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Reversed Field Pinch:
equilibrium, stability and transport
Piero Martin
Consorzio RFX- Associazione Euratom-ENEA sulla fusione, Padova, Italy
Department of Physics, University of Padova
Notes for the lecture at the European Ph.D. Course (Garching, 29 September 2008)
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Note for users
These slides are intended only as tools to accompany the
lecture. They are not supposed to be complete, since the
material presented on the blackboard is a fundamental part of
the lecture.
Relevant bibliography:
Freidberg, IDEAL MHD
Ortolani, IV Latin American Workshop on Plasma Physics
Escande, Martin et al, PRL 2000
and the references therein quoted
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Outline of the lecture
1) MHD equilibrium basics
2) 1d examples
1) Q-pinch
2) Z-pinch
3) Screw pinch
3) RFP equilibrium basics
4) RFP Stability
5) RFP dynamics and the dynamo.
6) Effects on transport
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A reversed field pinch exists: RFX-mod
The largest RFP in the world, located in Padova, Italy
A fusion facility for MHD mode control
a=0.459 m, R=2 m, plasma current up to 2 MA
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MHD equilibrium basics
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The MHD equilibrium problem
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Time-indpendent form of the full MHD equations with v=0
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Linear vs. toroidal configurations
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Magnetic flux surfaces
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Current, magnetic and pressure surfaces
The angle between J and B is in general arbitrary
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Rational, ergodic and stochastic
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Surface quantities
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One-dimensional configurations
Even if the magnetic configurations of fusion interest are toroidal, some physical
intuition can be obtained by investigating their one-dimensional, cylindrically simmetric
versions.
This separates:
– Radial pressure balance
– Toroidal force balance
For most configurations, once radial pressure balance is established, toroidicity can be
introduced by means of an aspect ratio expansion, from which one can then investigate
toroidal force balance.
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 pinch
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A simple example: -pinch
Configuration with pure toroidal field
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A simple example: -pinch
The sum of magnetic and kinetic pressure is constant throughout the plasma
The plasma is confined by the pressure of the applied magnetic field
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Experimental -pinch
Experimental -pinch devices among the first experiments to be realized
End-losses severe problem
A -pinch is neutrally stable, and can not be bent into a toroidal equilbrium
Additional field must be added to provide equilibrium
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Z-pinch
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Z-pinch
Purely poloidal field
All quantities are only functions of r
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Z-pinch
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In contrast to the -pinch,
for (LZW)
a Z-pinch
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gradient that provides radial confinement of the plasma
The Bennet pinch satisfies the Z-pinch equilibrium
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Bennet Z-pinch
Tension force acts inwards, providing radial pressure balance.
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Experimental Z-pinch
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Z-machine
The Z machine fires a very powerful electrical discharge (several tens
million-ampere for less than 100 nanoseconds) into an array of thin,
parallel tungsten wires called a liner.
Originally designed to supply 50 terawatts of power in one fast pulse,
technological advances resulted in an increased output of 290 terawatts
Z releases 80 times the world's electrical power output for about seventy
nanoseconds; however, only a moderate amount of energy is consumed
in each test (roughly twelve megajoules) - the efficiency from wall
current to X-ray output is about 15%
At the end of 2005, the Z machine produced plasmas with announced
temperatures in excess of 2 billion kelvin (2 GK, 2×109 K), even reaching a peak
at 3.7 billion K.
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The general screw pinch
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General Screw Pinch
Though the momentum equation is non-linear, the Q-pinch and Z-pinch forces ad as
alinear superposition, a consequence of the high degree of symmetry
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RFP equilibrium
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Tokamak and RFP profiles
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safety factor profiles in tok and RFP
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RFP B profile
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TOK to RFP q profile transition
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The reversed field pinch
Pinch configuration, with low magnetic field
The toroidal field is 10 times smaller than in a tokamak with similar current
Reactor issues: normal magnets, low force at the coils, high mass power density,
no additional heating
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Kruskal Shafranov limit for tokamak
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The reversed field pinch
Pinch configuration, with low
magnetic field
Bp and Bt have comparable
amplitude and Bt reverses
direction at the edge
Modes in RFP :
• low m (0-2)
Safety factor
• high n (2*R/a)
Bt (0)   Bt   B p (a)  Bt (a)
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The reversed field pinch
Pinch configuration, with low
magnetic field
Bp and Bt have comparable
amplitude and Bt reverses direction
at the edge
Most of the RFP magnetic field is
generated by current flowing in the
plasma
Magnetic self-organization
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..something on stability
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External Kink mode
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RFP stability diagram for m=1 modes
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RFP linear stability
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Modern technique: real time control of stability with
feedback coils
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Multi-mode control is a requirements for the RFP
q (r)
m=1, n =-5
Resistive Wall Modes
m=1, n =-6
m=1, n=-7
m=1, n=-8
m=1, n=-9
Tearing Modes
m=0, all n
Resistive Wall Modes
m=1, n > 0
r (m)
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RFX-mod: 192 active saddle coils, covering the whole
plasma surface
Each is independently driven (60 turns)
and produces br from 50 mT (DC) to 3.5 mT (100 Hz)
Power supply: 650 V x 400 A
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Feedback Control System Architecture on RFX-mod
Sensors: br, b, Icoil
b
192 power
amplifiers
outputs
192 I ref
ext
plasma
To control br (a)
50 ms
thin shell
Digital Controller
Each coil is independently controlled
Cycle frequency =2.5 kHz
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192 br
192b
inputs 192 Icoil
 576 signals
RFX-mod: 192 active saddle coils, covering the whole
plasma surface
Each is independently driven (60 turns)
and produces br from 50 mT (DC) to 3.5 mT (100 Hz)
Power supply: 650 V x 400 A
European Ph.D. course . - Garching 29.09.08)
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Feedback Control System Architecture on RFX-mod
Sensors: br, b, Icoil
b
192 power
amplifiers
outputs
192 I ref
ext
plasma
To control br (a)
50 ms
thin shell
Digital Controller
Each coil is independently controlled
Cycle frequency =2.5 kHz
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192 br
192b
inputs 192 Icoil
 576 signals
MHD stability feedback contro in RFX-modl
Full stabilization of multiple resistive wall modes in presence of a thin
shell (and RWM physics/code benchmarking)
Control and tailoring of core resonant tearing modes – mitigation of
mode-locking and smoother magnetic boundary
Test of new algorithms and models for feedback control
Design of mode controllers
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RFX-mod contribution to RWM physics and control
plasma current
Experiments can be designed to measure
very precisely growth rate dependencies
mode control
m=1,n=-6 mode amplitude
logarithmic mode amplitude
t [s]
mode control
o
Sophisticated algorithms are
developed to control single and
multiple RWM growth
o
Error Field Amplification
mode control
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Effect of the active control
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The reversed field pinch
Pinch configuration, with low
magnetic field
Bp and Bt have comparable
amplitude and Bt reverses
direction at the edge
Modes in RFP :
• low m (0-2)
Safety factor
• high n (2*R/a)
Bt (0)   Bt   B p (a)  Bt (a)
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RFP dynamics
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The reversed field pinch
Pinch configuration, with low
magnetic field
Bp and Bt have comparable
amplitude and Bt reverses direction
at the edge
Most of the RFP magnetic field is
generated by current flowing in the
plasma
Magnetic self-organization
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Non-linearity is built-in in RFP physics: an example

