1. Aline Jaimes Kreinovich, Vladik PhD. CYBER-ShARE Meeting Nov ‘08
Footprint Model
Aline Jaimes
Kreinovich, Vladik PhD.
CYBER-ShARE Meeting
Nov ‘08

2. OUTLINE Background Footprint definition Model description Conclusions
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3. Air flow in Ecosystem
Air flow can be imagined as a horizontal flow of numerous rotting eddies
Each eddy has 3D components, including a vertical wind component
The diagram looks chaotic but components can be measured from the tower
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4. Eddies at one point
covariance is a measure of how much two variables change together
The essence of the method is that vertical flux can be presented as covariance between measurements of vertical velocity, the up and down movements, and concentration of the entity of interest.
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5. Where is the flux coming from?
Footprint models and tower flux measurements is a first cut evaluation of the impact of biodiversity on net ecosystem CO2 exchange.
Issues related to footprint, spatial distribution of vegetation and net ecosystem exchange:
Representativeness of the integrated tower record
Appropriateness of conventional gap filling that does not consider the effects of wind direction on algorithm development.
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6. Until now estimates of annual NEE and its error have not considered the impact of wind direction, footprint and upwind vegetation. This effect can be a major source of bias error. (Kim at al. 2006)
One question that needs to be address is whether the temporal sum of net ecosystem CO2 exchange is equal to the net flux if the wind came from the same wind direction and was processed by the same patch.
To address this question we need to assess NEE for various wind direction sectors.
Because wind direction is not uniform, gap filling algorithms must be applied.
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7. Definition
Footprint. The contribution, per unit surface flux, of each unit element of the upwind surface area to a measured vertical flux.
The measured flux is the integral of the contributions from all upwind surface elements.
The flux footprint is the relative weight given each elemental surface flux.
The Plan view shows isopleths of the footprint function for a wind blowing from right to left, and the profile view shows the crosswind integrated flux footprint.
The footprint has its maximum directly upwind of the measurement location, decreases slowly upwind of the maximum, and decreases very rapidly downwind of the maximum. .
Figure 1. Plan and profile sketches of the flux footprint function f for a tower-based flux measurement. (Horst and Weil, 1994)
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8. Objective. Estimate flux footprint φ(x,y,z)
Flux portion at (0,0,z) caused by a unit point source at (x,y,0)
In our case z=zm =6m
Figure 1. (a) The crosswind integrated footprint f(x,zm). (b) Isopleths of the footprint. The solid lines depict the neutral case, the dashed and the dotted lines the stable and the instable case, respectively.
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9. Parameters describing the model
m – Describes how horizontal wind velocity depends on height z:
n – Describes how eddy diffusivity depends on the z:
How to empirically find m:
So,
- Vertical profile of the Reynolds-averaged wind velocity
- Constant in power-law profile of the wind velocity
- Vertical Profile of eddy diffusivity
- Von Karman constant =0.4
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10. Simplest case (Schuepp et. al. 1990)
Mathematical description:
n=1 , m=0
Wind velocity u(z)=U does not depend on height
Diffusivity K(z)=kz linearly increases with height
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11. Auxiliary parameters Shape factor r=2+m-n
Meaning: crosswind integrated concentration c decreases with height z
Schuepp case: r=1
Useful constant
Schuepp case µ=1
Eq. 12
c- crosswind integrated concentration.
r – shape factor
µ - (1+m)/r – constant
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12. Analytical formula for velocity ū(x) of the plume: ū(x)=
Eq. 18
Where Г(z) is the gamma function:
Is an extension of factorial n!=1·2·…·n to real numbers:
Г(n) = (n-1)!
Schuepp case: ū(x)=U
Gamma function is an extension of the factorial function to real and complex numbers.
ū(x) – Effective plume velocity
Г (x)– Gamma function
U – Constant of power of law profile wind velocity
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13. Analytical Model
Crosswind distribution function Dy (x,y) described horizontal distribution
Crosswind integrated flux footprint f(x,z) described vertical distribution
Ø(x,y,z) – flux footprint or vertical flux per unit point source.
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14. Crosswind distribution Function: Gaussian Model
where dispersion is:
Since ,
ū(x)~
Schuepp case: ū(x)~x
Typically :
σv - Constant crosswind fluctuation
σ(x) - Crosswind dispersion
ū(x)- Effective plume velocity
zm – Measurement height
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15. Maximum of a Footprint f(x,z)= Flux Length scale:
Crosswind integrated flux footprint or vertical flux per unit point source
f(x,z)=
Maximum is attained at:
Maximum footprint value is:
Eq. 22
ζ(z) – Flux length scale
Г – Gamma function
µ - Vertical profile of the Reynolds-averaged wind velocity
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16. Final Formula σ(x) – Crosswind dispersion r – shape factor
µ - Vertical profile of the Reynolds-averaged wind velocity
ζ (z) – Flux length scale
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17. Towards a more accurate description
In analytical model, we assumed power law
(Eq. 11)
A more accurate model:
where:
u* is friction velocity
zo Is roughness length
L is Obukhov length where:
Kinetic energy ,
is equal to the potential energy gz
The Monin-Obukhov Length is the height above ground, where mechanically produced (by vertical shear) turbulence is in balance with the dissipative effect of negative buoyancy, thus where Richardson number equals to 1: where u * is the frictional velocity, is the mean potential virtual temperature, is the perturbation scalar velocity' and θ * is a potential temperature scale (k).
a process that involves heat transfer (addition or loss of heat to the surroundings) is generally called diabatic.
u(z) - Vertical profile of the Reynolds-averaged wind velocity
U – Constant in power-law profile of the wind velocity
Ψm(z/L) – Diabatic integration of the wind profile
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18. Conclusions
Kormann & Meixner’s model is a simple approach to estimate footprint models.
Footprint models will allow estimation of net fluxes when wind direction is not uniform.
Therefore, improving development of gap filling algorithms for JER Station.
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19. Questions / Comments / Suggestions
Thank you
Questions / Comments / Suggestions
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