## 1. Introduction

During sea storms, marine structures and routing ships are exposed to severe wave conditions that are responsible for serious damages and occasional sinkings (Sand et al. 1990; Skourup et al. 1997; Socquet-Juglard 2005; Forristall 2007; Dysthe et al. 2008; Forristall 2011; Cavaleri et al. 2012). In this context, it is crucial for engineers, scientists, and seafarers to be able to predict the actual maximum sea surface elevation (or, equivalently, the maximum crest elevation or maximum wave height) a structure or a ship could encounter within a specific sea condition.

Traditionally, the sea surface elevation *η*(*x*, *y*, *t*) (where *x* and *y* are the Cartesian space coordinates and *t* is the time) is retrieved recording the time series *η*(*t*) by means of pointlike instruments (e.g., wave gauges, ultrasonic instruments, and buoys). In this context, the maximum expected wave height *H* and crest elevation *C* within a sea state have been defined by means of time extreme value analysis: *H*_{max} ≈ 2.0*H*_{s} and *C*_{max} ≈ 1.25*H*_{s} (where *H*_{s} is the significant wave height) have thus become rules of thumb for engineers and scientists (Dysthe et al. 2008). Although the probabilistic character of these estimates intrinsically accounts for the occurrence of even higher waves, occasionally the standard wave model framework based on the statistics of time records turned out to be poorly effective in describing extreme elevations, especially during short-crested sea states that are typical of storm conditions. Indeed, exceedances of the expected maximum crest heights as well as damages significantly above the corresponding level have been observed and a correct interpretation cannot be found even using second-order time wave statistics (Forristall 2005, 2006, 2007; Dysthe et al. 2008).

Recently, novel instrumentations [e.g., stereophotogrammetric systems (Shemdin et al. 1988; Banner et al. 1989) and radars (Dankert et al. 2003)] supported by increasing computing capabilities have allowed for the retrieval of 2D digital maps of the sea surface elevation *η*(*x*, *y*) evolving over time. The change of the domain of observation from time to space–time revealed that during short-crested sea states the maximum sea surface elevation gathered over an area is larger than that obtained at a single fixed point inside the area. Indeed, Fedele et al. (2013) observed an increase of the maximum sea surface elevation with the area by means of a stereophotogrammetric system [namely, the Wave Acquisition Stereo System (WASS) (Benetazzo 2006; Benetazzo et al. 2012)] deployed on top of the oceanographic tower “Acqua Alta” in the northern Adriatic Sea (Italy). Socquet-Juglard et al. (2005) and Forristall (2005) verified the same evidence by numerical simulations of short-crested sea states, while Forristall (2011) observed this during a wave tank experiment. Besides these, the European project MaxWave concluded that extending the analysis to the space domain results in many more individual waves observed, and hence the standard model criteria for extreme waves have to be modified (Rosenthal and Lehner 2008). Indeed, the occurrence of larger waves in space–time is strictly related to an increase of the number of waves associated with a genuine dimensional effect (Baxevani and Rychlik 2006; Fedele 2012).

To interpret the observations of sea state maxima over space and time, theoretical models have been derived from the analysis of multidimensional random fields: the Piterbarg theorem (Piterbarg 1996) and Adler and Taylor’s Euler characteristics approach (Adler 1981; Adler and Taylor 2007) have been applied to ocean wave statistics assuming that the sea surface *η*(*x*, *y*, *t*) can be modeled as a Gaussian random surface over the 2D space (*x*, *y*) and the time *t*. These theories opened the door to space–time extreme value analysis, as they allow one to estimate the exceedance probability distribution functions (EDFs) and the expected values of extremes over space and time. Adaptations of space–time extreme theories to wave analysis, that is, the Piterbarg theorem (hereinafter PT) and the Fedele (2012) model (hereinafter FM) based upon Adler and Taylor’s approach, have proved to be accurate in predicting synthetic (Forristall 2005, 2007; Krogstad et al. 2004; Socquet-Juglard et al. 2005), laboratory (Forristall 2011), and open sea maximum surface elevations over space–time (Fedele et al. 2013; Barbariol et al. 2014). Hence, these models could represent a change of paradigm in wave statistics and could support the explanation of the occurrence of extreme waves during storms, as demonstrated by Fedele (2012). Nevertheless, at present the diffusion of PT and FM is limited mainly because of the scarce availability of directional wave spectra, whose specific integral parameters are the inputs of space–time extreme models (Baxevani and Rychlik 2006; Fedele 2012).

Besides this, and most importantly, the coupling of such probabilistic models with the physical mechanisms suspected to be responsible for wave extremes’ generation has not been investigated yet. The spatial focusing due to current-induced refraction, the dispersive focusing, and the nonlinear focusing, that is, the so-called Benjamin–Feir instability, have been proposed as the main factors that contribute to the genesis of extremely high waves (Dysthe et al. 2008). In agreement with the conclusions of the European project MaxWave (Rosenthal and Lehner 2008), which analyzed several marine accidents in order to improve the understanding of the physical processes of extreme waves generation, the influence of the metocean forcings on the space–time extremes should be assessed in order to detect the favorable conditions for extreme waves’ occurrence. Indeed, the wind conditions (i.e., the wind speed and the fetch), the presence of an oceanic current, and the propagation onto a shallow-water environment intimately contribute to the sea state evolution and therefore could play a role in the generation of extreme waves. Such forcings directly affect the directional wave spectrum and, consequently, its integral parameters. Thus, analyzing how the spectral parameters are influenced by the metocean forcings provides an indirect assessment of the space–time extremes dependence upon them.

To this end, we propose two approaches aimed at computing the integral parameters of the directional spectrum required by the space–time extremes models, that is, the mean wave period *τ*, the components of the mean wavelength *L*_{x} and *L*_{y}, and the irregularity parameters *α*_{xt}, *α*_{xy}, and *α*_{yt}. The first approach is based upon the analytical integration of parametric directional wave spectra obtained by combining a deep-water frequency spectrum [i.e., Pierson–Moskowitz or Joint North Sea Wave Project (JONSWAP)] with the cos^{2} directional distribution function, representative of short-crested sea states. Two sets of formulas for the spectral parameters’ computation in fully developed (i.e., Pierson–Moskowitz) or fetch-limited (i.e., JONSWAP) short-crested sea states are obtained and discussed. Their dependence upon wind speed and fetch is then used in order to assess the wind conditions' effect on the space–time extremes, which were estimated by means of FM. In addition, the space domain size effect on the space–time extremes is assessed. The second approach makes use of the numerical integration of directional spectra routinely performed by spectral numerical wave models over arbitrary computational domains. By taking the computation of the cited integral parameters into the model, the storage of the output spectra at each computational grid node and time step, which may be highly resource demanding, is avoided. Therefore, version 40.85 of the Simulating Waves Nearshore (SWAN) model was adapted in order to develop an ad hoc version specifically suited to spectral parameters calculation that can be used in combination with theoretical space–time probabilistic models for wave extremes’ prediction in arbitrary conditions. Then, the effects of current- and depth-induced shoaling on FM space–time extremes are studied using this model. Before discussing the dependence of space–time extremes upon metocean forcings, FM performance is assessed by means of numerical simulations of Gaussian random seas, as in Forristall (2006).

