Solar Semidiurnal Tide in the Dusty Atmosphere of Mars

1. Introduction

One of the most dramatic phenomena on Mars is the occurrence of global or planet-encircling dust storms (e.g., Zurek 1982; Smith et al. 2002). Global dust storms generally occur during Southern Hemisphere summer after a period of localized episodic dust storm activity (Zurek 1982; Liu et al. 2003). Significant dust concentrations can extend up to 50-km altitude (Anderson and Leovy 1978; Clancy et al. 2003; Smith et al. 2003) and can persist for months after initial injection into the atmosphere (e.g., Conrath 1975; Martin and Richardson 1993). Solar heating is greatly enhanced due to absorption by dust, giving rise to significant modifications in the mean structure of Mars's atmosphere (e.g., Conrath 1975; Pollack et al. 1979; Smith et al. 2002), as well as tidal perturbations (e.g., Wilson and Hamilton 1996; Wilson and Richardson 2000; Smith et al. 2002).

This paper is primarily concerned with the sun-synchronous solar semidiurnal tide in Mars's atmosphere, with particular emphasis on the 60–200-km altitude region and on Southern Hemisphere summer conditions when amplified excitation of the tide at lower altitudes (0–50 km) occurs because of enhanced solar radiation absorption by globally elevated dust. This problem is of interest for several reasons. First, because of its relatively long wavelength, the semidiurnal tide has the potential to propagate into the thermosphere and produce significant density variations at aerobraking altitudes (ca. 100–170 km). Estimates of such effects are of interest, as existing measurements have inadequate local time coverage to delineate semidiurnal variations in Mars's thermosphere. In addition, molecular dissipation of the tide above 100 km will induce momentum and heat flux divergences that accelerate and heat the thermosphere, but the magnitude of these effects remain unknown. Breaking (convective instability) of the semidiurnal tide may, however, be a critical factor in determining how efficiently the tide couples the lower and upper atmospheres of Mars. Earlier work (Hamilton 1982; Zurek 1986; Zurek and Haberle 1988) suggests that the semidiurnal tide achieves convective instability by about 40 km and significantly modifies the zonal mean state of Mars's atmosphere. However, the implications of convective instability on the continued upward propagation of the semidiurnal tide, and on the zonal mean circulation and thermal structure of the atmosphere at higher altitudes, remains unexplored. Eddy mixing of the atmosphere that accompanies tidal breaking is relevant to minor constituent transport, establishment of the homopause (Leovy 1982), and therefore the planetary escape rate of light gases (Hunten 1973, 1974). It is the purpose of the present work to address the above issues pertaining to the semidiurnal tide in Mars's dusty atmosphere. In the remainder of this section, we introduce the tidal nomenclature used throughout the paper, and then set the problem into the broader context of solar thermal tides in planetary atmospheres, and of previous work relating to Mars.

Solar thermal tides are oscillations in temperature, density, pressure, and winds induced by the daily cyclic absorption of solar energy in an atmosphere or planetary surface. Assuming continuity in space and time around a latitude circle, solar thermal tidal fields are represented in the form

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where t = time (sols for Mars, where 1 sol = 24.6 h), Ω = rotation rate of the planetary atmosphere (= 2π sol⁻¹ for Mars), λ = longitude, n (= 1, 2, . . .) denotes a subharmonic of a sol, s (= . . . . −3, −2, . . . 0, 1, 2, . . . .) is the zonal wavenumber, and the amplitude A_n,s and phase ϕ_n,s are functions of height and latitude. In this context, n = 1 and 2 represent oscillations with periods corresponding to 1.0 and 0.5 sols, and hence are referred to as diurnal and semidiurnal tides, respectively. Eastward (westward) propagation corresponds to s < 0 (s > 0). At any height and latitude the total tidal response is obtained as a sum over n and s.

Rewriting (1) in terms of local time t_LT = t + λ/Ω, we have

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Tidal components with s = n yield local time variations that are independent of longitude, and thus correspond to solar radiation absorption by a zonally symmetric atmosphere or surface. From (1) such components correspond to a zonal phase speed C_ph = dλ/dt = −nΩ/s = −Ω, in other words westward-propagating at the same speed as the apparent motion or migration of the sun to a ground-based observer. These sun-synchronous tidal components are referred to as migrating tides. In response to the zonally asymmetric (longitude dependent) absorption of solar energy by a planetary atmosphere or surface, the local time structure of the atmosphere (at a given height and latitude) is dependent on longitude. In this case, Fourier representation must involve a range of zonal wavenumbers of both signs, corresponding to waves propagating to the east (s < 0) or west (s > 0) (Chapman and Lindzen 1970); these tidal components are commonly referred to as nonmigrating tides. From (2) it is evident that from sun-synchronous orbit (t_LT = constant) a tide characterized by a given s and n appears as a variation in longitude with wavenumber |s − n|. Conversely, a longitude variation observed from sun-synchronous orbit with wavenumber |s − n| can be produced by multiple combinations of s and n.

