The Present Position in the theory of Turbulent Diffusion

THE PRESENT POSITION IN THE THEORY OF TURBULENT DIFFUSION Sir Geoffrey Taylor Cavendirh Laboratory, Cambridge, England

When I was asked to name a subject on which I could prepare a talk for this Symposium I chose one which I thought might cover anything I could find to say, but on reading some of the more recent literature on turbulent diffusion I realized how far the subject has progressed since I was active in it. Perhaps the most instructive thing that I can do is to call to mind some of the ideas which were current thirty or forty years ago and compare them with those now prevailing. My own connection with the subject began in 1913 when I spent seven months as meteorologist on the old sailing whaler Scotia, measuring the distribution of temperature and humidity above the cold water of the Grand Banks of Newfoundland. These measurements were made by flying, from the quarterdeck, kites which had temperature, humidity and pressure recorders suspended within their frames. It was not possible to get to any great height with this equipment but the very rapid changes in temperature with height which occur in the first few hundred feet in this part of the world made the results of interest in spite of this limitation. The records showed that in most cases the temperature of the air was much higher at a few hundred feet than at sea level. This was clearly due to the fact that warm westerly winds blowing off the American continent during the summer were rapidly cooled when they reached the icebergladen waters off Newfoundland. When I came to work up the resuIts I found that in six cases I was able to trace roughly the paths by which the air which was being explored had reached the ship’s position. I was in fact lucky to be able to do this for the method depended on the chance that a ship which recorded wind force and direction happened to be in the neighbourhood of the place where I had calculated the air mass to be at various times up to three days before the ascent. I n this way I was able to connect the very pronounced bend in the temperature and humidity records, which marked the height to which the cooling had penetrated upwards with the thermal history of the particular air mass being examined. The rate at which this wave of cooling penetrated into the atmosphere could be accounted for by a transfer process of the same nature as 101

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molecular conductivity but much more vigorous. This idea was not new; i t had been used by many meteorologists. The only possibly new element that I introduced (Taylor, 1915) into it was the use of a definite “mixture length ”. Formally this is a perfectly definite conception. The rate of heat transfer and the mean temperature gradient could both be measured and their ratio determines a coefficient of diffusivity for heat. This has the dimensions of length x velocity. The transfer is in a vertical direction so that the relevant velocity is the vertical component, and this could be measured and the time-mean of its absolute value or the root-mean-square value determined. Thus the length which is defined as (Diffusivity/vertical turbulent velocity) is a measurable length . Some years later (Ithink in‘1920, though I have not been able to trace this reference) Prandtl independently introduced the idea of a mixture length and his name is usually associated with it. Though, as I mentioned, the idea that a mixture length can be defined and measured was brought out in my paper of 1915 I did not attempt to associate the magnitude of the turbulent velocity with this length. This was the new feature of Prandtl’s theory which assumed that the turbulent velocities at any point depend only on the mean rate of shear and the mixture length and were in fact proportional to the product of the two. This additional assumption was needed in order to make a mixture-length theory into a tool which could be used in interpreting hydrodynamic experiments, but even then a further assumption was needed, namely that the mass of fluid or “austauschyywhich carries the transferable property is unaffected by fluctuating pressure gradients or by molecular diffusibility during the period in which it preserves its identity. It is clear that if the transferable property with which one is concerned is momentum the fluctuating pressure gradients can alter the momentum of the transporting mass without necessarily causing it to mix with its surroundings. However, Prandtl made the assumption that fluctuating pressure gradients have no effect on the average transfer of momentum and the theory so produced is his well-known Momentum Transport theory. I n my 1915 paper I did not take that bold step, but I realized that if the whole field of flow, the mean as well as the turbulent velocities, were limited to two dimensions, vorticity would be a transferable property and would be unaffected by fluctuating or steady pressure gradients, if the effects of viscosity were neglected, and so give rise t o a theory of Vorticity Transfer. One of the main features of these mixture-length theories was that the transporting “ austausch ” carries all its transferable properties from a point in the field which is regarded as its beginning to another point at

