A decade-long climatology of terdiurnal tides using TIMED/SABER observations

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1 JOURNAL OF GEOPHYSICAL RESEARCH: SPACE PHYSICS, VOL. 8,, doi:./jgra.7, A decade-long climatology of terdiurnal using TIMED/SABER observations Y. Moudden and J. M. Forbes Received October ; revised 8 April ; accepted April ; published July. [] In this paper, we globally characterize the solar terdiurnal tide in the 8 km region of Earth s atmosphere through analysis of years of temperature measurements made by the Sounding of the Atmosphere using Broadband Emission Radiometry instrument on the Thermosphere-Ionosphere-Mesosphere Energetics and Dynamics spacecraft. The Sun-synchronous ( migrating ) component (TW), which is longitude-independent and achieves maximum amplitudes of order of K ( K) at 9 km ( km), not too different than the 7 K amplitudes that are typical of the migrating diurnal and semidiurnal in this region. Significant longitude variability ( %) in terdiurnal temperature amplitudes also exists, which is decomposed into zonal wave number components. The largest of these (TE, TW, and TW) reveal distinct seasonal-latitudinal and height versus patterns and interannual consistency. In addition, it is demonstrated that these particular components vary in ways that suggest that they originate from nonlinear interactions between diurnal and semidiurnal, specifically between DE and SW for TE, between DW and SW for TW, and between DW and SW for TW. We also demonstrate that the terdiurnal derived here are not influenced to any significant degree by aliasing due to the presence of other waves. Citation: Moudden, Y., and J. M. Forbes (), A decade-long climatology of terdiurnal using TIMED/SABER observations, J. Geophys. Res. Space Physics, 8,, doi:./jgra.7.. Introduction [] Atmospheric are generally expressed in the form A n,s cos(nt + s n,s ) where t is the time (days), is the rotation rate of the Earth = day, is the longitude, n (=,,...) denotes a subharmonic of a solar day, s(=...,,...,,,...) is the zonal wave number, and the amplitude A n,s and phase n,s are the functions of height and. The zonal phase speed is C ph = n. Therefore, s with s = n have phase speeds equal to the rate of westward migration of the Sun with respect to a ground-based observer and are thus called migrating. Transforming to a local time frame, one obtains A n,s cos(nt LT +(s n) n,s ), where it is clearly seen that migrating (s = n) are independent of longitude. Tides with zonal wave numbers s n are referred to as nonmigrating, and their superposition embodies the longitude dependence of any given tidal field. [] While most tidal studies are focused on diurnal ( h period) and semidiurnal ( h period) as they are generally stronger, there are a number of studies aimed Department of Aerospace Engineering Science, University of Colorado, Boulder, Colorado, USA. Corresponding author: Y. Moudden, Department of Aerospace Engineering Science, University of Colorado, UCB 9, Boulder, Colorado, 89, USA. (moudden@colorado.edu). American Geophysical Union. All Rights Reserved //./jgra.7 at understanding terdiurnal (TDTs). Ground-based observations have revealed some important information regarding TDTs. [] Teitelbaum et al. [989] used radar wind observations to illustrate the existence of a nonnegligible terdiurnal tide and suggested that the seen terdiurnal amplitudes result from a superposition of a tide directly generated by solar heating and one induced by nonlinear interaction between the diurnal and semidiurnal migrating. According to their theory [see also Teitelbaum and Vial, 99], such an interaction between two primary waves with frequencies n and n where = h and zonal wave numbers s and s, respectively, results in the generation of secondary waves with the sum and difference frequencies (n + n, n n ) and zonal wave numbers (s + s, s s ). Nonlinear interaction between the migrating diurnal tide (n =, s =)and migrating semidiurnal tide (n =, s = ) thus gives rise to the terdiurnal migrating tide (n =, s =)aswellasa secondary diurnal migrating tide (n =, s =). Teitelbaum et al. [989] conclude that it is necessary to invoke tidal interactions to explain the relatively high terdiurnal amplitudes in the northern hemisphere winter. Thayaparan [997] also noted significant terdiurnal amplitudes in the mesosphere and lower thermosphere, argued that the terdiurnal tide cannot be explained solely by solar heating, and suggested tidal interactions as a likely additional source. A study of equatorial terdiurnal is provided by Venkateswara Rao et al. []. This study refers to annual and semiannual oscillations and suggests the interaction of diurnal and

