ification of existing terms. Finally, Bjørnstadet al. apply a statistical method to their rawdata, and confirm the findings based on thetime-series method and the populationmodel.

We do not yet know how useful Bjørnstadet al.’s methods will be in identifying theimportant variables in other systems,including natural ones. Theory predicts6

how the dynamics of a population will beaffected by the strength of coupling betweentwo species. But it is not yet clear whethercoupling to more and more variables inex-orably increases the number of lags.

However, on the empirical front, thou-sands of sets of population data exist7.Although it would take a herculean effort toanalyse them all, breakthroughs may be inthe offing. The same group previouslyshowed8 that by increasing the depth of P.interpunctella’s artificial diet, the wasps’attack rate could be diminished, resulting ina weaker effect on the moth’s populationdynamics. Combining this system with thenew analysis techniques will provide anopportunity to test whether varying a habitatparameter affects the strength of coupling of the system. The prediction here is thatcoupling between moth and wasp popu-lation dynamics should decrease as dietdepth increases.

There are also broader implications. Thegroup previously found5 that when bothvirus and wasp confront the moth together,the simple generation cycles found in theone- and two-species systems are thrown

out of whack: the three-species systemexhibits transient cycles of longer periods,and eventually becomes extinct. If theimprints of one or both enemies on thesecycles could be found, we would be a stepcloser to knowing whether Bjørnstad et al.’stechniques can be applied to biodiversityand conservation research, where one mightwant to know how a species that is harmlessin one context can lead to the collapse of partof a community in another. These tech-niques could also be used to show how theuse of two natural enemies as biological pestcontrols yields outcomes that are qualita-tively different to the outcomes of usingeither enemy alone9. ■

Michael E. Hochberg is at the Institut des Sciencesde l’Evolution, Université de Montpellier II, 34095Montpellier, France.e-mail: hochberg@isem.univ-montp2.frArthur E. Weis is in the Department of Ecology andEvolutionary Biology, University of California,Irvine, California 92697, USA.e-mail: aeweis@uci.edu1. Bjørnstad, O. N., Sait, S. M., Stenseth, N. C., Thompson, D. J. &

Begon, M. Nature 409, 1001–1006 (2001).

2. Woiwood, I. P. & Hanski, I. J. Anim. Ecol. 61, 619–629

(1992).

3. Strong, D. R. Trends Ecol. Evol. 1, 39–42 (1986).

4. Turchin, P. et al. Nature 405, 562–565 (2000).

5. Begon, M., Sait, S. M. & Thompson, D. J. Nature 381, 311–315

(1996).

6. Tanner, J. T. Ecology 56, 855–867 (1975).

7. NERC Centre for Population Biology, Imperial College. The

Global Population Dynamics Database (1999).

http://cpbnts1.bio.ic.ac.uk/gpdd/

8. Begon, M., Sait, S. M. & Thompson, D. J. Proc. R. Soc. Lond. B

260, 131–137 (1995).

9. Murdoch, W. W. & Briggs, C. J. Ecology 77, 2001–2003 (1996).

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Turbulence is the chaotic and unpredict-able motion of fluids flowing at highrates. It plays a major role in many

processes from the environmental, for exam-ple cloud formation, to the technological,such as in industrial chemical reactors.Clearly, a deeper understanding of thisphenomenon would be beneficial, and inrecent years much progress has been made inthe fundamental theory underlying turbu-lence. But the ability to measure turbulenceexperimentally has not advanced at the samerate, making it difficult to verify the theoreti-cal developments.

On page 1017 of this issue, EberhardBodenschatz and collaborators1 report animportant technical improvement to the way in which turbulence is measured. Thisadvance may make a decisive contribution to

bringing experimental turbulence researchback on par with theory.

In their experiment, Bodenschatz and co-workers1 modified a detector from Cornell’selectron–positron collider. They used ‘siliconstrip’ detectors to optically image tracer particles (tiny transparent beads) in turbu-lent water flow. Compared with previouslyavailable techniques, this method offers un-precedented time resolution of up to 70,000frames per second. As a result, the researcherswere able to measure the acceleration of theparticles in turbulent water, discovering thatit can reach 1,500 times the acceleration ofgravity. The high time-resolution of themeasurements indicates that the accelerationis highly intermittent, reflecting the complexstructure of turbulent flow.

At present, the standard probe for turbu-

Turbulence

Go with the flowItamar Procaccia

Traditional devices for measuring turbulence have been unable to keep upwith the latest developments in theory. But detectors derived from high-energy physics may narrow the gap between experiment and theory.

