Wave Modulation in a Strong Tidal Current and Its Impact on Extreme Waves

Trygve Halsne aNorwegian Meteorological Institute, Oslo, Norway
bUniversity of Bergen, Bergen, Norway

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Alvise Benetazzo cInstitute of Marine Sciences, Italian National Research Council, Venice, Italy

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Francesco Barbariol cInstitute of Marine Sciences, Italian National Research Council, Venice, Italy

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Kai Håkon Christensen aNorwegian Meteorological Institute, Oslo, Norway
dUniversity of Oslo, Oslo, Norway

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Ana Carrasco aNorwegian Meteorological Institute, Oslo, Norway

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Øyvind Breivik aNorwegian Meteorological Institute, Oslo, Norway
bUniversity of Bergen, Bergen, Norway

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Abstract

Accurate estimates of extreme waves are central for maritime activities, and stochastic wave models are the best option available for practical applications. However, the way currents influence the statistics of space–time extremes in spectral wave models has not been properly assessed. Here we demonstrate impacts of the wave modulation caused by one of the world’s strongest open ocean tidal currents, which reaches speeds of at least 3 m s−1. For a bimodal swell and wind sea state, we find that most intense interactions occur when the wind sea opposes the tidal current, with an increase in significant wave height and spectral steepness up to 45% and 167%, respectively. The steepness modulation strengthens the second-order Stokes contribution for the normalized extreme crests, which increases between 5% and 14% during opposing wind sea and current. The normalized extreme wave heights have a strong dependence on the narrow-bandedness parameter, which is sensitive to the variance distribution in the bimodal spectrum, and we find an increase up to 12% with currents opposing the wind sea. In another case of swell opposing a tidal jet, we find the spectral steepness to exceed the increase predicted by a simplified modulation model. We find support in single-point observations that using tidal currents as forcing in wave models improves the representation of the expected maximum waves, but that action must be taken to close the gap of measurements in strong currents.

Significance Statement

The purpose of this study is to investigate how a very strong tidal current affects the surface wave field, and how it changes the stochastic extreme waves formulated for a space–time domain. Our results suggest that the expected maximum waves become more realistic when tidal currents are added as forcing in wave models. Here, the expected extremes exceed traditional model estimates, i.e., without current forcing, by more than 10%. These differences have implications for maritime operations, both in terms of planning of marine structures and for navigational purposes. However, there is a significant lack of observations in environments with such strong currents, which are needed to further verify our results.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Trygve Halsne, trygve.halsne@met.no

Abstract

Accurate estimates of extreme waves are central for maritime activities, and stochastic wave models are the best option available for practical applications. However, the way currents influence the statistics of space–time extremes in spectral wave models has not been properly assessed. Here we demonstrate impacts of the wave modulation caused by one of the world’s strongest open ocean tidal currents, which reaches speeds of at least 3 m s−1. For a bimodal swell and wind sea state, we find that most intense interactions occur when the wind sea opposes the tidal current, with an increase in significant wave height and spectral steepness up to 45% and 167%, respectively. The steepness modulation strengthens the second-order Stokes contribution for the normalized extreme crests, which increases between 5% and 14% during opposing wind sea and current. The normalized extreme wave heights have a strong dependence on the narrow-bandedness parameter, which is sensitive to the variance distribution in the bimodal spectrum, and we find an increase up to 12% with currents opposing the wind sea. In another case of swell opposing a tidal jet, we find the spectral steepness to exceed the increase predicted by a simplified modulation model. We find support in single-point observations that using tidal currents as forcing in wave models improves the representation of the expected maximum waves, but that action must be taken to close the gap of measurements in strong currents.

Significance Statement

The purpose of this study is to investigate how a very strong tidal current affects the surface wave field, and how it changes the stochastic extreme waves formulated for a space–time domain. Our results suggest that the expected maximum waves become more realistic when tidal currents are added as forcing in wave models. Here, the expected extremes exceed traditional model estimates, i.e., without current forcing, by more than 10%. These differences have implications for maritime operations, both in terms of planning of marine structures and for navigational purposes. However, there is a significant lack of observations in environments with such strong currents, which are needed to further verify our results.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Trygve Halsne, trygve.halsne@met.no

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 (Hs), (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 (N3D), which represents the sample size, and (vi) the narrow-bandedness parameter (ϕ*), which characterizes the width of the frequency spectrum. Barbariol et al. (2015) demonstrated how the linear (i.e., with steepness ε = 0) stochastic crest heights in space–time increased on a countercurrent due to the increase in N3D, 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 N3D 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 ϕ* is used as a measure.

