2.1. The Sandyduck’97 near-shore experiment and methods of data analysis

The SandyDuck’97 near-shore field experiment, hosted by the FRF at Duck, NC (figure 1a), was aimed at capturing a comprehensive snapshot of all possible aspects of near-shore wave, circulation and sediment transport processes, and involved 30 groups of investigators from universities and institutions in the US and Canada (see Birkemeier, Long & Hathaway (Reference Birkemeier, Long and Hathaway1997) for an overview). A comprehensive archive of the data collected, as well as a detailed description of the different components of the experiment may be found on the FRF's data servers (Hathaway & Birkemeier Reference Hathaway and Birkemeier2004). Wind data used here were provided by the FRF meteorological station, located at the seaward end of the pier, at $x=585.20$ m, $y=517.30$ m and $z=19.36$ m, in the FRF coordinate system (figure 1). Offshore wave data used here were obtained by the FRF directional Waverider buoy located near the 17 m isobath at the point $x=3.87$ km, $y=-2.35$ km in the FRF coordinates (30 10.10’ N, 75 42.04’ W). Near-shore bathymetric data were surveyed during the experiment using the CRAB (Coastal Research Amphibious Buggy) over a near-shore ‘minigrid’, with 2–3 m resolution along 18 profile lines parallel to the $x$-axis, spaced in the instrumented area at 25 m, and extending from dune base to 400 m offshore. At the near-shore array location, the bathymetry was essentially cylindrical, with nearly straight and parallel isobaths (figure 1b,c). The beach profile may be roughly divided into two segments: a steep near-shore segment, $100< x<250$, with a slope ${\approx }0.028$, typically intensely reworked by waves; and a milder offshore one, $x\approx 350$, with a slope $\approx 0.013$. Each of these segments exhibited a bar, located at $x\approx 150$ m and $x\approx 350$ in figure 1(c). The bars were always present, although at shifting positions and with varying heights and shapes. The hydrodynamic data discussed here were collected by the near-shore array, which recorded flow velocity and pressure at 2 Hz continuously from August to December 1997. The array data used here were recorded by a subset of 35 triplets of collocated electromagnetic current meters (UV) and bottom-mounted pressure (P) sensors distributed on six along-shore lines between approximately the 1 and 6 m isobaths (figure 1). The along-shore spacing of the instruments in the array was not uniform, in order to maximize the coverage and eliminate co-array (lag space) redundancies (see, e.g. Davis & Regier Reference Davis and Regier1977; Sheremet et al. Reference Sheremet, Guza and Herbers2005 for the near-shore array).

Figure 1. Configuration of the near-shore array experiment at Sandyduck’97. (a) Atlantic coast at USACE FRF, Duck, NC. The coastline is approximately $20^\circ$ west of north. In the FRF coordinate system used here, $y$-axis is along shore, the $x$-axis direction is cross-shore and elevation data are referenced to the 1929 National Geodetic Vertical Datum (NGVD). The origin of the FRF system is the intersection of a shore-parallel baseline with the southern boundary of the FRF property. The $y$ coordinate of the landward end of the pier is 516.6 m. (b) Distribution of instruments of the near-shore array (only the along-shore lines shown), overlaid on bathymetry map (surveyed on 14 August 1997). Circles represent collocated pressure (P) sensors and electromagnetic current meters (U cross-shore, and V along-shore component of the horizontal flow velocity). (c) Cross-shore bathymetry profile (14 August 1997) at all the cross-shore minigrid survey lines; mean slopes: ${\approx }0.028$ for $100< x<250$; ${\approx }0.013$ for $x>250$ m.

2.2. Directional asymmetry of edge-wave spectra

The analysis presented here covers observations from August to November 1997. Wind speed and direction, as well as offshore wave parameters measured by the Waverider buoy at 17 m depth (significant height, period and direction) are available from the data repository at a resolution of approximately 30 min (figure 2); here, we just report the data. The general metocean conditions during the SandyDuck’97 experiment were not very energetic (figure 2). A few mild storms were recorded, associated with winds ${\approx }15\,{\rm m}\,{\rm s}^{-1}$ out of the NE, that generated offshore waves with peak significant heights $\approx$2 m and with peak periods typically between 8 and 10 s. The most energetic storm on 19 October was characterized by northerly winds with a speed ${\approx }20\,{\rm m}\,{\rm s}^{-1}$, offshore significant wave height $\approx$4 m and peak period 10 s.

Figure 2. FRF wind (a,b) and offshore wave (c–e) parameters recorded between August and November 1997. Wind data were collected by a meteorological station located at the shoreline end of the pier; the wave data – by the Datawell Waverider buoy, located approximately 4 km offshore, near the 17 m isobath. (a) Wind speed; (b) wind direction; (c) wave significant height; (d) wave peak period; (e) wave peak direction. Wind records are from the FRF Met station. The direction of both wind and offshore waves is given in meteorological convention, clockwise and ‘coming from’, but with respect to the upcoast direction ($20^\circ$ west of north, see figure 1). Downcoast (blue markers) and upcoast (red markers) events correspond to wind direction aligned by ${\pm }20^\circ$ with the along-shore axis; offshore wave heights less than 1.5 m; and swell propagating nearly shore normal, within a ${\pm }40^\circ$ window.

In agreement with the analysis of Herbers et al. (Reference Herbers, Elgar and Guza1995), in most records swell direction and variance are the determining factors for the structure of the IG wave band. The example in figure 3 shows IG field recorded on 24 August, 05:00 (all times given are local time). This event was characterized by a weak sustained wind blowing in the upcoast direction, with speed ${\approx }1.9\,{\rm m}\,{\rm s}^{-1}$ and with an offshore swell of 0.5 m significant wave height, with a peak period of 8.3 s. Mean currents recorded at the near-shore array had maximum speed $0.11\,{\rm m}\,{\rm s}^{-1}$, and a variable direction, mostly onshore (figure 3a). The overall downcoast direction of the IG waves estimated at four lines of the near-shore array (figure 3c–f) matches the direction of the offshore swell. The location of the centre of mass (peak values) of the wavenumber–frequency spectrum (figure 3c–f), mostly in the trapped wave domain, suggests that edge waves were a significant component of the near-shore IG field.