## 2. Space–time extremes of a sea state

A space–time (hereinafter ST) extreme of a sea state is defined as the maximum sea surface elevation within a given time duration *T* and over a specified sea surface area *S* (Krogstad et al. 2004; Fedele 2012), assuming the wave field stationary over time and homogeneous over space. If *S* is larger than the characteristic sizes of waves (say, the wavelength by the wave crest length), then an ST extreme most likely corresponds to the elevation of the highest crest of a 3D wave group at focusing (Fedele 2012). It is unlikely to attain the actual elevation of this crest at a fixed point of observation (i.e., by taking into account a time-dependent sea surface elevation), unless the sea state is long crested or the chosen point is located exactly where the crest apex occurs. Hence, in short-crested sea states, wave extremes should be observed by means of instruments spanning the ST domain (e.g., radars or stereophotogrammetric systems) and predicted as maxima of multidimensional random fields. This requires that the ST features of the sea state or the directional wave spectrum are known.

### a. Space–time spectral parameters

*S*(

*σ*,

*θ*) and stem from the spectral moments

*m*

_{ijl}defined aswhere

*σ*is the radian relative frequency, and

*θ*is the direction of propagation, such that

*k*

_{x}=

*k*cos

*θ*and

*k*

_{y}=

*k*sin

*θ*are the components of the wavenumber vector

**k**. According to Baxevani and Rychlik (2006) and Fedele (2012), these features, which will hereinafter be referred to as “spectral parameters”, are the mean wave period

*τ*, the mean wavelength components

*L*

_{x}and

*L*

_{y}in a Cartesian reference frame (

*x*,

*y*), and the irregularity parameters

*α*

_{xt},

*α*

_{yt}, and

*α*

_{xy}expressed asThe average number of “waves” over a ST domain depends upon the size of the domain (i.e.,

*X*,

*Y*, and

*T*) relative to the average size of the waves (i.e.,

*τ*,

*L*

_{x}, and

*L*

_{y}), corrected for the space–time correlation which is accounted for by

*α*

_{xt},

*α*

_{yt}, and

*α*

_{xy}. In the time domain, the number of waves increases with

*T*or in the presence of smaller wave periods

*τ*. In the space domain, apart from larger

*S*and a shorter wavelength, the number of waves increase also by taking shorter wave crest lengths, that is, the more the sea state is short crested. If the reference frame is chosen such that the

*x*axis corresponds to the mean direction of wave propagation

*L*

_{x}and

*L*

_{y}represent the mean wavelength and the mean wave crest length, respectively. In general, this orientation can be achieved through a rotation of the directional spectrum, which does not affect the estimate of the ST extremes. Under this assumption, the short crestedness is given bywhich approaches 0 in long-crested conditions and 1 in short-crested sea states (Baxevani et al. 2003). While the first set of parameters (i.e.,

*τ*,

*L*

_{x}, and

*L*

_{y}) accounts for the geometry of the ST domain, the latter (i.e.,

*α*

_{xt},

*α*

_{yt}, and

*α*

_{xy}) describes the kinematic properties of the sea state. In fact, according to Baxevani and Rychlik (2006), the irregularity parameters are defined as the ratio between the highest crests velocity (i.e., the velocity of the specular points) and the sea state drift velocity (i.e., the principal velocity). Besides, the irregularity parameters account for correlation between space and time (

*α*

_{xt},

*α*

_{yt}) or space and space (

*α*

_{xy}) sea surface gradients. They can assume absolute values within 0 and 1 that correspond to the “confused” and “organized” sea state conditions, respectively, and influence the number of waves in the space–time domain. Indeed, the more the wave motion is organized, the smaller is the number of independent waves one has to expect in the space–time domain. For example, an organized sea with long-crested waves traveling toward the

*x*direction has

*α*

_{xt}= 1 and null

*α*

_{xy}and

*α*

_{yt}, whereas a confused sea with two wave systems traveling along the

*x*direction from opposite sides has all the parameters equal to zero. Therefore, as explained in section 2b, in the confused sea case the probability of encountering very large waves will be larger.

### b. Expected extremes

Theoretical models that predict maxima of the Gaussian sea surface elevation *η*(*x*, *y*, *t*) are PT and FM, which allow us to estimate the asymptotic EDFs of the sea state ST maxima and the expected values. The two models differ mainly on the treatment of the boundaries of the ST domain. In fact, PT assumes that maxima occur within the ST volume *V* = *XYT* (where *X* is the size along the *x* axis, *Y* is the size along the *y* axis, and *T* is the sea state duration over time *t*). Hence, the domain must be large enough to contain the characteristic sizes of the waves, that is, *T* ≥ *τ*, *X* ≥ *L*_{x}, and *Y* ≥ *L*_{y}. On the contrary, FM assumes that the maxima can also occur on the boundaries of *V*, that is, on its faces and edges, implying that domain sizes can also be smaller than *τ*, *L*_{x}, and *L*_{y}. Therefore, in order to entail the widest range of conditions, we will consider FM for the rest of the study, aware that results obtained and the following considerations are partially shared with PT (Barbariol et al. 2014).

*η*

_{ST}obey the following EDF:Here,

*M*

_{3},

*M*

_{2}, and

*M*

_{1}are the average numbers of waves within

*V*, on its faces (i.e.,

*XT*,

*YT*, and

*XY*) and on its sides (i.e.,

*X*,

*Y*, and

*T*), respectively, and are defined asand depend upon the spectral parameters [Eq. (2)] and the ST domain size (i.e.,

*X*,

*Y*, and

*T*). The asymptotic limit of Eq. (4) provides the Gumbel-like distribution of maxima, whose expected value

*h*

_{0}is the mode of the asymptotic distribution and is obtained as the solution of

*P*(

*η*

_{ST}/

*H*

_{s}≥

*h*) = 1;

*γ*

_{E}≈ 0.5772 is the Euler–Mascheroni constant; and

*β*(Fedele 2012), expressed aswhich is a statistical indicator of the geometry of the space–time extremes

*β*ranges from 3 to 1, which corresponds to the limiting cases of fully 3D wave extremes occurring within the

*V*and 1D extremes over a single (e.g., time) dimension, respectively. For

*β*values ranging between 3 and 1, extremes are likely to occur also on the boundaries of the domain (i.e., faces and sides).

