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Martian atmospheric Lee waves
ICARUS27,517-530 (1976)
Martian
Atmospheric JOSEPH
Laboratory
Lee Waves
A. PIRRAGLIA
for Planetary Atmospheres, Goddard Space Flight Center. Greenbelt, Maryland 20770 Received June 27, 1975; revised August 7, 1975
The Mariner 9 television pictures of Mars showed areas of extensive mountain lee ware phenomenon in the northern mid-latitudes during winter. In most cases the characteristic wavelength of the lee waves is readily observable and in a few cases the boundaries of the wave patterns, as well as the wavelength, are observed. The cloud patterns resulting from the waves generated by the flow across a mountain or crater are dependent upon the velocity profile of the air stream and the vertical stability of the atmosphere. Using the stability as inferred by the temperature structure obtained from the infrared spectrometer data, a two layer velocity model of the air stream is used in calculations based on the theory of mountain lee waves. The parameters that yield a pattern similar to that in a picture with a well-defined wave configuration are a lower 11 km deep air stream of 40m/sec and an upper air stream of 85m/sec. This calculation appears to be an upper limit of the wind speeds, with most of the pictures implying wind speeds of the lower layer to be less than 40m/sec. These results yield magnitudes generally in agreement with circulation models, in particular, the Leovy and Mintz (1969, J. Atmos. Sci. 26, 1167) two layer numerical model. Under not too different conditions they calculate winds of approximately 30 and 70m/sec in the lower and upper layers, respectively.
Atmospheric waves in the lee of terrestrial mountains are common and fairly well understood. The theory of mountain lee waves has been discussed in a series of papers by Scorer (1949, 1953,1954), among others, and in particular for an isolated peak by Scorer (1956) and Scorer and Wilkinson (1956). Several cases were observed during the Mariner 9 mission and interpretation of the patterns yields information on the wind profile with height. One of the objectives of the Mariner 9 mission to Mars was the investigation of the atmospheric thermal and dynamical behavior. The thermal structure was obtained for extensive, though not complete, coverage in local time and latitude in the earlier part of the mission and for somewhat less extensive coverage in the latter part of the mission (Hanel et al., 1972; Conrath et al., 1973). The dynamical state of the atmosphere can be investigated using the thermal structure or can be Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
inferred from apparent cloud motions (Briggs and Leovy, 1974). However, when the temperature data is supplemented by an additional observable, in particular, cloud patterns as in lee waves, winds can be inferred within certain constraints and serve as bounds on independently derived motion. This approach is discussed briefly by Briggs and Leovy (1974) and is the problem which will be addressed here. During revolutions I 79 t,o 220 of Mars the spacecraft was able to view the northern mid-latitudes during late winter. The pictures obtained during that period showed extensive waves in the clouds that comprise the edge of the polar hood (see Briggs and Leovy for a collection of cloud pictures). In many instances the cloud formations were obviously associated with atmospheric waves in the lee of mountains or craters. We shall consider two examples of wave patterns and attempt to determine the airstreams which, wher, used in calculations based on the theory of mountain lee waves with the vertical stability as inferred 517
518
*JOSEPH A. PIRRAGLIA
by t,he thermal st8ructure, yield wave pat,terns similar t’o those observed. We shall sket,ch 8corer’s theory repeating parts with minor modifications for continuit’y. A simple ext’ension of Scorer’s analysis will be used t’o introduce effects due t’o the horizontal dimensions of the crater generating the wave patterns. Rather than trying to reproduce the cloud patterns by calculating the vertical motions and condensation we shall calculat’e only the surfaces of constant phase. The wave patterns will be inferred by considering se&ions along the wave crest where t)he wave amplit’ude exceeds some reasonable but arbitrarily prescribed value. In most of the pictures the only easily observable parameter of the wave pattern is t,he wavelength. While the whole pattern is needed to define the airstream profile the wavelength alone does permit the establishment of limits on the velocity near the surface. This will become apparent in the course of our exposition. It will be shown that there is agreement bet’ween the observations and t,heorv and that’ in the lat&udes and season if t’he observations the theory is in agreement with independently obt’ained numerical resuhs. In a few instances wave patterns with the appearance of Mach cones give the impression of supersonic wind velocities but the analysis suggests ot,herwise. FREE MODES 0~ STATIONARY W'AVES The lee wave patt’ern is due to the discretje spectrum of waves generated by a crater ridge in an airstream. The discrete spectrum is the set of eigenmodes whose wa,venumbers are the solutions t’o t,he zero frequency dispersion equation which is derived from the homogeneous equation for the airstream. The free modes are infinite in lat’eral extent, being the type of wave generat’ed hy an infinitely long ridge not necessarily normal to the air&earn, and the relative amplitudes of the modes are obtained by expanding a ridge in terms of the stationar? waves with t)he ridge acting as a forcing function. The method is identical to a (Green’s function approach in which the
ridge is used as the source t)erm. A lee wave pattern of a crater is obtained by summing over a parallel set of ridges oriented at all angles between &tx/d from t’he normal t,o the airstream. The velocit’y, density, entropy and acoustic velocity of an undisOurbet1 airstream are designated by uO>pO, qO, and c0 respectivel)-. Using the approach of Eckart, (1960)the parameters associated wit’h the kinematics and dynamics of t,he aLtmosphere are defined by
where t’he prime indicates differentiation with respect to z. y and R are the ratio of the specific heats and the gas constant. N is the Brunt-V&is&l& frequency, r is a reciprocal scale height related bo the adiabatic changes in the vertical direct’ion, and /? is the stat’ic stability. Wit’h the stationary wave vector and horizont8al space vect’or defined by k and r t’he vertical velocity, II*, of a st’ationary wave is defined by rc =
W(p,c,)-1”
exp(ik.r+~j~~~2).
(2)
and assuming the acoustic velocity to be great’er than the stream veIocity, t,he homogeneous equat,ion for W is
(3) To obtain (3) it has also been assumed that t’he Rossby number is large and that t)he perturbation velocity of the stream is small compared to the stream velocity. One of the simplest’ models, and the one that, we shall use, is a t)wo layer model in which t)he velocity is uniform within a layer and in the same direction in both layers. The density decreases exponentiall> with altitude, the Brunt-V&is&l& frequency is constant and both are continuous across
519
MARTIAN ATMOSPHERIC LEE W-4VE
the interface of the two layers. In for the thermal st’ructure atmosphere in which we are interested, the fifth and sixth terms are much smaller than the first term in the brackets of (3) and may be neglected. subscripts 1 and 2 to identify quantities interface at z = 0, the surface at z = -h and let’ting 4 represent the angle between t’he wave vector and stream direct’ion we have from (3)
(a)
U’; + [(W/u~,)
(b)
x K, = 0, O > z ’ -hY 11’; + [ (N2/ug2) sec2 + - Ii21 x IV2 = 0,
The int,erface conditions
see2 4 - li2] (4)
z 2 0. are
(a)
uo, TV; = uo21i.;,
(b)
HV,/u,, = W,/u,z,
(5)
making the pressure and displacement continuous across the interface. At the surface the boundary condition is IV, (-h) = 0.
