the instrument suggests that the water vapor is outgassing 

from a source primarily external to the instrument while the CO2 is primarily 

located in the instrument, since its concentration decreased with time and 

no Doppler shift was measured. 

The winds obtained from the ATMOS spectra showed a similarity among 

the sunset occultations revealing consistent phase shifts and amplitudes. The 

source of this activity was studied from a theoretical point of view and the 

concept of the thermal wind and atmospheric tides was discussed. The study 

showed that models of the radiative transfer between the sun’s radiation, 

ozone and water vapor drive upper atmospheric circulation in a manner which 

can be considered to be consistent with the results of the ATMOS experiment. 

The regularity of winds measured during the sunset occultation indicates that 

mesospheric winds might indeed be induced by the diurnal and semidiurnal 

tides. The theory presented was found to be inadequate in describing the 

detailed structure of the winds since the amplitudes of the waves predicted by 

the theory grow exponentially with height due to the exponential decrease of 

the hydrostatic background density. The presence of an adiabatic instability 

in the theoretical model was shown to occur at about twelve scale heights or 

about 85 km above the earth’s surface. 
228 

The breakdown of the classical theory suggests that a more elaborate 

study into the nonlinear evolution of the tides at heights comparable to those 

studied by ATMOS is necessary. The response of the atmosphere to the 

heating of the ozone layer on a global scale suggests that upper atmospheric 

winds may contain much information regarding the latitudinal distribution of 

this species. A nonlinear theory of tides is probably need to better understand 

upper atmospheric wind patterns. The the vertical structure of atmospheric 

winds at each point in local time may contain information regarding the 

global distribution of the heating sources. 
229 

Appendix A. Scaling the Nonlinear Equations of 

Motion 

The dynamical equations in $111.2 can be used for many meteorological 

problems. If we wish to use them to describe motion over an entire sphere 

the dynamical equations (3.7-12) are used. The horizontal length scale is 

assumed be of the order of the radius of the earth a while the vertical scale 

is of the order of 100 km. The time scale will be of the order of the rotation 

period. Due to the latitude dependence of the Coriolis force a 28 sin 8, it 

can vary significantly over the horizontal scale. 

Due to the complexity of the fluid equations in spherical coordinates 

it useful to introduce several changes of variables. The latitude 8 can be 

transformed into the new coordinate 

where -1 < p < 1. Then the derivatives 

can be determined. It is also useful to redefine the horizontal velocity functions [Margules, 18931 as 
8 

t 

1 

I 

I 

I 

I 

1 

I 

230 

By substituting these definitions into the total time derivative (3.12) 

The continuity equation (3.10) becomes 

(A4 

and the horizontal momentum equations (3.7-8) are 

- + - - 20p v' + 20p&&v = --- a' + dG3, (A.7) Du' wu' 

Dt a Pa a4 P 

The vertical momentum equation (3.9) is 

Dw 1 1 18P 3.- 

Dt a l-pz P P 

---- (u" + va> - 20u' = --- -g+-. (A.9) 

We now wish to examine dynamics on the scales appropriate for global 

scale disturbances. Atmospheric oscillations can be classified by the time 

scale and horizontal distance scale over which the motions vary. In the case 

of planetary and tidal waves, the characteristic horizontal distance is of the 

order of the radius of the earth a. Since the horizontal extent of these kind 

of atmospheric disturbances is as large as it is, the Coriolis acceleration will 

vary significantly over the range of latitudes in which these kinds of global 

disturbances propagate. 

The problem is mathematically manageable by following a scaling technique similar to Pedlosky (1987). It is useful to introduce a change in notation. Dynamical fields will be represented in both dimensioned and nondimensioned form, a " superfix representing the dimensional form. For 
"Page missing from available version"
232 

Here, u', v', w, z, and t are dimensionless variables. The condition that the 

dimensionless velocities are much less than O(1) will be used in many of the 

arguments. The T* and t* derivatives become 

The total time derivative D/Dt* scales as 

where 

~a - a a a + W- + 0'- + -- Dt - at aZ ap 1--~2a4 

--- 

(A.12) 

(A.13) 

(A.14) 

(A.15) 

in terms of the dimensionless variables. 

The dynamics are superimposed on a hydrostatic background state. Let 

the pressure %(z) and density fi(z) represent this "standard" hydrostatic 

state and further assume that they depend on z* alone. Since we are interested in the magnitudes of the variables the neglect of horizontal temperature 

gradients is unimportant in the scaling transformations. The hydrostatic 

condition requires that 

-fig. 

dPh -= 

dz* (A.16) 

Let the background be characterized by a temperature Th(Z) and a constant 

mean molecular weight a. The background pressure and density are related 

by the ideal gas law (3.4) as 

(A.17) 
I 

8 

1 

a 

'1 

I 

I 

'8 

1 

8 

I 

:I 

8 

I 

Defining the scale height as 

RTh ( z ) 

= 

233 

(A.18) 

then 

% = phgH. (A.19) 

The hydrostatic condition (A.16) can then be written 

(A.20) 

Integrating this equation gives the background pressure field in terms of the 

surface value fi(0) and the scale height H(z') 

It is useful to introduce a dimensionless inverse scale height by 

The hydrostatic pressure is then 

(A.21) 

(A.22). 

in dimensionless units. By using the ideal gas law the hydrostatic density 

becomes 

(A.24) 

The pressure P' and density p+ must now be scaled appropriately and the 

quasi-geostrophic scaling given by Pedlosky (1987) is useful. In this scaling 
8 

8 

8 

I 

II 

I 

I 

t 

I 

234 

we define the total pressure and density as the sum of the hydrostatic value 

and a departure from this value given by "prime" notation 

The condition of quasi-geostrophic balance requires that the horizontal pressure gradients should be of the same order of magnitude of the Coriolis force. 

