radiative transfer, canopy radiation and turbulence

Lecture 4 Dirmeyer

CLIM 714Land-Climate Interactions

Radiative Transfer, Canopy Radiation and Turbulence

Paul DirmeyerCenter for Ocean-Land-Atmosphere Studies

Lecture 4 Dirmeyer


Radiative Transfer

Energy transfer between the surface and adjacent matter (above and below) is accomplished by various mechanisms:

• Radiative transfer Light (visible radiation) Heat (thermal or infrared radiation)

• Conduction to adjacent surfaces Sensible heat (air above) Heat diffusion (into the soil below)

• Latent heating Phase change of water (evaporation, transpiration,

sublimation, melting)• Mass transport (interception, throughfall, infiltration)

Lecture 4 Dirmeyer


The Electromagnetic spectrum

The chart below shows the common names given to regions of the electromagnetic spectrum, and their corresponding wavelengths.

Given the speed of light c = 2.9979×108 m s-1, we can relate wavelength and frequency as:

cwhere is the wavelength [m] and is the frequency [s-1].We will be predominantly concerned with radiation in the

infrared (IR), visible and ultraviolet (UV) ranges.

Primary reference:Liou, K.-N., 1980: An introduction to atmospheric radiation. International Geophysics Series, 26, Academic Press, 392 pp.

Lecture 4 Dirmeyer


Solid Angles

• Study of radiation requires the use of solid angles. A solid angle is the ratio of the subtended area σ of the surface of a sphere divided by the square of the sphere’s radius:

2r

Solid angle units are steradians [sr]. For a sphere of surface area r, the solid angle subtended by the entire sphere is 4, by a hemisphere is 2, etc.

Figure this page from Liou

Lecture 4 Dirmeyer


In spherical coordinates of zenith (colatitude) and azimuth (longitude) angles and , the differential elemental solid angle is:

Solid Angles

)sin)(( drdrd

ddrd

d sin2

So the differential solid angle is:

Figure this page from Liou

Lecture 4 Dirmeyer


Radiometric Quantities

Consider a differential amount of radiant energy dE in a time interval dt and in a wavelength interval from to +d, crossing an element of area dA oriented at an angle normal to dA. This energy can be expressed in terms of a specific intensity I by:

dtddAdIdE cos

I is the monochromatic intensity (or radiance) in energy per area per time per frequency per steradian.The monochromatic flux density (or monochromatic irradiance) is the normal component of I integrated over the entire spherical solid angle:

dIF cos

2

0 0

2

sincos),( ddIF

expressed in polar coordinates.

or:

Lecture 4 Dirmeyer


The total flux density of radiant energy or irradiance for all wavelengths is:

Radiometric Quantities

dFF0

IF

dAFfA

If the radiation is isotropic (i.e., the irradiance is the same in all directions), then this simplifies to:

and the total flux or radiant power is:

Symbol

Quantity Dimension Units

E Energy M L2 T-2 Joule

I Intensity, radiance M T-3 W m-2 sr-1

F Flux density, irradiance M T-3 W m-2

f Flux, power M L2 T-3 Watt

Summary:

* For monochromatic radiation, divide by unit wavelength (L-1 or m-1)

Lecture 4 Dirmeyer


Planck’s Law

The amount of radiation emitted by a blackbody (a perfect absorber and emitter) is uniquely determined by its temperature:

)1(2

/5

2

)( TKhce

hcTB

where Planck’s Constant h = 6.6262×10-34 J s,

and Boltzmann’s Constant K = 1.3807×10-23 J K-1.

This yields the smooth curves of intensity as a function of wavelength for an emitter at a given temperature.

On the next slide are shown normalized curves for the sun and earth (unnormalized the radiation from a body the temperature of the earth would be so feeble that it would not show up on the same plot with the sun), and several intensity curves for bodies with a range of surface temperatures typical of medium-sized stars like the sun.

Lecture 4 Dirmeyer


Blackbody curves

from Wallace and Hobbs:

Blackbody radiation curves at five different temperatures (K; see legend in upper right corner).

