Order and Disorder in Fluid Motion JERRY P

Proc. Natl. Acad. Sci. USA Vol. 92, pp. 6705-6711, July 1995 Colloquium Paper

This paper was presented at a coUloquium entitled "Physics: The Opening to Complexity," organized by Philip W. Anderson, held June 26 and 27, 1994, at the National Academy ofSciences, in Irvine, CA.

Order and disorder in fluid motion JERRY P. GOLLUB Physics Department, Haverford College, Haverford, PA 19041; and Physics Department, University of Pennsylvania, Philadelphia, PA 19104

ABSTRACT The development of complex states of fluid is devoted to two-dimensional systems where the underlying motion is illustrated by reviewing a series of experiments, geometry is essentially invariant with respect to translation in emphasizing film flows, surface waves, and thermal convec- a plane. The remarkable "Faraday" patterns that arise on a tion. In one dimension, cellular patterns bifurcate to states of free surface subject to vertical excitation are highlighted, spatiotemporal chaos. In two dimensions, even ordered pat- especially the novel quasipatterns that are similar to quasi- terns can be surprisingly intricate when quasiperiodic pat- crystals in solids. The subject of turbulence is outside the scope terns are included. Spatiotemporal chaos is best characterized of this article, but a brief discussion of transport and mixing statistically, and methods for doing so are evolving. Transport phenomena is given in Section V to show how mixing leads to and mixing phenomena can also lead to spatial complexity, complexity even in the presence ofvery modest disorder in the but the degree depends on the significance of molecular or velocity field. thermal diffusion. II. One-Dimensional Fronts and Spatiotemporal Chaos I. Introduction A large number of hydrodynamic systems experience symmetry-breaking instabilities as they are forced away from equi- Moving fluids exhibit an exceptionallywide range of dynamical librium. One of the earliest cases receiving attention was the phenomena, from highly ordered stationary periodic patterns Rayleigh-Benard instability in a convecting fluid layer. How- to disordered turbulent flows with a wide range of spatial and ever, simpler quasi-one-dimensional examples have been of temporal scales. At intermediate degrees of disorder, one finds interest recently because of their potential to provide clear states of spatiotemporal chaos that fluctuate in more restricted examples of bifurcations leading to spatiotemporal chaos. ways. These disparate phenomena have to be treated by An interesting one-dimensional case (2-4) can be obtained distinct methods-e.g., hydrodynamic stability theory for the by rotating a cylinder containing a small amount of fluid about ordered patterns, and statistical scaling theory for strong its horizontal axis. The fluid coats the cylinder, forming a film. turbulence. The intermediate states of spatiotemporal chaos However, the film sags, producing a linear "front" as shown in are still problematic from a theoretical point of view. In this Fig. la. This front is sustained against gravity by shear forces report, I describe some examples of the progressive develop- provided by the rotating inner wall of the cylinder. The name ment of complex fluid states from simpler ones, often through "rimming flow" is sometimes used for this phenomenon. The successive instabilities. Recent experiments have contributed front can be stable for a certain range of rotation rates (for a to a far richer picture of dynamical complexity in fluids than given fluid volume and viscosity). However, when the rotation was apparent only a few years ago. Because the literature on rate is increased, a cellular structure (Fig. lb) bifurcates pattern formation is quite voluminous and has been recently smoothly from the featureless "ground state." This instability reviewed in great detail (1), the focus here is on a set of has not been studied quantitatively with the Navier-Stokes experiments selected to illustrate the general ideas and mainly equations, because its spatial structure is fairly complex. On conducted at Haverford, with citations to related work. It the other hand, similar cellular fronts are well known in other should be read as an illustrated tour through some examples of hydrodynamic systems (5). The wavelength of the pattern is a complexity in fluids. function of the rotation rate, fluid volume, and film thickness. In the late 1970s many examples were found of temporal Furthermore, for some parameters the cellular front can chaos in fluid systems; the resulting complex time dependence appear as a stable state even though the flat front is never could be visualized as strange attractors in a low-dimensional stable. phase space. However, well-defined states of temporal chaos Though the problem of understanding the periodic pattern generally occur in systems that are tightly constrained spatially has not been fully resolved in this system, it is of interest to to reduce the number of degrees of freedom. More commonly, consider how the ordered pattern is transformed into a fluc- chaotic states emerge in extended systems and usually involve tuating state as the rotation rate is increased. Note that there many degrees of freedom. It is of great interest to determine are only a limited number of generic ways in which the periodic how these states of spatiotemporal chaos emerge from ordered pattern can be modified by a subsequent bifurcation, and these patterns and to consider methods of characterizing and ex- can be characterized by the symmetries that are broken. For plaining them. example, the reflection symmetry of the periodic structure I begin in Section II with the development of spatiotemporal could be broken, causing the cells to drift in one direction. chaos from an ordered pattern in a closed system where the Alternatively, the pattern could oscillate uniformly (in phase spatial variations occur mainly in one spatial dimension. The everywhere). The various possibilities allowed by symmetry effects of external noise on the development of ordered and have been enumerated (6). In the case at hand, the cells begin disordered patterns are considered in Section III. Section IV to oscillate in a way that doubles the wavelength: alternate maxima on the cellular front rise and fall periodically in time, The publication costs of this article were defrayed in part by page charge as shown in Fig. lc. A similar secondary instability (period- payment. This article must therefore be hereby marked "advertisement" in doubled oscillation) has also been noted qualitatively in a accordance with 18 U.S.C. §1734 solely to indicate this fact. convecting system (7). 6705 Downloaded by guest on October 2, 2021 6706 Colloquium Paper: Gollub Proc. Natl. Acad. Sci. USA 92 (1995) b a RL-°P ii £1

