## 1. Introduction

Stories of unexpectedly large waves rising out of nowhere and wreaking havoc in their paths have been reported throughout maritime history (see, e.g., Draper 1964, 1971; Slocum 1999; Smith 2006; Liu 2007) but these were invariably dismissed as part of maritime folklore, not to be taken seriously. In part, this early skepticism may have been due to the lack of understanding of the “randomness” of the ocean surface within the deterministic framework of nineteenth-century fluid dynamics, a frustration perhaps best captured by a remark ascribed to Lord Rayleigh (Kinsman 1965) that “the basic law of the seaway is the apparent lack of any law.”

It was not until after the Second World War that stochastic process theory was successfully introduced in ocean wave forecasting (Bates 1952; Kinsman 1965). Major advances in our theoretical understanding of ocean wave statistics followed (see, e.g., Hasselmann 1962; Kinsman 1965; Benney and Saffman 1966; Benney and Newell 1969) and, with the advent of modern computers, a rapid development of stochastic wave prediction models suitable for oceanic-scale wave forecasting became possible (e.g., WAMDI Group 1988; Tolman 1991; Komen et al. 1994; Booij et al. 1999; Janssen 2004; WISE Group 2007). In this statistical framework, extreme waves became a reality. After all, if—from a loose use of the central limit theorem—we assign a (near) Gaussian probability density function (pdf) to the ocean surface, such extremities must occur, only with low probability.

*O*(

*ϵ*

^{−2}) scale (where

*ϵ*is a characteristic wave steepness)—the Benjamin–Feir (BF) scale—and a much slower

*O*(

*ϵ*

^{−4}) scale—the Hasselmann scale (Hasselmann 1962; Annenkov and Shrira 2006). The slow scale is associated with resonant interactions, which maintain statistics close to Gaussian (Hasselmann 1962; Saffman 1967). The evolution on the fast BF scale occurs when the wave field is nonlinearly unstable, the random-wave equivalent of the well-known Benjamin–Feir instability process for periodic waves (e.g., Alber 1978; Onorato et al. 2001; Janssen 2003); physically, this instability is associated with near-resonant four-wave interactions that transfer wave energy across the unstable modes and—through phase coupling—can create coherent structures and strong deviations from Gaussian statistics (Onorato et al. 2001; Janssen 2003; Socquet-Juglard et al. 2005). The nonlinear instability of random waves has been studied extensively. Alber (1978) derived a stability criterion for narrowband random waves, which was later coined the Benjamin–Feir index (BFI) by Janssen (2003) and expressed aswhere

*ϵ*=

*k*

_{0}

*m*

_{0}

*m*

_{0}denoting the total variance of the wave field. The normalized spectral width Δ

_{ω}=

*β*/

_{ω}*ω*

_{0}, where

*β*is a measure of width of the wave frequency spectrum and

_{ω}*ω*

_{0}is the peak frequency. Consistent with the narrowband approximation, Mori and Janssen (2006) show that for a Gaussian-shaped spectrum the wave kurtosis (the normalized fourth cumulant) can be approximated in terms of the BFI aswhich, if applicable to oceanic waves, has some implications. For example, if we assume that a narrowband wave field is neutrally stable (BFI ≈ 1), then according to (2) its kurtosis would be approximately 1.8, a considerable deviation from Gaussian statistics.

However, although these narrowband relations [the BFI as a measure of stability and (2) relating BFI to kurtosis] seem to hold well for unidirectional waves in deep water (see, e.g., Janssen 2003; Onorato et al. 2004; Mori and Janssen 2006; Mori et al. 2007; appendix B), their physical relevance to realistic oceanic waves is not clear. First, BFI values in field observations, even for narrowband swells, are generally much lower than unity (also noted by Janssen 2003). Also, the relation between BFI and kurtosis [Eq. (2)] has so far not been convincingly corroborated with field observations [see, e.g., the kurtosis estimates in Fig. 1 obtained from buoy observations in 195-m depth off the North Carolina coast (Ardhuin et al. 2003)]. Second, laboratory observations (e.g., Waseda 2006) and numerical simulations (Onorato et al. 2002a; Socquet-Juglard et al. 2005; Gramstad and Trulsen 2007) of two-dimensional (directionally spread) wave evolution indicate that freely developing nonlinear wave fields do not retain high kurtosis values but instead relax to a near-Gaussian state, suggesting that in two-dimensional wave fields the nonlinear physics may be fundamentally different.

Wave nonlinearity is not the only potential mechanism for enhanced likelihood of extreme waves at sea. For instance, when in 1967 the Suez Canal closure resulted in an increase in shipping off the southeast coast of South Africa, this led to an alarming number of incidents involving extremely large waves (see, e.g., Mallory 1974). This regional “hot spot” was linked to the refractive focusing of northbound swell fields opposed by the Agulhas Current, a strong southward-flowing coastal current on the outer edge of the continental shelf (Smith 1976; White and Fornberg 1998). Such focusing effects, induced by either an ocean current or seafloor topography, provide a plausible explanation for the occurrence of regional hot spots of intensified wave energy. However, it does not explain the transient features of extreme waves that are generally reported (Draper 1964, 1971; Slocum 1999; Smith 2006; Liu 2007). Moreover, extreme waves are observed even in locations where no currents or strong topography are present (Haver 2004).

