A UATHEMATTC.A.L MODEL

OF PLAIII ROOI G.ROTITH

R¡ Pr HALE Be$c.r l[.4.

of the

Mathenatf oE Dopantnent

Unlversity of Ad.elalôe

Subnttted, as a thests for thed.egnee of lfiaster of Selence ln the

Unlverelty of AdelalôeOctoben 1965

by

CONTENTS

List of tables and figures

Llst of synbols anl theln Lmlte

Sunnary

Slgned statenent

AeknowLedgement

Chapten 1 .Ie!¡g1Pc.l!!9¡

page

ivv

vliixx

1

11

11

15

18

19

21

21

22

Ohapter

2.1

2'.2

213

2.4

Chapter

3.1

3.2

3.3

3.Lt

3,5

Chapten

l{.1

1112

l+.3

2 Nutr.ignt dllstnlþutigpe fon g nov_1ng poipt,slnlr

The general case

Special. casecl

Co on,ÊLlna te transfonmati ons

Investigation of the varlous d.lstnibutlons

3 Definltlon of noot surface_

Tlre (Vrt = oo) dietrlbutionÍhe bound.any cond,itlon

trfiathenatleal d.evelopnent of the bound.arycond.ltion

The bound.lng curve A = O

trrormuLation in carteslan coond.inates

l+ The noot pnofl-le for the V = O d.lstntbutlons

the d.lstributions

Clrcr¡l-an noot pnofiLes

Non-cireulan root pnoflles

¿r

7

I9

Page

Chapten 5 The noot onofile fon the (V.t = *)d.lstnlbutlon

5,1 The (vrt = -) tltotnlbutton

5.2 The encl polnts P ard. N

5,3 Pnopentles of the (vrt = oo) ctletntbutlon

5.4 Cholce of ctlffenentlal. eguatlon

5,5 Bound.lrrg cr¡rves fon soLutione of (lrg)516 Fr.¡¡rthen pnopentleo of ttre Eolutlon cu¡rves

5.7 Stantlng Boint fon nunenlcal lntegnatlon

5.8 Emplrlcal- nesulte

5.9 The fLr¡:c of nutnient

5.10 Nu¡nenl.cal results ueing N as stantlng polnt

Chapter 6 Bhe ngpt pnoftle Êon the (V.tì jlls_tnlbution

6.1 The (vrt) ôietnlbution6.2 Propentles of the (vrt) dtstnlbution6.3 Emplrleal. nesuLts

6.4 Proof of the resulte (6.4)

Chapter 7 Pnog4aJnninq Methgds. nunerlcal accuracv

7.1 Algebnalc equatlons

7.2 Diffenentlal equatJ.ons

713 Appnoxiuratlon of the complementany error fr¡nctlon

7.1+ Root area ar¡d. nutnient f].ow

7.5 NunenlcaL inaecunaey

Chapten I ExteFslo$er, gonelìualgn

Refenenceg

30

30

33

4l

l+1

l+¡+

46

I+7

56

62

6I+

6t+

66

58

70

71

73

76

76

79

80

1v

LIST Otr' TABTES AI{D FIGTJRES

Root pnoflle encl polnte - Ba¡raneter study 32

Coond.lrutes for the noot pnoflLe and. the 5l+bound,ing curveo

Nutnlent flow fon a pnoflle of glven d.lmensione 60

page

5.3

FTGÏ'RESæ2.1

3,1

3&¡J.33.1+

3.5t3.6

t.74.1

L+.2

5.1

5,2

5.3

5.4

5.5

5.6

6.1

(tn text)

Isopycnals

fsopycnals

Root proflle (fn text)(rn text)(in text)Rad.1l of eirsulan pnofllee - parameten stu{y

A non clrcular lntegral cutîve

Root pnoflle arrd bouncllng curveo-base parameters

Root proflle and. bouncllrrg cr¡rves-V varled

Root pnoflle anrL þound.lng cunves-q varled.

Root Bnoflle ard' bound.lng cïlrvêo-D varled.

Root pnofile arul bound.lng cllrv€s-M vanled.

Root proflle and. adjacent lntegnal- curves

Tine clepenclent root pnofiles

9

12

13

1¿r

15

19

23

29

¿19

50

51

5z

53

55

67

A

B

c

Ce

C

D

E

k

L

M

11

çL-

4nÐcå'

v

LTST OF SYIÍBOLSI AI{D TI{EIR UNITS

_ --9.- [ cn]I+rfD

$ rcn-11

Nutnient concentratlon [noJ.ee cm-3]

InitiaL unifonm value of c

Ç fd.imenslontesslvo

Dlffuelvity Icm" sec-Û]

Fraetional ðe letlon of nutnJ.ent [ôinensionless]

$ lcn-tl. Conetant of pnoportlona]-lty 1n

bound.any cond.ltion (eguation (f.t ¡ 1

Depletlon of nutnlent [noles Çm-e] (Cnapten 2)

Uptake paraneter Icn sec-1]

Untt vecton normal to root surface on root proflleln d.lrection of insneaslng C

Strength of fnstantaneous sink [uroles] (Chapter 2)

Strength of continuoue sfnk [no1es se"-1]

q [cr" seca]cb

Polan d.lstanee I en]

Tine Isec]

Veloclty of sir¡lç (tn atnectlon of r:egative z axls)

Iem eec-¿]

Canteeian coord.inates [ "t]

q.

tV

a

q.

xtY tz

q,

p

r

À

c

0

^p

vrAngle between lsopycnal ard cinculan anc

A¡¡gLe behrueen lsopyonal andl noot proflle

Cr-kC

cr'.'.þ;-kaca

z+ïtPolar angle, neasured fnom 0e

L r-.ft taimenelonlesel (crrapter 4)

^rF-rTr;-ït

ISOPYCT.IAI,: Thls tern 1s used to d.escnlbe a llne on

-

gurface Joining pointe of equal nutrlent concentnatl.on.

It corneeponcls to the leothernal of heat cond.uctlon.

(ft fs taken from Ptryelcal OceanognaphJi'r whene an

lsopycnal eu¡rface ln the Bea pessee through pointe

of equal saLlnlty).

v11

SUúM.ARY

In thts thesie, the actlon of a gnowlrrg pl-ant root1s representecl by a polnt sir¡I nenorrlng nutrient at a

constant nate as 1t moves through a homogeneous eoll atconstant veloclty. It ls assumed. that the nutnlentd.iffi¡ees through the eoll towands the elnk by a process of

llnean d.ifftrslon wlth constant ctlffirelvity, fn the neglon

of varylng nutnlent concentnatlon aroturÖ euch a novlng slnk¡

a rrnathenatical plant rootn 1s definedt by the cond.ltlon

that at each pofnt of lts surfaee, the rate of nutrlentuptake 1s pnopontional to the nutnient eoncentnatfon at the

point.

This d.eflnitlon of a pLant root ls shown to lead. to

a flnst ond.er dlffenentlal equatlon, soLutlone of whieh

have been founcl by numenlcal lnteg:eatlon. It transplnes

that ther.e ane two bound.lng su:rfaces between whlch the noot

eurface nust L1e, and that nuch of the noot surface Lles

extneneLy close to oræ of these bor¡nd.1ng eunfaces.

This nod.el of plant gnowth invoLves foun parameters:

the strength and. veLoelty of the novlng sink of nutnlent,the d.Íffirslvlty of the nutnlent 1n the soll and. the constant

of pnoportlonaLity 1n the uptake cond.ltlon. Nr¡merical

vaLuee of these paraneters have been pnovlctecl by plantbioLoglsts,

PJ.ant noots often conalst of a raass of flne halne,

the envelope of whleh le neferred. to as the rrroot haln

vlLtenveLopeil. Root sunfaces have been obtalned. for vanlous

pananeter conblnations, and. some of the results obtafnecL

do have shapes stnongly suggestive of a noot hair envelope.

the nutnlent d,lstr.iþutlon anound. the novlng slr¡k 1s

quasi etead.y fon lange t1rcr âÍrcl most of the numerlcal

nesul.te d.escribed. are eoncernecl with the correspond.lng

quasl stead.y root surfaces. Thene is also some d.iscussion

of the gnowth wlth time of the root sunfaces aesoclateal

wlth both stationany and. roovlng sinks of nutrlent.

fx

Thls thesÍe contalns no naterlal whtch has beenaeceptedl for the awand. of any other d.egree on d.iplonaln any Unlvenslty, and., to the þest of ny hnowled.ge

and. be11ef, contalns no naterfal prevlously publlsheclor wnltten by another pensonr except when due nefe¡'-ence ls nad.e 1n the text of the thesle.

R. P. IIAIrE

'E

ACKNOïtrTEDGEMDNT

-The work of thie thestE hae been earrled.

out und.er the supenvleíon of Pnofessor J.R.M. Railoltr

lllhe author wlshes to aelrnowLedge hls help and. guld.anoe,

both in t'Ìre d.eveloprnent of the theeisr artù ln lte flna].presentatlon,

Chaoten 1. Intnod.uctlon

Thts thesls ts concernedl with one nat]reuatical nod.el

for the growth of plant roote by tlæ uptake of nutnient fron

the eolI. The tnitial lmpetue for the 9t"¿y ca¡ne fnom

plant biologlste Ìvorl<ing at the Ttratte Agricultural Research

fnstltute of the Unlverslty of Adelatd.e whose e:çerimental

work 1e buildting up an ençlnlcaL of root growth (..g. Barley

te] ). It is hopeit that the moclel d.escribed., either !n ite

pneeent form or exter¡d.ecl as ind.icatect in Chapten Br w111 be

of eufficlent vah¡e to suggest fnultfr¡I new }lrree of e:qrenl-

mentation.

The actual growth of a plant noot 1s a complex bio-

chenj.cal process which involves the absorptlon of vanl'otls

nutnients fron ttre soiL and. thelr eubeeguent convension lnto

the ceII stnucture of the growlng noot. In genenalr the

noot gnowth will be cLepend.ent on the absonptlon of several

d,lfferent nutrlents fnon the sofl-. Iri ttre nod.e1 consid.ered.,

lt ls assr¡.necl ttrat root growth is due to the absonptlon of

one nutnient only¡ that the nutrient reashes the root sunface

by a Brocess of llnean cttffl¡sion with constant dllffuslvityt

ard that the soÍl can be regard.ed. as an inf inlte homogeneoug

leotropic ned.il¡n. The shape of the root surface Ís d.eter-

nlnefl by a bourrlany condltion whlch ie cllscussecl ln Chapter 5.

Z.

One mettrod. of procedurer whieh ls that adoptect by

Paesloura [Z], !s to nalre assumptions about the shape of

the noot and. then seek a solutlon of tü¡e cltfftrsion eq¡ation

which eatisfies the bor¡¡d.ary cond.ltlon on the root surfaee.

Howeverras exact solutions of the <liffueion equatj.on are

onÌy known fon extremely eimple borrrd.ary Bhapes, app¡3oxinate

solutions ïvould. almoet certalnly have to be usedt¡ The

adjustnent of such a eotution fon variations ln bou¡il.ary

shape urould., in generalr be far f¡rom trlvlal..

the pnocedure adoptecL he¡re is flfst to obtain a

solution of tTre diffuslon equatÍon basecl on the action of a

growLng root, and. tTren to clefi11e a noot sunface in the

nutrient distríbution so obtained,. It 1e assunecl that the

actlon of a plant root ean be represented. by a point Einkt

active from tlme t = 0e moving vfith constant velocity

thror.rgh the solL ancl removtng nutrient at a constant rate.

IhiE 1ead.s to an ercact er¡rresslon for the nutrient concentfa-

tíon C at points in tlre soi}. In this cListributlont a

surface at polnte of whlch C is oonstant (neferrect to aa

an rrisopycnaltr) ls d.efineil by an algebraic equatlon. The

root surface 1n tJrls d.lstrfbution 1s d.etermlnett by a d-iffer-

ential equatlon, as w111 be shov¡n tn Chaptër 3, AcÌJt¡st-

ments in tÌre ehape of the root surface so obtained' ean read.-

ily be effected. by alterlng the paranetens of the mod'el.

It wilI, in generaL, be necesgary to uae nuûterical" lntegna-

tion to d.eternine the noot surface, but thls is now I plso-

3'

ce¡lure wlth a ï¡ell knovm error ana1ys18. ThLe natfiematlcal

moctel of root groT'lth Ttrag flrst pnopOsetl by Anclerssen and.

