The Urban Canopy and the Plant Canopy

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1 The Urban Canopy and the Plant Canopy John Finnigan: CSIRO Australia Margi Bohm: University of Canberra Roger Shaw: U C Davis Ned Patton: NCAR Ian Harman: CSIRO Australia

2 Properties of turbulent shear flows over deep rough surfaces 1. Turbulence structure in plant canopies and the Roughness Sub Layer Current understanding of ISL eddy structure New evidence of RSL eddy structure How does it differ from the Inertial Sub Layer (Logarithmic layer) 3. A phenomenological model for Canopy-RSL turbulence 3. Turbulence structure in urban canopies and the Roughness Sub Layer How similar is it to that in a vegetation canopy? 4. Extending the model for large eddies in vegetation canopies to urban canopies When should it work and when should it fail? A new hypothesis 5. Conclusions

3 Urban canopies and plant canopies absorb momentum over a finite depth, not at a surface plane

4 Boundary layer Velocity Profiles Smooth wall Rough wall U U

Monin-Obukhov Similarity Theory (MOST) fails close to rough surfaces like plant or urban canopies Wind Potential temperature observations MOST Flow over tall

5 Monin-Obukhov Similarity Theory (MOST) fails close to rough surfaces like plant or urban canopies Wind Potential temperature observations MOST Flow over tall (plant) canopies does not conform to MOST to within 2-3 canopy heights this is the roughness sublayer. NB these are not errors in choice of d Chen and Schwerdtfeger (1989)

6 Flow statistics in the RSL are distinctly different from the ISL above: eddy coherence EOF spectra converge more rapidly than in the ISL, suggesting turbulence is dominated by coherent eddies with a distinct spatial scale (typically >80% of tke in first 5 eigenmodes) EOF analysis of wind tunnel canopy model: Finnigan and Shaw (2000)

7 Flow statistics in the RSL are distinctly different from the ISL above: correlations, higher moments, momentum and scalar exchange The correlation coefficient is ~0.5 in the neutrally stratified RSL/canopy as compared to ~0.35 in the ISL. Together with the smaller gradients for the same flux, this shows that RSL/canopy turbulence is in some sense more efficient at transport than that in the ISL. Skewnesses are O[1] and transport terms in covariance budgets are large-the turbulence is not in local equilibrium The turbulent Schmidt and Prandtl numbers are ~0.5 at the canopy- RSL interface compared with ~1.0 in the ISL. (Raupach, Finnigan and Brunet (1996))

8 Flow statistics in the RSL are distinctly different from the ISL above: ejection: sweep ratios Transport of momentum and scalars in the RSL/ canopy layer is dominated by the sweep quadrant whereas through the rest of the ISL, ejections dominate. Circles: almond orchard diamonds: cork oak plantation triangles: wind tunnel model canopy squares: DNS of urban blocks Ratios Q2/Q4 of the contributions to momentum flux from ejections and sweeps from a range of canopies

9 What is the origin of these differences? It is reasonable to assume that turbulence in the RSL must have a structure that is distinctly different from that in the ISL We will look first at the current thinking on eddy structure in the ISL over smooth walls This is obtained mainly from analysis of DNS data in boundary layers and channel flows. We will compare this with earlier and very recent analysis of eddy structure in vegetation canopies and RSLs

With a series of collaborators, Adrian has shown that the dominant, ejection-producing structures in the ISL are packets of head-up hairpin vortices The initial hairpin is presumed to formed by the

10 With a series of collaborators, Adrian has shown that the dominant, ejection-producing structures in the ISL are packets of head-up hairpin vortices The initial hairpin is presumed to formed by the upward deflection of a spanwise vortex line of the mean shear by a vertical gust. The initial HU hairpin generates an ejection that deflects another vortex line, eventually producing a packet aligned around an elongated low speed region (Gerz et al, 1994; Zhou et al, 1999, Tomkins and Adrian, 2003 )

What is the symmetry breaking mechanism that allows HU hairpins and hence ejections to dominate? In a homogeneous shear flow, head-up and head-down deflections are equally probable.

11 What is the symmetry breaking mechanism that allows HU hairpins and hence ejections to dominate? In a homogeneous shear flow, head-up and head-down deflections are equally probable. Near a wall in contrast, inward gusts are blocked so that outward deflections dominate. The larger the initial deflection, the more the vortex loop is stretched and rotated by the mean shear and its vorticity amplified. U(z) Questions remain about the mechanism of initial scale selection. Over a smooth wall, a single dominant scale does not seem to be strongly selected at any level. Schematics from Gerz et al. 1994

12 Structure of Canopy Turbulence Time-height traces from single towers in tall canopies give information about the x-z plane

13 Scalar ramps correlated through the depth of the canopy show wholesale flushing of the canopy airspace by large scale gusts Gao et al (1989), Camp Borden, Canada

14 Compositing shows that these ramps are signals of a scalar microfront compressed between downwind ejections and upwind sweeps Gao et al (1989), Camp Borden, Canada

15 Structure of Canopy Turbulence 3d structure from detailed wind tunnel simulations 1. Tombstones Raupach, Coppin and Le

