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  • Dou, X K., G. Scialom, and Y. Lemaître, 1996: MANDOP analysis and airborne Doppler radar for mesoscale studies. Quart. J. Roy. Meteor. Soc.,122, 1231–1261.

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  • View in gallery

    (a) Horizontal and (b) vertical cross sections of the simulated relative wind field (arrows) at 1 km and along the line (AB) [see location in (a)], respectively. Track 1 represents the relative flight track (relative to the moving reference frame of the squall line) of the airborne Doppler radar and track 2 illustrates the relative flight track of the aircraft that drops sondes. For Fig. 1b, isocontours represent the vertical velocity (isocontour every 0.5 m s−1).

  • View in gallery

    (a) Horizontal and (b) vertical cross sections of the retrieved relative wind field (arrows) without radiosounding (RS) data at order 5 at 1 km and along line (AB) [see location in (a)], respectively. For Fig. 2b, isocontours represent the vertical velocity (isocontour every 0.5 m s−1).

  • View in gallery

    Difference field of (a) u, (b) υ, and (c) w between the retrieved relative wind field without RS data at order 5 and the simulated relative wind field along lines (AB) (see location in Figs. 1a and 2a).

  • View in gallery

    (a) Horizontal and (b) vertical cross sections of the retrieved relative wind field (arrows) with RS data at order 5 at 1 km and along line (AB) [see location in (a)], respectively. For Fig. 4b, isocontours represent the vertical velocity (isocontour every 0.5 m s−1).

  • View in gallery

    Difference field of (a) u, (b) υ, and (c) w between the retrieved relative wind field with RS data at order 5 and the simulated relative wind field along lines (AB) (see location in Figs. 1a and 4a).

  • View in gallery

    CFAD of the retrieved relative wind field for (a) w, (b) υ, and (c) u with the MANDOPAS analysis and of relative simulated wind field for (d) w, (e) υ, and (f) u (contour every 10% of data per kilometer per meter per second).

  • View in gallery

    (a) Skew T–logp diagram and (b) hodograph (m s−1) of the composite RS for 22 February 1993.

  • View in gallery

    (a) Horizontal and (b) vertical cross sections of the retrieved relative wind field (arrows) and reflectivity (contours every 5 dBZ) for the real case with RS data at 1 km and along line (AB) [see location in (a)], respectively.

  • View in gallery

    (a) Horizontal and (b) vertical cross sections of the retrieved relative wind field (arrows) and reflectivity (contours every 5 dBZ) for the real case without RS data at 1 km and along line (AB) [see location in (a)], respectively.

  • View in gallery

    Vertical difference field of (a) u and (b) w between the retrieved relative wind field with RS data and the retrieved relative wind field without RS data along lines (AB) (see location in Figs. 8a and 9a).

  • View in gallery

    Horizontal difference field at 6-km altitude of (a) u and (b) w between the retrieved relative wind field with RS data and the retrieved relative wind field without RS data.

  • View in gallery

    Difference field of (a) u and (b) υ between the retrieved relative wind field with RS data along line (AB) of Fig. 8a and the RS field alone.

  • View in gallery

    Horizontal cross section of the pressure perturbation field (mb) at (a) 1 km and (b) 4 km.

  • View in gallery

    Horizontal cross section of the virtual potential temperature perturbation field (°C) at (a) 1 km and (b) 6 km.

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Three-Dimensional Variational Data Analysis to Retrieve Thermodynamical and Dynamical Fields from Various Nested Wind Measurements

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  • 1 Centre d’Étude des Environnements Terrestre et Planétaire, Velizy, France, and Centre National de Recherche Météorologique, Toulouse, France
  • | 2 Centre d’Étude des Environnements Terrestre et Planétaire, Velizy, France
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Abstract

The present study is devoted to a new analysis of wind measurements from dropsonding and/or radiosonding of Doppler information from multiple Doppler radar scanning and of other wind measurements (sodar, dynamical sensors on board aircraft, and instruments at ground) aimed at retrieving three-dimensional thermodynamical and dynamical fields both in clear air and in precipitating areas of mesoscale phenomena. This analysis, well suited to assimilate data from differing platforms specified at differing spatial/temporal resolutions, is based on the analytical and variational concept of the Multiple Analytical Doppler (MANDOP) analysis and thus is an extension of it. This new analysis presents many advantages, including the same as MANDOP and others well adapted for the verification or the initialization of a mesoscale cloud model. An application to simulated and to real data extracted from the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment database is presented in the paper.

Corresponding author address: Dr. Thibaut Montmerle, CETP-CNRS, 10, 12 av. de l’Europe, 78140 Vélizy, FranceEmail: montmerl@cetp.ipsl.fr

Abstract

The present study is devoted to a new analysis of wind measurements from dropsonding and/or radiosonding of Doppler information from multiple Doppler radar scanning and of other wind measurements (sodar, dynamical sensors on board aircraft, and instruments at ground) aimed at retrieving three-dimensional thermodynamical and dynamical fields both in clear air and in precipitating areas of mesoscale phenomena. This analysis, well suited to assimilate data from differing platforms specified at differing spatial/temporal resolutions, is based on the analytical and variational concept of the Multiple Analytical Doppler (MANDOP) analysis and thus is an extension of it. This new analysis presents many advantages, including the same as MANDOP and others well adapted for the verification or the initialization of a mesoscale cloud model. An application to simulated and to real data extracted from the Tropical Ocean Global Atmosphere Coupled Ocean–Atmosphere Response Experiment database is presented in the paper.

