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 PHYSICS OF PLASMAS 24, 012502 (2017) 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 October 2016; accepted December 2016; published online 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 confined 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 discharge Some FRC confinement properties are also included in the databases, as derived from global analyses3–5 based on the above formulae Powerful neutral beams have been used recently in the C2 and C-2U devices to heat and sustain FRCs.6,7 These plasmas 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 inaccurate 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.8,9 These calculations 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 databases 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 without neutral beam injection, is briefly reviewed in Section II The main effects of fast ions on the equilibrium of hybrid FRCs are presented in Section III A simple model of elongated hybrid FRCs is described in Section IV The main results are discussed in Section V, and they are summarized in Section VI 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, a) 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 (z ¼ 0) of elongated, purely thermal FRCs confined inside long cylindrical flux conservers.1 These relations permit to evaluate many FRC equilibrium parameters as functions of time with just a few measurements First, radial pressure balance is approximately p ỵ B2/ (2l0) ẳ 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 (r ¼ rw), where p is zero At the FRC field null (r ¼ R), one obtains kTR ¼ Bw =ð2l0 neR Þ; where TR is the total (T ẳ Te ỵ Ti) temperature and where ne ẳ ni is assumed TR can 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 hbi ¼ 1rs =rw ị2 =2; (2) where b ẳ 2l0p/Bw2 is the external plasma beta, rs is the FRC separatrix radius, and hbi is the volume-averaged external beta value Typical values of rs/rw are about one half, so that Eq (2) yields values of hbi close to unity Third, the separatrix radius rs can be estimated from excluded flux measurements The excluded magnetic flux is DU ¼Ðprw2Bw Uw, where Bw is measured by a probe and Uw ¼ B2prdr is the magnetic flux measured by a loop, both located at r ¼ rw The excluded flux radius is defined1,11 as rDU ¼ (DU/p/Bw)1/2 The integral in Uw is from r ¼ rs to r ¼ rw for an FRC One can approximate Uw ¼ p(rw2 rs2)Bw, for an elongated FRC with negligible plasma pressure outside of the separatrix, to obtain Author to whom correspondence should be addressed Electronic mail: mgtu@trialphaenergy.com 1070-664X/2017/24(1)/012502/7/$30.00 (1) 24, 012502-1 rs ¼ rDU : (3) Published by AIP Publishing 012502-2 Tuszewski et al Phys Plasmas 24, 012502 (2017) 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 interferometry 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 Table I Ð The field null density n* in Table I assumes that ndr is obtained from a single pass interferometer chord aligned along a diameter, and neglects plasma outside of the FRC The FRC volume Vs is 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 rDU axial pro12 file, in fair agreement with numerical simulations The FRC Ð thermal energy is defined as Et ¼ (3/2) pdVs The FRC magnetic flux estimate U* assumes a rigid rotor magnetic 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 species (s), one obtains Rms ns vs :rvs ị ẳ j B rp; (4) where j and p are the total plasma current and pressure, respectively The electric field cancels out of Eq (4), assuming the quasineutrality relation ne ¼ RnsZs Using Ampere’s Law, cylindrical coordinates, and assuming axisymmetry, the radial component of Eq (4) at z ¼ can be written as d=drp ỵ B2 =2=l0 ị ẳ Bz @Br =@z Bh =rị=l0 ỵ Rms ns vhs =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 MonteCarlo 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 pfR ỵ neR kTR aZ ị ỵ Rms ns vhs dr=r ẳ ẵBw =2l0 ịẵ1 þ dc dh ; (6) TABLE I Standard values of selected FRC Parameters Parameter rs hbi nR kTR Vs Et U Standard value (*) r* ¼ rDU b* ¼ (rDU/rw)2/2 Ð n* ¼ ndr/(2rDUb*) kT* ¼ Bw2/(2l0 n*) Ð V* ¼ prDU2 dz E* ¼ (3/2)[Bw2/(2l0)]b*V* U* ¼ (rDU3/rw) Bw where pfR is the fast ion pressure at