Three-dimensional wave-induced current model equations and radiation stresses

After the approach by Mellor (2003, 2008), the present paper reports on a repeated effort to derive the equations for three-dimensional wave-induced current. Via the vertical momentum equation and a proper coordinate transformation, the phase-averaged wave dynamic pressure is well treated, and a continuous and depth-dependent radiation stress tensor, rather than the controversial delta Dirac function at the surface shown in Mellor (2008), is provided. Besides, a phase-averaged vertical momentum flux over a sloping bottom is introduced. All the inconsistencies in Mellor (2003, 2008), pointed out by Ardhuin et al. (2008) and Bennis and Ardhuin (2011), are overcome in the presently revised equations. In a test case with a sloping sea bed, as shown in Ardhuin et al. (2008), the wave-driving forces derived in the present equations are in good balance, and no spurious vertical circulation occurs outside the surf zone, indicating that Airy’s wave theory and the approach of Mellor (2003, 2008) are applicable for the derivation of the wave-induced current model.


Introduction
Sea surface waves play a crucial role in ocean circulation, especially in driving shallow coastal current systems (Zheng et al., 2014). In the 1960s, Longuet-Higgins and Stewart (1964), hereinafter LHS64, proposed the concept of so-called wave radiation stress. Later Phillips (1977), hereinafter P77, stringently derived the wave-induced current equations. LHS64 and P77 lay a fundamental framework for theoretical analysis and numerical simulation of nearshore wave-induced current, but their works were limited to the horizontal current patterns. Dyhr-Nielsen and Sorensen (1970) noted that there exist the complex vertical circulation patterns driven by waves inside the surf zone. As pointed out by Zheng et al. (2014), the 3D nearshore wave-induced currents play important roles in the onshore and offshore mass transport and sediment movement. When propagating toward a shoreline, waves tend to be perpendicular to the shoreline due to wave refraction, which highlights the importance of the vertical cross-shore circulation. Hence, inside the surf zone, the disadvantages of a 2D horizontal wave-induced current simulation become more serious.
So far, many attempts have been made to set up the 3D wave-induced current model. Dolata and Rosenthal (1984) derived 3D radiation stress terms, but left off the effects from wave dynamic pressure. Through a sigma-coordinate kD 10 transformation, Xia et al. (2004), hereinafter X04, extended the depth-integrated radiation stress in LHS64 to include its vertical profile and incorporated it to 3D hydrodynamic equations for wave-induced current simulations. In recent years, Mellor (2003), hereinafter M03, has developed a set of concise and mathematically stringent approaches for the derivation of wave-current interaction equations, and the depth-dependent radiation stress in M03 has been widely used (e.g., Haas and Warner, 2009). However, Ardhuin et al. (2008) pointed out that there is a large discrepancy among wave action terms in M03 over a sloping bottom, which leads to a spurious circulation. They attributed the inconsistency in M03 to the fact that Airy's wave theory fails to accurately supply wave pressure and mass particle motion orbit over sloping bottoms. In the light of the test case by Ardhuin et al. (2008), Mellor (2008), hereinafter M08, further noted that, with the M03 formulation, unforced waves with bottom variations can produce mean currents even for deep water(say, , here k is the wave number and D, the water depth), which is physically unacceptable. Mellor (2008) attributed such a discrepancy to an incorrect calculation of wave pressure in M03 and revised the M03 equation sets by introducing a delta Dirac function at the surface for the phase-averaged wave pressure. However, the depth-integrated equations in M08 are still inconsistent with LHS64 and P77, leading to a spurious circulation . The revised expression for radiation stresses by Mellor (2008) results in a singularity at the surface, and the radiation stress becomes depth-independent beneath the surface in the wave propagation direction. Such a profile cannot be naturally derived from mathematical equations of Mellor. In the wave theory of Airy, wave velocity and pressure are continuous and derivative; hence, the radiation stresses associated with them should also be continuous and derivative. The delta Dirac function of Mellor is challenged by Aiki and Greatbatch (2013), hereinafter AG13. Based on the thickness-weighted mean approach, AG13 also developed a set of wave-driven current equations, which could reduce to the solution of Mellor (2011) without the delta Dirac function. The argument in AG13 caused Mellor to rethink various aspects of his 2003 paper. Mellor argued that the term in question is not singular, and corrected his expressions for radiation stresses by using an improved treatment of wave pressure. Mellor's finally corrected 2003 paper can be found at ftp://aden.princeton. edu/ pub/glm/corrected2003/.

