Estimating Wind Velocities in Mountain Lee Waves Using Sailplane Flight Data*

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R. P. MILLANE,G.D.STIRLING,R.G.BROWN,N.ZHANG, AND V. L. LO Computational Imaging Group, Department of Electrical and Computer Engineering, University of Canterbury, Christchurch, New Zealand

E. ENEVOLDSON AND J. E. MURRAY NASA Dryden Flight Research Centre, Edwards, California

(Manuscript received 24 December 2008, in ﬁnal form 4 May 2009)

ABSTRACT

Mountain lee waves are a form of atmospheric gravity wave that is generated by flow over mountain topography. Mountain lee waves are of considerable interest, because they can produce drag that affects the general circulation, windstorms, and clear-air turbulence that can be an aviation hazard, and they can affect ozone abundance through mixing and inducing polar stratospheric clouds. There are difficulties, however, in measuring the three-dimensional wind velocities in high-altitude mountain waves. Mountain waves are routinely used by sailplane pilots to gain altitude. Methods are described for estimating three-dimensional wind velocities in mountain waves using data collected during sailplane flights. The data used are the logged sailplane position and airspeed (sailplane speed relative to the local air mass). An algorithm is described to postprocess this data to estimate the three-dimensional wind velocity along the flight path, based on an assumption of a slowly varying horizontal wind velocity. The method can be applied to data from dedicated flights or potentially to existing flight records used as sensors of opportunity. The methods described are applied to data from a sailplane flight in lee waves of the Sierra Nevada in California.

1. Introduction drostatic mountain waves (typically 2–20 km), and they can extend far downwind of the mountain. The wave- Atmospheric mountain waves, or lee waves, are a guide is imperfect, however, and there is some propa- form of atmospheric gravity wave that is generated by gation upward and the wave loses energy. flow over elevated terrain in a stable, stratified atmo- Lee waves are of significant importance in meteorol- sphere (Corby 1954; Scorer 1997). Atmospheric gravity ogy. They can influence the vertical structure of wind waves arise from buoyancy forces when an air parcel is speed and temperature fields, cause fluctuations in wind vertically displaced from its equilibrium position, leading speeds in the lower atmosphere, and affect ozone con- to harmonic motion. Hydrostatic mountain waves form centration (Gill 1982; Sato 1990; Bird et al. 1997). Wave above the mountain and propagate vertically (Gossard momentum dissipation with height results in drag that and Hooke 1975; Durran 1990). However, appropriate affects the general circulation of the atmosphere (Gossard wind speed and temperature structures with altitude and Hooke 1975; Holton 1983; Medvedev and Klaassen can form atmospheric waveguides or ducts in which 1995). Turbulence associated with mountain waves can trapped or resonant lee waves can propagate horizon- produce strong vertical velocities, particularly downdrafts, tally (Durran 1990; Scorer 1949). Trapped lee waves that can be an aviation hazard (Doyle and Durran 2002; generally have shorter wavelengths than those of hy- Vosper et al. 2006). Strong waves can generate thin layers of turbulence because of wave breaking (Gossard and * This work is dedicated to Steve Fossett, a true adventurer. Hooke 1975; Chunchuzov 1996). Mountain waves also play a role in the vertical transport of aerosols and trace gasses. Our interest here is in measurement of the three- Corresponding author address: R. P. Millane, Department of Electrical and Computer Engineering, University of Canterbury, dimensional (3D) wind field structure in mountain lee Private Bag 4800, Christchurch 8140, New Zealand. waves. The wind field in the atmosphere can be measured E-mail: [email protected] using a variety of methods. Radiosondes give a trace of

