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Unformatted text preview: Jim}: 1955 J . W. HUTCHINGS 263 TURBULENCE THEORY APPLEED Ti} LARGE-SCALE ATMOSPHERIC PHENGMENA By J. W. Hwtchings New Zealand Meteorological Service (Manuscript received 5 August 1954) ABSTRACT An outline is given of a preliminary attempt to apply the methods which have proved serviceable in the treatment of small-scale turbulence to the wider problem of large-scale turbulence in the atmosphere. In particular, application is made of theoretical results giving the form of the correlation function for velocity fluctuations in the inertial range according to the theory of Kolmogoroif. It appears that some of these results may be Valid over a considerably greater range of eddy sizes than might at first have been antici- pated. Some attempt is also made to apply these methods to the determination of the correlation functions for larger-scale fluctuations of pressure and temperature in the atmosphere. Actual data taken at the earth’s surface and in the free atmosphere are used to test the validity of these deductions. 1. Introduction During the last few years, there has occurred a renewed interest in some of the fundamental problems of turbulence, and many important results have been obtained regarding the turbulent flows encountered in aerodynamic and hydraulic laboratories. These inves- tigations have naturally been concerned with turbu- lence on a relatively small scale, the scale being specified in advance by the dimensions of the appa- ratus used. Much of this work can be directly extended to the investigation of the turbulent fields occurring naturally in the atmosphere, and in this way many important properties of atmospheric turbulence have been obtained. This renewed interest in small-scale turbulence may in some measure be traced to the fruitful concepts first put forward by Kolmogoroff (1941) and later inde- pendently set out and amplified by von Weizsacker (1948) and Heisenberg (1948). The hypothesis ad- vanced by Kolmogoroff enabled specific quantitative predictions to be made about the values of many of the important statistical parameters which charac- terize an arbitrary turbulent flow, and made possible a comparison of theory with observation. This circum— stance has contributed much to the recent progress in this field. In meteorology, these advances in small-scale tur- bulence theory have been accompanied by a gradually increasing awareness of the importance of scale (Priest- ley and Sheppard, 1952) in dealing with atmospheric problems. There has been, too, a growing acceptance of the view that the atmosphere is essentially a large- scale quasi-horizontal turbulent flow-field and that many fundamental meteorological problems are ca- pable of being formulated and interpreted by means of turbulence theory. This View was first put forward by Defant (1921) for the surface phenomena of the temperate zones of the earth, but has since found wider application in a great variety of investigations concerning transport phenomena in the atmosphere. Indeed, it seems as if one of the most fruitful fields for the application of modern ideas on turbulence lies in the upper atmosphere, away from the disturbing effects of orography and free from the marked dissipative effects of small-scale turbulence in the boundary layer near the surface of the earth. ' Nevertheless, up to the present, there has been little or no attempt to make specific and numerical applica- tion of the results obtained in the domain of small— scale turbulence to the wider problem of large—scale quasi-horizontal atmospheric turbulence. On this scale, the general circulation of the atmosphere itself must be considered as the statistical mean flow. This article represents a preliminary attempt to treat the large- scale phenomena of the atmosphere along lines stem— ming from modern research in the theory of turbulence and may serve the following purposes: 1. To show how recent developments in small-scale turbulence theory can be applied to a much wider range of turbulence ele- ments than might at first have been thought possible; 2. To draw the attention of more meteorologists to recent developments in turbulence theory, and to suggest that much already existing statistical material can be given a new interpre- tation along these lines; and 3. To indicate the need for further extensions of turbulence theory in order to approach some of the important problems of dynamic meteorology — for example, the problem of geostrophic departures and the question of the transfer of energy between atmospheric motions of different scales. 2. The internal structure of a turbulent fluid Batchelor (1953) has recently given a comprehen— sive discussion of the modern approach to the theory of homogeneous and, in most applications, isotropic 264 turbulence. The structure of a turbulent fluid will therefore be discussed here only briefly. According to the View put forward by Kolmogoroff, any fully developed turbulent flow may be thought of as a superposition of pulsations or velocity fluctuations of different magnitudes, varying from what might be called changes in the mean 'flow down to the very smallest fluctuations belonging to the state of laminar flow. For any fluid flow, it is a well established prin— ciple that when the Reynolds number exceeds a certain critical value the flow becomes unstable for small dis- turbances, so that a superimposed flow in the nature of a pulsation or fluctuation with characteristic dimen- sions and velocity is generated. Kolmogoroff’s View