Leaky Wave Modes and Edge Waves in Land-Fast Ice Split by Parallel Cracks

Abstract

In this paper we consider flexural-gravity waves propagating in a layer of water of constant depth limited by a vertical wall simulating a straight coastline. The water surface is covered with an elastic ice sheet of constant thickness. The ice sheet is split by one or two straight cracks parallel to the coastline, simulating the structure of land-fast ice with a refrozen lead. Analytical solutions of hydrodynamic equations describing the interaction of flexural-gravity waves with the ice sheet and cracks have been constructed and studied. In this paper, the amplification of the amplitude of incident waves between the shoreline and cracks was described depending on the incident angle of the wave coming from offshore. The constructed solutions allow the existence of edge waves propagating along the coastline and attenuated offshore. The energy of edge waves is trapped between the coastline and ice cracks. The application of the constructed solutions to describe wave phenomena observed in the land-fast ice of the Arctic shelf of Alaska is discussed.

1. Introduction

Seasonal dynamics of sea ice cover and tides influence the formation of cracks, leads and ice ridges in the land-fast ice of frozen seas. These ice formations usually extend along the coastline. The inhomogeneous structure of ice cover affects the properties of waves propagating to the shore from the sea. A low-frequency swell propagates over long distances, causing flexural oscillations of Arctic pack ice without breaking the ice and with very little attenuation. The results of measurements of swell in Arctic pack ice are described in several papers. Table 1 summarises the results of measurements carried out in solid ice at a large distance from the marginal ice zones. Mahoney et al. [1] and Johnson et al. [2] observed waves on the Arctic shelf of Alaska more than 100 km from shore. Measurements taken before 2007 were shorter in duration than later measurements. However, since the 1950s, short- and long period oscillations in pack ice have been identified. Periods of measured waves varied from several seconds to several tens of seconds [2,3,4,5]. High-frequency oscillations with periods less than 10 s are usually associated with local effects. Waves with periods more than 15 s can be associated with swell propagating from marginal ice zones, or with deformations of the ice cover on scales of several hundred meters and more.

Waves with periods of 8– to 15 s are associated with resonant ones, which occur when the speed of atmospheric pressure fluctuations propagating with the wind is equal to the minimum phase speed of flexural-gravity waves [4,6]. However, this hypothesis was not confirmed until later, when a more accurate comparison of wind and wave measurements became possible. For example, Johnson et al. [2] observed an increase in the amplitudes of waves with periods 3.5 s and 14 s at wind speeds of greater than 20 m/s, while the resonance dictates the generation of waves with periods 7.8– to 10 s for typical properties of sea ice. The generation of waves of a certain frequency can also be explained by the instability of the air flow near sastrugas and ice ridges, leading to the formation of air pressure fluctuations with certain frequencies, similar to the von Karman vortex street [7]. Direct use of the Strouhal number 0.18–0.22 for sub-critical air flow near a fixed cylinder with dimensions close to those of ice ridges leads to vortex shedding periods of 1.7 to 8.5 s, which are in the range of periods of high-frequency oscillations measured by Johnson et al. [2].

Table 1. Characteristics of waves measured in the Arctic Ocean.

	Date	Location	Sea Depth, km	Ice Thickness, m	Wave Amplitude, mm	Wave Period, s
Crary et al., 1952 [3]	8–23 April 1951	74–75N 138–150W	3.4–3.8	–	0.5	5–40
Hunkins, 1962 [8]	24–26 July 1957 19–21 July 1958	80–85N 120–180W	>1	3	5	15–60
LeShack and Haubrich, 1964 [9]	23 January 26 March 1960 2 June 1961	76N 140–160W 77N 160W	3	1–3	0.5 <1 <3	20–60
Sytinskii and Tripol’nikov, 1964 [4]	Mar–May 1961 Mar–May 1962	86N, 165W 82N, 165W	>1	3	0.5	1–8 12–45
Bogorodsky and Smirnov, 1980 [10]	Mar–May 1979	81N, 144E	>1	-	-	10–15 25–35
Wadhams and Doble, 2009 [11]	Jan–Mar 2007	85–87N, 130E	3.5–4.2	2.1	-	14–33
Marsan et al., 2012 [12]	May 2007	88N, 90E	4.2	2.5–2.7	-	25–30
Mahoney et al., 2016 [1]	Jan 2015	72N, 156W	0.15	-	1.2–1.8	30–50
Johnson et al., 2021 [2]	7–15 Mar 2020	72N, 156W	0.15	1.2–3	15 2	10–20 2.7–3.8
Smirnov et al., 2023 [5]	9–18 Feb 2020	88.07N, 87.68E	4.3	1–1.5	<23 -	20–30 7–8

Table 1. Characteristics of waves measured in the Arctic Ocean.

	Date	Location	Sea Depth, km	Ice Thickness, m	Wave Amplitude, mm	Wave Period, s
Crary et al., 1952 [3]	8–23 April 1951	74–75N 138–150W	3.4–3.8	–	0.5	5–40
Hunkins, 1962 [8]	24–26 July 1957 19–21 July 1958	80–85N 120–180W	>1	3	5	15–60
LeShack and Haubrich, 1964 [9]	23 January 26 March 1960 2 June 1961	76N 140–160W 77N 160W	3	1–3	0.5 <1 <3	20–60
Sytinskii and Tripol’nikov, 1964 [4]	Mar–May 1961 Mar–May 1962	86N, 165W 82N, 165W	>1	3	0.5	1–8 12–45
Bogorodsky and Smirnov, 1980 [10]	Mar–May 1979	81N, 144E	>1	-	-	10–15 25–35
Wadhams and Doble, 2009 [11]	Jan–Mar 2007	85–87N, 130E	3.5–4.2	2.1	-	14–33
Marsan et al., 2012 [12]	May 2007	88N, 90E	4.2	2.5–2.7	-	25–30
Mahoney et al., 2016 [1]	Jan 2015	72N, 156W	0.15	-	1.2–1.8	30–50
Johnson et al., 2021 [2]	7–15 Mar 2020	72N, 156W	0.15	1.2–3	15 2	10–20 2.7–3.8
Smirnov et al., 2023 [5]	9–18 Feb 2020	88.07N, 87.68E	4.3	1–1.5	<23 -	20–30 7–8

The thinning of sea ice in the Arctic [13,14,15] and the reduction of ice-covered areas in the Arctic [16] influence increasingly large waves in the western Arctic waters [17,18]. The connection between the Atlantic Ocean and the Arctic wave climates is projected to strengthen due to the increase of swell influences [19]. Increasing dynamics of sea ice in the Arctic influences the formation of ice ridges and leads [20] leading to the excitation of flexural oscillations and elastic waves in sea ice [21]. Weather conditions may affect synchronous wave processes in remote areas of the Arctic Ocean. For example, flexural waves with dominant frequencies of 0.07 Hz and 0.28 Hz were measured on 13 and 14 March 2020 on the Arctic shelf of Alaska when wind speed exceeded 20 m/s [2]. At the same time flexural, waves with dominant frequencies of 0.03 Hz and 0.091 Hz were measured near the North Pole [22].

