Download Flow Measurements on Single and Merging Spheromaks atSSX

Flow Measurements on Single and Merging Spheromaks
atSSX
Jason Horwitz
Swarthmore College Department of Physics & Astronomy
March 19, 2007
2
Abstract
Results are presented from measurements of flow on spheromaks at the Swarthmore Spheromak
Experiment.
Mach probes of two different sizes, corresponding to two different methods of
analysis, are used to make these flow measurements. Flows of as high as 70
km
Is
are detected in the
single spheromak, as well as subsequent reversals of flow toward the edges of the spheromak. There
is little flow perpendicular to the magnetic fields in a single spheromak, confttming that SSX
spheromaks exhibit the property of frozen-in flux. Measurements during counter-helicity merging
of two axially aligned spheromaks tentatively confttm the presence of bidirectional jets due to
reconnection in the azimuthal direction. The azimuthal component of this outflow reaches speeds
of 10 kmI s during the expected time frame for reconnection.
3
4
Contents
1 Introduction
1.1 Previous Lab Results in Flow and Stability
1.2 Previous Lab Results in Flow and Magnetic Reconnection
7
8
10
2 Plasma Theory
2.1 Frozen-in Magnetic Flux
2.2 The Induction Equation
2.3 The Alfven Speed
2.4 Magnetic Reconnection Basics
2.5 Sweet-Parker Reconnection
2.6 The Spheromak
13
15
17
19
20
21
24
3 Mach Probe Theory
3.1 The Hudis-Lidsky Model
3.2 Problems with the Hudis-Lidsky Model
3.3 Magnetized Mach Probe Theory
27
4 The Experiment
4.1 Making a Spheromak
4.2 Mach Probe Specifics
39
41
43
5 Results
5.1
5.2
5.3
5.4
47
47
51
54
57
Calibrating the Mach Probe
Visualizing Mach Probe Data
Single Spheromak Results
Counter-Helicity Merging Results
30
33
35
6 Interpretation
6.1 Single Spheromak Flow
6.2 Flow During Counter-Helicity Merging
61
62
62
A Glossary
B Argument for Flow in MHD Equilibria
65
67
References
Acknowledgements
69
71
5
6
Chapter 1
Introduction
Plasmas are the most prominent state of visible matter in our universe. A gas becomes ionized and
becomes a plasma when it has enough energy such that electrons have been stripped off of ions in
the gas. The plasma is a complex dynamic state, in which charged particles are immersed in, moving
under the influence of, and therefore constantly regenerating a macroscopic electromagnetic field
determined by the positions and velocities of other charged particles in the plasma. Key parameters,
such as density, ion temperature, electron temperature, pressure, current, and drift are related in
ways that, at a macroscopic level, are often complex and heavily dependent on the geometry of the
plasma or the boundary conditions restraining the plasma. Nevertheless, as those interested in
alternative energy sources look forward to a functional fusion reactor, scientists the world over are
making efforts to create and better understand stable plasma formations in the laboratory.
One essential component for understanding the stability and general dynamics of a plasma is
flow, or average drift velocity. Specifically, there are two respects in which plasma flow is
particularly significant for the current trajectory of plasma research. The first is the case of stability.
It has been shown that there is a relationship between stability and the shear of flow in certain
plasma formations. These relations have been used to make marked progress in the creation of
hotter, longer-lasting plasmas. Perhaps this is most apparent in the case of H-mode tokamaks. The
characteristics of this mode include higher plasma energy and longer particle confmement times.
The transition to this higher confmement mode is accompanied by higher rotation of the plasma at
its edge, along with some changes in electricity and density [1]. Secondly, there is the case of
magnetic reconnection. Magnetic reconnection, to be described in greater detail later, is a
phenomenon in which two areas of plasma with opposing magnetic field lines collide, resulting in
7
the annihilation of the field lines along with bursts of particle acceleration and heating. Currendy
accepted models of magnetic reconnection predict high velocity bursts of outflow from the
reconnection region, and the ability to quantify this flow has profound consequences for the
interpretation of energy transport during this dynamic process.
1.1 Previous Lab Results in Flow and Stability
Over the past decade, plasma theory has taken a turn towards a magnetohydrodynamic analysis of
two intertwined fluids, corresponding to differing behavior between a general electron fluid and a
general ion fluid. Equilibrium analysis of flows in a two-fluid plasma yields the result that plasma in
a stable state requires the presence of significant sheared flows [2]. The Helically Symmetric
Experiment (HSX) at the University of Wisconsin at Madison provided one of the first verifications
for the plausibility of this two-fluid analysis. Using two separate Mach probes with a Gundestrup
design (the same diagnostic that will be used for the majority of results in this thesis), the scientists
at HSX verified that flows in their stellarator, in the presence of a rising current, evolved on two
separate time scales, one at 300
~s
and the other at 20
~s
or less [3]. These separate time scales
agree with neoclassical predictions dependent on two-fluid motion.
e
<:I
.o:z.
rE>
e e
0
Figure 1.1 A schematic for the z-pinch at two separate times. Plasma is forced out from the sides
of an electrode by magnetic pressure to eventually converge into a straight, pinched line of flowing
hot plasma. Figure from http://www.aa.washington.edu/ AERP IZaP
One particular plasma formulation in which shear flows are relevant is called the z-pinch,
shown in Figure 1.1. The Zap Flow Z-Pinch lab at the University of Washington has just such a
setup, and they have performed experiments to determine flow by measuring the Doppler shift of
impurity lines along several axial chords. During the brief period of stability in the z-pinch, they
found that flows at outer radii reach a maximum of approximately 110
8
km
Is> while flows
toward the
center of the machine are closer to 50
106
Sl.
I
km s .
The axial velocity shear dVr I dr during this period is 5 x
After this quiescent period, the shear notably drops to 3 X 106
Sl
and becomes unstable [4].
Another plasma configuration is the centrifugally confined plasma, such as that produced by
the recently-built apparatus at the Maryland Centrifugal Experiment (MCX) at the University of
Maryland. A centrifugally confined plasma maintains stability by rapid rotation in the presence of a
constant axial magnetic field and radial electric field. This is an example of flows due to the
presence of a non-zero E x B component in the plasma. Measurements were made of the rotational
velocity of the plasma along 5 different radial chords. Rotational flow was found to be 20
the center of the chamber. At r=15 cm, rotational flow grows to as high as 70
back to 20
km
Isnear the outside of the machine.
km
km
Isnear
Is> only to drop
MCX also found that axial velocity shear was as
high as 8 x 105 s\ concluding that the areas of high rotational flow contain a stabilized plasma [5].
These data imply that the bulk of centrifugally confined plasma is in the region of higher rotational
flow with little exchange with the other regions of the chamber (due to such high shear).
Figure 1.2. A diagram showing the experimental setup at the Maryland Centrifugal Experiment
(MCX). Note that the E x B force is coming out of the page here, and therefore we expect the
plasma to flow in this direction, resulting in a rotation. Figure from [6].
A configuration similar to the stellarator and tokamak described earlier, the reversed field
pinch (RFP), consists of a toroid of plasma containing both toroidal and poloidal magnetic fields,
driven by an exterior coil. The EXTRAP T2R device in Stockholm is capable of producing such a
configuration, and measurements were performed on an RFP using the same diagnostic as in the zpinch and MCX, Doppler spectroscopy of impurity ions. In the case of RFPs, fast rotation is
necessary to avoid loss of energy to the surrounding wall. This lab found that, towards the center of
a stable RFP, ions have a rotational velocity of up to 40
velocity reduces to a mere 10 km I s [7].
9
km
Is> while towards the edge, the rotational
1.2 Previous Lab Results in Flow and Magnetic Reconnection
In magnetic reconnection, as briefly described earlier, we see the dynamic result of sudden
annihilation of magnetic fields in plasma. In a process that will be described in detail later, during
magnetic reconnection, particles are accelerated outward perpendicular to the incoming fields , as in
Figure 1.3.
«
Figure 1.3 A simplified 2-D depiction of magnetic reconnection. Opposed magnetic fields enter
the figure from the top and bottom. As they break at the X point in the center, plasma is sent out
the sides in the same plane as the original magnetic fields.
A simplified theory, called the Sweet-Parker theory, predicts that this outflow will be equivalent to a
velocity dependent on the magnetic field, referred to as the Alfven velocity. The specifics of this
theory will be explained in more detail later. Observations at the Princeton Physics Plasma
Laboratory in the Magnetic Reconnection Experiment
(M~'0
showed that this was not a
satisfactory model. Measurements on local flow near the reconnection region by a Mach probe
showed equivalent opposing outflows of a mere 8 km Is> compared to a calculated Alfven velocity of
39
km
Is.
The PPPL attributed this difference to the presence of a large amount of electron pressure
just outside the reconnection region, which inhibited the outflow [8].
Finally, there is a plasma formation used to induce reconnection in the laboratory that does
not exhibit this buildup of electron pressure: the spheromak. Flows on the order of the predicted
Alfvenic outflow velocity were measured during the reconnection of two spheromaks at the
University of Tokyo. Ion Doppler spectroscopy data showed outflow velocities of 12
the Alfvenic speed was determined to be as low as 15
km
Is[9].
km
Is> whereas
It will become clear shortly what a
spheromak is and how the Swarthmore Spheromak Experiment (SSX) replicates reconnection in the
laboratory.
10
N ow that the importance of flow measurements in the study of magnetically confmed
plasmas has been contextualized, the next few sections of this thesis will introduce the reader to
some basic concepts about plasma and to some of the more complex theory behind local
measurements of flow. After these theory sections, I describe the specific plasma chamber at SSX
and the relevant diagnostics used during the period of time in which data was collected for this
thesis. Finally, results are presented concerning flow measurements taken on single spheromaks as
well as merging spheromaks, and the implications of these results are interpreted and discussed.
With the awareness that plasma physics tends to have a vocabulary all its own, I have included a
glossary as an appendix following the fmal sections in case the reader is unfamiliar with some of the
terms presented in this thesis. Another appendix summarizes a technical argument not covered in
the body of this thesis.
11
12
Chapter 2
Plasma Theory
Determining the equation of motion for anyone particle in a plasma is a simple task. Gravitational
forces can be ignored in the presence of strong electromagnetic fields, so Newton's law and the
Lorentz force law yield
dv
-
-
m- = q(E+vxB).
dt
(2.1)
Together with Maxwell's equations, which determine how the electric and magnetic fields change
according to the motion of charged particles, the motion of each particle is completely determined.
Unfortunately, these equations would have to be calculated for every particle-particle interaction to
microscopically analyze plasma motion, a task that even the fastest of computers cannot accomplish
with any appreciable amount of particles.
Rather, plasma theory has resorted to the use of models to approximate the behavior of
plasmas assuming certain conditions. A relevant model for predicting flows is one referred to as
magnetohydrodynamics (MHD). This model treats plasma as an electrically conducting fluid.
