## 1. Introduction

The primary purpose of the present study is to examine analytically and explain certain peculiarities of the tidal variations around the elongated Hopen Island located at Spitsbergen Bank in the Barents Sea. The trapping of the M_{2} tide around certain islands leads to a dipole structure in the sea level that rotates clockwise. This sea level difference travels around the island while the primary tidal wave determines semidiurnal variations of the mean water level around the island, as explained by Kowalik and Marchenko (2002, hereafter KM). This paper continues our research on the tidally trapped motion around islands started in KM. In KM, the reader will find an extended introduction to the topic of barotropic tidal wave trapped around islands.

Proudman (1915) constructed the first analytical solutions of tidal wave scattering by a coastline or small circular and elliptical islands. Huthnance (1974) analytically investigated the tidal currents over Rockall Bank in the North Atlantic Ocean, assuming nondivergent currents. The analysis was performed in polar and elliptic coordinates. Larsen (1977) and Reynolds (1978) analyzed the systematic variation in amplitude and phase around the Hawaiian Islands. They applied the scattering problem solution around the elliptical ridge and island to explain the observations. The analytical solutions for the tidal wave scattering around small islands were further developed by Lee and Kim (1999) by introducing bottom friction. The presence of bottom friction enabled them to extend the scattering theory to subinertial frequencies. Further analytical extension was achieved in KM. The model of an island in the ocean of a constant depth was found to be insufficient to explain the behavior of tidal waves in the region of the Pribilof Islands, Bering Sea. However, the inclusion of a sill changed the situation. The “skirt” around the Pribilof Islands played an essential role in forming the observed tidal features.

Numerical models of the tides in the Barents Sea and measurements indicated enhanced currents over Spitsbergen Bank due to the topographic amplification in the semidiurnal band of oscillations (Gjevik et al. 1994; Kowalik and Proshutinsky 1995; Padman and Erofeeva 2004; Chen et al. 2009). Over the entire region, the current vectors of M_{2}, S_{2}, and N_{2} tides showed a clockwise rotation. At the same time, tidal charts demonstrated the counterclockwise motion of the tide phases around the amphidromic point southeast off Hopen Island. Furevik and Foldvik (1996) suggested that this pattern of tidal motion might be described by a superposition of axially symmetric solution formulated in terms of the Bessel functions and Sverdrup wave (below the critical latitude) or complex wave (above the critical latitude).

The inertial period equals the M_{2} tidal period at the critical latitude (∼74.5°N). At Spitsbergen Bank, it crosses Bear Island, while Hopen Island is above the critical latitude. From our computation (Kowalik and Marchenko 2023), we have concluded that since Spitsbergen Bank straddles over the critical latitude, the presence of the critical latitude does not influence the trapping of the semidiurnal wave. The difference in the island shapes between the near-circular Bear Island and the elongated Hopen Island also demonstrates that the trapping is not dependent on the island’s geometry. To describe analytically the properties of trapped waves around an elongated island above the critical latitude, we solve the tidal scattering problem for an elliptically shaped island invoking elliptical coordinates.

The paper is organized as follows. In section 2, observations of ice drift and surface currents near Hopen Island are described. Basic assumptions for theoretical investigations and equations are formulated in section 3. Simple analytical solutions reproducing the structure of semidiurnal tidal waves in the vicinity of Hopen Island are constructed in section 4. An analytical solution describing the diffraction of semidiurnal tidal waves on the elliptic island surrounded by the sill is constructed and compared to the observations at Hopen Island in section 5. Natural modes of the water motion around the elliptic island with shelf are investigated in section 6. The main results of the investigation are summarized and discussed in section 7.

## 2. Observations of ocean currents near Hopen Island

Hopen Island is located in the northwest region of the Barents Sea, 115 nautical miles (n mi; 1 n mi = 1.852 km) east of Spitsbergen. The island’s length is approximately 30 km, and its maximal width is less than 2 km. Norwegian Hydrographic Service (2018) describes tides around the island in the following way: “the tidal stream, which changes at high and low water, rushes around the island with unusual strength and in some places, especially around the south and north points, it sets up a violent sea which can be dangerous to smaller vessels.”

Satellite images obtained using the Drift-Noise Polar Services application IcySea (https://icysea.app) show frequent formation of vortices near the south and north tips of Hopen Island (Fig. 1). Streamlines of drift ice following the surface current and wind drag are also visible in the island’s vicinity. With respect to the zonal direction of the tidal current, the vortices form on the island’s leeward side.

