Atmospheric Diffusion Shown on a Distance-Neighbour Graph ...eyink/Turbulence/classics/Richardson26.pdf · Richardson and D. Proctor), 1OI1 ~rn.~ sec:-l when the mean wind is taken

Atmospheric Diffusion Shown on a Distance-Neighbour Graph

Lewis F Richardson

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical andPhysical Character Vol 110 No 756 (Apr 1 1926) pp 709-737

Stable URL

httplinksjstororgsicisici=0950-12072819260401291103A7563C7093AADSOAD3E20CO3B2-0

Proceedings of the Royal Society of London Series A Containing Papers of a Mathematical and Physical Character iscurrently published by The Royal Society

Your use of the JSTOR archive indicates your acceptance of JSTORs Terms and Conditions of Use available athttpwwwjstororgabouttermshtml JSTORs Terms and Conditions of Use provides in part that unless you have obtainedprior permission you may not download an entire issue of a journal or multiple copies of articles and you may use content inthe JSTOR archive only for your personal non-commercial use

Please contact the publisher regarding any further use of this work Publisher contact information may be obtained athttpwwwjstororgjournalsrslhtml

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printedpage of such transmission

The JSTOR Archive is a trusted digital repository providing for long-term preservation and access to leading academicjournals and scholarly literature from around the world The Archive is supported by libraries scholarly societies publishersand foundations It is an initiative of JSTOR a not-for-profit organization with a mission to help the scholarly community takeadvantage of advances in technology For more information regarding JSTOR please contact supportjstororg

httpwwwjstororgSun Feb 24 085734 2008

Atm~osjhericDifztsion showiz on a Distance-Neighboz1r Graph

By Lewxs F RIOHARDSON

(Communicatedby Sir Gilbert Walker PR8-Received Novelnber 7 1925)

If the diffusivity K of a substance whose mass per volume of atmospfiore is Y be defvled by an equation of Picks type

a- a a~~tdiasl-as~- $-amp ax -- G k zz4+a$ ay i~ 6 X a y G a y i a z (1)

cc 9 z t being Cartesian co-ordinates and time 7z 5w being the components of mean velocity then the measured values of K have been found to be 02 cm2 see- in capillary tubes (Kaye and Labys Tables) lo5 see-f when gusts are smootlled out of the mean wind (Akerblom G 1Taylor IIwselberg etc) lo8 cm2 set-1 when the means extend over a time compaeablc with 4hours (L 3 Richardson and D Proctor) 1OI1 ~ r n ~ when the mean wind sec-l is taken to be the general circulation characteristic of the lati-tude (Defant) Thus the so-called constant K varies in a ratio of 2 to a billion The present paper records an attempt to comprehend all this range of diffusivity io one coherent sclileme

Lest the method which I shall adopt should strike the reader as queer and roundabout 1wish to justify it by showing ampst why some knotvrl s~ietltods are in difficulties

$12 Does the Brind possess a Velocity

This question a t first sight foolish improves on acquaintance A velocity is defined for example in Lanibs Dynamics to this effect Let Aa be the distance in tile 3 direction passed over in a time At then the x-cornpcnent of velocity is the limit of Az]At as At-gt 0 But for s n air particle it is not 0amp1Vio11s that ArAt attains a limit as At -gt 0

We may really have to clescribr the position z of an air pizdiclo by somtthintg rather lilcc iVeierstmss7s Rmction of whicla P Iileirr gives an entertaining description ( Anwendung der Differential and Integralr~chnungauf Geolazetsie Leipzig B G Teubner 1902) say

For references La publications set the table in $42

i 0 L I Richardson

shere li is inclependent of E aucl 9 is a positive ~nteger This gives a defin~te 1)osition s to the air particle because the series of co-efficients +B +- -k

-t lt onverges and makes s n continuous function of t but i t gives no limit to AzlAt because aq the se~im ITOCCC~S tuhe elementary waves ivhil~b becoming sl~orter beconic also deeper It is not suggested that these particubl periods and amplitudes iiliich Kle4n Lgthose for illustration have anything to

do wit11 Ihr wind A general and benntiEul thcory of Diffusion by Continuous Novements

had been given by C J TayZos It is aspressed in terms of velocity AbE~ough this theory of Taylors is available yet 1[ think i t will be a useful

adventtarc to try now to make a tlworr of diflusion without assuming that d t A t has L limit

$13 The Lay~angzanSpes$cubion adopted Xotntiogz fofo Means Time Rule qf ca Idenz

In view oi the foregoing conqiderdllonh irk 11s not think of velocity but only of wrio~ls hyphenated velocities such 24 ihe one-minute-velocity or the six- hours-velocity ihe words sttached by ihe hyplrcn indicating the value of At

Tire position of a particle is however a contimlous function of time The Lagrangian specification of fluid motion is applicable A particle at the point ab c ) a t time zcro is ay z ) at time t

Fol2owing Taylor a square bmclcet et 1 1vi14 be used to denote n mean value so tltat [A] is the mean of any quantity A The portion of space-time over which Ihc mean is taken will Le specified as occa~ion arises

Even if (r-a)has no derivaijvc wit11 respect to t yet [(s - a)] may have such a derivative Por instance this happens with thc Weierstrassian lunction mneampion~d above if the mean is 1~21~11 a time Let us thssume thatover [fs-a)]ha9 a derivative when taken ob-er either a space or a time for there is ~ i oevidence t o the c0ntr3~ry

fj 14 d i$en~chjor Nakural ilr2ealz V ~ l u e s

it first sight a good may of sreclfying diffusion would be to take the dis- pl~cornentsx -ay - b z -- c oi an air particle and to form meails of their powers lac1 prod~~etssuch as [z- a][(x- a)] [(r- 0 ) (y - h)] and the iike

Bat obstrvniion shows that th-e numerical values would depecd entirely upia ilom large a volume was included in the mcan To sec this inlagine that

Proc Iond Math Soc Ser 2 vol 20 Part 3 (1920)

Atmospheric D ~ ~ Z C S ~ O ~ Z Grapho n n Distnncc-Neig~bou~ 711

we could introduce just two nlolecules of acetylene and trace their wanderingk If initially they are 10-Qm apart i t seems likely judging by what is known ~ b o ~ i tnlolecular diffusion and by what one sees of the motion of smoke that after travelling for one second they would still be within 10 cnl of one another If howev~r the two nlolecules are initially lo f5 cm apart they may be caught in two gusts nloviitg in mtl~er different azinluths so that after one second their ceparatiu~ may have altered by several metres

Thus i t appears that if y is a co-ordinate directed lrorizontally a t right angles to the mean-wind so that [(y - b)]= 0 then the value obtained after a fixed

time for [(y - b)] would increase with the range of distance norlnal to the mind over ~vliich tho niearl $-as taken

Is there any type of mean that forms a natural standard We might try- (i) A mean over a volume so large that its exact size did not matter a limit

to [(y - b)21being attained TI-lid TI-ill not do because Defants researches shorn that no limit is attained within the volunie of the atlnosphere

(ii) A mean taken over a definite set of molecules Suppose that we were to

let loose a sphere 001 em in diameter of acetylene which has much the same densiry as air The sphere contains about 1013 molecules For the first fcw hundreclths of a second its rate of diflusioli will be the molecular one Ilt= 02 then micro-turbulence 15-ill spread i t less slowly then after a few seconds part may get caught in one of the gusts such as are shown by a pressure-tube anemometer wllilc another part may remain in a lull so that i t is torn asunder and gnsts scatter it Ii beiiig 10 Next sq~lalls of several minutes duration separate i t more rpidly Its rate of diffusion is now measured by K = lo8 Then one part gets into a cyclone and anotller remains behind in an ailti- cyclone and its rate of diffusion is measured by Defants value Ilt= 10 Finally i t is fairly uniformly spread throughout the earths atmosphere a t the rate of about one nlolecule of acetylene for every cnbe of surface air 70 metres in the edge

