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An Analytical Model for the Two-Scalar Covariance Budget Inside a Uniform Dense Canopy

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Abstract

The two-scalar covariance budget is significant within the canopy sublayer (CSL) given its role in modelling scalar flux budgets using higher-order closure principles and in estimating the segregation ratio for chemically reactive species. Despite its importance, an explicit expression describing how the two-scalar covariance is modified by inhomogeneity in the flow statistics and in the vertical variation in scalar emission or uptake rates within the canopy volume remains elusive even for passive scalars. To progress on a narrower version of this problem, an analytical solution to the two-scalar covariance budget in the CSL is proposed for the most idealized flow conditions: a stationary and planar homogeneous flow inside a uniform and dense canopy with a constant leaf area density distribution. The foliage emission (or uptake) source strengths are assumed to vary exponentially with depth while the forest floor emission is represented as a scalar flux. The analytical solution is a superposition of a homogeneous part that describes how the two-scalar covariance at the canopy top is transported and dissipated within the canopy volume, and an inhomogeneous part governed by local production mechanisms of the two-scalar covariance. The homogeneous part is primarily described by the canopy adjustment length scale, and the attenuation coefficients of the turbulent kinetic energy and the mean velocity. Conditions for which the vertical variation of the two-scalar covariance is controlled by the rapid attenuation in the mean velocity and turbulent kinetic energy profiles, vis-à-vis the vertical variation of the scalar source strength, are explicitly established. This model also demonstrates how dissimilarity in the emissions from the ground, even for the extreme binary case with one scalar turned ‘on’ and the other scalar turned ‘off’, modifies the vertical variation of the two-scalar covariance within the CSL. To assess its applicability to field conditions, the analytical model predictions were compared with observations made at two different forest types—a sparse pine forest at the Hyytiälä SMEAR II-station (in Finland) and a dense alpine hardwood forest at Lavarone (in Italy). While the model assumptions do not represent the precise canopy morphology, attenuation properties of the turbulent kinetic energy and the mean velocity, observed mixing length, and scalar source attenuation properties for these two forest types, good agreement was found between measured and modelled two scalar covariances for multiple scalars and for the triple moments at the Hyytiälä site.

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Authors and Affiliations

  1. Nicholas School of the Environment, Duke University, Box 90328, Durham, NC, 27708-0328, USA

    Gabriel G. Katul

  2. Department of Civil and Environmental Engineering, Duke University, Durham, NC, 27708-0328, USA

    Gabriel G. Katul

  3. CNR—Institute of Atmosphere Sciences and Climate U.O. of Lecce, Lecce, Italy

    Daniela Cava

  4. Department of Physics, University of Helsinki, P.O. Box 64, 00014, Helsinki, Finland

    Samuli Launiainen & Timo Vesala

Authors
  1. Gabriel G. Katul
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  2. Daniela Cava
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  3. Samuli Launiainen
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  4. Timo Vesala
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Correspondence to Gabriel G. Katul.

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Katul, G.G., Cava, D., Launiainen, S. et al. An Analytical Model for the Two-Scalar Covariance Budget Inside a Uniform Dense Canopy. Boundary-Layer Meteorol 131, 173–192 (2009). https://doi.org/10.1007/s10546-009-9361-y

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