Diurnal and semidiurnal solar thermal tides play an important role in determining the mean circulation and thermal structure of terrestrial planetary atmospheres (Venus, Earth, and Mars), in addition to their more obvious contributions to variability about the zonal mean state (see review by Forbes 2002). In Earth's atmosphere, solar tides begin to dominate flow patterns above about 80 km. At 95 km, migrating tides are the largest components, but nonmigrating tides are nearly as important and introduce significant longitude variability into the tidal structures (Forbes et al. 2003). In the region between about 100 and 150 km, molecular dissipation of solar tides induces zonal mean accelerations that strongly influence the mean circulation of that height region (i.e., Miyahara and Wu 1989; Angelats i Coll and Forbes 2002). On Venus, tides play a key role in maintaining superrotation of the atmosphere near the cloud tops around 65 km (i.e., Newman and Leovy 1992). However, although the solar semidiurnal tide is observed to propagate to the lower thermosphere of Venus (Schofield and Taylor 1983), its effects on the zonal mean structure of the Venusian atmosphere above the cloud tops is not known. Modeling studies for Mars (Lewis and Read 2003; Wilson and Hamilton 1996) suggest that thermal tides drive a superrotating (eastward) jet below about 30 km in the equatorial region, in much the same fashion that prograde winds are driven within the cloud region of Venus (Fels and Lindzen 1974). The jet is strongest near equinox and under conditions of high dust loading of the atmosphere. Retrograde (westward) winds occur at upper levels over the equator, presumably due to dissipation of the thermal tides as they propagate upward.

The thermosphere of Mars (ca. 100–250 km) is the recipient of a spectrum of diurnal and semidiurnal tides propagating upward from the lower atmosphere (Angelats i Coll et al. 2004) that appear to account for much of the longitude variability in total mass density in the aerobraking regime (100–170 km) inferred from Mars Global Surveyor (MGS) accelerometer measurements (Keating et al. 1998; Forbes and Hagan 2000; Wilson 2000, 2002; Forbes et al. 2003; Withers et al. 2003). Some of the most important tidal components include the eastward-propagating diurnal tides with s = −1, −2, and −3, each producing on the order of 15% density variation near 115 km (Angelats i Coll et al. 2004). Note from Eq. (2) and related discussion that these waves appear as wave-2, wave-3, and wave-4 longitudinal structures from sun-synchronous orbit. As on Earth, eastward-propagating tides have significantly longer vertical wavelengths than their westward-propagating counterparts, and hence penetrate to thermosphere altitudes much more efficiently (Ekanayake et al. 1997). The eastward-propagating semidiurnal tide with s = −1 may also contribute significantly to the wave-3 variation in thermosphere density, particularly at high latitudes (Wilson 2002; Withers et al. 2003). As shown in the general circulation model (GCM) simulations of Angelats i Coll et al. (2004), response of Mars's upper atmosphere to the vertically propagating component of the migrating diurnal tide is rather weak during solstice conditions, producing barely a 2% relative density perturbation at 115 km. In addition, as dust loading of the atmosphere increases, the semidiurnal atmospheric response at the surface and aloft increases in importance compared to the diurnal response (Wilson and Hamilton 1996; Wilson and Richardson 2000). There are several reasons for these differences in diurnal and semidiurnal tidal response. First, as noted above, the relatively short vertical wavelength (≈30–35 km) of the diurnal propagating tide makes it more susceptible to dissipation, thus hindering its propagation into the thermosphere. Second, under solstice conditions the latitude distribution of diurnal heating does not project effectively onto the first propagating diurnal tidal mode, which is symmetric about the equator. Third, under globally dusty conditions the vertical distribution of heating (0 to ∼40–50 km) projects well onto the vertical structure of the long-wavelength semidiurnal tide, but its projection onto the diurnal propagating tide tends to undergo changes of sign within the heating region, resulting in phase interference effects that diminish the atmospheric response aloft. However, quite large (∼20–45 K) diurnal temperature perturbations are possible within the heating region during planet-encircling dust storms (Smith et al. 2002), due to excitation of evanescent modes of the diurnal tide, which have very long vertical scales.

