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    Amplitudes of semidiurnal heating rates (W kg−1) as a function of height and latitude for dust opacities of (top) τ = 0.5, (middle) τ = 2.3, and (bottom) τ = 5.0, consistent with those in Zurek (1986) as described in the text. The latitude shape of the heating profiles is taken to be symmetric about the equator, consistent with the solar zenith angle dependence for the semidiurnal component of heating described in Leovy and Zurek (1979) for a subsolar point of −15° latitude. (The corresponding diurnal component of heating is much more asymmetric about the equator)

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    Comparison between computed relative semidiurnal surface pressure amplitudes (solid lines) and those experienced by the Viking 1 (22.5°N, 48°W) and Viking 2 (48°N, 134°E) landers (symbols) under similar conditions. The Viking data are mean values inferred from Wilson and Hamilton (1996).

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    Modeled zonal mean parameters vs height and latitude for low dust conditions in Mars's atmosphere, with zonal mean forcing only: (a) eastward wind, (b) southward wind, (c) temperature perturbation from global mean, and (d) vertical wind.

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    Modeled semidiurnal tide and zonal mean parameters vs height and latitude for low-dust conditions in Mars's atmosphere, with zonal mean forcing and semidiurnal tidal forcing: (a) semidiurnal tidal temperature amplitude, (b) eddy diffusion coefficient associated with tidal breaking, (c) zonal mean eastward wind, and (d) zonal mean southward wind. In comparison with Fig. 3, the zonal mean winds now include the effects of dissipation of the semidiurnal tide.

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    Same as Fig. 4, except for excitation of the solar semidiurnal tide corresponding to dust opacity of τ = 2.3. The zonal mean forcing is the same as that used in Figs. 3 and 4.

  • View in gallery

    Differences between modeled zonal mean parameters corresponding to simulations with (i.e., Fig. 5) and without (i.e., Fig. 3) semidiurnal tidal forcing for a dust opacity of τ = 2.3: (a) eastward wind, (b) temperature perturbations from zonal mean, (c) southward wind, and (d) vertical wind. These difference fields represent the wave-driven component of the zonal mean circulation.

  • View in gallery

    Fields associated with the semidiurnal tide in Mars's atmosphere as a function of height and latitude, corresponding to forcing with dust opacity of τ = 2.3: (a) eastward wind amplitude, (b) southward wind amplitude, (c) zonal mean zonal acceleration due to dissipation of the tide, (d) relative density amplitude, (e) temperature amplitude, and (f) temperature phase [longitude of maximum at 0000 universal time (UT)].

  • View in gallery

    (left) Semidiurnal temperature amplitudes and (right) eddy diffusion coefficients for a dust opacity of τ = 2.3, and n = 1 and n = 5 power laws in Eq. (3), respectively.

  • View in gallery

    (left) Amplitudes and (right) phases of (top) longitudinal wavenumber 2 and (bottom) wavenumber 3 relative density amplitudes at 125 km from phase I (filled circles) and phase II (open circles) of MGS aerobraking operations. The phase I data correspond to Ls = 180–300, and the phase II data correspond to Ls = 30–80, and times between 1800–1100 and 1700–1500 LST, respectively. The solid lines correspond to calculated amplitudes and phases (for a fixed local time of 1500 h) for the eastward-propagating diurnal tides with zonal wavenumbers (top) s = 1 and (bottom) s = 2 using the same zonal mean heating distribution as for the semidiurnal tide (Southern Hemisphere summer conditions). Vertical bars are 1σ uncertainty estimates based upon standard deviations of density residuals in 10° × 30° latitude × longitude bins at 125 km. Tropospheric forcing for these waves in the model has the horizontal structure of the first symmetric Hough mode (Kelvin wave) for the oscillation, and is set to produce a maximum 1-K temperature oscillation over the equator near 25 km.

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    Fields associated with the eastward-propagating diurnal tide with s = −1 (DE1) in Mars's atmosphere as a function of height and latitude, corresponding to Southern Hemisphere summer conditions and the modeled density perturbations depicted in Fig. 9: (a) temperature amplitude, (b) temperature phase (longitude of maximum at 0000 UTC), (c) eastward wind amplitude, (d) relative density amplitude, (e) zonal mean zonal acceleration due to dissipation of the tide, and (f) zonal mean zonal wind due to dissipation of the tide.

