LLNL-PRES-700107

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under contract DE-AC52-07NA27344. Lawrence Livermore National Security, LLC

Self-consistent modeling of

multiscale turbulence and

transportJeff Parker

Lawrence Livermore National Laboratory

Lynda LoDestro (LLNL), Frank Jenko (UCLA), Daniel Told (UCLA)Turbulence and Waves in Flows Dominated by Rotation, NCAR, Boulder, CO. August 16, 2016

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Talk Outline

• Background: fusion plasmas

• Gyrokinetic framework for turbulence and multiple-scale formulation

• Numerical method to solve a timescale-separated turbulence & transport system

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Magnetic Fusion Reactors

Plasma must be kept hot so that particles have enough energy to overcome Coulomb repulsion to fuse

Heat loss is primarily due to turbulence

Deuterium

Tritium Neutron14 MeV

Helium3.5 MeV

Poloidal field

magnet

Toroidal field

magnetVacuum

vessel

1021 reactions per second for 1 GW plant

Fusion plasmas: rotating, support a panoply of waves, turbulent

“Tokamak”

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The ITER Project

The ITER Project: A multinational effort to demonstrate the feasibility of fusion energy

Objective: 500 MW of fusion power, with a gain Q=10

Construction in Cadarache, France is underway!

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Gyrokinetic Theory: A theory of low-frequency fluctuations in magnetized plasma

Why use a kinetic description?

6D 5D and eliminate fast gyration timescale

Assuming gyration frequency >> frequencies of interest (turbulence):• Average over the gyromotion: 3D in space, 2D in velocity• Dynamics of charged rings or gyrocenters, not particles

Gyration time ~ 10 nsTurbulence or drift wave time ~ 10 𝜇s Ion collision time ~ 10 ms

Gyrokineticequation

Exact particle orbit

Averaged orbit

Phase-space distribution of particles

𝐹 = distribution of gyrocenters in reduced phase space

Brizard and Hahm, Rev. Mod. Phys. (2007)

Krommes, Ann. Rev. Fluid Mech. (2012)

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Gyrokinetics is widely used

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Towards Comprehensive Simulations of Fusion Plasma Turbulence

• State of the art turbulence simulations (using experimentally measured profiles):

– E.g., Global simulation of turbulence across almost an entire tokamak: 1024 x 32 x 48 x 96 x 64 grid points (Jenko et al., 2013)

• (At least) two goals from simulations:

– Understanding detailed physics

– Quantitative: predicting macroscopic profiles & total turbulent fluxes given heating sources

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Goal: Predict profiles and turbulence

Turbulent fluxes in the core are small enough that they result in long timescales for the evolution of macroscopic profiles, e.g., T(r)

Turbulence time ~ 10 𝜇s Energy confinement time ~ 1 s

Direct numerical integration capturing both turbulence and confinement time scales expensive!

Assuming a separation of timescales exists, how can we efficiently study the self-consistent evolution on the long timescale?

(Cross section of torus)Center line

Closed magnetic flux surfaces

radius

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An “equation-ful” approach

Here, assume one has derived explicit equations describing both the slow evolution and the fast evolution:

Some “equation-free modeling” approaches to multiscale problems do not require that equations are known for the macroscopic variables. E.g., Kevrekidis et al. (2003). Equation-free, coarse-grained multiscale computation: enabling microscopic simulators to perform system-level tasks Comm. Math. Sciences 1(4)

Multiscale gyrokinetics: an ordering scheme in which one can derive closed turbulence & transport equations directly from the 6D plasma Fokker—Planck kinetic equation and Maxwell’s equations.

Sugama and Horton (1997, 1998)Abel et al. (2013)

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Tem

per

atu

re

Implementing Multiscale Gyrokinetics:Initial Approaches with Local Turbulence Models

“Flux tube”

Local approach to simulating turbulence:• Based on a spatial scale separation between equilibrium and fluctuations• Typically, fix the equilibrium profiles (e.g., density, rotation, temperature) and their

gradients at some point• Allows convenient boundary conditions – usually periodic

Tokamaks Rotating Planets Astrophysical Discs

“Beta plane” “Shearing box”

Local Coupling Approach for self-consistent turbulence & transport:• Choose a few (N) radii as the “transport” grid• Run N local simulations and calculate the

turbulent flux• Uses fluxes in the transport equation to evolve

profiles on transport time scaleCandy et al. (2009), Barnes et al. (2010)

TRINITY –Barnes (2010)

Some regimes exist where local approximation is valid

Local simulations vs time

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Global Multiscale Modeling

nG = number of radial grid points in global simulationnL = number of radial grid points in local simulationN = number of distinct local simulations

Global approach: a single global turbulence simulation connected to a transport equation

• Conceptually simpler• Extra difficulties due to alternate numerical methods required, more

careful handling of boundary conditions• Not directly incorporating a separation of spatial scale

• Can capture physics that will be missed entirely in local approach• We will still assume existence of a separation of timescales

• Expected to be cheaper computationally. Computational cost estimate:

GOAL: Self-consistent calculation of macroscopic profiles in tokamaks using global gyrokinetic simulations

