Transport and Evolution of Ion Gyro-scale Plasma Blobs in Perpendicular Magnetic Fields
P.W. Gingell1 , S.C. Chapman1 , R.O. Dendy1,2
1. University of Warwick, Coventry, CV4 7AL, UK
2. Euratom/CCFE Fusion Association, Culham Science Centre, Abingdon, OX14 3DB, UK
Localised enhancements and depressions of plasma density and magnetic field strength (“blobs”) are common in the edge region of tokamaks. The existing fusion plasma literature on this subject focuses primarily on large blobs for which a fluid description is appropriate. However, the populations of blobs in the edge region may include some
whose characteristic lengthscales are on the order of the ion Larmor
radius. The dynamics and evolution of such blobs are unlikely to be
wholly capturable by fluid models, inviting the use of a hybrid model
for which we treat ions kinetically as particles and electrons as a massless fluid. Here, therefore, we report hybrid simulations of ion gyroscale blobs with parameters characteristic of medium-size tokamaks
such as MAST. We find that ion gyro-scale blobs are advected with the
background flow, and develop a twin-celled vortex structure. Asymmetry then arises from finite ion Larmor radius kinetics, manifesting in the
size of the internal vortices, the shape of tails forming from blob ejecta,
and the growth of a Kelvin-Helmholtz instability.
The outer regions of tokamak plasmas, near the last closed flux surface, are observed to generate coherent propagating blobs [1]. These
undergo radial transport, and can break free of the confinement region.
Blobs with sufficient radial velocity can strike the walls of the tokamak,
increasing local particle and energy fluxes, reducing divertor efficiency,
and increasing impurity levels. Polarising curvature drifts of ions and
electrons,
giving rise to20 an electric field that 40
may not be shorted out by
10
parallel motion under tokamak edge conditions, give rise in the clas5
10
20
sic picture [2] to E × B radial blob motion. Models for blob trans0
0
port
and evolution have
principally focused0 on fluid and multi-fluid
descriptions,
approached
both analytically and
−5
−10
−20 numerically; see, for example, Refs. [3, 4, 5]. For a recent review of edge plasma physics in
−10
−20
−40
tokamaks
[6]. 0 10 20
−10
−5
0 we5refer
10 to Ref. −10
−40 −20
0
20
40
Blobs are set up as flux ropes
with Gaussian spatial profiles for
enhancements of particle number
density, and depressions of magnetic field strength out of the simulation plane, Bz . The magnitudes
of the Gaussian enchancements and
depressions are chosen such that
blobs are set up in pressure equilibrium with the background flowing Magnetic field lines (black) and initial denplasma, balancing combined gas sity profile (colour) for a 10ρi blob.
and magnetic pressures.
The background plasma parameters are chosen to be approximately
characteristic of edge conditions in a medium-size tokamak such as
MAST:
10
0
10
x/ρi
30
0
20
x/ρi
40
60
Results
−5
0
−10
0
−20
Notable features of the evolution include:
−10
−20
10
15
20
x/ρi
−40
25
10
20
30
40
0
20
40
x/ρi
60
25
30
x/ρi
35
20
20
30
40
x/ρi
50
60
20
40
0
0
−10
−20
−10
−20
−40
35
40
x/ρi
45
50
80
30
40
50
x/ρi
60
70
z/ρi
5
100
4
50
ni/m−3
0
3
50
50
40
2
40
30
30
20
10
1
20
u
10
y/ρ
x/ρ
i
i
Number density, n0 = 1019 m−3
Magnetic fields strength, Bz,0 = 0.4T
Temperature, T0 = 4 × 106 K
Thermal gyro-radius, ρi = 8 × 10−3 m
Background flow speed, u0 = 0.2vA
20
40
60
x/ρi
80
Since an ion traverses much of the blob as it gyrates in the magnetic
field, we expect gyromotion to play a significant role in the dynamics
and evolution of both ion populations. The chirality of ion gyration can
thus break the symmetry of the evolution of an otherwise symmetric
blob.
Substituting the electric field derived from the electron momentum
equation into an ion momentum equation, we arrive at the following
!
