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# 托卡马克的平衡计算 - PowerPoint PPT Presentation

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### 托卡马克的平衡计算

2013.12.18 四室学术报告

Introduction
• Decompose the physics problem by the orders (time order and space order)
• Traditional decomposition of plasma physics (by time order): equilibrium, stability and transport
• Equilibrium is the basis for other problem
• Here the equilibrium means the state of force equilibrium
Introduction (cont.)
• Force balance equation (static momentum equation)
• Force balance equation in 2D form → Grad-Shafranov (G-S) equation (For axis symmetric, in (R,z) coordinate):
• Then the solution of the G-S equation describes the properties of the equilibrium
Equilibrium and poloidal field coils
• Poloidal field coils induct the ohmic plasma current and control the plasma shape
• On EAST
• PF1-PF6, center solenoid, mainly for the ohmic current
• PF7/9, elongation
• PF11,PF13, trianglarity
• PF5, PF7/9, PF11, divertor control

EAST PF coils and plasma

configuration

Properties of equilibrium
• Plasma configuration
• Embedded flux surface
• Plasma geometry
• Divertor configuration
• Profiles (functions of flux surface)
• : pressure
• : no direct physical meaning, but direct in G-S equation
• : safety factor, describe the pitch angle of magnetic field line
• : flux surface averaged parallel current
• , q and are not independent
Fixed boundary and free boundary equilibrium calculation
• Fixed boundary
• The plasma boundary is given, only calculate the plasma configuration inside the plasma
• Easy to calculate, useful for theory study
• Free boundary
• To calculate the configuration outside the plasma boundary
• The current in the PF coils is given
• Complicate but sometimes necessary
• A third kind
• Prescribe a non-fixed plasma boundary
Coordinate system
• Many kinds of coordinate system in tokamak study
• Two major coordinate systems: (R,z) coordinate and magnetic surface coordinate
• coordinate system
• Can handle the X-point

z

R

Mesh in (R,z) coordinate

Coordinate system (cont.)
• Flux surface coordinate system
• coordinate
• Easier, but cannot handle the X-point
• can be
• Orthogonal
• Equal arc length
• ……
• Some coordinate equivalence
• normalized toroidal flux
• normalized volume

Mesh in flux surface coordinate

Equilibrium construction and reconstruction
• Construction
• Generate an equilibrium from given profiles, plasma shape or current in PF coils, and other parameters
• Basis for tokamak design
• Basis for many theory study
• Reconstruction
• Find the experimental equilibrium from the diagnostic data
• Basis for experiments analysis
Equilibrium reconstruction with EFIT
• EFIT is the most popular code for equilibrium reconstruction. Maybe the most popular code in tokamak research area
• Assume a polynomial or spline profiles of P’ and FF’, then iteratively find the coefficient to minimize the error quality function
Different EFIT reconstruction constraints
• At present, EAST only has the magnetic diagnostics and limited kinetic diagnostics
• But we can add some constraints to the current profile
Magnetic diagnostic constraints
• All kinks of magnetic probe and flux loops

Strait (2007)

kinetic profiles on EAST
• Te and ne are from Thomson scattering
• Ti is from the XCS, but only central region data are available. So Ti is scale from Te and assume Ti=Te at the edge region
• First map the data to space, then fit them with tension spline
• Assume flat Zeff=2.5
• At present EAST has no NBI, so the fast ion contribution is neglected

Data and fitting profiles for 38300.3900

Edge current constraint for H-mode plasma
• For H-mode plasma, it is believed that at the edge region, the current is dominated by the bootstrap current
• Sauter bootstrap current model is used to calculate the bootstrap current. Bootstrap current calculation relies on the kinetic profiles (Te, Ti, ne, Zeff)
• Ohmic current

EAST 38300, 3900ms

Typical pressure and current profiles

of H-mode plasma at edge region

Bootstrap current at the edge region

Kinetic equilibrium reconstruction on EAST
• With the constrains of magnetic diagnostics, pressure profile, edge current profile, we achieved the kinetic equilibrium
• The current/q profiles at the central region are not reliable, though we have the global li constrain

38300, 3900ms

Pressure, current profiles and configuration from kinetic EFIT and magnetic EFIT

Equilibrium construction
• Lots of codes for equilibrium construction, most of them are fixed boundary codes
• EFIT, CORSICA/TEQ, TOQ, ESC, JSOLVER ……
• CORSICA
• CORSICA has both direct and inverse solver
• Inverse solver: coordinate, solve for R, Z, fixed boundary
• Direct solver: coordinate, solve for , free boundary
• CORSICA can easily change the plasma shape and profiles
Construct self-consistent equilibrium
• To construct a self-consistent equilibrium, the self-consistent plasma shape and profiles must be given
• Self-consistent profiles:
• Bootstrap current dominated edge current
• Self-consistent pedestal height and width, EPED model
• EPED model (peeling-ballooning model + kinetic ballooning model, ELITE+BALOO) has successfully predict the pedestal height and width
• This technic could be useful for EAST and CFETR