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.

Fig. 1.
Fig. 1.

A cartoon representation of the Moskstraumen tidal current during incoming tide (red colored arrows denote direction), which is located between the southern tip of the Lofoten Peninsula and the island of Mosken in northern Norway. The magenta dot shows the location of the ADCP. Red contours denote the land mask from the wave model. The bimodal sea state during one of our periods of investigation had easterly wind sea and a southwesterly swell component. During incoming tide, Moskstraumen takes the form of a broad, uniform current with eddies in its vicinity. Here the shape of Moskstraumen is exaggerated for illustrative purposes. Moskstraumen has been renowned for centuries for its strength and ferocity, and here the sea surface manifestation of Moskstraumen by Johannes Herbinius from 1678 is included.

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

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.

The Doppler shift equation
ω=σ+kU,
governs the shift in wave frequency. Here, ω is the absolute wave frequency as seen from a fixed point, σ is the intrinsic frequency (following the current), and k = (kx, ky) 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 k = |k| must change when exposed to a changing ambient current through the intrinsic frequency dispersion relation
σ2=gktanh(kh),
where g is the gravitational constant and h is the water depth. We define the effective current
Ueff=Uϑ,
where U = |U|, and the degree of opposition between the waves and the currents, ϑ, is computed by
ϑ=cos(θcθw).
Here, θc, θw 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.
Consider a wave train propagating along the x axis on deep water from an area with Ueff = u0 = 0 to an opposing current Ueff < 0. In the following, subscript 0 denotes the wave characteristics where Ueff = 0. In such a case, ω = σ0 = const. due to wave crest conservation. To compensate for the loss in the kU term of (1), there must be an accompanied increase in k. Increasing k implies a shortening in the wavelength λ = 2π/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)
Nt+(x˙N)=0.
Here, x = cg + U is the absolute wave group velocity vector, and cg=(k/k)(σ/k) the intrinsic group velocity vector. Solving (5) with respect to E for a constant current according to the above considerations we obtain [Phillips 1977, Eq. (3.7.11)]
EE0=c02c2(1+2Ueff/c).
Here, the impact by the currents is reflected in the denominator, and E increases toward the singularity Ueff → −c/2, implying that the waves have been blocked by the current. At the blocking point, Ueff = −c0/4. 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 Ueff > −c0/4. 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.
If we consider the wave steepness ε=Ek, since E is proportional to the square of the wave amplitude a, we can rewrite (6) (Rapizo et al. 2017)
εε0=(1kUeff/ω)31+2Ueff/c.
Here, the kUeff/ω term in the nominator expresses the effect by the current on the wave steepness, which can be recognized by considering (1):
kUeffω=ωσω=Δωω.
Thus, Δω < 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 Ueff = −1.0 m s−1 from a reference of Ueff = 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 Ueff and wave periods are given in Table 1. From the above considerations, both k and E are modulated simultaneously by wave straining, and consequently ε is very sensitive to the ambient current (Vincent 1979).
Table 1.

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, a0 = 1 m. Each row denotes different effective currents Ueff.

Table 1.

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 Hs = 4β, 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

