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    NSF–NCAR Gulfstream V research aircraft.

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    Aircraft racetrack B over the Sierra Nevada Range. The reference point on the northern leg is Independence, CA. Each racetrack has two ENE–WSW legs nearly parallel to the wind direction. See Fig. 3 for the terrain cross section.

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    Vertical section across the Sierra Range showing the terrain under each leg. The stratosphere is shaded. The GV flight altitudes and a typical wind profile are shown. The King Air flew shorter legs below 8 km.

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    Brunt–Väisälä frequency derived from Visalia and Lemoore upwind soundings during the RF: (a) large-amplitude wave cases (RFs 4, 5, and 10; see Table 1); (b) smaller-amplitude cases (RFs 6, 8, and 9). Note the strong static stability just above the tropopause, with weaker stability above and below.

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    As in Fig. 3, but for wind speed profiles. Note the decreasing wind speed in the stratosphere.

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    Averaged ozone profiles from the six wave cases (Table 1), from GV flight-level data. In the troposphere, the typical concentration is about 60 ppbv. The tropopause is marked by sharp vertical gradients in ozone concentration.

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    Geostrophy check for all racetracks in six flights. Geostrophic wind component parallel to xB is computed from the differences in geopotential between the two legs of the racetrack. This is plotted against the average wind along the legs. The gradient wind balance is also shown with different radii of curvature.

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    Flight-level vertical displacement from a stacked GV racetrack in three strong-wave flights: (a) RF 4, (b) RF 5, (c) RF 10 (see Table 1). The north leg is dashed and the south leg is solid.

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    An example static stability determination from two horizontal legs of a 12-km GV racetrack during RF 5 (IOP 6). Streamline displacement (2) is plotted against potential temperature. This example has the strongest stability found on any leg. The north leg is dashed and the south leg is solid. Straight lines are reference slopes.

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    Schematic of wave-deformed layers and level aircraft track.

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    Four conserved variable diagrams from Gulfstream flights: (a), (b) RF 5 and (c), (d) RF 10. Each set of plots shows two legs, 46 km apart, on the same racetrack (north legs dashed, south legs solid). Each plot shows (left to right) ozone, water vapor mixing ratio, Bernoulli-corrected speed (8), and cross-track wind speed vs potential temperature. These plots can be used to judge the degree of conservation and the horizontal vorticity in the flow. These selected legs are cleaner than the average leg.

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    Energy flux vs the dot product of mean wind and momentum flux for six cases (Table 1): (a) Gulfstream V and (b) King Air. Each point is a leg.

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    Momentum flux in the xB direction vs altitude for Gulfstream V south and north and King Air legs for the three strong-wave cases: (a) RF 4, (b) RF 5, and (c) RF 10. Each point is a leg. Only RF 10 shows a systematic gradient with height.

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    Log 10 of the EQR (EQR = PE/KE) vs altitude. Legs from the three big-wave cases are shown (GV data only). Theoretical curve is from a three-layer linear mountain wave model (Table 3) representing the tropopause inversion as a layer of more stable air between 11 and 13 km (shaded). Note the factor of 10 increase in EQR from just below to just above the tropopause.

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    Power spectra for vertical velocity for two legs at (a) 13 (b) 11 km, and (c) 9 km in RF 10. Note peaks at wavelengths of 14, 20, and 30 km. Reference lines show the 95%, 50%, and 5% confidence levels. Waves longer than 40 km are uncertain due to the short length of the measured wave train.

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    Phase diagrams for the spectrally dominant waves (from KEZ) for all legs on six Gulfstream flights (Table 1). (a) Vertical velocity and potential temperature (confirms orthogonality); (b) pressure and longitudinal velocity (confirms Bernoulli equation); (c) pressure and vertical velocity (describes wave dynamics: some waves propagate up or down and some are nearly trapped).

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    A wave property diagram for the dominant waves from RF 10 on 16 Apr 2006. The cospectrum between pressure and vertical velocity is plotted against wavelength. Three types of waves are present: upgoing waves (30 km), downgoing waves (20 to 30 km), and trapped short waves (11 to 14 km). The occurrence of these wave types changes markedly with altitude.

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Mountain Waves Entering the Stratosphere

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  • 1 Yale University, New Haven, Connecticut
  • | 2 NCAR*–UCAR, Boulder, Colorado
  • | 3 Naval Research Laboratory, Monterey, California
  • | 4 UCAR, Monterey, California
  • | 5 Desert Research Institute, Reno, Nevada
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Abstract

Using the National Science Foundation (NSF)–NCAR Gulfstream V and the NSF–Wyoming King Air research aircraft during the Terrain-Induced Rotor Experiment (T-REX) in March–April 2006, six cases of Sierra Nevada mountain waves were surveyed with 126 cross-mountain legs. The goal was to identify the influence of the tropopause on waves entering the stratosphere. During each flight leg, part of the variation in observed parameters was due to parameter layering, heaving up and down in the waves. Diagnosis of the combined wave-layering signal was aided with innovative use of new GPS altitude measurements. The ozone and water vapor layering correlated with layered Bernoulli function and cross-flow speed.

GPS-corrected static pressure was used to compute the vertical energy flux, confirming, for the first time, the Eliassen–Palm relation between momentum and energy flux (EF = −U · MF). Kinetic (KE) and potential (PE) wave energy densities were also computed. The equipartition ratio (EQR = PE/KE) changed abruptly across the tropopause, indicating partial wave reflection. In one case (16 April 2006) systematically reversed momentum and energy fluxes were found in the stratosphere above 12 km. On a “wave property diagram,” three families of waves were identified: up- and downgoing long waves (30 km) and shorter (14 km) trapped waves. For the latter two types, an explanation is proposed related to secondary generation near the tropopause and reflection or secondary generation in the lower stratosphere.

Corresponding author address: Ronald B. Smith, Yale University, P.O. Box 208109, New Haven, CT 06520-8109. Email: ronald.smith@yale.edu

This article included in the Terrain-Induced Rotor Experiment (T-Rex) special collection.

Abstract

Using the National Science Foundation (NSF)–NCAR Gulfstream V and the NSF–Wyoming King Air research aircraft during the Terrain-Induced Rotor Experiment (T-REX) in March–April 2006, six cases of Sierra Nevada mountain waves were surveyed with 126 cross-mountain legs. The goal was to identify the influence of the tropopause on waves entering the stratosphere. During each flight leg, part of the variation in observed parameters was due to parameter layering, heaving up and down in the waves. Diagnosis of the combined wave-layering signal was aided with innovative use of new GPS altitude measurements. The ozone and water vapor layering correlated with layered Bernoulli function and cross-flow speed.

GPS-corrected static pressure was used to compute the vertical energy flux, confirming, for the first time, the Eliassen–Palm relation between momentum and energy flux (EF = −U · MF). Kinetic (KE) and potential (PE) wave energy densities were also computed. The equipartition ratio (EQR = PE/KE) changed abruptly across the tropopause, indicating partial wave reflection. In one case (16 April 2006) systematically reversed momentum and energy fluxes were found in the stratosphere above 12 km. On a “wave property diagram,” three families of waves were identified: up- and downgoing long waves (30 km) and shorter (14 km) trapped waves. For the latter two types, an explanation is proposed related to secondary generation near the tropopause and reflection or secondary generation in the lower stratosphere.

