14( 

NEW YBRK UNIVERSITY 

DEPARTMENT OF APPUEO SCIENCE 

FACULTY OF ARTS AND SCIENCE 
FINAL REPORT 

THE MARTIAN CLIMATE & ENERGY BALANCE MODELS 

WITH C02/H20 ATMOSPHERES 

Martin 1. Hoffert 

Principal Investigator 

NY U/86- 1 54 

NASA GRANT NAGW-573 

March 1 1984 - March 31, 1986 

New York University 

Washington Square 

New York, N.Y. 10003 

I .. 

NEW YORK UNIVERSITY 

FACULTY OF ARTS AND SCIENCE 

DEPARTMENT OF APPLIED SCIENCE 
1 

SEQSONHL CYCLtS OF CO2 HND HIMOSPHERIC CIRCULATION ON MRRS: 

RNENERGYBRLRNCERPPRORCH 

Martin I. Hoffert, Vrjay K. Narayanan and Uarren H. Ziegler 

bY 

RBSTRRCT 

The analysis begins with a seasonal energy balance model (EBMI for 

Mars. This is used to compute surface temperature versus x = 

sin(latitude1 and time over the seasonal cycle. The core model also 

computes the evolving boundaries of the CO2 icecaps, net 

sublimational/condensation rates, and the resulting seasonal pressure 

wave. Model results are compared with surface temperature and 

pressure history data at Viking lander sites, indicating fairly good 

agreement when meridional heat transport is represented by a thermal 

diffusion coefficient D - 0.015 W m-2 K-l. Condensational wind 

distributions are also computed. 

Rn analytic model of Martian wind circulation is then proposed, as an 

extension of the EBM, which incorporates vertical wind profiles 

containing an x-dependent function evaluated by substitution in the 

equation defining the diffusion coefficient. This leads to a 

parameterization of D(x] and of the meridional circulation which 

recovers the high surface winds predicted by dynamic Mars 

atmosphere models (- 10 m s-l]. Peak diffusion coefficients, D - 0.6 W 

m-2 K-l, are found over strong Hadley zones - some 40 times larger 

than those of high-Latitude baroclinic eddies. When the wind 

parameterization is used to find streamline patterns over Martian 

seasons, the resulting picture shows overturning hemispheric Hadley 

cells crossing the equator during solstices, and attaining peak 

intensities during the south summer dust storm season, while 

condensational winds are most important near the polar caps. 
INTRODUCTION 

Data from Viking Mars orbiters and landers continues to provide a 

picture of a sister planet marked by profound seasonal variations in 

surface temperature, volatile gas exchange and occasional dust 

storms [Tillman, 19851. Some of the more dramatic events of Mars 

seasonal cycle are evoked in this descripitive passage [Haberle, 1986a): 

Fall has come to the northern hemisphere of Mars. In the mid-latitudes the 

mean temperature is dropping past - 70 degrees Celsius; at the north pole it has 

already reached - 123 degrees, cold enough to freeze carbon dioxide, the 

principal constituent of the thin Martian atmosphere. F1 cap of dry ice is now 

forming on the pole. Before winter ends ice will reach the 50th parallel. 

Meanwhile in the southern hemisphere winter has ended, and carbon dioxide is 

evaporating at the south pole. Along the edge of the retreating polar cap a 

sharp temperature contrast between ice and soil warmed by the spring sun is 

giving rise to strong winds. During the short but hot southern summer, as Mars 

makes its closest approach to the sun, the winds will lift dust off the surface in 

great swirling storms. Ultimately dust may envelop the entire planet. 

Much of the prior Viking data analysis, as well a current planning of 

future Mars science missions, is concerned with understanding 

meteorological seasonal cycles from observations by spacecraft 

orbiters and landers. Important related questions are whether and 

how such cycles have varied over Mars geological history, and the 

role they play in volatile gas exchange. 

Models at different complexity levels are instructive in such studies. 

For example, energy balance models [EBMs) with atmospheric transport 

neglected, or schematicallg depicted by thermal diffusion coefficents, 

have proven usefui in Understanding surface tnermai physics iieignton 

and Murray, 1966; Murray and Malin, 1973; Kieffer et al,, 1977; Toona 

&, 1980; Hoffert et aL, 1981; James and North, 1982; James, 1985), 
3 

whereas more computer intensive three-aimensionai general 

circulation models (GCMsl were historically used to model seasonal 

wind patterns. [Pollack et a1, 1976, 1981; Haberle et al,, 19791. f? 

simplification based on the nearlg zonally-symmetric Mars circulation 

was exploited by Haberle et & (19821, who used a two-dimensional 

dynamic circulation model, including dust storm effects; but extensive 

computations were still required. 

In this paper, simplified dynamics is taken a step further. Employing an 

integral method with assumed vertical atmospheric profiles, only the 

num eri ca 1 so lu t i on of the i nsola t i on-f orced, on e-di m ensi ona 1 uns t e a d y 

heat conduction equation is required. This allows circulation patterns 

for different seasons and orbital parameters to be rapidly assessed. 

The analysis begins with an energy balance approach to derive a core 

numerical model for surface temperature and carbon dioxide evolution 

over Mars seasonal cycle. Seasonally-varying temperature and 

pressure wave data from Viking landers are compared with the core 

model predictions to establish the realism of this part of the model. 

Next, new parameterizations of thermally driven and condensation 

winds, and baroclinic eddy transport effects, are developed. R test of 

our simple circulation model's surface velocity against existing GCM 

runs at north summer is presented. Finally, our wind model is employed 

to estimate the circulation streamline patterns over the four seasons 

on Mars. 

SURFACE ENERGY BALRNCE 

The core seasonal cycle computer model for Ts[x,t), where x is the 

sine of the latitude and t is the time, is conceptually similar to that of 

dames and North ii982j: it is derived from a zonaiiy-averaged energy 

balance in the spirit of the original Leighton and Murray [19661 model 

for seasonal volatile evolution on Mars, but includes the effect of 
4 

horizontal atmospheric transport on this balance. The core program 

yields as by-products the time-varying boundaries of north and south 

CO2 icecaps and the rates of mass fluxing to and from the atmosphere 

from phase changes over the seasonal caps. Model parameters 

adopted here are summarized in Table 1. 

