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An introduction to the soil-plant-atmosphere system
Gabriel Katul
Alpine Summer School, Course XXIII
Valsavarenche, Valle D’Aosta, Italy, (22 June – 1 July, 2015)
Tuesday, June 23, 2015
Lecture 4
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“The spread in width and depth of the multi-various branches of
knowledge during the last hundred odd years has confronted us
with a queer dilemma.
We are only now beginning to acquire reliable material for
welding together the sum total of all that is known into a whole;
but, on the other hand, it has become next to impossible for a
single mind to command more than a small specialized portion of
it.
I can see no other escape from this dilemma than that some of us
should venture to embark on a synthesis of facts and theories,
albeit with second-hand and incomplete knowledge of some of
them - and at the risk of making fools of ourselves”.
WHAT IS LIFE? ERWIN SCHRODINGER
DISCLAIMER
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The soil-plant-atmosphere - numerous interactions that share attributes
with molecular systems (high-dimensional).
Fundamental differences that prevent applications of statistical
mechanics:
1. The fundamental sub-macroscopic laws describing water movement
within the plant system are not entirely known (e.g., water flow at the
root-soil interface, in the xylem, and in the leaf );
2. Averaging the individual properties of these sub-macroscopic laws
may not provide a meaningful description for the next hierarchical level
because of nonlinear interactions and lack of scale separation (i.e.,
presence of significant variability at all scales);
3. The drivers of many macroscopic laws remain stochastic because the
soil-plant system is open to external environmental forcing such as
rainfall, temperature, and radiation.
Statement of the problem
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Variability in time
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A Strogatz like diagram of the dimensionality-nonlinearity problem
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Part 1: Brief introduction to the soil-plant-atmosphere
continuum
Part 2: Below ground processes
Part 3: Above ground plant processes
Part 4: General introduction to canopy flows
Part 5: Upscaling from leaf-to-canopy and beyond
Outline of General Lecture
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Part 2: Below ground processes
“We know more about the movement of celestial bodies than
about the soil underfoot”
Leonardo Da Vinci
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Bulk Soil Properties Defined
Solid
Water
Air
VTot
Mtot
Vair
Vwat
Vsol Msol
Mwat
Mair
VOLUME MASS
tot
wat
watair
airair
airwat
wat
tot
watair
tot
solb
V
V
VV
Vf
VV
VS
V
VVf
V
M
Bulk Density
Porosity
Degree of
Saturation
Air-filled
porosity
Soil moisture
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Representative Elementary Volume and the continuum hypothesis
• Continuum Assumption in Fluids
Scale
Density of
a Fluid
Scale
Density
independent
of scale
Macroscopic features such
as salinity or temperature
impact density
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State Equation
;0
z
q
t
dz
qin
qout
Rainfall P(t)
CONSERVATION OF WATER MASS:
1 equation, 2 unknowns
GROUND SURFACE
Mathematical Closure – Need to connect the flux to state variables.
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Fluxes and flow of water Transport laws:
Examples:
• Ohm’s law: i = (1/R) V – current is flux of electron, V = electric potential, R = resistance
• Fourier’s law: qh = - Kh (dT/dL) – flux of heat is proportional to the temperature gradient. Proportionality constant is thermal conductivity.
• Fick’s law: qc = - D (dC/dL), mass flux is proportional to concentration gradient, proportionality constant is molecular diffusion coefficient.