This diffusing dot is in a sense a natural standard In the theory of the tiiffusion of heat (see for cxarnple E W EIobson Encyk llath Wiss vol 4 1) 187)sometlling rather likc this is found to be useful h small dot of heat ih iinaginecl to spread out as time proceeds into an unbounded medium Tllis ciistribution of heat in space-time is talren as an clement like the point-charge in electrostatics from which more complicated distributions can be built up Chn we do likewise for cliffusion in the atmosphere Imagine a t an instant a gradient of concentration of acetylene in the atmosphere over an area measuring 100 km x 100 km Let earb cub of ccm edge begill bepararely to sprcatl

VOL cx-A 3 c

out in llte mmnner ilebcribccr ~love a rd ie us superpow tho separ~te spreadir~gs in tho h o p of fillding the flux of rlrass donn llic prridiertP c J f cor~c~cztra~ioi To tlo t h i ~we T~~OUJCInaturally consit lcr the change in a ahort lirrlt At 01 bay

1100 qecond But in this aliort tiwe each dot spreads n it11 J ~ ~ I P - I I I R L difil~hi-vity 02 cn~ei-l 80 any reuNs deducccl from the superposition oF the egects of the dots will corrcxspontl to rnolcclnlar cliiYtisii~itv and mill ignoxl fhp effects of eddies Tilib pictzlrc is false t o Natulc 80 1-errr~t coiiclutie rllitt

iiz the a~nosphere n sprendily do6 will $tot serve c t ~ wiich gerzercilail t ie i i~e l t f t -o~n disriDutions can be built rcp

$ 2 TIVE O F ~~EIU h1011121WINC ~ I F J ~ ~ C U L T I E YWIIICIT AVC~IIIJT H E 21111)

FVIiICII BrICIJT T14sI i )STANZ20E OE (PBJERSTB~)

The fundz-rncnsal idea of this p a p r i s t l ~ a ltbe m e ol digusion increases i i 4 h ihe diciarics apayt 40 state this carclully ieL 11s r e ~ e r tio the tv+oxn~~lecueof

dtceiylcnc 1tt loosc a t 1 -O tt the poiais (a b c) (a b c) At time t

let their po-ition be (2 y z2j (x2y2z2) The zco~riponent of thcjr separation 1s initially -- u and hecomes ti - x af time i Sov lei the reiese of n

of particles at t ~e sar~ic pidnbb be repeated inliy times iri stlc~ccssionaird

let [ 1 denote a olcnn talien oIver these suca~~s~ ivc Cow~xiclerthepairs

mean square sfthe cleviation oI (6- a) iron] its meal a l tnltl tiiczl i

[(zr- x2- --- I) -= [j]bay

BupposeJfor examplc that [x -- z is a liiloxneirc lhcn gusts svhicl~ may bc seen on a lalic or on a vorntieltl as s(ci~esof ~ ~ r f B r c iiurlacc a fraction of a kilo-metro 1015 1-ould affect icdiviilual incm7oer of thc ]air I zzseparately a d 8 0

wulrl Lend to irlcrease [ j J We shonld ge t i l e a oergamp ~f fec t ol sixcb gxsts if we proloipcl the time of averaging indefinitely Tlre time of averaging ninst nol be confn-eii ~s-ith t tlie time of flight It i au i ( ~-a~ag~2 0 have a p n i ~of

markcccl ~nulecules For iT inbtead iic5 eon~ideretl iuoiect~leb released one at a

ti lhen their mean square tlevjfit~oii roizi their xilean po5ition at t namely [(z - [G])-)]~ would depend on 2rger and larger eddies ils iEic time inciuded in ihe average [ ] IVI~ increasctl BO fhak110 Linzit TO the average svould bc attained lrntil ~ ~ e i o n i c But ivilcn molecules are released in p~ir6change were included this is not so For if a c j - c i o ~ I)ai+e oier tllc cit~ictso thzt the rvinci thnngees its dircctiori ibrouglz oil( or two right angles these changes will occur nearly simxltdnoo~i-ly at t1c tvo stations one kilometre apart so that tbcy 1~111not have rnurh effect on indivi~i~lalvalics of (z - x) nor on [jj1re liave a t last found in Cj1 a ma wl~icltattains a Iimit 2s the t h e oS a velaging is prolonged

Atmospheric Difftcsioi~07 a Distauzce-Nciyl~bourGruph 713

indefinitely and yet only brings in the effects of eddies comparable in diameter with [xl- x] or less

Direct observations of [j]haw not been made so far as I know but there is s mass of published evidence about turbulence which strongly suggests that [ j7 t is independent of t i l t is neither too small nor too large and that [ j ] t increases with the distance between the starting points

5 31 IlzLroduction

The failure of the dispersal of a point-charge to serve as a mathe-matical element from which the dispersal of an extended system may be built up7 appears to be intimately connected mith t21c fact that in the atmosphere the dispersal goes on in patches That is to say a small dense cluster of marked molecules represented by the dot in fig I which by molecula~ diff~rsion alone would spread through the successive spherical cluster shown in Ggs 2 and 3

actually seldom passes through the large spherical stage 3 because i t is first sheared into two detached clusters as suggestecl in fig 4 These are carried far from one another and are likely to be again torn into smaller pieces as fig 5 Meanwhile each of the torn parts is gradually spreading by molecular diffusion These diagrams are of course niercly illustrative fictions

As a preliminary to desc~ibing the stzccession of changes we must find out how to describe thc distribi~iioli a t a single iustagt The problem is rattler like tha t of finding sorne simple specifiaatioa of the rxtelit to which the populatio~t is divided bctncc~l cities to11-11s villages and isolated houses without making a

map for in the atmosyhere M map of ail the eddies ~ ~ ~ o u l dbe too bewilderiogly complicatctl MTe want something $hat gives US a genela1 measure of the spread of the n~olecnles thus serving the purpose of the standard deviatior oi the 1v11ole a g g ~ ~ t c of marked rnolecnles from its mean a~lci yet a t the same time informs us about he internal details of the cluster Both purposes will be served as will be sliown by the folloiving rrethori lvhich is not intender as L prmdical ob~ervntion but as n maihe1nat1c11 specifjcatiori

$32 DcJit~blonof the Statistic Q tlzc fecr)t X U P ~ ~ ~ P I per L~ngihojAT~ighi)otl)s

The air is supposed to contair a of lnarkcd mole cult^^^large ~ ~ u r n b e r They might foi rxample br acetylene For simplicity let ris confine acteniion to distribution of points on a stlaight line Take any mwrlted rnoiecde which for refererlre we will call 8 ITiii as origin divide the line by sectior~s a t positive and negitive integrdl n~ultjples of a uiiit h thus forming cells each of lcngth 11 Count the nurlrher of marke~l inolecules in cacah of these cells When a nlolccule is exactly on the partition betm-een a pain

of cells hag ol i t is attributed to each cell Let h -4 k denote tb r number in the cell betmeen 1 = 1~hand I ---- + 1)h wllere I is the distal~cc from B lnertsljred in thc positive seno and n is a n integer Let there be N ~nnrlcecl molecules altogether A riiolecule xnig11t conceivably be considered to bl its own neighbot~r a t zero distance but we do not innlce this coaventioil and therefore the snn of thc number ir Pile cells is W - 1 Kext repeat the performance with the origir~ at each one of the other marked molecules H C Iin tun] Then form thc mean

I n iliis ivay ve obtain a set of ampQhQociunutities Wzh 3 h wllich itre nunlbers of rnarked n~oiecules per length classiiic-dnccorcii~gto their distances

from other rriolcenlcs l~csedistjlr~coslying in the FRII~C 0 to h h to 2h 2h to 3h ant1 so 02 Xeesi- it il-ill be t~(cill o draw a (tlingra~nin n~hich tlte ordiuate is Q o r the rangc of abscism extentling f r ~ m1 = nlll io 1 =( 5 I) f2 Vc txay ncnv ctrop the suiliixes and regard Q a a f i l t ~ c t i ~ i ~ of 1 This (2 Q) dimgrarjl 11~snampry ii-~terest-ingprcpe~i-i~