Radok [t ] . they eneountenecl illfflcuLtles 1n the cletermina-

tion of tJle noot eurfaees by nrrnerieal integration¡and. nuch

of the original work of this thesls d.eals with these

d.ifficu].tl-es.

the d.evelopnent of thls thesls is as foll-o$Is¡

In Chapt er 2, exact erqrnessions are obtained. f or the nutrlent

distribution assoclated with a movirrg point slnk aniL for

various speciaL cases of this general result. In CÌupter Jt

the bound.ary eond.ltion is stateit ani[ a methofl for the d.eten-

nination of the root surface outlined.¡ and. tÌren 1n Chapters

l+ - 6¡ root surfaceg are d.iscussect for the d.ifferent dlstni-

butlons of Chapter 2. The computatfone have been perfonned'

on the IBM 1620 corqguter in the Conputing Centre of the

Unlvensity of Ai[e1ald.e, and. Chapter 1 d.eals with the various

nrrmerlcal pnoceduree uged.. tr'lnaLlyr trn Chapten 8, posslble

extenslons of tJre method. are cllscussedl.

Poesible ranges of numerlcal values for the para-

meters of tJre mod.e1 were supplieil. by the plant b1ologists,

ar:d. the values chosen for the examples of thls thesle allcame frc¡n wittrln these ranges. The obJect of tne thesis

is to d.emonstrate in broad. tenrns the feasibility of tÏre

proposed, moclelr âÍId. no d.etal]ed corparl-son hae as yet been

made between the nod.el ar¡d. empirical observatlons.

l+.

Ghapt-er. 2. Nutr¡ien! clletnlbutlons fgr) a .noviru¡ polnt s+t.E.

2.1. The senenal- case

Consid,er an lnflnite honogeneous lsotroplc med.lum inwhieh nutrient cliffuses by a process of llnear d.lffusion

with constant ttlffirsivity D. Then the concentl'atLon C"'

at ar\y point nust satisfy the ctlffusion equation

òa c';' ò" c"' òe ctt 1 òc'i' :F'rã5É-*æ=Dãfr, \2.1 t

where xe lt z are Cartesian coorcl.lnates refenrecl to

etationary axes, and. t 1s the tfune. As (2,1) fs the heat

equation with D replaclrrg thernal d.ifflrsivlty K, known

results for heat cond.uction sueh ag those ín Cans].aw and.

Jaeger t4l ean be appL1ed..

Suppose that initially the nutrient ls present at a

r.miforn concentratlon C;, ancl that at tine tt, A r¡nlte

of nutníent are sucld.enIy removeÖ from the ned.iun at the

polnt (oro, z'), Then the nutrlent d.epletion I at any

point (*ryru) at ar\y subsequent tine t > tr is given by

lOare].aw and. Jaeger [l+], P. 256, equation 10.2(2)1,

T, = e]Cp

(2.2)

will be glvenAs

by

L ls the d.epletion, the concentration C

C -Co -f,.

5.

sr4rpose now that instead of an lngtantaneous polnt

slnk a continuous polnt sink noves alor¡g the negatlve z

atcis with eonstant velocity V. T,et it be such that ltstarts to act at tfte origln at tlroe t = O an! that itwithctnaws nutrlent at a constant rate q.t' pern r:nlt tine.

Consldlen its action ln the interval t e tt to

t = tr + dtt, with dtr snall-. It rrt1-l be locatecl at

(0ror-Vtt ) appnoxlnately and. will wlthdnaw q."ftt units of

nutrLent. Then, bv ( 2.2), the d.epl-etion at (*ryrz) at

tine t >

dT, = :-o*'dt erq, [ - [:f + f +lz r.v!] \3-l I -vL! - B?rrúÇ¡-;'1:%'-- ¿ l"tt J') - t'

Thrrs the total ctepletlon at (xryrz) at tlme t due to the

strft actlng up to tine t wiIl be gíven by

I,= c¡

Í,ettlng

thls beeornes

Afrrol%lxe+13+(g+vt'ì31

l+D(t - tr )a

tt=t- #r 1.e,) T=(t-tt)-å ,

ï.=1

-6

orwith C =z+Tl,s p=Jffi

ir"nfi'e'qp [-Ëf* +yz +(z +vt -#)'Jfa",CI

r' =¡fiy5 exe t#l f,*"* t- {# - aþfu"

6.*

= ¡fiy; "* lå*. - p) I Ë*"*[- #,r" *l)"lu'.

the funther eubstltutlon

yleld.s now

fr=

Slnce, by deflnition,

I,=

where

f;t" * "rFiÇt, iåe. " = frt p, -!r),

$-lprrgDp "*[å(c-p)l L Ju' * g:

[t + ] erç( -s" )ds.¡{. g

ztlã

errc € =f, I?_r.ur,

6 "*[$cr - ùl"rr" ffi +ra¡

"*låcc-c)f h?l g e¡cp( -sa \d.e

/"" ..,#¡ (z,l)

2ñ'rAe the lntegnand. fn ( 2.3) ls an od.d. f\rnction oflower l1nlt can be talren as f

e - vtlr 60 tÌrat| ¿ffi l'negative throughout the range of lntegratlon.lhen the substltutlon

s = ,F4, wlth a > o, vtercts

L" =&"*[åcc + ùl f 'o'uþ+Vtz.Ñ

B¡ the

a 1g non-

7 a

=# "*låt( + P)luo," H '

The total clepletlon at (xrprz) at tine t is thue glven

byoY2D

oV2ÐL= erfc

and. hence the coneentratLon C

--g-SrrDp

o-Vt\

-l

)

z,lDt' / t

( 2.1+)

("Ie2Ð

("IC2Ð

#"5(Ëerrc

ep+Vt

-

z"rD{.

le glven by

_I,r

oV?D enfc o+Vt

-

oY?Ð+e

+e erfc

c;c

1o€r

ê z/tr

A= a

TUith C =c"'

trr 9= g-VO

this result maY be rewnltten

o+Vt

-

oY2Ð+eC=1-

where

A=

2.2 SpeeiaL cases

(a) v=O

z^Ft

Efut Ç=z+Vt, P ='\tm:rc a

Fon a statLonary einlç actlng at the origtn

startlng at tine t = 0¡ (2.4) becones

c = 1 - fr u"r" #, (2;5)

g.

[canslaw and. Jaeger t4] P. 261, equatlon 1o'4(2)J.

(bì . CgncentrF}loJlnear the s14r,, forjgnse ttrror large +' and. P < ( Vtt

erfe e +J't -r enfc(*) = o,zffi,

enfcP-vt-enfc(-æ)=2,2^Ñ

and (2.h) becones

c = 1 - å u*n[å,. - o)1, Q.6)

[Carsraw ard. Jaegen [J+] P. 267t equation 10.7(2)J.

(c\ ,Coneeptnation neaq tle eiElr f or lgnee t wh,en V=0

lilϡen both the conditlons (") and. (b) apply'

( 2.1+) þecouresAP

C=1- a

For eonvenience of referenee, (2.4) - (2.7)

wfl]- be refenreil to as

the (vrt) d.fetnibution (z.l+)t

the (v = ort) aistnibutlon (2,5),

the (vrt = -) d.istntbutlon (2.6),

the (V = o, t = oo) d.istribution (2.7).

2.1 Co-ond.inate tnansforrnatlons

The coorcll¡rate z only entens into (2.4) (2.7)

ln the form C = z + Yi,. For V / O, the ttletrlbutions

ane tTtu¡r simplified. if refernect to a moving coorclinate

syetem with orlgin at the noving sink. The polnt (xryrz),

9¿

refenredl to the onlglna} statlonafy axesr then becomes

(Xry¡f ) refenrefl to novlrrg â*es. In the new reference

systen, the (Vr,t r *) d.lstnibution (2,6) f" tlne lnèepenclent t

and. hence it is a quaei etea(y clletnlbution.

trrlÉ¡rê 2.1

-

It ls evlclent that the z-axÍs ls an axig of synnetry for

each of the d.lstrlbutlone (2.4) - Q,7). Thus they are

ad.equately d.eecribed 1n polar coordinates (rr0 ) wtth

originatthe¡novin€islnlt.ForthesakeofconvenÍence'0 will be measured. ae shown in Flgrrre 2t1 ,

2.-h InvestiFatloll of thp. varlqus gistnibutione

The (vrt) ana (vrt = *) dlstnfbutlons (2,4) ard'

(2.6) ane axlally syrnnetric but not spherically synnetrl'e¡

anil thin elongated. surfaces charaeteristlc of plant roote

night concelvaþ1y be d.efined. in then. The (Vrt = -)clistribution will be cleecnlbed. tn Clrapte¡. 3 in ord'er to

illr¡strate the choLce of borr¡rclary corrd.ltlon whÍch d.efines

a noot surface ancL a more d.etailect lnvestigatlon d.efenned'

to ChaBter 5. The (Vrt) öistnlbutlon whieh d.escnibes the

5

r

vJ

10.

transitlon fnom ttre lnltlal r¡niforn concentnation to t'he

quasl-eteaqy (V, t = oo) clfstnibutlon, w1lL be d.iscussed.

briefly tn Chapten 6.

The (v = ort) and. (v = o, t = *) dlstnlbutlons(2.5) and. (2,7) have sphenical synmetry about the onigln.

They w111 þe consldened. in Chapter l+ tn ond.en to d.emonetrate

the Ì¡se of the clefinltlon of the noot surface of Chapter J.

It ls evid.ent fnom (2.1+) (2,7) that rægatlve valrres

of C will be obtained. for sufficlently sûal1 values oî P.

As these can have no physical meanlngr subseq.uent consid.ena-

tlon le restricted. to the regfon whe¡re C > O.

11 .

Chaot 7 Definlt lon of Root Surfaee

?_.'t flhe lrr^ +. -l dI stnlbrrtlon a

Flgu:reo 3,1 ¡ 3r2, 3.3 ehov¡ leopycnal llrres fon the

(vrt = oo) distribution (z16)' the valr:ee of q. arrd- D

are ttre same in these cases, while Fig. 3r1 corresponls to

the snallest arrd. trtig. 3.3 to the largest value of V.

It can be seen that the leopycnals in tr'lgurea 3o2¡

5.3 have the elorrgated. shapes which one ex¡rects from plant

nootg. Other eonblnatione of the parametere leacl to even

more elongatect lsopycnals which ane not so easily displayecl

graphlcalIy.

In alL these caseÊ the lsopycnals appean to cnowd.

together ln front of the slnkf Qbviouslyr r¡¡henever the

isopycnals are most crowd.ed. the gnad.lent of the nutrient

Concentratlon w1ll be greatest, 1.ê.r the ratê of nutrlent

cllffusion will be a maxlmum.

1^2 The Bor:rrclarv Cond.itlon

As an appnolclnation, the plant blol.ogist assente

that the rate of nutrlent uptake at a polnt on ttre root

eurface 1s pnopontional to the nutnfent cor¡centnatlon at the

polnt. The accunacy of tJris assertÍon varles eons lclerably

wlth physlcal corrctitlone. ThiE assertion ls assuned. to be

precisely true by Passior,rra ll), and wllI lfkewlse be aesuned'

here. thle bor¡¡d.any conclltlon may be glven the foru

f'y

t,z

ø79

o,6

o'*

o,2

-o,2

tG*)

c - O,rl Cn o'lo

o'

lHo

c*o'8

)c (car,)

,,L

t,

cO. I

vt,ô&

| = 3, /0-6

-e,+

(Vrt = oo) dj"Etrilrutlon q = 10-s, Ð =,N0-6¡ \l = ]"10-6

o.¿

,r'l¡T¡g,$¿-ã"1 Jeopv.cg,ale

Cao

Ca o,7

t-4 l, tl

1,2

l,C

o,6

0,>

* o,a

C -O'6

ô,+

'i3 r

sO'8

I'Z

t.0

O'ß

C¡ ú.+ $O'6

0.3

ro.8

û,

o'( o,(Ò

o'lo,r

zOa

o,7

- 0.1

E,L

V: 3. /O-s V- ß. /o'r

(Vrt æ æ) d,letrlbublon. e = 1o-tr Ð = 1O-o

Fieure l.ä Jeews¡¡elå e F'3egäe-.fuå TpqpYg"À¡tlg

14.

ò0Fn = kc, (l.l)

whene n Ís the outwand. r¡¡rit nonma]. to the root sunface

(1.e., ln the d.ireetion of increasing concentration), and

k ls constant.

As each of the nutrtent tllstnlbutlons d.iscueeett lnChapter 2 is axlally syumetrLc, the noot surface will al.so

be axially eyrnnetrlc, and. so w111 be the surfaee of nevolu-

tlon obtained. by rotating a root' pnoflJ.e about the C axie

(see lrlgure 3.4). fn vlew of thls axlal synnetny, the

bonrd.ary cond.ltlon (3.1) wil-I also be eatisfleal on thlspnoflJ.e,

Fipnrne 3.1+ Root Pnoflle.

fn ord.er that the root surface w111 T¡e smooth ard

axlally syrnnetnlc, tlre proflle nust be horizontal at 1teòcðc^ãã=ãi=cr ard'

C¡ = kC. (3rZ)

Ae C¡ can be obtalned e:ç¡I1cltly by tllfferentla-tion, (3.2¡ will be an algebnaic equatlon lnvo1vlng rtï

F

o

i-r

N

end polnte PrN. Thr¡s t dt these polnte¡

the bourrd.any conöltÍon (3.1) becones

15'

and. the ptrysieal paranetsrg¡ It will have the eolutlon

r' = oP when 0 = Oo and. r = ON when 0 = 1800.

