Equilibrium properties of hybrid field reversed configurations

Equilibrium properties of hybrid field reversed configurations Equilibrium properties of hybrid field reversed configurations M Tuszewski, D Gupta, S Gupta, M Onofri, D Osin, B H Deng, S A Dettrick, K[.]

Equilibrium properties of hybrid field reversed configurations

M Tuszewski, D Gupta, S Gupta, M Onofri, D Osin, B H Deng, S A Dettrick, K Hubbard, H Gota, and TAE Team

Citation: Phys Plasmas 24, 012502 (2017); doi: 10.1063/1.4972537

View online: http://dx.doi.org/10.1063/1.4972537

View Table of Contents: http://aip.scitation.org/toc/php/24/1

Published by the American Institute of Physics

Articles you may be interested in

Dynamo-driven plasmoid formation from a current-sheet instability

Phys Plasmas 23, 120705120705 (2016); 10.1063/1.4972218

Kinetic modeling of Langmuir probe characteristics in a laboratory plasma near a conducting body

Phys Plasmas 24, 012901012901 (2017); 10.1063/1.4972879

Secondary fast reconnecting instability in the sawtooth crash

Phys Plasmas 24, 012102012102 (2017); 10.1063/1.4973328

Parallel electric fields are inefficient drivers of energetic electrons in magnetic reconnection

Phys Plasmas 23, 120704120704 (2016); 10.1063/1.4972082

Equilibrium properties of hybrid field reversed configurations

M.Tuszewski,a)D.Gupta,S.Gupta,M.Onofri,D.Osin,B H.Deng,S A.Dettrick,

K.Hubbard,H.Gota,and TAE Team

Tri Alpha Energy, Inc., P.O Box 7010, Rancho Santa Margarita, California 92688, USA

(Received 4 October 2016; accepted 2 December 2016; published online 5 January 2017)

Field Reversed Configurations (FRCs) heated by neutral beam injection may include a large fast

ion pressure that significantly modifies the equilibrium A new analysis is required to characterize

such hybrid FRCs, as the simple relations used up to now prove inaccurate The substantial

contributions of fast ions to FRC radial pressure balance and diamagnetism are described A simple

model is offered to reconstruct more accurately the equilibrium parameters of elongated hybrid

FRCs Further modeling requires new measurements of either the magnetic field or the plasma

pressure.Published by AIP Publishing [http://dx.doi.org/10.1063/1.4972537]

I INTRODUCTION

A Field Reversed Configuration (FRC) is a very high

beta compact torus formed without toroidal magnetic field.1,2

The equilibrium properties of elongated thermal FRCs

con-fined inside long cylindrical flux conservers can be estimated

to good accuracy with simple analytical formulae The main

FRC parameters are derived as functions of time, and are

included in experimental databases soon after each

dis-charge Some FRC confinement properties are also included

in the databases, as derived from global analyses35based on

the above formulae

Powerful neutral beams have been used recently in the

C-2 and C-C-2U devices to heat and sustain FRCs.6,7These

plas-mas are hybrid FRCs that have comparable thermal and fast

ion pressures The fast ions modify significantly the FRC

equilibrium, and the simple relations used up to now are

inac-curate A new analysis is required to infer the equilibrium

properties of hybrid FRCs from available measurements

Numerical simulations coupled to Monte Carlo codes

have been developed to model hybrid FRCs.8,9These

calcu-lations are sophisticated tools that can be used to model a

posteriori selected FRC data A relatively fast interpretative

FRC analysis has been recently proposed,10 but this model

does not include fast ions A simpler analysis of hybrid FRC

equilibrium would be useful to compile more accurate

data-bases of all FRC discharges The purpose of the present

paper is to motivate and start development of such a model

Standard FRC analysis, suitable for experiments

with-out neutral beam injection, is briefly reviewed in SectionII

The main effects of fast ions on the equilibrium of hybrid

FRCs are presented in SectionIII A simple model of

elon-gated hybrid FRCs is described in Section IV The main

results are discussed in SectionV, and they are summarized

in SectionVI

II STANDARD FRC ANALYSIS

After formation, most FRC parameters decay on relatively

long transport timescales Meanwhile, radial and axial FRC

equilibria require only a few Alfven transit times For example,

decay times are a few milliseconds in the C-2 device, while Alfven transit times are a few microsecond Hence, the FRC remains in equilibrium during its decay phase