2 solar local time (hrs) days of descending ascending Figure. Local times covered by TIMED at the equator between January and June. semidiurnal and gravity waves as important generation mechanisms. Ionospheric terdiurnal activity was reported by Gong and Zhou [] and Huang et al. [] who conclude that TDTs are as important as diurnal and semidiurnal over Arecibo. TDTs have also been detected in meteorological surface pressure observations. Ray and Poulose [] analyzed decades of surface pressure network observations to extract a terdiurnal signal and suggest that TDTs undergo larger seasonal variability than the diurnal and semidiurnal counterparts. [] Teitelbaum et al. [989], Thayaparan [997], and Venkateswara Rao et al. [] all used ground-based punctual observations, and their studies all refer to a single terdiurnal tide; thus, the use of tide in its singular form. While there are various terdiurnal that can possibly be distinguished by their different zonal wave numbers, a single local ground observation only views a single terdiurnal tide that is potentially a combination of a number of different terdiurnal. As with diurnal and semidiurnal, the dominant terdiurnal tide is the migrating one: Migrating here refers to the tide that follows the Sun in its apparent motion and thus has a zonal wave number equal to three. We abbreviate this tide as TW, where refers to its zonal wave number, W to its westward propagation direction, and T to its terdiurnal frequency. We abbreviate other similarly using E for eastward propagation, D for diurnal, and S for semidiurnal. The accompanying number always refers to the zonal wave number. The diurnal and semidiurnal migrating are therefore denoted DW and SW, respectively. [6] So while radar observations are capable of rendering a picture of the terdiurnal tide as a whole at a certain location, it is necessary to have a network of radars along the same zonal circle in order to distinguish different zonal wave numbers. The terdiurnal characteristics reported by the studies referenced above are an amalgam of characteristics that can be attributed to different individual terdiurnal. It is also likely that their conclusions are heavily influenced by the migrating tide which is always expected to dominate the nonmigrating ones. [7] Satellite observations generally offer better geographical coverage although they are lacking in temporal coverage. A global picture of the migrating terdiurnal tide was obtained by Smith [] using horizontal wind observations from the High Resolution Doppler Imager instrument onboard the UARS satellite. The results of Smith [] indicate that the terdiurnal tide can reach significant peak amplitudes and that tide-tide interactions are likely a factor in defining its structure and variability. Only a few satellite-based determinations of the terdiurnal tide exist. For instance, Forbes and Wu [6] revealed TW and TW in their MLS temperature data, and TW was a prominent component in the multiyear mean Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) temperature spectrum of Forbes et al. [8], although other terdiurnal tide components were significant in individual years. [8] Several works have investigated the terdiurnal tide using numerical models [Akmaev, ; Huang et al., 7; Du and Ward, ; Smith and Ortland, ]. Akmaev [], Huang et al. [7], and Smith and Ortland [] all argue that excitation of TW is due to some combination of direct thermal forcing and nonlinear interaction between DW and SW, although they disagree on the relative importance of these two mechanisms. On the one hand, Huang et al. [7] maintain that DW-SW nonlinear interactions are the predominant forcing for TW, while the Smith and Ortland [] model shows thermal forcing to be dominant for TW at middle and high s and that nonlinear interactions contribute more to low- TW. Akmaev [] emphasizes that nonlinear interactions are mainly important above 9 km during equinox periods. [9] Du and Ward [] provide a comprehensive study of the terdiurnal tide using the extended Canadian Middle Atmosphere Model (CMAM), a general circulation model extending up to about km. Based on a correlation analysis, this study concludes that DW-SW nonlinear interaction is unlikely to be a source of the migrating terdiurnal tide, TW. An especially valuable asset of the work of Du and Ward [] is that it represents the only comprehensive numerical study of nonmigrating teridurnal and includes results on terdiurnal with zonal wave numbers between and +. In section 6, we briefly summarize our data analysis results in the context of this model s predictions. [] In this paper, we provide a comprehensive study and a global view of migrating and nonmigrating TDTs using a decade-long record of Thermosphere-Ionosphere- Mesosphere Energetics and Dynamics (TIMED)/SABER observations. Special attention is given to data processing to minimize data shortcomings, artifacts, and other sources of error such as aliasing. We attempt to give a geographically global picture of the most significant TDTs to distinguish seasonal and interannual variability patterns and to provide a comparison to other with an emphasis on how they might be connected. The next section details data processing solar local time (hrs) 8 6 descending ascending days of Figure. Local times covered by TIMED at 8 ı N between January and June.

3 TE TE T TW TW TW TW 9S TW TW6 TW7 terdiurnal 9N Figure. altitude. Annual and decade ( ) amplitude average of different terdiurnal at 9 km methods. The migrating and nonmigrating terdiurnal are analyzed in the following two sections, respectively.. Data Description and Analysis Method [] The Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) is one of four instruments on NASA s TIMED satellite. TIMED collected stratosphere, mesosphere, and lower thermosphere data since its launch in. There is currently about years of data available ( ). For the SABER Version 7 temperature measurements analyzed here, each vertical profile begins with a climatological value from NCAR s TIME-GCM (Thermosphere Ionosphere Mesosphere-general circulation model), and a retrieval is performed from km downward. The retrieved temperatures begin to become independent of the climatology at about km (J. Russell, SABER Principal Investigator,, private communication), and this serves as a nominal upper limit for validity of the SABER temperatures [see also Mertens et al., ]. However, in some cases, we have found the temperatures above km to often represent a smooth extrapolation of those below, and in these cases, we have extended our plots to km. Remsberg et al. [8] provide a thorough overview of the v.7 temperature data quality from the lower stratosphere to the upper mesosphere and lower thermosphere. [] The TIMED orbit is circular with an altitude near 6 km and an eccentricity of 7 ı. In day, the satellite covers nearly (.87) longitude bands. For the purpose of our study, the coverage of longitude is very satisfactory since we are mainly concerned with zonal wave numbers in the range of. The critical feature of TIMED s orbit that we exploit is its precession relative to the Sun which is about ı (.9 ı ) per day. This precession means that the local times covered by the satellite slowly change at a rate of h per.8 days. This is illustrated in Figure which shows the Figure. Time-altitude cross section of equatorial (averaged between ı S and ı N) TW amplitudes. 6

