# Presentation: 103544

## Magnetic fluctuations and electron transport in a spherical toakmak

- By
- K.L. Wong
- In collaboration with R. Bell, S. Kaye,
- B. Le Blanc, D. Mikkelsen
- Plasma Physics Seminar at UC-Irvine
- Jan 26, 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
- Summary / Discussion

## Introduction_____________________________________

- Anomalous electron transport is an old subject, almost as old as magnetic fusion research itself
- While ETG turbulence is a popular 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
- 		- 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		 	       	  - energy dependence of e is crucial

## 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 Code_______________________________________

- Initial-value algorithm
- Flux tube code (ballooning limit)
- Calculate the growth rate and mode structure of the most unstable (fastest growing) eigenmode in a 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_____________________________________

## 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i 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i, 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

## Bench-mark with analytic theory__________________________________________

- 	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

## Nonlinear saturation _____________________________________

-    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 islands____________________________________

- Islands 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)

## 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 complicated. 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)

## e due to saturated microtearing modes_________________________________________________

- Put B/B=e/LT, get e = (e/LT)2 Rve(mfp/Lc)= (e/LT)2ve2/(eiq)
- Use parameters from #116313A11 at 0.9s, Lc= qR
-      Zeff  Te      ne    e(cm)   ve  LT(cm) ei(s-1)   q   eexp  etheo(cm2/s)
- _________________________________________________________________
- 0.35   2.31  820ev  7.2e13  1.36e-2      1.2e9   133         8.1e5     2.0   2.0e5     0.093e5
- 0.40   2.16  780     6.8        1.33          1.17        80          7.8         2.3   1.37         0.21
- 0.45   2.03   735     6.5        1.29          1.14        57          7.6         2.6   1.06         0.34
- 0.50   1.92  680     6.2        1.24          1.09        42          7.7         2.8   0.88        0.48
- 0.55   1.82   620     5.8         1.19          1.04        33          7.9         3.0   0.80        0.59
- 0.60   1.75   560    5.65        1.13          0.99        28          8.6         3.4   0.77        0.55
- 0.65   1.74  500     5.4          1.06          0.94        25          9.7         3.9   0.80        0.42
- 0.70   1.77  430     5.1          0.99          0.87        22          11.7        4.6   0.81       0.28
- 0.75    1.84 380     4.8          0.93         0.82        15          13.8        5.6   0.74        0.33

## 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

## 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 3X), better agreement with grad(ne) included.
- This is not surprising because of the low toroidal magnetic field
- More analysis (GEM simulation) is needed on more data before a definitive conclusion can be made
- This result may have some bearings on
- 		- high Te(0) in reversed central shear  (Levinton-ZI1.3) 		- fast cold-pulse propagation  (Tritz-NO1.5)
- 		- inefficient HHFW heating/CD at low k|| (Hosea-NO1.11)		- e~ BT-0.9 (Kaye-QP1.8)

## 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