# On the Observed Wind-Driven Circulation Response in Small Semienclosed Bays

Alejandro Cáceres-Euse, Veronica Morales-Márquez, Anne Molcard

► To cite this version:

Alejandro Cáceres-Euse, Veronica Morales-Márquez, Anne Molcard. On the Observed Wind-Driven Circulation Response in Small Semienclosed Bays. Journal of Physical Oceanography, 2023, 53 (9), pp.2245-2261. 10.1175/JPO-D-22-0224.1. hal-04482611

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# On the Observed Wind-Driven Circulation Response in Small Semienclosed Bays

ALEJANDRO CÁCERES-EUSE,ª VERONICA MORALES-MÁRQUEZ,ª AND ANNE MOLCARDª

ª Université de Toulon, Aix Marseille Université, CNRS, IRD, MIO, Toulon, France

(Manuscript received 29 October 2022, in final form 22 June 2023, accepted 27 June 2023)

**ABSTRACT:** This study analyzes horizontal and vertical wind-driven circulation responses in small semienclosed bays, the associated offshore dynamic conditions, and the relative importance of each term in the momentum balance equations using a multiplatform observational system. The observational platform consists of three ADCPs and a land-based radar monitoring the velocity field within the bay and in the contiguous offshore area. The wind-driven patterns in the bay can switch from a barotropic cyclonic or anticyclonic circulation to a two-layer baroclinic mode response as a function of the wind regime (its direction and magnitude). For the baroclinic mode, the vertical location of the inflection point in the velocity profile can vary according to the proximity of the boundary current to the entrance of the bay. The influence of offshore combined meteorological and marine conditions on the inner-bay dynamics is evidenced under moderate to strong wind conditions and is almost nonexistent under negligible wind. The momentum balance analysis as well as the nondimensional numbers evidence the impact of wind stress, coastline shape, stratification, and the nonlinear advective terms. Advection can be at the same order of magnitude as pressure gradient, Coriolis, or wind stress terms and can be greater than the bottom stress terms. The nonlinear terms in the momentum equations are frequently neglected when analyzing wind-driven circulation by means of in situ data or analytical models.

**KEYWORDS:** Mediterranean Sea; Currents; Atmosphere-ocean interaction; Measurements; Neural networks

## 1. Introduction

The wind shear stress exerted on the sea surface layer is one of the main drivers of the transport of dissolved and suspended materials (Sanay and Valle-Levinson 2005) and plays a decisive role in the circulation, stratification, and water properties in coastal bays (Largier 2020). Sea currents in coastal bays exhibit complex spatial circulation patterns that impact important ecological processes such as larval dispersal, transport of harmful algal blooms, and localized upwelling and connectivity with offshore (Largier 2020; Trautman and Walter 2021). In addition, the circulation response of bays to wind forcing involves a variety of physical factors, including wind stress magnitude and direction, Earth's rotation, density stratification (Jia and Li 2012), horizontal pressure gradients, tides, surface gravity waves, bottom friction, and bathymetric/geographic effects (Lentz and Fewings 2012).

The understanding of wind-driven circulation processes in coastal bays has been mostly addressed following three branches. A first branch uses primitive-based equation modeling to determine the vertical and/or the horizontal circulation patterns under realistic (Jia and Li 2012; Huang and Li 2017) and idealized conditions (Austin and Lentz 2002; Sanay and Valle-Levinson 2005; Li 2013). A second branch uses linear analytical models (Winant 2004; Liu and Weisberg 2005b; Reyes-Hernández and Valle-Levinson 2010; Hinata et al. 2010), where the momentum equations are simplified according to specific spatial-temporal scale assumptions [Rossby (Ro) number much smaller than 1], so that the analytical solution is used to identify the dominant processes driving currents. Finally, a third branch performs empirical

estimations from in situ data (Lentz et al. 1999; Ponte et al. 2012; AlYousif et al. 2021) in order to validate existent analytical models and to estimate the contribution from each term in the momentum equations locally. Based on these approaches, it has been possible to establish that wind-driven responses in coastal bays depend (mainly but not only) on the Ekman (Ek) and Wedderburn (W) numbers behavior (Lentz 2001; Valle-Levinson 2008; Reyes-Hernández and Valle-Levinson 2010; Ponte 2010). Thus, Ek and W provide insights about the circulation patterns in bays due to wind stress as well as the potential balances with the barotropic and/or baroclinic gradient contributions (horizontal pressure gradient component), with the bottom friction, with the Coriolis effects or a combination of all of these components.

Despite the relevance of wind stress on the circulation patterns in coastal bays, the most accepted theory describes a linear physical process with no advection involved (Lentz and Fewings 2012), with an important contribution from advection when the bay is exposed to outward shelf jets (wide-open bays) such that the inertia overcomes the wind stress (Valle-Levinson and Moraga-Opazon 2006; Largier 2020). Most investigations focused their attention on wind-driven circulation in the inner shelf zone along open coasts (Lentz et al. 1999; Lentz and Chapman 2004; Liu and Weisberg 2005a; AlYousif et al. 2021), within large semienclosed bays and away from the bay entrance (Ponte et al. 2012), in shallow enclosed lagoons (Jia and Li 2012; Melaku Canu et al. 2016), in a large and shallow semienclosed (almost closed) lake estuary with restricted openings to the coastal sea (Huang and Li 2017; Li et al. 2018; Huang and Li 2019), and in large semienclosed bays with several inlets formed by islands that protect the bay from open sea forcings (Li 2013; Li et al. 2019). Nonetheless, the wind-driven response in small semienclosed bays exposed to highly energetic open sea dynamics, and the role played by

<footnote>Corresponding author: Alejandro Cáceres-Euse, alejandro.caceres-euse@univ-tln.fr</footnote>
each term in the momentum equations are still poorly understood. In fact, there is scarce observational evidence of such complex circulation patterns and the effects produced by the geographical constraints.

This work presents advances in the understanding of wind-driven circulation responses in small semienclosed bays, through the development of a method that combines meteomarine (meteorological and marine) information from a multiplatform system. To this end, this study focuses on answering two main research questions: 1) how would be the horizontal and vertical structure of the circulation in the bay according to the wind regime and the associated offshore conditions, and 2) what is the role played by each term in the momentum equations as a function of the local wind regime?

The multiplatform system collects observations of offshore sea surface currents measured by a high-frequency radar (HFR) system and inner-bay current velocity profiles from acoustic Doppler current profilers (ADCPs) as well as wind, pressure, temperature, and density information surveyed in the study area from 14 December 2020 to 15 September 2021 (during a 9-month period). The combination of HFR and ADCPs together represents a powerful method to evaluate coastal circulation features (Liu et al. 2007; Paduan and Washburn 2013), since HFR provides an excellent horizontal coverage at the surface while ADCPs give a high vertical resolution at a single point. The region of interest for this study (northwestern Mediterranean Sea) was chosen for its rich meteomarine dynamics, including strong winds, a geostrophic boundary current, and complex geographical and bathymetric constraints.

This paper is structured as follows: in section 2 we describe the geographical configuration of the study site. Section 3 introduces the multiplatform system and the methodology developed to determine the linkage between winds and currents in the bay, and the horizontal and vertical structures of the wind-driven circulation. Results start with section 4, where the most relevant meteomarine conditions present offshore as well as within the bay are described according to the observations. In section 5 the time scale of linkage between wind and current inside the bay is determined, and the horizontal and vertical wind-driven responses together with the offshore dynamics analyzed. In section 6 a depth-averaged momentum balance is applied to the in situ data along with an analysis of the most relevant nondimensional numbers to reveal the contribution from each term in the momentum equations according to the wind regime. Section 7 concludes and highlights the findings obtained in this work.

