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Icarus 164 (2003) 14–32www.elsevier.com/locate/icaru

The effects of topographically-controlled thermal tides in the martiaupper atmosphere as seen by the MGS accelerometer

Paul Withers,a,∗ S.W. Bougher,b and G.M. Keatingc

a Lunar and Planetary Laboratory, University of Arizona, Tucson, AZ 85721, USAb Space Physics Research Laboratory, AOSS Department, University of Michigan, 2455 Hayward Avenue, Ann Arbor, MI 48109-2143, USA

c The George Washington University at NASA Langley, MS 269, Hampton, VA 23681, USA

Received 4 October 2002; revised 5 February 2003

Abstract

Mars Global Surveyor accelerometer observations of the martian upper atmosphere revealed large variations in density withduring northern hemisphere spring at altitudes of 130–160 km, all latitudes, and mid-afternoon local solar times (LSTs). This zoture is due to tides from the surface. The zonal structure is stable on timescales of weeks, decays with increasing altitude aboand is dominated by wave-3 (average amplitude 22% of mean density) and wave-2 (18%) harmonics. The phases of these haconstant with both altitude and latitude, though their amplitudes change significantly with latitude. Near the South Pole, the phawave-2 harmonic changes by 90 with a change of half a martian solar day while the wave-3 phase stays constant, suggesting diusemidiurnal behaviour, respectively. We use a simple application of classical tidal theory to identify the dominant tidal modes aresults consistent with those of General Circulation Models. Our method is less rigorous, but simpler, than the General Circulatioand hence complements them. Topography has a strong influence on the zonal structure. 2003 Elsevier Inc. All rights reserved.

Keywords: Mars, atmosphere; Tides, atmospheric; Atmospheres, dynamics

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1. Introduction

Our objectives in this paper are to understand theture of large variations in density with longitude observin the martian upper atmosphere with the Mars Global Sveyor (MGS) accelerometer and to identify the underlyphenomena that cause them. We present a quantitativdetailed analysis of this zonal structure as a function otitude, latitude, and local solar time (LST). We then uclassical tidal theory to identify the dominant mechaniscausing the zonal structure, and finally outline a simpletification for topography having a strong influence onzonal structure. We begin by quantifying the sol-to-sol (ais a martian solar day) variability, or weather, in the martupper atmosphere and placing some constraints on its o

Accelerometer data from MGS’s aerobraking phasesvealed an unexpected phenomenon: the upper atmospdensity at fixed altitude, latitude, LST, and season varie

* Corresponding author.E-mail address: [email protected] (P. Withers).

0019-1035/03/$ – see front matter 2003 Elsevier Inc. All rights reserved.doi:10.1016/S0019-1035(03)00135-0

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factors of 2 or more as a function of longitude (Keatingal., 1998). Corresponding atmospheric variabilities forEarth, measured by the Satellite Electrostatic Triaxialcelerometer at its reference altitude of 200 km altitude,on the order of 10% (Forbes et al., 1999). For referetypical atmospheric densities at 200 km altitude on Ein Forbes et al. are∼ 0.1 kg km−3 and typical atmospheridensities at 130 km altitude on Mars in this paper (Tableand 3) are∼ 1 kg km−3. No such variations can be observon Venus because its similar lengths of day and year doallow all longitudes to have the same LST on a subantimescale.

This zonal structure must originate in the lowermosphere; it cannot be created in situ. There are no zinhomogeneities present in solar heating, which powthe dynamics of the martian atmosphere, or in the upboundary of the atmosphere. There are many zonal imogeneities near the lower boundary of the atmospherecluding topography, surface thermal inertia, surface alband lower atmosphere dust loading, which may influethis zonal structure. Since the zonal structure must p

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Tides in the martian upper atmosphere 15

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agate through, and be affected by, the lower atmosphobservations of the zonal structure in the upper atmospmay reveal information about the properties of the lowatmosphere. This zonal structure is caused by atmosptides, which are global-scale atmospheric oscillations atriods which are subharmonics of a solar day (Forbes, 19

“Keating et al. (1998) initially found that the zonal struture intensified during the build-up of a regional dust stoand, based on the constancy of its phasing, suggested tmay be caused by topographically forced stationary waLater in the MGS mission measurements were made onnightside and Keating et al., (1999, 2000) detected a preversal of wave-2 from day to night which they attributednon-migrating tides (specifically the diurnal Kelvin wave(Keating, 2003, personal communication). Non-migrattides, disturbances that vary with time but are not locin phase with the Sun, were also suggested by (ForbesHagan, 2000; Joshi et al., 2000; Wilson, 2000; Forbeal., 2001). They are discussed in Section 5. The restrisampling in LST of the accelerometer data make thesetypes of disturbances impossible to distinguish observatally. Stationary waves are unlikely because they requireimplausible vertical profile of zonal winds in the lower amosphere if they are to be present in the upper atmospAll of these publications responding to Keating et al. (19predate the public release of the accelerometer dataset, hthey are based on the limited quantitative data presein that paper and short conference abstracts discussincomplete accelerometer dataset (Keating et al., 1999, 20Here we present an analysis of the complete acceleromdataset.

A previous publication has discussed the properties ozonal structure, including its responses to changes intude, latitude, and LST, using the complete acceleromdataset. Wilson (2002) observed that the zonal structurecreased in amplitude with increasing altitude and didchange in phase with changes in latitude. Wilson discuthe observation of Withers et al. (2000) that the zonal stture varied with half a sol changes in LST. Wilson did npresent quantitative results for how altitude or LST affecthe zonal structure, nor did he quantify changes in the zstructure on weekly timescales. Wilson only presented qutitative results for the strongest two components of the zostructure, wave-2 and wave-3. In this paper we present qtitative results for all these items, including the first foharmonic components of the zonal structure. Wilson copared his analysis of the MGS density data to GCM pretions that accurately reproduced some aspects of the zstructure, such as its amplitude and phase at 130 km anchanges with LST. He used them to identify the domintides causing the zonal structure. Unlike Wilson (2002),use our observations in conjunction with idealised classtidal theory to identify the dominant tides. In this longer pper we were able to go into more quantitative detail abthe accelerometer dataset and chose a less rigorous, bupler, theoretical approach to interpret our observations.

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also outline a simple justification for topography, rather thany other surface physical property, being the strongesfluence on the zonal structure.

The zonal structure and the tidal modes responsible halso been studied by other instruments onboard MGScluding the thermal emission spectrometer (TES), the MHorizon Sensor Assembly, and the Radio Science expment (Banfield et al., 2000, 2003; Smith et al., 2001a; Mphy et al., 2001; Bougher et al., 2001; Hinson et al., 20Tracadas et al., 2001).

The dynamics of the martian upper atmosphere areportant for several reasons. Firstly, great cost savingsgenerated when a spacecraft aerobrakes into orbit arMars, rather than using chemical propellant. Aerobrakis not possible without an understanding of the uppermosphere and improvements in this understanding ygreater cost savings. Secondly, theoretical models of theper atmospheres of Earth and Mars, regions which aresensitive to changes in solar flux, share many key featuUnderstanding the upper atmosphere of Mars helps devand verify these models, which can then be used, for exple, to monitor changes in solar flux in the Earth’s upatmosphere, an important measurement for terrestrialmate change. Thirdly, as we shall see later in Section 5.1dynamics of the lower and upper atmospheres on Marsstrongly coupled, so by understanding the upper atmospwe also learn about the lower atmosphere. The pubavailable accelerometer dataset from the MGS spacecra rich resource for studying the martian upper atmosp(Keating et al., 2001a,b). The results of this paper contribto a better understanding of the dynamics of the martianper atmosphere.

The remainder of this paper is divided into five parSection 2 provides background information on the MGScelerometer dataset. Section 3 examines short-term varity in the upper atmosphere. Section 4 presents observaof the zonal structure. Section 5 uses tidal theory to identhe phenomena causing the zonal structure. Section 6cludes.

2. MGS aerobraking

An aerobraking spacecraft, such as Mars Global Sveyor, passes through an atmosphere near the periapsisorbit and experiences an aerodynamic force which decrethe energy and semimajor axis of the spacecraft’s orbit wout the need for significant fuel consumption (FrenchUphoff, 1979; Eldred, 1991). Aerobraking has previoubeen used by Atmospheric Explorer-C and its successoEarth, the Pioneer Venus Orbiter at Venus, and MagellaVenus (Marcos et al., 1977; Strangeway, 1993; CroomTolson, 1994; Lyons et al., 1995). Aerobraking permitssitu atmospheric studies not possible from a typical spcraft orbit.

16 P. Withers et al. / Icarus 164 (2003) 14–32

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MGS carried an Accelerometer Experiment (Keatingal., 1998, 2001a,b). This accelerometer measured thedynamic forces on the spacecraft during an aerobraking pThe accelerometer readings have been processed to gate a “profile” of atmospheric density along each flight pthrough the atmosphere (Keating et al., 1998; Cancro e1998; Tolson et al., 1999, 2000; Keating et al., 2001aThis information was used by the spacecraft operations tin the hours immediately after the aerobraking pass tomodifications to MGS’s trajectory, changing the altitudethe next periapsis in steps of about 2 km by small expditures of chemical propellant at apoapsis, to achievedesired drag without exceeding heating rate thresholdsguide it safely to the desired mapping orbit. MGS’s orbit ding aerobraking was near-polar, with an inclination of ab93, and highly elliptical.