Instead of analyzing the motions of individual particles, the MHD model describes plasma dynamics
with parameters such as pressure, density, or flow velocity.
Most modern formulations of MHD actually describe the plasma as two separate fluids, an
ion fluid and an electron fluid. The following derivation was inspired by Falk [10] and Chen [11]. It
is reasonable to assume that fluids consisting of these two different types of particles will behave
differently, not only because they have different charges, but also because there is a significant
13
difference between their masses. When a conducting fluid of protons is in motion relative to a fluid
of electrons, there is an effective current density. If we defme the flow velocity
iT = Vi - ve , then
- = enV,- where e is the charge of a proton and n=ne=n; is the proton, or
this current density is J
electron, density. In this context, the Lorentz force law still applies to each of the fluids, but we can
consider it to be a force on a section of charged fluid immersed in an electromagnetic field. The
charge q of this section of fluid will be proportional to the product of the density of particles in the
section n and the charge of those particles e, so the force on the fluid is
-
FLorentz
= en(E- + V- X B).
(2.2)
There is a second force, according to resistive MHD, that works against this Lorentz force.
For simplicity, assume that the bulk of resistance to acceleration in the plasma is due to collisions
between particles. Electron-electron and ion-ion collisions can be ignored because they conserve
momentum, so only electron-ion collisions contribute to this resistivity. The collisional coupling will
be proportional to e2 and n 2 because it is happening between two charged particles. It will also be
proportional to the relative velocity of ions and electrons defmed earlier. The resistive force on a
section of fluid in a plasma is
F',esistive
2
= 1]e n
2V
(2.3)
Assuming that the only significant forces on a stable, charged fluid are the two opposing ones
shown in (2.2) and (2.3) and adopting the defmition of current density written above, we gain the
following equation of motion:
- - - -
E+ VxB=1]J,
(2.4)
often referred to as the resistive Ohm's law. The approximation that the only effective forces on a
plasma are the Lorentz force and a resistive force is called the resistive MHD approximation. The
proportionality constant that determines the extent to which ion-electron collisions contribute to the
resistive force is the essential resistivity of the plasma, 11.
There can be an even further approximation made under the name ideal MHD, which
assumes that the fluid is perfectly conducting, or that 11=0. In this case, (2.4) reduces to the ideal
Ohm's law:
- - -
E+VxB=O.
14
(2.5)
Note that the flow velocity in directions perpendicular to the field are immediately determined by
the ideal Ohm's law:
V..l
ExB
(2.6)
=--2-
B
2.1 Frozen-in Magnetic Flux
One of the most appealing results from ideal MHD is the idea of frozen-in magnetic flux. This
approximation implies that the magnetic field lines move along with the plasma itself, or, some
would prefer, the plasma sticks to the magnetic field lines as it moves. A proof attributed to Seshadri
[12] begins by integrating the divergence of the magnetic field over a volume of plasma V bounded
by the closed surface S*. According to Maxwell's equations, the divergence of a magnetic field is
always zero, so, invoking the divergence theorem yields the equation
fV.B(r,t)dV= f B(r,t)·dS*=O
v
(2.7)
S*
N ow, introduce a closed curve C which bounds an open surface S in a spatially and temporally
dependent magnetic field
B(r,t).
Assume also that this curve moves with uniform velocity
v(r).
It is not required that all parts of the curve move with the same uniform velocity, implying that as it
moves, the curve can get warped or undergo rotation and translation. Now defme the curve C and
surface S as C 1 and Sl' respectively, at time t and as C2 and S2 at time t
+~t,
where ~t is an
infmitesimal time interval. Defme the volume V in (2.7) as the infmitesimal volume bounded by the
surfaces Sl' S2' and a cylindrical surface traced by the curve over the time ~t that can be labeled Se.
As is apparent in Figure 2.1, the infmitesimal surface element
dSc = (v(r)~t) X dR.
The outwardly
directed magnetic flux through the entire cylindrical surface Se is then
f B(r,t)· dSc = f B(r,t) ·{v(r) X dR}~t = - f {vCr) xB(r,t)}· dR~t
(2.8)
where the last operation is a simplification from a vector formula. Now, recalling that the volume V
is the volume bounded by the three surfaces in Figure 2.1, it is clear that the closed surface S* from
(2.7) is actually S*=Sl+S2+Se. The total magnetic flux through this surface should equal zero, giving
f B(r,t)· dS - f B(r,t)· dS - f {vCr) X B(r,t)}· dR~t = 0
S2
15
(2.9)
where there is a negative sign in front of the second term because the surface vectors of Sz are
antiparallel to the outgoing vector of the closed surface S*.
Figure 2.1 Geometry of the infinitesimal volume enclosed by the surfaces Sj, Sz, and Sc.
The definition of the rate of change of the magnetic flux through a surface is
!£[I B(r,t)· dS] = lim ~[I B(r,t + l1t)· dS - IB(r,t). dS].
dt S
~HO l1t
S2
S,
(2.10)
Expanding B(r,t + l1t) in a Taylor series gives
!£[IB(r,t). dS] = lim[I~ B(r,t)· dS + ~{I B(r,t)· dS - IB(r,t). dS}].
dt S
~HO
l1t S2
S2
S,
at
(2.11)
Plugging (2.9) into (2.11) and evaluating at the limit reduces this to
a-
-
- I -B(r,t)·dS- I [v(r)xB(r,t)]·df.
-d [ IB(r,t)·dS]=
&S
s~
c
(2.12)
Applying Stokes' theorem,
I-
- I a-
--
-d [ B(r,t)·dS]= [-B(r,t)-Vx{v(r)xB(r,t)}]·dS.
dt s
s
at
(2.13)
Now recall the ideal Ohm's law (2.5) and Faraday's law,
-
aB
at
VxE=--.
16
(2.14)
Combining these, we obtain the relation
dB
--
- - Vx{VxB}= O.
(2.15)
dt
Thus, the integrand on the right side of (2.13) is equal to zero when ideal MHD is applied and,
consequently,
f-
-
d [ B(r,t)·dS]=O,
dt s
(2.16)
implying that the magnetic flux through any surface flowing with a plasma which satisfies the ideal
magnetohydrodynamic model is constant. This is exactly what it means for the magnetic field lines
to be frozen in to the plasma. Frozen-in flux can be useful in the laboratory (though ideal MHD,
and therefore this simplification, does not always apply) because it reflects how closely flow and
magnetic fields are tied together. In the right conditions, well-placed magnetic fields can be used to
induce desired flows in a plasma.
2.2 The Induction Equation
Returning to the resistive MHD model, which postulates the resistive Ohm's law (2.4), the
consequences of the extra resistivity term are not yet clear. One would assume that frozen-in flux
does not apply because ideal MHD was necessary for its proof. This is indeed the case, for the
following [13] will show that field lines in a resistive plasma undergo diffusion, causing changes in
flux, in addition to convection. Begin by taking the curl of the resistive Ohm's law:
v X (Ii + iT X B) = 1]V X J
(2.17)
if we assume that the resistivity remains constant throughout the plasma. Substituting relations from
Ampere's law, fl)
= V X 13, and Faraday's law (2.14), this becomes
dB
- -
1]
-
- - + V X (V X B) = - V X (V X B).
dt
flo
(2.18)
Invoking a vector field identity and the fact that the divergence of the magnetic field is zero, this
equation reduces to
- + _V2
1]
-dB = V X (V- X B)
B,
dt
flo
(2.19)
often referred to as the induction equation. This equation without the second term on the right is
exactly the same as (2.15), which was derived using the ideal Ohm's law. Thus, the first term on the
17
right can be attributed to magnetic field convection, which is behavior of the magnetic field insofar
as its flux is frozen in with the plasma flow. Along those same lines, if one were to eliminate the
convection term from (2.19), he would be left with a diffusion equation
djj = !Lv 2 jj
dt 110
'
(2.20)
implying that this term is responsible for the diffusion of magnetic fields. In addition, this relation
suggests that the higher the resistivity 11, the quicker the magnetic field will diffuse, which is perfectly
consistent with the idea that no diffusion occurs at 11=0.
A measure of the extent to which a magnetic field in a plasma convects or diffuses would be
a useful characteristic. It would essentially tell the experimenter to what extent ideal MHD holds to what extent the magnetic fields are frozen into the plasma. This measure is achieved by taking
the approximate ratio of the convection term to the diffusion term from the induction equation. In
order to generalize appropriately, the assumption is made that the vector operator V is
approximately equal to
1/L> where L is the length scale of the plasma.
The length scale is simply a
measure of the approximate size of the plasma (most laboratory plasmas have length scales of 1-10
m). With the length scale approximation, the ratio is
vB
convection "" ~
diffusion
1']B
l1oL2
(2.21)
The result is a dimensionless quantity called the magnetic Reynolds number, RM :
R _l1oLv
M -
1']
(2.22)
This relationship shows that the amount of convection of a field will go up compared to diffusion of
the magnetic field in a plasma as the flow of the plasma increases, a gratifyingly intuitive result which
explains why, in many plasma formations, faster plasmas lose magnetic energy more slowly and
therefore are more stable to a point. Past a certain speed, however, turbulence not accounted for by
resistive MHD begins to have a destabilizing effect.
Also note that the Reynolds number is dependent on an approximately defmed length scale
L. This length scale originated with the del operator, so when the magnetic field does not vary
18
considerably over small distances, L is larger, while if the spatial variation of the magnetic field is
high, L is smaller. Therefore, the Reynolds number is relatively high in the case of most stable
configurations of plasma. However, in the case of events such as magnetic reconnection, where the
distance over which the magnetic field reverses direction is small, the Reynolds number will be low
and the amount of magnetic field diffusion will be higher.
2.3 The Alfven Speed
Before continuing to a theoretical discussion of magnetic reconnection, it is appropriate to discuss
another parameter of plasmas: the Alfven speed. This is the speed that a portion of the plasma
would reach if all of its magnetic energy were converted to kinetic energy. Some may refer to it as
the MHD speed limit of a plasma because, in many plasmas, the only stored energy is in the form of
magnetic fields. The derivation simply requires equating magnetic energy density to kinetic energy
density:
1 2 1
2
- B =-pvA
2/10
(2.23)
2
where VA is the Alfven speed and B is the magnitude of the local magnetic field. The Alfven speed is
(2.24)
This speed goes up if the magnetic field is higher since there is more energy stored in the initial field
and goes down if the density is higher because there is more mass over which to distribute the
energy. The Alfven speed at SSX is approximately 100 km Is.
It is worth pointing out that a common parameter used to describe plasmas is the Lundquist
number, which is simply the magnetic Reynolds number with the velocity set to the Alfven velocity.
Essentially, the Lundquist number reflects the highest possible Reynolds number for an MHD
plasma:
/1o
LEfio
- -LV
- -S=
-A TJ
TJ
-
(2.25)
P
The Lundquist number at SSX is approximately one thousand, suggesting that the frozen-in flux
condition is pretty well satisfied for plasmas not undergoing magnetic reconnection.
