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Magnetic field perturbations due to a hole in a conducting wall near a time changingmagnetic fieldJ. W. Rudmin and J. R. Drake Citation: Journal of Applied Physics 48, 1032 (1977); doi: 10.1063/1.323724 View online: http://dx.doi.org/10.1063/1.323724 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/48/3?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of the magnetic lenses on the electron current due to near wall conductivity in a Hall thruster Phys. Plasmas 17, 070701 (2010); 10.1063/1.3469763 Experimental and theoretical study on effects of magnetic field topology on near wall conductivity in a Hallthruster Phys. Plasmas 16, 103504 (2009); 10.1063/1.3250289 Magnetization change due to stress change in a constant magnetic field on amorphous ribbons J. Appl. Phys. 81, 5796 (1997); 10.1063/1.364671 Magnetic field perturbations due to a hole in a conducting wall on the ZT40M reversedfield pinch J. Appl. Phys. 56, 2017 (1984); 10.1063/1.334244 Microwave Magnetic Field near a Conducting Perturbation J. Appl. Phys. 30, 1845 (1959); 10.1063/1.1735072
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Magnetic field perturbations due to a hole in a conducting wall near a time changing magnetic field
J. W. Rudmin* and J. R. Draket
University of Wisconsin, Physics Department, Madison, Wisconsin 53706 (Received 30 September 1976; in final form 10 November 1976)
Plasma confinement devices frequently employ a conducting wall to confine a pulsed magnetic field. Holes in the wall produce perturbations of the uniform field which can affect the plasma equilibrium and field topology. The magnetic field perturbation around such a hole has been calculated using the quasistatic approximation for various cases by analytic and numerical methods, and has been measured for one case experimentally. Simple analytic approximations are presented. The perturbations near a conducting interface are found to be of a much longer range than would be true for fields of the same symmetry in free space.
PACS numbers: 52.55.Ke
I. INTRODUCTION This case would apply to holes in the stainless-steel wall of the Wisconsin Toroidal Quadrupole.
Investigations of the effects of magnetic field perturbations on plasma confinement in axisymmetric poloidal fields have been investigated both theoreticallyl-6 and experimentally3,4,6,7,8 in recent years. The general conclusion of these studies is that field errors can enhance plasma transport across a poloidal field in three ways. First, they remove the azimuthal symmetry so that the canonical angular moment of an ion or electron about
In cylindrical coordinates (r, z, ¢), the vacuum-metal interface is at z = 0 with vacuum for z> 0 or r < a, and the region z < 0, r> a is filled with a material of conductivity a and permeability !lo' Then far from the hole
the axis of symmetry is not conserved. Second, they create magnetic surfaces which cross the poloidal field so that ·adiabatic particles such as slow electrons can reach the wall of the confinement device by flowing along the perturbed field lines. Third, they can create electrostatic potential structures which can Ex B drift plasma across the magnetic field. Even when the field lines are not carried to the wall, plasma equilibrium can be upset by the formation of magnetic islands.
One of the more common kinds of field perturbations occurs when a conducting wall is used to confine a pulsed pulsed magnetic field, and a circular hole is drilled through the wall to permit entry into the vacuum vessel. The perturbation fields far from the hole have been calculated for several cases and are presented here. A time dependence exp(iwt) of the fields is assumed with the quasistatic assumption that w« c/ L, where c is the speed of light and L is the length of any structure in the field.
Figure 1 illustrates the general configuration and the parameters involved in the calculation.
There are, in general, five parameters to be ordered. They are the hole radius a, the wall thickness T, the distance R of the observation pOint from the center of the hole, the skin depth a, and the height z from the surface of the conductor to the observation point. In these caluclations R is assumed large compared to a and a, i. e. , we are interested in the field far from the hole. With this assumption, the field is calculated from some special cases.
II. LAMINAR CURRENT FLOW
The first case calculated will be for laminar current flow, i. e. , J ~ = O. This applies when the skin depth is much greater than the wall thickness: 15= (2/ !law)2» T.
1032 Journal of Applied Physics, Vol. 48, No.3, March 1977
B = yBo exp(iwt) for z> 0,
B=yBoexp(kz+iwt) for z<O,
where
The far -field current denSity for z < 0 is
J a = -xHok exp(kz + iwt),
where
Ha = Bo! !la·
The total current is
(1 )
(2)
(3)
(4)
(5)
(6)
where J 1 is the perturbation current. The conditions on J 1 are that it satisfy
and
V 2J 1 =k2J ll
(J1+JO)'r=0 atr=a,
J1'Z=0 atz=O,
z
(7)
(8)
(9)
(10)
FIG. 1. A hole of radius a in a conducting wall of thickness T perturbing the currents J which are induced by an oscillating magnetic field.