(‘3)
Solving the set (4)) (5) ,and (6) under the condition that the wave energy decreases with height, that is the waves are trapped (Corby and Sawyer, 1958; Scorer, 1949), the e’igenvalue equation for the wavenumbers J%is obtained, V cot’ (vh) + (zco2/uo,)2 p = 0,
angle 4 to t)he direction of the airstream. The modes have t’he horizontal behavior, ?[, _ eik,(x cos @+Y sin 9) 3
(9)
where x is in the direction of the air&ream and y is normal to the airstream. If the waves are assumed to be generated by a ridge at an angle + to the stream and at a distance d from the origin of the x-y pla’ne, as shown in Fig. 1, we ha,ve, 2u_ eik,lr cos(fl-@)+dl =- eik,t 3 (10) with r and 6 as shown in Fig. 1. Solutions of t’he form (10) represent t,he free modes but in order to determine t’he amplitudes the source of the waves must be considered. In addition, we are interested in the vertical excursion of an atmospheric parcel not its vertical velocity since the clouds will presumably form or be more dense along the crests of t,he waves in the streamlines. The vertical excursion, &,, of a streamline for the wave of wavenumber ,$ is relat~ed t’o the vertical velocitj> by ik(z, 4) = (zc+/iku,)eik’ (11) to within a8constant, phase parameter. A ridge of assumed profile 5, (0 is described in terms of the st,abionary waves b> l,(f) = CI(1 + p/w-“2 = (d/n)
Re Ir eMkb+ ikcdk,
(12)
(7)
where (a) y2 = (X2 sec2 +/a$,) - L2 = 1: - Ii2, (b) $ z ,$’ - (AT2secz$/ui2) = iL2- 1:. (8) The Scorer parameters 1, and I, are t,he limits of the possible values of Ic. Solutions of (7) give t’he values of the wavenumber li, of the free modes of the airstream.’ The roots k, define the modes of the st’ationary waves with wave vectors at an 1 In considering only these modes we have neglected the modes that, are supercritical in both layers ; k c I,, 2,. These modes are of consequence at high altitudes and would have little effect on our results (Holmboe and Klieforth, 1957). The reference was brought to the attention of the author by Dr. Lawrence A. Schmid.
FIG. 1. Horizontal geometry of airstream, ridge and wave directions. The ridge is at a distance d from the z-y and r-6’ coordinate origin and its normal is at an angle 4 to the airstream uO. The wave vector k is normal t,o the ridge.
520
JOSEPH
A. PARRIGLIA
with Re designating the real part. Normalizing i$ to the amplitude of the displacement at the surface and using this with (12) which represents the boundary condition at z = -h when a ridge is in bhe air&earn we have
more detailed discussion of the k-plane integration.] Evaluation of the lee waves is straightforward and is simply the residue of the poles : &,r,e)
= 2iA 2
jniz
d+
-7712
ab,M-h) h(z) -
(, (z,r,B) = -
%l(z) P,(W)
7r
&, is the superposition of waves of all k’s and represents the vertical displacement) of a streamline due to the stationary waves with wave vectors perpendicular to the ridge and at, an angle qSto t,he airstream as shown in Fig. 1. If we no& consider a set of parallel ridges separated by 2d and take
where j= 1 when -h 0. Because of the eekrb term t’he mode with the smallest root will be dominant, and we shall consider it only and drop t(he summation and the r-subscript. Separabing the integral into parts containing E,+,and f,_ t’he integral is evaluated by t,he method of st’ationary phase where the stationary points c&, and $,_ are detined by
(1%)
(b)
; (ktL) <=0.
On constJantlphase surfaces defined by the c:onst)ant M. then integrate ~-12n-e tl AV<’
(I 3) over C#J front
~9/2 to (19)
1(;)
Using (1 X) and (II)), equations are obt~ainetl for 6’ and T in terms of the stationary points C& and +,_ :
[ (-_, t’, 0) is t’he superposition of and represents the complete solution for t’he displacement of the streamlines of HoU across a crater. The discrete free modes represent’ed by the solutions to (7) are at the poles of the integrand. Integration in the complex h--plane is chosen so t’hat the waves are downstream from the crater, the integration pat,hs being different depending upon whether fw or [,_ are less than or greater t)han zero. ISee Scorer for a
where (I/k)ak/&$ is obtained from the dispersion equation (7) and the upper and lower signs are associat’ed with the W and L subscripts, respectively. Equations (20) and (2 1) yield r and 0 along constant phase fronts where the phase is det’ermined b) the const’ant’ Al.
“‘(‘)
-kb(rik:w
+ (,ik:,.)
(
where waves
of’ all X-‘8 and directions
MARTIANATMOSPHERICLEE WAVE The stationary gives the result
phase evaluation
of (17)
112 1 eik,t,+i(n/4)
(22) where all the terms are evaluated at the stationary point & which equals & or & depending upon whether the wave is due to the windward or leeward rim of the crater and the sign of ~14 is + or - depending upon whether 82/a42 (k.$W,L) is greater than or less than zero. When the second derivative is equal to zero (22) is singular and the solution is similar to (22) but with the curly bracketed term replaced by condensation
l/3
J/3 1 eiktw+
i(a/6)
eik:L
with r (4/3) being the gamma function. If for take a2(ktW, L)i+2 ’ 0 we -41 = (6 + 8n - 1) n/4 and for a2(kEw L)/ a+’ G 0 take M = (6 + 8% + 1) x/4 with n=0,1,2 ,..., the constant phase surfaces are wave to windward and rims the Using (21), and amplitude and of wave be calculated of
DISCUSSION CALCULATIONS The displacement the lines to being upon the parameters also depend upon the parameters a and b which describe the crater ridge, as is evident in (15) and (22). As pointed out previously,
interference
521
i
f
524
JOSEPH A. PIRRAGLIA
waves fill a wedge shaped region whose edge makes an angle of approximately 18” with the apparent wind direction and the wave length of t’he transverse waves along the centerline of the pattern is approximately 6Okm. As implied by the stationary phase calculation there is construct’ive interference of waves of directions + associated with values of 6 inside the wedge and destructive int’erference outside the &edge. This is seen in (20) in which there is a maximum of the absolute value of 0 beyond which no solution exists. Using the model described in t’he previous seeCon by assigning values too N2, which depends upon the t’emperature profile, t,o the airstream speeds 740, and /do2 and to the depth h of the lower airsOream we attempt t’o construct t)he wave pattern which could generate the cloud configuration of Fig. 2. Unfort8unately. there is no temperat’ure dat’a available for the t,ime of the picture but’ other da& foi the same lat’itude and local time indicates that, t’he at’mosphere was substantially subadiabat’ic and in fact may have had temperature inversions. As an approximaBtc mean, in our models we shall use an isot’hermal atmosphere where from (lb), /3 = piSo s g/(c,Y’). Any stream velocities can then be considered to be normalized t’o t,he isothermal case and t’he velocit) associat,ed with any other value of p is obtained by multiplying the isothermal case by @/PiSo)+. 3 shows families of airstream
mately 30 and 45m/s and appropriate values of ua2 and
with
the
tiOm/s. In each part of the figure lines of const’ant
pattern of Fig. 2 is indicated portion of each sect’ion is the limit beyond which there is no transverse
bounded set, of airstreams produce the correct, angle and wavelengt,h. obvious one from the figure is the case with = 85m/s and 11,= 1 I km. %I = 40m/s, ~1.5~~ Any model with ua, bet,ween approxi-
c
t-1
100
km
FI(:. 4. Calculated wave patterns. l!art (a) is the wave system generated by a 100kmdiametcr crater in an air-stream with a 45m/s, 15km deep lower stream and a 92m/s upper strealn. Part (h) is with a 4Om/s, 11 km deep lower stream and 85m/s upper stream. Part (c) is with a 35m/s. 7.8 km deep lo\rer stream and 82.5m/s upper stream.
