A numerical model of the Martian polar cap winds

ICARUS 39, 151-183 (1979)

A Numerical Model of the Martian Polar Cap Winds ROBERT M. H A B E R L E AND CONWAY B. LEOVY Department of Atmospheric Sciences, University of Washington, Seattle, Washington 98195 AND

JAMES B. P O L L A C K Theoretical Studies Branch, Ames Research Center, NASA, Moffett Field, California 94035 Received August 17, 1979; revised April 4, 1979 An investigation of the Martian polar cap winds and their response to a variety of factors is carried out by a series of numerical experiments based on a zonally symmetric primitive equation model. These factors are the seasonal thermal forcing, mass exchange between polar caps and atmosphere, large-scale topography, and polar cap size. The thermal forcing sets up a circulation whose surface winds adjust to achieve angular momentum balance, with low-latitude easterlies and high-latitude westerlies. The maximum westerlies occur roughly where the horizontal temperature gradients are largest. This pattern changes when cap and atmosphere exchange mass. Coriolis forces acting on the net outflow or inflow produce easterlies at the surface during spring (outflow) and westerlies during winter (inflow). Topography appears to have a small effect, but cap size does play a role, the circulation intensity increasing with cap size. Peak surface winds occur when outflow or inflow is a maximum and are 20 m sec -1 during spring and 30 m sec -1 during winter for the northern hemisphere. The model results show that surface winds near the edge of a retreating polar cap are substantially enhanced, a result which is consistent with the Viking observations of local dust storm activity near the edge of the south polar cap during spring. The results also indicate that the surficial wind indicators near the south pole are formed during spring and those near the north pole during winter. The implication is that the high-latitude dune fields in the northern hemisphere are formed at a time when the terrain is being covered with frost. It is therefore suggested that the saltating particles are "snowflakes" which have formed by the mechanism proposed by Pollack et al. The model results for the winter simulation, which do not include transport by large-scale eddies, compare favorably with general circulation model (GCM) calculations. This suggests that the eddy transports may be less important than those associated with the net mass flow, and that 2-D climate modeling may be more successful for Mars than Earth. INTRODUCTION

Unlike their terrestrial counterparts, the Martian polar caps undergo dramatic changes in size, occupying 30% of the hemispheric surface area during winter and less than 1% during summer. A unique feature of Martian meterology is the occurrence of substantial meridional motions associated with the gain or loss of atmospheric mass due to sublimation or condensation of the polar caps. Leovy et al. (1973) have suggested that during southern hemisphere spring strong winds develop at the edge of the polar cap which, by generating local dust storms,

could increase the atmospheric dust opacity to a level sufficient to trigger a global dust storm. The Viking orbiter imaging experiments have subsequently found that there is a tendency for local dust storms to occur near the edge of the sublimating south polar cap (Peterfreund and Kieffer, 1979). In the northern hemisphere, the broad dark collar feature surrounding the residual cap was interpreted to be the result of a reduction in the surface albedo caused by enhanced wind erosion (Sagan et al., 1973), but has since been shown by Viking cameras to be a vast belt of dune fields (Cutts et al., 1976). Dunes form by saltation how ever--a pro151 0019-1035/79/080151-33502.22/0 Copyright© 1979by AcademicPress,Inc. All rightsof reproductionin any formreserved.

152

HABERLE, LEOVY, AND POLLACK

cess requiring strong w i n d s - - a n d the impli- equilibrium temperatures from a separate cation remains that winds associated with radiative-convective model described later. the polar caps are responsible for their The polar cap sublimation or condensation origin. rates are also computed from this model. In this paper we develop a three-level, The results of these calculations serve as zonally symmetric, primitive equation input for the dynamical calculations which model and carry out comprehensive numer- are based on the model described in the ical experiments which give insight into the next section. The results a r e presented relative importance of the large-scale pro- under two categories: "sensitivity s t u d y " cesses controlling the low-level polar cap and "seasonal simulation." The sensitivity circulation. These processes are (1) the sea- experiments examine the influences of cap sonally varying temperature discontinuity size, regional slope, and sublimation or at the edge of the polar cap, (2) the mass condensation mass flow in isolation, while exchange processes between polar cap and the seasonal simulation experiments include atmosphere, (3) the influence o f large-scale all processes. In the Discussion we point topography, and (4) the size o f the polar out the model's limitations and then use its cap. The first two result from changes in results to interpret certain observation. Our solar insolation; the frost-free regions, hav- conclusions are summarized in the last ing a low thermal inertia, can change tem- section. perature readily, while the polar cap temperature is fixed at the local frost point MODEL DESCRIPTION when phase changes take place. The third, Governing Equations topography, is ubiquitous on Mars, possessWe use it-coordinates and neglect coning relief amplitudes on the order of a scale tributions to momentum and heat transport height ( - 1 0 km) over both large and small by large-scale eddies. The primitive equahorizontal distances. The fourth, cap size, tions can then be written in the form may be important since at a given point in the Martian orbit it may vary from year to O(IIV)lOt year. = - v . ( n v v ) - 0(rI~V)/0~r Previous theoretical treatments of the Martian axisymmetric circulation relevant + H[2[I + (u/a cos 0)] sin 0(~ x V) to this work are those of Pirraglia (1975) and + /IF, (1) Burk (1976). Pirraglia found analytical solutions to the linear problem for a fixed tem- O(IIT)/Ot = - V . (IIV T) O(H6"T)lOtr perature distribution, and Burk used numerical techniques to study the diurnal winds + (KTto/tT)[1 -- (PT/Ihr)]-' near the polar cap edge. The approach + H ( F v + q), (2) taken here is to utilize the complete nonlinear equations to determine the seasonal OH/Ot = - 0 ( H v cos O)/a cos 000 -0(rI,~)/0~, (3) variation o f the middle- and high-latitude circulation. The nonlinear terms are particu06/Otr = IIRT/p. (4) larly important in the vicinity of the cap edge. In order, these are the horizontal momenWe examine the controlling processes in a tum equation, the thermodynamic energy series of numerical experiments, each of equation, the continuity equation, and the which is carried out in two parts. As our hydrostatic equation for an ideal gas. The modeling of the thermal drive is based on a independent variables are latitude 0, time t, simple Newtonian cooling formulation, we and ~r -= (p - p r ) / H , where H = Ps - PT, P first compute the spatially dependent is pressure, PT is a constant small -

MARTIAN POLAR CAP WINDS " t r o p o p a u s e " pressure, and ps is the surface pressure. The dependent variables are the eastward and northward components u and v of the horizontal velocity vector V, the or-coordinate "vertical velocity" 6- =- dor/dt, temperature T, the geopotential height $ of a sigma surface, the substantial derivative o f pressure to-= dp/dt, and the so-called terrain pressure II as defined above. Other symbols are 1~ the planetary rotation rate, the radius a o f the planet, the gas constant R, the specific heat at constant pressure cp, ~ the vertical unit vector, and V the spherical horizontal gradient operator on a sigma surface. The quantity F is the frictional stress per unit mass and consists of contributions by both horizontal and vertical momentum transfer due to small-scale eddies. FT is a horizontal thermal diffusion term a n d / t is the diabatic heating rate per unit mass. Equations (1) through (4) contain seven dependent variables. The set is closed by forming three auxiliary relationships; two from the continuity equation and one from the relation between pressure and or. Vertically integrating (3) through the troposphere gives the (terrain) pressure tendency equation

OII/Ot = - ( / [0(IIv cos O)/a cos O00]dtr

-(n6-)~=,.

(5)

At the " t r o p o p a u s e , " or = 0 and we apply the b o u n d a r y condition 6- = 0.6" also vanishes at the lower boundary (or = 1) except when phase changes take place at the surface. In this case there is an exchange of mass between atmosphere and ground which is accounted for by the second term on the right-hand side of (5). The computation o f this term is discussed in the section describing the radiative-convective model. The vertical velocity follows when the upper limit of the integration on (3) is changed to the or level of interest, II6" = - f o [O(Hv cos O)/a cos O00]dor

- or(OII/Ot).

(6)

153

Finally, an expression for to can be obtained with the aid of the definition of tr as to = II6-

+ or[(OII/Ot) + (v/a)(OH/O0)].

(7)

Equations (1), (2), and (4)-(7) form a complete set o f prediction and diagnostic equations which can be finite differenced and numerically solved.

Model Structure and Boundary Conditions In order to facilitate comparisons with the three-level general circulation model (GCM) calculations o f Pollack et al. (1976a, 1979), the model employs, insofar as possible, the same structure and computational techniques. The computational region is divided into three layers nonuniformly spaced in pressure and bounded above the or = 0 level by a dynamically inactive "stratos p h e r e " as illustrated in Fig. 1. The dependent variables are vertically staggered such that momentum, temperature, and geopotential are carried at layer midpoints (odd levels), while the pressure weighted vertical velocity (II6-) is carried at the layer interfaces (even levels). The nonuniform spacing increases the low-level resolution and the staggering facilitates the computation of vertical derivates. For Martian conditions the constant pr is taken to be 1 mbar and the initial surface pressure is 6.1 mbar. Also given in Fig. 1 are the mean values o f the pressure and height of the various levels. The vertical boundary conditions on 6" have been specified above. In addition, values of the momentum and temperature of the mass leaving or entering the atmosphere are needed when surface phase changes take place. We assume for both cases that the gas involved has a temperature equal to the local frost point and no relative angular momentum. The horizontal differencing employs a staggered grid with (u,v) grid points alternating with points at which II and T are carried and 6" and to are calculated. E x c e p t when otherwise specified, the spacing be-

154

HABERLE, LEOVY, AND POLLACK

STRAT.' LEVEL

o-

(mb)

(krn)

