Transport and evolution of ion gyro-scale plasma blobs in
perpendicular magnetic fields
P W Gingell, S C Chapman, R O Dendy, C S Brady
CFSA, Department of Physics, University of Warwick, Coventry, CV4 7AL
Blobs
Hybrid Models
The outer regions of tokamak plasmas, near the last closed
flux surface, are observed to generate coherent propagating blobs [1], which undergo 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. Models for blob transport
and evolution have principally focused on analytical and
numerical fluid and multi-fluid descriptions; see, for example, Refs. [2-4]. Here we pursue a hybrid method which
combines kinetic ions with fluid electrons. For a recent review of edge plasma physics in tokamaks we refer to Ref.
[5].
A hybrid model can capture the non-linear interaction between ion gyration
and plasma inhomogeneity; capture cross-scale coupling between ion gyro-scale
kinetic modes and fluid MHD-like modes; resolve phenomena on much shorter
timescales than fluid models; and study inhomogeneities in configuration space
and non-Gaussian ion velocity distributions.
The features of our hybrid code are as follows:
!
qi
∇B 2 (B · ∇) B
kTe∇n +
−
+ Ji × B ,
e
2µ0
µ0
1
E=−
en
(1)
Under these assumptions electric fields cannot arise from charge separation;
hence electric fields are purely motional in origin. This model allows for the
propagation of Alfvén and magnetoacoustic waves, ion cyclotron waves, lower
hybrid waves and whistler waves.
• 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 and velocity
components, e.g. Bx,y,z (x, y, t).
For our model we make the following assumptions:
• Inertia-less electrons, “me = 0”
• ∇ · E is negligible on length scales of interest, implying charge neutrality
ene = qini
• Collisionless plasma,
• Ideal, isothermal electron gas, ∇ · Pe = ∇p = kTe∇n,
Results
−10
10
Notable features of the evolution include:
10
−10
• Advection of the blow in the flow direction,
20
200
x 10
150
5
i
100
10
20
−40
40
i
10
0
−10
0
• Formation of twin-cell internal vortices,
0
x/ρ
20
−5
250
5
10
x/ρ
10
15
20
i
−20
0
x/ρ
20
40
20
x/ρ
40
60
i
20
0
0
−10
−20
−20
−10
20
−40
−20
40
0
10
x/ρ
20
30
i
0
i
4
50
40
40
30
• Asymmetry in all of the above, which becomes more pronounced for the
smallest blobs,
2
30
10
y/ρ
x/ρ
Ω
0
0
−5
−10
−20
−10
−20
−40
i
10
15
20
x/ρ
10
Magnetic field strength, Bz,0 = 0.4T
• Temperature, T0 = 4 × 106K
20
20
10
10
y/ρ
0
−10
−10
−20
−20
−30
−30
−40
−40
• Thermal gyro-radius, ρi = 8 × 10−3m
0
• Background flow speed, u0 = 0.2vA
20
40
60
x/ρi
80
25
30
x/ρi
10
35
5
0
0
−10
−20
−20
20
20
−40
30
20
40
x/ρ
60
80
50
60
20
40
0
−40
100
0
20
30
40
50
x/ρ
60
70
20
i
40
80
0
−20
i
40
60
x/ρi
20
i
0
40
x/ρi
−10
50
i
0
−20
45
60
Ω
0
−10
40
x/ρ
40
x/ρ
20
−5
35
20
40
i
10
30
100
0
5.8554t
0
Ω
30
5.3674t
30
40
10
20
i
40
30
x/ρ
Ω
5
−10
y/ρi
• Number density, n0 = 1019m−3
20
i
−5
40
20
Ω
Blobs are simulated as Gaussian enhancements in density
in combined pressure balance. The background plasma
parameters are chosen to be approximately characteristic of edge conditions in a medium-size tokamak such as
MAST:
10
Ω
Asymmetries are visible in both the trajectories of individual ions, and the
evolution of bulk properties such as number density:
4.0256t
Initial conditions with number density in colour and
flux-rope structure of magnetic field lines in black.
25
7.8072t
• Deflection of smaller blobs perpendicular to the background flow.
5.4894t
i
20
Ω
10
1
20
u
0
7.3192t
20
3.6596t
i
50
10
Ω
5
3.9036t
• Growth of the K-H instability in a tail formed from blob ejecta,
3
Ω
0
n /m−3
50
2.6837t
i
5
Ω
1.3419t
300
z/ρ
0
x/ρ
5
• Structures and gradients forming on the ion gyro-scale,
350
−5
Ω
Simulations
Ω
From the electron fluid momentum equation and Ampère’s Law, we arrive at
the equation used to update the electric field in the code:
Dispersion 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.
