*2.3. Transient Growth Analysis*

In order to study the linear stability of the SW profile and to determine the perturbations that grow the most on top of it within one pulsation period, we have performed transient growth analysis (TGA) for the parameter space at hand. This non-modal method returns the most dangerous perturbation in terms of energy growth out of all possible axial/azimuthal wavenumbers and pairs of initial (*t*0) and final (*tf*) times. To this end, the governing Equations (2) are linearised (LNSE). The LNSE and their adjoint counterpart are integrated forward and backward in time iteratively, until such an optimum is reached for each combination of *Re*, *Wo*, and *A*. During integration, the underlaying velocity profile develops in time (see, e.g., Figure 3a), but remains unchanged by the developing perturbations.

The LNSE and their adjoint are discretised using a Fourier–Galerkin ansatz in θ and *z* and a Chebyshev collocation method in *r*. Further details are given in Barkley et al. [23], and our TGA computations were undertaken using an in-house Matlab script.
