How can one study the collapses of societies “scientifically”?

Main Question in “Collapse”

J. Diamond’s 5 point Framework

1. Damage that people inadvertently inflict on their environment

2. Climate Change

3. Hostile Invasions

4. Decreased support by “friendly” neighbors

5. Societies responses to its problems

Extreme deforestation- Loss of wood, soil,agriculture, wildlife,Etc…

Anasazi – [U.S. Southwest]

Maya – [Central Americas]

Polynesian settlers – [Mangareva, Picairn, and Henderson Islands]

Vikings [Greenland]

Other collapses [first 2 points]

Regarding the first 2 points:

Scientific knowledge may assist in forecasting changes in the Earth's biotic environment - a logical first step.

All aspects of this forecasting problem are beyond the scope of a single course.

Water-Vegetation Interactions and Biodiversity in Changing Environments

Gaby Amilcare Ignacio

Spatial scales resolved

Four hierarchical scales:

•Leaf

•Plant

•Canopy

•Landscape

Canopy embedded in a turbulent air stream -extracting momentum and releasing/uptaking scalars at the solid-fluid interface.

Leaf Mass and Energy ExchangePart - I

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.

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 a simple sugar.

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+ + → +

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

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+ + → +

Photosynthesis: Basics

The sugar is then transported through tubes in the leaf to the roots, stems and fruits of the plants.

2 2 2 2CO H O light CH O O+ + → +

Leaf Scale: Photosynthesis

Photosynthesis: Basics

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.

Photosynthesis & Light

The light driving this reaction is known asPhotosynthetically 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+ + → +

Solar

Median Wavelengthof PAR ~ 550 nm

Photosynthesis & Radiation

The radiation per photon is given by

median

h ceλ

=Planck’s constant

346.63 10h J s−= ×

Speed of light 10 13 10c m s−= ×

34 8 119

7

(6.63 10 ) (3 10 ) 3.6 105.5 10

J s m se Jm

− −−

−

× × ×= = ×

×

Photosynthesis & Radiation

The energy content of a mole of photons

23 19

5

(6.023 10 ) (3.6 10 )

2.17 10

photons Jmol photon

Jmol

−× × × =

×

Photosynthesis & Radiation

From the Electro-magnetic Spectrum, PAR contains about 50% of the short-wave.

0.46PAR SR=

Photosynthesis & Radiation

Hence, the maximum photosynthetic photon flux density (PPFD) =

This estimate is close to the maximum observed PAR in clear sky conditions

2 1 3 2 15 2

1500 2.1 10 21002.17 10

mol molJ m s mol m sJ m s

µ− − − − −× ≈ × =×

Photosynthesis: Biochemical Models

Gross Photosynthesis to be minimum of 3 rates:

minE

c

s

JA J

J

⎧⎪= ⎨⎪⎩

Light-limited

Rubisco-limited

Sucrose-limited

Photosynthesis Models

( )*

*

*

2

( )

1

12

p m iE

i

m ic

oai c

o

s m

e PAR CJ

C

V CJCC KK

J V

α −Γ=

+ Γ

−Γ=

⎛ ⎞+ +⎜ ⎟

⎝ ⎠

=

G. Farquhar

Photosynthesis – Model parameters

( )*

*2p m i

Ei

e PAR CJ

C

α −Γ=

+ ΓLeaf absorptivity for PAR ~ 0.8

Maximum quantum efficiency ~ 0.08

Compensationpoint (ppm)

Photosynthesis – Model parameters

*( )

1

m ic

oai c

o

V CJCC KK

−Γ=

⎛ ⎞+ +⎜ ⎟

⎝ ⎠

Maximum carboxylation capacity (20-100 micromol m-2 s-1)

Photosynthesis – Model parameters

( )*1

2

in

i

CA

C

α

α

−Γ=

+

( )n iA f C=Mathematical Form

Photosynthesis - Model Parameters

max maxp mJ e PARα=

2 12100 mol m sµ − −:Leuning (2002)

Duke Forest

Photosynthesis – Model parameters

Pinus taeda

Photosynthesis – Model parameters

From Leuning (2002, PCE)

Cotton

ScotsPine

Tobacco

R. Leuning

Tleaf=25oC

( )*1

2

in

i

CA

C

α

α

−Γ=

+

Ci/Ca=0.66

Ci/Ca=0.66

Photosynthesis - Transpiration

Stomatal Conductance

Collatz et al. (1991): Semi-empirical

2

ns

A RHg m bCO

= +

In the absence of soil moisture stress

Duke Forest FACE-FACILITIES

Pinus taeda

Well-wateredconditions

Leaf Equation for CO2/H2O

( )*1

2

in

i

CA

C

α

α

−Γ=

+

2n

sA RHg m bCO

= +

( 2 )n s iA g CO C= −

Farquhar model

Collatz et al. model

Fickian diffusion

3 unknowns: An, gs, Ci

Leaf Transpiration

[ ]s i aTranspiration c g e e′= −

( )i le f T=If leaf stomata are saturated, then

Clausius-Clapeyron

Leaf equations: A parsimonious model for well-watered soils

*Given atmospheric state outside the leaf: CO2, PAR, ea (or RH) & Tl

*Given physiological properties of the leaf: Vcmax, m & b

Solve for: Ci, An, gs, and transpiration

Under-story

Naumberg et al. (2001; Oecologia)

sunfleck

MODEL TESTING

Plant Hydrodynamics

Plant Hydraulics -

Up to this point, the photosynthesisand leaf level processes are Discussed with little regards to Plant Hydrodynamics.

