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Land surface modeling: Basic concepts

Subhadeep Halder

Center for Ocean-‐Land-‐Atmosphere Studies and Dept. of Atmospheric, Oceanic and Earth Sciences

George Mason University

ICTP-‐IITM-‐COLA Targeted Training AcIvity (TTA) on "Modelling and PredicIon of Asian Monsoons: Improving Physical Processes", February 9 -‐ 20, 2015

Lecture -‐ 1 1

Acknowledgement

The lecture is a part of the course CLIM-‐714 of the Department of Atmospheric, Oceanic and Earth Sciences, GMU and taught by Prof. Paul Dirmeyer ([email protected], also at COLA)

Lecture -‐ 1 2 ICTP-‐IITM-‐COLA Targeted Training AcIvity (TTA) on "Modelling and PredicIon of Asian Monsoons: Improving Physical Processes", February 9 -‐ 20, 2015

•  A budget represents the balance of inputs and outputs of a conserved quanIty (a quanIty that can not be created or destroyed – only transported or transformed).

•  We assume conservaIon of certain types of ma[er at the land-‐atmosphere interface (e.g., water and carbon)

•  The rate of change in state S of a system is equal to the difference between input (flux into) and output (flux out of) the system:

•  is the rate of change of state S over Ime t

•  QI is the sum of fluxes into the system •  QO is the sum of fluxes out of the system


€

dSdt

=QI −QO

€

dSdt

Budgets at the L-‐A interface

Land vs. Ocean vs. Atmosphere: Heterogeneity

4

Water •  Flows (x,y: 1y; z: 102y) •  High heat capacity (4.2×106 J m-3 K-1) •  Moderate heat conductivity (0.6 J m-1K-1s-1) •  Dark (α=0.05) •  Evaporation at potential rate

Dry Soil •  Stationary (essentially) •  Low heat capacity (0.6-1.3×106 J m-3 K-1) •  Low heat conductivity (0.08-0.2 J m-1K-1s-1) •  Light (α=0.13-0.50) •  No evaporation

Wet Soil •  Water flows (x,y: 0-30d; z: 0-104y) •  Moderate heat capacity (2.2-2.9×106 J m-3K-1) •  High heat conductivity (0.8-1.7 J m-1K-1s-1) •  Not as light (α=0.1-0.4) •  Evaporation is a function of soil moisture

Vegetation •  Varies with time (species, density, color, coverage) •  Canopy creates microenvironment for radiation, heat exchange, interception of rain and snow •  Generally Dark (α=0.08-0.25) •  Transpiration controlled by photosynthesis, moisture stress


Lecture -‐ 1

•  At any point at the land-‐atmosphere interface:

•  P is precipitaIon •  E is evapotranspiraIon •  R is runoff •  ΔSW is change in water storage (on surface and in soil) •  is verIcally-‐integrated moisture flux divergence

•  is the change in verIcally integrated humidity (change in precipitable water)

Surface Water Balance

Lecture -‐ 1 5

€

P − E = R + ΔSW = − ∇⋅ (q! V )

z∫ − Δ q

z∫

€

Δ qz∫

€

∇ ⋅ (q! V )

z∫

Interface Land Atmosphere


Lecture -‐ 1

Water Balance for a Single Land Surface Slab, Without Snow

P = E + R + CwΔw/Δt + miscellaneous

Where: P = PrecipitaIon E = EvapotranspiraIon R = Runoff (effecIvely consisIng of surface runoff and baseflow) Cw = Water holding capacity of surface slab Δw = Change in the degree of saturaIon of the surface slab over the

Ime step Δt miscellaneous = conversion to plant sugars, human consumpIon, etc.

P E

w

Term on LHS comes from the atmospheric model – external “boundary condiIon”.

Terms on the RHS are determined by the land

surface model.

R

6 ICTP-‐IITM-‐COLA Targeted Training AcIvity (TTA) on "Modelling and PredicIon of Asian Monsoons: Improving Physical Processes", February 9 -‐ 20, 2015

Usually, a CombinaIon of Balances Is Considered

Lecture -‐ 1

Water balance associated with canopy intercepIon reservoir P = Eint + Dc +

ΔWc Δt

Eint = intercepIon loss Dc = drainage through canopy (“throughfall”) ΔWc = change in canopy intercepIon storage

P Eint

Dc

Wc

Water balance in a snowpack

P (snow) Esnow

M Wsnow

P = Esnow + M + ΔWsnow Δt

Esnow = sublimaIon rate M = snowmelt ΔWsnow = change in snow amount (“infinite” capacity possible)


More Balances…

Lecture -‐ 1

Water balance in a surface layer

Ebs = evaporaIon from bare soil Etr1 = evapotranspiraIon from layer 1 Q12 = water transport from layer 1 to layer 2 CW1 = water holding capacity of layer 1 ΔW1 = change in degree of saturaIon of layer 1

w2

w1

w3

Q12 Water

storage

M+Dc Ebs + Etr1 Rs M + Dc =

Ebs + Etr1 + Rs + Q12 + CW1ΔW1/Δt

Water balance in a subsurface layer (e.g., 2nd layer down)

Q12 = Q23 + Etr2 + CW2ΔW2/Δt

w2

w1

w3

Q12

Q23 water

storage

Etr2

Etr2 = evapotranspiraIon from layer 2 Q23 = water transport from layer 2 to layer 3 CW2 = water holding capacity of layer 2 ΔW2 = change in degree of saturaIon of layer 2

Note: some models may include an additional, lateral subsurface runoff term


Lecture -‐ 1

Water balance in the lowest layer

wn

Qn,n-‐1

water storage QD

Etr-n Qn,n-1 = QD + Etr-n + CWnΔWn/Δt

Etr-n = EvapotranspiraIon from layer n, if allowed QD = Drainage out of the soil column (baseflow)

A model may compute all of these water balances, taking care to ensure consistency between connecIng fluxes (in analogy with the energy balance calculaIon).

