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CHAPTER 5
Determination of magnetic field and measurement
location
This chapter deals with using MSTFit [ref] to re-construct magnetic fields in MST.
Magnetic fields enable the sample volumes required for electric field measurements to
be calculated. An ion trajectory modeling code utilizes the magnetic fields to compute
these measurement locations. In MST, the overall magnetic equilibrium is dependent
on the current profile because the poloidal magnetic field is completely generated by the
plasma current and the toroidal field is modified by the poloidal currents at the edge.
The general methodology for obtaining the magnetic field will be discussed in
section 1. In section 2, the process of incorporating the HIBP results to yield magnetic
equilibria will be discussed. In section 3, MSTFit equilibria reconstruction results in a
standard 373 kA standard discharge will be presented. Equilibra used to determine the
measurement locations in the electric field results discussed in this thesis will be
presented in this section.
5.1 Using MSTFit for equilibrium reconstruction
The MSTFit equilibrium reconstruction code is a nonlinear, fixed boundary
Grad-Shafranov equation solver for fitting RFP equilibria [ref]. From the view point of
the HIBP, the usefulness of MSTFit stems from the following: its ability to re-construct
a magnetic field profile at almost any time period during the discharge, its ability to
produce results in a short time, its ability to use a multitude of diagnostic inputs as
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constraints to mimic experimental conditions and its ability to output magnetic
equilibria averaged over a relatively small time window (0.2 ms).
MSTFit utilizes a variety of inputs as constraints to reproduce MST-RFP
equilibria. These constraints range from standard measured plasma parameters such as
F,  , I p , edge magnetic measurements from coil arrays to electron temperature and
density profiles. In addition, constraints from internal magnetic diagnostics (Motional
Stark Effect and Polarimetry) and an on axis value of the safety factor, q , are also used
[ref]. The experimental data from these constraints are generally obtained for the
particular discharge for which equilibrium reconstruction is sought. In some cases not
all data is available in the same shot. Due to restrictions on digitization rates of the
various signals, the data is averaged over a time period of 0.2 ms before it is input into
MSTFit. The time window is determined by the lowest digitization rate of a particular
input parameter. Each piece of information that is input can be assigned a particular
weighting in the code. Once an equilibrium is obtained, quantities that define it such as
F,  , I p , are compared to the experimental data that was used to obtain the fit. Other
critical parameters to be evaluated include the current profile, j(r), and the q profile
(safety factor profile).
These two quantities are related through the following
relationship [ref]:
r 2 B
q

aB
a  j
rB
(5.1)
The safety factor is also given in terms of the following relation [ref]:
q
m
n
(5.2)
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where, m and n refer to poloidal and toroidal mode numbers.
Evaluation of these latter two quantities is an important element for determining the
validity of the equilibrium generated. The q profile generated from MSTFit must have
an on axis q value that is at least 1/6. This is because the dominant core resonant mode
in MST is the m=1, n=6 mode [ref].
Evaluating the current profile goes hand in hand with examination of the q –
profile because the two quantities are related through equation 1. MSTFit results show
that the current profile is relatively broad when q is higher than 1/6 and peaks when q
is lower than this value. Plots of the current and q-profile are illustrated in figure 1. In
this case, the central current density is almost 300 A / cm 2 in the case when q is less
than 1/6 while it is on the order of 240 A / cm 2 when q is greater than 1/6.
q-profile
400
0.15
300
0.10
A
Peaked J|| profile
200
0.05
100
0.00
-0.05
0.0
0
0.1
0.2
0.3
 (m)
0.4
0.5
-100
0.0
q-profile
B
0.20
250
0.15
200
0.1
0.2
0.3
 (m)
0.4
0.5
Non peaked J|| profile
150
0.10
100
0.05
50
0.00
-0.05
0.0
0
0.1
0.2
0.3
 (m)
0.4
0.5
-50
0.0
0.1
0.2
0.3
 (m)
0.4
0.5
Figure 1. (a) Peak and (b) non-peaked current profiles and associated q profiles.
