Nonmigrating tides in the thermosphere of Mars

Jeffrey M. Forbes,1 Alison F. C. Bridger,2 Stephen W. Bougher,3 Maura E. Hagan,4

Jeffery L. Hollingsworth,5 Gerald M. Keating,6 and James Murphy7

Received 4 September 2001; revised 15 January 2002; accepted 4 June 2002; published 23 November 2002.

[1] The vertical propagation of nonmigrating (i.e., longitude-dependent or non–Sun-synchronous) solar diurnal and semidiurnal tides into the thermosphere of Mars isinvestigated through numerical simulation. The waves are generated in the NASA AmesMars general circulation model (MGCM) through solar radiative, topographic, andnonlinear processes using a comprehensive physics package and including a diurnal cycle.At an altitude near 70 km, zonal wave number decompositions of the diurnal andsemidiurnal tidal fields are performed, and each wave component is extended from 70 to250 km using a linear steady state global scale wave model for Mars (Mars GSWM).Conditions representative of aerocentric longitudes Ls = 30 (near equinox) and Ls = 270(Southern Hemisphere summer solstice) are considered. Modeled total relative densityvariations of order ±10–40% near 125 km are analyzed in terms of the zonal wave numbers(ks) seen from the Sun-synchronous perspective of the Mars Global Surveyor (MGS)accelerometer experiment, and yield reasonable agreement in amplitude and phase with thedensity measurements. The model indicates the twomost important waves responsible for ks= 3 to be the eastward-propagating diurnal and semidiurnal oscillations with zonal wavenumbers s = 2 (�15–40%) and s = 1 (�8%), respectively. The eastward-propagating diurnalcomponent with s = 1 (�15%) and the semidiurnal standing (s = 0) oscillation (�4–23%)are concluded to be the main contributors to the ks = 2 longitudinal density variation seenfrom the Mars Global Surveyor (MGS). The standing (s = 0) diurnal oscillation (�4–5%)and the westward-propagating semidiurnal component with s = 1 (�5–8%) emerge as themost likely contributors to ks = 1. Other waves that may make important secondarycontributions include the westward-propagating semidiurnal oscillations with s = 3(�4–6%) and s = 4 (�3–9%). In addition, above 100 km the wind and temperature fieldsassociated with the above waves represent�15–30% perturbations on the Sun-synchronouswind and temperature fields driven in situ by EUVand near-IR solar radiation absorption.Nonmigrating tides primarily arise from zonal asymmetries in wave forcing associated withMars’ topography; our results show for the first time that the dynamical effects of Mars’topography extend throughout the atmospheric column to Mars’ exobase (�200–250km). INDEX TERMS: 5707 Planetology: Fluid Planets: Atmospheres—structure and dynamics; 5739

Planetology: Fluid Planets: Meteorology (3346); 3384 Meteorology and Atmospheric Dynamics: Waves and

tides; 3369 Meteorology and Atmospheric Dynamics: Thermospheric dynamics (0358); KEYWORDS: Mars,

thermosphere, density, nonmigrating tides

Citation: Forbes, J. M., A. F. C. Bridger, S. W. Bougher, M. E. Hagan, J. L. Hollingsworth, G. M. Keating, and J. Murphy,

Nonmigrating tides in the thermosphere of Mars, J. Geophys. Res., 107(E11), 5113, doi:10.1029/2001JE001582, 2002.

1. Introduction

[2] The cyclic absorption of solar radiation by a rotatingplanet and its atmosphere naturally leads to oscillationswhich are subharmonics of the rotational period (T), i.e.,frequencies equal to 2pn/T where n = 1, 2, etc. If theexcitation of such oscillations is nonuniform in longitude,then continuity around the sphere suggests a Fourier seriesapproximation of the zonal asymmetries. The resulting‘‘tidal’’ oscillations in excitation and in the atmosphericresponse fields (i.e., temperature, winds, density) may beexpressed as follows:

X

n

X

s

An;s z; qð Þ cos n�t þ sl� fn;s z; qð Þ� �

ð1Þ

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. E11, 5113, doi:10.1029/2001JE001582, 2002

1Department of Aerospace Engineering Sciences, University of Color-ado, Boulder, Colorado, USA.

2Meteorology Department, San Jose State University, San Jose,California, USA.

3Lunar and Planetary Laboratory, University of Arizona, Tucson,Arizona, USA.

4High Altitude Observatory, National Center for Atmospheric Research,Boulder, Colorado, USA.

5NASA Ames Research Center, Moffett Field, California, USA.6NASA Langley Research Center, George Washington University,

Hampton, Virginia, USA.7Department of Astronomy, New Mexico State University, Las Cruces,

New Mexico, USA.

