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Kevin Durand May 17,2007

M agneto H ydro D ynamics: Numerical Simulation for Feedback Control of the Resistive Wall Mode(RWM). Kevin Durand May 17,2007. MHD. Dynamics of electrically conducting fluids Concern specific to plasma is that the equations give equilibrium and stability conditions

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Kevin Durand May 17,2007

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  1. MagnetoHydroDynamics:Numerical Simulation for Feedback Control of the Resistive Wall Mode(RWM) Kevin Durand May 17,2007 22.012 Fusion and Plasma Physics Seminar

  2. MHD • Dynamics of electrically conducting fluids • Concern specific to plasma is that the equations give equilibrium and stability conditions • Motivation: Avoid major disruptions • Navier-Stokes equations: Fluid Dynamics • Maxwell’s equations: Electromagnetism 22.012 Fusion and Plasma Physics Seminar

  3. MHD Complications • Resistive Wall Mode (RWM) • Edge Localized Mode (ELM) • Neoclassical Tearing Mode (NTM) 22.012 Fusion and Plasma Physics Seminar

  4. Resistive MHD • Ideal MHD vs. Resistive MHD • Resistive MHD is of concern to numerically simulate the Resistive Wall Mode (RWM) in advanced tokamaks • Resistive Wall Mode slowly grows to create instability in steady-state operation • Resistive MHD Extended model • This includes an extra term in Ampere's Law which models the collisional resistivity • Resistivity of the plasma serves as a diffusion constant 22.012 Fusion and Plasma Physics Seminar

  5. Computational MHD and Computation Fluid Dynamics(CFD) • CFD and MHD related: Both use Navier-Stokes but differ in other necessary state variables • CFD started in 60’s for NASA and military aircraft development • Both involve solving non-linear, 3-dimensional, complex differential equations simultaneously 22.012 Fusion and Plasma Physics Seminar

  6. Numerical Methods • Finite Volume Method (Most Popular) • Also can use Finite Element or Finite Difference • Discretize control volumes • Break up into a 2-dimensional grid or 3-dimensional mesh • State variables remain conservative via Finite Volume Method • Governing equations (Q) and fluxes leaving control volumes (F) 22.012 Fusion and Plasma Physics Seminar

  7. Simple CFD Example: Flow over an Airfoil • Matlab implementation of Navier-Stokes equations for a NACA 0012 airfoil at 10 degrees angle of attack Simulation Example 22.012 Fusion and Plasma Physics Seminar

  8. Simulation of RWM in an Advanced Tokamak • Much harder than previous CFD example • 3-D mesh + Resistive MHD model • In order to achieve high Beta plasmas in advanced tokamaks, we need to characterize system dynamics via simulation of unstable wall modes • Use Computational MHD 22.012 Fusion and Plasma Physics Seminar

  9. Computational MHD Simulation of the RWM MHD eigenmodes: Computed magnetic field perturbation for an MHD instability 22.012 Fusion and Plasma Physics Seminar

  10. Feedback Control of the RWM Instability • Stability achieved using dynamic compensation (feedback control) of active coils 22.012 Fusion and Plasma Physics Seminar

  11. System Dynamics Model 22.012 Fusion and Plasma Physics Seminar

  12. Problems • Sensor Noise at low frequency • Time delay of sensors • Minimize using internal poloidal and toroidal sensors • Control unstable modes and hopefully never enter modes unreachable using the controller • Propotional+Derivative Control (PID) • K_RWM= K+ (K_i)/s 22.012 Fusion and Plasma Physics Seminar

  13. Acknowledgments Professor Molvig Professor Friedberg Professor Hutchinson 22.012 Fusion and Plasma Physics Seminar

  14. Sources • “Feedback control of resistive wall modes in torodial devices”. Nucler Fustion, 44, pg 77-86, Dec. 2003. • “Active control of resistive wall modes in the large-aspect-ratio tokamak”. Nuclear Fusion, 42, pg 768-779, June 2002 . • http://www.elmagn.chalmers.se/elmagn/CEMgroup/MHD/ • http://flash.uchicago.edu/~emonet/astro/mhd/index.html • http://en.wikipedia.org/wiki/Magnetohydrodynamics • http://en.wikipedia.org/wiki/Computational_fluid_dynamics 22.012 Fusion and Plasma Physics Seminar

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