1. Introduction
The sun-synchronous, or migrating, diurnal tide is a major source of temperature and wind variability in the mesosphere and lower thermosphere (MLT) (Chapman and Lindzen 1970; Hays et al. 1994; Burrage et al. 1995; Vincent et al. 1998; Lieberman et al. 2007; Singh and Gurubaran 2017; Dhadly et al. 2018; etc.). Theory (Chapman and Lindzen 1970), as well as ground-based (Silber et al. 2017) and satellite (Xu et al. 2009; Riggin and Lieberman 2013; Gan et al. 2014; Ortland 2017) observations show that the time-mean amplitude of the tide in the MLT easily exceeds 10 K. Lindzen (1967) computed the amplitude of the migrating diurnal tide from classical theory (Siebert 1961) and found that it agreed well with the observations then available from the troposphere to the upper mesosphere, but concluded that his results “cannot not be extended to higher altitudes (i.e., into the lower thermosphere) because the actual tidal temperature profile will, at sufficiently great altitudes, become statically unstable.” Lindzen’s calculations suggested that instability would set in somewhere in the upper mesosphere; a subsequent study (Lindzen 1968, his Fig. 7) showed that, depending on local time, the diurnal tide could be expected to become unstable and “break” at altitudes between 85 and 95 km.
Spacecraft-based observations of the middle atmosphere, the region between the tropical tropopause and the lower thermosphere (∼16–120 km), have made it possible to observe migrating and nonmigrating tides and determine their global structure and seasonal and interannual variations. In particular, the Sensing of the Atmosphere using Broadband Emission Radiometry (SABER) infrared radiometer (Russell et al. 1994) on NASA’s Thermosphere, Ionosphere, Mesosphere Energetics and Dynamics (TIMED) satellite (Yee et al. 2003) has been measuring temperature in the middle atmosphere for over two decades, since January 2002. While several studies have used SABER to determine the structure and seasonal variability of the migrating tide (Zhang et al. 2006) and to compare the observations against numerical simulations (Gan et al. 2014), no study has examined the tide over the 20 years of extant observations or its behavior on daily time scales. The present study uses SABER observations processed using Salby’s (1982b) fast Fourier synoptic mapping algorithm, described in section 2, to obtain a synoptic spectrum of temperature from SABER’s asynoptic observations. The synoptic spectrum is then used to document, in sections 3 and 4, the mean structure of the migrating tide and its seasonal and interannual variability. Section 5 takes up the question of tidal breaking and illustrates by means of twice-daily reconstructions that the diurnal tide breaks frequently above 85 km, and that breaking has a major impact on the zonal-mean state of the upper mesosphere and lower thermosphere. The last section summarizes the results.
2. SABER data and processing
SABER is a 10-channel broadband (1.27–17 μm) infrared radiometer on board NASA’s polar-orbiting TIMED spacecraft. SABER observations of infrared radiance are used to retrieve profiles of temperature, geopotential height, and various chemical species, including ozone, water vapor, and carbon dioxide.
This study uses global temperature retrievals from the SABER level 2A, version 2.07 dataset for January 2002–December 2021, available at http://saber.gats-inc.com/data_services.php. The temperature retrieval and associated uncertainties are discussed by Remsberg et al. (2008) for version 1.07 data; updated uncertainty information for version 2.0 is available online at http://saber.gats-inc.com/temp_errors.php. SABER temperature data have an effective latitudinal resolution of ∼4°, zonal resolution of ∼6 wavenumbers, and vertical resolution of ∼2 km. SABER temperatures over the altitude range ∼17–105 km are used to study the behavior of the diurnal tide throughout the middle atmosphere. Because the TIMED orbit precesses slowly with respect to local time, a yaw maneuver (where the spacecraft rotates by 180° in azimuth) is performed approximately every 60 days to keep SABER from pointing directly at the sun. As a result, SABER observations are made alternately over the latitude ranges 53°S–84°N and 53°N–84°S during successive yaw cycles.
Although SABER measurements are asynoptic, Salby (1982a) has shown that the information content of combined ascending and descending orbit data is approximately equivalent to that of twice-daily synoptic observations. Spectra computed along asynoptic coordinates can be mapped to a synoptic spectrum using Salby’s (1982b) fast Fourier synoptic mapping (FFSM) algorithm. A comparison between the asynoptic Nyquist rectangle for SABER and the Nyquist rectangle of synoptic, twice-daily observations is shown in Fig. 1, where it can be seen that SABER asynoptic sampling resolves the westward-migrating diurnal tide (zonal wavenumber k = 1 and frequency σ = 1 cpd), as well as all of the eastward-propagating, non-sun-synchronous diurnal tides (σ = −1 cpd). In Fig. 1 and all other figures that show spectra, positive frequencies denote westward propagation and negative frequencies, eastward propagation.