Assuming a Gaussian probability distribution of η for every point within Γ, then the maximum individual crest height ηMAX can be defined in terms of an exceedance probability by a threshold z:
PST,MAX=Pr{ηMAX>z|(x,y,t)Γ},
where subscript ST stands for space–time. The STE model for ηMAX 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 (HMAX), 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.
In essence, the STEs are proportional to Hs, and subsequently modified by parameters that constitute the average number of waves (N3D), wave steepness (ε), and spectral bandwidth (ϕ*), which will now be introduced in that order (Benetazzo et al. 2021a). First, the average number of waves within Γ is (Fedele 2012)
N3D=XYDLxLyTm1αxt2αxy2αyt2+2αxtαxyαyt,
where Lx, Ly, Tm denote length scales associated with the mean crest length (in the 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
N2D=XDLxTm1αxt2+YDLyTm1αyt2+XYLxLy1αxy2,
N1D=XLx+YLy+DTm,
respectively. Studies have shown that the average number of waves within the interior of the space–time domain, i.e., N3D, 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):
A=1αxt2αxy2αyt2+2αxtαxyαyt,
which originates from the determinant of the covariance matrix of 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
αxt=m101m200m002,
αyt=m011m020m002,
αxy=m110m200m020,
where
mijl=kxikyjσlE(σ,θ)dσdθ.
From (10), it follows that N3D 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).
The degree of nonlinearity for a weakly nonlinear sea state is determined by the spectral steepness 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)
ε=βσ12g(1ν+ν2),
where σ1 = m001/m000 is the average angular frequency and
ν=m002m000/m00121,
is a spectral bandwidth parameter proposed by Longuet-Higgins (1975). For simplicity, the wave steepness is denoted by 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.
The bandwidth parameter ϕ* draws upon the quasi-determinism theory of Boccotti (2000) and characterizes the narrow-bandedness of the sea state. Formally, it stems from the autocovariance function for η(t),
ϕ(τ)=η(t)η(t+τ).
Here, τ and angle brackets 〈⋅〉 denote time lag and temporal mean, respectively. The narrow-bandedness is defined as
ϕ*ϕ(τ*)ϕ(0),
where
ϕ(τ)=E(σ)cos(στ)dσ,
and τ* is the time lag of the first minimum of ϕ(τ). Typical values for ϕ* are −1 for an infinitely narrow frequency spectrum and in the range [−0.75, −0.65] for wind-sea conditions (Boccotti 2000).
The expected maximum wave crest within a space–time domain Γ can be derived from the exceedance probability (9). Corrected to second order in ε, it is defined as (Benetazzo et al. 2021a)
η¯MAX,ST=β(h1+ε2h12)+βγ[(1+εh1)×(h12N3Dh1+N2DN3Dh12+N2Dh1+N1D)1],
where subscript ST stands for space–time and expected values are denoted by the overbar operator ()¯. Here, h1 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 η¯MAX,ST, i.e., with ε = 0, (Boccotti 2000),
H¯MAX,ST=β[h1+γ(h12N3Dh1+N2DN3Dh12+N2Dh1+N1D)1]×2(1ϕ*).
Here, the expected extreme height increases with decreasing ϕ*. Thus, H¯MAX is maximized for an infinitely narrow sea state where ϕ*=1 (Boccotti 2000).
In general, η¯MAX,ST in (23) increases with increasing β = Hs/4, N3D, and ε, and vice versa (Benetazzo et al. 2021a). Similar modulations are found for H¯MAX,ST in (24) through the parameters β, N3D, and |ϕ*| (|⋅| denotes the absolute value). Thus, (23) and (24) can be written in a simplified form as (Benetazzo et al. 2021a)
η¯MAX,STHs=Fη(ε,N3D),
H¯MAX,STHs=FH(|ϕ*|,N3D),
where Fη and FH denote the functional dependence with respect to the maximum 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 xi, yi can be computed if the sea surface elevation η(t, xi, yi) can be considered a stationary Gaussian process. Such an assumption generally holds for wave records with maximum duration 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 O(102)m, and smaller than the model grid size (Benetazzo et al. 2021b).

With regards to ambient currents, we consider mean flows with spatiotemporal variability much less than the characteristic length scales for ocean waves. This can be formalized by requiring (Peregrine 1976)
max|1UUt|σ,max|1UhU|k,
where ∇h denotes the horizontal gradient operator.

4. Model specifications, study region, and observations

a. Spectral wave model and oceanic current forcing

To assess the impact by Moskstraumen on the wave field, we used the WAM third-generation spectral wave model (Komen et al. 1994). In WAM, the sea state is modeled by solving the wave action evolution equation, i.e., a spectral representation of 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)
Nt+h(x˙N)+k(k˙N)=Sin+Snl+Sdsσ.
Here, ∇k is the wavenumber gradient operator and the wave kinematics on the left hand side are
x˙=dxdt=cg+U(t,x),
k˙=dkdt=khU(t,x).
Here, (29) is the advection of wave action density and (30) models refraction and the change in wavenumber components as the wave propagation direction is normal to the wavenumber vector. The wind input Sin and the wave breaking Sds together with the nonlinear quadruple wave–wave interaction Snl 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.

