Mars' surface pressure tides and their behavior during global dust storms

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 103, NO. E4, PAGES 8587-8601, APRIL 25, 1998

Mars' surface pressure tides and their behavior during global dust storms

Alison F. C. Bridger Department of Meteorology, San Jose State University, San Jose, California

James R. Murphy l San Jose State University Foundation, San Jose, California

Abstract. Simulations of Mars' atmospheric tides with the NASA-Ames Mars general circulation model are presented and analyzed. In an annual simulation, diurnal and semidiurnal tides in the model's surface pressure field are shown to compare well with tides observed by the Viking landers throughout the Mars year. Focused simulations of global dust storms during the northern winter season show a semidiurnal tide that responds very strongly to the increased dust load during the storms, again matching well with observations. Analysis of the structure and behavior of this tide during global storms verifies that it is dominated by the gravest symmetric Hough mode (H2), as has been noted previously However, it is shown here that additional symmetric 2 . . ' .... and asymmetric modes play an •mportant role •n accounting for the structure of the t•de and •ts variation during the dust storms. The phase advance observed between the Viking lander sites (earlier semidiurnal phase at the higher-latitude site) is explained by the presence of these higher- order modes. Finally, decomposition of the semidiurnal tide during a global dust storm into its component Hough modes 2 shows that the amplitude of the H 2 mode mirrors changes in the observed overhead visible dust opacity, while variations in higher-order modes are indicative of the zonal (and, possibly, longitudinal) distribution of airborne dust.

1. Introduction

The presence of thermally driven tides in the Martian atmosphere has been firmly established from analyses of surface pressure and wind measurements obtained by the two Viking landers [Leome and Zurek, 1979; Leovy, 1981; Zurek, 1980, 1981, 1982; Zurek andLeovy, 1981; Murphy et al., 1990]. Additionally, atmospheric temperatures derived from orbiting spacecraft indicate the presence of tidal oscillations aloft [Conrath et al., 1973].

Viking surface observations reveal the presence of both diurnal and semidiurnal tides, whose amplitudes and phases vary substantially through the Mars year (see the discussion in section 2, as well as •'lson and Hamilton [1996] (hereinafter referred to as WH)). Both tides were observed to amplify during the two global dust storms in the first Viking year (hereinafter referred to as the 77A and 77B storms (Leovy and Zurek[1979])). The semidiurnal tide becomes especially prominent during such storms. During the Vi- king mission, the largest semidiurnal amplitudes measured in both the surface pressure and wind fields occurred during the developing stages of the global dust storms. Previous analyses have indicated a strong dependence of the amplitude of this tide in the surface pressure field on the quantity of suspended dust at equatorial and subtropical latitudes [Leovy and Zurek, 1979; Zurek, 1981 ]. Using classical tidal theory, Zurek [ 1981 ] was able to estimate the increase in the amount of dust in suspension required to reproduce the observed diurnal and semidiurnal tidal amplitudes. These inferred dust optical depth values generally agreed with lander-based

•Also at NASA Ames Research Center, Moffett Field, California.

Copyright 1998 by the American Geophysical Union.

Paper number 98JE00242. 0148-0227/98/98JE-00242509.00

visible optical depth measurements obtained at the Viking lander 1 site (VL1)[Colburn et al., 1989].

Leomy and Zurek [ 1979] inferred the vertical distribution of dust- induced heating necessary to reproduce the observed dust storm enhancement of the semidiurnal tide relative to the diurnal tide at

VL 1. The elevated heating profile they deduced at the storm's peak is consistent with the vertical dust distribution determined during the decline of the 1971-1972 dust storm, as seen by Mariner 9 [Ander- son and Leovy, 1978].

The Martian atmospheric dust load, too, has been observed to vary substantially during the year and from year to year both in integrated dust content and its spatial distribution (horizontal and vertical)[Martin and Zurek, 1993; Zurek and Martin, 1993; Colburn et al., 1989; Martin, 1986; Anderson and Leomy, 1978]. Global dust storms have been observed to have their genesis in the southern hemisphere [Zurek and Martin, 1993], and they therefore induce large spatial variations in dust load on very short timescales. These variations in dust distribution induce variations in the ther-

mal forcing of the atmosphere, and in particular of atmospheric tides, since it is the suspended dust, via its absorption of solar radiation, which provides the thermal forcing for the tides.

Conversely, temporal variations in the latitudinal structure of thermally driven tides can be indicative of the variability in the structure of the thermotidal forcing, i.e., of the dust distribution. Ob- served spatial variations in the Mars surface tidal fields, and their evolution with time during dust storms, have not been closely examined to date. One goal of this paper is to show that such variations do occur (at least in the semidiurnal tide in the surface pressure field) and to relate them to variations in the spatial distribution of dust. Specifically, we present observations of temporal variability in the spatial structure of the observed surface pressure tidal field provided by the Viking landers, with a focus on the global dust storms of 1977. Additionally, we show how the identified varia-

8587

8588 BRIDGER AND MURPHY: MARS' TIDES AND GLOBAL DUST STORMS

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Figure la. Amplitudes (normalized by sol-mean pressure at the site) and phases (local time of first maximum) of the semidiurnal tide in the observed surface pressure field at Viking Lander 1 (VL1)(uppcr panel) and Viking Lander 2 (VL2)(lowcr panel) for years 1-4 of the Viking mission. The two global dust storms of the first Viking year (77A and 77B) began around sols 210 and 315 of year 1, respectively. Open circles indicate year 1; crosses indicate year 2; plus signs indicate year 3; and solid circles indicate year 4. Note the change of scale in the phase panels between the two sites.

tions can be related to temporal variations in latitudinal dust distribution in Mars' atmosphere.