## 3. Parametric wave spectral formulations

*S*(

*σ*,

*θ*) is decomposed into the frequency spectrum

*W*(

*σ*) and the directional distribution function

*D*(

*θ*) (assumed to be dependent on

*θ*only), according to

*U*

_{19.5}(19.5 m above the mean sea level) aswhere

*A*= 0.0081 is the Phillips constant,

*g*≈ 9.81 m s

^{−2}is the gravitational acceleration, and

*B*= 0.74.

*σ*

_{p}= 0.88

*g*/

*U*

_{19.5}) multiplied by a peak enhancement function (Holthuijsen 2007):with

*P*= 5/4. Parameters

*A*

_{JON},

*γ*,

*σ*

_{p}, and

*ϕ*provide the scaling of the spectrum, the peak enhancement, the position of the peak, and the width of the peak, respectively. It was found that the JON parameters’ relationships with the wind speed

*U*

_{10}(10 m above mean sea level) and fetch

*F*can be expressed in terms of power laws through the dimensionless peak frequency

*ν*= (

*σ*

_{p}/2π)

*U*

_{10}/

*g*and the dimensionless fetch

*A*

_{JON}= 0.032

*ν*

^{0.67},

*γ*= 5.87

*ν*

^{0.86}, and

*U*

_{10}and

*F*:with

*ϕ*equal to the mean value 0.08, according to Gran (1992). Typically,

*A*

_{JON}∈ [0.0081, 0.0032],

*γ*∈ [1, 7] with a mean value of 3.3, and

*ν*= (

*σ*

_{p}/2π)

*U*

_{10}/

*g*∈ [0.13, +∞). For a constant

*U*

_{10}, Eq. (11) shows that spectral parameters decrease with increasing

*F*, and they tend to the typical PM values as the sea state tends to fully developed conditions:

*A*

_{JON}→

*A*= 0.0081,

*γ*→ 1, and

*ν*→ 0.13.

*D*(

*θ*), we consider the cos

^{2}function, which depends on the direction

*θ*only (Holthuijsen 2007):To further simplify the analyses, without loss of generality, we assume a mean wave direction

*x*axis of a Cartesian reference frame. In this context,

*L*

_{x}represents the mean wavelength,

*L*

_{y}is the mean wave crest length, and

*γ*

_{s}≈ 0.58. The directional spreading of cos

^{2}is constant and equal to 31.5°, whereas it has been shown to be also frequency dependent in fetch-limited sea states (Ewans 1998). Nevertheless, we adopt the cos

^{2}distribution to compute the space–time extremes in short-crested sea states, and we assess (section 6a) the error with respect to a more realistic directional spreading function [i.e., cos2s with (Ewans 1998) frequency-dependent parameterization of the spreading parameter

*s*].

Directional spectra are finally obtained by combining one of the cited frequency spectra with the cos^{2} directional distribution function, in accordance with Eq. (8). Therefore, the PM-based directional spectrum (hereinafter called PM+cos^{2}) is obtained from Eqs. (9) and (12), while the JON-based spectrum (hereinafter called JON+cos^{2}) is obtained from Eqs. (10) and (12).

*σ*

^{2}=

*gk*; therefore,

*k*

_{x}=

*σ*

^{2}/

*g*cos(

*θ*),

*k*

_{y}=

*σ*

^{2}/

*g*sin(

*θ*), and the spectral moments [Eq. (1)] can be rewritten in terms of

*σ*only:where the spectral decomposition [Eq. (8)] has been used to split the directional spectrum

*S*(

*σ*,

*θ*) into

*W*(

*σ*) and

*D*(

*θ*). Since the cos

^{2}directional distribution function depends on

*θ*only, Eq. (13) can be rearranged in order to separate the integrals:Both the frequency spectra considered are proportional to

*σ*

^{−5}above the spectral peak. Then, when the exponent of

*σ*in Eq. (14) is equal to 4, that is, 2(

*i*+

*j*) +

*l*= 4, the integral over

*σ*is unbounded and spectral moments cannot be calculated (Ochi 2005). Given such spectral formulations, all the spectral parameters in Eq. (2), except

*τ*, are affected by the nonintegrability of the second integral of Eq. (14). Thus, we limit the upper bound of integration over

*σ*to a cutoff frequency

*σ*

_{c}. Since spectral parameters and ST extremes will depend to some extent on the choice of the cutoff frequency

*σ*

_{c}, we link

*σ*

_{c}to the physics of surface gravity waves, assuming it represents the higher frequency a harmonic wave of the spectrum could experience in the ordinary gravity waves range, that is, the gravity–capillary limit

*σ*

_{c}= 60 rad s

^{−1}(Holthuijsen 2007). A sensitivity analysis to assess the effect of the

*σ*

_{c}value on the ST extremes was conducted, and it is reported in section 3a.

### a. PM+cos^{2} spectral parameters formulae

*W*(

*σ*) =

*W*

_{PM}(

*σ*), according to Eqs. (9) and (12), Eq. (14) can be rewritten asand it can be integrated to provide the moments of PM+cos

^{2}:where Γ(1/4) and Γ(0,

*s*) are the Gamma and upper incomplete Gamma functions (Abramowitz and Stegun 1965), respectively, and

*s*= 1.296 × 10

^{−7}

*B*(

*g*/

*U*

_{19.5})

^{4}. The spectral parameters of PM+cos

^{2}are derived from Eqs. (16) and (2) asHere, spectral parameters have been also shown in a general formulation using the coefficients

*a*

_{1},

*a*

_{2}, and

*a*

_{3}that depend on the power

*m*of a general cos

^{m}directional distribution function [i.e., accounting for

*N*(

*m*) a normalization coefficient depending on

*m*]. In our context, if

*m*= 2, then the coefficients assume the values shown in the rightmost part of Eq. (17):

*a*

_{2}= 4, and

*L*

_{x},

*L*

_{y}, and

*α*

_{xt}depend upon the maximum cutoff frequency

*σ*

_{c}through

*s*. The function Γ(0,

*s*) can be calculated by using its upper limiting function

^{−1}), we verified that the third term of

*E*

_{1}(

*s*) can be neglected, being eight orders of magnitude smaller than the first two terms. Therefore, by rearranging we obtain a simplified expression for Γ(0,

*s*), which is explicitly dependent upon

*U*

_{19.5}and accurate to within a 10

^{−5}root-mean-square error [RMSE =

*τ*is not a function of

*m*or

*s*, meaning that in this context and by definition it is independent of the directional spreading of the spectrum and the cutoff frequency. The expressions obtained for

*m*

_{000}and

*τ*are consistent with results reported by Ochi (2005) for a spectral formulation with

*σ*

^{−5}tail. Besides, as

*m*

_{200}/

*m*

_{020}= 3 is independent of

*U*

_{19.5}. Consequently, the short crestedness associated with the cos

^{2}distribution (

*γ*

_{s}=

*L*

_{x}/

*L*

_{y}≈ 0.58) is recovered. In accordance with Baxevani and Rychlik (2006), the irregularity parameters are null, except for

