# Presentation: 210247

## BASIS AND APPLICABILITY OF  LIMITS: 

- PIPELINE PROTECTION AND THREAT INTERDICTION

**BASIS AND APPLICABILITY OF ******** LIMITS: **
**TO COMPACT STELLARATORS**

**A.D. Turnbull**
**ARIES-CS Project Meeting**
**September 15 2005**
**Princeton Plasma Physics Laboratory, NJ**

## Relevance of MHD  Limits in Stellarators is Not Well Understood

**MHD stability limits in Tokamaks is considered well understood:**
- Ideal MHD predicts stability limits, growth rates, and mode structures in many situations
- Fast, global instabilities are identified with disruptions and  collapse
- Localized and weakly growing instabilities are identified with benign MHD activity: Edge Localized Modes (ELMs), Sawteeth, etc.
**Stellarators however appear to violate MHD stability limits****: **
- Recent LHD and W7AS experiments exceeded predicted  limits
-  appears to be limited by a soft limit of degrading confinement:
****** ****limits in the tokamak sense have not yet been observed**
- Some correlation is still observed between mode onset and linear stability threshold	 Ideal MHD predictions do mean something
**But Stellarators and Tokamaks have the same underlying physics based on Maxwell’s Equations and Newtonian mechanics!**

## Tokamaks Provide Context for Understanding Role of MHD Stability in Stellarators

**Tokamak studies show importance of distinguishing different ******** limits:**
- Local ideal ballooning and interchange modes
- Global ideal internal modes
- Global ideal external modes
- Resistive interchange modes
- Equilibrium limits to 
**And suggest how to proceed:**
- Stability limits depend sensitively on the equilibrium details:
******	****Equilibrium characterization is crucial to identifying the problem precisely**
- Need to develop intuition for mode relevance in each case
- From experiments
- From nonlinear stability calculations
**What criteria can we use right now?**

## Tokamak Experience is Not So Different

**Tokamaks also routinely violate some MHD stability limits:**
- Limits are open to interpretation and are not always hard limits:
- Tokamaks routinely operate with q < 1, unstable to internal kink instability: 	⇒ Sawteeth
- Tokamak ballooning modes are not always devastating: ⇒ Soft  limit
- In H-mode Tokamaks also routinely reach intermediate n stability limits: 		⇒ Generally benign ELMs
**Tokamak stability limits depend sensitively on the equilibrium:**
- Not sufficient to fit equilibrium to global discharge parameters
- 	****** ****Stability can depend quite sensitively on profile details**
- In Stellarators the  profile is not normally measured at finite 
**Once it was measured, the q profile in Tokamaks was not what everyone thought it should be****!**

## Interpretation of Local Ideal Ballooning and Interchange Mode Limits 

**	****General consensus is that large Stellarator experiments routinely exceed local ideal ballooning and Mercier ******** limits**
**LHD Achieved ******** >**** ****4% ****(Sakakibara EPS 2004)****:**
- Heliotron configuration has a magnetic hill in the peripheral region
-  Violation of stability of ideal and resistive interchange modes is a concern but modes are not seen
**W7-AS: Achieved ******** > 3.4% ****(Zarnstorff IAEA 2004)****:**
- MHD activity in early medium  phase
- Predicted ideal MHD local stability limit  ~ 2%
******	****Should these limits be ignored in design studies?**

## Ignoring Local Limits is Consistent with Longstanding Tokamak Experience

**Large Tokamak experiments also routinely operate with axis q below 1.0 violating Mercier stability**
** ****This is identical to the situation in Stellarators**
**Ballooning instability in Tokamaks appears to cause confinement saturation so that profiles do not exceed the local ballooning limit**
**May be some differences in Stellarators but absence of accurate equilibrium reconstruction precludes a definite conclusion**
**Open question for Stellarators: how do local ballooning mode solutions relate to global modes:**
- Construction of global modes from local mode solutions may yield higher, more relevant  limits if local criteria are ignored (Ware EPS 04)
- It seems appropriate to construct such solutions and use those to determine the ballooning limit

