Antisymmetric (kink) mode (Eq. (15)) solutions for the non-magnetic case (B_z = 0) when α = 1 and k_y, V_0y = 0 (corresponding to the solution of Drazin 2002). Upper and lower panels: real and imaginary parts of frequency, respectively. Red solid lines correspond to unstable modes, and blue dashed lines to the damped modes. Imaginary solutions (and hence instability) vanish for k_zd > 1.8 and for k_zd → 0. The frequency and growth rate are normalised by k_zV_0z.