19
2.4 Magnetic Reconnection Basics
During magnetic reconnection, two regions of plasma with opposing magnetic fields approach each
other in a process in which their fields are annihilated, canceling each other out, and then
reconnected at lower energy. Plasma is forced out of the region at high speeds corresponding to the
leftover kinetic energy. Note that, for field lines to be broken, the frozen-in flux conditions
specified by ideal MHD must be broken. This suggests that we cannot use ideal MHD to analyze
magnetic reconnection, for if we did, we would predict that, as shown on the left in Figure 2.2, the
field lines would continue moving towards each other until there were an infinite curl of B around
the central reconnection region, resulting in an infinite current.
Magnetic reconnection is a complicated, three-dimensional process. The theory presented
shortly will be a 2-dimensional simplification that provides a rough approximation of what occurs
during magnetic reconnection in a resistive MHD plasma. Most laboratories, including our own,
attempt to induce magnetic reconnection in as predictable a way as possible, but ultimately, the
breaking of magnetic field lines is a difficult physical process to understand. The magnetic fields
undergoing reconnection are usually twisted and sheared, and the released energy actually
contributes to a lot of different processes, including but not exclusive to, bulk plasma motion,
electron and ion heating, and superthermal electron and ion acceleration [14].
y
RESIST E MHO
wi SWEET-PARKER
MODEL
IDEAL MHO
Figure 2.2 Magnetic reconnection. On the left is a depiction of two incident opposed magnetic
fields according to the ideal MHD formulation: reconnection does not occur. On the right is the
schematic for the Sweet-Parker model, which makes predictions about magnetic reconnection based
on resistive MHD.
20
2.5 Sweet-Parker Reconnection
One of the fIrst models for magnetic reconnection, now most useful for its simplicity, was released
by Sweet [1 S] and Parker [16], but here I will use a formulation by Priest [17]. It makes the
simplifying assumption of a two-dimensional geometry, as illustrated in Figure 2.2. The model
assumes a steady state, meaning that the inflow velocity and the outflow velocity remain constant.
There will be no non-zero time derivatives. In my derivation, I will use a coordinate system in
which plasma flows from positive and negative y into the reconnection region with magnetic fIelds
pointing towards positive and negative x, respectively. The outflows occur towards positive and
negative x with the outgoing fIelds aligned along positive and negative y, respectively. There is a
further assumption that reconnection occurs in a very thin layer so that the incoming magnetic fIelds
are essentially parallel to each other just outside the layer. The curvature of the fIeld lines
determines the size of this reconnection region. For the Sweet-Parker model to apply, width of the
layer 2L must be much greater than the thickness of the layer 2 i!. .
As one might expect, our derivation begins with the resistive Ohm's law (2.4). Outside the
reconnection zone, we assume that the incoming magnetic fIelds are approximately straight,
implying that the curl of the magnetic fIeld in this region is zero and, according to Ampere's law,
there is no current density in the region. The resistive Ohm's law then tells us that
(2.26)
and
(2.27)
at both positive and negative y outside the reconnection region. Here, v in is the inflow velocity of
the plasma and Bin is the magnitude of the incoming magnetic fIeld. In the reconnection layer, we
assume that the magnetic fIelds are annihilated so B=O. In this layer, Ohm's law gives us
Ez
= 1Jlz.
(2.28)
The current inside the reconnection layer is easily extracted using Ampere's law. Now imagine an
Amperian loop running around the boundary of the reconnection layer. Assuming that the outgoing
magnetic fIelds on the sides contribute a negligible amount to the line integral and again assuming
that the magnetic fIelds are straight along the x-axis, we can set the current density within the layer
equal to the line integral of the magnetic fIelds around the layer, providing the relation
B;n2L + B;n2L = JIJz2L2i!.
21
(2.29)
or
(2.30)
Recall the Sweet-Parker assumption that reconnection is a steady state, implying
-dB = -v X E- = 0, according to Faraday's law.
dt
If there is no curl of the electric field, then the
electric field outside the reconnection layer must be equal to the electric field inside the layer.
Combining equations (2.27), (2.28), and (2.30) provides an expression for the incoming velocity:
17
J10 f
v. =--.
In
(2.31)
Finally, the steady-state condition requires that there be no buildup of mass in the reconnection
layer. This tells us that the incoming mass flux of plasma must be equal to the outgoing mass flux of
plasma, or
(2.32)
where V out is the outflow speed and we assume constant density. Substituting (2.32) into (2.31) yields
(2.33)
Now, we must invoke a new equation from MHD, the fluid equation of motion [9]:
p[
d17
- at
+ (v . V)V] = J X B -
VP .
(2.34)
The left side of the equation is from Newton's law: p is the mass density and the bracketed factor is
the convective derivative of plasma velocity - acceleration. This convective derivative consists of
two components. The first component is the acceleration of fluid within the structure independent
of overall structural motion. The second component of the convective derivative comes into play
when the structure itself is accelerating as a whole in addition to accelerations within the structure.
This becomes particularly apparent when there is a velocity shear present. Even if a plasma is not
accelerating while it moves past a point in space, the velocity measured at that point will change
merely because of changes in velocity that have spatial dependence within the structure. The right
side of the equation is a little more straightforward. The first component corresponds to the
magnetic force on a current or, in this case, a flow of charged fluids. The [mal term predicts a force
22
based on plasma pressure. This is a common term in fluid equations, and it corresponds to the
tendency for fluids to move to areas of lower pressure.
In a steady state and assuming that pressure gradients are negligible, this equation becomes
(2.35)
Like when deriving the Reynolds number, we will approximate the gradient by 1/L. Applying (2.30),
the solution to this equation in the outflow region is
(2.36)
Combining (2.26), (2.27), (2.32), and (2.36), the outflow speed reduces to
V out
2
Bin
2
=--
(2.37)
l10P
which, when compared to (2.24) above, shows that the outflow velocity from the reconnection
region according to the Sweet-Parker model is equal to the Alfven speed. This is not surprising, for
the Sweet-Parker model does not account for the magnetic energy going into any process other than
kinetic outflow. The initial assumption that i!. < <L presumes that the outflowing magnetic field
would not be of significant magnitude compared to the inflowing magnetic field. With known
outflow, the inflow is predictable from (2.33):
V in
2
1J
l10L
= - - VA ·
(2.38)
N ow we can defme a new plasma parameter called the reconnection rate M: the ratio
between the inflow velocity during reconnection and the Alfven velocity. The Sweet-Parker
reconnection rate is
(2.39)
The term in the radical is just the inverse of the Lundquist number derived earlier (2.25), so this
reconnection rate reduces to
I
Msp
= JS.
23
(2.40)
2.6 The Spheromak
A spheromak is a stable configuration of plasma, which is notable for being, among other things,
force-free. That is to say that one need not run a large sustaining current like in the case of a
tokamak or centrifugal plasma in order to form a spheromak. It is essentially a donut of plasma
containing both toroidal and poloidal fields.
Plasma confined in
magnetic field
Polodial
Magnetic Field
Torodial
Magnetic Field
Figure 2.3 A rough image of a spheromak. The plasma is contained poloidal and toroidal magnetic
fields.
As a stable plasma, insofar as MHD is accurate, the spheromak should be in an equilibrium
state. This suggests that the spheromak as a whole has no bulk motion. Seeing that we are
interested in flows, this sounds odd, but that is only because we are referring to bulk flows rather
than internal flows. In an equilibrium state, we merely require that the magnetic fields and velocities
are constant in time. We do not require that the internal velocities of the plasma go to zero. A
more extensive discussion about why we expect some flow can be found in Appendix A. Another
consequence 0 f the spheromak being force- free is that it has low pressure [19]. Thus, returning to
MHD equation (2.34) and making these simplifications, the equilibrium equation becomes
JxB=O
(2.41)
(VxB)xB=O.
(2.42)
or, from Ampere's law,
24
Given no reason to generalize that either the magnetic field or the current density go to zero, it must
follow from this equation that the curl of the magnetic field is perpendicular to the magnetic field
itself. This implies that at equilibrium, the curl of the magnetic field is a multiple of the magnetic
field:
(2.43)
- - - 'I
...
."
I! :<f
."/
~
~
--
~
-
f
"
.
..:
:, .
f
h
,
-h
-
....r
,,
~
;:::.
... T
.;.
I
-
~
~
D ')l
.US·
62
Figure 2.4 Magnetic characteristics of a stable single spheromak inside the SSX chamber. The
views from the side of the chamber exhibit poloidal field characteristics while the cross-sections
below show the toroidal field. Figure from [18].
Let us introduce one final plasma parameter, the magnetic helicity K. This is a measure of
how twisted the magnetic flux tubes in a given plasma are. It is defined as follows:
K=
fA.BdV,
(2.44)
v
where A is the vector potential. It turns out that K is conserved in the approximation of constant
magnetic energy. In other words, K is relatively constant in plasmas like spheromaks, where the
magnetic energy is mosdy convective. It has been shown that solutions that minimize the energy for
a given K also predict a constant value for A in (2.43), making this equilibrium equation solvable.
The equilibrium solution for the magnetic field of a spheromak in a cylindrical container of height h
and radius a is given by [20] as
25
= -B)d1{lr)cos(kz),
Brp = B o IlJ1 {lr)sin(kz),
B z = B)daC1r)sin(kz)
Br
where kh=1t, la=3.83, and /1 =
(2.45)
e: J J.
3
+ (:
This suggests that the magnetic field gnes like a
half-period over the axis of the machine and like fIrst -order solutions of Bessel functions in the
radial direction. The validity of this model has been verifIed in SSX spheromaks [21], [22].
26
Chapter 3
Mach Probe Theory
When making local flow measurements in a plasma, there are several difficulties to be considered.
Primarily, it is necessary to differentiate between flow velocity and thermal velocity. Thermal
velocity is a Gaussian distribution of velocities manifested by an ensemble of particles. If we are to
determine the flow velocity by observing the velocity of individual microscopic particles, then we
must account for this thermal velocity and recognize its occurrence independently of a macroscopic
flow velocity in the plasma. Secondly, as is the case for all local measurements, caution should be
taken to perturb the plasma as little as possible and to account for any perturbations that are made
by the diagnostic in the plasma. Local flow measurements require that a diagnostic making the
measurement be immersed in the plasma and, the diagnostic having finite dimensions, perturb the
motion of the plasma that it is simultaneously trying to measure. This is particularly noticeable
during magnetic reconnection, when the diagnostic itself is occupying the reconnection region,
effecting the phenomenon in often unpredictable ways. A third difficulty follows from the charged
nature of the particles in a plasma. Because of the magnetic fields present throughout the plasma,
particles very rarely follow straight trajectories. Rather, they move throughout the plasma on orbits,
the size of which are determined by the magnitude of the electric and magnetic fields and the
velocity of the particle. This implies that, if a particle's velocity is measured at one point in time, it
could actually be the reverse of the overall drift velocity of that particle. The particle could be
moving backwards temporarily due to the orbital nature of its path. Any local flow measurements
will have to take into account that charged particles in a magnetic field, rather than free particles
moving in straight lines, are being observed.
