On the Turbulent Behavior of a Magnetically Confined Plasma near the X-Point

Abstract

We construct a model for the turbulence near the X-point of a Tokamak device and, under suitable assumptions, we arrive to a closed equation for the electric field potential fluctuations. The analytical and numerical analysis is focused on a reduced two-dimensional formulation of the dynamics, which allows a direct mapping to the incompressible Navier-Stokes equation. The main merit of this study is to outline how the turbulence near the X-point, in correspondence to typical operation conditions of medium and large size Tokamaks, is dominated by the enstrophy cascade from large to smaller spatial scales.

Symmetries of the Large Scale Structures of the Universe as a Phenomenology of a Fractal Turbulence: The Role of the Plasma Component

“…We clearly see that the energy was mainly concentrated in the first two modes, while the remaining part of the spectrum rapidly decayed as 1/k 2 . This was the result of the so-called condensation phenomenon, which is well-known in fluid dynamics [12]: while a direct cascade of enstrophy was observable and it corresponded to the 1/k 2 profile (see [54] for a detailed discussion), an inverse cascade of energy produced the lowest k values. This scenario, when referred to the cosmological setting, clarifies why we reliably apply the fragmentation due to eddy vorticity to two spatial scales only: de facto clusters of galaxies and galaxies.…”

On the Effects of Tokamak Plasma Edge Symmetries on Turbulence Relaxation

“…In this work, we consider a homogeneous and isotropic hydrogen-like plasma, and we adopt a two-fluid model. The following assumptions are implemented [8,10]: plasma quasi-neutrality (since the turbulence scale is much larger than the Debye length) (i.e., N i = N e ≡ N , with N i/e indicating the ion/electron number density), and thus we also consider p i = p e = p = N K B T (where the temperature T is equal for ions and electrons, in which K B is the Boltzman constant); the ion polarization drift velocity is taken as the only relevant contribution affecting the ortogonal current density; we neglect the diamagnetic effects (i.e., the pressure gradient effects on the electron and ion velocities); and the parallel ion velocity is negligible.…”

“…The time step has been set by ensuring the "energy" conservation (of O(10 −8 )) of the n = 0 mode. We recall that by ruling out the dependence on w in the whole scheme, the model is actually isomorphic to a 2D incompressible fluid [10], where the electric fluctuations correspond to the stream functions [12]. The relevant difference between the two schemes relies in the low spatial scale cut-off described above.…”

“…The basic features of the electrostatic turbulence (i.e., the magnetic fluctuations are neglected for sufficiently large values of the plasma's β parameter) are captured by the so-called Hasegawa-Wakatani model [6][7][8], which corresponds to a coupling between the dynamics of the number density and the electric vorticity (i.e., the Laplacian of the electric potential). In two dimensions, when the tokamak axisymmetric structure is extended to the turbulent fluctuations, the electric potential field dynamics is isomorphic to that of the stream function in the Euler equation for an incompressible fluid [9][10][11]. However, in such a limit, if the magnetic field is assumed to be along the z direction, then the drift coupling is absent, and the steady state of the system is associated with an enstrophy cascade and, in some cases, also responsible for an inverse energy cascade (known as the "condensation phenomenon"), which allows for the formation of large-scale eddies [12].…”

Effect of resonant magnetic perturbation on edge–core turbulence spreading in a tokamak plasma

“…Magnetic shear and flow shear are reported to be important for ITB formation, and the properties of turbulence spreading in the weak magnetic shear region [18] and through the strong flow shear region [19,20] have been reported. Furthermore, turbulence propagation throughout the magnetic island [21][22][23] and stochastic magnetic layer [24] have also received attention.…”