# Interaction of Ambipolar Plasma Flow with Magnetic Islands in a Quasi-Axisymmetric Stellarator

A. Reiman^1^, M. Zarnstorff^1^, D. Mikkelsen^1^, L. Owen^2^, H.
Mynick^1,^ S. Hudson^1^, D. Monticello^1^

## ^1^Princeton Plasma Physics Laboratory, Princeton, NJ 08543

*^2^Oak Ridge National Laboratory, Oak Ridge, Tennessee*

*email contact of main author: reiman@pppl.gov*

# 

### Abstract. A reference equilibrium for the US National Compact Stellarator Experiment is predicted to be sufficiently close to quasi-symmetry to allow the plasma to flow in the toroidal direction with little viscous damping, yet to have sufficiently large deviations from quasi-symmetry that nonambipolarity significantly affects the physics of the shielding of resonant magnetic perturbations by plasma flow. The unperturbed velocity profile is modified by the presence of an ambipolar potential, which broadens the profile and improves the shielding near the plasma edge. In the presence of a resonant magnetic field perturbation, nonambipolar transport produces a radial current, and the resulting jxB force resists departures from the ambipolar velocity and enhances the shielding.

**1. Introduction**

Resonant magnetic perturbations pose a threat to flux surface integrity
in toroidal magnetic confinement configurations. The width of the island
produced by a resonant perturbation scales as the square root of the
perturbation amplitude, so that even a relatively small resonant
magnetic perturbation at a rational surface can produce a substantial
magnetic island. Resonant perturbations can be shielded out at rational
surfaces by plasma flow.\[1,2\] This effect is believed to play a major
role in reducing the vulnerability of present day tokamaks to resonant
field errors, and an understanding of the effect will be important for
setting field-error tolerances for ITER. The flow shielding effect has
been studied systematically in tokamak experiments where externally
imposed magnetic field perturbations have been varied and their
penetration threshold determined.\[3,4,5\]

This paper considers the flow-shielding effect in a quasi-axisymmetric
stellarator. Quasi-axisymmetric stellarator configurations have drift
trajectories that look like those in an axisymmetric configuration, and
they allow undamped toroidal flow.\[6\] In the limit of perfect
quasi-axisymmetry, the flow shielding effect is predicted to look like
that in a tokamak having the same parameters. However, if we allow for
the presence of non-quasi-symmetric ripple in the field, the radial
transport is no longer intrinsically ambipolar, as it is in axisymmetric
configurations. This brings in an additional radial current which
modifies the physics of the flow shielding. The radial current produces
a **j x B** torque that resists externally induced changes in the flow
velocity and enhances the effectiveness of the shielding. It also
modifies the unperturbed rotation velocity of the plasma in the absence
of a resonant perturbation.

The work described in this paper focuses on a particularly interesting
regime of intermediate ripple amplitude, where the deviations from
quasi-symmetry are sufficiently large to substantially modify the
flow-shielding effect, but where the configuration is nonetheless
sufficiently close to quasi-symmetry that the flow damping in the
toroidal direction can be considered to be negligibly small compared to
that in the poloidal direction. A reference equilibrium for the US
National Compact Stellarator Experiment (NCSX) is calculated to be in
this intermediate regime\[7\], and the numerical calculations presented
in this paper focus on that NCSX reference equilibrium. The unperturbed
ambipolar velocity profile is calculated to be broader than the velocity
profiles seen in tokamaks, suggesting that it will provide greater
shielding for low order rational surfaces near the plasma edge, with
potential consequences for possible startup scenarios.

The three-dimensional NCSX device will have great flexibility for
controlling resonant magnetic field components and investigating their
interaction with plasma flow. Comparison of theoretical predictions with
experimental observations on NCSX, and with tokamak experiments having
comparable plasma parameters, will contribute towards the goal of being
able to reliably predict field error penetration thresholds.