MARTIAN
ATMOSPHERIC
parameters, [35, 83, 81, [40, 85, 111, and [45, 92,153 as indicated. These were chosen by interpolation in Fig. 3. Model [35, 83, 81 has wave crests with more curvature than those in Fig. 2 and model [45, 92, 151 are too flat while [40, 85, 111 is a reasonably good reproduction. The observed patterns of Fig. 2 lie between the two extreme models of Fig. 4 which yield velocities without very large differences so that the gross features of the airstream would be obtained in any case. The clouds would form along the cont’ours of Fig. 4 depending upon the shape
LEE
WAVE
525
of the ridge as manifested in the parameters a and b of (12). The amplitude of the waves are directly proportional to the height, a, of the ridge but depend upon the width parameter, b, as b[exp (bk,)] and because of t,he k, dependence vary along a wave. For shallower slopes the amplitudes of the transverse waves are dominant but’ as the slope is increased the amplitudes of the divergent waves increase relative to the transverse waves. The clouds in Fig. 2 appear to be along the transverse waves of the outer wedge, which is due to thewindwardridge,andalong the
FIG. 5. Mariner 9 television photograph taken during revolution 183. The crater on the left border is the same one shown in Fig. 2. The time of this picture precedes that of Fig. 2 by two Martian days and there are no transverse waves here as are evident in Fig. 2. 19
526
JOSEPH
A. PIRRAGLIA
waves of the inner wedge, which is due to t,he lee ridge. This suggests that the windward ridge of the crater is not as steep as the leeward ridge. The inner and outer wave systems are apparent in the pat’terns in Fig. 4. Figure 5 is an example of a less common wave pattern. This picture covers approximately the same area as Fig. 2 and t,he cloud pattern is due to t’he same crater but precedes that of Fig. 2 by one MarCan day. It is apparent that, the transverse wave system is not complete. This is because as predicted by (7) for the conditions of the airstream at that time t’here is no solut’ion for lowhen absolute values of 4 are less than some nonzero magnitude. The waves for those directions are below cutoff. The type of wave pattern in question is not’ defined on t.he diagrams of Fig. 3, but similar diagrams could have been constructed using the wavelength along the outer edge rather than the center of the patt’ern. In that case, they could be locat’ed on the wavelength-wedge angle plane and there lvould be no apparent difference from wave patterns with complete transverse waves. As before t’here is more than one model that will give the proper wedge and wavelength along some prescribed direct,ion. However, with only short segments of the waves observable it’ becomes difficult to use the contour of Dhe constant phase surface to choose the optimum parameters although one can determine maximum values on the lower stream velocity and depth. A candidate airstream is 120, 85,4] whose lee wave pattern is shown in Fig. 6. There is a good similarity between Figs. 5 and 6 especially in the first three waves in the lower part of the wake and in the characteristic streaks flaring out from the crater. The airstream of Fig. 6 followed by that of Fig. 4 by one day implies an upper layer of 85m/s, uniform in time, over a lower layer of 20m/s velocity increasing to 40m/s, and of 4km depth increasing to I I km. This behavior is plausible being suggestive of lower layer growt*h. The theory is essentially good only for far fields, i.e., kr large, but can in fact be used to give near field results for the free
FL<:. 6. Tile calculated wave system gerwratocl by a lower stream of ZOm/s and 3.5km depth and an upper stream of 85m/s blowing o\.cr il 100 km diameter crater.
modes while ignoring the continuous part, of the wave spectrum. An example of the agreement between theory and observat,ion is indicated by Fig. 7. This picture shows the first and second wave crest,s of the lee waves of a 45 km crater at 63 N, 347 W (this picture is 200” in longitude from, ant1 32 days prior to Fig. 2). The theory predicts that the first crest is 518 of a wavelength from the windward ridge of the crat,er. The wavelength between the t,wo crests is 45 km and the distance from the windward rim to the first crest is 28km which is in very good agreement with the theory. Returning to Fig. 3 for a moment and confining t’he discussion to patterns with complete transverse waves, the changing character of the wave pattern can be observed as a function of the airstream parameters. As the lower stream velocity increases the minimum depth of the layer must increase to produce patterns with complete transverse waves. For increasing lower stream depth the pattern becomes less dependent upon the upper stream velocity as, for example, along the h = 10 line for uO, = 20. As the difference between the upper and lower stream velocities decreases the pattern becomes less dependent upon the depth as is seen along t,he u 02 = 30 line for uoI = 20. B0t.h cases are intuitively evident. Of particular interest
MARTIdS
ATMOSI’HERIC
LEE
WAVE
527
528
JOSEPHA. PIRRAGLIA
to us is the observation that for a given lower layer velocity a lower limit can be placed on the wavelength by the constraint that k cannot exceed 1,) as seen from (8). k equals 1, where the contours of Fig. 3 touch the zero wedge angle line. The limit on Ic is established without the aid of Fig. 3 but the figure shows that for a given wavelength wide wedge angles infer small lower stream velocities. The implications from the observations will be discussed subsequently. NONUNIFORMSTREAMDIRECTIONS If the two streams of the model have different directions we have another parameter of the airstream that must be determined from theobservations. Because, for the most part, we cannot even determine three parameters no attempt, will be made to determine a fourth as
expressed by the angle between the upper and lower streams. However, calculations showing the types of patterns that are the result of nonuniform stream directions will be presented. If the angle between the two streams is 4 the eigenvalue equation for k: is, v
cot (vh) -
UOI cos(4 + $1 *IL= UOI
cos+ 1
(
0,
(2.3)
where (a)
v* = (N’ secZ +/t$,)
- k’,
(b)
p2 = k* - IN* set* (4 + $)/u&j,
(24)
with (23) and (24) used in the place of (7) and (8) the analysis proceeds as before. Using a model with a lower stream velocit’y of 20m/s and depth of 5km and an upper stream velocity of 80m/s the patterns were calculated for cases in which the angle between the streams was 4.5’
/ , I/,’
‘Y
Yo, = 20 m/s uo2 = 80
I’
m/s
h
: 5 Km
$I
-900
uo, = 40
m/s
uo2:
m/s
80
h
: 10 Km
*
=900
D
P’IG. 8. C’alculated wave patterns for two layer air streams in whicll tile upper and lo~vrr vclocitics Ilaw different directions. TIE speeds, lower laysI. depth and angle betwren the velocities of ttle 11ppcr and lower streams arc irldicakxl in each srrtion of the figurr.