  1 2
B
   V  B    B  0
t
 
  

B  Bo  B V  Vo  V
0
 
V B
►
map
J.M. Reynolds and C.R. Sovinec
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Electric field in the RFP
The RFP is an ohmically driven system: an inductive toroidal electric field, produced by
transformer effect, continuously feeds energy into the plasma
Ohm’s law mismatch: the electrical currents flowing in a RFP can not be directly driven by
the inductive electric field Eo


Ei  J
overdriven
..but stationary ohmic RFP are
routinely produced for times
longer than the resistive diffusion
time
underdriven
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The RFP dynamo electric field
An additional electric field, besides that externally applied, is necessary to sustain
and amplify the toroidal magnetic flux.
A Lorentz contribution v x B is necessary, which implies the existence of a
self-organized velocity field in the plasma.
Edynamo
  
E  Ei  Edynamo

 ~
Edynamo  v  b
Edynamo
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The old paradigm: Multiple Helicity (MH) RFP
the safety factor q << 1 and the central peaking of the current density combine to destabilize MHD resistive
instabilities.
For a long time a broad spectrum of MHD resistive instabilities ( m=0 and m=1, variable n ( “multiple
helicity” –MH – spectrum), was considered a high, but necessary, price to pay for the sustainment of the
configuration through the “dynamo” mechanism.
br spectrum

 
Edyn  v  b
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Turbulent dynamo: remarkable self-organization
An experimental and numerical database supports the MHD turbulent dynamo theory:
-2

