# Presentation: 810841

## MHD Stability in Stellarators

- Guo-yong Fu

## Outline

- Overview of ideal MHD stability
- MHD Stability in Stellarators
-    -3D stabilization
- Current Topics
-    -ballooning modes
- Summary

## Design of National Compact Stellarator Experiments (NCSX)

- Equilibrium
- +
- MHD Stability
- +
- Transport
- +
- coils

- optimization

## Energy Principle

******W = 1/2**

- d3x { **B**2 + [**B**- **B** (p/B2)]2  + (  )2 + **j**   x **B** - 2     p   }

- Field line
- Bending
- Shear Alfven

- Magnetic
- Compression
- Compressional
- Alfven

- Fluid
- Compression
- Sound
- waves

- Current
- Driven
- Kink mode
- Vertical
- mode

- Pressure
- Driven
- Ballooning
- Mode
- Mercier
- mode

## Mode representation

- Use flux coordinates (s,)
- Cylindrical:               single m and n
- Axisymmetric
-     tokamak:                 single n and multiple m
- 3D stellarator:          multiple n and m,
-                                     n=n+knp, k=...-2,-1,0,1,2,...

-  (s,)  = m,n(s)  sin(m- n)

## Plasma displacement of vertical, kink and ballooning modes

- Vertical                        kink                     ballooning
- m=1                             m=3                    multiple-m
- n=0                              finite-n                 high-n

## 3D Effects on MHD stability

- 3D
- shaping

- Rotational
- Transform
- Global magnetic
- Shear
- Local magnetic shear
- Magnetic curvature

- Vertical mode
- Kink mode
- Ballooning mode

## Vertical stability

******W = 1/2**

- d3x { **B**2 + [**B**- **B** (p/B2)]2  + (  )2 + **j**   x **B** - 2     p   }

- Consider a large aspect ratio current carrying stellarator and a uniform vertical
- plasma displacement (low beta):

- ~ (1+E) – (1-fi)(1+2) > 0    for stability
- or
- fi > (E2 - E)/(1+E2)               for stability
- where E is elongation and fi is fraction of
- external rotational transform.

- E

- G.Y. Fu, Phys. Plasmas 7, 1079 (2000)

## Vertical stability in NCSX

- G.Y. Fu et al., Phys. Plasmas 7, 1809 (2000)

## Stabilization of external kink mode by magnetic shear

## G. Y. Fu et al., Phys. Plasmas 7, 1809 ( 2000)

## Pressure perturbation has strong ballooning feature

## Local magnetic shear is enhanced by 3D shaping.

## Ballooning mode equation

******W = 1/2**

- d3x { **B**2 + [**B**- **B** (p/B2)]2  + (  )2 + **j**   x **B** - 2     p   }

| B   k2/B2 B  + (p / B2 (k x B)    =  0    |
| --- |

- In infinite-n limit, 3D MHD equations becomes a single 1D ballooning mode
- Equation along a field line.

## Ballooning second stability

- C.C. Hegna and S. Hudson, Phys. Plasmas 9, 2014 (2002)

## Can second stability region exist in stellarators ?

- QPS

- S. Hudson and C.C. Hegna,  Phys. Plasmas 11, 153 (2004)

## High-beta stable QPS have been found

- Ware et al, Phys. Rev. Lett,
- 89, 125003 (2002)

## Stability limit of ballooning modes:finite-n versus infinite-n

## Slide 19

## Summary

- 3D shaping generates external rotational transform and affects local magnetic shear and curvature;
- 3D shaping can be used to enhance MHD stability;
- Vertical mode is more stable in stellartors than in tokamaks;
- External kink mode can be stabilized via control of global and local magnetic shear;
- In NCSX, finite-n ballooning beta limit is higher than the infinite-n limit.

## Slide 21

## Slide 22

## Slide 23