Comparison of the Nyquist rectangle of synoptic, twice-daily sampling (black) and that of SABER asynoptic observations (red). The SABER Nyquist rectangle coincides closely with the synoptic rectangle but is rotated with respect to it; c0 is the precession rate of the SABER orbital plane. The SABER spectrum has a westward Nyquist limit of ∼1.008 cpd at k = 1, such that the migrating diurnal tide (k = 1, σ = 1 cpd) is resolved. Positive frequencies denote westward propagation and negative frequencies denote eastward propagation. Adapted from Salby (1982b). See text for details.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
The foregoing assumes that tidal variability inside the SABER Nyquist rectangle is not contaminated by aliases due to variability outside the Nyquist limits. The semidiurnal migrating tide, which is not resolved by SABER, is known to reach large amplitudes above about 100 km, making it the most important alias of SABER observations. However, variability due to the semidiurnal tide would alias the zonal mean, not the migrating diurnal tide (Salby 1982a). Thus, SABER measurements are expected to provide accurate estimates of the behavior of the tide, free from any substantial aliasing.
SABER observations are made continuously in time only over the latitude range ±53° due to the yaw maneuvers described above but this encompasses the latitudes of interest for the migrating diurnal tide. The yaw maneuvers cause SABER sampling locations to shift slightly between successive yaw cycles, so the data are interpolated to a common asynoptic coordinate that lies midway between the actual sampling locations in successive yaw cycles. The resulting very long data sequences, about 20 years, produce a spectrum with very fine bandwidth, approximately 0.000 28 cycles per day (cpd). This can be seen in Fig. 2, which shows the amplitude spectrum of k = 1 in the vicinity of σ = 1 cpd as function of altitude. The altitude is given in (number of) scale heights. The scale height (sh) is an isobaric coordinate, sh = ln(p0/p), where p is the SABER pressure in hPa and p0 is a reference pressure, taken to be 1000 hPa. The right-hand-side ordinate of this and other plots shows the equivalent geometric altitude in kilometers, computed from the hydrostatic equation using the time- and global-mean SABER temperature profile.
Synoptic spectrum of SABER temperature (log10|T|, in K) for zonal wavenumber 1 near 1 cpd at the equator as a function of altitude (scale height: left ordinate; geometric altitude: right ordinate). In addition to the westward diurnal harmonic at 1 cpd, three sets of sidebands encode amplitude modulation of the tide at semiannual (SA), annual (AN), and quasi-biennial (QB) frequencies. Positive frequencies denote westward propagation. See text for details.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
In addition to temperature, SABER geopotential height and ozone data are also used in this work. The geopotential height is used to calculate the zonal-mean zonal wind in order to compare the seasonal and interannual variations of the zonal wind and the migrating tide. SABER geopotential height is derived from SABER temperature via the hydrostatic equation, referred to daily 10 hPa geopotential values from the U.S. National Centers for Environmental Prediction. SABER geopotential height data are processed with the FFSM algorithm and the resulting synoptic spectrum is used to reconstruct the evolution of the geopotential height, as in Smith et al. (2017). Zonal-mean zonal winds are then obtained from the balance wind equation (i.e., the zonal-mean meridional momentum equation, neglecting zonal-mean advection, eddy flux divergence, and time tendency terms). As noted by Smith et al. (2017), their Eq. (2) and related discussion, the calculation of zonal-mean zonal winds from SABER zonal-mean geopotential is reliable up to about 85 km; above that altitude, the calculation becomes questionable for a number of reasons, among them the fact that eddy flux divergence terms are no longer negligible in the tropical lower thermosphere, where the amplitude of the migrating diurnal tide becomes very large.
The SABER ozone data are used to explore the relationship between the quasi-biennial signal of ozone in the stratosphere and the interannual variability of ozone in the MLT. SABER ozone is derived from emission in the 9.6 μm region of the infrared spectrum (see Smith et al. 2013, and references therein), and is available above ∼20 km. As with temperature and geopotential height, the FFSM algorithm is used to calculate the synoptic spectrum of SABER ozone, which can in turn be used to reconstruct the evolution of this chemical species.
3. Structure of the migrating diurnal temperature tide (DW1)
The structure of the diurnal westward-migrating temperature tide of zonal wavenumber 1 (hereafter DW1) is shown in Fig. 3. The structure is calculated using coherence analysis (Hayashi 1971) over the frequency band 0.994–1.006 cpd, which includes the westward diurnal harmonic, DW1, and its low-frequency variability up to the semiannual period (cf. Fig. 2). Figure 3a, shows the amplitude (K) and phase (degrees) of DW1, referred to a base point at the equator and 12 sh, as functions of latitude and altitude. Figure 3b shows amplitude and phase profiles at the equator, such that the behavior of these two quantities can be seen more precisely. The results shown in Fig. 3 are consistent with previous determinations of the structure of DW1 from SABER and other satellite-borne instruments (Lieberman et al. 2010; Gan et al. 2014; Liu et al. 2016; Gu and Du 2018, Sakazaki et al. 2018; etc.)
