## Abstract

The main objective of the present paper is the use of a constant volume balloon (CVB) as a tool to (i) study trapped lee waves and (ii) assess the forecasting capability of a nonhydrostatic numerical model. Then, CVB data obtained during the Pyrénées Experiment (PYREX) are compared with nonhydrostatic two-dimensional trapped lee waves simulated by the Meso-NH model. This model is a community research model based on the Lipps–Hemler form of the anelastic system, which has been recently developed by the CNRM of Météo-France and the Laboratoire d’Aérologie of Université Paul Sabatier in France.

To analyze how the CVB responds to lee waves, a simple CVB model is first applied to academic atmospheric stationary wave flows, analogous to those encountered during PYREX. This model takes into account the vertical velocity of the surrounding air, geometrical parameters of the balloon, and the atmospheric heating processes. Results show that the CVB reacts well to the atmospheric wave period, with a phase delay of only a few minutes.

Three CVB trajectories obtained during the third Intensive Observation Period of PYREX are then computed within Meso-NH 2D simulations from the balloon’s starting point, using the CVB model. The simulated quantities are compared to the experimental CVB data, focusing especially on the lee-wave vertical movements. The simulated lee-wave vertical velocities and amplitude are found to be in good agreement with the observations, as shown by the statistical analysis. The computed CVB heights deviate by less than 13% from the altitude of the measured trajectories. This comparison of the model output to the CVB experimental data demonstrates the good performance of the Meso-NH model in the prediction of the vertical excursions of the lee waves.

## 1. Introduction

The main objective of the Pyrénées Experiment (PYREX) (Bougeault et al. 1993) was to determine the different terms of the momentum budget modified by the mountain waves and secondary flows (local winds). In addition to other experimental tools (e.g., aircraft, glider, ST radar, lidar measurements), a large quantity of constant volume balloon (CVB) data was collected.

Several papers have already discussed the trapped lee waves observed during PYREX. Using aircraft data, the mean and turbulent structures of the lee waves in the central cross section have been analyzed by Attié et al. (1997). From the CVB data and stratosphere–troposphere (ST) radar measurements, Caccia et al. (1997) studied the nonstationarity aspects of the flow. Benech et al. (1994) and Tannhauser and Attié (1995) also studied the lee waves with linear models. Subsequently, the lee waves observed during several intensive observation periods (IOPs) have formed an ideal subject of study for nonhydrostatic numerical models (e.g., Elkhalfi and Carrissimo 1993; Satomura and Bougeault 1994). So far, the modeling results have been compared to the PYREX experimental data in the Eulerian framework.

In this paper, a different approach is proposed. This approach consists of simulating the airmass trajectories within numerical model forecasts (i.e., in a Lagrangian frame) and comparing them to measured ones derived from the CVBs. The CVBs, as airmass tracers, are an appropriate tool for investigating the kinematic and thermodynamic properties of the lee waves generated by mountains (i.e., length, amplitude, vertical velocities, height, duration, temperature, and mixing ratio).

The CVBs described in Benech et al. (1987) were instrumented to obtain airmass trajectories, and in situ measurements of pressure, temperature, and moisture in lee waves generated to the north of the Pyrénées during PYREX IOP3 are considered. The measured CVB trajectories are compared to the two-dimensional simulations from the French meteorological nonhydrostatic model Meso-NH (Lafore et al. 1998), focusing especially on the lee-wave vertical motions.

A brief description of the PYREX experiment and of the dynamic mean features of the lee waves documented by the CVBs during PYREX IOP3 is first presented. Second, the capability of the CVB to follow lee-wave systems is analyzed. In the third section, we use the output of the model Meso-NH to simulate the trajectories of the CVBs from their starting position by considering an equation that predicts their vertical position. Uncertainties in the computed trajectories are then quantified from the horizontal and vertical transport deviations statistical parameters. Finally, the measuring of the downstream decay of the wave amplitude in both balloon measurements and model data are investigated.

## 2. Description of PYREX field program and constant volume balloon data during lee-wave events

### a. PYREX field program

PYREX was launched by the French and Spanish meteorological services in 1990. The experimental field program carried out from 1 October to 30 November focused on the study of the influence of the Pyrénées Mountains on the atmospheric dynamic flow. The Pyrénées chain was chosen because of its relatively simple shape. The range extends along the French–Spanish border, from the Atlantic Ocean to the west to the Mediterranean Sea to the east (Fig. 1). The size of the range is about 400 km × 100 km and the crestline is around 3000 m in altitude. The experimental setup used in PYREX is described in Bougeault et al. (1993).

The present study focuses on IOP3, which extended from 1800 UTC 14 October to 1200 UTC 15 October 1990. The synoptic situation (e.g., Bougeault et al. 1993;Attié et al. 1997) was dominated by a deep trough over the eastern Atlantic Ocean, which induced a south to southwest flow over Spain and France above 800 hPa and was characteristic of the condition required to create a trapped lee wave. Upstream radio soundings that described the incoming airmass properties were taken at Zaragoza (Fig. 1a). The spatial characteristics of the lee waves at different times and heights were derived from aircraft data along the axis B1–B4 (Fig. 1) and measurements from CVBs that were launched from the Pic du Midi, a mountaintop in the northern center of the Pyrénées chain. In addition, two very high frequency (VHF) ST radars [one (St. Lary: SL) on the mean axis of the range and another (Lannemezan: LA) in the lee-wave field downstream] were used to document the temporal variations and vertical profiles of the vertical velocity (see Figs. 1a,b for the location of the sites). Horizontal wavelengths ranging from 7 to 14 km and a maximum amplitude of the air vertical velocity from 3 to 5 m s^{−1} were observed (e.g., Benech et al. 1994; Caccia et al. 1997). The lee waves were found to be far from stationary during their lifetime, but there were some periods, never longer than 1.5 h, during which the wave was found to be quasi-stationary (Caccia et al. 1997). During the PYREX IOP3, the CVBs flew for a duration of less than 2 h (Table 1) with a period of about 40 min when they crossed the lee-wave system. Therefore, the lee waves observed by the balloons can be considered as quasi-stationary.