The changing characteristics of MGS’s orbit during aebraking were quite complicated and controlled the data cerage of the accelerometer. As such, they strongly influethe data analysis that we are able to do in Sections 3 aWe first discuss the orbital characteristics for the case wperiapsis is away from the pole.

A typical aerobraking pass through the atmosphere laa few minutes and spanned several tens of degrees of lawith only small changes in longitude or LST; unlike plantary entry probes or landers, the flight path of MGS throthe atmosphere on each aerobraking pass is not close ttical. On a typical aerobraking pass, latitude has a rouquadratic dependence on altitude. The maximum altitudwhich the accelerometer measured atmospheric densitby the instrument sensitivity, was approximately 160 kThe minimum altitude at which the accelerometer measatmospheric density was that of periapsis, typically 110Periapsis altitude was rarely outside 100–120 km. It ochanged from one orbit to the next by a kilometre orin response to the non-spherically symmetric gravitatiofield of Mars, in addition to any deliberate trajectory moifications. As aerobraking progressed, MGS’s apoapsistitude steadily decreased and the parabolic aerobrakingthrough the atmosphere became flatter. A given altitude lwas crossed twice on each aerobraking pass, once desing towards periapsis at one latitude and longitude, andascending after periapsis at another latitude and longitTo distinguish between these two measurements at thealtitude and same orbit, we refer to the “inbound” and “obound” legs of an aerobraking pass.

Since the near-polar orbit was close to sun-synchronperiapsis LST changed only slowly between orbits. Persis longitude changed greatly between orbits as the protated every 24.6 h beneath MGS’s orbit. The change inriapsis longitude per orbit, relative to 360, was equal to theratio of MGS’s orbital period to the martian rotational priod.

Periapsis latitude changed slowly but steadily betworbits. Due to the oblateness of Mars, the orbit precearound in its orbital plane (Murray and Dermott, 1999). T

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caused periapsis latitude to change. The entire paraflight path also shifted in latitude as the orbit precessDue to this precession, we can analyse densities at a rof altitudes, but fixed latitudes, from the non-vertical aebraking passes (Section 4.3) and analyse densities at aof latitudes, but fixed altitudes (Section 4.4). To summarMGS had a slowly flattening, parabolic flight path throuthe atmosphere, travelled in either a north or south direcwith small changes in LST and longitude during a pass, wlarge changes in longitude due to the planet’s rotation, schanges in LST, and steady changes in latitude betweeriapses.

The picture is more complicated when periapsis is inpolar regions. This is discussed in Section 4.5.

Aerobraking took place in two Phases, 1 and 2, separby a hiatus containing the Science Phasing Orbits (Albeal., 1998, 2001). Phase 1 included orbits 1–201 from mSeptember, 1997, to late March, 1998, and Phase 2 inclorbits 574–1283 from mid-September, 1998, to early Fruary, 1999. This covers a range of martian seasons.Ls ,martian heliocentric longitude, is 0 at the northern sprinequinox, 90 at the northern summer solstice, 180 at thenorthern autumn equinox, and 270 at the northern wintesolstice. At the beginning of Phase 1,Ls = 180, periapsisoccurred at 30N and 18 h LST, then moved northwards aearlier in the day to reach 60N and 11 h LST at the end oPhase 1,Ls = 300. One “hour” of LST is equal to 1/24thof a sol, not 60 min of elapsed time. MGS flew from noto south through the atmosphere on each aerobrakingduring Phase 1. At the beginning of Phase 2, during themartian year atLs = 30, periapsis occurred at 60N and17 h LST, then moved southwards and earlier in the dacross 80S at 15 h LST. At the beginning of Phase 2, MGflew from south to north through the atmosphere on eaerobraking pass. Periapsis then reached its furthest(∼ 87S) near the south pole, crossed the terminator by ding through nighttime LSTs, and reached 60S and 02 h LSTby the end of Phase 2,Ls = 90. At the end of Phase 2MGS flew from north to south through the atmosphereeach aerobraking pass. When periapsis was near thepole, each aerobraking pass’s profile of atmospheric despanned a large range of LST. The behaviour of periaduring aerobraking is summarised in Fig. 1.

In this paper, we discuss two broad subsets of thecelerometer data, both from Phase 2 of aerobraking.Phase 1 data are complicated by significant changes inperiapsis latitude and LST between each orbit and bypresence of a regional dust storm, so we leave them foture work. Some work on the Phase 1 data has previobeen published (Keating et al., 1998; Bougher et al., 19The two subsets of data from Phase 2 are: (1) the southprogression from 60N to 60S at almost fixedLs and LSTat the beginning of Phase 2; and (2) the crossing of the spole from 60S back to 60S at almost fixedLs and twodifferent LSTs, half a sol apart, at the end of Phase 2.name these two subsets of data the “Daytime Preces


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Fig. 1. Parameters of periapsis during aerobraking: (a) periapsis date insince January 1, 1997 (DOY 1997), (b) periapsis latitude, (c) periapsisand (d) periapsis longitude; all plotted against heliocentric longitude,Ls .Phase 1 of aerobraking occured beforeLs = 360 and Phase 2 afterwardThe change from daytime to nighttime LSTs as periapsis reached its fursouth can be seen atLs ≈ 450. The Ls of perihelion and aphelion armarked on panel (a) for reference.

(subset 1) and the “Polar Crossing” (subset 2). The bouary between these two subsets is defined as when a spepoint on the aerobraking pass crosses the 60S boundary onthe dayside part of Phase 2. When discussing, for examdata at 130 km altitude from the inbound leg of aerobrakpasses, all aerobraking passes in which the latitude at 13on the inbound leg is south of 60S are classified in the Polar Crossing part of Phase 2. For data at different altituor from the other leg of the aerobraking pass, a slightlyferent group of aerobraking passes forms the Polar Crospart of Phase 2. As a rough guide, the first periapsistude southward of 60S occurred on orbit 1095. The daysiis defined in this paper as LSTs between 06 and 18 h.nightside is defined as LSTs between 18 and 06 h. In thelar night, the sun will be below the horizon on both daysand nightside.

Data from the Accelerometer Experiment is archivedthe Planetary Data System (PDS) (Keating et al., 2001This dataset contains density profiles from 800 aerobrapasses and data extracted from both the inbound andbound legs of 768 of them at 130, 140, 150, and 160altitude. The constant altitude data was used for this wo

3. Sol-to-sol variability

In this section we quantify sol-to-sol variability in the amosphere to test whether we can meaningfully comparesity measurements made at different longitudes on diffe

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days as if they were made simultaneously. Such compariwill be important in the remainder of this paper.

During aerobraking, the orbital period of MGS decreafrom over 30 h to less than two hours. Whenever the orbperiod was a submultiple of Mars’s 24.6 h rotational perirepeated density measurements were made along essethe same flight paths, that is longitude/altitude/latitudeone sol intervals. When the rotational period isn times theorbital period, then we say that MGS is in then:1 resonanceSuch a “resonance” typically lasted for several sols asorbital period decreased through the resonance conditiothe 1:1 resonance, an aerobraking pass traces the sametude/latitude/longitude path as the one before it. In the:1resonance, there are two different altitude/latitude/longitpaths 180 apart in longitude traced through the atmosphon each aerobraking pass, each traced by every secondIn the n:1 resonance, paths are 360/n apart, each traceby everynth aerobraking pass. If the outbound legs at, s130 km are considered, and the latitudes and longituof each orbit in then:1 resonance are plotted, then thform n clusters separated by 360/n of longitude. We willstudy the different density measurements taken in theserestricted altitude/latitude/longitude clusters during variresonances. If the atmosphere does not change from onto the next, then all the measurements in a given clushould be the same. Our measure of the sol-to-sol variity is the standard deviation of the density measuremena cluster, expressed as a percentage of the mean of thesity measurements in that cluster. For example, the num“50” implies a standard deviation equal to half of the mea

Orbits were assigned to then:1 resonance if their periodwere within 3% of the appropriate submultiple of the martsidereal day. With this criterion, the latitudes and longituat, say, 130 km on the inbound leg of each orbit in theonance formed well-defined clusters (typically 10 wide inlongitude and somewhat narrower in latitude) with enouorbits in each cluster to measure a meaningful atmosphsol-to-sol variability.