Corresponding author address: Ronald B. Smith, Yale University, P.O. Box 208109, New Haven, CT 06520-8109. Email: ronald.smith@yale.edu

This article included in the Terrain-Induced Rotor Experiment (T-Rex) special collection.

1. Introduction

The sharp boundary between the troposphere and stratosphere has been largely explained as the result of convective cloud-top entrainment (e.g., Staley 1960; Reid and Gage 1981; Held 1982; Sherwood and Dessler 2003). The tropopause is defined not only by the jump in temperature lapse rate but also by strong gradients in water vapor and ozone concentrations. In midlatitudes, the jet stream often has its maximum speed at the tropopause level. It is suspected that these properties of the tropopause may influence mountain waves propagating into the stratosphere. The sharp lapse rate and wind shear change at the tropopause may cause partial reflection and discontinuous aspects of wave structure. According to linear theory, the only wave properties that are likely to be continuous across the tropopause are the momentum flux (MF) and possibly the energy flux (EF; Eliassen and Palm 1961, hereafter EP61). Once the waves have entered the stratosphere, the greater static stability and decreasing wind speed and air density are likely to promote wave amplification and breaking.

Two well-known theoretical treatments of mountain waves in a variable medium define the range of possibilities. In the Wentzel–Kramers–Brillouin (WKB) approach to linear wave theory (Bretherton 1966; Broutman et al. 2003), medium variations are assumed to be gradual compared to the wave vertical scale (LZ = U/N ≈ 3 km). There is no partial wave reflection in this formulation, although trapped waves can be added separately. The wave amplitude and vertical wavelength adjust smoothly to each atmospheric layer encountered as the wave propagates upward. While this model neglects wave reflection, it captures the wave amplification due to decreasing wind and density aloft.

In the discrete layered approach to linear wave theory, property variations are assumed to be abrupt. Vertical displacement and pressure are matched across each interface and these interface conditions are used to find a self-consistent set of wave amplitude coefficients (EP61). In the layers below the discontinuity, both up- and downgoing waves will be present. In theoretical work, many authors (e.g., EP61; Klemp and Lilly 1975) considered partial reflections from inversions and the tropopause. In observational campaigns, however, no clear signature of partial reflection has yet been identified.

The observation of wave propagation across the tropopause with aircraft requires extensive flight duration at a high altitude. Over the last 40 yr, several field projects have used jet aircraft with capability for stratospheric wave observation. One pioneering study was the coordinated aircraft flights over the Front Range on 17 February 1970 described by Lilly and Kennedy (1973). Using four aircraft, the waves were surveyed from the surface up to 19 km, well into the stratosphere. The highest legs were flown by two Martin B-57 Canberra aircraft. Because the differential pressure method of vertical velocity determination was not available on all aircraft, some vertical motions were computed from gradients in potential temperature. Using these estimates for w, momentum fluxes were computed as a function of altitude. Typical values of MF = −200 × 106 dynes cm−1 = −200 kN m−1 were found up to a turbulent layer at 16 km. Above this layer, the MF seemed to decrease quickly. Other high-altitude wave observations have been reported by Lilly and Lester (1974) and Leutbecher and Volkert (2000).

Lilly and Kennedy (1973) included a discussion of phase relations between flight-level variables that might be associated with shear, especially influencing the perturbation horizontal velocity. In the present work, we extend their attempt to separate finescale background layering from waves in the atmosphere. Salathé and Smith (1992) discussed aircraft observations of small-scale temperature layering in the stratosphere.

A more recent observation of vertically propagating mountain waves was in the Pyrénées Experiment (PYREX). Stacked aircraft were used, with the DLR Falcon flying the highest legs near 13 km. In one event of southerly airflow on 15 October 1990, an MF profile was derived. This profile showed MF = −130 kN m−1 in the low troposphere, decreasing to MF = −40 kN m−1 in the upper troposphere and low stratosphere (Bougeault et al. 1997). Recent Alpine wave observations are reviewed by Smith et al. (2007).

In the present paper, we discuss new aircraft observations of Sierra Nevada mountain waves propagating across the tropopause into the stratosphere. We seek observational evidence of partial reflection, trapping, and nonlinear wave generation. Several new diagnostic tools are described in sections 3 through 7. An interesting case of reversed flux is discussed in section 8.

2. Flight track design

The Terrain-Induced Rotor Experiment (T-REX) project was carried out in March and April of 2006 with a primary objective of observing waves and rotors near the Sierra Nevada range and the Owens Valley in California (Grubišić et al. 2008, manuscript submitted to Bull. Amer. Meteor. Soc.). In the cases considered herein, two aircraft were used to probe at different altitudes. The University of Wyoming King Air was used to sample the atmosphere from the surface up to 8 km. The National Science Foundation (NSF)–National Center for Atmospheric Research (NCAR) Gulfstream V (GV) sampled from 9 km up to its maximum altitude of 14 km (Fig. 1). This was the first full research project to use the new GV (Laursen et al. 2006). Due to the long 2-h ferry flight to and from Colorado, the GV was more limited in duration and altitude in T-REX than it will be in future projects. Unfortunately, like the DLR Falcon in PYREX, the GV could not reach the middle stratosphere where mountain wave breaking is more common. In addition to the aircraft soundings and horizontal legs, there were frequent upwind balloon soundings from Visalia and Lemoore, California, and dropwindsondes released from the GV. These gave accurate wind and temperature profiles during each aircraft mission.

To make the GV flights as useful as possible, a standard wind-oriented 46-km-wide clockwise racetrack pattern (Fig. 2) was flown during each event, passing over two similar sections of the Sierras terrain. The racetrack pattern maximized the number of cross-mountain legs and minimized the time and space devoted to turns. The legs of the racetrack were flown very accurately, within 100 m horizontally, to insure comparability. The racetrack encircled Mt. Whitney (4421 m), crossing over the Sierras ridgeline near 3700 m (Fig. 3). Altitude changes were done in spiral ascents and descents at the eastern end of the racetrack. Repeated racetrack patterns and altitude changes were possible within a military control area (Joshua) over the Sierras, despite the heavy aviation traffic outside the area. Typically, flight missions were scheduled to coincide with the passage of baroclinic troughs bringing brief episodes of strong cross-barrier wind.

For maximum comparability, we limit our discussion to six events during which the wind direction was near 245°true north: our so-called Track B direction (Table 1). We computed statistics from 126 legs, including the north and south legs for each racetrack for each flight. Winds and position coordinates are rotated into a leg-aligned system for analysis (xB, yB). The origin of this coordinate system is Independence, California, on the north leg. As seen in Table 1, the six flights show considerable variability in the amplitude of vertical wind velocity and temperature fluctuations. In general, research flights (RFs) 4, 5, and 10 found strong waves while RFs 6, 8, and 9 found weak waves. While it may be just coincidence, the three strong wave events corresponded to the three strongest cross-mountain ridge-top winds, as determined by averaging 700-hPa data from upstream balloon soundings (i.e., Lemoore and Visalia). The data suggest a sharp ridge-top wind threshold of 15 m s−1 for large wave generation.