At a given polnt, the borrr¡dary cond'1tlon ( 1,1)

d.eflnes the d.ireetlon of the nornal to the proflLe (and

hence of the tangent to the pnofile). Thrrs acljacent polnte

on the proflle can be forrnö þy a process of lntegratioû.

Efthen of the end. points PrN Tvill" provlcle a Starting polnt

for thls proceso.

Along the leopyenals shovrn {n FigUres 3,1 , 3.2¡ 3r3¡

the value of $fi itt"oeases wlth 0, whll-e C remalns

constant (c.f . section 3,1). Or the root, profller by

d.eflnltion, $fi = XC. Thrrs , !1 a root pnofÍle can be for¡nd.

1n any of the clistrtbutlor¡s sÏrowrl in tr'igur€s 5.1¡ 3.2c 3r3¡

1t nust be even more elongated than the isopyenal that

passes thror¡gh its eniL Point N.

a.3 - Mathem3Ilea* developnent of-9he !.S¡rd.qrv co+ctitlgn

I n \?cL

( tt cl¿r- {.(>nç r.

\ -l"rofjle 'Âu

tsoþ;¡cnti I Ãot\ Þlcrql<a ng l.Ë - ¿o¡$1.

t 50

c

Flgure 3.5

r'r

trrLBn¡re 3.6

C.r

^ -òc"0 - òa

16.-

Denote the vector gnadíent of C by YC arld. let. lhen the d.lrectional d.erl.vatlve of C in the

d.lrection S nay be wrltten

# = lYcl eos B, ( 3Õ)

where B Íe the angle between * and. VCr and.

lycl = ^lffiW . slnce at a given polnt Yc is

always nonmal. to the lsopycnal through the poíntr if Ð ls

nonmal to the root pnofll.e, þ wlll also be the angle

between the proflle and. the isopyenal lsee FÍgure 3.5f.Fon þ É O, the noot profile w1lI have two possible

d.lrections¡orrê on each eld.e of the lsop¡lenali These âre

showfi as R.r% ln !'lgurte 3¿6t

Corobinlng the bor:¡"¡d.âr¡r sorrèltton $fi = AC wlth

(3.3) yieLd,s

cos B =ffi, (3.4)

whence

tanB=#, (3,0)

whene

A = Cra * C¡'tF - l*Q2 .

Along an lsopycnal, C = constantr so that

d.C = 0r

c

(1.e1

0

1¡êrr

C¡dr+CUd?=O, or itræ =- C¡ I

17.

At a glven polnt 1et ct be the angle between the lsopycnal

ar¡I the circul,an ane through the polnt, as shoÏrn ln tr'lgure

3.6.lhen

tana =-+#=.W.For the two posslble root profilee R

the incllnatlons to the circulan are wil]. be

respectivel,y, ar¡l go fon the profllee

- + åä = tan (ø ! P).

Using (l.g) ancl (Z.l) one fir¡ds

\,

ß,1)

a ancl

d ! þ,

crGor t kc,\,rÃ1 ctn

r cLO eræ-kpæ

Thr¡s, points of R" satlsfy the d.iffenentlal equationvl'vn

1 dr + kcJaFEa =ffifu ' í='3)

whl1e points of Ro satisfYCr Co

1dn-t' -ry/1 (3.s)7Fã =w;-iíræ '.

These are onitinany d.ifferentlal equations for the

d.etermfnatlon of the noot profÍle. If t enters e:çllcitly

lnto tTre C ctietributlon it wilJ- be an Ínilepend.ent varlable

in each cl.lfferential equation.

In genenal, only one of the d.1ffenent1al equatlons

(¡,9) and. (l,g) wlll yletd a root pnoflLe through both encl

a

18.

polnts P and. N. It wíII be shown that for the (Vrt = oo)

d,istriþutlon cliseussed fn Chapten 5, (3.9) lu appropriate.

In figr:re 3.5 the d.irection of incneasing C ls

lnd.icated by YC beeause the C d.lstribution arlses from

the aetion of a point slnk at the onigin. Along the rootpnofile R , the eoneentnation C wilL ilecrease a€t 0

increases, whlle along % ft will increase as 0 ltrcneasee.

This d.istinction between R" and. % is used in Chapter 5

to show that (5,8) cannot leacl to a root profiJ-e through P

as well as N.

i.lr The Bouncl-inc Cu¡rve A = 0

It is evid.ent that real values of ar/aO can only

be obtained. from the d.ifferential equations (3.0) ard (Z.g)

1f L> O. sy(3.6),

a=c¡2.3; -,.2ú

= lycln - !*c2.Thus, the condition A > O 1s eguivalent to

lycl > kc.

Along any rad.ial llne fron the sink, C t¡v1lI increasq

arrd. lyCl cleereaser so that A changes from positive tonegatlve. The points rvhere A = O w111 I1e on a cttrve

outsid.e of which 1t is inpossible to flnd. the noot profile,(Unren A < O, lyCl < kC arrd. no d.irectional clenivative of

C ean be equal to kC).

79.

BV ( 3.5)r ât polnts where A = O¡ one has P = 0.

In that case, Ra and, % eoincid.e, theln conmon d.irectlon

being that of the loeal- lsopyenal.

By the axial symnetry, one has on tfie (-axls

c0=0rand.henceA=G¡.2-!*c2.ThuÊ¡ldipointsofthe(-axiswhere A=0¡

C¡ = kC.

This 1e precisely the conditlon which locates the end. points

p an¿ N of the root profile (see seetion 3.2)r Éio that

the bound.ing curve A = O and the root Brofile have eommon

end. polnts.a.q , Eorrnulation 1n Carteslan GoorÈ1na3=q

t

Fleure 3.7

o

ALternatÍvely, the various nutrlent dlstrlbutione may be

speeífied by (*rC) coord.inates with eonsidl.eratlon restrlcted.

to the half plane x > O (see Flgwe 3r7). It was in these

coond.inates that And.erssen ar¡d. Rad.ok [t] mad.e the initlalformulatlon of the root mod.elr ârld. uruch of the nu¡rerical

computation of this thesls has þeen 1n then.

rç

x

20.

In these coordLnates, the er¡d. polnts P and N on

the C-axis ruill be d.etermineil. by the algebnalc equatlons

Cf = lrC and. Ce = -kC¡ reapectively, whiJ.e t'he d.ifferenttalequatlons whicÌr correspond. to (5.9) and. (3,9) can be shown

to be

-, C C- I kC^,6

ä = õié:ry't6r , (3'to)

vuhere

c* = #, cC = 88, a = c*2 * cc' - :r,zaz t

or

drc -%-þ[$ . (3.1,t)tr=d(A root profile with end. polnts on the (-axis ean

be obtained. by starting from one enil point and. integrating

with x as lnd.epend.ent variable using (3,1o7, then with C

as ind.epend.ent vaniabLe using (3.11) and., flnally, wlth x

again as indepenilent varlable using (f.1O).

21 .

ChapteI, h. The Boot Profile for tþe V = gglstl¡iÞtlong

b-l The d.lstnibutlons

In ttre (*r0) coonètnates cLefined 1n section 2.J¡

the (v = o¡t) d.lstnlbution (2.5) and the (v = o, t - -)ctlstrlbution (2.7) become, respeetivelyt

c=1-$errc-L (4,t)r z^Ff

c=1-*. (t+.2'¡

Eaclr of tLrese Olstributlons is lnd.epencLent of 0, arrd' thus

spherlcally synmetrLc.

$Iith C0 = O, the two èlfferentlal equatlons (¡.9)I

and (3.9) sfmpllfy tø

Lg* =-W (+.¡)rA0 kC

and.

+# =*W, (4.h)

where (4.1+) can be obtained from (4.1) by reversing tÏre

d.irectlon of increaelng 0a Thrrs, without loss of general-

ityr consld.enation can be restricted. to (4,J).

I+.2 Cincplar Loot proffLes

f,et the spherlcal surface on whictt

C¡ = kC (+.¡)

have rad.lus R. Then one solutíon of (4.3) will be

r=R=constantt

22.

1.e., One root profll-e 1s a Eemi-circle of rad.ius R. It

has been shoÌvn in sectlor. 3.2 that the c'oniLltlon (4.5)

cletermines the end points of the root pnofile. Obviouslyt

fon these speclal cases with spherical syrnrnetry it completely

d.etenmlnes the pnofÍle¿

Substltuting (4¿t) into (4.5) yield.s

# = r(r - $l (rr.e ¡

with the only posltlve solutlon

R* = fcl * 7ffi). (¿+.2)

thus the radius of Ûre semi-circulan noot profil-e fon the

(V = O, t = *) distnibution can be calculatecL 6l.irectly

fron (4.7). Substitution of (4.t1 lnto (l+.¡) yield.s norr

$[unre:= +:s- e]cp(- 6fttl = k[1 - * erfc *]'r- - Z^ffi, Jñ t+u.' r- 2^[út

(4'41

Solutions r. = R of (4.9) have been computed. nr¡mericall¡f

and. are shotrm in tr'igure 4.1 which also shows values of R-'

It is seen that R + R- as t lnereases, as wa6¡ to be

expected., slnce at polnts where r << zJfr', (l+.t) slnpllflesto (4.2¡, and. the conrespond.lng valuee of R shouLd. agree.

Thus the above mod.el of root growthr at least for

the case V = 0¡ impl.les an upper llmit to noot size.

h.< Non-cinc]rlgf root Pnofiles

In eection 3.2, 1t has been shown that the root

proflLe must pass thnough the end. polnts P and, N d'efinecL

lJ2€-J ø

/o

I'

.ôt

,32

1.2

.32

-t

.t

, O32

'ol

3.2

R (c-s¡loX s<<le-

tt' tor

R ( r^s)/oX scalz

7= /o-+

to to

Ð = /o-7

72æ. /Q- 6

¡<l = J,lO

Æoo : g'¿Ê

Ë! /, Of

zo7

Ræ=7'??

^ /,Ol

/?ú a /'ol

.t. _>?=10

?= to-6

(ime lse¿s) loX scale-

/o/ô.ot

/o

3'2

l.o

tô¡+

I

ø /o'z O,S16

?tme- fsec.r¡ /og scalc-

10 /0 /o4 lo lo-7 loto

ì"1 = lo = 3,2{

R çc^t1loq *rlo

-ê

V¡ = ta- * -0,8

ti^q- (sz¿s) /og sccrle-

fo toa' tor /o* l¿a taz lO6 lO /oto /on /Ðt' /d4

(V = ort) dletníbutionBase paraneterÉl g = 10-6, D = 10-6¡ ùl = 3"10-0

.32

,I

o32

'Ðl

tr'i¡iune h^{ Raciii of cincular rlrofiieÊ', * ÐarerllÉter gtuclv

24.

¡y (4.5). For eitt¡er of the above spherÍca1 d.lstributlons

these lfe at the same rad.ial clLstance R, Hot¡'leverr fon any

curve on whlch (4.3) fu satisfled., tÏre values of ar/aO v¡iJ.l

have one sign only. Thus, the only profile whictt can pass

tfrrough both P and. N rnr¡,st be senÍ-cineul-an.

It will be shown þelovr that there are other solu-

tions of (l+,1) whlch pass thnough only one of these enit

points¡ Such lntegral curves will not þe acc-e¡ltabLe as

noot pnofil-eg because they d.o not pass through both end.

polnts. Hor¡¡ever, slmilar curves have been obtalned. numer-

ical1y ln cases with V I O, and. they ane d.iscussed. here to

lndieate 'bhat their existence is not rnerely a spurious cons-

equence of nu¡rerical- inaceuracy. The d.erivation below w111

be restnicted. to the (V = O, t = -) d.istrlbution d.escribed.

bv ( 4,2).