100 YEARS AGOA simple workable, absolutely trustworthysystem is still urgently wanted for thedetection of criminals, and if the authoressof this book has succeeded she certainlydeserves the thanks of all the Governmentsof Europe… It so happened that about seven years ago the reviewer came to theconclusion that the external ear ought toyield some clue to the relationship of manand ape, and of one race of man toanother… To test the “criminal-mark”theory of Lombroso and many others, heexamined the ears of more than 800confirmed criminals, and of more than twothousand inmates of asylums for the insane,situated in parts of the country where hehad already examined the ears of the sane.Altogether the ears of more than 40,000people of different races and of differentmoralities, besides those of about 300 apesand anthropoids, were examined, but thetotal results of this elaborate investigationwere almost entirely of a negative nature…If the reviewer’s methods and observationsare correct, the confirmed criminal’s ear isthe ear of the average inhabitant of GreatBritain. Nor did the ears of the insane differ,on an average, from those of the peoplefrom which they were drawn, and if theauthoress had carried her observations over a number of men of genius or of highability, instead of drawing elaboratedeductions from single observations, shewould probably have arrived at a similarconclusion as to them.From Nature 21 February 1901.

50 YEARS AGOMiss Dorothea M. A. Bate, who died after abrief illness on January 13 at the age ofseventy-two, was for more than fifty yearsone of the outstanding personalities at theBritish Museum (Natural History). When onlyseventeen, and with neither qualification norencouragement, she started work in the BirdRoom as a voluntary worker; but herinterests lay chiefly in palaeontology inrelation to the Recent fauna, rather than inthe Recent fauna itself… During 1901–1902Miss Bate explored the caves of Cyprus andmade some notable discoveries, such as theremains of pigmy elephants, and soonextended her interest to cave deposits inCrete, the Balearic—where she discoveredthe unique ‘antelope’ Myotragus—Malta andSardinia, working meticulously andearnestly and always alone.From Nature 24 February 1951.

© 2001 Macmillan Magazines Ltd

lence research is the hot-wire anemometer.This consists of a thin wire that is heated bythe passage of an electrical current and iskept at a constant temperature by means of a feedback loop. The wire is placed at rightangles to the average flow of a turbulent fluid.The fluid cools the wire and, because thecooling effect depends on the fluid’s velocity— the faster the fluid flows, the more the wireis cooled — the velocity can be measured as afunction of time. As would be expected forturbulent fluids, an erratic time series isobtained.

Hot-wire technology has progressed inrecent years: for example, superconductingelements have been used to measure thevelocities of cryogenic helium turbulence2.But this technology gives only limited infor-mation on the structure of turbulence. Hot-wire anemometry is fundamentally limited:it can provide data only for a fixed point — in other words, for the point at which thewire is positioned. But theorists dream aboutmeasuring scale-dependent informationfrom points that move with the flow — theso-called ‘lagrangian’ trajectories (Fig. 1).

Turbulent flow is irregular and dis-ordered, with particles experiencing manydifferent velocities. At large scales, the

chaotic flow of the fluid demonstrates thepresence of high amounts of energy. Butinstability results in energy loss, which cas-cades nonlinearly down to smaller scales.This cascade continues down to the smallestscale where molecular friction comes in andenergy is dissipated by viscosity. The mostinteresting regime to study covers the inter-mediate scales, where the presence of a con-stant energy flux from large to small scalesestablishes a statistical equilibrium inside

the fluid. For theoretical calculations, themean velocity of the fluid is subtracted fromthe turbulent flow by a ‘galilean’ transform-ation. Theorists focus on the lagrangian trajectories in the moving frame of the fluid,and are interested in measuring the physicalphenomena in this lagrangian picture. Thetechnique devised by Bodenschatz and co-workers brings us closer to this dream.

Hot-wire anemometry would be quiteuseless for turbulence research had it notbeen for the ingenuity of G. I. Taylor, whopointed out that when the mean velocity of the fluid is very high, the time seriesmeasured at a point can be considered as a spatial cut through the turbulent field.This ‘Taylor frozen turbulence hypothesis’asserts that the turbulent field is sweptthrough the probe faster than the field canappreciably change, so the wire is taking aone-dimensional ‘snapshot’ of the velocityfield. This has been the basis for analysis ofturbulence data for decades, but besidesbeing only approximately true, recent theo-retical work has drawn attention to essentialfeatures of turbulence that are totally missedby measuring one-dimensional cuts (Box 1).

