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Vertical Shear, Diapycnal Shear and the Gradient Richardson Number
Version 1
: Received: 17 July 2024 / Approved: 17 July 2024 / Online: 17 July 2024 (12:41:08 CEST)
How to cite: Pelegrí, J. L.; Claret, M.; Sangrà, P. Vertical Shear, Diapycnal Shear and the Gradient Richardson Number. Preprints 2024, 2024071439. https://doi.org/10.20944/preprints202407.1439.v1 Pelegrí, J. L.; Claret, M.; Sangrà, P. Vertical Shear, Diapycnal Shear and the Gradient Richardson Number. Preprints 2024, 2024071439. https://doi.org/10.20944/preprints202407.1439.v1
Abstract
In vertical coordinates, (x,y,z), the gradient Richardson number Ri=N^2⁄S^2 is the square of the ratio between buoyancy frequency N and vertical-shear amplitude S, where N^2=-(g⁄ρ) ∂ρ⁄∂z and S^2=(∂u⁄∂z)^2+ (∂v⁄∂z)^2, with ρ potential density, (u,v) the horizontal velocity components and g gravity acceleration, so that a decrease in stratification brings a decrease in dynamic stability and vice versa. In the isopycnic coordinate system, (x,y,ρ), however, Ri=M^2⁄〖S_ρ〗^2 , is the ratio between 〖M^2≡N〗^(-2) and the squared diapycnal shear 〖S_ρ〗^2=(ρ⁄g)^2 [(∂u⁄∂ρ)^2+ (∂v⁄∂ρ)^2 ], so a decrease in stratification leads to an increase in dynamic stability and vice versa. The apparently different role of stratification arises because S and S_ρ are related through the stratification itself, S_ρ=S⁄N^2 or, equivalently, τ=〖t_o〗^2⁄t_d , which is interpreted as a natural oscillation time τ≡S_ρ that equals the buoyancy or oscillation time t_o=N^(-1) normalized by the ratio t_o⁄t_d , where the deformation time is t_d=S^(-1). Here we follow simple arguments and use field data from three different situations (island shelf-break, Gulf Stream and Mediterranean outflow) to endorse the usefulness of the isopycnal approach. In particular, we define the reduced squared vertical σ^2= S^2-N^2 and reduced squared diapycnal shear 〖σ_ρ〗^2= 〖S_ρ〗^2-M^2, which are positive for unstable conditions and negative for stable conditions. It turns out that Ri and σ^2 remain highly variable for all stratification conditions; in contrast, the mean 〖σ_ρ〗^2 values display a univocal tendency, its magnitude decreasing with stratification. We propose 〖σ_ρ〗^2 and 〖S_ρ〗^2 to be good indexes for the occurrence of effective mixing under high stratification conditions.
Keywords
vertical mixing; diapycnal mixing; isopycnic coordinates; Richardson number; flow instability
Subject
Environmental and Earth Sciences, Oceanography
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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