% From: "Andris Dimits" <dimits@elvis.llnl.gov>
% Date: Fri, 6 Aug 1999 16:21:38 -0700
% To: Hammett@pppl.gov, bcohen@llnl.gov
% Subject: Cyclone Paper

% This version includes more changes by Dimits
% search for ``%Dimits_7_30'' to locate these.
% Also, I have made an udpated version of figure 1 to change the halftone
% (result of saving color to BW) markers and lines to black
% (with dotted lines).

% This version includes a few small changes by Dimits
% search for ``%Dimits_7_29'' to locate these.


% Date: Thu, 8 Jul 1999 08:50:26 -0700
% From: Bruce Cohen <bcohen@sedona.llnl.gov>
% To: ntx-exec@acl.lanl.gov, cyclone@er.doe.gov, cyclone@science.doe.gov
% Subject: cyclone paper
%
% Attached is a complete version of the Cyclone paper, with all of the
% references cleanup up, figures dones, and in a state ready to be sent to
% the referees.  Since the last version distributed, the main changes have
% been in the final 2 sections, though there are some changes scattered
% elsewhere in the paper.  The present version incorporates the few small
% changes various people suggested:
%
% % updated refs. and wording per Parker 99-05-25
% % updated Weiland refs., per Weiland 99-09-25
% % added Rewoldt as co-author per Kritz circa 99-05-11
%
% A small group of us (Andris Dimits, Bruce Cohen, Greg Hammett, Bill
% Dorland) put a lot of work into this and iterated several times to converge
% on this version.  They don't think there is anything that should be too
% controversial, and hope that everything is worded diplomatically and
% clearly enough, with the right caveats, that everyone finds it acceptable.
% But if you have any comments or suggested changes, send them to me (Bruce
% Cohen) with a cc to Dimits and Hammett so we can agree on or discuss the
% changes.  Please send your changes within 2 weeks, by July 21 (or sooner if
% possible if we need to iterate more).  In the interest of getting something
% sent for publication soon, we are trying to avoid minor wordsmithing at
% this point.
%
% Thanks.
%
% Bruce, Greg, Andris, and Bill
%
%
% PS. Greg is posting the latest version of the paper on his
% web site in ps and pdf format, and the figures can be found there as
% well. See
% http://w3.pppl.gov/~hammett/work/cyclone/paper99b
%
%
% Here is the latex file:
%
% ******************************************************************

% Date: Fri, 2 Jul 1999 14:55:16 -0700
% From: Bruce Cohen <bcohen@sedona.llnl.gov>
% To: hammett@pppl.gov, dimits@elvis.llnl.gov
% Subject: latex file with my revisions to Greg's latest version of cyclone
% paper


% BC7x2    7/2/99 Reviewed and modified (slightly) by B. Cohen
%  Look for %BC7x2 to introduce changes or comments.


% This is the near final version from Greg Hammett, 99-06-29 (the only
% remaining thing I have to do is clean up some of the references).  It
% includes all of the things I think are important, and I would hope that
% everyone else will find it acceptable also.  I think it is ready to be
% send out for publication, though probably the co-authors should probably
% be given the opportunity to see it in its final form.  The main changes
% I've made (since Parker and Cohen last advertised a version to the
% co-authors) have been in the final two sections, and the final figure has
% been updated.  There are a few other changes that are sprinkled
% throughout the text (though I don't think these are major changes anyone
% would object to).  This version also includes suggestions made by a few
% people over the past couple of months.

% Includes many of the suggestions made by Dorland 99-06-29
% updated refs. and wording per Parker 99-05-25
% updated Weiland refs., per Weiland 99-09-25
% authors in alphabetical order, per Cohen 99-05-17
% added Rewoldt as co-author per Kritz circa 99-05-11
% with suggested changes by Hammett, 99-04-27
% included more suggestions by Dimits 27-Apr
% includes changes suggested by Dimits on 27-Apr
% includes changes by Cohen and Dimits on 99-04-26.

% based on the version mailed:
% Date: Tue, 20 Apr 1999 10:11:35 -0700
% From: Bruce Cohen <bcohen@sedona.llnl.gov>
% To: Hammett@pppl.gov
% Subject: Re: Cyclone paper latex file
% cc: bcohen@sedona.llnl.gov, nevins1@llnl.gov, dimits@elvis.llnl.gov


%Cyclone members,
%                Here is a draft of the text of the Cyclone paper for
%                discussion. The figures need to be cleaned and
%                collected in a consistent data format before I can
%                usefully transmit them. The text needs some discussion
%                of ITER implications which I cannot authoritatively
%                provide.
%
%Andris

%=======================================iaea98.tex==============================



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\bc Comparisons and Physics Basis of Tokamak Transport Models and
Turbulence Simulations \ec
%Dimits_7_30 Kinsey is at Lehigh - switch 6 and 7
\noindent
A.~M.\ Dimits,${^1}$ G.\ Bateman,${^2}$ M.~A.~Beer,${^3}$ B.I.\
Cohen,${^1}$ W.~Dorland,${^{4}}$ G.~W.~Hammett,${^3}$ C.~Kim,${^5}$
J.~E.~Kinsey,${^{2}}$ M.~Kotschenreuther,${^6}$ A.~H.~Kritz,${^2}$
L.~L.~Lao,${^{7}}$ J.~Mandrekas,${^{8}}$ W.~M.\ Nevins,${^1}$ S.~E.\
Parker,${^5}$ A.~J.~Redd,${^9}$ D.~E.\ Shumaker,${^1}$
R.~Sydora,${^{10}}$ J.~Weiland${^{11}}$\newline

\noindent
${^1}$Lawrence Livermore National Laboratory, P.O.\ Box 808, Livermore, CA
94550\newline
\noindent
${^2}$Lehigh University, Lehigh, PA 18015\newline
\noindent
${^3}$Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ
08540\newline
\noindent
${^{4}}$University of Maryland, College Park, MD 20742\newline
\noindent
${^5}$University of Colorado, Boulder, CO 80309\newline
\noindent
${^6}$Institute for Fusion Studies, Univ. of Texas, Austin, TX 78712\newline
\noindent
${^{7}}$General Atomics, Inc., San Diego, CA 92186-5608\newline
\noindent
${^{8}}$Georgia Institute of Technology, Atlanta, GA 30332-0225\newline
\noindent
${^9}$University of Washington, Seattle, WA 98195\newline
\noindent
${^{10}}$University of Alberta, Edmonton, Alberta, AB T6G2J1 Canada \newline
\noindent
${^{11}}$Chalmers University of Technology, S-412 96 Goteborg, Sweden\newline

\vspace*{0.25in}

\bc
Abstract
\ec
\small \rm

%Generic - What points do different people want to make?
%What will we agree on, what is supported by what we have?

The predictions of gyrokinetic and gyrofluid simulations of
ion-temperature-gradient (ITG) instability and turbulence in tokamak
plasmas as well as some tokamak plasma thermal transport models, which
have been widely used for predicting the performance of the proposed
ITER tokamak, are compared. These comparisons provide information on
effects of differences in the physics content of the various models
and on the fusion-relevant figures of merit of plasma performance
predicted by the models. Many of the comparisons are undertaken for a
simplified plasma model and geometry which is an idealization of the
% BD (1)
%plasma conditions and geometry in a fusion-reactor-relevant (DIII-D)
plasma conditions and geometry in a DIII-D H-mode
% BC7x2
%experiment.  Most of the models show good agreements in their
%experiment relevant to ITER. Most of the models show good agreements in their
% GWH7x6
experiment. Most of the models show good agreements in their
predictions and assumptions for the linear growth rates and
frequencies. There are some differences associated with different
equilibria.  However, there are significant differences in the
transport levels between the models. The causes of some of the
differences are examined in some detail, with particular attention to
numerical convergence in the turbulence simulations (with respect to
simulation mesh size, system size and, for particle-based simulations,
the particle number).  The implications for predictions of fusion
plasma performance are also discussed.

\noindent


\vspace*{0.25in}
\rm \normalsize

\noindent
PACS numbers: 52.35.Ra, 52.65.Tt,52.25.Fi

\pagebreak

%\vspace*{0.15in}
\rm \normalsize
\noindent 1.\ INTRODUCTION
\vspace*{0.15in}


We examine the physics basis and predictions of some tokamak plasma
thermal transport models which have been widely used for predicting
the performance of the proposed ITER tokamak~\cite{ITER}.  This topic
is of considerable importance and current interest since different
models in use give conflicting predictions on whether ITER will or
will not achieve thermonuclear ignition~\cite{ITER-IFSPPPL, ITER-MMM,
Mikkelsen98, ITERConnor, Kinsey97, Waltz97}.

The confinement in current tokamak experiments is generally believed
to be degraded primarily by turbulence driven by ``low-frequency''
microinstabilities (instabilities occurring on drift timescales). The
disagreements in Refs.~\citenum{ITER-IFSPPPL}-\citenum{Waltz97}
% aip.sty would automatically condense citation lists to a range,
% without needing to use citenum.
about predictions of ITER performance arise from using different models of
drift-instability-driven transport, which can be distinguished in
terms of their treatment of the detailed physics of
microinstabilities.

In order to build confidence in predictions of tokamak plasma
performance, the models used need to be tested. Here we rely
on tests involving cross checks between a number of
complementary models. A minimal condition for confidence in this case
is agreement between the models where they can be compared under the
same physical conditions. Where models disagree, a clear understanding
of the reasons for the disagreements is important. This may take the
form of knowledge that one of the models is being applied outside of
the range of validity of a derivation that underlies it.  Even when
this agreement is achieved, there is generally no guarantee that the
% BD (2)  %BC7x2 OKAY, (assume OKAY on all BD wording improvements
%unless otherwise noted.
%set of models will have predictive power, once applied outside the
set of models will have predictive power, if applied outside the
range of physical conditions for which they have been tested.


This evaluation process presents a great scientific challenge.  These
models are complex and contain many physical effects. The predictions
of the models should be compared at various levels, not only at their
final prediction, which is often a radial profile (e.g., density, flow
velocity, temperature), but also at the level of comparison of the
various submodels, theories and calculations.

In this paper we compare core transport predictions from the
following: the IFS-PPPL~\cite{IFS-PPPL,IFS-PPPL95} and Multi-Mode
(MMM)~\cite{MMM} transport models; flux-tube
gyrofluid~\cite{gyrofluid}, flux-tube gyrokinetic~\cite{DimitsPRL},
and global gyrokinetic~\cite{GlobalGK} turbulence simulations; linear
one-dimensional high-n eigenmode and initial-value
calculations~\cite{comprehensive} and their use in transport model
calculations. The MMM and IFS-PPPL models are two widely-used
transport models for ITER predictions. Also represented here are some
of the most advanced large-scale three-dimensional toroidal turbulence
simulations, and the most widely used high-n linear and quasilinear
calculations.


We first define some terms used here. The term ``gyrokinetic'' refers
to a kinetic model (i.e., one which evolves functions of
position-velocity phase-space variables and time), appropriate to
charged particles in a strong magnetic field, in which a
multiple-timescale perturbation expansion in the ratio of the
gyroperiod to the timescales of the phenomena of interest is
made~\cite{Frieman}. When appropriate, such a model is much more
efficient than one that tracks the full particle dynamics (including
the gyromotion). Gyrokinetic models retain ``finite-gyroradius''
effects (effects that arise when the scale of the gyro-orbit is
comparable to the spatial scale of the phenomena of interest)
non-perturbatively. This aspect distinguishes them from
``drift-kinetic'' models which either ignore or treat
finite-gyroradius effects perturbatively.  A ``fluid'' model of a gas
or plasma is one that evolves ``fluid variables'' which are functions
of position and time (and not particle velocity).
``Gyrofluid''~\cite{gyrofluid} models are a special class of fluid
models derived from the gyrokinetic equations, which similarly have
non-perturbative aspects to their treatments of finite-gyroradius
effects. Those discussed here can be called ``gyro-Landau-fluid''
models in that they also contain (fluid) models of
Landau damping and related processes.



%Redd Suggestion:describe what is meant by "fluid", "gyrofluid" and
%gyrokinetic" models, with a reference to a good introductory-level
%description of these things.
%Global
%Flux-tube


We present detailed comparisons of the various models using a
simplified physics problem which still contains the essence of core
transport in an ITER-like discharge.  A key point of comparison is the
predictions of the various models for the $\chii$, the ion thermal
diffusivity (ion thermal flux divided by the ion temperature
gradient).  Disagreements are found between the predictions of
different models for the ion thermal transport. We examine various
possible reasons for the disagreement.  Particular attention is paid
to linear mode growth rates and frequencies, ``zonal''
flux-surface-averaged flow damping, and noise due to particle
discreteness in the nonlinear gyrokinetic codes. Finally, the
implications for ITER performance are discussed.