uwũ,w
The problems in M03 and M08 still remain unsettled. As demonstrated in the present paper, the spurious circulation produced in M03 does not result from the treatment of wave pressure. Firstly, in the sigma coordinate equations, as baroclinic pressure gradients, the horizontal gradient term of radiation stresses should be split into an along-coordinate surface gradient component and a vertically corrective one, which are opposite in sign but large in magnitude over a sloping bottom. In X04 and M08, there is no vertical correction for the horizontal radiation stresses gradient term in the sigma coordinate. In M03, the vertical correction for the horizontal gradient of the phase-averaged wave pressure is only partly accounted for, omitting the vertical correction for the horizontal gradient of wave momentum fluxes. The latter is associated with the flux of the horizontal momentum across the sloping sigma coordinate level. As is also argued by , fluxes due to sloping iso-coordinates are often omitted in X04 and M08 or poorly approximated in M03. Because the wave momentum fluxes are larger than the phase-averaged wave pressure, the neglect of a vertical correction for the horizontal gradient of the wave momentum fluxes over a sloping bottom in M03 would produce a significant error. This is the reason why a monochromatic wave could produce mean currents even for deep water with a bottom slope in M03. Secondly, the development of the previous wave-induced current model equations is almost based on Airy's wave theory and, therein, the phase-averaged vertical wave momentum fluxes are nil (here, are the horizontal and vertical wave velocity, respectively). But, in shallow water with a sloping bottom, the vertical wave momentum fluxes are non-zero and are the major driving forces for nearshore currents (Rivero and Arcilla, 1995 are also neglected in M03 and M08. This is another reason why formulae of M03 and M08 produce spurious circulations in the numerical tests of Ardhuin et al. (2008) and . Thirdly, in the finally corrected 2003 paper of Mellor, the improved expression for the wave pressure is still open in question. That is, it does not exactly satisfy the sea surface condition . Moreover, such an expression gives a nil [see Eq. (34d) in the finally corrected 2003 paper of Mellor], here s is a vertical coordinate transformation, denoting its horizontal derivative. It means that the vertical correction for the horizontal gradient of the phaseaveraged wave pressure is omitted and a more serious error is existent in the corrected M03. As pointed out by Ardhuin et al. (2008), M03's formula is only valid if the sigma-coordinate vertical momentum flux is negligible (here is vertical velocity in the sigma coordinate). Unfortunately, the vertical flux is not negligible. It includes the flux of the horizontal momentum across the sloping sigma-coordinate level and the correlation of the wave's horizontal and vertical velocity . In sum, the inconsistency in the finally corrected M03 remains to be settled.
The problems in M03, as well as the importance of the wave action in driving currents, intrigues other scholars to develop wave-induced equations via other approaches. Especially, AG13 uses an equation system in the vertically Lagrangian and horizontally Eulerian coordinates. AG13 then takes a thickness-weighted mean on this equation system to obtain the wave-induced current equations. The formulae of AG13 fully include the effects of the wave's horizontal momentum flux, vertical momentum flux, and wave pressure, in which the wave driving mechanisms are more sound than the ones in M03. However, the calculation of the depth-dependent radiation stresses are not completed in AG13. AG13 should face the same problems as M03. Namely, AG13 needs to concisely treat the vertical wave momentum flux and wave pressure over a sloping bottom which Airy's wave theory fails to provide. It should be noted that the vertical Lagrangian-coordinate hydrodynamic equations [see Eqs. (1a) and (2a) in AG13] are similar in the mathematical form to the surface-following-coordinate equations of Mellor [see Eqs. (21) and (22) in M03], and the surface-following coordinate transformation in M03 and M08 could be similar to a vertically Lagrangian coordinate, so both the approaches could reach the same solution for the radiation stresses.
The inconsistencies or limitations in M03 and M08 led to a debate concerning the applicability of Mellor's approach and Airy's wave theory to the derivation of wave-induced current equations over a sloping bottom. The present paper found that only some modifications in the derivation of Mellor, as well as an introduction of the vertical wave momentum flux, would give an internally consistent equation system. Following the approach by Mellor (2003Mellor ( , 2008, by well treating wave dynamic pressure and vertical wave momentum fluxes, the present paper reports on an effort to repeat the derivation of the equations for 3D waveinduced current and radiation stresses. In the present equations, the wave driving forces are in good balance, and no spurious vertical circulation occurs outside the surf zone.

