An Analysis of the Effect of Topography on the Martian Hadley Cells

MARCH 2010 Z A L U C H A E T A L . 673

ANGELA M. ZALUCHA AND R. ALAN PLUMB Massachusetts Institute of Technology, Cambridge, Massachusetts

R. JOHN WILSON Geophysical Fluid Dynamics Laboratory, Princeton, New Jersey

(Manuscript received 4 March 2009, in ﬁnal form 17 September 2009)

ABSTRACT

Previous work with Mars general circulation models (MGCMs) has shown that the north–south slope in Martian topography causes asymmetries in the Hadley cells at equinox and in the annual average. To quantitatively solve for the latitude of the dividing streamline and poleward boundaries of the cells, the Hadley cell model of Lindzen and Hou was modified to include topography. The model was thermally forced by Newtonian relaxation to an equilibrium temperature profile calculated with daily averaged solar forcing at constant season. Two sets of equilibrium temperatures were considered that either contained the effects of convection or did not. When convective effects were allowed, the presence of the slope component shifted the dividing streamline upslope, qualitatively similar to a change in season in Lindzen and Hou’s original (flat) model. The modified model also confirmed that the geometrical effects of the slope are much smaller than the thermal effects of the slope on the radiative–convective equilibrium temperature aloft. The results are compared to a simple MGCM forced by Newtonian relaxation to the same equilibrium temperature profiles, and the two models agree except at the winter pole near solstice. The simple MGCM results for radiative– convective forcing also show an asymmetry between the strengths of the Hadley cells at the northern summer and northern winter solstices. The Hadley cell weakens with increasing slope steepness at northern summer solstice but has little effect on the strength at northern winter solstice.

1. Introduction seasonal variation, with Southern Hemisphere spring and summer as the favored seasons for large-scale dust The Martian atmosphere is affected by several asym- storms (Kahn et al. 1992). metries not present in the terrestrial one. One such Haberle et al. (1993) noted a factor of 2 difference in asymmetry, referred to as the topographic dichotomy, is Hadley cell intensity between the two solstices using the that the zonally averaged surface topography is at higher National Aeronautics and Space Administration (NASA) elevation in the Southern Hemisphere than the North- Ames Mars general circulation model (MGCM), which ern Hemisphere (Fig. 1). Another asymmetry exists they attributed to the variation in the solar constant be- because of Mars’ relatively eccentric orbit (the Martian tween these two seasons. However, Joshi et al. (1995) orbital eccentricity is 0.0934 whereas the terrestrial is found that when the solar forcing was held constant be- 0.0167), which causes the solar insolation to differ by tween the two seasons, a factor of 1.5 difference in Hadley 44% between perihelion and aphelion. Since Southern cell intensity was still present in the Oxford ‘‘interme- Hemisphere summer is near perihelion and thus North- diate’’ MGCM simulations. Wilson and Hamilton (1996) ern Hemisphere summer is near aphelion, the solar observed with the Geophysical Fluid Dynamics Labora- insolation varies greatly between the solstices. Atmo- tory (GFDL) MGCM that the zonal mean component of spheric dust, important radiatively, also shows a strong topography inhibited Hadley cell intensity during North- ern Hemisphere summer. Haberle et al. (1993) and Basu et al. (2006) remarked on the sensitivity of the Martian Corresponding author address: Angela Zalucha, Massachusetts Institute of Technology, 77 Massachusetts Ave., Room 54-1721, Hadley circulation to off-equatorial heating. Basu et al. Cambridge, MA 02139. (2006) also found that the relative strength of the circu- E-mail: [email protected] lation at the opposite solstices depends on dust loading.

DOI: 10.1175/2009JAS3130.1

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in the study by Richardson and Wilson (2002)—the zonal mean component of topography is the dominant factor in causing an asymmetric Hadley circulation. Both Richardson and Wilson (2002) and Takahashi et al. (2003) concluded that the cause of the asymmetry was an upslope (i.e., southward) shift in the peak heating. Takahashi et al. (2003) went on to state that the convective heating term was the main influence in this shift. In this paper, we expand the analysis of the effects of convection and the north–south topographic slope to include other seasons, most notably the solstices. We also not only use a simple MGCM, but also apply a modified version of the Hadley cell model of Lindzen and Hou (1988) that includes the effects of topography. To drive both of these models, we use a simple radiation scheme that assumes FIG. 1. Zonally averaged Martian topography measured by a gray atmosphere and no dust. While some aspects of the the Mars Orbiter Laser Altimeter instrument (Smith et al. 1999). Martian atmosphere may be poorly represented by this Martian topography slopes downward from the Southern assumption, we show that it captures the important aspects Hemisphere to the Northern Hemisphere. of Hadley cell dynamics in section 4d by comparing our scheme with the GFDL MGCM, which contains a nongray Webster (1977) realized that elevated regions on Mars radiation scheme. The gray radiation scheme has the ad- may act as heat sources for the adjacent atmosphere vantage of being analytical [such that it can be used in the because the (nondusty) Martian atmosphere is effectively modified Lindzen and Hou (1988) model] and it allows us transparent to solar radiation. Two recent studies have fo- to formulate a conceptually simple description of Hadley cused on the effects of the north–south slope in the zonally cell dynamics in the presence of a north–south slope. averaged topography on the Hadley circulation. Richardson In section 2, we develop the simple radiative transfer and Wilson (2002) noted in the GFDL MGCM results that model that we use to calculate equilibrium temperature. the annually averaged zonal mean circulation contained The effects of convection are included in our radiative– a stronger Northern Hemisphere Hadley cell, which also convective model but are not included in our ‘‘pure’’ radi- extended southward across the equator. They performed ative model. The equilibrium temperatures are used in two further experiments in which the argument of perihe- section 3, where we derive a model based on Lindzen and lion was shifted by 1808 (to test the effect of seasonal dif- Hou (1988) that predicts the latitude of the dividing ferences in the strength of the solar forcing) and in which streamline and the poleward extent of the Hadley cells in the zonal mean component of topography was removed the presence of nonzero topography. We use this model to (leaving only the mountain or ‘‘wave’’ component). The solve for the boundaries of the Hadley cells with and removal of the zonal mean component of topography cre- without the zonal mean component of topography and also ated two cells of nearly equal strength and shape, while the explicitly as a function of season [recall that Richardson and shift in the argument of perihelion produced little change Wilson (2002) concentrated their study on the annual av- from the full MGCM run. These results suggest that the erage, and Takahashi et al. (2003) focused on the equinox]. north–south slope in topography is important, but the In section 4 we present a simple MGCM and compare it to strength of the solar forcing is secondary. the modified Lindzen and Hou (1988) model in section 5. Similarly, Takahashi et al. (2003) found that in their own MGCM results at equinox the northern cell was stronger 2. Calculation of equilibrium temperature than the southern and extended across the equator into the Southern Hemisphere. They conducted three runs at per- In both the modified Lindzen and Hou (1988) model petual equinox in which variations in only one of the fol- (section 3) and the simple MGCM (section 4), the ex- lowing were included: topography, surface thermal inertia, ternal thermal forcing is applied through Newtonian re- and surface albedo. The runs with either surface thermal laxation to a radiative equilibrium state, represented by inertia only or surface albedo only did not match the con- the equilibrium temperature Teq. It should be stressed that trol run with all three parameters, but the run with to- Teq is a proxy for the diabatic heat source and does not pography did. Two subsequent experiments in which either correspond to a physical parameter that can be measured. only the zonal mean component of topography or only the We have deliberately chosen a conceptually simple radi- zonal wave component were included showed that—as ation scheme in order to more clearly understand its