Fig. 2.
Fig. 2.

Snapshots of Moskstraumen during maximum speed and its impact on the wave field. Panels denote (a) the current speed and direction, (b) Hs from W+C, (c) Hs from W, and (d) their relative difference according to (31). Black arrows in (b) and (c) denote the peak wave direction.

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 U10 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 < Hs < 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(fi, θ) (fi = σ/2π). Thus, a transformation had to be applied on the wave model output since it is given in absolute frequencies, fa = ω/2π. The transformation from an absolute to an intrinsic reference frame is presented in the appendix.

The relative difference for a variable X between the two wave models is defined as
RD(X)=XW+CXWXW.

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 |(1/U)(U/xi)|k11, where 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).

Fig. 3.
Fig. 3.

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 (xi = x2) and (right) x direction (xi = x1) 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, |(1/U)(U/t)|σ11, was fulfilled. To further support the stationarity condition, wave observations were analyzed by computing the variance β2, 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).

Fig. 4.
Fig. 4.

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 Hs 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 M2 modulation of the wind sea was well predicted by W+C, but the magnitude, and thus Hs, 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 Hs. 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 U10 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.

Fig. 5.
Fig. 5.

Comparing (left) snapshots of modeled and observed 2D spectrum, together with (right) the temporal evolution in the 1D spectra and Hs. 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 Hs 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.

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.

Fig. 6.
Fig. 6.

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 Hs and ε occurred when the wind sea was opposing the current, except for Hs during the first tidal cycle around 1200 UTC 23 January as mentioned in the previous section (Fig. 5g). The phase of the modulation in 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 Hs in W+C exceeded the W simulation by up to 44%, and ε by 167% (Table 2).

Fig. 7.
Fig. 7.

Temporal evolution in the key integrated spectral parameters (a) Hs, (b) ε, (c) N3D, and (d) |ϕ*| from W+C (blue line), W (orange line), and observations (Obs., green line). Observed sea state parameters are computed from the unidirectional spectrum and averaged over an hour. Black and magenta horizontal lines on top denote the phase of Moskstraumen, and the arrows its approximate east (right) and west (left) direction. Gray vertical bins denote when the current and wind waves were opposing with ϑ < −0.5 [see (4)].

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

Table 2.

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.

Table 2.

The average number of waves N3D from (10) also increased with currents opposing the wind sea due to the shift in frequency to shorter waves (Fig. 7c). Consequently, Lx, Ly, Tm all decreased (not shown). The impact by the degree of organization in the space–time wave field A on N3D 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 |ϕ*| (Fig. 7d). We recall that |ϕ*|0 implies a more broad-banded sea state, and that typical values for a wind-sea spectrum are in the range (0.65, 0.75). The lower |ϕ*| in W+C than in W during the first tidal cycle was a result of the swell and wind sea having a similar energy level due to Moskstraumen opposing the latter, which thus caused a broadening the 1D spectrum (Fig. 7d). For the last two tidal cycles, when U10 ≃ 3 m s−1, the energy on the swell and remaining wind sea was equally partitioned during outgoing tide, leading to the decrease in |ϕ*| in W+C. By contrast, for some of the intermediate opposing cycles, the strong shift in variance density to the wind sea components caused a more narrow-banded sea state (Figs. 5e,f). Furthermore, the observations and W+C model predictions show some similar fluctuations, but often have quite different values, which is to be expected since the energy distributions in the spectra were at times quite different (Figs. 5e–g).

4) Steepness modulation and simplified quasi-stationary model

The deep water spectral steepness in (18) can be rewritten
ε=βσ12g(1ν+ν2)=Ek1ψ,
where k1=σ12/g through (2), ψ ∈ [0.75, 1], and β=E. The global minimum for the parabolic function ψ is obtained for ν = 0.5, and similar values were obtained in W+C and W such that ψW+C/ψW1 (not shown). Thus, (32) can be further simplified, and we obtain the intermodel ratio
εε0(Ek1)W+C(Ek1)W,
which can be evaluated against the quasi-stationary model for steepness modulation (7).

The comparison is shown in Fig. 8 by using k1 from Tm01 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 εW+C/εW 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.

Fig. 8.
Fig. 8.

