## 1. Introduction

In the mesosphere and lower thermosphere (MLT), tidal waves and planetary-scale waves are the leading modes of variability and have important global effects on circulation and photochemistry (e.g., Forbes et al. 1993, 1997; Zhu et al. 2000). This makes the systematic and comprehensive studies of global change in the MLT from satellite measurements especially valuable. One of the fundamental problems in large-scale atmospheric dynamics is how the background zonal mean and wave fields interact with each other. The problem of interaction between the zonal mean state and tidal waves can be studied by theoretical or numerical modeling with realistic thermal forcing (e.g., Forbes and Hagan 1988; Forbes and Vial 1989; Wood and Andrews 1997; McLandress 2002a, b; Ortland 2005a, b). By means of sensitivity experiments with different parameter settings, such as the magnitude of the zonal wind and eddy viscosity, various aspects of the interaction between the zonal mean state and tidal waves can be investigated with analytical or numerical models. The interaction between tidal waves and zonal mean flow can also be studied by observational analysis (e.g., Lieberman and Hays 1994; Talaat et al. 2001). Given the observed tidal wind from the Upper Atmosphere Research Satellite High Resolution Doppler Imager (UARS/HRDI), Lieberman and Hays (1994) estimated the momentum deposition based on the derived velocity correlation terms that characterize the effect of tides on the zonal mean flow. When both satellite observations and numerical models are assimilated in a diagnostic analysis, additional physical insights or modeling parameters can be gained (e.g., Akmaev 1997; Zhu et al. 2005). Zhu et al. (2005) developed an algorithm for which the zonal mean and tidal temperature fields are derived simultaneously and without bias from the Sounding of the Atmosphere using Broadband Emission Radiometry (SABER) instrument onboard the Thermosphere–Ionosphere–Mesosphere Energetics and Dynamics (TIMED) satellite. The measured zonal mean fields are used as input in a linear spectral tidal model to simulate and compare with the measured seasonal variation of the migrating diurnal tide. The comparison of both models with observations consistently and conclusively shows the dominant effect of the zonal mean wind on the seasonal variation of the migrating diurnal tide in the MLT (Zhu et al. 2005).

Depending on how the problem is formulated, the interaction of the zonal mean state and tidal waves may also be categorized as an effect of the zonal mean state on the variability of tidal waves or an effect of wave forcing on the maintenance and variability of the zonal mean state. Early numerical modeling efforts that used thermal forcing and its variations as the main driver for producing seasonal variations of the diurnal tidal amplitude cannot fully reproduce the major characteristics of the observations (e.g., Groves 1982a, b; Hagan 1996; Hagan et al. 1997; Zhu et al. 1999) that indicate distinct peaks in wind amplitude for the diurnal tide around the equinox (e.g., Hays et al. 1994; Burrage et al. 1995; McLandress et al. 1996; Khattatov et al. 1997). As a result, most diagnostic studies of tides, whether based on numerical modeling or observations, have been concerned with how the background state will affect the variations of tidal waves. The effect of tidal and planetary waves on the zonal mean wind can be investigated by diagnosing the momentum budget or Eliassen–Palm (EP) flux divergence (e.g., Edmon et al. 1980; Palmer 1982; Andrews et al. 1987). In this case, we also need to quantify the wind components of the wave disturbances. Furthermore, the calculation of momentum flux often requires three components of the vector wind, whereas observations typically only give one or two components. Hence, certain approximations associated with the wave structure need to be adopted to derive the momentum deposition from observations. For example, the polarization relation based on classical tidal theory for individual tidal modes can be used to derive the vertical velocity from the measured meridional tidal wind (Lieberman and Hays 1994). However, classical tidal theory assumes zero background zonal wind, whereas it is now believed that the background zonal wind is a major cause of the seasonal variation of the diurnal tidal amplitude (McLandress 2002a; Zhu et al. 2005). To derive the EP flux divergence accurately, we also need to know the tidal temperature and background mean fields for a given tidal wind field. On the other hand, if the only measured field is temperature, we need to derive the corresponding wind fields before the EP flux divergence can be evaluated. A recently developed algorithm extracts the zonal mean and tidal temperature fields simultaneously from TIMED/SABER temperatures with significantly reduced aliasing between the two (Zhu et al. 2005). The main objective of this paper is to develop an algorithm that appropriately derives the tidal wind from the tidal temperature field, making it possible to evaluate the EP flux divergence from the satellite temperature observations.

Wave–mean flow interaction as measured by the momentum flux or EP flux divergence often leads to nonexternally forced variability in the middle atmosphere. The stratospheric quasi-biennial oscillation (QBO; e.g., Baldwin et al. 2001; Pascoe et al. 2005), mesospheric biennial oscillation (MBO; Burrage et al. 1996; Mayr et al. 1997; Sridharan et al. 2007), and stratospheric semidecadal oscillations (Mayr et al. 2007) are good examples of middle atmospheric variability, with periods different from external forcings such as the solar flux variations. Mainly because of its lower air density, the MLT has also been considered a region where the signatures of global change in the lower atmosphere will be amplified and can be used as good indicators for monitoring global change (e.g., Roble and Dickinson 1989; Akmaev and Fomichev 2000). Because the variability of solar activity is a primary natural forcing component and planetary-scale waves are the leading modes of variability in the MLT, characterizing and understanding MLT variability in both temperature and wind (induced by internal wave–mean flow interactions) will reduce uncertainties in determining global change caused by human activity in the lower atmosphere.