During each GV mission, the King Air flew legs within the Owens Valley and above the Sierras crest. In this study, only the higher cross-mountain King Air legs are included. These legs are not as long as the GV legs, but they give useful estimates of wave properties in the altitude range from 6 to 8 km. When two King Air flights occurred in an intensive observing period (IOP), only the one during the GV flight is analyzed (Table 1). Only one cross-mountain line is flown by the King Air, usually in between the north and south GV tracks.

The wind and stability profiles were similar for the six flights (Figs. 4 and 5). Each wind profile had strong positive unidirectional wind shear in the troposphere with a maximum wind speed of U ∼ 45 m s−1 at the tropopause near 11 km. Aloft near Z = 21 km, the wind weakened to less than 10 m s−1 or to zero in some cases. The frequent occurrence of weak winds and possible wave breakdown at 21 km during T-REX is consistent with the Sierras’ reputation. While the Sierras are a world-famous site for mountain waves (Grubišić and Lewis 2004), satellite observations show that the Sierras make little contribution to waves in the upper stratosphere and mesosphere (Jiang et al. 2003).

The temperature profiles showed some static stability just above mountain top from 3 to 4 km with N2 ≈ 0.0001 s−2, and weaker stability in the upper troposphere. Just above the tropopause, the static stability jumps to N2 ≈ 0.0005 s−2 for 2.5 km, and then decreases to about N2 ≈ 0.0002 s−2. We call this strongly stable layer the “tropopause inversion.”

To confirm the location of the tropopause, we show in Fig. 6 the average ozone distribution with height for each wave case, derived from GV soundings. The first sharp gradient can be used to identify the tropopause. There is considerable variation from case to case, but the lowest aircraft altitude of 9 km is always in the troposphere and the highest aircraft altitude of 13 km is always in the stratosphere.

Because of space limitations, we cannot provide a full description of the aircraft instrumentation used in T-REX. Both the GV and King Air deduce wind speed using a differential pressure on a nose cone or gust probe, together with acceleration sensors and GPS to obtain aircraft position and orientation (Brown et al. 1983). The GV achieved higher GPS-position accuracy using a differential correction from a ground station. Ambient atmospheric pressure is obtained from a static pressure port on the fuselage, corrected for airspeed, density, and orientation. Air temperature is obtained from a shielded Rosemount thermistor. Water vapor amount is obtained from an electrically chilled mirror dew-point detector for the middle troposphere, and (GV only) a dual wavelength absorption hygrometer (Spectrasensor Open Path Tunable Diode Laser) is used for the dryer air in the upper troposphere and low stratosphere. Ozone concentration is obtained on the GV with a chemiluminescence technique (Ridley et al. 1992). More instrument details are available from the NCAR and University of Wyoming flight facilities.

3. Flight level data and wave kinematics

One advantage of the racetrack path is that differences in GPS altitude and pressure between the two legs can be used to evaluate the degree of geostrophy in the mean flow. We computed the component of the geostrophic wind velocity along the xB direction using
i1520-0469-65-8-2543-e1
where ZGPS and Zp are the geometric and pressure altitudes. The Coriolis parameter is evaluated at 36.5°N ( f = 0.86 × 10−4 s−1) and the distance between legs is ΔyB = 46 000 m. Most of the contribution to (1) comes from the difference in GPS altitude (e.g., ΔZGPS ∼ 20 m) as the aircraft circles the racetrack at nearly constant pressure altitude (e.g., ΔZP ∼ 3 m). The exceptions are the highest flight legs (e.g., ZP ∼ 13 km) in strong waves, where the aircraft has difficulty maintaining an accurate pressure altitude.

In Fig. 7, we compare the observed wind speed, averaged for the two legs, with the geostrophic speed for all the racetracks in the six flights. For slower winds, the flow is slightly supergeostrophic, while for fast winds aloft, the flow is significantly subgeostrophic. We suggest that the strong subgeostrophy aloft is caused by streamline curvature and southward centripetal force in the upper-level troughs above the Sierras. To evaluate this hypothesis, we plot the gradient wind theory in Fig. 7 {i.e., UGEOS = U[1 + (U/fR)]}. From synoptic maps, we estimate the streamline radius of curvature to be about R = 800 km. Using this value for radius, the gradient wind theory agrees qualitatively with the observations. Further evidence for ageostrophy comes from the Naval Research Laboratory (NRL)–Coupled Ocean–Atmosphere Mesoscale Prediction System (COAMPS) nested numerical simulations of these cases. For the RF 10 (IOP 13) simulation, we found a similar degree of subgeostrophy, unconnected with the waves.

A useful wave variable is the vertical streamline displacement. As the flight legs are nearly parallel to the mean wind speed, and assuming steady flow, the vertical streamline displacement can be estimated by integrating the vertical velocity along the leg
i1520-0469-65-8-2543-e2
In Fig. 8 we plot the displacements for the three strong wave events. Data from the north and south legs are shown. We note that some of these displacements reach nearly 1 km. While some small waves are seen upwind of the Sierras crest, all the large waves are over the crest and downstream, supporting the idea that the waves are generated by the Sierras. Fair consistency over time is seen, supporting the steady-state assumption. The agreement between the north and south legs is fair to poor in Track B coordinates, indicating significant differences in the wave field over the lateral distance of 46 km.
As the wave advects air to the aircraft height from a range of altitudes, it is easy to estimate the static stability of the local atmosphere. In Fig. 9, potential temperature is plotted against displacement (2), so that the slope gives
i1520-0469-65-8-2543-e3
In the case shown, /dz ≈ 0.023 K m−1 near the θ = 340 K level. For weak waves, these plots are looping and nonlinear. For the larger waves, the plots are fairly linear and the slope estimates agree well with vertical profiles from radiosondes and dropsondes. Deviations from linearity suggest complex layering (e.g., Salathé and Smith 1992).

The displacement estimates provide one method for testing the linearity of the waves. We define the nondimensional nonlinearity parameter A = M/U. A typical low stratosphere value is A = (0.02 s−1)(1000 m)/ (35 m s−1) ≈ 0.6, so that nonlinear effects are important but not dominant at the observed altitudes. Further aloft, as the wind speed decreases, A exceeds unity and the wave will become more nonlinear.

4. Conserved variable diagrams

If the atmosphere is finely layered with conserved quantities and then disturbed by a gravity wave, a level flying aircraft will penetrate these layers repeatedly, in forward and reverse order as depicted in Fig. 10. Kinematically, this is similar to the encounter of a vertically saw-toothed aircraft track with horizontal layers, except that wave-induced winds and pressures will be present. In this section, we examine four conserved quantities: ozone, water vapor, cross-flow wind speed, and the Bernoulli function. Potential temperature (θ) is also conserved according to /dt = 0. Theta is used as the vertical coordinate for all tracer plots because it increases monotonically upward (Fig. 11). The range of theta is limited by the displacement amplitude of the wave.

Ozone and water vapor concentrations are conserved quantities (i.e., dqOZONE/dt = 0 dqW/dt = 0) over the advective time scales considered here (i.e., T ≈ 180 000 m/30 m s−1 = 6000 s). Ozone is well suited as a tracer for the stratosphere because it is abundant and finely laminated there. Water vapor is better suited for the troposphere or lowest stratosphere. Further aloft its concentration is too small for high-rate observation with GV instruments.

A useful dynamic tracer is the component of wind parallel to the wave crests (υB), that is, perpendicular to the horizontal wavenumber vector. Because the wave-induced pressure gradients are zero along the wave crests, this velocity component is conserved as parcels ride through the wave. We assume, and later confirm, that the wavenumber vector is nearly aligned with our Track B, thus υB is conserved.