%Motivation for choice of physics test problems, brief description
%of them

\vspace*{0.15in} \noindent
2. TEST PROBLEMS
\vspace*{0.15in}

We focus on the ``Cyclone DIII-D base case parameter set'' which
represents local parameters from an ITER-relevant DIII-D shot (shot
\#81499)~\cite{Greenfield}, at time $t=4000$ms., and minor radius
$r=0.5a$, where $a$ is the minor radius of the last closed flux
surface.  A concentric-circular-cross-section model equilibrium is
used, with $\ni=\ne$ and $\Te=\Ti$, where $\ni$ and $\ne$ are the ion
and electron densities and $\Te$ and $\Ti$ are the electron and ion
temperatures.  The parameter values in dimensionless form are $\etai
\equiv \Ln / \LT=3.114$, where $\Ln$ and $\LT$ are respectively
% BD (3):
% $\equiv \Ln / \LT=3.1$, where $\Ln$ and $\LT$ are respectively
the density and temperature gradient scale lengths, magnetic ``safety
factor'' $q \equiv r \Bt/R \Bp=1.4$, where $R$ is the major radius and
$\Bt$ and $\Bp$ are the toroidal and poloidal magnetic field
components, $\shat \equiv (r/q) dq/dr = 0.776$--0.796 (some minor
variations due to constraints in some codes), $R/\LT = 6.92$, and
% BD (3)
%components, $\shat \equiv (r/q) dq/dr = 0.78$--0.80 (some minor
%variations due to constraints in some codes), $R/\LT = 6.9$, and
${\epsilon} \equiv r/R = 0.18$.

% GWH ?? need to double check some of the unusual numbers above...

Additional simplifications which are made in all models discussed here
are (1) electrostatic fluctuations, (2) the electrons are taken to be
adiabatic, and (3) a single dynamical ion species (which represents
the ``bulk'' ions) is used.  Linear stability results from the
comprehensive linear gyrokinetic code of Kotschenreuther and from the
FULL code which include multiple-ion species and electromagnetic
effects are also discussed~\cite{comprehensive}.  These
simplifications match those in the nonlinear gyrofluid simulations
that underlie the IFS-PPPL model.  Scans have been made varying the
temperature gradient scale length while keeping other physical
parameters fixed. Additionally, cases have been compared in which each
one of $\shat$ and ${\epsilon}$ were set to zero, with the other parameters
held to the DIII-D base case values.  The first helps isolate
differences in the way magnetic shear is treated.  The various spatial
representations used become very similar in the limit of zero magnetic
shear.  The second helps isolate the effect of linear damping of
flux-surface-averaged poloidal flows.  The shear associated with these
flows is an important saturation mechanism and the physics of these
% BD (4)
%modes is under intense study (see further discussion below on radial
modes is under continuing study (see further discussion below on radial
mode damping).

We also included a case using parameters from TFTR L-mode shot 41309.
This was a case used in 1994 in a code comparison within the Numerical
Tokamak Project (NTP)~\cite{NTTP}.
% BD (5)
%These parameter values are $\etai =4$, $q=2.4$, $\shat = 1.5$--$1.6$,
%$L_T/R = 0.1$, and ${\epsilon}= 0.2057$ $\tau \equiv \Te/\Ti = 1.0$.
%Comparisons are also discussed in which Tokamak Project (NTP)~\cite{NTTP}.
These parameter values are $\etai = 4.0$, $q=2.4$, $\shat = 1.5$--$1.6$,
$R/L_T = 10.$, $n_i/n_e = \Ti/\Te = 1.0$, and ${\epsilon}= 0.2057$.
Comparisons are also discussed in which the purely radial modes in this
``TFTR L-mode NTP test case'' are suppressed.  The purpose of running these
cases was to verify previous NTP comparisons
%BC change for GWH
that showed gyrofluid and gyrokinetic values of $\chii$ which
differed by factors of 2 or less (the gyrofluid $\chii$ was higher
%BC7x2: NTTP GK vs. GLF vs. expt.
and the experimentally determined value was higher still by a factor slightly
less than 2).
Another earlier gyrofluid/gyrokinetic comparison\cite{Parker94}, which
looked primarily at slab geometry, found that there was good agreement in
the slab simulation $\chii$ though the saturation level of the RMS $\Phi$
differed by
40\%.
We also included a case using parameters from the
published result of Dimits~\etal.~\cite{DimitsPRL}, which had
parameters similar to the DIII-D base case.



%Brief description of the models involved:

\vspace*{0.15in} \noindent
3. DESCRIPTION OF MODELS
\vspace*{0.15in}

Simulations of a flux-tube sub-domain of the torus which neglect
global profile scale effects give a finite ion thermal diffusivity
$\chii$ and turbulent correlation lengths much shorter than the scale
of profile variation anticipated for ITER.  A reasonable conclusion
from this is that $\chii$ depends primarily on local plasma
parameters, if ITER-relevant values of the normalized gyroradius
$\rhoi/\LnT$ are used\cite{ gyrofluid, DimitsPRL, Waltz94}. Motivated by
this, all of the models discussed below with the exception of the
global gyrokinetic simulations assume that the simulation domain
covers only a thin radial extent compared to the plasma minor radius
so that quantities such as the density scale length, $(dn/dr/n)^{-1}$,
$\omega_*$, etc. are taken to be independent of minor radius.


%IFS-PPPL

The IFS-PPPL~\cite{IFS-PPPL,IFS-PPPL95} model is based on nonlinear
gyrofluid simulations~\cite{gyrofluid}, which predict the fluctuation
and thermal transport characteristics of toroidal
ion-temperature-gradient-driven (ITG) turbulence, along with
comprehensive linear gyrokinetic ballooning
calculations~\cite{comprehensive}, which provide accurate growth
rates, critical temperature gradients, and a quasilinear estimate of
$\chie / \chii$.  A key aspect of the IFS-PPPL model is an
interpolation formula which parameterizes both the gyrofluid
$\chii$'s, and calculations of the critical temperature gradients and
mixing-length predictions of $\chii$ from the more comprehensive
linear gyrokinetic ballooning code.  This linear
code~\cite{comprehensive} has full velocity-space dynamics including
resonances, trapped particles, Coulomb collisional pitch-angle
diffusion, etc. This corrects the somewhat inaccurate critical
temperature gradient and the neglect of non-adiabatic electron
physics in the gyrofluid simulations that were used as a basis for the
IFS-PPPL model.

%flux-tube Gf

The nonlinear gyrofluid simulations~\cite{gyrofluid} that underlie the
IFS-PPPL transport model use a ``gyrofluid'' reduction of the
gyrokinetic equations.  The resulting gyrofluid equations are evolved
in toroidal field-line-following coordinates.\cite{gyrofluid2} The
gyrofluid equations include toroidal effects (e.g., magnetic curvature
drive) and kinetic effects such as toroidal drift resonances, linear
and nonlinear FLR orbit-averaging, and parallel wave-particle
resonances, as well as nonlinearly
generated, fine-scale ($k_r \rhoi \sim 0.1$), sheared poloidal flows,
which play a major role in determining the saturation level for the
turbulence.
%Dimits reworking of GWH comments (Let's avoid this stuff-BC,WN)
%These treatments of resonances and sheared poloidal
%(``zonal'') flows distinguish the gyrofluid models from the Weiland
%ITG component of the MMM model discussed below, which does not include
%these effects.
%Endthedimits
In this paper, we focus on gyrofluid simulations with
an adiabatic electron response, since this was used in the simulations
on which the IFS-PPPL model is based. Gyrofluid simulation codes with
bounce-averaged nonadiabatic electrons have since been developed and
exercised~\cite{Beer}.

%Weiland

The philosophy underlying the Multi-mode transport model
(MMM)~\cite{MMM} has been to utilize a collection of
theoretically-derived transport models to predict temperature and
density profiles in tokamak plasmas and adjust the models as
necessary. As the models have improved, less adjustment has been
needed to fit the experimental data.  The 1995 Multi-mode model
combines the fluxes predicted by the Weiland ITG/TEM
model~\cite{Weiland89,Weiland} with those predicted by the
Guzdar-Drake resistive ballooning model~\cite{Guzdar}, with smaller
contributions from kinetic ballooning modes and neoclassical
transport~\cite{MMM98}.  The 1995 Multi-mode model was calibrated
against a small number of experimental discharges, and then held fixed
for all subsequent transport studies~\cite{MMM96}.

In the simulation of experimental plasmas, the transport predicted by
the 1995 Multi-mode model is typically dominated by the contribution
from the Weiland ITG/TEM model.
% BD (6)
%This is also the only contribution that is relevant to the Cyclone effort.
This is the only contribution that is kept in the comparisons for the
idealized Cyclone parameters in Figs.(1) and (3).
The Weiland ITG model is based on a fluid description in which all moments
that are driven by sources (i.e. fueling, heating) are included
self-consistently. The fluid moments that are not driven by sources
generally decay to zero. The model allows free energy exchange between
different transport channels, leading to pinch fluxes.  The transport
coefficients are derived by using quasilinear theory and a mixing-length
rule for saturation, which takes $k_\theta \rho_s = 0.316 $~\cite{MMM}. The
transport coefficients therefore have gyroBohm scaling. However, they have
been found to agree well with some non-gyro-Bohm L-mode and H-mode
experimental data~\cite{MMM,MMM98,MMM96,MMM99}. The Weiland model also includes
effects from the impurity profiles\cite{MMM}, fast ions, and $\Te \neq
\Ti$, and has been extended to include parallel ion motion and
electromagnetic effects~\cite{Weiland}.

% BD (7)
%The IFS-PPPL and MMM models are all basically gyroBohm-scaling models,
The IFS-PPPL and MMM models are both basically gyroBohm-scaling
models, though non-gyroBohm scalings can enter in several ways, for
example, marginal stability connections to edge boundary conditions,
change in particle fueling profiles or density profiles.  Recent
versions of the IFS-PPPL model (and the related GLF23
model\cite{Waltz97}) also add stabilizing {\EB} shear, which can
introduce additional non-gyroBohm scaling effects~\cite{Mikkelsen98,
Kinsey97, Waltz97, Scott96, Dorland96, Garbetshr}.

% GWH removed NonGB-Bateman references because issue is already dealt with
% in the previous paragraph, with references to the relevant MMM work.

%flux-tube GK

The gyrokinetic simulation codes, both flux-tube~\cite{DimitsPRL} and
global~\cite{GlobalGK}, solve the gyrokinetic Vlasov-Poisson system of
%Dimits_7_30 Changed Lee citation
equations~\cite{Frieman,WWLee83} (electrostatic limit) using
particle-based $\df$ methods~\cite{plin,Kotschendf,Parkerdf}  and
``four-point gyroaveraging''~\cite{WWLee}. A single fully toroidal
nonlinear gyrokinetic ion species with equilibrium temperature,
density, and velocity gradients is used. Adiabatic electrons with a zero
response to the flux-surface-averaged
potential~\cite{Cohen,DorlandThesis,Hammett93} are used in the present
comparisons, both in the flux-tube gyrokinetic and gyrofluid codes.
The lack of response of the electrons to the flux-surface-averaged
potentials is a key factor in the amplitudes to which these zonal flow
modes are
driven and therefore to the levels of turbulence and transport seen in
the gyrofluid~\cite{DorlandThesis,Hammett93} and
gyrokinetic~\cite{Cohen,DimitsIAEA94}
% GWH: In January I had suggested a reference to Dimits IAEA paper here,
% but I meant his earlier 1994 IAEA paper where he first showed in toroidal
% GK simulations a factor of 5 increase in chi when the zonal flows were
% suppressed... I don't think a reference to the later 98 IAEA is needed
% here, but you can add it back if you want.
simulations.  A low-$\beta$ concentric-circular-cross-section model
equilibrium is used here.

The flux-tube gyrokinetic simulations (like the gyrofluid simulations)
use a flux-tube domain (bounded by four magnetic field lines) of small
perpendicular extent, which spans one or more poloidal circuits in the
parallel direction.  The flux tube is taken to be periodic in the
toroidal direction, and periodic in the radial direction with a
toroidal offset such that a magnetic flux sheet is continuous across
the radial boundary. This prevents saturation of the turbulence by
profile relaxation. The field quantities are defined on a
quasiballooning-coordinate grid~\cite{fluidqbc}. The radial
differences, interpolation, deposition, and smoothing are formed using
shapes in {\it configuration} space (not ballooning-coordinate space)
that are independent of poloidal location~\cite{fluidqbc}.  This
combination of coordinates and shapes prevents grid stretching and
resolution loss in the presence of magnetic and velocity shear, and
allows a smooth implementation of the toroidal periodicity condition
across the parallel boundary for arbitrary profiles of the magnetic
safety factor $q(r)$. It contrasts with the flux-tube gyrofluid code
which uses direct discretization in ballooning coordinates.

