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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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 3 / 24
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 4 / 24
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.
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 5 / 24
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.
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 6 / 24
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 7 / 24
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 8 / 24
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 9 / 24
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 10 / 24
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∗
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 11 / 24
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 12 / 24
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.
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 13 / 24
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)).
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 14 / 24
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 15 / 24
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)
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 16 / 24
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)
Midplane (Z=0) B profiles at φ=180
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 17 / 24
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)
Midplane (Z=0) B profiles at φ=0
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)
Midplane (Z=0) B profiles at φ=180
BZ(t=0.85 ms)
Bφ(t=0.85 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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 18 / 24
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)
Midplane (Z=0) B profiles at φ=0
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)
Midplane (Z=0) B profiles at φ=180
BZ(t=1.3 ms)
Bφ(t=1.3 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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 19 / 24
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).
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 20 / 24
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
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 21 / 24
Emergence of q ≤ 1 surfaces within 10 cm. of themagnetic axis (0.33 cm)
A q=1/2 surface is shown here.
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 22 / 24
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)?
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 23 / 24
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 ?
Cihan Akcay (University of Washington) NIMROD simulations of HIT-SI APS 2010-NIMROD 24 / 24
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