(a) Amplitude (color contours) and phase (black line contours) of the migrating diurnal tide as a function of latitude and altitude obtained by coherence analysis over the frequency band 0.994–1.006 cpd for the period 2002–21. (b) As in (a), but for the amplitude (blue, in K, plotted on a log10 scale) and phase (red, in degrees) at the equator. The dashed line in (b) indicates the rate of amplitude growth with altitude for a conservative wave, exp(0.5·sh). The base point used for the coherence analysis is denoted by the + symbol.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
According to Fig. 3, the amplitude of DW1 becomes very large, ∼12 K, in the tropical MLT, above about 11 sh (∼80 km). The vertical wavelength at the equator, best appreciated in Fig. 3b, is about 25 km in the stratosphere, below about 5 sh, and again in the mesosphere, above 9 sh. This value is in good agreement with the vertical wavelength of the wavenumber 1, latitudinally gravest symmetric propagating mode (1, 1) of classical tidal theory (Chapman and Lindzen 1970). In the upper stratosphere and lower mesosphere, between about 5 and 9 sh (35–65 km), phase progression with altitude appears much slower and nonuniform because the tide there is a superposition of the main propagating mode and the nonpropagating, “negative equivalent depth” modes forced by latitudinally broad stratospheric ozone heating (Chapman and Lindzen 1970). The latitudinal structure above the middle mesosphere (∼10 sh, 70 km) shown in Fig. 3a is also consistent with linear theory, having maxima centered on the equator and at approximately ±30°–35° latitude, with nodes at about ±15°–20°. These correspond quite closely with the latitudinal structure shown in Chapman and Lindzen (1970).
Figure 3 shows that the amplitude of DW1 grows more or less continuously up to about 12 sh (∼85 km). Amplitude variations in this region might be attributable to the vertical variation of temperature (Sakazaki et al. 2013) or to interference between propagating and nonpropagating tidal modes. The dashed black line in Fig. 3b indicates the growth rate with altitude expected for a nondissipating wave, exp(0.5·sh). The amplitude of DW1 seen by SABER follows this curve more or less closely from 2.5 to 12 sh, suggesting that DW1 undergoes negligible dissipation in that range of altitude. Above 12 sh (∼85 km) the amplitude becomes nearly constant, which indicates dissipation. As noted earlier, Lindzen (1967, 1968) argued from calculations based on classical linear tidal theory that DW1 should become convectively unstable and “break” above 85 km. The prediction from linear theory appears consistent with SABER observations. This topic is revisited in section 5, where the breaking of DW1 in the MLT is illustrated, and its seasonal variation and its impact on the zonal-mean state is discussed.
4. Semiannual and quasi-biennial variability of DW1
To document the seasonal and interannual variability of DW1, the temperature field is synthesized between frequencies 0.994 and 1.006 cpd. As noted earlier (cf. Fig. 2), this band includes semiannual, annual and quasi-biennial low-frequency variability. Synthesis over this narrow range of frequency does not allow representation of the temperature field on a daily basis, so only its slowly varying amplitude is considered here.
a. Semiannual variability
The amplitude of DW1 is shown in Fig. 4 as a function of time and latitude at various altitudes, from the middle stratosphere (4 sh) to the lower thermosphere (14 sh). In the middle stratosphere (4 sh) and in the mesosphere (10 sh), the amplitude maximum moves between the Northern and Southern Hemispheres with the seasonal cycle. At the equator, this seasonal migration produces a semiannual variation with maxima near the equinoxes. In the upper stratosphere (7 sh) the tropically confined structure seen at lower and higher altitudes is replaced by a latitudinally broad structure that can be described in terms of the classical, nonpropagating modes excited by stratospheric ozone heating (cf. Fig. 3a and Fig. 1 of Gu and Du 2018). The seasonality of DW1 at this level is mainly annual, with maxima in the summer hemisphere. The influence of the nonpropagating modes appears to extend down into the middle stratosphere (4 sh), where an annual amplitude variation maximizing in local summer is seen poleward of about ±25°. In the lower thermosphere (14 sh), DW1 displays a semiannual variation at all latitudes, with maxima near the equinoxes; this contrasts with the behavior at lower altitudes (4, 10 sh), where the semiannual variability seen near the equator is the result of the seasonal migration of the amplitude maximum. The amplitude of DW1 at 14 sh has maxima in the tropics and in the subtropics, near ±30° latitude; from Fig. 3a, the subtropical maxima are in phase with each other (and out of phase with the tropical maximum), which is consistent with the gravest symmetric mode (1, 1) of linear theory.
Amplitude (in K) of the migrating diurnal tide as a function of latitude and time at different altitudes (indicated by the insets in each panel) synthesized over the frequency band 0.994 and 1.006 cpd.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
The variability of DW1 at the equator is predominantly semiannual at most altitudes. This can be appreciated more readily in Fig. 5, which shows the amplitude of DW1 (on a logarithmic scale) as a function of time and altitude. Below about 13 sh the semiannual variation is the result of the seasonal migration of the tidal amplitude maximum across the equator (cf. Fig. 4, at 4 and 10 sh). Previous studies have attempted to link the semiannual variation of DW1 to the seasonal variation of the background mean state either by demonstrating strong correlations between the two (Liu et al. 2016) or by numerical modeling in background states that include or exclude seasonal variations (McLandress 2002b; Ortland 2017). While there is no doubt that the amplitude of DW1 is well correlated with the semiannual oscillation (SAO) of the tropical zonal-mean zonal wind or with quantities derived from it, such as the potential vorticity gradient, correlations between DW1 and the SAO could arise simply because both exhibit seasonal variations caused by the annual cycle in solar or convective heating. Riggin and Lieberman (2013) showed that semiannual variability in DW1 is apparent at altitudes as low as the tropopause and argued that seasonal changes in the sources of tidal excitation can explain this seasonal variation “without invoking… nonlinear effects or wave-mean flow interaction.” The behavior of DW1 documented by SABER in the stratosphere and mesosphere is consistent with this view insofar as its amplitude maximum migrates seasonally, producing equinoctial semiannual maxima at the equator. In addition, the spectrum shown in Fig. 2 also supports Riggin and Lieberman’s (2013) assertion that semiannual variability in DW1 is already present near the tropical tropopause (although the lowest altitude documented by SABER is 2.5 sh, about 17.6 km).