The basic energy balance involves four types of zonally-averaged 

vertical energy flux per unit surface area in a ground/atmosphere 

column: 

where Hsto[x,t) is an energy sink by column heat storage, Hsub[x,t) is 

an energy sink to sublimation of an underlying CO2 frost, Hrad[X,t) is a 

net [sotar minus longwave) radiant energy source and Hhor[X,t] is an 

energy source from horizontal atmospheric transport. 

Thermal Storaae and Sublimation Hefit Sinks 

Heat is stored during the evolving seasonal cycle by the thermal 

inertia of the atmosphere and land. The atmosphere mixes fast enough 

relative to seasonal timescales for the entire column heat capacity to 

come into play. Its heat capacity per unit area is therefore Ca = pscph 

= 0.16 MJ m-2 K-l, from Table 1 values. 

- - 

The effective heat capacity of the underlying ground depends on how 

deeply the seasonal wave penetrates. with p9, c9, and A the ground 

density, specific heat per unit mass, and thermal conductivity, an 

annual heating wave of circular frequencij 

rad-s-’ produces a thermal wave penetrating to depth d 2 

[2A/[p9cgSIl1/2 The corresponding ground heat capacity per unit area 

is [Hoffert and Storch, 1979) 

= 2r;kp 2 1.06 x 
5 

.l I3 1 I3 where i = iPgcghJi' = iCv2ji' LL9 IS the ground thermal inertia. Notice 

that Cg scales with W1/2, whereas I is a property of surface materials 

only. f3 typical ground thermal inertia on Mars is (Pollack et aL., 1981) I z 

300 J m-2 K-l s-ll2, corresponding a heat capacity per unit area of Cg = 

1.30 MJ m-2 K-l for the seasonal wave. 

The total energy sink from column heat storage is therefore 

where C = Ca + C9 2 1.46 MJ m-2 K-l is the heat capacity per unit area of 

the atmosphere plus the thermallg interactive ground column. 

R unique feature of Mars surface physics over solid CO2 icecaps is 

sublimation to the gas phase of the main atmospheric constituent when 

T, > Tc Ips]. Conversely, the atmosphere condenses to form CO2 

icecaps when Ts < Tc Ips]. The pressure-dependence of the frost point 

temperature is approximately Tc[ps] = - BL/ln[ps/p*l, where p* z 9.74 

x 10l1 Pa is a pressure scale, and @L 2 3148 K is a temperature scale 

characterizing the latent heat of sublimation: L = R@L == 5.95 x lo5 J kg-l 

(James and North, 1982). Because of the weak logarithmic dependence, 

as surface pressure ps varies seasonally over its annual range 

(600-800 Pa, at the reference ellipsoid elevation), Tc varies over a 

relatively much smaller range (149-151 K]. For simplicity, a constant Tc 

= 150 K is assumed over carbon dioxide icecaps. The heat sink to 

sublimation phase change over CO2 icecaps is 

where rti(x,t] is the mass per unit time and per unit area injected to 

[removed from) the atmosphere bg phase changes at the surface. This 

term is nonzero only over CO2 ice when Ts I T,. 

Ra d I at I on Forti ng 

Seasonal climate cycles are driven by an insolation function 34, Ls], 

.. 
6 

where d> is the Latitude ana is ine neiiocentric iongituae of the sun - 

the angular displacement along Mars’ elliptic orbit measured from 

northern hemisphere spring equinox, Ls 0. The eccenticity e, obliquity 

i and heliocentric longitude of perihelion Lsp are parameters of the 

seasonal cycle, and are available for present epoch from Table 1. The 

diurnally- averaged incident solar radiation per unit area is given by 

[Berger, 19781: 

s[$ ,Ls) = {SOD + ecos[Ls-Lsp~)2~Hsin+sin~ + cos~cos~sinH1l/~~~l-e2]21, 

where S[Ls] is the solar declination angle given by sin8 = sini sinLs and 

H[$,Lsl is the hour angle for sunrise given bg cosH = - tanbtans; Here H = 

0 and H = -rf correspond to polar night and polar day conditions 

respectively . 

Because Mars accelerates along its orbit near perihelion, and 

decelerates near apehelion, Ls is nonlinearly related to elapsed time t 

from some reference point on the orbit. Neglecting terms of order e2, 

the time from northern spring equinox is related to Ls approximately 

by 

t (L,I == ‘cP/[2-rf1{ L, - 2e[sin[Ls - Lspl + sinLspll. 

It is also convenient to work with a stretched latitude variable 

proportional to the area of the planetary surface from the equator, 

x[+l= sin+. 

These relations,define S[x, tl for specified e, i and Lsp. 

A vertical column containing the atmosphere and thermally interactive 

ground is heated by a net radiative flux Hrad equal to the absorbed 
7 

soiar fiux Less the iongwave raaiation flux lost to space. This term is 

modeled by 

where a 2 5.67 x W m-* K-4 is the Stefan-Boltzmann constant and 

are planetary albedoes and emissivities, with different values over 

bare ground and ice. FI CO2 ice albedo of 2 0.8 is assumed over both 

caps in the program. This is a rough average of north and south cap 

CO2 albedos inferred from Viking data by Paige and Ingersoll [1985, Fig. 

51. FI corresponding ice emissivity of = 0.58 is needed to recover Tc 

zz 150 K at the poles. 

J-lori7ontal Heat TransDort 

Horizontal [meridional) energy transport arises when winds blow fluid 

parcels containing enthalpy and gravitational potential energy across 

latitude circles. The meridional energy flux per unit area is pv[cpT + 

gz], where p, v and T are the density, meridional velocity and 

temperature; cp is the constant-pressure specific heat, 9 the 

gravitational acceleration and z the altitude. The differential 

cross-sectional area perpendicular to the meridional energy flux can 

be written a[l - ~~]~/2dzdA, where A is the longitude and x = sin+. 

Rccordingly, a general, three-dimensional, atmospheric circulation 

creates an energy flow per unit time across a latitude circle of: 

Jo Jo 

Since a differential surface area element is dA, = 2na2dx, the net 

energy source per unit surface area into a column of surface area dA, 
8 

where 

is the horizontal diffusivity coefficient for meridional energy flux. 