Copyright 2002 by the American Geophysical Union.0148-0227/02/2001JE001582$09.00

23 - 1

where for Mars t = time in sols, l = longitude, s = zonalwave number, � = 2p sol�1 is the planetary rotation rate,An,s is the amplitude, fn,s is the phase (time of maximum atl = 0), and n = 1 for the diurnal tide, n = 2 for semidiurnaltide, etc. Note that the above expression represents a seriesof zonally propagating waves with phase speeds = �n�/s,so that s > 0 (s < 0) implies westward (eastward)propagation. To an observer fixed on the planet, waveswith s = n ‘‘migrate’’ westward with the apparent motion ofthe Sun. Waves with s 6¼ n can therefore travel faster thanthe Sun, travel opposite to the Sun’s apparent motion, or bestanding oscillations (s = 0). Rewriting in terms of localtime, each term in the series (1) has the form

An;s z; qð Þ cos n�tLT þ s� nð Þl� fn;s z; qð Þ� �

ð2Þ

It is clear that the migrating tides (s = n) are independent oflongitude. It is these components, therefore, that are excitedby the Sun’s radiation being absorbed by a longitude-invariant planetary atmosphere or surface. Nonmigratingtides (s 6¼ n), the subject of the present paper, are thosecomponents which capture the longitudinal dependence ofthe tidal oscillations.[3] Hereafter we will use shorthand notations to denote

diurnal (D) and semidiurnal (S) eastward (E)- and westward(W)-propagating oscillations as follows: DEs, SEs, DWs,SWs, where s is the absolute magnitude of the zonal wavenumber. Standing oscillations are represented as D0 and S0.Zonal wave number components of surface topography aredenoted with the integer m.[4] Nonmigrating tidal components in the lower and

middle atmosphere of Mars have been discovered in meas-urements [i.e., Conrath, 1976; Banfield et al., 2000; Wilson,2000] and are prominent dynamical components in Marsgeneral circulation models (MGCMs) [Joshi et al., 2000;Wilson and Hamilton, 1996]. Zurek [1976] also recognizedthe importance of nonmigrating components in his study ofthe diurnal tide on Mars. Much of the excitation for non-migrating tides is likely to occur as a result of nonlinearinteractions between radiative processes and the surface,including consideration of topographic height and rough-ness, surface thermal inertia and albedo. To first order, thebulk result of the above processes may be viewed as alongitudinal modulation of the westward-migrating solarradiation interacting with the near-surface atmosphere. Fig-ure 1 illustrates the basic concept. Here the surface modu-lation is characterized by a dominant zonal wave number m= 2, which represents the most important topographic zonalwave number component at low to middle latitudes onMars’ surface. It is easily seen how the 24-hour harmonicof the westward-migrating solar radiation interacts with s =2 topography to excite DW3 and DE1 tides, respectively.Similarly, the semidiurnal harmonic of the migrating Sun’ssolar radiation pattern interacts with s = 2 topography togenerate SW4 and S0 oscillations. DE1 is largely comprisedof the so-called diurnal Kelvin wave (DKW1), which is innear-resonance in Mars’ atmosphere, and as such a signifi-cantly amplified response is anticipated. Other possibleexcitation sources for nonmigrating tides on Mars includethe absorption of solar radiation by zonally asymmetric dustdistributions, and nonlinear coupling between migratingtides and stationary planetary waves. The latter mechanism

operates to produce significant SW1, SW3 [Angelats i Colland Forbes, 2002; Forbes et al., 1995; Portnyagin et al.,1998], D0 and DW2 [Hagan and Roble, 2001] oscillationsin the terrestrial atmosphere.[5] The generalized form of the above result is obtained

by multiplying the migrating (s = n) component of solarforcing of the form cos(n� t + sl � fn) by the wave numberm topographic component cos(ml � qm), yielding

cos n�t þ n� mð Þl� fn � fmð Þ½ ð3Þ

In the inertial frame, the wave numbers generated are thusequal to n ±m. In the local time reference frame, (3) reduces to

cos n�tLT � ml� fn � fmð Þ½ ð4Þ

Therefore, for solar forcing at any tidal frequency the zonalwave number m component of topography yields nonmigrat-ing tides that appear as ks =m features from Sun-synchronousorbit (tLT = constant). In a similar fashion, nonlinearinteraction between a migrating (s = n) tidal oscillation anda stationary planetary wave (s = m) produces the sum anddifference wave numbers (s = n ±m) at the same frequency asthe tidal oscillation. A few of the more likely wave numberinteractions and the resulting inertial and Sun-synchronouswave numbers that are generated are provided in Table 1.Throughout this paper we distinguish wave number featuresseen from Sun-synchronous orbit (ks) from the planetary-relative wave number (s) using these notations.[6] Only recently have data become available that suggest

the presence and importance of nonmigrating tides in thethermosphere of Mars. The existence of significant longi-tude variations in Mars’ thermospheric density was firstreported by Keating et al. [1998], based upon measurementsby the MGS Accelerometer instrument during Phase 1aerobraking. Phase 1 occurred during N. Hemisphere winterat midlatitudes between aerocentric longitudes Ls = 183–302� (September 1997–April 1998). They found the longi-tude variation of relative density (dr/r0) near 125 km to be

Figure 1. Schematic illustrating how solar radiation,interacting with dominant zonal wave number s = 2topography on a rotating planet, generates various wavenumber diurnal and semidiurnal oscillations which give riseto longitude-dependent tidal oscillations.