Amplitude (in K, log10 scale) of the migrating diurnal tide as a function of altitude and time synthesized over the frequency band 0.994 and 1.006 cpd.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
On the other hand, the comprehensive set of numerical experiments carried out by McLandress (2002a,b) demonstrated a clear dependence of DW1 on the zonal wind SAO, only a minor role for the seasonal variability of diurnal heating, and no apparent role for interactions between the tide and mesoscale gravity waves. The importance of the SAO is confirmed by the work of Ortland (2017), who found that the semiannual variation of DW1 amplitude was greatly enhanced by changes in the background winds, and emphasized the possible role of ozone variations associated with the SAO. McLandress (2002b) used a linearized model derived from the Canadian Middle Atmosphere Model (CMAM) to show that the zonal wind SAO is responsible for the semiannual amplitude variation of DW1 in the MLT at and above 70 km; and that the behavior of the tide at 70 and 98 km (∼10 and 14.5 sh) could be understood in terms of the behavior of the first symmetric (1, 1) and antisymmetric (1, 2) vertically propagating modes of linear theory. McLandress (2002b) argued that, at 70 km, both modes have substantial amplitude and interfere such as to produce an amplitude maximum in the winter hemisphere, whereas at 98 km the antisymmetric mode has small amplitude and the seasonal variation of DW1 is semiannual everywhere. This difference in the seasonal cycle of DW1 is consistent with SABER data (cf. Fig. 4 at 10 and 14 sh). McLandress (2002b) attributed the attenuation of the (1, 2) mode with altitude to stronger diffusive damping because that mode has a considerably shorter theoretical vertical wavelength (∼15 km) than the (1, 1) mode (∼25 km). However, he noted that “the dominant tidal damping mechanism in the CMAM is nonlinear interactions with resolved waves, which are not necessarily well represented by vertical diffusion.” Be that as it may, extensive breaking of DW1 at and above 12 sh (∼85 km) is documented in section 5. Such would be expected to produce substantial vertical mixing, and would be consistent with the notion that eddy diffusion plays a role in damping the (1, 2) mode.
While the semiannual variation of DW1 as seen by SABER supports the findings by McLandress (2002b) and Ortland (2017) that it arises from interaction of the tide with the tropical SAO, at least above 10 sh (70 km), it is also consistent with Riggin and Lieberman’s (2013) hypothesis that DW1 responds to seasonal changes in diurnal heating, since semiannual variability is clearly present well below the range of altitude affected by the SAO. In a study using an extended version of CMAM, Gu and Du (2018) also conclude that seasonal changes in heating contribute to the seasonal variability of DW1 in the stratosphere but cannot explain its semiannual variability in the mesosphere. On the other hand, Sakazaki et al. (2013) argued from linear numerical simulations that zonal wind variations (specifically, the seasonal march of the subtropical jets) can modulate tidal amplitude down to the middle stratosphere. Further sensitivity studies with numerical models will be required to resolve this ambiguity and determine more precisely what mechanisms, over what ranges of altitude, contribute to the semiannual variability of DW1.
b. Quasi-biennial variability
As discussed above and illustrated in Fig. 5, the seasonal variability of DW1 at the equator is semiannual throughout most of the middle atmosphere. A closer look at that figure also reveals the presence of interannual variability, especially in the middle and lower stratosphere. Some hint of such variability is also seen as higher altitudes, but it is obscured by the very strong semiannual variation. To better appreciate the interannual changes in DW1, Fig. 6 shows its amplitude synthesized over the narrower frequency band 0.9975–1.0025 cpd, which includes only variability slower than 1 year (cf. Fig. 2). The equatorial zonal-mean zonal wind derived from SABER geopotential (see section 2) and also filtered to periods greater than 1 year, is superimposed in the range of altitude 2.5–6 sh, where the tropical quasi-biennial wind oscillation (QBO) is present. It was noted in section 2 that zonal-mean behavior can be contaminated by aliasing of the semidiurnal migrating tide. However, the possibility of such aliasing would be important only above the upper mesosphere and is not a concern for the range of altitude of the QBO winds.
