Transcript – Physics results from experiment SSPX and simulation Bick Hooper

SSPX – Physics results from experiment
and simulation
Presented at
Princeton Plasma Physics Laboratory
Princeton, NJ
January 7, 2008
Bick Hooper
Fusion Energy Program, LLNL
UCRL-PRES-400218
Work performed under the auspices
of the U. S. Department of Energy by
Lawrence Livermore National
Laboratory under contracts W7405ENG-48 and DE-AC52-07NA27344.
This work was conducted with a large number of
collaborators, both experimentalists and theorists
The entire SSPX team contributed through SSPX operation, critique of the
NIMROD studies, and comparisons with experimental results
Collaborators include
Dave Hill
Harry McLean
Carlos Romero-Talamás
Simon Woodruff
Ken Fowler
Lynda LoDestro
Additional collaborators came from Caltech, U. Washington, FAMU, LANL,
U.C. Berkeley, U.C. Davis. We are members of NSF Frontier Science CMSO
NIMROD resistive MHD simulations conducted with Bruce Cohen and in
close collaboration with Carl Sovinec (Wisconsin)
Computer facility support provided by:
NERSC and Bill Meyer at LLNL
The Sustained Spheromak Physics Experiment
(SSPX)
Sustained Spheromak Physics Experiment
Thick wall (~.015 m) copper flux
conserver (dia=1 m).
Tungsten coating reduces
sputtering. Ti gettered to reduce
impurities
≥ 4 msec discharges. 500 kA peak
formation, 250 kA sustained
9 solenoidal coils provide flexible
vacuum field programming
Experiment –– Above a gun-current threshold the
spheromak field builds
Normalized gun current
 gun 
 0 I gun
gun
( I gun  gun current )
( gun  bias flux )

Flux conserver
  B  B
  0 j B
eigenvalues :  0 ,...
 0  9.9m -1
Threshold (simulation)


 gun  10.0 m –1
SSPX –– Summary of results
• Te = 500 eV
• nTe(peak) ~ B2
• e < 10 m2/s in spheromak core
• Magnetic fluctuation spectrum correlates well with q-profile
• Confinement has no (or weak) scaling with mass
• Energy confinement improves with Te and low-level
fluctuations
• Charge-exchange losses are not dominant
• Basic understanding of magnetic reconnection in the
spheromak
• NIMROD “whole-device” simulations agree well with many
experimental results
Whole-device model of SSPX includes power systems and
bias-coil magnets (not shown)
New, lower
resistance cables
Sustainment Bank 5kV, 120mF
SSPX
Modular Bank
(5 kV, 40 mF)  30
Formation Bank 10kV, 10mF
Whole-device, resistive MHD simulations of SSPX
discharges and physics have contributed significantly to
understanding experimental results
• Generally good agreement with experiment
• Predictive capability –– guide experiments and permit
exploratory modeling
• NIMROD's capabilities extended
• Resistive MHD physics studies in tokamaks and
alternative concepts complemented
Improved Match of NIMROD Simulations to Experiment
•
NIMROD: lam06
Igun (MA)
Shot 12560
Improved NIMROD simulations with
– Spitzer-Braginski resistivity and
parallel thermal conduction
– more detailed representation of
gun geometry
– careful match to current-drive
time history with experiment
give improved agreement with Bz and
Te in SSPX.
SSPX Bp (T) total
n=2 mode (au)
Te (eV)
SSPX
n=0
Simulation
•
As current ramps down, a large
amplitude n=2 mode is observed in
both experiment and simulation.
From: B. I. Cohen, APS invited talk, Nov. 2004
NIMROD Edge-Probe Fluctuation
History Is Similar To SSPX Data.
• During the strong drive period
t < 0.4ms the n=1 mode is active,
Bz,n1 /Bz ~ 7.3% , and the
closed field lines begin to form
after 0.4 ms.
lam06
n=1
n=1
n=2

n=2
n=1,2
• The n=2 mode emerges for
0.4ms < t < 2ms and gradually
subsides, but re-emerges at the
end of the simulation, > 4 ms and
Bz,n2 /Bz  7%
n=2
n=1,2

