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previous work on the wind and wind stress necessary to raise dust. We shall then examine the general-circulation winds needed to produce the required surface stress. Next we will look at some special processes for producing extreme winds. Finally, we shall discuss a feedback mechanism that can explain lateral growth from an initially small cloud that must, however, exceed a critical size and depth. 2. SURFACE WIND STRI’GSS AND DUST It is clear that the turbulent stress of the wind on the surface is the critical parameter determining whether dust particles of a given size wit1 be raised from the surface. The fundamental work for terrestrial conditions is that of Bagnold (1941). The basic parameter used is often called the friction velocity and is defined as

v* = V’Tp

(1)

where T is the stress of the wind on the surface and p is the air density. Bagnold reports that so long as the flow around the particles is fully turbulent, the threshold friction velocity needed to move a particle is V,,

=

A

_&

[ Pa

‘I8

d

1

(2)

where A is a dimensionless constant of value 0.1 in terrestrial air, g is the acceleration of gravity, p1 the density of the particle, p. the density of air, and d is the diameter of the particle. If the microscale flow is not fully turbulent, however, it becomes more difficult to move the dust. That is, A is no longer constant but increases in value. Bagnold reports that a Reynolds number defined as Re = V,td/v

(3)

where Y is the kinematic coefficient of viscosity, determines whether A is constant or not. He asserts that for Re 2 3.5, A = O-1. Below that value A is a function of Re and increases with decreasing Re. Thus a typical curve of V,t vs d shows a minimum required stress at some diameter. Particles of that size are the first to be lifted from the surface and no dust is raised until that minimum Vet is attained. The application of these results to Mars has been dealt with by Ryan (1964) and by Sagan and Pollack (1967). In particular, the results of Sagan and Pollack for a 5-mbar CO, atmosphere on Mars are that the most easily moved particles have a radius of about 270 ,um and the minimum V,,. is nearly 4 m se&. This is a large value of friction velocity by terrestrial standards and implies a wind above the boundary layer that is a very substantial fraction of the speed of sound on Mars. Unfortunately, Sagan and Pollack give no details of their calculations. We will re-examine this matter quantitatively. Equation (2) as used by Bagnold for Earth reduces to Vet = 149OA l/d

(4)

where cgs units are used. With this and Bagnold’s Fig. 28 it is possible to determine A(_Re) below the critical value of Re. Adopting the observation of Sagan and Pollack that Bagnold’s own values suggest no departure of A from 0.1 until Re < 2, we find the results

1551

MARTIAN WINDS AND DUST CLOUDS TABLE 1.

PROPERTIES

OF

A DEDUCED FROM BAGNOLD'S FIO. 28

(cmVi2~-l) 0*0156 040!93 09037

18.0 15-o 20.0

A

A’

Re

0.100 0.104 0.220

0900 -

2.00 1.00 0.50

in Table 1, where A’ is the derivative of A with respect to Re. These values may be represented by the polynomial

This is an exact fit to A and A’ at Re = 2.0 and to A at Re = O-5. It is also a very close fit to A at Re = 1.0. Equation (5), applicable for O-5 I Re I 2.0, and A = O-1 for Re > 2.0, constitute the best general statement about A(Re) that we can presently use to transfer terrestrial results to Mars. Equation (2), applied to a 5-mbar, 0” C, COz atmosphere at Mars becomes (in cgs units) V *t= 1.024 x IPA6

(6)

Thus, using Equations (31, (5) and (6) with v = 14.5 cm2sec-r for CO, we can determine Yet as a function of particle size on Mars. The result is given in Fig. 1. The most easily moved particles have a radius of 242 pm, in reasonable agreement with earlier results (especially since the minimum is broad and any particle from 200-300 pm is about equally movable). The minimum value of I/,, is, however, found to be 2.5 m set-r, nearly a factor of two lower than the resuh of Sagan and Pollack for the same pressure. A reduction in the necessary value of friction velocity by under a factor of two may be thought to be unimportant in view of the nature of the calculation. However, the

4r

‘\ ‘.

::

\

3

$ E .-9 2

\ \ \ t '\ I ,/

\ '\ \ '. ,/ ;< '. /' '3 '.\ /' '. 'z %_

‘\

'X\, -.

'. 2. I -‘IG

“-0.5 100

200

300

a. FIG. 1. THETH~HO~~~ON~~YR~~~

400

pm RAISE

OF PARTKLR

500

DUsTONMARsASAPUNCTION

VELOCZTY.

