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### 托卡马克的平衡计算

2013.12.18 四室学术报告

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• At present, EAST only has the magnetic diagnostics and limited kinetic diagnostics

• But we can add some constraints to the current profile

• 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

• 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

• 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

• 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