1.9518t
• We take the Darwin limit of Maxwell’s equations, neglecting displacement
current.
1.8298t
Blobs imaged in Alcator C-mod [1]. Blobs have also been
observed in DIII-D, NSTX, ASDEX Upgrade and MAST.
0
20
n/n0
40
n/n0
40
60
x/ρ
80
100
i
60
0
50
n/n0
100
Ion trajectories in the 10ρi (left) and 2ρi (right) blobs.
Number density evolution for 1, 4 and 10ρi blobs (left to right).
Kelvin-Helmholtz Instability
The difference in flow speeds on the upper and lower edges caused by
FLR symmetry breaking leads to the following bulk asymmetries:
A K-H instability forms at the shear boundary at the edges of blobs, and
becomes most apparent in the tail of the larger blobs. 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.
(2)
0.14
10
0.12
5
0.1
A
X
1
Dvα
nα ′ v α ′  × B
= enα vα −
mα n α
Dt
n ′
α
nα
+ (J × B − ∇pe) − ∇pα
n
15
i

y/ρ

0
0.08
−5
0.06
−10
0.04
The first term on the RHS represents an asymmetry present in plasmas with distinct ion populations in velocity space. The form of this
asymmetry is as follows:
Streamlines (black) over bulk
velocity (colour) for a 4ρi
blob. The vorticity cells on
the upper and lower edges
grow asymmetrically.
40
K-H instability growing both
more quickly and with a
larger scale size on the lower
edge of a 10ρi blob than the
upper edge.
20
i
0.16
20
y/ρ
The chirality of ion gyration can break the symmetry of the evolution of
an otherwise symmetric blob. The asymmetry can be described mathematically using the ion momentum equation [6]:
Bulk Asymmetries
u/v
Finite Larmor Radius
Symmetry Breaking
0
−15
0.02
−20
−20
20
25
30
35
40
x/ρ
45
50
55
60
−40
i
40
60
80
100
x/ρi
120
140
0.1
0
−0.1
−0.2
4tΩ
y/ρi
−0.3
−0.4
−0.5
Blob particles (blue) feel the E = −u × B electric field of the flow particles
(red), causing them to move up as they begin to gyrate, while the reverse
happens for the flow particles.
−0.6
2ρi
−0.8
−0.9
5tΩ
4ρi
−0.7
1ρi
0
5
10
15
20
x/ρi
25
30
35
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 [7].
−1
10
*
Trajectories of the centres
of mass for blobs of radius
1,2,4ρi. Blobs with radius
Rb ∼ ρi exhibit a small but
measureable downward deflection.
3tΩ
10
Γ
2tΩ
1tΩ
0
−2
10
−3
10
40
−1
10
Conclusions
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. Many of the phenomena described above are common to a purely fluid approach to blob
simulation, including:
• 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
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. FLR symmetry breaking plays a significant
role in the evolution of blobs, manifesting as:
• A deflection of the blob plasma in the u × B direction,
• A difference in size between the twin vortices generated internal to the blob,
• A difference in the growth rate of the Kelvin-Helmholtz instability between the upper and lower edges of the blob.
0
10
*
k
1
10
Acknowledgements
This work was part-funded by the EPSRC and the RCUK Energy Programme under grant EP/I501045 and
the European Communities under the contract of Association between EURATOM and CCFE. The views and
opinions expressed herein do not necessarily reflect those of the European Commission. We also thank the
EPOCH development team for their work on the PIC code adapted for this research.
References
1. Grulke O, Terry J L, Labombard B and Zweben S J 2006 Physics of Plasmas 13 012306
2. D’Ippolito D A, Myra J R and Krasheninnikov S I 2002 Physics of Plasmas 9 222-33
3. Aydemir A Y 2005 Physics of Plasmas 12 062503
4. Russel D A, D’Ippolito D A, Myra J R, Nevins W M and Xu X Q 2004 Phys. Rev. Lett. 93 265001
5. Krasheninnikov S I 2011 Plasma Phys. Control. Fusion 53 074017
6. Chapman S C and Dunlop M W 1986 J. Geophys. Res. 91 8051-55
7. Nagano H 1979 Planet. Space Sci. 27 881-4