The dashed line is the uncorrected result from Equation (6) and the solid line shows the result corrected for dependence of A on Reynolds number. The m~um is marked by a vertical bar. Dotted lines represent the indicated constant values of Reynolds umber.

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SEYMOUR

L. HESS

consequences for lander design are substantial in precisely this range of values; hence, it is worthwhile to be as precise as possible. There are several factors that can influence the choice of V,, with which to deal. These have been discussed by Sagan and Pollack and include the atmospheric pressure. Since Equation (2) involves air density, it is apparent that V,, is inversely proportional to the square root of pressure. Thus for a lo-mbar atmosphere the present calculations would be lowered to a minimum V,, of 1.75 m sec- l. Indeed, the preferred location for dust storms to originate is Hellas-Noachis, parts of which are known to be low compared to the mean elevation. It is certainly easier, from this point of view, to raise particles at certain areas within Hellas-Noachis but there are other low regions of the planet which do not seem to spawn storms, so there must be other factors at work. For the purpose of further calculations we shall assume that dust will lift off the surface when V, reaches 2-O m set-l which corresponds to a pressure of nearly 8 mbar. 3. THE PLANETARY

RESISTANCE

LAW

We wish now to relate our estimate of the surface stress (friction velocity) to the frictionally undisturbed flow at the top of the planetary boundary layer which may be assumed to be geostrophic and produced by the general planetary circulation. It is clear physically that the surface stress is produced by this geostrophic wind, G, and that V, and G must be related to each other. A difhculty in determining such a planetary resistance law lies in the fact that the connecting boundary layer (l-2 km thick) consists of two sublayers controlled by very different scale factors. The lowest one has a short length scale and the wind is represented under conditions of neutral stability by the well-known Prandtl logarithmic law. Above this is a thicker layer, with a much larger length scale, in which pressure-gradient force, Coriolis force and turbulent forces are all important. Under barotropic and neutrally stable conditions the wind here is represented by the Ekman spiral law. Only in the past few years has it become possible to overcome the problem of combining two inconsistent scales (see for example, the elegant solution by Blackadar and Tennekes, 1968). The results are: u/V, = i In (z/zJ

us/V* = $ln us/V* = -B/k

(2)

(7) - A] (9)

where u is the wind at height z near the surface, u, and v, are the two components of the geostrophic wind, k is the von Karman constant (0*4), z,, is the roughness parameter, f is the Coriolis parameter, and A and B are two constants of integration. Equation (7) is the Prandtl logarithmic law which applies in the lower boundary layer. Equations (8) and (9) together are the planetary resistance law determining G/V, and the angle between the surface and geostrophic winds. To utilize these results for our purposes we need the values of A and B. From the nature of the derivation these constants should be universal; there is nothing that makes them particular to Earth. Therefore, we may examine their terrestrial values and apply

MARTIAN

WINDS

AND DUST CLOUDS

1553

to Mars. Wipperman (1970) has examined the observational evidence and has argued that the best values are A = 0.9 and B = 4.5. Newer unpublished measurements support these values. With these and V, = 2-O m see-l, Equations (8) and (9) imply that G/V, w 30, a result which is not seriously dependent upon the adopted values of A and B. Thus a geostrophic wind of some 60 m se+ is required to produce a surface stress that will just raise dust particles of radius 200-300 ,um. This result is somewhat below the estimate by Sagan and Pollack of 300 km hr-l (82 m se+). The only numerical, general-circulation study for Mars presently available is that by Leovy and Mintz (1969). That model only rarely produces winds in excess of 60 m se+ at the top of the planetary boundary layer.? However, Leovy and Mintz neglected all topography while we now know that there is considerable relief to the Martian surface. The thermal effects of topography are quite capable of increasing these winds significantly as has been shown by Gierasch and Sagan (1971). Another factor that can change this result is the effect of static stability. The quantities A and B have the approximate values given only in the case of neutral stability. We do not know adequately what numbers to use for unstable conditions such as those that should prevail during part of daylight hours. However, we can be sure that the result of instability is to increase the friction velocity for a given geostrophic wind simply because the increased vertical mixing then will prevent the boundary layer from protecting the surface as effectively as it does in the neutral or stable conditions. Thus during daytime, a somewhat smaller geostrophic wind will suffice to produce the critical stress. It will be useful also to know the wind speed at the top of the logarit~ic layer as well as at the top of the entire planetary boundary layer that will produce a surface stress corresponding to Y*$ = 2-O m see-l. This can be done with Equation (7). The result is that a wind of 38 m se+ at a height of 50 m will produce the threshold stress if z, = 2.5 cm. This is insensitive to the choice of z0 since changing z, by a factor of two alters the calculated wind by less than 10 per cent. We conclude that general circulation winds on Mars are occasionally capable of raising dust from the surface but this capability is reduced in the summer (when large dust storms originate) because of lower winds then. It is, however, increased when the effects of topography are included and during the early afterno0n.t them