W2

W1

W3

Q12

Q23

Rs M Ebs

QD

P Eint

Dc P Esnow

Etr1 Etr2 Etr3

SIll More Balances…


PrecipitaIon P

•  Gekng the land surface hydrology right in a climate model is difficult largely because of the precipitaIon term. At least three aspects of precipitaIon must be handled accurately: –  SpaIally-‐averaged precipitaIon amounts (along with annual means

and seasonal totals) –  Small-‐scale (subgrid) distribuIon. –  Temporal variability and temporal correlaIons.

•  Otherwise, even with a perfect land surface model:

Lecture -‐ 1 10

Perfect land surface model

Garbage in

Garbage out


SpaIal Variability

•  PrecipitaIon varies on all spaIal scales, but atmosphere and land models operate in discrete “grid boxes”.

11

(mm/hr)

512

km

pixel = 4 km

0 4 9 13 17 21 26 30

R (mm/hr)

The image part with relationship ID rId4 was not found in the file.

0 4 9 13 17 21 26 30

R (mm/hr)

2 km

4

km

pixel = 125 m

Figures: Courtesy E. Foufoula-Georgiou. ICTP-‐IITM-‐COLA Targeted Training AcIvity (TTA) on "Modelling and PredicIon of Asian

Monsoons: Improving Physical Processes", February 9 -‐ 20, 2015 Lecture -‐ 1

Rainfall Variability

•  These two snapshots of precipitaIon intensity (mm/hr) have idenIcal area means.

•  The coefficient of variaIon of the top panel is 6.5 Imes greater than the bo[om (data on a 7-‐km grid).

•  Would these two events produce similar total runoff? Canopy evaporaIon? Change in soil moisture?

12

Grid box size of state-of-the-art • Climate change model • Seasonal forecast model • Weather forecast model


Lecture -‐ 1

PrecipitaIon: temporal correlaIons Temporal correlaIons are very important -‐-‐ but are largely ignored -‐-‐ in GCM formulaIons that assume sub-‐grid precipitaIon distribuIons. This is especially true when the Ime step for the land calculaIon is of the order of minutes. Why are temporal correlaIons important? Consider three consecuIve Ime steps at a GCM land surface grid cell:

time step 1 time step 2 time step 3 Case 1: No temporal correlation in storm position -- the storm is placed randomly with the grid cell at each time step.

Case 2: Strong temporal correlation in storm position between time steps.


Lecture -‐ 1

parameterization

EvapotranspiraIon E EvaporaIon from Open Water

•  EvaporaIon from open water occurs at the potenIal rate. •  Priestley-‐Taylor formulaIon for wet surfaces:

–  m is slope of saturaIon vapor pressure with temperature at surface temperature T (recall the Clausius-‐Clapeyron relaIonship).

–  RNET is net energy available, minus that which warms the water. –  γ is the psychrometric constant. –  Empirical observaIon has shown α ≅1.26.

•  EP is called the potenIal evaporaIon. In this formulaIon, there is only dependence on net radiaIon and temperature.

Lecture -‐ 1 14

€

m =desdT

€

EP =αmRNET

λv (m + γ)


The Aerodynamic Term

•  The Priestley-‐Taylor “fudge factor” α accounts for the aerodynamic aspect of evaporaIon. The Penman equaIon is slightly more sophisIcated and accounts for it directly:

–  Note that the vapor pressure deficit and aerodynamic resistance are expressed directly in this formulaIon.

–  Forms of this equaIon are commonly used in models for evaporaIon from wet surfaces or open water in models.

•  Note: Some LSMs neglect possible evaporaIon from open water enIrely, and some leave it to a lake model or parameterizaIon to handle.

Lecture -‐ 1 15

€

EP =mRNET + cpρ[es(TS ) − e(TA )]/ra

λv (m + γ )

parameterization


Bare Soil EvaporaIon •  Bare soil evaporaIon draws moisture only from the top layer of soil (that exposed to the air).

•  The simplest (and first) formulaIon of soil in an LSM was a “Bucket model” by S. Manabe*. It used a “beta” formulaIon for soil evaporaIon:

Lecture -‐ 1 16

€

E = βEP

* Manabe, 1969: Mon. Wea. Rev., 739‑774.

€

β = 0, w ≤ ww

β =w − ww

w f − ww

, ww < w < w f

β =1, w ≥ w f

Soil Wetness w 0 ww wf wsat

1 0

β

Field capa

city

Wil1

ng point

Satura1o

n

parameterization


Canopy IntercepIon, Loss and Throughfall

•  Moisture intercepted by the canopy can evaporate directly back to the air without reaching the soil.

•  Wet leaves (or the fracIon of canopy that holds intercepted moisture) are assumed not to transpire – transpiraIon only occurs from the dry leaves.

•  EvaporaIon of intercepted water is formulated the same as from an open water surface.

Lecture -‐ 1 ICTP-‐IITM-‐COLA Targeted Training AcIvity (TTA) on "Modelling and PredicIon of Asian Monsoons: Improving Physical Processes", February 9 -‐ 20, 2015

17

Capacity of bucket is typically a funcIon of leaf area index, a measure of how many leaves are present.

This works, but because it ignores sub-‐grid precipitaIon variability (e.g., fracIonal wekng), it is overly simple.

Sellers et al. 1986: J. Atmos. Sci., 505-531.

Koster and Suarez, 1992: J. Geophys. Res., 2697-2716.

parameterization

TranspiraIon

•  TranspiraIon is the loss of water vapor from plants as a result of their respiraIon of carbon dioxide. –  It is analogous to the water you lose by breathing as a consequence of taking in oxygen.

•  Plant leaves have Iny pores called stoma through which they “breathe” in CO2 and release O2 as waste. Water vapor is lost in the process. –  Loss of water is not advantageous – it is a consequence of plant respiraIon.

–  Plants try to opImize CO2 intake versus loss of water vapor.

-  Will be discussed in next lecture !


Lecture -‐ 1

parameterization


LimitaIons on Photosynthesis 1.  Availability of light 2. Maximum rate of the Rubisco enzyme (chemical) process 3.  Carbon compound export (C3) or PEP-‐Carboxylase limitaIon (C4) The rate of carbon flux into the plant can be modeled in the same way as transpiraIon out of the plant — as a diffusive flux through the stomata, regulated by stomatal conductance.

