## 1. Introduction

Strong wave field modulations are caused by periodic tidal currents and at river inlets (Baschek 2005; Tolman 1990; Guillou 2017; Zippel and Thomson 2017; Saetra et al. 2021; Ho et al. 2023; Chawla and Kirby 2002). The interaction between waves and tidal currents dictates the horizontal wave height variability and may cause dangerous sea states (Ardhuin et al. 2012; Masson 1996; Rapizo et al. 2017; Halsne et al. 2022). Such interactions are also linked to the generation of extreme waves, which poses a severe threat for maritime activities due to their random occurrence and abnormal size (e.g., Lavrenov 1998; Toffoli et al. 2011; Onorato et al. 2011). However, the influence of tidal currents on the extreme wave statistics have yet to be properly investigated. In this paper we demonstrate and discuss the impact by a strong tidal current in northern Norway on the short-term extreme wave statistics, by taking advantage of the recent implementation of space–time extremes in spectral wave models (Benetazzo et al. 2021b; Barbariol et al. 2017) in combination with tidal current forcing.

Extreme wave estimates for a given sea state have traditionally been computed using stochastic models defined for a single point in space over a certain time duration, such that the wave field can be considered a statistically stationary process. It is, however, recognized that the maximum sea surface elevation within a certain horizontal area is generally higher than what is measured in a single point (Forristall 2007, 2008; Krogstad et al. 2008; Fedele et al. 2011). Therefore, recent works have focused on extending the traditional time extreme approaches to take into account the three-dimensional space–time domain (e.g., Boccotti 2000; Fedele 2012; Fedele et al. 2013; Benetazzo et al. 2015). The maximum wave crests *η* and crest–to–trough heights *H* in such domains are referred to as the space–time extremes (STEs). Recent studies have found good agreement between observations of extreme waves and expected STEs based on higher-order Stokes waves (Benetazzo et al. 2015, 2021a; Fedele et al. 2017; Barbariol et al. 2019; Benetazzo et al. 2017). Although the impact by currents on extreme waves has been studied following deterministic approaches (e.g., Toffoli et al. 2011; Onorato et al. 2011; Hjelmervik and Trulsen 2009), currents in stochastic extreme wave models have been given little attention.

Expected extreme waves in short-term statistics are dictated by sea state parameters computed from the 2D wave spectrum. Four of these are particularly important in a space–time domain (Benetazzo et al. 2021a): (i) the significant wave height (*H _{s}*), (ii) the spectral steepness (

*ε*), which represents a measure of the nonlinearity of the sea state, (iii) the average number of waves within the space–time domain (

*N*

_{3D}), which represents the sample size, and (vi) the narrow-bandedness parameter (

*ε*= 0) stochastic crest heights in space–time increased on a countercurrent due to the increase in

*N*

_{3D}, caused by the frequency shift, and vice versa on cocurrents. They considered Pierson–Moskowitz and JONSWAP spectra and an idealized current in one direction. Consequently, the effect of current-induced refraction on the extreme crests was not taken into account. Moreover, the recent work by Benetazzo et al. (2015) takes into account weakly nonlinear random wave fields up to second order in

*ε*, which has not been analyzed in the presence of currents.

To the best of our knowledge, no previous works have addressed the influence by currents on the stochastic wave heights in space–time. Ying et al. (2011) investigated the role of currents on the traditional *time* extreme wave height distribution proposed by Longuet-Higgins (1957). They proposed to add a scaling term to the probability distribution due to the change in statistics caused by focal points due to current-induced refraction, which were derived based on the results by White and Fornberg (1998), such that the probability of extreme wave heights increases in caustics. However, since the stochastic extreme wave formulations considered here are formulated for space–time, we hypothesize that the local changes in wave statistics caused by currents are implicitly taken into account by the *N*_{3D} parameter. Furthermore, and building upon the quasi-determinism theory by Boccotti (2000), the maximum STE wave height depend on characteristic shape of the wave spectrum, also when reduced from the space–time to a single-point time domain, where the narrow-bandedness parameter

Our study takes place in the Loften Maelstrom (Fig. 1), locally referred to as *Moskstraumen* (“straum” is current in Norwegian), a very strong open-ocean tidal current which can at least reach a speed of 3 m s^{−1}. Moskstraumen, which the tidal current will be referred to hereafter, has been infamous for centuries for its strength and for the occurrence of large and steep waves (Gjevik et al. 1997; Moe et al. 2002). Saetra et al. (2021) presented the first simultaneous measurements of waves and currents in Moskstraumen, which they used to verify an ocean model representation of the tidal current. Halsne et al. (2022) used the wave observations and found a better agreement with the wave field predicted by a WAM spectral wave model forced with model currents than an identical model without currents. Here we use a similar setup, but take advantage of a more recent WAM version that includes the STE computations (Benetazzo et al. 2021b). We consider two periods with different met-ocean conditions, one where young wind sea opposes a broad uniform tidal current and another where swell opposes a tidal jet. Under these characteristic conditions, we assess the impact by wave straining (to be introduced later) and current-induced refraction, two important wave–current interaction (WCI) mechanisms, on the sea state by, among others, comparing against a quasi-stationary idealized theoretical solution. We evaluate the influence of Moskstraumen on the 2D spectrum, the key spectral parameters listed above, and ultimately on the STEs.

The paper is structured as follows. In the following section 2 we present the theoretical framework for currents effects on the wave field under quasi-stationary conditions. In section 3, the stochastic extreme wave formulations in a space–time domain is presented together with the homogeneity assumption for short term statistics. In section 4, we describe the study region and model specifications. The results for each of the cases are presented in section 5 and further discussed in section 6. Then our concluding remarks are presented in section 7.

## 2. Current-induced wave field transformation

Here we present the wave straining mechanism by first considering the impact by a horizontally uniform current on the wave variance density *E*, and second on wave steepness. We consider an ambient, quasi-stationary, current field **U**(*t*, **x**) ≃ **U**(**x**) = [*u*(**x**), *υ*(**x**)], where **x** is the horizontal position vector.