Mellor's approach
This paper only aims to derive the wave driving forces and wave-induced current equations. For simplicity, the baroclinic pressure gradient term is not included in the equations. Because of following the approach by Mellor (2003Mellor ( , 2008, most notations in M03 and M08 are used here. Let the xy-plane be the still sea level and the z coordinate be positively upward. The hydrodynamic equations are: , (u, v, w) are velocity components in x-, y-, and z-directions, respectively; p denotes the sea water pressure; the sea water density; is equal to 1 when are any triplet in the sequence, xyzxy, and is -1 in any other sequence; f is the Coriolis parameter; denotes the frictional force terms.
In the wave-current coexisting system, the kinematic components, such as the water surface elevation and current velocity, can in general be decomposed into a slowly varying process and a wave process , η =η +η, u α =û α +ũ α , w =ŵ +w, where the slowly varying motion works at the range of , and the surface wave motion works at . Owing to different existing ranges of slowly varying current and wave motion, how to linearly decompose into and must be settled. The coordinate transformation of Mellor (2003Mellor ( , 2008 successfully solved this problem.
In the approach by Mellor, Airy's wave theory is adopted, in which η = a cos ψ; (3a) where h is the still water depth; D, the depth beneath the slowly varying surface level, ; a, the wave amplitude; k, the wave number, ; , wave dynamic pressure; , the wave angular frequency; . Wave frequency and wave number satisfy the dispersion relation, .
In Airy's wave theory, the vertical motion orbit of a fluid particle in the steady state is Clearly, Eq. (4) is also a proper coordinate transformation, which has been used by Dolata and Rosenthal (1984) and Mellor (2003Mellor ( , 2008 for the derivation of the wave-induced current, and . Under such a coordinate system, the velocity fields can be conveniently expressed as: Note that, in the present paper, only the subscripts for coordinate transformation s denote partial derivatives, while subscripts for other variables denote the coordinate axes. And let us define a new s-coordinate vertical velocity: (7) Then the hydrodynamic Eqs. (1) and (2) can be rewritten as (Mellor, 2003): ∂s In Mellor's approach, taking the phase-averaging into Eqs. (8) and (9) yields the wave-induced equations. z * Note that, the coordinate transformation Eq. (5) follows the vertical motion of fluid particles in waves, and the balance position of a fluid particle is just its Lagrangian lowpass filtered height, so Eq. (5) should also be a properly socalled vertically Lagrangian coordinate in AG13.

Derivation of Cartesian-coordinate equations
Firstly, let us define a so-called Stokes drift velocity: Now, we decompose the constituent velocity Eq. (6) into Eqs. (8) and (9), and take the phase-averaging to them by use of the calculation method in M03 or in X04. In the equation of continuity Eq. (8), in combination with Eq. (6), all the phased-averaged velocities are, respectively: Let us further denote In momentum Eq. (9), the phase-averaged momentum fluxes are, respectively: u αw In Eq. (12), in the cases of deep water or the horizontal bottom, is nil, but in shallow water with a sloping bottom, it is non-nil (Rivero and Arcilla, 1995).
In the present paper, the vertical momentum equation is used to alternatively calculate water pressure. The vertical Cartesian-coordinate momentum equation is [ z,η +η ] Vertically integrating the above equation over , in collaboration with the surface kinematic boundary condition, and neglecting the horizontal advection terms that only cause small errors, one obtains: cosψ.
Under the s-coordinates, the water pressure can be expressed as: sinh (kD) cosψ.
Obviously, in the above equation, pressures due to slowly varying current and wave motion are, respectively: That is, in waters where waves and current coexist, dynamic pressure can also be decomposed into slowly varying pressure and wave pressure as the velocity fields: ,p ps z * which disagrees with M08. Thus, the phase-averaged and are obtained as: in which E is the wave energy and . In the s-coordinates, there is no relative motion between the mass particle and coordinate surface, and consequently, is nil. But there exists coupling between the wave pressure and coordinate surface motion, then is non-nil. The present expression for is the same as M03, but the latter is derived via the wave pressure Eq. (3d). Now, let us define (14) Then, the phase-averaged momentum Eq. (9) is: In Eq. (15), the depth integration of the wave action terms will result in .
The calculation of will be given later. In addition, in the depth-integrated momentum equations, the transfer of the momentum into the sea bed by wave pressure, similar to the momentum transfer by baroclinic pressure, is not negligible. If the sloping topography is taken into account, should be given as: sinψ.