Three simple relations are approximately valid at the midplane (z¼ 0) of elongated, purely thermal FRCs confined inside long cylindrical flux conservers.1These relations per-mit to evaluate many FRC equilibrium parameters as func-tions of time with just a few measurements

First, radial pressure balance is approximately pþ B2/ (2l0)¼ Bw2/(2l0), where p is the plasma pressure and B is the axial (z) magnetic field This relation holds at any radial location The constant right hand side is evaluated at the flux conserver (r¼ rw), where p is zero At the FRC field null (r¼ R), one obtains

kTR¼ Bw2=ð2l0neRÞ; (1) where TR is the total (T¼ Teþ Ti) temperature and where

ne¼ niis assumed TRcan be evaluated from Eq.(1)in most FRCs experiments, since neR and Bw are usually available from multi-chord side-on interferometry and from a wall magnetic probe, respectively

Second, axial balance between field-line tension and FRC plasma pressure yields the midplane “average beta condition”11

where b¼ 2l0p/Bw2 is the external plasma beta, rs is the FRC separatrix radius, andhbi is the volume-averaged exter-nal beta value Typical values of rs/rware about one half, so that Eq.(2)yields values ofhbi close to unity

Third, the separatrix radius rs can be estimated from excluded flux measurements The excluded magnetic flux is

DU¼ prw

2

Bw Uw, where Bw is measured by a probe and

Uw¼Ð B2prdr is the magnetic flux measured by a loop, both located at r¼ rw The excluded flux radius is defined1,11 as

rDU¼ (DU/p/Bw)1/2 The integral in Uwis from r¼ rsto r¼ rw

for an FRC One can approximate Uw¼ p(rw2 rs2)Bw, for an elongated FRC with negligible plasma pressure outside of the separatrix, to obtain

a) Author to whom correspondence should be addressed Electronic mail:

mgtu@trialphaenergy.com

The relations (1) to (3) permit (standard) estimates of

many FRC parameters These parameters can be obtained as

functions of time with just excluded flux and side-on

inter-ferometry measurements The parameters and their decay

rates are inputs to most experimental FRC databases, and to

global confinement models

The standard definitions (*) of the midplane separatrix

radius rs, hbi, nR, kTR, of the separatrix volume Vs, of the

FRC plasma thermal energy Et, and of the FRC magnetic

flux U are given in TableI

The field null density n* in TableIassumes thatÐ

ndr is obtained from a single pass interferometer chord aligned along

a diameter, and neglects plasma outside of the FRC The FRC

volume Vsis evaluated with an excluded flux array consisting

of probe/loop pairs at different axial positions The separatrix

length is often estimated as the 2/3 height of the rDUaxial

pro-file, in fair agreement with numerical simulations.12The FRC

thermal energy is defined as Et¼ (3/2)Ð

pdVs The FRC mag-netic flux estimate U* assumes a rigid rotor magmag-netic field

profile satisfying Eq.(2)

III HYBRID FRC ANALYSIS

A more accurate FRC radial pressure balance must

include fast ion pressure and other terms neglected in Eq.(1)