It rnlght be expected. that there exlsts a r:nlque sol-utlon of (l+,3) which passes tfrror-rgh an end. polnt with 3 = R¡

anil that it is the seni-clrcle with rad.ius R. Howevert

the cond.itions for a tmlque soLution for such a iLifferentialequation (".g., Ince [¡], P. Ø) inclr¡d.e a T,lpsehltz cond.1-

t1on. Sueh a eond.ltion wi]l- not be satisfled. at points

where P = R as the firnction

has a d.erivative v'¡ith respect to r which 1s urrþoultd.ed. nean

n=R¡

25'

Real values of åä can be obtained from (4.1) only

if c¡ > kcr so polnts where cr = lcc will form an outer

borrnd.any for soLutions of (4.¡), Obviously, this bound.ary

|s the curve A - O d.iscussed. in section 3.4r rr¡here r = Rr

as clefineiL above. Thr:s, for points on any curve satisfying

(4.¡) one must have r ( Re

For the d.istniþution (4.21, the d'ifferential equation

(4.¡) becomes

d.rd_0 k, n-A (4.9 )r- ¡

iltre shall seek a solutlon of (4.g) thnough the point

0 = o, r. = ït-i tet ). = # . Thenr by (4,7),

R." = $lr + li'îílÑl 'Introd.ucing

* = ${o * 1)r

(4.9) t"y be red.uced' to the form

(4.t o¡

(4.tt ¡ctud0

anil the startirrg point to 0 = Or Ìl = lñ-f r

By ( t+,1t';, åä will be defined and. continuous for

1<u<fi-liT,i¡ê¡r for (4.t z¡

A < r ( R...

By (4.21, t¡e nutrient concentration is zeno when

r = Ar and. (4.t21 cleflnee the region of physlcal lntenest

tu-

26,

ln whieh 1t ls posslble to flnd. lntegnal eurves satisfylng(4.t't ¡.

It can be seen from (4.tt ) tfrat at a polnt A where

u < ^ffiifi r # < o¡ Thus on the curve thnor-rgh A on

which (4.tt ) fs satisfied., if d is larger, u wil1 be

smaller and. hence åä w111 be numericatly larger¡ The

eurve thr-rs turns lnwards¡ and. # - - as u + 1 *¡i.e', the integr.al curve is orttrogonal to the circle ü = 1

(which 1s the isopycnal C = O) "

ï,et ü = U¡ 0 = @ be a general point on the

lntegnal curve passing through u - nffi-X, O = O,

Thenrbv ( 4.111,

-f (u-il¿u - =/"ur.hñffi Jo

aê¿lvOl

@=

where

fa=

¡/îf u dul-

JUifiã dur-Ju ffi = I¡. - Izt

(4.t1)

T 1---^-r- lP - 1

E -zarcs].n ----rñ- (4.14)d.vml_5ãv/\ v

The second. term of (4.t¡) may 1ce rewritten

lf.2 =+

du

+

(4.r ¡ )

t

27"

rrhere

For the special ease

l2

r 8nd'

1s the ineomplete elliptic integral of the finst kind.

(nv*¿ and Frl-ed-nan lll , p,B).

1

^1n

,

l- (4.te1

1S aS

(4. r a;

(4.t g)

a

J2

Thus for a general point (U, O) of the lntegraL

curve thror:gh u = .r/Taff,, 0 = O, by (4,1j) and (4.f h),

o = o/4 - L. arest" 4n-J- - rz, (4 .17)

whene Iz 1s given þv (4.15) o* (4.t61.

By (4.11), u d.eereases monotonically as 0

lncreases; then O will have a maxl-murn when

snall as (I+.lZ) will allow¡ 1.€.r when g = 1r

the eorrespond.ing value of @ 1s

o =o/4 - Tz',

By (4,r 7)r

U

o

where, bv (4,15), witho,uz = À + 1 - 2À sÍn2B,r /4

]ro'=/'dÉ-- JoffiIt can be aeen fron (4.19), that the value of lzt wtlld.eerease as h increases, that Ia. =

o/tr when À = O

and. T,zt .-+ 0 aÊ ^

increases. Thrr,s by (4.ta;, for an

lntegnal curve in the region (4.12), 0 ehanges by atmost o/tr as u d.ecreases from l-fT fu 1, i.ê.r âS

the lntegral curve movea froro the bountttng crllve

to the lsopycnal C = Or

28.

A=O

Ffgure 4.2 showe the solutlon cunve (4.t1) plottecl

fon the partlculan case A = 1, À = l+8. It can be seen

that the curve tr¡rns lror¡ard.s¡ âDd. that the nunerleal value

of U'/uU incneases rapJ-iLly wltTr clecreaeing r¡ Sfnce itreaches the lsopycnal C = 0, 1t 1s obvlouely unsrrltabl.e

as a noot pnoflJ-e.

¿1.,0

j,l

3.o

2,{

2,o

t.l

oç

o'zt

l.o

tc: o

29.

A=O

rnl-e1tal C vf ve-

I

\\\

o o.f o

(V = O, t = æ) dlgtrlbutlono A = 1, h = 4g"

SSgu¡p ¿rj3 .

2,o

30-

chapten 6. The Root Profile, {on the (v. t = -J dis.lribution

Fon large t at points near the movlng sink the

nutrient d.lstnlbution i.s d.escribed by the (Vrt = *) dls-tnlbution (2.6). Refemed. to porar coordj-nates vuith

onigln at the moving sink, (2.6) becomee

_ a u-Br(1 - cos 0),C=1-f,e-"'"' ul''Èr -, (¡.f)Ìrwhere B=ãb¡

T.et E=ê"-Bo(1-cogo)n

$.s)( ¡.¿+)

¡ (s,z¡Then C and its partial d.enivatlve nay be written

C =1-8,c¡ = Et+ + B(t - eos d)l 'CO = E Bn sln d,

crr = -E[# * t$ + nçr eos o )]" ] ,

Cr. - -EBar sin 0(1 eos o), (E.l)COO = EBr(eos 0 - Br sin30). (¡.4)

6.2 The end._ points P and, N

T,et p (*p, oo) and N (nxir l8oo) tu the poÍntswhere the root profile R meets respectively træ positiveanil. negative parts of tlre f axis. Fro¡o section 3.2¡

"p 1s the solution of

C¡ = kC (¡.g)with 0 = 0o, whtle rN 1s the solution when 0 = 1BOo.

(0.5 )

)(5.6

31.

lühen 0 = oo, 1t follor¡us firom (f.27 * (5.1+) that

c=1-*, c¡=#, (5.1o)

arrd. (5rg) gives

Aãürn-

Henee

op = $t., * yffi). (¡.tt)

Nr¡merical results obtained. from (5.tt¡ ane shovun inthe second. column of Table 5.1 . (tfre cTroice of M = Dlc

nathen than k as a base pararÞter for Tabre 5¡'l is ois-

cussed. in sectlon 5.9. )

It may be noticeil in TaþIe 5'1 that the *p values

remaj.n unchanged. as the parameter V ís varled'. Slnee

A, = q/t+nÐ is ind.ependent of V' the d.istribution (5.t0)

anfl hence tkre result (S.lt} are lrrlepenflent of V. Thr:st

at polnts urhere 0 = Oo (l.e.e on the posltive f-axisrd.lrectly behlncl the moving sÍnk) C is lnd.epend.ent of V.

In partieular, (5.t0) agrees wittr (4rZ) which d-eseribes the

(v = o, t = *) d.istrlbution, and. the erçresslons (5.tt1

and. (4.7) for op and. R*, respectively, are iiLentleal.

tr'igures 3r1 ¡ 3.2, 3.3 shoïu that C d.oes not d.epend-

on V at points on ti:e posltlve f-axls. In ttrese figtres

the only parameten vaniefl Ís the veloclty Vr and. it can be

seen that on eaclr figure the d.ifferent isopycnals have the

same intenceBts on the positive (-axis.

= k(.| - *).

32,

(vrt = *) d.lstnibution.Base parameters e = 1o-õrD = 10-6 rV = 3,10-6rM = Dk = 3,10-6.

.9326

.9249,9207

.9r+71

.921+9

.9182

,g690.9i21+9

.81 94,6269

,2940.791+1

.9249

.9729

.9971

.6164

.2412

.0387

,8570.2412?O0l+2

.21+'12

.2412

.2412,2412

.o123

.1O15

.2412

.42'11

.7549

.og69

.1434

.2063

,4zz5.11+31+

.0273

.0732

.11+31+

.2768

'49o1

.oB58

.1167

.1434

.1714

.2353

.20741.04879,2779

.55631.04877.9909

1.04871.04871 .04971'f O¿+87

. Bo56

.88561,04871.37473,2468

q. --O=

Q=

10-610-510-+

D=D=D=

10-61t610-7

V = B.10-6Y = 3.10-sV = 1.10-õY = 3.10-e

M = 1O-+

M = 10-6M = 3.10-6M = 10-6M = 1Q-"

Ç at N(or-o*)C at f (Orrr),NoPParaneten varied.

Table 5.1 Bqe! pn-ofile end points - parameten stuily.

33.

t4lhen o = 180o, (5.g) lead.s to

åc, + zer)e-zB" = ].(r -f "-8"), (5,12)

An explieit solutlon is not possible here because of the

e4ponential termsr Nurn€rical solutlons of (5.12) trave

bee¡r computeil, using Nev'¡tonr s rnettrod r ârrd. soilÞ resultsar.e shown 1n the ttrlrd. column of Taþl-e 5.1. The values

of the nutrlent concentratlon at the two end. points are

showr in colunns h and- I of the Table.

q.z Pnopentles of the (V.t = c"ì d.lstnibution

A nr-¡mber" of properties of the (V¡ t = -) d.ist¡rlbu-

tlon w1l-I now be established in the foru of theorems. ID

thë next seetlon these results $riII be uged to rejeet one

of the dlfferentlal equations (3.9) an¿ (lt9)¿ Later on,

in section 5.7t 1t v¡iII be shown that the end. point P isto þe prefernecl as a starting point fon the ileterminatlon

of the root proflle by nuueríea1 integrationr

,Thçoren,J. Along aII lsopycnals,,{n

Ëä.0 for o<

ir€r polnts on a given isopycnal are closer to the onigin

wlth incneaslng 0.

Proof

By d,efinition, an isopyenal is a line where

34.

C = coûstant. Thus for a d.lfferentla1 dlsplacenent

(a*, dá) along an lsopycnal,

d.C=0

i.e. C¡d'r + CUd? = 0r

otl

d-rd.a

cor./f a (5.t3)

Bv (5.1+) anð ( 5.0), for oo <

or co >

Wlrence, by (5.13),

d.rãã'

Theorem 2¿ Alorrg a given isopycnal in the d.irection ofæÍncreaelng 0

cot

VC Jc"' + Tz0

1s strietly monotonically increasing for Oo <

Pnoof.

--t,et e = lYCl'=CrB +

sy ( 5,4) and ( 5.5) t

Alorrg an lsopycnal, þy (5J), E

in G w111 be clue to changes inled. that

e = E2[82 + (f + e)' - zB eos 0 (B + t)] = E€H,

c0

2/¿,

ls constant and. char:ges

HÇ It is read.ily verif-

H¡ =-þf*+B(1 -cosa)1,Hd = 2B siir o(B + *).

(¡.rh)

35.

By Theonen 'l e fon Oo <

d.lsplacement along an isopyenal with A0 > O nust have

d.r < O, For Oo <

and. the correspond.lng change 1n H

Ë[H=Hoilr+HUdd

will be posltlve. Thrrs H¡ anÖ hence G and JG = llcltl-ncrease with 0 aLong an isopycnal for Oo <

Corollary. A = Co' * # cu' - Yzçz is strictly monoton-

ícally lncreasing along an lsopycnal in the d.inection of

increasing 0.

Th-eorem z. On a eurve where

nonctonleally increasi-ng with

A = coretante C ls strictly0 for oo <o <

Proof.

- As a=cra "Ç-!*c?t

Çocoo co'ar = z(crcrr +-r --FF -lr,zccr) , (5.15)

Bv (5.4) (E.l) ancl (5.15), it 1s evÍdent that foroo < 6 < l8oo

C¡>Ar<

Let Pr(rr ,0r) be ar¡y polnt on the curve A = A1 r

tet Pz(rr r0r) þe any point on the lsopycnal C = C1

th:nough P1 wlth Oo < 0t <

value of ^

at Pz. then, by tbe Corollany to Theorem 2¡

36.

ba > A+.

l,et Ps(rg ,0z) þo the polnt with polar ang].e 0z

at which A = 41. tret the Ya1ue of C at Ps be Cs.

By ( 0.17),rs 2 ?e,

lhen, by (5.t6),Cs>

As Pr(r¡. t?1_)r Pu(r.s¡02) ane pointe on the curve

A=A1 with 0p>

Corollar-v 1. The nutrlent concentrationsæ-the end. points P arriL N ane such that

Cp and. Cn at

cp<

Proof.æBy section 3.4r the end points P anÖ N 1ie on

the bounding curYe A = O¡

Since P and. N havepolarangles 0=Oor0 =1BOo¡

respecti-veIy, one has bY Theorem 3

cp . cN.