I

To illustrate tbe definition the lower part of fig 6 was obtained by making the prescribed counts on the linear cluster of 7 molecules niarked above The

gtucceseive distances between the nioleciiles are in tenths 2 4 6 8 10 12 and h ia unity

$ 33 Note on the Stey 1 ~

This elenlent of length should be chosen so that in the average Q tlzc eleine~it shall contain a considerable nwnber say a t least 100marked molecules over the values of Z where they are most crowded Otherwise random errors cf sampling might become apparent Even ii this is done for most of the diagram there may be other ranges of 1 where niarlcecl molecules are so scarce tliat sampling errors might become noticeable On the other hand if h were made too large the s5teps in the diagram might become too wide whereas we want the stairs to look like a curve These coniprornises are perfectly faniiliar in statistical work and are inevitable Although men- tioned here for completeness they are really of no importance as we may emily have a billion inolecules in the cluster So that in future me shall replace the stairs by a curve drawn througl~ the centre of each step In other words the mean number of neighbours per length like the density or the concentration attains a qnasi-limit when the element of space has a magni- tude lying in a certain range

$34 The (1 Q ) Diagram is Sym~netricnlabout the Q Axis

For the clistarice between every pair of molecules is counted twice as negative from one end as positive from the other Therefore if Q be expanded in powers of I only even powers can occur

5 35 As L)iflcsion y~oceeds the Area eurosed betweetz thc I B ~ i sarod the ( I Q j Curve re~naitzsconstant

For this area when ixpressed ir linit of I ~ i t lQ is ~inlplyone less than tlia

whole number W of lni~rlzeil -iioleuaic iiis is easiiy yrowci frcm the dclini-

tion of Q isncI by 11ypoihcsis IS rc-mains constant This property of tllc gri$i suggests t i~at CJ xmst satisfy s difTewuti~l

equation of the type

()--- 2 some furictiolz of 1 which attains --at

- $ 7 ) a limit as 2 -+ n eqml to that

1 abtaintic as Z f - - r~ dI

for iE eo

Tor if the distance between tlie c~xtren~c~~ioleculesti t tllc opposite euds of the cluster be IL then I ) i s zexo for all vahlrs of j l i greater than 1 L 1 arid amp is finite when Z ==amp 41 Thri hi extreme ~vjdth of the 1 Q curare

is twice the extreme diarnettr of tLc cluster llllr relatiom heiwen thc s t a ~ l d i ~ ~

~Ieviatiohs will be cliscnssed in $68

5 31 Ths Changiizy porn^ o j thc ( I Q) k)ictgrcriiamp as Dqft~sio)~proceetla

It is eviderlt irom the foregoing that if ihere is oniy olie linear cluster nud i t spreads along its line the ( I Q) graph must spreacl along bc I axis And as the area under it must be c0~lit211t its 11ICdll height in tle QI)-directiol must decrease

Let 11s consitle~ ar1othcr very simplc case (snalogoub to tllc melting of a

crystal) Suppose that initially-ainrkcc1 molecnlcs are equally spaceti intervals of one centimetre a l along tllc Iirre without hound in either direction $at will the (1 Q) graph Toolc iikc ) S o nlolccule vill Jlaw i neighbour

nearer than a centimetre SO Q is zero for 0 lt lt I At I == I em neighbonrs are indcfinitely ronimon and Q i infinite Again

there arc no ncighboars in the r n u q 1 lt 1 lt 2 anti gto or The graph consists of Lserlc of infinities oE Q at 1 1(1 2 3 4 r~n)with amp - O ewry~vhereeluc

No157suppose tlmt difukion fake place (an fro~ilFlickqe find tlro ( I ~ ~ E I I ~ P S equation Not from it alone -I-on Fictii eclnation i~ a sfwteruellt about tile gradients o i a ccirtiniioi~k rlnci run ol poition ~ i ~ l i e r r ~ ~ s vc hoe oniy pnrticlr9

Nor adclcd I)rcrmber 5

- ------- -

Rtmosphelhic Uzflusion on a Distance -ilreigh)o~~r 7 17Graph

widely selmrated Tf Ficlr5 equation is to he applied to this qpecial example i t w u l d have to be by the aid of an additioilal l~ypothesis derived from the theory of probability Instead I get the following from memories of snow-flalies falling of the Bro~~~nirnl inolion under a microzcope or of foam circulating on a millpond ITith these in mind i t is evident that the perfect regu- larity of the arrangcrnerit of marked molecules will soon be a little disturbed Molecules will seqi~ire iieig1llbours a little nearer and a little farther a m y than tho exact 1 2 3 4 cm and neighbours a t these distatlces will no longer be infinitely conunon Lliat is to say the infinities of amp mill be softened down into peaks with spreading bases In an early btage the bases will not join thcre will still be no neighboms a t distances sitcll as i I+ 21 cms In this stage the area enclosed betwcen each peak and the 1 axis must remain constant The forin of the intermediate curve shown in fig 7 is intended merely to suggest that flattening proceeds more rapidly is 1 is greater other-wise the curve is a guess Later (5 84 sect 67) some other cases will hhe discussed q~ant~itatively

SAMPLES OF DiSTRl8UTIONS EXTENDING BOTIiUAIS W I T H O U T C N C

O N A L INE

FINAL - -

Q T THE MEAN NUMBER O F Q

N E l G i i B O U R S PER LENGTH

AS A FUNCT iON O F

T H E DISTANCE APART I

5 38 T12Pinal State nfter Tho~oughD(8uskon

From tlle foregoing eve should expect the final state to be represented by s straight line parallel to the I axis This expectation is confirmed by con-siderations of probability For there seerns no reason why the number Q of nlarkect molecules per leilgth should have any depencie~lee upon the distance to any molecule wlzen the diqtribution is purely raiidonl

5 39 A P(ilzweof Co~~ce7ztrcctio7tas a Desu tjitirc Pden

la the preceding special example if we are to spcak of concentration at all me must talre a long ele~c~cnt of length say 1000 cns in older to 11ac-e a good

many i-narlrccl molecuici in it Then all that can be said about tile conceu tration is that it ma5 initidlly jnclepcndexlt of position and remained so always lceorciing to this iiev nothing happet~ed Hot- diflere~zt from the lively process shown on the (7 Q) diagram

The new theory 11-hich is intended l o apply to both eddy- aiici rnclc-xTar diflnsion ouglit to bc consistent with lTiuls ecluation in the special case of no eddies Let 11s no- explore this co17nection

lCTe have seen that tlre itlea of the coiicentratio~i o ~liarlreri niolecules given a9 a unction of position has in OPC Inbta11ee hignalljr Failed to deacrbc ihat xvliich we T-ish to cli5cnss Ficks pcpralion being I~ased on the jdca oi coneelltration as n Fnnction of pc~iliuo liir f~biledl-lwrr a h J3lre~vhere~ T - C

ha11 find both xicry uieRl1

In order to bring Pick equttion ixto our tlicovy re must suppost that the cnncentrxtjoi v definec1 to be the ~~ilv~beo ~nxrleil molecules per length is a contin-cous function of 17 jgtossesilng dori~a~tivesiiza2vax2

This supposition is a little ariii-icial But itt seenis Iiliely that AvlAz really nttains with sufFicient accuracy a yilasf-limit when IC iii ne id i~ rtoo large nor hcbo srnx Piclis ecp~ation is ihea