Lecture 4 Dirmeyer


Wien’s Displacement Law

The wavelength of maximum emission where:

0)(

TB

is inversely proportional to the temperature:

T

3

max

10897.2

Lecture 4 Dirmeyer


Stefan-Boltzmann Law

32

444

152

,)(hc

KbbTTB

If we integrate the Planck function across all wavelengths, it reduces nicely to:

Since blackbody radiation is isotropic, we can define the flux density (integrating over the entire spherical solid angle):

where = 5.67×10-8 W m-2 K-4.

4)( TTBF

where

Lecture 4 Dirmeyer


Kirchhoff’s Law

1 A

1 A

A body absorbs and emits energy at a given wavelength with equal efficiency. For a blackbody:

Where A is the absorptivity, and ελ is the emissivity. If the object is not a blackbody, then:

Scattering of radiation (e.g. reflection) is the main contributor.

Emissivity in the 5-5.5 μm

band primarily from MODIS

and ASTER, as used in the

ARPEGE GCM for surface

temperature assimilation

Lecture 4 Dirmeyer


Radiative Transfer Equation

dsIkdI

Radiation traversing a medium (e.g. air) will be weakened by its interaction with matter. The intensity of the radiation I, after traversing a distance ds, becomes I + dI, where:

D is the density of the matter traversed, and k is the mass extinction cross section (area per mass) at wavelength . The reduction dI is caused by scattering and absorption. Where scattering can be neglected (e.g., a blackbody), then k is the mass absorption cross section (or absorption coefficient).

Lecture 4 Dirmeyer


Radiative Transfer Equation

)exp()0()(1

01

s

dskIsI

At location s = 0, let I = I(0), and we can integrate the transfer equation to a distance s1:

Lecture 4 Dirmeyer


Beer-Bourguer-Lambert Law

1

0

s

dsu

ukeIsI )0()( 1

ukeTA 11

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

Then we have:

In fact, e-ku is the monochromatic transmissivity: T

For a non-scattering medium, the monochromatic absorptivity is:

If there is scattering, we define a monochromatic reflectivity (a.k.a. albedo): R, and:

1 RTA

Lecture 4 Dirmeyer


Radiation in the Atmosphere

Time average irradiance of solar radiation on a spherical earth (above).

Deviations from blackbody due to absorption by the solar atmosphere, absorption and scattering by the earth’s atmosphere (below).

Lecture 4 Dirmeyer


Absorption, Emission and GeometryThe distance between the earth and the sun is sufficiently large that the radiation coming from the sun can be treated as parallel. So, the earth intercepts radiation as a disk (or a circle) of radius re = 6.37×106 m. However, as a blackbody, it emits radiation in all directions from its surface. Thus, the earth presents an effective absorption area to the sun of re

2 but emits over an area of re2 (the surface area

of a sphere with radius re).At the earth's distance from the sun, the solar irradiance is given by the so-called "solar constant". It's value is not truly constant, but is about 1380 W m-2. This is the irradiance falling on a surface that is normal to the direction of solar radiation (e.g. a plane parallel to the surface of the earth at the equator at noon during the equinox). When the sun is not directly overhead, then the 1380 W of radiation are not falling upon a unit are 1 meter square, but upon a larger area:

Lecture 4 Dirmeyer


Net Radiation and Vegetation

RN = Net radiationS = Downward shortwave (solar) radiation at the surface = Net surface albedo (reflectivity) = EmissivityL = Downward longwave (thermal) radiation from atmosphere = Stefan Boltzmann constantTS = Surface temperature

1 is usually a good assumption (blackbody assumption) = 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 radiation balance at the land surface. Vegetation is particularly important in affecting the S term.

4)1( TLSRN

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

Lecture 4 Dirmeyer


•Photosynthesis

•Plants use light (photsynthetically active radiation; PAR) as a source of energy to drive the chemical reaction that feeds the plant.

•Photosynthesis combines water and carbon dioxide to produce sugar (basic fuel for all living organisms).

•Plants produce oxygen as a biproduct of photosynthesis ‑ oxygen is waste to the plant.

Biophysics

26126

22

6

66

OOHC

OHCOPAR

Lecture 4 Dirmeyer


Biophysics

•Transpiration

•In the process of plant respiration (taking CO2 from the atmosphere through the stomata), plants lose water vapor

•Plants are constantly trying to balance the benefit of gaining CO2 with the detriment of losing water

•The water loss removes heat from the land surface (latent heating)

GEHRN

This is the “other side” of the net radiation equation — the vegetation and soil play an important role in all of these terms. Photosynthesis particularly affects latent heat flux.