0

0

II&-

1.0

15.6

I

.213

u, v, qB,T

2. I

9.3

2

.426

l"[&

3.2

,5.6

3

.603

u,v,qB,T

4.1

3.5

4

.780

IIo-

5.0

1.8

5

.890

u,v,~,T

5.5

0.8

6

1.0 , / / / / / / / / /

6.1

0

TROP.,

rid'///////////

FIG. 1. Vertical structure of the model atmosphere illustrating the representation of the dependent variables in the three-layer troposphere.

tween (u,v) or ( H , / ) grid points is 2.5 ° (148 km) and lateral boundaries are set at (l-I,/) grid points at 27.5°N and 90°N. The lateral boundary condition is taken to be a vanishing mass flux, i.e., Hv = 0. In most cases the subtropical boundary was found to be far enough r e m o v e d to have a negligible influence on the circulation. In one experiment, however, it was necessary to move it further south. The selection of representative topography for use in a zonally symmetric model is difficult where the terrain exhibits significant longitudinal variations. It is made even more difficult when the data are incomplete. H o w e v e r , several features of the Martian terrain can be ascertained from the Mariner 9 data of Kliore et al. (1973) and Conrath et al. (1973). First, the higher latitudes (>45 °) of the northern hemisphere appear to have a greater longitudinal symmetry than those of the southern hemisphere. Hence, the northern hemisphere is more amenable to zonal averaging than the southern hemisphere. Second, the northern hemisphere terrain is characterized by a topographic trough roughly centered at 65°N flanked to the north by an elevated plateau some 3° in angular semidiameter, and to the south by a gentle incline to an elevation at

subtropical latitudes which is somewhat lower than that of the polar plateau. An ad hoc longitudinal average of these features is shown in Fig. 2, and will serve as the lower boundary for the topography experiments. The maximum elevation difference (valley floor to polar plateau) is 3.6 km with the maximum slope occurring on the poleward side of the trough ( - 7 8 ° N ) at a rise to fetch ratio of 1/240. Real Martian topography (longitudinal variations included) almost certainly has slopes which, in some locations, are two or three times greater than the maximum used here. H o w e v e r , the use o f these steeper slopes in preliminary experiments resulted in extremely noisy solutions indicating that the model limitations in handling topography had been exceeded. Forcing and Dissipation

Our goal is to isolate that component of the zonally averaged circulation which changes on time scales greater than a day but less than a season. This component responds to the daily averaged temperature field which is established rapidly by radiation and convection. This is not the temperature field which would be established by steady insolation at the mean rate, since the

MARTIAN POLAR CAP WINDS

155

400C

MAXM I SLOP~ UM

2000 E z 0

o

~

-2000

-4000

FIG. 2. Model version of the "zonally averaged" topography of the northern hemisphere. The zero elevation line is arbitrary.

diurnally varying convective boundary layer, which is an important part of the mean field, would not develop properly under steady-state radiative conditions. Furthermore, it is not possible to introduce the diurnal time variations into a large-scale zonally symmetric model without violating the longitudinal constraints. For these reasons we adopt the simple Newtonian cooling formulation ---- - ( r -

Te)/7-r,

(8)

where Te is a daily-average radiativeconvective equilibrium temperature and rr is a corresponding time constant. Both are derived from the radiative-convective model which deals explicitly with diurnal variations. This approach neglects the nonlinear interactions between temperature variations generated by radiativeconvective processes and those generated by advection, but it should yield results which are correct to first order. It couples the dynamics to the most important characteristics of any diabatic heating model: the equilibrium temperature and the relaxation time, and it is readily applicable in a zonally symmetric model. Under steady-state conditions the generation of kinetic energy is balanced by its dissipation, which takes place primarily through the vertical exchange processes of subgrid scale eddies. We model this dissipa-

tion with a Rayleigh friction parameterization, and the frictional stress for a threelayer model takes the form F1 = -(V~ -V3)/rD + (Fh)~, F3

---- ( V 1 - -

V3)/r D - ( V 3 - Vs)/TD + (Fh)3,

F5 = (Va - V~)/rD

-psu2(V~llVs])/Sms + (Fh)5, (9) where (Fn)k is the horizontal contribution at level k, rD a diffusive time constant, P5 the surface air density, 8rn5 the mass per unit area of the lowest layer, and u. = cd IV51 is the friction velocity corresponding to a drag coefficient cd. The third of these equations shows that the surface stress is assumed to be parallel to the low-level wind Vs. In general, a turning angle between the low-level wind and surface stress is expected, with rotation of the stress toward lower pressure with increasing stability, and toward higher pressure with increasing baroclinicity when geostrophic and thermal winds are from the same direction (see, for example, Vankatesh and Csanady, 1974). Thus, interpretation of the low-level wind requires consideration of both these effects. Our assumption here is that the internal dynamics are not sensitive to the turning angle. rD, the viscous coupling time constant, is assumed to be a constant equal to 5 days.

156

HABERLE, LEOVY, AND POLLACK

E x p e r i m e n t s h a v e shown that the results are nearly insensitive to r , for rD ~ 5 days. Smaller values s e e m to us to be unrealistic in the very stable a t m o s p h e r e we are considering. Also, to account for the difference in a t m o s p h e r i c stability, we s o m e w h a t arbitrarily assume Co to be equal to 0.030 in frost-free regions and 0.015 o v e r the polar cap as can be inferred from the boundarylayer studies of Deardorff (1972). Such a partitioning rather crudely represents the m o r e turbulent nature of the frost-free atm o s p h e r e due to daytime convection. Finally, the lateral diffusion t e r m s are (I'IFh)

=

(1/(a 2 cos 2

(9))(0/00)

[IIKH cos s O(O/O0)(V/cos 0)]

tions. This is m o s t important for our w o r k since we are concerned with the steadystate low-level wind distribution. The time integrations are p e r f o r m e d in 30-min sequences using a 3-min time step to satisfy the linear computational stability criterion. The initial step in e a c h sequence is the E u l e r - b a c k w a r d s c h e m e p r o p o s e d by Matsuno (1966), while the remaining steps use the centered or leapfrog scheme. The Matsuno scheme serves as a high-frequency filter and prevents the solution separation associated with the leapfrog scheme. At the end o f e a c h sequence provision is m a d e for the calculation of the source terms, i.e., heating and dissipation.

(10) RADIATIVE-CONVECTIVE MODEL

for m o m e n t u m and

In this section we briefly discuss the calculation of those quantities which serve as ( l I F T ) = ( 1 / ( a z c o s 0))(0/00) input for the dynamical model described in [rlKH c o s O(01"/O0)] (11) the previous section. These are the daily a v e r a g e d r a d i a t i v e - c o n v e c t i v e equilibrium for heat. T h e s e terms are introduced for t e m p e r a t u r e Te, the corresponding time numerical smoothing p u r p o s e s and use a constant rR, and the quantity (II--~)~=~. T~ diffusion coefficient KR = 5.0 x 104 m 2 and rR are functions of latitude, height, and sec -1 which renders the diffusive transports time o f year. Values o f important constant small c o m p a r e d to transport b y the mass used in these calculations are s u m m a r i z e d motions. N o t e that (10) is in angular in Table I. m o m e n t u m conserving form and assumes that the lateral m o m e n t u m flux by eddies TABLE I vanishes for solid b o d y rotation. CONSTANTS USED IN THE CALCULATIONS

Numerics We h a v e adapted the so-called " l o g theta" conserving space differencing s c h e m e of A r a k a w a et al. (1974). This scheme is used b y Pollack et al. (1979a) and has two features which compliment our purpose. First, by preserving the integral constraints on m e a n kinetic energy and m e a n - s q u a r e vorticity, long-period integrations can be p e r f o r m e d without the develo p m e n t of computational instability. This characteristic is particularly useful for three-dimensional simulations but is a desirable feature of any numerical model. Second, the s c h e m e c o n s e r v e s angular m o m e n t u m for statistically steady circula-

Parameter Frost-free albedo Polar cap albedo Surface gravity Initial m e a n surface pressure Soil thermal inertia Tropopause pressure Specific heat at constant pressure

Value 0.24 0.60 3.72 m sec -2 6.1 mbar 272 W m-2 sec-l~Z°K-~ 1 mbar

Gas constant

8.2 × 102 J kg-~°K -t 1.89 × 105 J k g - t ° K -~

Angular rotation rate

7.078 x 10-s see -1

Latent heat o f

sublimation Solar constant at semimajor axis

5.902 x lOa J kg-~ 6 × 102 W m -z

MARTIAN POLAR CAP WINDS In computing these quantities several general assumptions are made. First, we assume the Martian atmosphere to be cloudless and dust free with CO2 the only molecular constituent. In the northern hemisphere both cloudiness and aerosol content are probably greatest during winter--the former associated with the polar hood and the latter with global dust storms. In a crude sense we account for the influence of (COs) clouds on the thermal structure by not allowing the temperature to fall below the frost point. The influence of aerosols, however, is totally ignored and the consequences are assessed in the Discussion. Second, we assume a flat surface at which the atmospheric pressure is 6.1 mbar. Along a sloping surface the zenith angle and surface pressure can change. For the largescale topography used here, the zenith angle variations are negligible. Surface pressure variations, on the other hand, can be considerable. However, their influence on the temperature profiles and the subsequent changes to the dynamics is of second order and they are therefore neglected also. Assuming the motionless atmosphere responds to solar (qsr), infrared (qtr), and convective heating (qcv), the time dependence of Te at a level k is given by (OTe/Ot)k = (tlsr)k + (qir)k -Jr-(qev)k. (12)

We compute Te at grid points where the temperature is carried by time marching Eq. (12) in. 20-min steps for a period of 30 days. The initial atmospheric and ground temperatures are 150 and 200°K, respectively. Te is then determined to be the average value of T~ over the last day of the integration (day 30). The solar heating is computed from the expressions used by Blumsack et al. (1973) with the correction noted by Burk (1976). Though generally small we include heating by absorption of solar radiation for its influence on high-level temperatures. The infrared and convective contributions are based on the methods developed and described by Pollack et al. (1979). In their

