Carl Sovinec
and James Reynolds
University
of Wisconsin-Madison
American Physical
Society, Division of Plasma Physics
43rd
Annual Meeting
October 29-November
2, 2001
Long Beach,
California
Objectives
1.To
determine the influence of toroidal geometry on low-breversed-field
pinch configurations.
2.To
explore the influence of sheared flow on mode structure and on nonlinear
coupling in toroidal geometry.
Outline
I.Introduction
A.Background
B.Geometric
considerations
C.Modeling
II.Aspect
ratio scans
A.F-Q
B.Magnetic
fluctuations
C.Spectrum
width
III.Laminar
vs. single-helicity
A.The
role of viscosity
B.Varying
pinch parameter
IV.Initial
results with sheared flow
A.Enhanced
momentum transport
B.Magnetic
fluctuations
V.Conclusions
Background
·Most
numerical simulations and analytic computations for the RFP have been performed
in periodic linear geometry.
·This
is often a sound approximation:
·Since
q<1,
pressure gradients cannot stabilize tearing modes (assuming
p decreases
with r).[Glasser, Greene,
and Johnson, Phys. Fluids 18, 875 (1975).]
·Strong
nonlinear coupling among resonant fluctuations of different poloidal index
m
is a characteristic of standard RFP operation.[Ho
and Craddock, Phys. Fluids B 3, 721 (1991).]
·A
laminar version of the RFP dynamo, known as the "single-helicity state,"
exists at sufficient dissipation levels in periodic linear geometry.[Finn,
Nebel, and Bathke, Phys. Fluids B 4, 1262 (1992); Cappello and Escande,
PRL 85, 3838 (2000).]
·The
usual nonlinear coupling among different helicities is absent.
·Toroidal
geometry effects can make a qualitative difference in these conditions,
due to linear coupling of different m.
·The
stability of tearing modes may be influenced by flow shear.
·Toroidal
effects lead to a spectrum of m numbers in each mode.
·Differential
rotation of resonance surfaces affects the eigenvalue and spectral content
of tearing modes.
Geometric
Considerations
·Many
low-order helicities are resonant in a typical RFP q-profile.
·The
close spacing of the rational surfaces and the global nature of the dominant
tearing modes allow for strong nonlinear coupling in standard operation.
·A
well-established effect of toroidal geometry is that it leads to linear
coupling among different helicities with the same n-value.
·The
gradient operator contains
where
and
the 
terms
lead to the coupling.
·The
poloidal asymmetry of the equilibrium (mainly the Shafranov shift) also
leads to linear coupling.
·For
an RFP q-profile we can expect the strongest poloidal coupling to
occur between m=0 and m=1 helicities.
·The
unstable modes are predominantly 
helicity
resonant near r=0.
·The
corresponding (m=2,
n) helicity is not resonant for these
modes.
Modeling
To
investigate the electromagnetic activity, we solve the resistive MHD equations
in circular cross-section, toroidal and periodic linear geometriesusing
the NIMROD simulation code, http://nimrodteam.org.Specifying b ®0
for these studies:
·Density
is uniform, though flow is not incompressible.
·Resistivity
and viscosity are essentially uniform.
[
(mostly 2500 here) and
]
·Voltage
is dynamically adjusted to maintain the desired current.The
time-scale for the feedback is comparable to the tearing time to avoid
excitation of surface currents.
·Two-fluid
effects may be important.
·The
drift ordering is more realistic for RFPs than the MHD ordering even at
small b.
·Worth
further investigation.
·Numerical
parameters:
·Most
of the simulations reported here have 
.
·Some
of the simulations for laminar conditions have 
;
R/a=5
cases have 
.
·NIMROD
uses finite elements to represent the poloidal plane.Simulations
for the aspect ratio scan were run with a 48x48 or 64x64 (radial x azimuthal)
mesh of bilinear finite elements.
·When
parameters are scanned to achieve laminar states, a 16x24 or 16x32 mesh
of bicubic elements is used for a better representation of the magnetic
field.[See "Nonlinear Fusion Magneto-Hydrodynamics
with Finite Elements," Sherwood 2000, in http://nimrodteam.org/presentations.]
Aspect Ratio
Scans in Toroidal and Periodic Linear Geometries
·Results
on field reversal from dynamo action are similar in the two geometries,
even at very low aspect ratio.
a)
(b)
Comparison
of time-averaged reversal parameter (F) resulting from simulations
in (a) toroidal geometry and (b) periodic linear geometry at S=2500
and Pm=1.
·Magnetic
fluctuation levels are also comparable.
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geometry
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toroidal
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linear
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toroidal
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linear
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toroidal
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linear
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toroidal
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linear
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toroidal
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linear
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toroidal
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linear
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Results
are averaged over 1-2 tenths of a global diffusion time.
·Magnetic
energy spectra plotted vs. n and summed over m for the two
geometries are often nearly indistinguishable for standard multi-helicity
states.
a)
b)
Magnetic
fluctuation energy spectra for a) toroidal geometry and b) periodic linear
geometry showing the temporal average (red) and ± one standard deviation
(blue) for R/a=1.75,
Pm=1, Q=1.8.
·The
spreading of the magnetic spectrum with R/a reported by Ho, et
al. ["Effect of aspect ratio on magnetic field fluctuations in the
reversed-field pinch," Phys. Plasmas 2, 3407 (1995).], is also observed
in toroidal geometry.
·Each
Wn
is summed over m.
·Nonlinear
interaction seems to be more easily suppressed in toroidal geometry.
Simulation
results on 
for Q=1.6,
Pm=1
simulations.At
R/a=1.5, q(0)
is slightly greater than 1/3.
Laminar
RFP States
·As
viscosity is increased there is a transition to steady or near-steady states.