## Interpretation of Global Ideal Internal Mode  Limits

**Stellarators are beginning to distinguish physically relevant instabilities**
- W7-X studies considering “physically relevant” modes as those with growth rate above a finite cutoff
- LHD correlating observed modes with sufficiently large predicted radial width and growth rates above a finite cutoff
**This View is Also Consistent with Tokamak Experience**
**Tokamaks routinely operate with several weakly unstable ideal global internal modes**
- Ideal internal m/n = 1/1 mode generally weakly unstable if q < 1 ( > 1)
- 1/1 stability is dependent on a range of non-ideal contributions
- The 1/1 ideal instability is routinely ignored in stability calculations
- Weakly growing “infernal” modes localized in low shear regions are sometimes observed but typically saturate and then decay

## Stability to Physically Relevant Growth Rates Yields  Limit above 5% in W7-X

**W7-AS experiments saw high ******** quiescent phase after an earlier startup with noticeable MHD activity:**
- Study for W7-X compared CAS3D stability with W7-AS observations
**Physically Relevant Growth Rates Considered to be >  20 khz (20 ********s)**
****** ****Calculated ******** limit is 5.25% (limited by high m modes)**
****** ****For the low n modes only (m=14): calculated ******** limit ~ 6%**
**	****	****	****Physical growth rates versus ******

- C. Nuhrenberg IPP)

## Low m/n = 1/1, 2/3, and 2/5 Internal Modes Appear to Determine  Limit in LHD

**In LHD***** s*****everal MHD modes (m/n = 1/1, 2/3, 2/5) are excited in edge region and spontaneously stabilized in turn as ******** increases:**
- Profile flattening observed and contributes to MHD mode stabilization
- These modes limit the pressure gradient in the peripheral region
**Theoretical prediction suggests m/n = 1/1 mode has resonance around ******** = 0.9 and that this mode determines the ******** limit in LHD (Sakakibara EPS 2004)**
**Actual ******** limit appears to correlate with a ‘big enough’ mode:**
**‘****Big enough’ is defined by the radial mode width**

##  Limit in LHD Appears To Correlate with Predicted Mode Width ~ 5% Minor Radius

- Watanabe IAEA 2004

## Interpretation of Global Ideal External Mode  limits

**Several external mode types need to be distinguished:**
******  ****Global ******** driven modes****	****	********  Current driven peeling modes**
******  ****Current driven external kinks****	********   ELMs**
**Global ******** driven modes expected to result in a true ******** limit**
**Low n external kink and intermediate n peeling mode stability generally depends sensitively on edge conditions:**
**Tokamak experience shows a good equilibrium characterization is needed to fully compare experiment and theory predictions:**
- External “peeling mode” stability depends sensitively on vicinity of  mode rational surface in vacuum
- Similar  sensitivity to edge rational  is observed in LHD

## ELMs Are of Particular Significance Since They Are Observed in Stellarators And Tokamaks

**ELMs do not directly result in ******** limits in either case!**
**In Tokamaks ELMs appear to be primarily intermediate n ideal edge instabilities related to peeling modes:**
- Mode is driven by combination of bootstrap current and pressure gradient from steep edge pressure gradient
- Generally referred to as “peeling-balloning” modes
**In Stellarators it is not clear ELMs are related to the same ideal edge modes:**
- ELMs may be induced by resistive/ideal interchange modes
**Note: Intuition from simple models can be misleading:**
- Common intuition “peeling modes” are current driven modes related to finite edge q near a rational value
- In divertor case q   but “peeling modes” coupled to pressure driven modes still occur
- 	Classic current driven ”peeling modes” do not exist in diverted equilibria but a coupled pressure/current driven version does exist - the so-called “peeling-ballooning mode”

