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A computer-aided study of Z-pinch plasma transport reveals a high level of structured behavior. The equations asymptote into time and space separable forms, with the profiles of all the variables determined solely by the time dependence of the plasma current. Plasma profiles (normalized with respect to their values on the axis and to the plasma radius) are almost independent of the level of Bremsstrahlung radiation and thus can be determined using the self-similar pattern of a nonradiating plasma. Radiation causes the plasma column to collapse within a finite time and is accompanied by a rapid growth in the density and pressure. The temporal growth of the temperature initially follows that of the current, but if the current crosses the Pease limit, the temperature also rapidly grows.

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... Time–space separable self-similar equilibrium solutions for cylindrically symmetric Z-pinches of magnetized plasmas have been the subject of numerous works due to their possible role as attractors at long times. This tendency finds support in numerical simulations in general, according to which solutions of various initial value problems have settled into the self-similar Z-pinch at long times (see, e.g., Rosenau et al. (1989)). The self-similar solutions were constructed in the context of a model, more realistic than ideal magnetohydrodynamics (MHDs), accounting for effects of resistivity and thermal conductivity. ...

... The radiation effects which break the symmetry that underlies the self-similar solutions are usually neglected in the sub-Peasean limit (see, e.g., Sheehey and Lindemuth (1997) and references therein). Self-similar Z-pinch solutions with non-conserved line density were considered by Rosenau et al. (1989). However, conventionally the line density is assumed to be constant in order to avoid adding or removing mass from the system. ...

... The work by Lampe (1991) is based on the time– space separable self-similar Z-pinch solutions except for the temperature, which is assumed to be radius-independent. This is consistent with numerical simulations which for proper values of the system parameters show that the temperature profile is near-isothermal everywhere across the pinch except in the vicinity of the pinch edge (see, e.g., Rosenau et al. (1989)). In general, numerical simulations are required to determine Z-pinch radial profiles of the self-similar solutions. ...

The dynamics of thermally-isolated Z-pinches carrying power law in time total currents (${\sim} t^S$) in magnetized resistive plasmas is studied. Time–space separable self-similar solutions with cylindrical symmetry are considered. The non-dimensional variables are chosen in a way that makes the problem consistent with the moderately resistive magnetohydrodynamic (MHD) model. For $S\,{=}\,{-}\frac15$ and Lundquist number $Lu\,{>}\, 1.5$ a non-equilibrium solution is obtained in addition to the conventional solutions for either exact, $S\,{=}\,{\pm}\frac13$, or asymptotic, $Lu\,{=}\,\infty$, equilibria (the latter is homogeneously valid for long times only if $S\,{>}\,{-}\frac15$). The problem is treated asymptotically for high dimensionless thermal-conductivity, which is proportional to the square root of the ion/electron mass ratio. To obtain a closure condition for the leading-order isothermal solution, the first-order terms in the energy equation are invoked. Radial profiles are found explicitly which depend on $S$ for equilibrium, and on $Lu$ for non-equilibrium solutions. The multiplicity of the self-similar solutions is investigated.

Deuterium-fiber-initiated Z-pinch experiments have been simulated using a two-dimensional resistive magnetohydrodynamic model, which includes many important experimental details, such as cold-start'' initial conditions, thermal conduction, radiation, actual discharge current versus time, and grids of sufficient size and resolution to allow realistic development of the plasma. When the fiber becomes fully ionized (at a time depending on current ramp and fiber thickness), the simulations show rapidly developing [ital m]=0 instabilities, which originated in the corona surrounding the fiber, drive intense nonuniform heating and rapid expansion of the plasma column. Diagnostics generated from the simulation results, such as shadowgrams and interferograms, are in good agreement with experiment.

We study the nonlinear self-similar evolution of a cylindrical magnetic flux tube with two components of the magnetic field, axial and azimuthal. We restrict ourselves to the case of a plasma of low beta. Introducing a special class of configurations we call 'separable fields', we reduce the problem to an ordinary differential equation. Two cases are to be distinguished: (1) when the total field minimizes on the symmetry axis, the magnetic configuration inexorably collapses, and (2) when, on the other hand, the total field maximizes on the symmetry axis, the magnetic configuration behaves analogously to a nonlinear oscillator. Here we focus on the latter case. The effective potential of the motion contains two terms: a strong repulsive term and a weak restoring term associated with the pinch. We solve the nonlinear differential equation of motion numerically and find that the period of oscillations grows exponentially with the energy of the oscillator. Our treatment emphasizes the role of the force-free configuration as the lowest potential energy state about which the system oscillates.