To aid in our understanding of tidal oscillations ,in Mars' atmosphere (as evidenced by surface data), we have performed several experiments using the NASA-Ames Mars general circulation model (MGCM)[Pollack et al., 1990, 1993]. We first present an annual simulation designed to reproduce the first Viking year. Results are compared with Viking observations. Then, we focus on a number of simulated dust storm experiments in order to relate the response in the surface pressure tidal field to the dust-induced heating. This approach allows us to examine the response to spatially (longitudinally and latitudinally) and temporally varying dust distributions,

unlike the zonally symmetric dust distributions examined previously (e.g., by Leovy and Zurek [ 1979]). These results, too, are compared with Viking lander observations in order to explain the observed variations during the 1977 storms. As was mentioned above, it will be demonstrated that temporal variations in the latitudinal structure of the tide can be used to infer temporal variations in the latitudinal distribution of the suspended dust.

In the next section, we describe the observed surface pressure tidal fields (both diumal and semidiumal) at the Viking lander locations. This is followed in section 3 by an examination of the tides produced by the MGCM in the annual simulation. The results will be shown to compare well with observations. In section 4, we fo-

BRIDGER AND MURPHY: MARS' TIDES AND GLOBAL DUST STORMS 8589

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cus on simulations of dust storm events and compare model results with observations (with our focus now on the semidiumal tide). In section 4 we also examine the structure and variation of the tidal

response in terms of the underlying Hough modes. Finally in section 4 we compare observed and simulated latitudinal variations in tidal amplitude and show that they can serve as indicators of latitudinal variations in global dust distribution. Results are summa- rized in section 5.

2. Viking Surface Pressure Measurements

We begin with an examination of diurnal and semidiurnal variations in Martian surface pressures measured by the Viking landers. These variations will be compared with results from annual simulations of the MGCM to ascertain the model's ability to accurately generate thermal tides in the surface pressure field.

The two Viking landers provided time series of surface pressures, winds, and temperatures at northern subtropical (VL1 at

22.5øN, 48.0øW) and northem middle latitude (Viking lander 2 (VL2) at 48.0øN, 225.7øW) locations. VL1 operated for 3.3 Mar- tian years (landing at L s 96.9 ø, where L s refers to areocentric longitude; L s ranges from 0 ø to 360 ø over the year,. with northern spring equinox occurring at L s 0ø), and VL2 operated for 1.6 years (landing 45 sols (Martian solar days) later at L s 117.9ø). The surface observations have been the subject of numerous investigations deal- ing with thermal tides, baroclinic waves, slope-induced flows, and the structure of the boundary layer (see Zurek et al. [ 1992, chapter 26] for an extensive reference list and discussion of the Viking meteorology data; WH). A thorough discussion of the Viking pressure measurements is given by l•llrnan [ 1988].

The form of the observed pressure data used in this investiga- tion is "hourly" averaged values, with approximately four values per "hour" and 25 "hours" per sol. We calculate and display amplitudes (normalized by the sol-mean pressure at the site) and phases (local time of day of first maximum) of the diumal and semidiumal


harmonics over consecutive 5-sol composite intervals (sols 1-5, 6- 10, etc.) at both lander sites during the entire duration of the mission (Figure 1). Only those composites which possess at least one pressure measurement in each of the 25 "hours" of that composite are included in the figure.

Looking first at semidiurnal variations (Figure l a), the most prominent features are the amplitude enhancements at both sites at L s -205 ø and-270 ø of year 1 (open circles) and at VL 1 at L s -•205 ø of year 4 (filled circles). The first two events correspond to the onsets of the 77A and 77B global dust storms. The amplitude increases, and subsequent declines, mimic concurrent increases and declines in the overhead visible dust opacity [Colburn et al., 1989](optical depth measurements are not available for year 4). During Viking year 2 (crosses), an increase in the VL 1 optical depth was seen at L s -•210% but the VL 1 pressure data were too sparse during this time period to produce complete 5-sol composites. Us- ing a longer compositing interval (20 sols) and filling empty "hours" via linear interpolation, WH found a corresponding increase in the semidiurnal tidal amplitude at VL2 (but not at VL 1) around L s -210 ø of year 2. For year 3, the data are complete enough to indicate that no dust storms occurred. In all years, VL2 amplitudes are smaller than those at VL 1 due to the latitudinal structure of the tide (discussed further in section 4).

Also prominent in the VL 1 semidiurnal time series are the pronounced amplitude minima centered on L s -•90 ø in each of the four years. The observed dust optical depth minima occurred at a much later time (L s 143 ø) Colburn et al [1989], and orbital aphelion occurred earlier (L s 70ø). The amplitude minimum at this time was shown by WH to be caused by destructive interference between the westward traveling classical tide and an eastward traveling Kelvin

to the 77B storm is believed to be the result of destructive interfer-

ence at the VL 1 longitude between the classical westward propagating tide and an eastward propagating Kelvin wave excited by the developing dust storm [Zurek and Leovy, 1981 ].

As was the case with the semidiurnal tide, there is little year-to- year variation in diurnal amplitudes at VL 1 during northern spring and early summer. This includes the appearance of minima around L s -20% 60% and 120 ø. As with the semidiurnal tide, VL 1 amplitudes and phases show considerable variation near L s 140 ø during all four years. At VL2 just prior to this time, the amplitude of the diurnal tide is near zero, and during the same period phases are highly variable, suggesting interference with non classical modes.

Diurnal phases at the two lander sites differ considerably from one another. At VL 1, phases throughout the year are generally within 2 hours of 0600 LT, the theoretically predicted value. From late northern spring into midsummer the phase is earlier than 0600 LT. Then from late summer through midautumn there are significant variations (0400 LT through 1600 LT). The variations seen around L s 140 ø match those observed in the semidiurnal field. At L s -210 ø, the retarded phase (years 1 and 4) coincides with the dust storm events discussed above. At the start of the 77B storm, the VL1 diurnal phase rapidly advances, covering a complete cycle (25 "hours") in a 10-sol interval, indicating the interfering effect of the amplified eastward traveling Kelvin waves discussed above.