## Global Edge Stability Depends Strongly on Edge Conditions and Rational Edge Values

**Plasma Boundary Has a Significant Influence on MHD Stability in Heliotrons (N. Nakajima JIFT 2005):**
- Finite pressure gradient observed beyond LCFS
- Inward shifted configurations have narrowest stochastic layer
- Assuming average flux surfaces in stochastic region, configuration is predicted unstable for fixed boundary at  = 3%, but marginally stable for free boundary
- At high , growth rates decrease with increasing  due to boundary modification
**Plasma behavior is affected by the rational surface existing at the plasma boundary in H Mode in LHD (S.Morita EPS 2004):**
- Plasma edge behavior strongly affected by nearby  = 1 surface
**Proposed measure for the operational ******** limit in LHD from linear ideal MHD theory:**
**Maximum ******** occurs in a limited number of experiments where, for low n modes, ********/********A ****~10****-2**

## LHD Maximum  Reaches Value Where Predicted Low n Growth Rate Exceeds Critical Threshold

- (Watanabe IAEA 2004)

## Interpretation of Resistive Mode  Limits

**Observed edge MHD mode in LHD is thought to be resistive interchange mode (K. Toi EPS 2003):**
- Dominant mode at L-H transition of LHD plasmas is m=2/n=3
- Edge in magnetic hill (destabilizes resistive interchange) but high magnetic shear region (stabilizes ideal interchange)
**Resistive interchange in LHD appears to be much like their ideal counterpart:**
- Typically predicted to be unstable at low 
- Does not seem to be limit 

## Interpretation of Equilibrium  Limits

**Equilibrium ******** Limits are still a prime candidate for setting the operational ******** limit in Stellarators:**
- Both LHD and W7-AS observe equilibrium degradation at high
- Maximum  in W7-AS appears to be limited by changes in confinement and not MHD activity
**This is not necessarily in conflict with observations of MHD modes in Stellarator experiments at high ********:**
- The observed MHD may be a manifestation of the equilibrium degradation through island formation or:
- The equilibrium degradation (island formation) may be a manifestation of the approach to an unstable situation

## Equilibrium Degradation May Set W7-AS  Limit

-  PIES equilibrium
-    calculations indicate
-    fraction of good
-    surfaces drops with 
-   Drop occurs at higher
-     for higher ICC / IM

**Experimental ******** value**
**correlates with loss of**
**~35% of minor  radius to**
**stochastic fields or islands**

- M.C. Zarnstorff IAEA 2004)

## Summary: LHD and W7-AS MHD  Limit Status

**In LHD and W7-AS ******** values achieved significantly exceed the Mercier interchange limit:**
- Maximum volume-averaged  above 3.5% achieved in both
**In LHD ******** appears to be limited by an m/n = 1/1 ideal limit (Watanabe IAEA 04)**
**In W7-AS ******** appears to be limited by approach to the equilibrium limit (Zarnstorff IAEA 04)**
**In either case ideal stability plays a direct or indirect role:**
**Degradation of the equilibrium is strongly associated with approach to MHD stability limits**
**Strongly growing ideal modes appear to provide a direct limit**

## Stability Limits Can Depend Sensitively on the Equilibrium

**It is not normally sufficient to fit the equilibrium to just the global characteristics of Tokamak discharges:**
- One can obtain widely varying results depending on the form assumed for the current density and pressure profiles for similar global parameters
- Profiles need to be measured accurately and used in reconstructing the equilibrium for the stability calculations
**In Stellarators the equilibrium is believed to be known. But:**
- The  profile is often taken from the vacuum profile:
**	****	******** ****It may be different at finite ******
- The pressure profile is not known as a function of flux:
**At most it is measured as a function of space and the mapping to flux space needed for the equilibrium depends on the ******** profile**
- Given the sensitivity to the equilibrium, nested flux surfaces might be a poor approximation for stability even for small islands

