Overview of FRC-related modeling (July 2014-present) Artan Qerushi - - PowerPoint PPT Presentation

Overview of FRC-related modeling (July 2014-present)

Artan Qerushi

AFRL-UCLA Basic Research Collaboration Workshop

January 20th, 2015

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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

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Illustration of FRC configuration

θ−pinch formed FRC.

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FRC formation with RMF (Rotating Magnetic Field)

RMF-formed FRC.

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Original publications of modeling RMF-formed FRCs

• W. N. Hugrass and R. C. Grimm, A numerical study of the generation
• f 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).

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Illustration of 2D (r − θ) RMF model

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2D (r − θ) model equations and their numerical solution

Coupled equations for Bz(r, θ) and Az(r, θ) ∂Az ∂t = η µ0 ∇2Az + 1 neeµ0r ∂Az ∂r ∂Bz ∂θ

• −

∂Az ∂θ ∂Bz ∂r

• ∂Bz

∂t = η µ0 ∇2Bz + 1 neeµ0r ∂Az ∂θ ∂ ∂r ∇2Az − ∂Az ∂r ∂ ∂θ∇2Az

• Solve numerically by expanding Bz and Az 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).

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0D model equations

dne dt =

• k=1,Z

qini dnk dt = ne (nk−1Sk−1 − nk(Sk + αk) + nk+1αk+1) , k = 1, 2, · · · , Z − 1 dnz dt = ne(nZ−1SZ−1 − nZαZ), dTe dt =

• k=0,Z

nkLk(Te) All coefficients Sk, αk, Lk are functions of Te. 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.

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Test calculation for Oxygen

• R. A. Hulse, Numerical studies of impurities in fusion plasmas,

Nuclear Technol/Fusion 3, 259-272 (1983).

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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).

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Magnetized plasma closure (Introduction)

∂(ρj vj) ∂t = · · · − ∇·

↔

Πj +

• k
• Rjk

∂ǫe ∂t = · · · − ∇ · qj−

↔

Πj: ∇ vj +

• k

Qjk · · · indicate terms not related to collisions,

• Rjk represents transfer of momentum between the species j and k due

to collisions,

• qj represents flux of heat due to the temperature gradient ∇Tj and

Ohmic heating,

↔

Π represents the off-diagonal part of the pressure tensor, Qjk represents collisional energy transfer between the species j and k. This term is proportional to the temperature difference Tk − Tj. The purpose of a fluid closure is to provide explicit expressions for Rjk, qj,

↔

Π and Qjk in terms of nj, vj, Tj and the magnetic field strength.

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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.

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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).

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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.

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Extra Slides.

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Assumptions of 2D RMF model

Frequency condition for current drive ωci < ω < ωce. 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 = meνei nee2 Electron inertia is neglected; that is it is assumed that ω ≪ ωce, νei. The displacement current is neglected. This implies that only systems for which ωrp/c, where rp is the plasma radius, are considered.

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FRC formation with RMF (Rotating Magnetic Field)

Illustration of 2D (r − θ) RMF formation FRC.

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Test calculation for Iron

• R. A. Hulse, Numerical studies of impurities in fusion plasmas,

Nuclear Technol/Fusion 3, 259-272 (1983).

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