4. PROCESSES

PRODUCING

EXTREME

WINDS

Several processes are known here on Earth that can produce extreme winds under special conditions and there is reason to believe that some of these occur on Mars also. In addition, we have theoretical reason to believe that one process of litt1e significance here can become much more important on Mars. The terrestrial phenomena, beyond those discussed here in Section 3, that may occur on Mars seem all to be local in character and small in lateral scale. First there is a group that are related to the mechanical and thermodynamic effects of topography. These include complex downslope flows such as the Chinook winds of western North America and the t A specialstudy of the computer tapes produced by Leovy and Mintz has been performed by the U.S. Viking Project. This shows that the model produces winds in excess of 60 m se& at only 2-3 per cent of combined time and location over a 5-day period and in the latitude band lO”N-42”S at summer solstice of the northern hemisphere. $ This disagrees with the conchtsion of Golitsyn (1973) presented verbally at the 1972 COSPAR meeting. He used a threshold friction velocity of 4.0 m see-r and, as a result of this higher stress, concluded that general circulation winds are incapable of raising dust.

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foehn winds of the European Alps, as well as the strong winds that can blow in restricted channels between converging uplands. At present, we do not know enough about Martian topography on an appropriately small scale to form any quantitative estimates. The strength of certain local winds on Earth, however, can be sufficiently large that we cannot dismiss these phenomena as unimportant (i.e. the Chinook winds of Colorado are known to exceed 50 m se+ at only 2 m above the surface). Next, there is a set of terrestrial phenomena that are not caused by topography but are largely thermal in nature. These include, in order of decreasing scale, hurricanes, tornados and dust devils. We can immediately exclude hurricanes since they are known to depend upon the latent heat of condensation of water vapor as the source of energy, a source that is completely inadequate on Mars. We can likewise exclude tornados which, although they do not derive their energy primarily from condensation of H,O, do depend upon release of latent heat to produce special vertical, thermal instabilities. Again, the water-vapor content of the Martian atmosphere is inadequate to support this mechanism.? We are left with dust devils which are a consequence of strong, thermally produced, vertical instabilities. These are expected on Mars since the static instabilities there are predicted to be stronger than on Earth (Gierasch and Goody, 1968). The scale here is relatively small. Finally, the chief process peculiar to Mars stems from the much shorter radiative relaxation time in the atmosphere of Mars. This has been used by Gierasch and Sagan (1971) to predict an effect of large-scale topography on winds via a thermal rather than a mechanical effect. This process is subsumed here under our discussion of generalcirculation winds. 5. SCALING

ANALYSIS

AND GROWTH

We shall consider three scale of disturbances produced by a thermal inequality in order to determine if any of them contain a mechanism for growth to a much larger scale. Our motivation is the recognition that a closed feedback process is possible, in principle, in which a thermal gradient produces a pressure gradient, this generates wind which may raise dust, and the presence of dust can substantially affect the temperature (Gierasch and Goody, 1972) and its gradient. The three lateral scales may be defined loosely as: small, a few tenths of km; medium, a few tens of km; and large, a few hundreds of km. Small-scale disturbances, such as dust devils, are initiated by thermal inequalities but their characteristic velocities are determined directly by convergence of pre-existing vorticity into a small radius (conservation of angular momentum) rather than by the thermal gradient. Furthermore, they probably depart from hydrostatic equilibrium. Thus it is neither appropriate nor easy to do a scaling analysis with the temperature gradient. One can obtain a wind of almost any value by assuming a slow general rotation with convergence to a sufficiently small radius. Thus there is no difficulty in obtaining winds capable of raising dust and, indeed, dust entities on almost this scale have been observed from Mariner 9. Since these phenomena rapidly become mechanical rather than thermal in nature the feedback mechanism suggested above is broken and one does not expect lateral growth.$ Indeed, the strong convergence in such swirls confines the dust to a small radius rather than allowing it to grow. Of course, the t These exclusions are not meant to imply that disturbances of the scale and magnitude of hurricanes and tornados are impossible. If such entities exist on Mars other mechanisms than the terrestrial ones will have to be found to produce them. $ The medium scale is intented to be large enough to avoid this breaking of the feedback mechanism. Thus its assignment of tens of km is very loose.