A local increase in the amplitude of high-frequency waves can occur due to the interaction of waves with irregularities in the ice structure, for example, ice cracks and ice ridges of specific shape and geometry. At the same time, similar irregularities of a different shape affect the scattering of wave energy leading to a decrease in its amplitude. Several examples of such effects are further considered. The interaction of oncoming surface waves with rectilinear ice edge, ice cracks and ice ridges leads to reflection and redistribution of wave energy in space but does not lead to an increase in wave amplitudes along the path of wave propagation [23,24]. It was shown that several parallel cracks influence the formation of spectral zones with a very low wave transmission coefficient. In the limit of an infinite number of cracks or ice ridges wave transmission coefficients tend to zero within forbidden zones similar to the Brillouin zones [25,26]. The solutions of Fox and Squire [23] were constructed using the normal vertical modes. Marchenko [24] used the Wiener-Hopf technique [27].

The effect of wave amplitude amplification under a floating elastic ice sheet of finite width connected to a vertical wall was studied by Tkacheva [28]. Only sine waves propagating from the open sea perpendicular to the coastline were studied; the water depth was infinite. It was found that the increase in wave amplitude occurs for the resonant frequencies of incoming waves. The problem was solved analytically using the normal horizontal modes. Note that there is no increase in wave amplitudes under a floating elastic ice sheet of infinite width connected to a vertical wall [29]. Li et al. investigated the amplification of wave amplitude inside closed cracks in infinite floating elastic sheet [30]. The problem was solved analytically using the normal vertical modes.

The amplification of wave amplitude under a floating elastic ice sheet of finite width connected to a vertical wall is similar to the resonance of shelf waves or leaky wave modes considered in numerous papers [31,32,33]. It was found that the amplification of the amplitude of waves travelling from the open sea to the shoreline depends on the wave frequency [34] and incident angle [35]. The amplification effect is characterized by the coefficient of amplitude amplification equal to the ratio of the wave amplitude near the shoreline to the wave amplitude in the open sea. The dependence of the coefficient of amplitude amplification from the wave frequency may oscillate depending on the shape of the shelf. For the fixed wave frequency, the coefficient of amplitude amplification is greater than 1 for a range of the incident angles changing from 0^o (waves propagate in a normal direction to the shoreline) to Brewster’s angle [36] when it becomes equal to 1. For larger values of the incident angle, the coefficient of amplitude amplification is smaller than 1.

Stokes [37] described edge wave modes propagating along a shelf with a constant slope of the bottom and decaying with distance from the coastline. Ursell [38] studied solutions of hydrodynamic equations describing edge waves trapped near a submerged cylinder in a canal of infinite depth; the cylinder axis was perpendicular to the canal axis. Grimshow [39] and Evans and Mciver [40] studied edge waves traveling along sloping beaches of various geometries when the water depth approaches a constant value at large distances from the shore. Mollo-Christensen [41] constructed an analytical solution describing edge waves propagating along the shelf with a constant slope of the bottom in the presence of elastic ice sheet and alongshore current with constant velocity. It was shown that the amplitude of water velocity normal to shore could become large, and the resulting acceleration of the ice cover may cause the ice to ride up. Kuznetsov [42] and Martin et al. [43] studied edge waves propagating in unbounded water layer along underwater ridges. In above-considered examples, the energy of edge waves is trapped in shallow water regions where the phase velocity of waves is smaller than in surrounding regions.

Goldstein et al. [44] considered flexural-gravity edge waves propagating along a crack in an unbounded floating elastic ice sheet. The existence of the edge waves was confirmed by Evans and Porter [45]. Marchenko [46] studied flexural-gravity edge waves propagating along open water in a canal with an unbounded elastic ice sheet floating on the surface of shallow water. Ice cracks can be considered as a limit case for a canal of zero width. In these cases, the energy of edge waves is trapped in the vicinity of the open water region where the phase speed of waves is smaller than in areas covered with ice. Goldstein and Marchenko [47] considered edge waves on a shelf partially covered by a floating elastic ice sheet. In this case, the number of the edge wave modes where energy is trapped on the shelf depends on the width of the ice-covered area. The study was carried out in the shallow water approximation, taking into account the Coriolis force and assuming a rectangular step in the seabed. The excitation of edge waves near a crack and in an open water channel in an elastic ice sheet by local source of perturbations was investigated by Sturova [48] and Tkacheva [49,50]. The parametric resonance of two counter-propagating edge waves near an ice crack with an incoming flexural-gravity wave was studied by Marchenko [51].

In the present paper we consider flexural-gravity waves propagating in a water layer of constant depth limited by a vertical wall simulating a straight coastline. The water surface is covered by an elastic ice sheet of constant thickness. The ice sheet is split by one or two rectilinear cracks parallel to the coastline modelling structure of land-fast ice on the Arctic shelf of Alaska. The reflection of flexural-gravity waves from a straight coastline studied in papers [28,29] did not take into account the influence of ice cracks on waves. Edge waves for this type of land-fast ice have also not been studied. We consider analytical solutions of hydrodynamic equations describing the interaction of incoming flexural-gravity waves with the cracks and shore, and analytical solutions describing edge waves in which energy is trapped between the shore and cracks. The main goals of the paper are numerical investigation of constructed solutions and demonstration of wave effects in the coastal zone with application to the Arctic shelf of Alaska.

2. Basic Equations

We consider potential motions of ideal fluid below the elastic plate simulating the floating ice (Figure 1). Linearized equations are written as follows:

$$\left({\displaystyle \frac{{𝜕}^2}{𝜕x^2}}+{\displaystyle \frac{{𝜕}^2}{𝜕y^2}}+{\displaystyle \frac{{𝜕}^2}{𝜕z^2}}\right)\phi =0,z\in (-H,0),$$

(1)

$${\displaystyle \frac{𝜕\phi }{𝜕t}}+g\eta +p_w{\rho }_{sw}^{-1}=0,{\displaystyle \frac{𝜕\eta }{𝜕t}}={\displaystyle \frac{𝜕\phi }{𝜕z}},z=0,$$

(2)

$${\displaystyle \frac{𝜕\phi }{𝜕z}}=0,z=-H,$$

(3)

where $t$ is time, $x,y$, and $z$ are horizontal and vertical coordinates, $\phi$ is water velocity potential, $\eta$ is the elevation of water-ice interface, $p_w$ is water pressure below ice, $H$ is constant water depth, ${\rho }_{sw}$ is sea water density, and $g$ is the gravity acceleration. The zero value of $z$ corresponds to the sea surface. The Laplace Equation (1) follows from the mass balance of fluid of constant density. Equation (2) expresses dynamic and kinematic boundary conditions at the ice-water interface. Equation (3) specifies the condition of impermeability at the bottom. At rest, the ice-water interface coincides with the $z=0$ plane.