is based on the application of this principle to each superimposed flow, so that with increasing Reynolds number each superimposed flow in turn becomes 'un- stable and further disturbances are generated. Thus, at sufficiently high Reynolds numbers, fluctuations of all magnitudes are present. This expresses the well- established intuitive idea that the larger eddies of any turbulent motion break up into a series (or cascade) of smaller eddies, which in turn break up, i.e., pass their kinetic energy on to eddies of even smaller scale. Kolmogoroff further supposed that, as the energy passes down the system of eddies, the direct influence of the larger energy—containing eddies is gradually lost, so that all eddies smaller than a certain critical size have statistical properties which to a large extent are random. It follows, then, that as long as we deal only with eddies smaller than this critical size, the turbu- lence may be considered approximately homogeneous and isotropic. In Kolmogoroff’s terminology, we may say that any field of turbulence is locally isotropic and the range of eddy sizes over which thisrcondition is fulfilled may be called the range of local isotropy. Since almost all of the dissipation of energy by viscosity takes place in the smaller eddies, and since these must adjust themselves so as to convert all the energy they receive into heat, it is evident that the statistical state of the small eddies must depend entirely on two parameters: 1. The average rate at which energy is handed on to them from larger eddies (which is approximately identical with the total average rate of energy dissipation by viscosity, e); and 2. The kinematic viscosity, :1, which determines the rate at which the small eddies can convert kinetic energy into heat. As the Reynolds number increases, smaller and smaller eddies are brought into existence; and at very high Reynolds numbers, the larger eddies of the range of local isotropy will contribute very little to the dissi- pation of energy by viscosity. There will thus exist at the large-eddy end of the range of local isotropy a sub-range in which the statistical properties of the eddies are independent of v and depend only on c. This range of eddies is dominated purely by inertial JOURNAL OF METEOROLOGY VOLUME 12 forces and may be called the inertial range. In this article, we will be concerned almost entirely with con- ditions in the inertial range. 3. Correlation functions in the inertial range As we are considering conditions in the inertial range only, we may consider a field of turbulence which is approximately isotropic and statistically steady in time. It then proves convenient to introduce the longi- tudinal velocity correlation-function f (r) and the lat- eral velocity correlation-function g(r) for two points a distance r apart in any direction. With use of the bar symbol to denote averages over a sufficiently long period of time, these functions are defined by u.(A)_ 1MB) 2 “12 f0) = un(A$)t_:2ln(B), where up(A), 14,,(B), 14,,(A) and un (B) are the turbu- lent components of velocity at the two points A and B, respectively, measured parallel (suffix p) and nor- mal (suffix n) to the vector separation AB. In isotropic turbulence, 14—1,? = u_,,2 = 142. The introduction of the equation of continuity for an incompressible fluid leads, as was shown by von Karman and Howarth (1938), to the relation g = f + %r (if/61' between the functions f and g. If the tur- bulence is taken to be isotropic in two dimensions only, the same reasoning as employed by von Karman and Howarth leads to the relation g = f + r (if/6r between the functions f and g. These functions, f and g, may be called Eulerian space correlation-functions and either may be denoted by R(r). In the inertial range, R(r) must depend solely on e, and dimensional analysis shows that the form of R(r) is then given by u2|:1 — R(r)] ~ eiri. Introducing the above relations between f and g, we find that, a being a constant, f (r) , and g(r)= 1 — mg, and g(r) = 1 -— garg or = 1 —- gafi, according as the turbulence is taken to be isotropic in three Or two dimensions. The ratio (1 — f) / (1 — g) is thus 0.75 or 0.60, according as the turbulence is three- or two-dimensionally isotropic. In wind—tunnel turbulence, the turbulent fluctua- tions are normally only a few per cent of the mean speed of flow parallel to the sides of the tunnel, and it is a plausible assumption to consider that the turbu- lent field is simply carried down-stream past a single observation point at a constant rate U equal to the mean speed of flow. The space interval r then corre- sponds to a time interval 7- = r/ U, and it is possible and practically very convenient to define longitudinal and lateral velocity correlation-functions f ( -r) and g(r) by using values of the velocity components parallel JUNE 1955 J . and normal to the mean flow at a single point in space but separated by an interval of time (lag interval) 7-. These functions may be called Eulerian time correla- tion-functions, and either may be denoted by R(r). In these circumstances, a change of variable shows that R(r) and R(r) have the same form and that all the above relations are true when r is substituted for r. In other circumstances, this assumption of the equiva- lence of space and time correlations is much more questionable: but it will appear that in the particular case of large-scale atmospheric turbulence, the above assumption represents at least a crude approximation to actually occurring conditions. 