*α*

_{xt}, which expresses the correlation between the sea surface derivatives over the direction of propagation

*x*and the time

*t*. Therefore, wave motion is somehow organized along

*x*and a reduced number of exceedances of a certain threshold has to be expected inside the space–time domain and on its

*XT*boundary. Instead,

*α*

_{xy}and

*α*

_{yt}equal zero since cos

^{2}is symmetric about

To investigate the effects on the ST extremes of the cutoff frequency *σ*_{c} value, we performed a sensitivity analysis considered a PM+cos^{2} with wind speed *U*_{19.5} = 20 m s^{−1}. The cutoff frequency *σ*_{c} = 60 rad s^{−1} was modified to 30 and 90 rad s^{−1} (i.e., 60 rad s^{−1} ±50%) and 12.56 rad s^{−1} (2.0 Hz). ST extremes *T* = 100*τ* and four different areas *S* ranging from 1 m^{2} to 1000 × 1000 m^{2}. The variations with respect to the value associated with *σ*_{c} = 60 rad s^{−1}, expressed as *σ*_{c} = 12.56 rad s^{−1} and for *S* = 100 × 100 m^{2}, that is, when area sides *X* and *Y* are of the same order of magnitude as *L*_{x} and *L*_{y}. Concerning the spectral parameters, the maximum absolute variation with respect to the values obtained with 60 rad s^{−1} is 24.8% (for *L*_{x} and *L*_{y}) again when *σ*_{c} = 12.56 rad s^{−1}. We conclude that while spectral parameters are moderately affected by *σ*_{c}, the ST extremes are only slightly sensitive to this choice. In this context, the results of the sensitivity analysis support the choice of the capillary–gravity limit as an appropriate cutoff frequency, since, besides the physical meaning, it represents a practical selection within a range of frequencies that assure a rather stable estimate of

Effect of the cutoff frequency *σ*_{c} value on the ST extremes, expressed as the variation ^{−1} (i.e., 9.5 Hz).

^{2}, Eq. (17) can be rearranged accounting for Eq. (18), assuming the wind speed at 10-m elevation (i.e.,

*U*

_{10}≈ 0.93

*U*

_{19.5}within the near neutral conditions hypothesis) and taking the values of constants in place of symbols as

### b. JON+cos^{2} spectral parameters formulas

^{2}spectral parameters cannot be applied to JON+cos

^{2}, since

*W*

_{JON}(

*σ*) is not analytically integrable (Holthuijsen 2007). Therefore, spectral moments of JON+cos

^{2}have to be calculated in an approximate form. Yamaguchi (1984) obtained approximations for zeroth-, first-, and second-order moments of

*W*

_{JON}(

*σ*). Nonetheless, to compute the JON+cos

^{2}spectral moments up to the fourth order in frequency, we adopt the procedure used by Gran (1992) for JON and we adapt it to JON+cos

^{2}. According to Gran (1992),

*W*

_{JON}(

*σ*) can be approximated by a peak-enhanced wave spectrum consisting of two independent components: (i) a broadbanded component conforming to a PM spectrum with

*A*=

*A*

_{JON}and peak frequency

*σ*

_{p}(responsible for the low- and high-frequency tails of the spectrum) and (ii) a narrowbanded component (hereinafter NB) with density closely centered about

*σ*

_{p}. Moments of JON+cos

^{2}are therefore obtained as the sum of the two contributions:where

*m*

_{ijl,PM}are the moments of PM+cos

^{2}, and

*m*

_{ijl,NB}are the moments of the additional peak that are expressed in terms of a parameter

*δ*, which Gran (1992) estimated to beassuming

*ϕ*= 0.08 in Eq. (10). Parameter

*δ*depends upon the peak enhancement factor

*γ*, such that if

*γ*= 1 then

*δ*= 0 and if

*γ*= 7 then

*δ*= 1. Thus, using Eq. (21),

*m*

_{ijl,NB}can be written asAnalytic integration of Eq. (22) leads to the moments of NB:Hence, according to Eq. (20), the moments of JON+cos

^{2}arewhere

*A*=

*A*

_{JON}and

*U*

_{19.5}= 0.88

*g*/

*σ*

_{p}. Here, Γ(0,

*z*) is the upper incomplete Gamma function and

*z*= 1.296 × 10

^{−7}

^{2}, after Eqs. (24) and (21), is

^{2}spectral parameters are obtained from Eqs. (2) and (24), through Eq. (21):Here, as for PM+cos

^{2}, parameters are also shown in a general formulation where coefficients

*b*

_{1}and

*b*

_{2}add to the coefficients

*a*

_{1},

*a*

_{2}, and

*a*

_{3}obtained for PM+cos

^{2}. They all depend upon the power

*m*of

*D*

_{m}(

*θ*). For the specific case considered (i.e.,

*m*= 2),

*b*

_{1}= 5 and

*b*

_{2}= 6. Once again, the parameters

*L*

_{x},

*L*

_{y}, and

*α*

_{xt}depend upon the cutoff frequency through

*z*, that is, the argument of the upper incomplete Gamma function. A simplified expression for Γ(0,

*z*), which is explicitly dependent upon

*σ*

_{p}and accurate within a 10

^{−5}order RMSE is

Similar to what was obtained for PM+cos^{2}, the ratio *m*_{200}/*m*_{020} is independent of *σ*_{p} and *γ*_{s} ≈ 0.58. Besides, since we chose formulations of the JON parameters of Eq. (11) that account for the transition to fully developed conditions, if *δ* = 0 (i.e., *γ* = 1), then the PM+cos^{2} parameters of Eq. (17) are recovered (taking *σ*_{p} = 0.88*g*/*U*_{19.5}). The dependence of spectral parameters upon *U*_{10} and *F* was intentionally not achieved in Eq. (25) in order to provide equations that are as compact as possible. However, it can be easily obtained by using the parameterizations for *γ* and *σ*_{p} provided by Eq. (11).

Though derived analytically, Eq. (25) benefits from the approximation of Gran (1992) for the JON spectrum, and hence it is not exact. Thus, we estimated the error introduced in the computation of spectral parameters by such an approximation. To this end, we compared the moments and parameters computed analytically by means of Eqs. (17) and (25) with those computed numerically (i.e., after the numerical integration of a discrete spectrum, accounting for a high-frequency tail proportional to *σ*^{−5}). We assumed the numerical estimates as benchmarks since we verified that for a PM+cos^{2} spectrum (whose spectral parameter formulas are exact) the analytical and numerical estimates were exact within a few per thousand difference, which is the error introduced by the numerical integration technique. For JON+cos^{2}, in the range of *γ* between 1 and 7, we obtained differences between numerical and analytical results within 5% for spectral parameters of Eq. (2) and the significant wave height, *H*_{s} being the most affected parameter, since the JON approximation mostly affects the zeroth moment (i.e., *m*_{000}). Hence, thanks to the results of the sensitivity analysis on *σ*_{c} (section 3a), we expect much smaller errors on ST extremes.

