An Introduction to EnvironmentalBiophysics

Second Edition

Gaylon S. CampbellJohn M. Norman

An Introduction toEnvironmentalBiophysics

Second Edition

With 81 llIustrations

~ Springer

Gaylon S. Campbell Decagon Devices, Inc. 950 NE Nelson Ct. Pullman, WA 99163 USA

Library ofCongress CataIoging-in-Publication Data Campbell, Gaylon S.

John M. Nonnan University ofWisconsin College of Agricultural and Life Sciences Soils

Madison, WI 53705 USA

lntroduction to environmental biophysicsJG. S. Campbell, 1. M. Norman. -- 2nd ed.

p.cm. lncludes bibliographical references and index. ISBN 978-0-387-94937-6 ISBN 978-1-4612-1626-1 (eBook) DOI 10.1007/978-1-4612-1626-1 1. Biophysics. 2. Ecology. 1. Norman, John M. II. Title.

CH505.C34 1998 571.4--dc21 97-15706

ISBN 978-0-387-94937-6 Printed on acid-free paper

© 1998 Springer Science+Business Media New York Originally published by Springer Science+Business Media, lnc. in 1998 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher Springer Science+Business Media, LLC, except for brief excerpts in connection with reviews or scholarly anaJysis. Use in connection with any form of information storage and retrievaJ, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not theyare subject to proprietary rights.

9 8 7 6 5 4

springeronline.com

Preface to theSecond Edition

The objectives of the first edition of "An Introduction to EnvironmentalBiophysics" were ''to describe the physical microenvironment in whichliving organisms reside" and "to present a simplified discussion of heatand mass transfer models and apply them to exchange processes betweenorganisms and their surroundings." These remain the objectives of thisedition. This book is used as a text in courses taught at Washington StateUniversity and University ofWisconsin and the new edition incorporatesknowledge gained through teaching this subject over the past 20 years.Suggestions ofcolleagues and students have been incorporated, and all ofthe material has been revised to reflect changes and trends in the science.Those familiar with the first edition will note that the order of pre­

sentation is changed somewhat. We now start by describing the physicalenvironment of living organisms (temperature, moisture, wind) and thenconsider the physics of heat and mass transport between organisms andtheir surroundings. Radiative transport is treated later in this edition, andis covered in two chapters, rather than one, as in the first edition. Sinceremote sensing is playing an increasingly important role in environmen­tal biophysics, we have included material on this important topic as well.As with the first edition, the final chapters are applications ofpreviouslydescribed principles to animal and plant systems.Many of the students who take our courses come from the biolog­

ical sciences where mathematical skills are often less developed thanin physics and engineering. Our approach, which starts with more de­scriptive topics, and progresses to topics that are more mathematicallydemanding, appears to meet the needs of students with this type ofback­ground. Since we expect students to develop the mathematical skillsnecessary to solve problems in mass and energy exchange, we have addedmany example problems, and have also provided additional problems forstudents to work at the end of chapters.One convention the reader will encounter early in the book, which is

a significant departure from the first edition, is the use ofmolar units formass concentrations, conductances, and fluxes. We have chosen this unitconvention for several reasons. We believe molar units to be fundamen­tal, so equations are simpler with fewer coefficients when molar units

vi Preface to the Second Edition

are used. Also, molar units are becoming widely accepted in biologicalscience disciplines for excellent scientific reasons (e.g., photosyntheticlight reactions clearly are driven by photons of light and molar units arerequired to describe this process.) A coherent view of the connectednessofbiological organisms and their environment is facilitated by a uniformsystem of units. A third reason for using molar units comes from thefact that, when diffusive conductances are expressed in molar units, thenumerical values are virtually independent of temperature and pressure.Temperature and pressure effects are large enough in the old system torequire adjustments for changes in temperature and pressure. These tem­perature and pressure effects were not explicitly acknowledged in thefirst edition, making that approach look simpler; but students who delvedmore deeply into the problem found that, to do the calculations correctly,a lot of additional work was required. A fourth consideration is that useof a molar unit immediately raises the question "moles of what?" Thedependence of the numerical value of conductance on the quantity thatis diffusing is more obvious than when units ofm/s are used. This helpsstudents to avoid using a diffusive conductance for water vapor whenestimating a flux of carbon dioxide, which would result in a 60 percenterror in the calculation. We have found that students adapt readily to theconsistent use ofmolar units because ofthe simpler equations and explicitdependencies on environmental factors. The only disadvantage to usingmolar units is the temporary effort required by those familiar with otherunits to become familiar with ''typical values" in molar units.A second convention in this book that is somewhat different from the

