scholarly journals Green function for the Grad-Shafranov operator

2022 ◽  
Vol 19 (1 Jan-Jun) ◽  
Author(s):  
Julio Herrera Velázquez
Keyword(s):  
Magnetic Field ◽  
Green Function ◽  
Vector Potential ◽  
Elliptic Partial Differential Equation ◽  
Two Dimensions ◽  
Current Loop ◽  
Partial Differential ◽  
Azimuthal Component ◽  
Flux Function ◽  
The Magnetic Field

The Grad-Shafranov equation, often written in cylindrical coordinates, is an elliptic partial differential equation in two dimensions. It describes magnetohydrodynamic equilibria in axisymmetric toroidal plasmas, such as tokamaks, and yields the poloidal magnetic flux function, which is related to the azimuthal component of the vector potential for the magnetic field produced by a circular (toroidal) current density. The Green function for the differential operator can be obtained from the vector potential for the magnetic field of a circular current loop, which is a typical problem in magnetostatics. The purpose of the paper is to collect results scattered in electrodynamics and plasma physics textbooks for the benefit of students in the field, as well as attracting the attention of a wider audience, in the context of electrodynamics and partial differential equations.

Magnetic Field Imaging on Stacked Flash Memories by Laser Probing and SQUID Scanning

2017 ◽  
Author(s):  
K. Sanchez ◽  
G. Bascoul ◽  
F. Infante ◽  
N. Courjault ◽  
T. Nakamura
Keyword(s):  
Magnetic Field ◽  
Failure Analysis ◽  
Two Dimensions ◽  
Analysis Tool ◽  
Active Device ◽  
Current Path ◽  
3D Packaging ◽  
Magnetic Field Imaging ◽  
The Magnetic Field ◽  
Field Imaging

Abstract Magnetic field imaging is a well-known technique which gives the possibility to study the internal activity of electronic components in a contactless and non-invasive way. Additional data processing can convert the magnetic field image into a current path and give the possibility to identify current flow anomalies in electronic devices. This technique can be applied at board level or device level and is particularly suitable for the failure analysis of complex packages (stacked device & 3D packaging). This approach can be combined with thermal imaging, X-ray observation and other failure analysis tool. This paper will present two different techniques which give the possibility to measure the magnetic field in two dimensions over an active device. Same device and same level of current is used for the two techniques to give the possibility to compare the performance.

scholarly journals Ideal magnetohydrodynamic equilibria with helical symmetry and incompressible flows

1999 ◽  
Vol 62 (4) ◽  
pp. 449-459 ◽  
Author(s):  
G. N. THROUMOULOPOULOS ◽  
H. TASSO
Keyword(s):  
Electric Fields ◽  
Incompressible Flows ◽  
Elliptic Partial Differential Equation ◽  
Equilibrium Equations ◽  
Flux Function ◽  
Parallel Flows ◽  
Magnetic Surfaces ◽  
Ideal Magnetohydrodynamic ◽  
The Magnetic Field ◽  
Constant Mach Number

A recent study on axisymmetric ideal magnetohydrodynamic equilibria with incompressible flows [H. Tasso and G. N. Throumoulopoulos, Phys. Plasmas5, 2378 (1998)] is extended to the generic case of helically symmetric equilibria with incompressible flows. It is shown that the equilibrium states of the system under consideration are governed by an elliptic partial differential equation for the helical magnetic flux function containing five surface quantities along with a relation for the pressure. The above-mentioned equation can be transformed to one possessing a differential part identical in form to the corresponding static equilibrium equation, which is amenable to several classes of analytical solutions. In particular, equilibria with electric fields perpendicular to the magnetic surfaces and non-constant-Mach-number flows are constructed. Unlike the case in axisymmetric equilibria with isothermal magnetic surfaces, helically symmetric T = T(ψ) equilibria are overdetermined, i.e. in this case the equilibrium equations reduce to a set of eight ordinary differential equations with seven surface quantities. In addition, the non-existence is proved of incompressible helically symmetric equilibria with (a) purely helical flows and (b) non-parallel flows with isothermal magnetic surfaces and with the magnetic field modulus a surface quantity (omnigenous equilibria).

Static magnetic fields in vacuum

2018 ◽  
Author(s):  
J. Pierrus
Keyword(s):  
Magnetic Field ◽  
Problem Solving ◽  
Magnetic Fields ◽  
Vector Potential ◽  
Scalar Potential ◽  
Multipole Expansion ◽  
Magnetic Vector Potential ◽  
Magnetic Dipoles ◽  
Static Magnetic Fields ◽  
The Magnetic Field

Wherever possible, an attempt has been made to structure this chapter along similar lines to Chapter 2 (its electrostatic counterpart). Maxwell’s magnetostatic equations are derived from Ampere’s experimental law of force. These results, along with the Biot–Savart law, are then used to determine the magnetic field B arising from various stationary current distributions. The magnetic vector potential A emerges naturally during our discussion, and it features prominently in questions throughout the remainder of this book. Also mentioned is the magnetic scalar potential. Although of lesser theoretical significance than the vector potential, the magnetic scalar potential can sometimes be an effective problem-solving device. Some examples of this are provided. This chapter concludes by making a multipole expansion of A and introducing the magnetic multipole moments of a bounded distribution of stationary currents. Several applications involving magnetic dipoles and magnetic quadrupoles are given.

