Jc.x~1968 RALPH D. REYNOLDS, ROY L. LAMBERTH AND M. G. WURTELE 353Investigation of a Complex Mountain Wave SituationRALPH ID. REYNOLDS AND ROY L. LAMBERTItAtmospheric Sciences Office, White Sands Missile Range, N. Mex.^~) M. G. WURTEL~ University of California at Los Angeles (Manuscript received 27 February 1968)ABSTRACT A complex mountain lee wave was recorded by radar-tracked superpressure balloons at White SandsMissile Range on 6 May 1965 at a mean altitude of 3.5 km MSL; simultaneously, a very weak wave wasrecorded at 7 km. The lower complex wave showed variable wavelengths, amplitudes, and increasingvertical velocities with time. Several of the better existing mountain wave theories were tested against the data to determine whichtheory or theories, if any, could explain the physical cause of the particular features of the complex wave. It was found that existing theoretical models are too simplified to apply to the condition in the observedwave and explain only its grosset features. If our understanding of gravity waves is to be adequate toexplain quantitatively what we are capable of observing quantitatively, we must begin the analysis ofmore realistic models or turn to numerical integration of the relevant equations.1. Introduction The basic mission of the mountain wave study atWhite Sands Missile Range (WSMR) is to furnish theballistic meteorologist with a workable method for determining the occurrence, location, and strength of leewaves over the missile range. The WSMR mountain wave study has been in operation for five spring seasons and over 200 radar plots ofballoon flights have been obtained. Of these radar plots,30% showed no detectable wave, 50% showed a simplewave pattern, and 20% showed a complex wave pattern.A "complex wave pattern" is defined here as a series ofwaves in horizontal succession displaying variable wavelengths and amplitudes. Of these complex waves onlythe one that showed a well-defined wave system bothupwind of the San Andres Mountains and also downstream over the missile range was investigated in depth.This complex wave shows alterations in its wavelength7652k I 1.3 9.8 9.4 I [.312.1 10.4 0 I i I I . I 2 4 6 8 10 20 30 40 50 DIST~C[ Km Fro. 1. The cross section of two waves recorded on 6 May 1965 with the complex wave beingat a mean altitude of 3.5 kin. The wavelength of each wave is shown. The first wavelengthin the complex series is incomplete, the following waves being measured from trough-totrough positions. The upper wave at 7 km displays a very small amplitude.354 JOURNAL OF APPLIED METEOROLOGY400618 MST0645 MSTRADAR BALLOON TRACKS ~0710 MST 9 ~0705 MS' 6 ~ HEADOUARTERS HOLLOMAN tO 20 30 40 50 60 70 80 90 DISTANCE Km Fro. 2. Balloon tracks for 1 April 1964 showing the straight, level flow of a balloon prior toreaching the mountains. This straight line flow is characteristic of mountain wave balloontracks ,prior to reaching the mountains.and amplitude with time, and the vertical velocity increased from 2.8 to 5.0: m sec-L Most of the mountain wave theories give an adequateanswer for the location and intensity of a wave of simplesinusoidal type damping with time, but none of thetheories apply to complex waves per se. Since 20% of therecorded waves at WSMR are complex, it was decidedto test applicable lee-wave theories against this particular complex wave to determine which theory or theories,if any, could explain its observed features. One hypothesis that may be suggested by inspectionof the complex wave (the lower wave in Fig. 1) is thatthere is a superposition on a residual upstream wave,initiated by the mountain and resulting in resonance orreinforcement in the downstream waves. Accepting thish~vpothesis, we ask which of the current theories are applicable to this particular situation, and which theory ortheories may be used by the ballistic meteorologist toforecast the location and strength of complex lee waves.2. Background A mountain lee wave is a gravity wave that is causedby, and is stationary with respect to, some barrier in thefluid flow. The time scale of buoyancy has a characteristic frequency, the Brunt-~aisaala frequency,[(g/O)(O0/O~)~i, where g is the acceleration of gravityxnd 0 is potential temperature. The linearized mountain wave equation may be ex~pressed in the form O,w/ O~'+ (-'- ~,)w = o, (1)where L~ = co~/U~- 0~ U/0 z~-, w = vertical velocity, k = horizontal wavenumber, U= horizontal wind speed in the undisturbed flow, z = height. The mountain wave theory indicates that the