In contrast, the velocity component parallel to the wavenumber vector (uB) is not conserved. As parcels move through the wave, pressure gradients act to accelerate this component. In steady perfect compressible flow, the “mechanical” Bernoulli function
i1520-0469-65-8-2543-e4
with
i1520-0469-65-8-2543-e5
and K = p1/γ0/ρ0[1 − (1/γ)] is conserved following a streamline (Prandtl and Tietjens 1934). The subscript 0 indicates a reference state somewhere on the streamline. Note that we do not use the so-called dry static energy (Gill 1982) as a Bernoulli function because it is dominated by the internal energy and thus is partly redundant with theta (see appendix B).
Due to the low mach number of the flow and the fact that the parcel displacements are much less than the density-scale height, the relative variations in pressure along a streamline are generally small. Thus, we can linearize (5):
i1520-0469-65-8-2543-e6
where p′ is the difference between the pressure and the reference pressure. The error in the variable part of (6), with respect to (5), is less than 2% for a vertical displacement of 500 m. Dropping the constant term in (6) gives the familiar Bernoulli equation for incompressible flow,
i1520-0469-65-8-2543-e7
along a streamline. The Bernoulli function [(4) or (7)] cannot be evaluated without GPS altitude data for the last term (z). As the aircraft flies nearly on a constant pressure surface, the first term in (4) or (7) makes a relatively small contribution. In practice, a region of high pressure along a leg shows up in the data as a region of increased aircraft altitude. As a practical matter, the w term in (7) is also rather small. The dominant balance in (7) along a flight leg is between the (1/2)u2B and the gz terms.
For the plotting purposes, we use (7) to define a “Bernoulli-corrected” velocity (uC) as a conserved variable. It is the velocity that the air on that streamline would have if the pressure and altitude were not disturbed by a wave:
i1520-0469-65-8-2543-e8
where the barred values are leg means. If there were no shear upstream, changes in the terms inside the square bracket would cancel each other as the aircraft flies through the wave.

The confirmation that B1 (or uC) is conserved on a streamline can be seen when B1 is plotted against potential temperature (θ) for each leg (Fig. 11). While there is considerable scatter, there is a clear tendency for the repeated wave penetrations to overlie one another. Whenever the same streamline is encountered (i.e., the same θ), the same B1 is found as well.

In Fig. 11 we show four conserved-variable diagrams, comprising 8 of the 126 legs. The plots are aligned with potential temperature as a common vertical coordinate. Note that the curves repeat the same tracer profile as the aircraft flies through the wave, indicating that we are seeing the same layering over and over along the leg. In some of the conserved-variable diagrams, there is an apparent correlation between the chemical and dynamic variables, suggesting that the chemical and dynamical layering might arise from an ambient interleaving of distinct air masses.

We are not able to judge whether the wind layering has an influence on the mountain wave propagation. If the layers are finescaled compared to the vertical wavelength, the waves would only see average atmospheric properties. A typical vertical scale for mountain waves is LZ = U/N ≈ 2 km compared with a 100- to 300-m scale for the layers. However, the layers impact our ability to observe wave dynamics, especially the observed variations in horizontal wind. The layers might also impact wave breakdown by localizing shear instability.

5. Momentum and energy fluxes

An essential property of mountain waves is their flux of momentum and energy. We imagine a field of waves in two dimensions so that measurements along a single line can fully describe theses fluxes. Our Track B is neither necessarily perpendicular to the wave-phase lines nor parallel to the mean wind. Thus, in B coordinates, there may be two components to the wave momentum flux:
i1520-0469-65-8-2543-e9a
i1520-0469-65-8-2543-e9b
with units N m−1. The energy flux
i1520-0469-65-8-2543-e10
with units in W m−1, is associated with the propagation mechanism of the mountain wave. These quantities will be independent of leg length if the leg is long enough to contain all the large waves. Following EP61, in steady small-amplitude nondissipative flow, the energy and momentum fluxes are related by
i1520-0469-65-8-2543-e11
where U is the mean flow speed at the level of flux measurement. While the momentum flux (9) has frequently been computed from aircraft data, the energy flux (10) has never been directly computed from observations due to the lack of pressure data. Research aircraft usually measure static pressure, but this data were formerly used to obtain aircraft altitude. With no independent altitude data, a meaningful ambient pressure could not be obtained.

A solution to this long-standing observing problem has previously been found over the ocean where a radar altimeter can provide the aircraft altitude. Several authors have used the “D value” (i.e., the difference between pressure and geometric altitude) to diagnose pressure fields over the sea (Brown et al. 1981). The method has been used to study ageostrophic jet streams (Shapiro and Kennedy 1981), hurricane central pressure (Willoughby et al. 1989), and gap jets (Lackmann and Overland 1989; Smith and Grubišić 1993). However, such a method cannot be used over mountains, even if the terrain is known, because of uncertainty in the radar reflection point. Thus radar altimetry cannot be applied to the observation of mountain waves (but see the attempt by Shapiro and Kennedy 1982). The availability of GPS altitude data makes the computation of mountain wave EF possible for the first time.

As the aircraft altitude varies as it flies through the wave, we must correct the measured static pressure to a common level, assuming local hydrostatic balance
i1520-0469-65-8-2543-e12
In practice, as the aircraft nearly flies on a constant pressure surface, the first term in (12) makes the smaller contribution. The second term, derived from the GPS altitude data, is dominant in the calculation of (12). We used the mean leg altitude as the reference. When EF is plotted against MF for all the legs (not shown), a nice linear relationship is seen with a slope of about 38 m s−1, approximately the same as the average flight-level wind speed for the three large wave events, roughly verifying (11). The general magnitude of MF (100 to 200 kN m−1) agrees with values from Lilly and Kennedy (1973) (∼200 kN m−1) and PYREX (∼40 to 130 kN m−1) discussed in the introduction.

A more quantitative test of (11) requires that we take into account the variation in mean wind speed U from leg to leg. To do this, we take the vector product of mean wind velocity and MF for each leg so that the two sides of (11) can be directly compared (Fig. 12a). The fitted slope is slightly less than unity (i.e., 0.9), perhaps associated with wave nonlinearity, unsteadiness, or static pressure measurement errors. The EP61 relationship continues to hold for negative EF values (see section 8).

A similar flux comparison is shown for the King Air in Fig. 12b. It shows more scatter, probably due to the shorter legs catching only part of the wave field, but it still has reasonable agreement with (11). The lesser slope of 0.8 suggests that some of the momentum may be carried by turbulence rather than by waves. One strong negative EF value is seen from RF 5.

EP61 also proved that MF would be independent of altitude in steady nondissipative flow, even in the presence of background shear and lapse-rate variations. In Fig. 13 we test this prediction by showing MFxB as a function of altitude for the three strong wave flights. Each case is different and all have considerable scatter. In RF 4 (IOP 4), the south leg has 3 times the momentum flux of the north leg, and the north leg MF is more constant with height. The King Air data have similar MF values to the GV, but with even more scatter. There is no apparent MF shift at the tropopause.

