C syr, /3> : &/f> n, where n is the Kolmogorov microscale. The averaged dissipation rate was assumed to have a lognormal probability distribution with the variance of £n(e ) given by °lne = A + y ^ n(L/r) (1) r where y is a constant, L is an outer scale, and A depends on the large-scale nature of the turbulent flow. The logarithmic dependence on L/r describes the increasing intermittency of energy dissipation with decreasing scale size r. The presence of the parameter A in (1) con- tradicts the previous assumption of the independence of small-scale statistics from large-scale properties because A is expected to have different values depending on the macrostructure of the turbulent flow. The energy spectrum is predicted to have a k power law in the inertial range -5/3 instead of the original k power law. Measurements (Gibson and Masiello, 1972) indicate that u - 0.5, so that the predicted change in inertial range power law is quite small and difficult to detect experimentally. On the other hand, predictions for high-order moments of velocity derivatives show a strong effect from small-scale energy dissipation fluctuations, that is, from the logarithmic term in (1). Figure 6 of Williams and Paulson (1977) and fig. 6 of Champagne et 5/3 al. (1977) contain graphs of measured one-dimensional energy spectra multiplied by k . Such graphs are expected to show a k (i.e., approximately k ) behavior; however, the graphs — 1/18 appear more nearly constant in the inertial range than k would predict. In the middle of -3 -2 the observed inertial range (specifically, the middle means 2 x 10 < kn < 1 x 10 in the O _ <} _-| /I Q Williams and Paulson data and 4 x 10 < kn < 2 x 10 in the Champagne et al. data), a k curve can be fitted quite well; at lower wave numbers one expects the fall-off toward the — 1/18 energy-containing range to invalidate the k ' prediction. At the high-wave-number end of the inertial range the measured energy spectra must have been overestimated to be consistent with the k~ prediction (specifically by about 16% to 12% at kn = 5 x 10 ) . Measurements of — 1/18 energy spectra that are sufficiently accurate to reveal the k behavior are extremely difficult to obtain. The analogous theory to account for effects of fluctuations in both e and the rate of dissipation of scalar variance, x> on tne inertial-convective range of the scalar field was developed by Van Atta (1971) and Antonia and Van Atta (1975) . Analogous to e , the rate of dissipation of scalar variance averaged over a volume of dimension r is denoted by x > where r must lie within the inertial-convective range. The averaged dissipation rates e and x are assumed to have a joint-lognormal distribution. The variance of ln(e ) is given by (1), and the variance of £n(x ) is assumed to be given by o] = A + u 2,n(L/r) . (2) 2,nx r s s The interpretation of the quantities in (2) is the same as for (1). No assumption is made for the r-dependence of p, where p is the correlation coefficient of £n(e ) and in(\ )• Measure- ments by Antonia and Van Atta (1975) indicate that p increases with increasing r. By assuming u - u and ignoring the r-dependence of p, the inertial-convective range power law is k ~5/3+u( /3-p)/3. thig ig only a small change f rom tne original k prediction because ob- served values of p are close to 2/3. On the other hand, predictions for high-order moments of scalar derivatives and cross moments of velocity derivatives and scalar derivatives are sensi- tive to small-scale dissipation-rate fluctuations, that is, to the logarithmic terms in (1) and (2). Figure 5 of Williams and Paulson (1977) and fig. 6 of Champagne et al. (1977) contain 5/3 plots of measured one-dimensional atmospheric temperature spectra multiplied by k . Such / 9 / O \ / 9 plots are expected to show a k behavior in the inertial-convective range. Throughout -3 -2 the observed inertial-convective range (meaning the wave numbers 2 x 10 < kn < 2 x 10 ) there are no significant deviations from a constant value in the plots. This effect is con- sistent with the predictions from small-scale intermittency if p is very close to 2/3. We conclude that there are no observable effects of small-scale intermittency on the convective range of the temperature spectrum in air. This conclusion is the basis for neglecting the effects of small-scale intermittency for the convective range and the low-wave-number portion of the dissipation range in the following study of the effects of large-scale intermittency on the temperature spectrum. These small-scale