## Characterization of Experimental Stellarator Equilibria is Improving Rapidly

**Realization that accurate equilibrium reconstructions are needed in Stellarators is now becoming more widespread:**
- “In helical systems, the characteristics of MHD equilibrium, stability and transport with high  and large toroidal current are quite different from those in vacuum” (T. Yamaguchi EPS 2004)
- “The careful reconstruction of the equilibrium with applying asymmetrical profile is required for understanding of the mechanism of this mode stabilization [from profile flattening at high ]” (S. Sakakibara EPS 2004)** **
**New diagnostics are being developed and implemented at both LHD and W7-X for reconstructing pressure and current (********) profiles:**
**In future one can determine more precisely which modes actually exceed predicted limits !**
**One can then interpret the role of individual instabilities in determining operational ******** limit in Stellarators !**

## Physical Relevance Can be Studied by Considering Nonlinear Stability in Comparison With Experiments

**Existence of a nested flux surface equilibrium can be considered as either an equilibrium or a stability problem**
- Unstable equilibria with nested surfaces will evolve to a nearby non-nested surface state lower energy if physically possible
******	****PIES, HINST, NSTAB,...may be useful as nonlinear stability tools!**
**NSTAB nonlinear stability code exploits relation between equilibrium and stability by searching for bifurcated equilibria:**
- Existence of discontinuities ⇒ current sheet within nested flux surface approximation
- Current sheets resolved in reality by formation of islands
- Equilibria should be stable to profile preserving instabilities
- Nonlinear stability evaluated by employing a mountain pass theorem with the search for bifurcated equilibria
**Criteria appear to predict LHD and W7-AS ******** limits reasonably well**

## Some Important Distinctions  Exist Between Tokamaks and Stellarators

**Distinctions may produce superficially different behavior even if fundamentally MHD is valid in Tokamaks and Stellarators:**
- Current and pressure profiles may be quite different between Tokamaks and Stellarators
- Linear stability calculations generally assume nested flux surfaces
**	****⇒ ****In tokamaks this is normally an accurate assumption**
- In Stellarators nested surfaces may not exist !
- 	**⇒ ****Even non-nested surfaces ****might not exist: field may be stochastic !**
- Relative roles of current and pressure in driving MHD instability may mean different observed behavior
**Resolution requires testing predictions using discharge equilibria:**
**⇒**** ****Detailed measurements of stellarator ******** profiles are needed**
**Compact Stellarators may be more Tokamak-like than conventional Stellarators !**
**Finite average current may or may not reproduce more closely Tokamak-like MHD behavior**

## Conclusion: Linear Stability Predictions With Nested Surfaces Can Be Used as Guide if Interpreted Properly

**Distinction needs to be made between different mode types**
**Local stability criteria should probably be ignored:**
- There is little reason that infinite n should provide a physical limit
- Finite n corrections appear to be large given the difference between the global code limits and the infinite n localized limits
**Global MHD stability must be tested using reconstructed equilibria:**
- Need to use the measured equilibrium profiles
- May need to construct a non-nested flux surface equilibrium:
-  States with different prescriptions for the multiple values for p and j in different simply connected regions (islands etc.) are possible and may be physically accessible
- Flux surfaces might not even exist
-  Actual profiles will be determined by transport and topology
**MHD stability predictions need to be interpreted after testing using reconstructed equilibria against actual experiments**

## How Should We Proceed?  What Questions Remain?

**To proceed for ARIES-CS design:**
**Ignore local stability criteria**
**Check linear global stability (TERPSICHORE) as guide to approximate limit**
**Monitor linear stability predictions against nonlinear predictions (NSTAB)**
**Check flux surface quality (PIES)**
**Are nested surfaces a valid approximation for stability calculations:**
- Does linear instability of a nested flux surface equilibrium simply result in benign nonlinear evolution to a ‘nearby’ non-nested state?
- If nested surfaces are not valid, can the stability problem be formulated in terms of finding nonlinearly stable equilibria?
**Nonlinear consequences crucial for interpreting stability calculations:**** **
- Generally internal modes surrounded by a fairly robust and stable outer shell might be expected to be benign
- Is there a way to quantify this without the full nonlinear calculation?
**Further progress requires criteria to decide when linear instability of nested flux surface equilibria result in benign nonlinear evolution to ‘nearby’ states:**
**Requires direct comparison with experiments and nonlinear stability calculations**