Implementing Astronomical Potential and Wavelet Analysis to Improve Regional Tide Modeling

Abstract

This study aimed to accurately simulate the main tidal characteristics in a regional domain featuring four open boundaries, with a primary focus on baroclinic tides. Such understanding is crucial for improving the representation of oceanic energy transfer and mixing processes in numerical models. To this end, the astronomical potential, load tide effects, and a wavelet-based analysis method were implemented in the three-dimensional ROMS model. The inclusion of the astronomical tidal and load tide aimed to enhance the accuracy of tidal simulations, while the wavelet method was employed to analyze the generation and propagation of internal tides from their source regions and to characterize their main features. Twin simulations with and without astronomical potential forcing were conducted to evaluate its influence on tidal elevations and currents. Model performance was assessed through comparison with tide gauge observations. Incorporating the potential forcing improves simulation accuracy, as the model fields successfully reproduced the main features of the barotropic tide and showed good agreement with observed amplitude and phase data. A complex principal component analysis was then applied to a matrix of normalized wavelet coefficients derived from the enhanced model outputs, enabling the characterization of horizontal modal propagation and vertical mode decomposition of both $M_2$ and nonlinear $M_4$ internal tides.

1. Introduction

Tides in the Mediterranean Sea have been studied by several authors [1,2,3,4]. Although tidal amplitudes in this region are relatively small, they play a crucial role in amplifying certain basin modes of the Mediterranean Sea [5]. It was suggested that tidal energy may contribute to the energetic low-frequency circulation observed in the Mediterranean through mechanisms such as mixing and tidal rectification [6].

Despite advancements in numerical modeling, global models [7,8] and inverse methods [9] still struggle to accurately capture tidal characteristics across all regions, particularly over continental shelves. They have had some success in reproducing nodal locations and large-scale tidal structures, but they continue to exhibit significant amplitude inaccuracies in coastal and shallow regions [10]. Efforts to improve tidal simulations through data assimilation, such as incorporating tide gauge and altimetric data [11,12], have yielded some improvements, yet significant discrepancies persist. This is especially evident in the southern Mediterranean, where tide gauge and current measurements remain scarce, making model validation particularly challenging.

To address these limitations, this study investigates whether incorporating astronomical tidal forcing improves the accuracy of regional tidal simulations, particularly the amplification observed in the Gulf of Gabes. To this aim, we have implemented the perturbations of the astronomical tide-generating potential, and taken into account of the load tide in the Regional Ocean Modeling System (ROMS) [13]. We have conducted twin simulations with and without astronomical potential forcing to evaluate its impact. The model was forced at the four open boundaries using the tidal amplitudes of the five dominant tidal constituents ($M_2$, $S_2$, $N_2$, $K_1$, and $O_1$) for the Mediterranean Sea [14]. This study employs a high spatial resolution 1/32°, which enhances the accuracy of numerical simulations compared to previous models. By resolving finer-scale dynamics, we aim to highlight discrepancies between our regional ROMS simulations and global models, which are commonly used to filter tidal signals from altimetric data [15]. The improved accuracy of these simulations is not only relevant for scientific studies but also for practical applications such as navigation, coastal management, and marine pollution tracking, particularly in the Gulf of Gabes. A detailed representation of tidal currents is essential for simulating the spatial and temporal distribution of pollutants, offering valuable insights for environmental studies. Furthermore, since tides play a fundamental role in ocean mixing [16], their accurate representation enhances predictions related to ocean circulation, nutrient transport, and ecosystem dynamics. Moreover, understanding tidal processes requires accounting for their interactions with other oceanic dynamics across multiple spatial and temporal scales. In fact, ocean dynamics span a wide range of scales that interact nonlinearly, making it challenging to analyze them in isolation. Physical processes influence and are influenced by both their own scale and others, as seen in wind-tide interactions [17] and nonlinear tidal responses, particularly in mixing and internal tide formation. While improving regional tidal simulations is a primary goal of this study, accurately resolving tidal variability also requires advanced diagnostic tools. To better understand the generation and propagation of internal tides, we implemented wavelet decomposition [18] in the ROMS model as a key step toward achieving this objective. Indeed, wavelet analysis within numerical models enhances the identification of tidal components and their propagation while providing deeper insights into nonlinear processes.

The wavelet analysis method is increasingly used in meteorology and oceanography [18,19,20,21]. Previous studies have explored the integration of wavelet analysis with empirical orthogonal functions (EOFs) or principal component analysis (PCA) to study oceanic processes [22,23]. However, our approach differs by applying PCA directly to the complex wavelet coefficients of model-derived vertical velocity fields, allowing a physically coherent separation of modes and better tracking of their spatiotemporal evolution. This combined wavelet-PCA methodology yields Empirical Wavelet Orthogonal Functions (WEOFs), which provide improved insight into modal structure and propagation characteristics. To our knowledge, this application of WEOFs to internal tide diagnostics in a fully 3D, high-resolution Mediterranean configuration of ROMS represents a novel contribution. This is particularly valuable in regions where internal tides interact with background flows, leading to enhanced mixing and energy transfer. By combining wavelet decomposition with principal component analysis (WEOF), it is possible to extract coherent physical structures, separate internal tides by frequency bands (e.g., semi-diurnal and quarter-diurnal), and examine their generation and propagation mechanisms. Given these advantages, a key objective of this study is to apply wavelet decomposition to better characterize tidal variability and its role in ocean dynamics.