Summing equilibrium fluid momentum equations for all

spe-cies (s), one obtains

Rmsnsðvs:rvsÞ ¼ j  B  rp; (4)

where j and p are the total plasma current and pressure,

respectively The electric field cancels out of Eq.(4),

assum-ing the quasineutrality relation ne¼ RnsZs Using Ampere’s

Law, cylindrical coordinates, and assuming axisymmetry,

the radial component of Eq.(4)at z¼ 0 can be written as

d=drðp þ B2=2=l0Þ ¼ ðBz@Br=@z Bh2=rÞ=l0

þ Rmsnsvhs2=r: (5) The fast ion velocity is assumed sufficiently randomized that

the fast ions contribute to the pressure term p in Eq.(5)rather

than to the rotational pressure on the right hand side

Monte-Carlo simulations8,9 support this assumption, showing a

directed fast ion energy that is about half of the total fast ion

energy for C-2 cases Integrating Eq.(5)between the field null

and the wall, one obtains

pfRþ neRkTRð1  aZÞ þ Rms

ð

nsvhs2dr=r

¼ ½Bw2=ð2l0Þ½1 þ dc dh; (6)

where pfRis the fast ion pressure at the field null, and aZ, dc, and dhare positive dimensionless correction terms given by

aZ¼ ½TiRnjZj RnjTj=½neRkTR; (7)

dc¼ ð2=Bw2Þ

ð

dh ¼ ðBhR=BwÞ2–ð2=Bw2Þ

ð

ðBh2=rÞdr: (9)

The impurity term aZincludes all ion species j The contribu-tions of field line curvature and of possible toroidal magnetic field are given in Eqs (8) and (9), respectively, where the integrals are from the field null to the wall

Equation(6)can be rewritten as

kTR¼ ½Bw2=ð2l0neRÞð1 þ dc dhÞ=ð1  aZþ avþ afÞ;

(10) where avand afare positive dimensionless flow and fast ion terms given by

av¼ Rms

ð

nsvhs2dr=r

=½neRkTR; (11)

Equation(10)reduces to Eq.(1)for an elongated FRC with-out toroidal field, impurities, azimuthal rotation, and fast ion pressure

The correction terms aZ, av, and dhin Eq.(10)are rela-tively small for most FRCs Assuming equal temperatures for all ion species, Eq (7)yields aZP

fj(Zj 1)Ti/T, with

fj¼ nj/ne One estimates aZ 0.1 for either 4% Oxygen (Z 4) or 1% Titanium (Z  8), either concentration being consistent with C-2 Zeffmeasurements.13Assuming rigid azi-muthal rotation inside the FRC, one obtains av M2/2 from

Eq (11), where M¼ vhis/cs is a separatrix Mach number (cs2¼ 2kTR/mi) For C-2 FRCs, Doppler spectroscopy14 sug-gests M 1/3, hence av< 0.1 The terms aZand avhave com-parable magnitudes and tend to cancel each other in Eq.(10) Finally, one estimates dh (BhR/Bw)2/2 from Eq.(9) The cor-rection term dhis a few % for some measured values of Bh.15 The curvature term dccan be significant for short FRCs Numerical results from the Lamy Ridge equilibrium code16 are shown in Fig.1

The numerical results (black points) in Fig.1are consis-tent with dc 1/E2(red points) for an FRC inside a long flux conserver The FRC elongation E is defined as the ratio of the separatrix length to its midplane diameter The curvature correction term dcis small initially (E > 4), but may become large at late times if energy losses cause FRC axial shrinkage Fast ion pressure can modify significantly the FRC radial pressure balance Some C-2 and C2-U FRC data6,7 suggest values af 1 For such cases, Eq.(10)indicates that

TRmight be only about half of the value predicted in Eq.(1) Nonetheless, radial pressure balance is still a useful relation

If the fast ion pressure is measured, Eq.(10)permits an esti-mate of the total temperature of hybrid FRCs If the tempera-tures are measured, Eq.(10)yields an estimate of the fast ion pressure

TABLE I Standard values of selected FRC Parameters.

ndr/(2r DU b*)

pr DU 2

dz

3

/r w ) B w

Fast ion diamagnetism can also modify significantly the

FRC excluded flux analysis One can write generally

rDU2¼ rs2þ

ð

where b¼ B/Bw, and where the integral is from r¼ rs to

r¼ rw Equation(13)reduces to Eq.(3)if the open field lines

are elongated and have negligible plasma pressure (b¼ 1)

