Volume 6, Issue 11, November 1999
 SPECIAL TOPIC: NONLINEAR METHODS IN SPACE PLASMA PHYSICS


Selforganized criticality, multifractal spectra, sporadic localized reconnections and intermittent turbulence in the magnetotail
View Description Hide DescriptionIt has been suggested that the dynamics of the Earth’s magnetotail may be described by the stochastic behavior of a nonlinear dynamical system near forced and/or selforganized criticality(SOC). It has been further argued that multiscale intermittent turbulence of overlapping plasma resonances is the underlying physics that can lead to the onset and evolution of substorms. Such a description provides a convenient explanation of the localized and sporadic nature of the reconnection signatures and fractal spectra that are commonly observed in the magnetotail region. These concepts provide a new paradigm for the understanding of the everelusive phenomenon of magnetic substorms. In this review, we describe some of the basic physical concepts and mathematical techniques (such as the dynamic merging of coherent structures, nonclassical nonlinear instability, path integrals, the theory of the renormalizationgroup, lowdimensional chaos, selfsimilarity and scaling, fractals, coarsegrained helicity and symmetry breaking) that play a central role in the development of these new ideas.

The distribution of flares, statistics of magnetohydrodynamic turbulence and coronal heating
View Description Hide DescriptionIn this paper theoretical evidence in favor of the hypothesis that coronal dissipation occurs in bursts at very small spatial scales is presented. Each individual burst, though unobservable and energetically insignificant, is thought to represent the building block of coronal activity. In this framework, a large number of coherently triggered bursts is what appears as one of the many observed solar atmospheric events (i.e., blinkers, heating events, explosive events, flashes, microflares, flares,…). Histograms of such events, when computed, in terms of total energy, duration and peak luminosity appear to display powerlaw behavior. Simulations of the energy dissipation in the simplest possible forced magnetohydrodynamic(MHD) system, admitting reconnection events, indeed displays such kind of behavior: dissipative events of varying intensity, size and duration may be defined, whose distributions follow power laws. The meaning of cellular automaton models, introduced to describe the powerlaw statistics of observed energetic events on the Sun, i.e., solar flares, is then discussed. Finally, a minimal set of constraints necessary to render such automaton models more relevant for the description of dynamic phenomena described by magnetohydrodynamic equations is introduced.

Magnetohydrodynamic turbulence in the solar wind
View Description Hide DescriptionLow frequency fluctuations in the solar windmagnetic field and plasma velocity are often highly correlated, so much so that the fluctuations may be thought of as originating near the Sun as nearly perfect Alfvén waves. Power spectra of these fluctuations from to several Hz to suggest that the medium is turbulent. Near 1 AU, fluctuations below have a relatively flat slope and contain most of the energy in the fluctuating fields. From to the spectra exhibit a power law inertial range similar to that seen in ordinary fluid turbulence. At the highest frequencies, the rapid falloff of the power suggests that strong dissipation is occurring. From in situ measurements, it is clear that the fluctuations emanate from the solar corona. The turbulent cascade appears to evolve most rapidly in the vicinity of velocity shears and current sheets. Numerical solutions of both the compressible and incompressible equations of magnetohydrodynamics in both Cartesian and spherical geometry corroborate this interpretation. There are conflicting interpretations of observations suggesting that much of the power in magnetic fieldfluctuations resides in quasitwodimensional structures and simulations have helped to elucidate some of these issues.

Evidence for intermittency in Earth’s plasma sheet and implications for selforganized criticality
View Description Hide DescriptionIt has been proposed recently that a description of the magnetosphere as a system in a state of selforganized criticality would be fruitful for understanding (and predicting) the global response to solar wind input. In this paper it is shown that the proposed description fits the characteristics of magnetotailplasma flows and their variability. According to observations, the magnetotail is in a bimodal state: nearly stagnant, except when driven turbulent by transportefficient fast flows. The distributions of flows are in agreement with sporadic (intermittent) variability in the magnetotail. The variability may resemble hydrodynamic turbulence around a jet. The presence of turbulence alters the conductivity and the mass/momentum diffusion properties across the plasma sheet and may permit crossscale coupling of localized jets into a global perturbation. Burstyflowdriven turbulence is a physical process that may have an important role to play in the establishment of a state of selforganized criticality.