Lecture 4 Dirmeyer


The Plane-Parallel Simplification

• Usually we assume that locally the earth is flat.

• Radiation is divided into only two directions: UP and DOWN.

• Each direction is a proxy for all radiation in that hemisphere (i.e., within each hemisphere of 2 steradians, radiant energy is isotropic)

The plane is horizontal to the earth’s surface, and may represent any interface (between layers of the atmosphere, above and below a forest canopy, or even a single leaf in a vegetation canopy).

Lecture 4 Dirmeyer


Direct versus Diffuse Radiation• Shortwave radiation can be either direct (with a specific

source in a specific direction), or diffuse (coming from all directions).

• Direct radiation - (heliotropic)

• Emanates from the sun, which is typically treated as a point source of radiation, traveling as a beam.

• Can be absorbed, reflected, transmitted based on the specific location and geometry of clouds, mountains, leaves, buildings, etc.

• Diffuse radiation - (isotropic)

• Emanates from the entire hemisphere (above or below), and is scattered sunlight. e.g., the light coming from a clear blue sky (or a grey cloudy sky).

• Has no specific direction, and is typically treated as uniform.

• Thermal radiation (heat) is also typically treated as an isotropic diffuse radiation.

Lecture 4 Dirmeyer


Direct vs. Diffuse

In reality, diffuse radiation does vary across the sky, or the ground. And the sun is not a point source, but subtends a finite solid angle in the sky. For simplicity, the above assumptions are typically applied, and usually do not affect the net radiation balance by more than a few percent.

A “fisheye” lens view of the sky (the upper hemisphere) at noon. Both clear sky and clouds contribute diffuse downward solar radiation. The disk on the

right shows the idealization of the sky used in many radiative transfer calculations

Lecture 4 Dirmeyer


Direct Radiation in a Canopy

kLoL eII

LIkI

Analogous to the transfer equation, we can describe extinction by a plant canopy:

I = radiative flux, k = extinction coefficient, L = leaf area index.The extinction coefficient will depend on the orientation of the leaves:

Lecture 4 Dirmeyer


Direct Radiation in a Canopy

cos

Direct radiation comes from the direction of the solar zenith angle so we can define an inverse optical depth:

It interacts with the canopy depending on the orientation of the leaves. The projected area of the leaves can be represented as a function of the direction of radiation G which may be quite complex depending on the shape and structure of the leaves and plants. The extinction coefficient is defined as:

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

)(G

k

Lecture 4 Dirmeyer


Diffuse Radiation in a Canopy

dG

kdiffuse

1

0 )(

1

And the scattering coefficient = + is the sum of reflectance and transmittance. Most canopies have a fairly small scattering coefficient in the visible range (photosynthetically active radiation; PAR), and a very high scattering in the near infrared:

For diffuse radiation:

PAR Region: 0.2

NIR Region: 0.95

Lecture 4 Dirmeyer


Two-Stream Approximation

Assuming that the scattered (diffuse) fluxes are hemispherically integrated, we can split all of the fluxes of visible and near infrared radiation into 3 components:

• Upward diffuse flux I• Downward diffuse flux I• Incident direct flux Io

Lecture 4 Dirmeyer


is the mean leaf inclination angle relative to horizontal, and as is the single scattering albedo.

Two-Stream Formulation

kLo

kLo

ekIIdL

dI

ekIIdL

dI

])1(1[

)1(])1(1[

The equations for the change in diffuse flux as it penetrates the canopy (i.e. as a function of the LAI penetrated, where LAI is essentially a vertical coordinate within the canopy) are:

and o are the upscatter parameter for diffuse and direct radiation respectively:

)(1

]cos)([2

1 2

so ak

k

Lecture 4 Dirmeyer


Two-Stream

1oI

][ totalkLsoil eII

The pair of differential equations above can be solved given appropriate boundary conditions. Assuming a normalized incident solar radiation at the top of the canopy:

and at the bottom where L = Ltotal:

kLo

kLo

ekIIdL

dI

ekIIdL

dI

])1(1[

)1(])1(1[

Lecture 4 Dirmeyer


Leaf Area Index

This is the fraction of the surface area covered by leaf surfaces when viewed from above.