intermittency theories apply only to the inertial and inertial-convective ranges, but the strongest effects of dissipation-rate fluctuations on the shape of energy and scalar spectra are probably in their viscous and diffusive ranges, respectively. A theory of effects of small-scale intermittency on the dissipation ranges would, at the least, need to replace the logarithmic terms in (1) and (2) with some other function of r for scales r within the dissipation ranges. It is probable that ln(e ) and &n(x ) have a joint distribution that is different from joint-lognormal for r in the dissipation ranges. On the other hand, there is no conceptual difficulty in modeling the effects of large-scale intermittency on the dissipation ranges by using the parameters A and A . Although the above theories are restricted to the inertial and inertial-convective ranges, Kraichnan (1968) used a simple model to examine the effects of small-scale strain-rate fluctua- tions on the viscous-convective and viscous-diffusive ranges of the scalar spectrum. He found that the viscous-convective range is dominated by the frequent occurrence of moderate values of the rate-of-strain and that the k viscous-convective power law is unchanged. On the other hand, he found that the viscous-diffusive range is dominated by the infrequent occurrence of large values of the rate-of-strain and the spectral form is changed by the presence of strain- rate fluctuations. As r ■* L in (1) and (2), the variances tend to A and A . Since e and x T are tne dissipa- tion rates averaged over regions the size of the outer scale, it follows that A and A are measures of the intermittency of dissipation rates averaged over large-scale regions. Because of the relationship between derivatives of velocity and temperature and the dissipation rates e and x» tne typical strip-chart recordings of temperature and velocity derivatives given by Williams and Paulson (1977) (their fig. 1) illustrate this type of large-scale intermittency in the unstable atmospheric surface layer. These recordings correspond to a height of 2 m and a wind speed of about 6ms ; thus time intervals of about 0.3 s to 1.0 s correspond to large- scale regions. The strip-chart records show what might be described as highly active, moder- ately active, and relatively quiescent intervals of variable duration. 2. INTRODUCTION TO PRESENT STUDY Previous theoretical and experimental studies have concentrated on the effects of small- scale intermittency described by the logarithmic terms in (1) and (2) . The present study neglects the logarithmic terms and examines the effects on the scalar spectrum of the terms A and A ; these terms describe the influence of large-scale dissipation-rate fluctuations. As noted previously, the logarithmic terms cause only a small change in the inertial and inertial- convective range power laws for the energy and scalar spectra. Thus it is possible to neglect these effects and investigate the effects of large-scale intermittency alone. It must be kept in mind that only part of the problem is being treated, particularly for the dissipation ranges where small-scale intermittency is more important than in the inertial and convective ranges. For high-order moments and cross moments of velocity derivatives .and scalar derivatives it is not worthwhile to treat the large-scale intermittency separately because the contribution from small-scale intermittency is strong. The motivation for this paper is the theoretical interpretation of the propagation of light through the turbulent atmosphere. Fluctuations of optical refractive index in atmospheric turbulence are dominated by temperature fluctuations; the contributions of turbulent pressure and humidity fluctuations are negligible. Thus, this theoretical interpretation requires knowledge of the temperature spectrum, often at wave numbers higher than those in the inertial- convective range. Predictions for optical propagation in turbulence become less reliable depending on the variations in the shape of the temperature spectrum caused by large-scale intermittency. Because of this application to optical propagation, only results for the temper- ature spectrum in air are given here. However, the techniques presented may also be used for the energy spectrum or the spectrum of a scalar having a large Prandtl number; in the latter instance a viscous-convective range exists. Two models of large-scale intermittency are considered here. The first is called the two- state model. For this model, it is