Figure 3. Example of observations where the overall edge-wave propagation direction matches the direction of the offshore swell, but is opposite to the wind (24 August, 05:00). (a) Near-shore array lines (grey rectangles) used for the wavenumber–frequency spectrum analysis. Circles indicate the position of the pressure and velocity sensors used in the analysis. Arrows represent mean currents. (b) Spectral density of pressure variance at the along-shore lines. Inset: schematic of wave and wind directions. (c–f) Wavenumber–frequency spectra at four along-shore array lines. To facilitate the reading of the scales of the edge waves, rather than using $\omega$ and $k$, the plots are given in $f$ (frequency, reciprocal of the wave period), and $\kappa =k/2{\rm \pi}$ (reciprocal of the along-shore wavelength). Dashed lines represent the trapped-wave interval ${\pm }[k_{min}^{E},k_{max}^{E}]$ (${\pm }k_{min}^{E}$ – red; ${\pm }k_{max}^{E}$ – blue). The $f$-$\kappa$ domain outside the trapped-wave interval ${\pm }[k_{min}^{E},k_{max}^{E}]$ is shaded white; shaded region A: waves that can escape into the deep ocean; shaded regions B and C: up- and downcoast-propagating trapped waves that have the turning point offshore of the array line. The table provides relevant wave (offshore and near shore) and wind parameters, where $V_{max}$ is the maximum current speed; $x$ is the cross-shore position of the line; $U$ is the wind speed; $H_s$ is significant height, and $H_s^{IG}$ is the significant height of the IG frequency band).

While the direction of IG waves matches the swell direction in most cases, anomalies also occur, in which the strong IG directional asymmetry agrees with the wind, rather than the swell direction. To find such anomalies, ‘favourable’ metocean conditions are defined here as satisfying the following conditions: (i) wind direction predominantly along shore: within a $20^{\circ }$ centred on the along-shore axis, and, (ii) low energy swells, with offshore significant height ${<}1$ m and direction nearly shore normal (${\pm }40^{\circ }$). Although these constraints are somewhat arbitrary, they are not used here with any other purpose than to eliminate highly unfavourable events. These criteria identify 60 events (figure 2; 41 upcoast, and 19 downcoast) out of the total of 3809 as having potentially anomalous directionality. The preponderance of upcoast favourable events reflects the local wave climate and the orientation of the coast.

We present here two examples, showing upcoast- and dowcoast-propagating IG fields, respectively. In both cases discussed below the offshore swells are nearly shore normal. If wind did not play any role in the generation of the IG wave field, one would expect to see approximately directionally symmetric IG wavenumber–frequency spectra; i.e. low resolution maximum entropy estimates should show a spectral centre of mass (peak) located roughly at $\kappa =0.$ In fact, both cases exhibit strong directional asymmetry that agrees with the direction of the wind. We stress that, in both the cases, IG propagation direction is against the near-shore current.

The downcoast event shown in figure 4 occurred on 30 November, 16:00. The meteorological and offshore wave conditions during it were characterized by winds ${\approx }6\,{\rm m}\,{\rm s}^{-1}$ blowing downcoast, and offshore long swell of peak period 10.5 s and significant height 1.16 m, propagating nearly perpendicular to the coast (figure 4(b) and table). Mean currents at the near-shore array have a maximum speed $0.16\,{\rm cm}\,{\rm s}^{-1}$, flowing onshore at the deepest line ($x=385$ m) and rotating slightly upcoast at the shallower array lines (figure 4a). As before, the location of the centre of mass of the wavenumber–frequency spectrum suggests significant edge-wave content (figure 4c–f).

Figure 4. Example of observations of downcoast-propagating edge waves in the wind direction, with swells being normally incident (30 November, 16:00). Notations and conventions are the same as in figure 3.

The upcoast event shown in figure 5 occurred on 5 September, 11:00, during a ${\approx }5\,{\rm m}\,{\rm s}^{-1}$ upcoast wind and in the presence of an offshore short swell with peak period ${\approx }7$ s, significant height ${\approx }1.34$ m; (figure 5(a) and table). Mean currents at the near-shore array were less than $11\,{\rm cm}\,{\rm s}^{-1}$, flowing overall downcoast (figure 5a). The wavenumber–frequency spectra exhibit a combination, at different frequencies, of free and trapped power, the latter showing strong upcoast directional asymmetry.

Figure 5. Example of upcoast-propagating edge waves, matching the direction wind, with normally incident swells (observations on 5 September, 11:00). Notations and conventions are the same as in figure 3.

2.3. Preliminary conclusions

The events discussed above are examples of cases where the dominant propagation direction of the mostly trapped IG wave field is clearly correlated to wind direction, but seems disconnected from the direction of the swell (shore normal) and along-shore currents. We caution that this ‘associative’ logic might be misleading in general, because it assumes that the forcing (by either wind or waves) varies slowly and the IG wave system is at quasi-equilibrium. However, figures 4 and 5 represent only ‘1 h snapshots’ (processed time span) of the wave and wind fields. A directional misalignment might occur, for example, if the analysis window coincided with a major shift in swell direction, but before the IG wave field had time to respond. A quick examination of the details of the wind and wave fields before the events (figure 6), however, shows no significant shifts in the swell direction in the day before the selected events, which supports the suggestion that for these events wind, rather than swell, is the dominant forcing of the observed edge-wave wave field. Alternative interpretations of the reported asymmetry of edge-wave propagation are discussed in § 5.

Figure 6. Wind and wave parameters the day before the events analysed in figures 4 and 5 (same organization as in figure 2).

