Energetic ions can damp or destabilize tearing modes depending on the magnetic equilibrium configuration. Simulations and reduced analytic models show energetic particles damp the 2/1 tearing mode in monotonic q profiles but can destabilize it in configurations with an internal region of zero magnetic shear. The particle pressure contribution is modeled and enters the tearing stability criterion calculation. For an equilibrium with two internal pressure steps, the effect of energetic ions on tearing mode stability depends strongly on the magnetic shear at the internal step.

1. Effects of Shear on Energetic Ion
Interactions with Tearing Modes
M.R. Halfmoon1, D.P. Brennan2
1-University of Tulsa, Tulsa, OK
2-Princeton University, Princeton, NJ

2. Slowing Down Distribution of Energetic Ions
found Damping 2/1 Tearing Mode
Results from hybrid 𝛿𝑓 PIC, full MHD simulations in NIMROD show
damping. We are constructing a reduced model description to
understand the physics of this damping and/or destabilizing effect.

3. Simulations of reversed shear equilibrium configurations show a
high sensitivity to changes in 𝛽 𝑁 with the addition of energetic ions.
Reversed Shear Configurations have the 2/1 Tearing
Mode Destabilized in Simulations with Energetic Ions

4. Reduced Model of Effect on RWM Studied by Hu & Betti
• The energetic particle pressure contribution takes the form of a scalar
modification to the perturbed pressure, calculated by Hu & Betti in
2004.
• In their work, this was then placed into a δW calculation to determine
the stability of the resistive wall mode.
(Hu & Betti 2004)
• Did not take into account the interaction
between tearing modes and particles

5. Π𝑗 = −𝑁𝑗
𝑅
2
𝑑𝑇𝑗
𝑑𝑟
𝑣2 −
3
2
+
𝑙 𝑇 𝑗
𝑙 𝑁 𝑗
+ 2
𝑙 𝑇 𝑗
𝑅
𝑤 𝐸
𝑗
𝑤 𝐸
𝑗
+ 𝑣2 𝐻(𝑢)
• Where: 𝑤 𝐸
𝑗
=
𝜔 𝐸
𝜔 𝐵
𝑗 , 𝑙 𝑇 𝑗
= − 𝑇𝑗 𝑑𝑇𝑗 𝑑𝑟 , & 𝑙 𝑁 𝑗
= − 𝑁𝑗 𝑑𝑁𝑗 𝑑𝑟
𝜔 𝐵
𝑗
=
𝑞𝑣𝑡ℎ
2
Ω 𝑐 𝑅𝑟
, 𝐻 𝑢 = 2𝑠 + 1 +
𝐸(𝑢)
𝐾(𝑢)
+ 2𝑠 𝑢 − 1 −
1
2
• u is the pitch angle variable, q is particle charge, Ωc is the cyclotron
frequency, and s is the magnetic shear.
• The step function characteristic of the equilibrium pressure enters the
pressure integral through the temperature gradient in Π𝑗.
Terms Inside Pressure Integral Model the Dynamics of
High Energy Particles Interactions With Equilibrium
Fields

6. Goal: Construct Reduced Analytic Model of
Energetic Particle Effect on Tearing Mode
• Using an analysis similar to
Brennan & Finn '14 and Betti ‘98,
we approximate the equilibrium as a
set of step functions.
• 𝜓(𝑟) is calculated using the outer
layer equations with boundary
values at steps in equilibrium.
• The particle pressure affects the
outer region equation at the step in
pressure, just like Hu & Betti 2004.
• Δ’ is calculated by evaluating
𝜓′ 𝑎 𝑠
𝜓 𝑎 𝑠
and taking into account the
derivatives of the basis functions
aj apas rw rc
𝜑𝑗(𝑟) 𝜑𝑠(𝑟) 𝜑2(𝑟) 𝜑 𝑤(𝑟) 𝜑𝑐(𝑟)

7. Approximation of Magnetic Curvature Needed in
Cylindrical Tearing Model to Capture Trapped Particle
Dynamics
• In the high aspect ratio, circular cross section model, the
magnetic curvature has the form:
𝜿 = −
𝐵 𝜃
2
𝑟𝐵0
2 𝒓 −
𝑹 𝟎
𝑅0 + 𝑟 cos 𝜃
• Which can be written as a flux average and poloidally varying
term.
𝜿 ∙ 𝒓 = −
𝑟
𝑅0
2
1
2
−
1
𝑞2
, 𝜿 ∙ 𝒓 ~
= −
cos 𝜃
𝑅0
𝜿 = 𝜿 𝟎 + 𝜿 𝟏 cos 𝜃

8. Energetic Particle Contribution Enters Stability
Equation at Pressure Step
• Solving the ideal outer region, we obtain the jump conditions at
the 2/1 rational surface, including the current pressure steps.
𝜓′ 𝑎 𝑐
+
𝑚𝑗0
𝑎 𝑐 𝐹 𝑎 𝑐
𝜓 𝑎 𝑐 = 0
𝜓′ 𝑎 𝑠
= 𝛾𝜏 𝑟 𝜓 𝑎 𝑠
𝜓′ 𝑎 𝑝
+
2𝑚𝐵 𝜃 𝑎 𝑝 𝛽0
𝑎 𝑝
2
𝐹 𝑎 𝑝
(𝑚
1 − 𝛽𝑓𝑟𝑎𝑐
𝑎 𝑝 𝐹 𝑎 𝑝
− 𝜆𝛽𝑓𝑟𝑎𝑐) 𝜓 𝑎 𝑝 = 0
• For the ideal wall case: 𝜓 𝑎 𝑤 = 0
Particle Pressure