2. Materials and Methods

2.1. Numerical Modeling Approach

The hydrodynamic model used in this study is based on the Regional Ocean Modeling System (ROMS) ([24]; https://www.myroms.org/). ROMS is a three-dimensional, finite-difference model that solves the primitive equations in an Earth-centered rotating reference frame. It is a nonlinear model that incorporates the Boussinesq and hydrostatic approximations, making it well-suited for simulating oceanic circulation at various scales. The model employs terrain-following $\sigma$-coordinates in the vertical and orthogonal curvilinear coordinates in the horizontal, allowing for improved representation of coastal and shelf dynamics. ROMS utilizes a split-explicit, free-surface formulation, where the barotropic (fast) and baroclinic (slow) components of motion are solved separately. The barotropic momentum and surface elevation equations are advanced using short time steps, while baroclinic momentum equations are integrated with longer time steps. A two-way time-averaging procedure ensures that the barotropic mode remains consistent with the three-dimensional continuity equation. Additionally, ROMS incorporates parameterizations to account for surface gravity wave-current interactions, improving its ability to represent nearshore and coastal processes.

The model domain (Figure 1) extends from 9.7° E to 17.5° E and from 32.5° N to 39° N, covering the central Mediterranean region. The horizontal grid resolution is 1/32° (2.7 km) in both longitude and latitude. This high resolution, finer than the first internal Rossby radius of deformation (10 km; [25]), ensures an improved representation of small-scale circulation features and bathymetric variations. Vertically, the model employs 30 sigma-coordinate levels, allowing for better resolution of water column dynamics, particularly in coastal and shelf regions. To ensure numerical stability, the model employs an external time step is set to 3 s with an internal mode integration every 90 s, thereby satisfying the Courant-Friedrichs-Lewy (CFL) condition $\Delta t\le {\displaystyle \frac{\Delta s}{\sqrt{2gh}}}$, where $\Delta s$ is the minimum grid length.

The model bathymetry is derived from the Gebco database (https://www.gebco.net/data_and_products/gridded_bathymetry_data/, accessed on 15 December 2022), with bilinear interpolation used to map depth data onto the computational grid. Along continental boundaries, the bathymetry is set to a minimum depth of 1 m, ensuring realistic representation of shallow coastal areas. The model was initialized using the MEDATLAS 2002 monthly climatology of observed temperature and salinity fields [26].

2.2. Simulation Strategy

Tidal forcing was implemented by prescribing the elevations of the dominant tidal constituents in the region, specifically $M_2$, $S_2$, $N_2$, $K_1$ and $O_1$ [14], along the model domain’s four open boundaries. The tidal coefficients were derived from a two-dimensional gravity-waves model named MOG2D [8]. For each constituent i, the corresponding tidal elevation ${\zeta }_i$ is given by:

$${\zeta }_i=f_i{A}_{i}cos\left({\omega }_i(t-t_0)+V_i\left(t_0\right)+u_i-G_i\right)$$

(1)

where $A_i$ and $G_i$ are the amplitude and phase lag relative to Greenwich of the equilibrium constituent i, respectively. The parameters $f_i$ and $u_i$ are the nodal factors for amplitude and phase, and $V_i$ is the phase of the equilibrium constituent i at Greenwich time $t=t_0$ corresponding to the model time origin.

Bottom frictional stress was represented using a drag formulation with a quadratic drag coefficient $C_D$. Sensitivity tests, in which $C_D$ varied from ${10}^{-3}$ to $3\times {10}^{-3}$, led to an optimal value of $C_D=2\times {10}^{-3}$. This value provided the best agreement between simulated and observed amplitudes and phases at multiple coastal stations (Figure 1). The selected falls within the empirical range of ${10}^{-3}$ to $3\times {10}^{-3}$ reported by previous study [27] and aligns with values used in other numerical models [28,29].

We have implemented astronomical potential forcing in the ROMS model, a component that is generally neglected in regional modeling of barotropic tides. This addition is particularly relevant for our study domain, which is characterized by amplification phenomena in the Gulf of Gabes and a notable amplification of the barotropic tide amplitude. To assess the impact of astronomical potential forcing, we conducted twin simulations (hereafter $E1$ and $E2$). $E1$ was forced solely by tidal elevation variations at the open boundaries, whereas $E2$ included both tidal elevation variations at the open boundaries and tide-generating forces within the domain. For both numerical experiments, the model was integrated forward for 150 days. The final 30 days were subjected to harmonic analysis to extract amplitude and phase information for elevations and currents for the main constituents. Notably, the considered tidal constituents can be accurately separated through harmonic analysis over a one-month time series. To evaluate the model’s performance, we used tide gauge measurements from various sources [2,4,30]. The model’s amplitude and phase were compared with elevation time series recorded at fourteen coastal stations distributed across the model domain (Figure 1). Only stations with sufficiently long time series, enabling a reliable separation of different tidal constituents, were selected. Among these, SG is the only offshore tide gauge [31].

