the aerodynamic drag of grassland

The Aerodynamic Drag of GrasslandAuthor(s): F. PasquillSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 202, No. 1068 (Jun. 22, 1950), pp. 143-153Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/98519 .

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The aerodynamic drag of grassland

BY F. PASQUILL, Meteorological Office, London

(Communicated by 0. 0. Sutton, F.R.S.-Received 8 February 1950)

An improvised drag-plate apparatus, on the principle of that used by Sheppard (Iz947) on a concrete surface, but suitably modified in design, has been used for exploratory measurements of the aerodynamic drag of grassland. The grass cover was variable (1 to 15 cm. in height), and measurements were made at a number of positions (not simultaneously) in order to obtain an approximate representative value of the drag over a considerable area. Wind velocities were in the region of 500 cm./sec. and, judged in terms of the Richardson number, effectively adiabatic conditions of flow prevailed.

The drag (r0) and the simultaneous vertical distribution of wind velocity (u,,) up to a height of 2 m. were found to be expressible in terms of the law well established in the laboratory, i.e.

uz =z-dlogo ),

with k (von KarmAn's constant) = 0-37, d (the zero plane displacement) = 8 cm. and zo (the roughness parameter) = 0-66 cm. For reasons which are discussed the drag measurements are regarded as approximate, and the close agreement of the numerical value of k with the laboratory value of 0-4 is probably fortuitous. However, the general consistency achieved in this preliminary application suggests that the technique could be profitably developed for a critical investigation of the relation between drag and wind profile and its dependence on atmospheric stability.

In a brief discussion of previous work some evidence is now provided for the validity of the Reynolds formulation of the turbulent shearing stress. Attention is drawn to the application of the present results in treatments of the diffusion of matter in the lower atmosphere.

1. INTRODIUCTION

The relation between the drag of a surface and the distribution of fluid velocity in the boundary layer is of fundamental importance in problems of turbulent flow and plays an important part in treatments of turbulent diffusion in the lower atmosphere. Experimental researches by Nikuradse (I933) and Schlichting (I936) on fully developed turbulent flow through pipes and over flat plates with artificially roughened surfaces have established the following law, written here in the form customary in meteorological application:

= k/ loge zd) (1)

In this expression,

uz = fluid velocity at a distance z-from the pipe wall or plate, measured from the base of the roughness elements,

,, = shearing stress per unit area of the pipe wall or plate, p = fluid density, d = a 'zero plane displacement' associated with the finite size of the roughness

elements, zo= a constant for each type of roughness, in meteorological application termed

the roughness parameter, k = a dimensionless constant = 0-4.

[143



144 F. Pasquill

An equation of this form has been derived by Prandtl and von Karman on theoretical grounds (see Brunt I939, pp. 244-247), and the constant k is usually called von Karman's constant.

The above equation was first applied to the distribution of mean wind velocity near the earth's surface, in adiabatic conditions of flow, by Prandtl (I932), but while the functional form of the equation has since been widely observed to hold in such conditions (see Deacon 1949), the only direct evidence for its explicit validity is that obtained by Sheppard (1947) for a rather unnatural surface. Sheppard measured the drag exerted by the wind on a horizontal plate, floating in oil, under torsional control, and exposed in such a fashion as effectively to form part of an extensive concrete area of smoothness approximating to that of the floating plate. From simultaneous measurements of the vertical profile of wind velocity over the concrete surface it was demonstrated that an equation differing only slightly (see later) from that given in (1) was satisfied, with k = 046, a value which, in view of the unstable atmospheric conditions then existing, was considered to be in good agreement with that obtained in the laboratory.

The assumption of the explicit validity of the above law for airflow over grassland in adiabatic conditions, i.e. with negligible vertical gradient of potential temperature, with also the assumption that the eddy diffusivities for momentum and matter are identical, has been indirectly justified by recent satisfactory interpretations of data on the dispersal of smoke and vapour from artificial ground level sources (Calder 1949)

and of observations on the transport and distribution of natural water vapour (Pasquill I949a). However, direct observations of the relation between drag and wind profile and of the influence thereon of the thermal stratification of the atmosphere have yet to be made over a natural vegetated surface. The present paper describes some preliminary measurements, on grassland, which are first discussed in terms of equation (1) and are then considered briefly in relation to the general problem of aerodynamic drag and turbulent diffusion in the lower atmosphere.

