• Ball, F. K. 1960. Control of inversion height by surface heating. Quart. J. Roy. Meteor. Soc 86:483494.

  • Betts, A. K. 1973. Non-precipitating cumulus convection and its parameterization. Quart. J. Roy. Meteor. Soc 99:178196.

  • Betts, A. K. 1974. Reply to comment on the paper “Non-precipitating cumulus convection and its parameterization.”. Quart. J. Roy. Meteor. Soc 100:469471.

    • Search Google Scholar
    • Export Citation
  • Boers, R. and E. W. Eloranta. 1986. Lidar measurements of the atmospheric entrainment zone and the potential temperature jump across the top of the mixed layer. Bound.-Layer Meteor 34:357375.

    • Search Google Scholar
    • Export Citation
  • Carson, D. J. 1973. The development of a dry inversion-capped convectively unstable boundary layer. Quart. J. Roy. Meteor. Soc 99:450467.

    • Search Google Scholar
    • Export Citation
  • Carson, D. J. and F. B. Smith. 1974. Thermodynamic model for the development of a convectively unstable boundary layer. Advances in Geophysics Vol. 18A, Academic Press, 111–124.

    • Search Google Scholar
    • Export Citation
  • Cohn, S. A. and W. M. Angevine. 2000. Boundary layer height and entrainment zone thickness measured by lidars and wind-profiling radars. J. Appl. Meteor 39:12331247.

    • Search Google Scholar
    • Export Citation
  • Davis, K. J., N. Gamage, C. R. Hagelberg, C. Kiemle, D. H. Lenshow, and P. P. Sullivan. 2000. An objective method for deriving atmospheric structure from airborne lidar observations. J. Atmos. Oceanic Technol 17:14551468.

    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W. 1979. Prediction of convective mixed-layer entrainment for realistic capping inversion structure. J. Atmos. Sci 36:424436.

    • Search Google Scholar
    • Export Citation
  • de Franceschi, M., G. Rampanelli, D. Sguerso, D. Zardi, and P. Zatelli. 2003. Developement of a measurement platform on a light airplane and analysis of airborne measurements in the atmospheric boundary layer. Ann. Geophys 46:115.

    • Search Google Scholar
    • Export Citation
  • Driedonks, A. G. M. 1982. Models and observations of the growth of the atmospheric boundary layer. Bound.-Layer Meteor 23:283306.

  • Fedorovich, E. E. and D. V. Mironov. 1995. A model for shear-free convective boundary layer with parameterized capping inversion structure. J. Atmos. Sci 52:8395.

    • Search Google Scholar
    • Export Citation
  • Fitzjarrald, D. R. and M. Garstang. 1981. Vertical structure of the tropical boundary layer. Mon. Wea. Rev 109:15121526.

  • Garratt, J. R. 1992. The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.

  • Gryning, S. E., F. Beyrich, and E. Batchvarova. Eds.,. 1997. The determination of the mixing height—Current progress and problems. EURASAP Workshop Proc., Roskilde, Denmark, Risø National Laboratories, 157 pp.

  • Johansson, C., A. S. Smedman, and U. Högström. 2000. Critical test of the validity of Monin–Obukhov similarity during convective conditions. J. Atmos. Sci 58:15491566.

    • Search Google Scholar
    • Export Citation
  • Khanna, S. and J. G. Brasseur. 1997. Analysis of Monin–Obukhov similarity from large-eddy simulation. J. Fluid Mech 345:251286.

  • Rampanelli, G. and D. Zardi. 2000. Analysis of airborne data and identification of thermal structures with geostatistical techniques. Preprints, 14th Symp. on Boundary Layers and Turbulence, Aspen, CO, Amer. Meteor. Soc., 239–241.

    • Search Google Scholar
    • Export Citation
  • Rampanelli, G. and D. Zardi. 2002. Identification of thermal structure from airborne measurements in an Alpine valley with kriging technique. Preprints, 10th Conf. on Mountain Meteorology, Park City, UT, Amer. Meteor. Soc., 1.20–1.24.

    • Search Google Scholar
    • Export Citation
  • Seibert, P., F. Beyrich, S-E. Gryning, S. Joffre, A. Rasmussen, and P. Tercier. 2000. Review and intercomparison of operational methods for the determination of the mixing height. Atmos. Environ 34:10011027.

    • Search Google Scholar
    • Export Citation
  • Steyn, D. G., M. Baldi, and R. M. Hoff. 1999. The detection of mixed layer depth and entrainment zone thickness from lidar backscatter profiles. J. Atmos. Oceanic Technol 16:953959.