5.2 Utilization of the HIBP constraint
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The sensitivity of the HIBP ion beam trajectories to magnetic fields has been
used to determine current profiles and magnetic vector potential fluctuations in a
tokamak[ref]. In MST-RFP, HIBP results have been found to constrain the equilibrium
reconstructed with MSTFit. This section focuses on the details of generating HIBP data
for this purpose and examining the trajectories that result from these magnetic fields.
With further investigation, it is believed that HIBP constraints can provide critical
information about the current profile and its evolution in MST. The HIBP information
required for MSTFit is illustrated in figure 2.
Figure 2. Illustration of HIBP input and exit vectors used in MSTFit
The information required can be summarized as follows:

Primary beam velocity vector (direction and magnitude of the injected beam) at
point 1. Injection location and injection vectors are obtained from a computer
simulation program, which utilizes actual values of the sweep voltages applied
to deflect the primary beam [ref].
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
Secondary beam velocity vector at point 2 (puncture plane of the exit port.)
At point 2, the magnitude of the speed of the secondary beam is obtained from
computing the beam energy (obtained after determining the plasma potential).
However, determination of the secondary beam vector is generally more complicated.
This is because while the injected beam is on the order of 1cm in diameter and can be
localized, the emanating spray of secondary beam from MST is fan shaped and cannot
be localized at position 2 as shown in the figure. The shape of the fan beam is shown in
figure 3, where it is seen to span the entire exit port. The 7 different fan beams shown
in the figure result from sweeping the primary beam radially in the plasma. The fans
emanate from 7 different sample locations in the plasma. The collimating effect of the
applied electric field in the drift region (secondary beamline ) is also shown in the same
figure
Figure 3. Illustration of fan shaped secondary beam at exit port. Trajectory of a portion
of the fan hitting the detector after passing through the entrance aperture is also shown.
Since a portion of the fan beam enters the analyzer through the narrow entrance
slit, the knowledge of its landing point on the detector is equally critical for determining
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the secondary beam velocity vector. Because the HIBP detectors are divided into two
left and right sections as shown in figure 3, localizing the beam at the detector can only
be accomplished at one point: the center of the detector1. This is also shown in the
figure.
Figure 5.4 illustrates a simulation result showing the range of possible secondary
ion beam trajectories obtained by fixing the termination point of the secondary beam
(on the detector). A view from above of the secondary beamline and analyzer region is
shown in this figure.
Trajectories
Toroidal sweep
plate
Analyzer space
MST
Exit port
Secondary beamline
space
Detector line
View from above
Figure 5.4: View from above of secondary trajectories at the exit port, and in the
secondary beamline and analyzer regions.
The trajectories shown in shown in figure 4 lie on a horizontal plane at the exit
port. In this particular case, the midplane of the exit port is chosen as the horizontal
plane. In reality this plane can be displaced vertically as shown in figure 5.5. The task
1
The fact that the beam lands near the center of the detector is determined by examining if there is
approximately equal amount of secondary beam on the left and right detector plates.
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of choosing a secondary beam puncture point to determine the secondary beam vector is
very arbitrary. In general any one of the 5 points shown can be chosen at random.
Puncture plane
Horizontal plane used
for exit vectors
Possible choice of
puncture point
for this horzontal plane
Figure 5. Plot of the exit port and a number of possible puncture points for
secondary beam that arrive at the midplane of the exit port.
The procedure of reconstructing the magnetic equilibrium using the HIBP
constraint is summarized in figure 6. The steps leading up to the point of initial
evaluation of the magnetic field are the same as those discussed in section 2.
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Figure 5.6: Chronology of equilibrium re-construction using HIBP constraints
The equilibria generated from inclusion of HIBP information in MSTFit are
evaluated by examining two additional outputs from this routine. These outputs are
shown in figure 7. The important information to evaluate in these two figures are the
tangency of the primary and the secondary beam and the distance of closest approach
between them. The ion orbits shown in these figures result from trajectory computation
122
using the magnetic field reconstructed with HIBP information. The primary and the
secondary beams are shown in both figures.