Fourier
Ohm
Ficks
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Fluxes and important flows of water
• Darcy’s Law:
dL
dHKqw )(
Hydraulic conductivity
Energy gradients – that themselves
vary with soil moisture
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Fluxes and flow of water
• Energy = Kinetic + Potential
g
V
2
2
Pressure
(see text in ppt) Gravitational
(see text in ppt)
Potential Energy –
Related to the ability of this
energy to be used as work
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Fluxes and flow of water
Pgz
g
VH
2
2Total Energy
Head
(in m of Water)
Kinetic energy head
Usually small in soils
and ignored
Gravitational Potential –
z = distance from datum
g = gravitational acceleration
, = specific weight and density of water
Pressure potential –
Amount of energy needed to
Break the adhesive forces
between water and solids
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Fluxes and flow of water
Usage of Darcy’s law requires the:
• (i) Hydraulic conductivity function
• (ii) Soil water retention – describing the relation between the energy needed to pull the water held between soil particles by adhesion and the soil moisture
)(K
P)(
dL
dHKqw )(
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Fluxes and water flow
• Two fundamental
Hydraulic functions
Characterizing soil type
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Fluxes and important flows of water
32
)()(
)()(
b
s
ss
b
s
ss
KK
bKsss ,,,
see Clapp and Hornberger (1978) - for example
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State Equation with roots
1)(
)(
);,(
zKq
zSz
q
t
dz
qin
qout
Rainfall P(t)
S
Root-water uptake
Root water
uptake
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Forcing
• Boundary conditions: Rainfall or through-fall as a function of time
• Drainage fluxes, ground-water level (saturated conditions)
• Initial soil moisture state – rarely known a priori – though creative ways of assuming it available.
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Measurements – soil moisture
Local – time domain
reflectometry
Field scale – passive microwave
remote sensing
SMAP = remote sensing from space
Limited to top 10 cm, and 10’s of km resolution
http://smap.jpl.nasa.gov/
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Dynamic Responses
• Qualitative behavior of soil-plant system (i.e. presence of roots) as forced by variable rainfall and mediated by storage and losses to atmosphere (via root-water uptake).
• Example use of impulse-response via spectral analysis – precipitation Impulse
• Soil moisture data from Duke forest – case study.
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Broader implications of case study for
Soil moisture dynamics and climate
• Because of storage effects within the soil pores, the dynamics of soil moisture posses a memory that is often considerably longer than the integral timescale of many atmospheric processes.
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Background – Soil moisture dynamics and climate
• Hence climate anomalies can be ‘‘sustained’’ through land surface feedbacks primarily because they can ‘‘feed off’’ on this long-term memory.
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Experimental Results
Canonical findings across experiments are:
1) The amplitude of soil moisture variations decreases with
soil depth.
2) Soil moisture ‘memory’ across various geographic
regions increases for dryer states when compared to
wetter conditions.
3) Soil moisture is generally in-phase with precipitation at
long-time scales but can be out-of-phase for short time
scales.
Robock et al.,2000
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Qualitative Analysis of Soil Moisture Response to Rainfall Fluctuations
• Qualitative analysis – here explored by linking the spectrum of soil moisture content at time-scales ranging from minutes to inter-annual to the spectrum of the forcing variable - rainfall.
• Focus on a case study in which 8 years of 30-minute spatially and depth - averaged soil moisture time series sampled by TDR is available along with precipitation, throughfall, and eddy-covariance based evapotranspiration.
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Precipitation
Transpiration
Evaporation
Drainage
Through-fall
Soil Porosity Root-
Depth
RL
Dimensionless
( ) ( ) ( );rL t ET t D t ( ) ( ) / Ls t w t R
( ) ( ) ( )
L L
ds t L t p t
dt R R
( )( ) ( ) ( )
r
dw tp t ET t D t
dt
DEPTH-AVERAGED CONTINUITY
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4 rods per ring
1998-2005 – 8 years of 30 min. data
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Sample Time Series Measurements Duke Forest, Durham, NC
0 500 1000 1500 2000 2500 30000
5
10
15
20
25
30
Pre
cip
itati
on
(m
m)
0 500 1000 1500 2000 2500 3000-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
ET
(m
m)
0 500 1000 1500 2000 2500 30000.1
0.2
0.3
0.4
0.5
0.6
Time (days)
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Pi ~ 1280 mm y-1 [Measured]
Interception ~ 40% of P ~ 512 mm y-1 [See data below]
ET ~ 650 mm y-1 [Measured by EC]
Through-fall ~ Pi-Interception ~ 768 mm y-1 = P(t)
ET/Through-fall ~ 85%
L(t) ET ( ) ( ) ( )
L L
ds t ET s p t
dt R R
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Modeling Soil Moisture Dynamics: ET-s relation
ET/ETmax
S 1.0 0
1.0
Linear Model
Nonlinear
Model
Uniform Model
from Porporato et al. (2004)
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Models for Soil Moisture Dynamics Linearized model
1
( ) ( )( )
L
ds t p ts s
dt R
max1( ) ( )
L
ETs f s
R
Linear Model: f(s) = 1
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Spectral Analysis of Soil Moisture
Soil moisture spectrum Es(f)
Fourier-Transform:
( ) ( )
1( ) ( )
2
i f t
i f t
H f h t e dt or
h t H f e df
2
2 2
1
| ( ) |( ) ;s
P fE f
f
max1( )
L
ETs
R
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Phase Shifts (from Katul et al., 2007)
(1) By increasing the rooting zone depth (dr), the rainfall and
soil moisture variability become increasingly out-of-phase.