### b. Constant volume balloon data during lee waves

Several previous studies have demonstrated the capability of CVBs to visualize the lee-wave system above complex mountains such as, for example, the White Sands Missile Range (Reynolds et al. 1968) and the central Colorado Rocky Mountains (Vergeiner and Lilly 1970). The CVBs described in Benech et al. (1987) were operated during PYREX to obtain air trajectories and in situ measurements of pressure, temperature, and moisture. They consist of 2.08–2.8 m^{3} cylinders, 7 m in height and 0.6 m wide, made of transparent mylar and polyethylene films, 25 and 10 *μ*m thick, respectively, and equipped inside with a passive reflector. During PYREX IOP3, CVBs inflated with helium and pressurized from 20 to 40 hPa above ambient pressure were launched from Pic du Midi (PM, Figs. 1a and 1b) and subsequently tracked by a radar located in Lannemezan (Figs. 1a and 1b) with a 10-s sampling. The CVBs were also equipped with probes measuring air pressure, absolute temperature, and humidity. Their horizontal positions are deduced from the tracking radar. The vertical movements of the balloon were derived from the air pressure and temperature, using Laplace’s equation. The precisions of the measured meteorological parameters were estimated to be 0.2°C for temperature, 5% for humidity, and 1 hPa for pressure. The horizontal wind accuracy was 0.5 m s^{−1}. The accuracy of the observed balloon positions (i.e., height and horizontal location) is related to the precisions on the rough parameters involved in their calculation. Thus, the accuracy of the observed CVB height estimated from an error calculus on the Laplace’s equation mainly depends on the precision of the pressure sensor. On average, an absolute error of 20 m is found. The resolution and accuracy of the horizontal position of the balloons were of the order of 15 m for the distance and 0.1° for the angle. The errors in the observed balloon horizontal locations can reach 120 m at 50 km from the radar site.

The horizontal and vertical trajectories of the three successful CVB launchings during PYREX IOP3 are presented in Figs. 1a,b and 1c,d, respectively. Three horizontal wavelengths were observed (Figs. 1c and 1d). The CVB flight characteristics and the mean dynamic parameters of the lee waves documented along their paths are reported in Table 1. The three CVBs flew between 40 and 63 km within 3000–5000-m altitude in the direction 211°–218° for a duration of 57 min to 1 h, 28 min. The amplitudes of the lee waves were high between 0800 and 1000 UTC (indicated by the CVB b31), exceeding 1000-m vertical displacement (Figs. 1c and 1d), weakened at about 1200 UTC (b32). Unlike the balloon b31 and b32 observations, balloon b33 launched at about 1400 UTC traced out only one main lee wave. Nevertheless, at about the same time (1400 UTC), three trapped lee waves were still observed along the central cross section of the the Pyrénées, that is, along axis B1–B4 (Fig. 1a) from (i) the visible image of the National Oceanic and Atmospheric Administration (NOAA 11) advanced very high resolution radiometer (AVHRR) at 1417 UTC (Fig. 11; Bougeault et al. 1993) and (ii) the aircraft measurements (Fig. 12; Bougeault et al. 1993). Therefore, a local dissipation of the lee waves in the area where balloon b33 flew (i.e., in the western angle of the axis B1B4) can be suspected. In accordance with previous studies (e.g., Benech et al. 1994; Attié et al. 1997; and Caccia et al. 1997), the CVBs vertical velocities ranged from −5 to +5 m s ^{−1} and wavelengths from 8 to 10 km corresponding to wave periods of about 12 min were observed, which are in the same order of magnitude as the Brunt-Vaïsäla frequency periods (range from 8 to 10.5 min) deduced from the background airmass properties measured at Zaragoza. Such wave amplitudes and lengths have been previously recorded by Reynolds et al. (1968) and Vergeiner and Lilly (1970) using balloon-based trajectory studies above the central Colorado Rocky Mountains and the White Sands Missile Range, respectively.

## 3. Constant volume balloon modeling

To analyze how a CVB responds to lee waves, a simple CVB model that predicts its vertical motions is now presented.