    • Search Google Scholar
    • Export Citation
  • Stull, R. B. 1988. An Introduction to Boundary Layer Meteorology. Kluwer Academic, 670 pp.

  • Sullivan, P. P., C-H. Moeng, B. Stevens, D. H. Lenschow, and S. D. Mayor. 1998. Structure of the entrainment zone capping the convective atmospheric boundary layer. J. Atmos. Sci 55:30423064.

    • Search Google Scholar
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  • Tennekes, H. 1973. A model for the dynamics of the inversion above a convective boundary layer. J. Atmos. Sci 30:558567.

  • Vogelezang, D. H. P. and A. A. M. Holtslag. 1996. Evaluation and model impacts of alternative boundary-layer height formulations. Bound.-Layer Meteor 81:245269.

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  • Whiteman, C. D. 1990. Observations of thermally developed wind sytems in mountainous terrain. Atmospheric Processes over Complex Terrain, Meteor. Monogr., No. 23, Amer. Meteor. Soc., 5–42.

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  • Whiteman, C. D., T. B. McKee, and J. C. Doran. 1996. Boundary layer evolution within a canyonland basin. Part I: Mass, heat, and moisture budgets from observations. J. Appl. Meteor 35:21452161.

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  • Zilitinkevich, S. S. 1975. Comments on “A model for the dynamics of the inversion above a convective boundary layer.”. J. Atmos. Sci 32:991992.

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  • View in gallery

    Sketch of a vertical profile of potential temperature in the convective boundary layer developing over flat uniform terrain [adapted from Garratt (1992); notation for h0 and h2 follows Deardorff (1979)]

  • View in gallery

    (a) The smoothed vertical profile of potential temperature and the symbols used in the present paper, associated with (b) the vertical structure of the sensible heat flux

  • View in gallery

    Sketch of the basic functions f and g used in (2) to obtain the vertical profile

  • View in gallery

    Summary of criteria for the identification of standard meteorological parameters: (a) shaded areas are equal as a consequence of energy budget (Driedonks 1982), (b) linear potential temperature profile displaying the same lapse rate as the gradient of the real profile at the inflection point, and (c) evaluation of heights at which the best-fit profile is close enough either to the ML constant value θm (h0) or to the FA (h2)

  • View in gallery

    Sketch of the case of pure encroachment and its interpretation with (17) and (19)

  • View in gallery

    An example of the analysis of airborne data using the method outlined in the paper. Data were collected in the Adige valley, near the village of Besenello, south of the city of Trento (Italy) during the measurement flights. (a) between 0930 and 0946 LST 1 Oct 1999 and (b) between 0935 and 0948 LST 26 May 1999

  • View in gallery

    An example of the analysis of airborne data using the method outlined in the paper. Data were collected in the Adige valley, near the city of Trento (Italy), during the measurement flights (a) between 1023 and 1039 LST 2 Jul 1997 and (b) between 1039 and 1104 LST 22 Dec 2000

  • View in gallery

    An example of the analysis of airborne data using the method outlined in the paper. Data were collected in the upper Lakes Valley, near the village of Terlago, west of the city of Trento (Italy), during the measurement flight (a) between 1248 and 1304 LST 9 Sep 1998 and (b) between 1420 and 1452 LST 23 Sep 2001

  • View in gallery

    An example of the analysis of airborne data using the method outlined in the paper. Data were collected during the measurement flight (a) between 1217 and 1229 LST 24 Oct 1998 near the town of Riva del Garda, in the lower Lakes Valley, about 30 km southwest of the city of Trento (Italy), and during the measurement flight (b) between 0932 and 0958 LST 8 Jul 1999 in the Adige valley, near the city of Bolzano (Italy)

  • View in gallery

    An example of the analysis of airborne data using the method of the coupled fit using the vertical profiles of both θ and q. Vertical profiles of θ are shown: data (black dots), interpretation using single fit to the θ data (thin line), and interpretation using θq coupled fit (thick line). Data were collected during the measurement flight of (a) Fig. 6a, and during the measurement flight of (b) Fig. 8b

  • View in gallery

    An example of the analysis of airborne data using the method of the coupled fit using the vertical profiles of both θ and q. Vertical profiles of q are shown: data (black dots), interpretation using single fit to the q data (thin line), and interpretation using θq coupled fit (thick line). Data were collected during the measurement flight of (a) Fig. 6a, and during the measurement flight of (b) Fig. 8b

