AFTC PA Release# 15009, 16 Jan 2015 Overview of FRC-related modeling (July 2014-present) Artan Qerushi AFRL-UCLA Basic Research Collaboration Workshop January 20th, 2015 Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 1 / 18
Outline 1 FRC configuration Illustration of FRC configuration FRC formation with RMF 2 2D ( r − θ ) model of RMF-formed FRCs Original publications of RMF-formed FRCs Model equations and their numerical solution 3 Collision Radiative model 0D model equations Test calculations 4 Magnetized plasma closure for electron-ion-neutral mixture Magnetized plasma closure for electron-ion-neutral mixture Applied field modules 5 Conclusions Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 2 / 18
Illustration of FRC configuration θ − pinch formed FRC. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 3 / 18
FRC formation with RMF (Rotating Magnetic Field) RMF-formed FRC. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 4 / 18
Original publications of modeling RMF-formed FRCs W. N. Hugrass and R. C. Grimm, A numerical study of the generation of an azimuthal current ina plasma cylinder using a transverse rotating magnetic field, J. Plasma Physics 26 , 455-464 (1981). R. D. Milroy, A numerical study of rotating magnetic fields as a current drive for field reversed configurations, Phys. Plasmas 6 , 2771-2780 (1999). R. D. Milroy, A magnetohydrodynamic model of rotating magnetic field current drive in a field-reversed configuration, Phys. Plasmas 7 , 4135-4142 (2000). Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 5 / 18
Illustration of 2D ( r − θ ) RMF model Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 6 / 18
2D ( r − θ ) model equations and their numerical solution Coupled equations for B z ( r , θ ) and A z ( r , θ ) �� ∂ A z � � ∂ B z � � ∂ A z � � ∂ B z �� ∂ A z = η 1 ∇ 2 A z + − ∂ t µ 0 n e e µ 0 r ∂ r ∂θ ∂θ ∂ r � ∂ � ∂ ∂ B z = η 1 �� ∂ A z � ∂ A z � ∇ 2 B z + ∂ r ∇ 2 A z − ∂θ ∇ 2 A z ∂ t µ 0 n e e µ 0 r ∂θ ∂ r Solve numerically by expanding B z and A z in Fourier series and deriving 1D equations for the Fourier coefficients (coupled). Advance the equations for the Fourier coefficients in time using an ODE solver (method of lines). Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 7 / 18
0D model equations dn e � = q i n i dt k =1 , Z dn k = n e ( n k − 1 S k − 1 − n k ( S k + α k ) + n k +1 α k +1 ) , k = 1 , 2 , · · · , Z − 1 dt dn z = n e ( n Z − 1 S Z − 1 − n Z α Z ) , dt dT e � = n k L k ( T e ) dt k =0 , Z All coefficients S k , α k , L k are functions of T e . NIST (FLYCHK code) provides these coefficients for all the periodic table in the temperature range 0.5 [ev] to 100 [kev]. Have implemented in C++ the coefficients for H, He, O, Ar, Kr and Xe. Little work needed to add other elements if that is needed. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 8 / 18
Test calculation for Oxygen R. A. Hulse, Numerical studies of impurities in fusion plasmas, Nuclear Technol/Fusion 3 , 259-272 (1983). Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 9 / 18
Test calculation for Oxygen (time-dependent radiation loss) R. A. Hulse, Numerical studies of impurities in fusion plasmas, Nuclear Technol/Fusion 3 , 259-272 (1983). Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 10 / 18
Magnetized plasma closure (Introduction) ∂ ( ρ j � v j ) ↔ � � = · · · − ∇· Π j + R jk ∂ t k ∂ǫ e ↔ � = · · · − ∇ · � Π j : ∇ � v j + q j − Q jk ∂ t k · · · indicate terms not related to collisions, � R jk represents transfer of momentum between the species j and k due to collisions, � q j represents flux of heat due to the temperature gradient ∇ T j and Ohmic heating, ↔ Π represents the off-diagonal part of the pressure tensor, Q jk represents collisional energy transfer between the species j and k . This term is proportional to the temperature difference T k − T j . The purpose of a fluid closure is to provide explicit expressions for � R jk , � q j , ↔ Π and � Q jk in terms of n j , � v j , T j and the magnetic field strength. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 11 / 18
Magnetized plasma closure (literature) N. A. Bobrova et. al , Magnetohydrodynamic two-temperature equations for multicomponent plasma, Phys. Plasmas 12 , 022105 (2005). E. T. Meier and U. Shumlak, A general nonlinear fluid model for reacting plasma-neutral mixture, Phys. Plasmas 19 , 072508 (2012). E. M. Epperlein and M. G. Heines, Plasma transport coefficients in a magnetic field by direct numerical simulation of the Fokker-Planck equation, Phys. Fluids 29 , 1029-1041 (1986). A. Decoster, Fluid equations and transport coefficients in plasmas, in Modeling Collisions, Elsevier 1998. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 12 / 18
Magnetized plasma closure (results) E. M. Epperlein and M. G. Haines, Plasma transport coefficients in a magnetic field by direct numerical solution of the Fokker-Planck equation, Phys. Fluids 29 , 1029-1041 (1986). Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 13 / 18
Conclusions We have implemented in stand-alone C++ code ◮ The CR data for H, He, O, Ar, Kr and Xe. ◮ Magnetized plasma closure for electron-ion-neutral mixture. ◮ Applied field modules for the FRC experiment setup including the DC coils and the RMF antenna. The 2D r − θ model is currently implemented in modern fortran. A C++ version which uses the SUNDIALS ODE solver suite is being written to be included in our software framework (SMURF). Work on multi-dimensional fluid models using the Finite Element method and unstructured grids is ongoing and will be reported in the near future. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 14 / 18
Extra Slides. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 15 / 18
Assumptions of 2D RMF model Frequency condition for current drive ω ci < ω < ω ce . Infinitely long plasma cyclinder lying in a uniform magnetic field. All quantities are assumed independent of axial location z . The ions form a uniformly distributed neutralizing background of fixed, massive positive charges. The plasma resistivity, η , is taken to be a scalar quantity which is constant in time and uniform on space. In particular, η is assumed to be of the form η = m e ν ei n e e 2 Electron inertia is neglected; that is it is assumed that ω ≪ ω ce , ν ei . The displacement current is neglected. This implies that only systems for which ω r p / c , where r p is the plasma radius, are considered. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 16 / 18
FRC formation with RMF (Rotating Magnetic Field) Illustration of 2D ( r − θ ) RMF formation FRC. Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 17 / 18
Test calculation for Iron R. A. Hulse, Numerical studies of impurities in fusion plasmas, Nuclear Technol/Fusion 3 , 259-272 (1983). Distribution A–Approved for public release; distribution unlimited. Artan Qerushi (AFRL) FRC modeling overview (July 2014-present) January 20th, 2015 18 / 18
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