Fig. 2 shows the sol-to-sol variability for the 3:1 to 8:1resonances. These resonances occurred during the DaPrecession part of Phase 2. Crosses mark the latitudelongitude of each density measurement at a certaintude whose orbital period satisfies our resonance criteFewer crosses are plotted at higher altitudes becauseorbits do not have good density measurements at the haltitudes. This is especially common in the southern (wter) hemisphere, where observed densities were lowerat corresponding northern (summer) latitudes. Each rowcrosses on a panel in Fig. 2 corresponds to a differentonance. Then:1 resonance containsn clusters at equallyspaced longitudes. By counting the numbers of clustera given latitude in a panel of Fig. 2,n for that resonancecan be found. This does not work so well at the highertitudes where nothing is plotted for some clusters becanone of the orbits which belong to those clusters have vdensity measurements at those altitudes. On panel (a),


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Fig. 2. Latitudes and longitudes of clustered density measurements m130, 140, 150, and 160 km altitude on inbound and outbound legs durin3:1–8:1 resonances, plotted as crosses. The number adjacent to eachof crosses is the sol-to-sol variability for that cluster.

in the northernmost three clusters were collected over adays during the 3:1 resonance. During those same few dof the 3:1 resonance, data in the northernmost three cluson each panel were collected. A couple of weeks later,in the second row of four clusters were collected for epanel. On each panel, the same orbits contribute to, saynorthernmost and easternmost cluster. Then:1 resonance aany altitude is further north on outbound than inbound. Tlatitude of the outbound (inbound)n:1 resonance at a givealtitude is further north (south) than at any lower altituThe numbers adjacent to each cluster are our measure

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Table 1Ls , periapsis latitude, periapsis LST, and beginning and ending perinumber for the 3:1–8:1 resonances

Resonance Ls Latitude LST Beginning Ending() (N) (h) periapsis periapsi

3:1 48 45 16 645 6564:1 57 32 16 710 7255:1 64 16 15 784 8036:1 71 −4 15 864 9017:1 78 −28 15 963 9898:1 82 −45 15 1030 1057

sol-to-sol variability in that cluster. Table 1 gives the peapsis latitude, periapsis LST,Ls , and beginning and endinperiapsis numbers for each resonance. Periapsis numnwas thenth periapsis during the MGS mission.

1σ measurement errors in the individual density measments are about 3% or less over 130–150 km, increato 5% at 160 km (Keating et al., 2001b). These are negible compared to the sol-to-sol variability at 130–150altitude. At 160 km, the 5% measurement error is smathan the average sol-to-sol variability of 8–10%, sugging that we are actually observing sol-to-sol variability anot merely errors in individual measurements. The solsol variability decreases with increasing altitude. It avera15–20% at 130 km and 8–10% at 160 km. This is consiswith dissipative processes operating on upwardly propaing disturbances (Zurek et al., 1992; Forbes et al., 2001

Sol-to-sol variations in density due solely to short-tesolar flux variations have been observed in data from Phaof aerobraking (Keating et al., 1998). Keating et al. (19observed a 50% increase in density at 160 km altitude at32N and 57N near orbit 90, with no significant changedensity at 130 km altitude, at the same time as an incrin the extreme ultraviolet solar flux incident upon the mtian atmosphere. The decrease in sol-to-sol variability wincreasing altitude in this section, in contrast to the increin sol-to-sol variability with increasing altitude in the Keaing et al. (1998) example, suggests a mechanism beyonshort-term solar flux variations.

The sol-to-sol variability at the highest northern latitudis consistently below average. There is a wide variatiosol-to-sol variability over changes in longitude of less th60. Neither of these observations can be explained bylar flux variations alone. High sol-to-sol variability might bexplained by gravity waves preferentially propagatingwards from certain places when atmospheric conditionsfavorable. This is just one possible mechanism, but wenot investigate this any further in this paper.

Preflight estimates of the orbit-to-orbit variability, whicincludes effects from both sol-to-sol and zonal variabilitiof 35% were realistic (Stewart, 1987; Tolson et al., 199The sol-to-sol variability of 15–20% accounts for partthis, with variations in density with longitude accountingthe rest. For comparison, the Earth’s day-to-day variabat 200 km altitude can be on the order of 50% during m


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To answer the question posed at the beginning ofsection, it is reasonable to compare density measuremmade at different longitudes several days apart as londifferences in density smaller than or similar to the sol-sol variability of 15–20% are not automatically attributedzonal variations.

4. Observations of the zonal structure

4.1. Introduction to the zonal structure

We now move on from studying density measuremeat fixed longitude, altitude, latitude, and LST to studyidensity measurements at varying longitudes and fixedtude, latitude, and LST. Keating et al. (1998) discoveredlarge, regular variations in density with longitude existthe martian upper atmosphere. Similar variations on Eare significantly smaller (Forbes et al., 1999). Now thathave constrained how much sol-to-sol variability exists,can examine this zonal structure in density. In this secwe discuss our technique for fitting a model to the obserzonal structures.

When MGS’s orbital period is not in resonance with trotational period of Mars, reasonably complete longitudcoverage is obtained from a small set of consecutive odue to the changes in periapsis longitude between eacbit. As periapsis latitude precesses between each orbit,is a finite range of periapsis latitude in this subset ofdata. The same reasoning applies to building up a picof the zonal structure at a fixed altitude, say 130 kmthe outbound legs of aerobraking passes—reasonablyplete longitudinal coverage can be obtained over a narrange in measurement latitude. Fig. 3 shows outboundsities at 130 km altitude between 10S and 20S from theDaytime Precession part of Phase 2 of aerobraking.illustrates the zonal structure. LST (14.7–14.8 h) andLs

(77–80) are effectively constant for these measuremePeriapsis precessed southward between 10S and 20S withlarge changes in periapsis longitude between each periaThe longitudinal sampling is not built up in a regular patte(e.g., from east to west) and measurements that appesample the same longitude repeatedly are not from constive orbits. During this period, MGS travelled from southnorth on each aerobraking pass with its periapsis betw24S and 33S at altitudes of between 108 and 112 kThese data were taken during the 7:1 resonance, a perioof significant sol-to-sol variability as can be seen in FigDespite the variability in measurements at a given longitue.g., 90E, it is immediately apparent that regular variatioin density with longitude exist in the upper atmosphere

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Fig. 3. All outbound density measurements at 130 km altitude betweenSand 20S from the Daytime Precession part of Phase 2, crosses, waharmonic model fit to the data, solid line, and 1σ uncertainties about thfit, dotted lines. The data were collected over about a week, all at anof 15 h. Measurement uncertainties (not shown) are much smaller tharange in multiple measurements at any longitude. Zonal structure withpeaks in density at 90E and 250E can be seen.

Mars. Density varies by a factor of two over 90 of longi-tude, greater than the sol-to-sol variability.

Measurement uncertainties are not shown in Fig. 3.discussed in Section 3, they do not become comparabthe sol-to-sol variabilities until altitudes greater than 150 kWe use a least-squares fit to a harmonic model to chaterise the zonal structure. This model contains a consdensity term, an amplitude and phase for a sinusoid withcycle per 360 of longitude, which we label as the waveharmonic, and higher harmonics up to and including wa4. It has 9 free parameters and we call it a wave-4 moMeasurement uncertainties are not used to constrain thA wave-4 model was chosen because wave-3 modelssignificantly worse fits to the data in the Daytime Precesspart of Phase 2 of aerobraking and wave-5 models didhave significantly better fits. The error envelope plottedFig. 3 shows a 1σ uncertainty on what a single new obsevation at a given longitude might be. We define the phof a given harmonic as the longitude of its first peak eof 0. The phase of the wave-n harmonic must lie between 0and 360E/n. We generally normalized the zonally-varyinterms in each wavefit by their constant density term. Thiscilitates a comparison of the strength of the zonal strucbetween different seasons or altitudes.

Throughout this paper, we only attempted model fitsmore than 15 data points and only accepted the fit as gif there was a 90% probability that not all model paramters beyond the constant density term should be zero (Nand Wasserman, 1974). Bad fits generally occurred ingions where there were significantly fewer data points tusual, which might be due either to data dropouts or to a h


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rate of periapsis precession through a given latitude rangfewer than 16 data points were available, then we didattempt a fit.

4.2. Changes in zonal structure on weekly timescales

As periapsis latitude precessed between orbits, the eparabolic flight path through the atmosphere shifted withThis changed the latitude at which MGS passed through,130 km on its outbound leg.

If, as during the Daytime Precession part of Phaseaerobraking, periapsis precesses southward as MGS trfrom south to north during each aerobraking pass, thenthe 130 km measurement on the inbound leg, and latesame measurement on the outbound leg, will occur at a glatitude.

This gives us two separate opportunities, inboundoutbound, a few weeks apart, to study the atmosphericsity at a given latitude, altitude, season, and LST. Seaand LST are changing much more slowly than periapsisitude is precessing. Since periapsis longitude (which issame as the longitude of the rest of the aerobrakingaway from the polar regions) is continuing to change frone orbit to the next, a picture of the zonal structure agiven latitude range and altitude can be built up on theseseparate occasions. In this section we examine how theplitudes and phases of the harmonics making up the zstructure change between these two samplings and diwhether data from the two samplings should be combinekept separate.

Fig. 3 showsoutbound densities at 130 km altitude btween 10S and 20S from the Daytime Precession partPhase 2 of aerobraking. Fig. 4 showsinbound densities a

Fig. 4. As Fig. 3, but showing inbound data collected in the same latirange a couple of weeks earlier. The data were also collected over abweek, all at an LST of 15 h. The amplitudes of the two large peakschanged somewhat, but their phases have not. The zonal structure ison timescales of a couple of weeks.