In section 2, we introduce a spectral module that can be used to derive all the dynamical fields from TIMED/SABER temperatures. The technical details on how the module is applied to the diagnostic analysis of the wind fields of the tidal and planetary waves are explained in detail. The advantages and disadvantages of the two basic methods of finite difference and spectral expansion are further illustrated. Section 3 shows how the zonal mean wind, tidal temperature and wind, and EP flux divergence are all derived from the temperature measured by the TIMED/SABER instrument. Section 4 provides concluding remarks.

## 2. Diagnostic analysis of tidal winds and the Eliassen–Palm flux divergence based on a spectral module

Given zonal mean and tidal temperature fields derived from observations, as a first-order approximation the zonal mean wind can be calculated from the zonal mean temperature by the thermal wind equation and the tidal winds can be derived from the tidal temperature field by solving the linear wave equations. One critical issue in deriving wind components from temperature perturbations for tidal and planetary waves is eliminating the apparent singularities in the polarization relations derived from the steady wave equations. In the lower middle atmosphere, where the slowly moving planetary waves are the dominant components, the apparent singularities can be removed by replacing the wave polarization relations with the geostrophic relation near the critical latitudes (Randel 1987). When the same approximation is applied to the fast-moving tidal waves in the upper middle atmosphere, significant errors will be induced over a large latitudinal range. The common solution to this problem in numerical models with prescribed thermal forcing is to introduce damping and numerical viscosity, which can effectively eliminate the apparent singularities when the wind components are derived from the temperature or geopotential using the finite difference method. However, this strategy does not work well when the temperature perturbations are derived from satellite measurements, leading to significant error amplifications in the derived wind components. The spectral tidal module used in the Johns Hopkins University Applied Physics Laboratory JHU/APL spectral model (Zhu et al. 1999) takes a totally different approach to eliminate the apparent singularities. As a result, this makes it possible to accurately derive tidal winds and to diagnose wave dynamics (e.g., to diagnose the EP flux divergence) based on TIMED/SABER temperatures.

It should be pointed out that the interaction between zonal mean wind and tidal waves is a second-order, derived quantity with respect to the measured fields. For example, it is the meridional gradient rather than the magnitude of the zonal mean wind (McLandress 2002a; Wood and Andrews 1997) that has a significant effect on the seasonal variation of diurnal tides. It is also well known that the effect of wave disturbances on zonal mean flow depends on the divergence of the momentum flux or the EP flux rather than the flux itself (e.g., Andrews et al. 1987). Furthermore, unlike in the numerical model, the satellite observations are not simultaneous at different locations and usually cannot cover all the spatial and temporal grids of the wave components. As a consequence, diagnostic analysis of tidal winds and wave dynamics by using satellite data is expected to be sensitive to the accuracy and consistency of both the measurements and the processing procedures. We briefly describe two important issues regarding satellite observations that are closely associated with the diagnostic analysis.

First, the satellite observations often do not have complete coverage over a full wave cycle in longitude or time. Such incomplete sampling over a full wave cycle could lead to aliasing problems in digital signal processing when the discrete Fourier transform is used to derive wave components (e.g., Karl 1989). Sampling from satellite orbits is asynchronous in the space–time domain. Salby (1982) extended the sampling theorem of recovering the band limit signal into the space–time spectrum and examined the contaminations arising from irregular sampling. Palo et al. (1997) examined sampling and aliasing issues and estimated the errors for deriving global scale waves from ground-based radar observations. The traditional concepts of sampling and aliasing in Fourier analysis, as described in these studies, refer to aliasing among different frequencies or wavenumbers. Under special circumstances when a dominant wave component exists, the aliasing between the mean state and the wave component becomes an important issue. Zhu et al. (2005) developed a new algorithm that is able to derive the zonal mean and planetary-scale waves, such as migrating tides from satellite measurements, with significantly reduced aliasing between the two. This algorithm has been applied to TIMED/SABER temperatures in the upper middle atmosphere.

Second, the derived temperature fields from the satellite observations usually do not have full latitudinal coverage either. Among the two methods of deriving the wind vector from the temperature measurements, the finite difference is a local operator so that the tidal wind can be locally calculated from the temperature fields measured over any latitudinal range. On the other hand, the spectral method requires complete pole-to-pole latitudinal coverage of the temperature field to perform the spectral expansion. It is found that the shortcomings of the error amplification near the critical latitudes in the finite difference method cannot be easily overcome in diagnostic analysis of tidal waves. Therefore, we have to rely on the spectral method to accurately derive the wind vector from the measured temperature field. By appropriately filling the data gaps in the high-latitude regions, we are able to develop a spectral module for the diagnostic analysis of tidal winds.

