# Presentation: 103545

## Microtearing instabilities and electron transport in NSTX

- By
- K.L. Wong, S. Kaye, D.R. Mikkelsen,
- J.A. Krommes, K. Hill, R. Bell, B. Le Blanc,
- 34th EPS Conference on Plasma Physics
- Warsaw, Poland, July 2-6, 2007

## Outline________________________________________

- Introduction
- Properties of microtearing modes
- Calculation of unstable microtearing modes for NSTX experimental data
- Theoretical electron thermal conductivity due to microtearing instabilities in NSTX
- Comparison with TRANSP analysis
- Reversed magnetic shear plasma
- Summary

## Introduction_____________________________________

- Anomalous electron transport is an old subject, almost as old as magnetic fusion research itself
- While ETG turbulence is a possible candidate for the explanation in tokamaks, here for NSTX, we investigate the stochastic magnetic field effects from microtearing instabilities 1,2
- 	1. M.H. Redi et al., EPS (2003)
- 	2. D.J. Applegate et al.,Plasma Phys.(2004)

## Why is it important for NSTX ?___________________________________________

- Microtearing modes are thought to be important only at the edge of conventional tokamaks like D-III1 and C-MOD2
- They are stable in the interior of a tokamak where Te is high enough such that ei<*e
- They can be the most unstable mode in the interior of NSTX3, and should saturate at high amplitude due to the low magnetic field4
- N. Ohyabu et al., Phys. Rev. Lett. _58_, 120 (1987)
- J. Kesner & S. Migliuolo, Nucl. Fusion _39_, 163 (1999)
- 3.	M.H. Redi et al., EPS(2003)
- 4.	J.F. Drake et al., Phys. Rev. Lett. _44_, 994 (1980)

## Properties of microtearing modes_____________________________________

- High-m tearing modes (k||=0)
- Driven by Te gradient
- 	       - ’ is actually negative at high m (stabilizing)
- Perturbed electric field Er has odd parity, elongated along **B**		- different from the even parity for ITG modes
- Propagate in the electron drift direction (GS2: <0)
- 		- opposite to the ITG mode
- Perturbed magnetic field Br has even parity and creates 		  magnetic islands at q=m/n
- In slab geometry, instability requires (a) e=dlnTe/dlnne>0.3 	(b) Electron Coulomb collision rate e > *e

## Microtearing modes in toroidal geometry__________________________________________________

- In toroidal geometry, trapped particle effects come into play: (1) L. Chen (1977) - collisional detrapping is destabilizing - Krook e (2) Catto (1981) - Lorentz e
- This driving term has to compete with stabilizing effects like: (1) Field line bending effect (’ ≈ -2k< 0)    (2) Collisional effect of passing electrons (3) Ion orbit effects - stabilizing if dTi/dr < dTe/dr - Cowley (1986)
- When all stabilizing terms are retained, the trapped electron term is too feeble to drive microtearing instabilities in the interior of a realistic conventional tokamak - J. W. Connor (1990)

## The GS2 gyrokinetic stability code_______________________________________

- Initial-value algorithm
- Flux tube geometry (ballooning limit)
- Calculate the growth rate and mode structure of the most unstable (fastest growing) eigenmode in a preset range of wavenumber
- Get input parameters directly from TRANSP output file
- Ref: M. Kotschenreuther et al., Comp. Phys. Comm. 88, 128 (1995)
- 	    W. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000)

## Plasma equilibrium for NSTX #116313 at 0.9 s_____________________________________

## Mirnov coil signal spectrum__________________________________________

## Plasma Equilibrium____________________________________________________

## Results from GS2 code_____________________________________

- Microtearing modes can be the most unstable mode in the region r/a = 0.35 - 0.75
- In the region where Te has steep gradient (r/a=0.4-0.75), a broad spectrum of microtearing modes are unstable
-  ks is in the range of 0.3 - 1.0, comparable to ITG modes

## Cautionary Remark_______________________________________

- The GS2 code only finds the most unstable mode
- Microtearing modes may still be unstable at other values of ks , but they do not appear in the GS2 output because some other modes (usually the ITG) are more unstable
- ITG and microtearing modes have opposite k, which is reflected in the opposite signs of their real frequency

## Benchmark to analytic results__________________________________________

- 	Plasma parameters taken from TRANSP#116313A11, 0.9 s (NBI, H-mode)
-     e > 0.3 in the region 0.35 <  < 0.75
-    Choose  = 0.5 where:
- 	ne=6.5e13 cm-3, Te=650 ev, LTe=42 cm, Lne=78 cm, B=5 kG, Ti=800 ev, 			 e > *e is satisfied when microtearing modes are unstable according to the GS2 code

## Instability depends on ei__________________________________________________

## Nonlinear saturation _____________________________________

-    While the GEM code is still being developed, we rely on exiting nonlinear theory.		The nonlinear term causes an energy flow from short (unstable) to long wavelength modes which are stable. Growth and damping rates balance at  B/B ≈ e/LT ,
- 	and low magnetic field implies large e and high B/B.
- 	Ref: J. F. Drake et al., Phys. Rev. Lett. _44_, 994 (1980)

