5 results
Stability of an isolated pancake vortex in continuously stratified-rotating fluids
- Eunok Yim, Paul Billant, Claire Ménesguen
-
- Journal:
- Journal of Fluid Mechanics / Volume 801 / 25 August 2016
- Published online by Cambridge University Press:
- 25 July 2016, pp. 508-553
-
- Article
- Export citation
-
This paper investigates the stability of an axisymmetric pancake vortex with Gaussian angular velocity in radial and vertical directions in a continuously stratified-rotating fluid. The different instabilities are determined as a function of the Rossby number $Ro$, Froude number $F_{h}$, Reynolds number $Re$ and aspect ratio ${\it\alpha}$. Centrifugal instability is not significantly different from the case of a columnar vortex due to its short-wavelength nature: it is dominant when the absolute Rossby number $|Ro|$ is large and is stabilized for small and moderate $|Ro|$ when the generalized Rayleigh discriminant is positive everywhere. The Gent–McWilliams instability, also known as internal instability, is then dominant for the azimuthal wavenumber $m=1$ when the Burger number $Bu={\it\alpha}^{2}Ro^{2}/(4F_{h}^{2})$ is larger than unity. When $Bu\lesssim 0.7Ro+0.1$, the Gent–McWilliams instability changes into a mixed baroclinic–Gent–McWilliams instability. Shear instability for $m=2$ exists when $F_{h}/{\it\alpha}$ is below a threshold depending on $Ro$. This condition is shown to come from confinement effects along the vertical. Shear instability transforms into a mixed baroclinic–shear instability for small $Bu$. The main energy source for both baroclinic–shear and baroclinic–Gent–McWilliams instabilities is the potential energy of the base flow instead of the kinetic energy for shear and Gent–McWilliams instabilities. The growth rates of these four instabilities depend mostly on $F_{h}/{\it\alpha}$ and $Ro$. Baroclinic instability develops when $F_{h}/{\it\alpha}|1+1/Ro|\gtrsim 1.46$ in qualitative agreement with the analytical predictions for a bounded vortex with angular velocity slowly varying along the vertical.
Ageostrophic instability in rotating, stratified interior vertical shear flows
- Peng Wang, James C. McWilliams, Claire Ménesguen
-
- Journal:
- Journal of Fluid Mechanics / Volume 755 / 25 September 2014
- Published online by Cambridge University Press:
- 19 August 2014, pp. 397-428
-
- Article
- Export citation
-
The linear instability of several rotating, stably stratified, interior vertical shear flows $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\overline{U}(z)$ is calculated in Boussinesq equations. Two types of baroclinic, ageostrophic instability, AI1 and AI2, are found in odd-symmetric $\overline{U}(z)$ for intermediate Rossby number ($\mathit{Ro}$). AI1 has zero frequency; it appears in a continuous transformation of the unstable mode properties between classic baroclinic instability (BCI) and centrifugal instability (CI). It begins to occur at intermediate $\mathit{Ro}$ values and horizontal wavenumbers ($k,l$) that are far from $l= 0$ or $k = 0$, where the growth rate of BCI or CI is the strongest. AI1 grows by drawing kinetic energy from the mean flow, and the perturbation converts kinetic energy to potential energy. The instability AI2 has inertia critical layers (ICL); hence it is associated with inertia-gravity waves. For an unstable AI2 mode, the coupling is either between an interior balanced shear wave and an inertia-gravity wave (BG), or between two inertia-gravity waves (GG). The main energy source for an unstable BG mode is the mean kinetic energy, while the main energy source for an unstable GG mode is the mean available potential energy. AI1 and BG type AI2 occur in the neighbourhood of $A-S= 0$ (a sign change in the difference between absolute vertical vorticity and horizontal strain rate in isentropic coordinates; see McWilliams et al., Phys. Fluids, vol. 10, 1998, pp. 3178–3184), while GG type AI2 arises beyond this condition. Both AI1 and AI2 are unbalanced instabilities; they serve as an initiation of a possible local route for the loss of balance in 3D interior flows, leading to an efficient energy transfer to small scales.
