Axialsymmetrische Lösungen der magnetohydrostatischen Gleichung mit Oberflächenströmen

1957 ◽  
Vol 12 (10) ◽  
pp. 826-832
Author(s):  
L. Biermann ◽  
K. Hain ◽  
K. Jörgens ◽  
R. Lust
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Axial Symmetry ◽  
Circular Cross Section ◽  
Homogeneous Field ◽  
Electric Currents ◽  
Azimuthal Component ◽  
Axis Of Symmetry ◽  
The Given ◽  
The Magnetic Field

Es werden axialsymmetrische Lösungen der magnetohydrostatischen Gleichung betrachtet, bei denen das Gas einen Torus erfüllt, dessen Querschnitt im allgemeinen kreisförmig angenommen wird. Zur Vereinfachung wird angenommen, daß die elektrischen Ströme nur an der Oberfläche fließen. Für den Fall, daß diese Ströme gegeben sind, wird ein Verfahren angegeben, mit dem das äußere Magnetfeld berechnet werden kann, welches zusammen mit dem durch den gegebenen Strom erzeugten Feld die Grenzbedingungen erfüllt. Dies Verfahren wird für den Fall rein azimutaler Ströme mit einer Näherungsmethode und dann exakt durchgeführt. In erster Näherung genügt die Überlagerung eines homogenen Feldes parallel zur Symmetrieachse des Torus. Für den Fall gegebener meridionaler Ströme zeigt sich, daß die Randbedingungen nur durch Überlagerung eines zusätzlichen azimutalen Stromes erfüllbar sind, zusammen mit einem geeigneten meridionalen äußeren Magnetfeld. Im axialsymmetrischen Fall ist also stets mindestens eine azimutale Stromkomponente erforderlich.It is proposed to consider solutions of the magnetohydrostatic equation of axial symmetry in which the plasma is contained in a torus with circular cross-section. For mathematical simplification it is furthermore assumed, that the electric currents flow exclusively at the surface of the plasma. For the case, that these currents are assumed to be given, a method is outlined by which it is possible to calculate the exterior magnetic field which together with the magnetic field produced by the given currents, satisfies the boundary conditions at the surface. This procedure is carried through for the case of purely azimuthal electric currents, first by an approximate method, then by an exact method. To a first approximation it is sufficient to superpose an homogeneous field parallel to the axis of symmetry. For the case that the given currents are in a meridional planes, it is seen, that the boundary conditions can be satisfied only by the superposition of an additional current together with a suitable meridional exterior magnetic field. Therefore in the case of axial symmetry at least one azimuthal component of the electric current is necessary.

Axialsymmetrische Lösungen der magnetohydrostatischen Gleichung mit Oberflächenströmen II

1958 ◽  
Vol 13 (7) ◽  
pp. 493-498 ◽  
Author(s):  
K. Jörgens
Keyword(s):  
Magnetic Field ◽  
Cross Section ◽  
Circular Cross Section ◽  
Arbitrary Cross Section ◽  
Axially Symmetric ◽  
Equilibrium Configurations ◽  
Circular Case ◽  
Axis Of Symmetry ◽  
Hydrodynamic Equilibrium ◽  
The Magnetic Field

Axially symmetric magneto-hydrodynamic equilibrium configurations are considered where the plasma is contained in a torus of arbitrary cross-section and electric currents flow in the surface of the plasma only. It is shown that the magnetic field is uniquely determined by the values of the total current in azimuthal and in meridional direction respectively and by the gas pressure. A method is given to compute the location of discontinuities of the magnetic field outside of the torus. The singularities are found explicitly in the case of elliptic or circular cross-section. In the circular case only the magnetic field is regular everywhere outside except on the axis of symmetry.