**F**in the meridional plane are given byand the divergence is defined aswhere

*u*,

*υ*, and

*w*are the zonal (

*λ*), meridional (

*ϕ*), and vertical (

*z*) velocities;

*θ*is the potential temperature given by

*Te*

^{κz/H}, with

*T*being the temperature; (

*w*

*ρ*

_{0}is the basic state density;

*f*is the Coriolis parameter given by 2Ω sin

*ϕ*, with Ω being the earth’s rotation rate; and

*H*and

*R*are the mean scale height and gas constant, respectively. The subscripts

*t*,

*λ*,

*ϕ*, and

*z*denote time, longitude, latitude, and altitude derivatives, respectively; also,

*is the time derivative following the zonal mean flow,and*D

*X*′,

*Y*′, and

*Q*′ represent the forcing and numerical dissipation processes. All the other symbols in Eqs. (1)–(8) have their usual meaning and follow Andrews et al. (1987).

*u*′,

*υ*′,

*w*′, Φ′) and can be solved by different schemes for given thermal forcing and background states, as briefly reviewed in Zhu et al. (1999). The purpose of the current diagnostic study is to derive the dynamic fields of the tidal waves and the EP flux divergence based on TIMED/SABER temperatures. Because this is a

*localized*diagnostic analysis for a given temperature rather than a wave response to a remote thermal forcing, the effect of the right-hand-side forcing terms in (5)–(7) is expected to be small and those terms will be neglected. Mathematically, this means that the physical state can still be described by a wave equation within one cycle of oscillation. As a result, the procedure for solving Eqs. (5)–(8) becomes simpler. First, the geopotential (

*,*T

*T*′) are related through the hydrostatic equation:Second, the zonal mean wind (

*) can be derived from*u

*,*T

*) and the perturbation geopotential (Φ′) solved from (10b), the remaining three dynamical quantities (*u

*u*′,

*υ*′,

*w*′) can be derived by solving the linear perturbation Eqs. (5)–(8). Assuming a tidal wave solution of the formand substituting (11) into Eqs. (5)–(8), we have (Zhu et al. 1999)where

*s*and

*σ*are the zonal wavenumber and frequency, respectively. The detailed expressions for the coefficients in Eqs. (12)–(14) and the corresponding matrix coefficients for the spectral module shown below are given in Zhu (1997). For migrating diurnal and semidiurnal tides, (

*s*,

*σ*) are equal to (1, −0.5) and (2, −1), respectively. For 2-day waves, (

*s*,

*σ*) = (3, −0.25). The complex coefficients {

*A*

_{ij}} on the left-hand side of Eqs. (12)–(14) are functions of tidal wavenumber and frequency (

*s*,

*σ*) and background state (

*θ*

*). It can be shown by scale analysis that |*u

*a*

^{−1}

*θ*

_{ϕ}

*υ*′|/|

*θ*

_{z}

*w*′| ∼ 10

^{−4}|

*υ̂*/

*ŵ*| in the middle atmosphere. For migrating tides |

*υ̂*/

*ŵ*| ∼ 10

^{2}, so we have neglected

*a*

^{−1}

*θ*

_{ϕ}

*υ*′ in (7) while deriving (14), and the first-guess

*ŵ*(denoted

*ŵ*

_{1}) can be directly calculated from (14) for the derived Φ̂ based on the measured perturbation temperature. Such an approximation also decouples (12) and (13) from (14) so that we can solve Eqs. (12) and (13) for (

*û*,

*υ̂*), where both Φ̂ and

*ŵ*are considered to be known in the right-hand-side forcing terms. We have also added the biharmonic terms in (12) and (13) for suppressing the small-scale noise due to truncation when (12) and (13) are solved by a spectral module. Once

*û*and

*υ̂*are solved from (12) and (13), they are substituted into the continuity Eq. (15) to derive the second-guess

*ŵ*(denoted

*ŵ*

_{2}). The final or diagnostic solution for

*ŵ*is an equally weighted average of

*ŵ*

_{1}and

*ŵ*

_{2}.

The rationale of using such a solution procedure is as follows: In the diagnostic analysis in which the geopotential has been solved from the measured temperature field, the four Eqs. (12)–(15) contain only three unknowns (*û*, *υ̂*, *ŵ*). Therefore, Eqs. (12)–(15) constitute an overdetermined system. Because *û* and *υ̂* are of the same order of magnitude (Zhu and Yee 1999) and two Eqs. (12) and (13) [or (5) and (6)] are nearly symmetric with respect to *û* and *υ̂*, it is appropriate to solve (12) and (13) to obtain (*û*,*υ̂*). Simple scale analysis shows that |_{z}*w*′|/|2Ω*σu*′| ∼ 10|*ŵ*/*υ̂*| ∼ 0.1 in the middle atmosphere, suggesting that _{z}*ŵ* in (12) only makes a minor contribution to the horizontal tidal wind (*û*, *υ̂*). Such a correction term is included by using *ŵ*_{1} derived from the energy Eq. (14). Our numerical experiments show that *ŵ* is not sensitive to the relative weights between *ŵ*_{1} and *ŵ*_{2}, indicating an appropriate solution procedure to the overdetermined system (12)–(15).

