## Abstract

The seasonal variation of the propagating diurnal tide in the mesosphere and lower thermosphere is examined using results from a 2-yr simulation of the extended version of the Canadian Middle Atmosphere Model (CMAM). The CMAM is shown to be able to reproduce not only the observed semiannual amplitude variation of the tide in the lower thermosphere but also more subtle features such as amplitude maxima that are stronger in March/April than in September/October, a 4- to 6-h shift in phase between winter and summer in the Northern Hemisphere, and a weaker seasonal variation of phase in the Southern Hemisphere. Part I of this two-part series of papers investigates the importance of two of the mechanisms that have been proposed to explain the observed variation of tidal amplitude, namely, 1) interactions with small-scale gravity waves and 2) interactions with planetary-scale waves like the quasi–2-day wave. Analysis of the tidal momentum and thermodynamic budgets shows that the direct effects of parameterized gravity waves are not important, since the associated drag and vertical diffusion are found to be substantially weaker in magnitude than the wave–wave and wave–mean flow interaction terms. Nonlinear interactions with large-scale disturbances (possibly nonmigrating tides) are found to be an important mechanism to damp the diurnal tide in the lower thermosphere; however, the seasonal variation of these terms is of the wrong sign to explain the seasonal variation of the tidal amplitude. Although the CMAM exhibits a quasi–2-day wave at solstice, its overall impact on the tide is not found to be strong. The budget analysis points to the linear advection terms as being of particular importance in the seasonal variation of the tide.

## 1. Introduction

The sun-synchronous or migrating diurnal tide is arguably the largest and most persistent disturbance in the equatorial mesosphere and lower thermosphere (MLT). Although its general characteristics can be understood from classical tidal theory (Chapman and Lindzen 1970), its amplitude undergoes a strong semiannual variation which has not been successfully explained. The semiannual variation, which is characterized by strong tides at equinox and weak tides at solstice, was first observed on a global basis from *Upper Atmosphere Research Satellite* (*UARS*) wind measurements (Hays et al. 1994; Burrage et al. 1995; McLandress et al. 1996). It is a robust feature that also appears in multiyear ground-based radar observations (Vincent et al. 1988, 1998; Fritts and Isler 1994). Although a number of modeling studies have recently addressed this phenomenon, there is no general agreement as to the underlying cause.

The physical mechanisms that have recently been proposed to explain the semiannual amplitude variation of the diurnal tide can be broadly grouped into two categories: 1) those involving the effects of small-scale gravity waves through momentum deposition and turbulent diffusion, and 2) those involving the effects of large-scale disturbances through wave–wave interactions. Of these two, the role of small-scale gravity waves has received the most attention. Since the horizontal wavelengths of the vast majority of gravity waves are too small to be resolved by global models, their effects must be parameterized. Consequently, studies focusing on interactions between the diurnal tide and gravity waves require the use of a gravity wave parameterization such as that of Lindzen (1981) or Hines (1997a,b), for example.

There have been contradictory results regarding the impact of gravity wave momentum deposition (or drag) on the diurnal tide. Using the Lindzen parameterization, Miyahara and Forbes (1991) conclude that it results in a large reduction in amplitude, while McLandress (1998) and Mayr et al. (1998), using the Hines parameterization, find that the momentum deposition has an amplifying effect. (Mayr et al., in fact, attribute the semiannual amplitude variation of the tide in their model to this effect.) This discrepency appears to have been resolved by Akmaev (2001), who showed that the very strong damping found by Miyahara and Forbes was the result of an incorrect formulation of the Lindzen parameterization.

The turbulence and mixing that is generated by breaking gravity waves is treated in parameterization schemes using an eddy diffusion coefficient. Since the computed eddy diffusion coefficient depends strongly on the background atmosphere, seasonal variations of this coefficient could in principle be responsible for the seasonal variation of the tide. Meyer (1999) and Akmaev (2001) argue that larger values of the diffusion coefficient at solstice (when the zonal mean zonal winds are strongest) result in weaker diurnal tide amplitudes in the MLT region.