Bending deformations of ice are described by the equation of a thin elastic plate subjected to the action of water pressure [23]

$${\rho }_{si}{h}_i{\displaystyle \frac{{𝜕}^2\eta }{𝜕t^2}}+D{\left({\displaystyle \frac{{𝜕}^2}{𝜕x^2}}+{\displaystyle \frac{{𝜕}^2}{𝜕y^2}}\right)}^2\eta =p_w,D={\displaystyle \frac{E{h}_i^3}{12(1-{\nu }^2)}},$$

(4)

where $E$ and $\nu$ are the elastic modulus and the Poisson’s ratio of ice, and $h_i$ and ${\rho }_{si}$ are the sea ice thickness and density. All parameters characterizing ice properties are assumed to be constant.

It is assumed that the ice split by $N$ parallel irregularities of the ice sheet extended in the $y$-direction and is located at ${x=x}_j$, $j\in (1,N)$. Explicit analytical solution of Equations (1) and (2) describing periodic motions of sea water in time and in the $y$-direction is written in the form [24]

$$\begin{gathered} \phi =e^{i\theta }\left[\gamma \left(a_{+}{e}^{i{k}_{0}x}+a_{-}{e}^{-i{k}_{0}x}\right){\displaystyle \frac{\cosh \left({\lambda }_0(z+H)\right)}{\cosh \left({\lambda }_{0}H\right)}}+ \\ {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{\cosh \left(\lambda (z+H)\right)}{\kappa \left(\omega ,\lambda \right)\cosh \left(\lambda H\right)}}{\sum }_{j=1}^N{P}_j(k)e^{ik{(x-x}_j)}dk\right], \end{gathered}$$

(5)

$${\theta =k}_{y}y-\omega t,$$

(6)

$$\begin{gathered} \kappa \left(\omega ,\lambda \right)={\omega }^2\left(1+{\rho }_{si}{\rho }_{sw}^{-1}{h}_i\lambda \tanh \left(\lambda H\right)\right)-\lambda \tanh \left(\lambda H\right)\left(g+D{\left({\rho }_{sw}g\right)}^{-1}{\lambda }^4\right),\lambda = \\ \sqrt{k^2+k_y^2}, \end{gathered}$$

(7)

$$P_j\left(k\right)=c_{j0}+c_{j1}k+c_{j2}{k}^2+c_{j3}{k}^3,$$

(8)

where $\theta$ is the wave phase, $\omega >0$ is the wave frequency, and $k_y$ is the wave number in the $y$-direction. Arbitrary constants $a_{\pm }$ are determined from the radiation conditions at $\left|x\right|\to \mathrm{\infty }$. They determine the amplitudes of waves transmitting energy from the infinity to the shore. The existence of such waves is related to the existence of the real roots of the dispersion equation:

$$\kappa \left(\omega ,\lambda \right)=0$$

(9)

in complex plane $k$. Constant $\gamma$ is set to 1 if dispersion Equation (9) has real roots $k=\pm k_0$ in the complex plane $k$; otherwise, $\gamma$ is set to zero. Constant ${\lambda }_0$ is equal to $\sqrt{k_0^2+k_y^2}$.

From the second Equations (2) and (5) it follows that the elevation of water-ice interface is described by the formula

$$\begin{gathered} \eta ={{\omega }^{-1}e}^{i\theta }\left[-i{\gamma \lambda }_0\tanh \left({\lambda }_{0}H\right)\left(a_{+}{e}^{i{k}_{0}x}+a_{-}{e}^{-i{k}_{0}x}\right)+ \\ {\sum }_{j=1}^N\left\{{\displaystyle \frac{1}{2\pi i}}{\int }_{-\infty }^{\infty }\chi \left(\lambda \right)P_j(k)e^{ik{(x-x}_j)}dk\right\}\right],\chi \left(\lambda \right)={\displaystyle \frac{\lambda \tanh \left(\lambda H\right)}{\kappa \left(\omega ,\lambda \right)}} \end{gathered}$$

(10)

The integrand expression in Formula (5) has the asymptotic $O\left(k^{-2}\right)$ for $\left|k\right|\to \infty$ meaning that the velocity potential is an analytic function by $z\in (-H,0)$. The integrand expression in Formula (10) has the asymptotic $O\left(k^{-1}\right)$ for $\left|k\right|\to \infty$ meaning that the ice elevation determined by Formula (10) admits discontinuities of the function $\eta (t,x,y)$ and its spatial derivatives at ${x=x}_j$. Arbitrary constants $c_{jn}$, $n\in \left(\mathrm{0,3}\right)$ and $j\in (1,N)$, are specified from the contact-boundary conditions at the edges of the irregularities at ${x=x}_j\pm 0$. Further, we consider irregularities performed by cracks with free edges and coastline where the edge of the ice sheet is assumed to be clamped.

The qualitative shape of the dependence $\omega \left(\lambda \right)$ following from Equation (9) is shown in Figure 2. For sufficiently large range of the frequency $\omega$ dispersion Equation (9) has two real roots $\lambda =\pm {\lambda }_0$, four complex roots $\lambda =\pm {\lambda }_e$ and $\lambda =\pm {\lambda }_e^{\mathrm{*}}$, and an infinite number of pure imaginary roots $\pm {i\lambda }_j$, $j\ge 1$. Roots $\pm {\lambda }_0$, $\pm {\lambda }_e$, $\pm {\lambda }_e^{\mathrm{*}}$, and $\pm {i\lambda }_{\mathrm{1,2}}$ are shown by white circles in Figure 3a. Quantities ${\lambda }_j$ are positive real roots of the equation.

$${\omega }^2\left(1-{\rho }_{si}{\rho }_{sw}^{-1}{h}_i\lambda \tan \left(\lambda H\right)\right)+\lambda \tan \left(\lambda H\right)\left(g+D{\left({\rho }_{sw}g\right)}^{-1}{\lambda }^4\right)=0.$$

(11)

For sufficiently small frequencies $\omega$ each interval $\lambda H\in (\pi /2+j\pi ,\pi /2+(j+1\left)\pi \right)$ for $j\ge 0$ contains one root ${\lambda }_j$, which can be found by consequent iterations starting from the point $\lambda H=j\pi$. There is a frequency range where the root ${\lambda }_e$ becomes purely imaginary [52]. In this frequency range Equation (11) has three roots on the interval $\lambda H\in (\pi /\mathrm{2,3}\pi /2)$. This situation is not realized for the frequency range considered in the paper.