*ω*is the absolute wave frequency as seen from a fixed point,

*σ*is the intrinsic frequency (following the current), and

**k**= (

*k*,

_{x}*k*) is the wavenumber vector. Under quasi-stationary conditions, the number of wave crests is conserved in a fixed control volume [Phillips 1977, Eq. (2.6.2)]. This requires that the wavenumber

_{y}*k*= |

**k**| must change when exposed to a changing ambient current through the intrinsic frequency dispersion relation

*g*is the gravitational constant and

*h*is the water depth. We define the effective current

*U*= |

**U**|, and the degree of opposition between the waves and the currents,

*ϑ*, is computed by

*θ*,

_{c}*θ*denote the current direction and the wave direction, respectively, using the same convention. Thus, values of 1, 0, and −1 indicate following, perpendicular, and opposing, respectively.

_{w}*x*axis on deep water from an area with

*U*

_{eff}=

*u*

_{0}= 0 to an opposing current

*U*

_{eff}< 0. In the following, subscript 0 denotes the wave characteristics where

*U*

_{eff}= 0. In such a case,

*ω*=

*σ*

_{0}= const. due to wave crest conservation. To compensate for the loss in the

**k**⋅

**U**term of (1), there must be an accompanied increase in

*k*. Increasing

*k*implies a shortening in the wavelength

*λ*= 2

*π*/

*k*. In the presence of currents,

*E*is not a conserved quantity (Longuet-Higgins and Stewart 1964). However, the wave action density

*N*=

*N*(

**x**,

*t*) =

*E*/

*σ*is conserved and takes the general form (Bretherton and Garrett 1968)

**x**=

**c**

*+*

_{g}**U**is the absolute wave group velocity vector, and

*E*for a constant current according to the above considerations we obtain [Phillips 1977, Eq. (3.7.11)]

*E*increases toward the singularity

*U*

_{eff}→ −

*c*/2, implying that the waves have been blocked by the current. At the blocking point,

*U*

_{eff}= −

*c*

_{0}/4. According to (6),

*E*will increase when the waves are propagating into an opposing current, and decrease for a following current. The theory is valid in the absence of wave breaking and while

*U*

_{eff}> −

*c*

_{0}/4. In the absence of a clear naming convention, we denote the effect in (6) “wave straining,” by following Holthuijsen and Tolman (1991). Wave straining is the combined effect of the “concertina effect” (Ardhuin et al. 2017; Wang and Sheng 2018), referring to the change in wavenumber, and the accompanied “energy bunching” (Baschek 2005). It is similar to shoaling, which occurs when waves propagate from deep to intermediate and shallow waters.

*E*is proportional to the square of the wave amplitude

*a*, we can rewrite (6) (Rapizo et al. 2017)

*kU*

_{eff}/

*ω*term in the nominator expresses the effect by the current on the wave steepness, which can be recognized by considering (1):

*ω*< 0 implies a growth in (7), and the

*k*dependence denotes the sensitivity of wave straining to the initial wavelength. For example, for waves directly opposing a current

*U*

_{eff}= −1.0 m s

^{−1}from a reference of

*U*

_{eff}= 0, the increase in

*ε*is 26% and 95% for a

*T*= 12 s and a

*T*= 5 s period wave, respectively. A summary for different values of

*U*

_{eff}and wave periods are given in Table 1. From the above considerations, both

*k*and

*ε*is very sensitive to the ambient current (Vincent 1979).

Wave parameter modulation due to wave straining on steady currents according to (7). The left and right sides show the ratio in wave variables (subscript 0 means zero current) for a 5- and 12-s period wave, respectively. In all cases, *a*_{0} = 1 m. Each row denotes different effective currents *U*_{eff}.

## 3. Statistical models for extreme waves in a space–time domain

The zero mean sea surface elevation is denoted *η*(*t*, **x**). The sea state can be characterized by *H _{s}* = 4

*β*, where

*β*is the standard deviation of

*η*(

*t*,

**x**). Building upon the results by Fedele (2012), Benetazzo et al. (2015), and Boccotti (2000), we consider a 3D space–time domain Γ =

*XYD*, where

*X*and

*Y*are the size of the sides of a rectangular surface area and

*D*is the duration of a time interval (see Fig. 1 in Fedele 2012). Here fundamental properties of the STE models are described, with particular focus on the sea state parameters that are essential for the STEs.

### a. Expected extreme wave crests and heights

*η*for every point within Γ, then the maximum individual crest height

*η*

_{MAX}can be defined in terms of an exceedance probability by a threshold

*z*:

*η*

_{MAX}is based on the so-called “Euler characteristics” valid for

*n*dimensions (Adler and Taylor 2007), which was first reduced to

*n*= 3 and verified for ocean waves in a space–time domain by Fedele (2012), and thereafter further developed by Benetazzo et al. (2015) to take into account weakly nonlinear random wave fields up to second order in

*ε*. With regards to the maximum crest-to-trough wave height (

*H*

_{MAX}), we consider the linear quasi-determinism theory by Boccotti (2000) which takes into account the narrow-bandedness of the sea state. These maxima can be deduced from their expected value using integrated spectral parameters, provided that the sea state is temporally stationary and spatially homogeneous (Adler and Taylor 2007). Such assumptions may be altered in a rapidly varying tidal current and will be treated in the subsequent section.

*H*, and subsequently modified by parameters that constitute the average number of waves (

_{s}*N*

_{3D}), wave steepness (

*ε*), and spectral bandwidth (

*L*,

_{x}*L*,

_{y}*T*denote length scales associated with the mean crest length (in the

_{m}*X*and

*Y*direction) and the zero-crossing mean period, respectively. Furthermore, the average number of waves at the boundaries and at the edges of Γ reads

*N*

_{3D}, dominate over the others for large-sized space–time domains, and is therefore considered here (Fedele 2012; Benetazzo et al. 2021a). The degree of organization in the space–time wave field is characterized by the expression containing the square root in (10):

*η*(

*t*,

**x**) [see Eqs. (5)–(7) by Benetazzo et al. (2021a)]. The variables in

*A*are commonly referred to as the “irregularity parameters” (Baxevani and Rychlik 2006). These parameters can be computed from spectral moments as

*N*

_{3D}is maximized for

*A*= 1 and minimized for

*A*= 0. More details about the spectral distributions associated with

*A*are found in Baxevani and Rychlik (2006).

*ε*, which is related to the skewness coefficient of the sea state (i.e., the third-order moment of its probability density function). A characteristic

*ε*for deep water is (Fedele and Tayfun 2009)

*σ*

_{1}=

*m*

_{001}/

*m*

_{000}is the average angular frequency and

*ε*in (7) and (18), even though the first represents a monochromatic wave field and the second is a measure of the spectral steepness with finite bandwidth.

