T HERE ARE MULTIPLE PLASMA MODELS . Ensemble average of particles - - PowerPoint PPT Presentation

THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD FOR MULTI-FLUID PLASMA MODELING

Éder M. Sousa1 PI: Jean-Luc Cambier2

ERC INC.1, IN-SPACE PROPULSION BRANCH2, AIR FORCE RESEARCH LABORATORY EDWARDS AIR FORCE BASE, CA USA

RQ-West 6.1 Research Review, University of California, Los Angeles CA January, 2015

Distribution A: Approved for public release; distribution unlimited É.M. SOUSA (AFRL/RQRS) AFTC/PA clearance No. 15010, 16 January 2015 DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 1 / 11

OUTLINE

1 THE MULTI-FLUID PLASMA MODEL 2 THE DISCONTINUOUS GALERKIN FINITE ELEMENT METHOD 3 PROBLEMS OF INTEREST

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 2 / 11

THERE ARE MULTIPLE PLASMA MODELS.

3-Dimensions + 3-Velocities Evolve the particles position and velocity Ensemble average of particles distribution Distribution function: fs(x,v,t) Evolve the distribution function

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 3 / 11

PLASMA CAN BE MODELED AS A FLUID.

Take moments of the distribution function, fs(x,v,t)

ρs = ms

• fsdv

ρsus = ms

• vfsdv

ps = ρsTs = 1 3ms

• w2fsdv

hs = 1 2ms

• w2wfsdv

Bulk properties of the fluid are obtained Assumes local thermodynamic equilibrium and high collisionality There can be multi particles species in a plasma.

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 4 / 11

FLUID PROPERTIES AND FIELDS ARE COUPLED.

FLUID PROPERTIES:

∂ρs ∂t + ∇ · (ρsus) = Ss, ∂ρsus ∂t + ∇ · (ρsusus + psI) = ρsqs ms (E + us × B) + Rs ∂ǫs ∂t + ∇ · ((ǫs + ps)us) = ρsqs ms us · E + Qs, ǫs = ps γ − 1 + 1 2ρs|us|2

Ss, Rs, and Qs account for all collisional effects in the plasma

ELECTROMAGNETIC FIELDS:

∂B ∂t + ∇ × E = 0, ∇ · B = 0 1 c2 ∂E ∂t − ∇ × B = −µo

• s

qs ms ρsus, ǫo∇ · E =

• s

qs ms ρs

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 5 / 11

TEMPORAL MULTISCALES EXIT IN PLASMAS.

CHARACTERISTIC SPEEDS:

vcs =

• γ Ps

ρs , c

CHARACTERISTIC

FREQUENCIES:

ωps =

• nsq2

s

ǫoms , ωcs = qsB ms νsr = 4√πe4Z4nr log Λ 3√msT3/2

s

Example lab plasma (seconds):

1 ωpe = 10−14, L c = 10−9, 1 ωci =

10−8, L

vci = 10−5

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 6 / 11

ADVANTAGES OF THE MODEL

The Multi-Fluid Plasma Model (MFPM) is less computationally expensive than kinetic models Multi-fluid effects become relevant at small spacial and temporal scales The model includes finite electron mass and speed-of-light There is an inherent assumption of neutrality in MHD but not in the MFPM Neutrality does not allow local charge separation Displacement current effects are resolved in the MFPM Kinetic

CPU savings, LTE

− − − − − − − − − − − − − → MFPM

Generalization

← − − − − − − − − MHD

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 7 / 11

THE DISCONTINUOUS GALERKIN(DG) METHOD.

∂Q ∂t + ∇ · ← → F = S An ideal numerical method for the MFPM should: be high-order accurate capture shocks resolve high frequency

• scillations

not impose strict time-steps

POLYNOMIAL EXPANSION:

Q =

• i

civi ci coefficient vi Legendre polynomials

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 8 / 11

DG IS EASILY PARALLELIZABLE.

• Ω

vp ∂Q ∂t dV = Lp(Q) =

• Ω

vpSdV−

• ∂Ω

vp ← → F · dA +

• Ω

← → F · ∇vpdV Explicit Runge-Kutta time integration (2nd, 3rd or 4th order) CFL = c∆t/∆x ≤ 1/(2p − 1), p is the polynomial order

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 9 / 11

SOME APPLICATIONS OF INTEREST

FIELD REVERSED CONFIGURATION (FRC)

Formation, translation

HYDRODYNAMIC INSTABILITIES

Rayleigh-Taylor Kelvin-Helmholtz Additionally: Turbulence, Radiation, Plasma Reactions, Ionization and Recombination, LASER-plasma interactions, ...

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 10 / 11

SUMMARY

The Multi-fluid plasma model (MFPM) is implemented

less computational expensive than kinetic Model assumes Local Thermodynamic Equilibrium is more general than the MHD model

The discontinuous Galerkin (DG) numerical method is chosen

higher-order accurate (2nd-16th) resolves high frequency oscillations captures shocks

Applications of interest include the FRC, Hydro instabilities, Turbulence, etc ...

Thank You.

É.M. SOUSA (AFRL/RQRS) DISTRIBUTION A: APPROVED FOR PUBLIC RELEASE; UNLIMITED DISTRIBUTION 11 / 11