When carbon is represented in the model, the CO2 gradients can also affect rc. The rate at which photosynthesis "fixes" carbon in the plant (converts from gaseous CO2 to sugars) affects the CO2 concentraIon in the leaf, and thus the flux rate.

19

rc

SublimaIon (ice to vapor)

•  Latent heats of melIng and vaporizaIon of water must be traversed (recall: λs = λv + λm).

•  SublimaIon of a sikng snowpack is esImated much like evaporaIon from the land surface.

•  SublimaIon from blowing snow is much more efficient than sublimaIon from a snowpack. It involves its own parameterizaIons (e.g., G. Liston)

Lecture -‐ 1 20

λs ≅ 2.83x106 J/kg λv ≅ 2.5x106 J/kg Approximate because depends on T λm = 3.34x105 J/kg


•  For lateral moIon of water within a model grid cell, the terrain within the grid cell is ouen treated as an idealized hillslope.


21

10s to 100s of meters 10s to 100s of kilometers

–  Horton runoff: precipitaIon exceeds infiltraIon.

–  Dunne runoff: soil is saturated, cannot infiltrate.

–  Baseflow: lateral runoff beneath soil into river system.

MulI-‐layer soil model:

Runoff R

NeglecIng Lateral Flow

•  At GCM grid resoluIons, lateral flow is typically neglected. Why? A scale analysis shows… –  Typical “acIve” soil column considered in LSMs is only a few meters deep.

22

5m on a side, 5m deep

Top, bo[om, sides all have same area, water flows are approximately comparable.

50 km on a side, 5m deep

Top, bo[om, and internal horizontal planes have 10,000 Imes the area as the verIcal (side) faces, lateral flows are negligible compared to verIcal and internal flows.


Change in Storage

•  Depending on locaIon and climate, there can be mulIple water storage reservoirs: –  Soil moisture

•  Can exist anywhere there is soil exposed to air or supporIng vegetaIon •  Can be liquid, solid or vapor (typically soil water vapor is neglected in climate

applicaIons) –  Snow

•  Cold climates, surface storage (on top of soil) –  Ice (Glacier)

•  Historically treated as a specified boundary condiIon in weather/climate models because of its long Ime scales of variaIon – this is changing now.

–  Canopy intercepIon •  Where there is vegetaIon, equals precipitaIon minus throughfall

–  Surface water •  Ponding, lakes, rivers; treated with varying degrees of sophisIcaIon

–  Groundwater (Water Table) •  Treated separately from soil moisture, deeper, interacts with river channel

and may interact with soil moisture (vadose zone).


23

•  At the land-‐atmosphere interface:

–  RC is carbon respiraIon (uptake by plants), silicate weathering –  EC is carbon emission (microbe respiraIon, decomposiIon, wood burning, fossil fuels, wildfires, volcanic erupIons etc.)

–  ΔSC is change in carbon storage in the land (vegetaIon biomass, soil organic carbon, mineral formaIon, etc.)

–  is verIcally integrated carbon flux divergence –  is change in verIcally integrated atmospheric

carbon

Surface Carbon Balance


€

RC − EC = ΔSC = − ∇ ⋅ (C!

V )z∫ −Δ C

z∫

Interface Land Atmosphere

€

∇ ⋅ (C!

V )z∫

€

Δ Cz∫

Other Chemical Budgets

•  Once we consider the carbon cycle in our models, we find we must consider other chemical cycles that are important modulators of the biologic cycling of carbon and water. These are consItuents we call nutrients or ferIlizers. –  Nitrogen (#1) –  Phosphorus –  Potassium, Sulfur, Oxygen, etc…

•  Together, we refer to these as biogeochemical cycles.


25

Surface Energy Balance

•  At any point at the land-‐atmosphere interface:

•  FS is the downward flux of solar (shortwave) radiaIon •  α is the net surface albedo (shortwave reflecIvity) •  is the downward flux of thermal (longwave) radiaIon •  ε is the surface thermal emissivity •  σ is the Stefan-‐Boltzmann constant •  TS is the surface temperature •  H is upward sensible heat flux from the surface •  λ is the latent heat of vaporizaIon •  E is the surface evaporaIon •  G is heat flux into the ground •  miscellaneous includes energy associated with soil water freezing,

plant chemical energy, heat content of precipitaIon, etc.


€

FS (1−α) + FL↓ = εσTS

4 + H + λE +G + miscellaneous

€

FL↓

Modeling the Energy Balance for a Single Land Surface Slab, Without Snow

Si + Li = Sh + Lh + H +λE + CpΔT/Δt + miscellaneous

Where: Si = Incoming shortwave radiaIon Li = Downward longwave radiaIon Sh = Reflected shortwave radiaIon Lh = Upward longwave radiaIon H = Sensible heat flux λ  = Latent heat of vaporizaIon E = EvaporaIon rate

Cp = Heat capacity of surface slab ΔT = Change in slab’s temperature,

over the Ime step Δt miscellaneous = energy associated with

soil water freezing, plant chemical energy, heat content of precipitaIon, etc.

S S L L H λE

T

Terms on LHS come from “above” (atmospheric model). Strongly dependent on cloudiness, water vapor, etc.

Terms on RHS come from “below” (determined by the land surface model).


Energy balance of a vegetaIon canopy

S S L L H λE

S S L L H H

Tc

Energy balance in a surface layer

G12 = heat flux between soil layers 1 and 2

Energy balance in a subsurface layer

S S L L H λE

T2

T1

T3

G12 Internal energy

T2

T1

T3

G12

G23 Internal energy λm = latent heat of melIng

λs = latent heat of sublimaIon M = snowmelt rate GS1 = heat flux between bo[om of pack and soil layer 1

Energy balance in snowpack

T1

S S L L H λ sE

Tsnow λmM Internal energy GS1

Note: same symbols are used, but values will be different.

Other energy balances can also be considered


Sensible Heat Flux H

EquaIon commonly used in climate models: where: •  ρ is mean air density •  Cp is specific heat of air at constant pressure •  KH is exchange coefficient for heat •  |V| is wind speed at reference level •  TS is surface temperature •  TR is air temperature at reference level

€

H = ρCpKH V (TS −TR )


We commonly write this in terms of an aerodynamic resistance ra:

where: is effecIvely a conductance of heat between surface and air.

