NOTES AND CORRESPONDENCE. A Formulation of the Continuity Equation of MUSCAT for either Flat or Complex Terrain

Transcription

1 1556 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 17 NOTES AND CORRESPONDENCE A Formulation of the Continuity Equation of MUSCAT for either Flat or Complex Terrain MICHEL CHONG Centre National de Recherches Météorologiques, CNRS and Météo-France, Toulouse, France STÉPHANIE COSMA Laboratoire d Aérologie, CNRS-UPS, Toulouse, France 30 December 1999 and 21 April 2000 ABSTRACT The Mesoscale Alpine Programme (MAP) involved an ensemble of airborne and ground-based Doppler radars dedicated to the observation of precipitating systems over the Alps. The derivation of the three-dimensional wind fields from the multiple-doppler synthesis and continuity adjustment technique (MUSCAT) requires that orography-induced air circulation, in particular when solving the mass continuity equation, is taken into account. A formulation of this equation in its flux form is proposed, which has the advantage of applying to either flat or complex terrain and thus eliminating the need to explicitly evaluate the vertical velocity associated with the slope wind at the surface. Pseudo-Doppler observations of a pair of ground-based radars that were operated during MAP, deduced from a modeled pre-map case, are used to validate the proposed solution and to investigate the performances of the Doppler wind synthesis above mountainous regions. 1. Introduction The Mesoscale Alpine Programme (MAP; Binder et al. 1995) Special Observing Period (SOP, 7 September November 1999) that recently took place over the European Alps has involved an ensemble of airborne and ground-based Doppler radars. One of the main objectives was to derive the structure and evolution of the wind and reflectivity fields associated with orographic precipitating systems, in the context of a better knowledge of the physical processes of rain formation and maintenance over mountainous terrain. The derivation of the wind components from Doppler observations over such regions, however, needs to account for orographically induced air circulation, that is, the influence of slope winds. So far, most of the past studies on precipitating systems were dedicated to flat topography so that this problem was not really examined. In the scope of the MAP SOP Doppler data analysis, Georgis et al. (2000) proposed a variational approach to derive the vertical motions in the presence of complex topography. Their method is based on a first guess of the vertical Corresponding author address: Dr. Michel Chong, CNRM (CNRS and Météo-France), 42 Av. Coriolis, Toulouse Cedex, France. chong@meteo.fr. velocity obtained from the integration of the mass continuity equation, in which the surface boundary condition results from the hypothesis of a terrain-following airflow. The surface slope winds have also been considered by Tabary and Scialom (2000, manuscript submitted to J. Atmos. Oceanic Technol.) in the analytical wind retrieval proposed by Scialom and Lemaître (1990). The present paper is another contribution to the effort of incorporating orography effects in the Doppler data analysis. The integration of the mass continuity equation is implicitly performed in the multiple-doppler synthesis and continuity adjustment technique (MUSCAT) proposed by Bousquet and Chong (1998) for airborne Doppler radar data and extended to ground-based dual- Doppler radar observations by Chong and Bousquet (2000, manuscript submitted to Meteor. Atmos. Phys., hereafter CB). But this equation is discretized according to a flat surface, i.e., with the natural condition of no vertical velocity at the surface. This note proposes a formulation of the mass continuity equation that could be applied to either flat or complex terrain, without changing the basic principles of MUSCAT. Section 2 presents such formulation, while section 3 concerns an application to a modeled flash-flood episode that was investigated by Cosma and Richard (1998) and Stein et al. (2000) during the preparation phase of MAP (hereafter referred to as pre-map) American Meteorological Society