The spectacular progress made during the last few years in reaching high energy densities in fast implosions of annular current sheaths (fast Z pinches) opens new possibilities for a broad spectrum of experiments, from x-ray generation to controlled thermonuclear fusion and astrophysics. Presently Z pinches are the most intense laboratory X ray sources (1.8 MJ in 5 ns from a volume 2 mm in diameter and 2 cm tall). Powers in excess of 200 TW have been obtained. This warrants summarizing the present knowledge of physics that governs the behavior of radiating current-carrying plasma in fast Z pinches. This survey covers essentially all aspects of the physics of fast Z pinches: initiation, instabilities of the early stage, magnetic Rayleigh-Taylor instability in the implosion phase, formation of a transient quasi-equilibrium near the stagnation point, and rebound. Considerable attention is paid to the analysis of hydrodynamic instabilities governing the implosion symmetry. Possible ways of mitigating these instabilities are discussed. Non-magnetohydrodynamic effects (anomalous resistivity, generation of particle beams, etc.) are summarized. Various applications of fast Z pinches are briefly described. Scaling laws governing development of more powerful Z pinches are presented. The survey contains 36 figures and more than 300 references.

A solvable model is developed for the linearized sausage mode within the contest of resistive MHD. The model is based on the assumption that the fluid motion of the plasma is self-similar, as well as several assumptions pertinent to the long-wavelength limit. The perturbations to the magnetic field are not assumed to be self-similar, but rather are calculated. Effects arising from time dependences of the equilibrium, e.g., current rising as T, alpha ohmic heating, and time variation of the pinch radius, are included in the analysis. The formalism appears to provide a good representation of those modes that involve coherent sausage distortion of the entire cross section of the pinch, but excludes modes that are localized radially, and higher radial eigenmodes. for this and other reasons, it is expected that the model underestimates the maximum instability growth rates, but is reasonable for global sausage modes. The net effect of resistivity and time variation of the equilibrium is to decrease the growth rate if alpha somewhat< 1, but never by more than a fact of about two. The effect is to increase the growth rate if alpha somewhat>1. Keywords: Sausage instability; Magnetohydrodynamic Instability. (JHD)

A model for an expanding magnetic bubble or plasmoid is introduced, corresponding to a large aspect ratio torus, having one-dimensional (cylindrical) symmetry but with three dimensional expansion, with the length of the cylinder expanding in time in the same manner as the radius. This model has a general class of similarity equations in ideal magnetohydrodynamics (MHD) for spherical expansion. There are two parameters c, d characterizing the similarity solutions, depending on boundary conditions and conservation relations. These solutions exhibit either tangential discontinuities or shocks at the boundary, depending on the values of the constants c and d. Some of the solutions have magnetic fluxes within the bubble increasing with time, but with smaller or zero magnetic fields outside the bubble, requiring a shock and a dynamo in the shock region. The results of simulations of one class of solutions with a Lagrangian MHD code show good agreement. Some of the properties of fully toroidal solutions of the similarity equations are derived. This model has applications to a magnetic bubble from an accretion disk around an active galactic nucleus (AGN), appropriate to the phase in which the bubble has expanded to a size much greater than the disk field length scales but much smaller than any exterior scales. At this stage the magnetic reconnection and flux conversion stage associated with setting up the expanding bubble is completed. The model may also apply to a plasmoid formed in the solar corona. © 2004 American Institute of Physics.

This paper explores the electron-electron two-stream stability limit of a virtual cathode in spherical geometry. Previous work using a constant density slab model [
R. A. Nebel and J. M. Finn, Phys. Plasmas 8, 1505 (2001)
] suggested that the electron-electron two-stream would become unstable when the well depth of the virtual cathode was 14% of the applied voltage. However, experimental tests on INS-e have achieved virtual cathode fractional well depths ∼ 60% with no sign of instability. Here, studies with a spherical gridless particle code indicate that fractional well depths greater than 90% can be achieved without two-stream instabilities. Two factors have a major impact on the plasma stability: whether the particles are reflected and the presence of angular momentum. If the particles are reflected then they are guaranteed to be in resonance with the electron plasma frequency at some radius. This can lead to the two stream instabilities if the angular momentum is small. If the angular momentum is large enough it stabilizes the instability much the same way as finite temperature stabilizes the two-stream instability in a slab.