Unauthenticated | Downloaded 09/27/21 06:58 AM UTC MARCH 2010 Z A L U C H A E T A L . 675 effect on the results. Our experiments use two types of mation, no dust, no scattering, no solar absorption by the equilibrium conﬁgurations. The ﬁrst is referred to as pure atmosphere, and a gray atmosphere in the long wave, the radiative equilibrium. Assuming the Eddington approxi- solution to the radiative transfer equation is

( Q (f, L )[0.5 1 0.75t(p)] t t 4 o s 6¼ o sTeq,R(f, Ls, p) 5 (1) Qo(f, Ls)11 0.75to[po(f)] t 5 to,

22 where Teq,R is the pure equilibrium temperature, s is the where So 5 600 W m is the mean solar constant, A is Stefan–Boltzmann constant, f is latitude, Ls is the ecliptic the albedo (set to a constant value of 0.15), e 5 0.0934 is 1 longitude of the sun (in Mars-centered coordinates), the eccentricity, Lsp 5 2528 is the Ls of perihelion (located p is pressure, Qo is the daily averaged net solar ﬂux, t is near northern winter solstice), d is the declination of the the optical depth, and to is the optical depth at the sur- sun (in Mars-centered coordinates), and g is the hour face. Under the assumption of constant opacity, t( p) 5 angle of sunrise and sunset. Furthermore, cosg 52tanf ptoo/poo, where poo is a reference pressure set to the tand,andsind 5 sini sinLs,wherei 5 258 is the obliquity. mean surface pressure of 6 hPa and too (here taken to be The square of the term in brackets in (2) represents the 0.2) is the optical depth at poo. Likewise to[ po(f)] 5 correction to So due to the changing distance from the sun po(f)too/poo, where po is the height of the surface in in an elliptical orbit. For Earth, this quantity varies from pressure coordinates. Note that t increases downward. 0.968 to 1.069 between perihelion and aphelion, re- From Peixoto and Oort (1992, 99–100), spectively, whereas for Mars it varies from 0.837 to 1.217. "#The second prescribed equilibrium state is radiative– 2 S 1 1 e cos(L L ) convective equilibrium, in which the temperature above Q (f, L ) 5 o (1 A) s sp o s p 1 e2 the convective layer is the same as in the pure radiative equilibrium state, and the temperature within the con- 3(g sinf sind 1 sing cosf cosd), (2) vective layer follows an adiabat. Thus,

8 <>Q (f, L )[0.5 1 0.75t( p)] t , t o s t 4 4R/c sT (f, L , p) 5 t( p) p (3) eq,RC s :> Qo(f, Ls)[0.5 1 0.75tt(f)] t $ tt, tt(f)

where Teq,RC is the radiative–convective equilibrium tem- at constant pressure. The height of the convective layer is perature, tt is optical depth of the top of the convective calculated by requiring continuity of temperature and ra- layer, R is the specific gas constant, and cp is the specific heat diative flux (surface plus atmosphere) at tt.Explicitly,

ð "# t (t t ) o (t t ) sT4 (t ) exp o t 1 sT4 (t ) exp * t dt eq,R o m eq,R * m * tt ð "# (4) t (t t ) o (t t ) 5 sT4 (t ) exp o t 1 sT4 (t ) exp * t dt , eq,RC o m eq,RC * m * tt

2 where m 5 /3 is the average cosine of the emission angle (3) for Teq,R and Teq,RC, respectively, in (4) shows that tt and t* is a variable of integration. Substitution of (1) and is a function of only R, cp, and to (i.e., the height of the surface). The key difference between Teq,R and Teq,RC is that for the latter the equilibrium temperature aloft 1 Seasons on Mars are described by the ecliptic longitude of the (within the convective layer) depends on the height of sun, denoted by L . By deﬁnition, L 5 08 corresponds to the vernal s s the surface, whereas the former does not. equinox, Ls 5 908 to the Northern Hemisphere summer solstice, Ls 5 1808 to the autumnal equinox, and Ls 5 2708 to the Northern The Martian atmosphere follows a seasonal cycle in Hemisphere winter solstice. which atmospheric CO2 freezes out during the winter,

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FIG. 2. Zonally averaged pure radiative equilibrium temperature (K). Latitude is plotted on the x axis, pressure is plotted on the left y axis, and log pressure height is plotted on the right y axis, assuming a scale height of 11 km and a reference pressure of 6 hPa. Only temperatures below ;0.2 hPa (40-km log pressure height) are shown. From left to right, the columns are Ls 5 08,908, and 2708, respectively. From top to bottom, the rows are flat, full, and mean topography. Contour interval is 15 K. causing the surface pressure to vary about its annually phies are considered: flat topography at zero elevation averaged value by over 1 hPa (e.g., Hess et al. 1980). (‘‘flat’’), Mars Orbiter Laser Altimeter (MOLA) topog- Haberle et al. (1993) found that the mass change does raphy (‘‘full’’; Fig. 4a), and zonal mean MOLA topog- not significantly affect the zonal mean circulation except raphy (‘‘mean’’; Fig. 4b). For the pure radiative case, Teq at the lowest levels over the polar caps; therefore, we does not vary between the different topographies—it is have chosen to keep atmospheric mass constant. When as if the surface has been pasted onto the temperature

Teq falls below the CO2 condensation temperature Tco2, profile. Conversely, the surface modifies the equilibrium it is instantaneously reset to Tco2. From the Clausius– temperature aloft in the radiative–convective case. The Clapeyron equation, equilibrium temperature contours appear to be pushed up by the higher surface, changing the meridional tem- 1 1 R p perature gradient so that the equilibrium temperature is Tco2 5 ln , (5) T1 L p1 higher above an elevated surface. Our values for Teq,R (and hence Teq,RC) at high alti- where T1 5 136.6 K is the reference temperature at the tudes (not shown) are warmer and more isothermal than reference pressure p1 5 1 hPa (James et al. 1992), p is in the corresponding values for Teq,RC plotted in Joshi et al. hPa, L 5 5.9 3 105 Jkg21 is the specific latent heat of (1995) and Haberle et al. (1997). While these authors sublimation, and Tco2 is given in K. use different radiative transfer models, the discrepancy Figures 2 and 3 show zonally averaged Teq,R and is likely a result of the choice of too and the gray at- Teq,RC given by (1) and (3), respectively, for Ls 5 08, mosphere assumption, which is not fully accurate for Ls 5 908, and Ls 5 2708. The following three topogra- Mars. Hinson et al. (2008) observed in radio occultation

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FIG. 3. As in Fig. 2, but for zonally averaged radiative–convective equilibrium temperature (K).

data that the depth of the mixed layer varies with coordinates ( p/poo) is 0.44 (cf. our value of 0.51 for ﬂat elevation, an effect that is not captured in our model. topography).

Caballero et al. (2008) derive a radiative–convective Figure 5 shows Teq,R and Teq,RC at a constant pressure model for a semigray atmosphere, which they apply to level of 3.8 hPa or 5-km log pressure height, which is Mars. Their predicted height of the convective layer in s within the convective layer. It is evident that topography

FIG. 4. The topography of Mars: (a) the full topography measured by MOLA (‘‘full’’) and (b) the zonal mean MOLA topography (‘‘mean’’). Elevations are in km; contour interval is 0.5 km.