4. Application to large-scale atmospheric phenomena In recent years, the world-wide extension of the regular network of aerological soundings has demon- strated that at middle and higher levels the moving cyclonic and anticyclonic vortices, which are such a prominent feature of the circulation of the lower atmosphere in temperate latitudes, are replaced by a large-scale system of troughs and ridges in a strong and extensive westerly flow. Average annual or seasonal charts show that, with the exception of minor or local asymmetries, the mean flow at upper levels over large portions of both hemispheres is very nearly zonal, the whole circulation being dominated by vast cyclonic vortices located near the poles. It thus seems possible to consider many of the familiar features observed on daily or twice-daily upper—level synoptic charts as elements in a huge zonal turbulent flow. These turbu- lence elements will consist of motions with all different length-scales, ranging from the large slow-moving troughs and ridges, through the lesser and quicker- moving synoptic systems, down to the smallest fluc- tuations that can be revealed by the density of the aerological net used in constructing the charts. In these circumstances, the mean flow will correspond to an average of the individual (once, twice or even four times daily) patterns taken over a long period of time. The time of averaging will thus be measured in months or even years. This may be compared with averaging times of the order of minutes commonly used in wind- tunnel and small-scale atmospheric research. This (largely kinematical) view of the problem may be regarded as an extension of the ideas of Defant (1921) to the atmospheric circulation at higher levels and as an application of the philosophical viewpoint, first put forward by Richardson (1930) in connection with diffusion, that turbulence is essentially a compen- sation for smoothing and that, as a consequence, the kind of turbulence encountered is greatly dependent on the kind of smoothing (or averaging) employed. But these views are no doubt oversimplified; and to arrive at a more realistic conception of the applica- bility of turbulence theory to the large-scale atmos- W. HUTCHINGS 265 pheric phenomena, it is necessary to consider some of the more important aspects in which well defined differences exist between large- and small-scale turbu- lent motions. The most important of these considera- tions appear to be: 1. The use of the continuity equation for an incompressible fluid; this implies certain restrictions on the corresponding large- scale atmospheric flow; 2. The extent to which large-scale turbulence may be consid- ered sufficiently homogeneous and isotropic; 3. The influence of the Coriolis force on motions of larger scales; 4. Uncertainties regarding the sources of energy of the large- scale velocity fluctuations observed in the atmosphere; and 5. The assumption of the equivalence of space and time corre- lation-functions. This last consideration is not a difficulty in principle but is of some importance from the point of view of the data normally available for practical calculation of correlation functions. In the following sub-sections, these five questions will be briefly discussed. Assumption of incompresstbility.—-Recent theoreti- cal studies (Charney, 1948) have tended to show that the large-scale motions of the atmosphere at middle levels may be regarded as approximately horizontal and non-divergent. From a kinematic point of View, it would thus appear probable that many of the results obtained for the turbulent motion of a homogeneous incompressible fluid could be applied without serious modification to the problems of middle-level large- scale atmospheric turbulence. It would also seem probable that the SOO-mb level would be suitable for such applications. Homogeneity and isotropy in large scale turbulence.—~ It has already been emphasized that the application of Kolmogoroff’s hypothesis‘involves only the less stringent condition of local isotropy and does not require that the complete turbulent field be accu- rately homogeneous and isotropic. Some observational evidence regarding the extent to which the complete large-scale turbulent field in the atmosphere may be regarded as homogeneous and isotropic may be ob- tained from the work of Brooks et at (1950). Although only a limited amount of data was discussed, much evidence was put forward by Brooks et al to show that in the free atmosphere, away from the effects of orog- raphy, the distribution of the vector deviations of the instantaneous wind vectors from the long-term mean vector wind is approximately symmetrical and circu- lar. These results may be interpreted as indicating a certain measure of isotropy in actually-occurring large- scale atmospheric turbulence. The symmetrical circu- lar distribution implies that the large-scale zonal turbulent component of the wind, u, and similarly the meridional turbulent component, 2), are normally dis- tributed with equal standard deviations 0'“ and a,” respectively. Some independent computations of a“ 266 and 0,, together with the average westerly (U) and southerly (V) components of the wind at the 30,000-ft level are given in table 1 for Auckland, New Zealand (36°47’S, 174°38’E), and for Nandi, Fiji (17°45’S, 177°27’E). It will be seen that the relative magnitudes of a“ and 0., vary somewhat from month to month, but that at Auckland there does not appear to be any marked systematic difference between them, so that over a long period of time it might be expected that there would exist at the level an approximate equality