2 NOVEMBER 2000 NOTES AND CORRESPONDENCE Modified discrete form of the continuity equation The basic principles of MUSCAT (Bousquet and Chong 1998) that yield a simultaneous solution of the three wind components (u,, w) from at least a pair of Doppler observations lie in the minimization, in a least squares sense, of the function F: F(u,, w) [A(u,, w) B(u,, w) C(u,, w)] dx dy, (1) S such that F F F 0, 0 and 0. (2) u w The integral of (1) is performed over a horizontal surface S. Here, A, B, and C are expressions that represent the least squares fit of observed Doppler velocities to wind components, the least squares adjustment of the continuity equation, and the filtering of small-scale variations of the wind components, respectively. The Doppler adjustment includes a Cressman distance-dependent weighting function to account for noncollocated data and gridpoint values, permitting the interpolation of the polar coordinate radar data onto the Cartesian grid of interest, in the data fit. Bousquet and Chong (1998) noted that a separate interpolation produces averaging vectors that point in quite different directions and creates a sampling problem at short range. The result is an incorrect estimation of the average Doppler velocity and the associated beam-pointing angle. Incorporating the necessary interpolation into the least squares data adjustment allows the technique to consider all the available observed velocities and their actual orientation without any bias. Low-pass filtering of small-scale horizontal variations of u,, and w, is realized through the minimization of the second-order derivatives. The expression for the adjustment of the continuity equation u 1 w 0, x y z where is the air density, is based on an off-centered finite-difference scheme that is applied to rectangular grid boxes. This is a fundamental point which allowed MUSCAT to be solved in a two-dimensional framework. Indeed, such a differentiating scheme involves only two successive horizontal planes: the current plane for which solution for the wind components is searched and a previously investigated plane, the estimated components of which are used as boundary values for the implicit integration of the continuity equation. The first current plane above the surface involves some assumptions about the wind components at the surface. Horizontal components are assumed constant in the layer while vertical velocity is set to zero at the surface (flat topography conditions). In the presence of orography, neither this vertical velocity condition nor the differentiating scheme could be simply applied because of the terrain-induced effects that lead to nonzero vertical motion and because of the nonrectangular character of the grid boxes along the surface, respectively. An alternative to discretizing the continuity equation is to use the mass flux form or, in other words, to consider the balance of the mass transports throughout all the six faces delimiting an individual grid box (Fig. 1). This form allows the grid boxes to take any shape (i.e., above flat or complex terrain), and it can be expressed as (FU FV FW) 0, (3) where FU, FV, and FW stand for mass transported by u,, and w wind components through the surfaces normal to x, y, and z, respectively. Summation is over the two faces intersected by each wind component. For rectangular grid boxes intersected by a non-flat terrain (Fig. 1), the mass fluxes are estimated through the irregular faces, according to the following considerations. The rigid lid (boundary) condition turns into a zero mass flux through the bottom face, eliminating the requirement of explicitly evaluating the vertical velocity FIG. 1. Representation of a grid box intercepted by the surface topography.

3 1558 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 17 TABLE 1. Scan strategies of Monte Lema and Ronsard radars. Radar Elevation angles ( ) Monte Lema 0.3, 0.5, 1.5, 2.5, 3.5, 4.5, 5.5, 6.5, 7.5, 8.5, 9.5, 11.0, 13.0, 15.5, 18.3, 21.6, 25.3, 29.6, 34.5, 40.0 Ronsard 0.62, 1.23, 1.85, 2.46, 3.08, 3.69, 4.31, 4.92, 5.54, 6.15, 6.77, 7.38, 8.35, 9.67, 11.43, 13.89, 17.75, 24.35, 37.71, Azimuthal resolution ( ) Gate spacing (m) Number of gates Range interval (km) associated with a free-slip tangential velocity over orography. The mass transports through the other sides can be estimated under the following assumptions. Indexing the grid points above the surface as 1, 2, 3, and 4 (see Fig. 1), the mass flux at the top face can be evaluated as FW T (w 1 w 2 w 3 w 4 )L 2 /4, (4) where is the air density, L is the horizontal grid resolution, and the w s are the vertical wind components at the upper grid points. Through the lateral faces, we assume a mean air density ( ) and a linear horizontal variation of the wind components normal to the faces and of the topography. Moreover, we assume no vertical variation of these components down to the surface. This allows us to expand the mass flux in terms of the wind components at the upper grid locations. Thus, the mass flux # u(x) dy dz through the face containing grid points 1 and 3, for example, can be simply written as FU L[(h 1 /3 h 3 /6)u 1 (h 1 /6 h 3 /3)u 3 ], (5) where the minus sign accounts for the negative orientation of the normal to this face with respect to the Cartesian frame, h is the vertical distance between the grid point and the surface, and u is the normal horizontal wind component. Similar expressions can be formulated for the other lateral grid box boundaries, involving either u or components. Finally, the continuity adjustment term in (1) consists of the squared sum of all the mass fluxes described by (4) and (5), that is, the minimization of Eq. (3). When components u,, and w are assumed to be constant within a grid box, one can verify that the mass budget yields the expression of a free-slip tangential velocity, that is, hs hs w u (6) x y where h s denotes here the surface height. Compared to an explicit formulation of the continuity equation, the flux form offers more flexibilty. Using the explicit boundary condition (6) would require a careful discretization of that equation for grid boxes intercepted by the terrain, although both approaches would lead to similar results. 3. Application a. Data sampling and analysis domain In the following, numerically simulated Doppler observations (radial velocity and reflectivity) are used to test the flux form continuity equation of MUSCAT. We consider two ground-based Doppler radars involved in the MAP experiment: the Swiss Monte Lema operational radar located at 1625-m altitude, N, E (northern east side of Lago Maggiore), and the French Ronsard radar at 155-m altitude, N, E( 60 km to the west of Milano, Italy), both operating a scan strategy consisting of twenty 360 azimuth sweeps. Table 1 gives the corresponding elevation angles and the data sampling resolution used in this study. Most of the sweeps are concentrated to low-elevation angles. The pseudo-doppler observations are derived from the numerical simulations of a flash-flood episode that occurred on September 1993 over the Brig (Switzerland) region to the northwest of Lago Maggiore in northern Italy (Cosma and Richard 1998; Stein et al. 2000). For simplicity, we considered the model outputs at a given time so that no temporal variation was introduced in the sampled radar data, which could derive, for example, from advection effects during the scanning time. A linear distance interpolation of the model grid values surrounding each observation point is used. The Brig case simulation was performed with the mesoscale nonhydrostatic (Meso-NH) atmospheric simulation system, jointly developed by Centre National de Recherches Météorologiques and Laboratoire d Aérologie (for details, see Lafore et al. 1998) over a domain of 230 km 220 km centered on the Lago Maggiore region. A warm microphysical scheme (Kessler 1969) was deployed, and raindrops were assumed to follow the Marshall and Palmer (1948) distribution. This distribution was used to compute the equivalent radar reflectivity and the terminal fall velocity of precipitating particles (according to Liu and Orville 1969) that contributes to the Doppler velocity. Within the radar ranges of km (Table 1), the model grid resolution was km in the horizontal, while it was variable in the vertical, ranging from 60 m near the surface to 660 m at 10 km above the surface over 31 levels. The model Cartesian coordinate system was oriented at 4 clockwise from the north. To facilitate the comparison between the MUSCAT-derived and the