MARTIAN WINDS AND DUST CLOUDS

1555

dust is carried upward and is ultimately thrown outward but this is not responsible for any great increase in scale. An exception to this point of view would be a whole field of dust devils which could create a dust veil distinctly larger than any one of the responsible vortices. The consequences of such an event are properly considered under medium-scale flows. For medium- and large-scale motions one may utilize the radial equation of motion for an ideal, inviscid gas with steady-state and circular symmetry,

u---au ar

2

(10)

r

and the equation of hydrostatic balance,

RTalnP = --g.

(11)

aZ

Here, r is radial distance, u and v are the radial and tangential velocity components, f is the Coriolis parameter, R is the ideal gas constant, andp is the pressure. We can eliminate all reference to pressure by cross differentiation to obtain

(12) where a small term involving aT/az has been neglected. Equation (12) relates velocities and their derivatives to the horizontal temperature gradient. We note that in convergent rotary flows of the sort we are interested in, the two terms in parentheses are of the same sign. We take U as a characteristic velocity (whether radial or tangential), L as a characteristic lateral length, Has a characteristic vertical scale and AT as a characteristic temperature increment in the distance L. The three terms of Equation (12) then have magnitudes: I acceleration terms: 2V/LH II Coriolis term: fU/H III pressure-gradient

term: (g/L) (AT/T).

When the scale is small enough that Coriolis effects are negligible, the thermally induced pressure gradient results in accelerations and we balance I against III: U2 - igHAT/T. When the scale is large enough we expect quasi-geostrophic of the Coriolis and pressure-gradient terms, II against III:

(13) flow, i.e. the balance is that

LJ - UlfL) (.gH ATIT). These results become parallel if one introduces the Rossby number, Ro = U/fL. Then the geostrophic result is U2 - Ro gH AT/T.

(14)

Since the Rossby number expresses the ratio of accelerations to Coriolis force per unit mass, Ro greater than unity means Equation (13) is the dominant relationship while when Ro decreases somewhat below one, Equation (14) should begin to govern. In fact, Equation (14) is nothing more than the well-known thermal-wind equation of meteorology.

1556

SEYMOUR

L. HESS

Unless the Rossby number is very much lower than one, the values of U from Equations (13) and (14) do not differ appreciably. Thus we can calculate one order-of-magnitude wind for both cases. The value AT = 20” K is indicated by the calculations of Gierasch and Goody (1972) for the increase of temperature due to absorption of solar radiation by a dust cloud and is consistent with the radio occultation results from Mariner 9 when a large dust cloud was present. Thus an increment in temperature from cloudy to clear air of this value is appropriate. With a mean temperature of about 200” K, H must be about 8 km in order for U to reach the value of 38 m set-l we have estimated is required at the top of the Prandtl layer in order to raise dust. We then propose that some process, as yet unspecified, must create a cloud of dust of at least the order of 10 km in height and several tens of km in radius. In such a cloud we expect absorption of sunlight to create a horizontal thermal gradient adequate to generate winds capable of raising still more dust, thus causing the cloud to grow in size. Under this hypothesis the cloud could begin to grow while in the medium scale of size, under control of a balance between accelerations and pressure-gradient force. When it is sufficiently large, control shifts to a balance between Coriolis and pressure-gradient forces via the thermal wind law, Equation (14). One might suppose that when the cloud has grown large enough, the Rossby number will have diminished to the point where the winds are no longer adequate to raise dust and the process would stop. This is not the case since the length scale, L, is really the distance in which the chosen AT occurs. This will always be at the outer edge of the cloud and will not increase as the radius of the cloud grows. If we suppose, for example, that rotational effects begin to be important at Ro = $Cand the required wind is 40 m set-l withy= 1O-4 see-l then L is of the order 800 km. This hypothesis permits a cloud to grow indefinitely and so is consistent with the observation that Mars has dust storms at least hemispheric in size and often planet wide. However, there is at least one process that could stop the growth. The upward transport of dust will permit it to be conveyed laterally by the upper general-circulation winds and, in some cases, this could spread the temperature increment over a long distance. Then the self-generated winds would fall below the threshold value and no more dust will be raised. Of course, if the dust cloud grows to cover the entire planet, there will no longer be a contrast between cloudy and clear air and the self-generated winds discussed here will die out. In any case, when self-generation vanishes the remainder of the life of the cloud will, in the simplest case, be characterized by gravitational settling of the dust against viscosity and turbulent mixing. Two outstanding questions remain. What processes can create a cloud of the critical size and thickness? What is the time scale of the self-generating process described here? With respect to the first question, the following possibilities exist: (1) General circulation winds as discussed in Section 3 above. (2) Coexistence of a number of dust devils in a sufficiently large area. (3) Orographically induced strong thermal winds as described by Gierasch and Sagan (1971) in a basin or on a plateau of adequate dimensions. (4) Special orographic effects such as the Chinook wind and channel winds. We simply do not know enough yet to make a choice among these possibilities. With respect to time scale, one can distinguish between the response time of a dust cloud to solar radiation and the hydrodynamical time for growth of the cloud. Gierasch and Goody (1972) report that their radiative calculations give a rapid response with a scale of no more than 2-3 days. This is quite rapid enough to be consistent with observations. The dynamical processes probably should be considered in greater detail than attempted here in order to estimate the growth

