ATMOSPHERIC GRAVITY WAVES IN THE STRATOSPHERE OF MARS AND THE IONOSPHERES OF JUPITER AND SATURN

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1 ATMOSPHERIC GRAVITY WAVES IN THE STRATOSPHERE OF MARS AND THE IONOSPHERES OF JUPITER AND SATURN By DANIEL J. BARROW A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2011

2 c 2011 Daniel J. Barrow 2

3 To my family 3

4 ACKNOWLEDGMENTS I first offer my sincerest gratitude to my advisor, Dr. Katia Matcheva, for her unwavering help and support through the past five years. I thank our collaborator, Dr. Pierre Drossart, for his assistance, particularly with the observational aspect of the work. I thank my committee for their perspicacity. I also acknowledge NASA s Planetary Atmospheres program that, in part, funded this work. Finally and foremost, I express profound adoration for my wife, Joeva, whose love and support throughout has been nothing short of phenomenal; she is my perfect complement without which this work could not have been completed. 4

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Planetary Science Planetary Atmospheres Gravity Waves Overview Observations Overview of Work SPECTRAL PROPERTIES OF GRAVITY WAVES IN MARS LOWER ATMOSPHERE USING MARS GLOBAL SURVEYOR RADIO OCCULTATION MEASUREMENTS Background Theory and Observations of Gravity Wave Power Spectral Density in the Earth Atmosphere Data Analysis Results Spectral Shape of Gravity Wave Power Spectral Densities Latitudinal and seasonal dependence Topographical dependence Gravity Wave Seasonal Cycle Conclusions IMPACT OF GRAVITY WAVES ON THE IONOSPHERE OF JUPITER Background History of Observations Unresolved Questions in Jupiter s Upper Atmosphere Overview of Work Model of Jupiter s Upper Atmosphere Neutral Atmosphere Ion Chemistry Gravity Wave Theoretical Background Atmospheric Gravity Waves Wave-ion Interaction

6 Dynamical effects Chemical effects Analytical Solutions and Numerical Validation for Small Amplitude Waves Analytical Ion Response for Small Amplitude Waves Case I: Single ion, long lifetime Case II: Single ion, arbitrary lifetime Case III: Two ions, arbitrary lifetime for minor ion Case IV: Further generalizations Validation of the Numerical Model for Small Amplitude Waves Sample wave parameters Small amplitude results Large Amplitude Non-linear Simulations Phase relation between small and large amplitude waves Ion Density Perturbations Ion Flux Small amplitudes: Analytic Large amplitudes: Numerical Ion Column Density Variations Observability of Gravity Waves via H 3 + Emission Overview of H 3 + Emission Model Results Wave propagation direction Wave peak altitude Vertical wavelength Wave amplitude Latitude (magnetic field inclination) Summary GRAVITY WAVE EFFECTS IN THE REGION OF OBSERVED SATURN ELECTRON DENSITY PROFILES Background Model of Saturn s Upper Atmosphere Neutral Atmosphere Ionosphere Results Altitudes of gravity wave influence Model ion density perturbations Discussion DISCUSSION APPENDIX: CONTINUITY EQUATION COEFFICIENTS REFERENCES BIOGRAPHICAL SKETCH

7 Table LIST OF TABLES page 1-1 Comparison of the characteristics of the planets PSD slopes for each bin Reactions incorporated in the model Sample wave parameters Sample wave parameters

8 Figure LIST OF FIGURES page 1-1 Atmospheric temperature profile Oscillations in a stable atmosphere Theoretical gravity wave power spectral density Map of Mars Global Surveyor radio occultation data Sample temperature-height profile Latitude dependence of the power spectral density for northern spring Latitude dependence of the power spectral density for northern summer Latitude dependence of the power spectral density for northern autumn Latitude dependence of the power spectral density for northern winter Topographical dependence of the power spectral density Topography of mars in the analyzed region Seasonal dependence of the power spectral density Jupiter electron density observations Ion layering mechanism Ion frequencies Background neutral atmosphere Mixing ratios Photochemical production Ion lifetimes Background ion densities Gravity wave amplitudes Validation results Amplitude dependent perturbation phases Perturbed ion densities for wave A Perturbed ion densities for wave B

9 3-14 Wave-induced ion fluxes for wave A Wave-induced ion fluxes for wave B Wave impact on the background ionosphere Time dependent ion column density for wave A Time dependent ion column density for wave B Wave A induced horizontal variations in the ion column densities Wave B induced horizontal variations in the ion column densities Ion perturbation dependence on wave propagation direction Gravity wave peak altitude Vertical wavelength dependence of the emission contrast Amplitude dependence of the emission contrast Amplitude dependence of the emission contrast Magnetic field inclination dependence of the emission variation Magnetic field inclination dependence of the emission variation Saturn dawn electron density profiles Saturn dusk electron density profiles Model temperature profile Neutral atmosphere Mixing ratios Ion photoproduction rates Ion lifetimes Background ion density Ion frequencies Peak altitudes of gravity waves Model results for wave Model results for wave Model results for wave

10 A-1 Magnetic field and wave geometry

11 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ATMOSPHERIC GRAVITY WAVES IN THE STRATOSPHERE OF MARS AND THE IONOSPHERES OF JUPITER AND SATURN Chair: Katia I. Matcheva Major: Physics By Daniel J. Barrow May 2011 Atmospheric gravity waves transport energy and momentum from a source region to the location of wave dissipation and are understood to greatly affect the thermal and dynamical structures of the upper atmosphere. Atmospheric gravity waves have been studied extensively on Earth but due to observational limitations the study of these waves in the atmospheres of other planets has been minimal. This work is composed of three different parts that study gravity waves in different planets atmospheres. In the Earth atmosphere the spectral characteristics of gravity waves reveal a remarkable universality with respect to time of day, location, topography, and altitude. The first study in this work utilizes the nearly 20,000 temperature profiles from the Mars Global Surveyor (MGS) radio occultation measurements to calculate the power spectral density for gravity waves on Mars and compares them to the observations and theories for the Earth atmosphere. The second part of this work investigates the effects of gravity waves in the ionosphere of Jupiter. This work models a realistic ionosphere that maintains the chemistry and dynamics of multiple ion species in the presence of a propagating gravity wave. It is shown that a wave creates significant variations in the H + 3 column density across a horizontal plane of Jupiter. It is argued that the H + 3 thermal emission from Jupiter can be used to obtain observations of gravity waves in Jupiter s ionosphere 11

12 that would provide new information about gravity wave parameters that are valuable in understanding their contribution to the atmospheric energetics at these altitudes. The third part of this work modifies the gravity wave-ionosphere model for use on Saturn. The Cassini spacecraft has provided dozens of new electron density profiles of Saturn s ionosphere. Many of these profiles show sharp layering in the electron density profiles as well as a high degree of variability in the lower ionosphere. The model is used to assess the possibility that the layering and other structures are due to gravity waves and what parameters of waves are be required to reproduce these effects. 12

13 CHAPTER 1 INTRODUCTION 1.1 Planetary Science Planetary science is the study of the planets, moons, and any other body within the solar system and includes the origins and dynamics of planetary systems in general. These are studied not only to understand the many objects in our solar system but also to improve our understanding of our own world. Planetary science aims to understand the composition and geology of planets, the chemistry and dynamics of their atmospheres, as well as the dynamics and origins of the solar system as a whole. Because the sum of these systems is complex, planetary sciences is necessarily a highly interdisciplinary field. Planetary science encompasses many different fields including astronomy, geology, atmospheric sciences, the study of orbital dynamics and many more. In addition, comparative planetology is a powerful tool that planetary scientists employ regularly; one can hardly study an object without relating it to similar objects. Before the physical exploration of space the only planet that we had any relative understanding of was the Earth. As we have begun studying other celestial objects, a large amount of information has come from comparisons made to what we already understand. The terrestrial planets are best understood in context with the Earth, with their atmospheres and topography frequently compared to that of Earth. The giant planets are exotic compared to the Earth but are similar to each other and can be studied in comparison to one another. Recently, in roughly the past 15 years, the field of planetary science has exploded out of our solar system. Over 500 planets in other star systems have been discovered and there are hundreds more that have been observed and are awaiting confirmation. Many of these distant planets have sizes and densities similar to Jupiter and, while most have been observed to orbit their stars at distances less than 1 AU, they are only understood by considering them to be Hot Jupiters. With very little data about 13

14 each of these systems so far the models of these planets are necessarily based upon solar system planet models. Even more recently some direct measurements of the atmospheric composition of these planets have been made. With this data, both the composition and dynamics of these planets have begun to be studies, revealing exotic new worlds. As planetary science broadens its understanding about these myriad systems, we come to better understand our own system, our own planet, and our relative niche in the universe. 1.2 Planetary Atmospheres Planetary atmospheres are complex fluid dynamical systems comprised of several distinct layers. Figure 1-1 is the temperature profile of Jupiter s atmosphere observed in situ by the Galileo probe and illustrates the layers of an atmosphere. The lowest region of the atmosphere is the troposphere where the temperature decreases with altitude up to the tropopause. The thermal structure of the troposphere allows this region to be highly convective. It is in this region where clouds are present. Above the troposphere is the stratosphere, where the temperature becomes constant or increases slightly. This region of the atmosphere is stable; a vertically displaced parcel of air tends to return to its original altitude. In Earth s atmosphere, above the stratosphere, is the mesosphere where the temperature decreases due to radiative cooling of CO 2 molecules. Jupiter has no well-defined mesosphere. Above the stratosphere (or mesosphere for Earth) is the thermosphere where the temperature increases quickly with altitude due to absorption of solar radiation and other processes not well understood for the giant planets. Ionization due to solar radiation creates the ionosphere, containing ionized molecules, that roughly coincides with the location of the thermosphere, extending for Jupiter from about 300 km up through the thermosphere. There are several major, unreconciled inconsistencies between the observations of the giant planets and the models. Prominent among them is the thermal structure of the upper atmosphere; the temperatures in the thermospheres of the giant planets 14

15 are all much hotter than solar heating alone predicts. While the solar heating models properly reproduce the thermal structure of the upper atmospheres of the terrestrial planets, there appears to be a significant heating mechanism as yet unaccounted for in these atmospheres. Many ideas have been proposed to account for this heating such as low-latitude particle precipitation, energy transport via dynamical processes such as waves, or meridional transport of energy from the aurorae. These models can, in theory, reproduce the thermospheric temperatures observed. However, the parameters necessary for each of these mechanisms to reproduce the observed temperatures either are unrealistic or have not yet been observed to exist on the planet. Further, because this discrepancy arises not just for Jupiter and Saturn but also for Uranus and Neptune, it is prudent to seek a source that is common to all the planets. Energetic particle precipitation and energy transport from the aurorae are not nearly as strong for Saturn, Uranus, and Neptune as they are for Jupiter. The only process that is common to all four giant planets is dynamical heating from gravity waves (Yelle and Miller, 2004). The extent that gravity waves heat (or cool) the upper atmosphere is dependent on the properties of the extant waves as well as how often and for how long they are present (Matcheva and Strobel, 1999; Hickey et al., 2000). It is suspected that gravity waves are common to all the giant planets and thus possibly large contributors to the thermal and dynamical structure of these planets. To quantify the degree to which gravity waves contribute to the energy budget of these atmospheres there needs to be a better understanding of the parameters of the waves, how frequent they occur, and for how long they exist. 1.3 Gravity Waves Above the tropopause vertical motions are limited due to the stability of the atmosphere. An important phenomena in planetary atmospheres that transports of energy and momentum from the lower regions into the upper atmosphere are gravity waves. Gravity Waves are transverse waves, maintained by gravitational and buoyancy 15

16 forces, that propagate through atmospheres and are now well understood to greatly affect the thermal and dynamical structures of the upper atmosphere. Atmospheric density decreases exponentially with height so when a wave propagates upward its amplitude increases exponentially in order to conserve energy (if no wind shear is present). As the wave is dissipated and its energy and momentum absorbed into the upper atmosphere it can significantly alter the atmospheric state. The study of these phenomena is crucial to understanding the entire thermal and dynamical system Overview In a stably stratified atmosphere the environmental lapse rate, the change in atmospheric temperature with height dt o, is greater than the adiabatic lapse rate. The dz adiabatic lapse rate is the change in temperature a parcel of air would have if it were moved vertically and adiabatically through the atmospheric pressure gradient; it is defined as Γ = g c p, where g is the gravitational acceleration and c p is the specific heat at constant pressure. A parcel of air in a stable atmosphere that is forced upwards will expand adiabatically and become cooler than the surrounding air. The parcel then, more dense than the surrounding air, begins to descend back toward its equilibrium position. Its momentum may then carry it below its equilibrium position and the process repeats in reverse, creating an oscillation. This process is illustrated in Fig The frequency of oscillation of a vertically displaced parcel of air is given by ( g N = T o [ dto dz + g c p ]) 1 2. (1 1) The frequency N is called the buoyancy frequency or the Brunt-Väisälä frequency and is the the upper limit for gravity wave frequencies. Common gravity wave sources include airflow over topography, convection, and wind shear. As a gravity wave propagates vertically the surrounding atmospheric density decreases exponentially with altitude due to the hydrostatic nature of the atmosphere. To conserve energy the amplitude of the gravity wave must then increase 16

17 exponentially. This exponential increase in amplitude is countered by dissipative processes: molecular viscosity, thermal conduction, eddy diffusion, fast chemical reactions, ion drag, and radiative processes. If the wave amplitude exceeds a critical value, where the wave-induced temperature oscillations create gradients larger than the adiabatic lapse rate, the gravity wave will break and release its energy into the background atmosphere, often distant from the source. It is well recognized that gravity waves can have a significant effect on the thermal structure, chemical composition, and the dynamics of the upper atmosphere (Hines, 1974; Lindzen, 1981; Walterscheid, 1981) Observations On Earth there are a multitude of observations of gravity waves from instrumented aircraft, ground-based radars, lidars, airglow measurements, balloon observations, and rocket measurements. Despite the large amount of data, the effect of gravity waves on the global atmospheric circulation is still poorly understood. Better characterizations of gravity wave sources, dissipation, and drag are needed in order to create realistic parameterization schemes for the current global circulation models (Fritts and Alexander, 2003). The existing global circulation models for the atmospheres of other Solar System planets suffer from the same weakness as their terrestrial counterparts. Insufficient observations and the lack of proper wave analysis intensifies the problem. Mars global circulation models include only gravity wave parameterizations from orographic sources (Rafkin et al., 2001) whereas Creasey et al. (2006a) has demonstrated that gravity wave energy in the stratosphere is not well-correlated to topography. Jupiter global circulation models to date, while acknowledged, do not model the effect of atmospheric waves (Achilleos et al., 1998; Bougher et al., 2005). While limited, there is evidence of gravity waves in the atmospheres of other planets. For Mars, evidence of gravity waves has been observed by Mars Global 17

