Computation of Three Dimensional Tokamak and Spherical Torus Equilibria Page: 4 of 14
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Laplace's equation, V2X 0, has a unique answer that
is regular in region bounded by the external currents if
the normal component of the external magnetic field is
specified on the plasma boundary.
If the perturbed equilibrium is not supported by an ex-
ternal magnetic field, the plasma inertia associated with
an eigenfrequency w27- 26W2/ f dx3pl ,2 must supply
the required force. The DCON is used for finding per-
turbed equilibria because it minimizes SW, which gives
perturbed equilibria, rather than w1. For stable modes, a
minimization of w, is in effect a maximization of the iner-
tia, so the resulting displacements 2, n do not represent
perturbed equilibria.
Although imagining the plasma is surrounded by a
perfectly-conducting shell provides a compelling basis for
defining the external magnetic field associated with each
plasma perturbation, calculations are easier using a con-
trol surface that is located just outside the unperturbed
plasma surface, and this is the method used in IPEC.
A plasma displacement determines a magnetic perturba-
tion b V x ( x B), so IPEC uses the displacement of
the plasma boundary "n to determine a part of the per-
turbed magnetic field that is normal to the unperturbed
plasma boundary, b - n, and a part that is tangential to
the plasma boundary, n x b(P). Since the normal field
b - n is continuous across the plasma boundary and the
control surface, V - n then gives a unique vacuum field
outside the plasma, b( ), that vanishes at infinity. The
difference between the tangential field infinitesimally out-
side the control surface n x b( 0) and the tangential field
on the plasma side of the control surface n x b(P) deter-
mines an external surface current on the control surface,
po~x n x b(--) - n x b(P). Once Kx is known, the
externally produced normal magnetic field b' - n can be
found by V x b' poj? in vacuum.
Each of the M neighboring equilibria calculated by
DCON has a unique distribution of the external normal
magnetic field 6, - n, where 1 < i < M, that must be
produced by currents outside the plasma to sustain that
equilibrium. If an external magnetic perturbation, such
as that due to a magnetic field error b' - n, is specified on
the unperturbed plasma boundary, this perturbation can
be expanded as h n z c1b - n, with expansion co-
efficients c,. If this is done, the plasma displacement that
gives the perturbed equilibrium produced by the field er-
ror is (p, 0, <o) EM 1 cf(, 0, <p). This is the method
used by IPEC to find the perturbed equilibrium associ-
ated with a given magnetic field error. More information
about numerical implementation is provided in Sec. II
and theoretical consideration for the numerical result is
given in Sec. III.B. Information from a perturbed equilibrium
The Ideal Perturbed Equilibrium Code (IPEC) cou-
ples the DCON ideal MHD stability code with a routine
for relating a specific plasma displacement to a given
externally produced magnetic field b' and gives all infor-
mation that can be derived from an ideal MHD equilib-
rium. With magnetic surfaces Yo(p, 0, <o) in the unper-
turbed equilibrium, the perturbed equilibrium has sur-
faces ( , 0, <o) ro(t, 0, o) + ((, 0, so) - n, where nL is
the normal to the unperturbed magnetic surfaces.
Given the externally produced magnetic field on the
plasma boundary, particular things that IPEC can cal-
culate are: (1) The jumps in the perturbed magnetic field
tangential to the magnetic surfaces, or equivalently the
parallel current that is localized near the rational sur-
faces, which is required in ideal MHD to preserve the
magnetic surfaces. (2) The magnitude of the magnetic
perturbation throughout the plasma volume, 1b1 where
b = V x ( x B), or the variation of the magnetic field
strength within the magnetic surfaces.
The parallel current in the vicinity of the q = m/n ra-
tional surface is measured by the jump in the perturbed
magnetic field tangential to the magnetic surfaces, or
equivalently [11]AmnE [a6.v inn
a b - V@ Co(1)
where only the resonant component m and n is consid-
ered in calculation of the jump [- - -]. When a pressure
gradient exists at the rational surface q m/n, the inter-
pretation of the jump is subtle due to the large Pfirsch-
Schluter, j 11 /B, current that arises from B- V(j /B)
(B x Vp) - V(1/B2), near the rational surface. This
phenomenon is known as the Glasser effect [12, 13]. If
the Eq. (1) is interpreted as the jump across a fraction
of the radial coordinate Slp/p , then for cases we have
investigated, the current given by Amn is well behaved
if one calculates the current flowing in a narrow channel
but no narrower than approximately 10-3 of the plasma
radius a width comparable to the narrowest width at
which MHD could be a valid model, the gyroradius of
the ions or the electrons. If one ignores the region of
validity of MHD and calculates the current flowing in
narrower channels, interesting variations occur at 10-5
of the plasma radius.
The magnitude of Amn for a given external magnetic
field is a measure of how wide the magnetic island would
be if this localized current were to dissipate. If this lo-
calized current is not dissipated, then the island does not
open, but energy must come out of the plasma to the
maintain the current in the presence of resistivity. In an
MHD model, the flow velocity of the plasma dotted into a
force between the plasma and the magnetic perturbation
must balance the yj2 dissipation. Even when an island
does not open, there can be a significant drag between
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Jong-kyu Park, Allen H. Boozer, and Alan H. Glasser. Computation of Three Dimensional Tokamak and Spherical Torus Equilibria, report, May 7, 2007; Princeton, New Jersey. (https://digital.library.unt.edu/ark:/67531/metadc933722/m1/4/: accessed April 25, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.