A constant D - 0.015 w m- 2 K- 1' independent of season or latitude was 

assumed in our initial numerical calculations. As discussed below, this 

value gave reasonably realistic surface temperatures and seasonal 

pressure wave distributions in comparison with Vi king lander data. 

However, equations [5) and (7) are crucial links between any known 

circulation field and the thermal diffusivity. They will be employed 

subsequently to parametrize D[x] in terms of Mars wind circulation 

fields. 

flu m erica 1 Solution 

At constant D, substituting equations (21, (31, (4) and (6) into equation (1) 

gives an unsteadg "heat conduction on a sphere" partial differential 

equa t i on: 

Wiere n In 4n .nQ -, 4n T = L/U - I.U X IUW b - i.o?sP is a timescaie for norizonta 

diffusive heat transport. Numerical values are for D - 0.015 W m-2 K-! 
9 

Equation i8j is soivea numer-icaiig for Tsix, tj bg a Crsnk-i\iichoison 

finite-difference scheme with aTs/ax and a2Ts/ax2 approximated by 

centered differences at interior points, and one-side differences at the 

poles, where all space derivatives are evaluated at mid-timestep. The 

implicit Crank-Nicholson PDE solver employs computationally 

economical tri-diagonal matrices, is unconditionally stable, and has a 

second-order in space accuracy depending on p = At/[~Ax~l [Ferziger, 

1981, p. 1481. Normally, we want p L 1, corresponding to timesteps At < 

[rAx2]; that is, the number of timesteps per Mars year needed for a 

given level of accuracy increases with the square of the horizontal 

resolution. we use a grid of 40 points in x = sin4 [Ax = 0.05) as a 

compromise between small Ax desired to resolve icecap boundaries, 

and large Ax desired to minimize timesteps/year, and thus computing 

time, at given accuracy. To insure accuracy 1000 steps/year [At 

=IX~O-~T~] are used corresponding to p - 0.2. A D-coefficient 50 times 

larger would give p - 10. Some 10,000 steps/year at the same Ax are 

would be necessary to get p - 1 for an accurate solution. 

The program is initialized with a uniform annual mean temperature 

distribution and "spun up" for several Mars years to eliminate starting 

transients. It employs the following procedure to compute icecap 

boundaries and mass fluxes from sublimation/condensation over the 

caps: Normally hm = 0 in equation (61 whenever the computed Ts[x,tl > 

Tc. when the computed surface temperature droped below the frost 

point, the program sets Ts[x,t] = Tc. The time-dependent cap 

boundaries x~(t] and xs[t] are evaluated as the gridpoints in the 

northern and southern hemisphere, respectively, where equation (7) 

predicts Ts = Tc. Since space and time derivatives of temperature are 

small over the constant-temperature caps, the sublimation %ass 

source is simplg: 

, 

Balancing net radiant energy against phase change energy is an 

approximation dating back to the Leighton and Murray (19661 model. It 
10 

does seem justified however in Light of detaiiea stuaies of Mars 202 

cap energy budgets by Paige and Ingersoll (19853, who found small 

contributions by transport and storage. Rates of mass injection to the 

atmosphere by south and north cap sublimation are computed by 

integrating the mass fluxes per unit area over the respective cap 

areas, 

x,(tl +1 

P@tl = 2rrazJ k[x,t]dx, 

where negative values indicate mass removal by condensation. 

P?~[tl= 2rra2$ r~i[x,tldx, 

-1 xN(t1 

SERSONRL MODEL RESULTS RND VIKING DRTR 

Figure 1 shows the seasonal variation of pole-to-pole surface 

temperature computed from the energy balance model for Table 1 

parameter values. The results are expressed as constant surface 

temperature contours at ATs = 5 K intervals on the [x, Lsl plane. Rlso 

shown is latitude as a nonlinear vertical scale, 4 = $(XI, and time in 

sols from N. Spring equinox as a nonlinear horizontal scale, t = t[Ls]. 

The hemispheric asymmetry over the seasonal cycle is a consequence 

of maximum orbital distance from the sun [apehelion] and minimum 

orbital distance [perihelion] occuring presently at Ls = 70° and 250°, 

respectively. These are sufficently close to the southern winter and 

summer solstices to markedly reenforce the seasonal thermal contrast 

in the southern hemisphere and diminish it in the northern hemisphere. 

The coldest spots on the planet are the CO2 icecaps at = 150 K; the 

warmest is the southern hemisphere hot spot at = 240 K, extending as 

far south as + s 50 OS during southern spring and summer. Figure 1 

shows the computed seasonai bounciai-ies of the north and south 

carbon dioxide icecaps x$] and xs[t], assumed to form at the 

sublimation frost point temperature, Tc =j 150 K (The "wiggles" in the 
receeding cap boundaries are artifacts of the finite Ax resoiution of 

the model]. The CO2 caps are asymmetric with the southern cap 

extending more equatorward than the north during their respective 

winter maximums. 

The model predictions of seasonal temperature histories at the 

appropriate latitudes can be directly compared with observations of 

surface air temperature made by VL-1 and VL-2 landers during the 

Viking mission. The latitudes of VL-1 (22 ON - now the Thomas Mutch 

Memorial Station] and VL-2 (48 ON1 are indicated in Figure 1. Viking 

lander barometric pressure data at these sites exhibits the well-known 

seasonal CO2 pressure waves associated with icecap cycles. The 

existence of condensation/sublimation pressure waves on Mars was 

predicted a decade before Viking by Leighton and Murray (19661, who 

assumed an energy balance model with no atmospheric transport. 

I ander Temy>erRtures 

Figure 2 compares Mars surface temperatures measured at VL-1 and 

VL-2 over the seasonal cycle (Seiff, 19821 with predictions of the 

present energy balance model at the lander latitudes. 

The rapid temperature jumps computed during northern spring and 

summer near the VL-2 site are generated by abrupt sublimations of the 

receeding north CO2 polar cap over one spatial step (Ax = 0.051, which 

propagate south by atmospheric diffusion (Cf. Figure 11. These appear 

to be artifacts of the finite-difference resolution, and converge to a 

smooth curve as Ax --+ 0. UJith this minor exception, the agreement 

betiijeen the model and the lander observations in both phase and 

amplitude for the seasonal thermal waves is remarkably good at both 

sites. 