3.1. Formulation of the problem

To our knowledge the possibility of wind generation of edge waves has never been examined theoretically. Here, we address this gap by considering mathematical models of direct edge-wave excitation by wind. To this end we adopt the idealizations outlined in the previous section: we assume the coastline to be straight and infinite, the bathymetry to be cylindrical, i.e. we allow the water depth to depend only on the distance from the straight shore. We employ the Cartesian coordinate frame with the horizontal $y$ axis directed along the coastline, $x$ axis directed off shore and the vertical $z$ axis directed upwards. The edge waves are described in the shallow water approximation. We use the following notations: $u$ and $v$ are the offshore and the along-shore fluid velocity components, $H(x)$ is the undisturbed water depth, $p_a$ is total normal stress caused by the action of the atmosphere at the water surface and $\xi$ is the water surface elevation. Then the equations of motion can be written in the form

(3.1)\begin{gather} \frac{\partial u}{\partial t} + v\frac{\partial u}{\partial y} + u\frac{\partial v}{\partial x}+ g\frac{\partial \xi}{\partial x}={-}\frac{1}{\rho} \frac{\partial p_a}{\partial x}, \end{gather}

(3.2)\begin{gather} \frac{\partial v}{\partial t} + v\frac{\partial u}{\partial y} + u\frac{\partial v }{\partial x}+ g\frac{\partial \xi}{\partial y} ={-}\frac{1}{\rho} \frac{\partial p_a}{\partial y}, \end{gather}

(3.3)\begin{gather} \frac{\partial \xi}{\partial t}+\frac{\partial}{\partial y}v[\xi+H(x)]+\frac{\partial}{\partial x}u[\xi+H(x)]=0. \end{gather}

We confine our consideration to the linear setting. As shown by Longuet-Higgins (Reference Longuet-Higgins1969a), the effect of an oscillating shear stress at the water surface $T_a$ in a viscous fluid in the presence of a propagating surface wave is equivalent to that of the normal stress with ${\rm \pi} /2$ phase lag relative to $T_a$. Here, $p_a$ in ((3.1) and (3.2)) is a sum of the direct normal stress at the water surface and the contribution of the shear stress. On linearizing system ((3.1)–(3.3)), we consider an infinitesimal plane wave propagating along the coastline

(3.4)\begin{gather} \frac{\partial u}{\partial t}+ g\frac{\partial \xi}{\partial x}={-}\frac{1}{\rho} \frac{\partial p_a}{\partial x}, \end{gather}

(3.5)\begin{gather} \frac{\partial v}{\partial t}+ g\frac{\partial \xi}{\partial y}={-}\frac{1}{\rho} \frac{\partial p_a}{\partial y}, \end{gather}

(3.6)\begin{gather} \frac{\partial \xi}{\partial t}+ H(x) \left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)+u\frac{{\rm d}H}{{\rm d}\kern0.7pt x}=0. \end{gather}

We perform Fourier transform with respect to along-shore coordinate $y$ and time $t$. For each harmonic component specified by a particular wavenumber $k$ the wave velocities, the elevation of the water surface and the normal stress can be presented as products of a function specifying the cross-shore dependence which we mark by a hat symbol and ${\rm e}^{-{\rm i}(\omega t - ky)}$

(3.7)\begin{equation} \left( \begin{array}{@{}ll} u \\ v \\ \xi \\ p_a \end{array} \right) = \left( \begin{array}{@{}ll} \hat U(x) \\ \hat V(x) \\ \hat \varXi(x) \\ \hat P_a (x) \end{array} \right) {\rm e}^{-{\rm i}(\omega t - ky)}. \end{equation}

Edge-wave cross-shore dependence is determined by the boundary value problem

(3.8)\begin{equation} gH(x)\left(\frac{{\rm d}^2 \hat\eta}{{{\rm d}\kern0.7pt x}^2}-k^2 \hat\eta\right)+ g \frac{{\rm d}H}{{\rm d}\kern0.7pt x} \frac{{\rm d} \hat \eta}{{\rm d}\kern0.7pt x} + \omega^2 \hat\eta = \frac{\omega^2 \hat P_a (x)}{\rho g}, \quad \hat\eta(0)< \infty, \quad \eta(\infty)=0, \end{equation}

where

(3.9)\begin{equation} \hat\eta(x) = \hat\varXi(x) + \frac{\hat P_a(x)}{\rho g}. \end{equation}

To fix the idea we consider the simplest model of bathymetry. For the constant slope, $H(x)= \gamma x$, the explicit solution corresponding to the fundamental mode of edge waves is remarkably simple

(3.10a–c)\begin{equation} \hat\eta (x) = a{\rm e}^{{-}kx},\quad \hat P_a (x) = P_{a0} {\rm e}^{{-}kx}, \quad \omega^2 - \gamma g k = \frac{\omega^2}{\rho ga} P_{a0}. \end{equation}

It differs from the similar formulae for free edge waves in Appendix A by the presence of the term $({\omega ^2}/{\rho ga}) P_{a0}$ which accounts for the interaction with atmosphere. Since it is of the order of the air-to-water density ratio we can neglect its effect on the real part of the eigenvalue and present the wave frequency and the growth rate as follows:

(3.11)\begin{gather} \omega_0 = \sqrt{\gamma g k}, \end{gather}

(3.12)\begin{gather} \frac{\mathrm{Im} \omega}{\omega_0} = \frac{1}{2 \rho g} \mathrm{Im} \frac{P_{a0}}{a}. \end{gather}

At first glance the above problem set-up strongly resembles the formulation for the Greenspan (Reference Greenspan1956) resonance, where an edge wave in resonance with a moving pressure distribution is generated by a moving, given, atmospheric pressure disturbance. The key difference with our set-up is that, in contrast to Greenspan (Reference Greenspan1956), there is no prescribed atmospheric pressure disturbance, $P_{a0}$ is not given, but has to be found by solving the linear boundary value problem (3.8)–(3.12) in the absence of atmospheric forcing. Here, the challenge is to find and quantify instability mechanisms producing noticeable growth rates $\mathrm {Im}\, \omega > 0$. We identify three a priori seemingly plausible options which we examine:

1. (i) Edge-wave dispersion relation for any bathymetry allows the waves to have a critical layer in the wind flow. Therefore, the obvious candidate mechanism of edge-wave generation to be considered is an analogue of Miles’ mechanism of wind wave generation. This mechanism is examined in Appendix B for the main mode in the constant slope model; the growth rate is shown to be identically zero.

2. (ii) Since the edge waves are essentially quasi-horizontal motions, the shear stresses caused by the coupled atmospheric flow potentially might cause an instability if the turbulent viscosity in the boundary layer in the air is taken into account. This possibility is examined in Appendix C. It is shown that with viscous effects taken into account there is no instability in the system of edge waves coupled with atmospheric flow.