## 4. Numerical modeling

The numerical integration of arbitrary directional spectra, that is, in deep to shallow waters and accounting for generation, propagation, and dissipation processes, is routinely performed with spectral numerical wave models. In this context, we adapted the SWAN model (version 40.85) in order to calculate the integral parameters [Eq. (2)] of the directional wave spectrum as output variables, and we hereinafter call this implementation SWAN-ST. With this implementation, the storage of the output spectra at each computational grid node and time step, which may be highly demanding of resources, is avoided.

### a. The SWAN model

*S*(

*σ*,

*θ*) at each node of a computational domain by numerically solving the wave action density equation (Booij et al. 1999):Here,

*A*(

*σ*,

*θ*) =

*ρgS*(

*σ*,

*θ*)

*/σ*is the wave action variance density spectrum and

*ρ*is the water density. Equation (28) is a radiative, time-dependent transport equation that accounts for the wind input, the wave–wave interactions, and the dissipation phenomena both in deep and shallow waters. Processes characterizing wave propagation, such as shoaling, refraction, and wave–current interaction, are represented in the left-hand side of Eq. (28), while wind input, wave–wave interactions, and dissipations are included in the right-hand side of Eq. (28), that is, in the source term

*F*(

*σ*,

*θ*). To model the processes included in the source term of Eq. (28), SWAN adopts the Wave Model (WAM) cycle III (Hasselmann et al. 1988) and WAM cycle IV (Günther et al. 1992) formulations (SWAN Team 2011). In addition, SWAN includes depth-induced breaking, bottom friction, and triad wave–wave interactions in intermediate/shallow waters. In SWAN, the physical and spectral spaces are discretized; direction

*θ*is represented by

*N*bins

*θ*

_{n}(

*n*= 1, … ,

*N*), divided by a constant step Δ

*θ*, while frequencies are geometrically distributed in the prognostic range according to

*σ*

_{i+1}= 1.1

*σ*

_{i}between

*σ*

_{1}and

*σ*

_{Q}, that is, the minimum and maximum cutoff frequencies, respectively. Spectral moments of the spectrum are computed by adding a diagnostic tail proportional to

*σ*

^{−r}beyond

*σ*

_{Q}. Usually,

*r*equals 4 or 5, but

*r*= 5 is often preferred to resemble observed spectral tails (Pierson and Moskowitz 1964; Hasselmann et al. 1973; Forristall 1981).

### b. The SWAN-ST implementation

*x*,

*y*) and each time step. In accordance to what is routinely done by SWAN, the frequency domain is subdivided into prognostic (

*P*) and diagnostic (

*D*) ranges. Therefore, spectral moments are split into two contributions, such that

*σ*

_{1}≤

*σ*

_{q}≤

*σ*

_{Q}), the spectrum is numerically integrated to compute spectral moments according towhere

*λ*= ln(

*σ*

_{i+1}/

*σ*

_{i}). In the diagnostic range of frequencies (

*σ*

_{Q}<

*σ*< ∞), the wave spectrum is analytically integrated in a way thatwhere

*χ*=

*r*−

*l*− 2(

*i*+

*j*) − 1, and

*σ*

^{−5}when 2(

*i*+

*j*) +

*l*≠ 4, then

*χ*vanishes and

*m*

_{ijl,D}tends to be unbounded. Therefore, moments

*m*

_{200},

*m*

_{020}, and

*m*

_{101}are obtained by integrating up to a maximum cutoff frequency

*σ*

_{c}>

*σ*

_{Q}:where

*κ*= ln(

*σ*

_{c}) − ln(

*υσ*

_{Q}). The choice of

*σ*

_{c}is arbitrary since the chosen value can be directly put into Eq. (31). In this study, we imposed

*σ*

_{c}equal to the gravity–capillary limit, that is,

*σ*

_{c}= 60 rad s

^{−1}, for the reasons reported in section 3.

Equations (29)–(31) were implemented into the SWAN source code. To this end, two subroutines were written: one aimed at computing the spectral moments as described above and one aimed at computing the spectral parameters according to Eq. (2). In addition, the original source code was adapted in order to include the new subroutines and to allow the computation of five additional output variables: *L*_{x}, *L*_{y}, *α*_{xt}, *α*_{yt}, and *α*_{xy} (*τ* corresponds to the SWAN output variable *T*02). The subroutines and modules that were modified are those in charge of output variables’ initialization and output requests’ processing.

To validate the SWAN-ST implementation, a regression test was performed by first simulating the propagation of a PM+cos^{2} directional spectrum with SWAN-ST and then by comparing the spectral parameters computed by the model with the spectral parameters analytically obtained from Eq. (19). In fact, we expected a correspondence of the two sets of parameters within some differences introduced by the model numerics. To this end, we simulated the stationary 1D propagation of a PM+cos^{2} directional spectrum with *H*_{s} = 4 m imposed as boundary condition at the first node of a computational domain 500 km long with a 1-km resolution grid. The prognostic frequency range, consisting of 32 geometrically distributed frequencies, spans within 0.31 and 3.14 rad s^{−1} (i.e., from 0.05 to 1.00 Hz). Directions were discretized using 180 equally spaced values within the full circle [0, 2*π*) rad. To consistently compare the spectral parameters obtained from Eq. (19), without loss of generality, we imposed peak direction *θ*_{p} = 0 rad, which corresponds to the *x*-axis direction (^{2} is symmetric). No source term was included, since we were only interested in the propagative terms of Eq. (28). From the comparison between analytical and numerical results emerged that differences were smaller than 1% for all the spectral parameters. Thus, we concluded that the SWAN-ST implementation provides reliable estimates of the spectral parameters of Eq. (2).

Since most of the sea state energy/variance content is generally represented within the prognostic range of the wave spectrum, we also assessed the contribution of the diagnostic tail to the ST extremes *σ*_{c} = 60 rad s^{−1}. We adopted the same physical and spectral discretization used for the regression test as well as the same general setup (i.e., stationary 1D propagation without source terms). However, in order to evaluate the effect of the peak to maximum cutoff frequency ratio (*σ*_{p}/*σ*_{Q}, where *σ*_{Q} = 3.14 rad s^{−1}), we accounted for four different sea states with *H*_{s} = 1.0, 2.0, 4.0, and 8.0 m, pointing out that *S*: 1 × 1 m^{2}, 10 × 10 m^{2}, 100 × 100 m^{2}, and 1000 × 1000 m^{2} (time domain extension *T* was constant and equal to 1800 s). The results are presented in Table 2 in terms of variations *T*) and without (subscript *NT*) the spectral tail.