4 Figure. Ten year averages of (left column) TW, (middle column) SW, and (right column) DW presented in different cross-section plots: (top row) month-altitude cross section in the tropics (average between ı S and ı ), (middle row) -altitude cross section on the first of March, and month cross section at 9 km. One-sigma uncertainties in the TW amplitudes range between about K near 7 km to K near km. local times covered between ı and ı N (near the equator) at the beginning of. The ascending and descending portions of the orbit cover local times that are nearly h apart, and the consistent slope in Figure reflects the precession rate of ı /day or h/ days. In order to keep the same side of the satellite facing the Sun, TIMED makes a 8 ı yaw maneuver every 6 days around its nadir axis. Two of these yaws can be seen as a discontinuity in local time of about h. They occur at days 78 and of. These yaws that occur throughout the mission s lifetime induce a gap of h near noon and a h redundancy near midnight in local time coverage. The yaws mean that while it is expected that h local time is achieved in days based on the ı /day precession rate, only h are actually covered. [] Since terdiurnal are generally not as large as diurnal and semidiurnal, we seek here a method to extract terdiurnal over the shortest time span possible, so that the amplitude suppression that typically occurs over longer time spans (e.g., 6 days) due to variability in both amplitude and phase can be reduced. To adequately detect terdiurnal, we only need 8 h of continuous local time coverage, provided that aliasing by longer period diurnal and semidiurnal harmonics is small enough. In our analysis, the gaps induced by the yaw maneuvers are avoided by using selected8hcontinuous stretches of observations within this h local solar time (LST) period. Eight hours of continuous local time coverage is completed in nearly days. If both the ascending and descending nodes are used, then about days of observations are sufficient to cover 8 h of local time. As the ascending and descending nodes are not exactly at opposite local times, a period of slightly more than days is necessary (the length depends on ). 7

5 Figure 6. (left) Total terdiurnal tide amplitudes and (right) total nonmigrating terdiurnal tide amplitudes at (top) 9 km and (bottom) km altitudes during the month of September in K. [] Another issue is that the yaw maneuvers affect the observations of the polar areas. As Figure shows, SABER ceases to view the north polar region between the two yaws depicted in Figure. TIMED shifts its polar coverage between the two poles at every yaw maneuver, and each pole is covered for a period of 6 days at a time. Here again, since we only need days in observations to detect terdiurnal, it is still possible to use SABER data poleward of 6 ı despite the periodic 6 day gaps. [] While the use of 8 h local time windows to extract terdiurnal signals allows us to overcome the different gaps in data, it also introduces the risk of diurnal, semidiurnal, and planetary wave aliasing into the terdiurnal tidal determinations. This occurs because diurnal and semidiurnal, as well as the zonal mean, contribute to the variability seen within any day window that is analyzed for the terdiurnal tide. This issue is partially remedied by analyzing residuals from an 8 h sliding average, which suppresses waves with periods longer than 8 h. As we demonstrate in section A, aliasing by the diurnal tide is essentially eliminated by taking this step. However, because the semidiurnal tide is much closer in period to the terdiurnal tide, it can also alias into the terdiurnal tide at about % of the semidiurnal amplitude. It is important to emphasize, though, that such aliasing only occurs between terdiurnal and semidiurnal possessing common zonal wave numbers. Moreover, nonmigrating semidiurnal tidal amplitudes are sufficiently small that aliasing contributions to terdiurnal are minimal. For the TW, TE, TW, and TW waves that we analyze in the following sections, we show in section A that aliasing contributions to Figure 7. Time-altitude cross section of equatorial (averaged between ı S and ı N) amplitudes of TE. 8

6 Figure 8. Ten year averages of (top row) TE and (bottom row) DE presented in different crosssection plots: (left column) month-altitude cross section in the tropics (average between ı S and ı S), (middle column) -altitude cross section on the first of September, and -month cross section at km. One-sigma uncertainties in the derived TE amplitudes range between about K near 7 km to K near km. their amplitudes are estimated to be on the order of. K,. K,. K, and. K, respectively. Aliasing contributions to TW, on the other hand, can be as large as K due to the relatively large magnitude of SW ( K). It is also possible that DW, due to its large amplitude, could alias to some degree into TW. [6] In the following, we focus on TW, TE, TW, and TW because these terdiurnal are relatively unaffected by aliasing and because they possess seasonal-latitudinal behaviors that are well defined and repeatable from year to year. To derive the terdiurnal, a local time-longitude -D field covering 8 h of local time at any given height Figure 9. Time-altitude cross section of equatorial (averaged between ı S and ı N) amplitudes oftw. 9

7 Figure. Ten year averages of (top row) TW and (bottom row) DW presented in different crosssection plots: (left column) month-altitude cross section in the tropics (average between ı S and ı S), (middle column) -altitude cross section on the first of December, and -month cross section at 9 km. One-sigma uncertainties in the derived TW amplitudes range between about K near 7 km to K near km. and is first constructed using days of SABER temperatures. We then perform fast Fourier transforms within these day windows, moved forward sequentially at day intervals from the beginning of to the end of, for every and altitude. We also use a similar analysis to extract diurnal and semidiurnal that are used in different comparisons; except for these tidal determinations, we employ sequential 6 and day segments. [7] An error analysis was performed, which is described in section B. One-sigma uncertainties for the terdiurnal presented here are estimated to be of order of. K near 7 km and increase to about. K at km. Only terdi- Figure. Time-altitude cross section equatorial(averaged between ı S and ı N)amplitudes of TW.