assumed that the large-scale nature of the turbulence consists partly of agitated regions where the dissipation rates, averaged over an agitated region, are relatively large and partly of quiescent regions where the dissipation rates, averaged over a quiescent region, are relatively small. An example of a turbulent flow resem- bling the two-state model is the edge of a turbulent jet discharging into an exterior flow that is itself turbulent. Suppose that the turbulence from the jet has relatively large dissipation rates e and x> whereas the exterior turbulence has relatively small dissipation rates. A probe placed near the edge of the turbulent jet is alternately in the region of turbulent activity and in the relatively quiescent turbulent flow beyond the jet. The two-state model is simple enough that the results are easily interpreted. The second model assumes that the dissipation rates, averaged over large-scale regions, have the joint-lognormal probability distribution. 3. TWO-STATE MODEL It is assumed that one can identify two types of large-scale regions within a turbulent flow. The regions of type 1 occupy a fraction f of the total turbulent volume, and regions of type 2 occupy a fraction f- ; so that f + f 2 = 1. The average of the dissipation rates e and X over each region of type 1 yields the values e and Xy Likewise, the same average for type 2 regions gives e ? and x 9 - Tne Kolmogorov microscales in the type 1 and 2 regions are n ~| / A ^ -1 / A n = (v /e ) and n = (v /e ) , where v is the kinematic viscosity. The scaling parame- 1 1 ^ _i _3 _i -3 ters for the two regions are C = X-, H-, v and C 2 = x 2 n 2 v * The dissipation rates averaged over the entire turbulent flow are = f e ;l + f 2 e 2 and = f x X x + f 2 X 2 - The Kolmogorov microscale and scaling parameter determined from the entire-flow-average dissipation rates are ^ = ( v 3 /<£>) 1 and C" = ~ 1 v. The scalar spectrum measured in each region of type 1 and averaged together is I^Oc); for type 2 regions this gives r (k) . If the scalar spectrum measured over the entire turbulent flow is denoted by , then = f 1 r x (k) + f 2 r 2 (k) , (3) where the contributions from the interfaces between regions of type 1 and 2 are neglected. Equation (3) is valid only for k >> 2ir/L where L is a typical size of the large-scale regions. Consider a prototype turbulent flow that is as nearly homogeneous as possible in the sense that the dissipation rates averaged over any large-scale subsection of the flow do not differ greatly from the dissipation rates averaged over the whole flow. The effects of large-scale intermittency will be evident by comparison with this idealized prototype turbulent flow. A Kolmogorov microscale n and scaling parameter C are constructed using the dissipation rates averaged over the prototype flow. At high wave numbers the scalar spectrum measured in the prototype turbulence, T , and scaled with C is assumed to be a universal function, V = CT , P P P of the scaled wave number kn and the Prandtl number Pr. The dependence on Reynolds number of the scalar spectrum is ignored; such dependence is caused by small-scale intermittency. The need for the prototype flow and spectrum in the present study of large-scale intermit- tency has an analog in the small-scale intermittency theory. For example, in the small-scale theory it is assumed that if the locally averaged dissipation rate, e , has a given value then the structure function of longitudinal velocity fluctuations is given by n , , 2/3 2/3 D(r) a e r in the inertial range. This is then a prototype structure function corresponding to a partic- ular value of the averaged dissipation rate and is analogous to the prototype spectrum. Next, the structure function is averaged over a weighted superposition of e values; using (1) and the lognormal probability distribution for e gives tw v 2/3 2/3+y/9 T -u/9 cc < e > ' r M ' l , r which is the basic result of the small-scale intermittency theory for this structure function. In the present analysis, the temperature spectrum is also averaged over a weighted superposition of dissipation rate values. Thus, the present techniques for dealing with large-scale intermit- tency are not different in principle from those used in the small-scale intermittency theory. In the two-state turbulence model, the regions of type 1 and 2 are assumed to resemble the prototype flow so that if the spectra r , T„, and r are all evaluated at the same scaled wave number, then c. r, = r lip c 9 r = r . ii p These equations merely