Comparing ε/ε0 from (33) using W+C and W and the simplified wave straining model in (7). (top) The intermodel ratio, i.e., εW+C/εW (black line), and the output from (7) (orange line) using k1 computed from Tm01 in W. (bottom) Ueff from (3). Since the wave spectrum was bimodal (Fig. 6), we also added εW+C/εW 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 η¯MAX,ST/Hs increased when waves and currents were opposing and largely followed the curve of 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 Fη(ε = 0, N3D) in (25), demonstrate the contribution by N3D (see dashed lines Fig. 9a). The fluctuations, however small, in the linear η¯MAX,ST/Hs correspond to the N3D, 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 η¯MAX,ST/Hs (i.e., solid and dashed lines) imply that for an on-average high N3D,η¯MAX,ST is more sensitive to the nonlinear ε contribution than further increasing the number of waves due to currents.

Fig. 9.
Fig. 9.

Temporal evolution in expected extremes computed from the 2D spectra. Panels show (a) η¯MAX,ST/Hs from (23), (b) η¯MAX,VS/Hs using a domain of variable size (subscript VS), i.e., X = Lx, Y = Ly and D = 100Tm, (c) H¯MAX,ST/Hs from (24), and (d) H¯MAX,VS/Hs. Dashed lines in (a) and (b) denote the linear versions, i.e., η¯MAX,ST/Hs=Fη(ε=0,N3D). Labels and layout are similar to that in Fig. 7.

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 |ϕ*|. Following Benetazzo et al. (2021a), we define such a domain by forcing X = Lx, Y = Ly, and D = 100Tm. Then, only one wave, on average, is included in the horizontal space domain, and consequently N3D ≈ 100A. Such a choice also allows to assess the impact by A, ε, and |ϕ*| in different sea states. As mentioned, there was an unsystematic modulation in 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 |ϕ*|. The resulting η¯MAX,VS/Hs, with subscript VS for variable size, is given in Fig. 9b. Here the linear η¯MAX,VS/Hs [i.e., Fη(ε = 0, N3D) from (25)] were similar for W+C and W (see dashed blue and orange lines). Thus, the intermodel differences in the second order η¯MAX,VS/Hs were governed by 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 H¯MAX,ST/Hs had less of a systematic tidal modulation during the tidal cycles (Fig. 9c). That is, the extremes from W+C were sometimes lower during opposing wind sea and currents, and sometimes higher. We find a correlation coefficient with |ϕ*| of 0.74 in W+C, and consequently the modulation in H¯MAX,ST/Hs during the first and two last tidal cycles was due to the similar energy levels on the bimodal components (see Fig. 5). The predictions from the variable size domain, H¯MAX,VS/Hs, are shown in Fig. 9d, with values that to a large extent followed the fluctuations in |ϕ*|, now with a correlation coefficient of 0.97 for W+C (Fig. 7d).

2) Horizontal variability in expected extremes

When wind sea opposed Moskstraumen, η¯MAX,ST/Hs became most severe at the edges of the broad current (Figs. 10a,b). Here the intermodel ratio reveal a 10%–15% increase in η¯MAX,ST/Hs (Fig. 10b). Similar horizontal variability was also found for the other tidal cycles (not shown).

Fig. 10.
Fig. 10.

Horizontal variability in normalized extreme wave crests (a),(b) η¯MAX,ST/Hs and (c),(d) H¯MAX,ST/Hs during maximum incoming tide. The expected extremes from (top) W+C and (bottom) its ratio with W. Green dots denote the location of the ADCP. Contour lines indicate current speed (m s−1).

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

The H¯MAX,ST/Hs became most severe within the region of increased Hs (Figs. 10c,d), with a uniform horizontal distribution. We find an increase in H¯MAX,ST/Hs of about 5%–10% when adding currents as forcing (Fig. 10d). During the two last tidal cycles in the period of interest, the time series analysis showed a decrease in H¯MAX,ST/Hs during maximum opposing wind sea and current (Fig. 9c). In the field view, however, the decrease was confined to the northern part of the current, while further south a similar modulation and shape was obtained (not shown). At the southern part, the remaining wind sea was more dominating compared with farther north, as well as being more sheltered to the swell (not shown).