Since our main focus is on vertical coupling between the lower atmosphere and upper atmosphere during globally dusty conditions, we primarily restrict our attention to the migrating solar semidiurnal tide during the Southern Hemisphere summer season. Several previous investigations were devoted to the semidiurnal migrating in Mars's atmosphere and its interaction with the zonal mean flow (Hamilton 1982; Zurek 1986; Zurek and Haberle 1988) under similar conditions, but with emphasis on the atmosphere below about 40–60 km. These works used classical atmospheric tidal theory (Chapman and Lindzen 1970) to estimate tidal amplitudes that resulted from realistic thermal forcing, that is, heating rates that yielded tidal variations in surface pressure consistent with those measured by the Viking 1 and Viking 2 landers over a range of dust conditions (i.e., Zurek and Leovy 1981). In all cases tidal amplitudes easily reached unstable amplitudes by about 40 km. Dissipation of the waves gave rise to zonal mean momentum and heat flux divergences of major importance to Mars's momentum and heat budgets. It was furthermore established that the semidiurnal tide (as opposed to the diurnal tide) was of primary importance with regard to these coupling effects, for the reasons noted previously.

The above works did not consider propagation of the semidiurnal tide above the middle atmosphere. The only attempt to address this problem was in the work of Bougher et al. (1993) wherein they crudely estimated a semidiurnal tide lower boundary condition for the Mars thermosphere general circulation model (TGCM) at 100 km, and examined the thermosphere amplitudes that resulted. They did this for a range of boundary conditions that were thought to be representative of dusty and nondusty conditions in the lower atmosphere. They found dramatic changes in horizontal and vertical wind patterns, and temperature and atomic oxygen distributions, when the semidiurnal forcing was included at the lower boundary. However, the physical basis for their specific choices of boundary conditions did not originate in any realistic simulation of the semidiurnal tide in the lower atmosphere including breaking or mean wind interactions. Their study also did not examine the consequences of molecular dissipation of the tide on the zonal mean wind and temperature distributions.

The purpose of the present work is to gain further insight into this problem using a model that takes into account breaking of the tide due to convective instability, as well as the self-consistent interaction between the semidiurnal tide and the zonal mean flow in a global calculation extending from the surface to the upper thermosphere (∼250 km). We seek here to establish (i) the thermosphere temperature and density perturbations due to the solar semidiurnal migrating tide in the aerobraking regime (∼100–170 km) of Mars; and (ii) the zonal mean zonal wind, temperature, and meridional circulation changes induced by the breaking semidiurnal tide throughout Mars's atmosphere, for a range of global dust conditions. In addition, the parameterization for wave breaking that we employ yields estimates of eddy diffusivities throughout Mars's atmosphere. For comparison purposes, we also provide some results pertaining to the eastward-propagating diurnal tides with zonal wavenumbers s = −1 and s = −2. In the following section, we describe the model and various inputs assumed in the calculations, and the parameterization that is utilized to implement wave breaking in the model. This is followed in section 3 by a presentation of our results, and in section 4 by summary and conclusions.

2. The model

d. Breaking parameterization

Vertically propagating tides are basically internal gravity waves with periods that are subharmonics of a sol, modified by the rotation and sphericity of the atmosphere. Lindzen (1970) in fact provides a formalism for deriving “equivalent gravity waves” on a rotating plane that approximate the vertical propagation characteristics of the tidal waves. This similarity is also discussed in the work of Lindzen (1981), where he develops a parameterization for taking into account the effects of breaking (convective instability) on the vertical propagation of gravity waves, and the resulting turbulence (eddy diffusion) and acceleration of the mean flow. The basis of this approach is the hypothesis that the instability generates sufficient turbulence at the breaking level (z_break) to prevent further growth of the wave, and that the momentum carried by the wave will be deposited into the mean flow above z_break. Lindzen gives the following expression for the value of eddy diffusion required to prevent further growth of the wave:

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where k_x and k_y are the zonal and meridional wavenumbers of the wave, c is the zonal phase speed, H(z) and u(z) are the background atmosphere scale height and zonal wind, z is altitude, and N is the Brunt–Väisälä frequency. For the background flow shown in Fig. 3 and the phase speed of the semidiurnal tide, the second term is of order one-third or less times the first. Therefore, we drop the second term, and after some manipulation using relations in Lindzen (1981) arrive at the following approximation to (3):

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where k_z is the vertical wavenumber of the wave and ω is the wave frequency Doppler-shifted by the mean wind (see also Lindzen and Forbes 1983). As noted by Lindzen (1981), the neglected term in (3) is most important in the regime where u − c → 0. The semidiurnal tide does not come close to encountering a critical level in the wind distribution of Fig. 3. The zonal phase speed of the semidiurnal tide is approximately −240 cosϕ m s⁻¹ where ϕ is latitude. At the −100 m s⁻¹ peak of the westward jet near 50 km and −20° latitude, for instance, the zonal phase speed of the semidiurnal tide is about −226 m s⁻¹.