The NCSX, under construction at Princeton, is a quasi-axisymmetric
stellarator designed to combine favorable features of advanced tokamaks
with those of drift-optimized stellarators. The NCSX configuration has
been designed to have good flux surfaces, incorporating several layers
of defense against excessive magnetic island formation, but flow
shielding could nonetheless have an impact on flexibility and on
vulnerability to field errors. For the design of the NCSX, an
optimization code built around the PIES three-dimensional equilibrium
code was used in the coil-design process to reduce the magnitude of
resonant components of the magnetic field while preserving desired
engineering and physics properties. A series of calculations with the
PIES code showed that this coil design process, which targeted the
resonant components of the magnetic field in the NCSX reference
equilibrium, also greatly reduced the island widths for a range of
equilibria with varying profiles, betas, and coil currents.\[8,9\] The
NCSX design also incorporates two sets of trim coils to provide further
control over resonant magnetic fields. NCSX has also been designed to
have a monotonically increasing (=1/q) profile to give neoclassical
suppression of magnetic islands, and this is expected to further protect
against magnetic island formation. Nevertheless, to the extent that the
plasma flow shields out residual resonant magnetic field components at
rational surfaces, it will further improve the flexibility of the NCSX
device to generate a range of configurations with good flux surfaces,
and it will further reduce the vulnerability of the NCSX device to field
errors produced by finite tolerances in the construction and placement
of the magnetic field coils.

The calculations described in this paper have been done for a reference
= 4% NCSX equilibrium whose properties are extensively discussed in a
special volume of the journal Fusion Science and Technology devoted to
the NCSX physics design.\[10\]

# 

# 2. Shielding of Rational Surfaces by Plasma Flow in a Quasi-Axisymmetric Stellarator

In an ideal plasma, reconnection is prohibited and the flux surfaces
cannot be broken. A surface current is induced at the rational surface
that shields out resonant magnetic perturbations. In the absence of
plasma flow, the presence of even a small resistivity causes the surface
current to decay, and allows the resonant field to penetrate the
rational surface. If flow is present at the rational surface, a
localized current continues to be induced which partially shields out
the resonant component of the field. If the flow is sufficiently strong,
or the perturbation sufficiently weak, only a very small fraction of the
resonant field penetrates the rational surface.

The induced current at the rational surface interacts with the remnant
of the resonant field there to produce a **j** x **B** torque. This
electromagnetic torque opposes the motion of the plasma at the rational
surface, and acts to slow the flow. When the resonant perturbation
amplitude exceeds a threshold value, the torque is large enough to
locally suppress the plasma flow, allowing the resonant perturbation to
fully penetrate the rational surface.

Consider the case where a small perturbation of the magnetic field is
turned on in a stellarator plasma that initially has good flux surfaces.
Express the unperturbed magnetic field in magnetic coordinates: **B**~0~
= ~0~ x + ~0~ x , where **B**~0~ is the unperturbed field, and ~0~ is an
unperturbed flux function satisfying **B**~0~**·**~0~ = 0. Write **B** =
**B**~0~ + **B**, = ~0~ + . To first order we get (*n -* *m*) ~nm~ =
-(**B·**~0~ / **B**~0~**·**)~nm~. The nonresonant Fourier components
just introduce small ripples in the flux surfaces. If a resonant Fourier
component is present (one satisfying *n* = *m),* the flux surface is
broken and a magnetic island is produced.

The response of a rotating plasma at the rational surface to an
externally imposed resonant perturbation has been calculated
theoretically for a variety of regimes and under a variety of
assumptions.\[1,11,12\] These calculations have been done for either
slab or cylindrical geometry. Because the local induced current is
determined by the layer physics, these calculations are relevant for
shaped tokamaks and for stellarators.

The electromagnetic torque exerted on the rational surface by the
resonant perturbation is opposed by a viscous torque produced by the
plasma flow external to the surface. The threshold for resonant field
penetration is determined by the relative magnitude of the
electromagnetic torque and the viscous torque. While the physics
determining the magnitude of the electromagnetic torque is the same in
tokamaks and stellarators, the physics determining the viscous torque is
modified in a stellarator by the radial current produced by the
nonambipolar transport. In the absence of a resonant perturbation, this
radial current produces an ambipolar potential and a corresponding
contribution to the plasma flow. When a resonant perturbation is
imposed, the electromagnetic torque causes the flow velocity to deviate
locally from its ambipolar value. The radial current arising from the
resulting nonambipolar transport produces a **j x B** torque that
opposes the electromagnetic torque and enhances the effectiveness of the
shielding.

