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