[7]:
Dvα
1X
= enα vα −
nα0 v α0 × B
(3)
mα nα
Dt
n 0
α
nα
+
(J × B − ∇pe ) − ∇pα
n
The first term on the RHS represents an asymmetry due to the finite Larmor radius, while the following terms are usual for MHD. This asymmetry leads to momentum transfer between populations of ions with
different velocities, in our case the background flow and blob ions. The
form of this asymmetry is as follows:
0
−5
30
40
60
x/ρi
20
7.8072tΩ
10
7.3192tΩ
5
5.3674tΩ
5.8554tΩ
x/ρi
•10 Structures and gradients
forming
on the ion
20
40 gyro-scale,
•5 Advection of the blow
in the flow direction,
10
20
• Formation of twin-cell internal vortices,
0
0
0
• Growth of the K-H instability in a tail formed from blob ejecta,
−20
•−5 Asymmetry in all of−10the above, which becomes
more pronounced for
−10 the smallest blobs. −20
−40
20
150
Finite Larmor Radius Symmetry Breaking
20
3.9036tΩ
0
100
Number density for 1, 4 and 10ρi blobs (left to right) after approximately 5 gyroperiods.
0
From the electron fluid momentum equation,
20
40
0
20
n/n0
40
60
0
n/n0
50
n/n0
100
Kelvin-Helmholtz Instability
340
−10
320
−5
−0.6
0
5
−0.8
−1
0
15
20
25
−5
−3
10
260
−1
250
10
80
100
120
140
x/ρi
160
180
Number density plot showing K-H
instability growing more quickly on
the lower edge of a 10ρi blob after
approximately 10 gyroperiods.
0
10
k*
1
10
Theoretical growth rate Γ∗ of the K-H
instability on the upper (blue) and
lower (red) edges, assuming an incompressible plasma with a finite Larmor
radius. [8].
20
40
From the results collected so far, two avenues for further research im30
0
mediately present themselves for study in
both the short and long
term.
20
These include:
−20
10
• Quantitative study of the ability of smaller blobs, which can- −40 20 40 60 80 100
x/ρi
not be detected, to heat the bulk
40
1.6
plasma. Simulations have been
1.4
20
carried out to compare the ef1.2
fect of many smaller blobs with a
0
1
fewer larger blobs of the same to−20
0.8
tal mass.
• Replacing single species blobs
0.6
−40
and background with a D-T
20
40
60
80
100
x/ρi
40
plasma with α impurities, to Temperature
plot showing asymmetric
2
heating in the tail of a 8ρi blob.
examine the effect of popula20
1
tions with differing Larmor radii
within a single blob.
0
0
−1
−20
−2
EPSRC−40
and
20
This work was part-funded by the
the RCUK Energy Programme
40
60
80
100
under grant EP/I501045 and the European 40
Communities
under
the
contract
of
x/ρi
1.4
Association between EURATOM and CCFE. The views and opinions expressed
1.2
herein do not necessarily reflect those of the
20 European Commission. We also
thank the EPOCH development team for their work on the PIC code1 adapted
0
for this research.
0.8
References
4tΩ
4ρi
0
5
10
15
20
x/ρi
25
30
35
40
45
5
0.4
[1] Grulke, O., Terry, J. L., Labombard, B., and Zweben,
J. January
2006 Physics of
20
40 S. 60
80
100
1.02
Plasmas 13(1), 012306–+.
40
5tΩ
2ρi
0.6
−20
−40
[2] Krasheninnikov, S. I. May 2001 Physics Letters A 283, 368–370.
1ρi
10
−2
10
270
3tΩ
−0.4
310
290
2tΩ
1tΩ
0
−0.2
330
300
Future Work
50
0.2
−15
0
10
40
0
0.4
10
−1
The substantial asymmetry which we find develops in these evolving
features, which strengthens for smaller blobs, underlines the need to
study small scale blob propagation using a hybrid method. The emergence of structures and gradients on ion gyro lengthscales, both internal
to the blobs and at the boundary between the blob and external flowing
plasma, further demonstrates the value of such a model.
We also conclude that FLR symmetry breaking plays a significant role
in the evolution of blobs, manifesting as: a poloidal deflection of the
blob plasma in the u × B direction; a difference in size between the twin
vortices generated internal to the blob; and a difference in the growth
rate of the Kelvin-Helmholtz instability between the upper and lower
edges of the blob.
0
Bulk Asymmetries
−20
280
Disersion relation for a cold,
magnetised plasma (blue line)
parallel (right) and perpendicular (left) to the background
magnetic field, over the FFT
of thermal noise in a uniform
plasma run of the code.
Blob particles (blue) feel the E = −u × B electric field of the flow particles (red), causing
them to move up, while the reverse happens for the flow particles.
−25
y/ρi
Under these assumptions electric fields cannot arise from charge separation; hence electric fields are purely motional in origin. This level of
model description allows for the propagation of Alfvén and magnetoacoustic waves, ion cyclotron waves, lower hybrid waves and whistler
waves. However, since whistler waves are undamped at the electron
gyrofrequency, numerical damping must be applied to the electromagnetic fields to prevent their unstable growth.