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A Method to Determine the Capping Inversion of the Convective Boundary Layer

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  • 1 Dipartimento di Ingegneria Civile ed Ambientale, Università degli studi di Trento, Trento, Italy
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Abstract

A simple mathematical algorithm is proposed to decipher the thermal structure of the convective boundary layer by means of best-fit analysis of soundings or airborne measurements with a smooth ideal profile. The latter includes a constant-potential-temperature mixed layer, a strongly stratified entrainment layer, and a constant-lapse-rate free atmosphere. The resulting profile depends on five parameters amenable, through simple mathematical relationships, to physical variables defining the vertical structure of the layers. The method allows objective evaluation of parameters involved in the test profile and easy comparison of measurements with theoretically expected structure.

Corresponding author address: Dino Zardi, Università degli studi di Trento, Dipartimento di Ingegneria Civile ed Ambientale, via Mesiano 77, I-38050 Trento, Italy. dino.zardi@ing.unitn.it

Abstract

A simple mathematical algorithm is proposed to decipher the thermal structure of the convective boundary layer by means of best-fit analysis of soundings or airborne measurements with a smooth ideal profile. The latter includes a constant-potential-temperature mixed layer, a strongly stratified entrainment layer, and a constant-lapse-rate free atmosphere. The resulting profile depends on five parameters amenable, through simple mathematical relationships, to physical variables defining the vertical structure of the layers. The method allows objective evaluation of parameters involved in the test profile and easy comparison of measurements with theoretically expected structure.

Corresponding author address: Dino Zardi, Università degli studi di Trento, Dipartimento di Ingegneria Civile ed Ambientale, via Mesiano 77, I-38050 Trento, Italy. dino.zardi@ing.unitn.it

Introduction

Various simplified schemes have been proposed in the literature to model the vertical profile of potential temperature in the situation illustrated in Fig. 1. The latter reproduces the thermal structure of a convective boundary layer (CBL): the mixed layer (ML) below is connected to the stably stratified free atmosphere (FA) above through the entrainment layer (EL) or “interfacial layer” (Deardorff 1979). Key parameters to describe this structure are the potential temperature in the ML θm, inversion height h0, entrainment-layer depth Δh, inversion strength Δθ, and lapse rate in the free atmosphere γ.

Previous models (e.g., Deardorff 1979) capture the essential overall dynamics and diurnal evolution of the CBL without explicit reference to turbulence at smaller scales. In particular, the EL is modeled either as a layer displaying constant stability with rapid temperature variation between the ML and the FA (“first-order jump;” cf. Betts 1974; Deardorff 1979), or as a sharp temperature discontinuity (“zero-order jump;” cf. Ball 1960; Tennekes 1973; Betts 1973; Carson 1973; Carson and Smith 1974; Driedonks 1982). This approach allowed for a more comprehensive investigation of the various processes governing the diurnal evolution of an inversion-capped CBL (Zilitinkevich 1975; Tennekes 1973; Deardorff 1979; Fedorovich and Mironov 1985).

However, these theoretical results are not easily compared with either high-resolution model output (Khanna and Brasseur 1997) or field observations (Boers and Eloranta 1986; Cohn and Angevine 2000). A method for making such a comparison has been recently proposed using airborne lidar data (Davis et al. 2000). Likewise four different criteria to identify the CBL structure from large-eddy simulation models have been suggested by Sullivan et al. (1998).

On the other hand, making a precise estimate of many quantities, such as boundary layer depth (Vogelezang and Holtslag 1996; Gryning et al. 1997), is a crucial step not only for applications (e.g., estimate of mixing height for pollutant transport) but also for evaluating theoretical similarity solutions as these quantities enter as scaling variables (Stull 1988; Johansson et al. 2000).

In order to describe the vertical structure of a CBL from a sounding or model output in terms of the conceptual model sketched in Fig. 1, a best-fit analysis with a simple test vertical profile (such as the curve in Fig. 1) may seem like a reasonable procedure. However, the usual least squares approach, using a piecewise constant test curve where both heights (hs, h0, h2) and thermal structure parameters (Δθ, γ) are unknown quantities, does not lead to an analytical solution, but rather it requires an iterative search. It can be shown that in many cases multiple combinations of the parameters may get close to a minimum scatter of data around the test profile, but the solution producing the absolute minimum may not provide the most reasonable result from a physical viewpoint. On the contrary, a smooth test curve displaying regular merging of each layer into the adjacent ones, not only allows for a partly analytical solution of the minimization problem, but is also more likely to avoid ambiguity. Furthermore, as remarked by Deardorff (1979), a piecewise linear profile is a very rough approximation of the smooth transition from each layer to those adjacent to it, because it leaves out various finescale features. A similar strategy has been followed by Fitzjarrald and Garstang (1981) and more recently by Steyn et al. (1999) for the analysis of lidar backscatter profiles.