4 NOVEMBER 2000 NOTES AND CORRESPONDENCE 1559 FIG. 2. Terrain heights (km) associated with the wind synthesis domain of 118 km 118 km, and deduced from a digital terrain map. Grid resolution is km. The y axis of the Cartesian frame is at 4 clockwise from the north. Origin (0, 0) corresponds to N, 8.79 E. Also is reported the baseline connecting the Ronsard (to the south) and Monte Lema (to the north) radars (full circles), to the eastern side of Lago Maggiore (shaded zone). Shaded square indicates the position of Brig. modeled wind fields (reference fields), the 3D wind synthesis was performed within a domain of 118 km 118 km 10 km, with a grid resolution of km in the horizontal and 0.5 km in the vertical, the horizontal grid points matching those of the model in this domain of interest which is located to the southeast of Brig. The origin of the coordinate system was at N and 8.79 E. Figure 2 shows the surface terrain map associated with the so-specified domain. It indicates the complexity of the terrain in the proximity of the two radars, with sharp topography mostly to the northwest of the radar baseline. Terrain heights range from 70 to 4200 m. b. Results The horizontal and vertical influence radii of the Cressman weighting function used in the MUSCAT formalism (1) were set to respectively 2.5 and 1.2 km, while the low-pass filter was such that the cutoff wavelength was four times the grid resolution ( 7.5 km). Because of this filtering process, the Meso-NH parameters should be filtered before performing any comparison with the MUSCAT-derived fields. A Leise filter (Leise 1981) was applied at each level to these parameters, which could provide comparable reflectivity fields with the MUSCAT-derived ones, as will be seen hereafter. Figures 3a and 3c show the horizontal flow at 5-km altitude along with the reflectivity pattern, and the associated vertical velocity structure, respectively, as deduced from the application of MUSCAT, while Figs. 3b and 3d represent the corresponding reference fields as simulated by Meso-NH. The similar reflectivity patterns in Figs. 3a,b suggest that the procedure to filter out the small-scale features of the model outputs is consistent with the filtering used in MUSCAT. In most respects, the Doppler-retrieved horizontal flow duplicates the modeled flow in intensity as well as orientation. In particular, details of the wind intensification within the precipitation cores over the Alpine chain are well reproduced. It should also be noted that regions close to the radar baseline, where traditional dual-doppler analysis fails, are resolved without any bias as discussed in CB. Vertical motions (Figs. 3c,d) associated with the heavy precipitation ( 35 dbz) that occurred over either quasiflat (to the southeast) or complex topography regions (to the north-northwest) are quite comparable, although the continuity equation was solved in MUSCAT in a least squares sense with the hypothesis used to extrapolate the surface horizontal winds. This could explain the more variable retrieved vertical velocity. An alternative to reduce the apparent differences would be to use the variational integration method that Georgis et al. (2000) developed for complex terrain, subsequent to the MUSCAT processing. The combination of MUS- CAT with this procedure was effectively used in the real-time processing that was operated during MAP, the description of which is detailed in Chong et al. (2000). An implicit condition is that the horizontal flow and its associated divergence should be as accurate as possible. To analyze further the accuracy of the MUSCAT-derived wind components, Figs. 4 and 5 present two vertical cross sections of the flow and reflectivity structure and of the associated patterns of the u,, and w components, along respectively AA (west east section) and BB (south north section), as reported in Fig. 3a. These sections concern the main precipitation cores which were substantially enhanced by orographic effects. The west east cross section (Fig. 4) was associated with three cores occurring in the upwind slopes of the mountains at x 56, 28, and 22 km, respectively. The central core was the most intense, and was revisited in the south-north cross section (y 6 km in Fig. 5). Figure 5 also shows that the orographic influence exerted well upstream, in regions of relatively flat topography to the south of the main core, resulting in the observation of an upstream cell at y 38 km. The retrieved flow structure (wind vectors) in both vertical sections (Figs. 4a and 5a) strongly resembles the modeled flow (Figs. 4b and 5b), with a quite correct positioning of the updraft cores as revealed by the comparison of Figs. 4g and 5g with Figs. 4h and 5h, respectively. The globally similar patterns of vertical velocity suggest that the approximate flux form of the continuity equation in MUSCAT properly takes into account the orography-induced air circulation. More significant is the resemblance between the horizontal com-

