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
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.
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 86:483–494.
Betts, A. K. 1973. Non-precipitating cumulus convection and its parameterization. Quart. J. Roy. Meteor. Soc 99:178–196.
Betts, A. K. 1974. Reply to comment on the paper “Non-precipitating cumulus convection and its parameterization.”. Quart. J. Roy. Meteor. Soc 100:469–471.
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 34:357–375.
Carson, D. J. 1973. The development of a dry inversion-capped convectively unstable boundary layer. Quart. J. Roy. Meteor. Soc 99:450–467.
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.
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 39:1233–1247.
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 17:1455–1468.
Deardorff, J. W. 1979. Prediction of convective mixed-layer entrainment for realistic capping inversion structure. J. Atmos. Sci 36:424–436.
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 46:1–15.
Driedonks, A. G. M. 1982. Models and observations of the growth of the atmospheric boundary layer. Bound.-Layer Meteor 23:283–306.
Fedorovich, E. E. and D. V. Mironov. 1995. A model for shear-free convective boundary layer with parameterized capping inversion structure. J. Atmos. Sci 52:83–95.
Fitzjarrald, D. R. and M. Garstang. 1981. Vertical structure of the tropical boundary layer. Mon. Wea. Rev 109:1512–1526.
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 58:1549–1566.
Khanna, S. and J. G. Brasseur. 1997. Analysis of Monin–Obukhov similarity from large-eddy simulation. J. Fluid Mech 345:251–286.
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.
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.
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 34:1001–1027.
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 16:953–959.
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 55:3042–3064.
Tennekes, H. 1973. A model for the dynamics of the inversion above a convective boundary layer. J. Atmos. Sci 30:558–567.
Vogelezang, D. H. P. and A. A. M. Holtslag. 1996. Evaluation and model impacts of alternative boundary-layer height formulations. Bound.-Layer Meteor 81:245–269.
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.
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 35:2145–2161.
Zilitinkevich, S. S. 1975. Comments on “A model for the dynamics of the inversion above a convective boundary layer.”. J. Atmos. Sci 32:991–992.
APPENDIX
Minimizing the Functional S
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.
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
(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
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
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
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
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
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
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
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
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 θ–q 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
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 θ–q 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