state that when spectra are measured in different large-scale regions that have different mean dissipation rates and the spectra and wave number are made nondimen- sional by scaling with the corresponding dissipation rates, then these scaled spectra, plotted as functions of the scaled wave number, all lie along a single curve. This scaling law was previously assumed to apply to spectra obtained from records containing many large-scale re- gions, and the scaling law was known to depend on Reynolds number in the dissipation range because of small-scale intermittency effects. Here, this scaling law is assumed to apply to spectra from large-scale regions, and the deviations from the scaling law for scaled spectra obtained by averaging over many large-scale regions are found. Let be scaled by C and expressed as a function of k = kn . Then = C is the scaled spectrum obtained from measurements of T, e, and x averaged over the entire turbulent flow. Define r^) e c 1 r 1 (k 1 ) r 2 (k 2 ) = c 2 r 2 (k 2 ) k = kn, k = kn„ . Then, from (3) = f 1 £- r^k^ + f 2 |- r 2 (k 2 ) . (M The relationships between the scaled wavenumbers are k ± = k(/ £l ) 1/A = k [f 1 + f 2 (e 2 / £l )] 1/4 k 2 = k[f 1 (e 1 /e 2 ) + f 2 ] 1/4 . The coefficients in (4) are f C h [ h + h^2>^" i c 1 f 1 + f 2 (x 2 /x 1 ) c_ f 2 [f 1 (^ 2 > + f 2 ] 3M 2 C 2 f 1 (x 1 /x 2 ) + f 2 In the inertial-convective range, r^) = 6 k" 5/3 r 2 (k 2 ) = 6k~ 5/3 where 3 is the Oboukov-Corrsin constant. The value 3 = 0.72 is assumed here. The measured _ _ -5/3 value of 3 is given by 3 = (k) at k << 1; (4) then gives i [fAzJe 9 ) 1/3 + fJ[f.(eJ El ) 1/3 + f 9 (x 9 /x-,)] 3/3 = r f T i ( / m ^_^_J^_ . ( 5) [ 1 f 2 (x 2 /x l ) -' Thus 3 ^ 3 in general; however, 3 = 3 if e 9 = £-, regardless of the value of (Xo/Xi ) • I n fact, if (Xo/Xi) i s arbitrary and e„ = e , then x = x„ = x and Then (4) predicts that = T . P To make predictions using the two-state model one must specify values for f , f„, e„/e 1 , and x 9 /Xt ■ l n tne absence of experimental guidance the values of f and f „ are selected from the set (0.1, 0.5, 0.9), and the values of z~lz and X9/X1 are selected from (1.0, 10, 10 ). Combinations of these values give the 27 cases in table 1. The value 1.0 for 2 e»/e and Xn/X-, describes nonintermittent turbulence whereas the value 10 describes highly intermittent turbulence. The predicted values of 3/3 also appear in table 1. To determine the measured scalar spectrum, , from (4) one must first assume a form for r . The assumed form for V is that of model 4 of Hill (1978) using the parameters b = 1.9, + P P k /k = 0.072. Only temperature fluctuations in air are considered, so Pr = 0.72. The parame- ters b and k /k for the model 4 spectrum are obtained by fitting model 4 to the temperature spectrum measured in the atmospheric surface layer by Champagne et al. (1977). That this measured spectrum is already influenced by large-scale intermittency is bothersome; however, the predicted changes in spectral form caused by large-scale intermittency become evident neverthe- less. Experimental guidance as to the form of the prototype spectrum is lacking, but if it were Table 1 . --Parameter values for case study_of model predictions as well as predicted values of p/p and p /p Two -state Joint- •lognorraal Case (e^) (x 2 /x 1 ) f l f 2 P/P 2 a y OCT CT x y iye Symbol 1 1 1 0.1 0.9 1 1 2 1 10 0.1 0.9 1 1 3 1 100 0.1 0.9 1 1 4 10 1 0.1 0.9 1.08 0.48 1.11 X 5 100 1 0.1 0.9 1.32 1.91 1.53 A 6 10 10 0.1 0.9 0.98 0.48 48 0.95 7 100 10 0.1 0.9 1.004 1.91 95 1.11 8 10 100 0.1 0.9 0.97 0.48 95 0.81 9 100 100 0.1 0.9 0.97 1.91 1 91 0.81 10 1 1 0.5 0.5 1 1 11 1 10 0.5 0.5 1 1 12 1 100 0.5 0.5 1 1 13 10 1 0.5 0.5 1.29 1.33 1.34 X 14 100 1 0.5 0.5 2.25 5.30 3.25 A 15 10 10 0.5 0.5 0.91 1.33 1 33 0.86 16 100 10 0.5 0.5 1.06 5.30 2 65 1.34 D 17 10 100 0.5 0.5 0.83 1.33 2 65 0.55 * 18 100 100 0.5 0.5 0.83 5.30 5 30 0.55 19 1 1 0.9 0.1 1 1 20 1 10 0.9 0.1 1 1 21 1 100 0.9 0.1 1 1 22 10 1 0.9 0.1 1.17 0.48 1.11 23 100 1 0.9 0.1 2.04 1.91 1.53 X 24 10 10 0.9 0.1 0.89 0.48 48 0.95 25 100 10 0.9 0.1 1.30 1.91 95 1.11 A 26 10 100 0.9 0.1 0.63 0.48 95 0.81 □ 27 100 100 0.9 0.1 0.62 1.91 1 91 0.81 * known then the figures generated from the true prototype spectrum would look much like those -5/3 that follow. It is safe to assume that the true prototype spectrum has a k power law in the inertial-convective range; in this case, the predictions for the inertial-convective range spectral levels (i.e., the g/3 and g /S values in table 1) do not depend on the choice of