2.3. Potential Tide Implementation

In this study we incorporate the astronomical tide potential and the tidal load effect in the ROMS model to enhance its accuracy in simulating tidal dynamics at a local scale. The equilibrium tide, also known as the static tide, is directly proportional to the generating potential gradient. The surface of the oceans responds instantaneously to all tidal-generating forces reaching an equilibrium position. Under the hydrostatic assumption and neglecting ocean viscosity, equilibrium is reached when generating force balances the horizontal pressure gradient created by the surface slopes. Thus the displacement of the free surface at static equilibrium is given by:

$${\zeta }_{eq}={\displaystyle \frac{{\mathsf{\Pi }}_{astro}}{g}}$$

(2)

Taking into account the astronomical Earth tide and the perturbations of the gravitational field that it induces, the elevation due to the astronomical potential ${\zeta }_{astro}$ can be written as:

$${\zeta }_{astro}={\displaystyle \frac{(1+k-h){\mathsf{\Pi }}_{astro}}{g}}$$

(3)

where k and h are the Love numbers, estimated to be 0.3 and 0.6, respectively. For a given tidal wave constituent n, the associated elevation ${\zeta }_{astro}^n$ can be expressed as:

$$\begin{array}{cc}\hfill {\zeta }_{astro}^n& =(1+k-h){\zeta }_{eq}^n{f}_n{G}_{\nu }\left(\phi \right)cos\left({\omega }_n(t-t_0)+V_n\left(t_0\right)+u_n+\nu \lambda \right)\hfill \\ \hfill & \approx 0.7{\zeta }_{eq}^n{f}_n{G}_{\nu }\left(\phi \right)cos\left({\omega }_n(t-t_0)+V_n\left(t_0\right)+u_n+\nu \lambda \right)\hfill \end{array}$$

(4)

where:

• ${\zeta }_{eq}^n$ is the tidal equilibrium amplitude for a given constituent n;
• $V_n\left(t_0\right)$ is the phase of the disturbing celestial body (to which the tidal constituent pulsation ${\omega }_n$ is attributed) relative to the Greenwich meridian at the origin time $t_0$;
• $f_n$ and $u_n$ are the nodal correction factors for amplitude and phase relative to the tidal constituent n, respectively;
• $\nu$ is equal to 0 for long waves, 1 for diurnal waves, and 2 for semi-diurnal ones;
• $\lambda$ is the longitude and $\phi$ the latitude of a given point;
• $G_{\nu }\left(\phi \right)$ is a function of latitude that is equal to:

$$G_{\nu }\left(\phi \right)=\left\{\begin{array}{cc}{\displaystyle \frac{1-3{sin}^2\phi }{2}},\hfill & \mathrm{for}\mathrm{long}\mathrm{waves}(\nu =0)\hfill \\ sin2\phi ,\hfill & \mathrm{for}\mathrm{diurnal}\mathrm{waves}(\nu =1)\hfill \\ {cos}^2\phi ,\hfill & \mathrm{for}\mathrm{semi}-\mathrm{diurnal}\mathrm{waves}(\nu =2)\hfill \end{array}\right.$$

(5)

In the same way the elevation associated with the load effect for a given constituent n, ${\zeta }_{load}^n$,is expressed as:

$${\zeta }_{load}=A_{load}^n{f}_{n}cos\left({\omega }_n(t-t_0)-{\varphi }_{load}^n+V_n\left(t_0\right)+u_n\right)$$

(6)

where $A_{load}^n$ and ${\varphi }_{load}^n$ are the elevation load amplitude and phase lag for a given constituent n. These variables, which originating from the FES2014 model [7] and freely available at https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/global-tide-fes.html, accessed on 10 January 2023, were interpolated onto the model grid. The nodal correction coefficients for amplitude and phase at the model’s reference time were computed externally. We account for the time-varying tidal potential by implementing a dynamic update of the tidal potential forcing terms at each time step within the ROMS model. Two CPP options ASTRO_POTTIDE and LOAD_POTTIDE, were added to the model to account for the contribution of one or both tidal potential components. The tidal potential amplitude and phase are updated at each time step and ramped if necessary. The associated elevation ${\zeta }_{pot}$ is then added to the pressure gradient terms in the barotropic equation of motion:

$${\displaystyle \frac{\partial \overrightarrow{u}}{\partial t}}+\overrightarrow{u}.\overrightarrow{\nabla }\overrightarrow{u}+2\overrightarrow{\mathsf{\Omega }}\wedge \overrightarrow{u}=-g\overrightarrow{\nabla }(\zeta -{\zeta }_{pot})-{\displaystyle \frac{\overrightarrow{{\tau }_b}}{h}}+{\overrightarrow{D}}_{hu}$$

(7)

where $\overrightarrow{u}$ is the barotropic horizontal velocity, $\overrightarrow{\mathsf{\Omega }}$ is the Earth’s angular speed, h is the total water column, given by $h=H(x,y)+\zeta (x,y,t)$, ${\overrightarrow{D}}_{hu}$ is the horizontal diffusion and $\overrightarrow{{\tau }_b}$ is the bottom friction of the barotropic currents. Finally, ${\zeta }_{pot}={\zeta }_{load}+{\zeta }_{astro}$ represents the free surface elevation associated with the tidal potential including astronomical or and load effect.