Effect of the spectral tail contribution on the ST extremes estimate, expressed as the variation

We note in Table 2 that *H*_{s}, the maximum absolute *S* is in the same order of magnitude as *L*_{x}*L*_{y}, while the global absolute maximum is 5.7%, which points out that, in the tested ranges of *S* and *H*_{s}, the ST extremes are only slightly sensitive to the tail contribution, consistent with the Boccotti’s results. Additionally, differences are smaller for the highest sea states. In fact, *σ*_{p}/*σ*_{Q}, which is reasonable since higher sea states are associated with lower *σ*_{p}, and hence the prognostic range encompasses the most of the spectral variance, more than what happens for lower sea states. Furthermore, the largest areas (i.e., 100 × 100 m^{2} and 1000 × 1000 m^{2}) show the lowest variabilities over *H*_{s}. This can be explained by the means of the wave dimension *β*; indeed for large areas, *β* ≃ 3, and it seems to be unaffected by the tail contribution and by the significant wave height. Besides this, results of this sensitivity analysis allowed us to estimate which is the effect of neglecting the spectral tail, which is a reasonable choice when its shape is unknown or is not correctly represented by the available parameterizations.

## 5. Assessment of FM predictions

Preliminarily, ST extremes predicted by FM were compared to the ST extremes obtained from numerical simulations of Gaussian random seas with PM+cos^{2} and JON+cos^{2} directional spectra. This approach was used among the others by Forristall (2006) to show the agreement between simulated and predicted (according to PT) ST extremes. To generate a large number of realizations of the Gaussian process *η*(*x*, *y*, *t*) from a prescribed directional spectrum, we employed the Wave Analysis for Fatigue and Oceanography (WAFO) toolbox (WAFO Group 2011) for MATLAB, which has already been applied for simulations of wave extremes (e.g., Gemmrich and Garrett 2008). Hence, a PM spectrum and a JON spectrum were alternatively combined with the cos^{2} directional distribution to simulate a fully developed sea state and a fetch-limited sea state, respectively. The frequency spectra were imposed in order to simulate sea states with *H*_{s} = 1.0 m, *L*_{x} ≈ 14 m, and *L*_{y} ≈ 14 m. The frequency–direction domain was discretized using 7200 equally spaced frequencies between 2.8 × 10^{−4} (i.e., the frequency resolution) and 2 Hz and 180 directions with 2° resolution. For each sea state, we generated 100 independent realizations of *η*(*x*, *y*, *t*), with spatial resolutions Δ*x* = Δ*y* = 0.5 m and temporal resolution Δ*t* = 0.25 s. Then, from each realization we extracted the maxima *η*_{ST} over five ST volumes *V*_{j} (*j* = 1, 2, 3, 4, 5) with the same duration *T* = 1 hour and different areas *S*_{j} = *j*^{2}*L*_{x}*L*_{y}. In Table 3, the expected extremes

Assessment of FM model performance for two Gaussian sea states (PM+cos^{2} and JON+cos^{2}) through the comparison of simulated (given with standard deviation) and predicted

The numerical simulations provided ^{2}) and 0.98 m (JON+cos^{2}).

## 6. Results

To evaluate metocean forcings’ effects on ST extremes, analyses were conducted by the means of the analytical formulation and the numerical implementation of the spectral parameters discussed in sections 3 and 4. In particular, equations of the PM+cos^{2} and JON+cos^{2} spectral parameters were used to investigate the dependence upon the wind speed and fetch. In addition, in order to assess the spatial effect on ST extremes, the dependence of the space domain size was investigated. Afterward, the SWAN-ST implementation was employed to study the effects of current- and depth-induced shoaling.

### a. Wind speed and fetch effects

The ST extremes *U* = *U*_{10}) and fetch *F* were chosen such that *U* varied within 10 and 20 m s^{−1} [i.e., approximately the Pierson and Moskowitz (1964) experimental range], while *F* varied from 1 to 250 km. Thus, only the frequency part of the considered spectra was interested by the wind variability, whereas the cos^{2} distribution remained unchanged. Doing so, we achieved ST extremes of single independent sea states undergoing different wind conditions. The space domain size *S* varied from 1 to 10^{6} m^{2} in order to span a wide range of sea areas and assume the hypothesis of homogeneity was reasonably fulfilled. Focusing on the spatial contribution to *T* = 100*τ* in order to maintain a constant number of waves over the time domain while changing wind conditions [100 wave periods being a reasonable number to achieve meaningful statistical properties and stationarity of the sea states (Boccotti 2000; Holthuijsen 2007)].

The ST extremes ^{2} and the ST extremes ^{2} are shown in Figs. 1 and 2, respectively. In fully developed conditions (Fig. 1), *S* and reducing for increasing *U*. The dependence upon *U* seems weaker than that upon *S*; indeed, the whole range of the *S* for a given *U*, while modifying *U* for a given *S*, the maximum *S* and reducing for increasing *U* and *F*. In this context, *F* has a stronger effect than *U*. Indeed, a *F* from 1 to 250 km, whereas a variation of 10% at most is found modifying *U* from 10 to 20 m s^{−1}. As for *S* seems to be much more effective than those upon *U* and *F*. Comparing the bottom-right panel of Fig. 2 to Fig. 1 reveals that at *F* = 250 km, the sea states, especially those with the smallest *U*, are close to the transition to fully developed condition.

As shown in Figs. 1 and 2, the wave dimension *β* is mostly affected by *S*, approaching 1 for the smallest *S* and 3 for the largest areas. Conversely, *β* is weakly modified by *U* and *F*. Actually, *β* and *S*. Most likely, this is because of the average number of waves *M*_{3}, *M*_{2}, and *M*_{1} but particularly *M*_{3}. Indeed over a ST domain, as shown in Eqs. (6) and (7), *β* are principally governed by *M*_{3} is large, the more *β* tends to 3. This condition can be achieved by increasing *S* = *XY* for a given sea state or by reducing *τ*, *L*_{x}, and *L*_{y} and increasing *α*_{xt}, *α*_{xy}, and *α*_{yt} for a given *S*.

The dependence of the spectral parameters upon *U* and *F*, expressed through Eqs. (19) and (27), is displayed in Fig. 3 for fully developed sea states, as a function of *U*, and for fetch-limited sea states, as a function of the dimensionless fetch *i* = *x*, *y*). In fully developed conditions, *τ*, *L*_{x}, and *L*_{y} increase with *U*. The ratio *L*_{x}/*L*_{y}, that is, the short crestedness of the sea state, remains constant (*γ*_{s} ≈ 0.58) because the directional distribution of variance is not modified by wind in the analysis. Because of this, the irregularity parameters remain null, except *α*_{xt}, which slightly decreases with *U*, meaning that the sea state tends to be slightly more confused. Actually, the dependence of *α*_{xt} upon *U* appears to be weak, and hence for the reasons explained above *τ*, *L*_{x}, and *L*_{y} act as the driving factors for the reduction of *τ*, *L*_{x}, and *L*_{y} increase in fetch-limited conditions too; for smaller *α*_{xt}. This rapidly decreases within the smallest *τ*, *L*_{x}, and *L*_{y}.

