Horizontal diffusion in the atmosphere: A lagrangian dynamical theory

Horizontal diffusion in the atmosphere: A lagrangian dynamical theory

Discussion 194 AUTHOR’S REPLY Since Dr. Delumyea’s comments are more in the nature of a reply to comments on his paper (Delumyea and Fetel, 1979) ...

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Since Dr. Delumyea’s comments are more in the nature of a reply to comments on his paper (Delumyea and Fetel, 1979) than on the validity of my analysis, I will address what I see as the main issue: the applicability of box models to deposition velocity estimation. The purpose of tbe deposition velocity concept is to permit the estimation of deposition llux from fairly simple measurements of concentration and supporting information (e.g. wind speed and atmospheric stability). This means tbat the dependence of deposition velocity on the supporting information must he known and that the concentration measurements must be made under conditions when the deposition velocity is reasonably constant, over the time and space scales of the concantratiorr measurement. Tbe box model can he used to e&mate the deposition vetocity by relating the flux to the concantration di&ence across the box. However, as I pointad out the eoncentratian difftnaccisrmolliftbcfetchirsbott,w~~tolulEc uncertainty in the deposition velocity estimate. In addition tbe box model sbould only be apphed over time and space scales which do not contain larp varirtiotw in parmctm affecting the deposition velocity. Thus the box model has serious problems at short fetcbm and must he used under Tbe deposition vetocity~estimates af‘D&uxnyea &d MeI (1979) ranged from 0.01 to l.Sutt~-~, after @eeting those with [email protected] values. If these were all made under similar conditiooe, the range is not isbmdsmt %vitbmy npated unccrtoiatyin~~teOtPbOUt300*/*foranO.f~d L/H = 50. On the Otherhaad, if they were not made under similar conditmns they should not be [email protected] since the average flux is not equal to the product of the average deposition velocity and averssp ~tration. I agree with Dr. Dabunya that the field reaearcb team is bestquali6edtodecideancertainaspectaofdataqaahtyand should be a ffirly wide latitude in [email protected] data Sample . given . tammatmn could autainly lead to negative deposition %city estimates (if tbe downwind sample is contaminated). Were there no situations of upwind ooutamination leading to larger than actual estimates? However, with reasonable uncertainties in the concentration measurements (1OyJ it is not unlikely that negative estimates wilt be obtained. There is no more justi§cation to reject these estimates than there is to reject large cstirnattes.Italy, a wind speed of 11 mph (4.9 m s - ’ ) is probably not high enough to cause conditians conducive to possible upward fluxes (negative deposition v&city). Siinn and Slinn (1980) cite the results of Delumyea and Pete1 (1979), not to support a particular value of the’drag coefficient, but to support their turbulent transfer model of dry deposition. In fact any value between about 0.01 and 1 cm s- ’ (settheir Fig 2 with ad = 1 rmt) would support the model depending on the surfacc’laycr relative humidity and wind speed. Models are a useful tool to tield researchers, however their conditions for applicability must be recog&&, both in their use and in interpretation of results.

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Detumyea R. and Fctel R. (1979) Deposition velocity of phosphorus-containing particles over southern Lake Huron, April-fktoher 1975. Atmospheric Enuironmsnt 13, 287-294.

Slinn S. A. and Slinn W. G. N. (1980) predictions for particle deposition on natural waters. Atmospheric Environmenr 14, 1013-1016.

HORIZONTAL DIFFUSION IN THE ATMOSPHERE: A ~G~N~~AN D~AM~~AL THEORY lnatsaatplpcr(l982r)Dr.GiffordprrseDtDhis~~ndynamiml theory for horixomal di&aion in the atmosphere. This theory, basml on thaaobstion of the [email protected] sqrptloa, yielda most [email protected] his Bpration (13) whkh n*ctJ (with some small changes in notation)

