A Study on the 3d Inertial Instability Mechanism in the Sub-mesoscale Ocean

In the sub-mesoscale ocean vortices tend to be predominantly cyclonic [4]. Inertial instability (hereafter II), which is a centrifugal instability mechanism in the presence of the Coriolis force, is a destructive mechanism that acts only on anti-cyclones, and therefore is hypothesized to cause this asymmetry. Furthermore, since II is in fact the growth of the overturning vorticity, thus creating vertical mixing, it is expected to contribute to nutrient enrichment from the deep, affecting primary production and the oceanic carbon cycle. Linear stability analysis [6] shows that three-dimensional unstable modes of parallel shear flow (without curvature) may have stronger growth rates than the standard two-dimensional barotropic modes when the absolute vorticity is negative, thus II is more significant than shear instability in the destruction of strong anticyclonic shear. For circular vortices (with curvature), vortex columns are unstable to 3D perturbations where the generalized Rayleigh discriment is negative χ(r) ≡ ( 1 r∂r(rV ) + f ) (2V/r + f) < 0, where V (r) is the azimuthal velocity and f the Coriolis parameter, which implies that the region of instability is in the anticyclone periphery [2]. Axis-symmetry seems to be a proper simplification, as it was shown by [1] that the growth rates of axis-symmetric disturbances are larger than non axis-symmetric ones, and the latter are completely stable above a relatively low cut-off azimuthal wave number. Recent fully non-linear numerical simulations show that the area of the instability is breached [5] into the anticyclone core from the outside [3] in the nonlinear phase. However, for surface oceanic vortices, which are among the most energetic structures of the oceans, one should also take into account the thickness and the stratification of the thermocline. Both the shallow-water constraint and the stratification stabilize these intense anticyclones. The stratification induces a low vertical wave-number cutoff (smaller wavelength) [3, 5]. The small scale perturbations are more sensitive to the vertical dissipation and the growth rate of the unstable modes could be strongly reduced or vanish completely. We apply linear stability analysis to examine 3D II in a vertically confined and stratified, axis-symmetric, meanflow. We define the dynamical parameters that govern the stability, and find the parameters, which are both insensitive to different vorticity profiles and are relatively easy to estimate from laboratory experiments and oceanic in situ measurements. These are the eddy Rossby number, characterized by the maximal azimuthal velocity and the radius at which this velocity is reached, Ro = Vmax/rmaxf , which turns out to be a more suitable parameter than, say, the normalized vorticity of the eddy; the aspect ratio, δ = h/R (which is small δ << 1); the normalized stratification, S = N/f ; and the vertical Ekman number, Ek = ν/fh. In cases of strong stratification, we use the Burger number to combine both the aspect ratio and the stratification. We solve the problem analytically for a Rankine vortex (see figure 1a). For other vorticity profiles an eigenvalue decomposition is used, which we confirmed by recalculating for the Rankine vortex. We map the parameter space (see figure 1b), and corroborate our findings (see figure 2) with large-scale laboratory experiments studies, performed at the LEGI-Coriolis platform.