3. (iii) It is known (Longuet-Higgins Reference Longuet-Higgins1969b) that the effect of strongly wind-forced short free waves (of short gravity and gravity–capillary range) is equivalent to a viscous tangential stress at the surface. Since such short waves are strongly affected by orbital velocities of edge waves, these stresses are modulated by edge waves. Interaction of random free short waves with an edge-wave field might lead to self-excitation of a coherent edge wave. In the next section we examine such a ‘maser’ mechanism and show that, indeed, such self-excitation can occur. We will also provide rough estimates of its efficiency.

4.1. Edge-wave excitation by wind in the model with a quasi-harmonic wind wave

We start with examining the modulation of the amplitude and wavenumber of a short quasi-monochromatic wind wave by an edge wave. In the laboratory frame the local dispersion relation linking the short-wave frequency $\varOmega$ with the wave vector $\boldsymbol K (K_x, K_y)$ riding upon the edge wave with the account of the Doppler shift due to the orbital velocity $\boldsymbol U (U, V)$ is

(4.2)\begin{equation} \varOmega = \sqrt{gK + \sigma K^3} + UK_x + VK_y, \quad K = \sqrt{K^2_x + K^2_y}, \end{equation}

where $\sigma$ is the surface tension coefficient. In the reference frame following the edge wave with phase velocity $c$ this dispersion relation takes the form

(4.3)\begin{equation} \varOmega ={-}cK_y + \sqrt{gK + \sigma K^3} + UK_x + VK_y, \quad K = \sqrt{K^2_x + K^2_y}. \end{equation}

Recall that $U$ and $V$ are the edge-wave orbital velocity components

(4.4)\begin{equation} (U,V)= \frac{gka}{\omega} {\rm e}^{{-}kx} (-\sin ky, \cos ky ). \end{equation}

The standard WKB (Wentzel–Kramers–Brillouin) equations for the wavevector $\boldsymbol K (K_x, K_y)$ and position vector $\boldsymbol {r}$ of the short-wave wavepacket (e.g. Phillips Reference Phillips1977)

(4.5a,b)\begin{equation} \frac{{\rm d} \boldsymbol{K}}{{\rm d}t} ={-} \frac{\partial \varOmega}{\partial \boldsymbol{r}}, \quad \frac{{\rm d} \boldsymbol{r}}{{\rm d}t} = \frac{\partial \varOmega}{\partial \boldsymbol{K}}, \end{equation}

in our context take the form

(4.6)\begin{equation} \left. \begin{gathered} \frac{{\rm d}K_x}{{\rm d}t}=({-}K_x \sin ky + K_y \cos ky ) \frac{gak^2}{\omega} {\rm e}^{{-}kx}, \quad \frac{{\rm d}K_y}{{\rm d}t}=(K_x \cos ky + K_y \sin ky ) \frac{gak^2}{\omega} {\rm e}^{{-}kx}, \\ \frac{{\rm d}y}{{\rm d}t}={-}c + \frac{gak}{\omega} {\rm e}^{{-}kx} \cos ky + C_{gry}, \quad \frac{{\rm d}\kern0.7pt x}{{\rm d}t}={-}\frac{gak}{\omega} {\rm e}^{{-}kx} \sin ky + C_{grx}. \end{gathered} \right\} \end{equation}

Here,

(4.7a–c)\begin{equation} C_{gry} = \frac{K_y}{K}C_{gr},\quad C_{grx} = \frac{K_x}{K}C_{gr}, \quad \left(C_{gr} = \frac{{\rm d}}{{\rm d}K}\sqrt{gK + \sigma K^3}\right) \end{equation}

are the components of the intrinsic group velocity of the short wind waves. For simplicity only, consider a gravity–capillary wave, which initially has the wave vector directed along $y$: $\boldsymbol {K}(t=0) = (0, K_0)$. Then, in the linear approximation in the amplitude of the edge wave ($ka$), the solution to system (4.6) is

(4.8)\begin{gather} K_y = K_0 + K_0 \frac{gk^2 a}{\omega} \frac{\cos{ky}}{\omega - C_{gr0} k} {\rm e}^{{-}kx}, \end{gather}

(4.9)\begin{gather} K_x ={-}K_0 \frac{gk^2 a}{\omega} \frac{\sin ky}{\omega - C_{gr0} k} {\rm e}^{{-}kx}. \end{gather}

Here, $C_{gr0}$ is the group velocity of a short wave with the initial wave vector $K_0$

(4.10)\begin{equation} C_{gr0} = \frac{g + 3 \sigma K^2_0}{2 \sqrt{gK_0 + \sigma K^3_0}}. \end{equation}

Modulation of the short-wave amplitude can be easily found from the conservation of the wave action $N$

(4.11)\begin{equation} N = \frac{F}{\varOmega - (V - c)K_y - UK_x}, \end{equation}

where $\varOmega$ is given by (4.3), and

(4.12)\begin{equation} F= \frac{(\varOmega - (V - c)K_y - UK_x)^2}{2K}|A|^2, \end{equation}

is the intrinsic energy of the short wave with amplitude $|A|$. To the first order in $ka$ the Taylor expansion of (4.3) gives for the short-wave wave action

(4.13)\begin{equation} N = C_{f0} \left(1 +\varepsilon \frac{C_{gr0} - C_{f0} }{C_{f0}} \cos{ky} {\rm e}^{{-}kx} \right) |A|^2, \end{equation}

where

(4.14)\begin{equation} \varepsilon = \frac{gk^2 a}{\omega} \frac{1}{\omega - C_{gr0}k}, \end{equation}

and $C_{f0}=\sqrt {{g}/{K_0}+\sigma K_0}$ is the intrinsic phase velocity of the short wave. In the linear approximation in $ka$, (4.14) and (4.8) yield the following closed expression for the slope of the short wave:

(4.15)\begin{equation} (KA)^2 = (K_0 A_0)^2 \left( 1 + \frac{3C_{f0} - C_{gr0}}{C_{f0}} \frac{gk^2 a}{\omega(\omega - C_{gr0}k)} \cos ky {\rm e}^{{-}kx} \right). \end{equation}