It is well known in the numerical modeling of tides that the direct solution of an inviscid model [(5)–(8) or (12)–(15)] by finite difference in the meridional direction will encounter apparent singularities at which the determinant of the left-hand side coefficient matrix of the first three equations vanishes (Zhu et al. 1999). The same numerical problem with apparent singularities also occurs at the latitudes where Δ ≡ *A*_{12}*A*_{21} − *A*^{2}_{22} = 0 when (12) and (13) are solved for (*û*, *υ̂*). When Eqs. (12) and (13) are solved by the finite difference method, the errors in the right-hand-side forcing terms will be significantly amplified near the apparent singularities. The amplification results from the small differences in two large terms between two neighboring latitudinal grids in both the numerator and denominator. In numerical modeling, the geopotential field is solved self-consistently for a prescribed thermal forcing. This leads to a well-correlated field among neighboring latitudinal grids so that a relatively small numerical viscosity in the tidal model may effectively remove the apparent singularities (Δ ≠ 0) and lead to an accurate wind field in numerical modeling. On the other hand, the measured temperature field contains measurement and processing errors that are not well correlated among neighboring latitudinal grids. Furthermore, our algorithm (Zhu et al. 2005) derives temperature tidal waves at different latitudes independently. Hence, the error amplification due to the apparent singularities in the finite difference method cannot be removed effectively.

*u*

_{g},

*υ*

_{g}) is the geostrophic wind, (

*u*,

*υ*) is the wind derived from the linear wave equations, and

*ω*

*/(*u

*a*Ω cos

*ϕ*) is the relative angular velocity. Here,

*δ*provides a measure of the model approximation for momentum flux when the geostrophic balance is adopted to derive the wind vector from the temperature field. Setting

*σ*= 0 in (16), we recover Robinson’s (1986) formulation for stationary planetary waves.

Figure 1 shows the plots of *δ* for two different sets of planetary-scale waves for a specified zonal mean wind profile. In the case of stationary planetary waves (Fig. 1a), where *σ* = 0, *δ* quantifies the precision of the geostrophic approximation in comparison with the accurate gradient wind balance. We note that *δ* only slightly deviates from 1 and is insensitive to the zonal wavenumber *s*. Figure 1a suggests that the geostrophic wind approximation would yield a good approximation with errors less than 30% in the momentum flux except at high latitudes. The good approximation using the geostrophic wind, as shown in Fig. 1a, further confirms the feasibility of using the geostrophic wind approximation to replace the gradient wind balance near the critical latitudes, as adopted in Randel (1987) for stationary planetary waves. However, for migrating tides and fast-moving planetary waves (Fig. 1b), *δ* not only deviates greatly from 1 and changes rapidly with latitude but is also very sensitive to the wave frequency and wavenumber. The sensitivity of *δ* with respect to *σ* also implies that we cannot derive the wind field for a given spatial distribution (a snapshot) of the temperature field; it is necessary also to know the frequency (*σ*) in addition to the wavenumber (*s*) to apply the linear wave Eqs. (5)–(8) to derive the wind vector. Mathematically, introducing numerical damping near the critical latitude is equivalent to resetting the Coriolis parameter *f* in (5) and (6) so that Δ ≠ 0, which also removes the apparent singularity (e.g., Hitchman et al. 1987). Physically, such a technique is also equivalent to using an alternative and well-behaved approximation, such as the geostrophic approximation (e.g., Randel 1987), to eliminate the singularities. The fact that |*δ* − 1| is not only large but also changing rapidly with latitude suggests that this kind of technique of removing the apparent singularities may easily lead to inaccurate and sensitive results of EP flux divergence for tides and fast-moving planetary waves. Our extensive numerical experiments confirm this conjecture, and we find that the spectral module is the most appropriate technique for diagnosing the wind vector of the tidal waves from the measured temperature field.

*χ̂*and streamfunction

*ψ̂*, we first form a pair of vorticity and divergence equations from the momentum Eqs. (12) and (13). Next,

*χ̂*and

*ψ̂*are expanded by the associated Legendre polynomials (Zhu et al. 1999). The matrix forms for the spectral coefficients of the vorticity and divergence equations are (Zhu 1997; Zhu et al. 1999)where 𝗟

**Φ̃**is the forcing term derived from the two terms associated with the gradient of the geopotential in (12) and (13) and

**f̃**and

**G̃**are the forcing terms derived from −

*in (12). Detailed expressions and FORTRAN subroutines to calculate terms of the spectral module (17) and (18) are given in Zhu (1997). It is found that 𝗗 and 𝗘 are nearly singular for the diurnal tide and 2-day wave (Zhu et al. 1999). Specifically, for a static and frictionless atmosphere (classical tides) the matrix elements of 𝗗 and 𝗘 are given by (Zhu 1997)where*u

_{z}ŵ*μ*= sin

*ϕ*and

*P*

^{s}

_{n}(

*μ*) represents the associated Legendre functions of order

*s*with degree

*n*(Abramowitz and Stegun 1965). It can be shown that det𝗗 = det𝗘 = 0 for the diurnal tide and 2-day wave—that is, for (

*s*,

*σ*) of (1, −0.5) and (2, −1), respectively. Therefore,

**and**

*ψ̃***χ̃**are solved by directly inverting 𝗛 and indirectly inverting 𝗝:Both 𝗛 and 𝗝 are well behaved for an even number of truncations in the spectral expansion.

