The structure of canopy turbulence and its implication to scalar dispersion
Gabriel Katul1,2,3 & Davide Poggi3
1Nicholas School of the Environment and Earth Sciences, Duke University, USA
2Department of Civil and Environmental Engineering, Duke University, USA
3Dipartimento 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/IntroductionReview of turbulent flows inside canopies on flat terrain (momentum transfer).Turbulent transport of biologically active scalars (e.g. CO2 & H2O) 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)
2-5 hchc
Roughness sub-layerCanopy sub-layer
Constant Flux Layer Inertial Layer (Production = Dissipation)
w c′ ′ w u′ ′& ( )f z≠
Log – profile for mean velocity …Mixing length = k (z-d)
Nonlinearities in the dynamics - TopographyAirflow through canopieson complex topography:
TumbarumbaAU (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
Open channel
Test sectionFlow direction9 m
1 m
1m
From Poggi et al., 2004
FLUME EXPERIMENTSWeighted scheme
Rods positions
hwdr
Canopy sublayer
2h
2 cm
1 cm
h
+++++++++++++++
SECTIONVIEW
• PLAN• VIEW
wσuσ
Wind-Tunnel
Canonical Form of the CSL
THE FLOW FIELD IS A SUPERPOSITION OF THREE CANONICAL STRUCTURES
d
Displaced wall
Real wallREGION I
REGION II
REGION IIIBoundaryLayer
MixingLayer
TOP VIEWFlume Experiments
Flow Visualizations
Laser Sheet
Rods
The flow field is dominated by small vorticity generated by von Kàrmàn vortex streets. Strouhal Number = f d / u = 0.21 (independent of Re)
REGION I: FLOW DEEP WITHIN THE CANOPY
From Poggi et al. (2004)
xU
Kelvin-Helmholtz Instability
Mixing Layer
U2
U1
Canopy Flow - Mixing Layer
yω
Raupach et al. (1996)
Region – II: Kelvin-Helmholtz Instabilities & Attached Eddies
d
Displaced wall
Real wallREGION I
REGION II
REGION IIIBoundaryLayer
MixingLayer
REGION II: Combine Mixing Layer and Boundary Layer
LBL= Boundary Layer Length = k(z-d)LML = Mixing Layer Length = Shear Length Scalel = Total Mixing Length Estimated from an eddy-diffusivity
MLBL LLl αα +−= )1(
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 time-averaged velocity components within a horizontal plane embedded in the canopy sublayer (CSL).
From Kaimal and Finnigan (1994)
Dispersive Fluxes
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.
Previous studies found that dispersive fluxesare small compared to the Reynolds stresses (mainly for high frontal area index)
Spatial Variability and Dispersive Fluxes
RANS – Wilson and Shaw (1977)
DragCoefficient
From Poggi, Porporato,
Ridolfi, et al. (2004, BLM)
Model parameters
From Poggi, Katul, and Albertson (2004, BLM)
Methodology
Canopy environment – micro-meteorology
Simplified Scalar Transport Models – Biologically Active
PAR
dzS
Meteorological Forcing (~30 min)
Soil
Ta, RH, Caa(z)
••• ••
• •••
••
<U>
Time-averaged Equations
Time averaging ~ 30 minutes
At the leafStomata
Fickian Diffusion from leafTo atmosphere -
Fluid Mechanics
cSzq
tC
+∂∂
−=∂∂
∫∫= coo StztzpC ),|,(
,....,),|,( woo Utztzp σ>−−
bs
ic
rrCCzaS
+−
= )(ρ
CO2 Concentration (ppm)
z/h
Duke Forest ExperimentsCounter-Gradient Transport
Gradient-Diffusion Analogy?
1134.0'' −−−= smkgmgcw
Include all three scalars: T, H2O, and CO2
3 conservation equs. for mean conc.3 equations to link S conc. (fluid mech.)3 equations for the leaf state
3 scalars 9 unknowns (flux, source, and conc.)
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
( )*1
2
in
i
CA
C
α
α
−Γ=
+Farquhar model
Collatz et al. model
Fickian diffusion
3 unknowns: An, gs, Ci
2n
sA RHg m bCO
= +
( 2 )n s iA g CO C= −
Duke Forest FACE-FACILITIES
Pinus taeda
Well-wateredconditions
LightRepresentation
RandomPorousMedia
CrownClumping
Under-story
Naumberg et al. (2001; Oecologia)
sunfleck
T
CO2
H2O
Model ecophysiological parameters are independently measured using porometry (leaf scale).
