Statistical Analysis of Canopy Turbulence

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?

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:

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

Poor-Man’s version of the Navier-Stokes Equation

fxx

u

x

p

x

uu

t

ututututu

ii

i

ii

ii

i

2

2 1)()(2)()1(

Why Ensemble-Averaging is the appropriate operator?

Analogy borrowed from Frisch (1995)

Sensitivity to Initial Conditions

Solutions Statistically Identical

Introduction:Averaging

Monin and Yaglom (1971):

ExperimentNumber

u(t)

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

Ergodic Hypothesis:

t

E(u) or E(u2)

E [u,u(t+ ) ]

For stationary process: Ensemble average = time average

T

T

dttuT

uEei0

)(1

)(..

If :1) Ensemble Autocovariance decays as2) Individual Autocovariance decays as

T

Sampling Period and Eddy-sizes

A weaker version of these conditions:

Integral time scale of one realization is finite:

T

dttutuT

tutu0

)(')('1

)(')(')(

0

)(lim dI t ~ Energetic Scale

Lumley and Panofsky (1964): T>> tI

LIDAR EXPERIMENTSLidar Experiments at UC Davis - 1991

Ground(Bare Soil)

LIDAR

LIDAR Sight

3m

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.

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

FLUME DIMENSIONS

PLAN

Open channel

Test sectionFlow direction9 m

1 m

1m

FLUME EXPERIMENTS

Weighted scheme

Rods positions

hw

dr

Canopy sublayer 2h

2 cm

1 cm

h

+++++++++++++++

SECTIONVIEW

• PLAN• VIEW

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 IIIBoundaryLayer

MixingLayer

Vortex Pairing in Mixing Layers (Van Dyke, 1981)

Structure of Turbulence in Model Plant Canopy Lowest Layers in the Canopy:

Flow Visualizations

Laser Sheet

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.

Von Karman StreetsNASA’s EOS MODIS

Von Karman streets shedding off the Cape Verde Islands

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

0.01 0.1 1 10 100fd€€€€€€€€€€€€€€€€

u� 0.21

5. ´ 10-7

1. ´ 10-6

5. ´ 10-6

0.00001

0.00005

FwHfL d€€€€€€€€€€€€€€€€€€€u� 3 - 2� 3

z � h<1

z� h=1.1

z� h=1.9

0.01 0.1 1 10 100fd

€€€€€€€€€€€€€€€€u� 0.21

1.´ 10-6

0.00001

0.0001

- 2� 3z� h<1

z� h=1.1

z� h=1.9

Spectra of w

From Kaiman and Finnigan (1994)

-0.5 0 0.5 1 1.5 2 2.5Hz- dL� h0.25

0.75

0.5

1

lffe�h

é

é é é ééé

é

é

éé

é é

ã

ã ã ã ããã

ãã

ãã

ã

ã

HaL

a=0.0

a=0.45

LV=dr � 0.21Canopy top

1 2 3 4 5 6 7aHm- 1L

0.2

0.4

0.6

0.8

1

a

ì

ì

ì

ìììììììììì

REGION II: Combine Mixing Layer and Boundary Layer

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

diffusivity

MLBL LLl )1(

Re1

Re2

Spatial Averaging and Dispersive Fluxes:

Dispersive Fluxes =

wuwu /Consistent with: 1) Bohm et al. (2000)

2) Cheng and Castro (2002)

aa

% %

Roughness Density

Roughness Density

Roughness Density

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??

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

- 20

0

20

CO

2

T ime (minut es)

q' (

Kg/

m3)

- 0. 0001

0

0. 0001

0 2 4 6 8 10 12 14

u'w

' (m

/s)2

- 1. 2

- 0. 8

- 0. 4

0

0. 4

- 0. 3

- 0. 2

- 0. 1

0

0. 1

0. 2

w't

' (K

m/s

)

Fc

- 10- 505101520

- 4E- 005

- 2E- 005

0

2E- 005

4E- 005

6E- 005

w'q

'

T ime (minut es)

RN

(W

/m2)

- 35- 30- 25- 20- 15- 10- 5

0 2 4 6 8 10 12 14

Ramps: Stable Boundary Layer at z/h = 1.12 (Duke Forest)

Ramps: Cross-spectra

cospectrum

quad-spectrum

coherence

phase

0. 0001 0. 001 0. 01 0. 1 1- 0. 05

- 0. 04

- 0. 03

- 0. 02

- 0. 01

0

0. 01

u'w

'

0 . 0001 0. 001 0. 01 0. 1 10

0. 2

0. 4

0. 6

u'w

'

0 . 0001 0. 001 0. 01 0. 1 1f (H z)

- 180

- 90

0

90

180

u'w

'