%global GK

The global gyrokinetic simulations~\cite{GlobalGK} typically use a
domain which spans the whole tokamak volume, although simulations
using only an inner subvolume or a toroidal annulus have been
reported~\cite{Lin}. The field quantities are represented on a
radial-poloidal-toroidal mesh, and a fully nonlinear form of the
gyrokinetic equations is solved~\cite{Hahm} instead of the partially
linearized form of ref.~\cite{Frieman}. The important physics that the
global codes allow for is the full radial variation of gradient
quantities (e.g., temperature and density gradients, magnetic shear,
etc.).  These are generally a stabilizing effect, but get weaker in
larger tokamaks with larger $a/\rho$\cite{ Lin, Sydrhostar, Pkrhostar}.
In global gyrokinetic
simulations to date the simulation domain is bounded and no explicit
modeling of particle or thermal sources sinks is used. Thus profile
relaxation, including kinetic profile relaxation, in which spatial
gradients of ions in subregions of velocity space relax~\cite{Cohen},
may occur.



%Linear comparisons
%Figures: Show gamma, omega vs. k_theta,
%linfig.ps

%linear damping of purely radial modes and maybe radial mode damping test.


%Gamma vs. k_theta: Figure = merge of Beerlin.ps and newlincomp.ps

\vspace*{0.15in} \noindent
4. LINEAR COMPARISONS
\vspace*{0.15in}

Figure~(1) shows linear frequencies and growth rates as
a function of $\ktheta \rho_i$ obtained from several independent linear
and nonlinear codes (our convention is that the thermal gyroradius and
thermal speed are defined as $\rho_i = v_{ti}/\Omega_{ci}$ and
$v_{ti}=\sqrt{T_i/m_i}$).
Represented are the linear gyrokinetic codes of
Kotschenreuther and of Rewoldt (FULL code), the nonlinear gyrofluid
code of Beer and coworkers, and the nonlinear gyrokinetic codes of
Dimits (flux tube) and Sydora (global), as well as the fluid code of
Weiland. Very good agreement between the various codes is observed.
Agreement at this level is an important cross check of the codes since
all of the gyrokinetic codes should have the same linear physics (with
the exception that the global gyrokinetic code has radial variation of
profile quantities), and since the closure in the gyrofluid code is
designed to reproduce the gyrokinetic linear responses accurately.

Note that the ``GK (Dimits)'' gyrokinetic results are from a single flux-tube
simulation containing many growing modes. The growth rate for the $\ktheta
\rhoi = 0.1$ mode has quite a large uncertainty due to several possible known
effects. Since particles in the particle-based simulations constitute a
structure that does not have the same periodicity as the mesh, there is linear
coupling between different modes (both different toroidal mode numbers and
different radial wavenumbers or ballooning angles).
% BD (8) Delete the following sentence?  GWH: leave it in...
% BC7x2: agree with GWH on leaving it in
[This linear coupling is relatively small and gets weaker as more
particles are added.]  This mode has slow time variation, i.e., both
the expected $\omega$ and $\gamma$ are small. It is therefore slow to
reach its time-asymptotic linear behavior and is also susceptible to
the above effects.  For example, at a given $\ktheta$ there are many
different $\theta_0$ modes ($\theta_0$ is the ballooning parameter
value where $k_r=0$) in the simulation growing simultaneously, and it
can take time for the fastest growing mode to sufficiently dominate to
get an accurate growth rate.

% GWH: Andris and I had some discussions about the above
% paragraphs, and possible ways to improve the measured growth rate at low
% k.  I am not sure the linear particle-mesh coupling between different
% Fourier modes is the dominant cause of the present problem, but the above
% paragraph is acceptable to me for now.
% BC7x2: looks ok

Whether the small differences in the growth rates shown in Fig.~(2)
are important to the transport depends on the dynamics of the
saturated state. If the longer wavelength modes are primarily driven
nonlinearly, then these differences are likely to be unimportant. If
the correlation time of the long-wavelength modes is set by their
linear growth rates (as is implicit in $\gamma / k^2$-type mixing
laws) which assumes that the nonlinear driving does not set the
correlation times, then these differences may be significant since the
longer wavelengths dominate if one maximizes $\gamma / k^2$-type
mixing formulas over wavenumber.

Additional points of agreement between the flux-tube nonlinear
gyrokinetic code and the linear gyrokinetic code include the linear
critical temperature gradients for both the DIII-D base case
parameters and for ${\epsilon} = 0$ but with the other parameters as for
the DIII-D base case.

A second linear test is based on the linear damping of purely radial
modes of the electrostatic potential, i.e., modes which have no
variation within a flux surface.
% BD (9)
%A theory for the residual levels of these modes in the collisionless limit
%has been given by Rosenbluth and Hinton~\cite{RH}.
A linear theory for the residual levels of these modes in the collisionless
limit has been given by Rosenbluth and Hinton~\cite{RH}.
In this benchmark, the gyrokinetic code is
initialized with zero particle weights. A radially sinusoidal
potential with no variation within the flux surfaces, which represents
a near poloidal \EB\ flow, is imposed.  The particle weights evolve,
resulting first in a period of geodesic acoustic oscillations which
eventually damp. At late time, the net electrostatic potential is less
than the imposed potential.  For a circular cross-section equilibrium
in the large aspect-ratio limit, the theoretical prediction for the
ratio of the late-time net potential to the initial residual level is
given as the function $0.6h/(1.0+0.6 h)$ of the single parameter $h
\equiv \sqrt{\epsilon} / q^2$~\cite{RH}.

Figure~(2) shows the fractional residual \EB\ flows for two scans done
with the flux-tube gyrokinetic code. In one scan, $q$ is varied, while
in the other scan ${\epsilon}$ is varied. Very good agreement is
observed, lending confidence to both the Rosenbluth-Hinton theory and
the simulation.  The gyrofluid code, with the ``thesis
closure,''~\cite{Beer} which was used in the simulations that underlie
the IFS/PPPL formulas used to make the ITER projections of
Refs.~\cite{ITER-IFSPPPL}, appears to give \EB\ flows consistent with
the poloidal flow damping to zero~\cite{Beer}, i.e., in the
case of the setup here, the \EB\ flow damps essentially to zero,
although the short-time response, on the ion-transit timescale agrees
reasonably well with gyrokinetic calculations. In more recent
work\cite{Beer98},
improved gyrofluid closures have been developed for which the
long-time response agrees better with Ref.~\cite{RH}, and are discussed in
more detail in the next section.

%res_vs_kth.ps - time histories vs k_theta ; Shows typical behavior
%of flow damping tests - geodesic-acoustic oscillations and HRW residuals.
%flowresb.ps - residuals vs. HRW parameter; shows agreement with
%HRW theory, giving increased confidence in both the HRW theory and the
%gyrokinetic code.

%Nonlinear results:
%Report numbers, in text or in tabular form.  Have a brief
%description of how they agree/disagree.  Figure: Show chi_i vs. R/L_T
%(could show time trace of chi_i depending on length of paper.)

\vspace*{0.15in} \noindent
5. NONLINEAR COMPARISONS
\vspace*{0.15in}


Figure~(3) shows predictions from the various
simulations and models for $\chii$ vs.~$R/\LT$ for a scan about the
DIII-D base case parameters.

%Flux-tube GK
Two sets of flux-tube gyrokinetic results are shown.  The agreement
between these two sets is very good. The ``U.~Col.~Boulder'' results
were run with $4 \times 10^6$ particles, while the LLNL results were
obtained with $8$--$34 \times 10^6$ particles.  This accounts for the
difference at lower values of $R/\LT$.  Specifically, the LLNL code run
%BC7x2: punctuation: missing period above
with $4 \times 10^6$ particles also shows nonzero values of $\chii$
which are in agreement with the U.~Col.~Boulder code for $R/\LT <
6.9$.  A remarkably good fit to LLNL gyrokinetic results is given
by~\cite{DimitsIAEA98}
\bealn \chii \Ln / (\rhoi^2 \vti) \simeq 15.4
[1.0- 6.0 (\LT/R)], \eealn
which is shown in Fig.~(3).
This fit corresponds to an offset linear dependence of the {\it
thermal flux} on the temperature gradient $Q \propto (R/\LT -
R/\LTeff)$. Note that $R/\LTeff \simeq 6.0 > R/\LTcrit \simeq 4.0$,
where $\LTcrit$ is the temperature gradient scale length at which the
simulation is linearly marginally stable. This is a newly observed
phenomenon and is likely associated (see the discussion of ${\epsilon}$
% BD (11)
scans below) with undamped Rosenbluth-Hinton zonal \EB\ flows.~\cite{RH}

The linear critical temperature gradient $R/\LTcrit$ has been checked by
several different codes, and is found to be around 4.0 for this model
$s-\alpha$ equilibrium in both the flux-tube gyrokinetic particle code and
in Kotschenreuther's gyrokinetic code.  It is known that the quantitative
details of linear growth rates and critical gradients can be fairly
different in $\beta=0$ $s-\alpha$ model equilibrium (which makes large
aspect ratio approximations and has concentric circular flux surfaces)
versus a realistic numerically-calculated equilibrium with finite aspect
ratio (where even at zero $\beta$ there is still a Shafranov-shift of the
flux-surfaces due to the plasma current)\cite{Beer97,Redd}.  For the
$\beta=0$ $s-\alpha$ equilibrium used in these comparisons, the FULL code
gives a critical temperature gradient of about 3.7, while for a realistic
numerical equilibrium the critical gradients drops to 2.5\cite{Redd}.

%GFL

To be consistent, the 1994 IAEA version of the IFS-PPPL
model\cite{IFS-PPPL} is used to compare with the various simulations in
Fig.(3) because it also used adiabatic electrons in its parameterization of
$\chi_i$ and $R/\LTcrit$ (the 1995 version of the IFS-PPPL model in
Ref.\citenum{IFS-PPPL95} included linear gyrokinetic estimates of the
effects of trapped-electrons on $\chi_i$ and $R/\LTcrit$ as well as on
$\chi_e$).  The 1994 IFS-PPPL model predicts $R/\LTcrit = 3.1$ for this
case, but that is low because the IFS-PPPL model was based on
Kotschenreuther's linear gyrokinetic code using a more realistic
equilibrium than the $s-\alpha$ model used for the other codes in Fig (3).
The IFS-PPPL model was constructed to have a form $\chi = W_{NL}^{GF}
D_{mixing}^{GK} \propto G(R/L_T - R/\LTcrit)$, and so by construction
should go to zero at $R/\LTcrit$. Thus for a more consistent comparison
with simulations that assume a simpler $s-\alpha$ equilibrium, the IFS-PPPL
curve in Fig.(3) was shifted to the right from $R/\LTcrit=3.1$ to
$R/\LTcrit=4$.  Part of the reason the IFS-PPPL model was constructed in
this way was to attempt to correct for some known inaccuracies in the
gyrofluid prediction of the critical gradient and the growth rates near
marginal stability, where a slow growing residual mode can sometimes
persist in gyrofluid simulations below the gyrokinetic critical
gradient\cite{Waltz92,Beer94}.  [Improvements to the gyrofluid closures to
include frequency dependence or nonlinear effects, along the lines
suggested by Chang and Callen\cite{Callen} or Mattor and
Parker\cite{Mattor}, may help with this problem.]
% GWH: This essentially incorporates Scott Parker's 25 May 1999 suggestion,
% though I changed it slightly because the Nathan-Parker paper also makes
% important arguments justifying frequency-dependent closures.
Thus the IFS-PPPL curve is actually somewhat
below the gyrofluid simulations in Fig.(3) using the 1994 gyrofluid
closures\cite{Beer}.  The 1994-closure gyrofluid simulations also do not
%Dimits_7_29 replaced $R/\LTcrit$ with $R/\LT$ below - this may
%have been my typo.
show the roll-over at high $R/\LT$ that is in the IFS-PPPL model, which
would presumably be more apparent at other parameters\cite{Dorland96}.

