Analysis of the eigenspectrum of the MHD force operator
in the SIESTA equilibrium code
S. P.
1
Hirshman ,
2
Cook ,
C. R.
3
Sánchez
R.
1. Oak Ridge National Laboratory, 2. University of Wisconsin, 3. Universidad Carlos III de Madrid
Overview
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Mechanical analogue for eigenproblem
Magnetic confinement and subsequent heating of plasmas is one of the most viable
options for harnessing controlled fusion as an energy source.
Tokamaks and stellarators are the two most promising magnetic confinement devices, and
in order to confine the plasma there must exist a stable global magnetohydrodynamic
(MHD) equilibrium.
The system of MHD equations describes conducting fluids (such as plasmas) in the
presence of magnetic fields. MHD is the “simplest” description of a plasma and is used
to study global stability and equilibrium properties of the system.
SIESTA (Scalable Iterative Equilibrium Solver for Toroidal Applications) is a 3D
nonlinear ideal MHD equilibrium solver capable of resolving islands (broken magnetic
surfaces) and stochastic regions in confinement devices used for the study of fusionrelevant plasmas in an accurate and scalable manner.
SIESTA has been used to demonstrate formation of islands in both tokamak and
stellarator configurations.
Further work is being done to allow SIESTA to analyze more complicated equilibria
containing non-stellarator symmetric islands.
VMEC equilibrium is ideal MHD stable
B
p 
W  

dVol

 20   1
2
The Hessian matrix used in SIESTA is defined
below. The eigenspectrum for this matrix will be
discussed throughout the poster at both VMEC
and SIESTA equilibria.

2
 F
W
H     2


For quadratic energies, the Hessian is
essentially the coefficient of the
second-order term (with a negative
sign due to the conventions used in
the code. For a spring potential, the
Hessian would essentially be the
spring constant.
1 
W
2
k
2
2


W
H    2   kI

This Hessian is negative definite (has only
negative eigenvalues) for a completely stable
equilibrium.
x

k  H

m T
It can easily be seen from the linearized ideal MHD equations that a purely
parallel plasma displacement will result in zero contribution to the linear force.

 


F  J  B0  J 0  B  p



B     //  B0  0


1
J 
  B  0
0



Flux surfaces break and islands form to
find stable SIESTA equilibrium
A normal mode analysis on the equations of
motion gives the following expression:
x~e
 kx   mx
k
 
m
2
it



2
H   T
2
The following equation gives the
MHD eigenfrequncies:


2
det | H   T | 0
Nullspace structure for l=3 stellarator
Real eigenfrequencies correspond to numerically stable
modes while imaginary eigenfrequencies correspond to
numerically unstable modes.
CDXU axisymmetric VMEC equilibrium converged on
SIESTA mesh (101 surfaces)
CDXU SIESTA equilibrium w/ m=1, n=1 island (101
surfaces)
This result can be
confirmed by
studying the ideal
MHD equations.

   0
Summary
p    1 //  p0     p0 //   p0   //  0

 ~e
Leading to the normal mode eigenfrequency:
CDXU SIESTA equilibrium w/ m=1, n=1 island
(51 surfaces)
The normal mode analysis
for ideal MHD proceeds in a
similar fashion

it
x
CDXU axisymmetric VMEC equilibrium converged on
SIESTA mesh (51 surfaces)
Here incompressibility is used, ie


xequil
Numerically, we are worried about the eigenproblem in
 
which the inertia (T) matrix is set to unity:
T

  I
2
In this case the eigenvalue problem is given by:
H   
Solve for eigenfrequencies and eigenmodes.
Note that these are not the MHD normal modes as we have
set the inertia matrix to identity matrix.
The analytical nullspace
A stability analysis was performed on a high beta CDX-U equilibrium that is known
to be Mercier unstable.

The Hessian matrix
Total energy for a stationary plasma (magnetic +
internal)
SIESTA includes resistivity and non-ideal
perturbations to allow for tearing mode
instabilities
High beta CDX-U stability analysis


 


F  J  B0  J 0  B  p  0
These plots of the
nullvector dotted into
the magnetic field
show that a mode that
is parallel (red) or
antiparallel (blue)
everywhere
contributes nothing to
the MHD force
•SIESTA has successfully been used to resolve field-periodic islands in the equilibria of
several configurations, including the DIII-D tokamak and a 3 field-period stellarator.
•Work is underway on the removal of the stellarator symmetry requirement currently
implemented in the code. This will require a complex Fourier representation of field
quantities and will allow analysis of more complicated island structures and equilibria.
References
[1] S. P. Hirshman, R. Sanchez, and C. R. Cook, “SIESTA: a scalable iterative equilibrium solver for toroidal applications," Physics of Plasmas, vol. 18, no. 6, pp. 062504-1062504-13, 2011.
[2] S. P. Hirshman and J. C. Whitson, “Steepest-descent moment method for three-dimensional magnetohydrodynamic equilibria," Physics of Fluids, vol. 26, no. 12, pp. 3553-3568,
1983.
[3] M. D. Kruskal and R. M. Kulsrud, “Equilibrium of a magnetically conned plasma in a toroid," Physics of Fluids, vol. 1, no. 4, pp. 265-274, 1958.
[4] I. B. Bernstein, E. A. Frieman, M. D. Kruskal, and R. M. Kulsrud, “An energy principle for hydromagnetic stability problems," Proceedings of the Royal Society of London Series
A. Mathematical and Physical Sciences, vol. 244, no. 1236, pp. 17-40, 1958.
[5] J. Nocedal and S. J. Wright, Numerical Optimization. Springer, second ed., 2006.
[6] S. P. Hirshman, K. S. Perumalla, V. E. Lynch, and R. Sanchez, “BCYCLIC: a parallel block tridiagonal matrix cyclic solver," Journal of Computational Physics, vol. 229, no. 18,
pp. 6392-6404, 2010.
[7] Lawrence Livermore National Laboratory, VisIt visualization tool, 2011.
[8] A. Sanderson, G. Chen, X. Tricoche, D. Pugmire, S. Kruger, and J. Breslau, “Analysis of recurrent patterns in toroidal magnetic fields," IEEE Transactions on Visualization and
Computer Graphics, vol. 16, no. 6, pp. 1431-1440, 2010.
This research was supported in part by an award from the Department of Energy (DOE) Office of Science Graduate Fellowship Program (DOE SCGF). The DOE SCGF Program was made possible in part by the
American Recovery and Reinvestment Act of 2009. The DOE SCGF program is administered by the Oak Ridge Institute for Science and Education for the DOE. ORISE is managed by Oak Ridge Associated
Universities (ORAU) under DOE contract number DE-AC05-06OR23100. All opinions expressed in this paper are the author's and do not necessarily reflect the policies and views of DOE, ORAU, or ORISE.