It should be pointed out that there is also a disadvantage in the spectral method for the diagnostic analysis based on observations. The observed temperature fields derived from the algorithm do not have an entire latitudinal coverage for the spectral expansion. On the other hand, the finite difference is a local operator, so the tidal wind can be locally calculated from the temperature fields measured over any latitudinal range. Such an advantage in the finite difference method over the spectral method also motivated us to design various techniques (but failing to find a good one) to remove the apparent singularities in the finite difference method. The regions without observational data always occur at high latitudes and the poles (Zhu et al. 2005), so one needs to extrapolate the measured temperature field to fill the gaps at high latitudes before implementing the spectral method. Extrapolation is usually hazardous and often introduces unrealistic errors in numerical calculations (e.g., Press et al. 1992). Fortunately, the solution to the tidal wave equation has to satisfy the lateral boundary condition of zero perturbation in geopotential (Longuet-Higgins 1968; Zhu et al. 1999), which is equivalent to a zero perturbation in temperature. As a result, we can let the measured perturbation temperature near the high-latitude edges smoothly decay to zero at the poles. In other words, given the known value of zero at the poles, the filling of the perturbation temperature at high latitudes becomes an interpolation problem. The final diagnostic result at the mid and lower latitudes becomes insensitive to the details of the interpolation procedure. Also note that it is only the perturbation temperature (not the perturbation wind) that has to satisfy the zero lateral boundary condition at the poles (Longuet-Higgins 1968; Zhu et al. 1999).

## 3. Tidal winds and Eliassen–Palm flux divergence derived from TIMED/SABER temperatures

The data used in this paper are the temperatures from the SABER instrument onboard the TIMED satellite. SABER is a 10-channel radiometer that measures the limb emission from the surface to the lower thermosphere between 1.27 and 15.4 *μ*m (Mlynczak 1997; Russell et al. 1999). Temperature measurements are obtained from the emissions by CO_{2} co-vibrational bands at 15.4 *μ*m, which are in nonlocal thermodynamic equilibrium in the MLT (e.g., Mlynczak 1997, 2000; Zhu 2004). Six years of the latest publicly released version (1.07) of SABER temperatures from February 2002 to November 2007 were used for all the analyses in this paper.

We first use the algorithm developed in Zhu et al. (2005) to simultaneously derive daily zonal mean and migrating tidal fields. To eliminate the aliasing between the zonal mean and tidal component, we have to apply a regularity condition that appropriately selects the available grid points over a wave cycle of the local time for a migrating tide. One critical parameter that characterizes the trade-off between two types of aliasing and between the latitudinal coverage and the time-varying effect is the size of the moving window (Zhu et al. 2005). In this paper, we set the window size for analyzing TIMED/SABER temperatures to 61 days. This gives a near-complete and stable latitudinal coverage of the available grid points where the zonal mean and tidal fields are directly derived from the measurements (Fig. 2). Figure 2 shows that short-term variability still exists on the boundaries for a 61-day window to sample data. In addition, the local time and latitude grids selected based on the regularity conditions within the boundaries vary with time (Zhu et al. 2005). Outside the boundaries of the processed temperature fields, the zonal mean temperature is merged with the Committee on Space Research (COSPAR) International Reference Atmosphere 1986 (CIRA-86; Fleming et al. 1990) climatology, and the tidal temperature is allowed to decay to zero at the poles.