27
The diagnostic used to make local flow measurements at SSX is commonly referred to as a
Mach probe. The Mach probe is a variation on the Langmuir probe, a simple metal rod held at a
bias voltage and immersed in the plasma. The ions collected by the Langmuir probe can be used to
predict plasma density and temperature - this diagnostic has been used to measure these parameters
in plasma chambers for decades, though we will not be using it for that purpose here. Now imagine
that one side of the Langmuir probe is blocked and the probe is negatively charged, such that it
collects ions from only one direction. Right next to this "directed" Langmuir probe, on the other
side of the obstruction, is placed another Langmuir probe that collects ions from the other direction.
The resulting instrument is the simplest orientation of a Mach probe, as shown in Figure 3.1. Its
functionality is actually rather simple in principle. Let us imagine that we turn the Mach probe such
that one of the ion collectors (as I will refer to the biased Langmuir probes from here on) is facing
toward the direction of plasma flow, and the other ion collector is facing away from the plasma flow.
As ions move through the plasma, they will be caught by the ion collectors upon which they are
incident. The collector facing "upstream" (towards the flow) will collect more ions than the
collector facing "downstream." It is by taking the ratio of the current collected by these two probes
that we gain insight into the ion drift velocity in a plasma.
barrier
upstrea
co llector
FLOW :>
Figure 3.1. The simplest possible Mach probe consists of two ion collectors divided by a barrier.
The upstream collector collects more ions than the downstream collector, giving us the means to
approximate the flow.
28
Clearly, if there is no net drift velocity, the ratio of the two currents will be one: each
collector is gathering the same amount of current. However, when the plasma has a nonzero drift
velocity, how does this ratio deviate from one? This analysis proves to be rather difficult, and,
without making several assumptions, can become almost completely intractable. The theory behind
Mach probe analysis can be separated into two distinct regimes. The "magnetized" regime
corresponds to a larger probe, specifically one with a radius that is greater than the ion gyroradius.
The "unmagnetized" regime is exactly the opposite - the radius of a probe in the unmagnetized
regime is smaller than the ion gyroradius of the plasma. The results presented here include probes
of both sizes and, therefore, invoke the analysis for both regimes. The theory for each of these
models is relatively similar, though unequally well developed. For example, a rigorous treatment of
the magnetized regime has been presented with success [23], [24], but the unmagnetized regime is
still a controversial field, and analysis done thus far usually only extends to unique cases. It is also
worth adding that many of these analyses have been done using powerful computer simulations and
not with rigorous mathematical treatments. The electromagnetic forces determining the motion of a
very large ensemble of plasma particles are often too difficult to resolve without simplifications that
cripple the power of one's analysis.
Before we take a closer look at some of the models used to develop Mach probe theory, I
would like to clarify some crucial concepts. Each theory invokes the use of a Mach number rather
than directly referring to a drift velocity. The Mach number M is actually the ratio of the drift
velocity v to the sound speed in the plasma
Cs :
v
M=.-
(3.1)
Cs
or, put another way, the Mach number is the drift velocity normalized to the sound speed in the
plasma. Sound speed in a plasma is a function of temperature that determines roughly how quickly
particle waves move through the plasma. It is provided by the equation
(3.2)
where Te and Ti are the electron and ion temperatures respectively, y is the ratio of specific heats at
constant pressure and volume, respectively, familiar from thermodynamics, and tIl; is the ion mass. I
have absorbed Boltzmann's constant k into the statements of temperature here and will continue to
29
do so, as is the common practice in plasma physics, such that the "temperatures" listed are actually
in units of electron volts.
3.1 The Hudis-Lidsky Model
Until relatively recently, the Hudis-Lidsky model [25] had gone unchallenged as the method for
Mach probes in the unmagnetized regime. It consists of a one-dimensional analysis of the
differences in current between two oppositely directed probes. This model is only appropriate after
adopting four major assumptions. The fIrst one is that the orientation of the probe with reference
to the magnetic fIeld does not effect the results. This is basically the same as saying that we are
working within the unmagnetized regime. The orbits of the particles are not a factor in this theory.
The second assumption is a result of the fact that the Hudis-Lidsky model consists of a onedimensional analysis. That is, the geometry of the probe itself does not effect ion collection. There
is no turbulence unique to a certain probe shape as the plasma goes by. Another assumption states
that the plasma should be collisionless and high density. This assumption assures that we need not
consider collisions with other particles when determining the energy and motion of particles near
the probe. The [mal condition is similar to the second: there is no differentiation between the
plasma flow into and away from the probe. The probe is not blocking the motion of the plasma,
and therefore we can treat the velocity of the plasma as it approaches the upstream collector as if it
were the negative of the velocity of the plasma as it moves away from the downstream collector.
In more exact terms, we can sum up these assumptions in the form of relations between plasma
and probe parameters:
(3.3)
or the Debye length is much less than the radius of the probe. The Debye length is an important
parameter in plasmas. It determines the distance over which a potential immersed in the plasma will
act. Because a plasma consists of flowing particles, when a charge is immersed in the plasma, its
potential will be shielded since particles with opposite charges will be attracted to it while others are
repelled. It is possible to determine from Poisson's equation V 2 l/> = -
j( the rate at which the
potential drops off with distance from a charged body in the plasma [11]. The electric potential
decreases with distance x according to
30
Ixl/
l/J = l/Jo e - /AD
(3.4)
with the parameter An defmed as the Debye length:
(3.5)
where, again, Boltzmann's constant has been absorbed into the temperature Te. SSX has calculated
a Debye length of 3 J.lm, given an electron temperature of 20 eV and a density of 1014 cm 3 . The
assumption made in (3.3) is usually a safe one, considering that the Debye length in most laboratory
plasmas is on the scale of microns, while the radius of most Mach probes is on the scale of
centimeters.
Another assumption in terms of plasma parameters is
(3.6)
or the ion gyroradius is less than the radius of the probe. The ion gyrodradius is the radius traversed
by an ion in a magnetic field of magnitude B due to the Lorentz force. To approximate the
characteristic ion gyroradius for a plasma, we assume that the ions are moving at the ion thermal
velocity, giving an ion gyroradius of
miv-L
-J~mi
a - - - -'--'--i IqlB
- eB .
(3.7)
When the ion gyroradius is more than the radius of the probe, then we must take into account those
particles which hit the downstream side because they have curled around the probe due to magnetic
forces. Conversely, when the ion gyroradius is less than the probe, we do not have to account for
the effect of the magnetic field on individual particles. This is why the relation in (3.6) characterizes
the unmagnetized regime. The ion gyroradius at SSX is on the order of a centimeter. It is for this
reason that we chose to analyze the flow with two separate Mach probes, one with a radius smaller
than the ion gyroradius and the other with a radius larger than the ion gyroradius.
A third condition is that
(3.8)
or the ion temperature is less than the electron temperature. This condition is arguably one of the
most restrictive conditions for this model, considering that it is often not the case in laboratory
31
plasmas that ion temperatures are considerably less than electron temperatures. In fact, at SSX, the
ion temperature is approximately 60 eV while the electron temperature is approximately 20 eV: this
relationship does not hold. Finally,
Vi
> C s (at the sheath edge),
(3.9)
also known as the Bohm sheath criterion. The sheath is the region within one Debye length of a
potential in the plasma. In this case, it is a small region adjacent to each ion collector in which there
is a significant potential drop. The Bohm sheath criterion requires that the ions be moving at speeds
higher than the sound speed before they enter this sheath. This implies, according to the HudisLidsky model, that there must exist some presheath in which particles are accelerated to this required
speed before they enter the sheath to be collected by the ion collectors.
The Hudis-Lidsky model assumes that, an infInite distance from the probe, ions start at the
thermal velocity,
(3.10)
which is the velocity that an ion would travel if the average thermal energy (TJ were converted to
kinetic energy. Accounting for the one-dimensional motion of particles moving through the
presheath and sheath on either side of the probe, the model postulates the following formula for
received current:
(3.11)
A is the area of the exposed ion collector and Vd is the drift velocity of the plasma. Hudis and
Lidsky have determined a more specifIc value for C [25], but it is not necessary for the analysis
performed here. The plus and minus signs correspond to the upstream and downstream currents,
respectively. When we divide the upstream current by the downstream current, we fmd an
exponential relationship between the current density ratio and the Mach number.
(3.12)
Note that I have divided through by the area of the ion collector so that we are actually considering
the ratio of current densities rather than current. We would not want the size of our collectors to
effect the determined Mach drift. Simplifying the equation above, we have
32
·
4vt
*Vd
~=eSS
(3.13)
jdown
where I have separated out the ratio of drift velocity to sound speed so that we can put the
exponential in the form
jup
--=e
kM
(3.14)
jdown
where M is the Mach number and k is simply
(3.15)
3.2 Problems with the Hudis-Lidsky Model
The Hudis-Lidsky model was eventually satisfactorily debunked by Hutchinson [26] for several
reasons. First, the one-dimensional approach does not provide a satisfactory description of particle
behavior around a Mach probe. In the real conditions for a Mach probe, even at small ion
temperatures, ions will be drawn to the downstream side of the probe by being accelerated
transversely with the presheath potential, as shown in Figure 3.1. This common behavior could not
be predicted with a one-dimensional model that assumes all of the particle's energy is concentrated
into one-dimensional acceleration.
The next problem lies in the assumption that the Bohm sheath edge criterion (3.9) applies.
Based on the condition that ion density and electron density be equal at the edge of the sheath and
(3.11), Hudis and Lidsky project an ion density n i at the sheath radius rs of
_ (v, ±Vd )2
C,
2
(3.16)
where, as before, the plus and minus signs refer to upstream and downstream densities. Hutchinson
notes something odd about the implications of this relation. If the drift velocity is greater than the
thermal velocity of the plasma, the downstream current will actually increase as the drift velocity
increases. Not only is this not intuitive - it does not happen in the laboratory, implying a further
restriction on this model Vd <
Vt"
At SSX, the drift velocity of ions is often greater than their thermal
velocity.
33
\,
robe",
,
•
\
\
\
-
I
I
./
Do v stream
,.---- - ~--
-/
Velocity
/
\I
/
.
I
('
Figure 3.2 A two-dimensional depiction of particle behavior around a Mach probe in the
unmagnetized regime. Image from [5].
Hutchinson used a particle-in-cell (PIC) computational method to analytically derive the ion
distribution on a sphere moving through a plasma (relatively equivalent to a stationary sphere in a
moving plasma) [27] with the intention of comparing the situation to a Mach probe in a flowing
plasma. His findings agreed with the logarithmic relationship between the Mach number M and the
ratio of upstream and downstream currents, shown in (3.14), but he did not agree with the proposed
proportionality constant k shown in (3.15). His findings propose results for k corresponding to two
different temperature extremes. For T j < 3Te , he finds that k is constant:
kum T<3T
,
Ie
= 1.34.
