recent results from low s resistive mhd …...recent results from low s resistive mhd simulations of...

Recent results from low S resistive MHD simulationsof the HIT-SI experiment

C. Akcay, C. C. Kim

University of Washington

NIMROD APS meetingNov 6th, 2010Chicago, IL

Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 1 / 24

Highlights

NIMROD’s implicit advection algorithm has reduced thecomputation time by a factor of 10, which allows us to run moreinjector cycles in a reasonable amount of time.Results from low S resistive MHD simulations show production ofn=0 magnetic energy, toroidal flux and current during sustainment.The growth in n=0 occurs after 2 flat-top injector cycles.No separatrix formation observed during sustainment, consistentwith Taylor state calculations for a current amplification factorIinjItor≤ 1 1.

1G. Marklin, C. Hansen calculationsCihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 2 / 24

Outline

1 Highlights

2 The HIT-SI Experiment

3 Computational ModelInjector fieldsNIMROD implementation

4 Results from Resistive MHD Simulations


Helicity Injected Torus with Steady Inductive HelicityInjection (HIT-SI) is a current drive experimentThe spheromak plasma is formedand sustained by Steady Induc-tive Helicity Injection (SIHI) pro-vided by two semi-toroidal injectors

2 helicity injectors (X and Y)inject flux and current at aconstant rate and inductivelyinto the tank.The injectors have n=oddtoroidal symmetry andspatially rotated 90 forminimal couplingInsulating layer guaranteesinductive drive by preventingarcing to the flux conserverClosed flux device


Helicity injectors are not directly modeled in NIMROD

Injectors have a non-axisymmetric geometry.SIHI is implemented by applying flux and current B.C. at theappropriate locations on the annular regions of the HIT-SI tank.


The insulating layer is simulated as a highly resistiveedge layerThe resistive layer is approximatelyfive orders of magnitude more re-sistive than the plasma: ηedge

η ∼ 105

Finite Element Mesh

-1 0 1 2 3 4 5 6

x10-1

-3-2

-10

12

3x1

0-1

R

Z

1 mm thicknessResistivity profile is laiddirectly on the quad. points ofthe finite elements→ preventsovershootResistive layer covers theinjector mouths:

- turns the injectors into acurrent source andprevents line-tying.

- creates an electric fieldon the annulus consistentwith the locations of thevoltage gaps in theexperiment.


The injectors are modeled as a force-free, largeaspect ratio RFP

Axisymmetry→ MATLAB mimetic operator GS. solver 1 solves4∗ψ + λ2

injψ=0The cross-section is tailored to yield an eigenvalue ofλinj = 30 m−1 which corresponds to a solution with the reversalsurface at the wall (spheromak mode).

−0.3−0.2

−0.10

0.10.2

0.3

0

0.1

0.2

0.3

−0.1

−0.05

0

0.05

0.1

Injector half-torus usedin MATLAB for B.C.

Actual HIT-SI injector (courtesy ofJ. Rogers)

1Originally developed by G. Marklin, and modified by CA and CCK for quad-element meshes


In NIMROD B · n and E‖ must be prescribed on theannular surfaces to apply SIHI B.C.

A radial electric field ER is applied on each annulus to induce theinjector flux(Bz). Applying Faraday’s law to Bz with Eφ = 0 yields

ER =Rin

Bz =ωinjR

nBinj (1)

A call is made at the end of the magnetic field advance tooverwrite B · n |annulus=0 and update B according to flux B.C.An electrostatic E applied on each annulus gives rise to atangential B through (B =

∫dS× E + · · · ) with a net curl along z.