In Fig. 3, we show the daily zonal mean wind from 30 to 104 km that has been averaged over three latitudinal bands of 12° width centered at 28°N, the equator, and 28°S, respectively. The zonal mean wind is calculated from the measured zonal mean temperature according to the thermal wind approximation. The second-order gradient wind balance is used to derive the zonal mean wind at and near the equator (Fleming et al. 1990). In the upper mesosphere, say above 80 or 90 km where dissipation of tidal and planetary waves becomes significant, the gradient wind balance near the equator may no longer be a good approximation (e.g., Miyahara et al. 2000; McLandress et al. 2006). Inclusion of the wave drag terms in the meridional momentum equation is expected to lead to an improved approximation for the zonal mean wind near the equator. This will be explored in near future because other planetary scale waves such as 2-day waves are also included in the diagnostic study. Therefore, the zonal mean wind shown in Fig. 3, especially in the equatorial area above 90 km, can be considered a baseline analysis from TIMED/SABER temperatures. The most familiar and common features shown in the figure are those also shown in a middle atmosphere zonal mean wind climatology, such as CIRA-86 (Fleming et al. 1990), which include the midlatitude annual oscillation and the equatorial semiannual oscillation (SAO) of the zonal mean wind, with the jets in the Southern Hemisphere jets consistently stronger than those in the Northern Hemisphere. In addition, the figure also shows interannual variability characterized by a mesospheric biennial oscillation in wind strength. The MBO in zonal mean wind is most significantly shown at the equator, where the zonal mean winds between 55 and 75 km in the springs of odd years (2003, 2005, 2007) are consistently stronger than those in the even years (2002, 2004, 2006). Similar MBO features are also shown in the two midlatitude panels in which the magnitude of the peak zonal mean winds becomes slightly greater every 2 yr. A striking feature shown in Fig. 3 of the equatorial MBO is the downward propagation of the maximum zonal mean wind between 80 and 90 km in the springs of 2003, 2005, and 2007. This is a common feature for the stratospheric QBO and mesospheric SAO that illustrates the mechanism of wave–mean flow interaction between the zonal mean wind and upward propagating waves, which results from the divergence of the vertical momentum flux (e.g., Plumb 1977; Andrews et al. 1987).

Figures 4 and 5 show the amplitudes of the migrating diurnal and semidiurnal tides, respectively, at the same latitudinal bands as in Fig. 3. We immediately note that the short-term variability of the tidal amplitudes is ubiquitous at most spatial and temporal grids except near the midlatitude stratopause, where the strength of the migrating diurnal tides is mainly due to the localized excitation by ozone heating. Migrating tides at the equator are dominated by the diurnal component and the upward propagating modes, with water vapor heating in the relatively thick troposphere being the major excitation source. As a result, their strengths and variability in the MLT region will be subject to the changes of both the excitation sources and background states such as the zonal mean wind. Comparison of Figs. 4 and 5 indicates that the diurnal tide is generally stronger than the semidiurnal tide mainly because the diurnal component of the solar heating rate is much greater (say, by a factor of 3 at the lower-latitude region) than that for the semidiurnal component when the day–night solar heating rate is decomposed into its Fourier series (Zhu et al. 1999). Because the characteristic vertical wavelength for the semidiurnal tide is about twice that for the diurnal tide (e.g., Forbes 1995; Zhu et al. 1999), it is subject to less damping within a given altitude range to various dissipation processes while propagating upward. As a result, the semidiurnal amplitude in the lower thermosphere (Fig. 5) becomes comparable to or even greater than the diurnal amplitude (Fig. 4).

In Fig. 4, we note that there are also some indications of MBO in diurnal amplitude around 90 km at the equator. The tidal amplitudes in the springs of even years are greater than those in odd years except in 2006, when the peak amplitude of ∼15 K is about the same as its neighboring years of 2005 and 2007. Still, comparison of Figs. 3 and 4 for the two center panels at the equator suggests that the equatorial MBO in the zonal mean wind, especially around 90 km, could be partially driven by the diurnal tides through the wave–mean flow interaction. Figures 4 and 5 do not show any sign of the MBO in tidal amplitude at the midlatitude regions. The clear signatures of the MBO in the zonal mean wind at both the equator and midlatitudes (as shown in Fig. 3) led us to seek MBO signals in other tidal fields or components. Figure 6 shows the diurnal amplitudes of the vertical wind at the equator and the meridional wind at midlatitude regions. The figure shows that the relative strength in peak amplitude for the vertical wind is almost identical to that in temperature, as shown in Fig. 4. Specifically, it shows slightly greater amplitudes around 90 km in the springs of 2002 and 2004 than those in 2003 and 2005, but the amplitudes in 2006 and 2007 are nearly the same. We note from the energy Eq. (14) that the amplitude of the vertical wind is proportional to the temperature amplitude. However, we have also used the continuity Eq. (15) in deriving the tidal vertical wind with the horizontal wind solved from Eqs. (12) and (13). The fact that the relative strengths of the peak amplitudes between temperature and vertical wind are consistent also suggests that our approach to solving the overdetermined system (12)–(15) is appropriate.

The diurnal amplitude in the meridional wind at midlatitudes (Fig. 6) shows signs of the MBO in the lower thermosphere. The tidal amplitudes in even years are slightly higher than those in odd years in both hemispheres. It should be pointed out that in general the wave–mean flow interaction occurs through the wave momentum or heat flux divergence in both horizontal and vertical directions. Furthermore, the geostrophic constraint in the mid- and high-latitude regions also makes the forcing–response relation of the zonal mean state a nonlocalized process (e.g., Zhu et al. 1997, 2001). Therefore, the MBO signatures in both zonal mean wind and the diurnal amplitude in the meridional wind at midlatitudes (as shown in Figs. 3 and 6, although not at the same altitude) could be related through the wave–mean flow interaction.