DOI: 10.1175/2009JTECHA1274.1

Ó 2010 American Meteorological Society Unauthenticated | Downloaded 09/29/21 12:24 PM UTC 148 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 27 the horizontal wind speed and direction over the ascent for determining three-dimensional wind speeds using data path, which is determined by the wind profile with alti- recorded during routine sailplane flights. We focus on the tude. Vertical speeds can be estimated from radiosonde wind speeds in mountain waves because 1) the horizontal data by subtracting out the balloon ascent rate, although component of the wind speed is slowly varying in space this is somewhat error sensitive. Radiosonde soundings and time for which the method is most suitable; 2) sail- have been used to study mountain waves (Lane et al. plane pilots often fly in and explore mountain wave sys- 2000). Very high frequency (VHF) Doppler radar uses tems; and 3) three-dimensional lee wave wind speeds, processing of backscattered signals from inhomoge- particularly the vertical component, are not readily mea- neities in radio refractive index to estimate three- sured by other means. Some preliminary results have been dimensional wind speeds (Sato 1990; Green et al. 1979; reported by Millane et al. (2004). Ruster et al. 1986). This technique has good spatial and Our method makes use of relatively simple measure- temporal resolution and a range of up to 100 km. The ments that can be rather easily made from a lightly equipment used, however, is large and expensive. Satel- instrumented sailplane. The data used consist of post- lite scatterometer synthetic aperture radar (SAR) mea- flight logs of the sailplane GPS-derived position and surements of wind speeds are based on measurements of airspeed. Onboard pressure and temperature measure- the ocean surface roughness and appropriate processing ments can be used if available, although estimates of to derive surface wind speeds. SAR images have been these quantities, either from a nearby sounding or cal- used successfully to study gravity waves by their effect on culated from the altitude using a standard atmosphere near surface wind field variations (Vachon et al. 1994; model, would be sufficiently accurate. The horizontal Vachon et al. 1995; Chunchuzov et al. 2000). However, component of the wind velocity is determined from the this technique is suitable only over the ocean and for low- GPS-derived ground velocity and airspeed and does not level phenomena. require special flight maneuvers. The vertical component Aircraft are also used to measure wind velocities. The of the wind velocity is determined from the GPS-derived most comprehensive, as well as expensive, options are vertical speed and the sailplane sink-rate characteristics. highly instrumented, specialized research aircraft that The method has particular potential for research pur- generally use GPS, inertial navigation (INS), and mul- poses because 1) a skilled pilot can systematically ex- tihole pressure probes to derive ground velocities, plat- plore a lee wave system and 2) existing flight records form attitude, and vector airflow, from which the may be used as ‘‘sensors of opportunity’’ for subsequent 3D wind vector can be relatively easily calculated. Such analysis. However, some modification may be necessary systems have been used to measure mean and turbulent to compensate for the lower sampling rate of standard wind vectors (Bogel and Baumann 1991; Khelif et al. sailplane dataloggers used to produce existing flight re- 1999) and to measure winds in mountain waves (Doyle cords. Although our implementation is not in real time, et al. 2002; Smith et al. 2002). However, they provide a real-time system would be feasible. a limited facility because of their expense in terms of Calculation of the true airspeed from the measured capital outlay, maintenance, deployment, and opera- indicated airspeed is described in the next section. Al- tion. A number of less expensive options are available gorithms for estimating the horizontal and vertical wind based on light aircraft (Wood et al. 1997) or autonomous speeds are described in sections 3 and 4, respectively. aerial vehicles (AAVs; Holland et al. 2001; van den Results obtained by applying the methods to a sailplane Kroonenberg et al. 2008). These usually incorporate sim- wave flight are presented in section 5. Concluding re- ilar instrumentation to research aircraft but in some cases marks are made in section 6. use less expensive equipment, such as an inertial measurement unit (IMU) or a scalar airflow probe. However, 2. Airspeed calculation they are still relatively expensive, require considerable calibration, and the lower-instrumented units require The algorithm we describe requires the sailplane air- special flight maneuvers to derive the vector wind speed. speed (i.e., the sailplane speed relative to the air mass) as Sailplane pilots make extensive use of mountain lee input. The airspeed is generally measured by a pitot- waves as a source of energy with which to climb, and such static system, which measures the so-called indicated ind flights can cover large distances and altitudes (Reichmann airspeed (IAS), which we denote as ya . The IAS de- 2003). Indeed, sailplane flights provided some of the ear- pends on the air density, and the actual or true airspeed liest information on mountain waves (Whelan 2000). In (TAS), which we denote as ya, is given by spite of this, little systematic and quantitative work has 1/2 been done on using sailplane flights to study mountain ind r0 ind T P0 ya5 ya 5 ya , (1) waves. Here, we describe a novel and inexpensive method r T0 P