·Cappello
and Escande have established that this transition is more dependent on
the Hartmann number (
)
than the Lundquist number.
Transition
to laminar states in periodic linear geometry with R/a=4.[Cappello
and Escande, "Bifurcation in Viscoresistive MHD: The Hartmann Number and
the Reversed Field Pinch," PRL 85, 3838 (2000).]
·A
transition to laminar behavior also occurs in toroidal geometry as Pm
is increased.The following figure
shows the transition in the toroidal R/a=1.75, Q=1.8
case after Pm is
increased from 1 to 10.
·Plotting
energies vs. n (summed over m),
the spectrum shows suppression of nonlinear coupling.
·Poincaré
surfaces of section for B show that the final state is not single-helicity
in toroidal geometry.
a)
b)
Results
from a) toroidal geometry and b) periodic linear geometry with Pm=10,
R/a=1.75, Q=1.8.
·The
magnetic fluctuations in both configurations are dominated by the m/n=1/5
helicity, but the toroidal case has significant linear coupling among 
about
1/5.
for
the toroidal simulation.The contra-
and covariant components of B are defined in the straight field-line
coordinates wherej
is the toroidal symmetry angle.[diagnostic
developed with Tom Gianakon, LANL].
·Increasing
Pm to 100, the toroidal simulation loses reversal and a helical island
chain forms in the interior.
·In
simulations with Pm=10, Q=1.65,
and R/a=1.75, the final state is steady, but not single helicity,
even with linear geometry.
Linear Geometry
Energies vs. Time
Linear
Geom. SpectrumToroidal
Geom. Spectrum
·These
"quasi-single-helicity" states show small island structures [possibly by
having one sufficiently large perturbation Escande, et al.,
PRL 85, 3169 (2000)].
a)
b)
QSH
conditions in a) toroidal and b) periodic linear geometry.
·At
lower current (Q=1.4),
the pinch is not reversed, m=0 perturbations are not resonant, and
the toroidal magnetic topology is qualitatively similar to the linear geometry
result.
·The
linear geometry results are influenced by the changing relative importance
of the (1, -4) and (1, -5) modes, but single helicity occurs at both extremes
(with or without reversal).
Safety
factor (q) profiles for the linear geometry case as Q
is scanned from 1.4 (green line) to 1.8 (red line).
Like
the results obtained by varying viscosity, the Q-induced
formation of flux surfaces in toroidal geometry is correlated with the
loss of reversal, hence loss of resonance for m=0 fluctuations.
·At
larger R/a, surfaces of section for the two geometries are also
similar at F > 0.
a)
b)
Results
from a) toroidal geometry and b) periodic linear geometry with Pm=100,
R/a=3, Q=1.6.
Results
with Sheared Flow
·Having
established that toroidal geometry effects are strong enough to influence
the topology of laminar states, we would like to determine if they could
be used to improve confinement.
·Differential
rotation of rational surfaces alters the linear properties of the resonant
modes.If rotation reduces the growth
rate of MHD instability, it may result in lower fluctuation amplitudes
at saturation, if not complete stability.
·To
investigate this possibility numerically, a 
-directed
momentum source 
and
no-slip boundary conditions are used to induce an approximately parabolic
toroidal flow profile.
·
·The
flow profile develops consistently with the toroidally symmetric part of B.
·Our
RFP simulations are typically started from toroidally symmetric paramagnetic
pinch solutions.With 
,
the linear growth rates are essentially the same as the no-flow growth
rates.
·The
equilibrium is not reversed, so a dramatic change was not expected.
·These
growth rates may not reflect linear growth rates for equilibria with reversal.
Linear
m=1
growth-rate comparison for a Q=1.6,
Pm=10 toroidally symmetric paramagnetic pinch.The
momentum source has magnitude 
.
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 no
flow |
 with
flow |
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·The
temporal evolution of kinetic energy in the n > 0 Fourier components
differs little from the simulation without a momentum source.
Natural
logarithm of kinetic energy in each Fourier component as a function of
time (in resistive diffusion times) starting with small perturbations from
an unstable symmetric pinch.
·The
n=0
component shows a significant decrease in kinetic energy as the fluctuations
reach saturation.The presence of
finite-amplitude fluctuations increases the rate of momentum transport
to the wall.
Natural
logarithm of kinetic energy in the n=0 Fourier component as a function
of time starting from a paramagnetic pinch with momentum source.
·The
braking effect is also evident in the symmetric part of the toroidal flow
profiles.
Contours
of constant V_phi before saturation.
Natural
logarithm of kinetic energy in each Fourier component
For
t<0.4, 
For
0.4<t<0.64 
For
t>0.64 
·The
magnetic energy spectrum is only moderately affected by increasing the
momentum source.
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·The
flow distorts flux surfaces in the edge region; however, the magnetic topology
remains stochastic in the core.
Conclusions
1.In
typical multi-helicity RFP operation, toroidal geometry plays a minor role
in comparison to nonlinear coupling.
2.In
laminar conditions, toroidal geometry can make a qualitative difference.Conditions
producing nonstochastic magnetic field in periodic linear geometry may
have large regions of stochastic field in toroidal geometry.
3.All
toroidal simulations showing laminar conditions with good flux surfaces
are non-reversed states.The elimination
of
m=0 resonance seems to be essential.
4.Simulations
with an imposed source of toroidal momentum exhibit enhanced momentum transport
due to MHD fluctuations.A significant
reduction of magnetic fluctuations has not been observed so far.
Acknowledgement
The
authors wish to acknowledge the many important contributions of other members
of the NIMROD Team.
This
poster will be available through our web site, http://nimrodteam.org, shortly
after the meeting.