Then we have
Eq. (16) gives Thus, it can be seen that, after correctly calculating and , the vertically integrated Eq. (15) is consistent with P77.

Derivation of sigma-coordinate equations
Firstly, let us define a sigma-coordinate transformation (Blumberg and Mellor, 1987): , with at the sea surface and at the bottom, and let us denote After taking this sigma-coordinate transformation to the Cartesian-coordinate Eqs. (10) and (15), we will obtain the equations in the sigma-coordinates: If only the depth variation is taken into account, Eq. (18) can be rewritten as: we will also stringently derive Eq. (21).
Eqs. (19) and (20) can also be directly derived by the coordinate transformation of Mellor (2003): If taking such a coordinate transformation to Eqs. (8) and (9), we will obtain almost the same phase-averaged equations as M03. But the term must be carefully calculated. According to the definition of the vertical velocity in new coordinates, we have: In the above equation, , neglecting , then one obtains . However, Mellor (2003) of Mellor led to omitting the vertical correction of the horizontal gradient terms of the wave momentum fluxes, . However, the wave momentum fluxes, , is much larger than the wave pressure, , and is nearly exponentially decreased. In the waters with a sloping bottom, the omission of the vertical correction term, , will cause a significant error. In shallow waters, the vertical momentum flux, , is important, but Mellor (2003) also neglects it. As shown later, it is the neglect of in M03 that leads to a strong spurious circulation outside the surf zone in the test by Ardhuin et al. (2008). Mellor (2008) thought that the shortcomings in M03 were resulted from his incorrect calculation of the phase-averaged wave pressure, and he revised it as: S ασ which leads to a radiation stress profile of non-continuity at the surface and no vertical variation beneath in the wave propagation direction. Obviously, Mellor's revised radiation stress expressions are inconsistent with the vertical variation characteristics of wave. In the present author's opinion, the inconsistency in M03 is caused by the omission of the vertical momentum flux, , and has nothing to do with the calculation of the wave pressure. On the contrary, the formula for the wave pressure in M03 is rational and accurate.ũ In Airy's wave theory, the horizontal velocity and vertical velocity are 90° out of phase, which would produce a nil phase-averaged . But in the presence of the sloping bottom, wave energy dissipation, viscosity near bottom boundary layer and depth-varying current, the phase between and is altered, and is non-zero. Non-zero is thought to play an important role in driving the vertical near-shore circulations. So far, a variety of expressions have been suggested (see Svendsen and Lorenz, 1989;Deigaard and Fredsøe, 1989;Rivero and Arcilla, 1995;Zou et al., 2006). Among them, it seems that Zou et al. (2006) have given a more theoretically sound expression than others. Zou et al. (2006) first developed an asymptotical solution for a monochromatic wave propagating over a sloping bottom, and naturally obtained the vertical distribution of the phase-averaged wave vertical momentum flux , in which the effects of the bottom slope, bottom friction and wave breaking are fully taken into consideration. It reduces to the solution by Deigaard and Fredsøe (1989) for dissipative waves over a horizontal bottom in shallow water, and to the solution by Rivero and Arcilla (1995) for non-dissipative or breaking waves over a sloping bottom. However, the waveinduced current equations and radiation stresses in M03 and M08 are all developed in the framework of Airy's wave theory. It would be favorable to keep the internal consistency in each term of the radiation stresses if the estimates of allũw terms are based on the same wave theory. Under the assumption of irrotational wave motion and Airy's wave theory, Rivero and Arcilla (1995) gave an expression for , ς] Note that the asterisks in the coordinate axes are dropped out hereafter. Obviously, Eq. (23) satisfies the bottom kinematic boundary condition, but Eq. (23), as well as other solutions, fails to provide a nil surface momentum flux. This is irrational, especially for non-breaking wave. Hence, the present paper will also suggest an expression for , satisfying the kinematic boundary condition at the bottom and being nil at surface. Provided that over a sloping bottom the surface elevation and horizontal velocity can be approximated by Airy's wave theory, denoting , the vertical integration of the sigma-coordinate continuity equation over gives Thus, we have: Because the wave surface and horizontal velocity cannot be accurately given by Airy's wave theory over the sloping bottom, Eq. (24) is also non-nil at the surface. But if only the depth variation is taken into account, Eq. (24) can be rewritten as: and In the present paper, Eqs. (27) and (25) will be adopted XIA Hua-yong China Ocean Eng., 2017, Vol. 31, No. 4, P. 418-427 423 in the momentum equations, Eqs. (15) and (20), respectively.

