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Footprint prediction of scalar fluxes from analytical solutions of the diffusion equation

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Abstract

The use of analytical solutions of the diffusion equation for ‘footprint prediction’ is explored. Quantitative information about the ‘footprint’, i.e., the upwind area most likely to affect a downwind flux measurement at a given height z, is essential when flux measurements from different platforms, particularly airborne ones, are compared. Analytical predictions are evaluated against numerical Lagrangian trajectory simulations which are detailed in a companion paper (Leclerc and Thurtell, 1990). For neutral stability, the structurally simple solutions proposed by Gash (1986) are shown to be capable of satisfactory approximation to numerical simulations over a wide range of heights, zero displacements and roughness lengths. Until more sophisticated practical solutions become available, it is suggested that apparent limitations in the validity of some assumptions underlying the Gash solutions for the case of very large surface roughness (forests) and tentative application of the solutions to cases of small thermal instability be dealt with by semi-empirical adjustment of the ratio of horizontal wind to friction velocity. An upper limit of validity of these solutions for z has yet to be established.

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References

  • Alvo, P., Desjardins, R. L., Schuepp, P. H., and MacPherson, J. I.: 1984, ‘Aircraft Measurements of CO2 Exchange over Various Ecosystems’, Boundary-Layer Meteorol. 29, 167–183.

    Google Scholar 

  • Austin, L. B., Schuepp, P. H., and Desjardins, R. L.: 1987, ‘The Feasibility of Using Airborne CO2 Flux Measurements for the Imaging of the Rate of Biomass Production’, Agric. Forest Meteorol. 39, 13–23.

    Google Scholar 

  • Barr, S. and Kreitzberg, C. W.: 1975, ‘Horizontal Variability and Boundary-Layer Modeling’, Boundary-Layer Meteorol. 8, 163–172.

    Google Scholar 

  • Bean, B. R., Gilmer, R. F., Hartmann, R. E., McGavin, R. E., and Reinking, R. F.: 1976, ‘Airborne Measurement of Vertical Boundary Layer Fluxes of Water Vapor, Sensible Heat and Momentum during GATE’, NOAA Tech. Report ERL WMPO-36. NOAA Environm. Res. Laboratories, Boulder, Co, 83 pp.

    Google Scholar 

  • Calder, K. L.: 1952, ‘Some Recent British Work on the Problem of Diffusion in the Lower Atmosphere’, Proc. U.S. Tech. Conf. Air Poll., McGraw-Hill, New York, pp. 787–792.

    Google Scholar 

  • De Baas, A. F., Van Dop, H., and Nieuwstadt, F. T. M.: 1986, ‘An Application of the Langevin Equation for Inhomogeneous Conditions to Dispersion in a Convective Boundary Layer’, Quart. J. Roy. Meteorol. Soc. 112, 165–180.

    Google Scholar 

  • Desjardins, R. L., Brach, E. J., Alvo, P., and Schuepp, P. H.: 1982, ‘Aircraft Monitoring of Surface Carbon Dioxide Exchange’, Science 216, 733–735.

    Google Scholar 

  • Desjardins, R. L., MacPherson, J. I., Schuepp, P. H., and Karanja, F.: 1989, ‘An Evaluation of Airborne Eddy Flux Measurements of CO2, Water Vapor and Sensible Heat’, Boundary-Layer Meteorol. 47, 55–70.

    Google Scholar 

  • Durand, P., Druilhet, A., Hedde, T., and Bénech, B.: 1987, ‘Aircraft Observations of the Structure of the Boundary Layer over a Rugged Hilly Region (MESOGERS 84 Experiment)’, Annales Geophysicae 5B, 441–450.

    Google Scholar 

  • Dyer, A. J.: 1963, ‘The Adjustment of Profiles and Eddy Fluxes’, Quart. J. Roy. Meteorol. Soc. 98, 276–280.

    Google Scholar 

  • Dyer, A. J.: 1974, ‘A Review of Flux-Profile Relationships’, Boundary-Layer Meteorol. 7, 363–372.

    Google Scholar 

  • Gash, J. H. C.: 1986, ‘A Note on Estimating the Effect of Limited Fetch on Micrometeorological Evaporation Measurements’, Boundary-Layer Meteorol. 35, 409–413.

    Google Scholar 

  • Grossman, R. L. and Bean, B. R.: 1973, ‘An Aircraft Investigation of Turbulence in the Lower Layers of a Marine Boundary Layer’, NOAA Tech. Report ERL WMPO-291. No. 4. 166 pp.

  • Hacker, J. M.: 1982, ‘First Results of Boundary Layer Research Flights with Three Powered Gliders during the Field Experiment PUKK’, Beitr. Phys. Atmosph. 55, 383–402.

    Google Scholar 

  • Horst, T. W. and Slinn, W. G. N.: 1984, ‘Estimates for Pollution Profiles above Finite Area Sources’, Atmospheric Environment 18, 1339–1346.

    Google Scholar 

  • Leclerc, M. Y., Thurtell, G. W., and Kidd, G. E.: 1988, ‘Measurements and Langevin Simulations of Mean Tracer Concentration Fields Downwind from a Circular Line Source inside an Alfalfa Canopy’, Boundary-Layer Meteorol. 43, 287–308.