Several previous academic studies of CVB flight physics were performed to investigate the CVB behavior (among them, Vergeiner and Lilly 1970; Reynolds 1973;Tatom and King 1977; Nastrom 1980). It has been demonstrated that a CVB follows the horizontal motion of the air mass, whereas its vertical velocity is described by a simplified equation (when neglecting the terms of Coriolis, viscosity, and Basset, i.e., the term that gives the CVB “memory”) expressed as a function of the vertical velocity of the surrounding air, the geometrical parameters of the balloon, and atmospheric heating processes:

The terms I, II, III, and IV in Eq. (1) correspond, respectively, to the CVB vertical acceleration, the vertical acceleration of the air mass, the buoyancy force, and the aerodynamic drag. The indices a and b refer to the air and to the balloon, respectively. Here, *W* is the vertical velocity; *g* and *ρ* are, respectively, the gravity acceleration and the air or balloon density. In addition, *D*_{b}, *C*_{d}, *C*_{f}, and *υ*_{b} indicate, respectively, the apparent diameter, the aerodynamic drag coefficient, the induced drag (i.e., air shifted by the balloon) coefficient, and the volume of the CVB, which are all assumed to be constant.

### a. Response of the CVB to idealized stationary atmospheric waves

To compare the CVB data to the nonhydrostatic numerical model Meso-NH forecasts, we present the behavior of a cylindrical CVB shape, similar to those launched during PYREX, within idealized atmospheric flows. The atmospheric parameters acting on the balloon are the airmass wind vector and density. The prediction of the behavior of a CVB immersed in an atmospheric flow is then based on the three components of the wind vector and on the density field of the air mass that drifts with the balloon [see Eq. (1)].

#### 1) Description of the wave field

Following Tatom and King (1977), each of the three components of the wind speed vector can be determined as a superposition of a mean wind and sinusoidal pulsation including a wavenumber vector. For instance, the vertical component of the wind in space **OM** (*x, y, z*) and time *t* can be defined by the following analytical expression:

The dynamic field then depends on five parameters for each of the three wind speed vector components: the mean wind (*W*_{m}), the amplitude (*A*_{w}), and the pulsation of the wave (2*π*/*T* with *T,* the period), the wavenumber vector **k** (*k*_{x}, *k*_{y}, *k*_{z}), and the phase lag between the wind speed vector components (*ϕ*_{uw} referred to the *x* axis). The horizontal wind components *U*_{a} and *V*_{a} along the *x* and *y* axes, respectively, can also be defined by the same expression (2). The amplitude and the pulsation of the wave are the most sensitive parameters. The wavenumber vector that represents the horizontal or vertical advection can be considered as a “pseudo–phase lag” when one assumes it constant in time (i.e., stationary case). The mean wind speed vector components play only a role of horizontal or vertical advection of the wave. Therefore, they all are set to zero. Following Nastrom (1980), the airmass density field is defined as follows:

That is, it depends on four parameters: the amplitude of the density wave (*ξ*), its pulsation (2*π*/*T*), and phase lag (*ϕ*), with the vertical dynamic wave (*W*_{a}), and the airmass local density *ρ*_{al}(*z*) at the CVB flight level. For determination of the density *ρ*_{al}(*z*), the atmosphere is assumed to (i) be hydrostatic, (ii) follow a perfect gas law, and (iii) have a linear vertical thermal gradient. In such an atmosphere, the airmass density along the CVB path [i.e., *ρ*_{al}(*z*)] mainly depends on the vertical description of the airmass temperature. Here, *ρ*_{al}(*z*) is obtained by neglecting the horizontal thermal gradients (which are generally weak compared to the vertical one) and considering a Boussinesq assumption:

Here *T*_{a0} is the airmass temperature at the balloon flight equilibrium level, Γ = *dT*_{a}/*dz* is the temperature lapse rate, and *R*_{a} refers to the gas constant for the perfect gas. The density *ρ*_{b} of the CVB corresponds to the airmass density at the balloon’s equilibrium altitude.

Nastrom (1980), when studying the response of a balloon to atmospheric gravity waves, considered the phase lag (*ϕ*) between the wind vertical velocity and the density of the airmass wave to be 90°. This phase lag value was obtained from the polarization relation of the gravity waves, derived from the linear theory (e.g., Hines 1974). To be more general, as in Nastrom (1980), the amplitude of the air density *ξ* is approximated by the following expression according to the polarization relations:

where *γ* is the ratio of the specific heats.

Based on previous works (e.g., Reynolds 1973; Nastrom 1980) and the characteristics of the lee waves observed during PYREX IOP3, and assuming a stationary wave field, the following values of the parameters involved in the definition of the wave field are considered:*A*_{u} = *A*_{υ} = 3 m s^{−1}; *k*_{x} = *k*_{z} = *k*_{y} = 0, *W*_{m} = 0; *U*_{m} = *V*_{m} = 0; *T* = 12 min; *ϕ*_{uυ} = *ϕ*_{uw} = 0; *ϕ* = 90°; and Γ = *dT*_{a}/*dz* = −10^{−2} °C m^{−1}. The atmosphere modeled in this section is assumed to be a perfect gas. The parameters *R*_{a} and *C*_{p} that represent the perfect gas coefficients are set to 287 J kg^{−1} K^{−1} and 1005 J kg^{−1} K^{−1}, respectively. The airmass temperature (*T*_{a0}) at the CVB equilibrium level is set to 274 K, which corresponds to the mean temperature observed by the CVBs during PYREX IOP 3.

#### 2) Description of CVB parameters

The simulated CVB has a cylindrical shape. Its parameters are set equal to those of the CVBs used during the PYREX lee-wave experiment: that is, *C*_{d} = 0.44, *C*_{f} = 0.014, *D*_{b} = 0.27 m, *υ*_{b} = 2.8 m^{3}, *M*_{b} = 2.35 kg, and *ρ*_{b} = *M*_{b}/*υ*_{b} = 1.13 kg m^{−3} with *M*_{b}, the mass of the balloon.