Although both nonlinearity and refractive focusing have been identified as mechanisms for extreme wave generation, these processes are generally concomitant in the ocean and can potentially act together to create not only much larger average wave energy levels locally (focusing) but also an increased likelihood of extreme waves (nonlinear instability) in an already intensified sea state. The principal question we would like to address here is whether focusing of wave energy because of medium variations (e.g., currents or bathymetric variations) can force a stable random wave field into an unstable state where nonlinearity causes the development of non-Gaussian statistics (and an associated increase in extreme wave probability). This hypothesis, if confirmed, would provide a basis for understanding the observed transient character of extreme wave events in focal regions.

To address these questions, we develop an angular spectrum model for nonlinear waves and include the effects of a spatially varying medium (section 2) to describe the refractive effects of shear currents and seafloor topography. The model is suitable for wideband wave evolution in the half plane (forward-scattering approximation) and accounts for both nonresonant quadratic nonlinearity (bound waves) and near-resonant cubic nonlinear dynamics. Lateral medium inhomogeneities are treated through a scattering term, much in the same way as two-dimensional topography is treated in Janssen et al. (2006). We use standard pseudospectral techniques in our evaluation of refraction and nonlinearity to allow for efficient Monte Carlo simulations of the nonlinear evolution of wave statistics over an opposing shear current and an isolated shoal (section 3). In section 4 we discuss our findings and their implications, followed by conclusions in section 5.

## 2. A frequency-angular spectrum model

To study the evolution of nonlinear wave statistics we employ a frequency-angular spectrum model (see, e.g., Suh et al. 1990; Janssen et al. 2006) that is suitable for directionally spread random waves, propagating in the half plane of the positive principal coordinate (forward-scattering approximation). We account for weak lateral medium variations to include refractive focusing effects of an ambient current or seafloor topography on the evolution of wave statistics. The purpose of this study is not to discuss the intricacies of wave–current and wave–bottom interactions in great detail, but rather to provide a principal test of the effects of wave focusing on nonlinear wave statistics. Therefore, we will consider idealized conditions of waves over an opposing shearing current and wave propagation over an isolated bottom feature in otherwise deep water (section 3). In the present work, we include only the lowest-order terms for the lateral medium inhomogeneities to capture the principal refractive effects; higher-order extensions can be derived along the same lines (see, e.g., Suh et al. 1990; Janssen et al. 2006), but this is not pursued here.

*x*and

*y*as the principal and lateral coordinates in the horizontal plane, respectively. The vertical coordinate is

*z*, positive pointing upward from the still-water level. In anticipation of nonlinearity in the free-surface boundary conditions, which will result in quadratically coupled modes at second order in wave steepness [

*O*(

*ϵ*

^{2})] and near resonances at

*O*(

*ϵ*

^{3}), we write the wave field velocity potential, Φ, as the sum of primary components and second-order bound waves:Further, to exploit the fact that the medium variations are one-dimensional to leading order, we decompose the primary wave velocity potential, Φ

^{(1)}(

*x*,

*y*,

*z*,

*t*), as a Fourier sum over (absolute) frequencies and lateral wavenumbers, both of which are conserved in a stationary and (to leading order) laterally homogeneous mediumHere,

*ω*

_{1}=

*p*

_{1}Δ

*ω*and

*λ*

_{1}=

*q*

_{1}Δ

*λ*, where Δ

*ω*and Δ

*λ*denote the frequency and lateral wavenumber spacing, respectively. The numerical sub and superscripts on wave variables refer to frequency and lateral wavenumbers, respectively (e.g.,

We consider the wave field as a sum of forward-propagating plane waves, slowly modulated along the principal direction. The Wentzel–Kramers–Brillouin (WKB) approximation precludes the possibility of wave reflections and does not include exponentially decaying (evanescent) modes; the latter can be important locally in the near-field (a few wavelengths) of their generation source, but away from such regions the wave field can be accurately represented by the propagating modes alone (see, e.g., Stamnes 1986; Janssen et al. 2006). Lateral wave field variations are accounted for by the summation of the angular wave components and this representation includes rapid modulations such as those associated with wide-angle diffraction effects in a caustic region (Suh et al. 1990; Janssen et al. 2006) or other abrupt lateral variations of the wave field (Dalrymple and Kirby 1988; Dalrymple et al. 1989; Janssen et al. 2008).

