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Documented evidence of fungi associated with Mesozoic ferns is exceedingly rare. Three different types of fungal remains occur in a portion of a small, permineralised fern stem of uncertain systematic affinities from the Triassic of Germany. Exquisite preservation of all internal tissues made it possible to map the spatial distribution of the fungi in several longitudinal and transverse sections. Narrow, intracellular hyphae extend through the entire cortex, while wide hyphae are concentrated in the cortical intercellular system adjacent to the stele and leaf traces. Hyphal swellings occur in the phloem and adjacent cortex, while moniliform hyphae (or chains of conidia) are present exclusively in parenchyma adjacent to the stele. No host response is recognisable, but host tissue preservation suggests that the fern was alive during fungal colonisation. The highest concentration of fungal remains occurs close to the stele and leaf traces, suggesting that the fungi either utilised the vascular tissues as an infection/colonisation pathway or extracted nutrients from these tissues. This study presents the first depiction of fungal distribution throughout a larger portion of a fossil plant. Although distribution maps are useful tools in assessing fungal associations in relatively small, fossil plants, preparing similar maps for larger and more complex fossils would certainly be difficult and extremely arduous.
Three-dimensional direct numerical simulations are used to study the energy cascade rate in isothermal compressible magnetohydrodynamic turbulence. Our analysis is guided by a two-point exact law derived recently for this problem in which flux, source, hybrid and mixed terms are present. The relative importance of each term is studied for different initial subsonic Mach numbers
and different magnetic guide fields
. The dominant contribution to the energy cascade rate comes from the compressible flux, which depends weakly on the magnetic guide field
, unlike the other terms whose moduli increase significantly with
. In particular, for strong
the source and hybrid terms are dominant at small scales with almost the same amplitude but with a different sign. A statistical analysis undertaken with an isotropic decomposition based on the SO(3) rotation group is shown to generate spurious results in the presence of
, when compared with an axisymmetric decomposition better suited to the geometry of the problem. Our numerical results are compared with previous analyses made with in situ measurements in the solar wind and the terrestrial magnetosheath.
Objectives: Mild cognitive impairment is common in non-demented Parkinson disease patients (PD-MCI) and is considered as a risk factor for dementia. Executive dysfunction has been widely described in PD and the Verbal Fluency Tests (VFT) are often used for executive function assessment in this pathology. The Movement Disorder Society (MDS) published guidelines for PD-MCI diagnosis in 2012. However, no investigation has focused on the qualitative analysis of VFT in PD-MCI. The aim of this work was to study the clustering and switching strategies in VFT in PD-MCI patients. Moreover, these variables are considered as predictors for PD-MCI diagnosis. Methods: Forty-three PD patients and twenty normal controls were evaluated with a neuropsychological protocol and the MDS criteria for PD-MCI were applied. Clustering and switching analysis were conducted for VFT. Results: The percentage of patients diagnosed with PD-MCI was 37.2%. The Mann-Whitney U test analysis showed that PD-MCI performed poorly in different cognitive measures (digit span, Wisconsin Card Sorting Test, judgment of line orientation, and comprehension test), compared to PD patients without mild cognitive impairment (PD-nMCI). Phonemic fluency analyses showed that PD-MCI patients produced fewer words and switched significantly less, compared to controls and PD-nMCI. Concerning semantic fluency, the PD-MCI group differed significantly, compared to controls and PD-nMCI, in switches. Discriminant function analyses and logistic regression analyses revealed that switches predicted PD-MCI. Conclusions: PD-MCI patients showed poor performance in VFT related to the deficient use of production strategies. The number of switches is a useful predictor for incident PD-MCI. (JINS, 2017, 23, 511–520)
Physics has experienced several revolutions in the twentieth century that profoundly changed our understanding of nature. Quantum mechanics and (special, general) relativity are the best known and certainly the most important, but the discovery of the fourth state of matter – the state of plasma – as the most natural form of ordinary matter in the Universe, with more than 99% of visible matter being in this form, is unquestionably a revolution in physics. This discovery has led to the emergence of a new branch of physics called plasma physics.