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    Same as Fig. 10, except for the eastward-propagating diurnal tide with s = −2 (DE2).

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Solar Semidiurnal Tide in the Dusty Atmosphere of Mars

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  • 1 Department of Aerospace Engineering Sciences, University of Colorado, Boulder, Colorado
  • | 2 Department of Earth and Planetary Science, Kyushu University, Fukuoka, Japan
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Abstract

Vertical coupling due to the solar semidiurnal tide in Mars's atmosphere, and effects on zonal mean temperature and wind structures, are investigated using a numerical model. The model provides self-consistent solutions to the coupled zonal mean and tidal equations from the surface to 250 km. Breaking (convective instability) of the semidiurnal tide is parameterized using a linear saturation scheme with associated eddy diffusivities. 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. Results presented here primarily focus on globally elevated dust levels during Southern Hemisphere summer, conditions similar to those experienced by the Viking 1 and Viking 2 landers during the 1977 global dust storms. Semidiurnal temperature and wind amplitudes maximize in the winter hemisphere and exceed 50 K and 100 m s−1 above 150 km and are typically 10–20 K and 10–20 m s−1 at 50 km. 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.

Eddy diffusivities associated with the breaking parameterization reach values of order 103–104 m2 s−1 between 100 and 150 km, and can be of order 1–10 m2 s−1 between 0 and 50 km. Dissipation of the semidiurnal tide induces zonal mean westward winds of order 10–30 m s−1 below 100 km, and in excess of 200 m s−1 above 150 km. The corresponding temperature perturbations range between −20 and −70 K over most of the thermosphere, with 10–20-K increases in temperature at high winter latitudes between 50 and 100 km. All of the wave and zonal mean perturbations noted above represent very significant modifications to the thermal and dynamical structure of Mars's atmosphere.

Estimates are also provided for the eastward-propagating diurnal tides with zonal wavenumbers s = −1 and s = −2. These waves also have long vertical wavelengths and hence are capable of effectively coupling the lower and upper atmospheres of Mars. However, the perturbation and zonal mean effects of these waves are a factor of 2 or more smaller than those cited above for the semidiurnal tide under dusty conditions.

Corresponding author address: Jeffrey M. Forbes, Dept. of Aerospace Engineering Sciences, University of Colorado, Boulder, CO 80309. Email: forbes@colorado.edu

Abstract

Vertical coupling due to the solar semidiurnal tide in Mars's atmosphere, and effects on zonal mean temperature and wind structures, are investigated using a numerical model. The model provides self-consistent solutions to the coupled zonal mean and tidal equations from the surface to 250 km. Breaking (convective instability) of the semidiurnal tide is parameterized using a linear saturation scheme with associated eddy diffusivities. 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. Results presented here primarily focus on globally elevated dust levels during Southern Hemisphere summer, conditions similar to those experienced by the Viking 1 and Viking 2 landers during the 1977 global dust storms. Semidiurnal temperature and wind amplitudes maximize in the winter hemisphere and exceed 50 K and 100 m s−1 above 150 km and are typically 10–20 K and 10–20 m s−1 at 50 km. 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.

Eddy diffusivities associated with the breaking parameterization reach values of order 103–104 m2 s−1 between 100 and 150 km, and can be of order 1–10 m2 s−1 between 0 and 50 km. Dissipation of the semidiurnal tide induces zonal mean westward winds of order 10–30 m s−1 below 100 km, and in excess of 200 m s−1 above 150 km. The corresponding temperature perturbations range between −20 and −70 K over most of the thermosphere, with 10–20-K increases in temperature at high winter latitudes between 50 and 100 km. All of the wave and zonal mean perturbations noted above represent very significant modifications to the thermal and dynamical structure of Mars's atmosphere.