In the present paper the smooth-test-curve approach is adopted (section 2) and relationships between mathematical parameters defining the curve and the physical variables defining the CBL are determined. Tests with real data are provided (section 3), along with a discussion of the results and possible refinements of the method (section 4).

Outline of the method

Following Deardorff (1979), the lowest part of a horizontally homogeneous CBL (apart from the surface layer region) can be represented as a layer displaying a constant potential temperature θm up to a height h0, then a rapidly increasing profile in the EL and asymptotically approaching, above a height h2, the free atmosphere,
θ0zθ00γz,
where θ00 is a constant value of potential temperature at reference height (z = 0) and γ is the constant vertical gradient. The height h1 is where the vertical heat flux profile displays a minimum (Fig. 2). Such a structure may be easily reproduced by means of a linear combination of functions in the form
θzθmafηbgη
where
i1520-0450-43-6-925-e3
is the vertical displacement from a reference height l (to be related later to the CBL height) scaled with the EL depth Δh and c is a constant (to be specified later). The structure of f and g is sketched in Fig. 3: the function f provides the rapid transition from the constant value θm, as η ≪ 0, to a value of θ at the top of the EL, matching the FA above, while the function g provides the correct asymptotic behavior in accordance with (1) as η ≫ 0.
Accordingly, the following constraints have to be satisfied:
i1520-0450-43-6-925-e4
As will be clear later, the limit value of 1 in (5) and (6) is chosen for convenience, but any constant value would be acceptable. Asymptotic matching with the free atmosphere requires, using (1), (5), and (6),
i1520-0450-43-6-925-e7
which implies
bh.
Among the many possible analytic functions satisfying the above requirements, consider the following expressions:
i1520-0450-43-6-925-e9a
Equations (1), (2), (9a), and (9b), along with the required asymptotic behavior of θ(z) in the FA, imply
θ00θmaγl.
To link the parameters a, b, θm, l, and Δh to all of the physical variables identifying the atmospheric-layer structure as shown in Fig. 1, some criteria have to be chosen to compare the smooth profile of (2) with the usual conceptual schemes (Fig. 2). A first criterion based on the energy budget (Driedonks 1982) requires that the shaded areas in Fig. 4a be equal. A second criterion is based on the averaged stability in the EL: accordingly, the best approximation is a linear potential temperature profile displaying a lapse rate equal to the gradient in the real profile at the inflection point (Fig. 4b). A third criterion is based on evaluation of the heights at which the best-fit profile is close enough either to the ML constant value θm (h0), or to the FA (h2) (Fig. 4c). In the following, the latter criterion will be adopted.
The upper (h2) and lower (h0) limit of the EL can be calculated by imposing that the value of θ at that height must be very close to the asymptotic value, namely,
i1520-0450-43-6-925-e11a
This can be obtained by setting
i1520-0450-43-6-925-e12a
where ξ is a parameter related to the depth of the entrainment layer, setting the accuracy within which (11a) and (11b) are satisfied. For large values of ξ, the differences between θ(h2) and the asymptotic expression in the mixed layer and in the free atmosphere are very small, but this is verified only at heights very high in the FA or very close to the ground level. On the other hand, low values of ξ lead one to estimate a very thin EL, with Δh vanishing as ξ approaches 0. On the basis of many applications of expression (2) to real data, a value of ξ = 1.5 is recommended as a good compromise. This leads to a very strict fulfillment of (11a) and (11b). The actual depth of the EL can be calculated from (12a) and (12b) as
hh2h0ξch,
which implies c = 1/(2ξ) = 1/3.
The relationships between the parameters a, b, and l in (2) and (3), and physical quantities resulting from the above reasoning, are summarized in the following equations:
i1520-0450-43-6-925-e14
The evaluation of the inversion strength requires some explanation, because it has been variously defined in the literature. The potential temperature jump (Δθ) across the EL is
θθ00γh2θmab.
Betts (1974) suggests for the inversion strength the expression
θθ00γh1θm
assuming h1 to be both the height where the mixed layer is upper bounded by the EL (h0 in the present paper), and the height where turbulent heat flux displays a minimum. In fact, Deardorff (1979) shows that in general the two heights are different. Accordingly in the following, h0 will denote the height of the inversion base, whereas h1 will denote the heat flux minimum height (Fig. 2).
Straightforward application of (18) in the present case—in which a smooth temperature profile is adopted, using h0 instead of h1 (as stated above)—leads to
θθ00γh0θmab.
However, (19) would produce unphysical negative values for Δθ′ (see Fig. 5) in the limiting case of encroachment (cf. Garratt 1992, p. 151; Stull 1988, p. 454). According to Deardorff (1979), h1 is roughly halfway between h0 and h2. Thus, an optimal compromise should be reached by estimating h1 = (h0 + h2)/2 = l, and consequently
θθ00γh1θma.