2. INSTRUMENTAL DETAILS AND FIELD TECHNIQUE

The somewhat improvised apparatus used for the measurement of drag was built on the principle of Sheppard's drag-plate, with certain modifications of design to suit the application to a grassland surface. It consists of an aluminium pan floating in water in a slightly larger pan (see figure 1). In field operation the floating pan con- tains a close-fitting sample of the actual grassland surface, and the outer pan is embedded in the ground at the original position of the sample, so that the floating element of ground is exposed approximately in its natural state. Horizontal movement of the floating pan under the action of wind drag is controlled by a flexible metal strip (actually a pen arm of the type used in a barograph) which is attached to one side of the pan and rigidly held in a small clamping block at its other end. Deflexion of the pan is indicated by a long pointer attached to the opposite side of the pan. The control strip and pointer are shielded from the direct action of the wind by slender cases attached to the outer fixed pan. Adjustment of the height and level of the floating pan is made by altering the quantity and positions of brass weights



The aerodynamic drag of grassland 145

carried on a frame attached to the underside of the pan and, if necessary, by altering the quantity of water in the outer pan. This adjustment is made with the control strip unclamped; the latter is subsequently carefully clamped so that there is no resultant vertical force acting on the control strip.

Calibration of the apparatus was carried out in the laboratory by placing small weights in a light paper pan attached to the centre of the floating pan by a hair passing over a light pulley. The relation between the horizontal force applied in a direction normal to the plane of symmetry of the drag-plate assembly, and the deflexion of the pointer was found to be linear within satisfactory limits for a movement of the floating pan over a distance of about -A3 in. on either side of the central position, as would be expected in view of the small displacements involved. With the shorter

in -~,,T7U3

son;f, scl anairr olbt o dmig ,wn-hedn cae wih trnprn

s~ ~~~~~~ 6

oeation; -n levelling feet.

inches p w l

FIGURE 1. Drag plate-plan and section at plane of symmetry. a, exible metal strip rigidly held in clamping block at b or c; d, pointer, with horizontal and vertica adjustment at r (not shown); f, scale and mirror; , oil bath for damping; h, wind-shielding cases with transparent coapers; , false bottom to support ground sample (approx. 3 in. thick); j, water; e, adjustable weights for control of height and lewel of floating pan; 1, level of soil surface during field operation: m, lev7elling feet.

control strip (see figure 1), as used in the measurements discussed here, the corresponding overall movement of the pointer was approiimately 2bin. Outside this range the linear relation was distorted by surface-tension action in the narrow gap between the two pans. All subsequent measurements were contapneduwithin the range of linear response. Calibrations carried out after dismantling and reassembling the apparatus showed an extreme variation of + 3apr in the force/deflexion ratio. This degree of reproducibility was probably as good as could be expected from the rather crude mechanical features of the apparatus and was quite acceptable for the preliminary measurements envisaged. Damping of the movement of the plate, effected by a small metal vane carried on the pointer and dipping in an oil bath, was adjusted until criti'cal damping was achieved.

With the shorter control strip a pointer deflexion of 1 in. was produced by a force corresponding to 3 0 dynes/cm.2 over the whole horizontal area of the floating pan, acting normal to the plane of symmetry of the apparatus, and full response to an applied force was obtained in approxirnately 15 sec.

In field operation the drag-plate assembly was set up with its plane of symmetry across wind. The horizontal force acting at the centre of the floating plate, normal to

VOL. 202. A. IO



146 F. Pasquill

the axis of symmetry, is then -0A sin 0, where A is the area of the floating plate and 0 is the angle between the wind direction and the plane of symmetry. A light vane for recording wind direction, a portable distant-reading temperature gradient assembly (described elsewhere, Pasquill I9496) and masts carrying sensitive cup anemometers were set up close to and cross-wind of the drag plate.