Kcxt amp must LC redeliileci jn terms of v Fji clefi~iiiioliin tcrrni of ccrz-

tinuous eonrentrtion can bc nudc lo ngicrec ~ r i l i i that iiz icrms of yariiclcb ~ x c e p tas rcgqrds neigl2bours a c7Srse r s or cloeer than tie closcst pair O F par-ticles As a rnrniutJer of thiq ofieu r~rlirnport~iit discrepnnty the n e v Snnrtion will be denoted hy sniall (i For i~~siaice if each of (he articles i l l fig 7 vere replaced t y a s l i l d l l (Lot of C O ~ I ~ ~libbta~ice y ~ron ldhave an~ L U O ~ I S Cwn

tnfiility a t 1 -- 0 where amp i~ zr lo

$43 T h e DeJiniiiol~ijS q upplicable zrthcn ik D$jsilg ~Sz~bslanceis 07tlinerous ~ o tMoleczilar

Te take any point x on the h e at wd~ich tlc conccuiratior is v a fullctioii of 3 and tims only P t will solletlmes be written ( ~ c 1) TJre proceed frorrl j i

J a further distance 1 to 3 -$ 6 Let lhe concentrntioli a t r 4-1 he vl Rera

I tmospheric Uife~sionon a Distance-Nezghbozw G+apZi 7 19

id i k e v must be regarded as a function of -three independent vaciaules - Z and t and may accordingly be written (zI t ) The analogue of

4 in the previous rlijfinition is here simply vl regarded as a function of I ~ i ~ h i l e Thcn ijeeanse in taking the mean each parkick comes in once re is fixed as origin we must here form a weighted mean the weight being v We thus

is leaves q a function of I and t only If there is a limit there can only be one so that to a given frlnction v ( T )

only one function q ( I ) altdefined in this nray can correspond As ivc shall sce later the converse is not true

It generally happens thut the integrals in the nluinerator ancl denominator cltain their limits separaSily and when this occurs we can tvrite the definition in the simpler forin

+m N - [ v h (3)

J - ~0 that N is Lhe 1~4loitnnl~zher of marked particles as in the defiriition of 44

i41 Corrcspondenc~of A ~ e a s072 the (rv ) ancl ( I q ) Graphs zuhen the Areas are E i ~ z i f e

( - rn

ii I - --I -

-i (c) r (2 I ) dm dl r - li x -- - amp

Since the termini are incleyendent of one another changing the sequence $ i f integrations makes no difference to the result Integrate first with rcspecl lt o5 Then as the mnge is infinite the inner iittegral transforngts thu --

-I---a r -rri

Therefore on inserting this value of the inne intcgral

CP result which we 8bhsllokten require

5 45 The fies+e~JicrByz~ntioihJbr the ( d q GrupZz w h w the Dip~samp+zr i a

Fiislclf It miil now be proveti iat--

IYhen there i s no meavr motloil or cdclies than g the i~rdzt~rlimb~rof neigii

bours l ~ c ~ rlength is related to the tirnr 1 an4 tlic separation E by a tIjBerenit9

equation Pike Picks ivhich rclsica -i he c~o~rcrntrnlioav to tbe 1 iruc t and i i psition c but for q Itlv clifrnri~iy is donblc i tiat Sor v Hi is assrlmed tjtz

vlvcZ~vanishct a t hlfinity If 88 vv then igtetnrtra(3vlat - 3 ~ ~ a l a2viac2 i t u~ayblt ~ l i ( a2vUlla12

r 1Lusv I no Zorngrrappear sepwrately andthtir l~oc lac t7 satisfies the f i n c ~ ~ r

and on integraiirlg the I-eclurhtion wit11 $heprovi-ti that 61 axand anal boti vanish as z -++ GO r7nd - cn we obtain

Compare Ll~i with (2) antJ the theorervl i i prcnrti

3 4 -6 frheo~elzThe3 f l e c ~of a ioalr Moi~oulindcprozdeitto j x clisuppeurs Z L - ~ ~ ~ I

the number of hTeigi~botrsjet- Lclz~tiis selccltd to Sej3aratiau~and Tim

For liy tle deiinitjorr y is independe~tc-l ttlo cllojes of the origin sf LC s r i ~ k

so If thc mean ve1ooiiy tiis i idependint of cl vi (an gf t rid of ii by giving

Atnaouphe~+icDzJUsion on a Distance-Nciyhbour Graph 72 l

suitable velocity to the origin of z This can be done even if n is an arbitrary function of +ime

Po simplicity in $ 45 i t was assumed thav there was no nlean motion -4ctnaIly vIlen R has beer1 tle~ived from observntions of smoke or of balloons the mean velocity has customarily been talccn into account by using as the definition of K some equation more or lees cquivaleat to $11(I) it1 d i c h (ii 5 7ii appear

Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to

in which 21 does riot appear Incidentally i t is of interest to take the form (I)in place of fj45 (2) in fc~rniing

the 211-equalion MTe thlm obtain

E-k GJK2$-- 2 --a1 + 2 EL)at aJ a a ar2

and on integration xjth respect to x the term i n it ~~anjsbes htts the 8ameif value for all indefinitely great values of j x 1 Thus we arrive again a t eeqmttion (2) of the present section

$ 47 The Present Theory written Jar Dffuusion oft a 3ruight Lhte is applicable also to the PrGeclion of Ihree-dinte~tsionnl Diflusio~z 07 to this Stmight Line

For Jet p be the ncllnloer of marlred ~nolecules per volume and let them he diffrlsing according to

Kow project each molecule nornlally on to the x-axis afid let v be the r~unibec of projections per length Then

-tm + m

v = J p c7y (71 --a- -=

Let us nsuine that p wnishes a t infinity in ~ u c ha way s toomale v iinite On integrating (1)wit11 respect to y and z SO as to ~troci~lce an equation in 4

a -the term -( K 52) yieldsa t at

- 3 k7 and consequent explanations elscwhere were addcd on December 7

ancl we may usually safely nsdume that the intogrand 1-allisiies Thc cer111 a a i-(K -- 1 bcbnv~s sirniiarly Thus there results c1r amp I

Thh i s the interpretation that nzur t always br gi~ren to thc present theor- i)cgtforeif can he al)plieti to observ2tioni

9 2 XOS-FICRTANI~IPPUSIOX

$ 51 Gezeralisntion for Atmosjhenc Edtlzes

llThcaathe tiiftusion is molrc~zlar both equal ion5 5 46 (I)a d $46 (2) correctlj-tit~scribe it When the eddies of the free atmosphere cotnc into action neither of tflcsc tno equations describe the yhenome~ncorrectly but ~Eescas in

rcic ks ccjlration the defect appears to he incurable i t is very aiily relncdieil in t l ~ rnew equation now presented Thai is to say t llc chiel aclvantage ooE the -nrlabl~si and g to vdlich all the forcgoil~g is merely preparatory appears lien IT consider the effects of eddies For 2s ~ l r e a d y stated observation slrows that the r a i ~ of dil-Eusion incrcacch wit11 the sf1l)aration 1 of neigllbonrn 11-Pca 13 re present this by nriling

ti ould not n(ccsarily 7a1iisll nntl ils total number of part icles 1 ~ u l d~o iht ijxed

4f TP tvcgtreto motlify Ficks eqr~atlon by writing

that oilid mean thaL the diffusivity ilcpcndcd 911 positiou an rfreci altogetl~ea diRcrcxnt froln the one rcp~esentcd by P ( I ) rud one sirhiell ~x-ill riot be st~r(liedili this paper Instead the p p c r discuqses an al mosplierc in vhicll rhc c-fiffuiivitj 15 inclclwcideut of positiol~ but depidt i on separatiol~

1it i ia nexi consider some observ~tions vlilljch $lion- how I (1) tlepencls on i rt7~ 1 vohsercitions have bee11 made fur other purplxcs and are not quite ~ h a i

is ciesirable here The quantity usually n~easured has been K in the equation $ 11(I) or something equivalent

