theeducational, scientific SbClUS 5313111

and culturalorganization international centre for theoretical physics

international atomicenergy agency

SMR 1331/31.

AUTUMN COLLEGE ON PLASMA PHYSICS8 October - 2 November 2001

Introduction to NonlinearGyrokinetic Equation

G. Hammett

Princeton University, Plasma Physics Lab.Princeton, U.S.A.

These are preliminary lecture notes, intended only for distribution to participants.

strada costiera, I I - 34014 trieste italy - tel.+39 04022401 I I fax +39 040224163 - sci_info@ictp.trieste.it - www.ictp.trieste.it

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THEORY-BASED MODELS OF TURBULENCEAND ANOMALOUS TRANSPORT IN FUSION PLASMAS

G.W. Hammett, Princeton Plasma Physics Labw3.pppl.gov/~hammett

APS Centennial, Atlanta, March, 1999

Close collaboration with M.A. Beer (PPPL),W. Dorland (Univ. of Maryland), M. Kotschenreuther(Univ. of Texas), R.E. Waltz (General Atomics).

Acknowledgments: A. Dimits, G.D. Kerbel (LLNL),T.S. Hahm, Z. Lin, P.B. Snyder (PPPL),S.E. Parker (U. Colorado), and many others.

Supported at PPPL by DOE contract DE-AC02-76CH03073, computational resources at NERSC and the LANL ACL,

part of the Numerical Tokamak Turbulence Project national collaboration, a DOE HPCCI Grand Challenge.

Stable Pendulum

M

oco=(g/L)1/2

Unstable Inverted Pendulum

(rigid rod)

co=(-g/ILI)1/2=i(g/ILI)1/2=iy

Instability

Density-stratified Fluid

p=exp(-y/L)

Inverted-density fluidRayleigh-Taylor Instability

p=exp(y/L)

stable co=(g/L)1/2 Max growth rate y={gfL)m

"Bad Curvature" instability in plasmasInverted Pendulum / Rayleigh-Taylor Instability

Top view of toroidal plasma: Growth rate:

_ \SeB _ \ =L \RL~JRL

Similar instability mechanismin MHD & drift/microinstabilities

1/L= Vp/pinMHD,oc combination of Vn & VT

in microinstabilities.

plasma = heavy fluid

B = "light fluid"

geff = v centrifugal forceR

The Secret for Stabilizing Bad-Curvature Instabilities

Twi5"t to B CdrrieS

rwmx

fro* bcii. CurvaWe re*"i<M "to (food

UNSTABLE

GRAVITY

5i» iUr -b t-ovo a

v/

Cut-away view of tokamak turbulence simulation

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Waltz (General Atomics), Kerbel (LLNL), et.al., gyrofluid simulations. Similar pictures from gyrokinetc particle

simulations.

Lots more pictures at www.acl.lanl.gov/GrandChal/Tok/gallery.html.

Classic df

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In this ordering, the Vlasov equation reduces to a condi-/ ^ % tion on the zeroth-order parallel (relative to the magnetic

CO ^ field) electric field E» = 0, and the following kinetic equationo \ <j o .

for the zeroth-order distribution function of each species-£^

T

whwe es is the charge on species s, b is a unit vector in them*|?ietic field direction b=B/B,:. • T £ * C ( E X B ) / B 2 ,

Combining moments of this kinetic equation with Max-well's equations and taking the usual low Alfven speed limitv\<c2 yields Kulsrud's set of collisionless MHD equations:

(5)

.<6)8 - " • ^ • • j f • f J " • • • • • • • . . . . . -

i- - (? )

where p is the total mass density, .U— v f + «jb is tibevelocity, and P is the pressure tensor.

The above set of equations is exact to zeroth ordm in theexpansion parameter, but the kinetic equation itself, H| . •(!)•must be used to evaluate p8 and p± to close tflc system.Because lEq; (1) is difficult to solve directly, this system israrely employed without further simplification.

13

lem, it is necessary to introduce an additional ordering whichremoves the compressional Alfven time scale.

Another complication is the evaluation of the |£j||//fcuterms found in the Landau closures. As pointed out by Finnand Gerwin,18 the Landau damping must be evaluated alongperturbed field lines. Hence, for nonlinear calculations, trans-forming the closure to real space requires an integral alongthe perturbed field line. The numerical evaluation of thesenonlinear closures may be burdensome in some cases, asdiscussed in Sec. VII.