This can be written as 

p' 2Qu' = 0(2S1Uph), (A.27) 

and this should be of the order P'la. Then, P' = O(2nUafi). So the natural 

scaling for P' is 

P'=fi+ph*2QUaP (A.28) 

where P is dimensionless. To scale the density the vertical pressure gradient 

associated with P' is 

This suggests that p' be scaled as 

PI=.( 252Uaph gD ). 

Therefore we write 

p+=ph ( 1+- 2;rP) - 

By defining the Rossby number 

(A.29) 

(A.30) 

(A.31) U 

2Qa ' €E- 
235 

which is 1/2 since U = nu using this particular scaling, and by defining the 

parameter 

(A.32) 

the pressure and density can be written by using the hydrostatic condition 

(A.19) and (A.22) as 

P' = &,(l+eFAP), 

p* = fi(l+eFp). 

(A.33) 

(A.34) 

With the scaling parameters used above F x 0.9. 

By using these scaling laws the continuity equation (A.6) can be determined in terms of these dimensionless variables and the hydrostatic condition 

to be 

The horizontal momentum equations (A.7-8) become 

(A.36) 

1 BP 

e { E + Awv' + - (1 - Pa)% Dt 1 -p' 1 + eFp 

(A.37) 

The vertical momentum equation (A.9) is 

16 

A 8% 

1 

EA- (u" + v = AP - --(AP) - p 2Dw (1 + eFp) {€A 1 - p' - 
Figure A.l 

4 

A3: +- 2nuph' 

r xw 

/x /-' 

Spherical coordinate system. 

236 

(A.38) 

Equations (A.35-8) represent the continuity and momentum equations 

respectively. However, the system of equations cannot be solved without 

the thermodynamic relationship (3.11). This equation must be scaled in an 

appropriate manner consistent with (A.35-8). From (3.6) we have 

v= 1-1. A.3Y I 

= In 19h + -In( 1 1 + eFAP) - In( 1 + eFp) 7 

. , 
I 

8 

'8 

I 

1 

I 

8 

I 

I 

237 

1 

7 

N hdh + -EFAP - EF~ 

By using the approximation ln(1 + y) N y we have 

The fractional change of the potential temperature is 

If we define the dimensionless potential temperature perturbation 

1 

7 

d=-AP-p 

then the local potential temperature (A.41) becomes 

(A.40) 

(A.41) 

(A.42) 

(A.43) 

(A.44) 

The buoyancy frequency, described in Appendix B, of the hydrostatic 

state can also be defined by 

where the hydrostatic potential temperature is defined by using (3.6) 

(A.45) 

(A.46) 

By combining the hydrostatic condition (A.19), (A.22) and the relationship 

(A.47) 
I 

'I 

8 

18 

8 

1 

1 

:I 

.I 

'I 

238 

with (A.46) we obtain 

It is also useful to define dimensionless hydrostatic buoyancy term 

1 dA PGnA--- A dx 

so that 

9 

D N; = -P. 

(A.48) 

(A.49) 

(A.50) 

This relationship can be substituted into (3.11) and using the scaling laws 

in the previous two sections it can be shown that 

D9 KAT 

Dt gDn' EF- + (1 + EF9)PW = (1 +a)- 

If we define the dimensionless heating rate per unit mass 

(A.51) 

(A.52) 

and use the relationship (A.43) the thermodynamic relationship can be expressed in the form 

D9 

Dt EF- + (1 + EFB)P w = (1 + d'19)'H. (A.53) 

The fluid equations can be expressed in a still more fundamental form by 

a transformation of the vertical coordinate 

Then 

(A.54) 

(A.55) 
I 

8 

'I 

8 

1 

I 

I 

I 

I 

239 

This vertical coordinate allows us to express the hydrostatic background 

pressure (A.23) as 

fi(x) = %(O) e-=. 

The hydrostatic density (A.24) becomes 

pa(x) = -A(x) %(O) e-". 

9D 

The hydrostatic condition is then 

(A.47) becomes 

Ph& 

and the dimensionless hydrostatic buoyancy frequency (A.49) is 

dA 

dx 

fl= 4- -. 

The dimensionless total time derivative (A.15) is 

Da at a ut a 

Dt at ax all 1-~za4* - = - + Aw- +v - + -- 

(A.56) 

(A.57) 

(A.58) 

(A.59) 

(A.60) 

(A.61) 

We now have showed how the five dynamical equations behave under 

these scaling transformations. Since the scaling parameter A = D/a - 1/64 

is very small, terms involving it can be ignored to a first approximation. 