L _

C d 1J-2cn 40- z._

FIG. 1. Ordered and disordered states in a fluid flow. (a) A horizontal glass cylinder is rotated about its axis at angular velocity fQ; a small amount of oil coats the inner wall. Excess fluid forms a stable front at azimuthal position ¢(x, t) along the rising wall of the cylinder. (b) Spatially periodic pattern of cells on the front; the wavelength is about 1 cm. (c) Snapshot of time-periodic oscillating cells. (d) Strongly Position (cm) chaotic pattern with nucleation and annihilation of cells. FIG. 3. An alternative form of spatiotemporal chaos involving A small further increase in the rotation rate leads to a state traveling waves, but without regular cells. The cylinder contains less of spatiotemporal chaos: the oscillations lose their spatial and fluid than in Fig. 1. temporal coherence, and the oscillation amplitude becomes desirable in any case. A suitable model for problems of this nonuniform. The nonuniformities propagate through the sys- type (oscillatory cellular patterns) has been proposed (7). It is tem as traveling waves. This process is shown in Fig. 2, where basically a generalization of the well-known Ginzburg-Landau the local height of the front is shown with a gray scale, as a equation and is appropriate when the spatial variations of the function of space and time (plotted vertically downward). oscillation amplitude and phases are slow in space and time. It has become clear from a variety of experiments that there One difficulty that must be overcome to make such models are many different forms ofspatiotemporal chaos. Even for the useful is to determine appropriate values for the parameters flow described here, a substantially different form occurs when they contain so that they may be compared with experimental the fluid volume is slightly decreased. This case, first noted by data. This is not an easy task in a chaotic regime. An important Melo and Douady (3), is shown in Fig. 3. Here there are no goal would be for the model to reproduce the measured regular cells at all; localized traveling waves propagate through statistical properties of the spatiotemporal chaos. The corre- the system, interacting strongly when they collide. spondence between theory and experiment is not yet very close Models of Spatiotemporal Chaos. The hydrodynamic equa- for rimming flows. tions for rimming flows are largely intractable due to moving A number of widespread phenomena occurring in spatio- interfaces, cusps, and long time scales. Therefore, the use of temporal chaos have been elucidated through numerical stud- simplified model equations seems necessary, and might be ies of simplified model equations in one spatial dimension (1). Among these phenomena are the following: spatiotemporal 0 intermittency, in which "laminar" (or ordered) and "chaotic" I X (or disordered) domains coexist; phase turbulence, in which . the pattern amplitude is relatively constant while its spatial phase fluctuates; and defect-mediated "turbulence," in which the pattern irregularities are concentrated in localized defects 5 which are nucleated and annihilated at irregular intervals and locations. The initial bifurcations leading to these states have also been investigated in some model systems. However, many experimental phenomena, such as the fluid fronts described in this article, do not fit very well into these categories. Despite a considerable effort, we do not yet have an adequate phe- nomenology and, indeed, do not know whether there is a simple classification system for these complex processes. III. Noise-Sustained Patterns