CorollaTy 2. If an ísopycnal interseets the curve A = A1

at ("or0o), the isopyenal is further from or closer to

the origin than the curve A = A¿ according as 0 < 0o ott

o > oo,

Proof t

This fs implieit 1n the proof of Theoroo 5.

37.

lheonem h. Polnts on the curve I = Or where f = Ç - lc0r

are such that they

(") coincld.e wittr, polnts on the curve A = O when

0 = 0or lBOo but otherruise I1e lnsid.e 6 = Or

(¡) appnoach closen to the origÍn as 0 lncneasegr

Proof a

At potnts whene f = 0r one has A =

A=c¡z -ry-Êú¡.t 2

=r(c"+kc).+l:2uo

f

(")

a

a

Bv (5.5), Qo = o when o - oo, 18oo and. C 0

v¡i se.

Thr¡s, when d = Oor 18oo, A = o at the points ïuhene

I=0.As these are the points on the C-axÍe where Cr = kC¡

they ane also the end. points P anil N d.iscussed inChapten 5'

For 0loo and. o/1Boo, A>o where f =0.By ( 0.17)r ar <

thecunve[=0.(t ) For a d.ifferentlal ctlsplacement along the

curve I = 0¡

d.I = O¡

38.

1r €¡

1rê¡

I¡d.r+10A0=Or

d,rd.0

t0I¡ a ( l.t a¡

Slnce

IO = Ct¡ - kCOr fr = Crr - kOr ¡

one has, by (¡.r+) (5,Ð, fo:r o <

f0<0¡ I¡<Henee, þy (l.tA1 on the eurve I = o

åä<o ron oo<

Corollarv. Outslcte the eurve f = Or

-

ca < Icce

Pnoof.

By Theonen 4r I¡ <

(rr0) on the curve f = Os

I=Or-kC=0.Thr¡s , ãt a polnt (t' ,0) wíth r' ) Tr

r<fre¡

Cr < kC¡

Theonem 5. Points on a curve \ffhere A ls colls tant

approach closer to the orlgin withr increasing 0 fOr

0o<

Pnoof I

1rê¡

By ( 0.17) t

Since

39.

For a ctlfferentLal d.isplacement along A = constantt

d.A = O¡

Ard.r + AOdâ = Or

ao=-8..esd.0 U,lg)

A. <O.

A = C¡2 -4 - t*ú,

LAo = crcnd -ry - Êcco. (s,zo7

Substltutlng fnom (S.l) (5.81r one fÍnds

L0 = -2EBr sin 6[r"c + B[ 2B(1 cos 0) . $t 1 - 2 cos o)]J.(5.2t)

Obvlouslyr AO < 0 tf sin d >

and. (t - 2 cos o) >

1.e., if 600 < 0 <

Thr¡s in tJ:is range, ¡V (5.19)¡

åä<o'çommeq!: A emaller lower llmlt may be given þy

cos Oo = Bn eLnz 09 .

Bv (5.2o) and (Erl) (5.8), 1n the range o < 0 < l8oot

L0 1s certainly negative if COO <

(1.a1, when cos o < Bn elnao.

40.

Theor-em É,. If V < 2lÐ, polnts on the cunve where A = O

approach closen to the origLn wittr increaeing 0 foroo<

Proof a

(5.21) r,"y be ¡rewnltten

A0 = -2EBn sin 0[t2c-sn[þt1-cos o)]+sn(1-cos 0)GB$)I

= -2sBn sln 0[rrsc-n.c¡+BE(1-cos o)(gjÏ'+3)l

= -2EBr srn d[n(rc-c")+(rr-a)rc+Bp(1-cos o)(58+l)] .(5.22)

By d.efinitlon

Hencerwhere A=0r

A=c¡s.+-*c2.

¡-2aZ a 2Jt l-, - \,f

rt2uo

ro

Since C and. C¡

Hence, þy (5.22¡,are positiver kC à Cr whene A = O¡

A0 <o for oo<pnovided. k Þ B,

1¡€.r krS or v<2kÐ.

dnd.0

Comuent. ìIrilhen V > zkÐr by Theorem 5r points on the

curve A = O approach closen to the orlgin witlt increas-

ing o for6oo<o<1800.

As always Ar ( Or

<O onthecurve A=Oit folr-ows from (Srlg) that

for oo <

41 '5.1+ Cholce of Dlfferential Equatlon.

The bor'rnd'any ændltioî (3.2) which d-efines the noot

pnoflle has been shon¡n in sectloî 3.3 to give rise to the

two posslble d.lfferentlal equatlons (3.9) and. (3r9),

These lead. to solutlons urhich can be d.lstinguished by the

way C changes wlth inereaslng 0 3 narnelyr C d.ecneaeeg

along curvea obtalned fr"on (5.8), wh1lo it inereases alorrg

eurves obtairæd. frrom (3.9),

The end. polnts P ancl N, where the noot pnoflle

meets the axisrhave been d.etennined. in sectlott 3.2. They

have the poLar ang1es Oo a¡d lBoor respectively. By

Conollary 1 to Theorem 3, the vaLue of C at P ls less

than that at N. Thus it will be lnpossible to flnd. a

curve throrrgh botTr P anit N which satlsfies (3.8) t

þeeauee of the monotone behavlor-lr of C stated. above.

It follows froro this that the equation (¡.9) ean

be d.lscarcted. and. attentlon nestnlete¿ to (3.9). It stí}l

remains to be cleciiLed. vrhethen a curve through one end- polnt

and satisfylng (3,9) can pass tlrrorgh t}¡e other end. Bolnt.

6.Ã , Bgr¡ndlns clrves for solutlone of t3*2).

ConsÍd.er a polnt A on ttrre curve A = O. By the

nesults of section 5r4r the integral curve through A

satlsfylng (3.9) ls along the isopycnar through Q. By

Corollary 2 to lheoren 5 above, the lsopycnal through A

crosses the curve A = 0 from the outsid.e to the lnsid.e

LvZ.

with inereasing 0 for O <

lntegral eurve satisfying (3.9) passes thnough a poLnt

t(r*rO*) where A > O, points on the integral curve with

0t<(f,et it be assumed. that T{(rtr0ou) 1s tlre first

point on the integral curve with 0 >

Í.e., the first polnt where a < o. Then points on the

lntegnal curve adJacent to Ïli will þe on the isopycnal

through ilf. However, ât points with 0 . ?ri,tr, A ( 0r

whence f oll-ows a eontrad.lctÍon" )

Consid-er (3.9), vLz.e

+ - kc,,ra1 d.nn dL0

,a

whlch may be rewrj.tten

Now C¡ = kC

d.eflnitl on,

c 0 - ÉcP

1dn----r? (tøcr' - ÉÉ

+ kC^,6

f

I (5.23)

rIn the region L > o, the d.enor,rinaton of the rlght-

hand- siiLe of ( 5-23) i" positlve for oo <

At points on the curve satisfylng (5.ry), rvhen C¡ = kC¡

1È - "'rãõ'= vr

at points on the curve I = 0r sincer by

I = Cr - kC. By part (b) of Theorem hrat

l+3.

polntson f=O for O<d.rcl0

(0.

Tht¡s r ât a point where I = Or the integral curve satisfy-lng (3.9) w111 be circLlLar and. so ïri1I pass from the insÍd.e

to the outsid.e of the curve f = O wtth increasing 0,

If an integral curve satisfying (l.g) passes

through a point T(r'rdr) v¡hich is outsld-e the crrve

I = O, points on it for 0f <

outsld.e I = O ( cf . the eanlier argument fon the curve

A = O).

Let it be assumed. that the polnt f is both

outsld.e the curve D = O ar:d. insi.d.e the curve A = 0.

By part (.) of Theoren 4 sueh a polnt exists foro < ot <

the integral cupve tTrrough T w111 also lie betv,¡een the

two curves I = 0r and. A = O fon 0t <

Furtherras thecurves f =O and. A-0 coincid.eatN wlth 0 = 18Oo such an lntegnal curve thror:gh T mr.Lst

pass through N.

Astheeìfrves f =0 and. A=O alsoeolncid.eatPr the point f can be ch.osen arbitrarily close to p.

Then¡ by continulty, the integraJ. curve thnough p

eatisfying (S.g) will also pass thror¡gh N and. so ruil-l be

the requirecl noot pnofile.

ll4.q.6 Furthen Pnop_erties of the Solution Cr¡rves.

Sorne theorens concerning the behavÍour of lntegralcurves satlsfying (Z.g) will now be establlshed.. The choice

of end. point to conmence nr¡merical integratf on witl then be

mad.e in the following sectlon using these results.

fn the reglon between the bound.lng curves

A=O¡ # . o on curves satisfylng (3.g).

Pnoof.

OutslcLe f - O, C¡ < kC, by the Corollary toTheorem 4. Then, bV (5.23)¡

å;(0.Theorem B. In tTre reglon between tT¡e bound.iirg eurves

f = O arrl A = O¡ for given 0, the nr¡merical value of

åä increases with r on cunves satisfying (3.g).

Pnoof.

--In the notation of secti on 3,3, for the curve (3r9),

* åä = - tan (q - þ), (s.zt+)

whene

tan cl =c0

rCr (s.25)t

arrd.

tanÉ=# $.26¡a

Between the borrrrllng curves, by Theorem J¡ d.rd.0

45.

= tan (a - É). (5.27)bv(5,24)t a-É>o antt It a"lln ¿01

uslng (¡.4) and. (5.5), $'25) nay be written

tan ø = --Å sin o-*-"tì cos o)'whenee

3i r o¡ (5,2a)

nv (5.16) and (5117), c¡ >

and. A > 0 between the bound.lng curvesr 1t follows from

(s.ze¡ that

3€ . or ç5.2e)

conþfníng (5rzg) ard (5.297, r'¡e findò'É("-p)>

rt now forlovr¡s fnom (5.27) tr¡at l* åäl irr"*".ses wÍth F¡

anl tÀenefore certainly l*äl increases vrith r'r

Theorem a. In ttre region between the bound.irrg curvesr tT¡e

rad.laJ- d.lstance between two Ölfferent curves satisfylng(3.9) d.ecreases with increasing 0.

Pnoof.

T,et the sufflxes 'l and. 2 d.enote trrro eurves on

which (3.9) i" satlsfiecl and t]re first of these curves ]-ie

closer to the orlgin.By Theorem 8, for giveir 6 t

46.

i¡êr

slnce by Theoren 7 dnîë

the region between the

)

, (n ðo)

< o for curves satisfying (l.g) 1n

bound.ing eunves. Thr¡,s

' (*ä)

l. \ /z/¿n\\äãl

ð.' \fr(n" - r¿) < Or l.€', the rad.lal d.lstance between the

eurves d.ecneases v'rlth incneasing 0.

5.7 StartinE Point for Nu¡rerieal Integratio.n.

By Theorem p, aclJacent eurves on wlrich ( 3"9) 1s

satlsfLed Clraw closer together with increasing 0 t provld.-

ed. eactr of then l1es between the bounding curves I = O,

ar¡d. A = O. By the results of sectlon 5.5, they will both

rernain between the bouruling cunves wlth lncreasing 0 and.

v¡Í11- coincld.e at the point Nr where O = 'l8Oo.

Convensely, ln the d.lrectlon of decreaslng 0,

the curves will grow farther apart. It ls quite possible

that an integral eunve wÍÌl cross one or the other bourrd.-

ing eurve as 0 cleereases (cf . seetion 5.5). If thednFo'

crossed., eubeequent polnts of the lntegral- curve wt1l have

r = Cr - kc > o, and hence, bv ( 5,23) .åä

the dlrection of decreaslng 0 subseqrrent polnts r¡¡1Ll I1e

closer to the origln. By part (b) of Theoren 4r on the

curve A = O is clloseed., integration nust cease aa

aesumes neaL valueg only for A > 0. If I = 0 ls

47.

CUrVe I = 0r d.næ ( Or so that pofnts on this curve move

au/ay from the orlgln with d.ecneasing 0. As a consequence

such an integral curve eannot reaeh the end. point P¡

Theoretlcally there will be precisel-y one curve on

whieh (l.g) 1s satisfied. which passes through both P and.

N (ef . section 5.5). If tTris curve is sought nrrmericallVr

computation errors lllill }ead. to points on ad.Jaeent lntegral

cunves. It follows from the d.iscussion above that the

convergenee of ad.Jacent integral curves in the d.lrection of

increaslng 0 will reiluce sueh error if integration is inthe d.irection of lncneasing 0, rvhile it will nagnlfy

errors when in the opposite sense. The Ílanner 1n whieh

fntegral curves cross the bor:nrling eurveg further commend.g

integration in the d-irection of increasing 0.