first edition is the predominant use of conductance rather that resistance.Whether one uses resistance or conductance is a matter of preference,but predominant use of one throughout a book is desirable to avoid con­fusion. We chose conductance because it is directly proportional to flux,which aids in the development of an intuitive understanding of trans­port processes in complex systems such as plant canopies. This avoidssome confusion, such as the common error of averaging leaf resistancesto obtain a canopy resistance. Resistances are discussed and occasion­ally used, but generally to avoid unnecessarily complicated equations inspecial cases.A third convention that is different from the first edition is the use of

surface area instead of"projectedarea."This first appears in the discussionofthe leafenergy budget and the use of"view factors." Becausemany bio­physicists work only with flat leaves, the energy exchange equations forleaves usually are expressed in terms of the "one-sided" leaf area; this isthe usual way to characterize the area offlat objects. If the energy balanceis generalized to nonflat objects, such as animal bodies or appendages,tree trunks or branches, or conifer needles, then this "one-side" area issubject to various interpretations and serious confusion can result. Errorsof a factor of two frequently occur and the most experienced biophysi­cist has encountered difficulty at one time or another with this problem.We believe that using element surface area and radiation ''view factors"

Preface to the Second Edition vii

are the best way to resolve this problem so that misinterpretations do notoccur. For those interested only in exchangeswith flat leaves, the develop­ment in this book may seem somewhat more complicated. However, ''flatleaf' versions ofthe equations are easy to write and when interest extendsto nonflat objects this analysis will be fully appreciated. When extendingenergy budgets to canopies we suggest hemi-surface area, which is one­half the surface area. For canopies of flat leaves, the hemi-surface areaindex is identical to the traditional leaf area index; however for canopiesof nonflat leaves, such as conifer needles, the hemi-surface area index isunambiguous while "projected" leafarea index depends on many factorsthat often are not adequately described.One convention that remains the same as the first edition is the use

of Jlkg for water potential. Although pressure units (kPa or MPa) havebecome popular in the plant sciences, potential is an energy per unit massand the Jlkg unit is more fundamental and preferred. Fortunately, Jlkgand kPa have the same numerical value so conversions are simple.As with the previous edition, many people contributed substantially

to this book. Students in our classes, as well as colleagues, suggestedbetter ways of presenting material. Several publishers gave permissionto use previously published materials. Marcello Donatelli checked themanuscript for errors and prepared the manuscript and figures to be sentto the publisher. The staff at Springer-Verlagwere patient and supportivethrough the inevitable delays that come with full schedules. We are alsograteful to our wives and families for their help and encouragement infinishing this project. Finally, we would like to acknowledge the contri­butions ofthe late ChampB. Tanner. Most of the material in this bookwastaught and worked on in some form by Champ during his years of teach­ing and research at University ofWisconsin. Both ofus have been deeplyinfluenced by his teaching and his example. We dedicate this edition tohim.

G. S. CampbellJ. M. NormanPullman and Madison, 1997

Preface to theFirst Edition

The study of environmental biophysics probably began earlier in man'shistory than that of any other science. The study of organism­environment interaction provided a key to survival and progress.Systematic studyofthe science and recording ofexperimental results goesback many hundreds of years. Benjamin Franklin, the early Americanstatesmen, inventor, printer, and scientist studied conduction, evaporation,and radiation. One of his observation is as follows:

My desk on which I now write, and the lock of my desk, are bothexposed to the same temperature of the air, and have therefore thesame degree of heat or cold; yet if I lay my hand successively onthe wood and on the metal, the latter feels much the coldest, notthat it is really so, but being a better conductor, it more readily thanthe wood takes away and draws into itself the fire that was in myskin. I

Progress in environmental biophysics, since the observation ofFranklin and others, has been mainly in two areas: use of mathematicalmodels to quantify rates ofheat andmass transfer and use of the continuityequation that has led to energy budget analyses. In quantification of heat­and mass-transfer rates, environmental biophysicists have followed thelead of physics and engineering. There, theoretical and empirical modelshave been derived that can be applied to many of the transport problemsencountered by the design engineer. The same models were applied totransport processes between living organisms and their surroundings.This book is written with two objectives in mind. The first is to de­

scribe the physical micro environment in which living organisms reside.The second is to present a simplified discussion ofheat- and mass-transfermodels and apply them to exchange processes between organisms andtheir surroundings. Onemight consider this a sort ofengineering approachto environmental biology, since the intent to teach the student to calcu­late actual transfer rates, rather than just study the principles involved.