B. Cohen, Hooper, et al.
• The amplitudes of the magnetic
fluctuations and their temporal
history are very much like those
measured in SSPX.
NIMROD Simulations Show Good Flux Surfaces
and Electron Temperatures Similar to SSPX.
Magnetic Field Lines
NIMROD
•
t=2.25ms
Puncture Plot
Te(eV) contours
NIMROD
•
90
10
123
•
NIMROD simulations show regions
of good confinement (0.25<R<0.4)
surrounded by islands and chaotic
lines (0.15<R<0.25,0.4<R<0.48) and
then open field lines.
Electron temperature contours
align with the magnetic field lines.
Local flattening in the electron
temperature profile due to the
presence of islands
t=2.25ms
SSPX
•
Transition from good flux surfaces
to chaotic and then open field lines
leads to steep drop in Te
From: B. I. Cohen, APS invited talk, Nov. 2004
Spheromak formation
Magnetic reconnection
Magnetic field buildup
Experiment and NIMROD ––
Voltage spikes occur in both –– have the same effect on
building and sustaining the plasma
Same as
experiment
SSPX and NIMROD with identical
gun currents
The “bubble-burst” of plasma
from the gun is followed by
voltage spikes
Toroidal flux is converted into
poloidal flux at each spike ––
Reconnection
Magnetic oscillations (esp. n=1)
– driven by the gun current
– grow between voltage spikes
– relax at the reconnection event
NIMROD –– Reconnection events continue if the current is flattopped with gun > threshold
Simulation (shown) and experiment both
show voltage bursts when strongly driven:
here, gun > 2Xthreshold
 gun 
 0 I gun
 gun
 21.1 m Π1

