Introduction

Interactions between ocean currents and ocean surface waves have been investigated extensively in the last few decades. In particular, significant progress has been made in understanding how the Stokes drift of surface waves affects the near-surface turbulence (Skyllingstad and Denbo 1995;McWilliams et al. 1997, and many following studies), and depth-dependent currents in deep water (Holm 1996;Suzuki and Fox-Kemper 2016) and in finite depth to shallow waters (McWilliams et al. 2004;Ardhuin et al. 2008). Based on these new findings, significant efforts are being made to couple ocean models and surface wave models in deep water (e.g., Reichl et al. 2016;Zhou et al. 2023) as well as in finite depth coastal waters (e.g., Uchiyama et al. 2010;Olabarrieta et al. 2011;Kumar et al. 2012;Martins et al. 2022). In such coupled models, a statistical wave model is used to predict the wave spectrum, and the Stokes drift is integrated over the resolved wave spectrum and is passed to the ocean model. (In practice, the Stokes drift is usually constructed in the ocean model using a few wave parameters passed from the wave model.)

In contrast to the wave impacts on ocean currents, the impacts of ocean currents on the evolution of surface wave spectra have been investigated less frequently. In the wave action equation solved in the statistical wave models, the ocean currents appear in three different ways. First, the advection velocity of the wave action is modified by the surface current. Second, the wavenumber of a particular wave component is modified by stretching/compressing/turning by the horizontal shear of surface currents. Third, the wind forcing is determined by the difference between the wind speed vector (typically at a 10-m height) and the surface current vector. If the near-surface current is vertically uniform, including these current effects is straightforward. The impacts of vertically sheared Eulerian currents on the propagation of the wave action have been thoroughly investigated (Kirby and Chen 1989;Shrira 1993;Quinn et al. 2017;Ellingsen and Li 2017;Banihashemi and Kirby 2019). One remaining question is how the presence of other wave trains affects the propagation of the wave action (or the wave energy) of a particular wave component. In particular, does the group speed increase simply because of the advection by the Stokes drift integrated over the entire spectrum? To my knowledge, this question has not been addressed even though the near-surface Stokes drift is often comparable in magnitude to the near-surface Eulerian current. This question addresses one aspect of the nonlinear wave-wave interaction physics, but it focuses on the mutual modulation of wave propagation rather than the exchange of wave energy, which is frequently investigated.

In addition to the wave group velocity, near-surface currents also affect the wave phase speed. Understanding the current impacts on wave phase propagation is important because observed wave phase speeds are used for a variety of ocean current measurements. Calculating the impact of a vertically sheared Eulerian current on the wave phase speed is straightforward. Longuet-Higgins and Phillips (1962) investigated how the phase speed of a single wave train is affected by the presence of another aligned or opposing wave train. However, the impact of the entire wave spectrum (including waves propagating in all directions) on the phase speed of a particular wave component has not been investigated to our knowledge.

In this study, we first employ the standard multiple-scale perturbation analysis of a single wave train to examine how the wave nonlinearity modifies the wave phase and group speeds in section 2. In section 3, the impacts of an exponentially decaying Eulerian current, whose profile is set identical to the Stokes drift profile of the wave train, are examined. In section 4, the analysis is extended to include a second wave train, and the impacts of the second wave train on the phase and group speeds of the first wave train are investigated. The results are compared to the impacts of an Eulerian shear current with the same vertical profile as the Stokes drift of the second wave train. In section 5, the results of section 4 are applied to estimate how the phase and group speeds of a particular wave component are affected by the entire wave spectrum. Throughout this study, we focus on a simple case of deep water and statistically uniform wave and current fields.

Evolution equation of a single wave train in deep water

a. Amplitude equation Throughout this study, water density r is assumed constant and water viscosity and the Coriolis force are ignored. We define x and y as the horizontal coordinates and z as a vertical coordinate (positive upward) with z 5 0 at the mean sea surface. Let us introduce the dynamic pressure divided by r, defined as

where g is the gravitational acceleration, p tot is the total pressure, and 2rgz is the hydrostatic pressure without the waves.

The momentum equations then become (7)

u 5 y 5 w 5 p 5 0, z 5 2',

where (u, y , w) is the velocity vector in the (x, y, z) direction and z(t, x, y) is the surface elevation. The vertically integrated energy equation can be obtained from these equations as

with

u(e k 1 p)dz,

where E p is the potential energy, E k is the kinetic energy, E t 5E k 1 E p is the total energy, and (F x , F y ) is the energy flux in the (x, y) direction.