In accordance with $45 (B) P(l) if i t mere iuerely a constant would bc equal to 2K Actually many values of l co-operate in the diffusion but those w411cli are largest ilot only produce niost diffusion but also have most weight in determining K when K is calculated froill the mean sqttnred deviation from the mean Thus i t seems likely that we shall get the right order of rnagnitucle both for F (1) and 1 if we put P (I) equal to 2K anti 1equal to the standard deviation o l the articles from their mean As 1wries in the ratio I 109men very cr~ttlc cstiniates of F(1)show its rclation to I quite clearly

When K has been obtained from the variation of +rind with height 21lt is still assunied to be rougl2lp equal to F (1) and the corresponding I is talcen to bc the mean vertical separation of the anemometers that were used in finding the second derivatives of the wincl-components with respect to height TThen the observations were obtained from pilot balloons we Inuw that 1 cannot be less than the ver-tical rlisplaceinenl of a balloon between two sightings so that I will probably not be less than 100 metres On the other side 1cannot be greater than the height of the observation above the ground The mean of these two ciistances has been taken to be 1

The distance a t which molecular motion is the chief caue of ciifiusion in free air inay be roughly estiiriatecl in the following way -Suppose that a very thin lamina of marlred molecules coulcl be producecl in still air The concell-tration should be arranged to be greatest in the central sheet and to diminish towarcls the outer sheets according to the law of error The thickness of the lamina as measurecl by a standard deviation from the mean woixld then increase so that (standard deviation) -- d ( 2 K f ) w11ere t is the time from indefinite tllinncss

Now the value of K due to molecular diffusion is abont 017 cm2 see- Renee we have the following

time in seconcis I 0 IO4O01i001 j 01 1 1 thickness as measured by standarci deviation cnis O O 018 0 e058 018 058

These numbers show clearly that molecular diffusion is very effective when the bmirla is 001 cm thiclr and much leqs ~o IT-lien it i3 01 cm thick Nov if we iook at cigarette sinokc in the open air ancl ailc ourselves a t what separ a i-I O ~

molecular motion will procluce rather more eflect than the eddies it is 110

difficult to make a guess I put it a t 1 - 5 x 10-1n Tlic inlegral power of ten i s ~eailyall tlial uiatters

Phil Trans A vol 221 p G (1820)

-i22 Igt Y Richardson

r31 BLamp data are s~l~nmarisetiin the following tible ---

I IXcfc rc~iice k II

------ - -- I-_ ---I

~ C C - I C l l i

i i from nlolecalar diffasicn of oygcn into nilrcr_tgtn(Kayo

and irhy9s Piysi~ni nnri c o a t ) 1 7 x 0 1 5 x LO J+or P s o preceding discnssioii 1

K ai- 9 nietxcs above ground from sncmometsrs at lleigllts ~d 8 iii nnri 32 mrtns (128chmidi LVlra Aliid 1132 Y 10 15 s 10 Fiizb Ila rol 126 p _ 7 7 3 (1917)) 1 i 1 __

14 10

- -

Ilt from plot balloti~sa t fieight~ helwcon 100 and 800 metres (Taylor Phil Trans A rol 215 p 21 (1914) 6 x 10 rlsoHc~selberp ant1 iiverdrnp 1eipzig Geophps Inst r 8 Heft 10 (195))

Meteomogical Society nlomoirs No I )

VO~CZI I a h same referenr~ as last 5 x 10 1 5 l oG _____--~-lI-____-_---_~ I_ _ __l-_l_l___

1 Diiuioi due to cvcloriea ~ n a r d e d as devintio~sfrom t1r rwan circulation of tiic latitltde (Ueiant X e o Ant K I also (1921) Wicn Aliad Vls firtzb7 I ln u(bl 130 p 401 (192l))

Bijice when nob obstrtlcted by the gron~iclsmoke spreads about as rrillcll

iiorizorrtally as i t does vertically the obfiervations a t tho s~~lal lec valncs of I fhongnmaole in -the vertical c8n be treated its applicable to the horizontal Tli~iithe wllole collection is coherent

lilr logarihhms 01 K and I when potted on a graph (fig 8)are seen io lie eIxsi lo a line of siigEiL cur-wture Iis harcily ~ ro r thwhile to tliscuss cictails tmi-ii ohserwtioiis l i nw b9en rnntic ixl n manner appopriate for the clttii~minatIo~lof 3 (1) rat11e~ihru of K How such observatioizz could be obtaiued xvill be discusscil in 7

r1 1 ~ straight line on the logarithnlk diagram wh-icil eorreipoilils to K 2IT

0 2 k also fitx the ~bser-rra~tions almost 8s is the curve in the limitedi7-cl

i311gB between i == ~rietr ~+id1 -1 10 kiiometreu For ~-miliernaticd

qinipiieity tAis orrn-uiil i ~ l l lix riscd ia trle illusrielionswhiclt follow Trri3 in this range F ( I ) = 0 4 lUi a~proxin~a-i-elywheri tlre amit hre

cen iirrarlres and seconds

$ c 1 vroylt3r

The equation for thc changes in the (1 q ) graph is then

where the constant c ii of the order of 04 cn1~bec-2 This equation inxrnmarises the subject

$53 Arhalogy with $he DiJtcsion of Heat

The Rindamental equation S 52 (1) can be brought into touch with sornrl standard mathematical forms by changing the variable 1 to 1 7 L say

Por this transforms the equation into

which ia Po~xriers equation for the diffusion of heat in a homogeneous solid where the icothermal suriaces are col~centric spheres of racliu~o and the difusivity is ~ 9 The equivalence of the equations in or and 1 is complete except at E -= 0 where a source of sink might occur In choosing a solutiola we must be sure whethcr it mnkes the whole number of marked molecules independent of time

4 54 Norz-Piclcicm JdLfl~~~iolio j aiIniti~lP)irif-iirsttron n Line

in which A is Indeperrdent of t and or reprdseats a process in whic11 at 2 1-1 (1

all nrigbbours are Cndcfinitcly close znd as t h e proceeds they spread hint

conti~zaally The corrcspon(1ing valne of N is

1 -lt wL0 (2

jy = (1 [ --j ( ~ ~ z amp ) ) - - ~ ~ rtti FbK2 amp aF -gt - u- -x

Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n

Thus E is indepe~denl oi time as recjnireri anti there is no sowcc at I -= )

clszcpl at t -7 0 Pig 9 exl~ik~itsthis function in the special Eorrir

when E is git-en its observeied value of 04 CGS units At t - 0 the graph wollld consist of an infinity of q at I 0 aalil p = O-

elsewl-iere One graph shows the distribution 3t t = 100 seconds Nighbourr as distant as 4 metres are l ~ o ~ v not scarce Five minutes later a t t = 400

seconds ceighbours at $0 metres are noticeable In the corresponding distributioli in space the isopleths of (oncentration are perllel planes The standsrct deviation of the marked molecules from their n ~ c a n position ~yiii be in rertigaf e(X in $ 71

In the course of the theory we b e p n with conccitralion v gimri as 3 f u ~ i c -

tion of position z We -vished to know -I-~utbeenrne of the distribution a-~~r

diffusion occurred Eicks equation being contrary to the facts and no suitable adjustment 01 i t being in sight we had to change the variables fro111 (rc v) to separation 1 and mean number of neighbonrs per length g This

was easily done Then the ey~at ion $=E -a 1 h aq ) gave ihe changesat a i j $7

in the (1 q) graph But after all i t is v as a Ennction of x that me should like to kno-rv so that after the diffusion llas occurrecl we wish if possible to change the variables (1 q ) back again to (2v) This is not so easy and can only be done in part

SEPARPTi3M 1 IN METRES

L C 9

Ss with tile problem of integration a general method is lacking and so it seerns clesirable to give typical exainples and a variety of processes suited to different circu mslances

Ic mouId be too much to expect that the (1 p) graph iJiould give us noug ugh inCorlnation to allow the distribution of particles in spaw to be reconsir~~ttecl in all its details For the process of taking a mean has bcen used in forming

to get rid of n ~upzrxbundancc of detail The process is irre~~crsible We critlnot evolve the detail again from the mean

