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)