- 

f3 more formidable challenge to the model is its ability to reproduce 
12 

the conaensation/suDlimation seasonal pressure wave observed by 

the Vi king landers. Tillman (19851 has observed that interannual 

variations in the pressure wave at a given lander location are small. 

However, even after filtering out short-period oscillations associated 

with baroclinic "weather" disturbances, the VL-2 pressure wave of any 

given year is systematically higher than that at VL-1 [Seiff, 19821. This is 

mainly due to topographic differences at the two sites [VL-1 and VL-2 

are at elevations z1= -1.5 km and z2 = -2.5 km, respectively, relative to 

the Mars reference ellipsoidl. 

We therefore corrected the lander pressures pj[tl in Seiff (1982) to 

Mars' reference ellipsoid elevation by using the hydrostatic equation: 

~sj[tl =pj[tle~p[+Zj/hjItIl, where j = 1, 2. The effects of variations in 

scale height hj[tI = RTsj[t1/9 from different surface temperature 

histories Ts$tI at the two sites were included in the correction. Since 

the Zj'S are negative, surface pressures are lower than lander 

pressures in both cases, with a larger pressure correction for VL-2 [in a 

deeper valley]. The result, shown in Figure 3, is that the VL-1 and VL-2 

pressure curves become virtually indistinguishable when these 

corrections are made. Rlso shown as an alternate vertical scale in 

Figure 3 is the atmospheric column mass loading at the "surface" 

reference ellipsoid elevation, m = ps/g. 

To model the seasonal pressure uave, we assumed mass added by 

phase changes over both caps from equations [lo] is deposited in the 

atmosphere between the cap boundaries. Pressures over the caps are 

assumed to be maintained constant at the condensation frost point 

temperature Tc. The rate of change of atmospheric mass between the 

caps is then 

r'i~[t] 5 2fia2[XN - Xs)- pdz = r?~[tl + ~~[tl, [1 11 

at a Iff 

independent of x. For the special case where I'~N = - ~S the net flux 

into the system is zero; otherwise there is an increase or decrease of 

atmospheric mass. The surface pressure is assumed to adjust 
13 

"inslanlaneousiy" PLanetWiae to mass sources, and is found by 

numerical integration of 

The results of this calculation are shown in Figure 3, where they can 

be directly compared with the lander-observed pressure wave 

corrected to the reference surface elevation. The pressure peaks 

result from more rapid degassing by the subliming pole than gas 

absorption by the condensing pole. The stronger peak in southern 

summer a consequence of hemispherically asymmetric heating 

augmented by atmospheric transport. Indeed, a feature supporting the 

adopted D - 0.015 UJ m-2 K1, at least at high latitudes, is its recovery 

of the observed higher amplitude south summer pressure peak. 

James and North (1982) did not obtain the observed relative amplitudes 

of the seasonal pressure peaks with their "clear sky" diffususion 

found when diffusivity was increased in their model to values as high 

as D - 0.1 UJ m-2 K1 during 1977 global dust storms. Their inference is 

that D correlates with atmospheric opacity, increasing by orders of 

magnitude during large optical depth global dust storms. Such storms 

occurred in south summer of the first Viking year [Figure 21, precisely 

when high atmospheric heat transport is needed to rapidly sublime the 

south cap. But their correlation with optical depth may be fortuitous, 

since subsequent years when global dust storms did not recur exhibit 

essentiallg the same relative winter and summer peaks in the pressure 

wave [Tillman, 1985). By chosing a constant D - 0.015 W m-2 K1 - 

between James and North's "clear sky" and "optically thick" values - 

we get a fairly realistic pressure curve without invoking the dust 

storm connection. Even better agreement with the data, based on 

tuning the relevant paramers within observational constraints, seems 

possible, but uas not pursued. 

coefficient D - 0.004 w m- 21 K-, although acceptable agreement was 
14 

While the surface temperature curves at Viking lander latitudes are 

well represented, they may be too insensitive to transport effects to 

calibrate D. The seasonal pressure wave on the other hand seems 

mainly influenced by heat transport in the immediate vicinity of the cap 

boundaries. Clearly, it would be illuminating to if the thermal diffusion 

coefficient over the planet as a whole could be modeled in terms of 

dynamic atmospheric processes. This is the subject of the balance of 

this paper. 

MERIDIONAL CIRCULATION 

Unlike the earth's atmosphere for which extensive observational 

statistics on wind patterns are available [Oort, 19831, Mars atmospheric 

circulation is presently understood primarily from a few GCM 

simulations over restricted Ls ranges. Existing lander anemometer 

measurements [Tillman, 19851, surface wind streak patterns [Kahn, 

19831, and dust storm observations [Haberle, 1986bl provide only 

tantalizing hints of the realism of these simulations. At this point 

therefore, simplified, partly analytical, models are useful tools for 

- gaining insight into atmospheric circulation processes. 

R reasonably good approximation is that the time for the atmosphere 

to radiatively equilibrate with the seasonally evolving surface 

temperature is sufficient short - about 10 sols according to Pollack 

aG, (1981) - such that the circulation during the seasonal cycle of 

duration 670 sols is QuaSi-Sieady. The time, t, is then a parameter, so 

time-dependences of flow variables are omitted below for simplicity. 

It is also useful to distinguish between zonally-symmetric components 

of the circuiation; ana those which vary expiicitiy with the Longitude, 

A. Rny flow variable can be decomposed into such zonally-symmetric 

and longitudinally-varying components. Velocity and temperature, for 
15 

exampie, can be written: vix,zj + v'ix,z,Aj and Tix,zj + Tiix,z,Aj, where 

primes are departures from the zonal mean. Zonal averages around 

latitude circles of v' and T' individually vanish identically; but the v'T' 

product average [the meridional eddy flux) does not. On Mars, it is 

useful to further decompose v[x, zl into components from 

condensational winds, and thermally-driven cellular overturning in the 

meridional plane, 

For simplicity, a vertically uniform distribution of condensational wind 

v,[x] is assumed. The thermally-driven component VT[X,Z), reverses 

direction at higher altitudes forming closed Hadley-like cells, and is 

therefore altitude-dependent. It has the property JpvTdz = 0. Both 

components transport energy across latitude circles, but only the 

condensational wind transports mass. 

