Ion temperature gradient (ITG) driven instabilities develop when the ion temperature has a step enough gradient as compared to the density gradient. This instability is considered important in both tokamaks and stellarators producing fluctuations and transport at the ion scales. It is affected by the magnetic geometry and several branches (slab, toroidal, helical) can develop depending on it.
Here we show some examples of simulations of ITG instabilities in the stellarators TJ-II, LHD, and W7-X.
Simulations in TJ-II
In these simulations, the electrons are adiabatic and the ion and electron temperatures are artificially large to make the simulations cheaper computationally. The density and temperature profiles are ideal. Density and electron temperature profiles are flat while the ion density profile has a sharp gradient at the middle radius. The simulations are linear and started with numerical noise. As the simulation evolves the scales (modes) that are unstable start growing at an exponential rate and a clear structure, defined by the most unstable mode(s), appears.
Figure 1 shows the normalized power spectrum amplitude of the electrostatic potential versus m and n mode numbers, for an ideal simulation of ITG in TJ-II. The integral in radius is shown in the figure.
As the simulation evolves, the amplitude of the most unstable modes grows and a clear structure is defined in the Fourier plane. The modes are resonant and aligned along a straight line in the Fourier plane, which is defined by the value of the rotational transform.
linear simulation of ITG in TJ-II
In Figure 2, the electrostatic potential at the plane at toroidal angle phi=0 is shown in a linear simulation of ITG instability in TJ-II. The propagation of the perturbation in the counterclockwise (ion drift) direction is appreciated.
The ITG instability has been shown not to be relevant in TJ-II for the typical density and temperature profiles in TJ-II plasma scenarios, which have a quite flat ion temperature profile [Sánchez et al EPS 2014].
Simulations in LHD (under construction)
Simulations in W7-X (under construction)