The shear stress due to the wind-forced short wave is given by (4.1). The wave frequency in the linear approximation can be found from (4.3). To the first order in $ka$ we have

(4.16)\begin{equation} \varOmega = \varOmega_0 \left( 1 + \varepsilon \frac{C_{gr0}}{C_{f0}}\cos{ky}{\rm e}^{{-}kx}\right), \end{equation}

where $\varOmega = \sqrt {gK + \sigma K^3}$ is the frequency of the gravity–capillary wave with wavenumber $(0,K_0)$. The effective shear stress created as a result of interaction with the edge wave reads

(4.17)\begin{equation} \boldsymbol{\tau}_{wave} = 2 \rho \nu (KA)^2 \varOmega = 2 \rho \nu (K_0 A_0)^2 \varOmega_0 \left( 1 + \frac{3 gk^2 a}{\omega(\omega - C_{gr0}k)} \cos ky {\rm e}^{{-}kx} \right) \boldsymbol{y_0}, \end{equation}

where $\boldsymbol {y_0}$ is the unit vector in the $y$ direction. For the adopted simplest model with the constant slope bathymetry $H = \gamma x$ and the edge-wave dispersion relation $\omega = \sqrt {\gamma g k}$, we can now easily find the expression for the component of the shear stress acting upon the edge wave

(4.18)\begin{equation} \tau_{1y} = 6 \rho \nu (K_0 A_0)^2 \varOmega_0 \frac{gka}{\gamma (\omega - C^{(0)}_{gr}k)}. \end{equation}

Employing this expression for the shear stress we can finally obtain from (3.12) the normalized growth rate $\omega _1/\omega _0$ of the edge wave of frequency $\omega _0, \,\omega _0 = \sqrt {\gamma g k}$

(4.19)\begin{equation} \frac{\omega_1 }{\omega_0} = \frac{3 \nu (K_0 A_0)^2 \varOmega_0 k^2}{\gamma \omega^2_0 (1 - C^{(0)}_{gr}k / \omega_0)}=\frac{3 \nu (K_0 A_0)^2 \varOmega_0 k}{\gamma^2 g (1 - C^{(0)}_{gr}k / \omega_0)} . \end{equation}

On the basis of this formula a rough estimate of the edge-wave relative growth rate $(\omega _1 / \omega _0)$ for typical parameters ($\nu = 10^{-6}\,\textrm {m}^2\,\textrm {s}^{-2}$, $\gamma \sim 10^{-3}-10^{-2}$, $k \sim 10^{-2}\,\textrm {m}^{-1}$, $C_{gr0} \sim 0.5\,\textrm {m}\,\textrm {s}^{-1},$ ) yields

(4.20)\begin{equation} \omega_1 / \omega_0 \sim 10^{{-}2} (K_0 A_0)^2. \end{equation}

Since typical short gravity and gravity–capillary waves are often rather steep (e.g. Troitskaya et al. (Reference Troitskaya, Sergeev, Kandaurov, Baidakov, Vdovin and Kazakov2012), where $|A_0K_0 | \sim 0.23$), then $\omega _1 / \omega _0 \sim 10^{-3}$, which is quite substantial. However, the estimates strongly depend on the local conditions.

Obviously, the assumption of quasi-monochromatic short wind wave field adopted in the considered toy model is totally unrealistic, it was needed as a first step to fix the idea. The rough estimate it provides has a limited purpose, it suggests that edge-wave generation by wind via short wind waves is indeed feasible and, hence, a consideration of a more realistic model is warranted. Such a model is put forward in the next section.

4.2. Self-excitation of an edge wave interacting with a continuous spectrum of random short wind waves

Here, we consider a reasonably realistic model where short wind waves are described as an arbitrary continuous spectrum of nonlinearly interacting random weakly nonlinear waves, while the edge-wave field is assumed to be linear. The evolution of interacting random weakly nonlinear short gravity and gravity–capillary waves is described using the kinetic equation for the spectral density of wave action $N$ (e.g. Keller & Wright Reference Keller and Wright1975)

(4.21)\begin{equation} \frac{\partial N}{\partial t} + \frac{\partial \varOmega}{\partial \boldsymbol{K}} \frac{\partial N}{\partial \boldsymbol{r}} - \frac{\partial \varOmega}{\partial \boldsymbol{r}} \frac{\partial N} {\partial \boldsymbol{K}} = \beta[U,V]N + Int[N], \end{equation}

where the term $\beta N$ comprises energy input due to the interaction with wind and sinks due to dissipation, $\varOmega$ is the short-wave frequency specified by the dispersion relation (4.3), $\boldsymbol {K}$ is the short-wave vector and $Int[N]$ is the short-wave nonlinear interaction term. The full collision integral $Int[N]$, which for this range of scales has not been rigorously derived yet for the situations where both the triad and quartic interactions are important. The collision integral for the gravity range derived by Hasselmann (Reference Hasselmann1962) (e.g. Komen et al. Reference Komen, Cavaleri, Donelan, Hasselmann, Hasselmann and Janssen1994) is expected to be valid when the main effect is due to the waves of gravity range. There exist a number of parameterizations of the collision integral for the gravity–capillary range (see e.g. Kosnik, Dulov & Kudryavtsev Reference Kosnik, Dulov and Kudryavtsev2010, and reference therein). Here, instead of considering the full collision integral $Int[N]$, which for this range of scales remains to be properly derived, the nonlinear interaction of short gravity and gravity–capillary waves will be described by the simplest relaxation model commonly referred to as the $\tau$-approximation (e.g. Keller & Wright Reference Keller and Wright1975). This relaxation model is a commonly used approximation of the nonlinear interaction term based on the assumption that the spectrum of short waves differs little from the equilibrium spatially uniform spectrum $N_0(\boldsymbol {K})$ and is only slightly perturbed by the edge wave. The equilibrium spectrum $N_0(\boldsymbol {K})$ is determined by the balance between the input and relaxation specified by the right-hand side of the kinetic equation, i.e.