Gifford applies this equation to the width of puffsand Mater ~~(Giffonf,19~~~totbc~~of~~rneopuradovtr relatively short titms. I believe this application is wrong in concapt far masons given below. However, before reMring these, the equation near% exp&ation for any reader wbo has not studied Gifford% paper in some depth. Many readers will be familiar witb G. I. Tay%or’s&a&al expression for the width of a great ensemble of pa&&s released indepe&ntly from a Axed source into an airflow witb constant mean velocity and superimposed one-dimensional bomowus crosswind turbulence (see for example Pasquill, 1974). Expressing tbe width as a standard deviation u, of particle disp&eements ahout the mean wind direction, then T F -7 u; - v R(t-s)dsdt, (2) SI0 0 where T is the time of travel from the source, f and s are dummy time variabhts, and R is the wntpan timecorrelation of the eromwind hubulent velocity component u (w~~~~a~~~). fn 1968, I showed how Taylor’s analysis could be extended to study the CaM wben the eIl#JtnkIt? of emitted part* was limited to the subset whom initial value of u took a single value ooTIn the main table ofmy paper (whiih unfortunately was omttted in many journal copies due to a printer’s error) two relevant results were given: Tbe displacement of tbe antroid of these particles about the mean wind line was shown to be


T R(s)ds s 0


and the spread of the particles about their centraid by

Mependent of u, If we give R a rimgpa exponential form, as imp&itlyaammndinB.~~,R se-“*(risthe Lagrantime=at& than Equation (4) above becomes equal to (1) wban P$ in that squation is put xero. Tbe geacrrtimdandf*llK!q&on(l)isobfainedwhenweaddto (4) an extra ~nc&dnducwd easily from (3), required wben we broaden the ensem& of part&s under study to include l



F. A. (1982) AtrrParpheric Environment




the source within some rather small range of time (since we have to recognise there exists alongwind diffusion as well as cross-wind diffusion) and and during this time us probably varied. It is reasonable then to apply an equation, such as (l), which recognises this inevitable range in effective vc’s. Why then do I retain doubts? Chiefly because I do not 15) believe the ins~ntaneous plume considered by Gifford has [email protected] in (1) and (5) is to be inte~ret~ as the variance of the involved full ensembie averaging. Let me make thisclearer. If ensemble co’s about their own mean: we could identify those small sections of the continuous l$s= -(vo>s (6) plume that left the source with the same range of cc’s (i.e. the same v0 and 03~) and measure their crosswind widths a,,(i) the ensemble averaging taking place over the whole subset of and their displacement y,,(i) from the mean centre line of the particles. whole plume (where i identifies which section we are referring Equation (1). then, gives the spread of particles u: belongto, and runs from 1 to cc) then Dr. Gifford would choose to ing to a sub-ensemble of particles whose us’s have a variance compare Equation (1) with the average variance of all the [email protected] If vt2 = 0 then we return to Equation (4) for fixed ug, and sections, i.e. with if L$’ = ;” then all particles are included and we get Taylor’s result, Equation (2). I apologise for rather labouting this point of interpretation whereas the comparison should be with the total spread of all but in this case it is very important to be crystal clear. the particles belonging to all the marked sections taken Later in his paper Dr. Gifford makes the claim that together, i.e. with Equation (1) [i.e. his Equation (13)] should describe the average development of puffs and the instantaneouslymeasured width of a continuously-emitted plume, and goes on to compare the form of (1) with puff data and in later papers (1982b) with plume data collected by various groups. Indeed he purports to show that a very high degree of matching can be achieved. In spite of having an immense respect for any statements Figure 1 makes the point more clearly. I do not think the made by Dr. Gifford on this subject, I admit to retaining difference in our intuitions is a trivial one, and the reason some nagging doubts about the validity of this interpretation. that Dr. GiBord can achieve a reahively good matching with Essentially if he is right he has bridged a very important gap his data may be that compared with other formulations he has between, on the one hand, the long-term averaged plume, as an extra unspccif%d parameter ug’ whose value can be chosen given by our Equation (2), and, on the other, the puff or very at his discretion. short release (see Smith, 1961; Sawford, 1982). In conclusion, whilst I beiieve it is quite reasonable to The former is essentially a “‘one-particle” diffusion problem expect a continuous plume to exhibit accelerated growth over where every particle can be assumed to diffuse independently some section of its travel when sampled over some specified of every other particle, whereas the latter is a -multi-article” time interval which is small compared with its time of travel, I ditTusion problem where the inter-correlation between the am not convinced tha! Equation (1) is the appropriate motions of the particles involved is of the very essence of the equation, nor that theextensivedataexamined by Dr. Gifford problem. To claim that an equation such as (l), which is really demonstrates this accelerated growth phase, as a fundamentally one particle in character, can say anything consequence of the v-component, beyond reasonable doubt. about multiparticle diffusion is revolutionary and needs total justification. What arguments can be put forward to support F. 8. %.ilTH 3oundory Layer Research Branch this claim? They seem to be based on the follow~g notions: Me~e~ol~~a~ O&e (i) The ins~n~neously-rn~su~ phsme can be thought of Brack~ll as a succession of individual puffs, and each puff starts life at Berkshire the source with a burst of particles having a small range of vc RCI2 2SZ velocities (perhaps due to the inherent turbulence in the efRux U.K. process). (ii) We know from theory, and a few limited experiments in REFERENCES the field, that pulTs should exhibit an accelerating period of growth in which c, grows faster than linearly, and Equation Gifford F. A. (1982a) Horixontal diffusion in the atmosphere: (1) has this same property. a Lagrangian-dynamical theory. Atmospheric Enuironmenf (iii) The crosswind width ofa continuously-emitted plume 16, 505-512. measured at one moment must sample particles emitted from

