M=v+fx The absolute momentum is a conserved quantity in inviscid

M=v+fx.The absolute momentum is a conserved quantity in inviscid flow with no variations in the y  -direction (DM/Dt=0DM/Dt=0) and is often used as the determining factor for inertial instability, 1 which itself can be considered a form of SI in the limit where N2=0N2=0. Assuming thermal wind balance, the slope of the absolute momentum surfaces is equation(11) ∂Mdx∂Mdz=∂v∂x+f∂v∂z=f∂v∂x+f2M2=ff+ζM2,where again ζ=∂v/∂xζ=∂v/∂x learn more is the vertical component of the relative vorticity. If the initial PV is negative (unstable to SI), this implies that

equation(12) Ri=N2f2M4ff+ζM2.Then the isopycnal slope is steeper than that of the absolute momentum contours (which for brevity will henceforth be referred to as MM-surfaces), with equality when Ri=f/(f+ζ)Ri=f/f+ζ (neutral to SI). For an unstable

initial state one can also show that ff+ζ/M2>M2N2-fN1Ri-1+ζf, so that the MM-surface always lies within the SI-unstable arc. It is useful to begin by considering the energetics when parcels are exchanged along MM-surfaces. Haine and Marshall (1998) show that the change in potential energy ΔPΔP due to parcel exchange is given by equation(14) ΔP=ρ0N2Δy2ss-M2N2,where ΔyΔy is the horizontal distance of the parcel see more displacement and s   is the slope of the surface along which parcels are exchanged. Similarly, they also showed that the change in kinetic energy ΔKΔK by such an exchange is equation(15) ΔK=ρ0Δy2[f(f+ζ)-M2s]ΔK=ρ0Δy2ff+ζ-M2sand the total energy change, ΔE=ΔP+ΔKΔE=ΔP+ΔK, is equation(16) ΔE=ρ0Δy2ff+ζ-M2s+N2ss-M2N2.Factoring M2M2 out of the bracketed expression in (15), one has equation(17) ΔK=ρ0Δy2M2ff+ζM2-srevealing that there is no change in mean KE when parcels are exchanged along MM-surfaces. SI modes aligned with these surfaces thus grow purely via the extraction of background PE, forming a dichotomy with isopycnal-aligned modes, which grow purely via reduction of the geostrophic shear. One can extend this analysis to consider modes whose slope is between or around the isopycnals and MM-surfaces as well. Pregnenolone Substituting (9) into (16) reveals that ΔE=0ΔE=0

at the edges of the unstable arc; furthermore, Fig. 1 reveals that the extraction of energy smoothly transitions to zero as the edges are approached. Three “zones” thus exist: zone 1 contains all modes whose slope is steeper than the isopycnal, which grow by reducing the geostrophic shear but convert some of the extracted KE to mean PE in the background stratification; zone 2 lies between the isopycnal and the MM-surface, where both the background PE and KE are reduced; zone 3 lies between the MM-surface and the shallowest unstable slope, where the background PE is reduced but some KE is transferred back into the mean flow. A schematic of these zones appears in Fig. 2. The energetics of the unstable SI modes reveal that restratification is indeed possible in the absence of secondary Kelvin–Helmholtz instabilities.

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