#### 3) Academic PYREX IOP3 waves

Simulations are performed using the values of the atmospheric and CVB parameters defined above and for two wave amplitudes analogous to those observed during PYREX. Since the balloon’s response to atmospheric waves remains independent of its starting position (e.g., at the crest or at the trough of the wave), the simulated CVBs are initialized at the crest of the natural wave. Figures 2a and 2b present the temporal variations following the trajectory of a hypothetical CVB of the vertical velocity of the air mass (*W*_{a}) and the derived CVB vertical velocity (*W*_{b}) for two natural wave amplitudes (*A*_{w}) of 5 and 2 m s^{−1}, respectively. The hypothetical CVB reacts well to the air wave period, with a phase delay. In addition to that, the CVB vertical velocity is composed of odd harmonic periods of the input wave period (Nastrom 1980). The reasons of the phase lag and odd hump in the calculated balloon trajectory are mainly due to the drag term IV in Eq. (1). In fact, as already explained in Nastrom (1980), following the principle of harmonic balance, the form drag term IV in Eq. (1) can be approximated by a single sinusoidal term having the same period as the atmospheric disturbance. The Fourier expansion of the approximated function gives phase lag and odd harmonics in the calculated balloon vertical velocity with the first harmonic’s (i.e., fundamental) amplitude greater than those that follow. In the present study, we will only discuss the amplitude of the fundamental period and the phase lag between the wave vertical velocity and the balloon one. For *A*_{w} of 5 m s^{−1} (Fig. 2a), the values of the maximum and minimum amplitude of the balloon vertical velocity are 88% of the air values. Nevertheless, the CVB reacts to one period of the airmass oscillation with 1.7 min of delay (i.e., with 50° of phase lag). When the air wave amplitude decreases to 2 m s^{−1} (Fig. 2b), maximum and minimum values of *W*_{b} are 66% of that for *W*_{a} and the time lag increases to 2.5 min (i.e., 74° of phase lag).

Balloon response as a function of typical wave periods (T) for various typical amplitudes of air velocity (encountered during PYREX IOP3) and density [*ξ* given by Eq. (5)] and for two typical lapse rates (Γ) is carried out. The results are analyzed in term of (i) the ratio *A*_{wb}/*A*_{w} between the amplitude of the CVB (*A*_{wb}) and (ii) the wave amplitude (*A*_{w}) for T ranging from 5 s to 20 min every 5 s. Figure 3 displays the results for *A*_{w} values of 1, 2, and 5 m s^{−1} and for two lapse rates Γ of −10^{−2}° and 10^{−2} °C m^{−1}. In general for short-wave periods (i.e., 5 s⩽ *T* ⩽ 1 min) the CVB reacts perfectly well to the atmospheric disturbance; the ratio *A*_{wb}/*A*_{w} is close to 1 with a null phase lag between *W*_{a} and *W*_{b}. Around the CVB’s own period (which depends on the air stability condition; here 2 ⩽ *T* ⩽ 3 min), the atmospheric disturbance is amplified in the balloon response. For small *A*_{w} (and hence small *ξ*) and for large wave periods, the ratio *A*_{wb}/*A*_{w} falls off as 1/*T.* For *A*_{w} = 1 m s^{−1} and for Γ = 10^{−2} °C m^{−1}, it is 1.5 for *T* = 3 min and becomes lesser than 0.3 at *T* = 20 min. For large *A*_{w} (and hence large *ξ*), the ratio *A*_{wb}/*A*_{w} lineary decreases with a weak rate (*A*_{wb}/*A*_{w} < 1). For *A*_{w} = 5 m s^{−1} and for Γ = 10^{−2} °C m^{−1}, the ratio *A*_{wb}/*A*_{w} is 1.2 for *T* = 3 min. It is 0.7 for *T* = 20 min. The increasing of the temperature lapse rate shifts the maximum of the ratio *A*_{wb}/*A*_{w} toward the short periods. For large periods and large lapse rates, the wave amplitude is attenuated by the balloon. The null phase lag between *W*_{a} and *W*_{b} for short-wave periods, strongly increases from about 6 min and approaches an asymptote (Fig. 3b). Large values of a phase lag are obtained for small *A*_{w}. For large lapse rates, the phase lag increases with increasing period. The asymptotic values of the phase lag are found to be between 57° (≈3 min) and 93° (≈5 min) for *A*_{w} = 5 and 1 m s^{−1}, respectively. Nevertheless, as previously found for *T* = 12 min and for *A*_{w} = 5 m s^{−1} (the maximum wave amplitude observed during PYREX IOP3), the time lag is 1.7 min (i.e., 50°) (Figs. 2a and 3b). The cylindrical balloon shape response to the simulated academic wave vertical velocity amplitude presented in this study confirms qualitatively Nastrom’s (1980) findings. Thus, as expected, the spherical balloon considered in Nastrom’s work reacts better to atmospheric disturbances than the cylindrical CVB shape. As an example, for *A*_{w} = 2 m s^{−1} and for the adiabatic case (Γ = −10^{−2} °C m^{−1}), Nastrom (1980) found the ratio *A*_{wb}/*A*_{w} to be 1.05 for *T* = 14 min, while in the present study, it is about 0.6.