not just those with a single initial uc but those with a range of vo’s.

*,.a........... *.*-



So”r~/..~-~ ......._... c

Moon centre -line

Fig. 1. Boxes A and B have the same us properties but do not land upat the same place. Equation (1) is the total spread of al1 the particles in A, B and subsequent boxes, taken about their one combined centroid. Dr. Gifford seeks to compare (1) with the average spread within each box.



Gifford F. A. (1982b) Long-range plume dispersion: comparisons of the Mt. Isa data with theoretical and empirical ‘formulas. Atmospheric Enwironment 16, 1583-1586.* pa&mill F. (1974) Atwtosti Difision. 2nd edition D. I 123. ’ Ellis Ho&ood’ Ltd., dhichester. Smith F. B. (1%8) Conditioned particle motion in a homogeneous turbuknt field. ~t~s~r~ Env~on~nt 2,

The good agreement between (13) and data that I reported can in no way have bean cad by an arbitrary adjwstment of I+,, which parameter Dr. Smith seems to think I include as a kind of wild-card in the deck. Notice that if c * 1, u. _ 0 and in fact the vahtes of v. implied by the diffusion-data comparisons were all quite small. Neither the quality of the data comparisons nor the values of the large-scale Btmos491-508. pberic 7 that I deduced from (13) would change noticeably if vg were simply assigned the value zero, u priori. I Sawford B. L. (1982) Comparison of some different approximations in the statistical theory of relative dispersion. Q. JI preferred, however, to let this determination be made by the data. R. met. Sot. 108, 191-206. (3) In the random-force model, the intercorrelution beTaylor G. I. (1921) Diffusion by continuous movements. Proc. London Math. Sot. 20, 196. tween partick motions referred to by Dr. Smith occurs as a result of the correlation of their initial motions at the source. This determinea the quwdty c in (13) whi& in the general case treated by Lee and Stone (1982). is found to dapsnd explicitly on the fired-point (Eukrian) space and timecorrelations of the initial velocity, uo, over the source points. AUTHOR’S REPLY In the singk-Imrtkk case, of courae, these reduce to uf. The intercorrektion, like any mquinxi property of the partick My reply to Dr. Smith’s interestins comments is divided into motion, can readily be c&dated from the random-force four main points: [I) the relation, in tbe random-force theory, quations. since (see e.g. Brkr, IWO or Ba&Wor, 1952) of cockle dispemion to two, or rn~t~~~cle relative d#Usion equals (hvi& the fuBy~~~ Tayior dispersion; (2) the rok of the initial source parameter, tie or c, dkpersion minus the dot&k Mgral of the intercorrelation, in the theory and thedatacomparisons; (3)partickinter~rethe ktter is found, by subttwting (13) from (18), to have the lation and (4) the extent to which an approximation to form c (1 -e-q*. Thus far from ignoring intercorrelation, ensembk averaging 0~~3 in pm&al diffusion applkation. the random-force theoryprovidesan explicit way to calculate (1) In the random-force model, the equation for the its e&ct. spreading of two part&s about their an&d, i.e. relative (4) The question of the extit to which the spreading of ~~~ kas exactly the sama form as the sir&k-particle actual p&s and plumes coIIcspoIHIB to the ensaMe averagEquation 413) of Word (1%2a) and Smith’s Equation (1). ing of the thsanir, raisedin Dr. Smith’s final o(Jmment, is a This result has been known for many years now, ahbougb it good and in my opinioff an opan one. The point of view that I esemsnattab3VCtYf~to~~o~r,aadQeasily have take is that a very largegroup of particks ix emitted at demonstrated by repeating the steps kading to Equation (13) an instant at a source, for ins&nca a chimney or a small smoke for two partkka Roofs appear, for exampk, in thearticle by puff. I consider that these pa* all have the same initial Lin and Reid (1%3) in the Hcz&uc!