Wind speed *U* and fetch *F* have shown considerable effects on the spectral parameters, particularly on the geometric parameters *τ*, *L*_{x}, and *L*_{y}. Indeed, in fully developed conditions, *U* forces them to increase until they doubled (*τ*) and almost tripled (*L*_{x} and *L*_{y}) at 20 m s^{−1} with respect to the 10 m s^{−1} values. This has generally a direct effect on the average numbers of waves, and thus one could expect a direct effect on the ST extremes too. Nevertheless, the corresponding *U* is very weak compared to that of the spectral parameters, being at 20 m s^{−1} only 18% of the 10 m s^{−1} value. Similar considerations can be drawn for fetch-limited sea states. From the analysis, it emerges that the spectral parameters variations are damped out in Eq. (6), which leads to ST extreme estimates that are only slightly sensitive to variations in the number of waves, as noted by Holthuijsen (2007) for a time domain extreme statistics.

The cos^{2} function employed in this study provided a reasonable though simplified representation of the directional distribution in short-crested seas (Holthuijsen 2007). Indeed, it is unimodal and frequency independent, whereas realistic sea states have been proven to be bimodal (for *σ* > *σ*_{p}) and frequency dependent in the distribution of energy (Ewans 1998). To evaluate the error ascribable to the use of a constant spreading distribution in fetch-limited seas, we tested the use of a distribution which models a frequency-dependent spreading. To this end, we considered the cos2s directional distribution function with the parameterization proposed by Ewans (1998) for the spreading parameter *s*. Hence, the directional spreading (which is minimum at the peak frequency) increased both toward larger and smaller frequencies, in agreement with observations of directional spectra at sea. In this context, we compared the spectral parameters and the *A*_{JON} = 0.014, *σ*_{p} = 1.2 rad s^{−1}, *γ* = 2.0, and *ϕ* = 0.08) but having different directional distribution: a JON+cos^{2} and a JON+cos2s, assuming the latter as the reference. Spectral parameters were obtained by means of the numerical integration of the spectra, whose frequency-direction domain was discretized as in section 5. We tested a wide range of *S* (from 1 to 10^{6} m^{2}), observing significant differences in the spectral parameters *L*_{x} (14%), *L*_{y} (−23%), and *α*_{xt} (−7%). Nevertheless, such variations, though considerable, are not responsible for similar changes in the ST extremes, as previously shown in this section. Indeed, the ^{2} distribution are only 1% smaller than those obtained with the more realistic cos2s distribution. As an additional verification, we also compared the ST extremes computed from a JON+cos^{2} spectrum with the extremes retrieved from numerical simulations of Gaussian JON+cos2s sea states. To this aim, a set of 100 realizations *η*(*x*, *y*, *t*) of a JON+cos2s sea state was generated using WAFO (using the same domain discretization and JON spectrum of section 5), and the expected ST extremes detected over five ST volumes with the same duration (*T* = 1 hour) and different areas (*S*_{j} = *j*^{2}*L*_{x}*L*_{y} with *j* = 1, 2, 3, 4, 5, *L*_{x} ≈ 16 m, and *L*_{y} ≈ 19 m) were compared to the FM predictions of a JON+cos^{2} sea state sharing the JON spectrum. Such predictions, though obtained by means of a simplified directional distribution function, are in agreement with the simulations within a 1% error.

### b. Current- and depth-induced shoaling effects

We investigated the effects of current- and depth-induced shoalings on ST extremes by means of the SWAN-ST implementation. To analyze the contributions of complex processes (e.g., wave–current interactions) separately, in this paper we focused on the shoaling effect only, thus neglecting, for instance, the refractive phenomena induced by different wave–current directions and planimetric bottom variations. Shoaling is completely modeled within the propagative terms in the left-hand side of Eq. (28), and it could be altered by dissipative phenomena [e.g., wave breaking or bottom friction (Holthuijsen 2007)], and hence no source term was accounted for in Eq. (28). We performed stationary model runs over a 1D domain, imposing (at the deep-water boundary) the propagation of a PM+cos^{2} over an ambient current or over a sloping bottom, and we simulated the spectral parameters at each grid node of the model domain. The model setup used has already been described in section 4. We tested four different sea states with *H*_{s,0} = 1.0, 2.0, 4.0, and 8.0 m (subscript 0 referring to still or deep water conditions), and we computed the extremes over ST domains centered on the grid nodes. Since we already discussed the effect on *S*, herein the ST domain size for each sea state tested was fixed (i.e., *S* = *XY* = *L*_{x,0}*L*_{y,0} and *T* = 100*τ*_{0}). Unlike the analytical analysis of section 6a, using a numerical model we did not impose any constraint to the directional distribution function that was free to evolve under the different forcings effect, starting from an initial common spectral shape.

#### 1) Current-induced shoaling

Current-induced shoaling effects were investigated by imposing the transition from the propagation over still waters (i.e., current speed *V*_{x} = 0 m s^{−1} in the first half of the domain) to the propagation over a moving medium (i.e., *V*_{x} ≠ 0 in the last half of the domain). Eight different current speeds were tested, that is, −0.4, −0.3, −0.2, −0.1, 0.1, 0.2, 0.3, and 0.4 m s^{−1}, limiting the analysis to −0.4 m s^{−1}, which is the critical speed *V*_{x,c} = −*c*/4 over which the highest harmonic of the spectrum (i.e., with 1.0-Hz relative frequency and phase speed *c* = 1.56 m s^{−1}) is affected by energy blocking and energy reflection phenomena (Phillips 1977). The tail contribution to the spectral moments was not accounted for in this specific analysis because we could not control how the spectrum is modified by the current effect outside the prognostic frequency range.