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 ηMAX and HMAX were computed from a zero crossing analysis, and the η¯MAX,T and H¯MAX,T by taking the mean of the three realizations. This is a block maxima approach to assessing the expected maxima. These maxima will be independent and identically distributed under the assumption that the time series is statistically stationary (Coles 2001). Furthermore, values from the two bursts every hour were resampled to an hourly mean value. The expected extremes from W+C and W were computed over the same time interval. Since observations and model predictions were at times out of phase (e.g., Fig. 7), we applied the quantitative dynamic time warping method. Here, a distance metric ddtw 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 ddtw 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 η¯MAX,T from W+C mimic the increase from the observations at the M2 frequency when the wind sea opposed Moskstraumen (Fig. 11b), however often underestimating the magnitude. Note that here η¯MAX,THs, with a correlation of 0.95 and 0.99 for the observations and W+C, respectively. Even though the Hs 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¯MAX,T/Hs, there is also an underestimation in the ratio (not shown). Nevertheless, the trend is that the expected maximum wave crests increase when the wind sea opposes Moskstraumen. The observed extremes, according to the common definition of η/Hs > 1.25 (Dysthe et al. 2008), also occurred for the wind sea on counter currents (see red dots). For η¯MAX,T: ddtw,W+C = 2.96 and ddtw,W = 3.19.

Fig. 11.
Fig. 11.

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 H¯MAX,T, we find a similar tendency in M2 modulation between W+C and the observations (Fig. 11c). The ddtw,W+C = 3.93 and ddtw,W = 4.52, implying a better fit for the former. However, the observed extremes, according to the definition of H/Hs > 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 M2 modulation in the normalized H¯MAX,T/Hs (not shown).

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 cp/U10 ≃ 18/7.5 > 1, where cp 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.

Fig. 12.
Fig. 12.

(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 M2 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 Hs up to 90% (Fig. 13a). In addition to the increase in energy, the 2D spectrum also underwent significant directional broadening (not shown).

Fig. 13.
Fig. 13.

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) Hs, (b) ε/ε0 as in Fig. 8 (yellow shading shows the excess of the inter model ratio), (c) η¯MAX,VS/Hs, (d) |ϕ*|, and (e) H¯MAX,VS/Hs .

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 εW+C/εW, 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 η¯MAX,VS/Hs (note variable size VS) changed due to Moskstraumen, particularly under opposing swell and current situations, and the relative difference between W+C and W exceeded 10% (Fig. 13c). Similar changes were found in η¯MAX,ST/Hs, but we do not consider the N3D parameter since ε dominates the variability. The correlation between η¯MAX,VS/Hs and ε in W+C was 0.99. Following the reasoning about the impact by refraction on ε, the increase in Hs also constrain the ratio η¯MAX,VS/Hs.

For the H¯MAX,VS/Hs, the expected extremes followed the curve of |ϕ*|, with a correlation coefficient of 0.98 in W+C (Figs. 13d,e). During counterflow situations when refraction seemed to dominate, one may expect that the sea state would become broader in frequency due to the crossing sea state and nonlinear redistribution of energy across scales (Tamura et al. 2008; Rapizo et al. 2016). This appeared to be the case during certain tidal cycles (see around 5 January 2019 in Fig. 13d), but certainly not for all. However, when the sea state became broader, there was an accompanying decrease in H¯MAX,VS/Hs.

Maximum η¯MAX,ST/Hs were located at the edges of the tidal jet during maximum current speed at outgoing tide, and not within the current jet itself (left column Fig. 14). In the vicinity of the jet, we find a 5%–15% increase in η¯MAX,ST/Hs compared with W (Fig. 14b). The H¯MAX,VS/Hs was generally higher outside the current jet (Fig. 14c), and decreased around 5% within the area of strong currents (Fig. 14d). Similar horizontal variability was also found for the other tidal cycles (not shown).

Fig. 14.
Fig. 14.