## Overlapping resistive layers_________________________________________

- Resistive layers overlap when m > mo = q(2q’s)-1/2 or k>ko
- For shot 116313A11 at 0.9 s :
-    s=(2Te/mi)1/2/ci,  B=5kG,  a=65cm,  =r/a
-     q    Ti    Te    s  q’=dq/dr    mo   ko=mo/r
- _____________________________________________________________________________________
- 0.4       2.3    900ev 780ev    1.1cm   0.06cm-1     6.3       0.24cm-1
- 0.5       2.8    800     680        1.1       0.068           7.2       0.22
- 0.6       3.2    630     560         1.0       0.13             6.3       0.16
- 0.7       4.5    470     440        0.8       0.29             6.5       0.14
- * D.A. D’Ippolito et al., Phys. Fluids 23, 771 (1980)

## Global stochastic magnetic field______________________________________________

- When adjacent island chains are separated by KAM surfaces(as electron transport barriers), stochastic field lines are localized near the separatrix
- When adjacent island chains overlap, global stochastic magnetic field is expected ( usually the KAM surfaces are destroyed before islands overlap - the 2/3 rule )
- One can estimate the island width from (Br)mn and find that adjacent islands overlap
- Substantial electron transport should ensue when either adjacent resistive layers or island chains overlap
- 		- see D’Ippolito et al.,Phys.Fluids(1980)

## Electron thermal conductivity in stochastic magnetic field_____________________________________

-    When islands overlap, magnetic field lines become stochastic with diffusion coeff DM, the electron thermal conductivity in the collisional regime (mfp<< Lc) can be estimated by :
- 	         e = DMve(mfp/Lc) where DM≈ R|B|2/B2
- 	Ref:  A.B. Rechester & Rosenbluth, Phys. Rev. Lett. _40_, 38 (1978)
- 		T.H. Stix, Nucl. Fusion _18_, 353 (1978)
- 		B.B. Kadomtsev & O.P. Pogutse (IAEA,Innsbruck-1978)

## Field line correlation length_______________________________________

- A rigorous theory1 on plasma transport in stochastic magnetic fields is extremely complex. The precise formula for the field line correlation length Lc is unknown
- We use Lc=qR = field line connection length2,3 instead of the Kolmogorov-Lyapunov length
- For NSTX plasmas, the electrons are weakly collisional:  1 < Lc/ mfp < 10
- J.A. Krommes et al., J. Plasma Phys. _30_, 11 (1983)
- J.A. Krommes, private communication
- B.B. Kadomtsev et al., IAEA (1978)

## Comparison between eexp  and etheo________________________________________________

- Drake’s theory assumes Ln>>LT which is not strictly valid in the experiment
- In the region where Ln>LT (=0.45-0.75), the agreement between eexp  and etheo is not bad (within a factor 3)
- In the region ≤0.4, Ln< LT ≥80cm, the driving term from dTe/dr is too weak, and e may be determined by some other mechanism
- Get much better agreement (within 40%) in the entire range (=0.40-0.75) if we replace LT by L where L-1 = LT-1 + Ln-1

## e due to saturated microtearing modes_________________________________________________

- Put B/B=e/LT, get e = (e/LT)2 Rve(mfp/Lc)= (e/L)2ve2/(eiq)
- Use parameters from #116313A11 at 0.9s, Lc= qR
-      Zeff  Te      ne    e(cm)   Ln  LT(cm) ei(s-1)   q   eexp  etheo(cm2/s)
- _________________________________________________________________
- 0.35   2.31  820ev  7.2e13  1.36e-2       75      133         8.1e5     2.0     2.0e5         0.71e5
- 0.40   2.16  780     6.8        1.33             69        80          7.8        2.3     1.37           0.98
- 0.45   2.03   735     6.5        1.29             65        57          7.6        2.6     1.06          1.2
- 0.50   1.92  680     6.2        1.24             78        42          7.7        2.8     0.88          1.2
- 0.55   1.82   620     5.8        1.19             78        33          7.9        3.0      0.80         1.2
- 0.60   1.75   560    5.65       1.13             83        28          8.6         3.4      0.77         0.98
- 0.65   1.74  500     5.4        1.06             48        25          9.7         3.9      0.80         1.0
- 0.70   1.77  430     5.1        0.99              52       22        11.7         4.6      0.81         0.60
- 0.75    1.84 380     4.8        0.93             35       15        13.8         5.6      0.74         0.61

## Higher Te(0) when microtearing modes are stabilized by reversed magnetic shear_______________________________________________

- Growth rate at  =0.3, t=0.32s
- #115821 - normal shear
- #116960 - reversed shear

## Final Remark_________________________________________

-  	NSTX has the flexibility to operate in many regimes. What I show here is just one type of discharge that allows us to study the effect of microtearing modes. There are several ways to improve electron confinement in ST:
- 	 1. Raise BT
- - HHFW heating efficiency improves
- 	 2. Raise Te so that e < *e
- 		- can get Te(0) ~ 4 keV by HHFW heating
- 	 3. Reversed magnetic shear
- 		- High m modes stabilized and Te(0) becomes significantly 	    higher with same NBI power

## Summary_________________________________________

- Quantitative analysis of anomalous electron transport in NSTX is carried out - use no fudge factor
- In the entire region of strong Te gradient, e observed in experiment is in reasonable agreement with nonlinear microtearing mode theory (within 2X), better agreement with grad(ne) included.
- This is not surprising because of the low toroidal magnetic field. The high mode amplitude causes global stochastic magnetic field.
- Microtearing modes are stable in reversed shear plasmas where Te(0) is substantially higher.
- Existing nonlinear theory can be improved by electromagnetic gyrokinetic simulation (GEM code).