Tracking the centroid along with the 1σ, 2σ, and 3σ shadows, which represent the distribution of material for b₀/H₀ = 2/5 and Δx = b₀/40. The dashed curve represents the subsidence of the centroid, while the dotted line shows the analytical model prediction for the depth of the leading edge of the vortex ring from Sect. 2, z_lead ≡ z_centroid + b(z)/2. The solid line represents the bottom boundary of the simulation. We use dimensionless depth, ζ = z/b₀, and dimensionless time, τ = t/(b₀/g′)^1/2. The dashed-dotted line shows the penetration depth, z_p, computed using Eq. (26) for reference.