Let meridional diffusion effects from zonally-symmetric processes be 

represented bg a mean-flow diffusion coefficient Dm , those from 

eddies or waves in the east-west direction by an eddy diffusivity De. 

[Logically, Dm too can be considered an "eddy diffusivity" associated 

with the mean flow overturning eddies of the Hadley circulation in the 

x,z plane, but no confusion should result if the terms are understood.) 

It follows from equations [51 and (71 that the diffusion coefficient is 

expressible 

where 
The eddy component from unstable baroclinic waves can be estimated 

from Stone's [19721 baroclinic instability parameterization. Its value for 

the planet as a whole is approximately [Hoffert et al,, 19801 

- 

0.28R2psCpTs1/2[rad - r11'2 - Tcl 

IT 93/2a3f2 

De 

where the numerical estimate corresponds to Table 1 parameter 

diffusion coefficient value used in the core model [D - 0.015 W m-2 K-ll, 

and may characteristizes transport near the cap boundaries where 

baroclinic activity dominates [Hablerle, 1986bl. This may be why the 

seasonal pressure wave was modeled fairly well. 

- 0.018 W m-2 K-l, 

values and a Coriolis parameter f = 1.4 x 10- 41 s-. It is close to the 

In contrast to the terrestrial circulation, Martian winds are apparently 

dominated at low latitudes by the zonally-symmetric circulation 

[Haberle et &, 1982). UJe focus next on these components of the 

circulation, for which no dynamically-lin ked diffusion para met riza t ion 

has been developed as yet. 

Condenwtional wiM 

The mass flowing across a latitude circle carried by the condensational 

wind is: 

t$-$xl= 2na[l-x211/2 pvcdz. [161 

This quantity appears in the unsteady continuity equation integrated 

from the surface to the top of the atmosphere: 

JO Im 
17 

Substituting equations (111 and (161 into [17) yields an expression for the 

x-wise derivative of the condensational flow, afl,/ax = fl~/[xs - XN). 

LUe therefore use the linear distribution, 

which has the proper derivative, and satisfies the cap boundary 

conditions flc(x~] = - fl~ and fl,[xs) = fls. 

The condensational velocity distribution is obtained by integrating 

I equation [16) with the density profile p[z) sz pse-z/h, where h = ps/psg l 

I is the scale height: 

Condensational winds versus latitude at the equinoxes and solstices 

computed from equations (18) and [19) using the sources fl~ and fls 

from the core program are illustrated in Figure 4. The peak 

condensational wind magnitude, vc 5 0.4 m s-l, occurs near the south 

cap boundary in northern winter [Ls = 270°) as this cap sublimes under 

the influence of high southern hemisphere insolation and heat transfer 

to the cap boundary. This wind is comparable in magnitude and location 

to peak values computed by James (1985, Fig. 31. Rt this Ls, the 

condensational flow is everywhere northward, depositing mass in the 

atmosphere along the way, and contributing to the northern winter 

peak of Fig. 3, unrii what remains condenses at the north poiar cap. An 

opposite but weaker flow exists in northern summer [Ls = goo), when 

southward flow generated by north cap sublimation is absorbed by the 

I 
18 

south cdp. if subiirrisliori and coridensation rates were preciseiy equal 

at opposing caps, condensational winds would still blow, but seasonal 

pressure waves from this effect uould be absent. These winds are 

weaker at the equinoxes, as expected, but at Ls = 180°, vc reverses 

direction mid-planet to accomadate the model prediction of 

condensation at both caps during this time. 

The fact that average condensational winds are of order vc 0.2 m s-l 

raises an interesting question. The transit time for a gas parcel 

sublimed at one pole to cross the planet to is na/vc - 5.3~107 s- 600 

sols, so the time to physically transport condensational mass between 

VL-1 and VL-2 lander locations is [[48-22)/1801 x 600 sols - 87 sols. Why 

then is there no phase lag of this order in the pressure waves 

observed the two stations? Recall that both lander pressure curves, 

corrected to the reference ellipsoid and with baroclinic weather 

oscillations removed, are nearly coincident. One possibility is that the 

atmosphere rapidly adjusts on the diurnal timescale of internal 

gravity waves to create horizontally uniform surface pressures. The 

observational absense of a discernable phase lag seems in any event 

a sufficient rationale for the instantaneous pressure adjustment 

1 

assumed in equation (12). 

The zonally-symmetric thermally-driven circulation is composed of a 

mean merdional circulation, in which the merdional v(x, zl and vertical 

w(x,z) velocities are coupled by continuity, and a zonal mean wind 

u[x,z) and temperature T(x,z), which are coupled through the thermal 

wind relation. These components are in turn coupled through the 

Coriolis torque. Advection of relative angular momentum in inviscid 

models is also necessary to describe thermal forcing of the mean 

meridional circulation. 

For nearly inviscid models, the Hadley cell is limited in its latitudinal 

domain, which in turn limits the intensity of the zonal flow. A major 
19 

difference between terrestrial and Martian zonal mean flows is the 

the relative absense of eddy momentum fluxes on Mars, which makes 

the circulation nearly zonally symmetric over t 30° latitudes [Leovy 

et &, 1973). For example, a two-dimensional zonally symmetric 

circulation model, such as that employed by Haberle et al,, 1982) to 

study Mars dust storms, would probably not describe the earth's 

atmospheric circulation very well. Rs a consequence of the earth's 

land-sea contrasts eddy momentum fluxes are important at all 

latitudes, and need to be specified from observations in 

zonally-averaged models to recover a realistic circulation 

[MacCracken and Ghan, 19861. 

To model Dm an integral method was used in which a meridional wind 

shear of the form av/az = - G[x)/h is assumed. This is formally 

analogous to the zonal wind shear of the thermal wind equation, 

au/az = fi[xl/h, where XI = [gh/afTs)aTs/a$ [Washington and 

Parkinson, 1986). Rt this point is an unknown function to be 

determined in terms of Dm[X). However, it is well-known that latitudinal 

heating gradients drive the zonally-symmetric [u,v] circulation in 

general [Holton, 1979; Haberle, 1986b). In particular, it follows from 

equation (15) of Leovy et ~1, [19731 that for steady, inviscid flows in 

equatorial regions, and neglecting eddy momentum fluxes, Q[x) - - 

fr[x). We therefore assume as a crude approximation that 

in latitude bands of the Hadley circulation. 