Thus, the limiting case of encroachment (i.e., Δθ/γh1 ≈ 0; cf. Deardorff 1979) is recovered for vanishing a, when the only contribution of the curve g(η) in (2) survives (see Figs. 3 and 5). The case of pure encroachment will occur as a = 0, as a limiting case that is rarely met in real data. For this reason a value of a = 0.2 may be suggested as an upper limit for an encroachment condition.

The estimate of the parameters in (2) can be obtained by the usual least squares fit to the sounding data, requiring the functional
i1520-0450-43-6-925-e21
to be a minimum; here, θi is the value measured at height ηi = (zil)/cΔh, θ(ηi) is the value of potential temperature calculated from (2) at the same height, and N is the total number of data points. In particular, by minimizing S the optimal values of a, b, and θm can be related analytically to the values of l and Δh (see the appendix).

Application

The method presented above has been applied to data collected through airborne measurements in the atmospheric boudary layer, allowing for a very easy determination of the parameters identifying the atmospheric thermal structure at the sites of interest. Data have been collected in alpine valleys using an equipped motor glider during various measurement flights in the area surrounding the city of Trento in the Alps (northern Italy). Further details on the instruments and the measurements can be found in de Franceschi et al. (2003).

The vertically spiraling path performed by the motor glider produced an essentially vertical sounding. A few major deviations from a standard vertical sounding are related to horizontal displacements from a strictly vertical ascent and to the occurrence of significant cross-valley temperature gradients, in connection with different sidewall exposure to incoming solar radiation and related thermally driven flows (cf. Whiteman 1990). In spite of these drawbacks the method allows for efficient retrieval of a horizontally averaged basic vertical structure and provides the first step for subsequent analysis of local cross-valley perturbations, as shown in Rampanelli and Zardi (2000, 2002).

In Figs. 6–9 vertical profiles of potential temperature data from diurnal survey flights are shown. Figures 6, 7, and 8 display examples of strong and deep entrainment layers capping a shallow mixed layer. These are consistent with what is usually found in deep mountain valleys. A detailed analysis of physical mechanisms governing the diurnal evolution of thermal structure within valleys can be found in Whiteman (1990) and Whiteman et al. (1996). Figures 9a,b show how the method performs under “worst” cases, such as (a) “anomalous” CBL development (the so-called encroachment) and (b) a ground-based inversion. Both cases are relatively well captured using the proposed method. In fact, for Fig. 9a the algorithm produces a very small inversion strength (Δθ′ = −0.09 K), identifying this case as an encroachment, and the transition zone between h0 and h2 is well localized and meaningful. In Fig. 9b the flight was performed in the early morning: the height h0 turns out to be less than the height of the lowest measurement point of the profile, and h2 is only about 100 m higher. This suggests that the overall profile is characterized by a single layer, associated with a typical stable condition of a ground-based inversion. The values of the parameters h0, h2, Δh, θm, θ00, γ, Δθ, and Δθ′ obtained from the analysis of these data are reported in Tables 1 and 2. In the same tables the correlation coefficient of the best fit is also shown. The general mean vertical structure provided by the dataset is well captured by the resulting continuous profile and the values of parameters defining the vertical structure appear to be in a reasonable range.

Whenever vertical profiles of other variables, such as water vapor content q, are measured simultaneously with the thermal profile, a similar approach can be followed to design a shape function, such as (2), for q. This function, specifically shaped for the water vapor content structure, can be used to fit the q data and recover the stratification parameters that can be inferred from its profile. A separate fit for each profile (θ and q) is likely to produce different estimates of the same parameters, such as h0 and h2. To overcome this problem and to strengthen the estimate of the parameters, a simultaneous fit is recommended, possibly by minimizing the sum of the two functionals that calculate the distance of the theoretical profiles to the data. Obviously, each of these functions has to be normalized in order that there be an equal effect of the information associated with each profile (see the appendix). A comparison between the method of the coupled fit and the method of the single fit is shown in Figs. 10 and 11, and the results are given in detail in Tables 3 and 4. Notice that the differences between the results produced by the two procedures are very small. However, the coupled method uses a larger number of data points to estimate the inversion parameters l and Δh, and this obviously produces a more stable and reliable result. For this reason, when a coupled series of measurements (θ and q) are available, the application of the coupled procedure is recommended.

