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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
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- Journal:
- Journal of Fluid Mechanics / Volume 731 / 25 September 2013
- Published online by Cambridge University Press:
- 21 August 2013, pp. 418-442
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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.
Inertial nonlinear equilibration of equatorial flows
- BACH LIEN HUA, DENNIS W. MOORE, SYLVIE LE GENTIL
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- Journal:
- Journal of Fluid Mechanics / Volume 331 / 25 January 1997
- Published online by Cambridge University Press:
- 21 May 2009, pp. 345-371
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We explore the nature of inertial equilibration of equatorial flows in the presence of mean meridional and vertical shears of the basic state, with oceanic applications in mind. The study is motivated by the observational evidence that the subthermocline equatorial mean circulation displays nearly zero Ertel potential vorticity away from the equator, when taking into account the non-traditional horizontal component of the Earth rotation. This observed state precisely verifies the marginal condition for inertial instability: a linear analysis for the equatorial β-plane confirms that the usual condition of instability, namely that Ertel potential vorticity should be of opposite sign to the vertical Coriolis parameter, remains valid even when the traditional approximation is relaxed. Analytical linear normal modes reveal that a meridional shear of the basic state leads to a vertical stacking of equatorially-trapped zonal flows of alternate signs, with a new centre of symmetry located at the dynamical equator. A vertical shear of the basic state causes a meridional stacking of extra-equatorial zonal flows.
In an inviscid framework, a two-dimensional formulation is ill-posed and we resort to non-hydrostatic viscous simulations to determine the nonlinear normal forms of the system. The influence of a small-scale eddy diffusivity and a large-scale Rayleigh damping on the equilibrated vertical scale is determined numerically. The nonlinear equilibration occurs through a steady-state bifurcation from a basic state without jets to another steady state with secondary jets of alternate signs. The final state corresponds to eastward jets located on the geographic equator, while westward jets are located near the dynamical equator. These results are consistent with in situ observations of equatorial deep jets.
The analogy between the equatorial meridional shear flow and the cylindrical Couette–Taylor flow with an axial density stratification is detailed. There is a strong similarity in the general symmetries and nonlinear normal forms of the two problems. Similarly to the homogeneous Couette–Taylor flow, the gap width between the two cylinders is important for determining the axial scale of the secondary flow through the Reynolds number. For the equatorial problem, an upper bound for the height scale of inertial jets is such that the corresponding equatorial radius of deformation times √2 fits between the geographic and dynamic equators.
One of our main conclusions is that the raisond’être of the observed region of zero Ertel potential vorticity is to facilitate angular momentum exchanges between the two hemispheres and inertial deep jets are the byproducts of this angular momentum mixing.
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
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- Journal:
- Journal of Fluid Mechanics / Volume 610 / 10 September 2008
- Published online by Cambridge University Press:
- 08 August 2008, pp. 311-341
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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.
Surface kinetic energy transfer in surface quasi-geostrophic flows
- XAVIER CAPET, PATRICE KLEIN, BACH LIEN HUA, GUILLAUME LAPEYRE, JAMES C. MCWILLIAMS
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- Journal of Fluid Mechanics / Volume 604 / 10 June 2008
- Published online by Cambridge University Press:
- 14 May 2008, pp. 165-174
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The relevance of surface quasi-geostrophic dynamics (SQG) to the upper ocean and the atmospheric tropopause has been recently demonstrated in a wide range of conditions. Within this context, the properties of SQG in terms of kinetic energy (KE) transfers at the surface are revisited and further explored. Two well-known and important properties of SQG characterize the surface dynamics: (i) the identity between surface velocity and density spectra (when appropriately scaled) and (ii) the existence of a forward cascade for surface density variance. Here we show numerically and analytically that (i) and (ii) do not imply a forward cascade of surface KE (through the advection term in the KE budget). On the contrary, advection by the geostrophic flow primarily induces an inverse cascade of surface KE on a large range of scales. This spectral flux is locally compensated by a KE source that is related to surface frontogenesis. The subsequent spectral budget resembles those exhibited by more complex systems (primitive equations or Boussinesq models) and observations, which strengthens the relevance of SQG for the description of ocean/atmosphere dynamics near vertical boundaries. The main weakness of SQG however is in the small-scale range (scales smaller than 20–30 km in the ocean) where it poorly represents the forward KE cascade observed in non-QG numerical simulations.