Each series of observations was maintained over a period of 10 min. During this period readings of the drag-plate pointer were taken every 5sec. by an observer lying on the ground on the cross-wind side of the apparatus so as to impose as little as possible disturbance on the air flow. As anticipated the drag-plate deflexions showed considerable fluctuations, though no difficulty was experienced in taking precise readings at the above rate. Before and after each series a draught-proof cover was placed over the drag-plate and zero readings taken. The drag-plate deflexions were mainly in the region of 0 5 in., occasionally 1 in. or more, and were read to 001 in. Anemometers were operated simultaneously at heights of 200, 150, 100, 50, 37 5 and 25 cm., and in consecutive runs were interchanged over the levels 200 to 100 150 to 37l and 50 to 25 cm. The temperature profile was observed over the same height range, but only the differences over the interval 150 to 37*5 cm. and the absolute value at 200 cm. are employed here. Two 3 min. records of wind direction were taken, centred on 21 and 71 min. from the beginning of the observation, and from these a mean wind direction was obtained for each half of the observation period. In reducing the drag-plate results the mean deflexion for each half of the run was associated with the corresponding mean wind direction.

The site of the measurements was a level, well-exposed clayland pasture, with at least 150 yards of similar terrain upwind of the apparatus. The grass cover, however, was variable over distances of the order of yards, and ranged in height from 1 to 15 cm. With a surface non-uniformity on such a scale it was not to be expected that a single siting of the present drag-plate (of effective area approximately 250 sq.cm.) would provide a representative measurement of the drag over a substantial area. The same restriction applies to the vertical wind profile, particularly in the region close to the ground, and, indeed, previous observations on the same site (Pasquill 1950)

have demonstrated a local variation of wind distribution. In order therefore to obtain a closer approach to mutually representative drag and wind-profile data three adjacent areas were chosen on which the local grass length was mainly 1 to 5, 5 to 10 and 10 to 15 cm., and measurements were carried out with the drag-plate and wind- profile apparatus set up at each of these three localities in turn. Two 10 min. series of observations were made at each position, the whole process occupying approximately 4hr.

3. DisCussioN OF OBSERVATIONS AND RESULTS

The observations are presented in table I in the form of mean values for each 10 min. observation, except for the values of wind direction and 0, which correspond to the two 3 min. records of wind direction and are adopted as means for the component 5 min. periods. The value given for r0 is the average of the two values based on the mean drag-plate deflexions in these S min. periods. Before describing the results ultimately derived it is convenient to summarize at this stage the various limitations