*η*(

*t*),

*τ*and angle brackets 〈⋅〉 denote time lag and temporal mean, respectively. The narrow-bandedness is defined as

*ϕ*(

*τ*). Typical values for

*ε*, it is defined as (Benetazzo et al. 2021a)

*h*

_{1}is the normalized mode of the probability density function of the linear STE (see appendix A in Benetazzo et al. 2017), and the Euler–Mascheroni constant

*γ*≃ 0.5772 is obtained by following the asymptotic extreme distribution by Gumbel (1958). For a single point in space, i.e.,

*X*=

*Y*= 0, (23) reduces to the time-extreme model proposed by Tayfun (1980), which in turn reduces to the model proposed by Longuet-Higgins (1957) for

*ε*= 0. The expected maximum linear crest-to-trough wave heights within Γ can be computed from the linear

*ε*= 0, (Boccotti 2000),

*β*=

*H*/4,

_{s}*N*

_{3D}, and

*ε*, and vice versa (Benetazzo et al. 2021a). Similar modulations are found for

*β*,

*N*

_{3D}, and |

*F*and

_{η}*F*denote the functional dependence with respect to the maximum

_{H}*η*and

*H*, respectively.

### b. Sea state homogeneity under ambient currents

The wave spectrum *E*(*σ*, *θ*), and associated integrated variables, from a wave record at a single point *x _{i}*,

*y*can be computed if the sea surface elevation

_{i}*η*(

*t*,

*x*,

_{i}*y*) can be considered a stationary Gaussian process. Such an assumption generally holds for wave records with maximum duration

_{i}*D*= 15–30 min (Holthuijsen 2007, p. 56). Similarly, homogeneous means that variables are statistically invariant in space so that

*E*(

*σ*,

*θ*), computed over a duration interval

*D*, does not change within the area. Such an assumption generally holds for square areas with sides of about 10 wavelengths in the open ocean (Boccotti 2000, p. 251). In wave modeling, the homogeneity condition in space and time is satisfied by keeping

*X*and

*Y*within

*denotes the horizontal gradient operator.*

_{h}## 4. Model specifications, study region, and observations

### a. Spectral wave model and oceanic current forcing

*N*=

*N*(

*σ*,

*θ*;

**x**,

*t*) with a nonzero right hand side of (5), which in deep water takes the form (Komen et al. 1994)

*is the wavenumber gradient operator and the wave kinematics on the left hand side are*

_{k}*S*

_{in}and the wave breaking

*S*

_{ds}together with the nonlinear quadruple wave–wave interaction

*S*

_{nl}make up the source terms in (28).

Two 800-m resolution WAM simulations were carried out. The first included only wind forcing and lateral spectral boundary conditions from a coarser (4 km) outer wave model. Wind forcing was taken from the operational 2.5-km resolution Arome Arctic NWP model operated by the Norwegian Meteorological Institute, with further specifications given in Müller et al. (2017). The second run also included surface current forcing from MET Norway’s operational ROMS (Regional Ocean Modeling System; Shchepetkin and McWilliams 2005) model, also at an 800-m horizontal resolution. The ocean surface current was included in the wave kinematics, (29) and (30). The two model simulations are hereafter referred to as W and W+C, which stand for wind and wind + currents, respectively. These simulations are based on the same model setup as reported by Halsne et al. (2022), which provide more details about the forcing fields and wave model specifications. However, the WAM simulations were further extended by including the computation of STEs (Benetazzo et al. 2021b). Here, the dimensions of the space–time domain Γ = *XYD* were *X* = *Y* = 200 m, and the duration was *D* = 1200 s, after the general recommendations in Benetazzo et al. (2021b).

### b. Moskstraumen and characteristic met-ocean conditions

The Lofoten region is located within the belt of westerlies and thus characterized by westerly waves coming from the open ocean, which include local wind sea and the near constant presence of remotely generated swell. On the east side of the strait, we find the Vestfjorden basin, which is about 100 km wide in the east–west direction (Fig. 1). The Lofoten area is therefore not exposed to swell from the east, but will become subject to local easterly wind sea under certain synoptic situations.

There is an asymmetry in the flow field when Moskstraumen is flowing west and east (Børve et al. 2021). When flowing west, Moskstraumen takes the form of a narrow jet with eddies occurring in the vicinity regions with strong shear. Flowing east, Moskstraumen is much broader in extent, and thus characterized with a more uniform flow field. This flow field is exemplified in Fig. 1 but also seen in the 800-m ocean model (Fig. 2a). Even though the ocean model is able to provide a qualitatively good representation of Moskstraumen, it is incapable of resolving all the complex subgrid processes. For example, when the current turns from flowing eastward to westward at slack tide, the northern part turns first and then gradually further south, which results in an area of strong horizontal shear (Halsne et al. 2022). The gradual turning is resolved in the ocean model, but the timing and magnitude of the gradients are not always correct. The phases when Moskstraumen is flowing west and east are hereafter referred to as outgoing tide and incoming tide, respectively. An example of Moskstraumen during maximum speed at incoming tide, together with its impact on the wave field in WAM, is shown in Fig. 2.

Snapshots of Moskstraumen during maximum speed and its impact on the wave field. Panels denote (a) the current speed and direction, (b) *H _{s}* from W+C, (c)

*H*from W, and (d) their relative difference according to (31). Black arrows in (b) and (c) denote the peak wave direction.

_{s}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Snapshots of Moskstraumen during maximum speed and its impact on the wave field. Panels denote (a) the current speed and direction, (b) *H _{s}* from W+C, (c)

*H*from W, and (d) their relative difference according to (31). Black arrows in (b) and (c) denote the peak wave direction.

_{s}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Snapshots of Moskstraumen during maximum speed and its impact on the wave field. Panels denote (a) the current speed and direction, (b) *H _{s}* from W+C, (c)

*H*from W, and (d) their relative difference according to (31). Black arrows in (b) and (c) denote the peak wave direction.