Формирование изображений электрических сигналов преобразователя магнитного поля при гистерезисной интерференции для контроля металлов в импульсных магнитных полях

2020 ◽  
pp. 38-45
Author(s):  
В.В. Павлюченко ◽  
Е.С. Дорошевич
Keyword(s):  
Magnetic Field ◽  
Aluminum Plate ◽  
Thickness Variation ◽  
Magnetic Carrier ◽  
Electric Voltage ◽  
Total Strength ◽  
Magnetic Field Transducer ◽  
The Given ◽  
The Magnetic Field

Based on the developed methods of hysteresis interference, the calculated dependences U(x) of the electric voltage taken from the magnetic field transducer on the x coordinate were obtained. A magnetic carrier with an arctangent characteristic was exposed to a series of bipolar pulses of the magnetic field of a linear inductor of one, two, three, four, five and fifteen pulses. An algorithm is presented for the sequence of changes in the magnitude of the total strength of the magnetic field pulses on the surface of an aluminum plate, which provides the same amplitude of hysteresis oscillations of the electric voltage and makes it possible to obtain a linear difference dependence U(x) for wedge-shaped and flat aluminum samples. The results obtained make it possible to increase the accuracy and efficiency of control of the thickness of the object and its thickness variation in the given directions, as well as the defects of the object.

scholarly journals On the differential system govering flows in magnetic field with data inLp

1998 ◽  
Vol 21 (2) ◽  
pp. 299-305 ◽  
Author(s):  
Fengxin Chen ◽  
Ping Wang ◽  
Chaoshun Qu
Keyword(s):  
Magnetic Field ◽  
Initial Data ◽  
Boundary Conditions ◽  
Differential System ◽  
Existence And Uniqueness ◽  
Mhd Equations ◽  
The Earth ◽  
Initial And Boundary Conditions ◽  
Time Existence ◽  
The Magnetic Field

In this paper we study the system governing flows in the magnetic field within the earth. The system is similar to the magnetohydrodynamic (MHD) equations. For initial data in spaceLp, we obtained the local in time existence and uniqueness ofweak solutions of the system subject to appropriate initial and boundary conditions.

scholarly journals On the self-induction of electric currents in a thin anchor-ring

1912 ◽  
Vol 86 (590) ◽  
pp. 562-571 ◽  
Keyword(s):  
Cross Section ◽  
Circular Cross Section ◽  
The Self ◽  
Electric Currents ◽  
Fourth Term ◽  
Starting Point ◽  
Axis Of Symmetry ◽  
The Difference ◽  
Circular Axis ◽  
Mutual Induction

In their useful compendium of "Formulæ and Tables for the Calculation of Mutual and Self-Inductance," Rosa And Cohen remark upon a small discrepancy in the formulæ given by myself and by M. Wien for the self-induction of a coil of circular cross-section over which the current is uniformly distributed . With omission of n , representative of the number of windings, my formula was L = 4 πa [ log 8 a / ρ - 7/4 + ρ 2 /8 a 2 (log 8 a / ρ + 1/3) ], (1) where ρ is the radius of the section and a that of the circular axis. The first two terms were given long before by Kirchhoff. In place of the fourth term within the bracket, viz., +1/24 ρ 2 / a 2 , Wien found -·0083 ρ 2 / a 2 . In either case a correction would be necessary in practice to take account of the space occupied by the insulation. Without, so far as I see, giving a reason, Rosa and Cohen express a preference for Wien's number. The difference is of no great importance, but I have thought it worth while to repeat the calculation and I obtain the same result as in 1881. A confirmation after 30 years, and without reference to notes, is perhaps almost as good as if it were independent. I propose to exhibit the main steps of the calculation and to make extension to some related problems. The starting point is the expression given by Maxwell for the mutual induction M between two neighbouring co-axial circuits. For the present purpose this requires transformation, so as to express the inductance in terms of the situation of the elementary circuits relatively to the circular axis. In the figure, O is the centre of the circular axis, A the centre of a section B through the axis of symmetry, and the position of any point P of the section is given by polar co-ordinates relatively to A, viz.

Safety Factor for the New Exact Plasma Equilibria

2019 ◽  
Vol 74 (2) ◽  
pp. 163-181 ◽  
Author(s):  
Oleg Bogoyavlenskij
Keyword(s):  
Magnetic Field ◽  
Safety Factor ◽  
Axial Symmetry ◽  
Beltrami Equation ◽  
Exact Formula ◽  
Magnetic Star ◽  
Plasma Velocity ◽  
Plasma Equilibria ◽  
Magnetic Rings ◽  
The Magnetic Field