As in Fig. 5, but low passed to periods longer than a year. The amplitude contours are logarithmic (log10|T|). The zonal-mean zonal wind U in the region dominated by the tropical QBO (also low passed) is superimposed as black contours with 5 m s−1 spacing; dashed contours indicate negative values of U. Dashed vertical lines denote behavior associated with the QBO “disruption” events of 2016 and 2020. See text for details.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
Above the stratopause, there is an obvious relationship between the QBO and DW1; the amplitude of the tide is largest whenever the QBO wind is westerly in the lower stratosphere, below about 4–4.5 sh (∼27–30 km). The relationship is robust throughout the SABER period; note, in particular, how the QBO disruption events of 2016 and 2020 (Newman et al. 2016; Osprey et al. 2016; Kang et al. 2020; Anstey et al. 2021), indicated by dashed lines normal to the time axis, are reflected in the amplitude of DW1 in the MLT. Results similar to those shown in Fig. 6 have been obtained by Gan et al. (2014) from SABER temperature data for 2002–10 and Dhadly et al. (2018) using wind data for 2002–16 from the TIMED Doppler Interferometer (TIDI). Earlier studies using a shorter SABER record (Xu et al. 2009) could not determine whether the interannual variability of DW1 was truly quasi biennial, but the present results confirm the more recent work and leave no doubt that DW1 exhibits substantial quasi-biennial variability that appears to be linked to the stratospheric wind QBO. The quasi-biennial variability of DW1 can exceed 50% at some altitudes, as can be appreciated in Fig. 7, which shows DW1 amplitude filtered to periods longer than a year at three altitudes, from the upper mesosphere (10 sh) to the lower thermosphere (15 sh). For example, at 12 sh (Fig. 7, middle panel) the amplitude of DW1 at the equator varies from about 14 K when the stratospheric QBO wind in midstratosphere is westerly (cf. Fig. 6) to about 7 K when the QBO wind is easterly.
As in Fig. 4, but low passed to periods longer than a year.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
Quasi-biennial variability in DW1 could be caused by variability in heating sources or by wave–mean flow interaction. Most modeling studies discount changes in heating and emphasize instead the importance of variations in the background zonal wind (Hagan et al. 1999; Davis et al. 2013; Mayr and Mengel 2005; Dhadly et al. 2018; Pramitha et al. 2021a,b). There is a well-known and understood quasi-biennial variation in stratospheric ozone (see, e.g., Chipperfield et al. 1994; Tian et al. 2006; Park et al. 2017; Zhang et al. 2021); however, the ozone QBO is large in the middle stratosphere but negligible near the stratopause, where ozone heating is largest (Hagan et al. 1999). The interannual variability of SABER ozone and its relationship to SABER zonal-mean zonal wind is shown in Fig. 8 at the equator, between 3 and 7 sh (∼21–48 km). Both fields are low passed to periods greater than 1 year, and the time mean of the ozone field has been removed to emphasize its variability. There is a clear quasi-biennial variation in ozone between ∼4.5 and 5.5 sh (∼30–36 km, Fig. 8). Maxima occur when the shear of the zonal wind QBO is easterly in that range of altitude (and QBO westerlies are found below about 4 sh).
Evolution of SABER zonal-mean ozone (ppmv) anomalies with respect to the time mean (shaded contours, color bar) as a function of altitude and time at the equator. The SABER zonal-mean zonal wind is superimposed as black contours with a contour interval of 10 m s−1; negative values of the wind are denoted by dashed contours. Both fields are low passed to periods longer than 1 year.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
The relationship between the ozone QBO and the easterly shear zone of the zonal wind QBO arises from the fact that the QBO secondary circulation is upward in the region of easterly shear and will tend to reduce the abundance of odd nitrogen (NOx = NO + NO2) by advection of NOx-poor air from lower altitudes. NOx drives the principal catalytic loss cycle for ozone in the middle stratosphere, where ozone itself is in photochemical equilibrium (Brasseur and Solomon 1986). In the lower stratosphere, below ∼4 sh (27 km), a second quasi-biennial signal in ozone is present, disjoint from that in the middle stratosphere. In this case, ozone maxima occur along regions of westerly shear, where the QBO secondary circulation is downward. At these altitudes, the photochemical lifetime of the Ox family (O + O3) is long, Ox is dynamically controlled (Brasseur and Solomon 1986), and downwelling in the westerly shear zone of the wind QBO advects ozone-rich air from higher levels.
In midstratosphere (∼4.5–5 sh) the QBO variation of ozone shown in Fig. 8 is generally smaller than 2 ppmv peak to peak, which is about 20% of the time mean at 5 sh (∼11 ppmv). Ozone heating in this region is ∼3–4 K day−1. This is at most one-third of the maximum heating rate near the stratopause (∼12 K day−1), but by no means negligible. However, a 20% variation in ozone, which would lead to a commensurate variation in shortwave heating, is considerably smaller than the quasi-biennial variation in DW1 amplitude, which can be as large as 50%, as noted above in connection with Fig. 7. Perhaps more importantly, shortwave heating due to ozone in the stratosphere is latitudinally broad and will not project efficiently on the latitudinally narrow propagating mode of the diurnal tide (Chapman and Lindzen 1970). Thus, Hagan et al.’s (1999) conclusion that the QBO in stratospheric ozone is not responsible for the quasi-biennial variability of DW1 in the MLT is prima facie consistent with SABER observations. That would leave dynamical mechanisms, such as interaction of the tide with the background wind field (Hagan et al. 1999) or filtering of the spectrum of vertically propagating mesoscale gravity waves (Mayr and Mengel 2005), as the leading hypotheses for quasi-biennial variability of the diurnal tide.
