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An international journal of news from the stellarator community
Editor: James A. Rome Issue 160 February 2018
E-Mail: [email protected] Phone: +1 (865) 482-5643
On the Web at https://stelnews.info
Quasiaxisymmetry near
axisymmetry
The theoretical concept of quasisymmetry [1] explains
how a stellarator can overcome the lack of a continuous
symmetry in its magnetic field to achieve good confinement
of collisionless particle orbits: a quasisymmetric
equilibrium instead satisfies a hidden symmetry in special
magnetic (“Boozer”) coordinates. However, with some
notable exceptions, most work on quasisymmetric stellarators
has been based on numerical search algorithms, which
find these special equilibria by sifting through parameter
space. This approach, although the foundation of modern
stellarator optimization, might be improved by advancing
our basic understanding of the special set of stellarator
equilibria that have good particle confinement – more generally
referred to as “omnigenous” equilibria.
Garren and Boozer [2] performed an asymptotic expansion
of the equations of magnetostatic equilibria near the magnetic
axis, i.e., a large aspect ratio expansion. Based on
this expansion, they argued that quasisymmetry cannot be
satisfied globally and ultimately breaks down as the distance
from the magnetic axis increases. As an alternative
to satisfying quasisymmetry near the axis, they argued that
there should also be sufficient freedom to satisfy the condition
exactly on a single magnetic surface, at a finite distance
from the axis. However, several open problems
remained, such as the proof of the existence of such solutions,
how numerous they are, and how to construct them.
For practical purposes (e.g., to satisfy engineering constraints),
and also to satisfy sheer curiosity, it would also
be interesting to know of any geometric consequences of
quasisymmetry – are there any basic limits to what sort of
shapes are possible? The above problems also apply to the
generalization of quasisymmetry (omnigeneity).
In our recent paper [3] we propose a new approach to generate
quasiaxisymmetric (QAS) equilibria, a subclass of
quasisymmetry [4], by deforming axisymmetric equilibria.
We demonstrate the existence of a family of solutions near
an arbitrarily specified axisymmetric zeroth-order solution
that satisfy QAS on a single flux surface. Wes also prove
that QAS cannot be satisfied globally (when axisymmetry
is violated), supporting the conclusion of Ref. 2. The perturbed
solutions can be found numerically by applying the
condition of QAS as a (non-standard) boundary condition.
Importantly, the solutions are valid globally (even though
QAS is satisfied only on the outer surface), distinguishing
them from solutions that only satisfy MHD locally [5, 6].
The more general class of omnigenous solutions is also
shown to exist. Key parts of this work are summarized
below.
In this issue . . .
Quasiaxisymmetry near axisymmetry
A new approach is proposed to generate optimized
stellarator equilibria by deforming axisymmetric equilibria.
A large set of solutions that satisfy quasiaxisymmetry
(QAS) on a single magnetic flux surface
is identified. Solutions are independently verified with
a widely used MHD equilibrium solver, thereby
demonstrating that QAS solutions can be approximately
directly constructed, i.e., without using numerical
search algorithms. ............................................ 1
Fig. 1: Flux surface shape induced by quasiaxisymmetry-
preserving perturbation (assuming circular zerothorder
shape).
Stellarator News -2- February 2018
The first step is to reformulate the MHD equations in
terms of a coordinate mapping x, which is taken to be a
function of the flux , and the Boozer angles and . This
“inverse” problem formulation allows the condition of
QAS, namely that the field magnitude is independent of
the Boozer toroidal angle , to be directly enforced. The
coordinate mapping may be expressed in cylindrical coordinates
as
, (1)
where we note that the component in the -direction is
included for mathematical convenience. In terms of x the
magnetostatic equilibrium equation in a vacuum can be
written simply as
. (2)
The solution is expanded as ,
where x0 denotes the zeroth-order axisymmetric part, and
Eq. (2) is solved, order by order. Due to symmetry, the
toroidal angle is ignorable in the first-order equations, and
the mode number N is introduced as a free parameter. With
some manipulation of Eq. (2), the problem can then be
transformed into a remarkably simple form, a single second-
order elliptic partial differential equation for
P1R0Z0, the first-order part of the toroidal component
:
(3)
We note that the equation is differential in the zeroth-order
cylindrical coordinates R0 and Z0, and define the Grad-
Shafranov operator in these coordinates,
. (4)
The solution of Eq. 3 determines the entire magnetic field
solution at first order.The QAS condition (B0) at this
order, can be written as simply
(5)
Note that the equations are to be solved in the R0-Z0 plane
within a domain defined by the zeroth-order axisymmetric
solution. This means that, although the ultimate shape of
the outer surface (i.e., including the deformation) is a priori
unknown, the problem is still of the fixed boundary
type. A series of results follow rather directly from
Eqs. (3) and (5). First, it is found that the equations cannot
be both satisfied across the entire volume (R0-Z0) plane,
except for trivial solutions that preserve axisymmetry.
This fact is shown by solving Eq. (5), a first order ODE,
and substituting the result into Eq. (3). Second, it is noted
that Eq. (5) when applied on the boundary of the domain,
is an oblique-derivative boundary condition; this condition
was first studied by Poincaré [7]. The existence of solutions
in the two-dimensional case was proved in the 1950s
[8], and the results are summarized in [9]. For the particular
case here, two linearly independent solutions are guaranteed
to exist.