This term drives the injector current


Toroidal profiles (φ) of the injected fields with 11 (5injector–odd) modes

0 50 100 150 200 250 300 350 400−3000

−2000

−1000

0

1000

2000

3000

φ dependence of Bzinj

φ

Am

plitu

de

0 50 100 150 200 250 300 350 400−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 104

φ dependence of applied Eφ

φ

Am

plitu

de

0 50 100 150 200 250 300 350 400−3000

−2000

−1000

0

1000

2000

3000

φ dependence of applied ER

φ

Am

plitu

de

0 50 100 150 200 250 300 350 400−1000

−500

0

500

1000

1500

φ dependence of Efaraday

φ

Am

plitu

de

odd modes only

odd modes+n=0


Resistive MHD (rMHD) with uniform density and zero β

ρ

(∂v∂t

+ v · ∇v)

= J× B− ∇ · Π︸︷︷︸kin or iso_visc

(2)

∂B∂t

= ∇× (v× B)− η

µ0∇2B (3)

Table: Numerical parameters for low S simulations

Mesh (mx×my) 48x48Poly_degree 3 , 4

Toroidal modes 11mhdadv_alg ‘centered’

v_cfl 4.0divbd 2× 105


Physical parameters of the simulation and experimentclosely match (assuming Ti = Te = 10 eV)

Simulation ExperimentInjector flux (mWb) 0.47 0.6-1.4Injector current (kA) 10.6 10-20

ne (m−3) 3×1019 3×1019 (avg.)Resistivity (Ω.m) 1.5× 10−5 (elecd=11.7) 3.2× 10−5 (η‖)Viscosity (kg/m/s) 10−5 (kin_visc=100) 2.6× 10−5

Edge resistivity 8× 104ηpl ∞Edge thickness (mm) 1 ∼ 1

τinj (ms) 0.2 0.172τL/R (ms) 0.80 0.40 (η‖)τA (ms) 0.017 0.025

S1 45 16λinj (m−1) 30 15-20

Pulse length (ms) 1-2 4-10

1S here is approximately 100 times smaller than S∗


Constant helicity injection is sustained for 6 injectorperiods from t=0.1 to 1.3 ms

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−15

−10

−5

0

5

10

15Injector current vs. Time

I inj (

kA)

time (ms)

Injectors are linearly ramped-up for 0.1 ms, then run at a constantamplitude (flat-top) for 6 cycles from t=0.1 to 1.3 ms, followed by alinear ramp-down for another 0.1 ms.Injectors are shut-off at t=1.4 ms and the spheromak is left todecay for another 0.3 ms


n=0 magnetic energy rises rapidly after 2 injectorflat-top cycles (t=0.5 ms)

Magnetic energy of the higher even modes also grow.n=1 magnetic energy drops as n=0 rises.


n=0 mode undergoes an initial linear growth

0.3 0.4 0.5 0.6 0.7 0.8

−3

−2

−1

0

1

2

3

Time (ms)

Lo

g(E

mag

) (J

)

n=0n=1n=2n=3n=4n=5

0.4 0.45 0.5

−3

−2

−1

0

1

2

Time (ms)

Lo

g(E

mag

) (J

)

n=0n=1n=2n=3n=4n=5

The linear growth (γτA = 1.32) of n=0 precedes the rapid build-upand saturates at t=0.45 ms, followed by non-linear growth duringwhich n=0 magnetic energy is amplified 10-fold.n=0 growth drives higher even modes. n=2 and 4 also go througha linear growth stage, but at a decreasing linear growth rate(γ(2,3)τA = (0.88,0.86)).


Kinetic energy spectrum exhibits increased odd modeactivity during n=0 growth

0 0.5 1 1.5 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8Kinetic energy of toroidal modes at S = 50

Time (ms)

Kin

etic

Ene

rgy

(J)

n=0n=1n=2n=3n=4

0.45 0.5 0.55

−5

−4

−3

−2

−1

Time (ms)

Lo

g(K

inet

ic E

ner

gy)

(J)

n=0n=1n=2n=3n=4n=5

A marginally linear growth period is seen for the odd modes.In the aftermath of n=0 growth, kinetic energies of odd modesincrease and even modes decrease, a trend opposite to themagnetic energy spectrum


Plasma current (Itor ) reaches 9 kA

Current amplification ItorIinj∼ 0.85

Peak Itor ≈ 11 kA occurs at injector shut-off (t=1.4 ms)Current amplification up to a factor of 2 is seen in the experiment.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8−2

