Abstract
We address the flux footprint for measurement heights in the atmospheric surface layer, comparing eddy diffusion solutions with those furnished by the first-order Lagrangian stochastic (or “generalized Langevin”) paradigm. The footprint given by Langevin models differs distinctly from that given by the random displacement model (i.e. zeroth-order Lagrangian stochastic model) corresponding to its “diffusion limit,” which implies that a well-founded theory of the flux footprint must incorporate the turbulent velocity autocovariance. But irrespective of the choice of the eddy diffusion or Langevin class of model as basis for the footprint, tuning relative to observations is ultimately necessary. Some earlier treatments assume Monin–Obukhov profiles for the mean wind and eddy diffusivity and that the effective Schmidt number (ratio of eddy viscosity to the tracer eddy diffusivity) in the neutral limit \(S_c(0)=1\), while others calibrate the model to the Project Prairie Grass dispersion trials. Because there remains uncertainty as to the optimal specification of \(S_c\) (or a related parameter in alternative theories, e.g. the Kolmogorov coefficient \(C_0\) in Langevin models) it is recommended that footprint models should be explicit in this regard.
Highlights
The “flux footprint” is the zone of the surface upwind from an instrument that contributes to a measured vertical flux between the ground and the atmosphere: knowledge of that footprint confirms one is correctly attributing the measured flux to the region of the surface whence it derives
It will be shown that the penalty for adopting power-law profiles is not severe, but that there is good reason to base the footprint on a generalized Langevin model rather than the eddy diffusion paradigm: does the generalized Langevin model permit using the “true” Monin–Obukhov (MO) velocity statistics rather than a power-law representation, but more fundamentally the Lagrangian Stochastic Models (LSM), (i) allows the inclusion of the horizontal velocity fluctuations, long recognized as important (e.g. Rannik et al 2000), and (ii) corrects what Sawford (2001) has termed the “systematic failure of the diffusion approximation” in strongly inhomogeneous turbulence
Equivalently, the random displacement model, Random Displacement Model (RDM)) versus the “generalized Langevin” model (LSM). Addressing footprints of the former origin, note that the Kormann and Meixner (2001) analytical footprint, which is based on power-law profiles, is not very sensitive to the choice of reference height H and that it is exactly consistent with the solution given by the RDM, provided the latter is equipped with the same profiles
Summary
The “flux footprint” is the zone of the surface (mostly) upwind from an instrument that contributes to a measured vertical flux (e.g. of water vapour or carbon dioxide) between the ground and the atmosphere: knowledge of that footprint confirms one is correctly attributing the measured flux to the region of the surface whence it derives. Existing footprint models applicable to the atmospheric surface layer (ASL) are raw or modified solutions of an advection–diffusion equation, using power laws to represent profiles of mean wind speed u and eddy diffusivity Kc in lieu of the (actual) Monin–Obukhov profiles. It will be shown that the penalty for adopting power-law profiles is not severe, but that there is good reason to base the footprint on a generalized Langevin model (i.e. first-order Lagrangian stochastic model) rather than the eddy diffusion paradigm: does the generalized Langevin model (hereafter LSM) permit using the “true” Monin–Obukhov (MO) velocity statistics rather than a power-law representation, but more fundamentally the LSM, (i) allows the inclusion of the horizontal velocity fluctuations, long recognized as important (e.g. Rannik et al 2000), and (ii) corrects what Sawford (2001) has termed the “systematic failure of the diffusion approximation” in strongly inhomogeneous turbulence
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