## 2. Study area

Toulon Bay is a small semienclosed bay located on the southern coasts of France in the northwestern Mediterranean Sea. The bay is sheltered and enclosed by the Giens Peninsula (GIENS), the Saint Mandrier Peninsula (Mandrier), and several capes and islands [such as Porquerolles (POR), Port-Cros (CROS), and Levant (LEV) islands and Cape Carqueiranne] (see Fig. 1). It is an elongated bay and is less than 9 km long

in the longitudinal axis and less than 4.5 km long in the latitudinal axis. The entrance of the bay is connected to the Mediterranean Sea (offshore) through a V-shaped channel-type bathymetry, presenting a steep continental shelf that goes from tens of meters in the bay to several hundreds of meters offshore. The Toulon region is an urbanized zone with several socioeconomic activities such as fishery and tourism, the largest military port in France and the National Park of Port-Cros. The area has a monitoring system that measures the surface currents offshore and records meteomarine information in the bay. For instance, it has been observed that the circulation inside the Toulon Bay is highly influenced by the wind forcing (Dufresne et al. 2014), with some associated upwelling episodes observed (Mazoyer et al. 2020) depending to the wind direction. Conversely, in the offshore zone the well-known boundary Northern Current (NC) determines the main kinematic characteristics (Bourg and Molcard 2021), being part of the general cyclonic circulation of the western Mediterranean basin (Millot 1999). The study site can be considered as an ideal example for the aim of this work. It is exposed to a highly energetic offshore dynamic that might alter the inner-bay circulation through connectivity with the interior of the bay, the region is continuously subject to strong wind stress, the weather is dry with less than 550 mm of rain per year, and the bay has two small tributaries that can present episodic storm flow discharges lasting less than 2 days (Nicolau et al. 2012). To define a coastal bay as small, the size of the bay needs to be smaller than the spatial scale at which the rotation of Earth is dominant (Valle-Levinson 2008). Then, by using a characteristic length of the bay $L$ and the external Rossby radius of deformation $R_{\text{ext}}$ (Li et al. 2018), one can state that a coastal bay can be considered to be small if $L/R_{\text{ext}} \ll 1$, with $R_{\text{ext}} = \sqrt{g\tilde{h}/f}$ being a function of the gravitational acceleration $g$, water depth ($\tilde{h} = 30$ m) at the center of the bay, and Coriolis parameter $f$. In this study, $L = 4$ km and $R_{\text{ext}} = 172$ km, so the bay can be defined as small assuming no stratification effects. Up to this point, the definition of a small bay is formulated based on a barotropic flow assumption, so $R_{\text{ext}}$ does not account any baroclinic contribution. The stratification will be discussed later in the analysis when estimating the baroclinic Rossby radius of deformation.

## 3. Data and methods

### a. Offshore

In this study, the offshore data are composed of wind and surface sea currents. Information from the wind field at 10-m height ($U_{10}$) comes from the Application of Research to Operations at Mesoscale (AROME) wind model. AROME is a nonhydrostatic 3D high-resolution numerical model, operational at Meteo-France since December 2008. The size of the mesh is 1.3 km over France and the outputs available every 3 h (for details see http://www.umr-cnrm.fr/). The model has been assimilated at local scale on hourly basis using the Application Radar à la Météorologie Infra-Synoptique (ARAMIS) radar network (Doppler wind and precipitation) (Brousseau et al. 2016). On the other hand, the surface currents are surveyed
<figure><img src="image_2.png" /><figcaption>FIG. 1. Geographical and bathymetric configuration of the Toulon Bay and the offshore region, and the instrument deployment. Shown are the Porquerrolles (POR), Port-Cros (CROS), and Levant (LEV) islands and the Giens Peninsula (GIENS). The isobaths are in meters. The blue points represent the high-frequency radar coverage. The black circles are the stations with acoustic Doppler current profilers and weather and conductivity-temperature-depth instruments. The white diamonds are the pressure sensors. A map of the northwestern Mediterranean Sea is included in the lower-right corner with a red square box delimiting the study site. The photograph at the top is a panoramic view of the site and the mountain chain system located north of the bay. The black-and-white-outlined box (outlined in black on the bigger map) represents the center of the HFR domain.</figcaption></figure>

with a high-frequency wave radar (HF-WERA) system manufac-
tured by Helzel GmbH and operational since 2012 (http://hfradar.
univ-tln.fr/HFRADAR/squel.php?content=accueil). The HFR
system provides maps of total velocity currents in a Cartesian

projection every hour at 1 km spatial resolution approximately
(see Fig. 1, blue points). A complete description of the HFR per-
formance can be found in Guérin et al. (2019) and Dumas et al.
(2020), as well as the data collection, their processing, and
calibration. Gaps and noisy velocity values inside the HFR data coverage are regularly filled (and replaced) by means of the Data Interpolating Empirical Orthogonal Functions (DINEOF) gap-filling algorithm (Alvera-Azcárate et al. 2005). This is performed through determining the dominant modes of the data variability (from EOFs), so that these modes are used to fill gaps by an iterative method [see Bourg and Molcard (2021) for more detail].

## b. Toulon Bay

The field information in the bay includes two pressure sensors, three ADCPs, conductivity-temperature-depth (CTD) stations, and meteorological observations (wind speed, direction, rain, air temperature, and humidity). The pressure records are located in the inner bay (see Fig. 1, white diamonds P1 and P2). Both P1 and P2 are composed of two piezometric sensors, one immersed ($p_{sub}$) and one emerged ($p_{atm}$), so as to estimate the water level every 2 min as $\eta = \Delta p/\rho g$, where $\Delta p = p_{sub} - p_{atm}$, g is the gravity, and $\rho$ is the seawater density. This information is available through the operational HTM-NET network since 2013 (Hydrodynamique et Transport de Matière en Suspension-Niveaux dp'Eau et Températures; https://htmnet.mio.opusytheas.fr). On the other hand, the ADCPs (see Fig. 1, black circles), CTD, and meteorological observations were recorded starting from 14 December 2020 and extending through 15 September 2021. The ADCP moorings were deployed to measure the vertical velocity profile of the u and v components, with u positive eastward and v positive northward. The moorings on station 1 (A1) and station 2 (A2) are placed at 34- and 40-m depth oriented upward-looking from the bottom, measuring at 1 m interval from 2.8 and 3.7 m off the bottom to 4 and 7 m from the surface, respectively. Station 3 (A3) is located at 35-m depth oriented downward-looking measuring at 1-m intervals from 1.5 m off the surface to 5 m from the bottom. These upward- and downward-looking configurations do not yield to significant differences in the velocity records (Liu et al. 2007). The ADCPs on A1 and A3 are 600-kHz band, whereas the one on A2 is 300-kHz band. The CTD data were recorded with a SBE37-SM (manufactured by Sea-Bird Scientific) at $z = 1$, 5-, 15-, and 30-m depth below A3. The meteorological information was registered with a GMX500 weather station installed on top of A3 and used to estimate $U_{10}$ and the wind stress following the method in Large and Pond (1981). The ADCP, CTD, and meteorological information were collected every 15 min. These resultant time series were initially 3-h low-pass filtered to remove very high frequency variability and resampled hourly to be consistent with the offshore information. A summary of the data presented here is shown in Fig. 3, described in more detail below, for the inner-bay station A3.

