On Roughness Length and Zero-Plane Displacement in the Wind Profile of the Lowest Air Layer

Transcription

1 April Kyoiti 101 On Roughness Length and Zero-Plane Displacement in the Wind Profile of the Lowest Air Layer By Kyoiti Takeda Faculty of Agriculture, Kyushu University, Fukuoka (Manuscript received 1 December 1965, in revised form 11 February 1966) Abstract To describe the turbulent character in plant canopies, a simple model is assumed, and from conditions of the turbulent flow consisting at the boundary of two spaces, i. e. over and under a plant canopy, a relation is deduced which connects d (zero-plane displacement), zo (roughness length) and H (roughness height). It is made clear that when the wind rises, i. e. H decreases, there are several cases of variation of d against zo. The experimental results appearing very complicated at first sight is shown as expected from the theory. A diagram is given which illustrates the relation between d, zo, H and (u/v*)z=h and in which any rough surface is represented by a single point, being specified by not only the surface is smooth or rough, but also its roughness is dense or sparse. Also the so called skin-friction coeff=icient is discussed, and it is shown that the coefficient generally does not have a simple relation with d or zo, but it increases when the rough surface becomes more dense. 1. Introduction It is well known that the wind profile of surface layer in the case of the adiabatic condition is represented sufficiently by a formula as where u denotes wind velocity, z height above the ground surface,u*, frictional velocity, ic von Karman constant, and zo roughness length. But when the vegetation is tall, the equation (1) has been found better to be modified in the form : u = where d is called u*/klnz-d/zo a zero-plane (2) displacement in agreement with its actual meaning of representing the effect of the equivalent displacement of the ground surface upon the wind profile. Needless to say that zo was introduced as an integration constant representing the height where u=0. On the other hand d was introduced quite empirically to secure the validity of the logarithmic profile (1) in the space over the vegetation. Hence it seems that zo and d are independent of each other. For the sake of the recent accumulation of data of wind profile over rough surfaces, we are able to make some considerations concerning the variability of zo and d with other factors such as wind velocity and grass length. Thus Calder (1959), analysing the data of Deacon's measurements over various grasscovered surfaces on Salisbury plain, showed that d increased with the wind velocity at lm height, while zo decreased. Over shortgrass surfaces, d could effectively be taken as zero, but zo decreased as before, which was explained by the fact that the deflection of the long grass blades by the wind resulted in a `smoothing' of the surface. More recently, Tani (1960) has published extensive data of zo and d over paddy fields with about 90 cm grass length, which seem to show definitely that zo varies inversely with d, though their relations to the wind velocity at 1.5m height are far more complex than that of Deacon. A problem then arises : What is the interrelationship between zo and d? It is the purpose of this paper to deduce theoretically the relation, if any, between zo and d in the case of the adiabatic atmosphere.

2 102 Journal of the Meteorological Society of Japan Vol. 44, No Fundamental assumptions The space near the earth's surface is divided into two parts. The one is the space over the vegetation, and the other is in the vegetation or under the plant canopy. They may be designated as I and II respectively from up to down (Fig. 1). For the sake of brevity, the vegetation is assumed not to change its shape with time, so the case of the oscillation of vegetation as is seen on a wavy surface of plant field is excluded here. Next, the space II will be considered. Here we assume to hold the following fundamental equations :- for z<h, [II] d/dz(*/p) =C/HFu2 (7) K =ahu(1-f) (9) where F denotes the degree of the growth density of vegetation, being assumed to take the value between 0 (without vegetation) and 1 (completely dense growth). The first equation (7) describes a resistance law in the vegetation, and C is a constant of proportionality, H being introduced as a standard length making C non-dimensional. The second equation (8) indicates the vertical transport of momentum, and has the same form with (4) in the vegetation-free atmosphere. The third equation (9) may require some explanations. The fundamental expression for eddy viscosity is K=w'l'= ãw'2 l (10) Fig. 1. Vertical distribution of wind velocity. Then we have as fundamental equations in the space I as follows :- for z>h, where w denotes the vertical velocity, the term with ' the component of variation, and the one with _ the mean. In the space I, we can put ãw'2*u* (11) and l*z-d (12) [I]{d/dz(*/P)=0 (3) p/*=kdu/dz (4) K=u*l=ku*(z-d) (5) and get (5), though (5) has been used where v denotes frictional stress, p density of the air, K eddy viscosity, and l mixing length. As these equations are all familiar in the literature, the explanation of each of them may not be necessary. From (3) we have */p =const.*v*2 (6) Then from (4), (5) and (6), the equation (2) may be deduced. without any explanations. But in the space II, as it may be difficult to consider a representative velocity such as u*, it is assumed ãw'2*u(13) and, besides, l*(1-f)h (14) in agreement with the fact that eddies created in the space II are of the limited size varying inversely with the growth density of vegetation. a is another non-dimensional constant of proportionality. The resistance law (7) has the same form

