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Review of Mixing Length Estimates and Effects of Toroidicity in a Fluid Model for Turbulent Transport in Tokamaks

Journal article, 2016

Abstract—Basic aspects of turbulent transport in toroidal magnetized plasmas are discussed. In particular the
fluid closure has strong effects on zonal flows which are needed to create an absorbing boundary for long wave
lengths and also to obtain the Dimits nonlinear upshift. The fluid resonance in the energy equation is found
to be instrumental for generating the L–H transition, the spin-up of poloidal rotation in internal transport
barriers, as well as the nonlinear Dimits upshift. The difference between the linearly fastest growing mode
number and the corresponding longer nonlinear correlation length is also addressed. It is found that the
Kadomtsev mixing length result is consistent with the non-Markovian diagonal limit of the transport at the
nonlinearly obtained correlation length.
DOI: 10.1134/S1063780X16050184
1. INTRODUCTION
Turbulent transport in magnetic confinement systems
has been one of the central research areas in the
nuclear fusion research all the time since the research
started in the beginning of the 1950s [1–14]. Originally
it was thought that turbulent transport was
mainly caused by linear instabilities that were saturated
nonlinearly. Later it has become more and more
clear that background flows play an important role in
the dynamics of transport. As pointed out in [15]
sheared flows have two roles in turbulent transport.
The first is to create an absorbing boundary for the
driving eigenmodes at long wavelengths. This can be
achieved also for rather moderate flows since long
wavelength eigenmodes are more efficiently damped
by sheared (zonal) flows than shorter wavelength
modes. The second is to cause a significant damping,
sometimes leading to transport barriers, at the correlation
length of the turbulence.
The second case usually requires strong heating,
leading to steep transport barriers or strong drive of
toroidal rotation from neutral beams. Thus, there are
many cases where an absorbing boundary for long
wavelengths is a sufficient way of representing flows. A
very common method in turbulence simulations is to
apply an artificially large viscosity to damp out the turbulence
at larger wavelengths than would happen with
the natural viscosity. As long as there is only outgoing
flux toward shorter wavelengths (no reflections), this
does not change the situation around the source and
just saves computer time. This procedure is called
Large Eddy Simulation (LES). In our first simulations
of ion temperature gradient (ITG) turbulence, we
applied also an artificial damping at large wavelengths.
This damping was adjusted to give a minimum of the
transport and thus corresponded to absorbing boundary
also for long wavelengths. This gave a transport of
the order of the Kadomtsev mixing length level if we
used the wavelength of the fastest growing mode.
Thus, this works in a similar way to LES, but would
physically be due to zonal flows. This also means that
it is sufficient to be able to describe finite Larmor
radius (FLR) effects at the correlation length. The
main flows in the tokamaks can be divided into zonal
flows [5, 6] with zero real eigenfrequency and geodesic
acoustic modes (GAMs) with finite real eigenfrequency
[7–9]. When the real eigenfrequency of GAMs
is much lower than other frequencies in the dynamic
system we consider, they work effectively as zonal
flows. When the eigenfrequency of GAMs becomes
comparable to other frequencies, a careful investigation
of their roles gets important. The turbulence
responsible for transport in tokamaks typically has a
real eigenfrequency two orders of magnitude below the
ion cyclotron frequency. As seen both by mixing length
arguments [3, 10], renormalization [4], and experiments
[11], a typical turbulence level is given by
(1)
where the linear eigenvalue was written as
and kr is the inverse correlation length in the direction
1 The