the m prototype spectrum. Therefore, whether or not the model 4 spectrum is a good approximation to the prototype spectrum, the changes in spectral form caused by large-scale intermittency will be evident. The predicted are shown in the figures. Cases 1 to 9 use f = 0.1, f„ = 0.9 and are illustrated in figs. 1, 2, and 3. Cases 10 to 18 use f.. = 0.5 = f„ and are illustrated in figs, 4, 5, and 6. Cases 19 to 27 use f = 0.9, f„ = 0.1 and are illustrated in figs. 7, 8, and 9. The solid curves are from the two-state model; the dashed curves are the prototype spectrum; the symbols are from the joint-lognormal model discussed in the next section. Three types of figures are used. Figures 1, 4, and 7 are (k) versus k; this is the so-called g-plot because the flat portion of the curves at low wave numbers has the value g. The variations in 3 are evident from the variable spectral level in these figures. The bump in the temperature spectrum and the rapidly falling dissipation range are evident at the higher wave numbers. Figures 2, 5, and 8 show (g) "" (k) at wave numbers near the bump. All these curves must tend to unity at small wave numbers. Finally, figs. 3, 6, and 9 show the scaled dissipation - 2 ^ spectra (k) as functions of k. The area under each curve in these latter figures is one- half of the Prandtl number, i.e., 0.72/2. Not all cases given in table 1 have a corresponding curve on the figures. Cases 1, 2, 3, 10, 11, 12, 19, 20, and 21 have Eo/e., = 1; consequently, the dashed curves apply to all these cases. Cases 6, 8, and 9 have curves similar to those of case 7. Cases 15 and 18 have curves similar to those of case 17. Cases 22 and 24 have curves that more nearly resemble the dashed curve than do curves in cases 23, 25, 26, and 27. Let and ' be the spectra measured in turbulent flows having different large-scale i, a, structure. It would seem more attractive to formulate a theory that predicts from (i.e., predicts results of one measurement from another), than to predict from an idealized spectrum r . As shown in the Appendix, T can be eliminated from the two-state model so that P P is predicted from ' . The equations in the Appendix are, however, far more complicated than (4). 4. JOINT-LOGNORMAL MODEL Let e and x T be the dissipation rates e and x averaged over regions the size of the outer scale L. The joint-lognormal model is based on the assumption that the joint probability dis- tribution of e and x-r is bivariate lognormal. It is convenient to introduce the dimensionless 1 .D 1 "A A ' ' 1.4 - A A - 1.2 i 5 A A ^>\x -- /s 1.0 - \ / 1 \ v 08 4 t r> uo '-* 0.6 \ 0.4 a\ ~ 0.2 , ,1 , , Figure 1. — Predicted temperature spectra in air for cases 4, 5, and 7 showing (k) vs. k. Dashed curve is the prototype spectrum. Solid curves are from the two-state model with indicated case numbers listed in table 1. Symbols (also listed in table 1) are from the joint-lognormal model. 10-3 10-2 10~ 1 1.6 1.5 It* 1.4 1.3 LT5 1 01 1.2- 1.1 - 1.0 0.9 0.01 Figure 2. — Predicted temperature spectra in air for cases 4, 5, and 7 showing (6) _1 (k) 5/3 vs. k (see legend, fig. 1) Figure 3. — Predicted temperature spectra in air for cases 4. 5, and 7 showing (k) vs. k (see legend, fig. 1). Figure 4. — Predicted temperature spectra in air for cases 13, 14, 16, and 17 showing (k) vs. k (see legend, fig. 1) Figure 5. — Predicted temperature spectra in air for cases 13, 14, 16, and 17 showing (B) (k) vs. k (see legend, fig. 1). Figure 6. — Predicted temperature spectra in air for cases 13, 14, 16, and 17 showing (k) vs. k (see legend, fig. 1). 10 Figure 7. — Predicted temperature spectra in air for cases 23, 25, 26, and 27 showing (k) vs. k (see legend, fig. 1). Figure 8. — Predicted temperature spectra in air for cases 23, 25, 26, and 27 showing ($)~ (k) vs. k (see legend, fig. 1). Figure 9. — Predicted temperature spectra in air for cases 23, 25, 26, and 27 showing (k) vs. k (see legend, fig. 1). random variables x and y as follows: x = X L / (6) ^V (7) where an d are the mean values of x T an d £ T obtained by averaging over the entire turbulent flow. By using x and y in place of x T an d e T one avoids dealing with the logarithm of quantities having units. Furthermore, the means, variances, and correlation of £n(x) and £n(y) are independent of the units chosen for e and Xi whereas this is not true for the means, vari- ances, and correlation of £n(x T ) and 8,n(e ) . Li Li 11 The bivariate lognormal distribution for x and y is P(x,y) = N exp(-G) (8) N = o n 2 ,l/2 ZTTxya a (1-p ) x y -1 G = -/£n(x) - m \ 2 ^- [£n(x) - m ][£n(y) .