Lunar semidiurnal tide in the thermosphere under solar minimum conditions

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 118, 1 14, doi: /2012ja017962, 2013 Lunar semidiurnal tide in the thermosphere under solar minimum conditions Jeffrey M. Forbes, 1 Xiaoli Zhang, 1 Sean Bruinsma, 2 and Jens Oberheide 3 Received 17 May 2012; revised 27 October 2012; accepted 29 November [1] Renewed interest in lunar tidal influences on the ionosphere-thermosphere (IT) system has emerged in connection with recent studies of possible connections between stratospheric warmings and enhanced lunar tidal perturbations of the equatorial ionosphere. By virtue of its gravitational force, the Moon produces perturbations in the temperature, density, pressure, and winds throughout Earth s atmosphere. Lunar tidal winds in the dynamo region (~ km) can furthermore generate electric fields that map into the F-region and redistribute ionospheric plasma. Direct penetration (propagation) of lunar tides to F-region heights can also transport ionospheric plasma. Decades-long satellite data sets now exist that can provide a global perspective on lunar tidal oscillations, but this resource has not yet been exploited for this purpose. In this paper, we examine the global structure of the main M 2 (period = h) lunar tide through examination of temperatures measured by the Thermosphere Ionosphere Mesosphere Energetics and Dynamics SABER instrument at 110 km and densities at 360 and 480 km inferred from accelerometers on the CHAMP and Gravity Recovery and Climate Experiment satellites, respectively. Ten year mean SABER M 2 temperature amplitudes are of order 5 10 K while the corresponding density perturbations during the solar minimum period approach amplitudes of order 5% at 360 km and 10% at 480 km. The observed amplitudes are large enough to impose non-negligible day-to-day variability on the IT system. Global-Scale Wave Model simulations provide a theoretical and modeling context for interpreting these data, and moreover enable estimates of E- and F-region winds. Citation: Forbes, J. M., X. Zhang, S. Bruinsma, and J. Oberheide (2013), Lunar semidiurnal tide in the thermosphere under solar minimum conditions, J. Geophys. Res. Space Physics, 118, doi: /2012ja Introduction [2] Studies of the effects of the gravitational forcing of the Moon on the solid Earth, oceans, and atmosphere have a long history. In this paper we are interested in the effects of gravitational forcing throughout the atmosphere. Classic reviews on observational studies of atmospheric tides include those by Chapman and Lindzen [1970] for lunar tide signals in surface pressure, and Matsushita s [1967a, 1967b] reviews of lunar geomagnetic tides and tidal variations in the F-region ionosphere. Additional works [e.g., Stening et al., 1994, and references therein] examined wind measurements at altitudes (~ km) where meteor and MF radars have 1 Department of Aerospace Engineering Sciences, University of Colorado, Campus Box 429, Boulder, Colorado, Department of Terrestrial and Planetary Geodesy, Centre National d Etudes Spatiales, 18, Avenue E. Belin31401, Toulouse, France. 3 Department of Physics and Astronomy, Clemson University, 118 Kinard Laboratory, Clemson, South Carolina, USA. Corresponding author: J. M. Forbes, Department of Aerospace Engineering Sciences, UCB 429, University of Colorado, Boulder, CO USA. (forbes@colorado.edu) American Geophysical Union. All Rights Reserved /13/2012JA accumulated multiple decades of observations. Modeling work by Lindzen and Hong [1974] clarified the role of zonal mean winds and temperatures in controlling the atmosphere s response to tidal forcing, while Forbes [1982a, 1982b] emphasized the propagation of tidal perturbations into the thermosphere. Stening et al. [1997] later explored the consequences of anomalous background atmospheric conditions (e.g., stratospheric warmings) on lunar tide propagation. [3] Study of the lunar atmospheric tide is fundamentally interesting, because the lunar forcing is known reasonably well, and thus comparisons between numerical simulations and data provide important insights into the veracity of atmospheric models. Up until now, lunar tides were mainly derived from ground-based observations because long data records were required. Restriction to ground-based observations thus precluded truly global perspectives of lunar tidal effects in the atmosphere. However, there are now decadelong observations from satellites that in principle can provide such a global perspective. In many cases, these observations are also available at high altitudes where the lunar tidal variability in the more tenuous upper atmosphere might be more easily retrieved. This paper is motivated by the availability of such data, as well as the theoretical knowledge and numerical modeling capabilities that have accumulated over the past several decades. 1

2 [4] In this paper we seek to characterize and understand the seasonal-latitudinal structure of the lunar tide at several altitudes in the thermosphere. In the following section we describe data that will employ in this study from the TIMED (Thermosphere Ionosphere Mesosphere Energetics and Dynamics), CHAMP (CHAllenging Minisatellite Payload), and GRACE (Gravity Recovery and Climate Experiment) missions, and outline the methodologies that we use to isolate the lunar tidal signal. In section 3 we present the monthlymean lunar tide in the neutral temperature field between 50 latitude from TIMED-SABER (Sounding of the Atmosphere using Broadband Emission Radiometry) observations in the km height region and use a linear tidal model (The Global-Scale Wave Model or GSWM) with realistic tidal forcing and background atmospheric conditions to interpret the salient features of the observed temperature response. We furthermore employ GSWM predictions of lunar tidal perturbation densities to interpret those derived near 360 and 480 km from the CHAMP and GRACE satellites, respectively, and to assess our ability to model the lunar tidal response in the upper thermosphere. 