In RF 5 (IOP 6), the south leg MF again exceeds the north leg by factor of 3 or so (Fig. 13b). In this case, the King Air fluxes are near zero. There are even a few small positive values. Further aloft, the GV data show only negative values and no shift in MF value at the tropopause. This apparent vertical gradient in MF violates the Eliassen–Palm theorem. The wave may have been unsteady. Because our MF values are computed along a line instead of over an area, three-dimensional effects could also cause apparent violations of the Eliassen–Palm theorem.

The momentum flux profile for RF 10 on 16 April 2006 has a different character than the other two strong-wave flights (Fig. 13c). Below 8 km, the King Air shows large negative MF reaching −300 kN m−1 with much scatter. GV data in the upper troposphere and low stratosphere show smaller negative values. Above 12 km, the MF becomes positive. These are the points we saw in Fig. 12a with negative EF values and corresponding positive MF values. Such values would not be found in simple vertically propagating or trapped mountain waves. The north and south legs agree on this flux reversal, and there is little trend with time. This curious reversal of EF and MF presents a challenge to wave theorists and modelers. It suggests that wave energy is converging on the 12-km level from below and above. We will return to this question in section 8.

6. Wave energy density

a. Observations

Another important quantity in the diagnosis of gravity waves is the energy density (Gill 1982). Energy density is a quadratic quantity, like MF or EF, but with different properties. In the present situation, we integrate the energy contributions over the 180-km-long flight leg, defined as our coordinate xB, giving energy densities with units J m−2. The means are removed from all primed quantities. The horizontal kinetic energy (KEH) is the sum of the squared wind perturbations in the xB and yB directions:
i1520-0469-65-8-2543-e13
where
i1520-0469-65-8-2543-e14
i1520-0469-65-8-2543-e15
Recall from earlier discussions that only a portion of the horizontal KE is actual wave energy. The rest arises from vertical advection of dynamic layering. In fact, because of the orientation of our track, KEyB should be almost entirely due to layering. A valid analysis of horizontal wave kinetic energy should remove the layering part and use only the wave-induced horizontal velocities. To do this, we propose using the observed corrected pressure variations to compute KEH according to
i1520-0469-65-8-2543-e16
The vertical kinetic energy is
i1520-0469-65-8-2543-e17
The potential energy (PE) is a measure of the lifting of cold heavy air and depression of warm light air. If the background stability frequency is not known, we use
i1520-0469-65-8-2543-e18
If N2 is well defined, from the aircraft data itself or nearby balloon data, we can use
i1520-0469-65-8-2543-e19
We confirmed that (18) and (19) give similar but not identical values when N2 is evaluated from the upstream soundings.
By forming various ratios of these energy densities, we can identify wave properties in a way that is independent of wave amplitude (Table 2). The quantity
i1520-0469-65-8-2543-e20
describes the ratio of dynamic wave-induced horizontal kinetic energy (KHR) to the total horizontal kinetic energy measured by the aircraft. If there were no wind layering, KHR would be unity.
The ratio of vertical-to-horizontal wave kinetic energy is a measure of the relative magnitude of vertical-to-horizontal velocity perturbations. We define the kinetic energy ratio (KER) as
i1520-0469-65-8-2543-e21
A small value of KER indicates that the parcel oscillations are nearly horizontal and the waves are nearly hydrostatic. Values near unity would be nonhydrostatic. In the extreme example of a trapped wave at its level of maximum vertical motion (i.e., no horizontal motion), KER is infinite.
In many linear oscillating systems (e.g., pendulums, spring-mass systems, acoustic waves, etc.) the principle of equipartition applies; that is, the integrated kinetic and potential energies are equal (i.e., KE = PE). This principle applies to internal gravity waves as well. In trapped waves, the volume-integrated energy densities satisfy equipartition. In simple vertically propagating waves, or in the WKB approximation, the area-integrated KE = PE at each level in the atmosphere. To test the degree of equipartition in Sierra gravity waves, we define the equipartition ratio (EQR) for each flight leg as
i1520-0469-65-8-2543-e22
In appendix A, we prove that EQR will depart from unity whenever both up- and downgoing waves are present at a given level.

An overview of the energy ratios (20)(22) over the Sierras is given in Table 2. The ratio (KHR) of wave-induced horizontal speed variance to observed horizontal variance is roughly constant with a value of KHR = 0.5. Thus, for the big wave events, speed variance from layering is about equal to the wave-induced speed variance.

The ratio (KER) of vertical-to-horizontal speed variance is quite variable from case to case and with altitude. In the upper troposphere, KER is small, suggesting shallow particle orbits and hydrostatic flow. In the stratosphere, the value is significantly larger but never exceeds unity. This larger value is a surprise as one would expect a vertically propagating wave in the stable stratosphere to be even more hydrostatic. We believe that it arises from trapped waves in the stratosphere, riding on the tropopause inversion.

The equipartition ratio varies strongly with altitude (Fig. 14). In the upper troposphere, it is much less than unity, typically 0.2. Just above the tropopause, it exceeds unity, typically between 2 and 5. This is approximately a factor of 10 increase. In RFs 4 and 5, it stays large, while in RF 10 it drops back to a small value. In the next section we argue that this jump in EQR is an indication of partial reflection.

b. A linear model of tropopause impact on wave equipartition

To see how equipartition is influenced by the tropopause, we used a generic three-layer linear FFT model of steady mountain waves (EP61; Klemp and Lilly 1975; Smith et al. 2002). To mimic Fig. 4, the stability is increased to N = 0.02 s−1 between 11 and 13 km (Table 3). Above 13 km, the stability and wind speed are decreased by 30%, keeping the Scorer parameter fixed. No attempt is made to accurately represent the lower atmosphere. The waves are generated by a bell-shaped ridge lifted to a base altitude of 3 km.

The EQR predicted by linear mountain wave theory is shown in Fig. 14. Because the deviations from EQR = 1 are large, we use a logarithmic presentation. In the lower two layers, EQR oscillates strongly with altitude due to the presence of both up- and downgoing waves (see appendix A). The downgoing waves are caused by partial reflections at the z = 11 and 13-km interfaces. Note that there is some reflection at z = 13 km, even though the Scorer parameter has the same value above and below (Table 3), due to the discontinuity in wind speed. Above 13 km, where only upgoing waves are present, EQR is constant with a value of unity. This calculation confirms EQR as a useful measure of wave reflection. If the WKB approximation had been used, EQR would be forced to unity by the neglect of reflection.

7. Spectra and phase relationships for dominant waves

A further analysis of mountain waves can be accomplished with spectral decomposition. The spectral approach helps to identify dominant waves and to examine the phase relationships between different variables. To do this, we interpolate the 1-s flight-level data onto a 200-m spatial grid with approximately 900 points. We remove the mean, taper the ends of the record, and apply a Fourier transform to compute the power and cross spectra [NCAR Command Language (NCL) algorithm specxy_anal]. For the vertical velocity, the Fourier transform is
i1520-0469-65-8-2543-e23
or the equivalent discrete transform for equally spaced data points
i1520-0469-65-8-2543-e24
The power spectrum is
i1520-0469-65-8-2543-e25
and cross-spectrum between, say, vertical velocity and pressure is
i1520-0469-65-8-2543-e26
The real and imaginary parts of (26) are the cospectrum and quad spectrum.