18 Surveyor both in density variations of the upper atmosphere observed during aerobraking (Creasey et al., 2006b; Fritts et al., 2006) and in variations in the temperature profiles retried from radio occultations (Hinson et al., 1999; Creasey et al., 2006a). Variations in the retrieved temperature profiles from Jupiter yields evidence of gravity waves as well. These are seen in the temperatures retrieved from the Galileo probe in the upper atmosphere (Young et al., 1997; Matcheva and Strobel, 1999) as well as the stratosphere (Young et al., 2005) and in temperature profiles inferred from stellar occultations (French and Gierasch, 1974; Raynaud et al., 2003, 2004b). Possible evidence of gravity wave activity also exist in electron density profiles retrieved from radio occultations (Hinson et al., 1997; Matcheva et al., 2001) and from images of Jupiter s equatorial clouds (Arregi et al., 2009). There is also some evidence of gravity waves in Saturn s atmosphere (Cooray et al., 1998; Cooray and Elliot, 2003; Harrington et al., 2010). Since there are relatively few gravity wave observations, especially for the giant planets, it is difficult to know how often gravity waves occur, how long they persist, and what are the typical wavelengths. Without this information one cannot accurately calculate what are the net effects that gravity waves have on a planet s atmosphere. 1.4 Overview of Work This work is comprised of three parts, each of which studies gravity waves in a different planet s atmosphere. The first part compares the spectral power of gravity waves on Mars to those of waves observed in the Earth atmosphere. The observed power spectral densities of gravity waves on Earth in the saturated portion of the spectrum exhibit little variability with respect to location, time of day, underlying topography, and altitude. Temperature profiles retrieved from radio occultations of Mars lower atmosphere obtained by Mars Global Surveyor commonly exhibit vertical variations that are interpreted as gravity wave signatures. These variations are spectrally analyzed to obtain the power spectral density of gravity waves in Mars atmosphere and are compared to the observations in the Earth atmosphere. 18

19 The second part of this work examines the effects of gravity waves in the ionosphere of Jupiter. The wind shears produced by a propagating gravity wave in the ionosphere can induce significant modifications of the ionospheric structure. A two-dimensional, time dependent dynamical and chemical model of the jovian ionosphere that allows for an arbitrary wind field is constructed. This model is used to investigate the dynamical and chemical effects of a gravity wave propagating in the jovian ionosphere in order to better understand the physics of this phenomenon and to search for observable effects of waves. New observations of gravity waves will provide information about the types of waves present that will be helpful in understanding their contribution to the atmospheric energetics at these altitudes. The last part of this work modifies the wave-ionosphere model of Jupiter and applies it to Saturn. The Cassini spacecraft has provided many new observations of the electron density profiles in Saturn s ionosphere. Many of these profiles show sharp layering in the electron density profiles as well as a high degree of variability in the lower ionosphere. The model is used to assess the possibility that the layering and other structures are due to gravity waves and what wave parameters would be required to produce these effects. 19

20 Table 1-1. Comparison of the characteristics of the planets Earth Mars Jupiter Saturn Mean solar distance [AU] Mass [10 24 ] Equatorial radius [km] Mean density [g/cm 3 ] Equatorial rotational period [h] Equatorial surface gravity (m/s 2 ) Adiabatic lapse rate (K/km) Orbital sidereal period [yrs] Obliquity [degrees]

21 Altitude (km) Thermosphere Stratosphere Troposphere Stratopause Tropopause Pressure (bars) Temperature (K) Figure 1-1. Atmospheric temperature profile. Jupiter s atmospheric temperature profile observed in situ by the Galileo probe as an example of an atmospheric temperature profile (Seiff et al., 1996, 1997). The separate atmospheric layers are labeled and the pressure levels are given on the right axis for reference. 21

22 Figure 1-2. Oscillations in a stable atmosphere. T and z are the background temperature and altitude, T 0 and z 0 are the conditions at a reference level and the primed quantities are perturbations about the reference quantities. The quantities dt and g dz c p are the environmental and adiabatic lapse rates, respectively. 22

23 CHAPTER 2 SPECTRAL PROPERTIES OF GRAVITY WAVES IN MARS LOWER ATMOSPHERE USING MARS GLOBAL SURVEYOR RADIO OCCULTATION MEASUREMENTS 2.1 Background Atmospheric gravity waves are expected to be a common phenomenon in a stably stratified planetary atmosphere. On the Earth, gravity waves (GW) are responsible for much of the spatial and temporal variability of the atmosphere above the tropopause. They have been studied extensively using ground, air-born, and space-based observations since first described in detail in the early 1960s (Hines, 1960). In the last twenty years the study of atmospheric gravity waves has shifted from the detection and the analysis of individual wave modes toward modeling the waves as a continuous field of waves that propagate simultaneously through the atmosphere. The interaction (or the lack of such) of the individual wave modes gives rise to a wave field with a characteristic vertical wavelength power spectrum density. The properties of this power spectrum have been a subject of a large number of studies both observational and theoretical and will be discussed further in Section 2.2. Gravity waves have been observed in the atmospheres of almost all solar system planets that have a substantial atmosphere. They are identified using cloud tracking images as well as stellar occultations, radio occultations, and in-situ observations that lead to information about the thermal structure of the atmosphere with a reasonable vertical resolution. The observed small amplitude variations in the vertical temperature profiles are often interpreted as being induced by propagating atmospheric gravity waves. These occasional detections of individual wave modes have very limited planetary coverage and one cannot draw general conclusions about the importance of the waves for the planet s energy and/or momentum budget. Mars is the only exception. The Mars Global Surveyor radio science experiment has performed close to radio occultations in the period from January 1998 to June 2005, mapping the thermal structure of the Martian lower atmosphere. This extensive data set has been 23

24 successfully used to study transient eddies in the northern hemisphere of Mars (Hinson, 2006), seasonal and spatial variations of atmospheric gravity wave activity (Creasey et al., 2006a), as well as to observationally identify the presence stationary planetary waves and diurnal and semidiurnal Kelvin waves in Mars lower atmosphere (Hinson et al., 2003, 2008). The results from the MGS radio occultation experiment has also facilitated studies of variability in the structure of Mars ionosphere and upper neutral atmosphere (Bougher et al., 2001, 2004). Creasey et al. (2006a) analyzed the temperature profiles resulting from the MGS radio occultations and calculated gravity wave potential energy per unit mass based on the observed temperature deviation from a mean temperature profile. The vertical scales included in the analysis are between 2.5 and 25 km. They demonstrate that the gravity wave activity 1) has strong latitudinal dependence with maxima in the tropics, 2) shows a significant seasonal variation with enhanced power during the north summer versus north winter (especially south of the equator), 3) demonstrates a poor correlation with the underlying topography, and 4) has a bimodal wavelength distribution with vertical wavelengths of 8-10 km and km. The results presented in this work complement the above studies by analyzing the properties of the power spectrum density of the gravity wave field in the lower 40 km of Mars neutral atmosphere using the MGS radio occultation temperature profiles. The results are discussed in view of the existing theories of the observed terrestrial gravity wave power spectrum and wave propagation. A short review of atmospheric gravity waves, the properties of the observed power spectral density in the Earth atmosphere and the current competing theories are presented in Section 2.2. The MGS temperature profiles and the data analysis are detailed in Section 2.3. The results from these analyses are presented in Section 3.5 and a summary of the conclusions are discussed in Section

25 2.2 Theory and Observations of Gravity Wave Power Spectral Density in the Earth Atmosphere Gravity waves are buoyancy driven oscillations in a stably stratified atmosphere. In the hydrostatic limit they are non-compressive and have a frequency less than the buoyancy frequency N and larger than the planetary rotational frequency f for a given latitude. Sources of gravity waves include flow over variable topography, stable layer over a convective region, wind shear instabilities, and weather frontal systems. The temperature amplitude of a gravity wave in a conservative atmosphere with no wind shear increases exponentially with altitude as a result of decreasing atmospheric density and wave energy conservation. The growth of the amplitude is limited by instabilities as the local temperature gradient should not exceed the adiabatic lapse rate. The wave amplitude is modified in the presence of background wind gradient as well as a result of interactions between different wave modes. In a non-conservative atmosphere, dissipative processes that attenuate gravity waves include eddy and molecular viscosity, thermal conduction, radiative cooling, and ion drag. The dissipated wave energy and momentum are deposited into either the background atmosphere or other wave modes. The impact on the background state of the atmosphere is what makes this relatively small-scale phenomena to be of great importance to the dynamics and the energy budget of the atmosphere of a planet. The spectral properties of the gravity waves generated in the atmosphere is believed to be determined by the generating mechanism. The generated waves are rarely monochromatic and as the wave field propagates the waves are subject to interactions with the background atmosphere as well as with each other resulting in a power spectrum that differs from the originally generated wave field. The shape of the power spectral density (PSD) of the temperature and horizontal wind variations observed in the Earth atmosphere that are attributed to gravity waves shows a remarkable invariance with time, geographic location, and altitude (VanZandt, 1982). 25

26 Figure 2-1 presents a summary of the features of the vertical wavenumber power spectral density observed for gravity waves in the Earth atmosphere. The peak of the power is concentrated around a characteristic wave number m that divides the spectrum into two regimes: saturated and unsaturated. For wavenumbers smaller than the characteristic wave number (m < m ) the waves are unsaturated and are the spectrum is usually assumed to be proportional to m s where s = 1. Waves in this regime are considered to have retained the spectral properties produced by the source mechanism. Wave amplitude growth is limited by the requirement that the vertical temperature gradient not exceed the adiabatic lapse rate. Waves with larger vertical wavenumbers (smaller vertical wavelengths) produce perturbations with sharper temperature gradients for a given amplitude and so they will be attenuated at lower altitudes. The characteristic vertical wavenumber is proportional to the buoyancy frequency divided by the wind perturbation amplitude (m N/u ). It is height dependent and reflects the height dependence of the separation of the unsaturated and saturated regimes. At wavenumbers greater than m the spectrum takes on a shape proportional to N 2 /m 3. This wavenumber range is often referred to as the saturated regime although the actual physics that produces this spectral form is debated. There are several independent theories that all predict this same analytical shape for the power spectrum. Each of these theories fall into one of two categories: linear theories say that waves dissipate independently of one another which forms the observed spectral shapes whereas nonlinear theories state that the spectral shape is produced by wave-wave interactions. The first set of theories is based on the premise that individual waves are attenuated independently of other waves. Linear instability theory states that as the wave amplitudes exceed the critical level they becomes saturated and excess energy is deposited into the background as turbulence (Dewan and Good, 1986; Smith et al., 1987). The resulting spectral shape has a slope of 3 in the saturated regime. A 26

27 similar theory, saturated-cascade theory, states that instead of excess energy becoming turbulence the dissipated wave energy transfers energy to higher frequency waves via cascade processes. This theory expects the same spectral slope as linear instability theory. Hines (1991a) charged that these theories based on individual wave dissipation ignore the wave-wave interactions and therefore neglect important physics. Hines (1991b) suggested that the wave-wave interactions are important and that the smaller scale waves are affected by the wave system and are spread the toward larger m values (and eventually into the turbulent regime). Weinstock (1990) suggested a similar effect, that the amplitudes of all waves are limited by diffusion-like processes due to the motions of smaller scale waves. Both of these theories predict the same m 3 dependence in the saturated regime, although these theories claim that wave saturation is not the mechanism by which the spectral shape is formed. All of these theories predict that the spectrum of GWs becomes independent of the wave sources in this regime. At very large wavenumbers (m > m b ) the spectrum is representative of small-scale turbulence and proportional to m 5/3, where m b is the wavenumber at which this transition occurs. Observations of GWs in the Earth atmosphere demonstrate little variability in the slope of the PSD with respect to altitude, location, season, and time of day, with typical values of the slope between 2.5 and 3.0. The observed magnitude of the power of the PSD, however, does present variability with altitude (Beatty et al., 1992; Senft et al., 1993), season, and latitude (Hirota, 1984; Eckermann, 1995). Linear theories allow only for relatively small variabilities in the saturated regime so these differences are thought to be due to the unsaturated portion of the spectrum. Nonlinear theories, however, do allow for some variations in the saturated part of the spectrum (Hines, 1991b) so it has been argued that these variabilities are evidence in favor of doppler-spread theory (Hines, 1993). 27

28 In this analysis the MGS radio occultation measurements are used to investigate the gravity wave power spectra in Mars atmosphere and compare them to the observed and theoretical spectra for the Earth atmosphere. The temperature-height profiles derived from the MGS radio occultations are analyzed to obtain the power spectral density of gravity waves on Mars at different latitudes and times of the martian year. The resultant slopes and powers of the PSDs are compared for different latitudes and seasons as well as to typical results from the Earth. Mars axial tilt is 25.2 as compared to Earth s axial tilt of 23.4 so a season on Mars is analogous to a season on Earth. 2.3 Data Analysis Between 1998 and 2005 Mars Global Surveyor produced 19,820 radio occultation measurements of Mars lower atmosphere. Radio occultations are measurements in which a radio signal is sent from a spacecraft to a receiving station as the spacecraft either just dips behind or emerges from the planet such that the radio signal sweeps through all altitudes of the atmosphere. The radio signals are refracted by the atmosphere and these measured effects can be used to determine the atmospheric density profile. Other atmospheric properties can then be inferred, including pressure and temperature. These radio occultations from the Mars Global Surveyor provide temperature-height profiles of the lower atmosphere at different latitudes, longitudes, solar longitudes, and times of day (Hinson et al., 1999). Many of the temperature profiles derived from these measurements contain wavelike properties that are interpreted as gravity waves (Hinson et al., 1999). Figure 2-2 depicts the latitude and longitude of all the radio occultation measurements for each local (northern) season. This coverage enables the global analysis and comparison of gravity wave activity for different locations and times of year. While there is coverage for all latitudes and seasons, not all seasons have coverage at all latitudes and the number of observations vary with latitude, longitude, and time of the year. Therefore only direct comparisons of specific seasons and latitudes are made. 28