**3. Ambipolar Plasma Flow in NCSX.**

In this section we calculate the plasma flow velocity profile in the
absence of an external resonant perturbation for our reference = 4% NCSX
equilibrium.

Our transport model uses a set of one-dimensional transport equations to
solve for the temperature and self-consistent radial electric field,
E~r~, in cylindrical geometry (large aspect-ratio, circular
cross-section), with an assumed density profile. The model is described
in Ref. \[7\]. In the absence of a radial electric field, the ions are
lost more rapidly than the electrons, giving a net outward current. The
radial electric field, E~r~, builds up until it is sufficiently large to
equalize the radial flux of the ions and electrons, j~r~ = 0.

In steady-state, the ion momentum equation determines the component of
the flow perpendicular to the magnetic field: ![](media/image1.wmf).
There is in addition a component of the flow velocity aligned with the
magnetic field, ![](media/image2.wmf), and its magnitude is determined
by the relative flow damping in the poloidal and toroidal directions. As
in a tokamak, the damping in the poloidal direction is strong. The
configuration is sufficiently close to quasi-axisymmetry that the flow
damping in the toroidal direction is small. This implies that the flow
velocity in the poloidal direction can be taken to be zero to a good
approximation. (The relative viscosities have been estimated using the
analytic estimates of Coronado and Talmadge.\[13\].) Imposing the
constraint that the poloidal component of the velocity is zero, we get
![](media/image3.wmf).

The calculation thus far has not taken into account momentum diffusion.
We have taken the poloidal velocity to be zero because of the strong
poloidal damping, and we only need to consider the toroidal component of
the momentum diffusion equation. We consider the momentum diffusion
equation in a cylinder, where it takes the form:

![](media/image4.wmf). (1)

Here is the plasma density, is the (anomalous) momentum diffusivity
(![](media/image5.wmf)is the coefficient of perpendicular viscosity),
and ![](media/image6.wmf) is the current produced by nonambipolar radial
transport. In tokamak experiments, the anomalous momentum diffusivity
has been found to be approximately equal to the anomalous cross-field
thermal diffusion coefficient, and we assume that that is also the case
here. Stellarators often have experimentally determined thermal
diffusivities that are approximately radially constant (unlike many
tokamaks), and we will adopt the simple model of taking to be radially
constant in the following.

Equation (1) differs from the momentum diffusion equation in a tokamak
by the presence of the last term, which is nonzero when the flow
velocity on a flux surface is forced away from its ambipolar value.

Eq. (1) must be supplemented by boundary conditions at the origin and at
the edge. At the origin, regularity requires ![](media/image7.wmf). At
the plasma edge, the boundary condition is determined by the interaction
with neutrals and with the scrape-off layer, which produce a momentum
flow through the plasma edge equal to ![](media/image8.wmf), where *a*
is the minor radius and *R* is the major radius. The momentum flow is
equal to the total force exerted by the neutrals and scrape-off layer,
which are taken to act on a small region at the plasma edge. (We will
justify this approximation below.)

Near the plasma edge, momentum is transferred to the neutrals primarily
through charge exchange. Ionization reactions also must be taken into
account, because they serve to impart some of the momentum picked up by
the neutrals back to the ions. To estimate the rate of momentum transfer
to the neutrals we use the Degas code to do a Monte Carlo calculation
for a model axisymmetric geometry.\[14\] We expect the recycling to be
dominated by the inner midplane of the 60 deg. (bullet) cross section of
NCSX, so we adopt this geometry.