I I I i + + I

t -

-I 0 0

O C0 0 6 6 6 04 O I I 1 + +

00

00

X-ZA~~~~~~~~~~~~~~~~~~~~-

O -0

cs r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i

0

O 0-

t H

oS

Q 0 0- r- 0

6660~6

CO 0 CO O o o 0 ^ 5 ^ 0 0 Coj CO 4o 0 t - M O O ( O O O C

z ot 10.<4 2~

p i2 0 0 +?

~~~~~0 P

C)~~~~~~~~~~~~~~~~~~~C O) m CO oo CO CO CO 5 0 CO

O .0 o ?1

Q m

o a t-* o - 0 ' s * s

3.o * 0o0 co L- 01 00 Q o tO- -

0o t o =Z 00 =o t t-

0 0

* k S o - 0 01 C? tD4 Qo ' C O E

0 0 0 0 0



148 F. Pasquill

imposed by the improvised form of the drag-plate apparatus and the prevailing conditions of ground and weather.

The drag measurements refer to a small area of ground which had been freed from its surroundings. In doing so an artificial gap was produced between the test area and the undisturbed ground. Although nominally this gap was very small, approximately -in., in practice it was increased somewhat by the trimming of grass around the edges, a procedure which was necessary to avoid mutual interference of trailing grass blades projecting from the fixed and free edges of the gap. Even so, the artificial irregularity so produced was not greatly dissimilar to irregularities occurring naturally. It will be noted that the cases housing the control strip and pointer projected above the soil surface by about 4 cm. Since, however, these spurious obstacles to flow were on the cross-wind sides of the drag-plate and were for the most part submerged in the natural roughness elements, it seemed reasonable to accept this convenient arrangement for the preliminary measurements rather than to complicate the manipulation and adjustment for wind direction by embedding the cases in the ground. The proximity of the observer provided yet another source of disturbance of the flow conditions over the drag-plate, but as already mentioned this was minimized by the observer lying flat on the ground cross-wind of the apparatus. The limitations imposed by the non-uniformity of surface have been stated in the previous section. Finally, it should be noted that the derivation of the mean drag in the mean direction of the wind directly from the mean drag-plate deflexion requires that the wind direction should be constant apart from turbulent fluctuations, over the period of measurement. Most of the individual records showed an acceptably constant wind direction over the 3 min. periods, though as will be seen from table 1 the two values obtained in each O min. observation differ by as much as 240 in the worst case. For this reason, and as already mentioned, the drag was evaluated separately for the two component 5 min. periods, employing the appropriate values of 0, and averaged to give r0.

Turning now to the data presented in table 1 it is seen that the vertical temperature gradient, as indicated by the temperature differences over the height range 150 to 37 5 cm., varied from a moderate lapse at the beginning to a moderate inversion at the end of the series of measurements. It has previously been demonstrated, however, by Deacon (I 949), and by the present writer (Pasquill I949 a), that the effects of thermal stratification on processes of turbulent transport near the ground are controlled, not by the vertical temperature gradient alone, but by the ratio of the latter to the square of the vertical wind shear, as 'in the now familiar Richardson number,

Ki = g (UT/z+Fr) T (aU/az)2

where T is in ?K, F is the dry adiabatic lapse-rate, and u is in cm./sec. With high wind velocities and hence, usually, large values of au/lz, the departure of the above parameter from zero (the value which occurs with zero vertical gradient of potential temperature) may be quite small even with appreciable values of aT/lz. The approximate values of Ri at a height of 75 cm. have been computed from the temperature and wind velocity data appropriate to 37 5 and 150 cm. by assuming a linear variation




of temperature and wind velocity with the logarithm of the height and are reproduced in table 1. Owing to the fairly high wind velocities all values are less than about 0.01 numerically, with an algebraic mean of - 0O002. These values represent de- partures from 'adiabatic' or 'neutral' conditions of flow -which, in terms of the previously mentioned observations of the effect of thermal stratification, are of a very small order. It is thus considered justifiable to bulk the present series of observations as one sample of drag and wind profile data in effectively adiabatic conditions.

2*4 -

22 a

//

20 _ zX

1.0 ~ ~ ~ 0

X1.6 _ ..0;/

uo //}/~~~~~U/Uo 12--

I* 2

0.6 08 1.0 1~2 uz/u'oo

FIGURE 2. Velocity profile over grass 1 to 15 cm. high in effectively adiabatic conditiols. Curve, a, d=O. Curve b, d=8 cm.

It will be noted that the individual wind profiles, specified in terms of uzI1ioo so as to facilitate comparison of the various observations, exhibit considerable variation. Furthermore, graphs of the individual results, not produced here, display some de- partures from smooth u, log z profiles. These features are thought to be mainly a reflexion of genuine local variations in the wind distribution arising from the non-

uniformity of the surface. On the other hand, the magnitudes of | which

might have been expected to exhibit wide variations, are surprisingly constant, though there is an indication of a slight systematic increase with increase in local grass length.