8 Figure. Ten year averages of (top row) TW and (bottom row) SW presented in different crosssection plots: (left column) month-altitude cross section in the tropics (average between ı S and ı S), (middle column) -altitude cross section on the first of February, and -month cross section at km. One-sigma uncertainties in the derived TW amplitudes range between about K near 7 km to K near km. urnal of amplitude. K or more are displayed in the following figures.. Terdiurnal Migrating Tides [8] The migrating terdiurnal tide is the strongest among all terdiurnal. It is characterized by a zonal wave number equal to three and designated by TW since it is westward propagating. A comparison of TW s annual and decadal mean amplitude at 9 km with that of various nonmigrating TDTs is shown in Figure. The averages shown in Figure are obtained by averaging, at every and 9 km altitude, all the amplitudes of each TDT obtained during the decade. This figure shows that the mean TW has an amplitude that exceeds double the strongest nonmigrating TDTs. It also shows that TW peaks on average at the equator with a value near K. An altitude versus year view of TW s amplitude at the equator is shown in Figure. This plot reflects mostly the variability of TW in season and height. TW starts having sensible amplitudes at 7 8 km and reaches values near K upward of km. TW is organized into two main seasonal episodes during the first and second half of each year. TW has two distinct peaks that occur regularly each year above km near the spring and fall equinoxes. TW also tends to be stronger on average in the first half of the year between February and May and intensifies at altitudes as low as 7 km. It intensifies near the equinoxes and weakens near the solstices. It seems to be on average more active during the first half of the year than the second. Above km TW peaks several times with two distinct maxima that exceed K at the end of February and August. There is an apparent interannual modulation of TW below km that appears as strengthening during,, 6, and 8. [9] The repetitive nature of TW from year to year is also striking. The seasonal variability described in the previous paragraph repeats every year of the decade. Since the decade covers almost an entire solar cycle, and the amplitudes shown in Figure do not reflect any visible response to this cycle, we conclude that TDTs are not to a first degree affected by the solar cycle at these altitudes. A closer examination does not reveal any statistically significant dependence on level of solar activity. [] The salient features of TW described above share many characteristics with those of the diurnal and semidiurnal migrating. Figure demonstrates this by comparing three cross sections of all three migrating : a height versus month tropical cross section (average between ı Sand ı N) in the top row; altitude versus cross section on the first of March (middle row); and a month versus cross section at 9 km altitude (bottom row). All plots are averages over the entire period of. When we examine the SW and DW tropical amplitudes shown in the top row of Figure, we immediately see that TW shares

9 most of its characteristics with the other migrating. Both DW and SW have a semiannual cycle that makes them maximize near equinoxes and minimize near solstices. DW and SW also always appear to be stronger during the first half of the year. A strong possibility is that TW is produced by interaction between the diurnal and semidiurnal migrating. Obviously, this does not exclude the existence of other sources of TW but endorses the interaction of DW and SW as a major source. [] It is also important to note the importance of TW relative to DW and SW, especially at high altitudes. While DW dominates at altitudes between 7 and km with amplitudes that peak near K ( K is the shown year average), SW becomes the strongest migrating tide above km. TW certainly appears to be the third strongest tide after DW and SW and is thus expected to be an important source of diurnal variability in the 7 km altitude region. [] The other cross sections are shown in the middle and lower rows of Figure ; while they confirm the relationship between TW and the other migrating, they also show that TW is mostly a tropical tide below km and becomes a middle and high- tide above that. Figure. Reconstructed TW oscillations illustrating vertical phase structures of (top two panels) TW (at ı and ı for March ) and (bottom panel) TE (at ı for September ), in the form of height versus local time plots of amplitude at ı longitude.. Terdiurnal Nonmigrating Tides [] While individual nonmigrating terdiurnal tide components are usually not as strong as their diurnal or semidiurnal counterparts, the total terdiurnal amplitude can be at times strong enough to have a nonnegligible effect on the upper atmosphere, especially within localized regions around the globe. The total terdiurnal amplitude shown in the left column of Figure 6 is the amplitude that would be recorded by a ground-based instrument at any given location, blind to the contributions of different zonal wave number components. Note the significant longitude variability in the equatorial region at 9 km, ranging from a maximum value of 7 9 K in the Western Pacific to K over the Indian Ocean. And, although this is an equinox month, maximum amplitudes of 7 9 K occur over the South Polar region, while the North Polar region has amplitudes of only K. At km, the main Pacific maximum ( K) occurs over the Hawaiian Islands, and a second maximum ( K) occurs off the east coast of Argentina. Other nonpolar locations reflect amplitudes of order of 6 K, while the polar asymmetry seen at 9 km persists, albeit more confined to the poles. As our analysis allows us to discriminate zonal wave numbers, we are able to plot the same total terdiurnal amplitude while excluding the migrating one. The right column in Figure 6 shows the relative importance of nonmigrating TDTs which appears to be greater than two thirds of the total terdiurnal amplitudes depicted in the left column. The differences in the patterns between the left and right columns arise from differences in phase between the total nonmigrating tide (which varies with longitude) and the migrating tide (which is longitude-invariant). Note also that the total nonmigrating tide possesses a longitude-mean value of order of K ( 6 K) at 9 km ( km) at low to middle s. [] The strongest and most interesting nonmigrating terdiurnal tide uncovered in our analysis is TE. Its tropical amplitudes averaged over the period are shown in Figure7. It has sensible amplitudes at 9 km and can reach an amplitude of K between and km altitude. It reaches its maximum values in the Northern Hemisphere summer season. Figure8 shows a year average of TE amplitudes (top row) along with those of DE (bottom row). The left column in Figure 8 shows a tropical month-altitude cross section of TE (top) and DE (bottom), showing