By ignoring terms of order A as well as frictional terms, the five non-linear 

equations (A.35-8, 51) become 

d'-+(l+cFp){Ag- DP (A-2) Dt 
Du' I 1 ap e- - pv = - Dt 1 + eFpv' 

240 

(A.63) 

a 

az 

AP - -(AP) -p = 0, 

EF- + (1 + EFB)APw = (1 + eFB)'H 

(A.65) 

(A.66) 

D9 

Dt 

respectively. These five equations, along with (A.43) 

1 

7 9~-hP-p (A.67) 

are the nonlinear equation which describe global scale disturbances. 
1 

I 

1 

I 

’I 

I 

B 

8 

I 

8 

‘I 

241 

Appendix B. The Atmospheric Buoyancy 

Eequency 

The hydrostatic background temperature can sustain periodic free buoyancy oscillations, also known as ‘gravity waves’ [Houghton, 1986; Pedlosky, 

19871. Consider a fluid element A at the height ZA and another B at ZB 

where ZB = ZA + dz. Suppose the fluid element A is displaced upwards to 

the level ZB occupied by B. Let the displacement of fluid element A over the 

distance dz be slow enough that the pressure of the fluid element A continuously adjusts to its surrounding but rapid enough so that thermal dissipation 

and external heating of the fluid element is negligible. The fluid element is 

then undergoing an adiabatic displacement. Mathematically it can be stated 

that the potential temperature of the fluid element is preserved during the 

displacement. We can say that 

d9 = 0 (B.1) 

following the fluid element. From (3.23) this means that 

1 1 -dP = -dp. 7p P 

The change in density of the element A can then be determined 

Now the density of the fluid element B is 
242 

The difference in density of the displaced fluid elements A and the background 

fluid elements B is 

(PA +&A) - PB = --- - 5) dz. (:;E az 

The restoring force per unit mass caused by this density fluctuation is 

Since the restoring force is proportional to the displacement an oscillation 

with frequency N is implied 

N is called the Brunt-VGsX frequency, also known as the buoyancy or gravity wave frequency, and represents the natural frequency of vertical oscillation 

in a density stratified fluid. If N3 > 0 the atmosphere oscillates vertically 

about an equilibrium state and the atmosphere is adiabatically stable. However, if N2 < 0 the displacements of the fluid elements continue to grow and 

the atmosphere is unstable. 
243 

Figure 3.4.1 Fluid element A at z' is slowly raised to z' + dz'. 
I 

I 

I 

I 

I 

I 

1 

.I 

I 

I 

I 

'I 

.I 

,I 

244 

Appendix C. Method for Determining the 

Eigenfunctions Ozd( p) 

To determine the eigenfunctions of Laplace's tidal equation a decomposition described by Longuett-Higgins (1968) and Moura (1976) was found to 

be particularly useful for computational purposes. The streamline approach 

described by [Longuett-Higgins 1968; Moura, 19761 was found to be useful 

in determining the atmospheric tidal eigenfunctions and the implementation 

of this method is reviewed in this appendix. 

For convenience, we define the differential operator 

d 

2) (1 -p1) - 

, dP' 

Then (3.124-5) are written as the pair of equations 

ifu' -put = -isP, 

ifv'+pu' = -VP. 

Since 

n 

n 

v' = CiPnV,', 

n 

and by omitting the Q,S superscripts since they are understood at this point 

of the development, then it can easily be shown that for each mode n 
I 

I I 

'I 

fV'-pU' = VO. 

By taking 

-(C.5) d + -(C.6) 5 

dP 1 - pa 

we get the relation 

245 

(C.6) 

By taking 

d 5 

-(A.6) + -(A.5) dP 1 - pa 

we also find 

after defining the horizontal Laplacian 

At this point it is useful to introduce the velocity potential @ and stream 

function 9 [Longuett-Higgins, 19681 by the transformations 

(C.10) 

(C.11) 

Substituting these transformations into (C.7) and (C.8) we obtain the pair 

of equations 

(fVZ+s)9- (pv'+v)ip = 0, (C.12) 

( fV2 + a) @ - (pva + v) 9 = -vao. (C.13) 
246 

At this point we have two equations for the three unknown functions 9, \E, 

and 0. By using the expansions (C.4) and Laplace's tidal equation (3.149) 

in (3.129) it can be shown that 

(C.14) 

where it is understood that X is the eigenvalue. However, by using the transformations (C.10-I) 

Therefore 

v29 = XfO 

(C.15) 

(C.16) 

and 

(C.17) 1 V20 = -v49. Xf 

When this is substituted into (C.13) we find 

(C.18) 

This equation together with (C.12) constitutes the eigenvalue problem and 

is equivalent to solving the eigenvalue problem (3.149). 

In order to solve the system of differential equations (C.12) and (C.18) 

we expand the functions 9 and 9 over an orthogonal set. The "standard" 

representation of the associated Legendre functions PA satisfy the eigenvalue 

problem 

V'PA = -m(m + 1)PA. . (C.19) 

The "standard" representation is normalized as 

2 (rn+d)! 