The roots of dispersion Equation (9) in complex plane $k$ are related to the roots of dispersion Equation (9) in the complex plane $\lambda$ by the formula ${{\lambda }^2=k}^2+k_y^2$. The real root ${\lambda }_0$ corresponds to the real roots $\pm k_0$ in the complex plane $k$ if ${\lambda }^2\ge k_y^2$, otherwise, the real root ${\lambda }_0$ corresponds to pure imaginary roots $\pm {ik}_0$ in the complex plane $k$. This property is mapped in Figure 2 by the regions denoted as I and II. If $k_y<\lambda$ (Region I) then Equation (9) has two real roots ${\pm k}_0=\pm \sqrt{{\lambda }^2-k_y^2}$ in the complex plane $k$. If $k_y>\lambda$ (Region II) then Equation (9) has two pure imaginary roots ${\pm ik}_0=\pm i\sqrt{{k_y^2-\lambda }^2}$ in the complex plane $k$. The roots $\pm k_0\to 0$ for $\left|k_y\right|\to \lambda$. In addition to the roots $\pm k_0$ or $\pm {ik}_0$ dispersion Equation (9) has four complex roots ${k=\pm k}_e=\pm \sqrt{{\lambda }_e^2-k_y^2}$ and $k=\pm k_e^{\mathrm{*}}=\pm \sqrt{{\lambda }_e^{\mathrm{*}2}-k_y^2}$, and infinite number of pure imaginary roots ${k=\pm ik}_j$ with $j\ge 1$ and $k_j=\sqrt{{\lambda }_j^2+k_y^2}{>k}_y$. Figure 3b,c shows the locations of the real and complex roots of Equation (9) when $k_y$ is located respectively in Regions I and II.

The contact-boundary conditions at the crack edges specify that generalized shear forces ($N^{\pm }$) and bending moments ($M^{\pm }$) applied to the crack edges are equal to zero [53]

$$N^{\pm }\equiv \pm D{\displaystyle \frac{𝜕}{𝜕x}}\left({\displaystyle \frac{{𝜕}^2}{𝜕x^2}}+(2-\nu ){\displaystyle \frac{{𝜕}^2}{𝜕y^2}}\right)\eta =0,x=\pm x_j,j\in (1,N)$$

(12)

$$M^{\pm }\equiv -D\left({\displaystyle \frac{{𝜕}^2}{𝜕x^2}}+\nu {\displaystyle \frac{{𝜕}^2}{𝜕y^2}}\right)\eta =0,x=\pm x_j,j\in (1,N)$$

(13)

The number of the contact-boundary conditions $4N$ is equal to the number of unknown constants $c_{jn}$ in Formulas (5) and (10) with $n\in \left(\mathrm{0,3}\right)$ and $j\in (1,N)$.

Note, that $\lambda$ appears in Formula (7) in combination $\lambda \tanh \left(\lambda \right)$ or as ${\lambda }^4$. This fact allows us to avoid considering two-sheeted Riemann surface on which the function $\lambda \left(k\right)$ is determined. Integrals in Formula (10) are calculated using the residue method and Jordan’s lemma as follows

$${\displaystyle \frac{1}{2\pi i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\chi \left(\lambda \right)P}_j(k)e^{ik{(x-x}_j)}dk={\sum }_{l=1}^{\mathrm{\infty }}{\left.res\left[{\chi \left(\lambda \right)P}_j(k)e^{ik{(x-x}_j)}\right]\right|}_{k=K_l},x>x_j$$

(14)

$${\displaystyle \frac{1}{2\pi i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\chi \left(\lambda \right)P}_j(k)e^{ik{(x-x}_j)}dk=-{\sum }_{l=-1}^{-\mathrm{\infty }}{\left.res\left[{\chi \left(\lambda \right)P}_j(k)e^{ik{(x-x}_j)}\right]\right|}_{k=K_l},x<x_j,$$

$${\left.res\left[{\chi \left(\lambda \right)P}_j(k)e^{ik{(x-x}_j)}\right]\right|}_{k=K_l}={\left.{\displaystyle \frac{{\lambda }^2\tanh \left(\lambda H\right)}{\kappa \prime \left(\omega ,\lambda \right)k}}{P}_j(k)e^{ik{(x-x}_j)}\right|}_{k=K_l},\kappa \prime \left(\omega ,\lambda \right)={\displaystyle \frac{𝜕\kappa \left(\omega ,\lambda \right)}{𝜕\lambda }}.$$

(15)

were $K_{\pm 1}=\pm k_0$ in Region I, $K_{\pm 1}=\pm {ik}_0$ in Region II, $K_{\pm 2}=\pm k_e$, $K_{\pm 3}=\mp k_e^{\mathrm{*}}$, $K_{\pm 3}=\pm {ik}_1$, $K_{\pm 4}=\pm {ik}_2$, etc. (Figure 3b,c).

Formula (15) is obtained by the calculation of residues of the integrands in Formula (14), taking into account that $\chi \left(\lambda \right)$ has only simple poles $k=K_l$ in the complex plane $k$. For the analysis of the solutions in Section 5 and Section 6, poles $K_l$ are calculated numerically depending on the value of the wave frequency $\omega$ and the wave number $k_y$ for $1\le l\le 22$. The infinite series in Formula (15) are replaced by the sums of the first 22 terms. Taking into account 20 pure imaginary roots $K_l$ for $3\le l\le 22$ provided a good approximation of the infinite series in Formula (14). Note that the deviations of numerical solutions from the exact solutions given by Formulas (10) and (14) depend on the number of terms in the approximating sums and errors in calculating the poles $K_l$. Poles were calculated using the FindRoot program in Mathematica 5.4 with default settings.

For the derivation of Formula (14), the contour of integration $k\in (-\mathrm{\infty },\mathrm{\infty })$ is closed by a semicircle of the large radius located in the region $Im\left[k\right]\ge 0$ for $x\ge x_j$ and in the region $Im\left[k\right]\le 0$ for $x\le x_j$. Because of the exponential decay of the integrand functions, the integrals along the semicircles tend to zero when their radiuses tend to infinity. It is also assumed that in Region I the contour of integration shown in Figure 3b by a thin line passes below the root $k_0>0$, and above the root ${-k}_0<0$ in the region $k\in (-\mathrm{\infty },\mathrm{\infty })$. Such deformation of the contour of integration corresponds to the displacements of the roots $\pm k_0$ in the complex plane $k$ when a small dissipation is included in the system and supports the damping of the waves with wave numbers $\pm k_0$ in the direction of their propagation [27].