Corresponding author address: Dr. Thibaut Montmerle, CETP-CNRS, 10, 12 av. de l’Europe, 78140 Vélizy, FranceEmail: montmerl@cetp.ipsl.fr

1. Introduction

The rapid development in the last 20 years of numerical models, theoretical research, and new experimental tools (Doppler radar, etc.) have permitted an identification of the crucial role played by mesoscale circulation on the evolution of larger-scale structures through energy transfer and on convective motions through moisture supply. Tropical studies using ground-based Doppler radar have confirmed that trailing anvils associated with squall lines play an important role in their life cycle since they are implied in the feedback mechanism responsible for the squall line regeneration (e.g., Chong et al. 1987). Similarly, studies on kinematical and thermodynamical structures at the rear of fronts in stratiform precipitation have shown the important role of mesoscale circulations in rainband formation (e.g., Lemaître and Testud 1988; Lemaître et al. 1989). However, the study of dynamical processes leading to the mesoscale organization of precipitation has been rendered difficult because Doppler radar does not observe the airflow in clear air regions of these mesoscale circulations. During the Fronts and Atlantic Storm Tracks Experiment (FASTEX; A. Joly and Coauthors 1993, unpublished manuscript), this problem was addressed by performing dropsondings. These data were complementary to airborne Doppler radar observations performed by the NOAA P3-43 and the NCAR Electra aircraft in precipitating areas. This fact evidences the need for new analyses processing wind measurements from Doppler radar and concurrently from other wind measurements (dropsonding from aircraft, radiosonding, sodar measurement, and measurements by sensors onboard aircraft and from ground stations). Several solutions have been proposed in the past either to study mesoscale phenomena (see, e.g., Franklin et al. 1993) or to initialize nonhydrostatic numerical models (Lin et al. 1993). Sun et al. (1995) and Sun and Crook (1996) show two examples of storms where radar data and soundings have been combined in a variational analysis scheme using a cloud model and its adjoint.

The present paper is written in this context. It proposes a new analysis system called MANDOPAS (MANDOP Assimilation) based on the variational and analytical concept of the MANDOP (Multiple Analytical Doppler) analysis, analysis of data from several ground-based Doppler radars (Scialom and Lemaître 1990, hereafter referred to as SL; Lemaître and Scialom 1992), and/or airborne Doppler radars (Dou et al. 1996, hereafter referred to as D96), aimed at the retrieval of the three-dimensional mesoscale and convective-scale wind fields. The MANDOP analysis defines each of the three wind components as a product of three expansions in terms of an orthonormal functions series (each of these expansions depends on each spatial coordinate). Thus, the radial wind is also expressed in this analytical form. To retrieve the coefficients of the development, this analytical form of the radial wind is variationally adjusted to the observed ones under physical constraints, including an anelastic continuity equation and a lower kinematic boundary condition. Thus, the MANDOP analysis allows the three components of the wind to be analytically obtained. Practically, it leads to a matrix equation CB = A in which C depends on the radars and observations locations and on the used base orthonormal functions, A is a vector containing experimental information (radial winds), and B is the vector of unknowns (expansion coefficients) of the three wind components.

The present analysis rests on several advantages of the variational and analytical concept of the MANDOP analysis. As explained in SL, the retrieval process bypasses the interpolation, can use data points as densely or sparsely as they occur, and performs a sequential pass through all the data in any order, by simply accumulating terms in the matrices. It allows us to easily introduce other physical conditions or additional information of experimental nature under a variational form, expressing them in the same matricial form and adding them to the main matrix C and to the vector A. Moreover, the analytical formulation of the three-dimensional wind field provides a mathematically exact solution to the problem of correcting data for advection of the sampled phenomena, and one obtains an instantaneous analytical three-dimensional wind field. Finally, as explained in SL and shown in Protat et al. (1997b), the three-dimensional thermodynamic field can also be retrieved under the same analytical form as the wind field, introducing the analytical form of the three-dimensional wind field obtained by the MANDOP analysis in the nondissipative and stationary first-order anelastic approximation of the equations of motion. The coefficients of expansion of the wind are also those of the pressure and virtual “cloud” temperature perturbations field. Thus, the determination of the coefficients of expansion for the three-dimensional wind field (by the MANDOP analysis) allows a straightforward retrieval of these perturbation fields.

In section 2, we give the mathematical formulation of the analysis with special emphasis on its analytical and variational aspects. The analysis of data from different instruments calls for the solution of advection effects induced by nonsimultaneous measurements. This problem is addressed in section 3. Then, in order to perform an experimental test of the analysis accuracy, an application to simulated and real data collected during the Tropical Ocean Global Atmosphere Comprehensive Ocean–Atmosphere Response Experiment (TOGA COARE) is given in section 4. It shows the adequacy of the present analytical variational approach to retrieve three-dimensional mesoscale circulations in the precipitating part of convective system and also in its nonprecipitating environment. Section 5 lists possible extensions resulting from the analytical form of the wind field and associated dynamical and thermodynamical parameters obtained with MANDOPAS, including the verification and the initialization of a mesoscale model.