15 To what extent are spatial patterns in nonequilibrium systems affected by noise that is external to the system itself? The question is particularly important because these patterns occur 0 1U 2U 30 40 via instabilities; at the initiation of the instability, the patterns can be quite sensitive to noise, even to the small amount of Position (cm) noise that is of thermal origin (8). This noise sensitivity can FIG. 2. Space-time representation of the spatiotemporal chaos of persist even when the system is not maintained close to the Fig. ld. The local height of the front is encoded in a gray scale as a threshold of instability, provided that it is convectively unsta- function of position along the cylinder and time. The oscillation ble (9, 10). This situation typically occurs in open systems with amplitude fluctuates. net flow, so that when perturbations grow as a result of Downloaded by guest on October 2, 2021 Colloquium Paper: Gollub Proc. Natl. Acad. Sci. USA 92 (1995) 6707 instability, the growing structures are carried away. The perturbations expand only in a "moving" frame. Space/time structures can persist in the laboratory frame of reference, but only if sustained by some small input signal, even microscopic noise. They are often called noise-sustained structures, and they result from instabilities that are said to be convective rather than absolute. Several experimental examples are well documented (11, 12). Here I give another striking example and 90 100 110 120 also discuss the transition to spatiotemporal chaos in such an x (cm) open flow system. Consider a thin layer of fluid, perhaps 1 mm thick, flowing down a planar surface that is inclined relative to the horizontal by a few degrees. Such a flowing film will be unstable if the Reynolds number R is larger than a threshold value. Traveling waves then appear on the air/fluid interface, as shown in Fig. 4a. This photograph (13) was produced by imaging (onto a CCD camera) the fluorescence emitted from a low concen- 130 140 150 160 tration of dye in the fluid, under ultraviolet illumination. The x (cm) intensity is proportional to the local film thickness, with thick areas shown dark. In this example, the variations are about FIG. 5. Secondary instability of periodic wave fronts on an inclined 10%. surface. The instability leads to a chaotic state with irregularly spaced wave fronts. The interfacial waves are irregular in space and time, as can be seen from the space-time diagram in Fig. 4b. Here, the local to the film thickness respond primarily periodic signal. The waves are also on the center line of the film is shown as a larger at a given x than in the case without periodic forcing. function of downstream distance and time. The important fact However, the waves are unstable with to note is that the waves a regular respect to a have dominant frequency but that that causes the waves to combine in their strength at a given value of the downstream coordinate secondary instability pairs x is an irregular (14), as shown in Fig. 5b. This secondary instability is also of function of time. This characteristic is a direct the convective it is sensitive to noise. The consequence of the fact that these are noise-sustained struc- type; i.e., result is a tures form of spatiotemporal chaos in which the waves eventually (13). The local wave amplitude reflects fluctuations in into the noise intensity in the inception region of the waves, where aggregate large amplitude pulses that are irregularly they are quite small. spaced. If the viscosity of the fluid is not too large, then Regular periodic waves can be produced by imposing a small transverse instabilities can also occur; this leads to three- periodic driving signal in the inception region, as shown in the dimensional flows that are weakly turbulent (15). An example left part of Fig. Sa. The periodic driving signal is of larger is shown in Fig. 6, but they are not described in detail here. amplitude than the external noise, so that the growing waves IV. Two-Dimensional Patterns

Faraday Waves: Hexagons and Quasipatterns. When a container of fluid with a free surface is subjected to a vertical periodic oscillation of sufficient amplitude, the surface becomes unstable with respect to a pattern of standing waves oscillating at half the driving frequency. If the container is sufficiently large that its shape does not determine the pattern, the finite amplitude waves form an array of squares at low viscosity, and a pattern of lines at high viscosity. More complex patterns can be formed by adding a pertur- bation to the overall vertical so that the Downstream distance x (cm) oscillation, waveform, while still periodic, does not have a to twice b 30 40 period equal the 0 basic period of the drive. The reasons are subtle and are based on symmetry. Edwards and Fauve (16) were the first to appreciate this possibility and to demonstrate it experimentally. A I. striking pattern of regular hexagons can be created by adjusting