Drifting buoys (Oceanetic Measurements Ltd, Canada) were deployed on drift ice at Spitsbergen Bank in April 2018 to observe the ice motion (Marchenko 2022). One buoy survived the ice melt and kept transmitting GPS coordinates every 10 min until February 2019. In July 2018, the buoy drifted near Hopen Island (Fig. 2a). The buoy trajectory is subdivided into several fragments depicted by blue, black, yellow, and green lines. The blue line starts close to the island’s southern tip at 1850 UTC 16 July and continues for four M_{2} periods (4 × 12.42 h). The yellow line starts in the nearby location at 0150 UTC 20 July and continues for four M_{2} periods as well. The green line starts at the yellow point near the island’s north tip at 0330 UTC 22 July, and finishes after 30 M_{2} periods. Blue, yellow, and green dots mark the locations of the buoys after passing a single M_{2} period. Red dots mark the maximum speed of the buoy. Figure 2b shows the speed of the buoy as a function of time at the west side of the island. The initial points on the graphs correspond to the buoy’s speeds in the region 25°–25.5°E, 76.4°–76.5°N in Fig. 2a. The buoy circled the island two times. The starting yellow and green points almost coincide in Fig. 2a, but the speed of the buoy along the green line was lower than the speed along the blue and yellow lines. The green trajectory exhibits only partial trapping.

The buoy trajectory consists of many loops passed by the buoy in a clockwise direction. Similar buoy drift patterns were observed earlier in different regions of Spitsbergen Bank (e.g., Pease et al. 1995). Usually, buoys pass one round cycle over a time close to the semidiurnal cycle, as it is marked by blue, yellow, and green dots in Fig. 2a. In the open sea, when the wind drag is negligible, the buoys’ trajectories are close to circles. Increasing wind speed influences the resulting drift of buoys in a specific direction and changes the diameter of the loops.

To identify the influence of wind on the trapped motion around the island, we construct probability density functions of the wind direction and speed. The data from https://earth.nullschool.net at (76.6°N, 25.2°E) for 16–24 July 2018 were used. At this time, the buoy drifted near the island (see Fig. 2a). The colors of the paired histograms in Fig. 3 correspond to the colors of the buoy trajectory segments in Fig. 2. Because the green particle displays only a short trapping time and is dragged both to the east and west of the island, it is suitable case to observe the influence of the wind-driven motion. On 24 July strong southwest and west winds (30–40 km h^{−1}) pushed the buoy to the east, and until 28 July the buoy drifted in the area 76.4°–76.5°N and 26.0°–26.8°E (see Fig. 2a). However, later due to east wind (not shown in Fig. 3), the buoy drifted to the south tip of the island, and proceeded to drift to the west after passing the south tip of the island. This example shows that wind drag influences surface currents and the buoy drift, but it cannot explain high drift speeds and the trapped motion near the island tips as seen along the blue, yellow, and black trajectories.

Here, we describe the sea level and the buoy’s speed to be used later in section 4 for comparison with the analytical computation of sea level and velocity of an incident wave. The sea level is from Wolfram Mathematica TideData software, which calculates the data from the space altimetry and tide gauge stations. Figure 4a shows the sea level on 16–28 July 2018 at the point 76.4°N, 26.3°E located to the east of Hopen Island. Semidiurnal period of about 12.42 h is well recognized. Due to the two unequal tides per 24.84 h tidal day, the amplitude of the water surface elevation shows substantial diurnal inequality between 6 and 16 cm. Daily high and low tides slightly change amplitude at neighboring tidal days, and these changes are induced by the slight modulation caused by the fortnightly lunar tides.

The current velocities are calculated from the buoy drifting in the area 76.4°–76.5°N and 26.0°–26.8°E on 24–28 July 2018, where the depth is close to 100 m (Figs. 2a and 5). Blue and yellow lines in Fig. 4b show, respectively, the north and east components of the buoy velocity. The main period is close to M_{2}, and velocity amplitudes reach 0.5 m s^{−1}.

## 3. Basic equations and coordinate systems

*x*and

*y*are the Cartesian coordinates,

*t*is the time,

*u*and

*υ*are the components of water velocity,

*ζ*is the water surface elevation, and

*d*is the water depth. The actual water depth equals

*H*=

*dH*

^{*}, where

*H*

^{*}is the characteristic water depth (scale parameter). Dimensional variables denoted below by the subscript “dim” are related to the dimensionless variables in the following way

*t*

_{dim}is the time,

*x*

_{dim}and

*y*

_{dim}are the horizontal coordinates,

*u*

_{dim}and

*υ*

_{dim}are the water velocities averaged over the depth,

*ζ*

_{dim}is the water surface elevation,

*U*

^{*}is the characteristic amplitude of the velocity,

*f*is the Coriolis parameter, and

*g*is the gravity acceleration. Scale factors for the calculation of dimensional length and water surface elevation are

*ζ*,

*u*,

*υ*) = (

*ζ*

_{0},

*u*

_{0},

*υ*

_{0})

*e*, where

^{iωt}*ω*is the dimensionless angular frequency related to the dimensional frequency by the formula

*ω*=

_{d}*ωf*. Introducing the above solution into (1) we arrive at an equation for the sea level amplitude,

^{2}/∂

*x*

^{2}+ ∂

^{2}/∂

*y*

^{2}is the Laplace operator, and

*K*

^{2}=

*ω*

^{2}− 1 defines the dispersion equation.