(a) Poloidal view of HIBP ion orbit
(b) View of MST as seen from above torus
0.6
0.4
0.4
0.2
0.2
-0.0
0.0
-0.2
-0.2
-0.4
closest approach: 0.18 cm
-0.4
-0.6
1.0
1.5
2.0
1.0
1.2
1.4
1.6
1.8
2.0
Figure 5.7: HIBP ion orbit plots output from MSTFit.
The tangency condition arises because the secondary beam emanates
everywhere along the path of the primary in the plasma. Hence, qualitatively, the two
beams should be as tangent as possible. The distance of closest approach value should
be minimized for the same reason. In reality the secondary beam and the primary beam
at the intersection point must lay virtually on top of one another in figure 7. However,
the finite thickness of the HIBP primary beam allows certain tolerances in the distance
of closest approach in simulations. Typically any distance less than 1 cm is considered
reasonable because the primary beam has a diameter of at least 1cm.
The final step in the evaluation process is to determine whether the magnetic
field generated can reproduce the HIBP experimental results obtained.
Ion beam
trajectories are generated using this magnetic field and evaluated to determine whether
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they approach the entrance aperture of the HIBP analyzer. A more complete discussion
is given below.
5.2 Reconstruction of magnetic fields in MST plasmas
Equilibrium reconstruction in a high current (average ~373 kA) standard discharge has
been extensively preformed using a variety of internal constraints using data from
diagnostics such as polarimetry, MSE and HIBP [ref]. The techniques utilized in
obtaining these results have been used to reconstruct equilibria for the range of
experiments (low and high current standard and low current biased and locked
discharges) described in this thesis.
5.2.1 Equilibrium reconstruction in high current standard discharges
The results of equilibria obtained using a variety of permutations of the internal
magnetic constraints are summarized in table 1. All of the fits generated from this study
satisfy the evaluation criteria outlined in section 1.
Fit Polarimetry MSE HIBP
1
X
X
2
X
X
3
X
X
2
Table 5.1: Permutations of internal constrains utilized in equilibrium reconstruction.
The magnetic field, current and q profiles generated for fits 1 and 3 are shown in
figures 8 [ref] and 9. The on axis magnetic field in each case is between 0.38-0.39 T.
2
Fit 1 was generated by Ted Biewer
Fit 2 was generated by Jianxin Lei
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Noticeable changes in the two fits are in the value of the peak current density and q .
These quantities vary by approximately 8% and 10%, respectively between fits 1 and 3.
q
0.20
0.4
0.15
0.3
0.10
0.2
0.05
0.1
0.00
0.0
-0.05
0.0
0.1
0.2
0.3
 (m)
0.4
0.5
0.4
0.5
-0.1
0.0
B, B, |B|
0.1
0.2
0.3
 (m)
0.4
0.5
J||
250
200
150
100
50
0
0.0
0.1
0.2
0.3
 (m)
Figure 8. Profiles of safety factor, magnetic fields and parallel current density obtained
from MSTFit re-construction using polarimetry and MSE constraints
q
0.20
0.4
0.15
0.3
0.10
0.2
0.05
0.1
0.00
-0.05
0.0
B, B, |B|
0.0
0.1
0.2
0.3
 (m)
0.4
0.5
0.4
0.5
-0.1
0.0
0.1
0.2
0.3
 (m)
0.4
0.5
J||
250
200
150
100
50
0
-50
0.0
0.1
0.2
0.3
 (m)
Figure 9. Profiles of safety factor, magnetic fields and parallel current density obtained
from MSTFit reconstruction using HIBP and MSE constraints.
125
The differences between each equilibrium can be further quantified by using
them to compute HIBP ion orbits. The results from this simulation using fits 1 and 3
(table 1) are illustrated in figures 10 and 11. The poloidal cross section and the view of
the MST toroid from above are projected in these figures. The HIBP primary beam
injection condition is fixed in simulations in order to make a meaningful comparison.
Qualitatively, the secondary trajectories computed using both fields show that they
approach the entrance aperture of the HIBP analyzer. The main difference between the
two cases is in the location of the sample volumes, which are observed to vary by 0.76
cm.