A few examples of power spectra (25) are shown in Fig. 15. The highest spectral peaks are considered to be the dominant wavenumbers. Not every leg has a clear dominant wavelength. When a dominant wave is present, we can examine the phase relationships among different variables.

As a first step, we examine the phase of the potential temperature and vertical velocity. We assume that the background potential temperature Θ(z) is a function of altitude only and that /Dt = θt + x + w′Θz = 0. If a wave with vertical velocity w′ ∼ ŵei(σt+kx) distorts the theta field, then
i1520-0469-65-8-2543-e27
According to (27), if the wave is not growing or decaying [i.e., IM(σ) = 0, U > 0, and Θz > 0] then potential temperature lags w by 90°. We test this prediction by plotting the co- and quad spectrum for w and theta for each leg (Fig. 16a). The points cluster about a 90° phase shift, thus supporting the above assumptions. Figure 9 also supports this view. The scatter in Fig. 16a probably arises from finescale layering in potential temperature.
Another useful analysis examines the three-way balance expressed in the steady linearized horizontal momentum equation
i1520-0469-65-8-2543-e28
which with continuity ikû + imŵ = 0 gives
i1520-0469-65-8-2543-e29
In (29), if the background state is unsheared (i.e., Uz = 0), the perturbation velocity and pressure are in antiphase, as expected from linearized Bernoulli’s Eq. (13). Low pressure implies fast flow. If the shear dominates (with Uz > 0), the velocity will be delayed by 90° from the pressure (if m is real). In Fig. 16b we test (29) with a phase diagram for corrected pressure (10) and longitudinal velocity (uB). The scatter around 180° is significant, suggesting some influence of shear.

There is another important interpretation of the negative phase relationship between pressure and longitudinal velocity in Fig. 16b. This phase relationship proves that the wave is doing “pressure work” upstream. This must be the case if the wave is stationary in earth coordinates, propagating against the advective effect of the mean wind. The group velocity in fluid coordinates is oriented westward.

The third phase relationship of interest is between corrected pressure and vertical velocity (Fig. 16c). This relationship is not controlled by a simple conservation law like potential temperature or Bernoulli but rather relates to the nature of vertical wave propagation. The PW correlation is proportional to the EF. Trapped waves will have no PW correlation. In Fig. 16c we see a wide range of phase angles. Most of the points representing dominant waves have a positive (i.e., upward) EF, but negative EF and large quad spectra occur as well. This evidence for downward propagation and trapped waves is pursued in the next section.

8. The flux reversal on 16 April 2006 (RF 10, IOP 13)

We have left for last the interesting problem of the MF and EF flux reversals aloft for RF 10 on 16 April. All seven of the negative EF values in Fig. 12a come from RF 10. Under other circumstances, we would be tempted to ignore this case as a spurious outlier. In this instance, however, the reversal is seen in both fluxes MF and EF. It is persistent over 4 h of flight duration. It is found on both the north and south legs. With this amount of evidence it is hard to ignore this strange flux-reversal event. The only observational inconsistency is the sudden switch to normal fluxes (i.e., positive EF, negative MF) on the last leg of the mission at 14 km (an eastbound north track leg), just before leaving the Sierras area (Fig. 13c).

The spectral structure of the waves on this flight is also curious (Fig. 15). At z = 9 km, the waves are quasi periodic with a wavelength of 20 to 30 km dominating the KEZ power spectrum. These waves have positive EF and negative MF. At z = 11 km, the 20–30-km waves continue to dominate the KEZ, but a distinct 14-km wave is also seen in the KEZ power spectrum. At z = 13 km, the wave field changes dramatically. The 14- and 20-km waves dominate the KEZ power spectrum on most legs. The 30-km wave is gone.

To clarify this complex situation, we construct a “wave property diagram” (Fig. 17). By examining the W-power spectrum and PW-cospectrum for each leg, we identify the biggest and second-biggest wave. Each wave is plotted according to its wavelength and PW-cospectrum properties (recall Fig. 16c). Three clusters of wave types appear. First is a tight cluster of waves with a wavelength of 30 km and upward propagation (i.e., positive PW cospectrum). Second is a tight cluster with a wavelength of about 14 km and zero PW cospectrum. By its lack of vertical propagation, we classify it as a trapped wave. Third is a looser cluster of waves with wavelengths between 20 and 30 km and with downward propagation (i.e., negative PW cospectrum).

The vertical distribution of these waves is informative (note symbols in Fig. 17). At an altitude of 9 km, we find only the upgoing wave with a 30-km wavelength. This is the mountain- generated wave. At an altitude of 11 km, the 30-km upgoing mountain wave is still very evident. In addition, we see a trapped wave with a wavelength of about 14 km. At altitudes of 13 and 14 km, the 30-km upgoing wave is gone, or at least we cannot identify it. The 14-km wavelength trapped wave is present. In addition, an irregular downgoing wave is found with a wavelength from 20 to 30 km.

We interpret the short 14-km “fluxless” wave as a steady trapped wave on the stable tropopause inversion (Fig. 4). The idea of trapping is plausible, because the intrinsic frequency of the wave σI = Uk ≈ 2π(40 m s−1)/ 14 000 m = 0.018 s−1 is nearly double the stability frequency in the upper troposphere and at least slightly greater than N in the adjacent stratospheric layer. A simple theory for its wavelength can be constructed from the dispersion relation for a deep-water surface gravity wave σ2 = g′k/2, where the reduced gravity is g′ = gθ/θ) = N2Δz for the stable layer. For such a wave to stand steady against the wind, it must have a wavelength λ = 4πU2/g′. If U = 40 m s−1, N2 = 0.0005 s−2, Δz = 2.5 km (see Fig. 4), then λ = 16.1 km; not so different from the observed wavelength λ ≈ 14 km. Note that a trapped wave on the tropopause inversion would increase the EQR there (Fig. 14).

The generation mechanism of the short wave is unclear. Direct generation by the Sierras is unlikely, because the short wave was not found in the aircraft legs below the tropopause. Furthermore, the evanescent decay of a mountain forced short wave would be considerable by the time it reached the tropopause. More likely it is a mechanism by which the long mountain wave steepens at the tropopause inversion and weak nonlinearity generates a standing short wave. Such a transfer of energy from a long wave to a shorter wave is commonly seen in nature, and it does not require catastrophic or turbulent wave breaking. A good example is an undular bore. There, a smooth, long wave in shallow water steepens as it progresses. When a certain steepness is reached, a train of lee waves suddenly appears behind, generated by nonlinearity in the wave front. Even more common are the short trains of capillary waves that appear on surface gravity waves of moderate slope. As the trapped wave does not propagate vertically, we suspect that its generation region is within the tropopause inversion—right where the aircraft was flying.

The generation mechanism of the downgoing wave energy is also unclear. By the definition of “downgoing wave,” the generation region should be above the level of observation. Our attention is naturally drawn to the altitude range from 15 to 20 km (i.e., above the aircraft), where the wind and air density decrease significantly. As the waves propagate farther aloft and the mean wind speed decreases toward the critical level, they almost certainly become nonlinear. Wave steepening could generate a new set of waves with different wavelength. This process has been called “secondary generation” or “critical-level overreflection” in the literature. In section 3, we estimated the amplitude parameter to be A = 0.6 at the upper-GV-flight levels. If this wave continued to propagate aloft another 5 km, the wind speed would drop to 10 m s−1 and the density would drop by a factor of about 0.5. The amplitude parameter would climb to about A = 2.5 and nonlinear effects would dominate.