The MBO at the equator has been found previously in both the zonal mean wind and the temperature amplitude of the diurnal tides. Using 3.5 yr of UARS/HRDI-measured winds, Burrage et al. (1996) found the signature of a 2-yr oscillation in zonal mean wind at 82.5 km and attributed it to the selective filtering of small-scale gravity waves by the underlying winds they traverse (Mayr et al. 1997). Garcia and Sassi (1999) and Garcia (2000) showed that the upward propagating equatorial waves excited by deep convection can modulate the mesospheric SAO by the stratospheric QBO to produce the MBO signature in the UARS/HRDI-measured winds. However, the signature becomes less clear in a longer datasets of more than 7 yr, possibly because of the poor coverage during the latter part of the period (Swinbank and Ortland 2003). Sridharan et al. (2007) also found the mesospheric QBO signature near the equator in the zonal mean wind measured by a medium frequency (MF) radar over a 13-yr period and further correlated the signature with the stratospheric QBO in the same period. The equatorial MBO near 90 km has also been found previously in analyses of diurnal tides in TIMED/SABER-measured temperature (Talaat et al. 2007). It has also been attributed to the effect of the stratospheric equatorial QBO on the upward propagation of the tidal waves that are mainly excited in the troposphere. The downward propagation of the equatorial MBO shown in Fig. 3 suggests that the localized wave–mean flow interaction could also contribute to the observed MBO at the equator. We have used the term MBO in this paper because there is a distinct difference in oscillation period between the stratospheric QBO and MBO shown in Fig. 3. The stratospheric QBO has a varying period generally longer than 2 yr. This is because the stratospheric QBO is primarily driven by two sets of oppositely propagating waves of nearly equal wave strengths (Plumb 1977; Baldwin et al. 2001), such as the equatorial waves forced by standing convective heating (Holton 1972; Hamilton 1982). Under such a circumstance, the presence of the semiannual oscillation at high levels in the tropical atmosphere plays little role (Plumb 1977). On the other hand, we postulate that the MBO shown in Fig. 3 has a period of exactly 2 yr and precisely follows one of the westerly jets associated with the SAO in the MLT. In this case, waves that play a major role in producing the MBO, such as the migrating tides, are not required to have nearly equal strengths in both eastward and westward directions.

*T*|) and the zonal (|

*u*|), meridional (|

*υ*|), and vertical (|

*w*|) winds. Note that |

*T*| and |

*w*| are correlated proportionally very well, indicating again the appropriateness of our solution procedure for the overdetermined system (12)–(15). Also note from Fig. 7 that the maximum |

*u*| and |

*υ*| for the diurnal tide occur near the same latitude of about 30°, where the apparent singularities are approximately located. This makes the careful treatment of the apparent singularities in the diagnostic analysis especially important if the diurnal tidal winds make a major or important contribution to the wave field. Comparison between Figs. 7 and 8 show that |

*T*| and |

*w*| have similar magnitudes and meridional structure in the lower thermosphere for diurnal (

*s*= 1) and semidiurnal (

*s*= 2) tides. This is also consistent with a more detailed analysis of the mesopause temperature and altitude from TIMED/SABER temperatures that shows significant perturbations in both diurnal and semidiurnal components (Xu et al. 2007). On the other hand, the horizontal tidal wind |

*u*| and |

*υ*| for

*s*= 1 is far greater than that for

*s*= 2. Mathematically, this is related to the behavior of the apparent singularities. As a first approximation of assuming zero zonal mean wind and considering the classical tides, Eqs. (10), (12), and (13) suggest that the horizontal tidal wind is determined by a balance between the tidal temperature and its meridional gradient (Chapman and Lindzen 1970; Zhu et al. 1999):where Δ

_{2}= 2Ω

*a*(

*σ*

^{2}− sin

^{2}

*ϕ*). For the diurnal tide (

*s*= 1,

*σ*= −0.5), both numerators and the denominator Δ

_{2}in Eqs. (22) and (23) vanish at the critical latitudes where |

*σ*| = |sin

*ϕ*|. The horizontal tidal wind approaches a finite value at the critical latitudes (Zhu et al. 1999):On the other hand, the sums of two terms in the numerators of Eqs. (22) and (23) could still become very small for a similar meridional structure in |

*T*| for the semidiurnal tide, but the denominator for

*s*= 2 (and

*σ*= −1) will no longer be small at midlatitudes, as in the case for

*s*= 1 (and

*σ*= −0.5), to amplify the cancelled numerator terms, leading to a much smaller horizontal tidal wind for

*s*= 2, as shown in Fig. 8.

Figure 9 shows the tidal equatorial vertical and midlatitude meridional winds, respectively, at the local noon. Both the diurnal and semidiurnal components have been used to produce the composite wind. We have already shown in Figs. 7 and 8 and explained that the diurnal |*υ*| is far greater than the semidiurnal |*υ*| at midlatitudes. The top panel of Fig. 9 shows that the maximum northward wind occurs near 98 km and 75 km, indicating a vertical wavelength of ∼23 km for the diurnal tide. Figure 9 also shows how greatly the daily tidal wind at a fixed local time will change with time. The overall relative scale and magnitude of the variations are similar to those in the zonal mean wind, as shown in Fig. 3. This type of short-term changes will lead to type-I aliasing due to the time variation of a field when the asynoptic satellite observations are used to derive the mean and wave fields (Zhu et al. 2005). Both Figs. 3 and 9 suggest that although the type-I aliasing cannot be completely eliminated, the 60-day window used in the current algorithm to extract zonal mean and tidal wave fields is suitable for the diagnostic analysis.

