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In this paper we study the scaling relations for the triggering of the fast,or "ideal", tearing instability starting from equilibrium configurationsrelevant to astrophysical as well as laboratory plasmas that differ from thesimple Harris current sheet configuration. We present the linear tearinginstability analysis for equilibrium magnetic fields which a) go to zero at theboundary of the domain and b) contain a double current sheet system (the latterpreviously studied as a cartesian proxy for the m=1 kink mode in cylindricalplasmas). More generally, we discuss the critical aspect ratio scalings atwhich the growth rates become independent of the Lundquist number $S$, in termsof the dependence of the $\Delta'$ parameter on the wavenumber $k$ of unstablemodes. The scaling $\Delta'(k)$ with $k$ at small $k$ is found to categorizedifferent equilibria broadly: the critical aspect ratios may be even smallerthan $L/a \sim S^{\alpha}$ with $\alpha=1/3$ originally found for the Harriscurrent sheet, but there exists a general lower bound $\alpha\ge1/4$.

Onset of fast "ideal" tearing in thin current sheets: dependence on the equilibrium current profile