1- E- 2'

FIG. 4. (a) Onset ofwaves on a fluid film flowing down an inclined surface; the flow is to the right. The gray scale, produced by fluorescence imaging, is linear in the local film thickness. (b) Space-time representation of the film thickness along the center line of the film plane. Fluctuations in the strength of the pattern at a given downstream position x are a result of the amplification of noise by the instability. FIG. 6. Three-dimensional disordered waves on a flowing film at higher Reynolds number; the flow is toward the right. (Bar = 4 cm.) Downloaded by guest on October 2, 2021 6708 Colloquium Paper: Gollub Proc. Natl. Acad. Sci. USA 92 (1995) directly above the fluid and then photographing the surface with a camera located at the center of the ring. Thus, areas that are locally horizontal reflect light into the camera, whereas tilted regions do not. The exposure time is much longer than the period of the standing waves. The dark regions of Fig. 7 (the interiors of the hexagons) rise above (and then fall below) the hexagonal lattice. The oscillation is in phase everywhere. It is straightforward to establish that the hexagonal pattern consists of three standing waves with wave vectors oriented in such a way that they could be arranged to form an equilateral triangle. It is also possible to create inhomogeneous hexagonal patterns consisting of domains oriented differently in different parts of the fluid layer. In that case, point defects are created at the boundaries of the domains. Hexagonal patterns are found in many other nonequilibrium systems, including thermal convection (17) and reaction diffusion (18) systems. The most remarkable regular state found so far in the Faraday system is a quasipattern, a pattern with exact 12-fold rotational symmetry but only quasiperiodic translational symmetry. This means roughly that the pattern comes arbitrarily close to repeating, given a sufficiently large translation. Such patterns were previously created by Edwards and Fauve (16, 19), using two-frequency forcing. These patterns are similar (with regard to symmetry) to quasicrystals in the solid state. FIG. 7. Uniform hexagonal pattern of Faraday waves on a fluid surface, formed by vibrating the container at two commensurate An example, from the recent work of Edwards, Pier, and frequencies (W. S. Edwards, B. Pier, and J.P.G.). myself at Haverford, is shown in Fig. 8. It has been shown to consist accurately of two superimposed hexagonal patterns, the relative amplitudes and phases of two frequency compo- oriented at 30° with respect to each other. The 12-fold rota- nents. An example, obtained in a recent experiment by W. S. tional symmetry can be detected by holding the picture nearly Edwards and B. Pier at Haverford, appears in Fig. 7. The image at eye level and looking for lines that traverse the pattern. The is made by illuminating the surface with a circular ring of lights lack of translational periodicity can also be detected by visual