MARTIANATMOSPHERICLEE WAVE
529
suggest a large difference in the wind directions. With small differences and without isolated patterns it is not possible to determine the difference in the stream direction but, nevertheless, one may use the wavelengths to give an estimate of the winds in the lower layer. In most of the television pictures the wavelengths of the apparent lee waves are approximately 30 km and of the type shown in Figs. 2 and 4 rather than those of Figs. 5 and 6. Although the lack of many clear cases of wave patterns showing the wedge angle prevents a determination of the upper airstream velocity, an upper limit can be put on the velocity of the lower stream. From Fig. 3 it is apparent that wavelengths of 30km are associated only with airstreams with lower stream velocities less than 40m/s since the observations CONCLUSIONS suggest wide wedge angles and from the discussion in the previous section they are Regardless of the simple model used for associated with the smaller lower stream the airstream, the calculations appear to be velocities for a given wavelength. capable of reproducing the observed wave Pictures like Fig. 5 give the impression patterns. Insofar as the model is a reasonable approximation to the actual airof very high velocities but the results of the calculations shown in Fig. 6 and st,ream, were it not for the paucity of temperature data at the latitude and seacompared with Fig. 5 imply that there is no son of the observations, accurate assessneed to invoke extremely high velocities ments of the wind velocities could be made. to explain the particular wake pattern and, Since all our calculations were for a 200K in fact, the lower stream velocity is less isothermal atmosphere the velocities would than in the more common cases with transverse waves. have to be adjusted by the square root of the ratio of the stabilities, as was pointed The conclusion may be made that the out previously. If the temperature profile many observations of lee waves suggest approached an adiabatic state the velocithat the near surface wind speeds as t’ies as calculated would decrease toward implied by the lower stream velocities do zero. If there were a temperature inversion, not greatly exceed, and most of the time t’he velocities would increase, but to inare less than, 40m/s in the northern crease the velocities by 25% over those midlatitudes during the late winter except of the isothermal case would require an possibly on a very localized scale. This inversion of more than BK/km. Such an result is in agreement with the wind inversion is large and not very likely, magnitudes obtained independently, in making the possibility of velocities greater particular with t’he two level numerical by 25’/, improbable. model of Leovy and Mintz (1969) where The first wave in the clouds shown in they calculated the mean zonal winds at Fig. 2 is not normal to the apparent stream 50N and at winter solstice to be 30m/s direction and may be due to the nonuniat 3km and 70m/s at 13.5km. Winds of formity of the wind direction with altitude. this magnitude are in substantial agreeAs illustrated previously, the wave patterns ment with the winds implied by the lee are not symmetric about a horizontal line waves. when the windstreams have different A two layer model with a discontinuous directions but the wave patterns do not, velocity profile is subject to Kelvin-
and 90”. Similar calculations were made with a 1Okm deep lower stream of 40m/s. The results are shown in Fig. 8. The calculations indicate that a pair of streams with similar depth and wind speeds will yield shorter wavelengths for larger angles between the streams. However, the observed wave patterns generally are not suggestive of the type of patterns shown in Fig. 8 and the conclusions drawn from t’he parallel stream case of the previous section should be valid. The streams may be off parallel but not by the amounts used for the calculations of Fig. 8. If there were indications of greatly different wind direct’ions, without observations of isolated lee wave patterns it would be exceedingly difficult to determine four parameters of the airstream.
530
JOSEPH
A. PIRRAGLIA
Helmholtz instabilities at all wavelengths but the only waves apparent in the pictures are the extensive periodic trains of lee waves which are due to trapping in a waveguide formed by layers in which t,he Scorer parameter, 1 = Nsec+/?l,, allows propagation in a lower layer a,nd reflect’ion in an upper layer. However, a more realistic model would not have a velocit> discontinuit>-. A three layer model with uniform upper and lower velocities coilnect)ed cont’inuously through the varying middle laper is unstable t’o only a small band of wavenumbers which are dependent upon the Richardson number and an) Kelvin-Helm holtz instabilit8y wavclengtlr is likely to be far retnovetl from the ICY wave lengt,hs. Tn addition it permit8s t)rapping of the waves and is general enough t)o bc at’ lea,st a rough approximation t#o anv actual airstream with it monotonic p&file. The use of a tfvo Ia~.er model in a lee wave analysis is just’ified bjr t hr fact, that’ if t)he product oft he t8ransition layer thickness and the wavenumber is much less than unity the t*\\.o layer approximation in which the t’ransit’ion layer is absent’ is a good approximation fol the calculat~ion of lee waves in a continuously varying monotonic velocity profiles airstream. 0n~ interprebabion of the results baser1 upon the comments above irnplies a volocit!y profile of t’wo fairly tlist)inc$ levels connected by a transitSion layer of greater shear than eit’her of the atlj;ining layers. The velocities of the model approximat,e t’he average velocit8ies oft he upper and lower layers with the lower level wind velocit,ies being represent’at’ive of t,he wind velocit)ies above the surf;lce boundarp lay”‘. Another interpret’at’ion is t’hat of a layer of uniform velocity over a shear layer in which the velocities in the lower layer are not much different’ from the average of the layer. As before, in the lower layer t’he winds derived are averages and are not) surface winds. The models are a crude approximation to the real wind profiles with the results being representative of the magnitudes one might, find in t’he actual cases.