Figure 10. HIBP ion orbits in (a) poloidal and (b) toroidal planes computed using field
generated from MSE and polarimetry constraints
126
Figure 11. HIBP ion orbits in (a) poloidal and (b) toroidal planes computed using field
generated from MSE and HIBP constraints.
These simulations show that the field obtained using the HIBP (and MSE) yields
an HIBP ion orbit result that is consistent with that obtained using the polarimetry and
MSE diagnostics.
observations.
Furthermore, both are consistent with HIBP experimental
This is an important consequence from the point of view of
reconstructing magnetic fields in low current experiments because only internal
constraints from MSE and HIBP are available at these currents.
5.3.2 Reconstruction of equilibrium for electric field experiments and
determination of measurement locations
Equilibrium reconstruction was carried out using HIBP and MSE constraints for the
experiments described in this thesis. In particular, equilibria were generated for 383 kA
and 276 kA standard discharges. The equilibria were then used to determine HIBP
measurement locations required for electric field measurements.
In a 383 kA standard discharge equilibria were generated during three time
periods. These three time intervals are highlighted in the figure 12. The plot of the field
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reversal parameter is shown here to illustrate the exact times of the sawtooth crash
(14.5ms and 24.76ms), which serves as a time reference for equilibrium reconstruction.
Figure 5.12: Time periods of equilibrium re-construction during a sawtooth cycle.
These intervals correspond to: (a) after sawtooth crash, (m) mid-way through the
sawtooth cycle and (b) before sawtooth crash.
In particular, equilibria were reconstructed at 17.4ms, 20.5ms and 23ms during
the discharge shown in the figure 21. The standard parameters constraining MSTFit
were obtained for this shot and were averaged over 0.2 ms centered about these three
times. The value of the on-axis magnetic field, B , was inferred from the plot shown in
figure 133. The on axis magnetic field versus plasma current is plotted in this figure.
The magnetic field values depicted in this figure were averaged over a 3 ms time
interval at 2 ms after the sawtooth crash. The error in the measurement is on the order
of 10%. More generally, the equation describing the relationship between Bo and I p is
given by:
3
Data for this figure was provided by Dr. Gennady Fiksel
B_o (T)
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0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
100
200
300
400
500
Plasma current (kA)
Figure 5.13: MSE measurement of on axis B vs Ip for an F=-0.2 plasma
The profiles of magnetic fields, plasma current and safety factor obtained at
t=20.5 ms are plotted in figure 14. These quantities are qualitatively and quantitatively
similar at both 17.45 ms and at 23ms. The on axis magnetic field ranges from 0.38-0.39
T and the shapes of the profiles are also very similar; the variation in the peak plasma
current density is on the order of 4% of the peak value (245 A / cm 2 ).
q-profile
0.20
0.4
0.15
0.3
0.10
0.2
0.05
0.1
0.00
-0.05
0.0
B, B, |B|
0.0
0.1
0.2
0.3
 (m)
0.4
0.5
0.4
0.5
-0.1
0.0
0.1
0.2
0.3
 (m)
0.4
0.5
J||
250
200
150
100
50
0
-50
0.0
0.1
0.2
0.3
 (m)
Figure 14. Safety factor profile, magnetic field profiles and current profile obtained
using MSE and HIBP constraints at t=20.5 ms in discharge shown in figure 12.
129
HIBP trajectory simulations were carried out using these three magnetic fields.
The injection condition of the primary beam was obtained from experimental values of
sweep plate voltages that were recorded during this shot. The range of the sweep plate
values for the radial plates closest to MST was 2.6 kV to 6.8 kV. HIBP secondary
signal was detected for this range of radial sweeps.
Simulation results using the
magnetic field depicted in fig 14 (t=20.5ms) are illustrated in figure 15. The various
secondary ion beam trajectories result because the primary beam is swept in the
simulation.
Figure 5.15: HIBP ion trajectories generated using magnetic fields shown in
figure 5.14
The total separation between the measurement locations in each of the three
fields was also determined and is given in table 2. The total separation is obtained by
130
using the range of values of the sweeps that was used to detect secondary signal during
this discharge. The difference in the total separation, r , is on the order of 2.8 %.