## c. Wavelet coherence

To determine circulation features inside the bay that respond to wind forcing, it is necessary, initially, to define the time scales where wind and currents present the highest correlation according to their common powers. Therefore, temporal

scales where the wind and currents are not expected to be physically related (low correlated) can be removed in advance before determining any wind-driven circulation pattern. To do so, a wavelet transform coherence (WTC) analysis (Grinsted et al. 2004) was performed between the measured wind and the depth-averaged surface current (using the upper 5 m layer) from A3. WTC defines a cross correlation between two variables by identifying the main frequencies, phase relationships and time periods over which the variability of the two variables are linked to each other over time. In this case, WTC was carried out by components, so that the frequency bands at which the wind and currents covary were defined according to the zonal ($U_{wind}$) and meridional ($V_{wind}$) winds, and eastward ($u_{sc}$) and northward ($v_{sc}$) current components. Thus, WTC [$R_n^2(s)$] of two time series $x_n$ and $y_n$ is defined as

$$R_n^2(s) = \frac{|S[s^{-1}W_n^{XY}(s)]|^2}{S[s^{-1}|W_n^X(s)|^2]S[s^{-1}|W_n^Y(s)|^2]}, \quad (1)$$

where S is a smoothing operator, $W^X(s)$ and $W^Y(s)$ are the wavelet transforms for each time series for a wavelet scale s, $W^{XY}(s)$ is the cross-wavelet transform, and $s^{-1}$ is used to convert to power density. Here, $|W_n^X(s)|^2$, $|W_n^Y(s)|^2$ and $|W_n^{XY}(s)|^2$ are the wavelet and cross-wavelet powers (respectively). The complex argument expressed by arg[$W_n^{XY}(s)$] can be interpreted as the local relative phase between $x_n$ and $y_n$ in time-frequency space (shown by the arrows in Fig. 4, described in more detail below). The $R_n^2(s)$ has values between 0 (no correlation) and 1 (full correlation), with the relative phase relationship shown by arrows pointing right in-phase, pointing left antiphase, pointing upward 90° when $y_n$ leads $x_n$, and pointing downward 90° when $x_n$ leads $y_n$ [see Torrence and Compo (1998) and Grinsted et al. (2004) for more details].

## d. Self-organizing maps

The variability of the wind-driven circulation in the bay and the associated offshore conditions were explored by means of the self-organizing map (SOM). SOM is an artificial neural network method based on unsupervised learning, especially suited to extract patterns from large datasets (Kohonen 2001). SOM is a nonlinear pattern recognition and classification tool used to map high-dimensional input data onto a two-dimensional space, while preserving the topological relationships between the input data (Liu et al. 2006). SOM has been extensively applied to the oceanography field in order to recognize circulation features such as gyres and upwelling/downwelling processes at sea (Liu and Weisberg 2005b, 2007; Liu et al. 2007; Orfila et al. 2021; Morales-Márquez et al. 2021; Cáceres-Euse et al. 2022), as a tool to fill missing data in HFR systems (Hernández-Carrasco et al. 2018), or as a promising technique for ocean current forecasting (Vilibić et al. 2016).

The learning procedure consists of the presentation of the input data to SOM within an iterative process. The input data are clustered in units called neural units. The number of units is the same as the size of the SOM array (or lattice), where each unit has a weight vector that has the number of components equal to the dimension of the input sample data. In
<figure><img src="image_3.png" /><figcaption>FIG. 2. Wind (c) time series and (a) circular histogram from the AROME model, and surface current (d) time series and (b) circular histogram from HFR at the center of the HFR system coverage (black-outlined square in Fig. 1). Both records are in meteorological convention (i.e., shown as the direction from which wind or current is coming), and the color in (c) and (d) refers to direction, as labeled. This location was selected according to the Northern Current's most stable position and far from land constraints.</figcaption></figure>

each iteration the unit whose weight vector is the closest (the smallest Euclidian distance) to the presented input data is selected as the winner unit [best matching unit (BMU)]. Then, the BMU is updated along with its topological neighborhoods toward the input sample, and the topological neighboring units are modified based on a neighborhood function. Now to prevent any of the variables from dominating over the others during the SOM training, first, the input data are normalized with variance equal one, and then, when the training has finished the resultant neural units are denormalized and analyzed in terms of the original variable's units.

In this study, the batch version of the SOM algorithm (in Vesanto et al. 2000) was applied together with a linear initialization mode and the Epanechnikov neighborhood function “ep.” This setting allows one to use simultaneously all the input data to update the weight vectors, and provides lower quantization (QE) and topographic errors (TE) while keeping a computational efficiency (Liu et al. 2006; Mihanović et al. 2011). Additionally, a hexagonal lattice was defined to have equidistant neighbors avoiding the introduction of anisotropy artifact (Hernández-Carrasco and Orfila 2018), with a 2 × 3 (6 neurons) array determined after a sensitivity tests to find the minimum QE and TE. This 6-neuron array is large enough to clearly distinguish the circulation features and small enough to be visualized and interpreted.

# 4. Description of observations

## a. Offshore: atmospheric forcing and surface currents

Two circular histograms in Fig. 2, extracted at the center of the HFR coverage, depict the wind/current magnitude and their associated direction offshore. The weather offshore the Toulon Bay alternates between westerly-northwesterly (called mistral) and easterly winds, reaching values above 15 m s⁻¹ (see Fig. 2a). Mistral winds are cold and dry whereas easterly winds are warm and humid (Mazoyer et al. 2020). These strong wind conditions (>10 m s⁻¹) can be persistent during the year, being the strongest during winter (as shown in Fig. 2c). Conversely, calm weather conditions occur when the wind speed is lower than 5 m s⁻¹ and are observed in early spring and early summer. In agreement with the wind regimes, waves come from east-northeast (ENE) and west-northwest (WNW) from winter to fall, and reaching values up to 6 m of wave height. During summer the waves come predominately from WNW presenting the smallest wave height values, whereas the highest values are present during winter when coming from ENE (Cáceres-Euse et al. 2022).

For the surface currents, Figs. 2b and 2d show currents reaching values greater than 0.8 m s⁻¹, coming predominately from the east but oscillating between NE and SE according to a complex coastline constraint and atmospheric forcing. The NC can experience spatial and temporal variability of its magnitude and direction in form of undulations (latitudinal
<figure><img src="image_4.png" /><figcaption>FIG. 3. Meteomarine data measured in the inner bay at A3, showing circular histograms for (a) wind and (b) depth-averaged surface currents using the first 5-m layer (both histograms are in meteorological convention), (c) the wind time series after a 3-h low-pass filter and hourly averaging using the weather station (label WS), and (d) the temperature from CTD for 1-, 5-, 15-, and 30-m depths on A3 after a 3-h low-pass filter and daily averaging. Because of a malfunctioning of the CTD at 30-m depth on A3 from mid-May to early August, temperature on A1 and A2 measured at the bottom was depicted for descriptive purposes.</figcaption></figure>

migrations of the NC) due to the influence of (sub)mesoscale structures (Guihou et al. 2013; Bourg and Molcard 2021; Ourmières et al. 2011), wind forcing (Berta et al. 2018), and changes in the geostrophic balance in the region (Cáceres-Euse et al. 2022).

b. Toulon Bay: tides, atmospheric forcing, upper surface layer currents, and stratification

The tide in the bay presents a micromareal range with a maximum value of about 0.2 m (Rey et al. 2020). According to a harmonic analysis with nodal corrections performed to the water level (not shown here), using the T-Tide MATLAB toolbox (Pawlowicz et al. 2002), the bay has a semidiurnal-predominance tide as revealed by a form factor $(K_1 + O_1)/(M_2 + S_2) = 0.24$, and a relevant contribution from the semi-annual $S_{SA}$ component. The $S_{SA}$ conforms to an Earth-moon equilibrium system as well as to a larger-scale pressure variations during mistral or easterly wind seasonal events (Paugam et al. 2021).

Now, using the data collected inside the bay, Figs. 3a and 3c depict the wind variability, and Fig. 3b the upper surface currents. The wind follows a similar distribution to that observed

offshore, mostly alternating between westerly and easterly, and a bay subject to strong wind stress (above 0.1 Pa, according to estimations for $U_{10} > 10 \text{ m s}^{-1}$). However, considerable modification due to orography effects on the wind field are observed. For instance, easterly winds are generally weaker than those offshore, and mistral strong wind conditions are less often occurring and slightly changing from WNW to west being aligned with the longitudinal axis of the bay (for mistral condition wind speed also weakens). The changes in the mistral and easterly winds speed and direction can conform to the mountain chain system surrounding the bay at the north side (see Fig. A1 in the appendix).