The momentum transfer to the neutrals is localized at the plasma edge,
with the average momentum transfer rate in the zone .96 \< *r/a* \< 1
calculated to be about seven times as large as that in the zone .92 \<
*r/a* \< .96. The rate of momentum transfer to the neutrals can be
expected to scale roughly linearly with the plasma velocity. We write
this momentum transfer rate as ~n~ v~z~, where ~n~ is a coefficient to
be determined. For an edge velocity of 290 km/sec, the integrated
momentum transfer rate is calculated to be about 1.2 Newtons,
corresponding to ~n~ = 4 x 10^-6^ kg/sec. This gives the boundary
condition *a* dv~z~(*a*) / d *r* = - v~z~(*a*), with 2.

We next estimate the momentum transfer to the scrape-off layer. We
consider the case where there is a toroidal rail limiter. Field lines
outside the plasma edge intercept the limiter, with a

connection length of *L* R / , where .6 is the rotational transform at
the plasma edge. The ion mean free path is comparable to the connection
length. Particles outside the plasma edge

![](media/image9.wmf){width="2.775in" height="1.6833333333333333in"}

are lost to the limiter in a time *L* / v~ti~, so that the momentum loss
rate in the scrape-off layer is approximately v~z~ / . Combining this
with momentum diffusion, and adopting a slab approximation (which is
appropriate in the narrow scrape-off layer), we get
![](media/image11.wmf). The velocity decays exponentially as a function
of *r* in the scrape-off layer, v~z~(*r*) = v~z~(a) exp(-(r-a)/*l*). The
density obeys a similar equation, and it too decays exponentially in the
scrape-off layer. If the diffusion coefficients are equal, *l 1.6*
![](media/image12.wmf)*.* Momentum is dissipated in the scrape-off layer
at the rate ![](media/image13.wmf). The momentum transfer rate is again
of the form ~n~ v~z~, with the scrape-off layer contribution to ~n~
estimated to be roughly 0![](media/image14.wmf). The momentum transfer
to the scrape-off layer is sensitive to the value of at the edge. For
our assumed density profile *n*~e~(*a*) 1.5 x 10^19^ m^-3^, and we
calculate 18. For smaller values of *n*~e~(*a*), the momentum transfer
to the scrape-off layer is correspondingly smaller, with the total
momentum transfer rate bounded below by the contribution of the
neutrals.

Having determined the boundary conditions, we return to the solution of
Eq. (1). For this purpose, we must determine the dependence of j~r~ on
v~z~. The radial current vanishes when v~z~ has its ambipolar value,
corresponding to the ambipolar value of the electrostatic potential. We
adopt a simple linear approximation for j~r~, interpolating between the
values for E~r~=0 and for the ambipolar value of E~r~. The last term in
eq. (1) can then be written in the form ![](media/image15.wmf), where v0
is the ambipolar value of v~z~ (i.e. the value that v~z~ assumes when
E~r~ has its ambipolar value). Eq. (1) now assumes the linear form

![](media/image16.wmf) (2)

and can be solved numerically in a straightforward manner.

The top (solid) curves in Fig. (1) show numerical solutions for the
velocity profile for two different values of . Fig. (1a) corresponds to
2, the lower bound on momentum dissipation due to collisions with
neutrals. Fig. (1b) corresponds to 18, the estimate for momentum
dissipation in the scrape-off layer with the assumed value of
*n*~e~(*a*).

Because the ripple magnitude increases rapidly towards the plasma edge,
the flow velocity profile is broad relative to that in a tokamak. As the
*q* profile evolves during tokamak startup,

low order rational surfaces entering from the plasma boundary are
particularly vulnerable to resonant magnetic perturbations. The broader
velocity profile in NCSX will provide stronger

shielding for low order rational surfaces near the plasma edge, and this
will potentially impact the options available for startup scenarios.

**4. Viscous Torque on Rational Surfaces**

When a resonant magnetic field perturbation is imposed on a rotating
plasma, the resulting electromagnetic force slows the plasma rotation at
the rational surface. The electromagnetic force is balanced by a viscous
force exerted by the neighboring plasma on the rational surface, which
opposes the slowing of the plasma at the rational surface. As the
amplitude of the external perturbation is increased, the electromagnetic
force increases, and the rotation velocity of the plasma at the rational
surface decreases further. The magnitude of the viscous force on the
rational surface is determined by the momentum diffusion equation.