The linear profile VT[X,Z) = - G[x)[i - [z/h)l also satisfies zero mass flux 

across latitude circles, Jpv-ydz = o for p = pse -z/h, and produces a 

thermally driven surface wind flowing toward local hot spots [vs = 

VT[X,O) oc t3Ts/t3x). This wind reverses direction above the scale height 

corresponding to overturning Hadley Cells, but needs to be carefully 

interpreted at high altitudes where unrealistically large negative 

velocities are implied as z + 00. The linear profile seems a reasonable 
20 

first approximation in the lower troposphere [z < ZT), where Mars’ 

“tropopause” is at ZT = 40 km. The temperature at ZT is TT = 160 K. 

The scale height is expressible in either of the alternate forms 

h(x1 = - 

where typical planetary mean value is h = 11 km. The troposphere 

RTs(xI PS 

J (21) - 

9 Ps(X19 

lapse rate assuming linear temperature profiles is 

 XI E - - - N 

3~ Tskl - TT 

aZ ZT 

(221 

The planetary mean lapse rate in Mars’ stable atmosphere isr = 1.45 

x K m-l; whereas the adiabatic lapse rate is Tad = g/cp = 4.55 x 

10-3 K m-l > T[x). It is useful to define a dimensionless relative stability 

parameter, 

To summarize, the vertical structure of the atmosphere is 

approximated by the profiles, 

V[X,Z] % VJX] - G(~l{l- [z/h[xIlI. (261 

Substituting equations [21]-[26) into equation (15al and performing the 

required integrations gields: 
21 

We have departed in equation (271 from the Leovy et al, (19731 integral 

method in two ways: [il 6(xl is evaluated in terms of D~(x] as opposed 

to atmospherically-absorbed radiation and [iil the condensational wind 

effect is included. 

Neglecting condensational winds, and eliminating [x) between 

equations (201 and (271 with the help of equation (211, gives the 

following estimate of the zonally-symmetric thermal diffusion 

coefficient of Hadley cells: 

using Table 1 values. This is more than 30 times larger than the 

baroclinic eddy value, De, estimated previously. Of course, we would 

not expect this to apply beyond the x's bounding central Hadley cells, 

so a rather strong drop-off to values characterizing the cap region is 

implied. 

Both two and three-dimensional primitive equation general circulation 

models indicate that during solstice conditions cross-equatorial Hadley 

cells exist surface winds of order vs - 10 m s-l over latitudes within 2 

30° of the equator (Haberle et al,, 19821. Typical surface wind 

distributions, vs[x], during south summer computed with 

two-dimensional [ZD, Haberle et &, 1982) and three-dimensional (3D, 

Pollack et al,, 1976) dynamic models are shown in Figure 5. These 

dynamicallg explicit models are identical mathematicallg and 

numerically, except that &derivatives are set equal to zero in the 2D 

model, automatically suppressing any longitudinal waves or eddies. 

The closeness of the 2D and 3D results supports the assumed 

dominance of the zonally symmetric winds. These zonal mean 

mericknsl surface lijifids OR Mars BE substsnttally more intense then 

their terrestrial counterparts -the latter being of order vs - 1 m S1 

[Oort, 1983, p. 1571. 
22 

lUe sought to determine whether a D~[x] distribution could be found 

with peak values of Dmax - 0.6 W m-2 K-' which approximates the 

surface'wind distributions of the dynamic models. To test this we used 

Ts, aTs/ax, y and vc versus x from the core model at Ls = 90° - close, 

but not identical, to the GCM Ls = 106O conditions - and evaluated our 

model's corresponding surface velocity 

for various assumed functions D,(x]. 

Rn immediate finding was that surface wind distributions consistent 

with GCM results at all latitudes could not be obtained with a 

latitude-independent diffusion coefficient. R constant Om, matching vs 

- 10 m s-l over Hadley cell regions gave too large surface winds near 

the cap boundaries [where aTS/ax is large], while a small Dm could 

recover cap region surface winds, but not those of Hadley zones. 

Clearly , a la t i t udi na llu-v a ryi ng di f f usi on coe f f i cen t i s required. 

By experimenting with different functions, a reasonably successful 

diffusion coefficient distribution was found. As shown in Figure 5, the 

model produced a vs[xl distribution close to those of the dynamic 

models - almost as close as the 2D and 3D surface wind distributions 

are to each other - when the logarithm of D, varied parabolically 

with x between the cap boundaries. This was a very welcome finding, 

as it was not even certain a priori that a diffusivity approximation 

would appropriately describe the zonally-symmetric circulation. Rs it 

happened, we were able to take the results even a further step to 

compute the streamlines. 

In Figure 5 (TOP] the Dm(X) used can be compared with James and 

North's "clear sky" and "dust storm" diffusion coefficients discussed 
23 

previously, ana typical values of the baroclinic eddy component De. 

This D~[x) can be written 

where K = [IlnDmax /lnDminI - 11 and f[X] = [(X - Xs][X - XN]/(XN - XS]~]. 

Diffusivity is a minimum (Dmin = 0.015 W m-2 K-l] at the cap boundaries 

XN and XS, and peaks [Omax = 0.60 UJ m-2 K-l] midway between the cap 

boundaries. Near cap boundaries, D~(x] is sufficiently small that 

surface velocity is dominated by condensational winds, vs - 

vc[l+2;1']/;1'. For ;1' - 0.15, and vc - - 0.2 m s-l, this gives a southward 

surface velocity near the cap boundaries of vs - - 2 m s-l, close to 

values shown of Figure 5. Notice that the surface winds induced by the 

condensational winds exceed them by almost an order of magnitude, 

an effect produced in the dynamic models as well. 

Despite many approximations, the model reproduces GCM surface wind 

patterns near north summer solstice suprisingly well, including three 

overturning cells and condensation wind effects. Even details like the 

local surface velocity dip near the equator produced by the solar 

heating function are recovered. 

fln interesting result was that the diffusion coefficient is much larger 

across Mars Hadley cells, than at the cap boundaries. f3 comparable, 

but latitude-independent D - 0.6 W me2 K-l, will match observed zonal 

mean surface temperature distributions on earth [Hoffert et ~l,, 1983) - 

also surprising in light of the earth's 100 times greater surface 

pressure. In the terrestrial case, this must result from very different 

flow mechanisms including atmoc 

currents. 

