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RFP equilibrium
33
The reversed field pinch magnetic equilibrium
ORNL Colloquium – September 10th, 2009
RFX-RFP configuration
RFX coils
induction of plasma current
RFP configuration
toroidal magnetic field poloidal magnetic field
mean magnetic field radial
profiles
European Ph.D. course . - Garching 29.09.08) p.martin
Tokamak and RFP profiles
European Ph.D. course . - Garching 29.09.08) p.martin
safety factor profiles in tok and RFP
European Ph.D. course . - Garching 29.09.08) p.martin
RFP B profile
European Ph.D. course . - Garching 29.09.08) p.martin
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
European Ph.D. course . - Garching 29.09.08) p.martin
€
∇× r
B = μ(r)r B + μ0
r B ×∇p
B2
€
μ(r) = μo
r J •
r B
B2
European Ph.D. course . - Garching 29.09.08) p.martin
The reversed field pinch
Pinch configuration, with low magnetic field
Bp and Bt have comparable amplitude and Bt reverses direction at the edge
)()()0( aBaBBB tptt >>≈⟩⟨>
Resonances in RFP :
• low m (0-2)
• high n (2*R/a)Safety factor
European Ph.D. course . - Garching 29.09.08) p.martin
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 (dynamo) Magnetic self-organization (dynamo)
JBVE η=∧+
zr BVJE += ϑϑ η
EB
∧−∇=∂∂
t
0)( =rEϑ→=∂∂
0t
!!0)( =rJ ϑ
RFP dynamo 1
Bz B
Jz J
0)( ≠rJ ϑ
What we mean with “RFP dynamo effect ” 1/2
Ohm’s law
Induction equation
stationariety
!! inconsistency
at reversal
RFP dynamo 2
What we mean with “RFP dynamo effect ” : 2/2
to resolve the previous inconsistency we need an “additional” mean electric field with respect to the one provided by mean B and mean v fields, i.e. -within resistive MHD- the contribution by coherent modulation of B and v:
ϑϑϑ η >∧<−>><<+>>=<< BV~~
zr BVJE
Edynamo = < v /\ B >
Edynamo Edynamo allows us to balance Ohm’s lawjustifying that in stationary conditions:• less mean Jz is driven in the core • more mean J is driven in the edge then expected by externally applied E.
In other words:
About stability and its implication for RFP
44
The basic destabilizing forces arise from:
– Current density
– Pressure gradients, combined with adverse magnetic field curvature
The resulting instabilities are divided in two categories
– Ideal modes, i.e. instabilities which would occurr even if the plasma were perfectly conducting
– Resistive modes, which are dependent on the finite resistivity of the plasma
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
External Kink mode
European Ph.D. course . - Garching 29.09.08) p.martin
External Kink mode
Current driven kink
European Ph.D. course . - Garching 29.09.08) p.martin
m=1 kink in tokamak
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
Kruskal Shafranov limit for tokamak
European Ph.D. course . - Garching 29.09.08) p.martin
q (r)
Resistive Wall Modes
m=1, n=-7m=1, n=-8
m=1, n=-9
Resistive Wall Modesm=1, n > 0
m=1, n =-5
m=1, n =-6
m=0, all n
Tearing ModesTearing Modes
r (m)
MHD modes in RFP
European Ph.D. course . - Garching 29.09.08) p.martin
RFP stability diagram for m=1 modes
European Ph.D. course . - Garching 29.09.08) p.martin
RFP linear stability
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
European Ph.D. course . - Garching 29.09.08) p.martin
MHD stability: its implication on RFP self-organization and its
active control
55
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
..but stationary ohmic RFP are routinely produced for times longer than the resistive diffusion time
overdrivenoverdriven
underdrivenunderdriven
JEi
rrη≠
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 plasmaself-organized velocity field in the plasma.
EdynamoEdynamo bvE
EEE
dynamo
dynamoi
~×=
+=rr
rrr
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
bvEdyn
rrr×=
Wide k-spectrum bulging in the physical space
• …
A wide spectrum of m=0 and m=1 modes can produce severe plasma-wall severe plasma-wall interaction if the modes lock in phase and to the wall !interaction if the modes lock in phase and to the wall !
At the leading edge of active stability control
192 coils arranged in 48 toroidal positions cover the whole plasma surface
Each is independently driven (60 turns, 650 V x 400 A)
Digital controller elaborates real-time 576 inputs
RFX-mod has the best feedback system for real time control of MHD stability ever realized for a fusion device
Full stabilization of multiple RWMs and control of individual tearing modes achieved in RFX-mod and EXTRAP T2-RDemonstrates that a thick stabilizing shell is NOT needed
Strong integration between physics and control engineering key for success
ORNL Colloquium – September 10th, 2009
Feedback Control System Architecture
192 poweramplifiers
Sensors: br, b, Icoil
plasma
Each coil independently controlledDigital Controller: 7 computing nodes2 Gflop/s computing powerCycle frequency = 2.5 kHz
cycle latency (≤ 400 μs).
