Download Gerhardt_Measurements Poster_APS 2003

Measurements and Modeling of Biased Electrode Discharges in HSX
S.P. Gerhardt, D.T. Anderson, J.M. Canik, W.A. Guttenfelder, and J.N. Talmadge
HSX Plasma Laboratory, U. of Wisconsin, Madison
Key Points
4. Neoclassical Modeling of Plasma Flows
•IV Characteristics of the Biased Electrode.
•Evidence of Two Time Scale Damping in HSX.
•Neoclassical Modeling of Plasma Flows.
•Experimental Verification of Reduced Viscous Damping with Quasi-Symmetry
•Computational Study of Different Configurations
1. Structure of the
Experiments
3. Two Time Scales
Observed in Flow Damping
General Structure of Experiments
Simple Flow Damping Example
•
Solve the Momentum Equations on a
Flux Surface
•
mi N i
•
•
Mach Probes in HSX
•
•
•
6 tip mach probes measure plasma flow speed and
direction on a magnetic surface.
2 similar probes are used to simultaneously measure
the flow at high and low field locations, both on the
outboard side of the torus.
Data is analyzed using the unmagnetized model by
Hutchinson.
Looking  To The
Magnetic Surface
•
•
As the damping  is reduced, the flow rises more slowly, but to a
higher value.
Full problem involves two momentum equations on a flux
surface2 time scales & 2 directions.
U1,exp t   X 2 t  cos X 3 t 
•
Fit flows to models
U (t )  C 1  exp  t / 
•
s
2, fit
f
f
f
s
s
s
Model Fits Flow Rise Well
3
This allows the calculation of the radial electric field evolution:
 






U  1  e t /1 S1e  S3e  1  e t / 2 S2e  S4e



•
•


•
“Forced Er” Plasma Response Rate is
Between the Slow and Fast Rates.
Decay Time:
=33x10-6 seconds
Impedance: R=V/I=36 
C=  /R=8.9x10-7F
Accumulated Charge
from “Charging Current”:
Q=2.33 Coulombs
Voltage: V=310 Volts
C=Q/V=7.5x10-7F
Original Formulation of Capacitance/Impedance modeling published in J.G.
Gorman, L.H. Rietjens, Phys. Fluids 9, 2504 (1966)
Impedance is Smaller in the Mirror
Configuration
•
•
Current peaks at the
calculated
separatrix.
Electrode current
profile does not
follow the density
profileElectrode is
not simply drawing
electron saturation
current.
•
Mirror
•
QHS
Predicted Separatrix
Position
Impedance1/n
consistent with radial
conductivity scaling like n.
Consistent with both
neoclassical modeling by
Coronado and Talmadge
or anomalous modeling
by, for instance,
Rozhansky and Tendler.
•
•
1/n
•
Increasing the N4 Mirror Percentage is
Efficient at Increasing the Damping
10%
7.5%
5%
Mirror, slow rate

QHS, slow rate
40
Capacitance is Linear in
the Density
•Plasma dielectric
constant proportional
to density.
•Low frequency
dielectric constant:
1
2 1500  8.854 1012 2 1.2
 1F
ln 17.5 11
N4 Mirror (Standard)
+
+ + -
-
-
-
-
N12 Mirror
+
-
-
Broad N Mirror
+
-
+ -
+ +
-
-
+ +
-
-
+
-
-
+ -
+ -
+
Hill
+
+ + + +
+ +
+ +
+ + +
+
-
Percentage indicates the ratio of amp-turns in the two
coil sets.
Large Change in Surface Shape for
Deep Hill Configuration
QHS
Other Mirror Configurations Result
in Less Damping
+ + +
10% N4 Mirror
10% N12 Mirror
Hamada Spectrum
10% N12 Mirror
10% Broad N
QHS
16.66% Hill
Hamada Spectrum

B2
4 10 7  5 10 11 1.67 10  27  9 1016
 1500
.52
C
10% N4 (Standard) Mirror
Hamada Spectrum
Deep Hill Mode Leads to a Slight
Damping Increase
 o c 2
•Expected
capacitance:
2.5%
Aux. Coil Current
Direction (+ for
adding to the toroidal
field of the mail coils)
QHS, F
Positive Part of I-V Curve is Linear
Linear I-V relationship
Consistent with linear
viscosity assumption.
Very little current drawn
when collecting ions 
collect electrons in all
experiments in this
poster.
•H signal is linear in
the densityneutral
density approximately
constant.
•See Poster by J.
Canik for more details
on the H system and
interpretation.
•Flow decay time
becomes longer as
density increases.
13%
Two time scales/two direction flow evolution.
Radial Conductivity Has a 1/n Scaling
QHS Damping Does Not Appear
to Scale with the Neutrals
Scan of N4 Mirror Percentage
2. I-V Characteristics of Electrode Indicate Transport Limits the Current
Bias Waveforms Indicate a
“Capacitance” and an Impedance
QHS
Mirror, F
t 0


Many Different Configurations are
Accessible with the Auxiliary Coils
Fast Rates, QHS and Mirror
ExB flows and compensating Pfirsch-Schlueter flow will grow on the
same time scale as the electric field.
Parallel flow grows with a time constant F determined by viscosity
and ion-neutral friction.