Prior to studying the effect of current-induced shoaling on ST extremes, we analyzed the effects on *H*_{s} (Fig. 4); it increases in upcurrent conditions (15% at most, at −0.4 m s^{−1}), and it decreases in downcurrent conditions (9% at most, at 0.4 m s^{−1}), in agreement with current-induced shoaling (Holthuijsen 2007). Moreover, the higher *H*_{s} in still waters, the smaller the modifications under the current effect. A similar effect on ST extremes is depicted in Fig. 4; *H*_{s,0} show comparable variations of ^{−1}, the range of variations relative to the maximum is less than 1%), although the range of variations of the spectral parameters and *H*_{s} is larger, especially on an opposite current (at −0.4 m s^{−1}, the range of variations relative to the maximum is up to 10%). This is a consequence of *H*_{s} and not much sensitive to the number of waves’ variations. Results also indicate that the dimensional ST extremes, *H*_{s} and *H*_{s} increase has been already motivated, the *M*_{3}, *M*_{2}, and *M*_{1}), which is in turn caused by a decrease of the spectral parameters. This is confirmed by the results shown in Fig. 5, where *τ*, *L*_{x}, *L*_{y}, and *α*_{xt} decrease on an opposite current (*α*_{yt} and *α*_{xy} variations are negligible, and hence they are not plotted), indicating an average shortening of waves in time (16% at most on *τ*) and space (40% at most on *L*_{x}). The decrease of *τ* is in agreement with the frequency downshifting typical of shoaling (Holthuijsen 2007), and the decrease of *L*_{x} is consistent with the kinematics of waves on a moving medium (Phillips 1977). Also, *L*_{y} shorten on average (8% at most) and since *L*_{x} variation is considerably larger than *L*_{y} variation, then waves appear longer crested (*γ*_{s} < *γ*_{s,0}). In addition, the sea state tends to be more confused, as indicated by the decrease of *α*_{xt} (10% at most). Instead, on a following current, *τ*, *L*_{x}, *L*_{y}, and *α*_{xt} increase (Fig. 5), causing the average number of waves to reduce and in turn causing *L*_{x}, whose variation is comparable to the upcurrent decrease, the increase of the other spectral parameters is (at most) halved with respect to the decrease shown with opposite current. Moreover, waves on the following current are more short crested than in still waters (viz., *γ*_{s} < *γ*_{s,0}), though more organized (viz., *α*_{xt} > *α*_{xt,0}).

#### 2) Depth-induced shoaling

Depth-induced shoaling effects on spectral parameters and ST extremes were investigated by imposing the wave propagation over a sloping bottom whose depth, starting from a deep-water condition (i.e., 350 m), decreased to a minimum depth. The minimum depths, ranging from 2 to 125 m depending on the test, were chosen in order to observe the shoaling process in all the spectral components of the tested spectra.

As a first step, we analyzed the *H*_{s} variation toward the shallow depths. In Fig. 6, *H*_{s} initially decreases below the deep-water value and then grows above it consistently with the antibunching and bunching of energy peculiar of shoaling (Holthuijsen 2007). It is noteworthy that in the shallowest part, the remarkable *H*_{s} growth (up to almost 40% of *H*_{s,0}) should be realistically limited by shallow-water dissipation processes such as bottom friction and depth-induced breaking. The variation of *H*_{s} (Fig. 6). Indeed, *H*_{s} reduction and increment, respectively. Nevertheless, modifications induced by depth-induced shoaling on *H*_{s}, and as a consequence, dimensional maxima, *H*_{s}. All the sea states tested show similar tendencies, and as for current-induced shoaling, the variations experienced by the spectral parameters and *H*_{s} for different *H*_{s,0} are largely reduced for *H*_{s,0} are significantly reduced due to normalization and due to the slight sensitivity of *α*_{yt} and *α*_{xy} remain null). Indeed, approaching the shallow waters, at first *τ* decreases (up to 5% of the deep-water value *τ*_{0}) and then increases (up to 15% of *τ*_{0}) because of the spectral frequency shifting typical of shoaling: upshifting over the largest depths and downshifting over the smallest depths (Holthuijsen 2007). Parameter *L*_{x} monotonically decreases (40% of *L*_{x,0} at most) from deep to shallow waters due to the phase speed slow down, whereas *L*_{y} shows an initial slight decrease followed by an increase (80% of *γ*_{s} ≪ *γ*_{s,0}, and hence waves in shallow waters appear much more long crested than in deep waters. At the same time the sea state is more organized, with *α*_{xt} larger than in deep waters.

## 7. Conclusions

In this paper, we presented an analytical and numerical study aimed at discussing the influence of metocean forcings on the space–time extremes of short-crested sea states. In particular, the roles of the wind conditions and wave interaction with an ocean current and with the bottom were investigated by analyzing the effects on the space–time integral parameters of the directional spectrum. Space–time extreme

To investigate the wind conditions effect, we attained two sets of formulations for deep waters’ fully developed (Pierson–Moskowitz) and fetch-limited (JONSWAP) sea states to express the dependence of spectral parameters upon the wind speed and fetch. We pointed out that such formulations depend upon a cutoff frequency *σ*_{c} (imposed to ensure the boundedness of the higher-order moments) and upon the power *m* of the cos^{m} directional distribution function (herein *m* = 2). The cutoff frequency was influential on the spectral parameters but significantly less relevant for space–time extremes computation. In the future, the procedure herein used could be generalized to attain more general formulations that depend upon arbitrary cutoff frequency and directional spreading. Nevertheless, we assessed that the ^{2} directional distribution almost matches that one obtained using a more realistic directional distribution (with frequency-dependent spreading).

To investigate the effects of current- and depth-induced shoaling, we adapted the SWAN numerical wave model in order to store the relevant integrated spectral parameters at the nodes of the computational grid, as a resources-saving alternative to the storage of the spectra. Running a 1D model we found that the diagnostic spectral tail adds a small contribution to the space–time extremes, which could be neglected when the tail parameterization is not given for sure. In the future, the effects of wave steepness, which may be not negligible in the presence of strong opposing currents or approaching the shore, as well as the effects of bottom friction should be studied in the context of a higher-order wave model, and depth-induced breaking should be explored.

The main results presented in the paper are summarized as follows:

- Compared to the significant wave height
*H*_{s}, the increasing wind conditions turned out to have a weakening effect on the space–time extremes (both in fully developed and fetch-limited conditions). Indeed, we generally observed a reduction ofincreasing the wind speed and the fetch (for a given space domain size *S*) as a consequence of the spectral parameters’ variations inside the space–time domain. Hence, the wind effect on the dimensional extremesis to reduce the increment expected in presence of more severe wind conditions. As a matter of fact, the reduction of counteracts the increase of *H*_{s}, thus reducing the increment of. - The current-induced shoaling was found to amplify the space–time extremes in the presence of opposite currents and to reduce them in presence of following currents (for given
*S*and*H*_{s}) as a result of the spectral parameters modifications. Therefore, the upcurrent effect onis to further increase the increment expected due to the *H*_{s}contribution, and the downcurrent effect is to intensify the expected reduction. - The depth-induced shoaling exerted a weakening effect on the space–time extremes, since we observed a
reduction in the shallow depths caused by the spectral parameters variations. Such reduction counteracted the typical depth-induced increase of *H*_{s}toward the shore, thus reducing the increment ofexpected in shallow waters. - The space domain size
*S*had a strong influence on the space–time extremes, as expected. Indeed, a significant increase ofwas obtained enlarging *S*for a given sea state, providing evidence to support the role of the space domain size*S*in the prediction of extremes at sea.

## Acknowledgments

The research was supported by the Flagship Project RITMARE—The Italian Research for the Sea coordinated by the Italian National Research Council and funded by the Italian Ministry of Education, University and Research within the National Research Program 2011–2013. The authors gratefully acknowledge Prof. Francesco Fedele from GeorgiaTech (Atlanta, Georgia, United States) for useful comments. The SWAN model (version 40.85) was modified under the terms of the GNU General Public License.

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