157

treatment, the infrared is handled by an equivalent width formulation for COs at 15 /zm which is based on laboratory measurements and includes the dependence of absorption on temperature. The convective exchange algorithm is similar to that proposed by Deardorff (1972), but with the convective layer depth diagnostically determined rather than predicted. The surface heat flux (assuming free convection) is then computed and distributed linearly throughout the convective layer, vanishing at the top. In this manner the need to specify an " e d d y viscosity" profile is avoided. The ground temperature variations in frost-flee regions are modeled as in Leovy and Mintz (1969) using a soil thermal inertia and bolometric albedo representative of the average Martian surface (Kieffer et al., 1976) (see Table I). The method employs an approximate solution to the surface heat balance equation and includes a latent heating term when condensation or sublimation take place. Heat conduction into the ground is approximated by assuming a solution to the heat conduction equation driven by a prescribed surface temperature containing only the diurnal component. Over the polar cap the ground temperature is fixed at 149°K--the frost point of CO~ at 6.1 mbar. The condensation (sublimation) rate E is computed by assuming that the radiative losses (gains) balance the latent heating (cooling). Since (II~)~=l = - E g , where g is the gravitational acceleration, (I16-)~=1 is just the average value over the last day of the integration. It is computed at each grid point poleward of the cap edge. The boundary of the polar cap during the retreat phase is specified from the observations of Dollfus (1973). In one experiment, however, the cap is growing and we allow the model physics to determine the boundary since no precise inventory of solid CO~ is needed. In this case, the boundary is defined as that latitude closest to the equator where the dally averaged ground temperature is equal to the frost point. The time constants are evaluated in a

158

HABERLE, LEOVY, AND POLLACK 290 275

PRESENT M DEL

260 245 hi

230 FbA

2,5!

,,, 20C i.-185 17£

I

0

I

I

5

I

I

I

6

J

I

I

I

9

I

I

12

I

I

I

15

I

I

I

18

I

i

i

21

i

id

24

LOCAL TIME (MARTIAN HOURS}

F~G. 3. Diurnal ground t e m p e r a t u r e variation at the Viking I lander (22 ° N) during the early part o f the primary mission (Ls = 106°). Solid curve is for the present model. D a s h e d curve is for the Viking thermal model o f Kieffer et al. (1977).

natural fashion by monitoring the approach of the hourly temperatures to their equilibrium values. A time constant for each hour of the day is then obtained by noting the time at which the departure from equilibrium is e -1 of its initial value. The dynamical model then uses the averaged value of the 24 time constants. This procedure is performed at each grid point at which Te is computed. Thus, for a given time of year rr is a function of latitude and height.

calculation of subsurface conduction for a homogeneous soil. Major differences occur during midday and are less than 4°K. This is apparently due to our approximate treatment of conduction since we use an identical value for the thermal inertia and an albedo only 0.01 of a unit smaller. The associated variations in the atmospheric temperature profiles are shown in Fig. 4. As was found b y Gierasch and Goody (1968) only the lowest few kilometers are strongly time dependent. Above Radiative-Convective Model Examples this, the tropospheric lapse rate remains As the focus of this paper is on dynamics, nearly adiabatic throughout the day. Bewe do not present an exhaustive analysis of cause of the crude vertical resolution, the the model characteristics. Instead, a few amplitude of the air temperature variation representative cases have been selected for at the lowest grid point (AT6 ~ 30°K) is less brief discussion to illustrate the basic fea- than that observed (-50°K) by the Viking 1 tures of the model. For comparative pur- lander at a height of only 1.6 m (Hess et al., poses some of the sample calculations were 1977). The temperature discontinuity at the carried out for early northern summer, a air-ground interface, measured by T~T6, is time which corresponds to the early part of also shown and is largest shortly after noon when the surface heat flux and convective the Viking primary mission. Figure 3 shows the computed diurnal layer depth are both at their maxima (Fig. ground temperature variation at the Viking 5). Sutton et al. (1978) have estimated the 1 lander site (22°N, 48°W) f o r / ~ = 106° (L~ peak surface heat flux at the Viking 1 lander is the areocentric longitude and is measured site for this period to be from 13 to 28 W from northern spring equinox). Also shown m -2 depending on surface roughness and is a similar curve generated from the Viking the importance of molecular conduction. thermal model described by Kieffer et al. Our lower peak value (9.3 W m -z) is proba(1977), which employs an exact numerical bly due to the neglect of forced convection

MARTIAN POLAR CAP WINDS

20

159

[

1.5 O-

9 l0 t~ 27 Ld _J

015 (/3

I

I I 150 170 TEMPERATURE (°K)

130

190

1200 1800

~.~.-~-

0600~

( 210 11831

I 250

~, 230 ~, 11951 ~ (240)

(275)

FIG. 4. Diurnal variation o f a t m o s p h e r i c t e m p e r a t u r e profiles for the s a m e conditions o f Fig. 3.

N u m b e r s in parenthesis are the corresponding g r o u n d t e m p e r a t u r e s .

which is undoubtedly important to Martian boundary layer dynamics. However, our main purpose is to compute reasonably accurate temperature profiles with which to drive the large-scale dynamics. A more sophisticated approach is not likely to change our conclusions. The daily averaged radiative-convective equilibrium temperature profiles for midspring (L~ = 50°) are presented in Fig. 6. This is the thermal structure which drives the midspring circulation. Curves labeled A, B, and C are for the upper, middle, and lower

layers, respectively. At this season the latitude of the cap edge is at 69° where the horizontal temperature gradient at all levels is a maximum. Temperatures in both the frost-free and polar cap regions vary more with height than latitude. Over bare ground the lapse rate is nearly adiabatic while over the polar cap it is roughly isothermal. Also, the temperature of the upper polar atmosphere is at the local frost point (141°K) indicating that CO2 ice clouds would form if radiation was the only process at work. It is clear from these profiles that the horizontal

20

IO

//f~\\ / \

15

/ I /

"T"

E

\

?.5

\

vLO

/

"=~ Io tLIU

/

5

/

'/ /

\

\

I I

Z O

Oq

I

J

I

3

I I I

I~1

6

I I I I

9

I

IZ

I

I

15

I

"6 ~A

I

x~oc ~ u_o ~ wD bj r

2.5 ~

@

\ ~

18

J

I

~

I

21

LOCAL TIME (MARTIANHOURS) FIG. 5. Diurnal variation of convective layer depth (solid curve) a n d surface heat flux (dashed curve).

160

HABERLE, LEOVY, AND POLLACK

SENSITIVITY STUDY EXP I

w

i,i I.-

~

00

LIJ I.r_)

jpo C.L)

i,i

IH'O 20

30

qO

SO

6O

70

80

90

LRTITUOE

FIG. 6. D a i l y a v e r a g e d r a d i a t i v e - c o n v e c t i v e e q u i l i b r i u m t e m p e r a t u r e p r o f i l e s for n o r t h e r n h e m i s p h e r e m i d s p r i n g (L, = 50°). C u r v e s l a b e l e d A , B, a n d C, r e s p e c t i v e l y , r e p r e s e n t t h e u p p e r , m i d d l e , a n d l o w e r l a y e r s . H e a v y v e r t i c a l a r r o w o n the a b s c i s s a i n d i c a t e s the p o s i t i o n o f t h e p o l a r c a p e d g e .

temperature gradients which drive the polar middle layer. A somewhat surprising result circulation are mostly associated with the is that the middle layer takes longer to addiscontinuity at the cap edge rather than the just than the upper layer. Apparently, the latitudinal variations in isolation. upper layer radiates directly to space while The time constants for the midspring ex- the lower layer exchanges mostly with the periment are shown in Fig. 7. The values ground. The middle layer, however, exare comparable to those found by Goody changes with the other layers and cannot as and Belton (1967) for pure radiative pro- efficiently dispose of its energy and consecesses. A different functional dependence quently takes longer to reach equilibrium. exists for the frost-free and polar cap reNUMERICAL RESULTS gions due to the presence or absence of convective heat transport. Equatorward of In the beginning of this section we prethe cap edge the intense daytime convection sent results of a class of numerical experiacts to quickly establish the equilibrium profile and hence, the values are small (less T A B L E II than 4½ days), have virtually no latitudinal SENSITIVITY STUDIES dependence and are only weak functions of L, ~ao Phase changes Topography height. Over the cap interior, however, no Experiment convection takes place and radiation is the I 50° 69~ No No only process acting to establish equilibrium. lI 50° 55° No No The absence of convection has no effect on III 50° 69° Yes No IV 270° 45° Yes No the lower-layer time constants but has a V 50~ 69° No Yes dramatic effect on those of the upper and

MARTIAN POLAR CAP WINDS 20

~

MIDDLE LAYER

15

£ F-

z

161

I0

I-Q

~

5

I--

2Z5

I

37.5

I

475

I

57.,5 LATITUDE

I ~"LOWERI LAYER 67.5 77.5 ;0

FIG. 7. Daily averaged value o f the radiative--convective time c o n s t a n t s corresponding to the s a m e conditions o f Fig. 6.

ments which isolate the influence of the controlling factors on the polar cap circulation, i.e., the sensitivity study experiments. Their distinguishing features are given in Table II. At the end of this section we present the seasonal simulation results which highlight the changes in the spring circulation. In all, eight experiments are carried out. The integrations start from a resting isothermal atmosphere at 170°K and run for 32 Martian days to an approximate steady state. The results shown are time averaged over the last 2 days to filter out any small amplitude transient motions. The first experiment, hereafter referred to as the nominal case, is for the middle of spring when the edge of the polar cap is at 69°N. In this experiment no CO2 phase changes are allowed ((1-I6-)~=1 = 0) and the lower boundary is fiat. The results, therefore, represent the response to the thermal forcing alone. The second experiment is identical except that the cap edge is positioned at a more equatorward latitude (55°N) in order to assess the effect of polar cap size on the circulation. In the third and fourth experiments the polar caps are allowed to exchange mass with the atmosphere; losing mass in experiment Ill and gaining mass in experiment IV. In experiment V the variable topography of Fig. 2 is introduced. Except for experiment IV, all

sensitivity experiments correspond to incoming solar radiation at Ls = 50°.