*u*

_{0}and

*υ*

_{0}are expressed by the formulas developed in KM:

*r*,

*θ*) and elliptical coordinates (

*ξ*,

*η*), which are related to cartesian coordinates (

*x*,

*y*) in the following way:

*ξ*= const are ellipses with foci located at the points (±

*μ*, 0). The ratio of the major and minor axes of the ellipses along the

*x*and

*y*directions is equal to coth

*ξ*. The coordinate lines

*η*= const are hyperbolas.

The polar coordinates are further used to describe water motion generated by the semidiurnal tide near an amphidromic point. The water motion near the island surrounded by the elliptic sill (shelf) is studied in the elliptical coordinates. The island boundary coincides with the coordinate line *ξ* = *ξ*_{0}, and the shelf boundary is defined by the coordinate line *ξ* = *ξ*_{1} (*ξ*_{1} > *ξ*_{0}). Water depth in the deep domain outside the shelf equals *H*^{*} at *ξ* > *ξ*_{1} and at the shelf located at *ξ* ∈ (*ξ*_{0}, *ξ*_{1}) the depth is *H _{s}* <

*H*

^{*}. The water depth in dimensionless units is

*d*= 1 outside the shelf and

*d*< 1 on the shelf.

*u*

_{0,}

*and*

_{r}*u*

_{0,}

*, and elliptical coordinates*

_{θ}*u*

_{0,}

*and*

_{ξ}*u*

_{0,}

*are expressed by the formulas*

_{ξ}*J*=

*μ*

^{2}(sinh

*ξ*

^{2}+ sin

*ξ*

^{2}) is the Jacobian of the coordinate transformation.

*ω*

_{dim}=

*f*. The Coriolis parameter equals

*f*= 2Ω sin∅, where Ω = 7.2921 × 10

^{−5}rad s

^{−1}is the frequency of Earth’s rotation, and ∅ is the latitude. The frequency of semidiurnal tide M

_{2}equals

*ω*

_{M2}= 1.40 519 × 10

^{−4}rad s

^{−1}, therefore the critical latitude for M

_{2}tide is ∅

_{M2}=74.4717°. Since in the proximity to the critical latitude

*K*→ 0 it follows from (4), (7), and (8) that the current velocity goes to infinity when the sea surface has a nonzero slope.

The present paper investigates tidal currents near Hopen Island in the Barents Sea (Fig. 5). The island extends between (76.70609°N, 25.46224°E) and (76.45058°N, 24.93164°E), and is located above the critical latitude ∅_{M2}. The island length is approximately *L _{H}* ≈ 31.7 km, and the width is smaller than 2 km. In the middle point of Hopen Island, the Coriolis parameter equals

*f*= 1.415 × 10

^{−4}rad s

^{−1}, the dimensionless frequency of M

_{2}tide equals

*ω*=

*ω*

_{M2}/

*f*= 0.9931, and the value of

*K*

^{2}= −0.0138.

Representative depth in the open sea at the amphidromic point is close to 100 m, and representative M_{2} tidal velocity is about 0.1 m s^{−1} (Gjevik et al. 1994). Assuming *H*^{*} = 100 m and *U*^{*} = 0.1 m s^{−1} we find the scale factor *α _{ζ}* = 0.32 m. The scale factor

*α*in the middle point of Hopen Island equals 221 km. Expressed by the dimensionless variables, the island length equals

_{l}*L*/

_{H}*α*= 0.14. Elliptic coordinate lines specified by formulas (6) with

_{l}*μ*= 0.07 are given in Fig. 6. The coordinate line

*ξ*=

*ξ*

_{0}= 0.05 corresponds to the contour of an elliptic island with dimensions similar to Hopen Island.

## 4. Semidiurnal tidal waves near Hopen Island

To demonstrate the behavior of the semidiurnal M_{2} tide in the proximity to Hopen Island, we use amplitudes and phases of sea surface elevation computed by Kowalik and Proshutinsky (1995) (see Fig. 7). The main feature, the amphidromic point, is located to the south-southeast of Hopen Island (Gjevik et al. 1994; Kowalik and Proshutinsky 1995). A red line segment shows Hopen Island. Figure 5 depicts the approximate location of the amphidromic point at (75.7°N, 25.4°E), which sets this point above the critical latitude. The M_{2} tidal wave travels counterclockwise (CCW) around the amphidromic point. Using a superposition of explicit analytical solutions of Eq. (3) we aim to reproduce the distribution of cophase and amplitude lines of the M_{2}. The analytical solution will also describe the tidal ellipses and orbital circulation of water particles in the vicinity of Hopen Island.