(2) for long time scales (e.g., decadal), f0 and soil moisture
and rainfall variability become in-phase with each.
(3) Lowering ETmax, rainfall and soil moisture become out-of-
phase.
Consistent with linear phase shift analyses reported by Amenu
et al. [2005] (Illinois Climate Network stations).
1
max( ) tan ( / )rf f d ET
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Precipitation
Evapotranspiration
Soil moisture
Duke Forest Experiment – 8 years of 30-min. Data
1/f for pink noise, 1/f^2 for red noise)
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2| ( ) |
.
P f
Const
1 max
1 1
45L
ETR d
2
2 2
1
| ( ) |( )s
P fE f
f
=0.55 and 300LR mm maxET =0.9 mm h-1
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Qualitative Analysis – Systems Approach
•Simplified hydrologic balance suggests
that for white-noise precipitation, soil
moisture becomes red (decaying as f-2).
•Analytical model for memory
1 max
1 LR
ET
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Implications to climate
• If soil moisture memory (here ~ 45 days) is
>> 12 hours, then diurnal dynamics of
soil moisture do not contribute much
to the overall variance.
• 45 day memory is much larger than
those of many atmospheric processes.
Hence, climate anomalies can be
sustained through land-surface feedbacks
primarily because they can ‘feed-off’ on
this long-memory.
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Root Water Uptake
• Siqueira et al. (2008)
2 K
t z
1 zs
qr E
t r r r z
2
2
( ) ( )zq K
z z z
radial flow
vertical flow
( ) ( , )r r rRWU z K z r z
Boundary Condition
Use of scale
separation
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Root Water Uptake • Soil-plant model
features
– compensatory root water uptake
– water redistribution by roots (hydraulic lift)
(b) (c)
(a)
Review by Neumann and Cardon (2012)
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Soil moisture simulations
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From: Manoli, G., S. Bonetti, J.C.
Domec, M. Putti, G. Katul, and M.
Marani, 2014, Tree root systems
competing for soil moisture in a 3D
soil-plant model, Advances in Water
Resources, 66,32-42
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Future Research Trends
A single theory that predicts the hydraulic conductivity, gas diffusivity, solute diffusivity,
and electrical conductivity with scale is quite rare – but may remove the constraints
of the REV.
It may also allow down-scaling into sub-Darcian scale and open up a new perspective
on microbial and root-soil processes (Manzoni, 2015).
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Part 3: Above ground plant processes
“It is surely one of the triumphs of evolution that Nature discovered how to make highly accurate machines in
such a noisy environment”
(Phillips & Quake 2006)
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Photosynthesis: Basics
• Of all the organisms in the natural world, green plants are the only ones that manufacture their own food.
• This process is called photosynthesis and begins when light strikes the plant's leaves (both sunlight and artificial light can power this process).
• Cells in the plant's leaves, called chloroplasts, contain a green pigment called chlorophyll, which interacts with sunlight to split the water in the plant into its basic components.