The authors study the full system of Braginskii one-fluid transport equations with a self-consistent account of transport and dissipative phenomena in the screw-pinch geometry and derive exact self-similar solutions. The local and global plasma parameters-magnetic field, temperature and density profiles, magnetic Reynolds number, beta and pinch radius-are determined by a minimal set of external parameters, namely properly scaled magnitude of the applied axial field and an index related to the current time-evolution, and are computed as eigenfunctions and eigenvalues of the corresponding boundary value problem. The paramagnetic and diamagnetic properties of the axial field profiles are discussed in detail. Calculation of the m=1 growth rates demonstrates a relatively weak decrease of the maximum growth rate as the axial field amplitude increases although the range of unstable wavenumbers becomes substantially narrower. The solutions obtained describe plasma dynamics and profile structure in 'stabilized' pinch systems: Z-pinches with externally applied axial field and in ULQ pinches provided self-relaxation processes are suppressed.

The Z-pinch, perhaps the oldest subject in plasma physics, has achieved a remarkable renaissance in recent years, following a few decades of neglect due to its basically unstable MHD character. Using wire arrays, a significant transition at high wire number led to a great improvement in both compression and uniformity of the Z-pinch. Resulting from this the Z-accelerator at Sandia at 20 MA in 100 ns has produced a powerful, short pulse, soft x-ray source >230 TW for 4.5 ns) at a high efficiency of ~15%. This has applications to inertial confinement fusion. Several hohlraum designs have been tested. The vacuum hohlraum has demonstrated the control of symmetry of irradiation on a capsule, while the dynamic hohlraum at a higher radiation temperature of 230 eV has compressed a capsule from 2 mm to 0.8 mm diameter with a neutron yield >3 × 1011 thermal DD neutrons, a record for any capsule implosion. World record ion temperatures of >200 keV have recently been measured in a stainless-steel plasma designed for Kα emission at stagnation, due, it was predicted, to ion-viscous heating associated with the dissipation of fast-growing short wavelength nonlinear MHD instabilities. Direct fusion experiments using deuterium gas-puffs have yielded 3.9 × 1013 neutrons with only 5% asymmetry, suggesting for the first time a mainly thermal source. The physics of wire-array implosions is a dominant theme. It is concerned with the transformation of wires to liquid-vapour expanding cores; then the generation of a surrounding plasma corona which carries most of the current, with inward flowing low magnetic Reynolds number jets correlated with axial instabilities on each wire; later an almost constant velocity, snowplough-like implosion occurs during which gaps appear in the cores, leading to stagnation on the axis, and the production of the main soft-x-ray pulse. These studies have been pursued also with smaller facilities in other laboratories around the world. At Imperial College, conical and radial wire arrays have led to highly collimated tungsten plasma jets with a Mach number of >20, allowing laboratory astrophysics experiments to be undertaken. These highlights will be underpinned in this review with the basic physics of Z-pinches including stability, kinetic effects, and finally its applications.

New self-similar time-dependent z-pinch equilibria, valid for an unmagnetized plasma, are described. They assume a current rising as I alpha t13/, and thus exactly satisfy pressure balance. It is shown that these equilibria have a maximum radius, and that the equilibrium profiles differ considerably from well known magnetized plasma self-similar solutions. One-dimensional simulations show that, although the latter act as attractors the existence of two distinct classes of self-similar equilibria ensures, firstly, that convergence to the magnetized plasma self-similar state is not perfect, and, secondly, that the transition to the magnetized regime is accompanied by an increase in pinch radius. Variations of the value of the Coulomb logarithm also give rise to a slow expansion of the pinch.

Self-similar solutions are derived for a subsonic compression and expansion of dense Z-pinches taking into account all the essential dissipative processes for magnetized and nonmagnetized plasmas. The solutions reveal the radial profiles of MHD variables in the pinch dynamics. The minimum radius of a stationary deuterium pinch is obtained.

