i1520-0469-16-5-516 - 516 jOURNAL OF METEOROLOGY VOLUME 16...

Unformatted text preview: 516 jOURNAL OF METEOROLOGY VOLUME 16 A GENERALIZATION OF THE MIXING-LENGTH CONCEPT1 By J. A. Businger University of Wisconsin3 (Original manuscript received 2 May 1958; revised manuscript received 7 April 1959) ABSTRACT A concept for the mixing length in diabatic conditions is introduced and elaborated. The basic idea is that convective energy has effect on the mixing length but not on the size of the largest eddies. The theory devel- oped on this concept of the mixing length for the diabatic wind profile gives satisfactory agreement with observations over a wide stability range. 1. Introduction In the last ten years, several papers dealing with turbulent transfer in the lowest layer of the atmos- phere have been published. The progress made in that period is mainly experimental. Several techniques of collecting data were developed or improved. The two expeditions to O'Neill, Nebraska in 1953 and 1956, especially, resulted in a host of valuable data. Progress was made also on the mode of representation. Several studies, such as Deacon [7; 8]. Priestley [23], Monin and Obukhov [18], Businger [4], Lettau [16], and Inoue [12], were carried out to achieve in part a more comprehensive representation of the data in order to test several possible theories. Although unanimity has not yet been reached on this point, the different representations are more or less equivalent and in— dicate the existence of a unique and universal relation describing the turbulent structure of the surface layer (e.g., the wind profile in terms of a dimensionless height as a function of stability). On purely theoretical grounds, Batchelor [1] came to this same conclusion. However, until the present no satisfactory theory has been developed describing the observations in suffi- cient detail. The theories presented by Lettau [14; 15], Businger [4; 5], Halstead [10], Ogura [19], and Elliot [9] all introduce one or mere assumptions which have an apparent lack of physical basis. This situation is somewhat surprising because of the strong limitations on the problem imposed by the general assumption of a steady condition throughout the surface layer—that is, that all time derivatives of mean quantities are zero, and that the surface is homogeneous and has a constant roughness parameter. in sections 3 and 4 of this paper, a theory is developed based on a model of turbulence which makes possible 1 The research reported in this article has been made possible through support and sponsorship extended by the Geophysrcs Research Directorate, Air Force Cambridge Research Center, under contract No. AF 19(604)—1384. 2 Present affiliation: University of Washington. the introduction of a generalized mixing length. The agreement with observation, so farI is promising. Although the question of whether or not the coeffi— cients of eddy diffusivity for momentum and heat transport are equal is by no means settled, the equality of the two is assumed in most of the above mentioned theories and also in the present theory. It may be that this assumption is not true, but it simplifies the theoretical approach considerably and provides a result which eventually can be corrected because the analysis does not depend on it in any fundamental way. An empirical check of the equality of these coefiicients is obtained by comparing the curvature of the wind and temperature profiles from the O’Neill data [27]. Comparison shows that these curvatures are different, and therefore it is concluded that the coefficients must be different. It is very likely, how- ever, that the ratio of the coefficients is a function of the stability alone and does not affect the uniqueness of the above mentioned universal relation. Another simplification is made here with regard to the zero displacement for the logarithmic profile. This zero displacement is not taken into consideration, so the level 2 = 0 is assumed to be where the wind velocity, according to the extrapolated logarithmic profile, is zero. Therefore, the theoretical results cannot be trusted below a level of z = 1020, for instance. For a more detailed discussion of this matter, reference is made to Bjorgum [2]. 2. Dimensional characteristics and representation Since there is sufficient experimental as well as theoretical (Batchelor [1]) evidence that the Richard- son number, defined as aé Oz R1: / z gum/32y (1) (where 0 is potential temperature, at is horizontal mean wind, g is acceleration of gravity and Tm is the mean OCTOBER 1959 J . absolute temperature), determines the structure of the atmospheric turbulence in the surface layer, there must exist a unique relation describing this structure. Several studies [4; 7; 8; 12; 16; 18; 23] have been carried out in order to give a concise and compre- hensive representation of this relation. Although most representations are basically equivalent, because they can be derived from each other, preference can be given to some