European Ph.D. course. - Garching 29.09.08)p.martin Piero Martin Consorzio RFX- Associazione Euratom-ENEA sulla fusione, Padova, Italy Department of Physics,

European Ph.D. course . - Garching 29.09.08) p.martin

Piero Martin

Consorzio RFX- Associazione Euratom-ENEA sulla fusione, Padova, ItalyDepartment of Physics, University of Padova

Notes for the lecture at the European Ph.D. Course (Garching, 29 September 2008)

Reversed Field Pinch:

equilibrium, stability and transport


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


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


A reversed field pinch exists: RFX-modA reversed field pinch exists: RFX-mod

a=0.459 m, R=2 m, plasma current up to 2 MA

The largest RFP in the world, located in Padova, Italy

A fusion facility for MHD mode control


MHD equilibrium basics


The MHD equilibrium problem

Time-indpendent form of the full MHD equations with v=0

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.


Linear vs. toroidal configurations




Magnetic flux surfaces






Current, magnetic and pressure surfaces

The angle between J and B is in general arbitrary




Rational, ergodic and stochastic




Surface quantities






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.


pinch


A simple example: -pinch

Configuration with pure toroidal field








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




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







Z-pinch


Z-pinch

Purely poloidal field

All quantities are only functions of r


are needed to see this picture.QuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.


Z-pinch

In contrast to the -pinch, for a Z-pinch it is the tension force and not the magnetic pressure gradient that provides radial confinement of the plasma



The Bennet pinch satisfies the Z-pinch equilibrium


are needed to see this picture.QuickTime™ and a

TIFF (LZW) decompressorare needed to see this picture.


Bennet Z-pinch

Tension force acts inwards, providing radial pressure balance.






Experimental Z-pinch






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. QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.




QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.


The general screw pinch


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




RFP equilibrium


Tokamak and RFP profiles




safety factor profiles in tok and RFP




RFP B profile







TOK to RFP q profile transition




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


Kruskal Shafranov limit for tokamak






Bp and Bt have comparable amplitude and Bt reverses direction at the edge

)()()0( aBaBBB tptt >>≈⟩⟨>

Modes in RFP :

• low m (0-2)

• high n (2*R/a)Safety factor





Most of the RFP magnetic field is generated by current flowing in the plasma

Magnetic self-organization Magnetic self-organization


..something on stability











External Kink mode




RFP stability diagram for m=1 modes




RFP linear stability
















Modern technique: real time control of stability with feedback coils


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)

Multi-mode control is a requirements for the RFP


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


Feedback Control System Architecture on RFX-mod

192 poweramplifiers

Sensors: br, b, Icoil

plasma

Digital ControllerEach coil is independently controlled

Cycle frequency =2.5 kHzsignals

Icoil

b

br

576

192

192

192

=

ϕ

extb

refI192inputsoutputs

To control br (a) 50 ms thin shell


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


Feedback Control System Architecture on RFX-mod

192 poweramplifiers

Sensors: br, b, Icoil

plasma

Digital ControllerEach coil is independently controlled

Cycle frequency =2.5 kHzsignals

Icoil

b

br

576

192

192

192

=

ϕ

extb

refI192inputsoutputs

To control br (a) 50 ms thin shell


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


RFX-mod contribution to RWM physics and control

plasma current

m=1,n=-6 mode amplitude

t [s]

logarithmic mode amplitude

mode control

mode controlmode control

Experiments can be designed to measure very precisely growth rate dependencies

o Sophisticated algorithms are developed to control single and multiple RWM growth

o Error Field Amplification


Effect of the active control





)()()0( aBaBBB tptt >>≈⟩⟨>

Modes in RFP :

• low m (0-2)

• high n (2*R/a)Safety factor


RFP dynamics





Most of the RFP magnetic field is generated by current flowing in the plasma

Magnetic self-organization Magnetic self-organization


Non-linearity is built-in in RFP physics: an example

VVVBBB

BBVt

B

oorrrrrr

rrrr

δδ

σμ

+=+=

=∇+××∇=∂∂ 021

0

► J.M. Reynolds and C.R. SovinecJ.M. Reynolds and C.R. Sovinec

map BVrr

×


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

JEirr

η≠


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

bvEdynrrr

×=


Turbulent dynamo: remarkable self-organization

-8

-6

-4

-2

1000 3000 5000 7000

log b2

1n

t/τA

An experimental and numerical database supports the MHD turbulent dynamo theory:

the dynamo electric field is the dynamo electric field is produced by the coherent produced by the coherent interaction of a large interaction of a large number of MHD modes: number of MHD modes:

Multiple Helicity (MH) Multiple Helicity (MH) dynamodynamo

bvEd~~×=

r


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 ….


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 modesingle 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.


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:

Escande et al., PRL. 85 2000, Bonfiglio et al. PRL 2005

Helical symmetry of the magnetic equilibriumHelical symmetry of the magnetic equilibrium




– 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

MH QSH




– 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


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).

The wire is in an unstable equilibrium, and a small perturbation leads it to kink

Escande et al., PPCF 42, B243, 2000

II

BB

zz

BB

IIshellshell

BBzz


Resistive kink mode and dynamo: basic action

1. The azimuthal projection of the kinked current I 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!

BB

BB’’

IIshellshellII

BB

B’B’


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

Single and Quasi Single Helicity (QSH) in the experiment


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 MH


Properties of experimental QSH states

The k-spectrum of MHD modes is dominated by a single m=1 geometrical helicity

Dominant modeSecondary modes

QSH MH


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


Helical closed flux surfaces in the QSH plasma core

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

TTee (eV) (eV)SXRSXR


Lundquist number scaling is promising

S = τR / τA =

Dominant mode (m = 1, n = -7) Secondary modes (1,-8 to -15)

bdom

bsecd

5%0.2%

= = 25

b/B

(%)

b/B

(%)

S S

At higher current, when plasma gets hotter, the helical state is more pure At higher current, when plasma gets hotter, the helical state is more pure


X point

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 bdom /bsec increases

Magnetic axis


X point

bdom /bsec increases

New helical topology where the orginal axisymmetric axis is replaced by a helical magnetic axis


Extended transport barrier

(Escande et al PRL 2000)

New Axis


X point

bdom /bsec increases

New helical topology where the orginal

axisymmetric axis is replaced by a helical

magnetic axis

(Single Helical Axis)

From island to Single Helical Axis

QSHIsland SHAx

(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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European Ph.D. course. - Garching 29.09.08)p.martin Piero Martin Consorzio RFX- Associazione Euratom-ENEA sulla fusione, Padova, Italy Department of Physics,