Estimates are also provided for the eastward-propagating diurnal tides with zonal wavenumbers s = −1 and s = −2. These waves also have long vertical wavelengths and hence are capable of effectively coupling the lower and upper atmospheres of Mars. However, the perturbation and zonal mean effects of these waves are a factor of 2 or more smaller than those cited above for the semidiurnal tide under dusty conditions.

Corresponding author address: Jeffrey M. Forbes, Dept. of Aerospace Engineering Sciences, University of Colorado, Boulder, CO 80309. Email: forbes@colorado.edu

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
i1520-0469-63-7-1798-e1
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 An,s and phase ϕn,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 tLT = t + λ/Ω, we have
i1520-0469-63-7-1798-e2
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 Cph = /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 (tLT = constant) a tide characterized by a given s and n appears as a variation in longitude with wavenumber |sn|. Conversely, a longitude variation observed from sun-synchronous orbit with wavenumber |sn| 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

a. General description

The numerical model employed here is a quasi-non-linear time-dependent global numerical model that simulates the propagation of one or more linearly independent waves (forced within the model) interactively with the zonal mean flow (Miyahara and Wu 1989). All of the equations and boundary conditions for this model are fully detailed in Miyahara and Wu (1989), and thus are not repeated here. Briefly, the model consists of a zonal mean equation system and a perturbation equation system, obtained from a second-order perturbation analysis of the primitive equation system in log-pressure coordinates. The nonlinear zonal mean equations are coupled to the perturbation equations through zonally averaged eddy flux terms in the zonal and meridional momentum equations, and in the thermodynamic equation. Coefficients in the perturbation equation system are functions of the zonal mean winds and temperatures. The two equation systems are integrated interactively until a steady state is achieved. Note that the wave solution is affected not only by the zonal mean temperatures and winds induced by the solar heating, but also by the zonal mean field induced by momentum deposition due to the waves themselves. The nonlinear wave–wave interactions are neglected, as these appear in the high-order equation system (Miyahara and Wu 1989; Andrews and McIntyre 1976). The zonal mean quantities solved for in this system are actually departures from the global mean. The global mean atmosphere is approximated using the empirical atmospheric model of Stewart (1987) for Southern Hemisphere summer solstice and average solar and dust conditions. The corresponding temperature profile is characterized by a value near 230 K at the 6.1-mb surface, decreasing linearly to a middle-atmosphere (≈40–100 km) isothermal value near 160 K, then asymptotically increasing in the lower thermosphere (100–175 km) to an exospheric temperature of about 250 K above 200 km. Among other data, this model is designed to maintain consistency with the altitude of the 1.24-nbar level (the height of the ionospheric peak) in accord with radio occultation measurements from Mariner 4, 6, 7, and 9. As such, it provides realistic densities in the thermosphere where the vertically propagating tides of interest here undergo molecular dissipation.

The model was run with latitude and vertical resolutions of 5° and 0.1 scale heights, with lower and upper boundaries at the surface and approximately 250 km, which is close to the base of the exosphere for Mars. The model was integrated with a time step of 300 s, and converged within 40 days. The zonal mean equations were integrated forward without tidal forcing during the first 15 days, during which the zonal mean forcing ramped up to its steady-state value. At day 15, the tidal forcing was introduced, and it was ramped up to its steady-state value by day 25 of the total model integration. A time-dependent Rayleigh friction with initial value of 5 sol−1 throughout the domain and exponential time constant of 5 sols was employed in the zonal mean equation system to damp initial transients during the numerical integration.