2m + 1 (m - a)! 1; dp PAPA, = smm, (C.20) 
247 

and satisfies the recursion relations 

m-s+l 

pPA = K+1’ 2m+1 (C .21) 

(C.22) DP; = m(m - 8 + 1) (m + l)(m + SIP# m-1 - 

p;+1* 2m+1 2m+1 

It can be shown from these recursion relations that 

(m - l)(m + l)(m + 8) 

m(m + 2)(m - 8 + 1) - PA+l* 

(pva+v) PA = - 2m+1 p;-1 

(C.23) 2m+1 

We use the expansions 

(C.24) 

(C.25) 

With this representation and the above recursion relations then after a series 

of algebraic manipulations that (C.12) and (C.18) become the system of 

equations 

where 

(C .27) 

(C.28) 

(C.29) 

(C.30) 
I 

I 

I 

P 

# 

E 

1 

8 

E - 

(-;)I- 

- 

248 

Solving for \Em in (C.26) and substituting into (C.27) we get the equation 

where the auxiliary symbols 

are used. Then (C.31) becomes two sets of equations 

(C.31) 

(C.32) 

(C.33) 

(C.34) 

(C.35) 

(C.36) 

for the symmetric and anti-symmetric solutions respectively. If a f superscript denotes these two classes of solutions. Then (C.35-6) can be written 

as 

[c*Z- d*] 8* = 0. (C.37) 
249 

This is an eigenvalue problem where A* = -l/$ and d* is the matrix in 

brackets. The eigenvectors @* are the expansion coefficients in terms of the 

standard representation of the associated Legendre functions. However, in 

the form (C.35-6) the matrix d* is not symmetric. Therefore the eigenvectors 

will not be orthogonal. This is due to non-normalized form of the standard 

representation PA. From (C.16) and by using an expansion 

and by using (C.19) we have 

This can be written more compactly in matrix form as 

= -7 A: M*@* 

where Mi are the diagonal matrices 

, 

(C.38) 

(C.39) 

( C .40) 

(C .41) 

(C.42) 
I 

1 

8 

E 

1 

8 

8 

P 

I 

1 

I 

8 

1 

1: 

I 

'I 

.I 

I 

250 

and the matrix elements are given by 

Then (C.37) can be transformed into 

[c*Z - M*d*M*-'] e* = 0. (C.43) = 

In this form the matrix is still not symmetric because the standard representation is not normalized. Define the 'hormalized representation" $?; by 

I 

From (C.20) it is trivial to show that 

The eigenfunctions 0 can also be expanded upon these 

where the overbar 

malizing factor 

0 = 0,P; rnza 

(C.44) 

(C.45) 

(C.46) 

refers to the normalized representation. Define the nor- 

(C.47) 

so that (C.44) becomes 

PA = NmPk (C .48) 

and substitute into (C.46). Using (C.38) it becomes clear that the components in the standard and normalized representations are related by 

(C.49) 
In matrix form 

@* =po* 

T, &+a T,+a 0 

= 0 T,+a R,+4 T,+4 *** 

0 0 T,+4 &+6 

where the matrices N* are 

9 

251 

(C.50) 

(C.51) 
252 

.55) 

where 

(C.56) 2m+ 1 (m+ a + l)(m+ 8 + 2) Tm = -Qm 

Mm+2 Mm 4 2m + 5 (m - a + l)(m - a + 2)' 

Since these matrices are real-symmetric then the eigenvectors e* are orthogonal with real eigenvalues e* = -l/A* [Mathews, Walker, 19701. The 

eigenvectors are then normalized. The phases of the eigenvectors are chosen 

so that W(0) > 0 and dO-/dp(O) > 0 for the symmetric and anti-symmetric 

eigenfunctions respectively. 

Now that the eigenvectors 8* in the normalized representation have 

been determined we wish to compute Ut* and V'*. We use expansions over 

the standard associated Legendre functions 

(C.57) 

V'f = Vm r* Pm. (C.58) 

mha 

From (C.10-1) we see that Ut and V' are determined from 0 and g. Now 

a?; = N:i: (C.59) 

and since 

(C.60) 
I 

I 

8 

1 

8 

'11 

I 

1 

I 

253 

therefore 

f Nm -f Qm = -Aff-@,. M* (C.61) 

The stream function !D can be determined from (C.26) 

(C.62) 1 

qm = -- (pm+l@m+l+ qm-19m-1) Lm 

After substituting the standard representations of U', V', 9, and !I! into 

(C.10-1) it can be shown that 

(C.63) 

(m + 2)(m + 8 + 1) (m - l)(m - 8) v;f = - f + 92-1 2m+3 2m-1 

(C.64) 

The derivations in this section were used to obtain the Legendre function expansions for the eigenfunctions 0 and eigenvalues A of Laplace's tidal 

equation. This technique can be used for arbitrary values of f and 8. The 

differentid operator C is represented by a matrix and the eigenvectors and 

eigenvalues are obtained by using algebraic methods. Once the expansion 

coefficients 6, are determined the velocity eigenfunctions U' and V' are 

determined by using recursion relations. 

