Xavier Carton

Coastal upwellings, due to offshore Ekman transport, are more energetic at the western boundaries of the oceans, where they are intensified by incoming Rossby waves, than at the eastern boundaries. Western boundary upwellings are often accompanied by a local vortex field. The instability of a developed upwelling front and its interaction with an external vortex field is studied here with a three-dimensional numerical model of the hydrostatic rotating Navier-Stokes equations (the primitive equations). The baroclinic instability of the front leads to the growth of meanders with 100-200 km wavelength, in the absence of external vortex. On the $f$-plane, these waves can break into a row of vortices when the instability is intense. The $\beta$-effect is stabilizing and strongly decreases the amplitude of meanders. Simulations are then performed with a front initially accompanied by one or several external vortices. The evolutions in this case are compared with those of the unstable jet alone. On the $f$-plane, when an external vortex is close to the front, this latter sheds a long filament which wraps up around the vortex. This occurs over a period similar to that of the instability of the isolated front. Cyclones are more efficient in tearing such filaments offshore than anticyclones. On the $\beta$-plane, the filaments are short and turbulence is confined to the vicinity of the front. At long times, waves propagate along the front, thus extending turbulence alongshore. The initial presence of a vortex alley leads to a stronger destabilization of the front and to a larger cross-shore flux than for a single vortex, with many filaments and small vortices pushed far offshore. In the ocean, this cross-shore exchange has important consequences on the local biological activity.

The effect of a bottom slope on the merger of two identical Rankine vortices is investigated in a two-dimensional, quasi-geostrophic, incompressible fluid.
When two cyclones initially lie parallel to the slope, and more than two vortex diameters away from the slope, the critical merger distance is unchanged. When the cyclones are closer to the slope, they can merge at larger distances, but they lose more mass into filaments, thus weakening the efficiency of merger. Several effects account for this: the topographic Rossby wave advects the cyclones, reduces their mutual distance and deforms them. This alongshelf wave breaks into filaments and into secondary vortices which shear out the initial cyclones. The global motion of fluid towards the shallow domain and the erosion of the two cyclones are confirmed by the evolution of particles seeded both in the cyclones and near the topographic slope. The addition of tracer to the flow indicates that diffusion is ballistic at early times.
For two anticyclones, merger is also facilitated because one vortex is ejected offshore towards the other, via coupling with a topographic cyclone. Again two anticyclones can merge at large distance but they are eroded in the process.
Finally, for taller topographies, the critical merger distance is again increased and the topographic influence can scatter or completely erode one of the two initial cyclones. Conclusions are drawn on possible improvements of the model configuration for an application to the ocean.

In a barotropic model of an oceanic channel, bounded to the north by a straight coast indented by a Gaussian cape, the evolution of a coastal jet is studied numerically. In the absence of the cape, the barotropic instability of the jet is determined. In the presence of the cape, a regular row of meanders develops downstream of this feature, and becomes stationary for a particular range of parameters. The relevant parameters are the velocity and width of the jet, size of the cape, and beta effect. The formation of meanders occurs first via the instability of the jet, then via the generation of vorticity anomalies at the cape, which are advected both downstream by the flow and offshore by the radiation of Rossby waves. Once the meanders are established, they remain stationary features if the propagation velocity of the meanders (due to the dipolar effect at the coast) opposes the jet velocity and the phase speed of the wave on the vorticity front. Finally, a steady state of a regular row of meanders is also obtained via a matrix method and is similar to that obtained in the time-dependent case.

This paper addresses the instability of a two-layer coastal current in a quasigeostrophic model; the potential vorticity (PV) structure of this current consists in two uniform cores, located at different depths, with finite width and horizontally shifted. This shift allows both barotropic and baroclinic instability for this current. The PV cores can be like-signed or opposite-signed, leading to their vertical alignment or to their hetonic coupling. These two aspects are novel compared to previous studies. For narrow vorticity cores, short waves dominate, associated with barotropic instability; for wider cores, longer waves are more unstable and are associated with baroclinic processes. Numerical experiments were performed on the $f$−plane with a finite-difference model. When both cores have like-signed PV, trapped instability develops during the nonlinear evolution: vertical alignment of the structures is observed. For narrow cores, short wave breaking occurs close to the coast; for wider cores, substantial turbulence results from the entrainment of ambient fluid into the coastal jet. When the two cores have opposite-signed PV, the nonlinear regimes range from short wave breaking to the ejection of dipoles or tripoles, via a regime of dipole oscillation near the wall. The Fourier analysis of the perturbed flow is appropriate to distinguish the regimes of short wave breaking, of dipole formation, and of turbulence, but not the differences between regimes involving only vortex pairs. To explain more precisely the regimes where two vortices (and their wall images) interact, a point vortex model is appropriate.