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which it loses its identity and delivers the property to the surrounding fluid. According to the Prandtl theory therefore, the transfer of momentum and other transferable properties such as heat or water vapour are absolutely connected. The vorticity-transfer theory is incapable of dealing with any situation except one in which both the turbulent and mean velocity fields are two-dimensional. The only practical case in which these conditions approximately apply is that of flow in the wake behind a cylinder in a wind stream where the large-scale turbulence seems to have approximately a two-dimensional character analogous to the “K&rm&nStreet”. I n that case the difference between the transport of heat and momentum was found to be what was predicted according to the vorticity-transport theory (Taylor, 1932). On the other hand, the vortex lines close to a solid surface lie nearly parallel to the mean wind and these are the conditions in which Prandtl’s theory might be expected to apply so far as the relationship between the transport of momentum and heat is concerned. I n fact most experiments on flow near a solid surface seem to show that it does. Though mixture-length theories are not in general capable of giving a realistic description of the relationship between mass and momentum transport, they do lead to the expectation that transferable properties other than momentum will be transported in exactly the same way, and in particular that the diffusivity for water vapour and heat in the atmosphere will be the same. My observations on the whaling ship Scotia in 1913 showed that the sharp bends in the curves of distribution of temperature with height coincided with those of relative humidity, which was the quantity which the rather primitive hair hygrometer I was using gave. Reducing the results by means of tables of vapour pressure to give distribution of water-vapour content made the bends in the curves far less pronounced but I still convinced myself (perhaps wrongly) that they indicated that the diffusivity of water vapour and heat are identical. I realized that the molecular diffusivity of water vapour into air is accidentally not far from that of heat in air, but I did not believe that was likely to be the cause of the apparent equality of their turbulent diffusivities, and it was this that attracted me to the idea of a mixture-length theory. The chief difficulty in a mixture-length theory is to form a physical picture of the process by which a body of fluid carrying some transferable property discharges its load into the surrounding fluid. Taken literally the original mixture-length theories would envisage the unrealistic conception that a mas8 of fluid retains its load unchanged till at a certain instant it meets with a sudden catastrophe and disintegrates into fine threads or drops which then discharge their load by molecular processes.

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I n searching for a more realistic model I first thought that the function of turbulence in vertical transfer in the atmosphere might be regarded as an agency which caused thin vertical sheets to form. These would carry loads of transferable properties vertically and would discharge them by horizontal molecular diffusion as they passed upwards or downwards. Such a model would avoid the necessity for thinking of a catastrophic end, analogous to molecular impact, to the transporting body of fluid or “austausch” ;but it would suffer from the disadvantage that loads which had different molecular diffusivitieswould be carried different distances, so that the mixture length would depend on the molecular diffusivity of the property transported. It was my belief that this is not true that led me to reject that model. The model which I have just described was Eulerian in the sense that no attempt was made to follow particles of the diffusing matter or property, and it could even be regarded as describing a steady state. To devise a more realistic description of turbulent diffusion I had to base it on the Lagrangian conception in which attention is fixed on a particle of the diffusing matter which preserves its identity as it moves in the turbulent field of flow. I therefore tried to describe the diffusive properties of turbulence by defining the correlation between the velocity of a particle at one instant and that of the same particle at a later time (Taylor, 1921). I found in fact that the diffusion from a fixed point-source can be described when only this Lagrangian time-correlation function and the mean square velocity is known. I found that if this correlation vanishes after an interval of time, the diffusion for much longer periods proceeds as though it were due to a constant virtual coefficient of eddy diffusivity. Since, as I have already mentioned, a mixture length can be formally defined without reference to the physical process of mixture, the time-correlation function makes it possible to think of a mixture length roughly proportional to the average length of path which a particle traverses before its velocity becomes uncorrelated with that which it had at the beginning of the path. This method of describing diffusion makes it possible to define a mixture length without considering any physical processes, but it involves the assumption that the particles of the diffusing substance are indistinguishable so far as their motion is concerned from those of the transporting fluid. Since turbulent diffusion is mostly due to eddies of the larger sizes and the large part of the Lagrangian correlation-time curve is also due to them, one would only expect molecular diffusion to affect turbulent diffusion if the amount of material which could diffuse by molecular processes from a large-scale eddy were appreciable during the period over which its velocity preserves appreciable correlation with its initial velocity.