9. • Using the curvature approximation from before, we can
determine the poloidal harmonics of energetic particles.
𝑌𝑙
𝑗
=
−𝜋
𝜋
𝑑𝜃
2𝜋
𝑒−𝑖𝑙𝜃
( 𝑣2
𝝃⊥ ∙ 𝜿 +
𝑍𝑗 𝑒
𝑇𝑗
𝑍)
• Due to the poloidal dependence of 𝝃⊥, this reduces to:
𝑌𝑙
𝑗
= 𝑣2
[( 𝜉 𝑟 𝜅0 +
𝑍𝑗 𝑒
𝑇𝑗
𝑍)𝛿𝑙=𝑚 + 𝜉 𝑟 𝜅1(𝛿𝑙=𝑚+1+𝛿𝑙=𝑚−1)]
• Thus, the particle pressure has the form:
𝑝𝑗
𝑚
=
0
∞
𝑑 𝑣 𝑓0( 𝑣)
0
1
𝑑𝑢 K u Π𝑗 𝜎 𝑚 𝑣2
{ 𝜉 𝑟 𝜅0 +
𝑍𝑗 𝑒
𝑇𝑗
𝑍 𝜎 𝑚 + 𝜉 𝑟 𝜅1 𝜎 𝑚−1 + 𝜎 𝑚+1 }
• Which reduces to: 𝑝𝑗
𝑚
= 𝜆𝑝0
• Where 𝜆 is a constant of proportionality that contains all
information regarding the energetic particle population’s
response to external fields.
Curvature Simplifies Calculations

10. • For the m=2 mode, the BVP becomes:
𝜓′ 𝑎 𝑝
+
2𝑚𝐵 𝜃 𝑎 𝑝 𝛽0
𝑎 𝑝
2 𝐹 𝑎 𝑝
(𝑚
1−𝛽 𝑓𝑟𝑎𝑐
𝑎 𝑝 𝐹 𝑎 𝑝
− 𝜆𝛽𝑓𝑟𝑎𝑐) 𝜓 𝑎 𝑝 = 0
• For cases that have negative or zero shear in the low frequency
limit, the term Π𝑗 in the pressure integral has a pole when
H(u)=0.
𝜆 Plays a Role in Determining the Effect that Particles
Have on the Stability Criterion
𝑠 =
𝑟
𝑞
𝑑𝑞
𝑑𝑟
= 0 𝑠 =
𝑟
𝑞
𝑑𝑞
𝑑𝑟
= 2

11. Pressure Enters in Δ’ Calculation Indicating Damping
and Stabilizing Effect on Resistive Mode
βfrac= 0
βfrac= 0.125
βfrac= 0.25
βfrac= 0.40
• One limitation of our model is that the precession frequency is a fixed value
determined only by equilibrium quantities, and the mode itself is assumed to have no
rotation.
• For this configuration, Δ′ = 𝜑𝑠′ 𝑎 𝑠
+
𝜑 𝑠′(𝑎 𝑗+)𝜑 𝑗′(𝑎 𝑠−)
𝜑 𝑗′
𝑎 𝑗
+
𝑗0 𝑚
𝑎 𝑗 𝐹(𝑎 𝑗)
+
𝜑 𝑠′(𝑎 𝑝−)𝜑 𝑝′(𝑎 𝑠+)
𝜑 𝑝′
𝑎 𝑝
+
2𝑚𝐵 𝜃 𝑎 𝑝 𝛽0
𝑎 𝑝
2 𝐹 𝑎 𝑝
(𝑚
1−𝛽 𝑓𝑟𝑎𝑐
𝑎 𝑝 𝐹 𝑎 𝑝
−𝜆𝛽 𝑓𝑟𝑎𝑐)
∆1= 0
𝜑 𝑝′
𝑎 𝑝
+
2𝑚2 𝐵 𝜃
2
𝑝0(1+(𝜆−1)𝛽 𝑓𝑟𝑎𝑐)
𝑎 𝑝
3 𝐵0
2 𝐹(𝑎 𝑗)2 = 0

12. Consider an Equilibrium Configuration with a
Second Pressure Step Inside the Plasma Column
• Using an analysis identical to the
previous case, we calculate the basis
functions and resulting stability criterion
for an equilibrium with two pressure
steps and two current steps.
• In this calculation, each particle
pressure drive interacts with the mode
differently due to the geometric
configuration.
• The pressure drive in the negative
shear region leads to a trapped particle
resonance in the low-frequency limit.

13. Varying the Magnetic Shear at the Internal Step Greatly
Affects Stability
• These lines represent the marginally
stable values (Δ’=0) for varying
equilibrium pressure and particle
pressure contribution.
• Once the configuration reaches a
critical negative value, the effect of
particle resonances shifts from
stabilizing to destabilizing.

14. Δ’ Calculation Indicates a Destabilizing Effect for
Equilibria with Internal Step
For the equilibrium configuration with an internal pressure and current steps, particles
contribute to the growth of the 2/1 tearing mode.

15. Conclusions
• Ultimately, both simulations and analytic approximations have
shown energetic particles have a damping effect on the growth
of linear resistive tearing modes.
• This work has found that a destabilizing effect occurs due to
trapped particle resonance in a region of zero shear internal to
the rational surface radial position.
• Future studies will attempt to connect these analytic models to
simulation and experimental results by including finite rotation
and mode-mode interactions.