Thus P is to be preferred. aE the starting point

fon numerieal lntegratlon, and. it has been used. in obtain-

Íng the results cllscussed. 1n the next tv¿o sectioÌLs. Inpractiee curves obtained. by numerieal integration starting

at N qulekly leave the region between the bound.ing cüFrrês¡

Sueh integnal eurves are d.lseusseCL briefly in sectÍon 5.10.

5.8. Enpirical Results.

The root nod.el u¡rd.er consid.eration involves 4 para-

meters; the sink velocity Vr its strength gr the dlff-uslvlty D and. the uptalre parameter M = Dk, (see section

5.9). In Figure 5.1 , the bourrd.lng curves f = 0 ancL

A = 0 a¡rd. the noot pnoflle are eho¡¡n for the parameter

48.

vaLues Q = 1o-", D = 1o-6, 'l -- 3r1o-õ, M = 3'10-6' Flg-

urea 5r2 - 5.5 show the effeet on trre 5 curYes of variation

in eactr paraneter sepanately frorn thege base values. iVhene

less than 3 curYes are shownr 2 or more are ind'lstinguish-

able graphfcal-lY,

It can be seen ttrat in each case the root profile

d.raws e1tremely close to the outer bor:¡d.ing curve A = 0

and. renains elose with increasirig 0. Invariably the

nunerical results have sh.own that the curves converge with

inereasing 0. For the base val-ues of the parameters the

proflle has been d.eternined by numerical lntegration up to

0 = 176o. Tab1e 5.2 gives values of the coord.inate n for

the profile and the bound.ing curves f = o and. a = 0.

In ord-en to perform the numerieal integration over thle

range Ít was repeateöLy necessary to shorten the integratlon

step lerrgth ( to prevent entry lnto the region A < O at one

of the 'rsampllng polntsrr useÖ in the Ru¡ge Kutta integration

process ) .

Fr¡rther evid.ence of this tend.ency of lntegral cunves

to converge to the curve A = O is shown in Figf-lr e 5 16.

Integral curves on which (3.9) is satisfleiL, startir:g at

various points in the region where C >

The Values of the parameters are agaÍn the base values.

Jn view of the observed. eonvergence of the root

profile to ttre curve A = 0¡ the numerical integration ln

all other cases was stopped. when the coord.lnates of pointe

l+!:.

J (c-'¡

/,2

o.B

o'6

o,+

o.e

|"1 o

- o'Z

.â*O

þ raf ilz

o. ç T. (u"¡û

ill

{vot = .o) òlsts.lbutåon

q * lo'-t, Ð = xoan v * 5,10-se Ìí e Dk = 3*10"s

¡.;¡;s O'Z

l-=o

I¡lnrrre 5^{ Rse! pss[å1€.-N--Þss&*å

5ç,

t.tt

Á:Ot,2 I,L

Á=Ot,o

O,8

O,L

o,2. o.s- o.2-

-o¿ - 0'/

-5 V- 8, /Av- /o

(vrt =oo) distnlbution, q=1o-6, D = 10-6r M =5n104o

t,o

þ"o{,1¿

Qr

O,Ll

0.8

o,6

o,+

o,2

S'lpune U "2æ

l-- O

1.

g.

6,

5.

th

3.

2

A*O ,¡al,

.zD

,t2

'o.8

,O,+

-,OLl

A=O

7.

té

o ,t+

t-tl

-1= lo ' ''oe

o,,

(vrt =æ) dlstrlbutlon. D = 104, V=5"10-õ¡ ll'''3o1o4

Für}4Le-þgl

,o+ ,Ot

52,

Q,

6

t.

l.

¡O

g,l

7.î

Á-ori,.

7,

o'4

O,I

Á-O , þtolïc

o,2

-ol

-è4

Or( D- /o- 5

rfr¡loa neerrrþr e nd þial'¿a larqc¡l é.r a

{.rtoì 4 lo

b = /o'7

+,

t.

Qt

l,

O,l

(Vrt !'æ) ûfetrlþutlon. o= fO{, Y -3.10{, H = 5.1o'€

4tf¡r t'I

píJî.Tr.?.ri i¡ ''" 'il" 'T11.': ,ffIfTliTT!.tl r-fÍ n¡ 5 rr- i':! 'l :i:i J ir'

i*-',ti l '' í

C)l = lrl

Ir - .:.. 'L' !rj ' ç*ii T.i.;, ,' r.¡;1..: [ :.]- i-].r, T å .ì .. !. r: i

.l .. :,

()) ¡ ;; ¿,.,'

?*a/=bl-5-ol = bJ

l*+

t--l*Z,o *

a'õ

Tr?''

a

+¡rfo

'( ,ct

rllj...¿.|

r) o

.a .a)

1'a

}l,þ¡(g=¡ rrl*-y

8'o

o'l

0-Y(Lþ.tt

,'t) Ê

tt

s

,?

L

E'o

ü=v

ü't

1.01+87

1.1tlv31.17951.0865

.9519

.8228

.7124,6211.51+65

.l+345

.3510Õo16.26Q9.2303.2O7O

,1891,1752,1646,1565.1506,1)+66

,11+l+2

.1431+

1.04671 .04301.0246

,9893$261.8197.7107.6200.5456.l+34o.3566,3O'13

.2607

.2302

.2069

.1 890

.1752

..t645

.1565

.1506,11+66

,1U+2

1.0¿187

r iot e6

o.9255o.8231o.72630.6¿+1 Io.57030.51 01

o.l+595o.3802o.3221o.2781O.2l+57

o.2201o.20020.1 845

o.17229.1627o.15540.1 5OO

0.146fO.1 lt41

0.1 l+54

0

50

100

150

2oo

250

,oo350

4oo

5oo6oo

700Boo

9oo10001 100

1200

13001 4001 5001 600

1 700I BOo

Bound.ingeurve A = O

Root pnoflleBound.lng,curYe I = O

0 coord.lnaten coord.inate for

5l+¿

(Vrt = -) ctistnlbution.e = 1O-õr D = 10-6, Y = 3.10-6, M = 3.10-6.

Table 5.2Coordinateg for the-noot profile and the bourrctln¡q -cUnveS

55-

l.z

.A= o

/"!

l,o

o,?

o'8

o'7

o,6

o,{o o,l O,L

(Vrt = c,0) d.letnlbutiong = 1O"so D p 10'-6, V = 5,10-6, M = 1o10-6

3 o'

Flr¡ure 6"6

--

C= O\\

\

r

þrc{ilo

+

56.

on the two eurves agreed. to 3 or l+ slgnifícant figures.

All lntegral curves cotputed. have a connon chanact-

enlstic. The value of A eventually decreases along then

with lncreasing 0. It ean be shovun that when LO <

(whlch, by Theonem 5r is certalnly true for 0 >- 6Oo), the

fanily of curves A = cotrstant d.naw closer together with

lncreasing 0, If 1t ca¡r be ehown that A d.eeneases 1n

value along an lntegral curve beyond. a certaln polntr then

it will follow that the integnal curve must ind.eed. converge

to the outer bouncting curve A = O. .A.ttelryts to establish

ttris result analytically have not þeen suecessful to d.ate¡

The vanj-ous root pí'ofitee ghown 1n Figures 5.1 - 5.5

d.o not have the thin elongateê shape which chanactetlze

plant roots¡ However, the base parameters were drosen here

for graphical convenlence rathen than physical aectrracy.

The value D = 1O-7 more elosely nesembles pÏgrsical cond.i-

tions¡ ârrd. it ean be seen fnom Flgure 5.4 that a thin elong-

ated. shape has been obtalned. for this value of D.

ByTheoren6, H, O onthecurve A=O nearthe

end. point P 1f V > ?:jr¡D, i..€. if V > 2M. Flgures 5.1

5"5 glve graphical verlfication of this result.5.9 The Flux of nutrient

The flux of nutrient across the root surface at any

point on ft is given by

ø=D òcòn

5li[cf. Carslaw and. Jaegen [+], Plt equation 1 .5(2)).However, by definltion, at any point on the root surfacet

òcòn = kOr

and so

þ = Dkc = MCr (S.ll)

where

M=Dk.It ls thue evid.ent that M d.eternírres the reLatlon between

nutnient uptake and. nutrient concentratlon. M 1s neferred.

to as the uptaJre parar'leterr and. lt rathen than k has been

used as one of the base parametens for the stuil1es of thíe

chapten. llhis usage of M is in agneernent with that of

Passloura IZ].From (5Õl) ft follows that the total flow of

nutnlent across the noot surface ln r¡nit tine will be glven

by

F=llø*=ti/t*,il' tJ

whene the d.oubl-e integral is taken over the root surface.

A second. parameter stucl¡r was carried. out in ord.er

to investigate the flov¡ of nutrient in unÍt time aeross a

root sr.rrface of given d.lrnensions. The maximu¡r noot thick-ness (twlee the d.istance from the axls of syrmetny to the

root proflLe) was fixed. at or2 stÃtt ar¡d. separate paraneter

stud.ies mad"e f on lengftrs of 10 cil¡ and. 20 cIIt¡ It was

consid.ered. that the portion of the noot surface below the

58.

poÍnt of maximr:m ttilekness nlght corresporrl best to a

ptrysical noot. Accorëingly the nutrient fl-ow was computed.

for this portion only.

I'on eactr studSr, the value of the panameter D t¡ras

flxed. at 1O-8. It wilt be shot¡¡n below t'Ïrat tTre effect of

variations of D can be d,educect from the resulte obtalned.

for constant D. The remalnlng parameters were ehosen to

given the fixed. clÍmensions stated above¡

Fon the ttr. in pnofiles shown in Flgr¡¡:es 5.1 5.5t

1t can be seen tÏrat the interce¡rts "p ancl rN orl the

posltive and. negatlve axesr reelpecti-vely, are such that

rN<estimate of the total root length. ftie largest values of

oN obtained. were 0.006 with fp = 10 ar¡[ O.QO[ with

"p - 2Or so that op lnd.eed. provlcled. an excellent approx-

ination to the root lengthi

sv ( 5-11)

"P.Á;E)

1rê.r âB stateiL earllenr Pp is ind.epend.ent of V.

This result nay be rewrittenhzrurfl

Q=1+

a (tÕz)

59,.

ThuE, for flxecl values of D and. op, (5,32) d.etermines

q. as a fi¡netion of M. Conrespond.lng values are shoum

fn the first two eolurnns of Table 5,3, It can be seen

that the valuee of q- ane alnost uncÌranged. for large

variations in M. lhis arises as the snallest value ofMro-Dt oceurring in the table is 1OO, and. fi¡om (5J2), 1f

MoP

5">e * l+øDr" r

a result which is ind.epend,ent of Mo

Keeping all othen parareters fixed., the value of

V was d.eterrnlned. to give a root profile of næcimu.m

thickness Or2 crn. The cunve A = 0 with this pnoperty

was first obtained. by ad.Justing Vr and. then the integralcurve through P with thls V value was cornputed. untilit was extremely close to the curve A = 0. In eadr ease

eonsid.ened the lntegral curve was found to have come

extremely close to the curve A = 0 well above this point

of maximum thickness, ard hence the eurve A = 0 wae taken

as an aBproxínation to the noot profile over the portion of

intenest.Columne 4 and. 5 of Table 5.3 show the values of

"N and. of e at the point of maximum thickness. Fon

the lower portlon of the root, colunns 6 and. / give the

extneme values of C, coh.r¡nn I its surfaee area and. col-

60.

op = 10¡ *r"* = 0.1

1 0-81 0-+1 0-õ

.10{1 0-6

.1O-710-7

op = 20, xmax = O.1

(vrt = -) dlstnlbutlor. D = 1o-8

lû

1.961.971.971.gg1.gg2.404. 07

9.49.258.777,966.504.643.49

.ooo2

.002.O20.066.193.497.778

.016

.1,1+2

.627

.853

.952,ggo,gg8

.o051

.o052

.o056

.oo59

.oo58

.00¿+6

.oo27

31683.7O3.703.713.724.497.70

1.1.1.1.1.

257

257

257256255

1 .4721 .h741.5011.572'1 .7912,6695.422

1 .2521.21+4

tr'ron tip toFlowx 1013

Area*t"r,

N

C values at*t"xoN

(at*to"xVx1 06gx1 0e

19.1 8.¿+

17,-t+

15.8'13.o9,266.84

.o002

.o02

.o20

.066

.193

.4gB

.785

3.923.923.923,923.954,797.81

7.367.367.367.157.1+5

8.981 lr.9

.o32

.246

.767

.920,975.gg5,999

.o030

.oo31

.oo5h,OO35.oo34.oo26,o015

1 0-s10-41 0-õ

3.10-61 0-6

5.10-?10-7

2,5132.5132.5132.5132,5122.5O92.501

2,gly'42.91+9

3.OO23.1/.+5

3.581+5,347

10.95

tr'1owx1

Fnon tlp to xmt*A¡lea

oL2

C valr¡es atN l*'o

Catxn"xrNVx1 0õqx1 06M

Table q.ã .