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FIG. 6. Schematic of equilibrium temperatures for two different surface elevations (black and gray horizontal lines). The pure radiative equilibrium temperature (black and gray curve) is the same at a given level aloft (pa) for any surface height. The surface equilibrium temperature (Tg) in pure radiative equilibrium is lower for the elevated surface because of the reduced greenhouse effect, but this effect is negligible in the optically thin Martian atmosphere. The dashed lines are adiabats drawn in radiative–convective equi- FIG. 5. Equilibrium temperature at a constant pressure of librium. For this case, the equilibrium temperature at some level pa 3.8 hPa (5-km log pressure height), for Ls 5 (a) 08, (b) 908, and is warmer above the elevated surface. (c) 2708. The thin solid line is Teq,R (independent of topography), the heavy solid line is Teq,RC with flat topography, and the dashed solid line is Teq,RC with mean topography. radiative–convective equilibrium temperature profiles are being evaluated at different relative heights above changes the meridional gradient of Teq,RC (cf. the heavy the local surface (the primary heating element for the solid and heavy dashed lines). For flat topography, atmosphere), which dominates over the reduced green-

Teq,RC is offset to a higher temperature from Teq,R be- house effect at the higher altitude. In this way, the height cause in the former convection acts to redistribute heat of the surface modifies the heating profile of the lower from the surface into the convective layer. In pure ra- atmosphere. Moreover, the change in surface tempera- diative equilibrium, a temperature discontinuity exists ture due to the reduced greenhouse effect at higher ele- between the surface and the atmosphere directly above vations is again negligible, varying by less than 3 K for a the surface. The radiative–convective equilibrium tem- difference in elevation of 5 km. perature profile follows a dry adiabat that is continuous with the surface temperature. 3. Hadley cell model with topography Figure 6 shows a diagram of the two different types of

Teq. The gray and black curve is the pure radiative Held and Hou (1980) derived a model of the steady, equilibrium solution for Teq as a function of pressure nearly inviscid, Boussinesq, axisymmetric, Hadley cir- and is the same regardless of surface height. If the sur- culation in the absence of eddies. Diabatic heating was face is located at some elevation po, located at the level represented by Newtonian relaxation of temperature at of the solid black horizontal line, then the equilibrium a uniform rate a toward a prescribed distribution of equi- surface temperature is given by Tg (black text). If the librium temperature ue, which in their case was sym- surface is then located at some higher elevation (lower metric about the equator. Through consideration of the pressure), represented by the solid gray horizontal line, constraint of angular momentum conservation, they the equilibrium surface temperature (Tg in gray text) showed that the Hadley cell is conﬁned to a ﬁnite region will decrease because of the reduced greenhouse effect. about the equator; poleward of the edge of the cell, there In an optically thin atmosphere such as Mars, however, is no meridional circulation, and temperatures are in ra- this effect is extremely small, and the surface tempera- diative equilibrium. Using simple arguments, they were ture can be considered independent of surface height. able to solve for the location of the cell edge and other The dashed lines in Fig. 6, drawn for both surface el- characteristics of the circulation. evations, represent adiabats calculated such that the Lindzen and Hou (1988) solved the problem in the convective layer remains in energy balance with the rest nonsymmetric case, where the imposed equilibrium tem- of the atmosphere. Comparing the radiative–convective perature maximum is located on the summer side of the equilibrium temperatures at some pressure level pa within equator. They found strong sensitivity—with only a mod- the convecting layer shows that they are higher above est shift of the peak of ue off the equator, the rising branch the elevated surface than over the lower surface. The of the circulation shifted much further, and the resulting

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›F u 5 g , (8) ›z uo

where g is the acceleration of gravity at the surface, u is

temperature, and uo a constant reference temperature (note that in the Boussinesq approximation, temperature and potential temperature are the same); and the steady thermodynamic equation,

y ›u ›u 1 w 5 a(u u ), (9) a ›f ›z e

where y and w are the meridional and vertical velocities, respectively.

Now, consider the dividing streamline at f 5 f1. Assuming that the zonal ﬂow near the surface is weak, FIG. 7. Parameters in the model of Lindzen and Hou (1988) with the air rises out of the boundary layer with angular 2 2 a bottom topography zb(f) added. The x axis is latitude; the y axis momentum M0 5Va cos f1, from (6). Conservation of is height. Arrows indicate the direction of ﬂow. angular momentum then dictates that everywhere along this streamline (outside the boundary layer) M(f, z) 5

M0, where M0 is a constant; hence, within each cell and cross-equatorial cell is much stronger than the opposite along the top of the model at z 5 h, (6) requires that cell in the summer hemisphere. Recently, Caballero et al. (2008) revised the Lindzen and Hou (1988) result (cos2f cos2f) u(f, h) 5Va 1 . (10) using a more sophisticated, semigray specification of cosf equilibrium temperature. Their radiation scheme assumed a CO2-like absorber, where absorption is zero At z 5 zb(f), assume (again following Held and Hou) everywhere outside of a narrow band at a single wave- that u is much weaker than at the top, such that sub- length. Given the appropriate radiation parameters (i.e., tracting (7) applied at z 5 zb from that applied at z 5 h total broadband optical depth, pressure broadening, and and substituting from (10) gives equivalent bandwidth), they derive an analytical solution for the Hadley cell width, depth, energy transport, sinf V2a2 (cos4f cos4f) and mass flux. cos3f 1 Here we follow the analysis of Lindzen and Hou ›F ›F (1988), modifying the problem to permit varying height 5 1 ›f z5h ›f z5z of the surface and the equilibrium temperatures discussed b in section 2. A schematic is shown in Fig. 7. The Hadley › dzb ›F 5 [F(f, h) F(f, zb)] . (11) cells are confined between latitudes f1 and f2,withthe ›f df ›z z5zb dividing streamline at f1. The height of the surface as a function of latitude is given by zb(f). Nonaxisymmetric Using (8) and its vertical integral, and integrating in effects are ignored. The governing principles are as fol- f, we obtain lows: conservation of absolute angular momentum, ð ð g h g f dz 2 2 b M(f) 5Va cos f 1 u(f, z)a cosf, (6) u dz 1 u(f9, zb) df9 uo z uo f df9 b c at upper levels, where z is height, V is the planetary V2a2 cos4f 5 F^ cos2f 1 1 , (12) rotation rate, u is the zonal wind, and a is the planetary 2 cos2f radius; gradient wind balance, where the second integral is along the boundary starting 1 1 ›F 2V sinfu 1 tanfu2 5 , (7) at an arbitrary latitude fc, f9 is a variable of integration, a a ›f and F^ is a constant of integration to be determined. Next we apply the conditions invoked by Held and where F is geopotential; hydrostatic balance, Hou (1980). Since F must be continuous at the edges of