The intrusion of a density current along the coast of a rotating fluid
- Melvin E. Stern, John A. Whitehead, Bach-Lien Hua
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- Journal of Fluid Mechanics / Volume 123 / October 1982
- Published online by Cambridge University Press:
- 20 April 2006, pp. 237-265
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When light rotating fluid spreads over heavier fluid in the vicinity of a vertical wall (coast) a boundary jet of width Λ forms, the leading edge or nose of which propagates with speed ĉ along the coast. A certain fraction 8 of the boundary transport is not carried by the nose but is deflected backwards (detrained) and left behind the propagating nose. Theoretical and experimental results for Λ,ĉ, and δ are given for a quasi-equilibrium (constant-ĉ) regime. Over longer time intervals the laboratory observations suggest that the nose slows down and stagnates, whereupon the trailing flow separates from the coast and an intermittent boundary current forms. These processes may be relevant to the mixing of oceanic coastal currents and the maintenance of the mean current.
Equatorial inertial-parametric instability of zonally symmetric oscillating shear flows
- MARC D'ORGEVILLE, BACH LIEN HUA
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- Journal:
- Journal of Fluid Mechanics / Volume 531 / 25 May 2005
- Published online by Cambridge University Press:
- 18 May 2005, pp. 261-291
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This study revisits the problem of the zonally symmetric instability on the equatorial $\beta$-plane. Rather than treating the classical problem of a steady basic flow, it treats a sequence of problems of increasing complexity in which the basic flow is oscillatory in time with a frequency $\omega_0$.
First, for the case of a homogeneous fluid, a time-oscillating barotropic shear forcing may excite a subharmonic parametric resonance of inertial oscillations. Because of the continuous distribution of inertial oscillation frequencies, this resonance occurs at critical inertial latitudes $y_c$ such that $\beta y_c {=}{\pm} {\omega_0}/2$. Next the effects of stratification, characterized by Brunt–Väisälä frequency $N$, are taken into account. It is shown analytically (in the asymptotic limit of a weak shear) that the forced temporal oscillation leads to an inertial-parametric instability, when a resonance condition between the basic flow frequency and the sum of two inertio-gravity free-mode frequencies is met. This inertial-parametric instability has a well-defined inviscid vertical scale selection favouring the high-vertical mode $m_c{\sim}7.45m_0$, where $m_0{=}{\beta N}/{\omega_0^2}$ is the equatorial vertical mode characteristic of frequency $\omega_0$. The viscous critical shear of inertial-parametric instability is lower than the steady inertial instability one.
Finally, this type of setting naturally arises when the basic flow is considered to be an equatorial wave, so the problem is recast with the nonlinear adjustment of the vertically sinusoidal basic state of a zonally symmetric mixed Rossby–gravity (MRG) wave. Initial-value numerical simulations show that the same inertial-parametric instability exists leading to a resonant subharmonic excitation of free modes with vertical scales 7 and 8 times smaller than the basic-state wave. A simplified dynamical model of the instability is introduced, demonstrating that the oscillatory nature of the shear with height for the MRG wave necessarily implies a resonance between distinct vertical modes, the most unstable ones being modes 7 and 8 for a large enough Froude number of the MRG wave. The nonlinear action of the instability is described in terms of angular momentum and potential vorticity changes: a significant mixing due to the breaking of the excited high vertical modes creates a vertically averaged westward flow at the equator and extra-equatorial eastward flows. The ideas exposed may play a part in explaining layering phenomena and the latitudinal structure of the zonal flow in the equatorial oceans below the thermocline.
Lagrangian accelerations in geostrophic turbulence
- BACH LIEN HUA, JAMES C. McWILLIAMS, PATRICE KLEIN
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- Journal of Fluid Mechanics / Volume 366 / 10 July 1998
- Published online by Cambridge University Press:
- 10 July 1998, pp. 87-108
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A distinctive property of Lagrangian accelerations in geostrophic turbulence is that they are governed by the large and intermediate scales of the flow, both in time and space, so that the inertial part of the dynamics plays a much larger role than in three-dimensional turbulence where viscous effects are stronger. For the case of geostrophic turbulence on a β-plane, three terms contribute to the Lagrangian accelerations: the ageostrophic pressure gradient which often is the largest term, a meridional acceleration due to the β-effect, and an acceleration due to horizontally divergent ageostrophic motions. Both their spectral characteristics and patterns in physical space are studied in this paper. In particular the total accelerations field has an inertial spectrum slope which is identical to the geostrophic velocity field inertial slope.
The accelerations gradient tensor is shown to govern the topology of quasi-geostrophic stirring and transport properties. Its positive eigenvalues locate accurately the position of extrema of potential vorticity gradients. The three-dimensional distribution of tracer gradients is such that the vertical distribution is entirely constrained by the horizontal one, while the reverse is not true. We make explicit analytically their dependence on the three-dimensional accelerations gradient.