(4.22)\begin{equation} \beta[U, V]N^{(0)} + Int[N^{(0)}] = 0. \end{equation}

Consider a small spatially homogeneous perturbation $N_1(\boldsymbol {K})$ of the equilibrium spectrum $N = N^{(0)} + N^{(1)}$. In virtue of our assumption $N^{(1)} \ll N^{(0)}$, the kinetic equation can be linearized, which yields

(4.23)\begin{equation} \frac{\partial N^{(1)}}{\partial t} = \frac{\delta}{\delta N^{(0)}}\beta[N^{(0)}]N^{(1)}N^{(0)} + \beta[N^{(0)}]N^{(1)} + \frac{\delta}{\delta N^{(0)}}Int[N^{(0)}]N^{(1)}. \end{equation}

This equation predicts that all perturbations are decaying exponentially

(4.24)\begin{equation} N^{(1)} = N^{(1)}_0 (K) \exp (-\beta_r (K)t), \end{equation}

where $\beta _r (\boldsymbol {K})$ is the relaxation rate for the wave component with the wavevector $\boldsymbol {K}$. The inverse of $\beta _r (\boldsymbol {K})$ is the relaxation time $\tau$ which gave the name to this approach that originated in the kinetics of gases (Bhatnagar, Gross & Krook Reference Bhatnagar, Gross and Krook1954). The substitution of the spatially homogeneous solution (4.24) into (4.23) yields

(4.25)\begin{equation} -\beta_r N^{(1)}_0 = \left[ \frac{\delta}{\delta N^{(0)}}\beta[N^{(0)}]N^{(0)} + \beta[N^{(0)}] + \frac{\delta}{\delta N^{(0)}} Int [N^{(0)}] \right] N^{(1)}, \end{equation}

which means that the decay rate $\beta _r$ is the eigenvalue of the operator in square brackets. Equation (4.25) yields the approximation of the perturbed ‘collision integral’ we employ

(4.26)\begin{equation} \frac{\delta}{\delta N^{(0)}} Int [N^{(0)}] N^{(1)} = \left[ - \frac{\delta}{\delta N^{(0)}}\beta[N^{(0)}]N^{(0)} - \beta[N^{(0)}] - \beta_r \right] N^{(1)}, \end{equation}

which, strictly speaking, can be justified only for the spatially homogeneous perturbation. Following Keller & Wright (Reference Keller and Wright1975), we will use this expression for small, $N^{(1)},\,(N^{(1)}/ N^{(0)}\ll 1)$, weakly inhomogeneous perturbations of scales far exceeding those of gravity–capillary waves. Following Keller & Wright (Reference Keller and Wright1975), we also assume that $\beta _r$ is equal to the short-wave wind input $\beta _{wind} N^{(0)}$. Then, the kinetic equation for the perturbations of the wave action spectral density in the presence of edge waves takes the form

(4.27)\begin{equation} \frac{\partial N^{(1)} }{\partial t} + \frac{\partial \varOmega^{(0)}}{\partial \boldsymbol{K}} \frac{\partial N^{(0)}} {\partial \boldsymbol{r}} - \frac{\partial \varOmega^{(0)}}{\partial \boldsymbol{r}} \frac{\partial N^{(0)}} {\partial \boldsymbol{K}} ={-}\beta_{wind} (N^{(0)}) N^{(1)} , \end{equation}

were $\varOmega ^{0} = \sqrt {gK + \sigma K^3}$ is the unperturbed frequency of the gravity–capillary wave with wave vector $\boldsymbol {K}$. It is more convenient to use the polar form for $\boldsymbol {K}=(K\sin \theta, K\cos \theta$ characterized by its modulus ${K}$ and polar angle $\theta$. Making use of the dispersion relation (4.3), we re-write our kinetic equation (4.27) in the form

(4.28)$$\begin{gather} \frac{\partial N^{(1)} }{\partial t} + \frac{C^{(0)}_{gr}}{K}(\boldsymbol{K} \boldsymbol{\nabla}) N^{(1)} - \frac{1}{K} \frac{\partial N^{(0)} }{\partial K} \left( K_y \frac{\partial (\boldsymbol{U}\boldsymbol{K})}{\partial y} + K_x \frac{\partial (\boldsymbol{U}\boldsymbol{\cdot}\boldsymbol{K})}{\partial x} \right) \nonumber\\ - \frac{1}{K^2} \frac{\partial N^{(0)} }{\partial \theta} \left( - K_x \frac{\partial (\boldsymbol{U}\boldsymbol{\cdot}\boldsymbol{K})}{\partial y} + K_y \frac{\partial (\boldsymbol{U}\boldsymbol{\cdot}\boldsymbol{K})}{\partial x} \right) ={-} \beta_{wind} (N^{(0)}) N^{(1)}, \end{gather}$$

where,

(4.29a,b)\begin{equation} \boldsymbol{U}=(U,V), \quad \boldsymbol{U}=\boldsymbol{U_0}\exp({-{\rm i}(\omega t - ky) -kx}), \end{equation}

$C^{(0)}_{gr}= \textrm {d} \varOmega ^{(0)} / \textrm {d}K$ and $\beta _{wind}N^{(0)}$ is the input minus dissipation in the absence of edge waves. Since the problem is linear with respect to the edge-wave amplitude, it is convenient to present the fields in complex form. Then, taking into account (4.3)

(4.30)\begin{equation} \boldsymbol{U_0} = \frac{gka}{\omega} (i, 1). \end{equation}

We consider the perturbations of wave action spectral density $N^{(1)}$ in the form of a harmonic inhomogeneous wave repeating the structure of the orbital velocity field in the edge wave

(4.31)\begin{equation} N^{(1)} = N^{(1)}_0 \exp({-{\rm i}(\omega t - ky)-kx}). \end{equation}

Then, from the kinetic equation (4.27) we obtain the following expression for the perturbation amplitude $N^{(1)}_0$:

(4.32)\begin{equation} N^{(1)}_0 ={-} \frac{gk^2 a}{\omega^2} \frac{(K_y+{\rm i}K_x)^2}{(1- C^{(0)}_{gr}(k / \omega) (K_y + {\rm i}K_x) / K + \beta {\rm i} / \omega)} \frac{1}{K} \left(\frac{\partial N^{(0)}}{\partial K} + {\rm i} \frac{1}{K} \frac{\partial N^{(0)}}{\partial \theta} \right). \end{equation}

