This book, first published in 1998, treats turbulence from the point of view of dynamical systems. The exposition centres around a number of important simplified models for turbulent behaviour in systems ranging from fluid motion (classical turbulence) to chemical reactions and interfaces in disordered systems.The modern theory of fractals and multifractals now plays a major role in turbulence research, and turbulent states are being studied as important dynamical states of matter occurring also in systems outside the realm of hydrodynamics, i.e. chemical reactions or front propagation. The presentation relies heavily on simplified models of turbulent behaviour, notably shell models, coupled map lattices, amplitude equations and interface models, and the focus is primarily on fundamental concepts such as the differences between large and small systems, the nature of correlations and the origin of fractals and of scaling behaviour. This book will be of interest to graduate students and researchers interested in turbulence, from physics and applied mathematics backgrounds.

Gravitropism, the slow reorientation of plant growth in response to gravity, is a key determinant of the form and posture of land plants. Shoot gravitropism is triggered when statocysts sense the local angle of the growing organ relative to the gravitational field. Lateral transport of the hormone auxin to the lower side is then enhanced, resulting in differential gene expression and cell elongation causing the organ to bend. However, little is known about the dynamics, regulation, and diversity of the entire bending and straightening process. Here, we modeled the bending and straightening of a rod-like organ and compared it with the gravitropism kinematics of different organs from 11 angiosperms. We show that gravitropic straightening shares common traits across species, organs, and orders of magnitude. The minimal dynamic model accounting for these traits is not the widely cited gravisensing law but one that also takes into account the sensing of local curvature, what we describe here as a graviproprioceptive law. In our model, the entire dynamics of the bending/straightening response is described by a single dimensionless "bending number" B that reflects the ratio between graviceptive and proprioceptive sensitivities. The parameter B defines both the final shape of the organ at equilibrium and the timing of curving and straightening. B can be estimated from simple experiments, and the model can then explain most of the diversity observed in experiments. Proprioceptive sensing is thus as important as gravisensing in gravitropic control, and the B ratio can be measured as phenotype in genetic studies.perception | signaling | movement | morphogenesis

We show that the circular hydraulic jump can be qualitatively understood using simplified equations of the shallow-water type which include viscosity. We find that the outer solutions become singular at a finite radius and that this lack of asymptotic states is a general phenomenon associated with radial flow with a free surface. By connecting inner and outer solutions through a shock, we obtain a scaling relation for the radiusRjof the jump,Rj∼Q⅝v⅜g⅛, whereQis the volume flux,vis the kinematic viscosity andgis the gravitational acceleration. This scaling relation is valid asymptotically for largeQ. We discuss the corrections appearing at smallerQand compare with experiments.

We present an experimental study of a symmetric foil performing pitching oscillations in a vertically flowing soap film. By varying the frequency and amplitude of the oscillation we visualize a variety of wakes with up to 16 vortices per oscillation period, including von Kármán vortex street, inverted von Kármán vortex street, 2P wake, 2P+2S wake and novel wakes ranging from 4P to 8P. We map out the wake types in a phase diagram spanned by the width-based Strouhal number and the dimensionless amplitude. We follow the time evolution of the vortex formation near the round leading edge and the shedding process at the sharp trailing edge in detail. This allows us to identify the origins of the vortices in the 2P wake, to understand that two distinct 2P regions are present in the phase diagram due to the timing of the vortex shedding at the leading edge and the trailing edge and to propose a simple model for the vorticity generation. We use the model to describe the transition from 2P wake to 2S wake with increasing oscillation frequency and the transition from the von Kármán wake, typically associated with drag, to the inverted von Kármán wake, typically associated with thrust generation.