0 . 0001 0. 001 0. 01 0. 1 1- 0. 008

- 0. 006

- 0. 004

- 0. 002

0

0. 002

w't

'

0 . 0001 0. 001 0. 01 0. 1 10

0. 2

0. 4

0. 6

0. 8

w't

'0 . 0001 0. 001 0. 01 0. 1 1

f (H z)

- 180

- 90

0

90

180

w't

'

Ramps: Cross-spectra

cospectrum

quad-spectrum

coherence

phase

0. 0001 0. 001 0. 01 0. 1 1- 0. 1

0

0. 1

0. 2

0. 3

w'c

'

0 . 0001 0. 001 0. 01 0. 1 10

0. 2

0. 4

0. 6

w'c

'

0 . 0001 0. 001 0. 01 0. 1 1f (H z)

- 180

- 90

0

90

180

w'c

'

0 . 0001 0. 001 0. 01 0. 1 1- 2E- 007

- 1E- 007

0

1E- 007

2E- 007

3E- 007

4E- 007

w'q

'

0 . 0001 0. 001 0. 01 0. 1 10

0. 2

0. 4

0. 6

0. 8

w'q

'

0 . 0001 0. 001 0. 01 0. 1 1f (H z)

- 180

- 90

0

90

180

w'q

'

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

CO

2

T ime (minut es)

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. 4

0

0. 4

- 0. 3

- 0. 2

- 0. 1

0

0. 1

0. 2

w't

' (K

m/s

)

Fc

- 10- 505101520

- 4E- 005

- 2E- 005

0

2E- 005

4E- 005

6E- 005

w'q

'

T ime (minut es)

RN

(W

/m2)

- 30

- 25

- 20

0 2 4 6 8 10 12 14

Gravity Waves: Cross-spectra

cospectrum

quad-spectrum

coherence

phase

0. 0001 0. 001 0. 01 0. 1 1- 0. 0004

- 0. 0002

0

0. 0002

0. 0004

0. 0006

0. 0008

u'w

'

0 . 0001 0. 001 0. 01 0. 1 10

0. 2

0. 4

0. 6

0. 8

u'w

'

0 . 0001 0. 001 0. 01 0. 1 1f (H z)

- 180

- 90

0

90

180

u'w

'

0 . 0001 0. 001 0. 01 0. 1 1- 0. 0004

0

0. 0004

0. 0008

0. 0012

0. 0016

w't

'

0 . 0001 0. 001 0. 01 0. 1 10

0. 2

0. 4

0. 6

0. 8

1

w't

'

0 . 0001 0. 001 0. 01 0. 1 1f (H z)

- 180

- 90

0

90

180

w't

'

Gravity Waves: Cross-spectra

cospectrum

quad-spectrum

coherence

phase

0. 0001 0. 001 0. 01 0. 1 1- 0. 06

- 0. 04

- 0. 02

0

0. 02

w'c

'

0 . 0001 0. 001 0. 01 0. 1 10

0. 2

0. 4

0. 6

0. 8

1

w'c

'

0 . 0001 0. 001 0. 01 0. 1 1f (H z)

- 180

- 90

0

90

180

w'c

'

0 . 0001 0. 001 0. 01 0. 1 1- 1. 2E- 006

- 8E- 007

- 4E- 007

0

4E- 007

8E- 007

w'q

'

0 . 0001 0. 001 0. 01 0. 1 10

0. 2

0. 4

0. 6

0. 8

w'q

'

0 . 0001 0. 001 0. 01 0. 1 1f (H z)

- 180

- 90

0

90

180

w'q

'

Net Radiation

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

- 5 0- 4 0- 3 0- 2 0- 1 0

01 02 03 04 05 0

CO

2

T ime (minut es)

q' (

Kg/

m3)

- 0 . 0006- 0. 0004- 0. 000200. 00020. 00040. 00060. 0008

0 4 8 12 16 20 24 28

u'w

' (m

/s)2

- 1. 2

- 0. 8

- 0. 4

0

0. 4

- 0. 3

- 0. 2

- 0. 1

0

0. 1

0. 2

w't

' (K

m/s

)

Fc

- 10- 505101520

- 0. 0003

- 0. 0002

- 0. 0001

0

0. 0001

0. 0002

w'q

'

T ime (minut es)

RN

(W

/m2)

- 30

- 25

- 20

0 4 8 12 16 20 24 28

Autocorrelation function of temperature

Ramps Gravity waves

Net Radiation

secsec

sec

Time lag

Time lag

Time lag

RampsGravity Waves

Two End-members of stable CSL State

No TurbulenceWell-DevelopedTurbulence

Slightly Stable Flows

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.