*U*, along the principal direction (no lateral flow). We assume that the current is laterally homogeneous to leading order and we decompose it in a laterally averaged component

*U*

*Ũ*written aswhere we assume

*Ũ*(

*x*,

*y*,

*z*) is

*O*(

*β*) with

*β*≪ 1. The current is assumed stationary and spatially slowly varying such that (∂

*, ∂*

_{x}*, ∂*

_{y}*)*

_{z}*U*∼

*O*(

*β*,

*β*

^{2},

*β*

^{2}). From continuity, the vertical velocity

*W*is thus

*O*(

*β*) and the vorticity vector is

*O*(

*β*

^{2}). Upon setting

*O*(

*β*) =

*O*(

*ϵ*

^{2}), and because the highest-order terms in our wave model are

*O*(

*ϵ*

^{3}) cubic near resonances, we neglect the current vorticity and derive wave evolution equations from potential flow theory. Although we do not explicitly restrict

*U*

*ϵ*) and deep water, we find at the lowest order the familiar vertical structurewhere

*η*

*k*

_{1}is a wavenumber related to frequency,

*ω*

_{1}, and intrinsic frequency,

*σ*

_{1}, through the dispersion relationWeak lateral medium variations, resulting from either a spatially varying current field or a topographical feature, are included through higher-order scattering terms in the evolution equations for the frequency-angular components (along the lines of, e.g., Dalrymple et al. 1989; Suh et al. 1990; Janssen et al. 2006). The evolution of the frequency-angular components on account of such inhomogeneities, as well as cubic nonlinear forcing, is described through a solvability condition on the primary wave field (see, e.g., Chu and Mei 1970; Mei 1989, chapters 3 and 12; Liu and Dingemans 1989; and many others), which can be written as (see, e.g., Suh et al. 1990; Janssen et al. 2006, 2008)where the principal wavenumber component

_{1}

^{1}= sgn(

*ω*

_{1})

*k*

_{1})

^{2}− (

*λ*

_{1})

^{2}

*k̃*

_{1}(

*x*,

*y*), a two-dimensional correction to the (one dimensional) reference wavenumber

*k*

_{1}. We include only the lowest-order correction to

*k*

_{1}, which implies that formally

*k̃*

_{1}/

*k*

_{1}is assumed

*O*(

*ϵ*

^{2}). Explicit expressions for

*k̃*

_{1}(

*x*,

*y*) depend on the nature of the lateral inhomogeneity (viz., current or topography) and will be given below where they are needed. The nonlinear term in (8) accounts for near-resonant terms at the third order in nonlinearity and can be written asHere and in (8),

_{λ1}and

_{ω1}denote the discrete Fourier transform operator with respect to

*y*and

*t*, defined aswhere

*L*and

_{t}*L*denote the time and lateral extent of the domain, respectively. The

_{y}*ω*

_{1},

*λ*

_{1}) contribution of the nonlinear terms. Further, in (10) we used the notationThe off-resonant second-order modes Φ

^{(2)}, which are required to evaluate the cubic nonlinear terms in (10), can be obtained from the second-order forcing problem (e.g., Hasselmann 1962; Mei 1989; Janssen et al. 2006) and can be written aswhereHere,

*σ*

_{12}=

*σ*

_{1}+

*σ*

_{2}and the quadratic wave–wave interaction coefficient

*D*

_{12}

^{12}can be written aswhich is the deep-water asymptote of the expression in Janssen et al. (2006) with absolute frequencies (

*ω*) replaced by intrinsic frequencies (

*σ*) and use made of the dispersion relation (7).

*η*, including local second-order corrections, follows from the dynamic free-surface boundary conditionThe nonlinear wave model thus consists of the evolution equation (8) for the primary frequency-angular components of the velocity potential, the expressions (13), (14), and (15) for the second-order bound modes, and the explicit relation (17) for the free-surface elevation (including local second-order corrections).

## 3. Nonlinear evolution of wave statistics

*x*= 0 with a frequency-directional spectrum of the formFor the frequency spectrum

*ω*) we use a simple (double sided) Gaussian distribution, given for positive frequencies byHere,

*H*= 4

_{s}*m*

_{0}

*m*

_{0}is the sea surface variance),

*β*is a spectral width parameter, and

_{ω}*ω*

_{0}is the peak (angular) frequency. The normalized directional spreading function

*θ*) is parameterized as a wrapped normal distribution (e.g., Vincent and Briggs 1989; Mardia and Jupp 2000)Here,

*θ*represents the wave angle,

*θ*is the mean wave angle,

_{m}*N*is the number of harmonics in the series (set at 250 here), and

*σ*is the directional spreading parameter in radians. The phases are drawn randomly (uniform distribution) between 0 and 2

_{D}*π*, but the Fourier mode amplitudes (modulus) are taken deterministically from the spectral variance in each bin. Choosing the amplitudes deterministically instead of according to the theoretical Rayleigh distribution is numerically convenient (stability) and introduces only a very slight deviation from pure Gaussianity in the initial condition, which is of no concern here (see, e.g., Tucker et al. 1984; Janssen 2003).

We numerically integrate (8) along *x* using a variable step-size Runge–Kutta code (Matlab ODE45 routine) and ensemble average the results. For numerical efficiency, we solve the second-order bound waves (15) through an (approximate) spectral method instead of direct evaluation of the convolution (see appendix A). The convolution terms for the bound waves [Eq. (15)] require *O*(*N* ^{2}) operations (with *N* the number of spectral components), whereas the spectral implementation requires only *N* log_{2}*N* operations (see, e.g., Canuto et al. 1987; Bredmose et al. 2004; Janssen et al. 2006). The spectral approximation is excellent for narrowband waves (see appendix A). Verification with the full convolution (see appendix A) suggests that the accuracy of the spectral method for the evaluation of the bound modes is generally very good, even for wider-banded spectra than considered here.