between a“ and en. Conditions at N andi are less cer— tain, and more computations would be necessary to determine if the apparent systematic differences be- tween au and (7,, would persist over a long period of time. In this article, no further use is made of data for N andi. Batchelor (1953) has shown theoretically that, for small-scale turbulence in a homogeneous incompres- sible fluid, the effect of pressure forces on the turbulent motion is to transfer kinetic energy between the differ- ent directional components of the motion and thus lead to an approximate equality of turbulent kinetic energy in each direction. In the atmosphere, the ap- proximate equality of 0.. and (7,, may doubtless be attributed at least in part to the same process. Nevertheless, the results of Brooks et al (1950) imply that u and v are uncorrelated, thus leading (Hutchings, 1952) to zero meridional flux of absolute angular mo- mentum across any parallel of latitude. Actually, the theoretical necessity for a poleward flux of absolute angular momentum over most of the earth is well recognized, and practical calculations using observed wind data have amply demonstrated the reality of the flux and have also determined its approximate magni- tude. From these results, we may infer the magnitudes of the correlation coefficients between 14 and v at various levels in the atmosphere. In general, they appear to be less than 0.2 at most middle-latitude stations but may approach 0.3 at the higher levels for TABLE 1. Standard deviations and mean wind components (kn), 30,000 ft, Auckland and Nandi. Cu, an U V Auckland July 1950 39.4 25.2 25.0 14.1 July 1951 30.6 30.6 36.0 —0.3 July 1952 35.8 31.8 47.6 —0.6 July 1953 31.4 31.6 47.7 ——11.2 July 1950—1953 35.6 31.2 39.1 0.5 June 1952 31.8 36.4 73.3 —6.6 July 1952 35.8 31.8 47.6 —0.6 August 1952 26.0 21.9 21.1 2.6 June, July, Aug. 1952 38.0 30.6 46.6 ——1.5 Nandi July 1950 18.2 21.1 35.5 5.8 July 1951 16.2 19.8 35.2 5.1 July 1952 9.7 14.5 28.8 —-6.3 July 1950—1952 15.2 19.7 33.2 1.5 JOURNAL OF METEOROLOGY VOLUME 12 stations near 30 deg lat, across which parallel the maximum transport of absolute angular momentum takes place. It would thus appear likely that the re- sults for Auckland given in table 1 represent the greatest departures from isotropy likely to be experi- enced in the atmosphere. With regard to the amount of homogeneity to be expected in large-scale atmospheric turbulence, we may obtain some information from the charts of standard vector-deviation given by Brooks et al (1950). It would appear from these charts that the homo- geneity of large—scale turbulence varies considerably from level to level in the atmosphere and also varies somewhat with geographical location and season of the year. In general, large-scale turbulence appears to be most homogeneous at and above the ZOO-mb level. From the above considerations, we may conclude that, on the average, large-scale turbulence in the atmosphere is neither theoretically nor practically ex- actly homogeneous and isotropic; but at certain atmos- pheric levels and for areas which are not too large, the requirements for homogeneous isotropic turbulence will be approximately fulfilled. . Influence of the Coriolis force—As is well known, all atmospheric motions whose periodic times are of the order of half a pendulum day are strongly influenced by the Coriolis force arising from the earth’s rotation. Again, the variation of this force with latitude is known to exert an appreciable controlling influence on the large-scale flow patterns observed in the atmos- phere. However, the Coriolis force, although in the nature of an inertial force, depends linearly on the velocity and would not therefore be expected to con- tribute directly to the statistical decay of turbulent energy. As pointed out by Batchelor (1953), the effect of non-linear inertial forces is to spread the turbulent energy over an increasing range of wave-numbers. It may thus be anticipated that many of the results obtained for small-scale turbulence, particularly those of a kinematic nature, may be applied directly to the larger-scale problem. On the other hand, it would appear likely that important modifications in small- scale theory will be necessary before the results will be fully applicable to large-scale fluctuations of dy- namical quantities such as atmospheric pressure. Supply of energy to atmospheric motions.—Many basic questions regarding the sources of energy of the velocity fluctuations observed in the free atmosphere cannot as yet be answered with any certainty. In small-scale turbulence observed in a wind-tunnel and in mechanically produced turbulence in the boundary layer, there is no doubt that kinetic energy is com- municated directly to the mean flow and larger eddies by what may be regarded as an external agency. This energy is then passed down the system of smaller and smaller eddies and finally dissipated irreversibly into JUNE 1955 J . molecular motion (heat) by the smallest turbulence elements. In the free atmosphere, however, it is not certain that the large-scale atmospheric disturbances derive any major part of their kinetic energy from the mean flow, and indeed perhaps even the reverse is true. Very probably both processes take place. At times there is a transfer of energy from the mean flow to feed a young and growing disturbance, while in other circumstances the mean flow may be regenerated from time to time by the growth of particular large-scale disturbances which are able to transform most effi- ciently gravitational and thermal energy