_{s}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

### c. ADCP observations

Three months during winter 2019, concurrent wave and current measurements from a bottom-mounted Nortek Signature 500-kHz acoustic Doppler velocity profiler (ADCP) were available on the east side of Lofoten (see magenta dot in Fig. 1). Here, wave measurements at 2 Hz from a five-beam configuration (one vertical and four slanted) in burst mode were acquired every 30 min, and each burst lasted 17 min. Vertical current profiles were also acquired simultaneously at 2 Hz with vertical resolution of 2 m, together with average mode measurements made up by 60 samples every 10 min with similar vertical resolution. These measurements were presented by Saetra et al. (2021) and Halsne et al. (2022).

In addition to the subgrid processes, the complex environment with its irregular coastline and strong currents makes it challenging to obtain an accurate spatiotemporal collocation of the model data and the observations. Another source causing spatial shift in model grid point values is the interpolation of the ROMS current field onto the WAM model grid projection. Furthermore, the coastline in the two models are slightly different (not shown). In our analysis, we found better agreement in both phase and magnitude for wave parameters at model grid points in the vicinity of the ADCP location rather than in the exact location (not shown). We have selected the nearest grid point with 2D wave spectral output, about 2 km southwest of the ADCP location.

## 5. Results

In the following we first consider a period at the end of January 2019, where a local easterly wind sea from Vestfjorden was opposing the broad and uniform eastward current (Fig. 2). The situation lasted for about 5 days and was due to a high pressure ridge of about 1022 hPa located over Lofoten which set up wind speeds *U*_{10} of 3–12 m s^{−1} from east-southeast (not shown). Here, the sea state was bimodal with a local wind sea component together with a gentle southwesterly swell with 1 < *H _{s}* < 3 m (Figs. 1 and 2). This period was the only time during the 3-month ADCP deployment when easterly wind conditions lasted more than 2 days. This particular period was also investigated by Halsne et al. (2022). We then consider a period in early January with prevailing northwesterly swell opposing Moskstraumen, now shaped as a narrow jet on the offshore side at outgoing tide. Most emphasis is put on the first period, since there the observations are in the region of strong wave–current interaction.

The spectral parameters used for the extreme wave analysis should be computed from the intrinsic spectrum *E*(*f _{i}*,

*θ*) (

*f*=

_{i}*σ*/2

*π*). Thus, a transformation had to be applied on the wave model output since it is given in absolute frequencies,

*f*=

_{a}*ω*/2

*π*. The transformation from an absolute to an intrinsic reference frame is presented in the appendix.

*X*between the two wave models is defined as

### a. Sea state modulation in Moskstraumen

#### 1) Sea state homogeneity

The horizontal homogeneity condition in (27) is treated in Fig. 3, by using a representative wind sea peak period of 6 s. The ratios *i* = 1, 2 denote the *x* and *y* direction, respectively. The horizontal homogeneity condition was also satisfied during other stages in the tidal cycle, and during swell and tidal jet conditions (not shown).

Computing the criteria for the horizontal homogeneity according to (27) under maximum current speed for a *T* = 6 s period wave. Panels show the horizontal current gradient in the (left) *y* (*x _{i}* =

*x*

_{2}) and (right)

*x*direction (

*x*=

_{i}*x*

_{1}) on a logarithmic scale.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Computing the criteria for the horizontal homogeneity according to (27) under maximum current speed for a *T* = 6 s period wave. Panels show the horizontal current gradient in the (left) *y* (*x _{i}* =

*x*

_{2}) and (right)

*x*direction (

*x*=

_{i}*x*

_{1}) on a logarithmic scale.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Computing the criteria for the horizontal homogeneity according to (27) under maximum current speed for a *T* = 6 s period wave. Panels show the horizontal current gradient in the (left) *y* (*x _{i}* =

*x*

_{2}) and (right)

*x*direction (

*x*=

_{i}*x*

_{1}) on a logarithmic scale.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

For the stationarity condition in (27), time derivatives from the observed current using representative intrinsic frequencies for the swell and wind sea components are presented in Fig. 4. Also here, the criterion of a current field varying much slower than a characteristic wave scale, *β*^{2}, and its potential drift during the 17-min burst period for different stages in the tidal cycle. Here, no significant deviations were found during each burst period (not shown).

Computing the criteria for the slowly varying current assumption in time (27). (top) The time series of the measured current speed *U* and (middle) the time derivative for 7 tidal cycles. (bottom) The scaled time derivative of *U* on a *T* = 13 s period wave (orange) and a *T* = 6 s period wave (blue).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Computing the criteria for the slowly varying current assumption in time (27). (top) The time series of the measured current speed *U* and (middle) the time derivative for 7 tidal cycles. (bottom) The scaled time derivative of *U* on a *T* = 13 s period wave (orange) and a *T* = 6 s period wave (blue).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Computing the criteria for the slowly varying current assumption in time (27). (top) The time series of the measured current speed *U* and (middle) the time derivative for 7 tidal cycles. (bottom) The scaled time derivative of *U* on a *T* = 13 s period wave (orange) and a *T* = 6 s period wave (blue).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

#### 2) Tidal modulation of the wave field and the uni- and directional spectrum

When the easterly wind sea opposed the broad uniform current, the wave model with current (W+C) showed a region of increased *H _{s}* with a shape similar to the tidal current (Fig. 2b). The met-ocean conditions suggests that wave straining is the dominating mechanism for three reasons: that is, (i) its sensitivity to higher frequencies in (1), (ii) the horizontal current gradients are strongest at the edges of the broad current and more uniform in the center (not shown) and consequently less exposed to caustics compared with a narrow jet (Kenyon 1971; Dysthe 2001), and (iii) both the active wind forcing and the short-crested nature of the wind sea are working against the veering of the rays, and consequently the impact of refraction is more diffuse compared with a narrow swell spectrum (Rapizo et al. 2016; Holthuijsen and Tolman 1991). However, and even if the wave straining mechanism dominate in the model when the wind sea opposes Moskstraumen, it does not imply that wave straining was the dominating mechanism in the observations since there are processes like wave breaking and strong shear going on below scales of 800 m.

The observed and W+C unidirectional spectra had a similar relative variance distribution on the wind sea and swell components in the bimodal spectrum (Figs. 5e,f). The semidiurnal M_{2} modulation of the wind sea was well predicted by W+C, but the magnitude, and thus *H _{s}*, was at times off by about 1 m. There may be several reasons for such deviations, but the one around 1200 UTC 23 January (see red arrows in Fig. 5g) was due to the grid point resolution in the ROMS model. Here, Moskstraumen turned 180° prior to the observed current and opposed the swell, resulting in an increase in

*H*. Furthermore, the wave energy was at times located on lower frequencies in the model compared with the observations, as seen from about 1200 UTC 26 January and throughout the period in the lower panel in Figs. 5e and 5f. Here, the

_{s}*U*

_{10}decreased to about 3 m s

^{−1}in the atmospheric model (not shown). Another limitation with the measured 2D spectra was the cutoff in directional measurements at 0.2 Hz (Fig. 5c), related to the 500-kHz carrier frequency of the ADCP.

