Gyrokinetics & GS2, a physical picture of “bad-curvature”, flux-tube/global, geometry...

G.W. Hammett, Princeton Plasma Physics Lab http://w3.pppl.gov/∼hammett GS2 Users Workshop PPPL Jan 8-10, 2002

GS2 nonlinear gyrokinetic code http://gk.umd.edu http://gs2.sourceforge.net

Plasma Microturbulence Project http://fusion.gat.com/theory/pmp Definition of Poisson bracket

Bill wrote convective time derivative in the gyrokinetic equation including the nonlinearity as:

\frac{df}{dt} = \frac{∂f}{∂t} + \{χ, f\}

Simplifying to χ = Φ (the electrostatic potential), the “Poisson bracket” can be written as

\{Φ, f\} = \frac{∂Φ}{∂x} \frac{∂f}{∂y} − \frac{∂Φ}{∂y} \frac{∂f}{∂x}
= ẑ × ∇Φ · ∇f
= c\frac{E × B}{B^{2}} · ∇f
= v_{E×B} · ∇f

(1)

(once normalizations put back in).

Borrowing from previous slides:

http://w3.pppl.gov/∼hammett/fsp-gk.pdf

p.1-9: simple physical picture of ”bad-curvature” drive p. 15 small scale fluctuations p. 17 geometry start Aligning coordinate system with magnetic field reduces resolution requirements by a factor of ρ_{∗} = ρ/L. Very efficient. http://w3.pppl.gov/∼mbeer/ictp1.ps p.17/18 flux tube coordinates and annular toroidal wedge equivalent If flux tube has a toroidal width of 2π/n_{0}, and a radial extent from the q = q_{1} surface to the q = q_{1} + m_{0}/n_{0}, then the flux-tube domain is mathematically exactly equivalent to simulating a toroidal wedge of a toroidal annulus (a slice of a hollow Bundt cake). Toroidal periodicity on wedge means effectively simulating n = 0, n_{0}, 2n_{0}, 3n_{0}, . . .. I.e., n = 87 mode similar to n = 86 mode, can use coarser grid in k_{θ} = m/r = nq/r. Example: n_{0} = 10, radius from q = 2.2 to q = 2.3, k_{θ}ρ ≈ (m/r)ρ = 0.1, 0.2, 0.3, … for a/ρ = 400, or

n_{0} ≈ \frac{a}{ρ_{ref}} \frac{r}{aq} min(k_{θ}ρ_{ref})

Because turbulent eddies are so extended along field lines, while having a short correlation length perpendicular to the field line, it is more natural to think in terms of elongated flux-tube simulation domain. In this case these quantization/periodicity constraints aren’t really relevant in most cases of turbulent interest (see papers by Cowley, Kulsrud, and Sudan, and Beer, Cowley, Hammett, etc.).

p. 16: Key is to make the box more than a few decorrelation lengths in all directions so opposite sides of box are random with respect to each other. Then can assume statistical homogeneity: the statistics of the range of sizes of eddies that go in and out one side of the box is the same as at the other side of the box.

Use this “statistical similarity” to assume exact periodicity. Provides natural boundary conditions for small scale homogeneous turbulence.

Standard flux-tube simulations make further assumptions for simplicity: geometrical quantities vary along field line but are assumed to not vary much across the width of a thin flux tube. For example assume |B| is constant in directions perpendicular to the flux tube when calculating J_{0}(k_{⊥}v_{⊥}/Ω), etc. [Paper by Adil Hassam et.al. discuss extension of usual flux-tube geometry to include terms needed to get Stringer spin-up.]

Though constant gradients kept in terms like ω_{∗T} ∝ ∇(T), variations of equilibrium T_{0}(r) across box neglected if it appears without gradient (like in gyroradius for Bessel functions)

Though background temperature gradient is constant on scale of box, local ∇T (r, t) allowed to self-consistently fluctuate.

These are standard, widely-used, two-scale/periodic approximations, going back to ballooning mode theory, Hasegawa-Mima, homogeneous turbulence simulations in Navier-Stokes fluids, Balbus-Hawley shearing-box simulations of Magneto-Rotational-Instability in astrophysical accretion disks...

Flux-tube also natural for full implementation of “standard gyrokinetics”. Global codes are starting to investigate what are thought to be some of the most important finite ρ_{∗} corrections, but finite ρ_{∗} corrections in the gyrokinetic equation or in the assumed equilibrium have not been completely worked out.

?? Describe Waltz picture of ExB shear ρ_{∗} effects, showing gyro-Bohm, Bohm, etc. regimes...

E × B shear suppression of turbulence model based on BDT theory, Waltz approximation:

χ_{i} = χ_{GB} (1 − \frac{γ_{ExB}}{γ_{lin}} )

χ_{i} ≈ \frac{cT}{eB} \frac{ρ}{L} (1 − \frac{γ_{ExB}}{γ_{lin}} )

where γ_{lin} ∼ v_{t}/L, and

γ_{ExB} ∼ \frac{∂v_{ExB,θ}}{∂r} ∼ \frac{v_{t}ρ}{L^{2}}

(using enE_{r} ∼ ∂p/∂r), leads to

χ_{i} ∼ \frac{cT}{eB}ρ_{∗}(1 − Cρ_{∗})

Has gyro-Bohm, Bohm, worse-than-Bohm (Goldston ;-) ) regimes

Similar stabilizing effects if radial derivative of poloidal group velocity ω/k_{θ} ∼ ω_{∗}/k_{θ} included. Global codes are studying this.

Waltz-Candy source term seems reasoanble to me (though I haven’t checked all the details of it). Given time-averaged profiles T_{0}(r), n_{0}(r), etc. Microinstabilities would cause these profiles to relax to be uniform. If one is very close to marginal stability, only a little bit of relaxation would shut off the turbulence. There must be some source/sink terms (neutral beams, RF, radiation, chargeexchange losses), that maintains the profiles against turbulent diffusion. Waltz adds a time-averaged source to do this (beam heating, etc. is constant in time), whatever is required to match the time-averaged profiles T_{0}(r) etc. given by the experiments..

Profiles are allowed to fluctuate in time self-consistently.

Check: Waltz-Candy compared bounded simulation using compensating sources/sinks with periodic simulation using no sources/sinks and got good agreement. p.15: characteristics of: ITG (Ion Temperature Gradient driven instability) TEM (Trapped electron mode)

Antonsen et.al. picture of why negative magnetic shear is stabilizing:

Antonsen, Drake, Guzdar, Hassam, Lau, Liu, Novakovskii, PoP 96

B_{pol} ≈ \frac{B_{p}(r)}{(1 + \frac{r}{R} cos θ)(1 + Δ′ cos θ)}

Squeezing of flux surfaces by Shafranov shift Δ′ = dR_{0}/dr ≈ −(r/R)(β_{pol} + ℓ_{i}/2) can increase B_{pol} a lot.

Local magnetic shear ∝ (d/dr)B_{pol} ∝ (d/dr)β_{pol} leads to important parameter α = −Rq^{2}dβ/dr of the s − α model.

High α gives rise to a local negative magnetic shear.

Probably dominant stabilizing term in ST’s. See recent paper by Clarisse Bourdelle using GS2 to study various effects:

http://w3.pppl.gov/ hammett/gyrofluid/papers/2002/bp clarisse.pdf Emily Belli has studied other shaping effects, finds radial derivative of elongation is more important that elongation itself, etc.:

Belli, Hammett, Budny, Dorland, APS 02