3 April 1966 Kyoiti Takeda 103 with other author's expressions, but the eddy viscosity formula (9) is different (For some discussions concerning the resistance law and eddy viscosity formula in plant canopies, reference should be made to Cionco, 1965). = UHlv*, U11/2'(dU1/2/d*) z-h From (18) and (20), d and zo are solved at once in terms of H and U11]2 as follows : d=h(1-a/ku11/2), (21) - 3. Deduction of formulas Now let us consider what formulas and conclusions are derived from the above assumptions. From (7), (8) and (9) we have at where and once d2u/d*2-1/1-fdf/d*du/d*-Ĉ2f/1-fu= 0, (15) U =(u/u*)2,*= z/h (16) =2C/a. Ĉ2 (17) zo= HUll/2exp(-KU11/2), (22) and from (19) and (20) we have U11/2 U11/2'=1/a (23) As the right-hand side of (23) is constant, it shows that U1/2 or U 1/2' varies inversely with each other at z=h. Another relation which provides the principal equation of the following study will be deduced from (18) and (20) as follows :- The equation (15) reduces to a differential equation which prescribes the distribution of the wind velocity in plant canopy if the form In a special case of small roughness height of F (z) is given. But to solve (15) is generally not easy even for the case of simple or density d decreases and d=0 (25) functional expression for F (z), and it is not the purpose of the present paper. Some when solutions for specified simple forms of F (z) were already given (Takeda 1964, 1965) and U11/2= k/a. (26) more solutions will be given numerically elsewhere. a In this case we can obtain immediately from Here we consider some conditions which hold at the boundary between the space I and II, that is, at z=h. Three equations will be obtained from the condition of continuity of u, du/dz and z/p (or K). Thus we have from (2), (5) and (9), assuming (F). = H a relation which connects roughness length =0, and roughness height. From the equation (21), however, negative values of d may occur. Usually we do not consider negative U11/2=1/kIn1-*/*o, (18) U11/2'=1/k(1-*), (19) 11/2=Kk/a(1-*), (20) U where * (= d/h) and *o (zo/h) are non-dimensional zero-plane displacement and roughness length respectively, and U11/ 2 = (u/u*,)z-h 1/1-*ln1-*/*o=k2/a or 1/H-d (24) ln H-d/z (21) and (22) zo=hexp(-k2/a), (27) values of d, and treat the problem with d=0 however small the roughness height or density may be, and then (26) and (27) will be obtained also. But in such a case of small density, H should be considered not to represent the actual roughness height but an "effective" roughness height which is smaller and decreases with the decrease of density and makes the surface smooth in the limiting case.

