Computations of Vector Potential and Toroidal Flux and Applications to Stellarator Simulations
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1 Computations of Vector Potential and Toroidal Flux and Applications to Stellarator Simulations NIMROD Team Meeting Torrin Bechtel April 30, 2017
2 Outline 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations
3 Table of Contents 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations
4 Purpose To study magnetic topology evolution and plasma confinement in stellarators with 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
5 Beta Limits Have Been Studied with HINT2 Pressure profile is fixed as p = p 0(1 s)(1 s 4 ). At blue circle J B = p can no longer be satisfied on stochastic field lines 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
6 Equilibrium Beta Limit Observed to Depend on Conduction Anisotropy MHD equilibria are produced by heating from vacuum with zperiod limited Fourier spectrum. Beta limit is observed as time step crash at higher heating. Beta varies strongly with conduction anisotropy.
7 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, and increased poly degree. β varies by at most 3% with increased resolution. Tests with eqn model = tonly are consistent.
8 Table of Contents 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations
9 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 an artifical time A = c1 ( A) c2 ( A B). t This has the same form as the pertrubed magnetic field advance in NIMROD if the electric field is modified to have the form E = elecd [ B (B eq + ṽ)] with uniform elecd which gives B t = kdivb ( B) elecd [ B (B eq + ṽ)].
10 Equation Must Be Weighted Appropriately for Accuracy and Convergence The choice of coefficients dt, c 1, and c 2 will alter the matrix problem being solved. c = c 1 = c 2 is beneficial for matrix condition. c 1/dt reduces effect of artificial time (mass) but worsens matrix condition. B A is output to ensure sufficient accuracy. Solver has been fully implemented in nimplot mgt.f 90 under compute potential but is currently only used in 3D toroidal flux calculation.
11 Table of Contents 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations
12 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 Stokes Theorem. ψ = B ds = ( A) ds = A dl To compute A dl we need a path encircling a poloidal cross section. The differential equations defining a fieldline in 3D geometry are dr = dz = R dφ = dl B R B Z B φ B = dr = r dθ ( = dη ) 2D only. B r B θ B pol Choosing l = ˆθ we can use these equations to convert the integral to B θ ψ = A dˆθ = A θ r dθ = A θ B dl, where the path L is determined from the first 4 fieldline equations and θ is tracked to determine a stopping criteria.
13 Current Implementation Has Issues Toroidal flux should always be zero at the magnetic axis, but for some reason it appears to vary with the axis position.
14 Alternate Toroidal Flux Calculation The value of ψ can be computed by quadrature over triangles bounded by fieldline traces in a poloidal plane. This method also has pitfalls.
15 Comparison Shows... The flux function from fieldline tracing has been shifted down and both have been normalized in the second plot.
16 Table of Contents 1 Project Goals and Progress 2 Vector Potential Calculation 3 Toroidal Flux Calculation 4 Additional Computations
17 Behavior of Temperature on Closed Flux Surfaces is Not Intuitive On closed flux surfaces we expect, χ eff χ. However, the temperature profile in these regions is affected by changes in χ. This has prompted further investigation.
18 Simple Estimate of Effective Conduction In steady state the heat source, Q, balances the thermal conduction Q = (χ eff P). Integrating and applying the Divergence Theorem Q dv = (χ eff P) dv = (χ eff P) ds. Assuming a uniform heating source and that χ eff can be reduced to a scalar and choosing S to be a poloidal cross section we have χ eff = Q dv. P ds φ
19 Triangle Quadrature is Not Effective
20 Extra: Volume Triangulation
21 Extra: Other Triangulated Surfaces
22 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.
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