Finally, to prevent energy buildup at the high-end cutoff of the frequency domain, the nonlinear forcing term in (8) is computed up until component 2*ω*_{0}, whereas at higher frequencies the model accounts only for linear propagation and bound-wave contributions. In this manner the model thus absorbs energy cascading through the tail of the spectrum, which cumulatively is a small fraction of the initial wave energy for the propagation distances considered here. In appendix B, this model implementation is verified deterministically against observations of one-dimensional wave evolution in a flume (Shemer et al. 2001), and statistical simulations are compared to one-dimensional nonlinear statistical theory (Janssen 2003; Mori and Janssen 2006).

### a. Freely developing waves

To provide a context for our discussion of the combined effect of wave nonlinearity and refractive focusing, we first consider a nonlinear wave field evolving through a homogeneous medium (no topography or current; *U* = 0, *k̃*_{1} = 0). The initial two-dimensional wave field has a narrowband spectrum (*β _{ω}* = 0.025 rad s

^{−1}and

*σ*= 2°) centered around

_{D}*ω*

_{0}= 0.2

*π*rad s

^{−1}(peak period 10 s) and

*θ*= 0, and it has steepness

_{m}*ϵ*≈ 0.06 (

*H*= 5.9 m). For such small initial spreading, the BFI is a suitable measure of the (initial) stability of the wave field (Socquet-Juglard et al. 2005; Waseda 2006; Gramstad and Trulsen 2007) and we thus anticipate the wave field to be nonlinearly unstable (BFI ≈ 2.1 > 1). The spectral domain is discretized with Δ

_{s}*ω*= 0.0157 rad s

^{−1}and Δ

*λ*= 0.0031 rad m

^{−1}, and we evolve 80 realizations over 160 wavelengths

*L*

_{0}of the (initial) spectral peak component.

The nonlinear evolution of the random wave field is characterized by a rapid buildup of kurtosis over the first 20–30 wavelengths (Fig. 2) on account of the initial instability of the wave field. The kurtosis peaks at around 30 wavelengths after which the wave field returns, first rapid then gradual, to a near-Gaussian state. The directional spread at the peak of the frequency spectrum, which is computed from the spectral directional moments using standard definitions (see, e.g., Kuik et al. 1988; O’Reilly et al. 1996; Ardhuin et al. 2003), gradually increases from 2° initially to approximately 11.5° after 160 wavelengths. Although this directional spread is still relatively small for natural wave fields (see, e.g., O’Reilly et al. 1996), it continues to increase, albeit fairly gradually, after the instability has ceased and the statistics are already close to Gaussian.

*x*/

*L*

_{0}= 26 (inside the region of instability) indeed has much heavier tails than the Gaussian, whereas at

*x*/

*L*

_{0}= 160 the pdf is merely skewed toward positive values, thus differing from a Gaussian primarily because of skewness of the surface elevation on account of locally forced second-order bound waves. The pdfs computed from the Monte Carlo time series are in good agreement with the four-term Gram–Charlier expansion (Longuet-Higgins 1963), which for a zero-mean process can be written asHere,

*ζ*=

*η*/

*m*

_{0}

*R*depends on

*K*

_{3}and

*K*

_{4}, the coefficients of skewness and kurtosis, respectively (for algebraic details of

*R*, see, e.g., Longuet-Higgins 1963; Huang and Long 1980). If the sea state is linear (cumulants beyond the second are zero)

*R*is unity and the probability density function (21) is Gaussian.

The spectral evolution is characterized by a rapid widening in frequency space of the initial spectrum (Fig. 5), which effectively stabilizes the wave field. After this, the wave field continues to gradually widen in directional space (Figs. 2, 4) and the frequency spectrum develops an *ω*^{−4} tail (Fig. 5), as expected from theory (Zakharov and Filonenko 1966) and seen in other numerical simulations (e.g., Onorato et al. 2002b; Socquet-Juglard et al. 2005).

The evolution of the nonlinear statistics in two horizontal dimensions is quite disparate from unidirectional wave propagation (see Fig. 2). Although the initial instability effects are very similar, the two-dimensional wave field does not retain large kurtosis values but instead evolves to a near-Gaussian state (see Figs. 2, 3). The predicted strong deviations from Gaussianity in the first 40–50 wavelengths are on account of the unstable (unrealistic) boundary condition at *x* = 0. In nature, a freely developing swell field, gradually narrowing under the effects of dispersion, is unlikely to develop into such an unstable state, because nonlinearity continuously enforces a return to a stable state.