One of the difficulties of implementing (3) or (4) in a model is the sudden onset of eddy diffusion at z_break, which can lead to numerical difficulties. Usually, this is handled by applying some sort of exponential dependence below z_break to smooth out the eddy profile. Lindzen and Forbes (1983) introduce a plausible means of smoothing the sudden onset of turbulent diffusion at the breaking level that has a physical basis. The implementation is equivalent to a cascade law:

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where D_max is given by expression (4), T ′ is the perturbation temperature associated with the wave, Γ = (dT/dz) + (g/c_p) where T is the mean temperature, and n is a power. Physically, expression (5) implies that turbulence can be generated by a stable gravity wave via a cascade to smaller-scale waves whose vertical wavelengths are short enough to permit breaking. Utilization of (5) ensures that sufficient diffusion is generated to keep k_zT ′ < Γ. Lindzen and Forbes (1983) apply this to the diurnal tide in Earth's atmosphere, and demonstrate the insensitivity of the parameterization, in practice, to the choice of n. Smaller values of n tend to spread out the diffusion over a wider altitude regime. Using a sufficiently large value of n ensures that values of D within short distances of z_break are too small to have any consequence on the wave. In the present application we use a value of n = 3, and through a series of numerical experiments for n = 0 to n = 6 (detailed later) demonstrate the relative insensitivity to the choice of n.

Equation (5) was implemented in the model as follows. The model was first run with a sufficiently weak level of forcing that the semidiurnal tide did not break. A representative vertical wavelength of 95 km was estimated from the wave's phase progression with height below 100 km, leading to a D_max value of 3.25 × 10⁴ m² s⁻¹ from (4) for zero mean wind. The correct forcing was then introduced, and at each grid point and time step, D was calculated from (5) in accord with the current and local values of T ′, Γ, and u. The vertical wavelength for this latter run was checked a posteriori and found to remain close to 95 km. The fact that a vertically propagating tide can reach its limiting amplitude but still retain downward phase progression at its inviscid wavelength is consistent with previous numerical simulations (i.e., Forbes 1982a, b; Lindzen and Forbes 1983). Note also that 95 km is significantly less than the >200-km vertical wavelength of the fundamental semidiurnal tidal mode. This implies presence, at least, of the first antisymmetric and second symmetric modes with approximate vertical wavelengths of 80–90 km and 50–60 km, respectively, that are present due to mode coupling (Lindzen and Hong 1974) vis-à-vis interactions with the zonal mean wind field.

3. Results

Figure 4 illustrates results from a low-dust simulation similar to that displayed in Fig. 3, except in this case the solar semidiurnal tide is activated in the model. Perturbation temperature amplitudes range from about 5 K near 80 km to 40 K above 150 km. The zonal mean winds above 100 km and in the presence of the semidiurnal tide are shifted westward, reflecting deposition of westward momentum into the mean flow by the dissipating tide. The zonal mean meridional circulation is also modified, now extending more into the thermosphere. Eddy diffusion coefficients maximize between 100 and 150 km at winter middle latitudes at values of order 0.5–1.0 × 10³ m² s⁻¹. Note the tendency of the wave to propagate into the Northern Hemisphere where the winds are eastward, Doppler-shifting the tide to higher frequencies.

Figure 5 illustrates the same fields as Fig. 4, except for a dust opacity τ of 2.3, that is, representative of average conditions during the two global dust storms experienced by the Viking 1 and Viking 2 landers. Recall that the zonal mean heating is kept the same as for the previous simulation, so that all changes are due solely to the intensified semidiurnal tide. Semidiurnal temperature amplitudes are now of order 10–15 K near 50 km with maximum values of about 75 K above 150 km near +30° latitude. Changes in mean zonal and meridional winds are minor below 80 km. In the thermosphere, zonal wind and meridional winds have intensified to over 200 and 50 m s⁻¹, respectively, over the equator, with net temperature reductions throughout the thermosphere (not shown) ranging from −70 K below −30° latitude and −20 K above +30° latitude. Eddy diffusivities now exceed 1.2 × 10⁴ m² s⁻¹ at the peak near 30° latitude and 130 km. Eddy diffusivities of order 0.1–3.0 × 10³ are now spread over low latitudes in the Northern (winter) Hemisphere.

Results for high-dust conditions (τ = 5.0) are not sufficiently different from values in Fig. 5 to warrant illustration here. Values quoted above are increased about 20%–30%, and the Northern Hemisphere eastward jet maximum at 50 km is reduced to about 80 m s⁻¹. The reason that there are not profound changes in the thermospheric tidal response as one progresses from low- to high-dust conditions is that the tide has nearly achieved its convectively unstable limit under low-dust conditions. As the forcing is increased, therefore, the thermospheric response remains nearly the same. However, as dust heating increases, this unstable limit is achieved at progressively lower heights, so that the middle atmosphere does respond to increased dust loading. This effect is also seen in the eddy diffusivity levels.