€

H =ρCp (TS −TR )

ra

€

ra =1

KH V

€

KH V

Latent Heat Flux λE

•  Latent heat flux is the energy used to transform liquid water or ice into water vapor.

•  Latent heat flux from a liquid surface: λvE –  E = evaporaIon rate (flux of water molecules away from surface)

–  λv = latent heat of vaporizaIon ≅(2.501 -‐ .002361T)×106 J/kg

•  Latent heat flux from an ice/snow surface: λsE –  λs = latent heat of sublimaIon = λv + λm

–  λm = latent heat of melIng ≅ 3.34×105 J/kg


30

•  Typically defined as the surface energy imbalance:

–  is heat flux into the ground

–  ΔTSkin is the temperature gradient across the surface skin layer between land and air, auer all other flux terms have been accounted for.

–  In models, the “skin temperature” of the land surface is adjusted at each Ime step to provide the gradient consistent with the calculated surface energy imbalance.

Ground Heat Flux -‐ G


€

G = cpΔTSkinΔt

=ΔSΔt

Reservoir Input Output

€

G = cpΔTSkinΔt

=ΔSΔt

= FS (1−α) + FL↓ −H − λE −εσTS

4


Net RadiaIon and VegetaIon

RN = Net radiaIon (incoming at the surface) S↓ = Downward shortwave (solar) radiaIon at the surface α = Net surface albedo (reflecIvity) ε = Emissivity L↓ = Downward longwave (thermal) radiaIon from atmosphere σ = Stefan Boltzmann constant TS = Surface temperature ε ≈ 1 is usually an acceptable assumpIon (blackbody assumpIon) α = 10% ‑ 40% (~80% over fresh snow)

Sahara/Arabia 30‑35% Other deserts 20‑25% Dense forests 10‑15% (Compare to ocean 4‑5%)

This summarizes the radiaIon balance at the land surface. VegetaIon is parIcularly important in affecIng the S↓ term.

€

RN = S↓(1−α) +εL↓ −εσT4

Primary reference: Sellers, P. J., 1985: Canopy reflectance, photosynthesis and transpiration. Int. J. Remote Sensing, 6, 1335-1372.

32


Direct RadiaIon in a Canopy

kLoL eII

LIkI−=

Δ−=Δ

Analogous to the transfer equaIon, we can describe exIncIon (absorpIon) of radiaIon by a plant canopy:

I = radiaIve flux, k = exIncIon coefficient, L = leaf area index. The exIncIon coefficient will depend on the orientaIon of the leaves:

33

Two-‐Stream ApproximaIon


Direct RadiaIon in a Canopy

€

µ = cosθ

Direct radiaIon comes from the direcIon of the solar zenith angle θ so we can define an inverse opIcal depth:

It interacts with the canopy depending on the orientaIon of the leaves. The projected area of the leaves can be represented as a funcIon of the direcIon of radiaIon G(µ), which may be quite complex depending on the shape and structure of the leaves and plants. The exIncIon coefficient is defined as:

For a simple flat horizontal leaf, G(µ) = µ, so k = 1.

€

k =G(µ)

µ

34

•  This is actually a bit of an over-‐simplificaIon, because leaves sca[er light as well (if they didn’t they would appear black, not green, yellow, etc.).

•  The relaIonship for the exIncIon coefficient may be wri[en to include sca[ering:

where ωs is the sca[ering coefficient. In fact, the sca[ering is a funcIon of the leaf angle distribuIon, so may have its own complex dependence on µ.


Sca[ering

€

k =G(µ)

µ1−ωs

35


Diffuse RadiaIon in a Canopy

€

kdiffuse =1µ

µ =" µ

G( " µ )0

1

∫ d " µ

The sca[ering coefficient ω = α + τ is the sum of reflectance and transmi[ance. Most canopies have a fairly small sca[ering coefficient in the visible range (photosyntheIcally acIve radiaIon; PAR), and a high sca[ering in the near infrared:

For diffuse radiaIon:

PAR Region: ω ≈ 0.2

36

NIR Region: ω ≈ 0.95 PAR NIR

Note the “bump” in the green range!

The Planetary Boundary Layer (PBL)

Layer Processes Domain Free

Atmosphere Dynamic Dry convective adjustment

Turbulence closure

Mixed Layer

Thermal, Mechanical,

Coriolis

Mixed Layer Model Multi-Layer Model

AGCM

Constant Stress Layer

Thermal, Mechanical

Monin-Obukhov Similarity Theory

LSM

Molecular Layer Molecular

Land Surface

The PBL (shaded below) is the layer of the earth’s atmosphere between the earth’s surface and the free atmosphere (where the wind is essenIally geostrophic); it includes the surface fricIon layer (constant stress layer) and the mixed layer (Ekman layer).

Over land, the PBL height varies from 0 to >2000m with a strong diurnal cycle.


Lecture -‐ 1 37


Surface fluxes: Bulk transfer relaIons

aHopo

aHoo

aMoo

rcHrqqEruu

/)(/)(/)(

θ−θρ=

−ρ=

−ρ=τ

Here u is a non-‐direcIonal wind speed, and likewise the momentum flux τo is non-‐direcIonal. Here, (CD u) and its brethren are conductances that facilitate the rate of flux, given a certain gradient, and aerodynamic resistances are defined as:

The formulaIon is typically used in Land Surface Schemes (LSS), which is also the basis of turbulent fluxes within the Simple Biosphere (SiB) model and its so-‐called SiBlings.

uCr

uCr

HaH

DaM

1,1==

Primary reference: Garratt, J. R., 1992: The atmospheric boundary layer. Cambridge University Press, 316 pp.

38

Linking Land Surface and Boundary Layer

•  Surprisingly, research in these two areas have proceeded rather independently. –  Boundary layer research has focused on simulaIon of stable profiles, large eddy simulaIons of turbulence, and defining bulk properIes over various terrain – the land surface is usually treated as a boundary condiIon.

–  The link between land surface and climate, including feedback processes, has focused either on the immediate near surface processes (surface fluxes, screen-‐level meteorology) or the free atmosphere (e.g., precipitaIon) with the PBL as a “black box” in-‐between.