0

2

4

6

8

10

12

14

time (ms)

I TO

R (

kA)

Toroidal Current vs. Time

In=0

10.6 kA line

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

−10

0

10

Injector current vs. Time

I inj (

kA)

time (ms)


Early profiles (t=0.4 ms) exhibit a large n=1component, consistent with magnetic energy spectrum

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.01

−0.005

0

0.005

0.01

0.015

R (m)

B (

T)

Midplane (Z=0) B profiles at φ=0

BZ(t=0.4 ms)

Bφ(t=0.4 ms)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.015

−0.01

−0.005

0

0.005

0.01

R (m)

B (

T)


BZ(t=0.4 ms)

Bφ(t=0.4 ms)

0 0.5 1 1.5 20

5

10

15

20Magnetic energies by toroidal mode S =50

Time (ms)

Em

ag (

J)

n=0n=1


Profiles become mostly axisymmetric after rapidgrowth of n=0 (t=0.85 ms)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.03

−0.02

−0.01

0

0.01

0.02

0.03

R (m)

B (

T)


BZ(t=0.85 ms)

Bφ(t=0.85 ms)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.015

−0.01

−0.005

0

0.005

0.01

0.015

R (m)

B (

T)


BZ(t=0.85 ms)

Bφ(t=0.85 ms)

0 0.5 1 1.5 20

5

10

15


Time (ms)

Em

ag (

J)

n=0n=1


The n=1 distortion is still present at later times(t=1.3ms)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

R (m)

B (

T)


BZ(t=1.3 ms)

Bφ(t=1.3 ms)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7−0.02

−0.015

−0.01

−0.005

0

0.005

0.01

0.015

0.02

R (m)

B (

T)


BZ(t=1.3 ms)

Bφ(t=1.3 ms)

0 0.5 1 1.5 20

5

10

15


Time (ms)

Em

ag (

J)

n=0n=1


No separatrix formation observed during sustainment(SIHI)

Field lines are shortand connect theinjector mouths(t=0.825 ms)Puncture plots showthe volume is filled withstochastic lines duringsustainment

The configurationbegins to relax afterinjectors are shut-off(t=1.6 ms).


Nested flux surfaces do not form till the fluctuations(δ ≡

√(En=1

En=0)) get really small

Surface of Section

-1 0 1 2 3 4 5 6

x10-1

-3-2

-10

12

3

x10-1

R

Z

t=1.4 ms, δ = 0.22

Surface of Section

-1 0 1 2 3 4 5 6

x10-1

-3-2

-10

12

3

x10-1

R

Z

t=1.5 ms, δ = 0.023Surface of Section

-1 0 1 2 3 4 5 6

x10-1

-3-2

-10

12

3

x10-1

R

Z

t=1.6 ms, δ = 0.004

Surface of Section

-1 0 1 2 3 4 5 6

x10-1

-3-2

-10

12

3

x10-1

R

Z

t=1.7 ms, δ = 0.0017


Emergence of q ≤ 1 surfaces within 10 cm. of themagnetic axis (0.33 cm)

A q=1/2 surface is shown here.


Summary

We see a rapid growth of n=0 magnetic energy after 2 injectorcycles. Profiles become more axisymmetric, plasma current rises,but never exceeds injector current.The onset of n=0 growth corresponds to one energy (helicity)e-folding time⇒ Helicity balance: K (t) = Kinj(1− e−2t/τL/R )

n=1 magnetic energy decreases during n=0 amplification.No nested flux surfaces form during sustainment.How is the energy transferred from n=1 to n=0? Local 3-Dreconnection events (current layers)?


Future plans

Check for toroidal and poloidal convergence (currently beinganalyzed)Perform an S scan [10-100] at λinj = 30 and also at a lower λinj(15-25).Compare simulation results to experimental data and Taylor statecalculations.Turn on the Hall term. Run finite β ?


recent results from low s resistive mhd …...recent results from low s resistive mhd simulations of...

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