Interactions between the diurnal tide and planetary waves is the second of the two mechanisms proposed to explain the observed semiannual variation of the tide. Studies supporting this mechanism have involved the use of general circulation models. Norton and Thuburn (1999) suggest that interactions with the quasi-2-day wave, which is strongest at solstice, are responsible for the weak tidal amplitudes found at these times. McLandress (1997, hereafter M97) also noted a strong correlation between resolved wave activity and the diurnal tide amplitude in the Canadian Middle Atmosphere Model (CMAM) and suggested that nonlinear effects may be important, although a strong quasi-2-day wave signal was not found.

The aforementioned studies of M97, Mayr et al. (1998), Norton and Thuburn (1999), and Akmaev (2001), however, all employ models that involve complicated nonlinear feedbacks which make cause and effect difficult to unravel. Although they were all able to qualitatively reproduce the seasonal behavior of the diurnal tide amplitude in the MLT region, these studies provided only indirect evidence of the possible cause and consequently cannot be treated as conclusive. (It is also important to realize that since all geophysical fields vary on a semiannual basis at the equator, correlations between those variations and that of the tide prove nothing.) A more detailed analysis is therefore required to establish cause and effect. Since a complete tidal budget cannot be obtained from the observations, the best alternative is to examine simulations from a middle-atmosphere general circulation model that has minimal tuning and follows a first principles approach. The recently developed extended version of the CMAM (Fomichev et al. 2002, hereafter F02) shows very good agreement between the simulated tide and the *UARS* observations (Beagley et al. 2000). This model will therefore be employed.

In the first part of this two-part set of papers, the extended CMAM is used to investigate the seasonal variation of the propagating diurnal tide in the MLT region. After a short description of the model (section 2), a comparison with *UARS* observations is made, and the general features of the simulated tide are characterized (section 3a). A detailed analysis of the terms in the diurnal tide momentum and thermodynamic budgets is then presented. This includes a discussion of the effects of the physical parameterizations, which are referred to as the “diabatic” tendencies (section 3b), as well as an examination of the linear and nonlinear advection terms, which are referred to as the “adiabatic” tendencies (section 3c). In keeping with the points mentioned earlier, we focus on the effects of small-scale (unresolved) gravity waves and large-scale (resolved) waves in an attempt to ascertain their role in the semiannual amplitude variation of the tide. The conclusions and a brief summary follow (section 4). In the companion paper (McLandress 2002, hereafter Part II), a linear mechanistic tidal model is used to investigate the role of zonal mean zonal winds and tidal heating.

## 2. Model description

The extended CMAM is based on the standard version described in Beagley et al. (1997). The version reported on here has a T32 spectral resolution in the horizontal, corresponding to a grid size of roughly 6° in latitude and longitude. The model extends from the earth's surface to about 200 km, with 70 levels in the vertical and a resolution of ∼3 km throughout the middle atmosphere.

Since details of the physical processes in the extended CMAM are provided in F02, only a brief summary of the relevant parameterizations will be presented here. The model includes comprehensive solar and terrestrial radiative transfer schemes. Solar heating in the model occurs through absorption in the near infrared (by water vapor in the troposphere), in the ultraviolet (by ozone in the stratosphere and mesosphere), and in the extreme ultraviolet (by molecular oxygen in the thermosphere). While water vapor is predicted by the model, ozone and oxygen are not and so are prescribed using climatological values. Terrestrial radiation occurs primarily as a result of infrared emission by water vapor and carbon dioxide, with the latter including the effects of nonlocal thermodynamic equilibrium in the MLT region. Latent heating in the middle and upper troposphere is provided by the deep convection parameterization. Dissipation in the thermosphere is given by molecular diffusion and a simple ion drag parameterization. To avoid having to use an excessively short time step, enhanced horizontal diffusion in the thermosphere is employed; its direct impact on the diurnal tide, however, is very weak. The effects of unresolved orographic and nonorographic gravity waves are also included, with the latter treated using the Doppler spread parameterization (DSP) of Hines (1997a,b, 1999). The settings for the DSP, which are given in F02, are basically the same as the ones used in the standard version of the CMAM discussed in McLandress (1998), with the exception of gravity wave heating and diffusion, which are now included.