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Green plants are Earth's primary solar energy collectors. They harvest the energy of the Sun by converting light energy into chemical energy stored in the bonds of sugar molecules. A multitude of carefully orchestrated transport processes are needed to move water and minerals from the soil to sites of photosynthesis and to distribute energy-rich sugars throughout the plant body to support metabolism and growth. The long-distance transport happens in the plants' vascular system, where water and solutes are moved along the entire length of the plant. In this review, the current understanding of the mechanism and the quantitative description of these flows are discussed, connecting theory and experiments as far as possible. The article begins with an overview of low-Reynolds-number transport processes, followed by an introduction to the anatomy and physiology of vascular transport in the phloem and xylem. Next, sugar transport in the phloem is explored with attention given to experimental results as well as the fluid mechanics of osmotically driven flows. Then water transport in the xylem is discussed with a focus on embolism dynamics, conduit optimization, and couplings between water and sugar transport. Finally, remarks are given on some of the open questions of this research field.

It is shown numerically that the stability intervals for limit cycles of the circle map form a complete devil's staircase at the onset of chaos. The complementary set to the stability intervals is a Cantor set of fractal dimension D = 0.87. This exponent is found to be universal for a large class of functions.PACS numbers: 03.20.+i, 47.20.+m, 74.50.+r The transition to chaos in dynamical systems has been intensively studied by utilizing discrete maps. In particular, Feigenbaum' found that the bifurcation route to chaos, in which the transition is governed by one external parameter, is characterized by universal indices. Recently another route, namely through quasiperiodic behavior, has been studied by several authors. ' ' This transition can be established by varying two frequencies and may be studied by means of the so-called circle map, f(e) = e+n -(It/2~) sin(2~v).The ratio between the frequencies is given by the winding number WPC, n) = lim n-'[f" (0) -&].(2) n In numerical studies Shenker' found that when the ratio W approaches the reciprocal Golden mean the mapping exhibits unusual scaling behavior at the critical point (K =1) where chaos sets in. This transition has been elegantly treated by means of a renormalization-group technique by Feigenbaum, Kadanoff, and Shenker' and Rand eI; al. ' Our objective is quite different. We have studied the global mode-locking phenomenon at the critical point, K=1. A similar mode locking has been observed experimentally in a Josephson junction when an external frequency is applied. ' By numerical iterations of the mapping (1) at different values of 0 we have found evidence for mode locking of the mapping at every single rational value of W. Also, we find that the stability intervals for the different rational values fill up the whole 0 axis. The steps in the 8' vs 0 function thus form a comPlete devil's staircase, of a type which has been found in quite different contexts, such as the one-dimensional Ising model' and the Frenkel-Kontorowa model. ' The complementaryset (on the 0 axis) to a complete devil's staircase is a Cantor set of fractal dimension D smaller than or equal to 1. For the staircase of the circle map we have found D =0.87 but this number seems to be universal for a large class of functions. The dimension D is therefore a bona fide critical index characterizing the transition to chaos.The mapping (1) can be considered as the reduced (due to dissipation} one-dimensional map of a more general map of the plane onto itself. 'The iteration of (1) from a given starting point 8, converges towards a limit cycle or an aperiodic trajectory. A limit cycle is characterized by a rational winding number W =P/Q, where Q is the period of the cylcle and P is the number of sweeps through the unit interval [0; 1] in a cycle when the mapping (1) is considered modulo 1.We denote the interval in 0 for which the iteration converges to a limit cycle P/Q as AQ (P/Q). As shown by Herman' for 0&%&1 the iteration locks in to every single rational number in a finite interval 40. Ho...

We study laminar thin film flows with large distortions in the free surface using the method of averaging across the flow. Two concrete problems are studied: the circular hydraulic jump and the flow down an inclined plane. For the circular hydraulic jump our method is able to handle an internal eddy and separated flow. Assuming a variable radial velocity profile like in Karman-Pohlhausen's method, we obtain a system of two ordinary differential equations for stationary states that can smoothly go through the jump where previous studies encountered a singularity. Solutions of the system are in good agreement with experiments. For the flow down an inclined plane we take a similar approach and derive a simple model in which the velocity profile is not restricted to a parabolic or self-similar form. Two types of solutions with large surface distortions are found: solitary, kink-like propagating fronts, obtained when the flow rate is suddenly changed, and stationary jumps, obtained, e.g., behind a sluice gate. We then include time-dependence in the model to study stability of these waves. This allows us to distinguish between sub-and supercritical flows by calculating dispersion relations for wavelengths of the order of the width of the layer.