2.4. Wavelet Implementation

Harmonic analysis is commonly used to study internal tides by extracting tidal components, but it does not provide information about their temporal evolution. A more effective approach combines wavelet decomposition with principal component analysis (PCA), known as Wavelet Empirical Orthogonal Functions (WEOF) analysis [18]. This method integrates time-frequency wavelet analysis with spatial PCA to identify coherent physical structures, allowing for the separation of internal tides by frequency band (semi-diurnal and quarter-diurnal) and facilitating the study of their generation and propagation. By reconstructing the vertical velocity within a given frequency band, this approach helps identify the modal structure of internal tides, providing insights into their generation and propagation mechanisms.

We have implemented the wavelet analysis module [18] in ROMS model used in this study. The wavelet decomposition relies on the complex Morlet wavelet transform [21], retaining only the coefficients within the cone of influence (COI) to avoid edge effects. This analysis ensures both time and frequency localization. To isolate individual tidal constituents such as $M_2$ and $M_4$, the wavelet transform is applied in separate runs targeting narrow frequency bands. This selective approach avoids inter-frequency leakage and ensures consistent treatment of each tidal mode. This decomposition can be applied to a selected region of the model domain for dynamic variables such as sea surface elevation, as well as horizontal and vertical velocity components, allowing control over temporal and spatial resolution. PCA is then applied to a matrix of normalized wavelet coefficients, with the temporal mean removed, to extract dominant singular vectors and compute their variance. This enables the reconstruction of a given variable within a specific frequency band along with its dominant modes.

While the wavelet coefficients were normalized following standard energy conservation practices, we did not apply the amplitude bias correction proposed in [22]. That correction is particularly relevant for continuous spectrum comparisons across multiple frequencies, which is not the focus of the present study. Instead, our emphasis lies on identifying the spatial structure and vertical modal content associated with each individual tidal constituent, not on comparing absolute wavelet energy across constituents. This method effectively decouples processes across different time scales, improving our understanding of the physical mechanisms associated with each mode. When applied to internal tides, it allows for the separation of different tidal components, including non-linear ones, and aids in characterizing their generation and propagation, as demonstrated in the following section.

3. Results

3.1. Impact of the Astronomical Tide-Generating Potential

To evaluate the impact of potential forcing and the load tide on reproducing the principal tidal characteristics in our domain, we present in

Table 1. Observed amplitude ($A_{\mathit{ob}}$) and phase ($P_{\mathit{ob}}$), along with modeled amplitude ($A_{\mathit{E}1}$, $A_{\mathit{E}2}$) and phase ($P_{\mathit{E}1}$, $P_{\mathit{E}2}$) for the ${\mathrm{M}}_2$ constituent elevation at 14 coastal locations.

Stations	${\mathit{A}}_{\mathit{ob}}$ (cm)	${\mathit{A}}_{\mathit{E}1}$ (cm)	${\mathit{A}}_{\mathit{E}2}$ (cm)	${\mathit{P}}_{\mathit{ob}}$ (°)	${\mathit{P}}_{\mathit{E}1}$ (°)	${\mathit{P}}_{\mathit{E}2}$ (°)
La goulette	8	9.5	8.1	249.4	244.6	248.4
Sfax	41.6	16.5	37.1	76.1	29.1	62.3
Gabes	51.1	24.5	51.6	79.1	48.6	70.7
SG	4.8	0.5	4.6	50	23.1	57.1
Palermo	11.3	11.8	11.8	236.3	227.3	227.2
Mazzara	4.3	5.5	5.0	127.8	193.7	165.3
Po.Emp	4.9	1.5	4.6	73	131.6	91.5
Valetta	6	2.8	5.7	64	63.9	66.9
C.passero	6.7	4.7	6.3	61.6	69.3	67.3
Lampedusa	7.6	2.5	6.9	42.5	18.5	51.7
Catania	6.5	4.7	6.2	56.4	66.8	69.0
Tropea	14.6	12.3	13.9	242.2	226.8	228.6
Milazzo	12	12.2	13.0	234	227.4	229.2
Lipari	12	12.2	13.1	232	227.5	228.8

,

Table 2. Observed amplitude ($A_{\mathit{ob}}$) and phase ($P_{\mathit{ob}}$), along with modeled amplitude ($A_{\mathit{E}1}$, $A_{\mathit{E}2}$) and phase ($P_{\mathit{E}1}$, $P_{\mathit{E}2}$) for the $S_2$ constituent elevation at 14 coastal locations.

