## Benchmarks

This page lists some basic benchmarks that have been undertaken. A number of other benchmarks are detailed in the Manual.

### acknowledgement

We want to thank W. Dorland and M. Kotschenreuther for providing the GS2 code against which many of the benchmarks have been made.### ITG with adiabatic electrons

The standard benchmark for linear problems is the growth rate of the Ion Temperature Gradient mode (ITG) as a function of the poloidal wave vector kρs for the so-called Cyclone base case (safety factor q = 1.4, magnetic shear s = 0.78, inverse aspect ratio r / R = 0.19, normalized temperature gradient length R/LT = 6.9, normalized density gradient length R/LN = 2.2, electron to ion temperature ratio Te/Ti = 1, electro-static, and adiabatic electrons). The growth rate for this case is shown in the right panel of the figure above as a function of the normalized poloidal wave vector (kρs) for various values of the normalised ion temperature gradient (R/LT = 6.9, 8.28 10.35, 12.44, and 15.18) calculated with both GKW (red diamonds) and GS2 (blue line). As can be seen from the figure, good agreement is obtained. The left panel, furthermore, shows the comparison of the perturbed potential as a function of the coordinate along the field line (s) for both codes. Here the full line gives the real while the dashed line gives the imaginary part of the potential. The benchmark is published in the Main reference of the code.

### Zonal flow test

For nonlinear runs the proper response of the zonal flows is of utmost importance. The standard benchmark that addresses the physics is the zonal flow / geo-acoustic mode is the Rosenbluth-Hinton test which has an analytical solution. In this benchmark the initial condition is an ion density perturbation with a finite (small) radial wave vector, and no dependence on either the coordinate along the magnetic field s or the binormal coordinate ζ. The adiabatic electron response is used, keeping the correction due to the flux surface average of the potential. The density perturbation generates a potential perturbation and excites the so called geo-acoustic mode. This mode is damped and a small residual poloidal flow remains. The left panel of the figure above gives the potential perturbation as a function of time. The relevant parameters used for this benchmark are: safety factor q = 1.3, inverse aspect ratio r / R = 0.05, normalized poloidal wave vector kρs = 0.02, electrostatic, and collisionless. Rather large grid sizes (points along the magnetic field in one turn Ns = 128 and parallel velocity grid points Nvk = 128) are used to avoid the recurrence problem. The right panel of the above figure shows the comparison of the residual with both the Rosenbluth-Hinton as well as the Xiao-Catto calculation. The original Rosenbluth-Hinton theory is accurate to lowest order in the inverse aspect ratio only, while the Xiao-Catto calculation retains higher order terms. The numerical results (that do retain higher order terms) are in good agreement with the latter theory.