5 1560 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 17 FIG. 3. Horizontal cross section of the flow structure at 5-km altitude, as deduced from MUSCAT (left panels) and numerical modeling (right panels): (a) (b) horizontal wind vectors and reflectivity (dbz) pattern; (c) (d) vertical velocity (m s 1 ) pattern. Every other wind vector is plotted. Solid lines AA and BB in (a) indicate the positions where vertical cross sections are performed, at y 3.7 km and x 27.6 km, respectively. ponents deduced from MUSCAT (Figs. 4c, 4e, 5c, 5e) and the model (Figs. 4d, 4f, 5d, 5f), which were marked by strong shear zones. These figures clearly show the performances of MUSCAT in providing highly reliable horizontal wind components. An overall comparison can be summarized in Fig. 6, which shows the statistical distributions of the wind components and divergence from MUSCAT (Figs. 6a,c,e,g) and Meso-NH (Figs. 6b,d,f,h) with height, using the CFAD (contoured frequency by altitude diagram; Yuter and Houze 1995) representation. The number of common valuable points at each level was relatively stable, between 3200 and 3800 values at 2.5 z 8.0 km, and reaching 1800 values at 1-km altitude and 2200 values in the other levels. Figure 6 indicates that the statistical distributions are quite comparable. The CFADs of u,, w,and divergence are very similar in shape, value and location of the maximum occurrence, and contour spacing, although differences exist. Differences for u and are at the highest levels ( 7.5 km),

6 NOVEMBER 2000 NOTES AND CORRESPONDENCE 1561 FIG. 4. Vertical cross section of the flow structure along AA in Fig. 3a, as deduced from MUSCAT (left panels) and numerical modeling (right panels): (a) (b) in-plane wind vectors and reflectivity (dbz) pattern; (c) (d) nearly west east u-component (m s 1 ) pattern; (e) (f) nearly south north -component (m s 1 ) pattern; (g) (h) vertical velocity (m s 1 ) pattern. Every other wind vector is plotted in the horizontal. Terrain profile is also shown. resulting from the lower data sampling resolution from one of the two radars according to the relative position of the grid points. More sensible differences concern the vertical velocity. MUSCAT provides a broader distribution, probably due to the approximations discussed above. However, it is very satisfactory to observe that, except the light broadening, there was no significant deviation in the distribution shape that could be interpreted as an error amplification in resolving the continuity equation. In this case study, such an amplification would result in an increasing broadening with height of the w distribution that would make the vertically oriented contours to tilt to either left or right in a V-shape pattern. Finally, Fig. 6 ensures the validity of MUSCAT and its flux form continuity equation to synthesize wind fields over either flat or complex terrrain.