Unauthenticated | Downloaded 09/29/21 12:24 PM UTC JANUARY 2010 M I L L A N E E T A L . 149 where r, P, and T are the density, pressure, and temperature, respectively, and r0, P0, and T0 are the standard sea level values, 1.22 kg m23, 101 325 Pa, and 288 K, respectively. If the pressure and temperature are recorded during the flight, then ya is straightforwardly calculated using (1). Note that pressure data are often recorded indirectly as pressure altitude, which can be converted to pressure using standard equations. If temperature and/or pressure are not measured, then it is necessary to estimate T and/or P using a standard model of the atmosphere. Consider a single layer atmosphere (the troposphere) with a linear temperature gradient (lapse rate) 2G with respect to altitude z.A FIG. 1. Relationship between air velocity and ground velocity for single layer is generally sufficient for sailplane flight, a fixed airspeed. although multiple layers are easily incorporated. Using the hydrostatic equation, it is easily shown that the small, the difference between the airspeed and its hori- pressure varies with altitude as (Stull 2000) zontal component is very small. For convenience, we therefore do not distinguish between the airspeed and its P T a T Gz a 5 5 0 , (2) horizontal component, and we use y to denote both. P T T a 0 0 0 Note that because the airspeed is the sailplane speed relative to the air, the relationship between the airspeed where a 5 g/(RG), g is the acceleration due to gravity and its horizontal component is unaffected by flying in (9.81 m s22) and R is the gas constant [287 J (kg 8C)21 moving air. for dry air]. The lapse rate G is 9.88Ckm21 for dry We use the fundamental relationship or unsaturated air and about 5.58Ckm21 for saturated (moist) air. The standard atmospheric lapse rate is vg 5 va 1 vw (5) 6.58Ckm21. Using (1) and (2) shows that, if pressure but not to estimate the wind velocity from derived ground ve- temperature is recorded, ya can be calculated as locity and airspeed data. The ground velocity is calculated by differentiation of the GPS coordinates. At (1a)/2a ind P a particular time, the vector relationship (5) is shown ya 5 ya , (3) P0 in Fig. 1. Because only the magnitude of va is known, va is restricted to lie on a circle; therefore, there is a one- where a is calculated using an estimated value of G.If parameter family of solutions for vw. Therefore, there temperature but not pressure is recorded, then ya can be is insufficient information at a particular time point to calculated as uniquely determine the wind velocity. Consider now the case where two ground velocities, (1a)/2 ind T0 vg1 and vg2, and the two corresponding airspeeds, ya1 ya 5 ya . (4) T and ya2, are known in a region of constant wind velocity. The two corresponding vector diagrams as in Fig. 1 are

Finally, if neither temperature nor pressure is recorded, translated such that the origins of the vectors vg1 and vg2 then ya must be calculated using (4), and an estimated coincide as shown in Fig. 2. With this construction, the value of G must be used to calculate a and T. solution for the wind velocity must lie on both circles, and the one-parameter family of solutions reduces to 3. Horizontal wind velocity estimate two possible solutions at the intersections of the two circles. Referring to Fig. 2, the angle u is given by In this section, we describe an algorithm to estimate the horizontal wind velocity from the ﬂight data. The y2 1 c2 y2 u 5 a1 a2 horizontal components of the sailplane’s ground veloc- cos , (6) 2ya1c ity and air velocity (velocity of the sailplane relative to the air mass) and the wind velocity are denoted vg, va, where and vw, with magnitudes denoted yg, ya, and yw, respectively. Because the glide angle of a sailplane is very c 5kck5kvg1 vg2k, (7)

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FIG. 3. Vector relationships for a small change in wind speed dyw resulting from a small change in airspeed dya2. FIG. 2. Relationships between two ground velocities and two airspeeds in a region of constant wind velocity.

