First Fast-Ion D-Alpha (FIDA) Measurements and Simulations on C-2U
P. 1

Abstract
n  The goal of TAE’s experiments is to sustain through the of use of neutral beam heating and edge biasing an advanced beam-driven FRC for many milliseconds, i.e., well beyond the growth times of common instabilities [1].
n  Here we apply FPIC, a quasineutral hybrid code with fully kinetic ions, to study the growth rate of the toroidal n=1 mode and related kinetic effects versus different equilibrium parameters.
n  In particular we address how the n=1 growth rate scales versus S*/E and E. The mode strength is computed taking into account a full 3D mode decomposition of the FRC separating the physically distinct contributions, namely tilt, radial shift, ballooning and interchange modes and we study the role of each one of them.
FPIC Code
n  FPIC is an hybrid 3D code which includes the generalized Ohm’s law with finite electron mass.
n  The electron mass can be set to zero, yielding the usual Ohm’s law.
n  Faraday’s law, and Ampere’s law neglecting displacement current, are
also used in the field solver.
n  ACartesianstaggeredYeemeshisusedandshapedboundariescanbe straightforwardly included.
n  TheDey-Mittramethodforcut-cellboundariesonaYeemeshisusedto model any kind of boundary [2].
n  The full Lorentz equation for ions is used in the particle advance.
n  Esirkepov’scurrentdepositionschemeisusedtoobtainJi[3]
n  Finite resistivity is implemented for both the FRC inner “high density” and outer “low density” regions.
n  ThecodeisMPIparallelizedusingPETScanddomaindecomposition.
n  Particles can be pushed in time also with GPUs.
n  New diagnostic tools to “measure” the FRC geometric evolution are available.
n  Standardbenchmarktestshavebeencarriedoutandpassed.
FPIC Sample Run
§ Each FPIC simulation is initialized with an equilibrium computed with TAE’s code LR_eqMI. Conducting end conditions are applied [4].
§  No initial perturbation is assigned in the simulation.
§  No fast ions are included.
§  Plasma spin-up is observed.
Toroidal n = 1 Modes
The most relevant and investigated MHD instabilities in FRCs for the toroidal mode number n=1 are: tilt, radial shift, ballooning and interchange [5-8].
Numerical Results Simulation parameters and setup
Interchange n =1 m = 0 mode
§  After the initial rise the time evolution of the interchange mode is modest, slightly increasing for S* > 16 and slightly decreasing for S* < 16.
§  Its amplitude is strongly decreasing with increasing S*.
Tilt linear growth rate benchmark
The growth rate of the tilt instability predicted by MHD theory [10,11] is γmhd ≈ 2VAO / Ls , where VAO is the Alfven speed.
FPIC Study of the n=1 Toroidal Mode in FRC Plasmas
Francesco Ceccherini, Sean Dettrick, Dan Barnes, Laura Galeotti, and the TAE team
TRI ALPHA ENERGY, INC., P.O. Box 7010, Rancho Santa Margarita, CA 92688-7010
§ 
§ 
In order to identify these modes we calculated first the symmetric and anti- symmetric parts with respect to the midplane of the velocities v and v [9]
E = 3.06 equilibrium
E = 4.58 equilibrium
E = 5.46 equilibrium
i.e
vr- = vr(z) – vr(-z) vz- = vz(z) – vz(-z)
vr+ = vr(z) + vr(-z) vz+ = vz(z) + vz(-z)
S* = 16.0 Ls = 4.7 m
E = Ls
s
Bext = 0.11 T r = 0.48 m
S* = 11.9 L = 5.2 m
and then carried a full Fourier decomposition.
For our analyses we select the toroidal n = 1 mode and the axial mode m = 1 for tilt, radial shift and ballooning and m = 0 for interchange instability. Radial components have been summed up in all cases.
S*=rsωpi c
Radialsizeparameter
Ion plasma frequency
−3E/S* mhd
Initial Equilibrium
Ions have no poloidal velocity at t = 0
O-point
rz,
B
rs = 0.51 m
niO = 5.2 1019 m-3
ext
= 0.13 T
ext
S* = 16.3 Ls = 3.15 m
= 0.13 T
B
rs = 0.51 m
niO = 5.0 1019 m-3
Equilibrium S* has been modified through ion density and temperature
rescaling
Mesh size: 4 cm in each direction;
Particles: 60M
2r
Kinetic effects reduce the linear MHD growth rate as γ = γ
§  The linear evolution of the tilt mode very often begins when other
instabilities are already present and may have modified the equilibrium shape and parameters as E, Ls.