Copyright © 1977 American Institute of Physics 1032
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A laminar flow solution to these conditons is
J l = Hok (tt/y2) exp(kz + iwt)(x cos2<l> + y sin2<l»
for r> a, and
J l = xHrJ? exp(kz + iwt)
for r < a. Now if one defines
R = (y2 + Z2)1/2,
p=r!R,
~=z/R,
and
(11)
(12)
(13)
(14)
(15)
(16)
then to lowest order in a/ R, and to second order in ~, the vector potential was found to be
Ai = - JJ.oHott exp(iwt)[~ [1 - ~~ - ~2(1 - 3~2) J 2R 2
+ (x cos2<l> + y sin2<l>)(1 _ ~)2
(17)
Equation (17) was derived by integrating W 411" fJ / r dV over all space. The integrand was expanded in low-order powers of ~ and~, and logarithmic singularities were removed using integration by parts. Then considering only the zero-order terms in ~, the perturbation magnetic field far from the hole can be written
Bl=VXA1=VX1,
where
a2
sin<l> ( z) Xl = Bo exp(iwt) --v- 1 - (y2 + Z2)1!2 .
(18)
(19)
It is easily seen that this magnetic scalar potential satisfies V2 Xl = 0 for all z > 0, and that tangential Hi changes by the current per unit length as one crosses the surface. It is also interesting to note that Xl is not one of the usual spherical harmonics. In spherical coordinates (R, e, <I», it would be written
sin<l> 1 - cos e Xl- R sine (20)
The function (1- cose)/(sine) does satisfy Lengendre's equation for n = - 1, m = 1, and one could therefore write write
Xl- (sin<l>/R)P~f (cose). (21)
It also appears that the function is not expandable in terms of the usual spherical harmonics, and indeed falls off more slowly with increasing R than any spherical harmonic with a sin<l> dependence could do.
The perturbation due to a hole in a wall has sometimes been approximated as a pair of point dipoles. The justification for this procedure is that the perturbation currents form two adjacent coplanar oppositely flowing current loops and such a current configuration in free space gives a quadrupole field. Therefore it is of interest to compare this result with the magnetic scalar potential
1033 J. Appl. Phys., Vol. 48, No.3, March 1977
of such dipoles. If the dipoles are of strength M and are separated by ~y, the scalar potential is
3M JJ.o~Y rz sin<l> X2DP = 41T(y2 + Z2)5/2
For r«z, z»a,
whereas
Forz«r, r»a,
_ JJ.oHoa2 exp(iwt) sin<l> Xl - 2r '
whereas
(22)
(23)
(24)
(25)
(26)
Note that the field due to the hole is a much longer range field than that of the point dipoles, espechi.lly as one goes along the vacuum-metal interface.
Also note that the magnetic field of the dipoles in normal to the surface at the metal, which is very unlike the physical behavior of a time-changing field, whereas the field of the hole is more properly tangential.
III. NONLAMINAR CURRENT FLOW, HOLE RADIUS MUCH LESS THAN WALL THICKNESS
The previous discussion does not apply to hole such as probe ports on the large Wisconsin Octupole, since the hole radius in this case is much less than the wall thickness. To estimate this effect, the wall is approximated as being perfectly conducting and infinitely thick.
In this case we may write
B=VX
where
X=XO+X2'
Xo = Bor sin<l> exp(iwt) ,
and X2 is the perturbation magnetic scalar potential.
If one defines
a(r, z) = - x/[Bo sin<l> exp(iwt)],
then a must satisfy the following conditions
V 2 a = a/y2
aX I =0 3z ,,_0
r>a
lima=O for z>O, T~~
lima = r for r< a,
and
J.W. Rudmin and J. R. Drake
(27)
(28)
(29)
(30)
(31)
(32)
(33)
(34)
(35)
1033
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FIG. 2. The x component of the perturbation magnetic field near a 15-cm-diam hole as a function of x (cm), for various values of y and z (see Table 1.) Solid curve-measured; dashed-calculated from Eq. (41).
lim ex = O. r-O
(36)
Because of the sharp corner and mixed boundary conditions, an analytic solution was not found. However, a numerical solution was achieved with a digital computer, which in turn led to an analytic expression valid for R> 5a.