%Compare Flux-tube GK and GFL
For the DIII-D base case parameters (at $R/\LT=6.9$), the IFS-PPPL model
$\chii$ is
a factor of 2.7 larger than the gyrokinetic flux-tube results, and the 1994
gyrofluid simulations are a factor of 3.3 higher.  Also in Fig.(3) are
simulations labeled ``98 PPPL GFL'' that employ a recent neoclassical
improvement of the gyrofluid closure\cite{Beer98}. This improved closure
reduces the gyrofluid/gyrokinetic difference to a factor of 2 for the base
case, and it is able to reproduce some of the nonlinear upshift in the
effective critical gradient seen by the gyrokinetic code.  This improved
gyrofluid closure, and the possibility of further improved closures, will
be discussed more in the next section.
%
The differences between the $\chii$
vs. $R/\LT$ curves for the flux-tube gyrokinetic code and the IFS/PPPL
model can be characterized partly as a shift in $R/\LTeff$, which is a
strictly linear value in the case of the IFS-PPPL model, and partly
that $\chi_i$ shows a more gradual increase as the critical gradient
is exceeded in the case of the gyrokinetic model.
%%
%% GWH: The following is repeated almost verbatim in Sec. 7, and so I
%% suggest removing it from here:
%%
% It is important to
% note that these differences cannot be characterized by a simple ratio,
% or multiplication factor.
% %
% For example, lowering $R/\LT$ just a bit from 6.9 to 6 (where the
% gyrokinetic simulations vanish) causes the relative error to become a
% factor of infinity.  Instead of comparing the predicted $\chii$'s at fixed
% temperature gradient, it is experimentally more relevant to turn Fig.(3)
% on its side and compare the predicted temperature gradients at fixed
% heating power.
The flux-tube gyrofluid/gyrokinetic differences in Fig.(3) correspond to a
20-33\% difference in the local temperature gradient predicted at fixed
heating power, as discussed in more detail in Sec.(7).

%MMM and compare with flux-tube GK and GFL

The MMM model result (labeled ``Weiland QL-ITG'') agrees with the
flux-tube gyrokinetic result quite closely for the base case, although
comparisons have not yet been carried out for a wider range of
parameters.  The MMM model also gives a reasonable prediction for the
linear $R/\LTcrit$,
%GWH
though both the MMM and the IFS-PPPL model miss the nonlinear increase
in the effective $R/\LTcrit$ observed in the gyrokinetic simulations.
% EndGWH

The MMM model and the fit to the gyrofluid simulations have a linear
%scaling for the transport with $R/\LT-R/\LTcrit$, whereas the
%gyrokinetic simulations are reasonably well fit by a square-root
%BC/nevins 7x30 to clarify that there are two fits to GK data
scaling for the transport with $R/\LT-R/\LTcrit$.  In addition to the
fit given by Eq.(1), the gyrokinetic simulations are also reasonably well
fit by a square-root dependence on $R/\LT-R/\LTeff$ which is of the
same form as the IFS/PPPL model. However, the offset-linear $Q$ vs.
$R/\LT$ fit to the LLNL gyrokinetic results is a better fit than the
best power-law fit.  The MMM model result agrees with the gyrokinetic
results for the base case parameters.  The MMM model also gives a
reasonable prediction for the linear $R/\LTcrit$.

%Dimits - Can add FULL code results and comments - are k_theta
%assumptions similar to Weiland? If so, agreement between MMM and
%FULL is good - just a calibration difference?

%Global GK and compare with flux-tube GK
The global gyrokinetic results are 2.4 times lower than the flux-tube
gyrokinetic results for the base case.  The global code used the same
local dimensionless parameters, a realistic
%Mostly GWH version here
temperature profile, and a value of $\rho/a= 1/160$ that is comparable to
the actual DIII-D experiment but is somewhat large compared to values
achieved in larger tokamaks such as JET or TFTR, and even larger
compared to proposed designs such as ITER.  Variations of the
simulated tokamak size in global gyrokinetic
simulations~\cite{Lin, Sydrhostar, Pkrhostar} show that
$\chii/\chi_{\rm gyroBohm}$ increases as the simulation is made larger.
This is consistent with a theoretical picture that radial variation in
various profile and profile-gradient quantities (e.g., the diamagnetic
velocity) introduces stabilizing effects~\cite{Waltz97,Garbetshr}
that get weaker in larger tokamaks.
% BD (13)
%For very large simulation size, the global gyrokinetic simulations are
For very large $a/\rho$, the global gyrokinetic simulations are
expected to asymptote to the value of $\chii$ given by the flux-tube
gyrokinetic simulations.  Another difference in the global simulations is
that the purely radial mode is coherent, fairly stationary, and has a
radial scale comparable to the minor radius, whereas in the flux-tube
simulations, the radial modes are at shorter wavelengths (smaller than the
box size).  Recent studies indicate that this difference will also go away
as the global simulation domain is made larger and the profile variation
weaker~\cite{Lin,Parker98}.

In the DIII-D experiment that the parameters for Fig.~(3) were based on,
the measured $\chii=0.16$ (in the units of Fig.~(3)) and the measured
$R/\LT=6.9$.
%BD (14)
%This is very low compared to all of the simulations in Fig.~(3), but such a
%comparison is not very meaningful.
This is very low compared to all of the simulations in Fig.~(3).  Although
this is consistent with the general picture of strong ITG turbulence
forcing the plasma to be near marginal stability,\cite{ITER-IFSPPPL} one
cannot draw this conclusion based solely on the simulations presented here.
This is because the primary purpose of Fig.~(3) is to compare different
simulations with as similar a set of assumptions as possible, so a number
of factors that are important in experiments are not included (such as
non-adiabatic electrons, equilibrium rotation, and realistic geometry).
Several of our simulation methods have been used to study equilibrium-scale
sheared rotation\cite{Waltz94, DimitsIAEA94}, which can be particularly
important in DIII-D because of its unidirectional beam injection and
resulting high toroidal rotation speeds.  For example, while the IFS-PPPL
model looks pessimistic compared to the experimental measurement in
Fig.~(3), applying the full IFS-PPPL model\cite{Dorland96} including a
model of the stabilizing influence of equilibrium-scale \EB\
flows\cite{Waltz94, HahmBurrell} gives a predicted central ion temperature
which is actually somewhat above the measured temperature.
% GWH: See for example ``Kinsey 4/1/97'' plots linked to at
% http://www.er.doe.gov/production/cyclone/
However, there are some quantitative uncertainties in the standard models
of stabilization due to \EB\ flow that lead to uncertainties in the
predicted temperature profiles of order 10-30\%, comparable in magnitude to
the 20-33\% differences in the gyrofluid/gyrokinetic temperature gradients
described above.  Some of these effects are hard to distinguish: comparable
levels of agreement with experiments can be obtained with modified
transport models where the magnitude of the \EB\ flows is reduced
while the the coefficient of $\chii$ is simultaneously
reduced\cite{Kinsey97} (the two effects offset each other somewhat).
Nevertheless, there are a wide range of experiments indicating the general
importance of equilibrium-scale \EB\ flows on the
transport\cite{Scott96, Burrell97, Synakowski98, Ernst98}.

%========================================================================
%Briefly discuss possible reasons we identified for disagreement.
%========================================================================

\vspace*{0.15in} \noindent
6.  POSSIBLE REASONS FOR THE DISAGREEMENTS
\vspace*{0.15in}

We now address possible reasons for the disagreements between the
various nonlinear results discussed above.  We focus first on the
differences between the GK/GF flux-tube simulations.  This is
important since the gyrofluid models are an attempt to approximate the
gyrokinetic equations for the problems of interest here, and the
physical and numerical parameters have been otherwise matched in the
comparisons. The differences between global and flux-tube simulations
have already been discussed.


There are many things that might, in the absence of concrete data, be
viewed as possible causes for the differences found between the GK/GF
flux-tube codes. These include differences in linear growth rates and
critical gradients, differences in the linear damping rates or
residual levels of the radial modes, nonconvergence with respect to
system size and grid size, nonconvergence with respect to particle
number (which may lead to excessive particle noise either in the
radial modes or in the modes that have finite poloidal or toroidal
variation), and nonlinear wave-particle effects.

The differences in linear growth rates and critical
gradients have been addressed above for the Cyclone DIII-D parameters
and are probably not important.
% and appears to be ruled out.

\vspace*{0.15in} \noindent
6.1 Causes of gyrofluid discrepancies, and improved gyrofluid closures.
\vspace*{0.15in}

As noted above, the original closures used in the gyrofluid simulations which
underlie the predictions of Ref.~\cite{ITER-IFSPPPL} damp the
poloidal flow to zero in most cases, and therefore do not properly
reproduce the long-time residual levels predicted by
Rosenbluth and Hinton~\cite{RH}.
This is probably the main cause of the
gyrofluid/gyrokinetic differences near marginal stability, and may account
for about half of the differences in stronger turbulence regimes.  But
there are cases where the gyrofluid/gyrokinetic differences do not appear
to be attributable to the differences in the radial mode linear
dynamics. The dependence of $\chi_i$ on $r/R$ from gyrofluid and
gyrokinetic simulations is similar, as shown in Fig.~(4).
The residual Rosenbluth-Hinton (RH) zonal flows vanish in the
%BC/nevins 7x30 clarify zonal flow initial condition relation to r/R=0 limit
limit $\epsilon=r/R=0$ for the initial conditions considered, so any
differences in that limit can not be attributed to those flows.
However, the trapped ion drive of the turbulence also varies with
$r/R$, and so this does not completely isolate just the effect of
undamped flows. [The simulations in Fig.~(4) are for the TFTR-based
NTP test case described in Sec.~(2), and show a smaller discrepancy,
of about a factor of 2, than the Cyclone base case.]  However there is
evidence that the Rosenbluth-Hinton undamped component of the zonal
flows is a significant part of gyrofluid errors, particularly near
marginal stability.  The $R/\LT$ flux-tube gyrokinetic simulation scan
at ${{\epsilon}=0}$ in Ref.~\cite{DimitsIAEA98} showed that $\chii$
becomes non-zero once $R/\LT$ becomes slightly larger than the linear
${\epsilon}=0$ critical value. This contrasts with the scan done for
${\epsilon}=0.18$ (and therefore finite $h$) in Fig.~(3), which
indicates that the undamped Rosenbluth-Hinton flows play a role in the
departure of $R/\LTeff$ from $R/\LTcrit$. This is further supported by
the observation that in the cases where $R/\LTcrit < R/\LT <
R/\LTeff$, radial mode potentials develop stationary structures in
which the peak shearing rates are significantly greater than the
growth rate of the fastest growing ITG modes.  Thus, the undamped
Rosenbluth-Hinton flows significantly affect the behavior of $\chii$
near marginal stability.

The Rosenbluth-Hinton (RH) component of the zonal flows are linearly
undamped except by collisions.  The fact that a non-zero $\chii$ is observed
in these collisionless gyrokinetic simulations for $R/\LT > R/\LTeff$ is an
indication that nonlinear damping of the RH zonal flows by turbulent
viscosity is able to balance the nonlinear drive of these flows.  One might
expect that the turbulent viscosity would increase as $R/\LT$ increases, so
that the RH zonal flows would become unimportant relative to the other
components of zonal flows when the turbulence is sufficiently strong that
the turbulent damping rate of the RH component of the zonal flows becomes
comparable to the damping rate due to collisionless transit-time magnetic
pumping that affects the other components of zonal flows.
% BD (18)
However, certain types of turbulence can exhibit inverse cascades, and
these issues warrant more study.

As was noted above, more recent nonlinear gyrofluid simulations have been
completed using improved closures\cite{Beer98} that do allow for levels of
RH undamped zonal flows in rough agreement with Ref.~\cite{RH}.  As seen in
Fig.~(3), these simulations show a nonlinear upshift in the effective
critical temperature gradient, though not yet as large as the upshift in
the gyrokinetic simulations.  The improved closure used at present is able
to match the RH residual flow to within 20\% at $k_r \rho \sim 0.2$, where
the dominant contribution to the effective shearing-rate\cite{Hahm98} is
usually made, but the residual flow is about a factor of 2 low at very low
$k_r \rho$.  Future work will investigate further improvements of the
neoclassical
% GWH: rephrased the above to avoid an ambiguous ``we'', as suggested by
% Bruce Cohen.
treatment of the gyrofluid closures to better match the RH residual flows,
which should bring the gyrofluid simulations into better agreement with the
gyrokinetic simulations in Fig.~(3).  Other improvements to the gyrofluid
%BC7x2: removed an ambiguous ``we'' in next sentence with use of passive
model to be investigated include frequency-dependent
closures\cite{Callen,Mattor}. This may be particularly helpful in improving
the approximation of the branch cut in the toroidal kinetic response
function\cite{Waltz92,Beer94} and in improving the calculation of the
linear critical gradient and growth or damping rates of various modes.  An
improved frequency-dependent gyrofluid closure may account for most of the
remaining difference between the gyrofluid and gyrokinetic simulations in
these collisionless-ion adiabatic-electron comparisons.