Let us first consider a single deep water wave train:

with the linear dispersion relationship

where k is the wavenumber, v is the angular frequency, A is the complex wave amplitude, and () * denotes the complex conjugate of (). Although we are mainly interested in a wave field with a spatially/temporally uniform wave spectrum, the wave field is assumed to consist of multiple wave trains and each wave train may be modulated. Therefore, the amplitude A is allowed to vary slowly in space and time and is assumed to be small such that

We then perform the multiple-scale analysis expanded in e. First, all the variables are expanded as u 5 eu 1 1 e 2 u 2 1 …, y 5 ey 1 1 e 2 y 2 1 …, w 5 ew 1 1 e 2 w 2 1 …, p 5 ep 1 1 e 2 p 2 1…, z 5 ez 1 1 e 2 z 2 1 …:

Next, all the variables are assumed to depend on multiplescale variables, t, t 1 5 et, t 2 5 e 2 t, …, x, x 1 5 ex, x 2 5 e 2 x, …, y, y 1 5 ey, y 2 5 e 2 y, …, and z, z 1 5 ez (we do not need z 2 in this study), and all the derivatives are expanded as

All the perturbation analyses in this study were carried out using Mathematica, and the codes and the results are available in Hara (2024). Only the key results are presented below. The evolution equations of the wave amplitude A at the three successive orders in e are obtained as

and

Here, the subscripts denote the differentiation with respect to the subscripted variable. Equations ( 16)-( 18) can be combined to form a single evolution equation in terms of one set of slow variables T 5 et, X 5 ex, and Y 5 ey, by setting

The resulting equation is

This is a well-known equation originally derived by Dysthe (1979) and investigated by Stiassnie (1984), Trulsen and Dysthe (1996), and many others. This equation includes the slowly varying current u 30 as discussed by Trulsen and Dysthe (1996) and others in detail. This term does not contribute to the wave energy/action analysis below. It should be noted that in most previous studies, the coefficient of the term (vkA 2 A * X ) is 21/4 instead of 1/4. This difference arises because most previous studies carried out the perturbation analysis in terms of the velocity potential and defined A as the coefficient of the leading-order velocity potential. Here, we have obtained a different coefficient 1/4 because we have defined A as the coefficient of the leading-order surface displacement z 1 .

The well-known exact solution of a constant amplitude wave train, which satisfies (20), is given as

and z 5 |A|e ikx2ivt e 2e 2 (i/2)vk 2 |A| 2 t 1 c:c:

where "c.c." denotes the complex conjugate of the previous term. Its phase speed c is

that is, the wave phase speed has increased by e 2 c NL due to the wave nonlinearity from the linear wave phase speed c L .

b. Energy equation

The energy conservation equation can be obtained by multiplying (20) by A * /2 and adding its complex conjugate:

Unfortunately, this equation is not expressed in a flux form, and it is not possible to find the expression of the energy flux from this equation. Instead, the energy conservation equation (including the energy flux terms) can be obtained by directly introducing the solutions of u, y, w, p, z into the energy equation [( 9)] averaged over the short time scale t. The results are

with

where

and the overline denotes the average over t. It is straightforward to show that (25)-( 28) are consistent with (24), but the energy fluxes are clearly defined in ( 27) and (28). Note that the energy E T contains higher-order terms of O(e 2 ), and therefore, the fluxes F X and F Y also contain the fluxes of the higher-order energy terms. It is possible to obtain the budget of the leadingorder energy |A| 2 /2 only. First, we write the budget of the higherorder energy, which is the term multiplied by e 2 in (26), as

By subtracting (30) from the full energy ( 25)-( 28), we obtain

with

Again, (31)-(34) are consistent with (24).

If A is spatially uniform, the flux terms are simplified to

and the energy flux F X 0 can be expressed as

that is, the group velocity (the propagation speed of wave energy) has increased by e 2 c NL g due to the wave nonlinearity from the linear group velocity c L g . Even if A is spatially modulated, it can be shown that the terms proportional to e in (33) and (34) can be eliminated on average by simply redefining the wavenumber k of the wave train of interest (see appendix A). Therefore, the simplified energy flux [(36)] is approximately valid (on an average), provided the modulation of A occurs slowly with spatial scales longer than x 1 and y 1 . If the wave train is modulated with spatial scales of x 1 and y 1 , the energy flux is likely to be modified}even on average}because of the presence of terms proportional to |A X | 2 and |A Y | 2 in (33).

c. Wave action equation

Since the wave prediction models normally solve the wave action equation instead of the wave energy equation, we next derive the wave action equation. Although the definition of the wave action (N) needs to be modified in the presence of a sheared Eulerian current (Banihashemi and Kirby 2019), here, we use the standard definition N 5 E T 0 /V, that is, the leading-order wave energy E T 0 divided by the intrinsic frequency V related to the wavenumber through the linear dispersion relationship, because no background Eulerian current is included. This definition is also consistent with the quantity predicted in the wave models. After some algebra (appendix A), the wave action flux (F X N , F Y N ) for a uniform (or modulated slower than in x 1 , y 1 ) wave train can be obtained as

Notice that the impact of the wave nonlinearity on the action flux (c NL2 g ) is slightly less than the impact on the energy flux (c NL g ).