into kinetic energy. Whether, in a statistical sense, one type of transfer process can be said to be dominant is not at present known. In the same way, it is not at present known if and to what extent the smaller-scale atmos- pheric motions may be regarded as deriving all or part of their kinetic energy from motions on a larger scale. Nevertheless, from a fundamental point of View it is necessary to have some information regarding the energy sources present in large-scale turbulent flows, because without this knowledge it is not possible to determine beforehand the range of eddies to which Kolmogoroff's hypothesis applies or even to determine if such a range exists. Our present lack of knowledge thus prevents a strictly deductive approach, and it is necessary to proceed in an indirect fashion. Equivalence of space and time correlations.——The basic condition for the equivalence of space and time corre- lation-functions is that each turbulence element, as it moves past the point of observation, should undergo only relatively unimportant internal changes. This condition is probably fulfilled when, as in the wind tunnel, the magnitudes of the turbulent fluctuations are very small compared with the mean velocity of flow. The comparative smallness of the fluctuations no doubt also ensures that the translational speed of the turbulence elements approximates c105ely the mean velocity of flow. But the above basic condition is also capable of being fulfilled in cases Where, as in large- scale atmospheric turbulence, the turbulent fluctua- tions of velocity are of the same order of magnitude as the mean speed of flow and the translational speed of the turbulence elements differs considerably from that of the mean flow. The important condition that needs to be fulfilled is that the zonal advection of the velocity field should be more important than the effect of development in determining velocity changes at a fixed point. Although there are undoubtedly occasions on which this is not true, synoptic experience with upper-level charts tends to show that, on the average, this assumption may be a reasonable approximation to actually occurring conditions. Summary—From the above review of the difficulties likely to be experienced in the application of small- W. HUTCHINGS 267 scale turbulence theory to atmospheric phenomena on a larger scale, it appears that many simple and direct applications would be likely to give useful results but that it is not possible to say a priori that all the con- ditions for its strict application are fulfilled. Some doubt exists as to the full effect of the Coriolis force on turbulence of this scale, but the greatest uncer- tainty involved would seem to lie in our present lack of knowledge regarding the supply of energy to atmos- pheric disturbances of different scales. In view of these uncertainties, it seems clear that it would be more profitable to proceed in an inductive manner by making direct application of the theory to large—scale turbulence and then deciding, by a com- parison of the relevant theory with observational re- sults, whether or not there does in fact exist a range of atmospheric motions to which the theory may be reasonably applied. Indeed, such a procedure might yield indirect knowledge regarding the processes which govern the transfer of energy between atmospheric motions of different scales and might also provide information regarding the dimensions of atmospheric motions beyond which the geostrophic (Coriolis) con- trol plays a dominant part. 5. Correlation functions in the free atmosphere The most fundamental statistical functions charac- terizing the field of turbulent fluctuations of a par- ticular physical quantity are the power spectral- function for the quantity considered and the associated correlation function. This latter function is connected with the spectral function by means of the Fourier transform relations. Of the two kinds of function, the correlation function is the more easily computed from observed data. This section gives details of actually computed correlation functions for several variables of interest in large-scale atmospheric turbulence. Velocity fluctuations.—In the atmospheric case, the simplest functions that can be considered are the two- dimensional longitudinal and lateral correlation func- tions, f(T) and g(r), respectively. These may be de- fined by f(7) =@, and g(r) =@’ we) M) where u and 21 represent, respectively, the westerly and southerly components of the turbulent wind velocity, and T represents the variable time lag em- ployed. The bar symbol here represents a mean value taken over a sufficiently long period of time, while the turbulent components are represented by the devia- tions of the instantaneous wind components from their respective means. When the conditions for the equivalence of space and time correlation functions are satisfied, we may, in the inertial range, assume that [1 — 13(1)] ~ 1-}. 