Unauthenticated | Downloaded 09/27/21 06:58 AM UTC 680 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 67 ð ð the cells, both at z 5 h and zb (assuming that zb is con- h f g g dzb tinuous), it follows from (8) that the vertical integral of u Xe(f) 5 ue(f, z) dz 1 ue(f9, zb) df9. u z (f) u f df9 must be continuous there. Outside the edges of the cell, o b o c (21) there is no overturning circulation, and u 5 ue; hence, ð ð h h Thus, (16)–(19) constitute a set of equations in the four unknowns: F^ , f , f , and f . For a flat surface, X re- u(f1, z) dz 5 ue(f1, z) dz, (13) 1 Ð1 2 e z (f ) z (f ) h b 1 b 1 duces to the form 0 ue dz as in Held and Hou (1980) and Lindzen and Hou (1988). Variations in the height of with a similar expression evaluated at f 5 f2. Addi- the lower boundary impact the forcing function both tionally, since we assume no flow into or out of the cells, explicitly (since both terms involve zb) and implicitly (9) integrated across each cell gives (through their influence on equilibrium temperature). ð ð The relative importance of these contributions will be f h 1 discussed below. (u ue) dz cosf df 5 0, (14) f1 zb(f) The choice of h, taken to be the height of the tropopause on Earth, is not obvious for Mars. If the tropopause with a similar expression integrated between f2 and f1. is defined as the level where the static stability changes Equation (14) is a statement of no net diabatic heating drastically, no clear transition exists analogous to the within the Hadley cells. terrestrial troposphere and stratosphere. The tropopause The Hadley cells are characterized by weak horizontal could also be defined as the level to which convection temperature gradients; neglecting advection along the penetrates in radiative–convective equilibrium, but this bottom boundary, (9) says that u ’ ue there, so that (12) definition excludes the pure radiative case. While work can be written with MGCMs has shown that the Martian circulation ð extends to the mesopause, the mesopause hardly seems g h V2a2 cos4f u 5 F^ 2f 1 1 like a good definition since most of the mass transport in dz cos 2 uo z 2 cos f the Hadley cells is well below this level. Since in our b ð f simple MGCM (section 4) 50% of the mass flux is typi- g dzb ue(f9, zb) df9. (15) cally located below 15 km, we adopt that value for h. u f df9 o c Since the theory is assumed to be axisymmetric, only two of the topographies described above are relevant Substituting into (13) and (14) then prescribes the math- here: flat and zonal mean MOLA topography. The value ematical problem to be solved: of ue in (21) is evaluated using either the pure radiative equilibrium temperature given by (1) or the radiative– V2a2 F^ F(f , f ) 5 X (f ), (16) convective equilibrium temperature given by (3), re- 2 1 1 e 1 calling that the temperature must remain above the CO2 frost temperature. V2a2 F^ F(f , f ) 5 X (f ), (17) While the solutions to (16)–(19) depend solely on X 2 1 e e (f), it is only gradients of Xe that drive the circulation. ð "#ð f 2 2 f (If Xe is constant, the trivial solution is a cell of zero 1 ^ V a 1 F F(f1, f) cosf df 5 Xe(f) cosf df, width.) From (21), f 2 f 1 1 ð dX g h ›u (18) e 5 e dz, (22) "# df u z ›f ð ð 0 b f 2 2 f 1 ^ V a 1 F F(f1, f) cosf df 5 Xe(f) cosf df, from which we deduce the rather obvious conclusion that f 2 f (as in the case of a flat surface) the circulation is driven by (19) the vertically integrated temperature gradients. Topogra- phy enters both explicitly, in the integration limit, and where we have used the shorthand implicitly, through its impact on ue. Under the conditions of our calculations, both the surface temperature and the cos4f F(f , f) 5 cos2f 1 1 (20) depth of the convecting layer, while varying substantially 1 cos2f with latitude, are remarkably insensitive to the surface height variations. This insensitivity, a consequence of the and defined the forcing function: weakness of the greenhouse effect, can be exploited to

Unauthenticated | Downloaded 09/27/21 06:58 AM UTC MARCH 2010 Z A L U C H A E T A L . 681 arrive at a simple understanding of the role of topographic height variations on the forcing of the Hadley circulation. Now, ð ð ð h h R h C ›u ›u c ›u e dz 5 e dz 1 e dz, (23) ›f ›f ›f zb hc zb

C R where ue (f, z)andue (f, z) are the equilibrium temperatures within and above the convecting layer, respectively (and note that the latter is radiatively determined and independent of zb), and hc is the altitude of the top of the convecting layer. Within the convecting layer, and consistent with the Boussinesq approximation,

C S ue (f, z) 5 ue (f) G[z zb(f)],

S FIG. 8. (a) Forcing function given by (21) for Ls 5 08,908, and where ue(f) is surface temperature and G the adiabatic 2708. The black curve is flat topography and the gray curve is zonal lapse rate; hence, the gradient mean topography. Each curve has been normalized such that its value at the equator is subtracted from each point in the curve. ›uC ›uS ›z e 5 e 1G b (24) Regions where the temperature is equal to the CO2 condensation ›f ›f ›f temperature are not plotted. (b) Latitudinal derivative of the forcing function given in (a). The noise in the curves with topog- is independent of z. Therefore raphy is a result of the fact that the derivative of the measured topography is noisy. The net affect of topography is to reduce the ð h C S latitudinal derivative of the forcing function. c ›u ›u ›z e dz 5 D e 1G b , (25) ›f ›f ›f zb ative, and hence tends to influence the forcing function in where D 5 hc 2 zb is the depth of the convecting layer. S the same way as displacing the ue maximum into the As we have seen, both ue and D are, to an excellent Southern Hemisphere. At equinox, the second term is approximation, independent of zb, and so the surface small in the tropics and so the net effect of the topography topography enters (25) only through the term ›zb/›f. is to shift the effective thermal equator into the Southern Above the convecting layer, the contribution to (23) is Hemisphere. This is illustrated in Fig. 8, for which the ð ð ð h ›uR h ›uR D ›uR forcing function and its derivative have been calculated e dz 5 e dz 1 e dz. (26) by direct substitution of the radiative–convective equi- h ›f D ›f h ›f c c librium temperature distribution, with the topography of Fig. 1, into (21). The impact of the topography is Accordingly, dependence of (26) on zb is encapsulated in the second term. If the radiative lapse rate does not vary modest, but the analysis of Lindzen and Hou (1988) R leads us to expect that a small asymmetry in the thermal significantly with latitude, then ›ue /›f is independent of z (Fig. 2), so forcing will produce a much larger asymmetry in the ð circulation. At the solstices, the possibly competing ef- D ›uR ›uR ›uR fect of the two terms makes simple statements on the e dz ’ e (f, D)(D h ) 5 e (f, D)z . ›f ›f c ›f b basis of (27) more elusive. However, Fig. 8 shows that the hc net effect is weak at Ls 5 908, with only a slight weak- In total, then, substituting into (22) and (23), ening of the gross latitudinal gradient. At Ls 5 2708, the ð impact of the surface topography is somewhat greater, S h R R u0 dXe ›ue ›ue ›zb ›ue with an intensification of the gross gradient. ’ D 1 dz 1GD (f, D)zb. g df ›f D ›f ›f ›f Returning to (16)–(19), Fig. 9 shows the solutions for (27) f1, f1,andf2 for each of the two topographies with pure radiative or radiative–convective forcing and physical

Of the two contributions that depend on zb, the first constants appropriate for Mars. The solutions for flat depends directly on the local topographic slope, while topography for pure radiative and radiative–convective the second does not (but note that both contributions forcing are nearly equal, such that they lie on same curve are zero with flat topography, since then zb 5 0). For the in Fig. 9. Hadley cells are most sensitive to the meridi- most part, ›zb/›f , 0, so the first term is generally neg- onal gradient in equilibrium temperature, which is purely