3. Problems under Consideration

Five geometrical configurations of the computational domain modelling coastal zone are shown in Figure 4. Figure 4a shows a water layer of a fixed depth under continuous ice sheet bounded by a vertical shore at $x=0$. Figure 4b shows a water layer of a fixed depth under an ice sheet with a straight crack at $x=0$. Figure 4c shows a water layer of a fixed depth an under ice sheet with two straight cracks at $x=-L$ and $x=L$. Figure 4d shows a water layer of a fixed depth under an ice sheet with a straight crack at $x=L$ bounded by a vertical shore at $x=0$. Figure 4e shows a water layer of a fixed depth under an ice sheet with two straight cracks at $x=L_1$ and $x=L_2$ bounded by a vertical shore at $x=0$. Short names of the geometries of the coastal zone shown in Figure 4 are F, C, CC, FC, and FCC.

Contact-boundary conditions (12) and (13) must be satisfied at the crack edges. At the shoreline we consider the fixed edge of the ice sheet

$$\eta =0,{\displaystyle \frac{𝜕\eta }{𝜕x}}=0,x=0.$$

(16)

To satisfy the condition of zero horizontal velocity at $x=0$, $z\in (-H,0)$ we consider the symmetric potential of water velocity $\phi \left(t,x,y,z\right)=\phi \left(t,-x,y,z\right)$ in the region $x\in (-\mathrm{\infty },\mathrm{\infty })$ described by Formula (5) with constants $c_{ij}$ satisfying the symmetry conditions depending on the problem under the consideration

$$\mathrm{F},\mathrm{C}:x_1=0,c_{11}=c_{13}=0,a_{+}=a_{-},N=1.$$

(17)

$$\begin{gathered} \mathrm{CC}:x_1=-L,x_2=L,c_{10}=c_{20},c_{11}=-c_{21},c_{12}=c_{22},c_{13}=-c_{23},a_{+}= \\ {a}_{-},N=2. \end{gathered}$$

(18)

FC: $x_1=-L$, $x_2=0$, $x_3=L$, $c_{10}=c_{30}$, $c_{11}=-c_{31}$, $c_{12}=c_{32}$, $c_{13}=-c_{33}$,

$$c_{21}=c_{23}=0,a_{+}=a_{-},N=3.$$

(19)

$$\mathrm{FCC}:x_1=-L_2,x_2=-L_1,x_3=0,x_4=L_1,x_5=L_2,$$

(20)

$c_{10}=c_{50}$, $c_{11}=-c_{51}$, $c_{12}=c_{52}$, $c_{13}=-c_{53}$,

$c_{20}=c_{40}$, $c_{21}=-c_{41}$, $c_{22}=c_{42}$, $c_{23}=-c_{43}$,

$c_{31}=c_{33}=0$, $a_{+}=a_{-}$, $N=5$.

The number of unknown constants in the solution in Region I is one more than in Region II since $a_{\pm }=0$ in Region II. The contact-boundary conditions are used to calculate constants $c_{ij}$ depending on the amplitude $a_{+}$ of incident wave in Region I. In Region II the contact-boundary conditions lead to a system of homogeneous linear algebraic equations for the finding of constants $c_{ij}$. The system has nontrivial solution when the determinant $\mathsf{\Delta }(\omega ,k_y)$ equals zero. Any pair of real numbers $(\omega ,k_y)$ satisfying the equation

$$\mathsf{\Delta }\left(\omega ,k_y\right)=0,$$

(21)

corresponds to the frequency and wave number of an edge wave propagating along the cracks. The edge wave amplitude decays exponentially as $x\to \mathrm{\infty }$.

The maximum of absolute bending stress in an elastic plate reached at the plate surface and at the plate bottom is determined by the formulas [54]

$${\sigma }_n=0.5E{h}_i\left|{𝜕}_n^2\eta \right|,{𝜕}_n^2=n_x^2{\displaystyle \frac{{𝜕}^2}{𝜕x^2}}+2i{k}_y{n}_x{n}_y{\displaystyle \frac{𝜕}{𝜕x}}-k_y^2{n}_y^2,$$

(22)

where $\mathit{n}=(n_x,n_y)$ is the unit vector specifying the direction of the bending moment.

4. Dimensional and Dimensionless Variables

Representative scales of time ($t^{*}$), length ($l^{*}$), amplitude of velocity potential (${\phi }^{*}$), and bending stress (${\sigma }^{*}$) are determined by the formulas

$$t^{*}=\sqrt{{\displaystyle \frac{l^{*}}{g}}},l^{*}=\sqrt[4]{{\displaystyle \frac{D}{{\rho }_{sw}g}}},{\phi }^{*}={\displaystyle \frac{l^{*}{a}^{*}}{t^{*}}},{\sigma }^{*}={\displaystyle \frac{E{h}_i{a}^{*}}{2{l^{*}}^2}}.$$

(23)

Dimensionless variables denoted by subscript “$ds$” are determined by the formulas

$$t^{*}{t}_{ds}=t,l^{*}\left(x_{ds},y_{ds},z_{ds}\right)=(x,y,z),a^{*}{\eta }_{ds}=\eta ,{\phi }^{*}{\phi }_{dl}=\phi ,$$

(24)

where $a^{*}$ is a representative wave amplitude equal to 1 cm.

Numerical simulations were carried out with the following values of dimensional constants

$$H=15\mathrm{m},h_i=1\mathrm{m},L=50\mathrm{m},L_1=1\mathrm{km},L_2=1.1\mathrm{km},$$

(25)

${\rho }_{sw}=1030$ kg/m³, ${\rho }_{si}=920$ kg/m³, $E=3$ GPa, $\nu =0.3$.

From (23) and (25) it follows

$$t^{*}=1.059\mathrm{s},l^{*}=12.84\mathrm{m},{\sigma }^{*}=91\mathrm{kPa}.$$

(26)

Dimensionless values of lengths determined by (25) are

$$H_{ds}=1.17,L_{ds}=3.98,L_{1,ds}=77.87,L_{2,ds}=85.66.$$

(27)

According to the second Formula (4) the ice thickness $h_i$ is included in $D$. The dimensionless value of $D$ is equal to 1. The dimensionless Formula (7) is written as follows

$${\kappa }_{ds}\left(\omega ,\lambda \right)={\omega }^2\left(1+{\rho }_{si}{\rho }_{sw}^{-1}{h}_i{l^{*}}^{-1}\lambda \tanh \left(\lambda H_{ds}\right)\right)-\lambda \tanh \left(\lambda H_{ds}\right)\left(1+{\lambda }^4\right),$$

(28)

where $\omega$ and $\lambda$ are the dimensionless frequency and wave number.