### b. Comparison with the model Meso-NH lee waves

This section presents the CVBs trajectories calculated from their starting point using the CVB model [Eq. (1)] forced by the Meso-NH data. The meteorological parameters derived from the computed trajectories are compared to the measurements. The measured air vertical velocity requires the solution of the CVB motion equation using as inputs the parameters of the balloon, the thermodynamic parameters measured along the balloon path, and its path. In addition to that, the knowledge of at least one of the balloon internal parameters (i.e., pressure and/or temperature) along the trajectory is required. Such internal parameters were not measured during PYREX. Therefore, only the CVB vertical velocity and flight altitude are calculated within the Meso-NH forecasts from Eq. (1).

In the following sections, one should keep in mind that the CVBs were instrumented. Thus, airmass temperature, humidity, and both the balloon horizontal and vertical velocities were measured along their trajectories. Thermal parameters derived from the CVB flights are directly linked to the air mass ones. According to the CVB model, the horizontal wind components derived from the CVB instrumentation are representative of those of the air mass.

#### 1) Description of the model Meso-NH

Meso-NH (Lafore et al. 1998) is a community research model recently developed by the Centre National de Recherches Météorolgiques (CNRM; Météo-France) and the Laboratoire d’Aérologie of Université Paul Sabatier in France. Based on the Lipps–Hemler form of the anelastic system, the model allows research in a wide range of topics and atmospheric scales, in addition, validation tests have demonstrated that model behavior is correct over small-scale obstacles with steep slopes (up to 70%). The simulations of the PYREX IOP3 trapped lee waves were performed using a 2D version of Meso-NH, since observations described them as a two-dimensional phenomenon (Bougeault et al. 1993). The integration domain has 240 km in the horizontal (i.e., axis B1–B4 in Fig. 1) with 1-km grid spacing, and 20 km in the vertical divided into 60 stretched levels. The topographic relief is the averaging at 1 km of the actual relief measured by a research aircraft along the axis B1–B4 perpendicular to the Pyrénées range (Fig. 1c). This topographical section thus obtained represents 200 km and has been extended with 40 km of flat ground in order to avoid spurious reflection at the downstream lateral boundary. Between 12 km and the top of the domain an absorbing layer is used to avoid wave reflection at the model top. The one-dimensional version of the turbulence scheme is used. The surface roughness is fixed at 1 m for every point. The initial profiles of wind and temperature (Fig. 4a) describing the incoming airmass properties are the same as in Satomura and Bougeault (1994) and are adapted from the Zaragoza (Fig. 1a) sounding at 0600 and 1200 UTC 15 October 1990. The vertical profiles of the Scorer parameter *l*^{2} for these two soundings are displayed in Fig. 4b. The formula used for *l*^{2} calculation can be found, for example, in Tannhauser and Attié (1995). The two *l*^{2} profiles exhibit quite similar shapes and have the same order of magnitude below 6500 m of altitude. Two layers (i.e., 1500–3000 and 4000–6500 m) with high values of *l*^{2} at low levels were favorable to lee-wave formation. The quite similar variations of *l*^{2} for the two soundings on the midtroposphere imply that the temporal reduction of the static stability is offset by the reduced wind speed between 0600 and 1200 UTC. Consequently, the observed horizontal wavelength did not change greatly over this time period (Figs. 5 and 6).

The model produces forecasts (i.e., airmass wind speed, vertical velocity, and potential temperature) at 0800, 1000, and 1200 UTC using the Zaragoza sounding at 0600 UTC as both the initial and the lateral conditions and at 1400 UTC using the Zaragoza sounding at 1200 UTC as both the initial and the lateral conditions (Fig. 4a). These model outputs, which give a very good compromise between the computational cost and accuracy among the set of the simulations performed, are considered as representative of the three CVBs entire flight periods (Table 1).

#### 2) Simulation of the CVB trajectory within the Meso-NH forecasts

The trajectory model, recently used by Koffi et al. (1998) to analyze the transport mechanism of chemical tracers using data collected during the European Tracer Experiment, will be applied here to compute CVB trajectories by integrating the Meso-NH wind speed vector components. The well-known iterative scheme described by Petterssen (1956), reviewed by Seibert (1993), is used in this study. The vertical position of the balloon is derived from the differential Eq. (1). Euler’s backward scheme is used and an integration time step of less than or equal to 5 s is chosen to fulfill a stability criterion. This procedure allows the simulated balloon to rise under the buoyancy force influence from its starting level up to its equilibrium altitude (given by the balloon density *ρ*_{b}; see Table 1). At every position of the computed trajectory, two outputs of the model Meso-NH that bound the position of the balloon in time are linearly interpolated both in space and time. When two Meso-NH forecasts that bound the position of the balloon in time are not available, a simple interpolation in space is considered by using the nearest Meso-NH prediction in time. Along a simulated trajectory, the wind intensity, the vertical velocity, and the potential temperature of the air mass are calculated.

#### 3) Qualitative comparison of the calculated parameters with the measurements

The altitude, the wind intensity, the vertical velocity, and the potential temperature along both measured and simulated trajectories of the balloons b31, b32, and b33 are shown in Figs. 5, 6, and 7, respectively. The flight characteristics of the three balloons are reported in Table 1. As can be seen from Figs. 1a and 1b, the CVBs do not follow exactly the axis B1–B4 used for the 2D version of the Meso-NH model lee waves. Therefore, only the components of both measured wind intensity and horizontal trajectories referred to the axis B1–B4 are considered in the next sections.