This method presented in this appendix was used for the latitudinal structure of the linear wave theory 5111.4 and susequent calculations $111.5-6. 
254 

Appendix D. Method for Solving the Homogeneous 

Vertical Structure Equation 

In this appendix we present a method which is used to determine the 

eigenvalues and eigenfunctions of the homogeneous vertical structure equation (3.67) for the auxiliary temperature dependent function w defined in 

(3.68). Equation (3.67) can be written in the form 

where 

1 

4 kyx; 2) = wx - - 

is a complicated expression and plays the role of the refractive index [Chapman and Lindzen, 1970; Salby, 19791. In regions where IC2 > 0 we have 

oscillatory behavior of y while in regions where k2 c 0 the behavior of y is 

evanescent and reflections can occur [Salby, 19791. The homogeneous vertical structure equation (D.1) is solved with two boundary conditions. At the 

surface 2 = 0 we impose the condition that the vertical velocity w vanishes 

[Chapman and Lindzen, 1970; Salby, 19791. Using (3.44) this is expressed 

The upper boundary condition must satisfy a radiation/finite energy condition [Chapman and Lindzen, 1970; Salby, 19791. The radiation condition 

in the WKB approximation is discussed in Appendix E and we will use the 
255 

result for the upper boundary condition. From the WKB approximation, at 

some upper level atop the solution y must satisfy the boundary condition 

This condition along with (D.3) are used to determine the eigenvalues X of 

the system for the temperature profile described by w. 

To solve the eigenvalue problem we can use a finite difference approximation. We will divide the region 0 < a < atop into N + 1 levels with spacing A. 

The lower boundary is given by 20 = 0 and the upper boundary XN = xtop. 

In (D.l) the second derivative of y is approximated by 

The differential equation (D.1) can be expressed in terms of the difference 

equation for 1 5 m 5 N - 1 

Using the finite difference method the lower boundary condition (D.3) can 

be expressed in the form 

[-I+ A (& - i)] YO +y1= 0. (D.7) 

The upper boundary equation is derived from (D.4) and is 

YN-I + [-1 +A (ik - T)A dkldx yN = 0. 

The system of equations (D.6-8) can be cast into the form 

My=O 
m 

a 

1 

1 

I 

8 

I 

8 

I 8 

256 

where y is the column vector of y values 

(D.lO) 

and the matrix M is defined by the coefficients of the ym in (D.6-8). The 

equation (D.9) has a non-trivial solution if 

det[M] = 0. (D.ll) 

The eigenvalues of the system are those values of X in which (D.ll) is true. 

The eigenvalues X of the system can be determined by a search method 

in which the parameter X is varied and the magnitude of the determinant 

is evaluated. This technique was carried out for the ATMOS standard temperature profile described in 5111.5. Trial values of X were varied from -10 

through 40 and I det [MI I vs. X with a spacing AX of 0.2 is shown in Figure 

D.l. Two ‘spikes’ were found in the data at X N 8.88 and 13.18. These 

minima are magnified in Figures D.2-3. Increasing the resolution of AX to 

0.05 near A - 9 increases the sharpness of the first peak. This indicates 

that an eigenvalue has been detected. Near the eigenvalue X - 13 increasing 

the resolution of AA was not found to increase the sharpness of the peak. 

Indicate that there is no true eigenvalue near X - 13 and I will disregard it 

in this study. 

We see that a solution to the homogeneous vertical structure equation 

exists with a unique eigenvalue X - 8.9 and vertical structure function y(z). 

Corresponding to this eigenvalue is a multitude of eigenfunctions to Laplace’s 
I 

0 

I 

1 

257 

tidal equation (3.56) with various frequencies and azimuthal wave numbers 

for free oscillation. 
258 

t 1 m 

I 

-10 

I 1 I I I 1 I I I 

0 16 1= 40 

A 

Figure D.l Graph of values of det[M] u the parameter A WM varied between 

10 and 40. There are two distinct ‘spikes’ in the data at 8.88 and 13.18 

respectively. The first value corresponds to the free oscillating mode or Lamb 

mode. The second spike is ‘approximately’ an eigenvalue and it depends on 

the temperature profile [see Salby, 19791. 
259 

I 

7 9 11 

A 

Figure D.2 Region around A - 9 at higher dution. The magnitude of the 

minimum increases by a factor of 40 compared with Figure D.l u the search 

resolution AA is improved. Thir mggertr the premcnce of the eigenvalue. 

t R 

I I I I 

11 13 15 

A 

Figure D.3 Similar to Figure D.2 in the region around A - 13 at higher 

resolution. In difference to Figure D.2, the magnitude of the minimum doee 

not increase any further with increased search resolution AA. This suggests 

that the dip near 13.18 is an artifact of the calculation and probably doee 

not correspond to a true eigenvalue of the atmosphere. 
260 

Appendix E. The WKB Approximation 

One particularly useful approximation to the solution of equations of the 

form 

- @Y + k'(t)y = O dxa 

is to use the WKB approximation [Richmond, 1971; Mathews, Walker, 19701. 

This method is valid when k(x) varies sufficiently slowly with x so that the 

change in k in one wavelength is smaller than k itself. The form of (E.l) 

suggests solutions of the form 

Then (E.l) becomes 

- (4'')' + i4" + k' = 0. 

If we assume #' is small then to a first approximation 

so that 

4 N fldx k. 

The "smallness" of #' requires that 

A second approximation is determined by iteration. 

4" N fk'. 
261 

This is substituted into (E.3) to yield 

(4'')' N k2 f ik'. 