b. Tidal heating

Heating rates for three levels of atmospheric dust loading were formulated for this study, and are illustrated in Fig. 1. To isolate the effects of coupling between the lower atmosphere and thermosphere, other heat sources due to solar radiation absorption above 50 km are not included. The heating rates in Fig. 1 are based on those of Leovy and Zurek (1979) for low-dust conditions (normal-incidence optical depth τ ≈ 0.5), and those of Zurek (1986) for intermediate (τ = 2.3) and high (τ = 5.0) dust conditions. The low-dust heating distribution is mainly intended to simulate tidal thermal forcing produced by convective and radiative transfer from the ground, and includes a phase shift with height (not shown) as described in Leovy and Zurek (1979). The other two heating distributions are based on radiative calculations constrained by spacecraft observations of airborne dust distributions (Anderson and Leovy 1978) during clearing of the 1971 great dust storm and measurements of surface pressure changes during the 1977 storms (see below). It is assumed that the dust is distributed uniformly around the planet, that is, corresponding to so-called global dust storms, and is uniformly mixed up to 40 km. Intermediate- and high-dust conditions are meant to represent average and peak opacities, respectively, during two global dust storms experienced by the Viking 1 and Viking 2 landers during Southern Hemisphere summer conditions in 1977. The latitudinal distributions of heating are obtained by multiplying the above heating rates by zenith angle functions of the type illustrated in Fig. 9 of Leovy and Zurek (1979), normalized to unity at the equator. This zenith angle dependence is much more symmetric about the equator for the semidiurnal component of heating (shown in Fig. 1), whereas the diurnal component of heating maintains significant asymmetry about the equator (not shown).

The heating rates in Fig. 1 were designed to approximate the main features of the daily surface pressure oscillations observed at the Viking lander sites (Leovy and Zurek 1979; Zurek 1986; Zurek and Leovy 1981). To verify this, comparisons between observed (Wilson and Hamilton 1996) surface pressure amplitudes and those computed in our simulations (to be described below) are provided in Fig. 2. The numerical simulations capture the salient features of the observations, and it is therefore assumed that the same heating rates will also lead to a reasonable estimate of tidal fields and their consequences in the middle and upper atmosphere.

The details of how the heating rates were specifically inferred from Zurek's work are now described. Figure 1 of Zurek (1986), which illustrates noontime tidal heating rates at the equator, provides the main information needed. In this figure, Zurek explicitly illustrates the diurnal and semidiurnal heating rates for high-dust conditions (τ = 5.0). The diurnal heating profile for τ = 2.3 is also provided, and we assume the semidiurnal profile to have the same shape and the same maximum value relative to the diurnal component as τ = 5.0. Zurek (1986) also provides the noontime equatorial dusty 0.67J(III) profile from the Leovy and Zurek (1979) study. [In this notation a vertical profile of heating per unit mass (J) is denoted αsJ(I, II, III), where αs is the fraction of solar radiation absorbed by dust in the atmosphere, I denotes a low-dust vertical profile shape that mainly simulates surface heating, and II and III are two different vertical profile shapes that simulate the effects of atmospheric heating by airborne dust]. This provides an unambiguous means to calibrate the low-dust profile, designated 0.10J(I) in Leovy and Zurek (1979). Comparing Fig. 1 of Zurek (1986) with Fig. 10 of Leovy and Zurek (1979), a peak heating rate of 3.2 W kg−1 for 0.67J(III) implies a peak diurnal equatorial noontime heating rate (at the ground) of 1.34 W kg−1 for 0.1J(I). The relative amplitude of the semidiurnal heating for this low-dust case is given in Leovy and Zurek (1979), and we note that the correct sign in the exponential expression for J(I) as noted by Zurek (1986) is used here.

c. Zonal mean heating rates and background atmosphere

Although our emphasis in this paper is on the solar semidiurnal tide, it is necessary to introduce dissipative processes and a zonal mean wind distribution in order to provide some realism to the propagation conditions encountered by the tide. Given the virtual absence of direct wind observations in Mars's atmosphere, we adopt a simplistic approach guided by existing Mars general circulation models (GCMs) that have had some success in capturing the salient features of other observables in Mars's atmosphere, and on gradient winds below 60 km determined from temperatures measured by the Thermal Emission Spectrometer (TES) instrument on Mars Global Surveyor (Smith et al. 2001). Toward this end, we introduce zonal mean thermal forcing in the model that is antisymmetric about the equator with a cosθ dependence (θ = colatitude), consistent with Southern Hemisphere summer conditions. The vertical distributions of heating are Gaussian-shaped functions whose peak altitudes and amplitudes are chosen so that the zonal mean wind and temperature distributions simulated by recent GCMs for the middle atmosphere (Forget et al. 1999) and thermosphere (Bougher et al. 2000) are approximated (see below). For the radiative cooling we utilize the same Newtonian cooling parameterization detailed in Forbes et al. (2002). Above about 80 km, in the non-LTE region, (where LTE stands for local thermodynamic equilibrium) this is based on exact 15-μm cooling calculations over a range of reference temperatures and pressures, which are then used to choose an average profile that is representative of the range of temperature deviations from the mean reflected in the calculations. At lower altitudes, the rates closely approximate those illustrated in Barnes (1984, 1990) that are based on the detailed radiative calculations of Barnes (1983). The two cooling profiles are merged together in the 70–80-km height region.