We can thus conclude that the space of weakly nonaxisymmetric
QAS equilibria can be parameterized by (1)
the axisymmetric surface shape (2D function), (2) the
toroidal mode number N (to be interpreted as the fieldperiod
number), and (3) two complex numbers corresponding
to the solution space of the oblique-derivative
problem. To independently confirm the theoretical results,
these numerical solutions are used as input for the VMEC
equilibrium solver [10], coupled to the BOOZ_XFORM
code [11] which recovers the Boozer-coordinate representation.
For the purposes of this test, we assume N 2 and
take the zeroth-order flux surface to have a circular shape,
and an aspect ratio of 4. The strength of the perturbation is
controlled by the free parameter , and as an example, the
resulting flux surface shape for the case considered is
depicted in Fig. 1. The satisfaction of QAS is thereby confirmed
at the appropriate order; see Fig. 2.
x Rˆ = R + zˆZ + ˆ
x
2
------
+ x G
x
= x
x = x0 + x1 + 2x2 +
N 2 – 1P1 R0 2 0
= *P1.
0
* R0 R0
1
R0
------
R0
2
Z0 2
---------
·
= +
N 2 – 1P1 R0R0
P1 + = 0.
Fig. 2. Test of QAS at first order in size of non-axisymmetric
perturbation; note the measure
,
where is the Fourier amplitude of at the outer flux
surface. Solutions of the problem (labeled QAS) are compared
with a “control” deformation (nQAS), yielding 2 and
scaling, respectively, as expected; see Ref. [3] for further
details.
0.02 0.05 0.10 0.20 0.50
10- 4
0.001
0.010
0.100
Q
|Bˆ mn|2
m n 0
1 2
|Bˆ mn|2
m n
1 2
= ---------------------------------------------------
B ˆ
Stellarator News -3- February 2018
Omnigeneity, the general condition that collisionless particle
orbits are (radially) confined, is also considered as a
boundary condition for the deformation. Omnigeneity can
be restated geometrically as the condition that the distance
along a field line from two points of equal magnetic field
strength does not vary between neighboring field lines
[12]. The existence of such solutions, satisfying omnigeneity
on one surface, is also guaranteed for the associated
oblique derivative problem [9]. Finally, we note that the
QAS perturbations represent a freedom in the omnigenous
problem, as they can be superposed onto the omnigenous
solution without affecting the satisfaction of the condition
of omnigeneity. These encouraging results demonstrate
that it is possible to numerically construct omnigenous
solutions, using an approach similar to that used for QAS
ones, though this is left for future work.
References
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toroidal stellarators. Phys. Lett. A 129 (2):113 – 117,
1988. ISSN 0375-9601.
[2] D. A. Garren and A. H. Boozer. Existence of quasihelically
symmetric stellarators. Phys. Fluids B 3
(10):2822–2834, 1991.
[3] G. G. Plunk and P. Helander. Quasiaxisymmetric equilibria:
weakly nonaxisymmetric case in a vacuum.
ArXiv e-prints; submitted to J. Plasma Phys., January
2018.
[4] J. Nührenberg, E. Sindoni, W. Lotz, F. Troyon, S. Gori,
and J. Vaclavik. Quasi-axisymmetric tokamaks. In
Proc. Joint Varenna-Lausanne International Workshop
on Theory of Fusion Plasmas, pp 3–12, 1994.
[5] C. C. Hegna. Local three-dimensional magnetostatic
equilibria. Phys. Plasmas 7 (10):3921–3928, 2000.
[6] Allen H. Boozer. Local equilibrium of nonrotating plasmas.
Phys. Plasmas, 9 (9):3762–3766, 2002.
[7] Jules Henri Poincaré, Leçons de Mécanique Céleste,
Tome III, Théorie de Marees. Cours de la faculté des
sciences de Paris. Gauthier-Villars, Paris, 1910.
[8] I. N. Vekua. A boundary problem with oblique derivative
for an equation of elliptic type. Dokl. Akad. Nauk
SSSR (NS), 92, 1953.
[9] Carlo Miranda. Partial Differential Equations of Elliptic
Type. Springer-Verlag New York Inc., 1970. ISBN
978-3-642- 87775-9.
[10] S. P. Hirshman and J. C. Whitson. Steepest-descent moment
method for three-dimensional magnetohydrodynamic
equilibria. Phys. Fluids 26 (12):3553–3568,
1983.
[11] R. Sanchez, S. P. Hirshman, A. S. Ware, L. A. Berry,
and D. A. Spong. Ballooning stability optimization of
low-aspect-ratio stellarators. Plasma Phys. Control. Fusion
42 (6):641, 2000.
[12] John R. Cary and Svetlana G. Shasharina. Omnigenity
and quasihelicity in helical plasma confinement systems.
Phys. Plasmas 4 (9):3323–3333, 1997, http://
dx.doi.org/10.1063/1.872473
Gabriel G. Plunk and Per Helander
Max-Planck Institut für Plasmaphyisk
17491 Greifswald, Germany
E-mail: [email protected]