In Fig. 10, we show the acceleration in zonal mean wind contributed from the EP flux divergence as defined by Eqs. (1)–(4). The plot has been averaged over a three-day period [days 74, 75, and 76 of 2006 (March); i.e., it is calculated based on the same wave fields shown in Figs. 7 and 9]. Because EP flux divergence is a second-order quantity that involves taking the difference of two generally large terms, the distribution shows more spatial structure than the plots for the tidal temperature and wind fields (Figs. 7 and 8). The typical magnitude in the lower thermosphere around 95 km is about 10 m s^{−1} day^{−1}, which is consistent with previous calculations based on the wind measurements by UARS/HRDI (Lieberman and Hays 1994). Note that our spectral module includes all the meridional modes for both diurnal and semidiurnal tides, leading to more spatial structures than those calculated with one or two Hough modes.

Figure 11 shows the EP flux divergence induced acceleration as a function of time and latitude over three altitude ranges centered at 95 (top), 85 (center), and 75 km (bottom), respectively. It shows large spatial and temporal variability with peak values of EP flux divergence occurring in the midlatitude regions where the jet centers are usually located, indicating a strong interaction between tidal waves and zonal flow. Around 95 km, the typical peak values of acceleration or deceleration at the midlatitude regions by migrating tides are 10–20 m s^{−1} day^{−1}. The equatorial region shows a weak deceleration of about 5 m s^{−1} day^{−1} around 95 km. To illustrate the significance of the effect of the zonal mean flow on the derived tidal wind, we show in Fig. 12 the same EP flux divergence as Fig. 11 except that the zonal mean wind had been set to zero while deriving the tidal wind from the spectral module. We first note that Fig. 12 shows a similar pattern in magnitude and variability of the EP flux divergence as Fig. 11. This is because the major effect of the zonal mean wind on the tidal waves has largely been included and reflected in the derived tidal temperature based on the SABER measurements. The difference between Figs. 11 and 12 represents the additional effect due to the difference in numerical modeling of the tidal waves from the measured tidal temperature. Comparison of Figs. 11 and 12 shows that the largest difference occurs near the southern latitude where the zonal mean wind is the strongest (Fig. 3). With a zero zonal mean wind in the tidal model (i.e., assuming classical tides), the decelerations at 95 and 85 km and the acceleration at 75 km around 30°S are enhanced by approximately a factor of 2. A less significant but also noticeable change occurs at the northern midlatitudes. Therefore, it is necessary to include the effect of the zonal mean wind in tidal modeling to derive reasonably accurate wind fields and the associated dynamical quantities, such as the EP flux divergence.

## 4. Conclusions

In this paper, we have derived tidal winds from the measured tidal temperature by developing a spectral module that effectively eliminates the error amplification near the apparent singularities. Combined with a previously developed algorithm that extracts the zonal mean and migrating tidal temperature without bias (Zhu et al. 2005), we are able to conduct a diagnostic analysis of various dynamical fields based on TIMED/SABER temperatures. We have analyzed the zonal mean wind and migrating diurnal and semidiurnal tides over a 6-yr period from 2002 to 2007 with the latest publicly released version (1.07) of the SABER temperature. We have also diagnosed the EP flux divergence based on the derived zonal mean and tidal fields.

The derived zonal mean wind and diurnal tidal amplitude reveal new insights into the MBO that exists in the MLT in both equatorial and midlatitude regions. The equatorial MBO in the zonal mean wind is present in the entire mesosphere from 50 to 90 km. It is especially significant near the mesopause region between 80 and 90 km where the MBO zonal mean wind shows a downward phase propagation. Furthermore, the region of the downward phase propagation is also largely coincident with the equatorial MBO in the amplitude of the diurnal tide, indicating a possible mechanism of wave–mean flow interaction between the two. On the other hand, the newly discovered midlatitude MBOs in zonal mean wind and the meridional wind in diurnal tide occur at different altitudes, suggesting possibly a remote forcing–response relationship due to the geostrophic constraint.

The derived acceleration in zonal mean wind contributed by the EP flux divergence peaks at the midlatitudes, where the zonal jets are usually located. The typical acceleration or deceleration at midlatitude regions around 95 km is about 10–20 m s^{−1} day^{−1}.

## Acknowledgments

Helpful comments by two anonymous reviewers are greatly appreciated. This research was supported by the TIMED project sponsored by NASA under contract NAS5-97179, NASA Grant NNG05GG57G, and in part by NSF Grant ATM-0730158 to The Johns Hopkins University Applied Physics Laboratory.

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