268 There may be circumstances in which the above con- ditions are not satisfied, and in this case some infor- mation on the probable form of R(-r) may be obtained from a recent paper by Ogura (1953). Using a simple one-dimensional model of small—scale turbulence sug- gested by the results of von Weizsacker (1948), Ogura shows that, in the range where the Eulerian space correlation-function R(r) is given by [1 — R(r)] N r}, the Eulerian time correlation-function R(r) may be expressed in the form [1 — R(r)] ~ 7’”, m being a complicated function of U (JO—l, where U is the veloc- ity of the mean flow and (JV gives a measure of the magnitudes of the turbulent velocity components in the direction of the mean flow. The value of m lies between 2/3 and 1, the minimum value (m = 2/3) occurring when U >> (JV and the maximum value (m = 1) when U << (EV. It is at present not clear how far this simplified model of small-scale turbulence may be applied on a larger scale, but from these re- sults it is to be expected that there will always be some inaccuracy in assuming the equivalence of space and time correlation-functions, even when allowance is made for the regeneration and decay of the turbulence elements as is done in the model investigated by Ogura. Nevertheless, if we attempt to apply these results to large-scale atmospheric turbulence, it is clear that we must interpret U as .a mean translational speed for the large-scale atmospheric disturbances and not 'as a mean velocity of flow. We then know that above the 500-mb level U is approximately constant with height, while the intensity of turbulence as meas- ured by (fiafi increases considerably. It might then be expected that, if R(r) is identified with R(r), values 6 I2 /8 24 1' HRS. JO J‘ 4! FIG. 1. Longitudinal (westerly) (A) and lateral (southerly) (B) correlation functions at 500 mb, Larkhill. Circles denote observed values, full lines theoretical linear relation, slope 2/3. JOURNAL OF METEOROLOGY VOLUME 12 TABLE 2. Correlation functions, f and g; 500 mb, Larkhill, December 1944—February 1945. 0'“ = 29.8, m; = 21.4 kn. Time lag, 7: (hr) 6 12 18 24 30 36 48 f: 0.904 0.817 0.740 0.685 0.643 0.607 0.517 g: 0.741 0.566 0.413 0.315 0.235 0.228 0.077 (1—f)/(1—g): 0.37 0.42 0.44 0.46 0.47 0.51 0.52 of m near the theoretically expected value of 2 / 3 would be found at lower levels while those at higher levels would deviate towards the value m = 1. The calcula- tions in this section have therefore been made at two atmospheric levels, the 500-mb level (where we might expect values of m near 2/3) and the 30,000—ft level (at which values of m nearer 1 might be expected). The SOO—mb data used in this section consist of radar upper-wind determinations at 6-hr intervals for Larkhill, England (51°12’N, 1°48’W), during the win- ter period from December 1944 to February 1945. These data were taken from the Daily weather report ' of the Meteorological Office, London. The computed correlation functions, f and g, for this period are shown in table 2. . In fig. 1, the quantities l—f and 1—g have been plotted against 7', both on logarithmic scales, and com- pared with the theoretically expected linear relation ~ with slope 2/ 3. The agreement is clearly not perfect; but in view of the inaccuracy involved in substituting f(r) and g(1-) for f(r) and g(r), respectively, it is perhaps as good as could be expected with the data at hand. The results tend to indicate the probable appli- cability of Kolmogoroff’s hypothesis to a wider range of phenomena than might at first have been thought possible. It will be noted that in the above data the condition of complete isotropy (as judged by the ap- parent inequality of 0",, and 0,) is not particularly well fulfilled. If f and g have the form appropriate to the inertial range, the ratio (1 —f)/(1—g) should have the value 0.6; it may be seen from table 2 that this requirement is also not well fulfilled by these data. The data for 30,000 ft consist of radar upper—wind determinations at 6—hr intervals at Auckland, New Zealand (36°47’S, 174°38’ E) for the winter period from June to August 1952. These data were obtained from the original records of the New Zealand Meteoro- logical Service. The computed correlation functions, f and g, for this period are shown in table 3. Compu- tations were not continued beyond the 24—hr lag interval. TABLE 3. Correlation functions, f and g; 30,000 ft, Auckland, June—August 1952. 11.. = 38.0, a"; = 30.6 kn. ‘ Time lag, 7-: (hr) 6 12 18 24 f: 0.88 0.79 0.68 0.59 g: 0.78 0.61 0.43 0.31 (1—f)/(1—g): 0.55 0.54 0.56 0.59 JUNE 1955 J . The data in table 3 are plotted in fig. 2, in the same way as before. It will be seen that an approximately linear relation is given by the plotted points up to the 24-hr lag interval, and that the slopes of the two lines are identical. The value of the slope is approximately 0.88, which differs considerably from the 2/3 slope found in the 500-mb data from Larkhill. This devia- tion of the slope towards the value 1 is, however, in good agreement with the tendency to be expected if the model of small—scale turbulence proposed by Ogura (1953) is taken to be applicable to turbulence elements on a larger scale. If a relation of the form [1 —— R(r)] ~ 1’" is assumed, the reasoning of section 3 shows that the ratio (1 — f)/ (1 — g) should have the value 1/(m + 1), and for m = 0.88 this value is 0.53. It will be seen from table 3 that the computed values of the ratio (1 — f)/(1 — g) lie close to this value. Nevertheless, too much emphasis should not be placed on this result, as the data at the two levels in question have been obtained at different stations. For a full investigation, it would of course be desirable to study the variation of the index m with height by using data from different levels at the same station, and then repeating the procedure for several stations in different wind regimes. This course was not followed here, owing to the fact that the computed correlation coefficients for 30,000 ft at Auckland were already available and that it seemed more important to gain some experience with the form of the correlation func- tions at higher latitudes than to investigate the details of