Tlle origin of s is not represented by anything in the (1 I)graph

VOL CX-- 3 D

all Values of 1 can strue f o ~the Arzcnlber of flei~l7zbouisper Length

5 62 f i t ruery Eueiz 8u)tctto$f (1) vlhzch P gt ~ ~ k l i ) t ~ f o ~ l

Try for instance to imagine how tile

Iil pop~iacion coa~ld bo distrihnted on a

line so as to procluce a (1 amp) diagram tic fig 10 If you have 11eighbolr-sat

(1 tltsrty rlistnnees lying btrncen 5 and 8 F--- - h i some of them must have leigh I ~ ~ u r i --L----b

1 0 l i 3 ~ ( 3 i A ILL I 7 I it dist ices less tllsil 3 kw so the

(iiagra~nonid 1 1 ~ 1 7 ~t o have a cc~itral hump As drawn i-l cannot be in (1 Q) ciiagrbbrn

Howe~~er andwhen as usual q I~zs teen protIrlcerl by diflusioa from q g corresponded to v then if tlle maihc~natics fits the pllysics a it famp])I )PSrC

to do there m~zst exist a v~ corrcsp011ding tr) q

Given that v = b a quantity incirpcndent of z

Therefore v (x I ) = b also And by the definibion of $ 43

6 j)dt

q -- Linlit ----- == 6 fz b j rim

-n

So q is independent of I A5 to the converse see undtr Fourier series

-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore

See also a note at the end of 5 67

There inay be distribution of con~entration extending indefinitely in botr

directions for treating wilicli ITo~wier series will be suitable Let

v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1

$2 bei~g a positive integer and the constant A being so choser~ as to make v

everywhere positive Then it may be shonrn that

fi24 ( 0 4-A) -2- +L (-4 + B) cos nl + finite terms h

A + m e + m

q - Linllt amp A ( 0 -t- A) $ finite terms

1

The fundameiital wave-lengbh in q and v has been taken as 2n Any other value could be introduced by changing the units of x and I in the same ratio and the relations between A B and Cfi would remain as stated

In particular if q = Co simply then v = q is a solution and on account of the generality of the Fourier scries i t appears to be the only possible one

$67 The Correspotzdence of Biusion from Points orz the z and 1 Axes

Time is not involved in the connection between the position-concentration graph and t i e distance-neighbour graph but we can bring in time as an aid to finding the connection Suppose for example that the diffusing substance 19 initially concentrated i11 five masses each consisting of n particles near five equidistant points on the x-line a t intervals b of z Then there are neighbours near 1 = 0 amp b amp 2b 3b amp 4b but for no other values of 2 The relative numbers of neighbours in these five classes are easily counted

Eest suppose that Fickian cliffubion occurs Both graphs change but it

3 ~ 2

~ I C sailre irlitdnt they ~nrlsl a 1 n - a ~c ~ r ~ t ~ i j ) ~ l l dollc ilnuther [rig 11to ~honrsthe two grapl~s a t one instant

Now a give11 distribution 01 v ill J may i ~ esccn iro111tllc clefi~iltioliof p

t o lead to a ~rxliqueEunction g of 7 Thus 7 ( I ) t i o ~ stioi cIef)end011 1101~-v ((XI clmd to be but only on 1v41at I (sj i Ire 11aac arri-e(l a t the corrcsporcencc in thjs example by Ficlrian djffL~~lblr f3nt if he banle J ( x ) ilacb c ~ uprocIacci by non-4icliiln c1iRvsio11 or GI 1113- ot11r~ir-r- il woulcl correspoiitL to i h ~ baTlle y (1)

he sjtnpler prohien) ol $ 65 i u ~ yao b s o i t i(l ill this iayl

which is the lztl) ncinxe~t-corfl~~ic~nt cf thc q ( I ) r i i+tribt~liorial~orrtiis rueon l -L- 0

We shall treat only thc o t l c ~ I I I-llieli 3iil ltil the ~i~olrtc~it-ir~tt~g~iIsarc finite as occurs v11rilic11 7 a11d v vaaisli entirely in t l ~ cori-ter regions

Itlhcrefotjre ir aeco~danccwith t ~clcii~iiiio~iiri y in 5 43

X C - ~ oS out another o that uc mayi L 1 ~lililits 01 iiteg-rntion arc- inif~~c~r~tlei~t change the seclnence of inttgrntioils ~~tlroutchnngi~giiliythicg else Let us ilCeg~ateErst I5-itli respect tc i ~enlenlbciriingttli p (z) is izdepciirleri-i o f 2 LhereEore

8i1ice v i i the conce~~tration tst t l ~ c point ( L l ) the iniler integral is the iztlr ~nonieilt-coefficientof thr distributioll O F toncti~tration taken about thc point 0 7 1 cor~1paring equatiotlc ( I ) and (3) j l is seen that -

the i l f J 6 qf litr rlisttibzitiorr q ( 1 ) of) Y L O ~ ) L ~ Y L I - C O ~ $ C ~ ~ I ~ ~ theiylbo~irs ubout its c e ) t t ~ ~ 1 --= 0 is 112 qf the nthl i i ~ ~ l i ~ nzourteitt-eo$flicients of the distribiitioit of ncrred titolecttles t~~X3412(l)otr eoery mcrrk~d nwlecuJe in turil (4)

Since q (1)is an cveit fi~acstionof 1 i t ~nonlents of odd order about 1 = 0 all vaujqh Thnt ir

O - - pi - 11 - p pi etc ( 5 )

Iqor the even 111olnentsthe expression call be siinyliiied Uhe most interesting (lase is 71 -=2 Let r tbc centre of the clistribntioi~ v (x)he cTe61xeil as nxuul

11here is a fanliliar thcorcnl ill ~~~echailicscoilceriiii~g~tatliiof gymtion rountI t~ra l le laxes In like lllaniler it can be proved that iE 02be t11e secorld illonlent- coefficient of r(x)aboilt tljc ce11tic nailcI 0 be hat about any otherpoint r t i l c k l r

CJF2 - G ~ + (x - (7 )

KO+-the stai~darct diiations of q i I ) and v (s) are respectit-ely qpand a Flencc 1i-e have proved that the stnlidtcrd ampviatioj~ of thc q ( I ) area from its

rciztre 1 = 0 i s 2 liirtes the sta~zdard deviation of the ~izarlced molecules f ~ o l n their cen8e s~A glance a i ille di6tance-neigllbo~lr graph thus girrh us a good r~irpressionof the size of the ch~ster on the line

The skewtless of tlie v (x) distribution is not given 11s by the ( I q j grapli hecause slte-vness clepeiltls upoil a ~l-iome~lt But the higher evenof ocld order ~rio~rtcntscoultl he fomiti For the fourth niomc~~t-coefficient I filicl

Tlie proof of this is o~niltctl or h~e-jty

732 L E Itichardson

I t hrs aPrcsdy been prowl in 5 gt nrl t l S gtd ihar cl~itlcri1esc tilcni-cislnnre~ nrc haw q giwn by $54 (1)

So finti tile ~~cconrlmort~clrt rt i~lii~ll s t i ~ ~ f i l t d 4 6 8 ( l j i~ i r tas Ijcforci 3 l

Thc11 it nlay he illown that

Ncrw if se denote 1 p-i3c [XY k)y 3) (I - o

Also lor J I I ~ ~ I I I ~ ~the elicuiztjou of N vli~i~ivr-~ )fgllll ill 37 I ITC s l ~ d ilercj

0 886 (6

Vit11 t i r ~ i esil~ititutit)llit folloni t 11at

But by tlr I htampulvm( 5 68 (8))of tllcr yrc~~ioua$eci ion LT~thi-iridartl cii si~~ii~

of the innrlceeirnnleculps irolrl thcir Iirckil poiition on tho l l r lc i s ( j j $0