29 Each individual profile contains a temperature-height profile ranging from the Martian surface into the stratosphere. A typical temperature-height dataset is shown as red in Figure (2-3). The profiles generally exhibit wavelike features, as seen in Figure (2-3), superimposed on a background temperature and typically vary with height. The maximum height of the temperature profiles vary significantly, spanning altitudes from about 30 to 60 km. The vertical resolution of the analyzed profiles also varies significantly, spanning from 172 m to 1.26 km. The vertical resolution of the profiles determines the lower bound of observable vertical wavelengths. Therefore, this analysis will only consider wavelengths greater than 2.5 km, the Nyquist limit of the lowest resolution profiles. The number of available datasets is large so there is some freedom in the selection of profiles to be analyzed. Only nighttime profiles, taken between 2300 and 1100 local time, were analyzed. Nighttime hours were used to retain only those profiles that had a more stable atmosphere. The bottom 10 km of all the datasets was omitted to avoid boundary layer effects that might improperly assign power to specific wavelengths. Further, any profile that had a vertical range less than 25 km (15 km range after the bottom 10 km of data were cut) was not used. This omission was made in order to maintain an upper limit of wavelengths in the spectrum of 15 km. The vertical range of the profiles range from 30 to 60 km. Any profile that had an initial vertical range exceeding 35 km was limited to a range of 35 km in order to homogenize the datasets so that the profiles did not sample different altitudes. The profiles were then sorted into latitude bands of 20 width and four solar longitude bins representative of Earth s seasons (defined the same way as on Earth: 0 < L s < 90 is spring, 90 < L s < 180 is summer, etc. corresponding to the northern hemisphere). For simplicity in the analysis, each profile was linearly interpolated to a common vertical resolution of 500 m. To analyze the gravity wave spectrum of each profile the temperature perturbations are first calculated. A third-order, least squares, polynomial 29

30 fit was used to assign a background temperature to each profile. The background temperature was then subtracted from the data to yield the temperature perturbations. Fourier analysis was performed on the extracted normalized temperature perturbations T /T o to obtain the power spectral density for each profile. Due to the range and resolution restrictions of the profiles each PSD is difficult to interpret individually. The large number of datasets allow the individual PSDs to be averaged within latitudinal and seasonal bins to obtain zonally averaged PSDs for each season. The slopes of the PSDs were calculated with a linear, least-squares fit to the spectra for the wavenumbers corresponding to vertical wavelengths between 2.5 km < λ z < 15 km. 2.4 Results The MGS radio occultation experiment presents the unique opportunity to study the global morphology of gravity waves on a planet other than the Earth. It has been well documented for the Earth that the slope of the large m, saturated portion of the gravity wave power spectral density is generally invariant in latitude, altitude, season, and for different topographies. The results presented in this section examine the PSD slope of waves in Mars atmosphere for different seasons, latitudes, and topographies and compare them to the similar results for Earth s atmosphere. Slopes of the PSD could not be calculated for different altitude ranges due to the limitation of the available vertical range in most of the profiles. Finally, the total gravity wave power is calculated seasonally for two latitude bands, 55 to 75 and 55 to 75, that have observational coverage for almost all solar longitudes and this variability is compared to that observed GWs in the Earth atmosphere Spectral Shape of Gravity Wave Power Spectral Densities To determine the shape of the gravity wave power spectrum for different local seasons and latitudes the individual temperature profiles were Fourier analyzed for wavelengths between 2.5 km and 15 km. To investigate the seasonal and latitudinal dependencies of the spectral shape in the thousands of profiles the spectral data were 30

31 separated into latitudinal and seasonal bins and averaged. Each latitude bin covers 20 and includes profiles for all longitudes; there are four seasonal bins corresponding to local (northern) spring, summer, autumn, and winter Latitudinal and seasonal dependence The averaged power spectral densities for each bin are illustrated in Figures (2-4) through (2-7). It is apparent that all of the resulting power spectral densities have negative slopes in the range of 2.0 to 3.0. Comparing these results to the theoretical power spectral density for Earth suggests that the waves in the MGS datasets are in the saturated portion of the spectrum. Creasey et al. (2006a) pointed out for several sample temperature profiles that the temperature perturbations show evidence of wave saturation. This observation agrees with what is seen in the calculated power spectral densities. The slope of the averaged power spectral density for each season and latitude are tabulated in 2-1. The latitudes stated in this table correspond to the median latitude in the bin. The number N of datasets in the bin is also included. Most of the slopes fall between 2.0 and 3.0 which is consistent with terrestrial observations ( 2.5 to 3.0) and theoretical expectations ( 3) of the slope in the saturated part of the spectrum. The PSD slopes are, roughly, invariant for all latitudes and seasons. It is noted that the power at a given latitude for different seasons appears to have a seasonal variability; this will be further analyzed and discussed in Section Topographical dependence In the preceding analysis the bins are averaged over all longitudes and so any topographical variability is obscured. Winds over topography is one method of gravity wave generation; for regions with markedly different topographies it is expected that the generated spectra of waves will be disparate. To test the slope invariance for differing topographies two adjacent locations with very different terrain were chosen that had good radio occultation coverage for that 31

32 latitude and season. Figure (2-9) illustrates the topography of the chosen region. The data used is for the spring months and covers latitudes ranging from 30 to 50. One region is a flat plains area that spans longitudes from 170 to 220 and the other contains a mountainous region that spans longitudes from 220 to 270. The resulting PSDs of these locations are shown in Figure (2-8), revealing a slope of 2.04 for the flat terrain and a slope of 2.09 for the mountainous terrain. The slope for both regions is roughly the same and slightly shallower than the theoretically predicted 3 slope. Further, the magnitudes of the PSDs demonstrate that the mountainous region contains more power in the spectrum than does the flat region, as would be expected considering orographic wave generation. These results suggest that the slope of the saturated portion of the wave spectrum in Mars atmosphere is also invariant with respect to topography and that mountainous terrains generate more power in the wave spectrum than do flat terrains Gravity Wave Seasonal Cycle It was found in Section that the slopes of the PSDs for various latitudes and seasons are in accord with those observations for Earth. It was noted that the magnitude of the power in the spectra appeared to have a seasonal dependence. In this part of the spectra the waves are considered to be saturated and thus are not expected to have variability in power. To test this expectation the variation in total spectral power is investigated with respect to solar longitude. To obtain the variation in spectral power throughout a Mars year, latitude bands that had good coverage throughout the year were chosen. Latitudes from 55 to 75 and 55 to 75 were used for this purpose. The northern hemisphere latitude band has coverage throughout the entire year but the southern hemisphere band lacks coverage in southern winter. The integrated power over the entire wavelength range in the two latitude bands are shown in Figure (2-10) for solar longitudes at 30 intervals. The data for the southern hemisphere latitude band was shifted by 180 in order to overlay the 32

33 seasons for both plots. The results clearly show a strong seasonal dependence for both latitude bands with increased wave activity in the fall and early winter and low wave activity during late spring and summer. A similar, strong seasonal variability of the wave activity in the Earth atmosphere at high latitudes was observed by Hirota (1984) and further analyzed via simulation by Eckermann (1995). These data showed enhanced wave power in the late autumn/early winter months and diminished wave power in the late spring/early summer months. Eckermann (1995) demonstrated that the seasonal dependence of power should be due only to variations in the unsaturated (m < m ) portion of the spectrum and that there should be little to no variations in the saturated portion. The variations observed by Hirota (1984) were less pronounced than those in the simulations, so Eckermann (1995) concluded that the calculated power contained both saturated and unsaturated portions of the spectra. However, Hines (1993) has argued that the doppler spread theory of Hines (1991b) allows for power variations in the saturated part of the spectrum with altitude whereas the other gravity wave PSD theories do not. Eckermann (1995) suggested that this effect could also lead to seasonal variations in the saturated portion of the spectrum although his simulations did not include these possible effects. The results for Mars demonstrate that all of the observed waves appear to be limited to the saturated portion of the spectrum. However, it is not expected that there would be a distinct seasonal variation of power in this portion of the spectrum according to linear theories, however, these variations are observed and are prominent. It is probable that there are unsaturated waves, likely at lower altitudes, contained in the data which give rise to these seasonal variations. Unsaturated waves contained within the data would also result in the spectral power being diminished, especially at low wavenumbers, which would cause the slope to be slightly diminished. It was seen that the slopes of the PSDs were typically somewhat less than 3 which supports the possibility of some unsaturated waves within the datasets. 33

34 2.5 Conclusions Gravity waves on Earth have been observed and studied for decades. The relative ease of observing gravity waves on Earth has provided ample knowledge of global and seasonal gravity wave activity that can also serve as a comparison for similar studies on other planets. On Mars, the copious radio occultation measurements from Mars Global Surveyor provides a unique opportunity to systematically examine seasonal and global properties of gravity waves on a planet other than Earth. As a result of these analyses the following conclusions are reached: i) The gravity wave power spectral density (PSD) of the temperature fluctuations in Mars atmosphere exhibits a slope that does not depend on latitude, season, or underlying terrain. ii) The PSD slopes range mainly between 2.0 and 3.0 which coincides with the observed terrestrial values ( 2.5 to 3.0). iii) In contrast to the terrestrial observations the lower limit of the unsaturated GW regime (m < m ) is not detected; all spectra appear to be saturated. At the small wave number end the limited length of the acquired temperature profiles restricts the analyses. As a result, vertical wavelengths larger than 15 km cannot be considered, which may explain the lack of an unsaturated regime in the results. Further, the Nyquist limit on the resolution of the retrieved profiles limits the study to vertical wavelengths greater than 2.5 km; the results suggest that the transition to turbulence (m > m b ) occurs beyond this limit. However, the seasonal dependence of wave power and the tendency for the PSD slopes to be slightly less than the theoretical value of 3 suggests that there may be some unsaturated wave modes within the data. iv) The intensity of the gravity wave activity at high latitudes on Mars shows a clear seasonal dependence with a well-defined maximum during the onset of the winter and a minimum in the summer. In the Earth atmosphere a strong seasonal dependence is also observed that shows the wave power peaking in late autumn/early winter. This dependence is thought to be due to variations in the unsaturated portion of the spectrum only, although the doppler-spread theory does allow for variations in the saturated portion of the spectrum. 34

35 Further work should be done to assess the impact of the spectral properties of the observational noise on the derived GW spectral characteristics in order to evaluate the confidence of these results. Correlation studies with Mars weather phenomena can also be investigated, such as looking into the relation of gravity wave activity to local and global sandstorm events. 35

36 Table 2-1. PSD slopes for each bin Season Latitude (±10 ) N Slope Spring 80 0 Spring Spring Spring Spring 0 0 Spring Spring Spring Spring 80 0 Summer 80 0 Summer 60 0 Summer 40 0 Summer Summer Summer Summer Summer Summer Autumn Autumn Autumn Autumn 20 0 Autumn Autumn Autumn Autumn Autumn 80 0 Winter Winter Winter Winter Winter Winter Winter 40 0 Winter Winter

37 Figure 2-1. Theoretical gravity wave power spectral density. Schematic presentation of the vertical wavelength power spectral density (PSD) of the temperature fluctuations resulting from atmospheric gravity waves in the Earth atmosphere. 37

38 Figure 2-2. Map of Mars Global Surveyor radio occultation data. The locations of the radio occultations taken by Mars Global Surveyor in each of the four seasons are shown as dots. The highlighted regions (green) represent the latitude bands that are analyzed for seasonal variations. The blue box contains the data that are used to analyze the PSD slope dependence on topography. 38

39 Figure 2-3. Sample temperature-height profile. A typical temperature-height profile of Mars taken by Mars Global Surveyor. The red corresponds to the interpolated data and the green is the background temperature fit. 39

40 Figure 2-4. Latitude dependence of the power spectral density for northern spring. Power spectral density for different 20 latitude bands averaged over all seasons. Red lines represent northern hemisphere. Blue lines represent the southern hemisphere. The black line represents equatorial latitudes. 40

41 Figure 2-5. Latitude dependence of the power spectral density for northern summer. Power spectral density for different 20 latitude bands averaged over all seasons. Red lines represent northern hemisphere. Blue lines represent the southern hemisphere. The black line represents equatorial latitudes. 41

42 Figure 2-6. Latitude dependence of the power spectral density for northern autumn. Power spectral density for different 20 latitude bands averaged over all seasons. Red lines represent northern hemisphere. Blue lines represent the southern hemisphere. The black line represents equatorial latitudes. 42

43 Figure 2-7. Latitude dependence of the power spectral density for northern winter. Power spectral density for different 20 latitude bands averaged over all seasons. Red lines represent northern hemisphere. Blue lines represent the southern hemisphere. The black line represents equatorial latitudes. 43

44 Figure 2-8. Topographical dependence of the power spectral density. Power spectral density of gravity waves over very different topographical features. The green line covers flat terrains. The brown line covers mountainous topography. 44

45 Figure 2-9. Topography of Mars in the analyzed region. 45

46 Figure Seasonal dependence of the power spectral density. Seasonal variation of gravity wave power in two mid-high latitude bands, one in the northern hemisphere (red) and one in the southern hemisphere (blue). 46