-. . *. 

amunes 

In our flow model, equations 

information from the core EBP 

>heric eddy transport and ocean 

191 and [261-[281 together with 

, define the zonally-symmetric 
24 

meridional velocity field v[x,zJ at any time during the seasonal cycle. 

These patterns are more clearly visualized by looking at the 

streamline contours, than at the velocity field as such. 

The partly analytic nature of the model makes streamline calculations 

fairly straightforward: R streamfunction Q[x,z) can be defined by its 

partial derivatives aQ/bx = -2na2pw and aQ/az = 2na[l - x211/2pv 

such that continuity is identically satisfied: 

The streamlines - contours of constant mass flow per unit time across 

which there is no flow - are therefore available from 

@(x, Z] = Zna(1- x~I~/~ p[X,Z'IV[X,Z'I dz'. 

Substituting equations 

6 

(241 and [261 and integrating gives: 

Notice that the streamfunction at the top of the atmosphere equals 

the condensation mass flux Q(x,oo] = &(XI, and that the surface and 

poles are zero mass flux boundaries. Q[x,Ol= Q[kl,zl= 0. 

Figure 6 shows the meridional plane streamlines computed at Ls = 00, 

90°, 180° and 270°. The contours of constant tp have units of 

Megatonnes per second (1 Mt S1= lo9 kg s-l), and in the present sign 

convention are < 0 for clockwise circulations in the x,z plane. Rt north 

spring equinox we find north and south Hadley cells of roughly 

comparable intensity in each hemisphere with core flows centered at 

the scale height. A weak hemispheric asymmetry is produced even at 

equinoctal conditions because the ground thermal inertia produces 

as y m m et ri c surf a ce t em pera t ure di s t ri but i ons which "rem em ber" their 
25 

ni story. 

Fts the northern hemisphere warms during its summer solstice, the 

south main cell migrates northward, penetrating to some 30° N 

latitude, and intensifies to flow rates of - 18 Mts-l; Rt the same time 

the northern cell weakens, and a weak indirect cell is produced near 

the south cap boundary by condensational winds. The Hadley cells 

become more nearly balanced in the two hemispheres at northern fall 

equinox, although again there is some hemispheric asymmetry. Finally, 

during northern winter solstice [southern summer], the northern cell's 

boundary moves southward, and attains a core circulation of - 28 Mt 

s-l - the highest intensity of the seasonal cycle. Indeed, southern 

summer is the time of year on Mars when intense surface winds can 

attain the saltation velocities needed to raise particulate matter into 

the atmosphere. This is often a precursor of global-scale dust storms. 

CONCLUDING REMQRKS 

lUe have seen that the core energy balance model developed in the 

early part of this paper gave a reasonably good account of the 

seasonal surface temperature and pressure waves on Mars as 

recorded by the Viking landers. Moreover, our extension to wind field 

analysis of the energy balance approach gives qualitatively plausible 

descriptions of atmospheric circulation over all seasons, and 

inherently recovers GCM-predicted surface winds at the north summer 

calibration point. The wind field depends partly on the pole-to-pole 

surface temperature distribution, and partly on the parameterized 

diffusion coefficient distribution. 

More work is needed on modeling the diffusion coefficient distribution 

in terms of limitations on the boundaries of the Hadley circulation from 

angular momentum and stability constraints. Moreover, a feedback on 

the core model is suggested by the wind field analysis: In particular, 

internally-consistent calculations of surface temperatures at 

equatorial latitudes should emplou the DmIX) of the wind analysis in 
26 

the heat CWiduCtioii equation, as opposea the the constant D - 0.015 UJ 

m-2 K-' assumed here. In turn, surface temperatures computed with 

this Dm(X) feed back on the wind field. Prior to our wind analysis, we 

accepted the conventional wisdom of a largely decoupled lunar-like 

planet in which atmospheric transport feedbacks on surface 

temperature distributions are small and confined to polar cap 

boundaries. This assumption needs to be reexamined. 

In light of its computational economy compared to GCMs, applications 

of the present approach to related phenomena of the Mars seasonal 

cycle are promising areas of future research. For example, in the Mars 

seasonal hydrological cycle as modeled by James [19851, it is assumed 

that meridional transport of water vapor is accomplished by 

condensational winds plus some unspecified diffusionli ke process. Our 

wind model suggests Hadley Cell transport, in which the small scale 

height of H20 profiles would selectively favor transport by 

near-surface winds, might provide an explicit explanation of this 

"diffusive" transport. Viking orbiter data on vertically-integrated water 

vapor are available for testing such models. 

Rnother potential application is modeling of Martian dust storms. The 

zonally-symmetric circulations simulated by Haberle et al, (1982) at 

south summer conditions demonstrate that positive radiative 

feedbacks from dust raised by surface winds strengthens and expands 

the Hadley circulation. FI puzzling question is uhy global dust storms 

are not triggered every Mars year. The answer may be related to the 

interplay of the zonally-symmetric and baroclinic eddu components of 

the circulation [Haberle, 1986b). Rlthough as presently configured our 

model does not incorporate the radiative feedbacks needed to mimic 

such global dust storms, such extensions can be implimented in 

principle. A simplified representation of zonally symmetric and 

baroclinic eddy transports, including the influence of vertical thermal 

structure, might well provide usefui insignts into the compiex 

dynamics of dust storm events. 
27 

Ftriaiiy, the present approach seems well Suited to studies where runs 

over a large parameter space are needed. These include sensitivity 

studies, analyses of Mars paleocltmate when orbital parameters were 

different, and analysis of future scientific missions to Mars - where 

instrument design may require estimates of meteorological conditions 

over entire annual cycles. 

James and 

ACKNOWLEDGEMENTS 

CUe thank Robert Haberle, Phillip lames Tillrnan for 

stimulating discussions of the Mars seasonal cycles. This research was 

supported by the National Aeronautics and Space Administration, Mars 

Data Analysis Program, under NRSR Grant NAGW-573 to New York 

University. 

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444-456. 

33 3-3 4 2. 
30 

Annual penod, zp = ZnK? 