Several authors have used numerical simulations to study the nonlinear breaking of mountain waves in the stratosphere. The results of these studies vary somewhat according to the setup of the problem (e.g., Bacmeister and Schoeberl, 1989; Afanasyev and Peltier 2001; Franke and Robinson 1999; Doyle et al. 2005). All find wave breaking and some report a secondary generation of gravity waves from the breaking region.

Perhaps the closest analogy to our Sierras wave situation is the 2D formulation of Satomura and Sato (1999). They looked at the time-dependent development of waves from a simple bell-shaped ridge (h = 1.5 km and half-width 30 km) in a jet stream flow with a maximum wind of 29 m s−1 at 13 km. The tropopause and stratopause were put at 10 and 47 km, respectively. Unlike the T-REX case, there was no environmental critical level, although the wind speed decreased to 10 m s−1 at 25 km. The decreasing wind and air density, and increased static stability in the stratosphere-forced wave breaking to begin at altitude 21–24 km after 7 h. Secondary waves were soon found upstream and downstream, above and below the breaking region. The waves below (e.g., at 15 km) and downstream seem most relevant to our T-REX observations. With wavelengths of 5 to 9 km, they extend at least 30 km downwind. The authors argue that these waves appear to be trapped in the lower stratosphere, because their phase lines are vertical and they carry little MF. This fits with our identification of trapped short waves in the dataset. While many of the details of Satomura and Sato don’t match our 16 April 2006 event, we see many parallels too.

It is also possible that only linear reflection processes are involved in generating a downgoing wave. By chance, the aircraft might fly through a transient or localized beam of mountain waves, reflected downward by some discontinuity aloft. Over complex terrain, reflecting layers will produce a patchwork of up- and downgoing beams. We lean against this accidental explanation, because the reversed fluxes seem so robust in RF 10. Similar arguments were made by Smith and Broad (2003) concerning waves over Mt. Blanc.

9. Conclusions

Using the new NSF–NCAR GV, six Sierra Nevada mountain wave events were probed during March and April 2006. The aircraft dataset included 126 legs at altitudes spanning the tropopause. The dataset is unique in regard to flight-track strategy, aircraft performance characteristics, and instrumentation. The new GPS altitude measurement played a key role in our analysis. It allowed the computation of ageostrophy, wave energy flux, the Bernoulli function, and the wave kinetic energy. Of the six events, the 3 days with strongest winds had the largest waves. The waves mostly fit the simple mountain wave paradigm (i.e., mountain-generated, nearly steady, nearly two-dimensional, and nearly linear waves), but some clear violations of these assumptions were seen. One surprise was the subgeostrophy of the mean flow.

A challenging aspect of the fight-level data interpretation is the background dynamic and chemical layering in the atmosphere. Material layers are pushed up and down by the mountain wave, adding wind speed fluctuations to the flight legs. Layer structure was identified using conserved variable diagrams in which four conserved quantities (ozone, water vapor, Bernoulli function, and cross-flow speed) are plotted against potential temperature. The background wind speed variations double the apparent horizontal kinetic energy in the waves.

Using GPS altitude data, static pressure, and vertical velocity along the flight legs, the vertical energy flux was computed for the first time in a mountain wave. The energy flux observation confirmed the Eliassen–Palm energy-momentum flux relation and added confidence to the interpretation of wave sources.

Using the GV measurements of wind and temperature fluctuations, we computed wave energy densities. To identify partial wave reflection near the tropopause, we defined the equipartition ratio (EQR = PE/KE). The EQR increased suddenly by a factor of 10 across the tropopause, giving clear evidence of partial reflection. When only pure upward or pure downward waves are present, EQR = 1.

A dramatic violation of the simple mountain wave paradigm was the flux reversal on 16 April 2006. In this event, we found evidence for trapped waves on the tropopause inversion and downgoing waves from above the highest aircraft leg. We were not able to identify the generation mechanisms for these waves.

Our data suggest that mountain waves can be strongly modified as they enter the stratosphere. They can exhibit partial reflection, nonlinear generation of trapped waves, and downgoing waves from reflection or generation farther aloft, The new long-duration Gulfstream V, combined with an innovative flight track and new diagnostic methods, is well suited for observing the modification of mountain waves entering the stratosphere. Further analysis of these cases using wavelets will assist in wave group diagnostics. Numerical simulation may help in understanding the process of secondary wave generation.

Acknowledgments

The assistance of T-REX, University of Wyoming, and NCAR staff during the field phase was essential. Special thanks to the pilots who helped modify the flight-track designs to fit the aircraft and Air Traffic Control (ATC) capabilities. Strategic discussions with Joach Kuettner, Jim Moore, Richard Dirks, Brian Ridley, Laura Wang, Bill Randel, and others are much appreciated. The ozone data were provided by Ilana Pollock of NCAR. The ATC personnel in the Joshua Control Area were very helpful. RBS appreciates relevant discussions with Arnt Eliassen in 1977 and with Dale Durran recently. Three anonymous reviewers gave useful comments. The Yale group was supported by the National Science Foundation (ATM-112354 and -0531212). J. Doyle was supported by ONR PE-0601153N.

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APPENDIX A

Equipartition as a Measure of Partial Wave Reflection

While the wave momentum flux and wave drag are altered by downward wave reflection from various atmospheric layers, the resulting MF profile is independent of altitude in steady, nondissipative flow (EP61). Thus we turn to other measurable quantities to detect partial wave reflection. The EQR (22) is a good indicator of reflection because it is unity in any layer that contains only upgoing waves. To see this, consider the usual representation of a periodic mountain wave in a uniformly stratified layer:
i1520-0469-65-8-2543-ea1
i1520-0469-65-8-2543-ea2
where A and B are the coefficients of the upgoing and downgoing wave components and the vertical wavenumber m = (N2/U2k2)1/2 sgn(k). Other quantities such as horizontal velocity and vertical displacement can be found from û = (i/k)/dz and η̂ = (1/ikU)ŵ. The momentum flux, using Parseval’s theorem, is
i1520-0469-65-8-2543-ea3
For each wavenumber component,
i1520-0469-65-8-2543-ea4
which reduces to
i1520-0469-65-8-2543-ea5
because the difference of the middle two terms in (A4) is pure imaginary (i.e., the terms are complex conjugates). Not only is (A5) independent of altitude within the layer, but matching displacement and pressure at a layer interface gives constant MF between layers. Other quadratic quantities do not have this property of altitude independence. For example, the potential energy density is
i1520-0469-65-8-2543-ea6
For each wavenumber component,
i1520-0469-65-8-2543-ea7
In a similar way, the wavenumber contributions to the kinetic energy density KE = KEx + KEz are
i1520-0469-65-8-2543-ea8
i1520-0469-65-8-2543-ea9
Note that in (A7), (A8), and (A9), the sums of the middle two terms are real as they are conjugates. They contribute a strong height dependence to the energy density. Only if one wave is missing (i.e., A = 0 or B = 0) does the equipartition ratio
i1520-0469-65-8-2543-ea10
Thus any departure from (A10) at a given level indicates a mixture of up- and downgoing waves.