Consistency in the present equation system
6.1 Test for momentum balance of wave forces Ardhuin et al. (2008) designed a case with an ideal topography and non-breaking wave to test the balance among wave forces in formulae of Mellor (2003). For the convenience of the comparison with Mellor (2003) and Ardhuin et al. (2008), the same case is used here. In the test by Ardhuin et al.'s (2008), the topography profile is given as: , h 1 =6 m, h 2 =4 m and the maximum bottom slope . The offshore wave height is a 0 =0.12 m, with the angular frequency and the maximum wave steepness . The non-dimensional depth is 0.85<kD<1.1. Firstly, we check the consistency of Eqs. (23), (26), (27) for the vertical wave momentum fluxes. As shown in Fig. 1, the results from Eqs.  (27). It means that the three expressions are all comparable. Following Ardhuin et al. (2008), we test the balance of wave forces in the lowest order of the momentum equation: The wave-induced set-ups or set-downs are calculated following Longuet-Higgins and Stewart (1964). The vertical distributions of pressure gradient term and other wave F α4 F α3 force terms are shown in Fig. 2. Clearly, the vertical wave momentum flux term, , is much larger than , and it can not be neglected.
The summations of wave forces, , , , are shown in Fig. 3. is (-0.72)-1.23 times F eta , while is (-0.69)-1.14 times F eta and is (-0.07)-0.2 times . That is, when the vertical momentum fluxes are taken into account, the wave forces in the momentum equation are almost in balance.
is used in X04 and in the corrected M03, and , are used in the original M03 and the present study, respect-ũw gDε 3 gDε 3 Fig. 1. Snapshots of the profiles of vertical wave momentum fluxes for a slowly varying wave over a sloping bottom. Wave height is 0.12 m and wave angular frequency is 1.2 rad/s. The thick black line is bottom topography. All the fields are normalized by . In Figs. 2 and 3, the wave is the same as that in Fig. 1, and all the wave forces are also normalized by .

Fig. 2.
Wave forces over a sloping bottom.  ively. The three formulae share similar vertical circulation pattern outside the surf zone, but the former two produce strong spurious currents, which have been criticized by Ardhuin et al. (2008). Hence, the formulae in the original M03, the corrected M03 and X04 should be revised. In the definition of Ardhuin et al. (2008), the vertical momentum flux is mainly represented by . In their formulae, the evaluation of the vertical flux of momentum requires an estimation of the pressure and the coordinate transformation function s to first order in parameters that define the large-scale evolution of the wave field, such as over the bottom slope. Unfortunately, there is no analytical expression for and s at that order. Instead, they proposed a numerical correction method to calculate . However, their numerical method is not well suited for practical applications because very complex wave models are required . In the present paper, the inconsistencies in M03 and M08 are overcome after the vertical momentum flux, or , is introduced, thus, giving an acceptable remedy for M03 and M08.
6.2 A test case for wave-induced current over a slope The classical theory of the wave-induced current and many experimental data (e.g. Longuest-Higgins, 1970) show that the depth-integrated longshore currents are very weak outside the surf zone. So, it is rational to believe that the wave-induced vertical circulation outside the surf zone is also weak.  designed an idealized test to illuminate the drawbacks in Mellor's (2008) model. In their test case, a monochromatic wave, with a wave height of 1.02 m and a wave period of 5.24 s, propagates over a slope. The topography of the slope is the same as that in Fig. 1, but they added a symmetric slope back down to 6 m to allow periodic boundary conditions if needed. For such a wave period, the wave group celerity varies little from 4.89 to 4.64 m/s, producing a 2.7% increase of wave amplitude on the shoal. The wave shoaling over this slope is far from the breaking limit of 4-m depth. In their test, the Mellor's (2008) model yields a maximum horizontal velocity of 0.6116 m/s without the vertical mixing, and a maximum velocity of 0.1594 m/s with a vertical mixing coefficient of 2.8×10 -3 m 2 /s. Both velocities are much larger than the Lagrangian velocity of (-0.01)-0.025 m/s or the Stokes velocity of 0.01-0.05 m/s. For the convenience of comparing with , their test is used here to demonstrate the improvement of the present paper over Mellor's (2003) model. The present model area is 800 m in the x-direction with a horizontal grid resolution of 5 m and a vertical resolution of . The calculation period is 1 h with a time step of 0.1 s. The vertical mixing coefficient is set as 2.8×10 -3 m 2 /s. In the present model, the numerical scheme follows Xia et al. (2004) and the water levels at the open boundaries are set to zero. As shown in Fig. 4, for the test, the revised M03, M03 and the present one all produce a clockwise current pattern and an anticlockwise one over the upward slope and downward slope, respectively, which are similar to those calculated by . However the velocity magnitudes modelled by the three formulae are different from each other. The revised M03 produces the strongest current (see snapshot (a) in Fig. 4) with a maximum velocity of 0.207 m/s. The original M03 generates a maximum velocity of 0.129 m/s, which is close to that modelled by .  and  suggested that both M03 and M08 produce a spurious strong current outside the surfzone. However, the present one yields a maximum velocity of 0.04 m/s, which is the same order as the Lagrangian velocity or Stokes velocity. Hence, the present formulae provide at least a rational velocity in magnitude.
The formula tensor for the radiation stresses in the corrected M03 is:  (14). That is, the inconsistency in M03 is not improved at all. On the con- is zero, which increases the error of the model. This is why the maximal current in the original M03 is less than the one in the finally revised M03. Clearly, the introduction of the vertical momentum flux, , substantially improves Mellor's (2003) model systems.