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4. Experiments with a simple MGCM a. Description of the MIT MGCM We have converted the Massachusetts Institute of Technology (MIT) GCM to physical constants appropriate for the Martian atmosphere, which is assumed to

be entirely CO2 and to contain no dust. The dynamical core of the MIT GCM solves the fundamental equations of geophysical fluid dynamics in the hydrostatic approximation using the finite volume method on an Arakawa C grid (Marshall et al. 1997). The default configuration has no viscosity or vertical diffusion but uses an eighth-order Shapiro filter to remove gridscale noise. The horizontal configuration is a cube–sphere grid (Adcroft et al. 2004) with 32 3 32 points per cube face, equivalent to a resolution of 2.88 or 166 km at the equator. Note that the resolution of the MOLA dataset is 1/648 latitude 3 1/328 FIG. 9. Solutions to (16)–(19), for (a) f1, (b) f1, and (c) f2. The heavy solid line is the solution for flat topography for both pure longitude (Smith et al. 2001), which exceeds the hori- radiative and radiative–convective forcing, which differ by less zontal resolution for the simple MGCM. than 2%. The thin dashed line (barely visible as it is almost in- The vertical grid uses an h coordinate (Adcroft and distinguishable from the heavy solid line) is the solution for zonal Campin 2004) with 30 levels, and the grid spacing in- mean MOLA topography with pure radiative forcing, and the creases approximately logarithmically with height. Zero heavy dashed line is the same but for radiative–convective forcing. elevation corresponds to a pressure of 6 hPa (the observed annual and global average value), which is also a consequence of the meridional gradient in solar heating used as a reference pressure elsewhere in our calculations. in the absence of topography. Since the meridional gra- The top level is centered at a pressure of 0.000117 hPa, dient in solar heating is the same for both radiative which corresponds to a log pressure height of 119 km forcings [cf. (1) and (3)], their solutions are expected to (using the reference pressure above and a scale height of be similar. 11 km). This is above the level where nonlocal ther- For flat topography at equinox, symmetric cells are modynamic equilibrium effects may become important obtained, which agrees with the limit of Held and Hou (Lo´ pez-Valverde et al. 1998). We do not treat the top (1980). When topography is added in the pure radiative levels as realistically modeling the atmosphere in that case, f1 is shifted slightly upslope, while the poleward region, since a sponge layer is also located in the upper boundary of the winter cell (f1 for 2708,Ls , 3608 and levels (described below). Within each vertical level in- f2 for 08,Ls , 908) shifts downslope. The deviations tersecting the surface, the resolution of the topography is from the flat topography case may be attributed as the increased by inserting subgrids spaced at 10% of the full result of integrating over a different vertical range (zb to vertical grid spacing at that level (Adcroft et al. 1997). h instead of 0 to h). The external thermal forcing is specified by New- For the case of radiative–convective forcing and zonal tonian relaxation to a prescribed equilibrium tempera- mean MOLA topography, f1 is shifted a significant dis- ture following Held and Suarez (1994). Explicitly, this tance upslope, which qualitatively agrees with previous term in the energy equation is MGCM results. Again the poleward boundary of the ›T winter cell shifts downslope, but the boundary of the 5 k (T T ), (28) ›t T eq summer cell (f2 for 2708,Ls , 3608 and f1 for 08, L , 908) shifts upslope. The large shift in the solution s where T is temperature, t is time, k 5 ½ sols21 is the compared with the other cases is a result of the thermal T radiative relaxation rate,2 and T is the equilibrium effects of the slope on u . Differential convective heating eq e temperature, discussed in section 2. The choice of k is aloft associated with the slope moves the latitude of max- T based on a calculation using Eq. (6) of Showman et al. imum equilibrium temperature upslope and away from (2008) and is the same as the value calculated by Haberle the subsolar latitude. This change in the location of the et al. (1997) for clear-sky conditions. The assumption of maximum equilibrium temperature causes the boundaries of the cells to move relative to the cases with zero topography or without convective forcing. 2 A sol is a Martian solar day.

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fixed kT implies that radiation and convection (if im- sphere, but its maximum extent is not past 258. A slight plicitly present in Teq) operate on the same time scale. asymmetry between the strengths of the northern and Boundary layer friction is specified in the horizontal southern cells develops. For Ls 5 908 and pure radiative momentum equations by forcing (Figs. 10b,e,h), no significant difference arises ›u between the three topographies. The same occurs at 5 k (p)u, (29) L 5 2708 (Figs. 10c,f,i). ›t y s When convective forcing is allowed, a substantial dif- ›y ference arises at equinox between experiments without 5 k (p)y, (30) ›t y the north–south sloping component of topography (i.e., flat; Fig. 11a) and with it (i.e., full and mean; Figs. 11d where k is the wind damping rate and is defined by y and 11g, respectively). The flat topography produces p p symmetric cells, whereas the full and mean produce a k 5 k max 0, b , (31) y f p p stronger northern cell and shift the latitude of the di- o b viding streamline to about 2158 in the lower atmosphere where kf is the wind damping rate of the lower atmo- (cf. about 2208 in Takahashi et al. 2003). At Ls 5 908, sphere and pb 5 0.7po is the top of the boundary layer. the cross-equatorial Hadley cell is stronger when the 21 We set kf 5 1 sol , which is consistent with values in the north–south sloping component is removed (cf. Fig. 11b literature (e.g., Lewis et al. 1996; Nayvelt et al. 1997). with Figs. 11e,h). At Ls 5 2708 the correlation with Terms similar to the right side of (29) and (30) are also the presence of the north–south sloping component of included in the horizontal momentum equations in the topography is less apparent (cf. Figs. 11c,f,i). top three model levels as a sponge layer to avoid re- 21 c. Comparison of Hadley cell strengths flection off the model lid, with ky set to 9, 3, and 1 sols from uppermost to lowermost level, respectively. With pure radiative forcing, the strength of the Hadley cells at a given season varies little among the different b. Experiments topographies and does not appear to be correlated with

Each experiment was run at constant Ls for the fol- the presence of the north–south slope. A much different lowing seasons: Ls 5 08, Ls 5 908, and Ls 5 2708.Given scenario arises for radiative–convective forcing. Figure 12 the short radiative time scale of the Martian atmosphere shows the maximum magnitude of the zonally and time- (2 sols) and the lack of oceans with high heat capacity, the averaged streamfunction from each of the simple MGCM atmosphere is expected to respond quickly to changes in experiments with radiative–convective forcing. Specifi- radiative forcing compared with the length of a season cally, the maximum magnitude within each thermally (one quarter of the Martian year or 167 sols). As such, it direct cell above the boundary layer is plotted for each was unnecessary to integrate the model over several years topography (black symbols). to achieve a seasonal equilibrium. The model was run for For flat topography, the strengths of the northern and a total of 180 sols or 6 Martian months, where a Martian southern cells are symmetric at equinox, and the strengths month is defined to be 30 sols, and was fully spun up from at the opposite solstices are equal. For mean topogra- an initial rest state with Teq 5 200 K everywhere by the phy, at equinox the northern cell is enhanced, while the end of the second Martian month. A visual inspection of southern cell is reduced. It is generally recognized from the zonally and monthly averaged results (i.e., potential theory (e.g., Lindzen and Hou 1988) and terrestrial ob- temperature, horizontal and vertical velocities, and sur- servations (e.g., Peixoto and Oort 1992, their Fig. 7.19) face pressure) showed little variation from month to that moving the latitude of maximum equilibrium tem- month after the initial spinup. perature off the equator (as occurs because of changing Figures 10 and 11 show the simple MGCM results for seasons) produces a weak cell in the same hemisphere as zonally averaged mass streamfunction time averaged for the maximum and a strong cell that spans both hemi- sols 60–180 when forced with the pure radiative equi- spheres. The farther the maximum equilibrium temper- librium temperatures (Fig. 2) and radiative–convective ature moves off equator, the more pronounced this effect equilibrium temperatures (Fig. 3), respectively. With becomes.3 If the north–south slope is acting to shift the pure radiative forcing, equinox conditions (Ls 5 08), and flat topography (Fig. 10a), the cells are quite symmetric 3 about the equator. When the north–south sloping com- Compared with Earth, the difference between the equinoctial and solstitial circulations is much more dramatic, since the heat ca- ponent of topography is added (i.e., full and mean to- pacity ofMartian soilis less thanthat ofEarth’s oceans(Haberle et al. pographies; Figs. 10d and 10g, respectively), the dividing 1993). Without the mediating influence of the ocean, the latitude of streamline protrudes slightly into the Southern Hemi- maximum equilibrium temperature can move well off the equator.

