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 NOAA Technical Report ERL 385-AOML 26 
 
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 Sr 4TES 0* 
 
 A Transformation Relating 
 Temporal and Spatial Spectra 
 Of Turbulent Kinetic Energy 
 
 W. C. Thacker 
 
 February 1977 
 
 U. S. DEPARTMENT OF COMMERCE 
 
 National Oceanic and Atmospheric Administration 
 Environmental Research Laboratories 
 
Digitized by the Internet Archive 
 
 in 2013 
 
 http://archive.org/details/transformationreOOthac 
 
NOAA Technical Report ERL 385-AOML 26 
 
 rfggggW 
 
 ^IHENTOPC^ 
 
 A Transformation Relating 
 Temporal and Spatial Spectra 
 Of Turbulent Kinetic Energy 
 
 W. C. Thacker 
 
 Atlantic Oceanographic and Meteorological Laboratories 
 Miami, Florida 
 
 February 1977 
 
 > 
 
 a 
 o 
 
 c 
 o 
 
 o 
 
 U.S. DEPARTMENT OF COMMERCE 
 
 Juanita M. Kreps, Secretary 
 
 National Oceanic and Atmospheric Administration 
 Richard A. Frank, Administrator 
 
 Environmental Research Laboratories 
 Wilmot Hess, Director 
 
 Boulder, Colorado 
 
CONTENTS 
 
 Page 
 Abstract 1 
 
 1. INTRODUCTION 1 
 
 2. THE FROST-FREE TURBULENCE TRANSFORMATION 2 
 
 3. A COMPARISON WITH DATA 4 
 
 4. A COMPARISON WITH A DIMENSIONAL ARGUMENT 5 
 
 5. SHEAR DISPERSION 6 
 
 6. SUMMARY 8 
 
 7. REFERENCES 8 
 
 \ 
 
A TRANSFORMATION RELATING TEMPORAL 
 
 AND SPATIAL SPECTRA OF TURBULENT 
 
 KINETIC ENERGY 
 
 W. C. Thacker 
 
 ABSTRACT. A transformation is developed, based upon the scale dependence of turbulent 
 diffusion, that relates temporal and spatial spectra of turbulent kinetic energy. The basic 
 idea is that an eddy diffusivity is appropriate when scales of the flow smaller than a length 
 t and a time/ are unresolved. An expression similar to Heisenberg's for eddy diffusivity is 
 used to obtain the connection between ( and t necessary to transform temporal spectra into 
 spatial spectra. This transformation reveals a close connection between Webster's Site D 
 spectrum of turbulent kinetic energy in the ocean and Okubo's diagrams of oceanic mixing. 
 Furthermore, all spectra obtained from dimensional arguments satisfy this transformation. 
 
 1. Introduction 
 
 An important quantity in the theory of turbu- 
 lence \sE(k), the spatial spectrum of kinetic en- 
 ergy. To measure£(Ar) is difficult since it requires 
 sampling the velocity field at many spatial points 
 simultaneously. It is much easier to record a time 
 series of the velocity at one point, from which 
 <J>(w), the temporal spectrum of kinetic energy, 
 can be obtained. Therefore a transformation is 
 needed that will allowE(it) to be calculated if <J>(w) 
 is known. The usual transformation is based upon 
 Taylor's (1938) hypothesis of frozen turbulence, 
 which is valid only if there is a strong mean flow. 
 The purpose of this paper is to present a new 
 transformation that should be valid in the absence 
 of a mean flow. To stress the contrast with the idea 
 of frozen turbulence the term "frost-free turbu- 
 lence" is used. 
 
 The frost-free turbulence transformation is 
 motivated by the results of dye-diffusion experi- 
 ments in the ocean as summarized by Okubo's 
 (1971) diagrams. His first diagram, reproduced 
 here as Figure 1, illustrates that the spatial and 
 temporal scales of turbulence can be related. 
 This is certainly necessary if there is to be a trans- 
 formation that can relate $(oj) andE(A:). His sec- 
 ond diagram, Figure 2, shows the scale depen- 
 dence of the eddy diffusivity. An expression for 
 scale dependent diffusivity, such as Heisen- 
 berg's (1948) expression for eddy viscosity, is 
 central to this transformation. The transformation 
 
 is insensitive to the exact form of this expression 
 because <&(&>) and E(k) fall off rapidly with in- 
 creasing a; and k. 
 
