Numerical Studies of Transport in the Columbia Non-neutral Torus
Benoit Durand de Gevigney, Thomas Sunn Pedersen, Allen H. Boozer
Department of Applied Physics and Applied Mathematics, Columbia University
2- Electron drift trajectories integrator
1- Overview
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CNT is a non-axisymmetric experiment and electrons are confined on magnetic surfaces.
Boozer coordinates are used to study transport. (see 2)


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The electric potential in CNT is very strong:
- the EXB drift increases poloidal rotation and should prevent prompt losses of particles.
- however experiments indicate a large loss cone. This comes from the electric potential not being a
function of magnetic flux only (see 3)
Boozer coordinates: In CNT
At very low B-field passing particles can be lost (see 4)
d ∂H
=
dt
∂ 0
●
Equations of motion:
d
2 Wk 1 ∂B
=1  

dt
e B ∂ 0
d 0
2 Wk 1 ∂B
=−1  

dt
e B ∂
CNT is not an optimized stellarator
→ Lots of trapped electrons
● Trapped electrons cannot circulate poloidally
→ Prompt losses
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●
Equations of motion:
d
2 Wk 1 ∂B
=1  

dt
e B ∂ 0
d 0 d 
2 Wk 1 ∂B
= −1  
≃'
dt d 
e B ∂
Large variations in Φ across the plasma, eΔΦ/Te>>1 →
W k ∂ ln B
● Electrons at ψ lost only if
0a −


0
e  ' ∂ 0 max
→ Only thin layer close to ψa
GRAPH: Trajectory of deeply trapped electron
projected onto the (ψ,θ) plane.
SIMULATION: Load 1000 monoenergetic (Wk=4eV) electrons uniformly on a
surface and follow them until they leave confinement or they reach maximum time.
Keep track of the fraction of confined electron as a function of time.
W k ∂ ln B
d
and
'≫
≪1
e ∂
d
SIMULATION: Load 1000 monoenergetic (Wk=4eV) electrons uniformly on a
surface and follow them until they leave confinement or they reach maximum time.
1 eB 2 
H=
∥  B− ,=const
2 m
e
d  ∂ H d ∥ ∂ H
=
=−
dt
∂ ∥ dt
∂
0
0 
2

where a = flux enclosed by outermost surface

2 g
g = constant proportional to poloidal current G
4- Influence of magnetic field strength
2= ,0,  : Potential without conforming boundary
d 0 ∂ 
d

∂

2 Wk 1 ∂B
2 Wk 1 ∂B
Equations of motion:
= −1  

=− 1  

dt ∂
e B ∂
dt
e B ∂0
∂ 0
● Magnetic AND electrostatic trapping
→ More trapped electrons
● Non-zero radial ExB drift
→ Much more complex orbits
●
GRAPH: Trajectory of same GRAPH: Electrons jump from magnetic surfaces
deeply trapped electron as before. to magnetic surfaces by drifting on equipotential
Motion is much more complicated. and can find their way out of the plasma.
SIMULATION: Load 1000 monoenergetic (Wk=4eV) electrons uniformly on a
surface and follow them until they leave confinement or they reach maximum time.
All surfaces have almost
the same fraction of trapped
electrons (~60%)
Fraction of trapped
electrons still the same
●
All of these trapped
electrons are lost quickly
Electrons lost only if born
with ψ>0.85
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GRAPH: Left - fraction of confined electrons Right - corresponding home surfaces

2
GRAPH: Trajectory of same deeply trapped
electron as before projected onto the (ψ,θ)
plane. Poloidal precession appears clearly.
●
B  , ,
LOSS CONE: classical magnetic mirror configuration with ∣∣≤ 1−
= max
B max 
as coordinates
(in this part B=0.1T)
1= : Idealized potential with conforming boundary
●
Use
d 0 ∂ H
=−
dt
∂

 Here we use normalized magnetic coordinates:

a
3- Collisionless Orbits in Different Electric Potentials
0=0 : No potential

m
 ,0 , , ∥≡v∥ 
eB
Guiding center equations of motion can be derived from the Hamiltonian:
Variations in electric
potential on a single
magnetic surface
Electric potential on the ψ=0.5
surface. Red is -190V, blue is
-150V.
 × ∇
  =∇

B = ∇
0
Loose electrons up to ψ=0.5
ψ>0.7 loose more electrons
with than without potential
Loss cones at locations (ψ=0.8, θ=0, φ=0.2) with Φ (ψ) ιν δ ιφφε ρε ντ µ αγ νε τιχ φιε λδ σ.
As the B-field becomes very weak, passing electrons can be lost.
● Highly asymmetric, effect much stronger for counter-passing electrons. Resonances ?
● Could explain the anomalous dependence of neutral-driven transport a low B-field ?
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GRAPH: Left - fraction of confined electrons Right - corresponding home surfaces
LOSS CONE STRUCTURE:
GRAPH: Neutral-driven transport rate
divided by neutral pressure vs magnetic field
GRAPH: Left - fraction of confined electrons / Right - corresponding home surfaces
LOSS CONE STRUCTURE:
→ Below 3x10-2 T, this quantity depends on
B (approximately as B-1.5)
Recently discovered and
being investigated
Region covered
on the left
Graph: Loss cones at locations (ψ=0.8, θ=0, φ=0..2) and (ψ=0.9, θ=0, φ=0..2).
Passing-trapped separatrices are plotted.
Graph: Loss cone at location (ψ=0.9, θ=0, φ=0..2) (minimum of |B|).
Color (log scale) corresponds to the time it take an electron to drift out.
Effect of the ExB rotation is to confine electrons close to the trapped-passing separatrix
● Effect decreases with increasing ψ and increasing W
k
GRAPH: Left: Loss cone at locations (ψ=0.5, θ=0.2, φ=0.3).
Right: Passing-trapped separatrix at this same location.
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Can still see beneficial effect of ExB rotation on the sides
● Looks much more complicated → Good confinement region at the center
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5- Future work
Understand anomalous effect at low B-field
● Model the perturbation of the rods
● Include collisional effects in the code to quantify transport and confinement time
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