1. 1 On the “Do”s and “Don’t”s of Footprint Analysis in Difficult Conditions H.P. Schmid Indiana University, Bloomington IN, USA CO 2

2. 2 The earliest documented footprint-type idea: The “Effective Fetch” of Frank Pasquill (1972) Pasquill, F.: 1972. 'Some aspects of boundary layer description'. Q. J. R. Meteorol. Soc., 98, 469- 494. Frank Pasquill, FRS 1914 - 1994

3. 3 Effective fetch isopleths (C/Cmax = ½) dependent on height, stability and roughness

4. 4 In the time since Pasquill: Several Types of Footprint Models Analytical Stochastic (Lagrangian) Closure Models Large-Eddy Simulation applicability complexity For reviews of individual models, see: Schmid, Ag.For.Met., 113, 2002: 159-183. Foken & Leclerc, Ag.For.Met, 127, 2004: 223-234.

5. 5 Flux Footprint = spatial filter, “field of view” (convolution of the source distribution, Q S, with the footprint, f )  : scalar flux, F; or scalar concentration, c

6. 6 Concentration and Flux Footprint Models Governing equations in Eulerian analysis: * * following Finnigan (2004, AgForMet 127, 117-129); neglecting horizontal turbulent fluxes and pressure interactions. F:F: : advectiondiffusionforcing surface sources flux production rate (arises from c-gradient in turbulent flow). surface sources only in boundary conditions in inhomogeneous flow, may cause complex behavior of flux footprint

7. 7 Aad van Ulden (1978): Realistic analytic solution of advection-diffusion equation: based on power-law profiles fitting power-laws to similarity profiles M-O scaling widely used for analytic source area and footprint models (with some exceptions!) Van Ulden, A.P., 1978. ‘Simple estimates for vertical diffusion from sources near the ground’, Atmos. Environ., 12, 2125-2129. Analytical Footprint Models solution for crosswind integrated concentration,

8. 8 Paul Langevin (1872-1946) correlated part random part particle velocity Flux Distribution Continuous Point Source need large number of particles need flow and turbulence adaptable to vertically inhomogeneous turbulence (e.g., forest canopies) Lagrangian Stochastic Footprint Models Joseph-Louis Lagrange (1736-1813) based on the Langevin Equation:

9. 9 Dennis D. Baldocchi * * Baldocchi, D.D., Ag. For. Met. 85, 1997: 273-292. Forest Canopy LS-Footprint Models forward well-mixed LS model (2-D, 3D) parameterized turbulence/flow profiles vertically inhomogeneous turbulence includes streamwise diffusion

10. 10 Usage of Analytical & (forward) LS-Models motivated by spatial inhomogeneity (in the scalar field) assume horizontal homogeneity (in flow and turbulence) by the Inverse Plume Assumption Point Source flux plume from surface point source wind virtual wind Virtual Source inverted plume from virtual source Projected Footprint

11. 11 Alternatives for Inhomogeneous Flow Footprint computation based on full (Eulerian) flow models (plus scalar transport equation or LS-module): Closure Models LES Models Claude Louis Marie Henri Navier (1785-1836) George Gabriel Stokes (1819-1903) James W. Deardorff (born 1928) Monique Y. Leclerc (born...not long ago) Depending on resolution and closure / sub-grid scale treatment: can be made applicable to any complex condition can be computationally very intensive These models are not footprint models per se, but full flow models used to compute a footprint.

12. 12 Direct Footprint “Touchdown” Source Locations Sensor: continuous “backward release” point no Inverse Plume Assumption needed applicable in weakly inhomogeneous canopies Alternatives for Inhomogeneous Flow Backward LS-Model applicable in principle, but has never been done to date

13. 13 Objective: Examine Applicability of Footprint Model Types in “Difficult Conditions” “Difficult Conditions” ???  deviations from micrometeorological ideal: flat terrain homogeneous fetch low, homogeneous vegetation (if any) stationarity well-developed turbulence (MOST) topography patchy land-cover deep, multy-layer vegetation canopy instationarity weak turbulence; free convection

14. 14 Micrometeorologist’s traditional knee-jerk reaction: Stay away from it! Thou must provide flux data ! Flux Measurement in Difficult Conditions and Footprint Modeling in Difficult Conditions

15. 15 Heterogeneous Scalar Field (  LAI,  Bowen-Ratio) Heterogeneous Flow/Turbulence (disturbance, forest edges) Difficult Conditions: Patchy Land Cover “non-difficult” condition any footprint model applies analytic models have restriction to MOST “inverse plume assumption” (analytical, forward LS) does not apply full flow model needed case poorly understood

16. 16 Tall Trees Difficult Conditions: Deep Canopies Multi-Layer Understorey analytical models apply only if z m > 2h (Rannik et al. 2000) “forest” model better sensitive to turbulence profiles “forest” model needed sensitive to turbulence profiles “inverse plume assumption” (horizontal homogeneity) questionable

17. 17 Large Scale Topography Small Scale, Gentle Topography Difficult Conditions: Topography use footprint model only (with caution!) for small z m /h: local footprint use footprint model only for qualitative analysis full flow model is preferred use footprint model if terrain following flow can be assumed (stable conditions?) “inverse plume assumption”??? use footprint model only for qualitative analysis

18. 18 I.Know the site! II.Know the model! III.Know the assumptions! IV.Thou shall not use a model outside its applicability range! V.Thou shall not call it “footprint” if the model does not use unit source strength! VI.Thou shall not invert a footprint model to estimate a flux! VII.Thou shall not use a scalar footprint model for non- scalars! VIII.Thou shall exercise caution when using a footprint model with non-passive scalars! IX.Thou shall never blindly believe any footprint model result, but examine it in the context of the site (see I.)! X.Thou shall not complain that there are only nine commandments! The Footprint Modeling Commandments 10