We study the magnetohydrodynamic response of a plasma in the low solar atmosphere to a changing current system of a flaring
magnetictube, which contains a beam of fast non-thermal electrons. The local disturbances of a current system of a magnetic
tube when the beam is injected into it are estimated using the classical idea of a return current. According to this idea,
after injection of abeam, the total current density in a magnetic tube, which includes as well the current density of the
beam jb, should not change compared to the current density in the tube before the injection. In order to keep constant the total
current density j=j'+ jb, the current density of the magnetic tube j'in fact changes. This change is due to a return current jr.c.= –jb, which compensates the current density of the injected beam of fast electrons. At the same time, any changes of the current
density in the magnetic tube change the Joule heating and disturb the thermodynamic equilibrium of the system. Changing of
the plasma temperature destroys also the force balance and starts the process of complex dynamics of the whole plasma-magnetic
structure. The impulsive character of a beam injection causes two stages in the dynamic behavior of the tube. During the first
stage, characterized by the presence of a beam, the preliminary equilibrium state of a magnetic tube is disturbed and complex
dynamics of the plasma start in the region of the beam propagation. During the second stage, when the injection of the beam
is already over, the plasma and magnetic field continue to evolve from the disturbed state and gradually relax to an equilibrium
state. Various types of magnetic tube response onto injection of a beam of energetic electrons are studied using the dynamic
models of the magnetic tube (Khodachenko, 1996a; 1996b) built on the basis of known self-similar solutions of plasma MHD.
The model results are applied to the interpretation of observed flaring and burst phenomena.

This paper presents the results of computations of the behavior of the fiber-initiated high density Z-pinch (HDZP). Its purpose is twofold. One is to study the behavior of the physical system itself as an interesting controlled fusion experiment. The main result of this study is a demonstration of the relaxation of the full inertial behavior of the pinch to simplified self-similar behavior in which the forces on the system are in near balance. The moving free boundary and violent initial behavior of this configuration require careful treatment. This leads to the other purpose of the work, to use this realistic physical system as a test-bed for a general-purpose 1-dimensional code based on moving finite elements. A key step in accomplishing this goal has been the recognition that numerical stability of the discretized equations has required the use of nonconservative quantities as the fundamental dependent variables to be discretized. The main result of this work is a code which is capable of treating a very general class of nonlinear, time-dependent fluid equations.

Sarcopenia, the age related decline in skeletal muscle mass has dramatic consequences. It leads to impaired performance, increased vulnerability, frailty and an increased risk of falls. Various extrinsic and intrinsic factors contribute to the aetiology of sarcopenia. The aims of the present study was to analyse gender differences in the prevalence of sarcopenia and document gender differences in lean soft tissue mass in healthy elderly. 139 healthy subjects ageing between 59 and 92 years (x = 71.5 +/- 7.8), 77 females and 64 males, were enrolled in the study. Body composition was measured by means of dual energy X-ray absorptiometry. Additionally appendicular muscle mass (ASM) was calculated. While no linear decrease in lean soft tissue mass was found for both sexes, the prevalence of sarcopenia increased significantly with increasing age in females as well as in males. Significant gender differences in the prevalence of sarcopenia were found for people younger than 70 years and those older than 80 years. In the youngest age group (< 70 years) sarcopenia was found more frequently among women, while in the oldest age group (> 80 years) the opposite was true. It can be concluded that the prevalence of sarcopenia differs between the two genders however these differences are influenced by age.

A new type of self-similar solution of ideal magnetohydrodynamics (MHD) in the nonlinear stage of the undular model (k parallel to B) of the magnetic buoyancy instability (the ballooning instability in fusion plasma physics or the Parker instability in astrophysics) is found through MHD simulation and theory. The linear theory developed agrees well with the simulation in the early (linear) stage. The nonlinear stages of the instability in the simulation show the self-similar evolution. One of the solutions obtained from the nonlinear analysis has the characteristics of nonlinear instability in Lagrangian coordinates; the fluid velocity and the Alfven speed on each magnetic loop increase exponentially with time, because the loop is evacuated by the field-aligned motion of matter resulting from gravitational acceleration. In the later stage of the nonlinear evolution, the solution property changes from exponential to power-law time dependence. The latter corresponds to a force-free expansion solution. The later saturation of the velocity increment is also discussed.