the field null, and aZ, dc, and dh are positive dimensionless correction terms given by aZ ẳ ẵTi Rnj Zj Rnj Tj =ẵneR kTR ; dc ẳ 2=Bw ị Bz @Br =@zịdr; dh ẳ ðBhR =Bw Þ2 –ð2=Bw Þ ðBh =rÞdr: (7) (8) (9) The impurity term aZ includes all ion species j The contributions 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 kTR ¼ ẵBw =2l0 neR ị1 ỵ dc dh ị=1 aZ ỵ av ỵ af ị; (10) where av and af are positive dimensionless flow and fast ion terms given by ð (11) av ¼ Rms ns vhs dr=r =ẵneR kTR ; af ẳ pfR =ẵneR kTR : (12) Equation (10) reduces to Eq (1) for an elongated FRC without toroidal field, impurities, azimuthal rotation, and fast ion pressure The correction terms aZ, av, and dh in Eq (10) are relatively small for most FRCs AssumingPequal temperatures for all ion species, Eq (7) yields aZ fj(Zj 1)Ti/T, with fj ¼ 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 Zeff measurements.13 Assuming rigid azimuthal rotation inside the FRC, one obtains av M2/2 from Eq (11), where M ¼ vhis/cs is a separatrix Mach number (cs2 ¼ 2kTR/mi) For C-2 FRCs, Doppler spectroscopy14 suggests M 1/3, hence av < 0.1 The terms aZ and av have comparable magnitudes and tend to cancel each other in Eq (10) Finally, one estimates dh (BhR/Bw)2/2 from Eq (9) The correction term dh is a few % for some measured values of Bh.15 The curvature term dc can be significant for short FRCs Numerical results from the Lamy Ridge equilibrium code16 are shown in Fig The numerical results (black points) in Fig are consistent 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 dc is 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 For such cases, Eq (10) indicates that TR might 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 estimate of the total temperature of hybrid FRCs If the temperatures are measured, Eq (10) yields an estimate of the fast ion pressure 012502-3 Tuszewski et al Phys Plasmas 24, 012502 (2017) FIG Calculated values of dc as a function of FRC elongation E FIG Calculated fast ion density profile of a C-2U FRC Fast ion diamagnetism can also modify significantly the FRC excluded flux analysis One can write generally ð (13) rDU ẳ rs ỵ bị2rdr; where b ẳ B/Bw, and where the integral is from r ¼ rs to r ¼ rw Equation (13) reduces to Eq (3) if the open field lines are elongated and have negligible plasma pressure (b ¼ 1) For FRCs without neutral beam injection, the thermal plasma pressure decreases rapidly away from the separatrix, and rDU exceeds rs by only a few percent For FRCs with strong neutral beam injection, a significant fast ion pressure develops up to r ¼ 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 rs ¼ 0.3 m, rw ¼ 0.7 m, and Bw ¼ 0.087 T Examples of fast ion density and magnetic field radial profiles are shown in Figs and 4, respectively These midplane profiles are obtained at t ¼ ms from a 2-D (Q2D) numerical simulation9 of a C2-U FRC The fast ion density is zero at r ¼ and at r ¼ rt, and peaks at rm 0.24 m, a value close to rt/2 The fast ion density is substantial at the separatrix and outside of the FRC up to r ¼ rt The values of rm, rs, and rt are shown in Fig 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 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 rt are shown in Fig The normalized magnetic field b is lower than unity between rs and rt, because of the combined thermal and fast ion pressures For such a case, Eq (13) predicts that rDU should significantly exceed rs The Q2D simulation yields rDU ¼ 0.39 m and rs ¼ 0.29 m The former value can also be obtained from Eq (13) with a linear approximation between rs and rt for the Ð b profile of Fig The FRC magnetic flux U ( B2prdr from r ¼ R to r ¼ rs) is 1.7 mWb for the case in Fig The standard value (Table I) is U* ¼ 7.3 mWb, with rDU ¼ 0.39 m, rw ¼ 0.7 m, and Bw ¼ 0.087 T The large discrepancy between U and U* arises in part because rs is 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 reconstruct rs from the measured rDU value Equation (13) shows that rs can be calculated if the midplane open-field-line magnetic field radial profile is known If internal magnetic field FIG Calculated 15 keV proton orbits in the midplane of a C-2U FRC FIG Calculated Bz radial profile of a C-2U FRC 012502-4 Tuszewski et al Phys Plasmas 