Numerical modelling of hydraulic control, solitary waves and primary instabilities in the Strait of Gibraltar

Highlights

• Small-scale mechanisms in the Strait of Gibraltar simulated at very high resolution.

• Explicit simulation of primary shear instabilities in the area of the strait of Gibraltar.

• New non-hydrostatic non-Boussinesq version of the CROCO oceanic model.

Abstract

A two-dimensional, vertical section of the Strait of Gibraltar is simulated numerically with the nonhydrostatic/non-Boussinesq three-dimensional CROCO model to investigate details of small-scale dynamics. The proposed configuration is simple, computationally efficient and incorporates the configuration of sills characteristic of this region. Despite the shortcomings of a 2D representation, this configuration provides a realistic depiction of small-scale mechanisms in the strait during a typical tidal cycle: internal solitary waves generation and propagation, occurrence of hydraulic controls and hydraulic jumps at the sills and presence of active turbulent patches. In particular, the well-known eastward propagation of large amplitude internal waves is assessed using the Korteweg de Vries (KdV) propagation model for solitary waves.

As a step towards establishing a realistic three-dimensional Large Eddy Simulation (LES), the sensitivity of the configuration to various choices (e.g., resolution, amplitude of tidal forcing or numerical schemes) is investigated. Our analyses indicate that the representation of small-scale dynamics in the Strait of Gibraltar can be much improved by increasing resolution and relaxing the hydrostatic assumption. Further studies are necessary to grasp the mechanisms of mixing and/or stirring induced by this fine scale process.

Introduction

The Strait of Gibraltar connects two major basins: the Northern Atlantic and the Mediterranean Sea, over which evaporation exceeds precipitation and river run-off. To compensate the resulting loss, exchanges of mass and salt are required through the strait. Fig. 1 illustrates the rather complex exchanges occurring there. Inflowing Atlantic water is less salty (salinity $S_A\approx 36$) than the outflowing Mediterranean water ($S_M>38$), and spreads as a surface layer in the Alboran Sea. The interface between the two water masses is distorted by undulations that are not precisely periodic with regard to the tidal cycle but exhibit regularity in some areas. One of the paper objectives is to better understand the small-scale processes that lead to the Atlantic and Mediterranean water masses transformation in the vicinity of the Strait of Gibraltar.

To further illustrate the exchange between the Northern Atlantic and the Mediterranean, a very simple steady-state model can be expressed as a system of two basic conservation equations.

Volume conservation is expressed as :

$$Q_A+Q_M=E-P$$

while the conservation of salt requires:

$$Q_A{S}_A+Q_M{S}_M=0$$

where $Q_A$ is the Atlantic water volume flux (positive), $Q_M$ is the Mediterranean water volume flux (negative), both localized in the Strait of Gibraltar, and $E-P$ is the space-averaged Evaporation minus Precipitation (and river runoff) water budget integrated over the whole Mediterranean Sea. $E-P$ is positive. $S_A$ ($S_M$) stands for Atlantic (Mediterranean) water mean salinity and $S_M-S_A\approx 2$ (Bethoux, 1979). The water budget $E-P$ is positive in the Mediterranean due to excess evaporation that correspond to a yearly averaged loss of water of about 1 metre over the whole basin (Garrett et al., 1990).

A major dynamical feature in the Strait of Gibraltar is the so-called “flow criticality” usually characterized by the Froude number ($F$): it compares the internal wave phase speed with a flow characteristic velocity. Several definitions of the non-dimensional Froude number can be found in the literature: it can notably be defined for each layer, resulting in a composite number for the whole water column, as in Farmer and Armi (1988) or in Sannino et al. (2009b).

A “subcritical” (respectively “supercritical”) regime lies in the range of small (respectively large) values of the Froude number $F<1$ (respectively $F>1$), with an intermediate “critical” regime for $F\approx 1$. The upstream propagation of internal waves is inhibited for supercritical flow so that a hydraulic control occurs at the transition from subcritical to supercritical flow; it persists during periods and within regions of large Froude numbers. As such, the hydraulic regime at a given point will vary in time according to substantial currents variations occurring along the tidal cycle. It is, for example, well established that large amplitude solitary waves in the Strait of Gibraltar can develop due to the hydraulic control at Camarinal Sill (Farmer and Armi, 1988), making it a crucial process to represent.

Several analytical models have been proposed to investigate the hydraulic control in the Gibraltar region (Bryden and Stommel, 1984, Farmer and Armi, 1986, Garrett et al., 1990). The hydraulic control usually occurs in these models at Camarinal Sill (CS), Espartel Sill (ES), and Tarifa Narrows (TN), although the modelled hydraulic control location and frequency vary according to the model refinement :

• Farmer and Armi (1986)’s two-layer model accounts for the strait geometry (depth and width), the exchanged volumes ($Q_A$ and $Q_M$) and the salinity contrast ($S_A-S_M$). This simple model is able to simulate two hydraulic controls: the first one located by the sill, the other in the TN contraction, defining “maximal exchange regime” (further details are given below).

• In a slightly more elaborated model, the inclusion of entrainment between the two layers and the subsequent interfacial layer introduction modify the left-hand terms of Eqs. (1), (2) with the introduction of horizontal and vertical transports in the interfacial layer (Bray et al., 1995). Critical conditions are changed within such two interfaces model which may support two baroclinic modes and new hydraulic controls (Sannino et al., 2009b).

• Considering a three-dimensional flow, the definition of the control needs to account for cross-strait variations such as the tilt of the density interface in the latitudinal direction. In the maximal exchange solution, control in TN may induce the detachment of the surface layer from the northern coast (Sannino et al., 2009b).

