Computations of Vector Potential and Toroidal Flux and Applications … · 2017. 5. 10. · Project Goals and ProgressVector Potential CalculationToroidal Flux CalculationAdditional

Computations of Vector Potential and Toroidal Flux andApplications to Stellarator Simulations

NIMROD Team Meeting

Torrin Bechtel

April 30, 2017

Project Goals and Progress Vector Potential Calculation Toroidal Flux Calculation Additional Computations

Outline

1 Project Goals and Progress

2 Vector Potential Calculation

3 Toroidal Flux Calculation

4 Additional Computations


Table of Contents






Purpose

To study magnetic topology evolution and plasma confinement in stellaratorswith heating and eventually flow sources.

Goals:

Study high beta effects in toroidal, not helically symmetric plasmas

Studying magnetic geometries with a variety of stability properties

Perform rigorous convergence analyses

Benchmark with HINT2 code

Investigating the effects of plasma flow


Beta Limits Have Been Studied with HINT2

Pressure profile is fixed as p = p0(1− s)(1− s4).At blue circle J× B = ∇p can no longer be satisfied on stochastic fieldlines and pressure profile must be released → soft beta limit.At green circle hard beta limit is hit as axis is pushed into separatrix.

Y. Suzuki, et al. IAEA FEC 2008, TH/P9-19


Equilibrium Beta Limit Observed to Depend on Conduction Anisotropy

MHD equilibria are produced by heating from vacuum with zperiod limitedFourier spectrum.

Beta limit is observed as time step crash at higher heating.

Beta varies strongly with conduction anisotropy.


Thermal Conduction is Well Converged

Converged reference has 21 modes, 24x24 grid, poly degree = 5.

Separate tests have been run with decreased dt, increased nmodes, andincreased poly degree.

〈β〉 varies by at most 3% with increased resolution.Tests with eqn model = tonly are consistent.


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Potential is Computed Using iter solve

The equations for the potential in the Coulomb Gauge (for uniqueness)

∇× A = B, ∇ · A = 0,

are solved in NIMROD’s framework by formulating the problem in terms of anartifical time

∂A

∂t= c1∇(∇ · A)− c2∇× (∇× A− B).

This has the same form as the pertrubed magnetic field advance in NIMROD ifthe electric field is modified to have the form E = elecd [∇× B̃− (Beq + ṽ)]with uniform elecd which gives

∂B̃

∂t= kdivb∇(∇ · B̃)− elecd∇× [∇× B̃− (Beq + ṽ)].


Equation Must Be Weighted Appropriately for Accuracy and Convergence

The choice of coefficients dt, c1, and c2 will alter the matrix problem beingsolved.

c = c1 = c2 is beneficial for matrix condition.

c � 1/dt reduces effect of artificial time (mass) but worsens matrixcondition.

|B−∇× A| is output to ensure sufficient accuracy.

Solver has been fully implemented in nimplot mgt.f 90 undercompute potential but is currently only used in 3D toroidal fluxcalculation.


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Toroidal Flux is Integrated With lsode

In order to compare with HINT2 we need to know T (ψ).

We can compute ψ in 3D geometry using the vector potential, A, and StokesTheorem.

ψ =

∫ ∫B · dS =

∫ ∫(∇× A) · dS =

∮A · d`

To compute∮A · d` we need a path encircling a poloidal cross section. The

differential equations defining a fieldline in 3D geometry are

dR

BR=

dZ

BZ=

R dφ

Bφ=

dL

|B| =dr

Br=

r dθ

Bθ

(=

dη

Bpol2D only

).

Choosing ` = θ̂ we can use these equations to convert the integral to

ψ =

∮A · dθ̂ =

∮Aθr dθ =

∫Aθ

Bθ|B| dL ,

where the path L is determined from the first 4 fieldline equations and θ istracked to determine a stopping criteria.


Current Implementation Has Issues

Toroidal flux should always be zero at the magnetic axis, but for some reason itappears to vary with the axis position.


Alternate Toroidal Flux Calculation

The value of ψ can be computed by quadrature over triangles bounded byfieldline traces in a poloidal plane.

This method also has pitfalls.


Comparison Shows...

The flux function from fieldline tracing has been shifted down and both havebeen normalized in the second plot.


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Behavior of Temperature on Closed Flux Surfaces is Not Intuitive

On closed flux surfaces we expect, χeff ≈ χ⊥. However, the temperature profilein these regions is affected by changes in χ‖.

This has prompted further investigation.


Simple Estimate of Effective Conduction

In steady state the heat source, Q, balances the thermal conduction

Q = ∇ · (χeff ·∇P).

Integrating and applying the Divergence Theorem∫Q dV =

∫∇ · (χeff ·∇P)dV =

∮(χeff ·∇P) · dS .

Assuming a uniform heating source and that χeff can be reduced to a scalarand choosing S to be a poloidal cross section we have

χeff =Q∫

dV∮∂P

∂φdS

.


Triangle Quadrature is Not Effective


Extra: Volume Triangulation


Extra: Other Triangulated Surfaces


Extra: HINT2

Solves for MHD equilibrium by relaxing initial condition.

Toroidal coordinates make no assumption about magnetic geometry.

Uses 4th order spatial finite differencing and RK4.

Project Goals and ProgressVector Potential CalculationToroidal Flux CalculationAdditional Computations

Documents

Computations of Vector Potential and Toroidal Flux and Applications … · 2017. 5. 10. · Project Goals and ProgressVector Potential CalculationToroidal Flux CalculationAdditional