Layering and turbulence surrounding an anticyclonic oceanic vortex: in situ observations and quasi-geostrophic numerical simulations
- Bach Lien Hua, Claire Ménesguen, Sylvie Le Gentil, Richard Schopp, Bruno Marsset, Hidenori Aiki
-
- Journal:
- Journal of Fluid Mechanics / Volume 731 / 25 September 2013
- Published online by Cambridge University Press:
- 21 August 2013, pp. 418-442
-
- Article
- Export citation
-
Evidence of persistent layering, with a vertical stacking of sharp variations in temperature, has been presented recently at the vertical and lateral periphery of energetic oceanic vortices through seismic imaging of the water column. The stacking has vertical scales ranging from a few metres up to 100 m and a lateral spatial coherence of several tens of kilometres comparable with the vortex horizontal size. Inside this layering, in situ data display a $[{ k}_{h}^{- 5/ 3} { k}_{h}^{- 2} ] $ scaling law of horizontal scales for two different quantities, temperature and a proxy for its vertical derivative, but for two different ranges of wavelengths, between 5 and 50 km for temperature and between 500 m and 5 km for its vertical gradient. In this study, we explore the dynamics underlying the layering formation mechanism, through the slow dynamics captured by quasi-geostrophic equations. Three-dimensional high-resolution numerical simulations of the destabilization of a lens-shaped vortex confirm that the vertical stacking of sharp jumps in density at its periphery is the three-dimensional analogue of the preferential wind-up of potential vorticity near a critical radius, a phenomenon which has been documented for barotropic vortices. For a small-Burger (flat) lens vortex, baroclinic instability ensures a sustained growth rate of sharp jumps in temperature near the critical levels of the leading unstable modes. Such results can be obtained for a background stratification which is due to temperature only and does not require the existence of salt anomalies. Aloft and beneath the vortex core, numerical simulations well reproduce the $[{ k}_{h}^{- 5/ 3} { k}_{h}^{- 2} ] $ scaling law of horizontal scales for the vertical derivative of temperature that is observed in situ inside the layering, whatever the background stratification. Such a result stems from the tracer-like behaviour of the vortex stretching component and previous studies have shown that spectra of tracer fields can be steeper than $- 1$, namely in $- 5/ 3$ or $- 2$, if the advection field is very compact spatially, with a $- 5/ 3$ slope corresponding to a spiral advection of the tracer. Such a scaling law could thus be of geometric origin. As for the kinetic and potential energy, the ${ k}_{h}^{- 5/ 3} $ scaling law can be reproduced numerically and is enhanced when the background stratification profile is strongly variable, involving sharp jumps in potential vorticity such as those observed in situ. This raises the possibility of another plausible mechanism leading to a $- 5/ 3$ scaling law, namely surface-quasi-geostrophic (SQG)-like dynamics, although our set-up is more complex than the idealized SQG framework. Energy and enstrophy fluxes have been diagnosed in the numerical quasi-geostrophic simulations. The results emphasize a strong production of energy in the oceanic submesoscales range and a kinetic and potential energy flux from mesoscale to submesoscales range near the critical levels. Such horizontal submesoscale production, which is correlated to the accumulation of thin vertical scales inside the layering, thus has a significant slow dynamical component, well-captured by quasi-geostrophy.