Because ocean waves always exhibit some degree of directional spreading, this result—which confirms earlier findings with other models (Onorato et al. 2002a; Socquet-Juglard et al. 2005; Gramstad and Trulsen 2007)—suggests that freely developing swell fields in homogeneous media exhibit statistics that are close to Gaussian, which is in agreement with what is usually observed in the ocean. Thus, for freely developing wave fields in deep water and in absence of wind and medium inhomogeneities, nonlinearity is a determining factor for naturally occurring (stable) spectrum shapes, but is not expected to produce strongly non-Gaussian sea states.

### b. Waves in a focal zone

Ocean waves are generally not freely developing. Along their propagation paths, they are acted upon by winds, currents, and—on the continental margins—seafloor topography. In particular, spatial variations in current velocities and water depth can cause wave focusing; if the transformation is sufficiently fast to overcome nonlinearity, they can potentially force a wave field into an unstable state followed by the occurrence of large (positive) kurtosis values and an increased likelihood of extreme waves.

To investigate this hypothesis, we consider refractive wave focusing and the associated nonlinear evolution of statistics (in particular, we consider kurtosis values) for waves propagating against an opposing shear current (current refraction) and waves over a submerged shoal in otherwise deep water (bottom refraction). In both cases we let a narrow wave field propagate into a region with varying medium properties. The incident waves are the same as before (*σ _{d}* = 2°,

*ω*

_{0}= 0.2

*π*rad s

^{−1}, and

*θ*= 0), but the initial frequency spectrum is slightly wider (

_{m}*β*= 0.08 rad s

_{ω}^{−1}) and the steepness lower (

*ϵ*≈ 0.045, wave height

*H*= 4.6 m) so that the wave field is initially stable (BFI ≈ 0.5). The spectral domain is discretized with Δ

_{s}*ω*= 0.0157 rad s

^{−1}and Δ

*λ*= 0.0031 rad m

^{−1}, and we evolve 80 realizations over 80 wavelengths.

#### 1) Refractive focusing: Opposing shear current

*x*,

_{c}*y*)/

_{c}*L*

_{0}= (6.4, 6.4), (

*x*,

_{w}*y*)/

_{w}*L*

_{0}= (1.3, 1.9), and

*û*= l m s

^{−1}. The maximum laterally averaged current speed |

*U*

^{−1}; although

*U*

*Ũ*are of the same order here, both are relatively weak (

*U*/

*c*≪ 1) for the energetic part of the wave spectrum. From geometrical optics (Fig. 6), we estimate that, for a 10-s wave (the peak period of the random wave field), the current field induces a refractive caustic along the principal current axis (

*y*/

*L*

_{0}= 6.4) at around

*x*/

*L*

_{0}≈ 19.

*k*

_{1}is obtained from the dispersion relation (7), thus including the effect of the laterally averaged current on the wave dispersion characteristics (for small-angle wave–current geometry). The two-dimensional perturbation

*k̃*

_{1}can be expressed in terms of the laterally varying part of the current

*Ũ*[see Eq. (5)] asThe nonlinear angular spectrum model predicts a maximum wave height around

*x*/

*L*

_{0}= 19.5 along

*y*/

*L*

_{0}= 6.4 (Fig. 7, top panel), which is close to the caustic predicted by geometrical optics (Fig. 6). The directional spreading at the peak

^{1}

*σ*

_{θ}_{,p}increases rapidly in the caustic region from roughly 2° to 14° in and behind the focal region.

At the location where the wave height is maximum (*x*/*L*_{0} ≈ 19.5), the kurtosis dips down slightly, followed by a rapid buildup to a value exceeding unity at around *x*/*L*_{0} = 25. After this increase, kurtosis drops to fairly small values (near-Gaussian statistics) in about the same distance as required for the buildup. In this region of large positive kurtosis, the probability of the occurrence of large waves is considerably enhanced.

In contrast, in absence of the ambient current, kurtosis remains small throughout the domain (Fig. 7), consistent with the presumed initial stability of the wave field. Linear simulations including the current field (not shown) produce—as expected—near-zero kurtosis throughout, which confirms that the increase in kurtosis immediately following the caustic is the result of the nonlinear instability of the waves induced by the focusing current.

Example time series of the normalized surface elevation (Fig. 8) illustrate the difference in the wave field structure before and right after the caustic. Positive kurtosis values are reflected in the heavy tails of the probability density functions (Fig. 8); the Monte Carlo data are in good agreement with the Gram–Charlier expansion (21) for these skewness and kurtosis values.

This example illustrates that, in the presence of a focusing current, nonlinearity in the wave field can indeed result in strongly non-Gaussian statistics. Notably, in this example, the strongest deviations from Gaussianity do not coincide with the region of maximum wave height. Although nonlinear focusing effects will likely be strongest close to the maximum wave height, the higher-order correlations require a finite distance to develop; we return to that in section 4.

#### 2) Refractive focusing: Seafloor topography

*h*

_{0}is the surrounding depth (arbitrarily set at 500 m;

*k*

_{0}

*h*

_{0}≈ 20), the shoal center coordinates are (

*x*/

_{c}*L*

_{0}= 6.4,

*y*/

_{c}*L*

_{0}= 6.4), and the radii

*r*

_{1}= 3000 m and

*r*

_{2}= 2512 m, so that the minimum depth on top of the circular shoal is 12 m. Ray trajectories of a 10-s monochromatic wave over this topography indicate that a caustic occurs along the principal axis (

*y*/

*L*

_{0}= 6.4) behind the center of the shoal at around

*x*/

*L*

_{0}= 8 (Fig. 9).