5. Breaking of the diurnal tide
It was shown in Fig. 3 that the diurnal tide appears to propagate conservatively up to about 12 sh (∼85 km), that is, its amplitude grows approximately as exp(0.5·sh), as expected for a nondissipating wave; above 12 sh, the amplitude becomes constant, suggesting that the tide dissipates above that level. Over 50 years ago, Lindzen (1967, 1968) hypothesized that the migrating diurnal tide would become unstable and break in the lower thermosphere, and went on to compute the zonal-mean acceleration and eddy diffusivity induced by breaking (Lindzen 1981). Following Lindzen’s early work, the effects of tidal breaking were estimated by Hines (1972), Groves and Forbes (1985), and Vial and Teitelbaum (1986) based on theoretical models of various complexity; and by Lieberman and Hays (1994) and Lieberman (1997) using observations interpreted in the context of tidal theory. More recent studies (e.g., McLandress 2002a; Mayr and Mengel 2005; Lu et al. 2012; Yiğit and Medvedev 2017) have emphasized the impact of breaking mesoscale gravity waves on the tide rather than breaking of the tide itself. This might be due to the fact that observations that show tidal breaking have not been readily available. However, synthesis of the full spectrum of SABER data, which allows twice-daily temporal resolution, makes it possible to examine directly the potential temperature field for evidence of breaking.
The SABER potential temperature field, zonal mean plus zonal wavenumber 1, synthesized over the entire frequency range of the SABER synoptic spectrum on 2 days, 30 Mar 2012 and 14 Mar 2013. Above about 12 scale heights, there are extensive regions (enclosed by the red dashed contours) of reversed vertical gradient, indicative of convective instability, on both days.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
One might ask whether the behavior shown in Fig. 9 is confined to the equinoxes, when the amplitude of DW1 is considerably larger than it is at the solstices (Fig. 5). To examine the seasonal variability of breaking, two measures of convective instability were calculated: the number of instances per 5-day bin when the potential temperature gradient, sampled with twice-daily cadence, is reversed above 12 sh; and the average value (in K km−1), over the same 5-day bins, of the potential temperature gradient within the unstable regions. These diagnostics are shown in Fig. 10a and indicate that breaking is common at the equinoxes, about 3 times per 5-day period in Northern Hemisphere spring and twice per 5 days in fall, but much less frequent near the solstices (less than one event per 5 days). The average magnitude of the reversed potential temperature gradient is larger at the equinoxes (about −60 to −70 K km−1) than at the solstices (−45 K km−1). In addition, the average depth of convectively unstable regions (not shown) is considerably greater at the equinoxes (3–4 km) than at the solstices (∼2 km). Similar results are obtained off the equator, at latitudes within ±10°, where DW1 remains large (cf. Fig. 3). From these results, one may conclude that frequent breaking of DW1 near the equator is mainly an equinoctial phenomenon, although the periods when breaking is common span a couple of months around each equinox.
(a) Climatology (2002–21) of two diagnostics of tidal breaking above 12 sh: the number of potential temperature reversal events within 5-day bins (black) and the mean value of the temperature gradient reversals in each 5-day bin (blue). (b) Climatological seasonal cycle of DW1 amplitude at 12 sh; the blue bars indicate ±1 standard deviation from the climatological mean. The frequency and intensity diagnostics in (a) follow the seasonal cycle of the tide’s amplitude.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
Note also that breaking is more frequent, and potential temperature gradient reversals somewhat larger, during Northern Hemisphere spring than during fall. This behavior follows the climatological annual cycle of DW1 amplitude at 12 sh, shown in Fig. 10b, which also shows that the variability of the tide with respect to its climatology is very large. The standard deviation (blue bars in Fig. 10b) is comparable to the mean value throughout the year, and indicates that tidal amplitudes of nearly 30 K in Northern Hemisphere spring, and over 20 K in fall, are common on daily time scales. A hypothesis to explain the difference between the equinoctial maxima is lacking, but the behavior is reminiscent of the SAO in tropical zonal wind, whose amplitude is larger during the first half of the year (e.g., Garcia et al. 1997).
One might also ask, What are the consequences of tidal breaking for the climatology of the background atmosphere? SABER temperature data indicate that the zonal-mean temperature and zonal wind structure in the tropics are strongly affected by breaking. Figure 11a shows SABER zonal-mean temperature at the equator, as a function of altitude and time for 2010–15 (this is typical of the entire dataset; only 5 years are shown to enhance readability). The zonal-mean temperature is synthesized over the frequency range 0.0 to 0.0066 cpd, which includes periods longer than ∼150 days. A large semiannual oscillation dominates the MLT, with a warm-over-cold dipole seen between about 10 and 14 sh at the equinoxes. The temperature distribution in the meridional plane is shown in Fig. 11b for 30 March 2010, a time (indicated by the dotted line in Fig. 11a) when the dipole is particularly prominent. The vertical structure of the dipole is consistent with the secondary circulation forced by the breaking of DW1 near 12 sh; that is, the easterly zonal-mean zonal force due to breaking should induce upwelling below and downwelling above the location where it is applied. The secondary circulation is indicated schematically in Fig. 11b (red arrows), where it can be seen that the cold half of the dipole coincides with the upwelling anomaly and the warm half coincides with the downwelling anomaly.