The dispersion curve in dimensionless variables is shown in Figure 5a. Figure 5b shows dimensional wavelength versus dimensionless wave number. Vertical lines indicate the range of swell with periods from 6 to 30 s. Waves with periods from 1 s to 6 s correspond to flexural-gravity waves. Waves with periods from 30 s to 300 s correspond to infragravity waves, and waves with periods greater 300 s correspond to long-period waves. In numerical simulations we consider examples with dimensionless frequencies $\omega =0.01$ (long period wave, point 1 in Figure 5), $\omega =0.1$ (infragravity wave, point 2 in Figure 5), $\omega =0.5$ and $\omega =1$ (swell, points 3 and 4 in Figure 5), $\omega =3,4,5$ (flexural-gravity waves, points 5–7 in Figure 5).

5. Wave Reflection from Shore

Solutions of the wave reflection problem were studied with dimensionless frequencies $\omega$ equal to 0.01, 0.1, 0.5, and 1. It was found that maximum of the amplitude of bending stress is reached in the $x$-direction at the shoreline ($x=0$). From Formula (22) it follows that bending stress in the $x$-direction is proportional to ${𝜕}^2\eta /𝜕x^2$, where $\eta$ is determined by Formula (10). The value of ${𝜕}^2\eta /𝜕x^2$ for $t=0,y=0$ and $x=0$ is denoted as ${\eta }_{xx}$. The dependences of ${\eta }_{xx}$ from the incident angle ${\alpha }_i=\arctan {(k}_0/k_y)$ are shown in Figure 6 versus the incident angle of waves for different values of the wave frequency in Problems F, FC, and FCC corresponding coastal geometries F, FC, and FCC shown in Figure 4. The value ${\alpha }_i=0$ corresponds to waves propagating in the normal direction to the shoreline. For the analysis, the solutions are normalized to obtain the unit amplitude of waves for $x\to \mathrm{\infty }$. Here and further the subscript “$ds$” of dimensionless variables is omitted.

Figure 6a shows the dependence of ${\eta }_{xx}$ from the incident angle ${\alpha }_i$ in Problem F (reflection of wave from a vertical wall). The wave frequencies are shown in the figure. It can be seen that ${\eta }_{xx}$ was very close to 1 for $\omega =0.01$ and $\omega =0.1$ for relatively small values of the incident angle. The value ${\eta }_{xx}=1$ can be interpreted in the following way. Let us consider the stationary problem on the bending of a floating elastic plate due to vertical displacement of the fixed end at $x=0$. In the dimensionless variables the problem is described by the equations

$${\displaystyle \frac{d^4\eta }{d{x}^4}}+\eta =0,x>0;\eta ={\eta }_0,{\displaystyle \frac{d\eta }{dx}}=0,x=0;\eta \to 0,x\to \mathrm{\infty }.$$

(29)

The complex solution of (29) is expressed by the formula

$$\eta ={\displaystyle \frac{{\eta }_0}{2i}}\left[\left(-1+i\right)e^{-{\displaystyle \frac{1+i}{\sqrt{2}}}x}+\left(1+i\right)e^{-{\displaystyle \frac{1-i}{\sqrt{2}}}x}\right].$$

(30)

From (30) it follows that ${𝜕}^2\eta /𝜕x^2=-{\eta }_0$ at $x=0$. Therefore, the unit value of ${\eta }_{xx}$ corresponds to the bending stress in the static problem for ${\eta }_0=1$. Bending stress at the shoreline tends to 0 for ${\alpha }_i\to 90°$. Figure 6a shows that in Problem F waves of higher frequencies $\omega =0.5$ and $\omega =1$ influence smaller bending stresses at the shoreline.

Figure 6b shows the dependence of ${\eta }_{xx}$ from the incident angle ${\alpha }_i$ in Problem FC (reflection of the wave from a vertical wall when the ice sheet is split by a crack at $x=L_{1,ds}$). In this case the dependences ${\eta }_{xx}\left({\alpha }_i\right)$ look like it is shown in Figure 6a for $\omega =0.01$ and $\omega =0.1$. For higher wave frequencies $\omega =0.5$ and $\omega =1$ the dependences ${\eta }_{xx}\left({\alpha }_i\right)$ becomes nonmonotonic and include several local maxima and minima. The oscillations of ${\eta }_{xx}\left({\alpha }_i\right)$ are explained by the oscillations of the wave amplitude between the shoreline and the crack depending on the incident angle. Figure 7a shows the increase in wave amplitude for $x\in (0,L_{1,ds})$ in comparison with the unit amplitude of the wave for $x>L_{1,ds}$ with $\omega =1$ and ${\alpha }_i=30°$. Figure 7a shows decreasing of wave amplitude for $x\in (0,L_{1,ds})$ in comparison with the unit amplitude of the wave for $x>L_{1,ds}$ with $\omega =1$ and ${\alpha }_i=33.3°$. Figure 7 shows that the water surface elevation has a discontinuity at $x=L_{1,ds}$.

Figure 8 demonstrates that the amplitude of oscillations of ${\eta }_{xx}\left({\alpha }_i\right)$ becomes much higher in Problem FCC (reflection of the wave from a vertical wall when the ice sheet is split by two cracks at $x=L_{1,ds}$ and $x=L_{2,ds}$). The oscillations are not visible for low-frequency waves with the frequencies $\omega =0.01$ and $\omega =0.1$. The amplitude of oscillations is of about 0.1 when $\omega =0.5$ (Figure 8a) and increases up to 50 when $\omega =1$ (Figure 8b). Figure 8 illustrates that increasing of ${\eta }_{xx}$ is explained by the increase of the wave amplitude in the region $x\in (0,L_{1,ds})$. Figure 9a,b were constructed with the incident angles $9.15°$ and $18.6°$, whose values are closed to the local maxima of the function ${\eta }_{xx}\left({\alpha }_i\right)$. The local maximum at ${\alpha }_i=9.15°$ is greater than the local maximum at ${\alpha }_i=18.6°$ (Figure 8b). Therefore, in Figure 9 the wave amplitude in the region $x\in (0,L_{1,ds})$ is greater at ${\alpha }_i=9.15°$ than at ${\alpha }_i=18.6°$.