The reason why EQR variation is related to reflection may involve wave communication. If an entire unidirectional propagating-wave system must satisfy equipartition (A10), then each layer must satisfy (A10) independently. Reflections, on the other hand, allow different levels to compensate for local violations of (A10).

To test the utility of (A10) as detector of reflected waves, numerical simulations of 2D mountain waves were carried out using COAMPS with a sponge layer aloft. The computed velocity and temperature fields were used to evaluate PE and KE as functions of height. Initially, 20%–30% fluctuations in EQR were found, due to reflected waves. After improvements to the sponge layer, the fluctuations were reduced to 5%. Apparently, (A10) is a very sensitive detector of reflected waves.

APPENDIX B

Two Bernoulli Formulations

Two forms of the compressible Bernoulli equation are generally seen in the literature. From the momentum equation for adiabatic steady flow (e.g., Prandtl and Tietjens 1934; Aris 1962)
i1520-0469-65-8-2543-eb1
while from the energy equation
i1520-0469-65-8-2543-eb2
(e.g., Gill 1982; Batchelor 1967). Each quantity satisfies dB/dt = 0 for steady adiabatic flow. Using the expression for potential temperature θ = T(p/p0)R/CP, the first terms in each expression are related by d(CPT) = CP(T/θ) + dp/ρ, so that along an isentropic streamline d(CPT) = dp/ρ, making (B1) and (B2) identical, except for a constant. In the context of flight-level data analysis however, the two Bernoulli formulas are quite different. Along an aircraft path with nearly constant pressure, the first term of B1 barely changes. In our data, the static pressure variation along a leg seldom exceeded 100 Pa with mean density typically 0.25 kg m−3, so the pressure term variation was ≈±400 m2 s−2. In contrast, the first term in B2 changed significantly (≈±10 000 m2 s−2) along the leg. This occurred because the aircraft was cutting through theta surfaces, as in Fig. 10. The aircraft encountered parcels with strong variations in internal energy. If B2 is plotted against theta for an aircraft leg, the result is a nearly straight line with a slope of dB2/ = CP(p/p0)R/CP, where p is the average leg pressure. Such a plot contains no information about the wind field. For the present purpose, B1 is more useful because it describes how the air is being accelerated by pressure gradients and gravity. It can be plotted against theta to create a vertical profile of Bernoulli. Alternatively, the incompressible version of B1 may be used if the parcel displacement is much less than a scale height.

Fig. 1.
Fig. 1.

NSF–NCAR Gulfstream V research aircraft.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 2.
Fig. 2.

Aircraft racetrack B over the Sierra Nevada Range. The reference point on the northern leg is Independence, CA. Each racetrack has two ENE–WSW legs nearly parallel to the wind direction. See Fig. 3 for the terrain cross section.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 3.
Fig. 3.

Vertical section across the Sierra Range showing the terrain under each leg. The stratosphere is shaded. The GV flight altitudes and a typical wind profile are shown. The King Air flew shorter legs below 8 km.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 4.
Fig. 4.

Brunt–Väisälä frequency derived from Visalia and Lemoore upwind soundings during the RF: (a) large-amplitude wave cases (RFs 4, 5, and 10; see Table 1); (b) smaller-amplitude cases (RFs 6, 8, and 9). Note the strong static stability just above the tropopause, with weaker stability above and below.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 5.
Fig. 5.

As in Fig. 3, but for wind speed profiles. Note the decreasing wind speed in the stratosphere.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 6.
Fig. 6.

Averaged ozone profiles from the six wave cases (Table 1), from GV flight-level data. In the troposphere, the typical concentration is about 60 ppbv. The tropopause is marked by sharp vertical gradients in ozone concentration.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 7.
Fig. 7.

Geostrophy check for all racetracks in six flights. Geostrophic wind component parallel to xB is computed from the differences in geopotential between the two legs of the racetrack. This is plotted against the average wind along the legs. The gradient wind balance is also shown with different radii of curvature.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 8.
Fig. 8.

Flight-level vertical displacement from a stacked GV racetrack in three strong-wave flights: (a) RF 4, (b) RF 5, (c) RF 10 (see Table 1). The north leg is dashed and the south leg is solid.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 9.
Fig. 9.

An example static stability determination from two horizontal legs of a 12-km GV racetrack during RF 5 (IOP 6). Streamline displacement (2) is plotted against potential temperature. This example has the strongest stability found on any leg. The north leg is dashed and the south leg is solid. Straight lines are reference slopes.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 10.
Fig. 10.

Schematic of wave-deformed layers and level aircraft track.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 11.
Fig. 11.

Four conserved variable diagrams from Gulfstream flights: (a), (b) RF 5 and (c), (d) RF 10. Each set of plots shows two legs, 46 km apart, on the same racetrack (north legs dashed, south legs solid). Each plot shows (left to right) ozone, water vapor mixing ratio, Bernoulli-corrected speed (8), and cross-track wind speed vs potential temperature. These plots can be used to judge the degree of conservation and the horizontal vorticity in the flow. These selected legs are cleaner than the average leg.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 12.
Fig. 12.

Energy flux vs the dot product of mean wind and momentum flux for six cases (Table 1): (a) Gulfstream V and (b) King Air. Each point is a leg.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 13.
Fig. 13.

Momentum flux in the xB direction vs altitude for Gulfstream V south and north and King Air legs for the three strong-wave cases: (a) RF 4, (b) RF 5, and (c) RF 10. Each point is a leg. Only RF 10 shows a systematic gradient with height.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 14.
Fig. 14.

Log 10 of the EQR (EQR = PE/KE) vs altitude. Legs from the three big-wave cases are shown (GV data only). Theoretical curve is from a three-layer linear mountain wave model (Table 3) representing the tropopause inversion as a layer of more stable air between 11 and 13 km (shaded). Note the factor of 10 increase in EQR from just below to just above the tropopause.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 15.
Fig. 15.

Power spectra for vertical velocity for two legs at (a) 13 (b) 11 km, and (c) 9 km in RF 10. Note peaks at wavelengths of 14, 20, and 30 km. Reference lines show the 95%, 50%, and 5% confidence levels. Waves longer than 40 km are uncertain due to the short length of the measured wave train.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 16.
Fig. 16.

Phase diagrams for the spectrally dominant waves (from KEZ) for all legs on six Gulfstream flights (Table 1). (a) Vertical velocity and potential temperature (confirms orthogonality); (b) pressure and longitudinal velocity (confirms Bernoulli equation); (c) pressure and vertical velocity (describes wave dynamics: some waves propagate up or down and some are nearly trapped).

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Fig. 17.
Fig. 17.

A wave property diagram for the dominant waves from RF 10 on 16 Apr 2006. The cospectrum between pressure and vertical velocity is plotted against wavelength. Three types of waves are present: upgoing waves (30 km), downgoing waves (20 to 30 km), and trapped short waves (11 to 14 km). The occurrence of these wave types changes markedly with altitude.

Citation: Journal of the Atmospheric Sciences 65, 8; 10.1175/2007JAS2598.1

Table 1.

Track B flights used in wave analysis.

Table 1.
Table 2.

Energy ratios for the strong-wave flights (RFs 4, 5, and 10).

Table 2.
Table 3.

Parameters in the linear mountain wave simulations with a tropopause inversion.

Table 3.

* The National Center for Atmospheric Research is sponsored by the National Science Foundation.

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