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

“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

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

Variability in time

A Strogatz like diagram of the dimensionality-nonlinearity problem

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

Part 2: Below ground processes

“We know more about the movement of celestial bodies than

about the soil underfoot”

Leonardo Da Vinci

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

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

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.

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

Fluxes and important flows of water

• Darcy’s Law:

dL

dHKqw )(

Hydraulic conductivity

Energy gradients – that themselves

vary with soil moisture

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

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

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 )(

Fluxes and water flow

• Two fundamental

Hydraulic functions

Characterizing soil type

Fluxes and important flows of water

32

)()(

)()(

b

s

ss

b

s

ss

KK

bKsss ,,,

see Clapp and Hornberger (1978) - for example

State Equation with roots

1)(

)(

);,(

zKq

zSz

q

t

dz

qin

qout

Rainfall P(t)

S

Root-water uptake

Root water

uptake

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.

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/

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.

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.

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.

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

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.

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

4 rods per ring

1998-2005 – 8 years of 30 min. data

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)

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

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)

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

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

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

Precipitation

Evapotranspiration

Soil moisture

Duke Forest Experiment – 8 years of 30-min. Data

1/f for pink noise, 1/f^2 for red noise)

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

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

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.

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

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)

Soil moisture simulations

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

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).

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)

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 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

Photosynthesis: Basics

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

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 & 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

Solar

Median Wavelength

of PAR ~ 550 nm

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

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

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

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).

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

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.

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

Photosynthesis - Transpiration

Parts 3 and 4: Canopy turbulence and the upscaling from leaf-to-canopy

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?

Methodology Canopy environment – micro-meteorology

Simplified Scalar Transport Models – Biologically Active

dz Sc

q

Conservation of scalar mass

cSz

q

t

C

Soil

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

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

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

CO2 Concentration (ppm)

z/h

Duke Forest Experiments

Counter-Gradient Transport

Gradient-Diffusion Analogy?

1134.0'' smkgmgcw

From Katul et al. (1997)

James

Deordorff

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

Averaging & Canopy Turbulence

Navier-Stokes

equations are

averaged in

space and time

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 experiments

Velocity Measurements

Sampling Frequency = 300 Hz

Sampling Period = 300 s

Laser Doppler Anemometer

Giorgio Bidone hydraulics laboratory, DITIC Politecnico di Torino, Torino, Italy

wu

Wind-Tunnel

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)

TOP VIEW

Flume Experiments

Flow

Visualizations

Laser

Sheet

Rods

From Poggi and Katul (2006)

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)

Region – II: Kelvin-Helmholtz Instabilities &

Attached Eddies

d

Displaced wall

Real wall

REGION I

REGION II

REGION III Mixing

Layer

Boundary

Layer

xU

Kelvin-Helmholtz Instability

Mixing Layer

U2

U1

Canopy Flow - Mixing Layer

y

Raupach et al. (1996)

Region II: No co-existence between the two types of vortical structures

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.

RANS – Wilson and Shaw (1977)

Theories tested in flume and forested ecosystems

From Poggi, Katul, and Albertson (2004, BLM)

Flume Experiments

Field Experiments

Duke Forest

FACE-FACILITIES

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).

Comparison between

measured and modeled

mean CO2

Concentration

CO2 measured by a

10 level profiling

system sampled every

30 minutes.

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