Stations	${\mathit{A}}_{\mathit{ob}}$ (cm)	${\mathit{A}}_{\mathit{E}1}$ (cm)	${\mathit{A}}_{\mathit{E}2}$ (cm)	${\mathit{P}}_{\mathit{ob}}$ (°)	${\mathit{P}}_{\mathit{E}1}$ (°)	${\mathit{P}}_{\mathit{E}2}$ (°)
La goulette	3.0	3.6	3.0	274.4	271.1	271.9
Sfax	26.7	16.6	27.0	103.1	66.7	94.8
Gabes	36.4	24.2	38.3	107.3	85.8	90.2
SG	3.1	1.9	3.4	57	61.6	61.8
Palermo	4.3	4.5	4.4	259.6	246.7	249.8
Mazzara	1.8	1.7	1.9	102.8	184	141.7
Po.Emp	3.5	1.7	3.5	71.4	85.6	84.6
Valetta	3.7	2.3	3.7	71	78.5	74.5
C.passero	3.5	2.4	3.2	67.3	78.4	77.2
Lampedusa	4.9	2.6	4.8	56.6	68.2	64.3
Catania	3.5	2.5	3.2	61.9	77.8	74.3
Tropea	5.3	4.8	5.5	264.2	246.4	249.4
Milazzo	4.7	4.7	4.9	252.0	247.0	247.3
Lipari	4.5	4.7	4.7	254.0	247.0	247.0

and

Table 3. Observed amplitude ($A_{\mathit{ob}}$) and phase ($P_{\mathit{ob}}$), along with modeled amplitude ($A_{\mathit{E}1}$, $A_{\mathit{E}2}$) and phase ($P_{\mathit{E}1}$, $P_{\mathit{E}2}$) for the $K_1$ constituent elevation at 14 coastal locations.

Stations	${\mathit{A}}_{\mathit{ob}}$ (cm)	${\mathit{A}}_{\mathit{E}1}$ (cm)	${\mathit{A}}_{\mathit{E}2}$ (cm)	${\mathit{P}}_{\mathit{ob}}$ (°)	${\mathit{P}}_{\mathit{E}1}$ (°)	${\mathit{P}}_{\mathit{E}2}$ (°)
La goulette	3.0	3.8	3.7	195.7	215	210.8
Sfax	1.8	2.5	1.9	4.5	56.2	23.5
Gabes	2.5	2.6	2.2	349.3	316.9	356.3
SG	0.5	0.2	0.4	78	83.2	69.1
Palermo	2.9	3.8	4.1	199.2	185.6	187.5
Mazzara	3.5	3.2	4.2	85.5	118.5	114.9
Po.Emp	1.5	1	1.8	88.7	85.8	85.4
Valetta	1.1	1.1	1.4	28.5	14.5	16.7
C.passero	1.9	1.3	1.8	52.2	18.2	31.2
Lampedusa	0.6	0.9	0.9	346.6	309.8	359.1
Catania	1.6	1.3	1.8	42.1	27.8	37.6
Tropea	4.1	4.0	4.3	203.1	186.7	190.8
Milazzo	3.3	4.0	4.3	200	186.7	189.7
Lipari	3.1	4.0	4.3	199	187	189.6

the observed and modeled amplitudes and phases from the $E1$ and $E2$ numerical experiments for different tidal constituents. These results are compared to measurements from tide-gauge stations, whose distribution is shown in Figure 1. The computed amplitude and phase of the $M_2$ and $S_2$ tidal elevations for both experiments, $E1$ and $E2$, are presented in Figure 2. The general characteristics of the simulated semi-diurnal constituents $S_2$ and $N_2$, are similar to those of $M_2$ but with reduced amplitudes, particularly for $N_2$. The semi-diurnal constituents exhibit maximum amplitudes in the Gulf of Gabes, decreasing toward the north of the Strait of Sicily and the deep eastern region. The general patterns of amplitude and phase for $E1$ and $E2$ are similar, confirming the model’s ability to qualitatively reproduce the amplitude amplification in the Gulf of Gabes. However, the $E2$ amplitudes are closer to observational data, especially in the amplification region. The maximum $M_2$ amplitude in $E1$ (24.5 cm) is less than half of the observed value (51.1 cm), whereas in $E2$, it reaches 51.6 cm (Table 1), accurately reflecting the observed strong amplification in the Gulf of Gabes [30]. The same improvement is observed for $S_2$ constituent (Table 2).

Tidal elevations are predominantly controlled by semidiurnal components, as the maximum $K_1$ amplitude does not exceed 5 cm. A notable exception is observed over the Adventure Bank, particularly in the northwestern sector off Sicily and well away from the Gulf of Gabes, where amplitudes reach up to 4 cm. This supports the conclusion that tidal resonance within the basin primarily occurs at semidiurnal periods. The amplitude and phase distributions of the major diurnal $K_1$ constituent for simulations $E1$ and $E2$ (Figure 3) are nearly identical, with maximum and minimum values occurring at the same locations. However, the $K_1$ phase simulated in $E2$ exhibits better agreement with observations than that from $E1$ (Table 3). In Figure 4, we plot the amplitude and phase predicted by the two numerical experiments against observed values at tide gauge stations for the $M_2$ and $S_2$ constituents. The $E2$ predictions align more closely with observations than those from $E1$.

To provide a quantitative assessment, we computed the root mean square error ($RMSE$) between modeled and observed amplitudes and phases for each tidal constituent, along with the root mean square norm ($RMSN$):

$$RMSN=\sqrt{{\displaystyle \frac{1}{2n}}\sum _{i=1}^n\left(a_{oi}^2+a_{mi}^2-2a_{oi}{a}_{mi}cos(p_{oi}-p_{mi})\right)}$$

(8)

where n is the number of tide gauge stations, $a_{oi}$ and $a_{mi}$ are the observed and modeled amplitudes, and $p_{oi}$ and $p_{mi}$ are the observed and modeled phases.

