The efficient sustainment of a stable, high-β spheromak: modeling

ByTom Jarboe,

ToPSI-Center

July 29, 2015

Outline

• Physics basis of the effect of perturbations• The self-organized effect of flattening the j/B profile• Impact on large aspect ratio tokamaks • HIT-SI results• Summary

IDCD is a new relaxation mechanism for making j/B uniform in a stable symmetric equilibrium

This mechanism provides an explanation for:*– The level of field error (δB/B = 10-4) that spoils tokamak

performance.– The rate of poloidal flux loss in Argon induced disruptions in

DIII-D.– The change in plasma rotation direction with LHCD in the

edge on C-mod. – The toroidal current versus time, the injector impedance, and

the current profile of HIT-SI.

*T. R. Jarboe, B. A. Nelson, and D. A. Sutherland, “A mechanism for the dynamo terms to sustain closed-flux current, including helicity balance, by driving current which crosses the magnetic field”, to be published July 15, 2015 in Physics of Plasmas

4

An explanation of effect of perturbations on current starts with symmetric current

• Electron fluid is frozen to magnetic fields. (Two-fluid MHD)

• Current flow is also magnetic field flow.• For a stable equilibrium the magnetic

structure is rigid. It resists deformation. • Toroidal current is from electrons

frozen in nested solid shells that are rotating at different speeds.

• Symmetric flux surfaces allow free differential current flow.(Free means unobstructed by magnetic interference.)

Toroidal current in a torus with a hollow current profile

Sheared electron flow plus perturbations give current drive across closed flux surfaces

• Now add a magnetic perturbation (red) and differential flow is no longer free.

• If the perturbation is large enough the flow locks across flux surfaces (inner flux surfaces).

• If the perturbations are small enough the differential flow can symmetrize the perturbation and differential flow continues.

– A viscosity-like drag force will drive the current inside the symmetrize closed flux surface.

• Both effects are current self-organizing across closed flux surfaces towards uniform jtor/Btor.

• HIT-SI data indicate that the viscosity-like force per unit area needed to symmetrize is:

2( )

2 o

B

• Externally driving the edge results in Imposed Dynamo Current Drive (IDCD).*

*T. R. Jarboe et al., Nucl. Fusion, 52, 083017 (2012)

Viscous force on the flux surface area = Force require to drive the current throughout that flux surface volume.

22

2

( )2 2 ( ) 2

2

( )( )

rmso tor tor o

o

rmstor tor

o

BR r ne j E r R

Bj E rne

1/22

1/2

( )

4o i

tor o

l IB ner For disruptions

t

B j ner For steady state

For disruptions E j

For steady state E j

2( )( )

2 tor toro

BdA ne j E dvol

( ) ( ) ( )2 ;

4 4o i o i

o loop tord LI d l I d l I

R V Edt dt dt

IDCD effect is simple to estimate

IDCD equation agrees with radiative disruption* perturbations

• Disruption created by Argon injection.• Cold edge peaks the current until it is

unstable. Instability cools plasma.• The low edge current is maintained by the

Argon and drags down the plasma current:

• 1.5 MA profile is flattened in 1.2 ms

IDCD requires δB of 190G (at 2.045-6 s)

*P. L. Taylor et al., Phys. Rev. Lett, 76, 916-919 (1996)

( ) / t 0.92 /il I GA s

1/22 ( )

4o il IB nea

t

On a reactor and ITER the perturbation levels required to drive the current are a little higher than considered acceptable (confirming the effect). They are small.

Parameter Present tokamaks

ARIES-AT ITER

Itor (MA) 4.5 12.8 15.

Temp. (keV) 2 18 8.1

a (m) 1 1.3 2

L/R (s) 15 605 454

Brms /B 0.0004 0.0001 0.0001

• Current driving perturbations can also flatten the j/B profile which flattens the q-profile leading to poor performance often ending in disruptions.