FIG. 8. Quasipattern having 12-fold rotational order and quasiperiodic translational order (W. S. Edwards, B. Pier, and J.P.G.). This state is the hydrodynamic analog of a quasicrystal. Downloaded by guest on October 2, 2021 Colloquium Paper: Gollub Proc. Natl. Acad. Sci. USA 92 (1995) 6709 examination. More quantitatively, these features can be con- erties of the fluctuations about the averages. In particular, they firmed from an examination of the two-dimensional spatial found that the strength of the time-averaged pattern is deter- power spectrum, which contains more than 100 sharp spectral mined by amplitude and phase fluctuations about certain peaks. If the complexity of the pattern were simply due to regular base patterns. Structured average patterns in spatio- nonperiodicity, the peaks would be broadened. temporal chaos have also been found in rotating thermal The quasipatterns form quickly and, once formed, persist convection (24). These and other experiments imply that indefinitely. It is remarkable that such intricate but regular spatiotemporally chaotic states are not completely disordered; patterns can be formed as a result of instabilities in a homo- they often consist of a complex mixture of order and disorder. geneous nonequilibrium system. It seems possible that similar Rayleigh-Benard Spatiotemporal Chaos. Before leaving the states might be created in other systems, but this has not been topic of spatiotemporal chaos in two dimensions, I point out accomplished. some particularly stimulating experiments on Rayleigh- Faraday Waves: Statistics of Spatiotemporal Chaos. Spa- Benard convection, a system that has contributed tremen- tiotemporal chaos can also be created in the Faraday system. dously to what we know about complexity in fluid systems. This phenomenon has been studied by several groups, and the Morris et al. (25) succeeded in obtaining a uniform convecting transition process was reviewed recently (20). Fig. 9 shows a set layer of much larger aspect ratio than those studied previously. of five instantaneous images of disordered waves, obtained in These investigators have found that a chaotic pattern occurs a smaller cell than that of Fig. 7, and with a single forcing consisting of rotating spirals and other defects. The time scale frequency. The pattern consists of a disordered array of for fluctuations of the pattern decreases as a function of the patches of waves with local 4-fold symmetry. The pattern control parameter s, defined as the normalized distance above fluctuates with a correlation time that depends on the excita- the convective onset. For small s, the patterns are time- tion amplitude, but is typically about 1 s. dependent, but the curvature is rather small, and the patterns Gluckman et al. (21) have studied the time-averaged prop- are boundary-dominated. For larger s, the rotating spirals are erties of these fluctuating Faraday waves. They discovered that clearly independent of the boundaries. the apparently disordered images can have highly ordered time The patterns have been characterized by their correlation averages in containers that are considerably larger than the length (, which is found to have a power-law decay as a function correlation length of the pattern. The geometry of the aver- of E. Furthermore, the wavenumber band specified by - is aged images is related to the spatial symmetry of the bound- well within the range ofwavenumbers for which linear stability aries. For a circular container the instantaneous patterns have theory predicts stable rolls. Morris et al. (25) speculate that the a tendency toward radial alignment near the outer boundaries, stable roll state may exist but requires special initial and but the time-averaged image consists of concentric rings, as boundary conditions to be realized at large aspect ratios. shown in Fig. 9 Top Left. In a square container, the time This phenomenon has now been accurately reproduced by averages have square symmetry. The existence of structured direct numerical simulation (26) from the hydrodynamic equa- averages implies a degree of phase rigidity of the chaotic state tions, with only modest simplification to make the computation that had not been anticipated. The structured averages disap- tractable in a reasonable time. Simplified model equations pear at sufficiently high excitation amplitude and for contain- have also been used to describe spiral chaos (27). ers of sufficiently large horizontal extent. Related experiments have been carried out by Bosch et al. V. Transport and Mixing Phenomena (22). Gluckman et al. (23) recently showed how statistical studies of chaotic wave patterns could be used to infer prop- The mixing of a passively transported substance, such as a dye, can produce surprisingly complex structures, even if the flow is not turbulent. This phenomenon, known as chaotic advec- tion, was first identified by Aref (28). It has been studied both numerically and experimentally for a variety of systems (29), including, for example, periodically oscillating convection rolls (30). It is clear that a patch of impurity can be stretched and folded into a complex form by a flow that is strictly periodic. When molecular diffusion is incorporated as well, the net result is a dramatic enhancement of the effective diffusivity of material by the flow. In some cases such chaotic mixing processes give rise to distributions of material that can be modeled approximately as multifractals (31, 32). Mixing in turbulent flows is even more complex, and the spatial distribution of passively transported impurities is not fully understood. In this section, several fundamental issues concerning mixing phenomena in turbulent and weakly turbulent flows are highlighted. Geometrical Structure of Mixing in Thermal Turbulence. First we consider the geometrical patterns produced by mixing in thermal turbulence, far above the onset ofconvection. Given what has already been said about oscillatory periodic flows, one anticipates that dye mixing will produce complex patterns, and this expectation is borne out by experiment. I show in Fig. 10 the distribution of injected dye in thermal turbulence at about 17,000 Rc, where Rc is the Rayleigh number (nondimen- sional temperature gradient) at the threshold of thermal FIG. 9. Five instantaneous images of disordered Faraday waves in convection (33). The large Rayleigh number is obtained by a circular container, and a time-averaged pattern (Top Left) demon- using a cell whose height and width are comparable. The dye strating residual spatially periodic structure. Only the central region of is stretched and folded into a shape that becomes progressively the container is shown in each case. more complex with time, until diffusion eventually produces Downloaded by guest on October 2, 2021 6710 Colloquium Paper: Gollub Proc. Natl. Acad. Sci. USA 92 (1995)

a a ,j, afg9vz-.-1 ttftIl- X

11 >-* S --vf v% v \\ J

.x ......

