## 1. Introduction

The Stokes drift (Stokes 1847) is the difference between the Eulerian velocity in a point and the average Lagrangian motion of a water particle undergoing the orbital motion **u**_{w} of a wave field (Longuet-Higgins 1953; Phillips 1977),

Here the averaging ⟨⋅⟩ is over a period appropriate for the frequency of surface waves (Phillips 1977; Andrews and McIntyre 1978; Leibovich 1983). Conceptually, this can be written (LeBlond and Mysak 1978; van den Bremer and Breivik 2018) as

Stokes drift profiles have commonly been modeled from the assumption of unidirectional, monochromatic waves (see, e.g., Skyllingstad and Denbo 1995; McWilliams et al. 1997; Polton et al. 2005; Saetra et al. 2007). This yields a profile that has too weak shear near the surface and too strong Stokes transport deeper down. The Stokes drift profile under a Phillips-type (Phillips 1958) spectrum was explored by Breivik et al. (2014, hereafter B14), and later in more detail by Breivik et al. (2016, hereafter B16). The Phillips-type Stokes profile was found by B16 to improve both the shear and the transport compared to the earlier, empirical profile proposed by B14. Li et al. (2017) derived an analytical expression for the Stokes transport under the Phillips profile and its depth-averaged profile. The Phillips-type profile was shown to give a good approximation to the profile under an arbitrary spectrum, but it does not address the veering of the profile due to the presence of swell. Here we explore how a Phillips-type Stokes drift profile can be combined with a monochromatic profile or another Phillips-type Stokes drift profile for the swell component, thus providing an efficient way to parameterize a full three-dimensional profile. This has practical implications for the computation of the trajectories of drifting objects (Breivik et al. 2012a,b; Röhrs et al. 2015; Christensen et al. 2018) as well as for the fate of oil in the ocean (McWilliams and Sullivan 2000; Jones et al. 2016; Dagestad et al. 2018) and the drift of eggs and larvae (Röhrs et al. 2014; Strand et al. 2019). It also allows easier experimentation on wave model data to test the potential impact that crossing seas (e.g., swell and wind sea at significant angles to each other) might have on the Stokes production responsible for Langmuir turbulence (Van Roekel et al. 2012; Harcourt 2013; McWilliams et al. 2014; Harcourt 2015; Li et al. 2017; Ali et al. 2019; Breivik et al. 2019). Finally, the Coriolis–Stokes force in coupled models (Fan and Griffies 2014; Breivik et al. 2015; Staneva et al. 2017; Wu et al. 2019a) and the associated Stokes material transport (Wu et al. 2019b) are affected by the shape of the Stokes drift profile. The Stokes drift implementation in ocean models should ideally be based on the Fourier representation of Kenyon (1969), using a linear superposition of the contributions from discrete components of the two-dimensional wave spectrum. With few exceptions (Li et al. 2016), the Stokes drift is approximated based on a unidirectional and monochromatic representation of the wave field (see, e.g., Uchiyama et al. 2010). Implementing a combined swell and broadband wind sea profile as suggested here will allow better control over the Stokes drift profile and its impact on upper-ocean dynamics in realistic ocean models.

Superposition of the Stokes drift of different Fourier components means that, provided we have separated the spectrum into nonoverlapping swell and wind sea parts, we can also decompose the Stokes drift velocity in a swell component (sw) and a wind sea component (ws):

We are often faced with a situation where we either have access to integrated wave model parameters from one of the main spectral wave models in use today (Hasselmann et al. 1988; Booij et al. 1999; Tolman et al. 2014; ECMWF 2019) or we can make assumptions and construct an idealized wave field (McWilliams et al. 2014). The following discussion assumes that we have access to the integrated parameters significant wave height (*H*_{s}), swell and wind sea height (*H*_{sw}, *H*_{ws}), mean frequencies (

Integrated parameters required from a wave model (or assumed known) to calculate a combined Stokes drift profile.