## 3. Model results

Results from the last two years of the 3-yr simulation described in F02 are analyzed. Winds and temperatures are saved every 3 h and monthly mean migrating tides are computed using the method described in M97, modified as follows for the higher-frequency saving interval: the migrating diurnal tide component of a quantity *q* is expressed as

where *t* is local solar time (h) and Ω = 2*π*/(24h). The coefficients *A*_{q} and *B*_{q} are given by

where

The term *q*_{i,j} denotes the quantity *q* evaluated at the longitudinal grid point *λ*_{i} and the *j*th distinct universal (not local) time lying between 0000 (0h) and 2400 (24h) UTC. The overbar indicates a 1-month average and *M* denotes the number of longitudes. In the results that will be shown, amplitude and phase will be used. These are given by *C*_{q} = *A*^{2}_{q} + *B*^{2}_{q}/2 and *ϕ*_{q} = arctan (*B*_{q}/*A*_{q}), respectively.

### a. Winds and temperatures

Figure 1 shows the composite annual cycle of the diurnal tide amplitude and phase for the zonal and meridional wind components in the lower thermosphere from the CMAM and from the wind imaging interferometer on *UARS* (McLandress et al. 1996). The agreement between the simulated and observed tides is quite striking. Features which the model is able to reproduce are 1) the semiannual amplitude variation, with the March/April maximum exceeding that of September/October in the meridional component; 2) the annual variation of phase in the Northern Hemisphere, which exhibits a 4–6-h shift from winter to summer; and 3) the weak seasonal variation of phase in the Southern Hemisphere. The poorer agreement in the zonal wind component may be attributed in part to the effects of aliasing which are more problematic in the satellite data in the zonal direction (McLandress et al. 1996). Incidentally, the interhemispheric difference in the seasonal variation of the phase has not been commented on before in the *UARS* literature. The fact that it is reproduced in the model suggests that it is a robust feature that also needs to be explained.

Time series of the monthly mean amplitudes of the tidal meridional wind and temperature from the CMAM are shown in Fig. 2 at three different heights in the MLT region. The semiannual variation is clearly seen in the lower thermosphere and upper mesosphere (top two panels). In the lower mesosphere (bottom panel) a strong annual variation is seen, with the amplitude maxima situated in the winter hemisphere. Zonal-height cross sections of the tidal wind and temperature amplitudes for April and July are shown in Fig. 3. At equinox the amplitudes are quite symmetric about the equator, unlike at solstice when the structure is highly distorted.

### b. Diabatic tendency terms

Tendency terms that arise from the physical parameterizations in the model are referred to here as diabatic tendencies. Due to data storage restrictions they are computed at 18-h intervals, rather than every 3 h as is done with the winds and temperatures shown in the last section. Calculations are performed on a 32 by 64 latitude by longitude grid. The results are interpolated onto pressure surfaces and then projected onto the monthly mean migrating diurnal tide. Figure 4 shows the amplitude of the different diabatic tendency terms for April and July at the latitudes where the tidal winds and temperatures are strongest. The thick line denotes the sum of all the terms. Below 100 km the diabatic component of the momentum budget is dominated by the gravity wave drag (GWD) and eddy diffusion terms. Molecular diffusion becomes important at about 100 km and is by far the largest term in the region above. The effect of ion drag also grows exponentially with height, but at the altitudes shown here is still an order of magnitude smaller than molecular diffusion. In the thermodynamic equation, longwave (terrestrial) and shortwave (solar) radiation are the dominant diabatic terms, with the longwave radiation attaining a maximum value of ∼20 K day^{−1} near 110 km in April. Convective adjustment, which is also strongest near 100 km in April (attaining a value of ∼3 K day^{−1}), is nearly an order of magnitude larger than the eddy heat diffusion. However, its effect is still rather small in comparison to the longwave radiative term.