The numerical technique was to map all space into a finite region by using the change of variables
r=as/(2 -s) (37)
and
z = a(t -1 )/[t(2 - t)]. (38)
The differential equation in (r, z) was replaced by a finite difference equation in (s, t) and a relaxation algorithm was applied. The final result was
J.loHoa3 exp(iwt)r sin<I>
X2= 6(? + Z2)3 / 2 ' (39)
which is a longer range field than the two point dipoles, shorter range than Xl> and behaves properly near the metal-vacuum interface.
IV. HOLE RADIUS LARGER THAN WALL THICKNESS, NONZERO SKIN DEPTH LESS THAN WALL THICKNESS
This case might apply to a 6-in. port on the large octupole. In this case, the current flow is mainly laminar, since it is restricted to lying within a skin depth of the surface. However the perturbation current divides at the hole, half flowing on the inside surface of the wall and half flowing on the outside surface. Then the perturbation fields are reduced by a factor of 2, since the currents on the bottom surface produce small fields above the conductor and vice versa. Then from Eq. (19) the perturbation magnetic field is
(40)
Figure 2 shows a comparison between the measured perturbation field near an open 6-in. port in the large
1034 J. Appl. Phys., Vol. 48, No.3, March 1977
TABLE 1. Parameters for calculating the curves shown in Fig. 2.
Curve y (cm) z (cm)
A 8.5 5 B 8.5 15 C 16 5 D -8.5 5
octupole,8 and the field calculated from Eq. (40).
In this diagram, x is in the azimuthal direction around the Wisconsin large octupole. The parameters in Eq. (40) are w=1T/86 msec, a=7.6 cm, Rsin<p=y, and R = (x2 + y2 + Z2)1/2, where y and z are given for each curve in Table I. The x component of the perturbation field is shown, from Eq. (40),
Ex a2yx Eo = 2R5 (1. 5z-r).
As can be seen, the agreement is quite good, so Eq. (40) is confirmed to be applicable.
(41)
The difference in the magnetic field scaling between Eqs. (19) and (39) can be attributed to the different paths of the perturbation currents. In the case of Eq. (39), the perturbation current flows across the top of the hole (since the total current there must be zero), and then dives into the hole and returns by flowing in the walls of the hole. For the thin-wall case, Eq. (19), the current density in the sides of the hole would be too great, so the return current must flow in the z = 0 plane, and also in corresponding surface, z = - T, on the other side of the wall. Since the flow is restricted to a skin depth, this flow is laminar, and the scalar potential is thus of the form of Eq. (19).
V. CONCLUSION
Analytic approximations have been presented which give the magnetic fields produced by the induced currents flowing around holes in a conducting slab exposed to a uniform oscillating magnetic field. The approximations were achieved by both numerical and analytic techniques, and the approximation for one case was confirmed by measurements in a plasma confinement device. The perturbation fields were found to be of a much longer range then would be given by multipole expansions in unbounded space. These approximations can be used to estimate the effects of holes in plasma confinement devices upon the magnetic field topology and the plasma equilibrium.
ACKNOWLEDGMENTS
The authors wish to thank D. W. Kerst for support and guidance. The manuscript was written and typed at the University of Maryland using the support of the National Science Foundation. This work was supported by the United States Atomic Energy CommisSion, now the U. S. Energy Research and Development Administration.
*Present address: Physics Department, Florida Atlantic UniverSity, Boca Raton, Fla. 33431.
J.W. Rudmin and J.R. Drake 1034
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tPresent address: Department of Plasma Physics and Nuclear Fusion Research, Royal Institute of Technology, S-100 44 Stockholm, Sweden.
iD.W.Kerst, Plasma Phys. 4, 253 (1962). 2J . H. Williamson, J. Plasma Phys. 5, 229 (1971). 3T. C. Jernigan, Ph. D. thesis (Univ. 'of Wisconsin, 1971) (unpublished) .
4J. Rudmin and D. Meade (unpublished).
1035 J. Appl. Phys., Vol. 48, No.3, March 1977
5T. Ohkawa, Phys. Rev. Lett. 27,177 (1971). BT. Jernigan, J.Rudmin, andD.M. Meade, Phys. Rev. Lett. 26, 1298 (1971).
7Dale M. Meade and Raymond J. Fonck, Phys. Fluids 16, 1654 (1973).
8J.R. Drake and J.D. Greenwood, Bull. Am. Phys. Soc. 19, 876 (1974).
J.W. Rudmin and J.R. Drake 1035
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