47 CHAPTER 3 IMPACT OF GRAVITY WAVES ON THE IONOSPHERE OF JUPITER 3.1 Background Jupiter is the largest planet in our solar system and is comprised primarily of hydrogen and helium, 86.5% and 13.5% mole fraction respectively, yielding the designation of a gas giant. Jupiter contains more mass than all the other solar system planets combined and has a mass of nearly one-thousandth of the Sun. Jupiter is thought to have a dense core of heavy elements surrounded by liquid metallic hydrogen. Surrounding this is the gaseous envelope, its atmosphere, whose density and pressure decrease exponentially with radial distance from the center. Jupiter s equatorial radius, measured as the distance from the center of the planet to the 1-bar pressure level (atmospheric pressure at Earth s surface) is 71,492 km; its polar radius is almost 6.5% smaller. This oblateness is due to Jupiter s fast rotation rate. A jovian day lasts only about hours. Jupiter s distance from the Sun is, on average, 5.2 AU (1 AU is defined as the Earth-Sun distance). As a result, the solar flux incident on Jupiter s atmosphere is considerably less than that at Earth History of Observations In addition to a large number of ground-based observations, much of the detailed data of Jupiter comes from the several spacecraft that have traveled to the outer solar system. Pioneers 10 and 11 encountered Jupiter in 1973 and 1974, respectively. These spacecraft provided the first close-up photos of Jupiter as well as made observations of the atmosphere, magnetic field, and several of Jupiter s moons. In 1979 the two Voyager spacecrafts made flybys of Jupiter and provided further information about the Jovian system including discovering that Jupiter in fact had a thin ring system and observing the dynamics of Jupiter s Great Red Spot. Another notable flyby of Jupiter occurred by the Cassini spacecraft in 2000 while en route to the Saturnian System. The photos from Cassini had a very good resolution and were used to study the dynamics of the 47

48 atmosphere in great detail. Most recently to observe Jupiter close-up, the New Horizons spacecraft passed Jupiter in late 2007 and early 2008 on its way to Pluto. From 1995 to 2003 the Galileo spacecraft was injected into an orbit around Jupiter from where it studied the jovian system. The Galileo orbiter made detailed studies of Jupiter s atmosphere, many of the moons, its rings, as well as mapping Jupiter s substantial magnetic field. Included in the mission was an atmospheric probe that, in 1995, plunged into Jupiter s atmosphere making in situ observations including measurements of the atmospheric composition, pressure, and temperature down to a pressure level of 22 bars Unresolved Questions in Jupiter s Upper Atmosphere With the wealth of data obtained for Jupiter there are still several major unanswered questions. Jupiter s thermospheric temperatures are much hotter than they would be based on solar heating models alone. This is an outstanding problem that pervades all the giant planets and there has been much research dedicated to figure out what is the additional heating source. Energetic particle precipitation in the equatorial regions, heat transport from the auroral regions to low latitudes, joule heating, and heating due to dissipation of upward propagating gravity waves have all been suggested as possible mechanisms to enhance the thermospheric temperatures. All of the proposed mechanisms have either been shown to not heat the thermosphere enough or suffer a deficiency in available data that would determine to what extent the mechanism contributes to the thermal structure of the upper atmosphere. Yelle and Miller (2004) argue that gravity waves are the best candidate since all four giant planets show elevated thermospheric temperatures; while the other proposed heating processes may explain the case for Jupiter they would have little effect on the much more distant (lower solar input) giant planets or planets with weaker a magnetic field. Gravity waves are suspected to be pervasive in all these atmospheres. 48

49 Another major discrepancy between observations and the models is the ionospheric structure of Jupiter: the observed electron density peaks occur at higher altitudes and are smaller in magnitude than the expected values. Solar ionization, chemical, and diffusion models fail to account for observed electron density profiles. Majeed et al. (1999) was able to explain these structures by incorporating two additional effects. To reduce the electron density peaks he incorporated losses of H +, the major ion at these altitudes, with vibrationally excited molecular hydrogen, H2. To increase the altitude of the peaks he included specific winds to blow the plasma upwards along the magnetic field lines. Ions are constrained to move in the direction of the magnetic field at these altitudes due to the low collision rate of the molecules. The H + ions are long-lived at these altitudes so they can be significantly displaced by the winds, allowing the electron density peak to be moved considerably in this manner. By tweaking the vibrational temperature of H2 and the wind fields the observed ionospheric structure could be reproduced. Whether or not the parameters used in the model are physically plausible remains to be determined. Gravity waves can have a significant impact on the ionosphere. In addition to creating sharp layers in the ion densities, waves with large amplitudes can induce significant downward fluxes of ions. It has been demonstrated that a gravity wave propagating in an ionosphere can both reduce the magnitude of the electron density peak as well as increase its altitude (Matcheva et al., 2001). However, there are only a handful of observed gravity waves in Jupiter s atmosphere. It is difficult to know what the realistic effects of gravity waves are on the ionosphere without knowing the parameters of the extant waves as well as how often they occur Overview of Work The study in this chapter investigates the effects of atmospheric gravity waves on the vertical and horizontal structure of the ionosphere of Jupiter. The presented non-linear, two-dimensional model of the jovian ionosphere allows for spatially and 49

50 temporally varying neutral wind and temperature fields and tracks the time evolution of six ionospheric species: H +, H 2 +, H+ 3, He+, HeH +, and CH 4 +. An analytical approach is used to validate the model results for linear, small-amplitude waves and to elucidate the mechanisms that lead to perturbations in the density of the main ion species, H + and H 3 +. It is demonstrated that the long-lived H+ ions are perturbed directly by wave dynamics whereas short-lived ions such as H + 3 are perturbed by chemical interactions with other perturbed ion species. The non-linear model is then applied using larger gravity wave amplitudes that are consistent with observations. Atmospheric gravity waves propagating at high altitudes create layers of enhanced electron density similar to the system of layers observed during the J0-ingress radio occultation of the Galileo spacecraft (Figure 3-1). This study s best fit to the J0-ingress observation is achieved using an 82 min period forcing wave with horizontal and vertical wavelengths of 500 km and 60 km respectively, and peaks at 510 km above the 1 bar pressure level. Further investigated are the effects of the wave-induced ion flux on the background ionospheric structure and it is demonstrated that in the presence of a gravity wave the background density profiles of the H + and H + 3 ions are significantly modified. It is also found that the column density of H + 3 has variations that can exceed 10% as the wave propagates. Gravity waves have been proposed as possible explanations to several unanswered questions in Jupiter s atmosphere, however, currently there is little information about the spectrum of the present gravity waves, their amplitudes, frequency of occurrence, and direction of propagation. Without observational constraints the net effect of waves on the upper atmosphere cannot be correctly assessed. It is proposed in this study that wave induced variations in the H + 3 emission in Jupiter s ionosphere can be used as a measure of wave activity. The wave-ionosphere model is coupled to a radiative transfer model (Raynaud et al., 2004a) to examine the potentially observable parameters of the waves. It is found that the gravity waves strongly affect the lower ionosphere and can induce observable variations in the H + 3 emission. The wave parameter space is explored and 50

51 the magnitude of the effect for different directions of wave propagation, magnetic field orientations, and locations on the planet are examined. 3.2 Model of Jupiter s Upper Atmosphere A fully non-linear, two-dimensional, time dependent model of Jupiter s ionosphere is developed to investigate the effects of atmospheric gravity waves on the ion structure and dynamics. The model includes ion diffusion, chemistry, and ion-neutral drag and allows for spatial and temporal variations in neutral wind velocity and temperature. The ionospheric model is coupled with the atmospheric gravity wave model described in Section The vertical range of the model spans from 100 km to 1500 km above the 1 bar pressure level with a step size of 1 km. It is assumed that there is no net ion flux at the topside and that at the bottom boundary the ionosphere is in chemical equilibrium. In the horizontal direction periodic boundary conditions are imposed with 28 grid points within a single horizontal wavelength Neutral Atmosphere The dominant neutral species in Jupiter s atmosphere are H 2, He, and CH 4 with atomic hydrogen becoming important at high altitudes (Figure 3-4). Below the methane homopause, around 350 km above the 1 bar pressure level, eddy diffusion dominates and the atmosphere is well-mixed, shown by the mixing ratios of the neutral species in Figure 3-5. The eddy diffusion coefficient used in the model is cm 2 /s (Yelle et al., 1996). Above the homopause, where molecular diffusion is dominant, the species separate diffusively according to their individual scale heights and the CH 4 and He mixing ratios diminish quickly with altitude. H 2 is dominant up to roughly 3000 km at which point H becomes the main neutral component. The neutral densities are calculated using the mixing ratios for He and CH 4 at the bottom boundary of the model and are taken from the Galileo probe measurements (von Zahn et al., 1998; Niemann et al., 1996). The resulting neutral density profiles are in good agreement with the 51

52 composition model from Seiff et al. (1998). The steady state temperature profile used in the model is shown in Figure 3-4. It is an analytical fit to the temperature measurements taken in situ by the Galileo probe. The neutral waves are only small perturbations upon the background atmosphere and so second order effects on the composition of the neutral atmosphere are ignored. The background neutral composition and temperature are held constant throughout the numerical, ionospheric simulations. An important chemical reaction that maintains the ion chemistry in Jupiter s ionosphere is a reaction between H + ions and vibrationally excited hydrogen molecules: H 2 + H + H H. This reaction acts to reduce the magnitude of the H+ density peak and has been used to better model the observed electron density profiles on Jupiter. Following the work by Cravens (1987) and Majeed et al. (1999) the vertical density profile for H 2 is calculated using the expression N(H 2) = N(H 2 )e Eν ktν. (3 1) A value for E ν is adopted from Cravens (1987) and a vibrational temperature, T ν, of 1330 K from Majeed et al. (1999) is used to fit the Galileo J0-ingress electron density profile Ion Chemistry The time scales of the relevant chemical reactions and the ion dynamics in Jupiter s ionosphere span several orders of magnitude. The model needs to be able to resolve the time scale of fast chemistry and also maintain the wave propagation over several wave periods with available computational resources. A simplified but accurate chemical model for the processes of interest must be invoked so that the time-evolution of the system can be computed. The model accounts for six ion species, H +, He +, H 2 +, CH+ 4, HeH +, H 3 +, as well as electrons and maintains the chemical reactions listed in Table 3-1. This ionospheric model is very simplified compared to other jovian ionospheric models in 52

53 the literature (e.g. Kim and Fox (1991)) but this simplified model does reproduce well the structure of the H + and H + 3 ions which are the items of interest in this work. The H +, He +, H + 2, and CH+ 4 ions are created primarily via photoionization from solar extreme ultraviolet (EUV) radiation. The photochemical productions of these ions are calculated using the EUVAC solar spectrum model for the wavelength dependent solar radiation intensity (Richards et al., 1994). Photoionization cross-sections of H, H 2, He, and CH 4 are taken from Schunk and Nagy (2000) (see references therein). The model allows the solar zenith angle and incident intensity to vary diurnally or to be held constant with a diurnally averaged incident flux. The photoionization production rates are calculated using these data and the modeled neutral densities. Secondary electron ionization is included by adapting a parameterization scheme originally proposed for Saturn (Moore et al., 2009). The resulting photoionization rates are shown in Figure 3-6 and are in agreement with other similar calculations in the literature (e.g. (Kim et al., 2001)). Figure 3-7 shows the lifetimes of the various ions. In the lower ionosphere H + and H + 3 have the longest lifetimes and are expected to be significantly affected by the presence of atmospheric waves. H + is created primarily from photoionization processes and has recombination as its main loss mechanism. The long lifetime of the H + ion is due to its relatively slow recombination rate. The H + 3 ion is created primarily through the reaction H+ 2 + H 2 H H. There are two major loss processes that H + 3 undergoes. Near the homopause where CH 4 has a significant presence H + 3 is consumed via H+ 3 + CH 4 CH H 2. At higher altitudes where the CH 4 density becomes negligible and the electron density increases the major loss mechanism for H + 3 becomes recombination with electrons. In the transition altitudes between these two loss processes (roughly between 500 and 700 km) the H + 3 peaks sharply as shown in Figure 3-7. These changes in the lifetime of the H + 3 lifetime ion will 53

54 be shown to have a significant effect on the mechanism that drives the perturbations in the H + 3 ion density. N i t + (N i U i ) = P i + j i G ijn N n N j N i L in N n L ie N i N e (3 2) n To calculate the ion densities the continuity equation for each ion is solved n assuming charge neutrality (N e = i N i). The system of continuity equations for all ions is given by Equation 3 2 where N i is the number density of the ion to be solved, U i is the plasma (ion) velocity, and P i is the photochemical production of ion i. N j are the number densities of the other ions such that j i n G ijnn j N n represents the chemical productions of ion i from reactions of ion j with neutral n where G ijn is the reaction rate coefficient for these collisions. The term N i n L inn n are the losses of ion i via collisions with neutral n where L n is the reaction rate coefficient for these reactions. N i L ie N e are the losses of ion i via recombination with electrons where L ie are the corresponding recombination rate coefficients. The equilibrium solution for the ionospheric composition ( N i t = 0) is used without any forcing winds as an initial condition for the time dependent problem. This background structure of the ionosphere is shown in Figure 3-8. The H + and H + 3 ions are the most abundant ions above the homopause. The H + 3 ion is dominant up to 600 km with H + becoming dominant above this altitude. The CH + 4 ion is the only hydrocarbon ion calculated in this model even though other hydrocarbon ions have been shown in other works to have significant abundances below the homopause (e.g. Kim and Fox (1994)). Thus, the model results herein are only accurate above the homopause where hydrocarbon ions have little effect on the overall ion chemistry. 54

55 To solve the full two-dimensional, time dependent system of continuity equations, Equation (3 2) is rewritten as N i t + A N i in i + B i h + C N i i z + D 2 N i i h 2 + E 2 N i i z + F 2 N i 2 i h z = P i + G ijn N n N j (3 3) j,j i n where h is the local horizontal coordinate, z is the local vertical coordinate, and A i, B i, C i, D i, and F i are the resulting coefficients for each order of derivative (evaluated in Appendix 5). The second order derivatives arise from the derivatives in the ion diffusion velocity (described in Section 3.4.1). The wind field at this point is still arbitrary. The equation has been reduced to two dimensions. Without loss of generality the horizontal axis is taken to be in the direction of the horizontal propagation of the wave. The angle between the wave number vector and the direction of the local, horizontal component of the magnetic field is a variable parameter that allows a model wave to propagate in any direction. The time dependent equation is solved implicitly for terms containing vertical derivatives only and explicitly for those containing horizontal derivatives, including mixed derivatives. Since horizontal gradients result only from propagating gravity waves, the horizontal coordinate is set to have periodic boundary conditions. Equation (3 3) is solved for each ion independently using the values of ion densities from the previous time step with a time step of 50 seconds. The gravity wave model provides an input for the temperature and velocity perturbations as described in Section To obtain the magnetic field of Jupiter at different latitudes the VIP4 model (Connerney et al., 1998) is used. So long as the ion gyro-frequency is much greater than the ion-neutral collision rate only the direction of the magnetic field enters the equations. This relation is satisfied for altitudes above 430 km and illustrated in Figure