Through the year the VL2 diurnal phase varies between mid- night and noon. Early northern spring phases of-•1000 LT advance during late spring to 4)600 LT, and continue advancing to near mid- night by L s -•130 ø. Over the next -•25 sols the VL2 diurnal phase is highly variable. This is also the time during which amplitudes are near zero. At the end of this period, phases are around 0900 LT.

wave. These waves excited in our annual simulation (both diurnal: The most pronounced phase variations during northern autumn wavenumber 1 and semidiurnal wavenumber 2) are discussed further in section 3.

Despite the wide variation in atmospheric dust loading during northern autumn and winter (L s 180-360 ø) during the years observed by Viking [Colburn et al., 1989; Clancy et al., 1996], VL 1 amplitudes during the following spring and summer seasons are largely invariant. Finally, the short-term variations in amplitude (and phase) at L s -•140 ø seen in all four years were investigated by Jillman [1988]. He suggested that they are caused by interference with an atmospheric normal mode that amplifies at this time as the atmosphere's thermal structure passes through a resonant state.

Semidiurnal phases appear broadly similar at the two sites and vary significantly at both locations through the year. Both VL 1 and VL2 phases are late (relative to the theoretically predicted value of 0900 LT) during northern spring. At VL1 the phase steadily advances through summer so that by early autumn it is close to 0900 LT. Thereafter the phase retards steadily through northern winter, with the exception of a short period around L s -•270 ø (during the onset of the 77B storm), when the phase rapidly advances by 1 hour. We note that VL 1 phases appear very repeatable from year to year. At VL2, phases remain significantly retarded (relative to 0900 LT) until late summer, when there is an abrupt advance to around 0900

through winter coincide with the two dust storms but are not consistent (i.e., rapid phase advance at L s -•210 ø versus a less pronounced retardation around L s -•270ø).

In summary, the dust storm events most clearly delineated by this analysis are the two 1977 storms, which are marked by amplification in the diurnal and semidiurnal tides at both sites. Our abil-

ity to simulate these events in both the annual and dust storm experiments (sections 3 and 4) will be of particular interest.

3. Annual Simulation

The MGCM used in this study has been described in detail by Pollack et al. [ 1990, 1993]. Briefly, it is a primitive equation, finite difference model. The domain is global and extends to approximately five scale heights (0(50 km) (a high-top version of the model is available and was run to confirm some of our results reported in section 4). The resolution in all runs discussed here is 7.5 ø latitude by 9 ø longitude. Sigma coordinates are used in the vertical with 13 layers of varying depth. CO 2 is allowed to condense within the atmosphere and directly onto the surface. At solar wavelengths, multiple scattering calculations, which include the effects of CO 2 and suspended dust, are carried out to determine the wavelength-

LT. The VL2 phase remains at this time until late winter, when it integrated net solar flux at model pressure levels. In the infrared, gradually begins to lag. both gaseous CO 2 and dust are treated in two spectral intervals, one

Diurnal variations in surface pressures at both lander sites (Fig- covering the 15-mm vibrational fundamental of CO 2, the other cov- ure lb) exhibit evidence of the two global dust storms of the first ering the spectral domain outside the 15-mm fundamental. At the year (open circles) and also at VL 1 of the year 4 event at L s -•205 ø lower boundary, variable topography, thermal inertia, and albedo (solid circles). The 77A event at VL1 is not well defined by our are assumed. compositing, but clearer evidence of this storm is shown by WH In this section we examine the behavior of the diurnal and using 20-sol composites (their analysis also indicates amplification semidiurnal tides in the surface pressure field in an annual simula- at L s -210 ø of year 2 at VL2, but not VL1, as was noted in the tion of the MGCM. Our focus will be on the surface pressure field semidiurnal field). The sharp decline in VL1 amplitude just prior since this is very well documented by the Viking lander data set.


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The overhead visible dust opacity is assumed spatially uniform on a constant pressure surface, and in the vertical the dust distribution is prescribed following ½onrath [1975], with near-uniform mixing ratios in the first 11/2 scale heights (v=0.03 assumed) and rapid decay above. The dust loading is prescribed to vary in time, as was inferred from VL1 pressures during the first Viking year [Zurek, 1981]. We note that VL1 measurements of optical depth during dust storm peaks provided lower bounds on opacity.

Five-sol composites of amplitudes and phases of diurnal and semidiurnal fluctuations at the model's VL1 and VL2 grid points during this simulation (Figure 2) can be compared with observations (Figure 1). For the semidiurnal tide (we use the term "tide" in its broadest sense, embracing both Sun-synchronous and nonclas- sical modes), simulated amplitudes compare reasonably well with observations, with the following exceptions: Modeled peak values at VL1 during both dust storms are significantly lower than ob-

served; amplification at VL 1 during the 77A storm is too gradual; and the simulated amplitude peak at VL2 during the 77B storm occurs too early. Additionally, observed amplitude and phase variations around L s 145 ø, ascribed by Tillman [1988] to interference effects associated with transient wave activity, were not reproduced in the simulation.

At VL2, the simulated semidiurnal phase and its annual variation compares well with observations, although the simulated phase is 1-2 hours early around L s 0 ø. At VL1, the general character of the phase variation over the year is captured, as is the phase advance between VL1 and VL2 (note the change of scales in the phase panels between VL1 and VL2). However, the modeled phase lags observations by about 1 hour from L s 90ø-225 ø and leads by 0(30 min) late in the year. Differences between observed and modeled phases have been reported before [Zurek and Leovy, 198 l; WH], as have departures from theoretically expected values [Leovy and


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Zurek, 1979]. Additionally, we find that the phase differs from longitude to longitude (at the Viking latitudes) by as much as O(1 hour) at various times of the year (were this tide to be composed only of Sun-synchronous, classical modes, the phase would be the same at all longitudes). WH also note phase variations with longitude. For comparison with WH, we conducted an annual simulation with opacity (x) fixed at x=0.3 throughout the year (WH conducted a series of short-duration experiments with opacity x=0.3 for various times of the year (their Figures 21 and 22)). Our predicted semidiurnal phases at both VL 1 and VL2 are within 30 min of their values at all times.