 This transformation is obtained in two ways. 
 First 4>(w) and E(k) are related through the more 
 general spectral density, S (k, a>), which expresses 
 both spatial and temporal variations. A compari- 
 son is made with the frozen turbulence case, and 
 a more general transformation is suggested that 
 has the limits of frozen turbulence and frost-free 
 turbulence, depending upon whether advection 
 or diffusion dominates. Then a heuristic derivation 
 is given, based upon a mechanism for turbulent 
 mixing. The idea here is that the mixing is due to 
 shear dispersion on all scales. The results of a 
 two-layer model for the shear effect are iterated 
 over all scales to obtain expressions for the scale 
 dependence of turbulent diffusivity from which 
 the frost-free turbulence transformation follows. 
 
 Because it is difficult to measure E(k), it is 
 difficult to test the validity of this transformation 
 directly. Two indirect tests are discussed here. 
 The first is a comparison of a temporal spectrum 
 of kinetic energy of turbulence in the ocean ob- 
 tained by Webster (1969) with the diffusion data 
 displayed by Okubo (1971). It should be em- 
 phasized that no theory is presented for the ob- 
 served shape of this spectrum. That is a dynami- 
 cal problem, and the transformation discussed 
 here should be regarded as kinematical. The 
 
 1 
 
comparison shows that the two types of data are 
 in excellent agreement. The second test is pro- 
 vided by a dimensional argument. If it is assumed 
 that only one dimensional constant is important, 
 then consistent forms for E(k) and $>(&>) can be 
 obtained. Again, it should be emphasized that it is 
 unimportant whether such a dimensional argu- 
 ment can be applied to real data. What is impor- 
 tant is that the forms for £ (it) and $(oj) obtained 
 from the dimensional argument are indeed re- 
 lated through the frost-free turbulence transfor- 
 mation. 
 
 2. The Frost-Free Turbulence 
 Transformation 
 
 A turbulent velocity field can be considered 
 to be a random function of space and time. If the 
 turbulence is statistically homogeneous, iso- 
 tropic, and stationary, then the variance in the 
 velocity field (u 2 (x,t)) can be represented in 
 terms of spectral density, S(k,w). The temporal 
 and spatial spectra are obtained fromS(£,w) by 
 integrating over wavenumbers and frequencies, 
 respectively: 
 
 <£>(uj) 
 
 E(k) 
 
 dk S{k,o>) 
 
 du> S{k, oj) 
 
 (1) 
 
 Thus, <l> is related to E through S{k,w). 
 
 In general, a single frequency does not 
 correspond to a single wavenumber. Neverthe- 
 less, it is clear that high frequencies correspond 
 to high wavenumbers and low frequencies to low 
 wavenumbers. For example, oceanic motion with 
 a scale of hundreds of kilometers is expected to 
 correspond to time variations on the scale of 
 months, not seconds. Therefore, it should be 
 reasonable to assume that, for any wavenumber, 
 S(k,oj) is sharply peaked at a single frequency 
 and, for any frequency, S(k,co) is peaked at a 
 single wavenumber. This can be expressed in 
 two ways, 
 
 S(k,co) = E(k)b(u)-f(k)) 
 S(k, oj) = <t>(aj)8(k-g((o)) 
 
 (2) 
 
 which are equivalent if the functions/ and g are 
 the inverses of each other. The Dirac delta func- 
 tions can be considered as approximating more 
 general distributions with finite widths. 
 
 10" 
 
 1 1 ' 
 
 I 
 
 i 1 i i 
 
 
 ® Rheno 
 
 
 
 
 
 A 1964 V 
 
 
 
 t" / 
 
 
 D 1962 II 
 
 -North Sea 
 
 
 / 
 
 
 " O 1962 II 
 
 
 
 ® / 
 
 
 1961 I 
 
 
 
 
 10" 
 
 — 
 
 
 / — 
 
 
 # No 1 
 
 
 
 
 
 ■ ♦ No 2 
 
 
 
 1 
 
 
 A No. 3 
 - ♦ No. 4 
 
 Off 
 - Cape 
 
 
 ® 
 
 / 
 
 ^® 
 1 
 
 
 a No 5 
 
 Kennedy 
 
 
 10 12 
 
 - O No. 6_ 
 
 
 
 
 - © New York Bight 
 ® = a" 
 
 
 a' 
 
 r 
 
 
 - ® = b 
 
 
 