The condition (E.8) gives 

i k' 

2k 

#zfk+-- 

so that 

4zzt/dzk+slnk. i 

Then the two choices of sign yield two approximate solutions 

the general solution being 

y = - [,+e - ,-ir&k 4 

(E.ll) 

(E.12) 

For the case where k2(z) > 0 the solutions are oscillatory. In the case where 

kz < 0 the solutions will have an exponential behavior. 

Appendix F. The Upper Boundary Conditions 

The vertical structure equation (3.57) in the homogeneous region where 

31 = 0 is described by (3.67). At the upper boundary at ztop we can assume 

the solutions to be represented by the WKB approximation described in 

Appendix E. In this appendix we derive the relationship between dy/dz and 

y at the upper boundary appropriate to atmospheric gravity waves. 
262 

First consider the case where k2 > 0 in the WKB approximation (E.1). 

In this case there is an oscillatory boundary condition. The analysis is further simplified if we impose a ‘radiation condition’ that the energy flow is 

vertically upwards out of the heating region x < XN [Wilkes, 1949; Siebert, 

1961; Chapman and Lindeen, 19701. The radiation condition is specified by 

choosing c- to vanish. The reasons for this are discussed by Kat0 (1980) and 

he shows that for gravity waves, the energy propagation is opposite in direction of the phase velocity. If c- = 0 we have downwards phase progression 

which gives outward energy flow. The radiation condition assumes that there 

exist no energy sources outside of the integration region and that the energy 

propagates toward z = +oo from the lower atmosphere. As pointed out by 

Chapman and Lindzen (1970) this approximation neglects reflection of waves 

at greater heights which can be a source of disturbance. If this condition is 

used then our solutions above the top layer N are of the form 

c+ is’dzc Y=ze mN 

and 

For simplicity we define 

i k’ 

2k KGk+-- 

so that 

dY 

dx - = iKy 

represents the radiation condition. 
263 

In the case where IC2 < 0 we can write that IC = ik so that the WKB 

solution to (E.l) is given by 

The two solutions represent exponentially decaying and increasing amplitudes with height in addition to that predicted by (3.52). The exponential 

growth is assumed to be unrealistic in the context of this atmospheric model 

which demands c- = 0 unless a source of further attenuation exists at higher 

altitudes. If the solution at the upper boundary is specified by 

then 

Again we define 

we gives 
264 

Appendix G. Integrating the Vertical Structure 

Equation 

In this appendix we describe how the vertical structure equation (3.57) 

* can be solved. A method of solving second order ordinary differential equations with boundary conditions at different points is used which follows the 

integration scheme proposed by Chapman and Lindzen (1970). Two boundary conditions are specified at the surface of the earth and at altitudes where 

the heating term is zero. At the surface we assume that the vertical velocity 

of the atmosphere vanishes. At the upper ‘free’ boundary we assume that energy propagates outwards with no reflections. This is known as the radiation 

condi t ion. 

The method described in the previous two appendices describes how to 

obtain eigenvalues A? and eigenfunctions @; of the horizontal structure 

equation (3.56)’ i.e.- Laplace’s tidal equation. However, this is only half 

of the problem for understanding the three dimensional dynamics since the 

vertical structure must also be known for each eigen-mode Sy. The eigenvalue A? provides this coupling. The driving terms in (3.57) are the Hough 

components of the heating distribution 7i:(z). 

A fairly technique for integrating second order ordinary differential equations introduced by Richtmyer (1957) and described by Chapman and Lindzen 

(1970) was used. Consider an equation of the form 
265 

We wish to determine a numerical solution for Y(z) given the other functions 

F(z), G(z), H(z), and J(z). Suppose we wish to integrate this solution over 

the range xo < z < XN and this range is divided into N equal subintervals of 

width A. Then the derivatives of Y at each point zm can be approximated 

by 

(G.2) 

Ym+1- Ym-1 

2A ' 

Then the differential equation (G.l) can be cast into the finite difference form 

AmYm+1+ BmYm + CmYm-,= Dm (G.4) 

where 

If we introduce a change of variables 

then 

Ym-1 = %-lYm + Pm-1- (G.lO) 

Thus the difference equation (G.4) can be put into the form 

(G.ll) Am Dm - Pm-lcm 

ym+l -t Bm + %-1Cm' Ym = - 

Bm + 
I 

I 

I 

i 

I 

I 

I 

I 

1 

'I 

I 

'I 

:I 

1 

'I 

1 

I 

.I 

'I 

266 

By comparing this to (G.9) we see that 

(G.12) 

(G.13) 

If a0 and Po are known then all of the other a, and pm can be determined. The simplest lower boundary condition is usually chosen to represent 

a smooth spherical earth where the vertical velocity w vanishes at the surface ?: = 0 [Chapman and Lindzen, 19701. From equation (3.61) this can be 

expressed as 

In terms of finite differences 

and this can be rewritten 

Yo = [1- h (& - 91-l yl. 

Upon examination of (G.9) we can identify 

-1 

a. = [l-A(&-;)] , 

po = 0. 

(G.14) 

(G.15) 

(G.16) 

(G.17) 

(G.18) 

From (G.12-3) all of the other a, and pm are determined. 

After calculating all of the a, and Pm the function Y can be determined 

from (G.lO) provided that an upper value YN is given, since 
267 

(G.19) 

The result (G.19) implies that we need to know the value for the upper 

boundary YN to complete the integration. 