But Bodenschatz and co-workers offernew possibilities in following the detailedmotion of fluid particles. This progresspromises an exciting and fruitful interactionbetween theory and experiments in turbu-lence in the coming years. ■

Itamar Procaccia is in the Department of ChemicalPhysics, The Weizmann Institute of Science, Rehovot76100, Israel.e-mail: itamar.procaccia@weizmann.ac.il 1. La Porta, A., Voth, G. A., Crawford, A. M., Alexander, J. &

Bodenschatz, E. Nature 409, 1017–1019 (2001).

2. Tabeling, P. Phys. Rev. E 53, 1613–1621 (1996).

3. Bernard, D., Gawedcki, K. & Kupiainen, A. J. Stat. Phys. 90,

519–569 (1998).

4. Gat, O., Procaccia, I. & Zeitak, R. Phys. Rev. Lett. 80, 5536–5539

(1998).

5. Frisch, U., Mazzino, A. & Vergassola, M. Phys. Rev. Lett. 80,

5532–5535 (1998).

6. Arad, I. & Procaccia, I. in Proc. IUTAM Symp. Geom. Stat. Turb.

(ed. Kambe, T.) 175–184 (Kluwer, Dordrecht, 2001).

7. Celani, A. & Vergassola, M. Phys. Rev. Lett. 86, 424–427

(2001).

8. Arad, I., Biferale, L., Celani, A., Procaccia, I. & Vergassola, M.

(submitted).

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Box 1 A problem to be solvedOne of the theoretical problems that can be verified experimentally using the techniques developed byBodenschatz and co-workers1 involves an important aspect of turbulence: its role in dispersing contaminantssuch as smoke or radioactive elements in the air and pollutants in the ocean. The concentration of such acontaminant at a space point r at time t is denoted as T (r,t ). One may be interested in the expected value ofthe field at some point T (r3,t ) given the measurement of the concentration at two other points r1 and r2 at thesame time. Such expected values are related to ‘correlation functions’. In this example, we have a third-ordercorrelation function which is denoted as *T(r1,t ) T (r2,t ) T (r3,t )¤ where the averaging is over time.

Central to the statistical theory of turbulence is the finding that such correlation functions show aremarkable property called ‘scaling’. This means that if we increase all the distances between themeasurement points r1, r2 and r3 by a given factor l, then the value of the correlation function changes by amultiplicative factor lz3, where z3 is a characteristic exponent. If we consider higher-order correlationfunctions between four, five or n points, they have the same property but with exponents zn that depend onthe order of the correlation function. These scaling exponents zn are believed to be universal characteristicsof the small-scale structure of the turbulent velocity field, reflected in the structure functions of advectedcontaminant. Their theoretical calculation is much sought after in fundamental turbulence research.

In recent years there has been a fundamental shift in the theoretical approach to such characteristics ofturbulence3. It turns out that the nature of these exponents is related to subtle geometrical properties of groupsof lagrangian trajectories of tracer particles4,5. For example, to understand the exponent z3 one needs to focuson the dynamics of three tracer particles. Obviously, at any point in time three tracer particles define a triangle,which in turn is fully characterized by one length scale R (say the geometric mean of the lengths of its sides)and two angles, say u and f. When the three tracer particles are advected by the turbulent velocity field (seeFig. 1 on page 1017 for the lagrangian trajectory of one such particle), the scale R of the defined triangle andits shape (angles) change continuously (Fig. 1). If we rescale the triangle to size R41, the dynamics become a trajectory in the space of shapes6, the space of all triangles of size 1 (conveniently characterized by twoangles u and f). These dynamics have equilibrium distributions given by r(u,f). The deep and surprising new statement that can be made is that the three-point statistics are dominated by trajectories in which thechange in R is compensated by a change in shape so that R z3 r(u,f) remains invariant7.

Such ‘statistically preserved structures’ are crucial for the statistical theory, as they obviously come todominate the statistics8. Indeed, exponents such as z3 can be understood as the rescaling exponentscharacterizing precisely such special distributions: the correlation function *T(r1,t ) T (r2,t ) T (r3,t )¤ isproportional to R z3 r(u,f), where R is the geometric mean of the distances between the points r1…r3. Ofcourse, the same ideas apply to any order correlation function or structure functions with the appropriateshape dynamics, and they quickly become the new language used to discuss scaling phenomena in turbulenttransport7,8. I. P.

Figure 1 The lagrangian evolution of a group ofthree tracer particles. At any moment in time thetrio defines a triangle that is fully determined bya scale R, the Euler angles of its orientation inspace, and two internal angles. Theory focuseson ‘statistically preserved structures’ which aredetermined by the distribution on internalangles and the scale. These structures dominatethe statistical theory.

θ

ϕ

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