wavelength of any lee waves will lie between the maximumand minimum values of 2~r/- through the troposphere(Alaka, 1960). The mountain wave study field test precedures, using .especially adapted AMT/15 radiosondes flown on superpressure balloons, have been reported previously(Lamberth -t al., 1965; Reynolds and Lamberth, 1966).However, it should be pointed out that the balloon-detected wave cross sections shown in the illustrationswere taken from the-computer reductions of theAN/FPS-16 radar digital data, recorded on magnetictape at a rate of one data point per second and smoothedover 30-set intervals. Booker and Cooper (1965) have shown that superpressure balloons tend to follow the free airstream withgood fidelity while traversing mountain waves; so it is believed that the true airstream flow during these reportedmountain wave conditions is accurately depicted by theradar-tracked oscillations of the superpressure balloons.3. The complex wave and its associated parameters One of the more unusual aspects of the complex waveof Fig. 1 is the existence of a fairly well defined gravityJ~J~E1968 RALPH D. REYNOLDS, ROY L. LAMBERTH AND M. G. WURTELE 355J .? Fro. 3. 500-mb contours (dashed lines) showing a long-wave trough position centered overthe Great Basin area. The surface cold front is approaching White Sands Missile Range(WSD) from the west.wave upstream of the San Andres Mountains causedmost probably by mountains further west of WSMR.This is the only dominant pre-mountain wave that hasbeen recorded during our mountain wave study--the recorded air flow upstream of the San Andres has alwaysbeen smooth and level. The normal situation is shownin Fig. 2, in which the wave begins in the lee of the SanAndres. (To simplify references hereafter to any portionof the complex wave, the waves will be referred to aswave A for the first wavelength through wave E for theT\.1155 T RT BALLOON RIDGE R ~TROUGH T ...........9,~, ~, ~ ? ....Km MILES Fro. 4. Radar-balloon tracks for 6 May 1965 showing that the two balloons were close in timeand space. The steepest escarpment of the San Andres lies to the west of the Dry Lake whereSan Andres Peak rises more than 4000 ft above the valley floor.$$6 JOURNAL OF APPLIED METEOROLOGY Vor. u~x~E7:,oFAHRENHEIT TEMPERATURE SCALE FIG. 5. Temperature profiles showing dry adiabatic lapse rates in the layer above the surfaceinversion and between 5.9 to 7.2 km. The lapse rate is moist adiabatic between 3.0 to 4.9 km.Each specific stability regime above the surface boundary layer was used in the calculation of L2.last wave.) As shown in Fig. 1, wave B with amplitudeof 300 m in the complex series is exactly in phase withthe first ridge of the mountain and wave C shows a definite increase in amplitude to 580 m; this suggests thatresonance or reinforcement might be present. Wave Dhas an amplitude of 580 m, which increases in E to1165 m. This increase did not follow through to theupper levels where it is seen that the waves at 7 km arevery weak, with wavelengths of 10 and 12 km. The vertical velocities ranged from a maximum of1.0 m sec-~ in waves A and B to 2.9 m sec-~ in waves Cand D to a maximum of 5.0 m sec-~ near the last inflection point of wave E. l~rior to ~nd during the recording of the complexmountain wave, the sky was cloudless. The relative humidity went below the threshold of measurement. Along-wave trough was centered over Utah and westernArizona. A minor trough accompanied the cold frontmoving through central Arizona to Colorado (Fig. 3). The west-east cross section is shown in Fig. 1 for themountain wave and the mountain; the mountain rangeis orientated basically north-south. The wavelengthsshown are for the u component of the waves. The horizontal trajectories of the two balloons are plotted in Fig.4 where it may be seen that the balloons were close intime and space. A rawinsonde was released at 0300 MST, 12.8 kmsouth of the 1200 MST position of the balloon trackshown in Fig. 4. The temperature from this rz[winsondeis shown in Fig. 5 plotted on a skew T, log p diagram.There are seven discernible changes in the lapse rateand/or velocity profile from the surface to 200 mb. Asecond rawinsonde was released from the same positionas the 0300 release at 0900 MST. The only difference inJUNE1968 RALPH D. REYNOLDS, ROY L. LAMBERTH AND M. G. WURTELE 357the two observations was that a superadiabatic lapserate existed near the surface at 0900 MST.The normal wind profile, which in this case is also a t~.zonal wind profile, is