This article is organized as follows. Section 2 explores the degree to which swell and wind sea Stokes drift differ in strength and direction throughout the world’s oceans by defining a number of measures of relative Stokes drift and degree of crossing. In section 3 we recapitulate the properties of the Stokes drift profile and remind the reader of the features of a Phillips-type spectrum and its associated Stokes drift profile. In section 4 we derive the combined profile for a monochromatic swell component and a wideband wind sea spectrum for which we assume the Phillips profile. We also consider using a Phillips type spectrum for the swell. In section 5 we compare the combined profile against the Stokes drift profile calculated from the full two-dimensional (2D) spectrum of the ERA-Interim (Dee et al. 2011) wave model. Finally, section 6 discusses the practical use of such a combined profile and where in the world’s oceans the competing influence of swell and wind sea affects the Stokes drift profile the most.

## 2. Global swell and wind sea Stokes drift climate

For a directional wave variance density spectrum *E*(*ω*, *θ*) (m^{2} s rad^{−2}), the Stokes drift velocity in deep water is given by

where *ω* is the circular frequency, *z* the vertical coordinate (positive up, and negative below sea level) and *θ*. This can be derived from the expression for a wavenumber spectrum in arbitrary depth (Kenyon 1969), using the deep-water dispersion relation *ω*^{2} = *gk* (with *g* ≈ 9.81 m s^{−2} the acceleration due to gravity). The circular frequency spectrum is defined as

for which the Stokes drift speed profile becomes

From Eq. (5) it is clear that at the surface the Stokes drift is proportional to the third spectral moment [where the *n*th spectral moment of the circular frequency is defined as

While Eq. (5) yields the Stokes drift speed at different depths, the direction of the two-dimensional velocity vector, Eq. (4), can vary significantly, as the degree to which swell and wind sea directions deviate varies greatly throughout the world’s oceans. The extratropics exhibit a mix of local wind sea and swell (Semedo et al. 2011, 2015) whereas the swell-dominated tropics are mostly unaffected by wind sea. To quantify the relative impact of the swell and the wind sea and the degree of misalignment we now define and investigate a number of quantities.

### a. The Stokes balancing depth D_{b}

The first quantity we consider is the Stokes balancing depth *D*_{b} at which the swell and wind sea Stokes drift speed are equal in strength, found by solving the following equation for *z*:

The Stokes balancing depth is here used to investigate the regional differences in swell and wind sea contributions to the total Stokes drift. For this specific part of the analysis, the wind sea and the swell parts will each be represented by a monochromatic profile,

Using a monochromatic profile will yield a slightly larger balancing depth than if we employ a more complex parametric profile because it decays more slowly (B16). This is acceptable as it is the relative geographical differences that we wish to investigate, and those will remain the same. At *z* = −*D*_{b} we then have

Here,

The monochromatic swell wavenumber can be found from the linear deep-water dispersion relation,

The swell Stokes transport is proportional to the first spectral moment of the swell spectrum (using the circular frequency *ω*),

The monochromatic surface swell Stokes drift speed can then be found as

The same procedure gives us the monochromatic wind sea surface Stokes drift and wavenumber, with all swell quantities in Eqs. (10)–(12) replaced by their wind sea counterparts. These are readily available from wave models. For the mean frequencies, we have used the first moment,

The balancing depth can now be found,

Figure 1 shows the Stokes balancing depth for ERA-Interim, averaged over the years 2010–12, inclusively. Note that we set all cases where *D*_{b} < 0 to zero to reduce the effect of cases where there is no solution [i.e., where the swell height is smaller than the wind sea height and

### b. The Stokes depth ratio

A simpler estimate of the relative importance of the swell is the swell to wind sea Stokes *e*-folding depth ratio (not to be confused with the balancing depth *D*_{b}),

Here, the Stokes transport is defined as *D*_{ws} = 1/2*k*_{ws} and *D*_{sw} = 1/2*k*_{sw}, we can write this simply as

Figure 2 shows the ratio of the swell Stokes depth to the wind sea Stokes depth. We see that the areas where the swell Stokes *e*-folding depth is unusually large compared to the wind sea Stokes depth coincide with the regions where the balancing depth (Fig. 1) is large. It is also interesting to note that the depth ratio remains large in a larger region in the equatorial Pacific than what is found for the balancing depth (see the region between 180° and 135°W, south of the equator). In regions where the wind sea Stokes *e*-folding depth is shallow compared to the swell counterpart, i.e., where *k*_{ws} ≫ *k*_{sw}, the balancing depth approaches

which is independent of *k*_{sw}. This could mean that a decrease in the balancing depth *D*_{b} while *r*_{D} stays more or less the same could be caused by a decrease in the ratio *k*_{ws}.