105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243

r7ilkis for~lullrnablcs tlie ro2fioi~gt1lta i t 1 l o l a ~ ~ e i g l ~ k w ~ ~ r gtt h c cIitY~~ii~y ii t o

be determined froni the scatter of particles on a straight line Che particks will really inow in three dimensions but the formula can be applied to their projections on the line as mas shown in 3 47

Suppose now that the forinulze we have thus deduced froin the non-liickiar cliffasion of naighbours ~ r i t h difrlsivity zl4j represents the true secluence of events but that the observations have bcen reduced instead by the formr~la which is a necedsary consequence of the Yickian difksion of concentration with

what values will he oiotainctL for K Eliminating t between cc~uations (9) ilr~ci(10) it is founcl that

Ihis sho~rst bat Ilt obtained in tlris way -illinelease as the 413 power of the size oof thc scatter as iil fact K docs (see fig 8)

If me regard E (1) as what we wish to find froin publishecl values of I(then we must pnt

As a necehsarp preliminary to finrhng out that 8 (1) was nearly of the form c14P Imade in $52the guess that we should obtain the right order of magnitudu for E ( 1 ) and 1 by putting P (1) = 2K and 1=-ob These imply that F (an)-=

2K It is scen that the gucss is xi~lply justified But me may now revise the value 01 E from 0 4 to 0 4 x 312 =0 6 ~ r n ~

see-l This is only a mean value ro~~ghly applicable under average circum- stance in the range one nietre lt 1 lt 10 kilonletres A more detailedstudy will reveal variations of i O times or more in z according to the up-gradients of temperature ancl rr~ean-wind and other circumstances Even so s mill be remarkably more constant than the diffusivity K for concentration which as

we have seen varies with I about a billioil times

This formula was clctlnced by Einbtcin In conrlectioii wlth the Brownian motion ( Ann der Pliys vol 17 (1905)) Son~ethil~gllke it was employrd by G I Taylor fol gteducing the Scotla kite ascrXnts Trans A vol 216 11 10) Chu formula was given explicitly ancl much used by the prcsc~nt writer in iSorne Measurements of Atmospheric Turbulencc ( Phil Lranu A vd 321) In the latter paper there are two ~rldependent proofr of the fonnula one of which in Section I V is a correct deduction from pick^ equation the o t h r ~111 Section V is quite spoilt alas by a wrong s i p in

equation 3 p 9 and a rislcy assuinptioll about correlations This error affccts equation (32) on pp 15 ancl 7 of the aforp-aid paper hut +helest of the paper hold5 good indepnldentlp

734 1 I Richardson

Vanou5 observers (Dol~soli 13lc~hardson Roberts) Ilatl eparattAiy ylotccl t h ~ i-net that tllc width of an individual bra11 oi iit~oke fronl a point source xvhcn iricerured by its standard deviatio~ afront i t 5 ~ n ~ a r r l Lline is snvh that o roilghly pro1)ortionai to tl lt Thir ill~pljei that the diflusion i i Fiekiali in certazn hart ranges of 1 the difiusi vity for neiphbour~ I+ ( I ) being there independent of I When an enorillously wider raJlgcl of 1 - t i r oti~itlcrcil we ha1e ieen that amp (1) is proportioilal t o roughly

Tote added Deceiizb~r7 1925- The ob5clr L i~lorrscollectc~dby Richa~tlsonancl Proctor in the range a=- 3 krn to 86 l i ~ ~ I ill) thc lop( of tlic srncotli hi PI curve ill fjg 8

These apparently coiilrrttlict o r incts 13wv ppr1iapi be ~cconciled if we regzrd variations in K defined by 5 11( i ) as h ~ i ~ ~ qduv to variatioil5 ill the type ~f irlean chosen in forrning 11 2 17 in tllr sawe rcjoation As 1011g a tlie njear ii always taken over the ha~ne leir~tll ~rl(ttlrtr K I I I C I ~~ ~ 7 ~ 1 1hc Inore oric~i 17onstnntl

5 72 ilreory of a Seco~~cl uztp k(1) night Le abse~uedMethurl by zrjl I ( r ill( I l ~ A f ~ c v

I t bay just bee11 shoivrr that the ialile of c 111 ~ 1 = P ( I ) he cstracted fuciln observntions made a witlc mnycl of i 111 in opci-ation I opetlier TVllile tbic is posble matllematically ~t rc~eivd f r c i i ~ i thtt tanLpoint of prdctical physics to mix too rrlany ph~-llo~rle~la a x l I ~ili~cc~os+t~ily Eto aisulllrL that is ~ntiependelrt of I It vould hc beitcr to obctgtrvcgt w tici~aratt~lv or near each gtelected lo This can bc clonc- 1 ~a l)rccccgtliiclr ~ v i l lhe tlerlr ctl ronr t h e

non-Ficliian ecluatjon

S~~ppose arcJ inii itli- t oncclititt~ct111 t ~ othat thc nid~lit-ci ~noleculez plrtlle parallel lamina tlistnnt 1 froin one anothfr ancl similar t o one another The neighbour-distance diagran thrn ro~lsists oi tllrec sharp peaks ah suggested qualitatively in fig 12 The cclrtrai peak llo-i the ~ ~ e r y cl(c ntighhours

which each marked iilolecule has in its own lamina l i~c lateral peaks slmw tllc neighbours in the other lamina The spreading of the ccntral peak will go on wry much as if there were only a single laiilina ( 554 Q 71 ) and does not concern ti now If the lamin8 are observed during a tiirie such that they spreacl through only a sniall fraction of lo then we iilay regard F (1) as a con-tant anci equal to F (lo) in this short range of I and inay write accorclingly

An appropriate solutjoii of tllii eciuatlon is

whicli iinplies that the peak on the (2 q ) diagram has the form of tlie ~lor~lial curve of error q playing thc part of the freqx~ency The standard deviation of the curve from its mean lo is (213 (lo)l

In determii~ing the standanl cleviation from tile obiervatlons it is of course c~sential that the irldiviclual valueb of (1 - sliould be weiglltecl in the proper way On referring back to the definition of ampin 5 32 i t is see11 tliat we inust form every possible pair of niarkecl molecules one froin each lamina so obtaining a set of diitancei 1 Idet [ IQdenote the ineail for all illelllbers of the set Then in ordiiiary cire~arnstancrs 17IQ =- lo Tle next form tlie mean 5yuared deviation froin the l~iean and thus finti

In practice the ~narktd lrlolecules coultI be replaced by balloons for those -slues of lo which are ~ i a n y tinleb the clian~eier of u balloon This is lt u hecaubc F ( 1 ) increases notably with I C H Ley ]la ii~ventctla valve which a l l o ~ ~ s i balloon to rise to a pre-arranged hciqht ancl tllcn let5 out some gas so that the balloon ceases to nlo-e through the air That is tlie type of apparatus required

When there are only one or two observergt tliey could not alanipulate rliarly 1alloons a t once but they inight observe pairs of balloons on successive days ~ tus try to adapt the observation to thiq itustion Tinagine ampst for the gtake of the argument that the obserl-atiou with many balloons arranged

initially in two parallel lamin8 lo apart 1s made ancl let us denote (1 - - by I

Q u a ~ t +TournNoq- Meteor Boc 1) 247 (1911)

f o ~ sliorl Thus rnany values of p are obtainerl o ~ one occasion Xexl imagine that this observation were repeated once 8 clay for a year the time t being in all cases the same and n o t too smaiI nor too large I n this way a cIo1~1)lcqet of rsli~cs of p iobtnincci thus -

I1hc 1ItLRI1rlaI~gteof hx( lo) j)r ali tiim obserl-itioidis it (jO) == [pIQYthe double su 6iix cieaoting the doirble nlonn Q for rrtlars R for c o l l ~ ~ t n s We cannot proeeckd f~trtlwrwithout malri~~g ~is~tt1wptifi3 the gelzesctb w e a n of jithe flint wl 00tmiz

if we select at raizdom one vulue of p front etrch row cf the douhle srt mad tcrkt (he unean qf them This lookr pwsn bit