Effect of Eulerian shear current a. Evolution equations

We next investigate the advection of wave phase and wave energy by an Eulerian vertically sheared current. Let us add an exponentially decaying sheared current [U(z), V(z)], with a surface current magnitude e 2 U s and a vertical decay rate k s , U 5 e 2 U s e k s z cosu, V 5 e 2 U s e k s z sinu, (38) to the same wave train and repeat the same multiple-scale analysis. Note that similar analyses were carried out for more general current scaling and profiles by Kirby and Chen (1989) and many others as discussed in section 1. Here, we perform the analysis focusing on the exponentially decaying current of O(e 2 ) because it provides a simple analytical solution and it is consistent with the Stokes drift of surface waves as discussed later.

The resulting amplitude evolution equations become

and the combined equation is

In this equation, the effects of the sheared current appear in the three new terms, and their coefficients are all proportional to the imposed current. If multiple (exponentially decaying) currents coexist, their impacts on the wave train can be simply summed up. Therefore, this result can be applied to a more complicated current profile if its near-surface profile can be approximated by the superposition of exponential profiles of different magnitudes, directions, and decay rates.

The exact solution of a constant amplitude wave train becomes

Therefore, the phase speed c of a uniform wave train becomes

that is, it is modified by e 2 c CU due to the background alongwave current U. Notice that the cross-wave current V has no impact on c. The conservation equation of the leading-order energy of a uniform (or modulated slower than in x 1 , y 1 ) wave train becomes

and the group velocity vector (c gx , c gy ) of the wave energy becomes

and

Therefore, the group velocity vector is modified by both the along-wave current U and the cross-wave current V. Notice that the coefficients of c CU gx and c CU gy are different, that is, U and V affect the group velocity differently. The impacts of the sheared Eulerian current on the propagation speed of the wave action is also (c CU gx , c CU gy ) and is identical to those on the propagation speed of the wave energy (appendix A).

b. Linear shear profile

We may take a limit of k s , , k so that the current shear becomes approximately linear over a depth comparable to the surface wavelength. Then, the increase of the phase speed due to the sheared current becomes

that is, the wave phase speed is affected by the along-wave current U at a depth of 1/2k 5 l/4p, where l is the surface wavelength. This is a well-known result. In contrast, the increase of the group velocity becomes

and

Therefore, the group speed feels the surface value of U, but it feels V at a depth of 1/2k 5 l/4p. The along-wave and cross-wave currents have different impacts on the group velocity as discussed earlier.

c. Advection by Stokes drift

Let us examine whether the changes of the phase and group speeds due to the wave nonlinearity are consistent with the advection by a sheared Eulerian current that has the same vertical profile as its own Stokes drift. Since the sheared Eulerian current possesses vorticity but the Stokes drift of an irrotational surface wave does not, the effects of these two flows can be different. On the other hand, the mean (wave-averaged) advection speeds of a passive tracer by these two flows are identical. Therefore, if the wave phase (or wave group) behaves like a simple tracer, it may be advected by its own Stokes drift. If we set that the sheared current U has the same profile as the Stokes drift of the wave train itself, that is,

or

then the increase of the phase speed becomes

which is exactly equal to the phase speed increase c NL by the nonlinearity derived in ( 23). Therefore, it can be stated that the increase of the phase speed of the wave train nonlinearity is simply due to the advection by its own Stokes drift, as discussed by Pizzo et al. (2023). In contrast, the increase of the group speed by the sheared current is

which is smaller than the increase of the group speed due to the wave nonlinearity

derived in (36). Therefore, the increase of the propagation speed of wave energy is 7/6 times larger than the advection by a sheared current with the same vertical profile as its own Stokes drift. Interestingly, the propagation speed of the wave action is

and is consistent with the advection by its own Stokes drift.

Effect of another wave train a. Evolution equations

We now add a second wave train to the original wave train and repeat the same multiple-scale analysis. The second wave train is expressed as

with the linear dispersion relationship

where k 2 is the wavenumber, u is the wave direction, v 2 is the angular frequency, and B is the complex wave amplitude. We remove the sheared current from this analysis for simplicity, but it can be easily added.

The resulting amplitude evolution equations for the original wave train (amplitude A) at successive orders in e become

where the nondimensional coefficients C 1 -C 7 are the functions of s 2 /s (or k 2 /k) and u. Their general solutions are quite complex and are discussed in appendix B. The combined evolution equation becomes

If both |A| and |B| are constant, the exact solution of A is given as

Therefore, the phase speed becomes

that is, it is modified by c WV due to the second wave train. This result was obtained by Longuet-Higgins and Phillips (1962) for aligned (u 5 0) and opposing (u 5 p) wave trains only. The phase speed modulation in more general cases (u Þ 0, p) has not been investigated to our knowledge. The energy conservation equation can be obtained by introducing the solutions of u, y, w, p, and z into the energy equation [( 9)] averaged over the short time scale t. However, the resulting equations provide the budget of the total combined energy of the two wave trains and the higher-order terms. They are not useful for examining the energy budget of each wave train separately. We therefore construct the energy budget of the first wave train alone by multiplying (62) by A * /2 and adding its complex conjugate. The result is