IFrom a letter to John Lining, written April 14, 1757. The entire letter, along with otherscientific writings by Franklin, can be found in Reference [1.2].

x Preface to the First Edition

Numerical examples are presented to illustrate many of the principles,and are given at the end of each chapter to help the student develop skillsusing the equations. Working of problems should be considered as es­sential to gaining an understanding of modem environmental biophysicsas it is to any course in physics or engineering. The last four chapters ofthe book attempt to apply physical principles to exchange processes ofliving organisms, the intent was to indicate approaches that either couldbe or have been used to solve particular problems. The presentation wasnot intended to be exhaustive, and in many cases, assumptions made willseverely limit the applicability of the solutions. It is hoped that the readerwill find these examples helpful but will use the principles presented inthe first part of the book to develop his own approaches to problems, usingassumptions that fit the particular problem of interest.Literature citation have been given at the end ofeach chapter to indicate

sources ofadditional material and possibilities for further reading. Again,the citations were not meant to be exhaustive.Many people contributed substantially to this book. I first became inter­

ested in environmental biophysics while working as an undergraduate inthe laboratory of the late Sterling Taylor. Walter Gardner has contributedsubstantially to my understanding of the subject through comments anddiscussion, and provided editorial assistance on early chapters of thebook. Marcel Fuchs taught me about light penetration in plant canopies,provided much helpful discussion on other aspects of the book, and readand commented on the entire manuscript. James King read Chapters 7and 8 and made useful criticisms which helped the presentation. He andhis students in zoology have been most helpful in providing discussionand questions which led to much of the material presented in Chapter7. Students in my Environmental Biophysics classes have offered manyhelpful criticisms to make the presentation less ambiguous and, I hope,more understandable. Several authors and publishers gave permission touse figures, Karen Ricketts typed all versions of the manuscript, and mywife, Judy, edited the entire manuscript and offered the help and encour­agement necessary to bring this project to completion. To all of thesepeople, I am most grateful.

Pullman, 1977 G.S.c.

Contents

Preface to the Second Edition v

Preface to the First Edition ix

List of Symbols xvii

Chapter 1 Introduction 1

1.1 Microenvironments 31.2 Energy Exchange 31.3 Mass and Momentum Transport 41.4 Conservation ofEnergy and Mass 41.5 Continuity in the Biosphere 51.6 Models, Heterogeneity, and Scale 71.7 Applications 91.8 Units 9References 13Problems 13

Chapter 2 Temperature 15

2.1 Typical Behavior of Atmospheric and Soil Temperature 152.2 Random Temperature Variation 182.3 Modeling Vertical Variation in Air Temperature 202.4 Modeling Temporal Variation in Air Temperature 232.5 Soil Temperature Changes with Depth and Time 232.6 Temperature and Biological Development 262.7 Thermal Time 282.8 Calculating Thermal Time from Weather Data 302.9 Temperature Extremes and the Computation ofThermalTime 32

2.10 Normalization of Thermal Time 322.11 Thermal Time in Relation To Other EnvironmentalVariables 33References 34Problems 35

xii Contents

Chapter 3 Water Vapor and Other Gases 37

3.1 Specifying Gas Concentration 383.2 Water Vapor: Saturation Conditions 403.3 Condition of Partial Saturation 423.4 Spatial and Temporal Variation ofAtmospheric WaterVapor 47

3.5 Estimating the Vapor Concentration in Air 49References 50Problems 50

Chapter 4 Liquid Water in Organisms and their Environment 53

4.1 Water Potential and Water Content 534.2 Water Potentials in Organisms and their Surroundings 584.3 Relation ofLiquid- to Gas-Phase Water 58References 61Problems 61

Chapter 5 Wind 63

5.1 Characteristics ofAtmospheric Turbulence 645.2 Wind as a Vector 655.3 Modeling the Variation in Wind Speed 665.4 Finding the Zero Plane Displacement and theRoughness Length 68