Table 4. $RMSE$ and $RMSN$ between modeled and observed amplitude and phase for the main tidal constituents from $E1$ and $E2$ experiments.

	$M_2$		$S_2$		$N_2$		$K_1$
	$E1$	$E2$	$E1$	$E2$	$E1$	$E2$	$E1$	$E2$
$RMSE$ (cm)	10.3	1.5	4.6	0.5	2.2	0.7	0.6	0.5
$RMSE$ (°)	30.2	13.7	26.4	14.1	92.5	20.0	25.2	14.2
$RMSN$ (cm)	9.1	2.7	4.6	2.3	2.0	0.7	0.8	0.6

presents $RMSE$ and $RMSN$ values for $E1$ and $E2$. The $E2$ simulation provides a more accurate representation of amplitude and phase, with $RMSE$ values of 1.5 cm for $M_2$, 0.5 cm for $S_2$, and 0.7 cm for $N_2$. In contrast, $RMSE$ values for $E1$ are significantly higher (10.3 cm for $M_2$, 4.6 cm for $S_2$, and 2.2 cm for $N_2$). The $RMSE$ for $M_2$ in $E1$ corresponds to 20% of the maximum observed amplitude, while in $E2$, it is only 2.9%, demonstrating the substantial improvement brought by potential forcing. For the diurnal constituent $K1$, $RMSE$ values are similar in both experiments, with $E2$ showing a slight advantage for phase as simulated by $E2$ (Table 4). Modeled phases in $E2$ align more closely with observations, with $RMSE$ values below 20° for all constituents, whereas in $E1$, $RMSE$ ranges from 25.2° for $K_1$ to 92.5° for $N_2$. $RMSN$ values further support these findings, with a maximum of 2.7 cm in $E2$ compared to 9 cm in $E1$. The largest discrepancies are observed in the semi-diurnal constituents, particularly for $M_2$, partly due to its dominant amplitude.

3.2. WEOF Analysis of Internal Tides

3.2.1. Wavelet Decomposition

The reliability of the $E2$ simulation is reinforced by validation at 14 measurement stations across the domain, motivating further analysis using wavelet methods to investigate baroclinic tide propagation and its characteristics. Wavelet decomposition of the semi-diurnal $M_2$ and quarter-diurnal $M_4$ waves of the vertical velocity was performed using two separate simulations, both initialized with identical conditions and run at their respective tidal frequencies. In each case, the wavelet analysis started after model spin-up phase of the $E2$ simulation, as described in the Section 3.1. For the $M_2$ simulation, 72 complex wavelet coefficients were saved at hourly intervals, while for the $M_4$ simulation, 79 coefficients were stored every 15 min. The vertical velocity fields for each frequency were then reconstructed from their respective complex wavelet coefficient decompositions, allowing for the isolation of each individual tidal components. As shown in Figure 5a, the total vertical velocity near the primary $M_2$ internal tide generation site which is localised at (11.51° E, 37.48° N), is predominantly semi-diurnal. The decomposition enables a clear separation between the $M_2$ wave (Figure 5b) and the $M_4$ wave (Figure 5c). The $M_4$ wave results from nonlinear interactions within the governing equations of motion, specifically through self-interaction of the $M_2$ with itself.

3.2.2. WEOFs and Internal Tide Propagation

To identify the predominant modes of internal tide propagation, Principal Component Analysis (PCA) was performed on a matrix of normalized wavelet coefficients. This approach decomposes a multivariate, multidimensional field into Empirical Wavelet Orthogonal Functions (WEOFs). Only the most dominant singular values were retained for further analysis. As demonstrated by Liu et al. [23], the combined use of wavelet decomposition and EOF/PCA has proven effective in analyzing multiscale variability in oceanic processes. Our study builds on this conceptual foundation but applies the technique in a novel way: we perform PCA not on observational time series, but on spatial matrices of normalized wavelet coefficients extracted from a high-resolution ROMS simulation. By targeting narrow band frequency bands such as $M_2$ and $M_4$, we isolate internal tide signals specific to each tidal constituent and extract their dominant vertical modal structures. This modeling-based WEOF approach provides a physically grounded, three-dimensional perspective on the generation and propagation of internal tides in dynamically complex environments like the Sicily Strait.

For the wavelet decomposition of vertical velocity, three modes were extracted for both the $M_2$ and $M_4$ tidal constituents. The percentage of variance explained by each mode is summarized in

Table 5. Relative variances of EOFs on the reconstructed vertical velocity at $M_2$ and $M_4$ frequencies.

Mode	${\mathit{M}}_2$ (Variance, %)	${\mathit{M}}_4$ (Variance, %)
Mode 1	98.8%	78%
Mode 2	1%	19%
Mode 3	0.2%	3%

. The first mode is dominant for $M_2$, accounting for approximately 99% of the total vertical velocity variance, indicating a highly coherent baroclinic structure. In contrast, for $M_4$, although mode 1 still dominates, its explained variance is lower, allowing for the emergence of a second mode, which accounts for 19% of the total variance.