In the region of strong negative shear near +60° latitude and 120 km, the second term in (3) becomes comparable to the first. This would require increasing D_max in (5) by a factor of 2. Given the geographical localization and the uncertainties involved, we did not employ the more accurate version of (3) to revise this result. Moreover, ∂u/∂z → 0 above 140 km and the second term in (3) becomes small. Note also, that despite the large mean westward winds above 140 km in Fig. 5, the semidiurnal tide still does not reach a critical level. Even if a critical level were reached, in this altitude regime behavior of the tide is dominated by molecular dissipation and is relatively unaffected by the zonal mean winds.

The differences between fields illustrated in Fig. 3 (only zonal mean heating, no tidal heating) and Fig. 5 (tidal heating and zonal mean heating) are displayed in Fig. 6, to emphasize effects of the semidiurnal tide in the dusty Mars atmosphere on the zonal mean circulation. Consistent with the differences noted above, we note the following effects: Significant westward momentum is deposited into the thermosphere, resulting in ∼−100 to −200 m s⁻¹ changes in the zonal wind field of the Southern Hemisphere above ∼130 km; changes below 100 km are about an order of magnitude less. A net meridional flow of order 20–60 m s⁻¹ is generated by tidal dissipation in the thermosphere that is in the same direction as the diabatic circulation forced by differential heating between the hemispheres, and intensifies to about 100 m s⁻¹ near 120 km at middle northern latitudes. This wave-driven meridional flow is accompanied by rising (sinking) motions in the Southern (Northern) Hemisphere, with concomitant zonal mean temperature reductions of order 20–80 K over most of the thermosphere, and temperature increases poleward of 60° latitude between 50 and 100 km. The tendency for increased middle-atmosphere temperatures at high winter latitudes in the presence of the semidiurnal tide is also noticeable for low dust conditions (not shown). The intensification of the Hadley cell to high winter latitudes by tide-mean flow interactions has also been noted by Wilson (1997).

Figure 7 provides further insight into Mars's response to the semidiurnal tide under intermediate (τ = 2.3) global dust conditions. Perturbation zonal and meridional winds in Figs. 7a,b attain values of order 100 m s⁻¹ at middle and high latitudes in the thermosphere, and thus exceed the zonal mean winds at many locations under quiescent conditions. Zonal acceleration of the mean flow in Fig. 7c is of order 50–250 m s⁻¹ sol⁻¹ over most of the Northern Hemisphere between 75 and 175 km. Semidiurnal density perturbations in Fig. 7d are of order 40%–70% above 100 km, and thus contribute significantly to the structure and variability of density in Mars's aerobraking regime. Figure 7f illustrates our earlier claim that downward phase progression of the semidiurnal tide extends up to about 150 km at a rate consistent with a vertical wavelength of order 95 km, even in the presence of self-generated eddy diffusion. Note the abrupt transition to phase constancy with height above 150 km, due to dominance of molecular diffusion (time constant decreases exponentially with height).

The relatively large (50%–70%) density perturbations in Fig. 7 bring into question the assumption of linearity underlying the perturbation (tidal) equations, and deserve some comment. Our calculation of relative density amplitude is based upon the following expression:

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where ρ′ = perturbation density, ρ_o = background density, Φ′ = perturbation geopotential, g = acceleration due to gravity, H = scale height, T ′= perturbation temperature, T_o = background temperature, and all primed quantities are complex. This equation is derived from the linearized ideal gas law

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and assuming Φ′ = p′/ρ_o. Note that Φ′ and T ′ are solved for in the system of equations, but ρ′ is a derived quantity; moreover, from Fig. 7e, T ′/T_o ≤ 0.33. The double maxima and minimum in the vertical structure for ρ′ is due in part to interference effects between the two terms in the right-hand side of (6), whose relative amplitudes and phases change with height. The large density amplitude near 120 km is not surprising, as the wave is near-breaking at this location (see the derived eddy diffusion coefficient in Fig. 4b). As discussed by Lindzen (1970, 1971) and Lindzen and Blake (1971), if ρ′/ρ_o ≤ 1 in the region below which molecular viscosity and conductivity dominate (here, about 150 km), then the nonlinear terms in the equations will not play a dominate role. According to this criterion, then, we are barely within the limits where linear theory might apply. We cannot provide quantitative information on the errors involved in neglecting second-order perturbation terms in our solution for the tidal fields. However, we note the following. The dominant process limiting the amplitudes of tidal fields in the 50–150-km height region depicted in Fig. 5 is nonlinear feedback introduced into our equation system vis-à-vis Eq. (5); under these conditions neglect of other nonlinear effects in the equations may be justified. Further, we can estimate that if nonlinear terms in the equations were important, that they would probably limit the maximum amplitudes near 120 km to no less than 50% of those depicted; otherwise, at these amplitudes the assumption of linearity would be approximately valid.

