The structure of canopy turbulence and its implication to scalar dispersion Gabriel Katul 1,2,3 & Davide Poggi 3 1 Nicholas School of the Environment and Earth Sciences, Duke University, USA 2 Department of Civil and Environmental Engineering, Duke University, USA 3 Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Torino, Italy Seminar presented at the National Research Council, Institute of Atmospheric Sciences and Climate section of Lecce, Lecce-Monteroni, LECCE, ITALIA
Outline of the talk Motivation/Introduction Review of turbulent flows inside canopies on flat terrain (momentum transfer). Turbulent transport of biologically active scalars (e.g. CO 2 & H 2 O) on flat terrain. Flows and scalar transport inside canopies on gentle hills. Summary and conclusions
Introduction: Studies of turbulent transport processes at the biosphere-atmosphere interface are becoming increasingly hydrological and ecological in scope. Canopy turbulence no analogies to be borrowed from engineering applications (Finnigan, 2000).
Monin-Obukhov Similarity Theory (1954) Constant Flux Layer wc & wu f ( z) Inertial Layer (Production = Dissipation) Log profile for mean velocity Mixing length = k (z-d) Roughness sub-layer Canopy sub-layer hc 2-5 hc
Nonlinearities in the dynamics - Topography Airflow through canopies on complex topography: Tumbarumba AU (Leuning)
FLUME EXPERIMENTS To understand the connection between energetic length scales, spatial and temporal averaging, start with an idealized canopy. Vertical rods within a flume. Repeat the experiment for 5 canopy densities (sparse to dense) and 2 Re
FLUME DIMENSIONS PLAN Test section Open channel 9 m 1m Flow direction 1 m From Poggi et al., 2004
FLUME EXPERIMENTS PLAN VIEW Rods positions Weighted scheme + + + + + + + + + + + + + + + d r 2 cm 1 cm SECTION VIEW Canopy sublayer 2h h h w
σ w σ u Wind-Tunnel
Canonical Form of the CSL THE FLOW FIELD IS A SUPERPOSITION OF THREE CANONICAL STRUCTURES Displaced wall Boundary Layer Mixing Layer REGION III REGION II Real wall d REGION I
Rods Flow Visualizations TOP VIEW Flume Experiments Laser Sheet
REGION I: FLOW DEEP WITHIN THE CANOPY The flow field is dominated by small vorticity generated by von Kàrmàn vortex streets. Strouhal Number = f d / u = 0.21 (independent of Re) From Poggi et al. (2004)
Kelvin-Helmholtz Instability Mixing Layer U 1 U 2 Raupach et al. (1996) Canopy Flow - Mixing Layer U x ω y
Region II: Kelvin-Helmholtz Instabilities & Attached Eddies
Displaced wall Boundary Layer Mixing Layer REGION III REGION II Real wall d REGION I
REGION II: Combine Mixing Layer and Boundary Layer l = (1 α ) L BL + α L ML L BL = Boundary Layer Length = k(z-d) L ML = Mixing Layer Length = Shear Length Scale l = Total Mixing Length Estimated from an eddydiffusivity
From Kaimal and Finnigan (1994) Spatial Averaging & Dispersive fluxes Dispersive flux terms are formed when the time-averaged mean momentum equation is spatially averaged within the canopy volume. They arise from spatial correlations of timeaveraged velocity components within a horizontal plane embedded in the canopy sublayer (CSL).
Dispersive Fluxes Previous studies found that dispersive fluxes are small compared to the Reynolds stresses (mainly for high frontal area index) Bohm, M., Finnigan, J. J., and Raupach M. R.: 2000, Dispersive Fluxes and Canopy Flows: Just How Important Are They?, in American Meteorology Society, 24th Conference on Agricultural and Forest Meteorology, 14 18 August 2000, University of California, Davis, CA, pp. 106 107. Cheng, H. and Castro, I. P.: 2002, Near Wall Flow over Urban-Like Roughness, Boundary-Layer Meteorol. 104, 229 259.
Spatial Variability and Dispersive Fluxes
RANS Wilson and Shaw (1977)
Model parameters Drag Coefficient From Poggi, Porporato, Ridolfi, et al. (2004, BLM)
From Poggi, Katul, and Albertson (2004, BLM)
Methodology Canopy environment micro-meteorology
Simplified Scalar Transport Models Biologically Active Meteorological Forcing (~30 min) a(z) T a, RH, C a PAR S <U> dz Soil
Time-averaged Equations C t = q z + S c Time averaging ~ 30 minutes C ) = p( z, t zo, to Sc Fluid Mechanics p( z, t zo, to) > U, σ w,... At the leaf S c = ρ a( z) C r s i C + r Stomata Fickian Diffusion from leaf To atmosphere - b
Gradient-Diffusion Analogy? Counter-Gradient Transport Duke Forest Experiments z/h w' c' 1 1 = 0.34 mg kg m s CO 2 Concentration (ppm)
Include all three scalars: T, H2O, and CO2 3 scalars 9 unknowns (flux, source, and conc.) 3 conservation equs. for mean conc. 3 equations to link S conc. (fluid mech.) 3 equations for the leaf state 3 internal state variables (Ci, qs, Tl) 1 additional unknown - stomatal resistance (gs)
Farquhar/Collatz model for A-Ci, gs (2 eq.) Assume leaf is saturated (Claussius-Claperon q & Tl, 1 equ.) Leaf energy balance (Tl, 1 equ.) PROBLEM IS Mathematically tractable
