Combining Braginskii Closures with Magnetic …...Combining Braginskii Closures with Magnetic...

Combining BraginskiiClosures with MagneticPresheath Boundary

ConditionsNIMROD Team Meeting 2019-04-13

K.J. Bunkers & C.R. Sovinec

University of Wisconsin-Madison

Outline

1 Magnetic Presheath Model

2 Braginskii Model

3 Braginskii Calculations

4 Conclusions and Future Directions

K.J. Bunkers, C.R. Sovinec

Table of Contents


2 Braginskii Model




More detailed modeling of sheath physics provides a set ofboundary conditions that can be put in an MHD form.

A sheath boundary condition model has been successfullydeveloped for a fluid turbulence code.1

The edge can be divided into presheath, magnetic presheath(MPS), and sheath regions.

Figure of edge (not to scale) from Stangeby’s The Plasma Boundary of Magnetic Fusion Device (2000).

1Loizu, Ricci, Phys. Plasm. 19 (2012) 122307K.J. Bunkers, C.R. Sovinec

The boundary equations that are derived are adapted toNIMROD.

In the zeroth order (ε = ρs/L) cold ionapproximation , with V the ion velocity andcs =

√Te/mi the sound speed

n · Vwall = cs n · bwall , n · ∇Te = 0

Te

nqn · ∇n = n · ∇φ = −mics

qn · ∇(V · b) ∼ 0

n · J = qncs sinα(1− exp[Λ− η])

Here Λ = ln( mi

2πme) and η is the normalized

potential relative to the wall.

Strauss2has considered sheath compatibleboundary conditions and implemented aNeumann velocity boundary condition witheffects similar to this Chodura-Bohm criterion.

Magnetic presheathcoordinate directions.

2Strauss, Phys. Plasm. 21 (2014) 032506.K.J. Bunkers, C.R. Sovinec

The variation of the electric potential along the wall isshown to be negligible.

For a thin wall, the potential along the wall φ is governed by∂2φ∂x2 = − η

ρsJn,MPE where η is the electrical resistivity and Jn,MPE is the

current normal to the wall at the magnetic presheath entrance.

Using two Gaussians ∼ e(x−xc )2/λ so that the total normal current is zero,models the Jn,MPE from our forced VDEs well.

Using realistic values leads to potential variations shown below. Theylead to very small changes in ion velocities that can be safely ignored.

vE×B = 0.2 V(0.1 m)(1 T) ≈

2 m/s cs ≈ 1× 104 m/s


The varied boundary conditions affect the current decayand whether it rises at the beginning of the computation.

The plot shows four singletemperature computations andone two-temperaturecomputation with labels asdescribed above.

The two-temperaturecalculation uses an insulatingTe condition and Dirichlet Ti

condition.

The insulating Te

condition comes from theMPS theory.The Dirichlet Ti conditionis used assuming that ionsare cooled as theyaccelerate.


Table of Contents


2 Braginskii Model




The thermal diffusivities are calculated using the Braginskiimodel.

The Braginskii model should apply well in the sheath region where thelow temperatures have smaller mean free paths.

The thermal diffusivity forms are shown below with the magnetizationcoefficient xs = Ωsτs .χ‖,s =

Tsτs

ms

γ0,s

δ0,s

χ⊥,s =Tsτs

ms

γ1,s x2s + γ0,s

x4s + δ1,s x2

s + δ0,s

τe =12π3/2√me (kBTe )3/2ε2

0√2nee4 ln Λ

τi =12π3/2√mi (kBTi )

3/2ε20

nee4 ln Λ


Braginskii coefficients have huge variation over the plasmaregion.

The parallel diffusivity becomes extremely large in hightemperature regions, while the perpendicular peaks in arelatively low temperature region.

The ratio of χ⊥/χ‖ becomes extremely small in the mainplasma region.


There are some caveats to the Braginskii calculations.

The finite magnetization model for 0.3 < x < 30, has a κ⊥fractional error of up to 60%.3

Of minor concern is that the thermoelectric coefficients do notgo to the correct limit.

Be careful of γ − 1. Depending on implementation, may needto give input of (γ − 1)χ instead of χ.

The calculated Braginskii thermal coefficients are small, andprobably only applicable in the edge plasma.

3Epperlein and Haines, The Physics of Fluids 29, 1029 (1986)K.J. Bunkers, C.R. Sovinec

Can you see the dotted line?

κ⊥


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2 Braginskii Model




An equilibrium with a forced VDE is used to investigatethe effects of various boundary conditions.

The equilibrium pressure with poloidalflux contours for the central plasmaregion is shown in the upper right.

An example of the domains for resistivewall calculations is shown in the lowerright.

The VDE is created by turning off verticalstability coils and letting the inducedcurrents decay on the resistive wall.

For the first set of computations we usean anisotropic thermal conductivity modelwith k‖/k⊥ = 103.

A Spitzer resistivity model is used whereη(T ) = η0T

−3/2.


Braginskii coefficients are chosen so that they match acase corresponding to a physically realizable case.

The normalized coefficients have S = 1.3× 106 which is usedto choose a physically realizable case with

n = 1× 1019 m−3 R0 = 0.4 m a = 0.25 m B0 = 0.5 T

mi = 2mp = 3.3× 10−27 kg me = 9.1× 10−31 kg log Λ = 15

This leads to a peak temperature of 260 eV, resistive timeτr = 0.2 s, and Alfven time τA = 1.6× 10−7 s.

Thus previous fixed thermal coefficients corresponded toχ⊥ ∼ 300 m2/s.

The base Braginskii model yields at T = 1 eV thatχ⊥ ∼ 1× 10−2 m2/s. Generally χ⊥ & 1 m2/s so the model isadjusted by multiplying χ⊥ by 100 for Braginskii calculations.

Computations with the Braginskii closures presented here haveDirichlet conditions on Te and Ti .


With two-temperature Dirichlet conditions the Braginskiiresult decays more quickly than the fixed diffusioncoefficients case.

In both cases, electroninternal energy decays morequickly than the ion internalenergy.

Electrons have a higherparallel electron thermaldiffusivity in general.

The Braginskii case alsoshows a flatter current in theinitial stages of the VDE.


The Te insulating and Ti Dirichlet case shows an electronheating problem.

Electron heating tends tooccur in private flux regions.

This behavior is robust tochanging time steps,thermal diffusivities, andelectrical resistivity.

It also is occurring in stableequilibria (not forced VDE,as to the right).

This usually occurs within100τA.


The cause is likely due to electron velocity.

When calculating the electron velocity for separatetemperature calculations, the electron velocity depends onelementary charge e.

Ve = V − J

ne(1 + memi

)

Vi = V +Jmemi

ne(1 + memi

)

When using larger e (reducing J/(ne) contribution), thecalculations do not show the electron heating problem.


The large J that occur in this region during problematicperiods is consistent with this interpretation.

The J contours are shown below when the temperature isblowing up. For these calculations Vφ is normalized by Alfventimes and the sound speed should be about 0.05. SoJ ≈ 0.005 if moving near sound speed in cold region.


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2 Braginskii Model




Conclusions and Future Work

VDE calculations are sensitive to boundary conditions ontemperature, number density, and velocity.

Braginskii model appears to work, but the insulating electrontemperature boundary condition is posing difficulties.

The MPS condition seems fairly robust in axisymmetriccalculations.


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