Whether a particular term in the zonal (or meridional) wind equation is acting to locally alter the amplitude or phase of the tide can be characterized using an equivalent Rayleigh friction coefficient (ERF; e.g., Miyahara and Forbes 1991; Forbes et al. 1991). The ERF is intended to provide a qualitative understanding of the effects of different forcing terms; several caveats concerning its applicability are discussed below. The ERF is defined as the complex-valued quantity

where *û* and *F̂*_{u} are the coefficients of the zonal (or meridional) wind and right-hand-side forcing term in the corresponding momentum equation. Making use of the fact that *q̂* = (*A*_{q} − *iB*_{q})/2, the real and imaginary parts of *γ*_{u} are given by

where the coefficients *A* and *B* are given by (1). A similar analysis applied to the thermodynamic equation yields the equivalent Newtonian cooling coefficient.

For a purely dissipative forcing term, such as Rayleigh friction, *γ*_{u} is real and positive. Conversely, Re{*γ*_{u}} < 0 indicates amplification of the tide. If *γ*_{u} is purely imaginary then the forcing is acting to change the phase of the tide. Im{*γ*_{u}} < 0 implies a reduction in phase (i.e., the maximum wind will occur at an earlier local time). This implies a shortening of the vertical wavelength of the tide since the phase decreases with height as required for upward energy propagation, assuming that the forcing term lies above the source region of the tide. The converse occurs for Im{*γ*_{u}} > 0.

The above simple interpretation of the ERF is predicated on the frequency of the wave (*ω*) being greater than the computed “damping” coefficient (*γ*), and on the latitudinal structure of the wave being largely unaffected by the forcing. The former can be seen most easily in the case of a linear vertically propagating gravity wave (e.g., Andrews et al. 1987, p. 189) by including equal damping coefficients for momentum and temperature and deriving the dispersion relation. For |*γ*/*ω*| ≪ 1 it can be shown that only the disturbance amplitude will be changed if *γ* is real, and only the phase (i.e., the vertical wavelength) if *γ* is imaginary. As will later be seen the computed values of the ERF are much less than the frequency of the tide; consequently the above interpretation is valid to a first approximation. While the presence of a vertically localized forcing may cause partial reflection of an upward propagating wave [e.g., Gill (1982, p. 296), in the case of viscosity], this effect will be small or negligible if the vertical scale of the forcing is greater than that of the wave. In the case of the propagating diurnal tide, whose vertical wavelength is 25–30 km, the scale-height of the forcing must be of this length or greater for reflection not to occur. As will be seen later, the diabatic and adiabatic forcing terms generally exhibit a rather broad structure in the vertical to a first approximation. Small-scale variations superimposed on the large-scale structure will therefore not significantly alter the results.

Figure 5 shows the real part of the equivalent Newtonian cooling coefficient computed from the longwave radiation terms averaged between 30°S and 30°N. It is positive and large throughout the entire MLT region, indicating that longwave radiation is acting to damp the tidal temperature perturbation as one would expect (the imaginary part is much smaller and is not shown). The damping rate (*α*) exhibits two maxima. The smaller peak near 50 km with *α* ∼ (3 days)^{−1} corresponds to infrared cooling by O_{3}. The larger peak near 105 km with *α* ∼ (1 day)^{−1} results from nonlocal thermodynamic equilibrium effects of CO_{2} cooling, which become more important as the density decreases. Comparing the results for April and July indicates that the seasonal variation of the radiative damping rate is very weak as expected. Although infrared cooling is dependent upon the vertical scale of the temperature perturbation, the fact that the equivalent Newtonian cooling coefficient exhibits very little latitudinal variation in the MLT region (results not shown), even at midlatitudes where the diurnal tide is no longer propagating and its vertical wavelength has increased, implies that the scale dependence of *α* is rather weak at these heights. This indicates that the cooling-to-space term is dominant, in agreement with the results of Zhu (1993) and Fomichev et al. (1998).