The trajectory of the balloon b31 (flight period: 0753–0921 UTC; see Table 1) is computed using the Meso-NH output at 0800 and 1000 UTC. The CVB-measured variables and CVB variables computed in the Meso-NH fields are presented, in Fig. 5, along the balloon trajectory. While the simulated horizontal wind speed is too large (Fig. 5b), the simulated altitude and vertical velocity of the balloon fit the measurements reasonably well (Figs. 5a and 5d). The values of the minimum and the maximum amplitude of measured CVB vertical velocity are almost 100% of the corresponding simulated one within the first 20 km of the trajectory. After about 28 km of the traveled distance, the measured CVB had already stabilized at its equilibrium level, whereas the simulated CVB still exhibits significant oscillations. The calculated vertical velocity of the air mass and the CVB vertical velocity displays similar trends (Fig. 5c). In accordance with the previous academic study, the computed CVB vertical velocity is well predicted with a distance lag of 1.5 km (i.e., 2 min of a time lag at 12.9 m s^{−1} wind speed). The potential temperature measured along the CVB trajectory seems to indicate that the balloon followed different air parcels during its travel. Then, the balloon crossed the intense part of the lee-wave system (i.e., within 0–22 km from Pic du Midi) with a weak variation in the potential temperature (*θ* = 306.2 K ± 0.8 K). This suggests at least that when balloon b31 crossed the intense phase of the lee-wave system (0–20 km from Pic du Midi), the CVB behaves as a quasi-isentropic tracer by following almost an iso-theta surface (Fig. 5e), a result that confirms observations by Reynolds (1973) during the Mountain Waves Project. In fact, Reynolds (1973) showed that the CVBs follow an isentropic surface by underestimating the true wave crests by an average error of 5% and a true wave trough by errors averaging 30%. The increase in *θ* of 5 K after 22 km from the CVB’s launch site (Pic du Midi) is in great part explained by excursions of air from higher altitude crossed by the balloon as indicated by the variations of the observed mixing ratio along the balloon’s path (Fig. 5e). Between 0 and 22 km and after 35 km from Pic du Midi, the modeled potential temperature gives, on an average, a trend similar to the measured *θ.* The main differences between the model results and the observations appear between 22 and 35 km from Pic du Midi, with the zone subjected to important variations in the measured mixing ratio. Since the clouds are not included in the Meso-NH version used in the present study, the discrepancies can be partially ascribed to the presence of the clouds in those areas crossed by the balloon. Along the whole path of the measured trajectory (i.e., 63 km), the values of the observed and the simulated potential temperatures are of 307.7 K ± 1.5 K and 307.7 K ± 0.8 K, respectively. This indicates the fairly good agreement between the model data and the balloon measurements. Nevertheless, the most important variations are found in the measurements as shown by the standard deviations.

For balloon b32 (0953–1117 UTC), the trajectory is calculated by considering both the Meso-NH output at 1000 and 1200 UTC (Fig. 6). The forecasted wind intensity and wave amplitude are stronger than the measured ones (Figs. 6a, 6b, and 6d). The waves were disappearing in the zone observed by the balloon. The response of the real CVB is not well organized. Nevertheless, the envelope of the measured CVB vertical velocity (Figs. 6a and 6d) is in fair agreement with model predictions. The simulated CVB vertical velocity and the airmass vertical velocity obtained from the Meso-NH forecasts (Fig. 6c) exhibit the same morphology as the case of balloon b31. A distance lag of 1.5 km (or 2 min of time lag) is again found. The potential temperature weakly varies between 0 and 12 km from Pic du Midi. This segment (0–12 km) corresponds to the area of the intense part of the lee-wave system that was observed by balloon b32 (Figs. 6a and 6e). After 12 km from Pic du Midi, the increase in *θ* of about 4 K seems to be correlated to the descent of the dry air followed by the balloon as indicated by the variations in the mixing ratio (Fig. 6e). After 18 km from Pic du Midi, the balloon follows a quasi-isentropic surface as shown by the weak variation in the measured *θ.* Except over the segment 3–8 km from Pic du Midi, the simulated potential temperature varies in the same trend as the measurements. Along the whole path of measured b32 trajectory (i.e., 40 km), the modeled potential temperature (*θ* = 306.8 ± 0.7 K) is similar to the observations (*θ* = 306.8 K ± 0.6 K). Over the 3–8-km segment, there are important differences between modeled and observed data. This discrepancy between the two datasets can be partly ascribed to the presence of the clouds along the measured balloon b32 trajectory.

For CVB b33 (1331–1429 UTC), the Meso-NH output at 1400 UTC is considered as representative of the entire flight period of the balloon, assuming the flow to be stationary during the CVB flight. Thus, at every position of the simulated balloon path, only a spatial interpolation of the model Meso-NH data of 1400 UTC is considered. CVB and airmass modeled and measured dynamic parameters are shown in Fig. 7. The simulated CVB now goes slower than the measured one (Fig. 7b). The measured parameters documented by the CVB along the trapped waves system are well predicted. At later times, the simulated balloon shows significant lee wave amplitudes that are not measured. The discrepancies can be explained by two reasons. (i) As shown by the analytical study in section 3a, the drag of the CVB reduces the amplitude of the air waves. This effect is more sensitive when the vertical oscillations of the air mass become progressively smaller. (ii) As plotted in Figs. 1a and 1b, the mean path taken by the real CVBs are not parallel to the axis B1–B4 where three trapped lee waves were observed (see Figs. 11 and 12 in Bougeault et al. 1993).