    Google Scholar 

  • Leclerc, M. Y. and Thurtell, G. W.: 1990, ‘Footprint Prediction of Scalar Flux Using a Markovian Analysis’, Boundary-Layer Meteorol. (in press).

  • Lenschow, D. H., Pearson, R. Jr., and Stankov, B. B.: 1981, ‘Estimating the Ozone Budget in the Boundary Layer by Use of Aircraft Measurements of Ozone Eddy Flux and Mean Concentration’, J. Geophys. Res. 86, 7291–7297.

    Google Scholar 

  • Lenschow, D. H., Pearson, R. Jr., and Stankov, B. B.: 1982, ‘Measurements of Ozone Vertical Flux to Ocean and Forest’, J. Geophys. Res. 87, 8833–8837.

    Google Scholar 

  • MacPherson, J. I., Morgan, J. M., and Lum, K.: 1981, ‘The NAE Twin Otter Atmospheric Research Aircraft’, NAE Lab. Tech. Rep. LTR-FR-80, 21 pp.

  • McBean, G. A. and Peterson, R. D.: 1975, ‘Variations of the Turbulent Fluxes of Momentum, Heat and Moisture over Lake Ontario’, J. Phys. Oceanography 5, 523–531.

    Google Scholar 

  • Pasquill, F.: 1972, ‘Some Aspects of Boundary Layer Description’, Quart. J. Royal Meteorol. Soc. 98, 469–494.

    Google Scholar 

  • Philip, J. R.: 1959, ‘The Theory of Local Advection: 1’, J. Meteorol. 16, 535–547.

    Google Scholar 

  • Rao, K. S., Wyngaard, J. C., and Coté, O. R.: 1974, ‘Local Advection of Momentum, Heat and Moisture in Micrometeorology’, Boundary-Layer Meteorol. 7, 331–348.

    Google Scholar 

  • Rao, K. S.: 1975, ‘Effect of Thermal Stratification on the Growth of the Internal Boundary Layer’, Boundary-Layer Meteorol. 8, 227–234.

    Google Scholar 

  • Ripley, E. A. and Redmann, R. E.: 1975, Grassland, in J. L. Monteith (ed.), Vegetation and the Atmosphere, Vol. 2 (Case studies), Academic Press, London, pp. 351–398.

    Google Scholar 

  • Schmid, H. P. and Oke, T. R.: 1988, ‘Estimating the Source Area of a Turbulent Flux Measurement over a Patchy Surface’, Proc. Conf. on Turbulence and Diffusion, American Meteorol. Soc., San Diego, April 25–29, pp. 123–126.

    Google Scholar 

  • Schuepp, P. H., Desjardins, R. L., MacPherson, J. I., Boisvert, J., and Austin, L. B.: 1987, ‘Airborne Determination of Regional Water Use Efficiency: Present Capabilities and Initial Field Tests’, Agric. Forest Meteorol. 41, 1–9.

    Google Scholar 

  • Schuepp, P. H., Desjardins, R. L., MacPherson, J. I., Boisvert, J. B., and Austin, L. B.: 1989, ‘Interpretation of Airborne Estimates of Evapotranspiration’, in T. A. Black, D. L. Spittlehouse, M. D. Novak and D. T. Price (eds.), Estimation of Areal Evapotranspiration, Int. Assoc. Hydrol. Sci. Publ. no. 177, Wallingford, pp. 185–196.

  • Shwetz, M. E.: 1949, ‘On the Approximate Solution of some Boundary-Layer Problems’, Appl. Mathematics Mech. 13(3); (Moscow) (cited by Wilson, 1982).1982, ‘An Approximate Analytical Solution for the Diffusion Equation for Short-Range Dispersion from a Continuous Ground-Level Source’, Boundary-Layer Meteorol. 23, 85–103.

  • Sutton, O. G.: 1934, ‘Wind Structure and Evaporation in a Turbulent Atmosphere’, Proc. Roy. Soc. A146, 701–722.

    Google Scholar 

  • Van Ulden, A. P.: 1978, ‘Simple Estimates of Vertical Diffusion from Sources near the Ground’, Atmospheric Environment 12, 2125–2129.

    Google Scholar 

  • Wilson, J. D.: 1982, ‘An Approximate Analytical Solution for the Diffusion Equation for Short-Range Dispersion from a Continuous Ground-Level Source’, Boundary-Layer Meteorol. 23, 85–103.

    Google Scholar 

  • Wyngaard, J. C.: 1983, ‘Lectures on the Planetary Boundary Layer’, in D. K. Lilly and T. Gal-Chen (eds.), Mesoscale Meteorology — Theories, Observations and Models, Reidel, Dordrecht, pp. 603–650 and pp. 216–224.

    Google Scholar 

  • Wyngaard, J. C.: 1988, ‘New Developments in Dispersion Parameterization and Modeling’, 17th NATO/CCMS Int'l. Tech. Meeting on Air Poll. Modelling and its Application, Cambridge, England, Sept. 19–22.

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Schuepp, P.H., Leclerc, M.Y., MacPherson, J.I. et al. Footprint prediction of scalar fluxes from analytical solutions of the diffusion equation. Boundary-Layer Meteorol 50, 355–373 (1990). https://doi.org/10.1007/BF00120530

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