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On the “Do”s and “Don’t”s of Footprint Analysis in Difficult Conditions

H.P. SchmidIndiana University, Bloomington IN, USA

CO2

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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, FRS1914 - 1994

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

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In the time since Pasquill: Several Types of Footprint Models

• Analytical

• Stochastic (Lagrangian)

• Closure Models

• Large-Eddy Simulation ap

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ab

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For reviews of individual models, see:

• Schmid, Ag.For.Met., 113, 2002: 159-183.• Foken & Leclerc, Ag.For.Met, 127, 2004: 223-234.

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Flux Footprint = spatial filter, “field of view”

(convolution of the source distribution, QS, with the footprint, f )

: scalar flux, F; or scalar concentration, c

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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:

:c

advection diffusion forcing

surface sources

flux production rate(arises from c-gradient in turbulent flow).surface sources only in boundary conditionsin inhomogeneous flow, may

cause complex behavior of flux footprint

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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, :yC

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Paul Langevin(1872-1946)

correlated part

random part

particlevelocity

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:

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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

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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

windvirtual

wind

Virtual Source

inverted plume from virtual source

Projected Footprint

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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 applicableto 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.

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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

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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

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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

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HeterogeneousScalar Field

(LAI, Bowen-Ratio)

HeterogeneousFlow/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

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Tall Trees

Difficult Conditions: Deep Canopies

Multi-Layer Understorey

• analytical models apply only if zm > 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

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Large Scale Topography

Small Scale, Gentle Topography

Difficult Conditions: Topography

• use footprint model only (with caution!) for small zm/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

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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 Commandments10