61 .

1¡1nn 9 the total flow of nutrlent fn unit tiure tlrrough thls

surface. The coÍrputation of ttre area arrd. flour of nutnlent

wlll be d.iscusseCl in ChaPteF 7.

It can þe seen fnom Taþle 5.3 that the area of the

noot surface is vlrtual-ly unchangeil over the range M = 10-3

to M = 1g-o ancl that the lncreases which occutr v'¡hen

M = 3,1O-? ar¡t M - 1O-7 are }argely due to the lncrease

ln lengttr of tþe lower portion of the root' It can also

be seen that the total flow of nutrient 1n unit tine

incneases with M, but that it is insensitive to ehanges

in M.

A eomparison betvueen ttre results for op = 1O

arid. "p = 20 ind.icates that¡ apart from nN arrcl the

values of C, all othen entrles ane approxlnately llnear-

Iy depend.ent on rp, at least oven the rarrge rp = 10 to

Op = 20.

The effect of variations in the parameten D can

be d.ed.ucecl as follows. The (vrt = *) ctistnlbution (s.l)

may be written

O=1 t #'Ët.r - eos o)l . $,ll)The relation (5.32) between q. and. M fon fixed.

"pcan be given the form

oD

a $.t4)

62,

Bv ( 5Õ4), q/b must be eonstant f or constant */T, By

(Srll), the d.istribution of C is unaltered. if tlre para-

metens are varled. while lceeplng 4/Í, Ul¡ eonstant. Thrrs,

consld.er the parameten comblnatlons

(.) D=D1r g=Q1r V=Vr, l[=Ma

(t) D=fD1r g=TÇhr V=TVrr M=fM1 .

Both will lead. to ttre saürg d.lstributlon Cr each has the

Same value of V = M/'n and. so the noot prof iles for each

w111 be iilentlcal. llence the surface area wilL remain

unchangeil, wh1le the total flow of nutrlent in unlt tine

w111 be rnultipLied. by Y slnee, by (5.31)t the nut¡'ient

flux ø 1s given by

ø=MCr

5.10 Nwrerlqal resulss uglnq \ as startirrg potnt.

Ffgr.:re 5.6 exhibits the çonvergence of the varior¡^s

lntegral curves wittr lnereasing 0. Starting at N where

6 = lBOo and. lntegrating numerically in the cLirectlon of

d.ecreasing 0, it will be impoesible to reach P slnce

curves vrhieh are an appreeiable d.istance apart near P are

anbitnarlly elose togetþen nean IT* The few atterryts at

sueh integr.atlons end.ed. Ín faih:rer either þecause the eurve

entered. the reglon A < Or vrhere integratlon eould not

continuer or'else because the curve d.evlateit lnwarcls in the

nam.er of Figr:re 4.2.

63.

Flgu:re 4.2 has been d.erlved. 1n Chapter l+ when corr-

slderlng the spherically syurrnetrLc distnlbutlons with

l,l = 0. For these dlstnlbutlons CO = 0 ard. the curvea

I = O and A = O coinelde, and $t has been shown that an

lntegnal- curve !'¡1th one polnt lnsid.e A = O tÌutns lnward.

aDd. neaehes the lsopyenal- C = O. For the (Vr t = oo)

distnibutlon conslelerec[ 1n the pnesent e]rapter, the cunveÊ

I = O and. A = 0 cotne extremely close to eadt othen near

Nr and. nu¡nerlcal lntegratlon stantlng at N can qulekly

leail to polnts lnsid.e I = 0. By (5..23), values of U'/UU

wfIl then take the cr:nve inwand, in a nanner siurilan to that

d.Lscussed. in the slmple case of Chapten 4.

Equatlon ( 2,1+)

refe¡.red. to statlonary

at the movlng slnk, aÉt

d.lstrlbutlon becones

Â,.

"Br cosC = 1 -äo

6l+.

Ch3gteq Þ. Ehç Rggt PrgflLe, for.,tÏre (V.t) PrtFtrlÞutlon6.1 The (v.t) Dlstnibutlon.

wherer âs before, B =Thie may be rnewritten

deecribes the (vr t) d.letrlbutlon

axee. Refe¡.necl to axes wlth orlgln

d.eflneiL ln section 2.3¡ the (Vrt)

oleBnerf" g{p * u-Bru"f" #],?^Ft 2^F¡r

(6.t )u/*,

a.- "-Br(1

- cos 0)("2nru"1," J:J-J& + erfc " - Itc=.-fie \ zñi ?ffi

= 1 _ ê .-Bn(1 - cos 0) ¡,, (6.2)I - 2?"

r¡¡hene F is lnd.epend.ent of A )

The (vrt = *) iÉstnlbutlon alread.¡r consid.erecl in

Chapten 5 has (cf, (¡"t))

c=1-Su-n"{t eos u).2. (6Õ)

Thr¡s F d.etermlnes the d.iffenence þetween the (Vrt) an¿

(vrt = *) iÉstrlbutlons¡6.2^ Pnonenties of the (v.t) Distnibution.

Vanlous nesuLts have been established 1n Chapten 5

for the (Vrt = *) d.lstnibutlon, It will be suggested, that

as the (vrt) ¿fstnlbutlon approaches the (vrt = -) d.ls-

trlbution wlth lncreaslng t', nany of these nesults rrllL

)

65.

also apply to the (Vrt) üstributlon.Beeause F lnvolvee cornplementary eruor functions,

the eorrespondlns results are not nead.iJ-y establlshed.. As

an append.lx to ttris chapter, 1n seetion 6.1¡, the followingresults relatiirg to the (Vrt) d.istribtrtlon are established.l

These results permlt tlre establishnent of the

follow1ng theorens in a lranner lclentieal r¡¡lth that usect lnChapten 5.

,Thçoqen ïsopycnal l-inee d.naw closer to the onigln withlncneaslng 0.

îheonem îhe curve f = O, vrhere I = Cr - kC¡ appnoaehes

closen to the origin lvi th lncneaeing 0 ,

Theor-EE Thecurves I=0 and. A=O haveconnonend.

points on the e-axls and. all other points on I - 0 J.1e

lneid.e A = 0¡

T_heore+ If f is ar¡y point in tlre region between the

curves I = O and. A = 0, ar\y integnal eurve thnough Tsatlsfying (¡.g) remains outsid.e ttre eurve I = O aB A

lncreases to 1BOo.

Thw it seems probable that root profiles fon the

(Vrt) d,lstrfbutlon wilJ- have simllar charactenistlcs to

those fon the (V, t = Ò") d.ietnibution. Nu¡nerieal sol-u-

tlor¡s obtaineil. confinm thls resemblancer

lco\\" /

66.

6.3 Ernclrica]- nesults.

Figure 6,-1 eho\¡¡e the groÌvth óf a noot profile wlth

lnoreasing tlne fon a glven set of parameter val-uee. Itcan be seen that as the proflle d.eveJ-ope fnom a snaIl semi-

circular rreeed.rr, 1t lnfttally groÌus outwa.rd. in all cllrect-

lons. Even.tuatly hor¡'¡even the entlre proflle moves d.own-

wand., approachirrg a constant shape whlch moves clovmwanil

wlth constant veloclty V. Thls eonstant pnofile 1s that

obtaineit fon the (Vrt = oo) distrtbutlon v'¡hich is the ease

D = 1O-7 of Flgr¡re 5.1t. At firet sÍght, a root prof13-e

which grows d.ownv¡ard. seems rrnaceeptable, Hotteven lf the

lorruer part of the proflle 1s consJ.d.ened. to cornespond. to

the active part of a gnorvlrrg nootr ârLd. the upper pant of

the pnoflle is ignored.r the results are reasonable. Infact, pJ-ant biologlsts assert that the active tlp of a

Blant noot moves d.ovirnwand.s thnough the soil at appnoximate-

Ly constant speetL.

Íhe results strov¡n fn Figrrre 6.1 rvere obtained. on the

eompanatlvely s1oür ïBùi 1620 co¡ry)utere and. sevenal hou¡rs of

computer ti¡ne wene used. in ob taining them. A fullerÍnvestlgatlon has not been canrled. out because of the

couputer tine requireiL. Howev€r'¡ Figrre 6.1 il-lustratesthe groìffth of tTre noot proflle with tine r It is intend.ed.

at a later stage to uge a faster conputer fon a ¡nore

extensive stud.y of ttre (vrt) dlstributLon.

o,z

Jrsds

-o,

- o,4

[rr o)

o.l

ArZ

-o,2

6,É!rß

-o,4

ooo

A- o,þo(il,e 'otþtol'¡t+

'67 "

bo , þo(tlc

( = l¡, /g4

@r

-o'2

-o'+

r¡lod¡

-o,S

o

o

-o

-o,+

-Orê

- Or3

-1,o

-t,3

o,L

sie&

t.tO! tu_ü t= Z,/O(

(V, t) clletrlbutlon.

e = 1O-or D = 1O-?r Y = 3.1O-s, tt = r.104 (cunvee ane

it¡rawn relatlve üo etatlonary axee thnough tnltial Btr¡k Bosttlon)Flpr¡¡re 6.1

f'o

68.

6.1+. Pnoof of the results (6.1+).

By (6.2)

trl =ê2Bnenfc n+Vt

-

+ enfc

whenee

òFòn =Fr =2Be

2Bnenfe r+Vt

?^lDt,

expzffi, Jffi(6.5)

However¡for N>0,enfc x <

¡-tÉe. '--*lî ,

[Carsla'i'r aniL Jaeger [4] P.483] whenee

Fr<

sv ( 6.5)

2Br=2e .G.

But

G¡

arll since

Gr>Thr¡s it follows from (6.7) that

Frr >

("#* -')"*[ -'" fi*lll <0.

tr¡ = 2"28?ln errc }ffi - fu ",'e[ - r' fultIf f

( 6.6)

( 6.2)

= - è- .* [- ( n+vt]e I * 1 2( n+vt l"* [- ( r+vt ìt !

lñí, -t 4Ðt t Jffi qÐt -¿ 4Dt r

-B+"åF=-å*di*å'o'

(6.s)

69.

Arso bv ( 6.2)

c = 1 -* u-rn(l - cos o)rF, (6.9)

and.

so that, by (6,6)t

cr = * u-nr(1 - cos o)[nt* + B(i - coe 0)] - F.l

SLnce

F¡<lncreasing r,

Crn ( O.

Next, by (6.9)i

where equallty occure onlY when

fhue

=Bsin0

=Bslnd(t-C).ancl

Frr >

i.êrr

co = LzB sín d u-Br(1 - cos o).g , o,

A =Oo ar¡d. 0=1800.

ao

¡

€/3\ = -B sin 6 cr <an\n /\ffhere agaln equallty occurs 1f ard. onl¡¡ 1f d = 0o

0 = 1BOo"

70'Chapten 7. PJoErg¡nmtne Methods. SunerJ.S!¡\gggggg¡,.

7.1 AlÁrebnaic Eouatlons.

It has been shou¡n that the bourdlng cunves T = O

and. A = O and the root proflle lnterseet the axls of

synmetry at the Beme polnts P and, N. The eoord.lnates

r of P and. N are d.eternineil by the algebnalc equation

(3.27, vlz.cr = kc¡ (Z.t )

vr¡ith d = Oo anil 1BOo, respeetlvely.

[he rad.íaI d.istance of a polnt on the bounding

eurve I = 0 1s for given 0 the solutlon of the algebna-

1c equatÍon

t-Cr-kc=0. (t.z¡Similanly, fon glven 0,

A =Cr'

d.eternines the coord.inate

A=0.

a2uo'to - l*Cz = O (l.t)

n of a point on the curve

Each of the equatlons (1,17, Q,27, (7 Õ) has been

solveit numerically uslng irTewtonÌs method.. An lnltíalestimate of the solution Tuas supplied. and. Newtont s method.

Tvas used. repeatedly r:nt11 successlve solutlons d.lffered. by

less than a prescrlbed. tolenance. If this d.id not happen

within a specifled. nr:¡riber of lterationsr the latest iter-ated. vah¡e was accepted.. On the rare occasion-s when cotr-

vergence d.id, not oecu-r, lt was usually easy to teI1

71.whether the result was a rrnean mLgstr on faulty ctue to a poor

stanting value"

trron certain combirrations of the paranetens, (7.3)

had. two eolutions, one of whtch occumed. in the region

where C ls negatlve. It was thus neceasary tô test the

value of C to eneure that thÊ corneet eoLution had. been

obtal¡led. If the lncorceet solution r_ uras obtalned,

a new starting vaLue of r_ + 2(no - r_)¡ where re ruas

the nadial d.ietance to the curve C = 0, always provecl

effectlve.Successive polnte on elther bound.tng curve were

found. by changtng the values of d step by step and. using

the vaLue r obtalneët at the previous 0 value aB stant-ing va1ue for the new poLnt.