Comparing (left) snapshots of modeled and observed 2D spectrum, together with (right) the temporal evolution in the 1D spectra and *H _{s}*. Snapshots of 2D spectra are taken from 1100 UTC 25 January (see vertical dashed line). Output from the wave model forced with (a),(d) wind (W), (b),(e) wind and current (W+C), and (c),(f) the ADCP observations (Obs.). At 50-m depth, the ADCP cannot measure wave directions for frequencies above 0.2 Hz [see (c)]. A more complete frequency coverage is provided by the observed 1D spectrum [see (f)]. (g) The spectral significant wave height

*H*is shown for the observations (green line), W+C (blue line), and W (orange line). Red arrows around 1200 UTC 23 January denote the shift in the wave energy caused by the observed current turning before the model current, and the red arrows around 0000 UTC 27 January denote the different spectral wave energy distribution in the model vs the observations. Note that the color scale for the 2D spectra represents a scaled version of the 1D spectra, as the units are scaled by degrees.

_{s}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Comparing (left) snapshots of modeled and observed 2D spectrum, together with (right) the temporal evolution in the 1D spectra and *H _{s}*. Snapshots of 2D spectra are taken from 1100 UTC 25 January (see vertical dashed line). Output from the wave model forced with (a),(d) wind (W), (b),(e) wind and current (W+C), and (c),(f) the ADCP observations (Obs.). At 50-m depth, the ADCP cannot measure wave directions for frequencies above 0.2 Hz [see (c)]. A more complete frequency coverage is provided by the observed 1D spectrum [see (f)]. (g) The spectral significant wave height

*H*is shown for the observations (green line), W+C (blue line), and W (orange line). Red arrows around 1200 UTC 23 January denote the shift in the wave energy caused by the observed current turning before the model current, and the red arrows around 0000 UTC 27 January denote the different spectral wave energy distribution in the model vs the observations. Note that the color scale for the 2D spectra represents a scaled version of the 1D spectra, as the units are scaled by degrees.

_{s}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Comparing (left) snapshots of modeled and observed 2D spectrum, together with (right) the temporal evolution in the 1D spectra and *H _{s}*. Snapshots of 2D spectra are taken from 1100 UTC 25 January (see vertical dashed line). Output from the wave model forced with (a),(d) wind (W), (b),(e) wind and current (W+C), and (c),(f) the ADCP observations (Obs.). At 50-m depth, the ADCP cannot measure wave directions for frequencies above 0.2 Hz [see (c)]. A more complete frequency coverage is provided by the observed 1D spectrum [see (f)]. (g) The spectral significant wave height

*H*is shown for the observations (green line), W+C (blue line), and W (orange line). Red arrows around 1200 UTC 23 January denote the shift in the wave energy caused by the observed current turning before the model current, and the red arrows around 0000 UTC 27 January denote the different spectral wave energy distribution in the model vs the observations. Note that the color scale for the 2D spectra represents a scaled version of the 1D spectra, as the units are scaled by degrees.

_{s}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Snapshots of the temporal evolution of the modeled wave spectrum are shown in Fig. 6. Here, the wind sea broadened in direction and increased in frequency up to 0.3 Hz due to the opposing current (middle row). The spectra from the W simulation were stationary during incoming tide (top row). Clearly, the energy on the swell components reduced when propagating in the current direction (bottom row). Considering the unidirectional spectrum, both the energy and mean frequency level increased in W+C (rightmost column). The current speed reached 2 m s^{−1}, which exceeds the blocking velocity for the 5-s wave present in W, which according to (6) is −1.95 m s^{−1}.

Snapshots of the currents impact on the intrinsic wave spectrum during incoming tide (see Figs. 1 and 2). Rows show the normalized wave spectrum from the wave model forced with (top) wind (W), (middle) wind and currents (W+C), and (bottom) their difference. Columns on the left denote subsequent time steps, where the center column indicates the maximum current speed. The red arrows show the current direction, and the current speed (m s^{−1}) is denoted with red text. The rightmost column denotes the nonnormalized 1D spectrum.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Snapshots of the currents impact on the intrinsic wave spectrum during incoming tide (see Figs. 1 and 2). Rows show the normalized wave spectrum from the wave model forced with (top) wind (W), (middle) wind and currents (W+C), and (bottom) their difference. Columns on the left denote subsequent time steps, where the center column indicates the maximum current speed. The red arrows show the current direction, and the current speed (m s^{−1}) is denoted with red text. The rightmost column denotes the nonnormalized 1D spectrum.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Snapshots of the currents impact on the intrinsic wave spectrum during incoming tide (see Figs. 1 and 2). Rows show the normalized wave spectrum from the wave model forced with (top) wind (W), (middle) wind and currents (W+C), and (bottom) their difference. Columns on the left denote subsequent time steps, where the center column indicates the maximum current speed. The red arrows show the current direction, and the current speed (m s^{−1}) is denoted with red text. The rightmost column denotes the nonnormalized 1D spectrum.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

#### 3) Integrated spectral parameters

Time evolution in the key sea state parameters from (25) and (26) are shown in Fig. 7, and the intermodel differences are listed in Table 2. An increase in *H _{s}* and

*ε*occurred when the wind sea was opposing the current, except for

*H*during the first tidal cycle around 1200 UTC 23 January as mentioned in the previous section (Fig. 5g). The phase of the modulation in

_{s}*ε*was in general accordance with the observations with a correlation coefficient between

*ε*from W+C and the observations of 0.80. Note that the observed sea state parameters are computed from the intrinsic unidirectional spectrum. The

*H*in W+C exceeded the W simulation by up to 44%, and

_{s}*ε*by 167% (Table 2).

Temporal evolution in the key integrated spectral parameters (a) *H _{s}*, (b)

*ε*, (c)

*N*

_{3D}, and (d)

*ϑ*< −0.5 [see (4)].