No leaves at all: LAI = 0.

Imagine one giant leaf, covering everything like a blanket. The LAI would be 1.

An actual canopy has multiple leaves overlying any point of the surface. The LAI can exceed 1. In fact, for very dense canopies, it may be 5, 6, 7, or more. The LAI over an area is the average number of horizontal surfaces intercepted while traveling down through the canopy to the top to the soil:

Lecture 4 Dirmeyer


Reflectance

In general:

• In visible wavelengths, soil is dark, vegetation is darker.

• In near-infrared wavelengths, soil is dark, canopy is bright.

So LAI has a strong influence on the NIR albedo, but not so much on the visible albedo. The visible albedo is low, so that the fraction of photosynthetically active radiation absorbed by the plant (FPAR) is high. This optimizes the energy available for food production in the plant’s cells.

Lecture 4 Dirmeyer


This difference between albedos in PAR and NIR ranges allows us to define vegetation indices that can be used to estimate vegetation cover (and thus LAI) from remote sensing:

Simple ratio:

Vegetation Indices

v

N

aa

SR

vN

vN

aaaa

NDVI

Normalized difference vegetation index:

Lecture 4 Dirmeyer


Canopy Photosynthesis

totalV LkeFPAR 1

)(APARfPC

2/1)1()(

VV

Gk

APAR is the PAR absorbed by the green canopy. The fraction of PAR absorbed:

where V stands for visible, and :

Lecture 4 Dirmeyer


Canopy Photosynthesis

VN

N

kh

FPARaSR

2

2/1

2

)1(

1

NN

LhN

h

ea totalN

Meanwhile, the NIR is mostly reflected:

Since aV is relatively constant with LAI for a canopy over dark soils:

So vegetation index FPAR, and satellites may give us measurements of reflectance, photosynthesis, and transpiration.

Lecture 4 Dirmeyer


Friction

Laminar Flow

Smooth, Frictionless (slip condition)

Turbulent Flow

Laminar Flow

Rough (no-slip condition)

Earth’s surface is not smooth, frictionless, or inert. Thus there exist vertical wind speed, temperature, and moisture gradients.

•Vertical gradients in wind are caused by the "no-slip" condition at the interface between the fluid (atmosphere) and solid surface:

•Vertical gradients in temperature are caused by the different thermal and radiative properties of the surface and fluid.

•Vertical gradients in moisture are caused by the different water holding properties of the surface and fluid.

0V

Lecture 4 Dirmeyer


Surface Layers

•How are energy, momentum and moisture exchanged in the boundary layer over the surface of the earth?

• 0-10mm Molecular processes

• 10mm-10m Frictional processes (constant stress layer)

• 10m-2km Friction, pressure gradient, Coriolis (mixed layer)

•The atmosphere “sees” the fluxes from the surface as lower boundary conditions, but flux divergence in the vertical determines how the surface fluxes affect the circulation and the thermal structure.

As energy cascades to smaller scales, turbulence dissipates it. But turbulence can also transport energy across gradients — much more effectively than diffusion. This is key for energy fluxes between land and atmosphere.

Lecture 4 Dirmeyer


Land versus Ocean

1. Ocean has a much higher heat capacity than land.

2. Water “flows” while the land surface is fixed. Ocean can transport much heat laterally, land cannot.

3. Ocean, obviously, is wet (evaporation is not limited by lack of moisture). Land can be wet, dry, or somewhere in between (moisture limitations can impede evaporation).

4. The upper layer of the ocean is well mixed, so the surface characteristics are sufficient to define its interaction with the atmosphere, but soil has vertical structure and overlying vegetation. Heat conduction and moisture transport below the surface become important.

These facts have a bearing on the physical interactions between the surface and the atmosphere and on the manner in which drag coefficients are specified.

Lecture 4 Dirmeyer


The Planetary Boundary Layer (PBL)

Layer Processes Domain

Free Atmosphere

Dynamic Dry convective adjustmentTurbulence closure

Mixed Layer

Thermal, Mechanical,

Coriolis

Mixed Layer ModelMulti-Layer Model

AGCM

Constant Stress Layer

Thermal, Mechanical

Monin-Obukhov Similarity Theory

LSS

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 essentially geostrophic); it includes the surface friction 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. Over ocean, it is typically 100—700m thick.