Summary

A new technique to obtain the vertical structure of potential temperature from data collected within and above a CBL using light airplanes or vertical soundings has been introduced. The technique consists of a least squares fitting of data to a user-defined analytical expression. The adjustable parameters of this expression are amenable to the atmospheric variables generally used for the description of the stratification (i.e., inversion height, entrainment depth, mixed layer potential temperature, and free-atmosphere lapse rate).

Application of the technique to real data produces encouraging results. Furthermore, the conditions used for obtaining the suggested expression of the vertical profile are very general, and can be adopted to calculate other expressions for f(η) and g(η) as well.

As a final comment, we note that the suggested method assumes the database to be reasonably amenable to the basic structure of a capping inversion because it is most commonly found. More complicated structures, such as multiple inversions, could be only poorly reproduced by the method. Possible application of the proposed profile in simplified models for the diurnal evolution of the mixing height (Seibert et al. 2000) could improve the method introducing a more realistic capping inversion structure. An integration with similar methods for the identification of the CBL upper structure from other kinds of measurements (lidar, wind profilers, sodar, etc.) is possible, in order to obtain a more accurate knowledge of the physical variables.

Acknowledgments

Special thanks go to Massimiliano de Franceschi, for providing the experimental setup of the motor glider and technical support for the measurements, and to the motor glider pilots, Andrea Ferrari and Fabrizio Interlandi. This work has been partly supported by the Provincia Autonoma di Trento under Project PAT-UNITN 2001, and by granting a leave of absence to G. Rampanelli for completing his Ph.D. program. This work has been also partly supported by the Italian National Institute for Scientific and Technological Research on the Mountain under the program INRM2000.

REFERENCES

  • Ball, F. K. 1960. Control of inversion height by surface heating. Quart. J. Roy. Meteor. Soc 86:483494.

  • Betts, A. K. 1973. Non-precipitating cumulus convection and its parameterization. Quart. J. Roy. Meteor. Soc 99:178196.

  • Betts, A. K. 1974. Reply to comment on the paper “Non-precipitating cumulus convection and its parameterization.”. Quart. J. Roy. Meteor. Soc 100:469471.

    • Search Google Scholar
    • Export Citation
  • Boers, R. and E. W. Eloranta. 1986. Lidar measurements of the atmospheric entrainment zone and the potential temperature jump across the top of the mixed layer. Bound.-Layer Meteor 34:357375.

    • Search Google Scholar
    • Export Citation
  • Carson, D. J. 1973. The development of a dry inversion-capped convectively unstable boundary layer. Quart. J. Roy. Meteor. Soc 99:450467.

    • Search Google Scholar
    • Export Citation
  • Carson, D. J. and F. B. Smith. 1974. Thermodynamic model for the development of a convectively unstable boundary layer. Advances in Geophysics Vol. 18A, Academic Press, 111–124.

    • Search Google Scholar
    • Export Citation
  • Cohn, S. A. and W. M. Angevine. 2000. Boundary layer height and entrainment zone thickness measured by lidars and wind-profiling radars. J. Appl. Meteor 39:12331247.

    • Search Google Scholar
    • Export Citation
  • Davis, K. J., N. Gamage, C. R. Hagelberg, C. Kiemle, D. H. Lenshow, and P. P. Sullivan. 2000. An objective method for deriving atmospheric structure from airborne lidar observations. J. Atmos. Oceanic Technol 17:14551468.

    • Search Google Scholar
    • Export Citation
  • Deardorff, J. W. 1979. Prediction of convective mixed-layer entrainment for realistic capping inversion structure. J. Atmos. Sci 36:424436.

    • Search Google Scholar
    • Export Citation
  • de Franceschi, M., G. Rampanelli, D. Sguerso, D. Zardi, and P. Zatelli. 2003. Developement of a measurement platform on a light airplane and analysis of airborne measurements in the atmospheric boundary layer. Ann. Geophys 46:115.