 ® 
 / 
 
 
 O = c 
 
 Off 
 
 i 
 
 £ 
 
 i 10 11 
 
 ® = d 
 
 California 
 
 afi/ 
 
 
 — ' 
 
 e = e 
 
 
 *® 
 
 
 ^ <_> 
 
 » = f_ 
 
 
 1 
 
 
 b" 
 
 S Banana River 
 
 M* 
 
 
 
 - 6 $ 
 
 
 lO'o 
 
 _ v Manokin River 
 
 9 Aw 
 
 - 
 
 
 - 
 
 
 - 
 
 
 - 
 
 / 8 
 
 - 
 
 
 e& 
 
 ^ 
 
 
 109 
 
 
 
 
 
 * / 
 
 aP 
 
 
 
 o a 
 
 
 - 
 
 
 8 / 
 
 
 
 10« 
 
 / 
 
 
 - 
 
 
 I f 
 
 
 - 
 
 
 ® 
 
 
 
 
 Hourft /« 
 
 Day 
 
 Week Month 
 
 10? 
 
 I*/ , 
 
 1, , 
 
 ,1 , ,1 , ^ 
 
 100 km 
 
 - 10 km 
 
 - 1 km 
 
 100 m 
 
 103 
 
 10" 
 
 10 5 
 Time (sec) 
 
 10 6 
 
 10? 
 
 Figure 1. Variance of dye concentration (size of dye 
 patch) versus diffusion time (time elapsed since the 
 dye was introduced as a point source). (After Okubo, 
 1971.) 
 
 Using (1) with (2), transformations connect- 
 ing E and <P can be obtained: 
 
 E(g(oj)) 
 
 4>(w) 
 
 E(k) 
 
 <t>(f(k)) 
 
 (3) 
 
 g'(M)) 
 
 Thus, the frost-free turbulence transformation will 
 depend upon the form of the function/, its inverse 
 g, and their derivatives/' and g' . 
 
 Clues for the form of / can be found in 
 Okubo's (1971) dye diffusion diagrams. Figure 1 
 shows the relationship between the width of a dye 
 patch and the duration of the dye diffusion exper- 
 iment. This is the connection between space and 
 time scales that is to be expressed by/. The fact 
 that the slope of the line drawn through the data is 
 greater than one indicates that turbulent diffusion 
 
expression for the scale dependence used here 
 is 
 
 10? 
 
 106 
 
 105 
 
 103 
 
 102 
 
 T 
 
 © Rheno 
 A 1 964 V 
 D 1962 III 
 O 1962 II 
 O 1961 I . 
 
 # no. r 
 
 North 
 Sea 
 
 ♦ No 
 
 A NO. 
 
 <> No. 
 
 a No 
 
 O No 
 
 Off 
 
 Cape 
 
 Kennedy 
 
 B New York Bight 
 
 Off 
 California 
 
 ® = b 
 O = c 
 ® = d 
 e = e 
 a> = f_ 
 
 a Banana River 
 
 r 
 
 > 
 
 ®e/a 
 
 <^o 
 
 103 
 
 104 
 
 10 ! 
 
 106 
 
 10? 
 
 (cm) 
 
 Figure 2. A diffusion diagram for apparent diffusivity ver- 
 sus scale of diffusion. (After Okubo, 1971.) 
 
 is scale dependent. Okubo obtains a scale de- 
 pendent diffusivity K from these data using the 
 expression 
 
 P 
 
 2Kt 
 
 (4) 
 
 to relate the width of the dye patch / to the dura- 
 tion of the experiment t. Figure 2 shows K plotted 
 against ( . Eguation (4), plus an expression for the 
 scale dependence of K, gives the relation be- 
 tween spatial and temporal scales necessary to 
 define the functions/ and g. 
 
 It is easy to understand why turbulent diffu- 
 sion should be scale dependent. The spreading 
 of the dye patch can be due only to those eddies 
 that are smaller than the dye patch. Larger eddies 
 serve only to advect and to distort the dye patch. 
 At a later time when the dye patch is larger, larger 
 eddies are available to contribute their energy to 
 the mixing. Thus, the mixing proceeds faster as 
 the dye patch gets larger. 
 