2. Data, Model, and Method of Analysis 2.1. SABER, CHAMP, and GRACE Data Sets [5] Satellites provide exceptional spatial coverage for the study of global waves such as solar and gravitational tides. However, the slow precession in local time, which usually translates to months for a high-inclination satellite to cover a 24 h local time cycle, does introduce problems in terms of aliasing with trends in background mean conditions [Forbes et al., 1997], and temporal averaging of evolving tidal amplitudes [Forbes et al., 2008]. On the other hand, local time precession of a satellite-based measurement has an advantage in terms of distinguishing tides with very close oscillation periods, e.g., the 12.0 h solar semidiurnal tide and the h lunar semidiurnal tide, whereas can be problematic in the analyses of measurements made from the ground. This aspect of lunar tidal analysis of data from space-based platforms is discussed further in section 2.2. [6] The basic data to be employed in this study are from three satellites/instruments: TIMED/SABER, CHAMP/STAR and GRACE/SuperSTAR. The SABER Version 07 temperature measurements analyzed here cover March 2002 through March For this version of SABER temperatures, each vertical profile begins with a climatological value from NCAR s Thermosphere Ionosphere Mesosphere-General Circulation Model and a retrieval is performed from 140 km downward. The retrieved temperatures begin to become independent of the climatology at about 110 km [Russell, 2012], and this serves as a nominal upper limit for validity of the SABER temperatures (see also Mertens et al. [2001]; Remsberg et al. [2008]). Because we consider SABER data equatorward of 50 latitude where sampling is unaffected by yaw maneuvers, and relatively few data gaps exist, we have almost continuous (but asynoptic) coverage in UT and longitude. The CHAMP and GRACE total density data are introduced in Forbes et al. [2009, 2011] and the reader is referred there for relevant information on these data sets including additional references. With their higher orbit inclinations, CHAMP and GRACE measurements extend nearly pole to pole and have even longer local time precession rates than TIMED. In this study CHAMP densities are normalized to 360 km and GRACE densities are normalized to 480 km using the NRLMSISE00 model [Picone et al., 2002]. Because the model is independent of lunar time and residual densities are used, lunar tidal signals cannot be artificially introduced by this normalization process. Due to the obvious solar cycle influence at these altitudes, we use CHAMP and GRACE density data from 2007 to 2010 which is considered a solar minimum period Method of Analysis [7] The semidiurnal lunar tidal forcing (M 2 ) is the most significant of the tides that arise due to the gravitational potential between the Earth and the Moon [e.g., Chapman and Lindzen, 1970; Pugh, 1987]. Under the conditions that the Moon s orbit around the Earth is a perfect circle and in the same plane as Earth s equator, then M 2 would solely comprise the total lunar potential. But since the Moon s orbit has a small eccentricity and an angle with respect to Earth s equatorial plane, M 2 is not the only lunar-related periodicity. The gravitational potential between the Earth and the Sun represents a similar situation, and so the two pairs of gravitational potentials produce many sinusoidal forcing periodicities. The most significant ones are (in cm 2 s 2 ): (quasi-)diurnal lunar ( h) O 1 ¼ 6585:P2 1 ðθþsin (quasi-)diurnal solar ( h) P 1 ¼ 3067:P2 1 ðθþsin diurnal luni-solar ( h) sl r sl R ss r ss R t þ f t þ f K 1 ¼þ9268:P2 1 ðθþsin ½ s 0t þ fš large lunar elliptic semidiurnal ( h) N 2 ¼ 1518:P2 2 ðθþcos 2sL r m t þ 2f semidiurnal lunar ( h) M 2 ¼ 7933:P2 2 ðθþcos 2 sl r t þ f semidiurnal solar (12.00 h) S 2 ¼ 3700:P2 2 ðθþcos 2 ss r t þ f semidiurnal luni-solar ( h) K 2 ¼ 1005:P2 2 ðθþcos ½ 2 ð s 0t þ fþš where s 0 =2p/(sidereal day), s L R ¼ 2p= ð sidereal lunar month Þ, s S R ¼ 2p= ð sidereal yearþ. The angular speed of the Earth s self-rotation with respect to the Moon is s L r ¼ s 0 s L R,and with respect to the Sun is s S r ¼ s 0 s S R. In the Sun-Earth- Moon system, the Moon s elliptic angular speed is m =2p/ (anomalistic month). The associated Legendre Polynomials are P2 1ðÞ¼ 1:5sin2θ, θ whichpeaksat45 latitudes, and P2 2ðÞ¼ θ 3sin2 θ, which peaks at the equator with θ here being colatitude. Here t is UT and f is longitude. These were calculated by Siebert [1961] based on the work of Doodson 2

3 perspective is from satellites. Let the subscript e denote from the Earth and subscript s denote from the satellite perspective. Subtracting these two cases applied to s L = s S C, onegets s L s ¼ ss s þ sl e ss e (1) Figure 1. This figure illustrates the relation between solar local time (t), lunar local time t, and the lunar phase angle n for a point P on the Earth. The circle represents the plane of the Earth s equator with the center being the Earth s center. [1922] and Bartels [1957], and revisited by Chapman and Lindzen [1970]. [8] For investigations of gravitational tides, methods and techniques are particularly important, since extracting the M 2 lunar tide from surface observations historically suffered from aliasing with the solar semidiurnal tide. That is, if significant local time (orother)variationsoccurwithin the time interval that is being fitted for the lunar tide, then these variations can alias into the lunar tide determination, possibly rendering it invalid. This is compounded by the fact that the lunar tide is typically so much smaller than solar tides. From the ground, one cannot separate a 12.0 h oscillation from a 12.4 h oscillation without sufficiently abundant and frequent sampling, and data accuracy. However, it is well-known that from the perspective of a satellite platform the wave periods associated with the lunar and solar semidiurnal tides can be widely different (e.g., Ray and Luthcke [2006]; their equation (1) and Table 1 and example for GRACE). In the following, we describe how we use this advantage along with abundant sampling and detrending to characterize a relatively small semidiurnal tide in the upper atmosphere. [9] Figure 1 illustrates the relation between solar local time t, lunar local time t for a point P on the Earth (observed by a ground-based station or by a satellite) and the lunar phase angle n, after Sugiura and Fanselau [1966] and Chapman and Lindzen [1970]; that is, t = t n. Becausen, the lunar phase, is almost a linear function of universal time, the lunar phase speed is a constant for each sinusoidal mode; let it be denoted by C. Thens L = s S C,wheres L and s S denote the Earth s self-rotation frequency (corresponding to the s r s in the last paragraph when gravitational forcing is discussed from the ground-based perspective) with respect to the Moon (L) or to the Sun(S). From the perspective of a groundbased station, s S = 1 cycle/day. For M 2, C = 1 cycle/29.53 days, so 1/s L = days or h, which is confirmed as a lunar day or two times the M 2 period. Another practical where s L e is one of the gravitational modes diurnal