scholarly journals Linear Force-free Magnetic Fields and Coronal Models

1989 ◽  
Vol 42 (3) ◽  
pp. 317 ◽  
Author(s):  
CJ Durrant
Keyword(s):  
Boundary Condition ◽  
Magnetic Fields ◽  
Helmholtz Equation ◽  
Normal Component ◽  
Coordinate Systems ◽  
Mathematical Properties ◽  
Cylindrical Coordinate ◽  
Linear Force ◽  
Free Fields

The mathematical properties of linear force-free fields generated by the Helmholtz equation are reviewed, and the solutions in terms of spherical, cartesian and cylindrical coordinate systems are discussed. When only the normal component of the field on a single (photospheric) surface is available as a boundary condition, the solutions are not niquely determined. If further conditions are imposed, solutions may be unique or multiple or may not exist. The

Stability of force-free magnetic fields versus magnetic pitch

2002 ◽  
Vol 67 (2-3) ◽  
pp. 139-147
Author(s):  
Y. Q. HU ◽  
L. LI
Keyword(s):  
Magnetic Fields ◽  
Free Field ◽  
Plasma Phys ◽  
One Dimensional ◽  
Nonlinear Force ◽  
Linear Force ◽  
Free Fields ◽  
The Stability ◽  
Lowest Eigenvalue ◽  
Force Free Field

Starting from the one-dimensional energy integral and related stability theorems given by Newcomb [Ann. Phys (NY)10, 232 (1960)] for a linear pinch system, this paper analyses the stability of one-dimensional force-free magnetic fields in cylindrical coordinates (r, θ, z). It is found that the stability of the force-free field is closely related to the radial distribution of the pitch of the field lines: h(r) = 2πrBz/Bθ. The following three types of force-free fields are proved to be unstable: (i) force-free fields with a uniform pitch; (ii) force-free fields with a pitch that increases in magnitude with r in the neighbourhood of r = 0(d[mid ]h[mid ]/dr > 0); and (iii) force-free fields for which (dh/dr)r=0 = 0, Bθ α rm in the neighbourhood of r = 0, and (h d2h/dr2)r=0 > −128π2/(2m+4)2. On the other hand, the stability does not have a definite relation to the maximum of the force-free factor α defined by [dtri ]×B = αB. Examples will be given to illustrate that force-free fields with an infinite force-free factor at the boundary are stable, whereas those with a force-free factor that is finite and smaller than the lowest eigenvalue of linear force-free field solutions in the domain of interest are unstable. The latter disproves the sufficient criterion for stability of nonlinear force-free magnetic fields given by Krüger [J. Plasma Phys.15, 15 (1976)] that a nonlinear force-free field is stable if the maximum absolute value of the force-free factor is smaller than the lowest eigenvalue of linear force-free field solutions in the domain of interest.

scholarly journals Linear Force-free Magnetic Fields and Coronal Models

1990 ◽  
Vol 43 (6) ◽  
pp. 813
Author(s):  
CJ Durrant
Keyword(s):  
Magnetic Fields ◽  
Helmholtz Equation ◽  
Bessel Functions ◽  
Spherical Geometry ◽  
Radius Vector ◽  
Basis Vector ◽  
Radial Dependence ◽  
Unit Radius ◽  
Linear Force

I am grateful to Dr N. Seehafer for drawing attention to the fact that the basis vector for writing linear force-free solutions in spherical geometry is the radius vector r and not the unit radius vector r. In order to correct the results given in Section 3 of Durrant (1989). it suffices to replace the spherical Bessel functions iI. ni, hI by riI. rnI. rhI. The only effect of this replacement is to change the radial dependence of B8 and Bef> in the limit of large r to eiOCr fr. Thus the total magnetic density for the exterior volume r> R is unbounded. This result is now consistent with the fact that the cartesian components of B satisfy the Helmholtz equation.