For the upper surface layer currents, the strongest magnitudes come from WNW reaching values around $0.3 \text{ m s}^{-1}$, whereas the ones coming from the entrance of the bay are weaker with values below $0.2 \text{ m s}^{-1}$ and the current direction bins do not present a dominant-origin direction, covering the wingspan of the entrance of the bay. The surface currents from WNW are mostly associated with mistral wind forcing and upwelling favorable scenarios (Mazoyer et al. 2020). The currents coming from the entrance can be related to offshore conditions, that, enter into the bay induced by easterly winds
<figure><img src="image_5.png" /></figure>

FIG. 4. Wavelet transform coherence analysis between wind and surface currents by components. $U_{wind}$ and $V_{wind}$ for zonal and meridional components, and $u_{sc}$ and $v_{sc}$ for eastward and northward components. The black arrows represent the relative phase relationship between components, and the color map is for $R_n^2(s) \in [0, 1]$. Shown are WTC (a) between $U_{wind}$ and $u_{sc}$, (b) between $V_{wind}$ and $v_{sc}$, (c) between $U_{wind}$ and $v_{sc}$, and (d) between $V_{wind}$ and $u_{sc}$. The WTC was performed on the largest time span when gaps were shorter than 3 h (i.e, from 14 Dec 2020 to 30 Jul 2021).

with potential downwelling events. The connectivity between the inner-bay and offshore currents will be developed in the next section.

Based on the CTD records the time variation of temperature/ thermal stratification inside the bay is shown in Fig. 3d. Since a malfunctioning of the CTD at 30-m depth was observed during mid-May to early August, temperature information from two additional sensors on A1 and A2 were added for description purposes. From late fall to spring the water column is well mixed, with averaged temperature differences of around 1°C throughout the water from the bottom to the surface. During this time period the wind stress reaches its maximum values, and the water temperature presents the lowest. In late spring the wind stress weakens and the solar radiation increases, warming the upper half of the water column while the lower half remains colder strengthening the stratification. Then, in summer, oscillations between increased and decreased stratification are observed, possibly because of the combination of three main reasons: 1) a higher solar radiation warms the upper layer of the water column and strong mistral events cool the bottom water by advection from upwelled waters that come from offshore (as observed for early to mid-July), 2) easterly winds bring colder water from offshore decreasing the thermal stratification by cooling the upper-layer waters (as observed for late August), and 3) convective motion episodes during nighttime

that cool the upper water layer under calm wind conditions (Fischer et al. 1979).

## 5. Observed wind-driven circulation in the bay and offshore dynamics

### a. Time scales of linkage between wind and currents: wavelet coherence analysis

The four combinations of WTC by components are shown in Fig. 4: Fig. 4a illustrates the $R_n^2(s)$ value between $U_{wind}$ and $u_{sc}$, Fig. 4b is for $R_n^2(s)$ between $V_{wind}$ and $v_{sc}$, Fig. 4c is the $R_n^2(s)$ value for $U_{wind}$ and $v_{sc}$, and Fig. 4d is the $R_n^2(s)$ value for $V_{wind}$ and $u_{sc}$. As previously mentioned, the dominant zonal wind component present in the bay (mistral or easterly winds) is revealed also by $R_n^2(s)$, such that the highest correlation values are observed for $U_{wind}$ whether considering the eastward or northward current components (Figs. 4a,c). Conversely, the meridional wind component (Figs. 4b,d) does not present a strong or consistent coherency and phase relationship with the surface currents, only sporadic correlations occur at daily time scale that can be associated with sea breeze episodes, and some scattered correlations at weekly time scale with $R_n^2(s) < 0.6$. For Figs. 4a and 4c, the phase relationship is consistent (slowly varying along the scales), and the correlation is high during the full time span [$R_n^2(s) > 0.85$], being possible to establish that the surface current in the bay is physically
<figure><img src="image_6.png" /><figcaption>FIG. 5. Resultant SOM pattern representations of the wind-driven circulation in the inner bay, and their associated averaged offshore surface current and wind field conditions from the conditional averaging procedure. The black dash-outlined box depicts the Toulon Bay SOM features, with near-surface currents represented by red arrows, green arrows for middepth currents, blue arrows for near-bottom currents, and black arrows for the wind measured on A3. The averaged offshore conditions are those outside the black dash-outlined box with red arrows for HFR surface currents and black arrows for the AROME wind field. Here only the northernmost part of the HFR coverage and close to the bay entrance is utilized for the connectivity analysis. The percentage of occurrence of each neuron is located at the top center of the subplots.</figcaption></figure>

linked to the zonal wind forcing for time scales longer than 2.5 days. Particularly, the frequency band between 2.5 and 25 days shows the eastward component in-phase while the northward component out of phase 180°. The eastward component in Fig. 4a presents a fast response to the wind drag following the wind direction. Figure 4c reveals that the northward surface current response is deflected to the right of the wind 90° during zonal wind conditions. Thus, under this frequency band, mistral wind pushes the surface current offshore, whereas easterly winds induce the surface current to travel inward the bay creating favorable downwelling conditions.

This study deals with the assessment of wind-driven circulation in small semiclosed bays and the possible connectivity with the offshore dynamics. Therefore, all the variables considered hereinafter were low-pass filtered using a 60-h (2.5 days) cutoff frequency, according to the WTC correlation and relative phase analysis, and daily averaged. At the latitude of Toulon Bay, the inertial frequency is around 17.6 h, longer than the dominant semidiurnal tide component, such that the

resultant low-pass-filtered data represent the subinertial time scale variations.

b. Horizontal structure of wind-driven circulation patterns: Spatial SOM analysis

The joint SOM analysis was performed using the complete velocity profiles from the three ADCPs along with the recorded wind on A3. Now, the connectivity between the inner-bay and the offshore region is determined by a conditional temporal averaging applied to the HFR and AROME data. This temporal averaging is performed by finding HFR and AROME snapshot velocity fields with equal BMUs from the ADCPs SOM training (see Fig. A2 in the appendix for the BMUs time series). Therefore, each resultant neuron (from SOM) representing a circulation feature inside the bay can be associated with an averaged offshore surface current and wind field conditions. The 2 × 3 resultant SOM array and the associated averaged offshore conditions are shown in Fig. 5. The neurons are arranged such that the most dissimilar patterns
are located the farthest away from each other (e.g., neurons 1 and 6). The left-hand side of the lattice is populated by mistral wind-drive current responses, right-hand side by easterly wind-driven responses and the center of the lattice represents the transitional conditions (weak/calm winds) when wind is changing direction between mistral and easterly winds. Mistral wind-driven conditions present the strongest magnitudes both the wind and current, whereas the easterly conditions show weaker values. Conversely, easterly wind-driven patterns (N3 and N6) are the most often occurring with 37%, followed by mistral wind-driven patterns (N1 and N4) with 32% and transitional patterns with (N2 and N5) 31% of the time.

For the mistral wind-driven circulation patterns, N1 shows the strongest wind conditions along with large vertical shears on the current, a characteristic feature of predominately baroclinic modes. The upper surface water layer inside the bay as well as the one offshore are pushed southward by the wind action and deflecting to the right of the wind direction, consistent with an Ekman response to wind. On the other hand, N4 presents weaker and mostly zonal wind conditions (as compared with N1) with little influence on A3, whereas A1 and A2 show an anticyclonic circulation following the isobaths 30-40 m and land constraint. Additionally, little variation of the current with depth is also observed in N4, indicating a predominantly barotropic response to wind.

The transitional patterns N2 and N5 show a weakening mistral together with the weakest currents inside the bay and at the northernmost region of the HFR coverage (close to the entrance of the bay). Neither N2 nor N5 reveal a clear inward-bay flow from offshore through the entrance of the bay or outflow toward offshore from inner-bay (no connectivity). N2 and N5 are residual currents that obey to larger time scale signals associated with background circulation. In addition, during this period the NC is returning to its most stable position approaching to the coastal margin after been displaced southward by the strong mistral events previously mentioned [this NC behavior was observed and reported in Berta et al. (2018)].