The remainder of this section focusses on the effects of the parameterized gravity waves. Figure 6 shows the zonal mean eddy diffusion coefficient for April and July, averaged between 30°S and 30°N. In agreement with the studies of Akmaev (2001) and Meyer (1999), it is larger in July than in April. This arises solely as a result of changes in the background atmosphere (presumably the zonal mean zonal wind) since the source settings of the DSP are invariant. (Note that although the CMAM uses a local Richardson number–dependent diffusion coefficient, it is much smaller than the values shown here that result almost entirely from the DSP, aside from the 1 m^{2} s^{−1} background value seen below 40 km.) Nevertheless, the value of the diffusion coefficient at 80 km is about three times smaller than that shown in Akmaev (2001), which is consistent with the different parameter settings for the DSP used in the two studies. The choice of the parameter settings was governed by the zonal mean temperature in the summer mesopause, which becomes excessively cold if larger diffusion coefficients are employed, as explained in F02. If the CMAM were to simulate minor constituents like atomic oxygen (which it currently does not), the small values for the eddy diffusion coefficient might result in unrealistic atomic oxygen densities in the lower thermosphere. It is conceivable, therefore, that if larger diffusion coefficients were used (along with the necessary increase in the turbulent Prandtl number so that the resulting diffusion of heat was unchanged), the impact on the tidal winds would be greater than it is here.

The real part of the ERF computed from the eddy diffusion term shown in Fig. 4 is given by the dotted line in Fig. 7. It is positive, which indicates that the diffusion is acting to damp the tide. The imaginary part is much smaller and is not shown. The real part of the ERF computed from the gravity wave momentum deposition, which is also shown in Fig. 7 (dashed lines), is predominantly negative, meaning that GWD is acting to amplify the tide. However, since this effect is larger in July than it is in April, it is difficult to see how the larger tidal amplitudes in April could result from gravity wave momentum deposition as argued by Mayr et al. (1998). Figure 7 also shows that the net effect of the parameterized gravity waves (given by the solid line, which shows the ERF computed from the sum of the momentum deposition and eddy diffusion terms) is substantially reduced as a result of cancellation between the two terms. Thus, while changes in eddy diffusion are of the right sense to be consistent with the tidal amplitude variation (i.e., largest at solstice when the tide is weakest), the changes are largely canceled by the amplifying effect of momentum deposition for these particular parameter settings of the DSP. This effect is most clearly seen in the meridional wind component in July.

The large difference in the amplitude of the momentum deposition and eddy diffusion terms seen in Fig. 4, but not seen in the real parts of the ERF shown in Fig. 7, is a result of the large phase shift between the GWD and the tidal winds. This becomes apparent by examining the imaginary part of the ERF computed from the momentum deposition term, which is shown in Fig. 8. It is negative, which means that the GWD acts to reduce the phase of the tide and to decrease its vertical wavelength. A strong seasonal variation is also evident in the ERF, with larger values at solstice than at equinox. Comparing Figs. 7 and 8 indicates that the induced changes to tidal phase are larger than changes to amplitude. The amplitude and phase changes that are implied by this ERF analysis are consistent with the findings of McLandress (1998) and Mayr et al. (1998), who compare tidal simulations with and without the momentum deposition associated with the DSP. The idealized results shown in Fig. 9 of McLandress (1998) indicate that the changes in phase, however, are rather modest (∼2 h), which would explain why the real part of the ERF computed from the eddy diffusion term in Fig. 7 is still only a factor of 2 larger in July than in April despite the twofold increase in the diffusion coefficient in the solstice seasons (Fig. 6).

### c. Adiabatic tendency terms

The sum of the diabatic tendency terms shown in Fig. 4 must be balanced by the time tendency, advection, geopotential gradient and Coriolis terms. These terms are referred to here as the adiabatic tendencies. In order to assess the relative strength of the classical and nonclassical terms, the adiabatic tidal tendencies are separated into three components: 1) the classical terms, which involve only the time tendency, Coriolis, and geopotential gradient terms; 2) the linear advection terms, which involve only the products of zonal mean quantities and perturbations; and 3) the nonlinear advection terms, which involve only the products of perturbations. The tendencies are computed for each month at 3-h intervals in pressure coordinates on a 48° latitude by 96° longitude grid using centered finite differences, and the results projected onto the migrating diurnal tide. The time (*t*) and longitude (*λ*) derivatives in the classical terms are computed analytically from the prescribed form of the tide {i.e., exp[*i*(*λ* − *ωt*)]}. Since the adiabatic terms are not computed the same way as in the CMAM, some differences will occur. However, these differences are expected to be small for the large horizontal scales of interest here.