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Temporal evolution in the key integrated spectral parameters (a) *H _{s}*, (b)

*ε*, (c)

*N*

_{3D}, and (d)

*ϑ*< −0.5 [see (4)].

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Temporal evolution in the key integrated spectral parameters (a) *H _{s}*, (b)

*ε*, (c)

*N*

_{3D}, and (d)

*ϑ*< −0.5 [see (4)].

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Intermodel relative differences in integrated spectral variables and normalized expected extreme waves according to (31). The mean, maximum (max) and standard deviation (std) are given columnwise. The relative differences are given in percentage units (%). Values reflect the time periods in Figs. 7 and 9.

The average number of waves *N*_{3D} from (10) also increased with currents opposing the wind sea due to the shift in frequency to shorter waves (Fig. 7c). Consequently, *L _{x}*,

*L*,

_{y}*T*all decreased (not shown). The impact by the degree of organization in the space–time wave field

_{m}*A*on

*N*

_{3D}was less systematic during the tidal phases, which made the influence by the tidal current difficult to interpret (not shown). Less systematic differences were also found for the absolute narrow-bandedness |

*U*

_{10}≃ 3 m s

^{−1}, the energy on the swell and remaining wind sea was equally partitioned during outgoing tide, leading to the decrease in |

#### 4) Steepness modulation and simplified quasi-stationary model

*ψ*∈ [0.75, 1], and

*ψ*is obtained for

*ν*= 0.5, and similar values were obtained in W+C and W such that

The comparison is shown in Fig. 8 by using *k*_{1} from *T _{m}*

_{01}in W. Here, (7) gave similar results as the intermodel ratio. The spiky overshoots from (7) can be attributed to the lack of wave dissipation in the simplified model. Also, when comparing (7) against

*ε*

_{W+C}/

*ε*

_{W}from the partitioned wind sea part of the spectrum (using the spectral partitioning algorithm from https://github.com/metocean/wavespectra – accessed 17 August 2022), the “troughs” were also realistically captured, which were due to the lengthening of waves on following currents (see green lines). The troughs were not present for the full bimodal spectrum, since then the swell part opposed the current and consequently increased in energy. The similarity between (33) and (7) also suggests that wave straining was the dominating WCI mechanism.

Comparing *ε*/*ε*_{0} from (33) using W+C and W and the simplified wave straining model in (7). (top) The intermodel ratio, i.e., *ε*_{W+C}/*ε*_{W} (black line), and the output from (7) (orange line) using *k*_{1} computed from *T _{m}*

_{01}in W. (bottom)

*U*

_{eff}from (3). Since the wave spectrum was bimodal (Fig. 6), we also added ε

_{W+C}/

*ε*

_{W}from the wind sea partition of the spectrum (green line).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Comparing *ε*/*ε*_{0} from (33) using W+C and W and the simplified wave straining model in (7). (top) The intermodel ratio, i.e., *ε*_{W+C}/*ε*_{W} (black line), and the output from (7) (orange line) using *k*_{1} computed from *T _{m}*

_{01}in W. (bottom)

*U*

_{eff}from (3). Since the wave spectrum was bimodal (Fig. 6), we also added ε

_{W+C}/

*ε*

_{W}from the wind sea partition of the spectrum (green line).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Comparing *ε*/*ε*_{0} from (33) using W+C and W and the simplified wave straining model in (7). (top) The intermodel ratio, i.e., *ε*_{W+C}/*ε*_{W} (black line), and the output from (7) (orange line) using *k*_{1} computed from *T _{m}*

_{01}in W. (bottom)

*U*

_{eff}from (3). Since the wave spectrum was bimodal (Fig. 6), we also added ε

_{W+C}/

*ε*

_{W}from the wind sea partition of the spectrum (green line).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

### b. Extreme wave modulation

#### 1) Tidal modulation of extremes

The ratio *ε* with a correlation coefficient of 0.92 in W+C (Figs. 7b and 9a). Maximum values mostly coincided with the maximum of *ε*, and the W+C predictions exceeded W up to 14% (Table 2). The intermodel difference between the linear predictions in W+C and W, i.e., considering *F _{η}*(

*ε*= 0,

*N*

_{3D}) in (25), demonstrate the contribution by

*N*

_{3D}(see dashed lines Fig. 9a). The fluctuations, however small, in the linear

*N*

_{3D}, and the increase on counter currents is due to the wave lengths becoming shorter by the frequency shift, which is in line with the results by Barbariol et al. (2015). The offset between the linear and nonlinear

*ε*contribution than further increasing the number of waves due to currents.

Temporal evolution in expected extremes computed from the 2D spectra. Panels show (a) *X* = *L _{x}*,

*Y*=

*L*and

_{y}*D*= 100

*T*, (c)

_{m}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Temporal evolution in expected extremes computed from the 2D spectra. Panels show (a) *X* = *L _{x}*,

*Y*=

*L*and

_{y}*D*= 100

*T*, (c)

_{m}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Temporal evolution in expected extremes computed from the 2D spectra. Panels show (a) *X* = *L _{x}*,

*Y*=

*L*and

_{y}*D*= 100

*T*, (c)

_{m}Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Expected extremes over a domain of variable size were also analyzed to further elucidate their sensitivity to *ε* and *X* = *L _{x}*,

*Y*=

*L*, and

_{y}*D*= 100

*T*. Then, only one wave, on average, is included in the horizontal space domain, and consequently

_{m}*N*

_{3D}≈ 100

*A*. Such a choice also allows to assess the impact by

*A*,

*ε*, and

*A*during the tidal cycles, and the values were also often similar in W and W+C. Consequently, the tidal modulation was governed by the modulation in

*ε*and

*F*(

_{η}*ε*= 0,

*N*

_{3D}) from (25)] were similar for W+C and W (see dashed blue and orange lines). Thus, the intermodel differences in the second order

*ε*, now with a correlation of 0.98 in W+C, and consequently the most extreme conditions occurred when the wind sea opposed Moskstraumen.

The

#### 2) Horizontal variability in expected extremes

When wind sea opposed Moskstraumen,

Horizontal variability in normalized extreme wave crests (a),(b) ^{−1}).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Horizontal variability in normalized extreme wave crests (a),(b) ^{−1}).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Horizontal variability in normalized extreme wave crests (a),(b) ^{−1}).