Experiment I The results of this experiment embody the principle characteristics of the polar cap circulation and are discussed in greater detail than those of the other experiments. Figure 8 shows the evolution of the mean kinetic energy per unit mass as a function of time. A rather steady increase is observed with a weak tendency to asymptote by the end of the integration period. Clearly, a true steady-state condition has not been achieved. Other experiments approach equilibrium more rapidly. This is a consequence of the use of an invariant dissipation time constant in experiments for which the kinetic energy generation characteristics differ. However, in all experiments the qualitative nature of the circulation is established early (within approximately 2 weeks) and longer integrations are not performed since only minor quantitative changes would result. The meridional and zonal wind profiles are shown in Fig. 9. These curves, as" well as those to be presented, are labeled the same as in Fig. 6: A for the upper level, B for the middle level, and C for the lower level. To assist in visualizing the nature of the circulation, streamlines for this case

162

HABERLE, LEOVY, AND POLLACK

SENSITIVITY STUDY EXP I 500

l

l

,

l

I

i

!

l

i

I

!

i

l

J

I

l

,

|

i

I

l

w

l

i

I

i

l

l

,

I

l

i

~50

~00 N~SSO II

~oo LU Z LIJ

.in, '7

100

1

So ~

1 5

10

15

20

a , i a I i I , , I i i 25 ;30

a 3,5

TIME (DRY$) FIG. 8. M e a n kinetic energy per unit m a s s as a function o f time for the nominal sensitivity s t u d y e x p e r i m e n t (see Table II).

have been constructed from a mass streamfunction ~bdefined by the relationships 0~/0~ = - H v cos 0 and

0~/00 = Ilt~a cos 0 and are shown in Fig. 10. The mass streamfunction satisfies the continuity equation for steady-state conditions. Thus, mass flow in the meridional plane is proportional to the streamfunction gradient. The meridional circulation (Fig. 9a) is thermally direct, with low-level equatorward moving air rising over frost-free regions into the upper troposphere where all motion is poleward, then sinking over the polar cap interior. The strength of the meridional circulation is strongest at the edge of the polar cap (marked by the heavy arrow) where the temperature and pressure gradients are largest. The well-confined

character of this subsiding flow resembles an internal boundary layer in the meridional plane. Some of the return flow at higher latitudes takes place in the middle layer, just as it does for the poleward moving air just south of the cap edge. This pattern is associated with the strong horizontal heating gradients at the cap edge which produce maximum rising motions just south of the edge and maximum sinking motions just north of it. The role of the middle layer, therefore, is to assist in maintaining continuity by supplying air to these regions. At all latitudes the zonal wind component increases with height (Fig. 9b). The upperlayer wind is everywhere westerly with a maximum of 65 m sec-ljust poleward of the cap edge. In the middle and lower layers the zonal wind has an easterly component in the lower latitudes and a westerly component at high latitudes. At the surface (the lowestmodel layer) the easterly maximum is a few

MARTIAN POLAR CAP WINDS

SENSITIVITY

2.S

i

i

i

I

m

i

i

l

I

l

l

i

l

I

163

STUDY EXP I

~

J

l

l

I

i

i

i

l

I

~

l

i

l

I

i

!

i

i

I

a 2,0

1,5

(J ~U-II. 0

i

=_ nU~0"5 el ,~ 0o0 z

3: ~-O,S Z

l..4 ~-1,0 O~ -1.5

-2.0 -2.5

i

i

I

i

20

,

*

i

i

i

30

I

i

J

i

l

tO

I

i

i

i

,

50

I

,

i

i

,!rl,

60

,1

I

'

i

I

i

i

i

i

80

70

90

LflTITUBE

SENSITIVITY

100

"

w

l

l

i

~

i

I

l

l

I

l

i

i

i

I

I

STUDY EXP I i

i

I

I

!

!

1

I

I

i

w

r

i

I

'

|

i

i

I

l

b 80

6o

{J LI.I (sl

20

J--I

,T _J

~ -o -20

.

,

20

,

,

,

30

,

. . . . . . . . . . . . . .

~0

50

60

.,

70

. . . .

,

80

. . . .

/ •

90

LRT ] TUBE

FIG. 9. (a) Meridional wind profiles (positive poleward) and (b) zonal wind profiles (positive eastward) for the nominal experiment. Curves are labeled as in Fig. 6.

164

HABERLE, LEOVY, AND POLLACK

15.o / / /

/

~"\\\\

\~k

12.5 ~o.o I

\\

~7.5 I

\

I

ii

\\1\ 50

40

50

60 LATITUDE

70

~' 80

90

FIG. 10. Streamline pattern for the nominal experiment. The mass flow circulation in the meridional plane is in the direction indicated and is proportional to the contour gradient. Contour interval is 10 kg sec -3 except the dashed line which is for the -0.1 isopleth. It is included to emphasize the shallow nature of the reverse circulation at high latitudes. degrees e q u a t o r w a r d of the c a p edge with a as a consequence of strong sinking motion magnitude of 13 m sec -1 and the westerly and subgrid scale mixing. In computing the angular m o m e n t u m balm a x i m u m is poleward of the edge at 14 m s e c - L At all levels the zonal wind smoothly ance, however, it was found that the ada p p r o a c h e s zero at the boundaries indicat- j u s t m e n t was not complete, and that the ing that the location of the subtropical surface winds were exerting a net w e s t w a r d b o u n d a r y is not influencing the circulation. torque on the planet. O v e r the interior of T h e s e features can be interpreted as fol- the polar cap where the meridional velocity lows. Since at e a c h level the t e m p e r a t u r e is is small, the time scale for a vertical column e v e r y w h e r e decreasing poleward, the zonal to adjust to a balanced state rAM is wind must increase with height as required a p p r o x i m a t e l y by the thermal wind relationship and the rAM -- H~/(CdUDz, (13) greatest increase m u s t occur where the horizontal temperature gradients are where H is the scale height and ~ is the l a r g e s t - - j u s t inside the edge of the polar cap vertically a v e r a g e d zonal wind. Taking = 30 m sec -~, H = 10 kin, ca = 0.015, (see Fig. 11). The surface wind distribution is e x p e c t e d from a consideration of the an- and us = 10m sec -I, we get rAM -- 150days gular m o m e n t u m balance. For balance, a for a characteristic value o v e r the polar steady circulation must exert no net torque cap. O v e r lower-latitude regions rAM can be on the planetary surface, i.e., the surface a factor of 10 smaller. Thus, m o s t of the stress integrated along the lower b o u n d a r y final adjustment is taking place o v e r the must vanish. Thus, the surface winds must polar cap and we do not expect a totally change sign at least once with latitude. The balanced surface wind distribution for a ultimate distribution is determined by the 32-day integration. H o w e v e r , since the patconvergence of the angular m o m e n t u m flux tern of frost-free easterlies and polar cap due to the mean meridional motions which westerlies is real, we recognize that the adset up easterlies in the frost-free regions justed field will consist of slightly w e a k e r through the action of the Coriolis torque easterlies and slightly stronger westerlies. The thermal structure of this circulation and establish westerlies o v e r the polar cap

MARTIAN POLAR CAP WINDS

165

SENSITIVITY STUDY EXP I

220

.~200 ILl ,--.j I.-LU O.

~]LO0

160

I'~0

20

30

urO

50

60

70

80

90

LRT/TI.~E

FIG. 1 1. Temperature profiles for the nominal experiment.

is shown in Fig. 11. The subtropical temperatures are nearly equal to their radiative-convective equilibrium values and are characterized by weak horizontal temperature gradients and slightly subadiabatic lapse rates. In this part of the frost-free area the circulation has little influence on the temperature structure. However, near the cap edge advection of cold air from the polar cap interior controls the low-level temperatures. Indeed, the polar air travels some 1200 km south of the cap edge before losing its identity (its temperature becoming equal to that of its equilibrium value). This significant equatorward penetration of cold air is a consequence of the cooling power of the polar cap and the intensity of the meridional circulation. It resuits in the development of a low-level inversion which extends to about 500 km south of the cap edge. A low-level inversion also characterizes the polar cap atmosphere, but it is maintained by subsidence combined with weak diabatic heating at the midlevel.

Polar inversions over the winter polar cap have been observed by Mariner 9 (Hanel et al., 1972) and Viking (Kieffer et ai., 1977). No inversion over the remnant north polar cap has been observed (Kieffer et al., 1976). As the size of the polar cap decreases it becomes less effective in cooling the lowlevel air, particularly when the remnant water ice cap with its higher surface temperature is exposed. It is therefore likely that during a major part of the spring the polar atmosphere has a low-level inversion. At the start of the integration the surface pressure is initialized to 6.1 mbar and is independent of latitude. As the circulation evolves the surface pressure changes at a rate proportional to the vertically integrated divergence. The resulting surface pressure distribution is presented in Fig. 12. The B curve is the final distribution. The surface pressure is highest at the edge of the polar cap and decreases on either side with lowest values in the subtropics. In general, the pattern is consistent with the vertical motion field described above. The low-

166

HABERLE, LEOVY, AND POLLACK 6.q'O

SENSITIVITY STUDY EXP I i

i

i

i

I

i

i

g

i

I

i

,

i

,

i

i',

i

i

|

,

r

i

i

I

i

,

,

,

I

,

,

,

,

6.3S

6.30

W O~

/

.20 0.. hl L.)

~.~s (/I

6.10

6°00

i

'20

I

i

,

I

30

I

I

I

!