The zonal-mean zonal, meridional, and vertical winds and perturbation temperatures due to the heating rates specified above are illustrated in Fig. 3, for low-dust conditions. The main features of the zonal winds depicted in dynamical models of Mars's atmosphere (Théodore et al. 1993; Collins et al. 1997; Forget et al. 1999; Joshi et al. 1995; Bougher et al. 2000) are exhibited here. Zonal mean eastward (winter) and westward (summer) jets of order 60–80 m s−1 in the thermosphere, and ±100–120 m s−1 for the corresponding jets in the middle atmosphere, with some degree of closure of the middle-atmosphere jets above 70 km. The latter feature is caused by resolved-scale wave drag and to some extent parameterizations introduced into these models to account for subgrid-scale wave stresses. Given our inability to simulate the resolvable wave stresses accounted for in the GCMs, and the ad hoc nature of the gravity wave parameterizations, the same effect is accomplished here through introduction of a Rayleigh friction distribution peaking at 5.0 sol−1 near 80 km, the estimated breaking height for topographic gravity waves in Mars's atmosphere (Joshi et al. 1995). The middle-atmosphere westward jet is very similar in magnitude and latitudinal extent compared to the gradient winds estimated from MGS/TES temperatures (Smith et al. 2001), but the eastward middle-atmosphere jet maximum (≈120 m s−1) is 40 m s−1 less than that of Smith et al. (2001). Intrusion of the westward jet into the Northern Hemisphere, and the high-latitude location and intensification of the eastward jet in the Northern Hemisphere are also features exhibited in Mars middle-atmosphere models and gradient winds. The zonal mean temperature perturbations corresponding to these calculations are similarly consistent in broad features with those of the cited GCM results and with observations. For instance, the ∼20-K warming at 60–80 km in the high-latitude winter hemisphere is due to subsidence heating by the descending branch of a global Hadley cell (refer to Fig. 3) whose poleward motion is accelerated by Coriolis and centrifugal forces due to the westward mean winds intruding into the Northern Hemisphere. The latter feature results from conservation of angular momentum for a meridional flow driven by strong pole to pole heating gradient; an atmospheric particle can leave the high latitudes of the Southern Hemisphere, cross the equator at high altitude, and reach the high northern latitudes while conserving its angular momentum (Wilson 1997; Forget et al. 1999). The warming of the high-latitude middle atmosphere relative to radiative equilibrium values has also been noted from observational studies (Deming et al. 1986; Théodore et al. 1993; Martin and Kieffer 1979; Jakosky and Martin 1987; Santee and Crisp 1993; Smith et al. 2001). We have not made any attempt to reproduce anything but the salient features of the GCM and gradient wind patterns due to the uncertainties and limitations involved. We have also not attempted to reproduce GCM winds in the lower 10 km of the atmosphere, since these are inconsequential to the vertical propagation of the tidal disturbances into the middle and upper atmosphere of Mars, the primary focus of this paper.