Using (10) and the relationships dy 5 y du and y / and the two solutions for the wind speed are given by w a1 a2 sinu 5 c/sinb, where the angle b is shown in Figs. 2 and 3, ^ shows that the sensitivity is given by vw5 vg1 ya1R6u c, (8) s 5 cosecb, (11) where Ru denotes rotation by u and ^ denotes a unit vector. If the circles do not intersect (because of errors and b is easily calculated as in the data), then no exact solution can be obtained. In summary, given two ground velocities and the cor- y2 1 y2 c2 cosb 5 a1 a2 . (12) responding airspeeds in a region of constant wind ve- 2y y locity, the wind velocity can be determined up to a a1 a2 twofold ambiguity. Errors in the ground velocities will have a similar ef- Before discussing resolution of the twofold ambiguity, fect to errors in the airspeeds. For fixed Gaussian errors it is necessary to consider the sensitivity of the solutions in the data, s is proportional to the standard deviation of to errors in the measured airspeeds and ground veloci- the wind speed estimate. We make this assumption and ties. Referring to Fig. 2, the one-parameter family of treat s as a relative standard deviation. Note that s $ 1. solutions is reduced to two solutions only if the two For a pair of ground velocity and airspeed data, the circles have different centers (i.e., the two ground ve- sensitivity can then be calculated and used as a measure locities are different). The more different the ground of the suitability of the data for wind velocity estimation. velocities, the less sensitive the solutions are to errors in In a region of constant wind velocity (i.e., speed and the data. In fact, the two solutions for the wind velocity direction), multiple pairs of data at multiple time points are least sensitive when the circles intersect at 908. The can be used to find multiple sets of twofold-ambiguous sensitivity of the wind speed estimate to errors in the wind velocity estimates. The ambiguities can then be airspeed measurements can be derived as follows: con- resolved by choosing one of the two estimates from each sider the change dyw to the wind speed estimate yw re- pair of data such that the set of resulting estimates are sulting from a small change dya2 in the airspeed ya2, for most similar (or most consistent) among all choices. The fixed ya1 and ground velocities, as shown in Fig. 3. The measure of consistency is made more quantitative later. sensitivity s of yw to errors in ya2 is then s 5 dyw/dya2. A final unique wind velocity estimate is then obtained by Referring to Fig. 2 shows that averaging these consistent estimates. Because data at adjacent sample points (times) may be highly correlated, 2 2 2 ya2 5 ya1 1 c 2cya1 cosu, (9) every Nth datum is used for the analysis. The value of N used depends on the sampling period. and differentiating (9) with respect to u gives The horizontal wind velocity in high-altitude mountain waves will generally be a slowly varying function of dy cy sinu a2 5 a1 . (10) position and time. Our approach therefore is to partition du ya2 the flight path into a sequence of regions and to assume

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computationally feasible. Therefore, we select the m9, m solution pairs with the lowest sensitivity and calculate all 2m9 partitions of these pairs. We typically use m9 ’ 10, giving about 1000 partitions. The partitions are denoted

fAi, Big, where Ai and Bi are the two sets of a partition and i indexes the 2m9 partitions. The variance of the wind 2 speed vectors in the set A is denoted sA, and the labels 2 2 Ai and Bi for each i are assigned such that s , s . Ai Bi Hence, the set Ai contains the putative correct solutions, and Bi contains the incorrect solutions. The best (i.e., the most consistent) partition fAbest, Bbestg is that which gives the closest wind velocity estimates in the set Ai; that is, fA , B g 5 argmin(s2 ). (13) best best Ai fAi,Big FIG. 4. Partitioning of the ﬂight path into cylindrical regions. The weighted (by s22) average of the wind velocity es-