Poloidal flux
amplification
= 5.6
(extrapolates
to 6.7)
NIMROD simulations
NIMROD – Reconnection occurs when the amplitudes of modes
on the current column become large –– a “bursting” behavior
The n=1 mode dominates
the spectrum
The poloidal magnetic field and flux increase
with each event –– the temperature collapses
NIMROD – Magnetic surfaces are destroyed at each spike
• Magnetic surfaces may
form soon after the event
• The plasma heats until the
next event
• The magnetic lines become
stochastic and the plasma
cools rapidly
70 µs
NIMROD – Thin current sheets form and allow rapid
resistive reconnection
Current sheets are important
for reconnection:
• Resistive diffusion times
across the sheet are
comparable to
reconnection times
• Azimuthal structure is n=1
–– the sheet is diffuse
~ radians azimuthally
• Current reverses and large
flows perpendicular to B
are localized at the sheet
• Sheet forms near the meanfield separatrix
Trajectories of fieldlines which pass close to the current sheets are very
sensitive to their precise location –– as expected for generation of chaos
The mean-field generation is insensitive to the max. toroidal
mode number for n > 5
(=100 m2/s, n=5x1019 m–3)
NIMROD
SSPX
NIMROD (slowly decaying): Mode magnetic energy drops
rapidly with toroidal mode number (n ≤ 21) at high n
–– Most studies can use n = 5
kin_visc=1000 m2/s
kin_visc=100 m2/s
B_en_e3
B_en_e3
6.6069g-3(10/n)^89
1
0.1
10-5
0.01
-6
10
0.001
-9
10
10
2
par_visc=1.1x10 m /s
0.0001
6
1
10
Toroidal Mode No., n
0.1
0.001
0.01
0.0001
10-5
0.001
6
par_visc=1.1x10 m /s
-10
10
10-11
2
q-profiles depend slightly on viscosity
kin_visc=100 m /s
10-5
1
2
par_visc=1.1x10 m /s
2
2
kin_visc=1000 m /s
10-7
2
Mag. Energy (J)
10-8
Mag. Energy (J)
10-7
-6
n
-4
Kin. Energy (J)
0.0001
n
0.0001
Kin. Energy (J)
10-6
10-5
Р7
10
Kin. Energy (J)
Mag. Energy (J)
-9
0.001
-7
0.01
n=7
10
n
10
.0001252*(11/n)^4
1
-4
-5
6
Kv_n=21
0.001
n=7
n=5
0.01
10-8
v_kv_e1
v_ke_e3
Kv=e3_n=21
1
0.001
0.1
B_en_e1
0.004564(10/n)^7
v_kv_e3
Kv=e3_n=21 2:24:39 PM 12/31/06
kin_visc=10 m2/s
10-8
10
Toroidal Mode No., n
kin_visc=1000 m /s
0.0001
1
10-6
10
Toroidal Mode No., n
A transition from mode energies ≈ constant to a "cascade"
at high n
• Slope of the "cascade" is roughly proportional to
kinetic viscosity squared:
[index for (mode energy)1/2 ~ const.+log(kin_visc)]
Hypothesis: The transition occurs when viscosity is strong
enough to stabilize the modes
• The "cascade" results from mode coupling with
2
2
viscous damping (  v ~ n v )
Viscosity affects detailed time history but the generation of the
mean field is only weakly affected
Viscosity (100 m2/s and 1000 m2/s):
• More structure in the time
histories of current, field, etc.at
low viscosity
• However, the n=0 magnetic field
strength is insensitive to the
value of viscosity, dropping about
20% with an order of magnitude
increase in 
NIMROD:
(ntor,max=5, density =
5.0x1019
m–3)
Formation studies successfully reproduce the mean-field buildup
Resistive MHD is a good model for spheromak formation by helicity injection into
a flux-conserving geometry
Mean-field (azimuthally-averaged) parameters are generally reproduced well and
are relatively insensitive to assumed number of toroidal modes, viscosity, etc.
• Field ejection from the gun – timing and voltage response (compared to nearlyaxisymmetric experimental ejection)
• Magnitude of the gun voltage and current ≈ experimental values
• Azumuthally-averaged magnetic field ≈ experimental values
Detailed time histories are relatively sensitive to assumed number of toroidal
modes, viscosity and similar parameters
• Several parameters require low kinetic viscosity and ntor,max>1
– Sharp voltage spikes
– Structure in the time-history of the magnetic field
• Precise time histories (including fluctuation timing) differ from the experiment
These results suggest that field buildup is insensitive to the detailed physics in the
reconnection layer
Magnetic field sustainment
and
Flux Amplification
Sustained NIMROD discharges: Steady-state with
gun=20.3 m–1, and 13.6 m–1 (bias flux=51 mWb)
gun=20.3 m–1
gun=13.6 m–1
Gun
current
Gun
voltage
Note the different time scales
NIMROD – Poloidal flux amplification increases with gun above
threshold ––simulations show this clearly:
Flux Amplification = mag.axis/sep
Shown is the flux amplification in
a constant gun current pulse
Larger gun yields higher flux
amplification, consistent with
experiment
It is clear from the simulations and from a hyper-resistivity model that the
poloidal-field buildup above threshold is a balance between reconnection
and resistive losses.
We do not have a simple model that makes quantitative predictions
NIMROD Simulations Agree with Flux Amplification at
moderate gun in SSPX
• NIMROD predicts the flux amplification
in SSPX for current ≈ constant
• Experimental flux amplification scales
with gun as predicted by NIMROD.
200
Gun
Shot #
Curren
20111
t (kA)
8
3

2
Flux Amplification
Flux Amplification
400
gun
(m- 16
Driven 1
)
Not Driven
Shot 20111
 (m–1)
Gun Current (kA)
SSPX
3
NIMROD
Experiment
2

1
Exper. (GradShaf. analysis
Exper. (Bessel
Function model)
NIMROD
1
0
2
Extende
d
Formatio
Driven
n
SSPX – "Extended
Formation" (BesselFunction model)
Not
Driven
4
Time (ms)
6
8
8
10
12
gun (m-1)
14
16
Maximum amplification of the bias magnetic flux:
For strong drive, experiment and simulation differ
Poloidal flux:
6
5
Flux Amplification
Use Bessel function model and
measured Bp(edge)
standard formation
extended formation
slow building
nimrod
Shots with lower amplification
result from several causes
NIMROD
4
• Formation pulse too short to
reach saturation
3
• Lack of density control
2
• Other "kitchen physics"
problems
1
0
There appears to be a consistent
problem at high gun
0
5
10
15