- m ] a o x y x y X,n(y) - m \2~\ /2(l-p") 2 2 with the means, m and m , variances, a and a , and correlation coefficient p of £n(x) and x y x y £n(y) given by m = <£n(x)> x m = <£n(y)> y 2 2 a = < [£n(x) - m ] > x x 2 2 a = < [iin(y) - m ] > y y p = <[£n(x) - m ][dn(y) - m ]>/a a x y x y The angular brackets denote an average over the entire turbulent flow; this average is obtained by multiplying the quantity within brackets by P(x,y) and integrating over all x and y. The definitions (6) and (7) give = 1 and = 1, whereas (8) gives , ,12, = exp(m + tt a ) x 2 x = exp(m + -r- a ) ; y 2 y consequently, m = -a /2 x x (9) m = -a /2 y y (10) D S Let p and s be real numbers; the general moment is given by = exp[Tr p(p-l)a + pspo a + -=■ s(s-l)a ] 2 x x y 2 y (11) 12 The Kolmogorov microscale obtained by using the energy dissipation rate averaged over a 3 1/4 large-scale region is n = (v /e ) . The Kolmogorov microscale obtained by using the energy - 3 1/4 dissipation rate averaged over the entire flow is n = (v /) . The scalar spectrum ob- tained from a large-scale region is denoted by Y (k) and is a function of the random variables X T and e . The scalar spectrum obtained by an average over the entire turbulent flow is denoted by and is given by OO 00 = / / dx dy T L P(x,y) . (12) o o a. Equation (12) is valid only for high wave numbers, namely, k >> 2tt/L. The scaled spectrum r Li is obtained by scaling r with e and x T '■> that is, 9, -5/4 3/4 -1 . r L = V £ L X L r L ; % -1/4 - -1/4 - T is assumed to be a function only of the Prandtl number and of n k = y nk = y k, i_i J-j - — r\, where k = nk. The scaled spectrum is obtained by scaling using the entire-flow aver- aged dissipation rates < e T > and ; that is, Li J_i -i< -5/4 3/4 -1 _ = v . -5/4 3/4 -l By multiplying (12) by v < ^t > ' t ^ ie relationship between the scaled spectra is obtained: 00 OO = / / dx dy y~ 3/4 x T P(x,y). (13) O O The integration over x is performed by elementary methods after transforming to a new vari- able of integration X = (£n(x) - m )/o . The integration variable y is replaced by the argument a, _i /4 _ x x of T , namely by K = n k = y k. Then (13) becomes CO as a function of k if Y (K) is known. As in the two-state V X Li model, F (K) is an idealized spectrum characteristic of a turbulent flow which has the least possible intermittency at large scales. The spectrum used here for r (K) is the same as that used in the previous section; that is, T (K) is given by the dashed curves in the figures. Numerical integration is required to obtain from (14) . In the inertial-convective range, then (12) and (11) yield - B 1/3 k 5/3 (16) m L L where the measured value is given by m 2 2 1 0/0 = exp(^ a - 7 pa o ) . (17) m 9 y 3 x y The effects of large-scale intermittency on a measurement of Batchelor's constant, q, may be estimated in the same manner. If Pr >> 1, then a viscous-convective range exists where T L (k) = qx L v e L k (18) for k >> n • Inserting (18) into (12) and using (11) gives 1/2 -1/2 . -1 nQ x = a v k (19) wh ere q is given by V q = 6XP( I °l - \ P °x a y ) • Therefore, the measured value q differs from q. Large-scale intermittency also affects a measurement of the transitional wave number, k-% between the inertial-convective and viscous- convective ranges. Equating (16) and (19) at k = k gives -, * fo , n3/2 , 11 2,1 N nk m = (3/q) ex P (- 48 a y + ^ pa^aj , whereas equating (15) and (18) at k = k* gives 3/2 n L k* = (3/q) ; - 14 To calculate from (14), values of a and pa a are needed. The values chosen for 2 y x y a and pa a are the variance of £n(e) and the correlation of £n(£) and £n(x) as calculated from y x y the two-state model, namely, a 2 = f 1 [£n(F 1 )] 2 + f 2 [£n(F 2 )] 2 - [f^ntf^ + f 2 ^n(F 2 )] 2 pa a = f 1 £n(D 1 )£n(F.) + f„S>n(D„Hn(F ) xyll 1 11 l - [f fcn(F ) + f 2 £n(F 2 )][f 1 «,n(D 1 ) + f 2 Jln(D 2 )] where F 1 B (e 1 /e 2 )/[f 1 (E 1 /e 2 ) + f 2 ] F 2 = (e 2 /e 1 )/[f 1 + f 2 (e 2 /E 1 )] = (ej/e.^ D 1 E (x 1 /x 2 )/[f 1 (x 1 /x 2 ) + f 2 ] D 2 = (x 2 /x 1 )/[f 1 + f 2 (x 2 /x 1 )] = (X 2 /X 1 )D 1 ■ 2 The calculated values of a , pa a , and the values of g /g found from (17) are given in table 1. ^ y x y m The spectrum is calculated from (14) at several values of k for some of the cases in table 1. The results are plotted in the figures using symbols; the symbols also appear in table 1 for these cases. Cases 4 and 5 appear in figs. 1, 2, 3; cases 13, 14, 16, 17 appear in figs. 4, 5, 6; cases 23, 25, 26, 27 appear in figs. 7, 8, 9. All the cases in table 1 have p = 1.0, or 2 2 p is undefined because either a = or a =0. Data from the axis of symmetry of a heated x y turbulent round jet (Williams and Paulson, 1977) indicate that p = 0.7 is a reasonable choice. 