,

ls

-

-ls

le

Table 2Model fit parameters for measurements made between 10–20S during theDaytime Precession part of Phase 2. Uncertainties are 1σ

Parameters Inbound Outbound

Constant amplitude (kg km−3) 1.324±0.042 1.337±0.051Normalized wave-1 amplitude 0.054±0.045 0.165±0.055Normalized wave-2 amplitude 0.249±0.058 0.345±0.055Normalized wave-3 amplitude 0.204±0.059 0.177±0.071Normalized wave-4 amplitude 0.105±0.058 0.109±0.065Wave-1 phase () 298.7±47.7 260.5±18.3Wave-2 phase () 80.4±6.4 64.1±4.5Wave-3 phase () 103.0±5.1 109.6±7.2Wave-4 phase () 81.5±7.7 74.0±9.7

Fig. 5. Interval between inbound measurements at 130 km altitude at alatitude and later outbound measurements at the same latitude and afrom the Daytime Precession part of Phase 2.

130 km altitude between 10S and 20S from the DaytimePrecession part of Phase 2 of aerobraking. The inboundsurements, taken during the 6:1 resonance, were made abotwo weeks prior to the outbound measurements. Thein each figure were collected over a period of one weChanges in the zonal structure in this example are mTable 2 shows how the amplitudes and phases of the vaharmonics change. Only the wave-1 amplitude and waphase have changed in a statistically significant sense. Rthat the maximum value of the wave-n phase is 360E/n.

Fig. 5 shows the interval between repeated density msurements at 130 km altitude as a function of latitudethe Daytime Precession part of Phase 2. Fig. 1 showschanges in LST andLs are small on these timescales. Tallows us to characterise the zonal structure at a giventude, latitude, LST, andLs twice—with an interval on theorder of several weeks between the two characterisation

This leaves us with two options: we may either cobine inbound and outbound measurements to charactegiven altitude, latitude, LST, andLs once only or treat thestwo halves separately. Combining measurements givesdata points but introduces the issue of having half thecollected over a week, a week of no data, then a weecollecting the last half of the data. Wilson (2002) notes t


mettleunddat

icalal-dthu

andasby

eachn beleg.iffer-ue twithre,the

d tot theaer-s to

hav-und

den

de

and

ded foandin-

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arees.nifi-ithange

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the meridional variation of zonal mean density is the safor inbound and outbound, further notes that there is lidifference between the zonal structure observed in inboand outbound data, and then uses inbound and outboundtogether. Until we have carefully investigated the statistsimilarity of the inbound and outbound wavefits at a giventitude, latitude, LST, andLs we keep inbound and outbounmeasurements separate. Each data collection period iscontinuous.

4.3. Changes in zonal structure with altitude

In this section we characterise how the amplitudesphases of each harmonic in the zonal structure changefunction of altitude and see if this behaviour is affectedlatitude.

Density measurements are made at every altitude onleg of an aerobraking pass. Densities at one altitude cacompared with those at another altitude from the sameHowever, since the two densities are also measured at dent latitudes, the comparison does not isolate changes daltitude alone. Due to the precession of periapsis andit the entire parabolic flight path through the atmosphea density measurement at one altitude and latitude onoutbound leg of an aerobraking pass can be compareanother density measurement at a different altitude, busame latitude, from the outbound leg of a subsequentobraking pass. The precession of periapsis enables useparate variations due to altitude and latitude despiteing non-vertical aerobraking passes. Fig. 3 shows outbodensities at 130 km altitude between 10S and 20S. Orbitswhose data are shown in Fig. 3 also measured outboundsities at 140 km, but between 9S and 18S. Examining all140 km outbound data between 10S and 20S requires aslightly different set of orbits, and so on for other altitulevels.

Fig. 6 shows the zonal structure between 10N and 20Non the outbound leg during Phase 2 at 130, 140, 150,160 km. These data were taken between the 5:1 and 6:1resonances. We have shifted from 10–20S to 10–20N toshow the clearest example. The constant density termcreases monotonically as altitude increases, as expectethe background density structure in any atmosphere,the zonal structure tends to a zonal mean as altitudecreases. All the statistically significant peaks and trouappear fixed in longitude. The trough at 270E and peak a330E are no longer statistically significant at 160 km atude.

We have repeated this for other latitude bands betw60S and 60N using data from the Daytime Precession pof Phase 2. We generally find that the normalized wavamplitude decreases by about 50% from 130 km to 150and those for waves-2, 3, and 4 decrease by about 30%rate of change of normalized harmonic amplitude with atude is influenced by the nature of the atmospheric phenenon that we observe as zonal structure and will be discu

a

s

a

o

-

-r

e

Fig. 6. Density measurements at 130, 140, 150, and 160 km altitude bet10N and 20N for outbound measurements during the Daytime Precespart of Phase 2, plotted as crosses. Model fits to data from each altitudplotted as solid lines and 1σ uncertainties about each fit as dotted linMeasurement uncertainties (not shown) are much smaller than the ranmultiple measurements at any altitude and longitude. All data were tat an LST of 15 h. Measurements at each altitude level were takenabout a week, but this interval is offset by a couple of days betweenaltitude level and the next. All density measurements are associatedthe obvious altitude level; there are no pathological cases of, say, a 14density measurement lurking within the range of the 130 km measurem

further in Section 5.1. Wave-1 normalized amplitudessmall and not very statistically significant at higher altitudThe phase is meaningless when the amplitude is not sigcant, which is why its formal value changes erratically wincreasing altitude. The waves-2, 3, and 4 phases can chby up to 10–20 with increasing altitude, but are not systeatic in an eastward or westward sense within any restrilatitude region and are most consistent with no changphase with increasing altitude.

Decreases in normalized harmonic amplitude as altitincreases are evidence of dissipation. Classical tidal thewhich assumes a dissipationless medium, predicts that tnormalized amplitudes should increase as altitude incre(Chapman and Lindzen, 1970). Possible causes of this dpation include radiative cooling, wave–wave coupling, shinstabilities, and viscosity (Hooke, 1977; Forbes, 1995).

4.4. Changes in zonal structure with latitude

In this section we examine changes in the normalizedplitudes and phases of each harmonic in the zonal strucas a function of latitude, discuss whether the zonal strucis planetary-scale or localised, and quantify which harmics are dominant.


in-ationrth,

andfrom

uc-pleany10ylati-ces

from

ati-meereowshendja-atis-theonTheec-

edthis

crafoorlledtedwn,ity

erecen-ntsig. 7eta-in

thisude

ta,itudefew

r toted.ea-ese

eaknd a

isrt-sis-

sedhe-

plan-80

asesti-ve-2rop-ex-

Fig. 7. Contour plot of normalized fitted densities at 130 km altitude,bound leg, from the Daytime Precession part of Phase 2. The normalizhighlights the zonal structure. The LST of the data is 17 h in the nodecreasing to 15 h in the south. TheLs of the data is 30 in the north, in-creasing to 80 in the south. Contour intervals are 0.2 (dimensionless)negative regions (low densities) are shaded. The peaks and troughsFig. 4 can be seen between 10S and 20S.

In Section 4.1, we built up a picture of the zonal strture at 130 km using outbound data. In order to samenough longitudes we had to include data from so morbits that the measurement latitude precessed fromSat the first density measurement to 20S at the last densitmeasurement. By allowing periapsis latitude (and thetude corresponding to the measurement altitude) to prestill further, we can see how the zonal structure changesone latitude range to another.

Fig. 7 shows the zonal structure as a function of ltude for inbound data at 130 km altitude from the DaytiPrecession part of Phase 2. Wavefits similar to Fig. 4 wconstructed at five degree intervals with ten degree windin latitude, normalized by their constant density term, tmerged into a contour plot. There is overlap between acent latitude windows, so adjacent wavefits are not sttically independent. The overlap is included to smoothcontours. Fig. 7 should be compared to Fig. 1 of Wils(2002) which uses both inbound and outbound data.two figures are similar, as they should be according to Stion 4.2. Only wavefits from latitude ranges which yielda good fit, as discussed in Section 4.1, are included inplot. Latitude ranges with bad fits, 20–40S, include orbits911–961 which have poor or missing data due to spacecomputer problems. Since zonal fits in this range are pbecause of inadequate data, this region in Fig. 7 is fiby the interpolation of nearby fits. Uncertainties in the fitdensity as a function of latitude and longitude are not shobut the mean 1σ uncertainty is 20% of the constant dens

s

t

Fig. 8. Normalized amplitudes from the wave-4 harmonic fits that wmerged to create Fig. 7. The latitude of each set of harmonics is thetre of the 10 wide latitude band from which all the density measuremethat contributed to the harmonic fit came. Only every other fit used in Fis shown; including every fit increases the clutter without aiding interprtion. Gaps at 25S and 35S are due to bad fits which were not includedFig. 7.

term with most values within a few percentage points ofmean. Fig. 4 shows the uncertainty in the fit in one latitrange.

A contour plot similar to Fig. 7, but using outbound dacan also be constructed. Measurements at a given altand latitude are taken first on the inbound leg and then, aweeks later, on the outbound leg. A contour plot similaFig. 7, but at a different altitude, can also be construcThere is also a few days difference in time between msurements at a given latitude and different altitudes. All thsimilar contour plots have a high density peak at 80E, mostprominent in the northern hemisphere, a high density pat 250E, most prominent in the southern hemisphere, ahigh density peak at 330E. The peak at 330E is always thesmallest and the other two peaks are relatively large.