WH demonstrated that there is a considerable impact of topographically generated Kelvin waves on tidal amplitudes and phases at the Viking sites at certain times of the year. These modes inter- fere with Sun-synchronous tides to modify local tidal phases, the effects being largest when Kelvin wave amplitudes are largest. In short-duration runs with increased opacity (e.g., x = 1), WH found that phase advanced closer to observed values (the MGCM shows

similar behavior), with the effects being especially noticeable around L s 90 ø, at which time large-amplitude Kelvin waves were present in their simulations. Figure 3 shows amplitudes of eastward and westward propagating waves 1 and 2 as they vary with latitude and time during our annual simulation. Around L s 90 ø, there is an eastward propagating wave 2 with amplitude comparable to the Sun- synchronous component. The presence of this Kelvin wave accounts for the phase difference in the semidiurnal tide between the MGCM and observations at this time of year.

It is important to note that at the time of the two global dust storms, the amplitude of the wave-2 Kelvin mode is substantially lower than that of the classical tide. This is the period on which we will be focused (in section 4), and we note, too, that the simulated semidiurnal phases compare well with observations at this time.

Modeled diurnal tidal amplitudes at L s 90 ø are larger than observed at both Viking sites (Figures 1 b and 2b). This appears to be associated with the large-amplitude Kelvin wave generated in the


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Figure 3a. Amplitude of the eastward (upper panel) and westward (lower panel) propagating components of the semidiurnal tide in the surface pressure field of the annual simulation as a function of latitude and time (units are microbars).

model at this time (Figure 3b). Amplitudes at VL2 during the dust storm period are noisy but do indicate a response similar to observations. At VL1 just after L s 270 ø there is a sharp amplitude drop, mimicing the drop observed there prior to the 77B storm (Figure lb, open circles). However, the extent of the amplitude decay is underestimated, and the ensuing amplification is missed. The diurnal phase variation at VL 1 reproduces observations fairly well, in- eluding the phase advance just after L s 270 ø. As mentioned above, these amplitude and phase variations are believed to have resulted from destructive interference between the migrating diurnal tide and a Kelvin wave [Leovy and Zurek, 1981 ]. Figure 3b shows that the model's wave-1 Kelvin mode during this time period was comparable in amplitude to the classical tidal mode. The model's diurnal phase at VL2 broadly matches observations, at least after L s 180 ø. As was the case with the semidiurnal tide, there is notable longitudinal variation in the phase of the diurnal tide at both Viking latitudes, especially during the first half of the year.

Figure 4 shows a map of the instantaneous structure of the (normalized) semidiurnal tide in the surface pressure field during the

second global dust storm in the annual simulation. Tidal amplitudes maximize near the equator, which is expected in the annual - run since the dust is uniformly distributed (on a constant pressure surface). Some hemispheric asymmetry is evident. Results from a number of 50-sol simulations (not shown) indicate that this is associated with topographic, albedo, and thermal inertia variations, and with the subsolar point being south of the equator at this time of year. We note a pronounced phase advance with latitude in the southern hemisphere. A similar structure is also seen at higher northem latitudes; thus the simulated phase at VL2 is earlier than at VL 1, as indicated in Figure 2a and as observed (Figure 1 a). These structures also indicate that modes other than the dominant Hough mode must be present in the solution. A solution consisting of a single mode will have no phase variation with latitude. Only when additional modes are present (and with a phase shift between the modes) will the phase structures indicated in Figure 4 be present; this is elaborated upon in section 4.

The Martian tidal structure(s) shown in Figure 4 are comparable to those observed [Haurwitz, 1956] and modeled [Lieberman et al.,


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1994] for Earth's atmosphere. This is expected, given the similar rotation rates and large-scale structure of the tidal forcing on both planets. The amplitude of the tide modeled by the MGCM ranges from roughly 1% of the globally averaged surface pressure during northern summer, to 0(3%) during each global storm. These values also compare well with observations (O-4%)[Leovy, 1981 ]. A corresponding value for Earth is under 0.2% [Haurwitz, 1956], il- lustrating the relative importance of this tide in the dusty Mars atmosphere.

On the basis of our analyses, we believe that the MGCM satis- factorily reproduces the observed character of the semidiurnal tide in the surface pressure field (to the extent that the Viking lander data adequately represent this field). Our task now is to more closely examine changes in this tide during global dust storms and determine to what extent the changes can be utilized as indicators of global dust loading and distribution.

4. Dust Storm Experiments

We now discuss experiments specifically focused on global dust storm events, at which time the semidiurnal tide has its largest amplitude (hereinafter we focus exclusively on the semidiurnal tide).

These experiments are conducted using the MGCM coupled with the aerosol transport/microphysical model of Toon et al. [1988]. These models have been coupled for previous investigations of Martian dust storms [Murphy et al., 1995, 1996]. Briefly, winds calculated in the MGCM are input to the aerosol model to transport suspended dust. The evolving dust field in turn modifies the radia- tive forcing of the model atmosphere. This interactive coupling of the models allows for a self-consistent evolution of a suspended dust load.

