A preliminary assessment of Martian wind regimes

ZC~LRUS14, 312--318 (1971)

A Preliminary Assessment of Martian Wind Regimes PETER

G I E R A S C H 1 AND C A R L SAGAN

Laboratory for Planetary Studies, CorneU University, Ithaca, New York 14850 Received December 14, 1970 Elevation differences of the order of an atmospheric scale height are now known to exist abundantly over the Martian surface. The time-independent and frictionless thermal wind equation is solved for radiative-convective atmospheres at the level of scaling analysis, to determine the influence of topography on the mean wind, for large horizontal length scales. For small scales nonlinear slope winds dominate. MaYimum relief winds appear to occur in the lateral scale 200 to 1000 km. Winds produced by relief appear able on Mars to increase the mean winds by factors of 2-3 over the mean winds in the absence of relieL Thus winds at the half surface pressure level of 100-140 m/sec may not be uncommon on Mars. Such winds make the lifting of fine particles off the Martian surface very likely ; and place serious constraints on the times and places of unmanned landing operations on Mars. :[NTRODUCTIOI~T A p a r t from its intrinsic meteorological interest the question of Martian wind regimes impacts a n u m b e r of s t u d i e s - a m o n g t h e m explanations of seasonal and secular changes on Mars t h r o u g h windblown dust (cf. Sagan and Pollack, 1967, 1969), and considerations of the safety of landing vehicles such as Viking entering the Martian atmosphere. The t h e r m a l and d y n a m i c a l s t r u c t u r e o f the Martian atmosphere has been the subject of a n u m b e r of studies (Hess, 1950; Ohring and Mariano, 1969; Wen Tang, 1967; Gierasch a n d Goody, 1968; L e e r y a n d Mintz, 1969). L e e r y and Mintz p e r f o r m e d a general circulation e x p e r i m e n t with an atmosphere stratified into only two levels. Their calculations also exhibited the f o r m a t i o n of COe polar caps and strong tidal motions. Gierasch and G o o d y p e r f o r m e d a timed e p e n d e n t radiative-convective calculation of the t h e r m a l s t r u c t u r e of the Martian atmosphere, with high resolution in the vertical, revealing a deep diurnal t h e r m a l b o u n d a r y layer and d e v e l o p m e n t 1Permanent address: Institute for Geophysical Fluid Dynamics, Florida State University, TaUahassee, Florida 32306. 312

o f strong d a y t i m e turbulence t h e o u g h o u t the troposphere. All previous studies h a v e implicitly assumed t h a t topographical relief on Mars can be neglected, a n d a d o p t e d a s m o o t h lower b o u n d a r y to t h e atmosphere. B u t we now know (Sagan et al., 1967 ; PettengiU et al., 1969; Lincoln L a b o r a t o r y , 1970) t h a t the vertical relief on Mars exceeds the m e a n Martian scale height over lateral distance scales ranging from a few h u n d r e d kilometers to several t h o u s a n d kilometers. I t is of the greatest i m p o r t a n c e to determine w h a t effect this p r o m i n e n t Martian relief has on the behavior of the atmosphere. However, the interaction of topog r a p h y with atmospheric motions is an e x t r e m e l y complicated problem. Some degree o f success has been achieved in studying this interaction on the E a r t h (Queney, 1948; K a s a h a r a , 1966); b u t not enough is k n o w n for conclusions reached for the E a r t h to be e x t r a p o l a t e d to Martian conditions where the relief is c o m p a r a t i v e l y m u c h greater. On the E a r t h , m o u n t a i n ranges and continental relief are t y p i c a l l y a fraction of a scale height in vertical extent, and radiative time constants are longer t h a n in the Martian atmosphere. Flows over e x t e n d e d t o p o g r a p h y can

