7 Flux footprint

7.1 Flux footprint parametrizations

The calculation of a 2d flux footprint makes it possible to estimate the size of the surface that contributes to the measured flux. This also allows to analyze whether changes in the flux result from a change in the footprint (e.g. surface composition, vegetation, surface roughness). Here the flux footprint parametrization according to Kljun et al. (2015) is used.
The mathematical idea for deriving a flux footprint parametrization is to express the flux (\(F_c\)) as integral over the distribution of its sinks and sources (\(S_c\)) times a transfer function \(f\) (the flux footprint): \[ F_c(0,0,z) = \int\int S_c(x,y) f(x,y) \:dx\,dy \] By treating streamwise and crosswise velocity independently, the footprint can be expressed as product of the crosswind-integrated footprint (\(\overline{f^y}(x)\) which is then only a function of \(x\)) and a function expressing the crosswind dispersion (\(D_y\)) through \[ f(x,y) = \overline{f^y}(x)D_y. \] This assumption leads to a symmtric footprint in crosswind direction. For further derivations a concrete footprint model has to be applied, which is in Kljun et al., 2015 an advanced Lagrangian particle dispersion model (LPDM-B) based on 3d particle backtracking between surface and boundary layer height \(h\) that is valid for steady flows under all stabilities.
There are several other approaches to flux footprint estimation (e.g. Hsieh, Katul, and Chi (2000) and Kormann and Meixner (2001)), but the Kljun et al. (2015) model is the most common approach today.

#load Reddy package
install.packages("../src/Reddy_0.0.0.9000.tar.gz",repos=NULL,source=TRUE,quiet=TRUE)
library(Reddy)

#read in processed example data
dat=readRDS("../data/ec-data_30min_processed/processed_data_example.rds")

#select file
i=8 #daytime example

7.2 2D flux footprint estimate

7.2.1 Calculate 2D flux footprint estimate with `calc_flux_footprint`

The function calc_flux_footprintuses the 2d flux footprint parametrization (FFP) according to Kljun et al. (2015) to calculate the footprint based on measurement height zm, mean horizontal wind speed u_mean, boundary layer height h, Obukhov length L (calc_L), standard deviation of cross-wind component v_sd and either friction velocity ustar(calc_ustar) or surface roughness length z0 in a resolution given by nres. The boundary layer height can be taken from e.g. ERA5.

ustar=calc_ustar(dat$cov_uw,dat$cov_vw)
L=calc_L(ustar,dat$T_mean,dat$cov_wT)
zm=4.4
h=700

ffp=calc_flux_footprint(zm,dat$u_mean[i],h,L[i],dat$v_sd[i],ustar[i])
str(ffp)

List of 9
 $ xmax          : num 23
 $ x             : num [1:999] 4.38 5.17 5.96 6.76 7.55 ...
 $ fy_mean       : num [1:999] 3.43e-20 3.60e-10 5.56e-07 1.85e-05 1.37e-04 ...
 $ x2d           : num [1:1499, 1:999] 4.38 4.38 4.38 4.38 4.38 ...
 $ y2d           : num [1:1499, 1:999] -592 -591 -590 -589 -589 ...
 $ f2d           : num [1:999, 1:1499] 0 0 0 0 0 0 0 0 0 0 ...
 $ xcontour      :List of 9
  ..$ : num [1:1573] 6.76 6.75 6.72 6.7 6.69 ...
  ..$ : num [1:815] 7.55 7.48 7.42 7.38 7.35 ...
  ..$ : num [1:543] 8.34 8.25 8.16 8.08 8.03 ...
  ..$ : num [1:397] 9.13 9.03 8.91 8.81 8.73 ...
  ..$ : num [1:307] 9.92 9.78 9.64 9.52 9.43 ...
  ..$ : num [1:235] 10.7 10.7 10.5 10.3 10.2 ...
  ..$ : num [1:181] 11.5 11.3 11.2 11 11 ...
  ..$ : num [1:133] 12.3 12.2 12.1 12 12 ...
  ..$ : num [1:87] 13.9 13.8 13.7 13.7 13.7 ...
 $ ycontour      :List of 9
  ..$ : num [1:1573] -2.39 -2.37 -1.58 -0.79 0 ...
  ..$ : num [1:815] -4.56 -3.95 -3.16 -2.37 -1.58 ...
  ..$ : num [1:543] -6.14 -5.53 -4.74 -3.95 -3.16 ...
  ..$ : num [1:397] -6.85 -6.32 -5.53 -4.74 -3.95 ...
  ..$ : num [1:307] -6.95 -6.32 -5.53 -4.74 -3.95 ...
  ..$ : num [1:235] -6.48 -6.32 -5.53 -4.74 -3.95 ...
  ..$ : num [1:181] -5.45 -4.74 -3.95 -3.16 -2.37 ...
  ..$ : num [1:133] -3.37 -3.16 -2.37 -1.58 -0.79 ...
  ..$ : num [1:87] -1.91 -1.58 -0.79 0 0.79 ...
 $ contour_levels: num [1:9] 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1

Therein, fy_mean represents the crosswind-integrated footprint with coordinates x and xmax the location of the maximum footprint. (x2d, y2d, f2d) represent the 2d flux footprint and (xcontour, ycontour) the contour lines of the respective contour levels, which can be specified in the contoursargument in calc_flux_footprint. The output list ffp can be easily plotted using the function plot_flux_footprint, as shown in the following.

7.2.2 Plotting of flux footprint with `plot_flux_footprint`

The function plot_flux_footprint takes as input an object returned by calc_flux_footprint and plots the cross-wind integrated footprint and the 2d footprint.

plot_flux_footprint(ffp)

Flux footprint climatologies can be create as composite footprints by averaging over several FFP calculations, see the dedicated webpage https://www.footprint.kljun.net/. The flux footprint can then be plotted on an aerial photo or an ecosystem type classification, see e.g. Kljun et al. (2015) therein Fig. 5 or Pirk et al. (2023) therein Fig. 4.

References

Hsieh, C. I., G. Katul, and T. Chi. 2000. “An approximate analytical model for footprint estimation of scalar fluxes in thermally stratified atmospheric flows.” Advances in Water Resources 23 (7): 765–72. https://doi.org/10.1016/S0309-1708(99)00042-1.

Kljun, N., P. Calanca, M. W. Rotach, and H. P. Schmid. 2015. “A simple two-dimensional parameterisation for Flux Footprint Prediction (FFP).” Geosci Model Dev 8: 3695–3713. https://doi.org/10.5194/gmd-8-3695-2015.

Kormann, R., and F. X. Meixner. 2001. “An analytical footprint model for non-neutral stratification.” Boundary-Layer Meteorol 99: 207–24.

Pirk, N., K. Aalstad, Y. A. Yilmaz, A. Vatne, A. L. Popp, P. Horvath, A. Bryn, et al. 2023. “Snow-Vegetation-Atmosphere Interactions in Alpine Tundra.” Biogeosciences 20: 2031–47. https://doi.org/10.5194/bg-20-2031-2023.