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The idea that the increasing diffusivity of the atmosphere with increasingtime of diffusion could be represent,edby a correlation function which, while decreasing with time, retained a positive value so that its integral continually increased was, I think, first put forward by Sir Graham Sutton. Sutton (1932) also pointed out as an empirical fact that expressions involving error functions of space coordinates, associated with time factors differing from those appropriate to constant diffusivity might be used to represent dispersion from a fixed source. Roberts and others had earlier pointed out that these expressions were the solutions of a modified Eulerian diffusion equation in which the diffusion coefficient varied with the time since the diffusing material had been concentrated. It seems to me that this is an illogical conception. The one thing that seems to be agreed, whatever theory one may have about diffusion,is that diffusing distributions are superposable. If therefore you attempt to analyse the distribution of concentration from two source8 which started at different times by this method, it would be necessary to assume, in places where the distributions overlapped, that the diffusion constant had two different values at the same time and at the same point in space. It seems therefore that no physical meaning can be attached to the use of equations in which the coefficient of diffusion varies with the time of diffusion, even though the formulae produced by their use do represent adequately the concentrations in particular cases. This point was well understood by L. F. Richardson whose important paper (1926), “Atmospheric diffusion shown on a distance-neighbour graph”, initiated the modern approach to the subject. Richardson was a very interesting and original character who seldom thought on the same lines as his contemporaries and often was not understood by them. He began his paper with the provocative thesis that it would not be nonsensical to deny that an average called the velocity of a gas can be defined at all. In fact he did not use the conception of a contiiiuous velocity. He discussed diffusion in one dimension and layed down a statistical method for describing the number of particles, n, which are situated in unit length at distance 1 from each particle. He then showed that the principal agents in separating two particles are eddies of length comparable in size with 1 and he summed up the situation by showing that n must obey an equation of the form

K is therefore of the nature of a diffusion coefficient which depends on the separation of particles and does not suffer from the logical difficulty

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of attaching a meaning to a diffusibility which is a function of the time of diffusion. To find out how K vanes with 1 Richardson took a large number of recorded values of K . These had been determined by interpreting diffusion measurements over a limited range of I as being due to a constant diffusivity in that range. He found that for values of1 ranging from 0.05 to lo8 cm., K varied from 0 . 3 to 1011 cmZ/sec. and he plotted these on the logarithmic diagram reproduced in Fig. 1. It will be seen that if the lowest point, which refers to molecular diffusion, and the

s

0

LOG, (SEPARATION 1IN CM)

I0

FIU.1. highest point which refers to transfer over distances of thousands of kilometres are left out of consideration the straight line

K

(2)

=

-

0 2 14'3

is a very good approximation to the curve between 1 = lo2 and 1 = 106 cm. Since the curve shown here seems to contain all the observational data that Richardson had when he announced the remarkable law (2),i t reveals a well-developedphysical intuition that he chose as his index 2 instead of, say, 1* 3 or 1 4 but he had the idea that the index was determined by something connected with the way energy was handed down from larger to smaller and smaller eddies. He perceived that this is a process which, because of its universality, must be subject to some simple universal rule. It is perhaps rather surprising that he did not take the step which Kolmogoroff (1941) and Obukhov took fifteen years 'later, namely to express his equation non-dimensionally using only the two

-

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physical quantities which could be relevant to a universal rule regulating the handing down of energy, namely E the rate of energy dissipation and v the dynamical viscosity. The critical step taken by Kolmogoroff and Obukhov simply is to express the dimensions length and time in terms 4 time has dimensions of E and v . Length has dimensions v 3 J 4 ~ - 1 l and v 1 J a ~ - l Jso 2 , that if, as Richardson believed, the diffusion of a distribution of concentration u could be expressed by the equation (3)

the dimensions of A are 1 2 - m t - l . Expressed in terms of E and v, A has ) mthat it is only when m = g that the law of dimensions ~ - l + ( ~ / 4 ) m ~ - ( 1 / 4so diffusion does not depend on v. This step however was not taken for fifteen years. Richardson's equation ( 1) can be expressed non-dimensionally using Kolmogoroffs transformation. Thus writing t' = tv-1'2E1/2, 1' = l d 4 (3) becomes (4)