For the case in which the vertical structure equation is homogeneous 

(5111.5, Appendix D) the upper boundary value is arbitrary since there is a 

free constant of integration which can have any value. The upper boundary 

dynamics - i.e. radiation condition - is imposed on the solution during the 

determination of the eigenvalue A. In the case of the inhomogeneous equation (3.57) the upper boundary dynamics are imposed during the integration 

procedure. We use the WKB approximation (F.4) or (F.9) depending on 

whether k2 > 0 or IC2 c 0 at the upper boundary respectively. 

For the case where IC2 > 0 we have a radiation boundary condition and 

apply (FA) at the level N. Then (F.4) is approximately 

(G .20) 

Since, in this integration scheme, 

YN-1 = aN-1 YN + PN-1, (G.21) 

it can be shown that 

PN-1 

1 - aN-l - ~AK~' YN (G.22) 

This results then allows a full integration of the vertical structure equation 

below the top layer. 
I 

I 

I 

I 

I 

I 

I 

I 

e 

'I 

'1 

I 

,I 

I 

I 

I 

I 

1 

268 

For the case where ICz c 0 we have an evanescent boundary and apply 

(F.9) to the upper boundary level. Similar to the derivation of (G.15) we 

apply this condition at the level N so that 

Again using (G.16) we find that 

(G.23) 

(G.24) 

This final boundary condition allows the vertical structure equation to be 

integrated. 

From this appendix we can integrate the second order vertical structure 

equation (3.57) consistent with two boundary conditions specifying the behaviour of the atmosphere at great heights and at ground level. The upper 

boundary condition assumes that the heating of the atmosphere from below 

is eventually radiated outward by the tidal gravity waves (or, in the case of 

the homogeneous equation, any constant value will do). Although it does not 

assume sources of energy above the upper level, the inclusion of such sources 

can be built into theory by an appropriate adjustment of the upper boundary 

condition. The lower boundary is assumed to correspond to a flat surface 

which has no vertical component of motion and hence wind. Although this 

does not take into account features such as continents and possible tidal deformations of the oceans, they can be assumed to be negligable to at least 

O(A) in the scale of the dynamics and of secondary importance. 
269 

Appendix H. Energy Transport in Linear Wave 

Theory 

The linear wave theory presented in can be used to describe the physical 

structure of the velocity, pressure and density perturbations to the hydrostatic st ate generated due to homogeneous oscillations and inhomogeneous 

waves, particularly in the case of atmospheric thermal tides in which the 

periodic ozone and water water vapor heating of the atmosphere can drive 

upper atmospheric circulation. In this section we outline how these waves 

also transport energy through the atmosphere. In the absence of nonlinear 

turbulence and radiative dissipation the linear waves may transport the energy absorbed by the atmosphere outwards. In this section we examine the 

energy transport of these waves. 

For notational simplicity we will temporarily omit the " superscript to 

denote dimensioned quantities. The energy flux per unit mass S is given by 

[Kato, 19801 

s = Pu. (H.1) 

The divergence of the flux density is 

By using the momentum equation (3.1) and continuity equation (3.2) we 
270 

have 

v s = -p [E D ( pg ) + -- PDP - g .] . p2 Dt 

From the thermodynamic equation (3.11) it can be shown that 

-- p DP --- (’)-.. p2Dt 7-1Dt p 

Then (H.3) can be shown to be 

- -+‘P)+-V*S=H+g*u. 1 

Dt (‘uz” y-lp p 

We see that the energy per unit mass E is given by 

14’ 1P 

2 Y-1P 

. E=-+-- 

and (H.5) can be expressed 

DE 

Dt p - + v - s - pg . u = p. 

The energy equation (H.7) expresses how the energy of the wave is exchanged. Some of the heating is used to increase the energy density, some is 

transported, and some goes into doing work against gravity. Since the tides 

are periodic in time (and local time) we can average the energy equation over 

a sphere to obtain the average energy exchange as a function of altitude. Let 

us define the spherical average, which is also an average over one period, by 

The spherical average of (H.7) is then 

(P g) + (V * S) - (Pg u) = (P.) 
271 

The fields will be assumed to vary as 5111.6 

2 

f - f"" ,wJt+4)* (H.10) 

It is also understood that we use the real parts of the dynamical fields in 

(H.7) when taking these averages. If we denote the complex conjugate of a 

function f by ft then 

u=l 

(H.ll) 

Only products of non-conjugate and conjugate pairs will survive the averaging process. 

1 w = +f + ft)* 

If we write the pressure and density in the form 5111.3 

P = %(I+ P'), 

P = Ph(l+P') 

(H.12) 

(H.13) 

then the energy flux divergence is 

v.s = V*Pu=PV.u+u-VP 

= %(1+ P')V ' u + u * vq1+ PI) 

W 

= & [(l + P') V . u - -) + u - VP'.] (H.14) H 

The spherical average of (H.14) is then (omitting "uo" superscripts) 

(V.S) =&(P'(v.u-lfl)+u.vP'). H (H.15) 

Since the fields in (H.15) are the real parts of the solutions, in terms of the 

complex functions we have using (H.10-1) 