6. Edge Waves

Solutions describing edge waves were studied in the frequency range $\omega \in \left(\mathrm{0.5,4}\right)$. In the range of lower frequencies, the dispersion curves of edge waves approached very close to the dispersion curve of plane wave with $k_x=0$ propagating in the $y$-direction. The standard accuracy of simulations in Wolfram Mathematica software was not enough to investigate the solutions. Figure 10a shows points with frequencies and wave numbers of constructed edge waves in Problems C and CC corresponding to coastal geometries C and CC, as shown in Figure 4. They are practically coinciding with each other and sitting on the dispersion curve of a plane wave propagating in the $y$-direction. The wave numbers $k_{ew,y}$ of the edge waves are slightly greater than the wave numbers $k_y$ of the plane waves of the same frequencies. Differences in the frequencies of the plane waves and the frequencies of the edge waves with similar wavenumbers $k_{ew,y}$ are shown in Figure 10b, where blue and yellow points correspond to the edge waves in Problems C and CC.

The elevation of the ice sheet deformed by the edge waves propagating along a crack (Problem FC) is shown in Figure 11. Maximal amplitudes of the edge waves were set to the unit value. The amplitudes of the edge waves $\eta \left(x\right)$ with frequencies $\omega =0.5$, $1$, and $3$ decay respectively at the distances $x\approx 40000$ (Figure 11a), $x\approx 200$ (Figure 11c), and $x\approx 40$ (Figure 11d). In the dimensional variables these distances are equal to 513 km, 2.5 km, and 513 m. The elevation of ice sheet deformed by the edge waves propagating along two parallel cracks (Problem FCC) are shown in Figure 12. Maximal amplitudes of the edge waves were also normalized to the unit value. The amplitudes of the edge waves $\eta \left(x\right)$ with frequencies $\omega =0.5$, $0.8$, $1$, and $3$ decay respectively at the distances $x\approx 1000$ (Figure 12a), $x\approx 150$ (Figure 12b), $x\approx 70$ (Figure 12c), and $x\approx 30$ (Figure 12d). In the dimensional variables these distances are equal to 12.8 km, 1.9 km, 899 m, and 385 m.

Figure 13a shows the points on the plane $(\omega ,k_y)$ having the coordinates equal to the frequencies and wave numbers of the edge waves in Problems FC and FCC. They are also practically setting on the dispersion curve of a plane wave propagating in the $y$-direction. The wave numbers $k_{ew,y}$ of the edge waves are slightly greater the wave numbers $k_y$ of the plane waves of the same frequencies. Differences in the frequencies of the plane waves and the frequencies of the edge waves with the same wavenumbers $k_{ew,y}$ are shown in Figure 13b, where blue and yellow points correspond to edge waves in Problems FC and FCC.

The elevations of ice sheet deformed by the edge wave propagating along cracks parallel to the shoreline are shown in Figure 14a (Problem FC) and Figure 14b (Problem FCC). Black vertical lines indicate the crack locations. Maximal amplitudes of the edge waves were normalized to the unit value. In Problem FC the amplitude of the edge wave with frequencies $\omega =1$ and the edge waves with frequencies $\omega =3$, $4$, and $5$ decay respectively at the distances $x\approx 160$ and $x\approx 40$. In the dimensional variables these distances are equal to 2 km and 513 m. The shapes of the edge waves with the frequencies $\omega =3$, $4$, and $5$ are symmetric relative to the crack and similar to the shape of the edge waves propagating along a crack in Problem C (Figure 11).

The shapes of the edge waves propagating along two parallel cracks parallel to the coastline (Problem FCC) are shown in Figure 14b. Vertical lines indicate the crack locations at $x=L_{1,ds}$ and $x=L_{2,ds}$. Maximal amplitudes of the edge waves were also normalized to the unit value. The amplitude of the edge waves with frequencies $\omega =3.25$, $4$, and $5$ become close to zero near the edge of the second crack at $x=L_{2,ds}$. Thus, the energy of these waves is trapped between the shore and the second crack.

7. Discussion

The land-fast ice of the northern shelf of Alaska includes ice ridges and refrozen leads extended along the coastline (see Figure 1 and Figure 8 in [55]). Sea ice in refrozen leads is flatter and thinner than multiyear ice (see Figure 3 in [2]). Ice ridges can be grounded in some locations. The coastline looks straight. Local directions of isobaths, ice ridges and leads do deviate from the direction of the coastline, but generally maintain the average direction along the coastline. Refrozen ice in leads can be separated from multiyear ice by cracks. The idealized structure of land-fast ice considered in the paper is different from the structure of real land-fast ice. However, some of the effects studied in the paper may be relevant to real-life situations.

According to the results obtained here, a significant amplification of wave amplitudes in the land-fast ice may occur in the real situation due to the interaction of incoming waves with ice ridges, leads and cracks when the dimensionless wave frequency is greater 0.5, i.e., it may happen for swell with wavelength less 200 m and period less 12 s. Increasing the ice thickness to 1.5 m shifts the estimate of the swell period to less than 16 s. Such waves measured in different regions of the Arctic Ocean can propagate over long distances with small attenuation (Table 1). The amplification of swell amplitude on the shelf leads to increasing bending stresses in the land-fast ice. In our study maximal bending stresses occur at the coastline. This conclusion is based on the consideration of the fixed edge of the ice at the coastline supported by contact-boundary conditions (16). In reality, there is an ice foot partially setting at the bottom near the coastline. Therefore, it is of interest to estimate maximal bending stresses at some distance offshore from the ice foot. According to Formula (22) maximal bending stress caused by a periodic flexural wave is estimated with the formula ${\sigma }_{max}=0.5E{h}_{i}a{k}^2$, where $a$ is wave amplitude, $k=2\pi /l$ is wave number, and $l$ is wavelength.

Figure 15 shows the maximal bending stress calculated for three wavelengths $l=100$ m, $150$, and $200$ m versus wave amplitude $a$. The reasonable value of swell amplitude in the Arctic is 1 mm. Figure 15 shows that even if the amplification influences increase the wave amplitude 10 times, the maximal bending stress is smaller 100 kPa. Assuming the flexural strength of sea ice 300 kPa or higher [56] we may conclude that the level of amplification obtained in the paper can not influence the bending failure of land-fast ice by swell coming from the Arctic Ocean. The fatigue effect may influence the reduction of bending stress less than two times [57], but it is also not enough to break up the ice. Thus, the action of swell may lead to strengthening of land-fast ice [58].

The effect of redirection of waves caused by the interaction of incoming waves with grounded ice ridge was discussed in [2,55]. One of the discussed physical mechanisms explaining the effect is related to the generation of edge waves propagating along the lead. The frequency of the edge waves should be equal to the frequency of the incoming wave. In the present paper we demonstrated the existence of edge waves propagating along the cracks in land-fast ice. The energy of edge waves is trapped in relatively narrow region extended between the shoreline and the second crack. In the case of only one crack the energy of edge waves is trapped in much wider region extended offshore from the crack. We constructed edge wave solutions for relatively high wave frequencies ($\omega >3.25$) because of the numerical accuracy of the simulations. Using the shallow water approximation may help to investigate the properties of the edge wave of smaller frequencies.