In one of the two dust storm simulations discussed here (denoted ZCS), dust is injected through the model's lower boundary in a confined geographical region (45ø-108øW, 15øS-37.5øS) meant to crudely mimic the source region of both 1977 global dust storms inferred from observations [Briggs et al., 1979]. In a second experiment (denoted ZSS), the dust source fills the entire 15øS-37.5øS longitudinal corridor. The specified source in both cases provides a globally averaged visible dust opacity of x=5 for a 10-sol duration (the dust input rate is constant). The injected dust is assumed to be distributed across 10 particle size ranges spanning radii of 0.1-20 mm (and follows the dust distribution assumed by Toon et al. [ 1988]). Finally, each dust storm experiment runs for 50 sols from L s 270 ø with dust injected into an initially clear spun-up atmosphere. These


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Figure 4. Normalized semidiurnal surface pressure tidal field at L s = 275 ø in the annual simulation. The local time at 180øW is mid-

night. Contour interval is 0.01, and negative contours are dashed.

experiments are discussed in greater detail in Murphy et al. [ 1995]. Figure 5a shows the latitude-time variation of the zonally aver-

aged column-integrated visible dust opacity during experiment ZSS, and Figure 5b shows the corresponding time variations of the globally integrated opacity (solid line), the southern hemisphere opacity (dashed line), and the hemispheric opacity difference (dotted line). At sol 10, when the source is switched off, the global opacity has reached a value of 'c=2.6, with a value of x•.•-4 in the southern

hemisphere. By sol 20, hemispheric differences have become small. Quite high opacities remain near the south pole late in the run. We conducted an experiment with artificially high opacities imposed at high southern latitudes and found that these high values have no impact on our results. This is in line with Leovy and Zurek [ 1979], who showed that it is the dust distribution in low latitudes (40øS - 40øN) that is most important in determining tidal responses.

4.1. Tidal Amplitude and Phase Analysis

Figure 6 shows the amplitude and phase variations of the semidiurnal tide at the model's VL 1 and VL2 grid points during both simulations. Similarities noted with observations (Figure 1 a)(recall that the 50-sol time span covers the period L s 2700-296 ø) include a rapid increase in amplitude at VL 1 at the storm's onset (especially in ZCS); weaker amplification at VL2 than at VL 1; and phase advance with latitude between the two sites. The maximum amplitudes at VL 1 in these simulations are larger than those in the annual simulation (and are now comparable to observations at both lander sites). We attribute this to the spatially varying dust distribution (both vertically and horizontally) more accurately representing the thermotidal forcing in Mars' atmosphere during Viking year 1. We note, too, that the model reproduces observed semidiurnal surface pressure tidal amplitudes with somewhat lower global opacities than inferred by Zurek [ 1981 ], who assumed a globally uniform dust haze.

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Figure 6. Normalized amplitude and phase of the semidiurnal tide in the surface pressure simulated at the model's VL 1 grid point (upper panels) and VL2 grid point (lower panels) for dust storm simulations ZSS (left panels) and ZCS (right panels).

Figure 7 shows maps of the instantaneous tidal fields during sol 11 in both experiments (Figures 7a and 7b). Comparing them with the annual run (Figure 4), maximum amplitudes are now seen well south of the equator, which might be expected given the latitudinal distribution of the dust source. There is a pronounced SW-NE tilt to the structure through the southern hemisphere and to about the VL 1 latitude. Further north, the tilt becomes NW-SE (in agreement with the observed and modeled phase advance between VL 1 and VL2).

We conducted a number of experiments to quantify the role of dust distribution (as opposed to topography, season, etc.) in generating the horizontal structure of the modeled semidiurnal tide. In one experiment, dust was uniformly distributed in the region 60øS - 15øS (dust distribution remained constant in time with opacity 't=2.5). This choice was inspired by Leovy and Zurek [1979], who used classical tidal theory to estimate the latitudinal extent of a dust cloud needed to produce realistic tidal amplitudes at both VL 1 and

VL2. A dust cloud from 60øS - 15øS was the only case in which the dust-induced heating (thermotidal forcing) was not dominated by the gravest Hough mode syrnmetric about the equator (Hough modes are discussed in more detail in section 4.2). Instead, in this case, the forcing was dominated by the gravest equatorially asymmetric mode. As Figure 7c shows, the tide in this case has its maximum amplitude well into the southern subtropics, with strong phase advance toward the south pole. Both features are strongly reminis- cent of our dust storm experiments shown above (Figures 7a and 7b). By contrast, we were not able to reproduce these structures in experiments that featured horizontally uniform dust distribution, varying seasons, or varying topographic, etc., distributions.

The correspondence between observations and both theory and modeling efforts [Leovy and Zurek, 1979; Zurek, 1980, 1981; Zurek and Leovy, 1981; WH; and the experiments reported here], as well as the analysis discussed in sections 4.2 and 4.3, strongly suggest that structures similar to these would be seen in Mars' semidiurnal


10 e N

• o'

10 e $

LONGITUDE

Figure 7. Same as Figure 4 but for dust storm simulations (a) ZSS and (b) ZCS, both at sol 11 of the experiment, and (c) for a simulation in which dust is uniformly distributed in the region 60øS-15øS.

surface pressure tidal field (for example, using a network of sensors such as that proposed by Haberle and Carling [1996]). Fur- ther, the results suggest that these structures are generated by horizontally inhomogeneous dust distributions. This statement is particularly valid during northern winter, when Kelvin wave amplitudes are much smaller than Sun-synchronous tidal amplitudes (Fig- ure 3a). In section 4.2, we show how this correspondence can be used to estimate global dust distribution (to first order) from Viking lander pressure data.

At this time of year, although Kelvin wave amplitudes are small, significant longitudinal variations are still present (Figures 7a and 7b)(recall that prior to analysis, amplitudes are normalized at each grid point by sol-mean pressure at that grid point). It was mentioned above that longitudinal variations in amplitude and phase were found in the annual simulation. Closer inspection of normalized amplitudes at low latitudes in ZSS indicate a tendency for high values at preferred locations, especially just east ofTharsis. A dust storm experiment with no topographic, thermal inertia, or albedo variations showed no such behavior. A space-time harmonic analysis of the total surface pressure field in ZSS indicates that in addition to the westward migrating semidiurnal component, a weak

(O(10%) of the migrating component), standing, semidiurnal sig- nal is present. Its presence is not unexpected, since a similar component (0(5%)) is found in Earth's semidiurnal surface pressure tide [Haurwitz, 1956].