Above 150 km, molecular viscosity and conductivity dominate, and the solutions for u′, υ′, and T ′ become constant with height. In this regime, the assumption of linearity is more defensible despite the large density amplitudes. Consider the hydrostatic equation in the following form:

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where R is the gas constant and x = −ln(p/p_o). If T ′ oscillates with time but is constant with respect to height, then Φ′ increases linearly with height. According to (6), ρ′/ρ_o also varies linearly with height, as appears to be the case in Fig. 7d above 150 km. In fact, ρ′/ρ_o will exceed 1.0 above 220 km. However, the fact that the asymptotic solutions for u′, υ′, and T ′ are determined at altitudes (∼150 km) where linearity is valid, means that they are valid at all altitudes, even where ρ′/ρ_o > 1.0 (Lindzen 1970, 1971; Lindzen and Blake 1971).

Results for the more extreme dust conditions corresponding to τ = 5.0 do not differ appreciably from those depicted in Fig. 7, for the same reasons noted earlier in connection with Fig. 5; that is, once the semidiurnal tide already reached its convectively unstable limit, response to increased forcing is nearly absent in the thermosphere, but becomes more evident in the 50–100-km region as the tide breaks at progressively lower heights with increased forcing.

Similar considerations apply in Fig. 8, where we explore the dependence of our results on the choice of the power law in (5) for the τ = 2.3 dust opacity case. Figure 8 illustrates perturbation temperatures and eddy diffusivities corresponding to values of n = 1 and n = 5. Comparing temperature amplitudes between Figs. 5 and 8, there are not large differences in the salient features above 100 km where the primary effects on the mean circulation occur for dusty conditions. The greatest effect occurs between 50 and 100 km; as n increases from 1 to 5, maximum perturbation temperatures near 80 km in Fig. 8 increase from 10 to 30 K. Maximum zonal winds (not shown) are −121 m s⁻¹ in the thermosphere for n = 1 and −197 m s⁻¹ for n = 5, which can be compared with −180 m s⁻¹ for n = 3 in Fig. 5. Thus, while these differences cannot be described as small, our claim that the broad, salient conclusions of our work hold up for a range of power laws in (5) is supported. There is one exception, and that is in the eddy diffusivity in regions, where the semidiurnal temperature perturbation is far below its convectively unstable limit. From Fig. 8, it is obvious that the impacts of the cascade hypothesis are much more global in extent for smaller values of n.

4. Comparison with diurnal tidal responses

As noted in the introduction, the thermosphere response to the migrating diurnal propagating tide is weak, especially under solstice conditions, whereas the eastward-propagating tides are known to induce significant density variations in the lower thermosphere. While the present study is mainly focused on the migrating semidiurnal tide, it is nevertheless informative for comparative purposes to investigate what the potential impact of the eastward-propagating diurnal tides might be on the zonal mean circulation and thermal structure of the thermosphere. Toward this end, we provide herein a brief description of results for the eastward-propagating diurnal tides with s = −1 and s = −2, hereafter referred to as DE1 and DE2, respectively. Note from Eq. (2) and the discussion in the introduction that DE1 and DE2 appear as wave-2 and wave-3 longitude structures from near-sun-synchronous orbit.

In the spirit of a numerical experiment, and with the limited capabilities of our model, we take the following simple approach to arrive at a reasonably bounded solution. Excitation of DE1 and DE2 are introduced in the form of a tropospheric heat source with vertical dependence corresponding to the low-dust profile in Fig. 1, and horizontal structure corresponding to the first symmetric Hough mode (Kelvin wave) for the oscillation. The heat source is calibrated to yield a maximum 1-K temperature oscillation near 25 km, in rough accord with TES observations and GCM results (Banfield et al. 2003; Wilson 2000). We utilize the same zonal mean heating distribution as for the semidiurnal tide simulations (applicable to Southern Hemisphere summer conditions), but do not employ the breaking parameterization for these waves, as they do not come close to achieving convective instability.