shown in Fig. 6, calculated from 10.the 0300 MST rawinsonde.The m2 profile was computed noting the near-adia- 9.batic layers from 1.5-3 km and from 6-7 km. Values for/? were determined at the 3-, 6-, 7.2-, 9.2- and ll-km 0.levels, these being chosen such that a linear variationbetween consecutive points gave a good approximation.This profile up to 1! km has been plotted in Fig. 7.4. Quantitative application of theories Analytical models exist for constant L2 (Queney,1947), for discontinuous layers of constant L2 (Scorer,1949), for exponentially decreasing L~ (Palm andFoldvik, 1960), and for L2 decreasing in the troposphereand constant in the stratosphere (Wurtele, 1953). However, the profile of L2 in Fig. 7 as derived from real datais not easy to represent satisfactorily by any of thesemodels. The observed wave pattern shows the increasingwavelength and decreasing amplitude with height characteristic of decreasing L2, rather than the up-wind tiltcharacteristic of constant L~. So no attempt is made toapply the latter model. However, since the waves inquestion are observed at the 3.5-km level, it seems reasonable to attempt to explain the dynamics of the wavesby the stable layer from 3-6 kin, between the two neu -- 3IO 3 20 40 60 NORMAL W)ND PROFILE KNOTS) (U COMPONENT) Fro. 6. The normal wind profile for 6 May 1965 showing analmost linear increase with height. This is also a zonal wind profilesince the mountain range is orientated N-S. 0300 MST 6 MAY 1965'! FOLDVIK~ / /EXPONENTIALZERO (ADIABATIC) '\ZERO (ADIABATIC)0 I 10-8I I10-7 10-6L2 (m-2)SURFACE Fro. 7. The L2 profile based on the 0300 MST rawinsonde. Thedashed line represents the exponential curve fit to the two extreme points.tral layers. To this seg~nent of the profile, one may applythe Pahn-Foldvik model with reasonable accuracy. Let -~(z) = -2(z0)~-~-~%where the constant c is to be determined from the data.If we take z0 as 3 km, then c=0.20 km-~ for the 3-6 kmlayer. The eigensolution of Eq. (2) in this case is w"~J~/~FL(z)~, (3) t_ 6 _1where J is the Bessel function of the first kind. Freewaves exist if and only if In Gis case, L(0)= 1 k~~, so that k/c= 1.9, k=0.38(Jahnke and Emde, 1945, p. 152) and the wavelength isabout 16 kin. This is much larger than anything observed, and we conclude that a much deeper layer thanGe 3-6 km l~yer is involved in the dynamics. If the exponential profile is fit to the values of L2 at3 and 11 km, a value of c=0.133 is obtained. The freewaves are then given by J,.~,(7.5) = 0. (5)This yields two waves, k = 1.25 and 3.9 corresponding towavelengths of 37.6 and 12.0 km. The shorter of thesewaves is in reasonable ~greement, but -e modal is358 JOURNAL OF APPLIED METEOROLOGY -OLUME7clearly not to be expected to provide the significant details of the wave pattern, as can be seen by comparingthe observed L2 profile in Fig. 7 with the profile (dashedline) from the exponential model. A rough estimate of the maximum vertical velocityassociated with the 12-kin wavelength, which will predominate over the longer wave in the lower levels, maybe obtained from Foldvik (1962), i.e., 0.7 I-p (0) 1- [wl ..... = 2.54-, -hcU(o)l--I, (6) kp(z~)_Jwhere h is the mountain height andz,=!ln[. _L0_ 1 (7)c kL0--2.2cJis the height at which this maximum value of verticalvelocity is attained. Here h=0.91 km and z1=2.6 km(above the 3-km level); thus, th~ theory predicts a maximum value of 4 m sec-1 at an elevation of 5.6 km associated with a wavelength of 12 kin. The downstream intensification and increasing wavelength are equally difficult to explain. A given mountainshape will tend to excite certain waves to greater amplitudes than others. For example, the mountain profile k-)= 2' (8)where b is the "half-width" of the mountain, leads to thespectral representation for vertical velocity hUobkc-bk, (9)so that the maximum forcing is given to the wave forwhich bk= 1, or L= 2,rb. (10)For L~ 10 km, this requires a mountain less than 1.5 kmin half-width, which is a very steep escarpment. It should be noted that both theory (Foldvik andWurtele, 1967) and observation (Holmboe and Klieforth, 1957) agree that it is not the shape of the barrieras a whole that determines the "half-width" and therefore the forced waves, but only the lee-slope region. Therelevance of this fact to the present problem is thatwhen the balloon underwent amplification of its verticalvelocity, it was opposite the steepest part of the SanAndres lee escarpment. A possible ex~planation of the wave pattern may infact lie