# Presentation: 132011

## Explosive Instability, Fusion and MAST

- Princeton January 7th 2009
- Steve Cowley, *UKAEA Culham and Imperial*
- Eric Nardon, Howard Wilson, Omar Hurricane, Andy Kirk, Bryan Fong, Mehmet Artun, MAST team  ......

## Explosive Instability

- Dramatic eruptions are a feature of confined plasmas: *e.g. solar flares, tokamak disruptions and magnetospheric sub-storms.*
- Problem for fusion.
- Why do plasmas sometimes explode?
- I will argue that this is a nonlinear issue.
**Talk: **Introduction to the issues, the open questions and an **example **of a mechanism.
- Relation to Culham programme -- MAST.

## Passing Through Marginal Stability

- Moving slowly through marginal stability gives only moderate growth.  Let t=0 be the time of marginal stability.
- Approximate amplitude equation,
- The hybrid evolution time is,
- With non-linearity we can get fast singular growth. Or slow saturated growth.  What does happen and when?

## Nonlinear Evolution --Bifurcation

- Slow growth

- Critical amplitude for nonlinearity

## Bifurcation

- Explosive growth

- Critical amplitude for nonlinearity

- What is the energy landscape
- For MHD instabilities?

## Marginal Stability Models

- : 25 years ago Connor, Taylor and Turner proposed a
- marginal stability model.  The model supposes that the
- effect of the ballooning modes is to vastly increase transport
- bringing the pressure gradient back to the marginal value.
- : Sometimes in the pedestal before an ELM the plasma seems to be tied to the ballooning limit.

- Pressure profile
- without Ballooning
- modes

- Critical pressure
- profile.

- Actual pressure profile.

## Opening up the Tokamak

- ECE grating
- polychromator

##  TFTR Data    E. Fredrickson, et.al.Phys, Plasmas 3, 1996

- Contours of electron temperature.
- Ballooning mode (on top of kink mode)
- rotates past the detectors.

- Disruption of discharge.  Loss
- of plasma energy to wall --

*“**Linear” Theory: Bussac, *
*Pellat, Hegna, Callen, *
*Simulation, Park .......*

## JET Disruption.   James Paley and Philip Andrew.

- 200s

## Timescales

- The ballooning mode depends on three time-scales.  JET numbers.
- Transport time-scale.
- Typical growth rate of the ballooning mode.
- Diamagnetic - FLR time - drift time.

## Ballooning Mode

- Red lines q=4 field lines,
- Black lines q=2 field lines,
- Blue tube ballooning q=2 field line.

## ELMs on MAST Thanks to Andy Kirk for the ELM data

- Small Elms have very regular structure
- very little evidence for a radial expansion

- Type I

- Small

- Can release several % of the
- plasma energy in < 1ms
- On ITER: 20 MJ in 500s

## ELMs on MAST Thanks to Andy Kirk for the ELM data

## Filament propagation

- 100 kHz frame rate

**Filament propagation**

- MAST

- Filaments produce the peaks in mid-plane ISAT traces

- s

## ELMs Eruptions

- Observation of these filamentary structures has increased our knowledge of ELMs and allowed us to find mechanisms that can  mitigate them

## Slide 16

## Slide 17

## Nonlinear Ballooning Mathematics.

- Equations are expanded about marginal stability with:

- Displacement is fast varying across the field, slow varying along the field,
- slow in time and small (but finite). Derivation like *Connor, Hastie and Taylor*,
- 1978 but with nonlinear terms.  *Cowley and Artun 1995 (slab),*
*Hurricane, Cowley and Fong* 1998, *Wilson and Cowley* 2005.

## Nonlinear Ballooning Mathematics.

- Lowest order,                 incompressible across the field (filament).

- Order,                 marginal stability along field -- “Ballooning Equation”

- Distance from marginal stability
- = 1- 

- Envelope function determined in higher order

- Determined

## Envelope Equation

- A matching condition at high order yields an equation for A -- the behavior across **B**.