One cannot go further without knowing a value to take for E . Brunt made c -a~general average which would make (4) an estimate E = 5 ~ m ~ s e as become

All that can be said of (5) is that the factor 0 . 1 2 is of order 1 which is what one would intuitively expect from such a general non-dimensional equation. It would be interesting if one could make a more precise determination of this number. Attempts have been made to measure diffusion in a wind tunnel containing nearly isotropic turbulence for which E has been measured fairly accurately. Unfortunately a very large tunnel would be required to attain a good range of eddy sizes to which Kolmogoroffs principle could legitimately be applied. Another difficulty is that the turbulence in a wind tunnel is decaying as the diffusion proceeds. This difficulty could probably be overcome by a method due to Batchelor, who pointed out that observations show that, in a, wind tunnel, turbulent velocity decays approximately as t-112 while the scale increases as t1I2. If then one makes the assumption that these rules apply to all wave-lengths which are responsible for diffusion one can infer what the diffusion would

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have been if the turbulence were not decaying. Using this method one can perhaps determine the correlation curve which describes diffusion from a fixed source. Perhaps the greatest difficulty in using a wind tunnel to determine the constant in the universal diffusion equation may be to devise a method for measuring two-particle correlations. This might perhaps be done if one could set up apparatus for measuring the spread of concentration at each instant from the centre of gravity of the instantaneous distribution emitted from a fixed source but such measurements would not be easy and, as Batchelor has pointed out, the interpretation of them would not be as simple as appears from the simplicity of Kolmogoroff’s and Richardson’s expression. Measurements in the atmosphere even under the most favourable conditions could hardly be interpreted directly in any particular case, even if E and the distribution of concentration could be measured accurately, because the turbulence is certainly not sufficiently uniform over its depth to permit uncritical application of a theory which depends on uniformity as well as local isotropy. The reason why Kolmogoroff’s analysis seems to apply so well to Richardson’s synthesis of all observations of diffusion has been touched on by Batchelor and others but I think further theoretical work on that subject might be undertaken. I have neither the time nor the ability to describe the great amount of theoretical work that has been done on diffusion in recent years by Kamp6 de FBriet and others. I might however mention one set of papers by Batchelor (1949 and 1952) because they are so directly related to the work of Richardson. Richardson’s work is an analysis of the statistics of the distance between two particles wandering in one dimension. Batchelor extended this to three dimensions and took it as the case n = 2 in a set of statistical descriptions of the wandering of n particles. The case ?z = 1 describes diffusion of a single particle from a fixed source. These two cases have been described by some workers as Lagrangian and Eulerian representations. This seems to be an unfortunate way of describing them since the term Eulerian has usually been used to describe the distribution in space at one instant of time of the velocity field. It is only in the Eulerian system that one can relate a statistical description of a turbulent field of flow to E, the rate of dissipation of energy by viscosity. The simplest thing to describe in the Eulerian field is the mean square of the turbulent velocity, but to describe the scale of turbulence twopoint correlation between fixed points is needed and the description is very much simplified in the case when the turbulence is isotropic. In that case K&rm&n(1937) showed that only a single scalar function of distance between two points is needed to give the correlation between the