In addition to the viscous force on the rational surface, there is also
a direct **j x B** torque exerted by the current that arises from the
non-ambipolar transport. The total torque exerted directly by the radial
current is obtained by integrating the torque density across the
boundary layer at the rational surface. For the case considered here,
the viscous torque is estimated to be much larger than the torque
exerted directly by *j*~r~.

We again consider the reference NCSX equilibrium whose unperturbed
velocity profile we discussed in the previous section. The = 3/5
rational surface is of particular concern because of its low order and
because of its proximity to external perturbations. (It is located at
*r*/*a* 0.8.) The m=5, n=3 island proved to be the island that was the
most difficult to suppress in the NCSX coil design process. We consider
here the resonant mode penetration at the = 0.6 rational surface in the
presence of the ambipolar flow. Assuming that an externally generated
m=5, n=3 perturbation slows the rotation of the the rational surface, we
calculate the countervailing viscous force. We solve the momentum
diffusion equation for this purpose.

The solution of Eq. (2) is obtained under the assumption that the
electromagnetic force has slowed the rotation to a fraction of its
ambipolar value. Denote the velocity at the rational surface by v~s~,
and the unperturbed velocity at the rational surface by v~s0~. We
consider the case where v~s~ = v~s0~/2, and the case where v~s~ = 0.
Figures (1a) and (1b) show, respectively, the corresponding solutions of
Eq. (2) for 2 and 18. In each plot, the top curve corresponds to the
solution in the absence of an electromagnetic force, the middle curve
corresponds to the solution when the velocity at the = .6 rational
surface is slowed to half its ambipolar value, and the bottom curve
corresponds to the solution when the rotation at the rational surface is
entirely suppressed.

The viscous force exerted on the rational surface by the plasma flow is
given by ![](media/image17.wmf), where ![](media/image18.wmf) is the
jump in the radial derivative of the fluid velocity across the
associated boundary layer. For 2, we calculate *a*![](media/image18.wmf)
274 km/sec and 540 km/sec respectively for v~s~ = v~s0~/2 and v~s~ = 0.
For 18 we calculate *a*![](media/image18.wmf) 238 km/sec and 478 km/sec
respectively. Relative to v0, the velocity on axis for the unconstrained
velocity profile, we have *a*![](media/image18.wmf) 3.4 v~0~ and 6.8
v~0~ for 2, *a*![](media/image18.wmf) 3.5 v~0~ and 7.0 v~0~ for 18.

We compare these results with a simple tokamak model. Consider a
rotating tokamak plasma

driven by neutral beams with a uniform deposition profile, and take the
viscosity to be a

constant. The unperturbed velocity profile is quadratic, v = v~0~ (1 --
*r*^2^ / *a*^2^). Constraining v~s~ = v~s0~/2 at *r*~s~/*a* = .8 gives
*a*![](media/image18.wmf) = -( v~0~/2) (1 - *r*~s~^2^/*a*^2^) / (*r*~s~
ln *r*~s~) v~0~. Constraining v~s~ = 0 gives twice this value.

In a tokamak, the scale length of the velocity gradient is comparable to
the minor radius, so that the jump in dv / dr is of the order of v / a.
In a stellarator, the velocity gradient is determined by the magnitude
of the non-ambipolar **j x B** force, so that the gradient scale length
can be shorter, imparting greater stiffness to the flow velocity, and
enhancing the shielding effect.

**5. Resonant Mode Penetration Threshold**

In mode penetration experiments on tokamaks where the amplitude of the
external perturbation is gradually ramped up, it is found that the
rational surface first slows to some fraction of its initial rotation
frequency, and then abruptly ceases to rotate when the perturbation
amplitude exceeds a threshold value. The cessation of rotation is
accompanied by a complete penetration of the resonant perturbation at
the rational surface. This is consistent with the predictions of
theoretical calculations. The magnitudes of the viscous and
electromagnetic forces are functions of v~s~, and if v~s~ \> 0 it must
satisfy F~visc~(v~s~) = F~em~(v~s~). There is predicted to be a
threshold in the perturbation amplitude above which F~em~(v~s~) exceeds
F~visc~(v~s~) for 0 v~s~ v~s0~, so that v~s~ = 0 when the perturbation
amplitude exceeds this threshold.