39 Lecture -‐ 1 ICTP-‐IITM-‐COLA Targeted Training AcIvity (TTA) on "Modelling and PredicIon of Asian Monsoons: Improving Physical Processes", February 9 -‐ 20, 2015

Soil Horizons • There is a general verIcal structure to the soil that is the basis of most models of the verIcal soil column.

– O Horizon: organic ma[er, humus – A Horizon: topsoil, “root zone”, some organic ma[er, dynamic

– E Horizon: leached of water-‐soluble minerals, transiIon layer

– B Horizon: subsoil; “illuviated” accumulates minerals lost from “eluviated” zone.

– C Horizon: transiIon between soil and bedrock; new mineral soil formed.

• Not all soils have all horizons. Lecture -‐ 1 ICTP-‐IITM-‐COLA Targeted Training AcIvity (TTA) on "Modelling and PredicIon of Asian

Monsoons: Improving Physical Processes", February 9 -‐ 20, 2015 40

The Soil

Soil Texture •  For purposes of land surface modeling, we consider a simpler

categorizaIon based on composiIon with 3 or 4 components:


41

1.  Sand: hard round parIcles with voids between, do not clump.

2.  Silt: smaller grains than sand, smaller voids, easily carried in suspension by water.

3.  Clay: very fine parIcles, usually flat, compress to form nearly impervious sheets

4.  Organic Ma[er*

Texture and Soil ProperIes •  It is common in land models to define a small number of

texture classes, each with a set of parameters defining its ability to conduct or transfer heat and water in the verIcal.


42

•  An example is shown on the right clay

sand silt

Heat ConducIon in Soil •  In 3-‐dimensions:

•  This is usually expressed only in the verIcal:

•  Where:

•  The actual heat flux G into the soil [W m-‐2] at its surface is typically denoted as posiIve, even though it is downward:

•  The final form:


43

€

C ∂T∂t

=∇ ⋅ (k∇T)

€

C ∂T∂t

=∂∂z(k ∂T∂z) = k ∂

2T∂z2

+∂T∂z

∂k∂z

C = heat capacity of soil [J m-‐3K-‐1] T = temperature [K] k = thermal conducIvity of soil [W m-‐1K-‐1]

ρs = density of the soil [kg m-‐3] cs = mass specific heat of soil [J kg-‐1K-‐1]

€

C = ρScS

€

G = −k ∂T∂z

€

∂T∂t

= −1C∂G∂z

The ComplicaIon of k

•  Thermal conducIvity k is a strong funcIon of soil moisture w and also a funcIon of depth (e.g., soil horizons). –  Some models neglect the dependence on depth

•  Heat capacity C also varies with soil moisture content and depth [C(w,z)], but is more ouen neglected because its variance is orders of magnitude smaller than k(w,z)

•  Thus, we must parameterize k in order to simulate well

the transfer of heat into the soil.

44

parameterization


Lecture -‐ 1

The Bo[om Boundary

•  There is a boundary condiIon at the bo[om of the soil column that must be specified. Two commonly used boundary condiIons:

•  Zero heat flux – at the interface below the lowest soil layer N:

•  Constant temperature – the base soil temperature below

the soil column is set to a constant TBase (typically the annual mean air temperature at the locaIon):

•  Zero heat flux conserves energy, constant TBase does not!

45

€

qN +1/ 2 = 0

€

qN +1/ 2 = −kN +1/ 2TBase −TNZBase − ZN


•  We consider only the two largest terms: •  GravitaIonal potenIal: •  Matric potenIal: •  Pedotransfer funcIons are derived to calculate quanIIes like the

matric potenIal.


46

€

ψz = gρwΔz

€

ψ =ψz +ψm

€

ψm = parameterization

Difference relaIve to a reference height

Water ConducIon in the Soil

•  Soil water potenIal y describes the potenIal energy of water per unit volume relaIve to a reference (pressure, temperature, elevaIon), usually set to zero. In other words, it is the amount of work necessary to move the water verIcally in the soil matrix.

•  Units are potenIal/volume = force/area = pressure (pa). –  Engineers and hydrologists use potenIal/weight = distance (m), which they

call the hydraulic “head” h = ψ/(ρwg).

Water ConducIon in Soil

•  The predicIon equaIon is:

•  Splikng the potenIal term gives Richards EquaIon:

•  Expanding:


47

€

ρwg∂w∂t

= −∂∂z

kw∂ψ∂z

&

' (

)

* +

capillary acIon gravity

€

∂w∂t

= −∂∂z

kw1ρwg

∂ψm

∂z+1

&

' (

)

* +

,

- .

/

0 1

€

∂w∂t

= −1ρwg

kw∂ 2ψm

∂z2+∂ψm

∂z∂kw∂z

&

' (

)

* + −

∂kw∂z

Remember soil moisture is the reservoir of water

in the soil!

This w is dimensionless

All the ComplicaIons

•  Hydraulic conducIvity kw [ms-‐1] is a funcIon of soil moisture w itself (which varies in the verIcal) and with soil properIes (which vary horizontally and verIcally (soil horizons)), but the verIcal variaIon with soil texture is someImes neglected.

•  Matric potenIal is a funcIon of soil moisture and soil properIes as well.

•  These dependencies makes the non-‐linear parIal differenIal equaIon called the Richards EquaIon difficult to solve analyIcally in most cases.

•  In numerical models, the soil is broken into homogeneous layers, and the fluxes are solved at the layer interfaces, usually iteraIvely.


Lecture -‐ 1

ψm

ET Stresses Revisited

•  Remember canopy resistance rc? One of the factors controlling it is soil moisture, or more precisely the soil water potenIal ψ.

•  There is a potenIal between the soil water content and the roots of plants that allow water to diffuse into the roots – which feeds transpiraIon.

•  WilIng point – the potenIal below which plant roots cannot extract moisture from the soil – actually varies based on soil condiIons, vegetaIon properIes and even local meteorology, but ouen defined as ψWP = -‐1.5×106 pa.


49

•  Hydraulic redistribuIon also keeps rhizomes, soil microbes, and even shallow-‐rooted plants like grasses alive during dry spells, which benefits the tree as well.