*U*= 0). The wavenumber

*k̃*

_{1}is defined here aswhere

*k*

_{1}

^{h}is the local wavenumber satisfying the linear dispersion relation for finite depth [

*ω*

_{1}

^{2}=

*gk*

_{1}

^{h}tanh(

*k*

_{1}

^{h}

*h*)] and the deep-water reference wavenumber

*k*

_{1}=

*ω*

_{1}

^{2}/

*g*. The simulated wave height is maximum around

*x*/

*L*

_{0}= 8 (Fig. 10, top panel), consistent with the geometrical optics estimate (Fig. 9). The maximum wave height is followed by a peak in the kurtosis value, indicating an increase in likelihood of extreme waves at that location. The spread at the peak of the spectrum increases abruptly from roughly 2° to 25° in the focal zone and remains almost constant after that.

## 4. Discussion

A freely developing, directionally spread wave field, even when initially too narrowbanded to be stable, does not retain the high kurtosis values observed in unidirectional wave propagation (see Fig. 2 and appendix B). Instead, the initially unstable wave field evolves through a strongly non-Gaussian region, after which the statistics return to a near-Gaussian state. The remaining deviations from Gaussianity are due to the (locally forced) second-order bound-wave components, which affect the free-surface geometry but are without dynamical consequences. In other words, unidirectional waves can develop into a stable but strongly non-Gaussian state, but in our simulations such a state appears unavailable to directionally spread waves. From this, it would follow that in freely developing ocean waves, gradually narrowing under the effects of dispersion, nonlinear instability can be a determining factor in the spectral shape, but the statistics can be expected to remain close to Gaussian, in accordance with what is invariably observed.

We hypothesized that the transformation of a wave field in a focal zone can, if fast and strong enough to counter the stabilizing efforts of the nonlinear coupling, destabilize the wave field and result in strongly non-Gaussian features with associated increased likelihood of extreme waves (positive kurtosis). To test this, we considered the propagation of an initially narrowband (but stable) wave field through a focal region and modeled the evolution of the statistics through Monte Carlo simulations. Our examples obviously do not mimic the full complexity of ocean waves over natural seafloor topography or ocean currents; rather, they test the possibility of wave instabilities in a wave convergence zone. We showed two cases, an opposing current and a circular shoal, in which the focusing effects were strong enough for the waves to develop into a nonlinearly unstable state with high kurtosis values and associated increase in likelihood of extreme waves. Although the precise threshold for instability in random, directionally spread waves is unknown, numerical simulations with various initial conditions and medium variations (not shown) suggest a fairly abrupt transition between conditions for which such an instability occurs and conditions where the wave statistics remain close to Gaussian. For example, if we revisit the shoal case of section 3 but increase the water depth on top of the shoal from 12 to 30 m (*r*_{1} = 3000 m and *r*_{2} = 2530 m), there is still considerable focusing of wave energy (Fig. 11), but kurtosis values remain very small throughout the domain. BFI values for both the current-focusing and topography examples in section 3 exceed unity in the focal zone (Fig. 12), whereas the reduction of focusing for the case in Fig. 11 results in much lower BFI values. For these initially narrow (in directional space) cases, BFI ≈ 1 indeed appears a critical threshold for instability to develop (see Fig. 12). Because we only consider initially fairly long-crested wave conditions, this does not contradict the recent finding (Waseda 2006; Gramstad and Trulsen 2007) that, in general, stability also depends on the directional spreading, a variable of course not represented in the BFI, which after all is a normalized measure of nonlinearity relative to dispersion in a unidirectional wave field.

The fact that the location of maximum kurtosis along the center transect spatially lags the wave energy focal point (Figs. 7, 10) suggests that higher-order correlations induced by the wave nonlinearity in the focal zone require some distance to develop high kurtosis values in the wave field. In some ways, this is consistent with observations of nonlinear self-focusing effects in breaking waves (Babanin et al. 2007). Also, in the presence of refractive focusing, in particular for the case involving the topographic focusing, we note that changes in kurtosis are more abrupt than for typical homogeneous conditions (cf. the rates of change of kurtosis in Figs. 2 and 10). We suspect that the more rapid evolution of kurtosis seen in the cases including refraction is on account of the fact that the wave field in the focal zone is strongly inhomogeneous (see, e.g., Janssen et al. 2008), which will affect the nonlinear development of higher-order correlations and thus the evolution of the kurtosis. Note, however, that these (rapid) variations in kurtosis (or in wave height for that matter) are not on account of rapid variations of the individual wave components themselves (which would violate our assumptions) but rather the result of the coherent superposition of many wave components, with their mutual phase relations determined by the correlations in the wave field. This is somewhat similar, for instance, to the (linear) refractive focusing in a slowly varying medium in which the individual wave components are slowly varying but the refraction-induced correlations can result in rapid changes in the wave statistics near a caustic.

