Dispersion relation and eigenfunctions of the north-south ζ_z-symmetric columnar convective modes in the case of uniform rotation, no viscosity, and adiabatic stratification. (a) Dispersion relation of the north-south ζ_z-symmetric columnar convective modes in red points. For comparison, dispersion relation analytically derived using one-dimensional cylinder model by Glatzmaier & Gilman (1981) is overplotted in black dashed line. (b) Schematic illustration of flow structure of the mode. Red and blue volume rendering shows the structure of ℜ[ζ_z(r, θ)exp(imϕ − iωt)] for m = 6 at t = 0. (c) Meridional cuts of the m = 2 eigenfunctions for the velocity v(r, θ)exp[i(mϕ − ωt)], the pressure p₁(r, θ)exp[i(mϕ − ωt)], and the z-vorticity ζ_r(r, θ)exp[i(mϕ − ωt)]. The solutions are shown in the meridional plane at ϕ = 0 and t = 0 where v_r and v_θ are purely real and v_ϕ, p₁ and ζ_z are purely imaginary. The units of velocity, pressure, and vorticity are m s⁻¹, 10⁵ dyn cm⁻², and 10⁻⁸ s⁻¹, respectively. The eigenfunctions are normalized such that maximum of |v_ϕ| is 2 m s⁻¹. (d) The same as panel c but for m = 8.