**u**is the vector of the horizontal water velocity at a fixed point

*x*=

*x*

_{0}. In the right-handed reference frame (

*x*,

*y*), the positive and negative signs of

*w*denote, respectively, counterclockwise (CCW) and clockwise (CW) orbital motion of water particles located near the point

_{z}*x*

_{0}over the time interval

*t*∈ (

*t*

_{0},

*t*

_{0}+ Δ

*t*). The orbital motion of water particles located near the point

*x*

_{0}is expressed by the equations

*α*=

_{d}*U*

^{*}/(

*fα*), and

_{l}*u*

_{0}and

*υ*

_{0}are calculated at

*x*=

*x*

_{0}. The Lagrangian drift of water particles can be also described by Eq. (11). In this case

*u*

_{0}and

*υ*

_{0}are not any longer calculated at the fixed point but along a trajectory

*x*=

*x*(

*t*) and

*y*=

*y*(

*t*).

### a. M_{2} complex tidal waves

_{2}tidal frequency

*ω*= 0.9931 transmitting energy in the direction of the

*x*axis in an unbounded ocean of constant depth (

*d*= 1). In this section, the direction of the

*x*axis is not associated with the axis of Hopen Island. The following expressions describe the periodic solution of Eq. (3):

*A*is the amplitude of water surface elevation,

**x**is the vector with (

*x*,

*y*) coordinates, and

**k**denotes the wave vector with coordinates (

*k*,

_{x}*k*). When

_{y}*ω*< 1 it is evident that

*k*=

_{y}*k*

_{y}_{,±}is a pure imaginary quantity. In this case, the wave energy is transferred along the

*x*axis only. Solutions (12) are further referred to as M

_{2}complex tidal waves CTW+ and CTW−, where the symbols + and − correspond to the signs + and − in formula (12).

**u**= (Re[

*u*

_{0}], Re[

*υ*

_{0}]) traces an ellipse in the (

*u*

_{0},

*υ*

_{0}) coordinate system with the principal axes

*x*=

*x*

_{0}. Simple algebra demonstrates that the endpoint of

**u**travels the ellipse in CW direction when

*A*< 0, and in CCW direction when

_{u}A_{υ}*A*> 0 (Kowalik and Luick 2019).

_{u}A_{υ}In Fig. 8, the CW and CCW orbital motion regions are defined for the water particles in the plane (*k _{x}*,

*ω*). The orbital motion is induced by complex tidal waves CTW+ and CTW−. The regions are separated by the Kelvin waves’ dispersion curves

*ω*= ±

*k*. These waves propagate in the positive and negative directions of the

_{x}*x*axis. In the Kelvin waves the velocity component

*υ*= 0, and the orbital motion is reduced to oscillations along the

*x*axis.

### b. M_{2} tidal wave in the vicinity of an amphidromic point

*ω*

^{2}< 1 and were developed by Martin and Dalrymple (1994):

*A*is the sea surface elevation amplitude,

*I*

_{1}(|

*K*|

*r*) is the Bessel function of the first kind,

*θ*is the polar angle.

The Coriolis parameter in the amphidromic point equals *f* = 1.409 × 10^{−4} rad s^{−1}, the dimensionless frequency of M_{2} tide equals *ω* = *ω*_{M2}/*f* = 0.9973, and *K*^{2} = −0.0054. Solutions (14) are dubbed as M_{2} amphidromic tidal waves ATW+ and ATW−, where the symbols + and − correspond to the signs + and − in (14). Further, we consider only the first modes of ATW corresponding to *n* = 1. To explore the rotational motion caused by ATW, we made an estimate based on the above parameters. The motion induced by ATW− is CCW (*w _{z}* > 0) inside a circle with a radius of 1.48 and CW (

*w*< 0) outside the circle (Fig. 9a). Thus, the orbital motion is CCW inside a circle with the radius of 1.48 and is CW outside the circle (Fig. 9a). ATW+ induces the orbital CCW circulation for the arbitrary values of

_{z}*r*.

### c. Superposition of ATW− and CTW+ waves

*ζ*

_{C}_{0}and

*ζ*

_{A}_{0}are specified by (12) and (14), respectively, and

*c*∈ (0, 1) is the weight coefficient. Subscript “

*I*” means that (15) defines the incident tidal wave transmitting energy to Hopen Island.

Formulas (12) are rewritten to account for the arbitrary direction of the wavevector **k** of CTW+ in the frame of reference (*x*, *y*) related to Hopen Island (Fig. 10). The direction of the wavevector **k** and the weight coefficient *c* vary to adjust the location of the amphidromic point generated by the wave superposition defined by formula (15) to the observed amphidromic point A near Hopen Island (Figs. 5 and 7).

