Vertical Shear, Diapycnal Shear and the Gradient Richardson Number

In Cartesian coordinates x,y,z, the gradient Richardson number Ri is the ratio between the square of the buoyancy frequency N and the square of the vertical shear S, Ri=N2/S2, where N2=−g/ρ ∂ρ/∂z and S2=∂u/∂z2+∂v/∂z2, with ρ potential density, u,v the horizontal velocity components and g gravity acceleration. In isopycnic coordinates x,y,ρ, Ri is expressed as the ratio between M2≡N−2 and the squared diapycnal shear Sρ2=ρ/g2∂u/∂ρ2+∂v/∂ρ2, Ri=M2/Sρ2. This could suggest that a decrease (increase) in stratification brings a decrease (increase) in dynamic stability in Cartesian coordinates, but a stability increase (decrease) in isopycnic coordinates. The apparently different role of stratification arises because S and Sρ are related through the stratification itself, Sρ=S/N2. In terms of characteristic times, this is equivalent to τ≡Sρ=to2/td, which is interpreted as a critical dynamic time τ that equals the buoyancy period to≡N−1 normalized by the ratio td/to, where td=S−1 is the deformation time. Here we follow simple arguments and use field data from three different regions (island shelf break, Gulf Stream and Mediterranean outflow) to endorse the usefulness of the isopycnal approach. In particular, we define the reduced squared diapycnal shear σρ2=Sρ2−M2 and compare it with the reduced squared vertical σ2=S2−N2, both being positive (negative) for unstable (stable) conditions. While both Ri and σ2 remain highly variable for all stratification conditions, the mean σρ2 values approach Sρ2 with increasing stratification. Further, the field data follow the relation σρ2=1−Ri/N2Ri, with a subcritical Ri=0.22 for both the island shelf break and the Mediterranean outflow. We propose σρ2 and Sρ2 to be good indexes for the occurrence of effective mixing under highly stratified conditions.