The generation of the edge wave caused by the interaction of incident waves coming from offshore with grounded ice was not considered in the paper. Nevertheless, we may describe the structure of the solution describing this effect. In this case the velocity potential consists of two parts ${\phi }_1$ and ${\phi }_2$. The potential ${\phi }_1$ describing the incident and reflected waves is determined by the formula

$$\begin{gathered} {\phi }_1=e^{i\theta }\left[\gamma \left(a_{+}{e}^{i{k}_{0}x}+a_{-}{e}^{-i{k}_{0}x}\right){\displaystyle \frac{\cosh \left({\lambda }_0(z+H)\right)}{\cosh \left({\lambda }_{0}H\right)}}+ \\ {\displaystyle \frac{1}{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{dk}_x{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}d{k}_y\{{\displaystyle \frac{\cosh \left(\lambda (z+H)\right)}{\kappa \left(\omega ,\lambda \right)\cosh \left(\lambda H\right)}}{\sum }_{j=1}^N{P}_j(k)e^{ik{(x-x}_j)+i{k}_{y}y}\}\right], \\ {\theta =k}_{y}y-\omega t, \end{gathered}$$

(31)

where the coefficients $c_{ji}$ of the polynomials $P_j\left(k\right)$ introduced by Formula (8) depend on the wave number $k_y$. Formula (31) is the Fourier transform of solution (5), considering spatial changes in the potential ${\phi }_1$ along the $y$-axis caused by the waves scattered on the grounded ice. Symmetry conditions (20) are assumed to satisfy the condition of zero horizontal velocity in the normal direction to the shoreline at $x=0$.

The velocity potential ${\phi }_2$ describes the waves scattered on the grounded ice (Figure 16). To satisfy the symmetry condition at $x=0$ we consider two similar grounded ices set in symmetric locations with respect to the $y$-axis. The potential ${\phi }_2$ is a sum of the potentials ${\phi }_{s+}$ and ${\phi }_{s-}$ describing waves scattered on the grounded ices. The energy of scattered waves decay with the distance from the grounded ice. The potential ${\phi }_2$ is symmetric with respect to the axis $x=0$. In the case of the grounded ice of cylindrical shape, the potentials ${\phi }_{s\pm }$ are expressed by the series by the products of the Hunkel functions of type 1 describing radial dependence, sin-functions describing azimuthal dependence, and the vertical eigen modes describing the dependence of the potential from the vertical coordinate $z$ [59,60]. The coefficients of the series are found from the boundary conditions at the surface of the grounded ice and at the contact line of the ice sheet with the grounded ice. The total potential ${\phi }_1{+\phi }_2$ must satisfy these conditions.

The coefficients $c_{ji}\left(k_y\right)$ are found from the contact boundary conditions at the crack edges (12) and (13), where the ice surface elevation corresponding to the velocity potential ${\phi }_1{+\phi }_2$ is substituted. The system of linear algebraic equations with a forced term proportional to the amplitude of the incoming wave and the coefficients in the series determining the potential ${\phi }_2$ is obtained for the finding of the $c_{ji}\left(k_y\right)$. The coefficients $c_{ji}{(k}_y)$ are inversely proportional to the determinant $\mathsf{\Delta }\left(\omega ,k_y\right)$ of the corresponding system of homogeneous algebraic equations. From the dispersion Equation (21), it follows that for given value of the wave frequency $\omega$ the coefficients $c_{ji}\left(k_y\right)$ have two poles of different signs in the complex plane $k_y$ corresponding to the edge waves propagating in positive and negative directions of the $y$-axis. The amplitudes of the edge waves are determined by the calculation of integral (31) by the residue method in the regions $\mathrm{I}\mathrm{m}\left[k_y\right]>0$ for $y>0$, and $\mathrm{I}\mathrm{m}\left[k_y\right]<0$ for $y<0$.

Thus, the interaction of incoming waves with the grounded ice leads to the generation of the edge waves propagating along the cracks (Figure 16). The frequency of the edge waves is equal to the frequency of the incoming wave. This effect is also valid for land-fast ice with other type of linear irregularities admitting the existence of edge waves. For example, refrozen lead covered by thin ice or a canal with an open water surface with parallel edges are waveguides for edge waves [46,49]. It was shown that there are several modes of edge waves propagating along open water canal. Therefore, the interaction of incident waves with the grounded ice located near the lead or canal will influence the formation of edge waves of different modes. Deviations of the shapes of ice cracks and edges of refrozen leads from a rectilinear shape should affect the radiation of edge wave energy into surrounding waters covered with thicker ice.

In the absence of localized ice features edge waves are not generated by incident waves coming from offshore direction within the theory of linear waves even if the ice structure admits the existence of edge waves. It is because the wave numbers $k_y$ of incident and reflected waves are in Region I, and the wave number $k_y$ of edge waves of the same frequency is in Region II in Figure 2. Some redirections of incident and reflected waves may occur due to changes in ice thickness. The incident angle becomes larger (closer to the normal direction to the shoreline) in thinner ice. This effect is not significant for small variations in ice thickness.

8. Conclusions

We studied analytical solutions of hydrodynamic equations describing the interaction of flexural-gravity waves with land-fast ice with one and two cracks parallel to the straight shoreline. It has been found that the amplification of the wave amplitude in the coastal zone between the coastline and cracks depends on the angle of incidence of waves coming from offshore. The effect is significant for the waves with periods less than 16 s, ice thickness smaller than 1.5 m, and two cracks parallel to the coastline and extended from each other on 100 m. Considered solutions admit the existence of edge waves propagating along the coastline and attenuating offshore. The energy of edge waves is trapped between the shore and the ice cracks. The localization of wave energy between the coastline and cracks was found to be stronger in the case of two cracks parallel to the coastline.

Localized grounded ice ridges in the coastal zone may influence the partial redirection of the energy of incoming waves into the energy of edge waves propagating along the coastline. In the absence of localized ice features, edge waves are not generated by incident waves arriving from the offshore directions within the theory of linear waves, even if the ice structure admits the existence of edge waves. It is because the wave numbers of incident and reflected waves are in Region I, and the wave number of edge waves of the same frequency is in Region II in Figure 2. Some redirection of incident and reflected waves may occur due to changes in ice thickness. The incident angle becomes larger (closer to the normal direction to the shoreline) in thinner ice. This effect is not significant for slight variations in ice thickness.