*ω*= 0.9931 is the dimensionless M

_{2}frequency at the middle of Hopen Island, and the origin of the Cartesian reference frame (

*x*,

*y*) is placed at that location. CTW+ transfers energy in the negative direction of the axis

*θ*between the axes

_{k}*x*and

*θ*is counterclockwise with respect to the axis

_{k}*x*. The arrow in Fig. 10 shows the direction of CTW+ propagation coinciding with the direction of the vector −

**k**(negative direction of the axis

*x*,

*y*) in formulas (12) are replaced by

*r*and

*θ*in (14) are modified in the following way:

*x*

_{0}and

*y*

_{0}are found from the condition that the point with zero amplitude (the amphidromic point due to the wave superposition) coincides with the location of the amphidromic point A near Hopen Island (Fig. 9b).

The superposition of ATW− and CTW+ qualitatively reproduces the incident tidal waves in the vicinity of Hopen Island. Figure 11 shows real parts of the dimensional water surface elevation and water velocity as functions of time (*ζ _{I}*,

*u*,

_{I}*υ*) = (

_{I}*ζ*

_{I}_{0},

*u*

_{I}_{0},

*υ*

_{I}_{0})

*e*. The dimensionless velocity amplitudes

^{iωt}*u*

_{I}_{0}and

*υ*

_{I}_{0}are calculated by formulas (4). The computed amplitudes of the sea surface elevation and water velocities turned out to be in satisfactory agreement with those obtained from observation and presented in Fig. 4.

## 5. Semidiurnal tide diffraction by an elliptic island

Along with the general circulation around the amphidromic point described above, the analytical approach in this section will reproduce the trapped clockwise motion in proximity to Hopen Island. For this purpose, diffraction of an incident wave by an elliptic island with a shelf is considered. The Mathieu functions usually express the solution to the sea level oscillations in the elliptical coordinates. Fortunately, in proximity to the critical latitude, these functions can be developed into series of a small parameter, and solutions are expressed as cos and sine series for the angular coordinate (*η*) and as an exponential function for the radial coordinate (*ξ*).

*ω*= 0.9931. The island and the shelf boundaries coincide with coordinate lines

*ξ*

_{0}= 0.05 and

*ξ*

_{1}= 4

*μ*here

*μ*= 0.07 (see Fig. 6). The solution of Eq. (3) is constructed as a combination of an incident tide and a reflected wave

*ζ*

_{I}_{0}is given by (15) and (18). Functions

*ζ*

_{R}_{±}describes waves reflected from the shelf break and the island. They satisfy Eq. (3) and the boundary conditions

*u*

_{0,}

*is calculated by the first formula (8). Matching conditions (20) and (21) mean the continuity of the sea level and the conservation of the mass flux over the shelf break, respectively. Boundary condition (22) states the no-flow condition through the island. Conditions (20)–(22) are identical to the matching conditions used by KM at the shelf break and at the contour of the circular island.*

_{ξ}*ζ*

_{R}_{±}is constructed using Eq. (3) written in elliptic coordinates (6). In Eq. (3) the constant

*d*= 1 at

*ξ*>

*ξ*

_{1}(open water), and

*d*< 1 at

*ξ*

_{0}<

*ξ*<

*ξ*

_{1}(shelf). The Laplace operator written in the elliptical coordinates becomes

*ζ*

_{0}=

*F*(

*ξ*)

*G*(

*η*). This results in the following equation

*q*= (

*μK*/2)

^{2}

*d*

^{−1}.

*a*is the constant of separation.

Equation (25) has two linearly independent solutions for arbitrary values of constants *a* and *q*. We consider only periodic solutions of Eq. (25) with arbitrary real *q* existing when *a* = *a _{n}*(

*q*) (

*n*= 0, 1, …) and

*a*=

*b*(

_{n}*q*), (

*n*= 1, 2, …) (McLachlan 1964). Such solutions are called Mathieu functions of the first kind ce

*(*

_{n}*η*,

*q*) and se

*(*

_{n}*η*,

*q*). Linearly independent solutions of Eq. (26) corresponding to

*a*=

*a*(

_{n}*q*) (

*n*= 0, 1,…) and

*a*=

*b*(

_{n}*q*), (

*n*= 1, 2,…) are expressed in terms of the modified Mathieu functions of the third kind

*(*

_{n}*η*,

*q*) and se

*(*

_{n}*η*,

*q*) can be represented by Fourier series in the following way:

*K*

^{2}= −0.0138 and

*μ*= 0.07, we estimate

*q*≈ −1.7 × 10

^{−5}/

*d*. The depth at the shelf is greater than 10 m, and the open ocean depth is 100 m. It means that the coefficient |

*q*| ≈ 1.7 × 10

^{−5}at

*ξ*>

*ξ*

_{1}, (open ocean) and |

*q*| < 1.7 × 10

^{−4}at

*ξ*∈ (

*ξ*

_{0},

*ξ*

_{1}) (shelf). Further, we consider solutions of Eq. (3) with small negative values of

*q*, and set

*d*= 0.2, i.e., the shelf depth is 20 m.