Collaboration with Frank Jenko and Daniel Told (UCLA), developers of gyrokinetic code GENE

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Solving the transport equation

Shestakov, Cohen, Crotinger, LoDestro, Tarditi, Xu, J. Comp. Phys. (2003)

Consider the nonlinear transport equation as a paradigm model:

Turbulent flux that depends on the entire profile n(x). From now on, all details of turbulence will be hidden inside Γ[𝑛]

All other local terms and sources

Tokamak plasma turbulence: primarily diffusive, Γ ∼ −𝐷𝜕𝑥𝑛, but with a diffusion coefficient that can depend strongly on the profile gradient: 𝐷 ∼ 𝜕𝑥𝑛

𝑝 (in reality, temperature gradient)

A gradient-dependent diffusion coefficient leads to numerical instability that constrains the timestep to be extremely small even for semi-implicit integration.

To allow for large steps, use a fully implicit timestep.

How to solve this nonlinear equation? Enter the LoDestro Method

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The LoDestro Method: Numerical method for solving the implicitly-stepped transport equation

Shestakov, Cohen, Crotinger, LoDestro, Tarditi, Xu, J. Comp. Phys. (2003)

Key Elements of the LoDestro Method (more detail on next few slides) Represent turbulent flux as diffusive (+ possibly convective)

Picard iteration (no Newton steps) – No Jacobians or Jacobian-vector products

Average over iterates (exponentially weighted moving average) to stabilize the iteration

Computationally advantageous: A transport timestep may finish with a cost comparable to running a single standalone turbulence simulation

Works with either local or global simulations

Most efficient if transport equation is lower dimensionality than turbulent system so implicit timestep of transport equation is quick

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Represent Turbulent Flux as Diffusive and use Picard Iteration

Introduce a subscript representing iteration: when solving for the 𝑚th timestep, let 𝑛𝑚,𝑙 be the 𝑙th iterate. Represent the turbulent flux as diffusive:

This gives a tractable equation to solve for each iterate 𝑛𝑚,𝑙:

Picard iteration:• Diffusion coefficient

evaluated at previous iterate• Gradient at current iterate

The “heart” of the method

Nonlinear equation. How to solve it?

where

Flux computed in a separate turbulence simulation

Note: The turbulent flux Γ need not be represented as purely diffusive. Another scheme with convective flux:

If it converges, it doesn’t matter how you represented the turbulent flux: it’s the right answer

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Contrast with Newton Iteration

A Newton-type of iteration would Taylor expand the flux:

Proceeding in this way requires calculation of Jacobian terms 𝛿Γ/𝛿𝑛. Two problems:

• Computationally expensive to calculate Jacobians – extra runs of turbulence simulations for each forward difference

• Fluxes are intrinsically noisy due to statistical fluctuations of turbulence simulations if not run for extremely long time and time-averaged. Errors are amplified in the calculation of the Jacobian

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Iterate-averaging the flux for stabilization

Consider a model diffusion coefficient 𝐷 ∝ 𝜕𝑥𝑛𝑝. Analyze stability:

The iteration scheme is unstable for 𝑝 > 1 (Shestakov, 2003).

Stabilize the iteration by replacing 𝐷 (or Γ) with an iterate-averaged version 𝐷. Exponentially weighted moving average (equivalent to relaxation)

Stabilizing for 𝐴 < 1. Expect exponential convergence.

Picard iteration

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One solution: Split the turbulent flux into diffusive and convective contributions

What if the effective diffusion coefficient is negative or infinite?

Other Things

Iterating the transport equation concurrently with incremental evolution of the turbulence simulation

After a small update of profiles in turbulence code, compute new fluxes after only a couple eddy turnover times instead of waiting many eddy turnover times to equilibrate

𝑇𝑠𝑠 (time needed to reach statistically

steady state with fixed profiles)

Transport step Δ𝑡 (large)

Turbulencesimulation

Interval of a few eddy times

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Successful testing of the method

Model problem of plasma turbulence in simplified planar geometry: 2D Hasegawa-Wakatani model

Shestakov et al. (2003)

𝑛 − 𝑛𝑏𝑔

Successful comparison of the multiscale coupling method to a direct simulation of the turbulence system for long times

-handled regions of nondiffusive transport

(direct simulation)

(multiscale coupling)

Promising first results with 2D transport coupling:Rognlien et al., Contrib. Plasma Phys. (2004)Rognlien et al., J. Nucl. Mater. (2005)

In progress: coupling transport and 5D gyrokinetic simulations

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Conclusion

How do geophysicists and astrophysicists think about multi-timescale turbulence and transport problems?

• Numerical methods for problems with a separation of timescales?

The LoDestro Method: strategy of implicitly timestepping a nonlinear transport problem in which turbulent flux may depend strongly on profile gradients

Represent turbulent flux as a sum of diffusive and convective pieces Picard iteration, not Newton iteration. No calculation of Jacobians

Work is in progress to implement the LoDestro Method for self-consistent prediction of long-time, large-scale macroscopic profiles with global gyrokinetic simulations