5.5 Wind Within Crop Canopies 72References 74Problems 75

Chapter 6 Heat and Mass Transport 77

6.1 Molar Fluxes 786.2 Integration of the Transport Equations 796.3 Resistances and Conductances 796.4 Resistors and Conductors in Series 806.5 Resistors in Parallel 816.6 Calculation of Fluxes 81Problems 85

Chapter 7 Conductances for Heat and Mass Transfer 87

7.1 Conductances for Molecular Diffusion 877.2 Molecular Diffusivities 887.3 Diffusive Conductance of the Integument 907.4 Turbulent Transport 937.5 Fetch and Buoyancy 967.6 Conductance of the Atmospheric Surface Layer 977.7 Conductances for Heat and Mass Transfer inLaminar Forced Convection 99

7.8 Cylinders, Spheres and Animal Shapes 1027.9 Conductances in Free Convection 103

Contents xiii

7.10 Combined Forced and Free Convection 1057.11 Conductance Ratios 1057.12 Determining the Characteristic Dimension of an Object 1067.13 Free Stream Turbulence 108Summary of Formulae for Conductance 108References 110Problems 110

Chapter 8 Heat Flow in the Soil 113

8.1 Heat Flow and Storage in Soil 1138.2 Thermal Properties of Soils: Volumetric Heat Capacity 1178.3 Thermal Properties of Soils: Thermal Conductivity 1198.4 Thermal Diffusivity and Admittance of Soils 1238.5 Heat Transfer from Animals to a Substrate 126References 128Problems 128

Chapter 9 Water Flow in Soil 129

9.1 The Hydraulic Conductivity 1299.2 Infiltration ofWater into Soil 1319.3 Redistribution ofWater in Soil 1339.4 Evaporation from the Soil Surface 1369.5 Transpiration and Plant Water Uptake 1399.6 The Water Balance 144References 144Problems 144

Chapter 10 Radiation Basics 147

10.1 The Electromagnetic Spectrum 14810.2 Blackbody Radiation 14910.3 Definitions 14910.4 The Cosine Law 15610.5 Attenuation of Radiation 15710.6 Spectral Distribution of Blackbody Radiation 15910.7 Spectral Distribution of Solar and Thermal Radiation 16010.8 Radiant Emittance 162References 165Problems 165

Chapter 11 Radiation Fluxes in Natural Environments 167

11.1 Sun Angles and Daylength 16811.2 Estimating Direct and Diffuse Short-wave Irradiance 17111.3 Solar Radiation under Clouds 17311.4 Radiation Balance 17511.5 Absorptivities for Thermal and Solar Radiation 176

xiv Contents

11.6 View Factors 178References 183Problems 184

Chapter 12 Animals and their Environment 185

12.1 The Energy Budget Concept 18512.2 Metabolism 18912.3 Latent Heat Exchange 19012.4 Conduction of Heat in Animal Coats and Tissue 19412.5 Qualitative Analysis of Animal Thermal response 19712.6 Operative Temperature 19812.7 Applications of the Energy Budget Equation 20012.8 The Transient State 20112.9 Complexities of Animal Energetics 20212.10 Animals and Water 204

References 205Problems 206

Chapter 13 Humans and their Environment 209

13.1 Area, Metabolic Rate, and Evaporation 20913.2 Survival in Cold Environments 21113.3 Wind Chill and Standard Operative Temperature 21313.4 Survival in Hot Environments 21513.5 The Humid Operative Temperature 21913.6 Comfort 220References 222Problems 222

Chapter 14 Plants and Plant Communities 223

14.1 LeafTemperature 22414.2 Aerodynamic Temperature of Plant Canopies 22914.3 Radiometric Temperature of Plant Canopies 23014.4 Transpiration and the Leaf Energy Budget 23114.5 Canopy Transpiration 23314.6 Photosynthesis 23514.7 Simple Assimilation Models 23514.8 Biochemical Models for Assimilation 23914.9 Control of Stomatal Conductance 24114.10 Optimum Leaf Form 244

References 245Problems 246

Chapter IS The Light Environment ofPlant Canopies 247

15.1 Leaf Area Index and Light Transmission ThroughCanopies 247

15.2 Detailed Models of Light Interception by Canopies 25015.3 Transmission of Diffuse Radiation 254

15.4 Light Scattering in Canopies 25515.5 Reflection of Light by Plant Canopies 25515.6 Transmission of Radiation by Sparse Canopies--Soil Reflectance Effects 257