ACKNOWLEDGMENTS The ttuthor is indebted to Dr. B. J. Conrath for llis trrlpfut discussions and to tie and Dr. R. A. Hanet for critical comments. The Mariner 9 MTVS pllotoprapt>y has been provitlctl by t I)(% Na~tiorral Space Scirncr Data Cc:ntc>r.
Martian
Atmospheric JOSEPH
Laboratory
Lee Waves
A. PIRRAGLIA
for Planetary Atmospheres, Goddard Space Flight Center. Greenbelt, Maryland 20770 Received June 27, 1975; revised August 7, 1975
The Mariner 9 television pictures of Mars showed areas of extensive mountain lee ware phenomenon in the northern mid-latitudes during winter. In most cases the characteristic wavelength of the lee waves is readily observable and in a few cases the boundaries of the wave patterns, as well as the wavelength, are observed. The cloud patterns resulting from the waves generated by the flow across a mountain or crater are dependent upon the velocity profile of the air stream and the vertical stability of the atmosphere. Using the stability as inferred by the temperature structure obtained from the infrared spectrometer data, a two layer velocity model of the air stream is used in calculations based on the theory of mountain lee waves. The parameters that yield a pattern similar to that in a picture with a well-defined wave configuration are a lower 11 km deep air stream of 40m/sec and an upper air stream of 85m/sec. This calculation appears to be an upper limit of the wind speeds, with most of the pictures implying wind speeds of the lower layer to be less than 40m/sec. These results yield magnitudes generally in agreement with circulation models, in particular, the Leovy and Mintz (1969, J. Atmos. Sci. 26, 1167) two layer numerical model. Under not too different conditions they calculate winds of approximately 30 and 70m/sec in the lower and upper layers, respectively.
Atmospheric waves in the lee of terrestrial mountains are common and fairly well understood. The theory of mountain lee waves has been discussed in a series of papers by Scorer (1949, 1953,1954), among others, and in particular for an isolated peak by Scorer (1956) and Scorer and Wilkinson (1956). Several cases were observed during the Mariner 9 mission and interpretation of the patterns yields information on the wind profile with height. One of the objectives of the Mariner 9 mission to Mars was the investigation of the atmospheric thermal and dynamical behavior. The thermal structure was obtained for extensive, though not complete, coverage in local time and latitude in the earlier part of the mission and for somewhat less extensive coverage in the latter part of the mission (Hanel et al., 1972; Conrath et al., 1973). The dynamical state of the atmosphere can be investigated using the thermal structure or can be Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved. Printed in Great Britain
inferred from apparent cloud motions (Briggs and Leovy, 1974). However, when the temperature data is supplemented by an additional observable, in particular, cloud patterns as in lee waves, winds can be inferred within certain constraints and serve as bounds on independently derived motion. This approach is discussed briefly by Briggs and Leovy (1974) and is the problem which will be addressed here. During revolutions I 79 t,o 220 of Mars the spacecraft was able to view the northern mid-latitudes during late winter. The pictures obtained during that period showed extensive waves in the clouds that comprise the edge of the polar hood (see Briggs and Leovy for a collection of cloud pictures). In many instances the cloud formations were obviously associated with atmospheric waves in the lee of mountains or craters. We shall consider two examples of wave patterns and attempt to determine the airstreams which, wher, used in calculations based on the theory of mountain lee waves with the vertical stability as inferred 517
518
*JOSEPH A. PIRRAGLIA
by t,he thermal st8ructure, yield wave pat,terns similar t’o those observed. We shall sket,ch 8corer’s theory repeating parts with minor modifications for continuit’y. A simple ext’ension of Scorer’s analysis will be used t’o introduce effects due t’o the horizontal dimensions of the crater generating the wave patterns. Rather than trying to reproduce the cloud patterns by calculating the vertical motions and condensation we shall calculat’e only the surfaces of constant phase. The wave patterns will be inferred by considering se&ions along the wave crest where t)he wave amplit’ude exceeds some reasonable but arbitrarily prescribed value. In most of the pictures the only easily observable parameter of the wave pattern is t,he wavelength. While the whole pattern is needed to define the airstream profile the wavelength alone does permit the establishment of limits on the velocity near the surface. This will become apparent in the course of our exposition. It will be shown that there is agreement bet’ween the observations and t,heorv and that’ in the lat&udes and season if t’he observations the theory is in agreement with independently obt’ained numerical resuhs. In a few instances wave patterns with the appearance of Mach cones give the impression of supersonic wind velocities but the analysis suggests ot,herwise. FREE MODES 0~ STATIONARY W'AVES The lee wave patt’ern is due to the discretje spectrum of waves generated by a crater ridge in an airstream. The discrete spectrum is the set of eigenmodes whose wa,venumbers are the solutions t’o t,he zero frequency dispersion equation which is derived from the homogeneous equation for the airstream. The free modes are infinite in lat’eral extent, being the type of wave generat’ed hy an infinitely long ridge not necessarily normal to the air&earn, and the relative amplitudes of the modes are obtained by expanding a ridge in terms of the stationar? waves with t)he ridge acting as a forcing function. The method is identical to a (Green’s function approach in which the
ridge is used as the source t)erm. A lee wave pattern of a crater is obtained by summing over a parallel set of ridges oriented at all angles between &tx/d from t’he normal t,o the airstream. The velocit’y, density, entropy and acoustic velocity of an undisOurbet1 airstream are designated by uO>pO, qO, and c0 respectivel)-. Using the approach of Eckart, (1960)the parameters associated wit’h the kinematics and dynamics of t,he aLtmosphere are defined by
where t’he prime indicates differentiation with respect to z. y and R are the ratio of the specific heats and the gas constant. N is the Brunt-V&is&l& frequency, r is a reciprocal scale height related bo the adiabatic changes in the vertical direct’ion, and /? is the stat’ic stability. Wit’h the stationary wave vector and horizont8al space vect’or defined by k and r t’he vertical velocity, II*, of a st’ationary wave is defined by rc =
W(p,c,)-1”
exp(ik.r+~j~~~2).
(2)
and assuming the acoustic velocity to be great’er than the stream veIocity, t,he homogeneous equat,ion for W is
(3) To obtain (3) it has also been assumed that t’he Rossby number is large and that t)he perturbation velocity of the stream is small compared to the stream velocity. One of the simplest’ models, and the one that, we shall use, is a t)wo layer model in which t)he velocity is uniform within a layer and in the same direction in both layers. The density decreases exponentiall> with altitude, the Brunt-V&is&l& frequency is constant and both are continuous across
519
MARTIAN ATMOSPHERIC LEE W-4VE
the interface of the two layers. In for the thermal st’ructure atmosphere in which we are interested, the fifth and sixth terms are much smaller than the first term in the brackets of (3) and may be neglected. subscripts 1 and 2 to identify quantities interface at z = 0, the surface at z = -h and let’ting 4 represent the angle between t’he wave vector and stream direct’ion we have from (3)
(a)
U’; + [(W/u~,)
(b)
x K, = 0, O > z ’ -hY 11’; + [ (N2/ug2) sec2 + - Ii21 x IV2 = 0,
The int,erface conditions
see2 4 - li2] (4)
z 2 0. are
(a)
uo, TV; = uo21i.;,
(b)
HV,/u,, = W,/u,z,
(5)
making the pressure and displacement continuous across the interface. At the surface the boundary condition is IV, (-h) = 0.