2. Recall of important aspects of the MANDOP analysis

a. Analytical representation of the wind

The basic hypothesis of this MANDOP analysis is that the variations of the wind components U1, U2, and U3 with respect to each coordinate may be written as a product of three functions of each coordinate. For example, the analytical form Vi of the component Ui may be expressed as
i1520-0426-15-2-360-e1
with
i1520-0426-15-2-360-e2
where ailk are the expansion coefficients. The base of the orthonormal functions Filk and the order of expansion ni1 of the component i on the corresponding xl axis are chosen to obtain the best representation of the observations. Combining and rearranging Eqs. (1) and (2) leads to the following analytical form of the three wind components Vi:
i1520-0426-15-2-360-e3
with
i1520-0426-15-2-360-e4
where bKi is a product of three coefficients ai1k, ai2k", ai3k and Ki is given by the equation
Kikni2ni3kni3k
Coefficients k′, k", and k‴ range from 1 to ni1, ni2, and ni3, respectively. Here, gKi is the product of the three corresponding orthonormal functions:
gKi(x1x2x3Fi1kx1Fi2k"x2Fi3kx3
As explained in SL, these analytical expressions of the wind components U1U3 allow an analytical representation of the observed radial velocity.
The N coefficients bK are obtained by minimizing in the least squares sense the difference between the analytical form of the radial winds and the observed ones for all the experimental points (denoted r) obtained from all j radars (j = 1, 2, . . .) in the retrieval domain. This leads to the matrix equation (see SL)
CBA
in which
  • B is the N dimension vector of the unknowns bK;
  • C is an N × N dimension symmetric matrix, where the elements of CKK′ consist of analytical information (orthonormal functions through their products gK(x1, x2, x3) and are geometrically determined direction cosines MjK):
    i1520-0426-15-2-360-e8
  • and A is an N dimension vector, where the elements of AK contain experimental information data uj:
    i1520-0426-15-2-360-e9
Then, the unknowns bK are determined through inversion of Eq. (7) and a direct determination of the three wind components can be obtained using Eq. (3).

b. Additional constraints

The analytical form of the three wind components is constrained to satisfy statistically in the least squares sense the continuity equation and a ground level boundary condition (eventually taken into account the orography) for all experimental points of the domain. This leads to matrix equations
CB
where C′ is an N × N dimension matrix. Its elements CKK contain “analytical” information through divU and V3 (see appendix C of SL), and
CB
in which C" is an N × N dimension matrix. Its elements C"KK contain “analytical” information through V1, V2, and V3 (see appendix D of SL).

c. Mathematical formulation of the variational problem

The variational problem consists of finding the coefficients that give the best fit, in the least squares sense, to the radial wind in the whole domain, under the subsidiary conditions that the continuity equation and the boundary condition at ground level are satisfied by the analytical form of the wind field. This is expressed as the matrix equation
DBCλCλCBA
where λ′ and λ" are weighting factors whose evaluation is done following a standard deviation approach by taking the inverse of the square of the expected observational error (see SL).

d. Correction for advection

We seek an analytical form of the instantaneous three-dimensional wind field taken at time t0 (for example, midtime of the sequence). To find the coefficients of this analytical form, one must compare this analytical form to the observed wind. The problem is that these observations are not taken all at time t0 and thus, in particular, the advection (Cx, Cy) of the observed system must induce time evolution of the observed wind field. If one considers that the time evolution of the wind field observed at ground level between t0 and time t of an observation (at point x, y) is only due to the advection of the observed wind field, this comparison can be done only if we compare this observation at t with the analytical form displaced by −Cx(tt0) and −Cy(tt0). Therefore, the correction for advection simply consists in minimizing the difference between a radial wind observed at time t and the radial component of the analytical form of the wind field taken at the location it occupies at time t. Thus, this procedure provides a mathematically exact solution to the problem of correcting for advection, and one obtains an instantaneous analytical three-dimensional wind field (see SL and D96).

3. Extension of the analysis to other wind measurements

a. Case of radiosondings/dropsondings/wind profilers

As explained in SL, other physical conditions may be introduced in the process of the wind retrieval and also expressed as additional matrices. For instance, the analytical form of the horizontal wind component Vi may be assumed to be close to values obtained by radiosondings or dropsondings. The corresponding condition may be written as
i1520-0426-15-2-360-e13
for all the experimental points (denoted e) obtained from all the radiosounding or dropsonding j (j = 1, 2, . . .) in the retrieval domain with respect to the N coefficients bK. Here, Uij is the observed value from the additional (e.g., radiosonde) platform. We thus solve the following linear system of N equations with N unknowns bK, for K = 1, 2, . . . , N:
i1520-0426-15-2-360-e14
with the notations:
i1520-0426-15-2-360-e15a
System (14) with (15a)–(15c) is equivalent to the three matrix equations:
i1520-0426-15-2-360-e16a
in which, for example,
  • B is the N dimension vector of the unknowns bK; and
  • C1 is an N × N dimension symmetric matrix, where the elements of C1KK consist of analytical information [orthonormal functions through their products gK(x1, x2, x3)] concerning the first component. The nonzero terms of matrix C1 occupy the N1 first lines and columns of it. Here, A1 is an N dimension vector, where the elements A1K contain experimental information data uj. The nonzero terms of A1 occupy the N1 first lines.
Thus,
i1520-0426-15-2-360-e17a
Also, we have
i1520-0426-15-2-360-e17b
i1520-0426-15-2-360-e17c
The variational problem consists now in finding the coefficients that give the best fit, in the least squares sense, to the radial wind in the whole domain (under the subsidiary conditions that the continuity equation and the boundary condition at ground level are verified by the analytical form of the wind) and to the additional wind measurements performed by radiosonding/dropsondings. This is expressed as the matrix equation
i1520-0426-15-2-360-e18
with
i1520-0426-15-2-360-eq18
where λ1, λ2, and λ3 are weighting factors whose evaluation is done following a standard deviation approach as in Eq. (12). In the case of an identical confidence on each components, λ1, λ2, and λ3 are equal. In the case of measurements from different instruments we have to add matrix and vectors identical to C1,2,3 and A1,2,3, respectively, but with different weighting factors consistent with the accuracy on the wind measurements from each of these instruments. Then, the unknowns bK are determined through inversion of Eq. (18) and a direct determination of the three wind components can be obtained.