The meridionally broad nature of the zonal structureimmediately apparent. Together with its stability on fonightly timescales, discussed in Section 4.2, and its content behaviour with altitude at different latitudes, discusin Section 4.3, this implies that whatever atmospheric pnomenon is causing the zonal structure operates on aetary scale. At first glance, given the two large peaks 1apart, the wave-2 harmonic appears dominant.

Figs. 8 and 9 show the normalized amplitudes and phof the harmonic fit displayed in Fig. 7 over a range of latudes. The situation is more complicated than a mere wadominance. Wave-3 is dominant in the northern extratics and no single harmonic is dominant in the southern


t the

ter-

nicaventnalle fothe

entsof

eythe

ter-

har-e inve-3en--hee-eaknic

k into

nicednsesy isationntohere02;

iventhere ofuaremal-de.

s ated

gesthe

1%.ved

Fig. 9. Phases corresponding to the amplitudes in Fig. 8. Recall thamaximum value of the phase of the wave-n harmonic is 360E/n.

Fig. 10. As Fig. 7, but plotting only the wave-2 components. Contour invals are 0.1 and negative regions (low densities) are shaded.

tratropics or the tropics. Wave-1 is the weakest harmoover the entire range of latitude. Waves-2, 3, and 4 all hphases, all around 90E, which stay remarkably constaover a wide range of latitude. This constancy is additioevidence that a planetary-scale mechanism is responsibthe zonal structure. Similar conclusions are reached fromstudy of outbound data and/or different altitudes.

Figs. 10 and 11 show the wave-2 and wave-3 componof Fig. 7. They should be compared to Figs. 2a and 2bWilson (2002). The two pairs of figures are similar, as thshould be according to Section 4.2. The constancy of

r

Fig. 11. As Fig. 7, but plotting only the wave-3 components. Contour invals are 0.1 and negative regions (low densities) are shaded.

phases can be seen, as can the way in which these twomonics interfere to yield the apparent wave-2 dominancFig. 7. Overlaps between peaks in the wave-2 and waharmonics (i.e., constructive interference to give large dsities) occur at about 80E (mainly in the northern hemisphere) and 250E (mainly in the southern hemisphere). Twave-3 peak at 330E destructively interferes with the wav2 trough at the same longitude to give only a small pin density. The shift in the phase of the wave-3 harmowith latitude is responsible for the shift of the largest peathe zonal structure from 80E in the northern hemisphere250E in the southern hemisphere.

Fig. 8 shows erratic changes in normalized harmoamplitude with latitude, in contrast to the well-behavphases of Fig. 9. This may represent the individual respoby each tidal mode to whatever surface inhomogeneitcausing the zonal structure and/or the ease of propagthrough the lower atmosphere at this latitude. Insight ithis behaviour may come from new surface-to-thermospGCMs currently under development (Bougher et al., 20Angelats i Coll et al., 2002).

The total energy associated with each harmonic in a glatitude band is proportional to the product of the area oflatitude band and the square of the amplitude. A measuthe global strength of each harmonic is its root-mean-sq(rms) normalized amplitude, where the mean-square norized amplitude has been weighted by the cosine of latituUsing both inbound and outbound harmonic amplitude130 km altitude, we find that the wave-3 rms normalizamplitude of 22% is greatest. Wave-2 is the next stronat 18%, followed by wave-4 at 14% and wave-1 at 12%. T1σ uncertainties in the rms normalized amplitudes areWave-2 is not the most important harmonic in the obserzonal density structures; wave-3 is.


en-andde,

lack

partk re-m-

cs in

psispsisdero-erics isuthundlati-s onan-ernlPo

it to

psisPo-thelsoGS

ers ofS’s

t istlyme

, thuth

0 kmuthfur-legay toriaptheGS

om

ati-y, the

ce ofout

eseentsitudeeralpic-

iven

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and0is

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ouldimeare

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e-ents

andbe-

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Wilson (2002) used a GCM to model atmospheric dsities at 130 km altitude and below. He found wave-2wave-3 to be the strongest harmonics at 120 km altituwith wave-2 stronger than wave-3, and reproduced theof phase variation with changes in latitude.

4.5. Changes in zonal structure with local solar time

In this section we use data from the Polar Crossingof Phase 2 of aerobraking to examine the week-to-weepeatability of the zonal structure in polar regions, then copare the normalized amplitudes and phases of harmonidayside and nightside zonal structures in polar regions.

During the Daytime Precession part of Phase 2, perialatitude precessed southward with little change in periaLST from one orbit to the next, little change in longitufrom atmospheric entry to exit during an individual aebraking pass, and little change in LST from atmosphentry to exit during an individual aerobraking pass. Thishown in Fig. 1. Periapsis cannot continue to precess soward indefinitely. As periapsis continued to precess aroin the orbital plane, it reached an extreme southerntude, then precessed northward. Periapsis, which wathe sunward, daytime side of Mars, has shifted to thetisunward, nighttime side of Mars. The extreme southlatitude (∼ 87S) is set by the inclination of MGS’s orbitaplane. Since periapsis crossed the terminator during thislar Crossing, periapsis LST must change from one orbthe next.

As in the Daytime Precession part of Phase 2, perialongitude changes from one orbit to the next during thislar Crossing. Unlike the Daytime Precession behaviour,longitude of MGS during an individual aerobraking pass achanges during this Polar Crossing. The longitude of Mmust steadily track through all 360 during one orbit. At po-lar latitudes, MGS’s longitude will change significantly ovshort arcs of an orbit as MGS crosses the converging linelongitude. When periapsis occurs close to the pole, MGlongitude will change significantly over the short arc thathe aerobraking pass. MGS’s LST will change significanduring an individual aerobraking pass for exactly the sareason.

As periapsis precesses southwards towards the pole130 km altitude level on the inbound leg occurs to the soof periapsis and periapasis occurs to the south of the 13level on the outbound leg. That is, MGS travelled from soto north during an aerobraking pass before reaching itsthest south. The 130 km altitude level on the inboundreaches its furthest south, crosses the terminator from dnight, and moves northward before periapsis does. Pesis, in turn, does so before the 130 km altitude level oninbound leg. When periapsis was at its furthest south, Mtravelled from north to south on the inbound leg, then frsouth to north on the outbound leg.

When periapsis is exactly at its furthest south, the ltudes at which the inbound and outbound legs cross, sa

-

-

e

-

130 km altitude level are the same. This is a consequenthe reflection symmetry of an ellipse (such as an orbit) abits semimajor axis. With a near-polar orbit, the LSTs of thtwo points are about half a sol apart. Density measuremcan be made a few minutes apart at exactly the same latand altitude, half a sol apart in LST. By considering sevorbits close to when periapsis was at its furthest south, ature of the daytime and nighttime zonal structure at a galtitude and this altitude can be built up.

In practice, enough data to build up a picture of the zostructure at, say, 130 km takes so many orbits to acqthat there is a finite range in the latitude of the measments (as in Section 4.1). We use a 20 wide latitude rangeinstead of the usual 10, because there are fewer measuments per degree of latitude at this stage of aerobrakingbefore.

When periapsis is at its furthest south, the inboundoutbound legs cross the 130 km altitude level at about 7S.At this time, the 130 km altitude level on the inbound legmoving north, the same altitude level on the outbound lemoving south. To build up a picture of the zonal structur130 km between 50S and 70S, we must use data from thpreceding week for the outbound case and data from thelowing week for the inbound case. Despite this differeof about a week between the two sets of observation50–70S, can we compare them as if they were taken simtaneously? We addressed this problem for non-polar regin Section 4.2 and found that we could do so. Here we shexamine the week-to-week repeatability of both the daytand the nighttime zonal structure. We would like to compdaytime measurements at 130 km between 50S and 70Sfor inbound and outbound legs. We would also like to copare nighttime measurements at 130 km between 50S and70S for inbound and outbound legs. However, aerobing ended when the 130 km altitude level on the outboleg reached its furthest south and repeat nighttime meaments are not available.

There are three relevant subsets of data at 130 kmtude between 50S and 70S; inbound on the dayside witMGS travelling from south to north through the atmosphoutbound on the dayside with MGS travelling from soto north through the atmosphere, and inbound on the nside with MGS travelling from north to south through tatmosphere. Due to the cold temperatures and consequdecreased densities, nightside data at higher altitudes iavailable in useful quantities. Fig. 12 shows the intervbetween these three sets of measurements at a giventude. In the 50–70S latitude band, the 11 day interval btween inbound dayside and outbound dayside measuremis longer than the interval between inbound nightsideoutbound dayside measurements. The difference in LSTtween the nightside measurements at LST= 02 h and thedayside measurements at LST= 15 h is very close to hala sol. Figs. 13–15 show wavefits for the three sets of msurements at 130 km altitude between 50S and 70S. Eachof the three includes a couple of resonances, the 8:1 and


alti-sam

0des,mentd outoundound) fromaysid

ea-ween

f thebe-lts isrials ar

angecan

oundy si-ith

achbutve-f

gnif-ntlicat we

be-l fithementmea-hat inean.

