托卡马克的平衡计算

李国强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• For free boundary calculation• Can handle the X-point

Mesh in (R,z) coordinateR

z

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

Diagnostics Constraints Yield

Magnetic

Magnetic loops and probes

Ip, poloidal flux, external magnetic field

Plasma current, plasma shape, internal inductance, betap, edge current profile

Current

Motional Stark Effect or Li Beam, SXR…

Internal magnetic field or flux surface

Magnetic surface, current profile, safety factor profile

Kinetic

Thomson scattering, ECE, CER, XCS, FADI, nubeam calculation…

Pressure profile (from Te, Ti, ne, Zeff, Pf profiles)

Pressure profile

• 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

Typical pressure and current profilesof H-mode plasma at edge region

Bootstrap current at the edge region

EAST 38300, 3900ms

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

Thank you