For FRCs without neutral beam injection, the thermal plasma

pressure decreases rapidly away from the separatrix, and rDU

exceeds rsby only a few percent

For FRCs with strong neutral beam injection, a

signifi-cant fast ion pressure develops up to r¼ rt, the outer turning

radius of a full-energy fast ion For C-2 and C-2U FRCs, rt

exceeds largely rs because low magnetic fields (Bw< 0.1 T)

result in large fast ion orbits This is illustrated in Fig 2,

where 15 keV proton orbits (red curve) are calculated at

the midplane of an FRC with rs¼ 0.3 m, rw¼ 0.7 m, and

Bw¼ 0.087 T

Examples of fast ion density and magnetic field radial

profiles are shown in Figs.3and4, respectively These

mid-plane profiles are obtained at t¼ 1 ms from a 2-D (Q2D)

numerical simulation9of a C2-U FRC

The fast ion density is zero at r¼ 0 and at r ¼ rt, and peaks at rm 0.24 m, a value close to rt/2 The fast ion den-sity is substantial at the separatrix and outside of the FRC up

to r¼ rt The values of rm, rs, and rtare shown in Fig.3 The fast ion pressure and density radial profiles are similar, since the fast ion energy distribution has little spatial dependence The magnetic field in Fig 4 increases nearly linearly from 0.02 T at r ¼ rs to Bw 0.087 T at r ¼ rt The values

of R, rs, rDU, and rtare shown in Fig.4 The normalized mag-netic field b is lower than unity between rsand rt, because of the combined thermal and fast ion pressures For such a case,

Eq.(13)predicts that rDUshould significantly exceed rs The Q2D simulation yields rDU¼ 0.39 m and rs¼ 0.29 m The former value can also be obtained from Eq.(13)with a linear approximation between rsand rtfor the b profile of Fig.4 The FRC magnetic flux U (Ð

B2prdr from r¼ R to

r¼ rs) is 1.7 mWb for the case in Fig.4 The standard value (Table I) is U*¼ 7.3 mWb, with rDU¼ 0.39 m, rw¼ 0.7 m, and Bw¼ 0.087 T The large discrepancy between U and U* arises in part because rsis lower than rDU, and because the FRC internal magnetic field is lower than predicted by a rigid rotor satisfying Eq.(2)

IV HYBRID MODEL

A primary goal of any FRC hybrid model is to recon-struct rsfrom the measured rDUvalue Equation (13)shows that rscan be calculated if the midplane open-field-line mag-netic field radial profile is known If internal magmag-netic field

FIG 2 Calculated 15 keV proton orbits in the midplane of a C-2U FRC.

FIG 3 Calculated fast ion density profile of a C-2U FRC FIG 1 Calculated values of d c as a function of FRC elongation E.

FIG 4 Calculated B radial profile of a C-2U FRC.

measurements are not available, b can still be approximately

inferred from pressure measurements

The right hand side of Eq.(5)is relatively small outside

of the separatrix of a sufficiently elongated FRC Toroidal

field and rotational mass are mostly within the FRC, and

field line curvature is small at large radii for a straight flux

conserver Integrating the left hand side of Eq.(5)yields

In past FRC experiments without neutral beam injection,

Eq (14) has been used to infer b from measured midplane

radial density profiles, assuming isothermal plasmas.1,17This

method is inadequate for hybrid FRCs because

interferome-try is dominated by the thermal plasma density, and does

not permit to estimate the fast ion pressure that contributes

to b However, the peak fast ion pressure bfm of a hybrid

FRC can be inferred from a measurement of the bulk thermal

plasma pressure btm, since Eq.(14) implies bm¼ btmþ bfm

 1 inside the FRC where b  1 The open-field-line b radial

profile can be modeled as

b¼ btmexp½ðr  rsÞ=d þ ð1  btmÞðr=rmÞx½ðrt–rÞ=ðrt rmÞy;