~
~
Ed  v  b
log b21n
-4
the dynamo electric field is
produced by the coherent
interaction of a large number
of MHD modes:
-6
-8
1000
3000
5000
t/A
7000
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Multiple Helicity (MH)
dynamo
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A completely new view eliminates the old paradigm
For a long time….
….a broad spectrum of MHD resistive instabilities,
causing magnetic stochasticity, was considered a high,
but necessary, price to pay for the sustainment of the
configuration through the
“MULTIPLE HELICITY dynamo” mechanism ….
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A completely new view
A helical ohmic equilibrium is possible, with a single helicity dynamo, where all the work is done by a
single resistive mode (m=1, n=7 - opposite ordering wrt tokamak).
Experiments are coming ever closer to the theoretically predicted chaos-free helical ohmic equilibrium
This allows to retain the good features of self-organization without the past
degradation of confinement.
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A new approach to RFP dynamo: the Single
Helicity
Single Helicity (SH): the dynamo is driven by a single m =1 MHD resistive mode and
its harmonics:
Helical symmetry of the magnetic equilibrium
Escande et al., PRL. 85 2000, Bonfiglio et al. PRL 2005
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A new approach to RFP dynamo: the Single
Helicity
Single Helicity (SH): the dynamo is driven by a single m =1 MHD resistive mode and its
harmonics:
– Helical symmetry of the magnetic equilibrium
– Strongly reduced magnetic chaos in comparison to the standard multiple helicity (MH) RFP
m=1 mode spectrum
m=1 mode spectrum
QSH
MH
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A new approach to RFP dynamo: the Single Helicity
Single Helicity (SH): the dynamo is driven by a single
m =1 MHD resistive mode and its harmonics:
– Helical symmetry of the magnetic equilibrium
– Strongly reduced magnetic chaos in comparison to the
standard multiple helicity (MH) RFP
– It is expected to have a very strongly improved
confinement
Two orders of magnitude improvement in numerical loss time of a population of
test particles with respect to MH case (Predebon, White et al., PRL 2004)
The ohmic helical state retains all the good features of the RFP without the
problems connected with the high level of magnetic turbulence typical of the
MH scenario
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Resistive kink mode and dynamo: basic action
Plasma is approximated as a current carrying
wire placed on the axis of a cylindrical flux
conserver where some axial magnetic field Bz
is present due to the azimuthal current Ishell
(flowing in the flux container).
Ishell
Bz
The wire is in an unstable equilibrium, and a
small perturbation leads it to kink
I
Bq
Bz
Escande et al., PPCF 42, B243, 2000
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Resistive kink mode and dynamo: basic action
Ishell
Iq
B’
B
B
B’
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1.
The azimuthal projection of the
kinked current Iq has the same
direction as Ishell: growth of
instability.
2.
Solenoidal effect: B inside the
kinked wire increase
3.
Flux conservation: B’ outside
decreases
4.
Continuos growth force Ishell and
B’ to reverse. Saturation
5.
Final state: B’ in the outer
region is reversed!
Single and Quasi Single Helicity (QSH) in the
experiment
The Single Helicity state is theoretically predicted and partially understood, but physics
in the modeling is still not completed (Bonfiglio et al.PRL 2005)
• coupling with transport still missing..
• Dissipation coefficients (viscosity…) still unknown
• Toroidal effects… (coupling of m=1 modes and production of m=0)
In the experiments we observe Quasi Single Helicity (QSH) states
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Properties of experimental QSH states
The n-spectrum of MHD modes is dominated by a single m=1 geometrical helicity
Relative amplitudes of m=1 modes
QSH
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MH
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Properties of experimental QSH states
The k-spectrum of MHD modes is dominated by a single m=1 geometrical helicity
QSH
Dominant mode
MH
Secondary modes
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Dynamo electric field is produced in QSH by the dominant mode
We are observing the right mechanism!
Dynamo electric field toroidal spectrum
Piovesan et al. PRL 2005
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Helical closed flux surfaces in the QSH plasma core
Te (eV)
SXR
The “secondary” modes have amplitudes still too high for a global improvement of the plasma
performance and there is magnetic chaos outside the helical domain:
• Toroidal coupling
• m=0 modes
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Lundquist number scaling is promising
At higher current, when plasma gets hotter, the helical state is more pure
S = R / A =
Secondary modes (1,-8 to -15)
b/B (%)
b/B (%)
Dominant mode (m = 1, n = -7)
bdom
5%
bsecd = 0.2% = 25
S
S
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Topology change at high current:
from island to Single Helical Axis
Single Helicity states
experimentally discovered in
1998 (ppcf 98, prl 2000)
Exciting physics result
(theoretically predicted), but
relatively small volume of plasma
involved
X point
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Topology change at high current:
from island to Single Helical Axis
Magnetic axis
X point
bdom /bsec increases
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Topology change at high current:
from island to Single Helical Axis
Extended transport barrier
New Axis
New helical topology
where the orginal
axisymmetric axis is
replaced by a helical
magnetic axis
X point
(Escande et al PRL 2000)
bdom /bsec increases
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From island to Single Helical Axis
QSH
Island
SHAx
(Single Helical Axis)
New helical topology
where the orginal
axisymmetric axis is
replaced by a helical
magnetic axis
X point
bdom /bsec increases
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(Escande et al PRL 2000)
Lorenzini PRL 2008
Experimental confirmation of a helical equilibrium
With appropriate reconstruction of the dominant mode eigenfunction, we can build a helical flux
(r,u) = m(r,u) - nF(r,u)
considering the axisymmetric equilibrium and the dominant mode. (r and u = m-n are flux coordinates).
Lorenzini, Martines,
Terranova et al, 2008
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