Since $K_y = K \cos \theta$, $K_x = K \sin \theta$, we can present the above expression (4.32) in a more compact form

(4.33)\begin{equation} N^{(1)}_0 ={-} \frac{gk^2 a}{\omega^2} \frac{K^2 {\rm e}^{2{\rm i} \theta} }{(1- C^{(0)}_{gr}(k / \omega) {\rm e}^{{\rm i} \theta} + \beta {\rm i} / \omega)} \frac{1}{K} \left(\frac{\partial N^{(0)}}{\partial K} + {\rm i} \frac{1}{K} \frac{\partial N^{(0)}}{\partial \theta} \right). \end{equation}

Taking into account the relationship between the wave action spectral density $N$, the spectrum of surface elevations $F$ ($N=F/ \varOmega$) and the slope spectra $S=FK^2$ (4.33), we can readily obtain an expression for the perturbation of the slope spectra $S^{(1)}$ caused by the edge wave

(4.34)\begin{equation} S^{(0)} = S^{(1)}_0 \exp({-{\rm i}(\omega t - ky)-kx} ). \end{equation}

Then, the complex amplitude $S^{(1)}_0$ is

(4.35)\begin{equation} S^{(1)}_0 ={-} \frac{gk^2 a}{\omega^2} \frac{\omega K {\rm e}^{2{\rm i} \theta} } {(\omega - k C^{(0)}_{gr}{\rm e}^{{\rm i} \theta} + \beta {\rm i} )} \left(\frac{\partial S^{(0)}}{\partial K} - \frac{S^{(0)}}{K} \left( \frac{C^{(0)}_{gr}}{C^{(0)}_{ph}} + 2 \right) + {\rm i} \frac{1}{K} \frac{\partial S^{(0)}}{\partial \theta} \right), \end{equation}

and $C^{(0)}_{ph} = {\varOmega ^{(0)} }/{K}$. We can now calculate the spectral density of the perturbed tangential stress $\boldsymbol {\tau }^{(1)}$

(4.36a,b)\begin{equation} \boldsymbol{\tau}^{(1)} (\boldsymbol{K}) = \boldsymbol{\tau}^{(1)}_0 (\boldsymbol{K}) \exp({-{\rm i}(\omega t \!-\! ky)-kx}),\quad \left( \boldsymbol{\tau}^{(1)}_0 (\boldsymbol{K}) = 2\rho \nu S^{(1)}_0 \varOmega (\boldsymbol{y}_0 \cos \theta \!+\! \boldsymbol{x}_0 \sin \theta)\right), \end{equation}

where $\boldsymbol {y}_0$ and $\boldsymbol {x}_0$ are unit vectors in the along-shore and offshore directions. The integration of (4.36a,b) over the short-wave wavevectors yields the surface shear stress caused by the short-wave spectrum

(4.37)\begin{equation} \boldsymbol{\tau}_{10}= 2\rho \nu \int S^{(1)}_0 (\boldsymbol{y}_0 \cos \theta + \boldsymbol{x}_0 \sin \theta) \sqrt{gK +\sigma K^3} K\,{\rm d}K\,{\rm d} \theta. \end{equation}

For rough estimates we assume that the equilibrium short-wave spectrum is the saturated Phillips spectrum (Phillips 1958)

(4.38)\begin{equation} S^{(0)} = \frac{\alpha \cos (\theta - \theta_0)^2}{K^2}, \end{equation}

where $\theta _0$ is the wind direction with respect to the $y$ axis which can vary from $-{\rm \pi} / 2$ to ${\rm \pi} / 2$, while the Phillips constant $\alpha = 0.02$. Then, making use of the dispersion relation for edge waves, we find from (4.35) the complex amplitude of the perturbation of the short-wave field slopes in the form

(4.39)\begin{equation} S^{(1)}_0 = \frac{k a \omega}{\gamma} {\rm e}^{2{\rm i} \theta} \dfrac{\left( \left( 4 - \dfrac{C^{(0)}_{gr}}{C^{(0)}_{ph}} \right) \cos (\theta - \theta_0)^2 -{\rm i} \sin 2(\theta - \theta_0)\right) }{(\omega - k C^{(0)}_{gr} {\rm e}^{{\rm i} \theta}) + \beta {\rm i} } \frac{\alpha}{K^2}. \end{equation}

From the definition of tangential stress (4.36a,b) it follows

(4.40)\begin{gather} \boldsymbol{\tau}^{(1)}_{0} = \frac{2 \rho \nu \omega \alpha}{\gamma} ka \boldsymbol{\cdot} \end{gather}

(4.41)\begin{gather} \int^{K_{max}}_{K_{min}}\frac{{\rm d}K}{K} \int^{{\rm \pi} / 2}_{- {\rm \pi}/ 2} {\rm e}^{2{\rm i} \theta} \frac{ \left( 4 - \dfrac{C^{(0)}_{gr}}{C^{(0)}_{ph}} \right)\cos (\theta - \theta_0)^2 - {\rm i} \sin 2(\theta - \theta_0) }{(\omega - k C^{(0)}_{gr} {\rm e}^{{\rm i} \theta}) + \beta {\rm i} }\nonumber\\ (\boldsymbol{y}_0 \cos \theta + \boldsymbol{x}_0 \sin \theta) \sqrt{gK +\sigma K^3} \,{\rm d} \theta. \end{gather}

Since the spatial distribution of short-wave packets riding on edge waves is modulated by the edge waves, this creates the tangential stress which is phase linked to the edge wave and, thus, might lead to selection and self-excitation of the selected edge wave. The growth rate of the edge wave can be calculated employing (3.12), which yields

(4.42)$$\begin{gather} \frac{\mathrm{Im} \omega}{ \omega} =\frac{\nu \omega \alpha k}{ g \gamma^2 } \mathrm{Re} \int^{K_{max}}_{K_{min}} \frac{{\rm d}K}{K} \int^{{\rm \pi} / 2}_{- {\rm \pi}/ 2} {\rm e}^{3{\rm i} \theta} \frac{\left[ \left( 4 - \dfrac{C^{(0)}_{gr}}{C^{(0)}_{ph}} \right) \cos (\theta - \theta_0)^2 -{\rm i} \sin 2(\theta - \theta_0)\right] }{(\omega - k C^{(0)}_{gr} {\rm e}^{{\rm i} \theta}) + \beta {\rm i} }\nonumber\\ \times \sqrt{gK +\sigma K^3} \,{\rm d}\theta. \end{gather}$$

