rfp equilibrium 3. the reversed field pinch magnetic equilibrium ornl colloquium – september 10th,...

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


safety factor profiles in tok and RFP


RFP B profile


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


€

∇× r

B = μ(r)r B + μ0

r B ×∇p

B2

€

μ(r) = μo

r J •

r B

B2




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




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



External Kink mode

Current driven kink


m=1 kink in tokamak



Kruskal Shafranov limit for tokamak


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


RFP stability diagram for m=1 modes


RFP linear stability

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


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


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


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


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)



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.


Feedback rotation control principle

Perfect control

Incomplete controlExternal field

Plasma field

Total field≠0

External field

Plasma fieldTotal field=0


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


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


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


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


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



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


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


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)


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 ∝=


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)


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)


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


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.

terranova

il salto che vedi al reversal nella corrente di parallela è aspettato (conferma di Hirshman) ed è legato in aprte alla discretizzazione in S che si usa.

Flux surfaces

The flux surfaces obtained both in axisymmetric and helical configurations provide a good benchmark with present experimental observations and other numerical reconstructions.

endend

rfp equilibrium 3. the reversed field pinch magnetic equilibrium ornl colloquium – september 10th,...

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