The magnitude of F~em~ scales as the square of the resonant perturbation
amplitude, with the functional dependence of F~em~(v~s~) (i.e. the shape
of F~em~(v~s~)) independent of the amplitude. Our numerical solution
shows that F~visc~(v~s~) is well approximated by a linear function of
v~s~, as it is in a tokamak, so that while its amplitude may be quite
different, the functional dependence on v~s~ has not changed. It follows
that there is again a threshold value of the resonant perturbation
amplitude above which F~em~ dominates F~visc~, and that at the threshold
value v~s~ is the same as in the tokamak. The resonant mode penetration
threshold scales as F~visc~^1/2^.

To estimate the magnitude of the flow shielding effect for magnetic
islands in NCSX, we compare with a resonant mode penetration experiment
on DIII-D.\[4\] The DIII-D reference case has been chosen to have
similar parameters to those in our NCSX reference equilibrium. It has
3.7%, n~e~ 5 x 10^19^ m^-3^, and an ellipticity 1.8. Our reference NCSX
equilibrium has = 4%, n~e~ = 6 x 10^19^ m^-3^, and an average
axisymmetric component of ellipticity of 1.8. The magnetic field of both
the DIII-D reference shot and the NCSX reference case is 1.2 T. The
rotation frequency of the rational surface in the DIII-D reference shot
is about 12 kHz. For the NCSX case, the rotation frequency ranges from
about 9 kHz for 2 to about 7 kHz for 18. DIII-D has R 1.67 m and R/a
2.1, while NCSX has R 1.42 m and R/a 4.3. The experimentally observed
penetration threshold in the DIII-D reference case is
$\frac{B_{r\text{21}}}{B} \approx 4x\text{10}^{- 4}$.

**6. Discussion**

The physics determining the penetration of a resonant magnetic
perturbation in a stellarator differs from that in a tokamak due to the
presence of a radial current produced by nonambipolar transport. As the
electromagnetic force produced by the perturbation slows the rotation at
the rational surface, the radial current driven by the resulting
nonambipolar transport exerts a **j x B** force that resists departures
from the ambipolar velocity and enhances the shielding effect. The
unperturbed velocity profile is also modified in a stellarator. We have
focused here on a particularly interesting regime, corresponding to an
NCSX reference equilibrium, in which the configuration is sufficiently
close to quasi-symmetry that the viscous damping in the toroidal
direction is small, but the deviations from quasi-symmetry are
sufficiently large to produce a substantial ambipolar flow, and a
substantial modification of the flow-shielding effect. Because the
ripple magnitude increases rapidly towards the plasma edge, the flow
velocity profile is broad relative to that in a tokamak. The stronger
shielding for low order rational surfaces near the plasma edge will have
potential implications for startup scenarios.

A reference DIII-D shot with parameters similar to those of our
reference NCSX equilibrium has been reported to have a penetration
threshold of $\frac{B_{r\text{21}}}{B} \approx 4x\text{10}^{- 4}$.\[4\]
Calculations with the PIES code found that the resonant *m* = 5, *n* = 3
field component associated with an NCSX coil design algorithm that did
not explicitly target resonant field error reduction was
$\frac{B_{\text{rnm}}}{B} \approx 1\text{.}3x\text{10}^{- 3}$. This is
likely above the penetration threshold, even including the enhancement
of the shielding due to nonambipolarity, and a further coil optimization
using the PIES code to reduce the magnitude of the resonant field
components was a prudent step in the coil design process. To the extent
that the plasma flow shields out residual resonant magnetic field
components at rational surfaces, it will further improve the flexibility
of the NCSX device, and it will further reduce the vulnerability of the
NCSX device to field errors.

**Acknowledgment**

This work was supported by DOE contract DE-AC02-76CH03073.

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