gI J l I I-% ~ t I I

lU 4I>'vf'Pf 'P U t j~~~1- ,_,/ I It --' ~~~~~~~~~.... \ f- t 1.I 1t1

l 0 l' eeyXw- lll)iIlii- v---- X l\ \1 L- II/ 1,/z-' ala > II9 illJ (!lst \ '\ - t il ) %%B>>s %%Z41Il -z ,) _il,>t

-%%.. /.v'v'.%.. I' Av.JtLL./.I'...'A -.

I.\\\. 3.. II IIa. .,, FIG. 10. Dye mixing in thermal turbulence. Laser-induced fluo- * ill/la ,,- i /1 rescence is used to visualize the concentration field in the horizontal 1 //11 .' 'iii

mid-plane of a roughly cubic cell, 30 s and 58 s after injection. High ' ---. '111 %. ,,,1 concentrations appear dark. \v.\ *. '\. % %-.' ' . . - -- homogenization at long times. The interfaces between the . Iaaa - dyed and undyed fluid have been characterized statistically. t..l,ll ill--i ''i' 8 .ll jJ//.% Hit It is interesting to compare the dye observations with'the ' Ii JiI* %.i UI I/ri structures produced by the temperature field, which is also being transported and mixed by the flow. It has been possible to measure isothermal contours in cross sections of the flow by doping the fluid with chiral nematic liquid crystalline particles FIG. 11. Vertical cross section of thermal turbulence. An isother- whose color responds to the thermal field. Using broad-band mal contour at the mean temperature of the fluid is shown as a illumination and a color one can measure some of the camera, convoluted solid line, and the corresponding velocity field is repre- isothermal contours The same allow (33). particles simulta- sented by arrows. The Rayleigh number is R = 17,000 RC, and a and neous determination of the velocity field by particle image b are separated in time by 1.1 s. The isotherm shapes do not reveal the velocimetry, a method that statistically tracks particle veloci- fine convolutions seen in the dye distribution of Fig. 10. ties. An example of these measurements is presented in Fig. 11. is stretched and The solid> line shows the shape of the isotherm at the mean homogenized. But with significant probability, fluid can be transported over long distances in turbulence temperature of the convection cell; the arrows represent the without mixing, because the local strain rate varies widely. This velocity field. The main qualitative feature to note here is that phenomenon provides another example of the complexity of the isotherms are no more convoluted than the velocity field, mixing processes. The effect was first proposed theoretically by in contrast to the behavior of the dye distribution, which shows Pumir et al. (34). Confirming experiments were carried out by fine s'tructure. The reason is basically that the thermal diffu- Jayesh and Warhaft (35) and Gollub et al. (36). sivity is much larger than the molecular diffusivity of a dye. Spatial Power Spectrum of a Transported Field. It is well This causes the older finer scales to be smoothed out, so that known that the power spectrum of the velocity component along the stretching and folding process is hidden in the case of the some chosen direction in strong turbulence varies as k-5/3, where thermal field but visible for concentration fields. The level of k is the wavenumber for fluctuations along that axis. This complexity produced by mixing is thus strongly influenced by important scaling relation has been well confirmed experimen- diffusion. tally, as discussed in a recent review by Nelkin (37). It is also Probabillity Distributions of Temperature Fluctuations. expected that the spectrum of concentration fluctuations should Turbulence is a type of random process. However, it is very follow the same power law out to wavenumbers ofthe order ofthe different from a Gaussian random process. This fact can be inverse of the Koimogorov or dissipative scale. Beyond that nicely illustrated in mixing experiments. Suppose that one side wavenumber, the exponent ofthe power lawshould change to -1,