A sandpile model with dual scaling regimes for laboratory, space and astrophysical plasmas
View Description Hide DescriptionThere is increasing evidence that the Earth’s magnetosphere, like other macroscopic confined plasma systems (magnetic fusion plasmas, astrophysical accretion discs), displays sandpiletype phenomenology so that energy dissipation is by means of avalanches which do not have an intrinsic scale. This may in turn imply that these systems evolve via selforganized criticality(SOC). For example, the power law dependence of the power spectrum of auroral indices, and in situ magnetic field observations in the Earth’s geotail, indicate that the coupled solar windmagnetospheric system can to some extent be described by an avalanche model. However, substorm statistics exhibit probability distributions with characteristic scales. In this paper a simple sandpile model is discussed which yields for energy discharges due to internal reorganization a probability distribution that is a power law, implying SOC, whereas systemwide discharges (flow of “sand” out of the system) form a distinct group whose probability distribution has a well defined mean. When the model is analyzed over its full dynamic range, two regimes having different inverse power law statistics emerge. These correspond to reconfigurations on two distinct length scales: short length scales sensitive to the discrete nature of the sandpile model, and long length scales up to the system size which correspond to the continuous limit of the model. These are anticipated to correspond regimes accessible to both laboratory and astrophysical plasmas. The relevance of the emergence of distinct selforganized confinement regimes in space, astrophysical, and magnetic fusion plasmas is discussed. Since the energy inflow may be highly variable, the response of the sandpile model is examined under strong or variable loading.

The solarwind driven magnetosphere–ionosphere as a complex dynamical system
View Description Hide DescriptionThe solarwind driven magnetosphere–ionosphere system is a classic example of a complex dynamical system (CDS). The defining properties of a CDS are (1) sensitivity to initial conditions; (2) multiple spacetime scales; (3) bifurcation sequences with hysteresis in transitions between attractors; and (4) noncompositionality. Noncompositionality means that the behavior of the system as a whole is different from the dynamics of its subcomponents taken with passive or no couplings. In particular the dynamics of the geomagnetic tail plasma depends on its coupling to the dissipative ionospheric plasma and on the nature of the solarwind driving electric field over a suitably long (many hours) previous time interval. These complex dynamical system features are shown here in detail using the known WINDMI model for the solarwind driven magnetosphere–ionosphere (MI) system. Numerous features in the bifurcation sequence are identified with known substorm and storm characteristics.

Modeling the spatial structure of the high latitude magnetic perturbations and the related current systems
View Description Hide DescriptionPrevious input–output analysis of the electrojet indices has been generalized to a spatiotemporal dynamical model of the high latitude magnetic perturbation (HLMP) by using the spatially dependent measurements that can be provided by a ground magnetometer array of latitudinal coverage. A technique that utilizes the daily rotation of the Earth as a longitudinal, or local time, sampling mechanism is used to construct a twodimensional representation of the high latitude magnetic perturbations, both in magnetic latitude and local time, from the single latitudinal chain of magnetometers. Twodimensional static, linear dynamical and nonlinear dynamical models for the evolution of the spatial structure of the HLMP are constructed by including a coupling to the solar wind as the energy input. The nonlinear spatial model of the HLMP produces better predictions than the linear one, reducing the average error from 65 to 50 nT for the component. This can be taken as an indication that during intense activity, the HLMP and related current system evolves in a nonlinear fashion.