The interaction between two co-rotating vortices, embedded in a steady external strain field, is studied in a coupled Quasi-Geostrophic – Surface Quasi-Geostrophic (hereafter referred to as QG-SQG) model. One vortex is an anomaly of surface density, and the other is an anomaly of internal potential vorticity. The equilibria of singular point vortices and their stability are presented first. The number and form of the equilibria are determined as a function of two parameters: the external strain rate and the vertical separation between the vortices. A curve is determined analytically which separates the domain of existence of one saddle-point, and that of one neutral point and two saddle-points. Then, a Contour-Advective Semi-Lagrangian (hereafter referred to as CASL) numerical model of the coupled QG-SQG equations is used to simulate the time-evolution of a sphere of uniform potential vorticity, with radius $R$ at depth $−2H$ interacting with a disk of uniform density anomaly, with radius $R$, at the surface. In the absence of external strain, distant vortices co-rotate, while closer vortices align vertically, either completely or partially (depending on their initial distance). With strain, a fourth regime appears in which vortices are strongly elongated and drift away from their common center, irreversibly. An analysis of the vertical tilt and of the horizontal deformation of the internal vortex in the regimes of partial or complete alignment is used to quantify the three-dimensional deformation of the internal vortex in time. A similar analysis is performed to understand the deformation of the surface vortex.

The influence of an external strain (or shear) field on the evolution of two identical vortices is investigated in a two-dimensional incompressible fluid. Using point vortex modeling, two regimes of the vortex doublet (co-rotation and irreversible separation) are determined; the critical intensity of the large scale flow separating these two regimes for a given initial separation of vortices, is calculated. Finite-area effects are then considered for the vortices. The steady states of piecewise constant vortices are computed algebraically and numerically; positive strain (or shear) favors vortex deformation. This deformation has a dominant elliptical component. An elliptical model of two vortices confirms the point vortex model results for centroid trajectories, and the steady state model results concerning the influence of positive strain on vortex deformation. It also provides an estimate of critical merger distance in the presence of large scale flow. Finally, the finite-time, nonlinear evolution of the vortex doublet is simulated with a numerical code of the 2D vorticity equation. The various regimes (stationarity, merger, co-rotation, ejection) are classified in the plane of initial vortex separation and of external deformation. These regimes are analyzed, and the critical merger distance is evaluated for negative and positive external strain; the results are in agreement with the elliptical model prediction. Merger efficiency, defined as the ratio of final to initial vortex circulation, is computed; for the same initial distance, it is smaller for negative strain. It also depends in a more complex way of the initial vortex distance.

The parametric instability of a wall jet, with time-varying potential vorticity or transport of the baroclinic mean flow, is studied in a two-layer quasi-geostrophic model. This wall jet is composed of two superimposed strips of uniform potential vorticity, and the layer thicknesses are equal. The steady flow is stable with respect to short waves and its domain of linear instability grows with stratification. The time-dependent flow evolution is governed by a Hill equation which allows parametric instability. This instability indeed appears in numerical flow calculations. It is favored near the marginal stability curve of the steady flow. Near that curve, the evolution equation of the flow is calculated with a multiple time-scale expansion. This equation shows that for zero baroclinic transport of the mean flow, subcritical steady flows can be destabilized by flow oscillation, and supercritical steady flows can be stabilized by medium frequency oscillations. For finite baroclinic transport, this parametric instability vanishes in the limit of short waves or of long waves and narrow potential vorticity strips. Consequences for coastal flows in the ocean are drawn.