of a turbulent flow is maintained warmer than the other and the famous Batchelor prediction. It is one ofthe puzzles ofmixing that the fluid is mixed mechanically. At any given point within in turbulence that this prediction has not yet been either confirmed or seems the fluid, the temperature fluctuates about its local mean. conclusively disproved. Numerical evidence rather favorable (38), and a recent experiment on mixing in a thin What is the nature of these fluctuations? The thermal fluc- film has found evidence for Batchelor scaling (W. Goldburg, tuations can be strongly non-Gaussian, as can the velocity personal communication). On the other hand, experiments in fluctuations. An example is shown in Fig. 12, where the strong three-dimensional turbulence (39) have not so far detected probability distribution of temperature fluctuations is shown in the Batchelor regime. It is important to resolve the issue if we are flow at sever'al numbers. The non- grid Reynolds strongly to achieve a deep understanding of mixing in fluid flows. Gaussian character of the fluctuations (for the larger R) is evident. The tails of the distribution become much more VI. Concluding Remarks prominent as R is increased. What is the reason for this change in the statistical properties This brief tour illustrates some aspects of the development of of the fluctuations? As the fluid moves between two points, it spatial complexity in fluids. Even highly ordered states, such Downloaded by guest on October 2, 2021 Colloquium Paper: Gollub Proc. Natl. Acad. Sci. USA 92 (1995) 6711

a II I I~~~~~~~~~~~~~~ 1. Cross, M. C. & Hohenberg, P. C. (1993) Rev. Mod. Phys. 65, 851-1112.