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Photosynthesis: Basics
• Carbon dioxide enters the leaf through holes called stomata and combines with the stored energy in the chloroplasts through a chemical reaction to produce simple sugars.
• The biochemical reaction is often expressed as:
CH2O - represents the carbohydrate such as sucrose (e.g. sugar) or starch.
2 2 2 2CO H O light CH O O
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Photosynthesis: Basics
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Photosynthesis: Basics
• The sugar is then transported through tubes in the leaf to the roots, stems and fruits of the plants.
• Some of the sugar is used immediately by the plant for energy; some is stored as starch; and some is built into a more complex substance, like plant tissue or cellulose.
2 2 2 2CO H O light CH O O
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Photosynthesis: Basics
• Plants often produce more food than they need, which they store in stems, roots, seeds or fruit.
• We can obtain this ‘energy’ directly by eating the plant itself or its products (e.g. carrots, rice or potatoes).
• Photosynthesis is the first step in the food chain, which connects all living things
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Photosynthesis: Basics
• The oxygen that is released by the process of photosynthesis is an essential exchange for all living things.
• Forests have been called the "lungs of the earth" because animals and humans inhale oxygen and exhale carbon dioxide in the process of breathing, and plants take in carbon dioxide and give off oxygen in the process of photosynthesis
2 2 2 2CO H O light CH O O
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Photosynthesis & Light
• The light driving this reaction is known as Photosynthetically Active Radiation (PAR) – this is part of the solar radiation electromagnetic spectrum in the visible range (400-700 nanometers).
2 2 2 2CO H O light CH O O
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Solar
Median Wavelength
of PAR ~ 550 nm
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Photosynthesis: Biochemical Models
• Leaf photosynthesis to be minimum of 3 rates:
min
E
c c
s
J
f J
J
Light-limited
Rubisco-limited
Sucrose-limited
2 2 2 2CO H O light CH O O
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Photosynthesis Models
*
*
*
2
( )
1
1
2
p m i
E
i
m ic
oai c
o
s m
e PAR CJ
C
V CJ
CC K
K
J V
G. Farquhar
*
1
2
i
c
i
Cf
C
mathematical
Form
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Leaf equations for CO2
*
1
2
i
c
i
Cf
C
( )c s a if g C C
Farquhar model
Fickian diffusion
2 equations,
3 unknowns: fc, gs, Ci
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Empirical approaches
Approaches to ‘close’ this problem assume an empirical relation between gs and some environmental stimuli such as air relative humidity (RH) or vapor pressure deficit (D).
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Empirical formulations
1
1 21 1 2 1; 1c c
a a o
m m Dg f RH b g f b
c c D
'Ball-Berry' (Collatz et al., 1991) Leuning (1995)
Two well-known formulations that fit a wide range of data:
The Ball-Berry model was used to allow two-way interactions
between the biosphere and atmosphere in climate models (Sellers
et al., 1996).
Note the linear relationship between g and fc/ca
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Climate models and empirical formulations
• Climate projections for warming scenarios: usually constant relative humidity and hence exponentially increasing vapor pressure deficit (Kumagai et al., 2004).
• How stomatal conductance responds to changes in vapor pressure deficit (or air relative humidity) becomes critical in such two-way interactions within climate models.
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Optimization theories
• It has long been suggested that, at the leaf scale, natural selection may have operated to provide increasingly efficient means of controlling the tradeoffs between water vapor loss and carbon gain.
• DISCUSSED IN THE SPECIALIZED LECTURE
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Photosynthesis - Transpiration
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Parts 3 and 4: Canopy turbulence and the upscaling from leaf-to-canopy
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Up-scaling Problem in Biosphere-Atmosphere Exchange
• Given the state of the atmosphere
above the canopy, and given the
physiological, radiative, and drag
properties of the canopy:
• Can we predict sources, sinks,
concentrations, and fluxes within and
above the canopy?