representations above others. From a theoretical point of view, it is desirable to have a representation which is easy and reliable to derive from observations in order to be able to test the theory. The first representation of this kind is that of Deacon [7], who plots a dimensionless parameter [3, related to the curvature of the wind profile, defined as 26212/622 B = 612/62 (2) versus R1}. Although Deacon’s semi-empirical general- ized wind profile has not been found in theory, the [3 versus Ri relation is still of practical interest. In recent studies, this relation has been used to compare some theories with observation. A great advantage of both the parameters 6 and Ri is that they can be derived from wind and temperature measurements without using the surface roughness. However, this relation is difficult to visualize in terms of the wind as a function of height-—that is, in a more integrated form of the relation. Representations in this form are given by Monin and Obukhov [18] and also by Businger [5]. Monin and Obukhov represent the universal relation using a dimensionless Wind differ- ence Y and a dimensionless height gr as 66 a 0 Y = we — Us) and r = kg—fijf/afiui) (3) (where U = left/14* is the dimensionless wind, k is v. Karman's constant, u* is the friction velocity, and 20 is the roughness parameter). The representation Y = f (t) has the objection that it fails to show any detail for nearly neutral profiles; that means it is difficult to find the deviation from the neutral profile, as Lettau [17] has pointed out. This point is elab— orated in section 4. Businger [5] used a representation which is easy to interpret. Here, U was plotted as a function of 2/20 in a series of curves using the dimen- sionless stability number gee/62.20 5% = W TmBfi/dz ' 14* as a parameter. Although this representation gives good detail for the nearly neutral profiles, it is not as concise as might be desired. A better representation can be introduced using a dimensionless velocity A. BUSINGER 517 difference, AU; Z + Zn 20 AU=U-—ln (4) versus the dimensionless height g’ or versus Ri. An intermediate representation is given by Deacon [8]. He introduced a dimensionless coefficient of eddy transfer * K", 14* 17,3 — 2612/62 m and made a plot of 14%", versus Ri. In section (4), the u * . 0 n - reelprocal value of Km 15 used as dimensionless Wind gradient kzaii k P=—-——=———. 14* 62 Km In the following sections, both the representations 6 = f (Ki) (5) and AU = h(R1§) (6) will be used. 3. Model of turbulence under neutral conditions Turbulent structure under adiabatic conditions is now well understood. Prandtl's [20] derivation of the logarithmic law has found theoretical as Well as experimental support. Therefore, the concept of the mixing length must be regarded as a valuable tool, with an exact definition in the neutral case, as was pointed out by Businger [4]. An interesting fact, furthermore, as shown by Inoue [11], is that the logarithmic profile is consistent with the similarity hypothesis of isotropic turbulence when the mixing length represents the size of the largest vertical eddies. A consequence of this result is that in the equation of energy decay, derived by Calder [6], and which under steady adiabatic conditions can be written as d _._‘ at? ~—w’E-u’w’——D=0 (7) 62 62 ' (u’ and w’ are the horizontal and vertical components of the turbulent wind respectively, E is the total turbulent energy and D is the dissipation rate of turbulent energy into heat), the term 6(—‘zv’~E_)/az, representing the divergence of the turbulent energy, is negligible in comparison to the two others. The rate of decay of turbulent energy 6 from one wave number to the next is independent of the wave number and equal to the dissipation rate. This means that 103 D = e ~ ka0 ~ -— (8) 518 where 120 is the wave number of the largest vertical eddies; 'w = (Wfi and l is the mixing length. The equations (7) and (8), together with the equation for the flux of momentum pK— = — p We” (9) where the coefficient of eddy transfer K is defined by K = wl, (10) define the logarithmic wind profile, when the mixing length 1 ~ (l/ko) ~ 2. The proportionality constant which enters because of relation (8) is v. Karman’s constant. It should be remarked that the mixing length as introduced here is a Eulerian concept. Prandtl's original concept of mixing length was “Lagrangian." But, in his first development where it was related to u’w’, the transition to the Eulerian View was implied. Therefore, the explicit expression obtained for the mixing length in the neutral atmosphere, 1: k(z + 20), (11) is a Eulerian concept. (Note the analogy with the kinetic gas theory 1) It is of interest now to present a different argument in Prandtl’s original model. Consider an eddy with a vertical velocity 10’ with respect to its surroundings and a characteristic size 1’. The question is now how far in the vertical will this eddy travel before it loses its identity. That distance will then be the mixing length for the