23 - 2 FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS

mainly characterized by a zonal wave number ks = 2oscillation (±22%), with some contribution (±8%) from aks =1 component. These data and the corresponding empiri-cal fit are illustrated in Figure 2. Keating et al. [1998]tentatively interpreted these oscillations as stationary plan-etary waves. However, modeling studies [Hollingsworthand Barnes, 1996] indicate that stationary planetary waveson Mars maximize well below the thermosphere, especiallyfor s > 1. However, as can be inferred from Table 1, severaloscillations can give rise to a ks = 2 longitude variationwhen viewed from the near-Sun-synchronous orbit of MGS.These include DW3, DE1, SW4 and S0. The first symmetriccomponent of DE1 (the diurnal Kelvin wave with s = 1, orDKW1) is notable, since this oscillation is known to benearly resonant in Mars’ atmosphere and is thus subject to ahighly amplified response. This led Forbes and Hagan[2000] to investigate this wave as a possible explanationfor the data illustrated in Figure 2. Using a Mars globalscale wave model (Mars GSWM; see below) calibrated near25 km by output from the NOAA/GFDL MGCM [Wilsonand Hamilton, 1996], they calculated the global density,temperature and wind response to DKW1 forcing up to 250km. The computed relative density perturbations at 125 kmfor the same season and latitude as the accelerometermeasurements are also depicted in Figure 2, demonstratingthat DE1 is capable of explaining much of the ks = 2variation in the data. Wilson [2000] and Joshi et al.[2000] drew similar conclusions based on the appearanceof nonmigrating tide effects at the upper levels (�70 km) ofMars GCMs.[7] Recent examinations of Phase 2 aerobraking data

[Keating et al., 2000; Withers et al., 2000; Bougher et al.,2001] reveal the presence of a � ±20% ks = 3 component inaddition to the ks = 2 component. These measurementsoccurred from Ls = 38� to 93� when periapsis precessedfrom 60�N to 80�S on the dayside (1700 LT to 1400LT),and then from 80�S to 30�S near 0200 LT. The ks = 2 and ks= 3 longitude variations observed by MGS accelerometerare also observed in temperature measurements from theMGS Thermal Emission Spectrometer near 25 km [Wilson,2000]. Modeling results indicate that the oscillations are thesame, originating from zonal asymmetries in the diurnalforcing near the surface [Wilson, 2000]. As noted previ-ously, the ks = 2 longitude structure appears to be dominatedby DE1. The ks = 3 variation is most likely dominated byDE2 combined with SE1. The latter interpretation is con-sistent with a ks = 3 longitude variation in electron densityfrom MGS occultation measurements [Bougher et al., 2001]at the beginning of Phase 2 (Ls �30�; 60–65�N latitude),

which is in phase with nearly coincident MGS accelerom-eter measurements 12 hours earlier in local time.[8] The model calculation in Figure 2 was generated by

calibrating the lower-atmosphere thermal excitation ofDKW1 in the Mars GSWM to yield DE1 amplitudesapproximately equal to those in the NOAA GFDL MarsGCM [Wilson and Hamilton, 1996]. Mars GCMs such asthose developed at NOAA/GFDL, Laboratoire de Meteor-ologie Dynamique [Forget et al., 1999] and NASA Ames[Pollack et al., 1990; Haberle et al., 1993, 1997, 1999]include comprehensive nonlinear treatments of the radiativeand heat transfer processes that excite oscillations near theground, taking full account of realistic topographic varia-tions and interactions. The combined use of a GCM with amechanistic model by Forbes and Hagan [2000] was drivenby the fact that the GCM upper boundary lies below 100km. The Mars GSWM extends from the surface to 250 km,and can provide reasonable steady state approximations ofthe vertically propagating waves. In the present work, weuse an approach similar to Forbes and Hagan [2000] toinvestigate the thermospheric responses of the full range ofnonmigrating tides excited in Mars’ lower atmosphere: TheNASA Ames Mars GCM is utilized to provide the globaltidal fields up to about 70 km. The fields at 70 km are theninput as lower boundary conditions to the Mars GSWM,

Table 1. Wave Numbers s = n ± m Generated by (a) Solar-Radiation/Topographic or (b) Migrating-Tide/

Stationary-Planetary-Wave Interactions

(a) Migrating Solar RadiationFrequency Component, or

(b) Tide Frequency Component, n

Topographic Wave Number,Stationary Wave Wave Number,

Sun-Synchronous Wave Number, m, ks

Planetary-FrameWave Numbers (s)

n + m n � m

1 1 2 01 2 3 �11 3 4 �22 1 3 12 2 4 02 3 5 �1

Figure 2. Midlatitude density measurements (dots) andempirical fit (solid line) at 125 km from Mars GlobalSurveyor (MGS) accelerometer measurements [Keating etal., 1998] for orbits 90–110 near 1400 LST, compared withdiurnal Kelvin wave calculation (dashed line) by Forbesand Hagan [2000].

FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS 23 - 3

which is utilized to simulate the propagation of variouszonal wave number components of the diurnal and semi-diurnal tides into the dissipative thermosphere. This allowsus to examine the efficiency with which different non-migrating tidal components penetrate to thermosphericheights and produce potentially important effects. Recentresults from MGS accelerometer data provide some meansof evaluating our results near 125 km, giving furthercredence to our estimates for altitudes between 150 and250 km.[9] In the following section, we describe the essential

features of the NASA Ames Mars GCM and the MarsGSWM, and the inputs to these models. In section 3, wepresent a sufficient sampling of results to convey to thereader the overall importance of nonmigrating tides to thestructure of Mars’ thermosphere. We emphasize that theseoscillations are superimposed on a Sun-synchronous oscil-lation driven in situ in the thermosphere by EUV andnear-IR solar radiation absorption that is independent oflongitude. The last section contains our summary andconclusions.