313

MARTIAN W I N D R E G I M E S

probably not be treated as adiabatic for Mars, while they often can be so treated for the Earth. Holton's (1967) paper is particularly relevant to the present work. He considers thermally driven boundary layer slope winds over the Great Plains of North America. Although the physics of his discussion is slightly different from what we discuss presently because his work is a boundary layer study only, it is interesting t h a t his calculated wind magnitudes, when scaled up by the ratio of Martian to terrestrial slopes, agree with ours within an order of magnitude. Bearing these difficulties in mind, we attempt in the present paper a first, very approximate, study of this problem, on the level of a scaling analysis. Our primary motivation is to determine whether wind velocities might be much greater than has been calculated in the past on the assumption of no significant topographical relief. We will calculate the thermally driven winds for horizontal length scales L larger than ~1000 km. We employ the thermal wind equation and the assumption of approximate radiative-convective equilibrium to estimate temperature gradients and wind velocities. For small horizontal length scales (L <200 kin) we will find t h a t nonlinear slope winds dominate. The maximum wind speeds should occur over intermediate length scales, 1000 km > L > 200 kin, but because of the complexity of the equations we do not attempt a quantitative solution for this intermediate horizontal scale. Elsewhere, we estimate under which conditions certain topographically induced winds observed on E a r t h (for example, comer effects and obstacle winds) might occur on Mars ; and estimate crudely the influence of topography on the intensity of small scale thermal convection and on the incidence of dust devils. (Sagan, Veverka, and Gierasch, 1971). We do not attempt to discuss tidal phenomena, boundary layer effects, or condensation. These are all certainly important, but because of the difficulties they present, they are beyond the scope of the present paper. Our estimates are therefore far from complete. W h a t we have done is to isolate a few of the most

easily estimable phenomena which can be expected; the results derived on this basis alone seem to be quite striking. TOPOGRAPHY AND THERMAL W I N D S

We shall adopt the hydrostatic approximation and write the equations of motion in pressure coordinates. In a deep atmosphere this coordinate system provides the simplest formulation of the thermal wind equation. Neglecting time dependence and friction, the equations are (Eliassen, 1949): a¢ 2Dsin 0v + ~z = 0, (1)

-29sinOu + 0y = o,

(2)

O¢/Oh = R T ,

h =-In(p/p0),

(3)

(4)

where p = pressure, P0 = some reference pressure, h = height, measured in pressure scale heights, above level p = P0, R = gas constant for the atmosphere, T = temperature, ~b= geopotential height, x, y = east to west and south to north cartesian coordinates, u, v = eastward, northward wind components, 9 = rotation rate, and 0 = latitude. These equations are valid when the Rossby number, Ro = U ( 2 9 L s i n O ) -1, is small, where U and L are the characteristic wind speed and the horizontal length scale, respectively. I t will turn out t h a t Ro small places a restriction t h a t L be large; after estimating the quantities we shall return and evaluate just how large L must be. These equations are the ones which describe, to leading order, the relation between wind and temperature fields in extraequatorial regions of the terrestrial atmosphere. Solving for u and v in terms of T we obtain Ou

- -

1

---- 29 sin0 Ov

Oh = ~

1

R

OT Oy"

(5)

_ OT

~ ~"

(6)

We shall assume t h a t the temperature field is given, to a first approximation, by

314

PETER GIERASCHA-ND CARL SAGAN

local radiative-convective equilibrium. The assumption is not rigorously justifiable, but seems reasonable. Both Leovy and Mintz (1969) and Gierasch and Goody (1968) find it to be true in their models (for which Ro < 1). Radiative-convective temperature profiles in the Martian atmosphere are quite independent of surface pressure (Gieraseh, 1971), but do depend on surface temperature. The surface temperature variations however are quite decoupled from variations in relief (Sagan and Pollack, 1968). In the presence of topographic relief the thermal field will therefore be given by a function of the form T(x, y, h) = F[T,(x, y), h - h,(x, y)],

(7)

where T. and h, denote, respectively, surface temperature and surface geopotential height. We shall take this function to be of the form