•  This mechanism can also accelerate the moIon of rainwater down into the soil.


50

Hydraulic RedistribuIon •  Hydraulic redistribuIon is the mechanism by which some plants

redistribute soil water.

Dawson, 1993: Oecologia, 565-

Baseflow

•  Baseflow is the flow of water out of the soil column. This is a bo[om boundary condiIon analogous to that for heat. There are many ways to treat this as well.

–  Drainage – assumes no interacIon with a water table below the soil column. Water that drains out the bo[om is put in the river channel to be carried downstream. Usually some proporIonality to the soil moisture in the lowest layer, e.g.:

–  Drainage with slope – add a term that increases the baseflow as the terrain becomes steeper (α is average slope), e.g.:

–  2-‐way interacIon with water table (e.g. “TOPModel” approaches).


51 €

qN +1 = kNwN2B +3 sinα + kCwN€

qN +1 = kCwN

Hillslopes, InfiltraIon and Runoff •  For lateral moIon of water within a model grid cell, the terrain

within the grid cell is ouen treated as an idealized hillslope.

•  An alternaIve parameterizaIon is the VIC (Variable InfiltraIon Capacity) model:


52

10s to 100s of meters 10s to 100s of kilometers

–  Horton runoff: precipitaIon exceeds infiltraIon.

–  Dunne runoff: soil is saturated, cannot infiltrate.

–  Baseflow: lateral runoff beneath soil into river system.

MulI-‐layer soil model:

Liang et al., 1996: Glob. Planet. Change, 13, 195-.

Topographic Index

•  Topographic Index is a way to parameterize the large-‐scale infiltraIon and water table properIes in land models. Most common is the Compound Topographic Index:

•  A is the upstream area of the river basin draining through the grid box, and β is the slope of the terrain in the grid box.

•  Topographic indices are used to esImate the spaIal variaIon of water table depth, and thus the porIon of grid boxes that are saturated, experiencing Dunne runoff, baseflow rates, etc.

•  TOPModel is the standard for linking soil moisture and water table models.


53

€

CTI = ln Atanβ#

$ %

&

' (

Famiglietti and Wood, 1994: Water Resour. Res., 30, 3061-.

Ice in the Soil •  When water in the soil freezes, many things change…

–  Ice is essenIally staIonary – it will not drain and it will not rise by capillary acIon.

–  Ice had a different heat capacity and thermal conducIvity than liquid water, so thermal properIes change as well.

–  Unless condiIons are very cold for a long Ime, there will usually be both liquid water and ice in the soil column, making modeling of soil moisture and soil heat flux rather complicated.

•  In cold climates, permafrost (soil that is frozen throughout the year) exists below the top layers of the soil. Permafrost acts almost like bedrock, allowing li[le baseflow of water. Soils above permafrost ouen remain very wet.


54

Snow •  Snow is another complicaIng factor.

–  RadiaIve: Snow is highly reflecIve, increases surface albedo. –  Thermal: Snow is an effecIve insulator – prevents sensible heat flux

between soil and atmosphere. –  Hydrologic: Snow decouples soil moisture from atmosphere via latent

heat flux, while sublimaIng itself to the atmosphere. Also, snow melts, supplying water for infiltraIon and runoff days, weeks or months auer the precipitaIon occurred.

•  Snow can be modeled in a similar way as soil, with its own

heat and moisture budgets. –  The main difference – the thickness of the snow changes with

precipitaIon, melIng. –  Density varies with different snowfalls, and increases over Ime as

snowpack ages. Albedo, conducIviIes, diffusiviIes, heat and water capaciIes also vary over Ime and with depth in snowpack.


55


56

LSM development •  Improvement in the representaIon of snowpack, soil moisture,

vegetaIon and runoff.

•  Improvement in the parameterizaIon of evapotranspiraIon.

•  Linking the land surface and the planetary boundary layer.


Each grid box is

~50,000 km in area

Area-‐Averaged Rain Rates Accurate precipitaIon measurements are limited by the availability of rain gauges….

57

GPCP = Global PrecipitaIon Climatology Project • Coverage is very uneven across the globe – many regions have few or no gauges for hundreds of kilometers in any direcIon

• Some naIons have good internal rain gauge networks, but will not share their data.

• Others are too poor / unstable to maintain observaIons

1000

100

20

8

5

3

2

1.5

0.6

0.3

0

Figure: Koster et al. (2011) doi:10.1175/2011JHM1365.1.


Area-‐Averaged Rain Rates

How Good is the Es/mated SSM/I Rain Rate Climatology Data?

Estimated Nonsystematic Error

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14

Rain Rate (mm/day)

Perc

ent E

rror

(%)

F-14 F-13 TMI F-13+F-14 GPM F-13 AM

• The Special Sensor Microwave/Imager (SSM/I) is a 7-‐channel, 4-‐frequency, linearly polarized passive microwave radiometer system flown on several satellites

• Over oceans, no “truth” data available for validaIon – esImates only over land

• NonsystemaIc error includes sampling errors and random errors; sampling errors dominate

• DMSP F-‐13 and F-‐14 SSM/I, with similar sampling (orbits) have similar error

• The TRMM (Tropical Rainfall Measuring Mission) TMI (ThemaIc Mapping Imager) has a slightly lower error – limited to ±40°

• Combining F-‐13 & F-‐14 almost saIsfy the TRMM 1 mm/day and 10% for heavy rain

• GPM with 8 satellites would have had 50% less error than combining F-‐13 & F-‐14 – funding cuts limit mission to 1 satellite

… and by the inherent inaccuracies in satellite-‐derived precipitaIon data

58

Land-‐Ocean Exchanges

•  In our discussions we have neglected some major fluxes…

•  Rivers carry large amounts of water, and smaller but significant amounts of carbon (and other elements) from land to ocean.

•  Rivers are an important part of the global water and carbon cycles, but in this class we will focus on the land-‐atmosphere interacIons.

59

Figure: Washington Post, 8 November 2011.