. Theut the

r in

ase

rononic

Fig. 12. Interval between outbound dayside measurements at 130 kmtude and a given latitude and inbound nightside measurements at thelatitude and altitude (solid line) from Phase 2. The minimum near 7Soccurs when periapsis is at its furthest south. At more northern latituoutbound dayside measurements preceed inbound nightside measureAt more southern latitudes, inbound nightside measurements preceebound dayside measurements. Also plotted is the interval between outbdayside measurements at 130 km altitude and a given latitude and inbdayside measurements at the same latitude and altitude (dashed linePhase 2. Inbound dayside measurements always preceed outbound dmeasurements. South of 50S, the interval between day–night repeat msurements at a given latitude and 130 km altitude is less than that betday–day repeat measurements at that latitude and altitude.

9:1 for inbound on the dayside, the 10:1 and 11:1 for out-bound on the dayside, and the 11:1 and 12:1 for inbound onthe nightside. The normalized amplitudes and phases ovarious harmonics are shown in Table 3. A comparisontween the inbound dayside and outbound dayside resuconsistent with the results of Section 4.2 for more equatolatitudes—changes in normalized amplitudes and phaseminor.

Assuming that the nightside atmosphere does not chmore rapidly than the dayside atmosphere does, wecompare the phases of the inbound nightside and outbdayside wavefits as if the measurements were effectivelmultaneous. There is no way to test this assumption wthe MGS accelerometer data. The formal amplitude of eof the four harmonics has increased from day to night,with little statistical significance. The phase of the wa1 harmonic changes by approximately 90, the phase othe wave-2 harmonic changes by approximately 90, andthe phase of the wave-3 harmonic does not change siicantly. The wave-4 harmonic is not statistically significain the nightside wavefit. These phase changes have imptions for the nature of the atmospheric phenomenon tha

e

s.-

e

e

-

Fig. 13. All inbound dayside density measurements at 130 km altitudetween 50S and 70S from Phase 2, crosses, wave-4 harmonic modeto the data, solid line, and 1σ uncertainties about the fit, dotted lines. Tdata were collected over about a week, all at an LST of 15 h. Measureuncertainties (not shown) are much smaller than the range in multiplesurements at any longitude. The zonal structure is less pronounced tFig. 3 or others from the tropics, but is still significant above a zonal m

Fig. 14. As Fig. 13, but using outbound dayside density measurementsdata used here were collected about 11 days after those in Fig. 13, bweak zonal structure is still broadly the same.

observe as zonal structure and will be discussed furtheSection 5.1.

Bougher et al. (2001) found evidence for a zero phshift in a wave-3 zonal density harmonic at 64–67N be-tween 04 and 16 h LST in their comparisons of electand neutral density, but did not present a detailed harmbreakdown.


tside

Table 3Model fit parameters for measurements made at 130 km between 50–70S during Phase 2. Uncertainties are 1σ

Parameters Inbound dayside Outbound dayside Inbound nigh

Constant amplitude (kg km−3) 0.807±0.023 0.810±0.027 0.383±0.033Normalized wave-1 amplitude 0.069±0.040 0.083±0.047 0.275±0.125Normalized wave-2 amplitude 0.148±0.041 0.168±0.046 0.280±0.115Normalized wave-3 amplitude 0.066±0.040 0.134±0.046 0.202±0.118Normalized wave-4 amplitude 0.051±0.049 0.101±0.048 0.126±0.139Wave-1 phase () 154.8±33.0 207.1±32.2 107.7±24.2Wave-2 phase () 52.4±7.6 37.1±8.2 117.0±12.4Wave-3 phase () 108.3±11.5 117.3±6.7 114.8±10.8Wave-4 phase () 3.4±10.2 85.4±6.4 88.9±11.4

. Thezonal

r at-ericen-of ain

s inence

an

-e

eerver

tricns.daryphy,ateHa-outgen-ited

Ha-

di-

anated

an,r asxednics

in-onalidaleternsi-

at-a

des,

Fig. 15. As Fig. 13, but using inbound nightside density measurementsLST is 02 h. Densities have decreased from day to night, and thestructure is very different.

5. Modeling of the zonal structure

5.1. Constraints on tidal modes responsible for zonalstructure

The zonal structures observed in the martian uppemosphere can be studied using tidal theory. “Atmosphtides are global-scale oscillations in temperature, wind, dsity, and pressure at periods which are subharmonicssolar or lunar day” (Forbes, 1995). The dominant forcingthe martian atmosphere is solar heating. Tidal variationdensity at fixed altitude, latitude and season with dependon longitude and LST can be represented as (ChapmanLindzen, 1970; Forbes and Hagan, 2000; Wilson, 2000)

ρ(λ, tLST; z, θ,Ls)

(1)

=∑

σ,s

ρσ,s(z, θ,Ls)cos(σΩ(tLST − λ/2π)

+ sλ − φσ,s(z, θ,Ls)),

whereρ is density,λ is east longitude,tLST is local solar timein sols,z is altitude,θ is latitude,Ls is heliocentric longitude

d

or season,σ is the temporal harmonic (σ = 0,1,2, . . .), s isthe zonal wavenumber (s = . . . ,−2,−1,0,1,2, . . .), Ω =2π sol−1, andφσ,s is the phase.

Westward propagating tides haves > 0, eastward propagating tides haves < 0, and zonally-symmetric tides havs = 0. Migrating tides, which haves = σ , have zonal phasspeeds equal to that of the Sun as seen by a fixed obs(Forbes and Hagan, 2000). Non-migrating tides haves = σ .Only migrating tides are excited in a zonally-symmeatmosphere with zonally-symmetric boundary conditioIf asymmetries are present in the atmospheric bounconditions, such as albedo, thermal inertia, or topograthen their interactions with the migrating tides genernon-migrating tides (Forbes et al., 1995; Forbes andgan, 2000). Zonal inhomogeneities distributed throughthe lower atmosphere, such as dust loading, can alsoerate non-migrating tides. The non-migrating tides excby asymmetries of the form cos(mλ − φm), wherem is thezonal wavenumber of the asymmetry, are (Forbes andgan, 2000)

ρ(λ, tLST; z, θ,Ls)

(2)

=∑

σ,m

ρσ,m(z, θ,Ls)cos(σΩtLST ± mλ

− (φσ (z, θ,Ls) ± φm)).

Solar heating is well described by a combination ofurnal (σ = 1) and semidiurnal (σ = 2) harmonics. Whenthe migrating diurnal and semidiurnal tides interact withm = 1 asymmetry, such as topography, they are modulto form the s = 0 and s = 2 diurnal tides and thes = 1ands = 3 semidiurnal tides respectively (Forbes and Hag2000). All four of these non-migrating tidal modes appeazonal variations with wavenumber 1 as seen from the fiLST reference frame of MGS. Each of the zonal harmoidentified in the accelerometer dataset at fixed LST isfluenced by a near-surface asymmetry with the same zwavenumber and could be attributable to four possible tmodes. When we observe a harmonic in the acceleromdataset, which of the four possible tidal modes is respoble?

In the classical tidal theory of a highly idealizedmosphere, eachσ , s tidal mode can be decomposed intocomplete, orthogonal set of functions, called Hough mo


ithmi-here

,

r toe asal,

ifictlyinesere.ening

ts o

ently

ym-thes a

. 16

ese

veluef

escentpropa-

7.5

ldac-

ric,ivend-

ghtrav-p-

the1).

ria.e-

bovedese-

Fig. 16. Meridional structure of the eight lowest order Hough modes wσ = 1, s = −1. Asymmetric functions are shown in the southern hesphere only, symmetric functions are shown in the northern hemisponly. Asymmetric functions aren = −1 (solid line),n = 2 (dotted line),n = −3 (dashed line),n = 4 (dot-dash line). Symmetric functions aren = 1(solid line), n = −2 (dotted line),n = 3 (dashed line),n = −4 (dot–dashline).

that are each labeled with an indexn (Chapman and Lindzen1970). The meridional structure of eachσ , s, n Hough modeis separable from its vertical structure. Note that we refea disturbance with a given temporal and zonal structura σ , s tidal mode and to one with a given temporal, zonmeridional, and vertical structure as a specificσ , s, n Houghmode.

Roughly speaking, the meridional structure of a specσ , s, n Hough mode determines whether it will be efficienexcited by solar heating and its vertical structure determwhether it will be able to propagate to the upper atmosphIf none of theσ , s, n Hough modes that make up a givσ , s tidal mode can be efficiently excited by solar heatand propagate to the upper atmosphere, then thatσ , s tidalmode cannot influence the accelerometer measurementhe zonal structure.

For aσ , s, n Hough mode to contribute strongly to thzonal structure in the upper atmosphere, it must be efficieexcited by solar heating. Like solar heating, it must be smetric about the equator, be peaked at the equator rathan poleward of the tropics, and have no nodes (latitudewhich its amplitude is zero) too close to the equator. Figshows 8 Hough modes withσ = 1, s = −1. The observedwave-2 zonal structure could be attributable to any of th8 Hough modes.n ranges from 1 to 4 and from−1 to −4.Only then = 1 mode comes close to satisfying the aboexcitation criteria. It is the first (i.e., lowest absolute vaof n) symmetric mode for thisσ , s combination. For each o

f

rt

Fig. 17. Vertical wavelengths of the 16σ , s, n Hough modes that satisfy thexcitation criteria. Travelling waves are marked with a cross and evanewaves are marked with an asterisk. The dashed line represents thegation criterion—Hough modes with vertical wavelengths greater thanscale heights can propagate to the upper atmosphere.

the 16σ , s diurnal and semidiurnal tidal modes that coube influencing the wave-1 to wave-4 harmonics in thecelerometer dataset, only the first symmetricσ , s, n Houghmode satisfied the excitation criteria. The first symmetdiurnal, and eastward propagating Hough mode for a gzonal wavenumber−s is called a diurnal Kelvin wave anlabeled as DKs. Theσ = 1, s = −1, n = 1 mode that satisfies the excitation criteria above is DK1.