Coefficients for molecular diffusion of heat and momentum are consistent with the formulation in Miyahara and Wu (1989), with the numerical constants adjusted for Mars's chemical composition. The choice of eddy diffusion coefficient is subject to considerable uncertainty, as values used by 1D photochemical models most likely include the lumped effects of mixing by large-scale dynamics (Bougher and Roble 1991; Bougher et al. 1999, 2000). For instance, in his 3D dynamical model, Bougher obtained realistic results for lower thermosphere chemistry with eddy diffusion values of order 750 m2 s−1, whereas photochemical models utilize values in the range of 1–5 × 103 m2 s−1. Closer to the surface (i.e., <25 km) little information is available; estimates range between about 50–100 m2 s−1 (Krasitskii 1978; Krasnopolsky 1993; Leovy and Zurek 1979). Here, we assume a profile with values of 50 m2 s−1 from the surface to 25 km, increasing linearly with height to a value of 350 m2 s−1 at 75 km, and constant with height above 75 to 105 km where molecular diffusion begins to dominate, above which the eddy diffusion value decreases with height. Eddy diffusivities of these magnitudes are unimportant for the vertical propagation of the semidiurnal tide (see below), and so the details of this aspect of the model do not impact our results.

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 (zbreak) to prevent further growth of the wave, and that the momentum carried by the wave will be deposited into the mean flow above zbreak. Lindzen gives the following expression for the value of eddy diffusion required to prevent further growth of the wave:
i1520-0469-63-7-1798-e3
where kx and ky 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):
i1520-0469-63-7-1798-e4
where kz 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 uc → 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 zbreak, which can lead to numerical difficulties. Usually, this is handled by applying some sort of exponential dependence below zbreak 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:
i1520-0469-63-7-1798-e5
where Dmax is given by expression (4), T ′ is the perturbation temperature associated with the wave, Γ = (dT/dz) + (g/cp) 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 kzT ′ < 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 zbreak 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 Dmax 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 Dmax 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:
i1520-0469-63-7-1798-e6
where ρ′ = perturbation density, ρo = background density, Φ′ = perturbation geopotential, g = acceleration due to gravity, H = scale height, T ′= perturbation temperature, To = background temperature, and all primed quantities are complex. This equation is derived from the linearized ideal gas law
i1520-0469-63-7-1798-e7
and assuming Φ′ = p′/ρo. Note that Φ′ and T ′ are solved for in the system of equations, but ρ′ is a derived quantity; moreover, from Fig. 7e, T ′/To ≤ 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′/ρo ≤ 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:
i1520-0469-63-7-1798-e8
where R is the gas constant and x = −ln(p/po). If T ′ oscillates with time but is constant with respect to height, then Φ′ increases linearly with height. According to (6), ρ′/ρo also varies linearly with height, as appears to be the case in Fig. 7d above 150 km. In fact, ρ′/ρo 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′/ρo > 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′/ρo), 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, Ls, where Ls = 0 is northern spring equinox, Ls = 90 is northern summer solstice, Ls = 180 is northern autumn equinox, and Ls = 270 is northern winter solstice.) During phase I, extending from mid September 1997 to late March 1998, periapsis precessed from Ls = 180 at 30°N and 1800 local standard time (LST), to Ls = 300 at 60°N and 1100 LST. During the first part of phase II analyzed here, periapsis precessed from Ls = 30 at 60°N and 1700 LST to Ls = 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′/ρo 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:
i1520-0469-63-7-1798-e9
where a = planetary radius; θ = colatitude; u′, υ′, w′ are zonal, meridional, and vertical velocities; p is pressure; and x = −ln(p/po). 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.

Acknowledgments

This work was supported under Grant ATM-0346218 from the National Science Foundation to the University of Colorado. The authors thank Ms. Xiaoli Zhang for her assistance in preparing the figures, and the reviewers for providing excellent comments on the original manuscript. The MGS accelerometer data was acquired from the NASA Planetary Data System.

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Fig. 1.
Fig. 1.

Amplitudes of semidiurnal heating rates (W kg−1) as a function of height and latitude for dust opacities of (top) τ = 0.5, (middle) τ = 2.3, and (bottom) τ = 5.0, consistent with those in Zurek (1986) as described in the text. The latitude shape of the heating profiles is taken to be symmetric about the equator, consistent with the solar zenith angle dependence for the semidiurnal component of heating described in Leovy and Zurek (1979) for a subsolar point of −15° latitude. (The corresponding diurnal component of heating is much more asymmetric about the equator)

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 2.
Fig. 2.