Note: Floating Potential
Rises Much Faster
Than the Flows.
Mirror
Neutrals Only
6. Computational Study: Viscous Damping in
Different Configurations of HSX
t 0



t /  
U t   U E 1  e
e  BQ1 E 1  1  Q2 e  F t  Q2 e t / 
Total Conductivity
Viscosity Only
QHS Damps Less Than Mirror; Some Physics Besides Neoclassical and
Neutral Damping Appears to be Necessary to Explain the QHS Data.
Assume that the electric field, d/d,is turned on quickly
Er 0
 

 Er 0   E 1  e t /
QHS
Mirror

Formulation #2: The Electric Field is Quickly
Turned On.
•
1/in
QHS
4
•

Coronado and Talmadge Model
Overestimates the Rise Times
1/in
<B>=0 in net current
free stellarator, but not
a tokamak.
<B>= Boozer g=
2*1e-7*48*14*5361
=.7205
•
Measurements
Combination of neutral friction and viscosity
determines radial conductivity.
Mirror agreement is somewhat better.
t /  2
•

•
Define the radial conductivity as
 
J  
R
d d
2
t / 1

2 SS
Similar model 2 time scale / 2 direction fit is used to fit the
flow decay.
B / Bo  1   H cos4      M cos4 
Mirror :
1
d
t   d t  0  F1 1  et /1  F2 1  et / 2
d
d
t   C 1  exp  t /  sin    C 1  exp  t /  sin    U
U1, fit t   C f 1  exp  t /  f cos f   Cs 1  exp  t /  s  cos s   U1SS
U
Symmetry Can be Intentionally
Broken with Auxiliary Coils
 fˆ  C 1  exp  t /  sˆ
Tokamak Basis Vectors Can Differ from
those in Net Current Free Stellarator.
t /  2
S1…S4, 1 (slow rate), and 2 (fast rate) are flux surface quantities
related to the geometry.
Break the flow into parts damped on each time scale:
•
Convert flow magnitude and angle into flow in two directions:
Probe measures Vf with a proud pin.
QHS :
B / Bo  1   H cos4   
U
Flow Analysis Method
Predicted formf of flow rise f from modeling:
s

S  1  e S
S  1  e S
 1  e
U  1 e
QHS
Mirror
The “Forced Er” Model
Underestimates the QHS time
Original calculation by Coronado and Talmadge
After solving the coupled ODEs, the contravariant components of
the flow are given by:
t / 1
QHS
Mirror
Use Hamada coordinates, using linear neoclassical viscosities.
No perpendicular viscosity included.

U 2,exp t   X 2 t sin  X 3 t 
 X 

I sat    X 1 exp   2 .641  cos  X 3   .71  cos  X 3 
 2 

•
•
•
0
t0


U   jB 1  exp  t / nm t  0
 
•
•QHS Flow Rises and
Damps More Slowly

QHS Modeled Radial Conductivity agrees to
a Factor of 3-4
QHS Flow Damps Slower, Goes Faster
For Less Drive.
•Flow Goes Faster For
Per Unit Drive
Formulation #1: The External Radial Current
is Quickly Turned On.
 0 t0
dU
 F  U , F  
 jB t  0
Has solution dt
Need quantities like <e·e >, <e·e >, <e·e >,<||>, <| ·  |>.
Previous calculation used large aspect ratio tokamak approximations.
Method involves calculating the lab frame components of the
contravariant basis vectors along a field line, similar to Nemov.
Solve these with Ampere’s Law

•
•
We Have Developed a Method to Calculate
the Hamada Basis Vectors
t
 B U    B      mi N iin  B U 
  
  
 
t 

    4  J plasma      J ext   
  





Take a simple 1D damping problem:
•
t
c
 BP  U  
 J      BP      mi N iin  BP U 
  
g B  B  
  
 
mi N i
mn
•
Two time scales/directions come from the coupled momentum
equations on a surface.
•
•
•
5. Comparisons Between QHS and Mirror
Configurations of HSX and with Modeling
QHS
Scan of Hill Percentage
16.67% Hill
3.33% N4 Hill
13.33% Hill
QHS