The wind shear is in two parts: one due to the gradient of surface temperature, the other due to gradients of the surface elevation. The former is the familiar seasonal thermal wind which appears in Leovy and Mintz's calculations as strong westerlies (as high as 50 m/sec in the lower atmospheric layer) in the winter hemisphere, weak easterlies in the summer hemisphere, and moderate westerlies in both hemispheres at equinox. The latter is a new term (denoted here b y the subscript R for relief) whose magnitude we can estimate as follows: Assume that winds will typically be smallest near the ground and increase with height. The winds at mid-atmosphere level will then be given by, e.g., U~ ~- ½OUR~Oh. This gives us

1 R 0h. UR -- 4.Osin 0 R T , ~ ~ , 1

v a ~_ ~ s i n o R T , ~ [h - h.(x, yl];

(8)

i.e., the atmospheric temperature equals some mean diurnal surface temperature T.(x, y) at the ground, and decreases with height with the adiabatic lapse rate. This is consistent with the calculations of Gierasch and Goody and with the occultation observations reported b y Rasool, Hogan, Stewart, and Russell (1970). It is slightly inconsistent with the results of Leovy and Mintz ; they predict a lapse rate of slightly more than haft the adiabatic. Their value would reduce somewhat the wind speeds which we estimate below, but would not change our principal conclusions or the qualitative picture which will emerge below. We take T, to be only a function of the north-south coordinate y. Equations (5), (6), and (8) then give Oh

2~ sin 0 0 y - [ R R T , Oh. 2~Osin O % Oy '

av

R

R T , Oh.

8-h = 2-~Osin 0 Cp 0x"

(h - h.)

] (9) (10)

R Oh,

-~-~.

(11) (12)

We do not attempt to discuss the vertical structure of the wind field in any detail, but only to estimate a mean magnitude. The question of vertical structure and surface velocities involves the strength of vertical turbulent mixing, the development of instabilities within the wind field, and other extremely complicated processes. Surface winds may therefore be less than the numbers we present below, but this is a highly uncertain point. Because the radiative time scales are shorter and the diurnal temperature fluctuations are larger on Mars, we expect more turbulent mixing on Mars; therefore momentum should be carried down to the surface boundary layer more strongly on Mars than on the Earth. The question is sufficiently uncertain that a lander experiment, for example, should not be designed on the assumption that the velocities decline sharply towards the surface boundary layer. The direction of the relief wind, from Eqs. (11) and (12), is toward one's left as one faces uphill in the northern hemisphere, and reverse in the southern hemisphere where sin0 is negative. Figure 1 is a schematic representation of isotherms over

315

MARTIAN WIND REGIMES

J t

. _ _ s

p p , s / ~ 4 , ~,

]~'Io. 1. The dashed lines are isotherms. Horizontal temperature gradients imply thermal winds.

Seosonal Wind

I

Fz¢~. 2. Seasonal winds are augmented on poleward slopes of elevations and equatorward slopes of depressions. t o p o g r a p h y according to our assumptions, displaying the origin o f the horizontal t e m p e r a t u r e gradients which drive the t h e r m a l wind; Fig. 2 shows the direction of flow over an elevation or depression. Table I gives magnitudes of u R as a function of the large scale slopes. We have also e v a l u a t e d a Rossby n u m b e r b y assuming t h a t the slope e x t e n d s far enough so t h a t the total elevation difference equals one scale height. The R o s s b y n u m b e r reaches u n i t y for slopes o f a b o u t 1% (10 k m over 1000 km), and velocities are largest for the largest slopes. T h e largest measured slopes on Mars are TABLE I THERMAL WIND MAGNITUDE AND ROSSBY NUMBERS FOR DIFFERENT TOPOGRAPHIC ~LOPES AT LATITUDE30 °, ASSUMrSG Ts = 220°K a Slo~

UR(m/sec)

L(km)

Ro

0.003 0.005 0.010 0.020

26 40 80 160

3000 2000 1000 500

0.14 0.30 1.2 4.8

a The horizontal scale L is the distance which corresponds to one scale height of relief; Ro is based on L.