16 hm, Finnigan and Raupach (2000) 2. The Black Forest canopy

17 unet, Finnigan and Raupach (1994) 3. The waving wheet canopy

18 4. Water flume experiment over multiple hills covered with a deep canopy Poggi, D., Katul, G.G. and Finnigan, J.J., 2006

19 Finnigan and Hughes (2008) 5. Stably Stratified canopy model on a 2D Hill

20 3D Structure. The eddy structure of the RSL has been reconstructed from multi-point analysis of a WT model canopy and LES simulation of the same canopy using conditional sampling (compositing) and EOF analysis These are iso-surfaces of λ2, a measure constructed from the invariants of the du i /dx j tensor that captures aspects of the local vorticity and global rotation associated with the vortices (Jeong and Hussain, 1995). Data from canopy LES using NCAR code of Patton and Sullivan. Sampling triggered by p +

The dominant eddy structure of the Canopy-RSL has been reconstructed from multi-point analysis of a WT model canopy (wheet) and LES simulation of the same canopy using conditional sampling

21 The dominant eddy structure of the Canopy-RSL has been reconstructed from multi-point analysis of a WT model canopy (wheet) and LES simulation of the same canopy using conditional sampling (compositing) and EOF analysis These are iso-surfaces of λ2, a measure constructed from the invariants of the dui/dxj tensor that captures aspects of the local vorticity and global rotation associated with the vortices (Jeong and Hussain, 1995). Coincidence of λ2 and vortex core is seen in this plot of velocity vectors projected on the y-z plane

22 y-z slice through the composite eddy

40 u -w vectors from the same composite eddy on the x-z plane of symmetry

41 Convergence between the underlying ejection and overlying sweep produces a scalar microfront. Shear stress <u w > is concentrated between the hairpin legs The scalar is released from the canopy at a uniform rate (independent of local wind velocity). Within a structure, a sloping microfront is formed with high concentration below and in advance of the front, while low concentration follows and is above the microfront. Blue- λ2 isosurface Green-scalar microfront Red- u w sweep Orange-u w ejection

42 We have evidence that the Head-up and Head-Down hairpins are formed simultaneously as the linear instability theory suggests

Compositing using the high pressure trigger above the RSL produces a less coherent structure with the HU hairpin and ejection much stronger than the HD and sweep This composite is

43 Compositing using the high pressure trigger above the RSL produces a less coherent structure with the HU hairpin and ejection much stronger than the HD and sweep This composite is formed at z=3hc, about twice the height above the canopy where ejections first equal sweep contributions to Blue- λ2 isosurface Green-scalar microfront Red- u w sweep Orange-u w ejection

44 So what happens at the top of a canopy that is different to the ISL above in order to produce this distinct eddy structure? Unlike the boundary layer profile, the inflected velocity profile at canopy top is inviscidly unstable, leading to rapid growth and strong selection for a single scale, proportional to the vorticity thickness δ ω. Spanwise Stuart vortices develop which can be deflected into HU and HD hairpins. This is the mixing layer analogy (Raupach et al, 1996).

45 What happens at the top of a canopy that is different to the ISL above? Trains of transverse Stuart vortices are unstable and subject to a helical pairing instability which was described by Pierrehumbert and Widnall (1982). Non-linear development of the linear eigenmodes results in trains of Head-up and Head-down vortices spaced at twice the wavelength of the original Kelvin-Helmholtz inflection point instability

46 We have modelled a sequence of Stuart vortices in an inflected (tanh-1) velocity profile with an initial (isotropic) background turbulence intensity of 0.05% U

47 The top view shows the spanwise scale selection. Note, only the central part of the calculation domain is shown

48 Two opposing symmetry breaking mechanisms explain the shift from sweep dominance to ejection dominance as we move above the canopy The presence of the porous canopy allows HD hairpins to be deflected downwards -as long as their spanwise scale is <h c In the canopy-top shear flow, HD hairpins are stretched and rotated faster than HU hairpins so HD s dominate Further from the canopy top, large scale upward deflections become dominant again as downward deflections are blocked by the solid wall so that HU hairpins begin to dominate.

49 Vegetation canopy/rsl Eddy structure: Summary 1 The inflected velocity profile at the canopy top provides strong scale selection by an inviscid instability mechanism (unlike in a BL profile) The secondary instability produces trains of coherent spanwise vortices that are unstable to small perturbations The tertiary (helical pairing) instability preferentially produces HUs and HDs in pairs The convergence between the sweeps and ejections produces intense scalar microfronts

50 Vegetation canopy/rsl Eddy structure: Summary 2 Two symmetry-breaking mechanisms act in opposition to determine whether HUs or HDs dominate Near solid walls, large downward deflections are blocked so energetic HD hairpins cannot be formed. A porous canopy layer allows HD deflections of order h c near the canopy top. In a boundary-layer shear flow HD hairpins are stretched and rotated faster than HUs and so sweeps dominate ejections. Further from the canopy top the larger amplitude deflection of HUs overrides the greater local stretching of HDs and so ejections dominate