Figure 9 shows the amplitude of the sums of the classical, linear and nonlinear terms in the momentum and thermodynamic equations for April and July at the latitudes most closely corresponding to those shown in Fig. 4. Comparing the sum of the classical terms (thin solid lines) to the time tendency terms (crosses) gives an indication of the degree to which the dynamics are governed by the classical tidal equations. Overall, the time tendency term is ∼5 times larger, indicating that classical dynamics is accurate to about 20%. (In the classical case, the time tendencies in the momentum equations are exactly balanced by the Coriolis and geopotential gradient terms, which have not been individually plotted here. A similar argument holds for the thermodynamic equation.) Regions where the difference between these two curves is small indicate that classical dynamics is less accurate and that other terms like linear and nonlinear advection are important. Such a region exists near 85 km for the zonal wind component in July where the linear terms play an important role. The nonlinear terms for the zonal wind component and to a slightly lesser extent for temperature become more important higher up, while those for the meridional wind component appear to be important at all heights. The high degree of cancellation between these three sets of summed terms is illustrated by the thick solid line, which shows the total adiabatic tendency; it is generally one order of magnitude smaller than the largest of the summed adiabatic terms, and two orders of magnitude smaller than the time tendency term. Comparing the thick curves to the total diabatic tendency terms in Fig. 4 shows that a fairly reasonable balance between the two has been achieved.

At this point we can safely rule out the effects of parameterized gravity waves as the cause of the seasonal variation of the diurnal tide in the CMAM. Comparing Figs. 4 and 9 we see that the amplitude of the larger of the two gravity wave terms is still nearly one order of magnitude smaller than the larger of the summed adiabatic tendency terms. For example, in April at 80 km the zonal gravity wave momentum deposition term is only ∼10 m s^{−1} day^{−1} while the sum of the linear adiabatic terms is ∼80 m s^{−1} day^{−1}. In this region linear advection is clearly more important than GWD. It is perhaps worth mentioning in this context that in the tidal simulation discussed in M97 (which exhibited a semiannual amplitude variation in the upper mesosphere) a nonorographic GWD parameterization was not employed. This fact already demonstrated that the presence of parameterized nonorographic gravity waves is not a necessary requirement for generating the semiannual amplitude variation of the tide (see also McLandress 1998).

Before proceeding with a more detailed analysis of the nonlinear terms, it is useful to examine the planetary wave disturbances in the model. Figures 10 and 11 show the frequency (*ω*) spectrum of the meridional wind component in the MLT region at 19°N for July and April of the third year of the simulation. The spectra have been spatially averaged and are shown for the first four zonal wavenumbers (*m*). The migrating diurnal tide is seen in the left-hand panels (*m* = 1) at *ω* = −1 cycle day^{−1}. The large peak in July at *m* = 3 and *ω* = −0.6 cycle day^{−1} (a period of ∼40 h) is the quasi–2-day wave, which attains an amplitude of nearly 40 m s^{−1} in the summertime upper mesosphere. Note the complete absence of this wave in April. The spectral peaks occurring at other integer frequencies indicate nonmigrating tides.

The thick solid curve in Fig. 12 shows the real part of the ERF computed from the nonlinear adiabatic tendencies of Fig. 9. The fact that it is positive indicates that the nonlinear terms are acting to damp the tide. The imaginary part (not shown) is as large as the real part and exhibits an oscillatory pattern in the vertical, indicating the complex way in which the nonlinear terms affect the tide.

To identify which zonal wavenumbers are the dominant contributors to the nonlinear tidal forcing, the above analysis is repeated using different zonal wavenumber bands. These results are given by the other curves shown in Fig. 12. In April, wavenumbers one and two provide the largest contribution to the ERF. An examination of the frequency spectra (Fig. 11) suggests that interactions with the nonmigrating tides are responsible. A possible interacting triad is (*m* = 1, *ω* = −1), (1, −1) and (2, −2). In July, the relative contribution from the shorter horizontal wavelengths (*m* = 5–32) is greater than in April. Note also how the ERF computed from these higher wavenumbers exhibits a much smoother, more monotonic increase with height in July and April. These results, in conjunction with the behavior of the imaginary part of the ERF, suggest that the shorter wavelength disturbances in the CMAM are damping the tide and shortening its vertical wavelength.