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8 21 FIG. 10. Simple MGCM results for zonally and time averaged mass streamfunction with pure radiative forcing, Units are 10 kg s . Positive ﬂow is counterclockwise; negative contours are shaded. Latitude is plotted on the x axis, pressure is plotted on the left y axis, and log pressure height is plotted on the right y axis, assuming a scale height of 11 km and a reference pressure of 6 hPa. Only results below ;0.2 hPa (40-km log pressure height) are shown. From left to right, the columns are Ls 5 08,908,and2708. From top to bottom, the rows are ﬂat, full, 8 21 8 21 and mean topography. Note the different contour intervals (1 3 10 kg s for Ls 5 08 and 10 3 10 kg s for Ls 5 908 and Ls 5 2708).

maximum equilibrium temperature upslope, as indicated f 2 z 5 2 tanh , (32) by Figs. 3g and 5a, then at equinox a stronger northern bi b cell and weaker southern cell are expected for the mean topography case. where zbi is the height of the idealized topography For flat topography at Ls 5 908, the maximum equi- (in km), f is in degrees, and b controls the steepness of librium temperature is shifted far northward (Figs. 3b the slope (lower values correspond to a steeper slope). and 5b), producing no summer cell at all and a strong Figure 13 shows the simple MGCM results for Ls 5 908 winter cell. For mean topography, the strength of the for flat topography and the following three values of b: cell is reduced as compared with flat topography. For 40, 20, and 10. As slope steepness increases, the strength

ﬂat topography at Ls 5 2708, the latitude of maximum of the Hadley cell decreases. Figure 14 shows results equilibrium temperature is shifted far southward, pro- for Ls 5 2708 with the same topographies. In this case, ducing a strong ‘‘winter’’ cell in the opposite sense as at the steepness of the slope has little effect on the cell

Ls 5 908. However, the addition of zonal mean topog- strength. raphy at Ls 5 2708 does not change the strength of the Evidently, a circulation that flows downslope near the cell as significantly as at Ls 5 908. surface causes the Hadley cells to lose strength, but that To further illustrate this seasonal asymmetry, sev- strength is not regained for an upslope flow. This phe- eral more experiments were performed with radiative– nomenon further comes into play in comparing full to- convective forcing at Ls 5 908 and Ls 5 2708, which had pography and mean topography. In Fig. 12, the strengths the following idealized zonal mean topography: for full topography are always less than the strengths for

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8 21 FIG. 11. As in Fig. 10, but with radiative–convective forcing. Note the different contour intervals (4 3 10 kg s for Ls 5 08 and 8 21 15 3 10 kg s for Ls 5 908 and Ls 5 2708). mean topography at the corresponding season. As the and Wilson (2002) and Hinson and Wilson (2004). This circulation flows up and down the mountains and val- model has been used to examine tides and planetary leys, strength is lost as the air flows downward along waves (Wilson and Hamilton 1996; Hinson and Wilson the topography but not regained when it flows upward. 2002; Wilson et al. 2002; Hinson et al. 2003), the water This diminishing is observed when comparing flat and cycle (Richardson and Wilson 2002; Richardson et al. ‘‘wave’’ (i.e., full topography minus zonal mean; not 2002), the dust cycle (Basu et al. 2004, 2006), and cloud shown) topographies. For full topography, this moun- radiative effects (Hinson and Wilson 2004; Wilson et al. tain effect is superimposed onto the effect (if any) of the 2007, 2008). More recently, the physical parameteriza- mean sloping component. The effect is associated with tions have been adapted to the GFDL Flexible Model- the radiative–convective forcing, as runs with pure ra- ing System (FMS), which includes a choice of dynamical diative forcing show little difference in Hadley cell in- cores [finite difference, finite volume (FV), and spectral] tensity for any topography. and associated infrastructure. The Mars physics has been tested with all three dynamical cores and we have d. Comparison with the GFDL MGCM elected to use the finite volume model. To demonstrate that our simple radiation scheme The model uses the radiation code developed and captures the effect of topography on the Martian Hadley used by the NASA Ames Mars modeling group (Kahre cell, we compare our simple MGCM results to the GFDL et al. 2006, 2008). This code is based on a two-stream

MGCM, which uses a full (nongray) radiation scheme. solution to the radiative transfer equation with CO2 and The GFDL MGCM was originally based on the GFDL water vapor opacities calculated using correlated-k values SKYHI terrestrial GCM and an early version of the in 12 spectral bands ranging from 0.3 to 250 microns. model has been described in Wilson and Hamilton The two-stream solution is generalized for solar and in- (1996). Subsequent descriptions appear in Richardson frared radiation, with scattering based on the d-Eddington

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scheme (section 2) are not centrally important. Addition- ally, experiments (not shown) with the GFDL MGCM were performed with the same idealized topography used in Figs. 13 and 14. These experiments again showed that the

Hadley cell strength at Ls 5 908 is sensitive to the steepness of the topographic slope but shows little response to the

slope at Ls 5 2708. Quantitative differences in Hadley cell strengths between the simple MGCM and the GFDL MGCM are due to fundamental differences in the radiation schemes. The Hadley cell strengths in the simple MGCM are sensitive

to the values used for too and kt. The simple radiation code also fails to produce a difference in the Hadley cell strengths at the opposite solstices for ﬂat topography. The eccentricity of the Martian atmosphere creates a large difference in the solar constant; with more energy input into the atmosphere, the Hadley cells ought to be FIG. 12. Maximum magnitude (above the boundary layer) of the stronger. On the other hand, the boundaries of the cells zonally and time averaged mass streamfunction from the simple are in quantitative agreement between the simple and MGCM results with radiative–convective forcing (black symbols). The gray symbols are from Takahashi et al. (2003). The results are GFDL MGCMs and are less sensitive to the particular grouped by the various topographies. The squares are for Ls 5 908, aspects of our radiation scheme. This fact, along with the triangles are for Ls 5 2708, the circles are for the northern cell at qualitative agreement in the behavior of Hadley cell Ls 5 08, and the diamonds are for the southern cell at Ls 5 08. strengths between the two models, lends sufﬁcient cred- ibility to our simple radiation scheme. approximation at visible wavelengths and the hemispheric e. Comparison with previous simulations mean approximation at infrared wavelengths (Toon et al. 1989). The model has been validated with other MGCMs As wind data for the Martian atmosphere above the and the available data, and we consider it to be a robust boundary layer are extremely sparse compared to those representation of the Martian atmosphere. for Earth, we are left to compare the simple MGCM Simulations using the GFDL MGCM were carried out streamfunction results with previous models. The re- with 58 latitude 3 68 longitude horizontal resolution and sults for streamfunction strength from the MGCM of