56 3.3.1 Atmospheric Gravity Waves 3.3 Gravity Wave Theoretical Background Atmospheric gravity waves are buoyancy driven, non-compressible, transverse waves that propagate in a stably stratified atmosphere. The linear theory for hydrostatic gravity waves in a conservative atmosphere results in the following dispersion relation: kz 2 = k h 2N2 1, (3 4) ω 2 4H 2 where k z and k h are the vertical and the horizontal wave numbers respectively, N is the buoyancy frequency, ω is the wave frequency, and H = ( 1 ρ o ρ o z ) 1 is the atmospheric density scale height, where ρ o is the background mass density. In the hydrostatic regime the wave frequency ω is much smaller than the buoyancy frequency N. In this work the WKB gravity wave model presented in Matcheva and Strobel (1999) and expanded in Matcheva et al. (2001) (Equations (1) through (7) therein) is used by solving a set of linear wave equations using the Boussinesq approximation in a dissipative, rotating, background atmosphere with slowly varying temperature and zonal wind. The model includes eddy diffusion, molecular viscosity, and thermal conduction. At ionospheric heights the impact of ion drag on the wave motion can be significant. For atmospheric gravity waves with vertical wavelengths smaller than a scale height (λ z < H ) molecular viscosity and thermal conduction are the dominant dissipative factors that impact the wave amplitude. At 500 km above the 1 bar pressure level the pressure scale height is about 140 km. The effect of ion-drag is not included in the wave model. The vertical and horizontal gravity wave neutral wind perturbations, W n and V n respectively, and the gravity wave temperature perturbation T are for linear theory given by W n(x, y, z, t) = W n (z)exp(iφ) (3 5) V n(x, y, z, t) = W n (z) k z (1 + i )exp(iφ), (3 6) k h 2H k z 56

57 where and T Γ (x, y, z, t) = i ω + iβ W n(z)exp(iφ), (3 7) k 2 z = φ = k x x + k y y + z k h 2N2 ω( ω + iβ) 1 4H 2 z 0 k zr dz ω 0 t (3 8) ] [1 2 dh. (3 9) dz The parameters k z and ω are the gravity wave vertical wavenumber and frequency, respectively, and are complex if dissipation is present. The parameter β represents the dissipative effect of thermal conduction on the waves (Matcheva et al., 2001). The quantity φ is the phase of the perturbation. In the absence of dissipation β = 0, ω = ω and Equation (3 9) reduced to Equation (3 4) for an isothermal atmosphere. The amplitude of the wave induced vertical wind W n (z) is given by W n (z) = W n (z 0 ) ( ) 1 [ kzr (z 0 ) 2 z ( ) ] 1 exp k zr (z) z 0 2H k zi dz. (3 10) Here k zr and k zi are the real and imaginary parts of the vertical wavenumber respectively. The real part of the vertical wavenumber k zr is taken positive for a positive upward energy flux (downward wave phase velocity). The wave parameters k zr (z 0 ), k h, and W (z 0 ) are specified at the bottom of the modeled region at a reference altitude z 0. Equation (3 9) is then solved iteratively for the real and imaginary parts of k z and ω to calculate the amplitude and the phase of the propagating wave. The vertical scale of the gravity wave affects the dissipation rate and thus the altitude to which the wave propagates. Waves with large vertical wavelengths generally propagate to higher altitudes than waves with shorter vertical wavelengths Wave-ion Interaction Dynamical effects Figure (3-2) illustrates the interaction between a gravity wave and ions that give rise to layering in the ion density profiles. A gravity wave creates an oscillating perturbation 57

58 in the background neutral wind, shown in the figure as U n. The ions are dragged by the neutrals but are constrained to move along the magnetic field lines. The plasma velocity vector is the projection of the neutral velocity along the magnetic field line shown as U p. In the figure, where the neutral velocity is directed to the left, the plasma velocity is directed downward along the magnetic field lines. Where the neutral velocity is directed to the right the plasma velocity is directed upward along the magnetic field lines. This geometry gives rise to layers of ion flux convergence and divergence that yield layers of ion compression and rarefaction. This mechanism assumes that the motion of the ions is constrained to the magnetic field lines which is approximately true in the region where the ion gyro-frequency is greater than the ion-neutral collision frequency. For Jupiter this transition occurs just above the homopause, see Figure 3-3. At very high altitudes where the ionosphere becomes nearly collisionless the neutrals cannot drag the ions efficiently and the wave effect on the plasma distribution is minimal. The described mechanism for driving ionospheric plasma up and down along the magnetic field lines is especially optimized for midlatitudes where the magnetic dip angle is roughly 45. At the magnetic equator where the field lines are horizontal the ion-electron motion is confined to the horizontal direction. Regions of plasma convergence and divergence are formed along the horizontal magnetic field lines. Since gravity waves phase lines are tilted, these compressed/rarefied regions are not vertically aligned and the electron and ion density profiles do result in a periodic vertical structure. Thus, peaks in the ion/electron density due to a gravity wave are not prohibited from forming at the magnetic equator. This is consistent with electron density observations on Jupiter and Saturn where the lower ionosphere exhibits a layered structure in both mid and low latitudes (Fjeldbo et al., 1975, 1976; Hinson et al., 1997; Nagy et al., 2006; Kliore et al., 2009). 58

59 Chemical effects It is important to note that the mechanism illustrated in Figure 3-2 only applies to ions that have lifetimes on the order of or greater than the wave period. An ion that has a lifetime much shorter than the period of the wave does not exist long enough to undergo these perturbations significantly. However, short lived ions can demonstrate similar perturbation structures if they are involved in fast chemical reactions with longer-lived, perturbed species. In this situation, an ion that has significant gains or losses from a perturbed, longer-lived ion will mimic the perturbations in the longer lived ion if there are significant gains from the long-lived ion or else mirror the perturbations if there are significant losses due to the long-lived ion. In order to realistically model the effects of atmospheric gravity waves on Jupiter s ionosphere where dynamics competes with fast H + 3 chemistry the model ionosphere constructed in Section is used with the neutral wind variations of the forcing wave described in Section Before utilizing the model the physics in a few cases which use simplifying assumptions in order to allow the fully non-linear model to be tractable analytically are examined. The non-linear model is tested in the limit of small amplitude variations so that the equations can be linearized and solved analytically. First, the case of an ionosphere containing a single ion with infinite lifetime is solved. The equations are then generalized to a single ion with an arbitrary lifetime and then allow for influence from chemical reactions with another perturbed ion. Note that in all cases the neutral atmospheric gravity waves are well within the linear regime and no wave breaking is allowed. 3.4 Analytical Solutions and Numerical Validation for Small Amplitude Waves Analytical Ion Response for Small Amplitude Waves The continuity equations for each ion species i including chemistry and diffusion are given by Equation 3 2. To determine the ion velocity U i motion of the ions due to collisions and geomagnetic forces (neglecting other forces) is considered. The ion 59

60 equation of motion can be written d U i dt = ν in( U n U i ) + e m i ( U i B), (3 11) where the first term on the right is acceleration due to collisions with neutrals and the second term on the right is the acceleration due to geomagnetic forces. Here U n represents the neutral wind velocity, ν in is the ion-neutral collision frequency, m i is the mass of the ion, e is the elementary charge, and B represents the local magnetic field. If the accelerations, d U i, are small then the equation of motion can be rewritten dt U i = η 2 [η2 Un + η U n ˆl b + ( U n ˆl b )ˆl b ], (3 12) where η is the ratio of the collision frequency to the ion gyro-frequency and ˆl b is the unit vector associated with the direction of the local magnetic field (MacLeod, 1966). In the model the acceleration of ions is controlled by the wave motion. Gravity wave frequencies upper bound is the buoyancy frequency which is much smaller than both the ion gyro-frequencies and the ion-neutral collision frequencies in the range of the model. Figure 3-3 shows the relevant frequencies and justifies the use of Equation (3 12) in the model since the ion collision and gyro-frequencies are orders of magnitude larger than the allowable gravity wave frequencies. Figure 3-3 demonstrates that for altitudes above 430 km the ion-neutral collision frequency becomes much less than the ion gyro-frequency. In this regime η is small (η 1) so the ion velocity reduces to U i = ( U n ˆl b )ˆl b. With diffusion taken into account, the plasma velocity becomes U i = σ i [ ( N i N i + T p T p + ˆl z H p ) ˆl b ]ˆl b + ( U n ˆl b )ˆl b, (3 13) where σ i is the plasma diffusion coefficient, T p is the plasma temperature, H p is the plasma scale height, and ˆl z is the unit vectors pointing in the vertical direction. The 60

61 first term defines the ion diffusion velocity and the second represents the drag velocity, induced by the neutral winds, along the magnetic field lines. Equation (3 13) does not include the effect of any electric fields. In the case of strong ion-neutral coupling (η > 1) neutral winds can result in polarization electric fields that can map up to high altitudes along the magnetic field lines impacting the plasma motion even in the collisionless regime. This effect is observed in Earth s lower ionosphere which is dominated by long-lived metallic ions (Yokoyama et al., 2004). In contrast, Jupiter s lower ionosphere is dominated by fast chemistry (see discussion in Section 3.2.2) which does not allow for dynamically driven plasma accumulations and polarization electric fields. To solve Equation (3 2) for all ion densities analytically they must be linearized in terms of the ion densities and ion velocity. These variables are taken to be the sum of a steady background state, labeled with a subscript o, and a perturbative correction, denoted as a primed quantity: N i = N io + N i, U i = U io + U i. (3 14) Here U io = U id + ( U n ˆl b )ˆl b, the zeroth order ion velocity, consists of the ion diffusion velocity and the ion velocity contribution from the background winds. The perturbed ion velocity contains only the gravity wave induced ion velocities. Ignoring non-linear (second-order plus) perturbation terms, the linearized continuity equation for small amplitude perturbations of an ion i in an ionosphere containing multiple ions is given by ( t + ( U io ) + ( U io ) + [ L e N eo + ] ) L n N n N i = G ijn N n N j N io L ie N e (N iou i ). (3 15) j i n Perturbations of the photochemical production are due to small changes in the optical depth of the neutral atmosphere and are taken to be zero. The second and third n 61

62 terms in parentheses on the left hand side represent the effect of background winds and plasma diffusion on the ion perturbations. The two terms in brackets on the left hand side are losses in the perturbed quantity of ion i due to recombination with electrons and charge exchange reactions with neutrals, respectively. The first term on the right hand side represents gains of ion i due to charge exchanges between the perturbation in density of ion j and background neutral n. The second term on the right defines the losses of ion i due to recombination with the perturbed part of the electron density. The final term on the right hand side describes dynamical perturbations in the ion density as a result of perturbations in the ion wind velocity, U i. Equation (3 15) represents a system of equations, one for each ion species in the ionosphere. What follows are the analytical solutions for three simplified cases assuming no background winds and neglecting ion diffusion ( U io 0). The velocity field for the gravity wave propagating in the neutral atmosphere is given by Equations (3 5) and (3 6) Case I: Single ion, long lifetime First considered is the simplest case: a single ion with a long lifetime relative to the wave period such that perturbed chemical terms are negligible. This classic scenario is well understood and is described in the terrestrial literature by a number of authors (Hooke, 1970; Kirchengast, 1996). The continuity equations reduce to a single equation without chemical or diffusive terms, The long-lived, single ion case has the solution N i t + (N io U i ) = 0. (3 16) N i (x, y, z, t) = N io( U n ˆl b ) ω o [ ( 1 N io kr ˆl b i N io z + 1 ) ) 2H k zi (ˆl ] b ˆl z. (3 17) 62

63 The response of the ionosphere to a passing atmospheric gravity wave is highly non-isotropic. The magnitude of the induced ion perturbation strongly depends on the direction of wave propagation and the orientation of the local magnetic field. In the case of hydrostatic gravity waves with a small background ion density gradient the wave-induced ion perturbation, N i, is either in phase or 180o out of phase with the ion velocity perturbation, U i, depending on the direction of wave propagation. This phase relation will result in a vertical ion flux with a non-zero time average, F wz = N i U iz t. The wave induced ion flux becomes important for large amplitude waves (Matcheva et al., 2001). An ion density perturbation in phase with the vertical wind perturbation results in a positive net vertical ion flux while a phase difference of 180 o results in a negative net vertical ion flux (for further discussion see Section ). Case I is a good approximation for Jupiter s ionosphere a few hundred kilometers above the homopause where H + is the dominant ion and has a lifetime on the order of 10 6 seconds, much longer than typical wave periods that are on the order of 10 4 seconds (Matcheva et al., 2001) Case II: Single ion, arbitrary lifetime Considered next is an ionosphere containing a single ion that has an arbitrary lifetime due solely to recombination with electrons. Assuming local charge neutrality, N e = N i and N eo = N io, reduces Equation (3 15) to and has solution N i t + 2L en io N i + (N io U i ) = 0 (3 18) N i (x, y, z, t) = N io ( U n ˆl b )(ω o i2l e N io ) (ω 2 o + 4L 2 en 2 io ) [ ( 1 N io kr ˆl b i N io z + 1 ) ) 2H k zi (ˆl ] b ˆl z. (3 19) 63

64 In the limit that the ion lifetime is much longer than the wave period, the recombination rate is much smaller than the wave frequency such that L e N io ω o. In this limit the case II solution reduces to the previous solution for a long-lived ion (Equation 3 17). When the lifetime of the ion is much less than the wave period the recombination rate becomes the dominant term in the denominator. The ion perturbation amplitude is then limited by the ion lifetime rather than the time-scale of the wave since the ions recombine before being significantly displaced. Also, it can be seen in this limit that the complex ion perturbation has a dominant imaginary coefficient that yields a phase difference with the perturbations in the ion velocity that approaches 90 o as the ion lifetime approaches zero. This solution is well represented by the perturbations of H + for the entire vertical range of Jupiter s ionosphere, as will be demonstrated in Section Near and below the homopause the lifetime of H + diminishes rapidly as interactions with neutrals become frequent. As the ion s lifetime becomes shorter than the wave period the dynamical perturbation mechanism is of limited influence, thus the generalized case II applies to a greater vertical range than does case I Case III: Two ions, arbitrary lifetime for minor ion The effort in this work is to understand the physics behind the perturbations in the H + 3 ion. To do this one must consider the effects that multiple ions have on each other. Consider an ionosphere consisting of two ions, one dominant and one minor. It is assumed that the dominant ion (index j) has a very long lifetime such that its density is not readily affected via chemical reactions with the minor ion (index i) and so that its lifetime is much greater than the wave period. Equation (3 15) then reduces to a single equation for the minor ion i, N i t + [ L e N jo + n L n N n ] N i + [ N io L e n G n N n ] N j + (N io U i ) = 0 (3 20) 64