Plasma physics describes the coupling between electromagnetic fields and ionized matter (electrons, ions). Thus, it is based upon one of the four foundations of physics: the electromagnetic interaction whose synthetic mathematical formulation was made by the Scottish physicist J. C. Maxwell who published in 1873 two heavy volumes entitled A Treatise on Electricity and Magnetism. The discovery of the electron by J.J. Thomson in 1897 and the formulation of the theory of the atom at the beginning of the twentieth century have contributed to the first development of plasma physics. It was in 1928 that the name plasma was proposed for the first time by I. Langmuir, referring to blood plasma in which one finds a variety of corpuscles in movement. Experimental studies of plasmas first focused essentially on the phenomenon of electrical discharge in gas at low pressure with, for example, the formation of an electric arc. These studies initiated during the second half of the twentieth century were extended to problems related to the reflection and transmission of radio waves in the Earth's upper atmosphere (this was how the first transatlantic link was established by Marconi in 1901), which led to the discovery of the ionosphere, an atmospheric layer beyond 60 km altitude with a thickness of several hundred kilometers. As explained by the astronomer S. Chapman (1931), the ionosphere consists of gas partially ionized by solar ultraviolet radiation; therefore, it is the presence of ionospheric plasma which explains why low-frequency waves can be reflected or absorbed depending on the frequency used.
Ninety-nine percent of ordinary matter in the Universe is in the form of ionized fluids, or plasmas. The study of the magnetic properties of such electrically conducting fluids, magnetohydrodynamics (MHD), has become a central theory in astrophysics, as well as in areas such as engineering and geophysics. This textbook offers a comprehensive introduction to MHD and its recent applications, in nature and in laboratory plasmas; from the machinery of the Sun and galaxies, to the cooling of nuclear reactors and the geodynamo. It exposes advanced undergraduate and graduate students to both classical and modern concepts, making them aware of current research and the ever-widening scope of MHD. Rigorous derivations within the text, supplemented by over 100 illustrations and followed by exercises and worked solutions at the end of each chapter, provide an engaging and practical introduction to the subject and an accessible route into this wide-ranging field.
Turbulence is generally associated with the formation of vortices in a fluid. There are numerous experiences in daily life where one can note the presence of turbulence: the movements of a river downstream of an obstacle, the smoke escaping through a chimney, vortical motions of the air, or the turbulence zones that we sometimes cross by plane. Since it is not necessary to use powerful microscopes or telescopes to study turbulence one could conclude that it is probably not difficult to understand it. Unfortunately that is not the case! Although significant progress has been made since the middle of the twentieth century, several important questions remain unanswered and it is clear that at the beginning of the twenty-first century turbulence remains a central research topic in physics.
The first theoretical bricks of turbulence were laid from the moment physicists started to tackle the non-linearities of the hydrodynamic equations. As we will see, it is in this context that the first fundamental law of turbulence was established: this was the statistical law of Kolmogorov (1941). Nowadays the theoretical treatment of turbulence is partly based on numerical simulations which, accompanied by very powerful tools of visualization, allow us to tackle this difficult problem from a different angle and stimulate new questions. The purpose of this chapter is to present concepts and fundamental results on fully developed turbulence. This chapter is devoted to hydrodynamics, from which some foundations of the theory of turbulence have emerged. The two other chapters in this part of the book will be devoted to MHD turbulence.
What is Turbulence?
Unpredictability and Turbulence
It is not easy to define turbulence quantitatively because to do this one requires knowledge of a number of concepts that will be defined partly in this chapter.
Without going into the details, we can notice that the disordered – or chaotic – aspect seems to be the main characteristic of turbulent flows. It is often said that a system is chaotic when two points originally very close to each other in phase space separate exponentially over time. As we will see later, this definition can be extended to the case of fluids.
In our familiar environment, matter appears in solid, liquid, or gaseous form. This triptych vision of the world was shaken in the twentieth century when astronomers revealed that most of the extraterrestrial matter – namely more than 99% of the ordinary matter in the Universe – is actually in an ionized state called plasma whose physical properties differ fundamentally from those of a neutral gas. The study of this fourth state of matter was developed mainly in the second half of the twentieth century and is now considered a major branch of modern physics. A decisive step was taken in 1942 when the Swedish astrophysicist Hannes Alfvén (1908–1995) proposed the theory of magnetohydrodynamics (MHD) by connecting the Maxwell electrodynamics with the Navier–Stokes hydrodynamics. In this framework, plasmas are described macroscopically as a fluid and the corpuscular aspect of ions and electrons is ignored. Nowadays, MHD has emerged as the central theory to understand the machinery of the Sun, stars, stellar winds, accretion disks around super-massive objects such as black holes with the formation of extragalactic jets, interstellar clouds, and planetary magnetospheres. Also, when H. Alfvén was awarded the Nobel Prize in Physics in 1970, the Committee congratulated him “for fundamental work and discoveries in magnetohydrodynamics with fruitful applications in different parts of plasma physics.”