*a*(

_{n}*q*) and

*b*(

_{n}*q*) are calculated as power series of

*q*, and the first terms in these expansions are equal to

*n*

^{2}(McLachlan 1964). Functions sin

*nη*and cos

*nη*satisfy Eq. (24) with the accuracy of

*O*(

*q*). Numerical estimates of the Fourier coefficients show that

*nξ*and sinh

*nξ*if

*n*> 0. If

*n*= 0 the linearly independent solutions of Eq. (26) are reduced to a linear function of

*ξ*and a constant.

*q*the solution of Eq. (3) can be written as a series

*α*

_{0},

*α*

_{n}_{1},

*α*

_{n}_{2},

*β*

_{n}_{1}, and

*β*

_{n}_{2}are found from the boundary conditions.

The approximate solution (30) is valid when 2|*q*| cosh2*ξ* ≪ 1. For example, assuming 2|*q*| cosh2*ξ* = 0.1 and |*q*| = 1.7 × 10^{−4} we arrive at an estimate for the coordinate *ξ* < 3.7. From (6) it follows that Eq. (30) approximates the solution of Eq. (3) inside a circle with the coordinates *x*, *y* < 1.4. Figure 9b shows that this condition is fulfilled in the vicinity of the elliptic island including the amphidromic point A.

The values of *q* are different at the shelf and outside the shelf: *q* = *q*^{+} = (*μK*/2)^{2} at *ξ* > *ξ*_{1} and *q* = *q*^{−} = (*μK*/2)^{2}*d*^{−1} at *ξ*_{0} < *ξ* < *ξ*_{1}. Therefore, the constants *α _{n}*

_{1},

*α*

_{n}_{2},

*β*

_{n}_{1}, and

*β*

_{n}_{2}are also different in these regions. Further we mark these values by superscripts “+” for (

*ξ*>

*ξ*

_{1}) and “−”for (

*ξ*

_{0}<

*ξ*<

*ξ*

_{1}). Constants

*n*≥ 0) and

*n*> 0) because the reflected waves decay as

*ξ*→ ∞. From the boundary condition (22) integrated over the variable

*η*from −

*π*to

*π*and from the velocity in the elliptical coordinates (8) it follows that

*ξ*=

*ξ*

_{1}as Fourier series

*nη*(

*n*≥ 0) and sin

*nη*(

*n*> 0) on both sides of the equations we find a system of nonhomogeneous linear algebraic equations for calculation of constants

*α*

_{I}_{,}

*and*

_{n}*β*

_{I}_{,}

*are the “source” terms associated with the incident wave.*

_{n}*O*(10

^{−3}) of accuracy

*α*

_{I}_{,}

*and*

_{n}*β*

_{I}_{,}

*using the boundary conditions (20)–(22).*

_{n}Figure 12 shows calculated sea surface elevations, and in Fig. 13 velocity vectors are plotted at different phases of the semidiurnal tide. The sea level dipole rotating CW is formed near the island. Maximal amplitudes of the sea level and the strongest currents are found near the island tips O_{1} and O_{2}.

*u*and

*υ*are the complex components of the water velocity. The

*V*is strongly amplified at the island tips. The maximum water velocities are confined to boundary zones with the dimensionless radius of 0.05 near the tips. The dimensional radius of these zones is close to 10 km.

Figure 14b shows cophase lines and isolines of the dimensional sea surface elevation. The highest amplitude occurs near O_{1}. The lowest amplitudes are near O_{2}. The cophase lines highlight the CCW propagation of M_{2} tide at some distance from the island, while the dipole rotates CW around the island (Fig. 12).

An example of Lagrangian trajectory of a water particle trapped near the island, calculated for the initial position *ξ* = *ξ*_{1} − 0.5, *η* = 0.9*π* at *t* = 0, is given in Fig. 15a. The initial point of the trajectory is marked by a black dot. Simulations were performed from the initial time *t* = 0, to *t* = 64, i.e., for approximately 5.2 × 12.42 h of the dimensional time. Over this time interval the water particle traveled CW around the island. The trajectory includes two CW loops on each side of the island. The dimensional speed of the particle as a function of time (Fig. 15b) reveals the highest speeds of approximately 3 m s^{−1} near the points O_{1} and O_{2}, while for the rest of the trajectory the speed ranges between zero and 1 m s^{−1}. The shape of the trajectory and the particle’s speed are in good agreement with the observations shown in Figs. 2a and 2b.

*m*= 1, 2, 3, and 4, respectively. Simulations were performed for dimensionless time 80, starting from the initial time, or for approximately 12.6 tidal periods.

