# Presentation: 878280

## Finite Larmor Radius Stabilization of Ideal Ballooning Instabilities in 3-D plasmas

- J. G. Wohlbier* and C. C. Hegna
- Department of Engineering Physics
- University of Wisconsin
- Madison, WI 53706-1609
- *Los Alamos National Laboratory
- Los Alamos, NM
- Stellarator Theory Teleconference
- University of Wisconsin-PPPL
- March 11, 2004

## The nature of ideal MHD ballooning modes in 3-D systems differs qualitatively from ballooning modes in 2-D systems
Field-line dependence of ballooning mode eigenvalues
–  This typically corresponds to a global mode that is highly localized on the magnetic surface  ~ Can nonideal physics (e. g. FLR physics) more easily stabilize these localized modes in 3-D relative to 2-D systems?  
This work, include FLR effects in ballooning mode formalism of 3-D systems

- Field-line dependence of ballooning mode eigenvalues
- –  This typically corresponds to a global mode that is highly localized on the magnetic surface  ~ Can nonideal physics (e. g. FLR physics) more easily stabilize these localized modes in 3-D relative to 2-D systems?
- This work, include FLR effects in ballooning mode formalism of 3-D systems

**Motivation**

## Ideal MHD ordering and WKB-like formalism is used throughout

- For ideal MHD ballooning modes
- Use large **k** expansion
- 1/~ n  (“infinite-n theory”) large toroidal mode number
- Leading order solution leads to an ordinary differential equation for  along the field line, the ballooning equation

## Two-fluid physics brings in finite Larmor radius effects

- MHD equations modified by Hall-MHD terms in Ohm’s law and gyroviscosity
- Order such that FLR corrections enter
- Modified ballooning equation

## Modes correspond to quantizable action integrals 

- Action integrals of WKB trajectories are quantized
- Quantizable trajectories are actual MHD modes of the system.

## The inclusion of FLR physics in 2-D systems is straightforward

- In tokamaks, the kd quantization is trivial ---  toroidal mode number n is a good quantum number.  Local eigenvalues are independent of field line label,  ̨kis conserved along ray trajectories.
-  i* =  k (dp/d/ne = k *i is constant on WKB orbit equations.  Hence, 2 =  is conserved on WKB orbits and the frequency satisfies (Tang et al, 1980)
- For unstable local eigenvalue  < 0, stability is obtained if the criterion is satisfied

## In 3-D systems, the inclusion of FLR physics introduces complications

- In stellarators, local eigenvalues are generally functions of field lines, =(k, ) --- k and are no longer  constants on WKB rays. (Nevins and Pearlstein, ‘88)
- Only the ray equation changes,
- Given unstable mode (< 0) described by particular values (o, qo, ko, kqo), if
- Mode is stabilized

## Ideal ray orbits lie on topological spheroids in phase space labeled by (q,k,

## The projection of the ray equations into k space shows closed orbits - quantizable action

## The projection of the ray equations into kqq space shows multiple “timescales”

## Time scale separations typically allow for approximate integrability of the system

## Inclusion of FLR eliminates the topological spheroids --- integrability?

## Summary

- Inclusion of FLR effects into ideal MHD ballooning modes discretizes the spectrum.
- The inclusion of FLR physics on ballooning stability is complicated by the non-constancy of *i  ~ k  along the ray equations (“n” is not a good quantum number.)
- FLR stabilization is given by the criterion
-  kmax corresponds to peak value on periodic ray orbit
-  is the corresponding ideal MHD eigenvalue  = MHD