1O- 1 r 2. Melo, F. (1993) Phys. Rev. E 48, 2704-2712. 3. Melo, F. & Douady, S. (1993) Phys. Rev. Lett. 71, 3283-3286. 4. Vallette, D. P., Edwards, W. S. & Gollub, J. P. (1994) Phys. Rev. r E 49, 4783-4786. °/ \~~~~~~~~~~ 5. Michalland, S., Rabaud, M. & Couder, Y. (1993) Europhys. Lett. 1 0- 3 ;r 22, 17-22. . . . .~~~~~~~ 6. Coullet, P. & Iooss, G. (1990) Phys. Rev. Lett. 64, 866-869. r 7. Daviaud, F., Lega, J., Berge, J., Coullet, P. & Dubois, M. (1992) Physica D 55, 287-308. io-5 L 8. Rehberg, I., Rasenat, S., de la Toffe Judrez, M., Schopf, W., -6 -4 -2 0 2 4 6 Horner, F., Ahlers, G. & Brand., H. R. (1991) Phys. Rev. Lett. 67, 596-599. I I I 9. Huerre, P. & Monkewitz, P. A. (1990)Annu. Rev. FluidMech. 22, b 473-537. 1o-1 r 10. Deissler, R. J. (1987) Physica D 25, 233-260. 0I 11. Tsameret, A. & Steinberg, V. (1991) Europhys. Lett. 14,331-336. 12. Babcock, K. L., Ahlers, G. & Cannell, D. S. (1991)Phys. Rev. Lett. r II 67, 3388-3391. 13. Liu, J., Paul, J. D. & Gollub, J. P. (1993) J. Fluid Mech. 250, 1 0-3 r I 69-101. CP : 14. Liu, J. & Gollub, J. P. (1993) Phys. Rev. Lett. 70, 2289-2292. OD, r 15. Liu, J. & Gollub, J. P. (1995) Phys. Fluids 7, 55-67. 16. Edwards, W. S. & Fauve, S. (1994) J. Fluid Mech. 278, 123-148. 1 o-5 r I EI 17. Bodenschatz, E., de Bruyn, J., Ahlers, G. & Cannell, D. S. (1991) - 6 - 4 -2 0 2 4 6 Phys. Rev. Leu. 67, 3078-3081. 18. Ouyang, Q. & Swinney, H. L. (1991) Nature (London) 352, ST/a 610-612. 19. Edwards, W. S. & S. Acad. Sci. Ser. II FIG. 12. Distribution of temperature fluctuations in a grid flow at Fauve, (1992) C.R. Mec. two Reynolds numbers, 300 and 1850 In the first case, the Phys. Chim. Sci. Terre Univers 315, 417-420. (a) (b). 20. fluctuations form a Gaussian distribution, which appears parabolic on Gollub, J. P. (1994) in Turbulence, Weak and Strong, ed. Tabeling, the logarithmic scale. In the second case, the fluctuations are expo- P. (Plenum, New York). 21. B. J. & J. P. nential rather than Gaussian, and deviations far from the mean occur Gluckman, J., Marcq, P., Bridger, Gollub, (1993) with significant probability. Phys. Rev. Leu. 71, 2034-2037. 22. Bosch, E., Lambermont, H. & van de Water, W. (1994) Phys. Rev. E as the in are intri- 49, 3580-3583. quasipatterns Faraday waves, remarkably 23. Gluckman, B. J., Arnold, C. B. & Gollub, J. P. (1995) Phys. Rev. cate. I have pointed out that mixing produces complex struc- E 51, 1128-1147. tures when molecular diffusion is unimportant and that some 24. Ning, L., Hu, Y., Ecke, R. E. & Ahlers, G. (1993) Phys. Rev. Lett. aspects of mixing phenomena remain puzzling. 71, 2216-2219. Spatiotemporal chaos is a widespread phenomenon that 25. Morris, S. W., Bodenschatz, E., Cannell, D. S. & Ahlers, G. resists a comprehensive theoretical explanation. However, a (1993) Phys. Rev. Lett. 71, 2026-2029. wealth of experiments in various laboratories has provided 26. Decker, W., Pesch, W. & Weber, A. (1994) Phys. Rev. Lett. 73, novel ways of characterizing these states, often statistically. 648-651. Alternative approaches are still needed in order to discrimi- 27. Xi, H., Gunton, J. D. & Vifials, J. (1993) Phys. Rev. Lett. 13, nate adequately among the different types of disorder. Al- 2030-2033. 28. Aref, H. (1984) J. Fluid Mech. 143, 1-21. though the term "chaos" is frequently utilized in connection 29. Ottino, J. M. (1990) Annu. Rev. Fluid Mech. 22, 207-253. with deterministic nonlinear phenomena in extended systems, 30. Solomon, T. H. & Gollub, J. P. (1988) Phys. Rev. A 38, 6280- the standard methods of low-dimensional nonlinear dynamics 6286. are very difficult to apply experimentally here. Proposed 31. Ramshankar, R. & Gollub, J. P. (1991) Phys. Fluids A 3, 1344- generalizations to extended systems have been reviewed by 1350. Abarbanel et al. (40). Phase-space methods must overcome 32. Ottino, J. M., Muzzio, F. J., Meneveau, C. & Swanson, P. D. significant barriers because the amount of data required to (1992) Phys. Fluids A 4, 1439-1456. populate an attractor in a phase space of high dimensionality 33. Gluckman, B. J., Willaime, H. & Gollub, J. P. (1992) Phys. Fluids is very large. A 5, 647-661. It is 34. Pumir, A., Shraiman, B. & Siggia, E. D. (1991) Phys. Rev. Lett. 66, important to remember that states of spatiotemporal 2984-2987. chaos are not completely disordered: the local structures they 35. Jayesh & Warhaft, Z. (1992) Phys. Fluids A 4, 2292-2307. contain may provide a key to understanding dynamical com- 36. Gollub, J. P., Lane, B., Mesquita, 0. N. & Meyers, S. R. (1993) plexity in fluids. Phys. Fluids A 5, 2255-2263. 37. Nelkin, M. (1995) Adv. Phys. 43, 143-181. I am especially indebted to many collaborators, including Stuart 38. Holzer, M. & Siggia, E. D. (1994) Phys. Fluids 6, 1820-1837. Edwards, Bruce Gluckman, Jun Liu, and Doug Vallette; they provided 39. Miller, P. L. & Dimotakis, P. E. (1991) Phys. Fluids A 3, 1156- most of the figures. I appreciate the financial support provided by the 1163. National Science Foundation through Grants DMR-9319973 and 40. Abarbanel, H. D. I., Brown, R., Sidorowich, J. J. & Tsimring, CTS-9115005 for much of the work described in this article. L. S. (1993) Rev. Mod. Phys. 65, 1331-1393. Downloaded by guest on October 2, 2021