the variation of the index m with height. This latter investigation would gain more significance if attempted l-RK) 6 l? I! 24 T HRS. W. HUTCHINGS 269 in conjunction with the computation of both space and time correlations. Pressure fluctuations—One of the most important features of small—scale turbulent flow is the existence of turbulent fluctuations in fluid pressure accompany- ing the turbulent variations in velocity. In small-scale flow, it seems very plausible that these pressure fluc- tuations (p) are connected with the corresponding fluctuations in velocity through the Bernoulli equa- tion, and that we might write formally p = %p62, Where p is the density and c is the magnitude of the total turbulent velocity. Dimensional reasoning then shows that the correlation function h(r) for pressure fluctua- tions has the form [1 —— h(r)] ~ r4/3. In the same way as before, we may also write [1 - ~ 7-4/3. In the case of large-scale atmospheric turbulence, it is well known that the Coriolis force almost exactly balances the pressure gradient for motions whose period approaches half a pendulum day, so that it might be expected that the 74/3 law would cease to hold beyond a certain value of 1, this value being quite independent of the actual limits of -r determined by the inertial range. It might be, however, that in certain circumstances, for example in low latitudes or in a region where pressure changes are strongly influenced by orographic or topographic effects, the 74/3 law might hold over a range considerably greater than that indi- cated by considerations of geostrophic control. Owing to instrumental and other difficulties, accu- l 2 J 6 E T l-IRS. Id 24 $354.2“ 6072 90/21? FIG. 2. Longitudinal (westerly) (A) and lateral (southerly) (B) correlation functions at 30,000 ft, Auckland. Circles denote ob- served values, full lines to fit observed data, slopes 0.88. FIG. 3. Autocorrelation function, surface-pressure fluctuations, Buenos Aires, July 1943. Circles denote observed values, full line theoretical linear relation, slope 4/3. 270 rate data regarding short-period pressure fluctuations in the free atmosphere are difficult to obtain. There is more information available regarding surface-pressure changes, but even here published computations of lag correlation-coefficients have usually been available only for large lag intervals of one day or more. Very detailed computations (with lag intervals down to 1 hr over part of the range) have, however, been published by Dedebant et al (1953) for the surface—pressure fluc- tuations at Buenos Aires during July 1943. As this station is situated on the lee side of the Andes cor- dillera, it seems well suited for investigating whether or not anything resembling the theoretically antici- pated 4/3 law can be found in atmospheric data. With the values of h(r) taken from the above publication, fig. 3 shows the values of [1 —- h(r)] plotted against the corresponding values of 7, both on logarithmic scales. It will be seen that a linear relation gives a very satisfactory fit to the observations up to a lag interval of 60 hr, but beyond that a systematic devia- tion from the linear relation can be observed. The slope of the line drawn corresponds to the theoretical value of 4/3. Another station for which detailed com- putations of lag correlation coefficients have been carried out is Wellington, New Zealand (41°17’S, 174°46’E). As long as we are concerned with pertur- bations whose periods are less than half a pendulum day at the latitude of Wellington, it seems likely that the associated pressure changes would be of the Ber- noulli type and that these small-period fluctuations might approximate the 4/3 law. The relevant points for the pressure fluctuations at Wellington for the period 1 January 1941 to 31 December 1942 are shown in fig. 4, and it may be seen that an approximately linear law is followed up to a lag interval of about 18 hr, but that beyond this systematic deviations appear. The value for the slope of the line shown in 0" 02 0'/ v /- R00 0- 05 0025 00/! J 6 .9 lé‘ IJ' [82/24 3036‘ 46 1‘ HRS. FI_G. 4. Autocorrelation function, circles surface-pressure fluc- tuatlons at Wellington, crosses SOO-mb geopotential fluctuations at Larkhill. Full lines to fit data, slopes 1.5 and 1.0, respectively. JOURNAL OF METEOROLOGY VOLUME 12 fig. 4 is 1.5, which is only a very rough approximation to the expected value of 1.33. It is interesting to note that the length of a half pendulum day in the latitude of Wellington is approximately 18 hr. Some attempt has also been made to examine data from the free atmosphere, with use of the 6—hr values of the SOO-mb geopotential at Larkhill for the period from December 1944 to February 1945. This is the same period as was used for the velocity correlations, and the data have been taken from the Daily weather report of the Meteorological Office, London. The rele- vant points are also plotted in fig. 4 and seem to indi- cate an approximately linear law up to about the 24—hr lag interval. The length of the half pendulum day at Larkhill is about 16 hr. The indicated slope is 1.0, which represents a considerable deviation from the theoretical value. The reasons for this deviation are not entirely clear. Part of the deviation is no doubt due to the use of time correlations in place of space correlations, but it should also be emphasized that the determination of the correlation function for pressure fluctuations in the free atmosphere by means of the Bernoulli equation does not have a firm justification even when only fluctuations of small period are con- sidered. Pressure changes in the free atmosphere are the result of very complicated processes, and it is probable that the Bernoulli equation is applicable only in special circumstances. Temperature fluctuations.