 Those eddies smaller than the dye patch, 
 having wavenumbers greater than *jf, are para- 
 meterized by the diffusivity K As ( gets larger, 
 
 so does K Thus, K is a function of k 
 
 2tt 
 
 The 
 
 A 
 
 E(k')dk' 
 k' 2 
 
 1/2 
 
 (5) 
 
 where C is a dimensionless constant of propor- 
 tionality of order one. This is an expression quite 
 similar to that used by Heisenberg (1948) for 
 eddy viscosity, and is exactly that found by Tchen 
 (1973, 1975) and Nakano (1972) from their 
 dynamic theories of turbulence. A heuristic deri- 
 vation of this expression, based upon the idea 
 that the mechanism of turbulent mixing is shear 
 dispersion on all scales, is given below. 
 
 The parameter A", evaluated according to (5), 
 accounts for the effects of eddies with wavenum- 
 bers larger than k = —where / is the width of 
 the dye patch. The basic assumption made here 
 is that these eddies correspond to frequencies 
 
 greater than to 
 
 where t is the duration of 
 
 the dye experiment corresponding to the width <?. 
 Thus, (o is related to k and K by equation (4), 
 
 (D = TT- , Kk 2 . 
 
 (6) 
 
 Equation (6) expresses the relationship between 
 a> and k that is necessary to transform <l>(oo) into 
 E(k). This should be thought of as a statistical 
 assumption for several reasons. First, since, in 
 general, there is no one-to-one relationship be- 
 tween frequencies and wavenumbers, the rela- 
 tionship expressed by (6) must be statistical in 
 the sense that it is a "most likely" relationship. 
 Second, it is statistical since the parameter K is 
 assumed to represent the average effect of the 
 small scales. It is clear that (6) should be valid 
 only when the small scales can be described by 
 an eddy diffusivity. Finally, implicit in (6) is the 
 idea of ergodicity: a spatial average, a temporal 
 average, and an ensemble average should all be 
 equivalent. A time series of length / determines 
 O(crj) for ay > y^. Likewise, a spatial profile of 
 length / determines E(k) for k > =^. If the time 
 
 series is measured simultaneously with the dye 
 experiment, it seems most reasonable to relate w 
 and k according to (6). Clearly, the results for 
 each experiment should vary somewhat, but it is 
 reasonable to think of a most likely result that 
 represents the average of an ensemble of experi- 
 ments. It is in this way that (6) should be inter- 
 preted. 
 
By substituting (5) into (6), the expression for 
 / is determined; 
 
 /(*) = 77- 'Jt- 
 
 cC*n dk > 
 J, *' 2 
 
 (7) 
 
 The corresponding expression for g is found by 
 inverting/. The simplest way to do this is to take 
 advantage of the fact that E(k) is simply a trans- 
 formation of <!>(a;). This implies that A can also be 
 expressed as an integral of <t>{io) over frequencies 
 greater than w, and that expression can be used 
 in (6) to obtain g. To obtain that expression, first 
 
 write (5) in differential form as dK = - C E i k ,)f! i , 
 
 2Kk 2 
 
 and then use (6) and the identity E(k)dk = <S>(a>)da> 
 to obtain dK = - £- ®H d(t > . This can be inte- 
 
 grated to give 
 
 A 
 
 2tt 
 
 0) 
 
 ^{(o')d(o' 
 
 Now, (8) and (6) yield 
 C 
 
 g(v) 
 
 2tt 2 (i> 
 
 <$>(a)')du}' 
 (o' 
 
 (8) 
 
 (9) 
 
 Equations (7) and (9), together with (3), de- 
 termine the frost-free turbulence transformation. It 
 is simplest to write this in terms of A, 
 
 $(&))= 
 
 Eiky- 
 
 2,KE{^ 
 
 7TO) 
 
 -ce( 
 
 TTO) 
 
 4kK 2 
 
 A 
 
 4kK 2 <$>{TT- x Kk 2 ) 
 2TrK+C<S>(TT- l Kk 2 
 
 (10) 
 
 where A is given by (5) and (8), respectively. 
 Equations (5) and (8) will be discussed further in 
 section 5. Equations (10) will be compared with 
 results of experiments in section 3 and with re- 
 sults of dimensional arguments in section 4. 
 