frequencies listed in the prior subsection such as s O1 ¼ s L r sl R, s P1 ¼ s S r ss R, s K 1 ¼ s K2 =2 ¼ s 0, s N2 =2 ¼ s L r m=2 or s M2 =2 ¼ s L r ; ss s is the satellite precession rate and ss e is simply 1 cycle/day. As long as s L e 6¼ ss e = 1 cycle/day, we can get the s L s for a gravitational tide of interest without aliasing with s S s. That is, the difference between the periods of two gravitational modes from the ground-based point of view can be magnified if the two modes are viewed from a satellite. For example, a M 2 lunar day (1=s L s ) from either the ascending or descending leg of TIMED, CHAMP or GRACE is 23.7, 26.6, and 27.1 days, respectively, which are well apart from the corresponding solar day lengths (1=s S s ) of 120, 261, and 324 days, respectively, although s L e and ss e are very close (1.035 day vs day period as aforementioned) from the perspective of a ground-based station on the Earth. The calculation is based on equation (1) above where in most cases s S s has a negative value, which reflects a satellite precessing in the opposite direction that the Earth rotates with respect to the Sun. For instance for the TIMED satellite, 1=s L s ¼ 1= ð 1=120: þ 1=1:035 1Þ ¼ 23:7 days; for CHAMP, 1=s L s ¼ 1= ð 1=261: þ 1=1:035 1Þ ¼ 26:6 days; and for GRACE, 1=s L s ¼ 1= ð 1=324: þ 1=1:035 1 Þ ¼ 27:1 days. The negative sign in any of these results indicates that the lunar hour decreases as UT goes forward. The M 2 semidiurnal period is thus 11.86, 13.28, or from either the ascending or descending leg of the TIMED, CHAMP or GRACE satellite orbit, respectively, since the semidiurnal period is half the diurnal period of the same mode. As another example, an N 2 lunar day is (based on the N 2 frequency s L r m=2) solar days from a ground-based point of view, so its semidiurnal period is 1/( 1/ / )/2 = 8.23 days, 1/( 1/ / )/2 = 8.93 days, or 1/( 1/ / )/2 = 9.05 days, respectively, from the perspective of the TIMED, CHAMP, or GRACE satellite. Note, although a M 2 lunar day or a N 2 lunar day is mentioned in the above calculation, the diurnal mode of neither exists. As a brief summary, by taking advantage of slow satellite precession rates (including zero), the common aliasing problem between gravitational tides and solar thermal tides can be avoided to a significant degree. [10] We restrict our attention here to the largest gravitational tide excited in the atmosphere, the migrating (longitudeindependent) M 2 component. For the climatological study at hand, we use a binning and averaging method that yields composite lunar half-day variations of temperature or density from which the semidiurnal component can be derived. It is analogous to the method employed by Forbes et al. [2008] to determine solar tides from TIMED-SABER temperature measurements. In that study 60 days of data were required to form hourly-mean values during a composite 24 h solar day (including both ascending and descending parts of the orbit), thus enabling extraction of both diurnal and semidiurnal tides. The tidal analysis was performed on residuals from 60 day running mean temperatures in order to minimize aliasing due 3

4 to variations in the background (zonal-mean) temperature (see also Forbes et al. [1997]). Similarly, the lunar tide is superimposed on a background that evolves due to changes in the zonal mean and local solar time. In the present application each lunar half-day encompasses 11.85, 13.26, and days worth of measurements from the TIMED, CHAMP and GRACE satellites, respectively, during which the local time changes by about 2.37, 1.22, and 1.00 h. Our procedure for the lunar tide analysis of SABER data similarly begins with forming residuals, except in this case it is from the 12 day running-mean background temperature centered on that day, at any given height and latitude. Detrending the data this way helps to remove variations that could otherwise project onto the semidiurnal tide within the fitting interval. At each height and latitude we then bin 15 orbits per day of bi-monthly SABER temperature residuals over 9 years into twelve 1 h lunar local time bins centered at the 15th of each month. Least-squares sinusoidal fits are then performed to extract the semidiurnal lunar tidal signal for that month (see sections 3.1 and 3.2 for examples). Note that over 9 years we effectively average about = 675 data points (minus some missing points due to data gaps) in each 1 h bin, which greatly reduces random errors and averages out other variations not ordered in lunar time. Our attempts to extract the N 2 lunar component yielded amplitudes much smaller than M 2, so aliasing contributions from N 2 are considered to be small. The CHAMP and GRACE data were analyzed using the same method, except that density residuals from 14 day means were binned over the 4 year period. [11] Apart from aliasing with the solar tide which is addressed above, it is not unreasonable to assume that the lunar atmospheric tide might have a local time dependence, due for instance to day-night differences in propagation conditions. With the SABER binning described above, the five 12 day segments comprising each bimonthly period average out measurements from five evenly spaced different solar local times for each of the 12 lunar local hours. With CHAMP or GRACE, four 14 day segments for a given bimonthly period in each year from average out four evenly spaced different solar local times. In both cases, the four to five evenly spaced local times cover roughly a solar local time cycle. Therefore, the climatological results to be presented below represent local-time averaged lunar tides Global Scale Wave Model [12] The Global-Scale Wave Model [Hagan et al., 1995, 1999; Hagan, 1996; Hagan and Forbes, 2002, 2003] is the model adopted for the present study. The GSWM solves the linearized tidal equations; given the frequency, zonal wave number and excitation of a particular oscillation, and given a specification of the zonally-averaged atmospheric state, the height versus latitude distribution of the atmospheric response is calculated. The linear approximation is not considered to be a shortcoming of any significance in calculating the wave response to any given forcing. However, the linear approximation precludes excitation of some tidal oscillations by wave-wave interactions. The model includes in some form or another all other processes of known importance to the calculation of the global atmospheric tidal response: surface friction, mean winds and meridional gradients in scalar atmospheric parameters, radiative cooling, eddy and molecular