To validate the thermosphere response, we compare our DE1 and DE2 simulations with wave-2 and wave-3 relative density perturbations (ρ′/ρ_o), respectively, from MGS accelerometer measurements. Densities between 115 and 135 km were obtained from the NASA Planetary Data System (available online at http://starbrite.jpl.nasa.gov/pds/viewProfile.jsp?dsid=MGS-M-ACCEL-5-ALTITUDE-V1.1) during two phases of aerobraking, and normalized to a constant altitude of 125 km. Density residuals in 10° latitude bins were then fit with Fourier series in longitude to extract the amplitudes and phases of the wave components. The data cover season and local time as follows. (On Mars, season is defined according to Martian heliocentric longitude, L_s, where L_s = 0 is northern spring equinox, L_s = 90 is northern summer solstice, L_s = 180 is northern autumn equinox, and L_s = 270 is northern winter solstice.) During phase I, extending from mid September 1997 to late March 1998, periapsis precessed from L_s = 180 at 30°N and 1800 local standard time (LST), to L_s = 300 at 60°N and 1100 LST. During the first part of phase II analyzed here, periapsis precessed from L_s = 30 at 60°N and 1700 LST to L_s = 80 at 80°S and 1500 LST. Although phase II does not occur during the Southern Hemisphere summer conditions simulated by the model, these data are included since (i) the phase I data cover such a restricted range of latitudes; (ii) lower-atmosphere TES data at 25 km (Banfield et al. 2003) indicate DE1 and DE2 to exist at the 1-K amplitude level for a wide range of seasons; and (iii) some insight can be provided as to the magnitude of upper-level DE1 and DE2 effects for seasons other than Southern Hemisphere summer.

The model–data comparison is illustrated in Fig. 9. Observed phases correspond to the longitudes of maxima seen from MGS at the local times of measurement, while the model phases (calibrated to best fit the data by adjusting the phase of the heat source) are translated to the satellite frame assuming a single constant local time of 1500 LST. Shifting the data to conform to the single local time of the model would only require changes of order 10°–15° in the Southern Hemisphere data for phase II, and shifts less than 30° for the for phase I (under the assumption that the displayed oscillations purely reflect DE1 and DE2), and was therefore not implemented to maintain the integrity of the observational data. Several points are worth noting in connection with Fig. 9. First, the density perturbations predicted by the model are in reasonable accord with the observed amplitudes, although a better overall fit to the Southern Hemisphere summer (phase I) data would be obtained by a slightly larger response. However, any significant increases in the applied forcing would be inconsistent with the TES measurements near 25 km, and moreover, there are other waves that can contribute to the wave-2 and wave-3 features displayed in Fig. 9 (see the introduction). Second, there is reasonable consistency between the amplitudes and phases of the density perturbations between seasons (i.e., between the phase I and phase II data). It is concluded that the heat source in the model has been reasonably calibrated, so that the global tidal fields and effects on zonal mean temperature and wind structure in the model are credible. We now turn to illustrations of these effects.

Figure 10 illustrates results for DE1. In comparison to the intermediate-dust results for the migrating semidiurnal tide in Fig. 7, overall impact of DE1 on middle atmosphere and thermosphere wave dynamics is roughly a factor of three less than the semidiurnal tide. However, in terms of the low-dust semidiurnal tide (cf. amplitude of T ′ with that in Fig. 4), results are more comparable, at least within a factor of two. Note the asymmetry in all of the fields toward the Southern Hemisphere, due to favored propagation of the wave through westward zonal mean winds (cf. Fig. 3a). In terms of classical tidal theory, this distortion of the response can be accounted for by coupling into the first antisymmetric mode vis-à-vis mean wind interactions (Lindzen and Hong 1974). This mode is an inertia-gravity wave with vertical wavelength of order 50–60 km (Forbes et al. 2001), and accounts for the phase progression with height in Fig. 10b, which would not be apparent if the thermally forced near-barotropic Kelvin wave component of DE1 were solely present. The very low values of wave-induced zonal mean winds, of order 3–6 m s⁻¹ in the thermosphere, are particularly noteworthy compared to the −40 to −200 m s⁻¹ zonal mean winds generated in the thermosphere by dissipation of the low-dust and intermediate-dust semidiurnal tides, respectively. This effect is discussed below, in conjunction with the DE2 results.

Similar results for DE2 are presented in Fig. 11. In broad terms, the T ′, u′, ρ′/ρ_o amplitudes are of similar magnitude to DE1. Note the shorter vertical wavelength for the temperature perturbation, which reflects some combination of the first symmetric (Kelvin) and antisymmetric modes of DE2, with vertical inviscid wavelengths of order 110 and 40 km, respectively. Particularly obvious are the +20–80 m s⁻¹ zonal mean winds generated in the thermosphere due to molecular dissipation of DE2. The reason for the greater efficiency of DE2 in modifying the mean circulation (and thermal structure) may be understood in part through examination of the eddy flux divergence terms in the zonal mean equations. Taking a typical example, the eddy flux divergence term in the zonal mean zonal momentum equation appears as follows:

View Expanded

where a = planetary radius; θ = colatitude; u′, υ′, w′ are zonal, meridional, and vertical velocities; p is pressure; and x = −ln(p/p_o). DE1 is near-barotropic, and thus has larger vertical scales and smaller vertical velocities than DE2, and also possesses broader horizontal scales. These effects combine to produce zonal mean zonal accelerations 4 times greater for DE2 (Fig. 11) than DE1 (Fig. 10), and similar differences in the momentum and heat flux divergences in the other zonal mean equations.