### c. Stokes swell transport ratio

The third quantity that can shed light on the importance of the swell Stokes drift is the ratio of the swell Stokes transport to the total transport,

Figure 3 shows the degree to which the swell Stokes transport dominates the tropics. The reason the ratio of swell to total transport sometimes exceeds unity is that the total transport is reduced by directional spread, whereas the swell transport is calculated on the assumption of unidirectional waves, see the discussion in section 4 and Eq. (11). It is clear that the swell transport totally dominates the tropics and the subtropical regions outside the trade wind belts (Carrasco et al. 2014; Breivik et al. 2019). As expected, the swell transport ratio drops rapidly outside the latitude belt ±40°. The extratropics, and in particular the Southern Ocean, exhibits swell transport ratios around 50%. This is quite in line with what is known about the wave climate in general (Aarnes et al. 2017; Morim et al. 2019) and the swell climate in particular (Semedo et al. 2011, 2015).

### d. Degree of crossing Stokes drift

Finally, the degree to which swell and wind sea directions diverge can be used as a proxy for how much the effect of crossing seas should affect the Langmuir turbulence (Belcher et al. 2012; Van Roekel et al. 2012; McWilliams et al. 2014) in the turbulent kinetic energy (TKE) equation,

Here we have followed the notation used by Belcher et al. (2012); **u**_{H}. Primes indicate turbulent quantities, with *w*′ the vertical component. The Stokes production term (2) of interest here is similar in form to the Eulerian shear production but involves instead the shear of the Stokes drift. The buoyancy term (3) depends on the turbulent buoyancy fluctuations *b*′. Term 4 represents the turbulent transport and pressure correlation terms (Stull 1988; Kantha and Clayson 2000). Finally, term 5, *ε*, is the TKE dissipation term.

By defining the degree of crossing as

we get a dimensionless number *r*_{×} ∈ [−1/2, 1/2] that depends on the ratio of the cross product of the swell and wind sea over the total surface Stokes drift. Here,

## 3. Properties of the Phillips profile

The Phillips spectrum (Phillips 1958)

yields a reasonable estimate of the part of the spectrum which contributes most to the Stokes drift velocity near the surface, i.e., the high-frequency waves. Here *ω*_{p} is the peak frequency. We assume Phillips’ parameter *α* = 0.0083. The Stokes drift velocity profile under (19) is

An analytical solution exists for (20); see B14, their Eq. (11). Using the deep-water dispersion relation, Eq. (20) can be written as

Here erfc is the complementary error function and *αg*/*ω*_{p}.

Let us assume (B16) that the Phillips spectrum profile (21) is a reasonable approximation for Stokes drift velocity profiles under a general spectrum,

where *β* is a parameter. The profile differs from the monochromatic profile, Eq. (8), in the added error function term, which makes the shear stronger near the surface and the deep Stokes drift weaker. The total Stokes transport under Eq. (22) can be found (see appendix A of B16) to be

We can now determine the inverse depth scale *V* and the surface Stokes drift velocity *υ*_{0}. Both, as B14 argued, are normally available from wave models. Note that we still need to estimate *β*, but B16 found *β* = 1 to yield good agreement between modeled and parameterized Stokes drift profiles. In the following we will exclusively use this value. Note also that estimating the Stokes transport from a one-dimensional Stokes drift profile will overestimate it since the directional spreading of waves tends to cancel out some of the contributions. This effect is ignored by assuming all waves to be propagating in the same direction. The Stokes transport from a unidirectional profile should typically be reduced by about 17% (Ardhuin et al. 2009; B14). The spreading factor proposed by Webb and Fox-Kemper (2015) could be used to further correct the profile.