AbstractAn exact formula for the limit of the safety factor q at a magnetic axis is derived for the general up-down asymmetric plasma equilibria possessing axial symmetry, generalizing Bellan’s formula for the up-down symmetric ones. New exact axisymmetric plasma equilibria depending on arbitrary parameters α, ξ, bkn, zkn, where k = 1, ⋯, M, n = 1⋯, N, are constructed (α ≠ 0 is a scaling parameter), which are up-down asymmetric in general. The equilibria are not force-free if ξ ≠ 0 and satisfy Beltrami equation if ξ = 0. For some values of ξ the magnetic field and electric current fluxes have isolated invariant toroidal magnetic rings, for another ξ they have invariant spheroids (blobs) and for some values of ξ both invariant toroidal rings and spheroids (blobs). A generalization of the Chandrasekhar – Fermi – Prendergast magnetostatic model of a magnetic star is presented where plasma velocity V(x) is non-zero.

Magnetohydrodynamic Equilibria

2020 ◽  
Author(s):  
Thomas Wiegelmann
Keyword(s):  
Magnetic Field ◽  
Boundary Conditions ◽  
Lorentz Force ◽  
Plasma Flow ◽  
Plasma Pressure ◽  
Mhd Equations ◽  
The Sun ◽  
Planetary Magnetospheres ◽  
Mhd Equilibria ◽  
The Magnetic Field

Magnetohydrodynamic equilibria are time-independent solutions of the full magnetohydrodynamic (MHD) equations. An important class are static equilibria without plasma flow. They are described by the magnetohydrostatic equations j×B=∇p+ρ∇Ψ,∇×B=μ0j,∇·B=0. B is the magnetic field, j the electric current density, p the plasma pressure, ρ the mass density, Ψ the gravitational potential, and µ0 the permeability of free space. Under equilibrium conditions, the Lorentz force j×B is compensated by the plasma pressure gradient force and the gravity force. Despite the apparent simplicity of these equations, it is extremely difficult to find exact solutions due to their intrinsic nonlinearity. The problem is greatly simplified for effectively two-dimensional configurations with a translational or axial symmetry. The magnetohydrostatic (MHS) equations can then be transformed into a single nonlinear partial differential equation, the Grad–Shafranov equation. This approach is popular as a first approximation to model, for example, planetary magnetospheres, solar and stellar coronae, and astrophysical and fusion plasmas. For systems without symmetry, one has to solve the full equations in three dimensions, which requires numerically expensive computer programs. Boundary conditions for these systems can often be deduced from measurements. In several astrophysical plasmas (e.g., the solar corona), the magnetic pressure is orders of magnitudes higher than the plasma pressure, which allows a neglect of the plasma pressure in lowest order. If gravity is also negligible, Equation 1 then implies a force-free equilibrium in which the Lorentz force vanishes. Generalizations of MHS equilibria are stationary equilibria including a stationary plasma flow (e.g., stellar winds in astrophysics). It is also possible to compute MHD equilibria in rotating systems (e.g., rotating magnetospheres, rotating stellar coronae) by incorporating the centrifugal force. MHD equilibrium theory is useful for studying physical systems that slowly evolve in time. In this case, while one has an equilibrium at each time step, the configuration changes, often in response to temporal changes of the measured boundary conditions (e.g., the magnetic field of the Sun for modeling the corona) or of external sources (e.g., mass loading in planetary magnetospheres). Finally, MHD equilibria can be used as initial conditions for time-dependent MHD simulations. This article reviews the various analytical solutions and numerical techniques to compute MHD equilibria, as well as applications to the Sun, planetary magnetospheres, space, and laboratory plasmas.

Solitary Alfvén waveguide structures in a magnetized plasma

1980 ◽  
Vol 24 (1) ◽  
pp. 157-162 ◽  
Author(s):  
J. P. Sheerin ◽  
R. S. B. Ong
Keyword(s):  
Magnetic Field ◽  
Solitary Wave ◽  
Field Line ◽  
Magnetic Field Line ◽  
Axial Symmetry ◽  
Wave Structure ◽  
Magnetized Plasma ◽  
Alfven Wave ◽  
Wave Forms ◽  
The Magnetic Field

A nonlinear Alfvén wave structure with axial symmetry about the line of force of an ambient magnetic field is presented. The solitary wave forms a ‘ring’ shaped waveguide along the magnetic field line.

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