For the estimates we use the values of parameters from Sheremet et al. (Reference Sheremet, Guza, Elgar and Herbers2002) ($\omega = 0.088 c^{-1}$, $\gamma = 0.02$, $k = 0.037 m^{-1}$). If, for simplicity, we also assume that $\omega / k \gg C^{(0)}_{gr}$, although this assumption significantly underestimates the effect and is often not justified, then expression (4.42) takes a much simpler form

(4.43)\begin{equation} \frac{\mathrm{Im} \omega}{ \omega} = \frac{\nu \alpha k}{ \sqrt{g} \gamma^2 } \mathrm{Re} \left[ \int^{{\rm \pi} / 2}_{- {\rm \pi}/ 2} {\rm e}^{3{\rm i} \theta} \left( \frac{7}{2} \cos (\theta - \theta_0)^2 -{\rm i} \sin(\theta - \theta_0) \right) \,{\rm d} \theta \int^{K_{max}}_{K_{min}} \frac{{\rm d}K}{\sqrt{K}} \right]. \end{equation}

The integral with respect to $K$ in (4.43) logarithmically diverges at the upper and lower limits. Obviously, there is a viscous ‘cutoff scale’ specified by viscosity which is not explicitly taken into account in the presented model. Therefore, for our rough estimates we introduce the ‘cutoff scale’ $K=10^2\,\textrm {m}^{-1}$ as the upper limit of integration. Then, according to (4.43), for the above parameters and viscosity coefficient equal to the kinematic viscosity of water $\nu = 10^{-6} \,\textrm {m}^2\,\textrm {s}^{-1}$, $\mathrm {Im}\, \omega / \omega \sim 10^{-5}$. However, since a background small-scale turbulence is always present in the upper layer of the ocean, the use of turbulent viscosity is more appropriate. We estimate the eddy viscosity coefficient by the Prandtl formula $\nu = 0.4 w_* z$, where $w_*$ is the friction velocity in the water boundary layer linked to the air friction velocity by the condition of the equality of shear stresses $\rho _{air} u^2_* = \rho _{water} w^2_*$. We assume that the eddy viscosity for a wave with the wavenumber $K$ is $\nu = 0.4 w_* K^{-1}$. Then, from (4.43) follows

(4.44)\begin{equation} \frac{\mathrm{Im} \omega}{ \omega} = \frac{0.4 u_* \alpha k}{ \sqrt{g} \gamma^2 } \sqrt{\frac{\rho_a}{\rho_w}} \mathrm{Re} \left[ \int^{{\rm \pi} / 2}_{- {\rm \pi}/ 2} {\rm e}^{3{\rm i} \theta} \left( \frac{7}{2} \cos (\theta \!-\! \theta_0)^2 - {\rm i} \sin(\theta \!-\! \theta_0) \right) \,{\rm d} \theta \int^{K_{max}}_{K_{min}} \frac{{\rm d}K}{K^{3/2}} \right]. \end{equation}

The integral with respect to $K$ in (4.44) also diverges at the lower limit as $K_{min}\rightarrow 0$. The simplifications adopted in the model break down for insufficiently short wind waves. Longer short wind waves have larger group velocities and therefore are increasingly less sensitive to inhomogeneities created by the orbital velocities of edge waves. In our estimates we set the large-scale ‘cutoff’ to be equal to $4\,\textrm {m}^{-1}$, which corresponds to the wavelength of 1.5 m. For the friction velocity corresponding to a 10–12 $\textrm {m}\,\textrm {s}^{-1}$ wind and the above mentioned parameters of the edge wave from (4.44), expression (4.44) predicts quite substantial growth rates: $\mathrm {Im} \omega / \omega \sim 10^{-3}-10^{-2}$.

These estimates are quite crude and, therefore, the specific quantitative predictions should be treated very cautiously; a healthy scepticism would be appropriate. However, we could deduce from the formulae ((4.42) and (4.44)) the most basic qualitative features which we expect to be robust. Neither the specificity of the adopted bathymetry nor the choice of the main mode are important here. What matters most is the characteristic phase velocity of edge waves controlled in our simplest model by the bottom slope $\gamma$. For real bottom topographies the same role will be played by a ‘characteristic bottom slope.’ We expect the inverse quadratic dependence of the growth rates on characteristic bottom slope to be the most robust feature of this mechanism of edge-wave generation. Roughly, the milder the beach slope the more favourable are the conditions for the generation of an edge wave by wind. This prediction goes with a caveat; the bottom friction was ignored in the model and this oversimplification might preclude the generation of edge waves for particularly small bottom slopes. In its present form the model does not yield explicitly the distinguished preferred scale of edge waves. All things being equal, the excitation is expected to be stronger for shorter edge waves. The preferred scale of edge-wave excitation could be found by simulating directly equation (4.42) without extra simplifications, which goes beyond the scope of this paper and requires a dedicated work.

Implicit in the model is the key assumption of moderate wind. For winds too weak the steepness of gravity–capillary waves will be too small and since the mechanism is quadratic in steepness, their effect on edge waves will be negligible. On the other hand, for winds too strong the short waves might be breaking intensively and will be less sensitive to the inhomogeneity due to orbital velocities of the edge wave; the scale of inhomogeneity should be always smaller than the mixing scale due to wind. To quantify what winds are moderate enough and what are not, is a challenging task which goes beyond the scopes of the present work.

At present, there is a substantial gap between the maser model and observations. To bridge it concerted efforts from both ends of the divide are needed. From the observations we need detailed measurements of short wind waves, especially to determine whether there are phase correlations with the edge waves; difficult to provide accurate measurements of subsurface turbulence are necessary to find the scale-dependent turbulent viscosity for short wind waves. On the theoretical side the most crude approximation should be replaced by more accurate alternatives, of particular concern is the use of the long-wave cutoff. To this end, better models of short-wave interaction with the turbulent boundary layer are required. A novel kinetic equation which takes into account both the triad and quartet interactions should be derived and investigated numerically.