4 104 Journal of the Meteorological Society of Japan Vol. 44, No Some results obtained The equation (24) obtained here describes the relation between d, zo and H, and it may be remarked that it does not contain wind velocity or any factors about the growth density of vegetation. If we know or give two of them, remaining one will be determined. In Fig. 2 the relation between zo and d is illustrated taking H as a parameter, where the value of a=o.087 estimated from Tani's data is used. It is remarkable that curves are similar and that zo has a maximum for a certain value of d. Each starting point of when H=60 cm, corresponds to the case where the vegetation has the maximum density, i.e. the height z= H behaves like the earth's surface. Fig. 3 shows the relation between zo and H when d is given. In this case, it is clear that zo increases with H and does not show a maximum whatever the value of d may be. Also the third representation is easily obtained which shows the relation between d and H taking zo as a parameter, and is illustrated in Fig. 4. As in Fig. 2, zo decreases on either side of its maximum value when H is given. Generally speaking, the the curve, for instance, the point d=60 cmfig.a.relation upper side between now zo corresponds and d when H is to given. the Fig.3.Relation large between zo density of vegetation and the lower side to the small density. Fig. 4. Relation between d and H when zo is given. Some conclusions about the variability of d and zo with the wind velocity will be immediately given by making use of the figure. It may be conceivable that the height of vegetation decreases effectively with the wind velocity by the bending of vegetation in the lee side, though the oscillation is put out of Table 1. When the wind increases, i,e. H decreases :

5 April 1966 Kyoiti Takeda 105 consideration as assumed. Hence the point computed values of d and zo relating to the representing the rough surface (point L or S) will generally move to the left in the figure. But as the relation between d and zo is not simple, we have several probable cases arising from the location of the point or from the direction of the movement of the point in the figure as is shown in Table 1. Note that there are cases where d and zo not only vary in the opposite sense, but also only one of them varies while the other remains constant or both of them vary but in the same sense. 5. Comparison with the experiment The experimental results considered here is due to Tani (1960), who made a large number of wind distribution measurements over paddy fields of about 90 cm height and compiled Fig. 5. Changes of d and zo with u1-5 (wind velocity at 1.5 m height) over paddy fields (Tani, 1960). Fig. 6. Relation between d, H-d, zo and uh/u*. Straight lines descending from upper left to lower right are those of constant H. Generally speaking, a point falling near the lower left corner corresponds to a smooth surface, while falling near the upper right to a rough surface. Moreover, a point falling near the upper left corner corresponds to a surface of dense roughness, while falling near the lower right to a surface of sparse roughness.

6 106 Journal of the Meteorological Society of Japan Vol. 44, No. 2 wind velocity at 1.5m height. Curves given by him are reproduced here (Fig. 5), though original points showed a considerable dispersion especially in the case of small wind velocity. Among the paddy fields a few situated thousands of kilometers apart were included, but they all had a similar feature. In Fig. 4 they are represented by a single point 0 determined by H, d or z0. As the wind rises, the height H shows at first no remarkable change and the air looks like as if it can flow freely in the space of vegetation. But in spite of it, the computed d shows a decrease, while zo an increase untill the condition A is reached (Compare Figs. 4 and 5). In the next stage, created eddies at the plant surface have a predominant effect, and the air undergoes a large resistance to enter Into the space of vegetation as Tani remarked. Consequently the effective earth's surface is raised, and d increases while zo decreases until the condition B in Fig. 4 is reached. In this stage the bending of the plant will occur as the wind becomes large, and in agreement with the fact the point B comes to the left side of A. In the subsequent stage the wind blows stronger, and the bending of the vegetation becomes considerable, making d smaller and zo larger. This circumstance corresponds to B ~C in Fig. 4. Thus it may be concluded that the experimental results obtained by Tani is sufficiently explained by the present theory. 6. A rough surface diagram Another diagram which is only a slight modification of Fig. 4 but may be more useful for practical purposes is constructed and is illustrated in Fig. 6. As is seen the ordinate is d, but the abscissa in this case is H-d instead of H in Fig. 4. Straight lines descending from upper left to lower right are those of constant H. Also lines of constant H/v* are drawn as straight lines leaving u the origin (d=0, H-d=0) and rotating clockwise with the value of uh/v*. Of course, the same discussion as given in Fig. 4 are possible by making use of Fig. 6. The diagram may be called a rough surface diagram, because any rough surface is represented by a single point in the diagram, being specified by not only the surface is smooth or rough, but also its roughness is dense or sparse. Generally speaking, a point falling near the lower left corner corresponds to a smooth surface, while falling near the upper right to a rough surface. Moreover, a point falling near the upper left corner corresponds to a surface of dense roughness, while falling near the lower right to a surface of sparse roughness. 7. Skin-friction coefficient To know the effect of a rough surface upon the wind flow, it is customary to express the surface friction by means of a skin-friction coefficient CD, defined by the equation (Sutton 1953) *=1/2CDpus2 (28) where us is the wind velocity "near" the surface, i. e., at some convenient standard height. Various investigations have been done to estimate the value of CD and obtained values show a general increase as the vegetation becomes large. But in the case of very rough surfaces our knowledge is still insufficient. Now let us see the behaviour of the coefficient in the light of our theory. One of the ambiguities for the value of the coefficient is that the definition of the reference height for which the velocity is measured is not clear. Usually some convenient height, such as 100 cm or 200 cm has been adopted. But for a tall rough surface, a difficulty arises. Thus we come to a conclusion that we should not use an absolute height but a relative height. Thus if we use uh (the wind velocity at z=h) for us, skin-friction coefficient for various rough surfaces may be comparable with each other. If we put us =uh, we have immediately from (20) and (28) CD=2(a/kH/H-d)2=2/(uH/u*)2, (29 which means that CD varies inversely with ) (uh/v*) 2. From our diagram (Fig. 6), it is seen at once that CD does not have a simple