4.2. Analysis of Site-to-Site Amplitude Ratios

Thus far we have focused on single-site tidal amplitudes (and phases) and their variations in response to changing dust load. Ad- ditional information can be extracted by comparing tidal behavior at two (or more) sites.

The latitudinal structure of a given tidal harmonic (e.g., the classical Sun-synchronous semidiurnal tide) is given by a weighted sum of Hough functions (modes), each of which has a specific meridi- onal structure (Figure 8). In the case of Earth, the structure of the semidiurnal tide in the surface pressure field is largely described by the gravest Hough function symmetric about the equator (de-

noted H22 , where in Hnm,m is the zonal wavenumber and (n-m)is the number of nodes between the poles) [Haurwitz, 1956]. On Earth, a relatively small role is played by higher-order modes [Kertz, 1956], which feature an increasing number of nodes between the poles and are alternatingly symmetric or asymmetric about the equator (Fig-

ure 8). On Mars, the H22 mode also dominates the surface pressure semidiurnal tide [Leovy and Zurek, 1979; Zurek, 1980, 1981 ]. However, if this were the only mode present, the ratio of amplitudes between any two latitudes (e.g., the two Viking latitudes) would remain constant in time, regardless of the mode's amplitude. Only when the tide consists of two or more modes which are varying in time can the ratio of tidal amplitudes between two latitudes vary in time. We use this fact below to infer global dust load and distribution from surface pressure tidal amplitudes at the Viking lander sites.

The time history of the ratio of the observed semidiurnal amplitudes between VL 1 and VL2 is shown in Figure 9a for the second half of the first Viking year. Ratio values increase sharply from 1- 2 to -4 during the early phase of the storms and then fall to prestorm values over 10-20 sols. The relatively high values during the interstorm period may be associated with enhanced southern hemisphere suspended dust generated by the numerous local dust storms observed during this time [Kahn et al., 1992]. If only the gravest

symmetric Hough mode (H22 ) were present, the value of the ratio would be 4.6 at all times (based on the calculated structure of this mode at 1 ø resolution; if the ratio is calculated between the grid points in the model closest to the lander sites, namely, at 23øN and 45øN, it has the value 3.6). That the observed ratio differs from this value indicates the presence of additional modes in the tide. Fur-

].5

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/-

/ /

I

I !

-90 -75 -60 -45 -• -15 0 15 $0 45 60 75 go

LIlT l TUDœ

Figure 8. Latitudinal structures of the first five Hough functions for the semidiurnal tide computed at 1 ø resolution. Bold solid line

indicates H• 2; bold dashed line indicates H32' hatched line indi- ca 2 z . 2. ' res H 4; th•n solid line indicates H 5 , and thin dashed line indicates H62 .


VL I SOL.

Figure 9a. The ratio of the tidal amplitudes between the VL 1 and VL2 sites for the second half of the first Viking year. The quantity plotted is the ratio of the 5-sol running means of the normalized amplitudes at the two lander sites.

ther, the temporal variations in the ratio indicate that the amplitudes of one or more Hough modes must have varied with time during the dust storms.

The ratio of tidal amplitudes between the model's VL 1 and VL2 sites in experiments ZSS and ZCS is shown in Figure 9b. The sharp increase and rapid decrease in the ratio reproduces the general character of the observations very well. Peak simulated ratios are 0(4.6) between sols 5 and 10 (i.e., during the intensification stage of the storms), and values decay to 2-2.5 as the storms diminish. It is interesting to note that ratios constructed between the same latitudes but at different longitudes (but still separated by 180 ø longitude) may look quite different from those shown in Figure 9b (not shown). This is associated with the non-migrating components referred to above and reinforces the conclusion that results obtained

from the Viking lander sites may not be representative of all longitudes at those latitudes. In the next section, we relate these ratio variations to variations in modal structure of the tide and then, in turn, relate these to varying dust distribution.

4.3. Hough Modes

The ratio plots presented above indicate that the amplitudes of the Hough modes vary with time during dust storms, both observed and simulated. Decomposition of the semidiurnal tidal field into it Hough modes, and examination of the temporal variations of the modal amplitudes, allow further insight into the relationship between dust distribution and tidal structure.

The computed semidiurnal tidal fields (examples of which were shown in Figure 7) were projected onto Hough functions, yielding the amplitudes of the five gravest symmetric and asymmetric modes for the duration of each experiment discussed in section 4.1. Usu- ally in the application of classical tidal theory, heating rates (forcing) are determined as functions of height, longitude, and latitude. These are then projected onto the Hough functions to generate the forcing functions required to solve the vertical structure equation for the structure of each tidal mode. From this information, the complete structure of the desired tidal oscillation can be found (response). Our analysis differs in that we will use model-generated surface pressure tidal fields (response) to infer information about the distribution of the heating function (forcing) in the Mars atmosphere during dust storms. From this, we can infer (to first order) the distribution of the dust itself during a global dust storm.

The variations with time of the amplitudes of the first five Hough modes in the two dust storm experiments are shown in Figure 10 (the amplitudes of the higher-order modes are very small in these

experiments). Clearly, the gravest symmetric mode (S22 ) is strongly forced by the dust-induced heating in both ZSS and ZCS, reaching peak amplitudes between sols 10 and 15. The globally integrated opacity variation in ZCS is reproduced in Figure 10b (dotted curve). As has been reported elsewhere [e.g., Zurek, 1980, 1981 ], the am-

plification and decay of the S• component is concurrent with changes in global dust opacity, and changes in one can serve as a proxy for changes in the other.

o o •b •o 3b • so

SOL

Figure 9b. Same as Figure 9a but for experiments ZSS (solid line) and ZCS (dashed line).


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Figure 10. Variation with time of the amplitudes of the first five Hough modes in dust storm experiments (a) ZSS and (b) ZCS. In Figure 1 0b, the thin dotted line shows the variation of the globally integrated opacity during the simulation.