### a. The layer average of the Stokes drift profile under the Phillips spectrum

Li et al. (2017) found a closed-form expression for the integral of Eq. (22), i.e., the Stokes drift transport between the vertical level *z*_{0} and the surface,

See their appendix A for a full derivation. To find the average Stokes drift between a lower-level *z*_{0} and an upper-level *z*_{1} all that is needed is to use Eq. (24) twice to find

Here we have indicated that the chosen vertical range affects the averaging by writing

where *z*_{ref} is a reference depth.

### b. The shear under the Phillips profile

The shear under Eq. (22) is straightforward to find (B16),

which simplifies to

when *β* = 1. The latter (27) is of course particularly convenient and can readily be used to analytically calculate related quantities like the depth-averaged weighted Stokes drift shear (Kukulka and Harcourt 2017).

## 4. A combined Stokes drift profile for swell and wind sea

We will in the following impose the constraint (3) that the wind sea surface Stokes drift is determined as

This total surface Stokes drift is assumed to be found with a wave model or otherwise decided by the experimenter [see, e.g., McWilliams et al. (2014), who constructed a combined profile by dictating the wind sea and swell strength and direction]. There are no strong compelling reasons for determining the wind sea Stokes drift strength and direction from the swell and the total sea other than the fact that the swell direction is often more precisely known from climatology. We could thus equally well have chosen to determine the swell from the wind sea and the total Stokes drift.

### a. A monochromatic swell profile

Let us first assume that the swell is well represented by the monochromatic profile such that

Here

### b. Imposing swell and wind sea direction

The procedure above will lead to a wind sea Stokes drift that is not necessarily in the direction of the wind sea as it is dictated by the swell surface Stokes drift, which in turn is estimated from the assumption that the swell is monochromatic. In a real wave model, this is not the case, and the swell direction and strength will be composed of several Fourier components in the 2D spectrum. This may still work well in idealized cases (McWilliams et al. 2014), but is not desirable when we have access to model output of wind sea and swell directions. It is however possible to add the constraint that both the swell and the wind sea Stokes drift surface vectors are in line with the swell (*θ*_{sw}) and wind sea (*θ*_{ws}) direction. Our constraint (28) says that the surface Stokes drift must be preserved,

We can then work out the speed of the swell surface Stokes drift by taking the cross product of Eq. (30) and

This yields the following relation,

Here,

We now have enough information to estimate the swell and wind sea wavenumbers, either from Eq. (12) for the monochromatic or Eq. (23) for the Phillips Stokes drift profile [where we find *V*_{sw} from Eq. (11) and *V*_{ws} is computed in the same manner using corresponding wind sea quantities]. In the following analysis we use this method to determine the direction of the wind sea Stokes drift profile.

### c. A Phillips wind sea profile

Assume that Eq. (22) is a good approximation for the wind sea part of the Stokes drift profile, and let the surface Stokes drift from the wind sea part of the spectrum be defined as

This constraint, see Eq. (28), forces the sum of the swell and wind sea surface Stokes drift to sum to the total surface Stokes drift. The wind sea transport *V*_{ws} is determined similarly as the swell transport, Eq. (11),

Finally, the inverse depth scale (or wavenumber) of the wind sea profile is found from Eq. (23),

To ensure that the surface Stokes drift vector is preserved, the direction of the wind sea profile should be determined from Eq. (3),

This procedure allows us to estimate a combined profile with a directional veering due to the presence of swell. The parameters can all be estimated from standard output from atmosphere–wave reanalyses such as ERA-Interim (Dee et al. 2011) or regional wave hindcasts like NORA10 (Reistad et al. 2011; Haakenstad et al. 2020).

### d. Two Phillips profiles

As an alternative to the assumption that swell is well represented by a monochromatic profile, we can instead assume two Phillips profiles, one for the swell part and one for the wind sea part of the spectrum. This has the added complication that since we now assume a broadbanded spectrum (the Phillips spectrum) for the swell, we can no longer calculate the wavenumber from the swell peak frequency through the dispersion relation (10). However, following the procedure (31)–(34) and by computing the transports according to Eq. (11) allows us to calculate both the monochromatic and Phillips wavenumbers with equal ease.