 Equations (2) can be used to obtain the 
 frozen turbulence transformation also. For that 
 case, /(A:) = Uk and g{co) = oj/U, where V is the 
 mean velocity that advects the frozen turbulence. 
 Using (3), the frozen turbulence transformation is 
 given by 
 
 <!>( 
 
 
 E(k) = U<P(Uk) 
 
 It is possible to construct a more general 
 transformation that reduces to frozen turbulence 
 
 when advection dominates diffusion and to frost- 
 free turbulence when diffusion dominates advec- 
 tion. One possibility is through the use of the 
 function 
 
 f(k) = Uk + tT 1 Kk 2 
 
 and its inverse, where A' is given by (5) and (8), as 
 
 before. This leads to the transformation 
 
 E(k) = 
 
 {IttVK + 4kK 2 )<t>{Uk + TT-'Kk 2 ) 
 
 IttK + C<P(Uk + TT"'*-* 2 ) 
 
 If the trend is removed from the time series from 
 which <t> is to be obtained, then the information 
 concerning the mean velocity U is lost so the 
 transformation given in (9) should be used. 
 
 3. A Comparison With Data 
 
 A direct test of the frost-free turbulence trans- 
 formation given by equations (10) is impossible 
 since the data necessary to evaluate E(k) are un- 
 available. Nevertheless, long time series of the 
 velocity at one point in the ocean have been ob- 
 tained, so <P(oj) is available. Such a spectrum 
 from Site D (Webster, 1969) is shown in Figure 3. 
 This can be compared with Okubo's (1971) dye 
 diffusion diagrams to check whether the trans- 
 formation is reasonable. 
 
 It is possible to evaluate the diffusivity from 
 (8) numerically using the data in Figure 3. How- 
 ever, for simplicity, these data are approximated 
 by the formula 
 
 *(<a) ~ (o~ 4l \ 
 
 (ii: 
 
 This does not imply that there is any dynamic sig- 
 nificance to the exponent -4/3. This exponent 
 was chosen simply to represent the gross be- 
 havior of the spectrum over the entire range of 
 frequencies. The details of the tidal and inertial 
 peaks in the spectrum should contribute, at most, 
 shoulders to the curve of diffusivity versus fre- 
 quency scale. Substituting (11) into equation (8) 
 yields 
 
 A 
 
 .4/3 
 
 Using this in (6), with / = -^andf =-^—, gives 
 
 A: a) 
 
 a relation between spatial and temporal scales, 
 
 e 2 ~t m . d2) 
 
 This is exactly the behavior shown in Figure 1 . Of 
 course, the data in Figure 2 are well described by 
 
 D - P'\ 
 
 (13) 
 
 4 
 
Period, Hours 
 
 a 
 
 a> 
 
 a> 
 
 c 
 
 LU 
 
 480 
 
 96 24 
 
 
 8 
 
 3 
 
 1 .5 
 
 10 4 
 
 
 \ cJ 4/3 
 
 A / 
 
 P 
 
 I 
 
 I 
 
 10 3 
 
 
 V 
 
 
 
 aJ 5/3 
 
 
 10 2 
 
 
 
 
 
 
 10 1 
 
 
 1841 
 Site D 
 120 M 
 Jun. 24 — Aug. 
 
 11, 1965 
 
 \ \ 
 
 ■ 
 
 10° 
 
 
 
 
 
 !\ 
 
 .01 .1 1 
 
 Frequency, Cycles Per Hour (c.p.h.) 
 
 Figure 3. Kinetic energy density spectrum on a log-log 
 plot for a set of current measurements collected at 
 120-m depth. Minus-four-thirds and minus-five-thirds 
 slopes are indicated. (After Webster, 1969.) 
 
 obtained using equations (4) and (12), since 
 Okubo uses (4) to transform the data from Figure 
 1 to Figure 2. 
 
 In making this comparison, two equations were 
 used, (6) and (8). These are the two equations 
 that define frost-free turbulence transformation. 
 Equation (6) relates frequencies to wavenumbers 
 through the eddy diffusivity, and equation (8) ex- 
 presses the scale dependence of the eddy dif- 
 fusivity. The agreement found when these two 
 sets of data are compared in this way is evidence 
 that this transformation is indeed valid. 
 
 4. A Comparison With A 
 Dimensional Argument 
 
 A dimensional argument also provides a 
 transformation connecting <P(w) and E{k). This 
 argument is based upon the assumption that the 
 spectra depend upon only one dimensional 
 
 constant. For example, if this constant is e, the rate 
 of energy dissipation, then the dimensional 
 argument gives the familiar results for the inertial 
 subrange: E{k) - k~™ and <f>(aj) -- cu -2 . No 
 attempt is made here to argue that a single 
 dimensional constant is appropriate for the entire 
 spectrum shown in Figure 3. Perhaps it is 
 possible to divide the spectrum into subranges in 
 which a single constant is important, but that is 
 not assumed. The point is that if such a subrange 
 does exist, then the dimensional argument for the 
 subrange is in agreement with the frost-free 
 turbulence transformation. 
 