Lecture -‐ 1

Soil Taxonomy •  There is a thorough and complex taxonomy of soils, based on

composiIon and form, that includes classes, subclasses, etc. •  There are 12 USDA classes:


60

• Gelisols: permafrost soils • Histosols: organic soils • Spodosols: coniferous/boreal soils • Andisols: volcanic soils • Oxisols: wet weathered soils, e.g. under tropical forests

• Ver/sols: soils high in clay • Aridisols: dry climate, low organic ma[er, caliche layers

• Ul/sols: red clay soils • Mollisols: mid-‐laItude grassland soils, sand or limestone based, rich (“breadbasket” soils).

• Alfisols: typical soil under hardwood forests, rich, some clay

• Incep/sols: Largely unweathered soils lacking illuvial accumulaIon

• En/sols: Soils lacking any development (no E, B, and someImes C horizons).

ftp://ftp-fc.sc.egov.usda.gov/NSSC/Soil_Taxonomy/keys/ebook/Keys_to_Soil_Taxonomy_11th_Edition.pdf.


Turbulent Fluxes

€

τ x = −ρ % u % w = ρKM∂u∂z

E = ρ % q % w = −ρKV∂q∂z

τ y = −ρ % v % w = ρKM∂v∂z

H = ρc p % θ % w = −ρcpK H∂θ∂z

VerIcal fluxes of momentum, latent heat and sensible heat can be wri[en as:

where the turbulent stress (the second order terms in each) is related to the flux gradient in the verIcal. For example, for momentum, τ is the tangenIal fricIonal force. The total shear normal to the surface is:

Turbulence closure schemes like those applied in the Ekman layer are built around the expansion of the higher order terms to some level auer which derivaIves of the higher order moments → 0.

€

∂u /∂z

61

Turbulent mixing occurs over a length scale l that is a funcIon of the depth of the constant stress layer. k is the von Karman constant (∼0.4 as determined experimentally). l = k zc

Lecture -‐ 1

ICTP-‐IITM-‐COLA Targeted Training AcIvity (TTA) on "Modelling and PredicIon of Asian Monsoons: Improving Physical Processes",

February 9 -‐ 20, 2015

Near an interface (e.g., the surface of the earth) we must be concerned with the effects of the disconInuiIes in temperature and moisture, and the effect of the "no-‐slip" boundary condiIon on turbulent transfer. We can relate the viscosity to the length scale through a fricIonal velocity:

Flux-‐profile relaIonship

€

KM = u*l

u*2 =

τ oρ= [( !u !w )2 + ( !v !w )2 ]1/2

The fricIonal velocity is a funcIon of the horizontal surface stress: Disregarding the direcIon of the wind, and subsItuIng we have:

€

∂u∂z

=u*kz

62

€

kuu*

= ln zzo

"

# $

%

& '

IntegraIng in z, and choosing the constant C = -ln(zo) such that u = 0 at z = zo. zo is roughness length.


Drag Coefficients

€

CH =k 2

ln zzo

"

# $

%

& ' ln

zzH

"

# $

%

& '

Similar drag coefficients can be defined for heat and moisture (ouen chosen to be the same) based on a surface scaling length for heat that yields a zH analogous to zo:

63

€

CD =u*u

"

# $

%

& ' 2

=k 2

ln Z − dzo

"

# $

%

& '

2In bulk transfer relaIonships we can define a drag coefficient:

Monin-‐Obukhov Similarity Theory The turbulent fluxes of momentum, sensible heat and moisture are nearly constant with height in the constant stress layer. The M-‐O similarity theory states that when scaled appropriately the dimensionless mean verIcal gradients of wind, potenIal temperature and specific humidity are unique funcIons of a buoyancy parameter (ζ), where, is the M-‐O length scale

ζ = (z− d) / L

L = −u*3 / k g

θ

"

#$%

&'

HρCp

"

#$$

%

&''

(

)**

+

,--

u*2 =

τ oρ= [( !u !w )2 + ( !v !w )2 ]1/2

The fricIonal velocity is a funcIon of the horizontal surface stress: PosiIve values of the funcIon indicate stable atmosphere, whereas negaIve values indicate unstable condiIon.


Lecture -‐ 1 64

u*

Eventually Plants Die •  The carbon may remain in the land surface system as wood on the surface, or in the soil (woody material, peat, organic carbon parIcles, minerals) for some Ime.

•  There are biological and other chemical processes that can return this carbon back to the atmosphere.

•  In the Industrial Era (the Anthropocene*), humans have become a major converter of terrestrial carbon to atmospheric carbon through the burning of oil, coal, natural gas, wood, etc.


65

*Term coined by Eugene Störmer, but popularized by Paul Crutzen.

Carbon Emissions EC •  Natural terrestrial sources of carbon:

–  RespiraIon by animals and plants [plant chemistry of digesIng glucose (sugar) re-‐releases carbon]

–  Decay of organic Issues and ma[er by fungi and bacteria (the major component of soil respiraIon)

– Metamorphic rock formaIon (slow); volcanic erupIons (unpredictable) – these approximately balance silicate weathering.

– Wildfires •  Besides the burning of fossil fuels and wood, the producIon of cement is another large human carbon emission.


66

Downward Shortwave RadiaIon

•  Si is, basically and simply, sunlight. •  We usually make a simplifying assumpIon that the 3-‐dimensional complexity of sky and earth can be simplified to a 1-‐D problem where there is only “up” and “down”.

•  For downward (also called downwelling) shortwave radiaIon, integrate over the “upper” hemisphere:


67 €

FS = IS (θ,ϕ)cosθ sinθ dθ dϕ0

π2

∫0

2π

∫Azimuth angle Zenith angle Intensity of radiaIon

Direct and Diffuse RadiaIon

•  For shortwave radiaIon, we disInguish between direct beam and diffuse radiaIon.

•  Direct radiaIon is treated as uni-‐direcIonal (like a laser beam) directly from the disc of the sun.

•  Diffuse radiaIon is omni-‐direcIonal, from the whole sky (clear and cloudy).

•  Both have energy at a range of wavelengths.

•  IS(λ) is the intensity of radiaIon at wavelength λ. 68

€

FS = IS (Direct )(λ)dλShortwave∫ + IS (Diffuse )(λ)dλ

Shortwave∫


Spectral Bands

•  Shortwave radiaIon is defined to be across a range of the electromagneIc spectrum covering ultraviolet, visible and near-‐infrared wavelengths.