timates in Abest gives the best estimate based on the m9 that the horizontal wind velocity is constant in each re- pairs of solutions. The estimate is improved by adding gion. The wind velocity can then be estimated using the one of the solutions from each of the remaining m 2 m9 data within each region. We use cylindrical regions as pairs to Abest, one at a time in order of increasing sen- shown in Fig. 4, with the radius and half-height of the sitivity, such that the variance at each addition is mini- cylinder denoted rregion and hregion, respectively. The mized, and a new weighted average calculated. This parameters chosen will also depend on the spatial res- gives the final optimum partition fAopt, Boptg and the olution desired and the number of data points needed final wind velocity estimate. The wind velocity estimates within a region to obtain good estimates. The entire in Aopt will not generally all be independent, so the flight path is partitioned into such regions using a simple standard deviation of the final estimate s will be be- pffiffiffiffiffi w algorithm. Note that the cylinders abut each other either tween s and s / m. Because m is generally large, Aopt Aopt at their sides or on their ends, depending on the shape of we estimate sw as s /2. This will generally be a con- Aopt the flight path. servative estimate. We also calculate the quantity D 5 The wind velocity estimate in each region is found by s /s , which is a dimensionless measure of the dis- Bopt Aopt selecting pairs of data, using each pair to generate a pair crimination between the partitions Aopt and Bopt, with of solutions, and then selecting the members of each D 5 1 meaning no discrimination. Wind velocity esti- solution pair that are most consistent. A full solution to mates for which D , Dmin are discarded, and we gen- this problem would involve finding the solution from erally used Dmin 5 3. each pair that is the most consistent over all possible The algorithm we have described is applied to the data combinations of the pairs. For d data within a region, within each cylindrical region to determine the wind there are m 5 d2/2 pairs of data that give m pairs of wind velocity estimate at the center of that region. If the flight velocity estimates. An exhaustive search to find the most path in a particular region is relatively straight, then the consistent set of solutions would require considering ground velocities will be similar and the sensitivities will m d2/2 2 5 2 partitions of the estimates into two sets. Such be large. If the sensitivity is greater than smax for all pairs a computation is not feasible. We therefore limit the of data within a region, then no estimate is made for that search by using only those pairs with sensitivity s , smax region. and also limit the number of pairs m used to m , mmax. The GPS and airspeed data contain errors typically 21 We have found that smax ’ 1.5 and mmax ’ 100 are on the order of 10 m and 0.5–1 m s , respectively. The generally suitable. effect of the GPS uncertainty, in particular, is poten- The algorithm proceeds as follows for each region: all tially significant. The data are prefiltered before apply- pairs of data in the region are determined and the sen- ing the algorithm (see section 5), which provides some sitivity is calculated for each pair. The m pairs of data reduction in these errors. However, the effect of these with the lowest sensitivity s , smax and such that m , errors on the wind velocity estimates is ameliorated mmax are selected, and the m wind velocity solution pairs primarily by averaging the estimates in each region. For are calculated. For typical values of m used, however, an example, for the value m 5 100 typically used, a re- exhaustive search of all the 2m partitions is still not duction in the rms errors of the wind velocity estimates

Unauthenticated | Downloaded 09/29/21 12:24 PM UTC 152 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 27 pffiffiffiffiffiffiffiffi by a factor of approximately 100 5 10 is expected. For culated at each point. The bank angle is then calculated the results presented in section 5, with a data sampling by determining the centripetal force required to produce interval of 1 s, the standard deviations sv are usually the measured curvature. We found that typically less less than 1 m s21, which is consistent with these error than 5% of the vertical air velocity estimates are ex- estimates. cluded because of excessive bank. The second effect that needs to be considered is that of changing sailplane airspeed. Consider, for example, 4. Vertical wind speed estimate the case where the sailplane airspeed is decreasing. In The vertical component of the wind velocity is esti- this case, the kinetic energy of the sailplane is de- mated using the vertical component of the sailplane creasing; aside from the effects of drag, it is being converted to potential energy (i.e., the sailplane is climbing, velocity (relative to the ground) yyg, which is obtained by differentiating the GPS altitude. If the sailplane did not or descending more slowly, relative to its normal sink sink and the airspeed was constant, then the vertical rate at constant airspeed). The opposite occurs if the speed of the sailplane would equal the vertical wind sailplane airspeed is increasing. This additional vertical speed must be subtracted out when calculating the ver- speed yyv. However, to obtain the vertical wind speed, the sink rate of the glider in still air and the effects of tical wind speed. This effect can be significant, even for changes in airspeed must be taken into account. modest rates of change of airspeed. The additional ver- Consider first the sailplane sink rate, which refers to the tical speed component ye is calculated by equating the rates of change of kinetic and potential energy, giving speed at which it descends in still air. The sink rate ys is negative, because the altitude decreases with time. For y dy a particular sailplane, the sink rate depends on the air- y 5 a a , (15) speed, the weight of the sailplane, and the air density. The e g dt glide angle g, the angle between the flight direction and the horizontal in still air, is related to the sink rate and recalling that g is the acceleration due to gravity and dy /dt is the rate of change of airspeed. airspeed by sin g 5 ys/ya. Although the sink rate depends a on air density, g is a function of the IAS and is in- Incorporating these two effects, the vertical wind dependent of the air density. This information is provided speed is estimated as by the manufacturer in the ‘‘flight polar,’’ which is a plot y 5 y y y . (16) of the sink rate at sea level air density ys0 versus IAS for yw yg s e different sailplane weights. We denote this function by ind In practice, the effect of ys is fairly small except at high ys0(ya ). Because the glide angle for a particular IAS is airspeeds, but the effect of y can be significant. The independent of air density, the sink rate ys can be calcu- e lated at any altitude (air density) from the flight polar as vertical air velocity estimates are calculated using these calculations at each point in the flight path (except those y y 5 a y yind excluded because of excessive bank) and averaged over s ind s0( a ), (14) ya a suitable time window. where y is calculated from yind as described in section 2. a a 5. Results The flight polar applies to unbanked flight so that the above calculation of sink rate is less accurate when the The methods we have described were applied to data sailplane is turning. However, significant bank is typi- from a flight in lee waves of the Sierra Nevada in cally present during only a small proportion of a wave Southern California. This is flight 39 of the Perlan Pro- flight; because its effect is not straightforwardly in- ject (Carter et al. 2003). The flight began at California corporated, we simply calculate the bank angle a at each City (35.28N, 118.08E, altitude 750 m) at 2140 UTC position along the flight path and exclude points where 24 April 2003 (1340 LT). The sailplane was launched by a exceeds some specified maximum value amax to calcu- towing behind a powered aircraft and releasing from tow late the vertical wind velocity. We typically used amax 5 approximately 30 min after takeoff at an altitude of 308. The bank angle is calculated as follows: first, the approximately 3500 m. The pilots were E. Enevoldson flight path relative to the air is determined by subtract- and S. Fossett. The flight lasted 4.8 h and proceeded ing out the integrated horizontal wind velocity (de- along the Owens Valley to the east of the Sierra Nevada termined as in section 3) from the flight path over the to Big Pine and returned to California City. The flight ground. This removes the effect of wind drift. The cur- path is shown in Fig. 5. A plot of flight altitude versus vature of the flight path (relative to the air) is then cal- time is shown in Fig. 6. The maximum altitude was