Considering now the mean results for the whole series we may note that the wind velocities show a good approach to a smooth u, log z profile (figure 2a), which is, however, slightly concave to the log z axis. This is attributed to the zero-plane dis-



150 F. Pasquill

placement discussed previously (d in equation (1)). Trial plottings show that the best approach to a linear relation between u and log (z - d) is obtained with d =8 cm., a value which seems reasonably in accordance with the description of the grass cover. The profile is plotted in this form in figure 2 b, and from this we have

_ -= 0 466[log1o(z-8) + 0-182]. (2)

From equation (1) ?0 2OJQ)[logl0(z -d)-1logl10zo], (3) M4k100 p

100o kuloo lP)

which is satisfied by (2) if

z= 0-66 cm. and k J-= 0-466. (4)

From table 1 = 0-075, Uioo P

which with (4) gives k- 037

to be compared with theJaboratory value of 0 4. In view of the previously discussed limitations of the present measurements the

close agreement of the observed value of k with that established in the laboratory may be to some extent fortuitous. However, the satisfactory nature of this result and the general success achieved in manipulating the preliminary form of drag-plate give reason to believe that the present apparatus and technique could be profitably developed. With improvements in the design of the drag-plate, especially some increase in scale, and further reduction of the various factors possibly contributing to a distortionbof the air flow, and with a more rigid restriction to uniformity in ground and atmospheric conditions, it seems likely that the relation between drag and wind profile could be more critically examined and valuable data obtained on the out- standing problem of stability influence.

4. FURTHER DISCUSSION

Although the present measurements are to be regarded as approximate and exploratory in nature they provide data of a character hitherto not available on the subject of the horizontal shearing stress, r, in the lower atmosphere. This shearing stress, the surface value of which has been measured here, occupies a position of fundamental interest in all considerations of atmospheric turbulence. It is not proposed here to attempt a historical survey of the problem, but it is of interest to consider the present results in the light of certain previous observations and to note the application of the present results in treatments of the diffusion of matter in the lower atmosphere.

Comparisons of special interest are afforded by the work of Sheppard (I947), who measured the drag of an artificial surface effectively forming part of an area of concrete, and whose experimental technique, suitably modified, has been followed here, and by that of Scrase (I930), who evaluated the Reynolds stress (-r = -pu'w')




directly from cinematograph records of the mnovement of light vanes at two heights (1 5 and 19 m.) above downland.

As would be expected from the nature of the surface Sheppard's values of rO are appreciably lower than those given here for grassland. Direct comparison may be made by evaluating a drag coefficient, CD, defined by the relation

t0 = CDPU2.

The magnitude of CD depends on the level at which u is measured,* and taking this to be 200 cm. we have the following values of CD:

concrete 2*2 x 10-3, grassland 8*6 x 10-3,

for wind velocities respectively 421 and 522 cm./sec. It is also noteworthy that Sheppard related his drag and wind-profile observations

in terms of the equation following from the Rossby & Montgomery (I935) form of the mixing length theory of turbulence, with the boundary condition u = 0 on z = 0, i.e.

m = I loge( , z?) (5)

constancy of T with height being implied. In Sheppard's analysis no difficulty arose regarding the origin of z, because of the smooth nature of the surface, but in the present analysis it is immediately obvious that a reduction of the results in terms of equation (5) would introduce a serious ambiguity in the value of zo. This adds to previous indications (see Deacon I949) that in formulating a general wind-profile relation for rough surfaces it is necessary to introduce two independent arbitrary constants, i.e. z0, the roughness parameter, and d, the zero-plane displacement, both of which depend in a complex fashion on the geometrical form of the rough surface.

In Scrase's original paper a value of the shearing stress is given for the height of 19 in., and it is stated that a record taken at 1P5 m. gave a value roughly one-quarter of that at 19 m. Recently, however, Scrase has pointed out in a private communica- tion that the two measurements were of 1 min. duration and separated by an interval of half an hour. Since no simultaneous measurements of wind profile were made the two values of shearing stress cannot strictly be compared, though by appealing to an approximate wind-profile relation and adjusting the observed shearing stresses in accordance with the implied difference in wind velocity on the two occasions, Scrase now finds the 1 5 m. value to be roughly 0 4 time that at 19 m.

The foregoing variation of shearing stress with height, though approximate in view of the indirect nature of the comparison, still constitutes a striking contradiction to the familiar concept of the constancy of T with height. This constancy has been proved by Ertel (I933) to hold up to a height of about 25 m., and Ertel's proof has been examined and confirmed by Calder (I939). A satisfactory clarification of this