(‘3)
Solving the set (4)) (5) ,and (6) under the condition that the wave energy decreases with height, that is the waves are trapped (Corby and Sawyer, 1958; Scorer, 1949), the e’igenvalue equation for the wavenumbers J%is obtained, V cot’ (vh) + (zco2/uo,)2 p = 0,
angle 4 to t)he direction of the airstream. The modes have t’he horizontal behavior, ?[, _ eik,(x cos @+Y sin 9) 3
(9)
where x is in the direction of the air&ream and y is normal to the airstream. If the waves are assumed to be generated by a ridge at an angle + to the stream and at a distance d from the origin of the x-y pla’ne, as shown in Fig. 1, we ha,ve, 2u_ eik,lr cos(fl-@)+dl =- eik,t 3 (10) with r and 6 as shown in Fig. 1. Solutions of t’he form (10) represent t,he free modes but in order to determine t’he amplitudes the source of the waves must be considered. In addition, we are interested in the vertical excursion of an atmospheric parcel not its vertical velocity since the clouds will presumably form or be more dense along the crests of t,he waves in the streamlines. The vertical excursion, &,, of a streamline for the wave of wavenumber ,$ is relat~ed t’o the vertical velocitj> by ik(z, 4) = (zc+/iku,)eik’ (11) to within a8constant, phase parameter. A ridge of assumed profile 5, (0 is described in terms of the st,abionary waves b> l,(f) = CI(1 + p/w-“2 = (d/n)
Re Ir eMkb+ ikcdk,
(12)
(7)
where (a) y2 = (X2 sec2 +/a$,) - L2 = 1: - Ii2, (b) $ z ,$’ - (AT2secz$/ui2) = iL2- 1:. (8) The Scorer parameters 1, and I, are t,he limits of the possible values of Ic. Solutions of (7) give t’he values of the wavenumber li, of the free modes of the airstream.’ The roots k, define the modes of the st’ationary waves with wave vectors at an 1 In considering only these modes we have neglected the modes that, are supercritical in both layers ; k c I,, 2,. These modes are of consequence at high altitudes and would have little effect on our results (Holmboe and Klieforth, 1957). The reference was brought to the attention of the author by Dr. Lawrence A. Schmid.
FIG. 1. Horizontal geometry of airstream, ridge and wave directions. The ridge is at a distance d from the z-y and r-6’ coordinate origin and its normal is at an angle 4 to the airstream uO. The wave vector k is normal t,o the ridge.
520
JOSEPH
A. PARRIGLIA
with Re designating the real part. Normalizing i$ to the amplitude of the displacement at the surface and using this with (12) which represents the boundary condition at z = -h when a ridge is in bhe air&earn we have
more detailed discussion of the k-plane integration.] Evaluation of the lee waves is straightforward and is simply the residue of the poles : &,r,e)
= 2iA 2
jniz
d+
-7712
ab,M-h) h(z) -
(, (z,r,B) = -
%l(z) P,(W)
7r
&, is the superposition of waves of all k’s and represents the vertical displacement) of a streamline due to the stationary waves with wave vectors perpendicular to the ridge and at, an angle qSto t,he airstream as shown in Fig. 1. If we no& consider a set of parallel ridges separated by 2d and take
where j= 1 when -h
(1%)
(b)
; (ktL) <=0.
On constJantlphase surfaces defined by the c:onst)ant M. then integrate ~-12n-e tl AV<’
(I 3) over C#J front
~9/2 to (19)
1(;)
Using (1 X) and (II)), equations are obt~ainetl for 6’ and T in terms of the stationary points C& and +,_ :
[ (-_, t’, 0) is t’he superposition of and represents the complete solution for t’he displacement of the streamlines of HoU across a crater. The discrete free modes represent’ed by the solutions to (7) are at the poles of the integrand. Integration in the complex h--plane is chosen so t’hat the waves are downstream from the crater, the integration pat,hs being different depending upon whether fw or [,_ are less than or greater t)han zero. ISee Scorer for a
where (I/k)ak/&$ is obtained from the dispersion equation (7) and the upper and lower signs are associat’ed with the W and L subscripts, respectively. Equations (20) and (2 1) yield r and 0 along constant phase fronts where the phase is det’ermined b) the const’ant’ Al.
“‘(‘)
-kb(rik:w
+ (,ik:,.)
(
where waves
of’ all X-‘8 and directions
MARTIANATMOSPHERICLEE WAVE The stationary gives the result
phase evaluation
of (17)
112 1 eik,t,+i(n/4)
(22) where all the terms are evaluated at the stationary point & which equals & or & depending upon whether the wave is due to the windward or leeward rim of the crater and the sign of ~14 is + or - depending upon whether 82/a42 (k.$W,L) is greater than or less than zero. When the second derivative is equal to zero (22) is singular and the solution is similar to (22) but with the curly bracketed term replaced by condensation
l/3
J/3 1 eiktw+
i(a/6)
eik:L
with r (4/3) being the gamma function. If for take a2(ktW, L)i+2 ’ 0 we -41 = (6 + 8n - 1) n/4 and for a2(kEw L)/ a+’ G 0 take M = (6 + 8% + 1) x/4 with n=0,1,2 ,..., the constant phase surfaces are wave to windward and rims the Using (21), and amplitude and of wave be calculated of
DISCUSSION CALCULATIONS The displacement the lines to being upon the parameters also depend upon the parameters a and b which describe the crater ridge, as is evident in (15) and (22). As pointed out previously,
interference
521
i
f
524
JOSEPH A. PIRRAGLIA
waves fill a wedge shaped region whose edge makes an angle of approximately 18” with the apparent wind direction and the wave length of t’he transverse waves along the centerline of the pattern is approximately 6Okm. As implied by the stationary phase calculation there is construct’ive interference of waves of directions + associated with values of 6 inside the wedge and destructive int’erference outside the &edge. This is seen in (20) in which there is a maximum of the absolute value of 0 beyond which no solution exists. Using the model described in t’he previous seeCon by assigning values too N2, which depends upon the t’emperature profile, t,o the airstream speeds 740, and /do2 and to the depth h of the lower airsOream we attempt t’o construct t)he wave pattern which could generate the cloud configuration of Fig. 2. Unfort8unately. there is no temperat’ure dat’a available for the t,ime of the picture but’ other da& foi the same lat’itude and local time indicates that, t’he at’mosphere was substantially subadiabat’ic and in fact may have had temperature inversions. As an approximaBtc mean, in our models we shall use an isot’hermal atmosphere where from (lb), /3 = piSo s g/(c,Y’). Any stream velocities can then be considered to be normalized t’o t,he isothermal case and t’he velocit) associat,ed with any other value of p is obtained by multiplying the isothermal case by @/PiSo)+. 3 shows families of airstream
mately 30 and 45m/s and appropriate values of ua2 and
with
the
tiOm/s. In each part of the figure lines of const’ant
pattern of Fig. 2 is indicated portion of each sect’ion is the limit beyond which there is no transverse
bounded set, of airstreams produce the correct, angle and wavelengt,h. obvious one from the figure is the case with = 85m/s and 11,= 1 I km. %I = 40m/s, ~1.5~~ Any model with ua, bet,ween approxi-
c
t-1
100
km
FI(:. 4. Calculated wave patterns. l!art (a) is the wave system generated by a 100kmdiametcr crater in an air-stream with a 45m/s, 15km deep lower stream and a 92m/s upper strealn. Part (h) is with a 4Om/s, 11 km deep lower stream and 85m/s upper stream. Part (c) is with a 35m/s. 7.8 km deep lo\rer stream and 82.5m/s upper stream.