(3.1 7)
At the other extreme, for T j > 10Te , he finds a temperature-dependent k:
(3.1 8)
The ratio of ion and electron temperatures in SSX plasma lies close to the lower ion temperature
extreme, given that our ion temperatures vary from 60 to 80 eV and our electron temperatures vary
from 20 to 25 eV.
34
3.3 Magnetized Mach Probe Theory
In magnetized Mach probes, the ion gyroradius is smaller than the probe radius, while in
unmagnetized Mach probes, the opposite is true. The kinetic theory derived here can be attributed
to Chung and Hutchinson [24]. Let us begin by considering the sheath and presheath of a Mach
probe, as described earlier in reference to the Hudis-Lidsky theory. In the case of a magnetized
probe, we will assume that the magnetic field is aligned with the direction of flow of the plasma
(consistent with frozen-in magnetic flux), but allow for diffusion into and out of the pathway of
flow towards and away from the probe, as shown in Figure 3.3.
rro b·e
IE)
.
Oi ffus ive Ion SOiJ r'ce
,-. .....,-tl- -U .....t(v.z)~
Wf...
Par tild ,e
Ex.cha.nge
Figure 3.3 Schematic of the ion collection process assuming a one-dimensional analysis with
magnetic field in the same direction as flow. Ions are accelerated toward the probe in the presheath
region, and are given the freedom to diffuse in and out of this region. Figure from [24].
Calling f(z,vz>t) the one-dimensional ion distribution function, the Boltzmann equation for a fluid
provides
(3.19)
where C f is the collision operator and Sf is a volume source of ions, and V z and az are the respective
velocities and accelerations of the distributed particles, dependent on the Lorentz force. Assuming a
35
steady-state ion distribution (eliminating the time derivative) and a collisionless plasma, the
Boltzmann equation becomes
(3.20)
where m, q, and
<!>
correspond to ion mass, ion charge, and electric potential. Assuming a constant
total energy for a given ion E, consider the familiar energy relation
I
2
"2mvz + qljJ(z) = E.
(3.21)
It is clear from this equation alone that the relation between potential and velocity for each particle
will be a parabolic one given a position z. The rub, however, is to determine the spatial variation of
potential that satisfies these equations.
Assuming isothermal electrons, a Boltzmann relation implies that there is a spatially
dependent background density of electrons against which the ions move:
(3.22)
where n= refers to the density of the plasma outside of the presheath and T e corresponds to the
electron temperature. The electron and ion densities, insofar as they are responsible for the
potential, must follow Poisson's equation:
(3.23)
Here, I have assumed that the ions are simply protons, or that the plasma originated from hydrogen
gas, which is the case for SSx. Finally, we must account for the cross-field transport originating
from diffusion in and out of the presheath region. Consider a frequency W(z,v) which dictates the
rate at which particles move between the outer plasma and the presheath. The volume source of
ions Sf specified in equation (3.20) must depend on this frequency:
Sf
= W(z, v)[f=(v)- fez, v)].
(3.24)
The first term on the right-hand side corresponds to the ion distribution function of the exterior
plasma as a function of velocity. When multiplied by the rate at which particles move between the
outer plasma and presheath, it determines the relative amount of ions entering the flux tube from
the outside. The second term on the right corresponds to those leaving the tube. We are making
the assumption that a particle at position z with velocity v is just as likely to leave the tube as another
36
particle with the same position and velocity is to enter it; in other words, we are assuming that the
rate is the same for both the process of leaving the tube and entering the tube.
Once these equations have been established, they can be analyzed using numerical means.
For more details on how Chung and Hutchinson went about this, see [24]. The only further steps
taken before computational analysis were an assumption that the exchange rate between the exterior
plasma and the presheath is independent of particle velocity (W(z,v)=W(z)) and a
nondimensionalization of each of the plasma parameters in the above equations. In the end, it is
affttmed that there is a logarithmic relationship between the upstream and downstream currents in a
Mach probe and the flow of the exterior plasma. Assuming Ti = T e , the calibration constant k for a
Mach probe immersed in a plasma that satisfies the magnetized condition is
k m "T=Te
= 1.7.
37
(3.25)
38
Chapter 4
The Experiment
The Swarthmore Spheromak Experiment (SSX) lab is equipped with a chamber capable of
producing single stable spheromak formations or merging two axially-aligned spheromaks. All
measurements upon the plasma are made within a vacuum chamber, shown in Figure 4.1, consisting
of a stainless steel cylinder 0.60 m in diameter and approximately 1 m in length. The chamber is
kept at a vacuum of 2 x 10 7 torr by an oil-free cryogenic pump. Lining the inside of the chamber
are two cylindrical copper containers called flux conservers with inner radii of 0.20 m. These flux
conservers are placed side by side and separated by 2 cm at the midplane so that diagnostics may be
inserted into the midplane region. The plasma is contained inside these flux conservers over the
course of the shot, as they provide the high electrical conductivity necessary at the edge to allow for
stable boundary conditions. Additionally, having perfectly conducting boundary conditions in the
laboratory makes for a much simpler comparison to computer simulations.
The vacuum chamber is flanked by two coaxial plasma guns. The energy provided to the
guns during plasma formation comes from 2 large capacitor banks (the lab has a total of four). Each
of these banks have a capacitance of 0.5 mF and are capable of being charged to a voltage of 10 kV,
providing a maximum energy output of 25 kJ each. In practice, however, the capacitor banks are
only charged to about 5 kV in the process of creating a spheromak. There is a second group of
capacitors that drive current through coils surrounding each gun. These currents establish the
magnetic field necessary for a "stuffing" flux in each gun. This effect will be described in more
detail shortly.
39
Figure 4.1 The SSX plasma chamber. Several diagnostics are visible in this picture. The two small
cylindrical steel casings protruding toward us from near the top of the chamber are two of the
probes in the magnetic edge array. The aluminum bar on the far right of the picture holds the HeNe
interferometer setup, which sends a laser through a window on the side of the chamber. Also, on
the largest visible flange, both the soft x-ray and the IDS diagnostics are used to observe the plasma
from the edge of the midplane. The large green structure to the left in the back is one of the main
capacitor banks, and the steel cylinders protruding outward from either side of the chamber are the
coaxial guns.
Surrounding the midplane on the outside of the chamber, are two reconnection control coils
(RCC's). These toroidal coils contain a current that drives a magnetic field in a poloidal direction
through the center of the chamber. Originally installed to stabilize and hold apart spheromaks
during magnetic reconnection, the currents in these coils are now often used in alignment with the
current of a single spheromak in the chamber in order to stabilize it. The RCC's are powered by, of
course, another set of capacitors. Over the course of a shot (approximately 100 /ls), they produce a
steady magnetic field of 700 G at the central axis, in alignment with the poloidal fields of a
spheromak in the chamber.
In addition to the Mach probe, whose results are contained in this thesis, the SSX lab is
currendy working with several diagnostics. Other diagnostics that protrude into the chamber are
four magnetic probes, making up an array used to approximate the orientation of macroscopic fields
40
with high temporal resolution (10 MHz) at the edge of the chamber for minimal perturbation.
Optical diagnostics used recently include a soft x-ray (SXR) detector and a vacuum ultraviolet
(VUV) monochrometer. Analysis from these has been oriented towards determining timedependent electron temperature and model impurity levels within the plasma [28]. Results from a
HeN e-quadrature interferometer used to measure plasma density along a chord directly in front of a
plasma gun were presented by Gray [29]. Also, some interesting ion Doppler spectroscopy (IDS)
results concerning ion temperature and flow during reconnection were presented last year by my
colleague, Jerome Fung [13]. These were attained by an ion Doppler spectrometer with a CzernyTurner configuration focused along an adjustable chord through the midplane of the chamber.
In the summer of 2005, with the help of Michael Schaffer of General Atomics, SSX set up a
glow discharge cleaning (GDC) system to remove impurities from inside the chamber prior to taking
a new set of data. The motivation for such a system stems from the hypothesis that colder
impurities get stuck to the walls of the chamber over time and, when introduced into plasmas, have
an adverse effect on their stability. This new cleaning system has led to lower densities and higher
ion temperatures in our plasmas. The glow discharge is a plasma itself, consisting of partially ionized
helium. During the course of a glow, the pressure of helium in the chamber is maintained at about
100 millitorr while a constant DC voltage of +300 V is applied to the inner electrodes of each of the
plasma guns. At this voltage, helium plasma is forced out of the guns to scour any impurity buildup
on the inside of the chamber. Also, if any diagnostics protruding into the plasma are set at a bias
voltage (such as our Mach probe), they are cleaned as well, making GDC particularly important to
use directly after a new diagnostic has been introduced into the system. When the voltage on the
electrodes is reversed, such that the inner electrode is held at -800 V, the glow occurs inside the guns
(plasma is not expelled into the chamber). Each of these settings runs for at least an hour for a
thorough cleaning. Preliminary results suggest a marked increase in ion temperature to up to 100 eV
and a lowering of the plasma density by an order of magnitude to 1014 cm 3 as a consequence of the
new GDC system.
4.1 Making a Spheromak
The coaxial guns consist of a hollow cylindrical outer electrode, which is grounded, and an inner,
charged cylindrical electrode. Recall also that there is a coil surrounding the gun to induce a stuffing
41
flux as well as RCC coils surrounding the midplane of the chamber itself. There are plenty of
electronics running even before the plasma is created. Throughout this paper, I refer to the time at
which the main capacitor banks are discharged, or the time at which the voltage on the inner
electrode quickly rises, as t=O. Because they take the longest to reach the necessary current, the
RCC coils are always fired first, at approximately t=-50 ms, though this time is adjustable. Then, at
t=-25 ms, the capacitor banks controlling the coil surrounding the gun fire, setting off the stuffIng
flux. This stuffIng flux results in a magnetic field with a flux of around 1 mWb emanating out from
the inner electrode and around to the outer electrode in the radial direction. These are both fired
early because the amount of time it takes for their exterior magnetic fields to penetrate the metal
chamber is on the order of 10 ms.
(r
d)
Figure 4.2 The formation of a spheromak in a coaxial gun. First (a), hydrogen gas is puffed into
the end of the gun while a stuffIng field is present at the opposite end. Then, when the voltage is
applied to the inner electrode (b), current runs through the plasma to induce a toroidal field that
both contributes to the toroidal field in the eventual spheromak and forces the spheromak out of
the gun. (c) As the plasma passes by the stuffIng field, it stretches out the field lines and gains a
poloidal magnetic field. Finally, after about 20 j.ls, the spheromak is emitted from the gun (d).
Figure from [30].
