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

View Expanded

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

View Expanded

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 O₃. The larger peak near 105 km with α ∼ (1 day)⁻¹ results from nonlocal thermodynamic equilibrium effects of CO₂ 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 m² 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.