It is anticipated that the model will be useful for nonlin-ear numerical simulations. Some of the caveats involved inusing Landau closures in nonlinear simulations have beenextensively discussed in the gyrofluid literature,1112'23'2426"30

but these caveats are an area of ongoing research. There aresome regimes where certain nonlinear kinetic effects are notwell modeled by Landau-fluid closures.30 But we generallybelieve12'24'27'28 these closures will be adequate for strongerturbulence regimes where rapid decorrelation is occurringand the velocity space details of the distribution function arenot critically important.

It is hoped that the model will prove useful for simulat-ing both laboratory and astrophysical plasmas in the colli-sionless MHD regime. The model should be able to predictthe onset and structure of instabilities, as well as the heat andparticle transport caused by the instabilities.

ACKNOWLEDGMENTS

We would like to thank Dr. Mike Beer, Dr. StephenSmith, and Dr. Nathan Mattor for useful discussions aboutLandau closures.

We would like to acknowledge support from the U.S.Department of Energy (DoE) under Contract No. DE-AC02-76CH03073. P.B.S. acknowledges the support of the Na-tional Science Foundation (NSF) through the NSF GraduateFellowship Program. This work was also supported in parti

by the Numerical Tokamak Turbulence Project, part of theDoE High Performance Computing and CommunicationsInitiative.

!G. F. Chew, M. L. Goldberger, and F. E. Low, Proc. R. Soc. London, Ser._ A 236, 112 (1956).

2R. M. Kulsrud, in Proceedings of the International School of PhysicsEnrico Fermi, Course XXV, Advanced Plasma Theory, edited by M. N.Rosenbluth (North Holland, Varenna, Italy, 1962).

L 3R. M. Kulsrad, in Handbook of Plasma Physics, edited by M. N. Rosen-

^ bluth and R. Z. Sagdeev (North Holland, New York, 1983).4J. N. Leboeuf, T. Tajima, and J. M. Dawson, J. Comput. Phys. 31, 379(1979).

5R. D. Sydora and J. Raeder, in Cometary and Solar Plasma Physics, editedby B. Buti (World Scientific, Teaneck, NJ, 1988), pp. 310-364.

6G. Y. Fu and W. Park, Phys. Rev. Lett. 74, 1594 (1995).7H. Naitou, K. Tsuda, W. W. Lee, and R. D. Sydora, Phys. Plasmas 2, 4257(1995).

8M. D. Kruskal and C. R. Oberman, Phys. Fluids 1, 275 (1958).9M. N. Rosenbluth and N. Rostoker, Phys. Fluids 2, 23 (1958).

IOG. Hammett and F. Perkins, Phys. Rev. Lett. 64, 3019 (1990)."G. Hammett, W. Dorland, and F. Perkins, Phys. Fluids B 4, 2052 (1992).I2W. Dorland, Ph.D. thesis, Princeton University, 1993.13T. S. Hahm, W. W. Lee, and A. Brizard, Phys. Fluids 31, 1940 (1988).14A. Brizard, Phys. Fluids fc 4, 1213 (1992).I5Z. Chang and J. D. Callen, Phys. Fluids B 5, 1167 (1992).16Z. Chang and J. D. Callen, Phys. Fluids B 4, 1182 (1992).17A. Bondeson and D. Ward, Phys. Rev. Lett. 72, 2709 (1994).I8J. Finn and R. Gerwin, Phys. Plasmas 3, 2469 (1996).19M. Medvedev and P. Diamond, Phys. Plasmas 3, 863 f»9S). ) 1 % .20E. Gross and M. Krook, Phys. Rev. 102, 593 (1956).2IS. E. Parker and D. Carati, Phys. Rev. Lett. 75, 441 (1995).22S. I. Braginskii, in Reviews of Plasma Physics, edited by M. A. Leontov-

ich (Consultants Bureau, New York, 1965), Vol. 1, pp. 205-311.23M. A. Beer, Ph.D. thesis, Princeton University, 1995.24M. A. Beer and G. W. Hammett, Phys. Plasmas 3, 4046 (1996).25S. A. Smith, Ph.D. thesis, Princeton University, 1997.26C. L. Hedriek and J.-N. Leboeuf, Phys. Fluids B 4, 3915 (1992).27G. W. Hammett, M. A. Beer, W. Dorland, S. C. Cowley, and S. A. Smith,

Plasma Phys. Controlled Fusion 35, 973 (1993).28S. E. Parker, W. Dorland, R. A. Santoro, M. A. Beer, Q. P. Liu, W. W.

Lee, and G. W. Hammett, Phys. Plasmas 1, 1461 (1994).29J. Krommes and G. Hu, Phys. Plasmas 1, 3211 (1994).'°N. Mattor, Phys. Fluids B 4, 3952 (1992).

Phys. Plasmas, Vol. 4, No. 11, November 1997 Snyder, Hammett, and Dorland 3985

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