Fluxes shown aremeasured at the canopy scales
Fluxes at z/h=1Modeled Sc
Comparison between measured and modeled mean CO2Concentration
Sources and sinks and transport mechanics are solved iteratively to compute mean scalar concentration
CO2 measured by a 10 level profiling system sampled every 30 minutes.
Gravity Waves: Stable Boundary Layer at z/h = 1.12 (Duke Forest)
Uh
(m/s
)
0
1
2
3
4
-1
-0.5
0
0.5
1
w' (m
/s)
T' (K
)
-0.6-0.4-0.200.20.40.6
-50-40-30-20-10
01020304050
CO2
Time (minutes)
q' (Kg
/m3 )
-0.0006-0.0005-0.0004-0.0003-0.0002-0.000100.0001
0 2 4 6 8 10 12 14
u'w' (m
/s)2
-1.2-0.8-0.400.4
-0.3-0.2-0.1
00.10.2
w't
' (K
m/s
)
Fc
-10-505101520
-4E-005-2E-005
02E-0054E-0056E-005
w'q
'
Time (minutes)
RN (W
/m2 )
-30
-25
-20
0 2 4 6 8 10 12 14
Night-Time Nonstationarity
Uh
(m/s
)0
1
2
3
4
-1
-0.5
0
0.5
1
w' (m
/s)
T' (K
)
-2-1.5-1-0.500.511.52
-50-40-30-20-10
01020304050
CO2
Time (minutes)
q' (Kg
/m3 )
-0.0006-0.0004-0.000200.00020.00040.00060.0008
0 4 8 12 16 20 24 28u'
w' (m
/s)2
-1.2-0.8-0.400.4
-0.3-0.2-0.1
00.10.2
w't
' (K
m/s
)
Fc
-10-505101520
-0.0003-0.0002-0.0001
00.00010.0002
w'q
'
Time (minutes)RN
(W
/m2 )
-30
-25
-20
0 4 8 12 16 20 24 28
On Complex Terrain
~1 km
Data from SLICER over Duke Forest
Canopy height comparable to topographicVariability- the more difficult case.
MorningAfternoon U
Model for Mean Flow
Model topography has ONE mode of variability (or a dominant wave number responsible for the terrain elevation variance).
Model Formulation: 2-D Mean Flow
Continuity:
Mean Momentum Equation:
Two equations with two unknowns –after appropriate parameterization
Produced by the Hill
CanopyDrag
0=∂∂
+∂∂
zW
xU
),(1cd hzF
zwu
xP
zUW
xUU H×−
∂′′∂
+∂∂
−=∂∂
+∂∂
ρ
Finnigan and Belcher (2004)
Closure for Reynolds Stressmixing length insideCanopy – as before:
Closure as Drag Force:d d bF C aU U=
2' ' U Uu w lz z
∂ ∂= −
∂ ∂
Mean Flow Streamlines
Hill Properties:Four hill modulesHill Height (H) = 0.08 mHill Half Length (L) = 0.8 m
Canopy PropertiesCanopy Height = 0.1 mRod diameter = 0.004 mRod density = 1000 rods/m2
Flow Properties:Water Depth = 0.6 mBulk Re > 1.5 x 105
Polytechnic of Turin (IT) Flume Experiments for Momentum
Velocity MeasurementsSampling Frequency = 300 Hz
Sampling Period = 300 sLaser 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)
Hc = Canopy ht
Lc=Adjustment Length scale =1/(Cd a)
L=Hill half-length
FB04 = Analytical solutionof Finnigan and Belcher
Scalar Mass Transfer
1-equation with 3 unknowns:
Parameterize using First order closure and Ecophysiological Principles
Heaviside Step Function
Advection – topography induced
),( ccc hzS
zF
zCW
xCU H×+
∂∂
−=∂∂
+∂∂
C
cwFc ′′=
cS
Advective fluxes are opposite in sign
They are often larger than Photosynthesis (Sc)
From Katul et al. (2006)
⎟⎠⎞
⎜⎝⎛ +
∂∂
+∂∂
−=
=
dxdF
xCU
zCWS
dzdF
SdzdF
cc
c
cc (Flat Terrain)
⎟⎠⎞
⎜⎝⎛ +
∂∂
+∂∂
−=
=
dxdF
xCU
zCWS
dzdF
SdzdF
cc
c
cc (Flat Terrain)
LongitudinalVelocity
VerticalVelocity
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
Data from Lefsky et al. (2002) – BioScience, Vol. 52, p.28
Comparison between SLICER and field measurements
AcknowledgementsThe Fulbright-Italy Fellows Program
National Science Foundation (NSF-EAR, and NSF-DMS),
Department of Energy’s BER program through NIGEC and TCP.
uΔ
zkuT
vL
*≈
From Patton and Katul (2009)