This equation does not clarify the energy conservation in terms of the energy fluxes. However, we have already found the energy fluxes of a single wave train in ( 33) and ( 34). Therefore, it is possible to modify these fluxes to include the contributions of the second wave train, so that the resulting energy equation is consistent with (65). One possible form is

Notice that the last square bracket cannot be written in a flux form, that is, this term acts like a source/sink, indicating that the leading-order energy alone is not conserved. Only the total energy (including two wave trains and the higher-order terms) is conserved. However, if the wave amplitude |B| of the second wave train is not strongly modulated (changes slower than in x 1 , y 1 ), the last square bracket becomes negligible. More importantly, if we consider a spectrum of waves (as we do in section 5), that is, if many wave trains of random phase and amplitude exist in addition to the original wave train, the two wave amplitude terms}(|B| 2 ) X and (|B| 2 ) Y in the last square bracket}are summed for all the wave trains and the results become zero on average if the wave spectrum is spatially uniform. Without the last square bracket, (66) defines the fluxes of the leading-order energy (|A| 2 /2), even if the coexisting wave trains are modulated in the spatial scales of x 1 , y 1 .

If the original wave train is not strongly modulated (A changes slower than in x 1 , y 1 ), the energy equation is further simplified to

with

Therefore, the group velocity vector is modified by (e 2 c WV gx , e 2 c WV gy ) due to the second wave train. If the original wave train is modulated in the spatial scale of x 1 , y 1 , its energy flux is likely to be modified by its own modulation as discussed earlier. However, the enhancement of the group velocity by the second wave train remains the same, and the results of (e 2 c WV gx , e 2 c WV gy ) are applicable even in such cases.

b. Wave action equation

After some algebra (appendix A), the wave action fluxes for a uniform (or modulated slower than in x 1 , y 1 ) wave train can be obtained as

that is, the impacts of the second wave train on the wave action propagation and the wave energy propagation are identical.

c. Comparison with advection by Stokes drift

Let us examine whether the changes of the phase and group speeds due to the second wave train are consistent with the advection due to a sheared current that has the same vertical profile as the Stokes drift of the second wave train. Since the Stokes drift of the second wave train is expressed as

the equivalent sheared current can be described by setting

in the results obtained in section 3. Then, the changes of the phase and group speeds due to this sheared current are

and

For the sake of comparison, we also rewrite the modified phase and group speeds by the second wave train in terms of the surface Stokes drift magnitude U s as

and

Let us investigate the group speed modification c WV gx and c CU gx . In Fig. 1a, we compare c WV gx /U s (blue solid lines) and c CU gx /U s (red solid lines) for a range of v 2 /v between 0.2 and 5 (k 2 /k between 0.04 and 25) and at nine different wave angles u 5 0, p/8, 2p/8, 3p/8, … , p (lines from top to bottom). We first focus on the aligned case u 5 0. If the wavelength of the second wave train is much longer than the wavelength of the original wave train (k 2 , , k), it is expected that the wave energy is simply advected by the surface Stokes drift of the second wave train. Indeed, both c WV gx /U s (blue) and c CU gx /U s (red) approach 1 for k 2 /k , , 1. However, as k 2 /k increases toward 1, c WV gx /U s and c CU gx /U s gradually diverge, that is, the wave energy is not simply advected by the Stokes drift any more. While c CU gx /U s gradually decreases (the wave group feels the current at a deeper depth rather than at the surface) as expected, c WV gx /U s remains exactly at 1, that is, the group velocity increases exactly by the surface Stokes drift velocity of the second wave train, as long as k 2 is less than k. As k 2 /k approaches 1, c WV /U s becomes 4/3 times as large as c CU /U s which approaches 3/4. Interestingly, once k 2 /k exceeds 1, c WV gx /U s is discontinuous and jumps from 1 to 2 and then gradually decreases as k 2 /k increases further. Therefore, the presence of a second wave train with a slightly shorter wavelength significantly increases the group velocity of the original wave train. Its effect is twice as large as the effect of a second wave train with a slightly longer wavelength and is 8/3 times larger than the simple advection by the Stokes drift of the second wave train. As k 2 /k becomes very large, both c WV gx /U s and c CU gx /U s approach 0, that is, waves do not feel the effect of much shorter waves as expected.