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

The *H _{s}* (Figs. 10c,d), with a uniform horizontal distribution. We find an increase in

#### 3) Time extremes from observations

The single point observations presented here provide limited statistics due to the seldom occurrence of easterly wind situations. Moreover, the observations also include the signal of complex small-scale variability unresolved in the ocean and wave model (Fig. 5). Nevertheless, and despite such limitations, they can be used to consider the trends in the expected extremes during co- and counterflow situations. However, only the temporal extremes, i.e., with *X* = *Y* = 0, can be compared.

Stochastic time extremes from the ADCP observations were computed following the procedure outlined by Barbariol et al. (2019). That is, each 17-min burst, acquired twice per hour, was split into three equal subsegments, i.e., each with a duration of approximately 5.5 min. In each subsegment, the *η*_{MAX} and *H*_{MAX} were computed from a zero crossing analysis, and the *dynamic time warping* method. Here, a distance metric *d*_{dtw} is computed from a point-to-point matching of indices in a monotonically increasing sequence. Peaks that are out of phase will be matched if they are within a certain window size. Typical applications of the dynamic time warping method is found in automatic speech recognition, where sequences with different speeds can be matched. Lower values of *d*_{dtw} indicate shorter distances and a better fit.

A comparison between the model and observations is given in Fig. 11. Here, distinct local peaks in skewness occur for at least five out of the eight tidal cycles when the wind sea and current were opposing. The _{2} frequency when the wind sea opposed Moskstraumen (Fig. 11b), however often underestimating the magnitude. Note that here *H _{s}* from W+C and the observations were quite similar (Fig. 5g), the underestimation may be linked to energy being larger on the wind sea components in the observed spectra than the W+C (see the two first tidal cycles in Figs. 5e,f). From the normalized

*η*/

*H*> 1.25 (Dysthe et al. 2008), also occurred for the wind sea on counter currents (see red dots). For

_{s}*d*

_{dtw,W+C}= 2.96 and

*d*

_{dtw,W}= 3.19.

Comparing time series of expected time extremes from wave model against observations. (a) The skewness computed from the 17-min burst observations (green dots) and the hourly mean (solid black line); the expected maximum wave (b) crests and (c) heights in time (i.e., single point) in its dimensional form from the observations (solid black line), W+C (blue line), and W (orange line). Here, red dots show the cases where extreme events occurred according to the definition.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Comparing time series of expected time extremes from wave model against observations. (a) The skewness computed from the 17-min burst observations (green dots) and the hourly mean (solid black line); the expected maximum wave (b) crests and (c) heights in time (i.e., single point) in its dimensional form from the observations (solid black line), W+C (blue line), and W (orange line). Here, red dots show the cases where extreme events occurred according to the definition.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Comparing time series of expected time extremes from wave model against observations. (a) The skewness computed from the 17-min burst observations (green dots) and the hourly mean (solid black line); the expected maximum wave (b) crests and (c) heights in time (i.e., single point) in its dimensional form from the observations (solid black line), W+C (blue line), and W (orange line). Here, red dots show the cases where extreme events occurred according to the definition.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

For _{2} modulation between W+C and the observations (Fig. 11c). The *d*_{dtw,W+C} = 3.93 and *d*_{dtw,W} = 4.52, implying a better fit for the former. However, the observed extremes, according to the definition of *H*/*H _{s}* > 2, occurred when the swell partition opposed Moskstraumen during outgoing tide (see red dots in Fig. 11c). Moreover, the peaks were also here underestimated by the model, and there was no clear M

_{2}modulation in the normalized

### c. Opposing swell and tidal jet during outgoing tide

The other interesting case of characteristic wave and tidal current occurs when Moskstraumen is flowing westward. The Moskstraumen now takes the form of a narrow jet (Fig. 12a). Since swell conditions often prevail on the offshore side, the spectrum is often unimodal. A summary of the swell (wave age *c _{p}*/

*U*

_{10}≃ 18/7.5 > 1, where

*c*is phase speed) and tidal current conditions during a period in early January 2019 is given in Fig. 12. Unfortunately, no observations were available on the offshore side of the Lofoten archipelago.

_{p}(a) Summary of the predominant north westerly swell and tidal current conditions on the west side of Lofoten 2–8 Jan 2019. During outgoing tide, Moskstraumen takes the form as a narrow tidal jet, and (b) a snapshot of the swell and tidal jet interaction from W+C is, where blue arrows indicate peak wave direction. Wave rays are computed for a *T* = 13 s period wave. (c),(d) Time series of the unidirectional spectra from W and W+C [taken from the magenta/black dot in (b)] are shown.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

(a) Summary of the predominant north westerly swell and tidal current conditions on the west side of Lofoten 2–8 Jan 2019. During outgoing tide, Moskstraumen takes the form as a narrow tidal jet, and (b) a snapshot of the swell and tidal jet interaction from W+C is, where blue arrows indicate peak wave direction. Wave rays are computed for a *T* = 13 s period wave. (c),(d) Time series of the unidirectional spectra from W and W+C [taken from the magenta/black dot in (b)] are shown.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

(a) Summary of the predominant north westerly swell and tidal current conditions on the west side of Lofoten 2–8 Jan 2019. During outgoing tide, Moskstraumen takes the form as a narrow tidal jet, and (b) a snapshot of the swell and tidal jet interaction from W+C is, where blue arrows indicate peak wave direction. Wave rays are computed for a *T* = 13 s period wave. (c),(d) Time series of the unidirectional spectra from W and W+C [taken from the magenta/black dot in (b)] are shown.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

The tidal jet clearly modulated the spectrum at the M_{2} frequency, which led to a more energetic wave field compared with no current forcing (Figs. 12c,d). Solving the wave ray equations (29) and (30) numerically using the method by Halsne et al. (2023) and the tidal current field and bathymetry as input, the convergence of wave rays suggests that current-induced refraction was the dominating WCI mechanism (Fig. 12b). Moreover, the wave straining mechanism becomes less dominant the longer the waves are (Table 1), and here the peak period was at times 13 s. The wave field became much more energetic during these episodes with swell opposing the tidal jet, with an increase in *H _{s}* up to 90% (Fig. 13a). In addition to the increase in energy, the 2D spectrum also underwent significant directional broadening (not shown).