65 where it has been assumed that the electron density is equal to the density of the dominant ion, N eo = N jo and N e = N j. Here N j is the density perturbation of the dominant ion and is given by Equation (3 17). Solving for N i yields N i (x, y, z, t) = N io ( U n ˆl b )[ω o i(l en jo + n LnNn)] ω 2 o+(l e N jo + n L nn n) 2 [ ( kr ˆl b i 1 N io N io z + 1 2H k zi ) (ˆl b ˆl z )] [ N io L e n G nn n][iω o +(L e N jo + n L nn n)] ω 2 o+(l en jo + n LnNn)2 N j. (3 21) This solution for the perturbations of the minor ion i is separated into two parts: the first term is dependent on the wave velocity perturbations, U n, and the second term depends on the density perturbations in the dominant ion j, N j. The first term is the same solution as that in case II, Equation (3 19), except that the lifetime of the ion is not only dependent on the recombination rate, which now depends on the density of the other ion via N e = N i, but also on the losses due to charge exchange reactions with the other ion species. The second term describes the density perturbations of the minor ion due to chemical reactions with the density perturbations of the dominant ion and the density perturbations of the electrons since N j = N e. Case III manifests in Jupiter s ionosphere for the H + 3 ion. For most of the vertical range H + is the dominant ion that has a lifetime much greater than viable wave periods and acts to chemically perturb the density of H + 3 via gains due to [H+ ] + [H 2 ] + [H 2 ] [H + 3 ] + [H 2 ] and losses with electrons via [H + 3 ] + [e ] [H + 3 ]. The H + 3 lifetime varies dramatically with height and it will be shown that both the dynamical and chemical perturbation mechanisms play a significant role in perturbing the H + 3 ion in Jupiter s ionosphere. 65

66 Case IV: Further generalizations More general equations for the ion density perturbations of a particular ion can be derived. To include more ions one must account for additional perturbation terms of the primary ion due to perturbations in each other ion. If no one ion is dominant then the electron density, N e = j N j, can be treated as its own term whose perturbations give rise to perturbations in the ion of interest. Any more complicated case, however, will just be some combination of the physics described in the first three illustrative cases. To conclude, the perturbations in the density of a particular ion are created not only by the dynamical wave-forcing mechanism but also indirectly by reactions with wave-induced perturbations in the density of other ions. The effect that is dominant for a particular ion is dependent on the ion lifetime and background ion density which both vary with height. Thus, a given ion can be in different perturbation regimes at different altitudes. Further, different perturbation regimes have different phase relationships to the forcing wave velocity and thus can result in different net vertical ion fluxes Validation of the Numerical Model for Small Amplitude Waves In this section the ion responses resulting from small amplitude waves using the non-linear model presented in Section are compared to the analytical linear solutions presented in Section This is done not only to validate the model but also to gain insight into which perturbing mechanism is dominant for each ion at different altitudes. The results are presented only from the two major ions, H + and H 3 +, since all other ions have very short lifetimes in the lower ionosphere and any perturbations will be entirely due to chemical interactions with other perturbed species Sample wave parameters Since the timescales of the different chemical and dynamical processes in the ionosphere are important in assessing the impact of the waves on the ion distribution, two different wave period regimes are examined: a short-period wave (wave A) with a period on the order of the H + 3 lifetime and a long-period wave (wave B) with a period 66

67 much larger than the H + 3 lifetime. The sample wave parameters are detailed in Table 3-2. It is difficult to directly compare waves of different periods because they also have different vertical and horizontal wavelengths and dissipate at different altitudes. The parameters of the waves are chosen so that their amplitude peaks at the same altitude. In doing so the vertical wavelength for wave B is greater than that of wave A by roughly 50%. In this section the small amplitude values are used to ensure that the observed effects are entirely linear. Figure 3-9 illustrates the temperature and horizontal velocity amplitudes of the sample waves with respect to height. Both waves, by design, peak in the same region and have nearly the same amplitudes despite their very different wave periods. The figure demonstrates the amplitude values for the large amplitude wave. For the small amplitude waves in this section the figure is identical but has velocity and temperature values a factor of 10 3 smaller Small amplitude results Figure 3-10 illustrates the validation results for the fully non-linear model. The results from the numerical model are compared to the analytical solutions based on Equations (3 17), (3 19), and (3 21) for sample waves A and B. The top two panels (Figure 3-10a, b) show the response of the H + ions to the long and short period waves (wave A and B respectively) and compare the simulation to the result from Equation (3 17). In the region where H + lifetime is much longer the wave period (altitudes above 550 km) the analytic solution fits very well to the numerical simulations. The discrepancies are notable in the lower ionosphere below 550 km where the H + lifetime becomes comparable to or smaller than the wave period. The fit is much improved for H + if Equation (3 19) is used to describe the wave-ion interaction. Figure 3-10, panels c and d illustrate the very good fit for the entire vertical range, as this equation also accounts for effects due to fast chemical reactions which become important at these lower altitudes. 67

68 The bottom panels of Figure 3-10 (panels e and f) show the response of the H 3 + ions to the propagation of a small amplitude gravity wave. The nonlinear model is compared to the analytical expression Equation (3 21). The analytical solution takes into account both the dynamical forcing and chemical reactions with other perturbed ion species. For H + 3 there are three important chemical reactions: losses due to recombination with the perturbations in the electron density, [H + 3 ] + [e ] [H + 3 ], gains from H + perturbations via [H + ] + [H 2 ] + [H 2 ] [H + 3 ] + [H 2], and gains via [H + 2 ] + [H 2 ] [H + 3 ] + [H]. In order to solve for H+ 3 analytically the perturbations for the other ions, N j, must be known. The analytic solution for the H+ density perturbations is used for both the H + density perturbations and the electron density perturbations. The numerical solution of the H + 2 perturbations was used to avoid having to solve for all ion densities analytically in this limit. The structure and magnitudes of the H + 3 perturbations again show a very good agreement between the numerical and analytical solutions for both short and long period, small amplitude waves. In summary, these results demonstrate the response of the major ion species in Jupiter s lower ionosphere to a small amplitude gravity wave. It is shown that the analytical treatment of the wave-induced ion response in Section properly represents the relevant physics driving the perturbations. Further, the numerical model successfully maintains the dynamical and chemical perturbations due to a small amplitude gravity wave in Jupiter s ionosphere. With this validation of the non-linear model against small amplitude waves, the model can be confidently utilized to treat more realistic large amplitude gravity waves. 3.5 Large Amplitude Non-linear Simulations Realistic analytical solutions are only attainable for ion responses to small amplitude waves with temperature variations less than a kelvin or wind perturbations of only a few meters per second. In reality, wave amplitudes can be large at the peak altitudes. Temperature variations can be several tens of kelvins with wind perturbation velocities 68

69 that reach hundreds of meters per second. The resulting perturbations in the ion density can have amplitudes on the order of the background ion density itself. Note that the forcing atmospheric gravity wave is still linear with amplitudes less than the critical (breaking) values, however, the ionospheric response is very dramatic leading to ionospheric disturbances of large magnitudes, a phenomena well documented in the terrestrial ionosphere. The results from the non-linear numerical model described in Section 3.2 are first used to demonstrate some key differences between small and large amplitude waves. Several results are then presented using sample waves A and B using realistic amplitudes (Table 3-2, Large Amplitude) consistent with the Galileo observations (Young et al., 1997) Phase relation between small and large amplitude waves For a small amplitude wave the resulting ion perturbations are given by Equation This equation demonstrates that the phase relation between the perturbing wind field and the ion perturbations are either in phase or 180 out of phase. For upward propagating waves (downward propagating phase) the relation is 180 out of phase since k zr < 0. Figure 3-11 illustrates the phase relation between the ion perturbations and the forcing wave. Altitudes where the small amplitude, ion perturbations peak occur at the locations where the neutral wind velocity is maximum (negative); these altitudes are shown as thin, horizontal black lines. The mechanism for this is demonstrated in Figure 3-2. It shows the generation of regions of plasma convergence and divergence as the wave phase propagates downward. As the phase lines move downward, the plasma at a given location in the converging region increases in density until the maximum velocity phase line passes, at which point it enters the diverging region and the ion perturbation begins to decrease. Thus, for a small amplitude wave the maximum ion density occurs at the phase lines of maximum wind velocity. 69

70 The phase relation between the forcing winds and small amplitude waves does not hold true for large amplitude waves. For a large amplitude wave the ion fluxes are large enough to affect the background ion density. The location of maximum ion convergence is nearer to the center of the converging region, the locations where the wind velocity is zero. This relation is shown in Figure For large amplitude waves there are large ion fluxes and these large fluxes into the converging region deplete the ion density in nearby regions from where the ions are moving. The converging regions accumulate a significant portion of the ions in the region before the phase lines can pass and begin depleting this location. Thus, rather than the peak ion perturbation location occurring at the transition from the converging to the diverging region, as for small amplitude waves, the peaks occur nearer to the center of the converging region Ion Density Perturbations In order to calculate the wave-induce ion density perturbations, the diffusive equilibrium ion densities are first solved. Once the background (unperturbed) ionosphere is established, gravity wave wind perturbations are turned on and begin to affect the state of the ions. The initialization of the wave produces some transient effects that diminish with time until the quasi-steady state of the wave-ion interaction is attained. This typically takes about 5 wave periods to occur. In Section the theoretical expectation of the ion responses to the wind perturbations produced by a gravity wave were described in detail. Figures 3-12 and 3-13 demonstrate the model results for the perturbed ion densities in response to sample waves A and B, respectively, after a quasi-steady state is reached with inclination I = 47 (at 24 S latitude, the location of the J0-ingress radio occultation). The figures show all six ions; the corresponding electron density is omitted for clarity but is roughly equal to the dominant ion density. For comparison the observed electron density during the Galileo J0-ingress radio occultation is shown. Both propagating waves result in sharp layers of electrons below the main electron density peak. These perturbations 70

71 show remarkable similarity to observations of the electron density on Jupiter which reveal narrow peaks in the electron density below the main peak (Fjeldbo et al., 1975, 1976; Hinson et al., 1997). In the case of wave A a close fit to the J0-ingress electron density profile is attained. The simulations show layering in the longer lived H + and H + 3 ions as well as in the short-lived CH + 4 ions. The results for the H+ density perturbations fit well to the theoretical expectations and follow closely previously published large amplitude results of Matcheva et al. (2001) that use a more simplified chemical model. For the first time the response of the H + 3 ions to large amplitude waves can be addressed. Figures 3-12 and 3-13 show that the perturbations in the H + 3 density contain a more complicated structure, especially in the case of the long-period wave B. In the altitudes around 500 km the H + 3 perturbations appear to be in phase with the H+ perturbations with the peaks occurring at the same altitudes. Far above this region the H + 3 peaks are 180o out of phase with respect to the H + peaks. These two regions demonstrate the two different perturbation regimes due to the rapidly changing H + 3 lifetime with height. The H + 3 lifetime peaks at about 500 km and at this altitude has a lifetime that is comparable with the wave period. The H + 3 density in this regions is perturbed in the same manner as H +, via wave dynamics. Above this region the H 3 + lifetime decreases rapidly with height due to faster recombination reactions with the increasing electron density. At the higher altitudes, the electron density is roughly equal to that of the dominant H + ion and therefore maintains the same perturbations as that of the dominant ion. Since the H + 3 lifetime is short at high altitudes and is no longer perturbed significantly by the wave winds, the recombination with the perturbed electron density, [H + 3 ] + [e ] [H + 3 ], becomes the dominant perturbation mechanism. These chemistry-driven perturbations of H + 3 at the higher altitudes will be shown to be useful in looking for observable signatures for detection of gravity waves as described in Section

72 H + 2 There are also sharp peaks in the ion density of CH + 4 and to a lesser extent that of. These ions have lifetimes that are many orders of magnitude less than the wave period and are not expected to be perturbed by the dynamical wave-ion perturbation mechanism. Instead, these layers in the CH + 4 and H+ 2 densities arise primarily from chemical interactions with the perturbed H + density, namely [H + ] +[CH 4 ] [CH + 4 ] +[H] and [H + ] + [H 2] [H + 2 ] + [H] respectively Ion Flux Small amplitudes: Analytic The wave-induced ion flux mentioned in Section 3.4 arises from second order perturbations terms. For waves with very small amplitudes the ion flux can, in practice, be ignored, but for purposes of understanding the physics involved can be investigated analytically using the small amplitude ion perturbation solution. The significance of this flux becomes evident in the large amplitude simulations where it modifies the structure of the background ionosphere and results in large departures from the local chemical equilibrium Matcheva et al. (2001). The effect can be very big even if the forcing gravity wave has a modest amplitude since the plasma perturbations can be very large. This is of particular interest to 1D chemical models of giant planets ionospheres where chemical equilibrium is often assumed. The analytical solution 3 17 shows that the ion perturbation, N i, and the wave velocity perturbation, U n, are either in phase or out of phase by 180 o. For waves with downward propagating phases (upward energy transport), which is expected for waves generated deeper in the atmosphere, the perturbation and the ion velocity are 180 o out of phase. This results in a net downward flux of ions. The net flux is due to the downward pointing winds occurring in the regions where there is positive perturbation (more ions locally). Correspondingly, when the perturbation is negative (ion density diminished) the wave velocity is positive. In the presence of a downward propagating gravity wave a greater number of ions are forced downward while fewer ions are pushed 72