15.7 Daily Integration 25815.8 Calculating the Flux Density ofRadiation onLeaves in a Canopy 258

15.9 Calculating Canopy Assimilation from LeafAssimilation 259

15.10 Remote Sensing of Canopy Cover and IPAR 26415.11 Remote Sensing and Canopy Temperature 27115.12 Canopy Reflectivity (Emissivity) versus Leaf

Reflectivity (Emissivity) 27315.13 Heterogeneous Canopies 27315.14 Indirect Sensing of Canopy Architecture 275

References 276Problems 277

Appendix 279

Index 283

List of Symbols

A {mol m-2 S-l } carbon assimilation rateA(O) {C} amplitude ofthe diurnal soil surface

temperature

Aw plant available waterB {W/m2 } flux density ofblackbody radiation

c {m/s} speed oflight

c fraction ofsky covered with cloudcp {J mol- 1 C- 1 } specific heat ofair at constant pressure

CS{J kg- l C- l } specific heat ofsoil

Cj {mol/mol} concentration ofgas j in air

C {mol/kg} concentration ofsolute in osmotic solution

d {m} zero plane displacement

d {m} characteristic dimensionD {m} soil damping depthD {kPa} vapor deficit ofairD H {m2 Is} thermal diffusivitye {J} energy ofone photone {kg/MJ} radiation conversion efficiency for cropse {kPa} vapor pressure ofwaterea {kPa} partial pressure ofwater vapor in aires(T) {kPa} saturation vapor pressure ofwater at

temperature T

E {mol m-2 S-l } evaporation rate for water

E r {mol m-2 S-l } respiratory evaporative water lossEs {mol m-2 S-I } skin evaporative water lossf fraction ofradiation intercepted by a crop

canopy

fds fraction ofdownscattered radiation in aparticular waveband

Fa view factor for atmospheric thermalradiation

Fd view factor for diffuse solar radiationFg view factor for ground thermal radiationFp view factor for solar beam

xviii List of Symbols

Fj(z) {mol m-2 S-I } flux density of) at location zg {m/s2 } gravitational constant

gH {mol m-2 S-I } conductance for heat

gHa {mol m-2 S-l } boundary layer conductance for heat

gHb {mol m-2 S-l } whole body conductance (coat and tissue)for an animal

gHc {mol m-2 S-l } coat conductance for heat

gHr {mol m-2 S-I } sum ofboundary layer and radiativeconductances

gHr {mol m-2 S-1 } tissue conductance for heat

gr {mol m-2 S-I } radiative conductance

gv {mol m-2 S-I } conductancefor vapor

gva {mol m-2 S-l } boundary layer conductance for vapor

gvs {mol m-2 S-l } surface or stomatal conductance for vapor

G {W/m2 } soil heatflux densityh {m} canopy heighth {J s} Planck sconstanthr relative humidityH {W/m2 } sensible heatflux densityJw {kg m-2 S-I } waterflux densityk {W m- I C- I } thermal conductivityk {J/K} Boltzmann constantK canopy extinction coefficient

Kbe(1/!) extinction coefficient ofa canopy ofblackleaves with an ellipsoidal leafangledistribution for beam radiation

Kd extinction coefficient ofa canopy ofblackleaves for diffuse radiation

K m {m2 Is} eddy diffusivity for momentum

KH {m2 Is} eddy diffusivity for heatK v {m2 Is} eddy diffusivity for vapor

Ks {kg S m-3 } saturated hydraulic conductivity ofsoil

L {m2 /m2 } total leafarea index ofplant canopy

L oe {W/m2 } emitted long-wave radiation

L r leafarea index above some height in acanopy

L* sunlit leafarea index in a complete canopyr

m airmass numberM {W/m2 } metabolic rate

Mb {w/m2 } basal metabolic rateM j {g/mol} molar mass ofgas)

nj {mol} number ofmoles ofgas)

Pj {kPa} partial pressure ofgas)

Pa {kPa} atmospheric pressure

q {gig} specific humidity (mass ofwater vapordivided by mass ofmoist air)