Figure 16a corresponds to initial position (36a), where the particles are placed outside of the shelf (*ξ*= 2*ξ*_{1}). The values of *η* in (36a) specify the starting azimuthal coordinate of the particles. Figures 16a and 16b explore how a half-period shift in the initial tidal phase affects the particle trapping. The particles initially located at *η* = 0.5*π* (yellow) and *η* = 1.5*π* (red) are trapped by the island for both values of the initial phase, while trapping of *η* = 0 (blue) and *η* = *π* (green) particles depends on the initial phase. Figure 16c indicates that the trapping is not present when the particles start farther from the shelf (*ξ* = 4*ξ*_{1}). Figure 16d demonstrates that all trajectories initiated within the shelf (*ξ* = *ξ*_{1} − 0.1) exhibit the island trapping, but its characteristics might substantially differ.

## 6. Elliptic island with shelf: Eigenmodes of water oscillations

*ω*of trapped waves is not specified. Amplitudes of sea surface elevation are described according to (33) and (34) by the formulas

*n*= 1, 2, 3, … is the mode number representing azimuthal harmonics. Six constants

*nη*) and sin(

*nη*) lead to the homogeneous system of six linear algebraic equations to determine the six constants. The determinant of the system should be zero for the existence of a nontrivial solution. Solution of the determinant results in the dispersion relation between the frequency

*ω*and parameters

*ξ*

_{0},

*ξ*

_{1}, and

*d*. These parameters include gravity and Coriolis forces, the geometry of the island and the surrounding shelf.

Eigenfrequencies of the first (*n* = 1), the second (*n* = 2), the third (*n* = 3), the fourth (*n* = 4), and the fifth (*n* = 5) harmonics of the eigenmodes were calculated versus the water depth *d*. Eigenfrequencies of the first mode (*n* = 1) depicted in Fig. 17a explain dependence of the dispersion on the shelf width, specified by the value of *ξ*_{1}. Points N_{1}, N_{2}, N_{3}, N_{4}, and N_{5} in Fig. 17b belong to the dispersion curves constructed with *μ* = 0.07, *ξ*_{0} = 0.05, *ξ*_{1} = 4*μ*, and *d* = 0.2. These values are chosen to model the geometry of Hopen Island. The nondimensional frequencies in points N* _{n}* are

*ω*

_{1}= 0.173 (

*n*= 1),

*ω*

_{2}= 0.315 (

*n*= 2),

*ω*

_{3}= 0.427 (

*n*= 3),

*ω*

_{4}= 0.507 (

*n*= 4),

*ω*

_{5}= 0.563 (

*n*= 5). The dimensional periods of the eigenmodes are, respectively, 3 days (

*n*= 1), 1.64 day (

*n*= 2), 1.21 day (

*n*= 3), 1.02 day (

*n*= 4), and 22.06 h (

*n*= 5). The eigenfrequency of the fourth harmonic is close to the frequency of the diurnal tide, suggesting resonance in the diurnal band of oscillations, see Huthnance (1974).

As Fig. 17a illustrates, the eigenfrequency can reach the M_{2} frequency *ω* = 0.9931 for a very small shelf depth *d* and an extremely wide shelf (*ξ*_{1} ≫ *ξ*_{0}). We conclude that it is practically impossible to find eigenfrequencies close to the semidiurnal band for realistic islands and shelves. Even if such oscillations could be generated, they will be quickly damped by bottom friction due to small depth.

To compare dispersion curves for an elliptic island with the previous result for a circular island obtained in KM, we use the dispersion relation between frequency *ω* and the azimuthal wavenumber *n* of a free wave progressing CW around a cylindrical island with a surrounding shelf [see formula (13) in KM].

Eigenfrequencies of the first mode (*n* = 1) versus depth *d* are plotted in Fig. 18a. The island radius is *a* = 0.049, and the shelf break is located at *r* = *A*, where *A* = 0.05, 0.062, 0.082, 0.11, and 0.15. The values of *a* and *A* are chosen to demonstrate similarity of the dispersion curves in Figs. 17a and 18a. Location of the islands and shelf breaks are shown in Fig. 18b by solid and dashed lines for the elliptic and circular cases, respectively. The contour and the shelf break in the elliptic case are constructed for the following parameters, *μ* = 0.07, *ξ*_{0} = 0.05 (island) and *ξ*_{1} = *μ*, 4*μ*, 8*μ*, 12*μ*, and 16*μ* (shelf).

## 7. Conclusions

The present paper examined semidiurnal tide dynamics around the elongated Hopen Island. Observations and numerical computations (Kowalik and Marchenko 2023) suggest that the island traps M_{2} tidal wave as a dipole moving clockwise. Concurrently, the primary tidal wave is progressing counterclockwise around the amphidromic point at some distance from the island. To explain the trapping phenomenon, an analytical model includes both the elliptic island and the surrounding shelf. The analytical solution is subdivided into two steps. In section 4, the primary wave pattern is described by the incident wave as a superposition of axially symmetric solution expressed by the Bessel functions and complex wave (Furevik and Foldvik 1996). The sea level and currents generated by the incident wave compare well with the observed sea level and speed measured by the drifting buoy. Section 5 solves a diffraction problem for both incident and reflected waves in the elliptic coordinates corresponding to the geometry of Hopen Island. In the constructed analytical solution, the reflected wave’s amplitude is expressed by a series of the Mathieu functions.