b. Case of surface measurements

The analytical form of the horizontal wind components at surface must verify
i1520-0426-15-2-360-e19
where g denotes a summation on measurements at ground. Conditions (19) and (20) are equivalent to the matrix equations
i1520-0426-15-2-360-e21
in which CS" and CS are N × N dimension matrices.
The variational problem consists now in finding the coefficients of the analytical form of the wind field that give the best fit, in the least squares sense, to the radial wind in the whole domain (under the subsidiary conditions that the continuity equation and the boundary condition at ground level), to radiosondings/dropsondings, and to ground measurements or airborne in situ measurements. This is expressed by adding to the previous matrix E, the matrices CS", CS, and to the vector A*, AS", AS with weighting factors whose evaluation is done following, as previously shown, a standard deviation approach. This leads to the matrix equation
FBA
In conclusion, in this approach the radiosondings/dropsondings and ground measurements constraints are introduced in the process under a matrix notation and with a variational formulation. Practically, it leads to a matrix equation FB = A** in which F depends on the observations locations (x, y, z), on the locations (xj, yj, zj) of various radars and other instruments, on the base orthonormal functions, and on the advection speed. Thus, F is obtained by giving as input data of the C-building software the various locations of the orthonormal observations and of the radars, after giving the used advection speed and the expansion base. This formulation of the method is the most general one, allowing its application to all possible situations (dataset, number and types of instruments, types of scannings, types of bases, etc.).

4. Application to simulated and real data

These applications are an illustration of the capabilities of the variational analysis MANDOPAS to process wind data from various instruments involved in an experiment devoted to the study of mesoscale precipitating systems. Numerical tests are carried out to put in evidence the role of each constraint in the present analytical variational analysis and to estimate the robustness of the analysis. Simulated winds were chosen to represent a quasi-two-dimensional squall line with a three-dimensional wind field as close as possible to the observed one retrieved from real data hereafter. This application is carried out in the particular case of a Legendre polynomial base. This base is simply selected to illustrate the application. Let us recall that the analysis is not linked to a particular base and any other orthogonal function base may be used without reformulation of the analysis. However, as explained in SL, this base appears generally as the best choice adapted to the various encountered meteorological phenomena. The application to real data will be performed using airborne radar data gathered by the French dual beam antenna on NOAA’s P3 (43) aircraft and by the single beam radar on the P3 (42) aircraft during the 22 February case of TOGA COARE (Webster and Lukas 1992) and using a composite radiosonding (hereafter RS). This squall line has been described in detail by Jorgensen et al. (1997) and by Trier et al. (1996) and Trier et al. (1997) by means of numerical simulation. The application to real data will be done, as in SL, using a data reduction procedure averaging quasi-instantaneous (using a time classification process) data from each radar within a grid mesh, with a grid resolution consistent with the resolved small mesoscale motions. Let us recall, however, that in the MANDOP–MANDOPAS analyses, each of the measurements can be theoretically taken into account sequentially. But this scheme is difficult to apply to airborne radar data because of the huge amount of data, which would lead to a considerable computer time. The procedure used (space average on quasi-instantaneous measurements from each radar in each grid mesh leading to a single datum in each grid mesh at the mean position of these data) leads to a reduced dataset nearly regularly distributed in the retrieval domain for each radar. But as explained in SL, it is a particular adaptation of the MANDOP analysis for convenience. The normal way to use MANDOPAS analysis is to process all data in any order, without defining a grid mesh, simply accumulating terms in the matrices.

a. Principle

Let us first recall considerations about the cutoff wavelength of the method and the resolved characteristic scale of three-dimensional circulation by the analysis. These characteristics govern the way to process data with both high (airborne Doppler) and low spatial density (RS). This cutoff wavelength of λC is due, on the one hand, to the horizontal grid resolution used in the preprocessing (data reduction procedure) and, on the other hand, to the development order of the analytical winds in the given domain size. As for the horizontal resolution, if the spatial sample resolution is Δx, it means that no structure with a wavelength smaller than λC1 = 2Δx will be observed, according to Shannon’s theorem. Concerning the order of development, since components are developed up to the order n along all coordinates, a wave along each axis reaches zero at most (n − 1) times, thus implying a minimal wavelength λC2 of about 2/(n − 1) times the total explored domain. So, when applied to real cases, parameters n, L, and Δx will be chosen such that λC1λC2. For the present application, the horizontal resolution Δx = 2.5 km (1.5 km horizontally and 0.5 km vertically for real data) leads to a minimum observation wavelength λC1 = 5 km (3 km for real data processed in section 4c), and a development up to the order n = 5 on the domain size D1 of 100 km (50 km) leads to a λC2 of 50 km (25 km). Thus, the minimum observable wavelength is indeed imposed by the order of the development. If the method induces a cutoff wavelength λC, it means that the minimum observable scale is lc = 0.26λC (corresponding to half the power of the phenomenon, represented by, for example, a Gaussian function passing through the window resulting from the method cutoff in Fourier space). In the present case, the minimum observed scale lc is fixed by the development order and is found to be 13 km (4 km) for order 5.