    • Search Google Scholar
    • Export Citation
  • Driedonks, A. G. M. 1982. Models and observations of the growth of the atmospheric boundary layer. Bound.-Layer Meteor 23:283306.

  • Fedorovich, E. E. and D. V. Mironov. 1995. A model for shear-free convective boundary layer with parameterized capping inversion structure. J. Atmos. Sci 52:8395.

    • Search Google Scholar
    • Export Citation
  • Fitzjarrald, D. R. and M. Garstang. 1981. Vertical structure of the tropical boundary layer. Mon. Wea. Rev 109:15121526.

  • Garratt, J. R. 1992. The Atmospheric Boundary Layer. Cambridge University Press, 316 pp.

  • Gryning, S. E., F. Beyrich, and E. Batchvarova. Eds.,. 1997. The determination of the mixing height—Current progress and problems. EURASAP Workshop Proc., Roskilde, Denmark, Risø National Laboratories, 157 pp.

  • Johansson, C., A. S. Smedman, and U. Högström. 2000. Critical test of the validity of Monin–Obukhov similarity during convective conditions. J. Atmos. Sci 58:15491566.

    • Search Google Scholar
    • Export Citation
  • Khanna, S. and J. G. Brasseur. 1997. Analysis of Monin–Obukhov similarity from large-eddy simulation. J. Fluid Mech 345:251286.

  • Rampanelli, G. and D. Zardi. 2000. Analysis of airborne data and identification of thermal structures with geostatistical techniques. Preprints, 14th Symp. on Boundary Layers and Turbulence, Aspen, CO, Amer. Meteor. Soc., 239–241.

    • Search Google Scholar
    • Export Citation
  • Rampanelli, G. and D. Zardi. 2002. Identification of thermal structure from airborne measurements in an Alpine valley with kriging technique. Preprints, 10th Conf. on Mountain Meteorology, Park City, UT, Amer. Meteor. Soc., 1.20–1.24.

    • Search Google Scholar
    • Export Citation
  • Seibert, P., F. Beyrich, S-E. Gryning, S. Joffre, A. Rasmussen, and P. Tercier. 2000. Review and intercomparison of operational methods for the determination of the mixing height. Atmos. Environ 34:10011027.

    • Search Google Scholar
    • Export Citation
  • Steyn, D. G., M. Baldi, and R. M. Hoff. 1999. The detection of mixed layer depth and entrainment zone thickness from lidar backscatter profiles. J. Atmos. Oceanic Technol 16:953959.

    • Search Google Scholar
    • Export Citation
  • Stull, R. B. 1988. An Introduction to Boundary Layer Meteorology. Kluwer Academic, 670 pp.

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APPENDIX

Minimizing the Functional S

For the estimation of the parameters a, b, θm, l, and Δh in (2), the usual least squares method requires that the function
i1520-0450-43-6-925-ea1
be a minimum, where θi is the value measured at height ηi, θ(ηi) is the value of the vertical profile at the same height, and N is the total number of data. The minimum requirement can be imposed by setting
i1520-0450-43-6-925-ea2
then
i1520-0450-43-6-925-ea3a
Note that the terms (A3d) and (A3e) depend on the specific structure of the functions g and f, while by the definition provided in (2) we have
i1520-0450-43-6-925-ea4a
therefore, (A4a)–(A4c) become simply
i1520-0450-43-6-925-ea5
which upon substitution from (2) and rearrangements gives
i1520-0450-43-6-925-ea6
which is a linear symmetric algebraic system in the variables a, b, and θm. Once the system is solved, a, b, and θm are directly related to l and Δη, which are the only free parameters left to be varied to minimize S.
In the case of a coupled fit using vertical profiles of potential temperature and water vapor content, the function to be minimized is
i1520-0450-43-6-925-ea7
where qi is the q value measured at height ηi, q(ηi) is the value of the vertical profile of water vapor content at the same height, Θ and Q are two suitable scaling factors (suggested values are the span of θ and of q in the vertical profile under analysis). According to Stull (1988) and Garratt (1992), the shape function of the vertical profile of water vapor content can be assumed in the form
qzqmaqfηbqgη
with the functions f and g identical to those used for the θ profile. Obviously, the linear behavior of (A8) in the FA can produce negative values of water vapor when z is very large, but (A8) can be used as a linear approximation of the vertical profile for the lower region of the FA. The minimum requirement of S can be imposed by setting
i1520-0450-43-6-925-ea9
then
i1520-0450-43-6-925-ea10a
The terms (A10g) and (A10h), as in the single fit procedure, depend on the specific structure of the functions g and f, but even in this case we have
i1520-0450-43-6-925-ea11
therefore, (A10a)–(A10f) after splitting and rearrangements become simply
i1520-0450-43-6-925-ea12
These are two uncoupled linear symmetric algebraic systems with the variables a, b, θm and aq, bq, qm, respectively.