\vspace*{0.15in} \noindent
6.2 Nonconvergence with respect to system size and grid size
\vspace*{0.15in}

We have investigated and demonstrated convergence with respect to
system size and grid size for the flux-tube gyrokinetic and gyrofluid
simulations.  For the NTP test-case parameters, this
issue has been addressed in the flux-tube gyrokinetic
%Dimits_7_29 changed this - I think this is what is meant here:
%simulations~\cite{DimitsPRL}; and the parameters for the DIII-D base
%case are similar.  It was found that an increase in system
simulations~\cite{DimitsPRL}; and the results for the DIII-D base-case
parameters are similar.  It was found that an increase in system
size in the parallel direction made essentially no change in
%Dimits_7_30 new text here
%the simulation flux, while there was only a very weak dependence on
the simulation flux, while there was only a very small change between
simulation runs as if the perpendicular system size was increased or
decreased by a factor of 2 from the nominal value of $125 \rhoi$.
Similarly, it has been verified, both for the NTP test-case and
Cyclone DIII-D base-case parameters, that the parallel and
perpendicular grid sizes are adequate.
% GWH: add:
All of the simulations for the $R/\LT$ scans in Fig.(3) were done at fixed
system size.
%Dimits_7_29
Balancing the hydrodynamic and parallel
streaming frequences for the toroidal ITG modes
%Some theories
%BC7x2: Is ``Some theories'' ambiguous? Whose theories? Do we
%need a reference? Probably fine as is.
suggests that the turbulence shifts to longer
wavelengths at higher $R/\LT$, and so it may be useful to redo
%Dimits_7_29  eventually redo
the high $R/\LT$ simulations in a larger box size, which might cause the
$\chii$ to increase.
% some.
%Dimits_7_29 Add the following, but do we need this discussion at all? It seems
%to belabor a fine point. I will have a small-box-size Cyclone base run
%soon to make this argument more solid.
%Dimits_7_30 Changed text - checked base case at L=64 rho_i.
However, applying this scaling argument to the perpendicular-box-size
convergence check described above suggests that the box size that
was used is adequate even at $R/\LT=20$.
%
The gyrofluid simulations have also been tested for and appear to be
converged with respect to system size and grid size for the Cyclone
DIII-D base-case parameters, though more studies could perhaps be done.
A recent port of the gyrofluid code to the massively parallel T3E computer
will allow convergence checks at significantly higher resolution.
The comparisons made here between the
gyrokinetic and gyrofluid simulations were done at similar system
sizes, so this is very unlikely to be the cause of the difference.
The use of direct discretization in ballooning coordinates in the
gyrofluid simulations requires finer grid cells in the radial
direction than in the other perpendicular direction if magnetic shear
is present. Because of this and the fact that the mesh is explicitly
involved in representing the advection of the fluid fields,
establishing convergence with respect to grid size in the gyrofluid
%BC7x2: ``be be'' -> ``be'' in next line
simulations has been found to be more subtle and to impose more
stringent limits on the grid sizes in some cases than previously
thought.  Dimits has proposed an algorithm based on the periodicity of
discrete Fourier transforms that could help reduce the resolution
requirements in the gyrofluid code.

Next, we examine noise in the flux-tube gyrokinetic simulations, and
address the possibility of nonconvergence with respect to particle
number.
% BD (20)
Fig.~(5) shows $\chii$ vs. time from a particle number scan for the base case.
The simulations are for different numbers of particles ranging from $5
\times 10^{5}$ to $1.34 \times 10^8$, corresponding to 1 to 128 particles
per grid cell.  [These simulations used a $128 \times 128 \times 32$
(radial, poloidal, and parallel) grid, with finite-size particle filtering
to smooth out fields with $k_i > \sim 1/\Delta_i$, where $\Delta_i$ is the
grid spacing in the $i$'th direction.]  For $10^6$ or more particles,
$\chii$ at late time does not appear to change significantly with particle
number.
%BD (21)
The primary conclusion is therefore that $\chii$ appears to be
converged with respect to particle number for 2--4 or more particles per
grid cell for the base case parameters.
There is some random variation in the late time-averaged $\chii$
for the different cases.
%BC7x2: Suggest revision of
%When these random variations in $\chii$ (and
%somewhat larger random variations in the volume-averaged $\phi^2$) were
%observed in initial convergence studies over a smaller range of particle
%number, they caused some people to worry that there might be convergence
%problems.
%BC7x2 version:
% These random variations motivated convergence studies with respect to
% the number of particles.
% GWH7x6 version:
When these random variations in $\chii$ (and somewhat larger random
variations in the volume-averaged $\phi^2$) were observed in initial
convergence studies over a smaller range of particle number, questions
were raised that motivated convergence studies to the very large particle
number shown here, and motivated the additional scrambling tests
described below.
%
The particle number has now been varied over such a wide range,
and the random variations in $\chii$ are sufficiently small and show no
systematic dependence on particle number, that particle convergence does
not appear to be a problem.  This conclusion is made even more convincing
by the scrambling tests described below.  These random variations
are presumably just due to the sensitive dependence on initial conditions
of a chaotic system with long time-scale dynamics (for example, interactions
with low $k$ modes or zonal flows).  These small random variations
should average out over longer times or multiple realizations.
[The somewhat larger variations in volume-averaged $\phi^2$ are presumably
due to similar effects.  The $m=n=0$ zonal component of $\phi$ can have
quite large amplitudes at small $k_r$, but have little physical consequence
because their resulting shearing rate $\propto k_r^2 \phi_k$ is very
small.]
%
There is also some increase in the level of the
initial peak in $\chii$ which persists even if the scan is done
by increasing the initial weights (as the square root of the particle
number) so as to keep the initial mean noise level fixed.  The $5
\times 10^{5}$-particle case, which corresponds to one particle per
grid cell, shows secular growth in $\chii$ beyond Time $=700$.
% GWH: I thought the following change was a little clearer, but its no big
% deal to change back:
% BC7x2: I think this is ok. Andris will confirm.
This is probably due to a noise-driven runaway process in which the RMS
average particle weight, related to the detailed $\df$-particle entropy,
increases with
the time integral of $\chii$. The noise causes thermal transport
($\chii$), both of which increase together.


In order to further assess the impact of particle discreteness, the
following scrambling test~\cite{scramble} of the noise level was
performed. The gyrokinetic code was run saving restart files at
selected times. New restart files were formed from these by scrambling
the particle weight list. The gyrokinetic code was restarted from
these scrambled restart files.  After the restart, the temperature
gradient was reduced to slightly below the linear marginally
stable value in order to eliminate unstable ITG modes. The test was
done using $8 \times 10^6$ particles in the simulation.


Once the gyrokinetic code has run in the nonlinear phase longer than a
characteristic eddy turnover time, a typical simulation particle has
moved from its initial position farther than a characteristic eddy
radial scale. Eventually, simulation particles nearby in the $(\vx,
\vpar, \mu)$ phase space have weights that have large uncorrelated
components from which a physical density field must be
estimated. Physically this represents the failure of the particle
representation to resolve $\df$.
% BC7x2, minor rewrite of
%The ``worst-case'' interpretation of
%this uncorrelated component is as ``noise,'' although further work is
The ``worst-case'' interpretation is that this uncorrelated component is
``noise,'' although further work is needed to to establish a clear
interpretation.  The scrambling of the
particle weight list eliminates the physical signal, leaves a state
with a similar level of the uncorrelated component to the
pre-scrambling state, and therefore provides a measure of this
uncorrelated component.  The post-scrambling restarted simulations
were run long enough for stable geodesic acoustic fluctuations, which
are present immediately after the scrambling, to damp. The resulting
electrostatic potentials (or the shearing rates derived from them)
provide measures of the uncorrelated component of the signal in the
gyrokinetic simulation just prior to the scrambling.


Shown in Fig.~(6) are the time histories of $\chii$ and
the mean squared \EB\ shearing rate $(\LTi S / \vti)^2$ associated
with the flux-surface-averaged electrostatic potential, both in the
absence of scrambling and when the scrambling and gradient reduction
is done at three times during the run. Both of these quantities
decrease after the scrambling. The relative reduction is less the
later the scrambling is done, indicating a gradual buildup of
noise. However, even at the latest time, the post scrambling values
are down by an order of magnitude.  This indicates that the relative
impact of noise is small (or at most moderate at the latest time), and
supports the conclusion that the simulations are converged with
respect to particle number.
The noise shearing rate for 2 million particles (4 particles per cell)
would be a factor of 4 larger than shown in Fig.~6, and thus at late time
would be comparable to the pre-scrambling signal.  But much of that noise
shearing is at high $k_r$ and fluctuates rapidly in time, so it is less
effective than shearing by low $k_r$ modes\cite{Hahm98} and can be
ignored.

The numerical convergence of the particle codes with respect to
particles does seem to be sensitive to how close the system is to
marginal stability, however.  For values of $R/\LT$ somewhat lower than
% BD (22)
% the Cyclone DIII-D base case value, but above the linear marginal
the Cyclone DIII-D base case value, but well above the linear marginal
value of 4.0 (e.g., $R/\LT=5.3$ and 6.0), after the linear growth and
nonlinear saturation phases, the system evolves to stable states which
have radially dependent flux-surface-averaged temperature gradients
and \EB\ flows. In these situations the radial thermal flux asymptotes
to zero.
% BD (22)
%As many as 64 particles per cell are needed to observe this for $R/\LT=6.0$.
%Dimits_7_29
% What we know is that the number is somewhere greater than 32 but
% less than or equal to 64, so keep ``As many as...''
%64 particles per cell are needed to converge to this result
As many as 64 particles per cell are needed to converge to this result
for $R/\LT=6.0$.  Evidently, the stable nonlinear states become quite
delicate as a threshold value (larger than the linear critical value)
of the volume averaged temperature gradient is approached from below.

\vspace*{0.15in} \noindent
7. SENSITIVITY OF PREDICTED TEMPERATURE PROFILES AND FUSION GAIN TO
TRANSPORT MODEL VARIATIONS
\vspace*{0.15in}

The differences between the $\chii$
vs. $R/\LT$ curves for the flux-tube gyrokinetic code and the IFS/PPPL
model in Fig.(3) can be characterized partly as a shift in $R/\LTeff$,
which is a strictly linear value in the case of the IFS-PPPL model, and
partly that $\chi_i$ shows a more gradual increase as the critical gradient
is exceeded in the case of the gyrokinetic model. It is important to
note that these differences cannot be characterized by a simple ratio,
or multiplication factor.
%
For example, lowering $R/\LT$ just a bit from 6.9 to 6 (where the
gyrokinetic simulations vanish) causes the relative error to become a
factor of infinity.  Rather than compare the $\chii$'s at a fixed $R/\LT$,
one can instead turn Fig.(3) around and compare the predicted temperature
gradient at a fixed amount of heating power.  This way of characterizing
the gyrofluid/gyrokinetic differences is more relevant to experiments.
When heating power is added to a plasma, the temperature gradient on every
flux surface will rise until the resulting $\chii$ is large enough to
balance the rate at which the plasma is being heated.  The heat flux (or
power flow) across a given magnetic surface is given by $P = - n A \chi
\grad T$ in circular geometry, where $n$ is the density and $A$ is the
surface area.  This can be written as
%
\bealn
P = n A {\rho_i^2 v_{ti} \over L_n} {T \over R}
\, \, \hat{\chi}\left({R \over \LT}\right) {R \over \LT}
= P_0 \, \, g\left({R \over \LT}
\right).
\eealn
%
where $\hat{\chi}(R/\LT)$ is the normalized $\chii$ in the units of Fig.(3)
as a function of $R/\LT$, and $g(x)=\hat{\chi}(x)x$. For a given amount of
normalized heating power $P/P_0$, one can then solve this equation to find
the resulting temperature gradient $R/\LT = g^{-1}(P/P_0)$.  Carrying this
out for the IFS-PPPL curve in Fig.(3), and for the gyrokinetic flux-tube
results in Fig.3 (and Eq.(1)), we then take the ratio of these two
predicted temperature gradients to measure the relative error.
%
% Dimits 27-Apr email suggests rewording this section to make it clearer.
%
% where $\hat{\chi}(R/\LT)$ is the normalized $\chii$ in the units of Fig.(3)
% as a function of $R/\LT$. This equation can be inverted to find the
% (normalized) temperature gradient $R/\LT$ as a function of the normalized
% heating power $P_{norm}$, where $P_{norm} = P/ [n c T^2 \rho/ (e B L_n
% R)]$. Carrying this out for the IFS-PPPL curve in Fig.(3) results in a
% particular function of $R/LT = g_{GF}(P_norm)$, and for the gyrokinetic
% curve in Fig.(3) will lead to a
%
This ratio is plotted vs. normalized heating power $P/P_0$ in Fig.(7).  At
low heating power, the temperature gradient will be close to marginal
stability.  In this limit in Fig.(3), the IFS-PPPL linear critical gradient
$R/\LTcrit = 4$, which is 33\% low compared to the gyrokinetic nonlinear
critical gradient of $6$, explaining the result in Fig.(7) at low
power. At high heating power, the temperature gradients can pull away from
marginal stability, a regime where the differences in the IFS-PPPL and
gyrokinetic $\chii$'s are less and the predicted temperature gradients
differ by $\sim 20\%$. [The range of normalized heating power $P/P_0$ of 0
to 250 in Fig.(7) will cause the gyrokinetic predicted temperature gradient
to vary from $R/\LT = 6$ to 22.5, i.e., from close to marginal stability to
far above marginal stability.]  Thus we see that the temperature gradient
predicted by the IFS-PPPL model is only 20-33\% lower than the temperature
gradient predicted by the gyrokinetic simulations over a wide range of
heating powers. The fact that the predicted temperature gradient at fixed
heating power is less sensitive
% BC7x2: ``errors'' or ``model variations''?
%to errors
to model variations than the predicted $\chii$ at
fixed temperature gradient is a consequence of the critical gradient
feature of ITG turbulence.  Given the difficulties of the plasma turbulence
problem, a turbulence theory that predicts temperature gradients to within
20-30\% can be considered a significant achievement in many ways.  But the
fusion reaction cross-section scales as $\sim T^2$, and the resulting
fusion power feeds back to give more heating, so the performance of a
fusion device at high gain (near ignition) becomes fairly sensitive to the
transport.  Thus one would like to have even higher accuracy in the
transport model.