46 levels. Simulations were run for 50 sols. The CO2 Takahashi et al. (2003) are also plotted in Fig. 12 (gray condensation cycle was turned off, such that atmospheric symbols). Comparison of mean and full topographies at mass remained constant with a global mean surface equinox shows that that the northern cell is stronger for pressure of 6 hPa. Orbital eccentricity was set to zero to the mean topography case, in agreement with our results. eliminate differences in the solar constant between the Quantitatively, our simple MGCM predicts a weaker two solstice seasons, thus highlighting the asymmetric cell for full topography at Ls 5 2708, a stronger cell for inﬂuence of the topography. Surface albedo and surface full topography at Ls 5 908, and a weaker northern cell thermal inertia were held constant at 0.15 and 350 J for full and mean topographies at equinox. Quantitative m22 s21/2 K21, respectively. Column dust opacity was set disagreement between the two models arises from dif- to a globally uniform value of 0.3, a low-dust setting. The ferences in the radiation schemes between our model topographies considered were ﬂat and had idealized zonal and that of Takahashi et al. (2003). mean topography [given by (34)] with b 5 20. Finally, Joshi et al. (1995) stated that the Hadley cell

Figure 15 shows the results from the GFDL MGCM at Ls 5 2708 was a factor of 1.5 stronger than at Ls 5 908 with (top) ﬂat topography and (bottom) idealized zonal in their ‘‘intermediate’’ global circulation model (based mean topography for Ls 5 (left) 908 and (right) 2708. on the Oxford MGCM), which most closely corresponds The results shown are a 10-sol average centered on the to our radiative–convective forcing and full topography. solstices. At Ls 5 908, the addition of zonal mean to- Our simple MGCM also yielded a factor of ;1.5 for this pography suppresses the maximum strength of the setup, whereas Takahashi et al. (2003) obtained a factor

Hadley cell by 55%, whereas at Ls 5 2708 the addition of of ;3. Joshi et al. (1995) also noted that the effect was zonal mean topography strengthens the cell by only still observed even when solar insolation was kept con- 109%. The qualitative agreement with Figs. 13 and 14 stant for both seasons. However, they speculated that an indicates that the simpliﬁcations to the simple radiation enhanced circulation at Ls 5 2708 occurred because the

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FIG. 13. Zonally averaged mass streamfunction simple MGCM results for Ls 5 908, radiative–convective forcing, and idealized zonal mean topography. Units are 108 kg s21. Positive flow is counterclockwise; negative contours are shaded. Contour interval is 15 3 108 kg s21. Latitude is plotted on the x axis, pressure is plotted on the left y axis, and log pressure height is plotted on the right y axis, assuming a scale height of 11 km and a reference pressure of 6 hPa. Only results below ;0.2 hPa (40-km log pressure height) are shown. Results are shown for (a) flat topography (as in Fig. 11b), and (b)–(d) idealized zonal mean topography of increasing steepness (b equal to 40, 20, and 10, respectively). The maximum magnitudes for (a)–(d) are 161, 106, 80, and 77 3 108 kg s21, respectively; thus, streamfunction strength decreases as the topographic slope’s steepness increases. rising branch is located in an area of lower pressure (i.e., cell (one near solstice, two near equinox), the maximum higher elevation) and did not attribute the difference to magnitude of the zonally averaged mass streamfunction the thermal affects of the slope. (above the boundary layer) is found. Then, at the same pressure level as each maximum, the latitude where the streamfunction falls to within 1% of its maximum value 5. Comparison of MGCM and modified Lindzen is located. The northernmost 1% latitude is f and the and Hou (1988) models 1 southernmost is f2. When there is one thermally direct Figure 16 shows the solution to the modified Lindzen cell, f1 is equal to whichever of f1 or f2 is in the and Hou (1988) model replotted so that f1, f1, and f2 summer hemisphere. When there are two thermally di- for each topography and radiative forcing are displayed rect cells, the average of the two innermost 1% latitudes together. It is now apparent that as Ls approaches 908 is taken to be f1. Interpreting the boundaries of the cells and 2708, f1 approaches the boundary of the summer from the simple MGCM results can be difficult because cell in the summer hemisphere. The physical implication in some cases the latitude of the boundary varies with is that the summer cell disappears, which is what is ob- height (e.g., at the boundary in the winter hemisphere at served in the simple MGCM results (Figs. 10 and 11). solstice). Also, particularly near equinox on the pole-

Also plotted in Fig. 16 are the values for f1, f1, and f2 ward sides of the cells, multiple 1% latitudes may exist determined from the MGCM results for the appropriate when closed areas of circulation in the opposite di- topographies, seasons, and radiative forcings, including rection appear within the thermally direct cell. In this some runs at seasons not discussed in section 4. case the innermost 1% latitude is used. Near the solstice, The boundaries of the Hadley cells in the simple MGCM the streamfunction in the summer hemisphere decreases results are computed as follows. For each thermally direct very gradually, which causes the value derived for the

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FIG. 14. As in Fig. 13, but for Ls 5 2708, radiative–convective forcing, and idealized zonal mean topography. The maximum magnitudes for panels (a)–(d) are 163, 144, 140, and 140 3 108 kg s21, respectively; thus, streamfunction

strength is independent of the topographic slope’s steepness, contrary to at Ls 5 908 (cf. Fig. 13). boundary to vary slightly depending on the cutoff per- When vertical diffusion of momentum was not included, centage used (e.g., 1% versus 0.1%). inertial instabilities developed in the tropics that made

From Fig. 16, the agreement of f1 between the simple it impossible to determine f1 near equinox. Adding a MGCM results and the modified Lindzen and Hou (1988) constant diffusion of 0.0005 Pa2 s21 everywhere resolved model is generally good. The solutions for the poleward f1 but induced a Ferrel cell. boundary of the summer cell agree for pure radiative Figure 17 shows a plot of the modified Lindzen and forcing, but the simple MGCM results are offset in the Hou (1988) model solutions as in Fig. 16 but with the equatorward direction for radiative–convective forcing. axisymmetric simple MGCM results instead of the full The solutions for the poleward boundary of the winter 3D simple MGCM. The discrepancy in the poleward cell do not agree. The simple MGCM results are offset boundary of the winter cell remains, possibly due to the equatorward, and they do not follow the same shape as Ferrel cell created by adding diffusion. In runs without the modified Lindzen and Hou (1988) model solution. diffusion, the poleward boundary of the winter cell ex- One possible explanation is that in the simple MGCM tends nearly to the winter pole, in agreement with the results the latitudinal boundary of the winter cell varies modified Lindzen and Hou (1988) results. with height, and in fact extends further poleward at To further assess the role of eddies and the mean higher altitudes. Another explanation is that the simple circulation in the 3D simple MGCM simulations, we MGCM contains eddies, whereas the Lindzen and Hou decomposed the momentum budget as follows. Ignoring (1988) model does not, on account of its axisymmetric the effects of the boundary layer (i.e., mountain torque nature. and friction) on the momentum budget, the total de- Since the modified Lindzen and Hou (1988) model is rivative of the absolute angular momentum given by (6) axisymmetric, it is perhaps more sensible to compare it may be set equal to zero, to an axisymmetric MGCM. We ran our simple MGCM ›M u ›M y ›M ›M in axisymmetric mode, where the cube–sphere geometry a 1 a 1 a 1 v a fa cosfy 5 0, was replaced by grid points evenly spaced at 2.88 of ›t a cosf ›l a ›f ›p latitude along a single strip of longitude of width 2.88. (33)

