FLOW: an equilibrium code for rotating anisotropic axysimmetric plasmas

1. FLOW: an equilibrium code for rotating anisotropic axysimmetric plasmas L. Guazzotto, T. Gardiner and R. Betti Laboratory for Laser Energetics University of Rochester NSTX Theory Meeting Princeton Plasma Physics Lab. September 9-13, 2002,Princeton NJ

2. Outline The code Flow:►) The system of equations►) The numerical solution NSTX-like equilibria with toroidal flow NSTX-like equilibria with poloidal flow Conclusions

3. The relevant equations • Continuity: Momentum: Maxwell equations:

4. The previous system of equations can be reduced to a “Bernoulli” and a “Grad-Shafranov” equation: • Faraday’s law yields the plasma flow: The φ-component of the momentum equation gives an equation for the toroidal component of the magnetic field: The B-component of the momentum equation reduces to a “Bernoulli-like” equation for the total energy along the field lines: E. Hameiri, Phys. Fluids (1983)

5. The modified Grad-Shafranov equation  • Finally, the Ψ-component of the momentum equation gives a “GS-like” equation: W(ρ,B,Ψ) is the enthalpy of the plasma, and its definition depends on the description of the plasma (MHD, CGL, or kinetic) I(Ψ), Φ(Ψ), Ω(Ψ), H(Ψ), (p||/ Ψ), (W/ Ψ) are free functions of Ψ R. Iacono et al., Phys. Fluids B (1990)

6. The code input requires a “user friendly” set of free functions. • The input is a set of free functions representing quasi-physical variables D(Ψ) Quasi-Density Quasi-Parallel pressure P||(Ψ) P(Ψ) Quasi-Perpendicular pressure Quasi-Toroidal magnetic field B0(Ψ) Mθ(Ψ) Quasi-Poloidal sonic Mach number Mφ(Ψ) Quasi-Toroidal sonic Mach number

7. The numerical algorithm: the multi-grid solver • The Bernoulli equation is solved for  • The Grad-Shafranov equation is solved for Ψ using a red-black algorithm • If the system is anisotropic, the equation for Bφ is also solved • The procedure is repeated until convergence, then the solution is interpolated onto the next grid

8. NSTX-like equilibria with toroidal flow The following set of free functions is used as input to compute anisotropic equilibria with toroidal flow DC = 3.4*1019 [m-3] B0C = 0.29 [T] 0  MC  1 T = 9% I = 0.9 [MA] R0 = 0.86 [m] a = 0.69 [m] k = 1.9 M. Ono et al., Nucl. Fusion (2001)

9. MC = 0 MC = 0.4 MC = 0.8 MC = 1 4E+19 MC = 1 3E+19 MC Density [m-3] MC 2E+19 1E+19 0.5 1 1.5 R [m] The centrifugal force causes an outward shift of the plasma

10. 5E+19 Θ = 0 Θ = 0.2 Θ = 0.6 Θ = 1 4E+19 3E+19 Density [m-3] 2E+19 1E+19  0.5 1 1.5 R [m] The parallel anisotropy (p|| > p) causes an inward shift

11. MC=1,Θ=0 MC=0,Θ=1 5E+19 MC=1,Θ=0.2 MC=1,Θ=0.6  MC=1,Θ=1 4E+19 3E+19 Density [m-3] 2E+19 1E+19 R [m] 0.5 1 1.5 Flow and parallel anisotropyhave opposite effect

12. FLOW can also be applied to equilibria with poloidal flow • The free function determining the poloidal flow is M(Ψ) representing (approximately) the sonic poloidal Mach number (poloidal velocity/ poloidal sound speed). Poloidal sound speed = CsB/B. • Radial discontinuities in the equilibrium profiles develop when the poloidal flow becomes transonic (M(Ψ) ~ 1). • As the poloidal sound speed is small at the plasma edge, transonic flows may develop near the edge. • FLOW can describe MHD or CGL equilibria with poloidal flow. R. Betti, J. P. Freidberg, Phys. Plasmas (2000)

13. 130 Discontinuity Subsonic 120 Transonic 110 Poloidal sound speed 100 10 90 9 80 70 8 V [km/s] 60 7 50 6  [%] 40 Subsonic 5 30 Transonic 4 20 10 3 0 2 0.5 1 1.5 R [m] 1 0 0.5 1 1.5 R [m] NSTX-like equilibria with poloidal flow can exhibit a pedestal structure at the edge

14. Conclusions • The code FLOW can compute axysimmetric anisotropic equilibria with arbitrary flow. • MHD, CGL, and kinetic closures are implemented. At the moment, poloidal flow is described only by the MHD and CGL closures. • Fast-rotating anisotropic NSTX-like equilibria have been computed with FLOW. • Future work: develop a stability code for NSTX rotating equilibria computed with the code FLOW.