(a) SABER zonal-mean temperature at the equator as a function of altitude and time, low passed to periods longer than 150 days. A semiannual oscillation is evident in the upper mesosphere, between ∼10 and 13 sh. (b) The meridional distribution of zonal-mean temperature on 30 Mar 2010, the date indicated by the yellow dotted line in (a). The circulation forced by easterly acceleration (E) due to breaking of DW1 near 12 sh is indicated schematically by the red arrows.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
Smith et al. (2017, their Fig. 6) have documented a strong semiannual oscillation of the zonal-mean zonal wind in the tropical upper mesosphere (0.01 hPa, ∼12 sh), with easterlies at the equinoxes. The same result is obtained throughout the longer SABER dataset examined here (not shown). These equinoctial easterlies are geostrophically consistent with the temperature pattern seen in Fig. 11. As shown by Smith et al. (2017), the magnitude of the MLT easterlies varies between about −30 and −50 m s−1. The acceleration imparted by the breaking of DW1 has been estimated by Lindzen (1981) to be about −16 m s−1 day−1 near the equator from a simple “wave saturation” argument, and by Lieberman (1997) to vary between −2 and −16 m s−1 day−1 from an analysis of UARS/HRDI data, with the larger accelerations occurring at the equinoxes. As shown in the appendix, an acceleration of −16 m s−1 day−1 is consistent with the temperature anomalies shown in Fig. 11 (and the corresponding zonal-mean easterlies). This all suggests that breaking of the diurnal tide may be the major driver of the easterly phase of the SAO in the upper mesosphere. It is interesting in this context that the zonal-mean zonal wind cannot be calculated reliably from SABER geopotential height via the balance wind equation much above 12 sh, as mentioned in section 2 and explained in detail by Smith et al. (2017). Smith et al. noted that the contribution of eddy flux divergences to the balance wind equation could not be neglected in the tropical upper mesosphere, “where the migrating diurnal tide reaches large amplitude.” The present results suggest that eddy flux divergences might become large above 12 sh due to the breaking of DW1 at those altitudes.
6. Summary and discussion
This study documents the seasonal and interannual variability of the diurnal, westward-migrating temperature tide, DW1, using nearly 20 years of data (January 2002–December 2021) from the SABER instrument on board NASA’s TIMED satellite. The data were processed using Salby’s (1982b) fast Fourier synoptic mapping algorithm, which yields a synoptic spectrum with Nyquist frequency limits of approximately ±1 cpd and six zonal wavenumbers, as explained in section 2. In particular, DW1 is fully resolved by SABER and can be reconstructed from the spectrum at twice-daily cadence.
The results confirm previous studies of DW1 (Xu et al. 2009; Lieberman et al. 2010; Gan et al. 2014; Dhadly et al. 2018; etc.) as regards the overall structure of the tide and its temporal variability. The very long SABER record examined here allows excellent resolution of the low-frequency variability of DW1, and shows that it is predominantly annual and semiannual, but with a substantial quasi-biennial component (Figs. 4–7). The semiannual variation in DW1 has been attributed to the effect of the tropical zonal wind SAO (McLandress 2002b; Ortland 2017) and the results shown here are consistent with those studies in the MLT. However, semiannual variability is present in DW1 throughout the range of altitude sampled by SABER, down to 2.5 sh (∼17.5 km), well below the range of altitude where the tropical zonal wind is dominated by the SAO. In this sense, the behavior of DW1 documented here is also consistent with Riggin and Lieberman’s (2013) hypothesis that semiannual variability in shortwave diurnal heating in the troposphere can explain the behavior of DW1 amplitude without invoking wave–mean flow interaction.
The quasi-biennial variability of DW1 is related to the behavior of the QBO in tropical zonal wind, such that maxima in DW1 in the MLT are synchronous with the occurrence of QBO westerlies below about 4.5 sh. The variability is also closely associated with the quasi-biennial oscillation in ozone in the middle stratosphere (Fig. 8). However, the magnitude of the ozone QBO is generally no larger than 20% of the time mean, which is considerably smaller than the magnitude of the quasi-biennial variability of DW1 in the MLT (∼50%, Fig. 7). For this and other reasons discussed above, the present results appear consistent with Hagan et al.’s (1999) conclusion that quasi-biennial variations of DW1 in the MLT are unlikely to be driven by the ozone QBO.
A new finding of this study is the direct confirmation that DW1 dissipates due to convective instability (“wave breaking”) at and above the tropical mesopause (12 sh, ∼85 km), much as mesoscale gravity waves are known to do, but on a much vaster scale of many thousand kilometers. SABER data show that DW1 grows with altitude as expected from a nondissipating wave up to 12 sh, but growth ceases above that level (Fig. 3). On daily time scales, there are repeated instances when the vertical gradient of potential temperature becomes negative above 12 sh (Fig. 9). These events are frequent at the equinoxes, when the amplitude of DW1 is largest and relatively rare (and weaker) at the solstices, when the amplitude of the tide is smaller (Fig. 10). The possibility that DW1 breaks in the upper mesosphere was first proposed by Lindzen (1967, 1968). Lindzen (1981) went on to calculate the zonal-mean acceleration that would be produced (about −16 m s−1 day−1); this acceleration appears to be sufficient to drive the easterly phase of the mesopause SAO in zonal-mean temperature and zonal wind at the equinoxes (cf. Lieberman 1997). SABER zonal-mean zonal winds in the upper mesosphere are strongly easterly at equinox (Smith et al. 2017) and, as shown here, are associated with a cold-over-warm zonal-mean temperature dipole (Fig. 11) that is dynamically consistent with the zonal wind and with the secondary meridional circulation induced by dissipation of DW1 near the tropical mesopause. All in all, the evidence for convective breaking of DW1 obtained from SABER observations is clear and compelling.
