Effect of plasma flow on the equilibrium of an axisymmetric toroidal magnetic trap

1997 ◽  
Vol 58 (3) ◽  
pp. 421-432 ◽  
Author(s):  
ZH. N. ANDRUSHCHENKO ◽  
O. K. CHEREMNYKH ◽  
J. W. EDENSTRASSER
Keyword(s):  
Plasma Flow ◽  
Magnetic Trap ◽  
Plasma Equilibrium ◽  
Plasma Rotation ◽  
Stationary Plasma ◽  
Shafranov Equation ◽  
Nonlinear Vector ◽  
Analytical Expressions ◽  
Poloidal Rotation ◽  
Shafranov Shift

The effect of finite plasma rotation on the equilibrium of an axisymmetric toroidal magnetic trap is investigated. The nonlinear vector equations describing the equilibrium of a highly conducting, current-carrying plasma are reduced to a set of scalar partial differential equations. Based on Shafranov's well-known tokamak model, this set of equations is employed for the description of a kinetic (stationary) plasma equilibrium. Analytical expressions for the Shafranov shift Δ are found for the case of finite plasma rotation, where two regions of possible plasma equilibria are found corresponding to sub- and super-Alfvénic poloidal rotation. The shift Δ itself, however, turns out to depend essentially on the toroidal rotation only. It is shown that in the case of a stationary plasma flow, the solution of the Grad–Shafranov equation is at the same time also the solution of the stationary Strauss equation.

scholarly journals Nonlinear translational symmetric equilibria relevant to the L–H transition

2012 ◽  
Vol 79 (3) ◽  
pp. 257-265 ◽  
Author(s):  
Ap. KUIROUKIDIS ◽  
G. N. THROUMOULOPOULOS
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Plasma Flow ◽  
Sufficient Condition ◽  
Sheared Flow ◽  
Equilibrium Configurations ◽  
Shafranov Equation ◽  
The Impact ◽  
The Magnetic Field

AbstractNonlinear z-independent solutions to a generalized Grad–Shafranov equation (GSE) with up to quartic flux terms in the free functions and incompressible plasma flow non-parallel to the magnetic field are constructed quasi-analytically. Through an ansatz, the GSE is transformed to a set of three ordinary differential equations and a constraint for three functions of the coordinate x, in Cartesian coordinates (x,y), which then are solved numerically. Equilibrium configurations for certain values of the integration constants are displayed. Examination of their characteristics in connection with the impact of nonlinearity and sheared flow indicates that these equilibria are consistent with the L–H transition phenomenology. For flows parallel to the magnetic field, one equilibrium corresponding to the H state is potentially stable in the sense that a sufficient condition for linear stability is satisfied in an appreciable part of the plasma while another solution corresponding to the L state does not satisfy the condition. The results indicate that the sheared flow in conjunction with the equilibrium nonlinearity plays a stabilizing role.

Studying the Blood Plasma Flow past a Red Blood Cell with the Mathematical Method of Kelvin's Transformation

2014 ◽  
Vol 2 (1) ◽  
pp. 57-66
Author(s):  
Maria Hadjinicolaou ◽  
Eleftherios Protopapas
Keyword(s):  
Blood Cell ◽  
Blood Plasma ◽  
Stream Function ◽  
Plasma Flow ◽  
Red Blood Cell ◽  
Fluid Velocity ◽  
Prolate Spheroid ◽  
Mathematical Tool ◽  
Flow Past ◽  
Analytical Expressions

A mathematical tool, namely the Kelvin transformation, has been employed in order to derive analytical expressions for important hydrodynamic quantities, aiming to the understanding and to the study of the blood plasma flow past a Red Blood Cell (RBC). These quantities are the fluid velocity, the drag force exerted on a cell and the drag coefficient. They are obtained by employing the stream function ? which describes the Stokes flow past a fixed cell. The RBC, being a biconcave disk, has been modelled as an inverted prolate spheroid. The stream function is given as a series expansion in terms of Gegenbauer functions, which converge fast. Therefore we employ only the first term of the series in order to derive simple and ready to use analytical expressions. These expressions are important in medicine, for studying, for example the transportation of oxygen, or the drug delivery to solid tumors.

scholarly journals Modelling an isolated dust grain in a plasma using matched asymptotic expansions

2007 ◽  
Vol 73 (5) ◽  
pp. 793-810 ◽  
Author(s):  
N. ARINAMINPATHY ◽  
J. E. ALLEN ◽  
J. R. OCKENDON
Keyword(s):  
Asymptotic Expansions ◽  
Dusty Plasmas ◽  
Matched Asymptotic Expansions ◽  
Number Density ◽  
Plasma Parameters ◽  
Stationary Plasma ◽  
Dust Grain ◽  
Analytical Expressions ◽  
Single Set ◽  
Dust Plasma

AbstractThe study of dusty plasmas is of significant practical use and scientific interest. A characteristic feature of dust grains in a plasma is that they are typically smaller than the electron Debye distance, a property which we exploit using the technique of matched asymptotic expansions. We first consider the case of a spherical dust particle in a stationary plasma, employing the Allen–Boyd–Reynolds theory, which assumes cold, collisionless ions. We derive analytical expressions for the electric potential, the ion number density and ion velocity. This requires only one computation that is not specific to a single set of dust–plasma parameters, and sheds new light on the shielding distance of a dust grain. The extension of this calculation to the case of uniform ion streaming past the dust grain, a scenario of interest in many dusty plasmas, is less straightforward. For streaming below a certain threshold we again establish asymptotic solutions but above the streaming threshold there appears to be a fundamental change in the behaviour of the system.