Qp {J.Lmol m-2 S-I } PAR photon flux density

List of Symbols XIX

rH (m2 s mol-I) heat transfer resistance (1/gH)rv (m2 s mol-I) vapor transfer resistance (l/gv)R {J mol- 1 C- 1 ) gas constant

Rabs (W/m2 ) absorbed short- and long-wave radiationRd (jlmol m-2 S-l ) dark respiration rate ofleafR L (m4 S-l kg-I) resistance to waterflow through a plant

leafRn {W/m2 ) net radiation

RR {m4 S-I kg-I) resistance to waterflow through a plantroot

s (C- 1 ) slope ofsaturation mole fraction function(tl/Pa)

Sb (W/m2 ) flux density ofsolar radiation on ahorizontal surface

Sd (W/m2 ) flux density ofdiffuse radiation on asurface

Sp (W/m2 ) flux density ofsolar radiationperpendicular to the solar beam

Sr (W/m2 ) flux density ofreflected solar radiationSpo {W/m2 ) the solar constantSf (W/m2 ) flux density oftotal solar radiationt Is) timeto (hr) time ofsolar noonT(z) {C) temperature at height zT(t) {C) temperature at time tTd (C) dew point temperatureTe {C) operative temperatureTes {C) standard operative temperatureTeh {C) humid operative temperatureTo {C) apparent aerodynamic surface

temperatureTave (C) average soil temperatureTb (C) base temperature for biological

developmentTxi (C) maximum temperature on day iTni (C) minimum temperature on day iT {K) kelvin temperatureu* {m1s) friction velocity ofwindVm (fLmol m-2 S-2 ) maximum Rubisco capacity per unit leaf

areaw (gig) mixing ratio (mass ofwater vapor divided

by mass ofdry air)w {gig) mass wetness ofsoilx average area ofcanopy elements

projected on to the horizontal planedivided by the average area projectedon to a vertical plane

xx List of Symbols

Z {m} height in atmosphere or depth in soil

ZH {m} roughness length for heat

ZM {m} roughness length for momentum

Greek

a absorptivityfor radiation

as absorptivity for solar radiation

aL absorptivityfor longwave radiationf3 {degrees} solar elevation angle

0 {degrees} solar declination

~ {kPa/C} slope ofthe saturation vapor pressurefunction

E emissivityEac emissivity ofclear skyEa(C) emissivity ofsky with cloudiness cEs emissivity ofsurface

y {C- I } thermodynamic psychrometer constant (cpj"A)y* {C- 1 } apparent psychrometer constant

r* {mol/mol} light compensation pointr(t) dimensionless diurnal function for estimating

hourly air temperature

<P osmotic coefficient

4> {degrees} latitude

<PM diabatic influence factor for momentum

<PH diabatic influence factor for heat

<Pv diabatic influence factor for vapor<I> {W/m2 } flux density ofradiationK {m2 Is} soil thermal diffusivity"A {llmol} latent heat ofvaporization ofwater"A {fl m} wavelength ofelectromagnetic radiation1ft {J/kg} water potential1ft {degrees} solar zenith angle

\lim diabatic correction for momentum

\lIH diabatic correction for heatp {mol m-3 } molar density ofair

P leafreflectivity

Pb {kg/m3 } bulk density ofsoilH bihemispherical reflectance ofa canopy ofPb,cpy

horizontal leaves with infinte LA!

* canopy bihemispherical reflectance for diffusePb,cpyradiation and a canopy ofinfinite LA!

pt,Cpy (1ft) canopy directional-hemisperical reflectancefor beam radiation incident at angle \lIfor a canopy ofinfinite LA!

Pi {glm- 3 } density ofgas j in air

List of Symbols

e (degrees)

e (m3 m-3 )

T {day deg}T {s}T

T Is}Tb

Tb(ljI)

Tbt (ljI)

Td

~

W (S-1 }

XXI

angle between incident radiation and anormal to a surface

volume wetness ofsoilthermal timeperiod ofperiodic temperature variationssky transmittancethermal time constant ofan animalfraction ofbeam radiation transmitted by a

canopyfraction ofbeam radiation that passes through

a canopy without being intercepted byany objects

fraction ofincident beam radiation trans­mitted by a canopy including scattered andunintercepted beam radiation

fraction ofdiffuse radiation transmitted by acanopy

atmospheric stability parameterangularfrequency ofperiodic temperature

variations