diffusion, Rayleigh friction and ion drag. All parameterizations and properties of the background atmosphere are described for GSWM-02 in Hagan and Forbes [2002]. In GSWM-09 [Zhang et al., 2010], solar thermal forcings are updated with ISCCP (International Satellite Cloud Climatology Project) radiative heating and TRMM (Tropical Rainfall Measuring Mission) latent heating. New specifications of the mean zonal wind field derived from SABER geopotential data and the corresponding mean temperatures are also features of GSWM-09. In this study, we utilize GSWM- 09 and obtain the GSWM lunar tidal results by introducing the M 2 migrating lunar forcing in a manner identical to that described in Forbes [1982a, 1982b] and turning off the solar thermal forcing. All simulations shown here correspond to solar minimum conditions (F10.7 = 70). [13] A much less effective source of lunar tidal forcing on the atmosphere occurs as the result of vertical movements of the solid Earth and oceans, serving as a dynamic boundary condition for the tidal equations [Vial and Forbes, 1994]. While the lunar gravitational potential is known with reasonable precision, there are assumptions and uncertainties associated with specification of Earth and ocean tide effects on the atmosphere. The main motivation for including these effects in the GSWM would be to examine the longitudedependent lunar tides in the atmosphere, but since the focus of the present work is on the longitude-independent M 2 tide, Earth and ocean tidal forcing is not included in the GSWM simulations presented here. There is an M 2 component that arises in connection with Earth and ocean tidal forcing, but its influence on the atmospheric response is not completely straightforward to estimate. Based on discussion of this issue in Vial and Forbes [1994], GSWM lunar tidal responses in following sections could underestimate the total M 2 response by up to 30%, and may reflect some small phase effects as well. This level of uncertainty does not appreciably impact our use of the GSWM to diagnose the seasonal-latitudinal aspects of tidal responses presented in the following, as these arise solely as a result of changes in the background atmosphere. 3. Results 3.1. SABER and GSWM Lunar Tide Temperatures in the Lower Thermosphere [14] Figure 2 illustrates hourly and longitude means of the temperature residuals defined in section 2.1, for the bimonthly period centered on 15 January, and plotted vs. lunar local time between 0 and 12 h. Also shown are 12 h sinusoidal least squares fits to these data. We use 12 h fitting intervals (asopposed to a full lunar day) in order to double the data available for averaging. Each fitting interval is 12 days in length, so every bimonthly average includes five such intervals per year of observations, for a total of 45 intervals over the TIMED-SABER data set. In addition, the atmosphere is sampled on both ascending and descending portions of the orbit at about 15 longitudes per day. This accounts for the acceptably small (~1 K) standard deviations illustrated in Figure 2, in comparison with the derived wave amplitudes. A prominent feature of the results in Figure 2 is the asymmetric phase relationship between the lunar tide north and south of the 4

5 Figure 2. Zonal-mean lunar semidiurnal M 2 fits to multi-year binned SABER residual temperatures for January at altitudes of 90 km (left) and 110 km (right) and selected latitudes. The vertical bars are the standard deviations of the data within each hourly bin. equator. This may seem surprising given that the M 2 gravitational forcing is symmetric about the equator, but the reason is well known [Lindzen and Hong, 1974; Forbes, 1982] and is explained below. [15] The seasonal-latitudinal structures of lunar tidal temperature amplitudes from SABER observations and GSWM simulations that complement those in Figure 2 at 110 km are illustrated in Figures 3a and 3b. There are some common features between the observations and the model, namely the occurrence of minima around April and October, and in the vicinity of the equator during the November March and May September periods. Furthermore, during these nonequinox periods there is generally one maximum at low to middle latitudes on each side of the equator. During Northern Hemisphere summer SABER maxima of about 5 K (2 K) occur near + 20 ( 25 ) latitude as compared with the GSWM maxima, which are 12 K (7 K) and occur near +25 ( 10 ). During Northern Hemisphere winter, SABER maxima of 4 6K occur at 20 latitude whereas the GSWM maxima are of order 7 8K near 30 latitude and 4Knear+40 latitude. [16] One notes that the scatter in the data as well as the standard deviations of individual data points in Figure 2 vary with latitude and height. There is also some variability from 5

6 Figure 3. M 2 temperature amplitudes and phases from multi-year-mean SABER data and GSWM simulations. (a) SABER amplitudes versus latitude and month at 110 km. (b) GSWM amplitudes versus latitude and month at 110 km. Panels (c)-(f) are height vs. latitude depictions of amplitude for SABER and GSWM for January and July, respectively, and panels (g)-(j) are the corresponding phases. Note the different amplitude scales for panels (a) and (b) and for panels (e) and (f). month to month (not shown). If we take the displayed standard deviations (s) as a measure of uncertainty, then the corresponding 1s uncertainties in the amplitude and phase of the sinusoidal fit can be calculated using standard formulas. Doing this, we find typical 1s values at 110 km of K in amplitude and 0.3 h in phase where amplitudes exceed 0.8 K in Figure 3a, and 1s phase uncertainties up to 1.5 h at the higher latitudes where amplitudes are smaller. Amplitude (phase) uncertainties at 100, 90, and 80 km, respectively, are of order 0.25 K (0.3 h), K ( h), and ( h), but these must be weighed against the progressively smaller amplitudes at lower altitudes. For a normal distribution, 1s bounds 68% of the points about the mean value. Therefore, the data-model comparisons provided above must be tempered by these uncertainties which naturally arise 6