Collective phenomena in the inner magnetosphere
View Description Hide DescriptionThe Earth’s magnetosphere exhibits substantial complexity in many of its physical properties. Particle populations wax and wane and magnetic fieldsfluctuate on virtually all observed time scales, from less than 1 minute to many days. Much of the variability of the magnetosphere and its interaction with the ionosphere can be ascribed to the phenomena termed “substorms” and “storms.” Ample evidence is found that these geospace disturbances, though exhibiting eventtoevent differences, are remarkably repetitive and have basic underlying similarities. The ring current development, radiation belt particle changes, and basic substorm patterns suggest a high degree of coherence in the phenomena. Observations, modeling, and basic physical properties are discussed here that point to a relatively ordered, lowdimensional underlying dynamics in the magnetosphere. These results suggest that nonlinear processes and couplings introduce much of the observed complexity in magnetospheric particle and field changes.
 Top

 ARTICLES

 Basic Plasma Phenomena, Waves, Instabilities

Plasma–wall interaction in an oblique magnetic field: Model of the spacecharge sheath for large potentials and small Debye lengths
View Description Hide DescriptionAn analytical model of the spacecharge sheath around a planar wall is derived for the case of (i) uniform magnetic fieldB incident at any angle into the wall, (ii) large wall potentials (relative to the plasma temperature): and (iii) Debye length, much smaller than thermal Larmor radius of the attracted species, It is found that, irrespective of the angle of incidence: (i) The potential threshold for a magnetized sheath is (ii) the characteristic magnetic length in the sheath is much larger than and proportional to and the plasma flow into the sheath; (iii) the electric fieldE increases towards the wall and produces a noncycloidal plasma drift that breaks down magnetic insulation. Plasma dynamics in a magnetized sheath consists of an Ealigned region, a drift region, and a Baligned region. The drift region is relevant only for angles far from normal incidence and the density profile presents spatial oscillations there; for grazing incidence, the Baligned region is not found. Different scaling laws of the sheath thickness versus wall potential and incidence angle are obtained; in particular, the thickness of a magnetized sheath at parallel incidence is the local Larmor radius at the wall. The applicability of the model to experiments in the ionosphere is commented.

Two variational principles for incompressible plasmas
View Description Hide DescriptionA canonical Hamiltonian theory is used to investigate the stability of equilibria of incompressible plasmas with nonconstant density. Two different variational approaches are used and it is shown how to derive them systematically with the help of Clebschvariables. Both methods lead to an indefinite second variation of the Lyapunov functional for threedimensional equilibria. It is therefore shown that all threedimensional equilibria of the model are unstable.
 Nonlinear Phenomena, Turbulence, Transport

Theory of Langmuir wave generation in the presence of Alfvén wave turbulence in an electronpositron plasma
View Description Hide DescriptionThe plasmamaserinteraction of the test Langmuir wave in the presence of Alfvén waveturbulence is studied theoretically in a magnetized electron–positron (ep) plasma where the turbulence is produced by a temperature anisotropy. Langmuir waves are found to grow without the necessity of an electron (or positron) beam component. The growth takes place in a direction opposite to that of Alfvén waves.

The density and clustering of magnetic nulls in stochastic magnetic fields
View Description Hide DescriptionMany processes in astrophysical plasmas, such as magnetic reconnection and current sheet formation, are sensitive to the topology of the magnetic field. Magnetic nulls are special in a topological sense because they are connected to an infinite number of field lines. Complicated magnetic field geometries may have many nulls and many null–null lines. This work examines the statistical properties of zeros in a random divergencefree vector fieldmodel. An analytic expression for the density of nulls is obtained as a function of the dissipation and integral length scales of the random field and the exponent of the energy spectrum of the field. For a broad class of spectra which are similar to the spectra of turbulent cascades magnetic nulls form selfsimilar clusters. The fractal dimension of the set of nulls is computed as a function of the power law index of the random field. Complicated magnetic topologies may possess tangled webs of nulls and null–null lines. These webs of nulls may be the sites of current sheet formation and heating, and they may modify the properties of particle transport in turbulent plasma media.
 Magnetically Confined Plasmas, Heating, Confinement