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vectorial velocity at two points. The second differential coefficient of this function at the origin is all that is required to determine E (Taylor, 1935). This gives directly the rate of decay of the total amount of turbulent energy in isotropic turbulences but little else. Since it is theoretically possible, using the Navier-Stokes equations, to trace the changes in a field of flow which follow any given initial distribution of velocity, it might be thought possible, though no doubt beyond our power, to find average values at subsequent times for a set of initial conditions which themselves are only known through a number of averages. If it were possible to do this the dispersive power of turbulence would be one of the averages so calculated. This line of approach, however, fails at a very early stage. KBrmBn and Howarth (1938) showed that the rate of change of a two-point Eulerian velocity correlation can only be found if the values of certain three-point correlations are known, and the rate of change of a three-point correlation is only known when the values of some four-point correlations are known. Thus though the essential parameter E of the Richardson-Kolmogoroff universal law of diffusivity is known only when the second differential coefficients near the origin of the two-point correlations in the Eulerian description are known, theory gives no indication that there is any direct connection between the Eulerian two-particle velocity correlation at a given time and the Lagrangian correlation between the velocities of a particle at different times. Though it is difficult to connect experimental measurements of dispersion with measurements of E made under the conditions assumed in Kolmogoroffs theory and so to determine the constant in the universal diffusion law, there is one case in which a connection between the Eulerian and the Lagrangian methods of describing turbulence has been traced, namely that of dispersion in a turbulent flow through a straight pipe. For this reason it seems fitting to conclude with a short description of this case though it has only a very indirect bearing on atmospheric dispersion. The dispersion of soluble matter in laminar steady flow through a pipe is primarily due to laminar convection (Taylor, 1953), the flow in the middle of the pipe being faster than it is near the wall. If a concentrated mass of diffusible material is introduced into the flow the part of it which is at the centre of the pipe gets carried into the uncontaminated fluid. This produces a radial gradient of concentration which causes radial transport by molecular diffusion and so prevents the contaminating substance from spreading along the tube as rapidly as it would in the absence of molecular diffusivity. The result of this combined convection and molecular diffusion is to make the soluble material disperse relative

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TAYLOR

to a point which moves with the mean velocity of flow as though it were affected by a virtual coefficient of diffusion

where a is the radius of the tube, u the mean velocity, and D the COefficient of molecular diffusion. When the flow in a pipe is turbulent a similar mechanism occurs (Taylor, 1954), but the radial transfer of matter is due to turbulence. The magnitude of the radial transfer of momentum in a pipe is known. At radius r the turbulent shear stress is w/a, where T is the shear stress at the wall, r = a. The distribution of mean velocity is known as a result of numerous experiments. Thus the coefficient of turbulent transfer of momentum is known. It has already been pointed out that near a solid surface the coefficients of diffusivity for momentum and other transferable properties are found to be very nearly equal. If therefore it is assumed that they are actually equal the radial coefficient of diffusivity for transport of matter is known. I n this way, using the observed universal connection between the radial distribution of mean velocity and the surface stress, it was predicted (Taylor, 1954) that dispersion along a pipe relative to a point moving with the mean speed of flow would be as though it were due to a virtual coefficient of longitudinal diffusivity. (7)

KL = 10*1aw*,

where the surface stress T is p v f . Thus if a concentrated mass of some contaminant is introduced at time t = 0 into a stream flowing through a pipe it will spread so that the concentration is proportional to t-l/aexp( -z2/4K,t). This in fact is found to be the case (Taylor, 1954) and measurements of the longitudinal distribution of contaminant in straight pipes verified with considerable accuracy. The mean rate of dissipation of energy per unit volume is 2UT/a = EP so that E = 2 u v f / a ,where u is the mean velocity of the stream. For pipes with similar degrees of roughness v*/udepends only on Reynolds number which for large pipes is usually of order about &, Expressed in terms of f, Equation (7) can be written

K L = 10 - 1

(z)

113 a413 ~ 1 1 3 .

As would be expected on dimensional grounds (8) has the same form as Richardson’s and Kolmogoroff ’s universal diffusion coefficient, but the numerical value of the constant has been determined. I n a pipe for which ./v* = 20, KL = 2 -94a413~1/3. The number 2 . 9 4 is much greater than the

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number 0 . 1 2 which was found by very rough methods in t h e analogous universal diffusion law, b u t t h e diffusion in t h e pipe is essentially due t o convection which is modified by transverse turbulent diffusion. REFERENCES Batchelor, G. K. (1949). Aust. J . aci. Rea. 2,437. Batchelor, G. K. (1952). Proc. Camb. phil. SOC.48,345. Khnhn, Th. von. (1937). J. aero. Sci. 4, 131. Urntin, Th. von., and Howarth, L. (1938). Proc. ray. SOC. A 164, 192. Kolmogoroff, A. N. (1941). C.R.A d . Sci. U.R.S.S. 30,301. Richardson, L. F. (1926). Proc. roy. SOC.A 110, 709. Sutton, 0. G. (1932). Proc. roy. SOC.A 135, 143. Taylor, G , I. (1915). Phil. Trans. A 215, 1. Taylor, G. I. (1921). Proc. Lo&. math. Soc. 20, 196. Taylor, G. I. (1932). Proc. roy. SOC.A 185, 685. Taylor, G. I. (1935). Proc. roy. SOC.A 151,421. Taylor, G. I. (1953). Proc. Toy. SOC.A 219, 186. Taylor, G. I. (1954). Proc. roy. SOC.A 238, 446.