Lecture 4 Dirmeyer



z~0.1h

zo

qV Θ

Vo Θo qo

(Ts, qs) LANDOCEAN

Mixed Layer


(Ts, Vs)

τ Ho Eo

This layer near the surface is of primary concern for land-atmosphere interactions; it lies between the soil/vegetation models and the AGCM.

z = top of the constant stress layerzo = roughness length, h = depth of PBLV = Wind velocity = Potential temperatureq = Specific humidityNote:Vo = 0 over land (assumed), Vo = Vs over ocean

Lecture 4 Dirmeyer


Turbulent Fluxes

zKcwcH

zv

Kwv

zq

KwqEzu

Kwu

HppMy

WMx

Vertical fluxes of momentum, latent heat and sensible heat can be written as:

where the turbulent stress (the second order terms in each) is related to the flux gradient in the vertical. For example, for momentum, is the tangential frictional force. The 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 after which derivatives of the higher order moments → 0.

zu /

Lecture 4 Dirmeyer


Turbulent Fluxes

The far RHS terms are applied in the "Mixing Length" method (also known as "K theory").

KM, KH, and KW are the eddy viscosity, eddy thermal diffusivity and eddy diffusivity for moisture respectively. They represent the efficiency with which mixing (i.e., vertical fluxes) occurs for a given shear, and are a property of the fluid under consideration.

zKcwcH

z

vKwv

z

qKwqE

z

uKwu

HppMy

WMx

Turbulent mixing occurs over a length scale l that is a function of the depth of the constant stress layer:

l = k z

k is the von Karman constant (0.4 as determined experimentally).

Lecture 4 Dirmeyer


Near an interface (e.g., the surface of the earth) we must be concerned with the effects of the discontinuities in temperature and moisture, and the effect of the "no-slip" boundary condition on turbulent transfer.

We can relate the viscosity to the length scale through a frictional velocity:


luKM *

),(;

])()[( 2/1222*

yxo

o wvwuu

The frictional velocity is a function of the horizontal surface stress:

Disregarding the direction of the wind, and substituting we have:

kzu

zu *

Lecture 4 Dirmeyer



Czu

ku )ln(*

Integrating in z:

The constant of integration is chosen as C = ln(zo) such that u = 0

at z = zo. zo is called the roughness length. So:

zo 1—10 cm over a smooth surface like bare ground1—10 m over a varied canopy of tall trees.

Note: typically z is not measured from the ground, but from a reference level called the displacement height d which is the level of action of surface drag.

oz

z

u

kuln

*

Lecture 4 Dirmeyer


Displacement Height and Drag

ozdZ

uku

ln*

Where Z is the actual height above the ground. Where Z>>d, the displacement height can be ignored. In bulk transfer relationships we can define a drag coefficient:

Note: This is for “neutral” conditions (i.e. the vertical lapse rate of temperature is such that in the absence of diffusion, a parcel displaced vertically would heat/cool adiabatically to the temperature at its new elevation). In non-neutral conditions, this gets more complicated!

Note: typically z is not measured from the ground, but from a reference level called the displacement height d which is the level of action of surface drag. This may be far aloft in the canopy where vegetation is present. This is done to ensure a good fit to the logarithmic relationship:

2

22*

ln

o

D

zdZ

kuu

C

Lecture 4 Dirmeyer


Drag Coefficients

Ho

H

zz

zz

kC

lnln

2

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

So by analogy we can define the bulk transfer relations in terms of drag coefficients:

Here u is a non-directional wind speed, and likewise the momentum flux o is non-directional.

)(

)(

)(

oHpo

oHo

oDo

uCcH

qquCE

uuuC

Lecture 4 Dirmeyer


Aerodynamic Resistance

aHopo

aHoo

aMoo

rcH

rqqE

ruu

/)(

/)(

/)(

Finally, if we recognize that (CD u) and its brethren are conductances that facilitate the rate of flux, given a certain gradient, then we may define aerodynamic resistances:

So that:

This final formulation is typically used in Land Surface Schemes (LSS), and is 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.

radiative transfer, canopy radiation and turbulence

Documents