Eq. time
T=17.45ms
Measurement
15.96-31.8
range (cm)
r (cm)
15.84
T=20.5ms
T=23 ms
16.02-32.08
16.07-32.35
16.06
16.28
Table 2. Times when equilibria are generated and the corresponding separation in HIBP
measurement locations.
The average potential profile results, discussed in chapter 4, were obtained
through trajectory calculations using magnetic fields described in this section.
In
particular, since the potential profiles were divided into three time intervals, the
corresponding magnetic field for each time interval was used to determine measurement
locations in each case.
In low current discharges, equilibria were reconstructed using HIBP and MSE as
internal constraints during the middle of the sawtooth cycle for a representative standard
discharge. The particular time reference with respect to sawtooth crash was discussed
in chapter 4.3. In low current edge biased and locked plasmas, the magnetic equilibria
obtained for the standard discharge were used to compute HIBP measurement locations.
In all low current experiments, the HIBP primary beam energy and applied
sweep plate voltages were nearly identical. Additionally, HIBP secondary signal was
also detected for the same range of sweep voltages applied during each experiment
(discussed in chapter 4). The important plasma parameters such as plasma current and
field reversal parameter varied by less than 2% in these three cases. Hence, given these
factors and the sensitivity of the HIBP ion orbits to plasma current, it was assumed that
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equilibria should be very similar if not identical in each of these three cases. Another
possible reason for the similarity in the magnetic fields is that the magnetic fluctuation
levels which alter the mean-equilibrium fields are very similar in standard, biased and
locked discharges [ref]. Magnetic fluctuations in these discharges are typically 1.01.4% of the mean field value [ref].
5.3.3 Uncertainty in magnetic fields
Quantifying the accuracy of the magnetic fields described in this section is difficult for
several reasons. First, the various plasma parameters that are input into the MSTFit
code themselves are subject to experimental error ranging from 10-20%. Second, there
are uncertainties in the MSTFit program(10%).
Third, the data from the MSE
diagnostic is also subject to a 10% experimental uncertainty. And finally, the choice of
HIBP inputs, particularly the exit vector of the secondary beam is somewhat arbitrary.
However, for HIBP measurements of electric fields, the accuracy of the magnetic field
is important only from the point of view of determining the relative separation of the
sample volumes.
Given these uncertainties, one way to quantify the uncertainty in the magnetic
field is to assess the impact on HIBP measurements. Using magnetic fields generated
for plasma currents that are different by 3%, HIBP measurement locations can be
determined in each case. Results from trajectory simulations for I p = 383 kA were
discussed in section 5.3.3.
The measurement locations during the middle of the
discharge ranged from 16.02 –32.08 cm. Using the same injection conditions (radial
sweep voltage range 2.8 kV-6.8 kV) and the equilibrium generated for I p = 373 kA,
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trajectory computations show that the measurement location ranges from 15.67-30.57
cm. The overall change in measurement locations is 1.16 cm between the two cases.
This represents a 7.8% change in r resulting from a 3% change in plasma current.
5.5 Summary
The HIBP measurement of the radial electric field has been challenged by the lack of
complete knowledge of the actual magnetic fields in MST.
However, an RFP
equilibrium reconstruction code known as MSTFit has been used to determine
equilibria and measurement locations for the range of electric field experiments
discussed in this thesis. The inclusion of the HIBP constraints into MSTFit, although
somewhat ambiguous, showed that the HIBP can be used as a constraint in determining
the current profile. The equilibria reconstructed with and without HIBP showed that
measurement locations changed by approximately 0.75 cm.
When the magnetic
equilibrium was computed for plasma currents of 373 kA and 383 kA (3% variation),
the spatial separation between measurement locations was found to change by 1.16cm
or 7.8%. Equilibrium reconstruction was also carried out at various times during the
sawtooth cycle in a 383kA standard discharge.
This allowed calculations of the
evolution of the HIBP measurement locations and that of the radial electric field in high
current experiments. The technique established to implement HIBP as an internal
constraint was also used to generate equilibria in low current standard discharges.
When the approach described here is used, we obtain consistency (if not perfection)
between experimental conditions and the equilibria obtained.