* It follows from equation (1) that the drag coefficient relating to fully developed turbulent flow over a rigid rough surface is otherwise independent of wind velocity and is dependent only on the geometrical form of the surface. The same is not necessarily true of a vegetated surface, since the natural roughness elements may bend over in the wind, so leading to a smoothening of the surface as the wind velocity increases (see Deacon 1949). This effect would result in a diminution of CD with increase in wind velocity.



152 F. Pasquill

discrepancy has yet to be made. One possible explanation, which does not seem to have been considered hitherto, is that the shearing stress observed at 1-5 m., which is presumably dependent primarily on the nature of the surface in the upwind 50 m. or so, is perhaps not strictly comparable with that at 19 m., since the latter would be influenced by surface conditions at a much greater distance. Scrase's observations were made on Salisbury Plain with a level uninterrupted exposure in the upwind kilometre, but even so it is conceivable that the turbulent properties at 19 m. were affected to some extent by the remains of the large-scale eddy structure generated by major irregularities in the surface (houses, trees and topographical features in general) at considerable distances from the site.

A satisfactory understanding of the above points could only be provided by further observation, but for the present purposes it is thought reasonable to regard the value obtained at 15 m. as the more reliable figure for the shearing stress in the air immediately over grassland and the more appropriate figure for comparison with the present surface value. Scrase's measurement gives

-pu'w' at 1 5 m. = O-8 dynes/cm.2 for U - 468 cm./sec. at 1 5 m.,

while from the average of the present results

r = 1 4 dynes/cm.2 for -494 cm./sec. at 1-5 m.

Both observations were made over grassland, but it is not possible to say to what extent the surfaces were similar in detail. With this qualification the agreement is nevertheless very reasonable and may be regarded as some evidence for the validity of the Reynolds formulation of the turbulent shearing stress in the atmosphere near the ground.

Finally, it may be noted that the relation of the horizontal turbulent shearing stress to the vertical distribution of wind velocity specifies the eddy diffusivity K of the atmosphere. This provides a basis for treatments of diffusion of matter in the lower atmosphere, a general discussion of which has recently been given by Sutton (1949). In these applications the assumption is made that the eddy diffusivity for matter is identical with that for momentum as derived from the wind profile. Referring to the particular relation expressed in (1) and writing

au Ir = Io =pKaZ

we have K=kJk(T!)(z-d)- = Uj7%. (6)

The eddy diffusivity in adiabatic conditions over a natural rough surface is thus directly proportional to the height above the surface (zero plane displacement being applied) and to wind velocity, at some reference height z1, and is dependent on the roughness of the surface. For the surface involved in the present measurements of drag the numerical value K at 200 cm. for a wind velocity of 500 cm./sec. at the same height is calculated to be 2-5 x103 cm.2/sec. The form of K given in (6) has recently been shown by Calder (I949) to lead to solutions of the two-dimensional equation of




diffusion which are in excellent agreement with observations on the distribution of smoke and vapour from artificial sources at ground level, for adiabatic flow over downland. Furthermore, the same form of K leads to a simple expression for the local rate of evaporation from the ground in terms of the vertical distribution of wind velocity and vapour pressure, and this also has been confirmed by observations over grassland in adiabatic conditions (Pasquill I949 a, I950).

Acknowledgement is gratefully made to the Head of the School of Agriculture and the Director of the University Farm, Cambridge, for facilities provided during the course of this investigation, to the Director of the Meteorological Office for permission to publish the results obtained and to members of the Meteorological Office Unit at the School of Agriculture for assistance in the observational work. The writer is much indebted to Dr F. J. Scrase of the Meteorological Office who kindly made available additional and hitherto unpublished information concerning his shearing stress measurements, and to Professor 0. G. Sutton, F.R.S., for helpful discussion and criticism during the preparation of this paper.

REFERENCES

Brunt, D. 1939 Physical and dynamical meteorology. Cambridge University Press. Calder, K. L. 1939 Quart. J.R. Met. Soc. 65, 537. Calder, K. L. 1949 Quart. J. Mech. Appl. Math. 2, 153. Deacon, E. L. I949 Quart. J.R. Met. Soc. 75, 89. Ertel, H. I933 Met. Z. 50, 386. Nikuradse, J. 1933 Forschungsh. Ver. dtsch. Ing. no. 361. Pasquill, F. I949a Proc. Roy. Soc. A, 198, 116. Pasquill, F. I949b Quart. J.R. Met. Soc. 75, 239. Pasquill, F. 1950 Paper in course of publication in Quart. J.R. Met. Soc. Prandtl, L. 1932 Beit. Phys. frei Atmos. (Bjerknes Festschrift), 19, 188. Rossby, C. G. & Montgomery, R. B. I935 Pap. Phys. Ocean. Met., Mass. Inst. Tech. 3, no. 3. Schlichting, H. 1936 Ingen. Arch. 7, 1. Scrase, F. J. 1930 Met. Off. Geophys. Mem., Lond., no. 52. Sheppard, P. A. 1947 Proc. Roy. Soc. A, 188, 208. Sutton, 0. G. 1949 Atmospheric turbulence. London: Methuen.



the aerodynamic drag of grassland

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