§  Field line connection length is used to distinguish between regions inside and outside the FRC in the 3D case. The FRC radius and length can then be calculated on-the-fly.
§  Tilt growth rates have been measured for each run. Here we show tilt growth rates normalized to the MHD growth rate vs the kinetic scaling law using both the initial equilibrium parameters (left plot) and the on the fly parameters (right plot).
s
niO = 3.2 1019 m-3
Separatrix elongation
e
s
Normalized inventory
γ/γMHD
γ/γMHD
0.06μs 65.5μs 87.8μs 91.4μs
104.1μs 127.2μs
Poloidal velocity mode structure
Ion density during early phase of the mode
Acknowledgement
Internal tilt mode n=1 m=1 gives rise to (vr-, vz+), with vz+ predominant In highly elongated FRC, the vz+ dominates
vz+,vr-,vz-,vr+ versustime
S*= 12
Tilt saturation
Ballooning
S*= 16
Poloidal velocity mode structure
Ion density during early phase of the mode
In the early phase of all S* cases a dominant radial shift mode is present.
0.6 0.4 0.2
0
Radial shift mode n=1 m=1 gives rise to vz- (quite small) and predominant vr+
In the subsequent phase the tilt mode dominates for “high” S* values, while it compares with ballooning and radial shift for “low” S* values.
Particle inventory
Poloidal velocity mode structure
Ion density during early phase of the mode
S*= 20
S*= 40
1 0.9 0.8 0.7 0.6 0.5
When tilt reaches the end of the linear phase, a more vigorous particle loss rate from the FRC is established.
Ballooning mode n=1 m=1 gives rise to (vr+, vz-).
S* = 16
0 50 100 150 200
Poloidal velocity mode structure
Ion density during early phase of the mode
n  Adetailedstudyofthen=1toroidalmodeshasbeencarriedout
n  Results show that instabilities amplitudes are strongly dependent on the S* value and that significant non-linear interactions between the different modes are present.
References
Interchange mode n=1 m=0 gives rise to (vr+, vz-), with vr+ predominant
is observed. m a g n i t u d e
these modes very clearly inversely proportional to S*.
Ls Equilibriumiondensity
Tilt, radial shift and ballooning n=1 m=1 modes (E=4.58)
r
s
Radial shift
S*= 8
1 0.8 0.6 0.4 0.2 0
Equilibrium parameters 1 0.8
Punctual parameters
Tilt
v + versus v - and v - versus v + zrzr
time (tA)
In the (vr-,vz+) plane the slope of the tilt mode does not depend on the S* values. Related to the E value only [5].
In the (vr+,vz-) plot a transition from a first radial shift mode to a later ballooning mode
Summary
The o f
[1] M.W. Binderbauer et al., Phys. Plasmas 22, 056110 (2015)
[2] S.Dey et al., IEEE Microw. and Guided Wave Lett. 7, 273 (1997) [3] T. Z. Esirkepov, Comput. Phys. Commun. 135, 144 (2001)
[4] L. Galeotti et al.,Phys. Plasmas 18, 082509 (2011)
[5] E. Belova et al, Phys. Plasmas 8, 1267 (2001)
[6] Y. A. Omelchenko, Phys. Rev. E 92, 023105 (2015)
[7] E. Belova et al, Phys. Plasmas 10, 2361 (2003) [8] E. Belova et al, Phys. Plasmas 11, 2523 (2004) [9] N. Ohtani et al., Phys. Plasmas 10, 145 (2003) [10] C. Steinhauer, private communication
[11] S. Dettrick et al., BP12.00031, 57th APS-DPP Meeting 2015
is
0
E = 3.06
0.2
E = 4.58
0.3E/S*0.4
E = 5.46
0.5
0.6 exp(-3 E/S*)
0.1
0 0.1 E = 3.06
0.2
E = 4.58
0.3 E/S*0.4
E = 5.46
0.5 0.6 exp(-3 E/S*)
Very useful discussion with Loren Steinhauer are gratefully acknowledged


































































































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