The results at different u values are qualitatively similar. Both c WV gx /U s (blue) and c CU gx /U s (red) approach cosu for very small k 2 /k. They diverge toward k 2 /k 5 1. Once k 2 /k exceeds 1, c WV gx /U s becomes large before gradually decreasing to 0 at very large k 2 /k. While the transition from the k 2 /k , 0 regime to the k 2 /k . 0 regime is abrupt at u 5 0 and p, the transition becomes smoother away from these two limits. The result of c WV gx /U s with u 5 p/40 is also included (blue dashed line) to show how the abrupt jump of the u 5 0 line at k 2 /k 5 1 becomes smoother as u increases. In summary, the presence of a second wave train significantly increases the group velocity of the original wave train, much more than the simple advection expected by the Stokes drift of the second wave train. The effect is particularly strong if the second wave train propagates in a similar direction and its wavelength is slightly shorter. Next, in Fig. 1b, we compare the phase speed modification c WV /U s and c CU /U s for the same range of v 2 /v between 0.2 and 5 (k 2 /k between 0.04 and 25) and at nine different wave angles u 5 0, p/8, 2p/8, 3p/8, … , p. As discussed earlier, the aligned (u 5 0) and opposing (u 5 p) cases were studied by Longuet-Higgins and Phillips (1962). For the aligned case (u 5 0), both c WV /U s (blue) and c CU /U s (red) approach 1 when k 2 /k is very small, that is, waves are simply advected by the surface Stokes drift of much longer waves. As k 2 /k increases toward 1, c WV /U s and c CU /U s diverge. The former stays exactly at 1, but the latter decreases to 1/2. Therefore, the wave phase speed increase is twice as large as what is expected from the simple advection by the Stokes drift of the second wave train, if the second wave train has a similar wavelength. Once k 2 /k exceeds 1, c WV /U s suddenly starts to decrease from 1 but remains larger than c CU /U s . Unlike c WV gx /U s , which is discontinuous at k 2 /k 5 1, c WV /U s is continuous at k 2 /k 5 1. With very short waves (k 2 . . k), the impacts disappear and both c WV /U s and c CU /U s approaches zero as expected.

With different angles u Þ 0 of the second wave train, the overall trend is similar. For k 2 , , k, both c WV /U s and c CU /U s approach cosu. As k 2 /k increases toward 1, c WV /U s remains almost constant and diverges from c CU /U s before decreasing for k 2 . k. Compared to the u 5 0 case, the transition from the k 2 , k regime to the k 2 . k regime is smoother. In summary, when a second wave train of a similar wavenumber exists, the phase speed of the original wave train increases significantly FIG. 3. Two TC examples of group speed modification c WV g (red line) due to the entire spectrum and group speed modification c CU g (magenta line) due to the sheared current whose profile is identical to the Stokes drift profile integrated over the entire spectrum. Both are normalized by the peak group speed c gp (left axis) and plotted against angular frequency v (top axis) or angular frequency normalized by peak frequency v/v p (bottom axis). Blue line: surface Stokes drift U ST (v, 0) integrated up to v. Blue dashed line: surface Stokes drift U ST (3v, 0) integrated up to 3v. Cyan dashed line: Stokes drift at a depth of z 5 0.02/k integrated over the entire spectrum, U ST (', 0.02/k). Vertical black dashed line: peak frequency. Horizontal black dashed line: spectrally weighted average of c WV g .

(almost twice) more than the simple advection expected by the Stokes drift of the second wave train.

Application with a wave spectrum a. Method

In this section, the previous results are applied to a wave field with a directional frequency spectrum G(v 2 , u). For simplicity, we examine conditions where wind and waves are aligned, that is, the wind direction is u 5 0 and the spectrum is symmetric with respect to positive and negative u. We then estimate how the phase speed and the group speed of a particular wave component (with an angular frequency v and u 5 0) are affected by the presence of all the other waves. In the previous section, the coefficients C 1 and C 2 have been obtained as functions of v 2 /v and u. Therefore, the enhancement of the phase speed c WV (v) and the enhancement of the group velocity c WV g (v) by the entire wave spectrum are expressed as

and

Here, the factor 2 in these equations appears because the integrated spectrum is equal to 1/2 of the sum of the amplitude squared of all waves. We may also calculate the enhancement of the phase and group speeds by a sheared current whose profile is identical to the Stokes drift profile (integrated over the entire spectrum) as

and

For the sake of comparison, the Stokes drift at a depth 2z, integrated up to a frequency v, is calculated as

We first consider strongly wind-forced wave spectra under tropical cyclones, since the largest increase of the group speed c WV g (v) or the phase speed c WV (v) is expected in such conditions. The wave spectra have been simulated using the WAVEWATCH III (WW3) model (Tolman 2009) with the ST4 source terms forced by the COAMPS-tropical cyclone (COAMPS-TC) wind (Doyle et al. 2014) under Hurricane Ian (2022). Two examples of the simulated wave spectrum are chosen for this exercise. They have been validated against direct observations using the Spotter buoys (Davis et al. 2023) for the frequency spectra (Fig. 2) as well as for the directional moments (not shown). Note that the WW3 model predicts the full directional frequency spectra but the buoy observations provide the frequency spectra and the directional moments only. The significant wave height of the first example was 9.9 m observed and 9.4 m simulated, while the values of the second example were 10.4 and 10.0 m, respectively. Since the high-frequency tail of the WW3 is known to be overestimated (Reichl et al. 2014), it has been lowered to be more consistent with the observation. We have also confirmed that the estimated c WV g (v) values from the observation and the simulation are consistent (within 610%) if the same directional spreading is assumed. The WW3 directional frequency spectra are used in the following analysis.

