In the Quiescent H-mode (QH-mode), ELMs are replaced by a small but continuous long wavelength instability called Edge Harmonic Oscillation (EHO) which saturates non-linearly
at the plasma edge [1]. Recent analytical and numerical modelling suggest that EHOs correspond to the saturated state of external infernal (exfernal) modes [2, 3], driven unstable by
the pressure gradient in the pedestal region and the corresponding weakening of the safety
factor profile close to a rational surface. The interplay between these two effects defines the
parameter space for the excitation of exfernal modes, which can be modified by different
features such as edge magnetic shear, plasma shaping, the inclusion of a plasma separatrix
and the application of non-axisymmetric external magnetic perturbations. A detailed first principle investigation is presented on how such effects modify the linear and non-linear
parameter space of EHOs in static plasmas. For simplification, the study neglects plasma
flow, whose effect has already been treated in previous analytical work [2]. The linear stability analysis is performed analytically on a large aspect ratio tokamak, and numerically
using the KINX code. The non-linearly saturated 3D structure associated with EHOs is recovered in free boundary equilibrium calculations using the VMEC code [4] and in realistic
tokamak geometries. The obtained parameter space for excitation and saturation of exfernal
modes might offer possible routes to robustly access QH-mode.