2 2 The case p = 0.7, a =1.0, and a =1.0 has also been calculated and the results depart only x y slightly from the dashed curve; these calculated spectral values are only about 10% smaller than the dashed curve at the peak of the dissipation spectrum. The extent to which the predictions of the two-state and joint-lognormal models agree is found by making the following comparisons. In figs. 1, 2, and 3 compare the crosses with the case 4 curve and the triangles with the case 5 curve. In figs. 4, 5, and 6 compare crosses, triangles, squares, and asterisks with curves for cases 13, 14, 16, and 17, respectively. In figs. 7, 8, and 9 compare crosses, triangles, squares, and asterisks with curves for cases 23, 25, 26, and 27, respectively. The two-state and joint-lognormal models do not agree in detail, and one has no reason to expect that they would, but both models show the same trends away from the dashed curve. The joint-lognormal model shows generally greater deviations from the dashed curve than does the two-state model in the first 18 cases shown in figs. 1 to 6, whereas the opposite is true in the last 9 cases shown in figs. 7 to 9 . The figures show a trend toward the largest effects of intermittency being associated with the greatest e-intermittency but moderate X- intermit tency. 15 As shown in the Appendix, the prototype spectrum can be eliminated for the two-state model, thereby predicting one measured spectrum from another. For the joint-lognormal model no such elimination of the prototype spectrum seems possible. 5. CONCLUSION Two models, the two-state and joint-lognormal, are constructed to predict the effects of large-scale intermittency on measurements in high-Reynolds-number turbulence of the high-wave- number portion of the scalar spectrum. Analogous models for similar effects on the energy spectrum are easy to construct. Exemplary predictions for the temperature spectrum in air are given in the figures. The models are also applicable to large Prandtl number scalar advection. It would be interesting to see if the trends shown in the figures can be confirmed experi- mentally. One such experiment would be to measure the temperature spectrum at various positions transverse to the axis of symmetry of a heated, round, turbulent jet. Such an experiment must be at high Reynolds number because of the restriction that the models predict only the high- wave-number (k >> 2tt/L) behavior. It must be kept in mind that the models ignore the contribu- tion of small-scale intermittency. The small-scale intermittency may have a dominant effect deep in the dissipation range. 2 The temperature structure parameter C (obtained by an average over the entire flow) is related to the dissipation rates by 2 - -1/3 C T = 2.4 3 <£> The variation of 3 (or, in the notation of the joint lognormal model, the variation of g ) caused by large-scale intermittency implies an uncertainty in obtaining the structure parameter from dissipation rates or vice versa. The curves in figs. 2, 5, and 8 can be interpreted in 2 terms of scaling by C since (3) _1 (k) 5/3 = 2.4 k 5/3 /C 2 T . 2 The propagation of light through the turbulent atmosphere is described in terms of C , not the dissipation rates. Consequently, the variability of 3 does not influence the interpretation of optical propagation in turbulent air. What does influence this interpretation is the variability 2 of /C , that is, the variations of the spectral values as presented in figs. 2, 5, and 8. Moreover, the range of spatial wave numbers in these three figures often dominates the variance of log-intensity for short-path atmospheric propagation. The variations of spectral values in these three figures imply a lack of predictability for optical systems. Unfortunately, it is not yet established how variable the atmospheric dissipation rates are under various meteoro- logical conditions. 16 6. REFERENCES Antonia, R. A., and C. W. Van Atta (1975): On the correlation between temperature and velocity dissipation fields in a heated turbulent jet. J. Fluid Mech . 67:273-288. Batchelor, G. K. (1959): Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech . 5:113-133. ~ Batchelor, G. K. , I. D. Howells, and A. A. Townsend (1959): Small-scale variation of convected quantities like temperature in turbulent fluid. Part 2. The case of large conductivity. J. Fluid Mech . 