Abstract
This paper considers the ground area which affects the properties of fluid parcels observed at a given spot in the Planetary Boundary Layer (PBL). We examine two source-area functions; the “footprint,” giving the source area for a measurement of vertical flux: and the distribution of “contact distance”, the distance since a particle observed aloft last made contact with the surface. We explain why the distribution of contact distance extends vastly farther upwind than the footprint, and suggest for the extent of the footprint the inequalities: % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9Lq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9W8qr0-vr% 0-viWZqaceaabiGaciaacaqabeaadaqaaqaaaOabaeqabaGaamyvam% aalaaabaGaamiAaaqaaiabeo8aZnaaBaaaleaacaWGxbaabeaakiaa% cIcacaWGObGaaiykaaaacqGH8aapcaWG4bGaeyipaWJaamyvaKazaa% iadaGabaqaamaaDaaajqwaacqaaiaadIgacaGGVaGabmOEayaacaGa% aiilaiaabccacaGGVbGaaiiDaiaacIgacaGGLbGaaiOCaiaacEhaca% GGPbGaai4CaiaacwgaaeaacaWGubWaaSbaaKazcaiabaGaamitaaqa% baqcKfaGaiaacIcacaWGObGaaiykaiaabYcacaqGGaGaaeiAaiaabc% cacaGGHbGaaiOyaiaac+gacaGG2bGaaiyzaiaabccacaGGZbGaaiyD% aiaackhacaGGMbGaaiyyaiaacogacaGGLbGaeyOeI0IaaiiBaiaacg% gacaGG5bGaaiyzaiaackhaaaaajqgaacGaay5EaaaakeaaaeaacaGG% 8bGaamyEaiaacYhacqGH8aapcqaHdpWCdaWgaaWcbaGaamODaaqaba% GccaGGOaGaamiAaiaacMcadaWcaaqaaiaadIhaaeaacaWGvbaaaaaa% aa!7877!\[\begin{array}{l} U\frac{h}{{\sigma _W (h)}} < x < U\left\{ {_{h/\dot z,{\rm{ }}otherwise}^{T_L (h){\rm{, h }}above{\rm{ }}surface - layer} } \right. \\ \\ |y| < \sigma _v (h)\frac{x}{U} \\ \end{array}\] where U is the mean streamwise (x) velocity, h is the observation height, ΤL is the Lagrangian timescale, Σ v and Σ w are the standard deviations of the cross-stream horizontal (y) and vertical (z) velocity fluctuations, and ż is the Lagrangian Similarity prediction for the rate of rise of the centre of gravity of a puff released at ground.
Simple analytical solutions for the contact-time and the footprint are derived, by treating the PBL as consisting of two sub-layers. The contact-time solutions agree very well with the predictions of a Lagrangian stochastic model, which we adopt in the absence of measurements as our best estimate of reality, but the footprint solution offers no improvement over the above inequality.
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References
Carslaw, H. S., and Jaeger, J. C.: 1959 Conduction of Heat in Solids. 2nd edition, Oxford University Press. Oxford.
Churchill, R. V., Brown, J. W., and Verhey, R. F.: 1974, Complex Variables and Applications, Third Edition. McGraw-Hill, New York.
Dyer, A. J. and Bradley, E. F.: 1982, ‘An Alternative Analysis of Flux-Gradient Relationships at the 1976 ITCE’, Boundary-Layer Meteorol. 22, 3–19.
Gash, J. H. C.: 1986, ‘A Note on Estimating the Effect of Limited Fetch on Micrometeorological Evaporation Measurements’, Boundary-Layer Meteorol. 35, 409–413.
Gradshteyn, I. S. and Ryzhik, I. M.: 1980, Table of Integrals, Series, and Products, Academic Press, New York.
Hicks, B. B.: 1985, ‘Behaviour of Turbulence Statistics in the Convective Boundary Layer’, J. Climate Appl. Meteorol. 24, 607–614.