I

tl'O

I

i

~

i

I

50

¢

J

t

i

I

60

i

&

i

all

70

i

t

I

I

I

IlO

I

t

i

l

9i

LRTI TIJOE

FIG. 12. Surface pressure as a function of latitude for the nominal experiment. The A curve is the initial distribution and the B curve is the final distribution. pressure area centered on the pole is an indirect result of the stronger sinking motion at the c a p edge where the cooling rate is m u c h higher. Its existence helps drive a very w e a k low-level r e v e r s e cell at these higher latitudes. This can be seen f r o m a careful examination of Fig. 9a which shows low-layer southerlies and middle-layer northerlies poleward of 77.5 °. It is also evident in the streamline pattern of Fig. 10. The a p p e a r a n c e of a reverse cell is interesting since the r e v e r s e m e a n meridional cell in the E a r t h ' s t r o p o s p h e r e (the FerreU cell) is attributable to the action of largescale quasi-horizontal eddys. T h e s e are absent in our model, and our reverse cell is driven by a subgrid-scale eddy m o m e n t u m flux which varies with height. O v e r the polar cap interior there is vertical convergence of this flux in the middle layer and divergence in the lowest layer and this distribution m u s t be balanced by e q u a t o r w a r d m e a n flow in the middle layer and poleward mean flow in the lower layer. A similar fea-

ture, though of broader extent, was found by Schneider (1977) in his a x i s y m m e t r i c model of the terrestrial tropical circulation.

Experiment H This e x p e r i m e n t is for the same time of year as in the nominal case except the edge of the polar cap is now located at 55 ° . Thus the forcing profile is identical except in latitudes between 55 ° and 69 ° which are now frost covered. In this m a n n e r the effect of size o f the polar cap on the circulation can be isolated. This experiment is of interest since the position of the polar cap edge at a given L~ m a y v a r y f r o m year to year. The wind profiles for this case are shown in Fig. 13. Clearly, the greater latitudinal extent of the polar cap drives a m o r e intense circulation. The larger cap cools a greater portion of the polar a t m o s p h e r e and a stronger circulation is required to redistribute heat from the subtropics. This effect would be m a x i m i z e d when the polar cap and frost-free regions o c c u p y c o m p a r a b l e

MARTIAN POLAR CAP WINDS

167

SENSITIVITY STUDY EXP II

2.S

i

e

i

,

I

!

'

'

,

I

'

!

'

,

!

,

,

,

,

|

.

,

|

i

I

'

'

!

,

I

;

'

'

8 2,0

1.5

~u-~l,0

e,~ 0.5 O. u') r'~ 0.0 z .T ~k'O.5

~,?,-1.0 o~ uJ .r -1.$

-2.0

-2.5

30

20

~0

SO

60

70

80

N

LRTZTI~

SENSITIVITY STUDY EXP II 100 |

'

|

|

I

'

|

|

|

I

i

|

|

,

I

'

'

'

'

I

;

'

'

!

I

'

'

'

,

l

'

'

!

'

b i I10'

8O

L) I/.i E

a

'¢0

¢¢.i 4. Qrj

i

i~l

o0

-20

-u,L.0

|

20

l

l

,

|

l

30

e

J

,

l

|

|

|

q*o

l

l

l

l

~

l

50

l

l

l

l

l

l

60

l

l

l

,

|

|

l

70

i

|

|

Oo

LflTITLJDE

F]G. 13. (a) Meridional and (b) zonal wind profiles for experiment II.

90

168

HABERLE, LEOVY, AND POLLACK

SENSITIVITY STUDYEXP I I

~'0

i

i

!

,

i

,

i

t

I

,

,

,

,

I

i

i

i

i

I

,

i

,

,

I

i

i

t

i

I

!

i

,

i

I

|

,

,

,

I

'

,

,

i

I

,

,

,

i

~oo LLf

160

1~0 20

30

utO

I

50

i

,

i

i

|

60

t

i

i

i

I

i

|

i

70

i

I

80

i

i

i

|

90

LRTITUDE FIG. 14. Temperature profiles for experiment II.

areas of the hemisphere. In this case the maximum low-level meridional winds are 50% greater than in the nominal case. Increases in the low-level zonal winds, however, amount to only a few meters per second for both the easterly and westerly maximum. Not surprisingly, there is also a greater equatorward penetration of polar air at low levels (Fig. 14). The inversion develops at 41°N which is about 800 km south of the edge. As in the nominal case a polar inversion is also evident.

Experiment III Having established an understanding of the circulation response to the thermal structure, we now proceed to examine the influence on the circulation of a sublimating polar cap. The forcing profiles for this experiment are the same as those in the nominal case. At this time of year the cap is sublimating at close to its maximum rate, adding to the atmosphere approximately

3.0 x 1016 g of CO2 per day. Since no mass is allowed to escape through the vertical boundaries, an 18% increase in the model's tropospheric mass occurs over the 32-day integration period. In actuality, there would be a net mass transport across the subtropical " w a l l " so that our boundary condition there is somewhat unrealistic. H o w e v e r , this artificial wall has a negligible influence on the circulation at the latitude of the polar cap and even near the wall there is no evidence of large flow irregularities as would be expected if the wall were severely inhibiting the circulation. The important results are presented in Figs. 15-17. The major influence on the circulation is felt principally at low levels. By itself, the mass evaporating off the polar cap sets up an equatorward directed pressure gradient force which at low levels would complement the " b a c k g r o u n d " pressure gradient force (see Fig. 12) in frost-free regions and oppose it over the polar cap. Since the sublimation rate is large and increases toward

MARTIAN POLAR CAP WINDS

169

SENSITIVITY STUDY EXP III 8.0

| l l l l | l | l l | l l | l | l l l l | l l l l l l l l l l | l l

7.8

7.6 7.~

j.t

6,2 ~.8

I I I l l

I l l l l l l l l l l l l l l l l l l l l l

30

~0

SO

60

I l l l l l

70

U

M

~T1TLI~

FiG. 15. Initialandfin~sufface~essu~distributions~rexperimentIII. the pole, the resulting surface pressure distribution has an e q u a t o r w a r d directed gradient at all latitudes as is evident in Fig. 15. N e a r l y all the outflow appears to be confined to the lowest layer which can be seen by c o m p a r i n g Figs. 9a and 16a. In the latter, the low-level e q u a t o r w a r d flow is significantly enhanced, particularly in frost-free regions w h e r e the b a c k g r o u n d and outflow pressure gradients are in the same direction. The Coriolis torques which a c c o m p a n y the outflow are strong enough to p r o d u c e surface easterlies o v e r the entire polar cap (Fig. 16b), as the westerlies of the first experiment are no longer evident. The easterly m a x i m u m is 20 m sec -1, a 50% increase o v e r its value in the nominal case. Thus the influence o f a sublimating cap on the surface wind speeds near the cap edge is quite dramatic and almost certainly plays an important role in the dust raising potential o f the polar cap winds. A sublimating cap also affects the thermal structure which is shown in Fig. 17. The

mass being added to the a t m o s p h e r e has a t e m p e r a t u r e equal to the surface frost point (149°K) and is being t r a n s p o r t e d to frostfree regions faster than in the nominal case. As a consequence, there is a greater e q u a t o r w a r d penetration o f polar air which can lower the frost-fi'ee t e m p e r a t u r e s as m u c h as 15°K from their nominal values. The low-level inversion is now seen to develop at 54°N, which is 400 k m further south than for the nominal case. Thus the thermal effect of a sublimating cap is to broaden the region of m a x i m u m low-level t e m p e r a t u r e gradients which, in turn, has a broadening effect on the velocity field.

Experiment IV A natural extension of the previous experiment is to consider the effect a condensing (growing) cap would have on the circulation. As the previous e x p e r i m e n t was for a time o f m a x i m u m outflow, this experiment is for a time o f m a x i m u m inflow. This occurs at the winter solstice (/_~ = 270 ~)

HABERLE, LEOVY, AND POLLACK

170

SENSITIVITY STUDY EXP III 2.5

'

'

'

'

i

u

u

T"|

i

,

,

n

i'

I'

u

T

i

!

1

1

!

,

u

I

u

,

|

!

I

u

n

u

n

a 2.0

1.5

~u) l . 0

E

.~0.5 e-~ 0,0 X

Z ,r -1.5

-2.0

"2.S

|'

J l i i n n l l , J a l i , n n l n i n l | p a i i l l i o a 30 ~0 50 60 70 O0

20

L I I T I TUDE

90

I

SENSITIVITY STUDY EXP III tO0

l

,

i

i

i

i

i

i

r

l

i

i'

[

i

,

,

i

I

,

i

i

,

I

i

,

i

u

I

,

!

l

b 8O

¢.J tl.l U') .jr-

60

t-~ tFO l.lJ LIJ eL. t-t Z

2O

N

-0

-20

-qO

' ' 20

, n . . . . . . . . . 30 ~FO

' . . . . . . . . . 50 60 LRT

In . . . . 70

, .... 80

l TUDE

FIG. 16. (a) Meridional and (b) zonal wind profiles for e x p e r i m e n t III.