(15) where the first term is the thermal plasma pressure btand the

second term is the fast ion pressure bf

The thermal plasma pressure is assumed to decay

expo-nentially on open field lines from its maximum value The

fast ion pressure term in Eq.(15)is chosen to be zero at r¼ 0

and r rt, and to peak at r¼ rm The parameters x and y

determine the value of rmand the width of the fast ion

pro-file, respectively

Equation(15)yields b(r, rs) provided that btmand of the

decay length d can be estimated from thermal pressure

meas-urements, and that x, y, rm, and rtare chosen consistent with

separate numerical simulations Then, Eq.(14)yields b(r, rs),

and Eq.(13)yields rsfor a given value of rDU

For example, rDU¼ 0.35 m and btm¼ 0.5 are assumed

These values are those of a C-2 hybrid FRC in a Q1D

simula-tion8at t¼ 1.5 ms Choosing d ¼ 0.1 m and rt¼ 0.54 m is also

appropriate for this C-2 case Setting y¼ 4 yields a fast ion

full-width-half-maximum close to that in Fig.3 Adopting the

value rm¼ 0.24 m of Fig.3yields x¼ yrm/(rt– rm)¼ 3.2 One

calculates rs¼ 0.29 m by iterative procedure from Eqs.(13)to

(15) The normalized radial pressure and magnetic field

pro-files calculated by the model are shown between rs and rwin

Fig.5

Once rs is obtained, other FRC parameters can be

estimated more accurately, such as Vs V*(rs/rDU)2, and

Et E*(btm/b*)(rs/rDU)2 The model is robust to uncertainties

in d (rs¼ 0.29 6 0.01 m for d ¼ 0.10 6 0.02 m) Although

b(r) inside the separatrix is unknown, a fair approximation of

the FRC magnetic flux is

where bs¼ b(rs) is calculated by the model The values of rs

and U obtained by the Q1D simulation8at t¼ 1.5 ms, by the

present hybrid model and by standard analysis, are compared

in TableII

The hybrid model values in Table II, obtained with Eqs (13)to(16), are close to those of the Q1D simulation The standard values in Table II, obtained with formulae in TableI, differ substantially from the Q1D values

V DISCUSSION

Standard analysis of FRC equilibrium, reviewed in Section II, describes well elongated FRCs without fast ion pressure Although Eqs (1) to (3) are approximate, they have been verified within 10% in the FRX-C device.18 Neutron measurements and Doppler Spectroscopy, combined with Thomson Scattering, support the total temperature esti-mate T* Multichord side-on interferometry data validates the average beta condition, assuming an isothermal FRC plasma The midplane separatrix radius rs, apparent from end-on holography, is found close to the measured excluded flux radius rDU

The temperature T* is always approximate since ion impurities, toroidal magnetic field, and azimuthal flow con-tribute to radial pressure balance However, as shown in SectionIII, these effects are relatively small (<10%) in most FRCs that have few impurities (Zeff< 1.5), little (ideally zero) internal toroidal magnetic field, and relatively small plasma azimuthal rotations

Field line curvature can modify significantly the radial pressure balance of short FRCs, as illustrated in Fig 1 Curvature was negligible for most past FRCs (E > 5) formed

in long cylindrical coils with small end mirrors However, curvature may be important late in the discharges of C-2 and C-2U FRCs because of axial shrinkage

Substantial modifications of the FRC equilibrium occur when strong neutral beam injection creates comparable ther-mal and fast ion pressures To illustrate this point, selected FRC parameters computed by the Q1D code8 at t¼ 1.5 ms FIG 5 Model pressure and magnetic field radial profiles of a C-2 FRC.

