Canopy Spectra and Dissipation

1. Canopy Spectra and Dissipation John Finnigan CSIRO Atmospheric Research Canberra, Australia

2. The turbulent kinetic energy budget in the canopy

3. Canopy TKE BudgetVery high production rates of TKE are balanced by very large dissipation rates

4. Viscous Dissipation

7. The Canopy Dissipation Equation ++ Vortex stretching by turbulent strain field Vortex stretching by mean strain field Pressure, turbulent and dispersive transport of <> Destruction of <> by viscosity Vortex stretching in the strain fields around plants and their wakes Production/destruction of <> by coupling between local variations in mean viscous and Reynolds stresses Canopy waving; flux of –ve vorticity from solid surfaces + +/- - - ++ +/- +/- -

8. How can we measure and parameterize the dissipation? The problem: dissipation peaks at the Kolmogorov scale, in the atmosphere so we can’t measure eddies of this scale with conventional instruments. Instead we try to represent the dissipation in terms of the much larger and measurable eddies. To do this we need a model of the processes that transfer energy from the energy-containing to the dissipation scales. This is why Kolmogorov’s theory is so important for experimentalists-it relates  to the eddies in the ISR.

12. Kolmogorov Spectrum for free-air turbulence In the viscous subrange In the inertial subrange (ISR), Using dimensional and symmetry arguments, Kolmogorov (1941) found that in the ISR

15. Do measurements show the predicted form? Spectra measured in a spruce canopy by Amiro (1990) Canopy spectra measured at high Re generally show 11(k1) rolling off more rapidly than –5/3 in the low frequency end of the ISR but not all data show this. Isotropy in the ISR requires: And this is usually violated in canopy data

16. Measuring Spectra to compare with Kolmogorov and modified ISR theory

19. Comparison of the theory with spectra from the Moga experiment, a 10m high uniform Eucalyptus forest.

20. Canopy TKE Budget: estimating total dissipation The total dissipation is calculated as the sum of 2 terms: energy passing down the eddy cascade and work against drag by turbulence. The latter term represents the ‘spectral short-cut’. (Finnigan, 2000) Data from Brunet et al. (1996)

21. Implications for Modelling • In k-epsilon and higher order closure models the tke or variance equations must include the extra wake production by the mean flow working against drag. • The dissipation equation must include a re- parameterized eddy cascade term, plus WD(U, Cd), the term representing the work of the turbulence against drag. • The turbulent time scale used in all other parameterisations is formed from (Ayotte et al, 1999)

22. Conclusions • tke in canopies finds it way to dissipation scales via the eddy cascade and via interaction with the canopy elements • The assumptions of Kolmogorov theory for the form of E(k) in the ISR are violated because energy is continually lost as eddies do work against drag and the anisotropy of the drag ensures that the turbulence is at most axially symmetric, not isotropic • We can treat the two routes to the dissipation scales separately but we must derive a new form for E(k) in the ISR that reflects these extra processes • This form appears to predict the spectral levels and anisotropy of (two) canopy experiments quite well. • The total dissipation is the sum of the energy passing down the eddy cascade (reduced compared to free air) and the work of the turbulence against drag • This changes the turbulent time scale with implications for closure models.