A METHOD FOR THE TREATMENT OF A SLOPING SEA BOTTOM IN THE PARABOLIC APPROXIMATION

1996 ◽  
Vol 04 (01) ◽  
pp. 89-100 ◽  
Author(s):  
J. S. PAPADAKIS ◽  
B. PELLONI
Keyword(s):  
Boundary Condition ◽  
Helmholtz Equation ◽  
Pressure Field ◽  
Impedance Boundary Condition ◽  
Parabolic Approximation ◽  
Impedance Boundary ◽  
Local Boundary ◽  
Sea Bottom ◽  
Local Boundary Condition ◽  
Non Local

The impedance boundary condition for the parabolic approximation is derived in the case of a sea bottom profile sloping at a constant angle, as a non-local boundary condition imposed exactly at the interface. This condition is integrated into the IFD code for the numerical computation of the pressure field and implemented to test its accuracy in some benchmark cases, for which the backscattered field is negligible. It is shown that by avoiding the sloping interface, the results obtained are closer to the benchmark results given by normal mode codes solving the full Helmholtz equation, such as the 2-way COUPLE code, than those of the standard IFD or other 1-way codes, at least for problems that do not have significant backscattering effects.

A Test on Two Codes for Extrapolating Solar Linear Force-free Magnetic Fields

Solar Physics ◽  
2009 ◽  
Vol 260 (1) ◽  
pp. 109-124 ◽  
Author(s):  
Yiwei Li ◽  
Guoxiang Song ◽  
Junlin Li
Keyword(s):  
Magnetic Fields ◽  
Linear Force

A DISCUSSION OF THE “FAULT” AND “DIKE” PROBLEMS IN MAGNETOTELLURIC THEORY

Geophysics ◽  
1963 ◽  
Vol 28 (3) ◽  
pp. 487-490 ◽  
Author(s):  
J. T. Weaver
Keyword(s):  
Boundary Condition ◽  
Electromagnetic Wave ◽  
Current Density ◽  
Normal Component ◽  
The Earth ◽  
Source Field ◽  
Vertical Fault ◽  
Explicit Mention

In two recent papers appearing in Geophysics, d’Erceville and Kunetz (1962) and Rankin (1962) have dealt with the magnetotelluric theory for a plane earth which contains a certain type of vertical fault. In both cases the results depend on a boundary condition which requires the assumption that the normal component of current density vanishes at the surface of the earth. While d’Erceville and Kunetz confine their attention to the region below the surface and thereby avoid explicit mention of the source field, Rankin follows Cagniard (1953) by considering a plane‐polarized electromagnetic wave normally incident on the surface of the earth. In this case, the assumed boundary condition is not correct, as we shall see later; indeed, it actually leads to a contradiction.

scholarly journals Quantum Vacuum Effects in Braneworlds on AdS Bulk

Universe ◽  
2020 ◽  
Vol 6 (10) ◽  
pp. 181
Author(s):  
Aram A. Saharian
Keyword(s):  
Boundary Condition ◽  
Scalar Field ◽  
Energy Momentum Tensor ◽  
Normal Component ◽  
Momentum Tensor ◽  
Dirac Field ◽  
Curvature Radius ◽  
Perfect Conductor ◽  
Quantum Vacuum ◽  
Local Properties

We review the results of investigations for brane-induced effects on the local properties of quantum vacuum in background of AdS spacetime. Two geometries are considered: a brane parallel to the AdS boundary and a brane intersecting the AdS boundary. For both cases, the contribution in the vacuum expectation value (VEV) of the energy–momentum tensor is separated explicitly and its behavior in various asymptotic regions of the parameters is studied. It is shown that the influence of the gravitational field on the local properties of the quantum vacuum is essential at distance from the brane larger than the AdS curvature radius. In the geometry with a brane parallel to the AdS boundary, the VEV of the energy–momentum tensor is considered for scalar field with the Robin boundary condition, for Dirac field with the bag boundary condition and for the electromagnetic field. In the latter case, two types of boundary conditions are discussed. The first one is a generalization of the perfect conductor boundary condition and the second one corresponds to the confining boundary condition used in QCD for gluons. For the geometry of a brane intersecting the AdS boundary, the case of a scalar field is considered. The corresponding energy–momentum tensor, apart from the diagonal components, has nonzero off-diagonal component. As a consequence of the latter, in addition to the normal component, the Casimir force acquires a component parallel to the brane.

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