MARTIAN
ATMOSPHERIC
parameters, [35, 83, 81, [40, 85, 111, and [45, 92,153 as indicated. These were chosen by interpolation in Fig. 3. Model [35, 83, 81 has wave crests with more curvature than those in Fig. 2 and model [45, 92, 151 are too flat while [40, 85, 111 is a reasonably good reproduction. The observed patterns of Fig. 2 lie between the two extreme models of Fig. 4 which yield velocities without very large differences so that the gross features of the airstream would be obtained in any case. The clouds would form along the cont’ours of Fig. 4 depending upon the shape
LEE
WAVE
525
of the ridge as manifested in the parameters a and b of (12). The amplitude of the waves are directly proportional to the height, a, of the ridge but depend upon the width parameter, b, as b[exp (bk,)] and because of t,he k, dependence vary along a wave. For shallower slopes the amplitudes of the transverse waves are dominant but’ as the slope is increased the amplitudes of the divergent waves increase relative to the transverse waves. The clouds in Fig. 2 appear to be along the transverse waves of the outer wedge, which is due to thewindwardridge,andalong the
FIG. 5. Mariner 9 television photograph taken during revolution 183. The crater on the left border is the same one shown in Fig. 2. The time of this picture precedes that of Fig. 2 by two Martian days and there are no transverse waves here as are evident in Fig. 2. 19
526
JOSEPH
A. PIRRAGLIA
waves of the inner wedge, which is due to t,he lee ridge. This suggests that the windward ridge of the crater is not as steep as the leeward ridge. The inner and outer wave systems are apparent in the pat’terns in Fig. 4. Figure 5 is an example of a less common wave pattern. This picture covers approximately the same area as Fig. 2 and t,he cloud pattern is due to t’he same crater but precedes that of Fig. 2 by one MarCan day. It is apparent that, the transverse wave system is not complete. This is because as predicted by (7) for the conditions of the airstream at that time t’here is no solut’ion for lowhen absolute values of 4 are less than some nonzero magnitude. The waves for those directions are below cutoff. The type of wave pattern in question is not’ defined on t.he diagrams of Fig. 3, but similar diagrams could have been constructed using the wavelength along the outer edge rather than the center of the patt’ern. In that case, they could be locat’ed on the wavelength-wedge angle plane and there lvould be no apparent difference from wave patterns with complete transverse waves. As before t’here is more than one model that will give the proper wedge and wavelength along some prescribed direct,ion. However, with only short segments of the waves observable it’ becomes difficult to use the contour of Dhe constant phase surface to choose the optimum parameters although one can determine maximum values on the lower stream velocity and depth. A candidate airstream is 120, 85,4] whose lee wave pattern is shown in Fig. 6. There is a good similarity between Figs. 5 and 6 especially in the first three waves in the lower part of the wake and in the characteristic streaks flaring out from the crater. The airstream of Fig. 6 followed by that of Fig. 4 by one day implies an upper layer of 85m/s, uniform in time, over a lower layer of 20m/s velocity increasing to 40m/s, and of 4km depth increasing to I I km. This behavior is plausible being suggestive of lower layer growt*h. The theory is essentially good only for far fields, i.e., kr large, but can in fact be used to give near field results for the free
FL<:. 6. Tile calculated wave system gerwratocl by a lower stream of ZOm/s and 3.5km depth and an upper stream of 85m/s blowing o\.cr il 100 km diameter crater.
modes while ignoring the continuous part, of the wave spectrum. An example of the agreement between theory and observat,ion is indicated by Fig. 7. This picture shows the first and second wave crest,s of the lee waves of a 45 km crater at 63 N, 347 W (this picture is 200” in longitude from, ant1 32 days prior to Fig. 2). The theory predicts that the first crest is 518 of a wavelength from the windward ridge of the crat,er. The wavelength between the t,wo crests is 45 km and the distance from the windward rim to the first crest is 28km which is in very good agreement with the theory. Returning to Fig. 3 for a moment and confining t’he discussion to patterns with complete transverse waves, the changing character of the wave pattern can be observed as a function of the airstream parameters. As the lower stream velocity increases the minimum depth of the layer must increase to produce patterns with complete transverse waves. For increasing lower stream depth the pattern becomes less dependent upon the upper stream velocity as, for example, along the h = 10 line for uO, = 20. As the difference between the upper and lower stream velocities decreases the pattern becomes less dependent upon the depth as is seen along t,he u 02 = 30 line for uoI = 20. B0t.h cases are intuitively evident. Of particular interest
MARTIdS
ATMOSI’HERIC
LEE
WAVE
527
528
JOSEPHA. PIRRAGLIA
to us is the observation that for a given lower layer velocity a lower limit can be placed on the wavelength by the constraint that k cannot exceed 1,) as seen from (8). k equals 1, where the contours of Fig. 3 touch the zero wedge angle line. The limit on Ic is established without the aid of Fig. 3 but the figure shows that for a given wavelength wide wedge angles infer small lower stream velocities. The implications from the observations will be discussed subsequently. NONUNIFORMSTREAMDIRECTIONS If the two streams of the model have different directions we have another parameter of the airstream that must be determined from theobservations. Because, for the most part, we cannot even determine three parameters no attempt, will be made to determine a fourth as
expressed by the angle between the upper and lower streams. However, calculations showing the types of patterns that are the result of nonuniform stream directions will be presented. If the angle between the two streams is 4 the eigenvalue equation for k: is, v
cot (vh) -
UOI cos(4 + $1 *IL= UOI
cos+ 1
(
0,
(2.3)
where (a)
v* = (N’ secZ +/t$,)
- k’,
(b)
p2 = k* - IN* set* (4 + $)/u&j,
(24)
with (23) and (24) used in the place of (7) and (8) the analysis proceeds as before. Using a model with a lower stream velocit’y of 20m/s and depth of 5km and an upper stream velocity of 80m/s the patterns were calculated for cases in which the angle between the streams was 4.5’
/ , I/,’
‘Y
Yo, = 20 m/s uo2 = 80
I’
m/s
h
: 5 Km
$I
-900
uo, = 40
m/s
uo2:
m/s
80
h
: 10 Km
*
=900
D
P’IG. 8. C’alculated wave patterns for two layer air streams in whicll tile upper and lo~vrr vclocitics Ilaw different directions. TIE speeds, lower laysI. depth and angle betwren the velocities of ttle 11ppcr and lower streams arc irldicakxl in each srrtion of the figurr.