7 Figure 4. Top panel (a) shows the first four M 2 Hough functions normalized to a maximum value of unity: (2,2) (solid), (2,3) (dotted), (2,4) (dashed), and (2,5) (dashed-dot). Panels (b)-(d) show the vertical profiles of amplitudes from Hough function decomposition of the GSWM simulations for January, July and October, and panels (e)-(g) are the corresponding phases. in the extraction of a relatively small signal from an extended data set with other geophysical variability. There are also uncertainties in the model specification, which are discussed later on in this paper. [17] Comparing SABER and GSWM seasonal-latitudinal amplitude structures at a single altitude provides only a narrow perspective on the lunar tidal responses reflected in both the observations and model. Consequently, we compare the height vs. latitude structures of SABER and GSWM temperature amplitudes for January in Figures 3c and 3d and for July in Figures 3e and 3f. The corresponding phases are provided in Figures 3g 3j. The January SABER and GSWM amplitudes (Figures 3c and 3d) depict the exponential growth with height that is expected for verticallypropagating waves in the atmosphere. This growth and the overall amplitudes for GSWM are quite similar to those of SABER in the Southern Hemisphere. The Phase structures (Figures 3g and 3h) are also similar in the Southern Hemisphere, except that the contours are more compressed for SABER. Note that phases progress to later times as one moves downward in altitude ( downward phase progression ), consistent with upward group velocity and a 7

8 Figure 5. Zonal-mean background zonal winds (top three panels) and temperature (bottom three panels) used by GSWM-09 for January (left), July (middle) and October (right). wave source at lower altitudes according to atmospheric wave theory. Thus, due to the different phase gradients with height, while phase agreement is good near 80 km, by 110 km there is a 3 h discrepancy in phase between GSWM and SABER in the Southern Hemisphere. In the Northern Hemisphere, exponential growth is clearly slower during January for GSWM as compared with SABER, and in fact the SABER tidal structures are quite similar between hemispheres except between km where the amplitudes are small in any case. Focusing on the Northern Hemisphere phases during January, we see that at +20 latitude that GSWM undergoes a phase shift of about 5 h, from 6.0 h to about 1.0 h from 110 to 80 km. The SABER phases at +20 increase from about 1.0 h to + 6.0(or 6.0)h from 110 to 95 km, and continue to increase to 3.0 h at 80 km, for a total phase shift of 8 h for SABER. Again, the SABER data suggest presence of a shorter vertical wavelength wave than the GSWM. Note that there is a phase shift of about 6.0 h between hemispheres at middle latitudes for SABER (cf. Figure 2), whereas this phase difference is of order 3.0 h for GSWM. In July, the SABER amplitudes (Figure 3e) achieve much smaller maximum amplitudes (3 5K) as compared to GSWM (7 12 K, Figure 3f) in the km regime, and peak at a lower altitude. However, the relative amplitudes between Northern and Southern Hemispheres and the placement of the maxima in each hemisphere are about the same for SABER and GSWM, although SABER amplitudes peak in the km altitude regime compared to about 120 km for GSWM. If one views the SABER (Figure 3i) and GSWM (Figure 3j) phases diagonally from 110 km altitude and 50 latitude to 80 km and +50 latitude, one can see a slight compression of the phase contours in SABER relative to GSWM, but apart from that the phase structures share a lot of similarity. [18] The salient seasonal-latitudinal and vertical structure features noted above, and the relationships between them, can be explained in the context of linear tidal theory (see Chapman and Lindzen [1970] for details). First, consider the orthogonal Hough modes (Y n,s (θ)) of classical tidal theory, solutions of Laplace s Tidal Equation, where θ is latitude or colatitude, n = 2 denotes a tidal frequency that is twice Earth s rotation frequency, and s is a latitudinal index. A few Hough modes [usually denoted (n,s)] are depicted in Figure 4 for the lunar semidiurnal tide [Flattery, 1967]. These include the first and second symmetric modes (2,2) and (2,4), and the first two antisymmetric modes (2,3) and (2,5). Note that the (2,2) mode is very similar in latitudinal shape to that of the lunar M 2 geopotential. In practice the semidiurnal temperature field dt 2 (θ) at any given altitude can usually be closely approximated by the sum of these four functions: dt 2 ðθþ ¼ AðθÞcos½2Ωt fðθþš ¼ X4 s¼1 ¼ X4 s¼1 Y 2;s ðθþ a 2;s cos2ωt þ b 2;s sin2ωt A 2;s Y 2;s ðθþcos 2Ωt f 2;s where the a 2,s, b 2,s, A 2,s, f 2,s are single constants for each latitude-dependent Y n,s (θ). In fact, the structures in Figures 3a and 3b can actually be well approximated by only the sum of (2,2) and (2,3) modes (not shown). This makes it simple to understand the origins of the latitudinal structures in Figure 3, and the similarities and differences between them. (2) 8

9 Figure 6. GSWM M 2 tidal amplitudes of zonal wind (U, top left), meridional wind (V, top right), relative density (Δ r/r, bottom left) in percent, and temperature (T, bottom right) in January. For instance, note that the (2,3) mode maximizes at about 25 latitude with one hemisphere being out of phase with the other, and at these same latitudes the (2,2) mode is about 0.70 of its equatorial maximum value and is in phase between hemispheres. Furthermore, each mode is multiplied by a complex constant that determines its amplitude and phase. One can easily imagine that, depending on the relative amplitudes and phases of these complex constants, that a variety of latitudinal structures are possible, each one with maxima in the vicinity of 25 latitude, but covering a range of relative amplitudes and phases between hemispheres. For instance, the near-equal maxima in SABER amplitudes at 110 km in December January and the more asymmetric amplitude structures in June August (see Figure 3a), and the asymmetric structures for GSWM during both solstices in Figure 3b, can all be realized by combining the (2,2) and (2,3) modes in this way. (Note however that the existence of a Southern Hemisphere maximum closer to 10 latitude for GSWM during July is due to presence of the (2,4) mode with non-negligible amplitudes.) [19] Understanding the vertical structures and SABER- GSWM differences in Figure 3 requires some appreciation for the effects of the background atmosphere through which the lunar tide propagates between the lower atmosphere and the thermosphere, which in turn translates to the relative magnitudes and phases of the (2,2) and (2,3) modes. The GSWM represents a good vehicle for demonstrating these concepts. Consider the zonal-mean zonal winds and temperatures utilized by the GSWM in this study for January, July, and October in Figure 5, and the corresponding Hough mode decompositions of the lunar tidal temperature field provided in Figure 4. For January, the middle atmosphere winds between 20 and 90 km are