The distance lag of 1 km between the simulated CVB vertical velocity and the air vertical velocity is slightly weaker than the cases of balloons b31 and b32, which have smaller vertical velocities in the first part of the waves measured. This result is in agreement with the academic study performed in section 3a that shows that for strong air waves amplitude, the phase lag between *W*_{b} and *W*_{a} decreases.

## 4. Measure of uncertainties on the simulated quantities

The uncertainties on the modeled quantities presented here concern errors of (i) the computed CVBs trajectories in the framework of a Lagrangian approach [section 3b(2)] and (ii) the downstream decay of the wave amplitude in the balloon measurements by the model.

### a. Uncertainties on the simulated trajectories

The error sources regarding the computation of trajectories are related to (i) the trajectory numerical scheme and (ii) the uncertainties in the input meteorological data that depend mainly on the model’s formulation and its initialization. The trajectory numerical scheme errors result in (i) the truncation errors (when ignoring the higher-order terms of Taylor series), (ii) the spatio–temporal interpolation of the meteorological input, and (iii) the assumption on the vertical component of the wind vector. These error sources have been fully discussed in the literature and recently reviewed by Stohl (1998). It has been found that (i) a short time step of ⩽15 min ensures the truncation errors, and (ii) a minimum resolution of 6 h is necessary to minimize the errors due to temporal interpolations and high spatial resolution data (grid cell less than 1 degree) are obviously recommended. Bilinear, quadratic, and bicubic spatial interpolation schemes are all found to converge to the same solution for a grid size less than 1 degree (Stohl et al. 1995). Most studies agree now that the airmass trajectory using the air vertical velocity is the most accurate. In the present work, the temporal and spatial resolutions of the Meso-NH output are 2 h and 1 km, respectively. The time step used is less than or equal to 5 s. The temporal and spatial interpolation schemes of the input meteorological data are bilinear (then linear in the present study, i.e., 2D) in horizontal and linear in vertical planes. Therefore, the trajectory numerical scheme errors may be regarded as residual. Assessment of trajectory errors is mostly done in a statistical framework. A measure of trajectory errors that has been adopted by many authors in recent years is the absolute horizontal transport deviation (AHTD) and relative horizontal transport deviation (RHTD) and absolute vertical transport deviation (AVTD) and relative vertical transport deviation (RVTD) (Kuo et al. 1985; Stohl et al. 1995 among others). The AHTD or AVTD parameter corresponds to the difference in terms of distance (km) or altitude (m) between a measured trajectory and its corresponding computed one in time. The RHTD (RVTD) deviation is the AHTD (AVTD) normalized by the average length (altitude) of the two trajectories. General previous findings concerning trajectory errors linked to the interpolation methods (spatial and temporal resolution) and input forecasted data (i.e., model formulation and initialization) are that both AHTD and RHTD increase with travel time, though RHTD increases at a weak rate (e.g., Haagenson et al. 1990; Stohl et al. 1995). In the present work, the temporal and spatial resolutions of the Meso-NH output are 2 h and 1 km, respectively. Therefore, temporal and spatial errors introduced by wind interpolation may be regarded as residual. No special trend has been found for AVTD (RVTD).

These statistical parameters evaluated over each CVB path are reported in Table 2. The AHTD (RHTD) mean values over each CVB path are 11 km (27%) for b31, 24 km (58%) for b32, and 2 km (13%) for b33. The model predictions in the horizontal plane are then good for balloon b33, fair for balloon b31, and poor for balloon b32. The large errors in wind speed for b32 are possibly due to the fact that the background wind decreased between 0600 and 1200 UTC (see Fig. 4a), whereas in the model it remains constant and equal to the background wind at 0600 UTC. These model discrepancies are the same order of magnitude as those found in other studies using analogous methods of analysis (e.g., Stohl 1998; Koffi et al. 1998).

The values of AVTD (RVTD) averaged over each CVB path of 320 m (9%) for CVB b31, 238 m (7%) for b32, and 510 m (12%) for b33 demonstrate the good performance of Meso-NH in the prediction of the lee-wave vertical variations.

### b. Uncertainties on the downstream decay of the wave amplitude

To investigate how well the model handles upward leakage of gravity wave energy out of the wave duct (e.g., Keller 1994), calculations of the downstream decay of the wave amplitude in both balloon measurements and model data have been performed. Then, the horizontal variations of both measured and computed amplitude of the CVB vertical velocity (i.e., |*W*_{b}|) (shown in Figs. 5d, 6d, and 7d) have been fitted by a linear curve. In fact, the CVB flight times were about 1.5 h, the period during which the wave was quasi-stationary (Caccia et al. 1997); therefore the linear regression is then justified. The relevant statistical parameters to the linear correlations derived from both modeled and observed wave vertical velocity amplitude are summarized in Table 3. The slope of the linear regression curve is interpreted as the downstream decay rate of the amplitude of the lee-wave vertical velocity along the balloons paths. High and significant linear correlation coefficients (*P*_{c} < −0.5 and significant within 99%) are found for both CVB measurements and model data. The rates of the downstream decay of the wave amplitude in the three CVB’s measurements range from −4.9 to −6.1 × 10^{−5} s^{−1} with a mean value of −5.4 × 10^{−5} s^{−1} (Table 3). Except for CVB b33, the modeled and measured slopes are statistically equal as shown by the Snedecor and Alpin-Wech tests. The predicted leakages of the gravity wave out of the paths of balloons b31 and b32 are 5% and 6% less than the observations, respectively. While, along CVB b33’s trajectory, the simulated leakage of the waves is 80% greater than the measurements.

