Suprathermal ion transport in simple magnetized torus configurations K. Gustafson, P. Ricci, A. Bovet, I. Furno, and A. Fasoli Citation: Phys. Plasmas 19, 062306 (2012); doi: 10.1063/1.4725420 View online: http://dx.doi.org/10.1063/1.4725420 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v19/i6 Published by the American Institute of Physics. Related Articles Application of a three-dimensional model for a study of the energy transfer of a high-pressure mercury horizontal lamp Phys. Plasmas 19, 063504 (2012) Hyper-resistivity and electron thermal conductivity due to destroyed magnetic surfaces in axisymmetric plasma equilibria Phys. Plasmas 19, 062502 (2012) Quasilinear transport modelling at low magnetic shear Phys. Plasmas 19, 062305 (2012) Parallel transport of long mean-free-path plasma along open magnetic field lines: Parallel heat flux Phys. Plasmas 19, 062501 (2012) A coarse-grained kinetic equation for neutral particles in turbulent fusion plasmas Phys. Plasmas 19, 060701 (2012) Additional information on Phys. Plasmas Journal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors
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Suprathermal ion transport in simple magnetized torus configurationsK. Gustafson, P. Ricci, A. Bovet, I. Furno, and A. Fasoli Citation: Phys. Plasmas 19, 062306 (2012); doi: 10.1063/1.4725420 View online: http://dx.doi.org/10.1063/1.4725420 View Table of Contents: http://pop.aip.org/resource/1/PHPAEN/v19/i6 Published by the American Institute of Physics. Related ArticlesApplication of a three-dimensional model for a study of the energy transfer of a high-pressure mercury horizontallamp Phys. Plasmas 19, 063504 (2012) Hyper-resistivity and electron thermal conductivity due to destroyed magnetic surfaces in axisymmetric plasmaequilibria Phys. Plasmas 19, 062502 (2012) Quasilinear transport modelling at low magnetic shear Phys. Plasmas 19, 062305 (2012) Parallel transport of long mean-free-path plasma along open magnetic field lines: Parallel heat flux Phys. Plasmas 19, 062501 (2012) A coarse-grained kinetic equation for neutral particles in turbulent fusion plasmas Phys. Plasmas 19, 060701 (2012) Additional information on Phys. PlasmasJournal Homepage: http://pop.aip.org/ Journal Information: http://pop.aip.org/about/about_the_journal Top downloads: http://pop.aip.org/features/most_downloaded Information for Authors: http://pop.aip.org/authors
Suprathermal ion transport in simple magnetized torus configurations
K. Gustafson,a) P. Ricci, A. Bovet, I. Furno, and A. FasoliEcole Polytechnique Federale de Lausanne (EPFL), Centre de Recherches en Physique des Plasmas,Association Euratom-Confederation Suisse, CH-1015 Lausanne, Switzerland
(Received 20 December 2011; accepted 24 April 2012; published online 14 June 2012)
Inspired by suprathermal ion experiments in the basic plasma experiment TORPEX, the transport of
suprathermal ions in ideal interchange mode turbulence is theoretically examined in the simple
magnetized torus configuration. We follow ion tracer trajectories as specified by ideal interchange
mode turbulence imported from a numerical simulation of drift-reduced Braginskii equations. Using
the variance of displacements, r2ðtÞ � tc, we find that c depends strongly on suprathermal ion
injection energy and the relative magnitude of turbulent fluctuations. The value of c also changes
significantly as a function of time after injection, through three distinguishable phases: ballistic,
interaction, and asymmetric. During the interaction phase, we find the remarkable presence of three
regimes of dispersion: superdiffusive, diffusive, and subdiffusive, depending on the energy of the
suprathermal ions and the amplitude of the turbulent fluctuations. We contrast these results with
those from a “slab” magnetic geometry in which subdiffusion does not occur during the interaction
phase. Initial results from TORPEX are consistent with data from a new synthetic diagnostic used to
interpret our simulation results. The simplicity of the simple magnetized torus makes the present
work of interest to analyses of more complicated contexts ranging from fusion devices to
astrophysics and space plasma physics. [http://dx.doi.org/10.1063/1.4725420]
I. INTRODUCTION
We present a study of suprathermal ion dynamics in the
simple magnetized torus (SMT) configuration,1–3 in which a
vertical magnetic field, Bv, superimposed on a toroidal mag-
netic field, B/, creates helicoidal field lines terminating on the
vessel. In this configuration, turbulence driven by magnetic
curvature and plasma gradients causes the plasma to diffuse
radially, while it is lost to the vessel through parallel flows.
The SMT experimental setup, in which ions are subject to a
non-uniform, curved magnetic field and plasma turbulence, is
an ideal testbed for the study of the interplay of several phe-
nomena affecting suprathermal ion dynamics. Parameter scans
are easier and nonlinear dynamical behavior can be diagnosed
in greater detail when compared to fusion-prototype devices.