N l % (Pettengill et al., 1969). We e x p e c t t h a t the t h e r m a l wind e q u a t i o n will b r e a k down for slopes > 1 % or for relief of small horizontal extent. I n this and the following discussion the winds in question are the winds at some mean level in the atmosp h e r e - s a y half the surface pressure; t h e y are also the time-average winds, and gusts m a y make the i n s t a n t a n e o u s velocities on occasion larger. The Rossby n u m b e r is the ratio of acceleration t e r m s to Coriolis t e r m s in the m o m e n t u m equations. We shall a t t e m p t to gain insight into the n a t u r e of t h e r m a l l y driven winds over t o p o g r a p h y of a small horizontal scale b y considering n e x t a v e r y simplified model in which acceleration t e r m s alone balance the t h e r m a l l y induced pressure gradients, with Coriolis forces neglected entirely. SMALL HORIZONTAL SCALE TOPOGRAPHY

The thermally induced pressure gradients due to relief point up or down the slopes, with no c o m p o n e n t along contour lines. This follows from our a s s u m p t i o n t h a t the surface t e m p e r a t u r e is ind e p e n d e n t of altitude (Sagan and Pollack, 1968). F o r small length scales we can also

316

P E T E R GIERASCH AND CARL SAGAN

consider T S to be independent of latitude. In the case where pressure forces are balanced b y accelerations, it follows that motions will be primarily up or down slopes, and we shall therefore consider a two-dimensional flow with no variations along contour lines. The equations are, for steady frictionless flow (Eliassen, 1949), au Ou a¢

u~+w~+~=0, a¢

a--h= RT,

au Ow ax+~-w=O,

phere; its value is about 5 days for surface pressures of about 5 mb (cf. Goody and Belton, 1967). We make no attempt to solve these equations. B y equating orders of magnitude, however, one finds that the only consistent scaling results in the following estimates of wind speed and associated temperature fluctuations:

(13) U

~R RT~ \1/3 ,-LZlh,| ,

\% ra

(17)

/

(14)

AT~ (15)

where w = dh/dt and x now denotes the uphill direction. Winds will blow in the direction of the temperature gradients in this case (see Fig. 3). In the case of the thermal wind, the wind direction was along the isotherms. The energy equation contains advective terms which are proportional to the component of wind velocity a/on 9 temperature gradients. These terms m a y be important in the present case; if they are, radiativeconvective equilibrium will not obtain. We must therefore include the energy equation in the present model, and permit the possibility that the thermal field will be given b y a balance between advective influences and local radiative-convective influences. We shall use the approximate equation aT OT T~. -- T u ~ - + w Oh rR (16) where TE(h) is the radiative-convective equilibrium profile, and ~ is a radiative time constant characteristic of the atmos-

~

R,R2

j

,

(18)

where cp is the specific heat capacity at constant pressure, L is the horizontal scale of topography, and Ahs is the characteristic scale of topographic relief. These scalings represent dynamical control of temperatures; the other possibility, that T _ T~. in Eq. (16), is inconsistent unless r ~ < roughly 1 Mars day, in contradiction to the results quoted above. The results are insensitive to vertical scale. We also find

[R RTs

1 ~1/3,

(19)

where f = 2 • s i n 0 ; Ro must be greater than unity for the acceleration terms to be more important than the Coriolis ones as we have assumed. Table II gives values of these quantities for different L's, assuming AhB ~ 1 scale height, Leovy has pointed out to us that if rx were replaced b y a shorter convective time scale, which might be appropriate during the Martian daytime, velocities would be increased. We see the opposite trend to that exhibited in Table I. Here wind magnitudes increase with L while in Table I they

III I I i II IIIIIll

I1'/I t

/111¢11111111111111/ ~ o . 3. Schematic diagram of streamlines for small scale topography.