In the idealized case of classical tidal theory, a Houmode propagates vertically as either an evanescent or aelling wave (Chapman and Lindzen, 1970). In reality, daming in the martian atmosphere is quite strong for eventravelling waves (Zurek et al., 1992; Forbes et al., 200Fig. 17 shows the vertical wavelengths of the 16σ , s, n

Hough modes that satisfied the above excitation criteAn altitude of 130 km, representative of the acceleromter dataset, corresponds to about 15 scale heights athe martian surface. We assert that, of the Hough moin Fig. 17, the travelling Hough modes with vertical wav


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Table 4Hough modes satisfying the excitation and propagation criteria

Hough mode Zonal wavenumber Wavelengthat fixed LST scale heights

σ = 2, s = 1, n = 1 1 34 (Evanescentσ = 2, s = 3, n = 3 1 8.5σ = 1, s = −1, n = 1 (DK1) 2 25 (Evanescentσ = 2, s = 0, n = 2 2 18σ = 1, s = −2, n = 2 (DK2) 3 9.8σ = 2, s = −1, n = 1 3 14 (Evanescentσ = 2, s = −2, n = 2 4 24 (Evanescent

lengths greater than 7.5 scale heights (those which undno more than two complete cycles before reaching 130altitude) and all the evanescent Hough modes (whose sest vertical wavelength is 14 scale heights) can propagathe upper atmosphere. This propagation criterion suggthat only theσ , s, n Hough modes listed in Table 4 are plasibly contributing to the zonal structure. In cases where mthan one Hough mode could be contributing to a given zodensity harmonic, both could be operating simultaneouThe relatively rapid decrease in the normalized amplitudthe wave-1 harmonic with altitude, discussed in Sectionsuggests that its Hough mode has a shorter vertical wlength than those causing the other zonal harmonics.favours theσ = 2, s = 3,n = 3 Hough mode over theσ = 2,s = 1, n = 1 Hough mode in Table 4.

Observations at widely spaced LSTs can be used toject some of the possible Hough modes in Table 4. Diuand semidiurnal Hough modes, if contributing to the zostructure, will cause different changes in the phase ofmonics in the zonal structure over intervals of half a sol. Tresults of Section 4.5 (Table 3) lead us to conclude thawave-1 zonal density harmonic is not associated with onthe two possible Hough modes in Table 4, the wave-2 zdensity harmonic is attributable to theσ = 1, s = −1, n = 1Hough mode (DK1), the wave-3 zonal density harmonictheσ = 2,s = −1,n = 1 Hough mode, and the wave-4 zondensity harmonic, where present, to theσ = 2, s = −2,n = 2 Hough mode. The wave-1 zonal density harmoshows a phase change of about 90, midway between the 0change of a semidiurnal harmonic and the 180 change of adiurnal harmonic, so it cannot be explained by this simfied model. These preliminary conclusions are indepenof any previous modeling or observational work, haveglected the effects of winds, and are reexamined at theof Section 5.2.

5.2. Other tidal observations and theory

In this section we discuss previous modeling and obvational work on tides in the martian atmosphere tharelevant to Section 5.1 and then reexamine that section’sliminary conclusions in light of the additional information

Bougher et al. (2001) identified theσ = 2, s = −1 tidalmode as the most likely cause of the wave-3 harmonic

-

-

t

-

observed in both neutral density data and electron dedata at 64–67N latitude andLs = 70.

Joshi et al. (2000) used the results of HollingsworthBarnes (1996) to argue against the hypothesis of Keaet al. (1998) that stationary waves were responsiblethe zonal structure observed during Phase 1 of aerobing. They also suggested that the observed zonal strucould be due to diurnally varying tidal modes. Simulatioat 80 km altitude gave reasonable agreement with theserved phasing of the peaks.

Forbes and Hagan (2000) investigated theσ = 1, s = −1,n = 1 Hough mode (DK1). For simulated seasons applble to Phase 1 of aerobraking, they found that DK1 cogenerate wave-2 zonal structure in upper atmosphericsity. These simulations predicted an increase in normawave-2 amplitude with increasing altitude up to 200 kmLs = 270, unlike the decrease observed between 130and 150 km atLs = 30–90 in Section 4.3, and essentialno phase change with increasing altitude. In later simtions applicable to the observations in Section 4.3, Foet al. (2001) predicted that the amplitude in temperaturDK1 should be constant with altitude between 120 km200 km. Since density at a given altitude is sensitive tovertically integrated temperature below it, this implies tthe normalized wave-2 amplitude in the density shouldcrease from 120 km to 200 km.

Wilson has performed a series of tidal simulations (Wson and Hamilton, 1996; Wilson, 2000, 2002). Wilson aHamilton (1996) found that theσ = 1, s = −1,n = 1 Houghmode (DK1) had twice its usual amplitude duringLs = 60–150, a period which includes the end of Phase 2 of abraking. This may contribute to the noticeable increasthe normalized wave-2 amplitude south of 15N latitude inFig. 8 which is coincident with the arrival ofLs = 60,though simulations also show increasing amplitude tosouth at fixed season (Wilson, 2000). In a comparisoTES temperatures with GCM results using a vertically avaged temperature with a broad weighting centred on 25altitude, Wilson (2000) found that stationary waves, buttides, are too confined in latitude to cause planet-wide zstructure, that diurnal tides made larger contributions tsemidiurnal tides, and that each diurnal period, eastwpropagating tidal mode had a deep vertical and broad mional structure consistent with being dominated by thesymmetric Hough mode.

In simulations at 120 km altitude, Wilson (2002) fouthat the wave-2 zonal structure in the upper atmospherepredominantly due to theσ = 1, s = −1,n = 1 Hough mode(DK1), the wave-3 zonal structure was a combination ofσ = 1, s = −2, n = 2 Hough mode (DK2) and theσ = 2,s = −1 tidal mode, and that these two wavenumbers donate. He found the wave-3 zonal structure dominated bydiurnal mode in the tropics and by the semidiurnal min the extra-tropics, though he cautioned that those rewere sensitive to the (uncertain) details of damping inmodel. The wave-3 zonal structure has been observed


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dominated by semidiurnal tidal modes in high latitudeservations by Bougher et al. (2001) and our SectionThe simulated semidiurnal mode has little phase shift wchanging altitude, the diurnal mode has more. Wilson didaddress the wave-1 and wave-4 components of zonal sture. Theσ = 1,s = −1,n = 1 Hough mode (DK1) providea consistent description of the Viking surface pressure dTES temperatures at 25 km, and the wave-2 zonal denstructure at 130 km, and has also has been observed indata from Mariner 9 (Conrath, 1976; Wilson, 2002).

Banfield et al. (2000) analysed tidal signatures in TEStrievals of lower atmospheric temperatures up to an altitof ∼ 40 km and south of 30S latitude atLs = 180–390from Phase 1 of aerobraking and the Science Phasingbits. They examined the eightσ, s tidal modes with aσ, s, n

Hough mode shown in Fig. 17 that could cause wave-1wave-2 zonal structure and found that theσ = 1, s = −1tidal mode was largest, followed by theσ = 1, s = 0 tidalmode. The first of these two tidal modes is probably doinated by theσ = 1, s = −1, n = 1 Hough mode (DK1)discussed above. The second of these two tidal mode, wcauses zonal wave-1 structure, was identified in Wilsosimulations of temperatures centred on 25 km as the dinant mode in the zonal wave-1 structure (Wilson, 200All of Banfield et al.’s tidal amplitudes suggested increanorthward of 30S. Smith et al. (2001a) presented TES dfrom the later mapping mission, but did not repeat Bfield et al.’s detailed analysis of the tidal modes. Howeva mix of tidal modes is present in their data. Since we hfound wave-3 to be a much greater contributor to the zostructure in the upper atmosphere than wave-1, it wouldinteresting to repeat Banfield et al.’s work including wavemodes.

Banfield et al. (2003) have also analysed tidal signatuin TES retrievals from post-aerobraking data. Their Fig.is probably dominated by theσ = 1, s = −1 tidal mode andtheσ = 1, s = −1,n = 1 Hough mode (DK1). It shows a reatively weak amplitude during the season correspondinPhase 2 aerobraking. Like our Fig. 10, it shows highestplitudes at the equator and higher amplitudes in the southemisphere than in the northern hemisphere. The longiof the maximum is∼ 40E in their lower atmosphere daand∼ 80E in our upper atmosphere data, suggesting phchanges with height. Their Fig. 12 is probably dominaby theσ = 1, s = −2 tidal mode and theσ = 1, s = −2,n = 2 Hough mode (DK2). This is restricted to the tropicstheir observations and cannot be responsible for the stwave-3 harmonic which we observe in the extra-tropics (Fig. 11). In Banfield et al., the maximum amplitude of tσ = 2, s = −1 tidal mode is no more than half that of DKHowever, since it is not restricted to the tropics, it couldthe strongest contributor to the wave-3 zonal harmonic inupper atmosphere, which would be consistent with ourservations of a semidiurnal tidal mode for this harmonicthe southern polar regions.