The incident tidal wave generates mainly the first angular (azimuthal) harmonic at the shelf break, therefore, the diffraction problem was also simplified to the analysis of the first angular harmonics of the reflected wave. Because this is the lowest mode of oscillations, one would expect that it displays the strongest response to the forcing and a stable behavior, see discussion in KM. The derived analytical solution detailed the interaction of the incident waves with the island and trapping effects around the island.

The constructed solution shows that the sea surface elevation around the island is organized into a dipole structure rotating CW, and the maximum amplitude occurs near the south and north points of the island. Narrow regions of the amplified amplitude behave like boundary zones with the radiuses of about 10 km near the island tips; they are associated with the largest currents as well. The boundary regions correspond to observed locations of significant buoy drift. These currents also explain frequent formation of vortices clearly seen in satellite images.

Simulated Lagrangian drift of near-island water particles demonstrated the existence of trajectories confined to the island’s shelf. Shape and number of loops in these trajectories depend on the initial location of water particles and the tidal phase at the initial time. We managed to construct trapped trajectories with four loops circled by the water particles while traveling around the island. In this case observed trajectories and speed of a buoy are similar to the simulated trajectories and speeds of the Lagrangian particles.

Finally, we conclude that the interaction of the semidiurnal tide with Hopen Island causes high tidal currents near the south and north tips of the island and imposes a CW circulation around the island. These tidal features would govern drift of open ice. Fast moving ice floes could be dangerous to navigation. Strong currents may also force intense interaction of ice keels with the seabed.

Theoretical investigations in the present paper cannot explain all physical effects controlling drift of real floaters (buoys, floes) near Hopen Island. Many factors influence barotropic tidal currents calculated in the paper. The bathymetry near Hopen Island does not display elliptic symmetry (Fig. 5). The isobaths reproduce the island shape on the east side of Hopen Island, while on the west side, they deviate further to the west from the island. This shallow bathymetry influences the M_{2} tides around the island. The tidal wave along the west side propagates slower when compared with the east side. This introduces a phase difference into symmetrical dipole structures found via the elliptic island approximation, see Kowalik and Marchenko (2023).

Winds may destroy the trapped motion of floaters and force them to drift away from the island due to the wind drag and waves (Turnbull and Marchenko 2022). Hence, the barotropic tide can establish the trapped motion in shallow waters near Hopen Island only in calm weather.

Interaction of the tides with the continental slope in the Barents Sea generates internal waves with amplitudes of up to 10 m at the depth of 60–80 m (Marchenko et al. 2021). Currents caused by the internal waves modify the barotropic tidal currents predominantly in the deeper water and could be omitted in the considered problem.

Interaction of waves with islands could cause resonance when incident wave frequencies coincide with eigenfrequencies of island trapped waves. The performed analysis showed existence of eigen modes propagating in CW direction near an elliptic island with a shelf. The frequencies of the eigen modes are consistently below the semidiurnal frequency but can be close or even equal to the diurnal frequency (Huthnance 1974). Therefore, the general picture of M_{2} tide interaction with an island will not change due to the resonance; such phenomenon may only influence diurnal tide interaction with an island.

Field observations, analytical and numerical computations show that the most frequent pattern of tidally trapped sea surface around an island with a shelf is a dipole structure rotating clockwise. We related this pattern to the first azimuthal harmonic of waves trapped around the island. Our previous solution (KM) for a wave trapped around a cylindrical island surrounded by a sill and the present solution for the elliptical case demonstrate that the trapping and the dipole pattern occur both above and below the critical latitude, while the eigen modes of the trapped waves exist only in subinertial frequency range. This analytical validation of the trapping for a wide variety of the island geometry, from circular to elliptical, supports the assumption made by Dyke (2005) that the semidiurnal dipole pattern for an island is relatively immune to variations in geometry.

## Acknowledgments.

We are indebted to D. Brazhnikov, University of Alaska Fairbanks; I. D. Turnbull, C-Core Research Institute, St John’s, Canada; and to the reviewers for very valuable comments that greatly improved this paper. We gratefully acknowledge support for this work from the Research Council of Norway through the IntPart project Arctic Offshore and Coastal Engineering in Changing Climate, and the Petromaks2 project Dynamics of Floating Ice. We wish to thank Nataly Marchenko, The University Centre in Svalbard, for selection of satellite images for the paper.

## Data availability statement.

All data on trajectory of drifting buoy (Oceanetic Measurements Ltd, Canada) created or used during this study are openly available from the National Infrastructure for Research Data at https://doi.org/10.11582/2022.00002 as cited in Marchenko (2022).

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