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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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m agneto h ydro d ynamics numerical simulation for feedback control of the resistive wall mode rwm

MagnetoHydroDynamics:Numerical Simulation for Feedback Control of the Resistive Wall Mode(RWM)

Kevin Durand

May 17,2007

22.012 Fusion and Plasma Physics Seminar

slide2
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

mhd complications
MHD Complications
  • Resistive Wall Mode (RWM)
  • Edge Localized Mode (ELM)
  • Neoclassical Tearing Mode (NTM)

22.012 Fusion and Plasma Physics Seminar

resistive mhd
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

computational mhd and computation fluid dynamics cfd
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

numerical methods
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

simple cfd example flow over an airfoil
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

simulation of rwm in an advanced tokamak
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

slide9

Computational MHD

Simulation of the RWM

MHD eigenmodes: Computed magnetic field

perturbation for an MHD instability

22.012 Fusion and Plasma Physics Seminar

feedback control of the rwm instability
Feedback Control of the RWM Instability
  • Stability achieved using dynamic compensation (feedback control) of active coils

22.012 Fusion and Plasma Physics Seminar

system dynamics model
System Dynamics Model

22.012 Fusion and Plasma Physics Seminar

problems
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

acknowledgments
Acknowledgments

Professor Molvig

Professor Friedberg

Professor Hutchinson

22.012 Fusion and Plasma Physics Seminar

sources
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