Last, N3 and N6 show weak easterly wind conditions along with an NC close to the coast, such that the land constraints (islands, capes, and peninsula) induce the offshore current to turn right into the bay revealing a clear inflow pattern at the entrance. This is also observed in Molcard et al. (2021) by means of three-level-nested numerical simulations with zoom in the Toulon Bay. For N3 the offshore surface sea current enters by the east side between the isobaths 30-40 m following a cyclonic circulation pattern and exiting by the west side, mostly in a barotropic mode response (particularly for A2 and less A1). For N6 the surface current offshore has been strengthened by the easterly wind entering into the bay through A1 and A2 (west and east). A2 is predominately barotropic whereas A1 and A3 are predominately baroclinic, and the surface current on the three stations travel parallel to the wind direction, so that an Ekman response to wind is not clearly observed, potentially due to the combination of three main factors: an important coastline constraint that channelizes the incoming flow, along with the baroclinic pressure gradient contribution due to the horizontal density changes and/or advective effects when the NC enters into the bay, so that the Coriolis effect is overcome.

### c. Vertical structure of the wind-driven circulation patterns

The along- and cross-shore vertical profiles ($\hat{u}^*$ and $\hat{v}^*$) were determined by rotating A3 currents for each neuron 41° counterclockwise from north and normalizing them by the friction velocity ($u^*$). After this process, $\hat{u}^*$ and $\hat{v}^*$ are parallel to the axes $\hat{x}$ and $\hat{y}$ delimited by the black lines from the entrance of the bay (from the black square to the black circle) as shown in Fig. 6. For the upper panels, arrows pointing right show $\hat{u}^*$ traveling along positive $\hat{x}$. For the lower panels, arrows pointing right when the water flows southward (living the bay) and arrows pointing left when flowing inward the bay (entering the bay). The black square at the entrance was defined as the midpoint between A1 and A2, and the axes are aimed to be aligned with the main inflow/outflow directions for mistral and easterly wind following the along and cross-shelf directions. According to the normalization by neurons, the larger the normalized velocity profile the lower the wind stress effect, such as those observed for negligible wind conditions N2, N3, and N5. The $u^*$ tends to zero for negligible wind and increases for nonnegligible wind conditions.

The observed $\hat{v}^*$ is in agreement with the wind responses described in the previous subsection and reveals additional information about the velocity vertical profile; that is, the presence of an inflection point (or not) in the velocity profile and its vertical position according to the wind regime. N4 and N5 are represented by a barotropic mode response for moderate wind conditions (N4) or when the NC position is not close enough to the coastal margin (N5) for connectivity. For N4, the near-surface layer of the water column is still, while in the lower layer the currents travel inward the bay (10-30-m depth) reaching values up to $4 \text{ cm s}^{-1}$ (not shown in Fig. 6, but observed from data), potentially originated by the northward middepth flow in A1 (see green arrow Fig. 5) under the anticyclonic circulation. The vertical profiles for the transitional patterns N2 and N5 present a background current below $4 \text{ cm s}^{-1}$ (observed from data) along the water column traveling inward. During the cyclonic circulation pattern in N3, the profile shows that part of northward-rotated NC that enters into the bay reaches A3, revealing a weak two-layer baroclinic mode with the upper 26-m layer traveling inward the bay, and a bottom countercurrent going offshore. Last, N1 (strong mistral) and N6 (easterly wind) are depicted by a two-layer baroclinic mode, with the strongest current values for N1 and weaker for N6. N1 has an inflection point at 11-m depth that differentiates the upper water layer flowing offshore by the wind action and the lower layer going inward the bay. For N6 the inflection point has deepened to 20 m from a thick NC entering into the bay through the upper layer and a bottom countercurrent flowing offshore. The $\hat{v}^*$ profile in N6 is stronger than that in N3 as a result of the strengthened NC by the easterly wind offshore, phenomenon known as remote wind-driven forcing conditions (Largier 2020).

The vertical profiles in N1 and N6 are consistent with upwelling episodes and the intrusion of offshore waters into the bay observed from 3D numerical simulations in Mazoyer et al. (2020). So, N1 occurs when strong mistral shear can break the
<figure><img src="image_7.png" /><figcaption>FIG. 6. SOM pattern representation of the normalized along and cross-shore vertical profiles ($\hat{u}^*$ and $\hat{v}^*$) in the inner-bay for A3; $\hat{u}^*$ and $\hat{v}^*$ result from rotating A3 currents for each neuron 41° counterclockwise from north and dividing by $u_* = (\tau/\rho_0)^{1/2}$. The black lines in the top panel illustrate the alongshore and cross-shore directions defined for the depth-averaged momentum balance analysis.</figcaption></figure>

thermal stratification through upwelling as observed during winter and early to mid-July, whereas the downwelling in N6 is present when offshore currents reach the inner bay as seen for winter and late August–early September (according to a BMU analysis; Fig. A2 in the appendix). Thus, given the geometric configuration of this small semiclosed bay, the inflow/outflow masses of water will be balanced by the system through mass conservation, inducing wind-driven upwelling or down welling episodes according to strong mistral or easterly winds, respectively.

## 6. Depth-averaged momentum balance and analysis of nondimensional numbers

### a. Momentum balance

Assuming a hydrostatic flow approximation, with the atmospheric pressure set equal to zero, and the free surface

elevation much smaller than the water depth ($\eta \ll h$), the depth-averaged momentum balance equations can be defined (Lentz et al. 1999) as

$$ \frac{\partial \bar{u}}{\partial t} + \frac{\partial}{\partial \hat{x}} \left( \frac{1}{h} \int_{-h}^{0} \hat{u}^2 dz \right) + \frac{\partial}{\partial \hat{y}} \left( \frac{1}{h} \int_{-h}^{0} \hat{u} \hat{v} dz \right) - f \bar{v} = -\frac{1}{\rho_0} \frac{\partial P}{\partial \hat{x}} + \frac{\tau_{s\hat{x}}}{\rho_0 h} - \frac{\tau_{b\hat{x}}}{\rho_0 h} \quad \text{and} \quad (2) $$

$$ \frac{\partial \bar{v}}{\partial t} + \frac{\partial}{\partial \hat{x}} \left( \frac{1}{h} \int_{-h}^{0} \hat{u} \hat{v} dz \right) + \frac{\partial}{\partial \hat{y}} \left( \frac{1}{h} \int_{-h}^{0} \hat{v}^2 dz \right) + f \bar{u} = -\frac{1}{\rho_0} \frac{\partial P}{\partial \hat{y}} + \frac{\tau_{s\hat{y}}}{\rho_0 h} - \frac{\tau_{b\hat{y}}}{\rho_0 h}, \quad (3) $$

where $\hat{u}$ and $\bar{v}$ are the alongshore ($\hat{x}$) and cross-shore ($\hat{y}$) sub- inertial velocity components, respectively; $\tilde{u}$ and $\tilde{v}$ are the
<table><caption>TABLE 1. Standard deviation of the alongshore depth-averaged momentum balance (10⁻⁶ m s⁻²).</caption><thead><tr><th>Neuron</th><th>∂̃u/∂t</th><th>∂̃u²/∂x̃</th><th>∂̃u̇v/∂ẏ</th><th>ḟv</th><th>g<sub>∂x̃</sub>∂η/∂x̃</th><th>g<sub>ρ₀</sub>∫<sub>-h</sub><sup>0</sup>∂ρ'/∂x̃(1 + z/h) dz</th><th>τ<sub>s̃x̃</sub>/ρ₀h</th><th>τ<sub>b̃x̃</sub>/ρ₀h</th></tr></thead><tbody><tr><td>1</td><td>0.009</td><td>0.52</td><td>0.44</td><td>1.50</td><td>—</td><td>6.10</td><td>1.20</td><td>0.24</td></tr><tr><td>2</td><td>0.008</td><td>0.38</td><td>0.31</td><td>0.93</td><td>—</td><td>5.20</td><td>0.077</td><td>0.13</td></tr><tr><td>3</td><td>0.005</td><td>0.11</td><td>0.12</td><td>0.81</td><td>—</td><td>5.40</td><td>0.19</td><td>0.48</td></tr><tr><td>4</td><td>0.007</td><td>0.32</td><td>0.26</td><td>1.41</td><td>—</td><td>3.72</td><td>0.50</td><td>0.16</td></tr><tr><td>5</td><td>0.008</td><td>0.31</td><td>0.33</td><td>0.92</td><td>—</td><td>7.35</td><td>0.20</td><td>0.13</td></tr><tr><td>6</td><td>0.006</td><td>0.33</td><td>0.35</td><td>1.40</td><td>—</td><td>7.92</td><td>0.99</td><td>0.13</td></tr></tbody></table>

corresponding depth-averaged velocities; z is the vertical axis (positive upward); f is the Coriolis parameter; $ρ_0 = 1025$ kg m⁻³ is the reference density; $τ_s$ is the wind stress; $τ_b$ is the bottom stress; and $∂P/∂x̃$ and $∂P/∂ẏ$ are the depth-averaged horizontal pressure gradients.