Ageostrophic instability in a rotating stratified interior jet
- Claire Ménesguen, J. C. McWilliams, M. J. Molemaker
-
- Journal:
- Journal of Fluid Mechanics / Volume 711 / 25 November 2012
- Published online by Cambridge University Press:
- 28 September 2012, pp. 599-619
-
- Article
- Export citation
-
Oceanic large- and meso-scale flows are nearly balanced in forces between Earth’s rotation and density stratification effects (i.e. geostrophic, hydrostatic balance associated with small Rossby and Froude numbers). In this regime advective cross-scale interactions mostly drive energy toward larger scales (i.e. inverse cascade). However, viscous energy dissipation occurs at small scales. So how does the energy reservoir at larger scales leak toward small-scale dissipation to arrive at climate equilibrium? Here we solve the linear instability problem of a balanced flow in a rotating and continuously stratified fluid far away from any boundaries (i.e. an interior jet). The basic flow is unstable not only to geostrophic baroclinic and barotropic instabilities, but also to ageostrophic instabilities, leading to the growth of small-scale motions that we hypothesize are less constrained by geostrophic cascade behaviours in a nonlinear regime and thus could escape from the inverse energy cascade. This instability is investigated in the parameter regime of moderate Rossby and Froude numbers, below the well-known regimes of gravitational, centrifugal, and Kelvin–Helmholtz instability. The ageostrophic instability modes arise with increasing Rossby number through a near-degeneracy of two unstable modes with coincident phase speeds. The near-degeneracy occurs in the neighbourhood of an identified criterion for the non-integrability of the ‘isentropic balance equations’ (namely with the absolute vertical vorticity and the horizontal strain rate associated with the basic flow), beyond which development of an unbalanced component of the flow is expected. These modes extract energy from the basic state with large vertical Reynolds stress work (unlike geostrophic instabilities) and act locally to modify the basic flow by reducing the isopycnal Ertel potential vorticity gradient near both its zero surface and its critical surface (phase speed equal to basic flow speed).
Destabilization of mixed Rossby gravity waves and the formation of equatorial zonal jets
- BACH LIEN HUA, MARC D'ORGEVILLE, MARK D. FRUMAN, CLAIRE MENESGUEN, RICHARD SCHOPP, PATRICE KLEIN, HIDEHARU SASAKI
-
- Journal:
- Journal of Fluid Mechanics / Volume 610 / 10 September 2008
- Published online by Cambridge University Press:
- 08 August 2008, pp. 311-341
-
- Article
- Export citation
-
The stability of mixed Rossby gravity (MRG) waves has been investigated numerically using three-dimensionally consistent high-resolution simulations of the continuously stratified primitive equations. For short enough zonal wavelength, the westward phase propagating MRG wave is strongly destabilized by barotropic shear instability leading to the formation of zonal jets. The large-scale instability of the zonally short wave generates zonal jets because it consists primarily of sheared meridional motions, as shown recently for the short barotropic Rossby wave problem.
Simulations were done in a variety of domain geometries: a periodic re-entrant channel, a basin with a short MRG wave forced in its western part and a very long channel initialized with a zonally localized MRG wave. The characteristics of the zonal jets vary with the geometry. In the periodic re-entrant channel, barotropic zonal jets dominate the total flow response at the equator and its immediate vicinity. In the other cases, the destabilization leads to zonal jets with quite different characteristics, especially in the eastward group propagating part of the signal. The most striking result concerns the formation of zonal jets at the equator, alternating in sign in the vertical, with vertical scale short compared to the scale of the forcing or initial conditions.
A stability analysis of a simplified perturbation vorticity equation is formulated to explain the spatial scale selection and growth rate of the zonal jets as functions of the characteristics of the basic state MRG wave. For both types of zonal jets, the model predicts that their meridional scales are comparable to the zonal scale of the MRG wave basic state, while their growth rates scale as μ ∝ Fr |k|, where Fr is the Froude number of the meridional velocity component of the basic state and k its non-dimensional zonal wavenumber. The vertical scale of the baroclinic zonal jets corresponds to the dominant harmonic ppeak of the basic state in the fastest growing mode, given by ppeak≈0.55k2. Thus, the shorter the zonal wavelength of the basic state MRG wave, the narrower the meridional scale of the zonal jets, both barotropic and baroclinic, with the vertical scale of the baroclinic jets being tied to their meridional scale through the equatorial radius of deformation, which decreases as the square root of the vertical wavenumber. The predictions of the spatial scales are in both qualitative and quantitative agreement with the numerical simulations, where shorter vertical scale baroclinic zonal jets are favoured by shorter-wavelength longer-period MRG wave basic states, with the vertical mode number increasing as the square of the MRG wave period.
An Appendix deals with the case of zonally long and intermediate wavelength MRG waves, where a weak instability regime causes a moderate adjustment involving resonant triad interactions without leading to jet formation. For eastward phase propagating waves, adjustment does not lead to significant angular momentum redistribution.