We have also considered a typical ocean wave spectrum of fully developed seas (Pierson and Moskowitz 1964),

where v p is the peak angular frequency. With this spectrum, we have assumed a simple directional spreading of cosu for |u| # p/2 and no energy for |u| # p/2.

b. Group velocity

Let us examine the tropical cyclone wave fields first. In Fig. 3, the group speed enhancement c WV g (v) (red line) due to the entire spectrum are compared with the group speed enhancement c CU g (v) (magenta line) due to a sheared current whose profile is identical to the Stokes drift profile integrated over the entire spectrum. The right and top axes show the dimensional angular frequency v and dimensional values of c WV g and c CU g , while the left and bottom axes show v normalized by the peak angular frequency v p , and c WV g and c CU g normalized by c gp 5 v p /2k p (the group velocity at the spectral peak). The vertical black dashed line indicates the peak frequency.

In both examples, the actual group velocity increase c WV g at the spectral peak v p is almost twice as large as c CU g , which is the expected increase if the wave energy is simply advected by the integrated Stokes drift. The group velocity enhancement is around 0.3-0.4 m s 21 and is around 4%-5% of the linear group velocity. Although these values are likely to be smaller than the Eulerian surface currents, which is 0.6-0.7 m s 21 predicted by the MOM6 ocean model (Adcroft et al. 2019), they are not negligible. As v increases above v p , c WV g increases quite slowly and c WV g /c gp remains around 0.05 (red lines in Fig. 3). However, the group speed c g decreases inversely proportion to v, and the relative enhancement c WV g /c g becomes around 15% at v/v p 5 3 and around 50% at v/v p 5 10. The wave evolution equation of this study is formally invalid at such large v p /v, however. Fan et al. (2009) have investigated the surface wave evolution under a tropical cyclone by combining a wave prediction model (WAVEWATCH III) and wave observations using a scanning radar altimeter (SRA). Their results show that the predicted significant wave heights in some areas are significantly reduced by O(1) m and are more consistent with the observations if the surface Eulerian current of O(1) m s 21 is included in the wave prediction. The wave height reduction occurs mainly because a wave packet under a strong wind region propagates faster with the added surface current and spends less time forced by the strong wind before it overtakes the storm and stops growing. Their results suggest that including the additional enhancement c WV g in the wave simulation would decrease the significant wave height by additional O(0.3-0.4) m. Although this significant wave height reduction does not seem very large, the wave height reduction by 5% (say from 8 to 7.6 m) corresponds to decrease of the total wave energy by 10% and its impact is not trivial.

Next, the group velocity enhancement c WV g is compared with the Stokes drift magnitude. The blue solid line shows U ST (v, 0), that is, the surface Stokes drift integrated up to v. The surface Stokes drift integrated up to v p [U ST (v p , 0)] is very small, while the surface Stokes drift integrated over the entire spectrum [U ST (', 0)] approaches 0.4-0.5 m s 21 . If the surface Stokes drift is integrated up to 3v [U ST (3v, 0)], shown by the dashed blue line, the result is quite close to c WV g at all frequencies above v p . This suggests that c WV g is well approximated by U ST (3v, 0). Alternatively, the Stokes drift can be estimated not at the surface but at a certain depth. The cyan dashed line shows U ST (', 0.02/k), that is, the Stokes drift evaluated at a depth of z 5 0.02/k 5 0.0032l (0.3% of the surface wavelength) and integrated over the entire spectrum. The result is very close to U ST (3v, 0) (blue dashed line) and is a good approximation of c WV g as well. Since the results show increasing c WV g with v, they suggest (in principle) that it is necessary to evaluate the group velocity for each wave frequency separately to be included in wave prediction models. However, c WV g rapidly increases with v only up to v p , and it increases quite slowly above v p , by less than 0.1 m s 21 from v p to 10v p . Therefore, it may be acceptable to use a single c WV g value in the wave prediction model. In fact, we may define the spectrally weighted average of c WV g , so that it represents the total enhancement of the wave energy flux if it is multiplied by the total wave energy. The resulting value is indicated by the horizontal dashed line in the figure. It is evident that the weighted average of c WV g is almost identical to c WV g at the spectral peak, and this quantity is well approximated by the surface Stokes drift integrated up to 3v p or the total integrated Stokes drift evaluated at z 5 0.02/k p 5 0.0032l p . In summary, for strongly wind-forced conditions, it may be acceptable to include the enhanced group speed effect in wave models by simply adding c WV g (v p ) ≃ U ST (3v p , 0) ≃ U ST (', 0:02/k p ) to the Eulerian current.