Most other Hough modes quickly attain constant low amplitudes, but the asymmetric H• mode continues to amplify in both cases (bold dashed line), becoming the largest nondominant mode from sols 5-20 of both storms. Given the latitudinal distribution of dust,

together with the structure of the Hough modes (Figure 8), the presence of this mode is not unexpected. In addition, in the case of longitudinally confined dust forcing (ZCS), the next largest mode is the asymmetric H52 mode (thin solid line).

To illustrate the importance of these asymmetric modes in the simulated tide (and by inference the observed tide), we can com-

bine the results from our decomposition (amplitudes in Figure 1 0, together with phase information) with the Hough function structures and reconstruct VL 1/VL2 ratios. The result for experiment ZSS is shown in Figure 1 1. The solid line shows the reconstructed ratios when Hough modes H, 2 -H• 2 are used, and the dotted line shows the results when modes'-H22 -•'H2u are included. The results match both observed and simulated ratios well, with appropriate maximum and poststorm values (compare with Figure 9). The reconstructed values increase and decrease more slowly than the modeled point-to-point ratios and peak somewhat later. The observed ratios are based on locally varying values, which reflect both global tidal modal variations and local effects (see section 4.1). The reconstructed values involve only the global modes. Differences again highlight the fact that there are topographically induced local influences on the semidiurnal tide and that the Viking longitudes are not necessarily representative of all longitudes at those latitudes.

The dashed line in Figure 1 1 shows the ratio when only the sym-

metric modes H22 , H42, and H62 are retained. This result empha- sizes the role of the first two asymmetric modes in determining the values and variations of the ratio during global dust storms and suggests that the asymmetric modes are important in determining the surface pressure semidiurnal tidal structure. A further indication of the role played by these modes is illustrated using synthetic datasets constructed from a few Hough modes. Figure 12 shows maps from two such datasets (details in figure caption). With asymmetric modes included (Figure 1 2a), maximum amplitudes occur south of the equator, and there is phase advance from the northern hemisphere into the southern hemisphere; both features are present in our dust storm experiments (Figure 7). On the other hand, the synthetic field constructed with only symmetric modes (Figure 12b) features an amplitude maximum at the equator and phase advance with latitude in both hemispheres. This structure allows for the phase advance from

VL 1 to VL2, as observed. We should note that with only the H22 mode included, we would see maximum amplitudes along the equator and no phase tilt with latitude. The tidal fields generated in the developing stages of the simulated dust storms (Figures 7a and 7b) in fact have features characteristic of both synthetic cases but are

more clearly dominated by the presence of the asymmetric H• mode. After about sol 20 in the dust storm simulations, by which

time the amplitude of H• has fallen below that of Ha 2 , the tidal maps show more symmetry about the equator (not shown).

Our results indicate that Hough modes other than the dominant

H• mode, although small in amplitude, are important in explain- ing the structure and behavior of the observed tide. Previous stud-

o

SOL

Figure 11. Ratio of the semidiurnal tidal amplitudes between the VL 1 and VL2 latitudes reconstructed for experiment ZSS. Solid line indicates modes H22 -H62' dotted line indicates modes H22 -H121 ß dashed line indicates symmetric modesH22,Ha2,andH62 only.

8600 BRIDGER AND MURFHY: MARS' TIDES AND GLOBAL DUST STORMS

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LONG TUDE

•0' N

r" -- --' '• x

' ,,,,, ,,,

{ ;y//// ,'.1 • , '/;,_."" ;,,

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•igure 12. Same as Figures 4 and 7, but for s•thetic pressure fields. Each s•thetic field propagates wes•ard with time, simu- lating the migrating semidiurnal tide. The first case has

sin(2x)-O.2{H os(2x)+ sin(2x)} mb.r, the second hsv=]0+H sin(2x)+0.2{H os(2x)+H sin(2x)} mbar).

ies of the semidiurnal tide on Mars have tended to focus on the

symmetric modes only, largely due to the assumption of globally uniform dust hazes [e.g., Leovy and Zurek, 1979]. With this assumption, thermotidal asymmetry arises only from seasonal variations in the subsolar latitude. The presence in our simulations of the higher-order modes, especially of the asymmetric modes, accounts for the structural variations revealed by the ratios of observed VL 1/VL2 amplitudes. In addition, they explain the observed phase advance with latitude observed between VL1 and VL2. It should

be noted that the modes must be longitudinally phase lagged relative to each other to produce the desired phase structures (in the synthetic data sets, the lag is 22.5 ø . This value was suggested by the values calculated for experiments ZSS and ZCS, where HI lagged H2 2 by 0(30 ø) longitude, and H4 2 led H2 2 by O(15ø)).

Thus decomposing the tidal field into its Hough components and examining the variation of the modes as a global dust storm evolves

provide useful information. The temporal variation of the H22 mode closely matches that of the visible dust opacity. The presence of the higher-order modes, and in particular the relatively large amplitude

of m•, is indicative of an evolving dust distribution that is equatorially asymmetric. It is noteworthy that in experiments (e.g., the annual s,mulaUon) with dust distribution assumed spatially con-

stant on a constant pressure surface, the and other asymmetric modes are always weaker than the first few symmetric modes (not shown). Further, in the case of a dust source confined in longitude

and latitude (ZCS), m• and m• 2 are the largest nondominant modes. Thus the presence and relative amplitudes of these higher- order modes, deduced from observed surface pressures at a minimum of two sites, can yield information on the global dust distribu-

tion, its equatorial asymmetry, and perhaps its confinement to particular longitudes.

5. Summary

Analysis of diurnal and semidiurnal tidal oscillations in Mars' surface pressure field observed at the two Viking lander sites re- veals substantial variations in both amplitude and phase during the Mars year (see also WH). Particularly noteworthy are the large amplifications seen at both sites during the southern summer season of year 1 (L s -•200ø-300ø). These are associated with the global dust storms observed at this time.