 Suppose that the one important dimensional 
 constant Q has dimensions X"T h . Then E and <S> 
 must have the following forms in order to be di- 
 mensionally correct: 
 
 E(k)~Q "k 
 
 la 
 
 (-3 
 
 2 2Z> 
 
 <DM~e o)' 
 
 1 1 
 
 (14) 
 
 For the case where Q is taken to be e, the expon- 
 ents a and b are 2 and -3, respectively. 
 
 These forms can be seen to be consistent 
 with frost-free turbulence. Suppose that $(oj) ~ 
 oj-", where />=- (^-+1), and calculate E(k) 
 using the frost-free transformation (10). The result 
 is 
 
 E(k)~k y " +1 
 
 15) 
 
 which is exactly what is given in (14) when 
 
 lb 
 
 + 1 
 
 Dimensional arguments can also be made 
 for the diffusivity and for the width of the dye patch: 
 
 24 
 
 K~Q "t 
 
 ~( 
 
 2p 
 p+\ 
 
 (- 
 
 Q l t 
 
 2b 
 
 ~ t 
 
 p+\ 
 
 (16) 
 
 These results are also in agreement with frost- 
 free turbulence, as can be seen from equations 
 (5) and (6) with E{k) given by (15). 
 
5. Shear Dispersion 
 
 It is possible to derive the frost-free turbu- 
 lence transformation using the idea that the 
 mechanism for turbulent mixing is shear disper- 
 sion on all scales. Such a derivation should clarify 
 the idea of scale dependent diffusion that is 
 intrinsic to the transformation described here. 
 It should also clarify the manner in which the 
 advection of small eddies by large eddies is in- 
 corporated into the transformation. 
 
 Shear dispersion was discussed by Taylor 
 (1953, 1954) in the context of longitudinal disper- 
 sion in pipes. He found that an enhanced dif- 
 fusivity was needed to account for the dispersion 
 of contaminant introduced into the flow. This 
 enhanced diffusivity is appropriate in conjunction 
 with the cross-sectional average of the contami- 
 nant concentration. The enhancement is due to 
 the combined action of shear and cross-shear 
 mixing, features that are unresolved when we are 
 dealing with cross-sectional averages, and 
 whose effects are accounted for by the enhanced 
 diffusivity. 
 
 This shear dispersion relates to turbulent 
 diffusion in two ways. First, the eddy diffusivity 
 can likewise be considered as parameterizing 
 the details of the flow that are averaged out. The 
 value of the diffusivity appropriate to a given scale 
 is determined by those details of the flow that are 
 smaller than this scale. Second, the mechanism 
 of turbulent mixing is shear dispersion on all 
 scales. At any scale there are eddies that provide 
 shear, and there are smaller eddies that provide 
 mixing across this shear. As the scale is in- 
 creased, the eddy diffusivity must be enhanced 
 to account for the additional shear and cross- 
 shear mixing that is averaged out. 
 
 From the point of view of someone numerical- 
 ly modeling the flow, this is clear. Eddy diffusivity 
 is used to parameterize mixing due to sub-grid 
 scale motion. For a coarser numerical grid, a 
 larger value of diffusivity is needed to account 
 for the mixing that would be explicitly resolved 
 on a finer grid. 
 
 A simple two-layer model of shear dispersion can show how the diffusivity is enhanced as the details 
 of the shear are averaged over. This model is given by the eguations, 
 
 and 
 
 dt 
 
 dC 2 
 dt 
 
 + M, 
 
 dCj 
 
 dx 
 
 dC 2 
 dx 
 
 T 
 7 
 
 (C, - C,) + K 
 
 (C 2 -C 1 )+K 
 
 d 2 C, 
 dx 2 
 
 d 2 C x 
 dx 2 
 
 (17) 
 
 governing the contaminant concentrations C, and C 2 in the layers of fluid with velocities «, and u s . The 
 contaminant mixes from the more concentrated to the less concentrated layer with a mixing time T. 
 Longitudinal diffusion within each layer is described by a diffusivity K. If both T and K are due to eddies 
 that are unresolved in this two-layer description, then they should be related by 
 
 f 2 = 2AT, 
 
 (18) 
 
 where / is the thickness of the layers. 
 