Starting with the last two terms on the right-hand side (RHS) of Eqs. (2) and (3), the wind and bottom stresses are determined by applying a quadratic drag law on A3 along with the pertinent characteristic velocity. For instance, the $τ_s$ components use the wind drag coefficient defined as $C_d = 1.2 \times 10^{-3}$ for $U_{10} < 10$ m s⁻¹ and $C_d = (0.49 + 0.065U_{10}) \times 10^{-3}$ for $10 \le U_{10} < 25$ m s⁻¹ (Large and Pond 1981). For the $τ_b$ estimations, the bottom drag coefficient is defined as $C_D = 2.5 \times 10^{-3}$ (Liu and Weisberg 2005a) with the near-bottom velocity ($\hat{u}_b$ and $\hat{v}_b$) taken at the bottommost position in the ADCP profile. Now, to calculate the depth-averaged pressure gradient terms, first, P is decomposed into the barotropic contribution $P^s$ induced by the slope of the seawater level, and the baroclinic contribution $P^i$ due to the horizontal changes of density [given that $ρ(x, y, z) = ρ_0 + ρ'(x, y, z)$]. Thus, the first term on the RHS can be defined (Lentz et al. 1999) by

$$ \frac{1}{\rho_0} \frac{\partial P}{\partial \hat{x}} = \frac{1}{\rho_0} \frac{\partial P^s}{\partial \hat{x}} + \frac{g}{\rho_0} \int_{-h}^{0} \frac{\partial \rho'}{\partial \hat{x}} \left(1 + \frac{z}{h}\right) dz, \quad \frac{1}{\rho_0} \frac{\partial P^s}{\partial \hat{x}} = g \frac{\partial \eta}{\partial \hat{x}}, \quad (4) $$

with the corresponding equation for $∂P/∂ẏ$. The barotropic contribution is estimated using the low-pass-filtered water levels inside the bay (from P1 and P2 inside the bay). For the baroclinic part, CTD data for A1 and A2 are only available close to the bottom, so that the cross (and along)-shore changes for $P^i$ are evaluated using $ρ'$ records at the bottom for the three stations. The baroclinic estimations are performed when $ρ'$ is available for the three station simultaneously. The cross-shore gradients are determined using A3 and the midpoint values

between A1 and A2. The $ρ'$ and velocity values on the midpoint are determined with a weighted average with the local water depths on A1 and A2. The alongshore gradients are computed using the data on A1 and A2. Later, the time and space derivatives on both sides of the momentum equations are computed using a forward-difference scheme so that $∂/∂t = Δ/Δt$, $∂/∂x̃ = Δ/Δx̃$, and $∂/∂ẏ = Δ/Δẏ$ are determined according to the differences between the ADCPs/pressure sensors in time and space. The $Δt = 24$ h, $Δx̃$ is the alongshore distance (A1–A2 stations), and $Δẏ$ is the cross-shore distance (midpoint A1–A2 and A3 for velocity records and P1–P2 for pressure). Last, the contribution from each term in the depth-averaged momentum equation is evaluated according to their order of magnitude, that is, the standard deviation of the acceleration (Lentz et al. 1999) from all the snapshots with equal BMUs (see Fig. A2 in the appendix for the BMU time series) (a similar way to that applied to the HFR and AROME data in section 4). This estimation does not amount to a fully closed budget, but highlights the relevant and dominant aspects driving the momentum equation at the subinertial scale. In addition, it is important to remark that the alongshore pressure gradient term is not estimated, since it is needed a set of pressure sensors to be aligned with A1–A2 at the entrance of the bay in order to be consistent with the rest of the computed alongshore depth-averaged momentum terms.

The standard deviation of the depth-averaged momentum balance reveal that the local acceleration is the weakest term followed by the bottom stress; whereas the barotropic and baroclinic pressure gradient components are the greatest, with the Coriolis, wind forces and advection having an intermediate effect (see Tables 1 and 2). The cross-shore as well as the alongshore local acceleration determined by BMUs, from the SOM training, present a steady-state condition, that is, the unsteadiness term is two orders of magnitude lower than the

<table><caption>TABLE 2. Standard deviation of the cross-shore depth-averaged momentum balance (10⁻⁶ m s⁻²).</caption><thead><tr><th>Neuron</th><th>∂v̇/∂t</th><th>∂̃u̇v̇/∂x̃</th><th>∂̃v̇²/∂ẏ</th><th>ḟũ</th><th>g<sub>∂ẏ</sub>∂η/∂ẏ</th><th>g<sub>ρ₀</sub>∫<sub>-h</sub><sup>0</sup>∂ρ'(∂ẏ(1 + z/h)) dz</th><th>τ<sub>s̃ẏ</sub>/ρ₀h</th><th>τ<sub>b̃ẏ</sub>/ρ₀h</th></tr></thead><tbody><tr><td>1</td><td>0.005</td><td>0.61</td><td>1.02</td><td>2.42</td><td>7.89</td><td>5.19</td><td>2.16</td><td>0.25</td></tr><tr><td>2</td><td>0.003</td><td>0.34</td><td>0.56</td><td>1.35</td><td>6.28</td><td>3.04</td><td>0.053</td><td>0.34</td></tr><tr><td>3</td><td>0.004</td><td>0.11</td><td>0.15</td><td>1.25</td><td>6.89</td><td>2.22</td><td>0.05</td><td>0.078</td></tr><tr><td>4</td><td>0.005</td><td>0.29</td><td>0.64</td><td>2.37</td><td>7.29</td><td>2.95</td><td>1.32</td><td>0.19</td></tr><tr><td>5</td><td>0.004</td><td>0.34</td><td>0.57</td><td>1.52</td><td>9.18</td><td>2.94</td><td>0.26</td><td>0.31</td></tr><tr><td>6</td><td>0.005</td><td>0.26</td><td>0.59</td><td>1.18</td><td>8.82</td><td>3.32</td><td>1.93</td><td>0.24</td></tr></tbody></table>
rest of the terms in the momentum equations, as stated in Ponte (2010) at subinertial time scales. Now, for the bottom stress, the standard deviation is one order of magnitude smaller than the wind stresses under nonnegligible wind conditions (i.e., neurons N1, N4, and N6), so that neither the alongshore nor the cross-shore bottom friction forces are large enough to fully balance the wind stress. The low contribution from the bottom stress here observed is associated with the local water depth, which is always greater than 25 m for the three stations, such as the higher the local water depth the lower the bottom effect on the depth-averaged momentum balance.

