plasma simulation algorithm for the two-fluid plasma model · plasma simulation algorithm for the...

Aerospace & Energetics Research Program - University of WashingtonPlasma Dynamics Group

Plasma Simulation Algorithm for theTwo-Fluid Plasma Model

U. Shumlak, C. S. Aberle, A. Hakim, and J. J. Loverich

Aerospace and Energetics Research ProgramAeronautics and Astronautics

University of Washington, Seattle

American Physical Society - Division of Computational Physics August 2002


Abstract

Many current plasma simulation codes are based on the magneto-hydrodynamic (MHD) model whose derivation involves severalassumptions that severely limit its applicability, particularly for Halleffect physics. The two-fluid plasma model only assumes localthermodynamic equilibrium and, therefore, more accurately modelsthe appropriate physical processes. An algorithm is developed basedon an approximate Riemann solver for the two-fluid model. Thedevelopment of a preliminary one-dimensional algorithm based on thetwo-fluid plasma model will be presented. The results from the earlyverification and testing of the algorithm and comparisons to MHD testswill also be presented.


Plasmas may be most accurately modeled using kinetic theory. Theplasma is described by distribution functions in physical space,velocity space, and time, f(x, v, t). The evolution of the plasma is thenmodeled by the Boltzmann equation.

Background and Motivation

( )collisions

f f q f ft m tα α α α α

α αα α

∂ ∂ ∂ ∂+ ⋅ + + × ⋅ =∂ ∂ ∂ ∂

v E v Bx v

for each plasma species α = ions, electrons.

Coupled with Maxwell's equations this provides a complete description ofthe plasma dynamics.


● The Boltzmann equation is seven dimensional.

● As a consequence plasma simulations using the Boltzmann equationare only used in very limited applications with narrow distributions,small spatial extent, and short time durations. The seven dimensionalspace is further exacerbated by the high velocity space that is unusedexcept for the tail of the distribution or energetic beams.

● Boundary conditions are difficult to implement in kinetic simulations.

Kinetic Model


● Particle in cell (PIC) plasma model applies the Boltzmann equation torepresentative superparticles which are far fewer (107) than the numberof particles in the actual plasma (1020). The particles fill the sixdimensional phase space and are tracked in time.

● PIC simulations have similar limitations as simulations using kinetictheory due to statistical errors caused by the relatively fewsuperparticles.

● Boundary conditions can also be difficult to implement in PICsimulations.

Particle in Cell (PIC) Model


● Simpler plasma models are generated by taking moments of theBoltzmann equation and averaging over velocity space for eachspecies.

→ Two-Fluid Plasma Model

● The two-fluid equations are combined using simplifying assumptions toform the single-fluid MHD model.

� low frequency, zero electron mass, zero Larmor radii, zero Debyelength

● Plasma simulation algorithms based on the MHD model have beensuccessful in modeling plasma dynamics and other phenomena.

Single-Fluid (MHD) Model


The MHD model has limitations that are introduced in its derivation.

● The generation of non-solenoidal magnetic fields.

● The Hall effect and diamagnetic terms are often neglected. Theseterms represent the separate motions of the ions and electrons andaccount for ion current and the finite Larmor radius of the plasmaconstituents.

� Hall thrusters, MPD thrusters, Lorentz force thrusters

� Anode and cathode fall

Limitations of the MHD Model


In general the Hall terms are difficult to stabilize because they lead to thefaster moving whistler wave branch of the dispersion relation.

The Hall Effect

kz

0 1 2 3 4 5

ωτ

ωτ

ωτ

ωτ A

0

2

4

6

8

10

ωceτe = 0 (MHD)ωceτe = 1ωceτe = 2ωceτe = 10


Hall Effect Physics

● Semi-implicit techniques have been implemented to treat theHall effect terms.*

● The method corrects the ideal MHD evolution of the magneticfield and energy to account for the Hall effect.

● The operator uses a 45 point stencil in 3D.

● The method works for small Hall parameters but becomes slowto converge or unstable for large Hall parameters.

* Harned and Mikic, JCP 83, 1 (1989).


● The two-fluid model is derived by taking moments of the Boltzmannequation for each species.

● The model has the same dimensionality as the MHD model exceptthere are two fluids.

● The only approximation made is local thermodynamic equilibrium ofeach fluid but not with the other fluid.

● The model consists of governing equations for the continuity,momentum, and energy of the electrons and ions. Maxwell�sequations complete the model.

Two-Fluid Plasma Model


Conservation Form

( )

( )

( )

( )

+⋅

+⋅

+×−

+×+

=

+−

+

−−

+

−

⋅∇+

∂∂

e

eie

i

eii

eieee

eiiii

e

eee

i

iii

eee

ee

iii

ii

e

i

e

i

e

i

e

i

en

en

enme

enme

enp

enp

pme

en

pme

en

e

e

nn

t

REj

REj

RBjE

RBjE

j

j

Ijj

Ijj

j

j

jj

00

ε

εεε


Characteristics in 1D

● First order fluxes are split and upwind differenced to form a Roe-type,* approximate Riemann solver

● Hyperbolic equation set, so the flux Jacobian A have completesets of eigenvalues

±−−±=

e

e

e

ex

e

ex

i

i

i

ix

i

ix

mT

enj

enj

mT

enj

enj

35,,

35,λ

0t x t x∂ ∂ ∂ ∂+ = + =∂ ∂ ∂ ∂Q F Q QA

( ) ( )1 1 12

1 12 2i i k i i k ki

k

λ+ ++ = + − −∑F F F l Q Q r

* Roe, JCP 135, 250 (1997).