 The reason for considering this model is to illustrate the relationship of the mean concentration, 
 C =4- (Ci + Co), as determined by (17), and the solution of the advection-diffusion equation, 
 
 dC 
 dt 
 
 + u 
 
 dC 
 
 dx 
 
 A' 
 
 d 2 C 
 dx 2 
 
 which should be appropriate when the details of the two layers are averaged out. Here u =-i 
 is the average velocity and K* is the enhanced diffusivity. 
 
 Equations (17) can be combined to show thatC must satisfy 
 
 V dt 
 
 + u 
 
 dx 
 
 K 
 
 dx 
 
 r)H( 
 
 + u£—K 
 
 dx 
 
 dx 1 
 
 -(*.)•-£-] c-o, 
 
 (19) 
 
 -I- W, 
 
 (20) 
 
where Am 
 
 it would be the same as (19) with 
 
 u x - u 2 ). This is not exactly equation (19); however, if the first term were negligible, then 
 
 K* =K 
 
 1 
 
 (Au) 2 T. 
 
 (21) 
 
 Careful analysis (Thacker, 1 975) can show that ignoring this first term is equivalent to resolving only those 
 changes in C that are slower than the mixing time T and those spatial details thai are larger than a mixing 
 length x = {2K*T) 112 . Note that T and x are related according to equation (4), the equation that relates 
 length and time scales for the frost-free turbulence transformation. 
 
 On the other hand, for small changes in time, 
 the first term in equation (19) is important and the 
 second term is negligible. The reason for this is 
 that in a short enough time, a negligible amount of 
 mixing between the layers occurs. In this limit, 
 advection within each layer is important and an 
 enhanced diffusivity parameter does not apply. 
 However, each time a bit of contaminant crosses 
 to the other layer, its direction reverses. This gives 
 a long-term net effect of a random walk and 
 diffusion-like behavior. Thus, shear dispersion is 
 like either advection or diffusion, depending upon 
 the scale of the observation. If all of the details of 
 the flow are resolved, shear dispersion is 
 differential advection. But if these details are 
 ignored, which corresponds to filtering out high 
 frequencies and high wavenumbers, then shear 
 dispersion can be represented by an enhanced 
 diffusivity. 
 
 Equation (21 ) can be generalized to the case 
 of turbulent mixing. The shear of the two-layer 
 flow can be thought of as representing an eddy of 
 arbitrary scale in a turbulent flow and the mixing 
 as due to smaller eddies. If the resolution is 
 decreased, then the eddy that represented the 
 shear contributes to the mixing across the shear 
 of a still larger eddy. Thus, the difference dK = K* 
 - K can be thought of as the increase in eddy 
 diffusivity associated with a decrease in resolu- 
 tion. The factor (Am) 2 represents the energy in the 
 scale of the shear, so it should be proportional to 
 E(k)dk or <t>{w)doj. The mixing time T = ^j is re- 
 lated to the diffusivity through equation (4), I 2 = 
 2AT, if i = =£ is the scale of the shear. Thus, 
 
 Ac 
 
 (21) can be generalized to the differential equa- 
 tions 
 
 E(k)dk _ C <t>(uj)da) 
 2Kk 2 2tt o) 
 
 dK 
 
 (22) 
 
 These equations can be integrated to give 
 equations (5) and (8). 
 
 Equations (22) together with (6) are sufficient 
 to determine the frost-free turbulence transforma- 
 tion. To see this, differentiate equation (6) and 
 substitute from (21 ) to get an equation relating dw 
 and dk, 
 
 tt" 1 \2Kk - C 
 
 E(k) 
 
 2K 
 
 (23) 
 
 Now use (23) to eliminate d<o and dk from (22). 
 The result is exactly the transformation given by 
 equations (21). 
 
 The principal point to be seen from this 
 heuristic derivation is that the eddy diffusivity 
 parameterization should only be valid for an 
 appropriately averaged description of the flow. 
 Such an average should filter out high frequen- 
 cies and high wavenumbers, where the cut-off 
 values are related by equation (6). The result of 
 this averaging should be expressions like equa- 
 tions (5) and (8) for the diffusivity. It is the extent of 
 the averaging that determines what should be 
 resolved as advection and what should be 
 parameterized as diffusion. 
 