A Proposed Experiment of Direct Detecting of the Vector Potential within Classical Electrodynamics

1997 ◽  
Vol 11 (12) ◽  
pp. 531-540
Author(s):  
V. Onoochin
Keyword(s):  
Magnetic Field ◽  
Electromagnetic Field ◽  
Charged Particle ◽  
Vector Potential ◽  
Energy Exchange ◽  
Short Pulse ◽  
Classical Electrodynamics ◽  
Direct Influence ◽  
Ultra Short Pulse ◽  
The Magnetic Field

An experiment within the framework of classical electrodynamics is proposed, to demonstrate Boyer's suggestion of a change in the velocity of a charged particle as it passes close to a solenoid. The moving charge is replaced by an ultra-short pulse (USP), whose characteristics should depend on the current in the coil. This dependence results from the exchange of energy between the electromagnetic field of the pulse and the magnetic field within the solenoid. This energy exchange could only be explained, by assuming that the vector potential of the solenoid has a direct influence on the pulse.

Compressible magnetoconvection in three dimensions: planforms and nonlinear behaviour

1995 ◽  
Vol 305 ◽  
pp. 281-305 ◽  
Author(s):  
P. C. Matthews ◽  
M. R. E. Proctor ◽  
N. O. Weiss
Keyword(s):  
Magnetic Field ◽  
Shear Flow ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Nonlinear Behaviour ◽  
Two Dimensional ◽  
Horizontal Shear ◽  
Mean Flow ◽  
The Magnetic Field

Convection in a compressible fiuid with an imposed vertical magnetic field is studied numerically in a three-dimensional Cartesian geometry with periodic lateral boundary conditions. Attention is restricted to the mildly nonlinear regime, with parameters chosen first so that convection at onset is steady, and then so that it is oscillatory.Steady convection occurs in the form of two-dimensional rolls when the magnetic field is weak. These rolls can become unstable to a mean horizontal shear flow, which in two dimensions leads to a pulsating wave in which the direction of the mean flow reverses. In three dimensions a new pattern is found in which the alignment of the rolls and the shear flow alternates.If the magnetic field is sufficiently strong, squares or hexagons are stable at the onset of convection. Both the squares and the hexagons have an asymmetrical topology, with upflow in plumes and downflow in sheets. For the squares this involves a resonance between rolls aligned with the box and rolls aligned digonally to the box. The preference for three-dimensional flow when the field is strong is a consequence of the compressibility of the layer- for Boussinesq magnetoconvection rolls are always preferred over squares at onset.In the regime where convection is oscillatory, the preferred planform for moderate fields is found to be alternating rolls - standing waves in both horizontal directions which are out of phase. For stronger fields, both alternating rolls and two-dimensional travelling rolls are stable. As the amplitude of convection is increased, either by dcereasing the magnetic field strength or by increasing the temperature contrast, the regular planform structure seen at onset is soon destroyed by secondary instabilities.

Small-scale magnetic fields in turbulence: Saffman's approximation revisited

1980 ◽  
Vol 99 (3) ◽  
pp. 481-493
Author(s):  
Ralph Baierlein
Keyword(s):  
Magnetic Field ◽  
Numerical Evaluation ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Small Scale ◽  
Fluid Element ◽  
Relative Time ◽  
Fourier Decomposition ◽  
The Subject ◽  
The Magnetic Field

The subject is the small-scale structure of a magnetic field in a turbulent conducting fluid, ‘small scale’ meaning lengths much smaller than the characteristic dissipative length of the turbulence. Philip Saffman developed an approximation to describe this structure and its evolution in time. Its usefulness invites a closer examination of the approximation itself and an attempt to place sharper limits on the numerical parameters that appear in the approximate correlation functions, topics to which the present paper is addressed.A Lagrangian approach is taken, wherein one makes a Fourier decomposition of the magnetic field in a neighbourhood that follows a fluid element. If one construes the viscous-convective range narrowly, by ignoring magnetic dissipation entirely, then results for a magnetic field in two dimensions are consistent with Saffman's approximation, but in three dimensions no steady state could be found. Thus, in three dimensions, turbulent amplification seems to be more effective than Saffman's approximation implies. The cause seems to be a matter of geometry, not of correlation times or relative time scales.Strictly-outward spectral transfer is a characteristic of Saffman's approximation, and this may be an accurate description only when dissipation suppresses the contributions from inwardly directed spectral transfer. In the spectral region where dominance passes from convection to dissipation, one can generate expressions for the parameters that arise in Saffman's approximation. Their numerical evaluation by computer simulation may enable one to sharpen the limits that Saffman had already set for those parameters.

A formulation for computation of a class of collision-free plasmas in two dimensions

1982 ◽  
Vol 28 (1) ◽  
pp. 141-147 ◽  
Author(s):  
J. W. Dungey
Keyword(s):  
Magnetic Field ◽  
Dispersion Equation ◽  
Two Dimensions ◽  
Higher Moments ◽  
Time Step ◽  
Pressure Anisotropy ◽  
The Magnetic Field

The restrictions imposed are that the magnetic field is everywhere in the x direction, and that no quantity varies with x, but several interesting instabilities can still occur. After some discussion of objectives, a fluid-like formulation is pursued, in which the pressure anisotropy is retained, but higher moments neglected. It shows a resonance at twice the gyrofrequency, and for electrons the constraint on the time step would be unacceptable, so they should be treated more crudely. Then the dispersion equation shows only two modes, which appear sufficiently harmless for us to proceed to computations.

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