We next examine the results of the fully developed spectrum shown in

c. Phase speed

We next examine the increase of the phase speed of a particular wave component (angular frequency v, u 5 0) due to the same wave fields. The results from the two tropical cyclone examples are shown in Fig. 5, where the solid red lines show the actual phase speed increase c WV and the solid magenta lines show the phase speed increase c CU due to a sheared current whose profile is identical to the integrated Stokes drift profile. At the spectral peak v p , c WV is significantly larger than c CU , that is, the phase speed increase is significantly larger than the simple advection by the integrated Stokes drift. The values of c WV are roughly 0.2 m s 21 and 1% of the phase speed, which are not large. However, it increases to about 0.4 m s 21 at v/v p 5 10, which is close to the surface Stokes drift integrated over the entire spectrum. Since the phase speed is also inversely proportional to v, the relative phase speed increase c WV /c becomes about 6% at v/v p 5 3 and about 25% at v/v p 5 10. The value of c WV at and above the spectral peak is very well approximated by the surface Stokes drift integrated up to 1.5v, U ST (1.5v, 0), shown by the dashed blue line, or the Stokes drift evaluated at z 5 0.12/k 5 0.019l and integrated over the entire spectrum, U ST (', 0.12/k), shown by the dashed cyan line.

The results with the fully developed spectrum are shown in Fig. 6. The enhancement c WV at the spectral peak is small (around 0.05 m s 21 and 0.3% of c). At v/v p 5 10, it becomes about 0.15 m s 21 and 10% of c. Again, c WV is well approximated by U ST (1.5v, 0) or U ST (', 0.12/k).

Conclusions

In this study, the multiple-scale perturbation analyses have been conducted to investigate the propagation of a single wave train alone, as well as with a background Eulerian shear current, and with a second wave train. The analysis with a single wave train shows that the propagation speed of the wave energy is larger than what is expected from the advection by its own Stokes drift. However, the propagation speed of the wave action is consistent with the advection by its own Stokes drift.

The analysis with a second wave train clarifies how the phase and group speeds are modified by the presence of the second wave train. Their enhancements are significantly larger than what are expected from the simple advection by the Stokes drift of the second wave train. If the impacts are FIG. 6. Phase speed modification c WV (red line) due to the entire spectrum and phase speed modification c CU (magenta line) due to the sheared current whose profile is identical to the Stokes drift profile integrated over the entire spectrum, calculated with the typical fully developed wave spectrum (Pierson and Moskowitz 1964). Both are normalized by the peak phase speed c p (left axis) and plotted against angular frequency v (top axis) or angular frequency normalized by peak frequency v/v p (bottom axis). Blue line: surface Stokes drift U ST (v, 0) integrated up to v. Blue dashed line: surface Stokes drift U ST (1.5v, 0) integrated up to 1.5v. Cyan dashed line: Stokes drift at a depth of z 5 0.12/k integrated over the entire spectrum, U ST (', 0.12/k). Vertical black dashed line: peak frequency. integrated over the entire wave spectrum, the phase and group speeds of a particular wave train can be significantly enhanced, particularly in strongly wind-forced conditions. Our estimates using the simulated wave spectra (validated against the observations) under tropical cyclones suggest that the group speed at the spectral peak may increase by 0.3-0.4 m or 4%-5%, reaching to about 15% at 3 times the peak frequency. The integrated impacts on the phase and group speeds of a particular wave train are significantly larger than the simple advection expected from the Stokes drift integrated over the entire spectrum. These results suggest that including the enhancement of the group velocity by the presence of the entire wave spectrum in wave models may significantly impact wave predictions in strongly wind-forced conditions.

As a first step, this study has focused on a simple case of deep water and statistically uniform wave/current fields. It is desirable to extend the analysis to more general conditions of varying water depths and varying background currents, so that the effects can be fully incorporated in the wave prediction models. codes and results used in this study are available in Hara (2024).

APPENDIX A

Wave Action Equation

Here, the wave action equation is derived. Let us start with the truncated wave amplitude evolution equation

in the presence of the second wave train, and express A in terms of its amplitude |A| and phase P, A 5 |A|e iP : (A2)

Then, the real and imaginary parts of the evolution equation (divided by e iP ) yield

and

By multiplying (A3) by |A|, the truncated energy equation becomes

Noting that the wavenumber vector is allowed to be modulated to (k x 5 k 1 eP X , k y 5 eP Y ) from (k, 0), the terms proportional to e in (A5) simply represent the correction of the group speed (energy flux) because of the wavenumber modulation. This equation also suggests that for every wave train, we may define its leading-order wavenumber k such that the terms proportional to e in (A5) disappear on average. Next, we consider the wave action equation. When the wavenumber of the wave train is modulated, its intrinsic frequency V is also modulated, ) and the wave action is defined as N 5 |A| 2 /2V following the standard definition. If we multiply (66) by 1/V, we obtain