Modeling of the shear effects on the thermal ion transport in advanced tokamak scenarios
View Description Hide DescriptionPlasma shear effects on thermal ion transport in present advanced tokamak scenarios are analyzed. In order to overcome some limitations encountered from previous shear dependent models [e.g., V. Parail et al., Plasma Phys. Controlled Fusion40, 805 (1998), and references therein], a semiempiricalmodel combining the effects of the magnetic and rotation shears is proposed. These shear dependences are based on simple theoretical arguments from turbulence studies. The dominant stabilizing term of our shear correction is the shear in toroidal rotation. The predictive modeling of the formation and evolution of thermal ion internal transport barriers in relevant advanced scenarios of TokamakFusion Test Reactor [E. J. Synakowski et al., Phys. Plasmas 4, 1736 (1997)], Doublet IIID (DIIID) [B. W. Rice et al., Nucl. Fusion36, 1271 (1996)] and Joint European Torus (JET) [F. X. Söldner and the JET Team, Plasma Phys. Controlled Fusion39, B353 (1997)] with our shear correction is presented. A multimachine test of the model performed here provides a basis for predictive simulations of future experiments and for the optimization of advanced scenarios through plasma control.

Density fluctuation phenomena in the scrapeoff layer and edge plasma of the Wendelstein 7AS stellarator
View Description Hide DescriptionThe fluctuation of electron density in the scrapeoff layer (SOL) and edge plasma regions of the Wendelstein 7AS stellarator [H. Renner et al., Plasma Phys Controlled Fusion 31, 1579 (1989)] is investigated by the beam emission spectroscopy technique, observing fluctuations in the line emission intensity of a 50 keV accelerated Li beam. A recently developed numerical technique enables correct reconstruction of correlation functions (and as a consequence, fluctuation amplitudes as well) of the electron density fluctuations from correlations of beam light fluctuations. Depending on the plasma electron density, the radial range of the measurement is either limited to (at the highest densities) or can extend to about at the lowest densities. As the technique is nonperturbing to the plasma, several hundred shots were analyzed and a catalog of the observed phenomena was established. The results always show a change in the properties of the fluctuations at approximately the last closed flux surface. This observation might indicate differences in the turbulence drive in the SOL and the edge plasma. However, one cannot exclude that the same turbulence phenomenon results in different density fluctuations in the two regions due to the different magnetic topologies. The relevance of the observed phenomena to anomalous transport is discussed.

Stability studies and the origin of the mode in the SPHEX spheromak experiment
View Description Hide DescriptionOscillations with toroidal mode number are ubiquitous in helicity injected spheromaks and spherical tokamaks, and play a crucial role in current drive. It has been proposed that these arise from a current driven instability of the open flux tube. Stability calculations are presented to confirm this, and they are compared with experimental data from the Spheromak Experiment (SPHEX) [M. Rusbridge et al., Plasma Phys. Control. Fusion39, 683 (1997)]. The equilibria are modelled as piecewise constant μ profile forcefree plasmas with different values for the μ in the open and closed flux regions. A stability map in space is then calculated. The SPHEX experimental data is also reduced to the same space both as a culmination of direct single point measurements of μ and as a time history of the reconstructed equilibrium from a particular shot. The results show a favorable comparison of the stability map with experiment, both in magnitude and shape. The effect of inserting a central currentcarrying rod on the stability is also discussed.

Collisionless and resistive ballooning stability
View Description Hide DescriptionIt has been suggested [Kleva and Guzdar, Phys. Plasmas 6, 116 (1999)] that reconnectingballooning modes in which electron inertia replaces resistivity in a nonideal magnetohydrodynamic Ohm’s law can have substantial growth rates in the low collisionality regime. Numerical calculation, albeit necessarily at unrealistically large values of the collisionless skin depth, showed that strongly growing ballooning modes exist at beta values which are below the ideal beta limit. In order to investigate stability at more realistic values of the skin depth we exploit an analytic approach. As in the case of resistive ballooning modes, we find that inertial ballooning modes are stabilized by favorable average curvature effects at moderate values of the stability index for resistive ballooning. Instability only becomes possible close to the ideal stability boundary or at unrealistically large values of the toroidal mode number n (e.g., Another ballooning mode, the collisionless analogue of the Carreras–Diamond mode [Carreras, Diamond, Murakami, Dunlap et al., Phys. Rev. Lett. 50, 503 (1983)] can also be excited at larger values of the collisionless skin depth, but this mode is not valid for realistic parameters in a hot plasma.

