Statistical Analysis of Canopy Turbulence

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Statistical Analysis of Canopy Turbulence

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Problem Linking Measurements to Fluid Dynamics
Equations

• Turbulence is a stochastic phenomena
• Comparing measurements (e.g. time series) and
  model predictions can only be conducted
  statistically.
• What are the appropriate assumptions in such
  comparisons in field experiments?

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Outline

• Introduction Linking eddy-sizes, averaging
  operations through the ergodic hypothesis.
• Idealized setup Model canopy in a flume as a
  case study to highlight the relevant eddy-sizes
  and the spatial-temporal averaging.
• Real-world setup Stable flows near the
  canopy-atmosphere interface. No unique canonical
  form for the integral length - but several
  possible canonical forms are examined depending
  on external boundary conditions and inherent
  length scales.
• Conclusions

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Introduction

• Three types of averaging
• Spatial, Temporal, and Ensemble.
• Averaging equations of motion proper averaging
  operator is ensemble.
• Field experiments - typically provide temporal
  averaging.
• Key assumption when linking equations to
  measurements Ergodic hypothesis

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Poor-Mans version of the Navier-Stokes Equation
Why Ensemble-Averaging is the appropriate
operator?
Analogy borrowed from Frisch (1995)
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Sensitivity to Initial Conditions
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Solutions Statistically Identical
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IntroductionAveraging

• Monin and Yaglom (1971)

u(t)
Experiment Number
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Weakly Stationary Process

• Ensemble Mean independent of time
• Ensemble Variance independent of time
• Ensemble covariance - only dependent on time lag

Note Ensemble averaging is referred to as E
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Ergodic Hypothesis
T
E u,u(t )
For stationary process Ensemble average time
average
If 1) Ensemble Autocovariance decays as 2)
Individual Autocovariance decays as
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Sampling Period and Eddy-sizes

• A weaker version of these conditions
• Integral time scale of one realization is finite

Energetic Scale
Lumley and Panofsky (1964) Tgtgt
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LIDAR EXPERIMENTS

• Lidar Experiments at UC Davis - 1991

LIDAR Sight
Ground (Bare Soil)
3m
LIDAR
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ERGODIC HYPOTHESIS FOR CANOPY TURBULENCE

• ASL Ergodic hypothesis seems reasonable.
• CSL Two types of averaging are employed -
  spatial and temporal. What are the canonical
  length scales and how they affect both averaging
  operations.

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FLUME EXPERIMENTS

• To understand the connection between energetic
  length scales, spatial and temporal averaging,
  start with an idealized canopy (Finnigan, 2000).
• Vertical rods within a flume.
• Repeat the experiment for 5 canopy densities
  (sparse to dense) and 2 Re

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FLUME DIMENSIONS
1 m

• PLAN

9 m
1m
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FLUME EXPERIMENTS

• PLAN
• VIEW

SECTION VIEW
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Canonical Form of the CSL

• THE FLOW FIELD IS A SUPERPOSITION OF THREE
  CANONICAL STRUCTURES

Mixing Layer
Boundary Layer
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Structure of Turbulence in Model Plant Canopy

• Lowest Layers in the Canopy

Flow Visualizations
Laser Sheet
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Flow Visualization

• Flow visualization supports the hypothesis that
  the structure of turbulence in the deeper layers
  of the canopy is dominated by Von Karman streets
  periodically interrupted by sweep events from the
  top layers.

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Von Karman Streets

• NASAs EOS MODIS

Von Karman streets shedding off the Cape Verde
Islands
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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)

Spectra of w
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From Kaiman and Finnigan (1994)
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REGION II Combine Mixing Layer and Boundary
Layer
Re1
Re2

• LBL Boundary Layer Length k(z-d)
• LML Mixing Layer Length Shear Length Scale
• l Total Mixing Length Estimated from an
  eddy-diffusivity

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Spatial Averaging and Dispersive Fluxes

• Dispersive Fluxes

Consistent with 1) Bohm et al. (2000) 2)
Cheng and Castro (2002)
Roughness Density
Roughness Density
a
a


Roughness Density
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CSL Flows in Complex Morphology and Stability

• CSL flows for simple morphology and canopy
  density does have well-defined length and time
  scales (Ergodic).
• CSL flows for stable conditions in real canopies??

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Ramps Stable Boundary Layer at z/h 1.12 (Duke
Forest)
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Ramps Cross-spectra
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Ramps Cross-spectra
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Gravity Waves Stable Boundary Layer at z/h
1.12 (Duke Forest)
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Gravity Waves Cross-spectra
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Gravity Waves Cross-spectra
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Net Radiation
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Autocorrelation function of temperature
Gravity waves
Ramps
t 12 sec
t 30 sec
Time lag
Time lag
t 20 sec
Net Radiation
Time lag
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Well-Developed Turbulence
No Turbulence
Slightly Stable Flows
Gravity Waves
Ramps
Two End-members of stable CSL State
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Conclusions

• For ASL flows over uniform surfaces, the ergodic
  hypothesis is reasonable.
• For neutral flows within the CSL of simple
  morphology, the ergodic hypothesis is also
  reasonable.
• For stable CSL flows, too much contamination
  from boundary conditions (e.g. clouds or other
  disturbances) to sustain stationarity.