•  To make calculaIons manageable, we divide that spectrum into a number of discrete bands that we treat as having uniform properIes within each band.

•  I(b) is the intensity of radiaIon in band b.

69

€

S↓ = I (Direct )b∑ (b) + I (Diffuse )

b∑ (b)


Reflected Shortwave RadiaIon

•  Sh is the reflected shortwave radiaIon, also called upward or upwelling shortwave radiaIon.

•  Sh is related to Si by the net surface albedo α:

•  Albedo is also a funcIon of wavelength; net surface albedo is different for direct and diffuse radiaIon:


70 €

S↑ = αDirect (b) ⋅ IDirectb∑ (b) + αDiffuse (b) ⋅ IDiffuse

b∑ (b)€

S↑ =αS↓

b indicates spectral bands – typical to have only 2: UV+visible and IR


RadiaIve Transfer EquaIon

€

dIλ = −kλρIλds

RadiaIon traversing a medium (e.g. air) will be weakened by its interacIon with ma[er. The intensity of the radiaIon Iλ, auer traversing a distance ds, becomes Iλ + dIλ, where:

ρ is the density of the ma[er traversed, and kλ is the mass exIncIon cross secIon (area per mass) at wavelength λ. The reducIon dIλ is caused by sca[ering and absorpIon. Where sca[ering can be neglected (e.g., a blackbody), then kλ is the mass absorpIon cross-‐secIon (or absorpIon coefficient).

71


RadiaIve Transfer EquaIon

€

Iλ(s1) = Iλ(0) exp(− kλρ ds0

s1

∫ )

At locaIon s = 0, let Iλ = Iλ(0), and we can integrate the transfer equaIon to a distance s1:

72


Beer-‐Lambert-‐Bourguer Law

€

u = ρds0

s1

∫

€

Iλ(s1) = Iλ(0)e−kλu

€

Aλ =1−Tλ =1− e−kλu

Over a path length u, assuming the medium is homogeneous (i.e., kλ is constant along the path):

Then we have:

In fact, e-kλu is the monochromaIc transmissivity: Tλ For a non-‐sca[ering medium, the monochromaIc absorpIvity is:

If there is sca[ering, we define a monochromaIc reflecIvity (a.k.a. albedo): Rλ, and:

€

Aλ + Tλ + Rλ =1

73

ExIncIon: MathemaIcal RepresentaIon

•  ExIncIon is represented by the following differenIal equaIon:

•  I is the intensity, z is the distance travelled, and k is the exIncIon coefficient.

•  NoIce the minus sign – it indicates the intensity diminishes along the path.

Lecture -‐ 1

€

dI = −kIdz


ExIncIon: Along a Path

•  If we integrate the equaIon between z=0 and z=z1,

•  This is the classic “exponenIal decay”. •  For each increase of the exponent kz1 by one, the intensity will decrease by a factor of 1/e or to ≈37% of its previous value.

Lecture -‐ 1

€

I(z1) = I(0)e−kz1


ExIncIon: OpIcal Path

•  ExIncIon in fluids like the atmosphere can be highly variable depending on the type and density of the a[enuaIng consItuents (e.g., clouds, smoke, dust, chemical species, etc.).

•  In pracIcal terms, the path is represented not as a distance, but as the amount of a[enuaIng ma[er traversed.

•  For a single consItuent of density ρ:

•  u is usually referred to as the opIcal path length.

Lecture -‐ 1 €

u = ρdz0

z1

∫


ExIncIon: Beer-‐Lambert-‐Bourguer Law

•  We then have: •  Here we represent the intensiIes and exIncIon coefficient k as funcIons of wavelength λ because exIncIon typically varies greatly as a funcIon of wavelength.

•  is the monochromaIc (“one color”, i.e., one wavelength) transmissivity, and

•  is the monochromaIc absorpIvity.

Lecture -‐ 1

€

Iλ(z1) = Iλ(0)e−kλu

€

e−kλu

€

1− e−kλu


Lecture -‐ 1


February 9 -‐ 20, 2015

Two-‐Stream FormulaIon

€

µdI↓dL

= ωβI↑−[1− (1−β)ω]I↓+ωµk(1−βo)e−kL

µdI↑dL

= ωβI↓−[1− (1−β)ω]I↑+ωµkβoe−kL

is the mean leaf inclinaIon angle relaIve to horizontal, and as(µ) is the single sca[ering albedo.

€

θ

The equaIons for the change in diffuse flux as it penetrates the canopy (i.e. as a funcIon of the LAI penetrated, where LAI is essenIally a verIcal coordinate within the canopy) are:

β and βo are the upsca[er parameter for diffuse and direct radiaIon respecIvely:

€

β =12ω[α + τ + (α − τ )cos2θ]

βo =1+ µkωµk

as(µ)

78

Lecture -‐ 1


February 9 -‐ 20, 2015

Two-‐Stream

1=oI

][ totalkLsoil eII −+↓α↑=

The pair of differenIal equaIons above can be solved given appropriate boundary condiIons. Assuming a normalized incident solar radiaIon at the top of the canopy:

and at the bo[om where L = Ltotal:

€

µdI↓dL

= ωβI↑−[1− (1− β)ω]I↓+ωµk(1− βo)e−kL

µdI↑dL

= ωβI↓−[1− (1− β)ω]I↑+ωµkβoe−kL

79

Diurnal Cycle •  US Department of Energy

(DOE) has a facility in Oklahoma and Kansas for measuring the surface energy balance (ARM/CART*).

•  Single day’s hourly measurements at a wet (top) and dry (bo[om) site.

•  Note the relaIve magnitudes of fluxes and how they vary in Ime and between locaIons.


80

*Atmospheric RadiaIon Measurement (ARM) Program / Cloud And RadiaIon Testbed (CART)

Figure: Robock et al., 2002: J. Geophys. Res., 8846.

Diurnal Cycles of Surface Fluxes


81

Wet soils Unstressed ET β < 1 all day

TransiIon soil Semi-‐stressed ET β < 1 during morning only

Dry soils Stressed ET β > 1 all day

Typical July CondiIons Represented

Bowen RaIo: β = H/λvE

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