We investigate the formation and evolution of dipole vortices and their contribution to water exchange through idealized tidal straits. Self-propagating dipoles are important for transporting and exchanging water properties through straits and inlets in coastal regions. In order to obtain a robust dataset to evaluate flow separation, dipole formation and evolution, and the effect on water exchange, we conduct 164 numerical simulations, varying the width and length of the straits as well as the tidal forcing. We show that dipoles form and start propagating at the time of flow separation, and their vorticity originates in the velocity front formed by the separation. We find that the dipole propagation velocity is proportional to the tidal velocity amplitude and twice as large as the dipole velocity derived for a dipole consisting of two point vortices. We analyze the processes creating a net water exchange through the straits and derive a kinematic model dependent on dimensionless parameters representing strait length, dipole travel distance, and dipole size. The net tracer transport resulting from the kinematic model agrees closely with the numerical simulations and provides an understanding of the processes controlling net water exchange.

Knowledge of coastal ocean transport processes is vital for predicting human impact on the coastal marine environment. Coastal industry discharges pollutants and nutrients into the ocean. In order to understand the impact on the environment, we need coastal ocean circulation models to calculate concentrations and pathways of spreading. Setting up such models for a complex coastline requires a high level of understanding of nearshore transport processes in order to realistically represent these in the models. In shallow coastal regions with complex topography, tides are often a dominant driver of the ocean circulation and transport. In this study, we investigate the exchange process of tidal pumping through narrow tidal straits.

Tidal pumping is an important mechanism responsible for transport of water properties and particles like fish eggs, nutrients, and pollution between estuaries and the open ocean or in coastal regions with complex geometry in general

A sketch of the processes at play in water exchange by tidal pumping.

The propagation of dipoles has been studied for more than 100 years

Equation (

The circulation of the dipole vortices is an important parameter for determining the propagation velocity, and to determine the circulation it is vital to know the source of vorticity. A common assumption is that the vorticity is created in the viscous boundary layer

The net tracer transport through a tidal strait is commonly classified by the nondimensional Strouhal number,

Net tracer transport by tidal pumping is associated with

In this study, our aim is to understand how the geometric constraint of a tidal strait influences the effectivity of tidal pumping. We systematically perform 164 numerical simulations in an idealized tidal strait, varying the width and length of the straits as well as the amplitude of the tidal forcing. Although 3D processes may affect vortex flows

We use the Finite Volume Community Ocean Model (FVCOM)

The surface boundary conditions are

Left panel: the entire model domain with the peninsula attached to the eastern coast and the island located west of the peninsula. Red marks the area with initial tracer concentration equal to 1 m

The model domain is bounded by a semi-circled open ocean and a straight coastline on the eastern side (Fig.

Surface stress (Eq.

In order to investigate the geometric effects on the tidal pumping, we vary the width of the strait,

We simulate a homogeneous ocean over a flat bottom of 100 m depth. To avoid unwanted effects of boundary layers near a vertical wall, we use a sloping bottom at the innermost 600 m from the coastline inside the strait (Fig.

Inside the strait the resolution is 50 m along the coastline. Inside the focus region surrounding the strait the resolution linearly coarsens to 200 m with distance from the coast. The focus region is, in addition to the strait itself, the semi-circle (radius

The simulations are run for a total of 20 d. First, a 10 d spin-up is done before we introduce a passive tracer, which is simulated using the Framework for Aquatic Biogeochemical Models

By visual inspection we see that vortices form in all the different strait configurations. However, only a fraction of the straits produces self-propagating dipoles. Figure

Overview of all simulations performed in this study. Gray marks simulations in which self-propagating dipoles are formed. Panels

The temporal tracer and vorticity fields, with the corresponding stream function, are displayed for a 1 km wide and 4 km long strait in the left and right panel, respectively. The experiment is forced with a tidal wave of amplitude

We choose to show three examples in which the tidal forcing and the strait length are equal (

In the narrowest strait (

In the 4.5 km wide strait (Fig.

The temporal tracer and vorticity fields, with the corresponding stream function, are displayed for a 4.5 km wide and 4 km long strait in the left and right panel, respectively. The experiment is forced with a tidal wave of amplitude

In the widest strait (

The temporal tracer and vorticity fields, with the corresponding stream function, are displayed for a 12 km wide and 4 km long strait in the left and right panel, respectively. The experiment is forced with a tidal wave of amplitude

The three examples shown in Figs.

In the following, we go into the details of flow separation, vortex formation, and dipole properties. These topics are important for the understanding of how strait geometry affects flow dynamics and water exchange through narrow tidal straits.

The timing of flow separation depends on the flow dynamics at the strait exit. Here the balance between nonlinear advection and pressure forces leads to an adverse pressure gradient caused by the widening of the strait. The flow separates from the coastline when the adverse pressure gradient acts in the same direction as the friction and brings the velocity in the viscous boundary layer to zero

Separation time (

The formation of starting vortices and self-propagating dipoles occurs when the flow separates. The vorticity needed to form these vortices originates from the strong velocity front that is formed at the boundary between the newly separated flow and the reversed flow along the coast. At the time of flow separation, the velocity front immediately rolls up into a vortex. This process is illustrated in Fig.

For the simulations with strait widths of 1 and 4.5 km (upper and middle panels of Fig.