Next we consider the sensitivity of predictions of fusion power performance
to variations in the assumed transport model.
Both the 94 and 95 versions of the IFS-PPPL model are of the
form
%
\bealn
\chi_{\rm IFS-PPPL} = G\left({R \over \LT} - {R \over \LTcrit} \right) W
\label{eq:IFSfit}
\eealn
%
where $W$ and $R/\LTcrit$ are functions of various plasma parameters,
$G(x)=\min(x,x^{1/2})$ $\times H(x)$, $H(x)$ is the Heaviside function, and
the argument of $G$ is $x = R/L_T - R/L_{Tcrit}$ as shown.
%where $W$ and $R/\LTcrit$ are functions of various plasma parameters,
%$G(x)=\min(x,x^{1/2})$ $\times H(x)$, and $H(x)$ is the Heaviside function.
In order to fit the gyrokinetic flux-tube simulations shown in Fig.(3) and
described in Sec.(5), this was modified to be of the form of Eq.(1),
%
\bealn
\chi_{\rm GK} = \left[ {R \over \LT} - {R \over \LTeff} \right] {4.0 \over
R/\LT} W,
\label{eq:GKfit}
\eealn
%
where $R/\LTeff = R/\LTcrit+2$ is assumed, and the parameterizations of
$R/\LTcrit$ and $W$ given by the IFS-PPPL model are used. While this fits
Fig.(3) and matches Eq.(1) for the Cyclone DIII-D parameters, more detailed
studies need to be carried out to develop a gyrokinetic-based model which
has been tested over a wide range of parameters.  For example, the
nonlinear upshift in the effective $R/\LTcrit$ is probably not a constant
value of 2 in reality and should depend on various parameters (such as
collisionality, as demonstrated in recent gyrokinetic
simulations\cite{Lin99}).  But for now we will use these two equations,
Eq.(\ref{eq:IFSfit}) and Eq.(\ref{eq:GKfit}), to show the sensitivity of
the predicted fusion power gain to variations in the transport model that
are roughly of the magnitude represented by these two equations.

% GWH: moved this paragraph, otherwise unchanged:
Another possible fit to the gyrokinetic flux-tube simulations in Fig.(3),
which is not quite as good as Eq.(\ref{eq:GKfit}) but is more easily
compared with the IFS-PPPL model, is $\chi_{\rm GK2} = 0.8 G(R/\LT -
R/\LTeff) W$.  This corresponds to a rescaling and a shifting of the IFS-PPPL
model to an effective critical gradient $R/\LTeff=R/\LTcrit+2$.  The
resulting predictions are fairly similar to the predictions of Eq.(4) in
Fig.(8).  Using $R/\LTeff=1.5 R/\LTcrit$ would lead to slightly
($\sim$10\%) lower $Q$ predictions.

Fig.(8) shows the predicted fusion gain $Q=P_{\rm fusion}/P_{\rm aux}$ for
a particular reactor design, versus the boundary condition assumed for the
temperature at the top the pedestal caused by the H-mode transport barrier.
[In H-mode experiments the pedestal temperature can be much higher than the
separatrix temperature at the last-closed flux surface, so the distinction
between the two can be important.]  The assumptions in Fig.(8) are
described in more detail below.  Predictions are shown for the standard
IFS-PPPL(95) transport model, and the gyrokinetic-based version of this
model (given by Eq.(\ref{eq:GKfit}).  At a fixed pedestal temperature of 3
keV, $Q$ rises significantly, from Q=5.6 for the original IFS-PPPL model to
Q=13.5 for the gyrokinetic-based version.  However, the results are still
fairly sensitive to the assumed pedestal temperature boundary condition, so
there is a risk of $Q$ significantly below 10 even with the gyrokinetic
fit.  The gyrokinetic fit demonstrates primarily the sensitivity of the
results to a shift in $R/\LTeff = R/\LTcrit + 2$.  There are other possible
sources of uncertainty, so we also show the $Q$ curve predicted if the
gyrokinetic $\chii$ of Eq.(\ref{eq:GKfit}) is further reduced by a factor of
% BD (23)
% gyrokinetic $\chii$ of Eq.(\ref{eq:GKfit}) is further reduced by a factor of
% 0.5 (ignoring that this introduces problems when one tries to
% interpret existing L-mode data)\cite{IFS-PPPL95}.
% GWH possibility:
% 0.5 (putting aside the question of how such a model would compare with
% experiments).
%Dimits_7_29 BD's concern here seems legitimate, and I (personally) like
%GWH's diplomatic way of stating it.
%0.5.
0.5 (putting aside the question of how such a model would compare with
experiments).
In this case high $Q$ operation, $Q > 10$, can be achieved even at
fairly low pedestal temperatures.  These results are sensitive to the
achievable density.  The 3 cases just described use the standard assumption
in the 1996 ITER baseline scenario of a density 1.5 times the Greenwald
density.  Because of uncertainties about the achievable density, later ITER
designs considered lower density operating points, and the effect of
lowering the density to 1.15 times the Greenwald density is shown in
Fig.(8) for the most optimistic of these 3 cases ($\chi = \chi_{GK}/2$).

To be clear, the 94 version of the IFS-PPPL model\cite{IFS-PPPL} was used
in comparing with the adiabatic electron results of Fig.(3), while the 95
version of the IFS-PPPL model\cite{IFS-PPPL95,Dorland96} is used in
predicting the fusion reactor gain $Q$ in Fig.(8) because it includes
quasilinear/mixing-length estimates of the destabilizing effects of trapped
electrons that are thought to be important in real experiments.  The
trapped electrons cause an increase in $W$ and a drop in $R/L_{Tcrit}$.
Well above marginal stability, trapped electrons cause
roughly a factor of two increase in the 95 version of $\chii$ relative to
the 94 version, though this depends on parameters.  There is linear and
nonlinear evidence that non-adiabatic electrons can add significant
additional drive to ITG turbulence and lower the ITG mode critical gradient
(for example, see Refs.(\citenum{Beer, Sydrhostar, Beer97, Redd}), and it is
the 95 version of the IFS-PPPL model
% BC7x2: missing a ``that''
that has been more widely compared with
experiments.  As described in Sec. 5, the simulations in Fig.(3) used a
simplified concentric-circle equilibrium where the linear critical gradient
was about 4. The 94 and 95 IFS-PPPL models used a more realistic
equilibrium and so the 94 IFS-PPPL model predicts a lower $R/\LTcrit =3.1$.
The 95 IFS-PPPL model predicts that for the parameters of the DIII-D base
case, trapped electrons will lower $R/\LTcrit$ further to 2.1, while the
stabilizing influence of impurities ($Z_{eff}=2.37$), beams ($n_{beam}/n_e
= 0.05$), and $T_i/T_e = 1.183$ will raise the linear $R/\LTcrit$ back up to
4.2 (the most important of these 3 factors is the impurities).  These
various stabilizing influences will be weaker in ITER than in DIII-D, but
there may be various possible sources of uncertainty in the
IFS-PPPL parameterizations of these complicated effects that could be
investigated further.  This is beyond the scope of the present paper, but
Fig.(8) can be used as a rough guide to the sensitivity of the predicted
$Q$ to changes in the critical gradient $R/\LTcrit$ of 2 and to changes in
the coefficient of $\chii$ of a factor of 2.
%The GLF23 transport model\cite{Waltz97} shares some similarities to the
%IFS-PPPL model and is normalized to nonlinear gyrofluid simulations, but it
%uses a quasilinear/mixing-length numerical procedure (similar to techniques
%used in the Bateman-Weiland multi-mode procedures\cite{MMM}) to predict the
%various parametric dependences of $\chii$ instead of the analytic
%parameterization of the IFS-PPPL model.
The GLF23 transport model\cite{Waltz97} shares some similarities to
the IFS-PPPL model and is normalized to nonlinear gyrofluid
simulations, but it uses a quasilinear/mixing-length dispersion type
approach (similar to Bateman's implementation of the Weiland
model\cite{MMM}) to predict the various parametric dependences of
$\chii$ instead of the analytic parameterization of the IFS-PPPL
model.  Predictions of ITER by the GLF23 model show similar trends as
the IFS-PPPL model, including a strong dependence on the pedestal
temperature\cite{Kinsey97, Waltz97}.

The calculations in Fig.(8) were done with a standard type of transport
code; similar ITER-related calculations can be found in
Refs.\citenum{ITER-MMM, Mikkelsen98, ITERConnor, Kinsey97, Waltz97}.  The
plasma parameters used here are taken from the baseline scenario in use for
the design of ITER circa 1996: major radius $R=8.14$ m, midplane minor
radius $a=2.8$ m, elongation $\kappa_{95}=1.6$, triangularity
$\delta_{95}=0.24$, magnetic field
$B_{tor}=5.68$ Tesla, plasma current $I_p=21$ MA, $P_{\rm aux}=100$ MW of
auxiliary heating (assumed to be centrally deposited with a Gaussian
half-width of $r/a=0.1$, all deposited in the ions to maximize $T_i/T_e$).
beryllium impurities with $n_{Be}/n_e=0.02$ were assumed, and $n_{He}/n_e$
was determined by $\tau_{p*He}/\tau_E=10$.  The $q$ profile was chosen so
that $q$ on axis is 0.8 (lowering the central $q$ below 1 is favorable in
this transport model), and the midplane radius of the $q=1$ surface was
$r/a=0.43$.  Sawteeth are ignored. Neoclassical ion transport is included
but has little effect.
Particle transport and any associated convective heat transport is ignored,
and any other transport mechanisms that are sometimes thought to play a
role at high $\beta$ or near the edge of the plasma are neglected.
Brehmsstrahlung radiation is included, but any other line radiation or
charge exchange losses are ignored.
%
The calculations in Fig.(8) also include a small amount of favorable
elongation scaling found in Ref.\cite{Dorland96}, $\chi \rightarrow
\chi/(1+((\kappa-1) q/3.6)^2)$.  The effects of elongation and shaping will
be discussed below in more detail.  Equilibrium-scale sheared flows, which
are thought to get weaker in larger tokamaks, are neglected.

As in the 1996 ITER baseline scenario, a flat density profile is assumed
with $n_e=1.3 \times 10^{20}/$m$^3$, corresponding to 1.5 times the
Greenwald density limit.  Because of uncertainties about whether ITER could
operate at such a high density, later ITER designs considered lower density
operating points, and the effects of lowering $n_e$ to $0.98 \times
10^{20}/$m$^3$ (corresponding to 1.15 times the Greenwald density) are
shown for one case in Fig.(8).  This lower density operating point includes
$0.16\%$ argon for the operation of a radiative divertor, in addition to
the $2\%$ beryllium.

The nominal design goal of ITER was $P_{\rm fusion} = 1500$ MW, so in the
cases in Fig.(8) where $Q>15$ it would be possible to lower the auxiliary
power and $Q$ would rise further.  Because of the stiffness of the
temperature profiles, in cases with low $Q$ in Fig.(8) it is sometimes
possible to raise the $Q$ by lowering the auxiliary heating power (assuming
a fixed pedestal temperature), though this would not help on the power
loading and nuclear testing goals which require a certain level of $P_{\rm
fusion}$.

% GWH: Somehow describe the results diplomatically but objectively.