Thus, since the density of measurements from various instruments is not the same, the accessible characteristic scale by the analysis cannot be the same for the various datasets. To process data from both airborne radar and from dropsonding for instance, we have to choose an order of development consistent with the lowest density or resolution of any of the contributing datasets (i.e., RS). So, this leads to a severe degradation of the accessible characteristic scale in the airborne sampling area compared to the one accessible with the initial radar dataset. One way to resolve this problem is to process these data in two steps. First, the analysis is applied on airborne Doppler radar data and RS data with an order of development consistent with the “low-density” dataset. The obtained analytical form of the three-dimensional wind field is then used as an interpolated tool to fill in the dropsondings’ areas (outside the areas sampled by airborne radar) with interpolated data in order to increase the data density in these areas (to be comparable with the density of radar data). Then the analysis is applied on airborne data and on these interpolated dropsonding data with a higher order of development consistent with the resolution of an airborne radar dataset. This allows us to avoid the severe data degradation previously discussed and allows us to retain the small-scale circulation in the precipitating area (as with airborne radar data alone).

Note that this procedure is based on the hypothesis that the airborne sampling strategy is defined to cover totally the perturbed area, whereas dropsondings, located outside the airborne sampling areas, are done in nonperturbed zones or zones that can be adequately resolved with the low resolution of dropsonding data. This means that the perturbed regions correspond to precipitating areas scrutinized by the airborne Doppler radar.

b. Application to simulated data

The three-dimensional wind field used to perform these tests obeys the density continuity equation and the boundary condition at ground level. The detailed formulation of the simulated three wind components is given in the appendix.

The grid resolutions (Δx, Δy, Δz) are 2.5 km horizontally and 0.5 km vertically. The chosen advection speed is Cx = 0 m s−1 and Cy = 10 m s−1. All the simulations are performed with the addition of a random radial velocity error having 1.5 m s−1 standard deviation simulating the radar statistical error and 0.5 m s−1 standard deviation on radiosonde data.

Figures 1a and 1b give horizontal and vertical cross sections of the used simulated relative wind field defined by Eqs. (A1)–(A3). This simulated wind field represents a quasi-two-dimensional squall line with a maximum vertical velocity of about 3 m s−1 and horizontal velocities reaching 20 m s−1. It shows a classic dynamical structure including an ascending front-to-rear flow and a rear inflow in the stratiform region where the vertical motions are near zero.

This simulated convective system is sampled by one airborne Doppler radar equipped with a dual beam antenna. The direction of the flight was along the convective line, and the sampling was done at 3-km altitude only on one side of the virtual aircraft to obtain more data in the most interesting part of the line (see relative flight track 1 in Fig. 1a). The retrieved three-dimensional wind field is obtained using the MANDOP analysis with a high order of development (5 in the present case) consistent with the resolution of airborne radar dataset (see Figs. 2a,b). This field is in very good agreement with the initial simulated wind field (see Fig. 3). Differences less than 0.5 m s−1 are observed (with rms errors of ±0.3, ±0.1, and ±0.1 m s−1 for u, υ, and w, respectively).

Different tests on the ways to perform dropsonding have been done, particularly on the location, the number of dropsondes, and the release time (not shown). In this paper, we present results with only four noninstantaneous dropsondings (spaced apart by 15 and 17 min in duration) dropped by the same plane ahead and behind the simulated squall line (see relative flight track 2 and dropsonding locations in Fig. 1a). This case is a drastic test of the analysis since dropsonding are not done at the same time (45 min between the first and the last one), since each dropsonde lasts 17 min, and finally since only four dropsondes are used to define the inflow and outflow clear air regions. It allows us to estimate the capacity of the analysis to process nonsimultaneous measurements and to correct data from the induced advection problem. Afterward, the MANDOPAS analysis is applied on radar and dropsonde data with an order of development consistent with this low-density dataset (2 in the present case). The retrieved three-dimensional analytical wind field (not shown) is then used to fill up the regions outside the area sampled by the aircraft.

The final three-dimensional analytical wind field is retrieved by applying the MANDOPAS analysis on airborne data and on this interpolated field with an higher order of development consistent with the radar dataset (5 in this case).

The obtained wind field is displayed on Figs. 4a and 4b. The difference of retrieved velocities between the initial simulation and the wind field obtained with MANDOPAS along line (AB) (see Figs. 1a and 4a) is plotted in Fig. 5. They show the robustness of the analysis and the good accuracy of the retrieved wind field. Discrepancies less than 0.1, 1.5, and 0.5 m s−1 are obtained on u, υ, and w, respectively (with rms errors of ±0.1, ±1, and ±0.3 m s−1, respectively).

For the cross-line velocity υ, this maximum divergence occurs at the rear of the system, where υ reaches 15–20 m s−1. Thus, it represents less than 10% of these maximum values, which is lower than the accuracy of rawinsonde measurements. For the vertical velocity w, the maximum variation occurs also at the rear of the system, where the analysis is unable to reproduce the weak simulated updraft using only the dropsonding data. These divergences result primarily from the incapability of the assumed number of dropsondings to represent the nonlinear variation of the simulated horizontal wind field (and associated vertical motion) ahead and behind the squall line. Indeed, the area sampled by the radar does not entirely cover these important dynamical zones with strong horizontal and vertical gradient of horizontal wind. Figure 6 gives the CFAD [contoured frequency by altitude diagram, see Yuter and Houze (1995) for definition] for each of the three wind field components corresponding to the initial simulated wind field panels (Figs. 6a–c) and the retrieved wind field with the MANDOPAS analysis panels (Figs. 6d–f). They show essentially the same spatial distribution for each of the three wind components. This indicates the capability of the proposed analysis to process wind data from various source in order to extend the retrieval domain outside the region sampled by the Doppler radars.