Horizontal variability in normalized extreme wave crests (a),(b) η¯MAX,ST/Hs and (c),(d) H¯MAX,ST/Hs during maximum outgoing tide at 0100 UTC 6 January 2019. The expected extremes from (top) W+C and (bottom) its ratio with W. Green dots denote the location of the time series data at the offshore location. Innermost contour lines indicate current speed at 2 m s−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 η¯MAX

We find η¯MAX,ST to be more sensitive to the current-induced modulations in ε than those in N3D (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 η¯MAX,ST in young, short-crested sea states on horizontally homogeneous tidal currents. Under conditions where refraction is likely to dominate, however, the ratio εW+C/εW 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, η¯MAX,VS/Hs reached 1.31 during opposing wind sea and broad current (Fig. 9b) and 1.32 during the opposing swell and tidal jet (Fig. 13c). It seems plausible that the significant increase in Hs in the latter case constrained the ratio η¯MAX,VS/Hs. For the space–time extremes, the contribution from N3D increased η¯MAX,VS/Hs to 1.61 for the former (Fig. 9a) and 1.56 for the latter (not shown). The higher values for the former was due to the on-average shorter wind waves, such that more waves fitted into the space–time domain compared with the latter swell case. Therefore, our findings indicate that under conditions similar to the former, when wave straining is a prominent mechanism, a sea state exposed to more severe extremes can be reached.

The horizontal variability of η¯MAX,ST/Hs corroborate the findings by Hjelmervik and Trulsen (2009), suggesting that extreme waves become more severe at the edges of the current where the horizontal current gradients are strongest and Hs 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 η¯MAX,T from the wave model with current forcing and the observations, contrary to the wave model without current forcing (Fig. 11). Long-term single point observations in extreme environments are very rare in themselves due to the harsh conditions, and the ADCP measurements used here are the first of its kind in Moskstraumen (Saetra et al. 2021). Consequently, observations spanning both space and time under similar conditions are even more rare. Obtaining such measurements requires development and innovation in instrument deployment setups and operating methods.

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 H¯MAX

The modulation in |ϕ*| from the bimodal spectra, and its impact on H¯MAX,ST was at times difficult to interpret. Here, |ϕ*| was sensitive to the relative variance distribution on the swell and wind sea components in the spectrum, while becoming easier to interpret when the wind sea dominated, i.e., during the intermediate tidal cycles in the period (Figs. 5, 7, and 9). In their deterministic approach, Hjelmervik and Trulsen (2009) found that the amount of freak waves increased on uniform countercurrents for both narrow- and broad-banded sea states. Such an increase is difficult to conclude from our results since the bimodal partitions were at times propagating against each other (Fig. 6). However, when the wind sea dominated, we found an increase in H¯MAX,ST/Hs when it opposed the tidal current (Figs. 9c,d, and 10c,d).

The decrease in H¯MAX,VS/Hs during the opposing swell and tidal jet is contrary to the results of Ying et al. (2011) (Fig. 13e), which suggested that caustics caused by refraction increased the probability of extremes. Our results corroborate the findings by Hjelmervik and Trulsen (2009), which found it less likely to encounter extreme wave heights in the center of an opposing narrow tidal jet compared with its edges due to the reduction in the kurtosis, even though the wave heights were higher in the center (Figs. 14c,d). However, proper measurements are needed to further evaluate the impact by tidal jets and broad uniform currents on H¯MAX,ST/Hs. Moreover, such studies should also include areas subject to met-ocean conditions that are not present in the Lofoten area, like swell on collinear tidal jets.

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 Hs and spectral steepness ε increased.

The second-order non-Gaussian contribution through ε in the expected maximum space–time wave crests η¯MAX,ST increased when the wind sea in (i), and the swell in (ii), were opposing Moskstraumen. Consequently, the ratio η¯MAX,ST/Hs also increased, which was more sensitive to ε than the average number of waves within the space–time domain N3D 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 H¯MAX,ST corresponded strongly to the value of the narrow-bandedness parameter |ϕ*| during both (i) and (ii). The intermodel differences were very sensitive to the relative distribution of the variance density on the wind sea and swell components during (i). When the wind sea dominated, H¯MAX,ST/Hs increased when the waves opposed the tidal current, but vice versa when the wind sea and swell components had similar variance density. During (ii), H¯MAX,ST/Hs often decreased when the swell encountered the opposing tidal jet. Thus, our results suggest that the impact of strong tidal currents on the spectral shape is key for the accompanied modulation in H¯MAX,ST/Hs.