Once these two systems are solved, a, b, θm and aq, bq, qm are directly related to l and Δh, as in the case of the single fit to only one variable. This second procedure produces a single estimate for l and Δh for the vertical profiles of both potential temperature and water vapor content. An example is shown in Figs. 10 and 11 using the data displayed in Figs. 6a and 8b. The results are reported in Tables 3 and 4, for a comparison with the single-fit procedure.

Fig. 1.
Fig. 1.

Sketch of a vertical profile of potential temperature in the convective boundary layer developing over flat uniform terrain [adapted from Garratt (1992); notation for h0 and h2 follows Deardorff (1979)]

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 2.
Fig. 2.

(a) The smoothed vertical profile of potential temperature and the symbols used in the present paper, associated with (b) the vertical structure of the sensible heat flux

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 3.
Fig. 3.

Sketch of the basic functions f and g used in (2) to obtain the vertical profile

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 4.
Fig. 4.

Summary of criteria for the identification of standard meteorological parameters: (a) shaded areas are equal as a consequence of energy budget (Driedonks 1982), (b) linear potential temperature profile displaying the same lapse rate as the gradient of the real profile at the inflection point, and (c) evaluation of heights at which the best-fit profile is close enough either to the ML constant value θm (h0) or to the FA (h2)

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 5.
Fig. 5.

Sketch of the case of pure encroachment and its interpretation with (17) and (19)

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 6.
Fig. 6.

An example of the analysis of airborne data using the method outlined in the paper. Data were collected in the Adige valley, near the village of Besenello, south of the city of Trento (Italy) during the measurement flights. (a) between 0930 and 0946 LST 1 Oct 1999 and (b) between 0935 and 0948 LST 26 May 1999

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 7.
Fig. 7.

An example of the analysis of airborne data using the method outlined in the paper. Data were collected in the Adige valley, near the city of Trento (Italy), during the measurement flights (a) between 1023 and 1039 LST 2 Jul 1997 and (b) between 1039 and 1104 LST 22 Dec 2000

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 8.
Fig. 8.

An example of the analysis of airborne data using the method outlined in the paper. Data were collected in the upper Lakes Valley, near the village of Terlago, west of the city of Trento (Italy), during the measurement flight (a) between 1248 and 1304 LST 9 Sep 1998 and (b) between 1420 and 1452 LST 23 Sep 2001

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 9.
Fig. 9.

An example of the analysis of airborne data using the method outlined in the paper. Data were collected during the measurement flight (a) between 1217 and 1229 LST 24 Oct 1998 near the town of Riva del Garda, in the lower Lakes Valley, about 30 km southwest of the city of Trento (Italy), and during the measurement flight (b) between 0932 and 0958 LST 8 Jul 1999 in the Adige valley, near the city of Bolzano (Italy)

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 10.
Fig. 10.

An example of the analysis of airborne data using the method of the coupled fit using the vertical profiles of both θ and q. Vertical profiles of θ are shown: data (black dots), interpretation using single fit to the θ data (thin line), and interpretation using θq coupled fit (thick line). Data were collected during the measurement flight of (a) Fig. 6a, and during the measurement flight of (b) Fig. 8b

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Fig. 11.
Fig. 11.

An example of the analysis of airborne data using the method of the coupled fit using the vertical profiles of both θ and q. Vertical profiles of q are shown: data (black dots), interpretation using single fit to the q data (thin line), and interpretation using θq coupled fit (thick line). Data were collected during the measurement flight of (a) Fig. 6a, and during the measurement flight of (b) Fig. 8b

Citation: Journal of Applied Meteorology 43, 6; 10.1175/1520-0450(2004)043<0925:AMTDTC>2.0.CO;2

Table 1.

Parameters of the vertical profiles obtained from data shown in Figs. 6–7

Table 1.
Table 2.

Parameters of the vertical profiles obtained from data shown in Figs. 8–9

Table 2.
Table 3.

Parameters of the vertical profiles obtained from data shown in Fig. 10, and comparison between the single- and coupled-fit procedure

Table 3.
Table 4.

Parameters of the vertical profiles obtained from data shown in Fig. 11, and comparison between the single- and coupled-fit procedure

Table 4.
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