10 Figure. Reconstructed TW and TW oscillations illustrating vertical phase structures of (top panels) TW (at ı and ı for December ) and (bottom panels) TW (at ı and ı for December ), in the form of height versus local time plots of amplitude at ı longitude. that both nonmigrating reach their single yearly peak between July and October at km altitude. The middle column shows TE and DE amplitudes at the beginning of September averaged over years. These plots show that these are mostly tropical that maximize near the equator. The third column in Figure 8 shows the seasonal evolution of TE and DE at km altitude. DE is known to be an important nonmigrating tide, and the plots in Figure 8 show it to be an almost comparable tide to the migrating ones (Figure ). Theoretically, TE (n =, s = ) can arise as the sum by-product of the interaction between DE (n =, s = ) and SW(n =, s =). The time and space synchronization of the two shown in Figure 8 supports this hypothesis, which can only be confirmed through numerical modeling. Note that the rather pervasive presence of SW at nonnegligible amplitudes (middle column of Figure 6) is not inconsistent with this hypothesis. We also emphasize that DE does not alias into TE in our data analysis, due to their different zonal wave numbers. [] Another nonnegligible nonmigrating terdiurnal tide is the westward-propagating TW. The tropical amplitudes of TW are shown in Figure 9. TW intensifies on average in the northern fall/winter season in a relatively narrow period in the months of October December. Unlike TE, TW can reach relatively high amplitudes at lower altitudes; typically, TW can be as strong at 8 km as it is at km. The top row in Figure shows different year average depictions of TW. The tropical average between ı S and ı S confirms that TW is mostly a northern-fall tide although it moderately picks up during the northern winter. The middle panel (top row) in Figure shows the latitudinal structure of TW at the beginning of December. While TW is mostly tropical below km, it intensifies in the high s (6 ı ) above km and reaches relatively significant values above km. The last panel in the top row confirms that TW is mostly a northern-fall and tropical tide below km. Examining the same plots for DW in the bottom row of Figure shows that the two share many of the features discussed above while not possessing aliasing relationships. This may not be a coincidence since DW (n =, s =) has the potential to produce TW (n =, s =) when it interacts with the semidiurnal migrating tide, SW (n =, s =). [6] Semidiurnal nonmigrating are also capable of exciting terdiurnal nonmigrating. TW (n =, s =) shown in Figure could in principle be produced by the interaction of SW (n =, s = ) and DW(n =, s =). The tropical amplitudes of TW intensify near the northern winter solstice at maximum amplitudes over K. Figure, which shows time-height cross sections of TW (top row) along with those of SW (bottom row), allows us