For the advection, a differentiated response between the cross-shore and alongshore balances is present. For instance, the advective terms in the alongshore momentum balance seem to be less sensitive to the wind condition, being lower than the Coriolis force by a factor of 3, and wind stress terms (for nonnegligible wind regimes N1, N4, and N6), but higher than the bottom stress term. Conversely, the advective terms in the cross-shore direction are greater than those alongshore, varying according to the wind condition, reaching values as high as the Coriolis and wind stress terms, and being at the same order of magnitude as the horizontal pressure gradient contributions (see N1 in Table 2). The increment of advection in the cross-shore direction can be attributed to a channeling effect on the flow due to the V-shape type bathymetry and a reduction in the cross-sectional area at the entrance of the bay, such that a preferential flow across the isobaths is generated. Thus, the nonlinear advective term in small semia closed bays has a significant amplitude and plays a relevant role in the momentum balance, even reaching the same order of magnitude as the wind stress, Coriolis force, or the horizontal pressure gradients. Similar estimations were reported in Ponte et al. (2012) for Conception Bay, attributing the differences between the linear model in Winant (2004) and the in situ data to the nonlinear advection effects. In fact, Winant (2004) remarks that the linear models can depart from the in situ measures when the length scale of the bay ~ 10³ m so that the Ro will tend to approach 1 (L ~ 10³ m being the characteristic length scale of the Toulon Bay).

Now, for the horizontal pressure gradients, the barotropic and baroclinic components are the two leading terms in the depth-averaged momentum balance. Regularly one opposing the other (Lentz et al. 1999). The barotropic part can be up to twice as large as the baroclinic one for the cross-shore balance. For the baroclinic part alongshore, the pressure gradient is generally greater than that cross-shore and seems to be less sensitive to the wind regime. The P^i behavior alongshore can be associated with the proximity to the offshore water, which is highly influenced by horizontal baroclinic changes in the NC dynamics. Conversely, the cross-shore P^i reveals a close relation with the wind regimes, reaching its maximum value under upwelling conditions (N1 in Table 2) and the lowest for negligible wind (N3 in Table 2). Then, the cross-shore balance also shows the connectivity between the inner bay and the offshore region in agreement with the previous section. Upwelling (mistral) or downwelling (easterly winds) scenarios enhance the inflow/outflow with the offshore water so that the baroclinic

<table><caption>TABLE 3. Nondimensional numbers by neuron estimated at the center of the bay with data from A3; Ro is the Rossby number, Ek is the Ekman number, R_R is the curvature-based Rossby number, and S here refers to Burger number.</caption><thead><tr><th>Neuron</th><th>Ro</th><th>Ek</th><th>R<sub>R</sub></th><th>S</th></tr></thead><tbody><tr><td>1</td><td>0.30</td><td>1.85</td><td>0.12</td><td>0.58</td></tr><tr><td>2</td><td>0.004</td><td>0.23</td><td>—</td><td>1.55</td></tr><tr><td>3</td><td>0.01</td><td>0.33</td><td>—</td><td>0.62</td></tr><tr><td>4</td><td>0.11</td><td>1.12</td><td>—</td><td>0.57</td></tr><tr><td>5</td><td>0.01</td><td>0.40</td><td>—</td><td>0.62</td></tr><tr><td>6</td><td>0.09</td><td>1.04</td><td>0.23</td><td>0.54</td></tr></tbody></table>

pressure gradient increases (N1 and N6), whereas under weak or negligible wind conditions it is observed a weakened horizontal density change associated with little connectivity with the inner bay (or no connectivity).

## b. Nondimensional analysis

For the relevance of the advective component cross-shore under wind-driven circulation, observations in Ponte et al. (2012) and theoretical modeling in Lentz and Chapman (2004) and Austin and Lentz (2002) have reported similar responses. The role played by the nonlinear advection in the momentum equation can be assessed by $Ro = u_d/fB$ (Ponte et al. 2012), where B is half the width of the bay entrance, $u_d = \tau/\rho_0fh$ is the characteristic velocity necessary for Coriolis force to balance the wind stress under steady-state conditions (Cushman-Roisin and Beckers 2010), and $\tau$ is the wind stress magnitude. Table 3 depicts Ro for negligible and for nonnegligible wind conditions.

Under negligible wind $Ro \ll 1$ (below 0.01), the nonlinear advection terms can be neglected and the momentum equation simplified to a linear Coriolis-pressure gradient balance (geostrophic equilibrium) (see Table 2). For $Ro \ll 1$ the bottom stress term and the advection terms are similar, so that bottom effects can be discarded too. For nonnegligible wind conditions, when wind stress ranges from moderate to strong, $0.09 \le Ro \le 0.3$ [similar values were observed in Ponte et al. (2012)]. The nonlinear terms become relevant such that they can be as high as the wind stress and the Coriolis force, and surpass the bottom stress force. Similarly, higher $Ro$ values coincide with high $Ek = v_E/(fh^2)$ values (see Table 3), with $v_E = u_*\kappa h$, the friction velocity determined as $u_* = (\tau/\rho_0)^{1/2}$, and the von Kármán constant $\kappa = 0.41$. During moderate-to-strong wind regimes the wind stress dominates over Earth's rotation effects ($Ek \ge 1$), which in turn contributes to raise the advective terms. Additionally, the nondimensional Ekman depth [$\delta_E = \sqrt{2v_E/(fh^2)}$] is $\delta_E \ge 1$, and thus the semia closed bay can be defined as shallow and turbulent. In addition, low $Ro$ values are accompanied with low $Ek$, that is, wind stress decreases and so does advection; then, the bay behaves as a deep system ($\delta_E < 1$) and the flow dynamics is ruled by a geostrophic equilibrium. On the other hand, as it was discussed in the previous section, the alongshore current response at the entrance of the bay seems to be highly influenced by the offshore dynamics. The advective component in the momentum balance is always an order of magnitude smaller than the
baroclinic term, and smaller than the Coriolis one. The baroclinic as well as the Coriolis contributions are the dominant terms in the momentum equations following a thermal wind balance. The most notable wind stress effect is observed for the Coriolis term, where larger wind stress induces a larger Coriolis acceleration.

When the flow nearby the headlands of the bay entrance is strong enough, the centrifugal acceleration increases and (potentially) overcomes the Coriolis and the wind stress effects, such that the momentum equations can be expressed by a balance between the advection and the barotropic pressure gradients (Valle-Levinson and Moraga-Opazon 2006). This means that advection can rise due to the generation of vorticity when flow separation is dominant because of coastal constraint effects; for instance, the headlands at the entrance of Toulon Bay. Therefore, the hypothesis about the channeling effect and the preferential flow across the isobaths (increment of advection) can be assessed using the curved-based Rossby number $R_R$ (Valle-Levinson and Moraga-Opazon 2006). The $R_R = u_d/(f\hat{R}_i)$, with $\hat{R}_i$ being the radius of curvature of the coastline, and $R_R \ge 1$ when centrifugal acceleration plays an important role in the dynamics. Under mistral wind conditions (N1) $\hat{R}_i = 1100$ m measured at the Saint Mandrier Peninsula (Mandrier), and for easterly winds (N6) $\hat{R}_i = 550$ m at Cape Carqueiranne (see Fig. 1). Estimations of $R_R$ are performed for N1 using data from A3 and for N6 with data from A2 (see Table 3), given the geographical proximity and flow intensity. Thus, either for mistral or easterly wind regimes $R_R$ is below 0.25 (0.12 for mistral and 0.23 easterly wind), so that the increment in the advective component cross-shore in the momentum equation balance can be associated with the channeling effects when the flow approaches to the entrance of the bay and the reduction in the cross-sectional area induces an advective acceleration. Conversely, a contrasting pattern is observed offshore, where an additional estimation for $\hat{R}_i = 1400$ m at Porquerolles (POR) yields to $R_R = 1.8$ for N6, and, due to the islands, the flow direction is clearly modified toward the bay entrance by flow separation effects, generating a clockwise-rotation structure (see Fig. 5).