51 RSL model obtained by using length in MOST theory as an extra scaling LAI ~ 2 β = 0.31 n=17 Tumbarumba LAI ~ 3 β = 0.39 n=42 Duke LAI ~3.8 β = 0.28 n=27 Harman and Finnigan (2007, 2008) increasingly dense

52 Urban canopy statistics Q2/Q4 Castro, Cheng and Reynolds (2006) Various roughness types Bohm, Finnigan and Raupach (2009) Black Forest vs LES of Wheet WT canopy Finnigan, Shaw and Patton (2009), Coceal et al. (2006) Blocks vs vegetation

53 But, not all urban canopies have Q4>Q2 Diagonal Array of blocks Square Array of blocks Kanda, 2006 LES data

54 And not all urban canopies have r uw >0.35 Diagonal or Staggered Array of blocks Random array Square Array of blocks Cheng and Castro (2001) WT data

55 And not all urban canopies have r uw >0.35 The Black Forest WT model canopy of smooth bluff objects shows characteristics of canopy (K type roughness) and boundary layer (D type roughness) statistics Bohm, Finnigan and Raupach (2009)

56 A Hypothesis: Urban canopies that behave like plant canopies have the same statistical eigenmodes as attractors for their large eddies In vegetation canopies, the inflection point in the mean velocity is inviscidly unstable and generates energetic coherent eddies The wakes shed by plant elements are much smaller in scale than the eigenmodes generated by the inflected mean velocity profile Because of this, the inflected velocity profile is dynamically significant even though instantaneously it is distorted by plant wakes Under what circumstances could these dynamics be relevant to urban canopies, where the size of the element wakes is of the same order as the large eddies? We will examine the equation governing the relationship of a local velocity perturbation to the spatially averaged, inflected mean velocity profile

57 A note on the use of linear theory in fully turbulent flows Perhaps surprisingly, there is a long history of successful application of linear theory to explain the eddy structure of fully developed turbulent shear flows It has been applied to understand large-scale structures in free turbulent shear flows like mixing layers where the coherent eddies correspond to unstable eigenmodes (eg. Liu, 1988) And to boundary layer and channel flows where the coherent eddies correspond to singular neutral modes. We call this Rapid distortion Theory (Townsend, 1976; Hunt and Carruthers, 1990). In each case the central assumption is that the dominant eddy structures are determined by the interaction of the turbulence with the mean flow and that non-linear, turbulent-turbulent interactions act primarily to destroy the large eddies.

58 Averaging the velocity and Reynolds stresses in time and space x x

59 Equation for the local (in space and time) velocity perturbation Terms in black are considered in linearized approaches to analysing canopy large eddy structure such as hydrodynamic stability and rapid distortion theory. Terms in red represent the distortion of the local velocity perturbation by local variations in the time-averaged strain rates around buildings or blocks The non linear term in blue contains both the spatially averaged Reynolds stress and the local variations around this.

60 Mean Velocity and its spatial gradient in a canopy of blocks Coceal, Thomas and Belcher (2007)

61 Reynolds Shear Stress and its spatial gradient in a canopy of blocks Coceal, Thomas and Belcher (2007)

62 Hypothesis If the mean velocity gradients and Reynolds stress gradients vary on a scale that is similar or larger than the spatial scale of the inflexionpoint eigenmode, then we will not observe coherent eddies similar to those in vegetation canopies. Instead the turbulence will be dominated entirely by the eddies shed from individual obstacles and will have no universal features If the opposite is true, double hairpin type eddies will emerge and the characteristic features of vegetation canopy turbulence will be seen. The eigenmode size is linked to the vorticity thickness Urban canopies with large relative to element size as a result of varying building height or complex element geometry, may, therefore, have dynamically significant inflection point profiles Note that this hypothesis breaks down if the canopy elements are too sparse

63 Example from Black Forest

64 Conclusions The turbulence in and just above vegetation canopies is significantly more coherent than in the ISL or Log layer above and more efficient at transporting momentum and scalars. We can extract the space-time structure of canopy RSL eddies by compositing and we find that they have a double hairpin structure that is quite different to the Head-up hairpins or attached eddies generally assumed in smooth wall shear layers. A sequence of instability processes triggered by the inflected shear layer at the canopy top together with two opposing symmetry-breaking mechanisms successfully explains the eddy structure as well as a very wide range of observations in vegetation canopies

65 Conclusions In urban canopies, the crucial inflexion point is also present in the spatially averaged time-mean velocity profile but may not be dynamically significant Furthermore, not all urban canopy data show the universal features seen in vegetation canopies such as Q2<Q4 or r uw ~0.5 However, by comparing the magnitudes of terms in the equation for a local velocity perturbation, we have suggested a simple criterion to distinguish urban canopies that behave in a similar way to natural canopies from those that do not This criterion remains to be tested against the data.

Roughness Sub Layers John Finnigan, Roger Shaw, Ned Patton, Ian Harman

Roughness Sub Layers John Finnigan, Roger Shaw, Ned Patton, Ian Harman 1. Characteristics of the Roughness Sub layer With well understood caveats, the time averaged statistics of flow in the atmospheric