Flux Footprints: Key Concepts
What is a flux footprint and how is it represented mathematically?
A flux footprint is the up-wind surface area that contributes to the flux measured at a given sensor position. In two dimensions it is described by a source-weighting density \(f(x, y, z_m)\) that varies with stream-wise distance x, cross-wind distance y, and measurement height \(z_m\).
A common representation splits the footprint into a cross‑wind–integrated component \(f_y(x, z_m)\) and a cross‑wind distribution \(D_y(x, y)\).
The cross‑wind–integrated footprint is obtained by integrating the two‑dimensional footprint over the y‑direction:
where:
\(f(x, y, z_m)\) is the two‑dimensional footprint density (m –2),
\(x\) is the along‑wind distance from the sensor (m),
\(y\) is the cross‑wind distance (m), and
\(z_m\) is the measurement height (m).
How are footprint values calculated in modelling?
Several numerical approaches exist. In Lagrangian stochastic models a large ensemble of virtual particles is released at the sensor height and followed backwards in time until they reach the surface. The footprint value assigned to a surface element is proportional to the number of particle “touch-downs” within that element—optionally weighted by the particles’ properties (e.g. vertical velocity).
Two common implementations are:
1. Grid‑counting The up-wind area is discretised into grid cells; all touchdown events in a cell are counted and normalised to give the footprint density on that cell.
2. Kernel density estimation (KDE) Touchdown locations are treated as sample points of an unknown probability density. A kernel function (e.g. Gaussian, bi‑weight) is centred on each touchdown and the kernels are summed to form a smooth, continuous footprint field.
What meteorological parameters are crucial for footprint prediction models?
The following variables exert primary control on footprint size, shape and location:
\(z_m\) — measurement height
\(z_0\) — aerodynamic roughness length
\(u_{\text{mean}}\) — mean wind speed at \(z_m\)
\(h\) — boundary‑layer height
\(L\) — Obukhov length (stability)
\(\sigma_v\) — standard deviation of lateral velocity fluctuations
\(u_*\) — friction velocity
Three non‑dimensional groups appear repeatedly:
\(z_m/L\) (stability),
\(z_m/h\) (relative sensor height),
the wind‑speed profile \(u(z)\).
How does atmospheric stability affect the footprint?
Atmospheric stability modulates turbulent mixing and therefore alters both the extent and the peak position of the footprint:
Unstable / convective (\(L < 0\)) Strong vertical mixing broadens the footprint and shifts its peak closer to the sensor, while the tail extends further down‑wind.
Neutral (\(|L| \to \infty\)) Footprint size and shape lie between unstable and stable limits.
Stable (\(L > 0\)) Suppressed turbulence produces a narrow, elongated footprint whose peak lies further up-wind of the sensor.
These regime-dependent differences give rise to distinct footprint climatologies when long time-series are stratified by stability class.
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