] ~0

MARTIAN POLAR CAP WINDS

171

SENSITIVITY STUDY EXP Ill 2qO

l

l

l

I

i

l

l

m

I

l

.

l

l

l

!

l

i

rl

I

'

l

l

l

I

I

I

l

l

I

I

l

I

220

tU

II

180

! lqO 20

3o

~o

so

eo

70

oo

so

I_ATI11.I~

FIG. 17. Temperatureprofilesfor experimentIlL when the high-latitude radiative losses are at a maximum. The cap edge is determined by the radiative-convective model to be at 45°N. This is much closer to the subtropical boundary (27.5°N) than in the earlier experiments. Also, this boundary is now located in a region of strong horizontal temperature gradients and would arbitrarily constrain the circulation at these latitudes. For this experiment, therefore, the subtropical boundary is positioned in the southern hemisphere at 35°S by doubling the grid size and retaining the same number of grid points. Thus, some horizontal resolution is sacrificed. The expanded area offsets the 6 × 1016 g of CO2 the troposphere loses to the growing cap each day and at the end of the integration period the model troposphere has retained 87% of its initial mass. The most significant result is the dramatic change in the low-level circulation as shown in Fig. 18. As in the sublimation case, most of the inflow takes place in the lowest layer

since over most of the polar cap only the low-level wind has a significant southerly component. Furthermore, the meridional circulation is divided into two cells: an intense equatorial Hadley cell which reaches into the middle latitudes of both hemispheres and a weak reverse polar cell driven by the boundary-layer inflow. In this case, however, the Coriolis torques which act on the net mass flux produce westerlies at all levels throughout much of the northern hemisphere. The basic features of the model circulation are quite similar to the early general circulation experiments of Leovy and Mintz (1969) and the more recent calculations of Pollack et al. (1976a, 1979a). This is significant since it suggests that large-scale eddies, which are prominent features in those calculations, do not play a dominant role in determining the Martian zonally averaged flow at this season. For the present purpose the most in-

172

HABERLE, LEOVY, AND POLLACK

SENSITIVITY STUDY EXP IV l

i

i

!

I

I

l

I

i

I

i

!

a

tO

w

Qr~

a

-15

I

I -20

-u,'O

I

I -O

I

1 20

I

I • ~0

I

I 60

I

I 80

t 100

LATITUDE

SENSITIVITY STUDY EXP IV 180

'

I

,

I

I

I

,

I

I

b

w

i., -tO

.l~,lq, o_.

,

l -20

,

-0l

.

20|

'

' ~' q'O

t

60

'

80|

100

LAT]TLIDE

FIG. 18. (a) Meridional and (b) zonal wind profiles for experiment IV.

MARTIAN POLAR CAP WINDS

173

SENSITIVITY STUDY EXP IV

220

l,l.J l.---

I.--

\

160

lq'O I-.

-~0

-20

-0

20

~0

60

80

100

LATITUDE

FI6. 19. T e m p e r a t u r e profiles for e x p e r i m e n t IV.

teresting result is the strong band of surface tence of a warm band of high-level air near westerlies over the polar cap interior and the edge of the growing cap (Leovy, 1979). adjacent frost-free regions. The maximum is 33 m sec -1 at 60°N, polewards of which Experiment V This experiment is the same as the nomithe atmosphere is extremely barotropic, having virtually no vertical shear in the nal case except that the sloping topography zonal wind. The wind speeds in this region of Fig. 2 becomes the lower boundary. Only are obviously controlled by the net inflow wind profile results are presented and are rate which is a maximum at this time of shown in Fig. 20. year. We discuss the relationship these The basic features of the nominal circulastrong surface winds may have to be ob- tion are not significantly influenced by the served circumpolar dune fields in the next presence of topography. However, some section. differences are evident, particularly in Another interesting result of this experi- frost-free regions. There, both meridional ment is the upper-level temperature maxi- and zonal wind profiles exhibit a greater mum and its associated elevated inversion vertical shear, which, for the latter, results which forms in the vicinity of the cap edge in broader surface wind maxima over frostas shown in Fig. 19. This is produced al- free regions. At high latitudes the low-level most entirely by the substantial sinking mo- winds are equatorward, flowing downhill tion near the edge of the condensing polar over the polar cap. Very little change in the cap since the heating rate time constants maximum wind speeds and their location is vary little in this region. There is some evi- noted. dence in the Mariner 9 data for the exisWe interpret the influence topography

174

HABERLE, LEOVY, AND POLLACK

2.5

,

i

!

,

!

SENSITIVITY STUDY EXP V

.

.

,

i

i

,

i

,

.

i

.

,

,

.

i

,

.

!

,

i

.

,

,

!

i

!

!

!

a 2.0

1.5

,~, O.S G. (/'! r-~ 0.0 -0.5 c~ -L.5 -2.0

i

-2,S 20

l

t

i

i

30

,

I

I

i

i

~'0

l

I

,

I

i

50

I

I

I

I

I

eO

I

i

I c T ,

i

i

70

,

|

I

i

•

•

O0

•

90

LRTITLK3E

SENSITIVITY STUDY EXP V tO0 I I lI I l e W ; I l l e e | I I I I I IbI I l a l I I D V I I ; I ~ 00

6O

z

~0 o. z 2O ,,J ee

.... - .... , .... i .... , .... ~, .... , .... 20

30

~0

SO

60

70

90

LRTXTUDE FIc. 20.(a) Meridion~ and (b)zon~ wind profiles for experiment V.

1 90

MARTIAN POLAR CAP WINDS has on the frost-free circulation as being similar to that of an elevated heat source. The temperature decreases with height and the surface slope is negative (down toward the pole) south of the trough. Thus the temperature gradient along a level surface is increased and a stronger circulation results. Over that portion of the polar cap poleward of the trough, the low-level inversion and positive surface slope also tend to enhance the circulation, but the effect is minimized by the small vertical temperature gradients and strong sinking motion at the cap edge. Also, the results of several experiments not shown here indicate that the position of the cap edge with respect to topography does not appear to significantly influence the circulation. Thus, the baroclinicity and the condensation or sublimation mass flows near the cap edge have primary control over the circulation. Seasonal Simulation

The seasonal variation of the springtime circulation is now simulated by carrying out the integrations for the beginning (L~ = 0°), middle ( ~ = 5&), and end ( ~ = 90°) of spring. In each integration phase changes are allowed and topography is included. The forcing is appropriate for the time of interest and the results are shown in Figs. 21-23. The noisy character of the computed flow in Fig. 23 is numerical in origin and is due to the inability of the model to handle the differential heating and topography of this experiment when the polar cap is very small. The gross features of the circulation can, however, be ascertained. At the beginning of spring only part of the polar cap is sublimating since at higher latitudes deposition is still taking place. At this time a two-cell circulation develops with low-level air moving equatorward over frost-free regions and poleward over the polar cap. The Coriolis torques produce a rather broad band of surface easterlies in the subtropical cell and surface westerlies in the polar cell. As the season progresses and more of the polar cap sublimates, the

175

low-level circulation gradually becomes easterly at all latitudes. By midspring the cap edge temperature discontinuity and mass outflow are a maximum producing the strongest surface winds of the season - 2 0 m sec -1. It is at this time of spring that erosion of surface material would be most likely. Finally, by summer solstice the polar cap has receded to its residual size. Its cooling power and total sublimation have been reduced and it no longer drives an intense circulation. However, a narrow band of moderate surface easterlies near the cap edge is still evident. In our simulation of the summer solstice circulation we have assumed that the residual cap is composed of CO2 ice. However, Viking orbiters have established that H~O ice is the constituent (Kieffer et al., 1976). H20 ice has a higher equilibrium temperature and requires more energy to sublimate a unit mass. Had this been accounted for, the solstice circulation would be even weaker than that shown here. However, the circulation presented is representative of the last phases of the COs cap. DISCUSSION M o d e l Limitations

The results of the previous section are based on a model which oversimplifies the real situation. The assumption of longitudinal uniformity is quite restrictive. Important processes which operate in three dimensions, such as baroclinic instability and tides, are not represented or parameterized in our model. Baroclinic waves develop in a zonal current which becomes unstable to disturbances for wind shears which exceed a critical value. Our results indicate that strong shears exist during winter and spring making it likely that these waves are present. There is evidence for baroclinic waves in Viking data (Ryan et al., 1978; Tillman et al., 1979). The extent of their effect on the mean meridional circulation, however, is not clear. On Earth these disturbances play

176

HABERLE,

LEOVY, AND POLLACK

SEASONAL SIMULATION LS=00 2.s

,

.

,

,

I

g

i

,

,

[

i

,

,

,

. . . . . . . . .

,

[

,

,

.

.

I

.

,

.

,

I

,

.

,

.

I

.

,

,

. . . . . . . . . .

,

8

2.0

1,5

1,0 ,,zw

~j O,S uJ czl 0,0

z

;]c ~J-O.5

.,r

-I.S

-2.0

-2.5

. . . .

2MI

20

~l'O

1

,,

,, . . . . . . .

,

50

60

70

,

$0

90

LATITUDE

SERSONAL SIMULRTION LS=O0 100

80

6O (_.1 Lid U') Tw

u.i ill G. U') Z

2O

_1 rt7

I~l

-0

-20

.tF0

|

20

i

i

I

30

.

.

I

i

I

tl'O

i

.

i

.

I

50

t

1

|,t

i

I

60

.

i

.

.

i

70

J

|

i

i

i

O0

.

.

.

.

~0

LATITI.J~

FIG. 21. (a) Meridional and (b) zonal wind profiles for the seasonal simulation of the early spring circulation ( ~ = 0~).

MARTIAN POLAR CAP WINDS

177

SEASONAL $1MULATION LS=50 2.5 llil|llWlmlllllllillllll11111111W 1 a 2.0 1.5

__._. _._.__~~~

%o 0,$ t.