MARTIAN WIND REGIMES

TABLE II WIND ~.AG:NITUDE U, TEMPERATURE PERTURBATIONS A T , AND ROSSBY NUMBERS Ro, AS FUNCTIONS OF HORIZONTAL SCALE L, 1fOR SMALL SCALE TOPOORAPHY (km)

(m/see)

u

~,

(°K)

Ro

50 100 200 400

9 12 15 19

0.17 0.44 1.1 2.8

2.5 1.6 1.0 0.63

L

decreased with L. Therefore for fixed elevation differences over a varying horizontal length scale, L, we expect maximum winds to be reached between a few hundred and about 1000 km--unfortunately a range in which the Rossby number restrictions make our calculations break down. However, the best expectation is that very large topographical relief--on the order of a scale height--is not to be expected on very small lateral resolution scales of the order of 100 km or less. Accordingly the small scale topographic slope winds given in Table II are upper limits. INTERMEDIATE HORIZONTAL SCALES

It appears from the foregoing sections that the largest thermally induced winds can be expected for horizontal relief scales within the range 200 km < L < 1000 km. This conclusion must be accepted with caution; the simplifications we have made are extreme. Nevertheless, it seems possible that topographic thermally induced winds m a y be equal in magnitude to the seasonal winds predic.ted b y Leovy and Mintz. Since the topographic and seasonal winds m a y be additive, the full wind could be double the previously expected values. SOME CONSEQUENCES

Winds of some 80 m/sec just above the surface boundary layer appear to be required to raise fine dust from the surface of Mars (Sagan and Pollack, 1967, 1969). While some doubt has been expressed in the past (cf. Horowitz, Sharp and Davies,

317

1967) on whether such velocities are frequently achieved on Mars, these doubts were based on the magnitudes of the seasonal thermal winds in fiats. The foregoing discussion makes it quite likely that dust is raised frequently and under complex and diverse circumstances on Mars. Accordingly the possibility that albedo differences on Mars are connected with particle size differences, and the possibility that seasonal and secular variations in bright and dark areas are due to windblown dust (cf. Sagan and Pollack, 1967, 1969) must now be seriously considered. These questions and specific connections of winds with Martian topography are discussed in a companion paper (Sagan, Veverka, and Gieraseh, 1971). Here we restrict our attention to the question of the safety of unmanned landing vehicles upon entry through the Martian atmosphere. For example, the present design criteria of the Viking lander, intended for injection into the Martian atmosphere in Summer, 1976, specify a very great undesirability of landing through tropospheric winds as high as 40-60 m/sec. For other reasons potential Viking landing sites are restricted to ±30 ° latitude, but with the full 360 ° in longitude. Arrival is in Northern Summer. Comparison of Table I with present knowledge of Martian topography indicates certain regions to be avoided. For example, an extremely poor place to land would be the northern interior of the Hellas basin, which is just within the 30°S latitudinal limit. The seasonal thermal winds in the flats just outside of Hellas are westerlies with velocities of about 40-50 m/sec. The relief winds in Hellas at this time of year (Southern Winter) are clockwise as viewed from above with speeds of up to 90 m/sec, depending on whether the relief in this region approaches a scale height-there is at the present time no good information on the precise topography o{ Northern Hellas. Thus there appears to be a significant probability that the mean winds in Northern Hellas at the time of the Viking landing will be between 70 and 140 m/sec; this does not seem to be a desirable landing site.

318

PETER GIERASCH AND CARL SAGAN

There are other kinds of orographic winds. For example, there are indications from terrestrial experience (see the companion paper for discussion) that the polar edges of extensive ridges should be avoided. Thus there is an excluded region on the downhill polar side of Amazonis, near Nix Olympica, which might otherwise be of scientific interest. For the Viking aerodynamic braking to be effective, landing must be secured in regions of relatively high surface pressure. Therefore, the safest areas for Martian landings of systems such as Vil~ing appear to be summer and equatorial mean altitude to lowland fiats, such as Western Aethiopis-Cerberus, Moeris Lacus-Southern Thoth Nepenthes, Chryse-Xanthe, and southwestern Amazonis. The apparent connection between regions with dangerously high winds and regions of high scientific interest may not be coincidental, as is discussed further in the companion paper (Sagan, Veverka, and Gierasch, 1971).

ACKNOWLEDGMENT We are grateful to Conway Leovy for a thoughtful and constructive reading of an earlier version of this paper. This research was supported in part by NASA Contract No. NGR 33-010-098 and in part by the Atmospheric Sciences Section, National Science Foundation, under Grant No. GA 10836.

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