## 5. Comparison against ERA-Interim 2D Stokes drift profiles

Following the same procedure as B14 and B16, we here compare the full 2D Stokes drift profile calculated from the ERA-Interim (Dee et al. 2011) reanalysis against three profiles. The first is a Phillips profile, identical to that explored by B16, in the direction of the surface Stokes drift. The second profile is a combined monochromatic swell and Phillips wind sea profile, and the third is a combination of two Phillips profiles, one for the swell component and one for the wind sea. A full year of spectra from location 60°N, 340°W at 0000 UTC is used (location indicated in Fig. 3). The location is characterized by a mixture of swell and wind sea (Semedo et al. 2011) and is thus well suited for investigating the performance of the combined Stokes drift profile. It also coincides with the location used by B14 and B16 (as does the choice of ERA-Interim spectra for the year 2010). The ERA-Interim data used are the same as those used for the global statistics in section 2. The 2D spectra used for the calculation of the full Stokes drift profile have a directional resolution of 15° and cover the frequency range 0.0345–0.548 Hz with logarithmic spacing. The spatial resolution of the wave model component of ERA-Interim is approximately 110 km. We compute the 2D Stokes drift velocity vector at every 10 cm from the surface down to 30-m depth from the full spectra.

To illustrate the performance of the combined profiles, derived from the integrated parameters listed in Table 1, we start by choosing a rather demanding case where we have swell going in the northeast direction and wind sea propagating nearly diametrically in the west-southwest direction. Figures 5 and 6 give a 3D perspective and a 2D bird’s eye view of the Stokes drift profiles for the given situation, respectively. A short explanation to the figures is justified. First, the monochromatic swell profile is drawn in green in Fig. 5a. It can be seen to point almost diametrically opposite to the Phillips wind sea profile (blue). They combine to form the orange curve, which should be compared to the red curve that represents the full profile, calculated from the ERA-Interim 2D wave spectrum. For comparison we also include the unidirectional Phillips profile (purple). We see that the combined profile does a fairly good job at representing the total profile, and it is certainly an improvement over the Phillips unidirectional profile. Moving to the bird’s eye view (Fig. 5b), we see that the veering of the full profile is captured, but overdone (orange, fully drawn curve), by the combined monochromatic (swell, green) and Phillips (wind sea, blue) Stokes drift profile (Fig. 5a). This is improved significantly by exchanging the monochromatic swell profile with a Phillips-type profile (orange, dashed curve). This may seem surprising as we would expect the swell to be quite monochromatic. It is however important to keep in mind that real swell is not monochromatic, and we are after all trying to model a full 2D spectrum with a somewhat arbitrary separation of swell and wind sea performed by the wave model itself [see the discussion by ECMWF (2019), Breivik et al. (2019), and Strand et al. (2019) about the swell separation algorithm and its consequences for the swell statistics]. Figure 6 shows that the east and north components of the Stokes drift profile confirm the impression that representing the swell with a Phillips-type Stokes profile (dashed green curve) seems to do a better job than the monochromatic profile (green fully drawn curve). The total profile (orange dashed) is much closer to the red curve representing the total profile than the combination of a monochromatic swell profile with a Phillips wind sea Stokes profile.

In Figs. 7a–c we assess for the year 2010, like B16 did, the departure of the combined profiles from the full profile by calculating the normalized absolute difference as

Here *υ* represents the Stokes drift speed of the full 2D Stokes drift profile of the wave model and the Stokes transport

The reduction of (the nonnormalized) transport magnitude error,

is shown in Figs. 7d–f. This is an estimate of the departures in Stokes drift speed at each vertical level. As we can see, the speed is evidently much better represented by the combined profiles (Figs. 7e,f) than by the unidirectional profile (Fig. 7d).