2. Models and Model Inputs

[10] The NASA Ames MGCM is based on the meteoro-logical ‘‘primitive equations’’ that are time-dependent equa-tions in sigma coordinates for the horizontal flow andthermodynamic energy, as well as equations for massconservation and hydrostatic balance. In the simulations

utilized here, the model extends from the surface to roughly100 km altitude with vertical spacing between layers rang-ing from �10 to 20 m near the surface to �5 km at theupper boundary. Due to the artificial sponge layer in theupper regions of the model and neglect of non-LTE pro-cesses, results are anticipated to be realistic only to �80 km.The MGCM includes processes important for realisticallysimulating atmosphere-surface interactions, including:allowance for a complete diurnal cycle; a surface heatbudget; radiative heating (cooling) algorithms for CO2 gasand suspended aerosols (e.g., dust and/or water conden-sates); latent heat release associated with CO2 condensation;and heat exchange between the atmosphere and surface. Theboundary layer parameterization of the MGCM has beenimproved recently in that frictional forces and turbulentheating are treated as diffusion processes. The localRichardson-number and mixing-length dependent eddycoefficients are taken from a level-2 formalism for boundarylayer turbulence. Further details on the Mars GCM’s physicspackage are described by Pollack et al. [1990] and Haberleet al. [1993, 1997, 1999]. The model’s surface topographyhas been updated to that derived by the MGS Mars OrbiterLaser Altimeter (MOLA) [Smith et al., 1999] which is‘‘smoothed’’ to the horizontal resolution of the climatemodel (i.e., 7.5� latitude � 9� longitude). Fields for Mars’surface albedo and thermal inertia are presently based onvalues from the Mars Consortium data set.[11] The complexity of thermospheric processes to be

modeled has hindered the extension of current Mars GCMs

Figure 3. Zonal mean temperatures and zonal winds for Ls = 30.

23 - 4 FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS

to altitudes significantly above 100 km. (Angelats i Coll andForget (personal communication, 2002) recently reportextension of the CNRS/LMD model to 120 km.) Yet, itappears that the effects of upward-propagating thermal tideson the thermosphere of Mars are profound (see the Intro-duction). In addition, the complexity of GCMs often pre-cludes definitive interpretations of results seen in the modeloutputs. These limitations provide the basic motivation forthe work proposed herein — that of utilizing a mechanisticmodel in conjunction with the more complex and compre-hensive GCM to shed light on this issue. This mechanisticmodel, the Mars global scale wave model (Mars GSWM), isnow described.[12] The Mars GSWM solves the coupled momentum,

thermal energy, continuity and constitutive equations forlinearized steady state atmospheric perturbations on a spherefrom the surface to the thermosphere (�250 km). Given thefrequency, zonal wave number and excitation of a particularoscillation, the height versus latitude distribution of theatmospheric response is calculated. The model includes suchprocesses as surface friction, mean winds and meridionalgradients in scalar atmospheric parameters, parameterizedradiative cooling, eddy and molecular diffusion, and iondrag. The Mars GSWM is an extension to Mars of theterrestrial GSWM, which has been used to simulate planetarywaves, Kelvin waves and migrating and nonmigrating tidesin the Earth’s atmosphere [i.e., Hagan, 1996; Hagan et al.,1993, 1999; Forbes, 2000; Forbes et al., 2001a, 2001b;Hagan and Forbes, 2002a, 2002b].

[13] The background (zonal mean) zonal winds, temper-atures, and densities are prescribed as follows. Low-dust(visible opacity �0.3), Southern Hemisphere summer (Ls= 270), near-equinox (Ls = 30) and average solar activity(F10.7 = 130 at 1AU) conditions are assumed. Below 80km we utilize temperatures and mean zonal winds fromthe Ames Mars GCM, and above 80 km from the MarsThermospheric GCM, or MTGCM [i.e., Bougher et al.,2000]. A scheme is utilized to smoothly merge thesetemperatures and wind distributions across the 80-kminterface. Using a global mean pressure of 6.1 mb at amean reference level of z = 0 km, the hydrostatic equationis integrated upward to provide densities and pressures.The resulting temperature and wind distributions for Ls =30 and Ls = 270 are illustrated in Figure 3 and Figure 4,respectively. For Ls = 30, we note eastward jets in bothhemispheres in the middle atmosphere, with the SouthernHemisphere jet two to four times more intense than thatin the Northern Hemisphere. The thermospheric jets aremore nearly antisymmetric (i.e., equal and opposite signin the two hemispheres). The 25 m s�1 westward jet near45 km over the equator is probably due to dissipation ofthe diurnal propagating tide. For Ls = 270, the moreintense middle atmosphere eastward jet is in the NorthernHemisphere, with a � factor of two weaker jet in theSouthern Hemisphere. In the thermosphere, an antisym-metric jet system of the same sense also exists, but withthe �25% more intense jet occurring in the SouthernHemisphere.

Figure 4. Zonal mean temperatures and zonal winds for Ls = 270.

FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS 23 - 5

[14] The region of greatest uncertainty in the assumedzonal mean wind distributions occurs in the 80- to 120-km-height regime. Recent simulations using the LMDMars GCM (Angelats i Coll and Forget, personal commu-nication, 2002) with upper boundary at 120 km indicatesstrong retrograde winds above 80 km (especially duringnorthern winter). However, current models do not accountfor gravity waves due to nontopographic sources and

perhaps other thermospheric processes that may substan-tially alter this recent result. Concerning the presentsimulations, significant modifications of mean winds inthis regime are likely to alter the details of the non-migrating tide structures, but our conclusions concerningthe importance of nonmigrating tides in Mars’ thermo-sphere and even our identification of dominant wavemodes are likely to remain intact.

Table 2. Perturbation Northward Velocity Amplitudes at 200 km and Relative Density Amplitudes at 125 km for Various Zonal Wave

Number Components of the Diurnal and Semidiurnal Solar Tides for Aerocentric Longitudes Ls = 30 and Ls = 270a

Sun-Synchronous, ks

Diurnal Semidiurnal

s Ls = 270 Ls = 30 s Ls = 270 Ls = 30

v0(m s�1) at 200 km3 DE2 �2 12* 19* SE1 �1 19* 13*2 DE1 �1 15* 7* S0 0 9 28*1 D0 0 5* 8 SW1 1 12* 38*1 DW2 2 2 7 SW3 3 13* 82 DW3 3 7 4 SW4 4 6 16*

r0/r0 at 125 km3 DE2 �2 .15* .41* SE1 �1 0.08* 0.08*2 DE1 �1 .15* .15* S0 0 0.04 0.23*1 D0 0 .05* .04 SW1 1 0.05* 0.08*1 DW2 2 .01 .04 SW3 3 0.06* 0.042 DW3 3 .005 .02 SW4 4 0.03 0.09*

aVelocity amplitudes are in m s�1; s, zonal wave number. The left column gives the zonal wave number ks seen from a Sun-synchronous reference frame.All components that contribute 5% or more to the relative density variations at 125 km are asterisked.

Figure 5. Altitude versus latitude structures of wave fields for the DE2 oscillation at Ls = 30. (a)Amplitude of eastward wind (m s�1); (b) amplitude of northward wind (m s�1); (c) relative amplitude ofdensity; (d) phase of northward wind (time of maximum at zero longitude).

23 - 6 FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS

[15] Dissipation mechanisms in the Mars GSWM includeeddy and molecular diffusion and radiative damping. Theeddy diffusion rates are based on the study of Bougher et al.[2000] (see also references therein) above 85 km, and areassumed constant with height between 70 and 85 km.Formulae for viscosity and thermal conductivity coefficientsfor molecular diffusion of heat and momentum are the sameas those in the terrestrial GSWM, but modified in accordwith the chemical composition of Mars’ atmosphere. Below40 km, the radiative damping rates in the model closelyapproximate those illustrated by Barnes [1984, 1990] whichare based on the detailed radiative calculations of Barnes[1983]. These rates assume some dependence on dustopacity. Radiative damping is potentially problematic inthe region above 85 km where non-LTE effects assumeimportance [Lopez-Valverde et al., 1998], and CO2 coolingis nonlinearly dependent on temperature and pressure. In thecurrent work, we utilize a modified cool-to-space approx-imation to better approximate non-LTE cooling above 85km. It is important to note that the Mars GSWM computesperturbation fields on a zonal mean atmosphere. Therefore,some approach that deals with the perturbation temperaturesis all that is necessary, in contrast to GCMs that must utilizea full radiative transfer approach. In our approximation theNewtonian cooling coefficient, a, is defined as follows. Wefirst define

Qtot ¼ Qref þ a T � �Tð Þ ¼ Qref þ a�T ð5Þ

where T is temperature and Qref ¼ DTDt

� �ref

is a referencecooling calculated for a reference temperature profile �T (z).The remaining term represents the cooling due to theperturbation on this mean temperature. Using an accurate15-micron cooling calculation, we calculate cooling ratesfor various temperature departures from �T (z) and calculatea:

a ¼ �Q

�Tð6Þ

where �Q = change in 15-micron cooling rate due to a �Tperturbation. The family of curves so obtained wererelatively insensitive to �T except in the region of peakcooling, 100–120 km. In this height range a maximumcooling rate is chosen that lies within the typical range ofexpected �T’s. The resulting Newtonian cooling profilepeaks at a value of 5.2 sol�1 at 115 km, and is roughlyGaussian-shaped between 85 and 145 km with a width athalf-maximum of about 40 km.[16] The Ames GCM and GSWM were coupled together

at an approximate height of 70 km in the following manner.Ten sols of data from the GCM at Ls = 30 and Ls = 270 wereeach composited into a single sol with a grid resolution of7.5� latitude, 9.0� longitude and 1.5 hours in time. TheGCM fields were interpolated from pressure coordinates toa constant altitude of 70 km and the 3�-latitude grid of theGSWM. The diurnal and semidiurnal harmonics wereextracted, and for each temporal harmonic the longitude

Figure 6. Same as Figure 5, except for SE1 at Ls = 270.

FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS 23 - 7

dependence was Fourier-decomposed into zonal wave num-bers s = �4 through s = +4. This resulted in nine latitudinaldistributions of amplitude and phase of dynamical fields forthe diurnal tide, and nine for the semidiurnal tide. Thedynamical fields required for input into the GSWM con-sisted of the three velocity components and temperature.Vertical velocities in pressure coordinates (w) were con-verted to vertical velocities in height coordinates (w) usingthe following relation:

w ¼ @p

@tþ ~Vh � rpþ w

@p

@zð7Þ

where ~Vh is the horizontal wind vector, p is pressure, z isaltitude and t is time. Linearizing, assuming solutions of theform exp(ist) where s is frequency, and neglecting thesmaller advective terms, the complex forms of the perturba-tion (primed) vertical velocities are thus related:

w0 ¼ isp0 � w0

r0gð8Þ

where r0 is the unperturbed density and g is the accelerationdue to gravity. It is implicit in our approach that the variousmigrating and nonmigrating tidal components are linearlyindependent; that is, nonlinear interactions between thewaves can be neglected to first order. Based upon our

experiences with terrestrial atmospheric tides, this is areasonable approximation.

3. Results

[17] In this section, it is our objective to provide a broadoverview of the nonmigrating tidal response of Mars’thermosphere, and to convey to the reader the characteristicsof certain oscillations of central importance. As notedpreviously, two representative seasons were chosen foranalysis: Ls = 270 (Southern Hemisphere summer solstice),and Ls = 30 (near equinox). The former also correspondsroughly with the middle of Phase 1 MGS aerobraking, andthe latter with the beginning of Phase 2 MGS aerobraking,during which thermospheric density measurements weremade with the MGS accelerometer (see the Introduction).Table 2 provides a survey of the relative importance ofvarious zonal wave number components. Included aremaximum amplitudes for the perturbation northward windat 200 km altitude, and perturbation relative density at 125km, for those zonal wave numbers which give rise to zonalwave number ks = 1, 2 and 3 longitudinal structures fromSun-synchronous orbit (see Introduction). All componentsthat contribute 5% or more to the relative density variationsat 125 km are asterisked. In order of importance, theseinclude DE2, DE1 and SE1. DE1 appears to account formuch of the ks = 2 variation in density measured by theMGS accelerometer during Phase 1 of aerobraking

Figure 7. Same as Figure 5, except for S0 at Ls = 30.

23 - 8 FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS

(Figure 2). DE2 and SE1 have been suggested to account forthe large ks = 3 variation in density measured by MGSaccelerometer during Phase 2 [Keating et al., 2000; Witherset al., 2000; Bougher et al., 2001]. It also appears from Table2 that S0 may make important contributions to the ks = 2structure seen from Sun-synchronous orbit for Ls � 30. Thenext most important waves are SW1 and SW3, both of whichcan in principle be generated through nonlinear interactionbetween the migrating semidiurnal tide and the stationaryplanetary wave with s = 1. During Ls = 30, importantcontributions from SW4 can arise, and contribute to the ks= 2 density structure seen from Sun-synchronous orbit. Asnoted in connection with Figure 1, this wave can be generatedthrough interaction between the migrating semidiurnal solarradiation and the s = 2 component of topography.[18] Figures 5–7 illustrate some details of the most

important zonal wave components listed in Table 2, includ-ing height versus latitude structures of the amplitudes ofeastward and northward wind components, relative ampli-tude of density, and phase of the northward wind. Note thatsimilar figures are given by Forbes and Hagan [2000] andForbes et al. [2001b] for DE1 at Ls = 270, and the reader isreferred to these studies for related discussion of propaga-tion features of that wave component.[19] Figure 5 illustrates the wave fields for DE2 at Ls = 30.

The maximum eastward wind is about 70 m s�1 above 120km and over the equator. This is of the same order as the zonalmean and tidal winds in the low-latitude thermosphere duringequinox [Bougher et al., 1999], and therefore makes animportant contribution to the thermospheric circulation.According to classical wave theory (i.e., in the absence ofmean winds and dissipation) on a sphere [Longuet-Higgins,1968], the latitudinal shape of a Kelvin wave (i.e., the firstsymmetric eastward-propagating ‘‘gravity-type’’ mode) withs = 2 is nearly Gaussian-shaped about the equator with fullhalf-width of about 64� latitude and vertical wavelength�100 km. The eastward wind and relative density distribu-tions in Figure 5 are similar in latitudinal shape to the Kelvinwave, but distorted at middle to high southern latitudes sincethe mean zonal winds Doppler-shift the wave to zero phasespeed in this region. The northward wind distribution issimilar in shape to that expected for the Kelvin wave, i.e.,maxima near 33� latitude, zero near the equator, and 180�phase shift between hemispheres. This behavior contrastswith that for DE1 as described by Forbes and Hagan [2000],wherein the first antisymmetric propagating mode withmaximum over the equator dominated the northward windresponse in the thermosphere. In that case, the combination oftwo factors led to a significant thermospheric response for thefirst antisymmetric mode. First, the mean wind distributionfor Ls = 270 is highly asymmetric, causing indirect excitationof the first asymmetric mode via ‘‘mode coupling’’ [Lindzenand Hong, 1974]. In addition, the vertical wavelength of thatoscillation was�60 km, making it relatively unsusceptible todissipation until�150 km altitude. In the present case for Ls =30, the mean winds are much less asymmetric about theequator (eastward jets in each hemisphere), and the verticalwavelength of the second antisymmetric mode with s = 2 is�35 km, making it much more susceptible to dissipationabove 100 km.[20] Figure 6 illustrates the same wave fields for SE1 at Ls