73 upwards resulting in a time-averaged, downward ion flux. This flux alters the structure of the background ionosphere while a gravity wave persists as well as for a time afterward that depends on the ion lifetimes and diffusion timescales. The magnitude of the wave-induced ion flux can be determined by averaging the instantaneous vertical ion flux over a wave period, F wz = N i U t iz. The net vertical ion flux is calculated analytically using the real part of the ion perturbation density, Equation 3 17, and the perturbed ion velocity projection in the vertical direction U iz. This yields F wz = N io 2ω o ( V n cos I cos α W n sin I ) 2 (k h cos I cos α k zr sin I ) sin I (3 22) where V n and W n are the amplitudes of the wave perturbation velocities in the horizontal and vertical directions respectively, k h and k zr are the horizontal and the real part of the vertical wave number respectively, I is the local inclination of the magnetic field, and α is the angle between the horizontal wave number vector and the horizontal component of the magnetic field (Matcheva et al., 2001). This solution assumes that chemistry plays no part in the ion perturbations. Chemical interactions between species (described in Section ) can induce additional perturbations in a particular ion density that affects the magnitude and even the direction of the net ion flux. Short-lived ions do not have a large wave-induced net ion flux since they do not exist long enough to be significantly displaced. Instead, changes in the density profile of a long-lived ion due to the wave-induced net ion flux can have a drastic affect on the chemistry and thus the densities of the short-lived ions. This will be shown to be the case for H + 3 ions in the Jovian ionosphere Large amplitudes: Numerical In addition to driving large perturbations in the ion density (Section 3.5.2), gravity waves induce a net ion flux as described in Section This net ion flux alters the 73

74 structure of the background ion density and can significantly affect the net ion/electron content of the ionosphere. The magnitude and direction of the net ion flux, F wz = N i U t iz, is determined by the magnitudes and phase relationship between the ion density perturbation and the vertical plasma velocity perturbation. For H +, which has now been shown to abide by the dynamical forcing mechanism, the ion density and vertical plasma velocity perturbations are roughly 180 o out of phase (for an upward propagating wave). According to Equation (3 22), the H + flux should be negative for the entire range of the wave perturbations and so there should be a net downward flux of H + ions. Figures 3-14 and 3-15 illustrate the H + flux in the model for both short and long period sample waves. For both waves the net H + flux is shown to be negative for the entire range as expected from the small amplitude analytical solution. Figures 3-14 and 3-15 also show the net H + 3 flux throughout the vertical range of the model. Just above the homopause where the H + 3 lifetime peaks the ion perturbations are dynamically forced and the net H + 3 flux is downward, the same as for H+. At higher altitudes where the H + 3 lifetime is much shorter than the wave period, the wave-induced fluxes of H + 3 have little effect on its density profile since the H+ 3 ions recombine before being significantly displaced. Instead, there are net changes in the H + 3 density profile caused by chemical interactions with the wave-induced changes in the H + and electron density profiles. Figure 3-16 demonstrates the effect that a large amplitude gravity wave has on the background ionosphere. The figure shows the background ion density of H + and H 3 + with no wave present compared to the background ion densities after five periods have elapsed for both sample waves. The quasi-steady state background ion densities are calculated by averaging the densities across a horizontal wavelength. The H + density for both waves shows a significant depletion of ions above the wave peak and a somewhat more moderate gain of ions below the wave dissipation region. The number of ions is 74

75 not conserved in this process. The ions that are transported lower in the ionosphere are subject to quicker recombination reactions and so the net H + content of the ionosphere is diminished. This process has the effect of diminishing the magnitude and increasing the altitude off the main electron density peak (Matcheva et al., 2001). The effect is more notable for wave A and will be greater for larger amplitudes and for a wave that propagates for longer than just five wave periods. Figure 3-16 shows for both waves that there is a significant increase in the H 3 + density above where the wave peaks. The wave-induced net ion flux of H + 3 ions actually has very little effect on the restructuring of its ion density profile. Since the H + 3 lifetime is not much longer than a wave period at its peak, a net ion flux cannot effectively transport ions. The main cause for the region of enhanced H + 3 density is the reduction of the electron density which reduces the H + 3 recombination rate. As the H+ density, and thus the electron density, is diminished in this region the lifetime of H + 3 is increased, thereby enhancing the H + 3 density in the region. A gravity wave with parameters similar to either of the sample waves will yield an increase in the net H + 3 content of the ionosphere. Since H + is a long-lived ion these changes in the background ion densities will not only exist while the wave is present but will persist for hours after the wave has passed or dissipated. Figures 3-17 and 3-18 show the change of the total column density of H + and H 3 + over time for each sample wave. Both waves dramatically decrease the total H + column density as the wave persists while the H + 3 column density increases significantly for both waves. As the net downward ion flux of H + moves ions downward into regions dominated by fast chemistry, the H + ions are lost. A decrease in local H + density entails a decrease in the local electron density which leads to less recombination of H + 3 and thus an increase in the local H + 3 density. The change in the column density for both ions occurs very quickly for the first five wave periods due to the sudden initialization, numerically, of the wave wind field. It takes 75

76 roughly five wave periods for the atmosphere to reestablish a quasi-equilibrium state with the wave present after which there is still a modest change in the column density due to the wave-induced net ion flux Ion Column Density Variations Figures 3-17 and 3-18 also demonstrate variations in the column density that manifest as a sinusoid over the course of a wave period. A dynamical wave-ion interaction with no chemistry involved would have the gravity wave simply move ions back and forth and would not change the total ion content of the ionosphere. With chemistry accounted for, local wave-induced changes in the ion densities will result in changes in the local reaction rates. The gains and losses of each ion will then differ from equilibrium which will perturb the net ion content of the ionosphere over the course of a wave period. For long-lived ions, such as H +, these chemical effects are minimal. Ions whose timescales are much shorter than a wave period can yield significant variations in column density over a wave period. For H + 3 there are large chemically-induced perturbations at altitudes above 600 km. The local H + 3 density establishes chemical equilibrium very quickly as the H+ and electron density vary slowly over the wave period at these altitudes. The H + 3 column density needs not be conserved throughout a wave period since it is affected primarily by chemistry here rather than dynamics. As a result the H + 3 column density varies significantly over a wave period. Figures 3-17 and 3-18 show this effect; the H + 3 density variations are much grater than the density variations of the H + ion for both waves. The H + ions do undergo chemical interactions throughout a wave period as well, but because they have much slower reaction rates these column density variations are relatively small. As the wave phase propagates horizontally, the same variations that are observed in the ion column densities over time are also observed along the horizontal direction over a distance of a wavelength (Figures 3-19 and 3-20). The variations in the H

77 column density along the horizontal coordinate are much greater than those of H + and for the long period wave they are about 10%. Such large variations in the column density over the horizontal direction will have a large effect on the H + 3 thermal emission. These results suggest that in the presence of a gravity wave there will be significant variations in the H + 3 thermal emission along the horizontal direction which can potentially be observed with the use of a high resolution infrared spectrometer. This is the focus of Section Observability of Gravity Waves via H 3 + Emission Overview of H 3 + Emission Emission of H + 3 was first detected from Jupiter in 1988 (Drossart et al., 1989). The H + 3 ion is recognized as an important component in both heating and cooling processes in the upper atmosphere. It is also recognized as having substantial utility as a tracer to learn about the ionospheres of the giant planets (Miller et al., 1997). Infrared imaging of this ion has been used to constrain Jupiter s magnetic field parameters (Connerney et al., 1998), investigate the structure and dynamics of Jupiter s aurorae (Miller et al., 2000), and also to observe the conditions of Jupiter s non-auroral ionosphere (Miller et al., 1997). This work details the potential to observe the effects of gravity waves in the column emission H + 3 at Jupiter s mid to low latitudes. The infrared emission of H + 3 is a function of its vertical density profile and the vertical temperature profile for a given location. A radiative transfer model is coupled with the wave-ionosphere model to calculate the H + 3 column emission at each location along the horizontal phase of a propagating wave using the modeled, perturbed temperature and H + 3 density profiles Model Results The perturbations of the H + 3 density in Jupiter s ionosphere due to a propagating gravity wave give rise to horizontal variations in the H + 3 thermal emission from the planet which results in a contrast in the observed intensity along a horizontal wavelength. Observations of the H + 3 thermal emission from Jupiter s mid-latitudes, given sufficient 77

78 spatial resolution and signal to noise, can detect this contrast. These potential observations can reveal typical amplitudes, wavelengths, locations, directions of propagation, source regions, as well as the frequencies of occurrence of the observable waves, leading to a more complete understanding of the waves present in Jupiter s upper atmosphere. In this section wave and planetary parameters that most strongly affect the magnitude of the variations in the H + 3 thermal emission are investigated. These results will help to determine where on the planet will be most favorable to yield observations and waves with which parameters are most likely to be observed. The horizontal variations in the H + 3 emission give rise to significant contrasts in the emitted intensity. This contrast is calculated as the percentage difference between the maximum and minimum emission along a horizontal wavelength Wave propagation direction The direction of wave propagation relative to the magnetic field orientation is one factor that determines the magnitude of the ion perturbation. In the gravity wave-ionosphere perturbation mechanism the ion velocity is constrained to the magnetic field lines and is thus a projection of the neutral winds onto the magnetic field lines. Figure 3-2 assumes that the neutral velocity U n and the local magnetic field both have the same horizontal orientation. The ion velocity projections are maximum when the neutral velocity and the magnetic field have the same horizontal orientation and are minimum when the horizontal orientations are perpendicular. In this minimum case the ion velocities are due solely to the projections of the vertical component of the neutral velocity onto the magnetic field lines. In the hydrostatic limit, the vertical wind component of a gravity wave is small compared to the horizontal wind component. Figure 3-21 shows the relative magnitude of the ion density perturbation for the same wave propagating in different horizontal directions. The wave shown is a small amplitude wave with λ h = 1000 km and λ z = 130 km at the wave peak altitude. The inclinations are negative and so the data are for Jupiter s northern hemisphere. It is 78

79 apparent that the largest effect occurs for a wave propagating along the magnetic meridian. For small inclinations, mid-low latitudes, the effect is greatest for waves propagating away from the equator and is also shown in Section The remainder of the results deal only with waves that propagate along the magnetic meridian. Waves propagating off-meridian will have their resulting ion density perturbation amplitudes reduced as shown in Figure 3-21; the resulting emission contrast will be subject to the discussions in Section Wave peak altitude The magnitude of the H + 3 emission contrast also depends on the altitude of dissipation of the wave, or the altitude at which its perturbations on the background atmosphere are greatest. This altitude depends on the parameters of the wave. Figure 3-22 displays the altitude at which a wave peaks for vertical wavelengths up to 300 km and horizontal wavelengths up to 10,000 km. The background H + 3 density in Figure 3-8 shows a sharp peak in the altitude region between 550 to 650 km above the 1 bar pressure level. Most of the H + 3 thermal emission emits from this region. Gravity waves that peak in this range therefore create the largest effects on the H + 3 density perturbation and in turn the H + 3 emission contrast. The wave peak altitude is determined by the balance between the exponential growth of a wave with altitude and the increasing dissipative effects at higher altitudes. At high altitudes most of the dissipation is due to molecular viscosity. This dissipation is proportional to vertical shear of the wind. The vertical shear in the winds produced by a gravity wave is directly related to the vertical wavelength. Smaller vertical wavelengths yield larger vertical wind shears which result in more rapid dissipation of the wave and a lower peak altitude. Conversely, waves with larger vertical wavelengths peak at higher altitudes. Figure 3-23 illustrates the dependence of the H + 3 emission variation with respect to the vertical wavelength. For four horizontal wavelengths of 1000 km, 2000 km,

80 km, and km, waves with vertical wavelengths ranging from 50 km to 300 km are propagated in the model and the resulting percentage variation (peak to trough) in the H + 3 column emission is displayed. The wavelengths between the black lines yield waves that peak at altitudes between 550 and 650 km, corresponding to the background H + 3 density peak. It is evident that waves with these parameters result in the largest signatures Vertical wavelength In addition to the effect of the vertical wavelength on the wave peak altitude, the vertical wavelength itself has a large effect on the horizontal H + 3 emission variation. Since the effect is in the column emission, it is the integration of vertical variations that are observed. The observed horizontal contrast is diminished if the vertical wavelength is small and there more vertical phases to average out. The observed contrast is greater if the vertical wavelength is large which minimizes the vertical averaging. It was shown in Section that the emission contrast is greatest when a gravity wave peaks at altitudes near to the background H + 3 density peak. It is also demonstrated that the emission variation has a large dependence on the horizontal wavelength, longer horizontal wavelengths produce much greater emission variations. This is because larger horizontal wavelengths require larger vertical wavelengths to peak at the same altitude (Figure 3-22). Thus, for waves peaking at the same altitude (waves between the black lines in Figure 3-23) larger vertical wavelengths yield greater variations in the H 3 + emission variations Wave amplitude The observable effect in the H + 3 emission is very sensitive to the amplitude of the gravity wave. A larger wave amplitude has greater velocity perturbations which yields stronger forcing of the ions and larger ion density perturbations. If there are larger ion density perturbations then one might expect that there will be greater H + 3 column density variations resulting in a more pronounced contrast in the H + 3 emissions. This is true 80

81 in most cases, however, the model consistently demonstrates a significant departure from this expectation in the transition regime between small and large amplitude waves. The transition regime was defined in Section as amplitudes for which the projection of the ion velocities in the direction of the wave phase velocity ( U ir = U i ˆk r ) is approximately equal to the phase velocity, U ir V ph. Figures 3-24 and 3-25 illustrate the changes in the emission contrast with respect to amplitude at different latitudes for two different waves. The wave in Figure 3-24 has horizontal and vertical wavelengths of 1000 km and 130 km respectively, period of 96 min, and a phase velocity of V ph = 22.6 m/s. The wave in 3-25 has the same vertical wavelength but has a horizontal wavelength of 2000 km, period of 206 min, and a phase velocity of V ph = 10.5 m/s. Each color represents the wave propagating at a different latitude, with the magnetic field inclinations indicated. All but one of the plots demonstrate that the emission contrast is enhanced for a range of amplitudes in the transition regime. This regime occurs at smaller amplitudes for larger inclinations because U ir is greater at these higher inclinations. The one plot that does not show a peak in the emission, Figure 3-24 I = 2, has U ir that is smaller than V ph for amplitudes less than 30 K. The physical mechanism that gives rise to the enhanced emission contrast in the transition regime is not yet well understood. The effect is consistent in all numerical simulations thus far and is a promising result that suggests gravity waves may be more easily detected in the H + 3 thermal emission Latitude (magnetic field inclination) The magnetic field orientation strongly affects the wave-induced ionospheric perturbations and thus also the H 3 + emission contrast. The magnitude of the H+ 3 emission contrast for different magnetic field inclinations are presented in Figures 3-26 and 3-27 for four waves propagating northward along the magnetic meridian. The emission contrast with respect to magnetic field inclination demonstrates the similar 81

