This study investigates the linear and non-linear
instability of a buoyant coastal current flowing along a sloping topography.
In fact, the bathymetry strongly impacts the formation of meanders or eddies
and leads to different dynamical regimes that can both enhance or prevent the
cross-shore transport. We use the Regional Ocean Modeling System (ROMS) to
run simulations in an idealized channel configuration, using a fixed coastal
current structure and testing its unstable evolution for various depths and
topographic slopes. The experiments are integrated beyond the linear stage of
the instability, since our focus is on the non-linear end state, namely the
formation of coastal eddies or meanders, to classify the dynamical regimes.
We find three non-linear end states, whose properties cannot be deduced
solely from the linear instability analysis. They correspond to a
quasi-stable coastal current, the propagation of coastal meanders, and the
formation of coherent eddies. We show that the topographic parameter

We suggest the use of the parameter space (

Coastal currents can act either as a source of coherent eddies or as a
dynamical barrier to the offshore redistribution of coastal waters, thus
controlling the cross-shelf transport in a local or regional circulation. When
these currents are unstable, large meanders grow and can lead to the
formation of mesoscale eddies that capture and transport water masses
towards the open sea over large stretches of the coastline. Conversely,
stable coastal currents enhance the along-shore transport and strongly reduce
the cross-shelf transport. Hence, the formation and the propagation of
coastal eddies across the coastal shelf plays a significant role in the local
mixing of biogeochemical properties, in the dispersion of pollutants, and in
the redistribution of nutrient-rich coastal waters toward the oligotrophic
open sea

On the one hand, many coastal currents such as the Algerian Current

There exist numerous linear stability analyses of baroclinic currents flowing
over sloping topography which are based on layered models

Hence, for a geostrophic coastal current, the linear stability
analysis predicts that two different regimes of instability with distinct
wavelength selection can occur above the shelf bathymetry. When the geostrophic coastal current
is controlled by the baroclinic instability, the decrease
in the topographic parameter

However, it is
well known that the linear stability analysis is limited by its inability
to predict the final amplitude of unstable meanders. Furthermore,
non-linear processes may lead to larger or smaller structures than
the ones predicted by the linear analysis. Moreover, previous studies
have shown that a sloping topography has a strong impact on the non-linear
saturation of unstable surface flows

The primary goal of this study is to go beyond the linear stability analysis and investigate the non-linear impact of the sloping topography on the formation of coastal eddies, namely whether or not the current generates a significant non-linear cross-shore disturbance. To answer this question, we use an idealized model of a buoyant current with a continuous and non-uniform stratification. Indeed, the effects of a continuous stratification and the relevant set of dynamical parameters that govern the stability of geostrophic currents along a sloping bathymetry are not well established. Unlike standard linear instability analysis, the use of a full non-linear model allows us to discuss the end state of the instability, in other words the various regimes of formation of large-scale meanders or coastal eddies. We find that investigating the non-linear evolution shows three possible non-linear end states, while the linear analysis predicts only two types of instability – barotropic and baroclinic. A classification of the various non-linear end states provides a more direct comparison with surface oceanic observations, such as sea surface temperature (SST) images or sea surface height (SSH) maps, where only finite-amplitude perturbations or coherent eddies can be detected.

In Sect. 2, we present the initial state of the coastal current and the main dynamic and topographic parameters of the system. In Sect. 3, we investigate how different values of the bottom slope and depth affect the stability of a surface intensified current. We then show, in Sect. 4, the role played by the topographic parameter in controlling the amplitude of the unstable perturbations. If these perturbations reach a finite amplitude, large meanders or coherent eddies are formed. The characteristics of these final non-linear structures and their corresponding parameter space are given in Sects. 5 and 6. Discussions and conclusions are given in the final Sect. 7.

To investigate the effect of a sloping topography on meanders and eddy
formation in a buoyant coastal current, we employ the Agrif version

We use a periodic rectangular domain, with

The initial state consists of a steady geostrophic surface current flowing
along an idealized sloping bathymetry (Fig.

We use a linear equation of state and set salinity to a constant; thus, the
density stratification is a function of temperature only and is equal to

The width and the depth of the coastal jet are fixed by

Simplified configuration of a surface coastal current above an
hyperbolic tangent bathymetry. The along-shore velocity

The model solves the primitive equations with a split-explicit free surface, where short time steps are used to advance the surface elevation and barotropic momentum equation, and a larger time step is used for temperature and baroclinic momentum. We stick to the ROMS philosophy of removing explicit horizontal turbulent closure terms on both temperature (no diffusion) and momentum (no viscosity), and of letting the third-order upstream-biased advection operator handle the necessary dissipation at grid scale. We consider initial value problems with no external forcing, but only an initial white noise added to the velocity field. Consequently, the surface momentum, heat, and freshwater fluxes are set to zero.

For a flat bottom configuration, the dynamics of the coastal current
mainly depend on the Rossby (

In this study the intensity of the initial current is kept fixed with a
maximum surface velocity

To establish a contact point with two-layer theories, we introduce
the vertical aspect ratio parameter:

This is not the case in a sloping topography scenario. In fact, since the
current flows above a sloping bathymetry, a key dynamical feature is the
interaction of the current with the topographic Rossby waves. In the case of
a buoyant coastal current, the topographic Rossby waves propagate in the same
direction as the flow. The propagation speed of these waves is proportional
to the dimensionless topographic slope

The speed

We will see later that

In this section we present how different slopes and bottom depths can affect the growth rate and the non-linear saturation of the cross-shore perturbations, while keeping all the other parameters and the buoyant current constant.