~ der Physfk, and in velocity, vo. Tbk could be just&d, for instance, on the basis Cl&am‘s (1%2) note, botb [email protected] in my paper. that the Eukrian apace cc&a&m remains quite large over Extension ofthe [email protected] result to Edison point the chimney or puff width. Thus, I’m considering a source sources is dkc#sed in Gifford (1982b); and gene&z&on of small enough to be mgar&ad eff’ively as a point, but from all the above results to the case of sources consisting of an which many particles amerge at an instant, each with velocity arbitwily~~numbaot~thtueextmdediasprce vo. Accondimg to thefundamantal assumption of the randomand time was recantly providad by Lee and Stone (1982). force theory, each of thwe particles is being dkpemed by These latter X+&J show that, apart of course from a constant independent, random iwcei6mtions; and so I assume or, more initial souse-sixe term, an equation of exactly the same form accurateIy, I hope that such a clwter, consisting of an enormous nunWr of individual particks, does in some sense as Equation (131 governs the dispersion in the multipkpartick case also. The conclusion is that in all cases an approximatea t~~b~i~~ that thispoint of quation of the form of Equation (13) describes both singk view isnot theonlyonepoulbkandadntit that it isimp&ect. Yet it seems to me that as a pm&ml matter it is at least as particle diffusion and relative diffusion. (2) The average displacement, y, of the position of a satisfactory as any alternative that rm aware of. Since I consider a group ofpartkka whose initial velocities, particle, i.e. that of the ccntroid of the ensemble-averaged ug, are constant, the random-force model shows (by Equation cloud, is given by Equation (9) of Gifford (1982a). Since the total dispersion, relative to a fixed time or mean-wind axis, (9) of Gifford, 1%2a) that Bz = 7. Contently Dr. Smith’s equals the dispersion relative to the oentroid plus the Equation (7) is just quivaknt to Zfa in (If above. The initial dispersion of the centroid(Gifford, 19S9),Ihe raktive dispervelocity only afWs the mt. p, of the caneoid, not sion can be obtained by subtracting the square of (9) from its spreadi+ Thus, theinconsktencyreferred to in (7)and the (13); accompanying Rgun does not o(xxu, tiuse any group, or 2$x = r; -pl, (1) cluster, of particka rekned with a given velocity, u,,, has the same displacement. Dr. Smith refers to an ensemble whose where? = yz/2&:in the non&nensional notation I used. initial condition involves a mnge of +,-values caused by variations occurring over a time-range of emissions or by Thus, it is seen that the ~~~~ cloud spreading, r:a, rektive to the cent&d, i+e.the mktive diffusion, is given by irregukr source effect& To the extent that these arc properties of the (undisturbed) ambient flow at source locations that Equation (13) with c = 1. Indeed I found, by comparing extend in space or time, the random-force model can in Equation (13) with tropospheric puff- or relative-diffusion data over a wide range of diipersion times, that c = 0.996. principk account for them explicitly by specifying the quantity c (GiITord, 198tb; Lee and Stone, 1982) for any Comparisons with three indepemknt sets of short-range source. In practid appkdow, source effects, such as the plume data derived from photographs similarly produced explosive phase of even a amall puff reti, of plumevalues of c 2 0.96. Thus for all the comparisons I reported, e&c&, must dkturb the theetfective value oft in Equation (13fwasveryclose toone. I buoyancy and sta&e interpret this as good evidence that, on the one band, 113) ambknt Bow more or kas ma&&y and systematkatfy,near the source It seems iiaprtial not to mention unreaIistic, {with c = 1) dascribes rclrtive &&&on from paint souras and. on theother, that none of these data were;nRuenced by for calculating interm&&e and long-range dispersion, to try initial source-extension effaXs (sina c _ 1) to such a dagree to account for such near-&M effects in detail. For experimcntal, finite-duration, near-point releases of a passive tracer, a as to obscure their role as relative diffusion tracers.