in the effects of the rather pronounced variationof the topography in the direction normal to the windflow. In fact, the balloon tracks were over the most complex portion of the topography of the San Andres mountains. But a proper discussion of such a hypothesis requires abandonment of the two-dimensional assumptions employed exclusively so far in this paper. For applicable three-dimensional analyses we may refer to Scorer and Wilkinson (1956) and Wurtele (1957).Unfortunately, these models apply to isolated peaks,whereas the peaks of the San Andres are presumablyclose enough to influence each other. In any case, if weapply the formulas of Wurtele (1957), we see that weexpect maximum vertical velocities in crescent-shapedpatterns, concave downwind and centered about 5 kmdownwind of each peak. These patterns do not superimpose to give the observed intensification. No theory exists for a flow oblique to a ridge of profile similar to the San Andres.5. Conclusions If our understanding of gravity waves is to be increased, detailed L2 profiles and streamlines at multiplelevels are necessary. For this, the superpressure balloonis an excellent sensor, provided that adequate trackingfacilities are available. It is to be expected that once such data are gatheredthe simple theoretical models will be inadequate to explain many of the effects observed. In the instance discussed, the ]~2 profile, first, decreasing then increasingwith height, is not one for which a theory exists; thetopography of the barrier also introduced three-dimensional effects not tractable according to existing theory. Since the simple theoretical models have had muchsuccess in producing operationally applicable formulas,it seems probable that the analysis of somewhat morecomplex models might well be worth attempting. REFERENCESAlaka, M. A., 1960: The airflow over mountains. WMO Tech. Note No. 34 (WMO No. 98, TP. 43), World Meteorological Organization, Geneva, 135 pp.Booker, D. Ray, and Lynn W. Cooper, 1965: Superpressure balloons for weather research. J. Appl. Meteor., 4, 122-129.Foldvik, A., 1962: Two-dimensional mountain waves--A method for the rapid computation of lee wavelengths and vertical velocities. Quart. J. Roy. Meteor. Soc., 88, 271-285. , and M. G. Wurtele, 1967: The computation of the transient gravity wave. Geophys. J., 14, 161-185.Holmboe, J., and Harold Klieforth, 1954: Sierra Wave Project-Final Report. Contract No. AF 19(122)-263. Jahnke, E., and Fritz Emde, 1945: Tables of Functions. New York, Dover PUN., Inc., 304 pp.Lamberth, Roy L., R. D. Reynolds and M. G. Wurtele, 1965: Mountain lee waves at White Sands Missile Range. Bull. Amer. Meteor. Soc., 46, 634-636.Palm, E., and A. Foldvik, 1960: Contribution to the theory of two-dimensional mountain waves. Geophys. Norv., 21, No. 6, 1-30.Queney, P., 1947: Theory of perturbations in stratified currents with application to airflow over mountain barriers. The University of Chicago Press, Misc. Rept., No. 23.Reynolds, Ralph D., and Roy L. Lamberth, 1966: Ambient temperature measurements from radiosondes flown on constant-level balloons. J. A ppl. Meteor., ~, 304-307.Scorer, R. S., 1949: Theory of waves in the lee of mountains. Quart. J. Roy. Meteor. Soc., 75, 41-55.----., and Mary Wilkinson, 1956: Waves in the lee of an isolated hill. Quart. J. Roy. Meteor. Soc., 82, 419-427.Wurtele, M. G., 1953: Txvo new models in the study of lee waves. Univ. of California, Los Angeles, Sci. Rept. No. 4, Contract AF 19(122)-263.----, 1957: The three-dimensional lee-wave. Beitr. Phys. Atmos., 29, 242-251.

## Abstract

A complex mountain lee wave was recorded by radar-tracked superpressure balloons at White Sands Missile Range on 6 May 1965 at a mean altitude of 3.5 km MSL; simultaneously, a very weak wave was recorded at 7 km. The lower complex wave showed variable wavelengths, amplitudes, and increasing vertical velocities with time.

Several of the better existing mountain wave theories were tested against the data to determine which theory or theories, if any, could explain the physical cause of the particular features of the complex wave.

It was found that existing theoretical models are too simplified to apply to the condition in the observed wave and explain only its grosser features. If our understanding of gravity waves is to be adequate to explain quantitatively what we are capable of observing quantitatively, we must begin the analysis of more realistic models or turn to numerical integration of the relevant equations.