Age is a major determinant of osteoporosis, but the elderly are rarely assessed and often remain untreated for this condition. Falls, co-morbidities and co-medications compound the risk of fracture in senile osteoporosis. The prevalence of osteoporosis is expected to increase with increasing life expectancy, and the associated fractures - particularly hip fractures - will lead to significant demands on health resources. Treatment of senile osteoporosis can include pharmacological and non-pharmacological intervention. Calcium and vitamin D dietary supplementation is a relatively low-cost way of reducing the risk of fracture. Pharmacological interventions with risedronate, zoledronic acid, or teriparatide have been shown to reduce vertebral fracture risk in osteoporosis patients over the age of 75. Zoledronic acid has been shown to reduce fracture risk in frail patients with recent hip fracture. In the oldest old (patients over 80), strontium ranelate is the first agent with documented anti-fracture efficacy for both non-vertebral and vertebral fracture and documented sustained efficacy over 5 years. Falls prevention is an essential component of any strategy for decreasing fracture risk in old age. Currently, senile osteoporosis is under-diagnosed and under-treated, but age should not be a barrier to intervention.

The linear m=0 stability of the z pinch in the collisionless, large ion Larmor radius regime is examined using the Vlasov fluid model. The results reveal a strong equilibrium dependence. The uniform current density equilibrium shows a reduction in growth rate when the average ion Larmor radius is about one-fifth of the pinch radius. However, finite Larmor radius effects cannot in themselves produce a stabilized z pinch.

The equations governing the diffusion and transport of a fully collisional plasma across a strong magnetic field in a bounded domain are analyzed. Following a relatively short relaxation time, the diffusion exhibits universal properties independent of the choice of initial data. Mathematically this appears as a time asymptotic solution which is space-time separable. The temporal decay rate is a nonlinear eigenvalue which is found via the solution of a related eigenvalue problem. This determines the spatial distribution of both the particle density and the pressure. Some of the transport effects caused by Bremsstrahlung radiation, particle, and heat injection are considered, and the conditions under which the system evolves into an equilibrium are examined.

sing a simple model of a slowly diffusing plasma across a strong magnetic field, it is demonstrated that plasma mass and energy evolves from an initially given density and temperature distribution into isothermal state with a self-similar diffusion profile that depends only on its initial mass and energy.

A study of nonlinear plasma mass and energy diffusion reveals that the diffusion coefficients uniquely determine one of two diffusion patterns. After a short initial transient time, either an organized universal pattern, described by a time-space separable solution, is formed or transport is inhibited, allowing only a partial organization of diffusion into a universal pattern. Consequently, in the second case, unlike the first, the asymptotic shape of the solution will depend to some extent on its initial state.

The transport problem for tokamak plasmas, and particularly for noncircular cross section tokamaks, is complicated and can generally only be solved numerically. It is shown that under special circumstances time and space coordinates can be separated so that the transport problem is significantly simplified. These circumstances are close to those desired in experiments and the necessary assumptions regarding transport coefficients are in accordance with the best present day knowledge, namely, neo- or pseudo-classical-like heat transport and Spitzer-like resistivity. The time behavior of the solutions is particularly simple; the plasma current is proportional to t1/3 (where t is time) and the plasma temperature is proportional to t2/3

A new analytic, self-similar solution of the equations of ideal magnetohydrodynamics describes cylindrically symmetric plasmas conducting constant current. The solution indicates that an adiabatic Z pinch oscillates radially with a period typically of the order of a few acoustic transit times. A stability analysis, which shows the growth rate of the sausage instability to be a saturating function of wavenumber, suggests that the oscillations are observable.

Sausage and kink growth rates for a Z pinch are computed from ideal magnetohydrodynamic theory for an infinitely thin surface current sheath and for a surface current layer of finite width. The growth rate decreases with increasing layer width. Satisfactory agreement with experiment is obtained for reasonable width estimates based on magnetic field diffusion.

We have performed one-dimensional magnetohydrodynamic computer calculations of the formation and evolution of the solid deuterium fiber Z pinch. With use of tabulated atomic data base and ``cold start'' initial conditions, our computations show that current is carried by hot plasma which has been ablated from the solid fiber. The computations suggest that the experimentally observed instability growth may coincide with the complete ablation of the central fiber.