5. Summary and conclusions

A numerical model is used to explore the sun-synchronous (migrating) semidiurnal tidal response of Mars's atmosphere. Global dust opacity levels of 0.5, 2.3, and 5.0 are considered, typical of average values before and during dust storms experienced by the Viking 1 and Viking 2 landers in 1977 during Southern Hemisphere summer. The model provides self-consistent solutions to the coupled zonal mean and tidal equations from the surface to 250 km, and thus the modifications to the zonal mean thermal and wind structures due to tidal dissipation are also considered. Breaking (convective instability) of the semidiurnal tide is parameterized using a linear saturation scheme. The scheme is implemented using a power law (n = 3) of the form given by (5), and is physically consistent with a cascade from low to high wavenumbers, wherein the nonbreaking semidiurnal tide can lead to nonzero diffusivities vis-à-vis cascade to smaller-scale waves that do break. Thermal forcing in the model gives rise to surface pressure perturbations and middle-atmosphere zonal mean winds and temperatures that are consistent with available measurements and general circulation models.

The semidiurnal tide produces large temperature, wind and density perturbations in Mars's middle and upper atmosphere, with a nominal vertical wavelength of about 95 km below 150-km altitude. For example, for a global dust opacity of τ = 2.3:

• perturbation temperature amplitudes maximize at about 70 K above 150 km, while eastward and northward wind amplitudes achieve maximum values of order 100–150 m s⁻¹ between 90 and 150 km at middle latitudes in the Northern Hemisphere;

• at 50 km, temperature and wind amplitudes are typically 10–20 K and 10–20 m s⁻¹;

• perturbation densities are of order 50%–70% between 90 and 150 km, and thus contribute significantly to variability of the aerobraking regime in Mars's atmosphere; and

• eddy diffusivities associated with the breaking parameterization reach values of order 10³–10⁴ m² s⁻¹ at Northern Hemisphere middle latitudes between 100 and 150 km, and can be of order 1–10 m² s⁻¹ between 0 and 50 km.

Semidiurnal tidal amplitudes for low- (τ = 0.5) and very high– (τ = 5.0) dust opacities do not vary significantly from the above values in the thermosphere, since the semidiurnal tide is close to achieving its saturated value for low-dust opacities. The semidiurnal response between 50 and 100 km is, however, more sensitive to change in the semidiurnal forcing (cf. Figs. 4 and 5). Even during low-dust conditions in Mars's atmosphere, the eddy diffusion values quoted above begin to approach those required by photochemical models between the equator and middle latitudes in the winter hemisphere. As shown in Fig. 8, if the value of n in the power law of (3) is increased or decreased, the global distribution of the eddy diffusivities is correspondingly decreased or increased, respectively. Thus, our estimate of eddy diffusivity in these model simulations is the parameter subject to greatest uncertainty.

Dissipation of the semidiurnal tide also modifies the zonal mean temperature and wind structure of Mars's atmosphere, by virtue of the momentum and heat flux divergences associated with dissipation of the wave. For example, for τ = 2.3:

• wave-driven zonal mean westward winds are of order 10–30 m s⁻¹ below 100 km, and exceed 200 m s⁻¹ above 150 km at middle latitudes in the Northern Hemisphere;

• a meridional circulation is also forced, with northward winds of order 20–80 m s⁻¹ between 100 and 150 km, upwelling (∼5–20 cm s⁻¹) above 100 km in the Southern Hemisphere, and downwelling (20–80 cm s⁻¹) at middle to high latitudes in the Northern Hemisphere; and

• the corresponding temperature perturbations range between −20 to −70 K over most of the thermosphere, with 10–20-K increases in temperature above 60° latitude in the Northern Hemisphere between 50 and 100 km.

Comparisons are also made with simulations for the eastward-propagating diurnal tides with zonal wavenumbers s = −1 (DE1) and s = −2 (DE2), which give rise, respectively, to wave-2 and wave-3 longitude features in densities and other parameters measured from sun-synchronous orbit. Results for these waves are as follows:

• Perturbation temperatures, winds, and densities for DE1 and DE2 are comparable to those for the migrating semidiurnal tide under low-dust conditions.

• Dissipation of DE2 has a significant impact on the zonal mean circulation of the thermosphere, similar to that of the semidiurnal tide under low-dust conditions, while DE1 plays a relatively minor role in generating momentum flux divergences due to its more barotropic nature and broad horizontal scale.