Diurnal period, Td, 1 sol 

Eccentricity, e 

Obliquity [spin axis tilt], i 

heliocentric longtude of perihelion, Lsp 

Solar constant, So 

Planetaty radius, a 

Gra vi t a ti ona 1 acceleration, g 

Bare ground heat capacitu per unit area, Cg 

Bare ground thermal inertia, I 

Bare ground albedo, a9 

Bare ground emissivity, 

Ice albedo, ai 

Ice emissivity, ci 

Ice sublimation temperature, Tc 

Latent heat of CO2 ice sublimation, L 

Gas constant, R 

P Rtmosphere specific heat, c 

Mean surface pressure,& 

Mean mass per surface area, m =- ps/g 

Mean surface temperature,T, 

Mean surface density, ps = ps/RTs 

Mean scale height,h = RTs /g 

Tropopause height, ZT 

Tropopause temperature, TT 

Fkliabatic lapse rate, Tad = g/cp 

Mean tropospheric lapse rate, F = (TS - T~I/z-,- 

Mean stabiuty parameter, 7 = [Rigj[rad - 

-- 

-- 

- 

1 

5.94 x 107 

8.86 x lo4 

0.0934 

25.1 

250 

586 

3.39 x 106 

3.73 

1.30 x lo6 

300 

0.30 

1.00 

0.80 

0.58 

150 

5.95 x 105 

189 

830 

700 

188 

218 

0.0170 

1.1 x 104 

4.0 x 104 

4.55 x 10-3 

1.45 x 10-3 

160 

0.E 
31 

FIGURE CRPTIONS 

Figure 1. Mars seasonal surface isotherms computed from the present 

model. 

Figure 2. Seasonal variation of near-surface atmospheric temperatures 

at Viking lander sites (Seiff, 1982) compared with present energy 

balance model predictions. 

Figure 3. Seasonal variation of atmospheric pressure [reduced to Mars 

reference ellipsoid) at Viking lander sites compared with present 

energy model predictions. 

Figure 4. Latitudinal distribution of condensational uinds at north 

spring equinox [Ls = Ool, north summer solstice [Ls = go0], north fall 

solstice (Ls = 180°) and north winter solstice [Ls = 270°1. Wind directions 

are positive toward the North Pole. 

Figure 5. [TOPI. Diffusion coefficient distribution D(x1 used to match 

surface wind distribution of dynamic circulation models; [BOTTOM) 

Surface velocity distributions near north summer solstice. Present 

model results are for Dm= D[xl at L, = 90°. 

Figure 6. Streamline patterns on Mars over the four seasons from the 

present model. 
90 

60 

30 

n 

Q) 

Q) 

Q) 

L 

ET 

Q) 

v 

ZO . 

Q) 

TI 

=I 

Y 

Y 

W 

2 

.- 

-30 

-60 

-90 

0 30 60 90 120 150 180 210 240 270 300 330 360 

Aerocentric longitude of the sun, Ls (degrees) 

Time from N. Spring equinox, t (sols) 

I I I' 1 I I 1 I I I I I I 

0 100 200 300 400 500 600 670 

1 L Q1 

Q1 

.I= 

VI 

a 

E 

.r 

Q1 

I 

i L Q) 

r 

CI 

Q) 

.C 

E 

P 

i 

Figure 1 
220 

21 0 

200 

Y 

W 

If 

. 190 Q) 

L 

3 

w 

Q 

180 n. 

E 

170 c 

160 

150 

- - 

- - 

- 

- 

- 

- LANDER DATA (Seiff, 1982) 

V L - 1 2 2 ON 0.3 7 11111111111111111111I 

- - - 

- CO2 frost point --J. - - 

- - 

- 

- 

- I II Illill1 IIlIlltll,l Illllll I1 IIIIt' 

0 30 60 90 120 150 180 210 240 270 300 330 360 

Aerocentric longitude, L, (degrees) 

- 

I I II I I I I I I I I I 11 

0 100 200 300 400 500 600 670 

Ti me from N. Spring equinox, t (3013) 

Figure 2 
n 

reference ellipsoid 

0 30 60 90 120 150 180 210 240 270 300 330 360 

Aerocent ric longitude, L, (degrees) 

I I II I I I I I I 1 I I 11 

0 100 200 300 400 500 600 670 

Time from N. Spring equinox, t (sols) 

Figure 3 
0.5 

C 0.4 

E 0.3 

I 

0) 

W 

c, 

3, 0.2 w 

c 

.C 

3 0.1 c 

W 

c 

O 0.0 

a -0.1 

-0.2 

7 

W 

Q) 

c 

V 

c 

0 

- 

N. Cap condensing 

\- I 'I rLs= oo 

S. Cap condensing 

- 0.3 I1111 1111 11 11 I IIIII 

x = sin@ 

-90 -60 -30 0 30 60 90 

Latitude, 9 (degrees) 

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 

L!I 1 I I I 1 I I I I I I 1 I It 

Figure 4 
n 

1 

Y 

- 

';J 

E 

CI 

c 

Q) 

0 

Q) 

0 

0 

c 

. Y 

.C 

E 

C 

C 

0.4 

0.2 

0.1 

0.08 

0.06 

0.04 

0.02 

- 

- 

N strong dust storm value 

Ba rocli nic i nsta bili t y range 

- Present - 

- V 

- 

n 

I 

Q) 

- 

E 

v 

. 

V 

c 

.C 

? 

15 

o 2D (Haberle &, 1982) 

10 

5 

0 

-5 

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 

bl I 1 

n = sin@ 

Latitude, Q (degrees) 

I I I I I I I I I I II 

-90 -60 -30 0 30 60 90 

Figure 5 
N. Spring 

Eq ui nox 

S. Fall 

Eq ui nox 

20 

n 

E 

Y 15 W 

N 

a- 10 V 

3 

-cY 

=5 F 

N. Summer 

Solstice 

S. Winter 

Sol 3ti ce 

N. Fall 

Equi nox 

S. Spring 

Eq ui nox 

N. Winter 

Solstice 

S. Summer 

Solstice 

0 

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 

lJI I I 

x = sin@ 

Latitude, @ (degrees) 

I I I I I I I 1 I Ill 

-90 -60 -30 0 30 60 90 

Figure 6