7 1562 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 17 FIG. 5. As in Fig. 4, but for the cross section along BB in Fig. 3a. 4. Conclusions This paper has examined the performances of the multiple-doppler synthesis and continuity adjustment technique (MUSCAT) to provide 3D wind fields from Doppler radar data collected above mountainous regions. Some accommodation of the mass continuity equation, initially discretized in its derivative form for a flat surface, has been necessary to take into account the influence of the orography on the air circulation, namely, the slope winds that result in a nonzero vertical velocity at the surface. A flux form formulation has been proposed that has the capability of operating in either flat or complex terrain and avoids an explicit evaluation of the vertical velocity at the surface without changing the basic principles of MUSCAT. Numerically simulated Doppler radar observations have been used to test this flux formalism of the continuity equation in MUSCAT, by considering modeling results of a pre-map flash flood event. Two groundbased Doppler radars in the recent MAP SOP observing

8 NOVEMBER 2000 NOTES AND CORRESPONDENCE 1563 FIG. 6. Contoured frequency by altitude diagram (CFAD) of (a) (b) u, (c) (d), (e) (f) w, and (g) (h) horizontal divergence, as deduced from MUSCAT (left panels) and numerical modeling (right panels). Isolines represent the frequency (percentage) of observations, at each level, within bins sized by the variable interval (indicated on the top of each panel) and the vertical grid resolution. The u,, and w frequency distributions are contoured at intervals of 2% 2%, and 10% m 1 skm 1, respectively. Contour interval of 20% m 1 s is used for the divergence distribution. Only existing values from both MUSCAT and model are considered.

9 1564 JOURNAL OF ATMOSPHERIC AND OCEANIC TECHNOLOGY VOLUME 17 network have been selected to provide the simulated data sets. These radars were dedicated to the observation of precipitation over the Lago Maggiore region. The comparison of the retrieved wind field with the model reference field has revealed the effectiveness of MUS- CAT in overcoming the problem of orographically induced air circulation and in providing accurate flow description above mountainous regions. Finally, MUS- CAT s successful performance with synthetic data over the Lago Maggiore region is of significant interest for the postmortem analyses of the MAP Doppler radar data intensively collected during the fall of Acknowledgments. This study has been supported by the INSU/PATOM (Programme AtmosphèreetOcéan à Moyenne Echelle) Grant 98/03. REFERENCES Binder, P., and Coauthors, 1995: Mesoscale Alpine Programme: Design proposal. 65 pp. [Available from MAP Data Centre, ETH, Zürich, Switzerland.] Bousquet, O., and M. Chong, 1998: A multiple-doppler synthesis and continuity adjustment technique (MUSCAT) to recover wind components from Doppler radar measurements. J. Atmos. Oceanic Technol., 15, Chong, M., and Coauthors, 2000: Real-time wind synthesis from Doppler radar observations during the Mesoscale Alpine Programme. Bull. Amer. Meteor. Soc., 81, Cosma, S., and E. Richard, 1998: Simulations numériques de l épisode de précipitations intenses de Brig. Preprints, Atelier de Modélisation de l Atmosphère, Toulouse, France, Météo- France, Georgis, J.-F., F. Roux, and P. H. Hildebrand, 2000: Observation of precipitating systems over complex orography with meteorological Doppler radars: A feasibility study. Meteor. Atmos. Phys., 72, Kessler, E., 1969: On the Distribution and Continuity of Water Substance in Atmospheric Circulation. Meteor. Monogr., No. 32, Amer. Meteor. Soc., 84 pp. Lafore, J.-P., and Coauthors, 1998: The Meso-NH atmospheric simulation system. Part I: Adiabatic formulation and control simulations. Ann. Geophys., 16, Leise, J. A., 1981: A multidimensional scale-telescoped filter and data extension package. NOAA Tech. Memo. ERL WPL-82, 18 pp. [NTIS PB ] Liu, J. Y., and H. D. Orville, 1969: Numerical modeling of precipitation and cloud shallow effects on mountain-induced cumuli. J. Atmos. Sci., 26, Marshall, J. S., and W. Mck. Palmer, 1948: The distribution of raindrops with size. J. Meteor., 5, Scialom, G., and Y. Lemaître, 1990: A new analysis for the retrieval of three-dimensional mesoscale wind fields from multiple Doppler radar. J. Atmos. Oceanic Technol., 7, Stein, J., E. Richard, J.-P. Lafore, J.-P. Pinty, N. Asencio, and S. Cosma, 2000: High-resolution non-hydrostatic simulations of flash-flood episodes with grid-nesting and ice-phase parameterization. Meteor. Atmos. Phys., 72, Tabary, P., and G. Scialom, 2000: MANDOP analysis over complex orography in the context of the MAP experiment. J. Atmos. Oceanic Technol., submitted. Yuter, S. A., and R. A. Houze Jr., 1995: Three-dimensional kinematic and microphysical evolution of Florida cumulonimbus. Part II: Frequency distributions of vertical velocity, reflectivity, and differential reflectivity. Mon. Wea. Rev., 123,