A horizontal section of the total vertical velocity and the first WEOF at a depth of 100 m is shown in Figure 6. Specifically, it reveals propagation from the potential generation site toward the northeast, parallel to the trace of the vertical section. This pattern is more clearly observed in the first WEOF than in the total vertical velocity field. Additionally, $M_2$ internal tide propagates from the Maltese shelf break and reaches the southern Sicilian shelf. This propagation is also evident in the first mode of the $M_2$ vertical velocity horizontal section.

The total vertical velocity and the first WEOF along the vertical transect are shown in Figure 7. The total velocity along the transect does not reveal clear evidence of internal tide propagation. However, the first mode of the $M_2$ WEOF (WEOF 1) for the vertical velocity reveals the $M_2$ internal tide propagation from its generation site. In fact, the $M_2$ internal tides propagates from the western shelf break toward the north. By examining the repetition of maxima and minima in WEOF 1, the horizontal wavelength can be determined and associated with the theoretical mode. From the WEOF 1 plot, the estimated horizontal wavelength is approximately 70 km, which matches the theoretical value of the first horizontal baroclinic mode, as indicated in

Table 6. Table of horizontal wavelength propagation characteristics of internal tides for the semi-diurnal ($M_2$) and quarter-diurnal ($M_4$) components under August stratification, computed over an average depth of 400 m (mean depth of the vertical section from the radial origin to a distance of 100 km).

Mode	${\mathit{M}}_2$ (Wavelength, km)	${\mathit{M}}_4$ (Wavelength, km)
Mode 1	72	30
Mode 2	36	15
Mode 3	24	10

.

The WEOF 1 vertical profile near the main $M_2$ internal tide generation site (Figure 8, left) which explains 98% of the total variance, exhibits two distinct maxima at 50 m and 220 m, indicating the dominance of the first vertical baroclinic mode. Further offshore (Figure 8, right), the WEOF 1 profile evolves to exhibit characteristics consistent with the second baroclinic mode.

The generation of the $M_2$ tide also excites higher harmonics, including the $M_4$ internal tide. The vertical section of the reconstructed vertical velocity at the $M_4$ frequency does not exhibit a clear horizontal periodicity, likely due to complex modal interactions and bathymetric effects. However, the WEOF decomposition reveals more coherent spatial structures (Figure 9). Both WEOF 1 and WEOF 2 exhibit a horizontal wavelength of approximately 30–32 km, which is consistent with the characteristics of the first horizontal baroclinic mode. The first WEOF mode, which explains 78% of the total variance, is mainly confined below 200 m depth and extends over the first 100 m of the section. This pattern may be associated with topographically trapped internal wave modes. The second WEOF mode, accounting for 19% of the variance, shows a similar horizontal wavelength but is more pronounced in the upper layers, with extrema occurring around 150 m depth. This vertical structure suggests phase shifts, potentially linked to modal interference or partial reflection. Notably, the energy carried by these dominant modes diminishes around 100 km from the transect origin, indicating significant dissipation of the $M_4$ internal tide along its propagation path. The presence of the $M_4$ internal tide near the seafloor can be attributed to nonlinear interactions involving the dominant $M_2$ tide. In particular, bottom friction contributes through its quadratic dependence on tidal velocity. When the $M_2$ barotropic tide dominates, the quadratic bottom stress term generates energy at twice the fundamental frequency, thereby producing the $M_4$ harmonic. This mechanism is particularly effective over sloping topography, where strong tidal currents enhance bottom friction. As a result, $M_4$ internal tide energy is generated near the bottom and subsequently radiates upward. Due to its shorter horizontal scale, however, the $M_4$ component is more susceptible to local dissipation, which likely explains the observed decay of WEOF modes beyond the 100-km mark.

4. Discussion

This study aimed to accurately simulate the main tidal characteristics in a regional setting with four open boundaries, an important challenge (e.g., [9,32]). However, the primary objective was to gain insights into baroclinic tides. To achieve this, we used a high-resolution three-dimensional numerical model to simulate tidal elevations and currents, evaluating its performance against tide gauge measurements. In particular, we implemented astronomical potential forcing to assess its impact on the reliability of our local tidal simulations. To quantify this effect, we conducted twin simulations with and without astronomical potential forcing. Additionally, we integrated wavelet analysis in the ROMS model to better understand the generation and propagation of internal tides.

In both simulations, the model qualitatively reproduces the spatial distribution of tidal components. However, when only open boundary forcing is considered, the modeled elevation amplitude is significantly underestimated in the resonance region. Including astronomical potential forcing improves the simulation quality, as the model accurately captures the major features of the barotropic tide and shows reasonable agreement with observed amplitude and phase values. The $RMSE$ does not exceed 1.4 cm for the $M_2$ component and is about 0.5 cm for $K_1$. The maximum modeled amplitude (51.6 cm) closely matches the observed value (51.1 cm) in the resonance area. The $RMSE$ remains below 2.8% of the maximum observed amplitude but rises to approximately 20% without astronomical potential forcing. Nonetheless, comparisons between model results and observations at different measurement stations suggest that tidal characteristics outside the resonance region are only marginally affected by this forcing.