Solutions

1. Drive the edge current high and impose a perturbation profile that sustains the desired reversed-shear current profile.

2. Select a high performance equilibrium that has a uniform j/B (low aspect ratio)– Rigorously sustain the rock-stable profile by edge current drive

and repeated application of non-resonant perturbations of IDCD.– The method has been demonstrated on HIT-SI.

• Solves the sustainment problem. (IDCD is over two orders of magnitude more power efficient on a reactor than rf)

• High edge current prevent the edge from using perturbations to drag down the current in disruptions.

HIT-SI uses imposed fluctuations and an externally driven edge current to sustain the equilibrium

• Current amplification of 3.9, a new spheromak record

• 90 kA of toroidal current• Stable sustained equilibria

ohmically heat to the beta limit,*achieving the current drive goal

*B. S. Victor et al., Physics of Plasmas, 21, 082504 (2014)

HIT-SI produces equilibria stable to current driven modes

(a) Internal probes. (b) surface probes

• Data matches a very flat two step profile• First step is at the injector j/B and

wide enough to hold injector current.

• Second step is high enough to hold the toroidal current.

• Equilibria are stable.• The only significant fluctuations

observed are imposed.• The toroidal current versus time, the

injector impedance scaling with j/n of the spheromak, and the current profile data of HIT-SI are predicted by the IDCD model.

oj/B

The only significant fluctuations observed are imposed

Mode amplitudes vs time

Mode amplitudes minus the imposed perturbations vs time

n=1 amplitude and the injector current vs time

Toroidal current vs time

• During sustainment the n=1 is almost entirely imposed and the equilibrium is stable.

• HIT-SI is capable of testing MHD stability, which had been the problem with sustained spheromaks until now.

Gradients in j/B may inhibit instability• Imposed perturbations cause a force that inhibit

differential motion of rigid shells.• Conversely, differential motion of rigid shells cause

a force that inhibits perturbations, including eigenmodes of instabilities, preventing instability growth.

• The gradient in j/B is maintained at the current separatrix by keeping the injector j/B high.

• Differential current flow at the separatrix enforces symmetry and stability in the region.

Codes are not capturing the physics of IDCD

• Simulation has larger fluctuations indicating relaxation by instability

• Simulation has flatter Bpol near magnetic axis with q > 1

• Experiment has q < 1 and is stable

• For poloidally circular flux surfaces at the magnetic axis

1tor

polo

Bq

dBRdr

• Experiment uses imposed perturbations that sustain a stable equilibrium.

Bpol

Btor

Kyle Morgan

Two possible problems

• Once it is unstable the flux surfaces are not rigid. Magnetic interference does not drive current. Resorts to instability. (catch 22)

• Need to find the IDCD steady-state. (start with stable equilibrium)

• Electron fluid may not be sufficiently frozen in the magnetic field.

• May need a more accurate Ohm’s law. (Simulation electrons are not as frozen in because of more mass, more dissipation, or divB cleaning?)

• I think the first one is more likely and we have not worked on it. Initial conditions are very important for non-linear equations.

Conclusions

• Imposed perturbations allow the sustainment of kink-stable equilibria.

• Evidence of pressure confinement during sustainment is a spheromak first, with current gains of nearly 4.0, a new spheromak record.

• XMHD codes have a different profile of lower dBpol/dr at the magnetic axis and higher q (>1) than the experiment. Current probably penetrates by instability not imposed perturbations.

• May not be rigid enough for the effect because it is unstable? (catch 22)

• Electron fluid may not be sufficiently frozen in the magnetic field.

Problems getting initial conditions?

• Start with large Taylor state?• Pickup experiment after gain of 2 ?

Doing total Ohm’s law

• What are we missing?• Are we close to being able to do it?

• Two fluids?

19

An IDCD injector could be attached to a tokamak

NegativeIinj, ψinj

PositiveIinj, ψinj

Ion Doppler Spectroscopy confirms rigid motion

IDS geometry