5:134-139. Champagne, F. H. , C. A. Friehe, J. C. LaRue, and J. C. Wyngaard (1977): Flux measurements, flux-estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land. J. Atmos . Sci . 34:515-530. Corrsin, S. (1951): On the spectrum of isotropic temperature fluctuations in isotropic tur- bulence. J. Appl . Phys . 22:469-473. Gibson, C. H. (1968): Fine structure of scalar fields mixed by turbulence. II. Spectral theory. Phys . Fluids 11:2316-2327. Gibson, C. H. , and P. J. Masiello (1972): Observations of the variability of dissipation rates of turbulent velocity and temperature fields. In Statistical Models and Turbulence, Lecture Notes in Physics , vol. 12, M. Rosenblatt and C. W. Van Atta (eds.), Springer- Verlag, New York, pp. 427-453. Hill, R. J. (1978): Models of the scalar spectrum for turbulent advection. J_. Fluid Mech . 88:541-562. Kolmogorov, A. N. (1941): The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl . Akad . Nauk SSSR 30:301-305. Kolmogorov, A. N. (1962): A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. ^J. Fluid Mech . 13:82-85. Kraichnan, R. H. (1968): Small-scale structure of a scalar field convected by turbulence. Phys . Fluids 11:945-953. Oboukov, A. M. (1949): Structure of the temperature field in turbulent flow. Izv . Akad. Nauk SSSR , Ser. Geogr. i Geofiz. 13:58-69. Oboukov, A. M. (1962): Some specific features of atmospheric turbulence. J_. Fluid Mech . 13:77-81. Van Atta, C. W. (1971): Influence of fluctuations in local dissipation rates on turbulent scalar characteristics in the inertial subrange. Phys . Fluids 14:1803-1804. Williams, R. M. , and C. A. Paulson (1977): Microscale temperature and velocity spectra in the atmospheric boundary layer. J. Fluid Mech . 83:547-567. 17 APPENDIX: Elimination of Prototype Spectrum for Two-State Model Consider two turbulent flows for which the two-state model applies. Call one the primed flow and denote its parameters by f' f' (c'/e'), (xo/x-i')» C, C' C', the scaled spectrum by ' , and beta value g'. The other is called the unprimed flow with parameters f , f„, (e„/e 1 ) , (x 9 /x-,), C , C„, C, its scaled spectrum is , and beta value 3- It is convenient to define the following parameters: A ± = fjC'/Cj Pi fy — L^U ' ^ O A n /A f ' r' r l L 2 1 '"2 f ' r' H = [£[ + V 2 (e^/ep] 1/4 i'[f{ + t\ (xVxpi [f|( £ ;/e.p 1/3 + f^][f;(e'/sp 1/3 + f^( X */ X |)] fj (e[/e } 2 ) + f' 2 f] + f^ (epep 1/4 For simplicity let G(x) = V (x) . P If r = 1, then (4) gives ' = (A 1 + A 2 )G(x') where x 1 = x'H. Then (4) also gives = [f 1 C, r(zj>' + f. p- ']/(A 1 + a ) 2 C where f 1+ f 2 (e 2 / ei ) L f|+ fj (e 2 /ep 1/4 f 1 (e 1 /e 2 ) + f 2 L f|(e|/e 2 ) + f 2 1/4 This establishes the desired relationship between for r = 1. 18 On the other hand, if r ^ 1 no generality is lost in taking r < 1. Define a sequence of points {y . } by y 1 = x/h y i+l E ry i Choose the positive integer n so large that r x < 0.02 One can write a sequence of equations of the form of (4) for ' where the argument of the last G is r x, which is in the inertial-convective range. Then one can invert this sequence of a, equations to give G in terms of , that is, G(x) = B R n (r n x) 5/3 + A 1 "I R k ' 1 k=0 k+i (A-l) Of course, if x < 0.02 then one takes n = 0, and this expression reduces to r< \ a -5/3 G(x) = 3 x By inserting (A-l) into (4), is expressed in terms of by „,->.* c C ^n, n N -5/3 . c C „_,m - m -,-5/3 = f — BR (r x ) + f 2 — BR (r x 2 > f C _ n-1 L i i k=o k+i f„C m-1 , + — A" 1 I R k ' U 2 X k=0 k+i With x and x as in (4) , the positive integers m and n are chosen so that r x < 0.02 and r x <_ 0.02 . The sequences {E,.} and {£.} are given by 5 1 = x x /H and C 1 = x 2 /H, £ i + l = r? i and ? i+i = r? i 19 *U.S. GOVERNMENT PRINTING OFFICE: 1980-0-6 77-09 6/125 8 yiwiioktmbi ma^ck LABOR AT DRIES "N The mission of the Environmental Research Laboratories (ERL) is to conduct an integrated program of fundamental research, related technology development, and services to improve understanding and prediction of the geo- physical environment comprising the oceans and inland waters, the lower and upper atmosphere, the space envi- ronment, and the Earth. The following participate in the ERL missions: W/NI Weather Modification Program Office. Plans and coordinates ERL weather modification projects for precipitation enhancement and severe storms mitigation. NHEML National Hurricane and Experimental Meteor- ology Laboratory. 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