 It is interesting to note the similarity of the 
 ideas behind this heuristic derivation of equation 
 (5) and those of Tchen and of Nakano who obtain 
 the same expression for eddy viscosity. Tchen 
 (1973, 1975) uses a hierarchy of ensembles to 
 allow for a varying degree of resolution, which is 
 expressed here as a differential equation. His 
 memory chain corresponds to a hierarchy of time 
 scales that are associated with the hierarchy of 
 length scales. Nakano (1972) bases his theory 
 upon the idea that smaller eddies serve to damp 
 the larger eddies while larger eddies serve to 
 distort and advect smaller eddies. The damping 
 
is simply mixing of momentum, so his ideas are 
 the same as those used here. 
 
 Recently, McComb (1 974) obtained a similar 
 expression for eddy viscosity by modifying 
 Edwards' (1964) theory of turbulence. Their idea 
 is that small scale advection is like random 
 stirring, which is basically the same as the idea 
 presented here. In the two-layer model, the 
 random forces are provided by the shear and the 
 cross-shear mixing. This can be seen clearly from 
 Monm and Yaglom's (1971) derivation of equation 
 (20) as a Fokker-Planck equation for a Markov 
 process. 
 
 None of these dynamic theories has yet pro- 
 duced an equation, similar to (8), expressing the 
 eddy viscosity as an integral over the time spec- 
 trum. For both (5) and (8) to hold, there must be a 
 correspondence between frequencies and 
 wavenumbers, at least in the statistical sense that 
 an ensemble average should filter out high fre- 
 quencies and high wavenumbers. Tchen's theory 
 seems to be the closest to this. It would be in- 
 teresting to see if such an expression can be 
 obtained from these theories. If so, then a more 
 rigorous derivation of the frost-free turbulence 
 transformation should be possible. 
 
 6. Summary 
 
 The frost-free turbulence transformation pre- 
 sented here in equations (10) is based upon two 
 assumptions. The first is that there is a corres- 
 pondence between spatial and temporal scales 
 of the turbulence. This is justified by the results of 
 the dye diffusion experiments as summarized by 
 Okubo's (1971) diagrams and by the argument 
 that a mixing length and a relaxation time can be 
 assigned to the averaging process that results in 
 the eddy diffusivity. The second is that an ex- 
 pression such as (5) can be used to describe the 
 
 scale dependence of the eddy diffusivity. This is 
 justified by the generalization of the results from 
 shear dispersion and by the dynamical theories of 
 Tchen, Nakano, Edwards, and McComb. This 
 transformation successfully relates current re- 
 cords with dye diffusion data and agrees with the 
 results of dimensional arguments. Therefore, this 
 transformation appears to be valid and should be 
 useful as a working hypothesis for further studies 
 of turbulence. 
 
 7. References 
 
 Edwards, S. F. (1964): The statistical dynamics of homogeneous turbulence. J. Fluid Mech., 18:239-273. 
 
 Heisenberg, W. (1948): Zur statistischen Theorie der Turbulenz. Z. Physik., 124:628-657. 
 
 Monin, A. S., and A. M. Yaglom (1971): Statistical Fluid Mechanics , Vol. 1. MIT Press, Cambridge, Mass., 
 676-693. 
 
 McComb, W. D. (1974): A local energy-transfer theory of isotropic turbulence. J. Phys. A, 7:632-649. 
 
 Nakano, T. (1972): A theory of homogeneous, isotropic turbulence of incompressible fluids. Ann. Phys., 
 73:326-371. 
 
 Okubo, A. (1971): Oceanic diffusion diagrams. Deep-Sea Res., 18:789-802. 
 
 Taylor, G. I. (1938): The spectrum of turbulence. Proc. Roy. Soc. A, 164:476. 
 
 Taylor, G. I. (1953): Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. Roy. Soc. 
 A. 219:446-468. 
 
 Taylor, G. I. (1954): The dispersion of matter in turbulent flow through a pipe. Proc. Roy. Soc. A, 
 223:466-488. 
 
 Tchen, C. M. (1973): Repeated cascade theory of homogeneous turbulence. Phys. Fluids, 16:13-30. 
 
 Tchen, C. M. (1975): Cascade theory of turbulence in a stratified medium. Tellus, 27:1-14. 
 
 Thacker, W. C. (1976): A solvable model of shear dispersion. J. Phys. Oceanogr., 6:66-75. 
 
 Webster, F. (1969): Turbulence spectra in the ocean. Deep-Sea Res., 16 (Suppl.):357-368. 
 
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