The (nonnormalized) east and north components of the transport error are calculated as

Here *υ*_{i} represents either east or north components of the Stokes drift vector. These are shown in Fig. 8. The reduction in error in directional components is quite significant, and as for the normalized transport error, the component-wise error reduction is considerably greater for the combined Phillips Stokes profiles (about 40% reduction, and about 25% reduction for the combined monochromatic swell and Phillips wind sea Stokes profiles). Figures 7d–f and 8 together demonstrate that the combined profiles reduce errors in both the speed and the directional components of the Stokes drift, and, importantly, that combining two Phillips Stokes profiles reduces the error more than using a monochromatic profile for the swell component.

## 6. Discussion and concluding remarks

The combined profile is shown to reduce the overall error by about a third compared to a unidirectional Phillips profile in a location with mixed swell and wind sea conditions in the North Atlantic Ocean (60°N, 340°E, see Fig. 3). This is significant and it is thus a convenient alternative to calculating the full 2D profile from a wave model. We find significant differences between the Phillips and a monochromatic swell Stokes profile, and recommend using a combination of two Phillips Stokes profile.

The question of how such a combined profile will affect ocean models that rely on Stokes drift profiles for Langmuir turbulence and Coriolis–Stokes forcing can only be answered by running dedicated model experiments. It is still clear that the added degrees of freedom that the profile admits will allow much more complex Stokes drift profiles to be explored. This should allow easier experimentation on the question of whether crossing seas really do weaken the Langmuir turbulence (Van Roekel et al. 2012; McWilliams et al. 2014; Ali et al. 2019).

The global Stokes drift climate has been investigated using ERA-Interim reanalysis data. Two new quantities, the degree of crossing [Eq. (4)] and the Stokes drift balancing depth, i.e., the depth below which the swell Stokes drift is stronger than the wind sea Stokes drift [Eq. (13)], suggest that the region west of Central America exhibits a combination of strong swell at large angles to the local wind sea. The bands along 20°–30°S and to a smaller extent along 20°–30°N exhibit a high degree of crossing and can be expected to be very dominated by the large directional spread between the wind sea and the swell Stokes drift. We conclude that the degree of crossing is a useful quantity for identifying regions with significant deviation in the swell and wind sea Stokes drift direction. It is not enough to simply look at the deviation in wind sea and swell wave propagation direction, since the swell Stokes drift is in many cases so much weaker than the wind sea Stokes drift to be negligible. The balancing depth is also found to yield valuable information, but we note that it becomes noisy in the extratropics, something we attribute to the dominance of wind sea over swell. It is suited for the equatorial and subtropical regions where it clearly highlights regions where the balancing depth is small (i.e., where the strength of the swell Stokes drift rapidly overtakes the wind sea Stokes drift with depth). Further studies of the potential impact of crossing seas on Langmuir turbulence and the Stokes–Coriolis force demand ocean models that either take into account the Stokes drift profile from the full 2D spectrum of a wave model, or a combined parameterized Stokes drift profile as outlined in section 4.

We gratefully acknowledge support from the Research Council of Norway through the CIRFA project (Grant 237906) as well as the Copernicus Marine Environmental Monitoring Service (CMEMS) and Mercator Ocean through the WaveFlow Service Evolution project. We would also like to thank the two anonymous reviewers who with their comments made the paper a much better one. The data used are available through the ECMWF MARS archive. All parameters can be found with a standard request (experiment ID 1), except the Stokes drift which must be fetched from experiment FJVC.

# APPENDIX

## Seasonal Variation in Global Stokes Drift Parameters

The seasonal variation in swell and wind sea wave climate leads to significant changes in the seasonal Stokes drift climate. Figure A1 shows the seasonal impact on the balancing depth (see Fig. 1). The balancing depth shows a clear seasonal variation in the Indian Ocean, where the southwestern monsoon increases the Stokes drift balancing depth in the northern summer (less dominated by swell Stokes drift). In the northern subtropical Pacific Ocean the balancing depth is clearly reduced (becoming more dominated by swell Stokes drift) in the northern summer.

Figure A2 shows that the degree of crossing (see also Fig. 4) remains quite unchanged in the southern equatorial and subtropical band from 0° to 30°S, whereas the northern equatorial and subtropical band from 0° to 30°N shows a large degree of seasonal variation in the degree of crossing throughout all three ocean basins with much more crossing Stokes drift in the northern winter.

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