TABLE II Parameter comparison.

for a typical C-2 FRC are compared in TableIIIto their

cor-responding standard values

The parameters from the Q1D simulation have lower

than standard values The difference is due to fast ion

pres-sure, since the Q1D calculation does not include ion

impuri-ties, toroidal magnetic field, plasma rotation, and magnetic

field curvature The calculated thermal plasma pressure

nRkTRis lower than n*kT* by a factor of 2, suggesting

com-parable thermal and fast ion pressures inside the FRC The

Q1D thermal plasma energy (per unit length) is lower than

the standard value by a factor of 2 The Q1D simulation also

yields lower values of the separatrix radius and of the FRC

magnetic flux (rs¼ 0.29 m and U ¼ 1.6 mWb) compared to

standard values (r*¼ 0.35 m and U* ¼ 5.1 mWb), as already

mentioned in TableII

The standard total temperature T* 0.75 keV

underes-timates the value TR 1.1 keV obtained with Eq (1)

because n* 2.3 overestimates nR 1.6 Standard analysis

assumes a density maximum at r¼ R, while the calculation

shows a density minimum This hollowing effect is caused

by the fast ions

There is some evidence for T < T* in C-2 and C-2U

data The deuterium ion temperatures of some C-2 FRCs,

estimated by Charge Exchange recombination Spectroscopy

(CHERS), are shown in Fig.6

The blue and magenta points in Fig.6are CHERS data,

and the yellow curve is the pressure balance ion temperature

T* - Te, assuming an electron temperature of 100 eV The

magenta CHERS data, obtained near the FRC field null

(R 0.20–0.25 m), are consistent with T  T* at t  0.5 ms

and with T T*/2 at t  1.5 ms The latter suggests a

rela-tively large fast ion pressure (af 1)

Qualitatively similar results are obtained for oxygen

(OV) ion temperatures estimated from Doppler spectroscopy

Some C-2U Doppler data are shown in Fig.7as functions of

time, for cases with different injected neutral beam powers

Oxygen and deuterium ion temperatures are expected

to be comparable because energy equipartition times are relatively short (<0.1 ms) The measured oxygen ion tem-peratures (red) in Fig 7are close to the standard ion tem-peratures T* Te(yellow curve with Te¼ 100 eV), but tend

to lower at late times, especially for larger injected neutral beam powers (NBs¼ 4 and 6) This suggests higher fast ion pressures with higher neutral beam powers More accurate ion temperatures TR Te (purple curves with Te¼ 100 eV) offer a better comparison with the oxygen data The values

of TR are calculated from Eq.(1), with neRestimated from multi-chord interferometry.19

The fast ion pressure builds up on slowing down time scales (1–2 ms for C-2 and C-2U FRCs) Initially, fast ion pressure is negligible and standard FRC analysis should be adequate Later, the separatrix magnetic field is gradually reduced as the fast ion pressure increases, and rDUincreases above rsif resistive decay of the FRC magnetic flux is negli-gible Some rDU rises observed in C-2U discharges are shown in Fig.8

The fast ion pressure build-up obscures the FRC con-finement times The evolution of the FRC magnetic flux U cannot be inferred any more from excluded flux measure-ments because of the competing effects of rDUrise, resistive magnetic flux decay, and possible fast ion current drive The magnetic field must be measured to calculate U Future non-perturbing internal magnetic field measurements from either motional stark effect or multi-chord polarimetry may be

TABLE III Simulated and standard FRC parameters.

Parameter T R (keV) n R (1019m3) E t /L (KJ/m)

FIG 6 Deuterium ion temperature as function of time for a C-2 FRC.

FIG 7 Oxygen ion temperatures for C-2U FRCs with 2, 4, and 6 neutral beams.

