Using the local gyrokinetic code, GS2, to investigate global ITG modes in tokamaks. (I) s-$α$ model with profile and flow shear effects

Abstract: This paper combines results from a local gyrokinetic code with analytical theory to reconstruct the global eigenmode structure of the linearly unstable ion-temperature-gradient (ITG) mode with adiabatic electrons. The simulations presented here employ the s-${\alpha}$ tokamak equilibrium model. Local gyrokinetic calculations, using GS2 have been performed over a range of radial surfaces, x, and for ballooning phase angle, p, in the range -${\pi} {\leq} p {\leq\pi}$, to map out the complex local mode frequency, ${\Omega_0(x, p) = \omega_0(x, p) + i\gamma_0(x, p)}$. Assuming a quadratic radial profile for the drive, namely ${\eta_i = L_n/L_T}$, (holding constant all other equilibrium profiles such as safety factor, magnetic shear etc.), ${\Omega_0(x, p)}$ has a stationary point. The reconstructed global mode then sits on the outboard mid plane of the tokamak plasma, and is known as a conventional or isolated mode, with global growth rate, ${\gamma}$ ~ Max[${\gamma_0(x, p)}$], where ${\gamma_0(x, p)}$ is the local growth rate. Taking the radial variation in other equilibrium profiles (e.g safety factor q(x)) into account, removes the stationary point in ${\Omega_0(x, p)}$ and results in a mode that peaks slightly away from the outboard mid-plane with a reduced global growth rate. Finally, the influence of flow shear has also been investigated through a Doppler shift, ${\omega_0 \rightarrow \omega_0 + n\Omega^{\prime}x}$, where n is the toroidal mode number and ${\Omega^{\prime}}$ incorporates the effect of flow shear. The equilibrium profile variation introduces an asymmetry to the growth rate spectrum with respect to the sign of ${\Omega^{\prime}}$, consistent with recent global gyrokinetic calculations.