7 April 1966 Kyoiti Takeda 107 relation with zo or d, but grows larger when the roughess of the surface becomes more dense. 8. Some discussions The value of a (=0.087) was obtained from Tani's data on paddy fields and has been used throughout this paper. As for the other constant C, the value C=2.2 was estimated though the estimation was more difficult than a, and the value has not been used in the present paper. As the flow in actual cases is very complicated, our resistance law (7) and eddy viscosity formula (9) are considered rather not to represent physically defined laws but hold only in the average condition. Hence concerning the validity of these values the author has some doubt, and it is probable that some different values of a and C may be obtained for other types of vegetation such as shrubs or forests. We have considered only the case of the adiabatic atmosphere, because the effect of stability on d seems to be a very difficult problem. The author (Takeda, 1952) has ever presented a wind profile and discussed the variability of zo with stability, which was substantially the same with that of Laiktman (see, for instance, JIai xtmah, 1961). However, the subsequent authors on atmospheric turbulence in the surface layer have treated the problem with constant zo without any sufficient discussions for the variability of it with stability, and it seems to be fortunate that their formulas coincided fairly well with the experiment. But is it sure that the problem of zo variablity has come to a conclusion? Then, how about d? It may be noticed that in the limiting case of small growth density (F =0) our f unda mental equation of eddy viscosity (9) fails tc give (5) for vegetation-free space. It may seem to be a weak point of the theory at first sight. But in the actual case, as F decreases seriously H also decreases effectively as described, and reduces to H=0 when F=0, making the space II disappear and only the space I remains. Of course, making the theory more precise will be necessary, and at the same time the acquirement of many reliable experimental data of wind profile not only over various rough surfaces and weather conditions but also in wind tunnels is indispens able for the future development. References Calder, K.L., 1949: Eddy diffusion' and evaporation in flow over aerodynamically smooth and rough surfaces : a treatment based on laboratory laws of turbulent flow with special reference to conditions in the lower atmosphere. Quart. J. Mech. Math., 2, Cionco, R.M., 1965: A mathematical model for air flow in a vegetative canopy. Journ. appl. Meteor., 4, Sutton, O.G., 1953: Micrometeorology. New York, McGraw-Hill. Takeda, K., 1952: On the atmospheric turbulence, 4th paper. J. meteor. Soc. Japan, 30, : Turbulence in plant canopies. Journ. agric. Meteor., 20, : Turbulence in plant canopies (2). Ibid, 21, Tani, N., 1960: The wind on the cultivated field. Ibid, 16,

8 Journal of the Meteorological Society of Vol. 44, No. 2