ALBERT

Description: This study revisits the problem of the zonally symmetric instability on the equatorial β-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 ωo. First, for the case of a homogeneous fluid, a time-osci llating 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, such that βyc = ±ω o/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 mc ∼ 7.45mo, where mo = βN/ω0 2 is the equatorial vertical mode characteristic of frequency wo. The viscous critical shear of inertial-parametric instability is lower than the steady inertial instability one. Finally, this type of setting naturall y 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. © 2005 Cambridge University Press.

Description: When light rotating fluid spreads over heavier fluid in the vicinity of a vertical wall (coast) a boundary jet of width L forms, the leading edge or nose of which propagates with speed 6 along the coast. A certain fraction 8 of the boundarytransport is not carried by the nose but is deflected backwards (detrained) and left behind the propagating nose. Theoretical and experimental results for L, c, and 8 are given for a quasi-equilibrium (constant-c) 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. © 1982, Cambridge University Press. All rights reserved.

Description: 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 raison d'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.

Description: 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 quasigeostrophic 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.

Description: 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 ≈0.55 k2. 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. © 2008 Cambridge University Press.