2wt lav'+ 1 ia 

T Tap Ti-p= 
272 

-+-+-- aw 2w 1av' + --d) 1 ia +PI' ( - -(wP't 1 + W'P') 

az f rap r1-pa H 

V'BP't V48P' - - u'p't + - - u"p'+ -- + -- 

T1-p' fl+ T 8p f 8p 

1 it7 1 it7 

(H.16) 

We now wish to express this in terms of the dimensionless variables. From 

§III.lO 

n 

n 

n 

(H.17) 

n 

By using these transformations (H.16), the orthogonality of the Hough functions and (A.14) 

d A C E - [e-'A(wnP: + wLPn)] + O(A). (H.18) Ph(0) €FU (V4) = -- 

8 TO n dz 

In a similar manner the power lost to gravity can be evaluated 

(Pi74 = PhgEF(F'W) 

The term on the right hand side of (H.7) represents the energy expended in 

heating density perturbations 
'1 

i 

I 

I 

:I 

'I 

8 

I 

I 

2 73 

The total rate of change of energy following the fluid particles can be evaluated from (H.7) 

(H.21) 

The flux density can be computed from (S n) where n unit vector pointing 

outward from the spherical surface at level 2. 

(H.22) 

The energy sources and sinks given by (H.18-20) can be used to compute 

the spherically averaged energy exchange of the linear wave solutions. The 

energy flow of the linear waves are given by (H.22). We see that due to the 

orthogonality of the Hough functions that the total energy terms are the 

sum of the various component terms. This allows us to examine each mode 

separately. 

Appendix J. Including Radiative Damping into the 

Linear Wave Theory 

The primary absorption processes of solar radiation in the atmosphere 

are the ultraviolet absorption of water nee the surface and ozone in the 

stratosphere [Andrews, Holton, Leovy, 19871. The 15-pm band of COz is the 

primary source of infrared emission in the atmosphere and it provides the 
2 74 

medium by which the atmosphere radiates [Dickinson, 19841. The hydrostatic temperature distribution can be assumed to be produced by a balance 

between radiative and non-radiative processes - see Andrews, Holton, Leovy 

(1987) for details. We will assume that any temperature perturbation will 

lead to an increase or decrease in infrared cooling which tends to restore the 

atmosphere to its original temperature [Chapman and Lindzen, 19701. 

Including this type of dynamics into the linear wave theory 5111.4 requires 

only a slight modification. We can substitute (3.6) into (3.5) to obtain 

TDP M + -”. DT -=n-- 

Dt P Dt Cp 

Since the infrared cooling opposes the growth in temperature, following 

Chapman and Lindeen (1970) we include a term into the right hand side 

TDP M +-7i-a6T - DT = -- Dt P Dt Cp 

where a is the infrared cooling coefficient and 6T is the deviation of the 

temperature from the equilibrium temperature. This is known as the Newtonian cooling approximation [see Andrews, Holton, Leovy, 19871 and a is the 

Newtonian cooling coefficient. We can write (5.2) in the form 

DT TDP M -= n--+ -fi 

Dt P Dt Cp 

with the ‘modified’ heating 

CP 3-1 - -a6T. M 

(J.3) 

(J.4) 

When we include a Newtonian cooling term, the linear wave theory 5111.4 is 

still used with the replacement of N+ fi. 
275 

The vertical structure equation (3.57) requires that the heating be in 

the dimensionless representation of Appendix A. This is achieved with the 

transformation (A.52) 

where now we have introduced the dimensional '*' notation (see Appendix 

A). The temperature perturbation, described in Appendix K, is then 

where Th is the background temperature and T is the temperature perturbation function. Using (5.4) in (J.5) it can be shown using the definition of 

A and the vertical depth n described in Appendix A that the dimensionless 

heating is 

(J.7) 

where the dimensionless Newtonian cooling coefficient is defined 

Using (K.6) we can then write (J.7) in terms of the pressure and density 

perturbation functions P and p 

fi = e - a(AP - p). (J.9) 

The dimensionless heating 7?,, can be shown to be related to the vertical 

structure function yn for each Hough component of (J.9) using (3.60) and 

(3.62) 
I 276 

With this result, the vertical structure equation with Newtonian cooling can 

be derived. Substituting (J.10) for the heating function which drives (3.57) 

we obtain the following second order differential equation for the vertical 

structure function yn 

(J.ll) 
277 

Appendix K. Linearized Temperature Perturbation 

The ideal gas law (A.17) is a non-linear relationship between the pressure, 

density and the temperature. Using the notation of Appendix A for dimensional and nondimensional variables (A.33-4) describe the total pressure P' 

and density p' respectively in terms of the hydrostatic background Ph, pa 

and perturbation functions P, p 

P' = %(l+eFAP), 

p' = pi, (1 + EFP) 

where E and F are scaling factors aad A contains information regarding the 

hydrostatic temperature distribution with height. For simplicity, we can 

define a dimensionless background temperature perturbation function T by 

T' = T' (1 + EFT) (K.3) 

where T.' is the total temperature, Th is the background temperature. Using 

the ideal gas law 

p'RT' p' = - 

M 

and (A.17) for the background it can be verified that the dimensionless perturbation functions P, p and T are related to each other by 

AP = (K.5) p + T + eFpT. 

Linearizing and rearranging we obtain 

T=AP-p. (K.6) 
i I 

278 

II 

'I 

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