DISCUSSION REPORTER: 9. CORRSIN

0. a. SUTTON. When, for the first time, accurate data on diffusion were collected a t Porton, there was an urgent need for a technique which would allow different types of experiments t o be analysed and compared, and certain technical problems to be solved numerically. This situation could be compared with that which occurred in the development of the theory of heat. Fourier’s ThCorie awlytique de Ea ckuleur provided the mathematical machinery for solving explicit problems, but the philosophy lay more in the domains of thermodynamics and kinetic theory. I n atmospheric diffusion, the emphasis is now being placed more on the structure of a turbulent fluid and I believe this to be correct. An example of the way in which such studies illuminate the observations is given by the variation of the lateral and vertical dimensions of a cloud with time of sampling. It is known that the cross-wind dimension increases fairly rapidly, up to a point, with time of sampling, but that there is no such significant increase vertically. Panofsky’s data on the spectrum of turbulence throw considerable light on the matter. It is necessary also t o be sure that the mathematical models proposed do not obscure the real facts as revealed by measurement. Thus, although i t is tempting mathematically to use a model in which concentration from a continuous point source varies inversely as the square of the distance from the origin, the fact is that the power of the distance involved is significantly less than 2, and I regard this as important. Similarly, in the problem of the convective jet in calm air, it is not correct to assume that the radius of the jet increases as the first power of the height, although this greatly simplifies the mathematics.

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I n conclusion, I believe that for many of the applications of diffusion theory a tolerably satisfactory technique has now been evolved, and I particularly welcome the trend in research t o look more deeply into the far more difficult problems of the structure of turbulence. w.v. R. =us. I suggest that in general one might expect the kinematic viscosity v and molecular diffusivity D t o appear in the expression for turbulent diffusivity, given as

K

10*lau*

for longitudinal diffusionin pipe flow. The apparent unanimity among available experiments on the empirical coefficient (10* 1) may be attributable t o the fact that all are liquid cases with asymptotically large molecular Prandtl number vjD. c. H. BOSANQUET. I see no philosophical objection to a diffusion coefficient which varies with scale. If an eddy diffusion coefficient is regarded as proportional to the product of the square of eddy velocity and the averege time between direction changes, then the coefficient can be found by studying the relative motion of a particle pair. The action time of an eddy is approximately proportional to its linear dimensions. Further, a large eddy, affecting both particles in the same way, will gjve small relative velocities while large relative velocities may be produced by those eddies with linear dimensions of the same order as the particle separation. These two statements lead me to conclude that if eddies of all sizes are present and have the same associated velocity, the diffusion coefficient will be proportional to particle separation. For a cloud of particles, I suggest that the coefficient governing the motion of any particle relative to the centre of gravity of the cloud is proportional to its mean distance from all the other particles, i.e. to the radius of gyration of the cloud about the particle. This gives equations which are particularly easily solved for the two-dimensional case of a cylindrical cloud. The diffusion coefficient relative to a fixed origin is the sum of two terms, one of which governs the expansion of a cloud and the other the wandering of its centre of gravity. The sum of these will be a function of position only, so that it is unnecessary to postulate different coefficients occurring simultaneously at the same point. Near the ground large eddies cannot be present so that a cloud whose radius is comparable with its height will increase its radius a t a decreasing rate as i t expands, while small clouds will expand linearly with time. G . I. TAYLOR. The variation of diffusion coefficient with scale is not the point a t issue. The question is rather the use of a spatially varying diffusivity. I would emphasize that a local spatially variable diffusion coefficient used in a “diffusion equation” is an Eulerian concept, and hence cannot be directly applied to the Lagrangian phenomenon of turbulent dispersion from a fixed source.