The representation of the wave field as a sum of forward-scattering WKB modes is nonisotropic and thus restrictive for general ocean fields. Moreover, we included only the lowest-order (phase) corrections to account for lateral medium variations (see section 2), which generally results in underestimation of the actual focusing strength (Janssen 2006). However, the modeling approach presented here is an efficient and intuitive framework for the study of wave statistics in focal regions; if needed, higher-order approximations for the lateral medium inhomogeneities can be included (Suh et al. 1990; Janssen et al. 2006, 2008). Moreover, because we consider the evolution of the directional spectrum in space (rather than time), the model can be initialized with a point measurement (such as a buoy) and comparison with observations at other locations in the computational footprint can be made. This is of some practical importance, in particular for regions of variable currents or depth that are common on the continental shelf and in coastal areas. Also, the frequency-angular spectrum model can be readily extended to include varying depth (Dalrymple et al. 1989; Suh et al. 1990; Janssen et al. 2006) and shallow-water nonlinearity (Janssen et al. 2006) to study wave statistics over coastal topography and near the shore.

Extension of efficient models to finite and variable depth is an important step toward understanding cubic nonlinear dynamics and the associated statistics. Although for narrowband waves it is well known (see, e.g., Whitham 1974; Peregrine 1983; Janssen and Onorato 2007) that cubic nonlinear effects transition from a focusing (positive kurtosis) to a defocusing (negative kurtosis) regime when *kh* < 1.363, it is not clear what this implies for more realistic two-dimensional random waves propagating in areas with variable depth. In other words, extreme waves are not an exclusively deep-water phenomenon, a point perhaps best illustrated by the fact that the Draupner wave (Haver 2004), undoubtedly one of the best documented “freaks”, was observed in roughly 70-m water depth (*kh* ≈ 1.5). But despite the obvious importance for offshore and coastal engineering (offshore structures are almost exclusively situated in moderate water depths on the continental shelf), the nonlinear dynamics of random waves in variable depth, as well as the consequences for wave statistics and extreme wave events, are poorly understood.

Near the shore, nonlinear wave evolution is further complicated by the transition from a dispersive Stokes regime to a weakly dispersive Boussinesq regime (Janssen et al. 2006), with near resonance at the second order that allows a much faster [*O*(*ϵ*^{−1})] nonlinear evolution of the wave field. However, whether these shallow-water nonlinear dynamics play a role in coastal freak waves (Dean and Dalrymple 2002; Didenkulova et al. 2006) is unknown.

## 5. Conclusions

To study the effects of a focal region on nonlinear wave statistics, we have developed a frequency-angular spectrum model for waves in a slowly varying medium in which the lateral variations are weak. The model describes the forward propagation of slowly varying spectral components while accounting for quadratic and cubic nonlinearity. Monte Carlo simulations for freely developing, directionally spread random waves, in the absence of currents or topography, confirm that such wave fields, even if initially unstable, do not retain high kurtosis values but return to a near-Gaussian state instead. This behavior, at variance with what is seen in unidirectional waves, confirms previous findings by other authors using different models. To investigate the evolution of wave statistics in a focal region, we consider the propagation of an initially narrow (but stable) wave field through a caustic. If the focusing effects are strong enough, the waves are forced into an unstable state, followed by the development of strongly non-Gaussian statistics and an increased likelihood of extreme events (positive kurtosis). Although the waves are steepest (and most unstable) near the caustic, the maximum kurtosis values are found down-wave of that location. The observed nonlinear effects in a focal zone suggest that, in principle, the concomitant effects of focusing and nonlinearity can produce strongly non-Gaussian statistics in an already intensified sea state. Although we have considered idealized examples to test a principle, the coexistence of a focal zone with strong deviations from Gaussianity could explain the observation that extreme wave events, described as transient features of exceptional magnitude relative to their background, occur predominantly in regions where energetic swells encounter ocean currents and/or seafloor topography.

This research was performed while TTJ held a Research Associateship awarded by the National Research Council in the United States. We gratefully acknowledge the funding for this work provided by the National Science Foundation (Physical Oceanography Program) and the U.S. Office of Naval Research (Coastal Geosciences Program and Physical Oceanography Program). We thank Alex Babanin and Miguel Onorato for their useful comments and suggestions.

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# APPENDIX A

## An Efficient Approximation for the Second-Order Wave Field

To solve (15) through direct convolution is straightforward but computationally very intensive (for a large number of spectral components, say *N*, this convolution is nearly a factor *N* slower than the remaining terms in the evolution equation). Unfortunately, these off-resonant modes cannot be treated by spectral methods in an exact manner (see, e.g., Bredmose et al. 2005; Janssen 2006; Janssen et al. 2006), and instead we pursue an approximation. The purpose of this approximation is to reduce the number of operations from *O*(*N* ^{2}) to *O*(*N* log_{2}*N*).