OUTPUTS:
192 Iref
50 ms thin shell
576 INPUTS: 192br, 192b, 192Icoil
bEXT
ORNL Colloquium – September 10th, 2009
MHD stability feedback control
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
Test of new algorithms and models for feedback control
Design of mode controllers
RFX PERFORMANCE IMPROVEMENT
CONTRIBUTION TO THE GENERAL ISSUE OF MHD STABILITY ACTIVE CONTROL
EXPERIMENTAL PROPOSALS FOR 2009 FROM IPP (AUG), DIII-D, JT60-SA
ORNL Colloquium – September 10th, 2009
Steady progress in performance in a reliable device
Fully reliable MHD stability control system
no MHD active contro 2004
with MHD active control: 2006
upgraded MHD active control: 2008
spring 2009 - unoptimized
ORNL Colloquium – September 10th, 2009
Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009
The value of flexibility: high perfomance RFP,…but not only RFP
Exploration of high current RFP allows for the discovery of new physics, with structural changes
TOKAMAK
..but RFX can be run as a 150 kA Tokamak
A test bed forMHD feedback control
Full control of a (2,1) mode in a ramped tokamak
Follows an idea realized in DIII-D on a proposal by In, Okabayashi, et al (with RFX participation)
Okabayashi et al., paper EX/P9-5 2008 IAEA FEC, Geneva
RED: feedback OFF
BLACK: feedback ON
(Cavazzana, Marrelli, et al. 2009)
ORNL Colloquium – September 10th, 2009
Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009
RWM active rotation experiment: setup
2 control time windows:– FIRST: the mode is not controlled
– SECOND: the mode is initially feedback controlled with a pure real proportional gain. Gain scan performed (to obtain constant RWM amplitude)
The external field is always opposing the plasma error field with the same helicity and no net force is present to induce a controlled rotation.
Byproduct: simulation of feedback control systems with not enough power to cope with the growth of the selected instability.
Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009
Feedback rotation control principle
Perfect control
Incomplete controlExternal field
Plasma field
Total field≠0
External field
Plasma fieldTotal field=0
Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009
Complex gains (k+ i) can be used
Perfect control
Incomplete controlExternal field
Plasma field
Incomplete controlwith phase shift
Total field≠0
External field
Plasma fieldTotal field=0
External field
Plasma fieldTotal field≠0
Princeton Plasma Physics Laboratory Colloquium - June 4th, 20092008 IAEA Fusion Energy Conference, Geneva - P. Martin
Advanced RWM control and mode un-locking
Active rotation of non-resonant wall-locked RWM is induced by applying complex gains (keeping the mode at the desired constant amplitude)
RWM amplitude
RWM phase
Bolzonella, Igochine et al, PRL 08
Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009
The old storyFor a long time it was considered that….
– ….a q < 1 configuration like the RFP would have been intrinsically unstable,
– with a broad spectrum of MHD resistive instabilities,
– causing magnetic chaos and driving anomalous transport.
This was viewed as an interesting scientific case but a show-stopper for the RFP reactor ambitions
Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009
An emerging view for the RFPFor a long time it was considered that….
– ….a q < 1 configuration like the RFP would have been intrinsically unstable,
– with a broad spectrum of MHD resistive instabilities,
– causing magnetic chaos and driving anomalous transport.
This was viewed as an interesting scientific case but a show-stopper for the RFP reactor ambitions
Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009
Two strategies for chaos-free RFP: 1
Control of the current profile to stabilize tearing modes
Proof of principle experiment in MST to test RFP confinement and beta limits at the limit of negligible magnetic fluctuation (record values E and ) (most recent results in Chapman et al, IAE FEC paper EX/7-1Ra, to appear in NF 2009)
Toroidal mode number (~2R/a)
ampl
itude
The problem The solutionThe problem m=1 and m=0 modes
Toroidal mode number (~2R/a)
Dynamo modes active reduction
Pulsed Poloidal Current Drive (PPCD):
– the induction of a poloidal current at the plasma edge causes a dramatic reduction of the magnetic turbulence and STRONG PLASMA HEATING
plasmaVJ
It is TRANSIENT, but in RFX a quasi-stationary version has been implementedIt is TRANSIENT, but in RFX a quasi-stationary version has been implemented
Princeton Plasma Physics Laboratory Colloquium - June 4th, 2009
Density increased, fluctuations still reduced in MST (=17%)
Current drive
m = 0n = 1-4
Ip = 0.5 MA
x
x
xx
Tea
ring
ampl
itude
s (G
)<
n e>
(m
-3)
Chapman et al., IAEA 2008, tbp in Nucl. Fus.