Then, between 400 Jls and 700 Jls prior to the firing of the large capacitor banks, hydrogen
gas is puffed into the gun through high-speed valves that surround the part of the gun furthest from
the chamber. It has been projected that the amount of time delay between this puff of gas and the
firing of the banks has a profound effect on the density of the plasma because it determines the
overall amount of gas that enters the guns prior to ionization. Finally, the large capacitors are
discharged, resulting in a voltage on the inner electrode of approximately -5 kV. This voltage is high
enough to ionize the gas (it literally rips electrons off of protons) such that a plasma is formed.
Because an ionized plasma contains mobile charge, a current immediately flows between the
grounded outer electrode and the negatively charged inner electrode, illustrated in Figure 4.2b. As
this current flows down the inner electrode, a corresponding azimuthal magnetic field develops, as
42
according to Ampere's law, with two essential consequences. Not only does this field become the
toroidal field of the eventual spheromak, but it induces a force directly out of the gun, in the J x B
direction (where J is the radial current density in the plasma and B is the toroidal magnetic field).
Finally, as the spheromak is forced out of the gun, it encounters the stuffing flux, which is weak
enough to allow the plasma to eventually break through into the chamber but also strong enough to
induce a necessary component for the spheromak: a poloidal magnetic field. This formation process
is complete and the spheromak enters the chamber by approximately t=20 ).ls. The magnetic fields
in the plasma usually decay after another 80 J.ls, or when t = 100 J.ls.
Two of the main modes of our experiment, co-helicity and counter-helicity merging, require
that we have the ability to adjust the direction of fields in a spheromak. This is not easily
accomplished with the toroidal fields because of the electrical setup of the coaxial gun, but the
current in the stuffing coil, and therefore the stuffing flux, is easily reversed. This results in a change
in "handedness" of the magnetic fields within the spheromak. One refers to a spheromak as righthanded if the poloidal fields curl around the toroidal fields in the same way that a magnetic field
curls around current according to the right-hand rule. The spheromak is left-handed if the poloidal
field is in the reverse orientation with respect to the toroidal field. Thus, it is clear that we can create
either orientation of magnetic field in a spheromak simply by reversing the stuffing flux and,
therefore, the poloidal component of the magnetic field.
4.2 Mach Probe Specifics
There were two different Mach probes used to obtain the results presented here. They both
consisted of six ion collectors in a cylindrical boron nitride turret, referred to as a "gundestrup"
design [31]. The general schematic for the smaller Mach probe is presented in Figure 4.3. The
smaller Mach probe was built by Falk [8]. Since the probe was first built, the slits have been
elongated to increase the amount of received current. The entire probe, from its boron nitride tip to
the box containing the wire leads, is 75.9 cm. The cylindrical boron nitride sheath has a diameter of
0.65 cm with slits 0.05 cm wide. The tungsten rods, or ion collectors, are protruding from a stainless
steel rod of the same diameter as the sheath and are threaded into the sheath such that they are
aligned with the slits. The boron nitride sheath is sealed to the end of the stainless steel shaft using
TorrSeal. The central pin acts as a reference from which to bias the other ion collectors. It remains
43
close to the local voltage of the surrounding plasma. The tip of this reference pin extends from the
end of the Mach probe by about 1 mm. Each pin is welded to their respective leads using a gold pin
because of the difficulties involved in welding tungsten direcdy to copper. Note, in Figure 4.3, that
the vacuum seal does not occur at the tip of the Mach probe, but at a point exterior to the machine,
where the electrical leads are fed thru a vacuum weld. This means that, when the Mach probe is in
the chamber, the interior of the entire stainless steel rod is under vacuum.
referenoe pin
- - - - - ------r4=
--
-
feed-thru oonflat
fla ng e
,
bo ro n nitride
sheath
tungste n rods
electrical feedthru
(vacuum we ld)
Figure 4.3 A basic schematic of the Mach probe. Tungsten rods are inserted into a boron nitride
sheath at the end of the probe and the leads from these rods run down a stainless steel tube. The
feed-thru con flat flange seals to the outside of the chamber to maintain a vacuum. The mach probe
can move in and out of the chamber as well as rotate in this flange. The vacuum seal is made at a
plug at the furthest end of the stainless steel rod where the leads are passed through.
The large Mach probe is somewhat different than the smaller one in schematic, though it has
the same boron nitride sheath design and contains the same materials. It was originally engineered
so that it could both fit on our chamber and on the chamber at the Madison Symmetric Torus
(MST) lab in Wisconsin, but this collaboration has yet to be carried out. The probe stalk is
contained in a larger cylinder mounted on a track, on which the probe glides in and out, and is
allowed to rotate. It is this larger cylinder containing the stalk that is under vacuum when the probe
is attached to the machine. The boron nitride of this probe has a diameter of 1.6 cm and slits 0.05
cm wide. The length of the entire probe is 105 cm.
44
Figure 4.4 The tip of the larger Mach probe at two stages of construction. On the left, you can see
the empty boron nitride sheath and empty wire casing. On the right is the boron nitride sheath with
all of the tungsten rods inserted and their wire leads protruding.
Figure 4.5 The two fully assembled Mach probes. The probe on the left is visibly smaller than the
one on the right. On the right, the large Mach probe is retracted almost completely into the larger
cylinder containing its stalk.
The Mach probes are used separately and inserted into the same portal on the chamber.
Each is oriented along the radial axis of the chamber at the midplane. Before data is taken, six
capacitor banks are charged to -40 V, corresponding to the ion saturation current. The ion
saturation current occurs at diagnostics biased to a potential so negative that all negatively charged
particles are effectively repelled from the diagnostic and only a positive current is collected. Once
the capacitor banks are charged, they are isolated from anything other than the Mach probe. These
capacitor banks serve to provide the bias between the ion collectors and the central pin. Finally,
during the course of a shot, currents in each ion collector are measured by matched Pearson model
45
411 current transformers with an output of 0.1 V / A
± 1%. It is important that these transformers
be matched so that we can properly compare the currents between separate ion collectors.
46
Chapter 5
Results
5.1 Calibrating the Mach Probe
As the theory developed in the Mach probe section demonstrates, analysis of Mach probe data is
reduced to flnding a calibration constant k that determines the logarithmic relationship between the
ratio of opposing ion currents and the drift velocity of the plasma in which the probe is immersed.
This constant is closely tied to the geometry of the probe and dimensions of the probe relative to
the magnetic speciflcs of the surrounding plasma. It would be advantageous if, rather than
attempting to derive this calibration constant using still controversial analytical means and merely
approximate plasma parameters, we could calibrate the probe by comparing its output to a known
plasma drift velocity.
Mach probe
West mag probe
East mag probe
PLASMA FLOW
West gun
<'lres)
Figure 5.1 On the left is a diagram of the chamber when prepared for calibration. The Mach probe
is centered between two magnetic probes. The picture on the right shows the setup in the
laboratory. The large metal track in the middle supports the Mach probe while the two whitecapped cylindrical housings on either side contain wires for the magnetic probes.
47
This known plasma drift velocity can be determined by tracking the magnetic energy in the
chamber after a single spheromak is launched into the chamber from one side. As shown in the
figures on the previous page, we calibrate the Mach probe by taking data with the Mach probe at the
center of the chamber while simultaneously taking data with magnetic probes placed on either side
of the chamber. Recall that we are dealing with a plasma that has a Lundquist number of
approximately 1000, implying that the plasma will "stick" to the magnetic field lines reliably. We can
therefore assume that wherever an appreciable amount of magnetic energy is present within the
chamber, there is also plasma present. The magnetic probes used in our labs have a high time
resolution (less than 1 j.ls), allowing us to view the magnetic field with some precision. In this case, I
will use the total amount of magnetic field squared at each of the three-dimensional probes (B,2 +
B/
+ B z) to obtain a value proportional to the magnetic energy present at the probe. The result for
a single spheromak shot is shown below.
Figure 5.2 The total magnetic "energy" at each of the magnetic probes for one single spheromak
shot. The solid line corresponds to the West magnetic probe while the dashed line corresponds to
the East magnetic probe. The quick rise in energy makes it relatively easy to determine the time of
flight for the plasma between the two probes.
48
This figure makes it apparent that, when the plasma is initially present at either of the probes, the
magnetic "energy" at the probe is on the order of 106 G 2 • Noise on the order of 105 G 2 prior to the
sudden rise in the East probe suggests that the time of flight would be the difference between the
time at which the East probe reads a magnetic "energy" of 5xl05 G 2 and when the West probe reads
the same "energy." An average taken over twenty-five shots gives us a value for the initial drift
velocity of a spheromak out of the gun for our usual setup of 5 kV charge and 1 mWb of stuffing
flux:
(5.1)
Assuming an electron temperature of 25 eV [28] and an isothermal plasma, the sound speed in the
plasma is
C,
=
t~~' = 49. 0kmls·
(5.2)
Therefore, the initial Mach drift of the plasma at the center of the chamber should be
Mi =~=1.74+0.15·
(5.3)
Cs
For our flow experiments, we used two different Mach probes, one larger than the other.
Since k is dependent on the relative proportions of the Mach probe and the magnetic characteristics
of the plasma, these two probes should also have two different k values even though they exhibit the
same general geometry. We took calibration data with two different orientations for each of the
probes. In one orientation, one ion collector is facing directly upstream and another is facing
directly downstream, such that they are lined up with the cylindrical axis of the chamber. In the
other orientation, the Mach probe is rotated 180°, and we expect to see an inversion of the flows
measured using the original orientation for all 3 pairs of probes. For calibration purposes, however,
we are only concerned with the ratios of the upstream and downstream probes, shown in Figure 5.3
below.
49
Figure 5.3 The above two graphs show the logs of the ratio between the upstream and
downstream current densities averaged over 15 shots in the two different calibration orientations.
The error bars represent the standard deviation of the mean. The inversion reflects the fact that the
probe predicts a negative velocity when rotated 1800 from its original orientation.
50
Using our data from time of flight, it is clear that the initial flow velocity should occur at the
Mach probe around t = 30 J..ls. The log of the ratio for the large Mach probe at this time is 4.0 ± 0.5,
while the log of the ratio for the small Mach probe at this time is 3.3
± 0.7. Recalling the initial
Mach drift of the plasma out of the guns, we now have enough information to determine the
calibration constants.
k1g = 2.30 ± 0.35
(5.4)
= 1.90 ± 0.43
(5.5)
k sm
The value for k1g does not agree with Hutchinson's value for k for a magnetized probe (3.25) of 1.7.
However, this value was found by setting ion temperature equal to electron temperature, which is
not valid at SSx. It is harder to evaluate the value determined for
k.m because it is in the
unmagnetized realm. However, it is between the two values predicted for the two different
temperature extremes by (3.17) and (3.18). Considering that the temperature of our plasma also lies
between these extremes, unless we had some other way to approximate the value of the calibration
constant, this level of agreement is the best that we could hope for. Note that, also, the condition of
inversion has been satisfied within error in these two graphs. This was the case for each of the two
other pairs of probes during calibration, as well.
