Large-eddy structure and low-dimensional modelling of canopy turbulence or, Desperately seeking Big Eddy John J. Finnigan 1 and Roger H. Shaw 2 1 CSIRO, Atmospheric Research, Canberra, Australia 2 University of California, Davis, California

Contents Evidence for transport by coherent eddies in canopies Dynamics of the coherent structures-the Mixing Layer analogy An objective route to coherent structures-EOF analysis 3D structure of the eddy motion A dynamic model for the coherent eddies

Joint pdfs of wu and wc reveal both dominance by sweeps and intermittency of transport Data from Rivox forest, Gardiner (1994) Intermittency In the upper canopy ~90% of the momentum is transferred in ~5% of the time

Ensemble u-w-T and momentum and heat flux fields obtained from wavelet transforms triggering on ramps in a forest canopy (Collineau and Brunet, 1993)

Scalar ramps correlated through the depth of the canopy show wholesale flushing of the canopy airspace by large scale gusts. Data from Gao, Shaw and Paw U, (1989)

Composited velocity fields in the x-z plane LES data triggered on ramps

What have we learned directly from measurements? We know a good deal about the time evolution of transport events at a point. We know that the integral scales are large cf. the canopy height, that gusts of this size regularly flush the canopy and that the flushing events transport large amounts of both momentum and scalars. We know something about the shape of the gusts in the x-z plane from time-height plots of experimental data From conditionally sampled LES output we know a little about the three dimensional structure of the large eddies. We dont have a model for their dynamics

Coherent structures and predictive models Closure models (1st order, 2nd order, etc.) cant use information on eddy structure although it can be used to explain why they fail! Lagrangian transport models implicitly use the results on eddy scale but they cant be used to model the wind field Large eddy simulations dont need the information but can be tested against it To use our knowledge of coherent structure in predictive models we need a formulation where large eddies appear explicitly in the mathematics.

The Canopy- Mixing Layer Analogy Primary Instability: Kelvin-Helmholtz waves. Wavelength x is set by shear scale w. Clumped (Stuart) vortices retain initial wavelength. Thin sheet of vorticity between rollers is rapidly strained. Secondary instability in the vortex sheet leads to braids of streamwise vorticity that contain most of the total vorticity. Transverse spacing of braids is close to x.

Canopy-Mixing Layer Analogy (2). Linear perturbation models provide 3D eddy structure (eigenmodes) but may not be applicable to the fully turbulent case.

An objective approach to eddy structure: Empirical Orthogonal Functions (EOF) We are used to expanding turbulent fields in Fourier modes- sines and cosines- which are the eigenmodes of a vibrating string. EOFs are the eigenmodes (3D spatial patterns) that fit the actual turbulent flow as closely as possible in the sense that a smaller number of these eigenmodes must be added together to reproduce any given fraction of the turbulent kinetic energy than any other possible choice of spatial pattern.

EOFs are the eigenfunctions of the 2-point covariance field EOFs capture the spatial structure of the velocity field optimally in a least squares sense, (EOF=POD=PCA)

The original velocity fields, the two-point covariances and the turbulent stresses can be reconstructed from the EOFs * denotes complex conjugate (Lumley, 1981; Finnigan and Shaw, 2000)

Wind tunnel measurements of the 2-point covariance

We obtained the 2-point covariance field From measurements in an aeroelastic model canopy in a wind tunnel

In 1D just the first five EOFs capture most of the TKE in the canopy layer but convergence is slower in the surface layer

In 3D the first 5 eigenmodes account for 90% of the TKE and most of the structure in the covariance fields.

The characteristic eddy Because we constructed the empirical eigenvectors from time averaged covariances, we have lost information about the relative phases of the eigenvectors that make up the velocity patterns. That is we can reconstruct second moments but to reconstruct the velocity field that gave rise to them, we must add the information that was lost in the averaging process. A simple hypothesis is that the relative phases are those that make the resulting velocity pattern or eddy as compact as possible in space. With this simple assumption we can reconstruct the velocity and scalar fields of a characteristic eddy

We construct the 3D vector velocity field of the characteristic eddy in the WT from the first five EOFs with the weak assumption that an eddy is a structure that is compact in space. On the x-z plane we get: Rivox Forest Camp Borden

We have now repeated the EOF analysis on a detailed LES data set where u, v, w and scalar c were modelled Comparing the velocity patterns on the x-z plane we get: Wind tunnel LES

Velocity vectors in the y-z plane WT LES

characteristic eddy in the x-z plane from LES data

The 3D eddy structure reveals that the sweep transfer of uw and wc on the plane of symmetry is flanked by ejections. This pattern is a result of the double roller vortex structure of the characteristic eddy Distinct sweeps and ejections occupy the y-z plane

In the x-z plane of symmetry momentum and scalar are transferred by the same part of the eddy

And similarly in the y-z plane

Using EOFs to formulate a dynamic model for coherent structures in the canopy

Conclusions There is strong experimental evidence for the importance of large coherent structures in canopy turbulence The analogy between canopy flow and that in plane mixing layers provides a qualitative explanation for the origin of the eddies and suggests ways to scale eddy dynamics EOFs provide the 3D structure of canopy coherent structures without any a priori assumptions. We can see that they are double roller vortices with complex 3D structure Forcing the empirical eigenmodes to be compatible with the canopy flow equations generates a dynamic non-linear model for the coherent structures and suggests a new direction for canopy dynamics.