feasible, but sensitivity is an issue because of the low

inter-nal FRC magnetic fields

The e-folding decay time sEt of the thermal energy Et

can be significantly overestimated by the standard value sE*

For hybrid FRCs, E* is approximately the sum of the rising

fast ion energy Ef and of the decaying thermal energy Et

When Ef peaks, sE*exceeds sEt by a factor 1þ Ef/Et This

factor is about 2 for C-2 cases The time history of the fast

ion density must be measured to accurately assess the FRC

energy confinement Modeling of Fast Ion D-Alpha (FIDA)

measurements20may permit such an assessment

The simple hybrid model of Section IV requires some

knowledge of the thermal plasma pressure at the FRC

mid-plane The model is only valid for elongated FRCs because it

neglects axial variations and magnetic field line curvature

The FRC separatrix radius can be obtained, and the volume

and thermal energy content can then be estimated with

excluded flux array data Fast ion and thermal plasma

meas-urements would permit to construct more accurate models,

and to check the assumed b 1 (b  0) of the core plasma

The results of the hybrid model are in fair agreement

with numerical simulations8,9 of C-2 FRCs These

simula-tions combine fluid transport codes in 1 (r) and 2 (r, z)

dimensions with Monte-Carlo calculations of fast ions The

Q1D8 and Q2D9 codes yield very similar results for

elon-gated C-2 FRCs Hence, the 1-D hybrid model of SectionIV

appears adequate, and can be compared to either Q1D

(TablesIIandIII) or Q2D (Figs.3and4) numerical results

Q2D simulations of C-2 and C-2U FRCs show broad

fast ion axial profiles within the separatrix The coalescence

of fast ions around the field null (z¼ 0) seen in earlier 2-D

simulations21 is not observed in Q2D results This may be

explained by differences in neutral beam injection (z¼ 0.5 m

rather than z¼ 0), beam cross-section (0.1 m radius rather

than a pencil beam), beam divergence (1.2 degree half-angle

rather than 0 degree) Neutral beam injection yields some

decrease in FRC length in Q2D simulations

VI SUMMARY

The equilibrium parameters of thermal FRCs are well

estimated by the simple analytical formulae of standard

anal-ysis Strong neutral beam injection yields hybrid FRCs with

comparable fast ion and thermal plasma pressures The fast

ions contribute significantly to the radial pressure balance

and to the diamagnetism, and therefore modify the FRC equilibrium The temperature, separatrix radius, volume, energy, and magnetic flux of hybrid FRCs are overestimated

by standard analysis

New models are required to estimate the equilibrium parameters of hybrid FRCs A simple model of elongated hybrid FRCs is offered in Section IV This model requires only thermal plasma pressure information The magnitude of the fast ion pressure is inferred, and its radial profile is cho-sen consistent with numerical simulations The results of the model compare well with 1-D and 2-D numerical results The fast ion pressure builds up on slowing down time-scales and obscures FRC confinement analysis The time evolutions of the FRC magnetic flux and thermal energy remain largely unknown Spatially resolved measurements

of either the magnetic field or of the plasma pressure are required to quantify the equilibrium and confinement proper-ties of hybrid FRCs

1

M Tuszewski, Nucl Fusion 28, 2033 (1988).

2

L C Steinhauer, Phys Plasmas 18, 070501 (2011).

3

D J Rej and M Tuszewski, Phys Fluids 27, 1514 (1984).

4 J T Slough, A L Hoffman, R D Hoffman, R D Milroy, R Maqueda, and L C Steinhauer, Phys Plasmas 2, 2286 (1995).

5 S Okada, K Kitano, H Sumikura, T Higashikozono, M Inomoto, S Yoshimura, and M Ohta, Nucl Fusion 45, 1094 (2005).

6 M W Binderbauer, T Tajima, L C Steinhauer, E Garate, M Tuszewski,

L Schmitz, H Y Guo, A Smirnov, H Gota, D Barnes, B H Deng, M.

C Thompson, E Trask, X Yang, S Putvinski, N Rostoker, R Andow, S Aefsky, N Bolte, D Q Bui, F Ceccherini, R Clary, A H Cheung, K D Conroy, S A Dettrick, J D Douglass, P Feng, L Galeotti, F Giammanco, E Granstedt, D Gupta, S Gupta, A A Ivanov, J S Kinley,

K Knapp, S Korepanov, M Hollins, R Magee, R Mendoza, Y Mok, A Necas, S Primavera, M Onofri, D Osin, N Rath, T Roche, J Romero, J.