Princeton Plasma Physics Laboratory Colloquium - June 4th, 20092008 IAEA Fusion Energy Conference, Geneva - P. Martin
Two strategies for chaos-free RFP: 2
Toroidal mode number (n≈2R/a)
ampl
itude
The problem m=1 and m=0 modes
Toroidal mode number
n=7: the solution
Self-organized helical state: at high current the plasma spontaneously chooses a helical equilibrium where only one saturated mode is present, and sustains the configuration
This is potentially chaos-free and allows to retain the good features of self-organization without the past degradation of confinement.
For Ip > 1 MA this is the preferred state in RFX-mod, with strong electron transport barriers and improved confinement
Long periods with one large saturated m =1 mode
plasma current
density
Electron temperature
BLACK=DOMINANT MODE / color=secondary modes
SECONDARY MODES
DOMINANTMODE
ORNL Colloquium – September 10th, 2009
European Ph.D. course . - Garching 29.09.08) p.martin
Dynamo electric field is produced in QSH by the dominant mode
We are observing the right mechanism!
Piovesan et al. PRL 2005
Dynamo electric field toroidal spectrum
The 1st bifurcation: from MH to QSH
MH QSH
Quasi Single Helicity (QSH) states, where the mode n = -7 dominates, and the secondary mode amplitudes are reduced, are observed at medium current (0.5 MA < Ip < 1 MA)Escande et al, PRL 2000Cappello et al., PPCF 46 B313 (2004)
A typical feature is the appearance of a thin helical, thermal structure off-axis
ORNL Colloquium – September 10th, 2009
QSHi
2nd bifurcation at high current: from QSHi to SHAx
SHAx
single O-point
1st O-point
X-point 2nd O-point
The original axisymmetric axis is replaced by a helical axis as I > 1 MA
ORNL Colloquium – September 10th, 2009
The drive is mode amplitude: experiment & theory
Since the energy of secondary modes is particularly low in SHAx states, it results in a threshold on the ratio dominant/secondary
SHAx states appear when the amplitude of the dominant mode exceeds a threshold
R. Lorenzini et al., PRL 101, 025005 (2008)
~ 4% of the total B(a)
ORNL Colloquium – September 10th, 2009
D. F. Escande et al., PRL. 85, 3169 (2000)
Synergistic dependence on Lundquist number S
Dominant mode (m = 1, n = -7) Secondary modes (1,-8 to -15)
b/B
(%)
b/B
(%)
S S
Strongly leading towards chaos-free plasmas
At higher current, when plasma gets hotter, the helical state is more pure
2/1
2/3)0(
eeff
ep
A
R
nZ
TIS ∝=
ORNL Colloquium – September 10th, 2009
RFP helical states
66
X point and separatrix
Topology change at high current: from island to Single Helical Axis
• island-like structure
• predicted physics result
• strong T gradients
…but relatively small volume of plasma involved
In 2006:
Quasi Single Helicity states where reported:
• both the helical axis and the original axisymmetric axis were present
Te (keV)Te (keV)
ORNL Colloquium – September 10th, 2009
Single Helical Axis (SHAx) equilibrium at high current
The original axisymmetric axis is replaced by a helical magnetic axis thanks to the favourable S-scaling of the modes
Strong electron transport barriers
1/LTe ~ 20 m-1 e ~ 10-20 m2s-1
Temperature and density are constant on helical magnetic flux surfaces
Z (m
)Z
(m)
Te (keV)Te (keV)
Te (keV)Te (keV)
ORNL Colloquium – September 10th, 2009
Temperature and density are constant on helical magnetic flux surfaces
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 = mϑ-n are flux coordinates).
The assumption of good isobaric helical flux surfaces allows mapping of temperature profiles
ORNL Colloquium – September 10th, 2009
RFP axi-symmetric equilibrium
INPUT PARAMETERS:1/(s) = 0circular LCFS (fixed boundary)
Total magnetic field
Parallel current
VMEC adapted for RFP equilibria requires the use of the POLOIDAL FLUX to deal correctly with B reversal.
Ongoing work to use VMEC for RFP helical states collaboration with ORNL (S. Hirschmann) and PPPL (Boozer & Pomphreys)
RFP Helical equilibrium
Parallel current Total magnetic field
INPUT PARAMETERS:1/(s) = 0circular LCFS (fixed boundary)
Input 1/ profile is obtained by means of the field line tracing code ORBIT.
Flux surfaces
The flux surfaces obtained both in axisymmetric and helical configurations provide a good benchmark with present experimental observations and other numerical reconstructions.
European Ph.D. course . - Garching 29.09.08) p.martin
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