gun
Data: Wood, Hudson, et al.
20
-1
(m )
25
30
• The amplitude of the n=1 mode
becomes very large, perhaps
exacerbating other effects
Flux Amplification in an Extended Flux Conserver
A series of experiments have been conducted with the flux conserver
length increased from 0.5 to 0.6 m (L/R from 1 to 1.2)
Spheromak poloidal flux (magnetic
axis) in SSPX is determined from
Bz at the wall:
Bessel function model (midplane
probe): /Bz=332 Wb/T
NIMROD (probe near midplane):
/Bz=324±5 Wb/T
Correction (from NIMROD) for
SSPX Probe 9 position:
/Bz=354 Wb/T
NIMROD Flux Amp.:
FA=1+0.56(gun–7.5)
Data: Wood
Flux Amplification –– scaling with L/R
We have two threshold determinations from NIMROD, both of the form:
gun=1+0.56(–th)
In a cylindrical spheromak:
The tilt (n=1,m=1) mode is stable for
L/R < 1.67*
The approximate agreement with the
threshold (extrapolated to th=0)
suggests that:
• Coupling to the spheromak is
stabilizing for the column (n=1)
mode when the spheromak is
stable to the n=1, m=1 modes
Note: Scaling as thL=const. would
yield th=8.3 for L/R=1.2
*J. M. Finn and W. Manheimer, Phys. Fluids 24,
1336 (1981).
A. Bondeson, et al., Phys. Fluids 24, q682
(1981).
Studies of slowly-decaying plasmas
Te ~ 500eV in SSPX obtained by extending the formation
Extended formation
Te: Shot 17096 @ 1.50 ms
800
Lower density (less gas)
Te (eV)
600
525eV
400
X2 flux higher Bp edge
200
Similar fluctuations
0
0
0.1
0.2
0.3
Radius (m)
0.4
0.5
Optimum edge (= gun) and strong heating
produces high Te
edge 
0 I gun
gun
const
varied
Peak T vs 
e
500 
T vs. Radius at Optimum
e
edge

500
out_modes_high_e
ptsfit3_14546 11:17:41 AM 10/11/05
fc
400
400
Igun=260 kA
300
eV
eV
300
350 eV
200
230 kA
100
200
100
0
7
8
9

edge
10
-1
(m )
11
12
0
0
0.1
0.2
0.3
Radius(m)
Nimrod simulations show a similar optimum –
– associated with reduced mode activity
0.4
0.5
Equilibrium reconstructions –– optimal edge results in
slightly-peaked safety factor profile with 1/2 < q < 2/3
1  m  2
 q  
2 
n  3

Fluctuations grow when low-order mode-rational surfaces appear in plasma.
The safety factor in the simulation is similar –– but
not identical –– to that in the best SSPX shots
Separatrix
Nimrod
SSPX
Parameter sensitivities in slowly-decaying plasmas
–
Effects of mode amplitudes and q-profiles
Simulation parameter effects on evolution of magnetic
field and detailed time-history of temperature
The best SSPX shots have higher electron temperatures than
NIMROD!
Sensitivity to simulation parameters
The magnetic-field evolution is relatively insensitive to the
various simulation parameters
Temperature is relatively sensitive
• This appears to result from a feedback loop involving the
evolution of temperature
Temperature
Profiles of  and q
Fieldline quality (e.g. flux surfaces, magnetic islands and
stochasticity)
Energy confinement
It is an open question whether the experiment has a similar
sensitivity, e.g. to changes due to impurities
NIMROD has successfully modeled many experimental
results, providing insight into spheromak MHD physics
• Magnetic-field buildup by reconnection
– Buildup of the spheromak field is insensitive to simulation parameters
– The time history of the mean-field field, Te, etc. are sensitive to parameters
– Voltage spikes on the gun result from reconnection events
– Stochastic fieldlines during reconnection result in observed low Te during
magnetic buildup and sustainment
• Flux amplification at moderate gun agrees with experiment
– Amplification simulations (to date) suggest that the spheromak helps
stabilize the n=1 column mode
– Flux amplification at high gun is less in the experiment for reasons that are
not understood
• High Te results when flux surfaces close
– Surfaces form when magnetic fluctuations are low – Te is thus very
sensitive to the q-profile
Simulations helped guide the experiment and are being used to
explore alternative spheromak configurations
Backup slides
Limiting e is observed experimentally –– Likely due to
ohmic heating, but a stability limit cannot be excluded
500
400
vs /eV
TeTvs.
nk/PB
e
Shots 9895-10095
400
out_modes_050331.xls
20 nkT