H Schroeder, L Sevier, A Sibley, Y Song, A D Van Drie, J K Walters, W Waggoner, P Yushmanov, K Zhai, and TAE Team, Phys Plasmas 22, 056110 (2015).

7 M W Binderbauer, T Tajima, M Tuszewski, L Schmitz, A Smirnov, H Gota, E Garate, D Barnes, B H Deng, E Trask, X Yang, S Putvinski,

N Rostoker, R Andow, N Bolte, D Q Bui, F Ceccherini, R Clary, A.

H Cheung, K D Conroy, S A Dettrick, J D Douglass, P Feng, L Galeotti, F Giammanco, E Granstedt, D Gupta, S Gupta, A A Ivanov,

J S Kinley, K Knapp, S Korepanov, M Hollins, R Magee, R Mendoza,

Y Mok, A Necas, S Primavera, M Onofri, D Osin, N Rath, T Roche, J Romero, J H Schroeder, L Sevier, A Sibley, Y Song, L C Steinhauer,

M C Thompson, A D Van Drie, J K Walters, W Waggoner, P Yushmanov, K Zhai, and the TAE Team, in The Physics of Plasma-Driven Accelerators and Accelerator-Plasma-Driven Fusion: Proceedings of Norman Rostoker Memorial Symposium (AIP Publishing, Melville, New York, 2015),Vol 1721, p 030003-1.

8 S Gupta, D C Barnes, S A Dettrick, E Trask, M Tuszewski, B H Deng, H Gota, D Gupta, K Hubbard, S Korepanov, M C Thompson, K Zhai, T Tajima, and TAE Team, Phys Plasmas 23, 052307 (2016).

9

M Onofri, S Dettrick, D Barnes, T Tajima, S Gupta, E Trask, and L Schmitz, Am Phys Soc 59, UP8.00017 (2014).

10

L C Steinhauer, Phys Plasmas 21, 082516 (2014).

11

W T Armstrong, R K Linford, J Lipson, D A Platts, and E G Sherwood, Phys Fluids 24, 2068 (1981).

12

R L Spencer and M Tuszewski, Phys Fluids 28, 1810 (1985).

13 E Garate, N Bolte, D Gupta, H Gota, I Allfrey, J Kinley, K Knapp, and TAE Team, Bull Am Phys Soc 59, UP8.00009 (2014).

14

D Gupta, E Granstedt, R Magee, D Osin, M Tuszewski, and TAE Team, Am Phys Soc 59, UP8.00014 (2014).

15

B H Deng, J S Kinley, K Knapp, P Feng, R Martinez, C Weixel, S Armstrong, R Hayashi, A Longman, R Mendoza, H Gota, and M Tuszewski, Rev Sci Instrum 85, 11D401 (2014).

16

L Galeotti, D C Barnes, F Ceccherini, and F Pegoraro, Phys Plasmas

18, 082509 (2011).

17

R E Chrien and S Okada, Phys Fluids 30, 3574 (1987).

FIG 8 Excluded flux radius rises between 1 and 2 ms in C-2U discharges.

18 R E Siemon, W T Armstrong, D C Barnes, R R Bartsch, R E Chrien,

J C Cochrane, W H Hugrass, R W Kewish, Jr., P L Klingner, H R.

Lewis, R K Linford, L F McKenna, R D Milroy, D J Rej, J L.

Schwartzmeier, C E Seyler, E G Sherwood, R L Spencer, and M.

Tuszewski, Fusion Technol 9, 13 (1986).

19 O Gornostaeva, B H Deng, E Garate, H Gota, J Kinley, J Schroeder, and M Tuszewski, Rev Sci Instrum 81, 10D516 (2010); B H Deng, J.

S Kinley, and J Schroeder, ibid 83, 10E339 (2012).

20 W Heidbrink, Rev Sci Instrum 81, 10D727 (2010).

21 A F Lifschitz, R Farengo, and N R Arista, Nucl Fusion 42, 863 (2002).