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FIG. 6. Sailplane altitude vs time after takeoff.

The GPS coordinates and airspeed were low-pass ﬁl-

FIG. 5. Flight path shown in blue (displayed with SeeYou; tered with a cutoff frequency of 1 Hz. The ground ve- available online at http://www.seeyou.ws). The cross at the bottom locity was calculated, separately for the longitude and denotes the takeoff point at California City. Position as labeled is latitude components, by differentiating the ﬁltered po- relative to the takeoff point. The black line denotes the ridge line, sition data using central differences. The horizontal as described in the text. wind speeds were estimated as described in section 3

using the parameters rregion 5 2 km, hregion 5 100 m, 13 044 m. There was extensive middle cloud upwind of smax 5 1.5, mmax 5 100, m9510, N 5 5, and Dmin 5 3. the Sierra Nevada and lenticular clouds were present in Typical values obtained for sw were in the range of 0.2– layers up to 14 000 m. Large lee rotors were present in 1.0 m s21, and typical values for D were in the range of the Owens Lake area. 2–25. An example of the clustering analysis for one The sailplane was a production Glaser-Dirks DG- horizontal wind speed estimate is shown in Fig. 7. In this 505M (DG Flugzeugbau GmbH, Bruchsal, Germany) example, there are 100 pairs of data, and the partition specially equipped for high-altitude flight and flown into the sets A and B are shown by the circles and at an all-up weight of 805 kg (wing loading 44 kg m22). crosses, respectively. The partition gives a clear solution 21 In addition to the usual instruments, the sailplane was in this case, as shown with sw 5 0.6 m s and D 5 5. equipped with a modified Volkslogger GPS positioning Close-ups of two portions of the flight path and the system and pressure transducer (Garrecht Avionik corresponding horizontal wind velocity estimates are GmbH, Bingen, Germany), a Borgelt B-50 airspeed in- shown in Fig. 8. The flight segment in Fig. 8a is near the dicator (Borgelt Instruments, Toowoomba, Australia), center of Fig. 5, and that in Fig. 8b is near the top of and a Platinum resistance temperature detector (RTD) Fig. 5. Note also that estimates of the horizontal wind outside air temperature probe. GPS fixes were obtained velocity are obtained where the flight path is changing at 1-s intervals from the Volkslogger, and pressure re- direction but not where it is relatively straight. There- cordings were made at 8-s intervals. Airspeed and tem- fore, more estimates are obtained in Fig. 8b than in perature measurements were made at approximately Fig. 8a as a result of the more convoluted flight path in 2.5-s intervals. All data were merged into a serial data the latter than in the former. The horizontal wind ve- stream and recorded on a custom datalogger. All data locity is expected to be relatively constant in thin alti- (except GPS fixes) were linearly interpolated onto the tude layers and over fairly large horizontal areas. This is 1-s GPS time stamps post flight. The sailplane flight polar evident in Fig. 8, indicating stability of the method. was determined using a combination of the 45 kg m22 The estimated wind velocities for the whole flight flight polar from the DG-505M flight manual and mea- were collected into altitude bins 200 m thick and aver- surements made by comparison flights with another aged, separately for the climbing and descending por- standard sailplane. The sink rate varies between ap- tions of the flight, and the speed and direction are shown proximately 0.5 and 1.3 m s21 at 25 and 40 m s21 IAS, versus altitude in Fig. 9. Note the increasing wind speed respectively. with altitude and the fairly constant west-southwest