= 270. As noted in Table 2, this wave also appears as an s =

3 structure from Sun-synchronous orbit, and can account foras much as 8% density amplitude at 125 km, both at Ls = 30and Ls = 270. The first symmetric mode (Kelvin wave) for s= �1 has a Gaussian-like latitude shape with half-width ofabout 108� latitude, maxima in the northward wind at thepoles, and infinite vertical wavelength. These characteristicsdo not conform with the structures displayed in Figure 6.The density contours in Figure 6 indicate three maxima inlatitude, one near the equator and the other two in the ±45–65� latitude belts. This latitudinal shape is similar to that ofthe second symmetric mode of SE1, which has maxima overthe equator and near ±45�, and a vertical wavelength �100km. The meridional velocity expansion function for thiswave has a node at the equator, maxima at ±28� and zeroamplitude at the poles, which bears some similarity with thelatitude shape of the northward wind amplitudes in Figure 6.The asymmetries in the wind behaviors over the poles,combined with a range of vertical wavelengths in the phaseof northward wind, indicate that secondary contributionsexist from the first symmetric and antisymmetric modeswith � infinite vertical wavelengths, and the second anti-symmetric mode with lz � 60 km.[21] Wave fields for S0 at Ls = 30 are depicted in Figure 7.

The density distribution is similar to that of the first sym-

Figure 8. Latitude versus longitude composite of relativedensity variation at 125 km, including all asterisked wavecomponents listed in Table 2 for Ls = 30. Top: LT = 0200.Bottom: LT = 1400. This depiction excludes the zonal meanand migrating tidal contributions, which are constant withlongitude.

FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS 23 - 9

metric mode, which has maxima over the equator and at thepoles and a node near 30�. (Note that for s = 0 the funda-mental mode is antisymmetric.) The meridional wind shapefor this mode has nodes at the equator and poles, and maximanear ±36�, similar to the northward wind distribution inFigure 7. The asymmetries in the displayed wave fields arisedue to the secondary presence of the first and secondantisymmetric modes, which have very long vertical wave-lengths, consistent with the phase gradients in Figure 7.[22] A different perspective on the impact of nonmigrat-

ing tides on the structure of Mars’ thermosphere can beobtained by compositing the above wave components andexamining the resultant latitude versus longitude structuresat a given height. Figure 8 provides such a depiction for therelative density variation at 125 km and Ls = 30, for thelocal times of 0200 LT and 1400 LT, i.e., local times of dataacquisition from the MGS accelerometer. These contoursare obtained by superimposing all asterisked diurnal andsemidiurnal oscillations for Ls = 30 listed in Table 2 (a totalof 8 waves), and thus at a single local time reflect thepresence of ks = 1, 2 and 3 zonal components. (Note that inthis Sun-synchronous perspective the unprocessed observa-tions would also contain mean components consisting of thezonal mean and migrating tides; these are not present inFigure 8.) The ks = 3 wave number component is dominantbetween about ±45� latitude, and changes sign between0200 and 1400 LT. These behaviors indicate dominance ofDE2 shown in Figure 5, with a much smaller contributionfrom SE1. At latitudes poleward of about �45�, thelatitudinal structure in Figure 8 reverts to an ks = 2 zonal

structure, with no change in phase between 0200 and 1400LT. Comparison with Figure 7 and Table 2 points to S0 asthe primary contributor to this aspect of the structure, withsome contribution from DE1.[23] The 1400 LT results correspond closely to conditions

under which longitude structures were inferred from densitymeasurements made by the MGS accelerometer during thebeginning of Phase 2 aerobraking when periapsis precessedfrom the Northern to Southern Hemisphere near 1400–1500LT [Keating et al., 2000; Withers et al., 2000; Bougher etal., 2001]. Figures which delineate the modeled latitudeversus longitude contours of perturbation density for the ks= 2 and ks = 3 zonal wave components seen from Sun-synchronous orbit are depicted in Figure 9 (ks = 3) andFigure 10 (ks = 2), and in addition, Figure 11 illustrates thezonal wave number ks = 1 component. (The contours inFigure 8 represent the sum of values depicted in Figures 9,10, and 11, and the contributing waves to each of thesefigures can be identified by reading across in Table 2,beginning from the column labeled ‘‘Sun-synch ks.’’ Theks = 3 density variation depicted in Figure 9 at 1400 LT isabout ±10–30% with minima occurring between about 45�and 60� latitude; the ks = 3 density variations from the MGSaccelerometer are about the same magnitude with phaseoccurring about 30� east of those depicted in Figure 9. Themodeled ks = 2 component is illustrated in Figure 10.Poleward of �30� latitude, phasing is in good agreementwith the MGS ks = 2 density variation, but amplitudes areabout a factor of 2 too small. This suggests an underestimate

Figure 9. Sun-synchronous wave number ks = 3 contribu-tion to density variation in Figure 8.

Figure 10. Sun-synchronous wave number ks = 2contribution to density variation in Figure 8.