5.2 Visualizing Mach Probe Data
N ow that we know the calibration constants for each Mach probe, we can take data to determine the
flows in a plasma. Assuming cylindrical symmetry for the probe, we can use the same calibration
constant for each separate opposing pair of ion collectors. A look at the raw data in Figure 5.4
provides insight to the difficulties involved in analyzing the Mach probe. Notice how, at some points
in the data, the current density in a given ion collector will suddenly drop to zero or rise from zero.
Considering that the ion collectors are each facing nearly equal densities of plasma and that the
sudden drops to zero would correspond to unrealistically high plasma flows (because the flows are
determined by a ratio of current densities), it can be assumed that these portions of the data do not
actually correspond to measurements of flow of the plasma. Rather, there must be some electronic
source of these irregularities in the data acquisition process or in the interaction of the probe with
the plasma. We can only speculate as to the exact source, but it is clear that these portions of the
51
data should be ignored when we analyze the data for flows. Luckily, the irregularities occur
infrequently enough that results can be presented with confidence.
Figure 5.4 Six separate waveforms corresponding to the current densities in kA / m A2 for each of the
six ion collectors over the course of a single spheromak shot. Note how there are sudden drops and
rises that we speculate correspond to arcing, which render some sections of the data intractable.
Recall that there are six ion collectors, consisting of three pairs to determine the flow along
three different axes as shown in Figure 5.5. Because these three axes are in the 8-z plane, we cannot
make any measurements of radial flow, but the flows in the azimuthal and axial directions are
actually overdetermined. Using a simple coordinate transformation, we can resolve all three flows
into their axial and azimuthal directions. Also, the overdetermination in each direction provides a
measure of "error" that reflects how closely the individual measurements of flow agree with one
another.
52
z
Figure 5.5 Each black dot in the figure corresponds to an ion collector in the Mach probe. Each
opposing pair of ion collectors provides a measure of flow along an axis a, b, or c. The probe is
oriented in the machine to be aligned in the 8-z plane as shown.
Once the transformation is made, there is a clear depiction of the rotation of the plasma
(azimuthal component) as well as the axial flow through the center of the chamber. An example of
a typical result from the Mach probe over the course of a single spheromak shot is shown in Figure
5.6 below, along with a two-dimensional snapshot of flow right after the plasma has passed by the
probe in Figure 5.7, shown with relative current densities.
Figure 5.6 This graph is an example of the flow at the center of the chamber over the course of a
single spheromak shot. As we would expect, the axial flows are much more dynamic than the
azimuthal flows.
53
Figure 5.7 A snapshot of the flow 29 J..ls after plasma enters the chamber. Each of the six points
surrounding the origin are set at a radius relative to their respective current densities. The arrow
corresponds to the net flow. The error bars at the end of the arrow are a measure of error in each
direction showing how closely the distinct measurements of flow from each pair of ion collectors
agree. This graphing method was inspired by [3].
5.3 Single Spheromak Results
Both Mach probes were used to provide a radial profile of the flow at the midplane of the chamber
with interesting results. Measurements were made every two centimeters from the center to the
edge of the chamber, at a radius of 20 centimeters. Magnetic diagnostics show that, in the single
spheromak case, one spheromak is shot from one side of the chamber and reaches a relatively stable
state within the chamber for about 60 J..ls. When the spheromak is in this stabilized state, the
magnetic fields appear as in Figure 2.4. The two times during the shot that turn out to be most
notable occur right after the plasma passes by the probe and then 5-15 J..ls later. Radial profiles of
the flows in the chamber at these times as measured by both probes are shown below.
54
Figure 5.8 Radial profiles of drift velocity according to both the small and large probes at 28 J..ls
during a single spheromak shot, right after the plasma has started to pass by the probe. These drift
velocities are the averages of results from 10 different shots at each radial position.
55
Figure 5.9 Radial profiles of drift velocity according to both the small and large probes at 40 J..ls
during a single spheromak shot, after the plasma has stabilized in the chamber. These drift velocities
are the averages of results from 10 different shots at each radial position.
56
There are some notable similarities and differences between the flows according to these two
probes. At both times during the experiment and for both probes, the azimuthal velocity is very
low, though generally in the +8 direction. Early in the shot, both Mach probes agree that there is a
steady flow of approximately 70 kmI s for r > 10 cm in the axial direction. However, while the larger
Mach probe records flows of the same magnitude at r < 10 cm, the smaller Mach probe detects flow
of SO kmI s or less. The results at t=40 J.ls are even less consistent with each other. At r > 10 cm,
both probes agree that there is a reversal of flow at the edges such that, by this time, the flows are 40
I
km s
in the opposite direction. This reversal at the outer edge appears to be the result of a generally
linear shear of flow in the radial direction according to the large Mach probe, which shows the inner
radii maintaining an axial flow of SO km Is even 20 J.ls after the spheromak enters the chamber. The
smaller Mach probe tells a different story. Its results suggest that the axial flows at r < 12 cm are
actually very low by this time and even the flows near the center of the spheromak are reversed.
5.4 Counter-Helicity Merging Results
Recall that magnetic reconnection can be induced within the chamber by fIring two spheromaks
with opposing magnetic fIelds toward each other simultaneously. As discussed in the theory section,
radial and azimuthal flows are expected as a sign of magnetic reconnection. Flows were measured at
radii of 10 cm and 14 cm at the midplane by both the small probe and the large probe during these
counter-helicity merging shots. The data presented are averaged over SO shots total for the small
Mach probe and 100 shots total for the large Mach probe.
Looking at the flows from the small Mach probe, shown in Figure 5.10, the axial flows at
both of the radii at which measurements were made stay relatively close to zero, maxing out at
around 15 km Is in the E-7W direction. In general, the axial flow stays slightly negative throughout
the period shown. Concerning the azimuthal flows, the flow rises to about 20 kmI s in the positive
azimuthal direction at both radii between 40 and SO J.ls. Between SO and 65 J.ls, the azimuthal flow
at the outer radius drops to nearly zero while the azimuthal flow at the inner radius remains at 20-25
km/ s•
57
Figure 5.10 Drift velocity results from the small Mach probe during counter-helicity merging,
measured at radii of 10 em and 14 em, averaged over SO shots at each radius.
58
Figure 5.11 Drift velocity results from the large Mach probe during counter-helicity merging,
measured at radii of 10 em (bottom waveform) and 14 em (top waveform), averaged over 100 shots
at each radius.
59
Turning to the results from the large Mach probe during counter-helicity merging in Figure 5.11,
again, the axial flows are close to zero. The azimuthal flows are also very close to zero, except for a
period at 35 ± 5 s, over which time the azimuthal flow reaches a maximum of approximately 10 km /
in the +8 direction at r = 10 cm. At r = 14 cm, the azimuthal flow varies between 0 and 5 km /
during this time period.
60
s
s
Chapter 6
Interpretation
The fIrst observation that stands out from the results for each of the Mach probes is that, in some
crucial respects, they do not agree. Though some general conclusions can still be drawn from the
results attained here, this disagreement still requires some discussion. Recall, from Mach probe
theory, that it is appropriate to analyze Mach probe data differently depending on whether the probe
is in the magnetized or unmagnetized realm. The two probes used here are in two different realms,
so there are several possibilities that could explain this discrepancy. First, recall that it was
determined that the calibration constant for a Mach probe in the unmagnetized realm is highly
dependent on the temperatures of ions and electrons in the plasma. This would suggest that the
calibration constant for the smaller probe (unmagnetized) is changing over time and possibly even as
the Mach probe is moved radially. For example, it seems that there is more disagreement between
the probes at smaller radii. Perhaps this is because of differences in temperature that are only
reflected in the results of the smaller probe. The extent of these temperature changes and the effect
they would have on an unmagnetized analysis of a probe in an intermediate temperature range are
unclear. Does this mean that we should trust the results from the magnetized probe more?
Not necessarily, despite the fact that results shown here present many reasons to question
the dependency of the smaller probe. The calibration constant, according to one-dimensional
collisionless kinetic theory, should be a constant at k=1.7, independent of temperature and magnetic
fIeld strength. However, this calibration constant was established using a one-dimensional analysis
which assumed a steady direction of the magnetic fIeld in the direction of flow and perpendicular to
the probe stalk. This is a highly idealized situation, and doesn't at all resemble the orientation of the
61
magnetic fields in the plasma chamber, especially during highly energetic events such as magnetic
reconnection. This is not to say that the results from either of these probes is worthless, for the
calibration of each probe clearly passed the inversion test and they clearly demonstrate
measurements of flow, when disregarding electrical anomalies.
6.1 Single Spheromak Flow
After a single spheromak is released into the chamber, it passes the midplane of the chamber with a
radially constant axial flow of 70 km /
s.
Over the next 10-15 J.ls, a radial shear in the flow develops
until a reversal of flow is exhibited at the edges of the chamber. The flows at the edge of the
chamber during this period of time reach approximately 40 km / s in the opposite direction from the
original flows. At this point, the velocities remain relatively constant, decaying until the energy has
dissipated at about 100 J.ls. Throughout the shot, the azimuthal flow is practically zero, showing a
slight tendency toward the positive azimuthal direction.
These results are consistent with the frozen-in flux condition postulated by MHD for a
plasma that has a Lundquist number of 1000. The lack of radial shear in azimuthal flow reflects the
same lack of shear in the azimuthal magnetic fields of a spheromak at the midplane, as shown in
Figure 2.4. The radial shear in axial flow, which probably reflects plasma crashing into the opposite
end of the chamber and bouncing back around the edges of the chamber, follows the poloidal fields
in the spheromak. There is little apparent flow that acts perpendicularly to magnetic fields in the
spheromak.
6.2 Flow During Counter-Helicity Merging
It is not immediately clear exactly what to expect from flows in coounter-helicity merging. One
would assume that the axial flow at the midplane should remain at approximately zero, assuming
symmetry but allowing for some fluctuations. The data presented above roughly agrees with this
condition. In the azimuthal direction, we expect to see flows due to bi-directional jets from
magnetic reconnection. The geometry of the problem is as pictured in Figure 6.1, which depicts two
spheromaks coming together, focusing only in the r-z plane at r > O.
62
to roidal fields
po loidal fields
reoo nn ectlon plane
Into page
reoo nn ection plane
outofPage8
®
z
Figure 6.1 A schematic depicting the reconnection geometry of one side of two spheromaks in the
r-z plane. The fields of the bottom spheromak are tilted out of the page and to the left, while those
on the top spheromak are pointing into the page and to the right.