Our findings also indicate that astronomical potential forcing has a particularly strong impact on the semidiurnal frequency, which corresponds to the resonance frequency in the Gulf of Gabes [2]. These results suggest that potential forcing should be included even in local simulations, particularly in resonance regions. The accuracy of the semidiurnal tide simulation is higher than that of the diurnal tides. This is expected, as the model tuning was performed for $M_2$, the dominant tidal constituent in the study area. At the diurnal period, other local forcings, such as wind and atmospheric pressure, may influence the observations and thus contaminate the $K_1$ signal. Although tide gauge data may contain measurement errors, they provide valuable independent insights into the model’s performance. Discrepancies between model results and observations could stem from either model inaccuracies or uncertainties in observed tide estimates. Additionally, the misalignment between tide gauge locations and model grid points introduces further uncertainties. The primary sources of error in our model arise from uncertainties in boundary conditions and bathymetry. Indeed, the MOG2D model used for open boundary forcing in this study underestimates elevation amplitudes [8]. Our results show that even small modifications to the open boundary forcing can lead to significant changes in computed elevation amplitudes within the basin interior. This sensitivity study highlights the critical importance of accurately specifying tidal conditions along open boundaries. However, it is important to note that all results presented here were obtained without tuning the harmonic constituents derived from MOG2D. Incorporating more realistic tidal forcing and bathymetric data would likely improve the reliability of our results. Finally, despite some uncertainties, our simulations provide valuable insights that could aid in refining larger-scale tidal models of the Mediterranean Sea.

After improving the barotropic tide simulation, we focused on refining the characterization of the baroclinic response, particularly the generation and evolution of internal tides. To achieve this, wavelet analysis was integrated directly into the ROMS framework, providing a robust tool for examining the temporal and spatial variability of internal tide dynamics across different scales. A key numerical advantage of the wavelet-based approach implemented in ROMS is its flexibility to define a subdomain within the original simulation grid, with a spatial resolution equal to or lower than that of the parent domain. Additionally, it enables the extraction of model outputs at a temporal frequency different from the original model time step. This flexibility allows the large-scale dynamics resolved by the parent grid to be preserved while focusing the wavelet analysis on a smaller, targeted region. As a result, computational costs are significantly reduced, and the tidal signals of interest are resolved with greater accuracy within the subdomain, enhancing the overall quality and precision of the wavelet-based decomposition. Although one might consider running the model directly on a smaller regional domain, this approach is often constrained by the high sensitivity of the solution to the location of the open boundaries, requiring sensitivity tests to determine appropriate boundary conditions. By retaining the original parent model domain and applying wavelet analysis to a reduced area, we ensure that the internal tide dynamics are more accurately captured, while avoiding boundary-induced artifacts and preserving the overall accuracy of the simulation.

Wavelet-based decomposition of the vertical velocity successfully isolated the $M_2$ and $M_4$ internal tides, enabling the identification of their respective propagation paths and modal structures. The first WEOF mode captured nearly all the $M_2$ variance, revealing a north-eastward propagation from the main generation site, with a horizontal wavelength of approximately 70 km, consistent with the first baroclinic mode. Vertically, WEOF 1 exhibited two extrema and two nodes, matching the theoretical vertical structure of this mode. Farther from the generation site, a second vertical baroclinic mode emerged, indicating modal dispersion. For the $M_4$ component, the dominant mode propagated over shorter distances, with a horizontal wavelength of approximately 32 km and rapid dissipation beyond 100 km. The second WEOF mode of $M_4$ became significant at intermediate depths (150 m), suggesting more complex vertical energy structures. The successful isolation of the $M_4$ tide, generated through the self-interaction of the $M_2$ wave, highlights the utility of the method in capturing nonlinear internal tide dynamics and demonstrate the effectiveness of the wavelet approach in distinguishing internal tide modes and tracking their spatial variability across both horizontal and vertical scales.

Ultimately, this study highlights the critical role of incorporating astronomical potential forcing in enhancing the fidelity of regional tidal simulations, particularly in resonance areas such as the Gulf of Gabes. Accurately capturing tidal dynamics in such areas is essential for improving the predictive capabilities of ocean models, especially where nonlinear interactions and amplification effects are prominent.

The integration of wavelet-based diagnostics directly within the ROMS model further opens new avenues and significant methodological advance, enabling the analysis of internal tide generation, propagation, and modal structure with high spatio-temporal resolution. This approach not only deepens our understanding of tidal processes but also enhances computational efficiency by enabling focused analysis within targeted subdomains. As a result, it becomes possible to isolate and characterize key dynamical features of internal tides, including nonlinear components like the $M_4$ tide, that are often challenging to resolve using traditional methods. These developments contribute substantially to our understanding of both barotropic and baroclinic tidal processes in the Central Mediterranean, while also offering a versatile set of tools that can be applied to a wide range of regional and basin-scale studies.

Beyond academic interest, such tools have practical relevance for coastal management, marine spatial planning, and the assessment of environmental risks linked to tidal dynamics. Looking ahead, future research could benefit from coupling this advanced hydrodynamic framework with sediment transport, biogeochemical, or ecosystem models. This would allow for a more comprehensive evaluation of the role of tides in driving sediment resuspension, nutrient cycling, and tracer dispersion. These processes are particularly important in ecologically and economically sensitive areas subject to anthropogenic pressures such as dredging, pollution, or climate-induced changes in sea level and stratification.