24, 012502 (2017) 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 b ẳ bị1=2 : (14) 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.1,17 This method is inadequate for hybrid FRCs because interferometry 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 bm ẳ btm ỵ bfm inside the FRC where b The open-field-line b radial profile can be modeled as FIG Model pressure and magnetic field radial profiles of a C-2 FRC 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 Table I, differ substantially from the Q1D values b ẳ btm expẵr rs ị=d ỵ btm ịr=rm ịx ẵrt rị=rt rm ịy ; (15) V DISCUSSION where the first term is the thermal plasma pressure bt and the second term is the fast ion pressure bf The thermal plasma pressure is assumed to decay exponentially on open field lines from its maximum value The fast ion pressure term in Eq (15) is chosen to be zero at r ¼ and r rt, and to peak at r ¼ rm The parameters x and y determine the value of rm and the width of the fast ion profile, respectively Equation (15) yields b(r, rs) provided that btm and of the decay length d can be estimated from thermal pressure measurements, and that x, y, rm, and rt are chosen consistent with separate numerical simulations Then, Eq (14) yields b(r, rs), and Eq (13) yields rs for a given value of rDU For example, rDU ¼ 0.35 m and btm ¼ 0.5 are assumed These values are those of a C-2 hybrid FRC in a Q1D simulation8 at t ¼ 1.5 ms Choosing d ¼ 0.1 m and rt ¼ 0.54 m is also appropriate for this C-2 case Setting y ¼ yields a fast ion full-width-half-maximum close to that in Fig Adopting the value rm ¼ 0.24 m of Fig yields x ¼ yrm/(rt – rm) ¼ 3.2 One calculates rs ¼ 0.29 m by iterative procedure from Eqs (13) to (15) The normalized radial pressure and magnetic field profiles calculated by the model are shown between rs and rw in Fig 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 (rs ¼ 0.29 0.01 m for d ¼ 0.10 0.02 m) Although b(r) inside the separatrix is unknown, a fair approximation of the FRC magnetic flux is 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 estimate 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 contribute to radial pressure balance However, as shown in Section III, these effects are relatively small ( 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 thermal and fast ion pressures To illustrate this point, selected FRC parameters computed by the Q1D code8 at t ¼ 1.5 ms U ¼ ðpbs =4Þrs Bw ; (16) where bs ¼ b(rs) is calculated by the model The values of rs and U obtained by the Q1D simulation8 at t ¼ 1.5 ms, by the present hybrid model and by standard analysis, are compared in Table II TABLE II Parameter comparison Parameter Q1D Model Standard rs (m) U (mWb) 0.29 1.6 0.29 1.4 0.35 5.1 012502-5 Tuszewski et al Phys Plasmas 24, 012502 (2017) TABLE III Simulated and standard FRC parameters Parameter Simulation Standard (*) TR (keV) nR (1019 m3) Et/L (KJ/m) 0.55 0.75 1.6 2.3 0.7 1.4 for a typical C-2 FRC are compared in Table III to their corresponding standard values The parameters from the Q1D simulation have lower than standard values The difference is due to fast ion pressure, since the Q1D calculation does not include ion impurities, toroidal magnetic field, plasma rotation, and magnetic field curvature The calculated thermal plasma pressure nRkTR is lower than n*kT* by a factor of 2, suggesting comparable 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 The Q1D simulation also yields lower values of the separatrix radius and of the FRC magnetic flux (rs ¼ 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 Table II The standard total temperature T* 0.75 keV underestimates 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 r ¼ 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 The blue and magenta points in Fig are 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 relatively 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 as functions of time, for cases with different injected neutral beam powers FIG Deuterium ion temperature as function of time for a C-2 FRC FIG Oxygen ion temperatures for C-2U FRCs with 2, 4, and neutral beams Oxygen and deuterium ion temperatures are expected to be comparable because energy equipartition times are relatively short (