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FIG. 15. Zonally averaged mass streamfunction results from the GFDL MGCM, which uses a full radiation scheme. Units are 108 kg s21. Only temperatures below ;0.2 hPa (40 km) are shown. The top row is for ﬂat topography; the

bottom is for idealized zonal mean topography (b 5 20). The left column is Ls 5 908 and the right is Ls 5 2708. Positive ﬂow is counterclockwise; negative contours are shaded. Latitude is plotted on the x axis, pressure is plotted on the left y axis, and height is plotted on the right y axis. Contour interval is 15 3 108 kg s21. The maximum magnitudes for panels (a)–(d) are 117, 121, 65, and 131 3 108 kg s21, respectively. where l is longitude, v is the vertical velocity in pressure spatial eddies. We found that the second term was much coordinates, and Ma 5 a cosfu is the relative angular smaller than the other two; thus, it was neglected. Sim- momentum. Averaging zonally and in time and making ilarly, (35) may be written for the vertical transport of use of the continuity equation, angular momentum, where y is replaced by v.Thever- tical transport by the transient and spatial eddies is small, 1 › › leaving (cosf[ yM ]) 1 ([ vM ]) 5 fa cosf[y], a cosf ›f a ›p a (34) 1 › 1 › (a cos2f[u*y*]) 1 (a cos2f[u][y]) a cosf ›f a cosf ›f where the square brackets indicate a zonal average and › the overbar indicates a time average. The term [ yM ] 1 (a cosf[u][v]) 5 fa cosf[y], (36) a › may be further written as p

9 9 [ yMa ] 5 a cosf([u][y] 1 [u] [y] 1 [u*y*]), (35) where the term on the right-hand side is the planetary term. where primes indicate a departure from the time aver- Figure 18 shows the three terms in (36) calculated from age and asterisks indicate a departure from the zonal the simple MGCM results with radiative–convective average. The ﬁrst term on the right-hand side of (35) is forcing and full topography at Ls 5 08,908, and 2708. the meridional transport of angular momentum by the Within the Hadley cells, the planetary term is balanced steady mean circulation, the second is the transport by by the meridional transport by the steady mean circu- the transient circulation, and the third is the transport by lation during the solstices. At equinox, all three terms

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FIG. 16. Solutions to (16)–(19): (a) the solution for pure radiative forcing with ﬂat topography; (b) as in (a), but with zonal mean MOLA topography; (c) radiative–convective forcing with ﬂat topography; and (d) as in (c), but with

zonal mean MOLA topography. In each panel, the dashed line is f1, the solid is f1, and the dotted is f2. Also plotted are the boundaries of cells obtained from the simple MGCM results. The triangles are f1, the squares are f1, and the circles are f2.AsLs approaches 908 and 2708, f1 and the poleward boundary of the summer cell becomes indistinguishable, and the value of f1 is not plotted. are small within the Hadley cells. Outside the Hadley 6. Discussion and conclusions cells, the meridional transport by spatial eddies nearly balances the planetary term. The presence of eddies We have continued the investigation of the effect near the poleward boundary of the Hadley cell, together of a latitudinally varying slope on the Martian Hadley with the fact that the axisymmetric simple MGCM runs cells initiated in detail by Richardson and Wilson (2002) without diffusion (i.e., no spatial eddies), shows better and Takahashi et al. (2003). The problem was approached agreement with the Lindzen and Hou (1988) model and by revising the Hadley cell model of Lindzen and Hou suggests that by driving the extratropical Ferrel cell, (1988) to include topography, which allowed us to solve eddies may be important in determining the poleward for the latitudinal boundaries of the cells. This theory boundaries of the Hadley cells at equinox and in the was compared with the results from a simple MGCM. winter hemisphere at the solstices. Both models were thermally forced using Newtonian Walker and Schneider (2006) showed in idealized relaxation to one of two equilibrium temperature states: GCM simulations that the strength of the Hadley cell is one that included the effects of convection and one related to the eddy momentum ﬂux divergence. In Fig. that did not. In the case of the latter, almost no depen- 18, the spatial eddy term is greater in the northern dence on the topography was observed in either model.

(winter) hemisphere at Ls 5 2708 than in the southern When convective forcing was allowed, both models (winter) hemisphere at Ls 5 908. This effect is also seen predicted that the latitude of the dividing streamline is our simple MGCM simulations with ﬂat topography. shifted upslope at equinox. The effect of the slope on the Eddies may also be important in addition to the topog- radiative–convective equilibrium temperature aloft was raphy in determining the seasonal asymmetry in Hadley found to dominate over the geometrical effects of the cell strength. slope.

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FIG. 17. As in Fig. 16, but the results for the axisymmetric simple MGCM are shown instead of the 3D simple MGCM.

From the simple MGCM results, an analysis of the the Hadley cells. Eddies are stronger during north- strengths of the Hadley cells was carried out. At equinox ern winter (Ls 5 270) than southern winter (Ls 5 908). with radiative–convective forcing, the presence of the Eddies may play a role in determining the strengths and slope strengthens the southern cell and weakens the poleward boundaries of the Hadley cell, as also sug- northern cell as the latitude of maximum equilibrium gested by Walker and Schneider (2006) in idealized temperature moves upslope. At Ls 5 908 with radiative– GCM runs. convective forcing, the cross-equatorial cell is weakened Molnar and Emanuel (1999) considered the radiative– with sloping topography; at Ls 5 2708, the strength of the convective equilibrium case when absorbers, namely wa- cell is independent of slope. Further experiments with ter vapor, are present within the atmosphere. In the limit idealized zonal mean topography showed that as the in which the atmosphere is opaque to solar radiation, the steepness of the slope is increased, the strength of the temperature at any level aloft becomes independent of cell decreases at Ls 5 908. the height of the surface. The Molnar and Emanuel A discrepancy arose between our simple MGCM and (1999) study may be applicable to periods when the the modified Lindzen and Hou (1988) model results Martian atmosphere is experiencing very dusty condi- regarding the latitude of the poleward boundary of the tions (Viking Lander 1 measured optical depths in ex- Hadley cell in the winter hemisphere around the solstice cess of 4 during a dust storm; Colburn et al. 1989), since seasons. This difference could be caused by eddies, atmospheric dust is a strong absorber. Basu et al. (2006) since the simple 3D MGCM allows for their presence showed with the GFDL MGCM that as dust optical but the Lindzen and Hou (1988) model does not. Axi- depth increases, the ratio of the streamfunction maxi- symmetric simple MGCM runs without diffusion and mum at Ls 5 2708 to Ls 5 908 decreases, which implies hence without eddies produced better agreement with that dust inhibits the effect of the slope. It would be the modified Lindzen and Hou (1988) model. A break- interesting to use the modified Lindzen and Hou (1988) down of the momentum budget showed that the spatial model to study the effects of increased dust on the Hadley eddies are relatively weak within the Hadley cells at cell boundaries, provided the appropriate changes could the solstices but that eddies are relatively strong outside be made to our radiation scheme.

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2 22 FIG. 18. Simple MGCM results with radiative–convective forcing for the momentum budget derived in (36). Units are m s .Latitudeis plotted on the x axis, pressure is plotted on the left y axis, and log pressure height is plotted on the right y axis, assuming a scale height of 11 km and a reference pressure of 6 hPa. Only results below ;0.2 hPa (40 km log pressure height) are shown. From left to right, the columns are Ls 5 08,908, and 2708. From top to bottom, the rows are the meridional transport by spatial eddies [ﬁrst term on the lhs of (36)], combined meridional and vertical transport by the steady mean circulation [second and third terms on the lhs of (36)], and the planetary term [rhs of (36)]. Negative contours are shaded. Note the different contour intervals (100 m2 s22 for the top row and 200 m2 s22 otherwise).

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