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Methodology Canopy environment – micro-meteorology
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Simplified Scalar Transport Models – Biologically Active
dz Sc
q
Conservation of scalar mass
cSz
q
t
C
Soil
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Time-averaged Equations
At the leaf scale
Stomata
Fickian diffusion from leaf to atmosphere
Fluid Mechanics – p is the
Transition probability –
it varies with the flow field
coo StztzpC ),|,(
bs
ic
rr
CCzaS
)(
Conservation of scalar mass cSz
q
t
C
Amount of foliage
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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 conductance (gs)
BLUE PRINT OF THE MODELING FRAMEWORK
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Farquhar/Optimality solution for leaf-scale
(2 eq., fc=f(Ci), gs)
Assume leaf pores are saturated
(Claussius-Claperon – q & Tl, 1 equ.)
Leaf energy balance – (Tl, 1 equ.)
PROBLEM IS
Mathematically tractable
BLUE PRINT OF THE MODELING FRAMEWORK
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CO2 Concentration (ppm)
z/h
Duke Forest Experiments
Counter-Gradient Transport
Gradient-Diffusion Analogy?
1134.0'' smkgmgcw
From Katul et al. (1997)
James
Deordorff
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Modeling the fluid flow
• Navier-Stokes equations, which describe the conservation of fluid momentum, are very high dimensional.
• largest scale - 1 km ~ ABL
• smallest scale ~ 0.1 mm ~ viscous dissipation (or Kolmogorov scale).
• Some sort of averaging is required
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Averaging & Canopy Turbulence
Navier-Stokes
equations are
averaged in
space and time
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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
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Flume experiments
Velocity Measurements
Sampling Frequency = 300 Hz
Sampling Period = 300 s
Laser Doppler Anemometer
Giorgio Bidone hydraulics laboratory, DITIC Politecnico di Torino, Torino, Italy
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wu
Wind-Tunnel
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Canonical form of the CSL
THE FLOW FIELD IS A SUPERPOSITION OF THREE
CANONICAL STRUCTURES
d
Displaced wall
Real wall
REGION I
REGION II
REGION III Boundary
Layer
Mixing
Layer
From Poggi et al. (2004)
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TOP VIEW
Flume Experiments
Flow
Visualizations
Laser
Sheet
Rods
From Poggi and Katul (2006)
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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 inside the canopy
From Poggi et al. (2004)
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Region – II: Kelvin-Helmholtz Instabilities &
Attached Eddies
d
Displaced wall
Real wall
REGION I
REGION II
REGION III Mixing
Layer
Boundary
Layer
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xU
Kelvin-Helmholtz Instability
Mixing Layer
U2
U1
Canopy Flow - Mixing Layer
y
Raupach et al. (1996)
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Region II: No co-existence between the two types of vortical structures
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Fraction of time attached eddies and
Kelvin Helmholtz (KH) instabilities occupy
Region II – varies with leaf area density.
The basis of a mixing length model –
Linear superposition of attached eddies and
KH eddies based on leaf area density.
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RANS – Wilson and Shaw (1977)
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Theories tested in flume and forested ecosystems
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From Poggi, Katul, and Albertson (2004, BLM)
Flume Experiments
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Field Experiments
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Duke Forest
FACE-FACILITIES
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T
CO2
H2O
Fluxes shown
are measured
at the canopy
scales
Fluxes at z/hc=1 Modeled Sc Model ecophysiological
parameters are
independently measured
using porometry (leaf
scale).
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Comparison between
measured and modeled
mean CO2
Concentration
CO2 measured by a
10 level profiling
system sampled every
30 minutes.
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The fundamental barriers to progress in the soil-plant-
atmosphere system can be distilled to two main issues:
(a) we do not know how to describe microscopic laws
governing carbon and water movement in the soil-plant
system, and
(b) we do not know how to scale up, spatially and temporally,
these microscopic descriptions coherently, while preserving
the effects of nonlinearity and stochasticity.
The approaches reviewed here should be viewed as ‘initial
steps’ toward filling these knowledge gaps.
The soil-plant-atmosphere system
Thorny issues