-

Unambiguous identification of a particular Hough moin the martian atmosphere requires many observations.zonal wavenumber and period of the disturbance musmeasured using observations at varied longitudes and LThe meridional profile of the disturbance must be measuusing observations at varied latitudes. The vertical proof the disturbance must be measured using observatiovaried altitudes. Observations of the background state oatmosphere must be used in conjunction with modeling wbeyond the classical tidal theory to predict the behavioucandidate Hough modes as they are generated at the suand propagate upwards through the spatially and tempovarying lower atmosphere. We are not able to satisfythese requirements in this paper. Our identifications oftain Hough modes in this paper are based primarily uponobserved density variations with longitude at fixed LSTmany latitudes, the observed density variations with lontude near the South Pole at two LSTs half a sol apart,classical tidal theory. These restrictions should be unstood before the conclusions of this paper are acceptethe reader. We discuss ways to test our conclusions intion 6.

In summary, our conclusion that the wave-2 zonal strture is attributable to theσ = 1, s = −1, n = 1 Hough mode(DK1) is consistent with much observational and theoical work. Our conclusion that the wave-3 zonal structis attributable to theσ = 2, s = −1, n = 1 Hough mode isconsistent with theoretical work and observations at highitudes, but theoretical work suggests that theσ = 1, s = −2,n = 2 Hough mode (DK2), which we rejected based onobserved phase change at high latitudes, may be impoat low to mid-latitudes. Observational and theoretical win the lower atmosphere predicts that theσ = 1, s = 0 tidalmode dominates the zonal wave-1 structure, but we rejeit as a major influence in the upper atmosphere due toshort vertical wavelength. There has been little work relevto the wave-4 zonal structure. We make one modificatioour preliminary conclusions from Section 5.1. Previouslyjected, theσ = 1, s = −2, n = 2 Hough mode (DK2) maymake a significant contribution to the wave-3 zonal strucin the tropics.

5.3. Effects of the surface on zonal structure

The zonal structure in the upper atmosphere muscaused by a zonal asymmetry in the lower boundary cditions of the atmosphere interacting with solar heatingexcite a non-migrating tide. In this section we investigwhich zonal asymmetry might have the strongest influeon the zonal structure.

Dust loading is variable on relatively short timescalso it is unlikely to be responsible for the long-lasting, sble zonal structure. Information on the spatial distributiondust is available from archived TES results, but recent plications have focused on the dust distribution during d


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Fig. 18. Amplitudes, in km, of the various zonal harmonics in martopography as a function of latitude. Contour intervals are 0.6 km andues greater than 1.2 km are shaded.

storms and not during the relatively calm period of Phasof aerobraking (Smith et al., 2000, 2001a).

Continuing from Section 5.1, it seems reasonablethe meridional profile of the zonal asymmetry that, in cjunction with solar heating, can excite a given Hough mmust overlap significantly with the meridional profile of slar heating (peaked in the tropics, equatorially symmeand without any minima close to the equator) for that Homode to be efficiently excited. Using Mars Consortium dkindly supplied to us by Jim Murphy and MOLA topogrphy, we performed harmonic decompositions of topograthermal inertia, and albedo as a function of longitude atferent latitudes (Smith et al., 2001b). As shown in Fig.the wave-1 component of topography is low near the eqtor and highest in the extra-tropics. The wave-2 compoof topography has high amplitudes throughout the tropicgreatest at the equator, and decreases into the extra-trExcept for a narrow region of low amplitude at 30S, it isreasonably symmetric about the equator. The wave-3 cponent has high amplitude at the equator, is quite conthroughout the tropics and decreases rapidly polewar60. The wave-4 component of topography is low nearequator and highest in the extra-tropics. All amplitudesbelow 0.6 km in smooth terrain poleward of 60. The wave-2 and wave-3 components of topography satisfy the ovecriteria, but none of the components of albedo or thermaertia do. As discussed in Section 5.1, both these componof topography can generate Hough modes with the abilitpropagate into the upper atmosphere and appear as wand wave-3 zonal structure. This is consistent with the obvation in Section 4.4 that these harmonics are the strong

s.

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2

It has already been shown that topography, the rangwhich exceeds an atmospheric scale height, is the main cof the σ = 1, s = −1, n = 1 Hough mode (DK1) (Wil-son and Hamilton, 1996; Wilson, 2002). This justificatifor the dominance of wave-2 and wave-3 harmonics inzonal structure is intended to offer some physical insightcomplement to the rigourous results of more complete mels. It could be tested by repeating Wilson and Hamiltowork on theσ = 1, s = −1, n = 1 Hough mode (DK1) onthe other Hough modes that are possible contributors tozonal structure (Wilson and Hamilton, 1996).

6. Conclusions

Sol-to-sol variability, or weather, in the martian uppatmosphere is not well modeled by current climate mels. As martian climate models develop into weather mowith the goal of understanding and interpreting the resultcurrent and anticipated long-term, global-scale, atmosphmonitoring from orbiting instruments with high spatial atemporal resolutions, this variability with short spatial atemporal scales at high altitudes will provide a challengtest of their abilities. This variability is also significant fthe design of future aerobraking missions.

For the seasons and latitudes discussed here, the ssol variability at fixed longitude, altitude, latitude, LST, aseason is smaller than the longitudinal variability at fixaltitude, latitude, LST, and season. This repeatable vtion with longitude, or zonal structure, can cause densto change by a factor of two or more over less than 90 oflongitude. The zonal structure must be due to thermal tgenerated at or near the planet’s surface. It is remarkablesurface effects are so significant at altitudes of 150 km.

We have used the changes in the zonal structure withtude, latitude, and LST, together with a simple applicatioclassical tidal theory, to identify the tidal modes that havestrongest contributions to the zonal structure. The weekweek stability of the zonal structure and its similar behaviat all latitudes require a planetary scale, not localized, pnomenon. Theσ = 1, s = −1, n = 1 (wave-2, DK1) andσ = 2, s = −1, n = 1 (wave-3) Hough modes have beidentified as major contributors to the zonal structurethis is supported by previous theoretical work. Theσ = 1,s = −2, n = 2 (wave-3, DK2) Hough mode is predictedbe present. The wave-4 component of the zonal structuattributed to theσ = 2, s = −2, n = 2 Hough mode and thcause of the weak wave-1 component is uncertain. Wepresented a simple justification, based on the meridionalfiles of solar heating and of various harmonic componentopography, thermal inertia, and albedo, for why the wa2 and wave-3 components dominate the zonal structuresuggested that both are controlled by topography.

We believe that our conclusions are robust, althoughdiscussed in Section 5.2, these Hough modes haveidentified indirectly. The main ways in which they mig


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be tested are (a) observations of the zonal wavenumand periods of any zonal structure in the middle atmospwhich would test our conclusion that certain Hough moare propagating through the middle atmosphere, (b) uppemospheric observations at more varied longitudes and Lwhich, by identifying the zonal wavenumbers and perioof the zonal structure, would test our identification of ctain Hough modes in the upper atmosphere, and (c) detmodeling work, which would test our simplified ideas abovertical propagation of Hough modes. For consistency, thnew observations would be most useful if they occurrethe sameLs as the MGS accelerometer observations.

At the surface of the Earth, atmospheric tides are osmall-scale fluctuations masked by the effects of weaand only detectable in long duration records. At higher atudes, they are more prominent. On Mars, whose atmosphas a greater diurnal variation in solar heating per unit mthan Earth’s, tidal effects are greater. They are a majorture in the long-term pressure measurements at theViking landing sites and were predicted by the very first mtian climate model (Leovy and Mintz, 1969; Zurek et a1992). Tides have long been recognized as a basic feof the dynamics of the martian lower atmosphere; this woand others, demonstrates that they are important in theper atmosphere as well. As such, upper atmosphere mwhich do not have dynamic connections to lower atmosphmodels will be unable to reproduce observations or makecurate predictions.

The strength of a tidal mode in the martian uppermosphere depends on the winds and temperatures inlower atmosphere. If surface-to-thermosphere modelsreliably reproduce the upper atmosphere observations,only those lower atmosphere conditions which correspto the observed behaviour in the upper atmosphere aremissible. This offers a way to constrain the behaviour oflower atmosphere using only observations from the uppemosphere.

Accelerometers are the only scientific instrumentswill fly again and again into Mars orbit during the nedecade. As such, they offer a unique way to study thehaviour of the difficult-to-study upper atmosphere over inannual periods and the 11-year solar cycle. Their obsetions are difficult to compare to most current martian climmodels because of their high altitude, and this partiallyplains why so few recent publications have discussedcelerometer data, but they still have value. Acceleromdatasets will be invaluable in the future when, inevitabmodels reach such altitudes, and every opportunity shbe taken now to collect and preserve such datasets.

Acknowledgments

We thank Don Banfield, Jeff Forbes, Jill Hanna, Jim Mphy, Bob Tolson, John Wilson, and an anonymous revieThe studies of Hough modes in Section 5.1 were perforusing computer programs kindly provided by Jeff Forbes

-,

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-

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