characterized by strong asymmetry with jet maxima of order +45 ms 1 and 60 ms 1 in the Northern and Southern Hemispheres, respectively. In Figure 4, the Hough mode decomposition contains a strong (2,3) antisymmetric component in addition to the symmetric (2,2) and (2,4) modes. It is this (2,3) component that accounts for the significant asymmetry during November-March in the GSWM response at 110 km in Figure 3. The (2,3) mode arises as a result of the latitudinal distortion that the asymmetric mean winds produce in the tidal response; in the context of linear tidal theory this distortion is accommodated by exciting the (2,3) mode as a result of mode coupling [Lindzen and Hong, 1974]. Once this tidal mode is generated, it propagates freely into the thermosphere as an independent oscillation, carrying the signature of the middle atmosphere zonal jets to much higher altitudes. During July the (2,3) mode is even larger and the (2,2) mode is reduced, consistent with the even greater asymmetry in zonal mean winds depicted in Figure 5. This is reflected in the more distinct two-peaked structure during May July at 110 km, with low amplitudes at the equator. During October the zonal mean winds are much less asymmetric, with eastward winds jets in both hemispheres. In this case the (2,2) mode is dominant and little asymmetry is seen in the total tidal temperatures around the equinoxes (Figures 3a and 3b). Also, the overall amplitudes are smaller during equinox periods, consistent 9

10 Figure 7. Zonal-mean M 2 fits to multi-year binned CHAMP residual data for July at 360 km and selcted latitudes. The vertical bars are the standard deviations of the data within each hourly bin. with the seasonal-latitudinal patterns depicted in Figures 3a and 3b. [20] The latitude structures and vertical structures displayed in Figure 3 are in fact coupled to each other. This is because the (2,2), (2,3), and (2,4) modes each possess their own characteristic vertical wavelengths, which are roughly of order km, km, and km, respectively. Because these are dependent on both temperature and vertical temperature gradient, their local values can vary considerably in the mesosphere-lower thermosphere height region between km, as depicted in Figures 3e 3g. In particular, the theoretical vertical wavenumber for the (2,2) mode becomes imaginary between about km where the mean temperature decreases with height. This retards growth of the (2,2) mode with height, thus facilitating dominance of the (2,3) mode. It is clear that since the relative amplitudes and phases of the tidal modes vary with height, the aggregate vertical structures of amplitude and phase will vary with latitude, or equivalently, the horizontal structures will vary with height. As noted previously in connection with Figure 5, although the January and July zonal mean winds and temperatures are not seasonally symmetric, many of the salient features are, although they yield much different combinations of tidal modes (cf. Figures 3b an 3c). One can therefore appreciate the difficulty of obtaining more exact agreement between theory and observation in Figure 3, despite that fact that the lunar geopotential forcing is so well known. [21] There are a few reasons why better agreement may not be obtainable between the GSWM and SABER depictions of the lunar atmospheric tide. First, the SABER results correspond to a 10 year mean, and phase variability between half lunar day intervals within each bi-monthly period or from year to year can produce interference effects that tend to diminish the amplitude of the observed tide, and which are not necessarily uniform with respect to latitude or month. These phase differences can arise due to the lunar tide s sensitivity to the middle atmosphere zonal mean winds, a feature which we have already noted exists in the GSWM simulations, and which we presume carries over to the actual atmosphere. Second, the background temperature field between 20 and 100 km in the GSWM is constructed from 10

11 Figure 8. a 10 year mean climatology of SABER temperatures, and the winds are calculated from the same temperatures using the thermal gradient wind relationship. Inherent in this type of averaging is a smoothing of vertical and horizontal and wind gradients that are known to affect tidal propagation [cf. Mclandress, 2002]. Shortcomings in the temperature and wind specification are in turn manifested in the mixture of tidal modes, which as we have seen controls the aggregate horizontal and vertical structures of amplitudes and phases. [22] The horizontal and vertical wind components and perturbation density are additional outputs from the GSWM, which extends from the surface to 400 km altitude. Figure 6 illustrates the lunar tidal zonal and meridional wind amplitudes, percent relative density perturbation and temperature amplitudes, as a function of height and latitude for January. For all parameters, the maxima occur between about 110 and 120 km and on both sides of the equator, and within 30 for density and temperature and near latitude for the horizontal winds. The zonal wind amplitudes are of order ms 1 within the km dynamo region, which compares with ms 1 for the upward-propagating solar semidiurnal tide computed by the GSWM under the same conditions. The lunar tide is thus capable of producing dynamo electric fields that Similar to Figure 7 except for GRACE data at 480 km. 11 drive observable changes in the F-region ionosphere. The density perturbations are only 1 2.5% at 200 km, however, which suggests that extraction of the lunar tidal signal from, e.g., CHAMP and GRACE accelerometer-based measurements, might be difficult. This concern turns out to be unwarranted, however, as we show in the following section CHAMP, GRACE, and GSWM Lunar Tide Densities in the Upper Thermosphere [23] As noted previously, the CHAMP and GRACE data at 360 and 480 km, respectively, are analyzed using the same method as for the SABER data, except residuals from 14 day running means are used. Since vertically-propagating tides penetrate into the upper thermosphere with larger amplitudes during solar minimum vs. solar maximum (see Oberheide et al. [2011], for some recent examples), we consider only the solar minimum period for our analysis here. [24] Examples of fits to the CHAMP and GRACE density residuals are depicted in Figures 7 and 8, respectively for the July results. Note that the amplitudes are typically 2 5% for CHAMP and 5 10% for GRACE, and are sufficiently large compared to scatter and standard deviations that some confidence is warranted, especially in the case of GRACE.