~ 0.$

~1.0 -L5

/

-2.0 .~05

I l l l l l l l

20

I l i l l l l l l l l l l l l l ~ l l l l l l j l l l

30

qO

50

60

70

O0

90

~TITUI3E

IO0

SEASONAL SIHULATIEN LS=50 I

I

I

I

l

l

l

l

l

l

l

l

l

P

l

l

l

l

l

l

t

l

l

l

l

l

l

l

~

l

i

I

l

l

l

l

b $0

6O t.J

~o ILl

"1~0

I

20

'

I

i

I

30

|

I

I

I

q'O

I

I

•

I

I

50

I

I

I

i

I

60

•

I~1

70

l

i

I

I

I

80

I

I

I

1

t 90

LI:ITI TLJ~

FIG. 22. (a) Meridional and (b) zonal wind profiles for the seasonal simulation of the midspring circulation (L~ = 5(r),

178

HABERLE, LEOVY, AND POLLACK

SEQSONQL SIMULATION Ls=go 2.5

'

'

'

'

I

'

'

'

'

I

'

'

i

,

I

'

,

,

,

I

'

i

,

,

I

'

'

'

,

I

'

i

,

,

a

2.0 1.S f.) ,r w

,,~ 0.5 G. 03 C~ 0 . 0 Z

~1-0.5

i

i-e

] -1.5 -2.0

-2.5

I

I

,

I

30

20

,

i

i

i

i

'tO

,

l

i

i

I

i

SO

A I

,

I

60

i

l

l

i

I

i

i

I

70

L

I

,

80

l

l i I

90

LATITUDE

SEASONAL SIHULQTION LS=90 200

l l l l l l l l l l l l l l l l l l l l 1 , 1 1 | ; l l l l l l l l

b 8O

60

~

~o

-20

.~0

l l J l l ~

20

i l l l l l l l l l l l l l l i l l l i i l l l ! l l l

30

qO

SO

60

70

80

~0

LATITUDE

FIG. 2 3 . ( ~ M e r ~ i o n ~ c~eul~ion (& = 90°).

and (b) z o n ~ wind profilesfor the seasonal sim~ation o f t h e late spring

MARTIAN POLAR CAP WINDS a key role in maintaining the midlatitude westerlies. According to L e ovy and Mintz, however, the westerlies on Mars are maintained by the Coriolis torques acting on the net mass flux toward the growing cap and not the eddies whose transport is roughly balanced by the mean meridional circulation. Furthermore, heat transport by the waves is likely to be limited by the high radiative damping rates. Thus on Mars, the influence of baroclinic waves may be swamped by the polar cap inflow and outflow transports and by the radiative processes more effective than those that occur on Earth. The good agreement between the results of the winter solstice experiment and those of the three-dimensional GCM suggest that this is the case. It implies that modeling the zonally averaged circulation with a two-dimensional model may be more successful for Mars than Earth. Tidal motions are alSO eliminated in a 2-D model. For terrestrial studies the effect is generally inconsequential. For Mars, however, significant tidal amplitudes are expected from theoretical considerations (Zurek, 1976) and are present in modeling experiments of the general circulation. Strong tides have also been detected by Viking landers (Leovy and Zurek, 1979). Therefore, even though tidal amplitudes generally decrease poleward there may still be a significant component at high latitudes. To first order then, the results presented here should be viewed as a basic state upon which various tidal motions may be superimposed. The influence of diurnal heating and cooling on winds near the cap edge is treated in an accompanying paper (Haberle, 1979). Aside from the inherent weaknesses in the zonal averaging the model heating and dissipation are also based on assumptions which are critical to the nature of the results. A different heating distribution or dissipation scheme would lead to a different meridional circulation. Our heating rates are computed for a dust-free atmosphere and our treatment of friction is analogous to

179

an " e d d y viscosity" coefficient which increases with height. Given the complexity of the interaction between dust and atmosphere, it seems desirable to first understand the atmosphere's response without dust and our results are appropriate to this end. We can speculate about the influence dust may have on the circulation by considering how it affects the temperature structure. Dudiag winter,' t h e horizontal temperature gradients may actually be sharper at times than those computed here since dust at high latitudes is cut off from solar insolation and may be experiencing preferential removal (Cutts, 1973; Pollack et al., 1977). Viking measurements of the north polar atmospheric temperatures near winter solstice do show this tendency (Martin et al., 1978). Sharper gradients would drive a stronger circulation. During spring, however, a more uniform distribution of dust coupled with the exposure of high latitudes to solar insolation would lead to a smoother temperature distribution. Subsequently, our computed maximum wind speeds for the spring cases would be too high. The viscous parameterization is based on simplicity and intuition. It is not clear how friction should be treated in models of this type for the Earth's atmosphere much less that of another planet. Surely, the details of our results are sensitive to viscosity, particularly the meridional component at high latitudes. It would be difficult to conceive, however, that the use of a reasonable alternative to the parameterization used here would change a major conclusion reached in this study: that surface winds at middle and high latitudes are westerly during winter and easterly during spring. Surface Features and the Polar Cap Winds Observations have revealed a curious asymmetry between surface features of the northern and southern hemisphere. Of interest here are those features which can be used as reliable indicators of the surface wind direction. In the southern hemisphere

180

HABERLE, LEOVY, AND POLLACK

these are the long, bright and dark parallel streaks which emanate from craters found in the latitude range 60 to 75°S (Cutts, 1973). Their orientation suggests that they were formed during a period of strong southeasterly surface winds. In the northern hemisphere the extensive dune fields which surround the residual polar cap near 80°N can be used. A variety of types are present including longitudinal, transverse, and Barchan dunes, each of which are formed by depositional processes. The areas with a large population of Barchan dunes can be analyzed with confidence since these dunes are formed during periods of strong and steady winds. Ward (1978) has analyzed these areas and has concluded that the dune forming winds have a definite westerly component. A similar conclusion has been reached by Wolfe (1979). Thus the wind regimes of each hemisphere which are responsible for the surface features just described appear to have components of opposite sign; westerly in the northern hemisphere and easterly in the southern hemisphere. There is no reason to expect a qualitative difference between the springtime circulation of the northern and southern hemisphere. However, since southern hemisphere winter occurs near aphelion a larger cap develops which is later irradiated by stronger insolation. We can expect, therefore, a more baroclinic atmosphere and a greater mass outflow for the southern hemisphere spring. Indeed from Fig. 10b of Leovy et al. (1973), the maximum spring outflow for the south cap is nearly twice as large as it is for the north cap. Thus the springtime south polar cap winds could have saltation potential, which would explain the southeasterly surface streaks mentioned above and the local dust storms seen by Viking orbiters at this season (Peterfreund and Kiefer, 1978). If the northern hemisphere dune fields are products of the present climate regime, it does not seem possible to form them when the ground is unfrosted. These latitudes are exposed when surface winds are light, eas-

terly, and incapable of saltation. It appears then that the dunes are shaped during winter when strong surface westerlies are present (Fig. 18b) and frost is accumulating on the surface. In this case a candidate for the saltating particle would be a dust containing "snowflake" which has evolved according to the mechanism suggested by Pollack e t a l . (1977, 1979b). In their view suspended dust particles may be preferentially removed from the atmosphere by serving as condensation nuclei for water vapor and CO2 in the winter polar region. Modest amounts of water vapor freeze out first, followed by large amounts of CO2 until the particles are big enough to settle out. Such particles are more likely to saltate than dust because of their larger size ( - 5 0 / ~ m in diameter) and lower effective density (Pollack et al., 1976b). Then when spring comes, the COs sublimates leaving behind the dust in the form of dune fields. Once the dust is placed in this way, it stays there because its particle size is too small and wind speeds are too light for further motion to occur. The saltating snowflake model would also provide a partial explanation for the occurrence of extensive dune fields in the north polar regions and not in south polar regions or elsewhere on the planet. According to the model, the atmosphere must become cool enough for sustained COs condensation to take place and there must be substantial quantities of dust particles present. Lower latitudes are never cool enough to sustain the necessary condensation, thus leaving only the polar regions for consideration. But for the current precessional cycle, global dust storms occur only during northern hemisphere winter. Thus, only in the north polar regions would appreciable dune formation by saltating snowflakes be possible at present. However, it may still be difficult to form dunes in the south polar regions even during the depositional phase of the cycle since the geology of this region is partly characterized by pitted plains (Murray et al., 1972) which could serve as traps to migrating sand.

MARTIAN POLAR CAP WINDS Finally, we note that the model surface wind speeds are s o m e w h a t lower than those estimated to be n e c e s s a r y to initiate saltation (see for e x a m p l e , Pollack, et al., 1976b; Wood et al., 1974; Hess, 1973). H o w e v e r , we would not e x p e c t such model winds to be too close to the threshold value. Indeed, if our s y m m e t r i c model winds r e a c h e d or e x c e e d e d the threshold value unreasonably frequent eolian activity would be indicated. CONCLUSIONS

The zonally s y m m e t r i c polar cap circulation on Mars is controlled b y strong horizontal t e m p e r a t u r e gradients, mass exchange processes, and to a lesser extent by large-scale t o p o g r a p h y . During spring the t e m p e r a t u r e discontinuity at the edge o f the polar c a p drives an intense circulation which is proportional to the cooling power, or size, of the polar cap. Vertical motions are strongest in the vicinity o f the polar cap edge with rising motion o v e r frost-free regions and sinking motion o v e r the polar cap. This p r o d u c e s a surface pressure maxi m u m at the c a p edge which tends to w e a k e n the low-level e q u a t o r w a r d motion o v e r the cap interior. In the absence of p h a s e changes the surface winds adjust in angular m o m e n t u m conserving fashion with easterlies e q u a t o r w a r d o f the cap edge and westerlies poleward of it. This pattern changes for a growing or receding cap. In this case the net mass flow, which takes place in a shallow surface layer, and its a c c o m p a n y i n g Coriolis torque determine the sign of the low-level zonal wind component. For a net inflow, such as e x p e c t e d for winter, surface winds are westerly o v e r the entire c a p and parts of the adjacent frost-free area, while the springtime outflow case p r o d u c e s surface easterlies throughout the model domain. H o w e v e r , during the incipient retreat phase (late winter and early spring) a two-cell circulation is established characterized by an inflow at the higher latitudes of t h e polar cap and an outflow at the lower latitudes. T h e magnitude o f the surface wind is