23 - 10 FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS

of DE1 or S0 by the Ames GCM for Ls = 30. South of about�45�, a phase shift of almost 180� occurs in connectionwith the large density perturbations due to S0 at theselatitudes. The phase of the ks = 2 density variation fromthe MGS Accelerometer data does not change significantlywith latitude, suggesting that the S0 wave computed here isphased incorrectly, or overestimates the amplitude variationdue to this component. The ks = 1 density variation seen atfixed local times is also included in Figure 11, and amountsto roughly ±8–10%, confined mainly to the SouthernHemisphere for this season. An 8% ks = 1 amplitude wasquoted by Keating et al. [1998] for their MGS accelerom-eter analysis during Phase 1 aerobraking; this is consistentwith the magnitudes for Sun-synchronous ks = 1 listed inTable 2 for Ls = 270.[24] A similar depiction is provided in Figure 12 for the

northward wind component at 200 km and Ls = 270. Theidentification of particularly dominant wave numbers is notso obvious here. At 0200 LT a ks = 2 variation is obviousnear the equator, and at 1400 LT a ks = 2 variation isdominant at middle to high latitudes in the Southern(summer) hemisphere. Wind speeds are generally in therange 10–30 m s�1, which may be compared with diurnalvariations due to in situ heating of order 200 m s�1

[Bougher et al., 2000]; hence perturbations on the meri-dional flow due to upward-migrating nonmigrating tides areon the order of 5–15%. On the other hand, northward windcontours at 125 km and Ls = 30 are more typically in therange of 20–50 m s�1, whereas winds driven by in situheating are of order ±150 m s�1. Therefore, at this altitude

the nonmigrating tidal winds produce perturbations on thein situ driven circulation at the 15–30% level. Temperatureeffects (not shown) are a similar order of magnitude.

4. Conclusions

[25] Nonmigrating tides make important contributions tothe longitudinal structure of Mars’ thermosphere, and inparticular to the density field in the height regime of aero-braking operations. Tides generated in the NASAAmesMarsGCM and extended upward into the dissipative thermosphereusing the Mars GSWM account for much of the �10–40%longitudinal variations in density measured by the MGSaccelerometer. The model results indicate that the zonal wavenumber ks = 3 longitude variation seen from the Sun-synchronous orbit of MGS is most likely explicable in termsof the eastward-propagating diurnal tide with s = 2 (DE2) andsemidiurnal tide with s = 1 (SE1). The eastward-propagatingdiurnal component with s = 1 (DE1) and the semidiurnalstanding oscillation (S0) are concluded to be the maincontributors to the ks = 2 longitudinal density variation seenfrom MGS. The standing diurnal oscillation (D0) and thewestward-propagating semidiurnal component with s = 1(SW1) emerge as the most likely contributors to ks = 1. It isimportant to note that these wave components are super-imposed on the Sun-synchronous diurnal (DW1) and semi-

Figure 11. Sun-synchronous wave number ks = 1contribution to density variation in Figure 8.

Figure 12. Latitude versus longitude composite of north-ward wind variation at 200 km, including all asteriskedwave components listed in Table 2 for Ls = 270. Top: LT =0200. Bottom: LT = 1400. This depiction excludes the zonalmean and migrating tidal contributions, which are constantwith longitude.

FORBES ET AL.: NONMIGRATING TIDES IN THE THERMOSPHERE OF MARS 23 - 11

diurnal (SW2) oscillations generated below the thermosphereas well as in situ, and together combine to yield longitudedependences in the total thermospheric tidal field.[26] This broad survey of potential effects for near-

equinox and Southern Hemisphere summer solstice condi-tions points to a significant dependence of the nonmigratingtidal components on season. Further, elevated dust levelsand zonal asymmetries in dust distribution are likely tomodify the results illustrated here, as well as more realisticprescriptions of zonal mean winds, especially in the 80- to120-km-height regime. Future modeling efforts aimed atimproving agreement with middle atmosphere data, andproviding similar results for �Ls = 30� and more realisticdust distributions, are likely to significantly advance ourability to predict the aerobraking environment of Mars.

[27] Acknowledgments. J. M. Forbes received support under grantNAG5-8266 from the NASA Mars Data Analysis Program to the Universityof Colorado. The efforts of Xiaoli Zhang regarding some computationaltasks and preparation of the figures is greatly appreciated.

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�����������������������S. W. Bougher, Lunar and Planetary Laboratory, University of Arizona,

Tucson, AZ 85721, USA.A. F. C. Bridger, Meteorology Department, San Jose State University,

San Jose, CA 95192-0104, USA.J. M. Forbes, Department of Aerospace Engineering Sciences, University

of Colorado, Boulder, CO 80309-0429, USA. (Jeffrey.Forbes@colorado.edu)M. E. Hagan, High Altitude Observatory, National Center for Atmo-

spheric Research, P.O. Box 3000, Boulder, CO 80307, USA.J. L. Hollingsworth, NASA Ames Research Center, SJSUF, MS: 245-3,

Moffett Field, CA 94035-1000, USA.G. M. Keating, NASA Langley Research Center, George Washington

University, Hampton, VA 23606, USA.J. Murphy, Department of Astronomy, New Mexico State University, Las

Cruces, NM 88003, USA.

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