Leclerc, M. Y. and Thurtell, G. W.: 1990, ‘Footprint Prediction of Scalar Flux Using a Markovian Analysis’, Boundary-Layer Meteorol. 52, 247–258.
Legg, B. J., Raupach, M. R., and Coppin, P. A.: 1986, ‘Experiments on Scalar Dispersion within a Model Plant Canopy, Part III: An Elevated Line Source’, Boundary-Layer Meteorol. 35, 277–302.
Luhar, A. K., and Britter, R. E.: 1989. ‘A Random Walk Model for Dispersion in Inhomogeneous Turbulence in a Convective Boundary Layer’, Atmos. Envir. 23, 1911–1924.
Panofsky, H. A., Tennekes, H., Lenschow, D. H., and Wyngaard, J. C.: 1977, ‘The Characteristics of Turbulent Velocity Components in the Surface Layer under Convective Conditions’, Boundary-Layer Meteorol. 11, 355–361.
Pasquill, F.: 1972, ‘Some Aspects of Boundary Layer Description’, Q. J. Royal Meteorol. Soc. 98, 469–494.
Sawford, B. L.: 1985, ‘Lagrangian Statistical Simulation of Concentration Mean and Fluctuation Fields’, J. Climate Appl. Meteorol. 24, 1152–1166.
Sawford, B. L. and Guest, F. M.: 1987, ‘Lagrangian Stochastic Analysis of Flux-Gradient Relationships in the Convective Boundary Layer’, J. Atmos. Sci. 44, 1152–1165.
Schmid, H. P. and Oke, T. R.: 1990, ‘A Model to Estimate the Source Area Contributing to Surface Layer Turbulence at a Point over Patchy Terrain’, Q. J. Royal Meteorol. Soc. 116, 965–988.
Schuepp, P. H., Leclerc, M. Y., MacPherson, J. I. and Desjardins, R. L.: 1990, ‘Footprint Prediction of Scalar Fluxes from Analytical Solutions of the Diffusion Equation’, Boundary-Layer Meteorol. 50, 355–373.
Thomson, D. J.: 1984, ‘Random Walk Modelling of Diffusion in Inhomogeneous Turbulence’, Q. J. R. Meteorol. Soc. 110, 1107–1120.
Thomson, D. J.: 1987, ‘Criteria for the Selection of Stochastic Models of Particle Trajectories in Turbulent Flows’, J. Fluid Mech. 180, 529–556.
Weil, J. C.: 1990, ‘A Diagnosis of the Asymmetry in Top-Down and Bottom-Up Diffusion Using a Lagrangian Stochastic Model’, J. Atmos. Sci. 47, 501–515.
Wilson, J. D.: 1982, ‘An Approximate Analytical Solution to the Diffusion Equation for Short-Range Dispersion from a Continuous Ground-Level Source’, Boundary-Layer Meteorol. 23, 85–103.
Wilson, J. D., Clarkson, T. S., and Hadfield, M. G.: 1984, ‘Observations of Wind Flow and Tracer Gas Diffusion over Sand Dunes’, New Zealand J. Sci. 27, 237–242.
Wilson, J. D., Legg, B. J., and Thomson, D. J.: 1983, ‘Calculation of Particle Trajectories in the Presence of a Gradient in Turbulent-Velocity Variance’, Boundary-Layer Meteorol. 27, 163–169.
Wilson, J. D., Thurtell, G. W., and Kidd, G. E.: 1981, ‘Numerical Simulation of Particle Trajectories in Inhomogeneous Turbulence, III: Comparison of Predictions with Experimental Data for the Atmospheric Surface Layer’, Boundary-Layer Meteorol. 21, 443–463.
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Wilson, J.D., Swaters, G.E. The source area influencing a measurement in the Planetary Boundary Layer: The “footprint” and the “distribution of contact distance”. Boundary-Layer Meteorol 55, 25–46 (1991). https://doi.org/10.1007/BF00119325
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DOI: https://doi.org/10.1007/BF00119325