c. Application to real data

The data used in the present section were extracted from the TOGA COARE experiment data bank. During this experiment, very few dropsondings or RS have been done close to the sampled convective systems. This fact led us to select the 22 February 1993 data. Indeed, during the sampling of this case, the P3-42 located ahead of the precipitating system launched one dropsonde. Figure 7 gives the corresponding composite emagram and hodograph derived from the P3 flight level data and rawinsonde observation from Honiara, Guadalcanal (data courtesy of M.A. LeMone). This hodograph has been constructed from low-level in situ aircraft data (under 1.3 km) obtained immediately ahead of the precipitating system during its linear stage, from aircraft data 1 h later (1.3–6 km) and from a blend of the 1800 and 2400 UTC Honiara sounding, located 80 km northeast of the Doppler analysis area. It indicates a jet of 12 m s−1 centered at 2-km altitude. The squall line motion was nearly parallel to the low-level vertical shear vector. The propagation speed used (90° east from north, 12 m s−1) in the procedure for correcting for advection is estimated from visual correlation of successive radar reflectivity patterns (Jorgensen et al. 1994). The data coming from the Doppler radar of the two NOAA P3 aircraft processed in this paper have been taken during the most linear and mature stage of the squall line (between 2110 and 2120 UTC). The contribution of the terminal fall velocity of the hydrometeors Vt to the radial Doppler velocity is estimated through an empirical relation between the particle fall speed and the radar reflectivity factor (e.g., Protat et al. 1997a).

Then the MANDOPAS analysis is applied. The analytical form of winds and radial winds are variationally adjusted to the observed radial velocities (affected by a mean sampling time t relative to t0) and to the available RS wind according to the anelastic continuity equation and to the lower kinematic boundary condition. The used boundary condition will be w = 0 m s−1 at Z = 0 m (sea level). The deduced three-dimensional wind field is illustrated in Fig. 8. The horizontal relative wind at 1 km presented in Fig. 8a evidences a three-dimensional aspect of the squall line in particular in the northern part of the line. The leading edge of the squall line appears as a region of heavy convective showers with strong updrafts reaching 4 m s−1 (not shown). The vertical cross section displayed in Fig. 8b shows a classic dynamical structure including an ascending front-to-rear flow and a rear inflow characterized by a 4-km vertical extent in the region of heavy stratiform rain, which is suggested by the presence of a radar reflectivity bright band located at heights between 4 and 4.5 km (corresponding to the 0°C isotherm altitude) (e.g., Smull and Houze 1987). The line evolves during the aircraft sampling from a quasi-linear structure with a leading edge of high reflectivity (up to 40 dBZ) to a bow-shaped pattern near the center of the rainband (e.g., Montmerle et al. 1996; Trier et al. 1996; Trier et al. 1997; Jorgensen et al. 1997).

Figure 9 gives the three-dimensional wind field without RS information, and Fig. 10 shows the difference field for two selected cross-sectional along lines (AB) of Figs. 8a and 9a, respectively, between the cases with and without RS information. The vertical and horizontal difference field of vertical velocity (Figs. 10a and 11a, respectively) and cross-line velocity (Figs. 10b and 11b, respectively) put in evidence the fact that, as expected, adding RS measurements in the retrieval process does not significantly modify the internal circulation of the squall line deduced from airborne Doppler radar data but only extends the retrieval zone in the precipitation-free region. The only divergence for both wind components appears in altitude on the edge of the domain. This discrepancy results first from the fact that RS measurements at upper levels above the P3 flight altitude (deduced from an RS not taken at the same time and at the same location) do not accurately describe the near environment of the precipitating system. So it is slightly in disagreement with the radar observation. The discrepancy results also from the fact that 1) the performed aircraft flight track does not totally cover the perturbed region due in particular to attenuation of radar beam and 2) the flight plan was devoted to sample the convective part of the system. Indeed, this aircraft was flying just in front of the convective line and was sampling only the convective side. Thus, the region ahead of the convective line not sampled by the radar, the region without any reflectivity echoes, and the region outside the radar range (i.e., behind the convective system) are considered unperturbed and thus described only by the available RS.

Difference fields between the three-dimensional wind field retrieved with MANDOPAS and the RS field for the cross-line and the along-line velocities are shown in Figs. 12a and 12b, respectively. Except in the transition zone between the two sets of data, divergences less than 10% are made, which are lower than the error estimated from the RS. This illustrates the capability of the analysis to take into account extra RS measurements in order to extend the retrieval zone without modification of the retrieved internal circulation of the squall line.

5. Possible extensions

a. Retrieval of air parcel and particle trajectories

The analytical form of the wind component allows the three-dimensional path of air parcels and thus the three-dimensional airflow in the precipitating system to be readily obtained with great accuracy when the assumption of stationarity is done. This assumption could be made in the present case since the relative air flow is stationary during a time greater than the crossing of an air parcel throughout the system. As a parcel moves along its path, its next position is simply calculated from the wind components directly given by the value of its analytical representation at the current point and from the time step length that may be as short as required. Particle trajectories are obtained from wind components adding the particle vertical terminal velocity [deduced from reflectivity factor through an empirical relationship (Protat et al. 1997a)]. The deduced three-dimensional perspective airflow for the 22 February case in a larger domain (not shown) confirms the two-dimensional structure of the circulation except in the northern part of the convective line with streamlines starting and rising from the lowest layers, while those originating at middle levels descend. The inflow is confined to the front of the system.

b. Pressure and temperature retrieval

In addition to classic parameters directly related to the wind field such as vorticity, it is possible, as shown by Protat et al. (1997b), to retrieve physical parameters such as pressure or virtual “cloud” potential temperature perturbations (P′ and θc, respectively) with respect to a constant depending on the altitude, allowing the description of thermodynamic mechanisms implied in the mesoscale circulations. This is done introducing the analytical form of the wind field in the nondissipative and stationary first-order anelastic approximation of the equation of motion and performing the necessary integration/derivation using a symbolic software for manipulating formulas by computer. This leads to an analytical form of P′ and θc for the three-dimensional wind field, which means that the coefficients retrieved by the MANDOPAS analysis are the coefficients of both dynamic and thermodynamic fields. Thus, the analytical formulation of the wind field allows direct derivation of the analytical form of thermodynamic perturbation fields.