Approximate Riemann Solver for Two-FluidPlasma

● Overall solution is composed of the solutions to the Riemannproblem defined at the grid interfaces.

● Information propagates along characteristics.

t

x

vi+civi interface

State 1 State 2

vi-ci-ve -ve+ce-ve-ce


Electromagnetic Field Model*

Maxwell�s equations for the electromagnetic fields are required tocompute the RHS and complete the plasma model.

( )/0

/

i e o

o o i e

e n n

t

t

ε

ε µ

∇⋅ = −

∇⋅ =

∂ = −∇×∂∂ = ∇× − −∂

E

B

B E

E B j j

( )

( )

22

2

22

2

/o o i e o

o o o i e

e n nt

t

φφ µ ε ε

µ ε µ

∂∇ − = − −∂∂∇ − = − +∂AA j j

where the magnetic vector potential A and the electric scalarpotential φ are introduced.

* See related poster by Aberle and Hakim.


1-D Electromagnetic Field Model

For the 1-D electromagnetic field model, Maxwell�s equations aresolved

subject to the initial conditions

0

/ i e

t

o otε µ

∂ +∇× =∂

∂ −∇× = − −∂

B E

E B j j

The field equations are solved with a finite-volume, upwind scheme*where the equations are written as

* Shang and Fithen, JCP 125, 378 (1996).

( ) 0/i ee n n ε∇ = −Ei 0∇ =Biand

t xψ∂ ∂+ =

∂ ∂Q F


Treatment of the Two-Fluid Source Terms

( )1

1 12 2

n nn n ni i

ii itψ

+

+ −

− = − − +∆

Q Q F F

t xψ∂ ∂+ =

∂ ∂Q F

The two-fluid equation set contains source terms which result fromthe electromagnetic fields: the Lorentz force and Joule heating.

When the source terms are small, they are calculated explicitly andadded to the update equation.


Treatment of the Two-Fluid Source Terms

( )1

11 12 2

n nn n ni i

ii itψ

++

+ −

− = − − +∆

Q Q F F

( ) ( )11 12 2

1 k k nn k n n ki ii i ik i it t

ψ ψ++ −

∂ − − − = − − − + ∆ ∂ ∆ Q QQ Q F F

Q

When the source terms are large (large Lorentz force), they arecalculated implicitly with an iterative method.

The equation is solved using a symmetric Gauss-Seidel methoduntil k converges to n+1.

The source terms are linearized, and a local source Jacobian is defined.ψ∂ ∂Q


Normalization of the Two-Fluid Model

In the normalization process several important physical parametersare identified.

● Ionization state, Z. (Set to 1 for the simulations presented.)

● Ion / Electron mass ratio, mi/me. (Set to 1836 for the simulationspresented.)

● Ion Larmor radius, rLi, where .

● Debye length, λD.L Th c Thr v v m eBω= =


Benchmarks and Validation Tests

The algorithm described has been applied to several analyticalbenchmarks and validation tests.

● Langmuir Plasma Oscillations

● Debye Shielding

The algorithm has also been applied to the classical MHD shockproblem and reveals an effect of two-fluid physics and thetransition between a gas dynamic shock and an MHD shock.

● MHD Shock Problem


Langmuir Plasma Oscillations

Langmuir plasma oscillations are plasma responses to a displacementperturbation as the plasma maintains quasineutrality. In 1D, theequation of motion for electrons reduces to a second orderdifferential equation for the particle position.

where ωpe is the electron plasma frequency.

2 22

20

epe

e

d x n e x xdt m

ωε

= − = −


Langmuir Plasma Oscillations

t0.0 0.2 0.4 0.6 0.8 1.0

v xe

-8e-8

-6e-8

-4e-8

-2e-8

0

2e-8

4e-8

6e-8

8e-8

x0 50 100 150 200 250

(ne

- ne0

) / n

e0

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

vxe

-1.5e-7

-1.0e-7

-5.0e-8

0.0

5.0e-8

1.0e-7

1.5e-7

nevxe

Error in numericallycalculated ωpe is 0.27%.


Debye Shielding of a Charge Plane

Charge concentrations which would generate an electric field in othermedia are shielded by constituents in a plasma. The plasmaresponse is called Debye shielding and is described by ananalytical expression for the electrostatic potential generated by theparticle charge concentration.

where the Debye length is defined as

( ) 0 expD

xxφ φλ

= −

12

02DkTneελ =


Debye Shielding of a Charge Plane

Simulation withλD = 0.25.

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MHD - Gas Shock Transition

The coplanar Riemann problem (shock tube problem) has beenstudied extensively in gas dynamics with the Euler equations andin plasma dynamics with the ideal MHD equations.*

In this application a discontinuity is initialized in density, pressure, andtransverse magnetic field. The plasma also has a constantlongitudinal magnetic field.

The value of the ion Larmor radius is varied with (vs, vA constant):

� Large values rLi, rLe > L - the plasma is unmagnetized (limit is gasdynamic behavior)

� Small values rLi , rLe < L - the plasma is strongly coupled to themagnetic field (limit is MHD behavior)

� Intermediate values demonstrate the influence of two-fluid effects.

* Brio and Wu, JCP 75, 400 (1988).



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Summary

● The two-fluid plasma model relaxes some of the constrainingassumptions used in MHD and models more complete physics.

● We have developed a new algorithm based on a Roe-type approximateRiemann solver for the two-fluid plasma model.

● We have implemented an electromagnetic field solver and an implicittreatment of stiff source terms.

● The algorithm has been benchmarked to 1D analytical problems andproduces accurate results.

● MHD shock simulations reveal important physical insight anddemonstrate the importance of two-fluid effects.

● For copies of this poster, visit www.aa.washington.edu/cfdlab.

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