- Where

- and

- Average over toroidal direction

- Local instability drive, localizes mode

- Constants C1, C2, C3 and 0 are complicated
- averages over H() and equilibrium. *s* is usually 2
- but at low average curvature can be fractional derivative.

- Explosive Nonlinearity

## Finite Larmor Radius

- Singular
- behavior

- Poloidal width narrows 

- Radial width grows 

- Amplitude explodes

- Fong et. al. 1999

- FLR

## Origin of Explosive Nonlinearity

- _Two important effects_:
- Nonlinear reduction in bending force
- 	due to expansion of tube in
- 	low pressure region:
- 2.   Change in local magnetic shear with
- 	radius -- less twisting.  Howard Wilson
- 	has calculated C2 for JET and this effect
- 	is dominant.

## Fingers

- A

- 

- √(A2)

- Amax ~ (t0-t)-2

-  = (t0-t)

- ~ (t0-t)-3/2

- Pushed back down -- survival of the fittest

- Linear width set by FLR  ~ (i2L)1/3

## Detonation Scenario

- Instability grows in the narrow unstable region.
- Forms finger due to nonlinear explosive term.
- Grows explosively.
- Destabilizes the meta-stable region and widens in radius.
- Approaches singularity -- expansion breaks down and dissipation gets large*.*

## Simulation of Detonation Equation s  3/2, s=2 

- Simulation of Detonation Equation s  3/2, s=2

- Not a picture of field lines -- these are the magnetic surfaces.

-  -- Toroidal angle

- rRadial
- direction

- Fong 2000.

- Linearly unstable region

- Metastable region

## Bryan Fong’s 

- Bryan Fong’s
- Thesis 1999.
- Comparison of
- MHD direct
- Simulation and
- Asymptotic
- Theory with no
- Fitted parameters.
- Time evolution is
- also very similar.
- Cannot go further
- because of
- singularity
- Formation -- not
- slowing down.
- Ping Zhu has done
- some interesting work
- on the further nonlinear
- development.

## Nimrod Simulation.

**A.Y. PANKIN****1****, **
**G. BATEMAN****1****, **
**D.P. BRENNAN****2****,**
**A.H. KRITZ****1****, **
**S. KRUGER****3****, **
**P.B. SNYDER****4****, ****and the NIMROD **
**team**

## Ballooning, Peeling and Blobs

- Initiation of ELM coincides with MHD instability, **Peeling-Ballooning** -- Phil Snyder, Howard Wilson, Jack Connor.
- I would argue that the next stage is a nonlinear “explosive stage”.
- Extensive work on the motion of disconnected filaments, **Blobs, **probably describes the later stages -- J. Myra, D. Russell, S. Krashenikov, D. D’Ippolito
- Formation of fingers, narrowing, nonlinear growth and asymmetry has been seen in simulations -- M3D, Snyder, Dudson, Huysmans, Nimrod team, etc.

## Questions

- How does the filament form from the finger?
- Do the ends of the filament disconnect from the                              surface -- how?
- What fraction of the energy gets transported to                   divertor/wall?  Does the filament leak?
- How does the filament interact with the sheared flow?  How does it stop the flow
- What do the stochastic field coils do?

## Questions

- How does the filament form from the finger?
- Do the ends of the filament disconnect from the                              surface -- how?
- What fraction of the energy gets transported to                   divertor/wall?  Does the filament leak?
- How does the filament interact with the sheared flow?  How does it stop the flow
- What do the stochastic field coils do?

## All 12 coils fully installed 30/11/07

- Aluminium Alloy Housed in a boron-nitride shield

## Eric Nardon

-  Preliminary H-mode experiments show a clear effect on the ELMs
-  Remark: these experiments were done with only one NBI which limited the quality of the reference discharges
-  When increasing Icoil, the ELMs become compound, then dithering

- and Andy Kirk

## We need a better divertorfor DEMO even if we get ridof ELMS

## MAST UPGRADEA Chance to look atadvanced divertor solutionsSuper-X Kotschenreuther, Mahajan ....what happens to ELMs withthese divertors?