*σ*

_{0},

*λ*

_{0}= 0) and split the second-order potential amplitude into sum and difference interaction contributionsHere, the sum interactions, Φ

^{(2,+)}, are computed in the double-frequency range, nominally [3

*σ*

_{0}/2 … 3

*σ*

_{0}], and the difference contributions Φ

^{(2,−)}in the infragravity (below subharmonic) range, [0 …

*σ*

_{0}/2]. In this approximation, the second-order sum and difference contributions can be written aswhere

_{λ,ω}=

_{ω}{

_{λ}{}} and

_{1}

^{1}in (A3) is obtained through Taylor expanding the denominator in (A1) around (

*σ*

_{0},

*λ*

_{0}= 0), which, for the difference interactions

*C*

_{g},

_{0}=

*g*/(2

*σ*

_{0}). For the sum interactions,

*σ*= 20°,

_{D}*β*= 0.15 rad s

_{ω}^{−1}, and

*ω*

_{0}= 0.4

*π*rad s

^{−1}. The ambient (opposing) current velocity was set at

*U*

^{−1}and the spectral resolution is Δ

*ω*= 0.03 rad s

^{−1}and Δ

*λ*= 0.063 rad m

^{−1}. The good agreement between the convolution and spectral method (see Fig. A1) justifies the use of the narrowband spectral approximation to compute the bound-wave contributions.

# APPENDIX B

## One-Dimensional Verification of Evolution Model

### One-dimensional deterministic evolution

To verify our third-order model derivation and implementation, and to illustrate the implied wideband capability, we compare model simulations of wave evolution to observations of periodic wave groups propagating in relatively deep water reported by Shemer et al. (2001). The experiments were conducted in a wave flume with uniform water depth of 0.60 m. The positive *x* axis is in the direction of propagation, with the origin at the wave generator. For more detailed information on the experimental setup and the complete set of experiments conducted, we refer to Shemer et al. (1998, 2001).

*T*

_{0}= 0.9 s, of the formwhere

*ω*

_{0}= 2

*π*/

*T*

_{0}. The spectrum of this signal is characterized by a maximum at

*ω*

_{0}and sidebands at integer multiples of 2Ω

_{0}, with the two nearest to

*ω*

_{0}being the most significant. For the case considered here,

*k*

_{0}

*a*

_{0}≈ 0.21, where

*k*

_{0}is related to

*ω*

_{0}through the linear dispersion relation and

*a*

_{0}is taken (after Shemer et al. 1998) as the maximum amplitude of the carrier wave in a group close to the wave generator.

The model is initialized with the spectral components at *ω*_{0} and *ω*_{0} ± 2Ω_{0} of a time series of 18-s (i.e., 20 wave periods) duration observed at *x* = 0.245 m. Second-order components are included in the upwave boundary condition. We compute the evolution of an equidistant array of 65 frequencies with Δ*ω* = 0.35 rad s^{−1}.

In Fig. B1, we compare the observed (circles) and predicted (solid line) time series at *x* = 0.245 m (initial condition) and *x* = 8.425 m. The initially near-symmetrical wave groups develop strong left–right asymmetry of the envelope with steep fronts and gently sloping rears. The details of the nonlinear evolution are well represented in the model (see Fig. B1).

### One-dimensional nonlinear statistics

To illustrate the Monte Carlo simulation for one-dimensional wave evolution, we compare the evolution of an initially unstable wave field (BFI ≈ 2.1 > 1) to theoretical predictions (Janssen 2003; Mori and Janssen 2006). The initially narrowband wave field (with steepness *ϵ* ≈ 0.06, peak frequency *ω _{0}* = 0.2

*π*rad s

^{−1}, and

*β*= 0.025 rad s

_{ω}^{−1}) shows a rapid increase of the kurtosis (see Fig. B2) accompanied by spectral widening and a downshift of the spectral peak (not shown). Consequently, during this initial evolution, BFI values decrease and eventually the wave field stabilizes after about 60 wavelengths (after which location BFI values remain close to unity).

High kurtosis values are retained (Fig. B2), in good agreement with what is theoretically predicted by Eq. (2). The surface elevation pdf (Fig. B3) changes from near-Gaussian but skewed (second-order bound waves) to a strongly non-Gaussian pdf with heavy tails (Fig. B3, right panel) because of the high kurtosis built up in the nonlinear instability process. The truncated Gram–Charlier distribution function [Eq. (21)] is in good agreement with the simulated pdf, even for these large kurtosis values (which are formally outside of the validity range of that distribution).

Overall, the evolution of the wave field statistics confirms that the BFI stability criterion, based on narrowband unidirectional theory, captures the characteristics of the nonlinear evolution in the wideband unidirectional model and that theoretical kurtosis values from (2) (Mori and Janssen 2006) are in reasonable quantitative agreement with kurtosis values retained in the neutrally stable wave field (after roughly 60 wavelengths). This independently confirms earlier results (Janssen 2003; Mori and Janssen 2006; Mori et al. 2007) and provides some validation of our modeling approach.

^{1}

Directional spread is computed as before, but for laterally inhomogeneous wave fields the directional moments are taken over the absolute value of the coupled mode spectrum (or Wigner distribution) as defined in Janssen et al. (2008), instead of the variance density spectrum.