of the system at $t/t_0=625$, at the onset of the [[straight jet]{}]{}phase, is presented in Fig. \[Fig:BetaStand\]. The evolution of the free magnetic energy, $E_{mag}$ (defined earlier in Sect. \[sec:Model\]), and the kinetic energy, $E_{kin}$, for the runs at different $\beta$ are presented in Fig. \[Fig:BetaE\].

The first result of this parametric study is that, for all the values of $\beta$ tested, a [[helical jet]{}]{} always occurs eventually. Figure \[Fig:BetaDyn\] shows the existence of a higher-density/higher-temperature region related to helical upflows and the presence of an upward propagating nonlinear Alfvénic wave, as first introduced in PAD09 and discussed further in Sect. \[sec:Betadriver\] below.

We observe in all runs that the jet consists of a left-handed helical density structure. The specific density distribution (shown by the isodensity surfaces) in the different [[helical jets]{}]{}exhibits variations in the width and pitch of the helical density structure, which we discuss further in Sect.\[sec:Betamorpho\]. All of the simulations present a similar distribution of $v_x$ (red/blue color-coding of the isodensity surface) indicating plasma flowing away from the observer on the left side of the jet and toward the observer on the right. For each parametric run, this red/blue pattern on opposite sides of the jet characterises 