12 Figure 9. GRACE (top panels) and CHAMP (middle panels) M 2 amplitudes (left panels) and phases (right panels) of relative density compared with GSWM (bottom panels). The GRACE data correspond to an altitude of 480 km, and the CHAMP and GSWM results correspond to 360 km. Note also that the phases differ by about 3 h (or 9 h) between the two altitudes. It seems likely that GRACE leading CHAMP by 3 h is the correct solution, as this would be consistent with downward phase progression for an upwardpropagating wave. Although the phases of GSWM lunar tidal temperatures and horizontal winds at midlatitudes are roughly constant with height (not shown) above 300 km due to the effects of molecular diffusion, phases of perturbation densities are decreasing at the rate of about 1.2 h/100 km at these altitudes, about half the rate indicated by the CHAMP and GRACE observations. [25] Figure 9 provides latitude versus month depictions of the GRACE (480 km), CHAMP (360 km) and GSWM (360 km) M 2 perturbation density amplitudes and phases. We focus on the CHAMP-GSWM comparisons first. The CHAMP and GSWM results are similar in terms of the single maximum that occurs just south of the equator between November and March, and the double maxima that occur between May and September. The double maxima occur around 10 to 40 and + 30 to + 50 for GSWM and around 0 to 50 and + 40 to + 60, respectively, for CHAMP. The GSWM maxima are generally in the % range, whereas the CHAMP maxima are about %. However, while the phase of the CHAMP density perturbation during July decreases from h in the Southern Hemisphere to h in the Northern Hemisphere, the corresponding GSWM phase changes are more dramatic, from about h to h from Southern Hemisphere to Northern Hemisphere. If one considers the single CHAMP and GSWM maxima during the November to March period, there is a 1 2 h net difference in phase. These results are consistent with the Hough decomposition in Figure 4, which demonstrates a significant predominance of the (2,3) mode over the (2,2) mode during July, whereas these waves are more nearly equal in magnitude during January. These results tell us that the (2,3) mode generation and (2,2) mode suppression during Northern Hemisphere summer months may be too great in the GSWM, which in turn informs us that there may be shortcomings in the specification of middle atmosphere winds during this period. 12

13 It is also possible that the (2,3) mode may not be sufficiently dissipated within the GSWM. [26] The GRACE density perturbation maxima in the top panels of Figure 9 are greater than CHAMP, of order 10%, which is perhaps not surprising given the more tenuous atmosphere at 480 km. The South Hemisphere maximum between January and March in GRACE is shifted about 1 month earlier and southward compared with CHAMP. During June July a Northern Hemisphere maximum is the major feature in the GRACE results, but at CHAMP altitudes the major maximum is between 0-50 S latitude with a secondary maximum between N latitude. Similar to CHAMP, the GRACE phases are nearly constant with latitude, and as noted previously, lead CHAMP by about 3 h during July, consistent with an upward-propagating wave. The CHAMP- GRACE phase difference is less ( h) during January February, but becomes large (~7 h) in May. Similar to the interpretation of model-data results in Figure 3, uncertainties in the lunar tide fits (cf. Figures 7 and 8) need to be accounted for when drawing conclusions from, e.g., Figure 9. For CHAMP, we find that 1s values for amplitude are largest ( %) during August December and generally less than 1.0% between January July; 1s values for phases during these same periods are of order h and less than 0.6 h, respectively. For GRACE, 1s values for amplitude tend to maximize around February April and September December, but the total range ( %) over all months is not large. A similar pattern occurs for the GRACE phases, with maximum 1s values of 0.7 h between 0 40 latitude during the above months, but values generally of order h otherwise. Given these uncertainties, it appears that phase gradients are more reliable between January and July than during the rest of the year, and thus the large CHAMP-GRACE phase difference during May remains inexplicable. Visual inspection of the individual fits did not reveal any anomalies. The overall trend, though, is for the lunar tidal variations derived from GRACE measurementstoleadthosederivedfromchamp,consistentwith downward phase progression for an upward-propagating wave, although the phase gradient predicted by the GSWM between 300 and 400 km altitude (the upper boundary of the GSWM) is only about half the 3 h phase differences between 360 and 480 km indicated in Figure 9. We do suspect, though, that there might be some unanticipated effects reflected in these results that are attributable to the low solar minimum conditions within which these data were collected. For instance, Bruinsma and Forbes [2010] noted significant differences between CHAMP and GRACE density behaviors that they attributed to greater than usual winter Helium abundances at GRACE altitudes. Note that the minimum M 2 amplitudes depicted in Figure 9 for GRACE occur in the winter hemisphere. It is possible that He-O mutual diffusion may tend to damp tides at these altitudes similar to the way that O-N 2 mutual diffusion tends to damp vertically-propagating tides in the lower thermosphere [Forbes and Hagan, 1980]. The fact that the GRACE and CHAMP satellites might have been above and below the exobase, respectively, under these solar minimum conditions could also be a factor. 