Figures 13a,b and 14a,b present two horizontal cross sections of such pressure and temperature perturbation field deduced from the previous three-dimensional wind field. Figure 13a indicates a classic pressure perturbation field at 1-km altitude, with a positive value ahead of the convective line and a decrease toward the rear of the system, leading to a −1.2-hPa horizontal pressure change across the domain. At 4-km altitude, the pressure gradients appear oriented normal to the convective line (Fig. 13b). LeMone et al. (1984) and Roux (1985) have found similar results. The resulting vertical configuration of nonhydrostatic pressure perturbation thus seems to be favorable to development of upward vertical motions ahead of and within the convective line.

Figure 14a exhibits a warming at 1-km altitude on the eastern part of the domain well correlated with the reflectivity field corresponding to the region of maximum vertical motions (see 30-dBZ contours). This region takes a wave pattern resulting from the three-dimensional characteristic of the wind field in the northern part of the domain. Indeed, in this area is located a particularly strong subsident motion cooled by important evaporation of precipitation, as shown by the strong negative temperature perturbation. Typically, we observe warming and cooling of about 1.2 and −1.3 K, respectively. The warming is consistent with an adiabatic ascent of an air particle initially at the ground (see RS given in Fig. 7a). Furthermore, we can show that the cooling is consistent with the observed mean reflectivity. Indeed, to check that the rate of precipitation (R ≈ 1.8 mm h−1 corresponding to the observed mean reflectivity value of 27 dBZ, in the cores, through the Marshall–Palmer ZR relation) can sustain the necessary evaporation, we used the simple model proposed by Hardy (1963). This model predicts the evaporation rate of a Marshall–Palmer-type precipitation within an atmospheric layer at a given saturation deficit. The maximum saturation deficit (deduced from the RS) of the layer in which rain evaporates is 2.6 g kg−1. The output of the model (not shown) predicts a rate of evaporation of 0.7 ± 0.1 g kg−1 h−1 for a rate of precipitation of 1.8 mm h−1. The dry air enters the core system at a velocity of about 4 ± 1 m s−1 eastward and passes through the precipitation system in 42 ± 10 min (for a width of 10 km). This leads to an amount of evaporated water of about 0.47 ± 0.2 g kg−1. This evaporation rate corresponds to a cooling of 1.2 ± 0.3 K (using the adiabatic relation cpΔT = −LΔq, where cp is the specific heat at constant pressure, T the temperature, q is the specific humidity, and L the latent heat of condensation), which is very close to the observed one. At 6-km altitude (see Fig. 14b), the positive temperature region is slightly shifted westward. The slantwise ascent of the warm air is well observed on the vertical cross section given by Fig. 8b.

c. Verification and initialization of a nonhydrostatic model

The three-dimensional wind field and associated three-dimensional pressure and temperature perturbation fields, which is the output of the MANDOPAS analysis, can be used to verify and to initialize nonhydrostatic mesoscale models in order to obtain parameters that cannot be deduced from various measurements and to observe the temporal evolution of such an initial field. Indeed, the retrieved fields by this analysis overcome the following classic problems that often occur with such an objective.

  • The domain is totally filled;
  • A smooth transition is obtained between the perturbed areas sampled by airborne radars and the basic state defined by proximity soundings;
  • The filled wind field and the transition zone between the perturbed area and the base state verify the continuity equation. Indeed, the MANDOPAS analysis finds the optimal analytical wind field that both fits radar and RS measurements and verifies physical constraints (mass conservation and wind belonging to the plane tangent to the ground); and
  • Moreover, the analytical form leads to continuous space derivatives of the wind field. This will minimize classical rapid changes of the model fields in an attempt to adjust to a balanced state.

The procedure to initialize the model could be done as follows.

  1. Initialize the model with the analytical form of the three-dimensional wind field obtained by MANDOPAS;
  2. Introduce the analytical forms of pressure (Exner function Π, also called the nondimensional pressure) and temperature (the virtual potential temperature θυ) perturbation fields; and
  3. Specify the water substance field (rainwater mixing ratio derived from reflectivity, water vapor mixing ratio at saturation in ascending regions above the lifting condensation level and equal to the environmental mixing ratio outside, and cloud water mixing ratio equal to zero).

An attempt of such an initialization of the Meso-NH nonhydrostatic model developed by Meteo-France/CNRS and Laboratoire d’Aerologies (Lafore et al. 1997) is presently in progress and will be the subject of a future paper. Finally, two-dimensional outputs of the MANDOPAS analysis can also be used in microphysical retrieval techniques (e.g., Marecal et al. 1997).

6. Conclusions

The study previously developed describes a new analysis, MANDOPAS, for the retrieval of the three-dimensional mesoscale wind field from wind and radial wind measurements performed by various instruments (ground-based and airborne Doppler radars, wind profilers, radiosonding stations, dropsondes launched from aircraft, ground level stations, etc.). It gives the mathematical principle of the wind retrieval. This analysis, based on the variational concept of the MANDOP analysis, rests upon the expression of the wind components as products of expansions in series of orthonormal functions on each coordinate axis, thus allowing the radial wind itself to be analytically expressed. Then, the analytical forms of the winds and the radial winds are variationally adjusted to the observed ones by including in the minimization process the additional conditions satisfied by the wind, such as mass conservation and boundary condition at ground level. The coefficients allowing a complete description of the wind are thus derived by the processing. The filtering characteristics, as in the MANDOP analysis, are related to the order of expansion of the orthonormal functions.