The MHD description is not limited to astrophysical plasmas, but is also widely used in the framework of laboratory experiments or industrial developments for which plasmas and conducting liquid metals are used. In the first case, the emblematic example is certainly controlled nuclear fusion with the International Thermonuclear Experimental Reactor (ITER) in Cadarache. Indeed, the control of a magnetically confined plasma requires an understanding of the large-scale equilibrium and the solution of stability problems whose theoretical framework is basically MHD. Liquid metals are also used, for example, in experiments to investigate the mechanism of magnetic field generation – the dynamo effect – that occurs naturally in the liquid outer core of our planet via turbulent motions of a mixture of liquid metals. Most of the natural MHD flows cited above are far from thermodynamic equilibrium, with highly turbulent dynamics.
When a static equilibrium has been found (see Chapter 8), the next question that we have to address concerns the stability of this equilibrium. A part of the answer is given by the linear perturbation theory, which consists of analyzing the result of a small (i.e. linear) perturbation of the equilibrium. If the equilibrium is stable, the perturbation will behave as a wave that propagates in the medium; if it is unstable, the perturbation will increase exponentially.
In Figure 9.1, we present some unstable and stable situations arising from the example of a sphere placed in an external potential field. In case 1 a sphere is at the bottom of a well of infinite potential. In this position the sphere can only perform oscillations around its equilibrium position. These oscillations, once generated, are damped due to friction until the sphere reaches a static equilibrium position at the bottom of the potential well. This is a situation of stable equilibrium. In case 2 a sphere is placed a the top of a potential (a hill). In this case, a small displacement of the sphere is sufficient to move it to much lower potentials: this is an unstable situation that is often associated with a linear instability. The third case is that of a metastable state where the sphere is placed initially on a locally flat potential (a plateau): a small displacement around the initial position does not change the potential of the sphere. Finally, the last case (case 4) is that of a sphere placed in a hollow. This is an example of non-linear instability: the sphere is stable against small perturbations but becomes unstable for larger disturbances.
In plasma physics, the sphere in the previous paragraph corresponds to a particular mode of a wave and the shape of the potential can be a source of free energy. There are many energy sources in space plasmas. For example, the solar wind is a continuous source of energy for the Earth's magnetospheric plasma
which is never in a static equilibrium. The consequences of this energy input are the generation of large-scale gradients and the deformation of the distribution functions of particles at small scales.
In its primitive form the Kolmogorov theory states that the four-fifths law can be generalized to higher-order structure functions according to relation (11.33) by assuming self-similarity. Experiments and numerical simulations clearly show a discrepancy from this prediction (see Figure 11.10): this is what is commonly called intermittency. Even if intermittency remains a still poorly understood property of turbulence because it still challenges any attempt at a rigorous analytical description from first principles (i.e. the Navier–Stokes equations), several models have been proposed to reproduce the statistical measurements, of which the simplest is probably the fractal model, also called the β model, which was introduced in 1978 (Frisch et al., 1978). As we shall see, this model is based on the idea of a fractal (incompressible) cascade and is therefore inherently a self-similar model. However, because the structure-function exponents are not those predicted by the Kolmogorov theory, one speaks of intermittency and anomalous exponents. Refined models have also been proposed, and we will present in this chapter the two most famous models: the log-normal and log-Poisson models.
Fractals and Multi-fractals
The idea underlying the β fractal model is Richardson's cascade (Figure 11.7): at each step of the cascade the number of children vortices is chosen so that the volume (or the surface in the two-dimensional case) occupied by these eddies decreases by a factor β (0 < β < 1) compared with the volume (or surface) of the parent vortex. The β factor is a parameter less than one of the model to reflect the fact that the filling factor varies according to the scale considered: the smallest eddies occupy less space than the largest.
We define by ln the discrete scales of our system: the fractal cascade is characterized by jumps from the scale ln to the scale ln+1.We show an example of a fractal cascade in Figure 13.1: at each step of the cascade the elementary scale is divided by two.