% GWH: in response to Bruce's suggestion for a shorter version of the
% paragraph that had been here, here is my suggested shorter version:

To summarize the results of Fig.(8), at fixed pedestal temperature, the
gyrokinetic-based model achieves significantly higher $Q$ than the original
gyrofluid-based model, but the results are still sensitive to the
achievable density and to the assumed pedestal
temperature, which is fairly uncertain and could be less than 1.5 keV.
Some experiments on the largest tokamaks find that the H-mode
pedestal width scales linearly with the (poloidal)
gyroradius\cite{Hatae98,Cordey98}, while other experiments find more
optimistic scalings that are weakly dependent on or independent of
gyroradius\cite{Osborne98}.
Some simple theoretical models of H-mode pedestal scaling (such as
Refs.\cite{ITER-IFSPPPL, Perkins96}) give pedestal widths
proportional to the gyroradius $\rho$, consistent with some of the
largest tokamak experiments, while other models give
a $\rho^{2/3} R^{1/3}$ or weaker scaling.
Pedestal models with a strong dependence on gyroradius tend to predict
very low pedestal temperatures when extrapolated to regimes of high density
relative to the Greenwald density limit.
As stated in one review\cite{FESAC97}, ``While, given the
present state of knowledge, we cannot provide a reliable estimate of the
pedestal parameters in ITER \ldots, a pedestal temperature less than 1500
eV, perhaps much less, is a distinct possibility.''
% GWH: The more complete FESAC quote is below.
On the other hand, there are uncertainties both ways, and there is a
possibility that the pedestal temperature could be sufficiently high.
Also, there are various methods that may be able to improve the temperature
at or near the pedestal, such as pellet fueling, RF waves or low voltage
beams to drive sheared flows, or stronger plasma shaping.  Some of
the new ITER-RC designs that are presently under consideration have
significantly stronger plasma shaping (higher elongation and triangularity)
and may have much higher pedestal temperatures than the earlier 1996 ITER
design.

% As described in the FESAC review of the ITER physics
% basis (spring 97): ``While, given the present state of knowledge, we cannot
% provide a reliable estimate of the pedestal parameters in ITER, it must be
% stated that a pedestal temperature less than 1500 eV, perhaps much less, is
% a distinct possibility.  The severity of the consequences of such a low
% pedestal depend on the validity of the stiff marginal stability transport
% models, which predict rather poor performance (Q < 5) for ITER if the
% pedestal temperature is less than 3-4 keV.''

% GWH: The main points in the following section that I originally wrote
% for the end of Sec. 7 have mostly been moved to final section:
%
% In addition to the issues of the achievable density and pedestal
% temperatures, there are a number of other uncertainties in the turbulence
% simulations that need further investigation.  Electromagnetic effects,
% plasma shaping, and plasma-generated sheared flows can have significant
% stabilizing influences in some cases.  There are a range of semi-empirical
% or theory-based transport models that are less stiff (and thus less
% sensitive to the assumed pedestal temperature) than the original IFS-PPPL
% model or even than the gyrokinetic variant of Eq.(\ref{eq:GKfit}), and yet
% achieve similar levels of comparisons with steady-state
% measurements\cite{Mikkelsen98}. There is general evidence for
% critical-gradient types of transport models (it helps explain experimental
% sensitivity to edge conditions such as wall recycling and H-mode transport
% barriers, and the possibility of fast transient dynamics).


\vspace*{0.15in} \noindent
8. SUMMARY AND DISCUSSION
\vspace*{0.15in}

This paper has focused on the differences between the gyrofluid and
gyrokinetic simulations illustrated in Fig.(3) (for the simplified case of
adiabatic electrons, circular geometry, collisionless ions).  The resulting
sensitivity of the predicted temperature gradients are shown in Fig.(7),
and the sensitivity of the predicted fusion gain in Fig.(8).

A primary result of this paper is that detailed noise and other tests have
been carried out for the nonlinear gyrokinetic particle simulations, and
they appear to be well-resolved and correct for this parameter regime.  For
the Cyclone DIII-D parameters, at the fixed temperature gradient of
$R/\LTcrit=6.9$, the $\chii$ from gyrofluid flux-tube simulations are a
factor of 3.3 higher than the gyrokinetic flux-tube simulations (Fig.(3)).
However, turning Fig.(3) around to find the predicted temperature gradient
for a fixed amount of heating power, we find that these gyrofluid errors
lead to only a 20-33\% drop in the predicted local temperature gradient, as
shown in Fig.(7).
[This relative insensitivity is a general feature of critical-gradient
types of models near marginal stability.]  While these errors are relatively
small in one sense, because the fusion cross-section scales as $T^2$, and
because the resulting fusion power feeds back to give more heating, the
performance of a fusion device at high gain (near ignition) becomes fairly
sensitive to the transport model and one would like to have higher accuracy
than 20-33\%.  As shown in Fig.(8), modifying the IFS-PPPL model to better
fit the gyrokinetic simulations causes the fusion gain $Q$ to rise
significantly at fixed pedestal temperature.  But the results are still
sensitive to the assumed pedestal temperature, which is fairly uncertain.
%BD (27)
%There is still a risk of low $Q$, particularly at low density or low
%pedestal temperature.
There is still a risk of low $Q$, particularly if high density can not be
achieved or if the pedestal temperature is low.
% BD (28)
Of course, the uncertainties go both ways, and it remains possible that
the original ITER design may be adequate to achieve ignition.
Other sources of uncertainty which need better understanding, in addition
to the issues of the pedestal temperature and the achievable density,
include the effects of elongation and plasma shaping, collisional damping
of zonal flows, plasma rotation, the possibility of density peaking, and
fully electromagnetic simulations with non-adiabatic electrons.

It is important to note that the present gyrokinetic/gyrofluid comparisons
have been done in a simplified case of collisionless electrostatic ITG
turbulence with adiabatic electrons.
%BC/nevins 7x30 Because additional physics may change things will
%not assuage the critics who contend that we have not
%explained adequately differences in results with the current physics
%set. Also, additional physics make make differences worse. So remove
%sentence.
%Additional physics has the potential to reduce the differences.
One of the main causes of the
gyrofluid/gyrokinetic differences is that in the nonlinear gyrokinetic
simulations there is a significant nonlinear upshift in the effective
critical gradient, $R/\LTeff \approx R/\LTcrit + 2$, due to the generation
of an undamped component of zonal flow\cite{RH}, which then suppresses
further turbulence.  This nonlinear upshift is probably not a constant
value of 2 and future work should investigate how it depends on various
parameters.  Linear gyrokinetic calculations by Dorland have found that
this undamped component of the zonal flows may increase with elongation
(and thus make the gyrofluid/gyrokinetic disagreement worse, though the
other effects described next will tend to reduce the disagreement).  Recent
work by Z. Lin et.al.\cite{Lin99} shows that including a small but
experimentally-relevant amount of collisions can lead to enough damping of
the zonal flows to cause a significant increase in the heat conductivity
$\chii$, particularly near marginal stability.  This will tend to reduce the
effective critical gradient back towards its linear value and will cause
the gyrokinetic $\chii$ to rise to be closer to the gyrofluid-based IFS-PPPL
$\chii$, although the resulting $\chii$ near marginal stability will probably
depend strongly on ion-ion collisionality (an effect that is not in the
IFS-PPPL model or any other current transport model).  Recent work by
Diamond and others\cite{Diamond99,Lin99}indicates that the zonal flow
generation observed near marginal stability is related to inverse-cascades
and a resulting negative turbulent viscosity.  The collisionless nonlinear
gyrokinetic simulations presented here demonstrate that far above marginal
stability it is possible to reach a steady state where the nonlinear
generation of zonal flows is balanced by the turbulent viscosity, so a
non-zero $\chii$ can be achieved that presumably does not depend strongly on
ion-ion collisions.  An interesting topic for future work would map out the
transition between these two regimes of strong or weak collisionality
dependence.  Non-adiabatic electrons are known to limit inverse cascades
in some types of plasma turbulence, and so including non-adiabatic
electrons in gyrokinetic ITG simulations may further reduce the
gyrofluid/gyrokinetic differences.  Non-adiabatic electrons may also push
the plasma into stronger-turbulence regimes where the differences are less
important.

There has been some recent work\cite{Beer98} on developing improved
gyrofluid closures that incorporate some of the neoclassical effects needed
to model an undamped component of the zonal flows. As shown in Fig.(3),
this new gyrofluid closure eliminates about half of the original
gyrofluid/gyrokinetic difference, and helps isolate the source of the
differences.  Possible further improvements in the closures will be
investigated.

The IFS-PPPL and gyrofluid results in Fig.(3) are stiffer than the Weiland
and gyrokinetic results,
that is, they show a steeper increase in transport going above threshold.
Global gyrokinetics show yet lower transport levels, but are still
% BD (29)
%limited to values of $\rho/a$ lower than the regime of ITER.  It would
limited to values of $a/\rho$ lower than the regime of ITER.  It would
be interesting to undertake an $a/\rho$ scan about the DIII-D base
parameter set. Such a scan is probably now possible, even if not to
ITER-like values. The Weiland model gives transport levels in the same
range as the flux-tube gyrokinetic results for these parameters (though
like the IFS-PPPL model it misses the nonlinear upshift in the critical
gradient).  It is interesting that
% BD (30)
%the Weiland model agrees better with gyrokinetic simulations, since
the Weiland model agrees better with gyrokinetic simulations in this
circular limit, since
the Weiland model is based on a simpler fluid theory than the
gyrofluid model.

The elongation scaling is an important issue for determining the
implications for ITER or other reactor designs. The nonlinear comparisons
so far have been in simple circular geometry.  There are significant
differences in the transport models regarding elongation when scaled to
ITER plasmas.  The Multi-Mode model has an empirical elongation scaling
based on experimental data for the scaling of the confinement time with the
current, which leads to $\chii$ scaling asymptotically as ${\kappa}^{-4}$,
where ${\kappa}$ is the elongation
parameter.  The IFS-PPPL model has a much weaker elongation dependence,
based on initial linear gyrokinetic and nonlinear gyrofluid
calculations\cite{Dorland96} which saw little effect at moderate elongation
($\kappa < 1.6$) typical of the original ITER design.  However, with very
strong plasma shaping (very high elongation and triangularity, or at high
Shafranov shift and/or low aspect ratio) there may be a transition to a
regime of significantly improved
confinement\cite{Beer97,Kotschenreuther99}.  The shaping criterion needed
for this improvement needs to be studied in more detail.  Given the
constraints on the models described, the predictions for ITG transport in
an ITER plasma based on extrapolations from the gyrokinetic simulations and
the Weiland model are less pessimistic than the IFS-PPPL model predictions
based on gyrofluid simulations.  ITER predictions for the Weiland-based
%BC7x2: fixed a tex-typo [[ ]]
Multi-Mode model are in Ref.{\citenum{ITER-MMM}}.
It is interesting to note that there are a
number of different transport models with fairly different scalings on
various parameters (and different amounts of stiffness) that are
nevertheless able to achieve similar levels of comparisons with the
experimental profile database that has been developed for
ITER\cite{Mikkelsen98}.  For example, the IFS-PPPL model is able to follow
many of the main trends in this database from circular to elongated
tokamaks despite its weak elongation dependence because of other factors
which are also correlated with elongation (higher edge temperatures,
broader $q$ profiles, and toroidal rotation).  More detailed studies, such
as with controlled rotation scans or with perturbative heat pulse
propagation experiments, should help to better distinguish between
transport models.  More complete three-dimensional nonlinear simulations,
building on the
simulations done here, will also help in developing transport models that
can be used to predict and optimize the design of fusion reactors.

% GWH: as I described earlier in the text, there are a lot of known physics
% (such as equilibrium flows and trapped electrons) that are dropped in
% these idealized simulations in Fig.(3), so I don't think it is really
% relevant to compare them with experiments directly. Thus comment out the
% next section:
%
% It is also interesting to note that the gyrokinetic simulations give
% transport values less than the experimental value for the NTTP (TFTR)
% base case, but larger than the experimental value for the Cyclone
% DIII-D parameters. Some of this difference may be attributable to
% incomplete physics in the test cases we have we have used here for
% comparison purposes. Equilibrium plasma flows, electrons, impurities
% and realistic equilibria are certain to have a substantial impact on
% the simulation predictions. The gyrofluid simulation predictions are
% larger than the gyrokinetic predictions for both cases.

% BC7x2: The experimentally inferred values of chii for the TFTR NTP case
%and the DIII-D case are mentioned earlier in the paper.