11 to place Figure into context. SW undergoes a seasonal migration between the northern and southern mids across the equator, and TW seems to follow SW in this migration (see right panels in Figure ). At km during October, these waves have their maxima at middle s in the Southern Hemisphere. The centroid of this region of maximum amplitudes migrates toward and across the equator during November December, into middle s of the Northern Hemisphere during January, and back across the equator in February, and back into Southern Hemisphere middle s by March April. Both waves are relatively weak during May September. Note that this migration is also consistent with the two peaks seen in the tropical amplitudes of both SW and TW before and after January (see also Figure ). [7] While nonmigrating terdiurnal remain weak (there is on average a one to three factor) when compared with their migrating counterpart, they do illustrate the complex and interesting nature of the tidal population and do contribute, albeit moderately, to atmospheric variability. While DW and SW seem to be capable, when interacting with other nonmigrating diurnal and semidiurnal, of exciting a range of nonmigrating TDTs, we do not exclude the possibility of other sources. The analysis we presented here based on a year observation period clearly suggests the possibility of tidal interactions exciting nonmigrating TDTs and explaining their seasonal variability.. Phase Structures [8] In this section, we take a look at some typical phase structures for TW, TE, TW, and TW. We present here terdiurnal oscillations plotted in a height versus local solar time (LST) contour format at ı longitude for (a typical year). Figure comprises our first example, which illustrates TW at ı and ı during March, and TE at ı during September. Beginning with TE, we see that phase progresses in a quasi-monotonic manner to later times as height decreases, consistent with the presence of a single wave propagating upward from below with a vertical wavelength of about km. Based on our previous discussions, this would suggest that TE is produced below about 8 km by the nonlinear interaction between DE and SW and that exponential growth to its peak altitude of km (see Figure 9, top middle panel) contributes more to its local amplitudes in the 9 km height region than local forcing. [9] Turning now to the TW results in Figure, the situation becomes more complicated. While some evidence can be found for downward phase progression, the cellular features are much more pronounced. In addition, the phase progression near the equator is consistent with a vertical wavelength ( z ) of order of 7 km, while that at ı is of order of km. In the context of classical tidal theory, the latter would imply that at least two Hough modes of TW are contributing to the total amplitude and phase structure. The superposition of two modes with different vertical amplitude and phase structures can also produce cellularlike structures in the type of format we are employing in Figure. [] Similar plots are presented in Figure for TW (top) and TW (bottom) at the equator (left) and ı (right) for December. At ı, TW and TW exhibit quasi-monotonic downward phase progression, with average vertical wavelengths of order of 7 km and 7 km, respectively. At the equator for TW, downward phase progression with z 7 km is discernible amidst cellular structures similar to those discussed above, while for TW prolonged downward phase progression cannot be found. The possible reasons for these various types of phase structures and the differences between s are all attributable to various degrees of phase interference effects discussed in connection with Figure and need not be repeated here. 6. Summary and Concluding Remarks [] In this paper, temperature measurements from the SABER instrument on the TIMED spacecraft from through are analyzed to reveal the characteristics of the terdiurnal tide in the 8 km height regime of Earth s atmosphere. While there are a number of ground-based experimental studies that focus on the terdiurnal tide, this is the first to observationally delineate its longitude dependence, the nonmigrating tidal components that contribute to its longitude variability, and its variability from year to year. We also provide evidence that strongly suggests that individual terdiurnal zonal wave number components may arise, at least in part, as the result of nonlinear interactions between specific diurnal and semidiurnal tidal components. For instance, TW appears to arise as a result of DW interacting with SW; TE as a result of DE-SW interaction; TW from DW-SW; and TW from DW-SW. [] In the following, we attempt some comparison with the CMAM (Canadian Middle Atmosphere Model) results of Du and Ward [], since this is the only general circulation model comprehensive modeling study in existence that includes both migrating and nonmigrating terdiurnal tide components. In order to eliminate effects of interannual variations in phase on the data-based annual means and thus provide a more credible comparison with the model, we only utilize SABER data from (a typical year) in the annual-mean depictions in Figure and the CMAM comparisons to follow. Beginning with the migrating terdiurnal tide, the annual-mean height versus structure of TW revealed in Figure 6 is similar to that in the Du and Ward [] CMAM results in that there are maxima in each hemisphere separated by a shallow minimum near the equator, and the amplitudes are in the 8 K K range below km. However, the CMAM maxima occur between ı and ı in each hemisphere, whereas the SABER temperature maxima are centered more closely to 6 ı. [] In our analysis, we found TE to be one of the largest and most well-defined of the nonmigrating terdiurnal. It has a Kelvin wave-like (equatorial maximum and symmetric about the equator) structure confined within ı and similar to DE with almost half the amplitude. It appears likely that this wave is generated by nonlinear interaction between DE and SW. The annual-mean depiction of TE in CMAM does have an equatorial maximum of order of K, but the most prominent features are the maxima of order of K near 7 ı. Since our results are sparse poleward of 6 ı, our comparison with CMAM for

12 Input field TE TE T TW TW TW TW TW TW6 TW7 9S 9N Spectral analysis without filtering TE TE T TW TW TW TW TW TW6 TW7 9S 9N Spectral analysis with filtering TE TE T TW TW TW TW TW TW6 TW7 9S 9N Figure A. (top) Spectral analysis of dummy terdiurnal that have a fixed amplitude of K. Results from analysis without and with filtering of longer-period variations are shown in the middle and bottom plots, respectively. TE cannot be very definitive. Two other prominent nonmigrating terdiurnal to emerge from our analysis are TW and TW, and these are two of the more well-defined components in CMAM as well. In CMAM, TW and TW are confined within ı, have significant amplitudes only above km, and achieve annual-mean amplitudes of order of K at km. For SABER, we find that significant TW amplitudes (.. K) are achieved down to 8 8 km near the equator, whereas above km, the structures are larger (.. K) and more broad, extending beyond 6 ı while maintaining fairly symmetric behavior about the equator. For TW, latitudinal shapes and