1s so

vhich can be tll-tentlined by fl-yingn 1 j t ~ i ~oi 1alloons on each of the many CIc~y6 This illeasnre of atmospheric tiiffusiibr~is in agrc ennefit wit11 that t o which w r

vere led by n rpnrc11 lor a ~i i~tumlI r l r t b t l Y 2)

The at~tioplicr~c eqitation iiai been foullcl by wrlousciiRl+iriiy in P i ~ l i ~ ~nc~st igatorsto Increase from C12 t o iOiicrnec-l as the size of the clustal of diffusing particles increases froln 10 ro 10em The effect is d w to eddies of many sizcs acting together There is appmrtantiy no way of 1uodifyi1)g Eicks equatiorl in order to describe tlri3 pleno~nenon 13ut a new niathematical nlethod is bere developed in whic~ i~zstesdof thinking about concentration as a fllnction oi position tliillli aioill y ff-ic inearl number of acigl~bour per lengtli as a function of 1 their ciih~ancc apart Formal definition is given t o this idea and wriouh propertica of l rre inveestigsted For simplicity ouly clistrib11tionl on an unbonn~tc(eli -t might line are considered or projectio~s of three-dimensional ctistributions on I d the line T t the nlot~rnent of coneen tration v is ciescriberl by Ficlrs eqi~ttf~ollr

~vhrret is time z is distance jl is mean velocity and Ilt i i diffusivity

I f hovever tlre diffusion is non-Ficliian as in the atmosphere then the former of thew equations canfiot he gencraliicd hut the latter can taking thc form

A discussion of existing observations shons that a rough average value is I ( I ) == 06 l4I5 see-l for the atmosphere when I lies biltx~urren one nletre and 10knl The diffusion of a lamina is workecl out fron these principles The dmgrarn obtained in thih Kay exhibits the size of a cluster becauc it is ~ P O V C L ~

thnt the standard deviation of the ( I 1) area from its centre 1 -- 0 is 42 timcb

the ~ttancli~rtldeviation ol the cluster from its centroid Two lnethods are prepztred for the observation of 3 (1) by k~nlloons or iinoke h r ious allicd topics are examined

$9 I A T S C OF SYampI~~~OLS l t r1 VHEKI] ~ ~ C ~ R R ~ N G ITtl SICCTIOXS IBEY ARE DEFlXED

You have printed the following article

Atmospheric Diffusion Shown on a Distance-Neighbour GraphLewis F RichardsonProceedings of the Royal Society of London Series A Containing Papers of a Mathematical andPhysical Character Vol 110 No 756 (Apr 1 1926) pp 709-737Stable URL


This article references the following linked citations If you are trying to access articles from anoff-campus location you may be required to first logon via your library web site to access JSTOR Pleasevisit your librarys website or contact a librarian to learn about options for remote access to JSTOR

[Footnotes]

Eddy Motion in the AtmosphereG I TaylorPhilosophical Transactions of the Royal Society of London Series A Containing Papers of aMathematical or Physical Character Vol 215 (1915) pp 1-26Stable URL

httplinksjstororgsicisici=0264-3952281915292153C13AEMITA3E20CO3B2-D

httpwwwjstororg

LINKED CITATIONS- Page 1 of 1 -

NOTE The reference numbering from the original has been maintained in this citation list

Atm~osjhericDifztsion showiz on a Distance-Neighboz1r Graph

By Lewxs F RIOHARDSON

(Communicatedby Sir Gilbert Walker PR8-Received Novelnber 7 1925)

If the diffusivity K of a substance whose mass per volume of atmospfiore is Y be defvled by an equation of Picks type

a- a a~~tdiasl-as~- $-amp ax -- G k zz4+a$ ay i~ 6 X a y G a y i a z (1)

cc 9 z t being Cartesian co-ordinates and time 7z 5w being the components of mean velocity then the measured values of K have been found to be 02 cm2 see- in capillary tubes (Kaye and Labys Tables) lo5 see-f when gusts are smootlled out of the mean wind (Akerblom G 1Taylor IIwselberg etc) lo8 cm2 set-1 when the means extend over a time compaeablc with 4hours (L 3 Richardson and D Proctor) 1OI1 ~ r n ~ when the mean wind sec-l is taken to be the general circulation characteristic of the lati-tude (Defant) Thus the so-called constant K varies in a ratio of 2 to a billion The present paper records an attempt to comprehend all this range of diffusivity io one coherent sclileme

Lest the method which I shall adopt should strike the reader as queer and roundabout 1wish to justify it by showing ampst why some knotvrl s~ietltods are in difficulties

$12 Does the Brind possess a Velocity

This question a t first sight foolish improves on acquaintance A velocity is defined for example in Lanibs Dynamics to this effect Let Aa be the distance in tile 3 direction passed over in a time At then the x-cornpcnent of velocity is the limit of Az]At as At-gt 0 But for s n air particle it is not 0amp1Vio11s that ArAt attains a limit as At -gt 0

We may really have to clescribr the position z of an air pizdiclo by somtthintg rather lilcc iVeierstmss7s Rmction of whicla P Iileirr gives an entertaining description ( Anwendung der Differential and Integralr~chnungauf Geolazetsie Leipzig B G Teubner 1902) say

For references La publications set the table in $42

i 0 L I Richardson
























VOL cx-A 3 c












[(zr- x2- --- I) -= [j]bay








5 31 IlzLroduction










I















for iE eo












- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg



i 0 L I Richardson
























VOL cx-A 3 c












[(zr- x2- --- I) -= [j]bay








5 31 IlzLroduction










I















for iE eo












- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg












VOL cx-A 3 c












[(zr- x2- --- I) -= [j]bay








5 31 IlzLroduction










I















for iE eo












- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg














[(zr- x2- --- I) -= [j]bay








5 31 IlzLroduction










I















for iE eo












- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg






5 31 IlzLroduction










I















for iE eo












- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg










I















for iE eo












- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg

















for iE eo












- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg











for iE eo












- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg



- ------- -




O N A L INE

FINAL - -



AS A FUNCT iON O F
























+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg





















+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg









+m N - [ v h (3)



( - rn

ii I - --I -



-I---a r -rri



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg



















Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg






Tire see now that

---a v -+- fi---2)-- a 2 v K -at on ax

leads to








-tm + m

v = J p c7y (71 --a- -=



























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg


























------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg
















------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg






------ - -- I-_ ---I

~ C C - I C l l i




14 10

- -













$ c 1 vroylt3r











1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg













1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg







1 -lt wL0 (2


Putting

I$= -42~19

it is founci that

21 - oa j e - R 2 $ 2 t i ~ 621 - -n












L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg







L C 9





VOL CX-- 3 D













6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg















6 j)dt


-n


-- 1 (lt--iz d(27l)

Then

N iw d ] r -2

Also

Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg



Therefore




v (x) - Ao + 71

C -m

A cos r ~ x 4-B sin nx) (1) 71 -1




A + m e + m


1






3 ~ 2












O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg














O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg






O - - pi - 11 - p pi etc ( 5 )



CJF2 - G ~ + (x - (7 )





732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg



732 L E Itichardson






0 886 (6




105 O o --a 4 zF)i2- - i -1 [z[)Fjz

32 1 243













734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg














734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg



734 1 I Richardson



















1s so














[Footnotes]



httpwwwjstororg














1s so














[Footnotes]



httpwwwjstororg






1s so














[Footnotes]



httpwwwjstororg













[Footnotes]



httpwwwjstororg







[Footnotes]



httpwwwjstororg



Documents

Atmospheric Diffusion Shown on a Distance-Neighbour Graph ...eyink/Turbulence/classics/Richardson26.pdf · Richardson and D. Proctor), 1OI1 ~rn.~ sec:-l when the mean wind is taken