MARTIANATMOSPHERICLEE WAVE
529
suggest a large difference in the wind directions. With small differences and without isolated patterns it is not possible to determine the difference in the stream direction but, nevertheless, one may use the wavelengths to give an estimate of the winds in the lower layer. In most of the television pictures the wavelengths of the apparent lee waves are approximately 30 km and of the type shown in Figs. 2 and 4 rather than those of Figs. 5 and 6. Although the lack of many clear cases of wave patterns showing the wedge angle prevents a determination of the upper airstream velocity, an upper limit can be put on the velocity of the lower stream. From Fig. 3 it is apparent that wavelengths of 30km are associated only with airstreams with lower stream velocities less than 40m/s since the observations CONCLUSIONS suggest wide wedge angles and from the discussion in the previous section they are Regardless of the simple model used for associated with the smaller lower stream the airstream, the calculations appear to be velocities for a given wavelength. capable of reproducing the observed wave Pictures like Fig. 5 give the impression patterns. Insofar as the model is a reasonable approximation to the actual airof very high velocities but the results of the calculations shown in Fig. 6 and st,ream, were it not for the paucity of temperature data at the latitude and seacompared with Fig. 5 imply that there is no son of the observations, accurate assessneed to invoke extremely high velocities ments of the wind velocities could be made. to explain the particular wake pattern and, Since all our calculations were for a 200K in fact, the lower stream velocity is less isothermal atmosphere the velocities would than in the more common cases with transverse waves. have to be adjusted by the square root of the ratio of the stabilities, as was pointed The conclusion may be made that the out previously. If the temperature profile many observations of lee waves suggest approached an adiabatic state the velocithat the near surface wind speeds as t’ies as calculated would decrease toward implied by the lower stream velocities do zero. If there were a temperature inversion, not greatly exceed, and most of the time t’he velocities would increase, but to inare less than, 40m/s in the northern crease the velocities by 25% over those midlatitudes during the late winter except of the isothermal case would require an possibly on a very localized scale. This inversion of more than BK/km. Such an result is in agreement with the wind inversion is large and not very likely, magnitudes obtained independently, in making the possibility of velocities greater particular with t’he two level numerical by 25’/, improbable. model of Leovy and Mintz (1969) where The first wave in the clouds shown in they calculated the mean zonal winds at Fig. 2 is not normal to the apparent stream 50N and at winter solstice to be 30m/s direction and may be due to the nonuniat 3km and 70m/s at 13.5km. Winds of formity of the wind direction with altitude. this magnitude are in substantial agreeAs illustrated previously, the wave patterns ment with the winds implied by the lee are not symmetric about a horizontal line waves. when the windstreams have different A two layer model with a discontinuous directions but the wave patterns do not, velocity profile is subject to Kelvin-
and 90”. Similar calculations were made with a 1Okm deep lower stream of 40m/s. The results are shown in Fig. 8. The calculations indicate that a pair of streams with similar depth and wind speeds will yield shorter wavelengths for larger angles between the streams. However, the observed wave patterns generally are not suggestive of the type of patterns shown in Fig. 8 and the conclusions drawn from t’he parallel stream case of the previous section should be valid. The streams may be off parallel but not by the amounts used for the calculations of Fig. 8. If there were indications of greatly different wind direct’ions, without observations of isolated lee wave patterns it would be exceedingly difficult to determine four parameters of the airstream.
530
JOSEPH
A. PIRRAGLIA
Helmholtz instabilities at all wavelengths but the only waves apparent in the pictures are the extensive periodic trains of lee waves which are due to trapping in a waveguide formed by layers in which t,he Scorer parameter, 1 = Nsec+/?l,, allows propagation in a lower layer a,nd reflect’ion in an upper layer. However, a more realistic model would not have a velocit> discontinuit>-. A three layer model with uniform upper and lower velocities coilnect)ed cont’inuously through the varying middle laper is unstable t’o only a small band of wavenumbers which are dependent upon the Richardson number and an) Kelvin-Helm holtz instabilit8y wavclengtlr is likely to be far retnovetl from the ICY wave lengt,hs. Tn addition it permit8s t)rapping of the waves and is general enough t)o bc at’ lea,st a rough approximation t#o anv actual airstream with it monotonic p&file. The use of a tfvo Ia~.er model in a lee wave analysis is just’ified bjr t hr fact, that’ if t)he product oft he t8ransition layer thickness and the wavenumber is much less than unity the t*\\.o layer approximation in which the t’ransit’ion layer is absent’ is a good approximation fol the calculat~ion of lee waves in a continuously varying monotonic velocity profiles airstream. 0n~ interprebabion of the results baser1 upon the comments above irnplies a volocit!y profile of t’wo fairly tlist)inc$ levels connected by a transitSion layer of greater shear than eit’her of the atlj;ining layers. The velocities of the model approximat,e t’he average velocit8ies oft he upper and lower layers with the lower level wind velocit,ies being represent’at’ive of t,he wind velocit)ies above the surf;lce boundarp lay”‘. Another interpret’at’ion is t’hat of a layer of uniform velocity over a shear layer in which the velocities in the lower layer are not much different’ from the average of the layer. As before, in the lower layer t’he winds derived are averages and are not) surface winds. The models are a crude approximation to the real wind profiles with the results being representative of the magnitudes one might, find in t’he actual cases.
ACKNOWLEDGMENTS The ttuthor is indebted to Dr. B. J. Conrath for llis trrlpfut discussions and to tie and Dr. R. A. Hanet for critical comments. The Mariner 9 MTVS pllotoprapt>y has been provitlctl by t I)(% Na~tiorral Space Scirncr Data Cc:ntc>r.