181

substantially enhanced during those periods when the growth or retreat rate of the polar c a p is at or near its m a x i m u m . Our calculations indicate that the e n h a n c e m e n t is near a factor o f 2 for the northern hemisphere spring m a x i m u m . The influence of t o p o g r a p h y in the northern h e m i s p h e r e is m o s t evident in the unfrosted areas where it serves as an elevated heat source and subsequently increases the overall intensity of the circulation. Its influence is minimal o v e r the polar cap since there the strong sinking motion at the edge and w e a k lapse rates p r e v e n t any large increases in the low-level d o w n s l o p e c o m p o nent. F u r t h e r m o r e , the basic structure o f the polar cap circulation is insensitive to the relationship between polar cap size and topography. The circulations modeled indicate that the surficial wind indicators near the south pole were f o r m e d during spring and those near the north pole were f o r m e d during midwinter. This suggests that the recently d i s c o v e r e d circumpolar dune fields might be f o r m e d b y saltating snowflakes driven by the strong surface westerlies which develop at these latitudes during early winter. ACKNOWLEDGMENTS Several persons have provided enlightening discussions concerning the model itself and the dynamics it produces. These are Professors D. L. Hartmann, J. M. Wallace, and J. R. Holton at the University of Washington, and Dr. R. G. French and Professor P. J. Gierasch at Cornell. We are also grateful to Jim

Tillman and Donn Terry for their assistance in assembling the model on the Prime 400 computer. The lead author is especially grateful to Professor C. A. Riegel who has made this work possible. This research was supported by the Planetary Atmospheres program of the National Aeronautics and Space Administration under Grants NSG-2047 and NSG-7085. REFERENCES ARAKAWA, A., MINTZ, Y., AND COLLABORATORS (1974). The UCLA atmospheric general circulation model. (Available from the authors at the Department of Meteorology, University of California, Los Angeles, California 90024.) BLUMSACK, S. L., G1ERASCH, P. J., AND WESSEL, W. R. (1973). An analytical and numerical study of the

182

HABERLE,

LEOVY, AND POLLACK

Martia~r planetargl boundary :la~er~.oy¢r slopes. ~. Atmos. Sci. 30, 66-82. BURK, S. D. (1976). Diurnal winds near the Martian polar caps. J. Atmos. Sci. 33, 923-939. CONRATH, B., CURRAN, R., HANEL, R., KUNDE, V., MAGUIRE, W., PEARL, J., PIRRAGLIA, J., WELKER, J., AND BURK, T. (1973). Atmospheric and surface properties of Mars obtained by infrared spectroscopy on Mariner 9. J. Geophys. Res. 78, 4267-4279. CUTTS, J. A. (1973). Wind erosion in the Martian polar regions. J. Geophys. Res. 78, 4211-4221. CUTTS, J. A., BLASIUS, K. R., BRIGGS, G. A., CARR, M. H., GREELEY, R., AND MASURSKY, H. (1976). North polar region of Mars: Imaging results from Viking 2. Science 194, 1329-1337. DEARDORFF, J. W. (1972). Parameterization of the boundary layer for use in general circulation models. Mon. Weather Rev. 100, 93-106. DOLLFUS, A. (1973). New optical measurements of planetary diameters. IV. Size of the north polar cap of Mars. Icarus 18, 142-155. GIERASCH, P. J., AND GOODY, R. M. (1968). A study of the thermal and dynamical structure of the lower Martian atmosphere. Planet. Space Sci. 16, 615646. GOODY, R. M., AND BELTON, M. J. S. (1967). Radiative relaxation times for Mars. Planet. Space Sci. 15, 247-256. HABERLE, R. M. (1979). The influence of the Martian polar caps on the diurnal tide. Icarus 39, 184-191. HANEL, R., CONRATH, B., HOVIS, W., KUNDE, V., LOWMAN, P., MAGUIRE, W., PEARL, J., PIRRAGLIA, J., PRABHAKARA, K., SCHLACHMAN, B., LEVIN, G., STRAAT, P., AND BURKE, T. (1972). Investigation of the Martian environment by infrared spectroscopy on Mariner 9. Icarus 17, 423-442. HESS, S. L. (1973). Martian winds and dust clouds. Planet. Space Sci. 21, 1549-1557. HESS, S. L., HENRY, R. M., LEOVY, C. B., RYAN, J. A., AND TILLMAN, J. E. (1977). Meteorological r e sults from the surface of Mars: Viking 1 and 2. J. Geophys. Res. 82, 4559-4574. KIEFFER, H. H., CHASE, S. C., MINER, E. D., MUNCH, G., NEUGEBAUER, G., AND MARTIN, Z. (1976). Infrared thermal mapping of the Martian surface and atmosphere: First results. Science 193, 780-786. KIEFFER, H. H., MARTIN, T. Z., PETERFREUND, A. R., JAKOSKY, B. M., MINER, E. D., AND PALLSECONI, F. D. (1977). Thermal and albedo mapping of Mars during the Viking primary mission. J. Geophys. Res. 82, 4249-4292. KLIORE, A. J., LJELDBO, G., SEIDEL, B. L., SYKES, M. J., AND WOICESHYN, P. M. (1973). S band radio occultation measurements of the atmosphere and topography of Mars with Mariner 9: Extended mission coverage of polar and intermediate latitudes. J. Geophys. Res. 78, 4331-4352.

LEOV~Y, C. B, (197.9), M a ( t i ~ . n ~ t e r o l o g y . Ann. Rev. Astron. Astrophys. 17, 387-413. LEOVY, C. B., AND MINTZ, Y. A. (1969). Numerical simulation of the atmospheric circulation and climate of Mars. J. Atmos. Sci. 26, 1167-I 190. LEOVY, C. B., ZUREK, R. W., AND POLLACK, J. B. (1973). Mechanisms for Mars dust storms. J. Atmos. Sci. 30, 749-762. LEOVY, C. B., AND ZUREK, R. W. (1979). Thermal tides and Martian dust storms: Direct evidence for coupling. J. Geophys. Res. In press. MARTIN, T. Z., AND KIEFFER, H. H. (1979). Global thermal behavior of Mars in dusty conditions. II. 1-15 p.m band measurements. J. Geophys. Res. In press. MATSUNO, T. (1966). Numerical integrations of the primitive equations by a simulated backward difference method. J. Meteorol. Soc. o f Japan 4, 76-84. MURRAY, B. C., SODERBLOM, L. A., CUTTS, J. A., SHARP, R. P., MILTON,D. J., AND LEIGHTON,R. B. (1972). Geological framework of the south polar region of Mars. Icarus 17, 328-345. PETEREREUND, A. R., AND KIEFEER, H. H. (1979). Global thermal behaviour of Mars in dusty conditions. II. Local dust clouds. J. Geophys. Res. In press. PIRRAGLIA, J. A. (1975). Polar symmetric flow of a viscous compressible atmosphere. J. Atmos. Sci. 32, 60-72. POLLACK, J. B., LEOVY, C. B., MINTZ, Y., AND VAN CAMP, W. (1976a). Winds on Mars during the Viking season: Predictions based on a general circulation model with topography. Geophys. Res. Lett. 3, 479-482. POLLACK, J. B., HABERLE, R., GREELEY, R., AND IVERSEN, J. (1976b). Estimates of the wind speeds required for particle motion on Mars. Icarus 29, 395-417. POLLACK, J. B., COLBURN, D., KAHN, D., HUNTER, J., VAN CAMP, W., CARLSTON, C. E., AND WOLF, M. R. (1977). Properties of aerosols in the Martian atmosphere, as inferred from the Viking lander imaging data. J. Geophys. Res. 82, 4479-4496. POLLACK, J. B., LEOVY, C. B., GREIMAN, P., MINTZ, Y. H., AND VAN CAMP, W. (1979a). A Martian general circulation experiment with large topography: Transport properties and comparison with observations. Submitted. POLLACK, J. B., FLASAR, M. F., CARLSTON, C. E., PIDEK, D., AND KAHN, R. (1979b). Properties and effects of dust particles suspended in the Martian atmosphere. J. Geophys. Res. In press. RYAN, J. A., HESS, S. L., HENRY, R. M., LEOVY, C. B., TILLMAN, J. E., AND WALCEK, C. (1978). Mars meteorology: Three seasons at the surface. Geophys. Res. Lett 5, 715-815. SAGAN, C., VEVERKA, J., FOX, P., DUBISH, R., FRENCH, R., GIERASCH, P., QUAM, L., LEOERBERG,

MARTIAN POLAR CAP WINDS J., LEYlNTHAL, E., TUCKER, R., EROSS, B., AND POLLACK, J. B. (1973). Variable features on Mars. 2. Mariner 9 global results. J. Geophys. Res. 78, 4163-4197. SCHNEIDER, E. K. (1977). Axially symmetric steady state models of the basic state for instability and climate studies. II. Nonlinear calculations. J. Atmos. Sci. 34, 280-296. SUTTON, J. L., LEOVY, C. B., AND TILLMAN, J. E. (1978). Diurnal variation of the Martian surface layer meteorological parameters during the first 45 sols at two Viking lander sites. J. Atmos. Sci. 35, 23462355. TILLMAN, J. E., HENRY, R. M.', AND HESS, S. L. (1979). Frontal systems during passage of the Martian polar hood over the Viking lander 2 site prior to the first global dust storm. J. Geophys. Res. In press. VANKATESH, S., AND CSANADY, G. T. (1974). A

183

baroclinic planetary boundary-layer model, and its application to the Wangara data. Boundary-Layer Meteorol. 5, 459-473. WARD, A. W. (1978). Windforms and wind trends on Mars: An evaluation of Martian surficial geology from Mariner 9 and Viking spacecraft television images. Ph.D. thesis, The University of Washington. WOLFE, R. W. (1979). Orientations of eolian features in the north polar region of Mars: Preliminary assessment. Lunar and Planetary Science, Vol. X, 1364-1366. Lunar and Planetary Institute, Houston, Texas. WOOD, G. P., WEAVER, W. R., AND HENRY, R. M. (1974). The minimum free-stream wind speed for initiating motion of surface material on Mars. NASA TM X-71959. ZUREK, R. W. (1976). Diurnal tide in the Martian atmosphere. J. Atmos. Sci. 33, 321-337.