82 results that were seen in section For inclinations such that U ir V ph the emission contrast peaks sharply. For the large amplitude waves represented in these figures the sharp peaks occur near the equator where the magnetic field inclinations are small. It is seen in the geometry of Figure 3-2 that when the gravity wave phase lines are either parallel or perpendicular to the magnetic field lines there is no separation of the neutral and ion velocities. In these situation ion perturbations due to a gravity wave are non-existent. This is demonstrated in Figures 3-26 and The four northward propagating waves in these figures have phase line angle ranging from 4 to 8 above the horizontal. When the magnetic field inclination is equal to the phase line angle, south of the magnetic equator, the emission contrast nearly zero. The symmetric location north of the magnetic equator for the same northward propagating wave will have an angle between phase lines and magnetic field inclination of two times the phase line angle since the phase lines are unchanged but the magnetic field has gone from positive to negative. This geometry can give rise to large perturbations and large emission contrast. Therefore, there is a distinct asymmetry in the emission contrast about the equator. Waves that propagate away from the magnetic equator are expected to have a much greater potential observability. The complete latitudinal dependence of the emission contrast is complicated. In addition to the magnetic field inclination changing with altitude, which also affects the ion diffusion velocities, the photoproduction of ions is latitudinally dependent as the zenith angle of the incident solar flux changes with latitude. The ion densities profiles are altered at different latitudes. This latitudinal dependence of the H + 3 density peak affects the magnitude of the H + 3 density perturbations for a given wave and thus the emission contrast as well. 82

83 3.7 Summary This work (i)contributes to the theoretical understanding of how atmospheric gravity waves interact with ionospheric plasma by considering a simple analytical approach, (ii)assesses the effect of gravity waves on Jupiter s ionosphere through numerical modeling, and (iii)outlines a new detection method for gravity wave activity on Jupiter using H + 3 emission. A non-linear, time dependent, two-dimensional model of Jupiter s ionosphere is successfully constructed that incorporates a self consistent chemical model of the main ionospheric species. The model allows for a spatially and temporally varying wind field and tracks the time evolution of the ionospheric response in the presence of a gravity wave. In the small amplitude limit the non-linear model is validated with the analytic solutions of the linearized equations for the two main ions, H + and H 3 +. The non-linear model is then used to investigate the effects of short and long period atmospheric gravity waves on the ion distribution of the ionosphere. In agreement with previous investigations (Matcheva et al., 2001) it is demonstrated that atmospheric gravity waves in Jupiter s ionosphere can significantly perturb the density structure of the ions locally as well as alter the total electron/ion content of the ionosphere. It is shown that atmospheric gravity waves with amplitudes consistent with the Galileo observations give rise to layers in the electron density profile that agree well with the Galileo J0-ingress electron density profile. The resulting peaks have the correct magnitude, location, and vertical spacing as the ones observed. The different mechanisms through which ion densities are perturbed in the presence of a gravity wave were investigated. They can be perturbed directly by the wave wind field and also by chemical interactions with perturbed species. The H + ion is shown to be perturbed primarily through dynamical forcing while the dominant forcing mechanism for the H + 3 perturbations change with height: it is dynamically forced low in the ionosphere where the H + 3 lifetime peaks and it is chemically perturbed above this region. 83

84 Gravity waves induce a net ion flux. For the sample upward propagating waves examined, H + was shown to have a significant net downward ion flux that resulted in a net loss in the ion density. The net changes in the H + 3 density were shown to be due to the alterations in the ionospheric chemistry due to the wave-induced, net changes in the H + density structure. An interesting result that this model has provided is the large variation of the H + 3 column density across the horizontal direction in the presence of a gravity wave. High resolution spectroscopic observations of the H + 3 thermal emissions across the horizontal plane can potentially detect these observable parameters. The following gravity wave and magnetic field parameters have been shown to yield the largest variation in the H + 3 thermal emission: waves that have large amplitudes or amplitudes and locations that are consistent with the regime U ir = V ph, waves that propagate along the magnetic field lines (roughly north-south at mid to low latitudes), waves that have peak amplitudes around 600 km above the one bar pressure level, waves that have larger vertical wavelengths, and waves near to and propagating away from the magnetic equator. To maximize the likelihood of detecting gravity waves on Jupiter in this manner, and thus gain insights on the ensemble of waves on Jupiter, these factors can be utilized to optimize observations that may yield signatures of gravity waves. The results presented herein are based on the modeled and presented background atmosphere of Jupiter s upper atmosphere. The emission variation is highly dependent on the structure, magnitude, and altitude of the H + 3 density peak. All the results presented assume the same temperature profile and no background winds. In reality, these atmospheric properties vary with respect to location and time of day and therefore the gravity wave parameters that maximize the emission contrast will vary as well. Certain results, i.e. meridionally propagating waves away from the magnetic equator, are applicable in a general sense. In general, however, the parameters of waves that 84

85 maximize the emission contrast will depend on the structure of the ionosphere at the time and location of the observation. 85

86 Table 3-1. Reactions incorporated in the model Reaction Reaction Rate[SI units] Reference H H 2 H H (Anicich, 1993) H H H+ + H (Karpas et al., 1979) H He HeH+ + H (Smith and Futrell, 1976) H CH 4 CH H (Kim and Huntress, 1975) H CH 4 CH H (Kim and Huntress, 1975) H CH 4 CH H + H (Kim and Huntress, 1975) He + + H 2 HeH + + H (Schauer et al., 1989) He + + H 2 H He (Schauer et al., 1989) He + + H 2 H + + H + He (Perry et al., 1999) He + + H2 H + + H + He (Jones et al., 1986) He + + CH 4 H + + He + CH (Mauclaire et al., 1978) He + + CH 4 CH H + He (Mauclaire et al., 1978) He + + CH 4 CH He (Mauclaire et al., 1978) H + + H2 H H (Maurellis, 1998) H + + CH 4 CH H (Dheandhanoo et al., 1984) H + + CH 4 CH H (Dheandhanoo et al., 1984) HeH + + H 2 H He (Bohme et al., 1980) HeH + + H H He (Karpas et al., 1979) H CH 4 CH H (Bohme et al., 1980) CH H 2 CH H (Federer et al., 1985) CH CH 4 CH CH (Smith and Adams, 1977) H + + H 2 + H 2 H H [H 2 ] (Miller et al., 1968) H + + e H + hν ( 250 T e ) 0.7 (Bates and Dalgarno, 1962) He + + e He + hν ( 250 T e ) 0.7 (Bates and Dalgarno, 1962) H e H + H ( 300 T e ) 0.4 (Auerbach et al., 1977) H e H 2 + H ( 300 T e ) 0.65 (Canosa et al., 1992) H e H + H + H ( 300 T e ) 0.65 (Mitchell et al., 1983) HeH + + e He + H ( 300 T e ) 0.6 (Yousif and Mitchell, 1989) CH e CH 3 + H ( 300 T e ) 0.5 (Mul et al., 1981) CH e CH 2 + H + H ( 300 T e ) 0.5 (Mul et al., 1981) 86

87 Table 3-2. Sample wave parameters Sample Wave A Sample Wave B Period 1.36 h 5.96 h Hor. Wavelength 500 km 3500 km Peak Altitude 510 km 504 km Vert. Wavelength 60 km 93 km Small Amplitude K K Large Amplitude 21 K 20 K 87

88 Figure 3-1. Jupiter electron density observations. Galileo J0 ingress and egress radio occultation profiles of the electron density. 88

89 z Phase Line Plasma Convergence h U n B Plasma Divergence U p U p U n U n U p U p Plasma Convergence U n Phase Line B V ph Figure 3-2. Ion Layering Mechanism. The vertically alternating neutral wind field creates layers of plasma flux convergence and divergence. V ph denotes the wave phase velocity vector. The wave in the diagram is propagating along the magnetic meridian. The plasma velocity vector, U p, is the projection of the neutral velocity vector, U n, in the direction of the magnetic field vector B. 89

90 Gyro Collision Plasma Diffusion Buoyancy 1000 Altitude (km) Frequency (s ) Figure 3-3. Ion Frequencies. Ion gyro-frequency, ion-neutral collision frequency, and plasma diffusion time-scale for the major ion are shown in the region of the model. The buoyancy frequency is also included and represents the upper limit for gravity wave frequencies. 90

91 Temperature (K) H H 2 He CH 4 Temp. Altitude (km) Density (m ) Figure 3-4. Background neutral atmosphere. The background temperature (top axis) and the background neutral densities (bottom axis) used in the model. 91

92 He H Altitude (km) CH 4 H e-006 1e Mixing Ratios Figure 3-5. Mixing ratios. The fractional number densities of the four main neutrals in Jupiter s upper atmosphere. 92

93 H + H 2 + He + CH 4 + H 2 + (e-) 1000 Altitude (km) Production Rate ( m s ) Figure 3-6. Photochemical production. Photoionization production rates of H +, He +, H + 2, and CH + 4 and secondary electron ionization production rate of H

94 H + H3 + H2 + He + CH4 + HeH t N Altitude (km) wave A wave B 1 Jday Lifetime (s) Figure 3-7. Ion lifetimes. The chemical lifetimes of the ions included in the model are shown. The buoyancy period (thick solid line) and the length of one Jovian day are displayed for reference. The periods of sample wave A and B are labeled at the altitudes at which they peak. 94

95 e + H + H3 + H2 + He + CH4 + HeH Altitude (km) Density (m ) Figure 3-8. Background ion densities. Model steady state ion densities in the absence of atmospheric gravity waves. Here the model assumes diurnally averaged photochemical production. 95

96 Temperature (K) V h (wave A) V h (wave B) T (wave A) T (wave B) 1000 Altitude (km) Velocity (m/s) Figure 3-9. Gravity wave amplitudes. Temperature and horizontal velocity amplitudes of the large amplitude sample waves A and B. The small amplitude analogs of these waves are identical but with magnitudes three orders smaller. 96

97 A Wave A, H + B Wave B, H + Altitude (km) Altitude (km) Altitude (km) C E Wave A, H + Wave A, H D Wave B, H F Wave B, H N /N i io N /N i io Figure Validation results. Analytical solutions from Section 3.4 (red) and the numerical simulation result N i N io (green). Each line represents the perturbation at a horizontal grid point to illustrate the overall amplitude of perturbation. A) H + response to sample wave A, Equation B) H + response to sample wave B, Equation C) H + response to sample wave A, Equation D) H + response to sample wave B, Equation E) H 3 + response to sample wave A, Equation F) H+ 3 response to sample wave B, Equation

98 Velocity Perturbation Small Amplitude Wave Large Amplitude Wave 800 Altitude (km) Ion Perturbation (N /No) Figure Amplitude dependent perturbation phases. The phase of the ion perturbations relative the the forcing winds are amplitude dependent. At a given time and horizontal location, the wind phase is depicted as the black curve. The horizontal black lines show the phase lines for maximum (negative) winds. The green line shows that the ion perturbations for a small amplitude wave are 180 out of phase with the winds. The red line shows that the ion perturbations for a large amplitude wave (scaled down by a factor of 10) are 90 out of phase with the small amplitude perturbations at altitudes where the perturbations are large. 98

99 J0 Ingress e + H + H3 + H2 + He + CH4 + HeH 1000 Altitude (km) Density (m ) Figure Perturbed ion densities. Ionospheric response to the short period (wave A), large amplitude sample wave for a magnetic field inclination of 47. The electron density is omitted for clarity but can be assumed to follow closely the density of the dominant ion. 99

100 J0 Ingress e + H + H3 + H2 + He + CH4 + HeH Altitude (km) Density (m ) Figure Perturbed ion densities. Ionospheric response to the long period (wave B), large amplitude sample wave for a magnetic field inclination of 47. The electron density is omitted for clarity but can be assumed to follow closely the density of the dominant ion. 100

101 H H Altitude (km) Ion Flux (x10 m s ) Figure Wave-induced ion fluxes. The model calculated H + and H 3 + sample wave A are shown. ion fluxes for 101

102 H H Altitude (km) Ion Flux (x10 m s ) Figure Wave-induced ion fluxes. The model calculated H + and H 3 + sample wave B are shown. ion fluxes for 102

103 Initial H + Final H wa + Final H wb + Initial H3 + Final H3 wa + Final H3 wb 1000 Altitude (km) Density (m ) Figure Wave impact on the background ionosphere. Initial (solid lines) and quasi-steady state background H + and H 3 + ion densities for both sample waves A and B (short dash and long dash respectively). The background densities are calculated as the average density across a horizontal wavelength. H + is represented in black and H 3 + is represented in gray. 103

104 1.1 Column Density Variation H H 3 + Wave A Time (Wave Periods) Figure Time dependent ion column density. The H + and H 3 + ion column density variations over time are shown for sample wave A. The column densities are normalized to the initial, steady state values in the absence of a wave. 104

105 Column Density Variation H H 3 + Wave B Time (Wave Periods) Figure Time dependent ion column density. The H + and H 3 + ion column density variations over time are shown for sample wave B. The column densities are normalized to the initial, steady state values in the absence of a wave. 105

106 H H 3 + Wave A Column Density Variation Distance (Horizontal Wavelengths) Figure Horizontal variations in the ion column densities. Column density variations along the horizontal coordinate for the sample wave A. The column density is normalized to the quasi-steady state background ion column density. 106

107 H H 3 + Wave B Column Density Variation Distance (Horizontal Wavelengths) Figure Horizontal variations in the ion column densities. Column density variations along the horizontal coordinate for the sample wave B. The column density is normalized to the quasi-steady state background ion column density. 107

108 N W E I = -15 o S Figure Ion perturbation dependence on wave propagation direction. The wave used has a phase line angle of 8 and is propagating in each direction where the magnetic field inclination is 15. This is only a representative plot; these result depend heavily on the wave phase line angle, magnetic field inclination, and therefore latitude as well. 108

109 Figure Gravity wave peak altitude. The altitude at which the gravity wave amplitude is maximum for given vertical and horizontal wavelengths. The two sample waves are indicated by the letters A and B. 109

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