Ar¡ alternatlve fonmulation ln Canteslan coord.lnates

(xrf) nas been glven in Seetioî 3r5, Algebralc equations

1n x and. C slnllar to (7,1) anit (l,l) are read.ily ob-

talned. and. may be solved. in a s1n1lan naruler fon one eoord.-

lnate keeplng the othen flxed, usir,g Newtont s method..

In faet, nargr of the curves 1n Figr¡res 5.1 - 5.5 r¡ene obtainecL

1n this wây.

7.2 DifferentiaL Eouations.

In Chapter 3r tt has been ehown that the prescrlbed,

bou¡rd.any cond.itlon lead.s to the first ond,en d.iffe¡rentialeguations (l.A¡ , (3.9), (5.to¡ , (3.1'l), Nrrmerleat solu-tlons of these d.iffenential. equatlons have been obtainecl

72.

using the following forgth ord.en Runge Kutta J.ntegratlon

procedrme ([6], pr87)¡ clescribeÖ here fon r¡se Ïvith (*r0)

coord.inates.

T.,et(Rro)r(n+dÞro+do)besuccessivepointson the eurve

åä = r(r¡o).

Then

dÞ, = å tnr + 2lcr + zrtz + ka)r

where

ks = Eo r(nro),(n * Luo, o + åd@),

(n * åku o + åd@)'

ks =60 r(n+kz r@+ do).

If the nunerical integratlon 1s started at the

polnt P, the value of ke in the first integration cycle

is ze?o, (From section 3,2¡ t'tre root proflle is horlzonta].

at lts extreme polnts, Êo *ä = 0 at P.) lhe point A

at which kl 1s eomputed. will now be at the same rad'lal

d.lstance from the origln as P. rf the valrre of åä on

the bound.ing curve A = O is negative near P, tIæ point

a will be outsld.e a = o. This w111 lead. to a complex

value of k1 and. the lntegratlon pnoced.r.¡re cannot be

contlnued.

From the (vrt = *) d'istrlbution, lheoreu¡ 6

establlshee that fon A = O¡d.rcl0

is negative near P íf

f

=tof60k1

ka

V < 2kD. For such sets of the parameterst proflle curveg

only have k1 neal. tfeunve A = O.

73,

cafirot be oþtalned. by lntegration fno¡n Pr .{¡lproxinate

pnoflles ïvere obtalned. by stantlng at a polnt near P, iustlrwiêe the bound.lng curve A = Oç Ae Theonen 9 establlshee

the aclJacent lntegral eunves ônaw cJ.osen together uith in-cneaelng 0, the ennor 1n the etantlng value wlll- d.ecnease

ag 0 lncneases,

Fon lntegration in the coord.lnates (*rC) slmllard.lfflcultles arlse slnee lntegratlon startlng at P w111

# , o near P on the bound.tng

a

By cLefinltlon,

erf(x) =

enfc(x) =

It is easlly venlffect that

erfc(-x) =2- enfc(x). (7,6)

llhe beharrlour of enfc(x) with lnereaelng x ls shown inthe tabLe below.

It 0 1 ,17 1.82 2133 2.75

-+.2-u-dtr (z.¿r)

-t"d.t = 1 - enf(x).

Q,s)

enfc(x) 1 10-1 1O-3 10-e 10-4

x 3 .1 2 3 .46 3 ,77 4. 05 4.32

enfc(x) 1 o-ó 10-6 1o'? 1o-8 1 o-e

74.

tr'or the lnvestlgation of the (V = 0rt) and (Vrt)

d.lstributionsr ârr approximation to the functlon erfc(x)

which Ís accurate to machfne accunacy (8 d.ecimal figures)

ïvas requlreit . In vlew of (7 .6) , an approxirnatl on v¡as only

need.ecL for O < x ( oo¡ A number of approxirnatlons to

erf(x) are given 1n the literature. However these ane of

little value in conputirrg erfc(x) for large values of xt

as the id.entlty

enfc(x) =1 rerf(x)leacls to the loss of sevenal stgniflcant figures when

erf(x) 1s nearly 1.

Erfc(x) has the asynptotie expansion (Carslaw ar¡d.

Jaegen [4],P.481+)

1-*î.Mn -U#2 +...).(t.l)

As the erron in using a finlte nuriber of terms in (7,7) lsless than the numerical value of the first negleeted. term,

1t can be shown that (7.7) will approximate erfc(x) to

B slgnlficant figunes fon x à 4.2. Accord.ingly an

approprlate numben of terns of (1.7) were used. to appnoxirn-

ate erfc(x) for this range. The nurnber of terms nequlred

d.eereases as x increases; it 1s 19 when x = 4r2, and. Iwhen x = 610.

For 0 < x < 4.2, a nr¡nber of Taylor series elpans-

ions ïyere used- to approxlr¡ate erfc(x). lt is read.lly

venlfled. that the derivatives of erfe(x) satisfy the

75.

recurrence relatl0nr(n)qx) = -2x p(n-t )(*) - 2(n - z)p(n-2)(*),

vuhere f(x) = enfc (*). lhusr âs many ooeffleients of the

Taylor series as requlred. can be generated. fnom linown

value s of n(x) and. Ft (x).The range 0 < x < 4.2 wacr divided. into 21 equal

intenvals of length O.2¡ and values of e(x) and Ft(x)at the centre of each range vrere supplled.. Va1ues oferfc(x) were then eomputed. using at most 10 terne of the

appropnlate Taylor. series.fn terrns of computer storage, the nr¡mber of 4Z

conetants is und.esirably lange. Furttrer, 1n gçnena1,

a TayJ-or expanslon provld.es a poon approxlnating fr¡nctionbecause of the raplil. gr.owth of the erron with lnereasing

d.letance fnom tlre point of expanelon. In this ease,

howevenr the Taylor expansion pnovid.es a relativery quicJr

and. accurate means of approxirnating to I eigrr.lficant flgunesa funetlon whose value d.ecreases fnom 1r to 1O-8 over the

range of appnoximatf on.

sumrnlng upr erfc(x) was appz'oximated. nunenfcarlyby the follovring fonmulae I

x Þ l*.2 asym¡rtotie erçanslon (7.7),o < x < 4.2 Taylor serles e:q)anslons about

0.1¡ O.3t Q.5t r..r I1.1.

x<O enfe(-x) =2- enfc(x).In theony, these appnoxlrnatlons are aecu¡rate to I

76.

elgnlflcant figures; howeven, tn:neatlon ernorg somewhat

ned.uce the lnaccuraey. Tho langest ernor found ln 1oo

values, sanpled. over the nange -r3 < x <

the elghth eignlfteant flgure.7 .b Root area and. nutrient flo¡u.

The root surface obtafneil by t]re method. of thle

thesie ls surfaee of revol-utlon of the noot profile about

the axls of syrnnetny of the noot. AD estimate of ltssurface area has þeen obtained. by assumfng that the rootprofile is conposeil of stralght llne segmentsr If A ard.

R are two ad.Jaeent polnts on tJre proflle¡ the segment AR

then contributes dA = znIR to the total area, whene

1, = QR¡ arrcl f fs the nean d.lstance of the polnts I arrd.

R fron the axls of symmetny.

The flow of nutnlent in unit tlme over ô¿ wltl be

appnoxlmately dts = Uîd¿, (cf. section 5rg), where -C isthe nean of the nutnient concentnatlons at A and. R.

fhe total area and. total nutrient flow 1n unit tlrne

ean then be estimated. by surnning dA and. dF. The errorin such estfunates can be contnolled. by adJustirrg the sBac-

ing between the suceessive polnts used.,

7.5 Numerical fnaccu:raer¡.

In ad.clf tion to rrnormalrr sources of nu¡nerlcal ln-accuracy 1n obtainlng nesults cjn a computer, the foll_owlng

inaccunaeleE pecullan to the partietrlan problen vrere

observed.¡

Tl-a) . Floating polnt nr¡mbers on

the IBIII 1620 can l1e ar¡1rurhere betweer¡, lOes and. 1o-ee.

Hovrever, e22a c loee¡ and appfoximation to enfc(x) forlarge x ls given by ( 7.7) which lnvolves e-xz. Thus,

erfc(x) ls too small to be represented. 1n the nachine 1f

x > Jffi o 15, Erçnessions such as (6.1) contain the

term

eBr erfc r Lg! .2JÑ

T\Drlle the erfc facton may be very emallr the erponentiaL

facton nay be very large Bo that their prod.uct is not

negllglble. It was therefone necessary to pnovJ.d.e a

sfngLe appnoximation for such terms for lange values of

the angunent of enfe(x).b) eæ. For each of the nutnlent dis-

tributlone, C 1s computed fron an e4pression of tJle fonm

C=1-Erwhene E 1s the d.epletlon. ff C << 1¡ 1r€. E + 1,

there oceurs a lose of slgnlflcant flgures 1n corçutlr¡g

1 - Et This inaceuracy ie evid.ent in the flrst l1ne of

Table 3t where values of C as low as 0,0002 only justlfy

2 significant figures in the value for the total nutnient

flow.

c) rnitial- values of do/ao It has been sho'ì,un thata

the appropnfate cllfferentlal equatlon fon the d.eter¡nination

of the noot proflle ls

78.

ðndt0

-tsk t<c,ø Coo - lrâ CPIta

- ÉcF + -lrCy'A

At the enô poínt P¡ Cr = kC. Near P¡ C¡ s kC arld.

loes of accuracy oecurs ln th€ eornputatlon of the lntttalvarræs of Uo/Uu, Fotrtìrnatel.]lr ârrlr d.lsplaeement cf the

tnue root pnofile due to Euch errorg ned.ucee wlth l¡lcreaE-

ing 0 , aÊ ehoTvtÌ ¡rnerrl ously.

ölau

-iFl.-klq

:1

79.Chapter 8. Extensionsr Conclusion.

On the basls of a nu¡iben of simple assuu¡ltions, a

rnattrematical moclel of tTre growth of plant roots has been

fonmulated.¿ The inveetlgatlon of thls nod.el ln ltspresent fonm¡ as d.iscuseeil ln tJre preced.lng chapters, sugg-

ests that reasonable root profllee can þe obtalned. from it.One baeic assutption of the mod-el is tltat tlre

coefficient k ln the boundary cond.itlon

òcdIT =kC

is eonstant at all points of tÏre root srlrface. Ilovr¡ever,

Ít has been observed erperlmentally that only the 1ov¡en

part of the root surface is active ln tlre uptake of nutrlent¡The mocle1 eould. be rnod.ified. to allow fon thls phenomenon by

nalilng, k an assumed functlon of (t or of 0.

As the d.iffusion eguation is linear" in C, solu-tions eorrespor:dlng to separate poÍnt sir¡l;s nay be ad.ded..

Thus, a second. possible mod.lfication of 'che inoclel is to

represent the aetion of ttre root þy a number of noving

sinlcsr of varyirrg strength¡ al1 nrovS-ng i¡¡ith the same

velocity along the z-axic¡r

Empiricalljl¡ the ptant blologist observes a

bewilclering mass of fine root hairs, the outlÍne of whieh

eonstitutes th.e rrroot halr envelope¡'. rt is hoped. that as

empi:rical d.ata becomes available, the mathematicar mod.el- can

be mod.ifiecl to give a neasonable representation of thisbiological system.

80.

REF'ER,ENCES

1 . Arrilerssen R rS . âDd. Railok ü.R.M.

On the d.lffuslon into gnowing noots. (Papen

presented. 1n May 1961+ to Ar¡et. Math. Soc.).

2. Barley K.P.

The plant root as a eink fon water and. ¡tutrleil.ts¡

(Paper presented. in May 1961+ to Ar¡st. }rÍath. Socr).

3, Byrd. P.F. and. Friednan M.Ðr

I{a¡rdbook of elliptlc lntegrals for &rgineens and.

Ptryelclets. Sprlnger-Ven1age Berlln 1951+.

4. Canslaw, H.S. and. JaegêF JrCo

ConiLuetlon of Heat in so]-fd.s. 2nd. ed..

ClareniLon Press, Oxforct 1959.

5. Ince E.T,.

Ond.inany Dlfferential Equatl ons ¡

Longnan Green and. Co., T-,ond.on 1927.

6, Mod.ern Conputlng MethoiLs. 2nd. ed.. H.M.S.O. T,ond.on 1961 .

7. Passloura J.B.

A nathenatlcal mod.eJ. fon the uptake of ions f¡or¿ the

so1l solution. Plant and. Soll lg 1961 P.225-238.