Time series of key sea state parameters and associated expected extremes over a variable size domain during swell and tidal jet interactions. Panels show (a) *H _{s}*, (b)

*ε*/

*ε*

_{0}as in Fig. 8 (yellow shading shows the excess of the inter model ratio), (c)

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Time series of key sea state parameters and associated expected extremes over a variable size domain during swell and tidal jet interactions. Panels show (a) *H _{s}*, (b)

*ε*/

*ε*

_{0}as in Fig. 8 (yellow shading shows the excess of the inter model ratio), (c)

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Time series of key sea state parameters and associated expected extremes over a variable size domain during swell and tidal jet interactions. Panels show (a) *H _{s}*, (b)

*ε*/

*ε*

_{0}as in Fig. 8 (yellow shading shows the excess of the inter model ratio), (c)

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

The maximum increase in *ε* by the tidal jet exceeded the wind-only simulation (W) by a factor 2 (not shown). We observe that the analytical wave straining model (7) is mostly incapable of capturing the modulation in ratio *ε*_{W+C}/*ε*_{W}, as seen from the excess in yellow color shading in Fig. 13b. The excess can be understood if we consider *ε* as in (32) to be the product of wave amplitude and wavenumber only (i.e., skipping the finite bandwidth measure *ψ*). Then, the convergence of wave energy due to caustics leads to an increase in the wave amplitude part, while the wavenumber is less modulated. These results suggest that different WCI mechanisms may modulate the extreme wave crest statistics differently, due to the aforementioned sensitivity in *ε*. The ratio *N*_{3D} parameter since *ε* dominates the variability. The correlation between *ε* in W+C was 0.99. Following the reasoning about the impact by refraction on *ε*, the increase in *H _{s}* also constrain the ratio

For the

Maximum

Horizontal variability in normalized extreme wave crests (a),(b) ^{−1}.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Horizontal variability in normalized extreme wave crests (a),(b) ^{−1}.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

Horizontal variability in normalized extreme wave crests (a),(b) ^{−1}.

Citation: Journal of Physical Oceanography 54, 1; 10.1175/JPO-D-23-0051.1

## 6. Discussion

### a. Steepness modulation and the impact of Moskstraumen on ${\overline{\eta}}_{\text{MAX}}$

We find *ε* than those in *N*_{3D} (Figs. 9a,b and 13c). In both the cases considered, the expected extreme crests increased most when the waves were opposing Moskstraumen. Consequently, our results suggest that the simultaneous increase in the components of *ε* due to wave straining is an important factor in modulating *ε*_{W+C}/*ε*_{W} suggests that convergence of wave energy due to caustics provide an additional contribution to the steepness modulation (Fig. 13b), which indicates that the current-induced extreme wave modulations are sensitive to the underlying WCI mechanisms.

Interestingly, *H _{s}* in the latter case constrained the ratio

*N*

_{3D}increased

The horizontal variability of *H _{s}* has not reached its maximum value (left panels of Figs. 10 and 14).

In the case where our observations coincided with the region of strong wave–current interaction, we also found a similar trend in the tidal modulation of

Summarized, our results show that STE crests are very sensitive to the current-induced modulation in *ε*, and also suggests that including tidal current forcing in spectral wave models provides more realistic modulation of the expected maximum wave crests. Consequently, the expected maximum space–time wave crest parameters now available in spectral wave models can be useful in nearshore wave forecasting and in engineering applications like wave load analysis of tidal power facilities and other marine structures.

### b. Narrow-bandedness and ${\overline{H}}_{\text{MAX}}$

The modulation in |

The decrease in

## 7. Conclusions

We have investigated the wave, and extreme wave, modulation by one of the strongest open ocean tidal currents in the world, namely, the Moskstraumen in northern Norway. The study has considered the influence by Mosktraumen under two characteristic met-ocean conditions where (i) a bimodal sea state encountered a broad, uniform countercurrent, and (ii) a swell system encountered an opposing tidal jet. Methods and data include output from a spectral wave model with and without current forcing, accompanied by a simplified quasi-stationary model for wave steepness modulation, and in situ observations. The largest wave modulations occurred when the waves were opposing Moskstraumen, in both cases, where key parameters like the significant wave height *H _{s}* and spectral steepness

*ε*increased.

The second-order non-Gaussian contribution through *ε* in the expected maximum space–time wave crests *ε* than the average number of waves within the space–time domain *N*_{3D} from Eq. (26). We found a similar trend in tidal modulation when comparing time extremes from model and observations, although the model underestimated the magnitude of the expected extremes. Nevertheless, our results suggest that extreme wave crests in a time and space–time domain become more likely in the presence of a strong opposing tidal current, and that using tidal current forcing in wave models improves their estimates.

Current-induced modulations in the expected space–time wave heights

Our findings indicate that current-induced modulations in expected extremes are sensitive to the underlying WCI mechanism. For instance, wave straining will increase *ε* and *N*_{3D} for short waves encountering a broad countercurrent, i.e., similar to the conditions in (i), and a strong increase in *ε* and *H _{s}* are found during (ii), where refraction seemingly dominates. For the latter, however, the increase in

*H*constrains the ratios

_{s}## Acknowledgments.

This research was partly funded by the Research Council of Norway through the project MATNOC (Grant 308796). TH and ØB are grateful for additional support from the Research Council of Norway through the StormRisk project (Grant 300608). AB and FB acknowledge the contribution from the Korea Institute of Ocean Science and Technology in the context of the project ASTROWAVES.

## Data availability statement.

The ROMS model data are available from https://thredds.met.no/thredds/dodsC/sea/norkyst800m/1h/aggregate\_be. WAM and ADCP data are available from the Norwegian Meteorological Institute upon request.

## APPENDIX

### Transformation from Absolute to Intrinsic Frequencies

**k**= (

*k*,

_{x}*k*) and the velocity vector

_{y}**U**= (

*u*,

*υ*) must be considered. The wavenumber

*k*= |

**k**|, and the angle between

*k*and

_{x}*k*is

_{y}*θ*. The Jacobian

_{w}**k**,

**u**,

*k*, and

*c*must be computed for every direction

_{g}*θ*

_{w}_{,}

*in the discrete spectrum. The group velocity is defined*

_{j}**k**vectors must be considered such that

**k**

*=*

_{j}**k**

*(*

_{j}*θ*). Once the Jacobian is computed, the variance density for each frequency for each must be remapped from the absolute to intrinsic frequencies from the Doppler shift equation (1). The remapping was performed by using linear interpolation.

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