To conclude, a few remarks on the potential impact of tidal breaking on the generation of secondary gravity waves (Vadas 2007, 2013; Vadas et al. 2014; Karlsson and Becker 2016; Bossert et al. 2017; Yasui et al. 2018; Becker and Vadas 2018; Lund et al. 2020; etc.) are in order. The term “secondary waves” refers to waves that are generated in situ above the tropopause, either through instabilities of the background state or through large body forces associated with the breaking of “primary waves” (waves excited in the troposphere through mechanisms such as flow over topography, deep convection or frontal processes). Recent attention to secondary wave generation has been motivated in part by the notion that such waves might couple the middle atmosphere to electrodynamic phenomena in the ionosphere, such as traveling ionospheric disturbances and equatorial plasma bubbles (e.g., Vadas 2007; Vadas et al. 2014; Miyoshi et al. 2018; Liu 2020).
This study shows that DW1 breaks in the tropical upper mesosphere (Fig. 9), on spatial scales that can be as large as 100° of longitude (∼11 000 km) in the horizontal and several kilometers in the vertical, and that breaking takes place within a day (as deduced by inspection of individual breaking events). Such breaking might excite secondary gravity waves of long horizontal and vertical wavelength and fast phase velocity, able to survive the highly dissipative environment of the thermosphere (Vadas 2007). However, there is apparently no evidence that secondary gravity waves are generated by tidal breaking near equinox; instead, existing studies point to greater gravity wave activity at the solstices than at the equinoxes (e.g., Liu et al. 2017; Cullens et al. 2022). It is possible that waves excited by tidal breaking have spatial scales that are too large to be detectable using current methods for estimating gravity wave activity, which emphasize small- and medium-scale waves. Ideally, this problem could be investigated with high-resolution numerical models able to simulate the breaking of the diurnal tide observed by SABER and any secondary gravity waves that might be excited thereby.
Acknowledgments.
This paper is dedicated to the memory of Dr. M. L. Salby, colleague and friend, whose work and insights on the information content of asynoptic satellite data undergird the analysis presented here. I wish to thank Drs. A. K. Smith, R. S. Lieberman, W. J. Randel, and H.-L. Liu, as well as three anonymous reviewers, for their comments and critique of the original version of the paper. The National Center for Atmospheric Research (NCAR) is sponsored by the U.S. National Science Foundation (NSF). Computing resources were provided by NCAR’s Climate Simulation Laboratory, sponsored by NSF and other agencies. This research was enabled by the computational and storage resources of NCAR’s Computational and Information Systems Laboratory (CISL). This work was supported in part by NASA Grant 80NSSC19K1214.
Data availability statement.
The data used in this study are available for download from GATS, Inc.: http://saber.gats-inc.com/data_services.php.
APPENDIX
Forcing of Mesopause Temperature Anomalies by DW1
The temperature anomaly given by Eq. (A13) can be evaluated by assigning appropriate values to the various parameters. For the upper mesosphere, S = 3 × 10−3 K m−1 (e.g., Sica et al. 2007); the Newtonian cooling coefficient α near the mesopause is set to 0.2 day−1 (Wehrbein and Leovy 1982); and the equatorial value of β is 2.3 × 10−11 m−1 s−1. For F0 Lindzen’s (1981) value of −16 m s−2 day−1 is adopted. The horizontal and vertical scales can be estimated from the structure of the temperature SAO, shown in Fig. A1a, which was obtained by coherence analysis over the frequency band 0.0044–0.0063 cpd (period of 157–227 days). From that figure, the horizontal scale is about 12° of latitude, or L ∼ 1.33 × 103 km, and the vertical scale is about 12 km. With these choices, (A7) yields the equatorial temperature anomaly profile shown in Fig. A1b, which is similar to the vertical distribution and magnitude of the temperature SAO seen in SABER data.
(a) Structure of the zonal-mean temperature SAO observed by SABER, calculated by coherence analysis over the frequency band 0.0044 to 0.0063 cpd (period range: 157–227 days). Amplitude (in K) is denoted by the color contours and phase (in degrees) by the superimposed black line contours. The + sign denotes the base point for the coherence calculation. (b) Vertical profile of zonal-mean temperature anomaly at the equator calculated from Eq. (A12) using the parameters indicated in the inset.
Citation: Journal of the Atmospheric Sciences 80, 3; 10.1175/JAS-D-22-0167.1
While the zonal-mean temperature response derived from (A13) is sensitive to the choice of parameters, this simple analysis shows that a vertical profile consistent with the observed temperature SAO can be obtained using reasonable values for those parameters and a zonal-mean forcing of −16 m s−1. Incidentally, both Figs. 11 and A1a indicate that the mesopause temperature SAO does not propagate downward. In particular, Fig. A1a shows two nodes, near 11 and 12.5 sh, separated by a phase shift of ∼180° near 11.5 sh, with little phase change elsewhere in this range of altitude. On the other hand, at lower altitudes in the mesosphere, between about 7 and 10 sh, Fig. A1a indicates a more regular phase change with altitude, suggesting that the temperature SAO is a propagating phenomenon in that altitude range.
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