At the initial stage the coastal current is mainly along-shore; hence we can
assume that

Various metrics could be used to quantify the departure from the initial coastal current. We note here that, while this analysis does not take into account any anisotropic perturbation directed in the along-shore direction, we have checked that this does not change qualitatively the results presented in this study. Moreover, the focus here is on the generation of cross-shore transport by an unstable coastal current, as opposed to a stable along-shore flow that reduces the transport of water offshore. Therefore, we chose this metric to put emphasis on the cross-shore perturbations that may break the along-shore jet and lead to the formation of large meanders or coherent eddies.

The temporal evolution of the ratio in Eq. (

Time evolution of

We now present the impact that a variable

Figure

Now, we analyse the cases of a constant slope

A similar agreement with the two-layer case is found when we vary the water
depth above a sloping bathymetry: as we reduce

This initial set of results suggests that both the bottom slope and the water
depth have a strong impact on the non-linear stabilization of the along-shore
current. The impact of the sloping bathymetry is increased when the water
depth is reduced, and inversely for very large water depths. Extrapolating
this to the case of an infinitely deep ocean, we could expect to reduce or
even cancel the impact of the bottom slope. The combined effect of variables

Parameter space of the experiments performed with a 2 km grid resolution, where

In order to quantify more precisely the influence of a sloping bathymetry on
the stability of the coastal current, we plot in Fig.

Figure

As was shown in

In order to estimate the most unstable wavenumber, we perform, as in

Panel

In order to identify the nature of these two branches we determine the source
of kinetic energy of the perturbations for the instability. In this
simplified jet configuration, there are basically two source terms

We select the stage of exponential growth of KE

The relative vorticity (

We now present a possible interpretation on why

Same as Fig.

Same as Fig.

The discussion on

However, by using the topographic parameter alone we are not able to
distinguish between different non-linear end states when

We present in the next two sections the characteristics and parameter space of the different end states identifiable from the non-linear analysis.

We have shown in Figs.

From a linear stability perspective, the jet is unstable: waves grow
spontaneously from random perturbations, though always very slowly compared
to the other regimes. However, the wave growth does not last long and never
until the full breaking. The wave amplitudes get saturated at a level small
enough to be hardly competing with the background flow, causing the jet to be
barely changing in time (Fig.

A similar non-linear stabilization was found for coastal fronts and currents
with the two-layer laboratory experiments performed by

As stated in Sect. 2, we have also performed a few runs at the higher grid
resolution of d

The formation of coherent eddies from an unstable coastal current generally
results from the pinching off of large meanders. However, these meanders may
saturate at an intermediate stage and never lead to the generation of coastal
eddies. The value of the non-linear saturation parameter

However, the vertical structure is not universal and for a similar signature
at the surface these coherent eddies could have quite a different structure
in the deep layer. Two cases of coastal eddy formation are depicted in
Figs.

Hence, these examples show that two distinct mechanisms of linear instability, namely the baroclinic or the barotropic shear instability, can lead to the same non-linear end state: the formation of coherent eddies in the surface layer which are able to trap water mass in their core and escape from the coast. If we consider only the surface signature of coastal eddies, provided by standard remote-sensing measurements such as SST images or SSH maps, we can accurately identify the non-linear coastal eddy regime but hardly make any distinction between the underlying linear instability mechanisms.

In contrast to the coastal eddies regime, the formation of coastal meanders
corresponds to a non-linear stage where the parameter

Same as in Fig.

The vertical structure of such coastal meanders is quite different from the
coastal eddies regime. The vorticity is never in phase between surface and
bottom, but it is rather in phase quadrature (Fig.

We have shown that

Note that we have investigated here the range of small and moderate aspect
ratio parameter (

Diagram in the

In this paper we have studied the non-linear evolution of an unstable buoyant
current, flowing along a coastal slope, for various depths and sloping
topographies. The current, kept unchanged, is always linearly unstable. We
determined the properties of the linear instability (growth rate, wavelength)
from the direct integration of the primitive equations forward in time. The
properties of the linear stage (the exponential growth) match published
results

Snapshots of SST for the coastal eddies regime

Same as Table

The most interesting finding of this study is that

We have also shown that in addition to the topographic parameter

This work emphasizes the limitations of linear stability analysis to classify
eddy formation, because it does not account for the non-linear saturation
which is predominant for large negative

All the simulations performed in this study can be reproduced with the information in the text (domain geometry, resolution, Eqs. (1)–(4), and information in Tables 1 and 2).

For a continuous stratification

The first baroclinic deformation radius corresponding to this stratification
is

Vertical profile of the potential density anomaly

The authors declare that they have no conflict of interest.

We acknowledge helpful discussions with Andrew Stewart, and the anonymous reviewers for their useful comments. This work was funded by the ANR Astrid Project SYNBIOS (ANR 11 ASTR 014 01).Edited by: John M. Huthnance Reviewed by: two anonymous referees