eddy under consideration. To aHSWer this question, we consider the energy of the eddy in the z—direction, which will be proportional to pw’2l’3. This energy dissipates at a rate proportional to the eddy diffusivity of its surroundings K’, the gradient in energy from the eddy to its surroundings, which is proportional to w'2/l’ and the surface of the eddy ~ I”. The dissipation then is proportional to pK’w'2l’. It may be assumed that K’ is proportional to the - overall eddy diffusivity K. By the time all the energy is consumed, the eddy has lost its identity and has travelled its mixing length Z... In equation, pw’3l’3 wil/g Z," N --——"‘—-‘ ~ pK'w'w K ' (12) Averaging over all eddies, the left-hand side will give the mixing length in the Lagrangian form lg, and the right-hand side will be proportional to 'wlo2/ K (where lo is the predominant eddy size), provided the size distribution of the eddies is independent of the in- tensity of the turbulence, which is implied in equa- tion (8). When we now consider all the eddies passing by a point of observation, we may assume that on the JOURNAL OF METEOROLOGY VOLUME 16 average these eddies have travelled in the vertical over a distance which is proportional to la. The mixing length 1 obtained in this fashion is a Eulerian concept. So we finally obtain, using (10), ‘10qu2 lo2 l N U N —‘- N “" K I 01‘ l 1 1 (13) w ._ ———— ~ Z 0 [30 This argument may seem to be superfluous, but it will be shown in the next section that relation (12) provides a link to a generalization of the mixing-length concept. At the same time, it shows what assumptions are made in the present theory. 4. Generalization of the model for diabatic conditions The previous section emphasized that there is con— siderable agreement even in detail between Prandtl’s model and the exact theory as well as the observations. Therefore, it may be expected that a logical general— ization of this model to diabatic conditions would provide satisfactory results. Equation (7) is a special case of the more general equation for diabatic con- ditions (see [6]). a __ 612 ag __. —- w’E —— u’w’ ——+——w’0’ — D = 0, 62 az Tm D = wakn (14) where a is a proportionality constant indicating the efficiency with which density differences contribute to the turbulent energy. It was Ogura [19] who emphasized the usefulness of this equation in the derivation of the diabatic wind profile. Next, there is the equation for the heat flux (15) which is assumed to be similar to equation (9). Further, there is some evidence from an analysis of the power spectrum of the vertical wind component that the turbulence behaves isotropic in the range of high wave numbers [26] under all stability conditions. This behavior must be related to the dissipating structure which tends to be isotropic even in homo- geneous anisotropic turbulence, whereas the transition from homogeneous anisotropic to isotropic turbulence is very slow [28, p. 48]. The experimental evidence indicates that the turbulent energy from friction as well as from convection enters at the lowest wave numbers and that equation (8) is also true for diabatic conditions. The main difference between the diabatic and adiabatic case lies in the way the frictional and the convective energies enter into the turbulence. It OCTOBER 1959 J . may be said that the frictional energy enters the lowest vertical wave numbers in an isotropic way, because the horizontal turbulence extends to a lower frequency range than the vertical turbulence. However, the convective energy contributes only to the vertical wind component of the lowest vertical wave number. Considering again an individual eddy, we see that besides a dissipation of energy there is also a source of energy proportional to ffir9’l““w’, Tm where 0' is the temperature difference between the eddy and its surroundings. In this case, instead of (12), we find wlllg Oligo/ll2 ' Tm'w’ ln~ (16) K— When averaging over all eddies and following a similar line of reasoning as in deriving (13) and considering that 0’ may be replaced by - 1(69/62), we find 1 ( oz’g 63 1 )“i Tm 62 (how)2 . (17) It is easy to see that, in this more general case, 1 is no longer proportional to 2 because l/ko ~ 2 is still valid. The constant a’ is not the same as the a in equation (14) because it contains also the constant of pro— portionality between 1 / kg and 2. Under neutral condi~ tions, relation (17) should reduce to equation (11); therefore, by replacing 1/ko by k(z + 20), we find for (17) that assuming that in this case a is the same in (14) and (18). The equations (9), (10), (14), (15), and (18) now define the complete structure of the atmospheric surface layer. From these relations, it is possible to derive 3 R' e "‘ i 3m + %[(aRi)2 +4<1 - am} , 01f = ”—“i 1 —- aR’L 2 (19) using (1) and (3). From this equation, the relation (5) can be derived, considering that dlnRi dlnf :3! by combining (1), (2), and (3), by differentiation of (19). A consequence of the present theory is that for A. BUSINGER 519 R12 = — €10.43 = 4%, which is in agreement with Priest— ley’s findings in the region of free convection [22]. From equation (19), the wind profile or equation (6) can also be derived. Writing the symbol for the part of equation (19) in brackets and considering that (see (1) and (3)) a U 1 6g“ _ Ri ’ it may be seen that 3 U P f f or, when a function Q(§‘) = (1/§)(P — 1) is in- troduced, 6U 1 Ernst—FOG) and 5 r U = In — —I— Q(§')d§ {0 (n (20) where kgaé/aza) =k5 =——~——. {0 " I;u*aa/az Thus, equation (6) takes the form f AU: r Q(§)d§ (21) which can be obtained by graphical integration. The equations (20) and (21) indicate that U and AU are not only functions of f but also of {0. So it is necessary to use, in this case, a two-parameter representation. However, in most cases i“ _z+20 f0 20 >>1 and, because Q(§‘) is a limited positive function, i‘ 5‘ d 2 d . mac): foam: So equation (6) can be represented as f AU = f (and: (22) provided 2» 20. Combining equation (22) with equation (19), A U can be written also as a function of Ri. Various representations of the universal function as derived by the- present theory are illustrated in fig. 1. It is now possible to compare equation (22) with equation (3). Equation (3) can be written when 320 £25 JOURNAL OF METEOROLOGY VOLUME 16 FIG. 1. Various forms of the universal function as derived by the present theory. §>Oas I Y=hmr+j‘moa and when 3“ < 0 as r Y=meuo+fema a» When g“ is very small, the absolute value of the integral in the right-hand side of these equations is small in comparison with [In ZlfH. This explains, as was men- tioned in regard to equation (2), why detail is lost in Monins and Obukhov’s representation. 5. Comparison with experimental data The theoretical relationship between [i and aRi as given in fig. 1 is compared with two groups of data— i.e., a series of 260 hr of simultaneous wind and tem- perature profiles, measured at Shirley, New Jersey, and a similar series of about-230 hr measured at O’Neill, Nebraska in 1953 and 1956. The O’Neill ob- servations were carried out by the Johns Hopkins and MIT groups in 1953 and by the Texas A. and M. group in 1956. The data reduction was carried out by Lettau and the result was presented during the IUGG at Toronto in 1957. The curvature [3 and the Richard- son number were divided in seven stability classes, which are defined in section 7.4.3 of [27]. For the method of data reduction, applied reference is made to [16]. The comparison of the theoretical curve with the two groups of data is presented in figs. 2 and 3, respectively. A reasonable fit is obtained for a value a = 3 in fig. 2. In fig. 3, especially for nearly neutral conditions, a value of a = 10 would fit better. Also, the theoretical relation between AU and Ri has been compared with experimental results. To obtain ’81], Z + 20 AU = —— —— ln u* 20 from the profiles, it is necessary first to evaluate both u... and 20. The 20 was obtained by using the logarithmic profile under neutral conditions, and 14* was obtained by assuming the logarithmic profile to be valid to the lowest level of wind observation. The data represented in fig. 4 are derived from 1953 observations by the Johns Hopkins group for the 4-m level and from 1956 observations by the Texas A. and M. group for the 8- and 16-m levels, all at O’Neill, Nebraska. The total was divided in ten classes of overall stability. In the more stable classes, the determination of m was not accurate because of the large deviation from the logarithmic profile. The theoretical curve shows also in this case a reasonable fit with the presented data when or = 3. 6. Concluding remarks Although the present theory is not exact, a fairly accurate description is obtained of the structure of the turbulence in the atmospheric surface layer from extremely unstable to stable conditions, provided the steady state is maintained. In the very stable cases, the steady-state condition is usually not .fulfilled because the transfer processes OCTOBER 1959 J . l.50 I25 FIELD BARE [0.0], on. PLANTED WITH Rows or 050 SPINACH on com [Ami 3cm, V.V=4TO 8 «ma I =9T0 l2cm] WIND 5 TEMPERATURE LEVELS: a A v £1140, 80.50. AND 520 cm; 0 A v .180,l60,320,AND 640 cm, 025 -l.00 ~0.75 '050 ~0.25 A. BUSINGER 521 0.0 0.25 Pl FIG. 2. Comparison of the theoretical 5 versus Ri relation for a = 3 with observations obtained at Shirley, New Jersey. [.50 O'NEILL l953+l956 /3 versus Rt 0 2 =400 cm Az=200cm LIz . IOOcm 0.50 0.25 -l.00 ‘ 0.75 -0.50 ‘025 0.00.25 Ri FIG. 3. Comparison of the theoretical 6 versus Ri relation for a = 3 and a = 10 with observations obtained at O’Neill, Nebraska. become very slow. The result is a rather large scatter of the observed points in this region. It is probably not possible to obtain a more precise theory with Prandtl’s mixing length as a starting point. The constant of proportionality, :1, makes it p0ssible to adjust the theory to the observations. It is, of course, desirable to have a theory which predicts this constant and agrees ...
View Full Document