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 N3D for short waves encountering a broad countercurrent, i.e., similar to the conditions in (i), and a strong increase in ε and Hs are found during (ii), where refraction seemingly dominates. For the latter, however, the increase in Hs constrains the ratios η¯MAX/Hs and H¯MAX/Hs, and our results suggests that more severe extremes can be expected when wave straining dominate. However, more work is required to further understand the role of tidal currents on extreme waves. Such work should in particular involve more extensive measurement campaigns with simultaneous spatial sampling over longer time periods, and should also include areas where other combinations of characteristic tidal current flow fields and wave conditions occur.

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

Since the wave variance density is conserved, the transformation from an absolute to an intrinsic wave spectrum involves solving the Jacobian. For a 2D wave spectrum, the wavenumber vector k = (kx, ky) and the velocity vector U = (u, υ) must be considered. The wavenumber k = |k|, and the angle between kx and ky is θw. The Jacobian ω/σ becomes
σ+kUσ=1+ukxσ+υkyσ.
Inserting for the wavenumber components, we get
1+ukcos(θw)σ+υksin(θw)σ=1+ucos(θw)cg+υsin(θw)cg.
Using trigonometry, [cos(θw),sin(θw)]=k/k (i.e., the adjacent and opposite divided by the hypotenuse, respectively), we write
ωσ=1+1cg[ucos(θw)+υsin(θw)]=1+kUkcg.
The result is the same as that obtained in the WW3 user manual [see p. 14 in version 7.00 of The WAVEWATCH III Development Group (2019)], which also applies for the Jacobian fa/fi since
Ei(fi,θw)=2πEi(σ,θw)=2π(1+kUkcg)Ea(ω,θw)=(1+kUkcg)Ea(fa,θw).
To compute the Jacobian using the spectra from WAM, k, u, k, and cg must be computed for every direction θw,j in the discrete spectrum. The group velocity is defined
cg=ωk=nσk,
where n=0.5+[kd/sinh(2kd)]. Furthermore, the directionality for all discrete k vectors must be considered such that kj = kj (θw). 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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    • Export Citation
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    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Barbariol, F., and Coauthors, 2017: Numerical modeling of space-time wave extremes using WAVEWATCH III. Ocean Dyn., 67, 535549, https://doi.org/10.1007/s10236-016-1025-0.

    • Search Google Scholar
    • Export Citation
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    • Export Citation
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    • Export Citation
  • Benetazzo, A., F. Barbariol, F. Bergamasco, A. Torsello, S. Carniel, and M. Sclavo, 2015: Observation of extreme sea waves in a space–time ensemble. J. Phys. Oceanogr., 45, 22612275, https://doi.org/10.1175/JPO-D-15-0017.1.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Benetazzo, A., F. Barbariol, F. Bergamasco, L. Bertotti, J. Yoo, J.-S. Shim, and L. Cavaleri, 2021a: On the extreme value statistics of spatio-temporal maximum sea waves under cyclone winds. Prog. Oceanogr., 197, 102642, https://doi.org/10.1016/j.pocean.2021.102642.

    • Search Google Scholar
    • Export Citation
  • Benetazzo, A., and Coauthors, 2021b: Towards a unified framework for extreme sea waves from spectral models: Rationale and applications. Ocean Eng., 219, 108263, https://doi.org/10.1016/j.oceaneng.2020.108263.

    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
  • Chawla, A., and J. T. Kirby, 2002: Monochromatic and random wave breaking at blocking points. J. Geophys. Res., 107, 3067, https://doi.org/10.1029/2001JC001042.

    • Search Google Scholar
    • Export Citation
  • Coles, S., 2001: An Introduction to Statistical Modeling of Extreme Values. Springer Series in Statistics, Vol. 208, Springer, 225 pp.

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    • Search Google Scholar
    • Export Citation
  • Dysthe, K., H. E. Krogstad, and P. Müller, 2008: Oceanic rogue waves. Annu. Rev. Fluid Mech., 40, 287310, https://doi.org/10.1146/annurev.fluid.40.111406.102203.

    • Search Google Scholar
    • Export Citation
  • Fedele, F., 2012: Space–time extremes in short-crested storm seas. J. Phys. Oceanogr., 42, 16011615, https://doi.org/10.1175/JPO-D-11-0179.1.

    • Search Google Scholar
    • Export Citation
  • Fedele, F., and M. A. Tayfun, 2009: On nonlinear wave groups and crest statistics. J. Fluid Mech., 620, 221239, https://doi.org/10.1017/S0022112008004424.

    • Search Google Scholar
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