MARTIAN WINDS AND DUST CLOUDS

1557

rate. At least, one sees nothing as yet to suggest that the time scale will be so long as to militate against the process we suggest. 6. CONCLUSION

A feedback process (by which an initial dust cloud exceeding certain dimensions could modify its own temperature through absorption of sunlight and so create a thermal gradient which can induce winds strong enough to raise more dust) seems to be possible and plausible as an explanation of the very large dust clouds that occur on Mars. One of its bases is the heating of a cloud by solar radiation, so it is consistent that such clouds occur at southern hemisphere summer solstice when, because of the eccentricity of the orbit of Mars, the intensity of solar radiation is greatest. The preferred location of origin in the NoachisHellas region is somewhat equatorward of the latitude of maximum insolation but this area may well be dictated by peculiarities of topography that promote generation of initial clouds of the critical size. If this hypothesis is correct, the process is peculiar to Mars because the much greater density of Earth’s atmosphere would render the radiative time scale very much longer than on Mars. One would not expect an adequate thermal effect on a terrestrial dust cloud to develop before the cloud is dissipated by general winds or has settled out gravitationally. REFERENCES BAGNOLD,R. A. (1941). The Physics of Blown Sand and Desert Dunes pp. 265. Methuen, London. BLACKADAR,A. K. and TENNEKES,H. (1968). Asymptotic similarity in neutral barotropic planetary boundary layers. J. atmos. Sci. 25, 1015-1020. CAPEN, C. F. and MARTIN, L. J. (1971). The developing stages of the Martian yellow storm of 1971. Bull. LoweN Ohs. No. 157,7,211-216. GIERASCH,P. J. and GOODY, R. M. (1968). A study of the thermal and dynamical structure of the Martian lower atmosphere. Planet. space Sci. 16,615~646. GIERASCH,P. J. and GOODY,R. M. (1972). The effect of dust on the temperature of the Martianatmosphere. J. a tmos. Sci. 29, 400-402. GIERASCH,P. J. and SAGAN, C. (1971). A preliminary assessment of Martian wind regimes. Icarus 14, 312-318. GOLITSYN,G. (1973). On the Martian dust storms. Icarus. In press. LEOVY, C. and MINTZ, Y. (1969). Numerical simulation of the atmospheric circulation and climate of Mars. J. atmos. Sci. 26, 1167-1190. RYAN, J. A. (1964). Notes on the Martian yellow clouds. J. geophys. Res. 69, 3759-3770. SAGAN, C. and POLLACK,J. B. (1967). A windblown dust model of Martian surface features and seasonal changes. Smithsonian Astrophysical Observatory, Special Report 255, pp. 44. WIPPERMAN,F. (1970). The two constants in the resistance law for a neutral barotropic boundary layer of the atmosphere. Beitr. Phys. Atmos. 43, 133-140.

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