GOALS

The goal of the Edge
Simulation Laboratory (ESL) is to build an edge gyrokinetic code based on
continuum (as opposed to particle-in-cell) methods. The code will be runnable
as a 4-dimensional (axisymmetric) code on transport timescales, or as a full
5-D turbulence code. Target
applications include edge turbulence and transport, H-mode pedestal physics,
and ELM physics. The project will build on experience gained from the edge
kinetic code TEMPEST that has been in development at LLNL, as well as from the
core code GYRO developed at GA.

In the edge region of fusion
devices, the ion drift orbit width can be comparable to the pedestal width (or
temperature gradient scale length), and electron and ion mean free paths can be
comparable to or exceed scale lengths along a field line. In consequence, quantitatively
significant simulation requires *kinetic simulation*.
Furthermore, the kinetic simulation must be fully nonlinear (as opposed
to the Ò*d**f*Ó approach commonly used for core simulation, where the
distribution function is expanded about a Maxwellian), because the lowest-order
distribution function under these conditions is not Maxwellian and not known in
advance, and because turbulent fluctuation levels can be locally O (1) or
larger.

Since the timescales of
interest are long compared to the cyclotron period, *gyrokinetics* (in which the fast motion of gyration
about magnetic field lines is appropriately averaged) is still applicable. However, the orderings used to derive
the gyrokinetic equations used in core codes break down in the edge due to the
following important edge physics. First, the pedestal buildup and ELM cycles
require a large-amplitude time-dependent inhomogeneous electromagnetic field to
be self-consistently included in the gyrokinetic model. Secondly, the
characteristic turbulence scale lengths and drift-orbit widths become
comparable to the equilibrium radial gradient scale lengths. A set of generalized gyrokinetic
equations, valid for the edge, has been derived by Qin (PPPL) under the
auspices of the LLNL LDRD project. This formalism allows large-amplitude,
time-dependent background electromagnetic fields to be developed nonlinearly in
addition to small-amplitude, short-wavelength electromagnetic perturbations.
However, it remains to reduce these equations to a form convenient for coding,
and also to develop an appropriately ordered gyrokinetic Coulomb collision
operator.

We are adapting *continuum, *rather than particle-in-cell, techniques. While a PIC code is in some ways
simpler to construct, and is familiar to our group from years of experience
with core gyrokinetics, we believe that the continuum approach is preferable
for edge simulation. The issues
are particle noise, the computational cost of a suitable PIC-based collision
operator, and the relative ease of implementing implicit long-timescale
methods. The usual *d**f* approach is inappropriate for the edge,
as noted above, and hence one of the principal means of reducing noise in PIC
simulation is unavailable. Because we intend to use the kinetic code for both
turbulence and transport timescales (in, respectively, 5-dimensional and
4-dimensional instances), implicit techniques are essential to enable long
timesteps. In particular, implicit
techniques are needed to do any reasonably efficient simulation of turbulence
with kinetic electrons, and particularly when performing transport-timescale
simulations with nonlinear transport models (or with nonlinear transport
conceivably coming from the 5D instance of the kinetic code itself). Implicit methods are more
straightforward when the entire system is fluid-like equations. Also, we note
that core continuum gyrokinetic codes, such as GYRO, GS2 and GENE, have been
found to be very successful and productive, robustly able to provide a
physically comprehensive treatment (including all the key ingredients for core
turbulence: kinetic ions and electrons, electromagnetic fluctuations,
equilibrium and turbulence-driven ExB flows, and shaped flux surfaces). Finally, since PIC edge codes are being
developed by the Center for Plasma Edge Simulation (C.S. Chang et al.) and by a
group at Helsinki University (Heikkenin et al..), our continuum approach is
complementary.

We have chosen to develop a *new
code* rather than
retrofitting a core continuum gyrokinetic code such as GYRO, because of
complexities attached to edge simulation: the sizeable variation of the
electrostatic potential F along magnetic field lines, implying
significant variation of a particleÕs kinetic energy and particle trapping with
turning points that cannot be computed *a priori* (the solution for F
is needed), require different strategies for variable choices and
discretization; the divertor geometry implies a more complicated spatial
connectivity of regions than in the core; and the equations themselves need to
be fully nonlinear and generalized as noted above.

The code is
being constructed with *field-line-following coordinates *to reduce the grid requirements (since
edge fluctuations are strongly field-line aligned, and because the plasma
equilibrium is also characterized by much longer scale lengths along than
across magnetic field lines). A
significant challenge arises when the magnetic field evolves significantly
during the simulation, as it likely does during an ELM crash. We hope to
address this in the future, should funding allow, by implementing a dynamic
grid realignment strategy.