Photosynthetic Apparatus (~3.5 Billion years)

Land Plants (~350 million years; need for hydraulic apparatus)

From D. Christian (2004)

Photosynthesis and Terrestrial Plants

Plant Hydraulics: Background

The evolution of land plants required hydraulic adjustments that permitted the display of the photosynthetic machinery within a desiccating atmosphere.

A productive display of this machinery necessitates a vertical distribution of chlorophyll so that the light-use efficiency of the organism is increased relative to a concentrated display of chlorophyll.

Background:

This vertical distribution required certain “engineering solutions” to permit water to be supplied to leaves enclosed in an envelope that permits a controlled exchange of CO2 for water vapor.

Spatial Scales in Water TransportLeaf-Air ExchangeThrough Stomata

Xylem WaterTransport

Ecosystem scaleBiosphere-atmosphere Water Exchange

Meso-ScaleLandscape-AtmosphereExchange

Xylem – An Engineering Solution

Xylem has three main functions:1. Transport: They allow the passage of water from one conduit to another, but they must also prevent the spreading of air.

FromSperry

Xylem – An Engineering Solution

Xylem – An Engineering Solution

2. Storage

Xylem contains many living cells. These cells store large amounts of starch, which can be mobilized during leaf expansion.

Some of the cells release sugar into xylem conduits, which may play a role in embolism reversal

Xylem – An Engineering Solution

3. Structural Support:

Xylem conduits provide structural support to the plant body.

Data from Hubbard et al. (2001)

Relationship betweenstomatal conductance (gs) and photosynthesis (A) and plant hydraulicconductivity (KL).

KL = Soil to leaf

J. Sperry

From Hacke et al. (2000): Pinus taeda

Leaf pressure

Leaf-level Equations - Modified

2

( )ns leaf

A RHg m f bCO

ψ= +

leafψ

( )leaff ψCavitation occurs

1.0

Plant-Hydrodynamics

leafψ

soilψrootψ

xylemψ

To connect soil Water status to leafPressure, the root-xylem system must Be resolved.

Two-way interactions:

•Soil pressure -> leaf pressure,But leaf pressure affects transpiration And hence soil moisture/pressure.

•Atmospheric CO2 affects leafPhotosynthesis affects atmospheric CO2

Two-way Interactions

Relative humidity affects leaf Transpiration affects relative humidity

Transpiration & air temperature affectLeaf temperature photosynthesis

Leaf conductance and transpiration

A 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?

PARU,C,RH,T aa

RE

dzS

Atmospheric Forcing

Soil

)z(a

INPUT FORCING VARIABLES

Gs

cSzq

tC

+∂∂

−=∂∂

∫∫= coo StztzpC ),|,(

bs

ic

rrCCzaS

+−

= )(ρ

,....,),|,( woo Utztzp σ>−−

Time-averaged Equations

Time averaging ~ 30 minutes

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

Hokusai “The big wave”

The Fluid Mechanics?PART - II

jijii dtuxbdttuxadu Ω+= ),,(),,(

Linking concentrationto source strength:

Lagrangian Fluid Mechanics

,i ija b Determined from Thomson’s (1987) model -Require first and second momentsof the turbulent velocity

,....,),|,( woo Utztzp σ>−−

Oak-Hickory standDuke Forest

Modeling the Fluid Flow

Navier-Stokes equations, which describe the conservation of fluid momentum, are very high dimensional.largest scale - 1 km ~ ABLsmallest scale ~ 0.1 mm ~ viscous dissipation (or Kolmogorov scale).Some sort of averaging is required

Averaging & Canopy Turbulence

Navier-Stokes equations are averaged in space and time

Spatial Averaging and Canopy Effects on the Airflow

The spatial averaging excludes the solid plant parts, causing the averaging volume to be a “multiply-connected” space.

The effect of the solid plant part on the air flow can be formally described via closure models, analogous to standard closure models in turbulence

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

PLAN Open channel

Test sectionFlow direction9 m

1 m

1 m

FLUME EXPERIMENTS

Weighted scheme

Rods positions

hwdr

Canopy sublayer

2h

2 cm

1 cm

h

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

SECTIONVIEW

• PLAN• VIEW

How Leaf Area Density Affects the Flow Statistics

xU

Kelvin-Helmholtz Instability

Mixing Layer

U2

U1

Canopy Flow - Mixing Layer

yω

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

Structure of Turbulence in Model Plant Canopy

Lowest Layers in the Canopy:

Flow Visualizations

Laser Sheet

SHOW MOVIE

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.

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

Region – II: Attached Eddies /Kelvin-Helmholtz Instabilities

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(

inu w′ ′

outu w′ ′

2( )d dF C a z U=

a(z)

Toy Model for Momentum Transport

2( )d

du w C a z Udz′ ′

= −

1 equ. 2 unknowns

t

t

dUu w Kdz

dUK l ldz

′ ′ = −

⎛ ⎞= ×⎜ ⎟⎜ ⎟

⎝ ⎠

Toy Model for Momentum Transport

Toy Model for Momentum Transport

2( )t d

d dUK C a z Udz dz

⎛ ⎞− = −⎜ ⎟⎝ ⎠

2nd order nonlinear & non-homogeneous ODE

DragCoefficient

From Poggi, Porporato,

Ridolfi, et al. (2004, BLM)

Second-order closure models