Last, the modulation of the horizontal baroclinic pressure gradient cross-shore and the changes in temperature differences along the water column associated with the wind regime emphasize the importance of stratification effects inside the bay. The state of stratification in a water body subject to Earth's rotation as well as the baroclinic pressure gradients can be determined according to the Burger number $S = R_B/B$ (see Table 3) (Cushman-Roisin and Beckers 2010). The baroclinic Rossby radius of deformation is $R_B = Nh/f$, with the Brunt-Väisälä frequency $N$ estimated by the differences between the surface and the bottommost density records at the center of the bay (A3). The lower the $S$ values are, the weaker is the stratification in the bay, whereas the greater the $S$ is, the stronger are the stratification conditions. In Toulon Bay, moderate-to-strong wind regimes present the lowest $S$ values ($<0.58$), whereas negligible winds depict greater $S$ values with N2 representing the strongest stratification state ($S = 1.5$), according to the seasonal variability in the region (as shown in section 4). N2 occurs mostly during summer period, whereas

N3 and N5 take place when wind alternates between westerly and easterly (and vice versa). Now, under favorable upwelling/downwelling scenarios the stratification weakens inside the bay, the fluxes cross-shore increase, and the differences in density from the center to the entrance of the bay remain stronger than in cases of low wind regime. This circulation response is contradictory to that observed in Austin and Lentz (2002), where a weakening of stratification in the inner shelf causes reduction in the cross-shore circulation. The reason for these differences in the wind-driven circulation response can be explained by the nature of the semiclosed bay configuration. Upwelling/downwelling favorable conditions can induce a tilting in the pycnocline inside Toulon Bay as reported for open coasts in Austin and Lentz (2002), Lentz and Chapman (2004); nonetheless, the intersection of the pycnoclines with the surface/bottom (known as fronts) move seaward without reaching the entrance. The fronts are kept inside the bay, and the horizontal density changes between the inner bay and the offshore region increase.

## 7. Conclusions

This study analyzes the wind-driven circulation in a small semiclosed bay (the Toulon Bay) and its potential connectivity with the open seawater by combining offshore and inner-bay meteomarine data. The offshore information is composed of sea surface currents measured by HFR and wind fields, and the inner-bay data were collected by three ADCPs deployed between the entrance to the center of the bay along with CTD and meteorological observations. Then, here we developed a methodology that applies the wavelet transform coherence (WTC) analysis together with the unsupervised neural network self-organizing maps (SOM) to determine the time scales at which the wind and surface currents present their maximum common powers (the linkage between the wind and surface currents), and identify the wind-driven circulation features inside the bay with the associated averaged offshore dynamics. Thus, this methodology allows us to assess the bay responses to wind forcing such as the horizontal and vertical structures of the current, and the role played by each term in the momentum equations according to the wind regime, as well as the influence of offshore conditions in the inner-bay circulation.

The radiation stress term contribution in the momentum equation, generated by wave dissipation along its propagation axis [analyzed in Lentz et al. (1999) and Lentz and Fewings (2012)], has been omitted in this study, even if the wave height offshore can reach high values. As found from the 60-h low-pass-filtered wave height at the entrance and within the bay, exploratory estimations of the radiation stress fluxes (not shown here) are negligible when compering to the depth-averaged momentum balance present in Tables 1 and 2. At the entrance of the bay the wave parameters were taken from a calibrated WAVEWATCHIII simulation (see Lira-Loarca et al. 2022), while inside the bay the wave height was recorded on A3. The complex geographical configuration and sheltering effect by GIENS, the cape and islands result in a strong wave energy dissipation occurring before reaching the bay. In addition, the cross and alongshore axis defined here were located
deeper than the isobath 25 m, such that the momentum balance analysis performed in this study was always far from the surf zone.

The results observed here reveal that the circulation features inside the bay can vary from barotropic (either cyclonic or anticyclonic) to baroclinic, with potential connectivity between the offshore and inner-bay masses of water according to the wind conditions (direction and magnitude). For instance, moderate mistral or weak wind conditions induce barotropic circulation patterns, whereas strong mistral or moderate easterly winds show a two-layer baroclinic mode response mostly associated with upwelling or downwelling events. Conversely, negligible wind scenarios exhibit a background circulation inside the bay and no connectivity with the offshore dynamics, at least, based on the information provided by the sea surface currents offshore measured by the HFR system. This background circulation inside the bay can be explained by a geostrophic balance between the Coriolis and pressure gradient forces (both the barotropic and baroclinic) as observed through the momentum balance analysis in Tables 1 and 2. In contrast, the advective components in the momentum equations can be at the same order of magnitude as the pressure gradients, Coriolis, or wind stress terms under upwelling/downwelling events as well as greater than the bottom stress contribution.

With regard to the relevant role played by the advective terms in the momentum balance, an increment in the cross-shore advective component is straightened by the channeling effect on the flow at the entrance of the bay, when the flow is entering or leaving the bay, due to a reduction of the cross-sectional area. This finding opens the discussion concerning the need to include the nonlinear advection in the momentum equations when studying wind-driven circulation process in coastal bays. This term has been traditionally assumed negligible along open coasts as seen in Lentz et al. (1999), Winant (2004), Liu and Weisberg (2005a), Lentz and Fewings (2012), and Largier (2020). The contribution of the advective terms has been discussed in Lentz and Chapman (2004) for open coasts under upwelling conditions, and in Ponte et al. (2012) at the interior of a large semienclosed bay (Conception Bay). Nonetheless, to our knowledge, this is the first study presenting experimental evidence about the important contribution from the nonlinear advective terms in a small semienclosed bay and the connectivity with the offshore dynamics as function of the wind regime.

The major source of errors in the momentum balance analysis presented here come from the pressure gradient estimations [as reported in Liu and Weisberg (2005a)], both the $P^s$ and $P^i$ gradient estimations. Particularly, $P^i$ gradient computations (either alongshore or cross shore) were performed using density values at the bottom only, which can result in

underestimations of the baroclinic pressure gradient contribution because of the missing information through the water column, particularly close to the surface.

In addition to the physical processes revealed in this work concerning the wind-driven circulation in small semienclosed bays, a novelty in this work lies in applying WTC and a neural network method (more and more often applied to oceanography field) to a multiplatform system to identify and assess the relationship between offshore and inner-bay dynamics, and the estimation of the terms in depth-averaged momentum balance according to the wind regime. As future work, 3D high-resolution numerical simulations, surface sea currents from HFR systems, and inner-bay information will be combined to observe pollutant transport/concentration in small semienclosed bays according to the atmospheric conditions.

*Acknowledgments.* Author A. Cáceres-Euse is supported by the Postdoctorat PD22-260—Centre National d'Études Spatiales (CNES), and the surveyed meteomarine information in the study site have been funded by the research projects SICOMAR-PLUS and SINAPSI-Interreg Italia-Francia Marittimo 2014/2020. Author V. Morales-Márquez is supported by the MORHOC'H2 project, funded by the French "Agence Nationale de la Recherche" and "Direction Générale de l'Armement," Grant 21-ASM1-0003. The ADCP deployment and maintenance was performed by iXblue Company. The HFR system is part of the Mediterranean Ocean Observation Service for the Environment (MOOSE) French network. The authors thank Prof. V. Rey, D. Mallarino, and T. Missamou for the access to the HTM-NET network seawater levels and the Parc National de Port-Cros (PNPC) for its support and hosting of our radar transmitter in Porquerolles Island. We also thank D. Dumas, N. Bourg, and Prof. C-A. Guérin for the HFR data processing support. We are grateful for the comments from the handling editor and the three anonymous reviewers that helped to improve this paper.

*Data availability statement.* The HFR data presented herein are available upon request to the author (anne.molcard@univ-tln.fr). The quasi-real-time HFR maps (http://hfradar.univ-tln.fr/HFRADAR/squel.php?content=accueil) and the seawater level (https://htmnet.mio.osupytheas.fr) are available online.

# APPENDIX

## Additional Figures

Figure A1 shows typical mistral and easterly flows in the study region. Figure A2 can be used to analyze the BMU time series.
<figure><img src="image_8.png" /><figcaption>FIG. A1. Typical (left) mistral and (right) easterly wind fields. The color bar represents the topography elevation (m) for the study site.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>FIG. A2. BMU time series resultant from the SOM training applied to the 60-h low-pass-filtered and daily-averaged ADCP currents and wind.</figcaption></figure>

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