Discussion
It ought to be confessed that the limitations of Airy's wave theory may lead to some flaws in 3D wave-induced current models. Firstly, Airy's wave theory is based on the potential flow theory, but at the surface and bottom boundary layers, the sea water viscosity is not negligible. Wave motion induces oscillating velocities at viscous boundary layers, and the oscillating velocities produce stresses there (Longuet-Higgins, 1953), and the stresses at the boundary layers were thought of as one of main driving forces for the mass transport in waves (Longuet-Higgins, 1953;Huang, 1970;Bijker et al., 1974;Phillips, 1977). In all 3D wave-induced current models, the wave stresses due to the boundary layer viscosity are not taken into account. The flow at the boundary layer may play an important role in the vertical circulation, and such stresses ought not to be neglected in the 3D wave-induced current model.
Secondly, wave breaking is not taken as a driving force for currents. Inside the surf zone, the surface roller forms momentum flux, and its horizontal gradient drives flow. Such a driving force has been found by Svendsen (1984). Surface roller is regarded as a major force driving so-called undertow and plays a role in driving near-shore currents. Surface roller action is commonly included in analytical and numerical model of undertow and along-coast currents (see Svendsen et al., 1987;Deigaard and Fredsøe, 1989;Deigaard et al., 1991;Kuriyama and Nakatsukasa, 2000;Zheng et al., 2014). Such a force should also be taken into account in a wave-induced model.

Conclusionũω uw
Obviously, all the shortcomings in M03 and M08 are improved in the present paper. If the vertical wave momentum fluxes is included in the phase-averaged momentum equations, the inconsistencies in M03, such as strong spurious circulation outside the surf zone and mean currents in deep sea with a sloping bottom produced by the unforced waves, can all disapear. It is not the wave pressure expression in M03 that results in the inconsistencies, and thus it is unnecessary to revise it. The depth integration of wave forces in M08 is not consistent with LHS64. This arises from the neglect of the vertical wave momentum flux and the wave pressure term related to the s-coordinate level. In the present revised formulae, the wave forces in the lowest order momentum equation are approximately in balance. The depth-integrated radiation stresses of LHS64, based on Airy's wave theory, are practicable in most cases.
While Airy's wave theory is applied to derive 3D wave-current interaction, its main disadvantage lies in its inability to provide the vertical momentum fluxes in shallow water with the bottom variations. When an alternative estimate of the vertical momentum fluxes is introduced, Airy's wave theory and the approach of M03 and M08 appear to be acceptable for deriving 3D wave-current interaction.