Ion temperature gradient turbulence simulations and plasma flux surface shape
View Description Hide DescriptionA generalization of the circular local magnetohydrodynamic(MHD) equilibrium model to finite aspect ratio (A), elongation (κ), and triangularity (δ) has been added to a gyrokinetic stability code and our gyrofluid nonlinear ballooning mode code for ion temperature gradient(ITG)turbulence. This allows systematic studies of stability and transport for shaped flux surfaces with the same minor midplane radius label (r), plasma gradients, q, and α while varying A, κ, and δ. It is shown that the (linear, nonlinear, and sheared) terms in the equation of motion are unchanged from a circle at radius r with an effective field where is the toroidal flux, r is the flux surface label, and is the magnetic axis field. This leads to a “natural gyroBohm diffusivity” which at moderate to 3 is weakly dependent on shape (κ) at fixed Since and the label independent at fixed scales as with much weaker scaling at highq and stronger at lowq where increased κ is stabilizing. The generalized critical shear rate to be compared to the maximum linear growth rate is a flux surface quantity

Density measurements in exploding wireinitiated plasmas using tungsten wires
View Description Hide DescriptionCalibrated density measurements have been obtained of the coronal plasmas around exploding 7.5–40 μm Wwires carrying 15–120 kA per wire for 30–70 ns. Xray radiographs of the exploding wireplasmas using 2.5–10 keV photons from a MowireXpinch backlighter enabled measurements of areal densities of W ranging from equivalent to 0.03 μm of solid density W, to about The rapidly expanding (few mm/μs) coronal plasmas surrounding the slowly expanding (<1 mm/μs) residual wire cores have areal densities up to about Single 7.5 μm wires tested with 100 kA had as much as 90% of the initial wire material in the coronal plasma. Coronal plasmaWnumber densities were estimated to be up to a few times while core W densities as low as a few times were observed. With linear arrays of four (eight) 7.5 μm wires carrying 30 kA (15 kA)/wire, up to 35% (25%) of the initial Wwire material was in the plasma around and between the wires at 46–48 ns after the current started. Preheating the wires to drive off adsorbed gases and hydrocarbons increased the W mass in the coronal plasma and made it more uniform then when wires were not preheated.
 Inertially Confined Plasmas, Dense Plasmas, Equations of State

Measurement of the frequency and spectral width of the Langmuir wave spectrum driven by stimulated Raman scattering
View Description Hide DescriptionThomson scattering was used to measure the spectrum of Langmuir waves, in both frequency and wave number, driven below quartercritical density by a laser beam. These measurements were capable of detecting and identifying waves driven by stimulated Raman scattering(SRS) and also of detecting waves driven by other effects such as the bumpontail instability postulated by the enhanced Thomson scattering model of Raman emission. The observed Langmuir waves were consistent with SRS and not with other possible sources. The width in kspace of the measured Thomson scattering signals also has implications for the saturation amplitude of the Langmuir waves.

Physics of onedimensional capsule designs for the National Ignition Facility
View Description Hide DescriptionThis article describes a suite of 250, 280, and 350 eV copperdoped Be [Be(Cu)] capsule designs for the National Ignition Facility [Paisner et al., Laser Focus World 30, 75 (1994)] and compare these to previous Be(Cu) and brominedoped CH plastic [CH(Br)] capsule designs for 300 and 330 eV drives. These capsule designs are constrained to have the same deuteriumtritium (DT) fuel mass as the 300 and 330 eV designs so that differences in yield are due to differences in capsule compression before ignition. The onedimensional (1D) calculations show that the fuel reaches a maximum value when about 20–30 μm of ablator material is left behind, and this amount of ablator material provides the best tradeoff between maximizing the fuel the implosion velocity, and the calculated clean yield. The results of this paper add optimized 1D capsule designs that operate at drive temperatures of 250, 280, and 350 eV and they complement the established 300 eV CH(Br) ablator and the 330 eV Be(Cu) ablator designs.