Demonstration of Shafranov Shift by the Simplest Grad–Shafranov Equation Solution in IR-T1 Tokamak

2009 ◽  
Vol 29 (1) ◽  
pp. 73-75 ◽  
Author(s):  
A. Rahimirad ◽  
M. Emami ◽  
M. Ghoranneviss ◽  
A. Salar Elahi
Keyword(s):  
Equation Solution ◽  
Shafranov Equation ◽  
Shafranov Shift

On the accuracy of the symmetric ergodic magnetic limiter map in tokamaks

2010 ◽  
Vol 76 (5) ◽  
pp. 777-794 ◽  
Author(s):  
A. R. SOHRABI ◽  
S. M. JAZAYERI ◽  
M. MOLLABASHI
Keyword(s):  
Delta Function ◽  
Mapping Technique ◽  
Error Criterion ◽  
Flux Function ◽  
Energy Error ◽  
Chaotic Field ◽  
Shafranov Equation ◽  
Field Lines ◽  
Poloidal Flux ◽  
Shafranov Shift

AbstractA new symmetric symplectic map for an ergodic magnetic limiter (EML) is proposed. A rigorous mapping technique based on the Hamilton–Jacobi equation is used for its derivation. The system is composed of the equilibrium field, which is fully integrable, and a Hamiltonian perturbation. The equilibrium poloidal flux function is a solution of the Grad–Schlüter–Shafranov equation. This equation is written in polar toroidal coordinate in order to take into account the outward Shafranov shift. The static perturbation field breaks the exact axisymmetry of the equilibrium field and creates a region of chaotic field lines near the plasma edge. The new symmetric EML map is compared with the conventional (asymmetric) EML map which is derived by applying delta-function method. The accuracy of the maps is considered through mean energy error criterion and maximal Lyapunov exponents. For asymmetric and symmetric maps the approximate location of the main cantorus near the edge of plasma is determined with high accuracy by using mean energy error. The forward–backward error criterion is applied to show the relation between the accuracy of the symmetric EML map and the number of EML rings. We also report on the effect of the number of EML rings on the maximal Lyapunov exponent of the symmetric EML map.

Microscopic origins and macroscopic uses of plasma rotation

1995 ◽  
Vol 54 (1) ◽  
pp. 1-10 ◽  
Author(s):  
David Montgomery ◽  
Xiaowen Shan
Keyword(s):  
Kinetic Theory ◽  
Plasma Rotation ◽  
Poloidal Rotation

We consider the microscopic sources of toroidal and poloidal flow in confined magnetoplasmas and the effects of such flows on macroscopic stability. The central difficulty is satisfactory modelling of kinetic theory effects so as to permit their robust introduction into magnetohydrodynamic (MHD) codes. It is tentatively concluded that poloidal rotation, but not toroidal, is responsible for the computed stabilization and return to axisymmetry.

Diffusive time evolution of the Grad–Shafranov equation for a toroidal plasma

2021 ◽  
Vol 87 (3) ◽  
Author(s):  
Giovanni Montani ◽  
Matteo Del Prete ◽  
Nakia Carlevaro ◽  
Francesco Cianfrani
Keyword(s):  
Time Scale ◽  
Diffusion Time ◽  
Test Facility ◽  
Plasma Equilibrium ◽  
Toroidal Plasma ◽  
Divertor Tokamak ◽  
Analytical Properties ◽  
Shafranov Equation ◽  
Consistent Solution ◽  
Diffusive Time

We describe the evolution of a plasma equilibrium having a toroidal topology in the presence of constant electric resistivity. After outlining the main analytical properties of the solution, we illustrate its physical implications by reproducing the essential features of a scenario for the upcoming Italian experiment Divertor Tokamak Test Facility, with a good degree of accuracy. Although we find the resistive diffusion time scale to be of the order of $10^4$ s, we observe a macroscopic change in the plasma volume on a time scale of $10^2$ s, comparable to the foreseen duration of the plasma discharge by design. In the final part of the work, we compare our self-consistent solution to the more common Solov'ev one, and to a family of nonlinear configurations.

scholarly journals A mechanism of spark motion in inner acceleration region to investigate subpulse drifting in pulsars

2020 ◽  
Vol 496 (1) ◽  
pp. 465-482 ◽  
Author(s):  
Rahul Basu ◽  
Dipanjan Mitra ◽  
George I Melikidze
Keyword(s):  
Magnetic Fields ◽  
Radio Emission ◽  
Plasma Flow ◽  
Measurement Errors ◽  
Single Pulse ◽  
Phase Behaviour ◽  
Acceleration Region ◽  
Stationary Plasma ◽  
Different Types ◽  
Pulse Behaviour

ABSTRACT Coherent radio emission in pulsars is excited due to instabilities in a relativistically streaming non-stationary plasma flow, which is generated from sparking discharges in the inner acceleration region (IAR) near the stellar surface. A number of detailed works have shown the IAR to be a partially screened gap (PSG) dominated by non-dipolar magnetic fields with continuous outflow of ions from the surface. The phenomenon of subpulse drifting is expected to originate due to variable $\boldsymbol {E}\times \boldsymbol {B}$ drift of the sparks in PSG, where the sparks lag behind corotation velocity of the pulsar. Detailed observations show a wide variety of subpulse drifting behaviour where subpulses in different components of the profile have different phase trajectories. But the drifting periodicity is seen to be constant, within measurement errors, across all components of the profile. Using the concept of sparks lagging behind corotation speed in PSG as well as the different orientations of the surface non-dipolar magnetic fields, we have simulated the expected single pulse behaviour in a representative sample of pulsars. Our results show that the different types of drifting phase behaviour can be reproduced using these simple assumptions of spark dynamics in a non-dipolar IAR.

Sign in / Sign up

Export Citation Format

Share Document