Figure 13 shows a time series of the real part of the nonlinear ERF at 98 km for the meridional wind component computed using all 32 wavenumbers. Comparing this to the tidal winds themselves (Fig. 2a) indicates that the nonlinear terms are acting to damp the tide in the subtropics, and tend to be strongest when the winds are strongest. The negative values occurring poleward of ∼25° indicate that the nonlinear terms are acting to amplify the tide at these latitudes. This, in conjunction with the damping at low latitudes, indicates that the nonlinearities are horizontally smoothing the tidal structure in the lower thermosphere. Similar results are also found in the nonlinear tidal simulation of Wu et al. (1989).

Returning finally to the quasi–2-day wave (which is not particularly strong at the latitudes shown in Fig. 12). Figure 14 shows the real part of the nonlinear ERF computed using *m* = 1–2 and *m* = 1–4 at 24°N for July of the third year of the simulation. The inclusion of wavenumbers 3 and 4 has significantly increased the ERF at 90 km. That this is attributable to the quasi–2-day wave is clear from Fig. 10. The quasi–2-day wave is, however, not the whole story since the higher zonal wavenumbers also contribute substantially to the total ERF, as is seen by comparing the solid and dashed curves. Interestingly, the ERF for the meridional wind component (not shown) exhibits no significant enhancement when zonal wavenumbers three and four are included.

Figure 15 shows time–height contour plots of the nonlinear ERF and the corresponding monthly mean tidal amplitudes for both wind components. The top panel, which shows the meridional wind component at 20°S, indicates that the nonlinear damping is strongest when the meridional winds are strongest. Clearly, these terms are not responsible for the semiannual variation since their removal would mean an even further increase in the amplitude of the tide at equinox. The bottom panel shows results for the zonal wind component at 24°N. The seasonal variation of the nonlinear ERF is more complicated for this wind component: while there is still a tendency for larger values in equinox, there are also relatively large values in summer. Although the quasi–2-day wave appears to damp the tidal zonal wind at solstice it does not, however, appear to be strong enough to produce the observed seasonal variation of the tide.

The findings of Norton and Thuburn (1999) require some discussion here since they suggest a strong coupling between the diurnal tide and the quasi–2-day wave. In their Fig. 8 are shown results of three simulations in which the settings of the GWD parameterization are altered. While the first two simulations suggest that the presence of a strong quasi–2-day wave is what causes the weak tides, the third exhibits significantly different behavior in which the strong anticorrelation between the tide and quasi–2-day wave amplitudes is not nearly as apparent. While the tide still shows the relatively smooth seasonal growth and decay seen in the control experiment, the temporal behavior of the quasi–2-day wave is much more sporadic, is weakest when the tide is weakest, and grows in time along with the tide. This is not the behavior one would expect if the tidal amplitude was controlled primarily by the quasi–2-day wave.

## 4. Conclusions

The seasonal variation of the vertically propagating diurnal tide has been examined using the extended version of the Canadian Middle Atmosphere Model (CMAM). The simulated tide was shown to be in good agreement with satellite-based wind observations in the lower thermosphere. Not only is the CMAM able to simulate the semiannual variation of the tidal amplitudes, it is also able to reproduce more subtle features such as the stronger amplitude maximum in March/April, the 4- to 6-h phase shift between winter and summer in the Northern Hemisphere, and the much weaker seasonal variation of phase in the Southern Hemisphere.

The paper focuses on the seasonal variation of the tidal amplitude, which has received considerable attention in the recent scientific literature but has not been successfully explained. To help clarify the mechanisms that are (or are not) responsible for this variation, a diagnostic analysis of the diurnal tide momentum and thermodynamic budgets was performed. This involved an examination of the tendencies arising from the physical parameterizations (the diabatic tendencies), as well as those arising from the dynamical terms (the adiabatic tendencies). An equivalent Rayleigh friction coefficient (ERF) was used to characterize the extent to which a given term in the momentum equation changes the amplitude (or phase) of the tide.