Leaf Equation for CO2/H2O A n = α 1 ( * C ) i Γ C i + α 2 Farquhar model ARH n gs = m + b CO2 Collatz et al. model Fickian diffusion A = g ( CO2 C ) n s i 3 unknowns: An, gs, Ci
Duke Forest FACE-FACILITIES
Pinus taeda Well-watered conditions
Random Porous Media Light Representation Crown Clumping
Naumberg et al. (2001; Oecologia) sunfleck Under-story
Modeled S c CO 2 H 2 O Fluxes at z/h=1 Model ecophysiological parameters are independently measured using porometry (leaf scale). T Fluxes shown are measured at the canopy scales
Comparison between measured and modeled mean CO 2 Concentration CO 2 measured by a 10 level profiling system sampled every 30 minutes. Sources and sinks and transport mechanics are solved iteratively to compute mean scalar concentration
Gravity Waves: Stable Boundary Layer at z/h = 1.12 (Duke Forest) Uh (m/s) 1 0.5 0-0.5-1 T' (K) 50 40 30 20 10-10 0-20 -30-40 -50 q' (Kg/m 3 ) 0 2 4 6 8 10 12 14 Time (minutes) 4 3 2 1 0 w' (m/s) 0.6 0.4 0.2 0-0.2-0.4-0.6 CO2 u'w' (m/s) 2 0.2 0.1 0-0.1-0.2-0.3 Fc 6E-005 4E-005 2E-005 0-2E-005 0.0001-4E-005 0-0.0001-0.0002-0.0003-0.0004-0.0005-0.0006 0 2 4 6 8 10 12 14 RN (W/m 2 ) Time (minutes) 0.4 0-0.4-0.8-1.2 w't' (K m/s) 20 15 10 5 0-5 -10 w'q' -20-25 -30
Night-Time Nonstationarity Uh (m/s) 1 0.5 0-0.5-1 T' (K) 50 40 30 10 20-10 0-20 -30-40 -50 q' (Kg/m 3 ) 0 4 8 12 16 20 24 28 Time (minutes) 4 3 2 1 0 w' (m/s) 2 1.5 1 0.5 0-0.5-1 -1.5-2 CO2 u'w' (m/s) 2 0.2 0.1 0-0.1-0.2-0.3 Fc 0.0002 0.0001 0-0.0001-0.0002 0.0008-0.0003 0.0006 0.0004 0.0002 0-0.0002-0.0004-0.0006 0 4 8 12 16 20 24 28 RN (W/m 2 ) Time (minutes) 0.4 0-0.4-0.8-1.2 w't' (K m/s) 20 15 10 5 0-5 -10 w'q' -20-25 -30
On Complex Terrain Data from SLICER over Duke Forest ~1 km Canopy height comparable to topographic Variability- the more difficult case.
Model for Mean Flow Model topography has ONE mode of variability (or a dominant wave number responsible for the terrain elevation variance). Afternoon U Morning
Model Formulation: 2-D Mean Flow Continuity: U W + x z Mean Momentum Equation: = 0 Produced by the Hill U U x + W U z = 1 P ρ x + u w z F H ( z, d h c ) Two equations with two unknowns after appropriate parameterization Canopy Drag
Finnigan and Belcher (2004) uw ' ' = l 2 U z U z Closure for Reynolds Stress mixing length inside Canopy as before: F = CaUU d d b Closure as Drag Force:
Mean Flow Streamlines
Polytechnic of Turin (IT) Flume Experiments for Momentum Hill Properties: Four hill modules Hill Height (H) = 0.08 m Hill Half Length (L) = 0.8 m Canopy Properties Canopy Height = 0.1 m Rod diameter = 0.004 m Rod density = 1000 rods/m 2 Flow Properties: Water Depth = 0.6 m Bulk Re > 1.5 x 10 5
Velocity Measurements Sampling Frequency = 300 Hz Sampling Period = 300 s Laser Doppler Anemometer
FLUME EXPERIMENTS With Canopy Bare Surface
SPARSE: 300 rods m -2 DENSE: 1000 rods m -2
Red dye injected AFTER the Re-circulation zone
How is the effective mixing length altered by the re-circulation zone?
What happens to the second-order statistics? Data from all 10 sections for dense canopies
Topographically Induced Measured
LES Runs From Patton and Katul (2009)
H c = Canopy ht L c =Adjustment Length scale = 1/(Cd a) L=Hill half-length FB04 = Analytical solution of Finnigan and Belcher
Scalar Mass Transfer Advection topography induced Heaviside Step Function U C x + W C z = F c z + S c H ( z, hc ) 1-equation with 3 unknowns: C F = w c c Parameterize using First order closure and Ecophysiological Principles S c
df dz df dz c c = = S S c c W (Flat Terrain) C z + U C x Advective fluxes are opposite in sign + dfc dx They are often larger than Photosynthesis (Sc) From Katul et al. (2006)
Longitudinal Velocity Vertical Velocity df dz df dz c c = = S S c c (Flat Terrain) C W z + U C + x dfc dx
Summary and Conclusions: 1. Canopy sublayer can be divided into 3 regions that have dynamically distinct properties. These properties are sustained for gentle hills. 2. For gentle and dense canopies, experimental and analytical theories agree on the existence of a re-circulation zone. However, this zone is not a continuous rotor, rather oscillation between +ve and ve velocities. 3. No quantum jumps (like separation) exists in the turbulence statistics, unlike the mean flow. 4. Complex topography leads to breakdown in symmetry of concentration and flux variations.
SLICER = Scanning LIDAR Imager of Canopies by Echo Recovery From Lefsky et al., 2002
Comparison between SLICER and field measurements Data from Lefsky et al. (2002) BioScience, Vol. 52, p.28
Acknowledgements The Fulbright-Italy Fellows Program National Science Foundation (NSF-EAR, and NSF-DMS), Department of Energy s BER program through NIGEC and TCP.
T L u* k z v Δu From Patton and Katul (2009)