—If the grid of a wind tunnel is heated, turbulent fluctuations of temperature occur downstream. In contrast to the phenomenon of fluid pressure, there is no clear dynamical basis for dealing with fluctuations in temperature, and some special assumption appears to be necessary. For small— scale turbulence, Obukhoff (1949) has proposed that the fluctuations in temperature for a homogeneous incompressible fluid should, in the inertial range, 5 I? /8 24 1' HRS. J0 36 42 46 FIG. 5. Autocorrelation function, SOO-mb temperature, Lark- hi’lll. Circles denote observed values, full line to fit data, slope 0. . JUNE 1955 J . follow the same law as the fluctuations in velocity, i.e., the 2/3 law should apply. There appears to be only a very faint analogy be- tween these small—scale temperature fluctuations and those occurring on the larger scale, but it was never- theless thought worthwhile to compare some practical calculations with the above theoretical prediction. To reduce the error due to the use of time correlations instead of space correlations, data from the SOO-mb level in preference to the higher levels were used for these computations. The 500-mb temperature data for Larkhill, taken at 6-hr intervals, have been used over the same period as was used for wind and pressure correlations, that is, December 1944 to February 1945. The relevant points are shown in fig. 5 and show an approximately linear region of slope 0.7 from the 12-hr to the 48-hr lag interval. Computations were not con- tinued beyond the 48-hr lag interval. Although the data are not sufficient to allow definite conclusions to be drawn, these results are not inconsistent with the existence of a 2/3 law for the temperature fluctuations at this level. 6. General appreciation The work represents a first attempt to apply some of the results obtained in small-scale turbulence theory to the general problem of atmospheric turbulence on a larger scale. The conclusions must therefore of neces- sity be preliminary and tentative. Much further work will be necessary before the exact nature of the results can be ascertained and the conditions for their validity established. Many of the necessary refinements to the present crude approach will be possible only by further refinement and clarification‘of the basic ideas of tur- bulence theory itself. The theory of turbulence is at present in a very rudimentary state. In future prac- tical work in the atmosphere, more attention must be given to the determination of actual space correlations instead of time correlations. Only in this way will it be possible to avoid one of the basic difficulties in the treatment given in this article. Nevertheless it would seem, from the preliminary results obtained so far, that the application of turbu- lence methods to the interpretation of large-scale atmospheric phenomena is well worth further investi- gation on a more rigorous and extensive basis. W. HUTCHINGS 271 Acknowledgments.—-Grateful acknowledgment is made to Dr. J. F. Gabites of the New Zealand Mete- orological Service for details of the standard deviations of the 30,000—ft wind components at Auckland and Nandi, as well as for the velocity correlation-coeffi— cients at 30,000 ft for Auckland. Thanks are also due to the Director, Applied Mathematics Laboratory, Wellington, for details of the pressure correlations at Wellington, and to Miss E. Farkas for assistance with the computations in this article. The work is pub— lished by permission of the Director, New Zealand Meteorological Service. REFERENCES Batchelor, G. K., 1953: The theory of homogeneous turbulence. London, Cambridge Univ. Press, 197 pp. Brooks, C. E. P., C. S. Durst, N. Carruthers, D. Dewar, and J. S. Sawyer, 1950: Upper winds over the world. Geophys. Mem., No.85, 150 pp. Charney, J. G., 1948: On the scale of atmospheric motions. Geofys. Publ., 17, No. 2, 17 pp. Dedebant, G., R. Di Maio, and E. A. M. Machado, 1953: Las funciones aleatorias y su aplicacién a la meteorologia. Meteoros, 3, 140—173. Defant, A., 1921: Die Zirkulation der Atmosphare in den ge- ma'ssigten Breiten der Erde. Geograf. Amt, 3, 209—266. Heisenberg, W., 1948: Zur statistischen Theorie der Turbulenz. Z. Phys., 124, 628—657. Hutchings, J. W., 1952: A note on the distribution of free-air wind vectors about their mean. Quart. J. r. meteor. $05., 78, 105—106. Karman, T. von, and L. Howarth, 1938: On the statistical theory of isotropic turbulence. Prue. ray. Soc. London, A, 164, 192—215. Kolmogoroff, A., 1941: The local structure of turbulence in in- compressible viscous fluids for very large Reynolds num- bers. C. R. Acad. Sci. U.R.S.S., 30, 301—305. Obukhoff, A. M., 1949: Structure of the temperature field in turbulent flow. Izv. Akad. Nauk S.S.S.R., Ser. Geogr. Geofiz., 13, 58—69. Ogura, Y., 1953: The relation between the space and time corre» lation functions in a turbulent flow. I. meteor. Soc. Jap., 31, 355—369. Priestley, C. H. B., and P. A. Sheppard, 1952: Turbulence and transfer processes in the atmosphere. Quart. J. r. meteor. $00., 78, 488—529; Richardson, L. F., and J. A. Gaunt, 1930: Diffusion as a com— pensation for smoothing. Mem. r. meteor. Soc., 3, 171—175. Weizsacker, C. F. von, 1948: Das Spectrum der Turbulenz bei grossen Reynoldsschen Zahlen. Z. Phys., 124, 614—627. ...
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