The hydraulic control effect within the strait is illustrated in Fig. 1. The flow is initially subcritical in the Strait; the propagation of internal waves is not hindered at the interface between Atlantic and Mediterranean waters (denoted “a” in Fig. 1); then the tidal flood in the vicinity of the Camarinal Sill becomes supercritical. In the supercritical to subcritical transition, downstream of the sill, a “hydraulic jump” (“b” in Fig. 1) may occur.

Hydraulic jumps are large-amplitude depressions in the regions where hydraulic controls occur. There, intense mixing between the Atlantic and Mediterranean waters takes place as observed by Wesson and Gregg (1994). Shear flow instabilities can develop in the hydraulic jump of the Camarinal Sill (denoted “c” in Fig. 1).

The release of hydraulic jumps generates large-amplitude, non-linear, nonhydrostatic Internal Solitary Waves (ISW) trains (denoted “d” in Fig. 1) (Farmer and Armi, 1988). As the barotropic tide is constrained by the bathymetry, large vertical velocities appear and induce energy transfer to several normal modes of internal waves. Some observations in the Strait of Gibraltar identify the largest ISW amplitude to the first baroclinic mode; for which vertical velocities have the same direction throughout the water column and all isopycnal surface displacements are in phase. The signature of Mode 2 waves (the vertical velocity profile exhibits one node) has also been observed in the region of Gibraltar strait (Farmer and Armi, 1988, Vázquez et al., 2006). The internal waves propagate at the interface of Mediterranean and Atlantic waters.

As the strait flow varies at various timescales during the year, some deviation is expected in the occurrence of the hydraulic control in the strait. This may have a wide impact since local flow conditions combined with the above two conservation equations (1), (2) determine the relation between the volume fluxes, the evaporation minus precipitation budget ($E-P$) and the salinity difference ($S_A-S_M$) (Bryden and Kinder, 1991). Practically, an “overmixed” solution corresponds to a minimal salinity difference and a maximal exchange of water mass in the strait: it would thus constrict the formation of Mediterranean waters and diapycnal mixing over the Mediterranean basin (Bryden and Stommel, 1984, Garrett et al., 1990). Moreover, the small-scale processes occurring in the strait itself can directly modify the local characteristics of Mediterranean waters (García-Lafuente et al., 2011, Naranjo et al., 2015) and Atlantic waters (Millot, 2014). This can affect their characteristics as they enter respectively in the North Atlantic sub-basin and in the Mediterranean Sea.

To study the flow dynamics in the strait in further details, more realistic numerical modelling is of great help. Early attempts used two-layer models (Brandt et al., 1996, Izquierdo et al., 2001). The increase of computational power led to 3D modelling (Sannino et al., 2004) with increasing vertical and horizontal resolution, explicitly addressing the tidal cycle and flow characteristics. More recently, even nonhydrostatic models have been used (Sánchez Garrido et al., 2011, Sannino et al., 2014) to explicitly represent the ISW. Other configurations include the Strait of Gibraltar into a Mediterranean circulation model (Soto-Navarro et al., 2015). In this case, the increased resolution locally in the strait (Naranjo et al., 2014) – or the nesting of high-resolution grids within a coarse resolved regional model (Sannino et al., 2009a) – shows a clear impact on Mediterranean stratification and improves the representation of convective events in the northwestern Mediterranean basin.

The coastal and regional ocean modelling community model (CROCO¹ ) is based on a new nonhydrostatic and non-Boussinesq solver (Auclair et al., 2018) developed within the former ROMS kernel (Shchepetkin and McWilliams, 2005), for an optimal accuracy and cost efficiency. CROCO opens up new perspectives in terms of modelling of small-scale processes (Fox-Kemper et al., 2019, Lemarié et al., 2019). In this sense, the present study objectives are also numerical: we show that a new generation of nonhydrostatic ocean models can be used efficiently to simulate complex nonlinear, fine scale physics in a realistic but computationally-affordable configuration. The complete solution of Navier–Stokes equations is thus solved numerically for the very first time in a complex realistic regional configuration.

The present configuration of the Strait of Gibraltar is based on a classical lock-exchange initialization (Sannino et al., 2002). A 2D vertical section of the strait is adopted in order to reduce the number of parameters impacting the studied dynamics. This rather simple configuration is thus of weak computational cost and reduces the implementation burden; it allows to reach the horizontal and vertical scales of the largest turbulent structures observed in this area. In the strait, where most transverse dynamical feature are an order of magnitude weaker, our numerical approach is some kind of ersatz of a large-eddy simulation (LES² ), for which at least the generation process of primary instabilities is correctly represented. However, LES is a 3D concept as the route to molecular dissipation differs in 2D and 3D turbulence. The present study is focused on the description of the largest primary instabilities in the Strait of Gibraltar; as well as providing order of magnitudes for explicit simulations of these dynamics. Along with these physical aims, the relevance of the chosen numerical methods is a major concern. A quantified impact of the largest turbulent structures on the water masses is out of the scope of what is presented hereafter: it would require a fully three dimensional LES (also achievable with the CROCO model), in complement with dedicated relevant experimental measurements.

In Section 2, we present an overview of CROCO equations and the implementation for the 2D lock-exchange experiment. We describe the implementation of the bathymetry profile, water masses, and the exchange and tidal flows. In Section 3, we analyse the physics of the 2D configuration, comparing the model solution to already published data (e.g., in Farmer and Armi (1988)). Emphasis is then made on the hydraulic control (Section 3.2), the hydraulic jump (Section 3.3) and the mode-1 and mode-2 ISW (non-linear internal trains of solitary waves) propagation (Section 3.4). Last, the sensitivity to the tidal forcing amplitude and to the numerical choices are analysed respectively in Sections 4.1 Tidal regime, 4.2 Nonhydrostatic balance and numerical factors, with a focus on the fine-scales dynamics listed in Fig. 1.