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ﬂight path and the radiosonde location are geographically quite close. The rms differences between the sailplane-derived estimates of the wind speed and direction and the average of the two soundings are 6 m s21 and 88 (above 4000 m), respectively. In comparison, the rms differences between the two soundings are 3 m s21 and 98. Wind speed and direction estimates obtained from other aircraft-based systems have given standard deviations of 0.5–1.0 m s21 and 108 for AAVs (van den Kroonenberg et al. 2008) and 2ms21 for light aircraft (Wood et al. 1997). Overall then, the precision of the estimates we obtain is reasonable, given the variations in the radiosonde soundings, and comparable to other similar aircraft-based methods. The vertical wind speed along the ﬂight path was estimated as described in section 4. The maximum bank

angle was set to amax 5 308, resulting in 5% of the data points being excluded because of excessive bank. The rate of change of airspeed, used to calculate the cor-

rection ye described in section 4, must be calculated over a time interval longer than the time constant of the low- pass filter to obtain a good estimate. We found that using finite differences over a time interval of 8 s gave good results. The vertical wind speed estimates were thus obtained every 1 s and were low-pass filtered with a cutoff frequency of 0.2 Hz. Because the flight path traverses a small region of the total volume of airspace in which the wave exists, the estimates obtained are a sparse sampling of the wave structure. An inherent problem then is how to interpret this sparse information. Furthermore, although the wave structure is likely to be stable over periods on the order of 10–30 min, it may vary over hours. A denser FIG. 7. Example of partitioning for a horizontal wind velocity sampling in space and time may be feasible using special estimate: (a) optimum partition of the pairs of solutions into the purpose flights or multiple sailplanes, as described in sets Aopt (x) and Bopt (o) and (b) a close-up of the set Aopt. section 6. For this flight, we present the vertical wind speeds in small regions to show the local wave structure. direction. For comparison purposes, the wind speeds For fixed meteorological conditions, the wave struc- and directions recorded by radiosonde soundings at ture is, in principle, fixed relative to the topography. A Edwards Air Force Base (AFB), Edwards, California convenient reference point for a hill or ridge is the (approximately 120 km south of the center of the flight center of the ridge. For complex terrain, the situation path), at 1500 UTC 24 April 2003 and at Desert Rock, is more complicated. However, particularly for the Mercury, Nevada (approximately 180 km east of the northern part of this flight, the dominant topography is center of the flight path), at 0000 UTC 25 April 2003 are the eastern ridge of the mountain range, running north- also shown in Fig. 9. These soundings are approximately northwest to south-southeast, to the west of the flight 9 h before and at the time at the midpoint of the flight, path. We therefore take as a reference the center line of respectively. Inspection of the figure shows good con- this ridge, which we refer to as the ‘‘ridge line.’’ Points at sistency between the estimated wind speeds and the maximum altitude along this ridge were located manu- soundings, given the different locations and times and ally, and the ridge line was determined by spline in- keeping in mind that the radiosondes are drifting east. terpolation. The resulting ridge line is shown as the Note the good agreement between the wind speeds at black line in Fig. 5. low levels derived from the ascending flight path data The wave structure is most clearly evident in a plot of and from the Edwards AFB radiosonde data, where the the vertical wind speed versus distance downwind from