The first two terms can be rewritten as

From ( A4) and (A6), we may obtain

and

8k 3 |A| X |A| YY |A| 2 1 1 16k 3 |A| XXX |A| 2 1 8k 3 |A| XYY |A| 2 1 8k 3 P XX P X 1 1 4k 3 P XY P Y ) 1 O(e 3 ): (A10) By combining (A7)-(A10), the wave action equation is obtained as N T + v 2k N X + e 1 16k 2 (iA * A X 2 iA * A X ) X + e 2 1 8k 2 (iA * A Y 2 iAA * Y ) Y + e 2 7k 16 |A| 4 + C 2 2 k|A| 2 |B| 2 + 1 32k 3 |A X | 2 2 1 16k 3 |A Y | 2 2 1 32k 3 (A * A XX + AA * XX ) + 1 16k 3 (A * A YY + AA * YY ) X + e 2 C 3 2 k|A| 2 |B| 2 2 1 16k 3 (A * X A Y + A X A * Y ) + 1 8k 3 (A * A XY + AA * XY ) Y +e 2 C 4 + C 5 2 C 2 2 k|A| 2 (|B| 2 ) X + C 6 + C 7 2 C 3 2 k|A| 2 (|B| 2 ) Y 2 e 2 1 2 |A| 2 k 2 |A||A| X + C 1 k|B||B| X 2 1 16k 3 |A| X |A| XX |A| 2 + 1 8k 3 |A| X |A| YY |A| 2 + 1 16k 3 |A| XXX |A| 2 1 8k 3 |A| XYY |A| 2 1 8k 3 P XX P X + 1 4k 3 P XY P Y ) 2 e 2 P X 2k 1 16k 2 (iA * A X 2 iA * A X ) X 2 e 2 P X 2k 2 1 8k 2 (iA * A Y 2 iAA * Y ) Y O(e 3 ): (A11)

Since the terms proportional to e and the terms proportional to |A| 2 (|B| 2 ) X or |A| 2 (|B| 2 ) Y can be eliminated on average, the action equation for a uniform (or modulated slower than in x 1 , y 1 ) wave train is simplified to

and the wave action flux (F X N , F Y N ) can be expressed as ) is less than the impact on the energy flux (c NL g ). The impacts of the second wave train on the action flux and on the energy flux are identical. Following the same approach, it is straightforward to find that the impacts of the Eulerian shear current on the action flux and on the energy flux are also identical.

APPENDIX B

Evolution Equation with a Second Wave Train

As discussed in the main text, the evolution equation of the wave amplitude A in the presence of the second wave train (with the wave amplitude B) becomes

This perturbation analysis must be carried out separately:

• Case 1: u 5 0 (two waves propagating in the same direction).

• Case 2: u 5 p (two waves opposing each other).

• Case 3: u Þ 0 and u Þ p (all other cases).

In case 1, the analysis is further separated for v . v 2 and v , v 2 and the coefficients are given as

4 5 1 8 2 v 8v 2 2 3v 2 4v 1 3v 3 2 4v 3 , C 5 5 1 8 2 v 8v 2 1 3v 2 4v 1 3v 3 2 4v 3 , C 6 5 C 7 5 0, for v . v 2 , (B2) and C 1 5 v 2 v , C 2 5 2v 2 v , C 3 5 0, C 4 5 1 8 1 5v 2 4v 2 3v 8v 2 , C 5 5 1 8 1 5v 2 4v 1 v 8v 2 , C 6 5 C 7 5 0, for v , v 2 : (B3) The results for case 2 are C 1 5 C 2 5 2 v 3 2 v 3 , C 3 5 0, C 4 5 1 8 1 v 8v 2 2 3v 2 4v 1 v 2 2 4v 2 2 v 3 2 2v 3 , C 5 5 1 8 1 v 8v 2 1 3v 2 4v 1 v 2 2 4v 2 2 v 3 2 2v 3 , C 6 5 C 7 5 0, for v . v 2 , (B4) and C 1 5 2 v 2 v , C 2 5 2 2v 2 v , C 3 5 0, C 4 5 2 3 8 1 v 2 v 1 v 8v 2 , C 5 5 2 3 8 1 v 2 v 2 3v 8v 2 , C 6 5 C 7 5 0, for v , v 2 : (B5)

Note that Longuet-Higgins and Phillips (1962) carried out a similar analysis up to O(e)and derived C 1 for cases 1 and 2 only. For case 3, the coefficient C 1 becomes

with

The coefficient C 2 becomes

with

, (B9)

1 gk 3 [4v 5 1 (4 1 5 cosu)v 4 v 2 1 (23 1 4 cosu)v 3 v 2