% GWH: The next paragraph duplicates some things said already, but leave it
% as is for now:

Possible future tasks that may shed more light on the differences between
various transport models, and help develop more accurate transport models,
include more detailed comparisons between codes and experiments of
fluctuation spectra, and of the poloidal flow dynamics.  More complete
scans of $a/\rho$ with the
global codes with the same toroidal resolution as for the flux-tube
codes will be of great interest as such scans become possible.
Further comparisons which systematically remove the simplifications
made in the comparisons made here, including shaped equilibria,
equilibrium-scale sheared flows,
nonadiabatic electrons, and beam and impurity species are also
essential to making quantitative assessments of differences in
predictions of ITER performance between the models. Work is also under
way to design experiments that the turbulence simulations can model
with more certainty.

%Dimits: Any updates on where this went - e.g., any DIII-D shots?


\vspace*{0.15in} \noindent
ACKNOWLEDGMENTS
\vspace*{0.15in}

Several of us have had useful discussions on these topics with C.
%Dimits_7_30 Add WWlee to acknowledgments
Bolton, W.W.~Lee, Z.-H. Lin, F. Hinton, M.N. Rosenbluth, and
R.E. Waltz.  The ITG simulations were performed on the C-90 and T3E
computers at NERSC.  A. J. Redd and A. H. Kritz would like to thank
G. Rewoldt and W. M.  Tang for allowing them to use the FULL code, and
for useful advice.  This work was performed under the auspices of
U.S.~Department of Energy (DOE) by Lawrence Livermore National
Laboratory under contract No.~W-7405-ENG-48, by Princeton Plasma
Physics Laboratory under contract No. DE-AC02-76CH03073, and by
various other DOE contracts and grants, and is part of the Numerical
Tokamak Turbulence Project supported by the High Performance Computing
and Communications Program in the U.S.\ DOE.

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% ``Higher Order Geometric
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% GWH I think this could be replaced with Pkrhostar reference above, which
% is more accessible and complete and contains references to this and
% earlier work by Parker.  But it is fine with me to keep it both
% references...



% Hahm-Burrell PoP paper on ExB shearing rates in general geometry.
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% Burrell review article on importance of ExB flow:
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% "Formation and Structure of Internal and Edge Transport Barriers,"
% GWH: Above is broader review paper, below was primarily TFTR:
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% (1997).

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% IAEA 98 paper on role of ExB flow in a wide range of TFTR
% improved-confinement shots.

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% ``Shearing rate of time-dependent ExB flow''

\bibitem{scramble} The scrambling noise test was proposed originally
by M.\ Kotschenreuther, and valuable variations and refinements were
suggested by W.\ Dorland, G.W.\ Hammett and M.N.~Rosenbluth in private
communications dating between 1995 and 1997.


\bibitem{Hatae98} T. Hatae, Y. Kamada, S. Ishida, \etal., Plasma
Phys. Control. Fusion {\bf 40}, 1073 (1998).

\bibitem{Cordey98} J.G. Cordey and the JET Team,
% ``H-mode power threshold and confinement in JET H, D, D-T and T Plasmas''
{\it Proc. 17th Int. Conf. on Fusion Energy}, (Yokohama, 1998) (IAEA,
Vienna, 1999).
% Cordey's paper is IAEA-F1-Cn-69/EX7/1

\bibitem{Osborne98} T.H. Obsborne, K. H Burrell, R.J. Groebner, \etal.,
% ``H-mode pedestal characteristics in ITER shape discharges on DIII-D''
to be published in Plasma Phys. and Control. Fusion.

\bibitem{Perkins96} F.W. Perkins, D. Post, M.N. Rosenbluth, et al.,
in {\it Proc.\ 16th.\ Int.\ Conf.\ on Fusion Energy, 1996} (IAEA, Vienna,
1997), Vol. 2, p. 963.


% FESAC 97 review of the ITER physics basis.
\bibitem{FESAC97} Panel Report To Fusion Energy Sciences Advisory Committee
(FESAC) - ``Review of the International Thermonuclear Experimental Reactor
(ITER) Detailed Design Report'', April 18, 1997, Appendix D.I. \\
{\tt http://wwwofe.er.doe.gov/More\_HTML/REPORTAP
(for Microsoft Word),
http://w3.pppl.gov/$\sim$hammett/ark/1997/REPORTAP.pdf
http://wwwofe.er.doe.gov/More\_HTML/FESACREP.pdf}
%(GWH ??: will submit the full FESAC97 report including the appendix to the
%AIP or NTIS online archives so they can be cited in Phys. Plasmas.)

\bibitem{Lin99} Z. Lin, personal communication.

\bibitem{Diamond99} P. Diamond, personal communication.

\bibitem{Kotschenreuther99} M. Kotschenreuther, W. Dorland, Q.P. Liu,
M. Zarnstorff, R.L. Miller, Y.R. Lin-Liu,
% ``Attaining Neoclassical Transport ig Ignited Tokamaks''
{\it Proc. 17th Int. Conf. on Fusion Energy}, (Yokohama, 1998) (IAEA,
Vienna, 1999); to be published in Nucl. Fusion {\bf 39}, (1999).

% GWH: I now think we don't need to reference these:
% i \delta nonlinearity papers:
% \bibitem{TH} P.~W.~Terry and W.~Horton, Phys.~Fluids {\bf 26}, 106 (1983).
%
% \bibitem{Waltz82} Earlier Waltz paper pointed out $i \delta$ effects also,
% and even did 3D?

\end{thebibliography}

\vfill\eject


\bf \noindent
Figure Captions

\vspace*{0.3in}
\rm \noindent
Fig.~(1) $\gamma$ and $\omegar$ vs.\ $\ktheta$ for the Sydora (global)
and Dimits (flux-tube) nonlinear gyrokinetic codes, for the
Kotschenreuther and Rewoldt (FULL) linear gyrokinetic codes, the Beer
nonlinear gyrofluid code, and the Weiland fluid calculation.

\vspace*{0.3in}
\noindent
Fig.~(2) Residual \EB\ flow fraction vs.\ Rosenbluth-Hinton parameter
$h \equiv \sqrt{\epsilon}/q^2$. The points are results from the
Dimits~\etal.\ flux-tube gyrokinetic code and the line is the prediction
of Ref.~\cite{RH}.

\vspace*{0.3in}
\noindent
Fig.~(3) $\chii$ vs.\ $R/\LT$ from the gyrofluid code using the
1994 ``thesis closure''~\cite{Beer}, an improved 1998 gyrofluid closure,
the 1994 IFS-PPPL model~\cite{IFS-PPPL}, the LLNL and U.~Colorado
flux-tube and UCLA (Sydora) global gyrokinetic codes, and the MMM
model for the DIII-D base case.

\vspace*{0.3in}
\noindent
Fig.~(4) The dependence of $\chi_i$ on ${\epsilon} = r/R$ from gyrofluid
and gyrokinetic simulations.

\vspace*{0.3in}
\noindent
Fig.~(5) Normalized $\chii$ vs.\ $t \vti /\LT$ from gyrokinetic simulations
with particle numbers ranging from $5\times 10^5$ to $1.34 \times 10^8$,
corresponding to 1 to 128 particles per cell..

\vspace*{0.3in}
\noindent
Fig.~(6) Time histories of (a) $\chii$ and (b) the mean squared \EB\
shearing rate $(\LTi S / \vti)^2$ associated with the
flux-surface-averaged electrostatic potential, both in the absence of
scrambling and when restarts with scrambling and gradient reduction
are done at three times during the run. Cyclone DIII-D base-case
parameters were used.

\vspace*{0.3in}
\noindent
Fig.~(7) The ratio of the temperature gradient predicted by the IFS-PPPL
model to the temperature gradient predicted by the gyrokinetic flux-tube
simulations, versus the normalized heating power, based on the results in
Fig.(3).  Thus the gyrofluid/gyrokinetic differences in Fig.(3) correspond
to only a 20-33\% difference in the predicted temperature gradient at a
fixed amount of heating power.

\vspace*{0.3in}
\noindent
Fig.~(8) The predicted fusion gain Q vs. assumed pedestal temperature, for
the IFS-PPPL 95 model, for a modified model to fit the gyrokinetic flux-tube
results of Fig.3 (``GK fit''), and for a further reduction in $\chii$ by a
factor of 2 (``GK fit/2'').  These 3 cases are at 1.5 times the Greenwald
density.  Also shown is a lower density case at 1.15 times the Greenwald
density using the ``GK fit/2'' $\chii$.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\vfill\eject

\begin{figure}[ht]
\begin{center}
\epsfig{file=figs/cyc.fig1.ps, height=7.5in}
\end{center}
\vspace{-0.1in}
Fig.~(1) $\gamma$ and $\omegar$ vs.\ $\ktheta$ for the Sydora (global)
and Dimits (flux-tube) nonlinear gyrokinetic codes, for the
Kotschenreuther and Rewoldt (FULL) linear gyrokinetic codes, the Beer
nonlinear gyrofluid code, and the Weiland fluid calculation.
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsfig{file=figs/cyc.fig2.ps, height=5.5in, %
         clip=,bbllx=80,bblly=200,bburx=500,bbury=600}
\end{center}
\vspace{-0.1in}
\noindent
Fig.~(2) Residual \EB\ flow fraction vs.\ Rosenbluth-Hinton parameter
$h \equiv \sqrt{\epsilon}/q^2$. The points are results from the
Dimits~\etal.\ flux-tube gyrokinetic code and the line is the prediction
of Ref.~\cite{RH}.
\end{figure}


\begin{figure}[ht]
\begin{center}
\epsfig{file=figs/cyc.fig3.ps, height=8.0in,
         clip=,bbllx=150,bblly=100,bburx=540,bbury=710}
% angle=-90, but then have to make it much smaller to fit on page
\end{center}
\vspace{-0.4in}
\noindent
Fig.~(3) $\chii$ vs.\ $R/\LT$ from the gyrofluid code using the
1994 ``thesis closure''~\cite{Beer}, an improved 1998 gyrofluid closure,
the 1994 IFS-PPPL model~\cite{IFS-PPPL}, the LLNL and U.~Colorado
flux-tube and UCLA (Sydora) global gyrokinetic codes, and the MMM
model for the DIII-D base case.
\end{figure}


\begin{figure}[ht]
\begin{center}
\epsfig{file=figs/cyc.fig4.ps, height=5.5in,
         clip=,bbllx=75,bblly=300,bburx=450,bbury=650}
\end{center}
\vspace{-0.1in}
\noindent
Fig.~(4) The dependence of $\chi_i$ on ${\epsilon} = r/R$ from gyrofluid
and gyrokinetic simulations.
\end{figure}


\begin{figure}[ht]
\begin{center}
\epsfig{file=figs/cyc.fig5.ps, height=6.0in, angle=90,
         clip=,bbllx=100,bblly=300,bburx=500,bbury=600}
\end{center}
\vspace{-0.1in}
\noindent
Fig.~(5) Normalized $\chii$ vs.\ $t \vti /\LT$ from gyrokinetic simulations
with particle numbers ranging from $5\times 10^5$ to $1.34 \times 10^8$,
corresponding to 1 to 128 particles per cell..
\end{figure}


\begin{figure}[ht]
\begin{center}
\epsfig{file=figs/cyc.fig6.ps, height=6.0in,
         clip=,bbllx=50,bblly=100,bburx=500,bbury=700}
\end{center}
\vspace{-0.1in}
\noindent
Fig.~(6) Time histories of (a) $\chii$ and (b) the mean squared \EB\
shearing rate $(\LTi S / \vti)^2$ associated with the
flux-surface-averaged electrostatic potential, both in the absence of
scrambling and when restarts with scrambling and gradient reduction
are done at three times during the run. Cyclone DIII-D base-case
parameters were used.
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsfig{file=figs/cyc.fig7.ps, height=3.0in}
\end{center}
\vspace{-0.1in}
Fig.~(7) The ratio of the temperature gradient predicted by the IFS-PPPL
model to the temperature gradient predicted by the gyrokinetic flux-tube
simulations, versus the normalized heating power, based on the results in
Fig.(3).  Thus the gyrofluid/gyrokinetic differences in Fig.(3) correspond
to only a 20-33\% difference in the predicted temperature gradient at a
fixed amount of heating power.
\end{figure}

\begin{figure}[ht]
\begin{center}
\epsfig{file=figs/cyc.fig8.ps, height=3.0in}
\end{center}
\vspace{-0.1in}
Fig.~(8) The predicted fusion gain Q vs. assumed pedestal temperature, for
the IFS-PPPL 95 model, for a modified model to fit the gyrokinetic flux-tube
results of Fig.3 (``GK fit''), and for a further reduction in $\chii$ by a
factor of 2 (``GK fit/2'').  These 3 cases are at 1.5 times the Greenwald
density.  Also shown is a lower density case at 1.15 times the Greenwald
density using the ``GK fit/2'' $\chii$.
\end{figure}



\vfill \eject

\end{document}
\exit