Comparison between computed relative semidiurnal surface pressure amplitudes (solid lines) and those experienced by the Viking 1 (22.5°N, 48°W) and Viking 2 (48°N, 134°E) landers (symbols) under similar conditions. The Viking data are mean values inferred from Wilson and Hamilton (1996).

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 3.
Fig. 3.

Modeled zonal mean parameters vs height and latitude for low dust conditions in Mars's atmosphere, with zonal mean forcing only: (a) eastward wind, (b) southward wind, (c) temperature perturbation from global mean, and (d) vertical wind.

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 4.
Fig. 4.

Modeled semidiurnal tide and zonal mean parameters vs height and latitude for low-dust conditions in Mars's atmosphere, with zonal mean forcing and semidiurnal tidal forcing: (a) semidiurnal tidal temperature amplitude, (b) eddy diffusion coefficient associated with tidal breaking, (c) zonal mean eastward wind, and (d) zonal mean southward wind. In comparison with Fig. 3, the zonal mean winds now include the effects of dissipation of the semidiurnal tide.

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 5.
Fig. 5.

Same as Fig. 4, except for excitation of the solar semidiurnal tide corresponding to dust opacity of τ = 2.3. The zonal mean forcing is the same as that used in Figs. 3 and 4.

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 6.
Fig. 6.

Differences between modeled zonal mean parameters corresponding to simulations with (i.e., Fig. 5) and without (i.e., Fig. 3) semidiurnal tidal forcing for a dust opacity of τ = 2.3: (a) eastward wind, (b) temperature perturbations from zonal mean, (c) southward wind, and (d) vertical wind. These difference fields represent the wave-driven component of the zonal mean circulation.

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 7.
Fig. 7.

Fields associated with the semidiurnal tide in Mars's atmosphere as a function of height and latitude, corresponding to forcing with dust opacity of τ = 2.3: (a) eastward wind amplitude, (b) southward wind amplitude, (c) zonal mean zonal acceleration due to dissipation of the tide, (d) relative density amplitude, (e) temperature amplitude, and (f) temperature phase [longitude of maximum at 0000 universal time (UT)].

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 8.
Fig. 8.

(left) Semidiurnal temperature amplitudes and (right) eddy diffusion coefficients for a dust opacity of τ = 2.3, and n = 1 and n = 5 power laws in Eq. (3), respectively.

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 9.
Fig. 9.

(left) Amplitudes and (right) phases of (top) longitudinal wavenumber 2 and (bottom) wavenumber 3 relative density amplitudes at 125 km from phase I (filled circles) and phase II (open circles) of MGS aerobraking operations. The phase I data correspond to Ls = 180–300, and the phase II data correspond to Ls = 30–80, and times between 1800–1100 and 1700–1500 LST, respectively. The solid lines correspond to calculated amplitudes and phases (for a fixed local time of 1500 h) for the eastward-propagating diurnal tides with zonal wavenumbers (top) s = 1 and (bottom) s = 2 using the same zonal mean heating distribution as for the semidiurnal tide (Southern Hemisphere summer conditions). Vertical bars are 1σ uncertainty estimates based upon standard deviations of density residuals in 10° × 30° latitude × longitude bins at 125 km. Tropospheric forcing for these waves in the model has the horizontal structure of the first symmetric Hough mode (Kelvin wave) for the oscillation, and is set to produce a maximum 1-K temperature oscillation over the equator near 25 km.

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 10.
Fig. 10.

Fields associated with the eastward-propagating diurnal tide with s = −1 (DE1) in Mars's atmosphere as a function of height and latitude, corresponding to Southern Hemisphere summer conditions and the modeled density perturbations depicted in Fig. 9: (a) temperature amplitude, (b) temperature phase (longitude of maximum at 0000 UTC), (c) eastward wind amplitude, (d) relative density amplitude, (e) zonal mean zonal acceleration due to dissipation of the tide, and (f) zonal mean zonal wind due to dissipation of the tide.

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

Fig. 11.
Fig. 11.

Same as Fig. 10, except for the eastward-propagating diurnal tide with s = −2 (DE2).

Citation: Journal of the Atmospheric Sciences 63, 7; 10.1175/JAS3718.1

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