Sea surface height (left) and vorticity (right), with contours showing the corresponding streamlines, are shown at separation time. We show the fields in the three straits displayed in

Time series of the maximum magnitude of vorticity at the strait exit, defined as the area where

The initial vorticity of the vortices created during flow separation is an important parameter for determining their ability to form a dipole and the propagation velocity of the dipole that forms. Here, the vortices are represented by the radial profiles of Lamb–Oseen (LO) vortices

The vorticity distribution along a line intersecting the two vortices at each side of the strait at separation time. Panels

From the results shown in Fig.

The vortex core radius at separation time plotted against the separation time. The core radius is the mean radius of the two vortex cores formed at each side of the strait. Straits with self-propagating dipoles are marked in gray.

The theoretical velocity shear

We have shown that the flow separation coincides with a maximum in absolute value of vorticity and that the dipole is formed at the time of separation. The vorticity of the initial vortices is given by strait velocity divided by the core radius, and the initial core radius is nearly equal for all simulations. In the following section, we describe how dipole vortices are recognized and the determination of their propagation velocity.

To obtain dipole properties we track the initial vortices from the time of flow separation to the end of the tidal phase. The vortex centers are points of minimum surface elevation as seen in Fig.

A sketch illustrating the dipole tracking. (

The criteria are based on two simple principles. The first criterion is that a dipole will propagate normal to the line connecting the two vortices and therefore conserve the distance between them

When tracking the vortices we obtain the dipole propagation velocities, which, together with the tidal velocity and vorticity distributions, enable us to investigate the vortex properties.

Dipole properties, such as core radius (

Comparing the tracked dipole velocities to the theoretical velocities obtained from Eq. (

Dipole propagation velocity for a dipole formed in

The dipole velocities obtained from tracking (on the

Assuming the two vortices are of equal strength gives

The dipole propagation velocity is crucial when determining the transport properties of the dipole in relation to tidal pumping

To investigate the role of dipole vortices in generating net water exchange, we first quantify the effective tracer transport

The ability of the dipole to escape the return flow determines its contribution to water exchange through a strait

We follow the approach of

The position of the dipole relative to the sink radius at

The effective transport,

The dipole can only be an important contributor for water exchange if the strait is shorter than the tidal excursion. If the strait is longer than the tidal excursion, the water mass on one side of the strait will not be able to travel through the strait, with zero net tracer exchange as a result. In order to evaluate the effect of strait length we introduce the nondimensional length scale as

Idealized distribution of a tracer at

At

The effective transport

The resolution of our mesh varies from 50 m in the center of the strait to 20 km at the outer boundary. The Rossby radius is

Vorticity is created in the velocity front formed by flow separation. The simulated vorticity in the velocity front depends strongly on model resolution. However, the total production of vorticity with time is less dependent on resolution. This can be shown by integrating the vorticity over an area containing a segment of the velocity front. During a time

To study the effect of resolution, we have repeated a number of the simulations using finer mesh resolution. In the new simulations, the resolution at the coast is set to 10 m inside the strait. The other simulations presented in this paper have 50 m resolution at the coastline (see Sect.

Comparison between 10 and 50 m resolution simulations. Panels

The strait velocities, dipole propagation velocities, and the effective transport resulting from the high-resolution simulations are all similar to the results from the coarser simulations (Fig.

Even if the velocity, dipole propagation, and tracer transport are not very sensitive to mesh resolution, we clearly see that vorticity in the high-resolution simulations reaches larger values. Determining the separation time from the time of maximum vorticity is not a reliable method in the high-resolution simulations. There is still a significant vorticity increase at the time of separation, but the maximum vorticity now typically occurs at the time of maximum strait velocity. The separation times are therefore determined by visual inspection, and they are similar to the ones in the 50 m resolution simulations. Another interesting observation is that one side of the dipole may consist of two co-rotating vortices in the high-resolution simulations, while it is a single vortex in the coarser simulations. The theoretical dipole propagation velocity (Eq.

To understand why strait length is a restriction factor for dipole formation (Fig.

The dipole propagation velocity depends on the strength of the vortices set by their vorticity, and it is important to understand how the vorticity is generated.

The dipole formation is associated with a maximum in time of the absolute value of vorticity (Fig.

The velocity front rolls-up immediately after separation and creates the dipole vortices (Fig.

As shown by Figs.

A derivation of propagation velocity for a dipole connected to a jet is presented by

In this study, we have performed a total of 164 numerical simulations of an ideal tidal strait, investigating flow separation, dipole formation, and water exchange for different widths and lengths of the strait. We show that dipoles form and start propagating at the time of flow separation. The vorticity of the dipole vortices originates from the velocity front created by flow separation. The simulated dipole propagation velocity is twice as large as the propagation velocity derived for vortex pairs with no background flow

We derive two parameters

The kinematic model (Eq.

Model code is available at

OAN conceived the study, designed the model setup, and wrote the response to reviewers. EB did most of the numerical simulations and developed routines for dipole tracking and for analyzing the simulation results. Both authors have contributed to writing the paper, developing the kinematic model, and analyzing simulation results.

The authors declare that they have no conflict of interest.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank Pål Erik Isachsen for constructive scientific discussions and comments on the paper.

This research has been supported by the Norwegian Research Council (project no. 308796) and VISTA – a basic research program in collaboration between the Norwegian Academy of Science and Letters and Equinor (project no. 6168).

This paper was edited by Anne Marie Tréguier and reviewed by three anonymous referees.