13 Input field SE SE S SW SW SW SW SW SW6 SW7 9S 9N Spectral analysis without filtering TE TE T TW TW TW TW TW TW6 TW7 9S 9N Spectral analysis with filtering TE TE T TW TW TW TW TW TW6 TW7 9S 9N Figure A. (top) Spectral analysis of dummy semidiurnal that have a fixed amplitude of K. Results from analysis without and with filtering of longer waves are shown in the middle and bottom plots, respectively. (middle and bottom) The amplitudes from the spectral analysis are for terdiurnal resulting from analyzing the semidiurnal in the top plot. amplitudes for SABER and CMAM are similar. Amplitudes range between K and K between and km and extend to about 6 ı. Both latitudinal structures have a symmetric-like characteristic, but with some preference for larger amplitudes in the Southern Hemisphere at the higher altitudes. [] In terms of phase structures, we found it unusual to see downward phase progression that would be similar to that expected for a single tidal mode in the context of classical tidal theory. Instead, phase structures are generally more consistent with the superposition of at least two waves, one with a vertical wavelength in the range of km, and the other much longer. 6 [] We suspect what we are seeing might be the superposition of vertically propagating waves excited at lower altitudes with those excited in situ by nonlinear interactions between semidiurnal and diurnal, but verification must await the results of modeling studies that focus on the physics and consequences of tide-tide interactions. Appendix A: Aliasing Analysis [6] Section of this paper mentions the potential aliasing to terdiurnal that can occur due to the presence of diurnal and semidiurnal within the day analysis window, where only 8 h of local time are sampled. In order to

14 Input field DE DE D DW DW DW DW DW DW6 DW7 9S 9N Spectral analysis without filtering TE TE T TW TW TW TW TW TW6 TW7 9S 9N Spectral analysis with filtering TE TE T TW TW TW TW TW TW6 TW7 9S 9N Figure A. (top) Spectral analysis of dummy diurnal that have a fixed amplitude of K. Results from analysis without and with filtering of longer waves are shown in the middle and bottom plots, respectively. (middle and bottom) The amplitudes from the spectral analysis are for terdiurnal resulting from analyzing the diurnal in the top plot. assess the amount of aliasing that might occur, we apply the same data analysis process to fabricated SABER data where the existing are specified in advance. The input field has the same spatial and temporal distribution as SABER observations, but the value of the temperature field is set to match a chosen tidal population. This allows us to assess to what degree our data processing and spectral analyses faithfully reproduce the initial field. Our procedure is very similar to that of Tunbridge et al. [] in their analysis of the day wave. In their case, aliasing occurs because different waves project onto the same longitudinal wave numbers as seen from a Sun-synchronous satellite. In our case, we are concerned with waves of like zonal wave number projecting onto each other because the window length does not enable complete separation of 8,, and h harmonics within an 8 h local time window. [7] Figure A shows the analysis of an input field that only contains TDTs. The input field is shown in the top plot 7

15 Average semidiurnal amplitudes 8 6 9N SE SE SE S SW SW SW SW SW SW6 9S Aliasing from semidiurnal into terdiurnal.. TE TE TE T TW TW TW TW TW TW6 9S 9N Figure A. Actual aliasing values from semidiurnal into terdiurnal. The values shown in the top panel are the average semidiurnal amplitudes in at 9 km. The bottom panel shows the aliasing contributions to terdiurnal components due to the semidiurnal tidal amplitudes depicted in the top panel. The Y axis represent the amplitudes in K. with both migrating and nonmigrating TDTs having a uniform amplitude of K. The middle plot shows the result of the spectral analysis without filtering which reproduces the initial rather well although the initial amplitude of K is slightly diminished. Application of a filter intended to remove longer waves (bottom panel) does not have an effect on the TDTs analyzed amplitudes. This filter which consists of removing a sliding average from the original field Figure A. Spectral analysis of a K dummy TW (top) at the equator and (bottom) at 7 ı N. The X axis shows the years, and the Y axis represents the amplitudes in K. 8

16 Figure A6. (top row) SABER temperatures during and (bottom row) corresponding errors presented in different cross-section plots: (left column) month-altitude cross section in the tropics (average between ı S and ı S, (middle column) -altitude cross section on the first of March, and -month cross section at 9 km. is intended to filter any waves with periods longer than 8 h before applying the spectral analysis. The sliding average has a window of 8 h in local time. [8] When the input field contains only semidiurnal with a constant amplitude of K (top panel in Figure A) about half of the initial amplitude (. K) aliases into terdiurnal (middle panel in Figure A). Application of the filter reduces that to about one third (bottom panel in Figure ). The aliasing from diurnal illustrated in Figure A is equally important (middle panel), but the filter is much more effective in reducing aliasing from diurnal than semidiurnal, as shown in the bottom panel. The filter is even more effective in eliminating any aliasing from planetary waves (more than h period) for which we do not show any plots. We conclude that aliasing from diurnal and longer waves is insignificant. [9] These results show that the risk of aliasing is mostly present from semidiurnal, which are difficult to filter from the data since they have a period close to that of TDTs. They also show that about one third of the semidiurnal amplitudes transfers into TDTs of the same zonal wave number. [] Since the biggest risk of aliasing comes from semidiurnal, we conducted another experiment using actual semidiurnal amplitudes. As shown in Figure A aliasing from SW can create a false TW with amplitudes of up to K at 9 km. Given this risk, we steered clear from discussing TW in this paper as it can be strongly contaminated by aliasing. However, the aliasing contributions to TW, 9 TE, TW, and TW are on the order of. K,. K,. K, and. K, respectively. These are just representative values but are sufficiently small that we can conclude that aliasing from nonmigrating semidiurnal is largely insignificant. [] Figure A is intended to illustrate how the method fares poleward of 6 ı where major gaps in data are present. In comparison with the equator (top panel of Figure A) where the analysis returns a. K amplitude for an initial value of K, at 7 ı N, the retrieved amplitude is between. and K with some sporadic extreme values. While the method is generally less accurate at the edges of the gaps, it remains satisfactory. Appendix B: Random Error Analysis [] To estimate errors or uncertainties in the derived terdiurnal amplitudes, we first determine the geophysical variability (standard deviations) associated with the input temperature fields. Then, we run spectral analyses with different input temperatures with randomly selected errors consistent with the above standard deviations. The variability in the output spectra are then used to provide a measure of uncertainty in any given spectrum. Examples of these standard deviations in the input temperature field and in the TW, TE, TW, and TW amplitudes are provided in Figures A6 and A7, respectively. All plots in Figure A7 represent -altitude cross sections on the first of March. As for the temperature errors show in Figure A6, the dominating factor in spectral uncertainty is altitude. Uncertainties at other times of and other years remain

