Vertical Shear, Diapycnal Shear and the Gradient Richardson Number - Josep L. Pelegrí; Mariona Claret; Pablo Sangrà (Oceans)
In Cartesian coordinates <semantics><mfenced separators="|">x,y,z</mfenced>,</semantics> the gradient Richardson number <semantics>Ri</semantics> is the ratio between the square of the buoyancy frequency N and the square of the vertical shear S, <semantics>Ri=N2/S2</semantics>, where <semantics>N2=−<mfenced separators="|">g/ρ</mfenced><mtext> </mtext>∂ρ/∂z</semantics> and <semantics>S2=<mfenced separators="|">∂u/∂z</mfenced>2+<mfenced separators="|">∂v/∂z</mfenced>2</semantics>, with ρ potential density, <semantics><mfenced separators="|">u,v</mfenced></semantics> the horizontal velocity components and g gravity acceleration. In isopycnic coordinates <semantics><mfenced separators="|">x,y,ρ</mfenced></semantics>, <semantics>Ri</semantics> is expressed as the ratio between <semantics>M2≡N−2</semantics> and the squared diapycnal shear <semantics>Sρ2=<mfenced separators="|">ρ/g</mfenced>2<mfenced open="[" close="]" separators="|"><mfenced separators="|">∂u/∂ρ</mfenced>2+<mfenced separators="|">∂v/∂ρ</mfenced>2</mfenced></semantics>, <semantics>Ri=M2/Sρ2</semantics>. 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 <semantics>Sρ</semantics> are related through the stratification itself, <semantics>Sρ=S/N2</semantics>. In terms of characteristic times, this is equivalent to <semantics>τ≡Sρ=to2/td</semantics>, which is interpreted as a critical dynamic time <semantics>τ</semantics> that equals the buoyancy period <semantics>to≡N−1</semantics> normalized by the ratio <semantics>td/to</semantics>, where <semantics>td=S−1</semantics> 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 <semantics>σρ2=Sρ2−M2</semantics> and compare it with the reduced squared vertical <semantics>σ2=S2−N2</semantics>, both being positive (negative) for unstable (stable) conditions. While both Ri and <semantics>σ2</semantics> remain highly variable for all stratification conditions, the mean <semantics>σρ2</semantics> values approach <semantics>Sρ2</semantics> with increasing stratification. Further, the field data follow the relation <semantics>σρ2=<mfenced separators="|">1−Ri</mfenced>/<mfenced separators="|">N2Ri</mfenced></semantics>, with a subcritical <semantics>Ri=0.22</semantics> for both the island shelf break and the Mediterranean outflow. We propose <semantics>σρ2</semantics> and <semantics>Sρ2</semantics> to be good indexes for the occurrence of effective mixing under highly stratified conditions.