The present paper also deals with the application of the method to real cases and specifies how to operate in these cases where a great number of data with density and time scattering have to be taken into account. It illustrates how the analytical form of the wind implies data filtering and interpolating.

Thus, in conclusion, MANDOPAS analysis constitutes a new tool that will be used intensively to process airborne Doppler radar and dropsonding data gathered during the Fronts and Atlantic Storm Tracks Experiment in order to diagnose the multiscale interaction processes involved in frontal wave and/or secondary cyclogenesis.

Acknowledgments

The TOGA COARE experiment is the result of a cooperative work involving several laboratories, in particular U.S. and French laboratories. In addition to the contribution of the participating institutes, financial support was provided by the Institut National des Sciences de l’Univers (Programme Atmosphère Méteorologique).

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  • Marécal, V., Y. Lemaître, T. Montmerle, and F. Roux, 1997: Microphysical processes involved in a TOGA-COARE mesoscale convective system. Proc. 22d Conf. on Hurricanes and Tropical Meteorology, Fort Collins, CO, Amer. Meteor. Soc., 217–218.

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  • Yuter, S. E., and R. A. Houze Jr., 1995: Three-dimensional kinematic and microphysical evolution of Florida cumulonimbus. Part II: Frequency distribution of vertical velocity, reflectivity and differential reflectivity. Mon. Wea. Rev.,123, 1941–1963.

APPENDIX

Formulation of the Three-Dimensional Simulated Wind Field

The three-dimensional wind field used that obeys the density continuity equation and the boundary condition at ground level is given by
i1520-0426-15-2-360-ea1
with α = 2.5, β = 2α − 1, and x, y, and z are in kilometers. Here, z0 is the altitude of the ground level relative to the sea (equal to 0 in the present case), and x0 and y0 are given by
i1520-0426-15-2-360-ea4
where xmax, xmin, ymax, and ymin are the boundaries of the retrieval domain. Here, Hx and Hy are chosen to be 50 and 150 km, respectively, and Hz is 15 km.

Fig. 1.
Fig. 1.

(a) Horizontal and (b) vertical cross sections of the simulated relative wind field (arrows) at 1 km and along the line (AB) [see location in (a)], respectively. Track 1 represents the relative flight track (relative to the moving reference frame of the squall line) of the airborne Doppler radar and track 2 illustrates the relative flight track of the aircraft that drops sondes. For Fig. 1b, isocontours represent the vertical velocity (isocontour every 0.5 m s−1).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 2.
Fig. 2.

(a) Horizontal and (b) vertical cross sections of the retrieved relative wind field (arrows) without radiosounding (RS) data at order 5 at 1 km and along line (AB) [see location in (a)], respectively. For Fig. 2b, isocontours represent the vertical velocity (isocontour every 0.5 m s−1).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 3.
Fig. 3.

Difference field of (a) u, (b) υ, and (c) w between the retrieved relative wind field without RS data at order 5 and the simulated relative wind field along lines (AB) (see location in Figs. 1a and 2a).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 4.
Fig. 4.

(a) Horizontal and (b) vertical cross sections of the retrieved relative wind field (arrows) with RS data at order 5 at 1 km and along line (AB) [see location in (a)], respectively. For Fig. 4b, isocontours represent the vertical velocity (isocontour every 0.5 m s−1).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 5.
Fig. 5.

Difference field of (a) u, (b) υ, and (c) w between the retrieved relative wind field with RS data at order 5 and the simulated relative wind field along lines (AB) (see location in Figs. 1a and 4a).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 6.
Fig. 6.

CFAD of the retrieved relative wind field for (a) w, (b) υ, and (c) u with the MANDOPAS analysis and of relative simulated wind field for (d) w, (e) υ, and (f) u (contour every 10% of data per kilometer per meter per second).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 7.
Fig. 7.

(a) Skew T–logp diagram and (b) hodograph (m s−1) of the composite RS for 22 February 1993.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 8.
Fig. 8.

(a) Horizontal and (b) vertical cross sections of the retrieved relative wind field (arrows) and reflectivity (contours every 5 dBZ) for the real case with RS data at 1 km and along line (AB) [see location in (a)], respectively.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 9.
Fig. 9.

(a) Horizontal and (b) vertical cross sections of the retrieved relative wind field (arrows) and reflectivity (contours every 5 dBZ) for the real case without RS data at 1 km and along line (AB) [see location in (a)], respectively.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 10.
Fig. 10.

Vertical difference field of (a) u and (b) w between the retrieved relative wind field with RS data and the retrieved relative wind field without RS data along lines (AB) (see location in Figs. 8a and 9a).

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 11.
Fig. 11.

Horizontal difference field at 6-km altitude of (a) u and (b) w between the retrieved relative wind field with RS data and the retrieved relative wind field without RS data.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 12.
Fig. 12.

Difference field of (a) u and (b) υ between the retrieved relative wind field with RS data along line (AB) of Fig. 8a and the RS field alone.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 13.
Fig. 13.

Horizontal cross section of the pressure perturbation field (mb) at (a) 1 km and (b) 4 km.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

Fig. 14.
Fig. 14.

Horizontal cross section of the virtual potential temperature perturbation field (°C) at (a) 1 km and (b) 6 km.

Citation: Journal of Atmospheric and Oceanic Technology 15, 2; 10.1175/1520-0426(1998)015<0360:TDVDAT>2.0.CO;2

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