2
B
20 nkT  B2
350
300
200

T 
e
100
0
0
0.1
0.2
0.3
B
edge
(T)
200
e
T (eV)
250
300
Te (eV)
150
100
Q  nT  Pohmic  RI 2
50
nT
5%
BI
0
0 100 2 10-4 4 10-4 6 10-4 8 10-4 1 10-3 1.2 10-3
a
 RB2
2
2nk /B (1/eV)
0

Need external heat source to
differentiate transport limit
from pressure limit
0.4
0.5
Electron thermal diffusivity in core of SSPX is well
below Bohm, and scales as Te-5/2
SSPX core  vs. T Transport Scaling
e
1000
Power balance between ohmic heating and
radial transport yields thermal
diffusivity:
e
out_modes_good_data_5_opt
RR
Q 
100
Bohm
 nT dS
e
  Zeff Te3 / 2
2
m /s 10
2
 Pohmic 

j2 dV
Ti  Te
2
50(B/B)  /
s
 ~T
e
-5/2
e
2
 /
s
0.1
0
100
200
300
Core T (eV)
400


Te
16B
e
out_modes_good_data_5_opt 12:31:03 PM 10/17/05
3.5
500
1
1
1


Lc e R
3
~
DBohm 
4
s
e
2


dB
 RR vtheLc 
 B 
B/B vs. T
~Bp/B %
1
s
2.5
2
1.5
1
0.5

0
0
50 100 150 200 250 300 350 400
T (eV)
e
Using NIMROD for studying possible
advanced spheromak experiments
High flux amplification by Ramp-down of
gun flux and current
Active Bias Reduction (ABR)
Reduced power losses in edge plasma
and improving MHD stability
Reducing the bias flux and gun current by a factor of 10
increases the volume of good flux as the current column shrinks
Ramp down in 1 ms
Start
Finish
Reducing the bias flux and gun current by a factor of 10
decreases the amplitudes of the MHD modes by a large factor
Total mode energies
Amplitude at flux
conserver drops to ~ 10–3
of n=0
Topics for continuing spheromak simulations
Simulations and other modeling can continue to advance spheromak physics even if
the experiment ends:
• Use the SSPX data base to interpret results and improve physics
• Explore options for advanced experiments
• Strengthen comparative studies with RFP and FRC
Analysis of SSPX data
• Linear calculations to identify modes (ideal MHD, tearing)
• Finite beta –– is the limiting beta in SSPX due to MHD modes?
• Simulate multipulse discharges
– Reconnection physics
– Recovery of good confinement, high Te
• Explore effects of increased flux-conserver Length/Radius
• Clarify saturation of flux amplification and develop analytical approximation
– Calculate flux amplification at fixed Te (verify role of power balance)
– Compare with hyper-resistive model –– obtain hyper-resistive diffusion coefficient for
use in Grad-Shafranov model used in Corsica
• Model experimental limits of peak B/I (flux amplification) –– current distribution on cathode?
Continuing spheromak simulations
Analysis of spheromak opportunities
• Explore effects of geometry
– A new grid has been developed for exploring geometry changes but not yet used in
NIMROD
– Vary the gun radius –– Does it matter?
– Is there an optimum Length/Radius?
• Extend ABR calculations to more general geometries
• Demonstrate quasi-steady state with ABR and multi-pulse rebuilding of the magnetic field
• Do 2-fluid effects matter? (plasma rotation, reconnection)
• Explore auxiliary heating and current drive, e.g. with a simple model for neutral-beams
• Explore current-profile control –– establish requirements and examine options
The program presented on these two slides will take several man-years and substantial
computer time. The goal is to generate the maximum physics results from the SSPX
experiment and provide a substantial knowledge base for any future experiments