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FIG. 8. Wind velocity estimates for two flight segments, as described in the text. The length and direction of the arrows represent the estimated wind speed and direction, respectively. a particular point on the ridge line and at a fixed altitude. 0.7 m s21. The reason for this is unknown and did not The streamlines in such a plot are expected to approxi- occur with the analysis of data from other flights. mately follow an exponentially damped sinusoid; for A close-up of a 30-min flight segment that is approx- a constant horizontal wind speed, the vertical wind imately parallel to the ridge line is shown in Fig. 11a. The speed is also an exponentially damped sinusoid that is flight path is replotted in Fig. 11b as a function of dis- phase shifted relative to the streamlines. The only pri- tance downwind from the ridge line and color coded by marily downwind segment of significant length in this the derived vertical wind speed. The fight altitude versus flight is a 4-min segment near the maximum altitude of position north is shown in Fig. 11c. The results in Fig. 11b the flight, south of Lone Pine. A close-up of this flight show a strong vertical wind speed for positions between segment is shown in Fig. 10a; it extends approximately 6 and 12 km downwind from the ridge line, correspond- 20 km downwind, the altitude varies between 12 360 and ing to the leading edge of the wave. The short excursion 12 500 m, and it is downwind of a 6-km section of the of the flight path upwind to between 1 and 5 km down- ridge line. The estimated vertical wind speed versus dis- wind from the ridge line takes it out of the leading edge, tance downwind from the ridge line is shown in Fig. 10b. and weaker vertical speeds are evident. This is consistent Inspection of the figure shows a wave structure, and with expected behavior. a least squares fit of an exponentially damped sinusoid Derived vertical wind speeds for longer flight seg- to the data is shown in Fig. 10b. The fit is quite good ments are more difficult to interpret because of the and gives a wavelength of 10 km. A calculation of the complicated effects of complex terrain upwind of the buoyancy wavelength using the temperature and wind flight path. However, if the ridge cross section was uni- speed data (Stull 2000) gives values varying between form and conditions were time invariant, then the ver- 9 km at the ridge height of 3000 m and 18 km at an al- tical speed would depend approximately only on the titude of 12 km. Given the simplicity of the buoyancy altitude and the distance downwind from the ridge. We wavelength approximation, the varying wind speed with consider a 1-h flight segment that is around the maxi- altitude, and the complex terrain, the agreement is mum altitude of the flight and is downwind of a 50-km reasonable. We note that the vertical wind speed esti- length of the ridge line. The derived vertical wind speed mates appear to be offset from zero by approximately in this flight segment, as a function of distance downwind

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FIG. 10. Flight segment near the maximum altitude of the flight FIG. 9. (a) Horizontal wind speed and (b) direction estimates vs showing (a) the flight path relative to ground (ground height color altitude for the ascending portion of the fight (solid line) and the coded as shown and the black curve denotes the ridge line) and (b) descending portion (dashed line). The radiosonde soundings at the vertical wind speed estimates vs distance downwind from the Edwards AFB (dotted line) and Desert Rock (dashed–dotted line) ridge line and the fitted sinusoid. are also shown.

6. Conclusions from the ridge line and altitude is shown in Fig. 12, with Data collected during sailplane flights are potentially the vertical wind speed coded by color. An inspection of useful for atmospheric studies. Existing data or routine the figure shows the sailplane tracking the leading edge flights can be treated as ‘‘sensors of opportunity,’’ and of the wave with large vertical wind speeds, the leading there is also the potential of conducting flights specifi- edge tilted upwind with increasing altitude, a well- cally for this purpose. Basic flight data consisting of known phenomenon resulting from viscous drag (Stull logged position and airspeed can be used to estimate the 2000), and the expected decrease of wave strength with three-dimensional vector wind speed along the flight increasing altitude. A secondary weaker wave crest lo- path. This approach is suitable for estimating wind ve- cated approximately 10 km downwind is evident and is locities in mountain lee waves where the horizontal the same as that shown in Fig. 10. Hence, despite the component is assumed to vary slowly in space and time. sparse sampling and the complex topography and wave Methods for effecting this estimation have been de- structure, this plot shows some expected characteristics scribed. Implementation of the algorithm and applica- of the wave relative to the topography. tion to a sailplane wave flight show that it is effective,

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FIG. 12. Vertical wind velocity (color coded as indicated) vs distance downwind from the ridge line and altitude.

search grant, and NASA for support. Perlan Project flights are supported by the Steve Fossett Challenges. We also acknowledge our late friend Harman Halliday, who encouraged us and provided us with the first flight data in the early stages of this project.

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