• Advection in the direction of the background flow
• Development of a twin-celled internal vortex pattern
• Growth of a Kelvin-Helmholtz instability in the tail of the blob
Acknowledgements
y/ρi
and Ampère’s Law, we arrive at the equation we use to update the electric field in the code:
2
1 qi
∇B
(B · ∇) B
E=−
kTe ∇n +
−
+ Ji × B ,
(2)
en e
2µ0
µ0
A Kelvin-Helmholtz (K-H) instability forms at the shear boundary
at the edges of blobs. The instability allows for momentum transfer
between the two sheared fluids, contributing to the internal convection
pattern described above. The instability becomes most apparent in
the tail of the larger blobs, where a clear asymmetry is visible. Both
the growth rate and fastest growing mode scale size are seen to differ
between the upper and lower edge of the blob as a consequence of FLR
symmetry breaking.
y/ρi
(1)
Γ*
dve
ne me
= −ene (E + ve × B) − ∇ · Pe + µ
dt
20
10
3.6596tΩ
2.6837tΩ
5
For our model we make the following assumptions:
• Low frequency limit, so neglect displacement current in Ampère’s Law.
20
Blobs imaged in Alcator C-mod [1]
10
• Inertia-less electrons, “me = 0”
• ∇ · E is negligible on length scales of interest, implying charge neutrality
ene = qi ni
• Collisionless plasma, µ = 0
• Ideal, isothermal electron gas, ∇ · Pe = ∇p = kTe ∇n
15
x 10
10
15
20
x/ρi
25
30
35
40
Trajectories of the centres of mass for blobs
of radius 1,2,4ρi
Streamlines (black) over bulk velocity
(colour) for a 4ρi blob.
The twin-vortices deviate from the symmetric solution that would emerge from
a purely fluid description. The differ-
Blobs with Rb ∼ ρi exhibit a small but measureable downward deflection of up to a full
gyroradius over 5.5 gyroperiods, an order
of magnitude greater than the negligible deflection for larger blobs. This is a result of
ence in flow speeds between the upper
and lower sides of the blob causes the
vorticity cells to grow asymmetrically.
the downward deflection of the background
flow by the FLR symmetry breaking mechanism.
1.015
20
[3] D’Ippolito, D. A., Myra, J. R., and Krasheninnikov, S. I. January 2002 1.01
Physics of
Plasmas 9, 222–233.
0
[4] Aydemir, A. Y. June 2005 Physics of Plasmas 12(6), 062503–+.
1.005
|B|/B0
10
x/ρi
−40
−20
40
20
200
n/n0
5
−20
−10
20
5.4894tΩ
• It combines a particle-in-cell (PIC) algorithm for kinetic ions with an electron fluid.
• PIC codes subject weighted pseudo-particles to the Lorentz force in a continuous phase space, with E, B, J fields on a staggered grid.
• The code used is “2D3V” - a 2d simulation plane with 3 field components,
e.g. Bx,y,z (x, y, t).
0
4.0256tΩ
A hybrid model has numerous advantages, including the ability to capture the non-linear interaction between ion gyration and plasma inhomogeneity; cross-scale coupling between ion gyro-scale kinetic modes
and fluid MHD-like modes; the ability to resolve phenomena on much
shorter timescales than fluid models; and the ability to study inhomogeneities in configuration space and non-Gaussian ion velocity distributions.
The features of our hybrid code are as follows:
−20
250
•
•
•
•
•
0
−10
300
The preceding results demonstrate how the hybrid model (fluid electrons and fully kinetic ions) embodied in the code captures key elements
of the plasma dynamics and the evolution of the fields in ion gyro-scale
blobs. The parameter set chosen approximates to edge plasma conditions in a medium-size tokamak such as MAST. Many of the phenomena described above are common to a purely fluid approach to blob
simulation, including:
Ti/T0
−10
0
350
E ⋅ J/E0 ⋅ J0
−5
20
1.9518tΩ
0
x/ρi
40
10
1.8298tΩ
1.3419tΩ
5
Hybrid Physics
x/ρi
20
Conclusions
E × B/E × B
x/ρi
10
0tΩ
Simulations
0tΩ
Blobs
0tΩ
Abstract
−20
[5] Russell, D. A., D’Ippolito, D. A., Myra, J. R., Nevins, W. M., and Xu, X. Q.1 Dec 2004
Phys. Rev. Lett. 93(26), 265001.
0.995
−40
[6] Krasheninnikov, S. July 2011 Plasma Physics and
53(7),
074017.
20 Controlled
40
60 Fusion
80
100
x/ρi
[7] Chapman, S. C. and Dunlop, M. W. July 1986 Journal of Geophys. Research 91, 8051–
8055.
[8] Nagano, H. June 1979 Planetary Space Sci. 27, 881–884.