The goals of this paper have been to (1) present an annual simulation of these evolving tidal fields; (2) present simulations of the evolving tides (specifically the semidiumal tide) during simulated global dust storms forced by dust distributions that are spatially inhomogeneous; and (3) analyze the evolving semidiumal tide during these storms in terms of the underlying Hough modes and thus demonstrate that knowledge of these modal amplitudes can yield information on global dust load and distribution.

The simulations reported here have been made with the NASA- Ames Mars general circulation model [Pollack et al., 1990, 1993]. In the dust storm simulations, this is coupled to the aerosol transport/microphysical model of Toon et al. [ 1988], providing for a self- consistent evolution of a suspended dust load (the dust load in the annual simulation is spatially constant on a constant pressure surface and evolves in time following observations during the first Viking year).

The annual simulation with the MGCM successfully reproduces many of the observed attributes of both the diurnal and semidiurnal tides. Our focus in the dust storm experiments has been on the semidiurnal tide, since its observed counterpart responds very strongly to increased dust loading. The simulated variation of this tide during Viking year 1 matches observations well, although we do not adequately capture observed amplitude maxima during the dust storms. These maxima are better simulated in the dust storm

experiments, suggesting that a spatially varying dust distribution more accurately represents the tidal forcing that occurred during southern summer of Viking year 1. There are also discrepancies between observed and simulated phases in the semidiurnal tide at VL 1, particularly during northern summer-fall. WH noted a similar discrepancy in their simulations, which were chiefly focused on the northern summer season. During this season, eastward propagating Kelvin modes can attain amplitudes comparable to Sun-synchronous tidal amplitudes and can substantially modify local tidal amplitude and phase variations. We note that during the season on which our dust storm simulations are focused, Kelvin wave amplitudes are much smaller than Sun-synchronous tidal amplitudes.

The dust storm simulations reported here also successfully reproduce many features of the observations, including the amplitude variations at the two lander sites, and the phase advance from VL 1 to VL2 (-1 hour earlier at VL2). In these simulations, dust is injected into the atmosphere between latitudes 15øS and 37.5øS, in line with observations that global storms appear to begin in southern subtropical latitudes. Although the forcing is highly asymmetric about the equator, decomposition of the tide into its underlying Hough modes indicates that the response is dominated at all times

and in all cases by the H22 mode. The vertical wavelength of this mode is very long [O(150 km); Zurek, 1980], and thus the mode is efficiently forced by solar heating of the deep layer of dust.

Despite the dominance of the H22 mode, other modes play an important role in accounting for both the structure of the tide and the VL1/VL2 tidal amplitude ratios. The amplitude of the simu-


lated tide maximizes in the region 10ø-20øS (Figure 7), and examination of synthetic pressure fields (Figure 12) suggests that this is

due to the presence of the H32 mode. Additionally, the latitudinal phase structure of the simulated tide can be attributed to the pres-

ence of the and H4: modes. We believe that the observed phase advance of the tide between VL 1 and VL2 can be attributed to the presence of these higher-order modes during dust storms.

The ratio of semidiurnal tidal amplitudes computed from observed surface pressures at the Viking lander sites exhibits marked temporal variability at the time of global dust storms, with sharp increases during the intensification stages of both storms observed in 1977, followed by declines to prestorm values over the ensuing 10-20 sols. Our simulated dust storms reproduce these rapid changes in the ratio between the model's VL1 and VL2 grid points. We have shown that these variations indicate that the tidal response must consist of more than a single Hough mode (although a large fraction of the tide is accounted for by the gravest symmetric/_/.2 mode) and that the amplitudes of some or all modes must vary wit2'h time.

Decomposing the tidal field into its Hough modes and examining the modal amplitude variations during a global dust storm can be used to infer the global dust opacity (via the amplitude of H22 ) and its equatorially asymmetric distribution (via the amplitude of H32; the amplitude of H52 may also be indicative of longitudinal inhomogeneities in dust distribution).

Undoubtedly, a more extensive network of observing sites would give a clearer picture of the semidiurnal tide's structure [Haberle and Catling, 1996]. For example, such a network would show the degree to which observations of the semidiurnal tide at the two Vi- king lander sites are representative of all longitudes at these (and other) latitudes. We have developed an inversion algorithm which takes surface pressure semidiurnal tidal amplitudes obtained at a network of sites (e.g., the Viking landers sites or selected grid points from a simulation) and deduces the modal structure of the tide as it varies with time. Preliminary analysis (to be presented elsewhere) indicates that poor results are obtained with data from just the two Viking lander sites. Results are improved when we choose instead one site in each hemisphere. Much better results are obtained with a network of-•15 sites distributed globally at five to six latitudes and several longitudes. These preliminary findings, together with our results presented here, suggest that a network of sensors mea- suring surface pressures may be used to infer latitudinal (and possibly longitudinal) suspended dust distributions in the Mars atmosphere. As discussed above, time variations in the amplitude of the H22 mode would indicate global opacity changes, and those of 2 2

higher-order modes H' 3 and possibly H 5 would indicate the hemispheric asymmetry in the atmospheric dust loading.

Acknowledgements. We gratefully acknowledge R. Zurek for supplying code to compute the structure of the Hough functions for Mars, and for his insightful comments on an earlier version of the manuscript. We also acknowledge H. Houben for supplying the code used in decomposing pressure fields into spherical harmonics. We are also grateful to J. Schaeffer for his assistance in generating the experiments discussed here and R. Haberle for his encouragement and review of the manuscript. This work was funded by NASA's Planetary Atmospheres program, RTOP 154-20-80-16.

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A.F.C. Bridget, Department of Meteorology, San Jose State University, San Jose, CA 95192-0104. (e-mail: bridger•canali.arc.nasa.gov)

J.R. Murphy, San Jose State University Foundation, San Jose, CA 95172- 0130 (e-mail: murphy•canali.arc.nasa.gov)

(Received August 16, 1995; revised January 12, 1998; accepted January 22, 1998.)

Mars' surface pressure tides and their behavior during global dust storms

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