Challenging problems in kinetic simulation of turbulence and transport in tokamaks

Yang Chen

Center for Integrated Plasma StudiesUniversity of Colorado at Boulder

Outline

• Gyrokinetics• The δf-method for Particle-in-Cell simulation• Problems with kinetic electrons• A fluid electron model for energy particles

driven modes• Problems with long wavelength radial Er• Problems with transport time scale simulation

The task: understanding and predicting tokamak transport

• Neo-classical transport: collisional transport enhanced by toroida geometry

• Anomalous transport: induced by small scale turbulence (drift waves, micro-tearing modes)

GEM

• Is a semi-explicit δf particle-in-cell gyrokinetic turbulence code that includes full electron dynamics and shear Alfvenic magnetic perturbation, general magnetic field geometry and multiple species.

Gyrokinetic Equations

Gyrokinetic ordering

Reduce 6D to 5D

Eliminate high frequency waves

The δf-method

Conventional PIC (full-f):

Define weight:

δf method reduces the particle number by |δf/f|2

Outline

• Gyrokinetics• The δf-method for Particle-in-Cell simulation• Problems with kinetic electrons• A fluid electron model for energy particles

driven modes• Problems with long wavelength radial Er• Problems with transport time scale simulation

Problems with kinetic electrons

Use canonical momentum

To avoid time derivative

Ampere’s eqn:

Quasi-neutrality:

Direct implementation is numerically unstable due to Courantcondition

The split-weight scheme

Split-weight scheme to increase Δt:

(Manuiskiy and Lee 2000, ε=1). Originally thought to allow large Δt because electrons are almost adiabatic. Wrong!

Need an equation for

GK Poisson equation

Vorticity equation

Why is the split-weight scheme mode stable? by solving more equations. These eqns are compatible only for the physical modes. For grid-scale numerical modes the two sets of eqns are not equivalent.

Explicit finite difference in t still unstable

Outline

• Gyrokinetics• The δf-method for Particle-in-Cell simulation• Problems with kinetic electrons• A fluid electron model for energy particles

driven modes• Problems with long wavelength radial Er• Problems with transport time scale simulation

Mass-less fluid electronsElectron continuity equation:

Evolution equation for vector potential:

Ohm’s law:

Obtain from quasi-neutralityGyrokinetic thermal ions and alpha particles ->

Solve Ampere’s law backwards:

N=10 mode poloidal structure in ITER

Close the fluid electron model with particles

Ion terms

The complete Ohm’s equation is obtained by combining Ampere,Faraday’s law and velocity moment of GK equations:

Electron pressureElectron inertia

--- electron distribution from drift-kinetic equation

Closure scheme efficient for shear Alfven

Shear Alfven wave

ITG

For shear Alfven, closure scheme also allows larger ΔtHowever, the split-weight scheme more accurate for drift wavesBoth solve one additional equation to be stable

Outline

• Gyrokinetics• The δf-method for Particle-in-Cell simulation• Problems with kinetic electrons• A fluid electron model for energy particles

driven modes• Problems with long wavelength radial Er• Problems with transport time scale simulation

Problem with long wavelength Er

Radial Electric field is important for both neoclassical transport and for regulating turbulence as sheared zonal flows

Use MHD to determine the ion polarization density:

Quasi-neutrality condition:

But the ion gyrokinetic equation is only first order accurate!(Parra and Catto, 2008—2010)

To the first order guiding center variables are defined as

To the second order:

Littlejohn, J. Plasma Phys. 29, 111 (1983)

, etc.

Other problems with ion gyrokinetics

• GK equation not (completely) derived for -- the transport barriers or at the edge, equilibrium scale length not

much larger than ρi

-- cases of strong ExB flow,

• For ETG, , GK ions are valid but requires many points to do gyro-averaging

• For small devices (e.g. NSTX), time step of GK simulation

• It is possible to follow gyro-motion with comparable time steps

Solution: Return to Lorentz ions

• Quasi-neutrality with np double counts• Implicit δf algorithm developed (Chen & Parker POP 2009)• Being Used to study tearing mode (J. Cheng, to be submitted)• To be implemented in toroidal geometry

Outline

• Gyrokinetics• The δf-method for Particle-in-Cell simulation• Problems with kinetic electrons• A fluid electron model for energy particles

driven modes• Problems with long wavelength radial Er• Problems with transport time scale simulation

Challenges in transport time scale simulation

Current Transport Scalemodeling is based onmultiscale expansion

--TGYRO (based on GYRO)--TRINITY (besed on GS2)

Local fluxtube simulations to get transport coefficientsSolve 1D transport equation to evolve profiles

M. Barnes, POP 2010

M. Barnes, POP 2010

But scales are not well separated!

For ITER,

However, physical particle and energy sources have to be used,

and the Coarse Graining Procedure is needed to control noise

In principle, profile evolution is modeled if the simulationIs extended to transport time scale.

What is the turbulence problem, anyway?

• The turbulence problem is usually understood as: given density and temperature profiles, determine the anomalous transport coefficients.

• In practice, the only well-defined problem is the local problem:

• But if profile effects are important, i.e. no scale separation, how to define the turbulence problem?

• Or equivalently, what is the right source to use in a global simulation that prevents profile relaxation, so that a steady state is obtained?

Summary on code development

• Current features: unique algorithm for finite-beta/kinetic electrons. General equilibrium profiles and flux surface shapes, a hybrid GK ion/fluid electron model for TAEs, etc.

• Future work: 2-D domain decomposition, kinetic electron closure for the fluid electron model. Lorentz ion/drift kinetic electron in torus.