Long- and shortwave radiation were found to be the dominant diabatic terms in the thermodynamic equation. The equivalent Newtonian cooling coefficient derived from the longwave radiation indicated strong damping of about (1 day)^{−1} in the lower thermosphere. Only a very weak seasonal and latitudinal variation in the coefficient was found in the mesophere and lower thermosphere (MLT). Convective adjustment (the next largest diabatic heating term in the MLT region) was found to be an order of magnitude weaker than the longwave cooling term.

The momentum deposition and eddy diffusion arising from small-scale nonorographic gravity waves parameterized using the Doppler spread parameterization (DSP) of Hines (1997a,b) were the most important diabatic terms in the momentum equations. The amplifying effect of the momentum deposition in the DSP was offset by the damping effect of the diffusion, resulting in a much reduced net effect. No significant seasonal variation in the real part of the ERF for the combined gravity wave term was observed. This analysis suggested that the strong tides at equinox and weak tides at solstice are not the result of the parameterized gravity wave effects.

Analysis of the adiabatic tendency terms revealed that the nonclassical terms play a very important role in the tidal dynamics in the MLT region. Advection by the zonal mean winds in the mesosphere was found to be important for the zonal wind component, with the relative strength of the nonlinear terms increasing with height and becoming significant above 90 km. The nonlinear terms were found to be considerably larger for the meridional wind component throughout the entire MLT region. The fact that these terms are much larger than the gravity wave momentum deposition and eddy diffusion terms was further evidence that the small-scale (unresolved) gravity waves are not the key factor in the seasonal variation of the tidal amplitude in the CMAM.

Examination of the ERF computed from the nonlinear adiabatic tendencies in the momentum equations revealed that these terms are acting to damp the diurnal tide in the MLT region at low latitudes. Damping at these latitudes in the lower thermosphere occurs in conjunction with amplification in the extratropics, indicating a latitudinal broadening of tidal amplitudes. An examination of the spatial scales responsible for the nonlinear damping revealed that zonal wavenumbers one and two were the most important in April, with higher wavenumbers becoming important in July. It was suggested that interactions with nonmigrating tides were partly responsible for limiting tidal amplitudes in the lower thermosphere.

A large-amplitude quasi–2-day wave (westward propagating zonal wavenumber three disturbance with a period of ∼40 h) was found, with meridional wind amplitudes exceeding 40 m s^{−1} in the summer mesosphere near 90 km. The nonlinear ERF computed using only the first four zonal wavenumbers (which includes the quasi–2-day wave) exhibited a strong peak near 90 km for the zonal wind component, suggesting that the quasi–2-day wave acts to damp that component of the tidal wind. Surprisingly, no noticeable effect on the meridional component was found. Comparing the nonlinear damping coefficients to the tidal winds indicated that the damping was generally strongest where the winds were strongest. This argues against nonlinear interactions between the diurnal tide and other planetary waves as being the cause of the observed semiannual variation in tidal amplitude.

It is concluded that neither gravity wave–tide nor planetary wave–tide interactions cause the semiannual amplitude variation of the propagating diurnal tide in the lower thermosphere in the CMAM, and that other mechanisms must be responsible. The results show that the greatest departures from classical tidal dynamics occur as a result of the linear advection terms, in particular the linear terms in the zonal wind equation, which suggests that the effects of the zonal mean zonal winds are important. This, as well as the effects of seasonal variations in tidal heating, are explored in detail in Part II using a linear model that is forced with the CMAM tidal heating terms and uses as a basic state the CMAM zonal mean zonal winds and temperatures.

## Acknowledgments

The author expresses his thanks to Ted Shepherd for very helpful discussions and comments on several versions of the manuscript; to Victor Fomichev, Diane Pendlebury, Dave Ortland for helpful discussions; and to Jeff Forbes for a careful review. This research was supported by the Canadian Middle Atmosphere Modelling Project.

## REFERENCES

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_{2}band cooling in the middle and upper atmosphere for variable CO

_{2}concentration.

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## Footnotes

*Corresponding author address:* Dr. Charles McLandress, Dept. of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada. Email: charles@mam.physics.utoronto.ca