17 Figure A7. Uncertainties in amplitudes of different terdiurnal in a -altitude cross section on the first of March. The are from top to bottom: TW, TE, TW, and TW. basically similar to the ones shown in Figure A7. These uncertainties range between typical values of K at 7 km to. K at km with a typical value of. K at our reference altitude of 9 km. These values are to be compared with calculated amplitudes shown in Figures, 8,, and. For example, the left panels in Figure show TW amplitudes with values that range from K at 7 km to K at km, which are comfortably larger than our error estimates. A similar comparison for nonmigrating terdiurnal leads us to conclude that the amplitudes we display with amplitudes equal to or greater than. K are not error artifacts. [] Acknowledgments. This work was supported under Grant ATM- 979 from the National Science Foundation to the University of Colorado. [] Robert Lysak thanks the reviewers for their assistance in evaluating this paper. References Akmaev, R. (), Seasonal variations of the terdiurnal tide in the mesosphere and lower thermosphere: A model study, Geophys. Res. Lett., 8, 87 8, doi:.9/gl. Du, J., and W. E. Ward (), Terdiurnal tide in the extended Canadian Middle Atmospheric Model (CMAM), J. Geophys. Res.,, D6, doi:.9/jd79. Forbes, J. M., and D. Wu (6), Solar as revealed by measurements of mesosphere temperature by the MLS experiment on UARS, J. Atmos. Sci., 6, Forbes, J. M., X. Zhang, S. Palo, J. Russell, C. J. Mertens, and M. Mlynczak (8), Tidal variability in the ionospheric dynamo region, J. Geophys. Res.,, A, doi:.9/7ja77. Gong, Y., and Q. Zhou (), Incoherent scatter radar study of the terdiurnal tide in the E- and F-region heights at Arecibo, Geophys. Res. Lett., 8, L, doi:.9/gl88. Huang, C., S. Zhang, Q. Zhou, F. Yi, and K. Huang (), Atmospheric waves and their interactions in the thermospheric neutral wind as observed by the Arecibo incoherent scatter radar, J. Geophys. Res., 7, D9, doi:.9/jd8. Huang, C. M., S. D. Zhang, and Y. Fan (7), A numerical study on amplitude characteristics of the terdiurnal tide excited by nonlinear interaction between the diurnal and semidiurnal, Earth Planets and Space, 9, 8 9. Mertens, C. J., et al. (), Retrieval of mesospheric and lower thermospheric kinetic temperature from measurements of CO m Earth limb emission under non-lte conditions, Geophys. Res. Lett., 8(7), 9 9, doi:.9/gl89. Ray, R. D., and S. Poulose (), Terdiurnal surface pressure oscillations over continental United States, Mon. Wea. Rev.,, 6. Remsberg, R. E., et al. (8), Assessment of the quality of the version.7 temperature versus-pressure profiles of the middle atmosphere TIMED/SABER, J. Geophys. Res.,, D7, doi:.9/8jd. Smith, A. K. (), Structure of the terdiurnal tide at 9 km, Geophys. Res. Lett., 7, Smith, A. K., and D. A. Ortland (), Modeling and analysis of the structure and generation of the terdiurnal tide, J. Atmos. Sci., 8, 6. Teitelbaum, H., and F. Vial (99), On tidal variability induced by nonlinear interaction with planetary waves, J. Geophys. Res., 96,,69,78. Teitelbaum, H., F. Vial, A. H. Manson, R. Giraldez, and M. Massbeuf (989), Non-linear interaction between the diurnal and semidiurnal : Terdiurnal and diurnal secondary waves, J. Atmos.Terrest. Phys.,, Thayaparan, T. (997), The terdiurnal tide in the mesosphere and lower thermosphere over London, Canada ( ı N, 8 ı W), J. Geophys. Res.,,,69,78. Tunbridge, V. M., D. J. Sandford, and N. J. Mitchell (), Zonal wave numbers of the summertime day planetary wave observed in the mesosphere by EOS AURA microwave limb sounder, J. Geophys. Res., 6, D, doi:.9/jd67. Venkateswara Rao, V., T. Tsuda, S. Gurubaran, Y. Miyoshi, and H. Fujiwara (), On the occurrence and variability of the terdiurnal tide in the equatorial mesosphere and lower thermosphere and a comparison with Kyushu-GCM, J. Geophys. Res., 6, D7, doi:.9/jd9.