## 5. Conclusions

The constant volume balloon data obtained during the PYREX IOP3 have been compared with simulations performed by the 2D version of the nonhydrostatic model Meso-NH. Since the CVBs are not perfect Lagrangian tracers of the air mass, a CVB model is first used to show their capability to follow lee waves by calculating their vertical variations within idealized atmospheric stationary waves. Results show that CVBs react perfectly well to the wave amplitude and period measured during PYREX IOP3, with only a delay of a few minutes (∼2 min). The derived CVB fundamental period is equal to that of the air mass and the values of the maximum and the minimum amplitude of the balloon vertical velocity are 88% of the air mass. When the wave amplitudes become small, the ratio of the vertical velocity of the balloon (*W*_{b}) to that of the air mass (*W*_{a}) decreases with an increase in the time lag between *W*_{a} and *W*_{b}.

The analysis of the CVB experimental data show that (i) the vertical velocities of the lee waves observed during PYREX IOP3 range from −5 to +5 m s^{−1} with wavelengths from 8 to 10 km and (ii) the amplitudes of the lee waves were strong between 0800 and 1000 UTC on 15 October 1990, weakened at about 1200 UTC and the waves scattered in the area where the CVB flew between 1200 and 1400 UTC. Such wave amplitudes and lengths have been recorded by the CVBs above the White Sands Missile Range (Reynolds et al. 1968) and the central Colorado Rocky Mountains (Vergeiner and Lilly 1970). It is also found that the CVBs behaved as a quasi-isentropic tracer within the intense phase of the wave system, a result that confirms observations made by Reynolds (1973) during the Mountain Wave Project.

The comparison between CVB trajectories and those calculated from the Meso-NH forecasts shows that (i) the wind intensity observed in the lee-wave system is sometimes poorly predicted, (ii) the potential temperature of the air mass is in fairly well reproduced by the model Meso-NH, and (iii) the vertical velocity of the balloon is in good agreement with the observations.

The measure of the uncertainties in the calculated horizontal trajectories given by the AHTD (RHTD), which shows that the three computed trajectories deviated from the measured ones by 11 km (27%), 24 km (58%), and 2 km (13%), respectively for balloons b31, b32, and b33. These discrepancies are the same order of magnitude of those found in the literature. The large errors in wind speed for b32 are possibly due to the fact that the background wind increased after 0600 UTC, the time used to initialize the model. The vertical excursions of the calculated lee waves are quantified by the AVTD (RVTD) statistic parameters. AVTD (RVTD) mean values over each CVB of 320 m (9%), 238 m (7%), and 510 m (13%) for balloons b31, b32, and b33, respectively, are found.

An investigation of how well the model handles upward leakage of gravity wave energy out of the wave duct was performed and the predicted leakages of the gravity wave from balloons b31 and b32 were of 5% and 6% less than the observations, respectively. While, along the CVB b33 trajectory, the simulated leakage of the wave was 80% greater than the measurements.

Although in the model the upstream conditions at 0600 UTC were steady for the simulation at 1000 and 1200 UTC, the model fairly well reproduces the observed lee-wave amplitudes. The result that is in good agreement with the quite similar profiles of the Scorer parameter derived from the soundings at 0600 and 1200 UTC.

This work demonstrates the usefulness of the CVB for the study of the dynamic and thermal structures of lee waves generated by mountains. The CVB, a unique experimental tool that describes in a Lagrangian manner the amplitude of the lee waves, allows first for the description of the main characteristics of the waves (i.e., amplitude, vertical velocity, length) and second its data and the CVB model provide a severe test for mesoscale numerical models.

## Acknowledgments

The PYREX experiment was made possible by the participation of a large number of institutes and funding agencies. The participating institutes are CNRM, CRPE, LA, LAMP, LMD, LSEET, SA, EDF, France; INM, UV, UIB, Spain; and DLR, Germany. Funding was provided by Météo-France, INM, INSU (ARAT, PAMOS, and PAMOY programs), CNES, EDF, DLR, and Région Midi-Pyrénées. Much technical help was provided by CEV, ENM, and the French and Spanish Airforce and Air Control authorities. We would like to express our gratitude to J. C. André, D. Cadet, D. Guédalia, and A. Ascaso Liria for their help in the planning of this program. We also would like to express our deep appreciation to the many colleagues who have participated in the success of the experiment through enormous personal commitment. We are indebted to two anonymous reviewers for their valuable comments and suggestions.

Marc Georgelin died on 9 September 1999, within the very first days of the MAP field phase experiment, during which time the constant volume balloons were again deployed. Marc was a young scientist, bright, enthusiastic, and passionately fond of aerology. His memory will remain forever in our hearts.

## REFERENCES

## Footnotes

* Deceased.

*Corresponding author address:* Dr. Ernest N’Dri Koffi, Météo-France, CNRM/GMME/PI, 42 Avenue G. Coriolis, Toulouse, Cedex 31057, France. Email: Koffi@cnrm.meteo.fr