In this figure, the spheromaks contain opposing fields. The top spheromak is left-handed
(the poloidal fields move counterclockwise around a toroidal field going into the page in the -8
direction) and the bottom spheromak is right-handed (the poloidal fields move counterclockwise
around a toroidal field coming out of the page). Since the magnetic field lines are actually a
superposition of these two fields, the magnetic field lines about to reconnect from the bottom
spheromak are pointing to the left and out of the page, while the magnetic field lines about to
reconnect from the top spheromak are pointing to the right and into the page. This suggests that,
using the model of two-dimensional Sweet-Parker reconnection, the left part of the reconnection
plane is tilted out of the page and the right part of the reconnection plane is tilted into the page.
Therefore, the bi-directional jets from reconnection are not solely in the radial direction, as they
would be if the poloidal fields were the only ones present, but in the azimuthal direction as well. At
radii inside the reconnection region, some positive azimuthal flow should be recorded (+8 is
directed out of the page) , and at radii outside the reconnection region, some negative azimuthal flow
should be recorded.
In experiments performed at SSX last year, a bidirectional radial flow like that expected
during reconnection was detected using ion Doppler spectroscopy [13]. This radial flow occurred at
40 ± 10 j..ls, in agreement with other diagnostics that this is the time period during which
reconnection occurs. Data from the smaller Mach probe is difficult to parse. At both radii, positive
63
azimuthal flows occur on average, and they both seem to reach maxima of 2S
km / "
higher than we
would expect the azimuthal flows to be, at around SS J..ls, well after the expected average time period
for magnetic reconnection. The data during the period between 30 and 40 J..ls is broken up, though
it appears that there are positive azimuthal flows at a radius of 10 cm during this period, as we would
expect from the model explicated above. The larger Mach probe provides much nicer data. For the
measurements at r = 10 cm, a peak flow of 10 km / s occurs at 36 J..ls. The peak drops to
approximately no flow by t = 43 J..ls. The measurement at r = 14 cm seems to predict two separate
bursts of lesser magnitude. Each burst reaches a maximum flow of 7 km / s in the -8 direction, and
the bursts occur at 33 J..ls and 36 J..ls. There is no flow at this radius after t = 39 J..ls. These azimuthal
flows are in agreement with the expected bi-directional outflows due to magnetic reconnection.
64
Appendix A
Glossary
Force-free - A plasma configuration is described as force-free if it is not necessary to use
exterior energy sources in order to maintain the plasma. One of the most notable characteristics of
such a plasma is that the cross product between bulk currents and the magnetic field is always zero
Gyroradius - The gyro radius of ions in a plasma is defmed as the radius of ions in a plasma,
assuming that they are traveling at their average thermal velocity and given the strength of the
magnetic field in which they are immersed. More specifically, it is given by (3.7).
Magnetohydrodynamics (MHD) - This is a model popularly used to describe plasma that
treats the plasma as an electrically conducting fluid. This model focuses on analyzing the bulk flow
behavior of the plasma rather than the motion of individual particles. Some versions of this theory
include idealMHD, in which the fluid is assumed to be perfectly conducting, and resistive MHD, in
which the fluid is assumed to have a locally invariant resistance.
Poloidal- A component in the coordinate system used to describe toroids. This is the
direction which goes through the hole in the middle of the toroid and around the outside. It is
always perpendicular to the toroidal component.
Reversed field pinch (RFP) - A formation of plasma similar to the tokamak except for some
notable characteristics of the magnetic field. The toroidal field reverses direction as one moves
outward from the center of the toroid, and there is a non-zero poloidal field. This formation is
65
advantageous because it takes less energy to sustain than the tokamak, but it is vulnerable to less
predictable turbulence and non-linear effects.
Spheromak - A force-free toroidal formation of plasma and the focus of research at SSx. This
formation is characterized by the presence of both toroidal and poloidal fields, as well as its stability
without the imposition of exterior energy sources.
Stellarator - A formation of plasma, similar to the tokamak, in which the plasma is contained in
a nearly toroidal shape and driven by an exterior current. To avoid the buildup of magnetic energy
that is inherent toward the center of tokamaks, stellarators are twisted toroids, such that at one time a
given section of plasma is immersed in a higher magnetic field and at another it is in a lower
magnetic field.
Tokamak - A formation of plasma in which the plasma is driven around a ring containing
toroidal magnetic fields. These toroidal fields are maintained by strong electrical currents
surrounding the ring. Due to the discovery of higher confmement modes, this remains the most
promising formation for the prospect of fusion energy.
Toroidal- A component in the coordinate system used to describe toroids. If one were to
make concentric rings about the center of a toroid, the toroidal direction is defmed as the direction
along the path of these rings. It is always perpendicular to the poloidal component.
Z-pinch - A formation of plasm in which a straight, pinched line of flowing plasma is
maintained between two electrodes. The pinch occurs due to magnetic pressure contained in a
magnetic field surrounding the plasma.
66
Appendix B
Argument for Flow in MHD Equilibria
The following is an argument for the necessity of flow in any fonn of steady state
magnetohydrodynamic equilibria. In section 2.6, where spheromak theory is introduced, it is
protested that a spheromak in a steady state is not necessarily void of bulk flows. The following
derivation is intended as a supplement to that argument. This reductio was referred to but not fully
fleshed out in [32].
We begin with the resistive Ohm's law:
E+vXB=1]J
\
(B.l)
and assume there is absolutely no flow and a steady state. Thus, the above equation and Faraday's
law (V X E
dB
dt
= -) reduce to
- E=1]l
(B.2)
and
(B.3)
Now we invoke the spheromak force-free condition, discussed in section 2.6, and recall the
following formulas:
1xB=Q
VX B= AB = f.l) .
67
(B.4)
(B.S)
This is where the inconsistency occurs. The steady state Faraday's law (B.3) would imply that the
curl of the electric field is zero; however, if we take the curl of Ohm's law with no flow (B.2), we get
- 1]AB
VxE= VX(1]J)= V x -
(B.6)
110
which is clearly nonzero, since
1]A is a constant and the curl of the magnetic field is nonzero by
110
(B.S).
This argument has a couple key implications. The first is that, given MHD in a force-free
plasma formation, there is either necessarily some flow or the plasma is not in a steady state (that is,
its magnetic field is constantly changing). The latter is not witnessed in the laboratory in the case of
stable force-free plasmas, so we must settle on the former - that MHD dictates a constant flow in
order to satisfy Maxwell's equations and the steady-state condition. Another strong possibility is
that MHD is just too simplified, so we are using the wrong Ohm's law. An Ohm's law containing
more terms has been postulated with some success, and it seems that MHD is now developing into
a more complex theory, accounting for more factors that are integral to electromagnetic energy
storage and dissipation in plasmas [33].
68
References
[1] K.H. Burrell et al., Phys. Fluids B 2, 1405 (1990)
[2] H. Yamada et al., Phys. Plasmas 9, 4605 (2002)
[3] S.P. Gerhardt et al., Rev. of Sci. lnst. 75, 4621 (2004)
[4] U. Shumlak et al., Phys. Plasmas 10, 1683 (2003)
[5] ]. Ghosh et al., Phys. Plasmas 13, 022503 (2006)
[6] R.F. Ellis et al., Phys. Plasmas 12, 055704 (2005)
[7] M. Cecconello et al., Plasma Phys. Control. Fusion 48, 1311 (2006)
[8] S.c. Hsu et al., Phys. Plasmas 8, 1916 (2001)
[9] Y. Ono et al., Phys. Rev. Lett. 76, 3328 (1996)
[10] A. Falk, Dynamics ofField-Reversed-Configuration in SSX, Senior honors thesis, Swarthmore College
(2003)
[11] F.F. Chen, Introduction to Plasma P0;sics and Controlled Fusion Vol 1: Plasma P0;sics, Second
Edition, (plenum Press, N ew York, 1984)
[12] S.R. Seshadri, Fundamentals of Plasma P0;sics, (American Elvesier Company, New York, 1973)
[13] ]. Fung, High Resolution Flow and Ion Temperature Measurements with Ion Doppler Spectroscopy at SSX,
Senior honors thesis, Swarthmore College (2006)
[14] D. Biskamp, Magnetic Reconnection in Plasmas, (Cambridge University Press, Cambridge, 2000)
[15] P.A. Sweet, Electromagnetic Phenomena in Cosmical P0;sics (Cambridge University Press, New York,
1958)
[16] E.N. Parker,]. Geophys. Res. 62, 509 (1957)
[17] E. Priest and T. Forbes, Magnetic Reconnection: MHD Theory andApplications, (Cambridge
University Press, Cambridge, 2000)
69
[18] M. Landreman, The Three-Dimensional Structure ofMagnetic Reconnection in SSX, Senior honors
thesis, Swarthmore College (2003)
[19] P.M. Bellan, Spheromaks, (London: Imperial College Press, 2000)
[20] lB. Taylor, Rev. Mod. Phys. 58, 741 (1986)
[21] CG.R. Geddes et al., Phys. Plasmas 5, 1027 (1998)
[22] CG.R. Geddes, Equilibrium Studies on SSX, Senior honors thesis, Swarthmore College (1997)
[23] LH. Hutchinson, Phys. Rev. A 37,4358 (1988)
[24] K.S. Chung and LH. Hutchinson, Phys. Rev. A 38,4721 (1988)
[25] M. Hudis and L.M. Lidsky, l Appl. Phys. 41, 5011 (1970)
[26] LH. Hutchinson, Phys. Plasmas 9, 1832 (2002)
[27] LH. Hutchinson, Plasma Phys. Controlled Fusion 44, 1953 (2002)
[28] V. Chaplin, High Time Resolution Measurements ofElectron Temperature in the SSX Plasma, Senior
honors thesis, Swarthmore College (2007)
[29] T. Gray, Densiry Studies on SSX, Senior honors thesis, Swarthmore College (2001)
[30] T.W. Kornack, Magnetic Reconnection Studies on SSX, Senior honors thesis, Swarthmore College
(1998)
[31] C MacLatchy et al., Rev. Sci. Inst. 63, 3923 (1992)
[32] M.R. Brown, l Plasma Physics 57, 203 (1996)
[33] CD. Cothran et al., Geophys. Res. Lett. 32, L03105 (2005)
70
Acknowledgements
I would especially like to thank my advisor and the founder of the SSX lab, Dr. Michael Brown, for
his eagerness to share his expertise in this field and his willingness to let me playa big role in his
project. Also, I would be remiss if I failed to recognize his boundless patience with my working
habits.
Though David Cohen was not a constant presence in the SSX laboratory, his eagerness to
probe scientific issues was inspiring, and I appreciated having him as a colleague and an advisor.
Chris Cothran was also very important to the success of this project, as a mentor who was
willing to help me get my feet wet when I was new to the field, even when he was busy with a
multitude of his own projects.
Finally, I would like to thank all of the fellow students that have worked with me on this
project, whose unfailing kindness and diligence have helped make my experience at the Swarthmore
Spheromak Experiment both a fulfilling and a productive one. They are Brie Coeliner, Marc Chang,
Vernon Chaplin, Jerome Fung, and Victoria Swisher.
71