Moreover, SMT plasmas are well-understood after being sub-
jected to linear instability analysis4 and global turbulence sim-
ulations,5 making a thorough experimental study possible in
tandem with analytical progress.
Our work is inspired by fast ion experiments in
TORPEX,3 an SMT used for understanding basic plasma tur-
bulence phenomena. Recently, the TORPEX team has been
conducting suprathermal ion experiments6 with an emitter of
energetic lithium ions.7 The ion energy and beam orientation
are tunable. The current from the emitted ion beam is detected
by a double-gridded energy analyzer, which can be moved in
the plane perpendicular to the magnetic field to measure a
spatially resolved profile of the suprathermal ion current
density. This current density profile gives information about
the spreading of suprathermal ions due to the forces applied
by steady-state magnetic fields and fluctuating electric fields.
Expanding on results presented in Ref. 8, the goal of the
present paper is to construct a theoretical framework for
understanding the behavior of suprathermal ions in the SMT,
depending on ion energy and turbulence fluctuation level.
We also present an initial comparison with experimental
data from TORPEX. Due to its relative simplicity, the SMT
successfully disentangles factors that determine suprathermal
ion dispersion. Beyond interpretation of TORPEX mea-
surements, this new framework in the generality of the SMT
configuration is therefore useful to analyses of more compli-
cated configurations, ranging from fusion devices9,10 to
astrophysics11,12 and space plasma physics.13
Our study is based on numerical integration of the
charged-particle equation of motion in the SMT environment
for a range of injection energies and turbulence fluctuation
amplitudes. We consider suprathermal ions as tracer par-
ticles, such that they do not influence background fields.
This is a reasonable approximation in TORPEX because the
density of suprathermal ions is very small compared with the
plasma density. The tracer approach allows more efficient
computations by avoiding the recalculation of Maxwell’s
equations as the tracers propagate. We use SMT turbulence
simulations reported recently14,15 to provide the time-
dependent electric fields required to integrate realistic trajec-
tories with the full Lorentz equation of motion. Our primary
diagnostic tool is the time-dependent variance of suprather-
mal ion displacements, r2ðtÞ � tc, with which we measure
the dispersion of tracer ions.
We focus on fast ion spatial spreading in the direction of
the major radius, eR. We find that a nondiffusive model,
defined by c 6¼ 1, is necessary to describe the dispersion of
ions in our study. Our simulations show that, in fact, supra-
thermal ion dispersion in the SMT begins with a briefa)Electronic mail: [email protected].
Drift-reduced Braginskii equations20,44 are used for
plasma modeling in many contexts (see e.g., Refs. 2, 5, and
45). In the present paper, we consider kk ¼ 0 turbulence,
Ti � Te, and electrostatic fields. We assume Bv � B/ and
constant curvature. Bohm’s boundary conditions are used for
parallel flow at the sheath edge. The Boussinesq approxima-
tion for the polarization drift46 is taken as
r � nmi
eB
drU=B
dt
� �¼ nmi
eB2
d
dtr2U: (A1)
With these assumptions, the equations for the line-integrated
density, n(r,z), potential, Uðr; zÞ and electron temperature
Teðr; zÞ are
@n
@t¼ c
B½U;n�þ 2c
eR0B0
n@Te
@zþTe
@n
@z�en
@U@z
� �þDr2n
�rncs
R0
exp K�eUTe
� �þSn (A2)
062306-10 Gustafson et al. Phys. Plasmas 19, 062306 (2012)
@r2U@t¼ c
B½U;r2U�þ 2B0
cmiR0
Te
n
@n
@zþ@Te
@z
� �
þ�r4UþrcsmiX2i
eR0
1�exp K� eUTe
� �� �(A3)
@Te
@t¼ c
B½U; Te� þ
4c
3eR0B0
7
2Te@Te
@zþ T2
e
n
@n
@z� eTe
@U@z
� �
þ ker2Te �2
3
rTecs
R0
1:71exp K� eUTe
� �� 0:71
� �þ ST : (A4)
Here, Sn and ST are particle and heat sources,
r ¼ R=Lc ¼ D=ð2pLvÞ, and ½a; b� is the Poisson bracket.
Experimental values are used for R0 � 240 qs;D � 35 qs,
r ¼ 0:056, and K ¼ 3, with qs measured at the location of
suprathermal ion injection. Diffusion coefficients are similar
to experimental estimates, with ke ¼ 0:064 m2=s and
� ¼ 0:03 m2=s. The sources are chosen to mimic the electron-
cyclotron and upper-hybrid resonance heating in TORPEX.
The code15 used here to solve Eqs. (A2)–(A4) is based
on a previously developed algorithm.47 Simulations are
started with random noise and the constant plasma sources,
producing a steepening of the gradient, which provokes the
interchange instability and radial plasma transport balanced
by losses to the walls. A quasi-steady state is achieved, and
the data from this state are used as the input for the equation
of motion for tracer particles.
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