4. Summary and Conclusions [27] In this paper, we investigate the month-to-month climatology of the response of Earth s atmosphere to lunar gravitational (M 2 ) forcing using satellite-based observations and a steady state global wave model (the GSWM). The model forcing does not vary with time and is symmetric about the equator. Any deviations of the response from these characteristics are thus due to specification of background zonal-mean zonal winds in the model, and to a much lesser degree the background temperatures, both of which have some basis in observational data. [28] The GSWM is able to explain the salient characteristics of the seasonal-latitudinal distribution of lunar tidal temperature amplitudes revealed in the 9 year mean tidal climatology obtained from TIMED-SABER observations. These characteristics are in large part explicable in terms of the (2,2) and (2,3) Hough modes of linear tidal theory; the former is latitudinally symmetric and resembles the latitude shape of the M 2 forcing, while the latter arises as a result of mode coupling introduced by the zonal-mean zonal jets in the middle atmosphere. There are additional contributions due to the higher-order (2,4) and (2,5) modes, but these serve more to account for details than as drivers of the salient characteristics. Although the zonalmean winds are observationally-based, there are differences between the SABER and GSWM results that suggest that there is room for improvement in the background atmosphere specification. [29] Because we are primarily interested in the migrating (longitude-independent)m 2 lunar tide in this paper, the effects of solid Earth and ocean tides on the atmospheric lunar response [Forbes and Vial, 1994] were not considered. While these contributions, which are open to some uncertainty, are of primary interest with respect to the longitudinal variability that they might produce, it is possible that we could have underestimated the M 2 response in this paper by up to 30% by omitting them. However, this does not appreciably affect the conclusions that we draw with respect to the influences of middle atmosphere mean winds on the seasonal-latitudinal and vertical variations of the lunar tide in the thermosphere. [30] The lunar tide propagates into the thermosphere and achieves maximum amplitudes in the ionospheric E-region, and under solar minimum conditions propagates with sufficient strength to produce density perturbations at 360 and 480 km that are easily recovered from analyses of densities based on accelerometer measurements on the CHAMP and GRACE satellites. The GSWM captures many features of the seasonal-latitudinal characteristics of lunar tidal perturbations revealed by these data. However, there are features of the observed upper thermosphere density variations, such as near-constancy of phase with latitude during May September, which suggest that the (2,3) mode might be overestimated in the GSWM in comparison to (2,2) during these months. This may translate to a shortcoming in the specification of the zonal mean wind field in the middle atmosphere. At any given altitude it is important to recognize that any model discrepancies represent the integrated effect of such shortcomings introduced at much lower altitudes, compounded with the differential dissipation that the (2,3) and (2,3) modes experience as a result of their differing vertical wavelengths. Matching observed and modeled tidal amplitude and phase structures in the thermosphere thus remains a significant challenge. In addition, there are uncertainties inherent in the lunar tide 13

14 amplitudes and phases, which therefore limit our comparative conclusions to the most salient features reflected in both the data and model. [31] On the other hand, there is an alternate explanation for existence of model-data discrepancies that remains an important consideration for any interpretation of climatological lunar tidal analyses in the context of a numerical model. The observational results displayed in Figures 3 and 9 represent averages of the lunar tidal signal corresponding to many realizations of background wind conditions. Variability in background wind conditions would very likely change the phase of the (2,3) mode, which in fact owes its existence to the very presence of the background winds. The ensemble averages in Figures 3 and 9 could very well suppress the (2,3) mode to some degree due to phase cancellations effects, while the GSWM simulation adopted for comparison holds for one single climatological realization of the background wind field. The same applies, in fact, for any comparison between models and climatological mean data sets, or mean tidal fields that are constructed using data from slowly-precessing satellites [e.g., Forbes et al., 2008]. As our models grow in sophistication, and as long data sets become more plentiful, these considerations need to be taken into account. [32] Acknowledgments. The involvement of J. Forbes and X. Zhang in this work was supported by Grants NNX10AE62G from NASA and ATM from the NSF to the University of Colorado. J. Oberheide was supported by Grants NNXAJ13G from NASA and AGS from the NSF. Computational support was provided by the National Center for Atmospheric Research. References Bartels, J. (1957) Gezeintenkräfte, Handbuch der Physics Vol. XLVIII, Springer, Berlin, pp Bruinsma, S. L., and J. M. Forbes (2010), Anomalous behavior of the thermosphere during solar minimum observed by CHAMP and GRACE, J. Geophys. Res., 115, A11323, doi: /2010ja Chapman, S., and R. S. Lindzen (1970) Atmospheric Tides: Thermal and Gravitational, 200 pp., Gordon and Breach, New York. Doodson, A. T. (1922), The harmonic development of the tide generating potential, Proc. R. Soc. Lond. A 100, pp Flattery, T. W. (1967), Hough functions, Tech. Rept. 21, Dept. of Geophys. Sci., Univ. of Chicago, Chicago, Ill. Forbes, J. M. (1982a), Atmospheric Tides. I. Model description and results for the solar diurnal component, J. Geophys. Res. 87, Forbes, J. M. (1982b), Atmospheric Tides. II. The solar and lunar semidiurnal components, J. Geophys. Res. 87, Forbes, J. M., and M. E. Hagan (1980), Tidal dynamics and composition variations in the thermosphere, J. Geophys. Res., 85, , doi: /ja085ia07p Forbes, J. M., M. Kilpatrick, D. Fritts, A. H. Manson, and R. A Vincent (1997), Zonal mean and tidal dynamics from space: an empirical examination of aliasing and sampling issues, Ann. Geophys., 15, Forbes, J. M., X. Zhang, S. Palo, J. Russell, C. J. Mertens, and M. 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(2008), Assessment of the quality of the Version 1.07 temperature versus-pressure profiles of the middle atmosphere TIMED/SABER, J. Geophys. Res., 113, D17101, doi: /2008jd Russell, J. (2012), SABER Principal Investigator, Private Communication. Siebert, M. (1961), Atmospheric tides. Advances in Geophysics, Vol. 7. New York: Academic. Press, Stening, R. J., A. H. Manson, C. E. Meek, and R. A. Vincent (1994), Lunar tidal winds at Adelaide and Saskatoon at 80 to 100 km heights, , J. Geophys. Res., 99, 13,273 13,280. Stening, R. J., J. M. Forbes, M. E. Hagan, and A. D. Richmond (1997), Experiments with a lunar atmospheric tidal model, J. Geophys. Res., 102, 13,465 13,471. Sugiura, M., and Fanselau, G. (1966), Lunar phase numbers n and n 0 for years 1850 to Goddard Space Flight Center Report X Vial, F., and J. M. Forbes (1994), Monthly simulations of the lunar semidiurnal tide, J. Atmos. Terr. Phys., 56, Zhang, X., J. M. Forbes, and M. E. 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